Abstract

The purpose of this computer simulation study was to compare the excitation effect of Functional Electrical Stimulation (FES) via an extracellular point source in relation to the longitudinal position of the electrode above a denervated skeletal muscle fiber. In one case the electrode was placed in a middle position (a) and in the other case it was located near the muscle-tendon junction (b). The threshold current for each case was calculated by a Hodgkin-Huxley type model of a single denervated muscle fiber referred to the distance of the electrode in radial direction. Results showed that there is an optimal longitudinal position of the electrode at the muscle-tendon junction with strongest excitation effect. Threshold current increases roughly with 3^rd power in case (a) compared to 2^nd power in case (b)referring to the radial distance of the electrode.

1. Introduction

Denervated skeletal muscles have to be stimulated directly because motor neurons are not available. Each single muscle fiber must be excited to achieve a proper contraction of the whole muscle. The exciting effect of, for example, an extracellular point source near a denervated muscle fiber can be obtained by the classic activating function [1],

(1)

which was introduced to calculate the extracellular influence of the electrical field V_e along a nerve fiber x. Further studies showed [2, 3], that inhomogeneities along the fiber like changing fiber diameter, membrane geometry or intracellular conductivity lead to specific characteristics of the activating function.

The junction between skeletal muscle and tendon (Fig. 1) consists of hemidesmosomes at the end of each muscle fiber as a mechanical connection to the collagen bundles of fibers of the tendon. At this location a very strong inhomogeneity referring to the muscle fiber can be observed because there is no continuity between muscle fiber and collagen fibrils.

Fig. 1: Image of a muscle-tendon junction (enlargement 4000); 1…finger-shaped endings of muscle fibers, 2…fibrocyts, 3…collagen fibrils, 4…erythrocyte.

It can be assumed, that no intracellular current is able to flow across the barrier between muscle fiber and collagen fibrils of tendon tissue. This consideration can lead to a substantial contribution of activating effect at the terminal region of the muscle fiber (Fig. 1), where the classic activating function for quasi infinite fiber length is negligible small.

2. Methods

Geometrically the muscle fiber represents roughly a long cylinder with constant radius from one end to the other (Fig. 2). The denervated skeletal muscle fiber can be modeled by a Hodgkin-Huxley type model [4] adapted to muscle [5] and to denervated conditions [6].

Fig. 2: Scheme of a whole muscle fiber divided into equally spaced compartments (fiber segments) connected to the collagen fibrils of the tendon with intracellular potentials V_i.

If the fiber endings are stimulated by Functional Electrical Stimulation (FES) specific boundary conditions have to be considered [6], as no current passes the barrier between muscle fiber and collagen fibrils. This means, that the entire axial current of the first 1 or the last N segment has to pass the membrane, respectively. The membrane current at that locations can be described as

(2)

where e.g. V_i,1 and V_i,2 are the intracellular potentials of the first and the second compartment; R_i is the intracellular resistance between two compartments.

Including Equ. 2, the numerical form of the cable equation for the first and the last compartment can be written as

(3)

where a is the radius of the fiber, r_i is the intracellular resistivity, Dx is the length of the compartment, V is the membrane potential, V_e is the extracellular potential, c is the membrane capacity, i_ion is the ion current density through the membrane and finally i_T is the muscle fiber specific tubular current density [5, 6]. For the other compartments membrane voltage is calculated by

(4)

The differential ratio of the extracellular potential V_e is the excitatory part of Eqn. 3 and results in the terminal activating function

(5)

which is the first derivative of the extracellular potential along the muscle fiber.

3. Results

The simplest model of the FES of a denervated muscle fiber is a straight single fiber with a cathodic point source in middle position of the fiber length. The spherical symmetrical electrical field around the point source produces an extracellular potential distribution along the fiber [2] which results in the normalized activating function calculated by Eqn. 1 and shown in Fig. 3.

Fig. 3: Normalized activating function along a 20mm part of a muscle fiber evoked by a cathodic point source at central position in a radial distance of 1mm.

Fig. 4: Normalized terminal activating function dependent on the distance x_e of the point source to the muscle-tendon junction for 3 different distances z in radial direction; positive values for x_e means electrode position beyond the end of the fiber.

Fig. 4 shows the terminal activating function which is normalized to the maximum of the classic activating function |f_p,max|. The characteristic for a point source of distance z (1, 2 and 4mm) in radial direction of the fiber dependent on the longitudinal distance x_e to the end of the fiber can be observed. The electrode position x_e,max for the maximum of the terminal activating function depends on the radial distance z by the relation

(6)

In difference to the classic activating function f (Equ. 1) the terminal activating function f’ (Equ. 5) effects only in the first or the last segment of the muscle fiber; everywhere else it is zero. The effective activation f in the first or last segment can be derived by multiplication of f’ with the weight a/2r_iDx (comp. Eqn. 3 and 4).

Fig. 5: Amplitude I of cathodic threshold current of 1ms pulse width for a point source located above the center of the fiber (a) and at the optimal position above the end of the fiber (b) referring to its radial distance z. Solid lines indicate calculated results from the model and dashed lines indicate the fitted cubic (a) and square (b) trend.

The amplitude I of threshold current versus radial distance z of the electrode to the muscle fiber is shown in Fig. 5. It can be observed, that the amplitude of a cathodic current pulse must be higher for an electrode positioned above the center of the fiber than for an optimally positioned (Equ. 6) above the end of the fiber. Referring to the radial distance z of the electrode the results can be fitted by a cubic curve for the middle position and a square curve for the position above the end of the fiber (Fig. 5).

4. Summary and Conclusions

The classic activating function (Equ. 1) alone is useful for calculating the excitation of long nerve fibers. In contrast to typical muscle fiber stimulation in many FES applications the structure of an involved nerve fiber is homogeneous and seems to be of infinite length at the region of interest. Also for single muscle fibers the classic activating function can be used, but only in the case of a very low radial distance z (up to 1mm) of the electrode to the fiber. For larger distances z of about more than 2mm the exciting effect from an electrode placed over the end of the muscle fiber, calculated by the terminal activating function (Equ. 5) is stronger than the effect from an electrode positioned in the middle with equal amplitude.

In most cases for FES in denervated muscle the electrode is much farther than 2mm from the target fibers. For this reason the exciting effect at the muscle-tendon junction must be stronger than anywhere else along the muscle fibers.

5. References

[1] F. Rattay, ”Modelling and simulation of electrically stimulated nerve and muscle fibres.” A review. Math. Comput. in Simulation, 29:357-366, 1987

[2] F. Rattay, Electric nerve stimulation: theory, experiments and applications, Springer-Verlag/Wien, 1990.

[3] F. Rattay, P. Lutter, H. Felix, “A model of the electrically excited human cochlear neuron 1. contribution of neural substructures to the generation and propagation of spikes.” Hearing Research, 153:43-63, 2001.

[4] A.L. Hodgkin, A.F. Huxley, “A quantitative discription of membrane current and its application to conduction and excitation in nerve.” J. Physiol. Lond. 117:500-544, 1952.

[5] K. Henneberg, F.A. Roberge, “Simulation of propagation along an isolated muscle fiber in an isotropic volume conductor.” Ann. Biomed. Eng. 25:15-28, 1997.

[6] M. Reichel, Funktionelle Elektrostimulation denervierter Skelettmuskulatur – Modellbildung und Simulation, Dissertation TU-Wien, 1999.