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 Dglucose, 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(molwtofC)+12(molwtofH)+6(molwtofO)
= (6)(12)+(12)(1)+(6)(16) = 72+12+96 = 180
Glucose has the molecular structure
H ------= 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
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
=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 1m-mol of glucose = 180 mg. So 1mmol / 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.
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 50mg/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 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 120130 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 = - 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
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
=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 3U/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 6pmol / l = 1 U / ml. For example, 10 U / ml = 60 pmol / l. Since 6pmol/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 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
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
=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)G
where
P0=rate of glucose production by the liver at zero glucose
EG0P=glucose effectiveness for production at zero insulin
SIP=insulin sensitivity for production
U0=rate of glucose uptake by liver and cells at zero glucose
EG0U=glucose effectiveness for uptake at zero insulin
SIU=insulin sensitivity for uptake
Putting these together one has
(5.1)f(G, I)=R0 - (EG0 + SII)G
where
R0=P0 - U0 = net rate of glucose production by the body at zero glucose
EG0=EG0P + EG0U = total glucose effectiveness at zero insulin
SI=SIP + SIU = total insulin sensitivity
Topp uses the values
R0 = 864 mg/dl / day = 0.6mg/dl / min.
EG0 = 1.44/day = 0.001 / min.
SI = 0.72/U/ml /day = 5104/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 maximum
k = 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
= 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)
= 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 GGb and positive for GGb. Therefore U(G) is decreasing for GGb and increasing for GGb 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
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, , that is assumed to be proportional to the glucose concentration in the interstitium. Thus
=insulin-excitable tissue glucose uptake activity.
The units of 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)=– (G + )G + C
(6.2)=kaI – kb
=kb(SII – )
where
C=R0 = net glucose production at zero glucose concentration.
G=EG0 = glucose effectiveness, i.e. the insulinindependent 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 G with the parameter SG in version 2 of the minimal model below. It looks like SG is about 0.005 more than G. Bergman [12] reports that an average value of G 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 105/U/ml /min2.
kb=p2 = rate constant for decrease of . Pacini and Bergman [13] have an example where it is 0.02/min. 1/kb is the average time it takes for 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 2.3104 to 7.6104 in nondiabetic subjects with a mean of about 5104. For subjects with NIDDM the values are in the range 0.6104. Pacini and Bergman [13] report the value5104. Topp also uses this value; see section 5.
Note: In the papers and G are usually denoted by X and SG. We have used and G since X and SG are used in the second version of the model which follows.
Version 2. Let Gb, Ib, and b be the basal values of G, I, and . They satisfy the equations
(6.3)C=(G + b)Gb