Basic Applied Myology 19 (5&6): 237-242, 2009
Modeling the excitability of the denervated human thigh for the stimulation with surface and needle electrodes
Yvonne Stickler (1), Johannes Martinek (1,2), Martin Reichel (2), Frank Rattay (1)
(1) Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria; (2) Center of Biomedical Engineering and Physics, Medical University of Vienna, Austria.
Abstract
3D models of the human thigh based on CT data or with simplified geometries are combined with a muscle fiber compartment model for analyzing how excitation depends on electrode position, stimulus strength and rehabilitation status. Assuming idealized conditions, excitation at a fiber ending obeys a quadratic current distance law whereas it is cubic elsewhere, resulting in easier central spike initiation for electrode-fiber distances up to the fiber’s length constant (~1/3 mm). Realistic conditions like fiber tapering, reduced sodium channel density towards the fiber end or inhomogenities in extracellular conductances (e.g. fat-muscle mix) keep up central excitability.
Key Words: denervated muscle, muscle fiber, functional electrical stimulation, needle EMG, finite element model, activating function, threshold
Basic Applied Myology 19 (5&6): 237-242, 2009.
T
he complete denervation of muscles leads to changes in the muscle fibers as well as in the surrounding tissue. Concerning the excitability of long term denervated muscles the most important changes are reductions in fiber diameter, in muscle cross sectional area, and in electrical conductivity of the muscle tissue [3,9]. These changes can be partially reversed by intensive training with electrical stimulation over years [7,8]. Long disuse has led to very poor excitability. Therefore, in the early stages of training very long impulses are needed to actually produce single twitches. After the excitability has improved due to the training, titanic contractions can be achieved with trains of shorter impulses. Force training can start and patients can even regain stand up function [6].Stimulation needle electromyography is an invasive technique that provides valuable information about the status of denervated muscles like strength-duration relations, refractory period, and conduction velocity of single muscle fibers [5].
Concerning excitability, 3D finite element simulations of the thigh are a helpful tool for a quantitative analysis of threshold values and the fiber excitation process. Furthermore, a theoretical approach with a point source can provide additional information in a qualitative way. The aim of the following
simulations is to gain some insight into the relations between denervation, training effects, and the excitation process of denervated skeletal muscle fibers.
Materials and Methods
The effects of the alterations due to denervation are examined with the coupling of a 3D finite element model of the human thigh and a muscle fiber compartment model. The thigh is assumed to be axial symmetric and consists of compartments for bone, muscle tissue, and fat&skin, each of them having a different electrical conductivity (Fig. 1).
The electrical stimulation of the muscle tissue is simulated with the appliance of biphasic rectangular pulses of 5 ms per phase via two large surface electrodes. One geometric model, the poor trained thigh, represents a thigh in denervated condition after a short time of training, the other one, the well trained thigh with a comparatively larger cross sectional area, denotes an advanced state of training. Besides cross sectional area, a higher electrical conductivity of the muscle tissue and an increased muscle fiber diameter are assigned to the well trained thigh model to simulate the improvement of the muscle structure.
Threshold values are computed for muscle fibers in the rectus femoris and vastus intermedius region (Fig.
1) by coupling the finite element model to a muscle
Fig. 1 3D model of the poor trained and well trained human thigh with selected fiber paths (thick curves in the muscle tissue area) and two large surface electrodes. As consequence of the training, the enlarged muscle volume causes an increase of thigh radius as marked by arrow.
Fig. 2 Range of improvement of the muscle fibers in percent of threshold values when simulating a training effect with a 50% larger cross sectional area of muscle tissue, a 2.5 times higher muscle tissue conductivity and a 4 times increased fiber diameter. (a): 3D-view of the well trained thigh model with muscle fibers.
Distal fiber endings are highlighted with points and labelled with their range of improvement according to the table in (b). The arrow marks the fiber for which subsequent calculations on boundary conditions are made. (b) 2D-view of highlighted fiber endings projected onto the gray shaded part of the cross section in (a).
fiber compartment model described by Wallinga and coworkers [14].
Furthermore, excitation is analyzed with the help of the activating function [11]. For every compartment the driving force resulting from the external potential along the fiber is defined as the activating function [8,9]. Muscle fiber compartment models usually assume fibers to be homogeneous in terms of ion channel distribution and diameter and to be perfectly sealed at the end. These assumptions make the fiber endings likely to produce action potentials when electrically stimulated, at least for larger distances [12].
However, biopsies have often shown a tapering in muscle fiber diameter towards the tendon [1,10]. Also, ion channel distribution is known to be highly inhomogeneous. In denervated like in innervated fibers, there is a sharp decrease in voltage-dependent sodium channels towards the end of the fiber [2].
Additionally a current leakage at the fiber end might have an influence on the fiber excitation process [10].
In the following it is tested whether the appliance of these more realistic boundary conditions effects the
simulated excitation process in a qualitative or quantitative way. To investigate the excitation process along a fiber and strength-duration relations in the near field of the fiber, the geometry of the thigh is modeled with an inserted needle electrode and a reference electrode on the surface. By coupling the calculated potential distribution to the compartment model threshold values for straight fibers with increasing fiber-electrode distances are obtained. Due to the different geometric scales of thigh and needle electrode 3D finite element calculations are extensive. For a simpler analysis we substitute the needle by a point source in an infinite homogeneous medium and examine the excitation process in straight fibers.
Moving the electrode with constant distance along the fiber axis causes surprising threshold effects as consequence of the individual contribution of the fiber end compartment. Primarily, these phenomena do not depend on ion channel dynamics but on geometric relations. They are studied with linear analysis (that is with constant membrane conductivity) as well as with a simple muscle fiber model [4].
Basic Applied Myology 19 (5&6): 237-242, 2009
Fig. 3 Development of an action potential at the fiber end (bold line), propagating action potential at 2 mm distance (solid line) and applied biphasic voltage pulse (in Volt, dotted line). The insert shows the corresponding inward sodium current per cm2 of membrane area at the fiber end (bold line) and 2 mm away (solid line). (a):
Excitation of the fiber end region with constant sodium channel density. No excitation occurs when applying the same amount of voltage to a fiber with reduced sodium channel density at the fiber end (b), whereas a stronger pulse again leads to excitation (c).
Fig. 4 Development of an action potential at the fiber end (bold line), propagating action potential at 2 mm distance (solid line) and applied biphasic voltage pulse (in Volt, dotted line). The insert shows the corresponding current leakage per cm2 of membrane area at the fiber end.
Comparison of a fiber with sealed end condition (a) and a fiber with current leakage at the end (b). Superimposing of the realistic boundary conditions leads to 3 times higher threshold values (c).
Fig. 5 Isopotential surfaces in the thigh for a cathodic 1 Volt pulse with a detailed view of the needle and its surrounding area.
Fig. 6 Threshold values when applying biphasic (anodic/cathodic) 50+50 µs pulses. (a): 2D cross section at z=1 (1 mm above the needle tip) with the needle electrode cross section of radius 0.2 mm located in the origin and three fiber segments at 0.05, 0.8 and 1.6 mm shortest surface-to-surface needle-fiber distance.
Symbols on the 0.8 mm distant fiber indicate the three assumed fiber end positions corresponding to cases A (filled circle), B (square) and C (triangle) for all calculated fibers. (b) Thresholds for muscle fibers in case A (black line), case B (light gray line) and case C (gray line) computed for eight surface-to- surface distances from 0.05 to 1.6 mm. The symbols mark the thresholds for fiber ending positions corresponding to (a). Insert: 3D-view of the needle electrode containing the 2D-plane of (a). The lighter gray colored part of the needle corresponds to its active area, the darker one is part of the 3 cm long needle shaft.
Fig. 7 Straight fiber excited by a point source. (a) geometry and electric field, (b) subthreshold membrane response after a cathodic 100 µs pulse for central stimulation (case 1, gray curve, xel = 0.5 cm) and cases 2-4 where the left fiber end is shifted to positions as marked by vertical arrows, (c) extracellular potential and its first (d) and second derivative (e).
Fig. 8 Peak excitation values at the end of monophasic cathodic or anodic 100 µs pulses when the point source is moved along the fiber axis in the far field.
Fig. 9 Peak excitation values at the end of monophasic cathodic or anodic 100 µs pulses when the point source is moved along the fiber axis in the near field.
Simulations reflect the clinical findings that long term intensive electrical stimulation training with large surface electrodes leads in all patients to lower threshold values. Model analysis including single parameter variation demonstrates that the amount of improvement depends on the position of the fiber in the thigh (Fig. 2).
Single parameter variation reveals that the increase in fiber diameter is the main factor for the improved excitability. Higher electrical conductivity and increased cross sectional area can lead to lower as well as higher thresholds dependent on the position of the fiber in the thigh due to the fact that the corresponding shifts in equipotential surfaces change the regions of likely excitation.
The effects of different boundary condition assumptions were examined for a fiber beneath the fat&skin layer (marked with an arrow in Fig. 2a). The classical sealed end boundary condition with constant ion channel density and constant fiber diameter makes the fiber end region likely to produce action potentials (Fig. 3).
Assuming a decrease in sodium channel density to 10
% of its mid fiber value over the last several hundred micrometers makes the fiber less excitable, because it reduces the inward sodium current that depolarizes the fiber during the excitation process. Although the shape of the membrane voltage stimulated with a higher voltage pulse hardly resembles an ordinary action potential, the signal is able to propagate along the fiber (Fig. 3). Simulating a current leakage at the fiber end over a resistance of 0.4 kOhm cm2 leads to no significant changes: although the leakage counteracts the fiber depolarization, threshold is reached with nearly the same amount of voltage (Fig. 4.). On the contrary, a decrease in fiber diameter to half of its value towards the fiber end reduces the activating function and therefore leads to considerably higher thresholds. In case of superimposing the realistic boundary conditions it is still possible to initiate a spike at the fiber ending, but the required voltage pulse is about 3 times higher (Fig. 4.).
Muscle fiber excitation in the thigh was also modeled with a needle electrode with an active area of 5 mm2.
According to experimental procedures [5] the needle electrode is inserted into the thigh in the vastus intermedius region and a reference electrode is placed on the surface. Isopotential surfaces in the thigh were calculated for a cathodic 1 Volt pulse (Fig. 5).
The following investigations underline differences for spike initiation at the fiber end (in the first compartment) and anywhere else (central excitation) resulting in easier central excitation as long as the fiber-electrode distance is rather short. Thresholds for applied biphasic voltage pulses of 50 µs per phase were calculated for straight fibers lying in a (x/y) plane
Basic Applied Myology 19 (5&6): 237-242, 2009
1 mm above the needle tip (Fig. 6). Shortest surface-to- surface distances between needle and fiber vary from 0.05 to 1.6 mm. For all distances three different cases were considered (Fig. 6): Bypassing fibers with no ending close to the needle (case A), fibers with endings at the shortest surface-to-surface distance position (case B) and fibers that do not pass the needle but are terminating 1 mm lateral to the shortest surface-to- surface distance position to the needle (case C).
Despite the fact that fibers according to case A are activated in their central regions while case B fibers are most easily activated in their terminal compartment, the thresholds are almost identical. The values range between about 2.7 V for a distance of 0.05 mm and more than 80 V at 1.6 mm away (Fig. 6b). For short distances, thresholds for case A fibers are lower than for case C fibers. But with increasing distance, the thresholds for case A fibers become comparatively higher. Invasive techniques like stimulation needle electromyography have to limit applied voltage pulses in order to prevent tissue damage. Applying for instance a 40 Volt 100 µs pulse means that 1 mm is the greatest distance where central excitation (case A) occurs. By contrast, at a distance of 1.6 mm the needed threshold to trigger an action potential for fibers corresponding to case C is about 23 V and therefore well within the safety limits.
With increasing distance the electric field produced by a needle electrode resembles more and more that of a point source. Therefore we analyzed the excitability at the fiber ending in comparison to central spike inititiation also with a theoretical model [12]. We assume a homogeneous infinite medium with a straight finite fiber and a small spherical electrode which is under these assumptions physically equivalent to a point source. Figure 7 shows the electric field generated by a -100 µA pulse (a), the transmembrane voltage response (b), the extracellular voltage profile Ve along the fiber axis (c) as well as its first and second derivatives (d, e) which are proportional to the
‘terminal activating function’ and to the ‘activating function for the central compartments’ respectively, when the electrode and the fiber have a center-center distance of 1 mm and extracellular resistivity ρe = 0.45 kOhm cm. Note, the terminal activating function drives solely the left most and the right most compartment (but the far right fiber end is not shown). According to Fig. 7d it has a small influence for case 1 (small negative value at the left end of the gray curve in (b)), whereas it is much stronger in case 2. However, this strong negative excursion resulting from the terminal activating function in case 2 (comp. corresponding downwards arrow in (d)) causes less excitation under the electrode in comparison with case 1 and 3 (Fig.
7b). A much stronger excitation appears for case 4, because the terminal activating function has now the same polarity as the central activating function. The
differing x-distances from the electrode to the left fiber end result in different excitation profiles (Fig. 7b) and consequently the threshold currents for cases 1-4 are: - 4780 µA, -5580 µA, -4780 µA, and -2800 µA. Case 1 and 3 have the same thresholds because the small terminal activating function value cannot influence central excitation in case 1 and in case 3 the terminal activating function value is zero. The mismatch of terminal and central activating function contribution is clearly seen when the electrode is moved along the fiber axis starting from a position 0.2 cm in front of the fiber (Fig. 8). The region where the fiber is more sensitive to terminal excitation than to the central one depends essentially on the radial fiber electrode distance. The terminal influences decrease with the radial distance and lose their significance in the near field (Fig. 9). The border between near and far field is defined by the fiber’s length constant, which is 330 µm for the 40 µm diameter fiber and the presented model assumptions. For further analysis see Rattay 2008 [12].
Discussion
Simulations show that the training effects due to functional electrical stimulation improve the excitability of denervated muscle fibers. The amount of improvement depends on the position of the fiber in the thigh and the increased muscle fiber diameter is the main factor for better excitability. Applying more realistic assumptions about the fiber end conditions alters computed threshold values significantly.
However, simulations with reported values for sodium channel density and leakage [2,10] demonstrate that action potentials can indeed be initiated at the fiber ending under these assumptions. Further investigations of the membrane properties of the myotendinous region of denervated muscle fibers need to be done to make sure that all relevant factors involved in the fiber excitation process are taken into account. Mainly, capacitive effects and the increased membrane surface area at the myotendinous junction might also influence the excitation process in the fiber end region.
Simulations with a needle electrode reveal that thresholds rise rapidly with increasing distance and excitation with pulses within the safety limits is only likely to occur within a few millimetres of fiber-needle distance. Besides distance, the fiber end-needle position is crucial for excitation. For short distances excitation occurs centrally, while with increasing distance terminal excitation is comparatively easier.
However, other kinds of inhomogeneities like irregularities of the electric field as consequence of local disturbances of conductivities caused by, e.g., small regions of fat tissue also effect the fiber excitation process and can lower thresholds for central excitation significantly [13].
More detailed calculations with a point source model show that the fiber ending is very sensitive concerning
region. This sensitivity was explained with the help of the two types of activating functions, terminal and central. As a positive value of the activating function depolarizes a region, excitation is easy when both activating function types contribute with a positive sign. Therefore, the superimposed contribution of central and terminal activating function has its most depolarizing effect for fiber endings in certain positions near the electrode, a phenomenon that becomes more pronounced with increasing distance.
Further calculations with the needle electrode model are needed to obtain quantitative results and to incorporate possible effects of the needle geometry.
Acknowledgements
This study was supported by the Austrian Science Fund (FWF) research project P18848-N13.
Corresponding Author
Yvonne Stickler, DI - Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10 A-1040 Vienna, Austria
E-mail: yvonne.stickler@tuwien.ac.at References
[1] Boriek AM, Miller CC 3rd, Rodarte JR. Muscle fiber architecture of the dog diaphragm. J Appl Physiol. 1998; 84: 318-26.
[2] Caldwell J H, Milton R L. Sodium channel distribution in normal and denervated rodent and snake skeletal muscle. J Physiol 1988; 401: 145–
161.
[3] Eberstein A, Eberstein S. Electrical stimulation of denervated muscle: is it worthwhile? Med Sci Sports Exerc 1996; 28: 1463-1469.
[4] Henneberg K A, Roberge F A. Simulation of propagation along an isolated skeletal muscle fiber in an isotropic volume conductor. Ann Biomed Eng 1997; 25: 5-28.
[5] Hofer C, Forstner C, Mödlin M, Jager H, Mayr W, Kern H. In vivo assessment of conduction velocity and refractory period of denervated muscle fibers. Artif Organs 2005; 29: 436-439.
Fano G, Zanin ME, Podhorska-Okolow M, Protasi F, Carraro U. Long-term denervation in humans causes degeneration of both contractile and excitation-contraction coupling apparatus, which is reversible by functional electrical stimulation (FES): a role for myofiber regeneration? J Neuropathol Exp Neurol 2004;
63: 919-931.
[7] Kern H, Hofer C, Mödlin M, Forstner C, Mayr W, Richter W. Functional electrical stimulation (FES) of long-term denervated muscles in humans: clinical observations and laboratory findings. Basic Appl Myol 2002; 12: 291-297.
[8] Kern H, Rossini K, Carraro U, Mayr W, Vogelauer M, Hoellwarth U, Hofer C. Muscle biopsies show that FES of denervated muscles reverses human muscle degeneration from permanent spinal moto-neuron lesion. J Rehabil Res Dev 2005; 42: 43-53.
[9] Midrio, M. The denervated muscle: facts and hypotheses. A historical review. Eur J Appl Physiol 2006; 98: 1-21.
[10] Milton R L, Mathias R T, and Eisenberg R S.
Electrical properties of the myotendon region of frog twitch muscle fibers measured in the frequency domain. Biophys J 1985; 48: 253–267.
[11] Rattay F. Analysis of models for external stimulation of axons. IEEE-Trans. Biomed. Eng.
BME 1986; 33: 974-977.
[12] Rattay F. Current distance relations for fiber stimulation with pointsources. IEEE Trans Biomed Eng. 2008; 55: 1122-1127.
[13] Stickler Y, Martinek J, Hofer C, Rattay F. A finite element model of the electrically stimulated human thigh: changes due to denervation and training. Artif Organs, in press.
[14] Wallinga W, Meijer SL, Alberink MJ, Vliek M, Wienk ED, Ypey DL. Modelling action potentials and membrane currents of mammalian skeletal muscle fibres in coherence with potassium concentration changes in the T-tubular system.
Eur Biophys J 1999; 28: 317-329.
