a noninvasive electromagnetic conductivity sensor for biomedical applications

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 35, NO. 12, DECEMBER 1988 101 1

A Noninvasive Electromagnetic Conductivity Sensor for Biomedical Applications

LYNN W. HART, HARVEY W. KO, MEMBER, IEEE, JAMES H. MEYER, JR., DAVID P. VASHOLZ, AND RICHARD I. JOSEPH

Abstract-We present the theory and practice of using a simple coil sensor operated at high-frequency (1-10 MHz) to measure changes in conductivity in a biological sample. Explicit results are obtained for symmetric configurations that are useful for calibrating the device and for making order-of-magnitude estimates of the results of applying the technique to humans to monitor the onset and progress of brain edema. A simple model suggests that the technique may be able to successfully measure the onset of some medical complications using the technology described and characterized herein.

I. INTRODUCTION HERE are no techniques that allow routine ongoing T measurement of edema in the human. One-time esti-

mation of edema can be made by the CAT, PET, and MRI techniques, but it is extremely difficult to utilize them for continuous monitoring of changes because of the logistics and expense of moving the patient and doing the scan. There is a need for an accurate, inexpensive measurement technique which will allow the progression or regression of edema to be continuously estimated at the bedside.

To help solve this problem we have developed instrumentation and measurement techniques based on magnetic induction to make noninvasive measurements of changes in brain conductivity. We have found that pro- gressive changes in brain conductivity accompany the progression of brain edema in rabbit [ 11 ; we surmise that this will be the case with humans also. To validate the technique, in Section 11-A we make measurements using in vitro configurations and compare them with theory. In Section 11-B we develop and test prototype instrumentation that might prove suitable for inexpensive, continuous monitoring of brain edema. In Section I11 we develop an idealized model of conductivity changes in human brain during the progress of one type of brain edema, and calculate the response of the instrumentation to these changes. We conclude in Section IV that it should be fea- sible to monitor brain edema by the magnetic induction techniques developed herein. Of course, as with any new technique, considerable development must precede introduction into the clinic.

Manuscript received September 22, 1987; revised July 5 , 1988. L. W. Hart, H. W. KO, J. H. Meyer, Jr., and D. P. Vasholz are with

the Applied Physics Laboratory, The Johns Hopkins University, Laurel, MD 20707.

R. I. Joseph is with the Department of Electrical and Computer Engi- neering, The Johns Hopkins University, Baltimore, MD 21218.

IEEE Log Number 8823686.

Our magnetic induction device consists of a single coil acting as both an electromagnetic source and receiver operating typically in the frequency range 1-10 MHz. Os- cillating current driven through the coil acts as a source and creates an oscillating electromagnetic field. When the coil is placed in a fixed-geometric relationship to a conducting body of constant conductivity, the alternating electric field component generates electrical eddy currents in the conducting body that are proportional to the conductivity of that body. For more complex conducting bod- ies, the generated eddy currents are proportional to a geometry-dependent average of the conductivity throughout the body. More importantly, for certain biomedical applications, any change in the conductivity distribution creates a change in the induced eddy currents. These eddy currents generate a secondary magnetic field which is picked up by the same coil (acting now as a receiver) because of the mutual inductive coupling between the coil and the eddy currents in the conducting body. This mutual inductive coupling causes a change in impedance across the coil which can be measured directly or indirectly. Both direct and indirect measurements of the change in impedance across the coil are reported here for in vitro configurations; they are referred to as the “impedance meter” and “Colpitts oscillator” methods, and are discussed in Sections 11-A and B, respectively.

Brain edema is often induced by trauma to the skull, causing edema to propagate from the point(s) of trauma. A coil (or coils) placed flat against the head and centered about the point(s) of trauma may be ideal for detecting the onset of this type of brain edema. A rule of thumb is that the coil can sense conductivity changes in the head about as deep as the radius of the coil. Depending on the injury, coils with radii varying from about 2-5 cm would be useful for this application.

If brain stem injury occurs, edema may be induced in the white matter of the brain. The edematous white matter expands by displacing the surrounding grey matter. A clinical arrangement for detecting the onset of deep-seated brain edema of this type would be a coil placed like a headband around the head. This is the case for which an idealized detection model is developed in Section 111. This case is also the most difficult challenge for coil detectors because the coil radius is so much larger than the effective initial radius of edematous material.

The magnetic induction technique is in many ways the

0018-9294/88/1200-1011$01.00 O 1988 IEEE

1012 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 35, NO. 12, DECEMBER 1988

antithesis of a conventional imaging technique; it is extremely sensitive to changes in conductivity distribution integrated over a region of the head (including any change in the shape of the head). Measurement by magnetic induction has the advantage of not requiring the application of electrodes to the body, thereby avoiding the problems associated with time-varying, pressure-sensitive contact impedances. In addition, the electromagnetic field COU- ples into the deep-lying tissues without being screened by the insulating superficial layers such as skin and bone. Some have already used magnetic induction methods for biomedical purposes employing multiple sourceheceiver arrangements adapted from geophysical surveying, oceanographic instrumentation, and magnetohydrody- namics [2]. Our technique is related to these earlier magnetic induction methods but we use a singe coil (rather than multiple coils), which greatly simplifies the measurement geometry and electronics.

The electrical impedances of biological tissues, including the brain, have been directly measured at frequencies from dc to the microwave range [3]-[lo]. Most investigations have utilized contacting electrodes and Wheat- stone bridge techniques. The analytic approach varies from an ion species, plasma concentration, and mobility approach [ 111 to an image theory potentials approach reported by Maxwell [12]. The related “impedance camera” imaging technique is a noninvasive method typically using a distribution of potentials applied to multiple contact electrodes distributed about a conducting body [ 131, [ 141. By measuring the currents flowing between pairs of electrodes, it is possible to estimate the conductivity distribution using different algorithms [ 151, [ 161. The errors inherent in these algorithms, however, may make this technique insensitive to detecting small changes in conductivity. Perhaps different algorithms could be developed to enhance detection of spatially-averaged conductivity changes, which might be more useful as a brain edema monitor. Another possible brain edema monitor employs microwaves transmitted between antennas mounted on both sides of the head [17]. Changes in the spatially-averaged conductivity of the head are detected as phase shifts between the transmitting and receiving antennas. In fact, since the weighted spatial average of conductivity as seen by magnetic induction is different from that seen by the other related techniques, different noninvasive techniques might be used in a complementary fashion to enhance the detection and monitoring of brain edema.

11. EXPERIMENT A. Impedance Meter Method

Consider a coil source/receiver of N turns, radius R, and negligible wire diameter energized by a stable oscillation source. Fig. 1 shows such a coil centered with respect to cylindrical and hemispherical conductivity samples in the coordinate system shown. Theory is derived in the Appendix for the change in impedance seen by the coil in the presence of the conductivity sample; experimental

Cylinder 0- const. A

Coil m (N Turns)

Y

Z

A Coil

(N Turns) He rn i s p h e re

Y

X

Fig. 1. Coordinate system and coil placement relative to conductivity samples of cylindrical and hemispherical geometry.

results discussed in this section are compared with the theory in order to validate the measurement technique.

Fig. 2 is a block diagram of the impedance meter method of making noninvasive conductivity measurements of a conducting sample in the vicinity of a coil sourceheceiver. A commercial device is employed that uses the four-point method of measuring the impedance of the coil. The current passes through a complete circuit which includes the shield around the current leads to min- imize the chance of any mutual inductance between the measurement circuit and the coil. The circuit is basically a simple series RLC network where the capacitance can be tuned and the coil provides the inductance. (The two voltage probes are shielded, but the shields are not part of a complete circuit.)

A grounded copper strip (typically 0.01 m wide, 0.0004 m thick) is placed inside the coil to serve as a Faraday shield to prevent electric field pickup and the measurement of stray capacitance by the coil. This copper strip has a small break in it to prevent eddy currents from cir- culating. The coils were wound on plastic and fiberglass forms.

It is important that the connection between the shield box and the coil be as short as possible, otherwise the circuit will contain an anomalously high inductance which will make it difficult to tune the circuit. The leads to the coil should be twisted so that inductive pickup does not

HART et al. : NONINVASIVE ELECTROMAGNETIC CONDUCTIVITY SENSOR 1013

HP 4275A MULTCFREOUENCY

LCR METER COPPER STRIF SHIELD WITH

GAP SHIELD BOX

R C

VE$:I VOLTMETER

VECTOR AMMETER 1

Fig. 2. Block diagram of the impedance meter method of noninvasively measuring the impedance change of a coil when the coil is in the vicinity of an object of finite conductivity.

take place. The cables for the devices we built are about 1 m in length and are made of RG174 coaxial cable; the cables were secured so that stress took place on the cables and not on the connectors where it would cause impedance noise. The Hewlett-Packard 4275A LCR meter has provision for making open- and short-circuit measurements (using a fixture to replace the shield box and coil of Fig. 2) to determine the stray capacitance and inductance of the 1 m cables; the LCR meter then removes the effects of these cables from subsequent measurements.

For a series RLC network as shown in Fig. 2, we write for the impedance ( Z ) and change in impedance ( A Z ) of such a circuit

Z = R + j X ; (1)

(2 ) AZ = A R + j A X

where is the series resistance and X is the series reactance of the circuit, and w is the angular frequency of the current. The reactance X is given by

(3 ) 1

X = w L - - W C

where L is the series inductance and C is the series capacitance of the circuit. From the Appendix we rewrite (A84,

AZ = C2f (4)

which is the impedance change seen by the coil for a body of revolutio? located with its axis along the axis of the coil. Here I represents a six-fold dimensionless integral which (when evaluated for a particular geometry) is simply a positive definite constant that can be determined from Table 111. By comparing (2) with (4) and using (A8b) to define C,, we find that

A R = 327r3 X N2f2R3fAu; ( 5 )

A X = 647r4 X N2f3R3fq,Ac, ( 6 )

wheref = w/2 ?r is the frequency of the current, N is the number of coil turns, R is the coil radius, eo is the per-

mittivity of free space, and E , and U are the dielectric constant and conductivity of the body of revolution, respectively. We have made the substitutions U -+ A u and E , -+

A € , to account for cases where only changes in conductivity (or dielectric constant) of a body of revolution are measured, rather than the presence or absence of the body of revolution as was assumed in the derivation of (A8b).

The Hewlett-Packard 4275A multifrequency LCR meter used in this experiment does not measure (1) or (2) directly, 'but measures the magnitude of the impedance or change in impedance ( 1 2 1 or A 1 Z I ) and the phase angle ( 0 ) given by

e = tan-' ( x I R ) . We show below that it is possible to measure the change in the resistance ( A R ) by monitoring only changes in the magnitude of the impedance ( A 1 Z I ) for conditions of resonance. To do this, we first use (1) to write

(7)

Iz( = ( E 2 + x y 2 Then to first order,

R A E + XAX A l Z l =

Izl (9)

If the RLC circuit is tuned to resonance (by choosing the parameters in (3) so that X = 0), and a relatively large value of R is chosen for the shield box of Fig. 2 ( R = 5 Q, for example), then it follows that >> X and, using

- (8) and (91,

IZI = R ; (10)

A l Z l = A R . ( 1 1 ) Comparing (1 1) and ( 5 ) we see that

A 12 1 = 32 7r3 x 1O-l4N2f2R3fAu (12)

at resonance; the change in the magnitude of the impedance measured by the meter is entirely resistive and is directly proportional to the change in conductivity of the body of revolution. In other words, measurement of any


reactive changes in the circuit, including those due to stray capacitance, have been suppressed by tuning. For the experiments reported herein, a tuning capacitor in the series RLC circuit of Fig. 2 was used to achieve resonance. Res- onance was typically achieved (and remained stable) to within a fraction of one degree of phase ( 1 t9 1 < 1 O ).

When using (12) to make on-resonance impedance measurements, it is necessary to make one straightfor- ward correction to the data because of the existence of capacitive coupling between the copper shield and the coil. The electrical shielding of the coil is a result of this capacitive coupling, so it is necessary to the circuit. (If the grounded shield of Fig. 2 is removed, the coil impedance changes substantially in the presence of deionized water, indicating that stray capacitance is cormpting the measurement.) The EMF due to mutual inductance between the coil and shield generates a current (Z2) which flows from the coil to the shield ground via the aforemen- tioned capacitive coupling. The current Z2 returns to the coil through the vector ammeter (Fig. 2), so the vector ammeter measures a total current ( Zl - Z2 ) where Zl is the current flowing in the series RLC circuit of Fig. 2. At resonance, instead of measuring the impedance given by (lo), 1 Z 1 = V/Z1 where V is shown in Fig. 2, we actually measure I Z 1 = V / ( Zl - Z2), which can be re- written as 12 1 = R ( 1 - Z2/Z, ) -’. We define a correction factor ( F e ) as Fc = 1 - Z2 /II, which can be measured experimentally as

R Fc = - P I

where R is the (known) ac resistance of the circuit, and 12 I is the resistance measured by the impedance meter at resonance. The value of can be determined by measuring R in Fig. 2 and adding to this value the ac resistance of the coil wire. (The skin effect must be included to calculate the correct ac resistance of the coil wire.) Since we are interested in small conductivity-induced changes in coil resistance about a constant value of resistance, the above correction factor will apply to any small changes about this constant value of resistance as well as to the constant value itself. Therefore, we correct the measured changes in resistance (A I Z 1 ) by this correction factor as follows

AR = F,A(Z( (14)

to obtain the actual resistance change Ai?. Attempts to make the correction factor in (13) unity by

inserting a resistance in the line grounding the shield did not succeed because shielding effectiveness was also reduced. The measured capacitance between the coil and shield for the coils used in the experiments reported herein was between 50 and 130 pF; this capacitance proved suf- ficient to shield the coils from capacitively coupling to the environment.

Table I gives the characteristics of the eleven different coil source/receivers used in the measurements reported

TABLE I COIL CHARACTERISTICS

Coil No. R (m) N Inductance (wH)

1 0.1027 2 2.7 2 0.0784 2 2.0 3 0.0533 3 2.9 4 0.0312 4 2.3 5 0.0177 6 2.5

6 0.1028 3 5.5 7 0.0790 4 7.4 8 0.0534 5 7.1 9 0.0312 7 6.3

10 0.0175 1 1 7.3

11 0.0269 7 5.3

All coils are made with insulated number 26 copper magnet wire (0.1399 R/m, diameter of copper = 4.0498-4 m).

here. Coil numbers 1-5 were used for measurements at 10 MHz, coil numbers 6-10 were used for measurements at 4 MHz, and coil number 11 was used for measurements at both 4 and 10 MHz.

Fig. 3 shows a set of typical measurements using coil number 1 that are compared to the theory given by (12). Fig. 3 is a plot of A 12 1 versus Au in the notation of (12) where Au is the conductivity change between U = 0 (air alone) and a sphere or cylinder containing a saltwater mixture of calibrated conductivity. The spheres and cylinders were symmetrically located with respect to the coil; the parameter L is the cylinder half length. In other words, the configuration of Fig. 1 has been extended symmetrically about the plane of the coil for all the sphere and cylinder measurements reported here.

The saltwater mixtures were calibrated over the frequency range of interest by placing a sample in an elec- trostatically-shielded Jones cell and using the impedance meter to measure the resistance. The Jones cell was previously calibrated for conversion from resistance to conductivity. A temperature of 21 “C was maintained for all calibrations and measurements.

Each measured point in Fig. 3 is an independent determination of the slope A 12 I /Au, which, according to (12), should be a constant for the particular parameters listed. Each measured point was determined by comput- ing the mean of 15 separate measurements made by the impedance meter. The standard error was usually between about 0.1 and 0.2 m a for 15 measurements. (The least- significant digit on the impedance meter was 1 ma.) The measurement procedure consisted of making a background measurement in the presence of no conductivity for 15 samples, then a “calibration” (sphere or cylinder) measurement of 15 samples, followed again by a background measurement of 15 samples, next followed by another calibration measurement (at a different conductivity but identical geometry) of 15 samples, etc. The 15 background measurements made both before and after a calibration measurement were averaged together to give the total background measurement which was then subtracted from the calibration measurement. By this method, we hoped to account for any linear drift in the instrumenta-


300

200

100

0

f = 10 MHz R = 10 27 cm N = 2 A Sphere, r/R = 0.4713

Ave. Slope = 76.4 f 1.5 mW(S/m)

Cylinder, riR = 0.4372; U R = 0.7201 Ave. Slope = 107.4 + 1.4 mW(S/m)

0 1

Conductivity (Slrn)

Fig. 3. Typical measurements of coil-resistance change versus conductivity for symmetrically-located spherical and cylindrical conductivity samples using the impedance meter method.

tion that occurred between the two background measurements. The drift of the impedance meter was observed to be approximately 10-20 mQ over a 10 min period. Since the impedance meter samples at an approximately 3 Hz rate and the data are recorded using a Hewlett-Packard 9845A computer, it only takes about 6 s to accumulate 15 samples. Since successive calibration measurements and background measurements were made in quick succes- sion, there is little time available for drift to occur. Also, one of the calibration measurements was always made with a solution of deionized water (zero conductivity). This measurement (which was usually on the order of a few tenths of 1 m a after the appropriate background was subtracted) was itself subtracted from all of the other measurements to remove any effects of the container, stray capacitance, etc. The containers used were made of plastic or glass.

Average slopes (see the straight lines in Fig. 3) were determined by averaging, for a particular geometry, the independent slope measurements represented by individ- ual data points plotted as in Fig. 3. In Table I1 the measured average slopes A 1 Z I /Au with experimental standard deviation are given for many experimental geometries along with the corresponding theoretical slopes ( A 12 1 /A.) using (12). The “percent error” column is the percent error of this measured slope as compared to the theoretical slope determined from (12).

Since the theoretical slope determined from (12) is ab- solute (all parameters are known; all assumptions used in deriving the theory are well met for the range of experimental parameters used), the theory actually calibrates the output of the impedance meter. In fact, since no accuracy values are quoted by the manufacturer for the impedance meter operating in the A 12 1 mode (as opposed to the standard 12 I mode where accuracy values are specified), the “percent error” column of Table I1 actually defines

TABLE I1 COIL CALIBRATIONS

(a) 10 MHz Frequency

Correction Measurement Theory Coil No. r /R m Factor [mn/(S/m)] [mN(S/m)l B Error

1 0.4713 (sphere) 0.569 76.4 f 1.5 66.0 16 1 0.6130 (sphere) 0.569 260.2f 1.1 247.6 5 2 0.6177 (sphere) 0.862 129.8 f 2.4 114.3 14 3 0.9074 (sphere) 0.798 676.1 f 11.3 573.8 18

1 0.4372 0.7201 0.569 107.4f 1.4 99.6 8 2 0.5733 0.9442 0.904 167.0f 2.0 146.4 14 3 0.8421 1.387 0.821 642.3 f 6.5 549.0 17 4 0.4068 2.026 0.796 12.2 f 0.8 9.80 24 5 0.5354 3.470 0.743 11.2 f 0.8 12.3 -9

11 0.8854 2.051 0.344 481.3 f 6.7 477.4 1

Average: 11

(b) 4 MHz Frequency

Correction Measurement Theory Coil No. r/R m Factor [mn/(S/m)] [mN(S/m)l 5% Error

6 0.4369 0.7196 0.827 33.0 f 0.9 35.9 -8 7 0.5685 0.9364 0.738 89.6 f 1.2 92.6 -3 8 0.8417 1.387 0.720 225.0f 2.8 243.9 -8 9 0.4068 2.026 0.750 4.90f 0.09 4.80 2

10 0.5416 3.510 0.598 4.62f0.29 6.71 -31

11 0.8854 2.051 0.803 65.5 It 1.1 76.4 -14

Average: -10

the accuracy of the impedance meter method using the A I 2 I mode.

It is important to calibrate the impedance meter using cylindrical (or spherical) solutions of known conductivity over the range ( A 12 1 ) of interest prior to any biomedical measurements, even though Fc is known, so that the output of the meter can be shown to be linear in that range (or can, if necessary, be corrected back to linearity). For example, a coil that will be placed around a head should be calibrated for a range of impedance change ( A 12 I ) that includes the impedance change due to placing the head in the coil. This is important because the impedance meter can become nonlinear for high values of A 1 2 I ; in fact, Fig. 3 shows the beginnings of nonlinearity at the higher values of A 12 1 . B. Colpitts Oscillator Method

The relatively expensive impedance meter can be re- placed by an inexpensive oscillator for making noninvasive conductivity measurements. Fig. 4 is the circuit employed to test this idea, which consists of a marginally stable Colpitts oscillator feeding a demodulator and an integrator to the dc output, which was connected to a Hewlett-Packard 3455A digital voltmeter. The output from the digital voltmeter was recorded by a HP 9845A computer. An oscilloscope is useful to measure the ac voltage directly across the coil in order to tune the circuit (by adjusting the ‘‘oscillator set” resistor to obtain stable oscillation) and to measure the oscillation frequency. The oscillation frequency is determined by the inductance and capacitance of the tank circuit. The transistor with negative feedback provides stable voltage gain. A “dc out-

1016 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 35. NO. 12. DECEMBER 1988

COPPER STRIP SHIELD WITH

GAP

- Tank Circuit

Fig. 4. Block diagram of the Colpitts oscillator method of noninvasively measuring the impedance change of a coil when the coil is in the vicinity of an object of finite conductivity.

put” is extracted from the demodulator diode which re- flects any changes in oscillation amplitude.

The LC tank circuit of the Colpitts oscillator responds to changes in the complex impedance of a conductivity sample as given by (4), which we rewrite using (A8b) as

AZ = 32n3 X 10p’4N2f2R3[Au + j ( 2 n f ~ ~ A ~ , ) ] f (15) where we have again made the substitutions u + Au and E, -+ AE, as was done for ( 5 ) and (6). The imaginary part of (15) is independent of conductivity changes and is usually much smaller than the real part for most measurements of biomedical interest. In other words,

At a frequency of 10 MHz, the numerator of (16) has the value 0.045 S /m for water. For calibrations of the oscillator, we can think of the imaginary component of (15) as causing a small constant pickup for the u = 0 calibration (similar to pickup from stray capacitance) which is sub- sequently subtracted from all the other measurements.

Fig. 5 is an example of a calibration using coil numbers 3 and 4 (see Table I) with the parameters listed in the figure. Over the linear region a constant calibration factor having units of mV /(S /m) is experimentally obtained. Since the same calibration procedures and geometries are used as previously described for the impedance meter method, a theoretical change in impedance can be calculated as

(17) AZ - = 32a3 X 10-I4N2f2R3f A 0

where we have ignored the imaginary part of (15) because its effect, along with the effect of stray capacitance, has been determined from the u = 0 calibration and subtracted off. (Typically -0.1 mV is subtracted off.) The experimental calibration factor described above having units of mV/(S/m) can be divided by (17) to obtain a conversion factor having units of mV/mQ as shown in

R = 533cm,f = 6 78 MHz, N = 3 r/R = 0 842, UR = 1 387 Ave Slope = 77 2 + 0 9 mV/(S/m)

-7 a R = 3.12 cm, f = 7.84 MHz, N = 4

r/R = 0.599: UR = 2.37 Ave. Slope = 21.9 k 0.4 mV/(S/m) L

0 1 2

Conductivity (Slrn)

Fig. 5 . Typical measurements of Colpitts oscillator voltage change versus conductivity for two difference coil sensors using symmetrically-located cylindrical conductivity samples.

Fig. 5 . By this means the Colpitts oscillator is calibrated so that its final output over the linear range is converted to an impedance change of the coil. This mV/mQ calibration is valid for all measurements (using the same coil and Colpitts oscillator) in the linear range, even for non- symmetric coil-sample geometries. The mV /mQ calibrations obtained for different oscillators using the same coil often varied by factors of 2 or 3 or more, depending on transistor characteristics and “oscillator set” adjust- ments, but remained stable for periods of a day or more for a given choice of components and settings. The “best” oscillators have the largest mV /mQ calibration values consistent with acceptably linear output over the output range of interest. Note that the Colpitts oscillator of Fig. 5 becomes somewhat nonlinear at the highest values of conductivity for one of the coils. The calibration, however, can be used to correct the oscillator output back to linearity. For application to brain edema, a coil that will be placed around a head should be calibrated for a range of voltage change that includes the voltage change due to placing the head in the coil.

HART et a / ’ NONINVASIVE ELECTROMAGNETIC CONDUCTIVITY SENSOR 1017

- c E v

Q cn

m S

Q

C m

U)

K

0

.-

20

15

10

5

0

-5

* A1 - 61 * A2 -c 62 -L

0 1 0 2 0 3 0 4 0 5 0 6 0

Increase in Volume of White Matter (“A)

Fig. 6 . Calculated change i n resistance of a coil placed around a model of the brain where the volume andlor conductivity of the white brain matter is allowed to increase relative to the grey brain matter, mimicking a situation of brain edema. The model consists of spherical white matter (centered within the coil) surrounded by grey matter. The outer boundary of the grey matter does not change. The parameters associated with the various curves are given in the text.

111. EDE:MA MODEL Fig. 6 shows results of parameterizing a simple model

which is used to make order-of-magnitude estimates of the response of a coil sensor to the onset and progress of deep-seated brain edema such as might occur with a brain stem injury. The basic model is a circular coil placed around a head where we consider the white brain matter to be spherical and located concentrically with respect to the coil center. We plot the resistance change determined from (12) [or (17)] when the volume and/or conductivity of the white brain matter is allowed to increase relative to the surrounding grey matter, mimicking a situation of brain edema. The resistance change due to the whole head, before the onset of edema, is subtracted from each plotted point. The intrinsic conductivity of normal white brain matter is taken as 0.10 S/m, and that of normal grey brain matter as 0.23 S / m for the frequency range of 1-4 MHz. (Measurements indicate that these conductivity values are isotropic and do not change significantly over the frequency range of 1-4 MHz [l] , [4].) The conductivity and outer boundary of the grey brain matter never changes, although its space is invaded by the edematous white brain matter.

The human head has a diameter ranging from about 14- 20 cm which should be at least a factor of 3 smaller than the electrical skin depth of the head (defined in the Ap- pendix) to allow us to neglect exponential attenuation as we have done in this model. The human head has an integrated conductivity (measured by magnetic induction at f = 100 kHz) of 0.286 S / m [ 2 ] , which should remain relatively constant over the frequency range of interest [3], [4]. This means that at frequencies of 1, 2, 4, and 10 MHz the approximate electrical skin depth of the head is

94, 67, 47, and 30 cm, respectively. The baseline coil parameters chosen for the model are thereforef = 1 MHz with N = 20 turns and R = 10.27 cm. Recall that the inherent sensitivity of the coil to conductivity changes varies as the factor N 2 f 2 [see (12) or (17)] so one could have the same coil sensitivity by choosing f = 4 MHz with N = 5 turns and R = 10.27 cm on a small head (so as not to violate the factor of 3 rule mentioned at the be- ginning of this paragraph); of course, if a smaller value of R is used for this case (because the head is smaller), the intrinsic sensitivity of the coil increases. The other parameters are as follows:

Cases AI and BI: The white brain matter sphere increases in volume without any change in the intrinsic conductivity of the white brain matter. Two curves are shown for two initial volumes of white brain matter ( V, ) . For case Al , V , = 0.180 L; for case B1, V , = 0.113 L.

Cases A2 and B2: As the white brain matter sphere increases in volume, the intrinsic conductivity of the white brain matter increases linearly from 0.10 to 1 .O S / m over the range plotted. For case A2, Vw = 0.180 L; for case

The edematous brain conductivities chosen for cases A 2 and B2 above are based on conductivity measurements of normal and edematous tissue from a sectioned rabbit brain [I]. The curves labeled A I and BI represent a limiting case of minimum expected response of the coil source/ reciever where only volume and no conductivity changes occur in the white manner.

It is evident from Fig. 6 for cases A2 and B2 that if the early onset of brain edema is to be detected ( - 10 percent increase on the abscissa), then detection sensitivities on the order of 0.1 mn of resistance change are desirable in order to clearly see changes in resistance ranging from 1 to 3 mn. If the early onset of brain edema is to be detected for the limiting cases A 1 and BI , then detection sensitivities would have to improve an order of magnitude to 0.01 rnQ in order to see changes in resistance ranging from -0.1 to -0.3 mQ.

For time-domain monitoring of brain edema, changes in the instrumentation drift rate must be much less than the expected signal over the characteristic time period for edema onset, or else periodic background measurements must be made for the purpose of removing the drift. (The periodic background measurements could be made using a different coil switched into the same electronics to avoid having to disturb the coil around the head.) The drift of the impedance meter was a fairly linear 10-20 mil over a 10 min period. The drift of the Colpitts oscillator was a fairly linear 10-20 mV over a 10 min period. It would be desirable to substantially reduce this instrumentation drift. For the oscillator, this could be accomplished by using temperature-stable components, improved design, and temperature stabilization. The standard error for 15 measurements made in a time period of a few seconds was 0.001-0.01 mV for the Colpitts oscillator, which shows that if temperature stabilized electronics were used there is a potential for significant sensitivity. (The impedance

B2, V , = 0.113 L.


meter only measures to a resolution of 1 mil, so its sensitivity cannot be much improved.) Low-noise electronics could also be used in the Colpitts oscillator implementation, which should allow for improvements in sensitivity of several orders of magnitude if needed. Since broad- band Johnson noise (at room temperature) for the coils of Table I is calculated to be less than lo-’’ VI&, it is evident that sensitivity will be limited by electronics noise, biological noise, coil motion noise, etc., long before approaching the fundamental Johnson noise limit. Bi- ological noise could be caused by slight relative move- ments of brain tissue that occur over a time period comparable to the onset of brain edema.

To monitor deep-seated brain edema, the coil source/ receiver does not need to be circular, but zould be molded to the shape of the head, perhaps in a headband arrangement, making it more convenient to wear. For a noncir- cular coil, however, the ability to calibrate voltage changes of a Colpitts oscillator to units of milliohms by comparing calibration measurements to theory would be lost. The impedance meter method could be used, however, to directly measure the coil impedance change (in milliohms) using particular calibration geometries and conducting solutions, which calibration geometries and conducting solutions could then be repeated to obtain the output voltage of a Colpitts oscillator (in millivolts). A comparison of the outputs of the two devices would then give an experimental millivolts/milliohms calibration for the coil-oscillator combination. These calibrations should be done in the linear range of the impedance meter and Colpitts oscillator, and then extended to the nonlinear range if necessary to include the region of interest for monitoring brain edema.

IV. CONCLUSION A noninvasive method of monitoring a patient for the

onset and progress of brain edema has been described, and the feasibility of implementation has been explored using theory, models, instrumentation development, and sensor calibrations using in vitro configurations. A simple model of brain edema developed in Section I11 gives an idea of the worst-case measurement sensitivities required since they were calculated for the difficult case of deep- seated brain edema. It has been shown that the coil source/ receiver is a promising sensor for use in monitoring brain edema because there exists a relatively simple implementation (Colpitts oscillator method) which has the potential for considerable improvement in sensitivity (by orders of magnitude) before fundamental limitations are reached. The simple model of brain edema developed in Section I11 shows that even without any improvement, both the impedance meter and Colpitts oscillator methods of im- plementing a coil-based edema detector are potentially ca- pable of measuring the onset and progress of some forms of deep-seated brain edema (if a method for reducing the electronics drift is also implemented). Future measurements on animal models will help determine the extent of improvements needed in the proposed instrumentation.

APPENDIX Consider a coil source/receiver of N turns, radius R ,

and negligible wire diameter centered about the origin in the x, y plane of a rectangular coordinate system. If we consider only the quasi-static field range characterized by the condition that the distance from the source point to the field point is very much smaller than a free-space wavelength, then the z component of the magnetic induction at time t due to a coil current ( l e ’ ” ‘ ) at angular frequency w can be written in cylindrical coordinates using SI units as

B , ( ~ , z , t ) = (2 x 1 0 - ~ ) ~ 1 ~ j @ ~ i~ (R2 - pR COS 8) dB

0 ( p 2 + R2 + z’ - 2pR cos

where the Biot-Savart law has been applied to the coil to obtain this result (see [18] for a derivation). We definej = f i , p = and 8 = tan-’ ( y / x ) ; the sign of B, relative to I is determined by the right-hand rule. Using the integral form of Faraday’s law and the cylindrical symmetry of the problem, we can write the azimuthal (and only) component of the electric field as

where

If we now introduce a body of revolution of conductivity U and permittivity E generated by rotation about the z axis, we can write for the induced eddy current density in this body

Je = ( U + jwE)Ee (A3)

where the electric field boundary condition for EO (conti- nuity of the tangential component at the interface) is met by symmetry using (A2) both inside and outside the body of revolution. In (A3) we have also assumed that the electrical skin depth given by ( ~ p ~ u / 2 ) - ] / ~ is much larger than the body diameter where po is the permeability of the body of revolution which is generally that of free space ( p o = 4n x lop7 H/m). A circular element of induced eddy current in the body of revolution can be written as

d l = Je dz dp. (A4) We can now compute the “backscattered” z component of the magnetic induction B,, in the plane of the coil ( z = 0) due to these circular eddy currents by using (Al) to determine dB,, with the correspondences N + 1 , ZeJ’”‘ -+

d l R + p , z + z , 8 + CY, and p + h where h = in the z = 0 plane. Schematically, we then write

B,, = j dB,, where the integration is over the 2, p , p ’ , 8, and CY variables. Explicitly, using (A4), (A3), (A2), and (Al), we can write

HART er al. : NONINVASIVE ELECTROMAGNETIC CONDUCTIVITY SENSOR 1019

B,, = CI i:ydz i:dp i : d p ' j,ds j T d a 0 [ F , ]

( A 5 4 where r ( z ) is the radius of the body of revolution (as a function of z) which extends from z = L , to z = L 2 , and

C1 = - j ( 4 x w ( u +jw)NZej"' (A5b)

- ~ I R COS e) Fl = 3 j 2 (A5c)

(R2 + p'* + z2 - 2p'R cos 0 )

( p - h COS C Y ) X

(h2 + p 2 + z2 - 2hp cos 0 1 ) ~ " '

It is now simply a matter of applying Faraday's law to B,, over the plane of the coil to calculate the change in EMF (AE) of the coil as a receiver:

where R

@,y = io Bs,(27rh) dh. ( A6b )

We define an effective impedance change ( AZ ) in the coil that causes the same change ( AE) in EMF as

AZ = -AE/(ZejUr). (A7)

The minus sign in (A7) occurs because a negative change in EMF impedes current flow the same as a positive impedance change in the coil. Using (A7), (A6), and (A5) we write

Az = c2i ( A 8 4

where

C2 = 327r3 X 1 0 - ' 4 N 2 f 2 R 3 [ u + j ( 2 7 ~ f ~ ~ ~ , ) ]

1 = iz;: dz Sd," dp j: dp' i: dh dB

(A8b)

- s T d a 0 [ F ] (A&)

and

p ' ( i - cos e ) + z2 - 2p' cos

F = (1 +

X 3 1 2 . ( A 8 4 h ( p - h COS C Y )

h2 + p 2 + z2 - 2hp COS C Y )

In writing (A8c) we have switched to dimensionless integration variables by making the substitutions z + zR, p + pR, p' + p'R, and h --t hR. In writing (A8b) we have used the definitions E = (where c0 is the permittivity of free space and E , . is the dielectric constant of the body of revolution), and w = 27rf (where f is the frequency of

the coil source). The six-fold integral = AZ/C2 given by (A8c) is a dimensionless positive definite constant. It can be reduced to a two-fold form. The results of a numerical evaluation are presented in Table I11 for the cylinder and hemisphere cases of Fig. 1. The column labeled r / R in Table I11 applies to both cylinder and hemisphere; the hemisphere values of AZ/C2 are to the left of this column, and the cylinder values to the right, parameter- ized by values of L / R . The analytical and numerical de- tails of producing Table I11 will be published elsewhere.

It is possible to combine entries from Table I11 to represent more complex geometries by applying the fundamental theorem of integral calculus as shown schematically below for the case of a cylinder:

L2 / R l-2 l R

!L , /R dz j r , / , dp

* (j;'" dz - d z ) ( dp - i:/R d p )

dz sa/R dp - j;" dz j r ' IR 0 dp a

- 1:'"dz j n / K d p + j;'"dz j r ' l R d p ro

(A9)

where rl and r2 are the inner and outer radii of a cylindrical shell, and L, and L2 are the two values of z (positive or negative) corresponding to the endpoints of the cylindrical shell ( L , < b). The length of the cylindrical shell is (& - L l ) . Each of the four double integrals on the right side of (A9) corresponds (in principle) to an entry in Table I11 with the following caveat: if looking up a value of 1 L / R 1 in Table I11 to evaluate (A9) when the actual value of L / R < 0, take the table entry to be negative also. For example, if we extend the cylinder of Fig. 1 so that it extends equally on both sides of the coil, the value of AZ doubles by symmetry.

Similarly, for a hemispherical shell, we can write schematically

ri / R 11;; dp = j,"/R dp - 0 dp (A10)

where rl and r2 are the inner and outer radii of a hemispherical shell. Each of the two integrals on the right side of (A10) corresponds (in principle) to an entry in Table 111. Again by symmetry, if we extend the hemisphere of Fig. 1 (or a corresponding hemispherical shell) so that it becomes a sphere (or spherical shell) centered at the center of the coil, the value of AZ doubles. It is possible to combine shapes (cylinders, spheres, and cylindrical and spherical shells) to create more complex shapes, which can be treated analytically by multiple application of (A9) and (A10) so long as the requisite cylindrical symmetry is maintained.

1020 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 35. NO. 12, DECEMBER 1988

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ACKNOWLEDGMENT thereafter became a Charter Member in the Department of Submarine Tech- nology. His research has included low-frequency electromagnetic propagation and ocean magnetics as well as biomedical investigations. He has been a member of the Principal Professional Staff since 1980 where he presently conducts electromagnetics research sponsored by the U.S. Navy. His other areas of research have included magnetic properties of rare earth elements, transport properties of air pollutants through porous media, and

The authors wish to thank the s. s. Janney Fellowship Committee of The Johns Hopkins University Applied Physics Laboratory for providing fellowship support for the preparation of this paper.

131

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1131

REFERENCES D. M. Long and H. W. KO, “Quantification of brain edema by measurement of brain conductivity,” in Brain Edema. New York: Springer, 1985, pp. 632-637. P. P. Tarjan and R. McFee, “Electrodeless measurements of the effective resistivity of the human torso and head by magnetic induction,” IEEE Trans. Biomed. Eng., vol. BME-15, pp. 266-278, 1968. L. A. Geddes and L. E. Baker. “The specific resistance of biological material-A compendium of data for the biomedical engineer and physiologist,” Med. Biol. Eng., vol. 5 , pp. 271-293, 1967. H. P. Schwan and K. R. Foster, “RF field interactions with biological systems: Electrical properties and biophysical mechanisms,” Proc. IEEE, vol. 68, pp. 104-113, 1980. H. P. Schwan, “Determination of biological impedances,” in Phys- ical Techniques in Biological Research, vol. 6. New York: Aca- demic, 1963. pp. 321-407. J . B. Ranck, “Specific impedance of rabbit cerebral cortex,” Exp. Neurol., vol. 7 , pp. 144-152, 1963. M. Yedlin, H. Kwan, J . T. Murphy, H. Nguyen-Huu, and Y. C. Wong. “Electrical conductivity in cat cerebellar cortex,” Exp. Neu- rol., vol. 43, pp. 555-569, 1974. J . D. Kosterich, K. R. Foster, and S . R. Pollack, “Dielectric permittivity and electrical conductivity of fluid saturated bone,” IEEE Trans. Biomed. Eng., vol. BME-30, pp. 81-86, 1983. E. D. Trautman and R. S . Newbower, “A practical analysis of the electrical conductivity of blood,” IEEE Trans. Biomed. Eng. , vol.

S . E. Markovich, Ed., Internariotial Conference on Bioelectrical Impedance. New York: Ann. NY Acad. Sci., vol. 170, 1980, pp.

A. Van Harreveld, “The extracellular space in the vertebrate central nervous system,” in The Structure and Function of Nervous Tissue, vol. 4 . , G. H. Bourne, Ed. New York: Academic, 1972, pp. 447- 511. J. C . Maxwell, A Treufise on Electricity and Magnetism, vol. 1. New York: Dover, 1954, (originally 1891), p. 440. R. P. Henderson and J . G. Webster, “An impedance camera for spa- tiallv suecific measurements of the thorax.” IEEE Trans. Biomed.

BME-30, pp. 141-153, 1983.

407-836.

. .~

chemcial rate constant determinations in flames.

Physical Society. Dr. Hart is a member of Phi Kappa Phi, Sigma Xi, and the American

Harvey W. KO (S’67-M’74) received the B S degree in electrical engineering in 1967 and the Ph D. degree in electrophysics in 1973 from Drexel University, Philadelphia, PA.

He is currently a member of the Principal Pro- fessional Staff, The Johns Hopkins University Ap- plied Physics Laboratory Laurel, MD. Prior to joining the APL, he was a Trunk Design Engineer at the Bell Telephone Company and a Biomedical Engineer at the University of Pennsylvania Pres- byterian Medical Center, Philadelphia. His pres-

ent research interests address radar wave propagation and magnetometry He serves as the Director of the Bioelectromagnetics Laboratory and Pro- gram Manager in the Department of Submarine Technology at APL

James H. Meyer, Jr. was born in Boston, MA, in 1962. He received a dual B.S. degree in electrical engineering and bioengineering in 1985 from Syracuse University, Syracuse, NY, and is currently pursuing a Masters degree in computer sci- ence at The Johns Hopkins University, Baltimore, MD.

He worked at The Johns Hopkins University Applied Physics Laboratory, Laurel, MO. from 1985-1986 on biomedical engineering projects.

Eng:, vol. BME-25, pp, 250-254, 1978. [I41 K . A. Dines and R. J . Lytle, “Analysis of electrical conductivity

imaging,” Geophys., vol. 46, pp. 1025-1036, 1981. [I51 T . J . Yorkey, J. G. Webster. and W. J. Tompkins, “Comparing re-

construction algorithms for electrical impedance tomography,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 843-852, 1987.

[ 161 -, “An improved perturbation technique for electrical impedance imaging with some criticisms,’’ IEEE Trans. Biomed. EnR., vol.

Currently, he is employed with Radix Systems, Inc. as a Systems Development Engineer.

Mr. Meyer is a member of Tau Beta Pi and Eta Kappa Nu.

BME-34, pp. 898-901, 1987

Proc. IEEE, vol. 70, pp. 523-524, 1982.

Disrribured Circular Currmry.

David P. Vasholz was born in Milwaukee, WI, in 1943. In 1965 he received the B.S. degree in mathematics and physics from Valparaiso Univer- sity, Valparaiso, IN, and the Ph.D. degree in theoretical physics from the University of Wiscon- sin, Milwaukee, in 1970.

He did postdoctoral work in elementary parti- cle physics at the University of Arizona, Tuscon, from 1970-1971 and at the University of Florida, Gainesville, from 1971-1974. In 1975 he joined the Naval Coastal Svstems Center. Panama Citv.

[I71 J . C. Lin and M. J . Clarke, “Microwave imaging of cerebral edema,’’

1181 P. J. Hart, Universal Tables for Magnetic Fields of Filamentary and New York: Elsevier, 1967, ch. 2,

pp. 3-4.

Lynn W. Hart was born in Logan, UT in 1942 He received the B.S degree in physics from Brigham Young University, Provo, UT, in 1967, and the Ph.D. degree in physics from Iowa State University, Ames in 1971.

He worked as a Postdoctoral Fellow at Kent State University, Kent, OH, from 1971-1972, and at The Johns Hopkins University Applied Physics Laboratory (JHUIAPL), Laurel, MD, from 1972- 1973. In 1973, hejoined the SeniorTechnical Staff o f IHUIAPL in the Space Department and shortly

FL, where he conducted theoretical research in wave propagation through a random medium and in signal processing for passive acoustic arrays. In 1978 he joined the Senior Technical Staff in the Department of Submarine Technology, The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, where he has carried out research sponsored by the U.S. Navy centered upon small scale upper ocean dynamics, with particular emphasis on advection-diffusion and internal wave processes. He has been a member of the Principal Professional Staff since 1987, and is presently engaged in the study of optical propagation through a turbulent medium and in magnetic field calculations.

Dr. Vasholz is a member of the American Physical Society.


Richard I. Joseph was born in Brooklyn, NY, on May 25, 1936. He received the B.S degree from the City College of the City University of New York in 1957, and the Ph.D degree from Harvard University, Cambridge, MA in 1962, both in the theory of solitons and nonlinear evolution equations physics

From 1961 to 1966 he was a Senior Scientist with the Research Division of the Raytheon Com- pany Since 1966, he has been with the Depart- ment of Electrical Engineenng and Computer Sci- ence, The Johns Hopkins University, Baltimore,

MD, where he is currently the Jacob Suter Jammer Professor of Electrical Engineenng. During 1972 he was a Visiting Professor of Physics at the Kings College, University of London, on a Guggenheim Fellowship HIS research interests include electromagnetic theory, statistical mechanics, and

Dr Joseph is a Fellow of the American Physical Society

a noninvasive electromagnetic conductivity sensor for biomedical applications

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