Multi-Parametric Model Predictive Control Algorithm for Insulin Delivery Using Clinical Factors

Type 1 diabetes mellitus (T1DM) affects two million people in the United States [1]. The Diabetes Control and Complications Trial concluded that intensive insulin therapy was the best way to reduce diabetic complications such as blindness and amputations [2]. Effective therapy requires more than ten manual glucose measurements each day; such vigilance is burdensome to people with T1DM, and thus even if diabetic complications are reduced, this therapy does not constitute normal, healthy life, by definition [3].

Closed-loop insulin delivery could result in vast improvements in the quality of life of people with T1DM. With the advent of continuous subcutaneous (SC) glucose sensors and insulin pumps, closed-loop therapy should be possible, as systems combining hardware and software are becoming available [4]. An important challenge now is the development of a safe, reliable control algorithm [5].

The lag times of 1–2 h peak concentration and action associated with subcutaneous insulin delivery necessitate predictive control [6]. Additionally, a model-based controller allows further transparency because the model parameters then relate to physiological characteristics. Model Predictive Control (MPC) is therefore considered the ideal framework [7, 8, 9]. Because an “artificial pancreas” would have less computational power than a desktop computer, a practical solution to avoid both excessive online optimization and optimization failure is required. The reformulation of MPC as a multi-parametric programming problem allows offline optimization and thus both minimal online optimization and a guarantee of online stability [10]. Prior work in the field using multi-parametric MPC (mpMPC) has been limited to intravenous insulin delivery algorithms [11]. This work extends the application to subcutaneous insulin delivery algorithms, using a dynamic model to guarantee offset elimination.

Studies have attempted to model the effects of orally ingested carbohydrate (CHO) and subcutaneously administered insulin on plasma glucose concentrations [12, 13]. The procedures required to develop these models are too arduous to be practical on an individual basis, so less intensive protocols are required in practice. In order to avoid identifiability problems associated with simultaneous excitation of two inputs, an open-loop protocol consisting of two impulse response tests was implemented. Two models, each with three parameters were developed to identify parameters for second order transfer function models; these models were converted into state-space models for the mpMPC control algorithm.

Due to the difficulties in identifying a reliable dynamic model, additional safety constraints should be considered. Modern clinical therapy considers the effects of previously delivered insulin, so-called insulin-on-board (IOB) using pharmacokinetic models [14]. The IOB was included in a safety constraint used to prevent excess insulin delivery [15]. This safety constraint was included in the mpMPC algorithm for an optimal formulation.

Ten adult subjects from the UVa/Padova simulator [16] were the virtual cohort for the simulation study. The open-loop protocol included impulse response tests using a mixed meal (25 g CHO) and a subcutaneous insulin bolus (1 U) over 12 hours. The closed-loop protocol included three unannounced meals of up to 100 g CHO over 24 hours. Robustness was analyzed in the closed-loop simulations by perturbations in the control law parameters of up to 25%. The mpMPC formulation included output additive disturbance which eliminated offset in the case of intra-subject changes in insulin sensitivity.

The models developed accurately characterized the input-output data (coefficient of variation, R²>90%) and the impulse response tests and the form of the models were suitable for uniquely identifying parameters. These models provided adequate information for mpMPC algorithm to be effective, with the Average Daily Risk Rating [17] qualified as ‘low'. The mpMPC formulation allowed an optimal, constrained formulation to be implemented at minimal online computational cost. This algorithm will be implemented clinically in the future studies.

This work was supported by the Juvenile Diabetes Research Foundation (JDRF) grant 22-2007-1115, and the Institute for Collaborative Biotechnologies ICB.

Correspondence to:

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

frank.doyle@icb.ucsb.edu

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