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

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46 The Intermediated Model The differential equations that correspond to the compartment system in figure 6.1 are dM dt = a + (b + k)Ma − cM − f1MA (6.1) dMa = f1MA − kMa (6.2) dt dA dt = lMa − mA (6.3) The purpose of the CPH model was not to give a quantitative prediction of how the onset of T1D comes about, rather it was to illustrate that T1D is caused by multiple factors coming together in a synergetic way, and to show that the difference between onset and not onset depends on a threshold that incorporates these factors, i.e. the onset of T1D implies crossing a “threshold surface” that is spanned in the parameter space, of the model. This threshold was found through stability analysis and defined as f0 (Blasio et al., 1999, p.1684) f0 ≡ f1l (6.4) ck Where the system is guaranteed to be stable for f0 < 1. Marée et al. (2006) wants to illustrate that chronic inflammation can not occur for biologically reasonable parameter values in the CPH model, and thus the CPH model needs revision if it is to be used as a quantitative model. Therefore the DuCa model can be seen as an extension of the CPH model that has been modified to investigate a different hypothesis. In this section we are going to take a look at what we have dubbed the Intermediated Model (IM). This model bridges the gap between the CPH model and the DuCa model. The IM can be seen as a slightly modified CPH model and as a rudimentary version of the DuCa model. Marée et al. (2006) does not, however, tell the story of how to revise the IM model to transform it into the DuCa model. This implies that even though the DuCa model originates from the IM there is no straight line between the imperfections of the CPH model/IM model and the modifications done on the IM to reach the DuCa model. However the extension of the CPH model is quite straightforward since Marée et al. (2006) aims to investigate whether the effectiveness in phagocytosis can make the difference between a diabetic and a non-diabetic mice. As stated in section 5.1 this hypothesis is rooted in an observed difference in phagocytosis rates for Balb/c and NOD-mice, seen in the basal phagocytosis rate of the resting macrophages, but also in their ability to undergo an activation step, when the first apoptotic cell is engulfed. To investigate the effect of these differences they implement the clearance of apoptotic βcells done by the resting and activated macrophages into the CPH model, by including the terms f1MBa and f2MaBa. This modification addresses another peculiar aspect of the CPH model since there was no clearance of the antigens due to macrophage engulfment, but only a nonspecific decay. This seems odd since one should expect that upon engulfment of a protein, there is one less to activate the remaining resting macrophages. This leads us to the next modification, albeit of a more conceptual matter. Instead of monitoring the antigens – i.e. small protein fractions of the β-cells that underwent apoptosis, they look at the clearance of an apoptotic β-cell as a whole. These changes lead to the system of differential equations seen in 6.5 - 6.7, and the corresponding compartment can be seen in figure 6.2.
a ✓❄ ✏ ✛ M ✒ ✑ c ✛ ✛ f1 apoptosis ✻ l b k f1 ✓ ✲ Ba ✒ d ❄ ✏ ✓ ✲ Ma ✒ ✑ clearance ✲ f2 ❄✲ ✏ ✑ Figure 6.2 The figure shows the compartment system for the IM, as given by Marée et al. (2006). M, Ma, and Ba are the concentrations of macrophages, activated macrophages, and apoptotic β-cells respectively. An explanation of a, b, c, d, f1, f2 is given in table 5.1. Adopted with minor changes from (Marée et al., 2006, p.1270) dM dt = a + (k + b)Ma − cM − f1MBa (6.5) dMa dt = f1MBa − kMa (6.6) dBa dt = lMa − f1MBa − f2MaBa − dBa (6.7) Notice if we set f1MBa = 0, f2MaBa = 0 and Ba = A the system of equations reduces to the CPH model seen in 6.1,6.2 and 6.3. This is also presented in Marée et al. (2006) under the alias “the basic model”, and it undergoes an analysis which we will review and unfold. But first we will state what to expect from the behavior of the model from a physiological point of view. Model observations The biological frame plays a twofold role in the modelling process. It sets the limit for the dynamics the model should exhibit (e.g. no negative concentrations), but at the same time it works as a guideline for what kind of dynamics that, at least, should be 47
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- I, OM OG MED MATEMATIK OG FYSIK E
Page 3 and 4:
Exploring Treatment Strategies for
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Preface iii questions that were out
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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
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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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Page 51 and 52: 5.8 Discussion of Parameters 37 Fig
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Page 55 and 56: 5.8 Discussion of Parameters 41 tha
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Page 62 and 63: 48 The Intermediated Model expected
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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
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Page 88 and 89: 74 Bifurcation Diagrams and Analysi
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96 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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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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