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 num- ber 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 com- putational 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. 1
Transcript
Delay differential equation models in diabetes modeling: a
review
Athena Makroglou + and Iordanis KaraoustasDepartment of MathematicsUniversity of Portsmouth
1st Floor Lion Gate Bldg, Portsmouth PO1 3HF, UK
Jiaxu Li ∗
Department of MathematicsUniversity of Louisville
Louisville KY 40292, USA
Yang Kuang †
Department of Mathematics and StatisticsArizona State University
Tempe AZ 85287-1804, USA.
Abstract
Delay differential equation models can generate rich dynamics using minimum num-ber of parameters. This characteristic enables such models to play important roles ina growing number of areas of diabetes studies. Such areas include the insulin/glucoseregulatory system, intravenous glucose tolerance test (IVGTT), and insulin therapies.In this paper, an extensive review of such models is presented together with some com-putational results and brief summaries of theoretical results for the cases of models forultradian 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 classifiedinto two main categories. Type 1 diabetes which is juvenile onset and insulin-dependentand Type 2 diabetes which is adult onset and insulin-independent. Complications of thedisease include retinopathy, nephropathy, peripheral neuropathy, blindness The disease isaffecting hundreds millions of people worldwide (type 2 diabetes mellitus had an estimatedincidence of 151 million in the year 2000, which has motivated many researchers to studythe mathematical, computational and medical problems associated with it. Articles aboutthe prevalence and the problems of diabetes appear frequently in various media outlets.
NIH/NIEHS Grant P30ES014443.†[email protected]. Work is partially supported by DMS-0436341 and DMS/NIGMS-0342388.
1
Type 1 diabetes is considered to be the result of an immunological destruction of theinsulin-producing β-cells. According to Lupi and Del Prato (2008) ([12]), p. 560, the normalpancreas 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 usuallyconnotes resistance to the effects of insulin on glucose uptake, metabolism, or storage, dueto 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 Jaıdaneand Hober (2008) ([8]) (type 1 diabetes), and to Lupi and Del Prato (2008) ([12]) (type 2diabetes).
Treatment of type 1 diabetes is based on the administration of insulin of various typesin 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 dia-
betes. These include Ordinary Differential Equations (ODEs), Delay Differential Equations(DDEs), Partial Differential Equations (PDEs), Fredholm Integral Equations (FIEs) ( in theestimation of parameters problem), Stochastic Differential Equations (SDEs) and Integro-Differential Equations (IDEs). We refer for example to the review papers [13], [17], for moredetails about several such models and corresponding bibliography.
For information about numerical methods for solving delay differential equations werefer 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 thelate 1960s, 1970s and 1980s. Grodsky introduced the so called threshold hypothesis for thepancreatic granules according to which each granule secrets its insulin contents if glucoseis above a certain threshold level. Such recent papers (revisiting, modifying, extending thiswork but using ODEs mainly) include:
Pedersen, Corradin, Toffolo, Cobelli (2008) ([18]), (the paper describes also the currentstate of the art accompanied by rich bibliographical information. Their model includes thenotion of distinct pools of granules as well as various mechanisms, like priming, exocytosisetc, and it is claimed to be the first physiology-based one to reproduce the staircase exper-iment, which underlies derivative control, that is the pancreatic capacity of measuring therate of change of the glucose concentration).
Mari and Ferrannini (2008) ([14]) (the paper includes an interesting historical back-ground presentation too).
In this paper a review of some mathematical models in the form of delay differen-tial equations is given, accompanied by some computational results using Matlab andelements of their theoretical analysis. The organisation of the paper is as follows: Sec-tion 2 contains the description of the models and some computational results and briefsummaries of theoretical results. Two cases are covered here, the case of models forultradian oscillations of insulin (section 2.1) and the case of models used in diagnostictests (section 2.2). Concluding remarks are in section 3. The notation is kept as inthe original papers for easy reference. The Matlab function DDE23 was used for obtain-ing graphs of models in the form of DDE systems, see for example the tutorial http:
//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 areasof Biology and Medicine. Such areas include Epidemiology, Population Biology, Immunol-ogy, Physiology, Cell mobility.
Delayed effects often exist in the glucose-insulin regulatory system, for example, theinsulin secretion stimulated by elevated glucose concentration level, hepatic glucose pro-duction ([11], [19]). Therefore the delays need to be taken into account when modeling thesystems. General approaches include the technique of compartment-split by introducing ofauxiliary variables in ordinary differential equations (ODE) ([2], [19]), and modeling in delaydifferential equations (DDE) by using explicit time delays in either discrete or distributedforms (cf. [11], [10], [15], [16]).
The delays in the compartment-split approach as are classsified as “soft delays” by usingγ kernel that is an approximation of the Dirac kernel, while the explicit delays in modelsas “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 func-tions/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, modelsof 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 fromsubthreshold to stimulatory levels, with a rapid peak at 2-4 min (first phase), a decreaselasting 10-15 min (pulsatile insulin secretion) followed by a gradual increase within the nextcouple of hours (50-120 minutes), cf. Chew et al (2009) ([4]), (second phase, ultradianinsulin secretion).
As mentioned in Chew et al (2009) ([4]), ultradian oscillations have been seen after mealingestion during continuous enteral nutrition, and during intravenous glucose infusion.
Historically, it was in 1923 when rapid and slower oscillations in the peripheral concen-trations of glucose were reported by Karen Hansen and half a century later rapid oscillationsin the peripheral insulin concentrations were demonstrated.
As several authors mention, the precise mechanisms generating ultradian oscillationsare not fully understood yet and the two most common mechanisms mentioned are:
• instability of the glucose-insulin feedback loop, where the insulin oscillations entrainthese of the glucose
Rntrainment means the ability of a self-oscillating system when perturbed exogenouslywith a periodic stimulus, to adjust its period of oscillation to that of the stimulus. Theentrainment 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 modelthe 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. Thismodel was later simplified by Tolic et al (2000) ([20]). Several models based on this modelwere 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): 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 Tolic 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 utilizationand 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. Tolic,Mosekilde and Sturis (2000)([20]) simplified this model a little bit. This model has beenthe basis of several DDE models. It has the following form (cf. [20], p. 363)
where G(t) is the mass of glucose, Ip(t), Ii(t) the mass of insulin in the plasma and theintercellular space, respectively, Vp is the plasma insulin distribution volume, Vi is the ef-fective 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 indi-cates (exogenous) glucose supply rate to plasma, and x1(t), x2(t), x3(t) are three additionalvariables associated with certain delays of the insulin effect on the hepatic glucose produc-tion with total time td. f1(G) is a function modeling the pancreatic insulin production ascontrolled by the glucose concentration, f2, f3, f4 are functions for glucose utilization byvarious body parts (brain and nerves (f2), muscle and fat cells (f3, f4)) and f5 is a functionmodeling 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 reflectedby breaking the insulin in two separate compartments, and the other one is hepatic glucoseproduction delay which is fulfilled by the three auxiliary variables, x1, x2 and x3. Thismodel simulated ultradian insulin secretion oscillations numerically. For conclusions drawnfrom 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 descriptionof ultradian oscillations in human insulin secretion. The model equations are ([6], p.27)
dx
dt(t) = f1
(0.1z(t)
V3
)−
(E
V1+
1T1
)x(t) +
E
V2y(t),
dy
dt(t) =
E
V1x(t)−
(E
V2+
1T2
)y(t), (2.7)
dz
dt(t) = f3
(x(t− T )
V1
)− 0.1z(t)
V3f2
(y(t)V2
)+ (L− p0),
where x(t)(mU) is the amount of insulin in the plasma, y(t)(mU) is the amount of insulinin the interstitial fluid and z(t)(mg) is the amount of glucose treated as occupying onecompartment; V1(l), V2(l), V3(l) are the volumes of the plasma, interstitial fluid and theglucose 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 theenvironment (L = 100(mg/min), [6], p.28, corresponds to the normal delivery of glucose in150g/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)
where cx is the plasma concentration of insulin, cx = xV1
(µU/ml), cy is the remote com-partment insulin concentration, cy = y
V2(µU/ml), and cz is the glucose concentration
cz = 0.1zV3
(mg/dl). The authors mention that the form of the functions f1 − f3 is simi-lar to that proposed in [19], but with different parameter values which they obtained byleast squares fitting to published data. Initial conditions used in the numerical simulationsare, [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 alsopresented in the paper for a linearized system of DDEs. The claim is ([6], p. 31) that for verysmall and very large L values the steady state solution is stable and ultradian oscillationsdo not arise, but for moderate L values, the steady state solutions become unstable andperiodic oscillations of insulin and glucose occur.
Figures 2.1, 2.2 for plasma insulin and glucose concentrations correspond to Fig. 6 ofthe paper.
Figure 2.1: Plasma insulin concentration in µU/ml, Drozdov and Khanina (1995) DDEmodel
0 100 200 300 400 500 600 700 80016
18
20
22
24
26
28
30
32L=120 mg/min, delay T=25, Drozdov and Khanina (1995) DDE model
time t in min
c x=x(
t)/V
1: P
lasm
a in
sulin
con
cent
ratio
n in
µ U
/ml
Figure 2.2: Glucose concentration in mg/dl, Drozdov and Khanina (1995) DDE model
0 100 200 300 400 500 600 700 80090
95
100
105
110
115
120
time t in min
c z=0.
1z(t
)/V
3: g
luco
se c
once
ntra
tion
in m
g /d
l
L=120 mg/min, delay T=25, Drozdov and Khanina (1995) DDE model
6
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 thetime that the glucose concentration level is elevated to the moment that the insulin hasbeen transported to the interstitial space and becomes ‘remote insulin’ ([11], p. 726), thesecond one (τ2) has to do with the delay of the effect of hepatic glucose production measuredfrom the time that insulin has become ‘remote insulin’ to the moment that a significantchange of hepatic glucose production occurs ([11], p. 726). The model is formulated by theobservation of the Law of Conservation. The model is given by ([11], p. 726)
The form of the functions f1 − f5 is given by (2.1)-(2.5). Gin ≥ 0 is the glucose infusionrate (from meals, oral glucose intake, constant glucose infusion etc), di > 0 is the insulinclearance rate, [11], p. 724. Numerical simulations including bifurcation analysis are givenin the paper and comparisons are made to some existing models, namely these by [19], andtwo (new) variations of the [19] and [20] models.
See also [10] for analytical studies of the model. Based on the bifurcation analysis, theauthors suspect that the total time delay τ1 is critically responsible for the oscillation. Thetotal time delay is measured from the moment that the glucose concentration level startsto 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] andτ1 = 6, τ2 = 36, Gin = 1.35, di = 0.06 taken from Fig. 4 (right) of the same paper.
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 usedin [11] throughout the simulation time, so that external inputs like food uptake can besimulated too.
• They introduce two additional functions f6 and f7 to ‘provide the effects of hyper-glycemia’ ([3], 2nd page), with the rest of the functions f1 − f5 the same as in [11] (givenhere by 2.1-2.5).
• They introduce two additional parameters α, β for the purpose of ‘estimating thecondition of major dysfunction of diabetes’ ([3], 3rd page), (α for insulin release from thepancreas, β for the ability of insulin-dependent glucose utilization (IDGU)-small β valueindicates increasing severity of insulin resistance).
• They use f7(G(t)−330) for describing the kidney glucose excretion (KGE) rate abovethe 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).
where G(t), I(t) denote glucose and insulin concentrations respectively, Gm(t− tm),m ∈ Mdenotes the m−th exogenous food uptake at tm, u(t− tm) is a unit step function that hasthe 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
b2e−t2/(2b2).
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)
f6(G(t)) =1
1 + exp[γ((G(t)/(C3Vg))− C6)]
f7(G(t)− 330) = Sb +Sc − Sb
1 + exp[δ(((G(t)− 330)/(C3Vg))− C7)].
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
Figure 2.5: Glucose Concentration, θ1 = 16.2, θ2 = 35.1, Chen and Tsai (2009) DDE model
0 500 1000 15000
20
40
60
80
100
120
140
160
time t in min
Gin
(t)
(mg/
min
)
y 1(t)=
G(t
): G
luco
se c
once
ntra
tion
(mg/
dl) θ 1=16.2, θ 2=35.1, b=80, k=4300, Chen and Tsai (2009) DDE model
G(t)Gin(t)
2.2 Models used with diagnostic tests
A number of diagnostic tests have been developed to assess two indices important inmetabolic research known as insulin sensitivity and glucose effectiveness and also the β-cell function. Such diagnostic tests include: the OGTT (oral glucose tolerance test), theIVGTT (intravenous glucose tolerance test), tests based on the use of a clamp (the Hyper-glycaemic and the Euglycaemic Hyperinsulinaemic ones) and variations of them.
Insulin sensitivity is defined as the ability of insulin to enhance glucose effectiveness andglucose effectiveness, as the ability of glucose to promote its own disposal If one measuresglucose disposal by its rate of disappearance from the accessible pool Rd(t), then glucoseeffectiveness = ∂Rd(t)
∂G(t) |SS and insulin sensitivity = − ∂2Rd(t)∂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 bemeasured to calculate insulin secretion either by deconvolution or mathematical modeling.
The hyperglycaemic clamp starts with a glucose bolus to raise the glucose level abruptlyto a hyperglycaemic target value and then glucose is infused to maintain the glucose con-centration at that value. Hyperglycaemia stimulates biphasic insulin secretion and glucosedisposal to tissues is increased. After 2-3 hours, a steady state is reached in which insulinlevels vary according to the subject’s β-cell response.
The procedure for the Hyperinsulinaemic Euglycaemic Glucose clamp can be describedas follows: After an overnight fast, insulin is infused intravenously at a constant rate thatmay range from 5 to 120mUm−2min−1 (dose per body surface area per minute). Thisconstant 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 adiposetissue is increased, and HGP (hepatic glucose production) is suppressed. Under theseconditions, a bedside glucose analyzer is used to frequently monitor blood glucose levels at5-10 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 ofconstant 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 hepaticglucose production insignificant and thus negligible; and 2) the bi-phasic insulin secretioncaused 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, onlyone time delay should be considered.
There are several models associated with it (cf. the ODEs of the minimal model inBergman, Ider, Bowden and Cobelli (1979) ([2]), DDEs (cf. Panunzi, Palumbo, and DeGaetano (2007) ([16]), Giang, Lenbury, De Gaetano, Palumbo (2008) ( [7]), and the integro-differential equation models (cf. Palumbo, Panunzi, De Gaetano (2007) ([15])).
While the minimal model has its features widely used in research, it has several draw-backs in mathematics that were pointed out by De Gaetano and Arino (2000) ([5]). Theparameter fitting is to be divided into two separate parts: first, using the recorded insulinconcentration as given input data in order to derive the parameters in the first two equa-tions in the model, then using the recorded glucose concentration as given input to derivethe parameters in the third equation. However, the system is an integrated physiologicaldynamic system and one should treat it as a whole system and be able to conduct a single-step parameter fitting process. Secondly, some of the mathematical results produced bythis model are not realistic. Specifically, it can be shown that the minimal model does notadmit an equilibrium and the solutions may not be bounded. Finally, the non-observableauxiliary 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 discretedelay differential equation model to further study the short but complicated phenomenon.Although not able to confirm the conclusion analytically, the authors produced reasonablesimulation profile with experimental raw data statistically ([15], [16]).
A short introduction to integro-differential equation models used for modeling glucose-insulin 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, namelythe 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 ofintegro-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, Bow-den, 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 clinicsand extended in various applications ([13]). The model is an ODE model formulated in split
11
compartment and is given by
dG(t)dt
= G′ = −[b1 + X(t)]G(t) + b1Gb,
dX(t)dt
= X ′ = −b2X(t) + b3[I(t)− Ib],
dI(t)dt
= I ′ = b4[G(t)− b5]+t− b6[I(t)− Ib].
(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 activitythat 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] isthe 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 unitof insulin concentration excess over the baseline. b4 [(µU/ml)(mg/dl)−1min−1] is the rateof pancreatic release of insulin after the intake of the glucose bolus, per minute per unit ofglucose concentration above the “target” glycemia b5 [mg/dl]. b6 [µU/ml] is the first orderdecay rate for insulin in the plasma. b7 [µU/ml] is the plasma insulin concentration at time0, above basal insulinemia, immediately after the glucose bolus intake.
The minimal model contains minimal number of parameters ([2]) and it is widely usedin physiological research work to estimate metabolic indices of glucose effectiveness (SG)and insulin sensitivity (SI) from the intravenous glucose tolerance test (IVGTT) data bysampling 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 plasmainsulin concentrations following an IVGTT test, two without delay and two with delay oninsulin action (τi).
The authors ([16]) identified a special case (the model A in [16]) as the best model underthe Akaike Information Criterion (AIC), (‘without first order plasma glucose elimination(Kxg) and without delay on insulin action’). The model is given as follows.
dG
dt(t) = −KxgiI(t)G(t) +
Tgh
Vg,
(2.11)
dI
dt(t) = −KxiI(t) +
Tigmax
Vi
(G(t−τg)
G∗
)γ
1 +(
G(t−τg)G∗
)γ .
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 = KxgiIbGbVg, Tigmax = KxiIbVi
[1 +
(Gb
G∗
)γ]/
(Gb
G∗
)γ
,
where G∗ and I∗ are the steady state. The full description of the symbols and their unitsis given in Table 3 of [16].
The authors proved that the unique steady state (G∗, I∗) is globally asymptoticallystable when γ ≤ 1 under the condition
KxgiIbγ
1 +(
GbG∗
)γ ≤ Kxg + KxgiIb.
regardless of the length of the time delay. However this case only accounts for 3% ofthe experimental data the authors obtained. For more general cases, the authors haveshown that there exists a bifurcation point τ0 > 0 such that the steady state is locallyasymptotically stable when the delay is smaller than τ0. Apparently, delay dependentconditions for global stability are needed to improve the analysis.
The authors also demonstrated that the insulin sensitivity index can be obtained in thesame way as in the minimal model but it is more effective. The authors used raw data forparameter estimation (the GLS-Generalized Least Squares method described in the paper’sAppendix to obtain individual regression parameters and the WLS-Weighted Least Squaresmethod for the Minimal Model).
In a more recent paper (Giang, Lenbury, De Gaetano, Palumbo (2008) ([7]) the authorsconsider the most general of the 4 models of the family introduced in [16] (with first orderplasma glucose elimination (Kxg) and with delay in insulin action (τi)) and deal with the-oretical results concerning global stability under certain conditions on the parameters andthe effect of the delays τi, τg on the oscillatory behavior of the solutions when Kxg = 0. IfKxg = 0 and τi = 0 the DDE model of [7] reduces to that of [16], with parameters given inRemark 10 in [7].
The following graphs (2.6, 2.7) were produced using data from Remark 9 in [15] andinitial 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 discreteas the authors call it) delay DDE model, coincides with that of [16]. We can notice thatgraph 2.6 (insulin evolution) exhibits a derivative discontinuity at t = τg = 23.5 (notethat there is a typo in the paper’s Figure 10 (second graph), as Figure 12 and privatecommunications with one of the authors confirmed).
13
Figure 2.6: Glucose evolution (mM), Panunzi et al (2007) DDE model
0 20 40 60 80 100 120 140 160 1800
10
20
30
40
50
60
τ i=0, τ g=23.50, γ=2.52, Panunzi et al (2007) DDE model
time t in min
y1(t
): G
luco
se e
volu
tion
in m
M
Dg=1.833 mmol/KgBW
Dg=5 mmol/KgBW
Dg=10 mmol/KgBW
Figure 2.7: Insulin evolution (pM), Panunzi et al (2007) DDE model
0 20 40 60 80 100 120 140 160 1800
500
1000
1500
2000
2500
3000
time t in min
y2(t
): In
sulin
evo
lutio
n in
pM
τ i=0, τ g=23.50, γ=2.52, Panunzi et al (2007) DDE model
14
Palumbo, Panunzi and De Gaetano (2007) [15] considered a model that consists of asystem of two coupled integro-differential equations where different choices of the convo-lution kernels result in a number of different models including DDEs with constant delay.The model equations are
dG
dt(t) = −KxgG(t)−KxgiG(t)I(t) +
Tgh
Vg,
(2.12)dI
dt(t) = −KxiI(t) +
Tigmax
Vif(G(t)),
where
f(G(t)) =
(G(t)G∗
)γ
1 +(
G(t)G∗
)γ ,
G(t) =∫ τg
0wg(θ)G(t− θ)dθ,
I(t) =∫ τi
0wi(θ)I(t− θ)dθ
and the kernels wg : [0, τg] → R+, wi : [0, τi] → R+ are non-negative square integrablefunctions such that
∫ τg
0wg(θ)dθ = 1,
∫ τi
0wi(θ)dθ = 1,
and there exist Tg, Ti such that
∫ τg
0θwg(θ)dθ ≤ Tg < ∞,∫ τi
0θwi(θ)dθ ≤ Ti < ∞.
For the description of variables and units we refer to the paper.
In case (I), the integro-differential system reduces to a DDE system of two equationswith constant delays τg, τi,
dG
dt(t) = −KxgG(t)−KxgiG(t)I(t− τi) +
Tgh
Vg,
(2.15)dI
dt(t) = −KxiI(t) +
Tigmax
Vif(G(t− τg)).
15
In case (II) with τg = τi = ∞, using the linear chain trick (cf. Kuang (1993) ([9])), theauthors 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-differentialsystem has positive solution and that it is persistent. Local stability is shown for the DDEsystem (section 4.1) and the ODE system (section 4.2). Global stability is shown for theIDE system (Theorem 6), depending upon a condition on parameter values (Theorem 4),
KxgiIbγ
1 +(
GbG∗
)γ ≤ Kxg + KxgiIb.
The authors mention (Remark 7) that the above condition is not verified for particularvalues 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 itis reproduced below (figures 2.8, 2.9). The starting ODE values involving integrals werecalculated exactly. The Matlab function ODE45 was used for solving the ODE system.
16
Figure 2.8: Glucose evolution (mM), Palumbo, Panunzi, De Gaetano (2007) DDE model
0 20 40 60 80 100 120 140 160 1800
5
10
15
20
25
30
35
40
45
t (in min)
Glu
cose
evo
lutio
n (m
M)
gamma=2.66, Palumbo, Panunzi, De Gaetano (2007) model
Dg=1.833 mmol/KgBW
Dg=5 mmol/KgBW
Dg=10 mmol/KgBW
Figure 2.9: Insulin evolution (pM), Palumbo, Panunzi, De Gaetano (2007) DDE model
0 20 40 60 80 100 120 140 160 1800
500
1000
1500
2000
2500
3000
3500
4000
4500
t (in min)
Insu
lin e
volu
tion
(pM
)
gamma=2.66, Palumbo, Panunzi, De Gaetano (2007) model
Dg=1.833 mmol/KgBW
Dg=5 mmol/KgBW
Dg=10 mmol/KgBW
17
3 Concluding remarks
The time delays exist in the glucose-insulin regulatory system and other interactions ofthe real world including life sciences and control theory. It is natural to include explicitlythe delays in mathematical modeling. Although the delayed effects can be simulated byintroducing the auxiliary states in ordinary differential equation systems, as we have seenearlier, it is an approximation by using different kernels and also the number of equations inthe system is larger. Modeling by delay differential equations can not only more accuratelysimulate the real life problems, but also reduce the number of equations. Comparing toODE models, it is more challenging to analyze DDE models. However, a handful of existingbooks documents the theory of functional differential equations and their applications inbiology 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 andvaluable 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 SciencePublications, Clarendon Press, Oxford, 2003.
[2] R. N. Bergman, Y. Z. Ider, C. R. Bowden, C. Cobelli, Quantitative estimation of insulin sen-sitivity, 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 anddiabetic 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 stim-ulation, 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 andApplications, 343 (2008), 996–1006.
[8] H. Jaıdane, D. Hober, Role of coxsackievirus B4 in the pathogenesis of type 1 diabetes, Diabetesand 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 twodelays, SIAM J. Appl. Math., 67 (2007), 757–776.
[11] J. Li, Y. Kuang and C. C. Mason, Modeling the glucose-insulin regulatory system and ultradianinsulin secretory oscillations with two explicit time delays, J. Theor. Biol., 242 (2006), 722–735.
18
[12] R. Lupi and S. Del Prato, B-cell apoptosis in type 2 diabetes: quantitative and functionalconsequences, Diabetes and Metabolism, 34 (2008), 556–564.
[13] A. Makroglou, J. Li, Y. Kuang, Mathematical models and software tools for the glucose-insulinregulatory 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 effectiveapproach, 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 delay-differential 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-venousglucose tolerance test, Theoretical Biology and Medical Modelling, 2097, 4:35 (Open Accessarticle 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 glucose-stimulated 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 mechanismsunderlying ultradian oscillations of insulin and glucose, Am. J. of Physiol. Endocrinol. Metab.,260 (1991), E801–E809.
[20] I. M. Tolic, E. Mosekilde and J. Sturis, Modeling the insulin-glucose feedback system: thesignificance of pulsatile insulin secretion, J. Theor. Biol., 207 (2000), 361–375.
