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

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106 Expanding and Modifying the DuCa model II When the amount of β-cells do tend to 0, due to persistent inflammation or natural apoptosis, after a while so must the concentrations of cytokines, apoptotic- and necrotic β-cells as well as the active macrophages. III The concentration of β-cells in NOD-mice must die out quicker than in Balb/cmice. IV In the Balc/c-mice, the depletion of β-cells must be due to natural apoptosis only, after the neonatal phase. I is based on the fact that macrophages, cytokines and apoptotic-, necrotic- and healthy β-cells are not the only cell/protein-types that are involved in the pathogenesis of T1D, as T cells also play a significant part. In NOD-mice they enter the scene after about 4-5 weeks (Trudeau et al., 2003, p.219), and symptoms of T1D become apparent at about 30 weeks of age (Sreenan et al., 1999, p.989). II is a result of the causal relationship between the components of the model – if there are no apoptotic β-cells then resting macrophages cannot become activated, and no apoptotic β-cells can become necrotic – there will be some residual apoptotic cells after the healthy β-cells are gone, so to be very precise it is not until these cells have been phagocytized that all concentrations will tend to zero (except for the resting macrophages). Thus once the deleterious process of β-cells has been completed, the NOD-mice, too, will tend to what we called the HRS in the bifurcation analysis, i.e. (M, Ma, Ba, Bn, C) = ( a/c, 0, 0, 0, 0), but in this case it will obviously not be a sign of health, quite the contrary. III serves as a necessary, though not sufficient, condition for the validity of the model. IV is explained below. Because T cells become significant at 4-5 weeks of age, the expanded models, presented below, become more or less unrealistic at the same time; they will not include T cells. However, including several weeks more than 5 in our simulations (and in the adhering figures) makes us more able to determine if the models comply with the criteria. 11.2 First approximation to a governing equation – Model A At the neonatal stage the behavior of the β-cell mass in NOD-mice is a little more complex than a simple depletion from day one. The pancreas is still undergoing remodeling (Steer et al., 2006, p.262), e.g. the apoptotic wave. But even after these early events and after the perpetual inflammation has begun, data exists that suggests that the inflammation itself stimulates β-cell regeneration (Akirav et al., 2008, p.2883). This could be because some of the cytokines produced in the inflammatory process are antiinflammatory or benign growth factors (Souza et al. (2008)). This has been disputed by others, whose data indicates a more linear decline in the β-cell population after insulitis has been established (Akirav et al., 2008, p.2883). However a decrease in the β-cell population is not reserved for those destined to become diabetic. Everybody would be become diabetic at some point, given that they live long enough (Pociot (2009)), due to the natural occurring apoptosis. This is why we have the fourth criterion. One thing that all parts can agree on is that the β-cells are being removed, so as a first approximation we will use a simple linearly decreasing model for the β-cell population. The equation for the β-cell population, at this stage, is given by dB dt = −x1 (11.1)
11.2 First approximation to a governing equation – Model A 107 with the initial condition B(0) = 1 × 10 9 cells ml −1 . 1 Making the β-cell population decline linearly implies that the population will become negative, but it suffices as a first approximation. Notice that x1 is the naturally occurring apoptosis. By “naturally occurring” we mean that it will go unaffected by factors that will tend to increase the overall apoptosis. Factors that contribute to the overall apoptosis will be added as terms on their own. As a first approximation to the natural apoptosis we set x1 = 10 6 cells d −1 ml −1 , i.e. a million cells terminate through apoptosis every day. This means that if no other factors bare influence on the population, the β-cells will be totally depleted after 1000 days, in comparison a normal laboratory mouse (e.g. Balb/c or C57BL/6J) has a median lifespan of 30 months (Blüher et al., 2003, p.573). Another thing that has come to be generally accepted is that an apoptotic wave occurs at the neonatal stage in several bio-models (Rooman and Bouwens (2004)), including the NOD- and Balb/c-mouse. However we have to adjust the wave a little to fit into our program. We define W0(t) = W (t) B(0) (11.2) to be the adjusted apoptotic wave – Marée et al. (2006) could not make use a wave that was dependent on the concentration of β-cells since they did not include such a concentration. Adding (or rather subtracting) the wave, as given in equation 11.2, we get dB dt = −x1 − 4 × 107 exp(−((t − 9)/3) 2 ) B = −x1 − W0(t)B (11.3) B(0) Besides this addition we must include the term that describes the death induced by cytokines. Again we need to recast it to fit our model. We redefine the cytokine induced apoptosis to be A ′ maxC B (11.4) kc + C where A ′ max = Amax/B(0). All in all the change in the concentration of healthy β-cells is given by dB dt = −x1 − W0(t)B − A′ maxC B (11.5) kc + C Adding this equation to the DuCa model (cf. equations 5.6-5.10), and adding the term x1 to the equation describing the change in concentration of apoptotic β-cells, as well 1 Here we have chosen the initial condition to be in the high end as readers, endowed with a good memory, may agree to; cf. chapter 2.
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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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Page 79 and 80: 7 A Brief Introduction to Bifurcati
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Page 140 and 141: 126 Bibliography Christen, U. and M
Page 142 and 143: 128 Bibliography Notkins, A. L. and
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Page 154 and 155: 140 Additional Figures Figure B.2 P
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156 Additional Figures Figure B.25
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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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