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 -- C -- C -- C -- C -- C -- C = O

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.

We are interested in the concentration of glucose in the blood. Let

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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)

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

individual who only engages in light exercise requires 25 – 35 calories per hour on the average

for physical activity.

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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

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

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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

KG = - 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) KG = - lim ln( G(t) – Gb )

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

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

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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)

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 ml6 10-6 mol in one liter = 1 unit in one ml6 10-9 mol in one ml = 1 unit in one ml6 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

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

= Production - Uptake

= Secretion - Clearance

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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)

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

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= f(G, I) + PI

(4.1)= g(G, I)

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.

< 0 < 0 > 0 < 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

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)Gwhere

P0 = rate of glucose production by the liver at zero glucose

EG0P = glucose effectiveness for production at zero insulinSIP = insulin sensitivity for productionU0 = rate of glucose uptake by liver and cells at zero

glucoseEG0U = glucose effectiveness for uptake at zero insulinSIU = insulin sensitivity for uptake

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Putting these together one has

(5.1) f(G, I) = R0 - (EG0 + SII)G

whereR0 = P0 - U0 = net rate of glucose production by the

body at zero glucoseEG0 = EG0P + EG0U = total glucose effectiveness at zero

insulinSI = 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 = 2460 = 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) =

gU(G, I) = kI

where b = = -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 maximumk = clearance constant for insulin

The sigmoidal function is called a Hill function; see Topp [3 , p. 608]. Putting the above together we have

(5.2) g(G, I) = - kI

Topp uses the values

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= 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) = R – (E+SI)G

(5.4) = – kI

where R = R0, E = EG0, S = SI, and b = .

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) – kI = 0

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)

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= 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) = – SGb(I – Ib) – (E+SI)(G – Gb)

For (5.4) we have

- kI = – k[(I – Ib) + Ib]= – kIb – k(I – Ib)

Since (Gb, Ib) satisfies (5.6) we can replace kIb by . This gives

- kI = – – k(I – Ib)= – k(I – Ib)

So

(5.8) = – k(I – Ib)

Let

(5.9) U(G) =

=

= ( + )

= ( G + tan-1( ) )

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

(5.10) L(G, I) = U(G) +

Note that L(G, I) has a minimum at (G, I) = (Gb, Ib). One has

L(G, I) = + SGb(I – Ib)

= – (SGb(I – Ib) + (E+SI)(G – Gb)) + SGb(I – Ib)( – k(I – Ib) )

= – – SkGb(I – Ib)2

= – – SkGb(I – Ib)2

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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.

The units of X are the reciprocal of the units of time. The minimal model appears in the

literature in two slightly different forms.

Version 1. This version appears in [12] and is the following.

(6.1) = – (SG + X)G + C

(6.2) = kaI – kbX= kb(SII – X)

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whereC = R0 = net glucose production at zero glucose concentration.SG = 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 SG with the parameter SG in version 2 of the minimal model below. It looks like SG is about 0.005 more than SG. Bergman [12] reports that an average value of SG for people with NIDDM is 0.014. Topp uses the value 0.001 / min; see section 5 which is quite a bit different from Bergman's value.

ka = 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.

kb = p2 = rate constant for decrease of X. Pacini and Bergman [13] have an 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.

SI == 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.

Note: In the papers X and SG are usually denoted by X and SG. We have used X and SG since X

and SG are used in the second version of the model which follows.

Version 2. Let Gb, Ib, and Xb be the basal values of G, I, and X. They satisfy the

equations

(6.3) C = (SG + Xb)Gb

(6.4) kbXb = kaIb

Let’s make the change of variables

(6.6) X = X - Xb

Then the equations (6.1) and (6.2) become

(6.7) = – (SG + Xb + X)G + C

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(6.8) = kaI – kbX – kbXb

If we let

SG = SG + Xb

= SG + SIIb

= p1

and use (6.3) and (6.4) then these equations can be written as

(6.9) = – (SG + X)G + SGGb

(6.10) = 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) = – (p1 + X)G + p1Gb

(6.13) = p3(I – Ib) – p2X

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 SG.

For example, if SG 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 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

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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

whereγ = effect of an increment of glucose above the

threshold value h to increase the rate of secretion of insulin

h = threshold value of glucosen = fractional disappearance of insulin

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

to I(0). The equation = 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 =

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.

AIRglucose =

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

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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.

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.

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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

= =

Example. Gm = 15000 mg and VL = 52 dl = 288 mg/dl.

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 = 1. In particular, we shall only specify

f(t) for t 0 in the examples below. Let

tM = the time at which the meal is eaten

Then we have

PI = f(t - tM) = 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) =

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Then

PI =

where

r = rate at which glucose is entering each dl of blood over the period 0  t

T

= =

Example 1. = 288 mg/dl and T = 120 min, r = = 2.4 mg/dl/min.

Example 2. = 288 mg/dl and T = 300 min r = = 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

Thenf(t) =

Then

PI =

where

c = amount of glucose that is entering each dl of blood in each chunk

= =

Example. = 288 mg/dl, T = 15 min, n = 20 c = = 14.4 mg/dl and PI = .

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

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CHOG = rate of hydrolyzed carbohydrates that enters the stomach

then Puckett assumes

(7.1) CHOG = [ t U(t) - (t - 1) U(t - 1) - (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) = - 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) = - GA + 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

PI =

Using curve fitting techniques with actual data Puckett uses the values

T = 156.59 minP = 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

f(t) =

where

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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) = - y + x y(0) = 0

(7.5) = - x + 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)

wherex = rate at which glucose leaves the stomach for a meal of mass 4g(t) = rate of hydrolyzed carbohydrates that enters the stomach for a meal of

mass 4

Proposition 2.

f(t) =

Proof. Let’s first look at equation (7.5). If we take Laplace transforms of both sides we get

( ) = - ( x ) + ( g(t) )

Using the properties of Laplace transforms we get

s ( x ) – x(0) = - ( x ) + [ - - + ]

Using the fact that x(0) = 0, we get

(s + ) ( x ) =

( x ) =

x = -1( )= 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( )

Using partial fractions one obtains

= - +

So z = t - T + T e-t/T

x = (t - T + T e-t/T ) U(t) - (t - T - 1 + T e-(t-1)/T ) U(t - 1)

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- (t - T - 4 + T e-(t-4)/T ) U(t - 4) + (t - T - 5 + T e-(t-5)/T ) U(t - 5)

=

Next consider the equation (7.4). By a similar argument we get

y = v(t) U(t) - v(t - 1) U(t - 1) - v(t - 4) U(t - 4) + v(t - 5) U(t - 5)

where v = -1( )

Using partial fractions one obtains

= - + -

So v = t - T - P + e-t/T - e-t/P

y = (t - T – P + e-t/T - e-t/P ) U(t)- (t - T – P - 1 + e-(t-1)/T - e-(t-1)/P) U(t - 1)- (t - T – P - 4 + e-(t-4)/T - e-(t-4)/P) U(t - 4)+ (t - T – P - 5 + e-(t-5)/T - e-(t-5)/P) U(t - 5)

=

This proves the proposition since f(t) = . //

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

= Auwith

A =

where the partial derivatives are evaluated at (Gb, Ib).

Proposition 3. For Topp's model discussed in Section 5 one has

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(8.1) A =

The eigenvalues of A are negative real numbers if

(8.2) (E+SIb - k)2

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) = – kI

We have

= – (E+SI)

= – SG

=

= – k

Thus (8.1) follows. One has

det[A] = k(E+SI) +

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 22 matrix T are real if

[ trace[T] ]2 4 det[T]

If we write T = , then the eigenvalues of A are real if

(p – s)2 4rq

Thus the eigenvalues of A are real if

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(8.5) (E+SI - k)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) – kI = 0

From equation the second equation it follows that

(8.8) = kI

G2 =

(8.9) + G2 =

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 = 0.6 mg/dl / min.E = 0.001 / min.S = 5  10-4 / U/ml /min.b = 9 U/ml / min = 20000 (mg/dl)2

k = 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 =

=

In this case

(E+SI - k)2 = (0.001 + (0.0005)(10) – 0.3)2 = (0.006 – 0.3)2 = (- 0.294)2

0.086436and

= = = = 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.

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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 =

= and

(E+SI - k)2 = (1 + (1)(1) – 2)2 = 0and

= = = 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 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 insulin-mediated glucose uptake in normal subjects and in subjects with latent diabetes. J Clin Invest, 49, 2151-2160, 1970.

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[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.

[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.

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