Modeling Glucose Dynamics
Frank Massey
5.1. Carbohydrates, sugars and glucose. Carbohydrates are substances with the
general formula Cx(H2O)y; see Pauling [1, p. 585]. The simpler carbohydrates are called sugars,
and the complex ones are called polysaccharides. The simplest sugars are the monosaccharides;
see Wikipedia [2]. These include D-glucose, fructose (fruit sugar), ribose and galactose.
Slightly more complicated are the disaccharides. These include sucrose (table sugar), maltose
(malt sugar) and lactose (milk sugar). D-glucose, (or just glucose) (also called dextrose and
grape sugar) occurs in many fruits, and is present in the blood of animals. It is the body’s source
of energy. The cells combine glucose with oxygen to form carbon dioxide and water producing
energy. The formula for glucose is C6H12O6, so its molecular weight is
6 (mol wt of C) + 12 (mol wt of H) + 6 (mol wt of O)
= (6)(12) + (12)(1) + (6)(16) = 72 + 12 + 96 = 180
Glucose has the molecular structure
H
H
H
H
H
H
|
|
|
|
|
|
H -- C -- C -- C -- C -- C -- C = O
|
|
|
|
|
OH
OH
OH
OH
OH
Sucrose is ordinary sugar, obtained from sugar cane and beets. Its formula is C12H22O11
and its structure is somewhat complicated consisting of two rings (each containing one oxygen
atom), held together by bonds to an oxygen atom; see Pauling, [1, p. 574].
Important polysaccharides include starch, glycogen and cellulose. Starch, (C6H10O5)x,
occurs in plants, mainly in their seeds or tubers. Glycogen (animal starch) has the same formula
as (plant) starch. It occurs in the blood and internal organs of animals, especially the liver.
When complex carbohydrates are ingested, they are split up into simple sugars during digestion
and pass through the walls of the digestive tract into the blood stream, see [Pauling, 1, p. 604].
The liver converts these sugars into glycogen. Later the liver converts glycogen into glucose
when the glucose in the blood is low.
1
We are interested in the concentration of glucose in the blood. Let
G = G(t) = blood glucose concentration,
where t represents time. If we refer to a person's glucose concentration then it is assumed that
we are referring to the glucose concentration in the blood, unless otherwise stated. This is also
called the plasma glucose concentration. Similarly, sometimes we drop the word concentration
and just refer to a person's glucose. It should be clear from the context when we do this. The
most common units for measuring G is mg/dl, and unless stated otherwise values of G will be
expressed in these units. The fasting (or basal) value of G is denoted by
Gb
=
basal value of G
=
lim G(t)
t
Unless stated otherwise we shall assume a person is fasting when a limit as t tends to  is
evaluated, so Gb is an equilibrium value (or steady state value) of G during a period of fasting.
In practice 12 hours of fasting is usually considered sufficient to measure Gb. Normal values of
Gb are 80-120. Bergman [5] did a study of 18 lean and obese subjects and their Gb values ranged
from 85 to 109. There was one value of 85, 11 values in the 90’s and 6 values between 100 and
109. G may go up to 160-180 two hours after a meal and still be in the range 110-150 at
bedtime.
Another common unit for measuring G is m-mol/l. Since the molecular weight of
glucose is 180, it follows that 1 mol of glucose = 180 g and 1 m-mol of glucose = 180 mg. So
1 m-mol / l = 18 mg / dl. For example 90 mg/dl = 5 m mol/l.
5.2. Glucose Regulation. The body needs to regulate the glucose concentration so that it
is in a range that is good for the body. Consider a male weighing 70 kg = 154 lbs. He requires
about 2500 calories / day or 105 calories / hr on the average. Of this 65-70 calories / hr are
needed for basal metabolism, i.e. heart pumping action, brain electrical activity and kidney
filtration. Another 7.5 – 9 calories / hr are needed for thermogenesis, i.e. maintaining body
temperature during exposure to cold, digesting meals and reacting to stress. A sedentary
2
individual who only engages in light exercise requires 25 – 35 calories per hour on the average
for physical activity.
If the glucose is too low then the body doesn’t have enough glucose. This is called
hypoglycemia. If glucose falls below 70 mg/dl then a person starts to notice this and below
50 mg/dl brings about unconsciousness and death. If the glucose becomes too high then there is
a number of bad side effects for the body such as blindness, renal disease, vascular and heart
disease. This is called hyperglycemia. It is not known precisely above what level these bad
effects start to occur. Puckett [10] indicates that some people feel that 200 mg/dl is about where
the bad effects start to occur.
The body has a variety of methods to regulate glucose. They fall into two categories,
those that don’t involve insulin and those that do.
Non-insulin dependent methods:
1.
The liver and kidneys produce glucose. If glucose falls too low the liver
produces more from glycogen. If glucose becomes too high then the liver
produces less or converts some glucose back into glycogen. The glucose
produced by the liver is called hepatic glucose output (HGO).
2.
The cells use glucose to produce energy.
Insulin dependent methods:
1.
Insulin causes the liver to produce less glucose.
2.
Insulin causes the cells to use glucose faster.
Insulin is complex molecule produced by the -cells of the pancreas.
If a person’s glucose is higher than it should be the person is said to have impaired
glucose tolerance (IGT). Those whose glucose is at a dangerous level are said to have diabetes
mellitus.
Since glucose goes up after eating and then down again after the body has processed the
meal, it is not always obvious if a person has IGT or diabetes. A person could have IGT or
3
diabetes because their basal glucose value is too high or because the glucose level goes up to
high after a meal or because the glucose level doesn't return to the basal value fast enough or by
a combination of these. There are several common ways that doctors try to determine some or
all of these problems. The simplest is to measure Gb. If Gb is above 125 then the doctor usually
requests a 75-g oral glucose tolerance test (GTT or OGTT). For a normal person glucose
probably would be back to 120 - 130 two hours after drinking the glucose. For a person with
IGT it might be 180 or higher. If it is above 200 then the person probably has diabetes. Another
test used to measure glucose tolerance is the intravenous glucose tolerance test (IVGTT). This
test has both the advantage and disadvantage that it doesn't involve the body's absorption of
glucose from food. This test is more involved than the OGTT and it is used more for research
purposes than for diagnostic purposes.
If a person has IGT it may be due to a variety of reasons. For example, the body doesn’t
produce enough insulin or the body has lost its ability to use insulin. Bergman [7] reports that
Yalow and Berson [8] observed that insulin levels during the OGTT were elevated in obese
subject and hypothesized that this elevation was evidence of insulin resistance – an hypothesis
later confirmed by Reaven [9].
Since one reason a person could have IGT is because the rate at which glucose returns to
basal values after a meal is too slow. A parameter used to measure this is KG. This parameter is
usually measured in conjunction with the IVGTT. An older definition of KG is
d
KG = - dt ln( G(t) )
See Bergman [7, p. 7]. This definition seems to be based on the assumption that glucose declines
exponentially which, in fact, is not quite true. We shall use the definition
(2.1)
d
KG = - lim dt ln( G(t) – Gb )
t
which is based on a suggestion of Bergman [7, p. 7]. In section 5.8 we shall discuss the
linearization of G near its basal value. This linearization produces a matrix which describes the
4
glucose dynamics when G is near its basal value. The eigenvalues of this matrix are negative
numbers. - KG is the eigenvalue closest to zero.
5.3. Insulin. The metabolizing of glucose by the body is aided by insulin. Insulin is a
complex molecule with a molecular weight of about 12,000, see Pauling [1, p. 585]. Let
I =I(t) = blood insulin concentration
The most common units for measuring I are U / ml where U denotes a unit of insulin. The
value of U is discussed below. Let
Ib
=
basal value of I
=
lim I(t)
t
Ib is an equilibrium value of I in the same way Gb is an equilibrium value of G. Ib values vary
more than fasting glucose values. Bergman et. al. [5] did a study of 18 lean and obese subjects
and their Ib values ranged from 3 U / ml to 81. There was 6 values between 3 and 9, 6 values
between 11 and 17, 3 in the 20’s. The 3 higher values were 37, 68 and 81. After a meal I may
rise to 30-50 U / ml and higher values are not unusual.
Another common unit for measuring I is p mol / l, where p = pico = 10-12. It turns out
that 6 p mol / l = 1 U / ml. For example, 10 U / ml = 60 p mol / l. Since
6 p mol / l = U / ml one has
6  10-12 mol in one liter = 10-6 units in one ml
6  10-6 mol in one liter = 1 unit in one ml
6  10-9 mol in one ml = 1 unit in one ml
6  10-9 mol = 1 unit
Since the molecular weight is about 12,000
72 g  1 unit
5.4. Glucose – Insulin Modeling. The problem of making a mathematical model for
glucose – insulin kinetics has received a lot of attention. Most of the models are differential (or
5
differential – difference) equations for how the glucose and insulin concentrations in the blood
change with time. The simpler models try to use a single compartment for the glucose, although
they may use more than one compartment for insulin. The starting point for the simpler models
are equations of the form
dG
dt
=
Production - Uptake
dI
dt
=
Secretion - Clearance
Glucose is produced by the liver from glycogen and from the intestines from food. As
the glucose concentration rises, the production rate by the liver decreases. Also, it has been
observed that as the insulin concentration rises, this also causes the production rate by the liver to
decrease. Thus, for glucose we may have something like.
Production
=
fP(G, I) + PI
where fP(G, I) is some function of G and I which represents the internal production of glucose
and
PI = production of glucose from the intestines from food.
Insulin is secreted by the -cells of the pancreas. As the glucose concentration rises, the
production rate by the pancreas increases. Thus, for insulin we may have something like.
Secretion
=
gP(G, I)
where gP(G, I) is another function of G and I.
Glucose is removed from the blood by the cells and the liver which converts it back to
glycogen. As the glucose concentration rises, the uptake rate increases. Furthermore, it has been
observed that as the insulin concentration rises, this also causes the uptake rate to increase. Also
the amount of exercise that a person is doing should affect the uptake rate, but for the time being
we don’t incorporate this into the model. Thus, for glucose we may have something like.
Uptake
=
fU(G, I)
6
where fU(G, I) is another function of G and I.
Insulin is cleared by liver, kidneys and insulin receptors. For insulin we may also have
something like.
Clearance
=
gU(G, I)
where gU(G, I) is another function of G and I.
Putting these together we get
dG
dt
=
f(G, I) + PI
dI
dt
=
g(G, I)
(4.1)
where f(G, I) = fP(G, I) - fU(G, I) and g(G, I) = gP(G, I) - gU(G, I) are again functions of G and I.
These functions should have the following properties.
df
dG < 0
df
dI < 0
dg
dG > 0
dg
dI < 0
when G > 0 and I > 0. In the next few sections we look at some particular models that appear in
the literature. Before this let’s take a look at equilibrium values.
Equilibrium values. An equilibrium value for something that varies with time is a value
that the quantity approaches as time goes to infinity. Consider the system (4.1) when there is no
food intake so PI = 0. Then the equilibrium values Gb and Ib are the values of G and I that are the
solutions to the equations
f(G, I) = 0
g(G, I) = 0
5.5. Topp’s Model. One model that has received some attention lately is the model of
Topp et. al. [3]. Most of the equations in the model have been used previously by others.
However, Topp also models the variation of β-cell mass with time, something that had not been
done much previously. Thus we shall call this model “Topp’s model”. We shall ignore the
7
variation of β-cell mass with time and simply assume the β-cell mass is constant. For glucose
Topp uses the equations
fP(G, I)
=
P0 - (EG0P + SIPI)G
fU(G, I)
=
U0 + (EG0U + SIUI)G
P0
=
EG0P
SIP
U0
=
=
=
EG0U
SIU
=
=
rate of glucose production by the liver at zero
glucose
glucose effectiveness for production at zero insulin
insulin sensitivity for production
rate of glucose uptake by liver and cells at zero
glucose
glucose effectiveness for uptake at zero insulin
insulin sensitivity for uptake
where
Putting these together one has
(5.1)
f(G, I)
= R0 - (EG0 + SII)G
where
R0
=
EG0
=
SI
=
P0 - U0 = net rate of glucose production by the
body at zero glucose
EG0P + EG0U = total glucose effectiveness at zero
insulin
SIP + SIU = total insulin sensitivity
Topp uses the values
R0 = 864 mg/dl / day = 0.6 mg/dl / min.
EG0 = 1.44 / day = 0.001 / min.
SI = 0.72 / U/ml /day = 5  10-4 / U/ml /min.
Note 1 day = 2460 = 1440 min. The value of SI that Topp uses seems to agree with the value
Bergman uses; see the next section. On the other hand the value of EG0 does not. Bergman’s
value is more like 0.2 / min; see section 6. For his value of EG0 Topp cites Bergman, Phillips &
Cobelli [5] and Finegood [4]. However, it is not clear how Topp gets his value from these
sources.
For insulin Topp uses the equations
gP(G, I)
=
bG2
 + G2
8
gU(G, I)
where
b =
 =
 =
 =
=
kI

-cell mass
maximal secretion rate of -cells
a constant with the property that  is the value of G for which the
secretion rate is half its maximum
k = clearance constant for insulin
bG2
The sigmoidal function
is called a Hill function; see Topp [3 , p. 608]. Putting the above
 + G2
together we have
(5.2)
g(G, I)
=
bG2
- kI
 + G2
Topp uses the values
 = 43.2 U/ml / mg / day = 0.03 U/ml / mn
 = 20000 (mg/dl)2
k = 432 / day = 0.3 / min
Furthermore Topp assumes that for a normal person
 = 300 mg
Using this value one has
b = 
= (43.2 U/ml / mg / day) (300 mg) = 12,960 U/ml / day
= 9 U/ml / min
Thus, the equations for glucose and insulin are
(5.3)
dG
dt = R – (E+SI)G
(5.4)
dI
bG2
=
– kI
dt
 + G2
where R = R0, E = EG0, S = SI, and b = .
9
Let (Gb, Ib) be the equilibrium point that lies in the first quadrant, i.e. the solution to the
equations
(5.5)
R – (E+SI)G = 0
(5.6)
bG2
– kI = 0
 + G2
that lies in the first quadrant.
Proposition 1. The equilibrium point (Gb, Ib) is asymptotically stable.
Proof. We prove this by constructing a Liapunov function for the system. Substitute
G = (G – Gb) + Gb and I = (I – Ib) + Ib into the right hand sides of (5.3) and (5.4) and expand.
For (5.3) we have
R – (E+SI)G
= R – (E+SI) [(G – Gb) + Gb]
= R – (E+SI)Gb – (E+SI)(G – Gb)
= R – (E+S [(I – Ib) + Ib])Gb – (E+SI)(G – Gb)
= R – (E+SIb)Gb – SGb(I – Ib) – (E+SI)(G – Gb)
If we use the fact that (Gb, Ib) satisfy (5.5) we get
(5.7)
dG
dt = – SGb(I – Ib) – (E+SI)(G – Gb)
For (5.4) we have
bG2
- kI
 + G2
bG2
=
– k[(I – Ib) + Ib]
 + G2
bG2
=
– kIb – k(I – Ib)
 + G2
Since (Gb, Ib) satisfies (5.6) we can replace kIb by
bG2
- kI
 + G2
=
bGb2
. This gives
 + Gb2
bG2
bGb2
–
– k(I – Ib)
 + G2  + Gb2
10
=
b(G2 – Gb2)
– k(I – Ib)
( + G2)( + Gb2)
So
dI
b(G2 – Gb2)
=
– k(I – Ib)
dt
( + G2)( + Gb2)
(5.8)
Let
(5.9)
2
2
 b(G – Gb )
( + G2)( + G 2) dG
b

U(G) =
=
=
=
b
2
2
G – Gb dG
2
 + Gb2 
 +G
b
 + Gb2
b
 + Gb
2
2
2
G + 
 – Gb
(
dG
+

2
2 dG )
 + G
+G
(G +
 – Gb2 -1 G
tan (
) )


Note that the integrand is negative for G < Gb and positive for G < Gb. Therefore U(G) is
decreasing for G < Gb and increasing for G < Gb with a minimum at G = Gb.
Let the function L(G, I) be defined by
SGb(I – Ib)2
L(G, I) = U(G) +
2
(5.10)
Note that L(G, I) has a minimum at (G, I) = (Gb, Ib). One has
d
dU(G) dG
dI
L(G,
I)
=
+
SG
b(I – Ib)
dt
dG dt
dt
b(G2 – G 2)
b(G2 – G 2)
= – ( + G2)( +bG 2)(SGb(I – Ib) + (E+SI)(G – Gb)) + SGb(I – Ib)( ( + G2)( +bG 2) – k(I – Ib) )
b
b
= –
b(G2 – Gb2)(E+SI)(G – Gb)
– SkGb(I – Ib)2
( + G2)( + Gb2)
b(G + Gb)(E+SI)(G – Gb)2
= –
– SkGb(I – Ib)2
( + G2)( + Gb2)
11
The right side is negative for G  0 and I  0 so L(G, I) is a Liapunov function for the system
(5.3) and (5.4) in the first quadrant. //
Kyrtsos [17] has done quite a bit of work with Topp's model. He gave a different proof
of the stability of the equilibrium point (Gb, Ib) using Bendixson's criterion. He also did
numerical solutions of Topp's model combined with Puckett's model of food to glucose. We
shall say more about that in section 7. We shall also return to Topp's model in section 8 when
we talk about linearization.
5.6. The Minimal Model. One of the frequently used models is the “Minimal Model” of
Bergman, et al. This model consists of two parts. The first part (section 5.6.1) models the
change in glucose concentration given the insulin concentration and the second part (section
5.6.2) models the change in insulin concentration given the glucose concentration.
5.6.1 Glucose kinetics. The minimal model for glucose kinetics was introduced in [6].
This model is similar to the basic model discussed in section 5.4. However, it has a few
modifications. First, it is most often used to model the change in glucose concentration after an
IV glucose tolerance test that has been given after the person has been fasting. For that reason it
sets PI = 0. Second it is used to model changes in glucose concentration on a time scale of
minutes. In that case it seems that a more accurate model can be obtained by introducing the
insulin concentration in another compartment in addition to the blood. This is commonly
^,
interpreted as the interstitium. Since it is hard to measure this, we simply introduce a variable, X
that is assumed to be proportional to the glucose concentration in the interstitium. Thus
^
X
=
insulin-excitable tissue glucose uptake activity.
^ are the reciprocal of the units of time. The minimal model appears in the
The units of X
literature in two slightly different forms.
Version 1. This version appears in [12] and is the following.
(6.1)
dG
dt
=
^ )G + C
– (S^G + X
12
^
dX
dt
(6.2)
=
^
kaI – kbX
=
^)
kb(SII – X
=
R0 = net glucose production at zero glucose concentration.
=
EG0 = glucose effectiveness, i.e. the insulin-independent rate constant
of glucose to retard its own increase. Unless stated otherwise we shall
use the units / min. Bergman [12] reports in the abstract that an
average value is 0.026 in normal people while on p. 1516 he reports
that an average value is 0.021. In the case of the first value Bergman
might be confusing S^ with the parameter S in version 2 of the
where
C
S^
G
G
G
minimal model below. It looks like SG is about 0.005 more than S^G.
Bergman [12] reports that an average value of S^ for people with
G
ka
=
kb
=
SI
=
=
NIDDM is 0.014. Topp uses the value 0.001 / min; see section 5
which is quite a bit different from Bergman's value.
p3 = rate constant for flow of insulin from the tissues (blood ?) to the
interstitium. Pacini and Bergman [13] have an example where it is
10-5 / U/ml /min2.
^ . Pacini and Bergman [13] have an
p = rate constant for decrease of X
2
^
example where it is 0.02 / min. 1/kb is the average time it takes for X
to approach SII. If kb = 0.02 / min then 1/kb = 50 min.
ka
kb
insulin sensitivity. Unless otherwise stated we shall use units of
/ U/ml /min. Bergman [12] reports values that vary from 2.3  10-4
to 7.6  10-4 in nondiabetic subjects with a mean of about 5  10-4. For
subjects with NIDDM the values are in the range 0.6  10-4. Pacini
and Bergman [13] report the value 5  10-4. Topp also uses this value;
see section 5.
^ and S^ are usually denoted by X and S . We have used X
^ and S^ since X
Note: In the papers X
G
G
G
and SG are used in the second version of the model which follows.
^ be the basal values of G, I, and X
^ . They satisfy the
Version 2. Let Gb, Ib, and X
b
equations
(6.3)
C
=
^ )G
(S^G + X
b
b
13
^ =
kbX
b
(6.4)
kaIb
Let’s make the change of variables
(6.6)
X
=
^ - X
^
X
b
Then the equations (6.1) and (6.2) become
(6.7)
dG
dt
=
^ + X)G + C
– (S^G + X
b
(6.8)
dX
dt
=
^
kaI – kbX – kbX
b
SG
=
^
S^G + X
b
^S + S I
If we let
=
=
G
I b
p1
and use (6.3) and (6.4) then these equations can be written as
(6.9)
dG
dt
=
– (SG + X)G + SGGb
(6.10)
^
dX
dt
=
ka(I – Ib) – kbX
We have used the fact that
(6.11)
C = SGGb = R0
This form of the equations appears in Pacini and Bergman [13], Steil [14, p. 124] and
De Gaetano and Arino [15]. Using the p notation for the parameters the equations are
(6.12)
dG
dt
=
– (p1 + X)G + p1Gb
(6.13)
^
dX
dt
=
p3(I – Ib) – p2X
14
A typical value of Ib is 10 U/ml and SI is often about 5  10-4 / U/ml /min. So a
typical vaoue of SIIb is 5  10-3 / min. In this case SG would be about 0.005 / min more than S^G.
For example, if S^G were 0.021 / min then SG would be 0.026 / min. In the example of Pacini and
Bergman it is about 0.031 / min. Since Gb is about 90 mg/dl, this would make C = SGGb about
2.3 mg/dl / min. This is somewhat different from the value given by Topp [5] who gives the
value 0.6 mg/dl / min.
Bergman sometimes [e.g. 7] omits the term - kaIb in the right side of (6.10).
There are two related problems. The first is to solve (6.9) and (6.10) given the values of
the parameters SG, Gb, ka, kb, Ib, G(0), X(0) and I(t). The second is to estimate the parameters SG,
ka, kb, and G(0) given the values of Gb, Ib, X(0), G(t) and I(t). A frequently sampled intravenous
glucose tolerance test (FSIGT) gives the glucose and insulin concentrations in blood at
frequently spaced time intervals after an IV glucose infusion and can be used to determine the
parameters in this or other models.
5.6.2. Insulin Kinetics. For insulin kinetics Bergman [7] gives the following
(6.14)
g(G, I)
=
γ(G – h)t - nI
γ
=
h
n
=
=
effect of an increment of glucose above the
threshold value h to increase the rate of secretion
of insulin
threshold value of glucose
fractional disappearance of insulin
where
Here we are assuming G and I have their basal values Gb and Ib for t < 0 and at t = 0 a bolus of
glucose is given intraveneously causing a sudden rise in G to G(0). This causes a sudden rise in I
dI
to I(0). The equation dt = g(G, I) with g(G, I) given by (6.14) is assumed to hold for t ≥ 0.
Bergman defines two parameters related to the parameters in (2) and the values in G(0) and I(0).
φ1
=
I(0) - Ib
n [G(0) - Gb]
15
φ2
=
1000 γ
Typical values for 4.4 μU/ml-min / mg/dl and are 88 μU/ml / mg/dl / min.
Bergman [7] defines the following measure of insulin secretion during the IVGTT.
10
AIRglucose = 
 I dt
0
where time is measured in minutes.
Bergman [7] notes that decreased insulin sensitivity can be compensated for by increased
insulin secretion. In particular, what should determine if a person has IGT is the product of the
two, i.e.
Insulin-secretion  Insulin-sensitivity
In particular, in he defines
Disposition Index = AIRglucose  SI
5.7. Food to Glucose. How do the carbohydrates in a meal translate into a rate of
glucose entering the blood stream, i.e. into the term PI in (4.1)? There are various ways to model
this. To begin with, what parameters can we use to characterize a meal? One is the carbohydrate
content. Let
CHOM
= carbohydrate content of the meal
A typical meal may have between 50 to 100 g of carbohydrates. We shall use the value
CHOM = 90 g for the purposes of illustration in some examples below.
Not all the carbohydrates in a meal actually gets into the blood. Let
F
= fraction of meal carbohydrates that actually absorbs into the blood
The value of F depends on the actual kind of carbohydrates that one eats and other factors. We
shall use the value F = 1/6 for a normal person for the purposes of illustration in some examples
below.
16
From the parameters and F we can calculate the actual amount of carbohydrates from a
meal that get into the blood. Let
Gm
= amount of carbohydrates from a meal that actually absorbs into the blood
= F  CHOM
Example. CHOM = 90 g and F = 1/6  Gm = 15 g = 15,000 mg.
We are interested in how these carbohydrates affect the concentration of glucose in the
blood. One factor that affects this is the effective volume of the blood. We let
VL
= effective volume of the blood
This depends on the persons weight. One source assumes that the blood volume is proportional
to a person’s weight with proportionality constant 66 ml/kg. For example, a 80 kg person would
have a blood volume of VL = (66 ml/kg) (80 kg) = 5280 ml  5.2 l. We shall use the value
VL = 52 dl in the examples below.
If all the carbohydrate of a meal were to go into the blood all at once then the
concentration of the blood would rise by

=
Gm
F  CHOM
VL =
VL
Example. Gm = 15000 mg and VL = 52 dl   =
15000 mg
 288 mg/dl.
52
Fortunately, the carbohydrate doesn’t go into the blood all at once. We can relate PI to  by
means of a function f(t) that gives the rate at which one mg of food goes into each dl of blood.
f(t)
= rate at which one mg of food goes into each dl of blood assuming the meal
is eaten at time 0.

We should have f(t) = 0 for t < 0, f(t)  0 for t  0 and 
 f(t) dt = 1. In particular, we shall only
0
specify f(t) for t  0 in the examples below. Let
17
tM
= the time at which the meal is eaten
Then we have
PI
=  f(t - tM) =
F  CHOM
VL
f(t - tM)
Often one assumes tM = 0, in which case PI =  f(t). We shall assume this to be the case in the
examples below.
There are various possible types of functions f(t) we can use to model various possible rates at
which food from a meal enters the blood. We shall look at three.
Model 1. “A continuous constant rate model”. In this model we assume that the food
from a meal enters the blood continuously over a certain period of time. Let
T
= length of time it takes the body to absorb the food from a meal into the
blood
There doesn’t seem to universal agreement on what T should be. For a normal person T might
be some value between 2 and 5 hr. Once we have picked T then
f(t)
1
=  T
0
0tT
T<t
Then
0tT
T<t
r

0
PI
=
r
= rate at which glucose is entering each dl of blood over the period 0  t  T
where

= T =
F  CHOM
VL  T
288 mg/dl
Example 1.  = 288 mg/dl and T = 120 min,  r = 120 min = 2.4 mg/dl/min.
18
288 mg/dl
Example 2.  = 288 mg/dl and T = 300 min  r = 300 min = 0.96 mg/dl/min.
Model 2. “A discrete constant rate model”. In this model we assume that the food from
a meal enters the blood in equal discrete “chunks” that are uniformly spaced in time. Let
T
= length of time between chunks of food going into the blood
n
= the number of discrete chunks in which the meal is divided into
Then
n-1
f(t)
 1n (t – jT)
=
j=0
Then
n-1
PI
 c (t – jT)
=
j=0
where
c
= amount of glucose that is entering each dl of blood in each chunk

= n =
F  CHOM
VL  n
Example.  = 288 mg/dl, T = 15 min, n = 20  c =
288 mg/dl
20
= 14.4 mg/dl and
19
PI =
 14.4 (t – 15j).
j=0
Model 3. "Puckett's model". This model presented by Puckett [10] is a more detailed
model of the absorption of food into the blood stream. For simplicity, assume that a unit amount
of carbohydrate is eaten at time tM = 0. The first step is the hydrolyzation of the carbohydrates.
Puckett assumes that this takes place over a five minute period. If we let
CHOG
= rate of hydrolyzed carbohydrates that enters the stomach
then Puckett assumes
19
(7.1)
CHOG =
CHOM
[ t U(t) - (t - 1) U(t - 1)
4
- (t - 4) U(t - 4) + (t - 5) U(t - 5)]
where
U(t) = Heaviside function
The next step is flow of the food from the stomach to the small intestine. Puckett assumes that
this is a first order process, so that
(7.2)
dGG
1
=
dt
T GG + CHOG
GG(0) = 0
where
GG
= amount of glucose in the stomach
T
= time constant for gastric emptying
The last step is flow of glucose from the small intestine to the blood. Pucket assumes this is also
a first order rate process so that
(7.3)
dGA
1
F
dt = - P GA + P T GG
GA(0) = 0
where
GA = rate glucose is absorbed into the blood stream from the small intestine
TA
= time constant for absorption rate to equilibrate with gastric emptying
Finally
GA
PI = V
L
Using curve fitting techniques with actual data Puckett uses the values
T
P
= 156.59 min
= 48.66 min
Since (7.2) and (7.3) are linear equations, one can put this model into the framework of the
previous two models by letting
y
f(t) = 4
where
20
y
= rate glucose is absorbed into the blood stream from the small intestine
for a meal of mass 4 if all the meal carbohydrates are absorbed into the
blood
y is the solution to the system
(7.4)
dy
1
1
=
y
+
dt
P
P x
y(0) = 0
(7.5)
dx
1
1
dt = - T x + T g(t)
x(0) = 0
(7.6)
g(t) = t U(t) - (t - 1)U(t - 1) - (t - 4)U(t - 4) + (t - 5)U(t - 5)
where
x
g(t)
= rate at which glucose leaves the stomach for a meal of mass 4
= rate of hydrolyzed carbohydrates that enters the stomach for a meal of
mass 4
Proposition 2.
f(t) =

1
4 

T2
P2
e-t/T e-t/P
(T - P)
(T - P)
T2
P2
1+
e-t/T [1 - e1/T ] e-t/P [1 - e1/P ]
(T - P)
(T - P)
2
T
P2
-t+T+P+5+
e-t/T [1 - e1/T - e4/T ] e-t/P [1 - e1/P - e4/P ]
(T - P)
(T - P)
T2
P2
e-t/T [1 - e1/T - e4/T + e5/T] e-t/P [1 - e1/P - e4/P + e5/P]
(T - P)
(T - P)
t-T-P+
0t1
1t4
4t5
5t
Proof. Let’s first look at equation (7.5). If we take Laplace transforms of both sides we
get
dx
1
1
( dt ) = - T ( x ) + T ( g(t) )
Using the properties of Laplace transforms we get
1
1 1 e-s e-4s e-5s
s ( x ) – x(0) = - T ( x ) + T [s2 - s2 - s2 + s2 ]
Using the fact that x(0) = 0, we get
1
(s + T ) ( x ) =
( x ) =
1 - e-s - e-4s + e-5s
T s2
1 - e-s - e-4s + e-5s
1
T (s + T ) s2
21
x
=  (
-1
1 - e-s - e-4s + e-5s
1
T (s + T ) s2
)
= z(t) U(t) - z(t - 1) U(t - 1) - z(t - 4) U(t - 4) + z(t - 5) U(t - 5)
where
z
-1(
=
1
)
1
T (s + T ) s2
Using partial fractions one obtains
1
T (s
So
z
x
1
+ T ) s2
1
T
T
= s2 - s +
1
s+T
t - T + T e-t/T
=
= (t - T + T e-t/T ) U(t) - (t - T - 1 + T e-(t-1)/T ) U(t - 1)
- (t - T - 4 + T e-(t-4)/T ) U(t - 4) + (t - T - 5 + T e-(t-5)/T ) U(t - 5)
0t1
1t4
4t5
5t
 t - T + T-t/Te
1 + T e [1 - e1/T ]
=  - t + T + 5 + T e-t/T [1 - e1/T - e4/T ]
 T e-t/T [1 - e1/T - e4/T + e5/T]
-t/T
Next consider the equation (7.4). By a similar argument we get
y
where
= v(t) U(t) - v(t - 1) U(t - 1) - v(t - 4) U(t - 4) + v(t - 5) U(t - 5)
v
1
-1(
=
T (s +
1
)P
T
1
(s + P ) s2
)
Using partial fractions one obtains
1
1
1
T (s + T ) P (s + P ) s2
So
v
y
1
T+P
T2
P2
= 2 +
1
1
s
s
(T - P) (s + T )
(T - P) (s + P )
T2
P2
t - T - P + (T - P) e-t/T - (T - P) e-t/P
=
T2
P2
= (t - T – P + (T - P) e-t/T - (T - P) e-t/P ) U(t)
T2
P2
- (t - T – P - 1 + (T - P) e-(t-1)/T - (T - P) e-(t-1)/P) U(t - 1)
22
T2
P2
T2
P2
- (t - T – P - 4 + (T - P) e-(t-4)/T - (T - P) e-(t-4)/P) U(t - 4)
+ (t - T – P - 5 + (T - P) e-(t-5)/T - (T - P) e-(t-5)/P) U(t - 5)
=



T2
P2
e-t/T e-t/P
(T - P)
(T - P)
2
2
T
P
1+
e-t/T [1 - e1/T ] e-t/P [1 - e1/P ]
(T - P)
(T - P)
T2
P2
-t+T+P+5+
e-t/T [1 - e1/T - e4/T ] e-t/P [1 - e1/P - e4/P ]
(T - P)
(T - P)
T2
P2
e-t/T [1 - e1/T - e4/T + e5/T] e-t/P [1 - e1/P - e4/P + e5/P]
(T - P)
(T - P)
t-T-P+
0t1
1t4
4t5
5t
y
This proves the proposition since f(t) = 4. //
Kyrtsos [17] did numerical solutions of Topp's model with varying -cell mass using all
three of the above models of food to glucose. He assumed the person was given three identical
meals a day and he observed the effect of the person's glucose, insulin and -cell mass over a one
year period. In his simulations the -cell mass grew to accommodate the food input. However,
if the food input rate was too great the -cell mass declined.
5.8. Linearization. For a model of the form (4.1) the linearization about the equilibrium
point (basal value) ub = (Gb, Ib) is
du
dt = Au
with
A =
 df
dG
 dg
dG
df 
dI 
dg 
dI 
where the partial derivatives are evaluated at (Gb, Ib).
Proposition 3. For Topp's model discussed in Section 5 one has
(8.1)
A =
 – (E+SIb)
 2bGb
 ( + Gb2)2
– SG 
–k


The eigenvalues of A are negative real numbers if
(8.2)
(E+SIb - k)2 
8SkI(b - kIb)
b
23
Otherwise they are complex numbers with negative real part.
Proof. In this case f(G, I) and g(G, I) are given by (5.5) and (5.6), i.e.
(8.3)
f(G, I) = R – (E+SI)G
(8.4)
g(G, I) =
bG2
– kI
 + G2
We have
df
= – (E+SI)
dG
df
dI = – SG
dg
2bG
=
dG
( + G2)2
dg
dI = – k
Thus (8.1) follows. One has
det[A] = k(E+SI) +
2bSG2
( + G2)2
which is positive. Since det[A] is the product of the eigenvalues, it follows that the eigenvalues
are either real with the same sign or complex. Also note that
trace[A] = - (E+SI) - k
which is negative. Since trace[A] is the sum of the eigenvalues, it follows that the eigenvalues
are either negative or have negative real part. We want to determine when the eigenvalues are
complex and when they are real. In general, the eigenvalues of a 22 matrix T are real if
[ trace[T] ]2  4 det[T]
-p -q
If we write T =  r - s  , then the eigenvalues of A are real if
(p – s)2  4rq
24
Thus the eigenvalues of A are real if
(E+SI - k)2 
(8.5)
8bSG2
( + G2)2
In general this inequality might or might not hold, so the eigenvalues of A might or might
not be complex. Let’s restrict our attention to A at the equilibrium point. The equilibrium point
(Gb, Ib) is the solution to the equations
(8.6)
R – (E+SI)G = 0
(8.7)
bG2
– kI = 0
 + G2
From equation the second equation it follows that
bG2
= kI
 + G2
(8.8)
kI
G2 = b - kI
b
 + G2 = b - kI
(8.9)
Using (8.8) and (8.9) the inequality (8.5) for real eigenvalues becomes (8.1). //
Numerical example 1. Consider the values used by Topp [3]. They are
R
E
S
b
=
=
=
=
 =
k =
0.6 mg/dl / min.
0.001 / min.
5  10-4 / U/ml /min.
9 U/ml / min
20000 (mg/dl)2
0.3 / min
It is not hard to see that the values Gb = 100 and Ib = 10 are the equilibrium values. With these values one
has
A
 – (0.001+(0.0005)(10))
=  2(20000)(9)(100)
– 0.3
 (20000 + (100)2)2
=
 – 0.006
 0.04
– 0.05 

– 0.3
25
– (0.0005)(100) 


In this case
= (0.001 + (0.0005)(10) – 0.3)2 = (0.006 – 0.3)2
= (- 0.294)2
 0.086436
(E+SI - k)2
and
8SkI(b - kI)
b
8 (0.0005)(0.3)(10)(9 - (0.3)(10))
9
8 (0.0005)(0.3)(10)(6)
8 (0.0005)(0.3)(10)(2)
=
=
9
3
= 8(0.0005)(2) = 0.008
=
So the eigenvalues are real in this case. In fact they are about – 0.3 ≈ - 1/3.4 and – 0.013 ≈ - 1/77.
Numerical example 2. Consider the values R = 2, E = 1, S = 1, b = 4,  = 1 and k = 2. It is not hard to
see that the values G = 1 and I = 1 satisfy the equations (4) and (5) with these values of the parameters, so
these are the equilibrium values. With these values one has
A
 – (1+(1)(1))
=  2(1)(4)(1)
 (1 + (1)2)2
=
–2
 2
– (1)(1) 
–2


–1
–2
and
(E+SI - k)2
= (1 + (1)(1) – 2)2 = 0
8SkI(b - kI)
b
=
and
8 (1)(2)(1)(4 - (2)(1))
4
32
=
4
= 8
So the eigenvalues are complex in this case.
5.9 Bibliography.
[1]
Pauling, Linus. General Chemistry, 2nd ed. Freeman, 1953.
[2]
Wikipedia. http//en.wikipedia.org/wiki/Monosaccharide.
[3]
Topp, Brian, Keith Promislow, Gerda DeVries, Robert M. Miura and Diane T. Finegood.
A model of β-cell mass, insulin, and glucose kinetics: pathways to diabetes. J. theor.
Biol., 206, 605-619, 2000.
[4]
Finegood, Diane T. Application of the minimal model of glucose kinetics, In: The
minimal model approach and determinants of glucose tolerance (Bergman, R.N. &
J.C. Lovejoy eds), pp. 51-122. Baton Rouge: Louisiana State University Press, 1997.
[5]
Bergman, R.N., L.S. Phillips & C. Cobelli. Physiologic evaluation of factors controlling
glucose tolerance in man. Measurement of insulin sensitivity and -cell glucose
26
sensitivity from the response to intravenous glucose. J. Clin. Invest. (Journal of
clinical investigation) 68, 1456-1467, 1981.
[6]
Bergman, R.N., Y.Z. Ider, C.R. Bowden & C. Cobelli. Quantitative estimation of insulin
sensitivity. Am. J. Physiol. 236, E667-E677, 1979.
[7]
Bergman, R.N. The mininimal model: yesterday, today, and tommow, In: The minimal
model approach and determinants of glucose tolerance (Bergman, R.N. & J.C.
Lovejoy eds), pp. 51-122. Baton Rouge: Louisiana State University Press, 1997.
[8]
Yalow, R.S., S.M. Glick, J. Roth, and S.A. Berson. Plasma insulin and growth hormone
levels in obesity and diabetes. Ann NY Acad. Sci, 1965, 131, 357-373.
[9]
Shen, S.W., G.M. Reaven and J.W. Farquhar. Comparison of impedence to insulinmediated glucose uptake in normal subjects and in subjects with latent diabetes. J
Clin Invest, 49, 2151-2160, 1970.
[10]
Puckett, W.R. Dynamic modeling of diabetes mellitus. Univ. of Wisconsin – Madison,
1992.
[11]
Bergman, Richard N. and Jennifer C. Lovejoy, editors. The Minimal Model Approach
and Determinants of Glucose Tolerance. Baton Rouge (LA): Louisiana State
University Press, 1997. (This contains 16 papers presented at a symposium on the
minimal model in 1994. It is grouped into three sections. The first is concerned with
the principles of “the Minimal Model”. The second with the measurement of insulin
secretion using the Minimal Model and C-peptide and the metabolic pathways for
glucose metabolism. The third has clinical applications of the Minimal Model.
While most of the papers deal with diabetes, a few deal with insulin resistance in
people with cancer, aging and obesity.)
[12]
Bergman, Richard N. Toward Physiological Understanding of Glucose Tolerance:
Minimal Model Approach, Lilly Lecture, 1989. Diabetes, 38, 1512-1527, 1989.
[13]
Pacini, Giovanni, and Richard N. Bergman. MINMOD: a computer program to calculate
insulin sensitivity and pancreatic responsivity from the frequently sampled
intravenous glucose tolerance test. Computer Methods and Programs in Biomedicine,
23, 113-122, 1986.
[14]
Steil, G. In: The minimal model approach and determinants of glucose tolerance
(Bergman, R.N. & J.C. Lovejoy eds). Baton Rouge: Louisiana State University
Press, 1997.
[15]
De Gaetano, A. and O. Arino. Some considerations on the mathematical modelling of the
intra-venous glucose tolerance test. J. Math. Biol. 40, 136-168, 2000.
27
[16]
Kyrtsos, Christos T. The Effects of Carbohydrate intake on Plasma Glucose, Insulin and
Beta Mass levels for Normal and Type II Diabetic people. 2006.
28