nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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vi Contents Significance of the Apoptotic Wave . . . . . . . . . . . . . . . . . . . . . 33 Michaelis-Menten versus a Hill function . . . . . . . . . . . . . . . . . . 34 Implication of secretion of cytokines upon Ba phagocytosis . . . . . . . 35 Activation by phagocytosis of necrotic β-cells . . . . . . . . . . . . . . . 36 5.8 Discussion of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Reversible or irreversible activation? . . . . . . . . . . . . . . . . . . . . 38 The DuCa model with turnover of Ma . . . . . . . . . . . . . . . . . . . 41 6 The Intermediated Model 45 Model observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.1 Fixed Points of the IM Model . . . . . . . . . . . . . . . . . . . . . . . . 48 Stability of the fixed points . . . . . . . . . . . . . . . . . . . . . . . . . 50 Downfall of the IM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Phase plane analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A brief note on linearizing . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 The IM Including Crowding-Terms . . . . . . . . . . . . . . . . . . . . . 60 6.3 Recapitulation of The Analysis of the IM . . . . . . . . . . . . . . . . . 63 7 A Brief Introduction to Bifurcation Analysis and Numerical Methods 65 7.1 Generic Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 The Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.3 Biological Relevance of Bifurcation Analysis . . . . . . . . . . . . . . . . 69 7.4 Choice of Bifurcation Parameters . . . . . . . . . . . . . . . . . . . . . . 69 7.5 Locating the Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Evaluation of stability of the fixed points and detection of bifurcations . 71 8 Bifurcation Diagrams and Analysis 73 8.1 Codimension One Bifurcations . . . . . . . . . . . . . . . . . . . . . . . 73 8.2 Using f1 as the bifurcation parameter . . . . . . . . . . . . . . . . . . . 74 Basic features of the NOD bifurcation diagram . . . . . . . . . . . . . . 74 Eigenvalues, manifolds and behavior of the fixed points on the NODf1-USB 75 Eigenvalues, manifolds and behavior of the fixed points on the NODf1-LUB 79 Combining the analysis of the NODf1-USB and LUB . . . . . . . . . . . Hidden features in figure 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . 81 84 The Balb/c bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . 87 8.3 Using f2 as the bifurcation parameter . . . . . . . . . . . . . . . . . . . 88 8.4 Reducing the DuCa-model . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9 Discussion of the Bifurcation Analysis 97 9.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10 Conclusion 1 103 11 Expanding and Modifying the DuCa model 105 11.1 Including a Compartment of Healthy β-cells . . . . . . . . . . . . . . . . 105 Model criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.2 First approximation to a governing equation – Model A . . . . . . . . . 106 11.3 Modifying the governing equation – Model B . . . . . . . . . . . . . . . 110
Model B – Balb/c-simulation . . . . . . . . . . . . . . . . . . . . . . . . 110 Model B – NOD-simulation . . . . . . . . . . . . . . . . . . . . . . . . . 111 11.4 Adding β-cell growth – Model C . . . . . . . . . . . . . . . . . . . . . . 112 Model C – Balb/c-simulation . . . . . . . . . . . . . . . . . . . . . . . . 114 Model C – NOD-simulation . . . . . . . . . . . . . . . . . . . . . . . . . 115 12 Discussion of the Expanded Model 117 12.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 IL-1 induction of primary necrosis . . . . . . . . . . . . . . . . . . . . . 120 Adding T cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Adding treatment to Model C . . . . . . . . . . . . . . . . . . . . . . . . 121 13 Conclusion 2 123 Bibliography 125 A Mathematical Appendices 131 A.1 Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . 131 A.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . 132 A.3 The Hartman-Grobman Theorem . . . . . . . . . . . . . . . . . . . . . . 132 A.4 Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.5 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.6 Dimensionless Form of the Intermediate Model Including Crowding Terms136 A.7 Fixed Point for the IM Including Crowding Terms . . . . . . . . . . . . 137 B Additional Figures 139 B.1 Eigenvalue-Plots Used to Evaluate Stability of the Fixed Points Shown in Figure 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.2 Eigenvalue-Plots Used to Evaluate Stability of the Fixed Points Shown in Figure 8.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.3 Eigenvalue-Plots Used to Evaluate Stability of the Fixed Points Shown in Figure 8.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.4 Eigenvalue-Plots used to Evaluate Stability of the Fixed Points Shown in Figure 8.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B.5 Eigenvalue-plots Used to Evaluate Stability of the Fixed Points Shown in Figure 8.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 B.6 Stable Spirals for f1 near the Bifurcation Point . . . . . . . . . . . . . . 155 C matlab Code 157 C.1 Code for the Bifurcation Diagrams in Chapter 8 . . . . . . . . . . . . . 157 C.2 Code for the Newton-Raphson Method . . . . . . . . . . . . . . . . . . . 167 Code for Approximation of the Jacobian . . . . . . . . . . . . . . . . . . 168 C.3 The Function-File for Model C . . . . . . . . . . . . . . . . . . . . . . . 170 Simulation Code for Model C . . . . . . . . . . . . . . . . . . . . . . . . 171 Index 173 vii
Page 1 and 2: - I, OM OG MED MATEMATIK OG FYSIK E
Page 3 and 4: Exploring Treatment Strategies for
Page 5: Preface iii questions that were out
Page 10 and 11: List of Tables 5.1 summarizes the m
Page 12 and 13: x List of Figures 8.3 Phase space p
Page 14 and 15: xii
Page 16 and 17: 2 Introduction In the following dec
Page 18 and 19: 4 Introduction There are several sc
Page 20 and 21: 6 Introduction because I had to mak
Page 22 and 23: 8 Type 1 Diabetes and Its Etiology
Page 24 and 25: 10 Type 1 Diabetes and Its Etiology
Page 27 and 28: 3 Prospects for Therapy - A Mini Re
Page 29 and 30: 3.3 Gastrin 15 eration in situ Urus
Page 31 and 32: 4 Mathematical Modelling The langua
Page 33 and 34: 5 The DuCa Model In the following w
Page 35 and 36: 5.3 The DuCa Model - An Appetiser 2
Page 37 and 38: 5.4 The DuCa Model - Compartment Mo
Page 43 and 44: 5.6 Our Compartment Model 29 migrat
Page 45 and 46: 5.6 Our Compartment Model 31 a ✓
Page 47 and 48: 5.7 Discussion of Marée et al. (20
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46 The Intermediated Model The diff
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52 The Intermediated Model restrict
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54 The Intermediated Model We see t
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56 The Intermediated Model paramete
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58 The Intermediated Model ✻ Ma(t
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60 The Intermediated Model 6.2 The
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62 The Intermediated Model ✻ Ma(t
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7 A Brief Introduction to Bifurcati
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7.2 The Hopf bifurcation 67 the gen
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126 Bibliography Christen, U. and M
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128 Bibliography Notkins, A. L. and
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130 Bibliography Yu, J. and G. S. E
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132 Mathematical Appendices the com
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134 Mathematical Appendices in that
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136 Mathematical Appendices A.6 Dim
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140 Additional Figures Figure B.2 P
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142 Additional Figures B.2 Eigenval
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144 Additional Figures Figure B.8 P
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148 Additional Figures Figure B.14
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158 matlab Code w(r+1)=w(r)+dw; g=w
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160 matlab Code end V_11(r)=E(1); V
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162 matlab Code n=675; w_5=0; dw_5=
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164 matlab Code M=M’; % end s_7(r
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166 matlab Code w_7,V_34,’.b’,w
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168 matlab Code M=M’; % M must be
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170 matlab Code %options = odeset(
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172
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174 Index value, 73 Bifurcation Ana
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176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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