Modeling bone regeneration around endosseous implants

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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
Page 1 and 2: Modeling bone regeneration around e
Page 3: to my parents, Masha and Paulinka
Page 6 and 7: viSummaryentiate into other cell ph
Page 9 and 10: SamenvattingModelleren van botregen
Page 13: CONTENTSSummarySamenvattingvix1 Int
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
Page 57 and 58: 2.4. Numerical results 41or in the
Page 59 and 60: 2.4. Numerical results 43but such t
Page 61: 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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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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