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.
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
C
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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
 

 
 


1
32
d
d
d
d
b
b
Gt PGGtXtGt
t
Xt PItIPXt
t


(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
P
1
P
3
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).



 
 


1i
32
nf
d
d
d
b
b
PG GtXtGtg
t
Xt PItIPXt
t
 

(2)
where ginf is the infusion of glucose by a unit of volume.
Eq
dGt
The equilibrium points can be determined by making
uation (2) zero.



1i
32
0
0
be e
eb
GXG g
PI IPX
PG nf


  (3)
Then, at the steady-state


 
inf 1
3
inf 1
2
ebe
ebeb e

G
P
gXGPG
g
IIGPGG
P


(4)
Finally, the glucose infusion rate is determined by


3 inf
1
2
eb
e
P g
I
IP
GP



 (5)
The derivative of Equation (5) yields the insulin sen-
sitivity (SI)
2
3 inf
2
Iee
P g
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
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M. Tunç et al. / J. Biomedical Science and Engineering 4 (2011) 391-396 395
No
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.
)
c
model.
T
Group 1 (Hypoglycemic) Group 1 (Non-Hypoglycemic
Subject No. [min–1/(µU ml–1)] Subject No. [min–1(µU ml–1)]
SI
× 10–3
SI
× 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
gr
0
0
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
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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
S
Insulin
S
Insulin
ubject No. Secretion R2
[µU/dl]
ubject No. Secretion R2
[µU/dl]
1 7158.71
JBiSE
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
40
for the fall of plasma glucose
/dl has been investigated. T
for
th
other
so
uccessfully identi-
fie
ENTS
ıdıka Karakaş, M.D.,
o 60 minutes. The am insulin secreted for
1
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
([e-mail]), of University of California, Davis for pro-
viding the clinical data and guidance whenever needed.
REFERENCES
[1] International Diabetes Federation (2011) Diabetes Atlas,
ih.gov/pubmedhealth/PMH0001423
. and Chetouani, A. (2006) A critical review
System identi-
02
[6] C. (2002) The oral
redictive
[8] non, A. and Owens, D.
[9] lani, P., De Marzi, C.,
[10] ) Mathematical mod-
l Models for Glucose Insulin
available at http://www.diabetesatlas.org/content/eur-data
[2] National center for Biomedical Information (2011) Hy-
poglycemia Insulin shock, Low blood sugar, PubMed
Health, available at
http://www.ncbi.nlm.n
[3] Bergman, R.N., Phillips, L.S. and Cobelli, C. (1981)
Physiologic evaluation of factors controlling glucose tol-
erance in man. Journal of Clinical Investigation, 68,
1456-1467.
[4] Boutayeb, A
of mathematical models and data used in diabetology.
Bio-Medical Engineering, 5, 43.
[5] Chen, Y., Chen, Y. and Wienert, S. (2004)
fication of a nonlinear glucose insulin dynamics. Pro-
ceedings of the 5th World Congress on Intelligent Control
and Automation, 6, 5577-5581.
doi:10.1109/WCICA.2004.13438
Man, C.D., Caumo, A. and Cobelli,
glucose minimal model: Estimation of insulin sensitivity
from a meal test. IEEE Transactions on Biomedical En-
gineering, 49, 419-429. doi:10.1109/10.995680
[7] Lynch, S.M. and Bequette, B.W. (2002) Model p
control of blood glucose in Type 1 diabetics using sub-
cutaneous glucose measurement. The Proceedings of
American Control Conference Anchorage, 5, 4039-4043.
doi:10.1109/ACC.2002.1024561
Nguyen, H., Nguyen, D.K., Shan
(1997) Estimation of minimal model parameters with the
use of an adaptive observer for suprabasal insulin action.
Proceedings of the 19th annual International Conference
on Engineering in Medicine and Biology Society, Chi-
cago, 30 October-2 November 1997, 2146-2148.
doi:10.1109/IEMBS.1997.758778
Natalucci, S., Di Nardo, F., Staffo
Morosini, P. and Burattini, R. (2003) Glucose absorption
and insulin sensitivity from oral glucose tolerance test.
Proceedings of the 25th Annual International Conference
of the IEEE on Engineering in Medicine and Biology So-
ciety, Mexico, 17-21 September 2003, 2758-2760.
doi:10.1109/IEMBS.2003.1280488
Makroglou, A. and Kuang, Y. (2006
els and software tools for the glucose-insulin regulatory
system and diabetes: An overview. Applied Numerical
Mathematics, 56, 559-573.
[11] Van Riel, N. (2004) Minima
Dynamics-A Matlab Implementation. TUE, Eindhoven
University of Technology, Eindhoven.