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ultradian oscillations of insulin and models for diagnostic tests. 1 Introduction. Diabetes mellitus is a disease of the glucose-insulin regulatory system (see for ...
Delay differential equation models in diabetes modeling: a review Athena Makroglou + and Iordanis Karaoustas Department of Mathematics University of Portsmouth 1st Floor Lion Gate Bldg, Portsmouth PO1 3HF, UK Jiaxu Li ∗ Department of Mathematics University of Louisville Louisville KY 40292, USA Yang Kuang † Department of Mathematics and Statistics Arizona State University Tempe AZ 85287-1804, USA.

Abstract Delay differential equation models can generate rich dynamics using minimum number of parameters. This characteristic enables such models to play important roles in a growing number of areas of diabetes studies. Such areas include the insulin/glucose regulatory system, intravenous glucose tolerance test (IVGTT), and insulin therapies. In this paper, an extensive review of such models is presented together with some computational results and brief summaries of theoretical results for the cases of models for ultradian oscillations of insulin and models for diagnostic tests.

1

Introduction

Diabetes mellitus is a disease of the glucose-insulin regulatory system (see for example Fig. 1.1 in [13] for a picture of the plasma glucose-insulin interaction loops). It is classified into two main categories. Type 1 diabetes which is juvenile onset and insulin-dependent and Type 2 diabetes which is adult onset and insulin-independent. Complications of the disease include retinopathy, nephropathy, peripheral neuropathy, blindness The disease is affecting hundreds millions of people worldwide (type 2 diabetes mellitus had an estimated incidence of 151 million in the year 2000, which has motivated many researchers to study the mathematical, computational and medical problems associated with it. Articles about the prevalence and the problems of diabetes appear frequently in various media outlets. + [email protected] [email protected]. Work is partially supported by NIH/NIDCR Grant R01-DE019243 and NIH/NIEHS Grant P30ES014443. † [email protected]. Work is partially supported by DMS-0436341 and DMS/NIGMS-0342388. ∗

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Type 1 diabetes is considered to be the result of an immunological destruction of the insulin-producing β-cells. According to Lupi and Del Prato (2008) ([12]), p. 560, the normal pancreas contains approximately 1 million islets of Langerhans and each islet includes β-cells (60 − 80%), α-cells (20 − 30%), somatostatin (δ-cells) (5 − 15%) and pancreatic polypeptide (PP-cells). Type 2 diabetes is the result of insulin resistance. The term insulin resistance usually connotes resistance to the effects of insulin on glucose uptake, metabolism, or storage, due to excessive hepatic glucose production and defective β-cell function (cf. Lupi and Del Prato (2008) ([12]), p. 556). For more information about the pathogenesis of diabetes we refer for example to Ja¨ıdane and Hober (2008) ([8]) (type 1 diabetes), and to Lupi and Del Prato (2008) ([12]) (type 2 diabetes). Treatment of type 1 diabetes is based on the administration of insulin of various types in a number of ways. Insulin was discovered in 1921 by Bantig, Best, Collip and Macleod. Such insulin administration ways include subcutaneous injections and use of pumps. Cure of type 1 diabetes involves pancreas transplantation and islet transplantation. Many mathematical models have been developed for studying problems related to diabetes. These include Ordinary Differential Equations (ODEs), Delay Differential Equations (DDEs), Partial Differential Equations (PDEs), Fredholm Integral Equations (FIEs) ( in the estimation of parameters problem), Stochastic Differential Equations (SDEs) and IntegroDifferential Equations (IDEs). We refer for example to the review papers [13], [17], for more details about several such models and corresponding bibliography. For information about numerical methods for solving delay differential equations we refer for example to Bellen and Zennaro (2003) ([1]), see also the web page http://www.scholarpedia.org/article/Delay-differential_equations. Recently, several papers have appeared in the literature which show renewed interest in the models of insulin secretion introduced by G. H.Grodsky and his co-workers in the late 1960s, 1970s and 1980s. Grodsky introduced the so called threshold hypothesis for the pancreatic granules according to which each granule secrets its insulin contents if glucose is above a certain threshold level. Such recent papers (revisiting, modifying, extending this work but using ODEs mainly) include: Pedersen, Corradin, Toffolo, Cobelli (2008) ([18]), (the paper describes also the current state of the art accompanied by rich bibliographical information. Their model includes the notion of distinct pools of granules as well as various mechanisms, like priming, exocytosis etc, and it is claimed to be the first physiology-based one to reproduce the staircase experiment, which underlies derivative control, that is the pancreatic capacity of measuring the rate of change of the glucose concentration). Mari and Ferrannini (2008) ([14]) (the paper includes an interesting historical background presentation too). In this paper a review of some mathematical models in the form of delay differential equations is given, accompanied by some computational results using Matlab and elements of their theoretical analysis. The organisation of the paper is as follows: Section 2 contains the description of the models and some computational results and brief summaries of theoretical results. Two cases are covered here, the case of models for ultradian oscillations of insulin (section 2.1) and the case of models used in diagnostic tests (section 2.2). Concluding remarks are in section 3. The notation is kept as in the original papers for easy reference. The Matlab function DDE23 was used for obtaining graphs of models in the form of DDE systems, see for example the tutorial http: 2

//www.runet.edu/~thompson/webddes/tutorial.html for help with its use.

2

Models in the form of delay differential equations

Delay differential equations (DDEs) have been used as mathematical models in many areas of Biology and Medicine. Such areas include Epidemiology, Population Biology, Immunology, Physiology, Cell mobility. Delayed effects often exist in the glucose-insulin regulatory system, for example, the insulin secretion stimulated by elevated glucose concentration level, hepatic glucose production ([11], [19]). Therefore the delays need to be taken into account when modeling the systems. General approaches include the technique of compartment-split by introducing of auxiliary variables in ordinary differential equations (ODE) ([2], [19]), and modeling in delay differential equations (DDE) by using explicit time delays in either discrete or distributed forms (cf. [11], [10], [15], [16]). The delays in the compartment-split approach as are classsified as “soft delays” by using γ kernel that is an approximation of the Dirac kernel, while the explicit delays in models as “hard delays”. Apparently, modeling by explicit delays is more natural and accurate, although the analysis is usually harder ([5], [10], [15]). Models in the form of delay differential equations grouped according to their functions/purposes include: • Models used to analyze the ultradian insulin secretion oscillations, • Models used with diagnostic tests, • Models related to insulin therapies, • Models taking intracellular activity of β-cells into account. Due to limitations with respect to the number of references and the number of pages, models of the first and 2nd category are going to be presented here.

2.1

Ultradian insulin secretion related models

Insulin is released in a biphasic manner when the glucose concentration is raised from subthreshold to stimulatory levels, with a rapid peak at 2-4 min (first phase), a decrease lasting 10-15 min (pulsatile insulin secretion) followed by a gradual increase within the next couple of hours (50-120 minutes), cf. Chew et al (2009) ([4]), (second phase, ultradian insulin secretion). As mentioned in Chew et al (2009) ([4]), ultradian oscillations have been seen after meal ingestion during continuous enteral nutrition, and during intravenous glucose infusion. Historically, it was in 1923 when rapid and slower oscillations in the peripheral concentrations of glucose were reported by Karen Hansen and half a century later rapid oscillations in the peripheral insulin concentrations were demonstrated. As several authors mention, the precise mechanisms generating ultradian oscillations are not fully understood yet and the two most common mechanisms mentioned are: • instability of the glucose-insulin feedback loop, where the insulin oscillations entrain these of the glucose • existence of an intrapancreatic pacemaker. 3

Rntrainment means the ability of a self-oscillating system when perturbed exogenously with a periodic stimulus, to adjust its period of oscillation to that of the stimulus. The entrainment ability seems to have been lost in diabetic patients with type 2 diabetes. Two time delays exist in the glucose-insulin regulatory system, (cf. [19]). To model the ultradian insulin secretion oscillations in the regulation with time lags, Sturis et al (1991) ([19]) proposed a model in ODE utilizing the compartment-split technique. This model was later simplified by Toli´c et al (2000) ([20]). Several models based on this model were proposed consequently. Examples of such models are (see also Makroglou et al (2006) ([13])), Drozdov and Khanina (1995) ([6]), Li et al (2006) ([11]), Li and Kuang (2007) ([10]). The model in [19] and models in papers presenting extensions of it, like the models in [11], [20], make use of certain functions (f1 − f5 ) given below: f1 (G) = Rm /(1 + exp((C1 − G/Vg )/a1 )),

(2.1)

f2 (G) = Ub (1 − exp(−G/(C2 Vg ))),

(2.2)

f3 (G) = G/(C3 Vg ),

(2.3)

f4 (I) = U0 + (Um − U0 )/(1 + exp(−β ln

I(1/Vi + 1/(Eti )) )), C4

f5 (I) = Rg /(1 + exp(ˆ α(I/Vp − C5 ))).

(2.4) (2.5)

f1 (G): insulin production stimulated by glucose production, f2 (G): insulin-independent glucose utilization, f3 (G)f4 (I): insulin-dependent glucose uptake (mostly due to fat and muscle cells), f5 (I): glucose production controlled by insulin concentration. The values of the parameters may be found for example in [20]. The functions f1 − f5 are assumed to satisfy certain general assumptions by [10]. Here we present the DDE models by Drozdov and Khanina (1995) ([6]), Li, Kuang and Mason (2006) ([11]) model, Chen and Tsai (2009) ([3]), plus the Sturis et al (1991) ([19]) and Toli´c et al (2000) ([20]) models that formed the basis of the DDE models. Some more such models may be found in Makroglou, Li and Kuang (2006) ([13]). 2.1.1

Compartment-split ODE model proposed by Sturis et al (1991) ([19])

Based on two negative feedback loops describing the effects of insulin on glucose utilization and production and the effect of glucose on insulin secretion, the authors Sturis, Polonsky, Mosekilde and Van Cauter (1991) ([19]), developed a six dimensional ODE model. Toli´c, Mosekilde and Sturis (2000)([20]) simplified this model a little bit. This model has been the basis of several DDE models. It has the following form (cf. [20], p. 363) dG (t) dt dIp (t) dt dIi (t) dt dx1 (t) dt

= Gin − f2 (G(t)) − f3 (G(t))f4 (Ii (t)) + f5 (x3 (t)), Ip (t) Ii (t) Ip (t) − )− , Vp Vi tp Ip (t) Ii (t) Ip (t) = E( − )− , Vp Vi ti 3 3 dx2 = (Ip (t) − x1 (t)), (t) = (x1 (t) − x2 (t)), td dt td = f1 (G(t)) − E(

(2.6) 3 dx3 (t) = (x2 (t) − x3 (t)), dt td

where G(t) is the mass of glucose, Ip (t), Ii (t) the mass of insulin in the plasma and the intercellular space, respectively, Vp is the plasma insulin distribution volume, Vi is the effective volume of the intercellular space, E is the diffusion transfer rate, tp , ti are insulin 4

degradation time constants in the plasma and intercellular space, respectively, Gin indicates (exogenous) glucose supply rate to plasma, and x1 (t), x2 (t), x3 (t) are three additional variables associated with certain delays of the insulin effect on the hepatic glucose production with total time td . f1 (G) is a function modeling the pancreatic insulin production as controlled by the glucose concentration, f2 , f3 , f4 are functions for glucose utilization by various body parts (brain and nerves (f2 ), muscle and fat cells (f3 , f4 )) and f5 is a function modeling hepatic glucose production). The forms of the functions f1 , . . . , f5 are given in [20]. For the two time delays, one is glucose triggered insulin production delay that is reflected by breaking the insulin in two separate compartments, and the other one is hepatic glucose production delay which is fulfilled by the three auxiliary variables, x1 , x2 and x3 . This model simulated ultradian insulin secretion oscillations numerically. For conclusions drawn from the simulations we refer to [19]. 2.1.2

Single-delay DDE model proposed by Drozdov and Khanina (1995) ([6])

A single-delay DDE model is introduced by Drozdov and Khanina ([6]) for the description of ultradian oscillations in human insulin secretion. The model equations are ([6], p.27)     dx 1 0.1z(t) E E (t) = f1 + − x(t) + y(t), dt V3 V1 T 1 V2   dy E E 1 (t) = x(t) − + y(t), (2.7) dt V1 V2 T2     dz 0.1z(t) x(t − T ) y(t) (t) = f3 − f2 + (L − p0 ), dt V1 V3 V2 where x(t)(mU ) is the amount of insulin in the plasma, y(t)(mU ) is the amount of insulin in the interstitial fluid and z(t)(mg) is the amount of glucose treated as occupying one compartment; V1 (l), V2 (l), V3 (l) are the volumes of the plasma, interstitial fluid and the glucose compartment respectively, with values V1 = 3l, V2 = 11l, V3 = 10l, and T1 = 3 min, T2 = 100 min, are given parameters, L(mg/min) is the rate of glucose delivery from the environment (L = 100(mg/min), [6], p.28, corresponds to the normal delivery of glucose in 150g/day), p0 = 72 is a constant, E = 0.2/min, T is the delay in glucose production. The form of the functions f1 − f3 is (note that there is a small typo in the paper’s f2 formula in equation (9) which was easy to recover from the preceding calculations) f1 (cz ) = f2 (cy ) = f3 (cx ) =

210 , 1 + exp(a + bcz )

a = 5.21,

b = −0.03,

9 + 0.4, 1 + exp(7.76 − 1.772 ln(cy (1 + V2 /(ET2 )))) 160 , 1 + exp(0.29cx − 7.5)

(2.8)

where cx is the plasma concentration of insulin, cx = Vx1 (µU/ml), cy is the remote compartment insulin concentration, cy = Vy2 (µU/ml), and cz is the glucose concentration cz = 0.1z V3 (mg/dl). The authors mention that the form of the functions f1 − f3 is similar to that proposed in [19], but with different parameter values which they obtained by least squares fitting to published data. Initial conditions used in the numerical simulations are, [6], p. 28, cx (θ) = 30, −T ≤ θ ≤ 0, cy (0) = 20, cz (0) = 120. 5

Numerical results were obtained for a number of L and T values. Stability analysis is also presented in the paper for a linearized system of DDEs. The claim is ([6], p. 31) that for very small and very large L values the steady state solution is stable and ultradian oscillations do not arise, but for moderate L values, the steady state solutions become unstable and periodic oscillations of insulin and glucose occur. Figures 2.1, 2.2 for plasma insulin and glucose concentrations correspond to Fig. 6 of the paper. Figure 2.1: Plasma insulin concentration in µU/ml, Drozdov and Khanina (1995) DDE model L=120 mg/min, delay T=25, Drozdov and Khanina (1995) DDE model cx=x(t)/V1: Plasma insulin concentration in µ U /ml

32 30 28 26 24 22 20 18 16

0

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400 500 time t in min

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Figure 2.2: Glucose concentration in mg/dl, Drozdov and Khanina (1995) DDE model L=120 mg/min, delay T=25, Drozdov and Khanina (1995) DDE model cz=0.1z(t)/V3: glucose concentration in mg /dl

120

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400 500 time t in min

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2.1.3

Two-delay model proposed by Li, Kuang, Mason (2006) ([11])

The authors consider two time delays; one (τ1 ) to denote the total time delay from the time that the glucose concentration level is elevated to the moment that the insulin has been transported to the interstitial space and becomes ‘remote insulin’ ([11], p. 726), the second one (τ2 ) has to do with the delay of the effect of hepatic glucose production measured from the time that insulin has become ‘remote insulin’ to the moment that a significant change of hepatic glucose production occurs ([11], p. 726). The model is formulated by the observation of the Law of Conservation. The model is given by ([11], p. 726) dG (t) = Gin − f2 (G(t)) − f3 (G(t))f4 (I(t)) + f5 (I(t − τ2 )), dt (2.9) dI (t) = f1 (G(t − τ1 )) − di I(t), dt I(0) = I0 > 0, G(0) = G0 > 0, G(t) = G0 , t ∈ −[τ1 , 0], I(t) = I0 , t ∈ [−τ2 , 0], τ1 , τ2 > 0. The form of the functions f1 − f5 is given by (2.1)-(2.5). Gin ≥ 0 is the glucose infusion rate (from meals, oral glucose intake, constant glucose infusion etc), di > 0 is the insulin clearance rate, [11], p. 724. Numerical simulations including bifurcation analysis are given in the paper and comparisons are made to some existing models, namely these by [19], and two (new) variations of the [19] and [20] models. See also [10] for analytical studies of the model. Based on the bifurcation analysis, the authors suspect that the total time delay τ1 is critically responsible for the oscillation. The total time delay is measured from the moment that the glucose concentration level starts to increase to the moment that the insulin has been transported to the interstitial space. The following graphs (2.3, 2.4) use parameter values from Table 1, p. 727 of [11] and τ1 = 6, τ2 = 36, Gin = 1.35, di = 0.06 taken from Fig. 4 (right) of the same paper.

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Figure 2.3: Glucose Concentration, τ1 = 6, τ2 = 36, Gin = 1.35, di = 0.06, Li, Kuang, Mason (2006) DDE model τ 1=6, τ 2=36, Gin=1.35, di=0.06, Li, Kuang, Mason (2006) DDE model

y1(t)=G(t)− Glucose concentration in mg /dL

110 100 90 80 70 60 50 40 30

0

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1500

time t in min

Figure 2.4: Insulin Concentration, τ1 = 6, τ2 = 36, Gin = 1.35, di = 0.06, Li, Kuang, Mason (2006) DDE model τ 1=6, τ 2=36, Gin=1.35, di=0.06, Li, Kuang, Mason (2006) DDE model

y2(t)=I(t)−Insulin concentration in µU /mL

45 40 35 30 25 20 15 10 5

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1000 time t in min

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2.1.4

The two-delay model proposed by Chen and Tsai (2009) ([3])

Chen and Tsai (2009) ([3]) propose a modification of the two-delay model proposed by Li, Kuang and Mason (2006) ([11]), Li and Kuang (2007) ([10]), with respect to the following: • They use a variable glucose infusion function Gin (t) instead of the constant Gin used in [11] throughout the simulation time, so that external inputs like food uptake can be simulated too. • They introduce two additional functions f6 and f7 to ‘provide the effects of hyperglycemia’ ([3], 2nd page), with the rest of the functions f1 − f5 the same as in [11] (given here by 2.1-2.5). • They introduce two additional parameters α, β for the purpose of ‘estimating the condition of major dysfunction of diabetes’ ([3], 3rd page), (α for insulin release from the pancreas, β for the ability of insulin-dependent glucose utilization (IDGU)-small β value indicates increasing severity of insulin resistance). • They use f7 (G(t) − 330) for describing the kidney glucose excretion (KGE) rate above the urine threshold (330 mg/dl) ([3], 3rd page). • They perform least squares estimation of the parameters G0 , I0 , θ1 , θ2 , α, β, di , tm , m ∈ M , where m = 1, 2, 3 and tm are CHO (infused carbohydrate) times ([3], 7th page). The model equations are ([3], 2nd page) dG (t) = [Gin (t) + f5 (I(t − θ2 ))f6 (G(t))] − [f2 (G(t)) + f7 (G(t) − 330) + βf3 (G(t))f4 (I(t))] dt dI (t) = αf1 (G(t − θ1 )) − di I(t), dt where Gin (t) =

X

Gm (t − tm )u(t − tm ),

m∈M

where G(t), I(t) denote glucose and insulin concentrations respectively, Gm (t − tm ), m ∈ M denotes the m−th exogenous food uptake at tm , u(t − tm ) is a unit step function that has the unity value for t ≥ tm . Gin (t) is the overall effective exogenous food uptake. The form of Gm (t) is (paper, 3rd page) Gm (t; k, b) =

kt −t2 /(2b2 ) e . b2

k and b values are given in Table 3, 3rd page of [3]. The form of the functions f6 , f7 is ([3], Table 1, 2nd page) 1 1 + exp[γ((G(t)/(C3 Vg )) − C6 )] S c − Sb f7 (G(t) − 330) = Sb + . 1 + exp[δ(((G(t) − 330)/(C3 Vg )) − C7 )] f6 (G(t)) =

The units and values of all parameters appearing in the form of the functions f1 − f7 are given in Table 2, 3rd page in [3]. Values of the estimated parameters are given in Table 5, 9th page in [3]. Many computational results (graphs) are given for data and simulations corresponding to normal, type 1 and type 2 diabetic subjects. Here we present one graph obtained with b = 80, k = 4300 and parameters for type 1 diabetes and it corresponds to Fig. 9(a) of [3]. 9

Gin(t) (mg/min)

y1(t)=G(t): Glucose concentration (mg/dl)

Figure 2.5: Glucose Concentration, θ1 = 16.2, θ2 = 35.1, Chen and Tsai (2009) DDE model θ 1=16.2, θ 2=35.1, b=80, k=4300, Chen and Tsai (2009) DDE model

160 140 G(t) Gin(t)

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2.2

Models used with diagnostic tests

A number of diagnostic tests have been developed to assess two indices important in metabolic research known as insulin sensitivity and glucose effectiveness and also the βcell function. Such diagnostic tests include: the OGTT (oral glucose tolerance test), the IVGTT (intravenous glucose tolerance test), tests based on the use of a clamp (the Hyperglycaemic and the Euglycaemic Hyperinsulinaemic ones) and variations of them. Insulin sensitivity is defined as the ability of insulin to enhance glucose effectiveness and glucose effectiveness, as the ability of glucose to promote its own disposal If one measures glucose disposal by its rate of disappearance from the accessible pool Rd (t), then glucose ∂ 2 Rd (t) d (t) effectiveness = ∂R ∂G(t) |SS and insulin sensitivity = − ∂G(t)∂I(t) |SS , where G(t) denotes glucose concentration and I(t) insulin concentration and SS denotes the basal steady state. The OGTT involves ingestion of 75g of glucose and venous blood sampling. Glucose and insulin are measured at time zero, 30, 60, 90 and 120 minutes. Also C-peptide must be measured to calculate insulin secretion either by deconvolution or mathematical modeling. The hyperglycaemic clamp starts with a glucose bolus to raise the glucose level abruptly to a hyperglycaemic target value and then glucose is infused to maintain the glucose concentration at that value. Hyperglycaemia stimulates biphasic insulin secretion and glucose disposal to tissues is increased. After 2-3 hours, a steady state is reached in which insulin levels vary according to the subject’s β-cell response. The procedure for the Hyperinsulinaemic Euglycaemic Glucose clamp can be described as follows: After an overnight fast, insulin is infused intravenously at a constant rate that may range from 5 to 120mU m−2 min−1 (dose per body surface area per minute). This constant insulin infusion rate leads to a new steady state insulin level above the fasting level (hyperinsulinaemic). As a consequence, glucose disposal in skeleton muscle and adipose tissue is increased, and HGP (hepatic glucose production) is suppressed. Under these conditions, a bedside glucose analyzer is used to frequently monitor blood glucose levels at 5-10 minute intervals, while 20% dextrose is given intravenously at a variable rate to ‘clamp’ blood glucose concentration in the normal range (Euglycaemic). After several hours of constant insulin infusion, steady state conditions can be achieved for plasma insulin, blood 10

glucose and the glucose infusion rate (GIR). The intravenous glucose tolerance test (IVGTT) involves the intravenous administration of a bolus of glucose and the frequent sampling of glucose and insulin concentrations. Two noticeable differences in modeling IVGTT from the glucose-insulin regulation are that 1) the large bolus intravenous glucose infusion causes the time delay of the hepatic glucose production insignificant and thus negligible; and 2) the bi-phasic insulin secretion caused by the quick and direct stimulation of large bolus of glucose infusion in plasma. These distinguish the modeling rationale from the ultradian oscillation. In this case, only one time delay should be considered. There are several models associated with it (cf. the ODEs of the minimal model in Bergman, Ider, Bowden and Cobelli (1979) ([2]), DDEs (cf. Panunzi, Palumbo, and De Gaetano (2007) ([16]), Giang, Lenbury, De Gaetano, Palumbo (2008) ( [7]), and the integrodifferential equation models (cf. Palumbo, Panunzi, De Gaetano (2007) ([15])). While the minimal model has its features widely used in research, it has several drawbacks in mathematics that were pointed out by De Gaetano and Arino (2000) ([5]). The parameter fitting is to be divided into two separate parts: first, using the recorded insulin concentration as given input data in order to derive the parameters in the first two equations in the model, then using the recorded glucose concentration as given input to derive the parameters in the third equation. However, the system is an integrated physiological dynamic system and one should treat it as a whole system and be able to conduct a singlestep parameter fitting process. Secondly, some of the mathematical results produced by this model are not realistic. Specifically, it can be shown that the minimal model does not admit an equilibrium and the solutions may not be bounded. Finally, the non-observable auxiliary variable X(t) is artificially introduced to delay the action of insulin on glucose. An alternative and natural approach is to introduce explicitly the time delay in the model. To address these issues, De Gaetano and Arino proposed a so called dynamic model ([5]) with explicit delay in distributed form. Recently, Panunzi et al ([16]) built a discrete delay differential equation model to further study the short but complicated phenomenon. Although not able to confirm the conclusion analytically, the authors produced reasonable simulation profile with experimental raw data statistically ([15], [16]). A short introduction to integro-differential equation models used for modeling glucoseinsulin dynamics in IVGTT may be found in Makroglou, Li and Kuang (2006) ([13]). In this section we give the equations of one delay model associated with IVGTT, namely the Panunzi, Palumbo and De Gaetano (2007) ([16]) and the Giang, Lenbury, DeGaetano, Palumbo (2008) ([7]) models (section 2.2.2), and the equations of one model in the form of integro-differential equations, namely the Palumbo, Panunzi and De Gaetano (2007) ([15]). We start (section 2.2.1) with the equations of the minimal model (Bergman, Ider, Bowden, Cobelli (1979) ([2]) which is the first IVGTT model.

2.2.1

The minimal model proposed by Bergman et al (1979) ([2])

The first IVGTT model is the minimal model and it has been widely utilized in many clinics and extended in various applications ([13]). The model is an ODE model formulated in split

11

compartment and is given by  dG(t)   = G0 = −[b1 + X(t)]G(t) + b1 Gb ,   dt      dX(t) = X 0 = −b2 X(t) + b3 [I(t) − Ib ],  dt         dI(t) = I 0 = b [G(t) − b ]+ t − b [I(t) − I ]. 4 5 6 b dt

(2.10)

with initial conditions G(0) = b0 , X(0) = 0, I(0) = b7 + Ib . Here G(t) [mg/dl], I(t) [µU/ml] is the plasma glucose, insulin concentration at time t [min], respectively. X(t) [min−1 ] stands for the auxiliary function representing insulin-excitable tissue glucose uptake activity that is assumed to be proportional to insulin concentration in a “distant” compartment. Gb [mg/dl], Ib [µU/ml] is the subject’s baseline glycemia, insulinemia, respectively. b0 [mg/dl] is theoretical glycemia at time 0 after the instantaneous glucose bolus intake. b1 [min−1 ] is the insulin-independent constant of tissue glucose uptake rate. b2 [min−1 ] is the rate constant describing the spontaneous decrease of tissue glucose uptake ability. b3 [min−2 (µU/ml)−1 ]is the insulin-dependent increase in tissue glucose uptake ability, per unit of insulin concentration excess over the baseline. b4 [(µU/ml)(mg/dl)−1 min−1 ] is the rate of pancreatic release of insulin after the intake of the glucose bolus, per minute per unit of glucose concentration above the “target” glycemia b5 [mg/dl]. b6 [µU/ml] is the first order decay rate for insulin in the plasma. b7 [µU/ml] is the plasma insulin concentration at time 0, above basal insulinemia, immediately after the glucose bolus intake. The minimal model contains minimal number of parameters ([2]) and it is widely used in physiological research work to estimate metabolic indices of glucose effectiveness (SG) and insulin sensitivity (SI) from the intravenous glucose tolerance test (IVGTT) data by sampling over certain periods (usually 180 minutes). 2.2.2

The model proposed by Panunzi, Palumbo and De Gaetano (2007) ([16]) and by Palumbo, Panunzi and De Gaetano (2007) ([15])

In [16] the authors present 4 two-compartment models for the plasma glucose and the plasma insulin concentrations following an IVGTT test, two without delay and two with delay on insulin action (τi ). The authors ([16]) identified a special case (the model A in [16]) as the best model under the Akaike Information Criterion (AIC), (‘without first order plasma glucose elimination (Kxg ) and without delay on insulin action’). The model is given as follows. Tgh dG (t) = −Kxgi I(t)G(t) + , dt Vg (2.11) 

G(t−τg )



G∗ Tigmax dI   . (t) = −Kxi I(t) + dt Vi 1 + G(t−τg ) γ G∗

where G(t) = Gb for t ∈ (−∞, 0), G(0) = Gb + G∆ , I(0) = Ib + I∆ G∆ with G∆ = Dg /Vg . γ is a positive constant, which allows for Michaelis-Menten (γ ≤ 1) or sigmoidal dynamics 12

(γ > 1). The parameters Tgh = Kxgi Ib Gb Vg ,

Tigmax

  γ   γ Gb Gb / , = Kxi Ib Vi 1 + ∗ G G∗

where G∗ and I ∗ are the steady state. The full description of the symbols and their units is given in Table 3 of [16]. The authors proved that the unique steady state (G∗ , I ∗ ) is globally asymptotically stable when γ ≤ 1 under the condition Kxgi Ib γ  γ ≤ Kxg + Kxgi Ib . Gb 1+ G ∗ regardless of the length of the time delay. However this case only accounts for 3% of the experimental data the authors obtained. For more general cases, the authors have shown that there exists a bifurcation point τ0 > 0 such that the steady state is locally asymptotically stable when the delay is smaller than τ0 . Apparently, delay dependent conditions for global stability are needed to improve the analysis. The authors also demonstrated that the insulin sensitivity index can be obtained in the same way as in the minimal model but it is more effective. The authors used raw data for parameter estimation (the GLS-Generalized Least Squares method described in the paper’s Appendix to obtain individual regression parameters and the WLS-Weighted Least Squares method for the Minimal Model). In a more recent paper (Giang, Lenbury, De Gaetano, Palumbo (2008) ([7]) the authors consider the most general of the 4 models of the family introduced in [16] (with first order plasma glucose elimination (Kxg ) and with delay in insulin action (τi )) and deal with theoretical results concerning global stability under certain conditions on the parameters and the effect of the delays τi , τg on the oscillatory behavior of the solutions when Kxg = 0. If Kxg = 0 and τi = 0 the DDE model of [7] reduces to that of [16], with parameters given in Remark 10 in [7]. The following graphs (2.6, 2.7) were produced using data from Remark 9 in [15] and initial values those of the IVGTT model (equation (8) of the same paper), with I∆ = 56.97 pM/mM . We note that since Kxg = 0 min−1 , and τi = 0, this constant (or discrete as the authors call it) delay DDE model, coincides with that of [16]. We can notice that graph 2.6 (insulin evolution) exhibits a derivative discontinuity at t = τg = 23.5 (note that there is a typo in the paper’s Figure 10 (second graph), as Figure 12 and private communications with one of the authors confirmed).

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Figure 2.6: Glucose evolution (mM), Panunzi et al (2007) DDE model τ i=0, τ g=23.50, γ=2.52, Panunzi et al (2007) DDE model 60 Dg=1.833 mmol/KgBW Dg=5 mmol/KgBW

y1(t): Glucose evolution in mM

50

Dg=10 mmol/KgBW

40

30

20

10

0

0

20

40

60

80 100 time t in min

120

140

160

180

Figure 2.7: Insulin evolution (pM), Panunzi et al (2007) DDE model τ i=0, τ g=23.50, γ=2.52, Panunzi et al (2007) DDE model

3000

y2(t): Insulin evolution in pM

2500

2000

1500

1000

500

0

0

20

40

60

80 100 time t in min

14

120

140

160

180

Palumbo, Panunzi and De Gaetano (2007) [15] considered a model that consists of a system of two coupled integro-differential equations where different choices of the convolution kernels result in a number of different models including DDEs with constant delay. The model equations are dG ˜ + Tgh , (t) = −Kxg G(t) − Kxgi G(t)I(t) dt Vg (2.12) Tigmax ˜ dI f (G(t)), (t) = −Kxi I(t) + dt Vi where ˜

G(t) G∗

˜ f (G(t)) = ˜ G(t) =

1+ Z τg





G(t) G∗

γ ,

wg (θ)G(t − θ)dθ, 0

˜ I(t) =

Z

τi

wi (θ)I(t − θ)dθ 0

and the kernels wg : [0, τg ] → R+ , wi : [0, τi ] → R+ are non-negative square integrable functions such that Z

τg

Z

τi

wi (θ)dθ = 1,

wg (θ)dθ = 1, 0

0

and there exist Tg , Ti such that Z

τg

θwg (θ)dθ ≤ Tg < ∞, 0 Z

τi

θwi (θ)dθ ≤ Ti < ∞. 0

For the description of variables and units we refer to the paper. Kernel choices: (I)

wg (t) = δ(t − τg ),

wi (t) = δ(t − τi ),

(2.13)

(II)

αg2 te−αg t ,

αi2 te−αi t

(2.14)

wg (t) =

wi (t) =

In case (I), the integro-differential system reduces to a DDE system of two equations with constant delays τg , τi , Tgh dG (t) = −Kxg G(t) − Kxgi G(t)I(t − τi ) + , dt Vg (2.15) Tigmax dI (t) = −Kxi I(t) + f (G(t − τg )). dt Vi 15

In case (II) with τg = τi = ∞, using the linear chain trick (cf. Kuang (1993) ([9])), the authors transform the integro-differential system into a system of six ODEs (verified). Theoretical results presented in the paper are: It is shown that the integro-differential system has positive solution and that it is persistent. Local stability is shown for the DDE system (section 4.1) and the ODE system (section 4.2). Global stability is shown for the IDE system (Theorem 6), depending upon a condition on parameter values (Theorem 4), Kxgi Ib γ  γ ≤ Kxg + Kxgi Ib . Gb 1+ G ∗ The authors mention (Remark 7) that the above condition is not verified for particular values making physiological sense. Many simulation results are also included in the paper. Figure 11 of the paper is produced with parameter values given in Remark 11 and it is reproduced below (figures 2.8, 2.9). The starting ODE values involving integrals were calculated exactly. The Matlab function ODE45 was used for solving the ODE system.

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Figure 2.8: Glucose evolution (mM), Palumbo, Panunzi, De Gaetano (2007) DDE model gamma=2.66, Palumbo, Panunzi, De Gaetano (2007) model 45 Dg=1.833 mmol/KgBW 40

Dg=5 mmol/KgBW Dg=10 mmol/KgBW

Glucose evolution (mM)

35 30 25 20 15 10 5 0

0

20

40

60

80 100 t (in min)

120

140

160

180

Figure 2.9: Insulin evolution (pM), Palumbo, Panunzi, De Gaetano (2007) DDE model gamma=2.66, Palumbo, Panunzi, De Gaetano (2007) model 4500 Dg=1.833 mmol/KgBW 4000

Dg=5 mmol/KgBW Dg=10 mmol/KgBW

Insulin evolution (pM)

3500 3000 2500 2000 1500 1000 500 0

0

20

40

60

80 100 t (in min)

17

120

140

160

180

3

Concluding remarks

The time delays exist in the glucose-insulin regulatory system and other interactions of the real world including life sciences and control theory. It is natural to include explicitly the delays in mathematical modeling. Although the delayed effects can be simulated by introducing the auxiliary states in ordinary differential equation systems, as we have seen earlier, it is an approximation by using different kernels and also the number of equations in the system is larger. Modeling by delay differential equations can not only more accurately simulate the real life problems, but also reduce the number of equations. Comparing to ODE models, it is more challenging to analyze DDE models. However, a handful of existing books documents the theory of functional differential equations and their applications in biology and engineering. Interested readers can refer to Bellen and Zennaro (2003) ([1]), and Kuang (1993) ([9]). Acknowledgement: We wish to thank the anonymous referees for their detailed and valuable comments that contributed greatly to the improvement of this review paper.

References [1] A. Bellen, and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford Science Publications, Clarendon Press, Oxford, 2003. [2] R. N. Bergman, Y. Z. Ider, C. R. Bowden, C. Cobelli, Quantitative estimation of insulin sensitivity, Am. J. Physiol., 236 (1979), E667–E677, or Endocrinol. Metab. Gastrointest. Physiol. 5 (1979), E667–E677. [3] C. -L. Chen, H.-W. Tsai, Modeling the physiological glucose-insulin system on normal and diabetic subjects, Computer Methods and Programs in Biomedicine, 2009, in press. [4] Y. H. Chew, Y. L. Shia, C. T. Lee, F. A. A. Majid, L. S. Chua, M. R. Sarmidi, R. A. Aziz, Modeling of oscillatory bursting activity of pancreatic beta-cells under regulated glucose stimulation, Molecular and Cellular Endocrinology, 307 (2009), 57–67. [5] A. De Gaetano, O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol., 40 (2000a), 136–168. [6] A. Drozdov, H. Khanina, A model for ultradian oscillations of insulin and glucose, Mathl. Comput. Modelling, 22 (1995), 23–38. [7] D.V. Giang, Y. Lenbury, A. De Gaetano, P. Palumbo, Delay model of glucose-insulin systems: global stability and oscillated solutions conditional on delays, J. Mathematical Analysis and Applications, 343 (2008), 996–1006. [8] H. Ja¨ıdane, D. Hober, Role of coxsackievirus B4 in the pathogenesis of type 1 diabetes, Diabetes and Metabolism, 34 (2008), 537–548. [9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Math. Sci. Eng. 191, Academic Press, Boston, 1993. [10] J. Li and Y. Kuang, Analysis of a model of the glucose-insulin regulatory system with two delays, SIAM J. Appl. Math., 67 (2007), 757–776. [11] J. Li, Y. Kuang and C. C. Mason, Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722–735.

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[12] R. Lupi and S. Del Prato, B-cell apoptosis in type 2 diabetes: quantitative and functional consequences, Diabetes and Metabolism, 34 (2008), 556–564. [13] A. Makroglou, J. Li, Y. Kuang, Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview, Applied Numerical Mathematics, 56 (2006), 559–573. [14] A. Mari, E. Ferrannini, β-cell function assessment from modelling of oral tests: an effective approach, Diabetes, Obesity and Metabolism, 10 (suppl. 4) (2008), 77–87. [15] P. Palumbo, S. Panunzi, and A. De Gaetano, Qualitative behavior of a family of delaydifferential models of the glucose-insulin system, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 399–424. [16] S. Panunzi, P. Palumbo and A. De Gaetano, A discrete single delay model for the intra-venous glucose tolerance test, Theoretical Biology and Medical Modelling, 2097, 4:35 (Open Access article available from http://www.tbiomed.com/content/4/1/35. [17] R. Pattaranit and H. A.van den Berg, Mathematical models of energy homeostasis, J. R. Soc. Interface (2008) 5, 1119-1135. [18] M. G. Pedersen, A. Corradin, G. M. Toffolo, C. Cobelli, A subcellular model of glucosestimulated pancreatic insulin secretion, Phil. Trans. Soc. A, 366 (2008), 3525–3543. [19] J. Sturis, K. S. Polonsky, E. Mosekilde, E. Van Cauter, Computer-model for mechanisms underlying ultradian oscillations of insulin and glucose, Am. J. of Physiol. Endocrinol. Metab., 260 (1991), E801–E809. [20] I. M. Toli´c, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion, J. Theor. Biol., 207 (2000), 361–375.

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