Seminar by Emeritus Professor Andrea DeGaetano

Day 1: Modeling the glucose-insulin system for diabetes, problems and perspectives

PDF presentation followed by a Q&A session

We exemplify the application of mathematical and statistical techniques to the analysis of biomedical data by addressing a problem of practical importance in diabetology clinics, that of assessing the severity of Type 2 Diabetes Mellitus by estimating the degree of insulin sensitivity of resistance of a given patient. We showcase early attempts at modeling the glucose-insulin system and discuss the criteria by which a mathematical model of some physiologically relevant dynamical system is judged to be good or bad. We point to potential pitfalls in parameter estimation, with examples from widely diffuse medical literature. We try to understand what are the difficulties that arise when moving from the experimental diabetology lab to the analysis of data streams from common clinical practice. We progressively move from simpler Ordinary Differential Equations models to more mathematically advanced formulations: Fractional Differential Equations may be used to summarize, with an order that can in principle be estimated from data, different (presumably integer-order) interacting controls or influences upon the observed variable of interest, especially if the information content of the observations is relatively small compared to the complexity of the system of interacting variables. This is the case, for example, of trans-cutaneously measured glycemia, where besides glycemia itself (possibly decaying by first-order elimination) also unobserved factors (insulinemia, other hormones) may exert higher order effects. The problem is complicated by the fact that random events (food intake, exercise, emotions) may affect glycemia as well, leading to the formalization of the problem with Fractional Stochastic Differential Equations (FSDEs). We exemplify the use of simple FSDE models of glycemic control and undertake model parameter estimation in this framework. The difficulties in dealing with a numerically daunting task, while attempting to adhere to a physiological interpretation of the available data, are described.

Day 2: The nature of mathematical models

Blackboard presentation with Q& session

Mathematical modeling has become pervasive in applications, not only in physics or economy, but also in biomedicine and other ``soft'' sciences. To the conceptual formulation of a model there often follows its identification by statistical parameter estimation, given available observations. While the nature of the modeling process as well as its relationship with the attending statistical computations could both appear obvious to the practitioner, it may be useful to formalize them in a precise way. Insight into the process of (linear and nonlinear) model parameter estimation can be obtained from the description of the geometry of estimation in case space. The objective then is to also describe the geometry of modeling in the abstract, and to show how the correspondence between the conceptual context and the computational context can be formally represented.