Neuromodulation: Technology at the Neural Interface Received: May 30, 2013
Revised: November 24, 2013
Accepted: December 14, 2013
(onlinelibrary.wiley.com) DOI: 10.1111/ner.12163
Mathematical Model of Nerve Fiber Activation During Low Back Peripheral Nerve Field Stimulation: Analysis of Electrode Implant Depth Carsten Dahl Mørch, PhD*; Giang P. Nguyen, PhD*; Paul W. Wacnik, PhD†; Ole Kæseler Andersen PhD* Objectives: The lower back is the most common location of pain experienced by one-fifth of the European population reporting chronic pain. A peripheral nerve field stimulation system, which involves electrodes implanted subcutaneously in the painful area, has been shown to be efficacious for low back pain. Moreover, the predominant analgesic mechanism of action is thought to be via activation of peripheral Aβ fibers. Unfortunately, electrical stimulation also might coactivate Aδ fibers, causing pain or unpleasantness itself. The aim of this study was to investigate at which implant depth Aβ-fiber stimulation is maximized, and Aδ-fiber minimized, which in turn should lead to therapy optimization. Materials and Methods: A finite element model was used to estimate the electrical potential generated by a bipolar single-lead electrode implanted in the subcutaneous adipose tissue at depths of 5 mm to 30 mm below the skin surface. The model includes low back tissue; the epidermis, dermis, adipose, and muscle layers, and nerve fibers, which were programmed to branch randomly in the model in a fiber type-specific manner. Likewise, activation thresholds were specific to Aβ- and Aδ-fiber types and were estimated using a passive cable model. Results: The stimulus–response functions showed that the skin area covered by Aβ-fiber activation was larger than the area covered by Aδ-fiber activation at all depths and all intensities. The skin area covered by Aδ-fiber activation was largest when the electrode was modeled to have a superficial location (5 mm below the skin surface), while the skin area covered by Aβ-fiber activation was largest at lower depths. Conclusions: The present mathematical model predicts an optimal implantation depth of 10 to 15 mm below the skin surface to achieve activation of the greatest area of Aβ fibers and the smallest area of Aδ fibers. This finding may act as a guide for peripheral nerve field stimulation implant depth to treat low back pain. Keywords: Low back pain, nerve fiber modeling, subcutaneously implanted lead Conflict of Interest: Dr. Wacnik is an employee of Medtronic Inc.
INTRODUCTION
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Several methods for relieving chronic low back pain are used in daily clinical practice, including physiotherapy, nonsteroid antiinflammatory drugs, opioids, transcutaneous electrical stimulation (TENS), and implanted electrodes for spinal cord stimulation (SCS). SCS has proven to be effective for relieving pain (1,2) and has been shown to initiate several pain-relieving mechanisms via Aβ-fiber activation: a potential gating effect of incoming nociceptive stimuli in the dorsal horn of the spinal cord (3–5) and an activation of descending pain inhibitory pathways via supraspinal mechanisms (6). While SCS has proven effective in relieving lower limb pain following, for example, failed low back surgery, low back pain is sometimes more resistant to SCS (7). Peripheral nerve field stimulation (PNFS) has been introduced as a treatment for low back pain and, like SCS, is dependent on Aβ-fiber activation (8–10). In PNFS, one or more electrodes are implanted within or in the vicinity of the painful area via a small incision into the subcutaneous layers without identifying or exposing the target peripheral nerves. The electrodes are www.neuromodulationjournal.com
then tunneled to the implanted pulse generator placed distant from the pain area. PNFS electrode placement is therefore easier and safer to perform than direct stimulation of a specific peripheral nerve where risk of nerve damage from the surgery is present. Several recent case series studies showed pain relief after PNFS in
Address correspondence to: Carsten Dahl Mørch, PhD, Aalborg University, Frederik Bajers Vej 7A2–212, DK-9220 Aalborg, Denmark. Email: cdahl@ hst.aau.dk * Integrative Neuroscience group, Center for Sensory Motor Interaction, Department of Health Science and Technology, Aalborg University, Aalborg, Denmark † Neuromodulation Research, Medtronic Inc., Minneapolis, MN, USA For more information on author guidelines, an explanation of our peer review process, and conflict of interest informed consent policies, please go to http:// www.wiley.com/bw/submit.asp?ref=1094-7159&site=1 Financial statement: This study was supported by research grants to Drs. Mørch and Nguyen from Medtronic Inc. (Minneapolis, Minnesota, USA) and The Danish Council for Independent Research for Technology and Production Sciences.
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MODEL OF PERIPHERAL NERVE FIELD STIMULATION groups of chronic pain patients including low back pain patients with only few adverse effects such as electrode migration and electrode failure (8,9,12–14). PNFS can also be applied as an additional therapy in conjunction with SCS (15–19). PNFS is therefore a minimally invasive neuromodulatory method for low back pain relief. The placement of the PFNS electrode is based on the experience of the surgeon and the location and size of the painful area with only few rule-of-thumb guidelines for where and how deep the electrode should be placed to optimize pain relief. In general, electrodes are placed in the superficial part of the subcutaneous layer in the area of most intense pain (9). However, if the electrodes are implanted too superficially, the patient will experience pain from the stimulation and if the electrode is implanted too deep, the stimulation will be ineffective or cause uncomfortable muscle activation (20). It is difficult to predict which nerve fibers are activated by certain electrode placements. Mathematical modeling has been used to estimate nerve fiber activation in several areas of neuromodulation, for example, SCS (21,22), TENS (23), and small area cutaneous electrodes (24). These models are often built as two-part models. The electrical field generated by the electrode is estimated by a finite element (FE) model, and cable models are used to estimate the activation threshold, either as passive models (24) or active models containing nodes of Ranvier with Hodgkin–Huxley-like gating properties (25–27). These studies have increased the understanding of nerve fiber excitation and features of specific neuromodulation treatments. The aim of the present study was to construct a mathematical model describing nerve fiber activation during PNFS, which included a realistic model of the extracellular nerve potential for individual Aβ and Aδ fibers. We hypothesized that the quantity and type of nerve fiber activation is dependent on the depth of implantation and the thickness of the adipose subcutaneous layer.
Figure 1. An FE model used to estimate the extracellular potential generated by dipolar electrode implanted in the adipose tissue. The model consisted of four horizontal layers; epidermis, dermis, adipose tissue, and muscle tissue. The electrode was implanted horizontally at different depths. A tetrahedral mesh was generated. Colors indicating the mesh size (longest edge of each tetrahedral) shown to the right. FE, finite element.
was modeled at different depths (5 to 30 mm in steps of 5 mm) in the adipose tissue.
Electrical Current and Potential The electrical potential was modeled in a resistive volume conductor governed by Ohm’s laws and formulated by the Poisson equation:
METHODS
−∇ ⋅ (σ∇VFE ) = 0
A two-part model was used to estimate the nerve fiber activation during PNFS. The first part was to create an FE model to estimate the electrical field in the skin and subcutaneous tissue. The second part was to create a stochastically branching nerve fiber model to estimate the activation thresholds and area of activation of a set of cutaneous Aβ and Aδ fibers.
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Stochastic Nerve Fiber Model For each FE model, an average of 6,250 Aβ fibers and 12,690 Aδ-fibers were modeled as single fibers emerging from a random location in the lower boundary of the FE model. The number of simulated nerve fiber endings was selected so that the density of the Aβ-fibers endings in the dermis was approximately 60 endings per mm2 (32,33) and the density of the Aδ-fiber endings in the epidermis was approximately 160 endings per mm2 (34).
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FE Model of Human Low Back Skin and Implanted Electrode A three-dimensional FE model of the low back skin was built utilizing a software (COMSOL Multiphysics 4.3a, COMSOL, Stockholm, Sweden). The model represented a 5 × 5 cm area of the human low back skin consisting of four horizontal layers: epidermis, dermis, subcutaneous adipose tissue, and the muscle (Fig. 1). The thickness of the epidermal layer was modeled as 0.1 mm (28,29) and the thickness of the dermis as 2.4 mm (30,31). To investigate possible influences of adipose thickness on nerve fiber activation, the thickness of the adipose tissue was modeled from 10 mm to 50 mm in 10 mm steps. The thickness of the muscle layer was fixed at 10 mm. (Fig. 1). The electrode used in our model was based on the cylindrical or “percutaneous” Medtronic 3887 Pisces Compact electrode (Medtronic Inc., Minneapolis, MN, USA). The lengths of the active areas were 3 mm, the edge-to-edge distance between two active areas was 4 mm, and the diameter of the electrode was 1.3 mm (Fig. 1). The electrode lead was modeled as a hollow tube because the electrical potential inside the electrode does not affect the activation threshold of the nerve fibers. The subcutaneous electrode
σ: conductivity of the tissue and VFE: electrical potential calculated by the FE model. Stimulation was simulated as 0V potential clamped at the cathode, the anode set to 1V as boundary conditions. The electrical properties of the tissue were adopted from published literature (Table 1). To solve the model, the low back skin model was divided into a multi-resolution tetrahedral mesh, which was refined in the superficial parts of the skin (maximal element size of 0.2 mm) where the geometry of the model was small and at the electrode (maximal element size of 0.2) where the gradient of the electrical potential was large (Fig. 1). A set of convergence studies were performed to ensure that the mesh density of the epidermis and dermis layers and the electrode, the horizontal extent of the model, and the thickness of the muscle layer selected did not affect the electrical potential adjacent to the electrode.
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Table 1. Electrical Properties of the Tissue Were Adopted From Literature. Layer
Electrical conductivity (S/m)
Epidermis (35)
Horizontal: 0.95 Vertical: 0.15 Horizontal: 2.57 Vertical: 1.62 2.0 × 10−2 Across: 0.3 Along: 0.6
Dermis (35) Adipose (36) Muscle (36)
The electrical properties of the adipose tissue was considered isotropic, whereas the epidermis and the dermis had different electrical properties parallel (horizontal) to the skin surface compared with the electrical properties perpendicular to the skin surface. Further anisotropy is observed in muscle tissue as the electrical properties along the muscle fibers differ from the electrical properties across the muscle fibers.
Aβ Fibers The morphology of the Aβ fibers was modeled as previously reported (24), that is, a straight vertical line from the bottom of the model to the middle of the adipose layer. From the mid-adipose layer, the nerve fiber was allowed to branch with a probability of 10% at each node of Ranvier. The initial diameter of the Aβ fibers was modeled as 9 μm, but shrunk at each branch point with a factor assumed to be 0.98, resulting in thinner fiber in the distal end (32,33). Aβ fibers terminated at a random but at a uniformly distributed depth in the dermis (32) (Fig. 2a). Aδ Fibers The Aδ fibers likewise originated at the base of the model and followed a straight vertical line up to the dermal–epidermal junction. Here they turned to form the horizontal plexus (34). The vertical fiber diameter of the Aδ fibers was modeled at 5 μm, whereas the diameter of the horizontal plexus was set at 1 μm. Fourteen to 21 unmyelinated branches/mm were modeled from the horizontal plexus protruding into the epidermis and terminated at a random but at a uniformly distributed depth in the epidermis (37) (Fig. 2b,c). The nerve fibers were modeled as unmyelinated in the epidermis and, like the Aβ fibers, were allowed to branch with a probability of 10% at each node of Ranvier. Cable Properties The ratio of internodal space to fiber diameter and the ratio of axon and fiber diameters were adopted from McNeal (26) to be 100 and 0.7, respectively. The nodal length was modeled as 4 μm (32). Nerve Fiber Activation The membrane potential at each node of Ranvier was estimated for each nerve fiber using the methods described in Mørch et al. (24). The membrane potential at the node with the highest increase in membrane potential was divided by the activation threshold of 20 mV to obtain a scaling factor for the stimulation intensity needed to generate an action potential in the nerve fiber. The scaling factor was multiplied by the modeled stimulation intensity, that is, 1V used for generating the FE model. This method allowed calculation of the threshold for the individual nerve fiber without recalculating the external potential in the FE model.
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The Measure of Nerve Fiber Activation The skin area covered by activated nerve fibers was used as the measure of nerve fiber activation. To calculate the activation area, www.neuromodulationjournal.com
Figure 2. Examples of stochastically branching nerve fibers. a. An example of an Aβ fiber. b. An example of an Aδ fiber. c. A zoom-in of the fiber endings of the Aδ fiber shown in b. The nodes of Ranvier are depicted as dots and their colors indicate the estimated membrane potential in millivolts.
the location of nerve fiber being activated was projected onto the skin surface, and the smallest area containing the projected activation sites was estimated using alpha shapes. Alpha shapes are an expansion of the convex hull method that allows concavities. We set the curvature to an arbitrary size of 1 mm or more (38). The activation areas were calculated for both Aβ and Aδ fibers, and the ratio between the areas covered by Aβ-fiber activation and the area covered by Aδ-fiber activation was calculated to estimate stimulation efficacy.
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MODEL OF PERIPHERAL NERVE FIELD STIMULATION shunted through the muscle tissue when the electrode was implanted in the deeper parts of the adipose tissue (30 mm below the skin surface; Fig. 3c). The current was confined in the adipose tissue when the electrode was implanted in the middle of the adipose tissue (e.g. 15 mm below the skin surface; Fig. 3b). Location of Action Potential Generation The action potential was elicited in the vicinity of the cathode or at a small distance from the anode (a virtual cathode) in both the Aβ and Aδ fibers. For example, at a stimulation intensity of 5V, Aβ-fiber activation covered a larger area of the skin than was covered by Aδ-fiber activation: Aβ-fiber activation area at a given electrode depth: 5 mm; 146 mm2, 10 mm; 172 mm2, 15 mm; 156 mm2, 20 mm; 107 mm2, 25 mm; 46 mm2, 30 mm; 95 mm2 (Fig. 4, blue dots) and Aδ-fiber activation area: 5 mm; 61 mm2, 10 mm; 25 mm2, 15 mm; 26 mm2, 20 mm; 26 mm2, 25 mm; 25 mm2, 30 mm; 58 mm2 (Fig. 4, green dots). Stimulus–Response Function Clinically, paresthesia coverage of the painful area is desired for treatment by PNFS, thus activation of a large proportion of the Aβ fibers is desired. Further, to maximize pain relief, the stimulation intensity is increased to the highest amplitude possible before the electrical stimulation is perceived as painful. The present model showed that a larger area was covered by Aβ-fiber activation than Aδ-fiber activation for all depths of electrodes implanted in the adipose tissue (Fig. 5). Therefore, the ratio of the area covered by Aβ-fiber activation divided by the area covered by Aδ-fiber activation was larger than one (Fig. 6). For example, at stimulation intensities of 5V, the area of Aβ-fiber activation was 2.4, 6.9, 6.0, 4.0, 1.8, and 1.6 times the area covered by Aδ-fiber activation when the electrode was implanted at 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, and 30 mm, respectively. Thus, the largest separation between the stimulus–response functions were observed for electrodes implanted at 10–15 mm below the skin surface.
Figure 3. The electrical field generated by an electrode with a stimulation intensity of 1V. The electrical potential in the tissue is depicted by color on the vertical plane along the electrode. The current density is illustrated by black arrows. Implant depth of a. 5 mm, b. 15 mm, and c. 30 mm.
RESULTS
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DISCUSSION This study provides a mathematical low back skin model to investigate the activation threshold of Aβ and Aδ fibers during PNFS. The
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Electrical Field Figure 3 shows an example of the electrical field generated by the implanted electrode, which was estimated by the FE model. In the dermis and epidermis, the current was shunted when the electrode was implanted superficially (5 mm below the skin surface) as indicated by the black arrows (Fig. 3a). The current was to a lesser extent
Thickness of the Adipose Tissue A series of models were made with adipose tissue thicknesses ranging from 10 to 50 mm with the PNFS electrode placed 10 mm below the skin surface. The stimulus–response function of the Aβ fibers in the thinnest adipose layer was less steep than the models with thicker adipose layers, whereas the stimulus–response function of the Aδ-fibers was steeper for the thinnest adipose layer (Fig. 7). Only small differences were observed in the areas covered by Aβ-fiber activation, but a larger area of Aδ-fiber activation was observed in 10 mm adipose tissue. For example, at a stimulation intensity of 5V, the skin area covered by Aβ-fiber activation was smaller in the model with 10 mm adipose thickness than the models with thicker adipose layers (electrode depth; Aβ-fiber activation area; thus 10 mm; 131 mm2 and 20 mm; 152 mm2 and 30 mm; 153 mm2 and 40 mm; 144 mm2 and 50 mm; 155 mm2), whereas the area covered by Aδ fibers was larger in the model with 10 mm adipose thickness than the models with thicker adipose layers (electrode depth; Aδ-fiber activation area; thus 10 mm; 57 mm2 and 20 mm; 26 mm2 and 30 mm; 27 mm2 and 40 mm; 25 mm2 and 50 mm; 26 mm2).
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Figure 4. Location of action potential generation by stimulation at 5V viewed from above the skin. The location of the Aβ-fiber activation (blue dots) covered a larger area than the Aδ fibers (green dots) when the electrode was implanted at depths from 5–15 mm than when the electrode was implanted deeper. The red area indicates the cathode and the yellow area the anode. The areas covered by Aδ-fiber activation was largest when the electrode was modeled at depths of 5 mm or 30 mm, whereas the minor differences were observed in the areas covered by Aδ-fiber activation at other depths. The areas covered by Aβ-fiber activation was always larger than the areas covered by Aδ-fiber activation, so that areas covered by Aδ-fiber activation was always covered by Aβ-fiber activation also.
model indicated that Aβ fibers were activated at lower intensities than Aδ fibers, and that nerve fibers were activated in proximity to the cathode. The model also showed that an electrode implanted at 10–15 mm below the skin surface provided better separation between Aβ and Aδ fibers’ stimulus–response functions than electrodes implanted deeper or more superficial. In addition, the model predicts that the thickness of the adipose layer only effected Aδ-fiber area of activation (increased) and only in the thin (10 mm) layer model. This indicates possible difficulties to obtain sufficient Aβ-fiber activation for efficient PNFS in patients with thin adipose layers.
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The Model The present study applied the two-part approach used by previous studies (21,25–27), that is, combining an FE model with a cable model of the nerve fibers. The geometry of the nerve fibers was modeled as stochastically branching fibers to create an anatomically realistic feature for the different morphologies of Aβ and Aδ fibers. A passive nerve fiber model was used to estimate the activation threshold of the nerve fibers. The passive nerve fiber model does not simulate the active ion channel reaction during an action potential, and it is therefore limited to an estimation of the activation threshold. Discrepancies between activation thresholds estimated by active and passive fiber models have been reported (39). However, the passive and stochastically branching nerve fiber model used in the present study has previously been validated for cutaneous surface stimulation (24). www.neuromodulationjournal.com
In the model, it is assumed that an action potential would be generated in the node with maximal depolarization of the membrane. Yet the passive nerve fiber model does not facilitate calculations of action potential propagation. Action potential propagation could potentially be blocked by adjacent virtual or real anodes along the nerve fiber (40). However, experimentally anodal block has only been demonstrated with the use of cuff electrodes and other similar electrodes that are positioned longitudinally along the nerve fiber (41). In this study, electrodes were positioned longitudinally with the nerve fiber and therefore it is unlikely that their virtual/real anode is capable of blocking the action potential. The nerve fiber activation was quantified as the activation area by projecting the position of nerve fiber activation to the skin surface. This was done for easier translation to the area of paresthesia felt by the patients, although the translation should be done with caution as central aspects of perception are not modeled. Some examples of neuromodulation modeling include mathematical models of SCS, which have shown that dorsal column nerve fibers are most likely to be activated at nodes where collateral branches are perpendicular to the rostrocaudal dorsal column fibers (22). These findings have enabled optimization of electrode design and implantation for SCS. For deep brain stimulation, mathematical modeling has shown that electrode position and stimulation parameters can explain apparently contradictory findings that the stimulated nucleus was suppressed while input to projection nuclei was enhanced (42). The same nerve fiber model also proved feasible for modeling surface electrical stimulation as used in TENS treatment (23,27). We have previously shown, using a passive stochastically branching cable model, that small area surface electrodes
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Figure 5. The stimulus–response function of the Aβ fibers (solid line) and Aδ fibers (dashed line) for electrodes implanted at different depths in the adipose tissue. The best separation between activation of the two fiber types was observed when implanting the electrode at depths of at 10–15 mm below the skin surface.
Figure 6. Ratio of Aβ fiber area by Aδ fiber area. The area covered by Aβ-fiber activation was divided by the area covered by Aδ-fiber activation for electrodes modeled at 5 to 30 mm below the skin surface. The ratio was different between electrode depths and stimulation intensities.
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Figure 7. The stimulus–response function of the Aβ fibers (solid line) and Aδ fibers (dashed line) for different thicknesses of the adipose tissue. The electrode was implanted 10 mm below the skin surface in all models. All models showed similar stimulus–response functions although the stimulus response functions of the Aβ fibers were less steep in the model with the thinnest (10 mm) adipose tissue.
preferentially activate Aδ fiber (24). The latter study further implied that the intra-epidermal fiber endings play an important role in this inverted recruitment order of cutaneous afferents. Intra-epidermal fiber endings of the Aδ fibers were therefore included in the present study, but the present results showed that both Aβ and Aδ fibers were activated in close proximity to an implanted electrode even when the electrode was implanted superficially in the adipose tissue (Fig. 4). Therefore, intra-epidermal fiber endings are only likely to be activated if the electrode is implanted or migrates into the dermal layers.
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Nerve Fiber Activation When applying PNFS, electrical stimulation must be applied at intensities sufficient to activate a number of Aβ fibers to provide pain relief; however, as amplitudes are increased, pain and discomfort caused by Aδ-fibers activation follow. The present study shows steep stimulus–response functions for Aβ fibers, whereas the stimulus– response functions of the Aδ fibers are less steep (Fig. 5). Therefore, the model predicts that increasing the stimulation intensity will recruit relatively more Aβ fibers compared with Aδ fibers potentially leading to additional pain relief, but that the increase seems to plateau at higher intensities (Fig. 6). Increasing the stimulation intensity will obviously also recruit more Aδ fibers and eventually cause a painful or unpleasant experience. Experimental studies have reported great variability between patients for therapeutic stimulation intensities: 5.4V ± 2.5V (18), ranging from 2.1V to 5.7V (13), at pulse widths of 390 μsec ± 60 μsec (18) and 450 μsec (13). These individual variations may be caused by different adipose thickness or depth of implant, etc. The model does predict nerve fiber activation www.neuromodulationjournal.com
in the experimentally observed range, even though the simulated pulse length was longer than those used in clinical reports (13,18). Electrode Depth PNFS electrodes have been recommended to be placed in the painful area and in the superficial parts of the subcutaneous layers, that is, approximately 1 cm below the skin surface (9). More superficial implantation may add additional unpleasant or painful sensations that are unacceptable for the patients, and the electrode may even migrate through the skin surface, making electrode migration one of the complications of PNFS for the treatment of low back pain. These clinical recommendations are in accordance with the findings of the present model as the largest separation between the stimulus–response functions was observed for electrodes depths of 10–15 mm (Fig. 3).
Acknowledgments The work was supported by a grant from Medtronic Inc. and The Danish Council for Independent Research for Technology and Production Sciences.
Authorship Statement All authors substantially contributed to the design of the model, interpretation of the data, and several rounds of critical revision of the manuscript. The model implementation was done by Drs. Nguyen and Mørch. All authors approved the final manuscript.
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How to Cite this Article: Mørch C.D., Nguyen G.P., Wacnik P.W., Andersen O.K. 2014. Mathematical Model of Nerve Fiber Activation During Low Back Peripheral Nerve Field Stimulation: Analysis of Electrode Implant Depth. Neuromodulation 2014; 17: 218–225
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COMMENTS A superb piece of original modeling that reinforces clinical studies & findings. Paul Verrills, MD South Caulfield, Victoria, Australia
*** The paucity of elementary data relating to peripheral nerve field stimulation (PNFS) makes this paper most welcome. Like most discoveries in medicine there is inevitably a variable delay that may last years or decades before mechanisms are fully understood or, in some cases, never. With neuromodulation, ideas come fast and with the promise of clinical benefit, devices are frequently put to use without basic or clinical testing to determine their safety, effectiveness or even efficacy, let alone developing the associated skills to achieve best practice. The authors have carefully demonstrated the underlying reasons why choice of implanting depth below the dermis is so important to the separation of A beta from A delta fibers and hence achieving the maximum modulation of pain at minimal stimulating current thresholds. These data are even more germane to the justification of this variant of neurostimulation to those who hold our health purse-strings. Michael Stanton-Hicks, MD, DMED, MBBS Cleveland, OH, USA
Comments not included in the Early View version of this paper.
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Revised: November 24, 2013
Accepted: December 14, 2013
(onlinelibrary.wiley.com) DOI: 10.1111/ner.12163
Mathematical Model of Nerve Fiber Activation During Low Back Peripheral Nerve Field Stimulation: Analysis of Electrode Implant Depth Carsten Dahl Mørch, PhD*; Giang P. Nguyen, PhD*; Paul W. Wacnik, PhD†; Ole Kæseler Andersen PhD* Objectives: The lower back is the most common location of pain experienced by one-fifth of the European population reporting chronic pain. A peripheral nerve field stimulation system, which involves electrodes implanted subcutaneously in the painful area, has been shown to be efficacious for low back pain. Moreover, the predominant analgesic mechanism of action is thought to be via activation of peripheral Aβ fibers. Unfortunately, electrical stimulation also might coactivate Aδ fibers, causing pain or unpleasantness itself. The aim of this study was to investigate at which implant depth Aβ-fiber stimulation is maximized, and Aδ-fiber minimized, which in turn should lead to therapy optimization. Materials and Methods: A finite element model was used to estimate the electrical potential generated by a bipolar single-lead electrode implanted in the subcutaneous adipose tissue at depths of 5 mm to 30 mm below the skin surface. The model includes low back tissue; the epidermis, dermis, adipose, and muscle layers, and nerve fibers, which were programmed to branch randomly in the model in a fiber type-specific manner. Likewise, activation thresholds were specific to Aβ- and Aδ-fiber types and were estimated using a passive cable model. Results: The stimulus–response functions showed that the skin area covered by Aβ-fiber activation was larger than the area covered by Aδ-fiber activation at all depths and all intensities. The skin area covered by Aδ-fiber activation was largest when the electrode was modeled to have a superficial location (5 mm below the skin surface), while the skin area covered by Aβ-fiber activation was largest at lower depths. Conclusions: The present mathematical model predicts an optimal implantation depth of 10 to 15 mm below the skin surface to achieve activation of the greatest area of Aβ fibers and the smallest area of Aδ fibers. This finding may act as a guide for peripheral nerve field stimulation implant depth to treat low back pain. Keywords: Low back pain, nerve fiber modeling, subcutaneously implanted lead Conflict of Interest: Dr. Wacnik is an employee of Medtronic Inc.
INTRODUCTION
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Several methods for relieving chronic low back pain are used in daily clinical practice, including physiotherapy, nonsteroid antiinflammatory drugs, opioids, transcutaneous electrical stimulation (TENS), and implanted electrodes for spinal cord stimulation (SCS). SCS has proven to be effective for relieving pain (1,2) and has been shown to initiate several pain-relieving mechanisms via Aβ-fiber activation: a potential gating effect of incoming nociceptive stimuli in the dorsal horn of the spinal cord (3–5) and an activation of descending pain inhibitory pathways via supraspinal mechanisms (6). While SCS has proven effective in relieving lower limb pain following, for example, failed low back surgery, low back pain is sometimes more resistant to SCS (7). Peripheral nerve field stimulation (PNFS) has been introduced as a treatment for low back pain and, like SCS, is dependent on Aβ-fiber activation (8–10). In PNFS, one or more electrodes are implanted within or in the vicinity of the painful area via a small incision into the subcutaneous layers without identifying or exposing the target peripheral nerves. The electrodes are www.neuromodulationjournal.com
then tunneled to the implanted pulse generator placed distant from the pain area. PNFS electrode placement is therefore easier and safer to perform than direct stimulation of a specific peripheral nerve where risk of nerve damage from the surgery is present. Several recent case series studies showed pain relief after PNFS in
Address correspondence to: Carsten Dahl Mørch, PhD, Aalborg University, Frederik Bajers Vej 7A2–212, DK-9220 Aalborg, Denmark. Email: cdahl@ hst.aau.dk * Integrative Neuroscience group, Center for Sensory Motor Interaction, Department of Health Science and Technology, Aalborg University, Aalborg, Denmark † Neuromodulation Research, Medtronic Inc., Minneapolis, MN, USA For more information on author guidelines, an explanation of our peer review process, and conflict of interest informed consent policies, please go to http:// www.wiley.com/bw/submit.asp?ref=1094-7159&site=1 Financial statement: This study was supported by research grants to Drs. Mørch and Nguyen from Medtronic Inc. (Minneapolis, Minnesota, USA) and The Danish Council for Independent Research for Technology and Production Sciences.
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MODEL OF PERIPHERAL NERVE FIELD STIMULATION groups of chronic pain patients including low back pain patients with only few adverse effects such as electrode migration and electrode failure (8,9,12–14). PNFS can also be applied as an additional therapy in conjunction with SCS (15–19). PNFS is therefore a minimally invasive neuromodulatory method for low back pain relief. The placement of the PFNS electrode is based on the experience of the surgeon and the location and size of the painful area with only few rule-of-thumb guidelines for where and how deep the electrode should be placed to optimize pain relief. In general, electrodes are placed in the superficial part of the subcutaneous layer in the area of most intense pain (9). However, if the electrodes are implanted too superficially, the patient will experience pain from the stimulation and if the electrode is implanted too deep, the stimulation will be ineffective or cause uncomfortable muscle activation (20). It is difficult to predict which nerve fibers are activated by certain electrode placements. Mathematical modeling has been used to estimate nerve fiber activation in several areas of neuromodulation, for example, SCS (21,22), TENS (23), and small area cutaneous electrodes (24). These models are often built as two-part models. The electrical field generated by the electrode is estimated by a finite element (FE) model, and cable models are used to estimate the activation threshold, either as passive models (24) or active models containing nodes of Ranvier with Hodgkin–Huxley-like gating properties (25–27). These studies have increased the understanding of nerve fiber excitation and features of specific neuromodulation treatments. The aim of the present study was to construct a mathematical model describing nerve fiber activation during PNFS, which included a realistic model of the extracellular nerve potential for individual Aβ and Aδ fibers. We hypothesized that the quantity and type of nerve fiber activation is dependent on the depth of implantation and the thickness of the adipose subcutaneous layer.
Figure 1. An FE model used to estimate the extracellular potential generated by dipolar electrode implanted in the adipose tissue. The model consisted of four horizontal layers; epidermis, dermis, adipose tissue, and muscle tissue. The electrode was implanted horizontally at different depths. A tetrahedral mesh was generated. Colors indicating the mesh size (longest edge of each tetrahedral) shown to the right. FE, finite element.
was modeled at different depths (5 to 30 mm in steps of 5 mm) in the adipose tissue.
Electrical Current and Potential The electrical potential was modeled in a resistive volume conductor governed by Ohm’s laws and formulated by the Poisson equation:
METHODS
−∇ ⋅ (σ∇VFE ) = 0
A two-part model was used to estimate the nerve fiber activation during PNFS. The first part was to create an FE model to estimate the electrical field in the skin and subcutaneous tissue. The second part was to create a stochastically branching nerve fiber model to estimate the activation thresholds and area of activation of a set of cutaneous Aβ and Aδ fibers.
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Stochastic Nerve Fiber Model For each FE model, an average of 6,250 Aβ fibers and 12,690 Aδ-fibers were modeled as single fibers emerging from a random location in the lower boundary of the FE model. The number of simulated nerve fiber endings was selected so that the density of the Aβ-fibers endings in the dermis was approximately 60 endings per mm2 (32,33) and the density of the Aδ-fiber endings in the epidermis was approximately 160 endings per mm2 (34).
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FE Model of Human Low Back Skin and Implanted Electrode A three-dimensional FE model of the low back skin was built utilizing a software (COMSOL Multiphysics 4.3a, COMSOL, Stockholm, Sweden). The model represented a 5 × 5 cm area of the human low back skin consisting of four horizontal layers: epidermis, dermis, subcutaneous adipose tissue, and the muscle (Fig. 1). The thickness of the epidermal layer was modeled as 0.1 mm (28,29) and the thickness of the dermis as 2.4 mm (30,31). To investigate possible influences of adipose thickness on nerve fiber activation, the thickness of the adipose tissue was modeled from 10 mm to 50 mm in 10 mm steps. The thickness of the muscle layer was fixed at 10 mm. (Fig. 1). The electrode used in our model was based on the cylindrical or “percutaneous” Medtronic 3887 Pisces Compact electrode (Medtronic Inc., Minneapolis, MN, USA). The lengths of the active areas were 3 mm, the edge-to-edge distance between two active areas was 4 mm, and the diameter of the electrode was 1.3 mm (Fig. 1). The electrode lead was modeled as a hollow tube because the electrical potential inside the electrode does not affect the activation threshold of the nerve fibers. The subcutaneous electrode
σ: conductivity of the tissue and VFE: electrical potential calculated by the FE model. Stimulation was simulated as 0V potential clamped at the cathode, the anode set to 1V as boundary conditions. The electrical properties of the tissue were adopted from published literature (Table 1). To solve the model, the low back skin model was divided into a multi-resolution tetrahedral mesh, which was refined in the superficial parts of the skin (maximal element size of 0.2 mm) where the geometry of the model was small and at the electrode (maximal element size of 0.2) where the gradient of the electrical potential was large (Fig. 1). A set of convergence studies were performed to ensure that the mesh density of the epidermis and dermis layers and the electrode, the horizontal extent of the model, and the thickness of the muscle layer selected did not affect the electrical potential adjacent to the electrode.
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Table 1. Electrical Properties of the Tissue Were Adopted From Literature. Layer
Electrical conductivity (S/m)
Epidermis (35)
Horizontal: 0.95 Vertical: 0.15 Horizontal: 2.57 Vertical: 1.62 2.0 × 10−2 Across: 0.3 Along: 0.6
Dermis (35) Adipose (36) Muscle (36)
The electrical properties of the adipose tissue was considered isotropic, whereas the epidermis and the dermis had different electrical properties parallel (horizontal) to the skin surface compared with the electrical properties perpendicular to the skin surface. Further anisotropy is observed in muscle tissue as the electrical properties along the muscle fibers differ from the electrical properties across the muscle fibers.
Aβ Fibers The morphology of the Aβ fibers was modeled as previously reported (24), that is, a straight vertical line from the bottom of the model to the middle of the adipose layer. From the mid-adipose layer, the nerve fiber was allowed to branch with a probability of 10% at each node of Ranvier. The initial diameter of the Aβ fibers was modeled as 9 μm, but shrunk at each branch point with a factor assumed to be 0.98, resulting in thinner fiber in the distal end (32,33). Aβ fibers terminated at a random but at a uniformly distributed depth in the dermis (32) (Fig. 2a). Aδ Fibers The Aδ fibers likewise originated at the base of the model and followed a straight vertical line up to the dermal–epidermal junction. Here they turned to form the horizontal plexus (34). The vertical fiber diameter of the Aδ fibers was modeled at 5 μm, whereas the diameter of the horizontal plexus was set at 1 μm. Fourteen to 21 unmyelinated branches/mm were modeled from the horizontal plexus protruding into the epidermis and terminated at a random but at a uniformly distributed depth in the epidermis (37) (Fig. 2b,c). The nerve fibers were modeled as unmyelinated in the epidermis and, like the Aβ fibers, were allowed to branch with a probability of 10% at each node of Ranvier. Cable Properties The ratio of internodal space to fiber diameter and the ratio of axon and fiber diameters were adopted from McNeal (26) to be 100 and 0.7, respectively. The nodal length was modeled as 4 μm (32). Nerve Fiber Activation The membrane potential at each node of Ranvier was estimated for each nerve fiber using the methods described in Mørch et al. (24). The membrane potential at the node with the highest increase in membrane potential was divided by the activation threshold of 20 mV to obtain a scaling factor for the stimulation intensity needed to generate an action potential in the nerve fiber. The scaling factor was multiplied by the modeled stimulation intensity, that is, 1V used for generating the FE model. This method allowed calculation of the threshold for the individual nerve fiber without recalculating the external potential in the FE model.
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The Measure of Nerve Fiber Activation The skin area covered by activated nerve fibers was used as the measure of nerve fiber activation. To calculate the activation area, www.neuromodulationjournal.com
Figure 2. Examples of stochastically branching nerve fibers. a. An example of an Aβ fiber. b. An example of an Aδ fiber. c. A zoom-in of the fiber endings of the Aδ fiber shown in b. The nodes of Ranvier are depicted as dots and their colors indicate the estimated membrane potential in millivolts.
the location of nerve fiber being activated was projected onto the skin surface, and the smallest area containing the projected activation sites was estimated using alpha shapes. Alpha shapes are an expansion of the convex hull method that allows concavities. We set the curvature to an arbitrary size of 1 mm or more (38). The activation areas were calculated for both Aβ and Aδ fibers, and the ratio between the areas covered by Aβ-fiber activation and the area covered by Aδ-fiber activation was calculated to estimate stimulation efficacy.
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MODEL OF PERIPHERAL NERVE FIELD STIMULATION shunted through the muscle tissue when the electrode was implanted in the deeper parts of the adipose tissue (30 mm below the skin surface; Fig. 3c). The current was confined in the adipose tissue when the electrode was implanted in the middle of the adipose tissue (e.g. 15 mm below the skin surface; Fig. 3b). Location of Action Potential Generation The action potential was elicited in the vicinity of the cathode or at a small distance from the anode (a virtual cathode) in both the Aβ and Aδ fibers. For example, at a stimulation intensity of 5V, Aβ-fiber activation covered a larger area of the skin than was covered by Aδ-fiber activation: Aβ-fiber activation area at a given electrode depth: 5 mm; 146 mm2, 10 mm; 172 mm2, 15 mm; 156 mm2, 20 mm; 107 mm2, 25 mm; 46 mm2, 30 mm; 95 mm2 (Fig. 4, blue dots) and Aδ-fiber activation area: 5 mm; 61 mm2, 10 mm; 25 mm2, 15 mm; 26 mm2, 20 mm; 26 mm2, 25 mm; 25 mm2, 30 mm; 58 mm2 (Fig. 4, green dots). Stimulus–Response Function Clinically, paresthesia coverage of the painful area is desired for treatment by PNFS, thus activation of a large proportion of the Aβ fibers is desired. Further, to maximize pain relief, the stimulation intensity is increased to the highest amplitude possible before the electrical stimulation is perceived as painful. The present model showed that a larger area was covered by Aβ-fiber activation than Aδ-fiber activation for all depths of electrodes implanted in the adipose tissue (Fig. 5). Therefore, the ratio of the area covered by Aβ-fiber activation divided by the area covered by Aδ-fiber activation was larger than one (Fig. 6). For example, at stimulation intensities of 5V, the area of Aβ-fiber activation was 2.4, 6.9, 6.0, 4.0, 1.8, and 1.6 times the area covered by Aδ-fiber activation when the electrode was implanted at 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, and 30 mm, respectively. Thus, the largest separation between the stimulus–response functions were observed for electrodes implanted at 10–15 mm below the skin surface.
Figure 3. The electrical field generated by an electrode with a stimulation intensity of 1V. The electrical potential in the tissue is depicted by color on the vertical plane along the electrode. The current density is illustrated by black arrows. Implant depth of a. 5 mm, b. 15 mm, and c. 30 mm.
RESULTS
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DISCUSSION This study provides a mathematical low back skin model to investigate the activation threshold of Aβ and Aδ fibers during PNFS. The
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Electrical Field Figure 3 shows an example of the electrical field generated by the implanted electrode, which was estimated by the FE model. In the dermis and epidermis, the current was shunted when the electrode was implanted superficially (5 mm below the skin surface) as indicated by the black arrows (Fig. 3a). The current was to a lesser extent
Thickness of the Adipose Tissue A series of models were made with adipose tissue thicknesses ranging from 10 to 50 mm with the PNFS electrode placed 10 mm below the skin surface. The stimulus–response function of the Aβ fibers in the thinnest adipose layer was less steep than the models with thicker adipose layers, whereas the stimulus–response function of the Aδ-fibers was steeper for the thinnest adipose layer (Fig. 7). Only small differences were observed in the areas covered by Aβ-fiber activation, but a larger area of Aδ-fiber activation was observed in 10 mm adipose tissue. For example, at a stimulation intensity of 5V, the skin area covered by Aβ-fiber activation was smaller in the model with 10 mm adipose thickness than the models with thicker adipose layers (electrode depth; Aβ-fiber activation area; thus 10 mm; 131 mm2 and 20 mm; 152 mm2 and 30 mm; 153 mm2 and 40 mm; 144 mm2 and 50 mm; 155 mm2), whereas the area covered by Aδ fibers was larger in the model with 10 mm adipose thickness than the models with thicker adipose layers (electrode depth; Aδ-fiber activation area; thus 10 mm; 57 mm2 and 20 mm; 26 mm2 and 30 mm; 27 mm2 and 40 mm; 25 mm2 and 50 mm; 26 mm2).
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Figure 4. Location of action potential generation by stimulation at 5V viewed from above the skin. The location of the Aβ-fiber activation (blue dots) covered a larger area than the Aδ fibers (green dots) when the electrode was implanted at depths from 5–15 mm than when the electrode was implanted deeper. The red area indicates the cathode and the yellow area the anode. The areas covered by Aδ-fiber activation was largest when the electrode was modeled at depths of 5 mm or 30 mm, whereas the minor differences were observed in the areas covered by Aδ-fiber activation at other depths. The areas covered by Aβ-fiber activation was always larger than the areas covered by Aδ-fiber activation, so that areas covered by Aδ-fiber activation was always covered by Aβ-fiber activation also.
model indicated that Aβ fibers were activated at lower intensities than Aδ fibers, and that nerve fibers were activated in proximity to the cathode. The model also showed that an electrode implanted at 10–15 mm below the skin surface provided better separation between Aβ and Aδ fibers’ stimulus–response functions than electrodes implanted deeper or more superficial. In addition, the model predicts that the thickness of the adipose layer only effected Aδ-fiber area of activation (increased) and only in the thin (10 mm) layer model. This indicates possible difficulties to obtain sufficient Aβ-fiber activation for efficient PNFS in patients with thin adipose layers.
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The Model The present study applied the two-part approach used by previous studies (21,25–27), that is, combining an FE model with a cable model of the nerve fibers. The geometry of the nerve fibers was modeled as stochastically branching fibers to create an anatomically realistic feature for the different morphologies of Aβ and Aδ fibers. A passive nerve fiber model was used to estimate the activation threshold of the nerve fibers. The passive nerve fiber model does not simulate the active ion channel reaction during an action potential, and it is therefore limited to an estimation of the activation threshold. Discrepancies between activation thresholds estimated by active and passive fiber models have been reported (39). However, the passive and stochastically branching nerve fiber model used in the present study has previously been validated for cutaneous surface stimulation (24). www.neuromodulationjournal.com
In the model, it is assumed that an action potential would be generated in the node with maximal depolarization of the membrane. Yet the passive nerve fiber model does not facilitate calculations of action potential propagation. Action potential propagation could potentially be blocked by adjacent virtual or real anodes along the nerve fiber (40). However, experimentally anodal block has only been demonstrated with the use of cuff electrodes and other similar electrodes that are positioned longitudinally along the nerve fiber (41). In this study, electrodes were positioned longitudinally with the nerve fiber and therefore it is unlikely that their virtual/real anode is capable of blocking the action potential. The nerve fiber activation was quantified as the activation area by projecting the position of nerve fiber activation to the skin surface. This was done for easier translation to the area of paresthesia felt by the patients, although the translation should be done with caution as central aspects of perception are not modeled. Some examples of neuromodulation modeling include mathematical models of SCS, which have shown that dorsal column nerve fibers are most likely to be activated at nodes where collateral branches are perpendicular to the rostrocaudal dorsal column fibers (22). These findings have enabled optimization of electrode design and implantation for SCS. For deep brain stimulation, mathematical modeling has shown that electrode position and stimulation parameters can explain apparently contradictory findings that the stimulated nucleus was suppressed while input to projection nuclei was enhanced (42). The same nerve fiber model also proved feasible for modeling surface electrical stimulation as used in TENS treatment (23,27). We have previously shown, using a passive stochastically branching cable model, that small area surface electrodes
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Figure 5. The stimulus–response function of the Aβ fibers (solid line) and Aδ fibers (dashed line) for electrodes implanted at different depths in the adipose tissue. The best separation between activation of the two fiber types was observed when implanting the electrode at depths of at 10–15 mm below the skin surface.
Figure 6. Ratio of Aβ fiber area by Aδ fiber area. The area covered by Aβ-fiber activation was divided by the area covered by Aδ-fiber activation for electrodes modeled at 5 to 30 mm below the skin surface. The ratio was different between electrode depths and stimulation intensities.
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Figure 7. The stimulus–response function of the Aβ fibers (solid line) and Aδ fibers (dashed line) for different thicknesses of the adipose tissue. The electrode was implanted 10 mm below the skin surface in all models. All models showed similar stimulus–response functions although the stimulus response functions of the Aβ fibers were less steep in the model with the thinnest (10 mm) adipose tissue.
preferentially activate Aδ fiber (24). The latter study further implied that the intra-epidermal fiber endings play an important role in this inverted recruitment order of cutaneous afferents. Intra-epidermal fiber endings of the Aδ fibers were therefore included in the present study, but the present results showed that both Aβ and Aδ fibers were activated in close proximity to an implanted electrode even when the electrode was implanted superficially in the adipose tissue (Fig. 4). Therefore, intra-epidermal fiber endings are only likely to be activated if the electrode is implanted or migrates into the dermal layers.
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Nerve Fiber Activation When applying PNFS, electrical stimulation must be applied at intensities sufficient to activate a number of Aβ fibers to provide pain relief; however, as amplitudes are increased, pain and discomfort caused by Aδ-fibers activation follow. The present study shows steep stimulus–response functions for Aβ fibers, whereas the stimulus– response functions of the Aδ fibers are less steep (Fig. 5). Therefore, the model predicts that increasing the stimulation intensity will recruit relatively more Aβ fibers compared with Aδ fibers potentially leading to additional pain relief, but that the increase seems to plateau at higher intensities (Fig. 6). Increasing the stimulation intensity will obviously also recruit more Aδ fibers and eventually cause a painful or unpleasant experience. Experimental studies have reported great variability between patients for therapeutic stimulation intensities: 5.4V ± 2.5V (18), ranging from 2.1V to 5.7V (13), at pulse widths of 390 μsec ± 60 μsec (18) and 450 μsec (13). These individual variations may be caused by different adipose thickness or depth of implant, etc. The model does predict nerve fiber activation www.neuromodulationjournal.com
in the experimentally observed range, even though the simulated pulse length was longer than those used in clinical reports (13,18). Electrode Depth PNFS electrodes have been recommended to be placed in the painful area and in the superficial parts of the subcutaneous layers, that is, approximately 1 cm below the skin surface (9). More superficial implantation may add additional unpleasant or painful sensations that are unacceptable for the patients, and the electrode may even migrate through the skin surface, making electrode migration one of the complications of PNFS for the treatment of low back pain. These clinical recommendations are in accordance with the findings of the present model as the largest separation between the stimulus–response functions was observed for electrodes depths of 10–15 mm (Fig. 3).
Acknowledgments The work was supported by a grant from Medtronic Inc. and The Danish Council for Independent Research for Technology and Production Sciences.
Authorship Statement All authors substantially contributed to the design of the model, interpretation of the data, and several rounds of critical revision of the manuscript. The model implementation was done by Drs. Nguyen and Mørch. All authors approved the final manuscript.
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MODEL OF PERIPHERAL NERVE FIELD STIMULATION
How to Cite this Article: Mørch C.D., Nguyen G.P., Wacnik P.W., Andersen O.K. 2014. Mathematical Model of Nerve Fiber Activation During Low Back Peripheral Nerve Field Stimulation: Analysis of Electrode Implant Depth. Neuromodulation 2014; 17: 218–225
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COMMENTS A superb piece of original modeling that reinforces clinical studies & findings. Paul Verrills, MD South Caulfield, Victoria, Australia
*** The paucity of elementary data relating to peripheral nerve field stimulation (PNFS) makes this paper most welcome. Like most discoveries in medicine there is inevitably a variable delay that may last years or decades before mechanisms are fully understood or, in some cases, never. With neuromodulation, ideas come fast and with the promise of clinical benefit, devices are frequently put to use without basic or clinical testing to determine their safety, effectiveness or even efficacy, let alone developing the associated skills to achieve best practice. The authors have carefully demonstrated the underlying reasons why choice of implanting depth below the dermis is so important to the separation of A beta from A delta fibers and hence achieving the maximum modulation of pain at minimal stimulating current thresholds. These data are even more germane to the justification of this variant of neurostimulation to those who hold our health purse-strings. Michael Stanton-Hicks, MD, DMED, MBBS Cleveland, OH, USA
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Neuromodulation 2014; 17: 218–225
