Computational Fluid Dynamics

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Computational Fluid Dynamics
CFD is a general term used to account for the numerical solution of the governing equations of fluid flow (Navier-Stokes equations). These equations are solved for the unknown pressure, which varies with position and time, and for the three components (vectors) of blood velocity, each of which are functions of position and time.¹² The physical properties of blood, the fluid density and the fluid viscosity, are known when solving these equations. Although blood exhibits complex rheological properties, it can be approximated as a Newtonian fluid with a constant viscosity in large arteries. These equations were formulated as early as the 19th century, however, their solution only became possible with modern computing power and numerical methods.¹³ Typically, a CFD problem consists of flow in a certain vessel model which is subject to certain boundary conditions. The geometric model of the vessel is discretised into a number of smaller entities (finite volumes or finite elements), thus forming the nodes of a computational mesh, on which the unknowns are calculated. The discretisation of the governing differential equations results in systems of algebraic equations, whose solution gives the problem unknowns at the mesh nodes. In order to perform a CFD simulation of flow in a coronary vessel, a 3D description of the vessel lumen is required. Several methods have been used for this purpose, the most widely applied of which are coronary vessel reconstruction based on biplanar coronary angiography,^14,15 rotational coronary angiography,¹⁶ intravascular ultrasound and biplanar coronary angiography,¹⁷ optical coherence tomography (OCT) imaging,¹⁸ 3D quantitative coronary angiography¹⁹ and computed tomography coronary angiography.^20,21

Boundary flow conditions also need to be specified in order to solve the blood flow problem. Boundary flow conditions are mathematical relationships between the variables of interest, e.g. flow and pressure, defined on the boundaries (entrance and exit) of the vessel model.¹² Coronary flow and pressure at the coronary vessels are technically difficult to acquire both invasively and noninvasively due to the narrow lumens of the coronary vessels, the fact that they are embedded on the beating myocardium and due to the limited spatial and temporal resolution of applicable measurement techniques such as ultrasound, intravascular ultrasound and magnetic resonance imaging (MRI).²² Thus alternative methodologies are usually utilised such as methods that couple lumped parameter models of the microcirculation to the outflow boundaries,¹¹ generic boundary conditions are developed and applied to the arterial outlets,²³ or predescribed conditions are assumed.^24,25 The field of CFD has made substantial progress in the past two decades, taking advantage of the availability of fast supercomputer capabilities. In terms of the application of CFD to coronary flow, the main limitations arise from ambiguities associated with inflow boundary conditions, definition of the cardiac and artery motion, etc., result in uncertainties regarding the validity of computational results.