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[11] M. Rahal, J. Taylor, and N. Donaldson, “The effect of nerve cuff geometry on interference reduction: A study by computer modeling,” IEEE Trans. Biomed. Eng., vol. 47, pp. 136–138, Jan. 2000. [12] R. Merletti and L. Conte, “Advances in processing of surface myoelectric signals: Part 1,” Med. Biol. Eng. Comput., vol. 33, pp. 362–372, 1995. [13] H. Reucher, J. Silny, and G. Rau, “Spatial filtering of noninvasive multielectrode EMG: Part II—Filter performance in theory and modeling,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 106–113, Feb. 1987.

How Electrode Size Affects the Electric Potential Distribution in Cardiac Tissue

Fig. 1. The 40 mm 16 mm 16 mm slab of cardiac tissue with a 1 V potential difference between the two electrodes. The fibers are in the -direction.

Salil G. Patel and Bradley J. Roth* Abstract—We investigate the effect of electrode size on the transmembrane potential distribution in the heart during electrical stimulation. The bidomain model is used to calculate the transmembrane potential in a three-dimensional slab of cardiac tissue. Depolarization is strongest under the electrode edge. Regions of depolarization are adjacent to regions of hyperpolarization. The average ratio of peak depolarization to peak hyperpolarization is a function of electrode radius, but over a broad range is close to three. Index Terms—Bidomain, cardiac, electrode, stimulation.

I. INTRODUCTION The size of the stimulus electrode affects the excitation threshold in cardiac tissue [1]–[3]. In this paper, we calculate the distribution of transmembrane potential during electrical stimulation and examine how it depends on electrode size. Important questions arise when considering large electrodes. For instance, the electrical potential is constant over the surface of a circular electrode, but the current density is highest at its edge [4]. What are the implications of the large current density at the electrode edge for the transmembrane potential? In cardiac tissue, stimulation with a point electrode creates adjacent regions of depolarization and hyperpolarization [5]. Do such regions also arise for large electrodes? Previous simulations predicted a ratio of cathodal to anodal stimulus thresholds of about ten [6], whereas experiments often yield a ratio of about three. Is this discrepancy caused by the electrode size?

equivalent to a multidimensional cable model and can be represented by a network of resistors and capacitors [8]. The model consists of two coupled partial differential equations

(1) (2)

where is the ratio of membrane surface area to tissue volume (0.3 m ) and is the membrane conductance (1.65 S/m ) [9]. The intracellular and interstitial conductivity tensors, and , specify the anisotropic electrical properties in each direction: 0.1863 S/m, 0.0186 S/m, and 0.0745 S/m [10]. The calculation is for a passive membrane in steady state, and represents deviations of the transmembrane potential from rest. The myocardial fibers are straight and lie in the direction. The edges of the tissue are sealed

(3)

and the boundary conditions at the electrode-tissue interface are

(4) (5)

II. METHODS Our model describes a three-dimensional slab of cardiac tissue. Two circular electrodes, each of radius , are on opposite sides of the slab, with a potential difference of 1 V between them (Fig. 1). We use the bidomain model [7] to calculate the transmembrane potential, , and the interstitial potential, . The bidomain model is

Manuscript received October 6, 1999; revised May 15, 2000. This work was supported by the National Institutes of Health (NIH) under Grant RO1HL57207. Computing support was provided by the School of Engineering and Computer Science, Oakland University. Asterisk indicates corresponding author. S. G. Patel is with the G.W.C. School of Engineering, Johns Hopkins University, Baltimore, MD 21218 USA. *B. J. Roth is with the Department of Physics, Oakland University, 190 SEB, Rochester, MI 48309 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9294(00)08005-8.

where is the direction normal to the surface and is the voltage of the electrode. This formulation of the boundary conditions implies that the tissue is stimulated using a voltage source, not a current source. To determine and , we discretize the tissue into a grid of points and apply the method of successive overrelaxation. The dimensions of the tissue are 40 mm 16 mm 16 mm. By using even symmetry in the and directions and odd symmetry in the direction, we can calculate and over just one-eighth of the tissue. Thus, for 199 the space step is 0.10 mm parallel to the fibers and 0.04 mm perpendicular to the fibers (compared with a passive length constant of 0.43 mm parallel to the fibers and 0.17 mm perpendicular to the fibers). We studied electrodes with radii ranging from 0.125–8 mm.

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Fig. 3. Plot of peak depolarization, , and peak hyperpolarization, , vs. electrode radius, , for a) constant voltage (1 V), and b) constant current (0.1 mA). The error bars represent the difference between the results 199 and the results for 159. for

Fig. 2. The transmembrane potential on the surface of the tissue as a function of and for 3 different electrode sizes: 0.5, 1, and 2 mm. The electrode size is indicated by the dashed curve. The fibers are horizontal. The center 8 mm 8 mm area of the 40 mm 16 mm tissue surface is shown. The potential difference is 1 V, corresponding to a stimulus current of 1.15 mA ( 0.5 mm), 2.52 mA ( 1 mm), and 5.46 mA ( 2 mm).

III. RESULTS Fig. 2 shows the transmembrane potential versus and on the surface of the tissue for three different electrode radii: 0.5, 1, and 2 mm. For each electrode size, the tissue is depolarized (yellow) under the electrode and hyperpolarized (blue) in adjacent regions along the fiber direction. The depolarization is strongest under the edge of the electrode, and weaker under the center, particularly for the larger electrode sizes.

Fig. 3(a) shows the peak depolarization ( ) and peak hyperpolarization ( ) as a function of electrode radius. For radii greater than 0.5 mm, both polarizations decrease with increasing electrode radius. However, for very small electrode sizes both increase as the electrode radius increases. This increase reflects the increasing stimulus current. The stimulus current depends on the electrode radius because the electrodes are attached to a voltage source. A smaller electrode radius implies a larger total resistance and, therefore, a smaller stimulus current. The error bars in Fig. 3 represent the difference between the peak polarization calculated with 199 and with 159 and, therefore, indicate the approximate size of discretization errors in our calculation. Fig. 3(b) shows and calculated by adjusting the voltage between the electrodes so that the stimulus current is 0.1 mA (corresponding to a current source rather than a voltage source). In this case, the polarization is a strongly decreasing function of electrode radius. Fig. 4 contains a plot of the ratio versus electrode radius. This ratio has a minimum of 2.49 for 1.4 mm. It increases slowly for larger radii and increases more steeply for smaller radii. For the smallest radius examined ( 0.125 mm), is 7.8. (Analytical models suggest that for a point electrode and, therefore, the ratio , goes to infinity [5].) For the largest radius examined ( 8 mm), equals 3.3.


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Fig. 4.

Plot of the ratio of peak depolarization to peak hyperpolarization, , as a function of electrode radius, . The error bars represent the difference between the results for and the results for .

IV. DISCUSSION The change in size of the electrode has important ramifications for the transmembrane potential. In particular, the maximum depolarization, maximum hyperpolarization, and stimulus current depend upon the size of the electrode. The transmembrane potential is largest near the electrode edge. Thus, the transmembrane potential is largest where Wiley and Webster [4] found the current density to be largest. The transmembrane potential distribution always contains adjacent regions of hyperpolarization and depolarization, similar to the results of Sepulveda et al. [5]. Thus, phenomena that arise because of adjacent regions of opposite polarization—such as break excitation [6], [11], the dip in the anodal strength-interval curve [12], [13] and the quatrefoil reentry [14], [15]—all may occur for a wide range of electrode sizes. Some of our results are analogous to the observations by Rattay [16] of excitation of a nerve axon under a surface electrode. In previous calculations using the bidomain model, the ratio of cathodal to anodal stimulus thresholds was about ten [6], whereas in many experiments this ratio is approximately three [17]–[23]. The data in Fig. 4 may partly explain this discrepancy. The ratio of peak depolarization to peak hyperpolarization in our passive model corresponds roughly to the ratio of anodal to cathodal threshold. (This correspondence is valid for large electrodes. When the electrode size is similar to or less than the tissue length constant, liminal length and wave front curvature make excitation more difficult than the peak depolarization would suggest [2], [24]). In Fig. 4, becomes very large for small electrode sizes, but approaches the experimentally observed value of about three for intermediate and large electrode sizes. Therefore, the difference in the theoretical and experimental ratios of anodal to cathodal threshold may arise from electrode size. A previous calculation indicated that this difference might be caused by the influence of a conducting bath adjacent to the tissue surface [9]. Both factors play a role, but our present calculation provides an explanation for the difference between theory and experiment even when a conducting bath is not present. Our calculations are subject to certain numerical errors, of which the most important type is discretization error. The influence of space step size can be assessed from the error bars in Figs. 3 and 4. The errors are largest for smallest electrodes, where the electrode radius and the space step are similar. The step size is particularly important in these calculations because the rectangular, Cartesian grid does not match well the circular geometry of the electrode. We reduced the space step size until computational constraints imposed by the large ( ) grid

made further reductions impractical. Another computational limitation is caused by the sealed boundaries of the tissue. We limited our electrode radius to 8 mm or less in a tissue with a surface area of 40 mm 16 mm to minimize boundary effects. Even for our larger electrode sizes, the boundary had a small influence on our results. For 8 mm (the largest electrode and, therefore, the case most prone to boundary effects), we repeated our calculation for a node tissue (that is, we increased the tissue size in the direction from 16 mm to 24 mm). The ratio changed by about three percent. Our simulations are limited by the physical assumptions underlying our model. We assume a uniform fiber geometry. However, in cardiac tissue the fibers are not necessarily straight, and the fiber direction rotates with depth. None of the parameters used in this calculation are known exactly. The evidence for unequal anisotropy ratios in cardiac tissue is strong [10], but the tissue conductivities are uncertain to some extent and the transmembrane potential does change with the conductivity [8]. The assumptions of a passive membrane and steady-state stimulation imply that our results are only suggestive of what may occur in the real tissue, where the time-dependent and nonlinear behavior of the membrane modifies the tissue response. Moreover, for large membrane polarizations (several hundred millivolts) electroporation plays a role in determining the transmembrane potential distribution [25]. Finally, we ignore electrode polarization impedance in this calculation [26]. REFERENCES [1] F. W. Lindemans, R. M. Heethaar, J. J. Denier van der Gon, and A. N. E. Zimmerman, “Site of initial excitation and current threshold as a function of electrode radius in heart muscle,” Cardiovasc. Res., vol. 9, pp. 95–104, 1975. [2] F. W. Lindemans and J. J. Denier van der Gon, “Current thresholds and liminal size in excitation of heart muscle,” Cardiovasc. Res., vol. 12, pp. 477–485, 1978. [3] F. W. Lindemans and A. N. E. Zimmerman, “Acute voltage, charge, and energy thresholds as functions of electrode size for electrical stimulation of the canine heart,” Cardiovasc. Res., vol. 13, pp. 383–391, 1979. [4] J. D. Wiley and J. G. Webster, “Analysis and control of the current distribution under circular dispersive electrodes,” IEEE Trans. Biomed. Eng., vol. BME-29, pp. 381–385, May 1982. [5] N. G. Sepulveda, B. J. Roth, and J. P. Wikswo, Jr., “Current injection into a two-dimensional anisotropic bidomain,” Biophys. J., vol. 55, pp. 987–999, 1989. [6] B. J. Roth, “A mathematical model of make and break electrical stimulation of cardiac tissue by a unipolar anode or cathode,” IEEE Trans. Biomed. Eng., vol. 42, pp. 1174–1184, Dec. 1995. [7] C. S. Henriquez, “Simulating the electrical behavior of cardiac tissue using the bidomain model,” Crit. Rev. Biomed. Eng., vol. 21, pp. 1–77, 1993. [8] B. J. Roth, “How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle,” J. Math. Biol., vol. 30, pp. 633–646, 1992. [9] D. C. Latimer and B. J. Roth, “Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath,” IEEE Trans. Biomed. Eng., vol. 45, pp. 1449–1458, Dec. 1998. [10] B. J. Roth, “Electrical conductivity values used with the bidomain model of cardiac tissue,” IEEE Trans. Biomed. Eng., vol. 44, pp. 326–328, Apr. 1997. [11] J. P. Wikswo, Jr., S.-F. Lin, and R. A. Abbas, “Virtual electrodes in cardiac tissue: A common mechanism for anodal and cathodal stimulation,” Biophys. J., vol. 69, pp. 2195–2210, 1995. [12] B. J. Roth, “Strength-interval curves for cardiac tissue predicted using the bidomain model,” J. Cardiovasc. Electrophysiol., vol. 7, pp. 722–737, 1996. [13] J. A. Bennett and B. J. Roth, “Time dependence of anodal and cathodal refractory periods in cardiac tissue,” Pacing Clin. Electrophysiol., vol. 22, pp. 1031–1038, 1999. [14] B. J. Roth, “Nonsustained reentry following successive stimulation of cardiac tissue through a unipolar electrode,” J. Cardiovasc. Electrophysiol., vol. 8, pp. 768–778, 1997. [15] S.-F. Lin, B. J. Roth, and J. P. Wikswo, Jr., “Quatrefoil reentry in myocardium: An optical imaging study of the induction mechanism,” J. Cardiovasc. Electrophysiol., vol. 10, pp. 574–586, 1999. [16] F. Rattay, “Modeling the excitation of fibers under surface electrodes,” IEEE Trans. Biomed. Eng., vol. 35, pp. 199–202, Mar. 1988.


IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 47, NO.9, SEPTEMBER 2000

[17] C. McC. Brooks, B. F. Hoffman, E. E. Suckling, and O. Orias, Excitability of the Heart. New York: Grune and Stratton, 1955. [18] P. F. Cranefield, B. F. Hoffman, and A. A. Siebens, “Anodal excitation of cardiac muscle,” Amer. J. Physiol., vol. 190, pp. 383–390, 1957. [19] E. Dekker, “Direct current make and break thresholds for pacemaker electrodes on the canine ventricle,” Circ. Res., vol. 27, pp. 811–823, 1970. [20] R. Mehra and S. Furman, “Comparison of cathodal, anodal, and bipolar strength-interval curves with temporary and permanent pacing electrodes,” Br. Heart J., vol. 41, pp. 469–476, 1979. [21] O.-J. Ohm, H. Mitamura, E. L. Michelson, C. Sauermelch, and L. S. Dreifus, “Ventricular tachyarrhythmia initiation in a canine model of recent myocardial infarction. Comparison of unipolar cathodal, anodal, and bipolar stimulation,” Cardiology, vol. 74, pp. 169–181, 1987. [22] P.-S. Chen, “Ventricular fibrillation is not an anodally induced phenomenon in open-chest dogs,” Amer. J. Physiol., vol. 262, pp. H365–H373, 1992. [23] W. Krassowska, D. L. Rollins, P. D. Wolf, E. G. Dixon, T. C. Pilkington, and R. E. Ideker, “Pacing thresholds for cathodal and anodal high frequency monophasic pulses,” J. Cardiovasc. Electrophysiol., vol. 3, pp. 64–76, 1992. [24] A. T. Winfree, “The electrical thresholds of ventricular myocardium,” J. Cardiovasc. Electrophysiol., vol. 1, pp. 393–410, 1990. [25] F. Aguel, K. A. DeBruin, W. Krassowska, and N. A. Trayanova, “Effects of electroporation on the transmembrane potential distribution in a two-dimensional bidomain model of cardiac tissue,” J. Cardiovasc. Electrophysiol., vol. 10, pp. 701–714, 1999. [26] M. R. Neuman, “Biopotential electrodes,” in Medical Instrumentation: Application and Design, J. G. Webster, Ed. New York: Wiley, 1998, pp. 183–232.

Sensitivity and Versatility of an Adaptive System for Controlling Cyclic Movements Using Functional Neuromuscular Stimulation Edward C. Stites and James J. Abbas* Abstract—This study evaluated an adaptive control system (the PG/PS control system [2]) that had been designed for generating cyclic movements using functional neuromuscular stimulation (FNS). Extensive simulations using computer-based models indicated that a broad range of control system parameter values performed well across a diverse population of model systems. The fact that manual tuning is not required for each individual makes this control system particularly attractive for implementation in FNS systems outside of research laboratories. Index Terms—Adaptive control, cyclic movements, functional neuromuscular stimulation, neural networks, sensitivity analysis.

I. INTRODUCTION Functional neuromuscular stimulation (FNS) is a technique that can restore neuromotor function through the controlled application of electrical stimuli to induce muscle contractions [25]. These techniques have been used for a variety of rehabilitation applications, such as Manuscript received December 23, 1999; revised April 4, 2000. This work was supported in part by the Whitaker Foundation and the Neural Prosthesis Program of the National Institute of Neurological Disorders and Stroke under Contract N01-NS-6-2351. Asterisk indicates corresponding author. E. C. Stites is with the Department of Mathematics, University of Kentucky, Lexington, KY 40506 USA. *J. J. Abbas is with the Center for Biomedical Engineering and the Department of Physical Medicine and Rehabilitation, Wenner-Gren Research Building, University of Kentucky, Lexington, KY 40506-0070 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9294(00)08007-1.

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the restoration of hand grasp function to individuals with tetraplegia resulting from cervical-level spinal cord injury and the restoration of standing/stepping function to individuals with paraplegia resulting from thoracic-level spinal cord injury [8], [13], [15], [17], [23], [25]. The musculoskeletal system presents many challenges to the design of effective FNS controllers. It is a dynamic system that exhibits nonlinearities due to muscle fiber recruitment properties, muscle length-tension and force-velocity properties, and passive joint properties [4], [11], [27]. In addition, the system exhibits substantial time delays [4]. Furthermore, and perhaps most importantly, the system properties are unpredictable due to high variability across a set of individuals and high variability within each person due to fatigue, electrode location and other factors. FNS systems that use prespecified stimulation patterns for standing and stepping require the efforts of a trained therapist or engineer in a trial-and-error process that is time-consuming and unreliable. Several research efforts have utilized feedback control strategies in order to improve the quality and repeatability of posture and movement control systems [1], [21], [22], [26]. Other efforts have investigated the use of adaptive control techniques that are intended to customize the stimulation parameters for each individual. Some adaptive algorithms work in an off-line mode where the parameters are trained (i.e., adjusted) using input/output data that had previously been collected and stored [6], [18], [20]. Adaptive algorithms that work in an on-line mode can potentially adjust to changes such as those that would be exhibited as muscles fatigue [5], [7], [9], [10], [12], [14]. It should be noted that adaptive algorithms may be capable of automatically adjusting stimulation parameters, but they still require the a priori specification of the control system structure, such as model order, and of some control system parameter values, such as adaptation rate. In previous work, we developed and implemented an adaptive neural network algorithm for controlling cyclic movements that utilizes a twostage structure: the pattern generator (PG) and the pattern shaper (PS) [2]. This PG/PS control system is capable of automatically customizing stimulation parameters and adjusting them on-line. In computer simulation studies controlling single-segment [2] and multisegment movements [16] the PG/PS control system automatically adapted stimulation parameters to control a variety of model systems. The PG/PS system was also effective in experiments on human subjects with thoracic-level spinal cord injury for the control of cyclic isometric contractions [3] and for the control of swinging leg movements [24]. In these studies, the PG/PS control system performed very well and it shows great promise for applications in FNS systems for stepping, motor retraining, and/or exercise conditioning. As with any control algorithm, the PG/PS controller requires that certain parameter values be specified prior to operation. Ideally, the controller should perform well on a wide variety of systems and performance should not be sensitive to the selected values for control system parameters. Previous simulation and experimental studies [2], [3], [24] utilized a prespecified set of controller parameters. The purpose of the work presented here ws to evaluate the potential utility of the PG/PS controller in FNS systems by systematically investigating the effect of control system parameter selection on performance. An extensive series of studies using computer-based models was used to evaluate the sensitivity of control system performance to control system parameter values across a varied set of musculoskeletal systems. II. METHODS The PG/PS neural network controller was interfaced with a computer-based model of the swinging lower leg. The controller determined stimulation patterns for a pair of electrically stimulated muscles to generate cyclic movements of the leg (Fig. 1). System model parameter values were varied to assess the versatility of the PG/PS controller

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