Andrew Cookson

OpenCMISS: A multi-physics & multi-scale computational infrastructure for the VPH/Physiome project

The VPH/Physiome Project is developing the model encoding standards CellML (cellml.org) and FieldML (fieldml.org) as well as web-accessible model repositories based on these standards (models.physiome.org). Freely available open source computational modelling software is also being developed to solve the partial differential equations described by the models and to visualise results. The OpenCMISS code (opencmiss.org), described here, has been developed by the authors over the last six years to replace the CMISS code that has supported a number of organ system Physiome projects. OpenCMISS is designed to encompass multiple sets of physical equations and to link subcellular and tissue-level biophysical processes into organ-level processes. In the Heart Physiome project, for example, the large deformation mechanics of the myocardial wall need to be coupled to both ventricular flow and embedded coronary flow, and the reaction-diffusion equations that govern the propagation of electrical waves through myocardial tissue need to be coupled with equations that describe the ion channel currents that flow through the cardiac cell membranes. In this paper we discuss the design principles and distributed memory architecture behind the OpenCMISS code. We also discuss the design of the interfaces that link the sets of physical equations across common boundaries (such as fluid-structure coupling), or between spatial fields over the same domain (such as coupled electromechanics), and the concepts behind CellML and FieldML that are embodied in the OpenCMISS data structures. We show how all of these provide a flexible infrastructure for combining models developed across the VPH/Physiome community.

A novel porous mechanical framework for modelling the interaction between coronary perfusion and myocardial mechanics

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.

Multi-Scale Parameterisation of a Myocardial Perfusion Model Using Whole-Organ Arterial Networks

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.

Parameterisation of multi-scale continuum perfusion models from discrete vascular networks

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.

Toward GPGPU accelerated human electromechanical cardiac simulations

In this paper, we look at the acceleration of weakly coupled electromechanics using the graphics processing unit (GPU). Specifically, we port to the GPU a number of components of Heart—a CPU-based finite element code developed for simulating multi-physics problems. On the basis of a criterion of computational cost, we implemented on the GPU the ODE and PDE solution steps for the electrophysiology problem and the Jacobian and residual evaluation for the mechanics problem. Performance of the GPU implementation is then compared with single core CPU (SC) execution as well as multi-core CPU (MC) computations with equivalent theoretical performance. Results show that for a human scale left ventricle mesh, GPU acceleration of the electrophysiology problem provided speedups of 164 × compared with SC and 5.5 times compared with MC for the solution of the ODE model. Speedup of up to 72 × compared with SC and 2.6 × compared with MC was also observed for the PDE solve. Using the same human geometry, the GPU implementation of mechanics residual/Jacobian computation provided speedups of up to 44 × compared with SC and 2.0 × compared with MC.

Using coordinate transformation of Navier-Stokes equations to solve flow in multiple helical geometries

Recent research on small amplitude helical pipes for use as bypass grafts and arterio-venous shunts, suggests that mixing may help prevent occlusion by thrombosis. It is proposed here that joining together two helical geometries, of different helical radii, will enhance mixing, with only a small increase in pressure loss. To determine the velocity field, a coordinate transformation of the Navier-Stokes equations is used, which is then solved using a 2-D high-order mesh combined with a Fourier decomposition in the periodic direction. The results show that the velocity fields in each component geometry differ strongly from the corresponding solution for a single helical geometry. The results suggest that, although the mixing behaviour will be weaker than an idealised prediction indicates, it will be improved from that generated in a single helical geometry.

A spatially-distributed computational model to quantify behaviour of contrast agents in MR perfusion imaging

Graphical abstract

Myocardial perfusion distribution and coronary arterial pressure and flow signals:clinical relevance in relation to multiscale modeling, a review

Abstract Coronary artery disease, CAD, is associated with both narrowing of the epicardial coronary arteries and microvascular disease, thereby limiting coronary flow and myocardial perfusion. CAD accounts for almost 2 million deaths within the European Union on an annual basis. In this paper, we review the physiological and pathophysiological processes underlying clinical decision making in coronary disease as well as the models for interpretation of the underlying physiological mechanisms. Presently, clinical decision making is based on non-invasive magnetic resonance imaging, MRI, of myocardial perfusion and invasive coronary hemodynamic measurements of coronary pressure and Doppler flow velocity signals obtained during catheterization. Within the euHeart project, several innovations have been developed and applied to improve diagnosis-based understanding of the underlying biophysical processes. Specifically, MRI perfusion data interpretation has been advanced by the gradientogram, a novel graphical representation of the spatiotemporal myocardial perfusion gradient. For hemodynamic data, functional indices of coronary stenosis severity that do not depend on maximal vasodilation are proposed and the Valsalva maneuver for indicating the extravascular resistance component of the coronary circulation has been introduced. Complementary to these advances, model innovation has been directed to the porous elastic model coupled to a one-dimensional model of the epicardial arteries. The importance of model development is related to the integration of information from different modalities, which in isolation often result in conflicting treatment recommendations.

In silico coronary wave intensity analysis: application of an integrated one-dimensional and poromechanical model of cardiac perfusion.

Coronary wave intensity analysis (cWIA) is a diagnostic technique based on invasive measurement of coronary pressure and velocity waveforms. The theory of WIA allows the forward- and backward-propagating coronary waves to be separated and attributed to their origin and timing, thus serving as a sensitive and specific cardiac functional indicator. In recent years, an increasing number of clinical studies have begun to establish associations between changes in specific waves and various diseases of myocardium and perfusion. These studies are, however, currently confined to a trial-and-error approach and are subject to technological limitations which may confound accurate interpretations. In this work, we have developed a biophysically based cardiac perfusion model which incorporates full ventricular–aortic–coronary coupling. This was achieved by integrating our previous work on one-dimensional modelling of vascular flow and poroelastic perfusion within an active myocardial mechanics framework. Extensive parameterisation was performed, yielding a close agreement with physiological levels of global coronary and myocardial function as well as experimentally observed cumulative wave intensity magnitudes. Results indicate a strong dependence of the backward suction wave on QRS duration and vascular resistance, the forward pushing wave on the rate of myocyte tension development, and the late forward pushing wave on the aortic valve dynamics. These findings are not only consistent with experimental observations, but offer a greater specificity to the wave-originating mechanisms, thus demonstrating the value of the integrated model as a tool for clinical investigation.

Multiscale modelling of cardiac perfusion

To elucidate the mechanisms governing coronary blood flow in health and disease requires an understanding of the structure—function relationship of the coronary system, which exhibits distinct characteristics over multiple scales. Given the complexities arising from the multiscale and distributed nature of the coronary system and myocardial mechanical coupling, computational modelling provides an indispensable tool for advancing our understanding. In this work, we describe our strategy for an integrative whole-organ perfusion model, and illustrate a series of examples which apply the framework within both basic science and clinical translation settings. In particular, the one-dimensional reduced approach common in vascular modelling is combined with a new poromechanical formulation of the myocardium that is capable of reproducing the full contractile cycle, to enable simulation of the dynamic coronary and myocardial blood flow. In addition, a methodology for estimating continuum porous medium parameters from discrete network geometry is presented. The benefit of this framework is first demonstrated via an application to coronary wave intensity analysis, a technique developed to study time-dependent aspects of pulse waves invasively measured in the vessels. It is shown that, given experimentally-acquired boundary conditions the 1D model is capable of reproducing a wave behaviour broadly consistent with that observed in vivo, however, its utility is limited to a phenomenological level. The integrated 1D-poromechanical model on the other hand enables a mechanistic investigation of wave generation thus allowing the influence of contractile function and distal hemodynamic states on coronary flow to be described. In addition, when coupled with the advection-diffusion-reaction equation, the integrated model can be used to study the transport of tracers through the vascular network, thus allowing the dependence of noninvasive imaging signal intensities on the diffusive properties of the contrast agent to be quantified. A systematic investigation of the commonly used clinical indices and whole-organ modelling results are illustrated. Taken together, the proposed model provides a comprehensive framework with which to apply quantitative analysis in whole organ coronary artery disease diagnosis using noninvasive perfusion imaging modalities. The added value of the model in clinical practice lies in its ability to combine comprehensive patient-specific information into therapy. In this regard, we close the chapter with a discussion on potential model-aided strategies of disease management.