An analytical comparison of the information in sorted and non-sorted cosine-tuned spike activity.

Spike sorting is a technologically expensive component of the signal processing chain required to interpret population spike activity acquired in a neuromotor prosthesis. No systematic analysis of the value of spike sorting has been carried out, and little is known about the effects of spike sorting error on the ability of a brain-machine interface (BMI) to decode intended motor commands. We developed a theoretical framework to examine the effects of spike processing on the information available to a BMI decoder. We computed the mutual information in neural activity in a simplified model of directional cosine tuning to compare the effects of pooling activity from up to four neurons to the effects of sorting with varying amounts of spike error. The results showed that information in a small population of cosine-tuned neurons is maximized when the responses are sorted and there is diverse tuning of units, but information was affected little when pooling units with similar preferred directions. Spike error had adverse effects on information, such that non-sorted population activity had 79-92% of the information in its sorted counterpart for reasonable amounts of detection and sorting error and for units with moderate differences in preferred direction. This quantification of information loss associated with pooling units and with spike detection and sorting error will help to guide the engineering decisions in designing a BMI spike processing system.

Study of atrial arrhythmias in a computer model based on magnetic resonance images of human atria.

The maintenance of multiple wavelets appears to be a consistent feature of atrial fibrillation (AF). In this paper, we investigate possible mechanisms of initiation and perpetuation of multiple wavelets in a computer model of AF. We developed a simplified model of human atria that uses an ionic-based membrane model and whose geometry is derived from a segmented magnetic resonance imaging data set. The three-dimensional surface has a realistic size and includes obstacles corresponding to the location of major vessels and valves, but it does not take into account anisotropy. The main advantage of this approach is its ability to simulate long duration arrhythmias (up to 40 s). Clinically relevant initiation protocols, such as single-site burst pacing, were used. The dynamics of simulated AF were investigated in models with different action potential durations and restitution properties, controlled by the conductance of the slow inward current in a modified Luo-Rudy model. The simulation studies show that (1) single-site burst pacing protocol can be used to induce wave breaks even in tissue with uniform membrane properties, (2) the restitution-based wave breaks in an atrial model with realistic size and conduction velocities are transient, and (3) a significant reduction in action potential duration (even with apparently flat restitution) increases the duration of AF. (c) 2002 American Institute of Physics.

Simulation and prediction of functional block in the presence of structural and ionic heterogeneity.

Inhomogeneities in myocardial structure and action potential duration (APD) lead to dispersion of APD throughout the heart. APD gradients in the range of 20-125 ms/cm have been reported to produce functional block. In this study, a multicellular fiber model was used to examine the effect of structural and ionic inhomogeneities on the likelihood of premature stimuli to produce functional block. With the use of both the Fenton-Karma and Luo-Rudy phase II membrane models, functional block is found to occur in tissue with a maximum gradient <45 ms/cm and depends on the spatial extent. In general, the narrower the extent the larger the magnitude needed for block. A simple relationship for predicting block is presented that only requires information about the conduction velocity (CV) restitution properties of the tissue and the APD gradients. Analysis reveals that the effects of a steep CV restitution slope may be beneficial in overcoming intrinsic cellular heterogeneity for a single premature beat.

Influence of dynamic gap junction resistance on impulse propagation in ventricular myocardium: a computer simulation study.

The gap junction connecting cardiac myocytes is voltage and time dependent. This simulation study investigated the effects of dynamic gap junctions on both the shape and conduction velocity of a propagating action potential. The dynamic gap junction model is based on that described by Vogel and Weingart (J. Physiol. (Lond.). 1998, 510:177-189) for the voltage- and time-dependent conductance changes measured in cell pairs. The model assumes that the conductive gap junction channels have four conformational states. The gap junction model was used to couple 300 cells in a linear strand with membrane dynamics of the cells defined by the Luo-Rudy I model. The results show that, when the cells are tightly coupled (6700 channels), little change occurs in the gap junction resistance during propagation. Thus, for tight coupling, there are negligible differences in the waveshape and propagation velocity when comparing the dynamic and static gap junction representations. For poor coupling (85 channels), the gap junction resistance increases 33 MOmega during propagation. This transient change in resistance resulted in increased transjunctional conduction delays, changes in action potential upstroke, and block of conduction at a lower junction resting resistance relative to a static gap junction model. The results suggest that the dynamics of the gap junction enhance cellular decoupling as a possible protective mechanism of isolating injured cells from their neighbors.

Myocardial fiber orientation mapping using reduced encoding diffusion tensor imaging.

A precise knowledge of the myocardial fiber architecture is essential to accurately understand and interpret cardiac electrical and mechanical functions. Diffusion tensor imaging has been used to noninvasively and quantitatively characterize myocardial fiber orientations. However, because the approach necessitates diffusion to be measured in multiple encoding directions and frequently at multiple weighting levels, the required data set size may present a limitation on its acquisition time efficiency. Applying the principles of reduced encoding imaging (REI), four basic reconstruction schemes, keyhole using direct substitution, keyhole with baseline correction, symmetrically encoded REI with generalized-series reconstruction (RIGR), and asymmetrically encoded RIGR, are evaluated in terms of their accuracy in diffusion tensorfiber orientation mapping of excised myocardial samples. Results show that the performances of all REI schemes, at approximately 50% reduced encoding, are at least comparable with that of a control experiment consisting of proportionally reduced number of full k-space images. Moreover, although performances of the symmetrically and asymmetrically encoded RIGR schemes are similar, both methods provide significant improvements over the control experiment and the direct-substitution keyhole technique. These findings demonstrate the potential of the general REI methodology for diffusion tensor imaging and pave the way for modified schemes involving rapid imaging sequences or alternative k-space sampling strategies to achieve even better data acquisition time efficiency and performance.

A space-time adaptive method for simulating complex cardiac dynamics.

For plane-wave and many-spiral states of the experimentally based Luo-Rudy 1 model of heart tissue in large (8 cm square) domains, we show that a space-time-adaptive time-integration algorithm can achieve a factor of 5 reduction in computational effort and memory-but without a reduction in accuracy-when compared to an algorithm using a uniform space-time mesh at the finest resolution. Our results indicate that such an algorithm can be extended straightforwardly to simulate quantitatively three-dimensional electrical dynamics over the whole human heart.

A flexible method for simulating cardiac conduction in three-dimensional complex geometries.

In this article we present a method for creating a membrane-based computer model capable of representing a three-dimensional irregular domain. The spatial discretization of our model is based on the Finite Volume Method. In combination with a robust meshmaking tool, our method may be used to simulate conduction in an arbitrarily shaped, complex region. In this way, conduction in specific subregions of cardiac anatomy may be examined. The capabilities of this methodology are demonstrated through 3 examples. The first shows the influence of abrupt changes in tissue geometry on conduction parameters. The second highlights the ability of the model to incorporate interior boundaries and altered membrane properties. The final example shows the inclusion of a full description of the fiber architecture in a portion of the canine left ventricle. Future applications will make use of the model's capabilities to conduct investigations in complex regions including the atria.

Validation of three-dimensional conduction models using experimental mapping: are we getting closer?

The anisotropic material properties, irregular geometry, and specialized conduction system of the heart all affect the three-dimensional (3D) spread of electrical activation. A limited number of research groups have tried accounting for these features in 3D conduction models to investigate more thoroughly their observations of cardiac electrical activity in 3D experimental preparations. The full potential of these large scale conduction models, however, has not been realized because of a lack of quantitative validation with experiment. Such validation is critical in order to use the models to predict the electrical response of the myocardium to drugs or electrical stimulation. In this paper, a quantitative, experimental validation of paced activation in a 3D conduction model of a 3 cm x 3 cm x 1 cm section of the ventricular wall is presented. Epicardial and intramural pacing stimuli were applied in the center of a 528 channel electrode plaque sutured to the left ventricle in dogs. Unipolar electrograms were recorded at 2 kHz during and after pacing. Fiber directions within the tissue below the electrodes were estimated histologically and from pace-mapping. Simulated epicardial electrograms were computed for surface paced beats using our 3D bidomain model of the mapped tissue volume incorporating the measured fiber directions. Extracellular potentials and isochronal maps resulting from paced activations in both model and experiment were directly compared. Preliminary results demonstrate that our 3D model reproduces qualitatively such key features of the experimental data as electrogram morphologies and epicardial conduction velocities. Though quantitative agreement between model and experiment was only moderate, the validation approach described herein is an essential first step in assessing the predictive capability of present day conduction models.

Magnetic resonance myocardial fiber-orientation mapping with direct histological correlation.

Functional properties of the myocardium are mediated by the tissue structure. Consequently, proper physiological studies and modeling necessitate a precise knowledge of the fiber orientation. Magnetic resonance (MR) diffusion tensor imaging techniques have been used as a nondestructive means to characterize tissue fiber structure; however, the descriptions so far have been mostly qualitative. This study presents a direct, quantitative comparison of high-resolution MR fiber mapping and histology measurements in a block of excised canine myocardium. Results show an excellent correspondence of the measured fiber angles not only on a point-by-point basis (average difference of -2.30 +/- 0.98 degrees, n = 239) but also in the transmural rotation of the helix angles (average correlation coefficient of 0.942 +/- 0.008 with average false-positive probability of 0.004 +/- 0.001, n = 24). These data strongly support the hypothesis that the eigenvector of the largest MR diffusion tensor eigenvalue coincides with the orientation of the local myocardial fibers and underscore the potential of MR imaging as a noninvasive, three-dimensional modality to characterize tissue fiber architecture.

Bipolar stimulation of a three-dimensional bidomain incorporating rotational anisotropy.

A bidomain model of cardiac tissue was used to examine the effect of transmural fiber rotation during bipolar stimulation in three-dimensional (3-D) myocardium. A 3-D tissue block with unequal anisotropy and two types of fiber rotation (none and moderate) was stimulated along and across fibers via bipolar electrodes on the epicardial surface, and the resulting steady-state interstitial (phi e) and transmembrane (Vm) potentials were computed. Results demonstrate that the presence of rotated fibers does not change the amount of tissue polarized by the point surface stimuli, but does cause changes in the orientation of phi e and Vm in the depth of the tissue, away from the epicardium. Further analysis revealed a relationship between the Laplacian of phi e, regions of virtual electrodes, and fiber orientation that was dependent upon adequacy of spatial sampling and the interstitial anisotropy. These findings help to understand the role of fiber architecture during extracellular stimulation of cardiac muscle.

A finite volume model of cardiac propagation.

This paper describes a two-dimensional cardiac propagation model based on the finite volume method (FVM). This technique, originally derived and applied within the filed of computational fluid dynamics, is well suited to the investigation of conduction in cardiac electrophysiology. Specifically, the FVM permits the consideration of propagation in a realistic structure, subject to arbitrary fiber orientations and regionally defined properties. In this application of the FVM, an arbitrarily shaped domain is decomposed into a set of constitutive quadrilaterals. Calculations are performed in a computational space, in which the quadrilaterals are all represented simply as squares. Results are related to their physical-space equivalents by means of a transformation matrix. The method is applied to a number of cases. First, large-scale propagation is considered, in which a magnetic resonance-imaged cardiac cross-section serves as the governing geometry. Next, conduction is examined in the presence of an isthmus formed by the microvasculature in a slice of papillary muscle tissue. Under ischemic conditions, the safety factor for propagation is seen to be related to orientation of the fibers within the isthmus. Finally, conduction is studied in the presence of an inexcitable obstacle and a curved fiber field. This example illustrates the dramatic influence of the complex orientation of the fibers on the resulting activation pattern. The FVM provides a means of accurately modeling the cardiac structure and can help bridge the gap between computation and experiment in cardiac electrophysiology.

Paced activation mapping reveals organization of myocardial fibers: a simulation study.

INTRODUCTION: A three-dimensional bidomain model of a block section of both the right and left ventricular walls that included rotational anisotropy and fiber curvature was used to investigate potential distributions generated during paced activation mapping. Unlike previous large-scale tissue models, the extracellular stimulus was included. METHODS AND RESULTS: The model was used to test the hypothesis that information about the underlying tissue structure (surface fiber angle gradients, amount of fiber rotation per unit depth, and anisotropy) can be extracted from surface potential distributions during stimulation. Results from distributions during stimulation were compared to those obtained using the distributions during activation. To better correlate results to possible experimental measurements, the analysis was performed using a 21 X 21 grid of "electrode" sites, each separated by 1 mm. Fiber orientation was estimated from the surface data by: (1) curve-fitting the elliptical shape of the epicardial potential distribution during stimulation; (2) identifying the location of the potential maxima leading the wavefront during early activation; and (3) for epicardial stimuli, curve-fitting the elliptical shape of the activation isochrones. Results show that surface potential distributions from the stimulus can be used to estimate fiber orientation; however, the accuracy of the reconstruction is highly dependent on the amount of fiber rotation per unit depth. CONCLUSIONS: Extracellular potential data during and after stimulation is shown to reflect the organization of myocardial fibers and, as such, could be used to characterize the three-dimensional anisotropic electrical properties in situ.

Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: simulations in a three-dimensional bidomain model.

INTRODUCTION: A modeling study is presented to explore the effects of tissue conductivity, fiber orientation, and presence of an adjoining extracellular volume conductor on electrical conduction in cardiac muscle. Simulated results are compared with those of classical in vitro experiments on superfused thin layer preparations and on whole hearts. METHODS AND RESULTS: The tissue is modeled as a three-dimensional bidomain block adjoining an isotropic bath. In the thin layer model, the fibers are assumed parallel. In the thick block model, fiber rotation, curvature, and tipping are incorporated. Results from the thin layer model explain experimental observations that the rate of rise of the entire action potential upstroke is faster and the magnitude of the extracellular potential is smaller across fibers than along fibers in a uniformly propagating front. The simulation identified that this behavior only arises in tissue with unequal anisotropy in the two spaces and adjoining an extracellular bath. Simulated conduction and potential distributions in the thick block model are shown to well approximate experimental maps. The potentials are sensitive to changes in the fiber orientations. A slight 5 degrees tipping of intramural fibers out of the planes parallel to the epicardium and endocardium will lead to an asymmetry of the magnitudes of the positive regions. In addition, the introduction of fiber curvature leads to more realistic isochrone and extracellular potential distributions. The orientation of the central negative region of the extracellular potential is shown to be determined by the average of the fiber direction at the plane of pacing and the plane of recording. CONCLUSIONS: The simulations demonstrate the sensitivity of spread of activation and potential time courses and distributions to the underlying electrical properties in both thick and thin slabs. The bidomain model is shown to be a useful representation of cardiac tissue for interpreting experimental data of activation.

Spatial potential and current distributions along transvenous defibrillation electrodes: variation of electrode characteristics.

The therapeutic efficacy of an endocardial defibrillation lead system can be improved by controlling the profile of current delivery through a suitable choice of electrode characteristics, which include the length, radius, number of conductor elements, electrode resistance, and point of connection to the voltage source. Such control will minimize tissue and lead damage during long-term use. In this study, a semianalytical model was developed to study cylindrical electrodes of different constructions in an idealized electrolytic medium. Simulations were performed to investigate the effects of varying the electrode characteristics on the spatial voltage and current distributions and interelectrode resistance for cylindrical electrodes of different constructions. The results show that, for transvenous electrodes of realistic dimensions, the current distributions are determined largely by the edge effects. The edge effects increase as the aspect ratio of the electrode (length/radius) decrease. The multiple edges resulting from wrapping conductor elements over a nonconducting base are found to increase the nonuniformity and the current density over the conductor-covered surface. The model is used to demonstrate two techniques of controlling the current distribution. The first method involve modifying the electrode resistivity profile and point of connection. In the second approach, the electrode surface is covered with a thin film having a model-computed resistance profile. By using either methods to produce isocurrent electrodes, the interelectrode resistance is found to increase.

Linear algebraic transformations of the bidomain equations: implications for numerical methods.

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation L phi = F, various algebraic transformations can be expressed as PLQ-1 psi = PF, where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.

Simulating the electrical behavior of cardiac tissue using the bidomain model.

The complex microstructure of cardiac muscle comprised of coupled cells, enveloped by an interstitium made up of blood vessels, connective tissue, and fluid, presents some obvious problems to those interested in understanding the tissue as an electrical medium. One approach that has gained considerable favor in recent years views the tissue not as a discrete structure, but rather as two coupled, continuous domains: one for the intracellular space and the other for the interstitial space. For convenience, the averaged potentials and currents in both domains are defined at every point in space. The structure is partially preserved by assigning a conductivity tensor at each point. One advantage of using this space-averaged model is that the governing equations for the electric fields can be described by partial differential equations that on occasion lead to analytical solutions. This formal treatment of cardiac tissue as two coupled continua is referred to as the bidomain model. This article presents a mathematical description of the bidomain model and reviews the use of the model for simulating the electrical behavior of cardiac tissue.

Cardiac propagation simulation.

We have completed a range of membrane-based simulations of action potential propagation in two- and three-dimensional models of ventricular myocardium. The two-dimensional simulations included a bidomain representation of the myocardium which explicitly characterized the component volume conductors in the intracellular, interstitial, and extracellular spaces. With these simulations, we studied the contribution of the extracellular volume conductor to transmural myocardial propagation during depolarization. We also used two-dimensional bidomain simulations to study the effect of the interstitial volume conductor in the setting of planar myocardial depolarization with nominal and extreme tissue conductivities. Our three-dimensional simulations included a monodomain representation of the myocardium which characterized the three component volume conductors as a single lumped conductor. With these simulations, we examined the effects of the intramural rotation of the fiber axes on the timing and pattern of activation. To achieve practical solution times, we extended numerical techniques from previous reports and developed a range of new techniques applicable to this class of problems. Simulations of the depolarization wavefront used the nonlinear Ebihara and Johnson membrane equations for the fast sodium current as the membrane model. Simulations of the full action potential cycle combined the Ebihara and Johnson fast sodium current with the Beeler and Reuter membrane equations. Our results demonstrated that the individual volume conductors and the rotation of fiber axes have unique and identifiable consequences on the electrical activation in models of ventricular myocardium.

Examination of the choice of models for computing the extracellular potential of a single fibre in a restricted volume conductor.

The paper compares the rigorous and the conventional approximate line source solution of Laplace's equation used to evaluate the potential of a single cylindrical fibre. Particular attention is given to the solutions for a radially restricted circular cylindrical volume conductor. The effect of the extent of the volume conductor b on the difference between the potentials evaluated according to the different models is examined. For values of b larger than 10 times the fibre radius, the relative difference is less than 1 per cent and the values of b around 2 times the fibre radii, the error reaches as much as 17 per cent.

Modification of a cylindrical bidomain model for cardiac tissue.

Previous models based on a cylindrical bidomain assumed either that the ratio of intracellular and interstitial conductivities in the principal directions were the same or that there was no radial variation in potential (i.e., a planar front, delta Vm/delta rho = 0). This paper presents a formulation and the expressions for the intracellular, interstitial, extracellular, and transmembrane potentials arising from nonplanar propagation along a cylindrical bundle of cardiac tissue represented as a bidomain with arbitrary anisotropy. For unequal anisotropy, the transmembrane current depends not only on the local change of the transmembrane potential but also on the nature of the transmembrane potential throughout the volume.

Simulation of propagation along a cylindrical bundle of cardiac tissue--II: Results of simulation.

Previous evaluations of the cylindrical bidomain model of a bundle of cardiac tissue, have been obtained by using an analytic function for the transmembrane potential and assuming the activating wavefront through the bundle cross section is planar. In this paper, nonlinear membrane kinetics are introduced into the bidomain membrane and equal anisotropy ratios are assumed, permitting the transmembrane potential to be computed and its behavior examined at different depths in the bundle and for different values of conductivity and bundle diameters. In contrast with single fiber models, the bundle model reveals that the shape of the action potential is influenced by tissue resistivities. In addition, the steady-state activation wavefront through the cross-section perpendicular to the long axis of the bundle is not planar and propagates with a velocity that lies between that of a single fiber in an unbounded volume and a single fiber in a restricted extracellular space. In general, the bundle model is shown to be significantly better than the classical single fiber model in describing the behavior of real cardiac tissue.