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Journal of Theoretical Biology 252 (2008) 608620

Modeling a simplied regulatory system of bloodglucose at molecular levels

Weijiu Liu a, , Fusheng Tang b

a Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, USAb Department of Biology, University of Arkansas at Little Rock, 2801 S. University Ave., Little Rock, AR 72204-1099, USA

Received 13 June 2007; received in revised form 9 February 2008; accepted 13 February 2008Available online 23 February 2008

Abstract

In this paper, we propose a new mathematical control system for a simplied regulatory system of blood glucose by taking intoaccount the dynamics of glucose and glycogen in liver and the dynamics of insulin and glucagon receptors at the molecular level.Numerical simulations show that the proposed feedback control system agrees approximately with published experimental data.Sensitivity analysis predicts that feedback control gains of insulin receptors and glucagon receptors are robust. Using the model, wedevelop a new formula to compute the insulin sensitivity. The formula shows that the insulin sensitivity depends on various parametersthat determine the insulin inuence on insulin-dependent glucose utilization and reect the efciency of binding of insulin to its receptors.Using Lyapunov indirect method, we prove that the new control system is inputoutput stable. The stability result provides theoreticalevidence for the phenomenon that the blood glucose uctuates within a narrow range in response to the exogenous glucose input fromfood. We also show that the regulatory system is controllable and observable. These structural system properties could explain why theglucose level can be regulated.Published by Elsevier Ltd.

Keywords: Diabetes; Glucose regulation; Output feedback; Insulin infusion; Insulin sensitivity; Inputoutput stability

1. Introduction

In type 1 diabetes, the pancreas does not produce insulin,a peptide hormone is required for glucose uptake andendogenous glucose mobilization. Therefore, type 1 dia-betes patients have high blood glucose levels and needexternal insulin to assist glucose uptake and utilization.External insulin needs to be infused at an appropriate rateto maintain blood glucose within the narrow range from 80

to 120 mg/dl (4.4 to 6.7mmol/L).Mathematical control models of the regulatory system of

blood glucose are required for integrating a glucosemonitoring system into insulin pump technology to forma closed-loop insulin delivery system on the feedback of blood glucose levels, the so-called articial pancreas(Hovorka, 2006; Panteleon et al., 2006; Steil et al., 2006 ).To make this articial pancreas close to the natural

pancreas, numerous models have been proposed since thepioneering work of Albisser et al. (1947a, b) and Clemens etal. (1977) , including the linear model of Ackerman et al.(1965) and various compartmental minimal models pro-posed by Bergman et al. (1979, 1981, 1985) , Bertoldo et al.(2006) , Li et al. (2006a, b) , Man et al. (2004, 2005, 2007) ,Sturis et al. (1991) , Toffolo and Cobelli (2003) and Toffoloet al. (2006) . These models focused on the physiologyof glucose regulation. However, the regulation of glucose

level by insulin involves multiple feedback controls atthe molecular level. A mathematical model that describesthe dynamics of glucose regulation at the molecularlevel is still waiting to be developed, although such modelsas proposed by Sedaghat et al. (2002) for the insulinsignaling pathway have been seen in the literature. Thepotential for an automated closed-loop system to achieveround-the-clock glycemic control has not been fullyexplored and no one feedback control algorithm has beenuniversally accepted as optimal for insulin delivery ( Steilet al., 2006 ).

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www.elsevier.com/locate/yjtbi

0022-5193/$ - see front matter Published by Elsevier Ltd.doi: 10.1016/j.jtbi.2008.02.021

Corresponding author. Tel.: +1 501450 5684; fax: +1501450 5662.E-mail address: [email protected] (W. Liu).
http://www.elsevier.com/locate/yjtbihttp://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.jtbi.2008.02.021mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.jtbi.2008.02.021http://www.elsevier.com/locate/yjtbi

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the control system is inputoutput stable. This providestheoretical evidence for the phenomenon that the glucoselevel uctuates within a narrow range in response to theexogenous glucose input. In Section 6, we prove thatthe regulatory system is controllable and observable.These structural system properties could explain why the

glucose level can be regulated. Finally in Section 7, we raisean optimal insulin infusion problem that we could notsolve yet.

2. Mathematical model

The regulatory system of blood glucose is described inFig. 1 . We divide it into three subsystems: the transitionsubsystem of insulin and glucagon from plasma to cellularspace, the insulin and glucagon receptor binding subsys-tem, and the glucose production and utilization subsystem.

2.1. Insulin and glucagon transition subsystem

Plasma insulin does not act directly on glucose metabo-lism, but through remote cellular insulin ( Bergman et al.,1985 ). We use the model by Sturis et al. (1991) with amodication to describe the transitional delay process asfollows:

ds p j dt

k p j ;1s p j k

p j ;2s

p j u j ; j 1; 2, (1)

where s p1; s p2 denote the concentrations of plasma glucagon

and insulin, respectively. u1 stands for the glucagoninfusion rate (GIR) and u2 stands for the insulin infusionrate (IIR). The positive constants k p j ;1 are the transitionalrates, and k p j ;2 the degradation rates. Unlike the model bySturis et al. (1991) , we assume that the intracellular insulindoes not come back to plasma. This is similar to the modelby Bergman et al. (1985) .

2.2. Insulin and glucagon receptor binding subsystem

We assume that the receptor recycling is a closedsubsystem, that is, their synthesis is equal to theirdegradation. There was a detailed model for the insulinbinding subsystem developed by Sedaghat et al. (2002) .

Here we use the following simplied version:ds j dt

k s j ;1s j R0 j r j k

s j ;2s j

k p j ;1s p j V p

V , (2)

dr j dt


r j r j ; j 1; 2, (3)

where s1; s2 denote the concentrations of intracellularglucagon and insulin, r1; r2 the concentrations of glucagon-and insulin-bound receptors, and R01; R

02 the total concen-

trations of receptors, respectively. The positive constantsk s j ;1 are the association rates for glucagon and insulin tobind their receptors, k s j ;2 the degradation rates, and k

r j the

inactivation rates. V p is the plasma insulin volume and V is

the cellular insulin volume. k p j ;1s p j V p is the total amount of

the plasma glucagon or insulin transferred to the intracel-lular space and then k p j ;1s

p j V p=V is the concentration of

cellular glucagon or insulin.

2.3. Glucose production and utilization subsystem

Plasma glucose has two production sources: endogenoushepatic glucose produced by converting glycogen intoglucose in liver and exogenous glucose taken from food.We use the MichaelisMenton equation to model theprocess of conversion between glucose and glycogen cataly-zed by glycogen phosphorylase and glycogen synthase. Forexample, the MichaelisMenton equation for the glycogenphosphorylase is

v V gpmax g1K gpm g1

, (4)

where g1 is the concentration of glycogen, v is the reaction

velocity, V gpmax is the maximum velocity, and K gpm is theMichaelisMenton constant. We assume that the activationof glycogen phosphorylase and glycogen synthase byglucagon or insulin is proportional to the concentrationsof activated receptors. The inactivation of glycogensynthase by glucagon is modeled by 1 =1 k 2r1, wherek 2 is a positive constant called a feedback control gain.

Glucose utilization can be classied into two classes of utilizations: insulin-independent (by the brain and nervecells) and insulin-dependent (by the muscle and fat cells).We use the following function:

f 1g2 U b1 exp g2=C 2, (5)

to model the insulin-independent glucose utilization, whereg2 is the concentration of the plasma glucose and U b; C 2are positive constants. This function was proposed byTurner et al. (1979) and used by Li et al. (2006a) , Sturiset al. (1991) , and Tolic et al. (2000) . The graph of thisfunctions with U b 7:2 (mg/l/min) and C 2 144(mg/l)(these parameter values are adopted from Sturis et al.,1991; Tolic et al., 2000 ) is plotted in Fig. 2 (left), whichshows that the utilization is saturated when the bloodglucose level reaches about 1000 (mg/l).

The insulin-dependent glucose utilization is modeled bythe production of the following two functions:

f 2g2 g2C 3

, (6)

f 3s2 U 0 U m U 0

s2C 4

b

1 s2C 4

b , (7)

where C 3; C 4; b ; U 0, and U m are positive constants. Thesefunctions were suggested by Rizza et al. (1981) , Bergman(1989) , and Yang et al. (1989) , and used by Li et al.(2006a) , Sturis et al. (1991) , and Tolic et al. (2000) . Thefunction f 3 is an impact of insulin on the glucose

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utilization. Under this impact, the glucose utilizationdepends linearly on the glucose concentration. The graphof f 3 with b 1:77; C 4 80 (mU/l), U 0 4 (mg/l/min),and U m 94 (mg/l/min) (these parameter values areadopted from Sturis et al., 1991; Tolic et al., 2000 ) isplotted in Fig. 2 (right), which shows that the impact of insulin on the glucose utilization is saturated when theinsulin level reaches about 250 (mU/l).

The above analysis leads to the following glucoseproduction and utilization model:

dg1dt

k 1r21 k 2r1

V gsmax g2K gsm g2

k 3r1V gpmax g1

K gpm g1, (8)

dg2dt

k 1r21 k 2r1

V gsmax g2K gs

m g

2

k 3r1V gpmax g1

K gpm

g1

f 1g2 f 2g2 f 3s2 G in , (9)

where g1; g2 stand for the concentrations of glycogen andglucose, respectively; k s are the feedback control gains;V gpmax ; V

gsmax are the maximum velocity of glycogen phos-

phorylase and glycogen synthase, respectively; K gpm ;K gsm are

the MichaelisMenton constants of glycogen phosphor-ylase and glycogen synthase, respectively; G in stands for theexogenous glucose input rate from food. The termk 1r2=1 k 2r1V gsmax g2=K

gsm g2 describes the conversion

of glucose into glycogen and the term k 3r1V gpmax g1=K gpm

g1 describes the conversion of glycogen into glucose.

2.4. Complete model

For the reference convenience, we collect all aboveequations to form a complete system as follows:

ds p j dt

k p j ;1s p j k

p j ;2s

p j u j , (10)

ds j dt


s j ;2s j

k p j ;1s p j V p

V , (11)

dr j

dt k s

j ;1s j R0

j r

j k r

j r j ; j 1; 2, (12)

dg1dt

k 1r21 k 2r1

V gsmax g2K gsm g2

k 3r1V gpmax g1

K gpm g1, (13)

dg2dt

k 1r21 k 2r1

V gsmax g2K gsm g2

k 3r1V gpmax g1

K gpm g1 f 1g2 f 2g2 f 3s2 G in . (14)

In this control system, the natural output is the bloodglucose:

y g2. (15)

We need to distinguish the control variable G in fromu1; u2. While the glucagon secretion (or infusion) rate u1and the insulin secretion (or infusion) rate u2 areendogenous variables and can be manipulated internally

by pancreas, the glucose input rate G in from food is anexogenous variable and out of the control of pancreas.Therefore, in this control system, G in is a disturbance input.

Now the problem of regulation of blood glucose levelscan be formulated as a problem of disturbance attenuation:Determine the infusion rates u1 and u2 as a function of theblood glucose g2 such that g2 is maintained in the range80 mg =dlp g2p 120 (mg/dl) in response to the disturbanceinput G in .

3. Model validation

We validate the model (10)(14) by using the experi-mental data of Korach-Andre et al. (2004) (courtesy of Franc- ois Pe ronnet).

We consider the following feedback rates:

u1 G m

1 b1 exp a1g2 1000 , (16)

u2 Rm

1 b2 exp a2C 1 g2, (17)

where a1; a2; b1; b2; G m are positive constants. u2 is adoptedfrom Sturis et al. (1991) . The graphs of these functions inFig. 3 indicate that the glucagon secretion increases rapidlywhen the blood glucose level drops down to around

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0 500 1000 1500 20000

2

4

6

8

Glucose (mg/l)

f 1 ( m g

/ l / m

i n )

0 200 400 600 800 10000

20

40

60

80

Insulin (mU/l)

f 3 ( m g

/ l / m

i n )

Fig. 2. Insulin-independent glucose utilization (left) and impact of insulin on glucose utilization (right).

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800 (mg/l) and the insulin secretion increases rapidly whenthe blood glucose level rises up to around 1500 (mg/l).

To numerically solve the system, we need to estimatenumerous parameters in the system so that the system iscapable of simulating glucose dynamics. The estimation of these parameters is difcult since no universal methods canbe used. For instance, the nonlinear least-square techniqueof Marquardt (1963) cannot be employed because thesystem has eight equations and then eight functions on theright-hand side of the system and because no complete setsof data are available for all molecules in one experiment.Hence we here have to appeal to different means: adoptionof some existing parameters in the literature and numericalsimulations.

Most of these parameters are adopted from the literature

as indicated in Tables 1 and 2 . The parameters estimated inthis paper are as follows: k p1;1; k

p2;1; k

s1;2; k

r1; k i i 1; . . . ;5;

G m; a1, and b1 . The degradation rates of glucagon and itsreceptor k s1;2; k

r1 are assumed to be the same as the

respective rates for insulin. The maximum glucagoninfusion rate G m is selected so that it is much smaller thanthe maximum insulin infusion rate Rm . Thus we take G m 20 mU/L/min ( o R m 70 mU/L/min). After converting itto the unit of M/min, we have G m 20 6:945 10 12

5600 =3483 2:23 10 10 (the insulin molecular weight:5600g/mol and the glucagon molecular weight: 3483 g/mol).The constants a1 and b1 are selected so that the glucagonsecretion increases rapidly when the blood glucose level dropsdown to around 800 (mg/l).

A criterion for the selection of the feedback control gainsk i i 1; 2; 3 is that they should be selected such that theall equilibrium states of (10)(14) are positive. In addition,our numerical simulations show that they are robust. Thevalues of k 1 in the range from 10 5 to 10 6 and the values of k 2; k 3 in the range from 10 11 to 10 13 do not change muchthe dynamics of the system. To explain this mathemati-cally, we perform a sensitivity analysis about these threeparameters. Thus we construct the sensitivity system(Khalil, 2002, p. 99 ) by calculating the Jacobian matricesof (10)(14) with respect to the state variables and theparameters k 1; k 2; k 3, respectively (the matrices are too big

to show here). Solving the sensitivity system with thesystem (10)(14) together, we obtain the numerical solu-tions of the partial derivatives of the state variables withrespect to the parameters. The insulin sensitivity indicesk 1=s p2q s

p2=q k 1; k 2=s

p2q s

p2=q k 2; k 3=s

p2q s

p2=q k 3 with respect

to k 1; k 2; k 3 are plotted in Fig. 4 , and the glucose sensitivity

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0 1000 2000 30000

5

10

15

Glucose (mg/l)

u 1

( G I R ) ( m U / l / m i n )

0 1000 2000 3000 40000

20

40

60

Glucose (mg/l)

u 2

( I I R ) ( m U / l / m i n )

Fig. 3. Glucagon infusion rate (16) (left) and insulin infusion rate (17) (right).

Table 1Values of parameters of the model (10)(14)

Parameter Value Description

k p1;1 0.14 min1 Plasma glucagon transitional rate

k p2;1 0.14 min1 Plasma insulin transitional rate

k p1;2 0.3 min1 Plasma glucagon degradation rate

(Bastl et al., 1977; Emmanouel et al., 1978;Lefebvre et al., 1974 )

k p2;2 16 min1 Plasma insulin degradation rate

(Sturis et al., 1991 )k s1;1 6 107

M 1min 1

Glucagon association rate to its receptors(Kahn, 1976 )

k s2;1 4:167 10 4

mU =l 1min 1Insulin association rate to its receptors(Kahn, 1976; Sedaghat et al., 2002 )

k s1;2 0.01 min 1 Glucagon degradation rate

k s2;2 0.01 min 1 Insulin degradation rate ( Sturis et al.,1991 )

k r1 0.2 min 1 Glucagon receptor degradation ratek r2 0.2 min 1 Insulin receptor degradation rate ( Sturis et

al., 1991 )R01 9 10

13 (M) Total concentration of glucagon receptors

R02 0.52 (mU/l) Total concentration of insulin receptors(9 10 13 (M), Sedaghat et al., 2002 )

V gpmax 80 (mg/l/min) Maximum velocity of glycogenphosphorylase ( Winston and Reitz, 1981 )

K gpm 600 (mg/l) MichaelisMenton constant of glycogenphosphorylase ( Winston and Reitz, 1981 )

V gsmax 3:87 10 4

(mg/l/min)Maximum velocity of glycogen synthase(Nakai and Thomas, 1975 )

K gsm 67.08 (mg/l) MichaelisMenton constant of glycogensynthase ( Nakai and Thomas, 1975 )

k 1 8 105

mU =l 1Feedback gain for glycogen synthase

k 2 1 1012 M 1 Feedback gain for glycogen synthasek 3 4 1012 M 1 Feedback gain for glycogen

phosphorylase


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indices k 1=g2q g2=q k 1; k 2=g2q g2=q k 2; k 3=g2q g2=q k 3with respect to k 1; k 2; k 3 are plotted in Fig. 5 . These twogures show that the indices are small and then thefeedback control gains k i i 1; 2; 3 are robust. Thegures also predict that glucose and insulin levels decreaseas the gain k 1 of insulin receptors increases and that

glucose and insulin levels increase as the gains k 2; k 3 of glucagon receptors increase. In control engineering, thefeedback gain robustness is required in the feedbackcontrol designs. We speculate that cells may have alsodeveloped the robustness during their evolution.

Finally, we estimate the insulin transitional rate k p2;1using the experimental data of Korach-Andre et al. (2004) .Using their insulin and glucose data, we calculate

ys p2t ds p2

dt u2 k

p2;2s

p2

ds p2

dt

Rm1 b

2exp a

2C

1 g

2

k p2;2s p2

at t 0, 60, 90, 120, 150, 180, 240, 300, 360, 420, 480, 540.Then using the linear regression

ys p2 k p2;1s

p2,

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Table 2Values of parameters of the model (10)(14)

Parameter Value Description

V 11 (l) Volume of cellular insulin space(Tolic et al., 2000 )

V p 3 (l) Volume of plasma insulin space(Tolic et al., 2000 )

U b 7.2 (mg/l/min) Maximum velocity of insulin-independentglucose utilization ( Tolic et al., 2000 )

U 0 4 (mg/l/min) Minimum velocity of insulin-dependentglucose utilization ( Tolic et al., 2000 )

U m 94 (mg/l/min) Maximum velocity of insulin-dependentglucose utilization ( Tolic et al., 2000 )

G m 2:23 10 10

(M/min)Maximum glucagon infusion rate

Rm 70 (mU/l/min) Maximum insulin infusion rate(Tolic et al., 2000 )

C 1 2000 (mg/l) ( Tolic et al., 2000 )C 2 144 (mg/l) ( Tolic et al., 2000 )C 3 1000 (mg/l) ( Tolic et al., 2000 )

C 4 80 (mg/l) ( Tolic et al., 2000 )b 1.77 (Tolic et al., 2000 )a1 0.005 mg=l 1a2 1=300 mg=l 1 (Tolic et al., 2000 )b1 10b2 1 (Tolic et al., 2000 )

0 200 400 6000.8

0.6

0.4

0.2

0

t (min)

I n s u

l i n s e n s i

t i v i t y

i n d e x

t o k 1

0 200 400 6000

0.002

0.004

0.006

0.008

0.01

t (min)

I n s u

l i n s e n s i

t i v i t y

i n d e x

t o k 2

0 200 400 6000

0.5

1

1.5

t (min)

I n s u

l i n s e n s i

t i v i t y

i n d e x

t o k 3

Fig. 4. Insulin sensitivity indices k 1=s p2q s p2=q k 1;k 2=s

p2q s

p2=q k 2, k 3=s

p2q s

p2=q k 3with respect to k 1; k 2; k 3 , respectively. Insulin levels decrease as the gain k 1

of insulin receptors increases and insulin levels increase as the gains k 2; k 3 of glucagon receptors increase.

0 200 400 6000.2

0.15

0.1

0.05

0

t (min)

G l u c o s e s e n s

i t i v i

t y i n d e x

t o k 1

0 200 400 6000

1

2

3x 10 3

t (min)

G l u c o s e s e n s

i t i v

i t y i n d e x

t o k 2

0 200 400 6000.05

0

0.05

0.1

0.15

t (min)

G l u c o s e s e n s i

t i v i t y

i n d e x

t o k 3

Fig. 5. Glucose sensitivity indices k 1=g2q g2=q k 1; k 2=g2q g2=q k 2;k 3=g2q g2=q k 3 with respect to k 1; k 2; k 3 , respectively. Glucose levels decrease as thegain k 1 of insulin receptors increases and glucose levels increase as the gains k 2; k 3 of glucagon receptors increase.


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we obtain k p2;1 0:32. It is adjusted to 0.14 sincesimulations show that the value of 0.14 gives a betterresult. The glucagon transitional rate k p1;1 is assumed to bethe same as k p2;1.

The exogenous glucose input is the experimental data of Korach-Andre et al. (2004) , which is reproduced in Fig. 6 .

The data are converted into the glucose input rate(mg/l/min) by multiplying the data by 70 (kg) and dividingit by 6 (l) because the blood volume of a person with theweight of 70 (kg) is about 6 (l). The initial data are asfollows: s p10 1:4 10

11 (M), s p20 2 (mU/l), g10 200 (mg/l), and g2 0 918(mg/l).

Using MATLAB, we solve numerically the system(10)(14) and plot the results in Fig. 7 . The results showthat the glucose and insulin proles simulated by thesystem (10)(14) agrees approximately with the experi-mental data, although they do not match perfectly. Forinstance, they peak at 60 min as the data do and theyalmost match with the data 360 min later (300 for insulin).To further see the model accuracy, we analyze the averageerrors between the simulations and experimental data. Wedene the relative errors by

eg 1T Z

T

0

jg2t gd tjgd t

dt,

ei 1T Z

T

0

js p2t sd tjsd t

dt,

where g2; s p2 are the simulated glucose and insulin, and

gd ; sd are the experimental glucose and insulin. The relativeerrors of our model (10)(14) are as follows:

eg 0:09; ei 0:34.

To propose another feedback rate to test our model, welook at the process of pancreatic insulin secretion, which iscomplex and is not completely understood. Insulin secre-tion from pancreatic b-cells is subject to tight control byglucose, other nutrients, neurotransmitters, and hormones(Henquin, 2004 ) and is regulated by ATP-sensitive K

channels ( K ATP channel) in the plasma membrane ( Bergg-ren et al., 2004; Fridlyand et al., 2005; Hagren andTengholm, 2006; Herrington et al., 2006; Hinke et al.,2004; Landa et al., 2005; Li et al., 2006a ). The metabolismof glucose leads to an increase of the ATP/ADP ratio andthen closure of K ATP channels. The closure of K ATP channels leads to depolarization of the plasma membrane,opening of voltage-gated calcium channels, and an increasein cytosolic free calcium, to trigger insulin secretion.

Stimulatory hormones and neurotransmitters, such asglucagon-like peptide 1 (GLP-1) and acetylcholine, usuallypotentiate insulin secretion by a dual action. Theymoderately increase the triggering signal through complex,variable, but largely glucose-dependent mechanisms. Theyalso produce major amplifying signals, mainly throughactivation of protein kinases, in particular protein kinase A(PKA) and protein kinase C (PKC) ( Henquin, 2004 ).

The insulin secretion in response to the rise of glucoseconcentration is not immediate, but it takes certain timeranging from 5 to 15 min ( Cherrington et al., 2002; Li et al.,2006a; Sturis et al., 1991; Tolic et al., 2000 ). b -cells may bealso sensitive to a change of the glucose concentration.Thus we modify u2 and propose the following time-delayedderivative feedback rate of insulin infusion

udd ;2 aR m max 0 ;dg2dt

Rm

1 b2 exp a2C 1 g2t t ,

(18)

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0 120 240 360 480 6000

20

40

60

80

t (min)

E x o g e n o u s g

l u c o s e

( m g

/ l / m i n )

Fig. 6. Exogenous glucose input rate from the experimental data of Korach-Andre et al. (2004) (courtesy of Franc - ois Pe ronnet).

0 120 240 360 480 600456789

1011

t (min)

G l u c o s e

( m m o

l / L )

New modelExperiment

0 120 240 360 480 6000

20

40

60

80

100

120

t (min)

I n s u

l i n ( m U / l )

New modelExperiment

Fig. 7. Comparison of blood glucose (left) and insulin (right) dynamics simulated by the system (10)(14) and the feedback rates (16) and (17) with theexperimental data of Korach-Andre et al. (2004) (courtesy of Franc - ois Pe ronnet).


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where t 7 (min) is the delay time ( Li et al., 2006a )and a 0:003(min/(mg/l)) is a feedback gain constant.

In automatic control design, this derivative controlleraR m dg2=dt is a standard control mechanism to enhancethe sensitivity of a controller ( Ogata, 2002 ). Fig. 8 showsthat the simulated glucose and insulin proles with udd ;2 area little bit better than with u2 .

We now use the model to predict the sensitivity of theglucose regulation mechanism to small perturbations of insulin infusion. We rst consider the case of perturbationsof infusion magnitudes by doubling the infusion rate:

u2 2Rm

1 b2 exp a2C 1 g2,

and decreasing the rate by 50%:

u2 0:5R m

1 b2 exp a2C 1 g2.

Fig. 9 shows that, in both cases, glucose levels stay in thenormal range in most of time and do not changedramatically. This less sensitivity of the glucose regulationmechanism to the variations of infusion magnitudes maybe just the property of the model and does not reect thereal biological situation because experimental data indicatethat postprandial glucose pattern is sensitive to thevariations ( Panteleon et al., 2006 ).

The experimental study by Grodsky (1972, Fig. 7)showed that the insulin secretion as a function of glucose

concentration has a sigmoidal shape (S shape). Thus weconstruct the following rate:

u3 K 1arctan gmax a3 arctan a3g2 gmax , (19)

where gmax 2000(mg/l), K 1 24(mU/l), a3 0:0031mg =l 1. These parameters are selected so that the graphof the function has a sigmoidal shape, as shown in Fig. 10 .Fig. 11 shows that the glucose and insulin proles

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0 120 240 360 480 6004

5

6

7

8

9

10

11

t (min)

G l u c o s e

( m m o

l / L )

Without delayWith delayExperiment

0 120 240 360 480 6000

20

40

60

80

100

t (min)

I n s u

l i n ( m U / l )

Without delayWith delayExperiment

Fig. 8. Comparison of blood glucose (left) and insulin (right) dynamics simulated by the feedback rate (17) with the one simulated by the t ime-delayedderivative feedback rate (18).

0 120 240 360 480 6004

6

8

10

12

14

t (min)

G l u c o s e ( m m o

l / L )

Half infusionNormal infusionDouble infusionExperiment

0 120 240 360 480 6000

20

40

60

80

100

t (min)

I n s u

l i n ( m

U / l )

Double infusionNormal infusionHalf infusionExperiment

Fig. 9. Blood glucose (left) and insulin (right) dynamics when the magnitude of the rate (17) is doubled or decreased by 50%.

0 1000 2000 3000 40000

20

40

60

Glucose (mg/l)

u 3

( I I R ) ( m U / l / m

i n )

Fig. 10. Insulin infusion rate (19).


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simulated with u3 are close to the ones simulated with u2.Thus a small variation of infusion rate is acceptable.However, a big variation is not acceptable. To see this, weconsider the following two infusion rates

u4 K 2arctan gmax a4 arctan a4g2 gmax , (20)

u5 K 3 ln1 a5g2, (21)

where gmax 2500(mg/l), K 2 24 (mU/l), K 3 8 (mU/l),a4 0:008 mg=l 1, and a5 0:05 mg =l 1. Thegraph of u4 in Fig. 12 (left) show that u4 has a sig-

moidal shape while u5 (right of the gure) does not. Thesimulated glucose proles with these two infusionrates in Fig. 13 (left) indicate that u4 results in hypergly-cemia while u5 results in hypoglycemia. Thus the glucoseregulation mechanism is sensitive to the change of infusionshapes.

Evidently, the infusion rate u5 is bad since insulin is stillinfused with a large amount even though glucose levels arelow. Then it results in the hypoglycemia. Although u4 had asigmoidal shape, u4 is not sensitive enough to glucose levels

because a very small amount of insulin is infused when the

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0 120 240 360 480 6004

6

8

10

12

t (min)

G l u c o s e

( m m o

l / L )

Infusion u 2Infusion u 3Experiment

0 120 240 360 480 6000

20

40

60

80

100

t (min)

I n s u

l i n ( m U / l )

Infusion u 2Infusion u 3Experiment

Fig. 11. Comparison of blood glucose (left) and insulin (right) dynamics simulated by the feedback rate (17) with the one simulated by the feedback rate(19). Although these two controls are dened by different functions, their graphs are very close and then the resulted glucose and insulin proles are verysimilar.

0 1000 2000 3000 40000

20

40

60

Glucose (mg/l)

u 4

( I I R ) ( m U / l / m i n )

0 1000 2000 3000 40000

10

20

30

40

Glucose (mg/l)

u 5

( I I R ) ( m U / l / m i n )

Fig. 12. Insulin infusion rates (20) (left) and (21) (right).

0 120 240 360 480 600

5

10

15

20

t (min)

G l u c o s e

( m m o

l / L )

Infusion u 2Infusion u 4Infusion u 5Experiment

0 120 240 360 480 6000

50

100

150

200

t (min)

I n s u

l i n ( m U / l )

Infusion u 2Infusion u 4Infusion u 5Experiment

Fig. 13. Comparison of blood glucose (left) and insulin (right) dynamics simulated by the feedback rate (17) with the ones simulated by the feedback rates(20) and (21).


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glucose is at the very high level of 2000 (mg/l). Hence itresults in the hyperglycemia.

In conclusion, the model predicts that while thesigmoidal shape is critical for the design of the outputfeedback rates of insulin infusion, the inection location of the shape is also important. It should be located around

2000(mg/l) like u2 and u3.The above numerical results show that the simulatedproles have a sharp decrease after the rst peak, whichdoes not exist in the data. This might imply that themodeling of insulin signaling pathway was simplied toomuch and a detailed model for the pathway may benecessary. In addition, since the process of insulin secretionis complex as stated above, a simple function such as udd ;2probably is not powerful enough to describe the complexprocess. We may need to establish a mathematical modelfor the process and then the infusion rate will be given bythe model, rather than a function. Another possiblesolution of this problem is to model the glucose utilizationin tissue cells to improve f 3 since the impact ( f 3) of insulinon the insulin-dependent glucose utilization might beoverestimated. This scenario could tell that the modelingat the molecular level is more difcult than at the physio-logical level, but is needed for exploring the molecularregulation mechanisms.

4. Assessment of insulin sensitivity

The insulin sensitivity is dened in the steady states(Bergman et al., 1979, 1985 ). At the steady states of thesystem (10)(14), we have

G in U b1 exp g2=C 2

g2C 3

U 0 U m U 0 s2=C 4b

1 s2=C 4b !, (22)where denotes the steady state. The glucose effectivenessis dened as the quantitative enhancement of glucoseinfusion due to an increase in the plasma glucoseconcentration g2 (Bergman et al., 1979, 1985 ):

E G q G in

q g2

U bC 2

exp g2=C 2

1C 3

U 0 U m U 0 s2=C 4b

1 s2=C 4b !. (23)The insulin sensitivity is dened as the increment of glucoseeffectiveness with respect to the increment of the steadyintracellular insulin (not plasma insulin since it is theintracellular insulin that acts on glucose metabolism):

S I q E G

q s2

bU m U 0C 3 s2

s2C 4

b

1 s2C 4

b !2

. (24)

The dependence of S I on the parameters b ; U m; U 0; C 3; C 4is physiologically plausible since they determine the insulininuence on insulin-dependent glucose utilization. Since

the steady state s2 depends on the properties of insulinreceptors such as the total concentration R02 and theassociation rate k s2;1, S I also reects the efciency of binding of insulin to its receptors.

For the feedback infusion rates (16) and (17), we have s2 29:41899438 (mU/l) if we set g2 1000 (mg/l). Using

(24), we then obtain the insulin sensitivity S I 0:0006730656036 mU =l 1 min 1. For the feedback in-fusion rates (16) and (19), we have s2 44:57340577 (mU/l)and S I 0:0006911447233 mU =l 1 min 1. These indexare within the range (0.00010.00091) of insulin sensitivitywith the mean 0 :00043 0:00007 assessed by Bergmanet al. (1985) .

5. Inputoutput stability

Stability is a major concern in feedback control designfor automatic control systems in engineering because a

feedback control law not only can stabilize a system butalso can destabilize a system. We use Lyapunovs indirectmethod ( Khalil, 2002, Theorem 4.7 ) to investigate thestability of the system (10)(14). The equilibrium of thesystem depends on the steady state of glucose input G in.In order for the blood glucose g2 has the steady state100(mg/dl), the steady state G in of the system (10)(14)with the feedback infusion rates (16) and (17) and theparameter values given in Tables 1 and 2 must be equal to24.28422196 (mg/l/min). The blood volume of a personwith the weight of 70 (kg) is about 6 (l). Thus this inputsteady state gives the average exogenous glucose infusionrate: 24 :28422196 6=70 2:08 (mg/kg/min), which iswithin the normal range. In this case, using the Maple,we nd the following equilibrium

s p1 4:614147109 1011 M ; s p2 7:862490315 mU =l,

s1 1:75277312 10 10 M ; s2 29:41899438 mU =l,

r1 4:49606972 10 14 M ; r2 0:03007115957 mU =l,

g1 829 :6772512 mg =l; g2 1000 mg =l,

G in 24:28422196 mg =l=min .

The linearized system of the system (10)(15) at thisequilibrium is as follows

dxdt

Ax BG in; y Cx , (25)

where

x s p1 s p1; s

p2 s

p2; s1 s1; s2 s2; r1 r1; r2 r2,

g1 g1; g2 g2T ,

B 0 0 0 0 0 0 0 0 1 T ,

C 0 0 0 0 0 0 0 0 1 ,

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and A is the Jacobian matrix at the equilibrium. Using theMATLAB, we nd the following eigenvalues of A

0:4391404733886660:3068983470414610:219621388652266

0:2112425350974720:011149965727366 0:036624692581664 i 0:011149965727366 0:036624692581664 i

0:0101026449236320:002280206884769

266666666666664

377777777777775

.

Since the real parts of all eigenvalues are negative, itfollows from the inputoutput stability theorem ( Khalil,2002, Corollary 5.1 ) that there exist a constant r0 suchthat if an initial state belongs to the ball centered at theabove equilibrium point with the radius r0, then the system(2)(9) is small-signal nite-gain L p stable for each p 2 1; 1 . This implies that if the glucose input fromfood is slightly uctuating around G in 24:28422196(mg/L/min), then the blood glucose g2 is oscillating around100 (mg/dl).

For the feedback infusion rates (16) and (19), we canobtain the similar inputoutput stability results using thesame arguments above.

It could be an interesting mathematical problem to studythe inputoutput stability of the system (10)(14) withgeneric parameters. The difculty in this case is to calculatethe eigenvalues of the Jacobian A or construct a Lyapunovfunction. We used the Maple software to symbolically

calculate its characteristic polynomial, and waited for longtime, but could not get a result. This implies that thepolynomial is huge.

6. Controllability and observability

Controllability and observability are structural proper-ties of a dynamical system. Thus they may be used toexplain why the regulatory system of blood glucose can beregulated.

The system (25) is controllable if for any initial state x0and any desired state x f , there exists a control G in such thatxT x

f for some T 4 0. The system (25) is observable if

any initial state can be uniquely determined by the output yt g2 g2 (blood glucose) over 0; T for some T 4 0.

To check the controllability of (25), it sufces to examinethe rank of the Kalman controllability matrix ( Morris,2001; Ogata, 2002 )

C BjAB j jA6BjA7B .

Using the Maple software, we compute the determinant of the matrix

det C 6:825614968 10 79 .

Therefore C has the full rank of 8 and then the system (25)is controllable ( Morris, 2001 ). In the same way, we can

show that Kalman observability matrix

O C T jAT C T j j AT 6C T jAT 7C T

has the full rank of 8 and then the system (25) is observable(Morris, 2001 ). This structural property of controllabilityand observability could explain why the blood glucose can

be regulated.For the feedback infusion rates (16) and (19), we canobtain the same controllability and observability resultsusing the same arguments above.

It could be an interesting mathematical problem to studythe controllability and observability of the system (2)(9)with generic parameters. As mentioned before, we have adifculty in symbolically calculating the rank of theKalman controllability matrix in this generic case usingthe Maple software.

7. Optimal insulin infusion rate

The feedback infusion rates (16) and (17) are proposedon the basis of the physiology of pancreas and may not givethe optimal insulin infusion rate. To nd the optimal ratemathematically, we need to use the optimization technique.For this, we dene the functional

F u1; u2 Z 1440

0jg2t ; u1; u2 100j2

j u1tj2 j u2tj2dt, (26)

and then minimize it in the space L 20; 1440 L 20; 1440 :

minu1 ;u22L

20;1440 L

20;1440

F u1; u2, (27)

where g2t ; u1; u2 is the solution of the system (10)(14)corresponding to u1; u2. The functions um1 t; u

m2 tsatisfying

F um1 t; um2 min

u1 ;u22L20;1440 L20;1440F u1; u2

will give optimal glucagon and insulin infusion rates. It iswell known that an optimal control problem is difcult tosolve even for a linear system. Thus it can be expected thatthis optimal control problem is difcult to solve becausethe system (10)(14) is nonlinear and complex. Hence thisidea sounds good mathematically, but may not be veryuseful practically.

8. Discussion

We built up a new control system for the regulatorysystem of blood glucose at the molecular level. This controlsystem consists of three subsystems: the transition sub-system of insulin and glucagon from plasma to cellularspace, the insulin and glucagon receptor binding subsys-tem, and the glucose production and utilization subsystem.Since liver is one of three musketeers (muscle, liver, andpancreas) ghting to control glycemia ( Klip and Vranic,2006 ), we included more details about it in the system thanthe existing models. The glucagon and insulin signaling

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pathways and the process of conversion between glucoseand glycogen were included, but details in the pathwaysand the process were treated as black boxes.

The control system was validated by using the experi-mental data of Korach-Andre et al. (2004) . The numericalresults showed that the glucose and insulin proles

simulated by the system approximately agree with theexperimental data, although they do not match perfectly.For instance, they peak at 60 min as the data do and theyalmost match with the data 360 min later (300 for insulin).

The model predicts that while the sigmoidal shape of theinfusion rates is critical for the design of insulin infusionrates, the inection location of the shape around 2000(mg/l) is also important. It also predicts that the glucoseregulation mechanism is sensitive to the change of theshape and the inection location.

Based on our new control model, we developed a newformula to compute the insulin sensitivity S I that wasdened by Bergman et al. (1979) . The formula shows thatthe insulin sensitivity depends on various parameters thatdetermine the insulin inuence on insulin-dependentglucose utilization and reect the efciency of binding of insulin to its receptors. Therefore, such a dependence isphysiologically plausible. Our case studies showed that theindex calculated by using the formula is close to the onereported by Bergman et al. (1985) .

Using Lyapunov indirect method, we proved that thenew control system is inputoutput stable. The stabilityresult provides theoretical evidence for the phenomenonthat the blood glucose uctuates within a narrow range inresponse to the exogenous glucose input from food. We

also proved that the regulatory system is controllable andobservable. These structural system properties couldexplain why the glucose level can be regulated.

The simulated glucose and insulin proles have a sharpdecrease after the rst peak, which does not exist in theexperimental data. This might imply that the modeling of insulin signaling pathway was oversimplied and a detailedmodel for the pathway may be necessary. In addition, sincethe process of insulin secretion from pancreatic b-cells iscomplex, a simple function such as those proposed in thepaper probably is not powerful enough to describe thecomplex process of insulin secretion. We may need to buildup a control sub-system for the process with glucose as aninput and insulin as an output. Then the feedback rate willbe determined by the sub-system, rather than a function.

A number of open mathematical problems were left.These problems include the global existence and unique-ness of a positive solution, the global stability, the globalcontrollability and observability of the nonlinear controlsystem, and the optimal control of insulin infusion.

Acknowledgments

We thank Franc - ois Pe ronnet for sending us their glucoseand insulin data published in Korach-Andre et al. (2004) ,who spent more than one month on retrieving the data. We

gratefully acknowledge reviewer #3 of the paper forproviding critical, objective, and constructive commentsand references ( Albisser et al., 1974a, b ; Hovorka, 2006;Man et al., 2007; Panteleon et al., 2006; Steil et al., 2006 )who have corrected and improved greatly the old version of the paper. Indeed, the reviewer contributed a lot to the

paper and served as a anonymous advisor on the insulintopic. Liu was supported by the University ResearchCouncil Fund of the University of Central Arkansas. Tangwas supported by the Kathleen Thomsen Hall CharitableTrust Fund.

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