Development of mathematical models of everyday life factors on glycaemia, identifiable in outpatients with type 1 diabetes

Abstract

Insulin therapy for type 1 diabetes mellitus is used to manage and maintain near normal blood glucose. However, this therapy balances treating hyperglycaemia and avoiding hypoglycaemia, both of which have negative health consequences. Optimal insulin doses are often uncertain in everyday life. Factors that are often overlooked can noticeably alter the metabolism of glucose and insulin, which can confound glycaemic control. Four major categories of everyday life factors have been reviewed in depth in the literature: nutritional variability, psychological effects, physical activity and metabolic rhythms.

Physiological mathematical models have long been used to study, observe and control glycaemia. However, relatively little physiological modelling of everyday outpatient factors has been carried out. Clinical data of subjects with type 1 diabetes experiencing everyday life events such as exercise, meals, snacks and insulin boluses was used as the basis for model development. In particular, the inclusion of the appearance of insulin from subcutaneous infusion, and the effect of exercise on plasma insulin and glucose concentration have been modelled. The insulin system was modelled with multiple physiological compartments while the effect of exercise was initially modelled with a data-driven autoregressive technique before basis models were developed.

Practical identifiability was considered to be a mathematically limiting factor for model complexity and specificity given that high quality data is generally not available in the outpatient environment. Hence, model development strongly considered practical identifiability. For example, a number of analyses were employed to determine which of multiple options for the insulin model had the optimal complexity. The result was that one of the simpler models had the best compromise between fit, parameter robustness and prediction ability. Furthermore, since practical identifiability is a relatively new field with no formal analyses, two possible evaluation techniques were explored. One technique with an analytical a priori nature and the other using retroactive computational methods.

An in silico Monte Carlo analysis was carried out to test the potential model recovery of exercise, stress fatigue and insulin sensitivity in outpatient glycaemia. It was found that sparse, irregular and noisy data could be overcome as the data accumulated to provide a clearer picture of patient status. Variation in parameters decreased with increasing data according to the 1/√n rule, indicating that measurement error and other sources of noise introduced did not obscure parameter estimation. This proof of concept represents a pathway toward personalisable glycaemic models that can be fitted to the individual’s responses, and be used to predict their response to treatment. Ultimately, sound modelling of everyday life factors would improve the quality of life for sufferers of diabetes by improving control and decreasing the burden of disease management.