A solution method for the determination of cardiac potential distributions with an alternating current source

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using 'close' and 'wide' electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space. The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required. The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

A new approach to the determination of cardiac potential distributions: application to the analysis of electrode configurations

This paper presents a mathematical model and new solution technique for studying the electric potential in a slab of cardiac tissue. The model is based on the bidomain representation of cardiac tissue and also allows for the effects of fibre rotation between the epicardium and the endocardium. A detailed solution method, based on Fourier Series and a simple one-dimensional finite difference scheme, for the governing equations for electric potential in the tissue and the blood, is also presented. This method has the advantage that the potential can be calculated only at points where it is required, such as the measuring electrodes. The model is then used to study various electrode configurations which have been proposed to determine cardiac tissue conductivity parameters. Three electrode configurations are analysed in terms of electrode spacing, placement position and the effect of including fibre rotation: the usual surface four-electrode configuration; a single vertical analogue of this and a two probe configuration, which has the current electrodes on one probe and the measuring electrodes on the other, a fixed distance away. It is found that including fibre rotation has no effect on the potentials measured in the first two cases; however, in the two probe case, non-zero fibre rotation causes a significant drop in the voltage measured. This leads to the conclusion that it is necessary to include the effects of fibre rotation in any model which involves the use of multiple plunge electrodes.     

Mathematical modelling of intra- and extracellular potentials generated by active structures with short regions of increased diameter

The Hodgkin-Huxley (1952) model was used to calculate intracellular potentials by the method of Joyner et al. (1978). Extracellular potentials were estimated on the basis of a mathematical model proposed by us. It has shown that, irrespective of practical isopotentiality of the membrane of a local inhomogeneity, the latter affects extracellular potentials in two ways: 1) through changes in the potential profile in the region of the structure before the inhomogeneity; 2) through its own potential profile. The first effect is considerably greater than the second one, but the second is greater than the effect of the equal portion of the thin fibre. Increase in the diameter or length of an inhomogeneity is combined with such changes in the potential profile, that the effect of the inhomogeneity on the extracellular potential amplitude is practically independent of its actual size. The extracellular potential waveform substantially depends on the ratio of the diameters of the two parts of the structure and on the position of the inhomogeneity in relation to the sealed structure end. Registration of the positive-negative potentials having a large positive phase should not be considered as an indication of passive properties of the structure.     

A distributed-parameter model of the myelinated human motor nerve fibre: temporal and spatial distributions of action potentials and ionic currents

 A double cable model of the myelinated human motor nerve fibre is presented. The model is based on the nodal and internodal channels in a previous, two-component model of human motor axons (Bostock et al. 1991), added to a complex extended cable structure of nodal, paranodal and internodal segments. The model assumes a high-resistance myelin sheath and a leakage pathway to the internodal axolemma via the paranodal seal resistance and periaxonal space. The parameter values of the model were adjusted to match the recordings of threshold electrotonus in human motor fibres from Bostock et al. (1991). Kirchoff ’s current law was used to derive a system of partial differential equations for the electrical equivalent circuit, and numerical integration was performed with a fixed time increment and non-uniform spatial step sizes, in accordance with the complex structure of the fibre. The model calculations provide estimates of the spatial and temporal distributions of action potentials and their transaxonal and transmyelin components, both in different segments of the fibre and at different moments during action potential propagation. The distribution of transaxonal and transmyelin currents along the fibre and their contributions from different ionic channels are also explored. Received: 14 July 1994/Accepted in revised form: 4 April 1995     

Measurement of grouped intracellular solute osmotic virial coefficients

Models of cellular osmotic behaviour depend on thermodynamic solution theories to calculate chemical potentials in the solutions inside and outside the cell. These solutions are generally thermodynamically non-ideal under cryobiological conditions. The molality-based Elliott et al. form of the multi-solute osmotic virial equation is a solution theory which has been demonstrated to provide accurate predictions in cryobiological solutions, accounting for the non-ideality of these solutions using solute-specific thermodynamic parameters called osmotic virial coefficients. However, this solution theory requires as inputs the exact concentration of every solute in the solution being modeled, which poses a problem for the cytoplasm, where such detailed information is rarely available. This problem can be overcome by using a grouped solute approach for modeling the cytoplasm, where all the non-permeating intracellular solutes are treated as a single non-permeating “grouped” intracellular solute. We have recently shown (Zielinski et al., J Physical Chemistry B, 2017) that such a grouped solute approach is theoretically valid when used with the Elliott et al. model, and Ross-Rodriguez et al. (Biopreservation and Biobanking, 2012) have previously developed a method for measuring the cell type-specific osmotic virial coefficients of the grouped intracellular solute. However, the Ross-Rodriguez et al. method suffers from a lack of precision, which—as we demonstrate in this work—can severely impact the accuracy of osmotic model predictions under certain conditions. Thus, we herein develop a novel method for measuring grouped intracellular solute osmotic virial coefficients which yields more precise values than the existing method and then apply this new method to measure these coefficients for human umbilical vein endothelial cells.     

CRH-induced electrical activity and calcium signalling in pituitary corticotrophs

Pituitary corticotroph cells generate repetitive action potentials and associated Ca2+ transients in response to the agonist corticotropin releasing hormone (CRH). There is indirect evidence suggesting that the agonist, by way of complex intracellular mechanisms, modulates the voltage sensitivity of the L-type Ca2+ channels embedded in the plasma membrane. We have previously constructed a Hodgkin-Huxley-type model of this process, which indicated that an increase in the L-type Ca2+ current is sufficient to generate repetitive action potentials (LeBeau et al. (1997). Biophys. J.73, 1263-1275). CRH is also believed to inhibit an inwardly rectifying K+ current. In this paper, we have found that a CRH-induced inhibition of the inwardly rectifying K+ current increases the model action potential firing frequency, [Ca2+]i transients and membrane excitability. This dual modulatory action of CRH on inward rectifier and voltage-gated Ca2+ channels better describes the observed CRH-induced effects. This structural alteration to the model along with parameter changes bring the model firing frequency in line with experimental data. We also show that the model exhibits experimentally observed bursting behaviour, where the depolarization spike is followed by small oscillations in the membrane potential.     

Active electric properties of cardiac muscle

A model of electrical activity in the heart has been developed that treats the intracellular domain and the extracellular domain as electrical syncytia with anisotropic resistivities (bi-syncytial model). At the microscopic level, propagation is assumed to proceed primarily along the axes of individual cells. Considerations at the macroscopic level relate the transmembrane current to the intracellular and extracellular resistivity and the transmembrane potential. The result is a relationship between instantaneous extracellular potentials and cardiac action potentials.     

An Example of Information Retention in Rabbit Ventricular Muscle Fibres

The preparation was stimulated externally and transmembrane action potentials were recorded with intracellular microelectrodes. The relationship between the area of the first action potential after a pause in stimulation and the duration of the pause was examined. It was found that the area retained its dependence on the pattern of stimulation prior to the pause. These experiments confirm one of the predictions of a mathematical model (Gibbs et al., 1963) which describes the relationship between the area of action potentials and the pattern of stimulation.     

Electrical properties of skeletal muscle fibres of the flour moth larva Ephestia kuehniella

The electrical properties of the ventral longitudinal muscle fibres in the flour moth larva Ephestia kuehniella were investigated at rest and during electrical activity. The membrane resting potential was only partially dependent on the K-concentration gradient across the muscle membrane. The electrical constants λ, τ, R_m, R_i, and C_m were determined according to the equations for ‘short cables’ (Table 1). Current-voltage relationships of the muscle membrane were measured: they revealed anomalous as well as delayed rectification of the membrane. Stimulation of the muscle fibres with intracellular current pulses elicited graded action potentials in most fibres; in some fibres ‘all-or-none’ action potentials were generated. In contrast to graded action potentials these ‘all-or-none’ action potentials were propagated without decrement along the muscle fibre. Indirect stimulation of the muscle fibres resulted in large excitatory junction potentials which generally gave rise to action potentials.     

The retrograde frequency response of passive dendritic trees constrains the nonlinear firing behaviour of a reduced neuron model

Our goal was to investigate how the propagation of alternating signals (i.e. AC), like action potentials, into the dendrites influenced nonlinear firing behaviour of motor neurons using a systematically reduced neuron model. A recently developed reduced modeling approach using only steady-current (i.e. DC) signaling was analytically expanded to retain features of the frequency-response analysis carried out in multicompartment anatomically reconstructed models. Bifurcation analysis of the extended model showed that the typically overlooked parameter of AC amplitude attenuation was positively correlated with the current threshold for the activation of a plateau potential in the dendrite. Within the multiparameter space map of the reduced model the region demonstrating "fully-bistable" firing was bounded by directional DC attenuation values that were negatively correlated to AC attenuation. Based on these results we conclude that analytically derived reduced models of dendritic trees should be fit on DC and AC signaling, as both are important biophysical parameters governing the nonlinear firing behaviour of motor neurons.     

Interstitial potentials and their change with depth into cardiac tissue.

The electrical source strength for an isolated, active, excitable fiber can be taken to be its transmembrane current as an excellent approximation. The transmembrane current can be determined from intracellular potentials only. But for multicellular preparations, particularly cardiac ventricular muscle, the electrical source strength may be changed significantly by the presence of the interstitial potential field. This report examines the size of the interstitial potential field as a function of depth into a semi-infinite tissue structure of cardiac muscle regarded as syncytial. A uniform propagating plane wave of excitation is assumed and the interstitial potential field is found based on consideration of the medium as a continuum (bidomain model). As a whole, the results are inconsistent with any of the limiting cases normally used to represent the volume conductor, and suggest that in only the thinnest of tissue (less than 200 micron) can the interstitial potentials be ignored.     

The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.     

Current flow and potential in a three-dimensional syncytium

Analytical solutions are obtained describing the spread of potential in a continuous syncytium which is the three-dimensional analogue of the “cable” and “thin sheet” models. Two discrete (finite-element) models are also developed and their behaviour shown to agree with that of the continuous model. All three models are able to account for the peculiarities of passive responses to intracellular current injection in tissues such as cardiac and smooth muscle. The time course and spread of junction potentials in electrically coupled tissues is discussed.     

Estimating and eliminating junctional current in coupled cell populations by leak subtraction. A computational study

The quantitative characterization of ion channel properties in pancreatic β-cells under typical patch clamp conditions can be questioned because of the unreconciled differences in experimental conditions and observed behavior between microelectrode recordings of membrane potential in intact islets of Langerhans and patch recordings of single cells. Complex bursting is reliably observed in islets but not in isolated cells under patch clamp conditions. E. Rojas et al. (J. Membrane Biol. 143:65–77, 1995) have attempted to circumvent these incompatibilities by measuring currents in β-cells in intact islets by voltage-clamping with intracellular microelectrodes (150–250 MΩ tip resistance). The major potential pitfall is that β-cells within the islet are electrically coupled, and contaminating coupling currents must be subtracted from current measurements, just as linear leak currents are typically subtracted. To characterize the conditions under which such coupling current subtraction is valid, we have conducted a computational study of a model islet. Assuming that the impaled cell is well clamped, we calculate the native and coupling components of the observed current. Our simulations illustrate that coupling can be reliably subtracted when neighbor cells' potentials are constant or vary only slowly (e.g., during their silent phases) but not when they vary rapidly (e.g., during their active phases). We also show how to estimate coupling conductances in the intact islet from measurements of coupling currents.     

Simulation of ‘stationary’ SAP and SEP phenomena by 2-dimensional potential field modelling

In order to model the distribution of potentials in the hand due to antidromic SAP propagation and in the body due to afferent conduction of the median nerve volley, 2-dimensional matrices of the appropriate shape were constructed, each containing a ‘generator’ consisting of up to 3 ‘source’ and 3 ‘sink’ points. The value of the field potential at other sites was calculated using a finite difference method.It was shown that the potential gradient is virtually zero in matrix zones which are separated from the region containing the generator by a constriction in the boundary of the conductor. Points on the far side of the constriction remain virtually equipotential, at a level determined by the potential at the junction. This is naturally influened by the proximity of the generator, so that as the generator approaches the constriction a potential difference will develop between points on the far side, irrespective of their distance from the junction, and other remote parts of the matrix. In the context of human SAPs and SEPs, such factors may be of paramount importance in the generation of so-called ‘stationary’ or ‘far-field’ potentials.With additional postulates concerning the manner in which the SAP is attenuated by the termination of axons as it propagates through the hand, and the course taken by the median nerve volley between the arm and the neck, it was possible to model the majority of stationary SAP phenomena described by Kimura et al. (1984), and also the distribution and latency of the P9 SEP component following median nerve stimulation.     

A cylindrical model for studying subendocardial ischaemia in the left ventricle

In this paper a mathematical model of a left ventricle with a cylindrical geometry is presented with the aim of gaining a better understanding of the relationship between subendocardial ischaemia and ST depression. The model is formulated as an infinite cylinder and takes into account the full bidomain nature of cardiac tissue, as well as fibre rotation. A detailed solution method (based on Fourier series, Fourier transforms and a one dimensional finite difference scheme) for the governing equations for electric potential in the tissue and the blood is also presented. The model presented is used to study the effect increasing subendocardial ischaemia has on the epicardial potential distribution as well as the effects of changing the bidomain conductivity values. The epicardial potential distributions obtained with this cylindrical geometry are compared with results obtained using a previously published slab model. Results of the simulations presented show that the morphologies of the epicardial potential distributions are similar between the two geometries, with the main difference being that the cylindrical model predicts slightly higher potentials.     

The effect of boundary conditions on epicardial potential distributions

This study presents a comparison of semi-analytical and numerical solution techniques for solving the passive bidomain equation in simple tissue geometries containing a region of subendocardial ischaemia. When the semi-analytical solution is based on Fourier transforms, recovering the solution from the frequency domain via fast Fourier transforms imposes a periodic boundary condition on the solution of the partial differential equation. On the other hand, the numerical solution uses an insulation boundary condition. When these techniques are applied to calculate the epicardial surface potentials, both yield a three well potential distribution which is identical if fibre rotation within the tissue is ignored. However, when fibre rotation is included, the resulting three-well distribution rotates, but through different angles, depending on the solution method. A quantitative comparison between the semi-analytical and numerical solutiontechniques is presented in terms of the effect fibre rotation has on the rotation of the epicardial potential distribution. It turns out that the Fourier transform approach predicts a larger rotation of the epicardial potential distribution than the numerical solution. The conclusion from this study is that it is not always possible to use analytical or semi-analytical solutions to check the accuracy of numerical solution procedures. For the problem considered here, this checking is only possible when it is assumed that there is no fibre rotation through the tissue.     

Remodelling of collagen fibre transition stretch and angular distribution in soft biological tissues and cell-seeded hydrogels

The extracellular matrix in many biological tissues is adapted to its mechanical environment. In this study, a phenomenological model for collagen remodelling is introduced that incorporates angular remodelling (fibre reorientation) and the adaptation of the so-called transition stretch. This is achieved by introducing a local stress-free configuration for the collagen network by a multiplicative decomposition of the deformation gradient and the appropriate definition of the anisotropic free Helmholtz energy potentials and structure tensors. The collagen network is either treated using discrete fibre directions or a continuous angular distribution. The first part of the study illustrates the influence of force- and displacement-controlled loading on either stress- or deformation-driven remodelling processes in tissues with various degrees of fibre reinforcement. The model is then applied to recent experimental studies of collagen remodelling, specifically periosteum adaptation (Foolen et?al. in J Biomech 43(16):3168–3176, 2010), collagen gel (Thomopoulos et?al. in J Biomech Eng 127(5):742–750, 2005) and fibrin cruciform (Sander et?al. in Ann Biomed Eng 1–16, 2010) compaction. The model is able to capture the basic effects of an adapting transition stretch over time in the periosteal simulations, as well as the compaction and the development of structural anisotropy in the collagen and fibrin gels. The model can potentially be applied to elucidate structure–function relationships, better interpret in vitro experiments involving collagen remodelling, and help investigate aspects of certain pathologies, such as connective tissue contracture.     

Rate-dependent effects of lidocaine on cardiac dynamics: Development and analysis of a low-dimensional drug-channel interaction model

State-dependent sodium channel blockers are often prescribed to treat cardiac arrhythmias, but many sodium channel blockers are known to have pro-arrhythmic side effects. While the anti and proarrhythmic potential of a sodium channel blocker is thought to depend on the characteristics of its rate-dependent block, the mechanisms linking these two attributes are unclear. Furthermore, how specific properties of rate-dependent block arise from the binding kinetics of a particular drug is poorly understood. Here, we examine the rate-dependent effects of the sodium channel blocker lidocaine by constructing and analyzing a novel drug-channel interaction model. First, we identify the predominant mode of lidocaine binding in a 24 variable Markov model for lidocaine-sodium channel interaction by Moreno et al. Specifically, we find that (1) the vast majority of lidocaine bound to sodium channels is in the neutral form, i.e., the binding of charged lidocaine to sodium channels is negligible, and (2) neutral lidocaine binds almost exclusively to inactivated channels and, upon binding, immobilizes channels in the inactivated state. We then develop a novel 3-variable lidocaine-sodium channel interaction model that incorporates only the predominant mode of drug binding. Our low-dimensional model replicates an extensive amount of the voltage-clamp data used to parameterize the Moreno et al. model. Furthermore, the effects of lidocaine on action potential upstroke velocity and conduction velocity in our model are similar to those predicted by the Moreno et al. model. By exploiting the low-dimensionality of our model, we derive an algebraic expression for level of rate-dependent block as a function of pacing frequency, restitution properties, diastolic and plateau potentials, and drug binding rate constants. Our model predicts that the level of rate-dependent block is sensitive to alterations in restitution properties and increases in diastolic potential, but it is insensitive to variations in the shape of the action potential waveform and lidocaine binding rates.