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

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174 Index value, 73 Bifurcation Analysis, 65 Bifurcation Parameter choice of, 69 Bifurcation Value definition of, 66 Bio-Model, 21 Bio-model, 7 Biological System, 133 CD4 + , 8 CD8 + , 8 Chaos, 59 Characteristic Polynomial, 50, 66, 94 Closed Orbit, 66 Codimension of bifurcation, 65 Compact Subset, 59 Compartment Model, 21 Marée et al. (2006), 23 arrow fully drawn, 23 stippled, 23 discussion of, 28 apoptotic wave, 28 crowding terms, 28 Efflux, 22 inflow, 21 Influx, 22 Mathematical Modelling, 21 outflow, 22 Competition Lotka-Volterra model of, 26 active versus inactive macrophages, 26 modelling of, 26 Continuous Deformation, 59 Cooperativity, 35 Copenhagen Model, 2 CPH Model, 21, 45 is quantitatively incorrect, 54 Crowding Effect, 26 Crowding Rate, 25 Cytokine, 10, 45 and necrosis, 10 and proliferation, 10 cytotoxic, 19 Cytotoxicity, 10 Deactivation Step, 38 Dendritic Cell, 15, 18 Dendritic Cell (DC), 9 Determinant expansion of, 50 Diabetes, 1 etiology of, 7 subtype of, 11 DuCa Model, 19, 21, 46 expanding the, 105 Model A, 106, 108 Model B, 110 validating, 117 maximal interval of existence, 132 parameters, 27 uniqueness of solutions to, 132 with cMa limit cycle, 42 phase space, 42 solution curve of, 42 Efflux of active macrophages, 40, 41 of macrophages, 25 Eigendirection, 79 Eigenspace, 59 Eigenvalue, 70, 94 complex, 66, 75, 76 Eigenvector, 59, 67, 79, 94 Environmental Factors, 1 Enzyme, 35 Error-Tolerance, 70 Etiology, 1, 2, 13 Euler’s Formula, 67 Existence and Uniqueness Theorem, 25, 131 Feedback Loop, 19 Fixed Point, 54 and chronic inflammation, 97 definition, 48 nontrivial, 97 unstable, 97 Flow on the circle, 87
Gastrin, 14, 15 Global Bifurcation, 71, 83 GLP-1, 121 anti-apoptosis, 14 Glucagon-Like Peptide 1 (GLP-1), 2, 14 Glutamic Acid Decarboxylase (GAD), 8 Growth Rate, 133 Hartman-Grobman’s Theorem, 59, 132 Healthy β-Cells linear depletion of, 106 simulation of, 108 Healthy Rest State(HRS), 75 Heteroclinic Orbit, 83 figure 8.7, 83 Hill Function, 34 Hill function, 26 Homeomorphism, 59 Homeostasis, 7 Hopf Bifurcation, 71 degenerate, 68 on the USBf1-NOD, 77 point, 75 subcritical, 87 supercritical, 68 value, 87 Hopf Bifurcation Theorem, 77 Hyperbolic, 59 Hyperglycemia, 1, 7 Hypoglycemia, 2 Hysteresis, 69, 87 IDDM, 1, 2 IM including crowding terms flow equivalence to the DuCa, 81 Immune System Adaptive, 15 Implicit Function Theorem, 55, 132 Incretin Concept, 2, 8 Inflammation upper bound of, 81 Influx of macrophages, 25 Initial Condition, 66 Insulin, 1 first bolus of, 1 175 Insulin Dependent Diabetes Mellitus (IDDM), 1 Insulitis, 9, 106 Interferon (IFN), 10 Interleukin (IL) 1, 10, 120 2, 10 Intermediate Model (IM), 21 Irreversible Model NOD phagocytosis, 40 phase space M versus Ma, 42 Islet of Langerhans, 7, 9 Jacobian, 50, 59, 71 and general solution of ODE, 66 in Newton-Raphson, 70 La Barre, 2 Limit Cycle, 67, 77 detection of, 77 semi-stable, 43 Linear Differential Equation general solution of, 67 Lipschitz, 131 Lipschitz Continuity, 131 Lower Unstable Branch (LUB) NODf2, 90 NODf1 , 75 Macrophage, 9, 18 volume of, 99 Manifold, 59 stable, 94 stable, 79 NODf1-USB, 76 unstable, 79 of the NODf1-LUB, 83 of the NODf1-USB, 83 Mass Action Kinetics, 30 Mathematical Model, 16 Mathematical Modelling, 17 and etiology of T1D, 2 and medical sciences, 17 construction of, 17 GLP-1, 15 guidelines, 17
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- I, OM OG MED MATEMATIK OG FYSIK E
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Exploring Treatment Strategies for
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Preface iii questions that were out
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vi Contents Significance of the Apo
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List of Tables 5.1 summarizes the m
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x List of Figures 8.3 Phase space p
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xii
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2 Introduction In the following dec
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4 Introduction There are several sc
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6 Introduction because I had to mak
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8 Type 1 Diabetes and Its Etiology
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10 Type 1 Diabetes and Its Etiology
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3 Prospects for Therapy - A Mini Re
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3.3 Gastrin 15 eration in situ Urus
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4 Mathematical Modelling The langua
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5 The DuCa Model In the following w
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5.3 The DuCa Model - An Appetiser 2
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5.4 The DuCa Model - Compartment Mo
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5.6 Our Compartment Model 29 migrat
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5.6 Our Compartment Model 31 a ✓
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5.7 Discussion of Marée et al. (20
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5.7 Discussion of Marée et al. (20
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5.8 Discussion of Parameters 37 Fig
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5.8 Discussion of Parameters 39 Par
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5.8 Discussion of Parameters 41 tha
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5.8 Discussion of Parameters 43 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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102
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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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122
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Page 140 and 141: 126 Bibliography Christen, U. and M
Page 142 and 143: 128 Bibliography Notkins, A. L. and
Page 144 and 145: 130 Bibliography Yu, J. and G. S. E
Page 146 and 147: 132 Mathematical Appendices the com
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Page 150 and 151: 136 Mathematical Appendices A.6 Dim
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Page 154 and 155: 140 Additional Figures Figure B.2 P
Page 156 and 157: 142 Additional Figures B.2 Eigenval
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Page 162 and 163: 148 Additional Figures Figure B.14
Page 172 and 173: 158 matlab Code w(r+1)=w(r)+dw; g=w
Page 174 and 175: 160 matlab Code end V_11(r)=E(1); V
Page 176 and 177: 162 matlab Code n=675; w_5=0; dw_5=
Page 178 and 179: 164 matlab Code M=M’; % end s_7(r
Page 180 and 181: 166 matlab Code w_7,V_34,’.b’,w
Page 182 and 183: 168 matlab Code M=M’; % M must be
Page 184 and 185: 170 matlab Code %options = odeset(
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Page 190 and 191: 176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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