Photon density measured over a cut surface: implications for optical mapping of the heart

During optical mapping of the heart, the signal represents a weighted average of the transmembrane potential. Researchers sometimes illuminate the epicardium and record over a cut surface. My goal is to compare their signal to the photon density within an intact tissue. Does cutting the tissue affect the signal?     

Averaging over depth during optical mapping of unipolar stimulation

Numerical simulations have predicted the distribution of transmembrane potential during electrical stimulation of cardiac tissue. When comparing these predictions to measurements obtained using optical mapping techniques, the optical signal should not be compared to the transmembrane potential calculated at the surface of the tissue, but instead to the transmembrane potential averaged over depth. In this paper, the bidomain model is used to calculate the transmembrane potential in a three-dimensional slab of cardiac tissue, stimulated by a unipolar electrode on the tissue surface. For an optical decay constant of 0.3 mm and an electrode radius of 1 mm, the surface transmembrane potential is more than a factor of three larger than the transmembrane potential averaged over depth. Our results suggest that optical mapping underestimates the surface transmembrane potential during electrical stimulation.     

Correspondence between the location of evoked potential generators and sites of maximal sensitivity to stimulation

The potential recorded by a set of electrodes as an action potential traverses a small axonal segment is proportional to the transmembrane potential produced during stimulation of that axon segment by the same set of recording electrodes, under certain circumstances. First, the membrane must have a constant thickness which is so small that the difference between the surface area of the inner and outer surfaces is minimal. Second, all media must be linear. Third, there must be a monotonically increasing relation between the mean transmembrane potential induced by a stimulus and the maximum transmembrane potential. Fourth, as each axon segment depolarizes, the transmembrane current and change in membrane potential during this time are same. This principle remains true for magnetic stimulation and recording as long as currents generated at the boundaries between regions of differing conductivity outside the axon contribute minimally to the field at the axon. This allows the identification of the point at which an action potential generates a maximal extracellular potential as the point that is stimulated with the lowest threshold.     

Source-Field Relationships for Cardiac Generators on the Heart Surface Based on Their Transfer Coefficients

We consider three different types of equivalent sources over a closed surface enclosing all the electrical cardiac generators: the (in situ) potential, the (in situ) normal current density, and the (macroscopic) transmembrane potential on the heart surface. The last equivalent source, which behaves as a double layer, is derived from the bidomain (bisyncytia) model for anisotropic cardiac muscle. This model predicts that if ratios of intracellular to interstitial conductivity along all directions are equal, field potential can be calculated only using surface integrals. The volume integral arising from the tissue anisotropy of cardiac muscle vanishes in that case. For each type of source under study, we give the field potential in a bounded inhomogeneous volume conductor in the form of an integral equation. We also derive the conditions which the lead field (or the transfer coefficients) must satisfy. The in situ potential and normal current are related to the cardiac sources in a complex way, but their lead fields are independent of conductivity of heart muscle, whereas the transmembrane potential is directly involved as a source term, but the lead field depends on the anisotropy of the heart muscle.     

Potential distribution in three-dimensional periodic myocardium.II. Application to extracellular stimulation

Modeling potential distribution in the myocardium treated as a periodic structure implies that activation from high-current stimulation with extracellular electrodes is caused by the spatially oscillating components of the transmembrane potential. This hypothesis is tested by comparing the results of the model with experimental data. The conductivity, fiber orientation, the extent of the region, the location of the pacing site, and the stimulus strength determined from experiments are components of the model used to predict the distributions of potential, potential gradient, and the transmembrane potential throughout the region. Next, assuming that a specific value of the transmembrane potential is necessary and sufficient to activate fully repolarized myocardium, the model provides an analytical relation between large-scale field parameters, such as gradient and current density, and small-scale parameters, such as transmembrane potential. This relation is used to express the stimulation threshold in terms of gradient or current density components and to explain its dependence upon fiber orientation. The concept of stimulation threshold is generalized to three dimensions, and an excitability surface is constructed, which for cardiac muscle is approximately conical in shape. The numerical values of transmembrane potential and stimulation thresholds calculated using asymptotic analysis are in agreement with the results of animal experiments, confirming the validity of this approach to study the electrophysiology of periodic cardiac muscle.     

A comparison of noninvasive reconstruction of epicardial versus transmembrane potentials in consideration of the null space

We compare two source formulations for the electrocardiographic forward problem in consideration of their implications for regularizing the ill-posed inverse problem. The established epicardial potential source model is compared with a bidomain-theory-based transmembrane potential source formulation. The epicardial source approach is extended to the whole heart surface including the endocardial surfaces. We introduce the concept of the numerical null and signal space to draw attention to the problems associated with the nonuniqueness of the inverse solution and show that reconstruction of null-space components is an important issue for physiologically meaningful inverse solutions. Both formulations were tested with simulated data generated with an anisotropic heart model and with clinically measured data of two patients. A linear and a recently proposed quasi-linear inverse algorithm were applied for reconstructions of the epicardial and transmembrane potential, respectively. A direct comparison of both formulations was performed in terms of computed activation times. We found the transmembrane potential-based formulation is a more promising source formulation as stronger regularization by incorporation of biophysical a priori information is permitted.     

Discrete versus syncytial tissue behavior in a model of cardiacstimulation. II. Results of simulation

For pt. I see ibid., vol. 43, no. 12, p. 1129-40 (1996). The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the “periodic” bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small     

Study of transmembrane potentials on cellular inner and outer membrane--frequency response model and its filter characteristic simulation

Based on the multilayer dielectric model for a spherical cell, a frequency response model of transmembrane potentials on cellular inner and outer membranes is established with a simulating method. The simulating results indicate that transmembrane potential on the inner membrane shows first-order bandpass filter characteristic, while transmembrane potential on the outer membrane shows first-order low-pass filter characteristic approximately. It could be found that the transmembrane potential on the inner membrane is greater than that on the outer membrane, and can keep a higher value in the range from a center frequency to an upper cutoff frequency, which is desirable to induce intracellular electromanipulation. Both a discussion about an equivalent RC model of the cell and the experimental result are in agreement with the aforementioned conclusion. Therefore, the frequency response model could help to choose reasonable window parameters for the application of a nanosecond pulsed electric field to tumor treatment.     

Analysis of Intense, Subnanosecond Electrical Pulse-Induced Transmembrane Voltage in Spheroidal Cells With Arbitrary Orientation

Self-consistent evaluations of the transmembrane potential (TMP) and possible membrane electroporation in spheroidal cells arising from an ultrashort, high-intensity pulse are reported. The present study couples the Laplace equation with Smoluchowski theory of pore formation, and uses double-shell models. It is shown that the response of prolate spheroids is faster than that of the sphere, with the outer membrane reaching its steady-state value in about ${2} {mu}$s. The simulation result also shows that the TMP across an inner organelle could exceed the value across the plasma membrane at least over the first $hbox{0.4}; {mu }$s or so, indicating a possibility of intracellular, electromanipulation of cells. The TMP induced by pulsed external voltages is predicted to be higher in oblate spheroids in comparison to both spherical and prolate spheroidal cells. This occurs due to flattening of the surface area.      

The effect of plunge electrodes during electrical stimulation of cardiac tissue

The mechanism for far-field stimulation of cardiac tissue is not known, although many hypotheses have been suggested. This paper explores a new hypothesis: the insulated plunge electrodes used in experiments to map the extracellular potential may affect the transmembrane potential when an electric field is applied to cardiac tissue. Our calculation simulates a 10-mm-diameter sheet of passive tissue with a circular insulated plunge electrode in the middle of it, ranging in diameter from 0.05 to 2 mm. We calculate the transmembrane potential induced by a 500-V/m electric field. Our results show that a transmembrane potential is induced around the electrode in alternating areas of depolarization and hyperpolarization. If the electric field is oriented parallel to the myocardial fibers, the maximum transmembrane potential is 89 mV. A layer of fluid around the electrode increases the transmembrane potential. We conclude that plunge electrodes may introduce artifacts during experiments designed to study the response of the heart to strong electric shocks.     

Building maps of local apparent conductivity of the epicardium with a 2-D electrophysiological model of the heart

In this paper, we address the problem of estimating the parameters of an electrophysiological model of the heart from a set of electrical recordings. The chosen model is the reaction-diffusion model on the transmembrane potential proposed by Aliev and Panfilov. For this model of the transmembrane, we estimate a local apparent two-dimensional conductivity from a measured depolarization time distribution. First, we perform an initial adjustment including the choice of initial conditions and of a set of global parameters. We then propose a local estimation by minimizing the quadratic error between the depolarization time computed by the model and the measures. As a first step we address the problem on the epicardial surface in the case of an isotropic version of the Aliev and Panfilov model. The minimization is performed using Brent method without computing the derivative of the error. The feasibility of the approach is demonstrated on synthetic electrophysiological measurements. A proof of concept is obtained on real electrophysiological measures of normal and infarcted canine hearts.     

Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model

We have developed a new approach for imaging cardiac transmembrane potentials (TMPs) within the three-dimensional (3-D) myocardium by means of an anisotropic heart model. The cardiac TMP distribution is estimated from body surface electrocardiograms by minimizing objective functions of the "measured" body surface potential maps (BSPMs) and the heart-model-generated BSPMs. Computer simulation studies have been conducted to evaluate the present 3-D TMP imaging approach using pacing protocols. Simulations of single-site pacing at 24 sites throughout the ventricles, as well as dual-site pacing at 12 pairs of sites in the vicinity of atrio-ventricular ring were performed. The present simulation results show that the correlation coefficient (CC) and relative error (RE) between the "true" and inversely estimated TMP distributions were 0.9915 +/- 0.0041 and 0.1266 +/- 0.0326, for single-site pacing, and 0.9889 +/- 0.0034 and 0.1473 +/- 0.0237 for dual-site pacing, respectively, when 10 microV Gaussian white noise (GWN) was added to the BSPMs. The effects of heart and torso geometry uncertainty were also evaluated by shifting the heart position by 10 mm and altering the torso size by 10%. The CC between the "true" and inversely estimated TMP distributions was above 0.97 when these geometry uncertainties were considered. The present simulation results demonstrate the feasibility of noninvasive estimation of TMP distribution throughout the ventricles from body surface electrocardiographic measurements, and suggest that the present method may become a useful alternative in noninvasive imaging of distributed cardiac electrophysiological processes within the 3-D myocardium.     

The response of a spherical heart to a uniform electric field: abidomain analysis of cardiac stimulation

A mathematical model describing electrical stimulation of the heart is developed, in which a uniform electric field is applied to a spherical shell of cardiac tissue. The electrical properties of the tissue are characterized using the bidomain model. Analytical expressions for the induced transmembrane potential are derived for the cases of equal anisotropy ratios in the intracellular and interstitial (extracellular) spaces, and no transverse coupling between fibers. Numerical calculations of the transmembrane potential are also performed using realistic electrical conductivities. The model illustrates several mechanisms for polarization of the cell membrane, which can be divided into two categories, depending on if they polarize fibers at the heart surface only or if they polarize fibers both at the surface and within the bulk of the tissue. The latter mechanisms can be classified further according to whether they originate from continuous or discrete properties of cardiac tissue     

温度效应对声表面波传播的影响

通过引入材料的高阶热膨胀系数和高阶热弹性系数,对半无限大石英晶体内声表面波的传播方程进行了求解,得到了不同温度下的声表面波波速.计算结果表明,温度变化对AT切和ST切石英晶体的声表面波波速有显著影响.最后分析了ST切石英晶体的声表面波的温频特性,结果表明,由温度变化引起的频率漂移很明显.     

Patterns of and mechanisms for shock-induced polarization in the heart: a bidomain analysis

This paper examines the combined action of cardiac fiber curvature and transmural fiber rotation in polarizing the myocardium under the conditions of a strong electrical shock. The study utilizes a three-dimensional finite element model and the continuous bidomain representation of cardiac tissue to model steady-state polarization resulting from a defibrillation-strength uniform applied field. Fiber architecture is incorporated in the model via the shape of the heart, an ellipsoid of variable ellipticity index, and via an analytical function, linear or nonlinear, describing the transmural fiber rotation. Analytical estimates and numerical results are provided for the location and shape of the "bulk" polarization (polarization away from the tissue boundaries) as a function of the fiber field, or more specifically, of the conductivity changes in axial and radial direction with respect to the applied electrical field lines. Polarization in the tissue "bulk" is shown to exist only under the condition of unequal anisotropy ratios in the extra- and intracellular spaces. Variations in heart geometry and, thus, fiber curvature, are found to lead to change in location of the zones of significant membrane polarization. The transmural fiber rotation function modulates the transmembrane potential profile in the radial direction. A higher gradient of the transmural transmembrane potential is observed in the presence of fiber rotation as compared to the no rotation case. The analysis presented here is a step forward in understanding the interaction between tissue structure and applied electric field in establishing the pattern of membrane polarization during the initial phase of the defibrillation shock.     

声表面波器件温度敏感性研究

与内外应力一样,温度亦是影响声表面波器件工作性能的主要参数之一。文章分析了上前声表面波器件温度补偿的状况。利用波扰动方程分析研究了声表面波器件频率－温度敏感性与石英晶体各向异性参数之间的关系。从中得出石英晶体的高温度敏感和低温度第三物切型。实验验证取得了较好的效果。     

Lethal effect of electric fields on isolated ventricular myocytes

Defibrillator-type shocks may cause electric and contractile dysfunction. In this study, we determined the relationship between probability of lethal injury and electric field intensity ($E$ ) in isolated rat ventricular myocytes, with emphasis on field orientation and stimulus waveform. This relationship was sigmoidal with irreversible injury for $E ≫ hbox{50 V/cm}$ . During both threshold and lethal stimulation, cells were twofold more sensitive to the field when it was applied longitudinally (versus transversally) to the cell major axis. For a given $E$, the estimated maximum variation of transmembrane potential ( $Delta V_{max }$) was greater for longitudinal stimuli, which might account for the greater sensitivity to the field. Cell death, however, occurred at lower maximum $Delta V_{max }$ values for transversal shocks. This might be explained by a less steep spatial decay of transmembrane potential predicted for transversal stimulation, which would possibly result in occurrence of electroporation in a larger membrane area. For the same stimulus duration, cells were less sensitive to field-induced injury when shocks were biphasic (versus monophasic). Ours results indicate that, although significant myocyte death may occur in the $E$ range expected during clinical defibrillation, biphasic shocks are less likely to produce irreversible cell injury.      

Three-dimensional finite element solution for biopotentials:erythrocyte in an applied field

The use of the finite-element method in the analysis of bioelectric phenomena is demonstrated. The problem studied is three-dimensional steady current flow around an erythrocyte in an extracellular medium. The finite-element equations for the electrical field problem are derived and mesh generation and the use of a heat-conduction code for analysis are described. Spherical cell geometry, allowing an analytical solution, is also modeled to guide in mesh creation and error estimation for the case of erythrocyte geometry. The results are shown as contour plots of potential on the erythrocyte surface. The maximum transmembrane potential calculated for the erythrocyte is 22% lower than that of the sphere, a significant finding since spherical geometry is often used in studies involving the effect of applied electrical fields on cells