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Chapter 3
Diabetes Models
3.1 Introduction
The many complicated processes interacting in the body as it works to maintain healthy
glucose concentrations can be broken down into stages or compartments; furthermore, the
entire system can be modeled mathematically by describing the interactions between
these various compartments. Such models, called compartmental models, are usually
depicted by block diagrams. Most diabetes models contain more than one compartment,
with each compartment (a block in the diagram) described using some type of differential
equation. Before going specifically into the varied approaches and resulting models,
which formed the basis of our preliminary investigation, the following comment may be
illuminating.
Figure 3.1.1: Basic outline found in many mathematical models of diabetes.
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Many of the existing models of the glucose-insulin system contain two subsystems: (1) a
glucose subsystem, and (2) an insulin subsystem. Both glucose and insulin are removed
from the blood at a rate proportional to the product of their blood concentrations.
The removal of glucose and insulin represents their being utilized in powering the bodily
functions and thus exiting the system.
The subsystems vary in their sophistication
depending on the complexity of the model, especially the insulin subsystem, which
consisted of one to three compartments. Figure 3.1.1 shows a schematic for many of the
models examined. The model we will propose is a result of combining an insulin release
model with a model for the return of the body to basal values after a glucose bolus. This
chapter deals with the later.
The several and varied mathematical models for insulin-glucose system which were
studied [Topp, Hipzer, Li, Bergman] differ in that, some use a system of differential
equations, and others use a delay differential system introducing one or more delays [Li].
The most widely used mathematical model (based upon physiological research of the
metabolism of glucose) is called the minimal model (in future referred to as the Original
Minimal Model or OMM). In 1979, Bergman, Idler, Bowden, and Cobelli chose this
two-compartment model, after their study of seven mathematical models, as the best
model of the seven in terms of (1) identifiably of parameters, (2) meaning of parameter
values, and (3) goodness-of-fit. They deemed the model "minimal" believing that a
simpler model could not accurately estimate the data [Hipzer, pg16].
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3.2 Commonly used Minimal Models
The Original Minimal Model (OMM)
The basic purpose of the OMM (Figure 3.3.1) is to explain, mimic and help evaluate the
results of the intravenous glucose tolerance test (IVGTT). Since its creation the OMM
model has spawned several different models.
Figure 3.2.1: Block diagram of the Original Minimal Model.
The OMM consists of two compartments.
One compartment represents Glucose
concentration in the blood. The other represents insulin concentration a remote
compartment. [Hipzer]
The mathematical equations for the OMM are:
dG
 s1Gb - [s1  X(t)]G(t)
dt
dX
 p3(I p(t)  I b ) - p 2 X(t )
dt
(3.2.1)
[Bergman]
(3.2.2)
(For parameter definitions see Table 3.2.1, following.)
The remote insulin compartment, X(t), creates a delay in the effect that changing insulin
concentrations have on the glucose equation. Its value is affected by the blood insulin
concentration.
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Table 3.2.1
The glucose equation can be rewritten as follows:
dG
 -s1 (G(t) - G b ) - G(t)X(t)  - (glucose effectiven ess)(G(t) - G b ) - G(t)X(t)
dt
(3.2.3)
In this form we see that “ - s1 (G(t) - G b ) ” is a driving term*, furthermore it is a basalizing
term* driving the equation to basal value, G b (*See “Local Definitions” appendix H). The
other term, “ G(t)X(t) ”, always removes glucose as a result of its utilization in the
“remote compartment” which represents the interstitial tissues. The second equation can
be rewritten as:
p
dX
 -p2 (X(t) - 3 )(I p (t) - I b )  - p 2 ·(X(t) - insulin sensitivit y)(I p (t) - I b )
dt
p2
(3.2.4)
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This equation presumes that the “strength” of the X(t) term in the glucose equation varies
proportionally to the difference between its current strength and the insulin concentration.
Roughly, it suggests that the greater the “strength” the quicker X(t) is used up.
After an IVGTT is administered to a patient, the model is fitted to the data as well as
possible. From the best-fit model parameters an individual's glucose effectiveness ( p1 ),
and insulin sensitivity, (
p3
) are ascertained.
p2
The function I p (t) , which must be provided as an external function, may not be a
constant value. If it were not constant then one would see a lag in the effects of its
changes upon the glucose concentration. The function I p (t) is written simulate the
infusion of inter-venous insulin. That is, I p (t) is a data based function* (defined in
appendix H)
A good characteristic of the OMM is that it models the data with the smallest set of
identifiable, and meaningful parameters.
A shortcoming of the OMM is that I p (t)
(Plasma insulin concentration) is a data based function, and it must be known before one
can use the model.
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The Bergman’s Minimal Model (BMM)
The Bergman’s minimal model (below) is a three-compartment model.
One
compartment each is assigned to glucose and to insulin concentrations in the blood, and
the third to the non-observable auxiliary variable X(t) , which creates the delay in the
action of insulin on glucose [Li pg2].
The shortcomings in OMM are patched up in Bergman's model by introducing another
compartment for I p (t) and a third differential equation to model its concentration.
Figure 3.2.2: Block diagram for Bergman’s minimal model
The mathematical equations for the BMM are:
dG
 -s1 (G - G b ) - X·G
dt
(3.2.5)
dX
 p3 ( Ip - Ib ) - p 2 X
dt
(3.2.6)
dI p
dt
  (G - h )  t -  ( I p  I b 
(For parameter and variable definitions see table 3.2.2, following.)
(3.2.7)
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p3
p2
Here
is
the
insulin
sensitivity.
Of
interest
is the
term
“  (G - h)  t ” which indicates that if the glucose concentration is above
the
threshold,
then
insulin
is
added
to
the
system
at
a
rate
proportional to the product of its concentration above the threshold and
the time (= t).
Thus, it represents the secretion of insulin into the
blood as a time dependent function.
Where:
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Table 3.2.2
3.3 Simulations and Shortcomings of the Bergman’s Minimal Model
Anderson presents data representing the response of a normal glucose tolerant individual
to the IVGTT. Using the software package Matlab, which comes with a numerical solver
for differential equations, and the parameter values found by Anderson, both the original
data and the simulated data are graphed figure 3.3.1. The circled area indicates the
largest diversions of the simulated model from the actual data.
The initial insulin
concentration, I o  25 with a maximum of 132 at its peak, but Anderson’s estimated
values are I o  364 , which is about 200 above the data maximum. After about 20
minutes, the insulin simulation matches the data.
Figure 3.3.1: Simulation of the BMM. The red line is the graph of the actual data, and the blue line is the
simulation using Bergman’s minimal model and the parameters determined by Anderson using the
Bayesian method of parameter extraction [Anderson].
Shortcomings of Bergman’s Minimal Model
The BMM is designed to model the concentrations after an artificial glucose infusion. It
is a time dependent model; it does not start in response to the glucose bolus. After
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glucose is infused into the system the human body responds with the release of an insulin
bolus to compensate for the glucose. The BMM does not intend to model this response
by the body. In that sense it is not an insulin release model, but models the glucose
insulin dynamics in their return to basal values after the glucose injection.
The BMM does not have any mechanism to model the first phase peak of insulin release.
Thus, the initial values for the glucose concentration and insulin concentration, far from
the measured values, are ignored along with the first minutes. After about 15 minutes,
the BMM fits the data well. Work has been done to resolve these shortcomings.
3.4 The OGTT minimal model
Oral glucose minimal models developed for the estimation of insulin sensitivity from a
meal test or OGTT to some extent have dealt with the shortcomings of the BMM
[Caumo][Della Man][Natalucci]. The oral glucose model must also include glucose
release dynamics. The glucose release dynamics have been a major hurdle to overcome.
The new model we propose will also be fit to oral glucose data, and therefore oral
glucose model research follows.
The basic schematic used for these oral minimal models is a depicted in figure 3.5.1. The
known glucose dose of the IVGTT is replaced by the unknown rate of appearance of
orally ingested glucose Ra gut .
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Fig. 3.4.1: The oral glucose minimal model
The oral minimal model is described by the following equations.
R
dG
 SG (Gb  G (t ))  X (t )G (t )  a _ gut
dt
V
dX
 p2 ( S1 ( I (t )  I b )  X (t )
dt
(3.4.1)
(3.4.2)
The function Ra gut (t ) is the rate of appearance of orally ingested glucose at time t.
Because of lack of information on glucose in the gastro-intestinal tract during the OGTT
different methods have to be devised to model its release into the blood. A common
approach is to assume a linear spline approximation of the form
Ra gut
ki  ki 1

(t  ti 1 ) for ti 1  t  ti
ki 1 
ti  ti 1


0
otherwise

(3.4.3)
Alternatively, Natalucci has proposed a differential equation model based in the
assumption that Ra gut is proportional to the gut glucose concentration Ggut (t ) . He writes
Ra gut  kabcGgut (t )
(3.4.4)
including the extra differential equation
dGgut
dt
 Rge (t )  kabcGgut
(3.4.5)
The function Rge (t ) is the rate of gastric emptying. Schirra [Schirra] found that the
gastric emptying curve could be empirically fitted with a power exponentials of the form
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
e  ( kt ) . This implies that the rate Rge (t ) can be described by
Rge (t )  D k  t  1e
 kt 

(3.4.6)
The total amount of glucose ingested is D , and t is the time since the glucose was
ingested. The resulting model equations are
K G
dG
(3.4.1)
 SG (Gb  G (t ))  X (t )G (t )  abc gut
dt
V
dX
 p2 ( S1 ( I (t )  I b )  X (t )
(3.4.2)
dt
dGgut
(3.4.5)
 Rge (t )  kabcGgut
dt
These equations can be regarded as closest analogue in OGTT minimal models to the
OMM of the IVGTT. For complete model equations, parameter definitions, and initial
values see section 5.7 (Equations 5.7.1 to 5.7.4, Table 5.7.1, and a simulation in figure
5.7.1).