IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 9, NO. 4, DECEMBER 2001
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Interaction of Array of Finite Electrodes With Layered Biological Tissue: Effect of Electrode Size and Configuration Leonid M. Livshitz, Joseph Mizrahi, and Pinchas D. Einziger
Abstract—A hybrid scheme, combining image series and moment method has been utilized for the calculation of the intramuscular three-dimensional (3-D) current density (CD) distribution and potential field transcutaneously excited by an electrode array. The model permits one to study the effect of tissue electrical properties and electrode placement on the CD distribution. The isometric recruitment curve (IRC) of the muscle was used for parameter estimation and model verification, by comparison with experimentally obtained IRCs of functional electrical stimulation (FES)-activated quadriceps muscle of paraplegic subjects. Sensitivity of the calculated IRC to parameters such as tissue conductivity, electrode size, and configuration was verified. The resulting model demonstrated characteristic features that were similar to those of experimentally obtained data. The model IRCs were insensitive to the electrode size; however, the inclusion of the bone–fascia layer significantly increased the intramuscular CD and, consequently, increased the IRC slope. Of the different configurations studied, a four-electrode array proved advantageous because, in this case, the CD between the electrodes was more evenly distributed, providing better resistance to fatigue. However, due to the steeper linear portion of the IRC, this configuration suffered from a somewhat reduced controllability of the muscle. Index Terms—Electric field distribution, finite electrode array, functional electrical stimulation (FES), isometric recruitment curve, nonhomogeneous media, volume conduction.
I. INTRODUCTION
I
N many functional electrical stimulation (FES) applications, large and powerful leg muscles are stimulated [1]. Knowledge of the current distribution within the excitable tissues is an important factor to predict and control the muscle output. The difficulties encountered in applying field theory to biological tissues include their heterogeneity, anisotropy, and frequency dependency of their electrical properties [2]. Fortunately, most FES applications involve currents with significant frequency components well below 10 kHz. Under this condition, it is sufficient to deal with the stationary current in the quasistatic limit, which considerably simplifies the representation of the potential field and the resultant current distribution. In a recent work, we have suggested that when stimulating the quadriceps muscle, Manuscript received April 1, 2001; revised February 1, 2002. This work was supported in part by the Segal Foundation. L. M. Livshitz and J. Mizrahi are with Department of Biomedical Engineering, Technion—Israel Institute of Technology (IIT), Haifa 32000, Israel (e-mail:
[email protected]). P. D. Einziger is with Department of Electrical Engineering, Technion—Israel Institute of Technology (IIT), Haifa 32000, Israel. Publisher Item Identifier S 1534-4320(01)11422-1.
the thigh can be modeled as a multilayered medium, where the skin (itself composed of a number of layers), fat, and muscle are represented as flat, infinitely extended layers [3]. Stimulation can be achieved by using either transcutaneous or percutaneous electrodes [4], [5]. In the former case, the current passes through nonexcitable regions, such as skin, fat, and connective tissues, before reaching the muscle. One of the basic macroscopic properties of an artificially stimulated muscle is the isometric recruitment curve (IRC). The IRC of a muscle is defined as the static relation between the level of the activation stimulus and the force output when the muscle is held at a fixed length [6]. The shape of the IRC during FES is determined, among various factors, by the electric field distribution, location and distribution of the excitable elements, their excitation thresholds, and fatigue. Recent investigations have shown that the current densities beneath a stimulating electrode can be highly nonuniform and that severe burns may occur even with electrodes that are considered to be within the accepted conservative-area guidelines [7], [8]. These current nonuniformities depend on a number of factors including electrode placement, quality and uniformity of the electrode–skin interface, and effective electrical and thermal conductivities of the tissue immediately beneath contact [9]–[11]. Thus, a complete treatment of the problem is expected to be beyond the scope of analytical calculation [12]. This study deals with the three-dimensional (3-D) intramuscular potential and current density (CD) distribution due to stimulation by means of an array of electrodes. The effects of electrode size, electrode separation, array configuration, and tissue conductivities are treated. We make use of a hybrid solution method recently implemented by our group [13]. This method is based on a theoretical manipulation of a rigorous image series expansions scheme [3], [14] and on the numerical moment method [15]. The hybrid method is simple to implement and allows one to estimate the 3-D distribution of the potential, current, electric field, and power within the multilayer tissue, regardless of the number of layers and electrodes, with a relatively small computational effort. The potential promise of this proposed hybrid scheme lies within its flexibility and its being capable of efficiently handling 3-D problems in layered media, excited by finite electrode arrays of arbitrary (generally nonplanar) configurations. An isometric recruitment curve model was used for parameter estimation and model verification by comparison of the model prediction with experimentally obtained IRCs of FES-activated quadriceps muscle of paraplegic subjects. It is known that the
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defined via the convolution integral (3)
Fig. 1. Physical configuration for a layered biological tissue excited by an array of finite electrodes.
, Green’s function, can be The point-source response represented in an explicit closed-form expression known as the image series expansion, i.e., a collection of properly weighted and shifted point-source responses and a remainder term (collective image) by introducing a novel recursive scheme as preshould be replaced by sented in [14], [13]. We note that in every layer the complex conductivity is not satisfied. The pafor which the inequality rameter denotes the angular frequency corresponding to the electrode excitation. To simplify the notation, we allow to be complex in the remainder of the paper. C. Electrode Array
shape and slope of the IRC curve affect controllability of the muscle and that linearity of the rising portion of the IRC, with moderate slope being preferable [6]. Accordingly, this study also investigates the sensitivity of the IRC slope to parameters such as tissue conductivity, electrode size, interelectrode gap, and electrode configuration.
The total current of each electrode is obtained by integraover the election of the electrode current distribution trode surface (4) The uniqueness of the solution of system (1) in conjunction with the superposition principle leads to the following linear relation between the electrode currents and electrode voltages
II. FORMULATION A. Problem Statement The physical configuration of our problem, depicted in Fig. 1, consists of a stratified biological medium with boundaries sephomogeneous and isotropic layers. arated between the Each layer is characterized by its thickness, conductivity , and and for electrical permittivity , where . An array of rectangular . The evaluation of electrodes is placed in the first layer the electrodes’ current distributions and potentials is carried out within the quasistatic (low frequency) regime. B. Integral Equation Formulation Assuming that all of the electrode plates (see Fig. 1) are perfect conductors, i.e., constant potential patches, the potential of each electrode is specified. Thus, the problem constitutes a system of Fredholm integral equations of the first kind [13] for the electrodes’ current distribution
(1) , the th electrode current distribution can be where expressed as a superposition over all the electrode potentials
(2)
(5) where denotes the conductance (admittance) matrices of the electrode array. In view of Kirchhoff’s current law (KCL), the sum of all the electrodes currents must be zero, i.e., . Thus, one of the electrode potential (for example, ) should electrode potenbe expressible in terms of the remaining tials and the impedance matrix elements [13]. D. Moment Method The integral equation system in (1) can be inverted using the moment method with pulse base for the electrode current distribution and point match for the potential [15]. The discretized electrode potential is a linear transformation of the discretized current distribution via (6) , , The elements of and are given by . Note that only for and . The moment matrix is a square matrix specified by , representing the potential at the center of the its elements subsection due to uniform CD distribution on the subsection , given as (7) and represent the location of the observation and where source points, respectively. It can be readily verified that the . Thus, a discretization quantum is a square element of size
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Fig. 3. Physical configuration for a typical transcutaneous FES problem. The medium consists of five layers and is excited by an array of two or four electrodes.
III. CURRENT DISTRIBUTION WITHIN THE MUSCLE AND SIMULATION OF THE ISOMETRIC RECRUITMENT CURVE Fig. 2. Inversion of the integral operator in a two-step procedure. In the first step, the kernel (Green function) is expanded in image series. In the second step, moment matrix elements are calculated through explicit analytical integration of the image terms. Note that we use image expansion on two different occasions: first, to obtain the electrode CD distribution (using the image series expansion corresponding to the layer where the electrode array is placed), and second, to obtain, after moment matrix inversion, the potential at any layer using the image series expansion corresponding to that layer. Image series can be analytically differentiated to obtain the electric field and CD vectors. The relation between electrode current and potential via the impedance matrix can also be calculated. Because the impedance matrix depends only on the problem geometry, we must perform matrix inversion only once, and then study the electrode-array current-voltage relation for any given input voltage.
square electrode of size contains subdivisions (see Fig. 1). An explicit closed-form expression for the moment matrix element had been presented in [13]. of the conductance maFinally, the individual element trix in (5) is obtained by summation of the appropriate block elements of the inverse matrix
(8)
where
denotes elements of
.
E. Potential and Current Distributions Once the system (6) is inverted, i.e., solved for , the potential at the th layer is obtained via (2) through discretization of (3) as (9) -dimensional vector where are the components of the in (6) and is the corresponding th-layer Green’s function. and electric field are related via The CD and , respectively, and obtained through explicit (analytic) closed-form differentiation of and (i.e., term-by-term differentiation of the image series expansion). This is more accurate and stable than the numerical differentiation generally used in other solution schemes. A sketch of the computational algorithm is presented in Fig. 2.
A. Simulation Configuration The physical configuration for the model simulations of transcutaneous FES is presented in Fig. 3. The square electrodes, size 4 4 cm , were assumed to be perfect conductors and were . The layer thickdiscretized as nesses were set to be 0.4, 0.6, and 3 cm for the skin, fat, and muscle layers, respectively. A mesh step of 0.25 cm was used across the thickness and was found to numerically efficient yet small enough to avoid discretization errors. Typical conductivity – S/m and S/m for the values were skin and fat layers, respectively [16]. Muscle was assumed to be – S/m. The conisotropic, with conductivity of ductivity of the underlying muscle boundary (bone–fascia) was – S/m. Due to the wide variability assumed to be of the data found in the literature on skin and muscle conductivities, these quantities were subject to parameter estimation. Anisotropy was not taken into account in this study. However, the nature of the solution in terms of an integral formulation for a transversely anisotropic stratified medium was indicated by Grant and West [17]. The image method has already been successfully applied to simple anisotropic geometries [18]. The implementation of the image method in more complex anisotropic multilayered media is, however, a crucially important research topic. For validation of the model, we used experimental isometric recruitment results from three paraplegic subjects from previous studies conducted by our group. The detailed procedures of the preparation of the subjects, placement of the electrodes, and measurement protocols are as described in [19]. The IRC of a muscle is defined as the relation between the stimulus activation level and the output force when the muscle is held at a fixed length. The shape of the IRC during FES is determined, among various factors, by the electric field distribution, location and fiber size distribution of the excitable elements, and their excitation thresholds. The activation threshold, which is a measure of the interaction between the external electric field and the excitable fibers, is a complex property that, in this study, is also subject to parameter estimation using the experimentally obtained IRC. The force produced was calculated as the ratio of the number of active fibers to the overall number of fibers in the muscle. Although the quadriceps muscle slice was represented as an unbounded layer, it was assumed that the layer is effectively bounded, i.e.,
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Fig. 6. 3-D potential distribution and vector plot of the radial component of the electric field for a four-electrode array.
Fig. 4. 3-D potential distribution and vector plot of the radial component of the electric field for a two-electrode array.
Fig. 7. 3-D plot of the x component of the CD distribution for a four-electrode array simulation.
Fig. 5. array.
3-D plot of the x component of the CD distribution for a two-electrode
that only a finite part of this slice could generate a force. The width and length of this active part were taken to be 10 and 30 cm, respectively, and were based on MRI measurements made on paraplegic patients [19]. The amplitude of the stimulation was varied from 0 to 120 mA with step 10 mA. A typical IRC consists of three distinctive segments, the initial low-force dead zone, the steep central zone, and the leveled-off region. B. Potential and CD Distribution in Muscle Layer The simulations presented here were carried out after setS/m, ting the following parameters values: S/m; S/m; and S/m. array (one positive We first consider a two-electrode and one negative) placed longitudinally along the center line with a 20-cm distance between the electrode centers. Stimulation is done by a constant current of either 100 or 100 mA on each of the electrodes. The potential distribution at the fat–muscle interface is depicted in Fig. 4. The vector plot of the radial component of the electric field is presented at the bottom of the figure. Fig. 5 shows the variation of the component of the CD at the fat–muscle interface. Nonuniformity of the CD is easily noted. The contour plots at the bottom of Fig. 5 show equi-value CD plots.
The second case is a four-electrode array consisting of two pairs of electrodes placed symmetrically around the center line with distances of 20 and 5 cm between their centers in the longitudinal and lateral directions, respectively. Each pair of adjacent electrodes (short distance, Fig. 3) were of the same polarity. The current delivered by each electrode was either 50 or 50 mA. Fig. 6 shows the 3-D potential map and vector plot of the radial component of the electric field distribution at the cm). The crowding phenomenon fat–muscle interface ( corresponding to electrode edge singularity is self-evident. Fig. 7 shows the corresponding variation of the component of the CD at the fat–muscle interface. While the maximal CD value is less than half that of the two-electrode configuration (8 A/m versus 20 A/m ), the CD in the middle portion of the muscle tissue is evenly distributed (5 to 6 A/m ). The contour plots at the bottom of Fig. 7 show equi-value CD plots. C. Isometric Recruitment Curve Simulation The IRC curves depicted in Figs. 8–12 are an extension, to finite-size electrodes and to a five-layer medium (air, skin, fat, muscle, muscle boundary), of previous simulations [3]. The IRC was estimated using the following procedure. The first stage was the determination of the CD distribution within the muscle slice for both intramuscular and surface models. The second stage counted the elements within the muscle slice that escape the activation threshold for various values of input current. The muscle tissue was represented by an assembly of identical fibers
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Fig. 8. Best fit simulation of the IRC to experimental data. Square symbols represent experimental data and circle symbols represent model simulation.
Fig. 10.
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Effect of the lower-most layer conductivity on the IRC.
Fig. 11. Effect of the interelectrode distance on the IRC. Fig. 9.
Effect of electrode size on the IRC.
of approximately 50-mm length, each with the same force-generation characteristics. Pennation of the muscle was assumed to be negligibly small. Minimum root mean square error (rms) criterion between the experimental data and model solution was used to determine the best fit solution of the IRC problems. Fig. 8 presents the best fit between the model and the experimental recruitment data after parameter estimation. The following parameter values were obtained: S/m; S/m; S/m; and S/m. The minimal CD threshold was 3.3 A/m . The conductivities obtained lie within the physiological available data. Also, the CD corresponds to previously published results. The IRC presented in Fig. 8 serve as the reference solution for the following model simulations. Fig. 9 presents the IRC for various electrode sizes m, m, m, and m). It ( should be noted that the 0.001-m electrode case gives a similar solution for both CD distribution and IRC problems as for the point-electrode case [3]. Despite the broad range of sizes (1 mm to 4 cm), the model shows no substantial effect of the electrode size on the IRC shape. However, for electrode sizes comparable to the width of the quadriceps muscle ( 10 cm), the IRC shifts
toward the right side of the current axis and manifests an abrupt reduction of the maximal force. This is due to a more uniform CD, reducing the electrode CD below threshold. The effect of different conductivities of the lowermost layer on the IRC is shown in Fig. 10. The variations were from (semiinfinite case) to (poor conductive layer). The decrease in conductivity of the lowermost layer leads to an increase of the IRC slope and shifts the onset of the saturation region toward lower current values. Fig. 11 shows the IRC for different inte-electrode distances. It is noted that electrode separation has a complex effect on the IRC. Increasing the electrode separation from 5 to 20 cm leads to an increase of the maximal achievable force. However, a further increase of the separation to 30 cm displays a lower slope of the IRC without changing the maximal force. Thus, a higher current is required to attain this force, which may be regarded as a drawback. On the other hand, the resulting milder slope of the main part of the curve provides better controllability of the force compared to the higher slope of the 20-cm distance. Fig. 12 displays the IRC for three different electrode-array configurations. The total fed current is constant in all the configurations compared. The solid line represents the two-electrode array located along the center line. The dotted line rep-
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Fig. 12.
IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 9, NO. 4, DECEMBER 2001
Effect of electrode configuration on the IRC.
resents the case of four electrodes, each pair placed symmetrically around the center line, as shown in Fig. 3. In this case, the current per electrode is half the current of the two-electrode case. The four-electrode array model exhibits an extension of the dead zone and postponement of the saturation region and has a more uniform CD, as shown in Fig. 7. The slope in the central zone of this curve appears to be somewhat higher compared to the base case. The dashed line represents a case where only one of the two pairs of electrodes from the previous case is switched on with all the current flowing, while the other one is off. Shifting of the electrodes from the center line leads to a more linear curve and to a decrease of the IRC slope. These two features are often desirable for FES controllability. The achievable force, however, is 80% of the maximal recruitable force. IV. DISCUSSION This study outlines a hybrid model, based on the image series expansion and the moment method, for the electric field distribution within a biological tissue excited by an array of finite rectangular electrodes. Using the matrix element expression for annulus subsection, the proposed scheme can be used without modification to axisymmetrical electrodes (circular or ring shaped) [9]. Using relatively low computational power, the proposed procedure can be used as a reference model for the verification of numerical subroutines applied to either direct or inverse problems with arbitrarily placed electrodes. The proposed method is being used here for transcutaneous FES of muscles to calculate the intramuscular potential and CD distribution. A simple IRC model was used for parameter estimation by comparison with experimentally obtained IRCs of the quadriceps muscle. The model was further used to predict the effect of electrodes sizes and configurations. The general IRC shape appeared sigmoidal, in agreement with previous results [6], [19]. The success in matching the IRC experiments and IRC numerical simulation in a rather restricted value range of the parameters demonstrates the potential application of the model in FES problems. The CD distribution obtained in transcutaneous stimulation with two centrally located electrodes was found to be largely nonuniform near the stimulating electrodes and nearly uniform
within a large part of the muscle volume. This may lead to fast fatiguing of the fibers adjacent to the stimulating electrodes and make it more difficult to activate the middle portion of the muscle without exceeding the CD safety threshold near the electrodes. It has been shown, however, that if four electrodes, symmetrically placed about the center line, are used instead of two electrodes, the CD becomes more evenly distributed near the electrodes, while the CD in the mid-region between the electrodes remains without significant change. The model solutions obtained in this study for finite-size electrodes compared well with the previously published solution for point electrodes [3], except for a minor difference in the initial slope of the IRC. Under constant current stimulation conditions, the IRC curves were insensitive to the electrodes size, except when the electrodes became comparable in size to the lateral dimension of the muscle. In this case, there was a decrease in the average CD on the electrode, and, to a decrease in the maximum CD in the muscle bulk. The model IRC results also demonstrated the influence of tissue conductivity on the maximum force value. It was shown that, with low conductive muscle boundary, the curves were shifted to the left on the current axis and had a higher slope when compared to the IRC corresponding to a semiinfinite muscle slice. The inclusion of the bone–fascia layer significantly increased the IRC slope. Thus, ignoring this layer may lead to significant errors, resulting in an underestimation of the intramuscular CD or to an overestimation of the activation threshold. Apart from evening out the CD near the electrodes, the addition of a pair of stimulation electrodes enabled to reach the same maximal force, but with delayed muscle fatigue. This addition, however, led to an increase in the slope of the IRC, which may compromise controllability during FES. A more promising scheme is to use the four electrodes in a sequential manner, whereby at any given time only one pair is activated. However, because either of the activated pairs of electrodes is shifted with respect to the center line, only one part of the muscle exceeds activation threshold while the other part remains at rest, reducing fatigue of the entire muscle. It should be borne in mind, however, that in this configuration, only 80% of the achievable force is obtained when providing the full current to each pair of electrodes. The four-electrode array stimulation data demonstrate the trade off existing between higher force output and satisfactory controllability of the stimulated muscle. As shown in Fig. 12, the dead zone, in this case, is shifted to the right on the current axis and the saturation zone is shifted to the left when compared to the base case. Thus, not only does the linear part of the curve get steeper, but also a higher stimulation current is required for a given force level. The main advantage of the four-electrode configuration is that the stimulation current is less hazardous to the upper skin layers because it is more evenly distributed between electrodes. This seems to suggest that an electrode array would be preferable to increasing the electrode size. Future extension of the present model would necessitate detailed knowledge of the anatomy of the spatial axon enervation within the muscle and the distribution of the motor units. A first step toward the extension of the model could be addition of weight factors in the regions of the motor point to increase
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their effect, due to transmission. A typical example is the central region between the two electrodes where the one of the motor points of the quadriceps muscle is located. Another way is to use the probability function to describe the location of fibers with different diameters within the nerve bundles. Addition to the present model of a parameter that can take into account information on the muscle fiber population (slow versus fast) and its spatial location is possible but needs further anatomical and physiological data. REFERENCES [1] R. B. Stein, K. Momose, and J. Bobet, “Biomechanics of human quadriceps muscles during electical stimulation,” J. Biomech., vol. 32, pp. 347–357, 1999. [2] R. Plonsey, Bioelectrical Phenomena. New York: McGraw-Hill, 1969, pp. 129–139. [3] L. Livshitz, P. D. Einziger, and J. Mizrahi, “Current distribution in skeletal muscle activated by FES: Image-series formulation and isometric recruitment curve,” Ann. Biomed. Eng., vol. 28, pp. 1218–1228, 2000. [4] M. R. Neuman, “Biopotential electrodes,” in Medical Instrumentation Application and Design, J. G. Webster, Ed. New York, NY: Wiley, 1998. [5] J. P. Reilly, Applied Bioelectricity. New York: Springer, 1998. [6] W. K. Durfee and K. E. MacLean, “Methods for estimating isometric recruitment curves of electrically stimulated muscle,” IEEE Trans. Biomed. Eng., vol. 36, pp. 654–666, July 1989. [7] K. M. Overmyer, J. A. Pearce, and D. P. DeWitt, “Measurements of temperature distributions at electro-surgical dispersive electrode sites,” J. Biomech. Eng., vol. 101, pp. 66–72, 1979. [8] P. M. Caruso, J. A. Pearce, and D. P. DeWitt, “The effect of gelled-pad design on the perfomance of electrosurgical dispersive electrodes,” J. Biomech. Eng., vol. 104, pp. 324–329, 1982. [9] A. van Oosterom and J. Strackee, “Computing the lead field of electrodes with axial symmetry,” Med. Biol. Eng. Comput., vol. 21, pp. 473–481, 1983. [10] J. D. Wiley and J. G. Webster, “Analysis and control of the current distribution under circular dispersive electrode,” IEEE Trans. Biomed. Eng., vol. BME-29, pp. 381–385, 1982. [11] K.-A. Cheng, D. Isaacson, J. C. Newell, and D. G. Gisser, “Electrode models for electric current computed tomography,” IEEE Trans. Biomed. Eng., vol. 36, pp. 918–924, Sept. 1989. [12] T. Oostendorp and A. van Oosterom, “The potential distribution generated by surface electrodes in inhomogeneous volume conductors of arbitrary shape,” IEEE Trans. Biomed. Eng., vol. 38, pp. 409–417, May 1991. [13] L. Livshitz, P. D. Einziger, and J. Mizrahi, “A model of finite electrodes in layered media: An hybrid image series and moment method scheme,” ACES J. SI: Computational Bio-Electromagnetics, vol. 16, no. 2, pp. 145–154, 2001. [14] P. D. Einziger, L. M. Livshitz, and J. Mizrahi, “Rigorous image series expansions of quasistatic Green’s functions for regions with planar stratification,” IEEE Trans. Antennas Propagat., vol. 50, Oct. 2002. [15] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [16] K. R. Foster and H. P. Schwan, “Dielectrical properties of tissues,” in CRC Handbook of Biological Effects of Electromagnetic Field, C. Polk and E. Postow, Eds. Boca Raton, FL: CRC Press, 1986, pp. 26–95.
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[17] F. S. Grant and G. F. West, Interpretation Theory in Applied Geophysics. New York: McGraw-Hill, 1965, pp. 407–414. [18] P. Li and N. F. Uren, “Analytical solution for the point source potential in an anisotropic 3-D half-space. I. Two-horizontal-layer case,” Math. Comput. Modeling, vol. 26, no. 5, pp. 9–27, 1997. [19] O. Levin and J. Mizrahi, “EMG and metabolite-based prediction of force in paralyzed quadriceps muscle under interrupted stimulation,” IEEE Trans. Rehab. Eng., vol. 7, pp. 301–314, Sept. 1999.
Leonid M. Livshitz received the B.S. and M.S. degrees in electrical engineering from the Moscow State University of Railway Transport (MIIT), Moscow, U.S.S.R., in 1986 and the M.S. and Ph.D. degrees in biomedical engineering from the Technion—Israel Institute of Technology (IIT), Haifa, Israel, in 1997 and 2002, respectively. From 1986 to 1991, he worked as a Computer and Electrical Engineer at MIIT. His research interests include computational aspects of the interaction between applied electrical and magnetic fields with nerves and muscles, FES of muscles, and muscle excitation-contraction coupling.
Joseph Mizrahi received the B.Sc. degree in aeronautical engineering, the M.Sc. degree in mechanics, and the D.Sc. degree in biomechanics from the Technion—Israel Institute of Technology (IIT), Haifa, Israel, in 1967, 1970, and 1975, respectively. He is currently a Professor of Biomechanics in the Department of Biomedical Engineering, Technion—IIT. For 18 years, he was head of the Biomechanics Laboratory at the Loewenstein Rehabilitation Center in Israel. He also held several visiting positions, including with Harvard Medical School, Cambridge, MA, from 1989 to 1990, University of Cape Town, Cape Town, South Africa, in 1991, and Hong Kong Polytechnic University, Kowloon, Hong Kong, from 1998 to 1999. He is the principal author of some 200 publications, and holds several editorial responsibilities. His major research interests are in orthopaedic biomechanics and electrical stimulation of muscles.
Pinchas D. Einziger received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion—Israel Institute of Technology (IIT), Haifa, Israel, in 1976 and 1978, respectively, and the Ph.D. degree in electrophysics from the Polytechnic University, Brooklyn, NY, in 1981. Since 1981, he has been on the faculty of the Department of Electrical Engineering at the Technion—IIT. His main interests are electromagnetic wave theory, nonlinear wave phenomena, and bioelectromagnetics.
