Diabetes mellitus is a metabolic disease involving degradation of the body’s endogenous mechanisms for regulating concentration of glucose in blood plasma. In a healthy body, the pancreas produces insulin, a hormone which facilitates uptake of glucose into cells for use as energy. Where the pancreas’s insulin-producing cells are damaged (type 1 diabetes), or the body becomes severely resistant to insulin (type 2 diabetes), glucose plasma concentrations can reach harmful levels, a state labelled hyperglycaemia. Complications arising from diabetes and hyperglycaemia can cause organ failure, neuropathy, and, in severe cases, death. Diabetes treatment presents a significant health, social, and economic cost. In 2021, its total cost to New Zealand was NZ$2.1 billion in 2021, equivalent to 0.67% of gross domestic product for that year, a figure which is forecast to grow. Causes of diabetes include a variety of genetic and lifestyle factors, but its increasing prevalence is primarily due to increasingly energy-rich diets combined with more sedentary lifestyles in recent decades.

Many treatment approaches and diagnostic tests have been developed to assess and manage diabetes and its common co-morbidities. Treatment of fully progressed diabetes often involves insulin therapy, where analogues of human insulin are externally administered to replace or supplement poor endogenous action. Primarily, diagnostic tests aim to measure some metabolic aspect of an individual’s glycaemic system and coincide this with a development of diabetes or its precursors. Specific diagnostic metrics of interest include insulin sensitivity, how much plasma glucose levels change per unit of insulin; and endogenous secretion, the rate at which the pancreas produces insulin.

Developments in computation over the last 50 years have enabled creation of computer-simulated numerical models to represent human physiology. Glycaemic models such as the Intensive Control Insulin-NutritionGlucose (ICING) model use a compartment model to numerically simulate concentrations of glucose, insulin, and other quantities in various components of the overall glycaemic system. This model is well-validated in many experimental, clinical, and in-silico analyses, where it has been used for glycaemic control, assessments of insulin sensitivity and kinetics, and other metabolic treatments. However, it still presents various opportunities for improvement, especially regarding simplified modelling assumptions and identifiability of its parameters. This thesis explores various modelling and identifiability aspects of these models, and potential applications to novel phenomena in glycaemic control and diabetes care.

First, the existing ICING model is applied to two glucose tolerance test (GTT) trials, one with a small 1 U insulin modification and one with no insulin modification. Primarily, the trials were compared for the practical identifiability of insulin kinetic parameters representing hepatic clearances from plasma. Where a modelling approach is highly practically identifiable, clinical data provides unique, physiologically valid parameter values which produce computation simulations accurate to measurements. Where practical identifiability is poor, many combinations of parameter values, including non-physiological values, can provide an equivalently optimal fit to data, limiting the relevance and conclusions of the modelling approach. Identifiability analysis shows the trial with insulin modification yielded a domain of parameter values providing equivalently optimal fits which was 4.7 times smaller than the domain generated by the non-modified trial. This outcome shows insulin modification improves accurate assessment of subject-specific insulin kinetics and simulation, and suggests modification should be included in metabolic tests where accurate assessment of kinetics is a priority.

Mathematical analysis of model equations, combined with practical numerical analysis, suggests parameter identifiability is improved where time profiles of parameter coefficients are distinct from each other and from other model equation terms. To this end, potential modelling benefits are posed by the expansion of previously constant parameters into more complex profiles which may more completely represent their physiological action. A parameter representing hepatic clearance rate was re-structured as a time-varying sum of mathematical basis splines to explore this approach, where parameter identification techniques identify optimal weightings of individuals splines in the structure. To maintain physiological validity, this identification was constrained based on literature analysis of changes in hepatic clearance rate over time, and of correlation between increasing hepatic clearance rate and decreasing glucose concentration. This more complex model achieves better outcome insulin fits at the expense of greater computation and constraint requirements.

Improvements to existing parameter structure and identification approaches enable novel model expansions. Specifically, a potential loss dynamic reported for subcutaneous jet-injection insulin delivery can be modelled and identified by an expanded model structure, where this loss is not directly measurable nor quantified in a clinical setting. Thus, a new parameter is added to the existing model to represent a proportion of nominal insulin delivery lost at injection site. Where this dynamic possesses similar action to existing identified parameters, the enhanced time-varying structure allows robust per-trial identification of this novel parameter while maintaining or improving insulin simulation accuracy. This approach identified loss values of up to approximately 20% of a nominal 2 U dose in some patients. This identified loss proportion is consistent over a range of parameter identification protocols and values, demonstrating the robustness of this identification. Additionally, identified insulin sensitivity is shown not to vary significantly with identification of this loss factor, further validating that factor’s inclusion in the model and identification approach. Where modelling considered this potential loss and identified a loss of 5% or above, insulin fit accuracy is improved compared to the original case. Overall, this loss factor is shown to be present and quantifiable, and the model is shown to be expandable to novel loss dynamics without compromising fit accuracy or parameter identifiability.

The time-varying basis spline approach to modelling hepatic clearance is further justified and strengthened with an appropriate sensitivity analysis. While previous analysis considers the general improvement gained from time-varying parameter structures, an optimal configuration of spline quantity, placement, and polynomial order is desired. For the analysed data, 20 equidistant splines of 2nd or 3rd order yielded equally optimal results and reduced RMS error of insulin simulations by up to 29% relative to a constant-value parameter model. Importantly, these modelling improvements required no change to measurement or trial protocol and impacted identified insulin sensitivity by less than 0.05% at the optimum configuration. These results further validate the analysis of subcutaneous jet injection loss, finding a greater loss was identified with a more accurate timevarying clearance model, and this loss was invariant to changes in configuration. These outcomes, while related to a single dataset and modelling case, demonstrate a methodology to justify and analyse model modifications.

Overall, these analyses present various areas of improvement and development for glycaemic modelling. More accurate model predictions and parameter identification yields more precise assessment of diagnostic parameters and individual diabetic pathogenesis. Furthermore, the model’s expandability and its robustness in parameter identifiability allow novel treatments to be explored, analysed, and justified using numerical modelling techniques. While error, variability, and parameter trade-off will always pose challenges to modelling, the analyses and methodologies in this thesis ensure relevant and accurate modelling outcomes.