Estimating Sheets in the Heart Wall

Models of sheets in the heart wall play an important role in the visualization of myofiber geometry, in modelling mechanics and in cardiac electrophysiology. For example, the assumption of distinct speeds of propagation in the directions of myofibers, the sheets in which they lie, and the direction across them, can predict the arrival time of the conduction wave that triggers myocyte contraction. Almost all current analyses based on DTI data use the third eigenvector of the diffusion tensor as an estimate of the local sheet normal. This construction suffers from the limitation that the second and third eigenvector directions can be ambiguous since they are associated with eigenvalues that are quite similar. Here we present and evaluate an alternate method to estimate sheets, which uses only the principal eigenvector. We find the best local direction perpendicular to the principal eigenvector to span a sheet, using the Lie bracket and the minimization of an appropriately constructed energy function. We test our method on a dataset of 8 ex vivo rat and 8 ex vivo canine cardiac diffusion tensor images. Qualitatively the recovered sheets are more consistent with the geometry of myofibers than those obtained using all three eigenvectors, particularly when they curve or fan. Quantitatively the recovered sheet normals also give a low value of holonomicity, a measure of the degree to which they are orthogonal to a family of surfaces. Our novel fitting approach could thus impact cardiac mechanical and electrophysiological analyses which are based on DTI data.