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 Research Group, 1993) 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.

                The ideal treatment for controlling blood glucose levels in insulin dependent diabetic patients would be the use of an artificial pancreas which would have the following components: (a) a glucose sensor which monitors the blood glucose continuously with sufficient reliability and precision; (b) a computer which could calculate the necessary insulin infusion rates by an appropriate feedback algorithm; (c) an insulin infusion pump which would release the required amount of insulin into the blood. Safe delivery of insulin in this way requires reliable glucose sensors. Two types of sensors have been developed during the last 30 years, minimal invasive and non-invasive (Koschinsky and Heinemann, 2001). The non-invasive approaches are carried out using optical glucose sensors. These sensors work by directing a light beam through intact skin and measuring the properties of the reflected light that are altered either as a result of direct interaction with glucose (spectroscopic approach) or due to the indirect effects of glucose by inducing changes in the physical properties of skin (scattering approach). However, these optical sensors are not able to measure glucose with sufficient precision. On the other hand, minimally invasive sensors measure the glucose concentration in the interstitial fluid of the skin or in the subcutis. There is a free and rapid exchange of glucose molecules and interstitial fluid. Therefore, changes in blood glucose and interstitial glucose are correlated. However, there is a time delay between these changes varying from a few seconds to 15 minutes; which complicates the interpretation of measurement results (Roe and Smoller, 1998). The magnitude of this delay depends on factors such as the absolute glucose concentrations and direction of change. This delay shows intra and inter-individual variability. Furthermore, it has been found that the absolute values of interstitial glucose concentrations vary between 50 and 100% of the intravasal value.

                The wide-spread use of these glucose sensors is also complicated by biocompatibility issues and skin reactions. Also these sensors should be available at a reasonable cost in order to be applicable to insulin-dependent diabetic patients. The implementation of a closed loop system in daily life conditions requires these reliability, compatibility, cost and safety issues to be resolved.

                The aim of this paper is to develop an optimal control system. Optimal control is different from a closed loop feedback control where the desired operating point is compared with an actual operating point and knowledge of the difference is fed back to the system. Optimal control problems are defined in their time domain, and their solution requires establishing an index of performance for the system and designing the course (future) of action so as to optimize a performance index. Therefore optimal control allows us to make future decisions. Using optimal control theory we can minimize the deviations of blood glucose from non-diabetic levels, while penalizing the use of large amounts of infused insulin for safety. Swan (1982), Fisher and Teo (1989), Ollerton (1989), Fisher (1991) and Parker et al. (1999) applied optimal control theory to this problem. However, uncertainties in model parameters and variability among different individuals were not considered in these papers.

                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 (Simon et al., 1987; Bremer and Gough, 1999). Meals and exercise, the age and weight of the patient also affect the insulin/glucose dynamics. These daily and hourly fluctuations of patient parameters can create difficulties in continuous glucose control. These dynamic uncertainties affect the optimal insulin infusion profiles.

                The aim of this paper is to model 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 Merton and Samuelson (1990) described optimal investment rules developed for pricing options in financial markets, and Ito's Lemma (Ito, 1951; 1974) 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 encountered in other branches of science. For example, in chemical engineering literature, time-dependent uncertainties in batch processing and pharmaceutical separations were represented by Ito processes and time-dependent stochastic optimal control profiles were obtained. Using this approach, the performance of separation processes where stochastic optimal control was applied, has increased significantly as high as 69%. Using Ito processes, ideal and non-ideal systems were represented and thermodynamic parameter uncertainties associated with locally optimal parameter estimates as a result of nonlinear regression were addressed (Ulas and Diwekar, 2004; Ulas et al., 2005).

                This approach could also be extended to optimal glucose control in insulin dependent diabetic patients. The blood glucose profiles can be represented using Ito processes and stochastic optimal control profiles could be derived to achieve better treatment for diabetes. The results show that the hourly and daily variations of blood glucose in response to meals and insulin action can be modeled using this methodology and using stochastic maximum principle; optimal insulin infusion profiles can be computed. The stochastic optimal control profile results in fewer variations from the reference blood glucose value of 4.5 mmol/L as compared to the deterministic profile and could potentially be useful in preventing the complications of diabetes.

References:

Fisher M.E. and Teo K.L. (1989) ‘Optimal insulin infusion resulting from a mathematical model of blood glucose dynamics', IEEE Transactions in Biomedical Engineering 36:479-486.

Ito K. (1951) ‘On stochastic differential equations', Memoirs of American Mathematical Society 4(1).

Ito K. (1974) ‘On stochastic differentials', Applied Mathematics and Optimization, 4, 374

Koschinsky T. and Heinemann L. (2001) ‘Sensors for glucose monitoring: technical and clinical aspects', Diabetes Metab. Res. Rev. 17(2):113-23

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

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

Roe J.N. and Smoller B.R. (1998) ‘Bloodless glucose measurements', Crit Rev Ther Drug Carrier Syst. 15(3):199-241.

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.

The Diabetes Control and Complications Trial Research Group (1993) ‘The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin-dependent diabetes mellitus', New England Journal of Medicine 329: 977-986.

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. (2005), ‘Uncertainties in parameter estimation and optimal control in batch distillation', Computers and Chemical Engineering 29(8): 1805-1814.