Electrode systems for measuring cardiac impedances using optical transmembrane potential sensors and interstitial electrodes--theoretical design

Approaches for determining cardiac bidomain conductivity values: progress and challenges

Modelling the electrical activity of the heart is an important tool for understanding electrical function in various diseases and conduction disorders. Clearly, for model results to be useful, it is necessary to have accurate inputs for the models, in particular the commonly used bidomain model. However, there are only three sets of four experimentally determined conductivity values for cardiac ventricular tissue and these are inconsistent, were measured around 40 years ago, often produce different results in simulations and do not fully represent the three-dimensional anisotropic nature of cardiac tissue. Despite efforts in the intervening years, difficulties associated with making the measurements and also determining the conductivities from the experimental data have not yet been overcome. In this review, we summarise what is known about the conductivity values, as well as progress to date in meeting the challenges associated with both the mathematical modelling and the experimental techniques. Graphical abstract Epicardial potential distributions, arising from a subendocardial ischaemic region, modelled using conductivity data from the indicated studies.

Extracellular space attenuates the effect of gap junctional remodeling on wave propagation: a computational study

Ionic channels and gap junctions are remodeled in cells from the 5-day epicardial border zone (EBZ) of the healing canine infarct. The main objective of the study was to determine the effect of gap junctional conductance (Gj) remodeling and Cx43 redistribution to the lateral membrane on conduction velocity (theta) and anisotropic ratio, and how gap junctional remodeling is modulated by the extracellular space. We first implemented subcellular monodomain and two-domain computer models of normal epicardium (NZ) to understand how extracellular space modulates the relationship between Gj and theta in NZ. We found that the extracellular space flattens the Gj-theta relationship, thus theta becomes less sensitive to changes in Gj. We then investigated the functional consequences of Gj remodeling and Cx43 distribution in subcellular computer models of cells of the outer pathway (IZo) and central pathway (IZc) of reentrant circuits. In IZo cells, side-to-side (transverse) Gj is 10% the value in NZ cells. Such Gj remodeling causes a 45% decrease in transverse theta (theta(T)). Inclusion of an extracellular space reduces the decrease in theta(T) to 31%. In IZc cells, Cx43 redistribution along the lateral membrane results in a 29% increase in theta(T). That increase in theta(T) is a consequence of the decrease in access resistance to the Cx43 plaques that occur with the Cx43 redistribution. Extracellular space reduces the increase in theta(T) to 10%.

IN CONCLUSION

1), The extracellular space included in normal epicardial simulations flattens the Gj-theta relationship with theta becoming less sensitive to changes in Gj. 2), The extracellular space attenuates the effects of gap junction epicardial border zone remodeling (i.e., Gj reduction and Cx43 lateralization) on theta(T).

Linear electrode arrays for stimulation and recording within cardiac tissue space constants

In this paper, we document a fabrication process that yields linear arrays of rectangular platinum black electrodes spaced 25 mum apart with edge-to-edge separation of 20 microm. The spatial arrangement is therefore sufficiently fine to insure stimulation and recording within cardiac tissue space constants, as six electrodes with dimensions of either 5 x 100 microm2, 5 x 250 microm2, or 5 x 500 microm2 were positioned in a 130-microm2 span in the arrays. Despite the small electrode sizes and available surface areas, favorable impedance characteristics were identifed. Averages ranged from 111 kOmega to 146 kOmega at 0.5 Hz and from 14 kOmega 39 kOmega at 500 Hz. Differences in impedances between the electrode sizes tested were small. Potential differences (deltaphis) recorded using the two central electrodes during stimulation with combinations at separations of only 75 microm, 100 microm, and 125 microm had low signal noise. As a preliminary test of the use of these arrays for possible application to impedance measurements in cardiac tissue, the deltaphis recorded during stimulation were compared to deltaphis obtained from finite-difference simulations using an isotropic volume conductor model. Anticipated decays in deltaphi with widening electrode separation identified in those simulations matched the decays in the recorded deltaphis closely. These findings are significant because they suggest intracellular and interstitial microimpedance mesurements in heart experiments will be straightforward.

Construction and validation of a plunge electrode array for three-dimensional determination of conductivity in the heart

The heart's response to electrical shock, electrical propagation in sinus rhythm, and the spatiotemporal dynamics of ventricular fibrillation all depend critically on the electrical anisotropy of cardiac tissue. Analysis of the microstructure of the heart predicts that three unique intracellular electrical conductances can be defined at any point in the ventricular wall; however, to date, there has been no experimental confirmation of this concept. We report the design, fabrication, and validation of a novel plunge electrode array capable of addressing this issue. A new technique involving nylon coating of 24G hypodermic needles is performed to achieve nonconductive electrodes that can be combined to give moderate-density multisite intramural measurement of extracellular potential in the heart. Each needle houses 13 silver wires within a total diameter of 0.7 mm, and the combined electrode array gives 137 sites of recording. The ability of the electrode array to accurately assess conductances is validated by mapping the potential field induced by a point current source within baths of saline of varying concentration. A bidomain model of current injection in the heart is then used to test an approximate relationship between the monodomain conductivities measured by the array, and the full set of bidomain conductivities that describe cardiac tissue.

Bayesian analysis of fiber impedance measurements

The resistivities of microscale components of excitable tissue include the longitudinal intracellular and interstitial resistivities and the membrane resistivity. Measurements of these tissue micro impedances have rarely been obtained, mainly because of the lack of a satisfactory measurement system. Here we evaluate a possible strategy for obtaining such measurements, and begin with a simulation. In the model, a one-dimensional fiber was stimulated with closely space interstitial electrodes at four frequencies, and the voltage differences that occurred in response were recorded. We then considered the inverse question, asking if tissue micro impedances could be found from the voltage measurements plus additive noise. In so doing, we used a Bayesian interpretation of the measured data to find the probability that each of the longitudinal and transmembrane resistivity sets was their origin. The Bayesian procedure proved better suited for interpreting the measurements than was conventional least-squares analysis. It was better because all known data, including realistic noise specifications and a priori probabilities, were included in the defined procedure. The results show that the micro impedances were found satisfactorily using realistic parameters and noise levels. The overall quantitative evaluation is promising for future experimental measurements.

Laminar Arrangement of Ventricular Myocytes Influences Electrical Behavior of the Heart

The response of the heart to electrical shock, electrical propagation in sinus rhythm, and the spatiotemporal dynamics of ventricular fibrillation all depend critically on the electrical anisotropy of cardiac tissue. A long-held view of cardiac electrical anisotropy is that electrical conductivity is greatest along the myocyte axis allowing most rapid propagation of electrical activation in this direction, and that conductivity is isotropic transverse to the myocyte axis supporting a slower uniform spread of activation in this plane. In this context, knowledge of conductivity in two directions, parallel and transverse to the myofiber axis, is sufficient to characterize the electrical action of the heart. Here we present new experimental data that challenge this view. We have used a novel combination of intramural electrical mapping, and experiment-specific computer modeling, to demonstrate that left ventricular myocardium has unique bulk conductivities associated with three microstructurally-defined axes. We show that voltage fields induced by intramural current injection are influenced by not only myofiber direction, but also the transmural arrangement of muscle layers or myolaminae. Computer models of these experiments, in which measured 3D tissue structure was reconstructed in-silico, best matched recorded voltages with conductivities in the myofiber direction, and parallel and normal to myolaminae, set in the ratio 4:2:1, respectively. These findings redefine cardiac tissue as an electrically orthotropic substrate and enhance our understanding of how external shocks may act to successfully reset the fibrillating heart into a uniform electrical state. More generally, the mechanisms governing the destabilization of coordinated electrical propagation into ventricular arrhythmia need to be evaluated in the light of this discovery.

Possible four-electrode configurations for measuring cardiac tissue fiber rotation

A number of electrode configurations, based on the usual four-electrode probe, are analysed in relation to the effect that changes in cardiac fiber rotation have on the potentials measured. Simulations are carried out using a mathematical model and a new solution technique, based on Fourier series followed by a simple one-dimensional finite difference scheme. This electrode analysis leads to the proposal of an in-principle method for determining cardiac fiber rotation.

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.

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

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

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

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