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

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30 The DuCa Model a ✓❄ ✏ ✛ M ✒ ✑ c ❄ ✛ e1 ❄ f1 W (t) apoptosis ✻ Amax kc+C b ✻ f1 ✓ ✲ Ba ✒ ✲ d ❄ ✏ ✑ ✓ ✏ k ✲ Ma ✒ ✑e2 ❅ ❅❅❘ Figure 5.4 shows the non-necrotic part of our modified compartment model. M, Ma, and Ba are the concentrations of macrophages, activated macrophages and apoptotic β-cells respectively. An explanation of a, b, c, d, e1, e2, f1, f2, Amax, kc is given in table 5.1. Besides standardizing the use of arrows we have also included the apoptotic wave, and the crowding terms to make a compartment model that, albeit divided into two figures, agrees with the governing equations. Initially we wanted to rid the model of stippled arrows all t<strong>og</strong>ether, in order to be able to make a translation from the compartment model to a set of reaction equations, and from there to the set of governing equations. It is not impossible to get rid of the stippled arrows, but it requires extra compartments, that contain intermediate combinations of, say, active macrophages that have engulfed necrotic β-cells, and also we must introduce some extra parameters. In figure 5.6 we have picked out a part of the original compartment model and added a compartment containing active macrophages that have engulfed one or more necrotic β-cells. The macrophages will either secrete a cytokine, at rate k4, or merely clear the necrotic β-cell, at rate k5, without releasing any cytotoxic compounds – either way the active macrophage will return to its compartment, at rate k3. If k1 and k2 are equal it means that one macrophage engulfs one necrotic β-cell – if this engulfment is similar to the engulfment of apoptotic β-cells, this need not be the case 7 (Marée et al. (2005)). Upon engulfment the macrophage will either consume the β-cell quietly, at rate k5, or it will secrete a cytokine, at rate k4. We will not concern ourselves with the fate of the cytokines in this example. When we decompose the compartment system in figure 5.6 into mass action kinetic 7 In Marée et al. (2005) they observed that an activated macrophage could contain up to 7 apoptotic β-cells (that were at different levels of phagocytosis), and this did not appear to be an upper bound (Marée et al., 2005, p.542). f2 f2 ❄✲
5.6 Our Compartment Model 31 a ✓❄ ✏ ✛ M ✒ ✑ c ❄ ✛ f1 e1 ✠ ✛ ❄ f1 f1 b ✓ Bn ✏ ✒ ✑ ✻❅ ❅ d ✻ δ ✓ ✏ ✓ ✏ k α ✲ Ma ✲ C ✒ ✑ ✒ ✑ e2 ✻ ❅ ❅❘ f2 f2 ❄ ✲ α Amax kc+C Figure 5.5 shows the necrotic part of our modified compartment model. M, Ma, C and Bn are the concentrations of macrophages, activated macrophages, cytokines and necrotic β-cells respectively. An explanation of a, b, c, d, e1, e2, f1, f2, α, δ, Amax, kc is given in table 5.1. ✓ Ma ✏ ✒ ✑ ❅ k1❅ ❅ ❅❅❘ ✓ ✏ Bn ✒ ✑ ✒ ❅❅■ ❅ k3 ❅ ❅✓ ✒ k2 ❄ MaBn k5 ✏ ✑ k4 ✓ ✲ C ✒ ✏ ✑ Figure 5.6 shows the compartment system corresponding to equations 5.17 to 5.20. The active macrophages enter the MaBn-compartment at rate k1. The necrotic β-cells enter the same compartment at rate k2. The necrotic β-cell can either be broken down quietly, at rate k5 or yield a cytokine, at rate k4. After digestion of the necrotic β-cell, the active macrophage returns to the compartment containing these, at rate k3, i.e. k3 is a digestion rate. equations we get Ma + Bn MaBn MaBn MaBn k1 −→ MaBn (5.11) k3 −→ Ma (5.12) k4 −→ C + Ma (5.13) k5 −→ Bn + Ma (5.14) ❄ (5.15)
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 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: 5.6 Our Compartment Model 29 migrat
Page 47 and 48: 5.7 Discussion of Marée et al. (20
Page 49 and 50: 5.7 Discussion of Marée et al. (20
Page 51 and 52: 5.8 Discussion of Parameters 37 Fig
Page 53 and 54: 5.8 Discussion of Parameters 39 Par
Page 55 and 56: 5.8 Discussion of Parameters 41 tha
Page 57: 5.8 Discussion of Parameters 43 Fig
Page 60 and 61: 46 The Intermediated Model The diff
Page 62 and 63: 48 The Intermediated Model expected
Page 64 and 65: 50 The Intermediated Model Stabilit
Page 66 and 67: 52 The Intermediated Model restrict
Page 68 and 69: 54 The Intermediated Model We see t
Page 70 and 71: 56 The Intermediated Model paramete
Page 72 and 73: 58 The Intermediated Model ✻ Ma(t
Page 74 and 75: 60 The Intermediated Model 6.2 The
Page 76 and 77: 62 The Intermediated Model ✻ Ma(t
Page 79 and 80: 7 A Brief Introduction to Bifurcati
Page 81 and 82: 7.2 The Hopf bifurcation 67 the gen
Page 83 and 84: 7.3 Biological Relevance of Bifurca
Page 85: 7.5 Locating the Fixed Points 71 Ev
Page 88 and 89: 74 Bifurcation Diagrams and Analysi
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80 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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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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