title: Surrogate modelling of a patient-specific mathematical model of the left ventricle in diastole
Creators: name: Lazarus, Alan
Publisher: name: University of Glasgow
Year: 2022

description: Personalised medicine is a relatively new area of healthcare that uses patient-specific data at multiple scales, and different scientific models, to inform disease prognosis and treatment planning. Recently, there has been particular interest in the translation of mathematical models to the clinical setting. These models are usually implemented in the form of a computer code that relates a set of model parameters with a set of observable quantities. Often these parameters have a physiological meaning, and their estimation can provide information about the level of function or dysfunction of a particular physiological process. An important example is in modelling the behaviour of the left ventricle (LV) in diastole. This model relates cardiac tissue properties (the parameters) with the kinematic behaviour of the LV that can be observed from cardiac magnetic resonance images. The personalisation of this model to different patients depends not only on the parameters, but also on the geometry of the LV, which varies from patient to patient. Improved representation of the LV geometry, combined with improved modelling capabilities, has led to increasingly accurate and personalisable models that can better replicate the real world process. This increased model fidelity is accompanied by increased computational costs, which hinders the application of these models in the clinical setting. A natural solution to the problem posed by computational cost is to use statistical emulation. In emulation, we build a model that efficiently replicates the behaviour of the expensive simulator. Although conceptually a simple idea, the application of this methodology to mathematical models can be complicated. In the context of the LV model, this complexity is largely tied to the LV geometry. By its very principle, personalised medicine relies on the ability of the emulator to generalise to different LV geometries, meaning that the LV geometry itself must be treated as an input to the model. However, the high dimension of the LV geometry representation makes it incompatible with the statistical emulation framework. To resolve this issue, the work in this thesis uses a lowdimensional representation of the LV geometry to reduce the dimension of the input space of the model and construct a generalisable emulator of the LV model. Of primary interest is the efficient estimation of the parameters of the LV model, in a time frame compatible with the clinical setting. For this purpose, the generalisable emulator allows for the efficient use of Markov chain Monte Carlo, providing a measure of uncertainty in the parameters. A common problem in complex models, as is the case in the LV model, is the presence of weak practical identifiability. This manifests as large uncertainty in the posterior distributions of the parameters. In a Bayesian framework, this issue can be tackled using a more informative prior distribution. For the LV model, an informative prior that includes information from ex vivo studies is proposed, improving the estimation of the model parameters. Also motivated by the weak identifiability of the model, a new parameterisation of the model is considered. This involves a comprehensive sensitivity and inverse uncertainty quantification study that sheds extra light on the identifiability—both practical and structural—of the LV model. Finally, the problems posed by the measurement of clinical data, and the discrepancy between the model and reality, is considered and methods are proposed that account for this in the inference framework. Critically, the culmination of the work in this thesis highlights the problems that need to be resolved before the LV model can be applied in the clinical setting.