Computational evaluation of the dynamic minimal model for the root causes of hypoglycemia

doi:10.4236/jbise.2011.45049

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J. Biomedical Science and Engineering, 2011, 4, 391-396

doi:10.4236/jbise.2011.45049 Published Online May 2011 (http://www.SciRP.org/journal/jbise/

JBiSE

Published Online May 2011 in SciRes. http://www.scirp.org/journal/JBiSE

Computational evaluation of the dynamic minimal model for

the root causes of hypoglycemia

Murat Tunç, Sedat Şişbot, A. Kaya Gülkaya

Yeditepe University, Engineering Faculty, System Engineering Department, Istanbul, Turkey.

Email: [email protected], [email protected], [email protected]

Received 4 March 2011; revised 20 March 2011; accepted 8 April 2011.

ABSTRACT

This research is an attempt to validate how glu-

cose-insulin dynamic mathematical model facilitate

to identify the root causes for hypoglycaemia. The

purpose is to determine whether increased insulin

sensitivity or increased insulin secretion causes post-

prandial hypoglycemic (PPH) response, by linking

experimental patient data with dynamic mathemati-

cal model. For this purpose two groups, as hypogly-

cemic Group 1 and non-hypoglycemic Group 2, each

of which consists of 10 people, are formed. The oral

glucose tolerance test (OGTT) is carried out for each

person in the groups by measuring plasma glucose

and insulin concentrations at every 30 minutes for a

period of 5 hours. To distinguish the actual cause of

hypoglycemia, the glucose minimal dynamic model is

used. The model is executed in MATLAB platform

using patient data and the results showed that insulin

secretion is assumed to be the potential root cause for

the hypoglycemia.

Keywords: Simulation of Minimal Model; Evaluation of

Hypoglycemia; Insulin Sensitivity Analysis

1. INTRODUCTION

Diabetes Mellitus is a metabolic disorder that is charac-

terized by hyperglycemia defined as fasting plasma con-

centration being higher than 120 mg/dl. Approximately

3.8 million people corresponding to almost 5% of the

population have diabetes in Turkey [1]. The most com-

mon form of diabetes is type 2 diabetes mellitus. This

disorder results from dual abnormalities of insulin resis-

tance and relative insulin deficiency. The current concept

is that insulin resistance forces the pancreas to produce

excess insulin over time, this results a defect in insulin

secretion and leads to the elevation in blood glucose

(hyperglycemia). Long-term hyperglycemia triggers car-

diovascular diseases, chronic renal failure, retinal dam-

age and nerve damage.

Hypoglycemia refers to low plasma glucose concen-

tration which is associated by specific symptoms such as

shakiness, nervousness, changes in awareness. Depend-

ing on the population blood glucose levels below 70

mg/dl can be associated with clinical hypoglycemia.

Hypoglycemia that occurs after food intake is called

post-prandial hypoglycemia and this may precede the

development of diabetes [2].

The goal of this study is to determine, with the help of

clinical data and a convenient mathematical model,

whether the individuals who exhibit post-prandial hypo-

glycemia is due to increased insulin sensitivity or in-

creased insulin secretion. The glucose and insulin dy-

namics have long been studied by many researchers and

some methodologies been developed to quantify insulin

resistance and insulin secretion. Among these models the

minimal model [3] which was developed by Bergman

and co-workers almost three decades ago has been a

pivotal study for modeling the glucose-insulin kinetics.

Predicated on the minimal model, several other modified

models have also been developed [4,5].

The idea of the glucose tolerance test is to challenge

the homeostasis mechanism by a dose of glucose. It is

assumed that the subsequent rise and fall of the blood

glucose is due mainly to production of insulin in re-

sponse to hyperglycaemia and that the degree of insulin

response is mirrored in the behaviour of the blood glu-

cose. If the glucose load is injected intravenously, it is

called the intravenous glucose tolerance test (IVGTT).

Another approach is called oral glucose tolerance test

(OGTT) where a glucose dose is administered orally.

Oral glucose tolerance test or meal glucose tolerance test

is a method that can quantify insulin sensitivity under

normal life condition. It is also suitable for epidemiol-

ogical studies because the procedure is simple and cheap.

This test has been used widely to identify the subjects

who develop post-prandial hypoglycemia. Typically,

these individuals have low blood glucose levels ap-

proximately 3 hours after drinking the glucose. We

speculated that these individuals may be susceptible to

M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396

392

hypoglycemia because they are more sensitive to the

action of insulin. Alternatively, they may secrete more

insulin in response to the glucose drink.

The insulin sensitivity index, which quantifies insu-

lin ability to control glucose production and utilization,

is of primary importance in the assessment of glucose

regulatory system efficiency. Quantitative evaluation of

this index is usually accomplished with methods in-

volving an intravenous administration of glucose

and/or insulin, such as the glucose clamp or the intra-

venous glucose tolerance test. Difficulty in the intra-

venous administration and high (non-physiological)

levels of glycamia and insulinemia achieved during

these tests are limitations that need to be resolved.

Measurement of insulin sensitivity from oral tests, such

as a meal glucose tolerance test (MGTT) or an oral

glucose tolerance test would better reflect the normal

life [6]. Many authors investigated insulin sensitivity

during physical activity. The effects of the physical

activities on insulin sensitivity have been challenged,

for example, using model predictive control based on

minimal model [7] and the parameters of the model

have been determined by an adaptive observer [8].

There is a rich literature about glucose and insulin

dynamics. Among many dynamic model proposals,

Bergman’s minimal model has attracted much attention

due to its conceptual structure about the biological phe-

nomena. In this study, we have used Bergman minimal

model to investigate the changes in insulin sensitivity

and insulin secretion during a 5-hour oral glucose toler-

ance test. The minimal model is based on physiological

regulation scheme such that the model uses a glucose

compartment (G) and a remote insulin compartment (I)

controlling the glucose flux.

2. METHOD AND DATA

Oral glucose tolerance test in medical practice is the

administration of glucose to determine how quickly it is

cleared from the blood. The OGTT is usually used to test

for diabetes, insulin resistance, and sometimes reactive

hypoglycemia. The patient is instructed not to restrict

carbohydrates intake in the days or weeks before the test.

The test should not be done during an illness, as results

may not reflect the patient’s glucose metabolism when

healthy. A full adult dose should not be given to a person

weighing less than 43 kg, or exaggerated glucoses may

produce a false positive result.

The patient should have been fasting for the previous

8 - 14 hours. Usually the OGTT is scheduled to begin in

the morning as glucose tolerance exhibits a diurnal

rhythm with a significant decrease in the afternoon. A

zero time (baseline) blood sample is drawn. It is usually

a fasting blood or fasting midstream. The patient is then

given a glucose solution to drink. The standard dose is

1.75 grams of glucose per kilogram of body weight, to a

maximum dose of 75 g which should be consumed

within 5 minutes.

Blood is drawn at half an hour intervals for measure-

ment of glucose, and sometimes insulin levels. The in-

tervals and number of samples vary according to the

purpose of the test. For simple diabetes screening, the

most important sample is the 2 hour sample and the 0

and 2 hour samples may be the only ones collected.

A standard 2 hour OGTT is sufficient to diagnose or

exclude all forms of diabetes mellitus at all but the earli-

est stages of development. Longer tests have been used

for a variety of other purposes, such as detecting reactive

hypoglycemia or defining subsets of hypothalamic obe-

sity. Insulin levels are sometimes measured to detect

insulin resistance or deficiency.

In our study, the data set consists of twenty non-diabetic,

obese women who underwent OGTT. The subjects are

divided into two groups, as hypoglycemic Group 1 and

non-hypoglycemic Group 2, each of which consists of

10 people. The participants ingested 75 g of glucose

(Glucola™) at 0 min. The blood samples were obtained

at baseline and every 30 min thereafter for 5 hours. The

clinical studies were executed at the University of Cali-

fornia, Davis. The protocol was approved by the Institu-

tional Review Board.

The subjects remained supine in bed throughout the

testing to avoid confounding effects of physical activity

on blood glucose. The samples for glucose were col-

lected in sodium fluoride containing tubes on ice. Other

samples were collected either in serum separation tubes,

or in EDTA or heparin containing tubes. Glucose was

measured using hexokinase method in Poly-Chem Sys-

tem clinical chemistry analyzer (Cortlandt Manor, NY).

Insulin was measured using RIA kits from the Linco

Research Inc (St. Charles, MO) with cv of 8.2%. Prior to

data analysis, a glucose concentration less than 70

mg/dL was defined as hypoglycemia. The experimental

plasma glucose and insulin concentrations for Group 1

and Group 2 are given separately in Figure 1.

3. MATHEMATICAL MODEL AND

PARAMETER ESTIMATION

The majority of mathematical models proposed in the

literature were devoted to the dynamics of glucose-insulin,

including Intra Venous Glucose Tolerance Test, Oral

Glucose Tolerance Test and Frequently Sampled Intra-

venous Glucose Tolerance Test (FSIGT) [9]. Mathe-

matical models have been used to estimate the glucose

disappearance and insulin-glucose dynamics in general.

To represent glucose-insulin dynamics, various types of

mathematical models have ben suggested. These models e

M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396

393

(a)

(b)

Figure 1. Plasma glucose and insulin concentration during OGTT. (a) Group 1 experimental patient data- mean values; (b) Group 2

xperimental patient data-mean values. e

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M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396

394

may be classified as: 1) Ordinary differential equation

(ODE) models; 2) Delay differential equation (DDE)

models; 3) Partial differential equation (PDE) models; 4)

Fredholm integral equation (FIE) models; 5) Stochastic

differential equation (PDE) models and 6) Integro-dif-

ferential equation (IDE) models [10]. In this study, we

have used the minimal model developed by Bergman et

al. which has found broad acceptance to evaluate the

IVGTT records. To evaluate the insulin sensitivity from

OGTT based on classical Bergman’s minimal model, the

model of glucose absorption in the gut is coupled with

the minimal model.

The glucose minimal model is illustrated in Figure 2.

The model consists of two differential equations. Insulin

leaves or enters the interstitial tissue compartment at a

rate proportional to the difference between the plasma

insulin level, I(t), and the basal level, Ib; if the plasma

insulin level falls below the basal level, insulin leaves

the interstitial tissue compartment, and if the plasma

insulin level rises above the basal level, insulin enters

the interstitial tissue compartment. Insulin also disap-

pears from the interstitial tissue compartment via a sec-

ond pathway at a rate proportional to the amount of in-

sulin in the interstitial tissue compartment. Similarly,

glucose leaves or enters the plasma compartment at a

rate proportional to the difference between the plasma

glucose level, G(t), and the basal level, Gb; if the plasma

glucose level falls below the basal level, glucose enters

the plasma compartment, and if the glucose level rises

above the basal level, glucose leaves the plasma com-

partment. Glucose also disappears from the plasma

compartment via a second pathway at a rate proportional

to the amount of insulin in the interstitial tissue.

The change in glucose and interstitial insulin dynam-

ics can then be described as two differential equations as

 



 



Gt PGGtXtGt

Xt PItIPXt



(1)

where P1, P2 and P3 are the system parameters and de-

fined as

P1: The rate of insulin independent glucose disap-

pearance (min–1)

P2: The constant loss rate of remote insulin degrada-

tion (min–1)

P3: Insulin dependent increase in tissue glucose up-

take ability per unit of insulin concentration above the

basal insulin [min–2 (µU/ml)–1]

It should be noted that the initial conditions are as-

sumed to be G(0) = G0, X(0) = 0 and I(0) = I0

For determining the insulin sensitivity the glucose

clamp technique is used. Glucose clamp technique is a

I(t)

Plasma Insulin

X(t)

Interstitial Insulin

G(t)

Plasma Glucose X(t)

Figure 2. Glucose minimal model.

que that maintains a constant blood glucose level

techni

in human subjects by perfusion or infusion with glucose.

Applying this technique, the minimal model takes the

form given in Equation (2).







 



PG GtXtGtg

Xt PItIPXt

 



(2)

where ginf is the infusion of glucose by a unit of volume.

dGt

The equilibrium points can be determined by making

uation (2) zero.







be e

GXG g

PI IPX

PG nf





  (3)

Then, at the steady-state





 

inf 1

ebe

ebeb e



gXGPG

IIGPGG





(4)

Finally, the glucose infusion rate is determined by





inf







 (5)

The derivative of Equation (5) yields the insulin sen-

sitivity (SI)

inf

Iee

SIG P





 (6)

Equation (2) is a pair of nonlinear differential equa-

tio

IONS

nd Table 1.

ns which are solved near equilibrium point given in

Equation (4) and from Equation (2) to (5) there are five

parameter to be estimated, namely P1, P2, P3, Gb and Ib.

MATLAB® is used for implementation and the simula-

tion of the mathematical model with the patient data.

These five model parameters are estimated by using a

weighted Least-Squares algorithm [11]. The flowchart of

the program is shown in Figure 3.

4. RESULTS AND DISCUSS

The results obtained are presented in Figure 4 a

M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396 395

Input Glucose and Insulin

clinical data

Construct Time intervals for

differential equation

Set upper and lower bounds and

initial conditions

Fetch the first Clinical data

Execute ODE45 for diffrential

Equations for G(t) and X(t) and

store

Reached the

last data

Call lsqnonlin for stored

functions

End

Fetch the next

clinical data

Figure 3. Flowchart of MATLAB program to execute dynami

able 1. Insulin sensitivity, SI, for Group 1 and 2.

)

model.

Group 1 (Hypoglycemic) Group 1 (Non-Hypoglycemic

Subject No. [min–1/(µU ml–1)] Subject No. [min–1(µU ml–1)]

× 10–3

1 01 0. .473 18 097 566

2 0.406 05 2 1.2612

3 0.330 25 3

(a)

(b)

Figure 4. Glucose Disappearlation for Group 1 and

illustrates the simulated glucose disappearance

ance calcu

Group 2.

igure 4F

profiles for Group 1 and 2. And insulin sensitivities, SI,

calculated for each subject at each group are presented in

the Table 1.

Perusal of Table 1 shows that the mean values of both

MeSE 0.31 MeSE 0.

.091 234

4 0.1709 4 0.092 431

5 .0745 235 1.2813

6 0.279 65 6 0.402 22

7 0.310 89 7 0.086 094

8 0.756 14 8 0.447 14

9 0.162 99 9 0.031 563

10 0.2916 10 0.063 167

an ± 26 ± 0.06an ± 385 ± 0.488

oups are too close to each other. This closeness avoids

consideration of that the insulin sensitivity can be as-

sumed responsible for the plasma glucose concentration

to fall below 70 mg/dl. Therefore hyperglycemia is not

related to differences in insulin sensitivity. Next, the

total amount of insulin secreted between 0 and 60 min-

utes were compared. In order to consider the affect of

total amount of insulin secretion between 0 to 60 min-

utes to glucose absorption within 5 hours experimental

insulin data are examined. The data are fitted to high

order polynomial and then integrated at boundaries of 0

M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396

396

Table 2. Insulin Secretion for Groups 1 and 2.

ion for Group 2 Insulin Secretion for Group 1 Insulin Secret

Insulin

ubject No. Secretion R2

[µU/dl]

ubject No. Secretion R2

[µU/dl]

1 7158.71

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0.99 1 3786.8 0.99

2 5299.94 0.97 2 5989.

10 2005.10

Mean ± SE 6622 ± 99Mean ± SE 4034 ± 48

35 0.97

3 7369.55 0.98 3 2595.48 0.99

4 8620.19 0.98 4 4175.29 0.99

5 2201.43 0.97 5 2230.83 0.98

6 8774.19 0.99 6 7084.49 0.98

7 4195.63 0.99 7 3570.01 0.98

8 11645.2 0.99 8 4394.71 0.99

9 8947.93 0.96 9 4030.83 0.99

44 0.98 2482.54 0.99

2 6

tount ofGroup

and 2 are presented in Table 2.

6622 ± 992 µU/dl

for the fall of plasma glucose

/dl has been investigated. T

for

other

uccessfully identi-

fie

ENTS

ıdıka Karakaş, M.D.,

o 60 minutes. The am insulin secreted for

Table 2 shows insulin secretion between 0 to 60 min-

utes for Group 1 and Group 2 areand

34 ± 486 µU/dl respectively. The difference between

the mean values is not small. This may indicate that the

root cause of hypoglycemia may be due to the insulin

secretion.

5. CONCLUSIONS

In this study, the reason

concentration under 70 mghe

OGTT test has been applied to two patients groups;

Group 1 consists of ten Hypoglycemic subjects and

Group 2 consists of ten Non-Hypoglycemic subjects.

Bergman minimal model is used to identify whether

insulin sensitivity or insulin secretion is responsible

e fall of plasma glucose concentration under 70 mg/dl

in Group 2. The minimal model is simulated using MAT-

LAB® and results are presented in Table 1 and 2.

The results presented in Table 1 indicated that insulin

sensitivities in both groups are very close to each

that insulin sensitivity is not the major cause for the

hypoglycemia. The insulin secretion in the Group 1, on

the hand, is 1.5 times of the Group 1.

Assessment of insulin sensitivity and insulin secretion

using minimal model has therefore s

d the cause of hypoglycemic response as the increased

early secretory response during the first 60 minutes fol-

lowing the oral glucose load.

6. ACKNOWLEDGEM

Authors would like to thank to Prof. Dr. S

([email protected]), of University of California, Davis for pro-

viding the clinical data and guidance whenever needed.

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