In this work, a new framework is presented that is suitable to solve the cardiac bidomain equation efficiently using the scientific computing library PETSc. Furthermore, the framework is able to modularly combine different ionic channels and is flexible enough to include arbitrary heterogeneities in ionic or coupling channel density. The ability of this framework is demonstrated in an example simulation in which the three-dimensional electrophysiological heterogeneity was adjusted in order to get a positive T-wave in the body electrocardiogram (ECG).
Pyramidal GaAs structures on top of GaAs/AlAs distributed Bragg reflectors are investigated as candidates for true three-dimensional cavities with potentially low mode volume and high quality-factor. Different types of single and coupled resonators with base lengths of a few microns are realized using a combination of molecular-beam epitaxy, electron-beam lithography, and wet chemical etching. Embedded InGaAs quantum dots are utilized as light sources to verify the resonator modes. Furthermore, a spatially localized emission through the pyramid facets indicates the future possibility of coupling cavity modes to optical fibers. This could be interesting within the context of single photon emitters.
Conference Contributions (1)
M. Karl, G. Seemann, F. Sachse, O. Dössel, and V. Heuveline. Time and memory efficient implementation of the cardiac bidomain equations. In 4th European Conference of the International Federation for Medical and Biological Engineering, IFMBE Proceedings, vol. 22, 2008
Computer simulations can significantly improve comprehension of cardiac electrophysiology. A mathematical model for the simulation of complex cardiac electrophysiology is the bidomain model. A new tool, acCELLerate, was developed using the PETSc library  for a parallel time and memory efficient implementation of the bidomain equations enabling the computation of large scale cardiac simulations. It offers an extensible modular structure. The optimization of the cost-intensive solution of the elliptical part of the bidomain equation was achieved by analyzing several iterative Krylov subspace methods and preconditioners provided by PETSc. Best performance results were achieved by using a combination of minimal residual method (MinRes), conjugate residual method (CR) or conjugate grandient method (CG) as solver with adjusted successive over-relaxation preconditioning (SOR). A validation proved the authenticity of the new tool.
Student Theses (1)
M. Karl. Time and memory efficient implementation of the cardiac bidomain equations. Institute of Biomedical Engineering, Universität Karlsruhe (TH). Diplomarbeit. 2008
This work contributes to the development of a new tool for the simulation of cardiac electro- physiology. A few years ago, the simulation of such a large scale problem was not possible. Due to recent developments in hardware and software the problem is no longer insolvable. High performance computers and clusters are able to work on the problem by using paral- lelization. Several powerful libraries have been developed to enable parallel programming for the normal user.The new tool was implemented in parallel by using the PETSc library . To achieve extensibility of the program a modular structure was chosen. The bidomain equations constitute the mathematical basis of the implementation. Solving the bidomain equations numerically yields to a linear system of equations (LSE) which approximates the discretized problem.The first part of this work encloses the parallel implementation of the matrix generation for the LSE. Fundamentally, a Laplaces equation is to be solved approximately. Though, the weighting factor σ, a tensor described in section 4.4, complicates the structure of the sparse matrix in a way that a maximum number of 19 nonzero entries exist in each row of the matrix for the finite difference method.The solving of the LSE is the most expensive part of the numerical solution of the bidomain equations. On account of the size of problems in cardiac electrophysiology, a direct method can not be used. Alternatively, iterative methods such as Krylov subspace methods (KSP) enable the efficient solution of the sparse linear system. Preconditioning techniques improve the performance of the iterative solver. Therefore, the second part of this work deals with the investigation of several KSP solvers and preconditioners provided by PETSc. A pro- gram was implemented to optimize the iterative solution by using PETSc runtime options. Selected preconditioning parameters were adjusted to improve the time and memory per- formance of the solvers. In particular, the ASM and SOR preconditioner were investigated. Using ASM preconditioning turned out to be a bad choice which is surprising. The ASM domain decomposition method is known to be a capable preconditioning technique. One possible reason for the unsatisfactory results escpecially in time performance might be the way of how it is implemented in PETSc. Furthermore, because the PETSc library does not provide a relaxation factor for weigthing, no damping of the preconditioner is possible. After the disappointing results of ASM, the SOR method was investigated as precondi- tioning method. Adjusting two parameters of SOR, the preconditioned iterative method yielded to an execution time reduction of more than 50 % compared to the untuned ver- sion. Concerning the memory requirements, the SOR preconditioner was among the best methods. Although the SOR preconditioner achieved better and more stable results in time and memory performance, a general adjustment of the preconditioning parameters was not possible. Thus to conclude, the optimal parameter choice is highly problem dependent. The disparsities in the runtime analysis achieved within the computation on the three different geometries are a possible indication that the modeling contains inconsistencies. A different ratio between tissue and bath yielded to different optimal KSP solvers and different pa- rameters, respectively. Here, caution is advised, especially when the percentage of bath in the simulation block is relatively high.The third part of the results contains the validation of the program. By means of a publica- tion of Henriquez et al.  the simulation results computed by the new tool were validated. In addition, the results were compared to those of the former tool. Qualitatively as well as quantitatively the achieved simulation results were very similar. Disparities in excitation conduction velocity can be explained by the choice of different cell models. To investigate the scalability of the implementation, the exemplary simulation of the validation was also performed on a high performance computer: the Blue Gene/L of IBM, New York. As a re- sult the runtimes, speedups and efficiencies for different numbers of processors up to 2048 CPUs were computed. It was shown that the program is scalable and that the imlementa- tion was efficiently parallelized.Finally, a simulation in both ventricles of a human heart was computed to show the facili- ties of the new program.It is to be summarized that the new tool enables large scale simulations of cardiac electro- physiology within moderate time and memory requirements. Optimization of solvers and preconditioning parameters lead to efficient improvement of the performance. The exten- sibility of the program provides the possibilities to achieve even more realistic simulation results. Compared to the former program, time and memory requirements are reduced by a high factor. The simulation results are validable and very satisfying by comparison within the former tool and results in literature. New records in size and complexity of the simulation problems can be set within the implementation.