The SofTMech Statistical Emulation and Translation Hub

There have recently been impressive developments in the mathematical modelling of physiological processes. As part of a previously EPSRC-funded research centre (SofTMech), we have developed mathematical models for the mechanical and electrophysiological processes of the heart, and the flow in the blood vessel network. This allows us to gain deeper insight into the state of a variety of serious cardiovascular diseases, like hypoxia (a condition in which a region of the body is deprived of adequate oxygen supply), angina (reduced blood flow to the heart), pulmonary hypertension (high blood pressure in the lungs) and myocardial infarction (heart attack). A more recent extension of this work to modelling blood flow in the eye also provides novel indicators to assess the degree of traumatic brain injury.
What all these models have in common is a complex mathematical description of the physiological processes in terms of differential equations that depend on various material parameters, related e.g. to the stiffness of the blood vessels or the contractility of the muscle fibres. While knowledge of these parameters would be of substantial benefit to the clinical practitioner to help them improve their diagnosis of the disease status, most of the parameters cannot be measured in vivo, i.e. in a living patient. For instance, the determination of the stiffness and contractility of the cardiac tissue would require the extraction of the heart from a patient and its inspection in a laboratory, which can only be done in a post mortem autopsy.
It is here that our mathematical models reveal their diagnostic potential. Our equations of the mechanical processes in the heart predict the movement of the heart muscle and how its deformations change in time. These movements can also be observed with magnetic resonance image (MRI) scans, and they depend on the physiological parameters. We can thus compare the predictions from our model with the patterns found in the MRI scans, and search for the parameters that provide the best agreement. In a previous proof-of-concept study we have demonstrated that the physiological parameters identified in this way lead to an improved understanding of the cardiac disease status, which is important for deciding on appropriate treatment options.
Unfortunately, the calibration procedure described above faces enormous computational costs. We typically have a large number of physiological parameters, and an exhaustive search in a high-dimensional parameter space is a challenging problem. In addition, every time we change the parameters, our mathematical equations need to be solved again. This requires the application of complex numerical procedures, which take several minutes to converge. The consequence is that even with a high-performance computer, it takes several weeks to determine the physiological parameters in the way described above. It therefore appears that despite their enormous potential, state of the art mathematical modelling techniques can never be practically applied in the clinical practice, where diagnosis and decisions on alternative treatment option have to be made in real time.
Addressing this difficulty is the objective of our proposed research. The idea is to approximate the computationally expensive mathematical model by a computationally cheap surrogate model called an emulator. To create this emulator, we cover the parameter space with an appropriate design, solve the mathematical equations in parallel numerically for the chosen parameters, and then fit a non-linear statistical regression model to this training set. After this initial computational investment, the emulator thus created gives predictions for new parameter values practically instantaneously, allowing us to carry out the calibration procedure described above in real time. This will open the doors to harnessing the diagnostic potential of state-of-the art mathematical models for improved decision support in the clinic.

According to the British Heart Foundation (BHF), heart and circulatory diseases cause more than a quarter of all deaths in the UK, that is nearly 170,000 deaths each year, an average of 460 deaths each day or one every three minutes in the UK. There are around 7.4 million people living with heart and circulatory disease in the UK: 3.9 million men and 3.5 million women.

Mathematical modelling in cardiovascular physiology is a topical research area and has in principle paradigm-shifting potential for improving our understanding of a patient's cardio-vascular disease status, elucidating the nature of pathophysiological processes, improving patient-specific disease prognostication, and providing more accurate decision support for alternative treatment options. However, a major obstacle is the exorbitant computational cost of model calibration, as discussed in the "Summary" section. These are typically in the order of several weeks even on a high-performance computer, which currently renders state of the art mathematical models completely for the clinical practice.

The general impact of the proposed research hub is the fact that methodological improvements in statistical emulation will provide a decisive stepping stone towards enabling the use of state-of-the-art soft-tissue, electro-physiological and fluid-dynamic models for real-time decision making in the clinic and thereby harness their enormous potential for patient-specific disease prognostication. The emulation of soft-tissue mechanical models of the left ventricle of the heart will help assess the risk and treatment options for myocardial infarction (heart attack). The emulation of cardio-electrophysiological models will allow the monitoring of post-infarction scars to prevent sudden cardiac death. The emulation of fluid dynamic models for the pulmonary circulation system linked to the right ventricle of the heart will enable the non-invasive diagnosis of pulmonary hypertension, which is a major risk factor for stroke, heart failure and coronary artery disease. Endovascular drug delivery will be made more effective by emulating the patient-specific device-tissue-fluid interactions. And an extension of the cardiovascular modelling to the emulation of fluid dynamics in the human eye will allow the fast identification of traumatic brain injury, which will provide e.g. a clinical indicator for the "shaken baby syndrome".

To make specific progress towards these objectives, we will closely engage with the Scottish Pulmonary Vascular Unit at the Golden Jubilee Hospital in Clydebank, with the Cardiology Department at Queen Elizabeth University Hospital in Glasgow, and with NHS Scotland, as described in more detail in the "Pathways to Impact" section of this proposal.

The proposed research will also be relevant to companies that aim to deliver realistic simulation applications to explore real-world behaviour of complex systems particularly related to physiology, in that it will allow them to substantially reduce the computational complexity of inference and uncertainty quantification and thereby make their simulation systems applicable to decision-making in real time. A particular example is Dassault Systems, with whom the proposed research hub will closely engage. Moreover, the proposed research is relevant to companies that manufacture endovascular devices, like stents and drug-coated balloons, in that mathematical models of device-tissue-fluid interactions allow improvement of device design, and emulation is critical for fast patient-specific decisions. As a specific first step, the proposed hub will establish a collaboration with Terumo Aortic.

Statistical emulation is not only relevant to healthcare, but to the mathematical modelling of complex systems for safety-critical situations more generally. This includes e.g. early warning systems for tsunamis and volcanic activities, which will benefit from the methodological advancements made in the proposed research.