Statistical emulation of cardiac mechanics
Dirk Husmeier (University of Glasgow)
Friday 8th November 14:00-15:00 Sir Alwyn Williams Building, 422 Seminar Room
In recent years, we have witnessed impressive developments in the mathematical modelling of complex physiological systems. This provides unprecedented novel opportunities for improved disease diagnosis based on an enhanced quantitative physiological understanding. In a recent proof of concept study, we have shown that the biomechanical parameters of a state-of-the-art cardiac mechanics model have encouraging diagnostic power for early diagnosis of the risk to myocardial infarction (heart attack) and decision making related to alternative treatment options. However, estimating the biomechanical parameters non-invasively from magnetic resonance imaging (MRI) is computationally expensive and can take several weeks of high-performance computing time. This constitutes a severe obstacle for translational research, preventing uptake in the clinic and thwarting any pathway to genuine impact in healthcare. The problem is that state-of-the-art mathematical models of complex physiological systems are typically based on systems of nonlinear coupled partial differential equations (PDEs), which have no closed-form solution and have to be integrated numerically, e.g. using finite element simulations. This is not an issue for the so-called forward problem, where the objective is to understand a system’s behaviour for given physiological parameters. However, many physiological parameters cannot be measured noninvasively, and hence have to be estimated indirectly based on a quantitative measure of the discrepancy between model predictions and non-invasive measurements. This calls for thousands of numerical integrations as part of an iterative optimization or sampling routine, incurring computational run times in the order of days or weeks.
A potential way to deal with the high computational complexity and make progress towards a clinical decision support system that can make disease prognostications and risk assessments in real time, is statistical emulation. The idea is to approximate the computationally expensive mathematical model (the simulator) with a computationally cheap statistical surrogate model (the emulator) by a combination of massive parallelization and nonlinear regression. Starting from a space-filling design in parameter space, the underlying partial differential equations are solved numerically on a parallel computer cluster, and methods from nonparametric Bayesian statistics based on Gaussian Processes (GPs) are applied to multivariate smooth interpolation. When new data become available (e.g. myocardial strains from MRI scans) the resulting proxy objective function can be maximized (for maximum likelihood estimation) or sampled from (using Markov chain Monte Carlo) at low computational costs, without further computationally expensive simulations of the original mathematical model.
In my talk, I will compare different emulation strategies and loss functions, and assess the reduction in computational complexity. For large data sets, it is not computationally feasible to train a GP, as the computational complexity is of the order of the third power of the data set size, and I will compare various alternative paradigms for dealing with this issue. I will describe a proof-of-concept study, with encouraging results: While conventional parameter estimation based on numerical simulations from the cardiac mechanics model leads to computational costs in the order of weeks, our emulation method reduces the computational complexity to the order of the quarter of an hour, while effectively maintaining the same level of accuracy. However, there are still substantial hurdles to overcome in our endeavour to move this work forward towards personalised medicine and to develop a decision support system that can be used by clinical practitioners, which I will discuss.
If time permits, I will discuss an extension of this framework to uncertainty quantification in the fluid dynamics of the pulmonary blood circulation system, with applications to the diagnosis of pulmonary hypertension (high blood pressure in the lungs).