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

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List of Tables 5.1 summarizes the model-parameters and their values . . . . . . . . . . . . . . 27 5.2 different parameter values as found in the literature . . . . . . . . . . . . . 38 5.3 gives a comparison of the values as they are estimated in the most likely reversible and irreversible models in Marée et al. (2007). . . . . . . . . . . . 39 8.1 Overview of the eigenvalues of the DuCa model, with NOD parameters and f1 as the bifurcation parameter near the NODf1-USB. . . . . . . . . . . . . 80 8.2 Overview of the eigenvalues of the DuCa model, with NOD parameters and f1 as the bifurcation parameter near the NODf1-LUB. . . . . . . . . . . . . 81 8.3 Overview of the eigenvalues of the DuCa model, with NOD parameters and f2 as the bifurcation parameter near the USB. . . . . . . . . . . . . . . . . 91 8.4 Overview of the eigenvalues of the DuCa model, with NOD parameters, and f2 as the bifurcation parameter near the NODf2-LUB. . . . . . . . . . . . . 92 12.1 Summary of the expanded models. . . . . . . . . . . . . . . . . . . . . . . . 119 viii
List of Figures 2.1 Relation between the number of autoantibodies in the blood and how likely the person is to develop diabetes . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Dependence of treatment strategy on β-cell mass . . . . . . . . . . . . . . . 13 5.1 shows an example of a generic compartment model . . . . . . . . . . . . . . 22 5.2 shows the compartment system as given by Marée et al. (2006) . . . . . . . 24 5.3 dynamics of the Balb/c-mouse and NOD-mouse. . . . . . . . . . . . . . . . 28 5.4 the non-necrotic part of our modified compartment model . . . . . . . . . . 30 5.5 the necrotic part of our modified compartment model . . . . . . . . . . . . 31 5.6 shows the compartment system corresponding to equations 5.17 to 5.20 . . 31 5.7 Simulations of the DuCa model with a Hill function instead of the Michaelis- Menten function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.8 A plot based on the findings of Stoffels et al. (2004) that apoptotic as well as necrotic β-cells implies cytokine secretion . . . . . . . . . . . . . . . . . . 36 5.9 simulations of the DuCa model with activation due to necrotic cells . . . . 37 5.10 simulations of reversible and irreversible model with Balb/c phagocytosis parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.11 simulations of the reversible and irreversible model with NOD phagocytosis parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.12 shows simulations of the irreducible model with k = 0 . . . . . . . . . . . . 42 5.13 shows a simulation made with matlab of the reversible model where a term of −cMa has been added to equation 5.7 to model turnover of active macrophages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.14 In the figures simulations of the DuCa model with turnover of the active macrophages are presented . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1 A compartment representation of the CPH model. . . . . . . . . . . . . . . 45 6.2 Shows the compartment system for the Intermediate Model . . . . . . . . . 47 6.3 Sketch of the nullclines for the IM with dM/dt = 0. . . . . . . . . . . . . . . 57 6.4 Sketch of the nullclines for the IM with dM/dt = 0. . . . . . . . . . . . . . . 58 6.5 Sketch of the nullclines for the IM including crowding terms with QSS on M. 62 7.1 A supercritical (I), a subcritical (II) and degenerate Hopf bifurcation (III). 68 8.1 Bifurcation diagram for the DuCa model using f1 as the bifurcation parameter and NOD parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.2 Plot of the real and imaginary parts of the eigenvalues on the NODf1-USB 76 ix
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 8 and 9: vi Contents Significance of the Apo
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
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Page 47 and 48: 5.7 Discussion of Marée et al. (20
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Page 51 and 52: 5.8 Discussion of Parameters 37 Fig
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46 The Intermediated Model The diff
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48 The Intermediated Model expected
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50 The Intermediated Model Stabilit
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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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7.3 Biological Relevance of Bifurca
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7.5 Locating the Fixed Points 71 Ev
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74 Bifurcation Diagrams and Analysi
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98 Discussion of the Bifurcation An
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100 Discussion of the Bifurcation A
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106 Expanding and Modifying the DuC
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118 Discussion of the Expanded Mode
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120 Discussion of the Expanded Mode
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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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138
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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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