Automatic Insulin Delivery for Type 1 Diabetes Mellitus Using Offline Learning-Type Model Predictive Control (L-MPC)

Type 1 diabetes mellitus (T1DM) is an autoimmune disease manifesting as an absolute deficiency of insulin secretion. In 2000, the total number of people with T1DM was approximately 17.1 million, and this number is predicted to reach 36.6 million in 2030 [1, 2]. The chronic hyperglycemia (high blood glucose concentration) that results from T1DM causes a number of serious complications, such as cardiovascular disease, stroke, hypertension, retinopathy, nephropathy, and neuropathy.

Exogenous insulin is administered in order to reduce glycemia and sustain life; however, excessive insulin delivery results in hypoglycemia (low blood glucose concentration), which has an immediate danger much greater than that of hyperglycemia. Therefore, managing insulin delivery to achieve normoglycemia is a daily challenge.

The development of external insulin infusion pumps and the introduction of rapid-acting insulin analogues have made automated intensive insulin therapy feasible. Additionally, glucose sensor technology has improved significantly over the last few years and the reliable duration of in vivo sensors continues to increase [3, 4]. Continuous subcutaneous insulin infusion (CSII) pumps and continuous glucose monitoring (CGM) sensors are bringing a fully closed-loop artificial pancreas closer to reality.

Due to the time delay between insulin injection and glucose response, model predictive control (MPC) [5, 6] is one of the most promising candidates for closed-loop control of an artificial pancreas. However, MPC cannot improve control performance from day to day, in spite of the repetitive nature of glucose-meal-insulin dynamics over a 24-hour period. In literature [7], scholars used iterative learning control (ILC) to exploit the repetitive nature of glucose-meal-insulin dynamics for the first time, where the "updating law" of ILC was designed using MPC. This combination is named model predictive iterative learning control (MPILC). MPILC can learn an individual's lifestyle, thus allowing the control performance to improve from day to day.

To further improve the robustness of the learning schemes, another novel combination of ILC and MPC was proposed in literature [8, 9], where MPC determined the insulin delivery rate and ILC was used to adjust the set-point for MPC. This combination is termed as L-MPC. To the best of the authors' knowledge, these contributions are the first reported works on L-MPC.

These prior works on L-MPC involve some online optimization to get the "optimal" solution. However, the first generation of artificial pancreatic b-cells will have limited computational power; therefore online L-MPC should be transformed into an offline form. In this work, an offline L-MPC was proposed and applied in automatic insulin delivery.

To derive the model for control design, a 12-hour step response simulation was performed. A single-input single-output auto-regressive exogenous (ARX) model was used to approximate the relationship between insulin and glucose, where meal intake was considered as a disturbance. For mathematical convenience, the ARX model is transformed to a state-space model, and then an analytical solution for MPC can be obtained when there is no input constraint. In practice, there are constraints on the insulin delivery rate, e.g., it cannot be negative. The boundary values were used if the "optimal" input based on the offline solution exceeded the physical constraints.

To test the robustness of the proposed algorithm, it was implemented on ten adult subjects in the UVa/Padova diabetes simulator [10]. All subjects consumed three meals each day at {7:00, 12:00, and 18:00} with fixed amounts of carbohydrate {40g, 85g, and 60g}, respectively. The order of the ARX model was fixed as {2 and 1} for all subjects. There were four parameters in the proposed scheme: learning gain, prediction horizon, control horizon, and input variation penalty weight. The first three parameters were fixed as 0.5, 50, and 5, respectively. The input variation penalty weight was proportional to the subject's insulin sensitivity gain.

After ten days, L-MPC reduced the average tracking error (ATE) by 42% compared with MPC. In addition, L-MPC was robust to random variations in meal size within ±50% of the nominal value and meal timings within ±40 min. Future work will be clinical evaluation of the proposed scheme.

This work was supported by the Juvenile Diabetes Research Foundation (JDRF) grant 22-2006-1115.

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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