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

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ii Preface This thesis took its beginning when Professor Johnny Ottesen introduced me to a number of articles all of which used mathematical modelling to look at diabetes. Amongst the articles I found that an article called “Onset of Type 1 Diabetes - A Dynamical Instability” (Blasio et al. (1999)) was the most interesting. At the same time Lars Hervig Jacobsen was looking to pick a subject for his bachelor thesis. We got into contact with Professor, Dr. MD, Flemming Pociot, to discuss the aforementioned article, of which he is a coauthor, along with an article by Marée et al. (2006) called “Modelling the onset of Type 1 diabetes: can impaired macrophage phagocytosis make the difference between health and disease?” which we had found while searching for literature. The article by Marée et al. (2006) is a revised and expanded version of the model presented in Blasio et al. (1999). They modify the model from Blasio et al. (1999) to investigate another hypothesis than Blasio et al. (1999), but they also state that the model in Blasio et al. (1999) becomes physiologically unviable when estimated or experimentally obtained parameters are inserted in the model. We decided to gain insight into the writings of Marée et al. (2006) to see if they were correct in their assertions and to see if the revised model was a good foundation from which we formulate our own model. Acknowledgements This thesis would not have existed in the form it is in today had it not been for a number of people. I would like to thank: First of all my supervisor Professor Johnny T. Ottesen for all of his help with matlab, mathematical modelling, discussions about bifurcation diagrams, suggestions, his meticulous corrections, general inspiring interest and for establishing connection to the next person to whom I would like to extent my gratitude: Chief physician at Steno Diabetes Center Dr. MD. Flemming Pociot who was my external supervisor for the duration of my work. At the outset of this thesis I knew as much about molecular biology as the common walrus knows about accounting, but thanks to Dr. Pociot’s willingness to meet with Lars Hervig Jacobsen and I, to discuss articles and answer questions of varying depth the language of biology soon became familiar. B.Sc. Lars Hervig Jacobsen who defied nature and weird looks from people in the computer lab at imfufa, Roskilde University, to be able to work 24-7 on his bachelors thesis. I collaborated with him during the first half of the thesis. It made the process much more enjoyable and without him a lot of the matlab-based results might not have existed in the form they are in today. Approximately halfway through the process I was in the good fortune of receiving a grant from Mærsk. The grant was intended to allow one to go abroad for a semester, and thanks to the kindness of Dr. Tom Archibald and the mathematics department at the Simon Fraser University (SFU) in Vancouver, Canada, I was able to spend the fall-semester at SFU. This provided me with the privilege of having a meeting with Dr. Diane T. Finegood at her office on campus at SFU. There I was given the opportunity to discuss subjects from the article of Marée et al. (2006), which she coauthored, and ask questions of relevance in general. Dr. Finegood referred me to Dr. MD Bruce Vercherer of University of British Columbia (UBC), who provided me with answers to
Preface iii questions that were outside Dr. Finegood’s field of expertise. Dr. Vercherer’s answers were very helpful, and he also suggested that I might visit UBC to discuss my questions further and present my work. Another person from UBC who has shown me kindness in accommodating my questions is Dr. Leah Edelstein-Keshet, who is another coauthor of the Marée et al. (2006) article. She answered some of my questions regarding the mathematical aspects of the DuCa model, and also invited me to visit UBC. Unfortunately tempus fugit, so I never got to see UBC. Furthermore I would like to thank Dr. Ralf Wittenberg of SFU for letting me pop by his office at random, and (I assume) inconvenient, hours of the day to ask questions of mathematical nature. He was also very kind in answering my email. Finally I would like to thank my girlfriend for putting up with my long hours of working and being no fun at all, graduate student Eric and Ph.D. student Marc from the IRMACS center at SFU for letting me share their cubical and last but not least the researches who agreed to let me use their graphs in my thesis.
Page 1 and 2: - I, OM OG MED MATEMATIK OG FYSIK E
Page 3: Exploring Treatment Strategies for
Page 8 and 9: vi Contents Significance of the Apo
Page 10 and 11: List of Tables 5.1 summarizes the m
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Page 16 and 17: 2 Introduction In the following dec
Page 18 and 19: 4 Introduction There are several sc
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Page 22 and 23: 8 Type 1 Diabetes and Its Etiology
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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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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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74 Bifurcation Diagrams and Analysi
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98 Discussion of the Bifurcation An
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106 Expanding and Modifying the DuC
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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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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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