Modeling bone regeneration around endosseous implants

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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.
Page 1 and 2: Modeling bone regeneration around e
Page 3: to my parents, Masha and Paulinka
Page 9 and 10: SamenvattingModelleren van botregen
Page 13 and 14: CONTENTSSummarySamenvattingvix1 Int
Page 15: CONTENTSxv5.3.7 Treatment of the re
Page 18 and 19: 2 Chapter 1. IntroductionNecrotic p
Page 20 and 21: 4 Chapter 1. Introductionorganisms.
Page 22 and 23: 6 Chapter 1. Introductionenhances f
Page 24 and 25: 8 Chapter 1. IntroductionGómez-Ben
Page 26 and 27: 10 Chapter 1. Introductionstability
Page 28: 12 Chapter 1. Introductionand bioch
Page 31 and 32: 2.2. Biological model 15this chapte
Page 35 and 36: 2.3. Stability analysis 19The equat
Page 37 and 38: 2.3. Stability analysis 21χ = α m
Page 39 and 40: 2.3. Stability analysis 23Let us de
Page 41 and 42: 2.3. Stability analysis 25Remark 2.
Page 43 and 44: 2.3. Stability analysis 27where⎛
Page 45 and 46: 2.3. Stability analysis 29Lemma 2.1
Page 47 and 48: 2.3. Stability analysis 31Since k n
Page 49 and 50: 2.3. Stability analysis 332.3.4 Cor
Page 51 and 52: 2.3. Stability analysis 35The other
Page 53 and 54: 2.3. Stability analysis 37polynomia
Page 55 and 56: 2.3. Stability analysis 39This cond
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2.4. Numerical results 41or in the
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2.4. Numerical results 43but such t
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2.5. Conclusions 45ical simulations
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48 Chapter 3. Evolutionary cell dif
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3.4. Numerical simulations 71T= 6 d
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3.4. Numerical simulations 73T= 6 d
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3.5. Discussion and conclusions 75c
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3.5. Discussion and conclusions 77o
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CHAPTER 4Moving boundary model fore
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4.2. Recent mathematical models 81o
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4.3. Moving boundary model 834.3 Mo
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4.3. Moving boundary model 85Assume
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4.3. Moving boundary model 87concen
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4.3. Moving boundary model 89classi
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4.3. Moving boundary model 91zero).
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4.4. Model validation 93geometry of
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4.4. Model validation 95Table 4.1:
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4.4. Model validation 97Table 4.2:
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4.4. Model validation 99c m0.2c tot
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4.4. Model validation 101c m0.2c to
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4.4. Model validation 105ub=0.15 da
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4.4. Model validation 109t=42 days
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4.4. Model validation 111t=6 days0.
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4.4. Model validation 113the end of
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4.4. Model validation 115randomday
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4.4. Model validation 1174.4.3.2 Co
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4.4. Model validation 119L nz=10nz=
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4.4. Model validation 121log 10 err
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4.4. Model validation 123log 10 err
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4.5. Results and discussion 125Area
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4.5. Results and discussion 127t=15
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4.6. Conclusions 129bone formation
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132 Chapter 5. Numerical algorithmi
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134 Chapter 5. Numerical algorithmT
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136 Chapter 5. Numerical algorithm5
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138 Chapter 5. Numerical algorithmp
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140 Chapter 5. Numerical algorithmO
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142 Chapter 5. Numerical algorithmw
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144 Chapter 5. Numerical algorithmo
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146 Chapter 5. Numerical algorithmf
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150 Chapter 5. Numerical algorithmN
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152 Chapter 5. Numerical algorithmd
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156 Chapter 5. Numerical algorithmI
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158 Chapter 5. Numerical algorithmi
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160 Chapter 5. Numerical algorithmL
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166 Chapter 5. Numerical algorithmN
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168 Chapter 5. Numerical algorithmn
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170 Chapter 5. Numerical algorithmw
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172 Chapter 5. Numerical algorithmc
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180 Chapter 5. Numerical algorithma
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186 Chapter 6. Conclusions and outl
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188 Chapter 6. Conclusions and outl
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190 Appendix At=7 days0.5c m0.201.4
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192 Appendix Awhere ⃗n is the out
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194 Appendix BRemark B.1. Since the
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196 Appendix BCase 1b are valid.Cas
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198 AcknowledgmentTheda Olsder has
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BIBLIOGRAPHY[1] I. Abrahamsson, T.
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BIBLIOGRAPHY 203[20] P. Colella, D.
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BIBLIOGRAPHY 205[42] L. Geris, J. V
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BIBLIOGRAPHY 207[64] P. Mayer-Kucku
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BIBLIOGRAPHY 209[89] U. Simon, P. A
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