David Nordsletten

Coupling multi-physics models to cardiac mechanics

We outline and review the mathematical framework for representing mechanical deformation and contraction of the cardiac ventricles, and how this behaviour integrates with other processes crucial for understanding and modelling heart function. Building on general conservation principles of space, mass and momentum, we introduce an arbitrary Eulerian-Lagrangian framework governing the behaviour of both fluid and solid components. Exploiting the natural alignment of cardiac mechanical properties with the tissue microstructure, finite deformation measures and myocardial constitutive relations are referred to embedded structural axes. Coupling approaches for solving this large deformation mechanics framework with three dimensional fluid flow, coronary hemodynamics and electrical activation are described. We also discuss the potential of cardiac mechanics modelling for clinical applications.

An accurate, fast and robust method to generate patient-specific cubic Hermite meshes.

In-silico continuum simulations of organ and tissue scale physiology often require a discretisation or mesh of the solution domain. Cubic Hermite meshes provide a smooth representation of anatomy that is well-suited for simulating large deformation mechanics. Models of organ mechanics and deformation have demonstrated significant potential for clinical application. However, the production of a personalised mesh from patients anatomy using medical images remains a major bottleneck in simulation workflows. To address this issue, we have developed an accurate, fast and automatic method for deriving patient-specific cubic Hermite meshes. The proposed solution customises a predefined template with a fast binary image registration step and a novel cubic Hermite mesh warping constructed using a variational technique. Image registration is used to retrieve the mapping field between the template mesh and the patient images. The variational warping technique then finds a smooth and accurate projection of this field into the basis functions of the mesh. Applying this methodology, cubic Hermite meshes are fitted to the binary description of shape with sub-voxel accuracy and within a few minutes, which is a significant advance over the existing state of the art methods. To demonstrate its clinical utility, a generic cubic Hermite heart biventricular model is personalised to the anatomy of four patients, and the resulting mechanical stability of these customised meshes is successfully demonstrated.

Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics.

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain cardiovascular flow. Computational modelling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modelling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modelling and how their integration with medical imaging data is providing new pathways for translational cardiac modelling.

Coupling contraction, excitation, ventricular and coronary blood flow across scale and physics in the heart

In this study, we review the development and application of multi-physics and multi-scale coupling in the construction of whole-heart physiological models. Through an examination of recent computational modelling developments, we analyse the significance of coupling mechanisms for the increased understanding of cardiac function in the areas of excitation–contraction, coronary blood flow and ventricular fluid mechanical coupling. Within these physiological domains, we demonstrate and discuss the importance of model parametrization, imaging-based model anatomy and computational implementation.

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.

Simulating left ventricular fluid-solid mechanics through the cardiac cycle under LVAD support

In this study we have integrated novel modifications of the standard Newton-Raphson/line search algorithm and optimisation of the interpolation scheme at the fluid-solid boundary to enable the simulation of fluid-solid interaction within the cardiac left ventricle under the support of a left ventricular assist device (LVAD). The line search modification combined with Jacobian reuse produced close to an order of magnitude improvement in computational time across both test and whole heart simulations. Optimisation of element interpolation schemes on the fluid-solid boundary highlights the impact this choice can have on problem stability and demonstrates that, in contrast to linear fluid elements, higher order interpolation produces improved error reduction per degree of freedom. Incorporating these modifications enabled a full heart cycle under LVAD support to be modelled. Results from these simulations show that there is slower clearance of blood entering the chamber during early compared to late diastole under conditions of constant LVAD flow.

A finite-element approach to the direct computation of relative cardiovascular pressure from time-resolved MR velocity data

Graphical abstract Highlights ► Extraction of relative pressure from 4D MRI data sets. ► A novel workflow for determining relative cardiovascular pressure fields. ► Demonstration of the approach across a range of validation examples. ► Four subject specific cases showing agreement with published pressure differences.

A non-conforming monolithic finite element method for problems of coupled mechanics

In this study, a Lagrange multiplier technique is developed to solve problems of coupled mechanics and is applied to the case of a Newtonian fluid coupled to a quasi-static hyperelastic solid. Based on theoretical developments in [57], an additional Lagrange multiplier is used to weakly impose displacement/velocity continuity as well as equal, but opposite, force. Through this approach, both mesh conformity and kinematic variable interpolation may be selected independently within each mechanical body, allowing for the selection of grid size and interpolation most appropriate for the underlying physics. In addition, the transfer of mechanical energy in the coupled system is proven to be conserved. The fidelity of the technique for coupled fluid-solid mechanics is demonstrated through a series of numerical experiments which examine the construction of the Lagrange multiplier space, stability of the scheme, and show optimal convergence rates. The benefits of non-conformity in multi-physics problems is also highlighted. Finally, the method is applied to a simplified elliptical model of the cardiac left ventricle.

Inflow Typology and Ventricular Geometry Determine Efficiency of Filling in the Hypoplastic Left Heart

BACKGROUND Pediatric patients with hypoplastic left heart syndrome rely solely on the right ventricle, resulting in anatomic maladaptations that can significantly compromise diastolic efficiency and lead to heart failure. Clinical indices to evaluate diastole are generally derived from the adult left ventricle, limiting their relevance to patients with hypoplastic left heart syndrome. We investigated the effect of the ventricular cavity shape and tricuspid inflow typology on the filling dynamics to provide new directions of investigation for assessing diastolic function in these patients. METHODS Magnetic resonance imaging data were used to generate personalized mathematic models of 4 patients with different prognoses after stage I of the Norwood procedures. Two of these patients were also modeled after stage II. Numeric simulations were performed to analyze the interaction between blood flow and the myocardium during diastole. RESULTS The filling dynamics were characterized by the formation of an organized structure of swirling blood (vortex ring). This was strongly influenced by the ventricular shape and the timing of the E and A wave. Biphasic rather than fused inflows and more elliptical than spherical cavities were found to increase the intraventricular pressure gradients and the filling capacity by optimizing the energy transfer between blood flow and the myocardium. This resulted in a better flow propagation and higher tissue velocities and displacements. CONCLUSIONS The variations in the kinetic energy associated with the blood motion reflected the base-to-apex pressure difference and can therefore be used to quantify the efficiency of filling, providing a potential new metric to assess diastolic function in these patients.

Aortic relative pressure components derived from four-dimensional flow cardiovascular magnetic resonance.

To describe the assessment of the spatiotemporal distribution of relative aortic pressure quantifying the magnitude of its three major components.