Modeling bone regeneration around endosseous implants

Modeling bone regeneration around endosseousimplantsPROEFSCHRIFTter verkrijging van de graad van doctoraan de Technische Universiteit Delft,op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben,voorzitter van het College voor Promoties, in het openbaar te verdedigenop 13 november 2012 om 15:00 uurdoorPavel Aliaksandravich PROKHARAUMechanical Engineer, Applied Mathematician, Belarusian State Universitygeboren te Minsk, Belarus

Dit proefschrift is goedgekeurd door de promotor:Prof.dr.ir. C. VuikCopromotor Dr.ir. F.J. VermolenSamenstelling promotiecommissie:Rector Magnificus,Prof.dr.ir. C. Vuik,Dr.ir. F.J. Vermolen,Prof.Dr. J.M. García-Aznar,Prof.Dr. M.A. Zhuravkov,Prof.dr.ir. H.H. Weinans,Dr. A. Madzvamuse,Dr. L. Geris,Prof.dr.ir. C.W. Oosterlee,voorzitterTechnische Universiteit Delft, promotorTechnische Universiteit Delft, copromotorUniversidad de ZaragozaBelarusian State UniversityTechnische Universiteit DelftUniversity of SussexKatholieke Universiteit LeuvenTechnische Universiteit Delft, reservelidModeling bone regeneration around endosseous implants.Dissertation at Delft University of Technology.Copyright c○ 2012 by P.A. ProkharauThe work described in this thesis was financially supported by the DelftInstitute of Applied Mathematics (DIAM) of the Delft University of Technology.ISBN 978-94-6203-171-5Cover design by Maxim Molokov and Pavel Prokharau. Painting by MaryjaProkharava.Printed in the Netherlands by Wöhrmann Print Service.

to my parents, Masha and Paulinka

SummaryModeling bone regeneration around endosseous implantsPavel A. ProkharauThe present work is focused on mathematical modeling of bone regeneration.Various aspects of modeling are considered. In Chapter 2, a classicalsystem of partial differential equations (PDE’s) is analyzed, which is constructedto simulate bone healing around endosseous implants. The presentsystem is of the diffusion-advection-reaction type and is typical for mathematicalmodels for bone regeneration. The need of analyzing the PDE’sfollows from the appearance of the wave-like profiles, that are found in thenumerical solutions for the distribution of cells and of growth factors and,consequently, of newly formed bone matrix. Such predictions of the modelcontradict experimental observations. Hence it is critically important tounderstand why these wave-like patterns appear and how the model maybe modified in order to provide biologically relevant solutions. The linearstability analysis carried out around constant-state (i.e. homogeneous inthe physical space) solutions provides the answers to the stated questions.Stability of the constant-state solutions is determined by the values of themodel parameters. Explicit relations determining the stability region for theparameters are derived. It is concluded that if the model parameters havethe values outside of the stability region, then the constant-state solution isunstable and the exact solution of the current system will not converge toit. Hence formation of stable patterns is likely. The analytical results arevalidated by finite element simulations.Chapters 3 and 4 of the thesis are devoted to development of new approachesin modeling bone regeneration. In Chapter 3, the evolutionarydifferentiation model is introduced, that allows to incorporate the historyinto the differentiation of cells by defining an additional differentiation statevariable a ∈ [0, 1]. During bone regeneration, mesenchymal stem cells differv

viSummaryentiate into other cell phenotypes. It is known from experiments [13, 60, 97]that cells obtain new properties gradually during a finite period of time.The differentiation state variable a allows to track a gradual evolution ofcell states. Cell differentiation is simulated by an advection term in thegoverning equations, which is equivalent to modeling a certain velocity (differentiationrate) in the differentiation state space. Each cell is capable ofchanging its phenotype only after a finite (contrary to immediate) periodof time. With the immediate differentiation approach, which is commonlyused in classical models, it is only possible to consider just the initial andfinal states of differentiating cells. Cell differentiation is then modeled as animmediate change of the cell phenotype.The advantage of the new evolutionary approach is that the differentiationhistory is employed, i.e. the current state of cells depends on how thecells evolved before. Therefore the new approach has a potential to describecell differentiation more accurately, provided the model parameters are chosencorrectly. The most important advantage related to the present approachis the concept of the finite time of differentiation. In experiments, new boneformation is observed within the healing site only after certain time afterthe implant placement (at the end of the first week according to Berglundhet al. [14] and Abrahamsson et al. [1]). With the evolutionary differentiationapproach it is possible to calibrate the time of initiation of bone formationby setting the appropriate value for the differentiation rate u b of osteogeniccells into osteoblasts. Initiation of new bone release is related to the timeof appearance of secretorily active osteoblasts within the peri-implant region.If the rate u b is equal, for example, to 1/4 days −1 , then the minimaltime of differentiation of fully non-differentiated osteogenic cells (which arerecruited from the old bone surface) into osteoblasts is equal to four days.For the classical immediate differentiation approach, differentiation alwaysstarts immediately. Hence classical models usually predict formation of newbone already during the first hours after the implant placement.The present evolutionary approach is incorporated into another extendedinnovative model described in Chapter 4. The model is developed for earlystages of peri-implant osseointegration. It is known from experiments that,during endosseous healing, bone forms through a direct apposition on a preexistingsolid surface [24]. Therewith the concept of the ossification front orof the bone forming surface is considered. The bone forming surface is dealtwith directly in the present model, which is formulated as a moving boundaryproblem. The ossification front is modeled by the moving boundary ofthe computational domain containing the soft tissue. An explicit representationof the bone-forming surface is the main innovation of the proposedmodel, which distinguishes the current formalism from the classical modelsfor peri-implant osseointegration. Due to the finite time of differentiation ofosteogenic cells, no osteoblasts appear in the soft tissue region during firstfour days of simulation. New bone is not released and the initial geometry

Summaryviiis not changed within this time period. Hence during the first four days thesources of osteogenic cells and growth factors are modeled to be located atthe old bone surface and at the implant surface. Therefore, the evolutionarydifferentiation approach is an essential part of the present model. Theclassical immediate differentiation formalism would lead to an immediatemovement of the ossification front and to not being able to define the initialsources of cells and growth factors for the current moving boundary problem,since the old bone surface and the implant surface would not be theboundary of the physical (i.e. soft tissue) domain.The numerical method for the present mathematical model is describedin Chapter 5. The current problem has a number of characteristics whichmake the algorithm elaborate and complex. The constructed numericalmethod provides stable and non-negative solutions of the nonlinearly coupledsystem of the time dependent taxis-diffusion-reaction equations. Theadditional challenges, which are faced with at the stage of the construction ofthe algorithm, consist in the need to discretize the model equations withinthe irregular and time-evolving physical domain, and in the sensitivity ofthe mathematical model with respect to negative solutions. The presentlyproposed numerical approach is based on such methods as: the method oflines, finite volume method, level set method and the embedded boundarymethod. For coarse meshes, patterns are observed to develop. These patternsshould not be there and result from numerical errors. Since we observethat convergence is only reached at high mesh resolutions, we think that furtherresearch should point into the direction of adaptive mesh refinement,improved time integration (operator splitting methods) and/or higher-ordermethods, such as spectral or DG-methods (however the requirement of thesolution positivity is still essential for higher-order methods). The series ofnumerical simulations demonstrate the ability of the present osseointegrationmodel to predict various paths of new bone formation depending on thechosen parameter values.

SamenvattingModelleren van botregeneratie rond enossale implantatenPavel A. ProkharauHet huidige werk is gericht op de wiskundige modellering van botregeneratie.Een aantal aspecten van het modelleren worden beschouwd. In Hoofdstuk2, wordt een klassiek stelsel partiële differentiaalvergelijkingen (pdv’s) geanalyseerd,dat ontwikkeld is voor de simulatie van botingroei in enossaleimplantaten. Dit stelsel is van het diffusie-advectie-reactietype en is kenmerkendvoor de wiskundige modellen voor botregeneratie. De noodzaak van deanalyse van pdv’s volgt uit het optreden van de golfachtige profielen die inde numerieke oplossingen voor de distributie van cellen en groeifactoren enderhalve voor nieuw gevormd bot matrix waargenomen worden. Dergelijkevoorspellingen van het model spreken experimentele waarnemingen tegen.Daarom is het zeer belangrijk om te begrijpen, hoe deze golfachtige patronenverschijnen en hoe het model moet worden gewijzigd om biologisch zinvolleoplossingen te verkrijgen. De lineaire stabiliteitsanalyse die uitgevoerd rondeen constante oplossing (dat wil zeggen in de ruimte homogene evenwichtsoplossingen)beantwoordt deze vraagstelling. Of de constante oplossingenstabiel zijn, wordt bepaald door de waarden van de modelparameters. Explicieterelaties, die het stabiliteitsgebied voor de parameters bepalen, wordenin dit proefschrift afgeleid. Er wordt geconcludeerd, dat als de waarden vande modelparameters buiten het stabiliteitsgebied liggen, dat dan de constanteoplossing instabiel is en dat de exacte oplossing van het huidige systeemniet convergeert. Hierdoor is de vorming van stabiele patronen zeer waarschijnlijk.De analytische resultaten zijn gevalideerd door middel van eindigeelementen simulaties.Hoofdstukken 3 en 4 van het proefschrift gaan over de ontwikkeling vannieuwe aanpakken in het modelleren van botregeneratie. In Hoofdstuk 3is het evolutionaire differentiatie-model ingevoerd, waardoor het mogelijkix

xSamenvattingis om de voorgeschiedenis van de cellen mee te nemen door het definiërenvan een extra differentiatie-variabele a ∈ [0, 1]. Tijdens de regeneratie vanbot, differentiëren mesenchyme stamcellen naar andere cel fenotypes. Het isbekend uit experimenten [13, 60, 97] dat de cellen nieuwe eigenschappen geleidelijkin een eindige periode van tijd verkrijgen. De differentiatie-variabelea maakt het mogelijk om een geleidelijke evolutie van de cel te volgen. Celdifferentiatiewordt gesimuleerd door een advectie term in de beschrijvendevergelijkingen die een zekere snelheid (differentiatie-snelheid) in de differentiatietoestandsruimte modelleert. Elke cel is in staat zijn fenotype pas naeen eindig (in tegenstelling tot onmiddellijk) tijdsbestek te veranderen. Metde aanpak waarin differentiatie onmiddellijk optreedt, die meestal in de klassiekemodellen gebruikt wordt, is het slechts mogelijk de initiële en finalestaat van differentiërende cellen te beschouwen. Celdifferentiatie wordt dangemodelleerd als een instantane wijziging van het fenotype van de cel.Het voordeel van de nieuwe evolutionaire aanpak is dat de differentiatiegeschiedenis wordt gebruikt, dat wil zeggen de huidige stand van de cellenhangt af van hoe de cellen zich daarvoor ontwikkelden. Daarom biedtde nieuwe aanpak een mogelijkheid om celdifferentiatie nauwkeuriger te beschrijven,op voorwaarde dat de model parameters correct gekozen worden.De belangrijkste innovatie omtrent de huidige aanpak is het concept vande eindige tijd van differentiatie. In experimenten wordt de vorming vannieuw bot slechts na een bepaalde tijd na plaatsing van het implantaat inde genezende regio waargenomen (aan het einde van de eerste week volgensBerglundh et al. [14] en Abrahamsson et al. [1]). Met de geleidelijke differentiatieaanpak is het mogelijk om de aanvangstijd van botvorming tekalibreren door de juiste waarde voor de differentiatiesnelheid u b van osteogenecellen in osteoblasten te kiezen. Initiatie van de aanmaak van nieuwbot hangt af van het moment van ontstaan van botvormende osteoblastenbinnen het peri-implantaat gebied. Als de snelheid u b , bijvoorbeeld, gelijkis aan 1/4 dagen −1 , dan is de minimale tijd van differentiatie van volledigniet-gedifferentieerde osteogene cellen (die worden gerekruteerd uit het oorspronkelijkebotoppervlak) naar osteoblasten gelijk aan vier dagen. Voorde klassieke instantane differentiatie aanpak is de minimale differentiatietijd altijd gelijk aan nul. Daardoor voorspellen klassieke modellen vaak devorming van bot al gedurende de eerste uren na de plaatsing van het implantaat.De huidige evolutionaire aanpak is opgenomen in een ander uitgebreidinnovatief model, dat in Hoofdstuk 4 beschreven is. Het model is ontwikkeldvoor de vroege stadia van peri-implantaat osseo-integratie. Het is bekenduit experimenten dat tijdens de botgenezing bot vormt door middel van eendirect bijstelling op een reeds bestaande stevige ondergrond [24]. Daarmeewordt het concept van ossificatiefront of botvormend oppervlak beschouwd.Het botvormende oppervlak wordt direct meegenomen in dit model, datals een bewegend randwaardeprobleem geformuleerd wordt. Het ossifica-

CONTENTSSummarySamenvattingvix1 Introduction 11.1 Bone regeneration . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Study motivation . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Review of mathematical models for bone regeneration . . . . 41.4 Structure and subjects of the thesis . . . . . . . . . . . . . . . 92 Linear stability analysis 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Biological model . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 The simplified biological model . . . . . . . . . . . . . 182.3.2 Non-homogeneous perturbations . . . . . . . . . . . . 222.3.3 Stability of the system of two equations . . . . . . . . 252.3.4 Correspondence between the systems of two and threeequations . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.5 Stability of the system of three equations . . . . . . . 372.3.6 Parameter choice and stability . . . . . . . . . . . . . 392.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Evolutionary cell differentiation 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Differentiation model . . . . . . . . . . . . . . . . . . . . . . . 493.2.1 Flow of non-differentiated cells . . . . . . . . . . . . . 51xiii

xivCONTENTS3.3 Peri-implant osseointegration . . . . . . . . . . . . . . . . . . 543.3.1 Mathematical model . . . . . . . . . . . . . . . . . . . 543.3.2 Differentiation rates and tissue formation . . . . . . . 593.4 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 643.4.1 Model parameters and numerical method . . . . . . . 653.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . 673.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 754 Moving boundary model for endosseous healing 794.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Recent mathematical models . . . . . . . . . . . . . . . . . . 814.3 Moving boundary model . . . . . . . . . . . . . . . . . . . . . 834.3.1 Mature and immature cells . . . . . . . . . . . . . . . 834.3.2 Modeling approach . . . . . . . . . . . . . . . . . . . . 844.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 924.4.1 Numerical simulations . . . . . . . . . . . . . . . . . . 924.4.2 Model parameters . . . . . . . . . . . . . . . . . . . . 934.4.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . 984.4.3.1 Numerical solutions for various mesh resolution. . . . . . . . . . . . . . . . . . . . . . . 1084.4.3.2 Convergence against the mesh resolution . . 1174.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 1244.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 Numerical algorithm 1315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . 1325.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 1345.3.1 Positivity of the solution . . . . . . . . . . . . . . . . . 1345.3.2 Definitions of cells and vertices in the grid . . . . . . . 1365.3.2.1 Domain evolution . . . . . . . . . . . . . . . 1375.3.2.2 Treatment and reconstruction of active cells 1375.3.3 Level set equation . . . . . . . . . . . . . . . . . . . . 1395.3.4 PDE discretization in space . . . . . . . . . . . . . . . 1405.3.5 Advection-diffusion discretization . . . . . . . . . . . . 1455.3.5.1 Linear flux due to cell differentiation . . . . . 1455.3.5.2 Fluxes due to cell migration . . . . . . . . . 1485.3.5.3 Gradient approximation at cell edges . . . . 1535.3.6 Nonlinear system of conservation laws . . . . . . . . . 1585.3.6.1 The strictly hyperbolic case . . . . . . . . . . 1625.3.6.2 Nonconvex flux function . . . . . . . . . . . . 1665.3.6.3 Nonstrictly hyperbolic system . . . . . . . . 1705.3.6.4 Weakly hyperbolic system . . . . . . . . . . . 1725.3.6.5 Summary . . . . . . . . . . . . . . . . . . . . 176

CONTENTSxv5.3.7 Treatment of the reaction term . . . . . . . . . . . . . 1785.3.8 Boundary conditions . . . . . . . . . . . . . . . . . . . 1795.3.9 Time integration . . . . . . . . . . . . . . . . . . . . . 1805.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . 1836 Conclusions and outlook 1856.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 186AppendicesA Notes to the moving boundary model for endosseous healing189A.1 Cell source at the old bone surface . . . . . . . . . . . . . . . 189A.2 Fluxes at the moving boundary . . . . . . . . . . . . . . . . . 190B Notes to the numerical algorithm 193B.1 Determinant of the linear system . . . . . . . . . . . . . . . . 193B.2 Number of adjacent cells . . . . . . . . . . . . . . . . . . . . . 194Acknowledgment 197Curriculum vitae 199Bibliography 201List of publications 211

CHAPTER 1Introduction1.1 Bone regenerationOne of the most remarkable properties of bone tissue is its regenerativecapacity. Bone regeneration is a complex biological process, which occursstarting from the skeletal development and continuing in the form of boneremodeling and in the form of response to bone injuries, throughout theentire lifetime of organisms. In contrast to other tissues, damaged boneregenerates without formation of a scar tissue. The newly formed bone isusually remodeled to such a state, that its properties are almost indistinguishablefrom the properties of surrounding old bone, and a bone geometryin the healing site is usually restored to its initial shape [26, 31].Bone regeneration consists of a series of well-regulated biological processes,taking places in a certain pathway. Several cell types are involvedin these processes. Activity of cells is regulated by biological, chemical andmechanical environments around them. Fracture healing is one of the mostcommon cases of bone regeneration in a clinical setting. This process includesboth mechanisms, in which new bone can form: intramembranousossification and endochondral ossification [87]. The following three mainstages of bone healing can be distinguished: the inflammatory phase, bonerepair and bone remodeling [42].At the first stage, immidiately after the injury, blood, which leaks fromthe damaged blood vessels, coagulates and a blood clot is formed in thewound region. Platelets, which are found within the blood clot, becomeactivated and start to release growth factors, which regulate the processesof bone healing. Since the blood vessels are ruptured, the supply of nutrientsand oxygen to osteocytes, which are the living cells, situated within thebone matrix, ceases. The lack of nutrients and oxygen makes the cells die.1

2 Chapter 1. IntroductionNecrotic processes initiate inflammation. Inflammatory cells (macrophages,phagocytes, leukocytes) arrive in the wound site and remove a necrotic tissue.After this process, fibroblasts, mesenchymal stem cells (MSC’s) andendothelial cells, which originate from the periosteum (the vascularity-richexternal tissue layer of bone), from the marrow channel of the bone and fromthe soft tissues, which surround the bone including the muscles, migrate intothe healing region [11]. One of the mechanisms regulating cellular processesduring bone healing, is the influence of growth factors, which are releasedby activated platelets and by the aforementioned cells.MSC’s differentiate into osteoblasts, chondrocytes and fibroblasts underthe influence of different stimuli, for example, growth factors and the localmechanical environment in the tissue. Differentiated osteoblasts can synthesizenew bone matrix directly on a pre-existing surface without the mediationof the cartilage phase [87]. Such type of bone formation is called intramembranousossification. In this case, the apposition of new bone matrix on asolid surface takes place [24], and an ossification front or a bone-formingsurface is observed [1, 14]. Therefore, intramembranous ossification can bedescribed by a moving-boundary type of bone formation.The second mode of bone formation is referred to as endochondral ossification.During fracture healing, this type of ossification occurs in themiddle of the fracture area, in which MSC’s differentiate into chondrocytes.Chondrocytes proliferate and form cartilage tissue. Maturation of chondrocytestowards the hypertrophic chondrocytes is followed by the calcificationof cartilage. Chondrocytes undergo apoptosis and blood vessels grow intothe cavities, which were initially occupied by the chondrocytes. Osteoprogenitorcells are delivered into the cavities through a vascular network, andthey differentiate into osteoblasts. The mineralized extracellular matrix ofthe cartilage tissue serves as a scaffold, on which osteoblasts release newbone [11].Therefore during a reparation stage, new woven bone is synthesizedwithin the healing site. Right after the formation of new bone, a remodelingphase begins. A patch of newly formed woven bone is remodeled into maturelamellar bone, through its resorption by osteoclasts and through newbone synthesis by osteoblasts.1.2 Study motivationDespite regenerative properties of bone, there is a large number of relatedclinical problems, which are not solved at the present time. One of them is animpaired bone regeneration during fracture healing. For example, Einhorn[30] refers to Praemer et al. [76], when he notes, that “the healing of 5 to10 per cent of the estimated 5.6 million fractures occurring annually in theUnited States is delayed or impaired”. Another representative number is

1.2. Study motivation 3given by Audigé et al. [10]. They have done an observational study of thetreatment of 416 patients with tibial shaft fractures, and reported 52 (13%)cases of delayed healing or nonunion.The natural regenerative capacity of bone tissue is not sufficient in certainsituations, which appear in orthopaedic, oral and maxillofacial surgery,such as skeletal reconstruction of large bone defects created by trauma, infection,tumor resection and skeletal abnormalities, or cases in which boneregeneration is compromised due to diseases, like osteoporosis or avascularnecrosis [26]. For example, 3.79 million osteoporosis fractures were estimatedin the European Union in 2000, and direct medical cost for their treatmentwere around 32 billion euros [82].Einhorn [30] and Dimitriou et al. [26] review temporary treatment methods,which are aimed at promoting natural bone regeneration in situations,where this process is impaired or simply insufficient. Among the treatmenttechniques, which are often applied in clinical practice, the authors mentiondistraction osteogenesis and bone transport; bone-grafting methods, such asautologous bone grafts, allografts, and bone-graft substitutes or growth factors;non-invasive methods of biophysical stimulation, such as low-intensitypulsed ultrasound and pulsed electromagnetic fields; mechanical stimulation.These methods can be used separately or in combinations. However,the efficiency and the applicability of the considered techniques are stilllimited.In order to improve existing methods and to develop new approaches, abetter understanding of the processes, taking place during regeneration ofbone, is needed. A large number of experimental works is devoted to theinvestigation of the influence of various factors on the course of bone healing.In in-vivo experiments, the biological processes are observed in naturalconditions. However, obtaining temporal and spatial experimental dataoften becomes technically complicated. Contemporary non-invasive methodssuch as various imaging techniques can sufficiently enhance the acquisitionof the qualitative and quantitative information about the biology ofbone regeneration. Molecular imaging techniques, which are reviewed inMayer-Kuckuk and Boskey [64], can be used for real-time biology studiesof bone regeneration in living tissues. These techniques are classified intothree groups: nuclear imaging (among which are single photon computedtomography (SPECT) and positron emission tomography (PET)), opticalimaging methods (in particular fluorescence reflectance imaging (FRI) andbioluminescence imaging (BLI)) and magnetic resonance imaging (MRI).An alternative to in-vivo experiments is in-vitro studies. In these studies,it is possible to obtain a highly controllable and measurable environment, inwhich biological processes are observed. The disadvantage of this approachis that tissues and cells are separated from their natural environment. Somespecial attention should be paid to make the experimental settings correspondto natural conditions, in which bone regeneration takes place in real

4 Chapter 1. Introductionorganisms.As the examples of various directions of the experimental studies of boneregeneration, the following references can be mentioned. Abrahamsson et al.[1], Berglundh et al. [14] studied paths of bone formation around smoothand micro-rough endosseous implants, which were installed into the dogmandibles. A review of in-vivo experiments on the influence of mechanicalstimulation on bone healing is presented by Epari et al. [31]. The effectof a different micro-structure of an endosseous implant and of a mechanicalstimulation on bone ingrowth into the implant within an experimentalchamber, which was installed into a rabbit tibia, was investigated by Duycket al. [28, 29], Vandamme et al. [92, 93, 94]. The experimental works, inwhich the influence of growth factors on bone regeneration was studied, arereviewed in Lind [61]. In-vitro studies, related to bone regeneration, can befound, for example, in Gabbay et al. [33], Kasper et al. [54], Weinand et al.[96]. The aforementioned list of references does not give a full picture of theexperimental studies in the considered field. This list should be consideredjust as a short introduction into the topic.Study of bone regeneration with use of in-vivo or in-vitro experimentscan be very expensive, time-demanding or even impossible from ethical pointof view. Qualitative and quantitative data, which are possible to obtainfrom the experiments, are often limited. In this situation, a mathematicalmodel can deliver additional insight into various aspects of bone regeneration.Mathematical models and numerical simulations are valuable tools forrepresentation and analysis of complex multi-scale biochemical and biomechanicalprocesses. Simulations allow to investigate these cases, which areprohibitively difficult or impossible to consider in experiments. This makesmathematical modeling extremely valuable for, for example, the developmentof new treatment techniques and strategies.1.3 Review of mathematical models for bone regenerationCurrently there is a large number of mathematical models, which were developedto study different aspects of bone regeneration. The models arebased on various concepts and approaches and they differ in the types of theproblems, for which they can be used. Therefore, several criteria can be employedfor the classification of the models for bone regeneration. The modelscan be related to various forms of bone regeneration, for example, models forfracture healing, for bone regeneration around endosseous implants and/orfor distraction osteogenesis. Another way to classify the models, is to look atthe basic mathematical approaches used (continuous deterministic modelsand fuzzy-logic models). Other characteristics, in which models differ fromeach other, are, for example:

1.3. Review of mathematical models for bone regeneration 5• representation of biological tissues as linear elastic, visco-elastic orporo-elastic (biphasic) media;• choice of the mechanical stimulus (strain energy, hydrostatic stress,deviatoric strain, fluid flow etc.);• representation of the considered cellular processes (random diffusion,chemotaxis and haptotaxis for cell migration, immediate and evolutionarycell differentiation, etc.);• phases of bone regeneration represented (reparation phase and/or remodelingphase).The focus of this manuscript is on the processes taking place during thereparation phase of bone regeneration. In this work, bone remodeling isnot considered in detail. Bone remodels constantly throughout the entirelife of an organism. The time scale of this process is much larger thanthe time scale of the reparation processes [24]. For extended mathematicalmodels for bone remodeling see Doblaré and García-Aznar [27], García-Aznar et al. [34], Pivonka et al. [75], Ryser et al. [85] and references therein.Bone remodeling is out of the scope of the current work. Therefore, adetailed review of the models, mainly developed for the reparation phase ofbone regeneration, is given in the remainder of this section.Geris et al. [42] divide the models into three classes with respect to theessential mechanisms regulating the bone regeneration process. They distinguishmechanoregulatory models, bioregulatory models and mechanobioregulatorymodels. Such a classification has the advantage, that it reflectsthe chronological perspective of the mathematical modeling of bone regeneration.In mechanoregulatory models, the local mechanical environmentis considered as the only factor that influences the path of bone regeneration.On the contrary, in bioregulatory models only biochemical factors,like the influence of growth factors on various cell processes, are considered.Mechanobioregulatory models employ mechanical and biological effects simultaneously.An extended review of mathematical models for bone regeneration ispresented in Geris et al. [42]. The authors track the history of the developmentof several mathematical approaches to bone regeneration. Further,they outline the connections between various theoretical works and they listsome references to the original model formulations and to the numericalstudies carried out to verify applicability and to evaluate and compare theperformance of various approaches.First theoretical studies of bone regeneration were related to mechanoregulationof bone healing. Pauwels [72] defined a model for tissue differentiationdepending on local stresses and strains. Pauwels assumed, that deviatoricstrains promote formation of fibrous tissue and a hydrostatic pressure

6 Chapter 1. Introductionenhances formation of cartilage. Perren [73] and Perren & Cordey [74] presentedthe interfragmentary strain approach by defining the limiting (maximum)strains, which can be experienced by various tissues forming withinthe fracture site without rupturing.The aforementioned pioneering models were followed by two mechanoregulatorymodels, developed by Carter et al. [17, 18] and by Claes andHeigele [19]. In these models, biological tissues were considered as linearelastic materials, and the formation of bone, cartilage, fibrous tissue andfibrocartilage was related to a hydrostatic stress and to principal strains.In the approach by Carter et al. [17, 18], the history of a cyclic mechanicalloading was considered. Another tissue differentiation model was proposedby Prendergast et al. [77], who considered the biological tissues as poroelasticmaterials and who related differentiation into bone, cartilage or fibroustissue to different levels of a fluid flow and of a distortional strain. Huiskeset al. [47] estimated the numerical limits for qualitative relations, defined inPrendergast et al. [77]. A fuzzy-logic theoretical model was constructed byAment and Hofer [4], in which the mechanical stimulus was evaluated bymeans of a strain energy.The considered mechanoregulatory models were used in numerical simulations,in which various situations of bone regeneration were considered.The obtained results were usually compared to experiments. The mechanoregulatorymodels were used in numerical studies of fracture healing [15, 36,50, 51] , of bone regeneration near endosseous implants [8, 37, 38, 39] andof distraction osteogenesis [49, 52, 68].Purely mechanoregulatory models do not represent any biological processes,which take place during bone regeneration. Hence, these models areapplicable to studies, where only mechanical loading conditions are investigated.Another class of formalisms is represented by the bioregulatory models,which focus on representation of various biochemical and cell processes. In asimple approach by Adam [2, 3], a critical size defect in fracture healing wasdetermined from the the solution of a partial differential equation (PDE)defined for a growth factor concentration. More extended biological modelfor fracture healing was constructed by Bailon-Plaza and van der Meulen[11]. In this approach, a system of PDE’s was defined for seven variables:densities of MSC’s, of osteoblasts, of chondrocytes, concentrations of chondrogenicand osteogenic growth factors, and densities of connective/cartilageextracellular matrix and bone extracellular matrix. The authors representedthe processes of cell differentiation, proliferation, migration and death, synthesisand resorption of tissues. Some conclusions about a regulatory roleof growth factors in bone healing were drawn from numerical simulations.The model of Bailon-Plaza and van der Meulen [11] is rather populardue to its generality. The derivation of several other bioregulatory modelsis based on the approach due to Bailon-Plaza and van der Meulen [11].

1.3. Review of mathematical models for bone regeneration 7Geris et al. [40] extended the model of Bailon-Plaza and van der Meulen[11] by considering angiogenesis, by separating fibrous tissue and cartilagedensities, and by defining additional chemotactic and haptotactic terms forcell migration in the governing equations. This approach was applied tomodel fracture healing, including issues of atrophic nonunions [41, 45].Amor et al. [5, 6] adapted the model of Bailon-Plaza and van der Meulen[11] for peri-implant osseointegration. They disregarded chondrogenicgrowth factors, chondrocytes and cartilage, since the intermediate cartilaginousphase was not observed experimentally in bone healing occurring nearimplants [1, 14, 16, 21]. Chemotaxis of MSC’s was represented in equationsand a density of activated platelets was introduced as an additional modelvariable.A similar biological model was developed by Moreo et al. [67]. Thoughthe models of Amor et al. and Moreo et al. have several differences, ageneral representation of the main biological and cellular processes is verysimilar. The considered models allow to simulate bone regeneration aroundimplants with a different surface microstructure, which is represented implicitlythrough the values of model parameters and through initial andboundary conditions. A detailed comparison of the current two models isgiven in Chapter 4, in which a new moving boundary bioregulatory modelfor endosseous bone healing is derived.In contrast to purely mechano- and bioregulatory models, more complexmechanobioregulatory models allow to consider coupled effects of mechanoregulationand of biochemical factors on bone regeneration, in general.Further, an influence of a mechanical environment on cellular and biochemicalprocesses can be studied.This class of models is often derived from the bioregulatory and/or mechanoregulatorymodels, which were mentioned above. For example, a simplemechanobioregulatory model for normal fracture healing was developedby Lacroix et al. [58] and by Lacroix and Prendergast [57]. This formalismwas derived from the mechanoregulatory model of Prendergast et al. [77]and Huiskes et al. [47]. The authors made a first step to consider cellularprocesses together with mechanical factors by incorporating the randomwalk of MSC’s in their model.Bailon-Plaza and van der Meulen [12] introduced some mechanoregulatoryrelations into the coefficients of their previous model [11]. With a newmodel, the authors predicted beneficial effects of moderate, early loadingand adverse effects of delayed or excessive loading on bone healing. Mechanoregulatoryapproaches by Claes and Heigele [19] and Ament and Hofer[4] were used in a fuzzy logic model of Simon et al. [89] and Shefelbineet al. [88], in which such processes as vascularization, tissue destruction,intramembraneous and endochondral ossification, cartilage formation andbone remodeling were represented by a set of fuzzy logic rules, defined forthe various states of a mechanical environment within the healing cite.

8 Chapter 1. IntroductionGómez-Benito et al. [46] and García-Aznar et al. [35] constructed a mechanobioregulatorymodel in terms of a system of PDE’s, defined for densitiesof MSC’s, osteoblasts, fibroblasts and cartilage cells, and for volumefractions of the debris tissue, granulation tissue, fibrous tissue, bone and cartilage.Migration, differentiation, proliferation and death of cells, synthesisand damage of various tissues, and remodeling of bone were represented inthe model. These processes were assumed to depend on the second invariantof the deviatoric strain tensor. The authors applied their model to studythe influence of the intrafragmentary movement on the callus growth duringfracture healing.This approach was adapted by Reina-Romo et al. [83] to simulate distractionosteogenesis. One of the innovations introduced by Reina-Romo et al.[83], is that cell differentiation was determined by a history of mechanicalloading. A concept of cell plasticity, which was presented in Röder [84], wasincorporated into the model through the introduction of a maturation levelof MSC’s.The mechanoregulatory approach proposed by Prendergast et al. [77]was used in the models of Andreykiv et al. [9] and Isaksson et al. [53],which were developed for fracture healing. The biological processes, such asmigration, proliferation, differentiation and apoptosis of cells and productionand resorption of fibrous tissue, bone, and cartilage were assumed to berelated to the mechanical state of the tissues.Geris et al. [43] extended their previous bioregulatory model [40], bydefining dependencies for model parameters on mechanical stimuli, whichwere chosen according to Prendergast et al. [77]. This model was appliedto study impaired fracture healing. The authors checked predictions of themodel for various mechanoregulatory relations. They concluded that, if amechanical environment was assumed to influence angiogenesis alone, thenit was not possible to predict the formation of overload-induced nonunions.However, if both angiogenesis and osteogenesis were assumed to be effectedby the mechanical loading, then the overload-induced nonunion formationwas successfully predicted.The combined mechanobioregulatory models allow to consider test casesof both mechanical and biological approaches in the treatment of bone defects.For example, to investigate the effects of a mechanical stimulation andof the addition of growth factors. By use of these models it is also possibleto check the hypotheses for the influence of mechanical factors on variousbiological processes [42]. Mechanobioregulatory models are more general,compared to mechanoregulatory and bioregulatory approaches, and at thesame time, they are more complicated, due to a larger number of assumptions,involved in their formulation. The necessary effort to validate a modelincreases with the level of complexity of a model and the number of modelparameters therein. Therefore, it is not possible to choose the ’best’ modelor the ’best’ approach. The choice of a theoretical model should be deter-

1.4. Structure and subjects of the thesis 9mined by the most important characteristics of the considered problem, i.e.by a form of bone regeneration (for example, fracture healing or implantosseointegration), by a subject of the research, by available experimentaldata, etc.The issue of a proper model verification is even more critical in the caseof a poor cooperation between experimental researchers and developers oftheoretical models. Contemporary mathematical approaches are developedto accommodate the most important features in various applications of boneregeneration, which are known from experiments. Theoretical models allowto make some predictions, for example, about advanced treatment techniques.As it is reported by Geris et al. [44], there are no experimentalstudies on treatment strategies that were designed and optimized by usingmathematical models. In other words, an advance in bone regenerationresearch depends on the connection established between experiments andtheoretical modeling both on the stage of model construction and on thestage of model verification. The models have to be validated against experimentalsettings, that are different from the ones used on the stage of modelformulation [42].Bone regeneration is a multiscale process. In recent theoretical modelsa single time and space scale are considered. Cellular processes are representedon a tissue-level by means of a continuous approach. For example,random walk of cells is represented by a diffusion term in the governingequations. Development of multiscale models is another crucial issue ofbone regeneration research [42]. It is important to understand the mechanisms,in which experimental and theoretical knowledge can be transferredbetween the various levels. For instance, how to derive a proper representationof cell haptotaxis on tissue level from the knowledge of this processat the cellular level. Mechanical stresses and strains, which are often consideredas the main regulation factors, are represented in the recent modelson the tissue-level. In reality, tissue strains and stresses can differ frommechanical stimulation, experienced by cells [98]. Therefore, an advancedrepresentation of the mechanical behavior of biological tissues, i.e. moreaccurate models than linear elasticity or poroelasticity, and considerationof mechanical stimuli on various levels (the tissue-level, the cellular level,the intracellular level) is another possible direction for the future researchin modeling bone regeneration [42].1.4 Structure and subjects of the thesisAs it follows from Section 1.3, a large number of various approaches wasdeveloped in the field of mathematical modeling of bone regeneration. Inthe present work, various mathematical and numerical methods are usedto study several areas of modeling bone regeneration. In Chapter 2, a

10 Chapter 1. Introductionstability analysis is performed to study some characteristics of the behaviorof diffusion-taxis systems that are considered in the most of the recent(mechano)bioregulatory models. An advanced representation of graduallyevolving cell differentiation is presented in Chapter 3. This differentiationapproach is implemented into a new moving boundary model for intramembraneousossification, which is described in Chapter 4. A numerical algorithmfor the solution of this model in two dimensional spatial domain isdefined in Chapter 5. Further in this section, the mentioned subjects of thethesis are introduced in more detail.In most of the considered models, a continuous approach is employedand a governing system of PDE’s is defined for a number of unknowns,which can represent cell densities, growth factor concentrations, tissue volumefractions, mechanical stimuli, etc. Cell migration (if it is consideredfor a certain cell phenotype) is assumed to be caused by random walk ofcells, by chemotaxis and/or by haptotaxis. Random walk is represented byparabolic diffusion terms, and the cell flux due to chemo- or haptotaxis isrepresented by hyperbolic taxis terms. Other cellular processes, like proliferation,differentiation and death are usually introduced by reaction terms.The equations defined for the evolution of the growth factor concentrationscontain diffusion and reaction terms, which model, respectively, diffusion,and production and decay. Synthesis and resorption of various tissues arerepresented by reaction equations. Therefore, the systems of PDE’s thatconstitute continuous models for the reparation phase of bone regenerationare usually of taxis-diffusion-reaction type.Remark 1.1. In this work, the term “advection-diffusion-reaction system”is used to denote the equations, which contain taxis, diffusion and reactionterms. This terminology is adopted from Hundsdorfer and Verwer [48],which is a fundamental work about the numerical methods for the systemsof the considered type. In various systems, the advection terms representa flux of some quantity caused by an advection mechanism, i.e. the fluxdriven, for example, by the fluid velocity. In the case of chemotaxis, the fluxof cells is driven by a growth factor gradient. Therefore, advection can berelated to a taxis mechanism in a certain way.Some important characteristics of such systems, such as a stability of thesolution and a formation of ’wave-like’ profiles in the spatial distributionof cell densities and growth factor concentrations, are studied in Chapter2. Such wave-like profiles obtained in the solutions for bone regenerationmodels are unphysical and contradict experimental observations. Therefore,it is important to understand the reason of the appearance of these profiles.Remark 1.2. ’Wave-like’ patterns are also observed in solutions of advectiondiffusion-reactionequations, defined in the models for bacteria colonies [66,69, 91].As a particular example, a bioregulatory model by Moreo et al. [67]

1.4. Structure and subjects of the thesis 11is chosen, due to its sufficient simplicity and generality. First of all, theconsidered model contains diffusion and advection (taxis) terms, which areresponsible for the appearance of a wave-like profile in the solution. Suchterms are also included in similar bioregulatory models, developed by [5, 11].At the same time, the model of Moreo et al. [67] is simpler than anotherbioregulatory approach proposed by Geris et al. [40], in which angiogenesiswas represented. Mechanobioregulatory models are not mentioned here,since consideration of mechanical stimuli only complicate models and hasno added value for the present analysis.Some conditions for the formation of wave-like profiles are determinedfrom a linear stability analysis. These conditions are expressed in terms ofseveral quantitative relations for the model parameters. The linear stabilityanalysis, which is described in Chapter 2, provides some insight into thecharacteristic behavior of the bioregulatory models for bone regeneration,in which a diffusion-taxis mechanism of cell migration is incorporated.Another cellular process, which is of great importance for bone regeneration,is cell differentiation. In all continuous models, excluding the modelof Reina-Romo et al. [83], differentiation is represented by reactive terms,which correspond to immediate switch of cell phenotype. Reina-Romo et al.[83] considered the dependence of cell migration on the loading history byintroducing a maturation level of cells. The idea of a temporal transformationof cells into another phenotype is elaborated in Chapter 3, in which ageneral mathematical formalism is developed for the evolutionary cell differentiation.The considered approach can be applied to an arbitrary number ofcell types and to any set of factors influencing cell differentiation. The evolutionarymodel allows to consider a cell differentiation path, determinedby a history of mechanical and/or biochemical stimulation. The presentapproach results into a final time of differentiation, which can be importantin some applications. For instance, the boundary conditions for theperi-implant osseointegration model, which is described in Chapter 4, canbe applied, only if the time of differentiation of MSC’s into osteoblasts isfinal. An example of the application of the evolutionary approach for differentiationof MSC’s regulated by the mechanical environment, is given by asimple peri-implant osseointegration model described in Section 3.3. Numericalsimulations are carried out and results are compared with experimentaldata in Section 3.4.In Chapter 4, a new moving boundary model for bone healing aroundendosseous implants is presented. The model is derived to accommodate thefact, that synthesis of bone within a peri-implant region occurs in the formof intramembraneous ossification [1, 14]. Davies [24] describes this processas a direct apposition of new bone matrix on a pre-existing rigid surface.The evolution of the bone forming surface (ossification front) is incorporatedin the model directly through a movement of the boundary of the soft tissueregion, which is considered as a physical domain, where various cellular

12 Chapter 1. Introductionand biochemical processes are considered. The present approach conceptuallydiffers from the recent continuous models for endosseous bone healingmentioned in Section 1.3, in which bone tissue is represented by its volumefraction defined within the stationary physical domain. The moving boundarymodel is formulated in terms of the system of time-dependent nonlinearadvection-diffusion-reaction equations, defined within the physical domainthat evolves in time. The numerical algorithm, used to solve this challengingmathematical problem in two dimensional physical domain, is described inChapter 5. The solution procedure involves some methods, which are thefinite volume method, which is applied on an unstructured two dimensionalgrid, the embedded boundary or Cartesian boundary method and the levelset method, which is used to track the movement of the boundary of theirregular physical domain. In Chapter 6, final conclusions are drawn andsome directions for future research are given.

14 Chapter 2. Linear stability analysis2.1 IntroductionA stability analysis is performed for a biological model for peri-implantosseointegration, which was proposed in Moreo et al. [67]. This model allowsto take into account an implant surface microtopography. The resultspresented in Moreo et al. [67] were in agreement with experiments. Theauthors reported, that model can predict contact and distance osteogenesismodes of bone formation [24].From the numerical simulations, which were performed for a differentgeometry of the healing site, it was found that the system of equations,proposed in Moreo et al. [67], is characterized by appearance of a wave-likeprofile in the solution for a certain range of parameters. This profile wasinitially recognized in the solution of the model equations for the 1D domainof length 2.5mm (Figure 2.1b). This domain was chosen for the simulationsof bone formation near the cylindrical implant, located within the bonechamber, used in the experiments by Duyck et al. [29], Vandamme et al.[92, 93, 94]. The authors reported that in experiments, new bone was formedat all distances from the host bone, and integration of bone and implant wasachieved. That wave-like profile has not been noticed by Moreo et al. [67],since for the geometry used in his simulations, in which the distance fromhost bone to implant was 0.6mm, only a part of ’wave’ is visible in thesolution (Figure 2.1a), and a wave-like profile is not distinguishable. Forlarger domains, more ’waves’ appear in the solution. The solution for thedomain of length 5mm is shown in Figure 2.1c.s242(a)01.5 2x1050(b)1 2 3x86420(c)2 4 6xm2101.5 2x64201 2 3x4202 4 6xDay 300 Day 500 Day 1000Figure 2.1: Osteogenic cell m and growth factor 2 s 2 distributions at differenttime moments, obtained from the numerical solution of model equations,defined in Moreo et al. [67], for 1D axisymmetric domain with length (a)L = 0.6 mm and (b) L = 5 mmThe conditions, under which a wave-like profile appears, are studied in

2.2. Biological model 15this chapter. Such a wave-like profile in the solution for cell densities andgrowth factor concentrations is not realistic. In some cases it also leads toa wave-like distribution of bone matrix inside the peri-implant region. Thisdistribution is in contradiction with experimental observations, from whichit follows, that bone forms by deposition on the preexisting bone matrix,and no isolated bone regions appear [1, 14]. Thus, it is important to learn,under which conditions the model shows this unphysical behavior.The proposed approach is to study the linear stability of the constant solutionsof the system. As the full system of equations is large and extremelycomplicated for analytic derivations, an equivalent simplified system withsimilar properties will be defined.The phenomenon of a wave-like profile in the solution can be related tothe appearance of bacterial patterns in a liquid medium, described mathematicallyby similar systems of partial differential equations. The formationof bacterial patterns is studied in Miyata and Sasaki [66], Myerscough andMurray [69], Tyson et al. [91].In Section 2.2 the system of equations proposed in Moreo et al. [67] isreviewed. The linear stability analysis of the system is carried out in Section2.3. In Section 2.4 analysis results are validated with a series of numericalsimulations. Finally, in Section 2.5 some conclusions are drawn.2.2 Biological modelThe original model proposed in Moreo et al. [67] consists of eight equations,defined for eight variables, representing densities of platelets c, osteogeniccells m, osteoblasts b, concentrations of two generic growth factor types s 1and s 2 , and volume fractions of fibrin network v fn , woven bone v w , andlamellar bone v l . The above notations are introduced for non-dimensionalcell densities and growth factor concentrations, i.e. for those, related to somecharacteristic values. If ˆf and fc are notations of a dimensional variableand of its characteristic value, then a non-dimensional variable f is definedas f = ˆf/f c , f = c, m, b, s 1 , s 2 . The following characteristic values areproposed: c c = 10 8 platelets/ml, m c = 10 6 cells/ml, b c = 10 6 cells/ml,s 1c = 100 ng/ml, s 2c = 100 ng/ml. The model equations are:∂c∂t = ∇ · [D c∇c − H c c∇p] − A c c, (2.1)where D m and A c are the coefficients of random migration and death ofplatelets. The term ∇ · [H c c∇p] represents a “linear taxis”. It accounts forthe transport of platelets towards the gradient of the adsorbed proteins p,which is a predefined function of the distance from the implant surface d.

16 Chapter 2. Linear stability analysisAccording to Moreo et al. [67], it is defined asp(d) ={0.5(1 − d0.1), if 0 ≤ d < 0.1.0, if d ≥ 0.1∂m∂t = ∇ · [D m∇m − m(B m1 ∇s 1 + B m2 ∇s 2 )](+ α m0 + α ms 1+ α )ms 2m(1 − m) − (α p0 + α mbs 1)m − A m m,β m + s 1 β m + s 2 β mb + s 1(2.2)where the terms in the right-hand side represent random migration, chemotaxis,cell proliferation, differentiation into osteoblasts, and death respectively;∂b∂t = (α p0 + α mbs 1)m − A b b, (2.3)β mb + s 1where A b is the rate of osteoblast death;∂s( 1∂t = ∇ · [D αc1 ps1∇s 1 ] +β c1 + p + α )c2s 1c − A s1 s 1 , (2.4)β c2 + s 1where the terms in the right-hand side model random migration, growthfactor secretion and decay respectively;∂s 2∂t = ∇ · [D s2∇s 2 ] + α m2s 2β m2 + s 2m + α b2s 2β b2 + s 2b − A s2 s 2 , (2.5)where the first term in the right-hand side determines random migration,the second and the third ones – growth factor secretion, and the last one –decay;∂v fn= − α ws 2bv fn (1 − v w ), (2.6)∂t β w + s 2∂v w∂t= α ws 2β w + s 2bv fn (1 − v w ) − γv w (1 − v l ), (2.7)∂v l∂t = γv w(1 − v l ), (2.8)where terms containing coefficients α w , β w and γ model the substitution ofthe fibrin network by woven bone and the remodeling of woven bone intolamellar bone.Moreo et al. [67] proposed the following initial and boundary conditionsfor this equation set. Let Ω be a problem domain with the boundary ∂Ω,

2.3. Stability analysis 19The equations for platelets c and growth factor 1 s 1 (2.1) and (2.4), canbe solved separately from the other equations. That means, that the evolutionof the platelet density c(x, t) and growth factor 1 concentration s 1 (x, t)does not depend on the evolution of other biological and chemical speciesinvolved in the model. Equation (2.1) contains a term, corresponding to thedeath of platelets, but it does not contain a term, corresponding to the productionof platelets. Therefore, the total amount of platelets decays to zerowith time. The production of growth factor 1 s 1 is proportional to plateletsconcentration, and thus the production of s 1 also decays with time, whiledeath rate A s1 is constant in time. It can be proved, that the integrals ofplatelet density and growth factor 1 concentration over the problem domaintend to zero with time, if zero flux on the boundaries is considered. If negativevalues in the solution for c(x, t) and s 1 (x, t) are avoided (otherwise thesolution becomes biologically irrelevant), then it follows, that these functionstend to zero almost everywhere in the problem domain. Numericalsimulations confirm (Figure 2.2), that for a large time t the solution s 1 (x, t)is very close to zero.The stability analysis deals with the asymptotic behavior of the system,that is with the behavior of the solution for long time periods. Therefore,the simplified system is derived from equations (2.2), (2.5) and (2.3), andfrom assumption s 1 (x, t) ≡ 0 in the following form:∂m(∂t = ∇·[D m∇m − B m2 m∇s 2 )]+ α m0 + α )ms 2m(1−m)−(α p0 +A m )m,β m + s 2(2.14)∂s 2∂t = ∇ · [D s2∇s 2 ] + α m2s 2β m2 + s 2(m + b) − A s2 s 2 , (2.15)∂b∂t = α p0m − A b b. (2.16)Remark 2.2. In derivation of (2.15) it was assumed, that α b2 = α m2 andβ b2 = β m2 . These simplifying assumptions are in line with the values forα b2 , α m2 , β b2 and β m2 , proposed by Moreo et al. [67], which were introducedin (2.11).Remark 2.3. As it was mentioned, the concentration of s 1 becomes close tozero after a certain period of time. Then, differentiation of osteogenic cellsinto osteoblasts is roughly described by the term α p0 m, as this is done inequations (2.14), (2.16). This term turns to zero, if α p0 = 0, as was proposedby Moreo et al. [67]. The solution of (2.16), defined as b(x, t) = b 0 (x)e −A bt ,converges to zero with time. From a biological point of view, this means,that osteogenic cells stop to differentiate after a certain time period. Thereis no source of newly formed osteoblasts, and their amount decreases to zero,due to cell death.

20 Chapter 2. Linear stability analysisIf α p0 ≠ 0, then differentiation takes place also when s 1 is zero. Thisallows to obtain the solution for osteoblasts, which does not converge tozero, and hence, is more realistic from biological point of view. For thisreason, the parameter values in equation (2.12), as proposed by Moreo et al.[67], and the alternative values in equation (2.13) are considered.Moreo et al. [67] investigated the linear stability of the constant solutionsof the system, which is similar to system (2.14)–(2.16), against purelytemporal perturbations. In this paper, the system stability with respect toarbitrary perturbations (including non-homogeneous perturbations) is studied.Constant solutions z ′ = (m ′ , s ′ , b ′ ) of system (2.14)–(2.16) are derivedfrom the algebraic system:⎧ (α m0 + α ms ′ )2⎪⎨ β m + s ′ m ′ (1 − m ′ ) − (α p0 + A m )m ′ = 0,2α m2 s ′ 2β m2 + s⎪⎩′ (m ′ + b ′ ) − A s2 s ′ (2.17)2 = 0,2α p0 m ′ − A b b ′ = 0.Two solutions of the above system have been denoted by Moreo et al. [67]as:where• “Chronic non healing state”: z t = (0, 0, 0),• “Low density state”: z 0 = (m 0 , 0, b 0 ),m 0 = 1 − α p0 + A mα m0, b 0 = α p0A bm 0 . (2.18)From system (2.17), it follows, that root s ′ 2 can not be equal to −β m < 0or to −β m2 < 0. Vectors z − = (m − , s 2− , b − ) and z + = (m + , s 2+ , b + ) aredefined ass 2± = −a 1 ± √ a 2 1 − 4a 2a 02a 2, (2.19)m ± = A bA s2 (s 2± + β m2 )α m2 (A b + α p0 )= A s2(s 2± + β m2 ), b ± = α p0m ± , (2.20)χA bwhere⎧ (a 2 = A s2 1 + α )m,⎪⎨α m0(a 1 = 1 + α )m(β m2 A s2 − χm 0 ) + α mχ(m 0 − 1) + β m A s2 ,α m0 α m0⎪⎩a 0 = β m (β m2 A s2 − χm 0 ),(2.21)

2.3. Stability analysis 21χ = α m2 (1 + α p0 /A b ) , (2.22)and m 0 is defined in equation (2.18). They are the solutions of system(2.17), if s 2± /∈ {−β m ; −β m2 }. Therefore, depending on the values of modelparameters, system (2.17) can have two, three or four solutions.Remark 2.4. From the derivation of the expression (2.19), which is not givenhere, it follows, that at least one of the roots s 2+ and s 2− is equal to −β m , ifand only if A s2 (β m2 − β m ) = χ. And at least one of the roots s 2+ and s 2− isequal to −β m2 , if and only if (α m0 −α p0 −A m )(β m −β m2 ) = α m β m2 . For thechosen parameter values in equations (2.11), (2.12), (2.13), β m2 = β m > 0,α m > 0, χ = α m2 (1 + α p0 /A b ) > 0. Hence, s 2± ≠ −β m and s 2± ≠ −β m2 forthe considered parameter values, and system (2.14)–(2.16) has four constantsolutions z t , z 0 , z + and z − .It should be mentioned here, that for the existence of real s 2± the necessarycondition is:a 2 1 − 4a 2 a 0 ≥ 0. (2.23)This necessary condition is written in terms of the model parameters as:( (χ m 0 + α ) ( (m− A s2 β m + β m2 1 + α ))) 2mα m0 α m0(− 4A s2 β m 1 + α )( (m(β m2 A s2 − χm 0 ) = χ m 0 + α )mα m0 α m0a 2 1 − 4a 2 a 0 =where(+ χ+ χm 0 + α ) (mη − ηα m0(m 0 + α )m(η − 2ξ) + ξ 2 − ηα m0β m2 A s2 + χ α mα m0)) 2− ξ( (χ m 0 + α )) 2mα m0=(β m2 A s2 + χ α )m≥ 0, (2.24)α m0( (ξ = A s2 β m + β m2 1 + α ))(m, η = 4A s2 β m 1 + α )m. (2.25)α m0 α m0From (2.24) it is derived, that (2.23) is equivalent to:⎡ ( )√⎣ χ m 0 + αmαα m0≥ −A s2 β m m α m0+ η αmα m0χ,( )√χ m 0 + αmαα m0≤ −A s2 β m m α m0− η αmα m0χ.(2.26)The sign of s 2± depends on the sign of coefficients a 1 and a 0 (coefficienta 2 is larger than zero, which follows from its definition). Both roots will bepositive if a 1 < 0 and a 0 > 0 and if inequality (2.23) holds.For the parameter values in equations (2.11), (2.12) the constant solutionshave values: m 0 ≈ 0.9920, b 0 = 0; m − ≈ 0.0201, s 2− ≈ −0.0498,

22 Chapter 2. Linear stability analysisb − = 0; m + ≈ 0.9959, s 2+ ≈ 2.3898, b + = 0; and for parameter values(2.11), (2.13): m 0 ≈ 0.3253, b 0 ≈ 8.1293; m − ≈ 0.0012, s 2− ≈ −0.0245,b − ≈ 0.0290; m + ≈ 0.6623, s 2+ ≈ 42.9271, b + ≈ 16.5486.Remark 2.5. For the chosen parameter sets (2.11), (2.12) and (2.11), (2.13),growth factor 2 concentration s 2− is negative, which is unphysical. It isdesirable to avoid such a negative concentration of growth factor 2 in thesolution of the problem (2.14)–(2.16). Calculations show, that for the chosenparameter values there are two positive eigenvalues of the Jacobean ofthe equation system, linearized for the case of small purely temporal perturbationsnear the constant solution z − . Hence, the constant solution z −is unstable against temporal perturbations. It is possible to obtain nonnegativevalues in the numerical solution for s 2 , provided a sufficiently smalltime step and mesh size are chosen and positive initial values for concentrationsof cells and growth factor are considered.2.3.2 Non-homogeneous perturbationsIn this section, the stability of constant-state solutions of system (2.14)–(2.16) is analyzed. The present approach is valid for a domain in any coordinatesystem, for which eigenfunctions of Laplace operator can be found.In this paper, the examples of the eigenfunctions are given for domains in 1DCartesian coordinates and in 1D axisymmetric coordinates. The independentspace coordinate is denoted by x for both coordinate systems. Supposethat non-homogeneous perturbations m p (x, t), s 2p (x, t) and b p (x, t) are imposedon the constant solution (m ′ , s ′ 2 , b′ ). Then the solution is given in theform:⎧⎪⎨⎪ ⎩m(x, t) = m ′ + εm p (x, t),s 2 (x, t) = s ′ 2 + εs 2p (x, t),b(x, t) = b ′ + εb p (x, t),(2.27)where |ε| ≪ 1. Then, equations (2.27) are substituted into (2.14)–(2.16),and linearized with respect to small ε:⎧⎪⎨⎪⎩∂m p∂t[(=D m ∇ 2 m p − m ′ B m2 ∇ 2 s 2p + α m0 + α ms ′ 2β m + s ′ 2]− (α p0 + A m ) m p +α mβ m(β m + s ′ 2 )2 m′ (1 − m ′ )s 2p ,∂s 2p=D s2 ∇ 2 s 2p + α m2s ′ 2∂tβ m2 + s ′ (m p + b p )2[ αm2 β]m2+(β m2 + s ′ 2 )2 (m′ + b ′ ) − A s2 s 2p ,∂b p∂t =α p0m p − A b b p .)(1 − 2m ′ )(2.28)

2.3. Stability analysis 23Let us denote the problem domain as [x 0 , x 0 + L]. Assume, that on theboundaries the flux of cells and of growth factors is zero. Then, perturbationsare considered in the form:⎧∞∑m p (x, t) =C0 m (t) + Cn m (t)φ n (x),⎪⎨⎪⎩s 2p (x, t) =C s20 (t) +b p (x, t) =C b 0(t) +n=1∞∑n=1C s2n (t)φ n (x),∞∑Cn(t)φ b n (x).n=1(2.29)Functions C0 m (t), Cs2 0 (t), Cb 0 (t) represent purely temporal perturbations.Eigenfunctions φ n (x) satisfy equation ∇ 2 φ n (x) = −knφ 2 n (x) and consideredboundary conditions, i.e. zero flux on the boundaries: ∇φ n (x 0 ) = ∇φ n (x 0 +L) = ⃗0.If Cartesian coordinates are considered, then the function φ n (x) is givenas φ C n (x) = cos(k n (x − x 0 )), where k n = πnL , n = 1, 2, . . . . In this case k n isa wavenumber.In the case of axisymmetric coordinates functions φ n (x) have the formφ a n(x) = Y 0 ′(knx 0 )J 0 (k n x) − J 0 ′ (k nx 0 )Y 0 (k n x), where J 0 (k n x) and Y 0 (k n x)are Bessel functions, k n =wnx 0 +L and w n, n = 1, 2, . . . are positive real zerosof the function Φ(w) = −Y 0 ′(knx 0 )J 1 (w) + J 0 ′ (k nx 0 )Y 1 (w). Functions φ a n(x),n = 1, 2, . . . are not periodic. They can be roughly described as “waves”with variable in space wavelength and magnitude. For simplicity, k n will bereferred to as ’wavenumber’, also if it is introduced in functions φ a n(x).Remark 2.6. Perturbation modes φ n (x), n = 1, 2, . . . by their definition havepositive wavenumbers k n > 0. For the sake of generality, purely temporalperturbations are considered as perturbations of mode n = 0 with zerowavenumber k 0 = 0 and φ 0 (x) ≡ 1.Substitution of equations (2.29) into system (2.28) yields:⃗C ′ n(t) = A kn⃗ Cn (t), n = 0, 1, . . . , (2.30)where⎡C m ⎤n (t)⎢ ⃗C n (t) = ⎣ Cn s2 ⎥(t) ⎦ , n = 0, 1, . . . , (2.31)Cn(t)b⎛⎞A kn(1,1) A kn(1,2) 0α m2 s ′ 2α m2 s ′ 2A kn = ⎜⎝ β m2 + s ′ A kn(2,2)2β m2 + s ′ ⎟2 ⎠ , (2.32)α p0 0 −A b

24 Chapter 2. Linear stability analysiswhere(A kn(1,1) = α m0 + α ms ′ )2β m + s ′ (1 − 2m ′ ) − (α p0 + A m ) − knD 2 m ,2A kn(1,2) =α mβ m(β m + s ′ 2 )2 m′ (1 − m ′ ) + knB 2 m2 m ′ ,A kn(2,2) =α m2β m2 ( α p0 )1 + m ′(β m2 + s ′ − A s2 − k2 )2 AnD 2 s2 .bThen from (2.30):⃗C n (t) = e A kn t ⃗ C0n , n = 0, 1, . . . , (2.33)where C ⃗ n 0 define the perturbations imposed on the constant solution of thesystem initially at time t = 0:⎡ ⎤m p (x, 0) ∞∑⎣ s 2p (x, 0) ⎦ = ⃗C nφ 0 n (x).b p (x, 0) n=0Thus the solution of (2.28) is written as:⎡ ⎤m p (x, t) ∞∑⎣ s 2p (x, t) ⎦ = e A kn t C ⃗ 0n φ n (x). (2.34)b p (x, t) n=0The magnitude of perturbations ‖ C ⃗ n (t)‖ = ‖e A kn t C ⃗ 0n ‖ of mode n, willgrow in time, if at least one of the eigenvalues of the matrix A kn is a positivereal number or a complex number with a positive real part. And ‖ C ⃗ n (t)‖ willconverge to zero, if all the eigenvalues of A kn are real negative, or complexnumbers with the real part less than zero. If the matrix A kn has preciselyone zero eigenvalue, and other eigenvalues are real negative or complex witha negative real part, then small perturbations remain small for infinite timeperiod.It is not complicated to find expressions for the eigenvalues of A kn ,evaluated at the ‘chronic non healing state’ z t = (0, 0, 0) and ‘low densitystate’ z 0 = (m 0 , 0, b 0 ). For the constant solution z t eigenvalues of A kn are:λ 1t (k 2 n) = α m0 m 0 − k 2 nD m > 0, if 0 ≤ k 2 n < α m0m 0D m,λ 2t (k 2 n) = −A s2 − k 2 nD s2 < 0, λ 3t (k 2 n) = −A b < 0.(2.35)Therefore, if m 0 is positive, constant solution z t is unstable against purelytemporal √ perturbations and perturbations with a small wavenumber 0

2.3. Stability analysis 25Remark 2.7. For negative m 0 , ’chronic non-healing state’ z t will becomestable against perturbations with any wavenumber. Further the constantsolution z 0 will contain an unphysical negative concentration for osteogeniccells. Inequality m 0 = 1 − α p0+A mα m0< 0 implies, that differentiation anddeath of osteogenic cell dominate over their production. Therefore, thissituation is not relevant for the considered model of bone formation, andfurther m 0 > 0 is assumed a priori.For the constant solution z 0 = (m 0 , 0, b 0 ) matrix A kneigenvalues are:λ 10 (k 2 n) = −α m0 m 0 − k 2 nD m < 0,λ 20 (k 2 n) = α m2β m2m 0 (1 + α p0A b) − A s2 − k 2 nD s2 , λ 30 (k 2 n) = −A b < 0.(2.36)If expression α m2β m2m 0 (1 + α p0A b) − A s2 takes a positive value, which is true forthe current parameter values in equations (2.11), (2.12) and (2.13), then theconstant ( solution z 0 is unstable ) against perturbations with the wavenumberskn 2 < αm2β m2m 0 (1 + α p0A b) − A s2 /D s2 . The largest eigenvalue λ 20 correspondsto zero wavenumber k 0 , i.e. to the purely temporal mode of perturbation.The eigenvalues of the matrix A kn defined at points z − and z + can notbe found in such a trivial manner, as for the constant solutions z t and z 0 .They are obtained from the characteristic equation, which is a non-trivialcubic algebraic equation. Therefore, instead of analyzing the expressions forthe eigenvalues, which are extremely complicated in this case, a differentapproach is proposed, which based on a reduction of the considered systemof equations to two equations with similar stability properties.Remark 2.8. For the chosen parameter values, see expressions (2.11), (2.12)and (2.11), (2.13), s 2− is negative, hence the constant solution z − is biologicallyirrelevant in this case. Therefore, only the stability of the constantsolution z + and not of z − is analyzed. The stability analysis, being introducedfor z + , is not valid for the constant solution z − , if it contains thenegative value of growth factor concentration. Calculations also show, thatfor parameter values (2.11), (2.12) and (2.13), the constant solution z − isunstable against at least purely temporal perturbations.2.3.3 Stability of the system of two equationsTo simplify the stability analysis, system (2.14)–(2.16) is reduced to a systemof two equations. For this reduced system the assumptionb(x, t) = α p0A bm(x, t) (2.37)

26 Chapter 2. Linear stability analysisis made, instead of equation (2.16). The system is defined as:⎧∂m∂t ⎪⎨=∇ · [D m∇m − B m2 m∇s 2 )](+ α m0 + α )ms 2m(1 − m) − (α p0 + A m )m,β m + s 2∂s 2 ⎪⎩∂t =∇ · [D s2∇s 2 ] + α m2s 2(1 + α p0)m − A s2 s 2 .β m2 + s 2 A b(2.38)Substitution of (2.37) into equation (2.16), yields the condition ∂b∂t= 0,which is not true in general case. Therefore, system (2.14)–(2.16) and system(2.38) are not equivalent, and their stability properties are different ingeneral. However, it will be shown in Section 2.3.4, that there is a certainsimilarity (or correspondence) between the stability properties of the twosystems. This similarity is sufficient, to transfer important results, obtainedfrom the stability analysis for the system of two equations (2.38), onto thesystem of three equations (2.14)–(2.16).System (2.38) has constant solutions, that are analogous to those of system(2.14)–(2.16). They are: ˜z t = (0, 0), ˜z 0 = (m 0 , 0), ˜z + = (m + , s 2+ ),˜z − = (m − , s 2− ). Linearizing the system near the point (m ′ , s ′ 2 ), withm(x, t) = m ′ + εm p (x, t) and s 2 (x, t) = s ′ 2 + εs 2p(x, t), yields:⎧⎪⎨⎪⎩∂m p∂t∂s 2p∂t[(=D m ∇ 2 m p − m ′ B m2 ∇ 2 s 2p + α m0 + α ms ′ 2β m + s ′ 2]− (α p0 + A m ) m p +=D s2 ∇ 2 s 2p + α m2s ′ 2β m2 + s ′ 2+α mβ m(β m + s ′ 2 )2 m′ (1 − m ′ )s 2p ,(1 + α p0A b)m p[αm2 β m2(β m2 + s ′ 2 )2 (1 + α p0A b)m ′ − A s2]s 2p .Considering solutions of the form⎧m ⎪⎨ p (x, t) =⎪⎩ s 2p (x, t) =∞∑Cn m (t)φ n (x),n=0∞∑n=0C s2n (t)φ n (x),)(1 − 2m ′ )(2.39)and substituting them into system (2.39), for each n = 0, 1, . . . we arrive at:⎡dC m ⎤n (t) [ ]⎢ dtCm⎥⎣ dCn s2 (t) ⎦ = n (t)Ãk nCn s2 ,(t)dt

2.3. Stability analysis 27where⎛⎜à kn = ⎝ α m2 s ′ 2β m2 + s ′ 2à kn(1,1)(1 + α )p0A bà kn(1,2)à kn(2,2)⎞⎟⎠ ,(à kn(1,1) = α m0 + α ms ′ )2β m + s ′ (1 − 2m ′ ) − (α p0 + A m ) − knD 2 m ,2à kn(1,2) =α mβ m(β m + s ′ 2 )2 m′ (1 − m ′ ) + knB 2 m2 m ′ ,à kn(2,2) =α m2β m2 ( α p0 )1 + m ′(β m2 + s ′ − A s2 − k2 )2 AnD 2 s2 .bFirst the stability properties of system (2.39) are studied, and then it isdetermined, how they are related to the stability properties of the systemof three equations (2.28). Since s 2+ ≠ −β m2 , then from relation (2.20) itfollows that m + ≠ 0. Therefore, the matrix Ãk nevaluated in the point(m + , s 2+ ), can be simplified. From the first equation of system (2.17) itfollows: (α m0 + α )ms 2+(1 − m + ) − (α p0 + A m ) = 0. (2.40)β m + s 2+Then:(à kn(1,1)(m + , s 2+ ) = α m0 + α )ms 2+(1 − 2m + ) − (α p0 + A m ) − kβ m + snD 2 m( 2+(=2 α m0 + α ))ms 2+(1 − m + ) − (α p0 + A m )β m + s 2+( (− α m0 + α ))ms 2+− (α p0 + A m ) − kβ m + snD 2 m2+= − α m0 m 0 − α ms 2+β m + s 2+− k 2 nD m ,s 2+à kn(2,1)(m + , s 2+ ) = α m2s 2+(1 + α p0) = χ ,β m2 + s 2+ A b β m2 + s 2+where χ is defined in (2.22). From equation (2.20), it follows thatà kn(1,2)(m + , s 2+ ) =α mβ m(β m + s 2+ ) 2 m +(1 − m + ) + knB 2 m2 m += A s2α m β mχ(β m + s 2+ )β m2 + s 2+β m + s 2+(1 − m + ) + k 2 nB m2 m + .Everywhere in the calculations, presented in Moreo et al. [67] and in thispaper, the same values are used for the parameters β m and β m2 . So bothnotations β m and β m2 is used, though β m2 = β m is supposed below. Thenà kn(1,2)(m + , s 2+ ) =A s2α m β mχ(β m + s 2+ ) (1 − m +) + k 2 nB m2 m + ,

28 Chapter 2. Linear stability analysisα m2β m2Ã kn(2,2)(m + , s 2+ ) =(β m2 + s 2+ ) 2 (1 + α p0)m + − A s2 − kAnD 2 s2b=Therefore,β m2= A s2 ( − 1) − k 2 s 2+β m2 + snD s2 = −A s2 − k2+ β m2 + snD 2 s2 .2+Ã kn (m + , s 2+ )(−αm0 m 0 − αms 2+β m+s 2+− knD 2 mχ s 2+β m2 +s 2+−A s2s 2+A s2 α mβ mχ(β m+s 2+ ) (1 − m )+) + knB 2 m2 m +β m2 +s 2+− knD 2 s2Then the characteristic equation for the matrix Ãk n, evaluated at point(m + , s 2+ ), is given by:whereλ 2 (k 2 n) + b(k 2 n)λ(k 2 n) + c(k 2 n) = 0, (2.41)b(k 2 n) = −(Ãk n(1,1)(m + , s 2+ ) + Ãk n(2,2)(m + , s 2+ ))= knD 2 m + α m0 m 0 + α ms 2++ k 2 s 2+β m + snD s2 + A s22+ β m2 + s 2+s 2+= kn(D 2 m + D s2 ) + α m0 m 0 + (α m + A s2 ) ,β m + s 2+.c(k 2 n) =Ãk n(1,1)(m + , s 2+ )Ãk n(2,2)(m + , s 2+ )− Ãk n(1,2)(m + , s 2+ )Ãk n(2,1)(m + , s 2+ )(= knD 2 m + α m0 m 0 + α ) ()ms 2+k 2 s 2+β m + snD s2 + A s22+ β m2 + s 2+(− knB 2 m2 m + +A )s2α m β mχ(β m + s 2+ ) (1 − m s 2++) χ .β m2 + s 2+From equation (2.41) the eigenvalues of Ãk n(m + , s 2+ ) are determinedas:λ 1,2 (kn) 2 = − b(k2 n)± 1 b2 2√ 2 (kn) 2 − 4c(kn). 2 (2.42)It should be mentioned that{s 2+ > 0,m 0 > 0⇒ b(kn) 2 > 0. (2.43)Thus, the following lemma can be formulated.

2.3. Stability analysis 29Lemma 2.1. Suppose, that for the chosen parameter values, m 0 defined in(2.18) is positive, β m = β m2 and that there exists a real positive s 2+ definedin (2.19). Then the nature of eigenvalues of the matrix Ãk n(m + , s 2+ ) isdetermined by the sign of c(kn):2• if c(k 2 n) < 0, then one of the eigenvalues is positive and the other isnegative,• if c(k 2 n) = 0, then the matrix Ãk n(m + , s 2+ ) has one zero eigenvalueand one negative.• if c(k 2 n) > 0 then both eigenvalues are either negative, or complex witha negative real part.The wavenumbers which lead to growing perturbations are determinedby inequality c(k 2 n) < 0. c(k 2 n) can be written in the form:wherec(k 2 n) = γ 2 k 4 n + γ 1 k 2 n + γ 0 , (2.44)γ 2 = D m D s2 , (2.45)γ 1 = (D m A s2 + D s2 α m − χm + B m2 ) + D s2 α m0 m 0 , (2.46)β m2 + s(2+)s 2+s 2+γ 0 = A s2 α m0 m 0 + α m (2 − m + ) + α m (m + − 1) .β m2 + s 2+ β m + s 2+s 2+(2.47)Lemma 2.2. Suppose, that for the chosen parameter values, m 0 defined in(2.18) is positive, and that β m2 = β m . Then if there exists a real positives 2+ defined in (2.19), then γ 0 defined in (2.47) is non-negative.Proof. Since s 2+ > 0, then it is necessary to prove, thatα m0 m 0 + α ms 2+β m + s 2+(2 − m + ) + α m (m + − 1) ≥ 0.The previous inequality is simplified with use of equations(2.40) and (2.18):s 2+α m0 m 0 + α m (2 − m + ) + α m (m + − 1)β m + s(2+)s 2+s 2+= α m0 m 0 + α m (1 − m + ) − α m0 m + + α mβ m + s 2+ β m + s 2+( )s2++ α m0 m + + α m (m + − 1) = m + (α m0 + α m ) + α m − 1 ≥ 0.β m + s 2+(That is equivalent to m + (α m0 + α m ) ≥ αmβm. Considering (2.20), thistransforms to(β m + s 2+ ) 2 α m β m χ≥A s2 (α m0 + α m ) , (2.48)β m+s 2+)

30 Chapter 2. Linear stability analysiswhere χ is defined in (2.22). Next, it is proved, that inequality (2.48) holds.From equation (2.19) and assumption β m2 = β m it follows, thats 2+ + β m = −a 1 + √ a 2 1 − 4a 2a 02a 2+ β m ≥ − a 12a 2+ β m= − (α m0 + α m )(β m A s2 − χm 0 ) + α m χ(m 0 − 1) + α m0 β m A s22A s2 (α m0 + α m )+ 2β mA s2 (α m0 + α m )2A s2 (α m0 + α m )= α mβ m A s2 + χ(α m + α m0 m 0 ), (2.49)2A s2 (α m0 + α m )where a 2 , a 1 , a 0 are defined in (2.21). Since χ = α m2 (1 + α p0 /A b ) > 0 andm 0 is supposed to be positive, then from (2.26) it is derived that:(χ m 0 + α )√mα m≥ −A s2 β m + η α mχ. (2.50)α m0 α m0 α m0where η is defined in (2.25). Thus from (2.49) and (2.50) it follows:β m + s 2+ ≥ α mβ m A s2 + χ(α m + α m0 m 0 )2A s2 (α m0 + α m )≥ α mβ m A s2 − α m β m A s2 + √ ηα m α m0 χ2A s2 (α m0 + α m )√√ηαm α m0 χ=2A s2 (α m0 + α m ) = α m β m χA s2 (α m0 + α m )Thus inequality (2.48) holds, and consequently γ 0 ≥ 0.Remark 2.9. From the proof of Lemma 2.2 it follows, that γ 0 = 0, if andonly if a 2 1 − 4a 2a 0 = 0 which is equivalent for m 0 > 0 to(χ m 0 + α )√mα m= −A s2 β m + η α mχ. (2.51)α m0 α m0 α m0where η is defined in (2.25). In this case two constant solutions ˜z − and ˜z +coincide, since s 2− = s 2+ = − a 12a 0.It should be mentioned here, that under the assumptions of Lemma 2.2,c(0) = γ 0 ≥ 0. Then, Lemma 2.3 follows from Lemma 2.1.Lemma 2.3. If, for the chosen parameter values, m 0 defined in (2.18) ispositive, β m2 = β m and there exists a real positive s 2+ defined in (2.19),then for zero wavenumber k 0 , the matrix Ãk n(m + , s 2+ ) has either one zeroeigenvalue and one negative, or two negative eigenvalues, or two complexeigenvalues with a negative real part; and the constant solution (m + , s 2+ ) ofsystem (2.38) is stable against the purely temporal perturbations.

2.3. Stability analysis 31Since k n ∈ [0, ∞), then c(kn) 2 given in (2.44) can be considered as areal function of a real non-negative argument. It is a quadratic polynomial.The interval, where c(kn) 2 < 0, is defined by the roots of the polynomial.If this polynomial has no roots among non-negative real numbers, then for∀k n ∈ [0, ∞), c(kn) 2 > 0, since γ 2 defined (2.45) is positive. Thus, it isnecessary to find the conditions, for which the polynomial defined in (2.44)has at least one non-negative real root. The general formula for the roots ofthe polynomial is:The discriminant of the polynomial is:κ 2 1,2 = −γ 1 ± √ γ 2 1 − 4γ 2γ 02γ 2. (2.52)D γ = γ 2 1 − 4γ 0 γ 2 . (2.53)Since γ 2 > 0 and γ 0 ≥ 0 under the conditions of Lemma 2.2, the polynomialc(k 2 n) has either two real roots of the same sign as −γ 1 , which are differentif D γ > 0, and coincident if D γ = 0; or two complex roots with real part− γ 12γ 2, if D γ < 0.Theorem 2.1. Suppose, that for the chosen parameter values m 0 definedin (2.18) is positive, β m = β m2 and there exists a real positive s 2+ definedin (2.19). Then if D γ defined in (2.55) is positive, and γ 1 defined in (2.54)is negative, then ∃κ 1 , κ 2 ∈ R defined by expression (2.52), such that 0 ≤κ 1 < κ 2 , and the constant solution solution ˜z + = (m + , s 2+ ) of system (2.38)is unstable with respect to the perturbations with the wavenumbers k n ∈(κ 1 , κ 2 ). Otherwise, the constant solution ˜z + is stable.Proof. Let λ 1 (k 2 n) and λ 2 (k 2 n) be the eigenvalues of the matrix Ãk n(m + , s 2+ )defined in (2.42) and c(k 2 n) be defined in (2.44). Then:1. If D γ > 0, and(a) if γ 1 < 0, then ∃κ 1 , κ 2 ∈ R defined from expression (2.52), suchthat 0 ≤ κ 1 < κ 2 and:• for k n ∈ (κ 1 , κ 2 ): c(k 2 n) < 0, hence λ 1 (k 2 n) < 0 andλ 2 (k 2 n) > 0;• for k n ∈ {κ 1 ; κ 2 }: c(k 2 n) = 0, and λ 1 (k 2 n) < 0 and λ 2 (k 2 n) = 0;• for k n ∈ [0, ∞)/[κ 1 , κ 2 ]: c(k 2 n) > 0, and λ 1 (k 2 n), λ 2 (k 2 n) areeither real and negative, or complex with a negative real part;(b) if γ 1 > 0, then:i. if γ 0 > 0, then for ∀k n ∈ [0, ∞): c(k 2 n) > 0 and λ 1 (k 2 n), λ 2 (k 2 n)are either real and negative, or complex with a negative realpart;ii. if γ 0 = 0, then

32 Chapter 2. Linear stability analysis2. If D γ = 0, and• for ∀k n ∈ (0, ∞): c(k 2 n) > 0 and λ 1 (k 2 n), λ 2 (k 2 n) are eitherreal and negative, or complex with a negative real part;• c(0) = 0 and λ 1 (0) < 0 and λ 2 (0) = 0.(a) if γ 1 ≤ 0, then ∃κ 1 = κ 2 =√− γ 12γ 2≥ 0 , such that• c(κ 2 1 ) = 0, and λ 1(κ 2 1 ) < 0 and λ 2(κ 2 1 ) = 0;• for k n ∈ [0, ∞)/{κ 2 1 }: c(k2 n) > 0 and λ 1 (k 2 n), λ 2 (k 2 n) are eitherreal and negative, or complex with a negative real part;(b) if γ 1 > 0, then for ∀k n ∈ [0, ∞): c(k 2 n) > 0 and λ 1 (k 2 n), λ 2 (k 2 n) areeither real and negative, or complex with a negative real part.3. If D γ < 0, then for ∀k n ∈ [0, ∞): c(k 2 n) > 0 and λ 1 (k 2 n), λ 2 (k 2 n) areeither real and negative, or complex with a negative real part.Therefore, if D γ > 0, and γ 1 < 0, then λ 2 (kn) 2 > 0, if k n ∈ (κ 1 , κ 2 ), andthe magnitude of the perturbation modes having wavenumbers k n ∈ (κ 1 , κ 2 )grows monotonically after a certain period of time. Hence, the constant solution˜z + = (m + , s 2+ ) is unstable with respect to these perturbation modes.Otherwise, ∀k n ∈ [0, ∞) the eigenvalues of the matrix Ãk nare eitherreal non-positive numbers (the matrix Ãk ncan not have more than onezero eigenvalue) or complex numbers with a negative real part. Hence,initially small perturbations remain small during any period of time, oreven disappear when t → ∞, and the constant solution ˜z + is stable in thiscase.The parameters γ 1 and D γ , can be written in terms of the model parametersasγ 1 = (D m A s2 + D s2 α m − χm + B m2 ) + D s2 α m0 m 0 , (2.54)β m2 + s 2+s 2+(s) 2+2D γ = (D m A s2 + D s2 α m − χm + B m2 ) + D s2 α m0 m 0β m2 + s(2+s 2+s 2+− 4D m D s2 A s2 α m0 m 0 + α m (2 − m + )β m2 + s 2+ β m2 + s 2+)+ α m (m + − 1) .(2.55)

2.3. Stability analysis 332.3.4 Correspondence between the systems of two and threeequationsNext, the relation between the eigenvalues of the matrices Ãk n(m + , s 2+ )and A kn (m + , s 2+ , b + ) is determined, and the similarity between the stabilityproperties of systems (2.14)–(2.16) and (2.38), with respect to perturbationsabout the equilibria (m + , s 2+ , b + ) and (m + , s 2+ ), respectively, isdemonstrated. Let us define a matrix M kn :[ ]Akn(1,1) − λ AM kn =kn(1,2).A kn(2,2) − λA kn(2,1)From the definition of A kn it follows, that A kn(2,3) = A kn(2,1). Then⎡⎤A kn(1,1) − λ A kn(1,2) 0A kn − λI 3 = ⎣ A kn(2,1) A kn(2,2) − λ A kn(2,1)⎦α p0 0 −A b − λ⎡ ⎡ ⎤⎤0⎣ M kn⎦=⎢A kn(2,1)⎥⎣⎦ .α p0 0−A b − λThe determinant of this matrix is the characteristic polynomial of A kn :det (A kn − λI 3 ) = ( − A b − λ ) det(M kn ) + α p0 A kn(1,2)A kn(2,1). (2.56)From the definition of the matrices Ãk nand A kn , it follows thatà kn(1,1) = A kn(1,1), à kn(1,2) = A kn(1,2),(à kn(2,1) = 1 + α )p0AA kn(2,1), à kn(2,2) = A kn(2,2).bTherefore, the determinant of the matrix Ãk n− λI 2 and the characteristicpolynomial of the matrix Ãk nis⎛⎡⎤⎞)A kn(1,1) − λ A kn(1,2)det(Ãkn − λI 2 =det ⎝⎣⎦⎠A kn(2,1) + α p0A bA kn(2,1) A kn(2,2) − λ=det(M kn ) − α p0A bA kn(1,2)A kn(2,1).From (2.56) and (2.57) it is derived thatdet (A kn − λI 3 ) = ( − A b − λ ) det(2.57)(Ãkn − λI 2)− λ α p0A bA kn(1,2)A kn(2,1).(2.58)

34 Chapter 2. Linear stability analysisThen, the characteristic polynomials of the matrices A kn and Ãk n, whichare evaluated at the constant solutions (m + , s 2+ , b + ) and (m + , s 2+ ), respectively,are denoted by a cubic polynomial P 3 (λ) and a quadratic polynomialP 2 (λ) with respect to λ:P 3 (λ) = det(A kn (m + , s 2+ , b + )−λI 3 ); P 2 (λ) = det(Ãk n(m + , s 2+ , b + )−λI 2 ).Equation (2.58) can be written as:whereP 3 (λ) = ( − A b − λ ) P 2 (λ) − C(k 2 n)λ, (2.59)C(kn) 2 = α p0AA kn(1,2)(m + , s 2+ , b + )A kn(2,1)(m + , s 2+ , b + )b= α ()p0 α m2 s 2+ αm β mA b β m2 + s 2+ (β m + s 2+ ) 2 m +(1 − m + ) + knB 2 m2 m + .(2.60)If s 2+ > 0, it follows from (2.20) that m + > 0, and equation (2.40) yields:m + = 1 −< 1. Thus,α p0+A mα m0 + αms 2+βm+s 2+s 2+ > 0 ⇒ 0 < m + < 1 ⇒ C(k 2 n) > 0, ∀k 2 n ∈ [0, ∞). (2.61)Lemma 2.4. Suppose, that for the chosen parameter values m 0 defined in(2.18) is positive, and that there exists a real positive s 2+ defined in (2.19).If the matrix Ãk n(m + , s 2+ ) has one real negative eigenvalue ˜λ 1 < 0 andone real positive eigenvalue ˜λ 2 > 0, then A kn (m + , s 2+ , b + ) has one realpositive eigenvalue and either two real negative eigenvalues, or two complexconjugated eigenvalues with a negative real part.Proof. From the assumption of the lemma and from (2.61) it follows, thatC(k 2 n) > 0. Let Ãk n(m + , s 2+ ) have one real negative eigenvalue ˜λ 1 < 0 andone real positive eigenvalue ˜λ 2 > 0. The characteristic polynomial can bewritten as P 2 (λ) = (λ − ˜λ 1 )(λ − ˜λ 2 ). Then from equation (2.59)P 3 (λ) = ( − A b − λ ) (λ − ˜λ 1 )(λ − ˜λ 2 ) − C(k 2 n)λ = −λ 3 + (˜λ 1 + ˜λ 2 − A b )λ 2From (2.62) it follows:+ (−˜λ 1˜λ2 + A b (˜λ 1 + ˜λ 2 ) − C(k 2 n))λ − A b˜λ1˜λ2 . (2.62)P 3 (0) = −˜λ 1˜λ2 A b > 0 and P 3 (˜λ 2 ) = −˜λ 2 C(k 2 n) < 0. (2.63)Since P 3 (λ) is continuous, it follows from relation (2.63), that the polynomialP 3 (λ) has at least one real positive root λ 1 on the interval (0, ˜λ 2 ).

2.3. Stability analysis 35The other two eigenvalues λ 2 and λ 3 of A kn (m + , s 2+ , b + ) can be real(negative or positive) or complex conjugated numbers (as the coefficients ofthe polynomial are real). Then,P 3 (λ) = −λ 3 + (λ 1 + λ 2 + λ 3 )λ 2 − (λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 )λ + λ 1 λ 2 λ 3 ,(2.64)since this polynomial has λ 1 , λ 2 , λ 3 as its roots. As the coefficients at thesecond degree of λ in the two expressions for P 3 (λ) from (2.62) and (2.64)should be equal, it follows that λ 2 + λ 3 = ˜λ 1 + ˜λ 2 − A b − λ 1 . From (2.42) itis derived:λ 2 + λ 3 = −b(k 2 n) − A b − λ 1 < 0. (2.65)The above inequality holds, since it was mentioned in (2.43), that b(k 2 n) > 0,if m 0 > 0 and s + > 0. Thus, if two other eigenvalues are real, then from(2.65) it follows, that at least one of them is negative. Let us suppose λ 2 < 0.Thenlimλ→−∞ P 3(λ) = ∞,and P 3 (0) = −˜λ 1˜λ2 A b > 0. That means that on the interval (−∞, 0) thepolynomial P 3 (λ) does not change its sign, or changes it twice. Since P 3 (λ) iscontinuous, it follows from λ 2 < 0 that λ 3 also is negative. In the case, whenλ 2 and λ 3 are complex conjugated, their real part is λ re = (λ 2 + λ 3 )/2 0. Let Ãk n(m + , s 2+ ) have one zero eigenvalue and one real negativeeigenvalue, ˜λ 1 < ˜λ 2 = 0. Then the characteristic polynomial P 2 (λ) has theform P 2 (λ) = λ(λ − ˜λ 1 ). Then equation (2.59) impliesP 3 (λ) = ( − A b − λ ) λ(λ − ˜λ 1 ) − C(k 2 n)λ= −λ(λ 2 + (A b − ˜λ 1 )λ + (C(k 2 n) − ˜λ 1 A b )).(2.66)And eigenvalues of A kn (m + , s 2+ , b + ) are following:λ 1 = 0, λ 2,3 = −A b + ˜λ√1 ± (A b − ˜λ 1 ) 2 − 4(C(kn) 2 − ˜λ 1 A b ). (2.67)2Since C(k 2 n) − ˜λ 1 A b > 0 and A b − ˜λ 1 > 0, then from (2.67) it follows, thateigenvalues λ 2,3 are either real and negative (possible coincident), or complexwith a negative real part.

36 Chapter 2. Linear stability analysisLemma 2.6. Suppose, that for the chosen parameter values there exists areal positive s 2+ defined in (2.19). If Ãk n(m + , s 2+ ) has two real negativeeigenvalues, then A kn (m + , s 2+ , b + ) has either three real negative eigenvalues,or one real negative eigenvalue, and two complex eigenvalues with anegative real part.Proof. From the assumption of the lemma and from (2.61) it follows, thatC(k 2 n) > 0. Let Ãk n(m + , s 2+ ) have two real negative eigenvalues ˜λ 1 ≤˜λ 2 < 0. Then the characteristic polynomial P 2 (λ) has the form P 2 (λ) =(λ − ˜λ 1 )(λ − ˜λ 2 ). Then from equation (2.59)P 3 (λ) = ( − A b − λ ) (λ − ˜λ 1 )(λ − ˜λ 2 ) − C(k 2 n)λ = −λ 3 + (˜λ 1 + ˜λ 2 − A b )λ 2From (2.68) it follows:+ (−˜λ 1˜λ2 + A b (˜λ 1 + ˜λ 2 ) − C(k 2 n))λ − A b˜λ1˜λ2 . (2.68)P 3 (−A b ) = C(k 2 n)A b > 0 and P 3 (0) = −˜λ 1˜λ2 A b < 0. (2.69)Since P 3 (λ) is continuous, it follows from (2.69), that the polynomial P 3 (λ)has at least one root on the interval (−A b , 0). Thus it can be supposed, that−A b < λ 1 < 0. From (2.68) it follows, that for λ ≥ 0 the polynomial P 3 (λ)only takes values less than zero. That means, that P 3 (λ) has no non-negativereal roots P 3 (λ). Thus, if two other eigenvalues of A kn (m + , s 2+ , b + ) are real,they are also negative. Though it is possible, that the polynomial P 3 (λ) hastwo complex conjugated roots. Let us denote them as λ 2,3 = λ re ± iλ im .Then:P 3 (λ) = − (λ − λ 1 )(λ 2 − 2λ re λ + λ 2 re + λ 2 im)= − λ 3 + (λ 1 + 2λ re )λ 2 − (2λ 1 λ re + λ 2 re + λ 2 im)λ + λ 1 (λ 2 re + λ 2 im).(2.70)Since the coefficients at the second degree of λ in two expressions for P 3 (λ)(2.68) and (2.70) should be equal, it is derived that 2λ re = ˜λ 1 + ˜λ 2 −A b −λ 1 .As ˜λ 1 ≤ ˜λ 2 < 0 and −A b − λ 1 < 0, then λ re < 0. That is, if two eigenvaluesof A kn (m + , s 2+ , b + ) are complex, then their real part is less than zero.Lemma 2.7. Suppose, that for the chosen parameter values there exists areal positive s 2+ defined in (2.19). If Ãk n(m + , s 2+ ) has two complex conjugatedeigenvalues with a negative real part, then A kn (m + , s 2+ , b + ) has eitherthree real negative eigenvalues, or one real negative eigenvalue, and two complexeigenvalues with a negative real part.Proof. From the assumption of the lemma and from (2.61) it follows, thatC(k 2 n) > 0. Let Ãk n(m + , s 2+ ) have the complex conjugated eigenvalues witha negative real part: ˜λ1,2 = ˜λ re ± i˜λ im , ˜λ re < 0. Then the characteristic

2.3. Stability analysis 37polynomial P 2 (λ) takes positive values for ∀λ ∈ R and has the form P 2 (λ) =(λ 2 − 2˜λ re λ + ˜λ 2 re + ˜λ 2 im ). Then from equation (2.59), if followsP 3 (λ) = ( −A b −λ ) (λ 2 −2˜λ re λ+˜λ 2 re+˜λ 2 im)−C(k 2 n)λ = −λ 3 +(2˜λ re −A b )λ 2Equation (2.71) yields+ (−˜λ 2 re − ˜λ 2 im + 2A b˜λre − C(k 2 n))λ − A b (˜λ 2 re + ˜λ 2 im). (2.71)P 3 (−A b ) = C(k 2 n)A b > 0 and P 3 (0) = −A b (˜λ 2 re + ˜λ 2 im) < 0. (2.72)Since P 3 (λ) is continuous, it follows from (2.72), that the polynomial P 3 (λ)has at least one root on the interval (−A b , 0). Thus it can be supposed, that−A b < λ 1 < 0.From (2.71) it follows, that for λ ≥ 0 the polynomial P 3 (λ) takes valuesless than zero. That means, that P 3 (λ) has no non-negative real roots P 3 (λ).Therefore, if the two other roots of P 3 (λ) are real, they are also negative.Next, the possibility, that the polynomial P 3 (λ) has two complex conjugatedroots, is considered. The roots are denoted as λ 2,3 = λ re ± iλ im .Then:P 3 (λ) = −(λ − λ 1 )(λ 2 − 2λ re λ + λ 2 re + λ 2 im)= −λ 3 + (λ 1 + 2λ re )λ 2 − (2λ 1 λ re + λ 2 re + λ 2 im)λ + λ 1 (λ 2 re + λ 2 im).(2.73)Since the coefficients of λ 2 in two expressions for P 3 (λ) (2.71) and (2.73)should be equal, then 2λ re = 2˜λ re − A b − λ 1 . As ˜λ re < 0 and −A b − λ 1 < 0,it follows that λ re < 0. That is, if two eigenvalues of A kn (m + , s 2+ , b + ) arecomplex, then their real part is less than zero.2.3.5 Stability of the system of three equationsLemma 2.8. Suppose, that for the chosen parameter values, m 0 defined in(2.18) is positive, β m = β m2 and there exists a real positive s 2+ defined in(2.19). Then the constant solution z + = (m + , s 2+ , b + ) of system (2.14)–(2.16) is stable against purely temporal perturbations.Proof. From Lemma 2.3, 2.5, 2.6 and 2.7, it follows, that for zero wavenumberk 0 , the matrix A kn , evaluated at the constant solution z + , has either twonegative eigenvalues and one zero eigenvalue, or three real negative eigenvalues,or one real non-positive eigenvalue, and two complex eigenvalues witha negative real part.Theorem 2.2. Suppose, that for the chosen parameter values m 0 defined in(2.18) is positive, β m = β m2 and there exists a real positive s 2+ defined in(2.19). Then if D γ defined in (2.55) is positive, and γ 1 defined in (2.54) isnegative, then ∃κ 1 , κ 2 ∈ R defined by expression (2.52), such that 0 ≤ κ 1

38 Chapter 2. Linear stability analysisκ 2 , and the constant solution z + of system (2.14)–(2.16) is unstable withrespect to the perturbations having wavenumbers k n ∈ (κ 1 , κ 2 ). Otherwise,the constant solution z + is stable.Proof. The theorem is proved, by analogy with the proof of Theorem 2.1.In that proof all possible cases for the signs of the parameters D γ and γ 1 areconsidered, and the relations between the eigenvalues of the matrix Ãk nandthe wavenumber k n are determined for each case. From these relations, andfrom the relations between the eigenvalues of the matrices Ãk nand A kn ,stated in Lemma 2.4–2.7, it is possible to determine the correspondencebetween the eigenvalues of the matrix A kn and the wavenumber k n for thesets of signs of the parameters D γ and γ 1 .Therefore, it is obtained, that if D γ > 0, and γ 1 < 0, then ∃κ 1 , κ 2 ∈ Rdefined by expression (2.52), such that 0 ≤ κ 1 < κ 2 , and the magnitude ofthe perturbation modes having wavenumbers k n ∈ (κ 1 , κ 2 ) grow monotonicallyafter a certain period of time, since one of the eigenvalues of A kn ispositive. Hence, the constant solution z + = (m + , s 2+ , b + ) is unstable withrespect to these perturbation modes.Otherwise, ∀k n ∈ [0, ∞) the eigenvalues of the matrix A kn are eitherreal non-positive numbers (the matrix A kn can not have more than onezero eigenvalue) or complex numbers with a negative real part. Hence,initially small perturbations remain small during any period of time, oreven disappear when t → ∞, and the constant solution z + is stable in thiscase.The conditions on the parameters, stated in Theorem 2.2, can be formulatedin a compact form:{γ1 < 0,D γ = γ 2 1 − 4γ 0γ 2 > 0 ⇔ γ 1 < −2 √ γ 2 γ 0 . (2.74)From the proof of the theorem, it follows, that condition (2.74) is a necessarycondition for the instability of the solution z + , since it is equivalent to theexistence of the real positive numbers κ 1 and κ 2 . The necessary and sufficientcondition holds, if there exist wavenumbers k n ∈ (κ 1 , κ 2 ). From (2.52) itDfollows, that the length of the interval (κ 1 , κ 2 ) is equal to γD mD s2. If D γ issmall enough, then it is possible, that no wavenumber k n will lie inside theinterval (κ 1 , κ 2 ), and perturbations will not grow. In this case, the necessarycondition for the instability holds, but the solution is stable.The necessary instability condition (2.74), can be transformed into thesufficient stability condition by the substitution of the sign in inequality(2.74) by the opposite one:γ 1 ≥ −2 √ γ 2 γ 0 . (2.75)

2.3. Stability analysis 39This condition is formulated in terms of model parameters and does notdepend on the problem statement. This means, that a general instructionon the choice of the parameter values, which guaranties the stability of theconstant solution z + , can be formulated.The necessary and sufficient stability condition is opposite to the thenecessary and sufficient instability condition, which depends on the the wavenumbersk n . The set of the wavenumbers k n contains infinite numberof elements, and is determined by the domain size, by the coordinate systemand by the boundary conditions. Therefore, it is not possible to statethe necessary and sufficient condition in terms of the model parameters forthe general case. For different boundary conditions, coordinate systems ordomain sizes, these conditions have to be reformulated.2.3.6 Parameter choice and stabilityIn this subsection the choice of parameter values, providing stability of theconstant solution z + = (m + , s 2+ , b + ) of the system of three equations, isdiscussed.For the parameter values in equations (2.11), (2.12) and (2.13), the constantsolutions z t = (0, 0, 0), z 0 = (m 0 , 0, b 0 ) and z − = (m − , s 2− , b − ) areunstable, and the solution will not converge to these constant solutions.From a biological point of view, this is a favorable situation. Since, the’non-healing state’ z t contains zero concentrations of osteogenic cells andosteoblasts, the ’low density state’ z 0 corresponds to much lower concentrationsof osteogenic cells and osteoblasts, compared to those for z + , and theconstant solution z − contains unphysical negative concentrations of cells.For the chosen parameter value sets in equations (2.11), (2.12) and (2.11),(2.13), the sufficient condition (2.75) for the stability of z + does not hold.It is necessary to change the values of parameters, to guaranty the stabilityof the constant solution z + in general. It is proposed here, to vary valuesof parameters B m2 and D m . These two particular parameters are chosen,since:• their variation does not cause change of the values of the constantsolutions z t , z 0 , z − and z + (see equations (2.18),(2.19), (2.20));• from (2.35) and (2.36) it follows, that parameters B m2 and D m donot influence the stability of the constant solutions z t and z 0 againstpurely temporal perturbations. The stability of the constant solutionz − against purely temporal perturbations is determined from theeigenvalues of the matrix A k0 (m − , s 2− , b − ), see equation (2.32). Ask 0 = 0, this matrix does not depend on parameters B m2 and D m .For the considered parameter values in equations (2.11), (2.12) and(2.13), and for any B m2 and D m , z − is unstable against purely temporalperturbations. Therefore, variation of B m2 and D m , can provide

40 Chapter 2. Linear stability analysisthe stability of the constant solution z + , while the constant solutionsz t , z 0 and z − remain unstable;• calculations showed, that stability condition (2.75) is most sensitivewith respect to the parameters B m2 and D m . That is, the ratio of theinitial parameter value and the ultimate value of the parameter, whichsatisfies condition (2.75), is much smaller for B m2 and D m , comparedto the rest of the model parameters.The first quadrant of the plane (D m , B m2 ), which contains all possiblenon-negative values D m and B m2 , can be divided into three regions, withrespect to the stability of the solution z + :region R 1 : sufficient stability condition (2.75) holds, solution z + isstable;region R 2 : condition (2.75) does not hold, no wavenumbers k n lie inthe interval (κ 1 , κ 2 ), solution z + is stable;region R 3 : condition (2.75) does not hold, some of the wavenumbersk n lie in the interval (κ 1 , κ 2 ), solution z + is unstable.Configuration of regions R 2 and R 3 depend on the specified boundary conditions,on the coordinate system and on the domain length. In Figure 2.3these regions were plotted for the case of zero flux of m and s 2 on the boundaries,1D Cartesian coordinates, and the domain with length 0.6mm. Thislength is equal to the width of the domain, used in the numerical simulationsby Moreo et al. [67]. The values of the model parameters, given in (2.11),(2.12) (Figure 2.3a), and in (2.11), (2.13) (Figure 2.3b), were chosen.B m2, *0.1671 Unstable (R0.83)Unstable (R 3) Stable (R 2)Stable (R0.62)10.4γ 1=−2(γ 2γ 0) 1/2 0.6γ 1=−2(γ 2γ 0) 1/20.2 Stable (R 1)0.2 Stable (R 1)0 1 210 20 30 40 50D m, *0.133D m, *0.133B m2, *0.167Figure 2.3: Plot of the regions in the first quadrant of the plane (D m , B m2 ),where the constant solution z + is stable (R 1 , R 2 ) and unstable (R 3 ), for thecase of a zero flux of m and s 2 on the boundaries, 1D Cartesian coordinates,and the domain of the length 0.6mm. The rest of the model parameters areinitialized: (a) as in (2.11), (2.12), and (b) as in (2.11), (2.13).With use of (2.45), (2.46) and (2.47), sufficient stability condition (2.75)can be rewritten as follows:(1B m2 ≤D m + G 1 + 2 )√Dm D s2 γ 0 , (2.76)s 2+ + β m2 G 0

2.4. Numerical results 41or in the form⎧G 1D m ≥0, if B m2 ≤ ,s 2+ + β m2⎪⎨D m ≥B m2 (s 2+ + β m2 ) − G 1 + 2 γ 0D s2⎪⎩−2√γ 0 D s2G 2 0G 2 0(B m2 (s 2+ + β m2 ) − G 1 + γ )0D s2G 2 0, if B m2 >G 1s 2+ + β m2,(2.77)where G 0 = A s2s 2+s 2+ +β m2, G 1 = D s2sα m + α m0 m 2+ +β m20 s 2+), and γ 0 is definedin (2.47). Inequalities (2.76) and (2.77) determine the values of B m2 andD m , which ensure the stability of the solution z + .The following remark can be helpful for the solution of practical problems.Suppose, that initial values of model parameters do not satisfy sufficientcondition (2.75) for the stability of the solution z + . Then, it is possibleto guaranty the stability of z + in general case (i.e. for any set of wavenumbers,which are determined by problem statement), by decreasing the valueof B m2 , or increasing D m , until condition (2.76) or condition (2.77) is satisfied,respectively.A s2(2.4 Numerical resultsThe predictions from the linear stability analysis are validated against asequence of numerical simulations. The sufficient stability condition is consideredin the form of equation (2.76) and the parameter B m2 is varied.If the values of all parameters, except B m2 , are fixed, then the right partof inequality (2.76) can be denoted as the ultimate value Bm2 lim , such thatfor B m2 ≤ Bm2 lim small perturbations near (m +, s 2+ , b + ) are predicted not togrow with time. For B m2 > Bm2lim small perturbations of mode φ n(x) willgrow, if κ 1 < k n < κ 2 . If B m2 is close to ultimate value Bm2 lim , then theinterval (κ 1 , κ 2 ) is small, and it can happen, that no wavenumber k n liesinside this interval. In this case perturbations near the constant solutionwill not grow, in spite of the fact, that sufficient stability condition (2.76)does not hold.For the cases when the model parameters are initialized as in (2.11),(2.12) and (2.11), (2.13), the ultimate values are Bm2 lim ≈ 0.45716 · 0.167mm 2 /day and Bm2 lim ≈ 0.02481 · 0.167 mm2 /day.First, the parameter values (2.11), (2.12) are considered. If the problemdomain is a 1D interval x ∈ [1, 6] in Cartesian coordinates, and if a zero fluxof osteogenic cells m and growth factor s 2 is supposed, then the wavenumbersare determined as k n = πn/5 mm −1 , n = 0, 1, 2, . . . . Then for B m2 = 0.4572·0.167 mm 2 /day, which is larger than the ultimate value, still no wavenumberlies between κ 1 ≈ 4.2805mm −1 and κ 2 ≈ 4.3838mm −1 . Though, for B m2 =

42 Chapter 2. Linear stability analysis0.4573 · 0.167 mm 2 /day, the wavenumber k 7 ≈ 4.3982mm −1 ∈ (κ 1 , κ 2 ) =(≈ 4.2322mm −1 , ≈ 4.4339mm −1 ). If the parameter values (2.11), (2.13) arechosen, then for B m2 = 0.0249 · 0.167 mm 2 /day, the wavenumberk 6 ≈ 3.7699mm −1 ∈ (κ 1 , κ 2 ) = (≈ 3.6417mm −1 , ≈ 4.324mm −1 ).s22.38982.38980.9959(a)2 4 6x2.38982.38982.38980.9959(b)2 4 6x2.422.382.341.02(c)2 4 6x2102(d)2 4 6xm0.99590.9959110.99592 4 6x0.99592 4 6x0.982 4 6x02 4 6xDay 1Day 200Day 1000Day 2000Day 30000Day 100000Day 15000Day 30000Day 150000Day 20Day 1000Day 150000Figure 2.4: Solution of equations (2.14)–(2.16) in Cartesian coordinatesat different time moments. Small random initial perturbations near theconstant solution (m + , s 2+ , b + ) are considered. Zero fluxes of m, s2 on theboundaries are taken as the boundary conditions. Parameter B m2 takesdifferent values: B m2 = k · 0.167 mm 2 /day, (a) k = 0.3, (b) k = 0.4572, (c)k = 0.4573, (d) k = 1. The rest of parameters are initialized as in (2.11),(2.12).In Figure 2.4 the results of the numerical simulations are shown. Thesolutions were obtained with use of the finite element method. Linear 1Delements of size 0.02mm were used for the discretization in space. The implicitbackward Euler method, to prevent instabilities due to numerical timeintegration, and adaptive time stepping were used for time integration. Zeroflux of m, s2 on the boundaries was specified as the boundary conditions.To introduce the perturbations in the initial solution during simulations, thecorresponding constant solution value plus a small random number were assignedto every degree of freedom at time t = 0. From Figure 2.4 it follows,that for the values of B m2 less than the ultimate value, the numerical solutiontends to the constant solution (m + , s 2+ , b + ) with time (Figure 2.4a).And if the parameter B m2 is larger than Bm2 lim and such, that ∃k n ∈ (κ 1 , κ 2 ),then there is no convergence to the constant solution, and a wave-like profileoccurs in the solution (Figure 2.4c, d). However, if B m2 is larger than Bm2 lim,

2.4. Numerical results 43but such that no wave number lies inside (κ 1 , κ 2 ) yet, then the numerical solutionagain converges to the constant solution (m + , s 2+ , b + ) (Figure 2.4b).Thus, the predictions of the linear stability analysis are fully confirmed bythe numerical simulations.The linear stability analysis allows to assess the stability of the consideredconstant solution. From its stability it can be concluded, whether ornot small perturbations grow with time. The important conclusion can bemade, for cases in which perturbations are large: if the constant solutionis not stable, then the solution of the problem will never converge to thatconstant solution. Hence, the introduced linear analysis provides importantresults also for the case of large perturbations, since it allows to determinethe situation, in which the solution, which is constant in time and in space,can never be reached. However, if the constant solution is stable, it is stillunknown, how large initial perturbations behave, whether they disappear orprevail, or even grow.In reality, the deviations from the constant solution are large. The initialand boundary conditions, proposed by Moreo et al. [67] for the full system(2.1)–(2.8), were given in Section 2.2. When adapted to the simplified systemof three equations, initial and boundary conditions (2.9), (2.10) are rewrittenas:m(⃗x, 0) = 0.001, b(⃗x, 0) = 0.001, s 2 (⃗x, 0) = 0.01, ⃗x ∈ Ω. (2.78)⎧D s1 ∇s 1 · ⃗n = 0, D s2 ∇s 2 · ⃗n = 0, ⃗x ∈ ∂Ω, t ∈ (0, ∞)⎪⎨m = 0.2, ⃗x ∈ ∂Ω b , t ∈ (0, 14] [days][⃗x ∈ ∂Ω\∂Ωb , t ∈ (0, 14] [days],⎪⎩ (D m ∇m − mB m2 ∇s 2 ) · ⃗n = 0,⃗x ∈ ∂Ω, t ∈ (14, ∞) [days].(2.79)Initial conditions (2.78) are far from small perturbations near the constantsolution (m + , s 2+ , b + ).The simplified system (2.14)–(2.16), and the full system (2.1)–(2.8) weresolved numerically for initial and boundary conditions (2.9), (2.10) and(2.78), (2.79) respectively, and for several sets of parameter values. Someof the solutions for the full system (2.1)–(2.8) are plotted in Figure 2.5.The numerical simulations show, that if the parameter values are such, thatthe constant solution (m + , s 2+ , b + ) is stable, then the numerical solutionsof both systems for the unknowns m(x, t), s 2 (x, t), b(x, t) converge to thisconstant solution after a certain period of time (Figure 2.5a). Though, ifthe constant solution (m + , s 2+ , b + ) is not stable, then a wave-like profiledevelops in the solution for osteogenic cells and growth factor 2 and for parametervalues (2.11), (2.13) also in the solution for osteoblasts. For somevalues of the parameter B m2 the wave-like profile is steady (Figure 2.5b).

44 Chapter 2. Linear stability analysisThough, if B m2 is much larger than the ultimate value, then the waves inthe numerical solution are not steady, but moving (Figure 2.5c, d). This isin agreement with the stability analysis presented in Section 2.4.(a)(b)(c)(d)s2m50400.80.70.62 4 6x2 4 6x50400.80.62 4 6x2 4 6x60202102 4 6x2 4 6x20010.502 4 6x2 4 6xb20152 4 6x20152 4 6x402002 4 6x2002 4 6xDay 5000Day 10000Day 150000Day 5000Day 100000Day 150000Day 3000Day 80000Day 150000Day 3000Day 100000Day 150000Figure 2.5: Solution of equations (2.1)–(2.8) in axisymmetric coordinatesat different time moments. Initial and boundary conditions are chosen inthe form (2.9), (2.10), according to Moreo et al. [67]. Parameter B m2 takesdifferent values: B m2 = k · 0.167 mm 2 /day, (a) k = 0.0248, (b) k = 0.0249,(c) k = 0.04, (d) k = 0.2. The rest of parameters are initialized as in (2.11),(2.13).2.5 ConclusionsA simplified system of three equations is defined, which is characterized bythe appearance of a wave-like profile in the solution under the same conditions,as for the solution of the full system of eight equations. For theconsidered parameter values the simplified system has four constant solutions.The sufficient stability condition for one of the constant solutions,denoted as z + = (m + , s 2+ , b + ), is derived in terms of model parameters, bymeans of the linear stability analysis. If all constant solutions are unstable,then by changing the values of the model parameters B m2 and D m , it ispossible to make the solution z + stable, while three other constant solutionsz t , z 0 and z − remain unstable. The analytical predictions on the stability ofthe constant solution z + for various parameter sets are confirmed by numer-

2.5. Conclusions 45ical simulations, when starting from small perturbations near the constantsolution.In real simulations for the peri-implant osseointegration, initial conditionscorrespond to the large deviations from the constant solution. However,linear stability analysis provides important results also in this case.It allows to avoid such values of model parameters, for which all constantsolutions are unstable, and consequently, can not be reached. Linear stabilityanalysis makes it possible to determine parameter values, for which thesolution of the problem will never converge to the solution, which is constantin time and in space. This conclusion is confirmed by the numerical simulations,which evidence, that a wave-like profile appears in the solution, if allthe constant solutions are unstable. The numerical simulations also show,that if the solution z + is stable and z t , z 0 , z − are unstable, then numericalsolutions for unknowns m(x, t), s 2 (x, t), b(x, t) of the full and the simplifiedsystem converge to the constant solution (m + , s 2+ , b + ) after a certain periodof time, when starting with initial conditions proposed in Moreo et al. [67].Therefore, the numerical simulations demonstrate, that if the constantsolutions z t , z 0 , z − are unstable, then the stability of the constant solutionz + can determine the behavior of the solution of the whole system. Thatmakes it possible to determine the values of model parameters, for whichbiologically irrelevant solutions with a wave-like profile can be obtained.

CHAPTER 3Evolutionary cell differentiationIn this chapter an approach is developed, which allows to introduce theconcept of cell plasticity into the models for tissue regeneration. Opposedto most of the recent models for tissue regeneration, cell differentiation isconsidered as a process, which evolves in time, and which is regulated by anarbitrary number of parameters. In the current approach, cell differentiationis modeled by means of a differentiation state variable. Cells are assumedto differentiate into an arbitrary number of cell types. The differentiationpath is considered as reversible, unless differentiation has fully completed.Cell differentiation is incorporated into the partial differential equations(PDE’s), which model the tissue regeneration process, by means of an advectionterm in the differentiation state space. This allows to consider thedifferentiation path of cells, which is not possible, if a reaction-like term isused for differentiation.The boundary conditions, which should be specified for the generalPDE’s, are derived from the flux of the fully non-differentiated cells, andfrom the irreversibility of the fully completed differentiation.An application of the proposed model for peri-implant osseointegrationis considered. Numerical results are compared with experimental data. Potentiallines of further development of the present approach are proposed.The current evolutionary approach also is presented in Prokharau et al. [79].3.1 IntroductionCell differentiation, along with proliferation and migration, plays an importantrole at the early stages of the tissue regeneration process. A lot ofstudies have been devoted to the phenomenon of bone regeneration recently.In a number of studies various theoretical models were developed for frac-47

48 Chapter 3. Evolutionary cell differentiationture healing, peri-implant osseointegration and bone distraction (see Section1.3). In these models cell differentiation is considered as a crucial processin regulating the course of bone healing. In most of the mentioned works,the differentiation process is introduced into the constitutive equations bymeans of a logistic term. Such an approach corresponds to the situation,when a particular cell changes its type immediately.A more enhanced representation of cell differentiation is introduced inthe model by Reina-Romo et al. [83]. The differentiation process is assumedto depend on the mechanical stimulus and time. The differentiation evolutionis introduced by means of maturation state m i of the cells, beingreversible if m i < 1. Full differentiation into another cell type correspondsto the value m i = 1, and is assumed to be irreversible.In the first part of this chapter the conceptual mathematical model forthe evolutionary cell differentiation is derived. The basic assumptions forthis model are in line with those, proposed in Reina-Romo et al. [83]. Due tothe generality of the current model, it can be adapted and used for variousapplications.The main idea, characterizing the current approach, is that in the courseof differentiation, a cell of a certain type gains the properties of anothercell type gradually. Therefore, differentiation is considered as a process,which evolves in time, or in other words, the path of cell differentiation isincorporated into the model. The particular trajectory of the path can bedetermined by an arbitrary number of parameters, e.g. by biochemical andmechanical factors, by the level of differentiation of the cell at the currentmoment or by a random variable etc. The phenomenon, when cells changetheir differentiation path, depending on some parameters, is known as cell(or tissue) plasticity (see e.g. Röder [84]).In Section 3.2, the differentiation of mesenchymal stem cells (MSC’s) intoosteoblasts, fibroblasts and chondrocytes is considered. Due to the genericnature of the proposed approach, the present model can be modified for thecase of evolutionary differentiation of arbitrary cells into any number of celltypes.In Section 3.3, an example of the application of the current approach isgiven. A peri-implant osseointegration is modeled. The mechanical statein the interface tissue is chosen to be the main factor, which influences themesenchymal stem cell (MSC) differentiation. The series of numerical simulationsare carried out for the constructed model. The numerical method isdescribed and simulations results are presented and discussed in Section 3.4.Conclusions are drawn in Section 3.5.The present study is inspired by the work of Röder [84], where the completedifferentiation of a stem cell is considered as attainment of a certainvalue by some cell property a. The path of variation of property a is assumedto depend on the changing growth environment, to which the cellbelongs.

3.2. Differentiation model 493.2 Differentiation modelIn this section a new modeling approach for the differentiation of MSC’s isdeveloped. MSC’s are supposed to differentiate into osteoblasts, fibroblastsand chondrocytes. The most important innovating assumption of the currentapproach, is that cells are supposed to gain the properties of another celltype gradually, in the course of time, until a complete differentiation ofMSC’s takes place. Therefore, a certain finite time of differentiation can berelated to each MSC. The variation of the differentiation time is representedby means of the differentiation rate, which is assumed to depend on thestate of the environment, in which cells are situated. The proposed approachallows to consider the differentiation of MSC’s as partially stochastic process.However, for the numerical simulations, presented in this chapter, the MSCdifferentiation process is modeled only in a deterministic way.The model for cell differentiation is presented in two successive steps.First, the current stage of differentiation of MSC’s into a certain cell typei, where i ∈ {b, f, c}, is represented via a differentiation property a i . Thevalue b of the sub-index i corresponds to osteoblasts, f – to fibroblasts,c – to chondrocytes. The variables a b , a f , a c take values from 0 to 1.The value 1 implies, that a MSC has completely differentiated into therespective cell type, and the value 0 corresponds to the initial state of aMSC, which has not yet started to differentiate. The region, in which thevariables a b , a f , a c take their values, is referred to as the differentiation statedomain Ω a . The concept of differentiation properties a i is implemented inthe model, by considering the MSC density c as a function of location inphysical space ⃗x, of time t and of differentiation state coordinates a b , a f , a c .The process of differentiation corresponds to the variation of the value of thedifferentiation property a i , and it can be considered as a movement of theMSC within the region Ω a . This movement is determined by velocity u (theunderlined symbol denotes a vector in the differentiation state space), whichis assumed to depend on a number of parameters, being ψ 1 , . . . , ψ p , p ∈ N,and also on the differentiation state variables a i . Parameters ψ 1 , . . . , ψ pcan be defined, for example, as mechanical stimuli, biochemical factors oras a random variable, and they are functions of location in physical space⃗x and time t. For conciseness, the set of parameters is denoted as vectorΨ(⃗x, t) = (ψ 1 , . . . , ψ p ) ∈ Ω p ⊂ R p , where p is the number of parameters.Therefore, u = u(Ψ(⃗x, t), a b , a f , a c ).If no additional conditions on variables a i , i ∈ {b, f, c} are applied, thenthe differentiation state domain Ω a = [0, 1] 3 is a unit cube. In generalcase, when a cell is able to differentiate into n various types, Ω a is then-dimensional unit cube. For the considered model, the unknown functionc(⃗x, t, a b , a f , a c ) is defined within the domain Q = Ω×R + ×Ω a , which is a 6-dimensional region in the case of a two-dimensional domain Ω in the physicalspace, and a 7-dimensional region, for a three-dimensional domain Ω. For

50 Chapter 3. Evolutionary cell differentiationthe numerical simulations this can lead to extremely large computation timeand size of solution data.In the second step, the current model is modified, so that it becomes moreuseful for the practical applications. Certain constraints on variables a i areimposed, which allow to reduce the dimensionality of the problem domainby n − 1, where n is the number of cell types, into which the considered cellscan differentiate (n = 3 for the particular model for MSC’s differentiation).This restriction is formulated as follows.Assumption 3.1. A cell cannot have characteristics of the differentiationinto several cell types simultaneously. In other words, each cell can haveonly one non-zero differentiation property a i .a f01Ω a1 a b1a cPlot of the differentiation state domain Ω a , as defined in equa-Figure 3.1:tion (3.1)The differentiation state domain Ω a is then defined as:Ω a = {(a b , 0, 0), (0, a f , 0), (0, 0, a c ) : a i ∈ [0, 1], i = b, f, c} . (3.1)The region Ω a , given in (3.1), is depicted in Figure 3.1. This region canbe considered as a set of three one dimensional unit intervals situated onthe axes of Cartesian coordinates in a three-dimensional space, which haveone common point in the origin. The region Ω a lies in R 3 and it has a zeromeasure. In order to get a one-dimensional differentiation state domain, thefollowing technique is used. Instead of one unknown c, depending on threedifferentiation state variables a i , i ∈ {b, f, c}, three unknowns c i , i ∈ {b, f, c},depending on one differentiation state variable a are defined for the MSCdensity. The correspondence between c and c i can be determined in thefollowing way:c b (⃗x, t, a) = c(⃗x, t, a, 0, 0), c f (⃗x, t, a) = c(⃗x, t, 0, a, 0),c c (⃗x, t, a) = c(⃗x, t, 0, 0, a), for a ∈ Ω 1 a = [0, 1], ⃗x ∈ Ω, t > 0,where the one-dimensional differentiation state domain is denoted as Ω 1 a.Unknowns c i (⃗x, t, a), i ∈ {b, c, f}, denote the density of the MSC’s at thepoint ⃗x at the time moment t, differentiating into osteoblasts, fibroblasts or

3.2. Differentiation model 51chondrocytes, respectively, at the differentiation level a. These unknownshave the dimension of number of cells per unit volume, per unit of differentiationlevel. The differentiation path is determined by functions u i ,which denote the velocity of cell “motion” in the differentiation state domain,or the rate, at which a cell changes the value of its differentiationlevel a. In other words, if a fixed cell is considered, which differentiatesinto the cell type i ∈ {b, f, c}, and if its differentiation level is denotedas a cell (t), then u i = da celldt. The velocities u i depend on the parametersΨ(⃗x, t) = (ψ 1 (⃗x, t), . . . , ψ p (⃗x, t)) ∈ R p , on the differentiation level a, andon the specific phenotype i ∈ {b, f, c} the cell is differentiating to, i.e.u i = u i (Ψ(⃗x, t), a).The flux of MSC’s in the differentiation state domain is denoted asq i (⃗x, t, a) = u i (Ψ(⃗x, t), a) c i (⃗x, t, a).A positive value of the flux q i and of the velocity u i corresponds to the positivedirection of axis a. From a biological point of view, a positive velocityu i (⃗x, t, a), i ∈ {b, c, f} implies, that cells c i (⃗x, t, a) are gaining the propertiesof the phenotype i, and a negative rate u i (⃗x, t, a) means, that MSC’sare losing the properties of the cell type i. The process of differentiation isassumed to be reversible, except if the cell has achieved the maximum levelof differentiation a = 1, i.e. if the MSC has completely differentiated intoanother cell type. In this case the differentiation is irreversible, since a fullydifferentiated cell cannot become a MSC again [83]. Hence, the influx ofMSC’s at the boundary Γ 1 a = {(⃗x, a) : ⃗x ∈ Ω, a = 1} should be zero. Fromthis it follows:Proposition 3.1. If u i (Ψ(⃗x, t), 1) < 0, where ⃗x ∈ Ω, t > 0, i ∈ {b, f, c},thenc i (⃗x, t, 1) = 0. (3.2)Otherwise, reverse differentiation of cells of the type i ∈ {b, c, f} intoMSC’s would take place.3.2.1 Flow of non-differentiated cellsThe density of the fully non-differentiated MSC’s at the point (⃗x, t) is determinedby ∑ i c i(⃗x, t, 0). The “location” of the non-differentiated cells isthe part of the domain Ω × Ω 1 a boundary, where a = 0:Γ 0 a = {(⃗x, a) : ⃗x ∈ Ω, a = 0}.The magnitude of the flux ‖q i ‖ = ‖u i c i ‖, i ∈ {b, f, c} at Γ 0 a denotes eitherthe inflow or the outflow of c i , depending on the sign of the velocity u i . Apositive velocity u i at Γ 0 a, corresponds to the inflow of c i , and negative – tothe outflow.

52 Chapter 3. Evolutionary cell differentiationSuppose, that at a certain location ⃗x 0 and at the given time moment t 0the environment in the tissue is such, that non-differentiated MSC’s differentiateinto osteoblasts at a rate u 1 > 0. Hence, u b (Ψ 0 , 0) = u 1 > 0, whereΨ 0 = Ψ(⃗x 0 , t 0 ). Fully non-differentiated MSC’s are not biologically distinguishable,though they can be related to different densities c i . This means,that non-differentiated MSC’s, related to c f (⃗x 0 , t 0 , 0), also will differentiateinto osteoblasts with rate u 1 . They gain the properties of osteoblasts,and consequently will become related to c b . This implies, that the outflowof density c f , equal to u 1 c f , will take place at the point (⃗x 0 , t 0 , 0). Sinceoutflow at a = 0 corresponds to the negative differentiation rate, thenu 1 c f (⃗x 0 , t 0 , 0) = ‖q f (⃗x 0 , t 0 , 0)‖ = ‖u f (Ψ 0 , 0)c f (⃗x 0 , t 0 , 0)‖Hence,By analogy, it is derived, that= −u f (Ψ 0 , 0)c f (⃗x 0 , t 0 , 0).u f (Ψ 0 , 0) = −u 1 = −u b (Ψ 0 , 0). (3.3)u c (Ψ 0 , 0) = −u b (Ψ 0 , 0). (3.4)From this example, it follows, that the negative differentiation rate u i (Ψ, 0),i ∈ {b, f, c}, Ψ ∈ Ω p , corresponds to the situation, when non-differentiatedMSC’s are stimulated to differentiate into the cell type j, different from thecell type i.The path of cell differentiation is determined by the parameter vectorΨ ∈ Ω p and the maturation level a. It is supposed, that each value ofΨ ∈ Ω p corresponds to a certain state, for which non-differentiated MSC’sdifferentiate into a particular cell type i ∈ {b, f, c} and do not differentiateinto any other cell type.Remark 3.1. Randomness of the differentiation process, if needed, can betaken into account by considering one or more random variables ψ j , includedin the parameter set Ψ.The situation, when the non-differentiated MSC’s do not differentiate atall, is also allowed. From this, the following statement can be made.Proposition 3.2. For any Ψ ∈ Ω p ,• there exists exactly one i ∈ {b, c, f}, such that u i (Ψ, 0) > 0, andu j (Ψ, 0) = −u i (Ψ, 0) < 0 for j ∈ {b, c, f}, j ≠ i; or• u i (Ψ, 0) = 0 for all ∀i ∈ {b, f, c}.Suppose that u b (Ψ(⃗x 0 , t 0 ), 0) > 0, then the inflow of the density c i takesplace at the point (⃗x 0 , t 0 ). From Proposition 3.2, it follows, thatu f (Ψ(⃗x 0 , t 0 ), 0) = u c (Ψ(⃗x 0 , t 0 ), 0) = −u b (Ψ(⃗x 0 , t 0 ), 0) < 0.

3.2. Differentiation model 53If there is no external source of MSC’s, then this inflow can be formed onlyby the outflow of densities c f and c c . That means, thatHence,‖q b (⃗x 0 , t 0 , 0)‖ = ‖q f (⃗x 0 , t 0 , 0)‖ + ‖q c (⃗x 0 , t 0 , 0)‖.‖u b (Ψ 0 , 0)c b (⃗x 0 , t 0 , 0)‖ = u b (Ψ 0 , 0)c b (⃗x 0 , t 0 , 0)= ‖u f (Ψ 0 , 0)c f (⃗x 0 , t 0 , 0)‖ + ‖u c (Ψ 0 , 0)c c (⃗x 0 , t 0 , 0)‖= −u f (Ψ 0 , 0)c f (⃗x 0 , t 0 , 0) − u c (Ψ 0 , 0)c c (⃗x 0 , t 0 , 0),⇒ c b (⃗x 0 , t 0 , 0) = c f (⃗x 0 , t 0 , 0) + c c (⃗x 0 , t 0 , 0). (3.5)In general case, if u i (Ψ(⃗x, t), 0) > 0 for a certain i ∈ {b, f, c}, then theinflow of c i occurs at the point (⃗x, t, 0), which is supposed to be equal tothe total outflow of c j , j ∈ {b, c, f; j ≠ i}. Hence, the proposition can beformulated.Proposition 3.3. If u i (Ψ(⃗x, t), 0) > 0, where ⃗x ∈ Ω, t > 0, i ∈ {b, f, c},thenc i (⃗x, t, 0) = ∑ c j (⃗x, t, 0). (3.6)j≠iTo motivate the use of the boundary and initial conditions for c i due tothe differentiation state dimension, the following simple partial differentialequation (PDE) is considered:∂c i∂t + ∂ (u ic i )= 0. (3.7)∂aThe above equation describes the dynamics of the differentiation of MSC’sat a certain point in the physical space. Here, motility and proliferationof the cells are not considered. The equation illustrates, which boundaryconditions with respect to a should be specified, due to the presence of theterm ∂(u ic i )∂a, which represents cell differentiation. Consider the case whenu i is constant. Then, the lines in the (a, t)-plane are considered, over whichc i is constant, that is0 = d dt c i(⃗x, t, a(t)) = ∂c i∂t + ∂c i∂a a′ (t).From equation (3.7) and from the assumption u i = const, it follows thata ′ (t) = u i . Hence a(t) = u i (t − t 0 ) + a 0 is the equation of the characteristiclines, over which c i (⃗x, t, a(t)) = const. Plots of the characteristics a(t) inthe plane (a, t) are shown in Figure 3.2. It follows, that the solution c iat any point (t, a) ∈ [0, ∞) × [0, 1] is determined by the initial conditionc i (0, a) = g 0 (a), and by the condition c i (t, 0) = g b0 (t) on the boundary

54 Chapter 3. Evolutionary cell differentiationtu i > 0 t u i < 00(a)1a0(b)1aFigure 3.2: Projections of characteristics of the equation (3.7) on the plane(a, t) for different signs of velocity u i = u i (Ψ, a)(t, 0), if u i > 0 (Figure 3.2a), or by the condition c i (t, 1) = g b1 (t) on theboundary (t, 1), if u i < 0 (Figure 3.2b). In general case, i.e. if u i ≠ const,the condition will be required on the part of the boundary, where the influxtakes place, and no boundary condition should be specified on the outflowboundary (see e.g. LeVeque [59]). Therefore, for the existence and theuniqueness of the solution of (3.7):1. initial conditions at t = 0 should be specified;2. (a) if u i (Ψ, 0) > 0 (Figure 3.2a), then boundary conditions for c i ,defined in Proposition 3.3, are required on the boundary a = 0;(b) if u i (Ψ, 0) ≤ 0 (Figure 3.2b), then no boundary conditions for c ishould be prescribed on the boundary a = 0;3. (a) u i (Ψ, 1) < 0, then boundary conditions, defined in Proposition3.1, are required on the boundary a = 1;(b) if u i (Ψ, 1) ≥ 0, then no boundary conditions should be specifiedon the boundary a = 1.Therefore, in Propositions 3.1, 3.3, the boundary conditions are defined,which are necessary for the existence and the uniqueness of the solutionof partial differential equations, which contain the advection term ∂(u ic i )∂a,representing evolutionary cell differentiation. This differentiation term isincluded in the equations to model the evolution of the MSC densities, thatare derived in Section 3.3.3.3 Peri-implant osseointegration3.3.1 Mathematical modelIn Section 3.2, the generic mathematical representation of evolutionary celldifferentiation is described. Further, the validation of this approach is presented,and its advantages over the commonly used model for immediate celldifferentiation are highlighted.

3.3. Peri-implant osseointegration 55In order to look at the applicability of the current differentiation approach,it is incorporated in the simulations of bone regeneration duringperi-implant osseointegration. This specific example of an application ischosen due to the following reasons:1. In the current model, differentiation of MSC’s into osteoblasts andfibroblasts is considered, which is determined by the level of the mechanicalloading. Then, the ability of the model to represent evolutionarydifferentiation into several cell types under the influence theexternal factors can be demonstrated .2. Furthermore, a number of experimental and numerical studies havebeen devoted to the bone regeneration process. Hence, it will be possibleto compare the results of simulations with the available data.Further, the mathematical model for the peri-implant osseointegrationwill be derived. Bone regeneration is represented in the model as a series ofbiological events, such as:• migration and proliferation of MSC’s,• differentiation of MSC’s into osteoblasts and fibroblasts,• migration and proliferation of fibroblasts,• proliferation of osteoblasts,• fibrin network substitution by woven bone, and fibrous tissue,• woven bone remodeling into lamellar bone.Three cell types are considered: MSC’s, osteoblasts and fibroblasts. Thedensity of MSC’s, differentiating into chondrocytes, and the density of chondrocytesare not included into the model, since formation of cartilage tissue isnot observed in the experiments for the peri-implant osseointegration withinthe bone chamber [28, 29, 92, 93, 94].MSC’s are introduced into the model by variables c f , c b , which denotethe densities of MSC’s per unit of volume and per unit of differentiationlevel, normalized with respect to the limit density C of cells per unit volume.The unknowns c f , c b have dimension (unit of differentiation level) −1 . Thevariables c f , c b are functions of the space coordinate ⃗x ∈ Ω, of time t ∈ R +and of the differentiation level a ∈ [0, 1]. Non-dimensional model variablesb, f denote the densities of osteoblasts and fibroblasts per unit volume,normalized with respect to the limit density C. Unknown variables v n , v w ,v l , v f denote the volume fractions of fibrin network, woven bone, lamellarbone and fibrous tissue, respectively. They are functions of the physicalspace coordinates and of time.

56 Chapter 3. Evolutionary cell differentiationIt is assumed, that MSC’s tend to differentiate into cells of a certaintype, depending on the mechanical environment. Cells, originated from thedifferentiated MSC’s, provide the substitution of a fibrin network by therespective tissue type. Mechanoregulation of differentiation is introduced interms of the mechanical stimulus ψ. The stimulus ψ is evaluated, after themechanical environment in the interface tissue is determined. It is assumedthat the interface tissue is loaded at a low or moderate frequency and thatdeformations appearing in the peri-implant region are small. Therefore fluidmotion is neglected, and the interface tissue is modeled as an isotropic linearelastic medium. The elasticity properties of the medium are determined bythe mixture rule, using the volume fractions of various tissue types. Themechanical stimulus ψ is assumed to be defined by the octahedral shearstrain γ oct [50]:ψ = γ oct = 1 3√(ε1 − ε 2 ) 2 + (ε 2 − ε 3 ) 2 + (ε 1 − ε 3 ) 2 . (3.8)Assumption 3.2. It is assumed here, that within the regions with a low andmoderate level of mechanical stimulus, bone formation is promoted, whereashigh values of mechanical stimulus lead to the formation of fibrous tissue[18, 19, 77].The dynamics of the MSC densities c i , i ∈ {b, f} is described by theequation:∂c i∂t = ∇ s · (D c ∇ s c i ) − ∂ ∂a (u ic i ) + A c (1 − c tot − f − b)c i ,i ∈ {b, f}, (3.9)where c tot (⃗x, t) is the total density of MSC’s, normalized with respect to thelimit cell density C per unit volume, at the point (⃗x, t):c tot (⃗x, t) =∫ 10(c b (⃗x, t, a) + c f (⃗x, t, a))da. (3.10)The first term in the right-hand side of equation (3.9) represents randomwalk of MSC’s in the physical space. The symbol ∇ s denotes the nablaoperator in physical space, and the constant D c is the mobility coefficient.Proliferation of MSC’s is represented by the last term in the right-hand sideof equation (3.9). The constant A c is the rate of proliferation. Proliferationstops, if the normalized total cell density reaches its limit value, which isequal to one.The term − ∂∂a (u ic i ) in equation (3.9) corresponds to the change of MSCdensity due to the flux u i c i in the differentiation state domain Ω 1 a. The functionsu i = u i (ψ(⃗x, t), a) denote the differentiation rates. Mechanoregulationof the cell and tissue processes is introduced in the current model through

3.3. Peri-implant osseointegration 57the functions u i . This is done, by assuming a certain relation between themechanical stimulus ψ and differentiation rates u i . A relation should bechosen, such that the results, predicted by the model, are in agreementwith Assumption 3.2 concerning the mechanoregulation of tissue formationwithin the peri-implant interface. The choice of the functions u i (ψ, a) willbe described in the Section 3.3.2.Remark 3.2. In most of the recent models for bone regeneration, for examplein [6, 7, 11, 12, 35, 43, 46, 67], the process of differentiation of MSC’s isintroduced into the model equations by means of the reaction term, whichcan be written in the general form as −α i c. The variable c denotes thedensity of MSC’s per unit volume, and α i is the rate of differentiation ofMSC’s into cell type i, which can depend on a number of variables, such aschemical or mechanical stimuli. This representation implies, that the numberof MSC’s, differentiating into cell type i within the volume V duringtime period [t 0 , t 0 + dt], is equal to (∫ V α ic(x, t 0 ) dx ) dt. The rate of differentiationof MSC’s is determined by the total MSC density c at the currenttime moment t 0 , and not by the path of differentiation of cells, opposed towhat is done in the current work. In the classical models, there is always anonzero portion of MSC’s that will differentiate, even in the case of MSC’sthat did not yet inherit any property of the phenotype they are differentiatingto. Further, the entire differentiation is formally never completed withina finite time frame. The present evolutionary approach corresponds to afinal, bounded, time of cell differentiation, and further if the MSC’s wouldhave no property of the phenotype they are differentiating to, then differentiationis no longer immediate in the current approach. These are the maindistinctions of the current model from the mentioned models, which resultsfrom the consideration of MSC differentiation as an evolutionary process(see also the discussion in Section 3.5).A zero density of MSC’s is assumed within the peri-implant interface attime t = 0 [6]:c i (⃗x, 0, a) = 0 ⃗x ∈ Ω, i ∈ {b, f}. (3.11)Suppose, that ∂Ω is the boundary of Ω, ∂Ω b is the part of ∂Ω, correspondingto the bone surface, and ⃗n is an outward unit normal of ∂Ω. Itis assumed, that there is a source of MSC’s, with the differentiation levela ∈ [0, δ], on the bone surface ∂Ω b during the first two weeks [67]. Thissource is introduced into the model by specifying the influx h(t, a) ≥ 0 ofthe densities c i , i ∈ {f, b}, on ⃗x ∈ ∂Ω b , a ∈ [0, δ]. On the rest of the boundary⃗x ∈ ∂Ω, a ∈ [0, 1] a zero flux is assumed. Therefore, the boundary

58 Chapter 3. Evolutionary cell differentiationa1Ω azero flux of m iDifferentiation irreversibilitym i = 0, if v i < 0zero flux of f0m i = ∑ j≠i m j, if v i > 0influx of m iimplantsurfaceΩbonesurfacexFigure 3.3: Boundary conditions for the case of a one dimensional physicaldomain Ω. Additional source of MSC’s is shown at the bone surfaceconditions for the MSC densities c i , i ∈ {b, f} are defined as:⎧c i (⃗x, t, 0) = ∑ c j (⃗x, t, 0), if u i (ψ(⃗x, t), 0) > 0, ⃗x ∈ ¯Ω, t > 0,j≠i⎪⎨c i (⃗x, t, 1) = 0, if u i (ψ(⃗x, t), 1) < 0, ⃗x ∈ ¯Ω, t > 0, (3.12){−h(t, a), ⃗x ∈ ∂Ω b ,⎪⎩ − D c ∇ s c i · ⃗n s =0, ⃗x ∈ ∂Ω/∂Ω b , t > 0, a ∈ [0, 1],{0.01 · 1h(t, a) =δ (1 − a δ ), if t ∈ (0, 14), a ∈ [0, δ],0, otherwise,where ∇ s (·) denotes the gradient in the physical space and h(t, a) correspondsto the source of MSC’s on the bone surface. The parameter δ shouldbe related to the size of computational grid within the differentiation statespace. Therefore, in the current study, it is assumed, that δ = 0.1. In Figure3.3, the boundary conditions, defined in equation (3.12), are depicted for thecase of the one dimensional physical domain Ω.For conciseness, the fluxes of fully differentiated MSC’s are denoted viafunctionsQ i (⃗x, t) = u i (ψ(⃗x, t), 1)c i (⃗x, t, 1), ⃗x ∈ Ω, t > 0, i ∈ {b, f}. (3.13)They determine the increment of the density of osteoblasts and fibroblastsper unit of time, caused by the differentiation of MSC’s. The equations forthe dynamics of osteoblasts and fibroblasts are given as:∂b∂t = Q b + A b (1 − c tot − f − b)b, (3.14)∂f∂t = ∇ s · (D f ∇ s f) + Q f + A f (1 − c tot − f − b)f, (3.15)

3.3. Peri-implant osseointegration 59where D f is the coefficient of the fibroblast motility in the physical space, A band A f are the proliferation rates of osteoblasts and fibroblasts, respectively.It is straightforward, that the sum of all considered tissue volume fractionsis equal to one. Hence, the volume fraction of the fibrin network isdetermined by:v n = 1 − (v w + v l + v f ). (3.16)The evolution of the volume fractions of woven bone v w , lamellar bone v land fibrous tissue v f is described by the equations:∂v w∂t= α w bv n (1 − v w ) − γv w (1 − v l ), (3.17)∂v l∂t = γv w(1 − v l ), (3.18)∂v f∂t = α f fv n (1 − v f ), (3.19)where terms, containing coefficients α w and α f , correspond to the substitutionof the fibrin network by woven bone and fibrous tissue, respectively.The coefficient γ is the rate of the remodeling of woven bone into lamellarbone.The boundary and initial conditions for PDE (3.14), and the initial conditionsfor ordinary differential equations (3.15), (3.17)–(3.19) are given by:−D f ∇ s f · ⃗n s = 0, ⃗x ∈ ∂Ω, t > 0 (3.20){b(⃗x, 0) = 0, f(⃗x, 0) = 0,⃗x ∈ Ω. (3.21)v w (⃗x, 0) = 0, v f (⃗x, 0) = 0, v l (⃗x, 0) = 0,3.3.2 Differentiation rates and tissue formationIn this section the particular expressions for the differentiation rates u i ,i ∈ {b, f}, as a function of the mechanical stimulus ψ are defined in linewith Assumption 3.2 on mechanoregulation.First, it is investigated, how the magnitudes of the differentiation ratesinfluence tissue formation, described by the proposed model. It is assumed,that the differentiation rates u i , i ∈ {b, f}, do not depend on the level ofdifferentiation a. Hence,u i = U i (ψ), i ∈ {b, f}. (3.22)The following assumption is derived from the hypotheses in Proposition 3.2:U b (ψ) = U(ψ) = −U f (ψ), ψ ∈ [0, ∞). (3.23)A simplified problem is considered, that is derived from the full modelby assuming the mechanical stimulus ψ(⃗x, t) to be constant in the physical

60 Chapter 3. Evolutionary cell differentiationspace Ω and in time. The problem is defined by equations (3.9), (3.14)–(3.19) and by the initial and boundary conditions in equations (3.11), (3.12),(3.21),(3.20). The functions U b (ψ), U(ψ), U f (ψ) can be considered as constantparameters, denoted as U b , U, U f . From condition (3.23), it followsthat U b = U = −U f . The behavior of the solution of the simplified problemwith respect to the value of U is studied. The parameter U is assumed totake a range of values {U max −i∆U; i = 0, 1, . . . , N, ∆U = 2U max /N}. Theproblem is solved numerically for each value of U. As a measure of formationof bone and of fibrous tissue, the average volume fraction of woven andlamellar bone, v avgb, and the average volume fraction of fibrous tissue v avgfin the bone-implant interface at the time moment t = 63 days is chosen,respectively. Time t = 63 days is chosen according to the experiments byVandamme et al. [94]. A one dimensional axisymmetric physical domainΩ = [1, 3.5] mm is considered, in order to reduce the calculation time. Asource of MSC’s is assumed at the point r = 3.5 mm. A uniform rectangulargrid of 40 by 40 control volumes is constructed within the domain Ω × [0, 1].The governing equations are solved with the finite volume method and withthe explicit modified Euler time integration scheme. The considered parametervalues are described in Section 3.4.1.The relations between the average tissue volume fractions v avgband the values of differentiation rates U b and U fV ub (U b), v avgf= V uf (U f ). Their plots are given in Figure 3.4.and v avgare denoted as v avgf,b=1(a)v bavg0.50−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5U b1(b)v favg0.50−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5U fFigure 3.4: Plot of the average volume fraction of (a) woven and lamellarbone v avgband (b) of fibrous tissue v avgfin the peri-implant interface at thetime moment t = 63 days against the value of differentiation rate (a) U b and(b) U f , respectively

3.3. Peri-implant osseointegration 61MSC’s do not differentiate into osteoblasts, if u b < 0. Hence, for negativevalues of the parameter U b , the osteoblast density b is zero, and fromequation (3.17) it follows, that no bone forms in the peri-implant interfaceand v avgb= 0 (Figure 3.4a). According to the boundary condition, defined inequation (3.12), any MSC originated from the old bone surface has a differentiationlevel lower than δ = 0.1. Hence, some of the MSC’s will reach thedifferentiation level a = 1 at time t = 63 days, and consequently differentiateinto osteoblasts completely, only if the differentiation rate U b is largerthan U min = 1−δ63 ≈ 0.0143 days−1 . For U b ∈ [0, U min ] no osteoblasts andno bone tissue will appear in the peri-implant region within 63 days. For thevalues U b > U min , the amount of the bone, formed within the bone-implantinterface, grows with the increase of the parameter U b up to a certain extremumpoint Ubextr ≈ 0.24 days −1 , which follows from Figure 3.4a. Furtherincrease of U b leads to a bit smaller amount of bone in the interface.The relation between the parameter U f and the amount of fibrous tissuev avgfis plotted in Figure 3.4b. For negative U f and for U f ∈ [0, U min ], nofibroblasts appear within the peri-implant interface (f = 0), and no fibroustissue forms, i.e. v avgf= 0. The fibrous tissue volume fraction v avgfincreaseswith an increase of U f on the interval U f ∈ [U min , Ufextr ] and decreases forU f > Ufextr , Uf extr ≈ 0.1 days −1 .The decrease in volume fraction of bone and fibrous tissue, when thedifferentiation rates exceed the limits Ubextr and Ufextr , respectively, can beexplained by a coupled effect of cell migration, proliferation and differentiation.The inverse proportionality is more recognizable for the fibrous tissuedependence on the differentiation rate U f (Figure 3.4b). As it follows fromthe values of the motility and proliferation coefficients D c , D f , A c and A f(see Section 3.4.1), MSC’s migrate and proliferate within the healing sitefaster than fibroblasts. On the one hand, for smaller differentiation ratesU f , each cell remains a MSC for a longer time period. In this state, thecell migrates towards the implant surface and proliferates faster, that if itwere a fibroblast. This can lead to a higher total density of MSC’s withinthe peri-implant region for smaller U b after some time. On the other hand,if the supply of MSC’s is sufficient, then the larger the value of the differentiationrate U f , the more MSC’s differentiate into fibroblasts per unit oftime. For high rates U f > U extrf, the first effect prevails. However, for slow, the second effect becomes more important.differentiation, U f < UfextrNext, the following functions are defined: v avgwherei= V ψi(ψ), i ∈ {b, f},V ψi(ψ) = V ui (U i (ψ)), ψ ∈ [0, ∞). (3.24)A certain expression for the function U : [0, ∞) → U ⊂ R should be determined,such that U b (ψ) = U(ψ) = −U f (ψ), and such that the dependenciesv avgb= V ψb(ψ), vavgf= V ψf(ψ) will be in agreement with the Assumption 3.2

62 Chapter 3. Evolutionary cell differentiationabout mechanoregulation of tissue formation. The following relation is proposed(see Figure 3.5):⎧⎪⎨U(ψ) =(U maxb(U maxb− U 0 b ) sin ( ψπ2ψ 1)− U min ) cosU minδψ (ψ 2 − ψ),(U min − Ufmax ) sin+ U 0 b , 0 ≤ ψ < ψ 1,( )(ψ − ψ1 )π+ U min ,2(ψ 2 − ψ 1 − δψ)( (ψ − ψ2 − δψ)π2(ψ 3 − ψ 2 − δψ)ψ 1 ≤ ψ < ψ 2 − δψ,ψ 2 − δψ ≤ ψ < ψ 2 + δψ,)− U min ,ψ 2 + δψ ≤ ψ < ψ 3 ,⎪⎩− Uf max , ψ 3 ≤ ψ.(3.25)From the plot of the function U(ψ) (Figure 3.5) and from equations (3.22)–(3.23), (3.25), it follows, that the line of non-negative values of the mechanicalstimulus ψ is divided into five characteristic intervals. On the firstinterval [0, ψ 1 ], the rate of MSC differentiation into osteoblasts increasesfrom the value Ub 0 , which corresponds to a zero mechanical stimulus, to thevalue Ubmax . Abrahamsson et al. [1] and Berglundh et al. [14] observed formationof bone within the peri-implant region at the end of the first week.The value 0.2 days −1 is estimated for U max , so that osteoblasts, releasingbone matrix, will appear at the end of the first week. It is assumed, thatUb 0 = 0.1 days −1 . The threshold ψ 1 , corresponding to the maximal rateof differentiation of MSC’s into osteoblast Ubmax , is assumed to be equal to0.008. On the interval [ψ 1 , ψ 2 − δψ], the differentiation rate into osteoblastsdecreases monotonically to the value U min , for which no osteoblasts will beformed within 63 days. This time period is chosen for the numerical simulations,in line with Vandamme et al. [94]. The region [ψ 2 − δ, ψ 2 + δψ]is a ’transition region’. The function U(ψ) changes its sign at the pointψ = ψ 2 , and MSC’s stop differentiating into osteoblasts, and start differentiatinginto fibroblasts for ψ > ψ 2 . In other words, within the transitionregion, MSC’s change the ’direction’ of differentiation. The value of 0.05 isestimated for the threshold ψ 2 [50]. Note, that for ψ ∈ [ψ 2 − δ, ψ 2 + δψ],U b ≤ U min , U f ≤ U min , no osteoblasts and fibroblasts, and hence, no boneand fibrous tissue will appear after 63 days. This situation is biologicallyirrelevant. Therefore, the size of the transition region is chosen to be verysmall, and the following parameter value is used δψ = 10 −4 . The functionU(ψ) is very steep within the transition region, since it changes its valuefrom U min to −U min over a very small interval. For ψ ∈ [ψ 2 + δψ, ψ 3 ], therate of MSC differentiation into fibroblasts increases to its maximum valueU maxf= 0.1 days −1 , and then remains constant for ψ > ψ 3 . It is assumed,

3.3. Peri-implant osseointegration 63that ψ 3 = 0.07.U bmaxU b0U(ψ)U min0−U min−U fmax0 ψ 1ψ 2ψ 3ψFigure 3.5: Plot of the function U(ψ), related to the differentiation ratemagnitude, versus mechanical stimulus ψ, defined in equation (3.25)Equation (3.24) yields, that V ψ′b(ψ) = Vbu′ (U i(ψ))Ub ′ (ψ). The derivativeV ψ′b(ψ) has the same sign as the derivative Ub ′ (ψ), if Vu′b(U i(ψ)) > 0. FromFigure 3.5 it follows, that U i (ψ) ∈ [U min , Ubmax ], for ψ ∈ [0, ψ 2 − δψ]. It ismentioned in this Section 3.3.2, that the function v avgb= Vb u (U) increasesfor U ∈ [U min , Ubextr ] (see Figure 3.4a). The value for Ubmax was definedearlier, which was smaller than Ubextr . Hence, [U min , Ubmax ] ⊂ [U min , Ub extr ],and Vbu′ (U i(ψ)) > 0 for ψ ∈ [0, ψ 2 − δψ]. This means, that the function V ψbbehaves similarly to the function U b (ψ) on interval ψ ∈ [0, ψ 2 − δψ]: V ψbincreases on ψ = [0, ψ 1 ], has its maximum at ψ = ψ 1 , and monotonicallydecreases to zero on the interval ψ ∈ [ψ 1 , ψ 2 − δψ]. For ψ > ψ 2 − δψ,U b (ψ) < U min . As it is mentioned earlier, no osteoblasts will be formed tillday 63 for such values of the differentiation rate U b . Consequently, for thesevalues of ψ, V ψb (ψ) = V b u(Ub(ψ)) = 0.Let us consider the behavior of the function v avgf[0, ψ 2 + δψ], U f (ψ) = −U(ψ) < U min (see Figure 3.5). Hence, no fibroblasts= V ψf(ψ). For ψ ∈will appear at day 63, and V ψf (ψ) = V f u(Uf (ψ)) = 0. From equation (3.24) itis derived, that the derivative V ψ′ (ψ) is equal toV u′f (U f (ψ))Uf ′ (ψ). On the interval ψ ∈ [ψ 2 + δψ, ∞) the function U f (ψ) =−U(ψ) takes values, which lie inside the interval[U min , Ufmax ] ⊂ [U min , Uf extr ].For such values of U f (ψ), the derivative Vfu′ (U f (ψ)) is positive, which followsfrom Figure 3.4b. Consequently, the function V ψ′f(ψ) behaves similarly tothe function U f (ψ) = −U(ψ) for ψ > ψ 2 + δψ. It increases on the intervalψ ∈ [ψ 2 + δψ, ψ 3 ), and stays constant for ψ ≥ ψ 3 .f

64 Chapter 3. Evolutionary cell differentiationThe functions v avgbthe dependencies v avg= V ψb= V u(ψ) and vavgf= V ψf(ψ) can be derived from(u), obtained from numericalb b(u), vavgf= Vf usolutions, and from the function U(ψ), defined in equation (3.25). Thesefunctions are plotted in Figure 3.6. These plots are in line with the expec-0.4Time 14 days0.1v bavg0.2v favg0.05001ψ 1ψ 2ψTime 63 days1ψ 1ψ 2ψv bavg0.5v favg0.500ψ 1ψ 2ψψ 1ψ 2ψFigure 3.6: Plots of the functions v avgb= V ψb(ψ) and vavgf= V ψf(ψ) versusthe mechanical stimulus ψ for time t = 14 days and t = 63 days. Thefunctions are derived from the functions Vb u(u),V f u (u), and U(ψ), whichwere obtained earliertations. For the values of the mechanical stimulus ψ, lying in the interval[0, ψ 2 ), bone formation takes place, and for ψ ∈ (ψ 2 , ∞), fibrous tissue forms.The largest amount of bone is expected for ψ = ψ 1 . The promoting roleof mechanical stimulation of the level ψ 1 , if compared with no loading conditionsψ = 0, is more recognizable at early periods of bone regeneration.It is shown in Figure 3.6, that the amount of bone, corresponding to thestimulus value ψ 1 is by ≈ 50% larger than for a zero mechanical stimulus attime t = 14 days.3.4 Numerical simulationsA series of numerical simulations is carried out for the developed mechanobiologicalmodel. The chosen configuration of the bone-implant interfaceand the mechanical loading scheme correspond to those from the experimentsby [94]. In the experiments, a perforated chamber is installed intothe bone. The cylindrical implant is situated in the middle of the chamber.The perforations in the chamber allowed blood inflow and tissue growthfrom the surrounding area into the cavity of the chamber. The mechanical

3.4. Numerical simulations 658z, mm6ImplantΩBone1.50u d1 3 3.5r, mmFigure 3.7: The axisymmetric physical domain Ω, and the boundary conditionsfor the displacements u r and u z , which determine the distribution ofthe mechanical stimulus within Ωloading is applied by means of the axial movement of the implant inside thechamber.This experimental study is chosen for the verification of the currentmathematical model, due to the well controlled mechanical environment,achieved within the chamber. This is important for the present approach,since the field of the mechanical stimulus within the bone-implant interfaceis calculated, since it determines cell differentiation.3.4.1 Model parameters and numerical methodThe axisymmetric physical domain Ω, used for the present simulations, issketched in Figure 3.7. The computational model is defined by the elastostaticequations in Navier-Cauchy’s formulation and by equations (3.9),(3.14)–(3.19). Boundary conditions for the mechanical part of the model areshown in Figure 3.7. Constant vertical displacements of the magnitude u dare specified on the surface, attached to the implant. Initial conditions forthe cell densities and the tissue fractions are defined in (3.11), (3.21), andthe boundary conditions are taken from (3.12) and (3.20). The followingvalues for model parameters are chosen (according to a Moreo et al. [67] andb Andreykiv [7]):D c = 0.133 1dayA f = 0.1 1daya , D f = 0.5 1dayb , α w = 0.1 1dayb , A c = 0.5 1daya , γ = 0.06 1daya , A b = 0.5 1daya , α f = 0.06 1dayFor the limit cell density C the value of 10 6 cells/mm 3 is chosen [11]. Theprocedure of obtaining the numerical solution is outlined in Figure 3.8.Mechanical properties of the interface tissue are determined by the mixturerule, according to the distribution of the various tissue fractions at time t 0 .b ,b .

66 Chapter 3. Evolutionary cell differentiationStartInitial conditionst 0 = 0Mechanical stimulus distributionin Ω is foundMechanical properties of theinterface tissue are updatedSolution for cell densities and tissue volume fractionsis found for the time period t ∈ [t 0 , t 0 + ∆T ]t 0 < TSolution for m b and m fSolution for bSolution for fSolution for v w , v l and v fv n = 1 − (v w + v l + v f )t 0 = t 0 + ∆Tt 0 ≥ TEndFigure 3.8: Outline of the method of the numerical solutionElastic properties for the considered tissue types are given in Table 3.1 [39].The mechanical stimulus field is found from the solution of the elastostaticTable 3.1: Elasticity properties for the considered tissue types [39]FibrinnetworkWovenboneLamellarboneFibroustissueYoung’s modulus (MPa) 0.2 1000 6000 1Poisson’s ratio 0.17 0.3 0.3 0.17equations for the updated properties. Then, the solution for the biologicalpart of the model is obtained. After time period ∆T , the mechanical propertiesare updated again. The cycle repeats, until t 0 = T . For the currentsimulations ∆T = 1 day and T = 63 days.The linear elasticity equations are solved by use of the finite elementmethod. The equations for the evolution of MSC and fibroblasts densitiescontain advection terms. The finite volume method with the Koren flux limiteris used to discretize the partial differential equations (3.9) and (3.15) inthe physical space and in the differentiation state domain [48]. This methodprovides a converging non-oscillating non-negative numerical solution. Thepositiveness of the numerical approximation for the cell densities is criticalfor the present model. Negative densities could lead to negative tissue frac-

3.4. Numerical simulations 67tions and, consequently, to negative mechanical properties and, finally, tothe non-convergence of the solution. The Koren limiter for the flux at thecell interfaces is equivalent to the third-order upwind-biased discretizationscheme for smooth solutions. Time integration is performed with use of thesecond order modified Euler’s method.The physical domain Ω is meshed with rectangular cells (i.e. finite elementsor control volumes), with edge lengths dx = 0.125 mm and dy =0.25 mm. Twenty layers of the corresponding cuboid cells, with edge da =0.05 along the a axis, constitute the grid in the domain Ω × [0, 1]. Axialsymmetry of the physical domain Ω is considered. The configuration of theregion Ω is sketched in Figure 3.7. The discretized ordinary differential equationsare integrated by means of the second-order explicit modified Euler’smethod.3.4.2 Numerical resultsThe numerical simulations are carried out for different loading regimes,which are applied by specifying constant vertical displacements u d on theimplant surface (see Figure 3.7). Four levels of the displacements are considered:u d = 0 mm, u d = 0.03 mm, u d = 0.09 mm and u d = 0.24 mm.The spatial distribution of new bone matrix differs for the four consideredcases, as it follows from Figure 3.9.First, the numerical solutions for the displacements u d = 0 mm andu d = 0.03 mm are considered. Due to the chosen relations for the differentiationrates of MSC’s, no displacements imply a zero mechanical stimulusand a uniform value 0.1 day −1 of the differentiation rate u b within the entirebone-implant interface. The tissue deformations, corresponding to thedisplacements of the implant with magnitude 0.03 mm, are small enoughthroughout the whole simulation period, so that the differentiation rate u bis always positive (see Figure 3.10). This means, that in the consideredcases, all MSC’s differentiate only into osteoblasts. Hence, no fibrous tissueis formed in the peri-implant interface for the implant displacementsu d = 0 mm and u d = 0.03 mm. In the present model it is assumed,that a low level mechanical loading enhances differentiation into osteoblasts.Therefore the differentiation rate u b for the case u d = 0.03 mm is in generalhigher than in the case u d = 0 mm, when u b = 0.1 day −1 (Figure 3.10).This leads to an earlier appearance of osteoblasts within the bone-implantinterface for the displacements of the magnitude 0.03 mm. Hence, boneformation starts earlier in the case u d = 0.03 mm compared with no loadingconditions (see Figure 3.9). Another difference between the two consideredlevels of the loading, is the spatial distribution of new bone matrix. For thezero implant displacement case, the volume fraction of bone is larger nearthe implant surface than near the old bone surface for t ≥ 12 days, and thebone volume fraction is larger near the old bone surface for the displace-

68 Chapter 3. Evolutionary cell differentiationT= 9 daysx 10 −4 T=12 daysT=15 daysT=63 daysu d= 0 mm2.521.510.050.040.030.20.180.160.9620.9610.96u d= 0.03 mmT= 5 daysx 10 −68642T= 6 daysx 10 −4642T= 8 daysT=63 days0.020.0150.010.0050.9680.9640.960.956T= 5 daysx 10 −4T= 7 daysx 10 −3 T=10 daysT=63 daysu d= 0.09 mm2.521.510.5151050.10.060.020.9650.9550.945u d= 0.24 mmT= 6 daysx 10 −44321T= 8 daysx 10 −3 T=12 days151050.150.10.05T=63 days0.80.60.40.2000Figure 3.9: Plots of the volume fraction of woven and lamellar bone v w +v l in the peri-implant interface at different time moments for the implantdisplacements of the magnitude u d equal to 0 mm, 0.03 mm, 0.09 mm and0.24 mm

3.4. Numerical simulations 69ments u d = 0.03 mm for t ≥ 7 days. The different spatial distribution ofbone results from the respective behavior of cells.At the first place, it should be mentioned, that the MSC’s originatingfrom the bone surface initially have a differentiation level close to zero. Thesecells start to differentiate into osteoblasts and, simultaneously, migrate towardsthe implant. The migration of MSC’s takes place due to the higherMSC density at the bone surface at a low maturation level, which is providedby the inflow of MSC’s, specified at this boundary (see Figure 3.11).While MSC’s approach the implant surface, their maturation level increases.Hence, the average maturation level a avg of the MSC’s is higher near theimplant surface, than near the old bone surface (Figure 3.12). Due to alower total cell density at the implant interface, MSC’s proliferate faster inthis region, since the proliferation rate is assumed to be equal to A c (1−c tot ).The maturation level of daughter cells is assumed to be equal to the maturationlevel of their mother cell. Therefore, it is possible, that at some timemoment a number of MSC’s with a high maturation level becomes largernear the implant surface than near the old bone surface. This results ina higher density of osteoblasts and in a faster formation of bone near theimplant. Such a behavior is predicted for the case u d = 0 mm.As it was mentioned earlier, the differentiation rate u b is higher for thedisplacements u d = 0.03 mm, compared with no mechanical loading conditions.Therefore, MSC’s undergo fewer divisions till the moment, whenosteoblasts appear within the healing site and active bone formation starts.Despite the fact, that the average maturation level a avg is lower at the oldbone surface (Figure 3.12), the total number of MSC’s with high maturationlevel will be larger, due to a much larger total density of MSC’s inthe considered region compared with the area close to the implant surface(Figure 3.11). The effect of the MSC source at the old bone surface prevailsover the effect of a higher proliferation rate near the implant surface. As aresult, the volume fraction of bone is higher near the old bone in the caseu d = 0.03 mm.In a similar way, the bone volume fraction is larger near the old bonesurface for the implant displacements of the magnitude 0.09 mm. However,in this case a delayed bone formation on the implant surface is predicted.As it follows from Figure 3.9, new bone matrix starts to form approximatelyat day 5 predominantly in the region adjacent to the old bone surface. Theeffective tissue stiffness increases significantly in this region. The gradient ofthe tissue stiffness leads to the localization of deformations near the implantsurface. Thereby, high values of the mechanical stimulus appear near theimplant surface at day 6 (Figure 3.13). These values correspond to negativevalues of the differentiation rate u b . This means that MSC’s start todifferentiate into fibroblasts, i.e. u f > 0. The number of osteoblasts growsslowly, compared with the neighboring region, where u b > 0 and a largesource of osteoblasts is provided by differentiating MSC’s. This leads to a

3.4. Numerical simulations 71T= 6 days T= 9 days T=12 days T=15 daysu d= 0 mm0.50.450.40.750.70.650.850.80.750.70.850.80.750.70.65u d= 0.03 mm0.80.70.60.80.70.60.80.70.60.80.70.6Figure 3.12: Plots of the(average maturation level a avg of MSC’s, which is∫ 1calculated as a avg =0 c f (−a) da + ∫ )10 c b a da / ∫ 10 (c b + c f ) da, in theperi-implant interface at different time moments for the implant displacementsof the magnitude u d equal to 0 mm and 0.03 mmstimulusT= 5 days T= 6 days T=21 days T=22 days0.030.020.010.060.040.020.050.040.030.020.010.040.02u b0.180.160.140.120.10−0.10.150.10.0500.150.10.05Figure 3.13: Plots of the mechanical stimulus and of the differentiation rateu b in the peri-implant interface at different time moments for the implantdisplacements of the magnitude u d equal to 0.09 mm

72 Chapter 3. Evolutionary cell differentiationnarrow zone next to the implant surface with a lower volume fraction ofbone, which is well recognizable in Figure 3.9.The zone, where u f > 0, which is equivalent to u b < 0, and whereMSC’s differentiate into fibroblasts, is narrow. Due to random walk of cells,this region is densely populated by the MSC’s with a high level of differentiationinto osteoblasts, which migrate from the neighboring part of theperi-implant interface, where u b > 0 (see Figure 3.14). MSC’s, which differ-T= 6 days T= 9 days T=12 days T=15 daysu d= 0.09 mm0.80.70.60.80.70.60.80.70.60.50.80.70.6Figure 3.14: Plots of the(average maturation level a avg of MSC’s, which is∫ 1calculated as a avg =0 c f (−a) da + ∫ )10 c b a da / ∫ 10 (c b + c f ) da, in theperi-implant interface at different time moments for the implant displacementsof the magnitude u d equal to 0.09 mmentiate into fibroblasts within this narrow zone, migrate in their turn intothe neighboring region, and the direction of their differentiation is reversedto osteoblasts again. Active cell diffusion and a limited time period (approximately16 days), when u f > 0 at the implant surface, lead to that nofibroblasts are derived from MSC’s near the implant surface and no fibroustissue is formed for the displacements of the magnitude u d = 0.09 mm.Large implant displacements of the magnitude u d = 0.24 mm lead tohigh values of the mechanical stimulus near the implant surface, which correspondto a positive differentiation rate u f (Figure 3.15). The region, whereu f > 0 is larger compared with the analogous region, which appears for0.09 mm displacements. A larger size of the region leads to smaller valuesof the gradient of the density of MSC’s differentiating into fibroblasts. Hencefewer MSC’s migrate away from this region and more MSC’s differentiatecontinuously into fibroblasts. As a result, a non-zero density of fibroblastsis predicted in the middle of the implant interface at day 15 (Figure 3.15).Therefore, fibrous tissue is produced by fibroblasts and no bone formation ispredicted near the implant surface for the displacements u d = 0.24 mm. Inthe rest of the peri-implant interface the volume fraction of bone is close toone at day 63 (Figure 3.9). It should be noted, that the calculated density offibroblasts is much lower than the density of osteoblasts. From Figure 3.16ait follows that the average density of osteoblasts is equal to 0.82 mm −3 atday 63. While the maximal density of fibroblasts is 0.024 mm −3 at the sametime moment (Figure 3.15). The density f is low, since the region, where

3.4. Numerical simulations 73T= 6 days T=15 days T=32 days T=63 daysv u fffstimulus0.080.060.040.020.10−0.110−110−10.60.40.20.60.40.20.10.100−0.1−0.1x 10 −6x 10 −331521015x 10 −8 x 10 −338624120.50.40.30.20.10.10−0.10.020.0150.010.040.02Figure 3.15: Plots of the mechanical stimulus, of the differentiation rate u f ,of the fibroblast density f and of the volume fraction of fibrous tissue v fwithin the peri-implant interface at different time moments for the implantdisplacements of the magnitude u d = 0.24 mm

74 Chapter 3. Evolutionary cell differentiationu b > 0, occupies the most part of the peri-implant interface. It is even moreimportant, that u b > 0 near the old bone surface, where the total densityof MSC’s is the highest, due to the cell influx from the boundary surfaceduring the first two weeks. This leads to a high proportion of MSC’s differentiatinginto osteoblasts even in the regions, where u f > 0 (Figure 3.14),and few MSC’s differentiate into fibroblasts completely, as it is explained inthe previous paragraph.Therewith, the evolution of the average density of osteoblasts and of thebone volume fraction is shown in Figure 3.16. The average number of osteoblastsis maximal for the displacements u d = 0.03 mm and u d = 0.09 mm(Figure 3.16a), due to high differentiation rates of MSC’s into osteoblastswithin the entire bone-implant interface, excluding a very small area nearthe implant surface for u d = 0.09 mm (see Figures 3.10 and 3.13). Hence,the average volume fractions of the bone tissue for this loading regimes arehighest among the considered implant displacements (Figure 3.16b). At theend of the simulation period, the average amount of new bone tissue is comparablefor the displacements u d = 0 mm, u d = 0.03 mm and u d = 0.09 mm,and it is larger than for the large displacement conditions u d = 0.24 mm(Figure 3.16b).b avg10.5(a)00 20 40 60Time [days]v wavg +vlavg10.5(b)00 20 40 60Time [days]u d= 0 mm; u d= 0.03 mm; u d= 0.09 mm; u d= 0.24 mm;Figure 3.16: Plots of the average density of osteoblasts b avg (a) and of theaverage volume fraction of woven and lamellar bone vwavg + v avgl(b) againsttime. Averaging is introduced with respect to a spatial distributionVandamme et al. [94] determined the area fraction of bone tissue, averagedover the transverse sections, made at the level of the perforation in thebone chamber, after 9 weeks from the beginning of the experiments. In theexperiments, where the implant was not loaded, they obtained the bone areafraction (BAF) of 0.6403 ± 0.2381 (mean value ± standard deviation). Forthe displacements of the implant of the magnitude 0.03 mm and 0.09 mm,the measured values of BAF were 0.7509±0.2149 and 0.7347±0.2229, respectively.From the statistical analysis, Vandamme et al. [94] derived, that thebone area fraction was significantly higher for 0.09 mm implant displacementcompared with no displacement, and a significantly higher fractionof bone trabeculae was found for 0.03 and 0.09 mm implant displacement

3.5. Discussion and conclusions 75compared with the unloaded situation.The calculated values of the bone volume fraction are higher than thevalues obtained in the experiments. The numerical simulations yield, thatthe average values of the volume fraction of woven and lamellar bone in theregion of peri-implant interface, located at the level of the perforation, i.e.for r ∈ [1, 3.5], z ∈ [1.5, 6] (see Figure 3.7), are equal to 0.961, 0.968 and0.967, for the implant displacements of the magnitude 0 mm, 0.03 mm and0.09 mm, respectively. From Figure 3.17 it follows, that the average volumefraction of bone is lower for the displacements u d = 0 mm compared withthe displacements u d = 0.03 mm and u d = 0.09 mm. This difference is thelargest at the initial period of bone formation (3-4 weeks). Therefore, thev wavg +vlavg10.80.60.40.200 20 40 60Time [days]u d= 0 mm; u d= 0.03 mm; u d= 0.09 mm; u d= 0.24 mm;Figure 3.17: Plots of the average volume fraction of woven and lamellarbone vwavg + v avglin the region of peri-implant interface, located at the levelof the perforation, against timenumerical and experimental results are in partial agreement qualitativelyfor the considered levels of the implant displacements. The displacementsequal to 0.24 mm lead to the appearance of fibrous tissue. This is in linewith Assumption 3.2 concerning the mechanoregulation of tissue regeneration.A weak quantitative correspondence of the model predictions to theexperiments can be explained by the fact, that the current model is constructedin a rather simple form, since the main interest in this chapter, isthe representation of cell differentiation as a gradually evolving process, althoughseveral over biological processes like proliferation, motility and boneformation are modeled using classical formalisms.3.5 Discussion and conclusionsIn this chapter a model for peri-implant osseointegration is proposed, inwhich MSC differentiation is considered as an evolutionary process that is

76 Chapter 3. Evolutionary cell differentiationinfluenced by the mechanical state in the region of tissue formation. Themain distinction of the current model for peri-implant osseointegration is inthe description of the differentiation of cells. MSC’s are distinguished withrespect to their differentiation state. The differentiation state is evaluatedthrough the variable a, which takes a value from zero to one. In the commonapproach, used in the most of the recent models for bone regeneration, thedifferentiation level of MSC’s is not considered and cell differentiation is introducedinto the constitutive equations by means of the reaction term. Thiscan be interpreted, that only one state of the MSC’s is considered. Therefore,the evolutionary differentiation approach can be considered as continuous,and the common immediate differentiation approach, as discrete. The continuousapproach is likely to describe the process of cell differentiation moreaccurately, than the discrete model. It assigns a finite time of differentiationto each cell, which is in line with experimental observations [13, 60, 97],where cells gradually obtain the new properties in the course of time. Thehistory of cell differentiation is incorporated, so that the current state ofcells depends on how the cells evolved before. In the discrete approach thedifferentiation history or path cannot be modeled (see Remark 3.2).Reina-Romo et al. [83] applied the concept of cell plasticity in the modelfor the bone distraction. However, in this chapter a generic mathematicalmodel is defined, which allows to consider cell differentiation, as a processthat evolves in time. The current approach can be extended to an arbitrarynumber of parameters, which determine the differentiation rates of MSC’s,and to an arbitrary number of cell types, into which MSC’s differentiate.The mathematical model for evolutionary cell differentiation is presentedby means of a set of partial differential equations, boundary and initialconditions. The MSC differentiation is introduced into the model by addingone extra dimension to the problem domain. The differentiation rates areconsidered as functions of a number of parameters. A specific representationof the differentiation of fully non-differentiated MSC’s poses some conditionson the differentiation rates.Unfortunately, an example of an application, that is useful to directly validatethe present model for the evolution of the differentiation level of cells,was not found. Instead, the present differentiation approach is incorporatedinto a model for peri-implant osseointegration. This model allows to demonstratethe main characteristics of the evolutionary approach. The history ofMSC differentiation into two cell types is included in the model, and it isdetermined by the mechanical environment within the tissue. The presentapplication example also shows, how the current differentiation approachcan be used to represent cell differentiation, coupled with cell migration andproliferation, in one model.It is assumed, that a low and moderate level of mechanical loading promotesbone formation, whereas a high level of loading leads to the appearanceof the fibrous tissue within the bone-implant interface. On the basis

3.5. Discussion and conclusions 77of these assumptions and on the basis of the numerical assessment for the1D test problem, certain expressions for the differentiation rates of MSC’sare specified (see equations (3.22)– (3.23), (3.25)). The essential feature ofthese relations, is that the differentiation rates u b and u f do not exceedcertain thresholds Ubextr and Ufextr , which are determined in 1D simulations.This condition ensures, that the amount of a certain tissue type increaseswith the increment of the rate of differentiation of MSC’s into a respectivecell type. The simulations of peri-implant osseointegration within the bonechamber are carried out for four levels of loading.The obtained results are in a partial correspondence with the experimentalmeasurements, presented in Vandamme et al. [94]. Different pathsof bone formation are obtained for different levels of displacements. For noloading conditions, new bone matrix is formed faster at the implant surface,while for the displacements of the magnitude 0.03 mm, 0.09 mm and0.24 mm bone is released faster at the old bone surface. These two modesof bone formation are usually referred to as contact and distance osteogenesis.Moreo et al. [67] and Amor et al. [6] use their models for peri-implantosseointegration to study the situations, when contact and distance osteogenesisoccur. They related the two different paths of bone regeneration toa micro-roughness of the implant surface. This assumption is in line withthe experiments of Abrahamsson et al. [1]. The implant micro-roughnessis represented into the models implicitly by means of various growth factorsources at the implant surface, that correspond to the growth factor releaseby activated platelets.The predictions of the present model about different osseointegrationmodes follow from a coupled effect of MSC differentiation, migration andproliferation. The present model should be extended, in order to representthe conditions providing contact and distance osteogenesis in a more strictand clear way. This can be done, by adding a growth factor concentrationvariable into the model. A more complex model is not considered in thepresent chapter, since the focus of the current study is on evolutionary celldifferentiation approach. The current peri-implant osseointegration modelis used, to check the performance of the evolutionary approach as a part ofa more general bone regeneration model.As another potential direction of the model improvement, a variationof the mechanoregulatory relations for the differentiation rates, and a modificationof the mechanical stimulus can be considered. For example, dynamicalmechanical loading may be considered, and the tissue within theperi-implant interface may be introduced as a poro-elastic media. Anotheroption, is to consider differentiation of cells as a partly stochastic process,for example, by introducing a random variable into the relations for the differentiationrate. In this chapter the evolution of the differentiation processis introduced. Considering cell proliferation as an evolutionary process, canalso be a following step in improving the given model.

78 Chapter 3. Evolutionary cell differentiationOne feature of the evolutionary differentiation approach, which can beused to enhance the accuracy of the future models for bone regeneration, isthe finite time of cell differentiation. In reality, new bone appears in the periimplantinterface after some time from the implant placement. For example,in the experiments of Berglundh et al. [14] and Abrahamsson et al. [1], bonematrix started forming on the pre-existing surfaces at the end of the firstweek. This fact can be modeled with use of the evolutionary differentiationapproach. Due to a finite time of differentiation, osteoblasts, which releasebone matrix, can be assumed to appear after some time from the implantplacement. This issue cannot be modeled with the immediate differentiationapproach, since for this approach, bone would appear immediately.The present differentiation representation is incorporated into a movingboundary model, which is described in Chapter 4. For that model it isessential, that the time of differentiation of MSC’s into osteoblasts is finite.This condition is satisfied by the present evolutionary approach.

CHAPTER 4Moving boundary model forendosseous healingIn this section, a model for the early stages of peri-implant bone regenerationis developed. This model is able to capture some important characteristicsof endosseous healing, which are not incorporated in the existing models. Itis a well known fact, that during peri-implant osseointegration, bone formsonly by apposition on a pre-existing rigid surface, which initially consists ofthe implant surface and the old bone surface. In order to track the movementof the front of the newly formed bone, a moving boundary problemis formulated. Another important feature of the current model, is that celldifferentiation is considered as a gradual process, evolving in time and beinginfluenced by the presence of growth factors. Hence the evolution of celldifferentiation level is captured in the present approach.Some of the model parameters are taken from the literature and the restof them is estimated from numerical simulations. The numerical algorithm,developed for the solution of the model equations within a two dimensionalphysical domain, is described in Chapter 5. This algorithm allows to solvethe model equations within an irregular domain, which evolves in time. Inthe current chapter, only some general features of the numerical method areoutlined. A large number of numerical simulations is introduced, in orderto estimate the values of the model parameters. Results of the numericalsolutions are compared with experiments.4.1 IntroductionBone regeneration around implants and the ways of improving early mechanicalstability of the implants have been investigated by many researchers79

80 Chapter 4. Moving boundary model for endosseous healingrecently. A number of experimental in-vivo and in-vitro studies are availablein the literature [1, 14, 65, 80, 92, 93, 94]. The peri-implant endosseous healingconsists of the cascade of biological processes, which take place withinthe healing site after the implant placement [24, 63]. In Davies [24], thefour stages of implant osseointegration are distinguished. At the first stage,haemostasis, right after the placement of the implant into the bone, thewound site is filled with blood and a blood clot forms. At this stage bloodplatelets attach to the implant surface, and after being activated they startto release cytokines and growth factors. The second stage, referred as osteoconduction,involves recruitment and migration of differentiating osteogeniccells to the implant surface. At the third stage, named de novo bone formation,osteogenic cells differentiate into osteoblasts, which start to releasebone matrix at the implant surface. This bone formation occurs in the formof intramembranous ossification. New bone matrix is released by osteoblasts,attached to a rigid surface [1, 14, 65]. In other words, peri-implantossification occurs through a direct apposition of new bone matrix on thepre-existing surface [24]. Hence the concept of the newly formed bone frontor surface can be used. Due to the synthesis of bone, the bone front movesinto the region of the non-ossified tissue. In the case of successful osseointegration,a direct structural and functional fixation of the implant withinthe new bone matrix is established.Two modes of the bone front movement, called distance and contactosteogenesis, can be distinguished. For the first mode, bone forms initiallyat the old bone surface, and the bone front moves towards the implantsurface. The contact osteogenesis corresponds to the situation, when newbone matrix is released by osteoblasts at the implant surface itself, and thefront of newly formed bone moves from the implant towards the old bonesurface. In this case, the implant surface turns to be in a direct contact withnewly formed woven bone. Bone synthesis, taking place both at the old bonesurface and at the implant surface, leads to a faster biological anchorage ofthe implant, compared to distance osteogenesis mode [1, 32]. Therefore,contact osteogenesis provides better mechanical stability of the implant atthe early stages than distance osteogenesis.The osteoconduction phase is critical for the contact osteogenesis mode.Synthesis of bone at the implant surface is possible, only if enough differentiatingosteogenic cells are present near the implant, i.e. it is necessary,that osteogenic cells migrate constantly from the old bone surface towardsthe implant through the fibrin network, formed in the blood clot.Therefore, the first three stages of osseointegration result in the formationof woven bone, which provide the biological fixation of the implant.The fourth stage in peri-implant osseointegration is remodeling of the wovenbone into mature bone. The process of remodeling is slow, comparedto the processes, involved in the first three stages. In the present work,the second (osteoconduction) and third (de novo bone formation) stages of

4.2. Recent mathematical models 81osseointegration are considered. Bone remodeling, which follows after thewoven bone formation, is not considered in the current model. Accordingto Davies [24], osteoconduction and de novo bone formation are the crucialstages of osseointegration, since exactly at these phases, the implant is anchoredwithin the wound site by the newly formed bone matrix. Withoutsuch biological anchorage, successful osseointegration is impossible.The structure of this chapter is as follows. A brief review of severalrecent mathematical models for the peri-implant osseointegration is given inSection 4.2.The new model for the early stages of peri-implant bone regeneration isdeveloped in Section 4.3. The main innovation of this model is the movingboundary approach, used to represent the movement of the bone-formingsurface. It is derived to accommodate the aforementioned fact that bone onlyforms by apposition on the pre-existing rigid surface [24]. The peri-implantregion is divided into two domains: the first domain is occupied by the softtissue, and the second domain is filled with the new bone. The evolution ofthe osteogenic cell density and of the growth factor concentration within thefirst domain is modeled with partial differential equations. The number ofmature osteogenic cells and the concentration of growth factor at the boneformingsurface, which is the boundary of the soft tissue domain, determinesthe rate of new bone matrix release, and consequently the velocity of thebone-forming surface. Within the region, filled with new bone, the processof bone remodeling takes place, which is not considered in this work. Celldifferentiation is considered as a gradual process, which depends on thegrowth factors concentration. The approach, derived in Chapter 3, is usedto model the evolution of the cell differentiation level.Some basic features of the 2D numerical algorithm are outlined in Section4.4.1. A model validation procedure and initialization of the parametersare described in Section 4.4.2. In Section 4.4.3, a sensitivity analysisbased on the results of numerical simulations for various model parametersis presented. The main results of the numerical simulations are discussed inSection 4.5. Conclusions are drawn in Section 4.6.4.2 Recent mathematical modelsMathematical modeling is a useful tool for the investigation of peri-implantosseointegration. It allows to study the role of various processes, which takeplace during bone regeneration under the influence of numerous externaland internal factors. Mathematical models for bone regeneration aroundendosseous oral implants were developed by Amor et al. [5, 6], Moreo et al.[67]. Moreo et al. [67] reported that their model is able to predict differentmodes of bone regeneration (i.e. contact and distance osteogenesis), dependingon the degree of adhesion and activation of the blood platelets at the

82 Chapter 4. Moving boundary model for endosseous healingimplant surface, which is related to the implant surface micro-structure.Amor et al. [5] used the model by Bailon-Plaza and van der Meulen [11],initially created for bone fracture healing, and adapted it to model periimplantosseointegration. An enhanced model was presented in [6], wherethe implant surface micro-structure is taken into account, which determinesthe mode of osteogenesis (distance or contact).In the models by Amor et al. [6] and Moreo et al. [67], the platelet density,osteogenic cell and osteoblast densities, soft (fibrous) tissue and bonetissue volume fractions, and growth factor concentration are considered asunknown functions in space-time coordinates. Moreo et al. [67] defined twogeneric types of growth factors. The growth factors, related to the firsttype, are released by activated platelets (e.g. platelet-derived growth factor(PDGF), transforming growth factor beta (TGF-β), etc.). The growthfactors, related to the second generic type, are released by osteogenic cellsand osteoblasts (bone morphogenetic proteins (BMPs), TGF-β, etc.). Bothgrowth factor types cause chemotaxis of osteogenic cells and promote proliferationof osteogenic cells, while the first one enables differentiation intoosteoblasts, and the second one enables bone formation.Amor et al. [6] consider both mentioned growth factor types throughone unknown variable. In this model, growth factors cause chemotaxis ofosteogenic cells. Osteoblast differentiation is assumed to take place onlyunder the presence of growth factors.Moreo et al. [67] distinguish woven and lamellar bone, while Amor et al.[6] consider just one variable for the bone tissue volume fraction. The modelparameters, boundary and initial conditions, proposed for both models differand some of them – even considerably. Nevertheless, these two models havemuch in common. In particular, they were constructed to study differentpaths of bone regeneration around endosseous implants, depending on theimplant surface micro-structure. The models are capable to predict contactand distance osteogenesis mode. The switch mechanism between these twomodes is introduced in the models by means of the activated platelet density.The adhesion of platelets is assumed to depend on the implant surfaceroughness. Platelets produce growth factors, and growth factor concentrationdetermines osteoblast differentiation and bone tissue synthesis. Bonematrix release is proportional to the osteoblast density and ceases, if thevolume fraction of bone tissue exceeds one. It follows that bone is synthesizedby osteoblasts wherever these cells are found. It is also confirmed bythe simulation results, that were reported in Amor et al. [6] and Moreo et al.[67], that bone forms throughout the whole interface region simultaneously.However, it is released faster in the vicinity of old bone and implant surface,under certain conditions.

4.3. Moving boundary model 834.3 Moving boundary modelThe ideas and approaches, proposed by Moreo et al. [67] and Amor et al.[6], are used partially as the basis, on which the present model is built.The principal innovations are introduced in the current model, which allowto reflect some important characteristics of peri-implant osseointegration.These characteristics are taken from three basic principles, formulated byDavies [24], which are literally quoted.Hypothesis 4.1. Principles for endosseous healing1. “bone matrix is synthesized by only one cell: the osteoblast”, and “theosteoblast is irrevocably attached to the bone-forming surface”;2. bone can only be deposited by laying down matrix on a preexistingsolid surface. . . . the osteoblast is incapable of migrationaway from the bone surface, and the only method bywhich this surface can receive further additions (beyond thesynthetic capacity of a single osteoblast) is by the recruitmentof more osteogenic cells to the surface, which then differentiateinto secretorily active osteoblasts;3. bone matrix mineralizes and has no inherent capacity to“grow” . . . . Thus, once bone formation has been initiated,the matrix and the cells that have synthesized that matrixhave almost no ability to govern the ongoing pattern of bonegrowth on the implant surface.In line with the above hypotheses, the following variables for the newmodel are defined: the densities of immature and mature osteogenic cells (c iand c m , respectively), and the concentration of growth factor g.4.3.1 Mature and immature cellsThe concepts of immature and mature osteogenic cells are critical for thepresent model. First, it should be noted, that according to the stated principlesfor endosseous healing, the main characteristics of osteoblasts are thatthey are attached to the bone-forming surface and that they release bonematrix. Terms ’immature’ and ’mature osteogenic cells’ are used to introducethe differentiation of osteogenic cells in the model. Mature cells arethese cells, which have reached such a level of maturation (differentiationthreshold), that they are able to attach to the rigid surface and initiatebone matrix synthesis. In other words they are able to become osteoblasts.Mature osteogenic cells are mobile, until they become osteoblasts.Immature cells have to undergo the differentiation path for some time,until they reach the differentiation threshold, and only after that they will

84 Chapter 4. Moving boundary model for endosseous healingbe able to become osteoblasts. The osteogenic cell differentiation is introducedin the model in the way it was proposed in Chapter 3. Immatureosteogenic cells are distinguished according to their maturation level a (or’differentiation level’), which ranges from 0 to 1, and which is consideredas an additional independent coordinate in the problem domain. While animmature cell is differentiating into an osteoblast, its maturation level increases.If it reaches value 1, it is assumed that the cell has reached thedifferentiation threshold, and it becomes a mature osteogenic cell. Thiscell does not lose its mobility, as long as it has not become an osteoblast.Therewith, immature cell density c i is defined as a number of cells per unitvolume, per maturation level unit.4.3.2 Modeling approachLet the problem domain Ω be an approximation (e.g. in 1D, 2D or 3D) of thereal geometry of a bone-implant interface. It is assumed that initially thewhole peri-implant region is filled with soft connective tissue (fibrin networkof blood clot). The domain boundary ∂Ω is composed of the implant surface∂Ω i and the old bone surface ∂Ω b , i.e. ∂Ω = ∂Ω i ∪∂Ω b . The initial density ofosteogenic cells, and the initial growth factor concentration are equal to zero.A source of immature osteogenic cells and growth factors is assumed at theold bone surface ∂Ω b (Figure 4.1). At the implant surface ∂Ω i , a source ofΩ bΓ(t)∂Ω i Ω s = Ω ∂Ω b ∂Ω i Ω s ∂Ω bt = 0 t > 0Figure 4.1: Sketch of the problem domain Ω. The old bone and implantsurfaces are denoted by ∂Ω b and ∂Ω i respectively. Subdomains Ω b andΩ s correspond to regions within the healing site, filled with newly formedbone and soft tissue, respectively. They are separated by the bone-formingsurface, denoted by Γ(t). At t = 0, soft tissue occupies the whole periimplantspace Ω s = Ω, and the bone-forming surface is composed of theimplant interface and the old bone surface, i.e. Γ(0) = ∂Ω i ∪ ∂Ω bgrowth factor is considered, which corresponds to the release of growth factorby activated platelets, that are attached to the implant surface. Osteogeniccells, originating from the old bone surface, migrate into the peri-implantregion, and differentiate into osteoblasts under the influence of growth factor.

4.3. Moving boundary model 85Assume that at a certain time moment, there is a number of matureosteogenic cells within the healing site. Further, let some of them be situatedat the rigid surface, which initially consists of the implant and oldbone surface. It is assumed that these cells attach to the rigid surface andstart releasing bone matrix. These cell functions are characteristic for osteoblasts.Hence it is assumed that mature osteogenic cells, situated near therigid surface, immediately become osteoblasts. Due to bone production bythe osteoblasts, the front of newly formed bone starts moving. This frontforms the surface, onto which other mature cells can attach and synthesizebone matrix consequently, providing further bone front movement. Themobile bone-forming surface is the boundary between the soft tissue withinthe healing site and newly formed bone. In other words, the bone-forminginterface, denoted asΓ(t) ⊂ Ω, t ≥ 0, and Γ(0) = ∂Ω,divides the computational domain Ω into two regions Ω b and Ω s , which arefilled with bone tissue and soft tissue, respectively. Within region Ω b , boneremodeling starts. This is a long term process, and the present model isdeveloped for the second and third stages of bone regeneration. As it wasmentioned before, bone remodeling is not considered in the current model.Therefore, no processes, which take place within subdomain Ω b , are modeled.Though, it should be noted, that osteoblasts, trapped into the bone, becomeosteocytes.According to the third principle for endosseous healing, the formationof new bone tissue within the peri-implant interface is governed by the processes,taking place within the region Ω s . The following processes are consideredin the present model:1. osteogenic cell migration, caused by random walk and chemotaxis,directed towards the gradient of growth factor concentration;2. osteogenic cell proliferation and differentiation;3. growth factor diffusion and decay;4. growth factor release by osteogenic cells and activated platelets.The mathematical model consists of a set of partial differential equations(PDE’s), boundary and initial conditions. The equations are specified forthe unknown real-valued functions c i (⃗x, a, t), c m (⃗x, t) and g(⃗x, t), definedfor ⃗x ∈ Ω s , a ∈ [0, 1], t ≥ 0, where ⃗x is the coordinate vector in physicalspace, a is the maturation level of the immature osteogenic cells, i.e. a is thecoordinate in maturation or differentiation state domain, and t is time. Thefunctions c i , c m and g represent densities of immature and mature osteogeniccells, and concentration of growth factor, respectively.

86 Chapter 4. Moving boundary model for endosseous healingThe evolution of unknown variables is determined by the following PDE’s∂c i∂t = −∇ s · (−D c ∇ s c i + χ(g, c tot ) c i ∇ s g)} {{ }Migration− ∂ ∂a (u b(g)c i ) + A c (g)c i (1 − c tot ), (4.1)} {{ }} {{ }ProliferationDifferentiation∂c m∂t= −∇ s · (−D c ∇ s c m + χ(g, c tot ) c m ∇ s g)+ u b (g)c i (⃗x, 1, t) + A c (g)c m (1 − c tot ) , (4.2)∂g∂t = ∇ s · (D g ∇ s g)} {{ }Diffusion(+ E c (g)∫ 1)γ(a)c i dac m +}0{{ }Production−d g g}{{}, (4.3)Degradationwhere c tot = ∫ 10 c ida + c m is the total density of osteogenic cells per unit ofvolume. Equation (4.1) is defined for (⃗x, a, t) ∈ Ω s ×[0, 1]×R + , and equations(4.2), (4.3) for (⃗x, t) ∈ Ω s × R + . Operator ∇ s is the nabla operator, definedin physical space, e.g. in 3D physical space∇ s =( ∂∂x 1,∂∂x 2,)∂.∂x 3The parameter D c is the diffusion coefficient of osteogenic cells. The termsχ(g, c tot )c i ∇ s g and χ(g, c tot )c m ∇ s g in equations (4.1) and (4.2) representcell migration due to chemotaxis, where the coefficient χ(g, c tot ) depends onthe growth factor concentration and on the total density of osteogenic cellsin the current location. Cell proliferation is introduced by the terms, thatcontain coefficient A c (g), which depends on growth factor concentration. Itis assumed that a parent cell and the daughter cells, into which it divides,have the same level of maturation. The second term in the right-hand sideof equation (4.1) represents the increase of maturation level of immaturecells, due to cell differentiation (see also Chapter 3 for the derivation of thisterm). The parameter u b (g) is the differentiation rate (or maturation rate),and it depends on growth factor concentration. It can also be consideredas the velocity of cell “motion” in the differentiation state domain. Theterm u b (g)c i (⃗x, 1, t) in equation (4.2) corresponds to the increase of a numberof the mature cells due to the differentiation of the immature cells. Theterms in the right-hand side of equation (4.3) (in order from left to right)represent growth factor diffusion with constant coefficient D g ; growth factorproduction by cells with coefficient E c (g), depending on growth factor

4.3. Moving boundary model 87concentration; and growth factor decay, respectively. It was assumed thatgrowth factor is released by mature and immature osteogenic cells [25], andthat its production depends on the maturation level of immature cells. Thelast mentioned assumption is introduced by the function γ(a) under theintegral sign.The size and geometry of the subdomains change due to new bone formation.This process is introduced in the model by specifying the magnitudeof the normal velocity v n of the moving bone-forming surface Γ(t). That is,the variable v n denotes the normal component of the velocity of the pointsof the moving boundary, and is equal to the scalar product of the velocityvector and outward (with respect to domain Ω s ) unit normal of the boundarysurface. The magnitude of the front normal velocity v n is related tothe amount of new bone matrix, produced locally per time unit. As it wasmentioned, the new bone matrix is produced only by osteoblasts.dn cΓ(t)newbone matrixdn bΓ(t)dlosteocyteosteoblastsdlΓ(t + dt)soft tissue(a)soft tissue(b)Figure 4.2: Sketch of the bone release by osteoblasts. First, mature osteogeniccells, situated within the distance dn c from the boundary surfaceΓ(t), attach to it and become osteoblasts (a). After that, osteoblasts releasenew bone matrix. Osteoblasts are trapped by the released bone matrix andbecome osteocytes (b). The width of the layer of new bone matrix, releasedin time dt, is denoted as dn bLet us consider a small segment of the boundary surface Γ(t) of lengthdl (Figure 4.2a). Assume that mature osteogenic cells, situated within thedistance dn c from the boundary surface, attach to it and start to release bonematrix. In other words, these osteogenic cells become osteoblasts. A volumeof bone matrix, produced by one osteoblast per unit of time, is denoted byvolume unitsparameter V b . Then units of V b isday·cell. The number of osteoblasts,attached to the boundary segment dl is equal to N b = c m dl dn c . These cells

88 Chapter 4. Moving boundary model for endosseous healingwill produce the bone matrix of the volume dV = V b N b dt in time period dt.Since the new bone matrix is released, the bone-forming surface will changeits location from Γ(t) to Γ(t + dt) (Figure 4.2b). The volume dV of newbone, laid down on the boundary segment dl, corresponds to the shift ofthe boundary element dl by distance dn b = dVdlin normal direction, in timeperiod dt. The normal velocity v n is equal to the local shift of boundary innormal direction per unit of time. Hencev n = − dn bdt = − dVdl dt = −V b c m dl dn c dt= −V b dn c c m .dl dtThe minus sign appears, since the velocity v n is considered along the outernormal, with respect to the region of soft tissue Ω s , and quantity dn b is theshift along the inner normal. A new parameter P b = V b dn c is defined, andthe following relation for the normal velocity of the boundary is obtained:v n ( ⃗ X, t) = −P b c m ( ⃗ X, t), ⃗ X ∈ Γ(t), t > 0, (4.4)where constant parameter P b denotes the rate of the production of bonematrix by one osteoblast per unit of time, times the width of the layer ofosteogenic cells, attached to the bone-forming surface. The parameter P b[volume units]day·cellhas the units mm. The expression (4.4) for the normal velocityof the moving boundary has been derived, in which the normal velocity isassumed to be proportional to the local density of the mature osteogeniccells.Remark 4.1. The value of the parameter P b will be estimated, based on theresults of numerical simulations and on educated guesses. Parameters V band dn c were used to explain the physical meaning of the parameter P b andin the derivation of the relation (4.4) for velocity v n . Further, parametersV b and dn c will not be used.Remark 4.2. For convenience and computational efficiency, the consideredcell densities and growth factor concentration are nondimensionalized withrespect to a saturated cell density of 10 6 cellsml, and typical growth factorconcentration of 100 ngml [11]. That is, variables c i, c m and g are dimensionlessquantities. Consequently, the units of parameter P b should be redefined as(see equation (4.4)).mmdayThe initial position of the boundary is defined as:Γ(0) = ∂Ω b ∪ ∂Ω i . (4.5)The movement of the boundary is determined by its normal velocity, theexpression for which is given in equation (4.4).Remark 4.3. The term ’moving boundary problem’ is used for the formulatedequations, which constitute the present model. It is different from the

4.3. Moving boundary model 89classical Stefan problem. The term ’moving boundary problem’ is applied,since the partial differential equations within the evolving domain, whichchanges in time, are solved. The temporal evolution of the domain has tobe determined as a part of the solution. Time-dependent problems of suchtype are called moving boundary problems (see for example the introductionto the monograph by Crank [22]).The movement of the domain boundary, which corresponds to the ossificationfront, is given by equation (4.4). It is not derived from the principleof conservation of some quantity, as it is done in classical Stefan problems.Indeed, the ossification front moves due to the physical phenomenon, regulatedby the osteogenic cells, in which these cells are located on the bonesurface and deposit new bone matrix. Therefore, the domain boundary isgradually evolving due to the action of the cells. Some of these cells remainon the bone surface, whereas others are buried by the matrix as it is shown inFigure 4.2. Therefore, to describe accurately this phenomenon the temporalevolution of the bone surface is modeled. That is why it was assumed thatthe normal boundary velocity is proportional to the local concentration ofmature osteogenic cells. The boundary motion is effected only by the localcell density at the boundary itself. Mature osteogenic cells do not releaseany bone matrix away from a rigid surface, and consequently, they do nothave any influence on the movement of ossification front.The idea of the limit concentration at the moving interface, which is oftenused in classical Stefan problems, is not applicable in the present model,since even a low concentration of cells will provide some slow advance of theossification front.As it was mentioned, the initial cell densities and growth factor concentrationare assumed to be zero:c i (⃗x, a, 0) = 0, c m (⃗x, 0) = 0, g(⃗x, 0) = 0, ⃗x ∈ Ω, a ∈ [0, 1]. (4.6)The boundary conditions are derived from the following assumptions. Thesource of immature cells and of growth factor are considered at the old bonesurface ∂Ω b from t = 0 until time t cbone and t gbone , respectively. The timeperiod t gbone is set to 4 days [6]. The other time parameter t cbone is estimatedfrom numerical simulations, based on sensitivity analysis, presentedin Appendix A.1. The value t cbone = 1 day is chosen. For t > t cbone andt > t gbone , no source of osteogenic cells and of growth factor at the old boneboundary is assumed. As it was mentioned, during haemostasis phase, bloodplatelets attach to the implant surface, and after being activated, they startto release growth factors. This release of growth factor by activated plateletsis represented in the current model by means of a boundary condition.The influx of growth factor is defined as a time-dependent function g impl (t).This function is supposed to decay exponentially [6]. Since it is assumed todecrease rapidly, the influx is set to zero for max(t cbone , t gbone ) = 4 days.

90 Chapter 4. Moving boundary model for endosseous healingIn the current model, the movement of the domain boundary is supposedto start after a certain time period, larger than max(t cbone , t gbone ) = 4 days.Due to the boundary movement, a special treatment of the fluxes of themodel variables is necessary. Mature osteogenic cells, adjacent to the boneformingsurface, are assumed to become osteoblast. Hence according to theprinciples for endosseous healing, they become immobile and are trapped bythe newly formed bone. On the contrary, immature cells and growth factorare assumed to be pushed out by the moving surface, in a piston-like way(they are assumed not to be able to penetrate into the bone). According tothese assumptions, the conditions for the fluxes of the model variables areset at the domain boundary. The derivation of the particular expressions isgiven in Appendix A.2.Thereby, the boundary conditions are defined as follows:{(−Dc ∇ s c i + χ(g, c tot )c i ∇ s g) ( ⃗ X, a, t) · ⃗n( ⃗ X, t) = 0, ⃗ X ∈ ∂Ωi ,(−D c ∇ s c i + χ(g, c tot )c i ∇ s g) ( ⃗ X, a, t) · ⃗n( ⃗ X, t) = −c bone (a), ⃗ X ∈ ∂Ωb ,(4.7)for a ∈ [0, 1], t ≤ t cbone ;(−D c ∇ s c i + χ(g, c tot )c i ∇ s g) ( ⃗ X, a, t) · ⃗n( ⃗ X, t) = c i ( ⃗ X, a, t) v n ( ⃗ X, t), (4.8)for a ∈ [0, 1], ⃗ X ∈ Γ(t), t > t cbone ;for ⃗ X ∈ Γ(t), t > 0;(−D c ∇ s c m + χ(g, c tot )c m ∇ s g) ( ⃗ X, t) · ⃗n( ⃗ X, t) = 0, (4.9)for ⃗ X ∈ ∂Ωi , t ≤ max(t cbone , t gbone );−D g ∇ s g( ⃗ X, t) · ⃗n( ⃗ X, t) = −g impl (t), (4.10)−D g ∇ s g( ⃗ X, t) · ⃗n( ⃗ X, t) = −g bone , ⃗ X ∈ ∂Ωb , t ≤ t gbone ; (4.11)−D g ∇ s g( ⃗ X, t) · ⃗n( ⃗ X, t) = g( ⃗ X, t) v n ( ⃗ X, t), ⃗ X ∈ Γ(t), t > tgbone . (4.12)In the above formulas, ⃗n( ⃗ X, t) is the unit outward normal of boundary Γ(t)of subdomain Ω s . The constant parameter g bone and function c bone (a), respectively,represent the source of growth factor and of immature osteogeniccells at differentiation level a, both found at the old bone surface.Let us denote the total influx of immature cells by parameter C bone , i.e.C bone =∫ 10c bone (a) da. (4.13)It is assumed first that the cells, which are recruited from the old bonesurface, are completely non-differentiated (have maturation level equal to

4.3. Moving boundary model 91zero). Then function c bone (a) should be defined through the Dirac deltafunction δ(a), i.ec bone (a) = C bone δ(a), a ∈ [0, 1].The present mathematical model is constructed mainly to be used for thenumerical simulations of the osseointegration process. Then it can be usefulto smooth the Dirac delta function, so that the boundary condition (4.7) canbe incorporated into the numerical scheme exactly. Therefore, let us considersome small parameter δ a . Then the source of cells can be represented bysmoothed function{C bone · 2c bone =δ a(1 − aδ a), a ∈ [0, δ a ],(4.14)0, a ∈ [δ a , 1].This function satisfies condition (4.13). The parameter δ a corresponds to therange of maturation level of cells, that are recruited from the surroundingbone tissue, and is introduced in the model, in order to make it more suitablefor the numerical simulations.External sources of osteogenic cells and of growth factor are introducedthrough boundary conditions (4.7), (4.10) and (4.11), which are defined atsurfaces ∂Ω i , ∂Ω b for time less than t cbone , t gbone and max(t cbone , t gbone ),respectively. That means, that the implant surface ∂Ω i and the old bonesurface ∂Ω b should be a part of the boundary Γ(t) of the physical domainΩ s for t ≤ max(t cbone , t gbone ). Otherwise, conditions (4.7), (4.10) and (4.11)make no sense. A new parameter is introduced:t 0 = max(t cbone , t gbone ).In order to obtain initiation of bone formation only after time t 0 , we want oursolution to satisfy the following relation for the movement of the interfacev n ( ⃗ X, t) = 0, ⃗ X ∈ Γ(t), t ≤ t0 = max(t cbone , t gbone ). (4.15)The above equation is not imposed explicitly in the current simulations,however, the biological parameters are chosen in such a way, that equation(4.15) is satisfied a posteriori. From equations (4.5) and (4.15) it followsthat Γ(t) = ∂Ω i ∪ ∂Ω b , for t ≤ t 0 , i.e. the subdomain Ω s will keep its initialgeometry and occupy the whole peri-implant region at least until time t 0 .For the selected parameter values, time t 0 is equal to 4 days. Condition(4.15) means that bone does not start forming earlier than t 0 days. Valuet 0 = 4 days is in line with experiments [1], in which newly formed bone wasobserved only at the end of the first week. Condition (4.15) is accomplishedby an appropriate choice of the model parameters. In particular, an upperlimit u limbfor the differentiation rate u b is defined, i.e.u b ≤ u limb . (4.16)

92 Chapter 4. Moving boundary model for endosseous healingFrom condition (4.16), and since the initial density of immature and maturecells is zero, and since the maturation level of immature osteogenic cells,which originate from the source at the old bone surface, is not larger thanδ a (see equations (4.7) and (4.14)), it follows that no mature cells will derivefrom immature cells, at least until timeThat isIf the following relation holds:t lim = 1 − δ au lim . (4.17)bc m (⃗x, t) = 0, ⃗x ∈ ¯Ω s , t ≤ t lim . (4.18)u limb< 1 − δ at 0(4.19)then for time t lim , defined in equation (4.17), it holds t lim = 1−δa > tu lim 0 , andbfrom equations (4.18) and (4.4) it follows that v n ( X, ⃗ t) = 0, for X ⃗ ∈ Γ(t),t ≤ t 0 < t lim , and condition (4.15) holds.Remark 4.4. Note that two types of boundary conditions have been defined.Boundary condition (4.4) determines the evolution of the domain boundaryin time. Another set of boundary conditions, given by equations (4.7)–(4.12),is necessary for the uniqueness of the solution of governing equations (4.1)–(4.3), defined within the soft tissue region Ω s . Hence all the requirementsfor well-posedness (in terms of existence and uniqueness) of the movingboundary problem are fulfilled.4.4 Model validationIn this section, the present model is validated against experiments. Approximatesolutions are obtained from numerical simulations. The values ofthe model parameters are taken from the literature and are estimated fromnumerical simulations. A parametric study is carried out, and some conclusionsabout the influence of various parameters on the model predictions aredrawn.4.4.1 Numerical simulationsNumerical simulations are performed for a 2D axisymmetric physical domain.A 1D physical domain makes calculations much faster and muchsimpler. However, some important solution characteristics can not be representedin 1D simulations. For example, bone front propagation from thecorners, formed by the implant body and by the thread, has a large influenceon the final state of osseointegration and can only be modeled for a 2D

4.4. Model validation 93geometry of the physical domain. To have a full picture of the model characteristicsit is necessary to perform a parametric study for a 2D physicaldomain, despite it leads to a large computation time and to a complicatednumerical algorithm. The solution of non-linearly coupled PDE’s within anirregular 2D physical domain, which evolves in time, is a rather challengingtask. A derivation of the numerical approach designed for the solution of thepresent problem is a separate subject of the research. Therefore, Chapter 5as a whole is devoted to its description. At this moment it is only mentioned,that the governing advection-diffusion-reaction equations are solved with theuse of the finite volume method. The embedded boundary method is usedto capture an irregular geometry of a 2D physical domain and the level setmethod is applied to track the evolution of the physical domain. The explicitmodified Euler method is used for time integration. More details about thesolution technique are given in the Chapter 5.An approximate geometry of the initial healing site, formed by the threadof the implant and by the old bone surface, is determined from the imageof the endosseous implant given in Berglundh et al. [14] (Figure 4.3). The0.7z, mm0.460.2301.4 1.85r, mmFigure 4.3: Initial geometry of the healing site, bounded by the thread ofthe implant from the left and the old bone surface from the rightsizes of the initial uniform rectangular control volumes along r−, z− anda− axes are chosen to be equal to h r = 0.0045 mm, h z = 0.005 mm andh a = 0.05, respectively.4.4.2 Model parametersThe model parameters as such, and their magnitudes, determine the behaviorof the model. The values of the parameters, which provide physicalsolutions, are found in this section. It is necessary to specify certain criteriato assess biologically relevant solutions. A focus of the numerical studies presentedby Amor et al. [6] and Moreo et al. [67] is on the distance and contactosteogenesis modes, which are predicted by their osseointegration models formicro-rough and smooth implants, respectively. Therefore, a path of boneformation predicted by the present model for the implant surfaces with var-

94 Chapter 4. Moving boundary model for endosseous healingious magnitudes of the micro-roughness is the main criterion applied forthe choice of the model parameters. In the experiments by Abrahamssonet al. [1], contact osteogenesis was observed near a rough implant surface,and distance osteogenesis occurred near smooth implants. The possibility toquantify the micro-roughness of the implant surface is incorporated into thepresent model implicitly through the boundary conditions for the growthfactor concentration. The influx of growth factors at the implant surfacecorresponds to the release by activated platelets. It is known from experiments[56, 71] that the rate of adhesion of platelets, their activation andrelease of growth factors are higher near a micro-rough implant surface thannear smooth surfaces. Two types of implants are considered in the presentmodel: a micro-rough ’Sand-Blasted, Large Grit, Acid-Etched’ (SLA) implantand a smooth ’turned’ implant. A high value and a low value of thegrowth factor source are considered at the surfaces of a SLA implant and ofa ’turned’ implant, respectively. The micro-roughness of the implant is representedby the parameter E pl , which is proportional to the influx of growthfactors at the implant surface.A large number of numerical simulations is introduced in order to determinerelevant values of the model parameters. Due to a large number of theparameters a random selection of the parameters requires an enormous numberof simulations. Therefore, random search is not efficient in the currentcase. The following approach is used to choose the values of the parameters.At the first step, the ability of the model to predict distance osteogenesis forsmooth implants is tested. As a first approximation, the influx of growthfactors at the smooth implant surface is assumed to be very small and is setto zero: E pl = 0. The rest of the model parameters is estimated, such thata bone-forming front moves from the old bone surface towards the implant(distance osteogenesis mode).Bone formation near the two different implant types is modeled by varyingthe value of E pl , whereas all other model parameters are kept unchanged.Hence at the second stage, the value of E pl corresponding to a micro-roughSLA implant surface is chosen, such that new bone starts forming directlyat the implant surface (contact osteogenesis). If it is not possible to findsuch a value of E pl , then the parameters, which were estimated at the firststep, should be readjusted, until contact osteogenesis is implemented intothe model as well as distance osteogenesis.Finally, a limit value Epllim for the source of growth factor E pl near thesurface of the smooth turned implant is found, such that distance osteogenesismode is predicted for E pl ≤ Epl lim .The expressions for the model parameters are given in Table 4.1. Thevalues for the constant parameters of the model equations and of the boundaryconditions are specified in Table 4.2. In the right column of the table,the references are noted, from which the values of the constant model parametersare taken. No reference is given next to the parameters, which are

4.4. Model validation 95Table 4.1: Expressions for the dependent model parametersDependent parameter expressionDescriptionχ(g, c tot ) =⎧ χ⎨ 0 gKch 2⎩+ (1 − c tot), if c tot < 1g20, if c tot ≥ 1A c (g) = A c0(1 +u b (g) = u limb)gK p + gg 2K 2 u + g 2Osteogenic cell chemotaxisOsteogenic cell proliferationOsteogenic cell differentiationE c (g) =E c0 gK g + g Growth factor release by osteogenicγ(a) = a λc cellsg impl (t) = E pl e −λ pltGrowth factor release by activatedplatelets

96 Chapter 4. Moving boundary model for endosseous healingestimated from numerical simulations or based on educated guesses.Further, an initialization of each model parameter is described in detail.In order to justify the choice of values for the parameters, which areestimated from simulations, and in order to demonstrate, how various parametersinfluence the model behavior, a sensitivity analysis is performed.The main results of the sensitivity analysis are presented in Section 4.4.3.Remark 4.5. The model variables c i , c m and g are nondimensionalized (seeRemark 4.2). This fact is taken into consideration, when the parameters ofthe model are defined. Appropriate scaling of parameter values is appliedand suitable units of the parameters are chosen.The chemotaxis coefficient χ(g, c tot ) is defined as in [6] times the term(1 − c tot ), if the total cell density is less than saturation limit 1. Otherwise,chemotaxis is switched to zero. This switch prevents, that cells migratetowards regions, where the cell density has reached its saturation value. Inthe regions, where the saturation limit has been exceeded, no chemotaxistakes place, and cell migration is determined by cell diffusion. The value forthe constant parameter K ch is also taken from [6].The values of the chemotaxis coefficient χ 0 and of the cell diffusivitycoefficient D c are estimated from numerical simulations. See Section 4.4.3for a discussion on the choice of these parameters.The expression for the proliferation coefficient A c (g), and the values ofthe parameters A c0 and K p are taken from Moreo et al. [67].It is reported in Linkhart et al. [62] and Dimitriou et al. [25] that growthfactors enhance differentiation into osteoblasts. Therefore, it is assumed thatthe differentiation rate u b (g) depends on the growth factor concentration.For the proposed relation, u b (g) ≈ 0, if g ≪ K u , and it increases to the limitvalue u limb, if the growth factor concentration g grows. The choices of valuesfor the parameters u limband K u are discussed in Section 4.4.3.The expression for the growth factor release parameter E c (g) and the valuesof the parameters E c0 and K g are taken from [6]. It is assumed accordingto Dimitriou et al. [25], that growth factors are released by immature andmature cells. Furthermore, it seems to be necessary to introduce the dependenceof growth factor release on the level of maturation through the powerfunction γ(a). The exponent of this function is denoted by the parameterλ c ≥ 0. Various authors use different assumptions for growth factor releasein their models. For example, Moreo et al. [67] define the same productionrate for osteogenic cells (immature cells) and osteoblasts (mature cells). Forthe present approach, this assumption is equivalent to a zero value of theparameter λ c . On the other hand, Amor et al. [6] assume that growth factoris produced only by osteoblasts. Then an option λ c → +∞ will representthis situation in the present model. The choice of the value of exponent λ cis discussed in Section 4.4.3.As in [6], it is assumed that the number of platelets at the implant

4.4. Model validation 97Table 4.2: Values of the model parameters, and boundary condition parametersConstant parameter values Description Ref.D c = 0.0998 mm 2 /day Osteogenic cell diffusion coefficientχ 0 = 0.324 mm 2 /day Osteogenic cell chemotaxis coefficientK ch = 0.01 Growth factor concentration, related [6]to osteogenic cell chemotaxis coefficientA c0 = 0.25 day −1 Half of the maximal proliferation coefficient[67]K p = 0.1 Growth factor concentration, related [67]to osteogenic cell proliferation coefficientu limb= 0.2 day −1 Maximum cell differentiation rateK u = 0.01 Growth factor concentration, relatedto cell differentiation rateD g = 0.045 mm 2 /day Growth factor diffusion coefficient [6]d g = 100 day −1 Growth factor decay rate [6]E c0 = 570 day −1 Growth factor production rate by osteogenic[6]cellsK g = 1 Growth factor concentration, related [6]to growth factor production rateλ c = 2 Parameter, related to growth factorproduction rate by immature cellsg bone = 0.6 mm/day Influx of growth factors at the oldbone surfaceC bone = 0.07 mm/day Influx of cells at the old bone surfaceδ a = 0.1 Maximum maturation level of cells,recruited from the old bone surfaceInflux of growth factors, released byactivated plateletsE pl ≤ Epl lim = 0.32 mm/day – at the turned implant surfaceE pl = 204.8 mm/day – at the SLA implant surfaceλ pl = 1 day −1 Platelets degradation rate coefficient [6]P b = 0.01 mm/day Bone-forming surface velocity coefficient

98 Chapter 4. Moving boundary model for endosseous healingsurface decays exponentially with rate λ pl = 1 day −1 , which gives a halftime of ln 2 days. Hence the release of growth factors decreases in the sameway. The production rate E pl at t = 0 is estimated in Section 4.4.3.The value for the growth factor decay rate d g is defined as in Amor et al.[6].The maximum maturation level δ a = 0.1 of cells, recruited from the oldbone surface, is chosen to be of order of the length of the grid spacing in thea-coordinate, used in the simulations.The value of the parameter D g is taken from Amor et al. [6]. Thedependence of the solution on the value of D g is discussed in Section 4.4.3.4.4.3 Sensitivity analysisAs it is outlined in Section 4.4.2, at the first step of an estimation of theparameter values, a zero influx of growth factors is assumed at the implantsurface, i.e. E pl = 0. The values of the parameters are chosen, such thatdistance osteogenesis is predicted, i.e. a bone-forming front moves from theold bone surface towards the implant.Remark 4.6. The numerical solutions presented in this section are obtainedfor a fine mesh resolution. The sizes of the initial uniform rectangular controlvolumes along r−, z− and a− axes are equal, respectively, to h r = 0.0045mm, h z = 0.005 mm and h a = 0.05, respectively.The numerical solution of the model equations, which is obtained for theparameter values in Tables 4.1 and 4.2 and for E pl = 0, is plotted in Figure4.4. Distance osteogenesis is predicted by simulations, if a considerableamount of growth factors is only observed in the regions, situated close tothe bone-forming surface. Suppose, that bone starts forming at the oldbone surface and not at the implant surface. If the growth factors are ableto spread far from the bone-forming surface, then they can reach the implantsurface after a certain time. The growth factors cause cell differentiation andcell migration towards the implant due to chemotaxis. Mature osteogeniccells appear and bone formation is initiated near the implant. Hence distanceosteogenesis cannot be modeled, unless the growth factor concentration isalways low in the regions, which are not adjacent to the new bone surface.Since the sources of cells and of growth factors are considered only onthe old bone surface, a layer with a high cell density and a high growthfactor concentration is formed near the old bone. The thickness of this layerdepends on random walk and chemotaxis of the cells. Increasing the value ofthe cell diffusion coefficient D c and/or decreasing the chemotaxis coefficientχ 0 lead to a faster spreading of cells towards the implant. The cells releasegrowth factors and the width of the layer with a high cell density and a highgrowth factor concentration increases.Remark 4.7. Further, the region of the soft tissue domain with a high cell

4.4. Model validation 99c m0.2c tot0.2gt=7 days0.501.4 1.80.80.60.40.501.4 1.80.80.60.40.501.4 1.842.50.5t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5Figure 4.4: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 15, 24and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 0. The bone-forming surface moves from the old bone surface towardsthe implant (distance osteogenesis)

100 Chapter 4. Moving boundary model for endosseous healingdensity and a high growth factor concentration are referred to as “highdensity- high concentration layer”. Such a region is usually adjacent to theboundary of the domain. The thickness and the length of the layer canbe considered. The length of the layer corresponds to the length of theossification front.The value 0.75 · 0.133 = 0.0998 mm 2 /day is assigned to the diffusioncoefficient D c in Table 4.2, whereas 0.133 mm 2 /day is the value for the celldiffusion coefficient defined in Moreo et al. [67]. The value 0.133 mm 2 /dayyields a large width of the high density- high concentration layer, adjacentto the bone-forming surface, and mature osteogenic cells appear near theimplant body, i.e. at the middle of the implant surface, at day 21 (Figure4.5).A similar effect is reached, if the reference value of the chemotaxiscoefficient χ 0 is divided by two, i.e. if χ 0 = 0.5 · 0.324 mm 2 /days =0.162 mm 2 /days. In this case, bone starts forming at the implant surfaceat time t = 15 days (Figure 4.6).The growth factor diffusion coefficient D g effects the width of the highdensity- high concentration layer to a smaller extent. Increasing the valueof D g by two times yields the appearance of mature osteogenic cells in themiddle of the implant surface at day 24 (Figure 4.7).Decrease of the diffusion D g to the value 0.0225 mm 2 /days does notchange the path of bone formation significantly.Decrease of the value of the cell diffusion coefficient D c to 0.5 · 0.133 =0.0665 mm 2 /days, or increase of the reference value of the chemotaxis coefficientχ 0 by two times yield smaller length and width of the high densityhighconcentration layer compared with the solution for the reference valuesD c and χ 0 (see Figures 4.4, 4.8 and 4.9). For the decreased D c andincreased χ 0 , bone forms only from the old bone surface and the length ofthe ossification front decreases significantly at the end of the second week.As it is mentioned in Section 4.3.2, the value of the maximum cell differentiationrate u limbshould be less than 1−δa4days −1 = 0.225 days −1 .Abrahamsson et al. [1] observed, that bone formation started at the endof the first week. Hence the differentiation rate should be larger than1−δ a7days −1 ≈ 0.1285 days −1 . The estimated value of 0.2 days −1 is assignedto u limb. Simulations provide similar results for the rates 0.2 days −1and 0.224 days −1 . The smaller values 0.18 and 0.15 days −1 of u limbyield adelay of cell differentiation and, hence, of bone formation. A delay of boneformation is clearly observed in the numerical solutions. For example, themature cell density is maximal near the old bone surface and it reaches thevalue 0.6 at the time moments 7.9, 6.7 and 6 days for the values 0.15, 0.18and 0.2 days −1 of u limb, respectively, as it is illustrated in Figure 4.10.The parameter K u influences the cell differentiation in the regions with

4.4. Model validation 101c m0.2c tot0.2gt=7 dayst=15 dayst=21 days0.50.501.4 1.801.4 1.80.501.4 1.80.80.60.40.80.60.40.20.850.80.50.501.4 1.801.4 1.80.501.4 1.80.80.60.40.80.60.40.20.850.80.50.501.4 1.801.4 1.80.501.4 1.832142.513.83.63.4Figure 4.5: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 15 and21 days for the parameter values given in Tables 4.1 and 4.2 and for E pl = 0and D c = 0.133 mm 2 /days. New bone matrix starts forming at the implantbody surface at time 21 dayst=7 dayst=15 days0.5c m0.50.30.101.4 1.80.501.4 1.80.960.950.5c tot0.60.40.201.4 1.80.501.4 1.80.960.950.501.4 1.80.501.4 1.83214.484.45gFigure 4.6: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7 and 15days for the parameter values given in Tables 4.1 and 4.2 and for E pl = 0and χ 0 = 0.162 mm 2 /days. New bone matrix starts forming at the implantbody surface at time 15 days

102 Chapter 4. Moving boundary model for endosseous healingc m0.2c tot0.2t=7 days0.501.4 1.80.80.60.40.501.4 1.80.80.60.40.5g01.4 1.842t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842Figure 4.7: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 15 and24 days for the parameter values given in Tables 4.1 and 4.2 and for E pl = 0and D g = 0.09 mm 2 /days. Mature osteogenic cells appear in the middle ofthe implant surface at time t = 24 daysa low concentration of growth factors, since the differentiation rate u b is definedas u lim g 2bin Table 4.1. Small values of KKu+g 2 2u yield a large differentiationrate of cells even in the regions, where the growth factor concentrationg is considerably low. If K u is large, then active cell differentiation occursonly in the regions, where g is high. Sensitivity analysis shows, that thenumerical solutions are similar for the reference value 0.01 of K u , definedin Table 4.2, and for the small value 0.001. Increasing K u to the value 0.1yields that no mature cells appear within the peri-implant region and nobone forms, due to deficiency of cell differentiation.It follows from simulations that the values 1, 2 and 3 of the parameter λ cprovide similar numerical solutions. The value 2 is chosen as the referencevalue in Table 4.2.The values of the parameters g bone and C bone are chosen in such a way,that bone formation is initiated at the old bone surface at the end of thefirst week. Numerical simulations provide similar solutions for the referencevalue 0.6 mm/days of g bone and for values twice as small and twice as large.Multiplication of the chosen value 0.07 mm/days of C bone by two leads toa large width of the layer with a high concentration of growth factors anda high density of cells at the bone-forming surface. Hence a larger influx ofcells at the old bone surface leads to appearance of mature osteogenic cells

4.4. Model validation 103c m0.2c tot0.2gt=15 days0.501.4 1.80.80.60.40.501.4 1.80.80.60.40.501.4 1.842.51t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51Figure 4.8: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments 15, 24and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 0 and D c = 0.5 · 0.133 = 0.0665 mm 2 /days. The region, in whichmature osteogenic cells are present is much smaller, compared to the caseD c = 0.75 · 0.133 mm 2 /days, illustrated in Figure 4.4

104 Chapter 4. Moving boundary model for endosseous healingt=6 days0.5c m0.60.40.201.4 1.8c tot0.5 0.80.60.40.201.4 1.80.5g01.4 1.842.51t=7 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=9 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51Figure 4.9: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 6,7, 9, 15, 24 and 28 days for the parameter values given in Tables 4.1 and4.2 and for E pl = 0 and χ 0 = 0.648 mm 2 /days. The region, in whichmature osteogenic cells are present is much smaller, compared to the caseχ 0 = 0.324 mm 2 /days, illustrated in Figure 4.4

4.4. Model validation 105ub=0.15 days −1t=7.9 days0.501.4 1.80.60.40.2ub=0.18 days −1t=6.7 days0.501.4 1.80.60.40.2ub=0.2 days −1t=6 days0.501.4 1.80.60.40.2Figure 4.10: Plots of the mature cell density c m for the parameter valuesgiven in Tables 4.1 and 4.2 and for E pl = 0 and the values 0.15, 0.18 and0.2 days −1 of the differentiation rate limit u limb. Smaller values u limbyield adelay of cell differentiation and of bone formation, which are well observedin the present plotst=6 dayst=9 days0.50.501.4 1.801.4 1.80.80.60.40.20.50.501.4 1.801.4 1.80.80.60.40.80.60.40.20.50.501.4 1.801.4 1.832142c m0.60.40.2c tot0.2gt=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.8321Figure 4.11: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 6, 9and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 0 and C bone = 0.035 mm/days. The bone-forming surface moves fromthe corners formed by the implant thread and by the old bone surface

106 Chapter 4. Moving boundary model for endosseous healingat the implant body surface at day 18. Hence distance osteogenesis is notpredicted for a large value of C bone .Division of the reference value of C bone by two yields that the ossificationfront advances from the corners formed by the implant thread and by the oldbone surface (Figure 4.11). A similar path of bone formation is illustratedin Figure A.1 in Appendix A.1. The solutions plotted in the last figure areobtained for a small time span of the influx of osteogenic cells at the old bonesurface t cbone = 0.5 days. Therefore, the current path of bone formation isrelated to a weak source of osteogenic cells at the old bone surface.The value of the bone-forming surface velocity coefficient, P b , is estimatedin such a way that the amount of newly formed bone within theperi-implant region after four weeks of simulation is line with the experimentaldata of [1].Up to this point a zero source of growth factors at the implant surface wasassumed. This assumption is used in the numerical simulations, which arecarried out to estimate all necessary parameter values, except the value of theparameter E pl corresponding to a micro-rough SLA implant. For the SLAimplant, a value for E pl is chosen, such that bone formation is observed atthe entire implant surface (contact osteogenesis) at the end of the first week.Depending on the value of E pl , different paths are predicted by simulations,which are illustrated in Figures 4.12–4.15. In the case E pl = 3.2 mm/daysbone formation proceeds from the old bone surface and from the corners,formed by the implant body and by the thread (Figure 4.12).The values 4 · 3.2 mm/days and 16 · 3.2 mm/days of E pl yield the occurrenceof bone formation at the lateral sides, i.e. at the thread surface(Figures 4.13 and 4.14).For the reference value 64∗3.2 = 204.8 mm/days of E pl in Table 4.2, thebone-forming surface starts moving from the entire implant surface includingthe thread. The high density- high concentration layer is moving to the leftthe first ten days and becomes aligned with the implant body surface attime 11 days (Figure 4.15).The limit value Epllim is chosen is such a way, that for all E pl ≤ Epllimdistance osteogenesis is predicted by the model. The value 0.32 mm/daysof the parameter Epllim is determined from the simulations.Therewith, more than half of the model parameters are estimated numerically.The rest of the parameters is taken from previous models, inwhich some of the parameters are also estimated. The choice of the parametervalues is made in such a way, that the model solution is in linewith the experimental studies mentioned in this work. There is no experimentalevidence that the estimated parameter values correspond to theiractual values. The biological relevance of the proposed model parametersand, probably, more accurate parameter values, should be determined fromfurther experimental studies.

4.4. Model validation 107c m0.2c tot0.2t=7 days0.501.4 1.80.80.60.40.501.4 1.80.80.60.40.5g01.4 1.842.51t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51Figure 4.12: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 7, 15and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 3.2 mm/days. Bone formation occurs at the old bone surface and inthe corners, formed by the implant body and by the threadc mc gtot40.80.50.6 0.50.8 0.50.62.50.40.40.20.210001.4 1.81.4 1.81.4 1.8t=7 dayst=15 dayst=28 days0.501.4 1.80.501.4 1.80.80.60.40.20.80.60.40.20.501.4 1.80.501.4 1.80.80.60.40.20.80.60.40.20.501.4 1.801.4 1.8Figure 4.13: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 7, 15and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 4 · 3.2 mm/days. Bone forms at the thread surface0.542.5142.51

108 Chapter 4. Moving boundary model for endosseous healingt=7 daysc m0.80.50.60.40.201.4 1.8c tot0.50.80.60.40.201.4 1.80.5g01.4 1.842.51t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51Figure 4.14: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 19, 22and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 16 · 3.2 mm/days. Bone forms at the thread surface4.4.3.1 Numerical solutions for various mesh resolutionThe results of numerical simulations presented in Section 4.4.3 are obtainedfor a fine mesh resolution. The sizes of the initial uniform rectangular controlvolumes along r−, z− and a− axes are equal, respectively, to h r = 0.0045mm, h z = 0.005 mm and h a = 0.05, respectively. For conciseness, thepresent mesh grid is denoted by Ω h , where the subindex h refers to thechosen spatial resolution.Some interesting features about convergence of the numerical solutionsagainst the spatial resolution of the control volume grid are observed duringsimulations. Decrease of the mesh resolution within the physical space(r, z) can yield completely different path of bone formation predicted by thenumerical solutions.Consider the initial mesh Ω 2h with the following sizes of control volumes:h r = 0.009 mm, h z = 0.01 mm and h a = 0.05; and the mesh Ω 4h , such thath r = 0.018 mm, h z = 0.02 mm and h a = 0.05.The numerical solutions for the parameter values in Tables 4.1 and 4.2and for E pl = 0 and χ 0 = 0.648 mm 2 /days, which are obtained on themeshes Ω 2h and Ω 4h , are shown in Figures 4.16 and 4.17, respectively. Thelayer of high cell densities and of a high growth factor concentration nearthe old bone surface takes a wave-like appearance at the end of the firstweek. The wave length is approximately equal to 0.35 mm and 0.7 mm

4.4. Model validation 109t=42 days t=36 days t=28 days t=11 days t=7 daysc m0.80.5 0.60.40.201.4 1.80.501.4 1.80.501.5 1.80.501.5 1.80.501.5 1.80.80.60.40.20.80.60.40.20.80.60.40.20.80.60.40.2c tot0.80.5 0.60.40.201.4 1.80.501.4 1.80.501.5 1.80.501.5 1.80.501.5 1.80.80.60.40.20.80.60.40.20.80.60.40.20.80.60.40.20.50.5g01.4 1.801.4 1.80.501.5 1.80.501.5 1.80.501.5 1.842.514142.512.542.512.521.510.5x 10 −83t=63 days0.501.5 1.80.0850.080.501.5 1.80.0850.080.501.5 1.8642Figure 4.15: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 11, 28,36, 42 and 63 days for the parameter values given in Tables 4.1 and 4.2 andfor E pl = 64 · 3.2 mm/days. The high density- high concentration layer ismoving to the left during the first ten days and becomes aligned with theimplant body surface at time 11 days

110 Chapter 4. Moving boundary model for endosseous healingt=6 days0.5c m0.60.40.201.4 1.80.501.4 1.80.80.60.40.501.4 1.842.51c tot0.2gt=7 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=9 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.82.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.82.51Figure 4.16: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 6,7, 9, 15, 24 and 28 days for the parameter values given in Tables 4.1 and4.2 and for E pl = 0 and χ 0 = 0.648 mm 2 /days. The numerical solution isobtained on the mesh Ω 2h with the cell sizes h r = 0.009 mm, h z = 0.01 mmand h a = 0.05 along r−, z− and a− axes, respectively

4.4. Model validation 111t=6 days0.5c m0.60.40.201.4 1.8c tot0.5 0.80.60.40.201.4 1.80.5g01.4 1.842.51t=7 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=9 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.51t=24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.82.51t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.82.51Figure 4.17: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 6,7, 9, 15, 24 and 28 days for the parameter values given in Tables 4.1 and4.2 and for E pl = 0 and χ 0 = 0.648 mm 2 /days. The numerical solution isobtained on the mesh Ω 4h with the cell sizes h r = 0.018 mm, h z = 0.02 mmand h a = 0.05 along r−, z− and a− axes, respectively

112 Chapter 4. Moving boundary model for endosseous healingfor the meshes Ω 2h and Ω 4h , respectively. After that, the layer splits in itsmiddle and shrinks towards the corners formed by the old bone surface andthe implant thread. Consequently, the bone forming front moves from thecorners to the center of the healing site. Such a path of the ossification frontis completely different from the path predicted by simulations for the finemesh Ω h shown in Figure 4.9.Formation of the wave-like high density - high concentration layer nearthe old bone surface and its splitting are also observed for some other parametervalues and coarse meshes. Consider the values of the model parametersgiven in Tables 4.1 and 4.2 and the boundary condition E pl = 0. The numericalsolution obtained on the mesh Ω 4h is plotted in Figure 4.18. Thec m0.2c tot0.2gt= 7 days0.501.4 1.80.80.60.40.501.4 1.80.80.60.40.501.4 1.842.50.5t= 15 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5t= 24 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5t= 28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842.50.5Figure 4.18: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 7, 15, 24and 28 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 0. The numerical solution is obtained on the mesh Ω 4h with the cellsizes h r = 0.018 mm, h z = 0.02 mm and h a = 0.05 along r−, z− and a−axes, respectivelywave-like layer forms at the old bone surface at the end of the first week. At

4.4. Model validation 113the end of the second week the layer splits and bone is predicted to form inthe corners, which are formed by the old bone surface and by the implantthread, and then at the lateral surfaces, i.e. near the surface of the implantthread. The ossification path is different from the path obtained in the solutionfor the fine mesh Ω h (see Figure 4.4). For the present parameter values,the solution for the mesh Ω 2h is close to the solution on the mesh Ω h .Therefore, it can be concluded that insufficient mesh resolution can givelarge errors in the numerical solutions. It is important to assess the convergenceof the numerical solution against the mesh resolution. Therefore, asimplified problem is defined in order to study the convergence. The simplifiedproblem is derived from the original model by setting the velocity of theboundary of the physical domain to zero. Considering a constant physicaldomain makes the comparison of numerical solutions more straightforward,compared to the situation when the solutions, which are obtained on theevolving in time domains, should be compared. In order to reduce computationtime, the physical domain is simplified to the rectangular region¯Ω = {(r, z) ∈ [1.41, 1.85] mm × [0, 0.35] mm}. The part of the domainboundary (r, z) ∈ {1.85}×[0, 0.35] is assumed to correspond to the old bonesurface. The rest of the domain boundary is the implant surface.It was found from the numerical simulations, that the appearance ofa wave-like high density- high concentration layer is most sensitive withrespect to the mesh resolution is the x-direction. Further in this subsection,the solutions of simplified problem are compared for meshes with varioush r sizes. The initial meshes ¯Ω 4h , ¯Ω 2h , ¯Ω h and ¯Ω h/2 constructed within thesimplified physical domain ¯Ω have the respective values 0.02 mm, 0.01 mm,0.005 mm and 0.0025 mm of the linear size h r . The lengths of the controlvolumes along the z- and a axes are kept constant for the present meshes,such that h z = 0.0175 mm and h a = 0.025.Consider the numerical solution of the simplified problem for the parametervalues in Tables 4.1 and 4.2, for E pl = 0 and for P b = 0. The appearanceof wave patterns is studied. The patterns develop from initial perturbationsor inhomogeneity of the solution in the z− direction. For the full problem,perturbations appear due to irregular geometry of the physical domain Ω s .In the simplified problem, an inhomogeneity in the z− direction is imposedthrough perturbations of the boundary conditions at the old bone surface.The perturbations ˜c bone and ˜g bone are added to the original influx of cellsc bone (a) and of growth factors g bone , which are defined in Section 4.3.2 andin Table 4.2. Two modes of perturbations are considered. In the first case,˜c bone = 0.1 · c bone (a) sin( πz0.8), ˜g bone = 0.1 · g bone sin( πz0.8). (4.20)For the second type of perturbations, a random value of the magnitude from0 to 0.1 · c bone (a) and from 0 to 0.1 · g bone , respectively, is assigned to ˜c boneand ˜g bone at each boundary edge of control volumes, which coincides with

114 Chapter 4. Moving boundary model for endosseous healingthe old bone surface.Numerical solutions obtained for the mesh ¯Ω 4h predict formation of ahigh density-high concentration layer, which increases in width from thebottom to the top for the both types of perturbations (Figure 4.19). Theleft boundary of the considered layer has the form of a half wave with thewave-length approximately 0.7 mm. The wave mode is the same for therandom perturbations and for the ’sine’ perturbations. Therefore, it can beconcluded that the form of the initial perturbations does not directly determinethe mode of the current pattern in the high density - high concentrationlayer.Increase of the mesh resolution in the x-direction yields uniform widthof the considered layer near the old bone surface. The distribution of thegrowth factor concentration at time 9 days obtained for the meshes ¯Ω 2h , ¯Ω h ,¯Ω h/2 and for the perturbations defined in equation (4.20) is shown in Figure4.20. The layers of high cell densities have the same width as the layers ofa high growth factor concentration plotted in Figure 4.20.For a twice as large value of the chemotaxis parameter χ 0 , i.e. for χ 0 =0.648 mm 2 /days, the boundary layer with nonuniform width is obtained atday 7 for the meshes ¯Ω 4h and ¯Ω 2h , if the perturbations defined in equation(4.20) are added to the boundary conditions. The meshes ¯Ω h and ¯Ω h/2 witha finer resolution provide uniform solutions along the z− axis (see Figure4.21).Therefore, at this point the following conclusions can be made:Clause 1. If the parameter values in Tables 4.1 and 4.2 and E pl = 0 areconsidered, then• The solution of the full problem is characterized by the appearanceof the wave-like high density- high concentration layer with the wavelength ≈ 0.7 mm at the end of the first week, if the size of controlvolumes h r in r-direction is equal to 0.018 mm (Figure 4.18). If themesh resolution in the physical space is increased two times, then thehigh density- high concentration layer has almost a uniform length atday 7 (Figure 4.4).• The high density- high concentration layer in the solution of the perturbedsimplified problem has a wave-like profile with wave length≈ 0.7 mm, if h r = 0.02 mm at day 9 (Figure 4.19). The consideredlayer is uniform for the mesh with the resolution increased two, fourand eight times (Figure 4.20).Clause 2. If the chemotaxis coefficient is increased two times, i.e. χ 0 =0.648 mm 2 /days, then• The wave-like layer at the old bone surface appears in the solutionof the full problem at the end of the first week for the meshes Ω 4h

4.4. Model validation 115randomday 13sineday 90.40.2c mc tot g0.40.40.80.80.60.60.4 0.20.4 0.20.20.201.4 1.850.40.201.4 1.850.80.60.40.201.4 1.850.40.201.4 1.850.80.60.40.201.4 1.850.40.201.4 1.854242Figure 4.19: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g for the parameter values givenin Tables 4.1 and 4.2 and for E pl = 0, P b = 0 and for the two modes ofperturbations: random perturbations (upper row) and ’sine’ perturbationsdefined in equation (4.20) (lower row). The solutions are plotted at timet = 13 days for random perturbations and at time t = 9 days for the ’sine’perturbations. The numerical solution is obtained on the mesh ¯Ω 4hΩ 2h Ω h Ω h/20.44 0.44 0.440.220.220.2201.4 1.8501.4 1.8501.4 1.85Figure 4.20: Plots of the growth factor concentration g at the time momentt = 9 days for the parameter values given in Tables 4.1 and 4.2, for E pl = 0,P b = 0 and for the ’sine’ perturbations defined in (4.20). The numericalsolution is obtained on the meshes ¯Ω 2h , ¯Ω h , ¯Ω h/20.40.2Ω 4h Ω 2h Ω h Ω h/20.40.40.44442 0.2 2 0.2 2 0.24201.6 1.8501.6 1.8501.6 1.8501.6 1.85Figure 4.21: Plots of the growth factor concentration g at the time momentt = 7 days for the parameter values given in Tables 4.1 and 4.2 and forE pl = 0, P b = 0, χ 0 = 0.648 mm 2 /days and for the ’sine’ perturbationsdefined in (4.20). The numerical solution is obtained on the meshes ¯Ω 4h ,¯Ω 2h , ¯Ω h , ¯Ω h/2

116 Chapter 4. Moving boundary model for endosseous healing(h r = 0.018 mm) and Ω 2h (h r = 0.009 mm). The wave length of theleft boundary of the layer is ≈ 0.35 mm for Ω 4h and ≈ 0.7 mm forΩ 2h . The layer is uniform along the z-axis in the solution for the meshΩ h with the size h r = 0.0045 mm.• The wave-like layer at the old bone surface appears in the solution ofthe perturbed simplified problem at day 7 for the meshes ¯Ω 4h (h r =0.02 mm) and ¯Ω 2h (h r = 0.01 mm). The wave length of the leftboundary of the layer is ≈ 0.35 mm for ¯Ω 4h and ≈ 0.7 mm for ¯Ω 2h . Thelayer is uniform along the z-axis in the solution for the meshes ¯Ω h and¯Ω h/2 with the sizes h r = 0.05 mm and h r = 0.025 mm, respectively.Clause 3. From Clauses 1 and 2 it follows that the parameter values andthe mesh resolution in the x-direction, for which the present phenomenonis observed in the solutions of the full and the perturbed simplified models,perfectly correspond to each other. The wave characteristics of the leftboundary of the high density- high concentration layer in the solution of thefull problem are fully reflected by the solutions for the perturbed simplifiedmodel. Hence the perturbed simplified problem, which is derived in thissection, is suitable to study the phenomenon of the appearance of a wavelikehigh density- high concentration layer near the old bone surface.Clause 4. Formation of the high density- high concentration layer witha nonuniform width is likely to be related to large errors in the numericalsolutions obtained for a coarse mesh. Meshes with a fine enough resolutionprovide solutions with a uniform distribution of the cells densities and ofthe growth factor concentration along the old bone surface. The order ofconvergence of numerical solutions is studied in Section 4.4.3.2.Clause 5. Values of model parameters influence the appearance of a wavelikehigh density- high concentration layer. Such a layer is observed, forexample, in the numerical simulation for a large value of the chemotaxisparameter χ 0 for the mesh Ω 2h . For a twice as small parameter χ 0 , thenon-uniform layer forms in the solution on the coarser mesh Ω 4h but not onΩ 2h .Clause 6. A wave-like pattern formation and splitting of the high densityhighconcentration layer is observed in the solutions plotted in Figures 4.11and A.1, which are obtained on the finest mesh that can be afforded for simulationsat the present moment. The parameter values are defined in Tables4.1 and 4.2 and the parameters C bone and t cbone are decreased two times,respectively. Therefore, it is not clear, whether such a solution behavior ischaracteristic for the exact solution, or whether the numerical solution doesnot converge to the exact solution for the chosen mesh resolution.

4.4. Model validation 1174.4.3.2 Convergence against the mesh resolutionThe numerical errors and the order of convergence are estimated by comparingthe numerical solutions obtained for various meshes. Convergence isconsidered against the mesh resolution in each of the r-, z- and a-directionsseparately. In the series of simulations, the mesh resolution in one directionis varied, and the resolution in the other two dimensions is the same.The simulations considered in this section are performed for the parametervalues defined in Tables 4.1 and 4.2 and for E pl = 0 and P b = 0.The order of convergence with respect to the mesh resolution in thez-direction is obtained in the following way. Three meshes with variousnumbers n z of control volumes along the z-axis are considered. The sizes ofcontrol volumes along the r- and a-axes are fixed and are equal to 0.0025mm and 0.025, respectively. The sizes h r and h a are chosen to be as small aspossible, in order to make the error due to discretization in the r- and a- axescomparatively small. On the other hand, the considered sizes should providea reasonable computation time. The considered three meshes, referred to as¯Ω nz=10 , ¯Ω nz=20 and ¯Ω nz=40 , have 10, 20 and 40 control volumes along thez-axis, respectively. The length of the physical domain ¯Ω in the z-directionis 0.7 mm. Hence, the present meshes are characterized by the respectivesizes h z = 0.035 mm, h z = 0.0175 mm and h z = 0.00875 mm.Remark 4.8. The positivity and stability requirements impose severe restrictionson the time step size. Therefore, the time steps used for the simulationspresented in this section are very small. Comparison of the numerical solutionobtained for various time steps sizes shows, that the numerical errordue to time integration is several orders smaller that the errors due to spatialdiscretization. Therefore the time integration errors are not consideredbelow in the text.The numerical solutions obtained on the meshes ¯Ω nz=10 , ¯Ωnz=20 and¯Ω nz=40 are compared in the following way. The solution defined on a finermesh is projected onto the coarser mesh. The numerical solutions for thetotal cell density, for the mature cell density and for the growth factor concentrationare denoted, respectively, by C ⃗ totnz=20 (t), C ⃗ mnz=20 (t) and G ⃗ nz=20 (t),where the superscript nz = 20 refers to the resolution of the mesh, on whichthe present solution is defined, i.e. of the mesh ¯Ω nz=20 for the present notations.The solution on the mesh ¯Ω nz=20 can easily be projected onto thecoarser mesh ¯Ω nz=10 . Each control volume of ¯Ω nz=10 contains exactly twocontrol volumes of the mesh ¯Ω nz=20 . Hence the projected average value ofsome unknown (a cell density or the growth factor concentration) within acontrol volume of the mesh ¯Ω nz=10 is equal to the mean value of the (average)values of this unknown within two corresponding control volumes ofthe mesh ¯Ω nz=20 . The solution U ⃗ nz=40 (t), U ∈ { C ⃗ m , C ⃗ tot , G}, ⃗ is projectedonto ¯Ω nz=10 in a similar way. The projected solutions are denoted, respectively,by L nz=10nz=20 U nz=20 (t) and L nz=10nz=40 U nz=40 (t), U ∈ { ⃗ C m , ⃗ C tot , ⃗ G}, where

118 Chapter 4. Moving boundary model for endosseous healing2.52conv. order for c m conv. order for c tot conv. order for g1.50 5 10 15time210 5 10 15time21.5L 1norm L 2norm L infnorm0 5 10 15timeFigure 4.22: Plots of the estimated order of convergence with respect tothe mesh resolution the z-direction against time, which is derived from thesolutions obtained on the meshes ¯Ω nz=10 , ¯Ω nz=20 and ¯Ω nz=40 . Three typesof vector norms are considered: L 1 , L 2 and L ∞ .log 10 error for c m log 10 error for c tot log 10 error for gL 1normL 2normL infnorm−2−4−60 5 10 15−2−4−60 5 10 15−3−4−50 5 10 15−2−4−60 5 10 15−2−4−60 5 10 15−2−4−60 5 10 15−4−5−60 5 10 15−4−5−60 5 10 15−3−4−50 5 10 15log 10 ‖L nz=10nz=10 Unz=10 –L nz=10nz=20 Unz=20 ‖/‖L nz=10nz=10 Unz=10 ‖log 10 ‖L nz=10nz=20U nz=20 –L nz=10nz=40U nz=40 ‖/‖L nz=10nz=20U nz=20 ‖Figure 4.23: Plots of the order of relative errors between the numericalsolutions obtained on the meshes ¯Ω nz=10 and ¯Ω nz=20 (blue curve) and ¯Ω nz=20and ¯Ω nz=40 (green curve). Three types of vector norms are considered: L 1 ,L 2 and L ∞ .

4.4. Model validation 119L nz=10nz=N is the projection operator from the mesh ¯Ω nz=N , N = {20, 40}, ontothe mesh ¯Ω nz=10 . Note that Lnz=10 nz=10Unz=10 (t) = U nz=10 (t). The convergenceorder is estimated by comparing the solutions for three consecutivemesh resolutions. For example, the convergence order, which is determinedfor the total cell density c tot from the numerical solutions on the meshes¯Ω nz=10 , ¯Ω nz=20 and ¯Ω nz=40 , is calculated as‖L nz=10nz=10p = log Cnz=10 tot2‖L nz=10nz=20 Cnz=20 tot− L nz=10− L nz=10nz=20 Cnz=20 totnz=40 Cnz=40 tot‖‖(4.21)Derivation of equation (4.21) is based on the Richardson extrapolation andon the assumption that the numerical error obtained on the mesh with acell size h is equal to Ch p (see, for example, Vuik et al. [95]).The order of convergence, which is calculated in different vector norms,is shown in Figure 4.22. From Figure 4.22, it follows that the estimatedconvergence order is close to two most of the time. However the order candecrease to values less than one at certain time moments. The most likelyreason for such a fluctuation of the estimated convergence order is that themesh sizes h r and h a along r- and a-axes are not sufficiently small. Theerror from the discretization in the r- and a-direction is combined with theerror of discretization in the z-direction. This issue leads to jumps of theestimated order of convergence shown in Figure 4.22.The difference between the numerical solutions, obtained on the consideredconsecutive meshes, can be used to assess the order of the totalnumerical error related to the current mesh resolution in the z-direction.The logarithm of the relative error between the numerical solutions, that iscalculated in different vector norms, is plotted in Figure 4.23. In most ofthe plots in Figure 4.23, the relative errors are not larger than 10 −4 . Themaximum relative errors after time t = 5 days appear in the solution for c mand c tot and they are smaller than 10 −3.5 .By analogy, the convergence of the numerical algorithm in the other twospatial directions is derived. The convergence against h r is estimated fromthe numerical solutions obtained on the following meshes. The linear sizesh z and h a are equal to 0.035 mm and 1/160, respectively. Three values ofthe size h r : 0.01 mm, 0.005 mm and 0.0025 mm are used for the meshes¯Ω nr=44 , ¯Ω nr=88 and ¯Ω nr=176 .The convergence of the algorithm in the a-direction is derived for themeshes ¯Ω na=40 , ¯Ω na=80 and ¯Ω na=160 , which have, respectively, 40, 80 and160 elements along the a-axis. The sizes h r and h z are equal to 0.0025 mmand 0.035 mm, respectively.The estimated order of convergence against the grid sizes h r and h a isplotted in Figures 4.24 and 4.25, respectively. The corresponding relativeerrors are shown in Figures 4.26 and 4.27.The order of convergence in the r-direction is estimated between thevalues 2.2 and 2.6 after t = 6 days. It is larger than the actual order of

120 Chapter 4. Moving boundary model for endosseous healing43conv. order for c m conv. order for c tot conv. order for g4420 5 10 15time200 5 10 15timeL 1norm L 2norm L infnorm200 5 10 15timeFigure 4.24: Plots of the estimated order of convergence with respect tothe mesh resolution the r-direction against time, which is derived from thesolutions obtained on the meshes ¯Ω nr=44 , ¯Ω nr=88 and ¯Ω nr=176 . Three typesof vector norms are considered: L 1 , L 2 and L ∞ .conv. order for c m conv. order for c tot conv. order for g432100 5 10 15time32100 5 10 15time3210L 1norm L 2norm L infnorm0 5 10 15timeFigure 4.25: Plots of the estimated order of convergence with respect tothe mesh resolution the a-direction against time, which is derived from thesolutions obtained on the meshes ¯Ω na=40 , ¯Ω na=80 and ¯Ω na=160 . Three typesof vector norms are considered: L 1 , L 2 and L ∞ .

4.4. Model validation 121log 10 error for c m log 10 error for c tot log 10 error for gL 1normL 2normL infnorm−1−2−30 5 10 15−1−2−30 5 10 15−1−2−30 5 10 150−2−40 5 10 150−2−40 5 10 15−1−2−30 5 10 150−2−40 5 10 150−2−40 5 10 150−2−40 5 10 15log 10 ‖L nr=44nr=44U nr=44 –L nr=4488 U nr=88 ‖/‖L nr=44nr=44U nr=44 ‖log 10 ‖L nr=44nr=88U nr=88 –L nr=44nr=176U nr=176 ‖/‖L nr=44nr=88U nr=88 ‖Figure 4.26: Plots of the order of relative errors between the numericalsolutions obtained on the meshes ¯Ω nr=44 and ¯Ω nr=88 (blue curve) and ¯Ω nr=88and ¯Ω nr=176 (green curve). Three types of vector norms are considered: L 1 ,L 2 and L ∞ .

122 Chapter 4. Moving boundary model for endosseous healingthe accuracy of the numerical scheme, used for the spatial discretizationof the model equations (4.1)–(4.3), which is equal to two on the uniformrectangular grid. The errors related to the variation of the mesh resolutionin the r-direction (Figure 4.26) are much larger than those related to thesizes h z and h a (after time t = 7 days) plotted in Figures 4.23 and 4.27.The former errors prevail in the numerical solution for t > 7 days andhave the most influence on the estimated convergence order. Due to a largecomputation time, it is not possible to estimate the convergence order forfiner meshes with simultaneous refinement in all directions.The convergence of the algorithm against the resolution in the a-directionis very poor (Figure 4.24). The first possible reason for this, is that a largernumber of elements should be considered in the a-direction, in order toobtain better convergence estimates. The second potential reason, is thatthe errors due to discretization in r- and z- direction have a considerableinfluence on the behavior of the solution. Such a possibility follows fromthe order of the relative errors, plotted in Figures 4.23, 4.26 and 4.27. Theestimated errors related to the discretization along the r-axis are comparablewith the errors related to the discretization along the a-axis during the firstweek of simulation. The former errors become much larger than the latterones for t > 7 days. Further increase of the mesh resolution is likely toyield a larger order of convergence of the numerical solution, which wouldapproach the order of accuracy of the spatial discretization. However, it isnot possible to get a solution for finer meshes due to limited computationalresources.Remark 4.9. It follows from the performed numerical simulations that theappearance of the wave-like patters of the high density- high concentrationlayer is influenced in general by the mesh size h r . The slow convergenceagainst the mesh resolution in the a-direction is not likely to influence suchcritical solution characteristics as the earlier mentioned wave-like patternsand splitting of the high density- high concentration layer.An important conclusion, which should be drawn from the results of simulations,is that one should be very careful with choosing a mesh resolutionwhen using the present numerical approach for the considered model. Insome situations, it is not possible to assess the robustness of the solutionbased only on speculative conclusions about the biological relevance of thesimulation results.An analysis of the solution convergence against the mesh element size canreveal the hidden nature of large errors. Furthermore, a stability analysisof the mathematical model, as well as for the semi-discrete equations is ofrelevance.

4.4. Model validation 123log 10 error for c m log 10 error for c tot log 10 error for gL 1normL 2normL infnorm0−2−2−4−4−60 5 10 15 0 5 10 15−20−2−40−20 5 10 15−40 5 10 15−4−60 5 10 150−2−40 5 10 150−2−40 5 10 150−2−40 5 10 150−2−40 5 10 15log 10 ‖L na=40na=40 Una=40 –L na=40na=80 Una=80 ‖/‖L na=40na=40 Una=40 ‖log 10 ‖L na=40na=80U na=80 –L na=40na=160U na=160 ‖/‖L na=40na=80U na=80 ‖Figure 4.27: Plots of the order of relative errors between the numericalsolutions obtained on the meshes ¯Ω na=40 and ¯Ω na=80 (blue curve) and ¯Ω na=80and ¯Ω na=160 (green curve). Three types of vector norms are considered: L 1 ,L 2 and L ∞ .

124 Chapter 4. Moving boundary model for endosseous healing4.5 Results and discussionWe consider the solutions obtained for the reference values of the modelparameters given in Tables 4.1 and 4.2 in more detail. Two types of implantsare considered. The limit value Epllim = 0.32 mm/days is assigned to theparameter E pl in simulations for a turned implant.SLATurned0.60.40.20.60.40.201.4 1.6 1.8Time [days]:01.4 1.6 1.80 7 14 21 28 35 49 63Figure 4.28: Evolution of the soft tissue region, due to formation of newbone near a micro-rough SLA implant and near a smooth turned implant.The contours correspond to the boundary of the soft tissue region, whichshrinks, at time 0, 7, 14, 21, 28, 35, 49 and 63 daysRemark 4.10. It should be emphasized that the implant surface microstructureis not considered explicitly. The difference between turned andrough implants is modeled implicitly, by means of various values for thegrowth factor source at the implant surface. This feature is dealt with in asimilar way in the recent models by Amor et al. [6] and Moreo et al. [67].The evolution of the soft tissue region Ω s for a smooth turned implantand a micro-rough SLA implant is shown in Figure 4.28. The contours representthe boundary of the soft tissue domain at time 0, 7, 14, 21, 28, 35,49 and 63 days. The soft tissue region shrinks in both cases. However, thebone-forming surface moves in two different ways for a turned and SLA implant,respectively. The ossification front starts to move from the implantsurface if the micro-rough implant is considered, whereas it moves from theold bone surface till day 26 in the case of the smooth turned implant. Inother words, contact and distance osteogenesis are predicted for the roughand smooth implants, respectively. Contact osteogenesis leads to a slightlyfaster formation of new bone matrix, compared with distance osteogenesis.In Figure 4.29, the areas of newly formed bone for both implant types areplotted against time. The area of the healing site Ω is represented by thedotted line and the areas of newly formed bone near the SLA and turnedimplants are shown by the solid and dashed lines, respectively. From Figure4.29, it follows that approximately a half of the entire healing site is

4.5. Results and discussion 125Area, mm 20.20.1SLATurned00 20 40 60 80Time, daysFigure 4.29: The area of newly formed bone against time for contact anddistance osteogenesis, which are observed near the SLA and turned implants,respectively. The area of the healing site Ω is represented by the dotted linefilled by new bone matrix at the end of the fourth week near the roughimplant and near the smooth implant. In the case of the SLA implant, thelayer of a high concentration of growth factors and a high density of matureosteogenic cells, which is adjacent to the ossification front, reaches the oldbone surface at day 25 (see Figures 4.15 and 4.28). Hence bone formation isinitiated at the old bone surface at day 25 and the length of the bone formingsurface increases rapidly. This issue leads to the increase of the amountof new bone released per unit of time, which is reflected by the increase ofthe slope of the solid line in Figure 4.29 at time t = 25 days. An analogouseffect is observed for the turned implant at the day 27. The layer of a highcell density and of a high growth factor concentration reaches the implantbody surface at this time (Figure 4.31). Hence bone starts forming at theimplant surface and the rate of new bone formation increases rapidly.The SLA implant is predicted to be in a direct contact with a newlyformed bone at the end of the first week (Figure 4.28). It may be assumedthat a direct bone-to-implant contact may provide an earlier and betteranchorage of implants [1, 24, 32]. Almost the entire peri-implant region ispredicted to be filled with new bone matrix after nine weeks of simulationfor the both implant types.Two different modes of bone formation are observed, because of twodifferent sources of growth factors near the implant surfaces. Due to a strongsource on the micro-rough SLA implant, a high concentration of growthfactors near the implant surface (see the lower left plot in Figure 4.30) leadsto active migration of osteogenic cells from the old bone surface to theimplant, and, consequently, to active differentiation of immature cells intomature osteogenic cells near the implant, which release new bone matrix (seethe upper left plot in Figure 4.30). Cells at a high maturation level releasegrowth factors, hence the concentration of growth factors is maintained at a

126 Chapter 4. Moving boundary model for endosseous healinghigh level near the micro-rough implant surface during the first four weeks(see Figure 4.15).c mg0.50.501.4 1.801.4 1.8SLA0.20.150.10.053210.50.501.4 1.801.4 1.8Time 5 daysTurned0.140.10.060.02321Figure 4.30: Distribution of the mature cell density c m and of the growthfactor concentration g in the soft tissue region at time 5 days near SLAand turned implants. High concentration of growth factors leads to differentiationof immature osteogenic cells into mature cells at the SLA implantsurface. No growth factors and no mature cells are found near the turnedimplantIn experiments of Berglundh et al. [14] and Abrahamsson et al. [1], newbone formation was observed both on the SLA implant surface and on theold bone surface. However, the ossification front is predicted to move onlyfrom the SLA implant surface in the current simulations. The model parametervalues providing a better correspondence of the model predictions withexperimental results are not found. Adaptation of the treatment of diffusionand chemotaxis of the cells in the model, by introducing various functionaldependencies of the corresponding coefficients on the cell densities and/orthe growth factor concentration, is a potential direction of the improvementof the present model.A weak source of growth factors is considered at the smooth implant surface,and the concentration of growth factors is very low in this region duringthe first days in the simulation (see the lower right plot in Figure 4.30). Dueto a low growth factor concentration, cells are not attracted to the implantsurface by a chemotaxis mechanism. The cells are concentrated at the oldbone surface, due to a much larger source of growth factors in this area. Alow concentration of growth factors yields a slow cell differentiation at theimplant surface (see the dependence of the differentiation rate u b on g inTable 4.1). Cells with low maturation level release growth factors at a lowrate (see equation (5.3) and the expression for γ(a) in Table 4.1). Due tothe aforementioned factors and due to a high decay rate d g , the concentra-

4.5. Results and discussion 127t=15 daysc m0.50.80.60.40.201.4 1.8c tot0.50.80.60.40.201.4 1.80.5g01.4 1.842t=26 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842t=38 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.842t=42 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.8321x 10 −12t=63 days0.501.5 1.80.10.050.501.5 1.80.10.050.501.5 1.810Figure 4.31: Plots of the mature cell density c m , of the total cell density c totand of the growth factor concentration g at the time moments t = 15, 26, 28,38, 42 and 63 days for the parameter values given in Tables 4.1 and 4.2 andfor E pl = 0.1 · 3.2 mm/days, i.e. a turned implant considered. No growthfactors and no mature cells are found near the turned implant till day 26.Hence distance osteogenesis is predicted by the numerical simulations

128 Chapter 4. Moving boundary model for endosseous healingtion of growth factors and the density of mature osteogenic cells are closeto zero near the implant surface till day 26. As a result, no bone forms atthe surface of the smooth implant during this period (Figure 4.31).4.6 ConclusionsA general mathematical model for peri-implant bone regeneration is formulated,which is able to capture some important features, that are observedin reality. The model is defined for the immature and mature osteogeniccell densities and for the growth factor concentration. Cell differentiation isconsidered as an evolutionary process, regulated by the presence of growthfactors. Immature cells are distinguished with respect to the differentiationlevel, which is represented by the additional independent variable a, definedin the maturation space [0, 1]. Consequently, the differentiation path ofindividual cells can be considered.During peri-implant osseointegration, new bone is produced by osteoblasts,attached to a rigid surface [1, 14, 65]. Bone formation takes placeonly through a direct apposition of new bone matrix on a pre-existing rigidsurface [24]. That is, endosseous bone healing occurs in the form of intramembranousossification, which can be considered as a moving boundarytypeof bone formation. This effect is directly incorporated into the model,by the use of the concept of a bone-forming surface, which is defined as amoving boundary of a temporarily evolving computational domain. This isthe main innovation of the current model, that distinguishes the current formalismfrom the recent models for the peri-implant bone regeneration. Cellprocesses, like migration, differentiation and proliferation, are consideredwithin the region, filled with soft tissue and bounded by the bone-formingsurface. Osteoblasts are represented by the mature cells, which are situatedat the bone-forming surface. Bone formation is introduced through themovement of the boundary of the computational domain. Since bone is releasedby osteoblasts and boundary movement corresponds to the formationof new bone matrix, the velocity of the boundary movement is assumed tobe proportional to the local osteoblast density. The model is composed ofa system of partial differential equations, defined within the domain with amoving boundary, along with the set of initial and boundary conditions.In experiments of Abrahamsson et al. [1], contact and distance osteogenesisare observed for micro-rough SLA implants and smooth turned implants,respectively. It is assumed that this switch in osseointegration mode iscaused by different rates of growth factor release by activated platelets nearthe implant surfaces with different micro-structure [56, 71]. In the presentmodel, the release of growth factor at the implant surface is represented bymeans of boundary conditions.A number of the 2D simulations have been carried out. Two modes of

4.6. Conclusions 129bone formation are predicted for the smooth and micro-rough implants. Themicro-structure of the implant surface is modeled implicitly, by changingthe source of growth factors at the implant surface. Contact and distanceosteogenesis are modeled near the micro-rough SLA implant and the smoothturned implant, respectively. Two dimensional simulations allow to modelvarious paths of the ossification front corresponding to different sources ofgrowth factors at the implant surface.

CHAPTER 5Numerical algorithm5.1 IntroductionIn this chapter, a numerical approach for the solution of the time-dependentadvection-diffusion-reaction equations is described. The equations are definedfor a model of bone healing near endosseous implants in Chapter 4.The model is constructed to simulate early stages of bone regeneration.The equations model migration of osteogenic cells from the old bone surfaceto the implant surface, cell differentiation and proliferation. Theseprocesses are assumed to be regulated by growth factors. Diffusion, decayand release of growth factors by osteogenic cells are also taken into account.The unknowns in the model are the densities of immature and mature osteogeniccells and the concentration of growth factors. New bone is formedthrough apposition on a pre-existing surface [24]. The advance of the ossificationfront, which was observed in experiments by Abrahamsson et al.[1], Berglundh et al. [14], Meyer et al. [65], is modeled by the movement ofthe boundary of the physical domain.A robust method is constructed, which allows to get a numerical solutionin case, when the physical domain is defined in 2D axisymmetric coordinates.First, an appropriate discretization in physical space, maturation space andtime should be chosen, such that a stable nonnegative solution of the nonlinearadvection-diffusion-reaction equations will be obtained. The movementof the domain boundary, determined from the internal solution, is trackedwith use of the level set method. The embedded boundary method andsome auxiliary interpolation techniques are elaborated in order to adapt thefinite volume discretization, which is in general defined on the structuredrectangular grid, to the evolving irregular physical domain.Therefore, in Section 5.2, a short description of the model for peri-131

132 Chapter 5. Numerical algorithmimplant osseointegration is given. The numerical algorithm, developed forthe two dimensional physical domain is described in Section 5.3. The importanceof positivity of the numerical solution is outlined in Section 5.3.1.In Section 5.3.2 the construction of the computational mesh within the irregularphysical domain is presented. The level set function is used to trackthe temporal changes of the domain. The level set equation and the solutionmethod are presented in Section 5.3.3. The equations for averagedquantities, derived from the initial governing equations, are constructed inSection 5.3.4. The discretization of the advection-diffusion terms is consideredin Section 5.3.5, which demands the solution of the Riemann problem,defined for the system of hyperbolic equations. The Riemann solution isdeveloped in Section 5.3.6. The approximation of the reaction terms, theboundary conditions and the time integration of the discretized ordinarydifferential equations are discussed in Sections 5.3.7, 5.3.8 and 5.3.9, respectively.Final conclusions are drawn in Section 5.4.5.2 Mathematical modelFor convenience, the model equations from Chapter 4 are summarized in thissection. The model for bone regeneration, consists of three partial differentialequations (PDE’s), defined for the densities of immature and matureosteogenic cells, denoted as c i and c m , and for the concentration of growthfactors g. The peri-implant interface Ω is divided into two subdomainsΩ s and Ω b , which are occupied by soft connective tissue (fibrin network ofblood clot) and new bone respectively. Osteogenic cells and growth factorsare found within the soft tissue region. The boundary between subdomainsΩ s and Ω b is the bone-forming surface. This interface moves in time and isdenoted as Γ(t).The evolution of the unknown variables is determined by the followingPDE’s∂c i∂t = −∇ s · (−D c ∇ s c i + χ(g, c tot ) c i ∇ s g)− ∂ ∂a (u b(g)c i ) + A c (g)c i (1 − c tot ) , (5.1)∂c m∂t= −∇ s · (−D c ∇ s c m + χ(g, c tot ) c m ∇ s g)+ u b (g)c i (⃗x, 1, t) + A c (g)c m (1 − c tot ) , (5.2)(∂g∂t = ∇ s · (D g ∇ s g) + E c (g) c m +∫ 10)γ(a)c i da − d g g, (5.3)

5.2. Mathematical model 133where c tot = ∫ 10 c ida + c m is the total density of osteogenic cells per unitof volume. Equation (5.1) is defined for (⃗x, a, t) ∈ Ω s × [0, 1] × R + , andequations (5.2), (5.3) – for (⃗x, t) ∈ Ω s × R + . Operator ∇ s is the nablaoperator, defined in physical space, e.g. in cylindrical coordinates∂∇ s = ⃗e r∂r + ⃗e ∂θ∂θ + ⃗e ∂z∂z .The movement of the boundary Γ(t) of the physical domain Ω s is determinedby the following expressions for its initial location Γ(0) and thenormal velocity v n :{Γ(0) = ∂Ω b ∪ ∂Ω i ,v n ( ⃗ X, t) = −P b c m ( ⃗ X, t), ⃗ X ∈ Γ(t), t > 0.(5.4)The initial cell densities and growth factor concentration are assumed tobe zero:c i (⃗x, a, 0) = 0, c m (⃗x, 0) = 0, g(⃗x, 0) = 0, ⃗x ∈ Ω s , a ∈ [0, 1]. (5.5)The natural boundary conditions are imposed, by equating the sources h i ,h m and h g to the normal fluxes of the variables c i , c m and g:(−D c ∇ s c i + χ(g, c tot )c i ∇ s g) ( ⃗ X, a, t) · ⃗n( ⃗ X, t) = h i ( ⃗ X, a, t), (5.6)(−D c ∇ s c m + χ(g, c tot )c m ∇ s g) ( ⃗ X, t) · ⃗n( ⃗ X, t) = h m ( ⃗ X, t), (5.7)−D g ∇ s g( ⃗ X, t) · ⃗n( ⃗ X, t) = h g ( ⃗ X, t), (5.8)for X ⃗ ∈ Γ(t), a ∈ [0, 1], t > 0 and⎧0, a ∈ [0, 1], X⎪⎨⃗ ∈ ∂Ω i , t ≤ t c ,h i ( X, ⃗ −c bone · 2a, t) =δ a(1 − aδ a), a ∈ [0, δ a ], X ⃗ ∈ ∂Ω b , t ≤ t c ,0, a ∈ [δ a , 1], X ⃗ ∈ ∂Ωb , t ≤ t c ,⎪⎩c i ( X, ⃗ a, t) v n ( X, ⃗ t), a ∈ [0, 1], X ⃗ ∈ Γ(t), t > t c ,(5.9)h m ( ⃗ X, t) = 0, ⃗ X ∈ Γ(t), t > 0 (5.10)⎧− g ⎪⎨ impl (t), X ⃗ ∈ ∂Ωi , t ≤ t g ,h g ( X, ⃗ t) = − g bone , X ⃗ ∈ ∂Ωb , t ≤ t g ,⎪⎩g( X, ⃗ t) v n ( X, ⃗ t), X ⃗ ∈ Γ(t), t > tg .(5.11)In the above formulas, v n ( ⃗ X, t) is the normal velocity of the boundary Γ(t)of subdomain Ω s . The differentiation rate is assumed to be nonnegative.

134 Chapter 5. Numerical algorithmTherefore, the following natural boundary condition is specified at the inflowboundary, in order to provide uniqueness of the solution:u b (g( ⃗ X, t)) c i ( ⃗ X, a, t) = 0, ⃗ X ∈ Ωs , a = 0, t > 0. (5.12)The expressions and the values for the model parameters are defined inTables 4.1 and 4.2.5.3 Numerical methodIn this section a numerical method is described for a solution of the governingequations of the present osseointegration model in a 2D axisymmetricphysical domain.Consider the physical domain Ω s , defined in the 2D axisymmetric coordinates(r, z). For the current problem statement, the physical domain is ofirregular shape, and it evolves in time. In this case, the conventional finitevolume discretization of the governing equations on the structured rectangulargrid cannot be used in a straight-forward way. Therefore, irregularcells are used, in order to capture the irregular geometry. Further, the levelset method is applied to track the evolution of the problem domain.The method of lines forms the basis of the present numerical approach.First, the governing equations are discretized in the physical domain Ω s andin the differentiation state domain [0, 1], but not yet in time. The system ofthe PDE’s is reduced to a system of ordinary differential equations (ODE’s).Subsequently, a time integration technique is applied to solve the ODE’s.Remark 5.1. The governing equations (5.1)–(5.3) are solved within the softtissue region Ω s . No unknowns are defined in the subdomain Ω b , and thisregion is never considered during the solution procedure. Therefore, forconvenience, the terms ’physical domain’ and ’computational domain’ areused with respect to the soft tissue region Ω s further in the text.5.3.1 Positivity of the solutionThe initial densities of cells and the initial growth factor concentration arezero (see equation (5.5)). The outflow of c i , c m and g at the boundaries givenby the fluxes h i , h m and h g in equations (5.9)–(5.11) is nonpositive, if c i , c mand g are nonnegative. Hence the boundary conditions model the influx ofcells and of growth factors and they do not lead to negative solutions. Thenit can be shown that the governing equations (5.1)–(5.3) have a nonnegativesolution for the present initial and boundary conditions. Equation (5.1) canbe rewritten in the following form:∂c i∂t = D c∇ 2 sc i − χ(g, c tot ) ∇ s g · ∇ s c i − u b (g) ∂c i∂a− c i ∇ s · (χ(g, c tot ) ∇ s g) + A c (g)c i (1 − c tot ) , (5.13)

5.3. Numerical method 135The first term in the right-hand side is a diffusion term, the second and thethird terms model advection in the physical space and in the maturationspace with the velocities χ(g, c tot ) ∇ s g and u b (g), respectively. The last twoterms in the right-hand side are the reaction terms, which become equal tozero, if c i = 0. Therefore, neither of the terms in the right-hand side ofequation (5.13) can lead to a sign-change over time and hence there is noappearance of negative values in the solution for c i , if c i > 0 on Ω s at acertain time. The same conclusions can be drawn for equations (5.2), (5.3)providing a nonnegative solution for c m and g for the present initial andboundary conditions.Therewith, the exact solution of the considered problem should be nonnegative.However, it is quite common in practice, that numerical simulationsgive approximate solutions with negative values, even if an exactsolution is known to be positive. This issue does not have any critical meaning,as long as numerical solutions converge to the exact solution.A numerical algorithm described in this chapter provides nonnegativesolutions for the present model. The requirement on the positivity of approximatesolutions is essential in the current case due to the followingreasons:• Small negative values appearing in the numerical solutions may grow inmagnitude fast due to nonlinearity of the fluxes of cells. The taxis termis divided into advection and reaction terms in equation (5.13). The reactionterm (the fourth term in the right-hand side of equation (5.13))is responsible for potential growth of negative values. More insightinto the behavior of the system with nonlinear fluxes can be providedby considering a Riemann problem and by splitting the solution intoa set of waves. The Riemann solution is described in Section 5.3.6. InSection 5.3.6.4, it is shown that nonclassical delta shock waves mayform in the Riemann solution, if negative cell densities appear in thenumerical solution at some time moment. Due to cell diffusion, deltashock waves are smoothed and look like peaks and troughs growing inmagnitude. Large negative values in the approximate solution indicatethat convergence to a nonnegative exact solution is not achieved.• Furthermore, negative cell densities do not make any biological sense.• According to the present model a negative local density of osteoblastsleads to a reverse movement of the bone-forming front, i.e. to boneresorption. That is another unphysical consequence of negative approximatesolutions of the model.Therefore, a numerical algorithm should provide nonnegative values for thecell densities in order to ensure a convergence and biological relevance of thenumerical solution.

136 Chapter 5. Numerical algorithm5.3.2 Definitions of cells and vertices in the gridThe physical domain Ω s is defined in the 2D axisymmetric coordinate system(r, z). Let us consider some rectangular region [R 0 , R 1 ] × [Z 0 , Z 1 ], whichcontains region Ω s . Within this rectangular region, the uniform rectangularN r ×N z grid is constructed. The grid contains the set W of rectangular cells(or control volumes) W j,k ∈ W , (j, k) ∈ I c = {1, 2, .., N r } × {1, 2, .., N z },and the set Ψ h of cell vertices v j+12 ,k+ 1 ∈ Ψ h , (j, k) ∈ I v = {0, 1, .., N r } ×2{0, 1, .., N z }. The control volumes and the vertices are defined as W j,k =[r j−1 , r2 j+1 ] × [z2 k−1 , z2 k+1 ], (j, k) ∈ I c and v2j+12 ,k+ 1 = (r2 j+1 , z2 k+1 ), (j, k) ∈2I v , where r j+1 = R 0 + jh r , z2k+1 = Z 0 + kh z . Cell sizes along axes r2and z are equal to h r = (R 1 − R 0 )/N r and h z = (Z 1 − Z 0 )/N z . Thevertices from set Ψ h are the nodes of the rectangular volumes from setW . Edges ɛ j+1,k, (j, k) ∈ I c = {0, 1, .., N r } × {1, 2, .., N z } are defined as2the line segments with end-points v j+12 ,k− 1 and v2 j+12 ,k+ 1 , and edges ɛ2j,k+1 ,2(j, k) ∈ {1, 2, .., N r } × {0, 1, .., N z } are the line segments with end-pointsv j−12 ,k+ 1 and v2 j+12 ,k+ 1 .2In order to capture the irregular geometry of the problem domain, theCartesian boundary, or the embedded boundary approach is used (see forexample Colella et al. [20], Quirk [81]). The physical domain Ω s is superimposedon the Cartesian grid W , and the rectangular cells W j,k ∈ W aredivided into three types, depending on their location with respect to regionΩ s . The cells, which lie entirely within and outside region Ω s , are calledinner cells and outer cells, respectively. The cells, intersected by the boundaryof region Ω s , are defined as cut cells. For each cell, which is either aninner cell or a cut cell, the corresponding ’active’ cell is constructed, withinwhich the model equations are discretized. The indexes of active cells arethe indexes of all inner and cut cells, and they are denoted asI a = {(j, k) ∈ I c , such that µ(W j,k ∩ Ω s ) ≠ 0},where µ(W ) is the measure of a set W in the corresponding space, i.e. µ(W )denotes the volume, the area or the length of W , if the set W is defined in3D, 2D or 1D space, respectively. The active cells are constructed in thefollowing way:V j,k = W j,k ∩ Ω s , (j, k) ∈ I a , (5.14)and the set of the active cells is defined asV = {V j,k , (j, k) ∈ I a }.The edges of active cells, which lie on the grid lines, are defined as e j,k+12ɛ j,k+12∩ Ω s , and e j+12 ,k = ɛ j+ 1 2 ,k ∩ Ω s.The differentiation state space [0, 1] is divided into intervals [a l−12= lh a , l = 0, 1, ..., N a and h a = 1/N a . The three-l = 1, 2, ..., N a , where a l+12dimensional active cells are defined as V 3j,k,l= V j,k × [a l−12=, a l+1 ],2, a l+1 ], (j, k) ∈2

5.3. Numerical method 137I a , l = 1, 2, ..., N a . Hence cell V i,k is the orthogonal projection of cellVi,k,l 3 onto the plane (r, z). The set of the three-dimensional active cells isdenoted by V 3 = {Vj,k,l 3 , (j, k, l) ∈ I a 3 }, where Ia 3 = {(j, k, l), s.t. (j, k) ∈I a , l = 1, 2, ..., N a }. The faces of active cells Vj,k,l 3 , which correspond to theedges e j,k+1 and e2 j+1,k, are defined as e3 2 j,k+ 1 2 ,l = e j,k+ 1 × [a2 l−1 , a2 l+1 ] and2e 3 j+ 1 2 ,k,l = e j+ 1 2 ,k × [a l− 1 , a2 l+1 ].25.3.2.1 Domain evolutionFor the present model, the evolution of the problem domain should be determined,according to relation (5.4). The level set method is used for thispurpose, since it is very effective for those problems, where the temporalevolution of complex geometries should be determined, including changes intopology. Another advantage of the method, is that it can be applied on afixed Cartesian grid [86], and that there is no need to update the mesh everytime step. Further, it can be adapted for the embedded boundary method,which is used for the construction of irregular active cells near the boundaryof the physical domain.The boundary Γ(t) of the domain Ω s is found as the zero level of thelevel set function ϕ(⃗x, t) : [R 0 , R 1 ] × [Z 0 , Z 1 ] → R, which is approximatedin the nodes of the Cartesian grid Ψ h . Within region Ω s , the function ϕ isnegative, and in Ω b ϕ > 0.The equation for ϕ is given in Section 5.3.3. Next, it is considered howthe grid of the active cells is reconstructed, if the values of the level setfunction are known.5.3.2.2 Treatment and reconstruction of active cellsThe current geometry of the physical domain Ω s is determined by the valuesof the level set function at the grid vertices. Assume that the level setfunction is approximated by the values ϕ j+12 ,k+ 1 at vertices v2j+12 ,k+ 1 ∈ Ψ h2at some time moment t n . A vertex v j+12 ,k+ 1 ∈ Ψ h is referred to as an inner,2outer or boundary vertex, if the level set function is negative, positive or iszero at this vertex, respectively.A rectangular cell W j,k ∈ W is assumed to be an inner cell, if all itsvertices are either inner or boundary vertices. A cell W j,k ∈ W is an outercell, if all its vertices are either outer or boundary vertices, and at least oneof them is an outer vertex. If a cell contains at least one outer vertex andat least one inner vertex, then it is a cut cell (Figure 5.1).New boundary vertices are positioned on the edges of the cut cells, acrosswhich the level set function ϕ changes its sign. It is assumed, that the variableϕ can be approximated by a linear function along a cell edge. From thisassumption, the intersection of the cell edge with the zero level of function

138 Chapter 5. Numerical algorithmphysicaldomain Ω sinnercut cellscellsouter cellsFigure 5.1: Sketch of the grid of the control volumes. Cell types are determinedby the signs of the level set function ϕ at the cell vertices. At thevertices, denoted by the white circles with ’-’ sign, by the black circles andby the white circles, the function ϕ is negative, positive and equal to zero,respectively. Inner cells are the black squares within domain Ω s , cut cellsare the light gray squares, and outer cells are the white squares.ϕ can be found. The new boundary vertices are defined at the intersectionpoints. For the cut cell, shown in Figure 5.2, the level set function changesits sign across each cell edge. Hence four boundary vertices are added onthe edges of the cell.Each cut cell W j,k ∈ W corresponds to the active cell V j,k ∈ V , whichis defined as the intersection of cell W j,k ∈ W with region Ω s . The activecell V j,k is constructed as follows. Let us consider the new boundary points,defined at the intersection of the zero level of the function ϕ and the edgesof the cut cell W j,k (vertices E, F , G, H in Figure 5.2), and the verticesof the cut cell W j,k , at which the function ϕ is nonpositive (vertices Band D). These vertices lie on the boundary of the active cell V j,k . Theboundary of the cut cell W j,k is followed, for example, in a counterclockwisedirection, and the considered points are connected by line segments in thecorresponding order (vertices E, B, G, H, D, F in Figure 5.2). The activecell V j,k is defined as the area enclosed by the aforementioned line segments.The vertices, where the level set function is positive, are located outside thedomain Ω s . Hence they will lie outside the active cell V j,k .The computational domain, which represents the physical domain Ω s , isdetermined by the set of all active cells. Several limitations are introducedduring the reconstruction of active cells. For example, 1) new boundaryvertices are found by means of linear interpolation of the level set functionalong cell edges, and 2) line segments are used to connect adjacent vertices ofthe active cell boundary. Thereby, the initial uniform Cartesian grid shouldbe fine enough, so that the geometry of the domain Ω s can be represented

5.3. Numerical method 139DCDHCcutcellFactivecellGzero level offunction ϕABAEBFigure 5.2: Construction of the active cell EBGHDF , corresponding to therectangular cut cell ABCD. At the vertices, denoted by the white circleswith ’-’ sign and by the black circles, the level set function ϕ is negativeand positive, respectively. New boundary vertices E, F , G, H are theapproximated intersection points of the cell edges with the zero level offunction ϕ. The boundary of the cut cell ABCD is followed, for example,in a counterclockwise direction, and the new boundary E, F , G, H and thevertices B and D, where function ϕ is nonpositive, are connected by linesegments in the corresponding order. The active cell EBGHDF is definedas the area enclosed by the aforementioned line segments.correctly. It can be noted, for example, that any active cell, defined by us,will always be a simply-connected region. If the projection of the domain Ω son some rectangular cell is a multiply connected region, then the Cartesiangrid should be refined, in order to provide a reasonable approximation ofthe problem domain by the set of active cells.The vertices of active cells are either boundary or inner vertices. Thenthe edges of active cells are classified as follows. The edges, whose end-pointsare the boundary vertices, are assumed to coincide with the boundary of thephysical domain Ω s . Such edges are referred to as boundary edges. The restof the edges has at least one inner vertex and lies within the computationaldomain. Hence these edges are called inner edges.It should be noted here, that all inner edges are aligned with the coordinateaxes r, z, and lie on the lines of the Cartesian grid. The cell edges,which are defined if the Cartesian control volumes are cut by the boundaryof the physical domain Ω s , and which may not lie on the lines of the Cartesiangrid, will always correspond to the boundary of region Ω s . These edgesare considered as boundary edges.5.3.3 Level set equationThe evolution of the level set function ϕ is determined by∂ϕ∂t + v e|∇ϕ| = 0, (5.15)where v e is equal to the normal velocity v n , defined in equation (5.4), on theboundary Γ(t), i.e.v e = v n , for ⃗ X ∈ Γ(t). (5.16)

140 Chapter 5. Numerical algorithmOut of the boundary Γ(t), the value of the velocity v e is obtained througha continuous extension. The velocity v e is extrapolated in such a way, that∇v e · ∇ϕ = 0, ⃗x ∈ [R 0 , R 1 ] × [Z 0 , Z 1 ]. (5.17)In this case, the surface, which represents the distribution of the level setfunction, is not distorted with time, and |∇ϕ(t)| = 1 for t ≥ 0. This fact isimportant for the accuracy of the numerical solution of equation (5.15) [86].In the current simulations the field of velocity v e is constructed as follows.First, the values of the velocity v e are found at the boundary vertices.According to equation (5.4), these values depend on the density of matureosteogenic cells c m , which is found from the solution of system (5.1)–(5.3).The governing system (5.1)–(5.3) is solved with use of the finite volumemethod. Hence the internal solution for c m is obtained in terms of the approximatedaverage values Cmj,k within control volumes V j,k , (j, k) ∈ I a . Cellaverages Cmj,k can be considered, as the approximated (with second order accuracy)values of function c m in the centers of the control volumes V j,k . Thevalue c b m of the cell density at the boundary vertex is approximated by theaverage cell densities in the control volumes, adjacent to the current vertex:c b m =∑m ,(j,k)∈ ⃗ I adjw j,k C j,kwhere I ⃗ adj are the indexes of the active cells, adjacent to the consideredboundary vertex. An inverse distance weighting method is used for theinterpolation. Weights w j,k depend on the distances d j,k from the centers ofthe adjacent cells to the current boundary vertex in the following way:w j,k =1d j,k∑(l,m)∈ ⃗ I adj1d l,m.Given the values of the velocity v e at the boundary vertices, the boundaryvalue problem (5.16), (5.17) is solved. The Eikonal equation (5.17) is solvedwith use of the fast marching method [86].5.3.4 PDE discretization in spaceThe governing equations (5.1)–(5.3) are discretized in space by means of thefinite volume method. After the discretization, a system of ODE’s is derivedfor the vectors of the average values in the control volumes of the unknownfunctions c i , c m and g.The considered physical domain corresponds to the approximated geometryof the soft tissue region within the peri-implant interface, whichis defined in the 2D axisymmetric coordinates. Consequently, the operator’∇ s · (·)’, which appears in the governing equations (5.1)–(5.3), is the

5.3. Numerical method 141divergence operator in the 2D axisymmetric coordinates. In Section 5.3.2the construction of the grid of the active control volumes V j,k ∈ V is describedwithin the 2D axisymmetric physical domain. For convenience, inthe presentation of the discretization procedure, the current grid of the finitevolumes will be considered, as it were defined in the 2D Cartesian coordinates.For example, the calculation of the volumes of the computationalcells and the lengths of the cell edges becomes much simpler in Cartesiancoordinates. The axial symmetry of the bone-implant interface is taken intoaccount through the changes, introduced in the governing equations. Theaxisymmetric spatial operators are expressed through the Cartesian operatorsplus some additional terms. Therefore, two different operator formsof the governing equations will be considered: one for the axisymmetric coordinatesystem and one for the Cartesian coordinate system, so that thesolution, obtained in the 2D Cartesian coordinates satisfies the equations,which are defined for the 2D axisymmetric coordinates.In the 2D axisymmetric coordinates the divergence operator ’∇ s · (·)’ isgiven by∇ s · ⃗u = u r,r + u z,z + u rr ,and it is equal to the divergence operator in the 2D Cartesian coordinates(r, z), plus term urr. Therefore, the governing equations (5.1)-(5.3), whichwere defined initially in the 2D axisymmetric coordinates, will have thefollowing form in the 2D Cartesian coordinates:∂c i∂t = −∇C s · F c (c i , c tot , g) − ∂ ∂a (u b(g)c i )+ A c (g)c i (1 − c tot ) − F c r (c i , c tot , g), (5.18)r∂c m∂t= −∇ C s · F c (c m , c tot , g) + u b (g)c i (⃗x, 1, t)+ A c (g)c m (1 − c tot ) − F c r (c m , c tot , g), (5.19)r(∂g∂t = ∇C s · (D g ∇ s g) + E c (g) c m +∫ 10)γ(a)c i da − d g g + D g ∂gr ∂r , (5.20)where the operator ’∇ C s · (·)’ is the divergence operator, defined in the 2DCartesian coordinates. The gradient operator ∇ s (·) is the same for the2D axisymmetric and 2D Cartesian coordinates. Therefore, the gradientnotation ∇ s (·) is kept for the both coordinate systems. The cell flux isdefined as:[ ]Frc (c j , c tot , g)F c (c j , c tot , g) =Fc z = −D c ∇ s c j + χ(g, c tot ) c j ∇ s g, (5.21)(c j , c tot , g)

142 Chapter 5. Numerical algorithmwhere j = i, m.The average values of variables c i , c m and g are defined as follows:¯c j,k,li=¯c j,km =ḡ j,k =1µ(V 3j,k,l∫V ) j,k,l31µ(V j,k )1µ(V j,k )∫c i dv, (5.22)c m dv, (5.23)V∫j,kgdv, (5.24)V j,kwhere (j, k) ∈ I a , l = 1, 2, .., N a . In formulas (5.22)–(5.24) and further inthe text, the integration is performed over the cells Vj,k,l 3 and V j,k, consideredin the 3D Cartesian space (r, z, a) and in the 2D Cartesian space (r, z),respectively.Let us denote by u some abstract unknown function, defined in spaceand in time. The time derivative of the average value ū of function u insome volume V , which evolves in time, is determined by equation:dūdt = d ∫V u dvdt µ(V )= 1 ( ∫dV u dv − ū dµ(V ) ). (5.25)µ(V ) dt dtThe Leibniz integral rule is given by:∫ ∫du dv =dt VV∂u∂t∫∂Vdv + u v n ds. (5.26)∫The volumetric measure µ(V ) can be determined from the equation µ(V ) =vdv. Substitution of this relation and of formula (5.26) into equation (5.25)yields:dūdt = 1 (∫)∂uµ(V ) V ∂t∫∂Vdv + (u − ū)v n ds . (5.27)Equation (5.27) is used to derive the equations for the evolution of the averagesof the cell densities and of the growth factor concentration. Thevariables c i , c m and g replace u into equation (5.27), and volume V is replacedwith volumes Vj,k,l 3 and V j,k, (j, k) ∈ I a , l = 1, N a . The terms for c i ,c m and g, corresponding to the first term in the brackets in equation (5.27),are replaced with the relations, which are obtained from integration of theright-hand sides of PDE’s (5.18)-(5.20) over cells Vj,k,l 3 and V j,k. Finally, the

5.3. Numerical method 143following equations are obtained:d¯c j,k,lidt=( ∫1µ(Vj,k,l 3 ) − F c (c i , c tot , g) · ⃗n ds 3∂Vj,k,l∫3I(− u b (g) c i ( ξ, ⃗ a l+1 , t) − c i ( ⃗ )ξ, a2l−1 , t) dv 22V j,k∫+ A c (g)c i (1 − c tot ) − F c r (c i , c tot , g)dv 3Vj,k,l3r∫)+ (c i − ¯c j,k,l∂Vj,k,l3B i)v n − h i ds 3(5.28)d¯c j,kmdt( ∫1= − F c (c m , c tot , g) · ⃗n ds 2µ(V j,k ) ∂Vj,kI∫+ u b (g)c i ( ξ, ⃗ 1, t) dv 2V j,k∫+ A c (g)c m (1 − c tot ) − F c r (c m , c tot , g)dv 2V j,kr∫)+ (c m − ¯c j,km )v n − h m ds 2∂Vj,kB(5.29)dḡ j,kdt=( ∫1D g ∇ s g · ⃗n ds 2µ(V j,k ) ∂Vj,kI∫ ( ∫ 1+ E c (g) c m +V j,k0)γ(a)c i da − d g g + D gr∂g∂r dv 2∫)+ (g − ḡ j,k )v n − h g ds 2∂Vj,kB(5.30)The inner and boundary edges of cells V j,k are denoted as ∂Vj,k I and ∂VBj,k ,respectively. The corresponding faces of the 3D cells V j,k,l are defined as∂Vj,k,l 3J = ∂V j,k J × [a l− 1 , a2 l+1 ], J = I, B. (5.31)2Control volumes, which are situated in the middle of the computational domainΩ s , do not have boundary edges. For these cells, the last terms in theRHS’s of equations (5.28)–(5.30) are equal to zero. The vector ⃗n representsthe outward unit normal at the edges (faces) of the control volumes. Thevariables dv 3 , ds 3 , dv 2 and ds 2 , respectively, represent the differential elementsof volumes V 3j,k,l, of surfaces ∂V3Ij,k,land ∂V3Bj,k,l , of surfaces V j,k, and

144 Chapter 5. Numerical algorithmof line segments ∂Vj,k I and ∂VBj,k . Further, v n is the normal velocity of theboundary edges or faces, which is defined in equation (5.4).Further, the system of ODE’s with respect to the average values of thecell densities and the growth factor concentration is derived, which approximatesthe exact equations (5.28)-(5.30). Let us denote the approximateaverage values of the unknown densities and concentration by the respectivecapital letters: C j,k,li, Cm j,k and G j,k . The vectors of the unknownaverages are denoted as C ⃗ i = {C ⃗ i 3i ; ⃗i 3 ∈ Ia 3 }, Cm ⃗ = {C ⃗ im ;⃗i ∈ I a } and⃗G = {G ⃗i ;⃗i ∈ I a }. The system of the approximated ODE’s is written in thefollowing general form:dC ⃗ i 3idt= H M Ii( ⃗ C i , ⃗ C m , ⃗ G,⃗i 3 ) + H D i ( ⃗ C i , ⃗ G,⃗i 3 )+ H R i ( ⃗ C i , ⃗ C m , ⃗ G,⃗i 3 ) + H M Bi( ⃗ C i , ⃗ C m , ⃗ G,⃗i 3 ), (5.32)dC ⃗ imdt= H M Im ( C ⃗ i , C ⃗ m , G,⃗i) ⃗ + Hm( D C ⃗ i , G,⃗i) ⃗+ H R m( ⃗ C i , ⃗ C m , ⃗ G,⃗i) + H M Bm ( ⃗ C i , ⃗ C m , ⃗ G,⃗i), (5.33)dG ⃗ idt= H M Ig ( C ⃗ i , C ⃗ m , G,⃗i) ⃗ + Hg R ( C ⃗ i , C ⃗ m , G,⃗i) ⃗ + H M Bg ( C ⃗ i , C ⃗ m , G,⃗i). ⃗ (5.34)The terms in the right-hand sides of equations (5.32)-(5.34) are the approximationsof the respective terms in the right-hand sides of equations(5.28)-(5.30).As it was mentioned before, the current numerical approach is based onthe method of lines (MOL). The main feature of this method, which is oftenreferred to as the ”semi-discrete scheme“, is that the spatial discretization isseparated from the discretization in time. This allows to apply the principleof superposition to the discretization of the reaction, diffusion and advectionterms in the governing equations. In other words, each term can be approximatedindependently of the others. Therefore, the complicated system ofequations can be decomposed into a set of problems, that are easier to solve.The semi-discrete schemes for the scalar time-dependent advection-diffusion-reactionequations are described in a very clear and detailed way in thebook of Hundsdorfer and Verwer [48]. The authors put stress on equationswith linear flux functions, and nonlinear conservation laws are not studiedin detail in this work. The hyperbolic systems of nonlinear equations areconsidered thoroughly in the work of LeVeque [59], where the author describesthe fully discrete Direct Space-Time schemes for various types of thehyperbolic partial differential equations. The current numerical algorithm isconstructed in the framework of the principles and ideas, presented in thesetwo works.

5.3. Numerical method 1455.3.5 Advection-diffusion discretizationLet us consider the advection and diffusion terms in equations (5.18)–(5.20).The corresponding terms in the averaged equations (5.28)–(5.30) are givenby the integrals of the fluxes of the unknown variables c i , c m and g overthe inner faces ∂Vj,k,l 3I and V j,k × {a l+1 } and the inner edges ∂Vj,k I . The2considered integrals are approximated by the functions H M Ii, Hi D, HM Im andH M Ig from equations (5.32)–(5.34).The integrals are approximated by the value of the integrand in the centersof the corresponding face or edge times its area or length, respectively.Therefore, the approximate values of the unknown functions c i , c m , g andof their normal derivatives should be derived at the centers of interfaces ofcontrol volumes from the averages of the cell densities C ⃗ i , Cm ⃗ and of thegrowth factor concentration G. ⃗5.3.5.1 Linear flux due to cell differentiationThe general procedure of the discretization of hyperbolic equations in spacewith semi-discrete schemes involves two stages. First, piece-wise polynomialapproximations of the unknown functions are reconstructed from their averagevalues. The reconstructed piece-wise polynomial functions usually havejumps at the interfaces of the computational cells. This means, that thereare two states (values) of each reconstructed function at each inner interfaceof the control volumes. The reconstruction technique should be total variationdiminishing (TVD), in order to provide the numerical solution withoutspurious oscillations, which can also lead to negative values of the unknownvariables.At the second stage, the reconstructed unknown functions are used todetermine the values of the fluxes at the interfaces of the control volumes.In order to obtain a stable numerical approximation, the Riemann problemhas to be solved exactly, or with use of some approximate solver in the caseof large complex systems of equations.In the discretization of the multidimensional hyperbolic equations onregular Cartesian grids, the dimensions of the physical space can be split.That means, that the techniques, used to solve one-dimensional problems,can be applied for the spatial discretization in each individual coordinatedirection. Such a splitting can be applied due to the possibility of thesuperposition of spatial discretizations, when the method of lines is used[48].The current finite volume grid contains irregular active cells near theboundary of the physical domain Ω s . This irregularity requires a specialsolution reconstruction technique within the irregular cells and cells adjacentto them. This technique will be introduced in Section 5.3.5.2.Let us first consider the flux of immature cells due to differentiation. This

146 Chapter 5. Numerical algorithmflux should be approximated on the faces ( X, ⃗ a) ∈ V j,k × {a l−1 }, ( X, ⃗ a) ∈2V j,k × {a l+1 } of cells Vj,k,l 3 , (j, k, l) ∈ I a 3 . The flux vector is parallel to the2a-axis. The finite volume grid is uniform along the a coordinate axis. Hencethe method of the reconstruction of the solution on a 1D uniform grid canbe applied. The integral over the face of the control volume is approximatedby the values of the integrated functions in the center of the face. Hence itis not necessary to reconstruct the numerical solution for density c i withinthe whole cell. It is sufficient to approximate its value in the center of theconsidered cell interfaces.In order to justify the 1D reconstruction, applied in the considered situation,function ˜c j,ki(a, t) is defined, which depends only on a coordinate andtime, and which is equal to the average value of the function c i over thetransverse sections {( ξ, ⃗ a), s.t. ξ ⃗ ∈ Vj,k , a = const} of the cells Vj,k,l 3 , i.e.˜c j,ki(a, t) =1µ(V j,k )∫∫V j,kc i (r, z, a, t)dr dz.Sections a = a l+1 and a = a2l−1 coincide with the faces of cell Vj,k,l 3 . The2average values can be considered as the second order approximations of thefunction values in the face centers. Hence the “one-dimensional” function˜c j,ki(a, t) can be reconstructed, in order to determine the approximate statesof the immature cell density at the centers of the considered cell faces.Note, that∫1 al+ 12˜c j,kida =h a a l− 12∫1 al+∫∫12c i dr dz da = ¯c j,k,li.h a µ(V j,k ) a l− 1 V j,k2That is, the average value of the function ˜c j,kiover the interval [a l−1equal to the cell average of function c i over cell Vj,k,l 3polynomial approximation of ˜c j,kithe 1D uniform grid [a l−12, a l+12, a2 l+1 ] is2. Hence the piece-wiseis reconstructed from its average values on], l = 1, .., N a , given by C ⃗ i .The value of the function u b (g) on the midpoint of the considered cellfaces is approximated by value u b (G j,k ).The Riemann solution is straightforward in the considered case, sincethe flux is linear and it only depends on the unknown c i . The value ofthe coefficient u b (g) does not vary along a-axis, and it can be considered asconstant, when the flux is evaluated on faces V j,k × {a l+1 }. It is assumed2in the current model, that the differentiation rate u b is nonnegative (seeTables 4.1, 4.2). Hence the left state of ˜c j,kiis chosen at each interface, i.e.the value of the reconstructed function ˜c j,ki(a, t) for a → a l+1 − 0.2Therefore, the terms, which represent the flux of immature cells due todifferentiation, are approximated as follows:Hi D ( C ⃗ i , G, ⃗ (j, k, l)) = − 1 ()u b (G j,k ) C j,k,l+ 1 2i− C j,k,l− 1 2i, (5.35)h a

5.3. Numerical method 147where the states C j,k,l+ 1 2i, l = 2, ..., N a are determined asC j,k,l+ 1 2i= C j,k.li+ ψ(C j,k,liC j,k,l+1i− C j,k,l−1i− C j,k,li) (C j,k,l+1i)− C j,k,li, (5.36)for l = 1, .., N a . Function ψ(θ) is a ”limiter function“. It is responsible forthe TVD property of the reconstruction method. The choice of the limiterfunction determines the order of the reconstruction of the unknown functionfrom its average values. If ψ ≡ 0, then the order of approximation of C j,k,l+ 1 2iis one. By choosing the Koren limiter for a uniform grid,( ( 1ψ(θ) = max 0, min3 + 1 ))6 θ, θ, 1 , (5.37)a higher order approximation is obtained [48]. In fact, the Koren limiterprovides a third order of accuracy in the regions, where the function issmooth and where it does not have any extremes. Otherwise, a first orderaccuracy is achieved. However, since the surface integrals are evaluated atthe center points, and the values at the surface centers are approximatedwith the average values, the order of accuracy cannot be larger than two.Boundary condition (5.12) is used to define the flux at faces V j,k × {0}.ThenH D i ( ⃗ C i , ⃗ C m , ⃗ G, (j, k, 1)) = − 1 h au b (G j,k )C j,k, 3 2i.In order, to find the values C j,k, 3 2iand C j,k,Na+ 1 2i, the ’ghost cells’ are defined,such thatC j,k,0i= C j,k,1i, C j,k,Na+1i= C j,k,Nai, (j, k) ∈ I a .Constant extrapolation at the boundaries a = 0 and a = 1 is chosen, sinceit provides nonnegative solutions. Furthermore, it leads to the first orderaccuracy in the approximations of the states C j,k, 3 2iand C j,k,Na+ 1 2i.The function H D m in equation (5.29) approximates the inflow of maturecells due to differentiation of immature cells. The cell density conservationis maintained, if this inflow is equal to the flux of the density c i throughinterface ( ⃗ X, a) ∈ V j,k × {a Na+ 1 2}. That is,H D m( ⃗ C i , ⃗ C m , ⃗ G, (j, k)) = u b (G j,k )C j,k,Na+ 1 2i, (5.38)where the average cell density C j,k,Na+ 1 2iis obtained from equations (5.36)–(5.37).

148 Chapter 5. Numerical algorithm5.3.5.2 Fluxes due to cell migrationIn order to approximate the terms, which correspond to the fluxes of the celldensities due to migration, the values of the immature and mature cell densities,the values of the growth factor concentration and the normal derivativesof the cell densities and of the growth factor concentration have tobe determined at the centers of the inner edges and faces of the controlvolumes. The method, used to determine the values of the growth factorconcentration and the normal derivatives of the model variables is defined inSection 5.3.5.3. The approach, used to find the left and right states (values)of the cell densities at the centers of the inner edges and faces, is describedin this section.Consider first the density of mature osteogenic cells, defined within thetwo dimensional physical domain. In analogy to Section 5.3.5.1, function˜c k m(r, t) can be defined as the average value of the cell density c m over thesections S(r, k) = {(r, z), s.t. (r, z) ∈ V j,k , (j, k) ∈ I a }, where r and kare considered as fixed parameters, which characterize each section S(r, k).Therefore,∫˜c k 1m(r, t) =c m (r, z, t) dz. (5.39)µ(S(r, k)) S(r,k)If cell V j,k is a regular rectangular cell, then the average of ˜c k m is given by∫1 rj+ 12˜c k m dr = 1 ∫ rj+ 12 1h r r hj− 1 r r µ(S(r, k))2j− 12∫1=h r µ(S(r, k))∫V j,kc m dv =S(r,k)1µ(V j,k )c m dz dr∫V j,kc m dv = ¯c j,km . (5.40)This means, that the averages of the functions ˜c m and c m on regular rectangularcells are equal.Remark 5.2. Note, that equation (5.40) is valid, if and only if µ(S(r, k)) =const for r ∈ [r j−1 , r2 j+1 ].2Let us consider an inner edge e j+1,k.Assume that this edge is located far2away from the boundary of the physical domain, and that there exist activeinner (i.e. regular rectangular) cells V j−1,k , V j,k , V j,k+1 . Then the function˜c m has average values ¯c j−1,km , ¯c j,km , ¯c j+1,km on uniform intervals [r n−1 , r2 n+1 ],2n ∈ {j − 1, j, j + 1}, and the conventional one-dimensional TVD reconstructioncan be applied, so that the left state at the center of the edge e j+1,kis2given by:(C j+ 1 2 ,km (L) = C Cj,kmj,km + ψC j+1,km− C j−1,km− C j,km) (C j+1,km)− Cmj,k , (5.41)

5.3. Numerical method 149where the function ψ is a Koren limiter again, defined in equation (5.37). Ifcell V j+2,k is also a regular inner cell, then the right state is determined as:(C j+ 1 2 ,km (R) = C Cj+1,kmj+2,km + ψC j+1,km− C j+1,km− C j,km) (C j,km)− Cmj+1,k . (5.42)The situation is more complicated, if some of the cells in the three-cellstencil, which is used to approximate the left state or the right state atthe edge center, are irregular or do not exist at all (Figure 5.3). Consider,for example, the situation shown in Figure 5.3a. Assume that there existsz k+ 12V j−1,kV j,kV j+1,kz k− 12r j− 32r j− 12r j+ 12 r j+ 3 2(a)e j+ 12 ,kz k+ 12z k− 12e j+ 12 ,kV j,k V j+1,kr j− 12r j+ 12 r j+ 3 2(b)Figure 5.3: Situations, when the conventional one-dimensional reconstructionfor the uniform grid cannot be applied to approximate the solution atthe edge e j+1,k.Active cells are plotted in gray color.2an irregular active cell V j−1,k , which has a common inner edge e j−1,kwith2the cell V j,k . On irregular cells, the length of the section µ(S(r, k)) maynot be constant, hence, according to Remark 5.2, equality (5.40) does nothold in general. The value of ˜c m at point r j+1 is found with an approach,2which is similar to the 1D reconstruction on a cell-centered nonuniform grid,defined in Hundsdorfer and Verwer [48]. First ˜c m is approximated with aquadratic polynomial, and then a limiter is applied, in order to get a TVDreconstruction, which prevent spurious oscillations in the numerical solution.

150 Chapter 5. Numerical algorithmNote, thatC j,km≈ ¯c j,km ==∫1 rj+ 12µ(V j,k )r j− 1∫S(r,k)2∫1 rj+ 12=µ(V j,k )∫1 rj+ 12˜c k mµ(V j,k ) r j− 12r j− 12∫c m dz drS(r,k)µ(S(r, k))˜c k m drdz dr =∫1˜c k m dv. (5.43)µ(V j,k ) V j,kLet a 0 , a 1 , a 2 be the coefficients of the quadratic polynomial, which is usedto approximate ˜c k m:˜c k m ≈ a 0 + a 1 r + a 2 r 2 .From integration of this polynomial over the cells V n,k , n ∈ {j − 1, j, j + 1}and from equation (5.43), three linear equations are derived, which can beused to determine the coefficients a i , i ∈ {0, 1, 2}:a 0 + a 1 r n,kc+ a 2 I n,kzz= C n,km , n ∈ {j − 1, j, j + 1}, (5.44)where rcn,k is the r coordinate of the center of cell V n,k , and Izzn,k is thesecond moment of area of cell V n,k about z-axis, i.e. Izz n,k = 1 ∫µ(V n,k ) V n,kr 2 dv.The determinant of the matrix of system (5.44) is not equal to zero (seeB.1), hence coefficients a i , can always be uniquely expressed in terms of thegeometric characteristics rcn,k , Izzn,k of the control volumes, and approximateaverage values Cm n,k . Evaluating the quadratic polynomial at point r j+1 for2the values of coefficients a 0 , a 1 and a 2 , obtained from the solution of thelinear system (5.44), the following expression is obtained:˜c k m(r j+12where+ ψ r 0, t) ≈ Cmj,k(Cmj,kC j+1,km− C j−1,km− C j,kmψ r 0(θ, r e , j 1 , j 2 , j 3 , k) =, r j+1 , j − 1, j, j + 1, k2((r j 1,kc − r j 2,kc )r e + (I j 2,k+I j 1,kzz r j 2,kc − I j 2,kzz r j 1,kc + θ) (C j+1,kmzz − I j 1,kzz((r j 3,k)r 2 ec − r j 2,kc )r e))+(I j 2,kzz − I j 3,kzz )re 2 + I j 3,kzz r j 2,kc − I j 2,kzz r j 3,kc(/ (I j 2,kzz − I j 1,kzz )(r j 3,kc − r j 2,kc ) + (I j 3,kzz − I j 2,kzz )(r j 1,kc − r j 2,kc ))− Cmj,k , (5.45)). (5.46)The quadratic approximation, given by formula (5.45), provides a third orderof accuracy for smooth solutions. Since a TVD approximation is needed,

5.3. Numerical method 151the function ψ r 0 has to be limited in the regions, where ˜c m changes rapidly.A limiter function is given by:ψ r irr(θ, ·, ·, ·, ·, ·) = max(0, min(ψ r 0(θ, ·, ·, ·, ·, ·), θ, 1)),so that 0 < ψirr r < 1, and 0 < 1 θ ψr irr (θ, ·, ·, ·, ·, ·) < 1. This means, that thelimiter ψirr r provides a TVD reconstruction of the solution [48]. Note, thatthe limiter function ψirr r , which is defined for irregular cells, is equivalent tothe Koren limiter function (5.37), if the geometric characteristics rcj,k , Izzj,kof uniform regular rectangular cells are substituted into expression (5.46).Therefore, for the present case, the left state of the mature cell density c mat the edge e j+1,kis approximated as2C j+ 1 2 ,km(L)+ ψ r irr= Cmj,k(Cmj,kC j+1,km− C j−1,km− C j,km, r j+1 , j − 1, j, j + 1, k2By analogy, the right state is determined as) (C j+1,km)− Cmj,k . (5.47)C j+ 1 2 ,km (R) = Cj+1,k m+ ψ r irr(C j+2,kmC j+1,km− C j+1,km− C j,km, r j+1 , j + 2, j + 1, j, k2) (C j,km)− Cmj+1,k . (5.48)If the inner edge e j+1,kis very close to the boundary, then, for example,the active cell V j−1,k and/or the active cell V j+2,k may not exist (Fig-2ure 5.3b). In this case, the first order piece-wise constant reconstruction ofthe solution for the density c m is used and its left and/or right states aredefined as:C j+ 1 2 ,km(L)= Cm j,k , C j+ 1 2 ,km(R)= Cm j+1,k . (5.49)This is equivalent to the constant extrapolation for the ’ghost’ cells, outlinedin Section 5.3.5.1.Each inner edge e j+12 ,k corresponds to n a inner faces e 3 j+ 1 ,k,l, l = 1, ..., n a.2The derivation of the formulas for the left and right values of the densityc i at the centers of these inner faces is very similar to the derivation of them(L) , Cj,k m(R). The only difference is that integrationto a l+1 is performed additionally. Due to a2homogeneity of the mesh in the a-direction, this integration does not leadto any modifications in the derived formulas. Six different situations of themutual location of regular and irregular active cells and inner edge e j,k areformulas for the states C j,kin the a-direction from a l−12represented by the formulas (5.41), (5.42), (5.47), (5.48) and (5.49). Foreach situation, the following equations for the values of the immature cell

152 Chapter 5. Numerical algorithmdensity in the centers of faces e 3 j+ 1 2 ,k,l, l = 1, ..., n a are given, respectively,by:C j+ 1 2 ,k,li (L)C j+ 1 2 ,k,li (R)= C j,k,li= C j+1,k,li+ ψ+ ψ(C j,k,liC j+1,k,li(C j+2,k,liC j+1,k,li− C j−1,k,li− C j,k,li− C j+1,k,li− C j,k,li) (C j+1,k,li) (C j,k,li)− C j,k,li, (5.50))− C j+1,k,li, (5.51)C j+ 1 2 ,k,li(L)+ ψ r irr= C j,k,li(C j,k,liC j+1,k,li− C j−1,k,li− C j,k,li, r j+1 , j − 1, j, j + 1, k2) (C j+1,k,li)− C j,k,li,(5.52)C j+ 1 2 ,k,li (R)+ ψ r irr= C j+1,k,li(C j+2,k,liC j+1,k,li− C j+1,k,li− C j,k,liC j+ 1 2 ,k,li(L), r j+1 , j + 2, j + 1, j, k2) (C j,k,li)− C j+1,k,li,(5.53)= C j,k,li, C j+ 1 2 ,k,li(R)= C j+1,k,li. (5.54)The values of c m and c i at the horizontal inner cells edges e j,k+1 and2faces e 3 j,k+ 1 2 ,l are determined with use of limiters ψ and ψz irr in the same wayas for the vertical edges and faces. The only difference is that the cells V j,n ,Vj,n,l 3 , n ∈ {k − 1, k, k + 1, k + 2}, are considered instead of the cells V m,k,Vm,k,l 3 , m ∈ {j − 1, j, j + 1, j + 2} in the stencil, used in the computation ofthe states of the variables. Further, the limiter ψirr z is used instead of thelimiter ψirr r , which is defined as follows:ψ z irr(θ, ·, ·, ·, ·, ·) = max(0, min(ψ z 0(θ, ·, ·, ·, ·, ·), θ, 1)),ψ z 0(θ, z e , k 1 , k 2 , k 3 , j) =((z j,k 1c − z j,k 2c )z e + (I j,k 2((z j,k 3rr − I j,k 1rr)z 2 e+I j,k 1rr z j,k 2c − I j,k 2rr z j,k 1c + θ c − z j,k 2c )z e))+(I j,k 2rr − I j,k 3rr )ze 2 + I j,k 3rr z j,k 2c − I j,k 2rr z j,k 3c(/ (I j,k 2rr − I j,k 1rr )(z j,k 3c − z j,k 2c ) + (I j,k 3rr − I j,k 2rr )(z j,k 1c − z j,k 2c )),

5.3. Numerical method 153where zcj,kthe second moment of area of the cell V j,k about the r-axis, i.e.is the z coordinate of the center of the cell V j,k , and I j,krrisIrr j,k =∫1µ(V j,k ) V j,kz 2 dv. The left and right states of the unknown variables are, infact, the lower and upper states, respectively.5.3.5.3 Gradient approximation at cell edgesIn this section, the approach is described to approximate the normal derivativesof the densities of immature and mature cells and of the growth factorconcentration and the value of the growth factor concentration in the centersof the inner edges and faces of the control volumes.First, the expressions for the normal derivative and the value of thegrowth factor concentration g at the centers of the inner edges are derived.Consider a vertical inner edge e j+1,k.The normal derivative is equal to2partial derivative ∂g∂r . Since the edge is inner, the cells V j,k, V j+1,k , whichare adjacent to it, are active cells. If these cells are regular rectangularcells, then their centers and the center of the edge e j+1,khave the same()2z-coordinate, equal to z k−1 + z2 k+1 /2, which is denoted as z j+ 1 2 ,ke . The2edge center lies at the middle of the line segment, which connects the cellcenters. From a Taylor series expansion, it follows thatand∂g∂r (r j+ 1 2g(r j+12, z j+ 1 2 ,ke, z j+ 1 2 ,ke) = g(rj+1,k c) = g(rj+1,k c, z j+1,kcr j+1,kc, z j+1,kc2) − g(rcj,k , zc j,k )− r j,kc) + g(r j,kc, z j,kc )+ O(h 2 r),+ O(h 2 r).Hence the approximate normal derivative ∂g j+ 1 2 ,k∂rand the state G j+ 1 2 ,k ofthe function g at the center of the edge e j+1,kare determined as2j+∂G1 2 ,k = Gj+1,k − G j,k∂r rcj+1,k − rcj,k , (5.55)G j+ 1 2 ,k = Gj+1,k + G j,k. (5.56)2The equation for the evolution of the growth factor concentration is adiffusion-reaction equation. The flux function for the growth factor concentrationonly depends on the normal derivative of the growth factor concentration.Equation (5.56) is not a TVD approximation, in contrast to thereconstruction of the cell densities at the inner edges and faces. However,due to the presence of the purely diffusive flux of the growth factor concentration,the TVD property is not needed for the considered approximation.

154 Chapter 5. Numerical algorithmThe value of the growth factor concentration on the inner edges is consideredas fixed during the solution of the Riemann problem for the fluxes of the celldensities, which is presented in Section 5.3.6. Despite the reconstruction ofthe solution for function g does not contain a TVD property, this will notlead to the appearance of spurious oscillations in the numerical solution.The right-hand side of equation (5.55) approximates the directional derivativealong the vector, which connects the centers of the cells V j,k , V j+1,k .If this vector is parallel to the r-axis, then this directional derivative coincideswith the normal derivative at the inner edge. If the considered vectoris not parallel to the r-axis, then the directional derivative along one morevector, is needed, which is not parallel to the first one, in order to obtainat least a first order of accuracy for the normal derivative at the considerededge. To accomplish this, a third point is defined, on which the value of thefunction is known. The third point is chosen among the centers of the activecells, adjacent to the end-points of the edge e j+1,k, excluding the cells V j,k,2V j+1,k . There can be from four to one such cells. This statement followsfrom the definition of inner edges and active cells in Section 5.3.2.2 (see adetailed proof in B.2). If there are some adjacent active cells besides cellsV j,k , V j+1,k , then such a cell is chosen, that the perimeter of the trianglewith vertices at the centers of the considered cell and cells V j,k , V j+1,k isthe smallest among all adjacent cells. Suppose, that the third point is chosen.The general formula for the average value of the gradient over somearbitrary volume V is defined as∫1∇g dv = 1 ∫g⃗n ds,µ(V ) V µ(V ) Swhere S is the boundary surface of the volume V with the outer unit normal⃗n. This formula is applied for the triangle with the vertices at the centersof the three chosen cells. If the cell averages of the function g within thethree chosen cells are denoted by G 1 , G 2 , G 3 , and the r- and z- coordinatesof the cell centers are denoted by r 1 , r 2 , r 3 , z 1 , z 2 and z 3 , then the normalderivative ∂g∂r (r j+ 1 2, z j+ 1 2 ,ke ) is approximated as:j+∂G1 2 ,k∂r= (G1 + G 2 )(z 2 − z 1 ) + (G 2 + G 3 )(z 3 − z 2 ) + (G 3 + G 1 )(z 1 − z 3 )(r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 . (5.57))

5.3. Numerical method 155The partial derivative with respect to z coordinate is determined as followsj+∂G1 2 ,k∂z= − (G1 + G 2 )(r 2 − r 1 ) + (G 2 + G 3 )(r 3 − r 2 ) + (G 3 + G 1 )(r 1 − r 3 )(r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 .)(5.58)Then the value of the growth factor concentration g in the center of edgee j+1,kis determined from linear interpolation:2G j+ 1 2 ,k = G 1 + ∂G j+ 1 2 ,k (r j+ 1 2 ,ke∂r− r 1 ) + ∂G j+ 1 2 ,k (z j+ 1 2 ,ke − z 1 ), (5.59)∂zwhere (r j+ 1 2 ,ke , z j+ 1 2 ,ke ) are the coordinates of the center of edge e j+1,k. 2If a horizontal edge e j,k+1 is adjacent to regular rectangular cells V j,k ,2V j,k+1 , then the following formulas are used:j,k+∂G1 2∂zG j,k+1 − G j,k=zcj,k+1 − zcj,k , G j,k+ 1 G j,k+1 + G j,k2 = . (5.60)2If at least one of the cells V j,k , V j,k+1 is irregular, then the third adjacentcell is chosen in the way, as it is done for the vertical edges. The partialj,k+ 1 2 ∂G j,k+, 1 2derivatives ∂G∂r ∂zare approximated with the right-hand sidesof equations (5.57), (5.58), in which variables G 1 , G 2 , G 3 , r 1 , r 2 , r 3 , z 1 ,z 2 and z 3 are defined in the same way. The value of g at the edge center(r j,k+ 1 2e , z j,k+ 1 2e ) is determined asG j,k+ 1 2= G 1 + ∂G j,k+ 1 2 j,k+(r 1 2e∂r− r 1 ) + ∂G j,k+ 1 2 j,k+(z 1 2e − z 1 ), (5.61)∂zThe derivation of the normal derivatives of the immature and maturecell densities is analogous to the approach for the derivatives of the growthfactor concentration. If a horizontal edge e j,k+1 and faces e 32j,k+ 1 2 ,l or avertical edge e j+12 ,k and corresponding faces e3 j+ 1 ,k,l, l = 1, 2, ..., N a, are2adjacent to regular rectangular cells, then the following formulas are used,respectively:j,k+∂C 1 i 2 ,l = Cj,k+1,l i∂zz j,k+1c− C j,k,li− z j,kc,j,k+∂C 1 m 2∂zCmj,k+1=zcj,k+1− C j,ki− zcj,k, (5.62)j+∂C 1 i 2 ,k,l = Cj+1,k,l i∂rr j+1,kc− C j,k,li− r j,kc,j+∂C 1 m 2 ,k = Cj+1,k m∂rr j+1,kc− C j,ki− r j,kc, (5.63)

156 Chapter 5. Numerical algorithmIf at least one of the cells adjacent to the considered edge is irregular,then the third adjacent cell is chosen in the way, as it was describedabove. If the cell averages of the functions c i and c m within the chosencells are denoted by C 1,li, C 2,li, C 3,li, and Cm, 1 Cm, 2 Cm, 3 respectively, andthe r- and z- coordinates of the cell centers are denoted by r 1 , r 2 , r 3 ,z 1 , z 2 and z 3 , respectively, then the linear approximations of the normalderivatives ∂c i∂r (r j+ 1 2∂c m∂z(r j,k+ 1 2, z j+ 1 2 ,ke , a l ), ∂cm∂r (r j+ 1 2e , z k+1 ) are given by:2, z j+ 1 2 ,ke ), ∂c 1 i∂z (rj,k+ 2e , z k+1 , a l ) and2∂Ĉi∂rj+ 1 2 ,k,l = ( (C 1,li+ C 2,li)(z 2 − z 1 ) + (C 2,li+ C 3,li)(z 3 − z 2 )+ (C 3,li+ C 1,li)(z 1 − z 3 ) ) / ( (r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 ) ) ,(5.64)j+ 1 2∂Ĉm,k = ( (Cm 1 + C 2∂rm)(z 2 − z 1 ) + (Cm 2 + Cm)(z 3 3 − z 2 )+ (Cm 3 + Cm)(z 1 1 − z 3 ) ) / ( (r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 ) ) , (5.65)j,k+ 1 2∂Ĉi,l = − ( (C 1,li+ C 2,li)(r 2 − r 1 ) + (C 2,li+ C 3,li)(r 3 − r 2 )∂z+ (C 3,li+ C 1,li)(r 1 − r 3 ) ) / ( (r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 ) ) ,(5.66)j,k+ 1 2∂Ĉm= − ( (Cm 1 + C 2∂zm)(r 2 − r 1 ) + (Cm 2 + Cm)(r 3 3 − r 2 )+ (Cm 3 + Cm)(r 1 1 − r 3 ) ) / ( (r 2 − r 1 )(z 3 − z 1 ) − (z 2 − z 1 )(r 3 − r 1 ) ) . (5.67)The linear approximations of the normal derivatives on the unstructuredmesh may yield negative values in the numerical solutions for the cell densitiesin the regions, where the cell densities have values close to zero andwhere the gradients of the densities are high. As it is mentioned in Section5.3.1, the current model is very sensitive with respect to negative valuesof the cell densities. Hence the present numerical approach should providenonnegative approximate solutions for the densities. Therefore, the linearapproximations in equations (5.64)–(5.67) should be limited in the followingway:∂C i∂r⎛j+ 1 2 ,k,l = max ⎝min⎛⎝ ∂Ĉi∂rj+ 1 2 ,k,l ,C j+1,k,lir j+1,k,lc− r j,k,lc⎞⎠ , −r j+1,k,lcC j,k,li⎞⎠ ,− rcj,k,l(5.68)

5.3. Numerical method 157∂C i∂z∂C m∂r⎛j+ 1 2 ,k = max ⎝min⎛j,k+ 1 2 ,l = max ⎝minj,k+∂C 1 m 2∂z⎛⎛⎝ ∂Ĉm∂r⎝ ∂Ĉi∂z⎛ ⎛= max ⎝min⎝ ∂Ĉm∂zj+ 1 2 ,k ,j,k+ 1 2 ,l ,j,k+ 1 2,C j+1,kmr j+1,kcz j,k+1,lcC j,k+1,li− r j,kc− z j,k,lcC j,k+1mz j,k+1c− z j,kc⎞⎠ , −⎞⎠ , −⎞⎠ , −r j+1,kcC j,kmz j,k+1,lcz j,k+1c− r j,kcC j,k,liC j,km⎞⎠ ,(5.69)⎞⎠ ,− zcj,k,l(5.70)− z j,kc(5.71)The limiting procedure introduced above is equivalent to the assumption,that the outflow of the cell density caused by random walk and taking placethrough the boundaries of control volumes is bounded from above. If the outflowis bounded by some limit value, an appropriate time step size providingnonnegative numerical solutions can be chosen as described in Section 5.3.9.The upper boundary for the outflow of the cell density per unit length isdetermined in a straightforward way for a uniform rectangular mesh. TheC−Coutflow caused by random walk is equal to D adj c h, where C is the averagedensity over the considered control volume, C adj is the average density inthe adjacent control volume and h is the linear mesh size. Then the outflow⎞⎠ .Cis less than or equal to D c hif the solution is nonnegative. The limitingformulas (5.68) – (5.71) provide the values of the outflow of the cell densitieson the unstructured mesh, which are bounded from above by some limitvalue related to the average cell densities within the considered control vol-Cumes. The limit value is determined in an analogous way as the value D c hfor a uniform mesh. The linear gradient approximation given by formulas(5.64)–(5.67) for the unstructured mesh can yield a nonzero outflow fromthe control volumes with zero average cell densities. In this case, negativevalues of the cell densities can appear in the numerical solution at any timestep. Therefore, the limiting of the linear flux approximation is crucial forthe positivity of the numerical solution.The first stage of the derivation of the fluxes at the inner edges andfaces, which are represented by the terms H M Ii, H M Im and H M Ig in equations(5.32)–(5.34), is outlined in Sections 5.3.5.2 and 5.3.5.3. At this stage, thetwo states of the cell densities, the values of the growth factor concentrationand the normal derivatives of the cell densities and of the growth factorconcentration at the centers of the inner edges and faces are derived. Thesequantities are used for the initialization of the Riemann problem at each

158 Chapter 5. Numerical algorithminner edge or face. As it was mentioned earlier, the Riemann problem issolved at the second stage, in order to provide a stable discretization ofthe advection-diffusion terms. Therefore, the exact Riemann solution forthe considered flux functions is obtained in Section 5.3.6. This Riemannsolution provides the fluxes at the inner edges and faces.5.3.6 Nonlinear system of conservation lawsAt the first stage of the approximation of the fluxes of the cell densities atthe inner edges e ⃗ j and faces e3 ⃗j 3, the left and right states of the cell densitiesC ⃗ jm(L) , C⃗ j 3i(L) , C⃗ jm(R) , C⃗ j 3i(R) , the values of normal derivatives ∂C ⃗j 3 ⃗ji∂x, ∂Cm∂x and∂G⃗j∂x and the values of the growth factor concentration G ⃗j were determinedat the centers of the considered edges and faces. The vectors ⃗j and ⃗j 3 arethe indices of the considered inner edge and of the corresponding inner faces,respectively. The variable x is used to denote the independent coordinate ror z, which corresponds to the normal direction at the considered edge orface. For the horizontal edges and faces, x = z is used, and for the verticaledges and faces, x = r is employed.At the second stage of the approximation, a one-dimensional Riemannproblem is solved. The problem is defined by one-dimensional equations,which are derived from the governing equations (5.18)–(5.19). The solutionof the Riemann problem for the system of hyperbolic PDE’s provides a stablediscretization of the initial governing system. The diffusive parabolic terms∂−D 2 c i ∂ c , −D 2 c m∂x 2 c and the reactive terms from equations (5.18)–(5.19) do∂x 2not contribute to a wave-like behavior of the solution. Such a behavior is themain characteristic of the hyperbolic equations. Therefore, these terms areskipped in the present analysis. The flux of cells is considered only in thenormal direction with respect to the considered inner edges or faces. Thefollowing equations are obtained:∂c i∂t + ∂∂x∂c m∂t(χ(g, c tot ) c i∂g∂x)= 0, (5.72)+ ∂ ()∂gχ(g, c tot ) c m = 0, (5.73)∂x∂xRecall that the total osteogenic cell density is defined as c tot = ∫ 10 c i da+c m . If equation (5.72) is integrated over the interval a ∈ [0, 1], and equation(5.73) is added to it, then the equation for the total cell density is obtained:∂c tot∂t+ ∂∂x(χ(g, c tot ) c tot∂g∂x)= 0. (5.74)Instead of obtaining a Riemann solution for system (5.72), (5.73), whichis defined for the unknowns c i , c m , two systems can be considered. The

5.3. Numerical method 159present systems consist of either equations (5.74), (5.72) or equations (5.74),(5.73), which are defined with respect to the unknown variables c tot , c iand c tot , c m , respectively. The new systems are more convenient to solveand to analyse than the initial system (5.72), (5.73). A solution for theunknown c tot is easily obtained from equation (5.74), which is a nonlinearscalar conservation law similar to Burger’s equation for c tot ∈ [0, 1]. Thesolution procedure for the two new systems (5.74), (5.72) and (5.74), (5.73)is the same. Equation (5.72) becomes identical to equation (5.73), if thevariable c m is substituted in place of the variable c i .Therefore, a Riemann problem is solved for system (5.74), (5.73), withthe initial conditions:c tot ={c L tot, x < 0,c R tot, x > 0,c m ={c L m, x < 0,c R m, x > 0,(5.75)where c L tot, c L m and c R tot, c R m are the left and right states of the variablesat the considered inner edge. The normal derivatives of the growth factorconcentration and the value of the growth factor concentration in equations(5.73), (5.74), are assumed to be constant and equal to their values at theinner edge, which are denoted as ∂g ↓∂x and g ↓ , respectively. If, for example,a vertical inner edge e j+1,kis considered, then2∑N ac L tot = h a C j+ 1 2 ,k,li(L)+ C j+ 1 2 ,km(L) ,l=1∑N ac R tot = h a C j+ 1 2 ,k,li(R)+ C j+ 1 2 ,km(R) ,l=1c L m = C j+ 1 2 ,km(L) , cR m = C j+ 1 2 ,km(R) ,↓∂g∂x= ∂G j+ 1 2 ,k , g ↓ = G j+ 1 2 ,k .∂rThe variables C j+ 1 2 ,k,li(L), C j+ 1 2 ,k,li(R), C j+ 1 2 ,km(L) , 1 Cj+ 2 ,km(R) , ∂G j+ 1 2 ,k∂r, G j+ 1 2 ,k , which areused in the current formulas, are defined in Sections 5.3.5.2, 5.3.5.3. System(5.74), (5.73) can be rewritten in the form:where∂⃗q∂t + ∂ f∂x ⃗ (⃗q) = ⃗0, (5.76)[ ] [ ]q1 ctot⃗q = = ,q 2 c m⎡⎤[ ]f1 (q 1 )χ(g ↓ ↓∂g, q 1 ) q 1⃗f(⃗q) == ⎢∂x⎥f 2 (q 1 , q 2 ) ⎣χ(g ↓ ↓∂g⎦ ., q 1 ) q 2∂x

160 Chapter 5. Numerical algorithmLet us assume first, that c tot ∈ (0, 1]. This assumption is taken into accountfrom here up to Section 5.3.6.1 inclusive. The situation c tot ∈ [0, ∞) isconsidered in Sections 5.3.6.2–5.3.6.4.The Jacobian of the flux function f(⃗q) ⃗ is equal to[ ]χ ↓ (1 − 2q 1 ) 0J =− χ ↓ q 2 χ ↓ ,(1 − q 1 )where χ ↓ =areχ 0 g ↓ ∂g ↓Kch 2 +g↓2 ∂x . Its eigenvalues and the corresponding eigenvectorsλ 1 = χ ↓ (1 − 2q 1 ), λ 2 = χ ↓ (1 − q 1 ), (5.77)[ ] [q1⃗r 1 = , ⃗ 0 r 2 = .q 2 1]The Hugoniot-locus and the integral curve [59] for the waves from thefirst family coincide. These curves are the straight linesq 2q 1= const. (5.78)The Hugoniot locus is the location of all states ⃗q ∗ , which can be connectedto some fixed state ⃗q by a shock wave from the corresponding wave family.It is determined from equation⃗f(⃗q ∗ ) − ⃗ f(⃗q) = s(⃗q ∗ − ⃗q), (5.79)where s is an arbitrary scalar value, denoting the shock wave speed. Fortwo states ⃗q and ⃗q ∗ , which lie on the line (5.78), the following relation holdsf(⃗q ∗ ) − f(⃗q) =⎡⎣ χ↓ (q1(1 ∗ − q1) ∗ ⎤− q 1 (1 − q 1 ))χ ↓ q ⎦2(q ∗q1(1 − q1) ∗ − q 1 (1 − q 1 ))1= χ ↓ (1 − q ∗ 1 − q 1 )Hence the speed of the shock waves from the first family is equal to[ ]q∗1 − q 1q2 ∗ .− q 2s = χ ↓ (1 − q ∗ 1 − q 1 ) . (5.80)The integral curve is the location of all states, which can be connected witha given state by a rarefaction wave. It is defined as a curve, which is tangentto the eigenvector from the corresponding family at each point. In equation(5.78), the integral curves are defined as contours of the function ξ(⃗q) = q 2q 1.Since the gradient of ξ is orthogonal to eigenvector ⃗ r 1 , equation (5.78) is thecorrect definition for the corresponding integral curves.

5.3. Numerical method 161Hence the mature cell density can be considered as the total densitytimes a weight function w m :c m = c tot w m .Then equation (5.78) is equivalent to w m = const. This means, that wavesfrom the first family connect the states of the variables with a constantweight. The density of mature cells changes along lines (5.78), for examplefrom state c m to c ∗ m, only as a result of variation of the total cell densityfrom c tot to c ∗ tot. The weight fraction remains constant, that is c m = w m c totand c ∗ m = w m c ∗ tot. In Figure 5.4, the lines of the first family are plotted asthe radial lines.q 1 = c totc R totc L totq 2q 1= wmLr ⃗2q 2⃗q L = wmR⃗q ∗⃗ q Rq 1⃗r 1q 2 = c mFigure 5.4: Hugoniot loci and integral curves for system (5.74), (5.73), whichare represented by the straight dashed linesFor the second wave family, referred to as the 2-waves, it can be derivedthat∇λ 2 · ⃗r 2 = (λ 2 ) ′ q 1 · 0 + 0 · 1 = 0.This means that the characteristics from the second family have a constantslope λ 2 along a simple wave from this family, and this field is linearlydegenerate [59]. The 2-waves are the contact waves, i.e. they do not showany nonlinear behavior (formation of shocks and rarefaction waves). TheHugoniot loci for the contact waves are defined as straight lines, which areparallel to the corresponding eigenvector (in the current case to vector ⃗ r 2 ).The equation for these lines isq 1 = const.The two states, connected by a 2-wave have the same total cell densityc tot = c ∗ tot. The density of mature cells varies due to a variation of theirweight fraction, i.e. c m = w m c tot and c ∗ m = w ∗ mc tot .The first field is genuinely nonlinear, if∇λ 1 · ⃗r 1 = −2χ ↓ q 1 ≠ 0.

162 Chapter 5. Numerical algorithmThe genuine nonlinearity implies, that the waves in the considered familypropagate as shock waves or as rarefaction waves. If χ ↓ = 0, then the flux ofcells due to chemotaxis is zero. In this trivial situation, a Riemann solution isnot needed, since the hyperbolic terms in the governing equations are equalto zero. Further, it is assumed, that χ ↓ ≠ 0. Note, that from assumptionsc tot ∈ (0, 1] and χ ↓ ≠ 0 it follows, that the 1-wave, associated with theeigenvector ⃗ r 1 , is genuinely nonlinear.5.3.6.1 The strictly hyperbolic caseFrom equation (5.77), it follows that the eigenvalues λ 1 and λ 2 are equal, ifand only if q 1 = 0. Hence the eigenvalues λ 1 and λ 2 are distinct and system(5.74), (5.73) is strictly hyperbolic, if c tot ∈ (0, 1]. For a general strictlyhyperbolic system, if the left and right states ⃗q L and ⃗q R are situated sufficientlyclose to each other in the phase plane, then they can be connectedby the intersecting Hugoniot loci and integral curves, as it is shown in Figure5.4 for the present system of two equations. In this case, the Riemannsolution consists of classical nonlinear and/or linear waves. The qualitativestructure of the solution is determined in a simple way, which is described,for example, in LeVeque [59].However, if ⃗q L and ⃗q R are located far away from each other, then theymay not be connected by intersecting Hugoniot loci and integral curves. Inthis case, a Riemann solution cannot be represented as a set of classicalnonlinear and/or linear waves. Instead of classical shock waves, which arealso referred to as Lax shocks, nonclassical delta shocks or singular shockscan form [see, for example, 23, 90]. Delta shocks waves are characterizedby appearance of singularities in the solutions, which are expressed throughthe Dirac delta function. Classical Lax shocks are always intersected bythe characteristics from other wave families. This feature distinguishes Laxshocks from nonclassical shocks. For a particular system of two nonlinearhyperbolic equations, which was considered in Tan et al. [90], the characteristicsof the second family go into the 1-delta shock from the left and fromthe right. A situation, when system (5.74), (5.73) is weakly hyperbolic andwhen delta shock waves appear in the Riemann solution, is considered inSection 5.3.6.4.Under the current assumption c tot ∈ (0, 1], Hugoniot loci and integralcurves of the first family always intersect any integral curve from the secondfamily. Hence any two states ⃗q L and ⃗q R , such that c L tot ∈ (0, 1] and c R tot ∈(0, 1], can be connected by Hugoniot loci and integral curves in the phaseplane (Figure 5.4). A Riemann solution is composed of a 1-shock or a 1-rarefaction wave and a 2-contact wave. It is easy to demonstrate, that the1-shock wave is a Lax shock.A nonlinear wave in the solution of scalar equation (5.74) for the totalcell density c tot coincides with a 1-wave in the Riemann solution of system

5.3. Numerical method 163(5.74), (5.73). Therefore, the shock wave of the first family (if it appears)splits the x − t plane in two parts, so that the total cell density is equal toc L tot and to c R tot to the left and to the right from the shock wave, respectively.The shock wave speed s is equal to χ ↓ (1 − c L tot − c R tot). The 2-characteristicshave the slopes λ 2 (c L tot) = χ ↓ c L m(1 − c L tot) and λ 2 (c R tot) = χ ↓ c R m(1 − c R tot) to theleft and to the right from the shock, respectively. Hence(λ 2 (c L tot) − s)(λ 2 (c R tot) − s) = χ ↓2 c L totc R tot > 0. (5.81)Inequality (5.81) implies, that the 2-characteristics always enter the 1-shockfrom one side and leave the shock from the other side, that is they intersectthe shock. Hence the 1-shock wave is a Lax shock, if c tot ∈ (0, 1] and χ ↓ ≠ 0.The number of characteristics, which impinge on a 1-shock will be equal tothree: two characteristics of the first family from the left and from the rightof the shock and one set of 2-characteristics at some side from the shock.Assume thatλ 1 > λ 2 . (5.82)Consider a 1-shock wave, which has a speed s. It follows from the entropycondition [59], that λ 1 (c R tot) < s. Hence λ 2 (c R tot) < λ 1 (c R tot) < s. Inequality(5.81) yields that λ 2 (c L tot) < s. Therefore, 2-characteristics intersect a 1-shock wave from right to left. It follows from condition (5.82) that the 2-characteristics intersect a 1-rarefaction wave from right to left (Figure 5.5).Hence the 2-contact wave always lies to the left from the nonlinear shockor rarefaction wave from the first family, if inequality (5.82) holds. Theintermediate state ⃗q ∗ lies between the first and second waves (Figure 5.5c, d).The left state q ⃗L = [c L tot, wmc L L tot] T is connected to ⃗q ∗ by the 2-wave. Hencethe state ⃗q ∗ should lie on the same integral curve from the second familywith the state q ⃗L . By analogy, the state ⃗q ∗ lies on the 1-integral curve(or 1-Hugoniot locus, since they coincide), which passes through the rightstate q ⃗R= [c R tot, wmc R R tot] T . In the phase plane, shown in Figure 5.4, sucha structure of the solution looks like a thick poly-line, which connects thestates q ⃗L , ⃗q ∗ and q ⃗R. It follows, that q1 ∗ = cL tot and q2 ∗ = wR mc L tot. Our aimis to determine the value of the flux function f 2 (⃗q) = χ ↓ (1 − q 1 )q 2 at theedge between two control volumes. The flux function is evaluated for thestate ⃗q ↓ = [q ↓ 1 , q↓ 2 ]T on the considered edge, which is found from the solutionof the Riemann problem. The values q ↓ 1 , q↓ 2 can be determined from thecharacteristic portrait of the solution in the x − t plane, which is shown inFigure 5.5. For the considered Riemann problem, the location of the edgebetween the control volumes corresponds to the ray x t= 0 for t > 0 in thex − t plane.Remark 5.3. If rays x t= const are mentioned further in the text, it is impliedthat t > 0.

164 Chapter 5. Numerical algorithm1−rarefactiont1−characteristics10.50−2 0 2x(a)1t2−characteristics10.50−2 0 2x(b)11−shock⃗q ∗ q ⃗R⃗q ∗ 0.50.5 q ⃗R⃗q L ⃗q L00−2 0 2 −2 0 2xx(c)(d)ttFigure 5.5: Solution structure in the x − t plane for a strictly hyperbolicsystem. The thick lines denote the edges of the 1-rarefaction wave or the1-shock wave and the thick dashed lines denote the 2-contact wave. The thinlines are the characteristics of the first (a), (c) and second families (b),(d).The 2-contact wave lies to the left from the nonlinear 1-wave, since it isassumed that λ 2 < λ 1 everywhere

5.3. Numerical method 165For the present system it is possible to derive simple general formulas,which provide the values of c tot and c m at x t= 0. The discontinuities ofthe variables c tot and w m are considered separately, since they are carriedby two separate waves. The 1-wave carries the discontinuity in c tot . Thestate c ↓ tot at the ray x t= 0, corresponding to the considered inner edge, canbe determined from the Riemann solution for the scalar nonlinear equation(5.74) [59]:c ↓ tot = ⎧⎪⎨⎪ ⎩argargminc L tot ≤ctot≤cR totmaxc R tot ≤ctot≤cL totf 1 (c tot ),f 1 (c tot ),if c L tot ≤ c R tot,if c R tot ≤ c L tot.(5.83)The 2-contact wave carries the discontinuity in the weight fraction w m . Forthe considered case λ 1 > λ 2 , the speed of this wave is equal to λ 2 (c L tot). Thenthe state wm ↓ at the ray x t= 0 is equal to wm L or to wm, R if the eigenvalueλ 2 (c L tot) is positive or negative, respectively. If the 2-wave speed is zero,i.e. λ 2 (c L tot) = 0, then according to the current assumption, the 1-wave willlie to the right from the 2-wave and, hence, from the ray x t= 0. Thismeans, that c ↓ tot = cL tot, and that the 2-wave speed coincides with eigenvalueλ 2 (c ↓ tot ). Note, that f 2(c tot , c m ) = c m λ 2 (c tot ) = w m c tot λ 2 (c tot ). Hence ifthe second eigenvalue λ 2 (c L tot) is zero, then irrespectively of the states wmLor wm R assigned to wm, ↓ the flux function on the inner edge will be zero:f 2 (c ↓ tot , w↓ mc ↓ tot ) = f 2(c L tot, wmc ↓ L tot) = wmc ↓ L totλ 2 (c L tot) = 0.Further, the following general formula for the state wm ↓ will be used:{wLwm ↓ m , if λ 2 (c ↓ tot=) ≥ 0,wm, R if λ 2 (c ↓ tot ) < 0. (5.84)To justify equation (5.84) for the case λ 1 > λ 2 , it should be shown thatthe eigenvalues λ 2 (c L tot) and λ 2 (c ↓ tot ) have the same sign or are equal to zerosimultaneously. For conciseness, the following notations are introducedλ L i = λ i (c L tot), λ R i = λ i (c R tot), λ ↓ i = λ i(c ↓ tot ), i = 1, 2.Suppose that a 1-shock wave has a speed s > 0 and lies to the right fromthe ray x t = 0. Then c↓ tot = cL tot and λ L 2 = λ↓ 2 .Assume that the 1-shock has a speed s ≤ 0. Condition (5.82) yields that2-characteristics intersect the 1-shock wave from right to left. This meansthat λ L 2 < s ≤ 0 and λR 2 < s ≤ 0. The eigenvalue λ 2 is negative everywhere.Hence λ L 2 < 0 and λ↓ 2 < 0, irrespectively of the states cL tot or c R tot assigned toc ↓ totİt should be mentioned that a zero shock speed s = 0 leads to an ambiguityin the definition of c ↓ tot . The total cell density is discontinuous at theray x t = 0. Formula (5.83) provides two values cL tot and c R tot for c ↓ tot , sincef 1 (c L tot) = f 1 (c R tot). (5.85)

166 Chapter 5. Numerical algorithmNote thatf 2 (c tot , c m ) = χ ↓ c mc totc tot (1 − c tot ) = χ ↓ w m c tot (1 − c tot ) = w m f 1 (c tot ).The value of the flux f ↓ 2 is defined uniquely in this case, since the weightfraction w m does not change through the 1-shock wave and since equality(5.85) holds.For a 1-rarefaction wave, it follows from the entropy condition that λ L 1 0, then the rarefaction fan lies to the right from the ray x t = 0,and c L tot = c ↓ tot . Consequently, λL 2 = λ↓ 2 .If λ R 1 ≤ 0, then c↓ tot = cR tot. Further, λ L 2 < λL 1 < λR 1 ≤ 0, and λ↓ 2 = λR 2 λ 2 . Alternatively, if λ 1 < λ 2 , then a 2-wave will befaster than a 1-wave. Hence the speed of the contact wave, which carries thediscontinuity in the weight fraction w m is equal to λ R 2 . By analogy, it canbe proved that the eigenvalues λ R 2 and λ↓ 2 have the same sign or are equalto zero simultaneously, if λ 1 < λ 2 . This proof will justify formula (5.84).5.3.6.2 Nonconvex flux functionIn a more general case c tot ∈ (0, ∞), the solution structure becomes morecomplicated. The flux of the total cell density{χ ↓ c tot (1 − c tot ), c tot ≤ 1,f 1 (c tot ) =(5.86)0, c tot > 1,is not a convex or a concave function in this case. The Riemann solution ofthe scalar equation (5.74) for the total cell concentration Riemann problemhas the form of the compound wave, i.e. the wave, which is composed of anumber of shocks, rarefactions and/or traveling waves. The structure of thesolution is determined from the boundary of the convex hull, constructedfor the given flux function [59]. If the left and right states in the Riemannproblem are such that c L tot < c R tot, then the convex hull is constructed for theset {(x, y) : c L tot ≤ x ≤ c R tot, y ≥ f 1 (x)}, and if c L tot > c R tot, then the convexhull is constructed for the set {(x, y) : c R tot ≤ x ≤ c L tot, y ≤ f 1 (x)}.Remark 5.4. The flux of the mature cell density is defined as:{χ ↓ c m (1 − c tot ), c tot ≤ 1,f 2 (c tot , c m ) =0, c tot > 1.(5.87)

5.3. Numerical method 167Let us consider the case c L tot < c R tot. Suppose, that c R tot > 1. Otherwise,due to the monotonicity of the entropy solution of the Riemann problem,the solution will be in the interval (0, 1], and this situation was consideredbefore. If c L tot ≥ 1, then the Riemann solution will be constant in time, sincef 1 ≡ 0 everywhere. That is the jump between c L tot and c R tot will not move.Therefore, it is supposed that 0 < c L tot < 1 < c R tot. Then the convexhulls can have three possible configurations, which are shown in Figure 5.6.For the situation illustrated in Figure 5.6a, the compound wave consists of0−0.05f 1(c tot)0−0.05f 1(c tot)−0.1−0.1−0.15−0.15−0.2−0.2−0.250 L ctot*c totRc tot−0.250 Lc totRc tot(a)(b)0.250.20.15f 1(c tot)0.10.0500 Lc tot(c)*c tot =1R ctotFigure 5.6: Three possible configurations of the convex hull, if c L tot < 1 < c R tot.The convex hull is constructed for the set {(x, y) : c L tot ≤ x ≤ c R tot, y ≥ f 1 (x)}.It is denoted by the filled area, with the boundary shown in the solid line.the rarefaction wave, which connects the left state c L tot to an intermediatestate c ∗ tot, and of the shock wave, connecting the states c ∗ tot and c R tot. Theintermediate state is determined from the conditionf 1 (c ∗ tot) ′ = f 1(c R tot) − f 1 (c ∗ tot)c R tot − . (5.88)c∗ totThe convex hull boundaries, which are shown in Figures 5.6b, c, correspondto the single shock wave between states c L tot and c R tot and to the compoundwave, which consists of the shock and the linear contact wave, respectively.The state c ↓ tot on the inner edge of the control volumes can be foundfrom the general formula (5.83), which is also applicable for the case of a

168 Chapter 5. Numerical algorithmnonconvex flux function [59]. Subsequently, it is proved that equation (5.84)for the weight fraction w ↓ m is correct in the current case.Consider a single shock wave between the states 0 < c L tot < 1 < c R tot(Figure 5.6b), which appears if χ ↓ < 0. Then the shock speed is given by:s = f 1(c R tot) − f 1 (c L tot)c R tot − cL tot= −χ ↓ cL tot(1 − c L tot)c R tot − cL tot> 0,andλ L 2 − s = χ ↓ cR tot(1 − c L tot)c R tot − cL tot< 0, λ R 2 − s = χ ↓ cL tot(1 − c L tot)c R tot − cL tot< 0. (5.89)It follows from condition (5.89) that the characteristics of the 2-nd familyintersect the 1-shock from right to left. Hence the 2-wave speed is equal toλ 2 (c L tot) = χ ↓ (1 − c L tot) < 0 and lies to the left of the ray x t= 0. It followsthat wm ↓ = wm. R Since the shock speed is positive, the 1-shock lies to theright of the ray x t= 0, and c↓ tot = cL tot. Therefore, formula (5.84) providesthe correct value of wm, ↓ since λ 2 (c ↓ tot ) = λ 2(c L tot).Consider the compound wave, which consists of the rarefaction and shockwaves (Figure 5.6a). This wave corresponds to the situation, when c L tot < c R totand χ ↓ < 0. Therefore, λ 2 −λ 1 = χ ↓ c tot < 0 for c tot ∈ (0, 1]. The rarefactionwave connects the states c L tot and c ∗ tot, which lie in the interval (0, 1]. Hencethe 2-characteristics intersect the 1-rarefaction wave from right to left. Thecharacteristic portrait of the present Riemann solution is shown in Figure5.7. The shock wave front coincides with the right edge of the rarefactiont0.51−characteristics10−2 0 2x(a)t0.52−characteristics10−2 0 2x(b)Figure 5.7: Characteristics of the first family and of the second family in thex−t plane are shown in thin unbroken lines in plots (a) and (b), respectively.The wedges of the rarefaction fan are shown in solid dashed lines. The 2-contact wave front is denoted by a solid unbroken line. The 1-shock wavecoincides with the right edge of 1-rarefaction wave. The 2-wave front lies tothe left from the compound 1-wavefan (see equation (5.88)). Then the 2-characteristics also intersect the shock

5.3. Numerical method 169wave between the states c ∗ tot and c R tot from right to left, sinceλ 2 (c ∗ tot) − s = χ ↓ cR tot(1 − c ∗ tot)c R tot − c∗ tot< 0, λ R 2 − s = χ ↓ c∗ tot(1 − c ∗ tot)c R tot − c∗ tot< 0.This means that the 2-contact wave lies to the left of the compound 1-wave. The speed of the 2-wave is equal to λ L 2 . To justify formula (5.84)for the present case, it is necessary to show, that the eigenvalues λ L 2 and λ↓ 2have the same sign or are equal to zero simultaneously. This is proved asfollows. Since χ ↓ < 0, the right edge of the rarefaction fan has the slopeλ 1 (c ∗ tot) = s = χ ↓ (1 − c ∗ tot − c R tot) > 0. If λ L 1 > 0, then c↓ tot = cL tot. If λ L 1 ≤ 0,then λ ↓ 1 = 0 and c↓ tot = 0.5. Further, λ↓ 2 < λ↓ 1 = 0 and λL 2 < λL 1 ≤ 0.tt10.5010.50c ∗ tot = 1 c R tot > 1c L tot < 1−2 −1 0 1 2x(a)c ∗ tot = 1 c R tot > 1c L tot < 1−2 −1 0 1 2x(b)Figure 5.8: The solution to the Riemann problem, which consists of theshock wave (dashed line) and the stationary wave (thick solid line), includesthree states: c L tot, c ∗ tot = 1 and c R tot > 1. The 1- and 2-characteristics areshown in thin solid lines in plots (a) and (b), respectively. The derivativeof the flux function f 1 (c tot ) is discontinuous at the point c tot = 1. As aresult, the 1-characteristic lines are not defined in the wedge between theshock wave front and the contact wave front. The 2-characteristics intersectthe 1-shock wave. The 2-contact wave is stationary and coincides with thestationary 1-contact waveThe third situation appears, if the compound wave consists of the shockwave between the states c L tot and c ∗ tot = 1 and of the linear contact wave,which connects the states c ∗ tot = 1 and c R tot > 1 (Figure 5.6c). The presenttype of the 1-wave appears for 0 < c L tot < 1 < c R tot, if χ > 0. The linear waveis stationary, since λ 1 = f ′ 1 = 0 for c tot ∈ (1, c R tot), where f 1 (c tot ) is given inequation (5.86). The derivative of the flux function f 1 (c tot ) is discontinuousat the point c tot = 1. Consequently, the 1-characteristic lines are not definedin the wedge between the fronts of the 1-shock wave and of the 1-contact

170 Chapter 5. Numerical algorithmwave. The solution of the Riemann problem in the x − t plane for theconsidered case is plotted in Figure 5.8. The 2-characteristics intersect the1-shock wave, since s = χ ↓ (1 − c L tot − c ∗ tot) < 0, λ L 2 = χ↓ (1 − c L tot) > 0 andλ L 2 (c∗ tot) = 0. The discontinuity of the weight fraction w m propagates alongthe ray x t= 0. The general formula (5.83) for the total cell density at theinner edge of the control volumes provides multiple solutions c tot ∈ [1, c R tot],which corresponds to the stationary discontinuity between states c ∗ tot = 1and c R tot > 1 in the solution for the total cell density at the inner edge (seeFigure 5.8). This does not give us any problem, since the flux of maturecells is approximated, and f 2 (c m , c tot ) = 0 for c tot ∈ [1, c R tot].The situation when c L tot > c R tot is symmetrical to the case c L tot < c R tot. Conclusions,which can be made in this case, are analogous to the conclusions,which were stated for the case c L tot < c R tot.5.3.6.3 Nonstrictly hyperbolic systemIt was assumed in Section 5.3.6.2, that c tot ∈ (0, ∞). For c tot ∈ (1, ∞) theJacobian is a zero matrix and the eigenvalues λ 1 and λ 2 are both equal tozero. Hence the considered system is not strictly hyperbolic for c tot ∈ (1, ∞).The zero eigenvalues have a two dimensional eigenspace. Therefore, the algebraicmultiplicity of the eigenvalues is equal to their geometric multiplicity.Systems with this property are referred to as nonstrictly hyperbolic systems.At the points of the phase plane, where the system becomes nonstrictly hyperbolic,there exists an infinite number of eigendirections. The Riemannsolution for the nonstrictly hyperbolic system may be more complicated,compared to a strictly hyperbolic case [59]. For c tot ∈ (1, ∞) the first andsecond fields are linearly degenerate. The waves in these fields are stationarycontact waves. That is, the 1-wave and the 2-wave coincide in the regions,where c tot ∈ (1, ∞). This leads to the structure of the Riemann solution asdescribed in Section 5.3.6.2.Another point of the phase plane, where system (5.73)–(5.74) is nonstrictlyhyperbolic, is (c tot , c m ) = (0, 0). The eigenvalues λ 1 and λ 2 areequal to χ ↓ and have a two dimensional eigenspace. An infinite number ofthe integral curves of the first family passes through the point (0, 0), andany state (c tot , c m ) ∈ (0, ∞) × R can be connected to the state (0, 0) by a1-wave. Assume that the left state ⃗q L is [0, 0] T . Depending on the sign ofχ ↓ and on the value of c R tot, the 1-wave, which connects ⃗q L and ⃗q R can havefour configurations shown in Figure 5.9. Formula (5.83), which determinesthe value of c tot at the ray x t= 0, is valid in all considered cases, since thisidentity is derived for a scalar equation (5.74).The 1-wave is a rarefaction wave if χ ↓ < 0 and c R tot ≤ 1 (Figure 5.9a). Itis shown in Section 5.3.6, that the weight fraction is constant along a 1-wavein the region c tot ≤ 1. Hence the value of c ↓ m is determined by the expression

5.3. Numerical method 1710−0.05f 1(c tot)0−0.05f 1(c tot)−0.1−0.1−0.15−0.15−0.2−0.2−0.25Lc tot =0R ctot−0.25Lc tot =0* ctotRc tot(a)(b)0.250.250.20.20.150.150.10.10.05f 1(c tot)0Lc tot =0Rc tot 1(c)0.050Lc tot =0f 1(c tot)(d)* Ctot=1R ctotFigure 5.9: The structure of the 1-wave in the case 0 = c L tot < c R tot, whichis determined from the convex hull reconstruction. The convex hull is constructedfor the set {(x, y) : c L tot ≤ x ≤ c R tot, y ≥ f 1 (x)}. It is denoted by thefilled area, with the boundary shown by the solid line.

172 Chapter 5. Numerical algorithmc ↓ m = c ↓ totc R mc R tot. (5.90)If χ ↓ < 0 and c R tot > 1, then the 1-wave consist of a rarefaction wave and ashock, which connect the states ⃗q ∗ and ⃗q R (Figure 5.9b). From the Rankine-Hugoniot conditionit is derived, that c∗ mc ∗tot⃗f(⃗q R ) − ⃗ f(⃗q ∗ ) = s(⃗q R − ⃗q ∗ ),= cR m. Therefore, the weight fraction is constantc R totthrough a shock and rarefaction part of the compound 1-wave, and c ↓ m isdetermined by equation (5.90). The 1-wave is a pure shock, if χ ↓ > 0 andc R tot ≤ 1 (Figure 5.9c). Then ⃗q = ⃗q L = [0, 0] T to the left of the shock frontand ⃗q = ⃗q R to the right of the shock. If χ ↓ > 0 and c R tot > 1, then the1-wave is a compound wave, consisting of a stationary non-moving shockand a stationary contact wave (Figure 5.9d). In this case, ⃗q = ⃗q L = [0, 0] Tfor x t < 0 and ⃗q = ⃗qR for x t > 0. At the ray x t= 0, the total cell density isdiscontinuous. The flux function f 2 is equal to zero, if it is evaluated at theboth sides of the discontinuity. Formula (5.83) provides multiple values forc ↓ tot : c tot ∈ {0} ∪ [1, c R tot). Any of these values provide a zero flux f 2 , if c ↓ m isbounded. Therefore, equations (5.83) and (5.90) give a correct value of theflux f 2 at the inner edge.The 1-wave structure is analogous in the symmetric case c L tot > c R tot = 0,c R m = 0 for the opposite signs of χ ↓ . The state c tot is determined fromexpression (5.83), and c ↓ m is given by:c ↓ m = c ↓ totc L mc L tot. (5.91)5.3.6.4 Weakly hyperbolic systemAccording to Section 5.3.1, the present numerical algorithm should providenonnegative solutions for the cells densities. In this case, the total cell densityc tot is equal to zero, if and only if c i = 0 and c m = 0. The definitionof the states c ↓ tot and c↓ m given in Sections 5.3.6.1–5.3.6.3 is sufficient to determinethe flux f 2 at the cell boundaries if a numerical solution for the celldensities is nonnegative. Nonnegativity of the solution is crucially importantfor the current model, since negative values of the cell densities maygrow in magnitude, leading to divergence of the numerical solution. Thisphenomenon is discussed further in this section.Assume that the cell densities can take negative values. In this case, azero total cell density does not imply zero densities of immature and matureosteogenic cells. Therefore, the states c tot = 0 and c m ≠ 0 can be considered.

5.3. Numerical method 173At these states, the eigenvalues λ 1 and λ 2 are equal to χ ↓ and have justone eigenvector [0, 1] T . Therefore, the algebraic multiplicity is larger thangeometric multiplicity. The Jacobian matrix is defective and the system isweakly hyperbolic. One of the characteristics of a weakly hyperbolic system,is that the Riemann solution may not be represented as a set of classicalrarefaction, shock and/or contact waves. In some situations nonclassicalshock waves, which are referred to as delta shock waves or singular shocks,can appear in the solution.The delta shocks can arise in various hyperbolic systems under certaininitial and boundary conditions. These waves contain singularities, whichcan be expressed by the Dirac delta function. If delta shocks appear, the solutionsof the hyperbolic systems are defined in a generalized form. Variousauthors [23, 55, 59, 70, 90] propose multiple approaches to construct generalizedsolutions for different hyperbolic systems. Most often the generalizedsolution is derived as a limit of some set of regularized viscous solutions.For the present hyperbolic system, a delta shock wave appears, for example,if χ ↓ > 0 and c L tot = 0, c L m ≠ 0 and c R tot > 0. To understand the natureof this delta shock wave, let us consider a Riemann problem for c L tot = ε,c L m ≠ 0, 0 < c R tot ≤ 1. Let us look at the solution behavior for ε → 0. Thesolution consists of a shock wave, connecting the state ⃗q L to an intermediatestate ⃗q ∗ = [c R tot, c R c L mtot ε] T . The contact wave connecting the states ⃗q ∗and ⃗q R lies to the right of the shock wave and propagates along the rayxt = χ↓ (1 − c R tot) (Figure 5.10). The shock speed is equal to χ ↓ (1 − c R tot − ε).Therefore, if ε → +0, then the shock wave approaches the contact wave,and the density c ∗ m between these waves tends to infinity.Therefore, the 1-characteristics impinge on the delta shock from the leftand from the right. The 2-characteristics impinge on the delta shock fromthe left and are parallel to it from the right. According to [23], the Riemannsolution for c m for the present case is defined in the distributional form:c m = c L m + (c R m − c L m)H(x − st) + χ ↓ t ( c R totc L m − c L totc R m)δ(x − st),where H(·) is the Heaviside step function, δ(·) is the Dirac delta function ands is the shock speed, which is equal to χ ↓ (1 − c L tot − c R tot). If χ ↓ > 0, c L tot = 0,c L m < 0 and 0 < c R tot < 1, then the mature cell density tends to minus infinityon the ray x t = χ↓ (1 − c R tot). Due to a random walk of cells, represented bythe diffusive terms in the governing equations, the singularity of the deltashock is smoothed in the solution of the full system. A negative densitygrows in magnitude in time. This issue can be demonstrated explicitly forthe following simplified problem.

174 Chapter 5. Numerical algorithm11−characteristicst0.501c L tot = εc L m ≠ 0c ∗ m = cR c L mtot εc R tot > 0−2 −1 0 1 2x(a)2−characteristicss = χ ↓ (1 −ε −c R tot)t0.50λ R 2 = χ↓ (1 −c R tot )−2 −1 0 1 2x(b)Figure 5.10: The solution to the Riemann problem, which consists of the 1-shock wave (dashed line) and the 2-contact wave (thick solid line), includesthe left state c L tot = ε, c L m ≠ 0, the right state c R tot > 0 and c R m and theintermediate state c ∗ tot = c R tot, c ∗ m = c R totc L mε. The 1- and 2-characteristicsare shown in thin solid lines in plots (a) and (b), respectively. The 2-characteristics intersect the 1-shock wave. The difference between the speedsof the 1-shock and the 2-contact wave is equal to χ ↓ ε

5.3. Numerical method 175Large negative solutions Consider the following simplified equationsdescribing the migration of cells in the 1D physical space x ∈ R:∂c tot∂t∂c m∂tAssume that+ ∂ (∂c tot−D c∂x ∂x + χ 0 g ↓)∂gKch 2 + g↓2 ∂x (1 − c tot)c tot = 0 (5.92)+ ∂ (∂c m−D c∂x ∂x + χ 0 g ↓)∂gKch 2 + g↓2 ∂x (1 − c tot)c m = 0 (5.93)χ 0 g ↓ ∂gKch 2 +g↓2 ∂x≡ const = ˆχ > 0. The functionĉ tot (x) =1exp{− ˆχ D cx} + 1is the stationary solution of equation (5.92), such that ĉ tot → 0 if x → −∞,ĉ tot → 1 if x → ∞ and ĉ tot (0) = 1 2. The derivative of the stationary solutionis given bydĉ totdx = ˆχ ĉ tot (1 − ĉ tot ).D cIf ĉ tot is substituted into equation (5.93), then the equation for the maturecell density becomes the linear diffusion-advection-reaction equation:∂c m∂t+ ˆχ(1 − ĉ tot ) ∂c m∂x = D ∂ 2 c mc∂x 2 + ˆχ2 ĉ tot (1 − ĉ tot )c m , (5.94)D cwhich can be easily solved numerically. Suppose that chemotaxis prevailsover random walk of cells, that is ˆχ D c>> 1. The stationary solution ĉ tot forthe case ˆχ D c= 30 is plotted in Figure 5.11a. The derivative dĉtotdxis largenear x = 0 and its maximal value is dĉtotdx (0) = ˆχ4D c. Hence the reaction termin the right-hand side of equation (5.94) is large near x = 0, and negativevalues of c m grow in magnitude in this region, since ĉ tot ∈ (0, 1) and inparticular for ĉ tot = 1 2. In Figure 5.11b, the numerical solution of equation(5.93) is plotted for different time moments. The solution is obtained forthe spatial domain x ∈ [−1, 1] and for the following initial and boundaryconditions: c m (x, 0) = ĉ tot (x)/2 − 0.1, c m (−1, 0) = ĉ tot (−1)/2 − 0.1 andc m (1, 0) = ĉ tot (1)/2 − 0.1. The minimal value of c m grows in magnitudefrom −0.1 at t = 0 to −0.56 at t = 1. The point of minimum lies to theright from x = 0 due to the advection term in the left-hand side of equation(5.94).Therefore, this example shows how small negative values of the cell densitycan grow significantly in magnitude in time, due to a nonlinearity of theflux function. Hence negative values of the cell densities should be avoidedduring simulations at all times in order to get a convergence of the numericalsolution.

176 Chapter 5. Numerical algorithmcmĉtot10.5(a)0−1 −0.5 0 0.5 1x(b)0.40−0.4−1 −0.5 0 0.5 1xt=0;t=0.5;t=1;Figure 5.11: (a) Plot of the stationary solution ĉ tot of equation (5.92) forthe case ˆχ D c= 30; (b) Plots of the solution c m of equation (5.94) for thetime moments 0, 0.5 and 1 days. Negative values of c m grow in magnitudein time near the point x = 0, where the derivative of ĉ tot is large5.3.6.5 SummaryTo summarize, the fluxes due to cell migration are approximated as follows.First, the left and right states of the immature and mature cell densitiesare found at the inner edges of the 2D control volumes and at the faces ofthe 3D control volumes, corresponding to these edges, as it is described inSection 5.3.5.2. The left and right states of the total cell concentration areobtained by adding the total density of immature cells and the density ofmature osteogenic cells. The considered states of the immature, mature andtotal cell densities are denoted as c L i,l , cR i,l , cL m, c R m, c L tot and c R tot, respectively.The index l ranges from 1 to N a . It is used to denote the states of the immaturecell density at N a inner faces of the 3D control volumes, correspondingto each inner edge of the 2D control volumes.The normal derivatives of the immature and mature cell densities andthe normal derivative and the value of the growth factor concentration atthe inner edges and at the corresponding inner faces are determined in Section5.3.5.3. These quantities are denoted as ∂c i,l∂xtively.↓,∂c m∂x↓,∂g∂x↓and g ↓ , respec-Then the variables c ↓ tot , c↓ m, c ↓ i,lare defined from the Riemann solution,

5.3. Numerical method 177which is determined by formula (5.83) and by equations:⎧c ↓ tot w↓ m, if c L tot ≠ 0, c R tot ≠ 0,⎪⎨c ↓ m =c ↓ c R mtotc R totc L m, if c L tot = 0, c R tot ≠ 0,c ↓ totc⎪⎩L , if c L tot ≠ 0, c R tot = 0,tot0, if c L tot = 0, c R tot = 0,c⎧⎪ ↓ tot w↓ i,l , if cL tot ≠ 0, c R tot ≠ 0,⎨ c ↓ c R i,lc ↓ toti,l = c R , if c L tot = 0, c R tot ≠ 0,tot⎪ ⎩c ↓ c L i,ltotc L tot, if c L tot ≠ 0, c R tot = 0,0, if c L tot = 0, c R tot = 0.The weight fraction w ↓ m is given by equation (5.84), and w i,l is defined in asimilar way:w ↓ i,l = {wLi,l , if λ 2 (c ↓ tot ) ≥ 0,w R i,l , if λ 2(c ↓ tot ) ≤ 0,where the left and right states of the weight fractions are determined in thefollowing way:wm S = cS mc S , wi,l S = cS i,ltotc S , S ∈ {L, R}.totThe functions f 1 and λ 2 , which are included in the mentioned equations,are defined in equations (5.86) and (5.95):λ 2 (c tot ) ={χ ↓ (1 − c tot ), c tot < 1,0, c tot ≥ 1,(5.95)where χ ↓ =χ 0g ↓ ∂g ↓Kch 2 +g↓2 ∂x .Finally the flux of mature osteogenic cells at the inner edge, and theflux of immature cells at the corresponding inner faces are approximated asfollows:Fm ↓ ↓∂c m= −D c + f 2 (c ↓ tot∂x, c↓ m), F ↓↓i,l = −D ∂c i,lc + f 2 (c ↓ tot∂x, c↓ i,l ),where the flux function f 2 (·, ·) is given by equation (5.87).The present formulas are valid only if solutions for the cell densitiesare nonnegative. For negative cell densities, singular delta shock waves

178 Chapter 5. Numerical algorithmcan appear in the solution to the Riemann problem, from which the givenrelations are derived. One of the situations, when delta shocks form, isdescribed in Section 5.3.6.4. Although the equations describe some of thequalitative nature of the solutions, the present formulas are not justified andhence they may give wrong approximation of the fluxes, if negative valuesof the cell densities are admitted.5.3.7 Treatment of the reaction termThe reaction terms in equations (5.28)–(5.30) are approximated by functionsHi R, HR m and Hg R , respectively. The integrals over the control volumes Vj,k,l3and V j,k are substituted by the values of the integrated functions at thecenters of the control volumes. The fluxes of the cell densities c i and c mand of the growth factor concentration g depend on the gradients of thesefunctions, which are approximated at the inner edges and faces of the controlvolumes. Hence the values of the r-components of the fluxes at the centersof control volumes are obtained through the interpolation of the normalflux values, determined at the centers of the inner edges of the 2D controlvolumes V j,k , {j, k} ∈ I a , which are parallel to the z-axis, and at the centersof the corresponding faces of the 3D control volumes V 3j,k,l , {j, k} ∈ I 3 a . Theconsidered normal fluxes were approximated using the principles outlined inSections 5.3.5 and 5.3.6.The following interpolation technique is applied. Each regular rectangularcell V j,k has two vertical, i.e. parallel to the z-axis, edges e j−1,kand2e j+1,k. The center of the cell V j,k lies in the middle of the line segment,2which connects the centers of the considered edges. Hence the approximatevalues F j,k,li, Fm j,k , Fgj,k of the r-components of the fluxes corresponding tothe variables c i , c m and g in the centers of cells Vj,k,l 3 and V j,k are definedas:= F j− 1 2 ,k,liF j,k,li+ F j+ 1 2 ,k,li,2Fp j,k = F j− 1 2 ,ki+ F j+ 1 2 ,ki, p ∈ {m, g},2(5.96)where F j− 1 2 ,k,li, F j− 1 2 ,km , F j− 1 2 ,kg and F j+ 1 2 ,k,li, F j+ 1 2 ,km , F j+ 1 2 ,kg are the normalfluxes of variables c i , c m and g at the corresponding vertical faces and edgesof control volumes Vj,k,l 3 and V j,k.An irregular cell V j,k can have either one or two vertical edges. If thecell V j,k has one vertical inner edge, for example edge e j−1,k,then the flux2at the cell center is determined through the following extension:F j,k,li= F j− 1 2 ,k,li,F j,kp = F j− 1 2 ,ki, p ∈ {m, g},(5.97)

5.3. Numerical method 179If there are two inner vertical edges e j−12 ,k and e j+ 1 ,k,adjacent to the active2irregular cell V j,k , then the inverse distance weighting interpolation is used:F j,kpF j,k,li= w j,k1 F j− 1 2 ,k,li+ w j,k2 F j+ 1 2 ,k,li,= w j,k1 F j− 1 2 ,kp + w j,k2 F j+ 1 2 ,kp , p ∈ {m, g},w j,k1 =d j,k2d j,k1 + d j,k , w j,k2 =2d j,k1d j,k1 + d j,k ,2(5.98)where d j,k1 and d j,k2 are the distances from the center of cell V j,k to its verticalinner edges e j−12 ,k and e j+ 1 ,k,respectively.2Therefore, the following expressions for the reaction terms are defined:Hi R ( C ⃗ i , C ⃗ m , G, ⃗ (j, k, l)) = A c (G j,k )C j,k,li(1 − C j,kH R m( ⃗ C i , ⃗ C m , ⃗ G, (j, k)) = A c (G j,k )C j,km (1 − C j,kHg R ( C ⃗ i , C ⃗ m , G, ⃗ ( ∑N a(j, k)) = E c (G j,k ) Cm j,k + h al=1tot ) − F j,k,li, (5.99)r j,kctot ) − F mj,k, (5.100)r j,kc)a l−1 + a2 l+12C j,k,li2− F gj,krcj,k(5.101)where A c (g) and E c (g) are defined in Table 4.1, and the averages of the totalcell concentration are determined as followsC j,ktot = Cj,k m∑N a+ h al=1C j,k,li, (j, k) ∈ I a .,5.3.8 Boundary conditionsThe fluxes of the immature and mature cell densities c i and c m and of thegrowth factor concentration g at the boundary Γ(t) of the physical domainΩ s and at the corresponding boundary Γ 3 (t) = Γ × [0, 1] of the 3D domain,which includes the maturation space, are represented by the integrals overinterfaces ∂Vj,k,l 3B and ∂Vj,k Bapproximated by functions H M bin equations (5.28)–(5.30).i, H M bmThe integrals areand H M bg in equations (5.32)–(5.34)through the values of the integrands at the centers of the boundary facesand edges. Each boundary face and edge is adjacent to exactly one activecontrol volume, otherwise they will lie between active control volumes and,hence, they will not lie on the boundary of the physical domain. The valuesof the unknown cell densities and of the growth factor concentration are

180 Chapter 5. Numerical algorithmapproximated at the centers of the boundary edges and faces by their averagevalues in the adjacent control volumes. The velocity at the center of theboundary edge and at the centers of all corresponding 2D faces is definedas the mean value of the normal velocities at the vertices of the considerededge, which were determined in Section 5.3.3.5.3.9 Time integrationOrdinary differential equations (5.32)–(5.34) are integrated with the use ofthe explicit modified Euler method (also called explicit trapezoidal rule). Itis a two stage method. The grid of control volumes is updated at each stage.Since an explicit time integration is used for the advection-diffusionreactionequations, the time step size should be chosen correctly. Due tothe presence of the parabolic diffusion terms, severe restrictions on the timestep size should be set, in order to provide a stable numerical scheme. Thereason, why an implicit time integration method is not used, is that implicitmethods are not efficient for the solution of the equations of the consideredhyperbolic type, if no dominance of the diffusion terms is observed, and ifa positivity of the solution should be kept. An explicit time integrationis usually more efficient than implicit methods, if a nonnegative numericalsolution of hyperbolic equations has to be obtained. Another advantage ofthe explicit scheme, is that it is much more convenient for the implementationinto the computer code and for its verification, especially if nonlinearlycoupled equations defined within the evolving in time physical domain areconsidered.Let us first estimate the order of the time step size, determined by thestability restrictions. Such an estimate can be derived for the linear 1Dadvection-diffusion equation∂u∂t + a∂u ∂x = ud∂2 ∂x 2with constant coefficients d, a, in the following way. The eigenvalues ofthe matrix obtained from the discretization of the advection-diffusion termstimes the time step should lie within the stability region of the consideredtime integration scheme. If the modified Euler method for the time discretization,the third order upwind biased discretization for the advectionterm, and the second order central scheme for the diffusion term are considered,then the following restrictions are obtained for the time step τν = τ|a|hτd< 0.874, µ = < 0.209, (5.102)h2 where h is the size of uniform 1D control volumes. The given conditions aresufficient, but not necessary. That is, the time step size is underestimated

5.3. Numerical method 181by inequalities (5.102). The preliminary estimates for the considered nonlinearmultidimensional problem are obtained, if the diffusion coefficient ofosteogenic cells D c is substituted for the parameter d, and the parameter ais replaced with the derivative of the total cell flux, caused by chemotaxis,with respect to the total cell density, which can be estimated in the followingway:∣| (χ(c tot , g)c tot ∇ s g) ′ c tot| =χ 0 (1 − 2c tot )g∇ s g ∣∣∣∣ Kch 2 + ∼ χ 0g2 h .Then inequalities (5.102) yield:τ < 0.874 h2χ 0,τ < 0.209 h2D c, (5.103)where h is the characteristic linear size of the mesh.Another restriction on the time step size is applied in order to keepnonnegativeness of the numerical solution. If regular rectangular 2D controlvolumes in the center of the physical domain are considered, then thefollowing constraint on the time step size can be derived:τ

182 Chapter 5. Numerical algorithmThe combined control volumes are constructed in the following way. Thecharacteristic linear size of an irregular computational cell is defined as theminimal ratio of its volume to the length of the inner edges. The approximationsH M Ii, H M Im , H M Ig of the advection-diffusion terms are inverselyproportional to this characteristic linear size. If the considered ratio becomestoo small, then very small time step should be chosen in order toprovide stability and positivity of the numerical solution. Hence the consideredlinear size of the irregular cell is used to determine, whether the cellshould be united with its neighbors.Let us consider some irregular active control volume V j,k at the boundaryof the physical domain. It is united with some of its neighbors, if the minimalratio of its volume to the length of its inner edges is smaller than limit h irr .If the neighbor, with which the considered control volume is united, is part ofanother combined cell, then that combined cell is included in the combinedcontrol volume, which is constructed around cell V j,k (see Figure 5.12).Figure 5.12: Sketch of the combined cells, which are constructed near theboundary of the physical domain. Each combined cell consists of severalactive control volumes, which are shown in one colorThe value of the limit h irr is chosen to be equal to 0.4h, where h is theminimal linear size of the regular rectangular control volumes in the middleof the computational domain. From constraint (5.106), it follows that thetime step should be proportional to the square of the linear size of the mesh.Due to the presence of small irregular cells near the boundary of the physicaldomain, the following time step size{τ = h irr hτ c min14 (D c + χ 0 ) , 0.209 ,D c}14D g(5.107)is chosen, where τ c is some constant parameter. For the parameter valuesdefined in Table 4.2, we obtain:{1min4 (D c + χ 0 ) , 0.209 } [ ]1days, = 0.590D c 4D g mm 2 .For the current problem statement, the numerical simulations, carried outfor the parameter value τ c = 0.9 and for the different mesh resolutions,

5.4. Discussion and conclusions 183provided nonnegative solutions for the cell densities and the growth factorconcentration, in which no spurious oscillations were recognized. The numericalsolutions, obtained for the parameters τ c = 1 and τ c = 0.5, wereclose to each other. Hence we assumed, that the chosen time-step size issmall enough for stability and positiveness of the numerical solution.5.4 Discussion and conclusionsThe aim of the current work is to develop the algorithm, which allows toobtain the numerical solution in a two dimensional physical domain for theperi-implant osseointegration model, defined in Chapter 4. The consideredmodel is formulated in terms of three coupled nonlinear time-dependentadvection-diffusion-reaction equations, which are defined within the evolvingin time domain.The method of lines is used as a basis of the current numerical approach.The discretizations in time and space are separated. This allows to applysuperposition in the discretization of the advection, diffusion and reactionterms. Since a moving boundary problem is considered, the efficiency ofthe method, which is used to construct the computational mesh, is of greatimportance. In order to avoid a complete remeshing at every time step,the embedded boundary method is applied on the initial regular rectangulargrid. This means, that only the control volumes, adjacent to the boundary ofthe domain, change their shape. The evolution of the domain is tracked withthe level set method, which is easily implemented on the initial rectangulargrid. The irregular geometry of the physical domain is determined from thevalues of the level set function at the nodes of the rectangular grid. Thecomputational mesh consists of the regular rectangular control volumes inthe middle and of the irregular computational cells near the boundary ofthe domain.Special attention should be paid to the discretization of the hyperbolicterms from the governing partial differential equations. The finite volumemethod is known to be efficient for this type of problems. With this method,a Riemann problem should be solved at the inner edges of the control volumes.The Riemann problem is defined for the nonstrictly hyperbolic systemof equations without genuine nonlinearity, which is analogous to nonconvexflux functions in scalar hyperbolic equations [59]. In this chapter, the exactRiemann solution is defined in the simple form.For the discretization of the advection-diffusion terms, the values of thegradients of the total cell density and of the growth factor concentrationshould be approximated at the inner edges of the control volumes. Sincethe control volumes, which are located near the boundary of the domain,can be of an irregular shape, the approach for the gradient approximationon the irregular grid is constructed.

184 Chapter 5. Numerical algorithmThe explicit trapezoidal method is used for the time integration of theordinary differential equations, which are obtained after the discretizationof the PDE’s in the physical and maturation spaces. The explicit method ischosen, since the positivity of the solution is very important for the currentmodel. First of all, negative densities and concentration do not make anysense from biological point of view. Second, it is shown that negative valuesof the cell densities can grow in magnitude due to nonlinearity of the fluxfunctions. Hence even small negative values of the cell densities can causebad convergence of an approximate solution to the exact solution. Anotheradvantage of the explicit methods, is that they are easy to implement andthat they are more convenient on the stage of the verification of the solutionof the complex numerical model. A disadvantage of the explicit methods isthat we have to use very small time steps, in order to obtain positive andstable numerical solutions. This leads to large computation times, especiallywhen a fine mesh in the physical domain is considered.A weak point of the current algorithm is the large computation time,needed to obtain the numerical solution, when a fine mesh resolution isused. In future, the development of an efficient time integration method,which will reduce the time of computations, could improve the approach,proposed in the present work. An operator splitting method can be usefulin this case as discussed in Section 6.2.

CHAPTER 6Conclusions and outlook6.1 ConclusionsExtended and deep knowledge of the main processes occurring in a realsystem, which can be obtained, for example, from experiments and observations,are of great importance on the stage of construction of a theoreticalmodel for the considered system. It is also critically important to knowthe characteristics of the developed mathematical model in order to justifythat the initial assumptions incorporated in the model are representedby the current mathematical formalism in a correct way. It can happenthat solutions of the mathematical model have some features, which werenot expected initially. Such a situation is considered in Chapter 2. It isfound that biologically irrelevant wave-like patterns appear in the solutionfor a continuous bone regeneration model. A stability analysis allows todetermine the relation between the parameter values and a characteristicbehavior of the solution. It is demonstrated by combining analytical andnumerical approaches that for a certain range of the parameter values thesolution for the cell densities and the growth factor concentration convergesto unphysical wave-like profiles. However, if the parameter values are withina stability region, then biologically irrelevant characteristics of the solutiondisappear, and a final homogeneous distribution of cells and growth factorsis predicted.Modeling can be considered as a representation of complex phenomenain a rather simple form by taking certain assumptions and focusing on thoseaspects, that are supposed to have the most significant impact on a processthat is considered. Therefore, the main processes occurring in bone regenerationare modeled by a number of various mathematical formalisms. Differenttheoretical approaches yield different results and allow to study bone185

186 Chapter 6. Conclusions and outlookregeneration from various perspectives. In Chapters 3 and 4, the modelsfor cell differentiation and bone formation, which are conceptually differentfrom classical approaches, are proposed. Several new features of bone regenerationare represented with these approaches, which cannot be simulatedwith the already existing models.The evolutionary differentiation model described in Chapter 3 allowsto incorporate a history of cell differentiation, so that the current state ofcells depends on how the cells evolved before. In contrast to an immediatedifferentiation formalism, which is commonly used in classical models, thenew approach assigns a finite, bounded, time of differentiation to each cell.The finite time of differentiation corresponds to a gradual gaining of newproperties by cells in the course of time, which is in line with experimentalobservations. Another advantage of the evolutionary differentiation model,is that bone formation within the peri-implant region can be predicted tostart only after some time from the implant placement, as it happens inreality.Due to its advanced characteristics, the evolutionary approach is an essentialpart of a moving boundary problem for endosseous bone healing,which is presented in Chapter 4. A moving boundary-type of bone formationaround endosseous implants, which is well observed in experiments, is directlyincorporated into the present model. The ossification front is modeledby the moving boundary of the computational domain, which correspondsto a soft tissue region. An explicit representation of the bone-forming surfaceis the main innovation of the proposed model, which distinguishes thecurrent formalism from the recent models for peri-implant osseointegration.The robust numerical algorithm for the solution of a complex mathematicalproblem is developed and presented in Chapter 5. The main challengeswere to capture the irregular and time-evolving geometry of the problemdomain, and to construct a stable and positivity preserving discretizationof the nonlinearly coupled system of time dependent taxis-diffusion-reactionequations for this domain. Such approaches based on the method of lines,the finite volume method, the level set method and the embedded boundarymethod are incorporated in the present elaborate numerical algorithm.6.2 RecommendationsA general comment, which concerns most of the recent models of bone regenerationand, in particular, the two models presented in this work, is thatfurther model validation is necessary in order to assess an applicability anda robustness of the proposed theoretical approaches. This can be done bycomparing observation data obtained for various experimental settings withthe corresponding simulation results performed for the current continuousmodels.

6.2. Recommendations 187For a validation of the evolutionary differentiation approach, the presentapproach is incorporated into a mechanoregulatory model. Mechanoregulatoryexpressions for differentiation rates are derived from numerical simulationsand based on the educated guesses. In order to verify and to evaluatethe expressions for the differentiation rates directly, corresponding experimentalstudies should be done. One of the potential research directions, relatedto the present study, is an experimental and theoretical investigation ofhow cells gain properties of some phenotype in time under influence of variousfactors (for example, mechanical loading or an influence of growth factorsand other chemicals). The mechanoregulatory model, which is described inChapter 3, is derived in a rather simple form, since it is used mainly tocheck the performance of the evolutionary differentiation approach. Therefore,there is a lot of space for further improvement of the model. Forexample, an additional unknown for the growth factor concentration can beincorporated into the formalism, in order to consider bioregulatory effectsoccurring during bone regeneration. A mechanical stimulus can be adaptedto a dynamical loading of the healing tissue, which then can be modeled as aporo-elastic medium. Considering cell differentiation as a partly stochasticprocess can be a following step in extending the present model. Based onthe model for cell differentiation, a similar evolutionary approach for cellproliferation is another potential line of the research.The aforementioned directions for future research can be also consideredfor the improvement of the moving boundary model presented in Chapter4. A further justification and adjustment of the model parameters andformalisms, which are used to model cellular and biochemical processes,can be made by comparing paths of the ossification front during endosseousbone healing for various experimental settings. Making a diffusion coefficientdependent on the local cell density and on the growth factor concentrationand analogous modification of the rest of the parameters can deliver newinteresting results, which were not observed at the present moment. Adisadvantage of all these extensions is the increase of the model complexityand of the number of parameters that are barely known, or very hard todetermine or to estimate.The numerical algorithm, developed for the solution of the moving boundaryproblem is based on an explicit time integration scheme, which yieldsan enormous calculation time, especially if the control volumes used forthe spatial discretization are small. In this situation, an operator splittingmethod can be useful. Applying an implicit time integration for diffusionand reaction terms and an explicit integration scheme for hyperbolic taxisand differentiation terms will allow the use of larger time steps, such that thestability of the numerical solution will be maintained. However, an implicitintegration can violate a positivity of the solution, which is critical for theconsidered problem. Therefore, further research on the efficient and robusttime integration method for the present continuous model is needed.

188 Chapter 6. Conclusions and outlookAn interesting behavior of the numerical solutions was obtained on themeshes with various resolution in Chapter 4. The high density- high concentrationlayer, which was located at the old bone surface, had a wave-likepattern on coarse meshes. The aforementioned pattern disappeared withthe increase of the mesh resolution. The issue of the pattern appearance inthe numerical solution, which is dependent on the sizes of the mesh elementsand on the model parameters, can be a subject of a theoretical research. Anelaborated stability analysis may unravel the reasons of such a behavior ofthe solution and provide some guidelines on the choice of mesh resolutionand of the parameter values.The current numerical scheme operates on a uniform initial rectangularmesh. It was concluded in Chapter 4, that it is necessary to use very finemeshes within the physical domain in order to get convergent numericalsolutions. Small element sizes lead to a large number of time steps, to abnormalamount of the required memory and hence to a huge CPU time. Infact, an enhanced mesh resolution is only necessary in the regions, where thegradients of the cell densities and/or of the growth factor concentration arehigh. Therefore, small mesh elements are only needed within a small part ofthe physical domain: at the boundary between the high density - high concentrationlayer located near the ossification front and the rest of the physicaldomain, where the cell densities and the growth factor concentration areclose to zero. In such a situation, adaptive mesh refinement can significantlyreduce the computational time and required computer resources. Therewith,an incorporation of the adaptive mesh refinement technique into the presentnumerical algorithm is a potential direction for the following studies. Wealso think that spectral or discontinuous Galerkin methods may be worthinvestigating with respect to accuracy issues. We remark that the use oflimiters will probably still be necessary in order to grant positivity of thenumerical solution.

APPENDIX ANotes to the moving boundarymodel for endosseous healingA.1 Cell source at the old bone surfaceThe parameter t cbone denotes the time period, during which the influx ofimmature non-differentiated cells takes place near the old bone. In Abrahamssonet al. [1] new bone formation is recognized at the end of the firstweek after the implant placement. Therefore, the value t cbone = 1 day isestimated from numerical simulations, so that new bone formation is observedat the old bone surface at time t = 7 days, if no growth factor sourceis considered at the smooth implant surface.Decrease of t cbone to value 0.5 days, leads to the issue, that fewer cells arerecruited from the old bone surface. Bone formation starts in the corners,formed by the old bone surface and by the tread of the implant. No boneforms in the middle of the old bone surface (Figure A.1).On the other hand, for the value 2 days of the parameter t cbone , newbone does not form. This happens, due to the extensive source of nondifferentiatedcells at the old bone surface. The differentiating osteogeniccells are squeezed from the old bone surface by non-differentiated cells. Cellswith low level of differentiation release little growth factor (see equation(4.3)). A low concentration of growth factor near the old bone surface leadsto a slow cell differentiation in this region (see the expression for u b (g) inTable 4.1). Therefore, the density of mature osteogenic cells is zero, andno new bone forms near the old bone surface within the considered timeframe.189

190 Appendix At=7 days0.5c m0.201.4 1.80.80.60.40.5c tot0.2gt=9 days0.501.4 1.80.80.60.40.20.501.4 1.801.4 1.80.80.60.40.80.60.40.20.50.501.4 1.801.4 1.832142t=28 days0.501.4 1.80.80.60.40.20.501.4 1.80.80.60.40.20.501.4 1.8321Figure A.1: Plots of the mature cell density c m , of the total cell densityc tot and of the growth factor concentration g at the time moments t = 7, 9and 28 days for the parameter values given in Tables 4.1 and 4.2 and fort cbone = 0.5 days. The bone-forming surface moves from the the corners,formed by the old bone surface and by the tread of the implantA.2 Fluxes at the moving boundaryNext, the flux of some quantity is considered, which can be, for example,cell density, which is denoted as c, near the moving boundary. Cellularprocesses, like differentiation and proliferation, are disregarded. Let thedensity be defined within some region Ω c (t), which evolves in time and hasthe moving boundary ∂Ω c (t). The movement of the domain boundary isdetermined by the velocity field ⃗v( X, ⃗ t), X ⃗ ∈ ∂Ωc (t), t > 0. Since theflux of cell density is studied, assume, that the evolution of the density isdetermined only by cell migration. Then, the cell dynamics can be describedby the conservation law:∂c∂t (⃗x, t) + ∇ · ⃗F (c, ⃗x, t) = 0, ⃗x ∈ Ω c , t > 0, (A.1)where F (c, ⃗x, t) is the flux density vector field. From application of theLeibniz integral rule to equation (A.1) and from the divergence theorem, it

A.2. Fluxes at the moving boundary 191follows:∫ ∫d∂cc dV =dt Ω c Ω c∂t∫∂Ω dV + ∫c(t)∫+ c ⃗v · ⃗n dS = −∂Ω c(t)∫c ⃗v · ⃗n dS = −∂Ω c(t)∇ · ⃗F dVΩ∫c⃗F · ⃗n dS + c ⃗v · ⃗n dS.∂Ω c(t)(A.2)Let us denote the projection of the velocity ⃗v on the normal ⃗n asv n ( ⃗ X, t) = ⃗v( ⃗ X, t) · ⃗n( ⃗ X, t),where X ⃗ ∈ ∂Ω c (t), t > 0 and ⃗n is the outward unit normal to the surface∂Ω s .Assume first, that cells are pushed by the moving boundary in a pistonlikeway. In this case, the total number of cells within the domain will notchange, i.e.∫dc dV = 0.(A.3)dt Ω c(t)The velocity of cells at the moving boundary ∂Ω s , projected on the normalto the boundary, will be equal to the normal velocity of the boundary v n .Hence, the flux through the surface element ⃗n dS will be equal to the localcell density times the velocity in direction ⃗n times the area of the surfacedS:⃗F (c, ⃗ X, t) · ⃗n( ⃗ X, t) dS = c( ⃗ X, t) v n ( ⃗ X, t)dS,or⃗F (c, ⃗ X, t) · ⃗n( ⃗ X, t) = c( ⃗ X, t) v n ( ⃗ X, t),(A.4)where X ⃗ ∈ ∂Ω c (t), t > 0, ⃗n is the outward unit normal to the surface ∂Ω c (t).Equation (A.4) is the natural boundary condition for the cell density underinterfacial movement. Further, from the equations (A.2) and (A.4) it follows,that the condition (A.3) is satisfied.Suppose now, that the cells, adjacent to the boundary, attach to it andbecome immobile. They do not move together with boundary, but theyare trapped by it. Hence, it is assumed, that they do not change theirposition, i.e. their velocity and, consequently, their flux, with respect to thecoordinates of the physical space, become equal to zero, after the cells areattached to the domain boundary:⃗F (c, X, t) = ⃗0, ⃗ X ∈ ∂Ωc (t), t > 0. (A.5)If the cells are trapped by the moving boundary, then the total cell numberwithin the domain Ω s will decrease. This fact can be derived from equations(A.2), (A.5) and from equation (4.4), where the normal velocity v n isdefined as non-positive variable, i.e.⃗v · ⃗n = v n ≤ 0,

192 Appendix Awhere ⃗n is the outward normal. This means, that the boundary movesinwards the domain (domain shrinks). Consequently,∫ ∫dc dV = c ⃗v · ⃗n dS ≤ 0.(A.6)dt Ω c ∂Ω c(t)Neumann boundary conditions are formulated for the flux through theboundary surface. Thus, condition (A.5) should be rewritten in the form:⃗F (c, X, t) · ⃗n(X, t) = 0, ⃗ X ∈ ∂Ωc (t), t > 0, (A.7)where ⃗n is the outward unit normal to the surface ∂Ω c (t).The boundary conditions, formulated in Section 4.3.2, are derived fromthe equations (A.4) and (A.7) and from the assumptions, made in Section4.3.2. The flux density field ⃗ F is substituted by expressions(−D c ∇ s c j + χ(g, c tot )c j ∇ s g) , j = i, m,for immature and mature cell densities, and by the expression −D g ∇ s g, forgrowth factor concentration, respectively.Relation (A.7), used as a boundary condition for the mature cell density,is derived from a rather rough assumption, that these cells do not changetheir position in physical space, after they attach to the moving boundary. Inreality, some osteoblasts move simultaneously with the bone-forming surface,and a part of them becomes completely surrounded by the new bone matrix[63]. The mechanism of movement of the osteoblasts, which attach to thesolid surface and which release new bone matrix, seems to be complicatedand not well studied in the literature. In future, the present model can beimproved by a more accurate representation of the relative movement of themature osteogenic cells and the bone-forming surface.

APPENDIX BNotes to the numerical algorithmB.1 Determinant of the linear systemLinear equations (5.44) can be written in the matrix form:A ⃗a = ⃗ C,where⎡⎢A = ⎣1 r j−1,kc1 r j,kc1 rcj+1,kThe determinant of matrix A is equal to:Izzj−1,kIzzj,kIzzj+1,k⎤⎥⎦ .det(A) = (Izz j,k − Izzj−1,k)(r j+1,kc− r j,kc )+ (I j+1,kzz− Izz j,k )(rcj−1,k − rc j,k ). (B.1)In this appendix, it is proved that det(A) ≠ 0.Let us define a new coordinate system (r ′ , z ′ ), such that r ′ = r − r 0 andz ′ = z − z 0 , where r 0 , z 0 are some real numbers. The r ′ -coordinate of thecenter of any area V is equal tor ′ c = r c − r 0 ,(B.2)where r c is the r-coordinate of the center in the old coordinates. The secondmoment of the area is modified in the new coordinate system in the followingwayI ′ zz = 1µ(V )∫Vr ′2 dv = 1 ∫(r − r 0 ) 2 dv = I zz − 2r 0 r c + rµ(V )0, 2 (B.3)Vwhere I zz is the corresponding moment in the old coordinates.193

194 Appendix BRemark B.1. Since the second moment of area is non-negative, then I ′ zz ≥ 0for any real r 0 . If it is assumed, that r 0 = r c , then from equation (B.3) itfollows, thatI zz = I ′ zz + r 2 c ≥ r 2 c ,i.e. the second moment of area is greater or equal to the square of thecorresponding coordinate of its center.From relations (B.2), (B.3), it follows that the determinant of the matrixA is invariant with respect to the parallel shift of the coordinate system.Hence, without loss of generality, it can be assumed, that the coordinateorigin is situated in the middle of the line segment, which connects the+r j+1,kccenters of cells V j−1,k and V j+1,k , i.e. rj−1,k cR > 0. Further, the following parameters are introduced:2= 0 or −rcj−1,k= r j+1,kc =r m,kmax =max r, r m,kmin =(r,z)∈V m,kmin r, r m,k(r,z)∈V|max| =m,kmax |r|,(r,z)∈V m,km ∈ {j − 1, j, j + 1}.(B.4)From Figure 5.3a it follows, that−R = r j−1,kchence r j,k|max|< R. Then,I j,kzz =From Remark B.1 it follows, thatI m,kzz< r j,kmin < rj,k c < rmax j,k < rc j+1,k = R, (B.5)∫1r 2 dv ≤ (r j,kµ(V j,k )|max| )2 < R 2 .V j,k≥ (rcm,k ) 2 = R 2 > Izz j,k , m ∈ {j − 1, j + 1}. (B.6)Inequalities (B.5) and (B.6) yield, that the determinant of matrix A is lessthan zero. Hence, there exists a unique solution for linear system (5.44).B.2 Number of adjacent cellsConsider for example an inner vertical edge e j+1,k,which lies within the2physical domain Ω s . Hence, rectangular cells of the initial mesh W j,k , W j+1,klie within the domain, at least partially. Then, there exist active cells V m,k =W m,k ∩ Ω s , m ∈ {j, j + 1}. Further, all possible situations of the locationof the vertices of the inner edge e j+12 ,k with respect to domain Ω s will bedetermined, and the number N c of active cells, exclusive cells V j,k , V j+1,k ,adjacent to these vertices will be found in each case. The vertices of edgee j+12 ,k can have at most 4 adjacent active cells, exclusive cells V j,k, V j+1,k

B.2. Number of adjacent cells 195(see for example Figure B.1a), hence N c ≤ 4. It will be proved, that thisnumber is also greater than zero.Case 1a. Assume, that one of the vertices of the edge e j+1,k,for examplevertex v j+212 ,k− 1 , is inner, i.e. the level set function is negative at this2vertex. Then, besides cells V j,k , V j+1,k , there exist two active cells V j,k−1 ,V j+1,k−1 , adjacent to vertex v j+12 ,k− 1 . If vertex v2j+12 ,k+ 1 is also an inner2vertex, then there exist two active adjacent cells V j,k+1 , V j+1,k+1 , and N c = 4(Figure B.1a).z k+ 12z k− 12r j− 1 r 2 j+ 1 r 2 j+ 3 2(a)z k+ 12z k− 12r j− 12r j+ 12(b)r j+ 32z k+ 12z k− 12r j− 12r j+ 12(c)r j+ 32z k+ 12z k− 12r j− 12r j+ 12(d)r j+ 32z k+ 12z k− 12r j− 12r j+ 12(e)r j+ 32z k+ 12z k− 12r j− 12r j+ 12(f)r j+ 32z k+ 12z k− 12r j− 12r j+ 12(g)r j+ 32Figure B.1: Possible locations of the inner edge e j+1,kand of the adjacent2cells V j,k , V j+1,k with respect to the physical domain Ω sCase 1b. In case, when vertex v j+12 ,k+ 1 is a boundary vertex, then, besidescells V j,k , V j+1,k , there may exist from zero to two active cells, adjacent2to this vertex (see for example Figures B.1b–d). The number N c ranges fromtwo to four.Case 1c. If vertex v j+12 ,k+ 1 2is an outer vertex, then the additional boun-and v j+12 ,k− 1 . This new2dary vertex is defined between vertices v j+12 ,k+ 1 2vertex is the second vertex of the inner edge e j+1,k. Only cells V j,k, V j+1,k2are adjacent to the new boundary vertex (Figure B.1e). Hence, N c = 2.Case 2. Suppose, that vertex v j+12 ,k− 1 2inner, then vertex v j+12 ,k+ 1 2symmetrical to Case 1c.has to be an inner vertex.Case 3a. Assume, that vertex v j+12 ,k− 1 2vertex v j+12 ,k+ 1 2is inner. If vertex v j+12 ,k+ 1 2is outer. Since, edge e j+12,kisThis situation isis a boundary vertex.Thencannot be an outer vertex, since the considered edge e j+12 ,kis an inner vertex, then the conclusions from

196 Appendix BCase 1b are valid.Case 3b. Suppose, that vertex v j+12 ,k+ 1 2is also a boundary vertex. Let usnow consider vertices v j−12 ,k+ 1 and v2 j+32 ,k+ 1 . If they are boundary vertices,2then vertices v j−12 ,k− 1 and v2 j+32 ,k− 1 cannot be outer vertices. Otherwise,2cells W j,k , W j+1,k would be outer cells, and there would not exist activecells V j,k , V j+1,k . This contradicts to the initial assumption. Hence, theconsidered cells V j,k , V j+1,k are regular rectangular cells (Figure B.1f). Inthis situation, the normal derivatives and the value of the growth factorconcentration are obtained with formulas (5.55), (5.56). Further, it is notnecessary to define one more adjacent cell to approximate the gradients ofthe unknown functions.Case 3c. Assume, that at least one of the vertices v j−12 ,k+ 1 and v2 j+32 ,k+ 1 ,2is inner (Fig-is not a boundary vertex. If, for example, vertex v j−12 ,k+ 1 2ure B.1g), then cell V j,k+1 will be an active cell. Hence, N c ≥ 1.Case 3d. If, for example, vertex v j+32 ,k+ 1 is outer (Figure B.1g), then2vertex v j+32 ,k− 1 should be an inner vertex. Otherwise, there would not exist2active cell V j+1,k . It follows then, that cell V j+1,k−1 is active, and N c ≥ 1.Therefore, it has been proved for all possible situations, when at leastone of cells V j,k , V j+1,k is irregular, that 1 ≤ N C ≤ 4. The proof for thehorizontal inner edges is analogous.

AcknowledgmentAt the end of my four year research at Delft University of Technology, Iwould like to acknowledge the people who helped me a lot in various aspectsof my work and made my stay in Delft an exciting experience.First of all, I would like to thank my supervisor Prof. Kees Vuik forgiving me an opportunity to work on the present project in the Group ofNumerical Analysis at Delft Institute of Applied Mathematics, and for hiswise guidance and valuable suggestions during our meetings. Thank youKees, for your support and understanding from my first days in Delft tillthe end of my project. It was an honor for me to do a research under yoursupervision.It was a great pleasure to me to work with Dr. Fred Vermolen, my dailysupervisor. Fred, informal and trusting relations and the confidence that Ican rely on you were extremely important for me to adapt fast in the Netherlandsand to feel comfortable at my new job at the Delft University. Yoursupport, coaching and encouragement have helped me to remain optimisticin several complicated situations concerning my research and to progressin my work. I enjoyed our talks during out weekly meetings and during anumber of formal and informal events in Delft and Amsterdam. The tripsto Zaragoza, Madrid and Edinburgh were exciting. It was a fun to walk inthese cites accompanied by you and to go to bars in the evenings.The present research was done together with Prof. José Manuel García-Aznar from the University of Zaragoza, Spain. Our cooperation was extremelyvaluable for me and resulted into a number of interesting researchachievements published in several journal articles and in the present thesis.Manu, I appreciate a friendly atmosphere your created during our meetings,your enthusiasm and willingness to help with various complicated aspects inthe research. I am happy that I had a possibility to work with you.I would like to acknowledge the financial support of my stay and workin Delft provided by the Delft Institute of Applied Mathematics. I wouldlike to thank Agentschap NL for the sponsored collaboration between DelftUniversity of Technology and the University of Zaragoza.197

198 AcknowledgmentTheda Olsder has managed with a huge amount of formal proceduresrelated to my arrival and stay in the Netherlands. I appreciate her supportand readiness to find a way out of any complicated situation.I am also grateful to Dr. Wim van Horssen for his help with severaladministrative questions.I am grateful to my first supervisor at the Belarusian State UniversityProf. Michail Zhuravkov. He played an important role for my decision tocontinue study at the PhD level.I am thankful to Dr. Valery Gaiko, who actually told me about theopen PhD position at Delft University of Technology when I was in Minskconsidering various places to continue my education.It was a fun and pleasure to share the office with the cheerful, friendly andenthusiastic guys Tijmen and Reijer, and then Xiaozhou and Gido. Duringmy study in Delft I met a lot of pleasant and interesting people whom I wouldlike to express my gratitude: the former and present PhD students Denis,Elwin, Rohid, Paulien, Liangyue, Bowen, Jing, Miranda, Fahim, Abdul,Ibrahim, Joost, Lech, Miriam, Alexandr, Timofiy, the post doc fellow Sergey,the permanent staff members of the the Numerical Analysis group Jenifer,Domenico, Duncan, Martin van Gijzen, Prof. Kees Oosterlee, Guus Segal,Niel Budko, Fons Daalderop, Peter Sonneveld, our secretaries Deborah andMechteld, Prof. Sybrand van der Zwaag from the aerospace faculty, EtelvinaJavierre from Centro Universitario de la Defensa in Zaragoza.I am thankful to Maxim Molokov and to Masha for their help with designof the cover for the present thesis.I would like to acknowledge my friends in Belarus and in the Netherlandswho added colors to my life and with whom I had a lot of fun during myfree time.I express my deepest gratitude to all my large family and in particularto my parents. Despite a large distance to home I always experienced theirsupport and encouragement during these four years. Masha and Paulinkaare the source of inspiration in my work. I thank them for their patienceand care which were of enormous importance to me.

Curriculum vitae• September 2008 - 2012: Delft University of Technology, The Netherlands,PhD student in Numerical Analysis, supported by Delft Instituteof Applied Mathematics, subject: Modeling bone regenerationaround endosseous implants, supervisor: Prof.dr.ir. Kees Vuik, dailysupervisor: Dr.ir. Fred Vermolen, Department of Applied Mathematics,Faculty of Electrical Engineering, Mathematics and Computer Science• July 2007- July 2008: Engineer, Belgorkhimprom, Minsk, Belarus• 2002- July 2007: Belarusian State University, Minsk, Belarus, Diplomawith honors in Applied Mathematics and Mechanics• 1991-2002: Gymnasium #4, Minsk, Belarus• Born on 01-09-1984 in Mahiliou, Belarus199

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List of publications• P. Prokharau and F. Vermolen. Stability analysis for a peri-implantosseointegration model. J Math Biol, pages 1 – 32, 2012. ISSN 0303-6812. doi: 10.1007/s00285-012-0513-1• P. Prokharau, F. Vermolen, and J. M. García-Aznar. Evolutionarycell differentiation approach in a peri-implant osseointegration model.In Proceedings of II International Conference on Tissue Engineering,pages 149–156, 2011.• P. A. Prokharau, F. J. Vermolen, J. M. García-Aznar. Model fordirect bone apposition on pre-existing surfaces, during peri-implantosseointegration. J Theor. Biol. 304: 131–142, 2012. ISSN 0022-5193. doi: 10.1016/j.jtbi.2012.03.025• P. A. Prokharau, F. J. Vermolen, J. M. García-Aznar. A mathematicalmodel for cell differentiation, as an evolutionary and regulated process.Accepted for publication in Comput. Meth. Biomech. Biomed. Eng.• P. A. Prokharau, F. J. Vermolen, J. M. García-Aznar. Numericalmethod for the bone regeneration model, defined within the evolving2D axisymmetric physical domain. Accepted for publication in Comput.Meth. Appl. Mech. Eng.211