Safety constraints in an artificial pancreatic beta cell: an implementation of model predictive control with insulin on board

Abstract

Background: Type 1 diabetes mellitus (T1DM) is characterized by the destruction of pancreatic beta cells, resulting in the inability to produce sufficient insulin to maintain normoglycemia. As a result, people with T1DM depend on exogenous insulin that is given either by multiple daily injections or by an insulin pump to control their blood glucose. A challenging task is to design the next step in T1DM therapy: a fully automated insulin delivery system consisting of an artificial pancreatic beta cell that shall provide both safe and effective therapy. The core of such a system is a control algorithm that calculates the insulin dose based on automated glucose measurements.

Methods: A model predictive control (MPC) algorithm was designed to control glycemia by controlling exogenous insulin delivery. The MPC algorithm contained a dynamic safety constraint, insulin on board (IOB), which incorporated the clinical values of correction factor and insulin-to-carbohydrate ratio along with estimated insulin action decay curves as part of the optimal control solution.

Results: The results emphasized the ability of the IOB constraint to significantly improve the glucose/insulin control trajectories in the presence of aggressive control actions. The simulation results indicated that 50% of the simulations conducted without the IOB constraint resulted in hypoglycemic events, compared to 10% of the simulations that included the IOB constraint.

Conclusions: Achieving both efficacy and safety in an artificial pancreatic beta cell calls for an IOB safety constraint that is able to override aggressive control moves (large insulin doses), thereby minimizing the risk of hypoglycemia.

Figure 1.

An illustration of nonlinear insulin action curves. The action curves differ from each other by their duration of action and are being used by CSII pump bolus wizards to estimate the available insulin in plasma, modified from Walsh and associates. These curves are the result of pharmacodynamic–pharmacokinetic studies and are specific to the insulin type.

Figure 2.

An ARX-based MPC on subject 2 with and without the IOB constraint. The 24 h scenario starts at 7:00 AM at steady state followed by a protocol of three meals (8:00 AM, noon, and 6:00 PM) with 20, 40, 70 g of CHO, respectively. The MPC used a prediction horizon of 400 time steps, a control horizon of 5 time steps, and a weighting of unity for both insulin delivery rates and glucose tracking error. Panel (A) describes glycemic trajectories (continuous line) with a fixed hard constraint that exhibits risky behavior by crossing the hypoglycemic threshold. The precarious results are prevented by the controller incorporating the IOB constraint as presented in panel (B). Moreover, the MPC with the IOB constraint keeps glycemia, marked by a continuous curve, above 100 mg/dl without any risk of hypoglycemia. The IOB constraint on the rate of insulin administration is described by the dashed curve in panel (D), while the implemented insulin rate of the control moves is represented by circles in both panels (C) and (D). Panel (D) shows that the rate of the injected insulin is frequently constrained by the empirical values of the CF and I:C to prevent potential hypoglycemia.

Figure 3.

An ARX-based MPC on subject 10 with and without the IOB constraint. The 24 h scenario starts at 7:00 AM at steady state followed by a protocol of three meals (8:00 AM, noon, and 6:00 PM) with 20, 40, 70 g of CHO, respectively. The glucose trajectories with and without the IOB constraint are presented in panels (A) and (B), respectively. The controller moves with or without the IOB constraint are presented in panels (C) and (D), respectively. The dashed lines represent the values of hyperglycemia and hypoglycemia. The controller that incorporated the IOB constraint shows a more conservative behavior in panel (D) than the controller that was missing the IOB constraint in panel (C).

Figure 4.

Clinical results from a closed-loop trial, presenting the use of IOB constraint to prevent an overdose of insulin. The glucose trajectory, the controller set point of 100 mg/dl, and the insulin delivery rate are presented in the upper panel as constant curve, dotted curve, and dashed curve, respectively. The IOB constraint, insulin for correction, and allowed insulin amount above basal are presented in the lower panel as constant curve, dashed–dotted curve, and dotted curve, respectively. As can be seen from the lower panel, the allowed insulin delivery rate above basal is the amount needed for correction minus the IOB, or zero if the results is negative.

Figure 5.

An ARX-based MPC on subject 2 using the ARX model derived from subject 1 data. The 24 h scenario starts at 7:00 AM at steady state followed by a protocol of three meals (8:00 AM, noon, and 6:00 PM) with 20, 40, 70 g of CHO, respectively. The glucose trajectories with and without the IOB constraint are presented in panels (A) and (B), respectively. The controller moves with or without the IOB constraint are presented in panel (C) and (D), respectively. The dashed lines represent the values of hyperglycemia and hypoglycemia. As depicted, the IOB constraint overrides the control moves that produced hypoglycemia otherwise.

Figure 6.

An ARX-based MPC on subject 9 using the ARX model derived from subject 9 data in the presence of measurement noise. The 24 h scenario starts at 7:00 AM at steady state followed a protocol of three meals (8:00 AM, noon, and 6:00 PM) with 20, 40, 70 g of CHO, respectively. The glucose trajectories with and without the IOB constraint are presented in panels (a) and (b), respectively. The controller moves with or without the IOB constraint are presented in panel (c) and (d), respectively. The dashed lines represent the values of hyperglycemia and hypoglycemia. As depicted, the IOB constraint “jumps” between insulin action curves at the presence of the noise but overall maintains good control and prevents hypoglycemia and the unstable behavior introduced otherwise by the hard constraint control.