Multi-resolution geometric modeling of the mitral heart valve leaflets

Abstract

An essential element of cardiac function, the mitral valve (MV) ensures proper directional blood flow between the left heart chambers. Over the past two decades, computational simulations have made marked advancements toward providing powerful predictive tools to better understand valvular function and improve treatments for MV disease. However, challenges remain in the development of robust means for the quantification and representation of MV leaflet geometry. In this study, we present a novel modeling pipeline to quantitatively characterize and represent MV leaflet surface geometry. Our methodology utilized a two-part additive decomposition of the MV geometric features to decouple the macro-level general leaflet shape descriptors from the leaflet fine-scale features. First, the general shapes of five ovine MV leaflets were modeled using superquadric surfaces. Second, the finer-scale geometric details were captured, quantified, and reconstructed via a 2D Fourier analysis with an additional sparsity constraint. This spectral approach allowed us to easily control the level of geometric details in the reconstructed geometry. The results revealed that our methodology provided a robust and accurate approach to develop MV-specific models with an adjustable level of spatial resolution and geometric detail. Such fully customizable models provide the necessary means to perform computational simulations of the MV at a range of geometric accuracies in order to identify the level of complexity required to achieve predictive MV simulations.

Appendix

1.1 Non-uniform data structure

We used an NUFFT algorithm that first interpolates (oversamples) the data on a dense Cartesian grid using truncated Gaussian kernels and then applies a standard FFT (Greengard and Lee 2004). This approach is a subset of NUFFT algorithms known as gridding algorithms. It has been shown that the effect of Gaussian kernels to interpolate the data on a regular grid can be removed using the convolution theorem. Thus, applying Gaussian gridding to perform Fourier analysis does not introduce interpolation errors if the data is uniformly distributed over a rectangular domain (one 2-D period). However, because MV attributes like the scalar fields representing geometric details are defined over irregular domains, gridding-based NUFFT algorithms fail unless the shape of the domain is accounted for.

1.2 Irregular domains

The deviation fields denoting the MV geometric details have free-form top and bottom boundaries (Fig. 16). These boundaries, corresponding to the MV annulus and free edge respectively, do not coincide with the top and bottom boundaries of the superquadric parametric domain. While the gridding step in the NUFFT algorithm oversamples the known function values on a periodic Cartesian grid, the data for deviation fields is defined on a subdomain of the entire periodic domain (Fig. 16a). This causes implicit zero-padding of the function values (Fig. 16b).

Applying FFT on the zero-padded data results in a Fourier analysis that is drastically different from the actual frequency content of the MV geometric details. This is due to the Gibbs phenomenon (Gottlieb and Shu 1997), which can pollute the entire spectrum with noise and is a direct artifact which is a direct artifact of a discontinuity (jump from actual function values to zero) at MV boundaries. Following this, all the high-frequency pollution in the reconstruction of geometric details can be attributed to the spectral leakage caused by the effect of windowing (Bernstein et al. 2001), which in our case models the MV boundaries. Expanding the bandwidth to attenuate the Gibbs phenomenon (Gottlieb and Shu 1997) causes two problems: (1) the computational cost of the 2-D Fourier analysis process increases quadratically with the number of frequencies and (2) faithful reconstruction of the geometric details becomes intractable as a result of the high-frequency components polluting the power spectrum. We addressed the latter issue by imposing sparsity penalization to the objective function for recovering the Fourier coefficients. Consequently, we were able to eliminate the spurious frequency components caused by the irregular boundaries and recover a clean spectrum through the implementation of iterative NUFFT penalized with the sparsity constraint Eq. (7). Applying fast harmonic analysis (NUFFT) with accelerated convergence (Sect. 2.4.2) enabled us to perform very fast image in-painting to recover the MV surface details in the 2D superquadric domain.