The heart is a complex three-dimensional structure in which the biophysics of the cardiac action potential and the mechanics of muscle cell contraction interact to produce efficiently coordinated ventricular pumping.
The aim of this research is to develop and experimentally validate an accurate three-dimensional model of regional cardiac mechanics and electrophysiology and their mutual interactions. Three-dimensional finite element (FE) models of the heart are being developed that include accurate descriptions of ventricular anatomy and myofiber architecture, the resting and contractile mechanical properties of myocardium, and the cellular dynamics of action potential propagation. To analyze the biological basis of electromechanical interactions in the intact heart, theoretical models of cardiac excitation-contraction coupling and mechanoelectric feedback will be incorporated into the continuum framework. The coupled models involve large-scale computations and are being impl emented on the Cray T3E parallel supercomputer by exploiting the structural parallelism of the underlying physical problem. These models will be used to investigate basic questions such as how stretch-activated ion channels affect conduction patterns in the intact heart, and how altered pacing sequences affect ventricular pumping efficiency. In summary, the goals of this project are: * To implement a parallel continuum model of coupled cardiac mechanics and electrophysiology * To develop computational framework to integrate cellular properties to the tissue and organ levels * To investigate ventricular mechanoelectric feedback in anatomically accurate simulations and in an experimental model Progress in 1997 Serial finite element (FE) models for electrical propagation have been fully integrated with our mechanical models and validated on single processor workstations. The electrical propagation model presently implements the two-variable FitzHugh-Nagumo (FHN) model of wave propagation in three dimensions. The more biophysically detailed Luo-Rudy (LR) model of transmembrane potential is being incorporated into the 3-D model. This quantitative detail is necessary to relate the cellular dynamics of activation to macroscopic shape and the evolution of the propagated wave. Furthermore, the LR model will be extended to include variables that couple the mechanical and electrical states of the tissue by including strain-dependent currents mediated by known stretch-activated channels. Future possibilities include incorporating the most recent extensions of the LR model, which simulate the dynamics of calcium transport within the cell and additional transmembrane ionic currents. In the coupled mechanical-electrical model, the two problems share the same physical domain, but the electrical problem requires a more fine-grained finite element discretization because of the finer spatial scale of electrical activation. Significant progress has been made in parallelizing our finite element methods for three-dimensional wall stress analysis using MPI, porting them to the Cray T3E and testing them. In this implementation, a single mechanical element is assigned to a single processor, which also maintains its own local copies of the global solution and residual vectors. The computation of the local finite element Jacobian is executed in parallel. The resulting element stiffness matrices are not assembled into a global stiffness matrix, but instead reside on each processor, making this an """"""""element-by-element"""""""" formulation. The nonlinear system of global equations is solved using a full- or modified Newton algorithm. At each iteration of the Newton algorithm, the linear system is solved using the restarted Generalized Minimum Residual (GMRES) iterative method, which requires one global matrix-vector multiplication per iteration. Our parallel program uses GMRES codes implemented in the """"""""Parallel Iterative Methods"""""""" package, allowing us to treat the global matrix as an operator, further exploiting the element-by-element formulation by executing the matrix-vector multiplication in parallel. This new program was tested using a 3D model of the canine LV. Previously, obtaining the stress and strain solutions required 60 minutes on an Indigo 2 workstation with a 150 MHz R4400 processor. Using the restarted GMRES method with a Krylov subspace dimension of 100 and left diagonal preconditioning, the same results are obtained in 5.2 minutes using sixteen 300 MHz Alpha 21164 processors on the Cray T3E. This 92% reduction in solution time will be a significant benefit when analyzing coupled mechano-electrical problems. The speedups using both full and modified Newton iteration schemes are shown in Figure 1. Figure 1. Speedup for the compressible 3D canine left ventricle finite element model on the SDSC Cray T3E. """"""""FN"""""""" denotes full Newton algorithm; """"""""MN"""""""" is modified Newton. Plans for next year * Laminar sheet architecture * Use detailed ionic models with stretch-activated channels in continuum framework to investigate mechanisms of mechanoelectric feedback * Web interface for remote users and transparent supercomputing (with Gribskov and Bourne) * Facilitate integrative physiological modeling collaborations (with Ten Eyck) * Add new visualization tools (with Olsen) * Conduct validation experiments on regional mechanoelectric feedback in intact rabbit heart The electrical FE domain will be realized by further spatial refinement of the mechanical elements to generate multiple electrical finite elements within the mechanical subdomain. In total, both the mechanical and electrical problems are distributed across all of allocated processors. In each processor the computation time will be split between alternatively solving each problem and updating the global solution vector after each step. This process automatically balances the load and reduces interprocessor communication to a single vector write/read at each step. Since the program has been implemented on the Cray T3E using MPI, which is available for a variety of parallel platforms and families of workstations, the software is highly portable, facilitating the distribution of our final software product to other investigators. Future work will include adaptively discretizing the electrical problem domain using multilevel methods, so that regions of electrically quiescent tissue (requ iring less spatial refinement) will use fewer finite elements than active regions. The fully 3D finite element model of the rabbit ventricular geometry has been enhanced to capture more of the observed
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