IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-22, NO. 6, NOVEMBER 1975

513

Electrolytic Tank Solution of Electric Field Associated with Two Circular Disks on the Surface of an Isotropic Homogeneous Medium JERRY H. GOLD, MEMBER, IEEE,

AND

Abstract-The electric field associated with two disks on the surface of a semi-infinite isotropic homogeneous medium is found on a plane perpendicular to the disks and passing through their centers. To an accuracy which is primarfly determined by the fiite dimensions of the electrolytic tank, the solution is presented as families of curves. In this presentation, the magnitude of the electric field (normalized in tenns of the distance of the field point from the surface, conductivity of the medium, and applied power) and the angular orientation of the field vector are plotted as functions of the ratios of the linear dimensions which characterize the geometry of tlhe system to the distance of the field point from the surface.

JOHN C. SCHUDER,

MEMBER, IEEE

disks, the distance between their centers, and their position on the surface of the body in order to minimize the power required to achieve a given radio-frequency electric field at the internal position where the diode is located, and what is the magnitude of this required power? Since irregular body surface geometry and lack of tissue homogeneity make it very difficult to achieve a satisfactory solution to the problem as stated, we attempt to gain some quantitative insight into the factors involved by considering an idealized model in which the circular disks are placed on the plane surface of a semi-infmite isotropic homogeneous medium. Essentially, the problem is one of obtaining a solution to Laplace's equation subject to the appropriate boundary conditions. But even the idealized model is refractory to an analytical solution in closed form. Furthermore, because the field is a truly three dimensional one which cannot be reduced by symmetry considerations to a two dimensional field, a digital computer solution using numerical relaxation methods is awkward and computationally expensive. Consequently, we have chosen to solve the problem by means of an electrolytic tank. This paper describes our experimental arrangement and presents the derived data in graphical form. It is believed that our format of data presentation and the conclusions which can be drawn from the data wiJl prove generally useful to investigators involved in a variety of muscle and nerve stimulation studies as well as to those contemplating the use of capacitiveconductive coupling for the transmission of radio-frequency energy into the body. Although the general concept of using electrolytic tanks in the solution of field problems is widely appreciated, literature relevant to the design and construction of such systems and to the consideration of sources of error is somewhat limited. Consequently, reference is made to four papers which we have found particularly useful in the design of our experimental system [31-[6].

INTRODU.CTION AN electrode arrangement in which two circular disks are positioned on the surface of the body is widely used in the transmission of internally generated electrical energy to external receiving apparatus and in the transmission of externally generated electrical energy to regions within the body. Electrocardiography and electroencephalography are examples of procedures in which surface electrodes are used for the outward transmission of electrical energy while extemal cardiac pacing and ventricular defibrillation are examples of procedures in which surface electrodes are used in the inward transmission of electrical energy. Our own interest in the use of circular surface electrodes in the inward transmission of energy has been in conjunction with the analysis of a system in which selective stimulation of internal biological material is achieved by means of a very small implanted solid-state diode which generates a localized pulsed DC field when a pulsed radio-frequency electric field is established in the tissue by electrodes on the surface of the body [11. In this system, satisfactory cardiac pacing has been achieved in large dogs with implanted diodes as small as 4.1 X 10-3 meters in length and 1.8 X 10-3 meters in diameter [2]. But the size of the diode unit which can be utilized in such an application is inversely related to the magnitude of the radio-frequency field. Thus, it becomes a matter of conPRELIMINARY CONSIDERATIONS siderable interest to relate the radio-frequency electric field at the position of the diode to the geometry of the surface elecIn this paper we shall focus our attention upon points in a trodes which establish the internal field and to the radio- plane perpendicular to the plane of the disks and passing frequency power supplied to these electrodes. Stated more through their centers. The nomenclature which we shall use in succinctly, we ask what should be the radii of the surface describing our model is indicated in the sketch of Fig. 1. The angular orientation of the electric field at the point Q is deManuscript received August 2, 1974; revised February 6, 1975. This scribed by the angle 0 which the field vector makes with the work was supported in part by the National Heart and Lung Institute horizontal line. The rationalized meter-kilogram-second sysunder Public Health Service Grant HL-15439. The authors are with the Thoracic and Cardiovascular Section, De- tem of units is used exclusively in this paper. In scaling data from our tank to the idealized model, we use partment of Surgery, University of Missouri, Columbia, Mo.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NOVEMBER 1975

514

pressed as

SURFACE ELECTRODES

R=

X /~ \>4-e

AIR

Ro

VIo

VOI cosq -

(6)

When values of V/VO and of (I cos 4)/Io from (2) and (3) are substituted into (6), R/Ro becomes -r< /Q R _ rodo (7) Ro ad Fig. 1. Geometric relationship between disks and field point in idealized which can be written as model. (8) Rod = Roorodo. the subscript 0 to refer to the tank and unsubscripted symbols Equation (8) tells us that the value of Rud which we evaluate to refer to the idealized model. The ratio of the power input in our tank applies equally well to any corresponding geomein watts to the electrodes in the idealized model to the input try in the idealized model. In the idealized model, the R value in (8) is to be interpreted as being a shunt resistance in a parin our tank can be expressed as allel equivalent circuit in which the other element is a capaciP VI cos tance which accounts for the displacement current. ISOTROPIC HOMOGENEOUS MEDIUM

dI_,

where the V's represent the rms potential differences applied to the disks in volts, the I's represent the rms electrode currents in amperes, and 0 is the phase angle between the current and voltage phasors in the idealized model. Since in our tank the current and voltage are in phase, the term involving the phase angle is absent in the denominator of (1). Now in going from the tank to the idealized model, let all of the linear dimensions be changed by the same ratio. That is, let- a/ao = e/eo = r/ro = dldo. Under such a transformation Ed

V

(2)

Eodo

Vo

and I cos q

Eod2 (3) E0a0dO where the a's are conductivities expressed in mhos per meter and the E's are the magnitudes of the electric fields in rms volts per meter at any set of corresponding points in the idealized model and the tank. While both E and Eo are individually functions of position, for the transformation under consideration their ratio is independent of position. Equations (2) and (3) are easily derived by means of simple line and surface integrations. When the values of V/VO and of (I cos 0)/Jo from (2) and (3) are substituted into (1), we have

Io

P

E2ad3

PoEo od3 which can be written as Eol/2d3/2

p1/2

_

E .0 al/2d3/2

p1/2

(4)

5c

Equation (5) tells us that the value of (Er1/2d3/2)/P1/2 which evaluate in our tank applies equally well to any corresponding point in the idealized model. The ratio of the resistance in ohms as measured between the electrodes in the idealized model and in our tank can be exwe

EXPERIMENTAL APPARATUS

Probe and Tank Our electrolytic tank, which is constructed of plywood, is shown in Fig. 2. The electrolyte region is nominally 1.6 meters in length, 0.8 meters wide, and 0.4 meters deep. A rigid aluminium I-beam which serves as a track for the probe assembly also supports an assembly for accurately positioning brass disks on the surface of the surface of the electrolyte. The tip portion of the probe consists of two platinumiridium wires;' each slipped through thin-walled glass sleeves having an outside diameter of 1.73 X 10-3 meters and a length of 30 X 10-3 meters. The tips of the electrodes project about 10- meters beyond the insulating sleeves and are separated from each other by approximately 5 X 10-3 meters. For the electrolyte concentration used in our study, this probe has an output resistance in the neighborhood of 40 X 103 ohms. The shank of the Plexiglas probe serves to join the proximal ends of the glass sleeves to a Plexiglas bearing block and a large circular Plexiglas protractor disk which is accurately inscribed in degrees. The disk electrodes, which are constructed of brass and have their vertical surfaces covered with insulating tape, have radii, e, of 0.0125, 0.025, 0.050 and 0.10 meters. The rectangular cross-sectioned bar that serves to position the Plexiglas rods which hold the disks has accurately machined holes which provide for distances, a, of 0.01375, 0.025, 0.0275, 0.05, 0.055, 0.10, 0.11,0.15, and 0.20 meters. In operation, the tank is filled to the desired depth with distilled water to which approximately 64 grams of CuSO4 5H20 are added. This concentration of copper sulfate has a conductivity in the neighborhood of 0.01 mhos per meter. The accurate determination of electrolyte conductivity and the precise calibration of the field probe are accomplished by means of a rectangular Plexiglas box which contains parallel plate electrodes at both ends and an access hole for the shank of the field probe along one of its sides. When this box is placed in the tank and held by the same rectangular crosssectioned bar which ordinarily supports the disks, a well-

515

GOLD AND SCHUDER: ELECTRIC FIELD TANK SOLUTION

EXPERIMENTAL PROCEDURE

Fig. 3. Circuit arrangement.

defined one dimensional field is established when the end plates are energized. Circuits The circuit arrangement is shown in Fig. 3. The 1000 hertz sinusoidal energizing signal is obtained from an oscillatorpower amplifier combination which has a power output capability of 15 watts. When switch S3 is closed, the potentiometer functions as part of a Wagner ground system which is used to bring the electrolyte in the immediate vicinity of the probe to ground potential-thus substantially eliminating commonmode signals to the probe measurement instruments. Electrode voltage and electrode current are measured with a HewlettPackard 400D vacuum tube voltmeter. (True electrode voltage is the observed voltage minus a small drop in the current monitoring resistor.) To prevent conflicts in grounds, these measurements are made with switch S3 open. Switch SI is always in the off position when probe measurements are made. The Hewlett-Packard 3740 voltmeter system which consists of a 34702A multimeter, a 34740A digital display, and a 34720A battery module has an input resistance of 11.1 megohms. It is operated from battery power and is floating with respect to ground. The Tektronix 503 oscilloscope is operated in its differential mode. With the X10 attenuation probes, the input resistance from each probe tip to ground is 10 megohms. Switch S2 is closed while the Wagner ground system is being adjusted. At other times it is open.

After the conductivity is determined and the probe is calibrated in a one dimensional field, data are collected with the disk electrodes. The detailed data in the present paper were generated with an electrode voltage of 30 rms volts, a probe depth of 0.05 meters, and with 18 different e-a combinations. For each of the e-a combinations, the input power and resistance are determined from electrode voltage and electrode current values, and the magnitude of the field and its angular orientation are found at distances of 0, 0.05, 0.10, 0.15, and 0.20 meters from the neutral plane by rotating the probe and observing the maximum probe voltage and the angle associated with the voltage null. The magnitude of the field in volts per meter is found by dividing the probe voltage by the calibrated value of tip separation in meters. The direction of the field is perpendicular to the null angle. The values of (Eal/2d3/2)/pl/2 and of Rad presented here are based upon the average values derived from three coinpletely independent runs using the procedure outlined above. Auxiliary data, which we have used to verify the validity of our scaling procedure and for estimating the error introduced by virtue of the finite dimensions of our electrolytic tank, have been generated with an applied electrode voltage of 30 volts for a probe depth of 0.025 meters and with an applied electrode voltage of 60 volts for a probe depth of 0.10 meters. The auxiliary experiments have involved two independent runs each through nine e-a combinations at a probe depth of 0.025 meters and through nine e-a combinations at a probe depth of 0.10 meters. RESULTS Our experimental results are concisely summarized in Fig. 4 through Fig. 8. Although these families of curves were derived with a probe depth of 0.05 meters, they are to be interpreted as being perfectly general except for the inaccuracies introduced by the finite dimensions of the electrolytic tank. Since

0- .0

b _

.05

I

*. 275

Fig. 4. Relations between (Ea1/2d3/2)/Pl/2 and rid and between 0 and rld for disk electrodes having an eld ratio of 1/4.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NOVEMBER 1975

516

.150

-

e =2

I.100.

a/d

b o .050

o4

>/ =

3~~~~~~~~~~ 2.2

I

4

I

/d

a/d

en)

//// z

4~~~~~

15 b1 0

C) iooz

100-

50

q0

Fig. 5. Relations between (EalI2d3/2)1P112 and rld and between 0 and rld for disk electrodes having an eld ratio of 1/2.

Fig. 7. Relations between (Eacl12d3/2)/Pl/2 and rid and between 0 and rid for disk electrodes having an eld ratio of 2.

2.5 -

e/d

2.0 -

__> t~0/4

_

1.5 b

Er1.0

2

-

0.5 -

o04C 21,

CI

2

-, aid

.- o 3

~~~~~2

0 2

4

Fig. 8. Relation between Rod and aid for disks having eld ratios of 1/4, 1/2, 1, and 2.

.150

Fig. 6. Relations between (Ea l/2d3/2)/Pl/2 and rid and between and rid for disk electrodes having an eid ratio of 1.

1/2

=X3

0 C.100

for some applications one may be particularly interested in the e/d I neutral plane of symmetry between the two disks, in Fig. 9 .050 02 we have replotted the rid = 0 data points from the graphs of o Figs. 4 through 7. 1/2 1/4 Values of (Eal/2d3/2)/pl/2 and of 0 derived for identical e/d, aid, and rid conditions, as well as values of Rod derived 2 3 4 a/d for identical eld and aid conditions, with d taking on values of 0.025, 0.05, and 0.10 meters, are found to be relatively inde- Fig. 9. Relation between (Ekla2d3 2)/P1I2 at points on neutral plane (r O) and aid for disks having eid ratios of 1/4, 1/2, 1, and 2. pendent of d. These results tend to confirm the validity of our scaling scheme and suggest that the families of curves in Figs. 4 through 9 constitute a reasonably satisfactory solution for the appropriate graph of Figs. 4 through 7, with interpolation if case of disks on the surface of a semi-infinite region. necessary, (Eal/2d3/2 )/pl/2 and 0 are found. Entering the graph of Fig. 8, Rad is found. Since d is known, the derived DISCUSSION AND CONCLUSIONS value of (Euli2d3/2)iPli2 allows us to solve for any one of The families of curves are very convenient to use in the anal- the variables E, a, or P in terms of the other two. That is, if ysis of systems with given geometries. For example, if d, e, a, we know the conductivity, the power required to achieve a and r are given, we calculate eid, aid, and rid. Entering the given field or the field which a given power level will establish w

0

0

517

GOLD AND SCHUDER: ELECTRIC FIELD TANK SOLUTION

is immediately apparent. Furthermore, the derived value for Rad allows us to find the resistance looking into the disk pair when the conductivity is known. In the design of systems, one ordinarily tries to achieve specific results under certain stated or implied constraints. As a simple example, suppose we wish to achieve a field E at a given depth d in a medium with a conductivity a with the least possible expenditure of power. What geometry should we employ? In looking over the curves in Figs. 4 through 7, we observe that several of them peak at nearly the same maximum value of (Eal/2d3/2)/P1/2 with perhaps the peak at eld = 2 aid = 2, and rid = 2 being very slightly higher than the others. On this basis, the minimum expenditure of power is required when e = 0.5d, a = 2d, and r = 2d. From Fig. 5, one finds the angular orientation of the field to be 0 = 850. If we wish, we can further observe that since (Ea1/2d3/2)1P1/2 = 0.132 at the point in question, the required power is given by 57.39E2ad3 watts. And furthermore, by entering the graph of Fig. 7 with eid = and aid = 2, we find a Rad value of 0.92 which means that the resistance looking into our disk pair is given by 0.92/ (ad) ohms. In addition to allowing us to solve specific analysis and design problems, the curves of Figs. 4 through 8 permit some interesting generalizations. If, for example, there are no constraints upon the size of the disks or their positions on the surface, that is, if we are free to select eid, aid, and rid so as to maximize (Eal/2d 3/2)/P/2, then it is apparent that the power required to achieve a given field strength is proportional to d3 and the field strength for a given applied power is proportional to d-3/2 In many cases, one may be more interested in the component of the electric field along some particular direction than in the magnitude of the field vector per se. For example, one may be interested in that component of the field which is parallel to the plane surface on which the disks are positioned. If such is the case, new families of curves may be generated from those shown in Figs. 4 through 7. To do this, one selects an appropriate number of points along each curve and at each of the points selected one finds the absolute value of the product of (Eal/2d3/2 )/Pl1/2 and cos 0. These derived values of (Eha1/2d33/2 )/Pl1/2, where Eh represents the horizontal component of electric field in rms volts per meter, are then plotted as functions of eid, aid, and rid. Since this is not a simple scaling procedure, the shapes of the derived curves will, in general, differ drastically from the shapes of the original curves. If instead of being interested in the horizontal component of the field one is interested in the component along a line making an angle a with the horizontal, the cos 0 term is replaced by cos (0 - a). The normalization scheme used in Figs. 4 through 7 and in Fig. 9 is particularly valuable in studying those problems in which interest is focused upon operation at a particular depth below the surface of the body and in which the power level per se is an important consideration. If, for example, we are working at a fixed depth and at a fixed power level, Figs. 4 through 7 and Fig. 9 directly indicate the relative field strengths at various eld, aid, and rid locations. If, on the other hand, the surface electrodes are being energized from a constant voltage source, it may be desirable to

a/d

Fig. 10. Relation between (Ed)/V at points on neutral plane (r = 0) and aid for disks having eld ratios of 1/4, 1/2, 1, and 2. .150

0

.100 e /dcl/2

b

V e/d

.0501

1/4

~~~~~~e/d=I

2

a/d

3

4

Fig. 11. Relation between (Ead 2)/(I cos 4) at points on neutral plane (r = 0) and ald for disks having e/d ratios of 1/4, 1/2, 1, and 2.

use a different normalization scheme. New families of curves may be easily generated from the families of curves in Figs. 4 through 7 and Fig. 9 by making use of the data in Fig. 8. To do this, one selects an appropriate number of points along each of the curves in Figs. 4 through 7 and Fig. 9 and at each point selected one finds the ratio of (Eal/2d3/2)IP1/2 to (Rad)1/2 where Rad is found for the relevant eid and aid coordinates from Fig. 8. These derived values of

Eal/2d 3/2

Ed

pl/2(R d)1/2 pl/2R1/2

Ed

(9)

V

are plotted as functions of eid, aid, and rid, thus generating new families of curves whose ordinates are given by Ed/V. If we are working with a constant current source, yet another normalization scheme, derived by multiplying (EOl/2d3/2)/ p1/2 by (Rad)1/2, may be convenient. In this scheme, the new ordinates are given by

(Eal/2d3/2)(Rad)1/2 Eard2Rl/2 p1/2

p1/2

Ead2 Icos4

(10)

where 0 is the phase angle between applied electrode voltage and the electrode current. In terms of the parameters which describe the medium, 0 is given by the arc tangent of (coe)/a where co is the angular frequency in radians per second and e is the permittivity in farads per meter. The results of replotting the data of Fig. 9 in terms of (Ed )/V and of (Ead2)/(I cos C) are shown in Figs. 10 and 1 1

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-22, NO. 6, NOVEMBER 1975

From these families of curves, it is evident that one's choice of electrode geometry may very well depend upon the nature of the energizing source. Despite the geometrical irregularities and tissue inhomogeneities which are often present in biological applications of the twin-disk electrode system, we anticipate that the data as presented in this paper will prove valuable in the preliminary analysis and design of such systems. REFERENCES [11 J. C. Schuder and J. H. Gold, "Localized DC field produced by

diode implanted in isotropic homogeneous medium and exposed to uniform RF field,," IEEE Transactions on Biomedical Engineering, vol. BME-21, pp. 152-163, 1974.

[21

J. H. Gold, H. Stoeckle, J. C. Schuder, J. A. West, and J. A. Holland, "Selective tissue stimulation with microimplant," Transactions American Society for Artificial Internal Organs, vol. 20, pp. 430-435, 1974. [31 P. A. Einstein, "Factors limiting the accuracy of the electrolytic plotting tanks," British Journal ofApplied Physics, vol. 2, pp. 4955, 1951. [41 K. F. Sander and J. G. Yates, "The accurate mapping of electric fields in an electrolytic tank," The Proceedings of the Institution ofElectrical Engineers, vol. 100, pt. II, pp. 167-175, 1953. [51 P. A. Kennedy and G. Kent, "Electrolytic tank, design and applications," The Review of Scientific Instruments, vol. 27, pp. 916927, 1956. [6] E. R. Hartill, J. G. McQueen, and P. N. Robson, "A deep electrolytic tank for the solution of 2- and 3-dimensional field problems in engineering," The Proceedings of the Institution of Electrical Engineers, vol. 104, pt. A, pp. 401-411, 1957.

Computer Classification of Pneumoconiosis from Radiographs of Coal Workers ERNEST L. HALL,

MEMBER, IEEE,

WILLIAM

Abstract-The accurate categorization of profusion of opacities in radiographs of coal workers is a significant medical problem. In this study, the feaslbility of computer classification of profusion was investigated. Standard pattern recognition techniques were used except for the spatial moments which were computed as measurements of the texture patterns. A normal-abnormal classification was performed on 178 zonal samples and resulted in a training classification rate of 99% and a testing rate of 97%. A four category classification was also performed for the zonal samples with a correct classifilcation rate of 84%. The zonal decisions were used to obtain overall film profusion. The results of this classification compared favorably with readings by radiologists. This study provides positive evidence for a quantitative approach to the classification of profusion. The significance of this study with respect to the understanding and measurement of lung pathology from radiographs is that an alternative or supplement to the presently used visual analysis is demonstrated.

COAL WORKERS' PNEUMOCONIOSIS T HE DIAGNOSIS of coal workers' pneumoconiosis (CWP) is dependent on a history of exposure to coal dust and the presence of certain distinctive features on a chest radioManuscript received March 12, 1974; revised October 21, 1974, and March 21, 1975. This work was supported in part by the National Institute of Occupational S fety and Health, the Health Services and Mental Health Administration, the Department of Health, Education, and Welfare under Contract PLD-5527-73, and the National Science Foundation under Grant GK-38308. E. L. Hall was with the Department of Diagnostic Radiology, School of Medicine, Yale University, New Haven, Conn. He is now with the Department of Electrical Engineering, University of Southern California, Los Angeles, Calif. W. 0. Crawford, Jr., is with the Department of Diagnostic Radiology, School of Medicine, Yale University, New Haven, Conn. F. E. Roberts is with the Perkin Elmer Corporation, Norwalk, Conn.

0.

CRAWFORD, JR.,

AND

F. ERIC ROBERTS

graph [1 . The severity of the condition is indicated by the extent, profusion, and character of opacities observable on the chest radiograph. The chest film is not only the most reliable method for diagnosing the disease during life, but also the only means of assessing progression. In addition, there is an almost linear relationship between the coal content of the lungs and the roentgenographic category [2] as determined by the ILO U/C International Classification of Radiographs of Pneumoconiosis.

In 1969, the Federal Coal Mine Health and Safety Act was enacted. One of the provisions of this law required that each of the approximately 130,000 coal miners in the United States be given the opportunity to have a chest roentgenogram that was to be followed by a repeat examination in three years and at five year intervals thereafter. The law also specified that the radiographs were to be coded according to either the ILO 1968 or the U/C Classification [3]. In addition, the Social Security Administration has been reviewing claims for disability due to CWP by approximately 200,000 coal workers or their survivors. The chest radiographs of these claimants are also being interpreted by the ILO U/C system with the compensation decision strongly dependent on the category. Thus the chest radiograph is of paramount importance in the diagnosis, investigation and management of CWP and in awarding compensation due to CWP. The ILO U/C 1971 provides a comprehensive and semiquantitative description of the radiographic changes of all the principal features in the chest. A summary outline of the classification is shown in Table I. The Short Classification is intended primarily for clinical use, while the Complete Classifi-