Analysis of glucose-insulin-glucagon interaction models.

Transcription

1 C.J. in t Veld Analysis of glucose-insulin-glucagon interaction models. Bachelor thesis August 1, 2017 Thesis supervisor: dr. V. Rottschäfer Leiden University Mathematical Institute

2 Contents 1 Introduction 3 2 Glucose homeostasis Glucose blood concentration Diabetes Glucose-insulin-glucagon interaction models Minimal model Dynamical model Model developed at the CHDR Other models Analyzing model developed at the CHDR Modeling infusion of somatostatin as an on-off function Different time periods introduced Consequences of modeling infusion of somatostatin as an on-off function Critical points in time periods 1 and Critical points in time period Nondimensionalization of the model Considering the model on different time scales Model without infusion terms Improving the model developed at the CHDR Changing the values of the parameters Comparison with other models Comparison with Minimal model Comparison with Dynamical model Conclusion 46 References 47 7 Appendix I 49 8 Appendix II Main file original model Function file original model Main file model when somatostatin is modeled as an on-off function Function file model when somatostatin is modeled as an on-off function Main file model when infusion terms are removed Function file model when infusion terms are removed Code to determine eigenvalues in time periods 1 and Code to determine eigenvalues in time period Code for the time scale analysis

3 1 Introduction Glucose is a simple sugar that serves as the main source of energy for most cells in the human body. Glucose molecules are delivered to cells by the circulating blood. Therefore, it is important that the blood glucose level is held relatively constant. This process is called glucose homeostasis. The hormones insulin and glucagon play an important role in maintaining the blood glucose level relatively constant. Diabetes is a disease in which the glucose-insulin regulatory system does not work properly. The cause of this disease is still unknown. In order to better understand this disease and to develop anti-diabetic drugs, many mathematical models have been developed. In general, these models are systems of differential equations, describing the glucose and insulin blood concentration. The aim of this thesis is to analyze these models. First, a basic understanding of glucose homeostasis is needed. Therefore in the first chapter we give a short introduction to the working of insulin and glucagon. Second, models that describe the glucose-insulin-glucagon interaction are given and compared with each other. There exist many models, so only the most important ones are discussed [4, 7, 11, 18]. One of these models is a model developed at the Center for Human Drugs Research (CHDR) in Leiden [18]. This model is analyzed thoroughly, because this model is relatively recent and has the almost unique property in that it also describes the glucagon blood concentration. Most models only consider the glucose and insulin blood concentration. The model developed at the CHDR has not been analyzed yet. The analysis is done in association with thesis supervisor dr. V. Rottschäfer. It consists of determining the critical points and their stability and nondimensionalizing the model in order to detect small and large parameters. Furthermore, the model is considered on different time scales. By doing this, we are able to better understand the behavior of the system. The analysis of the model is concluded by observing whether the model is able to describe the natural glucose-insulin-glucagon interaction. In its original form, the model describes the glucose-insulin-glucagon interaction after an infusion, which contains glucagon, insulin and glucose. When removing all terms associated with this infusion, we can use numerical simulations to see whether the model is able to describe the natural situation. In the last chapter certain ideas are presented to improve the model developed at the CHDR. These ideas arose during the analysis of this model and from other models described in this thesis. However, it is not possible to test whether these ideas actually improve the model, because no data is available to us. So, the ideas are future research suggestions. 3

4 2 Glucose homeostasis This chapter aims to explain how the glucose blood concentration can increase or decrease and which hormones play an important role in this process. Also the concept of the so-called internalization of glucagon receptors is explained. Furthermore, information about diabetes, a disease in which the glucose-insulin regulatory system does not work properly, is given. 2.1 Glucose blood concentration There are many factors and hormones that cause an increase or decrease of the glucose blood concentration. Important factors that increase the glucose blood concentration include glucose absorption after eating a meal and the production of glucose by liver cells. Important factors that decrease the blood glucose level include the transport of glucose into cells for use as a source of energy and the loss of glucose in urine [14]. Blood glucose levels are restored to normal levels primarily through the action of two hormones, namely insulin and glucagon. Insulin is secreted by β-cells in the pancreas when the glucose blood concentration is high. Insulin stimulates most body cells to increase their rate of glucose uptake from the blood. This results in a decrease of the glucose blood concentration. Insulin also stimulates the formation of glycogen from glucose. Glycogen is a readily mobilized storage form of glucose, which is stored in liver and muscle cells. Because glucose in blood is used to form glycogen, the glucose blood concentration decreases [8]. Glucagon is secreted by α-cells in the pancreas when the glucose blood concentration is low. Glucagon stimulates the breakdown of glycogen to glucose in liver and muscle cells. This results in an increase of the glucose blood concentration [8]. All these effects are illustrated in Figure 1, reproduced from [6]. Figure 1: Influence of insulin and glucagon on glucose homeostasis. 4

5 As mentioned above, glucagon stimulates the breakdown of glycogen to glucose in liver and muscle cells. This is explained in more detail in order to be able to understand the models described in Chapter 3. When glucagon is secreted, it binds to glucagon receptors on membranes of liver cells. A receptor is a chemical group on a membrane of a cell that has an affinity for a specific molecule or chemical group [3]. When glucagon is bound to glucagon receptors, the glucagon receptors are absorbed into the liver cells. This process of absorption of membrane receptors into the cells is called internalization of receptors [2]. After the internalization, processes are started in the liver cells to break glycogen down to glucose. All these effects are illustrated in Figure 2, reproduced from [13]. Figure 2: Glucagon receptor internalization. 2.2 Diabetes Diabetes is a group of metabolic diseases. In this disease the glucose-insulin regulatory system does not work properly. Either the pancreas does not produce enough insulin or the cells of the body do not respond properly to the insulin produced [1]. There are different types of diabetes. The most occurring one is diabetes type 2. In this type the cells of the body do not respond properly to the insulin produced and too few insulin is produced by the pancreas. This type of diabetes can be caused by excessive body weight and not enough exercise. However, it can also be caused by inheritance. Another type of diabetes is diabetes type 1. In this type the β-cells producing insulin are destroyed by the immune system. Patients with diabetes type 1 have to take insulin injections in order to ensure a proper glucose blood concentration [1]. The cause of this type of diabetes is unknown. In order to better understand the pathophysiology of diabetes, several models have been developed. These models can serve as a basis for drug development. 5

6 3 Glucose-insulin-glucagon interaction models The relationship between glucose, glucagon and insulin has been studied by many researchers. During the last three decades several models have been formulated to describe the homeostasis of glucose. These models can be useful to assess anti-diabetic drug effects. In this chapter several glucose-insulin-glucagon interaction models are described and compared with each other. There exist many models, so only the most important ones are discussed. 3.1 Minimal model The first model describing the homeostasis of glucose was developed by Bergman et al [4]. It is called the Minimal model. This model is widely used in research to estimate glucose effectiveness and insulin sensitivity from intravenous glucose tolerance test (IVGTT) data [12]. During this test a certain amount of glucose is injected into the bloodstream of a human being and the glucose and insulin blood concentrations afterwards are measured for a period of about three hours [7]. In order to describe those glucose and insulin blood concentrations, the minimal model has been proposed. The following variables and parameters are used in the minimal model [7]: t =[min] time; G(t) =[mg/dl] glucose blood concentration at time t; I(t) =[µui/ml] insulin blood concentration at time t; X(t) =[min 1 ] function representing the glucose uptake activity by cells, depending on the insulin blood concentration; G b =[mg/dl] normal glucose blood concentration level of a human being; I b =[µui/ml] normal insulin blood concentration level of a human being; p 0 =[mg/dl] theoretical glucose blood concentration at time 0 after a certain amount of glucose; p 1 =[min 1 ] rate constant expressing the decrease of glucose caused by glucose uptake by cells, independent of the insulin blood concentration; p 2 =[min 1 ] rate constant expressing the spontaneous decrease of glucose uptake activity by cells; p 3 =[min 2 (µui/ml) 1 ] rate constant expressing the increase of glucose uptake activity by cells per unit of insulin blood concentration excess over normal insulin blood concentration level; p 4 =[(µui/ml) (mg/dl) 1 min 1 ] rate constant expressing the production of insulin per unit of glucose blood concentration above p 5 ; p 5 =[mg/dl] level of glucose blood concentration at which the pancreas releases insulin; 6

7 p 6 =[min 1 ] rate constant expressing the decrease of insulin; p 7 =[µui/ml] theoretical insulin blood concentration at time 0 above the normal insulin blood concentration level, after a certain amount of glucose. The minimal model is given by [4]: dg(t) = X(t)G(t) p 1 (G(t) G b ), dt G(0) = p 0, (1) dx(t) = p 2 X(t) + p 3 [I(t) I b ], dt X(0) = 0, (2) di(t) = p 4 [G(t) p 5 ] + t p 6 [I(t) I b ], dt I(0) = p 7 + I b, (3) where [G(t) p 5 ] + is given by max{0, G(t) p 5 }. The model is composed of two separate parts. The first part consists of the equations (1) and (2) and describes the glucose blood concentration. The term p 1 (G(t) G b ) describes the decrease of glucose caused by the glucose uptake by cells per unit of glucose blood concentration excess over normal glucose blood concentration level. This decrease is independent of the insulin blood concentration. Here, X(t) represents the glucose uptake activity, which depends on insulin. So, the term X(t)G(t) describes the uptake of glucose by cells, which depends on the insulin blood concentration. The variable X(t) is introduced to mimic the time delay of the effect of insulin on glucose. The glucose uptake activity decreases linearly, as described by the term p 2 X(t) and increases when the insulin blood concentration is above normal level, as described by the term p 3 [I(t) I b ]. For this first part, the insulin blood concentration is to be regarded as a known forcing function. The second part consists of equation (3) and describes the insulin blood concentration. The term p 4 [G(t) p 5 ] + t describes the increase of the insulin blood concentration. The insulin blood concentration only increases, when the glucose blood concentration is above a certain level p 5. Otherwise, this term is equal to 0. The multiplication by t is introduced to express the hypothesis that the rate of pancreatic secretion of insulin proportional is to the time elapsed from the glucose stimulus, see [7]. The last term in equation (3) models the decrease of insulin per unit of insulin blood concentration excess over normal insulin blood concentration level. For this second part, the glucose blood concentration is to be regarded as a known forcing function. As mentioned by De Gaetano et al. [7], the model parameters are determined in two steps: using the insulin blood concentration as input data, the parameters in the first two equations are derived and using the glucose blood concentration as input data, the parameters in the last equation are derived. While the above minimal model has been very useful in physiological research, [7] argues that it has the following three drawbacks associated with it. First of all, the parameter fitting is divided into two separate parts, as described above. However, because the glucose-insulin interaction is a coherent system, it would be better if the parameter fitting is a single-step process. Secondly, some of the mathematical results produced by this model are not realistic. For example, the minimal model does not admit an equilibrium and for certain values of parameters the solutions are unbounded. Thirdly, the non-observable variable X(t), used to describe the delay of the action of insulin on glucose, is introduced artificially. It would be more natural to explicitly introduce the time delay in the model. Finally, the assumption that the rate of pancreatic secretion of insulin is proportional to the time elapsed from the glucose stimulus seems questionable [7]. 7

8 3.2 Dynamical model To overcome the drawbacks of the minimal model, a new model was introduced in 2000 called the Dynamical model [7]. In this model the two parts of the minimum model are coupled, the delay is represented explicitly and non-observable state variables are deleted. Also the assumption that the rate of pancreatic secretion of insulin is proportional to the time elapsed from the glucose stimulus is not used. The following variables and parameters are introduced in the dynamical model [7]: t =[min] time; G(t) =[mg/dl] glucose blood concentration at time t; I(t) =[pm] insulin blood concentration at time t; G b =[mg/dl] normal glucose blood concentration level of a human being; I b =[pm] normal insulin blood concentration level of a human being; b 0 =[mg/dl] theoretical increase in glucose blood concentration over normal glucose blood concentration level at time 0 after a certain amount of glucose; b 1 =[min 1 ] rate constant expressing the decrease of glucose caused by glucose uptake by cells, independent of the insulin blood concentration; b 2 =[min 1 ] rate constant expressing the decrease of insulin; b 3 =[pm (mg/dl) 1 ] theoretical increase in insulin blood concentration per (mg/dl) increase in the glucose blood concentration at time 0 due to the injected amount of glucose; b 4 =[pm 1 min 1 ] rate constant expressing the decrease in glucose blood concentration per pm of insulin blood concentration; b 5 =[min] length of the past period whose glucose blood concentrations influence the current pancreatic insulin secretion; b 6 =[pm (mg/dl) 1 min 1 ] rate constant expressing the production of insulin per (mg/dl) of average glucose blood concentration throughout the previous b 5 minutes; b 7 =[(mg/dl) min 1 ] constant increase in glucose blood concentration due to constant baseline liver glucose release. The dynamical model reads [7]: dg(t) = b 1 G(t) b 4 I(t)G(t) + b 7, dt di(t) = b 2 I(t) + b t 6 G(s)ds, I(0) = I b + b 3 b 0, dt b 5 t b 5 G(t) G b t [ b 5, 0), G(0) = G b + b 0. In this model the changes in glucose blood concentration depend on the insulin independent uptake of glucose by cells, described by the term b 1 G(t), the insulin dependent uptake of glucose by cells, described by the term b 4 I(t)G(t) and on baseline liver glucose production, 8

9 described by the term b 7. The changes in insulin blood concentration depend on a spontaneous constant-rate decay, modeled by the term b 2 I(t) and on pancreatic insulin secretion, modeled by the term b6 t b 5 t b 5 G(s)ds. In this model, the effective pancreatic secretion at time t is considered to be proportional to the average value of glucose blood concentration in the b 5 minutes preceding time t. In the dynamical model a few assumptions are made that may not be realistic or necessary, as denoted by [11]. For example, the assumption that the effective pancreatic secretion at time t is proportional to the average value of glucose blood concentration in the b 5 minutes before time t is questionable. Therefore, Li et al. [11] proposed another model, in which the time delay was incorporated in another way. The model includes the dynamical model as a special case and is formulated as [11]: dg(t) dt di(t) dt = b 1 G(t) b 4I(t)G(t) αg(t) b 7, = b 2 I(t) + b 6 G(t b 5 ), I(0) = I b + b 3 b 0, G(t) G b t [ b 5, 0), G(0) = G b + b 0, α 0, 1 = half-saturation constant. α This model has two notable differences in comparison with the dynamical model. First of all, the insulin dependent glucose uptake is assumed to take the form, which has a maximum G(t) αg(t)+1 capacity b4 α. The reason for this is due to the limit of time and the capacity of insulin s ability of digesting glucose. Furthermore, the effective pancreatic secretion at time t is assumed to be affected by the glucose blood concentration in b 5 minutes preceding time t, instead of the average glucose blood concentration in that period. 3.3 Model developed at the CHDR In the models described above, only the glucose and insulin blood concentration are considered. Recently, another model has been developed at the Center for Human Drug Research (CHDR) in collaboration with the Mathematical Institute in Leiden [18]. This model describes the glucoseinsulin-glucagon interaction during a glucagon challenge. During this challenge the endogenous release of insulin and glucagon is blocked by infusion of somatostatin and simultaneously a high dose of glucagon and physiological dose of insulin is infused. The glucose, insulin and glucagon blood concentrations afterwards are measured for a certain period. In order to describe the time course of these concentrations, a new model has been proposed. In comparison with other models, this model also includes the so-called internalization of glucagon receptors, see section 2.1 for more details. This internalization of glucagon receptors is included in the model, because in recent years there has been an increased interest in the glucagon receptor as a therapeutic target to treat diabetes patients [10]. 9

10 The following variables and parameters are used in the model [18]: G(t) =[g] amount of glucose in blood at time t; I(t) =[mu] amount of insulin in blood at time t; A(t) =[ng] amount of glucagon in blood at time t; R(t) = internalization of glucagon receptors at time t; t begin = begin of infusion; t end = end of infusion; k in,g =[g (ng min) 1 ] rate constant expressing the production of glucose per ng glucagon; k in,a =[ng g (min) 1 ] rate constant expressing the glucose dependent production of glucagon; k in,i =[mu (g min) 1 ] rate constant expressing the production of insulin per g glucose; k in,r =[(g min) 1 ] rate constant expressing the increase of the internalization of glucagon receptors per g glucose; k out,i =[(min) 1 ] rate constant expressing the decrease of i, where i = A, G, I, R; k out,gi =[(mu min) 1 ] rate constant expressing the decrease of glucose per mu insulin. The infusion of somatostatin, which blocks the endogenous secretion of glucagon and insulin, is modeled by an inhibitory function S i (t). This function equals 1, except during the inhibition. The inhibitory function is given by: S i (t) = 1 S max,i H(t; t begin, t end, α i ) i = A, I. S max,i denotes the maximal inhibition of i by infusion of somatostatin (i.e., 0 S max,i 1). The function H(t; t begin, t end, α i ) is given by: 0 for 0 t t begin H(t; t begin, t end, α i ) = 1 e αi(t t begin) for t begin < t < t end (1 e αi(t tbegin) )e αi(t tend), for t end t <, where α i > 0 is the rate at which the inhibition of i reaches its maximum value. The infusion of glucagon, insulin and glucose is given by functions Q i (t): { Q max,i for t begin t t end Q i (t) = i = A, G, I, 0 else, where Q max,i denotes the maximal infusion of i. 10

11 The model is given as follows [18]: dg(t) dt da(t) dt di(t) = k in,g A(t) R(t) k out,gg(t) k out,gi G(t)I(t) + Q G (t), (4) = k in,a G(t) S A(t) k out,a A(t) + Q A (t), (5) = k in,i G(t)S I (t) k out,i I(t) + Q I (t), (6) dt dr(t) = k in,r G(t) k out,r R(t). (7) dt In this model the amount of glucose in blood depends on the insulin independent uptake of glucose by cells, the insulin dependent uptake of glucose by cells, the infusion and the production of glucose, which depends on the amount of glucagon in the blood. The amount of glucagon in blood depends on the infusion and the production of glucagon, which decreases when the amount of glucose in blood is high. It also decays with a constant rate. The changes in the amount of insulin in blood are caused by a certain decay, the infusion and by pancreatic insulin secretion. The pancreatic insulin secretion increases when the amount of glucose in blood is high. The internalization of glucagon receptors increases when the amount of glucose in blood is high and decays with a constant rate. Data of 36 healthy subjects during a glucagon challenge has been collected, see [18]. Fitting the model to this data yields parameter estimates as mentioned in [18]. These values of the parameters are given in Table 1. Parameter Value Parameter Value t begin 180 S max,i 1 t begin 360 α k in,g 275 α k out,g Q max,g k out,gi Q max,a 240 k in,a 559 Q max,i 7.94 k out,a G(0) 4.20 k in,i 2.09 A(0) k out,i I(0) 17.7 k in,r 655 R(0) k out,r Table 1: Values of parameters as determined in the original article. In Figures 3, 4, 5 and 6 results of numerical simulations of the model are shown based on values of parameters as described in Table 1. The code for the numerical simulations is given in section 8.1 and section

12 Figure 3: Numerical simulation of the amount of glucose in blood. Figure 4: Numerical simulation of the amount of glucagon in blood. Figure 5: Numerical simulation of the amount of insulin in blood. 12

13 Figure 6: Numerical simulation of the glucagon receptor internalization. In Figure 3, we plot the amount of glucose. We can see that during the infusion the amount of glucose increases. This is due to the infused amount of glucagon. After the infusion the amount of glucose decreases to normal level. In Appendix I, data of four typical healthy subjects is shown, reproduced from [18]. When comparing Figure 3 with this data, we conclude that our simulation is pretty good. In Figure 4, the amount of glucagon is plotted. It can be seen that the amount of glucagon increases rapidly during the infusion, because the infusion contains a high dose of glucagon. Because somatostatin does not block the internal production of glucagon instantly, the graph shows a peak at the beginning of the infusion. After the infusion the amount of glucagon decreases and returns to normal level. When comparing Figure 4 with the data as described in Appendix I, we can see that our simulation differs from the data. The profiles of subjects 1001 and 2901 do not show a peak at the beginning of the infusion. Furthermore, the profiles of subjects 2901, 801 and 501 are much higher or lower than our simulation. Figure 5 describes the amount of insulin. In the graph three peaks appear. These peaks do not show up in the data. The first two peaks appear, because the working of somatostatin is modeled in such a way that the internal production of insulin is not blocked and unblocked instantly. The last peak occurs, because of the high glucose blood concentration directly after the infusion. However, according to the data the amount of insulin in blood should be almost constant before and during the infusion. In Figure 6, we plot the glucagon receptor internalization. We can see that during the infusion the glucagon receptor internalization increases. This increase is caused by the high amount of glucagon in blood. After the infusion the glucagon receptor internalization decreases slowly. Because no data is collected of the glucagon receptor internalization, nothing can be said about the reliability of the graph. 3.4 Other models Besides the models described above, many other models have been developed. Many of them are described in an overview by Makroglou et al. [12]. Some of these models are briefly discussed in this paragraph: 1. In 1991 a six-dimensional ODE model was formulated [17]. The first differential equation describes the mass of glucose in the plasma, the second differential equation describes 13

14 the mass of insulin in the plasma and the third differential equation describes the mass of insulin in the intercellular space. The last three equations are associated with certain delays. There are two time delays in the system, namely the glucose triggered insulin production delay and the hepatic glucose production delay. This model has been the basis of several other models that include delays. 2. In 2007 Silber et al. developed a comprehensive model estimating glucose effectiveness and insulin sensitivity from IVGTT data [15]. In comparison with other models, this could be used for the analysis of the effects of anti-diabetic drugs on the glucose-insulin system from a more mechanistic point of view. This model describes the glucose and insulin blood concentration in different compartments in order to get a more mechanistic view of the glucose-insulin system. So, the glucose and insulin blood concentration are considered in both a central and a peripheral compartment. Furthermore, certain effect compartments are included to control the glucose production and insulin secretion. However, this model contains many parameters and is therefore less easy to use and understand. 3. In 2010 Silber et al. extended the previously developed model of [15] to include the oral glucose tolerance test (OGTT) [16]. During this test a human being drinks a certain amount of glucose and the glucose and insulin blood concentrations afterwards are measured for a certain period. 4 Analyzing model developed at the CHDR In this chapter the model developed at the CHDR and described in section 3.3 is analyzed. This model has been chosen to be analyzed thoroughly, because this model is relatively recent and has the almost unique property in that it also describes the glucagon blood concentration. In the first part of this chapter the infusion of somatostatin is modeled as an on-off function in order to be able to determine critical points and their stability. In the second part the model is nondimensionalized in order to detect small and large parameters. The influence of these small parameters on the system are examined. Furthermore, the model is considered on different time scales. This is useful for our understanding of the behavior of the system. At the end of this chapter all terms in the differential equations associated with the infusion are deleted. By doing this, we are able to see whether the model is capable of describing the natural glucose-insulinglucagon interaction. 4.1 Modeling infusion of somatostatin as an on-off function Different time periods introduced In the model several step functions are introduced. These step functions separate the system in three different time periods: the time before, during and after the infusion. The behavior of the system differs in each time period. Therefore, the model is analyzed in each time period separately. Originally, the functions describing the working of somatostatin were modeled as on-off functions [18]. Because Van Dongen et al. encountered numerical problems, they decided to model the working of somatostatin by the function S i (t), as described in section 3.3. However, these numerical problems do not appear when Matlab is used. Therefore, we model the working of 14

15 somatostatin as an on-off function in this paragraph. The time periods are as follows: 1. Time period 1: From t = 0 to t = t begin. During this time period Q i (t) equals 0 for i = A, I, G. Furthermore, S i (t) equals 1 for i = A, I. 2. Time period 2: From t = t begin to t = t end. During this time period Q i (t) equals Q max,i for i = A, I, G. Furthermore, S i (t) equals 0 for i = A, I. 3. Time period 3: From t = t end to t. During this time period Q i (t) equals 0 for i = A, I, G. Furthermore, S i (t) equals 1 for i = A, I. Note that the conditions in time period 1 are equal to the conditions in time period 3. Therefore, the behavior of the system in these time periods is the same Consequences of modeling infusion of somatostatin as an on-off function Here, we compare the numerical simulations when the working of somatostatin is modeled by an on-off function with the numerical simulations when the working of somatostatin is modeled by function S i (t). In Figure 7 we plot on the left-hand side the amount of glucose when the infusion of somatostatin is described by function S i (t) and on the right-hand side the amount of glucose when the infusion of somatostatin is described by an on-off function. In a similar way, Figures 8, 9 and 10 plot the amount of glucagon, insulin and the internalization of glucagon receptors respectively. The code for the numerical simulations is given in section 8.3 and 8.4. Figure 7: Amount of glucose in blood with infusion of somatostatin modeled by S i (t) (left-hand side) and infusion of somatostatin modeled by an on-off function (right-hand side). 15

16 Figure 8: Amount of glucagon in blood with infusion of somatostatin modeled by S i (t) (left-hand side) and infusion of somatostatin modeled by an on-off function (right-hand side). Figure 9: Amount of insulin in blood with infusion of somatostatin modeled by S i (t) (left-hand side) and infusion of somatostatin modeled by an on-off function (right-hand side). 16

17 Figure 10: Glucagon receptor internalization with infusion of somatostatin modeled by S i (t) (left-hand side) and infusion of somatostatin modeled by an on-off function (right-hand side). In Figure 7, we can see that the graphs describing the amount of glucose are almost the same in the time periods 1 and 3. However, in time period 2 the graphs differ. When the infusion of somatostatin is modeled as an on-off function, the amount of glucose in blood during the infusion is much less. This is due to the fact that the internal production of glucagon is blocked instantly. When describing the infusion of somatostatin by S i (t), glucagon is still produced at the beginning of the infusion. That results in more glucose during the infusion. In Figure 8 the amount of glucagon is plotted. We can conclude that the graph on the left-hand side is almost the same as the graph on the right-hand side in the time periods 1 and 3. However, in time period 2 the graphs differ. First of all, the amount of glucagon is much less when the infusion of somatostatin is modeled as an on-off function. As described above, this is due to the fact that the internal production of glucagon is blocked instantly. Secondly, the peak in the graph of glucagon at the beginning of the infusion disappears when the infusion of somatostatin is modeled as an on-off function. This is also due to the direct blocking of the internal production of glucagon. The amount of insulin is plotted in Figure 9. The graphs are almost the same, except for two peaks. When modeling somatostatin as an on-off function, the first two peaks disappear. The first peak disappears, because the internal production of insulin is blocked instantly during the infusion. The second peak disappears, because the internal production of insulin is unblocked instantly. When modeling the infusion of somatostatin by S i (t), it takes time before the insulin production is at normal level. Therefore, the amount of insulin is very low for a short time. That results in a peak. In Figure 10, we can see that the shapes of the graphs describing the glucagon receptor internalization are almost similar. Because the amount of glucagon is lower when modeling the infusion of somatostatin as an on-off function, the glucagon receptor internalization is lower as well. Describing the infusion of somatostatin by an on-off function has several advantages in comparison with describing the infusion of somatostatin by the function S i (t): 1. The model becomes easier and is therefore easier to study. 2. The graph describing the amount of insulin fits the data, as described in the original article 17

18 [18], more precisely. The data does not show any peak and when the infusion of somatostatin is modeled by the function S i (t), three peaks appear in comparison with one peak when the infusion of somatostatin is modeled as an on-off function. However, there are also several disadvantages in comparison with modeling the infusion of somatostatin by the function S i (t): 1. The graph describing the amount of glucose fits the data, as described in the original article [18], less precisely. 2. The graph describing the amount of glucagon fits the data, as described in the original article [18], less precisely. In Appendix I we can see that subjects 801 and 501 have a short, high amount of glucagon at the beginning of the infusion. That is modeled when the infusion of somatostatin is described by S i (t). However, the data varies a lot, so for some subjects it will be better to model the infusion of somatostatin as an on-off function. 3. It is unlikely that somatostatin will block the endogenous production of insulin and glucagon instantly and completely. The working of somatostatin is modeled more gradually when S i (t) is used in the model. The above arguments are based on the assumption that the values of the parameters do not change when the infusion of somatostatin is described by an on-off function instead of function S i (t). However, it is plausible that by changing the values of parameters the graphs on the right-hand side might fit the data better than they do now Critical points in time periods 1 and 3 In this section, we study the fixed points of the system in the time periods 1 and 3. In the time periods 1 and 3 Q i (t) equals 0 and S i (t) equals 1. Then, the system of differential equations reads: dg(t) A(t) = k in,g dt R(t) k out,gg(t) k out,gi G(t)I(t), da(t) = k in,a dt G(t) k out,aa(t), di(t) = k in,i G(t) k out,i I(t), dt dr(t) = k in,r G(t) k out,r R(t). dt 18

19 The critical points can be determined by setting dg(t) dt in the following equations: = da(t) dt = di(t) dt = dr(t) dt = 0. That results 0 = k in,g A R k out,gg k out,gi GI, (8) 0 = k in,a G k out,aa A = k in,a k out,a G, (9) 0 = k in,i G k out,i I I = k in,ig k out,i, (10) 0 = k in,r G k out,r R R = k in,rg k out,r. (11) Substituting the expressions for I, A and R in equation (8), gives us: k in,g k in,a k out,r k out,a k in,r G 2 k out,g G k out,gik in,i G 2 k out,i = 0. (12) This equation is hard to solve explicitly. Therefore, G is determined from this equation for the values of the parameters as described in the original article [18]. Those values are given in Table 1. In that case the equation reads: This has solutions: 1.22 G G G2 = 0. G and G Using relations (9), (10), (11) then leads to two critical points: K 1 = ( 4.42, , 18.6, ) and K 2 = (4.24, , 17.9, ). Note that K 1 is not interesting for this model, because the components are negative. The components of K 2 are positive, so this point is interesting for the model. Next, the stability of K 2 is determined. The Jacobian matrix of the system of differential equations equals: k k out,g k out,gi I(t) in,g R(t) k out,gi G(t) k in,ga(t) R(t) 2 J = k in,a G(t) k 2 out,a 0 0 k in,i 0 k out,i 0. k in,r 0 0 k out,r Using Matlab (see section 8.7), the eigenvalues of the Jacobian in K 2 are determined. eigenvalues are equal to: The λ 1 = , λ 2 = , λ 3 = 0.114, λ 4 =

20 Because the real parts of the eigenvalues are negative, this critical point is stable, as follows from the theory for linear systems, see [5, pg. 386]. In the next paragraph the critical point K 2 is plotted in the graphs. The critical points and their stability are determined based on values of parameters as described in the original article. Now, the general case is examined. For that we will analyze equation (12). We will show that this equation has one positive root. From that result we can conclude that there is always one critical point, which has only positive real components. Setting f(g) = k in,gk in,a k out,r k out,a k in,r G 2 k out,g G k out,gik in,i G 2 k out,i, we determine df dg : df dg = 2k in,gk in,a k out,r k out,a k in,r G 3 k out,g 2 k out,gik in,i G. k out,i Note that all parameters are positive per definition. Thus, for G > 0 df dg < 0 holds. That means f is always decreasing. Furthermore, k in,g k in,a k out,r lim f(g) = lim G 0 G 0 k out,a k in,r G 2 k out,g G k out,gik in,i G 2 =, k out,i k in,g k in,a k out,r lim f(g) = lim G G k out,a k in,r G 2 k out,g G k out,gik in,i G 2 =. k out,i Because f is continuous, it can be concluded that f has one positive root. That means that there is one critical point, of which the first component is real and positive. If the first component is real and positive, then from equation (11) it follows that the last component is also real and positive. Note that from equations (9) and (10) it can be concluded that the second and third component of critical points are always real and positive. So, there exists a critical point, which has only positive real components. The stability of this critical point is very hard to determine analytically Critical points in time period 2 In this section, we study the fixed points of the system in time period 2. In time period 2 Q i (t) equals Q max,i and S i (t) equals 0. Then, the system of differential equations reads: dg(t) dt da(t) dt di(t) = k in,g A(t) R(t) k out,gg(t) k out,gi G(t)I(t) + Q max,g, (13) = k out,a A(t) + Q max,a, (14) = k out,i I(t) + Q max,i, (15) dt dr(t) = k in,r G(t) k out,r R(t). (16) dt 20

21 Two of these differential equations can be solved analytically. Lemma 4.1. In time period 2, A(t) is equal to: A(t) = Q max,a k out,a + c 2 e k out,at, where c 2 R can be determined from the initial conditions. Proof. We start with equation (14) and use separation of variables. Integrating, we get In conclusion, A(t) equals: A(t) = Q max,a k out,a 1 da(t) = 1, Q max,a k out,a A(t) dt log(q max,a k out,a A(t)) = t + c 1, c 1 R, k out,a Q max,a k out,a A(t) = e k out,at k out,a c 1, A(t) = e k out,at k out,a c 1 Q max,a k out,a. + c 2 e k out,at, where c 2 R can be determined from the initial conditions. Lemma 4.2. In time period 2, I(t) is equal to: I(t) = Q max,i k out,i + d 2 e k out,i t, where d 2 R can be determined from the initial conditions. Proof. This proof is similar to the proof of Lemma 4.1. The critical points can be determined by setting dg(t) dt in the following equations: = da(t) dt = di(t) dt = dr(t) dt = 0. That results 0 = k in,g A R k out,gg k out,gi GI + Q max,g, (17) 0 = k out,a A + Q max,a A = Q max,a k out,a, (18) 0 = k out,i I + Q max,i I = Q max,i k out,i, (19) 0 = k in,r G k out,r R R = k in,rg k out,r. (20) Substituting these expressions for I, A and R in equation (17), gives us: k in,g Q max,a k out,r k out,a k in,r G Multiplication by G, gives us: k out,g G Q max,ik out,gi G k out,i + Q max,g = 0. (k out,g + Q max,ik out,gi )G 2 + Q max,g G + k in,gq max,a k out,r = 0. k out,i k out,a k in,r 21

22 Setting a = k out,g + Q max,ik out,gi k out,i b = Q max,g c = k in,gq max,a k out,r k out,a k in,r, this results in the equation: Solutions of this equation are: G 1 = b b 2 + 4ac 2a ag 2 + bg + c = 0. and G 2 = b + b 2 + 4ac. 2a When we choose the values of the parameters as in the original article [18], we find that: So, a and b = and c G and G Using relations (18), (19), (20) then leads to two critical points: K 1 = ( 5.99, , 16.0, ) and K 2 = (6.04, , 16.0, ). Note that K 1 is not interesting for this model, because some components are negative. The components of K 2 are positive, so this point is interesting for the model. Next, the stability of K 2 is determined. The Jacobian matrix of the system of differential equations equals: k k out,g k out,gi I(t) in,g R(t) k out,gi G(t) k in,ga(t) R(t) 2 J = 0 k out,a k out,i 0. k in,r 0 0 k out,r Using Matlab (see section 8.8), the eigenvalues of the Jacobian in K 2 are determined. eigenvalues are equal to: λ 1 = , λ 2 = , λ 3 = 0.114, λ 4 = Because the real parts of the eigenvalues are negative, this critical point is stable, as follows from the theory for linear systems, see [5, pg. 386]. The In Figures 11, 12, 13 and 14 the critical points are plotted in the graphs describing the amount of glucose, glucagon, insulin and the internalization of glucagon receptors respectively. In Figures 12 and 13 also the analytic solutions, as determined in Lemma 4.1 and 4.2, are plotted in time period 2. 22

23 Figure 11: Height of the critical points in the time periods 1 and 3 (blue) and in time period 2 (green) in the graph describing the amount of glucose. Figure 12: Height of the critical points in the time periods 1 and 3 (blue) and in time period 2 (green) in the graph describing the amount of glucagon. The analytic solution is equal to A(t) = e 0.114t and is plotted in time period 2 (black). Figure 13: Height of the critical points in the time periods 1 and 3 (blue) and in time period 2 (green) in the graph describing the amount of insulin. The analytic solution is equal to I(t) = e 0.496t and is plotted in time period 2 (black). 23

24 Figure 14: Height of the critical points in the time periods 1 and 3 (blue) and in time period 2 (green) in the graph describing the glucagon receptor internalization. We can see that some solutions become stationary in a short time (see Figure 12 and 13), whereas for other solutions it takes a long time before they become stationary (see Figure 11 and 14). A way to exploit this property is to consider the system on different timescales. That is done in section 4.3. In Figures 12 and 13, we can see that the analytic solution is exactly the same as the numerical simulation. So far, we determined the critical points and their stability based on values of parameters as given in Table 1. Now, the general case is considered. As mentioned above, G 1 = b b 2 + 4ac 2a and G 2 = b + b 2 + 4ac, 2a a = k out,g + Q max,ik out,gi k out,i b = Q max,g c = k in,gq max,a k out,r k out,a k in,r. Note that all parameters are positive per definition. Thus, a, b, c > 0. From this, the following can be concluded: G 1 = b b 2 + 4ac 2a < b b 2 2a = 0, G 2 = b + b 2 + 4ac 2a Moreover, G 1 and G 2 are real-valued, because b 2 +4ac > 0. So, in time period 2 there are always 2 real-valued critical points. For one of them G is positive and for the other one G is negative. When G is positive, A, I and R are positive as well. So there exists a critical point, which has only positive real components. The stability of this critical point is determined. The eigenvalues of the Jacobian of the system > 0. 24

25 are equal to: Here, W is equal to: λ 1 = k out,a, λ 2 = k out,i, λ 3 = 1 2R 2 (Z + R 2 W ), λ 4 = 1 2R 2 ( Z + R 2 W ). And Z is equal to: W = k 2 out,gr 2 + 2k out,g k out,gi IR 2 2k out,g k out,r R 2 + k 2 out,gii 2 R 2 2k out,gi k out,r IR 2 4k in,g k in,r A + k 2 out,rr 2. Z = (k out,g + k out,r + k out,gi I)R 2. Because all variables and parameters are positive, λ 1 < 0 and λ 2 < 0 hold. Furthermore, Z > 0 holds. Lemma 4.3. There are two cases for λ 3 : 1. When W 0, λ 3 is negative and real-valued. 2. When W < 0, λ 3 is complex and has a negative real part. Proof. Consider the case that W 0. In that case R 2 W is a real number and R 2 W 0 holds. Because Z is a positive real number, it follows that λ 3 is negative and real-valued. Consider the case that W < 0. In that case R 2 W is a positive complex number and has no real part. Because Z is a positive real number, it follows that λ 3 is complex and has a negative real part. Lemma 4.4. There are two cases for λ 4 : 1. When W 0, λ 4 is negative and real-valued. 2. When W < 0, λ 4 is complex and has a negative real part. Proof. Consider the case that W > 0. In that case: R2 W = R W, = R kout,g 2 R2 + 2k out,g k out,gi IR 2 2k out,g k out,r R 2 + kout,gi 2 I2 R 2 2k out,gi k out,r IR 2 4k in,g k in,r A + kout,r 2 R2, < R kout,g 2 R2 + 2k out,g k out,gi IR 2 2k out,g k out,r R 2 + kout,gi 2 I2 R 2 2k out,gi k out,r IR 2 + kout,r 2 R2, = R R 2 (k out,g + k out,gi I k out,r ) 2, = R 2 k out,g + k out,gi I k out,r, < (k out,g + k out,r + k out,gi I)R 2, = Z. 25

26 So, R 2 W < Z, and hence, it follows that λ 4 is negative and real-valued. If W equals 0, R 2 W is equal to 0. It follows that λ 4 is negative and real-valued. Consider the case that W < 0. In that case R 2 W is a positive complex number and has no real part. Because Z is a positive real number, it follows that λ 4 is complex and has a negative real part. Corollary Every positive critical point in time period 2 is stable. Proof. This follows directly from Lemma 4.3, Lemma 4.4 and from the theory for linear systems, see [5, pg. 386]. As follows from Lemma 4.3 and Lemma 4.4, λ 3 and λ 4 are complex when W < 0. That leads to the following Lemma: Lemma 4.5. λ 3 and λ 4 are complex, when the following relation holds: ( k in,rg ) 2 Q max,i (k out,g + k out,gi k out,r ) 2 Q max,a < 4k in,g k in,r. k out,r k out,i k out,a Proof. As follows from Lemma 4.3 and Lemma 4.4, λ 3 and λ 4 are complex when W < 0. When W < 0, the following equations hold: k 2 out,gr 2 + 2k out,g k out,gi IR 2 + k 2 out,gii 2 R 2 + (21) k 2 out,rr 2 2k out,gi k out,r IR 2 2k out,g k out,r R 2 < 4k in,g k in,r A, (22) R 2 (k out,g + k out,gi I k out,r ) 2 < 4k in,g k in,r A. (23) Note that in the equation above R, I and A are components of a critical point. So, substituting expressions (18), (19) and (20) in (23), we can conclude that λ 3 and λ 4 are complex, when the following relation holds: ( k in,rg ) 2 Q max,i (k out,g + k out,gi k out,r ) 2 Q max,a < 4k in,g k in,r. k out,r k out,i k out,a When using the values of the parameters that were used in the original article [18], we find that the right-hand side is equal to and the left-hand side is equal to Note that the values differ with a factor 5.6. That means that when the parameters are changed slightly, the eigenvalues can become complex. When two eigenvalues are complex, only the graph describing the amount of glucose shows special behavior. In Figure 15 the graph describing the amount of glucose is shown, when the parameters k out,g and k out,r are changed in the values 1.24 and 1.78 respectively. In this case, two eigenvalues are complex. In Figure 15, we can see that the graph oscillates slightly, because of the complex eigenvalues. However, the oscillation is small, so in this specific case complex eigenvalues do not have a great impact on the model. There are good reasons to assume that even in the general case complex eigenvalues do not have a great influence on the system. As follows from Lemma 4.5, one of the fastest ways to get complex eigenvalues is to increase the Q value of k out,r and to take care that k out,g + k max,i out,gi k out,i k out,r stays small. In the specific case above, k out,r is therefore multiplied by 3000 and k out,g is multiplied by 975. Multiplying 26

27 Figure 15: Amount of glucose in blood when k out,g and k out,r are changed in the values 1.24 and 1.78 respectively. Now, two eigenvalues are complex. these parameters by such high factors caused relatively high imaginary parts. We have seen that even in that specific case the oscillation is small, so we can assume that complex eigenvalues do not have a great influence on the system at all. 4.2 Nondimensionalization of the model In this paragraph the model is nondimensionalized. Nondimensionalization means that the problem is transformed into dimensionless form. This is a way to determine whether a term in a differential equation is large or small in comparison with other terms in the differential equation. It is important to know how all the terms in the problem compare and for this we need the concept of scaling. The variables are scaled by using characteristic values. The first step in nondimensionalizing a problem is to introduce a change of variables. So, in our case: t = t c s, G = g c u 1, A = a c u 2, I = i c u 3, R = r c u 4. In the above expression, g c is a constant that has the dimension of mass. It is called a characteristic value of the variable G. It is going to be determined based on the graph of G. In a similar way, the following holds: Variable dimension characteristic value dimension characteristic value t time t c time G mass g c mass A mass a c mass I mass i c mass R dimensionless r c dimensionless 27