Oblique dipole layer potentials applied to electrocardiology

We study the properties of the potential field generated by an oblique dipole layer. This field arises, for instance, in describing the potential elicited by a depolarization wavefront spreading in the myocardium when a dependence of the potential on the cardiac fiber orientation is introduced. The representation of cardiac bioelectric sources by means of an oblique dipole layer leads to a mathematical structure which generalizes the classical solid angle theory used in electrocardiology, which has been challenged by recent experimental evidence, and links models previously proposed with a view to adequately reproduce the potential observed in experiments. We investigate also the relationship between our model and an intracellular current model and we derive potential jump formulae for some models which account for the anisotropic structure of the myocardium. The potential generated by an oblique dipole layer is considered both for unbounded and bounded domains. In the latter case an integral boundary equation is derived and we study its solvability. A numerical procedure for solving this integral equation by means of the finite element method with collocation is outlined.     

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations

In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.     

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.     

Unequivocal solution of the one-dimensional inverse problem of electroencephalography for point sources

A point sources system placed in a homogeneous volume conductor with a plane boundary is proved to be uniquely determined by its surface potential field. For the nonlocal sources the inverse problem solution is nonunique.     

Volume Conductor Fields of Action Currents

The formulation of Lorente de Nó for the potential distribution in a volume conductor due to a contained bioelectric source is reviewed. The aforementioned formula is based on data taken from the bioelectric source in air (excised). This work is extended to the derivation of a formula which depends upon quantities available from in situ measurements. The latter formula appears to simplify to the form

[Formula: see text]

where J_n is the normal current density flow across the surface S_o that bounds the bioelectric source. The simplification is considered in some detail for a hypothetical action potential on a circular cylindrical axon, and is shown to be adequately substantiated for this case.

     

Synthesis of voltage-sensitive optical signals: application to panoramic optical mapping

Fluorescent photon scattering is known to distort optical recordings of cardiac transmembrane potentials; however, this process is not well quantified, hampering interpretation of experimental data. This study presents a novel model, which accurately synthesizes fluorescent recordings over the irregular geometry of the rabbit ventricles. Using the model, the study aims to provide quantification of fluorescent signal distortion for different optical characteristics of the preparation and of the surrounding medium. A bi-domain representation of electrical activity is combined with finite element solutions to the photon diffusion equation simulating both the excitation and emission processes, along with physically realistic boundary conditions at the epicardium, which allow simulation of different experimental setups. We demonstrate that distortion in the optical signal as a result of fluorescent photon scattering is truly a three-dimensional phenomenon and depends critically upon the geometry of the preparation, the scattering properties of the tissue, the direction of wavefront propagation, and the specifics of the experimental setup. Importantly, we show that in an anatomically accurate model of ventricular geometry and fiber orientation, the morphology of the optical signal does not provide reliable information regarding the intramural direction of wavefront propagation. These findings underscore the potential of the new model in interpreting experimental data.     

Coupled map lattice simulation of epileptogenesis in hippocampal slices

Bioelectric activity of a nervous tissue and its synchronization with formating epileptiform bursts are simulated by a coupled map lattice. The functional units of the map located in the lattice sites represent neural masses which consist of current sources and sinks. The sources lead to depolarization of neurons, and sinks provide hyperpolarization. The map describes a single variable – the bioelectric potential. This potential is created by the interplay of all current sources and sinks in the neural masses. The neural masses are diffusively coupled with each other both by electrotonic influence and synaptic coupling. Both mechanisms mentioned are suggested to be essential for the formation of synchronous bursts. The transition from chaotic activity to bursts was studied. Received: 30 September 1997 / Accepted in revised from: 11 March 1998     

Bioelectric potential changes in the style of Lilium longiflorum Thunb. after self- and cross-pollination of the stigma

Two different types of bioelectric potential changes have been registered in the style of Lilium longiflorum cv. Arai 5 after different treatments of the stigma. Self-pollination induces a bioelectric potential change different from the potential change induced by cross-pollination. Removal of the stigma or apllication of killed compatible pollen induced a bioelectrical response similar to the potential change recorded after cross-pollination. No bioelectric potential change was recorded after application of pollen of Petunia hybrida, pollen of Haemanthus katherinae, killed self-pollen, or no treatment at all.The mean generation time of the bioelectric potential change was 338 min after the treatment. The translocation velocity varied between 1.2 and 5.4 cm h^-1. Both responses are similar to the potential changes registered in Mimosa pudica after different types of stimulation and to the potential changes generated by the phytochrome in Avena coleoptiles. The translocation of the signals is discussed in relation to the models on the translocation in Mimosa. The relation between the bioelectric potential changes and the incompatibility reaction in Lilium longiflorum is discussed.     

A Mathematical-Physical Model of the Genesis of the Electrocardiogram

A fundamental problem of cardiac electrophysiology is that of relating quantitatively the electrical activity within the heart to the complete timevarying potential distribution at the body surface. A new numerical method is described for the calculation of the surface potential on an irregularly shaped closed external surface due to an arbitrary source distribution in a medium containing regions of different conductivity, subject to the appropriate boundary conditions. The method is intended to provide an exact theoretical analysis of the experimental data acquired by A. M. Scher and others who have been mapping the pathways of ventricular depolarization in dogs and other animals. In anticipation of the above research program, a number of exploratory computations are reported. For example, the surface potential distribution has been calculated for a cylinder of human torso cross-section with a hemispherical dipole layer current source in approximate heart position and orientation and containing “lungs” of conductivity different from that of the surrounding medium. Under certain conditions, when lung-like inhomogeneities are introduced, a simple dipole source can generate a potential distribution having the multiple maxima and minima characteristic of higher multipole sources.     

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes.

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.     

A reference curve for axial bioelectric potentials in rabbit tibia

A high coefficient of variation is characteristic of the bioelectric potentials recorded from living long bones. As a consequence, the data collected from animal and human experiments are difficult to use in practical ways. A distribution curve for bone bioelectric potentials has been calculated using polynomial regression analysis to process the voltage values recorded on the whole length of rabbit tibiae, with reference electrodes positioned in three different points of the bone. The distribution curve so obtained is presented to fulfill the need for a reference curve for the bioelectric potentials recorded from rabbit tibial surfaces.     

食蚊鱼的生物电场特征

文章采用活体记录的方法测量了食蚊鱼(Gambusia affinis)的生物电场。实验分单尾鱼、两尾鱼同向和两尾鱼反向三组测量, 每组10 个重复。结果表明:单尾鱼的生物电场表现为头负、尾正的偶极子直流电场,头部相对电势为(242.4) V, 尾部为(211.6) V, 且头部附近产生1-3 Hz 与呼吸频率对应的交流呼吸电场, 大小为(4.20.8) V。两尾鱼生物电场测量表明, 其直流电场均大于单尾鱼(P0.05); 两尾鱼同向靠近时产生的交流呼吸电场显著大于单尾鱼(P0.01), 而反向靠近时产生的呼吸电场显著小于单尾鱼(P0.001)。这表明两条鱼不同方向靠近时, 可通过呼吸作用改变交流呼吸电场的大小。此种现象对于依靠感知交流呼吸电场来摄食的被动电感受鱼类是不利的。       

The bioelectric field of the catfish Ictalurus nebulosus.

The variability of the bioelectric field of the electrosensitive catfish, Ictalurus nebulosus, was investigated by recording the potential variation occurring when the fish passed a stationary electrode, and by recording the field of a stationary fish by a 15-electrode array. A good first order approximation of the recorded field of a 20 cm long fish is a dipole dc source with the source and sink about 7 cm apart, carrying a current of about 1 microA in water with a specific resistivity of 3.3 kohm cm. At 5 cm distance from the dipole axis such a source generates an electric potential swing in the order of 50 microV in free space, head negative, tail positive. Superimposed on the basic component are respiration related fluctuations, and fluctuations related to the activity of the alimentary canal, gills, and skin. Novel stimuli, or stressors like investigators approaching the aquarium, evoke sudden increases in field strength. which last about 15 min. Demineralization of the aquarium water causes changes in field strength and reversal of field polarity. The administration of food causes field variations in the vicinity of the anal opening. The bioelectric field shows diurnal fluctuations of 100 microV. The peak is at about 04:00, the dip at 14:00. The fluctuations of the bioelectric field are sufficiently strong and specilic to serve as electrical stimuli to other electrosensitive catfish. It is suggested that the field changes allow a simple form of electrocommunication. i.e. inform conspecifics about some physiological properties of the field source. The cellular mechanisms underlying the fluctuations of the bioelectric field are homeostatic processes mediated by ion pumps and ion channels.     

Non-Uniform Dispersion of the Source-Sink Relationship Alters Wavefront Curvature

The distribution of cellular source-sink relationships plays an important role in cardiac propagation. It can lead to conduction slowing and block as well as wave fractionation. It is of great interest to unravel the mechanisms underlying evolution in wavefront geometry. Our goal is to investigate the role of the source-sink relationship on wavefront geometry using computer simulations. We analyzed the role of variability in the microscopic source-sink relationship in driving changes in wavefront geometry. The electrophysiological activity of a homogeneous isotropic tissue was simulated using the ten Tusscher and Panfilov 2006 action potential model and the source-sink relationship was characterized using an improved version of the Romero et al. safety factor formulation (SF_m2). Our simulations reveal that non-uniform dispersion of the cellular source-sink relationship (dispersion along the wavefront) leads to alterations in curvature. To better understand the role of the source-sink relationship in the process of wave formation, the electrophysiological activity at the initiation of excitation waves in a 1D strand was examined and the source-sink relationship was characterized using the two recently updated safety factor formulations: the SF_m2 and the Boyle-Vigmond (SF_VB) definitions. The electrophysiological activity at the initiation of excitation waves was intimately related to the SF_m2 profiles, while the SF_VB led to several counterintuitive observations. Importantly, with the SF_m2 characterization, a critical source-sink relationship for initiation of excitation waves was identified, which was independent of the size of the electrode of excitation, membrane excitability, or tissue conductivity. In conclusion, our work suggests that non-uniform dispersion of the source-sink relationship alters wavefront curvature and a critical source-sink relationship profile separates wave expansion from collapse. Our study reinforces the idea that the safety factor represents a powerful tool to study the mechanisms of cardiac propagation in silico, providing a better understanding of cardiac arrhythmias and their therapy.     

Active muscle fiber as an equivalent cardiac current generator

Equations are derived describing potentials due to an active muscle fiber in an infinite medium in terms of two surface integrals—one of the propagated action potential and the other of the membrane current density, both integrals being taken over the surface of the muscle. These equations are incorporated into an equivalent cardiac current generator in which the left ventricle (i.e. the current source) is represented by a three-dimensional wedge and the thorax (i.e. the volume conductor), by a homogeneous circular cylinder. Since this current generator expresses the body surface potentials in terms of the membrane current density and the membrane potential at any point on the surface of the electrically active muscle fiber, the calculated ECG can be correlated with theactual sources within the heart. This equivalent cardiac generator possesses many of the physical and physiological properties of cardiac muscle. The equations were evaluated numerically on a digital computer. The results indicate that equivalent cardiac current generators of this type can yield clinically significant results and that further research is necessary to investigate their properties fully.     

High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue

We magnetically imaged the magnetic action field and optically imaged the transmembrane potentials generated by planar wavefronts on the surface of the left ventricular wall of Langendorff-perfused isolated rabbit hearts. The magnetic action field images were used to produce a time series of two-dimensional action current maps. Overlaying epifluorescent images allowed us to identify a net current along the wavefront and perpendicular to gradients in the transmembrane potential. This is in contrast to a traditional uniform double-layer model where the net current flows along the gradient in the transmembrane potential. Our findings are supported by numerical simulations that treat cardiac tissue as a bidomain with unequal anisotropies in the intra- and extracellular spaces. Our measurements reveal the anisotropic bidomain nature of cardiac tissue during plane wave propagation. These bidomain effects play an important role in the generation of the whole-heart magnetocardiogram and cannot be ignored.     

The magnetic field of a single axon. A comparison of theory and experiment.

The magnetic field and the transmembrane action potential of a single nerve axon were measured simultaneously. The volume conductor model was used to calculate the magnetic field from the measured action potential, allowing comparison of the model predictions with the experimental data. After analyzing the experiment for all systematic errors, we conclude that the shape of the magnetic field can be accurately predicted from the transmembrane potential and, more importantly, the shape of the transmembrane potential can be calculated from the magnetic field. The data are used to determine ri, the internal resistance per unit length of the axon, to be 19.3 +/- 1.9 k omega mm-1, implying a value for the internal conductivity of 1.44 +/- 0.33 omega -1 m-1. Magnetic measurements are compared with standard bioelectric techniques for studying nerve axons.