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Closed-Loop Control of An Artificial Pancreatic Beta Cell Using Multi-Parametric Model Predictive Control

Matthew W. Percival1, Eyal Dassau1, Howard Zisser2, Lois Jovanovic2, and Francis J. Doyle III1. (1) Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106-5080, (2) Sansum Diabetes Research Institute, 2219 Bath St, Santa Barbara, CA 93105-4321

Diabetes mellitus is a disease characterized by insufficient endogenous insulin production, leading to a poorly regulated plasma glucose concentration. Chronic hyperglycemia can lead to macro- and micro-vascular complications, e.g., stroke, heart attacks, blindness, and kidney disease. An indication of the quality of glycemic regulation, and thus the likelihood of such complications is given by the quantity of glycosylated hemoglobin (A1C). Successful control consists of normalizing glycemia, and thus reducing A1C to below 6.0% [1].

Type 1 diabetes mellitus (T1DM) is an autoimmune disease that results in the destruction of the pancreatic beta cells found in the islets of Langerhans; these cells are responsible for the production of insulin, one of the hormones that regulate plasma glucose concentration. In order to regulate glycemia, insulin must be delivered exogenously. Current therapeutics involve manually taking a capillary blood sample (a finger stick measurement) to determine glycemia, then injecting an appropriate amount of insulin depending upon the subject and the situation, e.g., at meal time. For this method to be effective, this process must be repeated up to 12 times daily. Such a high level of discipline is burdensome, and thus patient adherence is typically lacking. As a result, glycemic control is inadequate and the complications of T1DM occur. As an indication of the worldwide scale of the problem, there are 1.5 million people with T1DM in the United States alone [2].

Recent technological advances have brought continuous real-time glucose sensors and continuous insulin infusion pumps, which in combination with rapid-acting insulin analogs provide a framework for a closed-loop controller suitable for use in ambulatory conditions [3]. Currently, however, robust control algorithms for such a device are not yet available. One of the main challenges is the large lag time before subcutaneously delivered insulin affects the plasma glucose concentration. For this reason, model predictive control (MPC) has been identified as a highly promising controller structure, and has been applied in a clinical environment [4]. The MPC framework also allows for the explicit inclusion of constraints [5], which are pertinent to the physical capabilities of the insulin pump.

In order to mitigate the computational problems involved with solving large optimization problems online, multi-parametric programming techniques can be applied which lead to the solution of a single optimization problem offline; the online problem thus reduces to reading a lookup table, and the evaluation of an affine function [6]. Such techniques are germane to closed-loop devices for managing glycemia, since the first commercially available artificial beta cells will no doubt have limited computational capacity. This technique has previously been used to develop a multi-parametric MPC (mpMPC) algorithm in the context of diabetes, albeit for intravenous insulin delivery [7].

For the regulatory agencies, any device that administers a drug automatically, such as an artificial beta cell is considered high risk. In this application, the Food and Drug Administration requires that the algorithm is safe without any abnormalities that can be related to optimization and that a risk analysis on the outcome of such a device and a system validation to ensure its efficacy and safety [8]. This implementation provides mathematical stability since the only online calculation is the evaluation of an affine function.

A distinctive feature of mpMPC is that it provides the necessary ability to ensure controller stability to different inputs as required by the FDA and demonstrated on several applications in [12] without the need to compromise the controller design to a less sophisticated one such as PID or unconstrained MPC. This implementation also provides means to analyze the algorithm before implementation, which is critical to satisfying FDA requirements concerning risk analysis.

This study sought to address the issue of glycemic control using subcutaneously delivered insulin in the presence of meal disturbances. The physiological models of glucose insulin kinetics and subcutaneous insulin absorption presented by Hovorka et al. [4] and Wilinska et al. [9] were linearized, and an mpMPC controller was formulated using the Multi-Parametric Toolbox, available for MATLAB [10], and interfacing software, YALMIP [11].

Simulations were performed with measured disturbances of 10-40 g CHO meals. In order to test for robustness, mismatches of 25% were introduced to the CHO measurement and of 15 minutes to the disturbance timing. Under these conditions of uncertainty, the hypoglycemia and hyperglycemia were avoided for meal sizes of up to 35 g CHO.

The use of a disturbance model is of paramount importance in this application, due to the large lag time for insulin action. Without such a model, controller performance is inadequate under conditions of model uncertainty similar to those likely to be experienced with intra-subject variation. Model identification may need to be performed on a daily basis to ensure that the controller had the most up-to-date model of subject dynamics. Further evaluation of the control law will take place in an advisory mode capacity, using clinical data.

This work was supported by the Juvenile Diabetes Research Foundation (JDRF) grant 22-2006-1115, the Institute for Collaborative Biotechnologies ICB, and the Otis Williams Fund at the Santa Barbara Foundation.

Correspondence to:

Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106-5080

frank.doyle@icb.ucsb.edu

References:

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