Lexicographic derivatives of nonsmooth glucose-insulin kinetics under normal and artificial pancreatic responses

Abstract

Parametric sensitivities of nonsmooth glucose-insulin kinetics models are investigated in this article. In particular, we study several variations on a model of the so-called Intravenous Glucose Tolerance Test (IVGTT), which measures a subject’s insulin response to glucose over time and yields data useful for characterizing and diagnosing diabetes. Nonsmoothness refers here to continuous behavior that is punctuated by “switches” (physiological events) arising due to biochemical and control thresholds being crossed. Insulin secretion from a typical pancreatic response and insulin infusion from an external device (for diabetic patients) are both considered here. The presence of nonsmoothness means that classical sensitivity theory and associated numerical methods are ill-equipped to handle these models. Motivated by this, lexicographic directional differentiation and lexicographic “sensitivity functions” are used, which are recently developed tools in nonsmooth analysis. Using this new approach, we derive nonsmooth sensitivity systems associated with the considered glucose kinetics models, whose unique solutions provide local sensitivity information analogous to classical forward parametric sensitivity functions. Moreover, the theory is practically implementable and the resulting sensitivity functions are computationally relevant, as they can be supplied to dedicated nonsmooth numerical methods (e.g., optimization). The nonsmooth sensitivity systems are solved numerically, giving new insights into the model, and thus into the dynamics of glucose kinetics under the simulated conditions. It is hoped that the results here may be used to improve the design of assessments for diabetes (e.g., the IVGTT) and its treatment.