M O D E L I N G T H E O N S E T O F T Y P E 1 D I A B E T E S by RICHARD ALEXANDER KUBLIK B.Sc.(Hons.), The University of Alberta, 2003 B.A., The King's University College, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA April 2005 © Richard Kublik, 2005 Abstract A neonatal wave of apoptosis (programmed cell death) occurs normally in the pancreatic /3-cells of mice and rats. Previous collaborative work by members of Diane T Finegood's experimental lab at Simon Fraser University and Leah Edelstein-Keshet's mathematical biology group at the University of British Columbia demonstrated that macrophages from non-obese diabetic (NOD) mice become activated more slowly and engulf apoptotic cells at a lower rate than macrophages from control (Balb/c) mice. It has been hypothesized that this low clearance could result in secondary necrosis of the /3-cells, escalating inflammation and the self-antigen presentation that later triggers autoimmune, type 1 diabetes. We here investigate whether this hypothesis could offer a reasonable and simplified explanation for the onset of type 1 diabetes in N O D mice. We quantify variants of the Copenhagen model, developed by Freiesleben De Blasio et al (1999, Diabetes 48, 1677), based on parameters from N O D and Balb/c experimental data. We demonstrate that the original Copenhagen model fails to explain observed phenomena within a reasonable range of parameter values, predicting an unrealistic all-or-none disease occurrence in both strains. However, if we assume that the activated macrophages produce harmful cytokines only when engulfing necrotic (but not apoptotic) cells, then the revised model becomes qualitatively and quantitatively reasonable. Further, we show that known differences between N O D and Balb/c mouse macrophages kinetics are large enough to account for the fact that an apoptotic wave can trigger escalating inflammatory response in N O D , but not Balb/c mice. In Balb/c mice, macrophages clear the apoptotic wave so efficiently, that chronic inflammation is prevented. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgments vii Co-Authorship Statement viii Chapter 1. Introduction 1 1.1 A Crash Course in Basic Immunology 1 1.2 Pathogenesis of Type 1 Diabetes 3 1.3 The Role of Apoptosis in the Initiation of Type 1 Diabetes 4 1.4 Macrophages 5 1.5 Previous Models of Type 1 Diabetes 7 1.6 Goals of Modeling, and the Current Study 7 Chapter 2. The Copenhagen Model of Type 1 Diabetes 10 2.1 Original Description of the Copenhagen Model 10 2.2 A Mathematical Formulation of The Copenhagen Model 11 2.3 Limitations of the Copenhagen Model 14 Chapter 3. A n Extension of the Copenhagen Model 15 3.1 The Basic Model 15 3.2 The Simplest Minimal Model 17 3.3 A Second Informative Variant 20 3.3.1 Engulfment of Apoptotic Bodies by Resting Macrophages 24 3.4 Analysis of The Complete Basic Model 25 Chapter 4. Parameter Values 32 4.1 Macrophage Activation and Engulfment Rates 32 4.2 Volumes and Sizes 34 4.3 Macrophage Density and Flux Rates 35 4.4 Non-specific Removal of Apoptotic /3-cells 37 4.5 Calculated Parameters 37 Chapter 5. A More Detailed Model for the Initiation of Type 1 Diabetes 41 5.1 Additions to the Model 42 5.1.1 Secondary Necrosis 42 5.1.2 Cytokines 43 iii 5.2 Model Equations 44 5.2.1 Stability of the Rest State 45 5.2.2 Reduction of the System 46 5.3 New Parameter Values 47 5.4 Simulations of the Reduced Necrotic Model 48 5.5 Bifurcation Analysis of the Reduced Necrotic Model 55 Chapter 6. The Neonatal Wave of /?-cell Apoptosis 64 6.1 Description of the Wave 64 6.1.1 Methods to Determine Apoptosis 65 6.1.2 Quantification of the Wave 66 6.2 The Wave as a Triggering Event for Type 1 Diabetes 67 6.3 Simulations of the Apoptotic Wave 68 Chapter 7. Discussion 73 7.1 Summary and Conclusions 73 7.2 New Questions 75 7.3 Limitations of Our Model 77 7.4 Future Work 79 Bibliography 81 Appendix A . Immunological Terms 85 Appendix B. Dimensionless Formulation of the Model 88 Appendix C. XPP Code 91 C.l Simulations of the Basic Model 91 C.2 Simulations of the Reduced Necrotic Model 92 C.3 3D Reduced Necrotic Model 93 C.4 Simulations of the Full Necrotic Model 94 iv List of Tables 3.1 Parameter Definitions 17 3.2 Parameter Groupings in Simplest Model 21 4.1 Estimates for the Parameter b 39 4.2 Parameter Values for the Basic Model 39 5.1 Parameters in the Extended Model 49 6.1 Percent Apoptosis at peak of neonatal apoptotic wave 67 B.l Non-dimensional Variable Scaling 89 B.2 Non-dimensional Variables 90 v List of Figures 2.1 Schematic Representation of the Copenhagen Model 13 3.1 Schematic Representation of Basic Model 18 3.2 Schematic representation of Simplest Model 19 3.3 Four Cases 23 3.4 Nullclines for basic model, with arbitrary parameter values 29 3.5 Null-surfaces for basic model, with arbitrary parameter values 30 3.6 Null-surfaces for basic model, with arbitrary parameter values 31 5.1 Schematic Representation of the Most Detailed Model 45 5.2 Nullclines of reduced necrotic model (NOD) 50 5.3 Trajectory from one initial condition in N O D phase plane 51 5.4 Nullclines of reduced necrotic model (Balb/c) 52 5.5 Trajectory from one initial condition in Balb/c phase plane 53 5.6 Null surfaces of 3D variant of Extended Model (NOD) 54 5.7 Bifurcation of Reduced Necrotic Model, with respect to parameter g = f\ 55 5.8 Different dynamics obtained through bifurcation with respect to g = f\ 56 5.9 Unstable Limit cycle seen in Bifurcation with respect to g = f\ 57 5.10 Bifurcation of Reduced Necrotic Model, with respect to parameter fz 59 5.11 Different dynamics obtained through bifurcation with respect to fz 60 5.12 Unstable Limit cycle seen in Bifurcation with respect to fa 61 5.13 Bifurcation of Reduced Necrotic Model, with respect to parameter e\ = ez 62 5.14 Different dynamics obtained through bifurcation with respect to e\ = ei 63 6.1 Simulations of the Necrotic Model, with the Apoptotic wave 71 6.2 Simulations of the Necrotic Model, Showing a Sub Threshold Apoptotic wave 71 6.3 Simulations of the Necrotic Model, Showing Limit Cycle Behavior 72 vi Acknowledgments I would like to thank my supervisor, Professor Leah Edelstein-Keshet, for allowing me to take part in this project. I have learned a great deal from her through the course of my Master's degree. I would also like to thank Dr. A F M Maree, and the members of Professor Diane Finegood's lab for their involvement and assistance on the project. I also wish to acknowledge the organizations that provided funding for my studies: the Mathematics of Information Technology and Complex Systems (MITACS) and the Natural Sciences and Engineering Research Council of Canada (NSERC), without who's support this would not have been possible. vii Co-Authorship Statement A version of this thesis has been submitted for publication. Athanasius F.M. Maree, Richard Kublik, Diane T. Finegood, and Leah Edelstein-Keshet. (2005) Modelling the onset of type 1 diabetes: can impaired macrophage phagocytosis make the difference between health and disease? Philosophical Transactions of the Royal Society. This project was initiated by discussions between Dr. Diane Finegood (SFU) and Dr. Leah Edelstein Keshet (supervisor). It was then investigated by PDF Marek Lab§cki and revisited by Dr. A.F.M. Maree who had also been a PDF with Dr. Keshet. When Richard Kublik came into the project, the first extension of the Copenhagen model had already been developed. R. Kublik worked extensively on the analysis of this model, determining and understanding the conditions associated with the existence and stability of the steady states. Following this analysis, R. Kublik began to search the literature for parameter estimates to use in the model. These parameter estimates varied widely, and it was not clear which values to incorporate into the model. Drs. Keshet and Maree made the final choice of parameter estimates and went on to extend the model to include more details. Early analytic work proved to be un-informative, leading to mainly numerical exploration of the most detailed model. Drs. Keshet and Maree performed some preliminary numerical simulations and closely guided R. Kublik in the re-maining simulations. A l l simulations presented in the thesis were done by R. Kublik. R. Kublik then researched the neonatal wave of apoptosis, and determined the equation to simulate this wave for inclusion into the model. Through all stages of the project, Dr. Finegood provided helpful discussions and clarified many of the biological details. Leah Edelstein-Keshet Richard Kublik ..T viii Chapter 1 Introduction Type 1 diabetes, also known an juvenile or insulin dependent diabetes, is the result of the destruction of the insulin producing pancreatic /3-cells by the body's own immune system. The pathogenesis of type 1 diabetes in humans is difficult to study: currently there are no non-invasive methods to determine the amount of /3-cell death in the pancreas and clinical symptoms of diabetes are not obvious until most of the /3-cells have been destroyed [39]. In order to study the early development of type 1 diabetes, researchers have relied heavily on experimental animals that spontaneously develop diabetes with the hope that lessons learned from these animal forms of the disease may be applied to human patients. One commonly used experimental animal is the non-obese diabetic (NOD) mouse. This animal was discovered and selectively bred during the 1970s, resulting in an inbred strain with 60-80% of the female mice and 20-40% of male mice developing overt diabetes by 210 days of age [19, 40]. The pathogenesis of type 1 diabetes in N O D mice will be described in section 1.2. In this study, we provide a model for the early stages of type 1 diabetes in N O D mice, and contrast the evolution of the disease in N O D mice to the events in the Balb/c control strain mouse. 1.1 A Crash Course in Basic Immunology The immune system is comprised of two sub systems: the innate and adaptive immune systems. The innate immune system is responsible for the early response to infection. The innate immune system remains the same throughout an individual's lifetime, not improving 1 with exposure to infection [14]. A n important cell type in the innate immune system is the macrophage, a large phagocytic cell that clears and digests intra-cellular debris as well as dead and dying cells. Dendritic cells are also an important part of the innate immune system, collecting debris as macrophages do, but also processing the engulfed cellular proteins into antigenic peptides. These peptides are small fragments of the protein engulfed by the dendritic cell that are presented to cells of the adaptive immune system to trigger an adaptive response. Thus dendritic cells serve as a link between the innate and adaptive immune response. The adaptive immune response gets its name from its ability to respond more quickly to repeated attacks by an infective agent. After the initial exposure, cells of the adaptive immune system require 4-7 days to be primed. They then proliferate to form a response [14]. The response to a repeated infection by the same pathogen however, is much quicker due to the existence of "memory" cells that are triggered more rapidly on a second encounter with pathogenic antigen. The adaptive immune system is made up of two types of cells called B and T lymphocytes (also called B and T cells). In their naive state, B and T lymphocytes circulate through the body, waiting for an activating signal from a dendritic cell. Once activated, the lymphocytes proliferate and exhibit specific behaviors. Activated B cells produce antibodies that neutralize pathogens and prepare them for engulfment by macrophages. Activated T cells kill cells displaying antigen they recognize, either directly, or by sending a signal for the cell to undergo apoptosis-a form of programmed cell death. In the case of type 1 diabetes, T cells specific for the insulin producing /3-cells are activated and then migrate to the pancreatic islets of Langerhans, where the /3-cells are located, and destroy the /3-cells. In type 1 diabetes, an initial injury to the pancreatic /3-cells leads to the recruitment of macrophages and dendritic cells into the islets of Langerhans, where the /3-cells are concen-trated. Both cell types engulf dead cells and extra cellular debris. Macrophages are "activated" upon engulfment of apoptotic cells and begin to secrete chemical mediators, called cytokines. Some cytokines act as messenger signals, recruiting more macrophages to the tissue, and signal-ing to other immune system cells. One side effect of cytokine release by activated macrophages is the damage these cytokines cause. Macrophages digest the engulfed material, performing a 2 "garbage collection" function. Dendritic cells, on the other hand, process the engulfed matter into peptides that are then displayed on the cell surface. Dendritic cells then migrate to the lymph nodes, where they present the antigenic peptides to T-cells. In the lymph nodes, any T-cells that recognize the peptides displayed by the dendritic cells become activated, prolif-erate and migrate to the site of infection. In the case of type 1 diabetes, the site targeted by lymphocytes and other immune cells is the the pancreatic islets of Langerhans. This infiltration of the islets by immune cells is referred to as insulitis. Once in the islets, the T-cells attack and destroy any cell displaying peptides that it recognizes, i.e. the /3-cells. A brief glossary of the terms described here is given in Appendix A. 1.2 Pathogenesis of Type 1 Diabetes Insulitis in the N O D mouse occurs in two stages: a "benign" stage seen in all mice, followed by a "malignant" stage seen only in the mice that go on to develop type 1 diabetes [39]. The benign stage of insulitis is seen beginning around two weeks of age, when macrophages and dendritic cells infiltrate the pancreas and collect near the islets. The macrophages remain at the edge of the islets while dendritic cells invade. By 6-10 weeks of age the second, malignant stage of insulitis develops. /3-cell-specific T-cells begin to infiltrate the islets and destroy more yS-cells. At this point, macrophages are still concentrated at the edge of the islets and dendritic cells are found in close proximity to the T-cells [5]. It is currently unknown if /3-cell destruction takes place gradually or whether the /3-cells are destroyed just before the onset of clinical symptoms [39]. Overt diabetes develops around 12 weeks of age, when approximately 95% of the /3-cells have been destroyed [13,21] In type 1 diabetes, T-cells play the dominant role in the /3-cell destruction. However, at this point, we are not concerned with describing the mechanisms by which the T-cells are activated or their interactions leading to almost complete destruction of the pancreatic /3-cells. Rather, we focus our attention on the very earliest stages of the disease, on the activity of macrophages and their role in creating conditions that ultimately lead to T-cell activation. 3 The fact that not all N O D mice develop type 1 diabetes suggests that there is more to the initiation of the disease than just genetic susceptibility. Al l N O D mice have identical genes, demonstrate autoimmunity and develop lymphocyte infiltration of the pancreas, yet only a fraction of the animals develop the disease. The difference between those that develop type 1 diabetes and those that do not is that in the first case, the lymphocytes invade the islets, while in the second case the infiltrating cells remain at the edge of the islets [19]. Merely having auto-reactive T-cells is apparently not enough to cause the autoimmune destruction of the /3-cells, these T-cells must first be activated. There are many possible triggering events that lead to activation of the /5-cell specific T-cells. Anything that damages /3-cells and causes antigen to be presented to T-cells can serve as a trigger for the development of type 1 diabetes: viral infection, physical damage, exposure to toxins are all potential triggers for this process. 1.3 The Role of Apoptosis in the Initiation of Type 1 Diabetes Apoptosis, programmed cell death, is seen in many organs during development, and is thought to be involved in remodeling of tissues. Apoptosis is known to be important in the nervous system [4] and has also been observed in organs such as the heart, kidney and adrenal cortex [23,37]. A developmental wave of apoptosis has also been observed in the pancreas of neonatal rodents. This wave, we believe, has a significant impact on the development of type 1 diabetes. It has been estimated that during this apoptotic wave, at least 60% of the pre-existing /3-cells in the pancreas undergo apoptosis [41]. The pancreatic wave of /3-cell death peaks at about 10 days of age, and has been observed in many different animals. A more complete description of the wave is given in Chapter 6. In animals that spontaneously develop diabetes, the onset of insulitis appears to be fixed at approximately 15 days of age - immediately following the apoptotic wave. It was suggested by Trudeau et al [41] that this temporal association may be more than coincidental, that the wave of apoptosis is, in fact, a trigger for the initiation of type 1 diabetes. How can this be? Apoptosis is generally regarded as a non inflammatory form of cell death [6], yet in the case of type 1 diabetes, the /3-cells die as a result of the local tissue inflammation 4 that follows the monocyte infiltration. Ren and Savill [34] present a hypothesis for how large scale apoptosis may trigger an inflammatory response. When macrophages engulf apoptotic material, the inflammatory response is generally inhibited [3, 34]. However, when the amount of apoptotic material exceeds the macrophage clearance capacity, the uncleared apoptotic cells loose their structural stability and the plasma membrane swells and bursts. This process, called secondary necrosis leads to the release of cellular contents into the surrounding tissue. It is believed that macrophages that engulf necrotic material initiate an inflammatory response, though the details of this remain controversial. Once the inflammatory response is initiated, macrophages induce apoptosis in more /3-cells, continuing the cycle of /3-cell destruction. At this time, antigen presenting cells (called dendritic cells) in the tissue engulf apoptotic and necrotic cells, process and present the antigenic peptides in the lymph nodes where T-cells specific to /3-cell proteins are activated. These /3-cell specific T-cells then infiltrate the pancreatic islets and efficiently kill the remaining /3-cells, resulting in type 1 diabetes. 1.4 Macrophages In the case of diabetes-prone animals, a hypothesis exists that susceptibility stems from apop-totic /3-cells remaining in the pancreas too long and triggering an inflammatory response. In a previous study in this group, it has been shown that macrophages from N O D mice are not efficient at phagocytosis when compared to Balb/c and other control strains [22, 30]. This inef-ficient clearance leads to a build-up of apoptotic cells in the tissue, and many of the apoptotic cells then undergo secondary necrosis, known to trigger an inflammatory response from the immune cells. Resident macrophages are supposed to clean cellular debris from the tissue. In addition to their role as phagocytes, macrophages secrete cytokines that attract other macrophages to the tissue. These cytokines have been shown to be essential for activation of /3-cell cytotoxic T-cells in N O D mice. If macrophages are removed from N O D mice (through treatment with silica, toxic to macrophages), the mice do not develop diabetes [15]. Macrophages do not always produce and secrete cytokines. However, cytokines are only released following some 5 "activating" stimulus when macrophages take on additional duties, and are termed "activated macrophages." The term "activated macrophage" is not completely descriptive, as there are multiple stimuli for activation, each resulting in a functionally different macrophage. We will distinguish between resting and activated macrophages. However some caution is required, as the term "activated" has been used in many different ways in the literature. Mosser [24] describes the "classically activated macrophage." Classically activated macro-phages are able to kill inter-cellular pathogens and secrete inflammatory mediators [24, 31]. This "classically activated macrophage" requires two activating signals: the cytokine inter-feron (IFN)-y to prime the macrophage for activation and some mediator to stimulate the macrophage to produce tumor necrosis factor that then serves as the second activating stim-ulus. Macrophages activated in this manner are not more phagocytic than the non activated macrophages. However, they up-regulate M H C class II production and have enhanced ability to present antigen and kill pathogens. These classically activated macrophages are identified via their production of nitric oxide (NO) [24]. Stein et al [38] describe another activation phenotype. Macrophages activated in response to interleukin (IL)-4 behave much differently than the classically activated macrophage: they are unable to produce and secrete N O and are less able to kill intracellular pathogens. These "alternately activated macrophages" are thought to serve a regulatory and recovery function as opposed to a killing function: promoting cell growth, collagen formation, and tissue repair, while inhibiting T-cell proliferation [24]. A third type of activated macrophage is the so-called type II-activated macrophage, named for its ability to preferentially induce the Th2 (anti-inflammatory) adaptive immune response. The type II-activated macrophage is similar to the classically activated macrophage in that it requires two signals to be activated: the ligation of a specific (FcyR) receptor on the macrophage combined with some macrophage stimulatory signal. These signals combine to halt production of interleukin-12 and increase secretion of interleukin-10 which has been shown to have anti inflammatory properties [24] 6 Maree et al [22] quantified the phagocytic ability of macrophages from N O D and Balb/c mice in vitro. They discovered that following the first engulfment of an apoptotic cell, subsequent engulfment occurred at a higher rate. Macrophages having engulfed at least one apoptotic body were considered to be activated. In our work, we consider activated macrophages to be those with increased phagocytic capa-bilities and the ability to produce and secretes cytokines. 1.5 Previous Models of Type 1 Diabetes One of the earliest theoretical treatments of the initiation of type 1 diabetes is the so-called Copenhagen model, developed by Nerup et al [26]. In the Copenhagen model, the cycle of /5-cell destruction is initiated and fueled by the release of /S-cell antigen into the tissue. This free antigen is taken up by macrophages and dendritic cells, triggering the production and release of cytokines. These cytokines further stimulate macrophages, and lead to the formation of oxygen radicals (eg. N O , ) that damage /3-cells releasing more antigen into the tissue. The Copenhagen model was given a mathematical formulation by Freiesleben De Blasio et al [9]. This model considered the interactions of /3-cell antigen with resting and activated macrophages. The model was presented as a qualitative study, supporting the claim that susceptibility to type 1 diabetes is due to an underlying instability in the system, rather than a single triggering stimulus. The model predicts a steady state solution, corresponding to a non-inflamed condition. In healthy individuals this healthy state is stable, and the authors demonstrated that, under certain conditions, the healthy state may become unstable, leading to autoimmunity. 1.6 Goals of Modeling, and the Current Study The main goal of the Copenhagen model was to propose a qualitative hypothesis. Here, we extend the Copenhagen model and gather parameter values from the experimental literature to conduct quantitative analysis. Our goal is not to model type 1 diabetes in its full complexity. 7 Rather, we hope to provide a simple explanation of its initiation and answer two specific questions: 1. Are the differences in macrophage function observed by Maree et al [22] sufficient to explain why some N O D mice develop diabetes while all Balb/c mice remain healthy? 2. Can the normal, developmental wave of /3-cell death observed in the pancreas of neonatal mice serve as a triggering event that initiates the inflammatory response in N O D mice? Experimental observations suggest the answer to both questions above is yes. But using exper-imental observations, it is difficult to distinguish temporal details of interacting cells. Through the use of a mathematical model, and well-defined assumptions, the nature of interactions between cells in the pancreas can be quantified and better understood. Since the publication of the Copenhagen Model in 1999, some of the parameter values in the model have been determined experimentally, allowing a model to make quantitative predic-tions which can be checked experimentally. Since not all N O D mice develop the chronic inflammation of the pancreas that leads to diabetes, any model of the early stages of the disease should be able to account for a healthy state for all animals and a potentially inflamed state for N O D mice only. This distinction was incorporated into the Copenhagen model, and in terms of mathematical modeling, translates into a requirement that a stable healthy rest state exists for both N O D and Balb/c mice. The chronic inflammation preceding type 1 diabetes in N O D mice will correspond to a second stable steady state where the densities of all cells in question are elevated. As none of the Balb/c mice develop the disease, this "inflamed" steady state should not exist in the case of Balb/c mice. In the N O D case, the healthy and inflamed steady states are separated by a saddle point and its separatrix, a threshold separating the regimes attracted to the healthy steady state from the regions attracted to the chronic inflammation steady state. A triggering stimulus in the model then becomes anything that moves the system into the basin of attraction of the inflamed state. Any sub-threshold stimulus would not trigger chronic inflammation, but would be resolved by the immune system. 8 We will demonstrate that the Copenhagen model gives an incomplete picture: when biolog-ically reasonable parameter values are incorporated into the Copenhagen model it fails to provide the required behavior. Only when the model is extended to include other reasonable features, such as necrosis, do both the qualitative and quantitative aspects of the model agree with experimental observations. 9 Chapter 2 The Copenhagen Model of Type 1 Diabetes Nerup et al [26] suggest that any model describing the pathogenesis of type 1 diabetes should explain not only why the /3-cell destruction is initiated and sustained, but also how the autoim-mune response is specifically directed toward the /3-cells. 2.1 Original Description of the Copenhagen Model In what has come to be known as the "Copenhagen Model," Nerup et al [26] presented the first comprehensive hypothesis regarding the pathogenesis of type 1 diabetes. Based on experimental observations, the authors proposed a model where the destruction of /3-cells is due to a self-perpetuating inflammatory state, triggered and fueled by release of /3-cell antigens from damaged cells. According to the Copenhagen model, anything that damages /3-cells (toxins, chemicals, cytotoxic cytokines) causes /S-cell proteins to be released into the tissue. Free proteins are then engulfed by antigen presenting cells (macrophages, monocytes, and dendritic cells), processed into antigenic peptides, and displayed on an MHC-II complex. Engulfment of /3-cell proteins also stimulates the antigen presenting cells to produce and secrete chemical factors called cytokines, notably interleukin-1 (IL-1) and tumor necrosis factor (TNF), that trigger the production of cytokines by helper T-cells in the tissue. One of these lymphokines, interferon (IFN)-y, further stimulates macrophages, monocytes, dendritic cells and natural killer cells in the islet to produce more interleukin-1 that may then cause additional damage to the /3-cells. In the presence of tumor necrosis factor and interferon, interleukin-1 induces free radical formation (eg. N O , O p within the islet. The free radicals are toxic to 10 /3-cells, thus causing more /3-cell damage, releasing more dead /3-cell material and sustaining the cycle of /3-cell destruction. [9, 25, 26]. In the Copenhagen Model, the initiation, progression and limitation of /3-cell destruction is assumed to ultimately depend on /3-cell antigen release: the cycle of /3-cell destruction begins with the initial release of /3-cell antigen and subsides when antigen release stops [26]. In the early onset of the disease, this dependence on /3-cell antigen may account for the heterogeneity seen in the pancreas: destruction of /3-cells is limited to the islets where the initial damage was inflicted. Later, the primed immune response spreads to other islets. Nerup et al [25] describe the rapid acceptance of their hypothesis as uncritical, claiming that it soon became dogma that type 1 diabetes was a classic organ-specific autoimmune disease. Uncritical though its acceptance may have been, there is little experimental evidence contra-dicting the Copenhagen model. The pivotal role given macrophages in the model as antigen presenters, accelerators of the immune response and producers of cytokines [25] has been sup-ported through morphological studies: it has been shown that macrophages are the earliest immune cells to infiltrate the islets at the onset of type 1 diabetes [39,48], and are essential to the disease process [48]. Other studies have confirmed the /3-cell cytotoxicity of certain cytokines [6, 33,48]. 2.2 A Mathematical Formulation of The Copenhagen Model The Copenhagen model was formalized in 1999 by Freiesleben De Blasio et al [9] as a set of ordinary differential equations. This model examines the interactions between macrophages, /3-cell antigen and T-cells leading to the onset of type 1 diabetes. Within the context of the Copenhagen model, the early stages of type 1 diabetes (insulitis, inflammation leading to /3-cell destruction), are triggered by a perturbation away from a stable rest state. In the case of healthy individuals, a perturbation from complete health (manifested as release of /3-cell proteins into the tissue by damaged /3-cells) will be corrected by the immune system and the inflammation quickly resolved. In individuals susceptible to developing type 1 diabetes, this 11 initial inflammation is not resolved, but rather, due to an underlying instability of the system, leads to self perpetuating /3-cell destruction and eventually to flow-blown diabetes. The pathogenesis of type 1 diabetes is comprised of two stages: a non-lymphocyte dependent initial phase, and a T-cell mediated amplification and perpetuation of /3-cell destruction [25]. Accordingly, Freiesleben De Blasio et al present two variations of their mathematical model, a simple version which only considers the interaction of macrophages and /3-cell antigen and a more detailed version that also includes the role of T cells. The current study focuses on the very early stages, before T cell involvement, and so builds on the simpler variant of the Copenhagen model described below. In their model, Freiesleben De Blasio et al consider only the interactions occurring within the pancreatic islets of Langerhans, depicted as a single, open compartment through which macrophages are free to circulate. Due to their central role in the Copenhagen model, macrophages are modeled explicitly in both resting and activated forms. The density of resting and activated macrophages, measured in cells m l - 1 , is given by the variables M and Ma respectively. The concentration of jS-cell antigen is measured in units per ml and given by the variable A. Resting macrophages circulate through the volume with constant influx (/ cells m l - 1 d a y - 1 ) and efflux (c day - 1 ) rates. Macrophages become activated following uptake of /3-cell antigen, at a rate g ml cel l - 1 day - 1 . Once activated, macrophages produce and se-crete cytokines that are cytotoxic to the /3-cells. These cytokines stimulate the release of /3-cell antigen into the tissue, at a rate / day - 1 . Thus, antigen is introduced into the system at a rate proportional to the density of activated macrophages, and it decays at a constant rate, m day - 1 . The cytokines released by activated macrophages also act as a signal to recruit more resting macrophages into the volume. Specifically considered in the Copenhagen model is the IL-1 and TNF-a induced recruitment of resting macrophages, at a rate b day - 1 . In the absence of stimulation by /3-cell antigen, activated macrophages return at a rate k d a y - 1 to the resting state. The Copenhagen model is represented schematically in Figure 2.1, and expressed explicitly in 12 equations (2.1). M = J + (k + b)Ma - cM - gMA, (2.1a) Ma = gMA-kMa, (2.1b) A = IMa-mA. (2.1c) Figure 2.1: Schematic representation of the Copenhagen Model, (equations 2.1): Resting Macrophages (M) circulate through the volume with a constant influx (/) and efflux (c). Macrophages become activated following contact with /3-cell antigen (A) at a rate g. Activated macrophages (M„) release cytokines which recruit more resting macrophages to the volume (b) and damage /3-cells, causing more antigen to be released (I). Activated macrophages revert to the resting state at rate (fc). Antigen decays at a constant rate (m). (Based on Figure 3 in [9]). Analysis of the model given in equations (2.1) shows there is a steady state solution at (M,Ma,A) = (l/c,0,0) corresponding to a healthy individual with no inflammation. It is the stability of this steady state that determines whether an individual will be susceptible to type 1 diabetes or will remain healthy. Following a non-dimensionalization of the model, the authors derived a condition for stability of the healthy steady state: h-fk<l. (2.2) The parameter grouping fo can be thought of as the amount of secondary antigen produced by the primary /3-cell destruction, and is similar to the Ro value found in epidemic disease 13 models. In this case, the healthy state will be unstable when, on average, the death of a single /3-cell leads to the death of more than one additional /3-cell. It is this feedback mechanism that determines whether an individual is susceptible to type 1 diabetes. In cases where /o > 1, the healthy state is unstable, and as soon as there is any perturbation from complete health the system will move to unchecked inflammation. Freiesleben De Blasio et al point out that it is not the perturbation (caused by disease, viral infection, chemical exposure, etc.) that determines whether one will develop the disease, but rather, this underlying instability. It is interesting to note that the feedback parameter /o, and consequently, the stability of the healthy state, does not depend on the cytokine-induced recruitment of resting macrophages (b). 2.3 Limitations of the Copenhagen Model The Copenhagen model is presented in [9] as a "proof of principle," supporting the authors claim that there is no single cause of type 1 diabetes susceptibility, but that the disease is the result of an underlying instability resulting from the interactions of many components. In this sense, the Copenhagen model has been successful and has provided a good qualitative description of mechanisms that may be responsible for susceptibility to type 1 diabetes. The Copenhagen model, however, is limited to a qualitative description. At the time of its formula-tion, no reasonable estimates for the parameter values of the model existed. The Copenhagen model is a first approximation, and considers only a few of the processes taking place during the early stages of type 1 diabetes, thus it provides a simplistic view. The authors acknowledge this, and stress that the feedback parameter, /o does not include all possible causes of instability. The Copenhagen model is not suited to a quantitative exploration of the events leading to development of type 1 diabetes. As Freiesleben De Blasio et al were only concerned with results when the healthy steady state is stable, the model is not well behaved when the healthy state is unstable. In the case when /o > 1 numerical simulations of the Copenhagen model are unbounded, leading to unrealistic behavior. These difficulties can be addressed by adding terms to the model that limit the density of the resting and activated macrophages in the volume, taking physical spatial restrictions into consideration. 14 Chapter 3 An Extension of the Copenhagen Model To study quantitatively what occurs during the very earliest stages of type 1 diabetes, some of the deficiencies of the Copenhagen model must be addressed. Relevant parameter values are taken from the literature, and the model is adapted to include more details. Macrophage crowding effects are included in this extension of the Copenhagen model, ensuring that any numerical simulations remain bounded and cell densities do not exceed reasonable levels. 3.1 The Basic Model In order to build on the work done by Freiesleben De Blasio et al [9], we use the Copenhagen Model as a starting point, making modifications and additions to the model in order to address our specific questions. As we are only considering the very early stages of type 1 diabetes, before T-cells become activated, we expand on the simplest version of the model (without T-cells) presented in Freiesleben De Blasio et al [9]. We add crowding terms to the equations for the resting and activated macrophages to ensure that any predicted level of inflammation will remain finite. Trudeau et al [41] hypothesized that the neonatal, developmental wave of /3-cell apoptosis may trigger the initiation of type 1 diabetes. As we are interested in determining whether this wave of apoptosis is sufficient to trigger the initiation of type 1 diabetes, we expand the Copenhagen model by explicitly considering the density of apoptotic /3-cells (Ba). Recent experimental results by Maree et al [22] show that macrophages engulf apoptotic /3-cells at a basal rate that becomes accelerated following engulfment of the first apoptotic body. We associate these 15 states with the resting and activated forms (in contrast to other interpretations of activation in the literature). Our model assumes that both resting and activated macrophages are able to engulf apoptotic /3-cells, possibly at different rates (/i and f2 respectively). Once activated, macrophages induce further /3-cell damage, mediated through the release of cytokines and oxygen radicals. This damage occurs at a rate (I) proportional to the density of activated macrophages. Apoptotic /3-cells are removed from consideration of the current model at a non-specific rate, d. This removal may be due to clearance by scavenger cells other than macrophages (dendritic cells) or the apoptotic cells may undergo secondary necrosis, and no longer fall under the scope of the model. Formulating these assumptions mathematically, and combining these with the equations of the Copenhagen model, we obtain: M = J + ik + fyMa-cM-gMA-eiMiM + Ma), (3.1a) Ma = gMA - kMa - e2Ma(M + Ma), (3.1b) A = r]Ba-5aA, (3.1c) Ba = Ma- fiMBa- f2MaBa - dBa. (3.1d) In this extension of the Copenhagen Model, the terms e\M(M + M„), e2Ma(M + M„) represent a forcing out of resting and activated macrophages from the volume due to crowding by the total density of macrophages. These terms were not included in the original formulation of the Copenhagen model by Freiesleben De Blasio et al. This was fine for their purpose of demonstrating that there is an underlying instability leading to development of type 1 diabetes. However, as the current study is interested in quantitative understanding of the early onset of type 1 diabetes, we include these space limitations to ensure our simulations remain bounded. The remaining parameters carry the same meaning as in the Copenhagen model, summarized in Table 3.1. In this model, the density of antigen is no longer proportional to the density of activated macrophages. Instead, we assume that antigen is released at a constant rate by the apoptotic /3-cells (rj), and decays, or is cleared at a constant rate (5a). A quasi-steady state for the antigen results in the antigen concentration merely "following" the concentration of the apoptotic /3-cells: A * 16 Parameter Meaning Units / normal macrophage (M) influx cells m l - 1 day - 1 c macrophage efflux rate day - 1 b recruitment rate of M by Ma day - 1 I Ba apoptosis induced per Ma day - 1 d Ba nonspecific decay rate day - 1 k Ma deactivation rate day - 1 8 M activation rate ml cel l- 1 day - 1 h basal phagocytosis rate per M ml cel l- 1 day - 1 h activated phagocytosis rate per Ma ml cel l- 1 day - 1 crowding by resting and activated macrophages cells day - 1 Table 3.1: Definitions of parameters of the basic model We can then rescale the variables and make the substitution A ~ Ba- Many groups have measured apoptotic cell density in neonatal mice and rats, detailed in Chapter 6. Through the quasi steady state assumption on A , we reduce the model to correspond better with what can be observed. This simplification also eliminates a variable for which little or no information is available. Following this substitution we obtain the model shown schematically in Figure 3.1 and given in equations (3.2). M = J + (k + b)Ma - cM - gMBa - e\M(M + Ma), (3.2a) Ma = gMBa-kMa-e2Ma(M + Ma), (3.2b) Ba = lMa - fiMBa - f2MaBa - dBa. (3.2c) In the analysis of this model, we will look at two simplified versions, from which enough insight is gained to understand the full model given here. 3.2 The Simplest Minimal Model We begin with a fixed population of macrophages in the tissue. In this case, macrophages can only alternate between the resting and activated states. First, we assume that only the activated macrophages engulf apoptotic /3-cells (/i = 0) (We will consider the case f\ £ 0 later). These simplifications lead to the model represented in Figure 3.2 and presented explicitly in equations (3.3). 17 non specific decay (d) Figure 3.1: Schematic Representation of Basic Model: Resting macrophages (M) circulate through the tissue with a basal influx / and efflux c. Resting macrophages can also become activated at a rate g, and return to the resting state at a rate k. Activated macrophages (M„) recruit more resting macrophages to the tissue, at a rate b, and induce apoptosis in the /3-cells (I). Apoptotic /3-cells are engulfed by resting and activated macrophages (at rates f\ and fa respectively), as well as by other cells and processes (d). M = kMa-gMBa, (3.3a) Ma = gMBa-kMa, (3.3b) Ba = lMa - f2MaBa - dBa. (3.3c) In this case, the equations describing the resting and activated macrophage densities are analogous to the equations for a susceptible-infected-susceptible (SIS) type epidemic model, and our analysis follows similar methods. Since no macrophages enter or leave, the total number in the volume is conserved and we let M = M + M f l = Constant. 18 non specific decay (d) Figure 3.2: The model with a fixed population of macrophages that alternate between the resting and activated states (at rates g and k). Activated macrophages cause /3-cells to undergo apoptosis at a rate / and engulf the apoptotic bodies at a rate f2. Apoptotic /3-cells are also removed through non-specific clearance at a rate d. Where now M is used to denote the total number of macrophages. Eliminating M = Mr— Ma from equations (3.3) leads to Ma = g(M-Ma)Ba-kMa, (3.4a) Ba = lMa-f2MaBa-dBa. (3.4b) This system can be studied in the phase plane. We find that there is always a steady state at (Ma, Ba) = (0,0), that corresponds to a physiological state where there are no apoptotic /3-cells. We refer to this as the un-inflamed, or healthy steady state. Under specific conditions, described below, an additional equilibrium exists at: IgM-kd kMq kJ2 + lg g(M - Ma) This additional equilibrium corresponds to an inflamed, or diseased state, in which the density of both apoptotic /3-cells and activated macrophages is elevated. For this reason, it only makes biological sense if the densities are non-negative. This steady state is always positive provided 19 IgM - kd > 0. (The denominator of the Ma term is always positive, as all parameter values are assumed to be positive. The denominator of the Ba term is positive since the number of activated macrophages is smaller than the total number of macrophages in the volume, i.e. M„ < M). It can be seen that the condition for the second steady state to exist is equivalent to ro = Ml£>l. (3.5) This is analogous to the threshold condition for the spread of a disease (ro > 1). The parameter VQ can be viewed as the basic reproductive parameter for the inflammation, and is equivalent to the "basic feedback" parameter defined in Freiesleben De Blasio et al [9] (see (2.2)). The interpretation of our parameter, however, is slightly different - as a number of secondary activations of macrophages per macrophage in the volume. (Multiplying this parameter group by macrophage density (M) leads to a dimensionless number that is the number of secondary activations caused by one activated macrophage during its activation period.) The various parameter groupings that appear in our analysis and an interpretation of their meaning is summarized in Table 3.2. When ro > 1, each macrophage that becomes activated will in turn activate more than one additional macrophage. In this case, the number of activated macrophages will continue to rise, leading to a sustained, elevated level of inflammation. The parameter ro can be affected by changing any one of the parameters g, I, k, d, but if all these parameters remain unchanged, ro can be affected by adjusting the density of macrophages in the volume (M). We can therefore use ro to determine a critical level of macrophage density, M c . If M > Mc then the system will tend to elevated inflammation. Conversely if M < M c the system will be stable in the healthy state. Note that this is similar to an epidemic becoming endemic only if the susceptible population is large enough. 3.3 A Second Informative Variant More complicated variants of the model arise as soon as the volume is considered to be open, with macrophages free to circulate into and out of the volume. One variation that has 20 parameter meaning dimensions l /c lie normal macrophage (M) residence time normal M density in tissue t M l/d j/d basal turnover time of apoptotic /3-cells (Ba) influx of M during Ba turnover time t M l / ( / 2 M) bMa b/fi phagocytosis time of one Ba inflammation-induced flux of M inflammation-induced flux of M during engulfment time t M/t M glc c/g number of M activated per Ba during residence time number Ba needed to activate one M during residence time 1/B„ Ba 1/k l/k average time that M„ spends in activated state number B„ produced per Ma during activation time span t Ba/Ma g/d gl kd fraction M activated per Ba during Ba turnover time fraction M activated per Ma V(BaM) 1/Ma M— Number of secondary activations per Ma Table 3.2: Parameter Groupings and their interpretation in the context of the model presented here. proved to be informative is obtained by removing the crowding terms and assuming that resting macrophages do not engulf apoptotic /3-cells. This is the variant obtained by setting fa = e\ = e2 - 0 in equations (3.2), i.e. with the equations: M = / + (k + b)Ma - cM - gMBa, (3.6a) Ma = gMBa-kMa, (3.6b) Ba = lMa - f2MaBa - dBa. (3.6c) When solving for the steady states, we immediately obtain the healthy state, (M,Ma,Ba) = (J/c, 0,0), but the existence of other steady states is not so obvious when looking at all three equations. A consideration of time scaling and dimensional issues (see Appendix B) allows the use of a quasi-steady state (QSS) assumption to reduce the system to two equations. This simplifies analysis through use of phase plane methods. 21 A quasi-steady state assumption on M (M = 0 in (3.6a)) gives: M = J + ( k + b)Mg c + gBa Substituting this expression for M into equations (3.6b, 3.6c) gives a reduced system that we can study in the Ma-Ba phase plane: M„ = Jl±£±mi)B,-M,, (3.7a) Ba = IMa - f2MaBa - dBa. (3.7b) Solving for the nullclines in this reduced system, we obtain: ( kr\ M i » ) o / ! > t s : ' ( 3 8 a ) B.nullcline: B. - (3*) These are both monotonic Michaelian curves with horizontal asymptotes C M = (kc/gb) and CB = 1/fz for the M a and Ba nullclines, respectively. Both curves go through the origin, where their initial slopes are SM = (kc/gl) and Sg = (l/d). When looking in the phase plane (Figure 3.3), there are four possible configurations of the Ma and Ba nullclines, only one of which (d) reasonably captures the idea of "inducible inflammation" resulting from a super-threshold stimulus of some type. This inducible inflammation is key to understanding why NOD mice develop type 1 diabetes while Balb/c mice do not. We are looking for conditions under which the healthy state will be stable for both NOD and Balb/c mice - accounting for the fact that not all NOD mice develop the disease, and under which a sufficiently large stimulus will push the NOD (but not Balb/c) system to sustained inflammation. We consider each case explicitly to understand how the system will respond if any of the conditions for inducible inflammation are violated. The four possible nullcline configurations are shown in Figure 3.3 and characterized below: 1. C M > CB, SM < SB (Figure 3.3a): This case corresponds to an unstable rest state and a stable inflammatory state in the first quadrant. It is not biologically reasonable, as all cases lead to inflammation. In terms of the parameters, the inequalities in this case are: kc I kc I gb h' gJ d' 22 (c) (d) Figure 3.3: Four possible nullcline configurations: (a) Case 1: stable inflammation, (b) Case 2: rampant inflammation, (c) Case 3: persistent health, (d) Case 4: inducible inflammation. Combining these two inequalities, we obtain b_ kc I h < gl < d' 2. CM < Cg, SM < SB (Figure 3.3b): In this case, there is only one steady state at ( M A , BA) = (0,0), and it is always unstable, so rampant (unbounded) inflammation always occurs. In terms of the parameters, the inequalities are: kc I kc I h' gJ d' or, by rearranging, we obtain: kc b kc J gb fi gl d' 3. C M > C B , S M > Sg (Figure 3.3c): Again there is only one steady state at (M f l , BA) = (0,0). However, in this case, it is globally stable, so there is never any inflammation. In terms of the parameters, the inequalities are: kc I kc I gb fi gJ d' 23 or, by rearranging, we obtain: kc b kc } gl > h' gl > d ' 4. CM < CB, SM > Sg (Figure 3.3d): This is the case of "inducible inflammation." Here we have a stable healthy state (MAR BA) = (0,0) and a saddle node in the positive quadrant. That saddle node will create a threshold for inflammation. In terms of the parameters, the inequalities for this case are: This is the only case that can capture the idea of inflammation resulting from a super-threshold stimulus of some type. This case occurs whenever: The condition for case 4 to occur is equivalent to: Note that the term (gl/ck) is the same as the fo parameter described by Freiesleben De Blasio et 3.3.1 Engulfment of Apoptotic Bodies by Resting Macrophages The assumption that resting macrophages do not engulf apoptotic bodies greatly simplified the preceding analysis. This assumption, however, is not biologically realistic and leads to the "dimintehing effector cell" phenomenon: when the inflammation subsides, macrophages return to the rest state and cease to engulf apoptotic bodies. Complete resolution of the inflammation requires some mechanism to remove the remaining apoptotic bodies. In the case of our model, this mechanism is the non specific clearance of apoptotic bodies (d). This results in the condition (3.9) being overly restrictive and leads to a huge overestimate of the value of d when the parameters values are quantified. kc I kc I ti gJ d' al [9]. 24 This is addressed by allowing resting macrophages phagocytosis ability (f\ + 0). This makes one of the inequalities considerably "softer," namely SM < SB that ensures the stability of the resting state. This inequality is equivalent to (kc/gl), or (kcd/gjl) < 1. The condition for stability of the healthy state now becomes: ckd „ kf\ — > 1 - — =>J- < " (3.10) c gl - kfi Implicit in this condition is the requirement that gl > kf\, as J/c will always be positive. 3.4 Analysis of The Complete Basic Model We now consider the analysis of the model given by equations (3.2). When quantifying macrophage engulfment rates Maree et al [22], observed that after the first engulfment the macrophages became activated and engulfment rates increased. In line with these observa-tions, we assume that g = f\, which just means that resting macrophages become instantly activated as a result of their first engulfment of an apoptotic /3-cell. For the moment, assuming negligible crowding effects, e\ = 0, the system of equations (3.2) has a healthy rest state given by (M,Ma,Ba) = (//c,0,0), with a basal density of resting macrophages, J/c, and no apoptotic /3-cells or activated macrophages. This essentially remains true even when crowding effects exist (i.e. when 0 < e\ <K 1). In this case, the density of resting macrophages at the healthy rest state is M = ~° + ^ + 4 g J This can be approximated by terms in the Taylor series expansion (about e\ = 0): so that M w J/c provided e\ « c 2 / J , which is generally the case (see Chapter 4). We can define a threshold condition for the basic model, analogous to ro defined by equation (3.5) for the simplest model, representing the tendency for a low level of inflammation to be amplified. Close to the resting steady state (J/c, 0,0), inflammation grows if Ma > 0, i.e. 25 gMBa - kMa > 0, or simply gMBa/kMa > 1. About the rest state we define r 0 = gMBa/kMa > 1, and approximate M « J/c, Ma « B«(/iM + fiMa + d)/l « Ba{J\M + d)/l, since near the equilibrium f2MaBa ~ 0. Substituting these approximations into the expression for ro we obtain the condition for accelerating inflammation: glJ ro (3.11) cdk + kfiJ Freiesleben De Blasio et al [9] scale their model so that J = d = 1 and omit engulfment by resting macrophages (f\ - 0). In this case, ro = gl/ck - fo, corresponds exactly to their equivalent threshold parameter. The definition and interpretation of ro is closely linked to the stability of the healthy rest state. When e\ is small, the approximation of resting macrophage density by //c holds, and we obtain the Jacobian Matrix (evaluated at the rest state (//c, 0,0)): Jac (c + 2e^) (k + b)-eiL -k-ez -si si 0 o / -{M+d)\ In the case e2 <K ck/J (generally true, see Chapter 4) the Jacobian can be approximated by: •(c + 2e,l) (k + b)-e,}- -g\ 0 -k gl 0 / -{filc+d)\ In order for the rest state to be stable, we require that all the eigenvalues be negative. One of the eigenvalues, - [c + 2e\ j), is always negative. The remaining eigenvalues will be negative whenTrace(/[2x2]) < 0 and Det(/[2x2]) > 0/ where: jac J[2x2] = -* g'c J ~(h\+d)\ The trace is always negative, and the determinant will be positive provided 0 < kfj- + kd-glt }_ kd _ kd ^~c < gT^kfl ~ glj^ky (3.12) 26 Note that this condition can only be satisfied if the denominator is positive, leading again to the requirement that gl > kf\, or / > k. This is complementary to the condition for growing inflammation (ro in equation (3.11)), and is equivalent to condition (3.10). Analysis of the non trivial steady states of equations (3.6) becomes cumbersome due to the multiple nonlinearities. (Note that even when e\ «: c 2//, the second term of the Taylor expansion for M can only be ignored close to the healthy rest state.) A simplified system, in which the crowding effects are ignored (e\ = e2 = 0) provides some further insight. This system allows for at most one non trivial positive equilibrium, and whenever the healthy state is stable, this second equilibrium must be a saddle point. The stable manifolds of this saddle point act as a separatrix, dividing the healthy regime from the inflamed regime. The simplification eliminates a finite high density (or inflamed) equilibrium (leading to unbounded inflammation), but only slightly alters the existence and location of the separating saddle point. Under the simplifying assumptions, the saddle point satisfies the equations: M = / + (Jfc + b)Ma - cM - gMBa = 0, (3.13a) Ma = gMBa -kMa=Q, (3.13b) Ba = lMa - fiMBa - f2MaBa - dBa = 0. (3.13c) We begin by adding equation (3.13a) to equation (3.13b): 0 = M + M f l = J + bMa-cM => M = - + -M„. (3.14) c c Solving equation (3.13b) we obtain: gM Adding equations (3.13b) and (3.13c) together gives: „ kMa B° = -zn- (3-15) 0 = Ma + B\ - (/ - k)Ma - (f2Ma + d)Ba (since g = ft) (I ~ k)Ma Setting (3.15)=(3.16) gives: kMq = (I - k)Mq gM f2Ma + d' 27 one solution of which is Ma = 0 (the stable rest state). Otherwise, when Ma 0, we obtain: (f2Ma + d\ k We can now solve for Ma by setting (3.14)=(3.17): l + b_Ma = (f2Ma+d\k c ' c"'u \ l-k jg We can now solve for the values of M, Ma, Ba at the saddle point. Using equations (3.14), (3.15), and (3.18) we obtain: M _ _ m - m _ kdc-jgd-k) kdc-ig(i-k) ~ bgd - k) - ckf2' M a - bg(l - k) - ckf2' 3 a - g(bd - ff2) • ( 3 - 1 9 ) As this saddle point will only exist for NOD mice, we can now make a distinction between the two strains. Combining the condition for stability of the healthy state (equation (3.12)) with the requirement that the saddle exists for NOD mice (with M , Ma, Ba > 0) we obtain: for N O D : ]- < < (3.20a) forBalb/c: 1 < IT^Tv (3-20b) d g(l - k) This ensures that both strains will have a healthy rest state, but only NOD mice can have chronic inflammation. It is important to note that the distinction can be made by changing only the parameter f2 between the two strains (i.e. f2 (NOD) < fi (Baib/c)) The results of the above analysis can be seen numerically by looking at the Ma-Ba phase plane. We begin with the system given in (3.2) and use a QSS assumption on M . We then have the system: -(eiMa + c + gBa) + V ( e i M f l + c + gBa)2 + 4ei(J + (k + b)Ma) = 2e\ ' ( 3 ' 2 1 a ) Ma = gMBa-kMa-e2Ma{M + Ma), (3.21b) Ba = lMa - ftMBa - f2MaBa - dBa. (3.21c) 28 i L i I i I i I i I 0 1 2 3 4 M a Figure 3.4: Plot of nullclines for equations (3.21), using arbitrary parameter values. The horizontal axis gives the density of activated macrophages, Ma, and the vertical axis gives the density of apoptotic /3-cells, Ba. The M„ nullcline is given by the red curve, and the Ba nullcline is given by the black curve. The dashed blue curve shows the separatrix. Here we see that any sufficiently large increase in the concentration of Ma or Ba will result in the system entering the basin of attraction for the inflamed steady state. This figure was produced using XPP, and the .ode file is given in Appendix C. l . The parameter values used to create this figure are: / = 0.01 cells m l - 1 , c = 1 day - 1 , b = 1 day - 1 , / = 1 day - 1 , d = 0.5 day - 1 , e\ =0.1 cells day - 1 , e2 = 0.1 cells day - 1 , g = 1 ml cel l - 1 day - 1 , ft = 1 ml cel l - 1 day - 1, f2 = 0.01 ml cel l - 1 day - 1 , k = 0.3 day - 1 Figures 3.4, 3.5, 3.6 show that it is possible to find parameter values for which this model predicts a case of "inducible inflammation" in NOD mice. Figure 3.4 shows the Ma and Ba nullclines in the phase plane, following a QSS assumption on M . Figures 3.5,3.6 shows the null-surfaces in the full phase space of the system. These figures demonstrate that this extension of the Copenhagen model captures qualitatively the dynamics we believe to be important in the early stages of type 1 diabetes The inequalities in (3.20), however, are quite restrictive and cannot be satisfied using biologically reasonable parameters, as demonstrated in Chapter 4. 29 Figure 3.5: Null surfaces for the basic model (3.2), using arbitrary parameter values. The Ma and Ba axes are labeled on the plot, and the M axis is in the vertical direction. Here the M null-surface is blue, the Ma null-surface is green and the Ba null-surface is red. The steady states of the system given in (3.2) are visible as the points where all three surfaces intersect: the stable resting and inflamed states, with an unstable saddle point separating them. The two small images are a stereoscopic view of the larger image, providing the illusion of 3 dimensions. This figure was produced using MAPLE, with parameter values: / = 0.01, c = 1, b = 1, / = 1, d = 0.5, C] = 0.1, e2 = 0.1, g = 1, / i = 1, f2 = 0.01, k = 0.3 30 Ma Figure 3.6: Here we see the null-surfaces of (3.2) as if looking "up from below." The Ma and Ba axes are labeled on the plot, and the M axis points into the page. The M null-surface is blue, the Ma null-surface is green and the Ba null-surface is red. Again all three steady states of the system can be seen. This figure was produced using MAPLE, with parameter values: / = 0.01, c = 1, b = 1, / = 1, d = 0.5, ei = 0.1, e2 = 0.1, g = 1, fx = 1, f2 = 0.01, k = 0.3 31 Chapter 4 Parameter Values Estimates for the parameter values used in the current study were taken from the literature where available, and estimated otherwise. 4.1 Macrophage Activation and Engulfment Rates Various defects in the macrophages from NOD mice have been observed. Of most relevance to the current study is the decreased phagocytic ability of NOD macrophages as compared to Balb/c macrophages [1, 22, 27, 30]. Maree et al [22] combined in vitro experiments with math-ematical modeling to quantify the rates of engulfment and digestion of apoptotic thymocytes by macrophages from NOD and Balb/c mice. Macrophages were cultured in complete medium and apoptotic cells were introduced at a ratio of 5:1 apoptotic cells to macrophages. Subsets of macrophages were examined at intervals of 5, 10,20,30 minutes and 1,2,4,6,8 hours to determine the number of apoptotic bodies engulfed. Results were recorded as the number of macrophages M , having i engulfed apoptotic cells.The kinetic model presented by Maree et al is based on determining how the number of macrophages M , changes. The model can be represented schematically as: Mo k M i ^ M 2 ^ • • • k MN k • • • (4.1) kd kd kd kd kd In this model, ke is the rate of binding and engulfment of apoptotic cells by macrophages - considered a single, irreversible process. It is assumed that digestion of apoptotic cells occurs in a saturation regime, i.e. that apoptotic cells are digested sequentially, in the order 32 of engulfment. The rate of digestion, k^, corresponds to the rate at which macrophages move from class j to class / - 1 by a single digestion (of one internalized apoptotic cell). This model and variants of it were fitted to the experimental observations. It was determined that for Balb/c mice, the best fit to the data came from a variant of the model with an "activation" step. In this variant, macrophages move from class 0 to class 1 at a rate ka. Following this initial engulfment, macrophages become activated and proceed to engulf apoptotic cells at an increased rate ke. It is assumed that digestion occurs at a constant rate and that activation is immediately reversible, i.e. any cell returning to class 0 will engulf the next apoptotic cell at a rate ka. Using this model, and extensive parameter fitting methods, the authors determined that ka = 2 x 10 - 5 ml cell - 1 day - 1 for Balb/c macrophages. This corresponds to the parameter g = f\ in our model. The rate of engulfment by activated macrophages (/2 in our model) was determined to be ke = 5 X 10 - 5 ml cell - 1 day - 1 for Balb/c macrophages. For NOD mouse macrophages, the authors determined that ka = ke = 1 x 10 - 5 ml cel l - 1 day - 1 , i.e. these macrophages do not display any clear 'activation' step, a fact of primary importance for our model. The rate of digestion, kd = 25 day - 1 , was determined to be the same for both NOD and Balb/c macrophages. Other studies have included models of macrophage engulfment. Wigginton and Kirschner [47] present a model of human infection with Mycobacterium tuberculosis. In this model, the authors consider three classes of macrophages: resting macrophages which can become infected with the M . tuberculosis bacterium, or activated to fight the infection. Here, activated macrophages are defined to be those capable of effectively engulfing and killing the bacteria. It is assumed that activated macrophages produce oxygen radicals and may cause some tissue damage as a result of their killing activities. In the absence of sufficient stimulus, activated macrophages will return to their resting state. The authors showed that the cytokine IL-1 acts to down regulate the immune system, including deactivation of macrophages. Using fluorescence labeling and flow cytometry data from another source, Wigginton and Kirschner calculated the maximal deactivation rate of macrophages to be k = 0.4. 33 The value given by Maree et al [22] for the digestion rate of apoptotic cells by macrophages provides only an upper bound for the macrophage deactivation rate. In the model scheme (4.1), transitions to the right (by engulfment of cells) occur at a rate /26a while transitions to the left (by digestion of engulfed cells) occur at a rate k^. Thus, at quasi steady state, M 2 = AMi, and the number of macrophages in class n is Mn = AM„_i, where A = /^ Ba/fcd- In terms of our model, activated macrophages are any macrophages with at least one engulfed apoptotic cell, i.e.. Ma = (Mi + M 2 + M 3 -1 ) - M i ( l + A + A 2 H ) ~ M i 1/(1 - A). Only the macrophages in class M i can revert to the rest state. These macrophages make up a fraction M\/Ma — (1 - A) of activated macrophages. Thus, an estimate for the deactivation rate is k « fcd(l - A) = k^ - jjBa. In the case fzBa/kd 1 the deactivation rate k becomes much smaller than the digestion rate kd = 25 day - 1 . Moreover, studies have shown that complete digestion of apoptotic material does not signify an instantaneous return to the rest state, and that macrophage capacity to engulf apoptotic bodies is increased following uptake of apoptotic cells on the previous day [20, 30]. The timescale of macrophage deactivation following digestion of all apoptotic bodies is controversial, but is likely on the order of 1-2 days. For the current study, the macrophage deactivation rate is taken to be k = 0.4 day - 1 . 4.2 Volumes and Sizes According to van Furth [45], the blood volume of a mouse weighing 25 g is 3 ml. A human pancreas weighs between 60-100 g, and so has a volume of roughly 100 ml. The islets of Langerhans account for 1% of the pancreas by mass (or volume), implying that for humans, the total islet volume is approximately 1 ml. A typical human weighs 75 kg, while a mouse weighs around 25 g. Thus the ratio of mouse to human masses (and volumes) is 0.33 X 10"3. If the ratio of pancreas to body size is the roughly same in mice and humans, then a human pancreas of 100 ml corresponds to a mouse pancreas of 3 x 10~2 ml, but the density of cells should be about the same, and 1 ml of human islets scales to 3 x 10 - 4 ml total mouse islet volume. 34 In the mouse, each individual islet is approximately 150 um in diameter, corresponding to a volume of 1.77 X l O ^ m 3 « 1.8 X 10~6ml [8,35]. An islet contains roughly 500-1000 /3-cells, thus the /3-cell density is in the range of 4 x 108 cells m l - 1 . 4.3 Macrophage Density and Flux Rates There are many estimates in the literature for flux rates and densities of macrophages, some of which are presented here. 1. For their model of human infection with M. tuberculosis, Wigginton and Kirschner [47] searched the literature for parameter estimates to use in their model. Some of these parameters correspond to parameters in the Copenhagen model and our extension of it. For their model of M . tuberculosis infection, Wigginton and Kirschner used the bron-choalveolar lavage (BAL) fluid (used to wash the inside of the lungs and collect cells and secretions from the respiratory tract) as a reference space and parameter values are recorded in cells m l - 1 . In the BAL fluid, in the absence of infection, resting macrophages are present at a density of 3 - 4 X 105 cells m l - 1 , with a basal influx rate of 330 - 440 cells m l - 1 day - 1 . In the model, it is assumed that macrophages do not leave the BAL fluid, but die at a rate 0.011 day - 1 . Following infection with M . tuberculosis, resting macrophages are recruited to the BAL fluid by activated macrophages (at rate 0.03 - 0.05 day - 1) and also by resting macrophages (at a rate 0.01 - 0.07 day - 1). Of these recruitment terms, only the recruitment by activated macrophages has an analogue in our model. 2. Charre et al [2] examined macrophage accumulations in the pancreas of NOD mice along with control strains. The authors considered two specific subsets of macrophages, with slightly different roles. During the first three weeks of life, the densities of both types of macrophages were between 4-15xl0 7 cells m l - 1 . This density, however, is representative of the whole pancreas and is not limited to the density of macrophages within the islets, making it likely that this is an overestimate of macrophage density. 3. Pilstrom et al [33] counted the number of cells infiltrating the pancreatic islets of Langer-35 hans in NOD mice. The pancreas was removed from the animal, and the islets of Langerhans were isolated and then disassociated. Infiltrating cells (macrophages as well as lymphocytes) were then isolated from the islet suspension in order to measure cytokine production levels. Normally, the authors were able to isolate 1 - 2 X 104 infiltrating cells per pancreas at 6-7 weeks of age, increasing to 5 - 15 X 104 cells per pancreas from 12 weeks on. Infiltrating cells made up about 20 % of the islet cells at 6 weeks and increased to 75 % by week 12. This is due to more macrophages and lymphocytes being recruited to the islets as well as a decrease in the number of endocrine islet cells due to damage. The ratio of macrophages to lymphocytes in the infiltrating cells is not given. However, a rough estimate for the density of resting macrophages may be obtained using this data. Taking the average cell count at 6 weeks (1.5 X 104) and dividing by the total islet volume (3 x 10 - 4 as determined above) we get a resting macrophage density of m = 5 x 107 cells m l - 1 of islet. Note this figure is significantly different than that obtained by Wigginton and Kirschner, but agrees with that from Charre et al. It is also important to note that this is at best a very crude estimate, as the authors were interested in acquiring the infiltrating cells for other purposes and not in obtaining an accurate count of macrophages. 4. Van Furth et al [45] quantified the production of monocytes and their kinetics in mice. By labeling newly formed monocytes with 3H-Thymidine, which is incorporated into newly synthesized DNA, the authors were able to measure the rate of production of monocytes in the bone marrow, and track their movement from the bone marrow into the blood and ultimately to the tissues. "Normal" mice, with no inflammation, produce monocytes in the bone marrow at a rate of 6.5 x 105 cells hr 1 and have 1 x 106 monocytes circulating in the peripheral blood. These circulating monocytes leave the blood randomly at a rate 0.04 h _ 1 and enter the tissue where they develop into macrophages. Using this information, we can calculate the rate at which macrophages enter the pancreatic islets of Langerhans. Assuming monocytes leave the blood and enter all tissues uniformly, and assuming total volume of mouse tissue to be approximately 20 ml (based on a mouse weight of 25 g, 36 and 3 ml of blood) we calculate the influx of macrophages into the pancreas to be: / = (1 x 106 cells) (0.04b-1 ) 1 2000 cells m l - 1 h _ 1 20 ml Converting this to a rate per day we obtain: J = 5 x l 0 4 cells m l - 1 day"1. (4.2) 5. Macrophages are present in all tissues of body, sometimes near the blood vessels, other places close to the epithelial cells or vascular endothelial cells [31]. The rate of turnover of macrophages is not well known, but under normal conditions is estimated to be between 4 and 15 days [31,42, 43,44]. This leads to a turnover rate in the range 0.07 - 0.25 day - 1 . We approximate c « 0.1 day - 1 . 6. Using the estimate for macrophage influx (4.2) along with the estimate for the rate of macrophage turnover, we obtain a resting macrophage density of 4.4 Non-specific Removal of Apoptotic /?-cells van Nieuwenhuijze et al [46] investigated the kinetics of leakage of DNA fragments (called mi-crosomes) from apoptotic Jurkat cells in vitro. The leakage of nucleosomes from apoptotic cells is an indication that the cell has become necrotic. The authors determined that nucleosomes entered the culture medium approximately 1-2 days following the initiation of apoptosis. For the basic model we take this to be the residence time of apoptotic cells in the tissue, and set the non specific clearance rate d = 0.5.. 4.5 Calculated Parameters 1. The Copenhagen model is based on the assumption that during the inflammatory re-sponse, there is additional recruitment of resting macrophages into the tissue due to the 37 presence of activated macrophages. This recruitment is represented by the parameter b. We now consider what is an appropriate range for this parameter. In the extension of the Copenhagen model, presented and analyzed in Chapter 3, the total density of macrophages (both resting and activated) at the saddle point satisfies: d(Ma+M) X A —— = } + bMa - cM « 0. at If M is the total density of macrophages, let (p be the fraction that are in the activated state, i.e.: Ma = (f)M, M = (1 - (f))M,0 <cp<l. Then: / + bcj)M - c(l - cp)M = 0 <l-4>\ J <P I (pM c (p \ m Here the fraction inside the braces represents the ratio of uninflamed macrophage density (J/c) to total macrophage density M at the saddle point. Table 4.1 gives values of b/c, and b, corresponding to specific assumptions about the fraction of total macrophages that are activated at the saddle point. The total macrophage density at the saddle point is not known, but it should be between 5 x 105 and 1 X 108 cells m l - 1 (since, due to crowding effects the total macrophage density is not expected to ever become much larger than 1 x 108, and at values smaller than 5 x 105 chronic inflammation would always occur). From this, we choose b = 0.09 day - 1 as a reasonable estimate for the recruitment rate of resting macrophages by activated macrophages. 2. There remains one parameter to estimate, /, the rate of induced /3-cell apoptosis caused by activated macrophages. We wish to choose a value for / that will demonstrate that reduced macrophage engulfment is a sufficient difference between NOD and Balb/c mice to account for the fact that NOD mice may develop chronic inflammation while Balb/c mice remain healthy. To estimate /, we use the conditions found in (3.20), combined with 38 M 1 X 106 l x l O 7 l x l O 8 fraction MJM calculated resulting value of <P b/c b b/c b b/c b 0.8 — — 0.187 0.019 0.244 0.024 0.7 — — 0.357 0.036 0.421 0.042 0.6 — — 0.583 0.058 0.658 0.066 0.5 — — 0.900 0.090 0.990 0.099 0.4 0.250 0.025 1.375 0.137 1.487 0.149 0.3 0.667 0.067 2.167 0.217 2.317 0.232 0.2 1.500 0.150 3.750 0.375 3.975 0.397 0.1 4.000 0.400 8.500 0.850 8.950 0.895 Table 4.1: Estimates calculated for the parameter b, based on assumptions regarding the total macrophage density and the fraction of activated macrophages at the saddle point. the parameter estimates determined here (see Table 4.2). The required existence of a stable healthy rest state for both strains implies that k Jg (5 x 104)(2 x 10"5) (l-k) k (l-k) dc (0.5)(0.1) Jg (5 x 104)(1 x 10 - 5) dc (0.5)(0.1) = 20 Using Balb/c parameters, (4.3a) = 10 Using NOD parameters. (4.3b) Parameter Balb/c NOD units source / 5 x l 0 4 cells m l - 1 day - 1 [45] c 0.1 day - 1 [31, 42, 43, 44] b 0.09 day - 1 est. d 0.5 day - 1 [46] k 0.4 day - 1 est, [47] 1 x 10 - 8 cells day - 1 est. s = h 2 x 10~5 1 x 10 - 5 ml cel l - 1 day - 1 [22] h 5 x 10 - 5 1 x 10 - 5 ml cel l - 1 day - 1 [22] Table 4.2: Estimates for parameter values used in numerical simulations of the Basic Model. Parameters are defined in Table 3.1 We also require an inflamed steady state for NOD mice, corresponding to a state of chronic inflammation. This additional steady state exists when k bg _ (0.09)(1 x 10 - 5) < = 0.9. (4-4) (/ - k) f2c (1 x 10-5)(0.1) These two conditions (4.3, 4.4) are contradictory and both cannot be satisfied! Either all 39 mice develop chronic inflammation (if k/(I - k) < 20 for Balb/c or < 10 for NOD), or all mice remain healthy (if k/(I -k) > 20). Furthermore, for NOD parameters, we have / = ! ^ = l x l t f< 9 x l ( ? = _*°» * d 0.5 l x l O - 5 fa violating condition (3.20a) by an order of magnitude. It is now clear that given the clear-cut range of values of /, g, fa, fa , d, b, NOD mice could never enter the state of chronic inflammation in a biologically reasonable range of parameters of the Copenhagen model, regardless of any other strain specific differences (in the values of c, k, or /). It is also evident that differences in macrophage function alone, in the context of the Copenhagen model cannot account for the chronic inflammation in NOD mice. In the next chapter, the model is expanded to include the role of necrosis. 40 Chapter 5 A More Detailed Model for the Initiation of Type 1 Diabetes We have seen that, in the context of the original Copenhagen model, reduced macrophage clearance in NOD mice is not sufficient to account for NOD mice being susceptible to type 1 diabetes while Balb/c mice remain healthy. While it is possible for the Copenhagen model to capture the bistability we believe to be responsible for the susceptibility of NOD mice, it does not have this property in a biologically reasonable parameter regime that distinguishes between normal and diabetes prone animals. The Copenhagen model, however, is a first approximation of the early stages of type 1 diabetes pathogenesis. Freiesleben De Blasio et al demonstrated that an underlying instability could be the cause of type 1 diabetes susceptibility, and their original model considers only a few details to demonstrate this point. We now extend the Copenhagen model, adding several key details to capture more fully the interactions leading to susceptibility to type 1 diabetes. In the original Copenhagen model, cytokine mediated damage to |3-cells is treated as a direct effect of the macrophages themselves. We now incorporate the mediators themselves, introducing non linearities into the model that more closely approximate the biology. We also consider the "remains" of apoptotic cells that remain uncleared. The necrotic material left in the tissue may play an important role in accelerating the inflammatory response, making its inclusion in a model logical. 41 5.1 Additions to the Model In the original Copenhagen model, it is assumed that tissue damage (induced apoptosis of /3-cells) is caused directly by the activated macrophages. This damage, however, is actu-ally caused indirectly by the activated macrophages through the production and secretion of cytotoxic cytokines and reactive oxygen species (eg. NO - ) . In addition, apoptosis has generally been considered a non-inflammatory form of cell death, and it has been demon-strated that when macrophages engulf apoptotic material the production and release of pro-inflammatory cytokines is inhibited [3,34]. When apoptotic cells are not cleared quickly enough (by macrophages) they undergo secondary necrosis. Macrophage engulfment of necrotic ma-terial then results in the production of pro-inflammatory cytokines [3, 34]. 5.1.1 Secondary Necrosis Apoptotic cells are characterized by condensed chromatin and degraded DNA. Fragmentation of the cell then leads to the formation of apoptotic bodies [6]. Apoptosis is a physiological way to eliminate cells that no longer have a biological function [1], and allows for rapid clearance by phagocytes in an orderly manner, possibly bypassing inflammatory consequences [3]. Necrosis, on the other hand, is a form of cell death associated with pathological tissue injury and is a rapid disorganized swelling of the cell followed by cellular rupture [3]. Necrotic cells are characterized by an acute loss of cellular homeostasis and disruption of cellular organelles, leading to leakage of cellular contents. One current hypothesis is that necrosis induces an inflammatory response [6]. Phagocytosis of apoptotic bodies does not lead to the production or release of pro-inflammatory cytokines by activated macrophages. Rather, engulfment of apoptotic bodies is believed to have anti-inflammatory results [1]. If apoptotic cells are not efficiently cleared, tissue damage may result [34]. Uncleared apoptotic cells undergo secondary necrosis and engulfment of the necrotic cells by activated macrophages triggers release of pro-inflammatory cytokines. Necrotic cells, however, may not be enough to cause an inflammatory response. Cocco et 42 al [3] demonstrated that necrotic cells enhance macrophage activation but are not sufficient to initiate it - necrotic material causes previously activated macrophages to secrete inflam-matory cytokines. Cocco et al [3] also demonstrated that macrophages bound apoptotic and necrotic cells to comparable extents and that the binding of apoptotic and necrotic cells occurs through non-competing mechanisms. This will be reflected in the extended model presented in this chapter - where we assume that only necrotic cells trigger the release of cytokines, and macrophages engulf apoptotic and necrotic cells with the same rate constants. 5.1.2 Cytokines Cytokines are small, soluble proteins, produced and secreted by numerous cell types. Once secreted, cytokines diffuse through the tissue altering the behavior or properties of the cells they come in contact with [14]. In the Copenhagen model, the effect of cytokines is not considered. Cytokine mediated cell damage and recruitment of resting macrophages is taken to be a direct action of activated macrophages. It was demonstrated in Chapter 4 that the Copenhagen model is an oversimplification. We extend the model, incorporating the fact that cytokines are secreted when macrophages come in contact with necrotic cells, and that it is the cytokines that damage the /3-cells. Following engulfment of necrotic material, macrophages have been shown to secrete inter-leukin (IL)-1,6, tumor necrosis factor (TNF)-a, interferon (IFN)-y as well as reactive oxygen free radicals (eg. N O - ) that are known to be cytotoxic to /3-cells [14,48, 33]. Ideally, the effects of each of the macrophage-secreted cytokines would be considered sepa-rately as their effects interact in a complex way. For example, IL-12 secretion by macrophages is enhanced by IFN-y and inhibited by IL-10 [47]. At this time, such details have not been suffi-ciently quantified to be accurately included in a model. As a first approximation, we consider a generic chemical substance, referred to as "cytokine," representing the net inflammatory effect of all the cytokines involved. 43 5.2 Model Equations In the extended model, two new variables are introduced: Bn, the density of necrotic /3-cells (measured in cells ml - 1 ) , and C, the concentration of a generic cytokine (measured in nM). In this model, the equations for the macrophage densities (3.2a,b) remain unchanged from the basic model. The equation for apoptotic /3-cells (3.2c), however, is slightly changed. Due to the finite number of cytokine receptors on /3-cells, the effect of cytokines is assumed to be Michaelian, saturating to some maximum level (A m a x ) as the cytokine concentration increases. This induction of apoptosis is approximated by replacing the term lMa in equation (3.2c) with a Michaelis-Menten saturating function of cytokine concentration, with maximal rate A m a x , and half-max cytokine concentration kc. The decay of apoptotic /3-cells, d, is now a transition from apoptosis to secondary necrosis. Necrotic /3-cells (Bn) are formed when apoptotic cells undergo secondary necrosis, and are cleared by resting and activated macrophages. Based on the observations of Cocco et al [3], it is assumed that macrophages engulf necrotic cells at the same rate as apoptotic cells: f\ and f2 for resting and activated macrophages respectively. Activated macrophages produce cytokines following contact with necrotic /3-cells, these cytokines are released at a rate a, and decay at a rate 5. The full model is presented in Figure 5.1 and equations 5.1. M = ] + {k + b)Ma-cM-gMBa-eiM{M + Ma), (5.1a) Ma = gMBa-kMa-e2Ma(M + Ma), (5.1b) B'„ = dBa - fiMBn - f2MaBn, (5.1c) Ba = jrf§-fiMBa-f2MaBa-dBaf (5.1d) C = aB„Ma-bC. (5.1e) This system has a steady state at (M, Ma, B„, Ba, C) = (J/c, 0,0,0,0) corresponding to the healthy state. As we require this steady state to be stable, we now look at the eigenvalues of the system to determine under what conditions the healthy state will be stable. 44 recruitment (b) Figure 5.1: The most detailed model: Inflammatory cytokines (C) are secreted by activated macrophages upon contact with necrotic /3-cells {B„). These cytokines cause apoptosis of /3-cells with Michaelian kinetics (maximum rate A m a x , half max concentration kc). Apoptotic and necrotic /3-cells are cleared at the same rates by macrophages. 5.2.1 Stability of the Rest State We find the Jacobian matrix of the detailed model to be: -c - gBa - e i (2M + Ma) k + b- e\M 0 -gM 0 gBa - e2Ma -k - e2(M + 2Ma) 0 gM 0 Jac - -flBn -flBn -fiM-f2M„ d 0 -flBa -flBa 0 -frM-ftMa-d •^ max^ c 0 aBn aMa 0 -6 45 At the healthy rest state, (J/c, 0,0,0,0), the Jacobian reduces to: Jac = -c-l£\\ k + b-ei1-0 0 0 -k-e2[ 0 0 0 0 -h'c 0 0 0 0 max Since this is a diagonal matrix, the eigenvalues are exactly the values on the diagonal. Since all these are negative for all parameter choices, we see that the healthy state is always stable. 5.2.2 Reduction of the System Determining the existence and stability of additional steady states in the full, five equation system is difficult. For this reason we simplify the model through quasi steady state (QSS) assumptions. We begin with a QSS on the concentration of cytokines ( C « 0), leading to a CQSS ~ -^BnMa. Substituting this approximation for C into the equation for apoptotic /3-cells (5.1d) reduces it to: A Ii A / I (5.2) Ba = *ma%lnhA; - ftMBa - f2MA - dBa, kb + BnMa where fcf, = (b/a)kc. This effectively replaces the constant rate of macrophage induced apoptosis, per macrophage in the basic model, /, with the term AxnaxBn kb+BnM/ and means that damage to the /3-cells is now a saturating function of M H . The reduced model, following the QSS on C, retains the features of the full model given in (5.1), and has only two parameters, A m a x and ky, not found in the basic model (3.6). 46 A further reduction, assuming a QSS on Bn results in the system: M J + (k + b)Ma - cM - gMBa - exM{M + Ma), (5.3a) Ma gMBa-kMa-e2Ma(M + Ma), AmaxdBaMa , . (5.3b) kb(fiM + f2Ma) + dBaMa - ftMBa - f2MaBa - dBa. (5.3c) The difference between this model (now referred to as the "reduced necrotic model"), and the basic model given in equations (3.2) lies in the /3-cell damage term. In this model, the damage induced rate of apoptosis of /3-cells is very small when both M„ and Ba are small. Thus the positive feedback effect for low level inflammation of the pancreas is negligible. (This addresses an unrealistic aspect of the basic Copenhagen model.) Furthermore, this "effective damage function" saturates with respect to both Ma and Ba, with a maximal rate of induced apoptosis A m a x . This means that the highest damage rate per Ma remains at intermediate levels and ensures that /3-cell destruction is bounded. The limited damage caused per Ma is a reasonable restriction, as without it, a very rapid and complete destruction of the /3-cell mass could occur even before the adaptive immune system becomes involved. This scenario is never observed in NOD mice and therefore cannot be realistic. 5.3 New Parameter Values In this chapter, four new parameters have been introduced: a, 5, A m a x , and kc. Ideally, the values of these parameters could be determined through identification and quantification of the mechanism of induced damage and the dose-response behavior of apoptosis caused by cytokines of other cytotoxins. These details, however, are not yet fully understood and parameterized, hence we use a generalized chemical substance, C, whose level at inflamed conditions far exceeds its half-max level kc. To parameterize the extended model we make the following rough estimates: At the chronic inflammation steady state in NOD mice, we take as reasonable ballpark esti-mates, cell densities of M » 5 x 106 and (Ba + B„) = 2 x 105 cells m l - 1 . In NOD mice, clearance of apoptotic and necrotic cells occurs at the same rate by resting and activated macrophages 47 (/i = fi = f ~ 1 X 10 5cells ml May 1 (see Table 4.2). Taken together, we see that fM{Ba + Bn) ~ (1 x 10~5)(5 x 106)(2 x 105) = 1 x 107 cells ml" 1 , or about 2.5% of the /3-cell mass is cleared from the islets on a daily basis. At this steady state, production and clearance of apoptotic cells must balance. This leads to •^maxC kr + C ~ 1 x 107. Assuming that the maximum /3-cell mass destruction due to the innate immune system is around 5% per day leads to an estimate of A m a x = 1 X 107 cells m l - 1 day - 1 . We set the cytokine concentration relative to an arbitrarily chosen level for half-max induced apoptosis, kc = 1 nM. Only the relative levels of maximal C and kc are important to the model, with the provision that during the chronic inflammation state the Michaelian apoptosis term should be half saturated. The ratio a/6 determines the amount of cytokine accumulation. If we assume a short cytokine decay time of 1 hour, i.e. 5 « 25 day - 1 , the cytokine secretion rate should be in the ballpark of a « 5 X 10 - 9 nM cell"2 day - 1 . Then in NOD mice, during the chronic inflammation stage, a„ „ / 5 X 1 0 " 9 \ , . . . . ^ C v i n 6 , C = -BnMa « ^ 2 5 j (1 x 10J)(5 x 10b) = InM, the required level for inducing a half-maximal rate of apoptosis. We have estimated the new parameter values using information about the NOD mouse. As we are attempting to account for the differences between NOD and Balb/c mice using only the clearance rates, we assume the parameters involved in cytokine secretion calculated here (shown in Table 5.1) apply to both strains. 5.4 Simulations of the Reduced Necrotic Model Numerical simulations of the reduced necrotic model (5.3) were performed using the parameter values given in Tables 4.2 and 5.1. We begin with phase plane analysis of the reduced necrotic model, following a QSS assumption 48 Symbol Meaning value units C toxic cytokine level nM B„ density of necrotic cells cells m l - 1 A m a x maximal cytokine-induced /3-cell apoptosis rate 2 x l 07 cells m l - 1 day - 1 kc cytokine concentration for half-max apoptosis rate 1 nM a rate cytokine secretion by Ma due to B„ 5x10' nM cell 2 day - 1 5 cytokine turnover rate 25 day - 1 Table 5.1: Definitions of the parameters of the extended model (Equations 5.1) together with estimates of their values. on M , i.e. with the equations: -(eiMa + c + gBa) + V(eiMfl + c + gBa)2 + 4ei(J + (k + b)Ma) M = , (5.4a) 2ei Ma = gMBa-kMa-e2Ma(M + Ma), (5.4b) = ir , f u A T ^ w R A/f - fiMBa - flMaBa - dBa. (5.4c) kb{fXM + f2Ma) + dBaMa Figure 5.2 shows the Ma-Ba phase plane, and the nullclines obtained using NOD parameter values. In this case, the system has three steady states: two stable steady states corresponding to perfect health and chronic inflammation separated by a saddle point. The separatrix (dashed line) is given, showing that a large enough perturbation in either Ma or Ba (or some combination of the two) will push the system toward chronic inflammation. Also included in the figure are trajectories from various initial conditions, with arrows showing the direction of motion along the trajectory. In the top plot, all initial conditions shown reach the inflamed steady state, moving quickly to the Ba nullcline (evidenced by the vertical lines) and following it to the equilibrium. One initial condition (marked with a * in the top plot) follows an interesting trajectory: because the high density of activated macrophages quickly clears the relatively few apoptotic /3-cells, the trajectory moves quickly to the portion of the Ba nullcline on the horizontal axis, and then moves toward the origin. As the density of activated macrophages decreases, the trajectory rises steeply to the upper branch of the Ba nullcline and follows it to the inflamed steady state. This may be due to cytokine remaining in the tissue (produced when Ma density was very high) that continues to trigger /3-cell apoptosis. Once the Ma density drops below a certain threshold, there is insufficient clearance to prevent a rapid increase in 49 Ba Figure 5.2: The dynamics of activated macrophages (M„) and apoptotic /}-cells (B„) (measured in cells ml - 1 ) for equations (5.3) showing nullclines, using the parameter values, for NOD mice, given in Tables 4.2 and 5.1. In both plots, the horizontal axis gives the density of activated macrophages and the vertical axis shows the density of apoptotic /3-cells. The Ma nullcline is shown in red, and the Ba nullcline is black. The top plot shows the shape of the nullclines over a large range of values for Ma and Ba. The bottom plot is a close up view showing the healthy steady state and the separating saddle point. The separatrix is given by the dashed blue line. A sufficiently large perturbation in Ma or B„ (or a combination of both) will push the system over the separatrix and toward chronic inflammation. A small perturbation from complete health (i.e. that does not cross the separatrix) will be resolved and with the system returning to the healthy steady state. This figure was was produced with XPP, using the.ode file given in Appendix 50 e density of apoptotic /3-cells and the system ultimately moves to the inflamed steady state, time plot showing the evolution of similar initial conditions is given in Figure 5.3. Ba Figure 5.3: This plot shows a time course of the levels of activated macrophages (Ma) and apoptotic /3-cells (Ba) corresponding to the trajectory marked with a * in Figure 5.2 is shown. We see that the initially elevated densities of both apoptotic /3-cells and activated macrophages quickly reaches a value close to 0. Before the inflammation is completely resolved, damage to the /3-cells (possibly by cytokine remaining in the tissue) triggers a renewed macrophage response and ultimately leads to chronic inflammation. The bottom plot of Figure 5.2 is a zoomed in view of the healthy state, showing the saddle point and the separatrix dividing the regimes of health and chronic inflammation. Initial conditions above the separatrix move to the inflamed steady state, while initial conditions below the separatrix are attracted to the healthy rest state. Figure 5.4 shows the Ma-Ba phase plane and nullclines obtained with parameter values repre-sentative of Balb/c mice. Only one steady state, the stable healthy state exists, demonstrating that all Balb/c mice will remain healthy. Using the necrotic model, it is possible for reduced macrophage clearance alone to account for the NOD mice developing type 1 diabetes, while Balb/c mice remain healthy. Again, we see that trajectories move almost vertically to the Ba null-cline and follow it to the rest state. In most cases shown, the density of activated macrophages 51 Ba 2e+05 1.5e+05 le+05 -1 1 1 i • r >- 1 1 - Ba'=0 \ 1 * / ' c . i Ma'=0 — i U e _ i 1 J . 50000 M a le+05 1.5e+05 2e+05 M a Figure 5.4: Nullclines for equations (5.3) as in Figure 5.2, but using the parameter values, for Balb/c mice, given in Tables 4.2 and 5.1. In this case only the healthy steady state exists, so any perturbation from complete health will be resolved. This figure was was produced with XPP, using the.ode file is given in Appendix C.2. 52 is sufficiently large that the apoptotic bodies are quickly cleared, after which the macrophages deactivate and the system returns to the rest state. One case (marked with a * in both plots) shows the density of apoptotic /3-cells rising steeply before the inflammation is resolved. A time plot showing the evolution of similar initial conditions is given in Figure 5.5. Ba M a Figure 5.5: In this plot, a trajectory, similar to the trajectories marked with a * in Figure 5.4 is shown. In this case, the initial activated macrophage density (M„) is not sufficient to imme-diately clear the apoptotic jS-cells (Ba), leading to a transient, elevated level of inflammation. However, as more macrophages become activated, the increased phagocytic capacity clears the apoptotic cells, and the system returns to the healthy rest state. Figure 5.6 shows the null surfaces of the 3 equation model, obtained using NOD parameter values. Two sets of stereoscopic images are presented, showing two views of the null-surfaces. In both sets, the inflamed steady state can be seen, though the healthy steady state and the saddle point are not visible. 53 Figure 5.6: Stereoscopic visualization for the null surfaces of the model given by equations (5.3). Here the M null surface is blue, the Ma null surface green and the Ba null-surface is red. In both sets of plots, the M axis is in the vertical direction. In the top set, the Ma axis on the left and the Ba axis is in the back. In the bottom set, the Ba axis is in front, and the Ma axis behind. These plots were generated with MAPLE using the parameter values (for NOD mice) from Tables 4.2 and 5.1 54 5.5 Bifurcation Analysis of the Reduced Necrotic Model We now consider more closely the effect of phagocytic rates by the resting and activated macrophages (f\, ji). Numerical bifurcation diagrams are plotted using XPP/AUTO. Parameter values are taken from Tables 4.2, 5.1. Ma 6e+06 Figure 5.7: Bifurcation of the 3 dimensional reduced necrotic model, as given in equations 5.3 (showing equilibrium solution of M„), with respect to the parameter governing macrophage ac-tivation and basal engulfment rate, g = f\. Solid line represents stable equilibrium, dashed line represents unstable equilibrium. The Fold (FB), Hopf (HB) and Homoclinic (HQ bifurcation points are labeled. The open circles represent an unstable limit cycle. Dynamics corresponding to regions of this bifurcation diagram are shown in Figure 5.8. This figure was produced with XPP/AUTO and the parameter values given in Tables 4.2,5.1. The .ode file is given in Appendix C.3. Figure 5.7 shows a bifurcation diagram with respect to the parameter g = fi (the rate of activation and engulfment by resting macrophages). From the diagram we see that if resting macrophage engulfment and activation is sufficiently high, only the healthy steady state will exist. As clearance and activation become more impaired (g = f\ decreases), the saddle point and inflamed steady states emerge at a fold bifurcation. The inflamed steady state is not stable immediately. For the range 2.6 x 10 - 5 < g, f\ < 3.4 x 10 - 5 ml cel l - 1 day - 1 only the rest state is 55 le+06 2e+06 3e+06 ~4e+06 ~5e+06 6e+06 0 Ma le+06 2e+06 3e+06 4e+06 5e+06 6e+06 Ma (C) (d) Figure 5.8: Phase plane diagrams for the model of equations (5.4) illustrating the dynamics corresponding to several regions in the bifurcation diagram given in Figure 5.7. In all cases, the trajectories move quickly to the Ba nullcline (evidenced by the near vertical start to the trajectories) and then follow the Ba nullcline to the destination steady state, (a) g = f\ = 1 x 10~5 ml cell _ 1day _ 1: classic bistability (see also Figure 5.2). (b) g - f\ = 2.163 x 10"5 ml cell - 1day _ 1: unstable limit cycle. In this case, most initial conditions return to the healthy state. One initial condition inside the limit cycle is plotted, this evolves to the inflamed steady state. Plots demonstrating this effect are given in Figure 5.9. (c) g = f\ = 3 x 10 - 5 ml cel l - 1 day - 1: 3 steady states; the inflamed steady state is unstable, and all initial conditions eventually reach the healthy rest state, (d) g = f\ =3.7 x 10 - 5 ml cell - 1 day - 1: only the healthy rest state exists, and all trajectories evolve to the stable rest state. Note that the value of the bifurcation parameter used here does not always agree with the bifurcation diagram given in Figure 5.7, due to the QSS assumption on M . This figure was produced using XPP, the .ode file given in Appendix C.2 and the parameter values in Tables 4.2 and 5.1. 56 200 300 400 time (days) 500 600 J—i—I i I i L 0 10 20 30 40 50 60 70 80 90 100 time (days) (a) (b) Figure 5.9: Time dependent solutions for the reduced necrotic model of equations (5.3), for g = f\ = 2.5605 x 10 - 5 ml cell - 1 day - 1 . These plots corresponds to the region between the Hopf and homoclinic bifurcations shown in Figure 5.7. (a) Initial conditions (M,Ma,Ba) -(5 x 105,2.9 x 106,2.4 x 104) are inside the unstable limit cycle and evolve in an oscillatory way to the inflamed steady state, (b) Initial conditions (M, M„, Ba) = (5 x 105,3.1 x 106,2.9 x 104) are outside the unstable limit cycle, yet in the basin of attraction of the inflamed steady state. In this case, an initially large inflammatory response is resolved, and the system returns to the healthy rest state. stable. The inflamed state becomes stable at a sub-critical Hopf bifurcation (with an unstable limit cycle terminating in a homoclinic bifurcation). For values of g sufficiently below the Hopf bifurcation value, the necrotic model exhibits classic bistability between the healthy and inflamed steady states where a large enough stimulus will cause the switch from health to disease. The existence of the Hopf bifurcation has two important consequences: 1. The possibility of chronic inflammation requires more than the existence of the inflamed steady state. Thus the conditions are more stringent, and the parameter range under which chronic inflammation occurs is reduced. 2. In the region between the fold and homoclinic bifurcations, dynamics can be observed in which the system initially exhibits a large inflammatory response following the initial inflammatory stimulus. On a longer time scale, the solutions will return to the healthy rest state, forming a loop about the inflamed equilibrium. This is demonstrated in Figure 5.9(b). For small deficiencies in macrophage clearance, the system will never end up at the inflamed (still unstable) equilibrium. For slightly more impaired macrophages (with 57 clearance rates between the value at the Hopf and homoclinic bifurcations) the system will only go to the inflamed equilibrium when the initial inflammation is not too large. One set of initial conditions that show this behavior is given is Figure 5.9(a). Figure 5.8 shows plots of the reduced 2D system (obtained by putting M on QSS (5.4)) that demonstrate the dynamics at various values of g = f\, showing the results of each bifurcation. These plots capture qualitatively the features of the bifurcation diagram. However, due to the reduction of the system through the QSS assumption, the values of g at which the plots in Figure 5.8 are made no not correspond exactly with the values observed in the bifurcation diagram shown in Figure 5.7. In Figure 5.8(a), the system exhibits classic bistability (also shown in Figure 5.2) where any stimulus above a certain threshold (determined by the separatrix) will move the system to the inflamed steady state and any sub-threshold stimulus will be resolved and the system will return to the healthy rest state. The plots in Figures 5.8(c) and (d) show dynamics that always return to the healthy rest state, in the case where the inflamed steady state is unstable (c) and the case where only the healthy equilibrium exists (d). The plot in Figures 5.8(b) and 5.9 show the dynamics between the Hopf and Homoclinic bifurcations: Some initial conditions inside the unstable limit cycle will move toward the inflamed state, while initial conditions outside the limit cycle (but still above the separatrix) make a loop around the inflamed steady state and return to the healthy equilibrium. The same conclusions may be drawn from a bifurcation with respect to f2, shown in Figure 5.10. In this figure, the unstable limit cycle has not been plotted, as it is very similar to that given in the bifurcation with respect to g = f\ (Figure 5.7). The interesting difference between the two bifurcation diagrams is what happens to the value of the inflamed steady state (shown for Ma) as the bifurcation parameter decreases. As g = f\ decreases, activation of macrophages decreases, and accordingly, the inflamed density of activated macrophages decreases as well. However, as f2 decreases, the decreased phagocytosis ability of activated macrophages means that apoptotic bodies will not be cleared effectively, leading to a buildup of apoptotic /3-cells. This buildup of B„ will be augmented through increased levels of necrotic cells (as a result of reduced clearance), and the release of more cytokines, maintaining apoptotic cell density. This 58 Ma le+06 9e+05 -8e+05 -7e+05 -6e+05 -5e+05 -4e+05 -3e+05 -2e+05 -le+05 -4 - - - . ' ' r - - , 2e-05 4e-05 6e-05 8e-05 0.0001 Figure 5.10: As in Figure 5.7 but showing a bifurcation for the model in (5.3) with respect to the parameter for the engulfment rate of activated macrophages, f2. cycle of death will lead to increased activation of macrophages, evidenced in Figure 5.10 by the steeply rising value of the inflamed steady state as the parameter / 2 decreases. Similar dynamics are observed to those seen in the bifurcation with respect to g = f\, and these are presented in Figures 5.11 and 5.12 using the 2 dimensional reduced necrotic model (5.4). The phase plane plots given in Figure 5.11(b) and (c) show cases where all initial conditions progress to the healthy state: when the healthy state is the only (and globally stable) steady state (b), or the inflamed state is unstable as in (c). Figures 5.11 and 5.12 give the dynamics between the Hopf and homoclinic bifurcations: most initial conditions will move to the healthy equilibrium (possibly making a loop around the inflamed state) while initial conditions inside the unstable limit cycle will move in an oscillatory way to the stable inflamed steady state. Figure 5.13 shows a bifurcation with respect to the macrophage saturation parameters (e\ = e2). Similar to the bifurcations along the phagocytosis parameters, we see a region of classic bistability for 2.2 X 10 - 9 cells day - 1 < e\ = e2 and a region where the inflamed state is unstable and any sufficient perturbation from health results in a large initial response before the system 59 (c) (d) Figure 5.11: Phase plane diagrams illustrating the dynamics corresponding to different regions in the bifurcation diagram given in Figure 5.10. (a) f2 = 1.732 x 10 - 5 ml cell - 1 day - 1 : unstable limit cycle. In this case, most initial conditions return to the healthy state. One initial condition inside the limit cycle is plotted, this evolves to the inflamed steady state. This can be seen also in Figure 5.12. (b) f2 = 9.8 x 10~5 ml cell - 1 day - 1: Only the healthy rest state exists, and is globally stable, (c and d) f2 = 9 x 10 - 5 ml cell - 1 day - 1: 3 steady states, inflamed steady state is unstable, all initial conditions eventually reach the healthy rest state, (d) gives a representative time plot, showing the evolution of similar initial conditions to those in (c) marked with a *. The dynamics corresponding to classic bistability are shown in Figure 5.2. Again, due to the QSS assumption on M , these plots capture qualitatively the behavior at various points on the bifurcation diagram in Figure 5.10, but at different values of the bifurcation parameter f2. 60 100 time (days) 10 15 time (days) 20 25 (a) (b) Figure 5.12: Plots of 2 dimensional reduced necrotic model (5.4), for f2 = 1.732 x 10 - 5 ml cel l - 1 day - 1 . These plots corresponds to the plot in Figure 5.10(a). (a) Initial conditions (M f l, Ba) = (2.66 x 105,5.2 x 104) are inside the unstable limit cycle and evolve in an oscillatory way to the inflamed steady state. The top plot shows the density of activated macrophages (M„) and the bottom plot shows the density of apoptotic /3-cells (Ba). (b) Initial Conditions (Ma, Ba) = (2 x 106,4.4 x 104) are outside the unstable limit cycle, yet in the basin of attraction of the inflamed steady state. In this case an initially large inflammatory response is resolved and the system returns to the healthy rest state. returns to the healthy equilibrium (shown in Figure 5.14(a)). This may be due to the fact that when the crowding effects are very small, many macrophages enter the tissue and the combined effort is capable of clearing any level of inflammation. As the crowding factors increase, clearance is no longer able to eliminate the apoptotic and necrotic jS-cells, and the chronic inflammation state becomes stable. As in the previous bifurcations discussed, the transition of the inflamed steady state from unstable to stable occurs at a sub critical Hopf bifurcation. Examples of the dynamics observed in the parameter range between the Hopf and Homoclinic bifurcations are given in Figure 5.14: (b) shows sample trajectories in the 2 dimensional phase plane, (c) and (d) give time plots of initial conditions inside and outside the unstable limit cycle. We have demonstrated that, in the context of the extended model presented in this chapter, reduced macrophage clearance rates are sufficient to cause the distinction between NOD and Balb/c mice. In the next chapter, we consider the neonatal wave of /3-cell apoptosis, and ask whether it is sufficient to trigger initiation of type 1 diabetes. The apoptotic wave is 61 4e-08 Figure 5.13: Bifurcation of reduced necrotic model (showing equilibrium solution of M„), with respect to the macrophage saturation parameters, e\ = e2. Note that the bottom panel shows a magnified view of the stable rest state and the separating saddle point. Solid line represents stable equilibrium, dashed line represents unstable equilibrium. The open circles represent the unstable limit cycle. Dynamics in various parameter regimes are shown in Figure 5.14. This figure was produced using XPP/AUTO, and the parameter values given in Tables 4.2, 5.1. The .ode file is given in Appendix C.3. incorporated into the model by adding a term, W(t), to the equation for the apoptotic yS-cell density (Bfl). Thus equation (5.Id) becomes BA = W(t) + - f\MBa - f2MaBa - dBa. Chapter 6 is concerned with determining the exact form of the function W, and provides simulations of the necrotic model with the apoptotic wave included, showing that, in fact, the wave may be a triggering event for type 1 diabetes. 62 200 300 time (days) J i-t- •« I i lla I i I i I • I 0 10 20 30 40 50 60 70 80 90 100 time (days) (C) (d) Figure 5.14: Dynamics corresponding to regions in the bifurcation diagram of Figure 5.13. (a) ei = e2 = 1X10 - 9 cells day - 1: there are 3 steady states, with the inflamed steady state unstable. Al l initial conditions eventually reach the healthy rest state, (b) e\ - e2 = 4.775 x 10 - 9 cells day - 1: unstable limit cycle: in this case, most initial conditions return to the healthy state. One initial condition inside the limit cycle is plotted. This evolves to the inflamed steady state, (c) and (d) plots of the 3 dimensional reduced necrotic model (5.3) showing the effect of the unstable limit cycle in (b). In these plots e\ = e2 = 2.18 cells day - 1 , (c) The evolution of initial conditions (M,Ma, Ba) = (5 x 105,1.5 x 106,4.4 x 104) inside the unstable limit cycle, (d) shows the evolution of initial conditions (M,Ma, Ba) = (5 x 105,7 x 105,1.6 x 104) outside the unstable limit and the return of the system to the healthy rest state. See Figure 5.2 for the dynamics in the range of classic bistability. As in Figures 5.7 and 5.10, the plots presented here capture qualitatively the dynamics shown in the bifurcation diagram, even though the value of the bifurcation parameter may not exactly agree. 63 Chapter 6 The Neonatal Wave of (3-cell Apoptosis Apoptosis, a programmed form of cell death has been implicated in the remodeling of organs and tissue during development. Remodeling is demonstrated in the loss of inter-digital tissue in mammalian limbs during fetal development. It has been seen in the nervous system, kidney, heart and adrenal cortex during the neonatal period [23, 37]. Apoptosis during the development and remodeling of the pancreas was first suggested by Finegood et al [7] on the basis of experimental observations combined with data analysis using a balance equation for net cell flux. This neonatal, developmental wave of /3-cell apoptosis has been postulated to be a triggering event for the development of type 1 diabetes. 6.1 Description of the Wave A neonatal wave of /3-cell apoptosis has been observed in rodents susceptible to type 1 diabetes. This wave has been described and quantified in mice (NOD, Balb/c [30,41] and C57BL/6 [29]) and rats (Bio-Breeding - diabetes prone (BBdp) and resistant (BBdr), [27, 41], Wistar [32] and Sprague-Dawley [37, 41]) and is qualitatively the same in all cases studied. Trudeau et al [41] used a balance equation for /3-cell mass dynamics to estimate the net /3-cell flux (expressed as %/day) and presented the results for Sprague-Dawley rats as representative of all rodents: in the Sprague-Dawley rats, a control strain that does not develop type 1 diabetes, /3-cell mass increases almost linearly until about 100 days of age [7], aside from a plateau in the growth rate from 5 to 20 days of age. During development, the rate of /3-cell replication decreases from 10% per day at birth to approximately 2% per day in adult rats. Trudeau et al [41] predicted 64 that the plateau in /3-cell mass growth is due to a wave of net cell death. (Net cell death being the difference between absolute cell death and the sum of replication and neogenesis i.e. death-[neogenesis+replication] ). The authors predict that the wave of net cell death peaks around 12 days of age with net cell loss approximately 9% per day. A l l studies of this /3-cell apoptotic wave agree in their qualitative descriptions; it is only in the quantification of the wave that discrepancies arise. Because apoptosis is a very rapid process, often only a few minutes elapse between the onset of apoptosis and the cell being reduced to a cluster of apoptotic bodies [29]. The apoptotic debris is cleared by macrophages and other scavenger cells within a few hours [22,29]. Thus, it is very difficult to measure the rate at which apoptosis occurs. This reason, along with differences between murine strains and methodologies used in different labs, may explain the different estimates of the magnitude of the wave. 6.1.1 Methods to Determine Apoptosis As an actual rate of /3-cell apoptosis cannot be directly measured, it must be inferred from observable data. In all studies considered here, the pancreas was removed from the animal, fixed, and slices observed under a microscope. Apoptotic cells were identified using various techniques: O'Brien et al [29] identified apoptotic cells under light microscopy based on the presence of previously identified morphological features common to apoptotic cells. Other studies identified apoptotic cells based on histological staining using the TUNEL method for labeling DNA strand breaks [27, 41], propidium iodide staining to visualize condensed or fragmented DNA [37], or a commercially available assay called the "Apoptag in situ apoptosis detection kit" [32]. In all studies the level of apoptosis observed was recorded as a fraction of the total number of cells. In some cases, additional staining for insulin confirmed that the apoptotic cells were in fact /3-cells [27, 37, 41] but in other cases, this double staining was inconclusive [32]. 65 6.1.2 Quantification of the Wave The studies of the neonatal wave of apoptosis considered here all show a significant increase in the % apoptotic cells observed between days 5 and 20 compared to either the newborn (day 1) or the adult animal. Additionally, all studies observed the largest fraction of apoptotic cells at approximately 10 days of age (9 days for NOD, 12 days for Balb/c [41], 5-10 days for BBdr and BBdp rats [27], 13 days for Sprague-Dawley rats [37], 14 days for Wistar rats [32]). The main difference noted between different studies of apoptosis is the stated size of the maximum fraction of apoptotic cells that the authors present as their results. This fraction varied from 0.17% (O'Brien et al [27], using TUNEL staining and BB rats) to 17% apoptotic cells (Trudeau et al [41] using TUNEL staining and NOD mice). Between these two extremes, Scaglia et al recorded 3.5% apoptosis in Sprague-Dawley rats using propidium iodide staining to show condensed or fragmented nuclei. Petrik et al [32] observed a peak of 13% apoptosis in Wistar rats (supporting the observations made by Trudeau et al) using the Apoptag in situ apoptosis detection kit. Petrik et al note that while it is likely the apoptotic cells counted were /3-cells (based on location in the center of the islets), other cell types could not be excluded, as insulin staining proved to be inconclusive. When we compare the results collected in Table 6.1, it is clear that the results by Petrik et al and Trudeau et al are outliers and we do not consider these to be accurate quantifications of the apoptotic wave. Other measurements of apoptosis, published by O'Brien et al [27, 29], do not describe the neonatal wave of apoptosis, but may serve as a comparison. A wave of apoptosis, similar to the neonatal wave, is induced with injections of streptozotocin. Timing for the experiment began with day one being the day of the first stz treatment. At the peak of this induced wave, 2.7% apoptosis was observed on day 5 [29]. O'Brien et al [30] present data for many different strains of mice, showing the fraction of apoptotic /3-cells present in the islets of neonates. These observations were made in 2 week old animals, from sections that were fixed immediately after the animal was killed. Apoptosis was visualized using TUNEL staining. A summary of these various observations is given in Table 6.1. 66 Strain % Apoptotic Peak of Observed Visualization Source cells observed Apoptosis (day) Method Balb/c (F) 0.21 14 TUNEL [30] Balb/c 9 11 TUNEL [41] C57BL/6 (F) 0.22 14 TUNEL [30] C57BL/6 (F) 2.7 5 morphology [29] NOR (F) 0.29 14 TUNEL [30] Idd5 (F) 0.31 14 TUNEL [30] NOD (M) 0.33 14 TUNEL [30] NOD (F) 0.57 14 TUNEL [30] NOD 17 9 TUNEL [41] BBdp 0.15 5-10 TUNEL [27] BBdr 0.07 10 TUNEL [27] Wistar rat 13.18 14 Apoptag [32] Sprague-Dawley rat 3.5 13 propidium iodide [37] Table 6.1: Percent apoptotic /3-cells observed in islets of neonatal mice and rats. There has been some discussion that differences in the observed fraction of apoptotic cells may be a result of impaired macrophage function in the spontaneous forms of type 1 diabetes, such as the NOD mouse. The main macrophage deficiency of relevance to the current discussion is the reduced phagocytic ability of NOD and BBdp macrophages [22, 27, 30, 41] (i.e. reduced values of / i , / 2 in our model). This has been shown directly through experimental observations [22, 27, 30]. Trudeau et al [41] demonstrate that the difference in visualized apoptotic cells between NOD and Balb/c mice can be accounted for if the clearance by NOD macrophages is reduced by 50% compared to the Balb/c macrophages. 6.2 The Wave as a Triggering Event for Type 1 Diabetes It has been suggested that the wave of /3-cell apoptosis is a possible triggering event for the de-velopment of type 1 diabetes. While apoptosis is not considered an inflammatory form of cell death [1], a high concentration of apoptotic cells, combined with reduced macrophage phago-cytosis results in apoptotic cells becoming secondary necrotic cells, which may contribute to inflammatory processes [41]. O'Brien et al [29] demonstrate that /3-cell apoptosis precedes development of type 1 diabetes, with insulins closely following the induction of apoptosis. In-67 sulitis has also been observed to follow apoptosis in spontaneous models of diabetes, observed in NOD mice [28] as well as accelerated models of type 1 diabetes [12,18]. In all cases, the onset of insulitis (the lymphocyte infiltration of the pancreatic islets) seems fixed at approximately 15 days, shortly after the neonatal wave of apoptosis. This temporal association, while not conclusive, suggests that there may be a causal relationship between the wave of apoptosis and the development of type 1 diabetes. 6.3 Simulations of the Apoptotic Wave The current study is concerned with testing the hypothesis that the neonatal wave of /3-cell apoptosis can serve as a triggering event for the development of type 1 diabetes. Due to the many different observations of the wave, it is not clear how to accurately capture the dynamics of /3-cell apoptosis. Furthermore, the observed apoptosis is measured as a percent apoptosis, and in our model W(t) carries units of cells m l - 1 day - 1 . In the extended version of the Copenhagen model, equations (5.1), the transition from the healthy state to the inflamed state is due to some threshold stimulus. As a result, the most important feature of the apoptotic wave (for the purposes of our model) is the magnitude of the wave, W(t), at its peak. However, note that the observations recorded in Table 6.1 are not rates, but percent apoptotic cells. We are interested in demonstrating that the wave is a sufficient trigger for the development of type 1 diabetes, that it is a large enough stimulus to push the system to chronic inflammation. For this reason, we want the peak of our simulated wave to correspond to the smallest observed wave. This is difficult as no rates of apoptosis have been measured and we must estimate these rates based on the fraction of apoptotic cells observed. If the smallest wave evidenced by observations is sufficient to trigger chronic inflammation, then any larger wave will be as well. To model the neonatal wave of apoptosis, the observed fraction of apoptotic cells (reported as % apoptosis) must be converted to a rate of apoptosis (cells m l - 1 day - 1). As a first approximation, we model the neonatal wave as an exponential function, with the peak of the function, W m a x , 68 occurring at time t m a x and the width determined by parameter Ww: W(t) = W m a x exp The parameters f m a x and Ww are chosen to match the width and location of the approximate wave to the data recorded in [41]. We choose t m a x = 9, Ww = 3, and thus model the apoptotic wave as where W m a x is the rate of apoptosis in cells ml 1 day at the peak of the neonatal wave. One method to calculate W m a x is given below. Before calculating the rate of apoptosis at the peak of the wave, we need to know how many apoptotic /3-cells are observed. Thus it is necessary to convert the maximum fraction of apoptotic cells observed (as % apoptotic) into an absolute number of apoptotic /3-cells. For this we need an estimate for the density of /3-cells in the pancreatic islets. Recall from Chapter 4, the /3-cell density in the pancreatic islets is approximately 4 X10 8 cells m l - 1 . Using a value of 0.15% apoptotic cells at the peak of the wave [30] we calculate that a density of (4xl08)(0.0015) - 6xl0 5 apoptotic /3-cells m l - 1 would correspond to the value of B„ in our model at the peak of the neonatal wave of apoptosis. Any rate of apoptosis calculated should be consistent with this observed density of apoptotic /3-cells at the peak of the apoptotic wave. A coarse approximation of the maximum rate of apoptosis may be estimated using the percent apoptotic cells observed. If we assume that during the initial part of the wave, the flux of apoptotic /3-cells (B„), is dependent only on the maximum rate of apoptosis, the basal decay of apoptotic material and clearance by resting macrophages, we can approximate the flux of apoptotic /3-cells as: ^ * W m a x - R B f l , where R = d + is the decay and clearance mechanisms effective before macrophages are activated. At the peak of the apoptotic wave, the flux of apoptotic /3-cells is zero = o] and W(f) = W m a x exp (6.1) so W m a x = RBa. 69 We can then solve for the maximum rate of apoptosis using the observations of O'Brien et al [30] and our estimates for the parameters. At the peak of the wave, O'Brien et al observed 6 x 10 5 apoptotic jS-cells m l - 1 (corresponding to Ba). Using the parameter values estimated in Chapter 4, we calculate R = 0.5 + (1 x 10 - 5 ) (5 x 10 4)/0.1 = 5.5 day - 1 . Substituting these values into the above equation for W m a x , we obtain: W m a x ~ (5.5)(6 x 10 5 ) = 3.3 x 10 6 cells ml" 1 day"1. For our simulations of the neonatal wave of apoptosis, we take W m a x = 3.3 X10 6 cells m l - 1 day - 1 . Thus the neonatal wave of apoptosis is approximated as: W(t) = 3.3 x 10 6 exp (- ( ^ f ) • (6-2) Simulations of the full necrotic model, including the wave of apoptosis are given in Figures 6.1 - 6.3. In Figure 6.1, time plots show the response to a simulated wave of /3-cell apoptosis, as described in equation (6.1). We see that the apoptotic wave is sufficient to trigger chronic inflammation in NOD mice, but that the Balb/c mice clear the inflammation and return to the healthy rest state. Figure 6.2 shows the response in the case of NOD mice to a small, sub threshold wave (WM 3 X = 3205 cells ml _ 1 day _ 1 ) . In this case the NOD macrophages are able to clear the inflammation and the system returns to the healthy rest state. Figure 6.3 shows limit cycle dynamics obtained for f2 = 8.478 ml cel l - 1 day - 1 . In this case a relatively small apoptotic wave (WM a x = 34 000 cells m l - 1 day - 1) pushes the system inside the unstable limit cycle (solid lines in all plots), and the system evolves to the inflamed steady state. A larger wave ( W M 3 X = 52 000 cells m l - 1 day - 1) shows the effect of being outside the unstable limit cycle (dashed lines in all plots). In this case a large response is initiated that ultimately clears the inflammation and the system returns to the rest state. 70 Figure 6.1: Log-density vs time plots from simulations of the full necrotic model, given in equations (5.1). These plots were generated using XPP, with the .ode file given in Appendix C.4, the parameter values found in Tables 4.2 and 5.1 for NOD and Balb/c mice , and the simulated apoptotic wave as given in (6.2). Shown are the cell densities (log cells ml - 1 ) for resting and activated macrophages (M and Ma respectively), apoptotic and necrotic /3-cells (Ba, Bn) and the concentration of the generic cytokine (C) (in nM). We see that in NOD mice (a) chronic inflammation occurs following the wave of apoptosis, while in Balb/c mice (b) the system returns to complete health. le+06 time (days) Figure 6.2: Log-density vs time plots from simulations of the full necrotic model, as in Figure 6.1. In this case, a small wave (WM S X - 3205 cells ml _ 1 day _ 1 ) of /3-cell apoptosis does not result in chronic inflammation, but rather, is cleared by the macrophages and the system returns to the healthy state. 71 6e+05 5e+05 4e+05 3e+05 2e+05 le+05 0 1 I 1 1 J -1 — M 1 — M* - — M a _ - - M * a — 7 V i , i , i , 1 , 1 0.07 70000 60000 50000 40000 30000 20000 10000 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 0 50 100 150 200 250 300 time (days) time (days) Figure 6.3: Plots of the necrotic model showing oscillatory behavior, obtained for f2 = 8.478 ml cel l - 1 day - 1 . In all plots, the solid lines correspond to simulations with Wmax = 34 000 cells m l - 1 day - 1 , and dashed lines correspond to W m a x = 52 000 cells m l - 1 day - 1 . In the case of the smaller value for W m a x , the system is moved inside the unstable limit cycle, and evolves in an oscillatory way to the inflamed steady state. A larger value of W m a x places the system on the outside of the limit cycle, and so following a brief elevated response, the inflammation is cleared and the system returns to the healthy state. This figure was created using XPP (with the .ode file in Appendix C.4) and the parameter values from Tables 4.2 and 5.1 for NOD mice (except for f2 = 8.478 ml cell _ 1day - 1.). 72 Chapter 7 Discussion 7.1 Summary and Conclusions This study has been concerned with creating a simple model that explains the different de-velopmental outcomes observed in NOD and Balb/c mice. We were specifically interested in determining whether the reduced macrophage clearance observed in NOD mice could alone account for the the susceptibility to type 1 diabetes in NOD mice, while the Balb/c mice all re-main healthy. We also considered the role of a naturally occurring wave of apoptotic cell death in the pancreas of all neonatal rodents, and explored whether it could trigger the initiation of type 1 diabetes in susceptible animals. To explore these questions and describe the events in the early stages of type 1 diabetes onset, we used the Copenhagen model as a starting point. After adapting the model for our quantitative study, and incorporating apoptotic /?-cells explicitly, we showed that this adaptation of the Copenhagen model qualitatively captured the dynamics we believe important in the early stages of type 1 diabetes. A search of the literature resulted in reasonable estimates for many of the parameters in the model, and analysis of the model equations provided conditions that were used to estimate the remaining parameters. We discovered that our adaptation of the Copenhagen model could not, under any reasonable parameter regimes, explain why some NOD mice develop chronic inflammation of the pancreatic islets of Langerhans while all Balb/c mice remain healthy. The differences in macrophage clearance are insufficient to account for the distinction between NOD and Balb/c mice either alone or in conjunction with other strain specific parameter differences. 73 In an attempt to understand what key details were left out of the Copenhagen model, we extended our model to include other reasonable details. In the extended model, we considered uncleared apoptotic cells that undergo secondary necrosis. We assumed that the necrotic (but not apoptotic) /3-cells have a profound stimulatory effect on the inflammatory response by the activated macrophages. We also altered the mechanism of induced /3-cell apoptosis: we considered explicitly the cytokines and chemical substances directly responsible for the damage to /3-cells. These substances, considered together as a toxic factor, are produced when activated macrophages come in contact with necrotic /3-cells and proceed to damage /3-cells in a saturating way. The saturating assumption is essential to avoid unrealistically rapid destruction of the 8-cell mass: without this assumption, the pancreatic /3-cells would be destroyed before the T-cells could be activated. As this is never observed, it can not be realistic. Following these additions, we have a model of the early stages of type 1 diabetes, in which the reduced clearance by NOD macrophages is sufficient to explain NOD susceptibility to type 1 diabetes. In all the models considered in this study, macrophages have two roles. Through secretion of cytokines and harmful chemicals, macrophages increase the level of inflammation in the tissue and lead to the destruction of /3-cells. Macrophages also play an important anti-inflammatory role through the engulfment of apoptotic /3-cells. The difference between the original Copen-hagen model (and our first extension of it) and the extended necrotic model, is that in the Copenhagen model, the damage to /3-cells occurs at a rate proportional to the density of acti-vated macrophages. In the necrotic model, the damage to /3-cells saturates with respect to the density of activated macrophages. It is this saturation that is important. In the Copenhagen model, as the density of activated macrophages increases, the engulfment of apoptotic cells is never enough to overcome the inflammatory effects of cytokine release. In the case of the necrotic model, because the damage to /3-cells saturates as macrophage density increases, the influx and activation of sufficient macrophages leads to a situation where the net effect of macrophage activity in anti-inflammatory. Note that this is only the case for Balb/c mice. In the NOD mice, the reduced macrophage engulfment rates are not sufficient to overcome the inflammatory effect of the cytokines, rather a balance is reached, resulting in chronic inflam-74 mation. In addition to the bistable behavior of the model, leading to the threshold separated regimes of health and chronic inflammation, bifurcation analysis of the reduced necrotic model shows the existence of other dynamical features, including limit cycle behavior in some parameter ranges. The existence of these limit cycles demonstrates that the mere existence of an inflamed steady state is not enough to cause susceptibility to type 1 diabetes, and thus the parameter ranges under which chronic inflammation occurs is reduced. As shown in Chapter 5, the existence of unstable limit cycles surrounding a stable inflamed equilibrium alter the threshold dynamics slightly. In some cases an inflammatory stimulus triggers a large inflammatory response, clearing the inflammation and the system returns to the healthy state. A slightly larger stimulus places the system inside the unstable limit cycle where it is attracted to the inflamed state. Into this extended model, we then included an approximation of the neonatal wave of /3-cell apoptosis to study its role in the pathogenesis of type 1 diabetes. We determined that in the context of our model, the neonatal wave of apoptosis is of significant magnitude to trigger the chronic inflammation in the NOD mice, supporting the hypothesis of Trudeau et al [41]. 7.2 New Questions Work on this project has lead to questions for future study: 1. This study implicitly assumes that all mice involved have /3-cell specific T-cells, and that if Balb/c mice would develop the chronic inflammation of the pancreatic islets, initiation of type 1 diabetes would follow. The only feature that saves Balb/c mice from this fate is the effectiveness of their macrophages. What would happen to Balb/c mice if the clearance capacity of the macrophages was exceeded? The model predicts that any level of inflammation will be resolved, due to the global stability of the healthy rest state. It is not clear that this truly reflects the biology. What would happen if apoptosis was induced in Balb/c mice (through injection of streptozotocin) to such a level that the 75 fraction of observed apoptotic cells in the pancreas was comparable to that seen in NOD mice at the peak of the neonatal wave (i.e. if 0.15% apoptotic cells could be observed)? Is the distinction between recovery and chronic inflammation dependent on the ratio of apoptotic cells to macrophage clearance ability, or is it more dependent on the relative tendency of macrophages to induce apoptosis and to clear apoptotic debris? (In the model, increasing Amax can lead to a case where Balb/c mice also become susceptible to the chronic inflammation) 2. It would also be interesting to explore the differences between NOD mice that develop type 1 diabetes and those that do not. Is this distinction due to differences in macrophage activity? What accounts for the greater susceptibility of female NOD mice - are the macrophages from male NOD mice more effective at phagocytic clearance? It would be interesting to compare the phagocytic ability of macrophages from adult NOD mice and see if any difference exists between the macrophages in mice that have developed the disease and mice that have not. 3. Can we determine what is causing the defects in NOD macrophage clearance? Is this due to defective receptors or cellular processes? Discussions with the experimental group in Diane Finegood's lab suggest that these questions may be answered through gene analysis or proteiomic analysis. Another tool that may provide some answers about macrophage receptor deficiencies is FACS (fluorescence activated cell sorting) analysis, a procedure based on antibody staining of cell surface receptors to determine strain dependent differences in macrophage receptors. 4. Do other differences exist between the macrophages from NOD and Balb/c mice? In the model, the major difference is the relationship between the clearance capabilities of the macrophages and the cytokine-induced apoptosis in /3-cells that seems important. Is the difference between NOD and Balb/c mice due entirely to the reduced clearance (even though this is a plausible explanation), or could it be the case that the NOD macrophages secrete more harmful cytokines in addition to their reduced clearance ability? There is some evidence in the literature that this may be the case, though it remains controversial. 76 Further exploration of this issue will be important for understanding all factors leading to type 1 diabetes susceptibility. 5. Why is autoimmunity in NOD mice restricted to the pancreas? Developmental waves of apoptosis have been observed in other organs of the developing rodent, notably in the kidney [4], yet these animals do not develop the chronic inflammation observed in the NOD pancreas. 6. The ultimate goal of any research into the dynamics and factors responsible for the development of type 1 diabetes is to understand and eventually halt this progression in human patients. There is some evidence of deficient phagocytosis by macrophages in human patients [11, 17]. A development apoptotic wave, similar to that seen in mice, seems to occur in humans as well [16]. It would be very interesting, once enough data has been gathered, to conduct a quantitative study, much like the one presented here, to determine the what effect the reduced macrophage clearance and wave of jS-cell apoptosis have in the pathogenesis of type 1 diabetes in human patients. 7.3 Limitations of Our Model In this study, we expanded and improved on the Copenhagen model to address some of its shortcomings. The Copenhagen model provides only a qualitative description of the early stages of type 1 diabetes pathogenesis, we wanted to study these events in a quantitative way. We obtained reasonable estimates for the parameters in the model and included other details to obtain a model that describes quantitatively as well as qualitatively the early stages leading to type 1 diabetes. In order to ensure all solutions of the model remain bounded, saturation terms were included in the equations for resting and activated macrophages. These terms represent a "forcing out" of macrophages due to crowding. While these forces are a reasonable inclusion into the model, it is likely only a crude approximation of the biology. It may be that other cell types (including cells not considered in the model) contribute to crowding and forcing out of macrophages from 77 the tissue. The Copenhagen model is very limited in the scope of cell types and interactions considered. This remains true of even the most detailed model presented here. Likely other cell types (dendritic cells for example) play an important part in the early stages leading to chronic inflammation, and even the cell types considered here may interact in a more complicated manner than captured by the model. For example, we have greatly simplified the behavior of the macrophages. We have assumed that macrophages do not die, and that they have un-limited capacity to engulf apoptotic cells. A more realistic approach would consider death of macrophages (both resting and activated) and incorporate a saturating effect into the phago-cytosis terms. We have also simplified things by assuming that macrophages exist in only two states: resting and activated. Likely the variety of behavior exhibited by macrophages is much more rich. Similarly we have considered an idealized situation in which there is a clear distinction between the macrophage response to apoptotic and necrotic cells. At this stage, the biological details are still controversial. The inclusion of the generic "cytokine," representing all cytokines and chemical secreted by activated macrophages, is only a "proof of principle." At this time we have demonstrated only that it is plausible that apoptosis is induced in a nonlinear response to secreted cytokines. Indeed, all the parameters involved in the variable C, have been estimated in order to show that this is a reasonable explanation. The model also simplifies the issue by considering all cytokines together as one general substance, and does not consider the role of cytokines in the recruitment of resting macrophages. It is not clear that these are reasonable approximations, as these cytokines are known to be responsible for the recruitment of resting macrophages and have been shown to interact with each other in a complicated and not completely understood way. It may be, as in the case of the Copenhagen model, that once more realistic parameter values are determined the model will no longer provide the expected dynamics. In such a case further details may need to be included. In addition to the specific details mentioned above, all of the models presented in this study 78 deal with an idealized situation. By creating a model using ordinary differential equations, we have assumed that the pancreatic islets are homogeneous, and that spatial distribution of cells or structural features of the islet do not play a significant role. We have also assumed that all individuals of a given strain are identical, modeled with the same parameter values. A more realistic model would also allow for the possibility that parameter values are not constant in time. It is possible that as the animals age the macrophage residence time and rates of activation and deactivation may change. The rate of non-specific clearance of apoptotic cells may also be affected by age. We have also simplified the models presented here by assuming an unlimited pool of healthy /3-cells. This assumption is reasonable, since during the early stages of the disease, when our model is applicable, the damage to /3-cells caused by macrophages, or cytokines reduces the total number of /3-cells by only a small fraction. Still, a more realistic model should include healthy /3-cells explicitly. Even with the limitations of our model, it provides insight into the early stages of type 1 diabetes development. Considering as few details as possible, we are able to account for the difference between health and susceptibility to chronic inflammation, suggesting that we have captured the important details of the biology. The model's predictions also correspond well with experimental observations. Thus our model seems a reasonable first approximation to the underlying causes of susceptibility to type 1 diabetes. 7.4 Future Work The pathogenesis of type 1 diabetes is a complicated process, and in this study we concerned ourselves with the details of a relatively short period, and the actions of a very few types of cells known to be involved. In the future, it would be beneficial to incorporate this model into a larger study that considers further stages of type 1 diabetes development. The model presented here could be coupled to a model describing insulitis, as well as activation and cytotoxicity of T-cells, and thus present a picture of the complete pathogenesis. Even before expanding on the current model, refinements can be made. Experiments to verify 79 and refine the parameter estimates given here would ensure that this model is a reasonable approximation to the details considered, and would strengthen further the hypotheses tested here. In addition to checking the parameter estimates found in the literature, experiments to determine the parameters involved with the cytokine variable are important. Some experi-ments of this type have already been done, and the results need to be taken from the literature and incorporated into the model. Specifically we are interested the rates of production and decay for the cytokines involved in the early stages of type 1 diabetes, as well as determining the cytotoxic effect of these cytokines on the pancreatic /3-cells. Using a dose-response curve in place of the Michaelian saturating curve currently used would bring the model and its predictions much more closely in line with the actual biological processes taking place. 80 Bibliography [1] H. Beyan, L.R. Buckley, N . Yousaf, M . Londei, and R.D.G. Leslie. A role for innate immunity in type 1 diabetes? Diabetes/Metabolism Research and Reviews, 19:89-100, 2003. [2] S. Charre, J.G.M. Rosmalen, C. Pelegri, V. Alves, P.J.M. Leene, H.A. 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Wigginton and Denise Kirschner. A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. The Journal of Immunology, 166:1951-1967, 2001. [48] Ji-Won Yoon and Hee-Sook Jun. Cellular and molecular roles of /3-cell autoantigens, macrophages and T cells in the pathogenesis of autoimmune diabetes. Arch Pharm Res, 22(5):437-^47, 1999. [49] Ryogo Yui and Tsuneo Fujita. Frontiers in Diabetes Research: Lessons from animal diabetes II, chapter II.3, pages 112-116. John Libbey & Co., 1988. 84 Appendix A Immunological Terms 1. Antigen: a macromolecule, usually a protein or polysaccharide that triggers immune cells and immune responses. Antigens are usually substances that are foreign to the body (from viruses, bacteria or transplanted organs), but may also be from cells of the body (auto antigens) (wikipedia on-line encyclopedia: www.wikipedia.com) 2. Antigen Presenting Cell: phagocytic cells that engulf antigen f ound in the body. Antigen presenting cells cut and process the antigen into small fragments, called peptides, which are then displayed on the cell surface to activate T-cells. The most prominent antigen presenting cells are called dendritic cells [14]. 3. Apoptosis: a programmed form of cell death, in which the cell activates an internal death program [14]. Apoptotic cells are characterized by DNA degradation, condensation of chromatin and protein cleavage [6]. 4. /3-cell: an endocrine cell, found in the pancreas. /3-cells are responsible for the production of insulin and are the target of self-reactive T-cells in the pathogenesis of type 1 diabetes. 5. Cytokines: proteins made by a variety of cells to affect the behavior of other cells [14]. These proteins are mostly soluble and diffusible, acting over a limited range from the point of release. Cytokines have a variety of functions, ranging from recruiting cells to areas of tissue damage [9] to damaging tissue cells and triggering apoptosis [33]. 6. Islets of Langerhans: small isolated clusters of endocrine cells, found in the pancreas containing the insulin producing /3-cells. In the NOD mouse, the pancreatic islet is a 85 round cellular mass, covered by a sheath of thin connective tissue [10]. An is composed of a number of cell types, but /3-cells predominate: approximately 70% of the cells in the islet of Langerhans are /3-cells [10, 49]. 7. Insulitis: a term coined by J. V. von Meyenburg, referring to the inflammatory lymphocyte infiltration of the pancreatic Islets of Langerhans [36]. 8. Lymphocyte: a class of white blood cells with receptors to recognize a specific antigen. There are two main types of lymphocytes: B and T cells, responsible for the adaptive im-mune response. Each individual lymphocyte has a unique set of receptors that recognize only one antigen [14]. 9. Lymphokine: cytokines secreted by T-cells are sometimes referred to as lymphokines [14] 10. Macrophage: large phagocytic cells, important in the innate (non-adaptive) immune response. Macrophages derive from monocytes and perform a number of roles: anti-gen presenting cell, clearance of cellular debris, and as an effector cell in cell mediated immunity [14]. 11. M H C complex: (Major Histocompatibility Complex) a cluster of genes that encode a set of membrane proteins called MHC molecules. There are two classes of MHC molecules: MHC-I present cytosolic antigen peptides to CD8 + T-cells, MHC-II present antigen peptides degraded intra-cellularly to CD4 + T-cells [14]. 12. Necrosis: a pathological form of cell death, following chemical or physical injury. Necrotic cells are typified by a loss of cellular homeostasis, swelling and rupture of the plasma membrane [6]. Cell death by necrosis has been shown to promote an inflam-matory response [3]. Related to necrosis is secondary necrosis, observed in vitro when apoptotic cells are not removed by phagocytes. The apoptotic cells are unable to regulate cell volume and eventually swell and burst [34]. 13. T-cell: a subset of lymphocytes, defined by their development in the thymus. T-cells are 86 further classified: / • CD4 + T-cells respond to peptides presented on MHC-II molecules and activate macrophage and B cell responses to antigen. • CD8 + T cells respond to peptides presented on MHC-I molecules and mature into cytotoxic cells that can kill any cell displaying antigen it recognizes. [14] 87 Appendix B Dimensionless Formulation of the Model To find a dimensionless form of the model given in Chapter 3, we express the equations (3.2) (with e, = 0) in terms of the dimensionless variables, t*, AT, Ma*, Ba*, defined by: t = zt\ M = MM*, Ma = MaMa*, Ba = BaBa*, where t, M, Ma, Ba are the original dimensional variables of the model, and T, M , Ma, Ba are the dimension carrying constant scales used. Following this substitution, we obtain: > dM IF dMa* dt* dBa* dt* +(z(k+b))^Ma* - [e y^k* -1 (TgBa) M 1 1 \MBa = (TgBa)^MBa - (T/C)AV, Ma Tl Ma \Ma - (Tf\M)M*Ba* - (zf2Ma)Ma*Ba* - {xd)Ba\ (B.la) (B.lb) (B.lc) We now scale the variables to simplify the equations, setting the boxed coefficients to 1. Solving for the dimension carrying constants we obtain: r = 1/c, the normal residence time of resting macrophages, as our time scale, and set M = J/c - the resting macrophage density in uninflamed tissue. Solving for B„, we obtain Ba = c/g, the number of apoptotic /3-cells required to activate one resting macrophage during its residence time. Solving for Ma, we obtain Ma = c2/gl. This quantity can be factored as (c/g)(c/l) and the meaning deciphered by understanding each factor. One factor, c/g = Ba is already understood. The other factor is best understood if we first consider the reciprocal, l/c. The parameter / is the rate of induced apoptosis per activated macrophage, and l/c is the residence time of resting macrophages. Put together, l/c is the rate of induced apoptosis per activated macrophage during the macrophage residence time. 88 It then follows that c/l is the number of activated macrophages required to induce apoptosis in one /3-cell during the residence time of resting macrophages (x). Putting the two factors together, we interpret M„ as (the number of apoptotic /3-cells needed to activate one resting macrophage) (the number of activated macrophages needed to induce apoptosis in one /3-cell) or simply the number of apoptotic /3-cells required to trigger apoptosis in one additional /3-cell during the resting macrophage residence time. These scalings are summarized in Table B.l . Unit Carrying Chosen Meaning Quantities Scales X 1/c Normal residence time of resting macrophages M lie Normal M density in tissue Ba c/g Number of Ba needed to activate one M during residence time Ma c2/gl Number of Ba needed to trigger apoptosis in one additional /3-cell during each time unit x Table B.l : Variable Scaling for non-dimensional form of model. By scaling the variables as in Table B.l and dropping the *'s we obtain: = l + (k + b)^JMa-M-MBa, dM ~dt dMa dt dBa dt = Ma-g4\MBa hi Ma, MBa-^JMaBa-(^Ba. (B.2a) (B.2b) (B.2c) If we now give each of the parameter groupings a label (defined in Table B.2) the system becomes: ^ = \ + {K + E,)Ma-M-MBa, (B.3a) ^ = yMBa-^Ma, (B.3b) ^ = Ma-pMBa-qMaBa-ABa. (B.3c) When we substitute realistic parameter values into these equations (see Table 4.2) we see that a quasi steady state assumption on M is justified. 89 Non Dimensional . Parameter Meaning Value Parameter Grouping (non dimensional) (see Table 4.2) K increase in M due to deactivation of Ma 0.02 recruitment of M by Ma 4.4 X 10"3 y (4) increase in Ma due to activation of M 205 k decrease in M f l due to deactivation of Ma 4 (f) engulfment of Ba by M 50 n (f) engulfment of Ba by Ma 0.2 A (?) nonspecific decay of Ba 5 Table B.2: Non Dimensional Variable Definitions. 90 Appendix C XPP Code C.l Simulations of the Basic Model # basic.ode # for simulations of the basic extension of the Copenhagen Model. # Parameter Values: par J=Q.®1, c=l, b=l, 1=1, d=0.5, el=0.1, e2=8.1, par g=l, f l = l , f2=Q.©l, k=0.3 # Equations: #M on QSS: M =C-(el*Ma+c+g,'tBa)+sqrtCCel*Ma+c+g,vBa)"2+4''tel,v(J+Ck+b)*Ma)))/C2*el) Ma'=g*M*Ba-k*Ma-e2*Ma*(M+Ma) Ba'=l*Ma-fl*M*Ba-f2*Ma*Ba-d*Ba @ bell=Q @ XP=Ma, YP=Ba @ xlo=0, ylo=0, xhi=4, yhi=2 91 C.2 Simulations of the Reduced Necrotic Model # reducednec.ode # for simulations of the reduced necrotic model extension of the # Copenhagen Model. # Parameter Values: par J=5e4, c=0.1, b=0.Q9, d=0.5, k=0.4 par el=le-8, e2=le-8 par Amax=2e7, kc=l, alpha=5e-9, delta=25 # For NOD Mice #par g=le-5, fl=le-5, f2=le-5 # For Balb/c Mice par g=2e-5, fl=2e-5, f2=5e-5 # Equations: kb=(delta/alpha)*kc #M on QSS: M =C-(el'vMa+c+g*Ba)+sqrt((el,vMa+c+g'vBa)"2+4>,rel*(J + (k+b)*Ma)))/(2 , vel) Ma' =g'vM*Ba-k*Ma-e2*Ma'v (M+Ma) Ba' = CAmax*d*Ba*Ma)/Ckb',f (f l*M+f 2'vMa)+d*Ba*Ma) - f l*M*Ba-f 2'vMa*Ba-d'vBa @ bell=0 @ maxstor=2Q®00OOQ0, bounds=lel0, nmesh=400, meth=gear @ XP=Ma, YP=Ba @ xlo=0, ylo=0, xhi=6e6, yhi=2.5e5 92 C.3 3D Reduced Necrotic Model # 3dnecrotic.ode # for simulations of the reduced necrotic model extension of the # Copenhagen Model. # For NOD Mice par g=le-5, f2=le-5 # Parameter Values: par J=5e4, c=0.1, b=©.©9, d=©.5, k=0.4 par e=le-8 par Amax=2e7, kc=l, alpha=5e-9, delta=25 # Equations: kb=(delta/alpha)*kc Ma'= g*M*Ba-k*Ma-e*Ma*(M+Ma) M' = J+(k+b)*Ma-c*M-g*M*Ba-e*M*(M+Ma) Ba' = (Amax*d*Ba*Ma)/(kb* (g*M+f2*Ma)+d*Ba*Ma) -g*M*Ba-f 2*Ma*Ba-d*Ba # I n i t i a l Conditions i n i t M=5e5, Ba=20000 @ bell=© @ maxstor=2Q008Q0, bounds=lel0, nmesh=400 @ total=300, meth=gear @ NPL0T=3 @ XP=t, YP=M, XP2=t, YP2=MA, XP3=t YP3=Ba @ xlo=0, ylo=0, xhi=300, yhi=5.5e5 # AUTO STUFF @ NTST=20 @ NMAX=100000, NPR=100000, DS=0.02, DSMIN=0.000001, DSMAX=2000 @ PARMIN=0, PARMAX=1, normmax=lel2 @ autoxmin=0, autoymin=Q, autoxmax=4e-5, autoymax=6e6 done 93 C.4 Simulations of the Full Necrotic Model 0 # necrotic.ode # For simulations of the F u l l Necrotic Extension of the # Copenhagen Model # Parameters par J=5e4, c=0.1, b=0.09, d=0.5, k=0.4 par el=le-8, e2=le-8 par Amax=2e7, kc=l, alpha=5e-9, delta=25 par g=le-5, fl=le-5, f2=le-5 # Equations: #The Apoptotic Wave: par Wmax=3.4e6 W(t)=Wmax*expC-(Ct-9)/3)"2) M' = J+(k+b)*Ma-c*M-fl*M*Ba-el*M*(M+Ma) Ma' = fl*M*Ba-k*Ma-e2*Ma*(M+Ma) Bn' = d*Ba-fl*M*Bn-f2*Ma*Bn Ba' = W(t)^Amax*Cy)/(kc+Cy)-fl*M*Ba-f2*Ma*Ba-d*Ba Cy' = alpha*Bn*Ma-delta*Cy # I n i t i a l Conditions i n i t M=5e5 & bell=0 @ maxstor=2Q00000®, bounds=lel0, nmesh=400 @ meth=gear @ total=3Q0 @ nplots=5 @ XP=t, YP=M, XP2=t, YP2=Ma, XP3=t, YP3=Ba, XP4=t, YP4=Bn, XP5=t, YP5=Cy @ xlo=0, ylo=0, xhi=300, yhi=5e5 94
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Modeling the onset of type 1 diabetes Kublik, Richard Alexander 2005
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Title | Modeling the onset of type 1 diabetes |
Creator |
Kublik, Richard Alexander |
Date Issued | 2005 |
Description | A neonatal wave of apoptosis (programmed cell death) occurs normally in the pancreatic β-cells of mice and rats. Previous collaborative work by members of Diane T Finegood's experimental lab at Simon Fraser University and Leah Edelstein-Keshet's mathematical biology group at the University of British Columbia demonstrated that macrophages from non-obese diabetic (NOD) mice become activated more slowly and engulf apoptotic cells at a lower rate than macrophages from control (Balb/c) mice. It has been hypothesized that this low clearance could result in secondary necrosis of the β-cells, escalating inflammation and the self-antigen presentation that later triggers autoimmune, type 1 diabetes. We here investigate whether this hypothesis could offer a reasonable and simplified explanation for the onset of type 1 diabetes in NOD mice. We quantify variants of the Copenhagen model, developed by Freiesleben De Blasio et al (1999, Diabetes 48, 1677), based on parameters from NOD and Balb/c experimental data. We demonstrate that the original Copenhagen model fails to explain observed phenomena within a reasonable range of parameter values, predicting an unrealistic all-or-none disease occurrence in both strains. However, if we assume that the activated macrophages produce harmful cytokines only when engulfing necrotic (but not apoptotic) cells, then the revised model becomes qualitatively and quantitatively reasonable. Further, we show that known differences between NOD and Balb/c mouse macrophages kinetics are large enough to account for the fact that an apoptotic wave can trigger escalating inflammatory response in NOD, but not Balb/c mice. In Balb/c mice, macrophages clear the apoptotic wave so efficiently, that chronic inflammation is prevented. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2009-12-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079412 |
URI | http://hdl.handle.net/2429/16525 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2005-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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