Preliminary investigations of circular volume conductors suggested that farfield potential magnitude declines progressively slower with increasing radial distance from a current source and follows a cosine function with angular displacement of the recording electrode from the electrical generator’s axis. Using circular volumes of 6 differing radii, the mathematical relationship between angle, radii, and far-field potential amplitude is determined. Previous theoretical relationships of amplitude versus dipolar spacing, current, and distance from a dipole generator in a bounded volume conducting medium are verified for the near-field. Far-field potentials in circular volumes are found to become constant at radii greater than 75% of the bounded volume’s radius. Additionally, an adjoining volume conductor acts simply as a passive fluid-filled electrode (wick electrode) to the circular volume containing the generator until the intercompartmental opening to the circular volume exceeds 20% of its circumference. This finding was clinically supported by recording similar P9 somatosensory-evokedfar-field potentials generated caudal to the foramen magnum from various portions of the cranium, whose connections to the torso, foramen magnum, and neck, average 6.2% and 17.8%, respectively. Finally, 3 circular volume conductors were connected in series by channels less than 20% of the volume conductor’s circumference. Both adjoining circular volumes were equipotential to the far-field potential present at the boundary of the first circular volume containing the dipole generator. This observation supports the clinical finding of far-field potential transmission through multiple human bodies in conductive Contact. 0 1992 John Wiley & Sons, Inc. Key words: far-field potentials action potentials stationary potentials virtual dipoles MUSCLE & NERVE 15:949-959 1992

FAR=FIELDPOTENTIALS IN CIRCULAR VOLUMES: THE EFFECT OF DIFFERENT VOLUME SIZES AND INTERCOMPARTMENTAL OPENINGS DANIEL DUMITRU, MD, and JOHN C. KING, MD

Approximately 20 years have elapsed since farfield potentials were first described clinically in auditorys39 and somatosensory-evoked potentials.’.’ Far-field potential observations in both somatosensory-evoked and peripheral nervous system investigations offer insight into far-field potential characteristics. Additionally, mathematical substantiation of far-field potential principles are provided through computer m ~ d e l i n g . ‘ ~ Investigators have confirmed many of the clinical From the Department of Rehabilitation Medicine, University of Texas Health Science Center at San Antonio, San Antonio, Texas. Address reprint requests to Daniel Dumitru, MD, Department of Rehabilitation Medicine, University of Texas Health Science Center at San Antonio, 7703 Floyd Curl Drive, San Antonio, TX 78284-7798. Accepted for publication February 1, 1992 CCC 0148-639X/92/080949- 11 $04.00 0 1992 John Wiley 8, Sons, Inc.

Circular Far-Field Potentials

far-field potential observations through animal research and proposed the leadingkrailing dipole model and “wick electrode” (fluid-filled electrode) effect to explain the generation and observation of far-field potentials in volume conductors.3910Muscle far-field potentials were also predicted by computer-modeling techniques7 and clinically documented in thus unifying far-field theory for both nerve and muscle action potential generators. The correlation between mathematical models and biologic tissue far-field production is presently investigated utilizing constant current dipole generators and mapping of the associated near-field and far-field voltage distribution in circular and spherical volumes. It is first necessary to create, under controlled laboratory conditions, far-field potential production in simple volume conductor^.^,^^,^^ The simplest volume to consider is a sphere with a cen-

’’

MUSCLE & NERVE

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949

trally located dipole. l2 Experimental manipulation of electrodes within a fluid-filled solid bounded sphere is technically difficult. If correlations can be obtained with a circular volume of limited depth representing a central section through a sphere or a slice of a cylinder, then modeling becomes much easier. Such a model simulating a coronal slice through a sphere or a perpendicular slice of a cylinder can be formed by a circular bounded volume of limited depth filled with a 0.9% saline solution containing a constant current dipole source/sink in this volume's center. It is mathematically predicted that the electric potential flux from a dipole becomes approximately radially oriented at distances beyond three times the dipolar spacing.*' The potential measured in this region is inversely proportional to the cross-sectional area subtended by the dipole's radially oriented flux.*' For a sphere, this results in the measured potential decreasing proportional to l/r2 where r is the radius from a volume's center." For the shallow circular volume, the area subtended by the radially oriented flux is described by 2 r r (circumference of a circle) times the depth (h), and thus the measured potential would be anticipated to decrease inversely proportional to this area, 2nrh, or proportional to llr. The validity of these relationships in both spherical and shallow circular volumes is presently investigated. Preliminary investigations5 posed several relevant questions. First, does the mathematical relationship describing far-field otential generation in infinite spherical volumesg apply to far-field potentials produced in volume conductors with different radial boundaries, particularly regarding appropriate correction factors for the boundary's effect?*' Second, is there a critical opening relationship in circular volumes of different circumferences separating two volume conductors where the "wick electrode" effect is degraded by an intervolume current spread? Third, what clinical relevance of far-field potential production in circular volume conductors can be deduced? Finally, can the clinical observation of far-field potentials extending through multiple individuals be reproduced experimentally? The present investigation was designed to explore these questions as they pertain to far-field potential observations in circular-volume conductors. MATERIALS AND METHODS Instrumentation. A Neuropack 2 (Nihon Kohden Corp., Tokyo, Japan) electrodiagnostic instrument with an amplifier input impedance of 200 MR,

950

Circular Far-Field Potentials

amplifier sensitivity of 500 +V to 5 mV/cm, average sampling frequency of 50 kHz, sweep of 1 ms/ cm, and highhow filter settings of 20,000 Hz and 1.O Hz, respectively, were utilized. A constant current square wave pulse generator of 1.0 ms duration was delivered 5.0 ms following the initiation of the cathode ray tube (CRT) sweep. Subdermal electroencephalographic platinum/iridium needle electrodes (Grass Corp., Quincy, MA) were employed for both the stimulating and recording electrodes. All electrodes were secured in a vertical orientation and completely submerged in the volume conducting fluid. The E-1 (active) and E-2 (reference) electrode insulated leads were connected to the amplifier such that a negative potential difference resulted in an upward CRT trace deflection. Ten trials of each recording were averaged. Volume Conductor. Six volume conductors with different radii (Rb = radius to the volume's boundary), similar in design to those previously described, were constructed.5 A 0.9% saline solution (conductivity 0.015 R-' * cm-') acted as the volume-conducting medium and remained at 25°C throughout each experiment. Single Volume Conductor: Angular and Radii Measurements. A circular volume with an R b of 27.5

cm, filled with normal saline to a depth of 2.5 cm, served as the primary volume conductor. The dipole generator was located in the center of this volume and oriented such that the cathode was toward an arbitrary radius designated as the zero axis. A current intensity of 2.0 mA ? 0.1 mA through a cathodekinode interelectrode separation of 1.0 cm & 0.05 cm was used. Within the primary 27.5-cm volume conductor, 5 additional secondary circular volumes were constructed of a nonconducting (Sculpey; Polyform Products Inc., Schiller Park, IL) material ( > l o MR impedance) with respective Rb values o f 4, 8, 12, 16, and 20 cm -t 0.2 cm. Only one secondary volume was present at a time for its respective field measurements. Potential electric field magnitudes were measured for the 4-, 16-, and 27.5-cm R, volumes clockwise along the 0" (E-l/O"), 30" (E-1/30"), 45" (E-1/45"), and 60" (E-1/60") axes in 5.0.mm increments beginning 3.0 cm (3 times the dipolar spacing) from the circle's center (2.5 cm from the cathode at the 0" axis) and extending radially to the volume conductor's boundary (Fig. IA). The 8-, 12-, and 20-cm Rb volumes were similarly measured, but only along the 0" axis. The E-2 electrode was placed on the zero potential line mid-

MUSCLE & NERVE

August 1992

*

FIGURE 1. (A) A small (Rb = 8.0 cm) circular volume conductor is depicted within a larger ( R b = 27.5 cm) circular volume. The four angles of measurement (E-1/0”, E-1/30”, E-1/45”,and E-1/60”) utilized to record potentials generated in the circular volume conductor are also shown. E-2 and G represent the reference and ground electrodes respectively. (B) A secondary circular volume is shown with approximately 20% of its circumference removed). The small solid circles (0) represent the multiple E-1 recording locations.

way between the cathode and anode at the boundary’s rim counterclockwise to the zero degree axis of each secondary volume, and the ground electrode was located 20” counterclockwise from this position (Fig. 1A). All electrodes were perpendicular to the plane of the circular volume and in contact with the nonconductive bottom. Spherical Volume to Circular Volume Correlation. A spherical volume with a nonconductive boundary 0.1 cm was compared with an at Rb = 8.6 cm 8.6 cm +- 0.1 cm Rb circular volume with 2.5 cm and 8.6 cm depths of conducting medium. Each was measured with the above stimulation and dipole parameters and along the 0” axis at 0.5-cm intervals beginning 3.0 cm from both the circles’ and the sphere’s center. In the sphere, the E-2 and ground electrodes were in the same plane as the stimulating anode, cathode, and E-1 electrode, which intersected the sphere’s center point. The sphere was constructed with a 5.9-cm radius opening located 5 cm above this plane and parallel to it for measurement access. The potential magnitudes were measured as described in the previous section.

*

Single Volume Conductor: Variable Dipole Spacing, Current, and Conductive Medium Depth. In the

27.5-cm R b circular volume at a 2.5-cm normal saline depth, measurements were made at r (radial distance from the volume’s center) values of 4.0, 8.0, 16.0, and 27.5 cm 0.1 cm along the zero degree axis for dipole spacings of 0.5, 1.0,

*

Circular Far-Field Potentials

2.0, and 4.0 cm 0.05 cm, and respective stimulating dipole currents of 0.2, 1.0, and 5.0 k 0.1 mA. Because the depths used in this investigation are essentially arbitrary and directed by the material available, it is necessary to document the relationship between potential magnitude and depth of the volume conductor. In circular volumes with R, of 8.9 cm and 27.5 cm utilizing a 2 mA, 1 ms current pulse with an anodelcathode separation of 1.0 cm, near- and far-field potentials were recorded for normal saline depths of 1.0, 1.5, 2.0, and 2.5 cm. For the circular volume of R, = 8.9 cm, the additional depth of 3.0 cm was also investigated. Serial measurements were made beginning 3.0 cm from the cathode along the 0” axis extending in 0.5 cm increments to the boundary, along the 0” axis. Dual Volume ConductorNariable Opening. Three separate concentric secondary circular volumes were independently constructed within the primary circular volume (R, = 27.5 cm) with the following R, values: 6.8, 13, and 19.6 cm k 0.2 cm (Fig. IB). T h e cathodelanode delivered a constant current pulse of 3.6 mA while utilizing an inter0.05 cm. The electrode separation of 3.0 cm central volume was first filled to a depth of 1.5 cm with the 0.9% saline solution and visually inspected for structural integrity by observing for fluid leakage. The surrounding circular “doughnut”-shaped volume was then filled to a depth of 1.5 cm with the saline solution. An impedance meter demonstrated a lack of electrical continuity ( > I 0 MR) between the two volumes along the circle’s boundary. The E- 1 electrode was sequentially located on both sides of the secondary volume’s boundary, along the 0” axis, at the rim of the larger volume for the same zero axis, and at 6 additional sites within the “doughnut” volume conductor (Fig. 1B). The E-2 and ground electrodes were placed in each central volume as noted above in the previous section. The central volume’s boundary was sequentially removed in 0.5-cm increments symmetrically along the zero axis. Specifically, 0.25 cm was deleted on either side of the central axis incrementally. A far-field potential measurement was recorded within the small and large volumes, respectively, at the above designated sites, prior to the removal of the secondary volume’s next 0.5-cm portion. This process was continued until there was a change noted between the central and surrounding volume conductors’ far-field poten-

*

MUSCLE & NERVE

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951

scribed (Fig. 2). The zero axis line of the first volume also arbitrarily defined the zero axis of the two adjoining circular volumes which aligned with the bridges. Serial E-1 far-field potential magnitude recordings were performed at the zero axis rim of the first container, and along the boundary of the two adjoining volumes at: o", 90", 180", 360", and in their centers (Fig. 2).

E-2

I

t

-

. .

. .

FIGURE 2. Three circular volume conductors are connected in series with each other and filled to a depth of 1.5 cm with 0.9% saline solution. The small solid circles (a) within the three volume conductors represent the multiple E-1 recording locations. E-2 and G are the reference and ground electrodes, respectively.

Ten skeletal human skull circumferences were measured along the coronal plane through the vertex and foramen magnum. Additionally, the foramen magnum's diameter through this coronal plane was recorded. Second, referential upper extremity somatosensory-evoked potentials were performed on 5 normal volunteers. Four subdermal electroencephalographic needle electrodes (Grass Corp., Quincy, MA) were placed subcutaneously on the subjects at C3', C4', FpZ', and OZ (international 10-20 system) and referenced to the right knee (subdermal electrode utilized). The impedance for all electrode montages was less than 7 k f i in all subjects. The left median nerve was excited with a 200-ps current

Clinical Correlation.

tial magnitudes. An additional incremental 20 cm of boundary removal was also studied. Three nonconcentric adjacent circular volumes (27.5 cm Rb) were connected at their rims by a 6.0-cm long and 10.0-cm wide bridge (Fig. 2). All the volumes were filled to a depth of 1.5 cm with the 0.9% saline solution. One of the end volumes contained the cathode, anode, E-2, and ground electrodes as previously de-

Triple Volumes.

01 0

5

10

15

20

25

30

3.253

01 0

5

10

20

15

25

30

2.251

C

Scm)

0

5

10

15

r (cm)

20

25

30

0

5

15

10 f-

20

25

30

(cm)

FIGURE 3. Graphic representation of near- and far-field potential magnitudes with respect to distance from the center of the current generator. Note how the potential amplitude's rate of decline diminishes as the boundary is approached particularly at greater than 75% of the volume's radius (far-field). This relationship is similar for all bounded volumes and the four angles measured. Angular displacements from the dipole generator are: A = 0"; B = 30";C = 45"; and D = 60".

952

Circular Far-Field Potentials

MUSCLE & NERVE

August 1992

pulse at an intensity yielding a moderately vigorous thumb twitch. High- and low-frequency filters were set at 500 Hz and 10 Hz, respectively, at a sensitivity of 2.5 pV/cm. Two trials of 500 averages were obtained for each individual montage. The P9 far-field potential was identified and compared in latency and amplitude for the four recording positions in all subjects. The neck and coronal head diameter and circumferences of these individuals were also determined. RESULTS Single Volume Conductor: Angular and Radii Yeasurements. The far-field polarity and morphol-

ogy of all responses for the six circular volumes was a negative square wave as anticipated given the recording and dipole generator electrode montages. All electric potentials values and formulae should be understood to be preceded by a negative sign due to the dipole generator's orientation and recording montage. Sequential electric field magnitude measurements revealed the characteristic rapid near-field decline and subsequent leveling off of the far-field close to the volume's b ~ u n d a r y This . ~ trend was noted irrespective of the volume's radius or measurement angle (Fig. 3). The magnitude settled within 5% of its final voltage at an r value greater than 75% of the radius of the various volumes' boundary (Rd. These values at greater than 75% R,, appeared very constant within our measurement error, and thus, the potential is considered to be a far-field potential. The far-field potentials' magnitudes are inversely proportional to R , and varied about the volume's circumference by the cosine of the angle (8)from the 0" axis (Fig. 3) as predicted.1232o The rapid decline of the near-field potentials in the circular volumes is inversely proportional to r and directly proportional to cosine 8, given a correction factor for a finite boundary ( R J . For a bounded s here, where potentials otherwise decline by llr , a boundary induced correction factor of (1 + 2 (r/Rb)')is mathematically derived.20x21 A correction factor for the bounded circular volume, where potentials otherwise decline by llr, is deUsing this correction facrived to be (1 + (r/Rb)2). tor, our best fit (by least-squares method) equation took the form o f

(see below) to both I (dipolar current) and d (dipolar spacing), and inversely proportional to h (the depth of the conducting medium), while V is the measured potential. Use of the above equation demonstrates excellent curve fit (Fig. 4) to the measured data in all circular volumes noted in Figure 3. The best fit constant, k, iterated to 8.54 mV-cm (root mean square error = 0.061 mV> for all bounded circular volume radii at a depth of 2.5 cm. The predicted k, using k = I d I 2 ~ u h (see DISCUSSION) is (2 mA)(1 cm)/((2)(3.14)(0.015 a-1 cm-')(2.5 cm)) which equals 8.49 mV-cm. Spherical Volume to Circular Volume Correlation.

The potentials measured (Table 1) conformed well to the theoretically derived equation for a bounded sphere2': V

=

( I d cos €)/4m~)(l/r~)(l + 2 (r/4J3)

for r > 3d, where I = dipolar current and u = conductivity of the medium (0.015 O-1cm-').20 The constant, {Zd/4nu}, obtained by empirically best fitting the data, was 10.3 mV-cm2, which agrees closely with that calculated: (2 mA)(1.0 cm)/(4(3.14)(0.015 W'cm-') mV-cm2

7

=

10.6

L

. ..._. ......L (4cm)

4

4

3

Y

................................

10

0

0

15

20

(27 5cm)

25

30 1

5

r (cm) where 8 = the angle from the zero degree axis of the dipole, and k is a constant that is proportional

Circular Far-Field Potentials

FIGURE 4. Comparison between measured data (solid line) in volume conductors of different boundaries (Rb noted in parentheses) at a 2.5-cm depth with predicted values (dashed line) using the equations noted in the text.

MUSCLE & NERVE

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953

Table 1. Spherical/cylindricaI/circular potential magnitudes (mV). R,= Sphere

r (cm) 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.6

8.6 cm

Cylinder

Circle

Measured

Best fit

Predicted

Measured

Measured

Best fit

Predicted

1.26 0.97 0.78 0.65 0.56 0.52 0.47 0.45 0.43 0.42 0.42 0.42

1.24 0.95 0.77 0.65 0.57 0.52 0.48 0.45 0.44 0.43 0.42 0.42

1.28 0.98 0.80 0.67 0.59 0.53 0.49 0.47 0.45 0.44 0.43 0.43

1.87 1.47 1.20 1.03 0.90 0.82 0.75 0.70 0.67 0.65 0.65 0.65

3.27 2.87 2.57 2.40 2.27 2.20 2.13 2.07 2.03 2.03 2.03 2.03

3.22 2.86 2.62 2.43 2.30 2.20 2.13 2.08 2.04 2.02 2.01 2.00

3.17 2.83 2.58 2.40 2.27 2.17 2.10 2.05 2.02 1.99 1.99 1.97

Comparison of potential magnitudes (mV) measured at various sites (r) from the center or center axis of each respective volume. The sphere's k for best fit is 10.3 m V c m while the predicted k is 10.6 mV.cm. In the circular volume, the best fit k is 8.60 mV%m and 8.49 m V a n for the predicted value. Best fit and predicted values were not attempted for the cylindrical volume (see text).

In the circular volume of similar Rb to the sphere (8.6 cm) and at a depth of 2.5 cm, the best empiric fit equation is:

where k = 8.60 mV.cm (Table 1). This k is within 1.3% of the predicted 8.49 mV.cm2 calculated from k = Z d/2nuh. With an 8.6-cm radius circle filled to an 8.6-cm depth, results recorded along the bottom became intermediate between the decrementing pattern of l/r for the circular volume and l/r2 for a sphere (Table 1).

with I = 2.0 mA and d = 1.0 cm, revealed that amplitude relationships in both the near- and farfields to be exactly inversely proportional to depth (Fig. 5 ) . This also means that amplitudes measured are inversely proportional to the area subtended by the electric potential flux since the area subtended is described by 2nrh (area of a flat cylinder's rim), where h = depth of the conductive medium. Far-field potential magnitude and morphology recorded along

Dual VolumesNariable Openings.

Table 2. Variable dipole moment potential magnitudes (mV). Single Volume Conductor: Variable Dipole Spacing, Current, and Conductive Medium Depth. In the cir-

cular volume, Rb = 27.5 cm, testing at r of 4, 8, 16, and 27.5 cm revealed near- and far-field potential amplitudes that varied directly with both dipole spacing and current magnitude, similar to the theoretically derived relationships for a sphere (Table 2).'2,20 For a given interelectrode separation, if the current increased by a given factor, the potential also increased proportionately (Table 2). Also, if the dipolar spacing doubled, with the current remaining constant, the potential doubled. This relationship held for all currents (0.2, 1.0, and 5.0 mA) and dipolar spacings (0.5, 1.0, 2.0, and 4.0 cm) tested whether in the near-field region or in the far-field. Depths of 1.0, 1.5, 2.0, and 2.5 cm in the circular volume of R, = 27.5 cm, and 1.0, 1.5, 2.0, 2.5, and 3.0 cm in the circular volume of Rb = 8.9 cm,

954

Circular Far-Field Potentials

lnterelectrode separation (cm)

I = 0.2 mA r = 4.0 8.0 16.0 27.5 I = 1.0 mA r = 4.0 8.0 16.0 27.5 I = 5.0 mA r = 4.0 8.0 16.0 27.5

0.5

1.o

2.0

4.0

0.15 0.08 0.05 0.04

0.29 0.14 0.09 0.08

0.60 0.30 0.19 0.17

1.27 0.60 0.37 0.33

0.77 0.38 0.23 0.21

1.50 0.75 0.47 0.41

3.17 1.58 0.97 0.88

6.67 3.13 1.90 1.73

3.80 1.93 1.13 1.oo

7.33 3.73 2.20 2.00

15.3 8.00 4.83 4.33

34.0 15.7 9.50 8.50

Potential magnitudes arising from various dipole moments for 4 separate recording sites (r) in centimeters from the center of a circular volume with an Rb = 27 5 cm

MUSCLE & NERVE

August 1992

8-

8-

6-

6-

(1.Ocm)

(1.5cm)

5E

w

4.

4-

(2.0cm)

>

(2.5cm) (3.0cm)

2.

2-

0‘

0’

0

0.20

r

-’ (ern-')

0.40

.20

.10

.30

r

.40

(cd)

Figure 5. Comparison of potential magnitude in the near-and far-field at various depths for an R, of 27.5 cm (A) and 8.9 cm (B). Note how the potential magnitudes for a particular curve in a given bounded volume are inversely proportional to depth with respect to each (larger) is due to the boundary effects. other. The flattening of the curves at small r

-’

the zero axis on either side of the volume’s boundary (Rb = 6.8 cm, circumference = 42.7 cm), at openings from 0.5 to 8.0 cm, are identical to the potential recorded at all regions in the surrounding “doughnut”-shaped portion of the volume (Table 3). The potential morphology (negative square wave) and amplitude remained constant throughout the outer concentric volume and correlated with the waveform observed at the rim of the smaller circular volume at the zero axis until the opening reached 8.5 cm or approximately 20% of the total circumference (Table 3). At an opening of 8.5 cm, the potential in the volume surrounding the smaller circle was less than that recorded for the zero axis at the former boundary of the smaller circular volume, which also had de-

clined compared with its previous value. As the opening continued to increase, the magnitude of the far-field potentials observed along the zero axis at the boundaries of both the smaller and larger volumes progressively declined. The morphologies of all potentials maintained the appearance of a square wave. During all measurements, the entire “doughnut”-shaped volume remained equipotential. This relationship of the far-field potentials recorded at the boundary of the circular volume to an increasing opening in a circumference of 81.7 cm was compared with that observed above. The isopotential relationship across the boundary continued until the opening in the inner circular volume reached 16.0 cm or 20% of the secondary volume’s

Table 3. Variable opening potential magnitudes (mV) Opening (cm)

0

(A) Circumference = 42.7 mm 18.3 Amp 1 0.0 Amp 2 (B) Circumference = 81.7 mm Amp 1 11.2 0.0 Amp 2 (C) Circumference = 123.2 mm Amp 1 3.7 Amp 2 0.0

2.0

8.5’

10.0

12.0

16.0*

18.0

20.0

18.3 18.3

18.0 17.3

17.7 16.3

11.2 11.2

11.2 11.2

11.2 11.2

11.2 11.2

11.2 10.7

11.0 10.4

10.9 10.3

3.7 3.7

3.7 3.7

3.7 3.7

3.7 3.7

3.7 3.7

3.7 3.7

3.7 3.7

23.5”

25.0

3.7 3.5

3.6 3.4

Far-field potential magnitude (mV) recordedpst within the boundary of the smaller circular volume (Amp 1) and at the outer boundary of the surrounding “doughnut”-shaped volume (Amp 2j conductor along the zero axis for the three circular circumferences Potentials recorded at additional sites within the surrounding volume are not shown, as they were always equipotential with the outer boundary potential *Denotes the opening size when a disparity between the two far-field potentials was first noted A representative sample of the total data obtained is Drovided in the above table

Circular Far-Field Potentials

MUSCLE & NERVE

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955

total circumference (Table 3). For a secondary volume of circumference 123.2 cm, similar findings of constant far-field potential magnitudes and morphologies were noted for the recording locations along the zero axis at the inner rims of both volumes until an opening of 23.5 cm. This opening represents 19% of the inner volume's circumference. All far-field potentials declined once the intercompartmental opening reached approximately 20% of the volume's circumference (lable 3). Triple Volumes. Three large circular volume conductors (Rb = 27.5 cm) were connected in series

and filled to a depth of 1.5 cm with a 0.9% saline solution. The far-field potential recorded along the zero axis at the boundary of the circular volume containing the far-field generator was compared with the designated recording sites in the two adjoining circular volume conductors (Fig. 2). All regions of the two connected circular volumes were equipotential with the zero axis potential of 3.6 mV recorded at the 0" boundary in the first circular volume. T h e morphology (square wave) and magnitude (3.6 mV) of all potentials were indistinguishable. T h e mean coronal circurnference measurement for the ten human skulls was 43.8 cm, with a mean foramen magnum diameter of 2.7 cm. -1he foramen magnum diameter, therefore, is 6.2% of the total skulls' circumference in the coronal plane. T h e subjects' neck diameter corresponded to 17.8% of their heads' coronal measurement. As the foramen magnum and neck diameter represents an opening less than 20% of the skull's circumference, the human skull should be isopotential and act as a wick electrode to far-field potentials generated in the body. Utilizing a Student's t-test arid Pearson's cot-relation coefficient, the P9 far-field potential demonstrated similar latencies and amplitudes ( P ? 0.3, r = 0.9) within an individual for the four different referential skull montages in all 5 subjects.

Clinical Correlation.

DISCUSSION

Far-field potentials in an infinite spherical volurne,I2 or finite circular v o l ~ m e sare , ~ mathematically modeled to follow a cosine function with respect to magnitude decline as the recording electrode is angularly displaced from the generator source axis at distances beyond three times the interdipole spacing.20 Preliminary investi ations in a circular volume support this assertion.' It is im-

956

Circular Far-Field Potentials

portant to explore the influence a boundary has on the potential field decrement as biologic systems are not infinite volumes. This may be accomplished by varying the radius of a circular volume filled with a good volume conducting solution such as 0.9% saline and generating far-field potentials. Serial measurements from the circular volume's central region extending out to the boundary along various angles should provide information to establish these relationships.12 Data obtained in this study confirm that farfield potentials in circular volumes do rapidly decline in magnitude as one proceeds out from the center toward the boundary zone, where recorded potentials level off and become nearly constant (far-field potentials) with radial distance (Fig. 3 ) . The nature of this decline, however, varies from that of a sphere. In a sphere, the electrical potential flux produced by a dipolar current source and sink in conductive medium is described by:

Y

=

(I d cos 0/u)

where Y = potential flux in V.cm2, I is the dipolar current (mA), d is the dipolar spacing (cm), u is the conductivity of the medium (W'cm-') and cos 0 is the cosine of the angular displacement of the recording electrodes from the electrical generator's axis.'3321T h e electrical potential ( V ) is described by the flux divided by the total area subtended by this electrical flux or: V

=

WA

= (I d Cos € ) / ~ ) ( = (I d cos 0/47~u)(l/?)

1/4~~)

where A is the area of the sphere (4nr2) subtended, and r is the radius from the center of the dipole pair. This equation describes the result derived for a dipole in an infinite medium.22 To bound the infinite volume conducting medium to that of a finite sphere, whose center is at the dipole center, requires the radial current at the boundary to become zero at a nonconductive boundary. A boundary factor must be added to the infinite medium equation such that its rate of change with respect to r (i.e., its derivative being proportional to the radial current) will negate the infinite medium e uation's derivative with respect to r, at r = R,. 20'1,22 This causes the bounded sphere equation to become

or

v = (I d cos 0 / 4 7 ~ (l/r2 ~ ) + 2dRb3) = (I d cos0l4.sra)(l/r2)(1+ 2(r/R,)"). MUSCLE & NERVE

August 1992

In a flat circular volume with a centrally located dipole, the radially oriented flux is subtended by a perpendicular area described by 2mh, where r is the radius from the center and h is the depth of the medium. This is the cross-sectional area of the circular volumes’ imagined conductive medium rim at any radius, r. The radial electrical potential flux is ap roximately perpendicular to this rim for r > 3dYo and r >> h. With such a subtended area, A, V

= ?/A = (I = (I d cos

d cos 8 / a ) ( 2 ~ r h ) 8/2~uh)(l/r)

for an infinte such planar volume. A factor must be added to this infinite flat circular volume equation to meet the boundary condition of no radial current at a nonconducting circular boundary which results in V = (1 d cos 8/21~uh)(l/r + r/Rb2) (Zd cos W 2 ~ u h ) ( l / r ) (+ l (r/Rb)‘)

=

When evaluated at r = R,, this equation allows the rate of change of the potential with respect to r, the derivative which is proportional to the radial current, to become zero. This discussion is a simplified conceptual derivation of boundary correction factors applied to general infinite volume equations for spherical and flat circular volumes. More rigorous formal mathematical derivations are available for the case of the sphere.’3220-22 The above equations for both the spherical and flat circular volumes were found to well describe the behavior of the measured findings (Fig. 4,Table 1). They also allow one to convert the results found in flat, easily manipulated circular volumes to the results of the more difficult to measure three-dimensional spherical model. The direct dependence of the measured potential on both current and dipolar spacing as well as the inverse relationship to conductive medium depth in circular volumes is verified. The calculated results using the above equations agree very closely with the k values obtained from empirically best fitting the measured data (Fig. 4,Table 1). The behavior of the potential decline versus radial distance along the floor in a tall cylinder is found to be intermediate between that of a sphere (diminishing proportional to l/r2) and that of a flat circular volume (diminishing proportional to Ur). Using spherical coordinates and considering

Circular Far-Field Potentials

the vertical incident angle of the flux to the sidewalls, one might be able to derive predictive mathematical equations, however, this is beyond the scope of this investigation. Action potentials produced by nerve and muscle fibers are constant current sources.22 As demonstrated clinically in both neural” and muscular tissue^,^ increases in the number of fibers depolarized leads to an elevation in the magnitude of recorded near-field and far-field potentials. Intuitively, this finding is understandable from the perspective that additional excitable tissue generates more current which, in turn, results in a larger transient dipolar moment imbalance at a boundary region. This temporary elevation of unbalanced dipole moments yields larger net far-field potentials. In the present experiment, the dipole moment of the current source can be increased by keeping the current constant and increasing the separation between the anode and cathode or maintaining a constant anode/cathode distance, but raising the current strength. This investigation documented that both these possibilities result in propagating larger near-field and far-field potentials. Although this finding is not identical to adding generator sources, it does indirectly support the clinical finding of elevations in both near- and far-field potential amplitudes associated with larger dipole moment imbalances for both nerve and muscle. One characteristic of far-field potentials is little change in magnitude with local electrode positioning. From the above derived and empirically best fit equations, one can calculate that, at radii beyond 0.725 R, for the circular volume or 0.806 Rb for the spherical volume, the potential will be within 5% of its final value. Our data are consistent with the far-field potential becoming constant, within measurement error, beyond these radii (Table 1). Consecutively larger openings in the volumes’ defining boundary along the zero axis revealed several interesting findings. The “wick electrode” effect was substantiated by the recording of waveforms identical in morphology and magnitude at the inner boundary of the opening within the circular volume compared with all regions of the surrounding volume conductor. That is, the “doughnut”-shaped volume became equipotential throughout its extent to the far-field potential recorded at the boundary of the circular volume. The shape of the volume conductor acting as the wick electrode most likely does not have to be a circular volume of equal size, but may be any surrounding volume, provided it is a good volume

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conductor, as demonstrated by the equipotential recordings in the “doughnut”-shaped volume. There appears to be a critical ratio of circular boundary opening length compared with the circumference of the volume conductor containing the far-field generator. As long as the channel connecting the two volume conductors is less than 20% of the circular volume’s circumference, the wick electrode effect is maintained. At 20% of the circular volume’s circumference and beyond, the wick electrode effect deteriorates, probably due to interchamber current spread. Even though the channel opening extended considerable distances away from the zero axis of the circular volume, as long as the channel width was less than 20% of the circumference, the surrounding volume continued to be equipotential with the zero axis value only. These findings have potential clinical implications. Anatomic measurements on a limited but representative number of human skulls and subject’s necWhead relationships revealed that the foramen magnum and neck diameters are less than 20% of the skull and head’s coronal circumference. Because it is unknown whether the head utilizes the foramen magnum or neck as its connection to the torso for detecting far-field potentials, both the neck and foramen magnum’s ratios were determined. One may anticipate that the head, therefore, should act as a spherical volume conductor and wick electrode to detect far-field potentials generated outside this volume. Further, referential recordings at any position along the cranium should record extracranially generated far-field potentials as occurring at the same latency and with the same magnitude. Indeed, a referential scalp-to-knee montage in a limited number of subjects demonstrated that a P9 potential was produced with similar latencies and amplitudes irrespective of the scalp recording location. The correlation between circular volume conductors and spherical human volume conductors, such as the cranium, suggests that they may both act as wick electrodes with respect to far-field potentials generated at a distance. The “wick electrode” effect, in which a volume conductor under certain conditions acts as a fluidfilled electrode and conveys far-field potentials essentially instantaneously over large distances without decrement, was also demonstrated in circular volumes. Specifically, a circular volume connected through a small channel to the zero axis of the far-field generating volume, became equipotential with the far-field potential observed at the generating volume’s b ~ u n d a r yThis . ~ finding confirmed

958

Circular Far-Field Potentials

the ability of volume conductors to convey farfield potentials over relatively large regions without decrement. Three separate circular volumes with sequentially larger openings to a surrounding “doughnut”-shaped volume conductor were constructed to investi ate this relationship. Yamada et al.” demonstrated that far-field potentials generated in one person could be recorded in a second individual by electrically connecting them at the wrist with electrolyte paste. This experiment clinically suggested that several volume conductors (people) in series could act as wick electrodes with respect to far-field potentials. We investigated three circular volumes with circumferences of 173 cm connected by channels 10.0 cm wide by 6.0 cm long. As the channel width was only 5.8% of the circumference, findings in this investigation suggest that these parameters would allow the two serially connected circular volumes to act as wick electrodes to the farfield generating volume. In essence, the circular volume in immediate contact with the volume conductor containing the current generator is comparable with the human body (Yamada et al. experiment24) in which the far-field potential was generated. The third circular volume in continuity with the middle volume acted as the second person in the previously described human study. The two circular volumes became equipotential with the far-field potential detected at the boundary of the volume conductor generating the far-field potentials. Although circular volumes were utilized in this investigation, the principles of fluid-filled electrodes or “wick electrodes” appears to confirm the Yamada’s et al.24finding of far-field potentials being observable across different volume conductor’s through limited regions of physical contact. T h e experimental findings of mapping nearand far-field potentials corroborate a number of theoretical, experimental, and clinical findings. Far-field production using simple constant current generating sources is substantiated in both circular and spherical volumes. T h e mathematical formulae theoretically derived for a sphere12v20 describing far-field potential behavior and observed in frog nerves’* has been verified to follow the derived predictions using nonbiologic substrates. Although different near- and far-field decrement rates are noted for spheres ( l l r 2 )and circular volumes (lh), the qualitative findings are similar, predictable, and allow circular volumes to be used as empirical models of far-field potential generation. An unexpected finding when incrementally increasing the connection between t w o volume

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conductors is that the volume without the current source acts as a wick electrode to the adjoining volume until the circumferential opening between the two volumes reaches approximately 20%. It is interesting to note that the passive volume conductor (without the current source) became equipotential to the voltage along the zero axis, and did not acquire other angular voltages even though the opening enlarges to 20% of the current containing volume. We can only speculate that the opening confines all other potentials for an unclear reason. Elevations in both near- and far-field potentials with increasing dipole moments in the circular field generators are corroborated. The wick electrode effect is substantiated for the clinical findings of connecting multiple individuals together in that the subjects, like our circular volumes, behave as wick electrodes and become equipotential with the boundary potential. The human skull is also shown to become equipotential with respect to the P9 far-field potential, similar to the circular volumes again acting as a wick electrode. A number of the reported far-field potential findings have been reproduced in this investigation with current sources iu simpe circular volumes. These observations suggest that easily manipulated flat models can be correlated to three-dimensional results, providing a potentially useful research tool for investigating the various body compartment interactions as they relate to far-field potential production. For a more formal derivation of the equation describing the flat circular volume interested readers are encouraged to write the authors.

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Circular Far-Field Potentials

5. Dumitru D, King JC: Far-field potentials in circular volumes: evidence to support the leadingltrailing dipole model. Muscle Nerve 1991;14:981-989. 6. Dumitru D, King JC: Far-field potentials in muscle: A quantitative study. Arch Phys Med Rehabil 1992;73:270274. 7. Gootzen THJM: Muscle Fibre and Motor Unit Action Potentials: A Biophysical Basis f o r Clinical Electromyography. Gravenhage, CIP-Gergevens Kininklije Bibliotheek, 1990, pp 1332. 8. Jewett DL, Romano MN, Williston JS: Human auditory evoked potentials: Possible brain stem components detected on the scalp. Science 1970;167:1517- 1518. 9. Jewett DL, Williston JS: Auditory-evoked far-fields averaged from the scalp of humans. Brain 1971;94:681-696. 10. Jewett DL, Deupress DL: Far-field potentials recorded from action potentials and from a tripole in a hemicylindrical volume. Electroencephalogr Clin Neurophysiol 1989; 72:439-449. 1 1 . Jewett DL: The leadingltrailing dipole model as a means of understanding generators of far-field potentials. Clin Evoked Potentials 1990;7:9- 13. 12. Jewett DL, Deupree DL, Bommannan D: Far-field potentials generated by action potentials of isolated frog sciatic nerves in a spherical volume. Electroencephalogr Clin Neurophysiol 1990;75:105- 117. 13. Johnk CTA: Engmeering Electromagnetic Fields and Waves. New York, Wiley, 1975, pp 26-77. 14. Kimura J, Yamada T: Short-latency somatosensory evoked potentials following median nerve stimulation. Ann NY Acad Sci 1982;388:689-694. 15. Kimura J, Yamada T , Shivapour E, Dickens QS: Neural pathways of somatosensory evoked potentials: Clinical implications. Electroencephalogr Clin Neurophysiol 1982;36: (~~pp1)328-335. 6. Kimura J, Mitsudome A, Yamada T, Dickens QS: Stationary peaks from a moving source in far-field recordings. Electroencephalogr Clin Neurophysiol 1984;58:351-36 1 . 7. Kimura J, Kimura A, Ishida T, Kudo Y, Suzuki S, Masafumi M, Matsuoka H, Yamada T: What determines the latency and amplitude of stationary peaks in far-field recordings. Ann Neurol 1986;19:479-486. 8. Nakanishi T : Action potentials recorded by fluid electrodes. Electroencephalogr Clin Neurophysiol 1982;53:343345. 9. Nakanishi T: Origin of action potential recorded by fluid electrodes. Electroencephalogr Clin Neurophysiol 1983;55: 114- 115. 20. Nunez PL: Electric Fields of the Brain: The Neurophysiology of EEG. New York, Oxford University Press, 1981, pp 82-83. 21. Plonsey R: Bioelectric Phenomena. New York, McGraw-Hill, 1969, pp 212-215. 22. Plonsey R: Introductory physics and mathematics, in: MacFarlane PW, Veitch Laurie TD (eds): Comprehensive Electrocardiology. New York, Pergamon, 1988, vol 1, pp 41-76. 23. Stegeman DF, Van Oosterom A, Colon EJ: Far-field evoked potential components induced by a propagating generator. Electroencephalogr Clin Neurophysiol 1987;67: 176- 187. 24. Yamada T, Machida M, Oishi M, Kimura A, Kimura J, Rodnitzky RI: Stationary negative potentials near the source vs. positive far-field potential at a distance. Electroencephalogr Clin Neurophysiol 1985;60:509- 524.

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