Effect of a perfusing bath on the rate of rise of an action potential propagating through a slab of cardiac tissue

Experiments show that the rate of rise of the action potential depends on the direction of propagation in cardiac tissue. Two interpretations of these experiments have been presented: (i) the data are evidence of discrete propagation in cardiac tissue, and (ii) the data are an effect of the perfusing bath. In this paper we present a mathematical model that supports the second interpretation. We use the bidomain model to simulate action potential propagation through a slab of cardiac tissue perfused by a bath. We assume an intracellular potential distribution and solve the bidomain equations analytically for the transmembrane and extracellular potentials. The key assumption in our model is that the intracellular potential is independent of depth within the tissue. This assumption ensures that all three boundary conditions at the surface of a bidomain are satisfied simultaneously. One advantage of this model over previous numerical calculations is that we obtain an analytical solution for the transmembrane potential. The model predicts that the bath reduces the rate of rise of the transmembrane action potential at the tissue surface, and that this reduction depends on the direction of propagation. The model is consistent with the hypothesis that the perfusing bath causes the observed dependence of the action-potential rate of rise on the direction of propagation, and that this dependence has nothing to do with discrete properties of cardiac tissue.     

Estimation of Cardiac Bidomain Parameters from Extracellular Measurement: Two Dimensional Study

Cardiac tissue conductivity measurements can be used to assess the electrical substrate underlying normal and abnormal wavefront propagation. We describe a method of solving the inverse cardiac bidomain model to estimate average longitudinal and transverse intra and extra-cellular conductivities and fiber angle relative to an electrode array placed arbitrarily on the epi- or endocardial surface. A Newton–Raphson reconstruction method and two Tikhonov-type regularizations were able to stably identify conductivities and fiber angles in tissue models having anisotropies similar to those in real cardiac tissue. The reconstruction methods were tested with data from increasingly realistic two dimensional cardiac bidomain models and performed well both when measurement noise was added, and when simulated experimental and forward model matching was diminished. This approach may be a suitable basis for continuous monitoring of myocardial condition in-vivo via a catheter based electrode array.     

Incorporating Histology into a 3D Microscopic Computer Model of Myocardium to Study Propagation at a Cellular Level

We introduce a 3D model of cardiac tissue to study at a microscopic level the relationship between tissue morphology and propagation of depolarization. Unlike the classical bidomain approach, in which tissue properties are described by the apparent conductivity of the tissue, in this “microdomain” approach, we included histology by modeling the actual shape of the intracellular and extracellular spaces that contain spatially distributed gap-junctions and membranes. The histological model of the tissue was generated by a computer algorithm that can be tuned to model different histological changes. For healthy tissue, the model predicted a realistic conduction velocity of 0.42 m/s based solely on the parameters derived from histology. A comparison with a brick-shaped, simplified model showed that conduction depended to a moderate extent on the shape of myocytes; a comparison with a one-dimensional bidomain model with the same overall shape and structure showed that the apparent conductivity of the tissue can be used to create an equivalent bidomain model. In summary, the microdomain approach offers a means of directly incorporating structural and functional parameters into models of cardiac activation and propagation and thus provides a valuable bridge between the cellular and tissue domains in the myocardium.     

Activation Dynamics in Anisotropic Cardiac Tissue via Decoupling

Bidomain theory for cardiac tissue assumes two interpenetrating anisotropic media--intracellular (i) and extracellular (e)--connected everywhere via a cell membrane; four local parameters sigma(i,e)(l,t) specify conductivities in the longitudinal (l) and transverse (t) directions with respect to cardiac muscle fibers. The full bidomain model for the propagation of electrical activation consists of coupled elliptic-parabolic partial differential equations for the transmembrane potential upsilon(m) and extracellular potential phi(e), together with quasistatic equations for the flow of current in the extracardiac regions. In this work we develop a preliminary assessment of the consequences of neglecting the effect of the passive extracardiac tissue and intracardiac blood masses on wave propagation in isolated whole heart models and describe a decoupling procedure, which requires no assumptions on the anisotropic conductivities and which yields a single reaction-diffusion equation for simulating the propagation of activation. This reduction to a decoupled model is justified in terms of the dimensionless parameter epsilon = (sigma(i)(l)sigma(e)(t) - sigma(i)(t)sigma(e)(l))/(sigma(i)(l) + sigma(e)(l))(sigma(i)(t) + sigma(e)(t)). Numerical simulations are generated which compare propagation in a sheet H of cardiac tissue using the full bidomain model, an isolated bidomain model, and the decoupled model. Preliminary results suggest that the decoupled model may be adequate for studying general properties of cardiac dynamics in isolated whole heart models.     

The electrical potential produced by a strand of cardiac muscle: A bidomain analysis

Analytic expressions are derived relating the transmembrane potential to the intracellular, interstitial and external potentials in a cylindrical strand of cardiac muscle lying in a saline bath. The bidomain model is used to account for the anisotropy and interstitial space in the tissue. The implications of this model for interpreting potential data from strands of cardiac muscle are discussed.     

Extracellular Measurement of Anisotropic Bidomain Myocardial Conductivities. I. Theoretical Analysis

The passive electrical properties of cardiac tissue, such as the intracellular and interstitial conductivities along the longitudinal and transverse axes, have not been often measured because intracellular electrodes are usually needed for these measurements. In this paper, we present a theoretical analysis of two myocardial models developed to estimate these properties by analyzing potentials recorded with a pair of extracellular electrodes while injecting alternating current between another pair of electrodes. First, the cardiac tissue is represented by a standard bidomain model which includes a membrane capacitance; second, this model is modified by adding an intracellular capacitance representing the intercalated disks. Numerical solutions are computed with a fast Fourier transform algorithm without constraining the anisotropy ratios of the interstitial and intracellular domains. We systematically investigate the effects of changes in the bidomain parameters on the voltage-to-current ratio curves. We also demonstrate how the bidomain parameters can be theoretically estimated by fitting, with a modified Shor's r algorithm, the simulated potentials along the longitudinal and transverse axes for different frequencies between 10 and 10000 Hz. An important finding is that the interelectrode distance must be similar to the myocardial space constant so as to obtain frequency dependent measurements. © 2001 Biomedical Engineering Society. PAC01: 8719Nn, 8719Hh, 8716Uv, 0230Uu, 8716Ac     

Estimation of the Bidomain Conductivity Parameters of Cardiac Tissue From Extracellular Potential Distributions Initiated by Point Stimulation

A method for determining the bidomain conductivity values is developed. The study was generated because the different sets of measured conductivity values reported in the literature each produce significantly different bidomain simulation results. The method involves mapping the propagation of the electrical activation of cardiac tissue, initiated by point stimulation, via extracellular electrodes. A time-dependent bidomain model is used to simulate the electrical phenomena. The optimum set of conductivity values is achieved by minimizing the difference between the bidomain model output and the measured extracellular potential, by means of inverse techniques in parameter estimation least-squares and singular value decomposition. The method is validated with synthetic data with added random noise. Other parameters in the model such as membrane capacitance and fiber angle can also be estimated. The method takes a different approach to the conventional four-electrode technique, as it does not require the small electrode separation needed to separate the extracellular current from the intracellular.     

A planar slab bidomain model for cardiac tissue

A fully three-dimensional model of the ventricular or atrial free wall will involve a planar geometry of finite thickness. The governing equations for the interstitial and extracellular potential of a planar slab of cardiac tissue comprised of parallel fibers undergoing uniform plane-wave activation are presented. A comparison with a bidomain of cylindrical geometry with the same half-thickness shows that the potentials in the planar bidomain (as a function of depth) approach core-conductor behavior more quickly.     

The Effect of Conductivity on ST-Segment Epicardial Potentials Arising from Subendocardial Ischemia

We quantify and provide biophysical explanations for some aspects of the relationship between the bidomain conductivities and ST-segment epicardial potentials that result from subendocardial ischemia. We performed computer simulations of ischemia with a realistic whole heart model. The model included a patch of subendocardial ischemic tissue of variable transmural thickness with reduced action potential amplitude. We also varied both intracellular and extracellular conductivities of the heart and the conductivity of ventricular blood in the simulations. At medium or high thicknesses of transmural ischemia (i.e., at least 40% thickness through the heart wall), a consistent pattern of two minima of the epicardial potential over opposite sides of the boundary between healthy and ischemic tissue appeared on the epicardium over a wide range of conductivity values. The magnitude of the net epicardial potential difference, the epicardial maximum minus the epicardial minimum, was strongly correlated to the intracellular to extracellular conductivity ratios both along and across fibers. Anisotropy of the ischemic source region was critical in predicting epicardial potentials, whereas anisotropy of the heart away from the ischemic region had a less significant impact on epicardial potentials. Subendocardial ischemia that extends through at least 40% of the heart wall is manifest on the epicardium by at least one area of ST-segment depression located over a boundary between ischemic and healthy tissue. The magnitude of the depression is a function of the bidomain conductivity values.     

Analysis of Electrode Configurations for Measuring Cardiac Tissue Conductivities and Fibre Rotation

This paper describes a multi-electrode grid, which could be used to determine cardiac tissue parameters by direct measurement. A two pass process is used, where potential measurements are made, during the plateau phase of the action potential, on a subset of these electrodes and these measurements are used to determine the bidomain conductivities. In the first pass, the potential measurements are made on a set of ‘closely-spaced’ electrodes and the parameters are fitted to the potential measurements in an iterative process using a bidomain model and a solver based on a modified Shor's r-algorithm. This first pass yields the extracellular conductivities. The second pass is similar except that a ‘widely-spaced’ electrode set is used and this time the intracellular conductivities are recovered. In addition, it is possible to determine the fibre rotation throughout the tissue, since the bidomain model used here is able to include the effects of fibre rotation.In the simulation studies presented here, the model is solved with known conductivities, on each of the two subsets of electrodes, to generate two sets of ‘measured potentials.’ Conductivities are then recovered by solving an inverse problem based on the measured potentials, to which various levels of noise are added. For example, simulations in the first pass are performed using an electrode spacing of 500 μm, for a situation where the longitudinal and transverse space constants are 769 and 308 μm, respectively. These give very accurate average percentage relative errors for the longitudinal and transverse extracellular conductivities, over five simulations with 1% noise added, of 0.3 and 0.2%. Twenty-five second pass simulations, on a 1 mm grid, yield average percentage relative errors of 3.8, 2.6 and 1.4% for the corresponding intracellular values and the fibre rotation angle, respectively.     

A Bidomain Model Based BEM-FEM Coupling Formulation for Anisotropic Cardiac Tissue

A hybrid boundary element method (BEM)/finite element method (FEM) approach is proposed in order to properly consider the anisotropic properties of the cardiac muscle in the magneto- and electrocardiographic forward problem. Within the anisotropic myocardium a bidomain model based FEM formulation is applied. In the surrounding isotropic volume conductor the BEM is adopted. Coupling is enabled by requesting continuity of the electric potential and the normal of the current density across the boundary of the heart. Here, the BEM part is coupled as an equivalent finite element to the finite element stiffness matrix, thus preserving in part its sparse property. First, continuous convergence of the coupling scheme is shown for a spherical model comparing the computed results to an analytic reference solution. Then, the method is extended to the depolarization phase in a fibrous model of a dog ventricle. A precomputed activation sequence obtained using a fine mesh of the heart was downsampled and used to calculate body surface potentials and extracorporal magnetic fields considering the anisotropic bidomain conductivities. Results are compared to those obtained by neglecting in part or totally (oblique or uniform dipole layer model) anisotropic properties. The relatively large errors computed indicate that the cardiac muscle is one of the major torso inhomogeneities. © 2000 Biomedical Engineering Society. PAC00: 8719Nn, 8719Hh, 8719Ff, 8710+e, 8717Nn     

On the Computational Complexity of the Bidomain and the Monodomain Models of Electrophysiology

The bidomain model, coupled with accurate models of cell membrane kinetics, is generally believed to provide a reasonable basis for numerical simulations of cardiac electrophysiology. Because of changes occurring in very short time intervals and over small spatial domains, discretized versions of these models must be solved on fine computational grids, and small time-steps must be applied. This leads to huge computational challenges that have been addressed by several authors. One popular way of reducing the CPU demands is to approximate the bidomain model by the monodomain model, and thus reducing a two by two set of partial differential equations to one scalar partial differential equation; both of which are coupled to a set of ordinary differential equations modeling the cell membrane kinetics. A reduction in CPU time of two orders of magnitude has been reported. It is the purpose of the present paper to provide arguments that such a reduction is not present when order-optimal numerical methods are applied. Theoretical considerations and numerical experiments indicate that the reduction factor of the CPU requirements from bidomain to monodomain computations, using order-optimal methods, typically is about 10 for simple cell models and less than two for more complex cell models.     

On the Passive Cardiac Conductivity

In order to relate the structure of cardiac tissue to its passive electrical conductivity, we created a geometrical model of cardiac tissue on a cellular scale that encompassed myocytes, capillaries, and the interstitial space that surrounds them. A special mesh generator was developed for this model to create realistically shaped myocytes and interstitial space with a controled degree of variation included in each model. In order to derive the effective conductivities, we used a finite element model to compute the currents flowing through the intracellular and extracellular space due to an externally applied electrical field. The product of these computations were the effective conductivity tensors for the intracellular and extracellular spaces. The simulations of bidomain conductivities for healthy tissue resulted in an effective intracellular conductivity of 0.16S/m (longitudinal) and 0.005S/m (transverse) and an effective extracellular conductivity of 0.21S/m (longitudinal) and 0.06S/m (transverse). The latter values are within the range of measured values reported in literature. Furthermore, we anticipate that this method can be used to simulate pathological conditions for which measured data is far more sparse.     

Dependence of cardiac strength-interval curves on pacing rate

The purpose of the research is to determine how the pacing rate affects the strength-interval curve in cardiac tissue. Computer simulations are used to calculate the cathodal and anodal strength-interval curves. The tissue is represented by the bidomain model with Beeler-Reuter membrane properties. The strength-interval curves shift to shorter intervals as the pacing rate increases. However, the shape of the strength-interval curve, including the separation into ‘make’ and ‘break’ sections and the presence of a ‘dip’, is insensitive to pacing rate.     

A Finite Volume Method for Modeling Discontinuous Electrical Activation in Cardiac Tissue

This paper describes a finite volume method for modeling electrical activation in a sample of cardiac tissue using the bidomain equations. Microstructural features to the level of cleavage planes between sheets of myocardial fibers in the tissue are explicitly represented. The key features of this implementation compared to previous modeling are that it represents physical discontinuities without the implicit removal of intracellular volume and it generates linear systems of equations that are computationally efficient to construct and solve. Results obtained using this method highlight how the understanding of discontinuous activation in cardiac tissue can form a basis for better understanding defibrillation processes and experimental recordings.     

Modelling passive cardiac conductivity during ischaemia

The results of a geometric model of cardiac tissue, used to compute the bidomain conductivity tensors during three phases of ischaemia, are described. Ischaemic conditions were simulated by model parameters being changed to match the morphological and electrical changes of three phases of ischaemia reported in literature. The simulated changes included collapse of the interstitial space, cell swelling and the closure of gap junctions. The model contained 64 myocytes described by 2 million tetrahedral elements, to which an external electric field was applied, and then the finite element method was used to compute the associated current density. In the first case, a reduction in the amount of interstitial space led to a reduction in extracellular longitudinal conductivity by about 20%, which is in the range of reported literature values. Moderate cell swelling in the order of 10–20% did not affect extracellular conductivity considerably. To match the reported drop in total tissue conductance reported in experimental studies during the third phase of ischaemia, a ten fold increase in the gap junction resistance was simulated. This ten-fold increase correlates well with the reported changes in gap junction densities in the literature.     

Development of a model for point source electrical fibre bundle stimulation

A model is presented for determining the excitation (transmembrane) potentials on nerve and muscle fibres in a cylindrical bundle from an external point source electrode at the surface and within the preparation. The fibre bundle is considered to be immersed in an infinite isotropic conductive medium and is idealised as an infinitely extending cylinder. This cylinder is initially represented as an isotropic monodomain. A subsequent degree of complexity introduces anisotropy in the monodomain, and finally the bundle is represented as an anisotropic bidomain comprised of the interstitial radial and longitudinal conductivities, the intracellular longitudinal conductivity and the fibre membrane between the two domains. In this latter model, electrical coupling from extracellular to intracellular space is included by means of the bidomain formulation. Computational aspects are discussed, and preliminary results for prescribed conditions are presented.