IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I . JANUARY 1991

A Semiclosed-Loop Algorithm for the Control of Blood Glucose Levels in Diabetics

Michael E. Fisher

Abstract-In this paper, a theoretical analysis of the control of plasma glucose levels in diabetic individuals is undertaken using a simple mathematical model of the dynamics of glucose and insulin interaction in the blood system. Mathematical optimization techniques are applied to the mathematical model to derive insulin infusion programs for the control of blood levels in diabetic individuals. Based on the results of the mathematical optimization, a semiclosed-loop algorithm is pro- posed for continuous insulin delivery to diabetic patients. The algo- rithm is based on three hourly plasma glucose samples. A theoretical evaluation of the effectiveness of this algorithm shows that it is superior to two existing algorithms in controlling hyperglycemia.

A glucose infusion term representing the effect of glucose intake re- sulting from a meal is then introduced into the model equations. Var- ious insulin infusion programs for the control of plasma glucose levels following a meal are then assessed. The theoretical results suggest that the most effective short-term control is achieved by an insulin infusion program which incorporates an injection to coincide with the meal.

I. INTRODUCTION HE past two decades has seen a great deal of effort ex- T pended in the design of insulin infusion devices for the con-

trol of blood glucose levels in insulin dependent diabetes mellitus patients. These devices deliver insulin either intrave- nously or subcutaneously, but both delivery methods have so far proved to be less than perfect [ 11. Intravenous infusion has suffered from problems with indwelling catheters while contin- uous subcutaneous infusion has yet to prove its superiority over more conventionally administered subcutaneous infusion. Nevertheless, we take the optimistic view that future research (see, for example, [2], [3]) promises the eventual resolution of problems associated with insulin delivery devices.

The development of insulin infusion programs has generally proceeded along two fronts: open-loop methods and closed-loop methods. Open-loop programs deliver a predetermined amount of insulin to the patient and these have become increasingly more sophisticated in an attempt to obtain normality in a dia- betic’s response to changing plasma glucose levels (see, for ex- ample, [4]-[SI). Along with the development of open-loop methods there has been a simultaneous development of closed- loop methods [9]-[ll], sometimes referred to as an artificial beta cell or artificial pancreas. These devices require continuous monitoring of blood glucose levels and can involve quite so- phisticated and costly apparatus.

An alternative approach to open- or closed-loop methods are semiclosed-loop methods based on intermittent blood glucose sampling. In a sense, these are a compromise between open- loop systems and fully closed-loop systems and are particularly

Manuscript received June 8, 1989; revised February 27, 1990. The author is with the Department of Mathematics, The University of

IEEE Log Number 9042378. Western Australia, Nedlands, Western Australia 6009.

appealing because of their simplicity and lack of expense. Such systems have been investigated in [ 121-[15]. Clinical studies with two of these systems, which are based on three hourly plasma glucose readings [ 131, [ 141, show them to be reasonably successful in practice.

In this paper, we undertake a theoretical analysis of the con- trol of plasma glucose levels in diabetic individuals using the simple mathematical model of the dynamics of glucose and in- sulin interaction in the blood system developed by Bergman er al. [ 161-[ 181. Mathematical optimization techniques are used to calculate optimal insulin infusion programs for the correction of hyperglycemia based on the theoretical model. The relative merits of various insulin infusion programs such as single in- jections, continuous infusion, and a combination of both single injection and continuous infusion for the control of blood glu- cose levels in diabetic individuals are then assessed. Based on the results of the mathematical optimization for the Bergman model, a semiclosed-loop algorithm is proposed for insulin de- livery to diabetic patients. This algorithm is based on three hourly plasma glucose samples and combines a single injection with continuous infusion of insulin. Computer simulation is then used to theoretically evaluate the effectiveness of this algo- rithm. The results show that, in terms of the Bergman model, the algorithm compares favourably with the algorithms pro- posed by Chisolm er al. [13] and Furler et al. [14].

A glucose infusion term representing the effect of glucose intake resulting from a meal is then introduced into the model equations. Three insulin infusion programs incorporating single injections, continuous infusion, and a combination of both are assessed using mathematical optimization techniques. The the- oretical results suggest that the most effective short term control is achieved by an insulin infusion program which incorporates an injection to coincide with the meal.

11. THE MATHEMATICAL MODEL

The model we shall study is the so-called “minimal model” of Bergman er al. [16]-[18]. The model consists of a single glucose compartment, whereas plasma insulin is assumed to act through a “remote” compartment to influence net glucose up- take. Bergman er al. claim that among the models studied which satisfy certain validation criteria this model has the minimum number of parameters. While still classified as a very simple model of the dynamics of the interaction of blood glucose and insulin this model still retains some compatibility with known physiological facts and has now been validated in a number of clinical studies [18]-[20]. In this model, G ( r ) and Z ( t ) repre- sent the differences of plasma glucose concentration and free plasma insulin concentration, respectively, from their basal val-

0018-9294/91/0100-0057$01 .OO 0 1991 IEEE

58 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I , JANUARY 1991

ues. The model equations are

G = -PIG - X ( G + GB) + P ( t )

x = -p2x + p31

I = -n(I + I,) + u ( t ) / V I (2.1)

with initial conditions G ( 0 ) = Go, X ( 0 ) = Xo and Z(0) = Io. In (2. l) , P ( t ) and U ( t ) are the rates of infusion of exogeneous glucose and insulin, respectively, and X ( t ) is proportional to the concentration of insulin in the remote compartment. The units of measurement will be taken as mmol per liter of plasma for G and mU per liter of plasma for 1. The constants G, and 1, are basal values of plasma glucose concentration and free plasma insulin concentration, respectively. V, is the insulin dis- tribution volume and n is the fractional disappearance rate of insulin.

The model equations (2.1) actually correspond to those used by Ollerton [ 151. Furler et al. [ 141 yse the same equations with additional equations describing the effect of insulin antibodies on the insulin dynamics. Values for the model parameters pl , p 2 , andp, are estimated by Bergman et al. in [17] in a study of diabetic and normal human subjects and in [16] for a group of dogs. Based on [17], Furler et al. give three alternative sets of values for p , , p 2 , and p 3 . Values they use for normal subjects are

pI = 0 . 0 2 8 , ~ ~ = 0.025, p 3 = 0.000013. (2.2)

For diabetic (glucose resistant) subjects they argue that the value of pI is significantly reduced and so for theoretical purposes they use the set of values

= 0, p2 = 0.025, p3 = O.ooOo13. (2.3)

The other model parameters are also discussed in [ 141 and, for a person of average weight, they use the values

V, = 12 L, n = (5/54)/min,

G, = 4.5 mmol/L, I, = 15 mU/L. (2.4)

The values of V, and n they use are obtained from [21], while the value of Gs corresponds approximately to the basal plasma glucose concentration found in normal individuals. The value of I, is derived in [14] from the investigations of Home et al. [22] and is typical of free insulin levels of controlled diabetic subjects under steady-state conditions.

The steady state in the model corresponds to a constant in- sulin infusion rate of U = nV,Z, mU/min which, for the param- eter values (2.4), corresponds to 1 U/h. This is consistent with observations of the infusion rates that are required to maintain steady-state plasma glucose levels of severe diabetics at the basal values for normal subjects.

111. OPTIMAL CONTROL IN THE ABSENCE OF GLUCOSE INFUSION

In this section we examine various insulin infusion programs, based on the model equations (2.1), for correcting an initial state of hyperglycemia. As a means of comparing the effective- ness of the various programs we will use the performance cri- terion

J ( u ) = i o T G 2 ( t ) dt (3.1)

TABLE I COMPARISON OF THREE INSULIN INFUSION PROGRAMS FOR THE MODEL

(2.1)

Go = 10.5 mmol /L mmol/ L

insulin ( U ) J ( u ) insulin ( U ) J(u)

i) Injection/basal infusion 1.91 + 6 683 2.98 + 6 2219 ii) Optimal hourly infusion 7.89 1168 8.39 4065

iii) Injection/hourly infusion 3.29 + 4.33 454 4.57 + 4.10 1595

Go = 5.5

where [0, TI is the time interval under consideration. Other performance criteria have been used in the literature. For ex- ample, Swan [23] includes a term proportional to the square of the insulin infusion rate to obtain closed-loop solutions for the linear Ackerman model [24]. For the Bergman model, closed- loop solutions are not obtainable and (3.1) seems to be an ap- propriate performance criterion.

The problem we consider is the minimization of J ( U ) subject to the system equations (2. l ) , the corresponding initial condi- tions, and the constraints

0 5 u ( t ) 5 U,,,, for all t E [O, TI (3.2)

where U,,, is a constant. This problem cannot, in general, be solved analytically. However, open-loop solutions can be ob- tained using numerical techniques. One numerical algorithm which can be used to solve these problems is the general opti- mal control program MISER developed by Goh and Teo [25]. The solution procedure used by this program partitions the in- terval [ 0, T ] into equal subintervals and approximates the con- trol u ( t ) by a constant function on each subinterval of the partition.

We have used the MISER program to solve the problem for the parameter sets (2.3) and (2.4) together with a variety of initial conditions. The numerical results show that, as the par- tition of [ 0, TI becomes finer and finer, if we remove the upper bound U,,, on U ( t ) , the optimal control for this problem in- cludes an impulse control at t = 0. This suggests that a com- bination of impulse control at t = 0 and continuous infusion is in fact the true optimal control for the optimal relaxed control problem in which the control is not bounded above. For a dis- cussion of optimal relaxed control problems the reader is re- ferred to [26].

In Table I we compare the values of the performance criterion J over a 6-h period resulting from three insulin infusion pro- grams for the initial plasma glucose levels of 10 mmol/L (Go = 5.5) and 15 mmol/L (Go = 10.5). These three pro- grams are i) a single injection at t = 0 plus infusion at the basal level of 1 U/h, ii) optimal infusion, constant over hourly pe- riods, iii) a single injection at t = 0 followed by optimal hourly infusion of insulin. The insulin entries in Table I correspond to the initial injection plus 6 U for program i), the total amount infused over the 6-h period for program ii), and the initial in- jection plus the total amount infused following the injection over the 6-h period for program iii).

Fig. l(a) shows the plasma glucose responses for these three infusion programs for an initial glucose level of 15 mmol/L together with the plasma glucose response when only the basal insulin level ( 1 U/h ) is administered. Fig. l(b) shows the in- sulin infusion rate profile for program iii) where the initial glu- cose level is 15 mmol/L. A comparison of the three programs show that they all use similar total amounts of insulin while

- 1 - -

59

0

FISHER: CONTROL OF BLOOD GLUCOSE LEVELS IN DIABETICS

I GO)

d

basal infusion /

injection +optimal hourly infusion II \\\ I \ ' \

insulin infusion , ,5 1 rate (U/hour)

time (hours) 0

3 6

(b) Fig. 1. (a) Plasma glucose profiles for three insulin infusion programs. (b) Insulin infusion rate profile for program iii). In both cases, Go = 10.5 mmol/L and T = 6 h.

'

giving different plasma glucose profiles. The two programs in- corporating an injection are far superior in terms of the objec- tive function J ( U ) , although an interesting feature of program iii) is that it fails to retum the plasma glucose level to the basal level inside 6 h.

IV. A SEMICLOSED-LOOP ALGORITHM Because of the complexity and expense of fully closed-loop

insulin infusion devices, the development of a simple semi- closed-loop system based on, say, three hourly plasma glucose readings is especially appealing. Such systems have been de- veloped by Chisolm et al. [13] and Furler et al. [14] with the insulin delivered by a computer-assisted insulin infusion sys- tem. Ollerton [15] has also used theoretical techniques to derive semiclosed-loop insulin infusion algorithms based on three hourly and 10 min plasma glucose readings.

The systems of Chisolm et al. and Furler et al. have been used with some success for short term therapy of diabetic in- patients. The algorithms on which these systems are based sup- ply a constant insulin infusion rate over a three hour period calculated from a simple piecewise linear graph on the basis of a three hourly plasma glucose reading. The first algorithm (Chisolm et ul.), which we shall refer to as Algorithm 1, deliv- ers insulin at the rate of 0.5 U/h for plasma glucose levels below 4 mmol/L and 2.5 U/h for levels above 8 mmol/L, with a linear transition between these rates for plasma glucose

TABLE I1 VALUES OF THE PERFORMANCE CRITERION J( U ) FOR ALGORITHMS 1, 2,

AND 3

Initial Plasma Glucose Level (mmol/L)

8 10 12 15 20

Algorithm 1 5284 7600 6563 8347 17092 Algorithm 2 1220 2859 5041 8134 16790 Algorithm 3 294 672 1197 2213 4375

levels from 4 to 8 mmol/L. The second algorithm (Furler et al.), which we shall refer to as Algorithm 2 , delivers insulin at the rate of 0.5 U/h for plasma glucose levels below 2 mmol/L and 2.5 U/h for levels above 12 mmol/L, with a linear tran- sition between these rates for plasma glucose levels from 2 to 12 mmol/L. These two algorithms have been evaluated theo- retically in [ 141 using the Bergman model and Algorithm 2 was found to be superior in controlling hyperglycemia.

The results of solving the optimal control problem formulated in the previous section clearly show that a combination of an impulse with infusion held constant over fixed time periods re- sults in a far superior correction of a hyperglycemic state than control based purely on constant infusion. Based on the results from solving this optimal control problem for various initial values of plasma glucose levels we have constructed a simple semiclosed loop algorithm for plasma glucose control. This al- gorithm, which we shall call Algorithm 3, uses the glucose reading in the beginning of each three hour period and consists of two parts. Firstly, if the plasma glucose reading is greater than or equal to 6 mmol/L (a somewhat arbitrary amount) then an injection is given followed by constant infusion for the 3-h period at the rate required to maintain plasma insulin at its basal value in the steady state, that is, U = nV,Z, mU/min. The size of the injection is given by the formula

U = G(0.41 - 0.0094G) U. (4.1 )

This formula was obtained by a least-squares fit of the expres- sion

U = G(u - bG) U

to the data obtained from solving the optimal control problem for various initial values of plasma glucose levels. Secondly, if the plasma glucose reading is below 6 mmol/L then insulin is delivered at a constant rate over the next 3 h which is

(4.3)

This formula is very close to that obtained from linear regres- sion based on solving the optimal control problem for various initial values of plasma glucose levels.

The effectiveness of Algorithm 3 was then assessed from a theoretical viewpoint using the model and the parameter values (2.3) which correspond to a patient with relatively severe dia- betes. The performance criterion (3. l ) was used as a means of comparing the performances of this algorithm with those of Al- gorithms l and 2. The time period considered was 24 h. In all cases, Algorithm 3 was found to be vastly superior to the other two, not only in the values of J obtained, which are shown in Table 11, but also in the increased stability (speed with which any oscillations are damped out) of plasma glucose levels. Fig. 2(a) and (b) shows the theoretical plasma glucose profiles re-

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60 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. I , JANUARY 1991

(b) Fig. 2. Plasma glucose profiles over a 24-h period for Algorithms 1 , 2,

and 3. (a) Gn = 5.5 mmol/L. (b) Gn = 10.5 mmol/L.

sulting from the three algorithms for initial glucose values of 10 mmol/L (Go = 5 . 5 ) and 15 mmol/L (Go = 10.5), respec- tively.

V. CONTROL OF PLASMA GLUCOSE LEVELS FOLLOWING A MEAL

Here we examine means of controlling the plasma glucose level following an infusion of exogenous glucose. In the model we shall assume that oral glucose infusion commences at t = 0 prior to which plasma glucose and insulin are at their fasting levels.

The term P ( t ) in (2.1) will be taken to represent the rate at which glucose enters the blood from intestinal absorption fol- lowing a meal. In oral glucose tests with normal subjects, the aim is for the model to produce the desired effect of the plasma glucose level rising quite rapidly (from the rest level) to a max- imum in less than 30 min and then falling to the base level after about 2-3 h. There is some evidence to suggest that the exact form of P ( t ) for nondiabetics is not important provided the pre- viously stated aim is met. A function which produces the de- sired behavior in the model is

P ( t ) = B exp ( - k t ) , t 2 0 (5 .1) provided appropriate values are chosen for the constants B and k . If we use the values

B = 0.5 k = 0.05

I

2 - injection + optimal hourly infusion

rime (hours) 6 0 I

- 2 L

Fig. 3. Plasma glucose profiles for 6 h following a meal for three insulin infusion programs.

then, with the parameter values (2.2) which represent normal subjects, we obtain the desired effect.

The optimal control problem to minimize (3.1), subject to (3.2) was now solved numerically using the basal values of plasma glucose and insulin as the initial conditions. The time period chosen was 6 h and the parameter values were again given by (2.3). The results of three insulin infusion programs are il- lustrated in Fig. 3. These three programs are i) a single injec- tion at t = 0 plus infusion at the basal level of 1 U/h, ii) optimal infusion, constant over hourly periods, and iii) a single injec- tion at t = 0 followed by optimal hourly infusion of insulin. The results clearly show that, of the three programs considered, the most effective short term control is achieved by an insulin infusion program which incorporates an injection coincident with the meal.

VI. DISCUSSION The results obtained in this paper suggest that the most effec-

tive insulin infusion programs are those which include an in- sulin injection. Similar results were also obtained in [27] from a theoretical analysis of the linear Ackerman model [24]. Al- though the present conclusions are based only on a theoretical analysis of the Bergman model, they are consistent with many clinical studies published in the literature which suggest that an appropriate method of treatment for diabetes ketoacidosis is to give a bolus of insulin followed by infusion [9], [28].

The simple algorithm we have proposed here for the correc- tion of hyperglycemia in the absence of a meal uses a combi- nation of injections and constant infusion which are determined by three hourly plasma glucose readings. A theoretical com- parison using the Bergman model shows it to be superior to the algorithms proposed in [ 131 and [ 141. The algorithm suffers from the same deficiency as these earlier algorithms in that it is based on fixed values of the model parameters pl , p 2 , and p 3 . These parameters will, of course, vary from subject to subject. An effective implementation of the algorithm would therefore re- quire some means of compensating for differences between diabetic subjects. This could be achieved by estimating the pa- rameters pl , p 2 , and p 3 for each patient or by incorporating a “patient factor” into the algorithm as in [14].

A complete insulin infusior, program would need to include within it a mechanism for handling meals. Chisolm et al. [13] have incorporated an open-loop meal program into their overall

FISHER: CONTROL OF BLOOD GLUCOSE LEVELS IN DIABETICS 61

insulin infusion program. Our results and also those in [27] sug- gest that an injection should be given which coincides with the meal. We acknowledge that these results are still very much at the preliminary stage and require further investigation before they can be implemented, together with the results of Section IV, in a complete method of treatment. They are, however, consistent with much of the work that has been conducted on physiological responses to glucose infusion. Grodsky [29], [30], for example, has shown that a normal physiological response to a meal is a two or three phase insulin release consisting of a large “bolus” followed by a gradual infusion. In other clinical studies it has been shown that effective control can be achieved with programs which include a large bolus [4], [12] or several pulses 181 of insulin timed to rouszhlv corresuond with meals.

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to the oositions of Let

Michael E. Fisher was born in Yorkshire, En- gland, in 1945 and immigrated to Australia in 1958. He received the B.Sc. degree with first class honors in 1967, and the M.Sc. and Ph.D. degrees in 1971 and 1982, respectively, in mathematics from The University of Western Australia, Nedlands.

He has been employed in the Department of Mathematics at The University of Western Australia since 1970, first as a Teaching Assis- tant, then as a Senior Tutor, and was promoted

;turer in 1985 and Senior Lecturer in 1990. His current research interests include practical aspects of optimal control theory and the dynamics and control of mathematical models in ecol- ogy and biomedicine.