Progress has been made in two areas over the past year. First, we have completed initial trials of a parallel implementation of the finite element (FE) method on the Intel Paragon, and we are currently working with Ken Steube at SDSC who is porting the FE codes for cardiac mechanics and electrical activation to the Cray T-3D at SDSC. Secondly, we have completed the measurement and reconstruction of a fully 3D model of rabbit heart that includes right and left ventricular anatomy and the distribution of muscle fiber angles throughout-the wall. Because the mechanics models require relatively few, expensive high-order elements, over 95% of the total single-processor solution time is used in the data-parallel integration of the element matrices The opposite is true of conventional low-order finite elements in which the equations are easily computed but the systems of equations are large. Moreover, the message sizes are deterrm~ned by the size of the element stiffness matrices which are modest (<40 kWords). In krelinunary experiments on the Intel Paragon we obtained almost linear speedups in the element computations when we distributed them between available processors. These tests used a single processor to factorize the global stiffness matrix, so that speed-ups in total solution time departed from linearity and flattened off between 16 and 32 parallel processors, which is consistent with expectation, given the 20:1 ratio of parallel to scalar computation times in these models. The next step will be to implement the linear solution algorithm in parallel. Since effective parallel sparse direct matrix solvers are still being developed, we will begin by testing parallel iterative sparse matrix solvers like the preconditioned conjugate gradient solver, MP-PCGPAK2, developed at SDSC. In the electrical models, the memory requirements of the large systems of equations and the compute times required to invert them are the major obstacles to improved performance. We plan to use an iterative sparse matrix solver for the electrophysiological analyses. Recently, we developed new techniques to measure geometric contours and 3-D distributions of muscle fiber angle in the perfusion-fixed rabbit heart. By completely embedding the entire heart in quick-setting dental rubber (polyvinylsiloxane), and sectioning it into 10-12 rings, we obtained over 8,000 epicardial and endocardial contour points on the left and right ventricles, and over 12,000 measurements of local fiber angle in over 3,000 sections distributed throughout the wall. In the first model, 24 high-order (bicubic-linear) prolate spheroidal finite elements (288 degrees of freedom) were fitted to these measurements with an RMS error of +0 .38 mm and +170 in the fiber angles. These fitting errors will be further reduced by refining the mesh to 40 high-order elements.
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