Validation of a parameterized, open-source model of nerve stimulation

Abstract

Background:

Peripheral nerve stimulation is an effective treatment for various neurological disorders. The method of activation and stimulation parameters used impact the efficacy of the therapy, which emphasizes the need for tools to model this behavior. Computational modeling of nerve stimulation has proven to be a useful tool for estimating stimulation thresholds, optimizing electrode design, and exploring previously untested stimulation methods. Despite their utility, these tools require access to and familiarity with several pieces of specialized software.

Objective:

To simplify and streamline the computational modelling process into one application, which would increase accessibility significantly.

Methods:

We developed an open-source, parameterized model with a simple online user interface that allows user to adjust up to 36 different parameters (https://nervestimlab.utdallas.edu).

Results:

The model accurately predicts fiber activation thresholds for nerve and electrode combinations reported in literature. Additionally, it replicates characteristic differences between stimulation methods, such as lower thresholds with monopolar stimulation as compared to tripolar stimulation. The model predicted that the difference in threshold between monophasic and biphasic waveforms, a well-characterized phenomenon, is not present during stimulation with bipolar electrodes. In vivo testing on the rat sciatic nerve validated this prediction, which has not been previously reported.

Conclusion:

The accuracy of the model when compared to previous experiments, as well as the ease of use and accessibility to generate testable hypotheses, indicate that this software may represent a useful tool for a variety of nerve stimulation applications.

Introduction

Peripheral nerve stimulation has emerged as an effective treatment for a variety of disorders. Examples include vagus nerve stimulation for epilepsy, depression, and stroke, tibial nerve stimulation for bladder control, sacral nerve stimulation for constipation, occipital nerve stimulation for migraines, and hypoglossal nerve stimulation for sleep apnea [1-7]. In each application, the stimulation effects can be influenced by several variables such as the nerve diameter, electrode geometry, fascicle positioning, amount of insulation, and tissue conductivity. One of the most useful tools to predict how each of these variables will affect nerve activation is computational modeling. It can be used to estimate effective parameters in cases where in vivo testing is not feasible, compare and optimize electrode designs, translate results across nerves and species, predict which fiber populations are responsible for a therapeutic effect, and explore previously untested methods of nerve stimulation.

Computational models have proven useful for multiple applications, such as optimizing electrode placement for spinal cord stimulation, estimating fascicle selectivity of intrafascicular electrode designs, predicting which fiber types are activated during vagus nerve stimulation, and applying current steering to deep brain stimulation [8-11]. Despite their apparent utility, computational models require access to and familiarity with several pieces of specialized software [12]. Thus, it may be possible to expand the field by condensing this process into an easy to use, open-source format to facilitate use. In this study, we develop and validate an open-source, parameterized model with an online, front-end interface that can be used to easily simulate a wide variety of nerve and electrode configurations.

To validate the model, we compared the simulated dose-response curves to in vivo data reported in previous literature. Three studies using a variety of nerve stimulation paradigms in multiple species were selected for comparison [13-15]. We then used the model to generate predictions that we tested empirically in vivo on the rat sciatic nerve. Finally, we demonstrated how differential fiber type recruitment could produce higher-order effects commonly observed with nerve stimulation, including inverted-U dosing phenomena. Our results indicate that this open-source model represents a useful tool for researchers to estimate the effects of peripheral nerve stimulation and offers a starting point for optimization of electrode design and parameter selection.

Methods

Online Front-End Interface

To increase accessibility of the model, a front-end interface was developed and hosted online at https://nervestimlab.utdallas.edu. The interface allows users to adjust each of the parameters by either typing in values or adjusting sliders. The model can then be run with the inputted parameters by hitting the “solve” button, and dose-response curves will be returned in approximately 1-5 minutes. Additionally, the “estimate” button can be used to quickly approximate model results by interpolating between similar previous models. Sigmoid curves are fit to the normalized dose-response curves of previous models using the function

$$Y=\frac{1}{1+e^{-(x-x_{0.5})}∕k}$$ (1)

where x is current, x_0.5 is the current value corresponding to the 50% response value, and k is a slope factor related to the steepness of the curve. For a new model that has not been previously run, the values for x₀.5 and k are linearly interpolated using the 30 most similar models. If the new model lies outside the range of previously run models, nearest neighbor interpolation is used instead.

The front-end interface was connected to the model, which was hosted on an Ubuntu server, using the Python web framework Flask and the Python web server Gunicorn [16].

Computational Model

The computational model was creating using the open-source programs Gmsh, FEniCS, and Python [17-19]. A link to the source code on GitHub is available on the webpage. All aspects of the nerve and electrode geometry, the electrical properties of the various materials, the fiber diameters, and the stimulation waveform were parameterized with users being able to adjust the values through the front-end interface. The list of parameters can be found in table S1.

The 3-dimensional cylindrical structure of the nerve, electrodes, and insulation were created and meshed in Gmsh 4.0 after using the pygmsh package for Python to write the Gmsh script [20]. The created mesh was then imported into a Python program utilizing the FEniCS package to compute the electric potentials generated inside the nerve. This was done by assuming a quasi-static system with no enclosed charge, which allows the electric potential to be expressed through the Laplace formulation.

$$\nabla \cdot \sigma \nabla V=0$$ (2)

Dirichlet boundary conditions were applied to the radial surfaces of the cylindrical model while Neumann boundary conditions were applied to both ends. Unit voltage was applied to all cathodic electrodes simultaneously and the current exiting the model was measured. The same was then done with all anodes. The solutions were linearly scaled to 1 mA and added together giving the result as if +1 mA was applied to the cathodes and −1 mA was applied to the anodes [21]. The resulting electric potential distribution was applied to a passive axon model to create dose-response curves of fiber activation as a function of both current and voltage. This passive model was created by fitting 3-dimensional curves to data from an active mammalian axon model in NEURON [22]. The fitted curves determined the magnitude of a variable, such as the electric potential, present at the node of Ranvier that was necessary to induce an action potential in a fiber with a known diameter using a stimulation waveform with a known phase-width. Curves were created for four different waveform types: cathode-leading biphasic, anode-leading biphasic, cathodic monophasic, and anodic monophasic. Additionally, five different variables were selected: electric potential, current density in the axial direction, current density in the radial direction, total current density, and the activating function. The fitted functions took the form of the fundamental Weiss formula [23].

$$Rheobase=A_1{e}^{-K_{1}D}+C_1$$ (3)

$$Chronaxie=A_2{e}^{-K_{2}D}+C_2$$ (4)

$$Threshold=Rheobase\left(1+\frac{Chronaxie}{PW}\right)$$ (5)

where A, K, and C are constants, D is the fiber diameter in micrometers, and PW is the phase-width in milliseconds. Values for the constants were obtained for each of the five variables and for each type of waveform for a total of 20 fitted curves (Tables S2, S3). In addition, for biphasic pulses, the effect of the interphase delay on the threshold was modeled using a simple reciprocal function.

$$Ratio=\frac{0.1}{(Delay+0.1)}$$ (6)

$$Threshold=MT+Ratio\ast (BT-MT)$$ (7)

where MT is the monophasic threshold and BT is the biphasic threshold.

Rat Sciatic Nerve Stimulation

All handling, housing, stimulation, and surgical procedures were approved by The University of Texas at Dallas Institutional Animal Care and Use Committee. Nine Sprague Dawley female rats (Charles River, 10 to 15 months old, 300 to 800g) received sciatic nerve stimulation using either monophasic waveforms, biphasic waveforms, or both. All animals were housed in a 12:12 h reverse light-dark cycle. Rats were anesthetized using ketamine hydrochloride (80 mg/kg, intraperitoneal (IP) injection) and xylazine (10 mg/kg, IP) and given supplemental doses as needed. Once the surgical site was shaved, an incision was made on the skin directly above the biceps femoris. The sciatic nerve was exposed by dissecting under the biceps femoris. The gastrocnemius muscle was separated from skin and surrounding tissue. Cuff electrodes, described in detail previously, were then placed on the sciatic nerve with leads connected to an isolated programmable stimulator (Model 4100; A-M Systems™; Sequim, WA) [24]. For monopolar stimulation, only one lead was connected to the stimulator with a subcutaneous needle placed in the animal’s back serving as the other electrode. The nerve was left in place underneath the biceps femoris and the cavity was kept full of saline at all times to ensure that the cuff would be operating in a uniform medium with conductance similar to tissue. The Achilles tendon was severed at the ankle and affixed to a force transducer using nylon sutures. The foot was clamped and secured to a stereotaxic frame to prevent movement of the leg during stimulation and to isolate recordings from the gastrocnemius muscle. Stimulation was delivered through the A-M Systems™ Model 4100. Stimulation consisted of 0.5 second trains of both monophasic and biphasic pulses (10-500 μs phase-width, 20-2000 μA) at 30 Hz. Stimulation phase-widths and intensities were randomly interleaved. The minimum and maximum current values were manually set in each experiment to ensure that the range of values included the entire dose-response curve. Stimulation was delivered every 15 seconds and each parameter was repeated in triplicate. Voltage traces were recorded using a digital oscilloscope (PicoScope1 2204A; Pico Technology; Tyler, TX). The force of muscle contraction was recorded through a force transducer (2kg EBB Load Cell; Transducer Techniques; Temecula, CA) which was connected to an analog channel on an Arduino1 Mega 2560. All components were integrated using MATLAB. Data was sampled at 100 Hz.

Results

Tripolar Cuff Electrode

To validate the model, we compared thresholds and dose-response curves to data obtained from previous literature. As an initial comparison, we selected a simple tripolar cuff electrode tested on the rat sciatic nerve reported in a previous study [13]. Fiber activation thresholds predicted by the model closely matched those reported in vivo using this cuff electrode (Fig. 1). The thresholds at multiple phase-widths were evaluated to generate strength-duration curves for the specific cuff electrode geometry. There was no significant difference between the in vivo thresholds and the thresholds predicted by the model at any phase-width (Fig. 1b, unpaired two-tailed t-tests with unequal variance, 0.05 ms: p=0.811, 0.1 ms: p=0.970, 0.5 ms: p=0.480, 1.0 ms: p=0.600). Additionally, the predicted dose-response curves closely matched the representative example curves selected for two different experimental subjects (Fig. 1c).

Figure 1: Model accurately replicates results with tripolar cuff electrode.

a) Schematic showing the tripolar cuff electrode tested in-vivo and recreated in the model. b) Example dose-response curves from both the in-vivo study and the model. Dose-response curves generated in the model closely resemble those measured in-vivo. c) The activation threshold of the modeled nerve is similar to the in-vivo threshold at each phase-width tested.

Monopolar Electrodes vs Tripolar Electrodes

Stimulation with monopolar cuff electrodes has been reported to activate nerve fibers at lower thresholds than stimulation with tripolar cuff electrodes on the cat sciatic nerve [14]. We tested this finding in the model and found concurring results. Dose-response curves generated in the model aligned with those selected from a representative experimental subject (Fig. 2e,f). Mean thresholds and gains, which represent the steepness of the dose-response curve, were not reported for each type of stimulation in the given study. However, the ratio of these values for monopolar and tripolar stimulation conditions were reported. We replicated this calculation for values obtained with the model. The ratio of the threshold for monopolar and tripolar stimulation was similar in the model as compared to the in vivo data (Fig. 2g, unpaired two-tailed t-tests with unequal variance, threshold ratio: p=0.087). However, the gain ratio between the two was found to be slightly higher in the model as compared to the in vivo data (Fig. 2h, unpaired two-tailed t-tests with unequal variance, gain ratio: p=0.006). Despite this difference, there is still significant overlap between the two distributions.

Figure 2: Monopolar stimulation is more efficient than tripolar stimulation.

a) Schematic showing stimulation with a monopolar electrode configuration. b) Schematic showing stimulation with a tripolar electrode configuration. c) Representative dose-response curves from one experimental subject receiving stimulation with a monopolar electrode configuration. d) Representative dose-response curves from one experimental subject receiving stimulation with a tripolar electrode configuration. e,f) Inverted and normalized versions of the same dose-response data as found in (c) and (d). Dose-response curves generated in the model closely match the in-vivo data. Different colored curves represent different fascicles in the same nerve. g) Histogram distributions of the ratio between the monopolar stimulation threshold and tripolar stimulation threshold for both in-vivo data and model data. No significant difference was found between the two distributions. h) Distributions of the ratio between the gains (steepness of the curve) for monopolar and tripolar stimulation. A significant difference was found between the two distributions, but there is still significant overlap.

Intrafascicular Electrodes vs Cuff Electrodes

Comparisons of intrafascicular stimulation to cuff electrode stimulation have demonstrated that intrafascicular electrodes have significantly lower thresholds [15]. To determine whether our model reflected these observations, we modeled intrafascicular stimulation using the transverse intrafascicular multichannel electrode (TIME) and compared it to a standard tripolar cuff electrode. Again, the model predicted responses similar to those reported in vivo. There was no significant difference between the in vivo data and the model when comparing the intrafascicular thresholds (Fig. 3c, unpaired two-tailed t-test, p=0.2), the intrafascicular saturation point (Fig. 3c, unpaired two-tailed t-test, p=0.15), the cuff electrode thresholds (Fig. 3d, unpaired two-tailed t-test, p=0.23), and the cuff electrode saturation point (Fig. 3d, unpaired two-tailed t-test, p=0.38).

Figure 3: Intrafascicular stimulation model results resemble in-vivo data.

a) Schematic showing stimulation with an intrafascicular electrode. b) Schematic showing stimulation with a tripolar cuff electrode. c) Threshold and saturation currents for both in-vivo data and the model using intrafascicular electrodes. No significant difference was found. d) Threshold and saturation currents for both in-vivo data and the model using a cuff electrode. No significant difference was found.

Monophasic vs Biphasic Waveforms

Multiple studies have demonstrated that monophasic stimulation waveforms activate nerve fibers at a lower stimulation threshold than biphasic waveforms when using a monopolar electrode [25-28]. The model confirmed this result, but also predicted that for bipolar configurations, monophasic and biphasic waveforms have similar thresholds. We tested this prediction in vivo by examining activation of the rat sciatic nerve. Our findings confirm that when using monopolar stimulation, the monophasic waveform threshold is significantly lower than the biphasic stimulation threshold (Fig. 4c). However, when using bipolar stimulation, the thresholds are similar (Fig. 4d). Additionally, the effect is larger with lower phase-widths. There was a significant difference in the thresholds of monophasic and biphasic waveforms with three out of the four phase-widths tested when delivering monopolar stimulation (Fig. 4c, paired two-tailed t-tests, 0.01 ms: p=0.005, 0.025 ms: p=0.003, 0.1 ms: p=0.033, 0.5 ms: p=0.157), but no significant differences were found for bipolar stimulation (Fig. 4d, paired two-tailed t-tests, 0.01 ms: p=0.213, 0.025 ms: p=0.673, 0.1 ms: p=0.192, 0.5 ms: p=0.484). Additionally, the model predicted that the difference between monophasic and biphasic pulses on a monopolar electrode can be removed if a suitably long delay is introduced between the two phases of the biphasic pulse (Fig. 4e), which agrees with previously reported results [29]. The finding that monophasic and biphasic waveforms have similar thresholds when using bipolar electrodes has not been previously reported, validating the utility of the model in identifying and testing new features of stimulation.

Figure 4: Monophasic and biphasic waveforms are similar in bipolar electrode configurations.

a) Schematic showing stimulation with a monopolar cuff electrode. One of the electrodes inside the cuff was selected as the cathode and a distant, subcutaneous needle was used as an anode. b) Schematic showing stimulation with the same cuff electrode, but using a bipolar electrode configuration. c) Difference in thresholds between monophasic and biphasic waveforms at various phase-widths when using a monopolar electrode around the nerve. The left end (triangle) of each line represents the biphasic threshold and the right end (circle) represents the monophasic threshold. Monophasic thresholds were significantly lower than biphasic thresholds at phase-widths of 0.01 ms, 0.025 ms, and 0.1 ms. d) Difference in thresholds between monophasic and biphasic waveforms for bipolar electrodes around the nerve. No significant difference was found at any phase-width. e) Effect of interphase delay on the threshold of a biphasic pulse. Y-axis represents the percent reduction in threshold from that of a standard biphasic pulse with no interphase delay.

Higher Order Effects

Previous studies show an inverted-U relationship between stimulation intensity and some higher-order effects of vagus nerve stimulation, including enhancement of synaptic plasticity and facilitation of memory [30-37]. This non-monotonic effect cannot be described simply by direct activation of a single population of fibers in the vagus nerve. While the mechanisms that underlie this effect are unknown, it has been proposed that differential activation of two fiber populations with differing thresholds and opposing actions could explain the inverted-U. We sought to evaluate the viability of this explanation by comparing two identical models with different fiber diameter distributions (Fig. 5). We leveraged previous observations that the range of effective intensities is wider at shorter widths, which would increase the difference in threshold between the fiber populations [28]. Our modeling data replicated this effect, such that differential activation of the simulated fiber populations resembles the inverted-U seen in vivo at both phase widths tested. While these findings do not directly implicate differential fiber populations as the driver of the inverted-U observed with VNS, they illustrate that the model can be used to corroborate and make testable hypotheses related to higher order effects of nerve stimulation.

Figure 5: Possible mechanism responsible for the inverted-U effect present with vagus nerve stimulation.

a) A model of stimulation of the rat vagus nerve with a simple bipolar cuff electrode and a phase-width of 100 μs. The inverted-U effect can be explained by activation of two distinct fiber populations: one with a large diameter and low-threshold that increases plasticity, and one with a small diameter and high-threshold that limits plasticity. b) The same model as in (a), but with a 500 μs phase-width. c) Inverted-U function in (a) overlaid with in-vivo data from six studies using phase-widths of 100 μs [33-37]. The modeled curve closely resembles the in-vivo data in demonstrating that this mechanism is a feasible explanation for the inverted-U. d) Inverted-U function in (b) overlaid with in-vivo data from three studies using phase-widths of 500 μs [30-32]. Just as in (c), the modeled curve closely resembled the in-vivo data. In-vivo data points were connected using piecewise cubic interpolation.

Discussion

In this study, we created and validated an online, parameterized, open-source tool to improve accessibility to computational modeling of nerve stimulation. The model was created by combining the open-source programs Gmsh and FEniCS and hosted online through a Python web server. We validated the model by comparing the predicted threshold and dose-response curves to three prior studies utilizing a variety of nerve stimulation paradigms and species. In all cases, results predicted by the model were similar to those reported in vivo. We report that the model aligns with previously reported data in conditions using a simple cuff electrode model, a cuff electrode model comparing monopolar to tripolar stimulation, and a model comparing stimulation with intrafascicular electrodes to cuff electrodes. Finally, we demonstrated how differential fiber type recruitment can produce higher-order effects commonly observed with nerve stimulation, including inverted-U phenomena.

Multiple studies demonstrate that monophasic pulses activate nerve fibers at significantly lower thresholds than biphasic pulses when using a monopolar electrode [25-28]. Our model replicated this finding, but also predicted that there is no significant difference in thresholds when using bipolar electrodes. In vivo testing on the rat sciatic nerve confirmed this result (Fig. 4). Further testing with the model suggests that this is due to the difference between anode leading and cathode leading biphasic pulses. For a monopolar electrode, anode leading biphasic pulses have lower thresholds than the more typically used cathode leading pulses [38]. Thus, when cathodic monophasic pulses are compared to cathode leading biphasic pulses as previous studies have done, there is a large difference in thresholds. However, when cathodic monophasic pulses are compared to anode leading biphasic pulses, the difference is much smaller. This finding, combined with the fact that bipolar electrodes will have both anodic and cathodic leading pulses present, one on each pole, explains why the choice of waveform has little impact on threshold when using bipolar electrodes. This behavior has not been previously reported to our knowledge, validating the utility of this tool in generating testable hypotheses.

Beyond confirming previous results, the model can also be used to make testable hypotheses about the effects of various stimulation applications. For example, the intensity of vagus nerve stimulation has been shown to exhibit an inverted-U relationship with efficacy [30,33-36]. One possible explanation for this phenomenon is that activation of low threshold, large diameter fibers promotes VNS effects, whereas activation of a second population of smaller diameter fibers at a higher threshold limits these effects. We show that the model can be used to demonstrate the validity of this explanation, although it may not account for the full complexity of the inverted-U phenomenon. More importantly, this demonstrates that the model may be useful to predict and evaluate higher-order effects of nerve stimulation implementations.

The use of open-source software and an accessible front-end interface for the model offers several advantages. First, our implementation allows for a wide variety of nerve stimulation paradigms to be created and tested without requiring researchers to learn specialized software and create a de novo model each time. Next, the use of open-source software aids in standardization of modeling assumptions, which can vary between studies and significantly impact the results. Moreover, open-source software allows flexibility for further customization to test geometries and configurations that the current model cannot replicate. Creation and testing of new electrode implementations will also aid in improving the model as new in-vivo data can be used to fine-tune the model parameters.

While the accuracy of the model is generally high and concordant with reported data, there is still some discrepancy compared to in vivo measures. These differences likely arise from both variance innate to biological experiments as well as oversimplification of values used for the geometric and electrical properties of the nerve. Despite these limitations, we posit that the model can still be reliably used as a starting point to estimate effective stimulation parameters, compare and optimize different electrode designs, and explore previously untested methods of nerve stimulation.

In an effort to balance reduction of complexity in the model with inclusion of commonly used features, we imposed a number of limitations on the model parameters. This may reduce the precision of the results and prevent certain stimulation methods from being tested. For example, the requirement of the nerve to be circular does not allow for modeling of the flat interface nerve electrode (FINE) [39]. More detailed parameters, specifically in the nerve geometry, would allow for greater customization and precision. Additionally, only one active axon model was used to create the passive model, and it only accounts for myelinated fibers. Other more recent axon models may exhibit slightly different results [40]. The passive model for each variable was also created from the same monopolar electrode model around a rat sciatic nerve with no insulation. This could result in reduced passive model accuracy when extrapolated to other types of models such as bipolar, tripolar, or intrafascicular configurations. We found that the passive model using voltage retained the most accurate results when used with the electrode configurations seen in this study. While increasing the complexity of the model would improve precision, we elected to restrict these parameters in order to promote simplicity and broad applicability. Further developments that increase the complexity of the model may improve its accuracy. Additionally, future studies comparing monophasic and biphasic waveforms may further clarify how various conditions affect stimulation outcomes with each type.