Effect of a perfusing bath on the rate of rise of an action potential propagating through a slab of cardiac tissue

Experiments show that the rate of rise of the action potential depends on the direction of propagation in cardiac tissue. Two interpretations of these experiments have been presented: (i) the data are evidence of discrete propagation in cardiac tissue, and (ii) the data are an effect of the perfusing bath. In this paper we present a mathematical model that supports the second interpretation. We use the bidomain model to simulate action potential propagation through a slab of cardiac tissue perfused by a bath. We assume an intracellular potential distribution and solve the bidomain equations analytically for the transmembrane and extracellular potentials. The key assumption in our model is that the intracellular potential is independent of depth within the tissue. This assumption ensures that all three boundary conditions at the surface of a bidomain are satisfied simultaneously. One advantage of this model over previous numerical calculations is that we obtain an analytical solution for the transmembrane potential. The model predicts that the bath reduces the rate of rise of the transmembrane action potential at the tissue surface, and that this reduction depends on the direction of propagation. The model is consistent with the hypothesis that the perfusing bath causes the observed dependence of the action-potential rate of rise on the direction of propagation, and that this dependence has nothing to do with discrete properties of cardiac tissue.     

Electrode guarding in electrical impedance measurements of physiological systems—a critique

The application of the guard ring has been advanced as a method for focusing the applied current in performing electrical-impedance measurements of volume conductors. The objective is to obtain impedance and/or its variation at localised regions of heterogeneous tissue. The paper discusses the capabilities and limitations of both guarded and unguarded measurements and shows that the application of guarding does not, normally, fulfil the capabilities being ascribed to it. A comparison is made of the guarded system under constant-current and constant-voltage constraints with an unguarded electrode system, and it is shown that the guarded system does not have any significant advantages.     

Mechanism for polarisation of cardiac tissue at a sealed boundary

If current is flowing in cardiac tissue, and if the myocardial fibres approach a sealed boundary at an angle, then the tissue within a few length constants of the boundary is polarised. This polarisation occurs when the cardiac tissue has different anisotropy ratios in the intracellular and extracellular spaces. This new mechanism of tissue polarisation is demonstrated using a simple, analytical model, and it is shown quantitatively that this polarisation can be nearly as large as that occurring near an electrode.     

Mechanism for polarisation of cardiac tissue at a sealed boundary

If current is flowing in cardiac tissue, and if the myocardial fibres approach a sealed boundary at an angle, then the tissue within a few length constants of the boundary is polarised. This polarisation occurs when the cardiac tissue has different anisotropy ratios in the intracellular and extracellular spaces. This new mechanism of tissue polarisation is demonstrated using a simple, analytical model, and it is shown quantitatively that this polarisation can be nearly as large as that occurring near an electrode.     

A planar slab bidomain model for cardiac tissue

A fully three-dimensional model of the ventricular or atrial free wall will involve a planar geometry of finite thickness. The governing equations for the interstitial and extracellular potential of a planar slab of cardiac tissue comprised of parallel fibers undergoing uniform plane-wave activation are presented. A comparison with a bidomain of cylindrical geometry with the same half-thickness shows that the potentials in the planar bidomain (as a function of depth) approach core-conductor behavior more quickly.     

A comparison of two boundary conditions used with the bidomain model of cardiac tissue

In the bidomain model, two alternative sets of boundary conditions at the interface between cardiac tissue and a saline bath have been used. It is shown that these boundary conditions are equivalent if the length constant of the tissue in the direction transverse to the fibers is much larger than the radius of the individual cardiac cells. If this is not the case, the relative merits of the two boundary conditions are closely related to the question of the applicability of a continuum model, such as the bidomain model, to describe a discrete multicellular tissue.     

The importance of blood resistivity in the measurement of cardiac output by the thoracic impedance method

Twenty simultaneous pairs of cardiac output values from patients who did not have valvular abnormalities were obtained by the radioisotope method and the electrical-impedance method of Kubicek et al. (1966). If a standard value of 150Ω-cm was assumed for the resistivity of each patient's blood, the mean value for the impedance cardiac output was 14·5% high compared with the mean radioisotope value. In this study the patient's haematocrits ranged from 20 to 48%. Inserting the appropriate value of the resistivity for each patient into the stroke volume equation of Kubicek from the data of Geddes and Sadler (1973) made the mean impedance value 10·3% low compared with the mean isotope value. The use of our measured resistivity data made the mean impedance cardiac output value 21·5% lower than the mean isotope value. The correlation coefficient between the impedance and isotope techniques was 0·61 for the standard value of resistivity of 150Ω-cm. Using the resistivity data of Geddes and Sadler (1973) the correlation became 0·87, and with our data it was 0·88.     

Variability of impedivity in normal and pathological breast tissue

The impedivity of six groups of breast tissue is measured between 0.488 kHz and 1 MHz using a hand-held probe, ensuring a constant geometry factor, and a microcomputer-controlled impedance spectroscopy system. 120 spectra are collected in excised tissue samples from 64 patients undergoing breast surgery. Each spectrum consists of 12 frequency points. The mean m, the standard deviation s, and the ‘reduced standard error’ (ε=s/(m N)) of the magnitude and the phase angle of the impedivity are calculated at each frequency for all groups of tissues. The variability at low frequency (f<10 kHz) is attributed to the dispersion in measurement errors. This contributed to the choice of 32 kHz as the lower limit of measurement frequency in the constructed electrical impedance tomograph. The collected data also show that frequencies larger than 1 MHz are needed for the bio-electrical characterisation of breast tissue. In the frequency range used in electrical impedance tomography the reduced standard error of impedivity in breast tissue is about 0.1 or less. The lowest dispersions are observed in the adipose tissue, carcinoma and fibro-adenoma.     

The electrical potential produced by a strand of cardiac muscle: A bidomain analysis

Analytic expressions are derived relating the transmembrane potential to the intracellular, interstitial and external potentials in a cylindrical strand of cardiac muscle lying in a saline bath. The bidomain model is used to account for the anisotropy and interstitial space in the tissue. The implications of this model for interpreting potential data from strands of cardiac muscle are discussed.     

Measurement of body fluid volume change using multisite impedance measurements

Previous electrical impedance studies that predict body fluid volume and body fat used a single measurement site between the wrist and ankle Calculations based on an electrical model of the thorax suggest that with a single wrist-to-ankle measurement only the electrical characteristics of the legs and arms will contribute significantly to a prediction of the total body fluid volume. In this study, the impedance of the arms, legs and trunk were measured separately at 100 kHz and the values were correlated with weight change that occurred on 11 patients undergoing haemodialysis. Using multiple linear regression analysis combining data from the arms, legs and trunk gave a correlation coefficient of 0·87, whereas the correlation coefficient for measurements between the wrist and the ankle was 0·64. These results suggest that multisite impedance measurements will allow more reliable body fluid and body fat determinations to be made.     

Modelling passive cardiac conductivity during ischaemia

The results of a geometric model of cardiac tissue, used to compute the bidomain conductivity tensors during three phases of ischaemia, are described. Ischaemic conditions were simulated by model parameters being changed to match the morphological and electrical changes of three phases of ischaemia reported in literature. The simulated changes included collapse of the interstitial space, cell swelling and the closure of gap junctions. The model contained 64 myocytes described by 2 million tetrahedral elements, to which an external electric field was applied, and then the finite element method was used to compute the associated current density. In the first case, a reduction in the amount of interstitial space led to a reduction in extracellular longitudinal conductivity by about 20%, which is in the range of reported literature values. Moderate cell swelling in the order of 10–20% did not affect extracellular conductivity considerably. To match the reported drop in total tissue conductance reported in experimental studies during the third phase of ischaemia, a ten fold increase in the gap junction resistance was simulated. This ten-fold increase correlates well with the reported changes in gap junction densities in the literature.     

Use of electrical impedance for continuous measurement of stroke volume of a skeletal muscle-powered cardiac assist device

This study describes the use of electrical impedance Z to continuously measure the stroke volume SV of a skeletal muscle-powered ventricle (SMV). An SMV was constructed surgically in four anaesthetised dogs. The rectus abdominis (two dogs) or latissimus dorsi (two dogs) muscle was wrapped around a compressible pouch, the ends of which were connected to a saline-filled (0.9 per cent) mock circulation. The motor nerves to the muscle were stimulated to produce tetanic contractions at a rate of 10 min-1. Z was measured between brass sleeve electrodes within the end conduits of the pouch. To derive a simple expression relating pouch volume V to Z, the pouch was represented as two truncated cones with their bases joined. For V ranging from 53 to 103 ml, the relationship between Z and 1/square root of V was nearly linear; i.e. Z = m(1/square root of V) + b. Impedance-derived stroke volume SV (delta Z) was calculated using this linear approximation and the impedance measured just before and after muscle contraction. The stroke volume SV (EM) ejected by the pouch during muscle contraction was measured with an electromagnetic flowmeter. The linear regression coefficients ranged from 0.99 to 2.55; the correlation coefficients ranged from 0.90 to 0.98. In general, SV(delta Z) tracked SV(EM) very well, although SV(delta Z) tended to overestimate SV(EM).     

The effect of applied pressure on the electrical impedance of the bladder tissue using small and large probes

There are a number of studies using electrical impedance spectroscopy, a minimally invasive technique, as a tissue characterizing method with different probe sizes (usually with larger probe diameters than that used in this work). In urinary bladder studies the probe size are limited to 2 mm diameter, in order to pass through the working channel of the cystoscope to measure the impedance inside the urinary bladder. Thus, bio-impedance of the human urothelium can only be measured using a small sized probe for in vivo studies. Different pressures were applied with this probe and it was demonstrated that increasing the applied pressure over the probe would increase the measured electrical impedance of the bladder tissue. Therefore, the effect of applied pressure on the resulting electrical impedance was considered in this study (all of the measurements were taken on points that had benign histology). An excessive amount of the applied pressure beyond the first visible indentation (first recordable reading) pressure has a significant effect on the impedance of the bladder tissue (p < 0.001). Then, to reduce the effect of pressure on the measured bio-impedance, the effect of a larger probe (10 mm diameter) was considered (p < 0.001). Increasing the probe contact area is one way to reduce the pressure effect on measurements; however this is difficult in practice in the in vivo situation.     

头部组织电阻抗成像研究进展

脑部疾病和脑功能活动期间常伴随脑组织电阻抗的变化,利用电阻抗成像技术可以对大脑疾病和脑功能活动进行临床诊断和监护。首先对人体头部组织阻抗测量技术的优缺点及其在头部组织阻抗成像上的应用前景进行简介,然后重点介绍了几种基于磁场测量的电阻抗成像方法,最后给出了目前头部阻抗成像研究存在的问题及该领域下一步的研究方向。     

Two-dimensional Chebyshev pseudospectral modelling of cardiac propagation

Bidomain or monodomain modelling has been used widely to study various issues related to action potential propagation in cardiac tissue. In most of these previous studies, the finite difference method is used to solve the partial differential equations associated with the model. Though the finite difference approach has provided useful insight in many cases, adequate discretisation of cardiac tissue with realistic dimensions often requires a large number of nodes, making the numerical solution process difficult or impossible with available computer resources. Here, a Chebyshev pseudospectral method is presented that allows a significant reduction in the number of nodes required for a given solution accuracy. The new method is used to solve the governing nonlinear partial differential equation for the monodomain model representing a two-dimensional homogeneous sheet of cardiac tissue. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions and the equation is enforced at the Gauss-Lobatto grid points. Spatial derivatives are obtained using the fast Fourier transform and the solution is advanced in time using an explicit technique. Numerical results indicate that the pseudospectral approach allows the number of nodes to be reduced by a factor of sixteen, while still maintaining the same error performance. This makes it possible to perform simulations with the same accuracy using about twelve times less CPU time and memory.     

On the Passive Cardiac Conductivity

In order to relate the structure of cardiac tissue to its passive electrical conductivity, we created a geometrical model of cardiac tissue on a cellular scale that encompassed myocytes, capillaries, and the interstitial space that surrounds them. A special mesh generator was developed for this model to create realistically shaped myocytes and interstitial space with a controled degree of variation included in each model. In order to derive the effective conductivities, we used a finite element model to compute the currents flowing through the intracellular and extracellular space due to an externally applied electrical field. The product of these computations were the effective conductivity tensors for the intracellular and extracellular spaces. The simulations of bidomain conductivities for healthy tissue resulted in an effective intracellular conductivity of 0.16S/m (longitudinal) and 0.005S/m (transverse) and an effective extracellular conductivity of 0.21S/m (longitudinal) and 0.06S/m (transverse). The latter values are within the range of measured values reported in literature. Furthermore, we anticipate that this method can be used to simulate pathological conditions for which measured data is far more sparse.     

生物阻抗的测量方法

通过对生物阻抗的测量来监测生理功能和检测病理事件 ,已成为近年来研究者非常关注的一个研究方向。本文介绍三种生物阻抗的测量方法 ,它们分别是阻抗断层摄影成像技术 ( electricalimpedance tomography,EIT)、基于磁共振成像 ( magnetic resonance imaging,MRI)的方法和阻抗谱测量 ( impedance spectrometry)技术 ,并分别简要介绍了各自的应用     

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.     

Three-dimensional pseudospectral modelling of cardiac propagation in an inhomogeneous anisotropic tissue

Various investigators have used the monodomain model to study cardiac propagation behaviour. In many cases, the governing non-linear parabolic equation is solved using the finite-difference method. An adequate discretisation of cardiac tissue with realistic dimensions, however, often leads to a large model size that is computationally demanding. Recently, it has been demonstrated, for a two-dimensional homogeneous monodomain, that the Chebyshev pseudospectral method can offer higher computational efficiency than the finite-difference technique. Here, an extension of the pseudospectral approach to a three-dimensional inhomogeneous case with fibre rotation is presented. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions, and the monodomain equation is enforced at the Gauss-Lobatto node points. The forward Euler technique is used to advance the solution in time. Numerical results are presented that demonstrate that the Chebyshev pseudospectral method offered an even larger improvement in computational performance over the finite-difference method in the three-dimensional case. Specifically, the pseudospectral method allowed the number of nodes to be reduced by ≈85 times, while the same solution accuracy was maintained. Depending on the model size, simulations were performed with ≈18–41 times less memory and ≈99–169 times less CPU time.