A Model of Nerve-Bundle Fibre-Stimulation using Implantable Cuff Electrodes

Transcription

1 UNSW GSBME 1 A Model of Nerve-Bundle Fibre-Stimulation using Implantable Cuff Electrodes Joseph Radford Abstract The aim of this study is to demonstrate radial selectivity when stimulating axons that innervate skeletal muscle. A model was constructed using COMSOL Multiphysics, consisting of a 3-D rotationally symmetrical cylindrical geometry to simulate the electric field resulting from electrode stimulation and two 2-D edges to simulate the action potential in the nerve fibre. Selectivity is achieved using an anodal block to stop large fibres from propagating their action potential. A trapezoidal pulse from the cathodic cuff electrode raises the electric potential, exciting the larger axons then the smaller ones. The larger axons also propagate faster as conduction velocity depends of radius, creating a spatial discrepancy. The anodal block is a smoothed square anodic pulse from an anodic cuff which is some distance from the cathodic cuff. It hyperpolarises the axons as the faster action potentials reach it, arresting them, and switches off before the slower action potentials reach it, letting them through. The model clearly demonstrates the differences in conduction velocity, and the spatial discrepancy between the action potentials as they reach the second, anodic cuff. However, there were many difficulties dealing with the attempted hyperpolarisation. These include the requirement for a reasonably long area of the axon to be hyperpolarised for the required amount of time. As the electrodes get longer and the anodic pulse gets stronger, a cathodic affect on the fringes of the plates is observed which can set off further action potentials. I. INTRODUCTION In functional electrode stimulation (FES), nerve fibres are recruited by raising them above their threshold, creating an action potential that then propagates to the muscles they innervate, providing some level of function for people suffering from spinal cord injuries, stroke, and other ailments that stop the conscious, uninhibited use of their muscle[1]. There are two issues that need addressing. The first is the surgical implantation of the electrodes. If the internal geometry of the nerve was well documented, it would be possible to utilise and design the electrodes for current steering [2] to activate the nerves required. However, during surgical implantation of either cuff electrodes or a helical coil, it is very difficult to implant it so that it aligns to the specific design for the particular internal geometry. This geometry will also be different across patients and locations in the body. Therefore, the stimulation needs to work no matter how the cuff is arranged rotationally with respect to the nerve. The second issue is that of nonphysiological recruitment. There are two groups of nerves: those of a greater diameter that innervate the type II fibres, and those of a smaller diameter that innervate the type I fibres[3]. In able-bodied humans, type I are recruited first as they are slow twitch and fatigue resistant. Type II fast twitch is only recruited if greater strength or speed is needed. However, under basic FES (a single cathodic stimulus), the type II fast twitch are recruited first. Due to their larger radius, they have a lower action potential threshold, and they also occupy a greater volume, making them more likely to exist in an electrical field above their threshold[1]. Type II fibre activation causes the user to become fatigued more quickly. Therefore, the two requirements of the model are that it works independently of the rotational position of the electrodes, and that it reverses the typical nonphysiological recruitment. A model that can do this is the anodal block method[4], [5], [6]. Utilising different threshold levels and conduction velocities based on the differing radii of the axons of interest[7], a spatial gap occurs between the action potentials for larger and smaller axons. At a distance from the original cathodic stimulating cuff, a second cuff applies anodic stimulation that hyperpolarises the axons. As with cathodic stimulation, the axons of a larger diameter are hyperpolarised first[5]. As the action potentials in the larger diameters arrive, they are arrested. Then, the anodic stimulation is switched off before the smaller axon action potentials arrive, allowing them to pass and activate the type I muscle fibres. As a result, the user experiences less fatigue from the FES system, and there is greater reliability as the process is independent of the rotational location of the surgically implanted electrodes. II. DESCRIPTION OF THE MODEL A simple cylindrical geometry was used to map and simulate the voltage fields in the nerve outside of the axon (the endoneurium, perineurium, epineurium and connective tissue), while a 2-D edge was employed to simulate the Hodgkin- Huxley model in the axons. The diagram is shown in Figure 1 (all units are in mm and ms). A. Volume Conductor While the conductivities of all the parts of the nerve are known to vary in mammals[8], [9], both in magnitude and anisotropy, there is little achieved in separating the outer nerve into many domains of different electrical characteristics. This would lend further accuracy to the way the voltage is distributed within the volume, but with the geometry left at a simple cylinder there is no point in doing this. Therefore, a simple one domain model was created, with radius of 1.5mm, conductivity σ = 1S/m and length of 20mm, as suggested by Laforet et al. [4].

2 UNSW GSBME 2 There is also the axoplasmic resistivity, ρ i which determines the speed of the action potential along the axon. Here we have taken the value as 60Ω.cm. The ionic current and membrane currents and the transmembrane voltage are governed by the activity of the ions, which are described by equations (2) - (13), taken from the literature. a) Potassium: Fig. 1. Basic wireframe of the geometry The electrodes are modelled as 2-D boundaries on the surface of the cylinder. As per the conditions, there are two rings of electrodes. In each ring there is are three sources and one ground, in an attempt to achieve a more uniform field. The first ring is an excitatory cathodic impulse. This raises the transmembrane voltage within the cylinder, and interacts with the PDE model (see section II-B). Its purpose is to raise the membrane potential above threshold, to start the action potential. The second ring is an anodal block, in that it lowers the transmembrane voltage within the cylinder. The intention is to hyperpolarise certain axons so action potentials cannot propagate past this point[4], [5], [6]. Finally, the membrane current from the 2-D axon within the domain must also be accounted for, as this will have an effect on the electric field within the 3-D domain. From the Hodgkin-Huxley model[7] we have equation (1) i m = dv m + i ion (1) The parameters are given in Table I. V m is the membrane potential, V i V e where V i is determined from the edge PDE and V e comes from the 3-D voltage mapping geometry. i ion is the total ionic current whose value is obtained using the method describe in the next section. B. Hodgkin-Huxley PDE The activity of the axon within the cylinder is modelled using the classic Hodgkin-Huxley model[7]. This is achieved by creating a 2-D edge inside the 3-D geometry, following a weak form of the PDE. The parameters are given in Table I. TABLE I PARAMETERS FOR THE HODGKIN HUXLEY MODEL IN THE 2-D EDGE[7] Parameter Value 1.0µF/cm 2 V Na 115mV V K 12mV V L mV g Na 120mS/cm 2 g K 36mS/cm 2 g L 0.3mS/cm 2 b) Sodium: g K = g K n 4 (2) dn = α n(1 n) β n n (3) V m + 10 α n = 0.01 exp V m+10 (4) 10 1 Vm β n = exp (5) 80 g Na = m 3 hg Na (6) dm = α m(1 m) β m m (7) dh = α h(1 h) β h h (8) V m + 25 α m = 0.1 exp V m+25 (9) 10 1 Vm β m = 4 exp (10) 18 Vm α h = 0.07 exp (11) 20 1 β h = exp V m (12) c) Total: The total ionic current is given by i ion = g K (V m V K )+g Na (V m V Na )+ g L (V m V L ) (13) Equation (1) can be expressed using Poisson s form: i m =.( σ e V ) i m = r a 2ρ i 2 V (14) where r a is the radius of the axon (6.5µm r a 10µm) Thus dv m = 1 (i m i ion ) (15) Which completes the system of PDEs to be evaluated, equations (4), (8), (9) and (15). III. METHOD - SETUP IN COMSOL MULTIPHYSICS A. Geometry The volume conductor is a cylinder with diameter 1.5mm and height 20mm. The first ring of electrodes are between x = [1, 2] and the second are between x = [6, 7]. This gap is suggested by Laforet et al. [4]. The electrodes themselves are in groups of four, each starting on one of the four longitudinal

3 UNSW GSBME 3 edges of the cylinder as a 2-D polygon. They are then revolved by one radian. The axons are defined as points on the y z planes at ± radius 3 along the y axis. They are then extruded the length of the nerve bundle. They have radii of r small = 6.5µm and r big = 10µm, the values reflecting the extremes between the nerves that innervate type I and type II fibres respectively [3]. These are the dependent variables that create the spatial discrepancy that the model exploits. B. Physics 1) Electric Currents: The conductivity is left at its default value of 1S/m to simplify computation and the relative permittivity is also left at 1. The entire bundle is insulated except, of course, for the electrodes. The initial value of the domain is 0V. The normal current density of the cathode is while the anode is n.j = I stim s(t) A/m 2 (16) n.j = 10I stim r(t) A/m 2 (17) I stim is kept at about 100µA/mm 2 or 100A/m 2, which means each source electrode puts out 100µA maximum. It was found that the anode needed to have a magnitude at least 10 greater then the cathode to observe any hyperpolarisation effects. s(t) is an asymmetrical wave, that gradually raises the electric potential so that the larger axons are stimulated first and essentially have a head start over the smaller axons so that the anodal block has a bigger time difference to work in. It rises linearly to I stim over t = 1ms and stays there until t = 2ms and then it drops to zero (see Figure 2). Fig. 3. Hyperpolarising waveform from the anode (scaled by 10I stim ) instantaneous rise time) between t = 1ms and t = 3ms (see Figure 3). The 2-D edge also acts as a current source; this is the membrane current given in equation (1). 2) PDEs: The PDEs are given in weak form, to allow for the coupling between 2-D and 3-D in the software. While the model is based on the equations of Hodgkin-Huxley, scaling changes were made for faster computation. The full Hodgkin-Huxley equation is describe by the PDEs of V i, n, m and h. The equations are changed into their weak PDE form: 0 = weak L (18) 2 Ω The V i terms comes from (15) dv m + i ion i m = 0 ( dvi dv ) e + i ion + r a 2 V = 0 2ρ i Which is then turned into its weak form: r a 2ρ i ( ( dvi dv ) ) e + i ion test (V i ) + ( dvi dx testdv i dx + dv i dy testdv i dy + dv ) i dz testdv i dz (19) The rest of the weak expressions come from equations (4), (8) and (9): Fig. 2. Stimulating waveform from the cathode (scaled by I stim ) r(t) is a pulse wave, smoothed somewhat to allow for easier computation (as there are derivative terms) and a realistic stimulation (inherent reactiveness in any system prevents an dn α n(1 n) + β n n test(n) (20) dm α m(1 m) + β m m test(m) (21) dh α h(1 h) + β h h test(h) (22)

4 UNSW GSBME 4 The PDE edge also requires boundary conditions. To avoid any tedious action potentials propagating backwards, a Dirichlet boundary was placed at x = 0. This was to keep the internal voltage at 60mV at one end so no propagation could occur in that direction. Also, the other end of the edge (x = 20) had a Neumann boundary condition of zero flux. There are two sets of this PDE, one for the big axon and one for the small axon. They had to be separated as the size of the radius effects all of the variables. axon. A gap is observed at 10mm, which can be exploited using an anodal block. This gap is a result of the different conduction velocities in the axons, demonstrated in Figure 4, as well as the gradual cathodic stimulation exploiting the different threshold levels. Figure 5 shows how the larger axon has already produced an action potential at t = 1.5ms, while the smaller axon has not. This is a result of the trapezoidal stimulation wave. C. Meshing The meshing of the volume conductor was coarse, especially along the x axis. The meshing within the edges was kept at 0.1mm, essentially forming the Nodes of Ranvier[3], [7]. D. Study Time steps of 0.125ms were taken for a total of 10ms. These numbers may seem strange as the model has been scaled somewhat, so that the conduction velocity is actually much slower (about 1m/s) then normal for a myelinated fibre. This is fairly irrelevant. While numbers can be scaled, this model aims to observe the effect of the concept of anodal blocking, rather then giving a hugely accurate quantitative portrayal of the timing of the process in the body. It is important that the study solves the model directly and not iteratively. Iterative solving would put the solver into an infinite loop. IV. RESULTS Recordings were taken of transmembrane potential vs time at a point on the axis, as well as the length along the axis for fixed time. There is a clear spatial discrepancy at x = 10mm after the action potential has been created. Figure 4 displays what occurs when there is no anodal block in place. Fig. 5. Transmembrane potential along axons when t = 1.5ms The setup of the hyperpolarisation can be shown as the transmembrane voltage as a function of distance along the axons s x axis while time is held constant. Figures 5 and 6 show the voltages along the two axons. Again, the dashed line is smaller axon and the solid line is the larger axon. Note the problematic positive fields (V m 60mV ) either side of the anodic cuff. Fig. 4. Transmembrane potential vs time at x = 10mm when no anodal block is in place The dotted line represents the action potential of the smaller axons, while the solid line shows the potential of the larger Fig. 6. Transmembrane potential along the axons when t = 2ms It appears that this has been successfully hyperpolarised, and the solid line will not be able to propagate any further. Unfortunately, this is not the case. In this simulation the

5 UNSW GSBME 5 anodal block appears to be ineffective as both action potentials continue to propagate to x = 10mm as shown in Figure 7. VI. DISCUSSION Changes occurred many times over the course of the simulation being designed. These included adapting ideas from the literature, deliberately moving away from these ideas, and making alterations based on the current simulation results. A. Domain conductivities A complex geometry was devised to accurately map the voltage fields, based on Zariffa et al[8]. This considered the varying conductivities of all the components of a nerve (endoneurium, perineurium, epineurium and the connective tissue). Obviously, this made a very bulky model with many different domains and boundaries to consider, and also made meshing for each domain difficult. This was abandoned in favour of a one domain model, used in many models[10], [11], [12]. A two domain model was considered[4] but even this slowed down simulations. Fig. 7. Transmembrane potential vs time at x = 10mm when anodal block has been attempted However, there is a slight arrest in the action potentials. At t = 2.5ms, a slight dip is observed in the axon s potential. But it is not enough as the hyperpolarisation does not occur over a long enough area. This slight dip can be observed in Figure 8. Fig. 8. Transmembrane potential along the axons when t = 2.5ms Note that the times may not portray those normally observed in the human body as this is a scaled model. V. CONCLUSION This model shows the spatial discrepancy between the action potentials that can be exploited by an anodal block to produce selectivity. This model s anodal block is not successful in portraying the selectivity that is desired, but it does display the general idea of the concept as it does suggest some hindering of the potentials in Figure 8. B. Hodgkin-Huxley Equations The scaled expressions used were rejected at one stage in favour of the original Hodgkin-Huxley equations[7]. However, simulation time took far too long so the scaled versions were reverted back to. Interestingly, most models that have simulated the anodal block in the past have used an adapted variation of the McNeal model[13] for the axon. However, these models were made on older software that required manual discretisation as well as a simpler method to solve in the 3-D domain. It can be seen in equation (19) that the Hodgkin-Huxley equation has been adapted to take into consideration the gradient along the y and z domains, whereas when it first appeared in the literature it was just a one dimensional model along the x axis. C. Placement of electrodes The anode was initially placed further from the cathode, to allow the full affect of the action potential to be observed. However, this wasn t working particularly well, so the placements from Laforet et al. [4] were used. This gives a close hyperpolarisation, and while it doesn t display a full action potential being arrested, it actually gives a more accurate representation of how this could work effectively in the body. There is also the option of laying the electrodes out in a helical coil. An advantage of doing this is to lay the electrodes out along the same side. Preliminary modelling showed that this requires less current to be injected for the potential to be raised sufficiently. This is better because it is less likely for there to be damage to tissue, even though the currents are for extremely short periods. However, there is an interesting unwanted side affect of this arrangement, where there is a positive potential surrounding the negative potential either side of the electrodes. This positive field can be great enough to set off the action potential. This is much worse then the cathodic effect for the ring arrangement (seen in Figures 5, 6 and 8, either side of the anodic electrodes located at x = [6, 7], where the transmembrane potential is V m 60mV ). Therefore, the ring layout is better as it can set up a greater anodic potential without a large cathodic effect on the fringes of the plates.

6 UNSW GSBME 6 D. Stimulating wave functions Some smoothing needs to be simulated so that the software can handle the current transients. Furthermore, smoothing will occur in the real situation. While a rapid pulse is suggested in some models[4], there are advantages to having a slightly gradual one. This of course is that it sets the axons with larger diameters off first, then those with a smaller one. This is advantageous for obvious reasons in a model that employs an anodal block. Waveforms are also the subject of further investigation in the hopes of decreasing the required current needed to be injected[5]. E. Merits This model clearly displays the spatial discrepancy in the case of nonphysiological recruitment. This can be exploited using the anodal block. F. Limitations The hyperpolarisation effect was not shown properly. Assuming the model equations and characteristics were implemented correctly, this comes down as an issue with the amount of nerve that was hyperpolarised. Due to the conduction process of saltation, the hyperpolarisation must occur over many nodes. This was difficult to achieve with the electrodes, due to the phenomenon previously described of the raised potential along the fringes of the anodic cuff link. Even when the cuff was lengthened (which will become limited in reality) the raised potential became even more problematic. The scaling of the model is also misleading. While it allows for easier computation for the COMSOL Multiphysics software, the timing of the events is not an accurate representation of what occurs in the human body. This also leads to inaccuracies of the velocity values, even though spatial values are fairly similar to what is to be expected in the human body. VII. RECOMMENDATIONS There are some features of this model that would be useful in future simulations while other areas need a different approach. The trapezoidal cathodic pulse is very effective in creating a greater gap between the larger and smaller radius axons. Any waveform that features a somewhat gradual rise will be able to set the larger axons off first. The anodal block mechanism itself has proved its usefulness in other models[4], [5], [6], and hints here that it is effective, even though it didn t work completely. The use of the Hodgkin-Huxley model is an improvement over past models[5], [6] using the McNeal model. Applying this model on a 2-D edge within a 3-D volume allows for faster computation while maintaining accuracy. While the McNeal model has its uses, the Hodgkin-Huxley completely describes the behaviour of the axons, thus gives a more accurate representation in the model. Future models should improve on the anodal cuff. It does not hyperpolarise a large enough area to completely arrest the action potentials according to the Hodgkin-Huxley model. Testing in this model found creating a longer electrode caused greater positive fields on the fringes, setting off other action potentials. Further investigation into electrode configurations and injected current levels is required. Something that may improve on this is that the components of the nerve bundle closest to the axon have a lower conductivity[4], [8] to what was used in this model. Proper scaling will also benefit future models. Accurate spatial and time models will allow for greater understanding and improve practical realisations of this concept. REFERENCES [1] C. L. Lynch and M. R. Popovic, Functional electrode stimulation, IEEE Control Systems Magazine, pp , Aprial [2] W. Grill, C. Veraart, and J. Mortimer, Selective activation of peripheral nerve fascicles: Use of field steering currents, in Engineering in Medicine and Biology Society, Vol.13: 1991., Proceedings of the Annual International Conference of the IEEE, pp , oct-3 nov [3] C. L. Stanfield and W. J. 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