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

Abstract

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 patient’s 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.

Introduction

The discipline of computational physiology provides tools to quantitatively describe and analyse physiological behaviour across a range of time scales and anatomical levels using mathematical and computational models (Smith et al., 2007). Recent advances have enabled multi-level and multi-physics models, which link molecular, subcellular and cellular functions to whole organ performance (Lee et al., 2009). These developments, accompanied by computational progress, have recently been organised into international initiatives such as the International Union of Physiological Sciences sponsored Physiome and Virtual Physiological Human projects (Hunter and Borg, 2003).

Within these programmes the heart is arguably the most advanced current exemplar of this approach (Bassingthwaighte et al., 2009). As such, it represents an active research focus among the international community, and an important application field for developing diagnostic and planning support tools. As part of these developments, quantitative physiological models have the potential to make a significant impact in the management of cardiovascular diseases in the clinic. To realise this potential, one of the central challenges is the personalisation of computational models to individual patients. This study focuses on one of the fundamental aspects involved in this clinical personalisation, the customisation of mechanical meshes to the patient’s anatomy captured from medical imaging data.

Computational meshes define a geometric representation of anatomy, providing the domain for solving the mathematical description of physiological processes. They are required to be both geometrically accurate and to provide good computational stability. For many physiological simulations, the computational domains are described using linearly interpolated tetrahedral meshes. This is primarily due to the availability of tools for automatic meshing of complex geometries, their conceptual simplicity and their well characterised properties (Lohner, 1997). Nevertheless, linear interpolation schemes in tetrahedral elements can introduce significant numerical errors in calculating solutions for the important class of incompressible soft tissue deformation simulations (Pathmanathan et al., 2009).

An alternative solution is the use of cubic Hermite meshes, which provide an efficient representation of the mechanical state of an organ as well as improved accuracy (Zienkiewicz and Taylor, 2000). In a Hermite mesh shape is encoded, not only with the 3D Cartesian coordinates of nodes, but also with the derivatives of shape versus local finite element coordinates. This enables a much more compact and smooth C¹ continuous representation (both the function and its first derivative are continuous) and thus a reduction of the computational cost in simulations. For these reasons, cubic Hermite meshes are a popular choice for the simulation of heart mechanics. Using this approach recent contributions have provided insight into the behaviour of the electro-mechanical coupling of the heart, specifically the Frank–Starling law (Niederer et al., 2010), the effects of resynchronization therapies in heart failure patients (Kerckhoffs et al., 2009), and the estimation of passive diastolic ventricle mechanic parameters from MRI images (Wang et al., 2009).

There are two broad approaches for the personalisation of geometrical meshes: the direct construction of a mesh from segmented images (Fernandez et al., 2004, Lohner, 1997) or the customisation of a mesh from an existing mesh model (Barber et al., 2007, Couteau et al., 2000, Fernandez et al., 2004, Sigal et al., 2008). Whereas the literature for linear meshes is extensive (Bucki et al., 2010, Couteau et al., 2000, Gibson et al., 2003, Lohner, 1997, Sigal et al., 2008), its translation to meshes with higher order of interpolation, like cubic Hermite meshes, remains to be developed. This is evidenced by the fact that no automated tools are currently available for the construction or personalisation of these meshes. Furthermore, while stability and convergence of mechanical simulations is dependent on the regularity and quality of mesh elements, there are currently no mesh generation tools that guarantee these properties. There is thus a need for a fast, accurate and robust cubic Hermite personalisation method.

This article presents a generalist solution for cubic Hermite mesh customisation that applies a predefined high quality template and an image registration step with a novel cubic Hermite warping technique. The key contribution is the variational warping step after image registration, which achieves an optimal (in terms of accuracy) description of the shape using the high order interpolation functions. Results quantify the accuracy and robustness of such a solution in a synthetic workbench, and illustrate the importance of using a variational warping technique when working with high order but low resolution meshes. Finally, a clinical cardiac application is chosen to illustrate the performance of this method, reporting accuracy, mechanical quality and computational cost. The choice of a heart as the shape for the study is justified by its potential clinical impact and its complexity. Simulating heart biomechanics is arguably one of the most complex single body problems of human muscular physiology, due to its topology, shape and the large strains that occur during the heart cycle. A preliminary version of this work can be found in (Lamata et al., 2010), and an illustration of the personalisation process is provided in the Supplementary Video available in the online version of this manuscript (URL).