1. Threshold Current vs. Conductivity
Computational Modeling of Peripheral Nerve Stimulation
P. Krastev, B. Tracey, M. Williams, NeuroMetrix, Inc, Waltham, MA, USA
Peripheral nerve stimulation (PNS) is used in regional anesthesia
I. INTRODUCTION
II. COMSOL MODEL
Threshold Current vs. Conductivity
(PW = 100 mS)
Typical bioelectric modeling studies simplify the field equations, ignoring tissue capacitance and assuming tissue properties are frequency-independent. We sought to avoid these assumptions in order to investigate the impact of capacitive and frequency-dependent effects.
Model of an arm. The model includes skin and muscle, as well as surface electrodes and a needle tip. COMSOL is used to solve the Helmholtz equation (exact solution) for the potential within the arm.
III. RESULTS
Threshold Current vs. Resistivity
In PNS, a needle is used for both electrical stimulation and drug injection. The needle is advanced slowly towards the target nerve while current pulses are applied. PNS is used to identify the nerve by verifying that the correct muscles twitch when the nerve is stimulated. When a twitch is seen at target stimulus (generally mA) the nerve has been localized and anesthetic is injected.
Needle
(point source)
Return electrode
Nerve is assumed parallel to x-axis
Boundaries are electrically insulated
In plots above, tissue resistivity is varied over the same ratio as the impedance changes in [1]. Right and left plots are identical except data is presented vs. conductivity (s) or resistivity (1/s). Lower plots are for isotropic tissue, and can be compared to elbow measurements. Upper plots are for anisotropic muscle tissue with transverse to longitudinal conductivity ratio of ½ [2], and can be compared to axilla measurements.
While widely used, PNS suffers from several drawbacks. The threshold current needed to elicit response when close to nerve is variable, and the causes for this variability are not well understood. Also, visual observation of muscle twitch as an indicator of needle to nerve distance is subjective.
A recent paper [1] hypothesized that current thresholds should be related to tissue impedance. Impedance can be measured during the procedure, so establishing a link between impedance and threshold could lead to improved procedures.
Anisotropy effect on threshold currents: threshold vs. conductivity ratio (sL/sT)
Rotation effect on threshold currents: rotation is around Z-axis
We use a frequency-domain solution: the source spectrum is found from the FFT of the stimulus waveform, the field is calculated at each frequency, and the time-dependent field along the nerve is found from inverse FFT. Because there are thousands of FFT bins, a brute-force solution is infeasible. We use an interpolation approach and verify that the solution converges by iteratively refining the frequency grid. This gives a computational speedup of roughly 100x. The “sweep” feature in COMSOL turns out to be particularly useful for this calculation.
Ref. [1] suggested that threshold variations in the axilla were caused by muscle anisotropy. The effects of varying the degree of anisotropy and the relative angle between muscle and nerve are shown above. Variations in the ratio have a stronger effect than variations in the angle between muscle and nerve. The overall variability in threshold currents due to anisotropy is sufficient to explain the observed variability reported in [1].
Threshold currents for nerve excitation are modeled using COMSOL and the Frankenhauser-Huxley equations for nerve response. Their dependence on distance and tissue properties agrees with measurements. The model confirms the anti-correlation between threshold currents and impedance, and also confirms that there is little dependence on tissue property when the needle touches the nerve. The model shows a 1/resistivity dependence which is predicted by theory. Studies of muscle anisotropy suggest it is possible that the variability seen in axilla thresholds can be explained by variations in the anisotropy of muscle conductivity.
[1] A.R. Sauter, M.S. Dodgson, H, Kalvoy, S. Grimnes, A. Stubhaug, O. Klasstad, “Current threshold for nerve stimulation depends on electrical impedance of the tissue: a study of ultrasound-guided electrical nerve stimulation of the median nerve”, Anesthesia & Analgesia, Vol. 104(4), 2009, pp
[2] C. Gabriel, S. Gabriel and E. Corthout, “The dielectric properties of biological tissues: I. Literature survey”, Phys. Med. Biol, Vol. 41, 1996, pp
[3 ] B. Frankenhaeuser and A.L. Huxley, “The action potential in the myelinated nerve fiber of Xenopus Loevis as computed on the basis of voltage clamp data,” J. Physiol., Vol 171, 1964, pp
IV. CONCLUSIONS
Tissue property frequency dependence
(Muscle, transverse, from [2])
Magnitude of COMSOL field along nerve vs. frequency
R=3 mm, 30 frequencies
Field solutions vs. # of COMSOL frequencies
R=3 mm and 8.7 mm
The figure above presents results of a clinical study of threshold currents in 29 patients. A statistically significant anti-correlation is found between threshold current and impedance. The anti-correlation is stronger when the distance between the needle tip and the nerve is larger. There is considerably more scatter in the axilla than in the elbow. The authors make a conjecture that this might be due to a larger muscle tissue conductance anisotropy (more muscle in the axilla than in the elbow, which has more fat).
Our goal is to use COMSOL and mathematical models of nerve response to confirm the observed relationship between impedance and threshold, and to test the hypothesis that muscle anisotropy causes variation in current thresholds.
The potential distribution calculated with COMSOL is used as an input in a model for nerve conduction threshold calculation implemented in MATLAB. The model uses a nerve fiber distribution and solves the F-H equations [3].
COMSOL 2009, Boston, MA