Numerical study of POD-Galerkin-DEIM reduced order modeling of cardiac monodomain formulation.

The three-dimensional cardiac monodomain model with inhomogeneous and anisotropic conductivity characterizes a complicated system that contains spatial and temporal approximation coefficients along with a nonlinear ionic current term. These complexities make its numerical modeling computationally challenging, and therefore, the formation of an efficient computational approximation is important for studying cardiac propagation. In this paper, a reduced order modeling approach has been developed for the simplified cardiac monodomain model, which yields a significant reduction of the full order dynamics of the cardiac tissue, reducing the required computational resources. Additionally, the discrete empirical interpolation technique has been implemented to accurately estimate the nonlinearity of the ionic current of the cardiac monodomain scheme. The proper orthogonal decomposition technique has been utilized, which transforms a given dataset called 'snapshots' to a new coordinate system. The snapshots are computed first from the original system, and they encapsulate all the information observed over both time and parameter variations. Next, the proper orthogonal decomposition provides a reduced order basis for projecting the original solution onto a low-dimensional orthonormal subspace. Finally, a reduced set of unknowns of the forward problem is obtained for which the solution involves significant computational savings compared to that for the original system of unknowns. The efficiency of the model order reduction technique for finite difference solution of cardiac electrophysiology is examined concerning simulation time, error potential, activation time, maximum temporal derivative, and conduction velocity. Numerical results for the monodomain show that its solution time can be reduced by a significant factor, with only 0.474 mV RMS error between the full order and reduced dimensions solution.

Finite difference neuroelectric modeling software.

This paper describes a finite difference neuroelectric modeling software (FNS), written in C and MATLAB, which can be executed as a standalone program or integrated with other packages for electroencephalography (EEG) analysis. The package from the Oxford Center for Functional MRI of the Brain (FMRIB), FMRIB Software Library (FSL), is used to segment the anatomical magnetic resonance (MR) image for realistic head modeling. The EEG electrode array is fitted to the realistic head model using the Bioelectromagnetism MATLAB toolbox. The finite difference formulation for a general inhomogeneous anisotropic body is used to obtain the system matrix equation, which is then solved using the conjugate gradient algorithm. The reciprocity theorem is utilized to limit the number of required forward solutions to N-1, where N is the number of electrodes. Results show that the forward solver only requires 500 MB of random-access memory (RAM) for a realistic 256×256×256 head model and that the software can be conveniently combined with inverse algorithms such as beamformers and MUSIC. The software is freely available under the GNU Public License.

Finite difference iterative solvers for electroencephalography: serial and parallel performance analysis.

Currently the resolution of the head models used in electroencephalography (EEG) studies is limited by the speed of the forward solver. Here, we present a parallel finite difference technique that can reduce the solution time of the governing Poisson equation for a head model. Multiple processors are used to work on the problem simultaneously in order to speed up the solution and provide the memory for solving large problems. The original computational domain is divided into multiple rectangular partitions. Each partition is then assigned to a processor, which is responsible for all the computations and inter-processor communication associated with the nodes in that particular partition. Since the forward solution time is mainly spent on solving the associated matrix equation, it is desirable to find the optimum matrix solver. A detailed comparison of various iterative solvers was performed for both isotropic and anisotropic realistic head models constructed from MRI images. The conjugate gradient (CG) method preconditioned with an advanced geometric multigrid technique was found to provide the best overall performance. For an anisotropic model with 256 x 128 x 256 cells, this technique provides a speedup of 508 on 32 processors over the serial CG solution, with a speedup of 20.1 and 25.3 through multigrid preconditioning and parallelization, respectively.

Orthogonal field calibration analysis for myocardial electrode arrays used in defibrillation studies.

Mapping the myocardial electric field during a defibrillation pulse requires the recording of potential differences between electrodes. The field is then calculated from these quantities and the corresponding calibration matrix. One straightforward calibration technique involves alignment of a known electric field along each of the orthogonal axes of an electrode array and recording the resulting potential differences. However, no results have been reported to support the efficacy of this technique. This study performs a detailed error analysis including a one-to-one comparison to a precision calibration technique, and quantitatively establishes the efficacy of the orthogonal field technique.

Calibrated current divider network for precision current delivery during high-voltage transthoracic defibrillation.

The design of a calibrated resistive-network current divider for precision current delivery during transthoracic defibrillation shocks is presented together with test results. The current divider presents a constant 50-ohm load to the defibrillator and thus maintains a constant pulse shape. Current is selected before the shock by setting three rheostats using a computer-generated calibration table. Following each shock, the data acquisition and display software updates the calibration table based on the measured value of transthoracic resistance. Over a range of 15-27 A, the root-mean-square (rms) error for delivered versus selected current was 0.48% for a 45-ohm resistive load, and 0.71% for a 100-ohm load. These test results were confirmed by animal experiments. Over 3 dogs, the rms error was 0.49% from 15-27 A and not greater than 1.5% over the entire 8-44 A range.

Simplified calibration of single-plunge bipolar electrode array for field measurement during defibrillation.

In an earlier study, the authors presented a calibration technique for a triaxial bipolar electrode array (EA) that used 72 data points collected during a global sweep of the electric field vector relative to the EA axes. Although necessary for the initial characterization of the EAs, this data requirement has to be significantly reduced for the technique to become a practical tool. Therefore, in the present study, an analysis is performed to determine the relation between the number of data points used in the calibration and the mean root-mean-square error. The analysis shows that 18 data points can produce results nearly identical to those obtained with the 72-point calibration, thus reducing the required amount of data fourfold.