Leonid M. Livshitz

Interaction of array of finite electrodes with layered biological tissue: effect of electrode size and configuration

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.

Rigorous image-series expansions of quasi-static Green's functions for regions with planar stratification

A novel image-series expansion scheme for quasi-static Greens function in n+1 layered media is obtained by expanding the frequency-dependent Hertz potential in finite expansions and remainder terms. The expansions utilize a unique recursive representation for Greens function, which is a generic characteristic of the stratification, and are explicitly constructed for n/spl les/3. While results for 0/spl les/n/spl les/2 are given for reference only, the expansion scheme for a double-slab configuration, n=3, is quite general and outlines the procedure for n>3, without any increase in the complexity. The expansion-remainder terms can be made negligibly small for sufficiently large summation indices in the quasi-static limit, leading to rigorous image-series expansion. The image-series convergence is accelerated by including a collective image term, representing a closed-form asymptotic evaluation of the series-remainder integral. Thus, the proposed computational procedure can be used as a simple tool for producing analytical data for testing numerical subroutines applied to direct problems such as electrical simulation of muscles in the biomedical field and inverse problems, such as electromagnetic imaging.

Rigorous Green's function formulation for transmembrane potential induced along a 3-D infinite cylindrical cell

The quasi-static electromagnetic field interaction with three-dimensional infinite-cylindrical cell is investigated for both intracellular (IPS) and extracellular (EPS) current point-source excitation. The induced transmembrane potential (TMP), expressed conventionally via Greens function, may alternatively be expanded into a faster-converging representation using a complex contour integration, consisting of an infinite-discrete set of exponentially decaying oscillating modes (corresponding to complex eigenvalues) and a continuous source-mode convolution integral. The dominant contributions for both the IPS and EPS problems are obtained in simple closed-form expressions, including well documented special mathematical functions. In the IPS case, the dominant modal contribution (of order zero)-an exact solution of the well-known cable equation-is explicitly and analytically corrected by the imaginary part of its eigenvalue and the source-mode convolution contribution. However, the TMP along a fiber was shown to decay at infinity algebraically and not exponentially, as predicted by the classic cable equation solution. In the EPS case, the dominant contribution is expressed as a source-mode convolution integral. However, for a long EPS distance (e.g., >10 cable length constant) the order-one-modes involved in the convolution is not a solution of the cable equation. Only for shorter EPS distance should the cable equation solution (i.e., the order zero dominant mode) be included in addition to the modes of order one. For on-membrane EPS location, additional modes should be included as well. In view of our EPS result, we suggest that the cable equation modeling existing in the literature and related to functional electrical stimulation for EPS problems, should be critically reviewed and corrected.

Current distribution in skeletal muscle activated by functional electrical stimulation: image-series formulation and isometric recruitment curve.

AbstractThe present work develops an analytical model that allows one to estimate the current distribution within the whole muscle and the resulting isometric recruitment curve (IRC). The quasistatic current distribution, expressed as an image series, i.e., a collection of properly weighted and shifted point-source responses, outlines an extension for more than three layers of the classical image theory in conductive plane-stratified media. Evaluation of the current distribution via the image series expansions requires substantially less computational time than the standard integral representation. The expansions use a unique recursive representation for Greens function, that is a generic characteristic of the stratification. This approach permits one to verify which of the tissue electrical properties are responsible for the current density distribution within the muscle, and how significant their combinations are. In addition, the model permits one to study the effect of different electrode placement on the shape and the magnitude of the potential distribution. A simple IRC model was used for parameter estimation and model verification by comparison with experimentally obtained isometric recruitment curves. Sensitivity of the model to different parameters such as conductivity of the tissues and activation threshold was verified. The resulting model demonstrated characteristic features that were similar to those of experimentally obtained data. The model also quantitatively confirmed the differences existing between surface (transcutaneous) and implanted (percutaneous) electrode stimulation.

Generalized cable equation model for myelinated nerve fiber

Herein, the well-known cable equation for nonmyelinated axon model is extended analytically for myelinated axon formulation. The myelinated membrane conductivity is represented via the Fourier series expansion. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficients, known as Hills equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the entire periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent nonmyelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Hills equation is explicitly integrated for the specific choice of piecewise constant membrane conductivity profile, thereby resulting in an explicit closed form expression for the transmembrane potential in terms of trigonometric functions. The Floquets modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model, provided that the periodic membrane point-wise passivity constraint is properly modified. Indeed, the modified condition, enforcing the periodic membrane passivity constraint on the average conductivity only leads, for the first time, to the inclusion of the nerve fiber activation modes in our novel model. The validity of the generalized transmission-line and cable equation models for a myelinated nerve fiber, is verified herein through a rigorous Greens function formulation and numerical simulations for transmembrane potential induced in three-dimensional myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.

Augmentation of Dilated Failing Left Ventricular Stroke Work by a Physiological Cardiac Assist Device

Abstract: A novel physiological cardiac assist device (PCAD), otherwise known as the LEVRAM assist device, which is synchronized with the heartbeat, was developed to assist the left ventricle (LV) in chronic heart failure (CHF). The PCAD utilizes a single cannula, which is inserted in less than 15 s through the apex of the beating LV by means of a specially designed device. Blood is withdrawn from the LV into the PCAD in diastole and is injected back to the LV, through the same cannula, during the systolic ejection phase, thereby augmenting stroke volume (SV) and stroke work (SW). CHF with dilated LV was induced in sheep by successive intracoronary injections of 100‐μm beads. The sheep (92.2 ± 25.9 kg, n= 5) developed stable CHF with increased LV end‐diastolic diameter (69.4 ± 3.3 mm) and end‐diastolic volume (LVEDV = 239 ± 32 mL), with severely reduced ejection fraction (23.8 ± 7.6%), as well as mild‐to‐moderate mitral regurgitation. The sheep were anesthetized, and the heart was exposed by left thoracotomy. Pressure was measured in the LV and aorta (Millar). The SV was measured by flow meters and the LV volume by sonocrystals. Assist was provided every 10 regular beats, and the assisted beats were compared with the preceding unassisted beats, at the same LVEDV. The PCAD displaced 13.6 ± 3.4 mL, less than 8% of LVEDV. Added SW was calculated from the assisted and control pressure‐volume loops. The efficiency, defined as an increase in SW divided by the mechanical work of the PCAD, was 85.4 ± 16.9%. We conclude that the PCAD, working with a small displaced blood volume in synchrony with the heartbeat, efficiently augments the SW of the dilated failing LV. The PCAD is suggested for use as a permanent implantable device in CHF.

Generalized transmission-line model for myelinated nerve fiber

The validity of the generalized transmission-line and cable equation models for myelinated nerve fiber, is verified herein through rigorous Greens function formulation for transmembrane potential induced in a 3D myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.

Force, Sarcomere Shortening Velocity and ATP-ASE Activity

UNLABELLED We have tested the hypothesis that the transition rate (G) of the cardiac XB from the strong force generating state to the weak state is a linear function V of the sarcomere (VSL); furthermore, we tested whether the ATPase rate of the two isoforms of myosin can be held responsible for the difference between V0 of rat cardiac trabeculae containing V1 isomyosin versus those containing V3 isomyosin. METHODS V1 isomyosin was induced by thyroid hormone treatment of the rats for 2 weeks, V3 isomyosin by PTU treatment for 1 month. Force was measured with a strain gauge in trabeculae from the rat right ventricle in K-H solution ([Ca]o=1.5 mM, 25 degrees C). Sarcomere length (SL) was measured with laser diffraction techniques. Twitch force at constant SL, and the force response to shortening at constant VSL (0-8 microm/s; deltaSL 50-100 nm) were measured at varied time during the twitch. RESULTS The force response to shortening consisted of a fast initial exponential decline (tau = 2 ms) followed by a slow decrease of F. The instantaneous difference (deltaF) between isometric force (FM) and the declining force depended on shortening duration (deltat), VSL and instantaneous FM: deltaF = G1 x FM x deltat x VSL x (1-VSL/VMAX), where VMAX is the unloaded VSL and G1 was 6.15 +/- 2.12 microm(-1) (mean +/- s.d.; n=6). deltaF/FM was independent of the time onset of shortening. G1 of V1 and V3 trabeculae did not differ. V0 of V1 and V3 trabeculae differed 2-2.5 fold, as did both the ATPase rate and the velocity of actin sliding in a motility assay of the myosin purified from V1 or V3 hearts. The temperature dependence of the ATPase rate (Q10: 4.03 and 4.33, respectively; n.s.) was similar to that of V0 that has previously been reported for predominantly V1 trabeculae. Cross-linking of actin to myosin with the short chain cross linker EDC increased the ATPase rate of the two isomyosins (200-fold and 600-fold respectively) to exactly the same final level and reduced their Q10 by 50%. CONCLUSION The linear interrelation between deltaF and VSL is consistent with feedback, whereby XB kinetics depends on VSL. This feedback provides an integrated description of cardiac muscle mechanics and energetics. The results, also, suggests that it is unlikely that the hydrolytic domain of the cross bridge determines V0 and warrant ongoing experiments to investigate the role of the actin binding domain of the XB in cardiac sarcomere kinetics. In order to further investigate the role of the actin binding domain, we have expressed chimeric cardiac myosin, co-assembled with MLC, by mutual substitution of actin binding loop on alpha MHC and beta MHC.

Transmission-Line Model for Myelinated Nerve Fiber

Herein, the well-known cable equation for non-myelinated axon model is extended analytically for myelinated axon formulation. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficient, known as Hills equation. Hills equation exhibits periodic solutions, known as Floquets modes. The Floquets modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model