(243b) Optimal Control of Diabetes Mellitus under Time Dependent Uncertainties

20.8 million people in the U.S. suffer from diabetes, which has many complications such as heart disease and stroke, high blood pressure, kidney disease, nervous system disease and amputations. The hormone insulin has many functions in the body; most importantly it influences the entry of glucose into cells. The lack of insulin prevents glucose from entering the cells and be utilized, which leads to excess blood sugar and excretion of large volumes of urine, dehydration and thirst. The current treatment methods for insulin-dependent diabetes include subcutaneous insulin injection or continuous infusion of insulin via an insulin pump. The former treatment requires patients to inject insulin four to five times a day. The amount of injection is usually determined by a glucose measurement, an approximation of the glucose content of the upcoming meal and estimated insulin release kinetics. The continuous insulin infusion pump allows for more predictable delivery due to its constant infusion rate into a subcutaneous delivery site. Keeping the blood glucose levels as close to normal (non-diabetic) as possible is essential for preventing diabetes related complications. Ideally this level is between 90 and 130 mg/dl before meals and less than 180 two hours after starting a meal. The Diabetes Control and Complications Trial (DCCT) followed 1441 people with diabetes for several years. This trial concluded that the patients who followed a tight glucose control program were less likely to develop complications such as eye disease, kidney disease and nerve disease, than the ones who followed the standard treatment, because the former group had kept the blood glucose levels lower.

Knowing this fact, we can ask, what is the nature of continuous glucose control in order to minimize the deviations of blood glucose from a preset level, while penalizing the usage of large amounts of infused insulin?  Optimal control theory was applied to this problem by Swan (1982), Fisher and Teo (1989), Ollerton (1989) and Parker et al. (1999). Given the daily and hourly fluctuations of insulin in body can create difficulties in continuous glucose control. In this work, we use a novel approach based on stochastic optimal control theory (Diwekar, 2003).  

The optimal control problem formulation involves finding the set of insulin infusion rates {u_k} that minimizes a cost criterion which is the deviations of blood glucose from a preset level over time, where a ?minimal model? is used to represent insulin/glucose dynamics, which means that the model uses the smallest number of parameters and yet it satisfies certain validation criteria. This model was developed by Bergman et al. (1985). In this model, insulin is assumed to affect glucose levels via a remote compartment X and physiological parameters become ?lumped? together in the resulting equations. The subject-dependent model parameters are glucose effectiveness, which is the net effect of glucose by itself, at basal insulin, to normalize the glucose concentration within the extracellular glucose pool and insulin sensitivity index, which represents the insulin dependent increase in the net glucose disappearance rate. These subject-dependent model parameters are obtained by a data fitting procedure.

The success of optimal control method depends on the accuracy of the model; therefore, the inherent uncertainties in the patient need to be addressed. If the uncertainties are omitted and if the model cannot accurately represent the glucose and insulin dynamics, this can lead to significant performance degradation. Significant variability of relevant parameters among patients and within a given patient during the course of the day or week has been reported in literature. For example, cyclic variations in plasma insulin and glucose levels were observed in subjects undergoing continuous enteral nutrition (Simon et al., 1987). Meals and exercise, the age and weight of the patient also affect the insulin/glucose dynamics.

This paper focuses on modeling these uncertainties by a novel approach and incorporating them into formulations of optimal control. Time-dependent uncertainties are commonly encountered in finance literature. Dixit and Pindyck (1994) and Metron and Samuelson (1990) described optimal investment rules developed for pricing options in financial markets, and Ito's Lemma to generalize the Bellman equation or the fundamental equation of optimality for the stochastic case. This new equation constitutes the base of the so called Real Options Theory. Although such a theory was developed in the field of economics, it was recently applied to optimal control problems in batch processing and pharmaceutical separations, where time-dependent uncertainties were represented by Ito processes. 

An Ito process is a stochastic process, x(t), whose increment, dx, is represented by the equation:

                                                dx = a(x, t)dt + b(x, t)dz                                               (1)

where dz is the increment of a Wiener process, and a(x, t) and b(x, t) are known functions. By definition, E(dz) = 0 and (dz)² = dt where E is the expectation operator and E(dz) is interpreted as the expected value of dz.

Ito processes and Real Options Theory allows us to obtain time-dependent stochastic optimal control profiles. Using this approach, the performance of separation processes where stochastic optimal control was applied, has increased significantly as high as 69%. Using Ito processes, we were able to distinguish between ideal and non-ideal systems and locally optimal and globally optimal parameter estimates (Ulas and Diwekar, 2004; Ulas et al., 2005).

This approach could also be extended to optimal glucose control in insulin dependent diabetic patients. The patient parameters can be forecasted from previous data using Ito processes and stochastic optimal control profiles could be derived to achieve better treatment for diabetes.

The preliminary results using Ito processes, for representing insulin/glucose dynamics show that this method could easily be applied to blood glucose control and optimal insulin delivery profiles could be obtained in the face of static and dynamic uncertainties. 

REFERENCES

Bergman R., Finegood D., and Ader M. (1985), ?Assessment of insulin sensitivity in vivo', Endocrine Reviews 6: 45-86.

Bremer T. and Gough D.A. (1999), ?Is blood glucose predictable from previous values? A solicitation for data', Diabetes 48: 445-451.

Dixit A.K. and Pindyck R.S. (1994), Investment under Uncertainty,
Princeton
University Press, Princeton, NJ.  

Diwekar
U. (2003), Introduction to Applied Optimization, Kluwer Academic Publishers, Netherlands.

Fisher M.E. and Teo K.L., ?Optimal insulin infusion resulting from a mathematical model of blood glucose dynamics', IEEE Transactions in Biomedical Engineering 36:479-486.

Merton R.C., and Samuelson P.A. (1990), Continuous-time Finance, Blackwell Publishing, CambridgeMA

Ollerton R.L. (1989), ?Application of optimal control theory to diabetes mellitus'. International Journal of Control 50: 2503-2522.

Parker R.S., Doyle III F.J. and Peppas N.A. (1999), ?A model-based algorithm for blood-glucose control in type I diabetic patients', IEEE Transactions in Biomedical Engineering 46(2): 148-157

Simon G., Brandenberger G., and Follenius M. (1987), ?Ultradian oscillations of plasma glucose, insulin and c-peptide in man during continuous enteral nutrition', Journal of Clinical Endocrinology & Metabolism 64: 669-674.

Swan G.W. (1982), ?An optimal control model of diabetes mellitus', Bulletin of Mathematical Biology 44: 793-808.

Ulas S. and Diwekar U.M. (2004), ?Thermodynamic uncertainties in batch processing and optimal control', Computers and Chemical Engineering 28(11): 2245-2258.

Ulas S., Diwekar U.M., and Stadtherr M.A. (2005a), ?Uncertainties in parameter estimation and optimal control in batch distillation', Computers and Chemical Engineering 29(8): 1805-1814.