Mathematical modelling of cardiac perfusion

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Cardiovascular disease is the leading cause of disability and death in the UK and worldwide. The British Heart Foundation (BHF) estimates it has a £19B annual economic impact. Structural impairment such as Ischaemic heart disease, even when treated in time, can lead to residual angina within one year. One of the key problems is to understand the myocardial blood perfusion after the treatment. Myocardial perfusion cardiovascular magnetic resonance (CMR) imaging [1] uses first- pass contrast-enhanced imaging to invasively detect the defects in the coronary circulation (e.g. stenosis, blockage, microvessel rarefaction) to assess myocardial ischaemia, myocardial viability, and cardiac function without the need for ionizing radiation. It is particularly valuable for the evaluation of patients with moderate to stable ischaemic heart disease, and patients with persisting angina. However, the large variations in coronary vessel networks and contrast agent properties limit imaging technologies; it is also difficult to pin down the exact mechanism of perfusion deficiency from perfusion CMR imaging alone. The development of a computational framework by coupling a poroelastic myocardial model with a multi-scale coronary flow and a scalar transport model for freely- diffusive contrast agents, will enable the efficient and accurate mapping of biomechanical factors (e.g. permeability tensor) in a spatial-temporal approach [2]. This PhD project will focus on developing mathematical models of a coupled myocardial-coronary circulation model. The large arteries/veins obtained from in vivo imaging (i.e. coronary angiograms) will be used to model blood flow in large vessels using one-dimensional (1-D) equations. For each terminal vessel, a tree of small vessels will be developed using a constrained constructive optimization technique [3] that minimises the total blood flow resistance to the perfused territory. The 1-D large arteries/veins are coupled with the optimized trees of small vessels using the SofTMech's structured-tree based network model [4] for quantifying myocardial blood perfusion driven by the pressure supplied by the large coronary arteries. For the microcirculation, we shall consider the myocardium as poroelastic [5]. 3-D patient-specific left ventricle geometry will be derived from MRI data using a well-established approach developed by the group [6]. For the large and small blood vessels, the blood flow is affected by the intramyocardial pressure (IMP) external to the blood vessels due to myocardial contraction. For the poroelasticity modelling, blood supply and drainage to the myocardium will be obtained from the coupled large-vessel/small-vessel blood flow. We shall use our in-house developed novel robust finite-element method for poroelasticity [7] to simulate the nonlinear deformation of the three-dimensional left ventricle coupled with the coronary circulation, by solving the 1-D blood flow model together with the Darcy flow inside the heart wall.