Mathematical Model Of The Intra Venous Glucose Tolerance Test Using Modified Minimal Model

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to analyze the results
of glucose tolerance. .tests 111
common Iy ｵｳｾ＠ ｾ､＠
·
humans. In this paper. a mathematical model for dcscnh111g the

ｾｳ･､＠

lucose infusion rate is introduced . The modified minimal model was
to study a few sets of published data including healthy humans
and type 2 diabetes. The gluc0se-i nsulin system is a dynamic

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insulin system.

588

Nurullaeli, Agus Kartono and Kiagus Dahlan
Introduction

Human bodies need to maintain a glucose concentration level in a
narrow range (70-110 mg/di or 3.9-6.04 mmol/l after overnight fast) . lfone's
glucose concentration level is significantly out of the normal range, then this

person is considered to have a plasma glucose problem: hyperglycemia or
hypoglycemia. Diabetes mellitus, or simply, diabetes, is characterized by
high blood glucose levels resulted from defects in the glucose-insulin
endocrine metabolic regulatory system, in which either the pancreas does not
ｲ･ｬｾｳ＠
insulin or the cells do not properly use insulin to uptake glucose.
. Diabetes mellitus has become an epidemic disease in the sense of life
st)rle. To diagnose whether or not an individual subject is already a diabetic
or has the potential to develop such disease, the so-called metabolic portrait,
including the insulin sensitivity index and glucose effectiveness, of the
subject needs to be sketched. To this end, several glucose tolerance tests have
been developed and applied in clinics and experiments. The fundamental idea
of such tests is to examine the response of insulin, called insulin sensitivity,
after a large amount of glucose is infused into one's body. A commonly used
protocol is the intravenous glucose tolerance test (IVGTT).
Type 2 diabetes (formerly called non-insulin-dependent diabetes
mellitus, NIDDM, or adult-onset diabetes), a more widespread metabolic
disorder, is primarily characterized by insulin resistance, relative insulin
deficiency and hyperglycemia. Some cases of type 2 diabetes also appear to
be an autoimmune disease where the immune system attacks the P-cells,

decreasing the function of producing insulin, while other type 2 diabetes
cases may simply result from excessive body weight that strains the ability of
the P-cells to produce sufficient insulin.

In the procedure of IYGTT, overnight fast is required for the subject, and
then the subject is given a bolus of glucose infusion intravenously, for
example, 0.33g/kg body weight or 0.5g/kg body weight of a 50% solution,
and is administered into an antecubital vein in approximately 2 min. To
observe the metabolic regulation between the glucose and insulin, within
the next 180 min, the plasma glucose and serum insulin of the subject

Mathematical Model of the Intravenous Glucose Tolerance Test . . . 589
arc sampled frequently at the time marks 2', 4', 6', 8', JO', 12', 14', 18',
21', 24', 30', 35', 40', 45', 50', 60', 70', 80', 90', 100', 120', 140', 160' and
180'. According to the rich information constituted in the sampled data,
appropriate analysis can reveal the metabolic portrait.
One popular approach of the analysis is as follows : (I) fonnulate or
choose a well-formulated kinetic model based on physiology; (2) estimate
the parameters of the IVGTT model with experimental data, and then (3) the
parameter values are used to obtain physiological infonnation, for example,

the insulin sensitivity and glucose effectiveness.
The intravenous glucose tolerance test (IVGTT), as an important
experimental procedure for the estimation of glucose effectiveness and
immlin sensitivity, has been widely used, because it is relatively easy to
measure, and its analysis generates a lot of information. The test includes
injecting a glucose bolus and then collecting a set of plasma glucose and
insulin samples over a period of 3-5 h. For diabetic patients, it is sometimes
necessary to have an exogenous insulin infusion because of insufficient
endogenous insulin secretion despite stimulation by the glucose injection.
The glucose-insulin system as an in\egrated dynamic system m
physiology, however, should be described mathematically as a whole. When
an integrated dynamic system is split into two interacting sub-systems, and
then, the system parameters are optimized by fitting the measured data
separately, the generated parameters cannot be thought as optimal ones for
the whole system. In the glucose-insulin physiological system both glucose
and insulin have feedback effects on each other via pancreatic response and
stimulation. It is worth to obtain optimized model parameters from measured
plasma glucose and insulin concentrations simultaneously in the time course
by using the regression method. This single-step parameter-fitting process
could generate a real optimal approximation to the dynamical integrated

glucose-insulin system without losing the interaction infom1ation implicitly
contained in the measured concentration profiles.
The modified minimal model approach yields estimates of insulin
sensitivity (S 1 ) and glucose effectiveness (S(i ). Insulin sensitivity represents

590

Nurullacli, Agus Kartono and Kiagus Dahlan

Mathematical Model of the Intravenous Glucose Tolerance Test . . . 591

a measure of the dependence of fractional glucose disappearance. The latter

in plasma; p 2 and p 3 are the fractional transfer coefficients of insulin into
and out of the remote compartment where it acts to accelerate glucose uptake

term represents a measure of the fractional ability of glucose to lower its own
concentration in plasma independent of increased insulin at basal insulin, to

or decrease glucose production. G0 is the glucose concentration one would

obtain immediately afler glucose injection if there were instantaneous mixing

normalize its own concentration within the extracellular glucose pool. This
normalization occurs after glucose administration and may involve glucose-

of glucose in the extracellular fluid compartment. Gh and 1,, arc the pre-

mediatcd inhibition of hepatic glucose output and/or accelerated glucose
disposition.

injcction values of glucose and insulin. Insulin sensitivity, represented as the
insulin sensitivity index, S / is calculated as p 3 / P2 . Glucose effectiveness is

The Modified Minimal Model from the IVGTT

the ability of glucose, at basal insulin, to normalize its own concentration.
This latter parameter is represented as SG and is equal to p 1 as defined in

The dynamic insulin and glucose responses to glucose injection were

equation ( 1), where

analyzed as previously described lo yield the individual parameters of the

I,.

minimal model. The modified minimal model, which we proposed as the
simplest representation that can account for the dynamics of glucose during
the IVGTT, is described by the following differential equations:

dG(t)

-;ff= P1(Gh - G(t))- X(t)G(t), G(to) =Go.
dX(t)

-;ff= -p2X(t) + p3(/(t)- Jb), X(to)

= 0,

dl(t)
(ff= y(G(t) - a,,)1 - k(/(1)- 1,,) if G(t) > Gb, 1(1 0 ) = 10,

､ｾｴＩ＠

= -k(J(t)-1,,)

if G(t) < G,,, /(1 0 )

= 10 ,

(1)

(2)

JI

modified minimal model, which estimates the model parameters from the real

I!

data. This program is based on the non-linear least-squares estimation
method of Marquardt. /(1) is submitted to the program modified minimal

!;

model, which predicts a glucose time course, G(t), which fits data G(t) as

I
I

closely as possible in the least-squares sense. In the course of the titting,
the model yields X(t), an estimate of X(t), as well as estimates of

I.

parameters Pl· P2· p3 and Go. S1(P3/p2), S