We present a method to efficiently simulate coronary perfusion in subject-specific models of the heart within clinically relevant time frames. Perfusion is modelled as a Darcy porous-media flow, where the permeability tensor is derived from homogenization of an explicit anatomical representation of the vasculature. To account for the disparity in length scales present in the vascular network, in this study, this approach is further refined through the implementation of a multi-compartment medium where each compartment encapsulates the spatial scales in a certain range by using an effective permeability tensor. Neighbouring compartments then communicate through distributed sources and sinks, acting as volume fluxes. Although elegant from a modelling perspective, the full multi-compartment Darcy system is computationally expensive to solve. We therefore enhance computational efficiency of this model by reducing the N-compartment system of Darcy equations to N pressure equations, and N subsequent projection problems to recover the Darcy velocity. The resulting 'reduced' Darcy formulation leads to a dramatic reduction in algebraic-system size and is therefore computationally cheaper to solve than the full multi-compartment Darcy system. A comparison of the reduced and the full formulation in terms of solution time and memory usage clearly highlights the superior performance of the reduced formulation. Moreover, the implementation of flux and, specifically, impermeable boundary conditions on arbitrarily curved boundaries such as epicardium and endocardium is straightforward in contrast to the full Darcy formulation. Finally, to demonstrate the applicability of our methodology to a personalized model and its solvability in clinically relevant time frames, we simulate perfusion in a subject-specific model of the left ventricle.

No abstract

The strong coupling between the flow in coronary vessels and the mechanical deformation of the myocardial tissue is a central feature of cardiac physiology and must therefore be accounted for by models of coronary perfusion. Currently available geometrically explicit vascular models fail to capture this interaction satisfactorily, are numerically intractable for whole organ simulations, and are difficult to parameterise in human contexts. To address these issues, in this study, a finite element formulation of an incompressible, poroelastic model of myocardial perfusion is presented. Using high-resolution ex vivo imaging data of the coronary tree, the permeability tensors of the porous medium were mapped onto a mesh of the corresponding left ventricular geometry. The resultant tensor field characterises not only the distinct perfusion regions that are observed in experimental data, but also the wide range of vascular length scales present in the coronary tree, through a multi-compartment porous model. Finite deformation mechanics are solved using a macroscopic constitutive law that defines the coupling between the fluid and solid phases of the porous medium. Results are presented for the perfusion of the left ventricle under passive inflation that show wall-stiffening associated with perfusion, and that show the significance of a non-hierarchical multi-compartment model within a particular perfusion territory.

A method to extract myocardial coronary permeabilities appropriate to parameterise a continuum porous perfusion model using the underlying anatomical vascular network is developed. Canine and porcine whole-heart discrete arterial models were extracted from high-resolution cryomicrotome vessel image stacks. Five parameterisation methods were considered that are primarily distinguished by the level of anatomical data used in the definition of the permeability and pressure-coupling fields. Continuum multi-compartment porous perfusion model pressure results derived using these parameterisation methods were compared quantitatively via a root-mean-square metric to the Poiseuille pressure solved on the discrete arterial vasculature. The use of anatomical detail to parameterise the porous medium significantly improved the continuum pressure results. The majority of this improvement was attributed to the use of anatomically-derived pressure-coupling fields. It was found that the best results were most reliably obtained by using porosity-scaled isotropic permeabilities and anatomically-derived pressure-coupling fields. This paper presents the first continuum perfusion model where all parameters were derived from the underlying anatomical vascular network.

In this paper we numerically simulate flow in a helical tube for physiological conditions using a co-ordinate mapping of the Navier-Stokes equations. Helical geometries have been proposed for use as bypass grafts, arterial stents and as an idealised model for the out-of-plane curvature of arteries. Small amplitude helical tubes are also currently being investigated for possible application as A-V shunts, where preliminary in vivo tests suggest a possibly lower risk of thrombotic occlusion. In-plane mixing induced by the geometry is hypothesised to be an important mechanism. In this work, we focus mainly on a Reynolds number of 250 and investigate both the flow structure and the in-plane mixing in helical geometries with fixed pitch of 6 tube diameters (D), and centreline helical radius ranging from 0.1D to 0.5D. High-order particle tracking, and an information entropy measure is used to analyse the in-plane mixing. A combination of translational and rotational reference frames are shown to explain the apparent discrepancy between flow field and particle trajectories, whereby particle paths display a pattern characteristic of a double vortex, though the flow field reveals only a single dominant vortex. A radius of 0.25D is found to provide the best trade-off between mixing and pressure loss, with little increase in mixing above R = 0.25D, whereas pressure continues to increase linearly.

Experimental data and advanced imaging techniques are increasingly enabling the extraction of detailed vascular anatomy from biological tissues. Incorporation of anatomical data within perfusion models is non-trivial, due to heterogeneous vessel density and disparate radii scales. Furthermore, previous idealised networks have assumed a spatially repeating motif or periodic canonical cell, thereby allowing for a flow solution via homogenisation. However, such periodicity is not observed throughout anatomical networks. In this study, we apply various spatial averaging methods to discrete vascular geometries in order to parameterise a continuum model of perfusion. Specifically, a multi-compartment Darcy model was used to provide vascular scale separation for the fluid flow. Permeability tensor fields were derived from both synthetic and anatomically realistic networks using (1) porosity-scaled isotropic, (2) Huyghe and Van Campen, and (3) projected-PCA methods. The Darcy pressure fields were compared via a root-mean-square error metric to an averaged Poiseuille pressure solution over the same domain. The method of Huyghe and Van Campen performed better than the other two methods in all simulations, even for relatively coarse networks. Furthermore, inter-compartment volumetric flux fields, determined using the spatially averaged discrete flux per unit pressure difference, were shown to be accurate across a range of pressure boundary conditions. This work justifies the application of continuum flow models to characterise perfusion resulting from flow in an underlying vascular network.

Graphical abstract

From basic science to translation, modern biomedical research demands computational models which integrate several interacting physical systems. This paper describes the infrastructural framework for generic multiphysics integration implemented in the software CHeart, a finite-element code for biomedical research. To generalize the coupling of physics systems, we introduce a framework in which the geometric and operator relationships between the constituent systems are rigorously defined. We then introduce the notion of topological interfaces and define specific operators encompassing many common model coupling requirements. These interfaces enable the evaluation of weak form integrals between mesh subregions of arbitrary finite-element bases' orders, types, and spatial dimensions. Equation maps are introduced which provide abstract representations of the individual physics systems that can be automatically combined to permit a monolithic matrix assembly. Flexible solution strategies for the resulting coupled systems are implemented, permitting fine-tuning of solution updates during fixed point iterations, and subgrouping where several problems are being solved together. Partitioning of coupled mesh domains for optimal load balancing is also supported, taking into account the per-processor cost of the entire coupled problem within the graph problem. The demonstration of the performance is illustrated through important real-world multiphysics problems relevant to cardiac physiology.