(574o) Blood Glucose Regulation with Stochastic Optimal Control for Insulin-Dependent Diabetic Patients

            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.

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