Optimal meal time after bolusing for Type 1 diabetes patients under meal uncertainties

The focus of this paper is on the characterization of the uncertainties in the evolving states of a diabetic model, to permit a study of the impact of the time interval between insulin bolusing and meal initiation on hypo- and hyperglycemic events. A polynomial chaos based approach is used to characterize the independent uncertainties in the initial condition and meal size. Galerkin projection of the resulting equations reduce the stochastic differential equations to a set of deterministic equations. This forms the framework to optimize for the post bolusing time to initiate the meal. Two cost functions are considered which correspond to the postprandial hypo- and hyperglycemic excursions of the blood glucose. Numerical results from the minimal Bergman model suggest a 13 and 14 minute interval between bolusing and the initiation of the meal.