(36e) Pre-Filtering The Insulin Input Improves The Quality Of Empirical Models Identified From Clinical Type 1 Diabetes Data

Type 1 diabetes mellitus is a disease characterized by insufficient regulation of blood glucose levels resulting in high and low glucose concentrations (hyper- and hypoglycemia, respectively). Each extreme has deleterious health implications; severe hypoglycemia can present immediate risks such as diabetic coma and insulin shock while hyperglycemia is known to cause a variety of long-term complications including eye, nerve, and kidney disease [1]. The achievement of sustained normal glucose levels, or normoglycemia, realized through intensive insulin self-therapy greatly reduces these health risks but requires a high degree of attention from the individual with diabetes. An ?artificial pancreatic β-cell? is a medical device consisting of a continuous glucose sensor, an implantable insulin pump, and a control algorithm which automatically regulates blood glucose and thus frees a diabetes subject from the daily onus of diabetes self-management.

Central to an artificial pancreatic β-cell device is the control algorithm which determines the optimal insulin infusion rate based on past insulin-glucose dynamics. In a model predictive control (MPC) framework, the algorithm also uses model predictions of future glucose trends. The performance of an MPC algorithm depends strongly on the quality of the model used [2]; thus, the identification of accurate predictive models is a key step in the development of an artificial pancreatic β-cell.

Modeling the insulin input as impulses, which can be interpreted as the rate of insulin delivery into subcutaneous tissue (SC), can give poor estimates of model parameters such as steady-state gains. Thus, the effect of pre-filtering the insulin input is explored. The filtering is accomplished using low-order physiologically based submodels [3] that describe subcutaneous-to-intravenous insulin absorption (IV) and the consequent intravenous insulin concentration (I). The three insulin input forms are described below:

• SC: This input form is simply the rate of subcutaneously infused insulin. Boluses have been effectively treated as impulses, and thus scaled such that they were delivered over the length of one sample, which was 5 min. For example, a 4 U bolus is modeled as a delivery rate of 48 U/h for one sample (i.e., an impulse of magnitude 48). This timeframe is quite reasonable given the current delivery capabilities of current insulin pumps.

• IV: This input form is the insulin absorption rate into the blood. It is obtained by filtering the SC input through a second-order transfer function describing subcutaneous-to-intravenous insulin absorption. This transfer function has a gain of unity and identical time constants of 55 min (time-to-maximum absorption rate).

• I: This input form is the concentration of insulin in the blood. It is obtained by filtering the IV input through a first-order transfer function. This transfer function has a gain dependent on the subject's weight and a time constant of 7.2 min. Thus the overall transfer function from the SC input to the I input is third-order and has a subject-specific gain.

These physiological submodels are discretized and incorporated into autoregressive exogenous input (ARX) models identified from the subject data to produce ARX models which describe the effects of meals and subcutaneously infused insulin on blood glucose. Thus, using either pre-filtration method effectively results in an overall physiological-empirical hybrid ARX model.

Low-order ARX models of various orders and time delays were identified from data from four subjects with type 1 diabetes. The data included continuous (5-min) glucose measurements, insulin pump records, and subject-reported times and carbohydrate (CHO) content of meals. Metrics were then calculated from these models to ascertain which model(s) were most likely to make the most accurate long-range predictions of future glucose trends and thus contribute to effective MPC performance. The metrics used to assess the quality of the identified models were: 1) R² values for a range of prediction horizons, comprising 6, 12, and 24 steps (30, 60, and 120 min, respectively), 2) steady-state model gains for both the insulin and meal inputs, which are good measures of the sensitivity of the model to changes in these inputs, and 3) t_1/2, the time-to-50% change due to a unit step change in the subcutaneous insulin infusion rate, a measure of the dynamics associated with the insulin input. A ?good? model was assumed to be one that predicted the data accurately (i.e., had high R² values) and had reasonable gains and insulin dynamics. From extensive clinical experience, it was determined that a ?reasonable? gain describing the change in glucose due to a change in subcutaneous insulin infusion rate was approximately -25 to -75 mg dl^-1 U^-1 h, highly dependent on the insulin sensitivity of the subject. Similarly, the gain describing a change in glucose due to a meal was approximately 100 to 500 mg dl^-1 (g CHO)^-1 min, again, highly subject-dependent.

Table 1 shows the modeling results for two subjects. Values shown are means ± standard deviations. While pre-filtering resulted in either a marginal improvement or no improvement at all in prediction capability for the three prediction horizons studied in this research, it did result in a substantial increase in the magnitude of the insulin gain (closer to reasonable values), a modest increase in the magnitude of the meal gain, and a substantial increase in the effective time constants associated with the insulin input. The increased (and more realistic) sensitivity of the models to the insulin input as a result of pre-filtering this input will result in better MPC performance compared to using models identified from data not pre-filtered. In particular, models more sensitive to the insulin input will tend to avoid producing aggressive controller actions which could lead to hypoglycemia.

This joint UCSB-SDRI research project is sponsored by the National Institutes of Health, grant R21-DK069833-02, and the Juvenile Diabetes Research Foundation, grant 22-2006-1115.

References

[1] Centers for Disease Control and Prevention. National diabetes fact sheet: general information and national estimates on diabetes in the United States, 2005. Atlanta, GA: U.S. Department of Health and Human Services, Centers for Disease Control and Prevention (2005).

[2] Parker RS, Doyle FJ III. Control-relevant modeling in drug delivery. Adv Drug Deliv Rev. 2001;48:211-228.

[3] Hovorka R, Canonico V, Chassin LJ, Haueter U, Massi-Benedetti M, Federici MO, Pieber TR, Schaller HC, Schaupp L, Vering T, Wilinska ME. Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes. Physiol Meas. 2004;25:905?920.