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.     

Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.     

Feasibility of cardiac microimpedance measurement using multisite interstitial stimulation

This study was designed to test the hypothesis that analyses of central interstitial potential differences recorded during multisite stimulation with a set of interstitial electrodes provide sufficient data for accurate measurement of cardiac microimpedances. On theoretical grounds, interstitial current injected and removed using electrodes in close proximity does not cross the membrane, whereas equilibration of intracellular and interstitial potentials occurs distant from electrodes widely separated. Multisite interstitial stimulation should therefore give rise to interstitial potential differences recorded centrally that depend on intracellular and interstitial microimpedances, allowing independent measurement. Simulations of multisite stimulation with fine (25 microm) and wide (400 microm) spacing in one-dimensional models that included Luo-Rudy dynamic membrane equations were performed. Constant interstitial and intracellular microimpedances were prescribed for initial analyses. Discrete myoplasmic and gap-junctional components were prescribed intracellularly in later simulations. With constant microimpedances, multisite stimulation using 29 total electrode combinations allowed interstitial and intracellular microimpedance measurements at errors of 0.30% and 0.34%, respectively, with errors of 0.05% and 0.40% achieved using 6 combinations and 10 total electrodes. With discrete myoplasmic and junctional components, comparable accuracy was maintained following adjustments to the junctions to reflect uncoupling. This allowed uncoupling to be quantified as relative increases in total junctional resistance. Our findings suggest development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.     

Quantal currents and potential in the three-dimensional anisotropic bidomain model of smooth muscle

The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.     

Theory for the formation of intercellular junctions based on intramembranous particle patterns observed in the freeze fracture technique

A heterogeneous continuum theory of biological membrane interactions is presented which takes account of the spatial inhomogeneity of intramembranous particle patterns observed in ultrastructural studies of junctional complexes using the freeze-cleaved technique. The theory attempts to explain (i) how electrostatic and electrodynamic forces between particles of different biochemical composition in the same and opposing membranes might give rise to the specialized particle configurations characteristic of tight and gap junctions, (ii) how the spatial non-uniformity of the membrane proteins quantitatively modifies the local long range molecular level force field between adjacent membrane bilayers and (iii) how membrane elastic stresses and the modified molecular level forces combine to determine the equilibrium configurations of the various junctional complexes. The mathematical problem is highly non-linear since the molecular forces are a rapidly varying function of the local membrane spacing which is a priori unknown. The simplified dimensionless boundary value problem to determine the junction geometry has been reduced to a fourth order quasi one-dimensional equation with split end point conditions. The equation is singular in the vicinity of the junction complex and special solution techniques had to be employed. The numerical results are in reasonable quantitative agreement with electron microscopic observations of tight, gap and venous junctions.     

Towards a computational method for imaging the extracellular potassium concentration during regional ischemia

We investigate the possibility of using body surface potential maps to image the extracellular potassium concentration during regional ischemia. The problem is formulated as an inverse problem based on a linear approximation of the bidomain model, where we minimize the difference between the results of the model and observations of body surface potentials. The minimization problem is solved by a one-shot technique, where the original PDE system, an adjoint problem, and the relation describing the minimum, are solved simultaneously. This formulation of the problem requires the solution of a 5 × 5 system of linear partial differential equations. The performance of the model is investigated by performing tests based on synthetic data. We find that the model will in many cases detect the correct position and approximate size of the ischemic regions, while some cases are more difficult to locate. It is observed that a simple post-processing of the results produces images that are qualitatively very similar to the true solution.     

Effect of axially coordinated pi-acid ligands on photovoltaic properties of Zn-tetraphenylporphyrin

Following our earlier observations that the well known doping effect of oxygen and water on electrical properties of porphyrin and phthalocyanine films may be attributed to a pi-acid axial interaction throughout the film in the case of PdTPP, we have compared Zn-TPP films supported on transparent n-doped SnO₂ electrodes which had been treated with several pi-acids in contact with an electrolyte to give photoelectrochemical cells. Photovoltages obtained in contact with a series of solution couples were used to obtain approximate photo flat band potentials. The doped films were examined by magnetic circular dichroism (MCD) spectroscopy so that the electronic effect of the dopant could be diagnosed. It was found that pi-acid dopants cause shifts to low energy in the band which indicates “hole stabilization” in the order pyridine < CO < triphenylarsine. The potentials of zero photopotential ‘E_FB’, correlate approximately with spectral shifts. It is concluded that manipulation of axial ligand dopants is a promising method for design of metal porphyrin and perhaps phthalocyanine films with desired photovoltaic properties.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

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.     

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.     

Periodic solutions: a robust numerical method for an S-I-R model of epidemics

 We describe and analyze a numerical method for an S-I-R type epidemic model. We prove that it is unconditionally convergent and that solutions it produces share many qualitative and quantitative properties of the solution of the differential problem being approximated. Finally, we establish explicit sufficient conditions for the unique endemic steady state of the system to be unstable and we use our numerical algorithm to approximate the solution in such cases and discover that it can be periodic, just as suggested by the instability of the endemic steady state. Received: 1 September 1995 / Revised version: 30 April 1997     

Microdomain Effects on Transverse Cardiac Propagation

The effect of gap junctional coupling, sodium ion channel distribution, and extracellular conductivity on transverse conduction in cardiac tissue is explored using a microdomain model that incorporates aspects of the inhomogeneous cellular structure. The propagation velocities found in our model are compared to those in the classic bidomain model and indicate a strong ephaptic microdomain contribution to conduction depending on the parameter regime. We show that ephaptic effects can be quite significant in the junctional spaces between cells, and that the cell activation sequence is modified substantially by these effects. Further, we find that transverse propagation can be maintained by ephaptic effects, even in the absence of gap junctional coupling. The mechanism by which this occurs is found to be cablelike in that the junctional regions act like inverted cables. Our results provide insight into several recent experimental studies that indirectly indicate a mode of action potential propagation that does not rely exclusively on gap junctions.     

Microdomain Effects on Transverse Cardiac Propagation

The effect of gap junctional coupling, sodium ion channel distribution, and extracellular conductivity on transverse conduction in cardiac tissue is explored using a microdomain model that incorporates aspects of the inhomogeneous cellular structure. The propagation velocities found in our model are compared to those in the classic bidomain model and indicate a strong ephaptic microdomain contribution to conduction depending on the parameter regime. We show that ephaptic effects can be quite significant in the junctional spaces between cells, and that the cell activation sequence is modified substantially by these effects. Further, we find that transverse propagation can be maintained by ephaptic effects, even in the absence of gap junctional coupling. The mechanism by which this occurs is found to be cablelike in that the junctional regions act like inverted cables. Our results provide insight into several recent experimental studies that indirectly indicate a mode of action potential propagation that does not rely exclusively on gap junctions.     

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 discrete method for the identification of parameters of a deterministic epidemic model

We study a problem of identification of the parameters for a deterministic epidemic model of the Kermack-McKendrick type. Particular emphasis is put on the analysis of the conditions of numerical stability of the method of integration used to calculate the solutions of the system of differential equations which describe the model. The numerical method can be regarded as a discrete model which reproduces the basic qualitative properties of the continuous model, which are positivity of the solutions, points of equilibrium, and the “threshold theorem.” This allows us to identify the parameters with good reliability, by means of an iterative procedure to minimize the functional which is the measure of discrepancy between the data observed and the data obtained from the discrete model. The initial estimate of the parameters is obtained by a direct method applied to the discretized system of equations.     

Ion transfer during electrolyte osmosis through a charged porous membrane

In an earlier investigation (I) concerning the osmotic flow of an electrolyte through a charged porous membrane it was shown that in order to determine the osmotic reflection coefficient for the process a solution of the associated ion transfer equations is required. In I, previously unpublished approximate formulae for the required variables were quoted. The current paper presents the derivation of these solutions. The investigation considers the solution of the one-dimensional form of the coupled Poisson/convection-free Nernst-Planck equations subject to boundary conditions derived in I. Both equations and boundary conditions contain unknown parameters which are evaluated as part of the solution. Exact numerical and approximate analytical solutions are derived for the intrapore electrostatic potential and ion concentrations and for the unknown ion fluxes. Formulae are given for the electric current generated in the process and for the electrolyte factor in the osmotic reflection coefficient.     

Radiation Response of Connexin43-Transfected Cells in Relation to the “Contact Effect”

Some cell lines grown for only two cell doublings as multicell spheroids develop a form of resistance to killing by ionizing radiation that has been called the “contact” effect. While our previous results have implicated a role for higher order chromatin structure in the contact effect, another possible explanation is the presence of intercellular gap junctions that might facilitate communication between cells grown as spheroids and thereby enhance the ability of cells to resist or recover from radiation damage. To examine the role of gap junctions in the contact effect, rat glioma C6 and mouse EMT6 cell lines were transfected with a gene encoding the gap junctional protein connexin43. While C6 glioma cells are deficient in gap junctional communication, cells from spheroids were nonetheless more resistant than monolayers to killing by ionizing radiation, and the contact effect was present to a similar extent in the three transfected clones. For mouse EMT6 cells, radiosensitivity was similar whether cells were grown as monolayers or spheroids. Transfection of EMT6 cells with connexin43 increased gap junctional communication but did not promote development of a contact effect. Tumor volume doubling time in SCID mice increased significantly for one transfected clone; however, doubling timein vitrowas also increased relative to the EMT6 parent. We conclude that extensive gap junctional communication is not a requirement for the increased radiation resistance observed when some cell lines are grown as spheroids.     

Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations

We present numerical solutions for the one-dimensional Nernst-Planck and Poisson system of equations for steady-state electrodiffusion. Commonly used approximate solutions to these equations invoke assumptions of local electroneutrality (Planck approximation) or constant electric field (Goldman approximation). Calculations were performed to test the ranges over which these approximate theories are valid. For a dilutional junction of a 1:1 electrolyte, separated from adjoining perfectly stirred solutions by sharp boundaries, the Planck approximation is valid for values of kappa dL greater than 10, where 1/kappa d is the Debye length of the more dilute solution. The Goldman approximation is valid for kappa cL less than 0.1 where 1/kappa c is the Debye length of the more concentrated solution. These results suggest that the modeling of electrodiffusive flows in and near membrane ion channels may require numerical solutions of this set of equations rather than the use of either limiting case.     

Decoupled time-marching schemes in computational cardiac electrophysiology and ECG numerical simulation

This work considers the approximation of the cardiac bidomain equations, either isolated or coupled with the torso, via first order semi-implicit time-marching schemes involving a fully decoupled computation of the unknown fields (ionic state, transmembrane potential, extracellular and torso potentials). For the isolated bidomain system, we show that the Gauss-Seidel and Jacobi like splittings do not compromise energy stability; they simply alter the energy norm. Within the framework of the numerical simulation of electrocardiograms (ECG), these bidomain splittings are combined with an explicit Robin-Robin treatment of the heart-torso coupling conditions. We show that the resulting schemes allow a fully decoupled (energy) stable computation of the heart and torso fields, under an additional hyperbolic-CFL like condition. The accuracy and convergence rate of the considered schemes are investigated numerically with a series of numerical experiments.     

Smart platform for the analysis of cupula deformation caused by otoconia presence within SCCs

In this paper, we first briefly describe the mechanical model of cupula deformation with the appropriate analytical solution. Then, we present the numerical solution of the same problem and compare it with the analytical one. Besides, we provide another numerical solution based on the Finite Element Method procedure, in an effort to encompass a more realistic approach to the problem such as considering the real geometry of the SCCs and the obstruction of the fluid flow during head movement due to the presence of otoconia. As a result, we obtain fifty solutions for a head rotation angle in a range from 0° to 120°, taking into account that such a manoeuvre lasts exactly 3?seconds. In the end, we present a mobile client–server application for visualising the finite element solutions in a way that is convenient for the clinical practice.     

Protein phosphorylation and hydrogen ions modulate calcium-induced closure of gap junction channels.

The regulation of the cell-to-cell pathway formed by gap junctions seems to involve the interaction of the junctional channels with either calcium or hydrogen ions, as well as protein phosphorylation and calmodulin. These mechanisms of junctional regulation have been considered to act independently on specific sites of the gap junction protein; however, the possibility that they may be interrelated has not been adequately explored mainly due to the difficulties involved in simultaneous measurement of intracellular cations and protein phosphorylation. To further understanding of mechanisms regulating gap junctions, we have internally perfused coupled lateral axons from crayfish with solutions containing different calcium and hydrogen concentrations under conditions favoring phosphorylation, while monitoring the junctional conductance. We found that calcium ions regulate cell communication probably through a direct interaction with the channel protein. Phosphorylation and low pH do not alter junctional conductance themselves, but appear only to modulate the effects of calcium, possibly by altering the affinity of the channel for calcium. We propose that a combination of free intracellular calcium and protein phosphorylation form an important physiological mechanism regulating intercellular communication.