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ANALGESIA

The Effect of Injectate Conductivity on the Electric Field with the Nerve Stimulator Needle: A Computer Simulation

From the Department of Anaesthesia, Addenbrooke’s Hospital, Cambridge, UK.

Address correspondence and reprint requests to Dr. Ari Ercole, Department of Anaesthesia, Box 93, Addenbrooke’s Hospital, Hills Road, Cambridge, CB2 2QQ, UK. Address e-mail to ae105{at}cam.ac.uk.

Abstract

BACKGROUND: Electrical neural stimulation is commonly used to localize neural structures and place local anesthetic for regional anesthesia. The sharp tip of the stimulating needle gives rise to an electric field which is highly localized. The electrostatic effect of the injected solution on the field distribution and strength has not previously been modeled.

METHODS: The three-dimensional electric field around a 0.7 mm diameter, un-insulated, hollow needle with a 30 degrees bevel was calculated in silico using a hybrid finite difference/impedance network method to solve the Laplace equation. The surrounding tissue was assumed to be electrically uniform. A sphere of injectate centered on the bevel was modeled as a region of conductivity differing from that of the bulk tissue.

RESULTS: The electric field strength was highly concentrated at the needle tip and decayed rapidly with distance r approximately as 1/r^1.7. It was demonstrated that the electric field in the immediate vicinity of the needle tip was greatly reduced in the presence of solutions with conductivity greater than that of the surrounding tissue. A 1.5 mm radius region of conducting solution (equivalent to saline or local anesthetics) reduced the field by 31%. The same volume of a relatively insulating injectate, such as dextrose solution, led to a field enhancement of approximately 15%.

CONCLUSIONS: The electric field magnitude in the vicinity of the needle tip decayed more slowly with distance than predicted by Coulomb’s law. This was independent of the presence of injectate. The near instantaneous abolition of muscle twitch with injection of small volumes of local anesthetic is consistent with an electrostatic effect, rather than a pharmacological or mechanical one. The change in field strength depended upon the volume of the injectate and its conductivity relative to that of the surrounding tissue. In this simulation, even tiny volumes of injectate lead to significant changes in field and therefore threshold current, which may have clinical implications.

Reliable and safe regional anesthesia is critically dependent on the accurate placement of the local anesthetic in relation to the nerve to be anesthetized. Electrical peripheral nerve stimulation remains a popular method for achieving this. Electric fields (and therefore the resulting current densities) are highly localized around sharp points, such as the needle tip. If a motor response can be elicited even at low stimulation currents, then the needle must be in close proximity to the nerve and reliable anesthesia should be obtained when the local anesthetic is injected. This rapid decay of field with distance from the needle tip has been demonstrated both experimentally in electrophoresis models,1,2 by numerical simulation of one-dimensional insulated and un-insulated needles1,3 and, more recently, for a number of realistic needle tip geometries.4

It is observed clinically that the twitch is abolished immediately on injection of even a very small volume of local anesthetic solution, and this is often regarded as an indicator of successful needle placement.5 The mechanism of this effect is unclear. Diffusion of local anesthetic into neural tissue is not instantaneous and in any case it is apparent under ultrasound that the nerve is not instantly surrounded by injectate and so action potentials cannot be blocked immediately. Furthermore, it has been shown2,6 that injection of saline rather than local anesthetic also abolishes motor response, and that the twitch may be restored, or even augmented, by the injection of 5% dextrose solution (an electrical insulator). These results suggest an electrical rather than pharmacological or mechanical effect. Electrophoresis experiments2 have led to the suggestion that conducting solutions, such as local anesthetic or saline, spread around the tip and along the shaft of the needle, dissipating the field and also rendering the insulation ineffective. However such models are not spatially quantitative and provide no explanation for the augmentation phenomenon seen with injection of 5% dextrose solution.

The physics of the interaction of the electric field around the needle with the injectate has not been previously studied. The tiny dimensions of the problem and complex, heterogeneous nature of biological tissue serve, respectively, to challenge and confound any experimental approach. Computer modeling, in contrast, is increasingly recognized as a useful methodology in anesthesia research7 as it offers a method of reducing the system of interest to its most basic constituents. In this investigation, numerical solutions for the fundamental electrostatic equations are presented for a simplified model of a needle surrounded by injectate of varying conductivity and uniform tissue.

METHODS

The needle geometry adopted is shown in Figure 1. The 30 degrees bevel is representative of typical regional block needles used in clinical practice. The shaft was assumed to be un-insulated. The needle was arbitrarily set as having negative polarity to mirror clinical practice in which cathodic stimulation leads to lower threshold currents.8

View larger version (16K): [in this window] [in a new window]   Figure 1. Needle geometry assumed in the model. The needle is assumed to be a perfect conductor. The shaft is uninsulated and the conductivity of the lumen contents can be chosen independently of that of the surrounding tissue.

The electric potential and thus electric field strength were calculated using a hybrid approach based on a finite difference method similar to that of Davis et al.4 but solved iteratively on a three dimensional impedance network. The latter refinement allows a spatially varying conductivity to be accommodated. Further mathematical details are given in the Appendix. The simulations were made on a regular, three-dimensional cubic lattice 201 nodes wide (i.e., 201³ nodes in total). The mesh size was 0.025 mm. The center of the needle bevel face corresponded to the center of the lattice.

The effect of the injectate was modeled as a spherical region around the center of the 30 degrees bevel face of different conductivity to the surrounding tissue. Additionally, the conductivity of the contents of the needle lumen could be adjusted independently. Electrical conductivity of biological materials varies, and order of magnitude estimates are therefore appropriate. The electrical conductivity of bulk tissue was assumed to be isotropic and of the order of 1 mS cm^–1. This value is representative of fat, muscle, and neural tissue.9,10 The conductivity of 0.9% sodium chloride solution was taken to be approximately 10 mS cm^–1.10 A direct resistance measurement was performed which showed that the conductivity of 0.5% l-bupivacaine solution (Abbot Laboratories, Abbot Park, IL) was approximately 100 mS cm^–1. The conductivity of deionized water on the other hand is typically of the order of 1 µS cm^–1.

The simulations were performed on a 1.5 GHz single-processor Intel platform. Mirror symmetry about the needle’s mid-saggital plane was exploited to reduce the number of calculations required. Nevertheless, the computational overhead was significant. All the software was written in the Java programming language which has reliable memory management and performs acceptably for this sort of numerical computation with the current generation of just-in-time compilers. Mathematical details are given in the Appendix.

To characterize the effect of the different injectates more succinctly and facilitate a quantitative comparison, the electric field magnitude |E| was assumed to vary as a function of distance directly ahead of the needle tip r (up to the edge of the injectate sphere) according to an equation of the form;

The parameters and were determined by nonlinear curve fitting using Levenberg-Marquardt optimization without data transformation. The form of equation (1) was chosen because for = 2 it reduces to Coulomb’s law for a point (i.e., infinitely sharp) source. For = 0, however, it describes the uniform field seen within a parallel plate capacitor (i.e., an ideal "blunt" case). Since the maximum field is mathematically undefined, the fitted value of is a useful measure with which to compare the electric field strength in the region surrounding the needle tip. Thus and describe the degree of field localization and the absolute field strength, respectively.

RESULTS

The calculated electric field magnitude in the needle mid-saggital plane (top row) and in the plane of the needle bevel (bottom row) is demonstrated in Figure 2. The center panel shows the case of a needle in uniform tissue with a conductivity of 1 mS cm^–1. It is clearly appreciated that the field is most strongly concentrated at the very tip of the needle with weaker contributions along the leading edges. There is also a weak field at the back of the bevel and around the shaft of the needle. The applied potential was –1 V, and therefore the fields are directed towards the needle. Fields of nearly 4 V mm^–1 were found in the immediate vicinity of the needle tip surface, a value which scales linearly for other applied needle potentials. This is only slightly less than that found in Davis et al.,4 the discrepancy being easily accounted for by the lack of needle insulation. Some granularity is evident around the tip in the bevel plane views which is a numerical artifact. The field rapidly decreases to zero within the lumen which acts as a Faraday cage. Similarly the field within the needle wall is zero as is to be expected since the metal represents an isopotential.

View larger version (50K): [in this window] [in a new window]   Figure 2. Plots of the absolute electric field in the mid-saggital plane (top row) and oblique plane of the bevel tip (bottom row). The side panels show the effect of 1.5 mm radius spheres of injectate with conductivity as indicated. The needle voltage was –1 V. The conductivity of the surrounding tissue was 1 mS cm–1. The needle lumen is modeled as having the same conductivity as the immediate surroundings. Since the needle polarity was negative, the electric field is oriented towards the needle.

Further calculations were performed assuming a 1.5 mm radius sphere of injectate centered on the middle of the 30 degrees bevel. This was the largest diameter sphere that could be accommodated without edge effects becoming significant. The left and right panels of Figure 2 illustrate the effect of injections of 1 µS cm^–1 and 100 mS cm^–1 solutions. These cases correspond to low conductivity (water or 5% dextrose) and high conductivity (local anesthetic), respectively. In each case, the lumen contents are assumed to be the same as that of the injected solution.

For the high conductivity case, the electric field magnitude can be seen to be significantly reduced at the tip and also at all points around the end of the needle. The converse is true for the low conductivity case, where the field around the needle end and within the injected solution is enhanced. Although the magnitude of the electric field in the vicinity of the needle tip varies with the conductivity of the injected solution, its distribution seems similar. This is demonstrated in Figure 3 where the field strength is plotted against distance along a line ahead of the needle tip for the high, low, and uniform conductivity cases described above. The field declines monotonically with distance except for a step which occurs at the boundary between injected solution and the surrounding tissue.

View larger version (10K): [in this window] [in a new window]   Figure 3. Calculated absolute electric field strength as a function of distance ahead from the needle tip. The discontinuity approximately 1 mm from the needle tip (corresponding to 1.5 mm from the center of the bevel) coincides with the edge of the injectate sphere.

The three curves of Figure 3 differ in magnitude but have broadly the same shape. This implies that the effect of the injected solution is not confined merely to the very surface of the needle but affects the field everywhere in its vicinity. As expected, the fitted value of (Eq. 1) was found to vary with injectate conductivity. This relationship is shown in Figure 4. With the surrounding tissue taken to have conductivity of 1 mS cm^–1, it can be seen that injectate of conductivity corresponding to local anesthetic (100 mS cm^–1) caused a 31% reduction in the electric field. Conversely, injection of water (1 µS cm^–1) leads to a field enhancement of approximately 15%. This correlates with the clinical observations2,6 described earlier. The exponent was found to be between 1.6 and 1.7. This was less than the value of two predicted by Coulomb’s law and was independent of the injectate. The electric field magnitude was found to decay with radial distance from the needle tip in a quantitatively similar way.

View larger version (10K): [in this window] [in a new window]   Figure 4. Values of (units of V mm(–1)) determined by non-linear fitting as a function of injectate conductivity. Again a 1.5 mm radius sphere was modeled and bulk tissue conductivity of 1 mS cm–1 was assumed.

DISCUSSION

The results presented predict that the electric field around a stimulating needle is inversely related to injectate conductivity. That is, solutions such as 5% dextrose, which were less conductive, enhanced the field, whereas more conductive solutions, such as local anesthetics, diminished the electric field. Although the in vivo case is likely to differ quantitatively due to the assumptions made in the model, this is consistent with the clinical observation that injection of any conductive solution leads to abolition of muscle twitch, whereas insulating solutions cause an apparent reduction in threshold current and augment nerve stimulation. The simulations demonstrated that this can be explained as a purely electrostatic effect. The highest electric field strengths are localized at the sharpest edges of the needle and decay with distance less rapidly than predicted by Coulomb’s law. This finding could be experimentally verified using an electrolysis tank with scale model needles. The presence of conducting solution around the tip effectively creates a blunt end, electrically speaking, to the needle. Clinically this is manifested as an increase in threshold current; if the stimulation current were to be increased, both the needle voltage and the resulting electric field would change in proportion and muscle twitches would return. The converse situation is true for the injection of relatively insulting solutions. Since the conductivity gradient between needle body and injectate is greater, the electric field strength is stronger for a given needle potential.

Any mathematical model must make certain assumptions in order to be computationally tractable and the system presented here is no exception. The surrounding tissues have been taken to be structurally uniform. Clearly this is an approximation, since real nerves are typically bundled with other structures and surrounded by connective tissue. These tissues may additionally be electrically anisotropic9 with conductivity varying with direction by an order of magnitude or so. Since it is the ratio of tissue to injectate conductivity that enters the calculation, and even the order of magnitude changes do not change significantly (Fig. 4), the effect of anisotropic conductivity is likely to be minimal.

The injectate has been assumed to adopt a spherical shape centered on the needle bevel. This is justifiable for the small volumes of injectate of interest: in this case the mechanical effects of tissue distension would not yet be significant and the spherical geometry has minimum surface tension and therefore minimizes energy. For larger volumes, experience with ultrasound-guided nerve blockade suggests that the injected volume is initially approximately circular or slightly ellipsoidal in cross-section before rapidly redistributing along fascial planes. Violation of these assumptions would be primarily expected to redistribute the electric field distribution at the local anesthetic/tissue interface, concentrating it at points where the surface is most curved. Unless the injectate adopts a grossly nonspherical conformation, the effect at the needle tip would be of second-order and so the spherical approximation seems reasonable.

Although electrostatic modeling of insulated needles is possible,4 the detailed field distribution at the tip depends crucially on the position of the insulation, which is difficult to measure accurately. The needle in this study was therefore assumed to be un-insulated. Although atypical of needles used clinically, this assumption makes the simulations more parsimonious and makes little difference to the field profile ahead of the needle tip11 but it does change the field around the shaft.3,12 Since we have been interested only in the field around and ahead of the tip, the insulation can be safely omitted from the calculation without loss of generality. Furthermore, the gel electrophoresis experiments of Tsui et al.2 suggest that injectate spreads around the insulation making it ineffective. The results here clearly demonstrate that, while this may or may not be the case in vivo, it is not necessary to explain twitch abolition or augmentation.

The results presented may have practical implications for situations where the nerve localization has been traumatic or difficult such that multiple attempts have been required. Blood is an electrical conductor with conductivity similar to saline, and it too would be expected to cause a similar reduction in field. Even without actual intravascular placement, the calculations presented suggest that only a tiny amount of blood at the needle tip could cause a significant apparent increase in threshold current. Thus, if the procedure has been locally traumatic, the needle would need to be closer to the nerve than normal for muscle twitches to be apparent. This may make nerve localization more difficult and potentially hazardous.

The model presented here is a DC model and assumes the impedance network to be purely resistive. In reality, nerve stimulators operate in a pulsed fashion and real biological tissues exhibit frequency-dependent impedance behavior, which may have significant reactive elements. However, this is unlikely to be significant at the low frequencies at which stimulation is performed.10

In this study, a model of nerve stimulator needles has been presented which predicts that the electric field at the tip can be either reduced or augmented simply by the injection of a small volume of solution of either high or low conductivity respectively. Clearly a number of simplifications have been necessary. Ultimately, however, the power of mathematical models as an aid to understanding complex systems depends on the selection of the most parsimonious models which accurately represent clinical experience.

APPENDIX

The electric potential at Cartesian position (x, y, z) may be calculated by solution of the Laplace equation,

Equation (A1) is solved using the finite difference method with successive over-relaxation to aid convergence. This is similar to the method of Davis et al.4 except that the differing conductivities of tissue and local anesthetic solution are incorporated as follows. The space is divided into a regular cubic lattice. For each node (i, j, k), the residual, R, is calculated as,

where,

In equations (A2a) and (A2b), (i, j, k) represents the local conductivity.

This residual represents the conductivity-weighted disequilibrium contribution to the potential from the surrounding nodes. The electric potential is calculated iteratively by replacing the value for each node at the n-th iteration by

In equation (A3), is a parameter selected empirically to optimize convergence.

There are two boundary conditions of note. Firstly, since the needle body is a perfect conductor, its surface is forced to be an isopotential at –1 V. Secondly, no current can flow perpendicular to the surface of the skin (the plane given by z = 0). This is equivalent to the boundary condition,

Without this boundary condition, the presence of the needle shaft at the z = 0 discontinuity causes an electric field distortion artifact which may lead to inaccuracies.

Having confirmed that the approach to convergence was essentially monotonic, calculations were stopped once the fractional change in all node potentials with iteration was less than one part in 10⁴. As the simulation progresses, the influence of the needle on the surrounding nodes propagates to the edge of the injectate sphere. In turn, the presence of the injectate begins to affect the field back at the needle tip as convergence is approached. Thus simulations of several thousand iterations were required for convergence which is greater than the 600 required in Davis et al.4

Once the three-dimensional electric potential has been found, the electric field strength vector, E, can be calculated at postprocessing from the gradient of using the equation,

The electric field is a vector quantity and determines the effect on the peripheral nerve being stimulated. The absolute magnitude of the electric field |E| calculated from the components E_x, E_y, and E_z by evaluation of the Euclidean norm,

Finally the electric field magnitude in any particular plane of interest could be calculated by linear interpolation as required.

Footnotes

Accepted for publication May 16, 2008.

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