Intrathoracic measurements
electrical impedance from an esophageal
MARK M. MITCHELL AND RONALD S. NEWBOWER Bioengineering Unit, Department of Anesthesia, Massachusetts General Hospital, Boston, Massachusetts
MITCHELL, MARK M., AND RONALD S. NEWBOWER. Intrathoracic electrical impedance measurements from an esophageal probe. Am. J. Physiol. 236(3): R168-R174, 1979 or Am. J. Physiol.: Regulatory Integrative Comp. Physiol. 5(2): R168R174, 1979.-The sensing of intrathoracic electrical impedance from an esophageal probe may allow relatively noninvasive monitoring of cardiac and respiratory functions of particular interest in anesthesia and intensive care. We have obtained a partial solution of the intrathoracic current-field problem for impedance measurements made from a four-terminal linear array of electrodes located in the esophagus. It allows prediction that aortic root motion will exceed aortic distension as a major determinant of the cardiac intrathoracic esophageal impedance signal. This pl’ediction was confirmed for a specific carefully selected and placed electrode array in anesthetized dogs. In general, motions of organs will be more important than volume changes in affecting the esophageal impedance signal. Thus, timing information (preejection period and left ventricular ejection time) is available from electrodes on an esophageal probe, but cardiac output information appears to be inaccessible for fundamental reasons.
physiological modeling; esophageal electrocardiogram; vascular physiology of the dog; systolic time intervals; ance cardiogram
cardioimped-
probe
02114
objective is to exploit these advantages and to develop measurement techniques applicable to anesthesia monitoring, where placement of esophageal probes is a practical and common strategy. Preliminary results indicated that there are large variations in intrathoracic impedance, as measured from a linear array of electrodes in the esophagus, which are associated with both cardiac and respiratory cycles (X-23). The nature of the cardiac signals suggested that systolic time interval information could be calculated precisely from the combination of impedance and ECG signals. The accessibility of cardiac output information was unclear. This paper presents a theoretical solution of a portion of the intrathoracic current-field problem for electrical impedance measurements made from a linear four-terminal array of electrodes located in the esophagus. We demonstrate the observability of thoracic aortic displacement from the esophageal intrathoracic electrical impedance (EIEI) signal and test the theoretical model in experiments on dogs. Extrapolation from these observations suggests that the timing of cardiac events is accessible but not the related flows. Although useful respiratory information appears to be available from EIEI (21, 22), we do not analyze that situation here. The
THEORETICALMODELANDSOLUTION
there has been extensive interest in measuring the time-varying electrical impedance of The determination of physiological parameters from various regions of the body. Yet the determination of impedance measurements is the inverse of the somewhat important physiological indices from such signals has simpler problem of predicting the observed signal from proven difficult. In particular, there have been many a given anatomic geometry and electrode configuration. attempts to derive cardiac and pulmonary parameters (2, Although this latter problem is still quite complicated, it .i$, .$-2,.p$.)-a -wli ?a ~~~~~L~m~~~*us vw~m-m (.& .lT) -tkcem&ica~~ -f.ias‘b m3iqae -md TmiTrpletely sp3cified -sefrom analysis of the transthoracic impedance signal. But l&ion that can be derived by the appropriate application the various models and theories developed to account for of Maxwell’s equations. Once this solution is obtained, it observed thoracic electrical impedance signals have is necessary to demonstrate that the desired physiological either represented unconvincing simplifications of the parameters will be observable, in the mathematical sense, relevant anatomy (10,ZO) or have defied analytic solution from the predicted signal. The theorems of modern con(4, 6,9). These difficulties have hindered development of trol theory assure us that this will not necessarily be the acceptable physiological measurements based on this case (1). sensing technique. The impedance signal will have a strong mathematical Recently efforts have been made in this laboratory to dependence on the specific electrode placement and on use an esophageal sensing location to circumvent some the inherent anatomic geometry. The desire to be nonof the problems associated with transthoracic impedance invasive severely restricts manipulation of these parammeasurements. An esophageal location offers specific eters. The problem is to discover by inductive logic, technical simplifications. For certain electrode geome- intuition, or by any other means a practical electrode tries, the current flows predominantly through the organs configuration from which the desired information can be of interest and not through or around the thoracic cage. obtained. FO ROVERFORTYY
EARS
0363-6119/79/0000-0000$01.25
Copyright
0
1979 the American
Physiological
Society
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
R169
IMPEDANCE
In general, we are interested in sensing physiological parameters reflecting changes in size, shape, or position of the heart, lungs, or major vessels. Anatomic considerations and exploratory studies (21-23) led us to focus on use of a linear four-terminal array ~2 electrodes placed behind the heart in the portion of the esophagus adjacent and closely parallel to the descending aorta (5). We chose to first isolate and carefully model the interaction between the aorta and the esophageal electrode array, because of its relative simplicity. For purposes of computation, we idealize the relevant anatomy as shown in Fig. 1. The lumen of the aorta is represented by a circular cyclinder of radius R, which is infinite in the axial z dimension. The midline of the esophagus is assumed to be parallel to the axis of the cylinder and displaced from the near wall of its lumen by a distance, A. The conductivity of blood within the cylinder is 02 and the average conductivity of all other intrathoracic contents is ol. B and C describe the spacing of the four electrodes along the midline of the esophagus. The outer pair of electrodes is driven by a IO-kHz constant-current source, The resultant voltage is sensed with the remaining inner pair of electrodes. The intrathoracic impedance is determined by the ratio of the voltage to the current, This impedance can be derived directly by the application of Maxwell’s equations and by solution using a conformal transformation and the method of images. Details are presented in the APPENDIX. The result is that
i
L x
-SECTION
MIDLINE Of ESOPHAGUS
Of
AORTA
1
FIG. 1. Idealized geometry of descending aorta and four electrodes of an esophageal electrical impedance probe. Conductivity of blood is m2 and average conductivity of intrathoracic tissue is 01.
Z-OHMS 65.00
-r
60.00
z=&[fi+(=J .
55.00 LEFT TO R 0.01 R 0.1 R 0.5 R 0.8 R 1.0 R 2.0 R INFlNlTY
50.00
1
(1)
( d(2A + A2/R)2 + (B + C)” 1 J(~A + A2/R)2 + (B - c)”
45.00
)I
where 2 is the esophageal intrathoracic electrical impedance, in Q; A is the distance of near wall of aortic lumen from midline of esophagus, in cm; R is the radius of aortic lumen, in cm; 2B is the spacing between outer electrodes driven by current source, in cm; 2C is the spacing between inner electrodes from which voltage is measured, in cm; ol is the average conductivity of intrathoracic contents, in l/%cm; and 02 is the conductivity of blood within aortic lumen, in I/% cm, Figure 2 graphically represents evaluation of the above solution for the experimentally employed values of electrode spacing, B and C, with appropriate values for the conductivity of dog blood, 02 = l/(155 Qmcm) (7), and the average conductivity of the surrounding intrathoracic tissue, 01= l/(600 Q cm). The value for ~1is determined by computation from the base-line EIEI, knowledge of CT~, reasonable estimates of A and R, and known values of B and C. The derived value is consistent with estimates based on invasive measurements (21). From anatomic and ultrasound studies, we find that, for a dog of 25-32 kg, the aortic radius, R, is about 0.8 cm and AR, the change in aortic radius with left ventricular ejection, is 0.04-0.08 cm (14, 16). Furthermore, the distance of the near wall of the aortic lumen from the l
40.00 35.00
RIGHT
30 _00
t-+--1 .50 2.00 A-CENTIMETERS esophageal electrical impedance, (Z), as a funcdisplacement of near wall of aortic lumen from geometric parameters of electrode array (B oI and oz. For this plot, B = 2 cm, C = I cm, 02 = l/(155 %cm).
0.50 FIG. 2. Intrathoracic tion of aortic radius (R), esophageal midline (A), and C), and conductivities a-cm), and (31 = l/(600
1.00
midline of the esophagus, A, is about 0.5 cm and AA, the change in A with ejection, has a negative value of about 0.1 cm (14, 18). The model predicts that the change in EIEI due to AR = 0.08 cm will be -0.41 fi and that due -0.1 cm will be -5.08 Q (see Fig. 2). Because of toAA= the nonlinearity of Eq. 1, the total change in EIEI for both effects simultaneously will not be exactly the sum of these changes, but will be -5.41 CL Notice that the predicted change in EIEI for AA is more than 10 times as large as for AR. This indicates that the major determinant of the impedance signal associated with ejection will be aortic motion, and not aortic distention. Because of the specific nonlinear form of the solution, this would be true even if AR were considerably larger than AA.
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R170
Mm M.
It is important to note that this conclusion is not sensitive to such characteristics of the geometry as the cylindrical shape of the aorta. In general, changes in the proximity of any volume conductor to a four-terminal linear array of electrodes should influence the impedance signal more than similar changes in size of the conductor, EXPERIMENTAL
METHODS
AND
RESULTS
EKG
FOR
19
REFERENCE
ELECTRODES
IMPEDANCE
FOR (1
AND
R. S. NEWBOWER
former isolation. The resultant potentials were measured with a Princeton Applied Research lock-in amplifier model 128, followed by a Hewlett-Packard model 8802A medium-gain amplifier. Femoral and aortic pressures were measured with Hewlett-Packard model 880% pressure amplifiers and model 1280 transducers. HewlettPackard model 881lB electrocardiograph amplifiers were employed for both the standard lead II and esophageal electrocardiograms. All data were recorded on a HewlettPackard model 7758A chart recorder and a model 3900 FM magnetic tape system. Radiographs were taken to document the positions of the esophageal probe and the aortic root catheter. Relative position of the esophageal probe could also be determined from the esophageal electrocardiogram. This was recorded sequentially from each of the first 19 esophageal probe electrodes, using the 20th electrode, located in the neck, as the reference. A typical series of electrocardiograms obtained in this fashion is shown in Fig. 4. It demonstrates a positive P wave and QRS complex for electrodes near the diaphragm and progressive inversion of the polarity of the P wave and the QRS complex as the sensing electrode moves up the probe. Those with higher numbers, located in the neck, have completely negative P waves and QRS complexes. With l-cm spacings, a unique electrode could always be found from which the ECG recording demonstrated both a biphasic P wave and biphasic QRS. The esophageal probe could be repositioned reliably to within 0.5 cm of this reference position, using only the electrocardiogram from a fiducial electrode as a guide. Once the probe was placed in that fashion, a tetrapolar array of impedance electrodes was selected, so that the proximal voltage sensing electrode was the fiducial ECG electrode and the other three were spaced 1 cm proximal and 2 and 3 cm distal to it. The chosen array was verified by x ray to be located adjacent to the middle third of the descending thoracic aorta, the region in which the esophagus is in closest proximity to the aorta. This was also
The measurements of EIEI that are presented here were obtained from four mongrel dogs, weighing between 25 and 32 kg, anesthetized with intravenous thiopental. The induction dose of 10 mg/kg was supplemented as necessary to maintain adequate surgical anesthesia. After tracheal intubation, the dogs were ventilated with room air at a tidal volume of 600 ml from a Harvard animal ventilator. The femoral artery was cannulated and arterial pressure was monitored. In addition, a 7-French pressure-measurement catheter was placed through a cutdown in the neck into the right carotid artery and passed retrograde into the aortic root, just distal to the aortic valve. The pulse propagation delay for this catheter was previously measured to be less than 5 ms, The aortic root pressure was used as a reference for detecting the onset and end of left ventricular ejection. A lead II electrocardiogram was obtained from needle electrodes in the limbs. An esophageal electrical impedance probe was fabricated from an Argyle no, 18 Salem sump tube with 20 electrodes of Emerson and Cumming, Eccobond 57C conductive epoxy, as shown in Fig. 3. It was passed down the esophagus so that the most distal electrode, designated no. 1, was positioned at the level of the diaphragm. The electrodes were connected through a switch box in such a way that any electrode could function as a driven electrode or as a sensing electrode for the impendance measurement and/or as a sensing electrode for the esophageal electrocardiogram. The lOO+A, lo-kHz constant-current source for the impedance measurements consisted of a 400-kti series resistance driven by a Hewlett-Packard 242A oscillator through transELECTRODE
MITCHELL
CM
.
ELECTRICAL
CENTERLINE
SPACING)
4
FIG. 3. Experimental probe for measurement electrical impedance ographic potentials.
EKG RECORDER
’
esophageal of intrathoracic and electrocardi-
A SWITCH 60X
I
’ c
10 KHZ CONSTANT CURRENT SOURCE
ELEC TRJCAL IMPE DANCE MEAS UREMENT
1 PROBE IS FABRICATED #18 NASO-GASTRIC CONDUCTIVE EPOXY
FROM TUBE WITH ELECTRODES
i
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
IMPEDANCE
proximity to the descending aorta. No direct measurements of aortic displacement and distension were made in our work. However, given previous studies of these variables (14, 16, 18), it is reasonable to expect that they will closely follow aortic root pressure. The impedance signal observed in our work is also very similar in time dependence to the aortic root pressure. Given the expetted amplitudes of aortic displacement and distension for the blood pressures observed (14, 16), our theoretical model predicts an impedance change, associated primarily with aortic displacement, consistent with the observed 4.X2 amplitude. Taken together, all these observations support our inference that, for this electrode array position, the EIEI signal is primarily a product of aortic displacement due to left ventricular ejection. Other electrode arrays and locations give signals whose shapes are suggestive of ventricular and atria1 volume and position changes, and of valvular motions. A complete catalog of these is complex and beyond the scope of this paper, However, the physical principles of sensing other volume conductors from a linear array of small electrodes will be the same and relative motion with
LlJA~~~
&A&&
4
9
14
19
0.5
R171
SEC
4. Esophageal electrocardiograms from a dog. Trace I is from electrode 1 located at level of diaphragm. Subsequent tracings are taken in sequence from electrodes of corresponding numbers. Reference electrode is, in all cases, eZe&ode 20 located in the neck. FIG.
confirmed by anatomic dissection after the animal was killed. The probe’s location is indicated in Fig. 5. The outer electrodes were driven by the current source and the inner pair used to sense the derived voltage, using the Princeton Applied Research lock-in amplifier. This amplifier measured the component of the voltage signal that was in phase with the driving current. The result was used to compute the real or resistive component of the impedance. The phase shift of the voltage signal due to capacitive-reactance effects was generally negligible and was not used in determining the impedance. Figure 6 shows a typical signal measured in the manner just described. The peak-to-peak amplitude of the cardiac-related modulation of the impedance signal is 4.5 0, which is 10% of the base-line impedance. There are two sharp, easily identified inflection points coincident with the onset and termination of left ventricular ejection, and the shape of the impedance wave form is very similar to that of the aortic pressure. The systolic time intervals can be derived solely from the esophageal elecFIG. 5. A line drawing made from a right lateral chest x ray of a dog Elecp. L-L!,a~~-ao~~-~rQ~t~~~r. 0 )--in&e. - trwr&egrx -MC! -the -~-I-E-I-s&al --z&s -&own -& --FJg.4. --Wikk!-SSophpIg9~LpEQb I-20, starting at distal end. Electrodes The preejection period for this dog was 80 ms and the trodes on probe are numbered 7, 8, 10, and 11 (heavy shaded) are used for EIEI measurement. left ventricular ejection time was 170 ms with a heart Electrocardiogram is sensed between electrodes 10 and 20, separate ---~-_--~ ~~-~~~ rata-&-g@ vr -&&&&$T~?,p G-Y-CIWW Qa_ rra- Jlinlc2__03?a - rvAr&alr l&VI zLr~~z& _ -refe-en~~-&Qrt~&~ &v&&-+rr-t~le-rr&;~e~a~~~n~~~p~~~~~~~~-~ro~e -v rrrithinm__n_rl\rmalm limits for the dog (19). and thoracic aorta (3), heart (4), lungs (5), and some of the pulmonary DISCUSSION
The electrode array employed in this study was positioned in that portion of the esophagus which is in close
arteries and veins (6) are shown. Also indicated are superior vena cava (7), sternum (8), spine (9), and tangential edge of the diaphragm (16). Most distal electrode is located at level at which esophagus traverses diaphragm and only appears to be subdiaphragmatic here because of curvature of diaphragm.
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- =~~
_
R172
M.
ElECTROCARDIOGRAM
AORTICPRESSURE
ESOPHAGEAL IMPEDANCE
0.2
0.4
0.6
0.8 1.0 SECUNCS
FIG, 6. Esophageal electrocardiogram (from lead 10 with reference electrode 20), aortic root pressure, and intrathoracic esophageal electrical impedance (from electrodes 7, 8, 10, and I I) recorded frGm a dog during apnea at functional residual capacity. Vertical lines mark onset of QRS complex and beginning and end of left ventricular ejection. Preejection period and left ventricular ejection time are determined to be 80 and 170 ms, respectively. Base-line intrathoracic esophageal change in electrical impedance 20, is 46.5 s1 and the peak-to-peak impedance with left ventricular ejection is 4.5 G2,
respect to the esophagus will, in general, be more important than diameter or volume changes in determining impedance effects. Thus, in general, timing information will be more readily accessible than volume information, no matter what the esophageal location of the electrodes. Figure 6 demonstrates how the preejection period and left ventricular ejection time can be determined solely from the esophageal electrocardiogram and the appropriate fiducial points of the EXE1 signal. The aortic root pressure tracing gives us an independent determination of the onset and end of left ventricular ejection and was used to validate the systolic time intervals as determined from the electrical measurements. In the dog, we found that the appropriate, easily identifiable, aortic signal could be obtained despite a 05cm variation in the longitudinal position of the electrode array within the esophagus. We can expect that in the human subject a similar impedance signal will be obtainable over a larger range of electrode array positions because the human descending thoracic aorta and esophagus are more nearly parallel over longer segments of their lengths and all relevant dimensions are greater. The fractional changes in impedance with cardiac function are much larger with esophageal electrodes than with surface electrodes. The fractional change in thoracic impedance with left ventricular ejection is generally on the order of 0.2% when measured from the surface (12). When the esophageal electrode configuration described here is used, the fractional change is 540%. This results in a much improved signal-to-noise ratio and makes the recording and processing of the impedance signal easier and more reliable. It appears that shunting of current
M.
MITCHELL
AND
R. S. NEWBOWER
fiow through the thoracic wall and other structures that are not of interest is minimized (al).’ In general, there are several practical reasons for exploring the placement of an electrode array on an esophageal probe for the measurement of variations in intrathoracic electrical impedance. The location is one that is generally available. The esophageal stethoscope is commonly employed during general anesthesia. It could be replaced by a simple multifunction probe capable of monitoring not only heart and breath sounds, but also esophageal temperature, electrocardiogram, and electrical impedance (21, 22). In the intensive-care setting, it would be feasible to instrument the esophagus of a crititally ill patient with such a probe in place of a routine nasogastric tube. We are hopeful that by extension of the modeling techniques described, an even better understanding of the nature of the origins of the intrathoracic electrical impedance signal can be obtained. Although it seems unlikely that an esophageal electrical impedance signal can be found whose major determinant is stroke volume or cardiac output, we are encouraged that at least systolic time intervals and possibly other useful physiological parameters can be reliably determined in such a simple fashion. APPENDIX Details
ofMethod
of Solution
of the Model
The exact solution of Maxwell’s equations for our model of the esophageal electrode array and the thoracic aorta can be obtained as follows. The simplified geometry assumed is shown in Fig. 1, where the cylinder is assumed to be infinite in the z dimension. We assume that the materials are isotropic, nondispersive, and independent of other parameters such as temperature and pressure (8). Furthermore, materials outside the aorta and inside the aorta are assumed to be identical except for their conductivities, which are CT]
and~ respectivelywhere~~~ o, 2,
We seek a stationary, quasi-ztatic solution (8). In practice, the measurements are made at 10 kHz, but this frequency is sufficiently low to allow treating this as a static problem. I3y conservation of charge
@ v-J=-z=o where J is the electric current density, charge density, in C/m’. Furthermore, constitutive relations become
in amperes; Maxwell’s
(Al) and p is the electric equations and the
VxE=O
M2)
VxH=J
L43)
v-D=0
(A4)
V*B = 0
b45)
J=uE where
--
e is the electric
field,
in V/m;
H is the magnetic
(A@ field,
in A/m;
~-
’ It is difficult to estimate what contribution the concepts described in this paper will make to understanding surface recordings from circumferential band electrodes. The small amplitude of the cardiac signal as observed from the surface implies substantial cancellation of opposing effects and significant shunting of currents. The physics modeled in this paper may, however, apply to a certain extent to arrays of spot electrodes on the surface, or to band electrodes viewed as a linear superposition of arrays of spot electrodes.
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
D is the electric displacement, in C/m2; B is the magnetic in Wb/m2; and 0 is the conductivity, in 1/Q* m. Therefore, if
a, the scalar
electric
at the sources, V-J=
potential
function,
then
(A81
I [6(x
2 - B) - 8(x + A, y, z + B)]
n*(Jp is continuous
an impulse at the point (x’, y’, at the surface between regions
- J]) = 0 and there
: j
V
1
1 .
t-c
I
I B
/
of surface (All)
is, E tangential is continuous Clearly, this system is equivalent to an electrostatic problem of the same geometry if we replace 0 by E, J by D, and I by 4, where I is the current source magnitude, q is a charge, and l is the electrical permittivity. So we can use the method of images to solve this problem (15). The geometry in the (x, y, z) coordinate system is still intractable. This can be circumvented by employing a conformal transformation (31. Let W = x + iy, and T = u + iv, where i is the square root of -1; then a conformal transformation W = F(T) is a regular function which maps tu, u) + (x, y) in such a way that angles are preserved, and therefore so is the flux through corresponding areas and potentials at corresponding points (3). The conformal transformation of use in this application is of the bilinear form, generally
where, a, b, c, and d are complex employed is
+ d)
constants.
W2) The
specific
transform
W = (AZ/T)
W3)
of the near wall of the aortic lumen as shown in Fig. 7. This is equivalent + u2);
y = -A’v/(u”
from to
+ f));
v = -y/(A2($
l
-
+ v2)
7
I
-
FIG. 8, Method of images used to solve Maxwell’s equation for transformed geometry. Source I and -I are reflected with appropriate scale factor (02 - aJ/(oz + oJ, in plane perpendicular to u-axis through u = A’/2R.
frame. We need to derive an expression for the difference in potential between the two sensing points as a function of the geometry parameterized by A, R, B, C, and the current source magnitude, I. This requires establishing the two image sources shown in Fig, 8. The potential for a point source at the origin is (3)
the
and inversely u = x/(A2(;1c2
0;'
(AlO)
is no accumulation
that
W = (aT + b)/(cT
+C
Ir +
(A9
Ii x (E, - El) = 0
x = A2u/(u2
+B
.
+ A,y,
where A is the distance midline of the esophagus
I
l +
0;:
where
where A(#‘, y’, z’) is the Dirac function, z’) = (0, 0, 0). The boundary conditions of conductivity u1 and uz are
that is, J normal charge and
T
L47)
va=o except
7
flux density,
VQe-E defines
R173
IMPEDANCE
@ b, y, 21 = Therefore we find that region of interest is
I
(A141
4 7m Jx* + y2 + 2
by superposition,
the total
potential
for the
+ s;-‘,,
The transformation (x, y, z) t (u, v, z) is also conformal, and preserves angles, flux densities, and potentials. The interior of the aorta becomes the right half-space, u < A2/2R as shown in Fig. 7. We can then proceed with solution of the problem in the transformed
if, =-
The
1
I 4m
d(u
sensed
voltage
+ A)2 + v2 + (z - B)’
-1 + Jju+A)“-+rr’-q;-+B)’
is c
7. Bilinear conformal transformation, W = (AZ/Z’), which maps geometry (x, y) on Left into geometry (u, u) on right, and preserves angles of intersection and flux densities through corresponding areas.
v=-
FIG,
&dz
= a, (-A,
0, C) - 0 (-A,
I -c
Remembering
that
0, -C)
(Al61
B > C > 0, we find that
Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (129.186.138.035) on January 11, 2019.
RI74
M. @ (-A,
0, C) = 4
(-A,
0, -C)
(A171
and therefore V= Because
2 = V/I,
the solution
2 if, (-A,
0, C)
(Af@
is
M.
MITCHELL
AND
R. S. NEWBOWER
Special thanks for help and guidance are given to Professor Jin Au Kong of the Massachusetts Institute of Technology Electrical Engineering Department and to Drs, Jeffrey B. Cooper, Michael C. Long, Richard J. Kitz, and James H. Philip, and to Mr. Josh Tolkoff of the Anesthesia Department at the Massachusetts General Hospital. This work was supported in part by Grant GM-15904 from the National Institute of General Medical Science, The MACSYMA facilities of the Massachusetts Institute of Technology Mathlab group (supported by the Defense Advanced Research Projects Agency work order 2095 under the Office of Naval Research Contract NOOO14-75-C 0661) were used to correct and verify the mathematics used herein. M. M. Mitchell acknowledges support, during the course of this work, by the Anesthesia Research Training Grant GM-01273. His present address: Dept. of Anesthesiology, University of California, San Diego, San Diego, California 92103. Received
19 December
1977; accepted
in final
form
29 September
1978.
REFERENCES 1. ATHANS, M., AND P, L. FALB. Optimal ControZ. New York: MCGraw, 1966. 2. BAKER, L. E., AND L. A. GEDDES. The measurement of respiratory volumes in animals and man with use of electrical impedance, Ann, NY Acad, Sci, 170: 667-668, 1970. 3. BINNS, K. J., AND R. J. LAWRENSON. Analysis and Computation of Electric Mugnetic Field Problems. London: Pergamon, 1963. 4. BONJER, F. H., J. W. VAN DEN BERG, AND M. N. J. DIRKEN. The origin of the variations of body impedance occurring during the cardiac cycle. Circulation 6: 415-420, 1952. 5. BRADLEY, 0. C., AND T. GRAHAME. Topographical Anatomy of the Dog. London: Oliver & Boyd, 1948. 6, COOLEY, W. L. The parameters of transthoracic electrical conduction. Ann. NY Acad. Sci. 1970: 702-713, 1970. 7. GOLLAN, F., AND R. NAMON. Electrical impedance of pulsatile blood flow in rigid tubes and in isolated organs. Ann. NY Acad. Sci. 170: 568-576, 1970. 8. KONG, J. A. Theory of Electra-Magnetic Waves. New York: Wiley, 1975. 9. KROHN, B, G., E. F. DUNNE, H. HANLSH, 0. MAGIDSON, AND J. H. KAY. T,he basis of the electrical impedance cardiogram. Ann. NY Acad. Sci. 170: 714-723, 1970. 10. KUBICEK, W. G., R. P, PATTERSON, AND D. A. WITSOE. Impedance cardiography as a non-invasive method of monitoring cardiac function and other parameters of the cardiovascular system. Ann. NY Acad, Sci. 170: 724-732, 1970, 11. LUEPKER, R. V., J. R. MICHAEL, AND J. R. WARBASSE. Transthoracic electrical impedance: quantitative evaluation of a non-invasive measure of thoracic fluid volume. Am. Heart J, 85: 83-93, 1973. 12. NYBOER, J. Electrical Impedance Plethysmography. Springfield, IL: Thomas, 1959. 13. NYBOER, J., S. BAGNO, A. BARNETT, AND R. H, HALSEY. Radiocar-
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21. 22.
23.
diograms-the electrical impedance changes of the heart in relation to electrocardiograms and heart sounds. Am. Sot. Chin. Invest. 19: 773, 1940. OLSON, R. M., AND D. K. SHELTON. A nondestructive technique to measure wall displacement in the thoracic aorta. J. Appl. Physiol. 32: 147-151, 1972. PANOFSKY, W. K., AND M. PHILLIPS. Classical Electricity and Magnetism. London: Addison-Wesley, 1955. PATEL, D. J., F. M. DEFREITAS, J, C. GREENFIELD, AND D. L. FRY. Relationship of radius to pressure along the aorta in living dogs. J. AppL Physiol. 18: llll-1117, 1963. POMERANTZ, M., F. DELGADO, AND B. EISEMAN. Clinical evaluation of transthoracic electrical impedance as a guide to intrathoracic fluid volumes, Ann Surg. 171: 686-691, 1970. PRATT, R. C., A. F. PARISI, J. J, HARRIXGTON, AND A. A. SASAHARA, The influence of left ventricular stroke volume on aortic root motion. Circulation 53: 947-953, 1976. RASMUSSEN, J, P., B. SORENSEN, AND T. KANN. Evaluation of impedance cardiography as a non-invasive means of measuring systolic time intervals and cardiac output. Acta Anaesthesiol. Stand. 19: 210-218, 1975, SOVA, J. Cardiac rheometry: impedance plethysmography of the human trunk as a method for measurement of stroke volume and cardiac output. Ann. NY Acad. Sci. 1970: 577-593, 1970. SCORDATO, R. Thoracic Impedance Measurements from the &zophagus (thesis). Boston, MA: MIT, 1975. SCORDATO, R. E., J. B. COOPER, R. S.NEWBOWER, AND M. C. LONG, Esophageal electrical impedance probe (Abstract). Proc. Ann. AAMI Meeting Atlanta 10: 71, 1976. WEAVER, L. Measurement of the Be-Ejection Period from the Esophagus (thesis). Boston, MA: MIT, 1976.
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electrical impedance from an esophageal
MARK M. MITCHELL AND RONALD S. NEWBOWER Bioengineering Unit, Department of Anesthesia, Massachusetts General Hospital, Boston, Massachusetts
MITCHELL, MARK M., AND RONALD S. NEWBOWER. Intrathoracic electrical impedance measurements from an esophageal probe. Am. J. Physiol. 236(3): R168-R174, 1979 or Am. J. Physiol.: Regulatory Integrative Comp. Physiol. 5(2): R168R174, 1979.-The sensing of intrathoracic electrical impedance from an esophageal probe may allow relatively noninvasive monitoring of cardiac and respiratory functions of particular interest in anesthesia and intensive care. We have obtained a partial solution of the intrathoracic current-field problem for impedance measurements made from a four-terminal linear array of electrodes located in the esophagus. It allows prediction that aortic root motion will exceed aortic distension as a major determinant of the cardiac intrathoracic esophageal impedance signal. This pl’ediction was confirmed for a specific carefully selected and placed electrode array in anesthetized dogs. In general, motions of organs will be more important than volume changes in affecting the esophageal impedance signal. Thus, timing information (preejection period and left ventricular ejection time) is available from electrodes on an esophageal probe, but cardiac output information appears to be inaccessible for fundamental reasons.
physiological modeling; esophageal electrocardiogram; vascular physiology of the dog; systolic time intervals; ance cardiogram
cardioimped-
probe
02114
objective is to exploit these advantages and to develop measurement techniques applicable to anesthesia monitoring, where placement of esophageal probes is a practical and common strategy. Preliminary results indicated that there are large variations in intrathoracic impedance, as measured from a linear array of electrodes in the esophagus, which are associated with both cardiac and respiratory cycles (X-23). The nature of the cardiac signals suggested that systolic time interval information could be calculated precisely from the combination of impedance and ECG signals. The accessibility of cardiac output information was unclear. This paper presents a theoretical solution of a portion of the intrathoracic current-field problem for electrical impedance measurements made from a linear four-terminal array of electrodes located in the esophagus. We demonstrate the observability of thoracic aortic displacement from the esophageal intrathoracic electrical impedance (EIEI) signal and test the theoretical model in experiments on dogs. Extrapolation from these observations suggests that the timing of cardiac events is accessible but not the related flows. Although useful respiratory information appears to be available from EIEI (21, 22), we do not analyze that situation here. The
THEORETICALMODELANDSOLUTION
there has been extensive interest in measuring the time-varying electrical impedance of The determination of physiological parameters from various regions of the body. Yet the determination of impedance measurements is the inverse of the somewhat important physiological indices from such signals has simpler problem of predicting the observed signal from proven difficult. In particular, there have been many a given anatomic geometry and electrode configuration. attempts to derive cardiac and pulmonary parameters (2, Although this latter problem is still quite complicated, it .i$, .$-2,.p$.)-a -wli ?a ~~~~~L~m~~~*us vw~m-m (.& .lT) -tkcem&ica~~ -f.ias‘b m3iqae -md TmiTrpletely sp3cified -sefrom analysis of the transthoracic impedance signal. But l&ion that can be derived by the appropriate application the various models and theories developed to account for of Maxwell’s equations. Once this solution is obtained, it observed thoracic electrical impedance signals have is necessary to demonstrate that the desired physiological either represented unconvincing simplifications of the parameters will be observable, in the mathematical sense, relevant anatomy (10,ZO) or have defied analytic solution from the predicted signal. The theorems of modern con(4, 6,9). These difficulties have hindered development of trol theory assure us that this will not necessarily be the acceptable physiological measurements based on this case (1). sensing technique. The impedance signal will have a strong mathematical Recently efforts have been made in this laboratory to dependence on the specific electrode placement and on use an esophageal sensing location to circumvent some the inherent anatomic geometry. The desire to be nonof the problems associated with transthoracic impedance invasive severely restricts manipulation of these parammeasurements. An esophageal location offers specific eters. The problem is to discover by inductive logic, technical simplifications. For certain electrode geome- intuition, or by any other means a practical electrode tries, the current flows predominantly through the organs configuration from which the desired information can be of interest and not through or around the thoracic cage. obtained. FO ROVERFORTYY
EARS
0363-6119/79/0000-0000$01.25
Copyright
0
1979 the American
Physiological
Society
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
R169
IMPEDANCE
In general, we are interested in sensing physiological parameters reflecting changes in size, shape, or position of the heart, lungs, or major vessels. Anatomic considerations and exploratory studies (21-23) led us to focus on use of a linear four-terminal array ~2 electrodes placed behind the heart in the portion of the esophagus adjacent and closely parallel to the descending aorta (5). We chose to first isolate and carefully model the interaction between the aorta and the esophageal electrode array, because of its relative simplicity. For purposes of computation, we idealize the relevant anatomy as shown in Fig. 1. The lumen of the aorta is represented by a circular cyclinder of radius R, which is infinite in the axial z dimension. The midline of the esophagus is assumed to be parallel to the axis of the cylinder and displaced from the near wall of its lumen by a distance, A. The conductivity of blood within the cylinder is 02 and the average conductivity of all other intrathoracic contents is ol. B and C describe the spacing of the four electrodes along the midline of the esophagus. The outer pair of electrodes is driven by a IO-kHz constant-current source, The resultant voltage is sensed with the remaining inner pair of electrodes. The intrathoracic impedance is determined by the ratio of the voltage to the current, This impedance can be derived directly by the application of Maxwell’s equations and by solution using a conformal transformation and the method of images. Details are presented in the APPENDIX. The result is that
i
L x
-SECTION
MIDLINE Of ESOPHAGUS
Of
AORTA
1
FIG. 1. Idealized geometry of descending aorta and four electrodes of an esophageal electrical impedance probe. Conductivity of blood is m2 and average conductivity of intrathoracic tissue is 01.
Z-OHMS 65.00
-r
60.00
z=&[fi+(=J .
55.00 LEFT TO R 0.01 R 0.1 R 0.5 R 0.8 R 1.0 R 2.0 R INFlNlTY
50.00
1
(1)
( d(2A + A2/R)2 + (B + C)” 1 J(~A + A2/R)2 + (B - c)”
45.00
)I
where 2 is the esophageal intrathoracic electrical impedance, in Q; A is the distance of near wall of aortic lumen from midline of esophagus, in cm; R is the radius of aortic lumen, in cm; 2B is the spacing between outer electrodes driven by current source, in cm; 2C is the spacing between inner electrodes from which voltage is measured, in cm; ol is the average conductivity of intrathoracic contents, in l/%cm; and 02 is the conductivity of blood within aortic lumen, in I/% cm, Figure 2 graphically represents evaluation of the above solution for the experimentally employed values of electrode spacing, B and C, with appropriate values for the conductivity of dog blood, 02 = l/(155 Qmcm) (7), and the average conductivity of the surrounding intrathoracic tissue, 01= l/(600 Q cm). The value for ~1is determined by computation from the base-line EIEI, knowledge of CT~, reasonable estimates of A and R, and known values of B and C. The derived value is consistent with estimates based on invasive measurements (21). From anatomic and ultrasound studies, we find that, for a dog of 25-32 kg, the aortic radius, R, is about 0.8 cm and AR, the change in aortic radius with left ventricular ejection, is 0.04-0.08 cm (14, 16). Furthermore, the distance of the near wall of the aortic lumen from the l
40.00 35.00
RIGHT
30 _00
t-+--1 .50 2.00 A-CENTIMETERS esophageal electrical impedance, (Z), as a funcdisplacement of near wall of aortic lumen from geometric parameters of electrode array (B oI and oz. For this plot, B = 2 cm, C = I cm, 02 = l/(155 %cm).
0.50 FIG. 2. Intrathoracic tion of aortic radius (R), esophageal midline (A), and C), and conductivities a-cm), and (31 = l/(600
1.00
midline of the esophagus, A, is about 0.5 cm and AA, the change in A with ejection, has a negative value of about 0.1 cm (14, 18). The model predicts that the change in EIEI due to AR = 0.08 cm will be -0.41 fi and that due -0.1 cm will be -5.08 Q (see Fig. 2). Because of toAA= the nonlinearity of Eq. 1, the total change in EIEI for both effects simultaneously will not be exactly the sum of these changes, but will be -5.41 CL Notice that the predicted change in EIEI for AA is more than 10 times as large as for AR. This indicates that the major determinant of the impedance signal associated with ejection will be aortic motion, and not aortic distention. Because of the specific nonlinear form of the solution, this would be true even if AR were considerably larger than AA.
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R170
Mm M.
It is important to note that this conclusion is not sensitive to such characteristics of the geometry as the cylindrical shape of the aorta. In general, changes in the proximity of any volume conductor to a four-terminal linear array of electrodes should influence the impedance signal more than similar changes in size of the conductor, EXPERIMENTAL
METHODS
AND
RESULTS
EKG
FOR
19
REFERENCE
ELECTRODES
IMPEDANCE
FOR (1
AND
R. S. NEWBOWER
former isolation. The resultant potentials were measured with a Princeton Applied Research lock-in amplifier model 128, followed by a Hewlett-Packard model 8802A medium-gain amplifier. Femoral and aortic pressures were measured with Hewlett-Packard model 880% pressure amplifiers and model 1280 transducers. HewlettPackard model 881lB electrocardiograph amplifiers were employed for both the standard lead II and esophageal electrocardiograms. All data were recorded on a HewlettPackard model 7758A chart recorder and a model 3900 FM magnetic tape system. Radiographs were taken to document the positions of the esophageal probe and the aortic root catheter. Relative position of the esophageal probe could also be determined from the esophageal electrocardiogram. This was recorded sequentially from each of the first 19 esophageal probe electrodes, using the 20th electrode, located in the neck, as the reference. A typical series of electrocardiograms obtained in this fashion is shown in Fig. 4. It demonstrates a positive P wave and QRS complex for electrodes near the diaphragm and progressive inversion of the polarity of the P wave and the QRS complex as the sensing electrode moves up the probe. Those with higher numbers, located in the neck, have completely negative P waves and QRS complexes. With l-cm spacings, a unique electrode could always be found from which the ECG recording demonstrated both a biphasic P wave and biphasic QRS. The esophageal probe could be repositioned reliably to within 0.5 cm of this reference position, using only the electrocardiogram from a fiducial electrode as a guide. Once the probe was placed in that fashion, a tetrapolar array of impedance electrodes was selected, so that the proximal voltage sensing electrode was the fiducial ECG electrode and the other three were spaced 1 cm proximal and 2 and 3 cm distal to it. The chosen array was verified by x ray to be located adjacent to the middle third of the descending thoracic aorta, the region in which the esophagus is in closest proximity to the aorta. This was also
The measurements of EIEI that are presented here were obtained from four mongrel dogs, weighing between 25 and 32 kg, anesthetized with intravenous thiopental. The induction dose of 10 mg/kg was supplemented as necessary to maintain adequate surgical anesthesia. After tracheal intubation, the dogs were ventilated with room air at a tidal volume of 600 ml from a Harvard animal ventilator. The femoral artery was cannulated and arterial pressure was monitored. In addition, a 7-French pressure-measurement catheter was placed through a cutdown in the neck into the right carotid artery and passed retrograde into the aortic root, just distal to the aortic valve. The pulse propagation delay for this catheter was previously measured to be less than 5 ms, The aortic root pressure was used as a reference for detecting the onset and end of left ventricular ejection. A lead II electrocardiogram was obtained from needle electrodes in the limbs. An esophageal electrical impedance probe was fabricated from an Argyle no, 18 Salem sump tube with 20 electrodes of Emerson and Cumming, Eccobond 57C conductive epoxy, as shown in Fig. 3. It was passed down the esophagus so that the most distal electrode, designated no. 1, was positioned at the level of the diaphragm. The electrodes were connected through a switch box in such a way that any electrode could function as a driven electrode or as a sensing electrode for the impendance measurement and/or as a sensing electrode for the esophageal electrocardiogram. The lOO+A, lo-kHz constant-current source for the impedance measurements consisted of a 400-kti series resistance driven by a Hewlett-Packard 242A oscillator through transELECTRODE
MITCHELL
CM
.
ELECTRICAL
CENTERLINE
SPACING)
4
FIG. 3. Experimental probe for measurement electrical impedance ographic potentials.
EKG RECORDER
’
esophageal of intrathoracic and electrocardi-
A SWITCH 60X
I
’ c
10 KHZ CONSTANT CURRENT SOURCE
ELEC TRJCAL IMPE DANCE MEAS UREMENT
1 PROBE IS FABRICATED #18 NASO-GASTRIC CONDUCTIVE EPOXY
FROM TUBE WITH ELECTRODES
i
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
IMPEDANCE
proximity to the descending aorta. No direct measurements of aortic displacement and distension were made in our work. However, given previous studies of these variables (14, 16, 18), it is reasonable to expect that they will closely follow aortic root pressure. The impedance signal observed in our work is also very similar in time dependence to the aortic root pressure. Given the expetted amplitudes of aortic displacement and distension for the blood pressures observed (14, 16), our theoretical model predicts an impedance change, associated primarily with aortic displacement, consistent with the observed 4.X2 amplitude. Taken together, all these observations support our inference that, for this electrode array position, the EIEI signal is primarily a product of aortic displacement due to left ventricular ejection. Other electrode arrays and locations give signals whose shapes are suggestive of ventricular and atria1 volume and position changes, and of valvular motions. A complete catalog of these is complex and beyond the scope of this paper, However, the physical principles of sensing other volume conductors from a linear array of small electrodes will be the same and relative motion with
LlJA~~~
&A&&
4
9
14
19
0.5
R171
SEC
4. Esophageal electrocardiograms from a dog. Trace I is from electrode 1 located at level of diaphragm. Subsequent tracings are taken in sequence from electrodes of corresponding numbers. Reference electrode is, in all cases, eZe&ode 20 located in the neck. FIG.
confirmed by anatomic dissection after the animal was killed. The probe’s location is indicated in Fig. 5. The outer electrodes were driven by the current source and the inner pair used to sense the derived voltage, using the Princeton Applied Research lock-in amplifier. This amplifier measured the component of the voltage signal that was in phase with the driving current. The result was used to compute the real or resistive component of the impedance. The phase shift of the voltage signal due to capacitive-reactance effects was generally negligible and was not used in determining the impedance. Figure 6 shows a typical signal measured in the manner just described. The peak-to-peak amplitude of the cardiac-related modulation of the impedance signal is 4.5 0, which is 10% of the base-line impedance. There are two sharp, easily identified inflection points coincident with the onset and termination of left ventricular ejection, and the shape of the impedance wave form is very similar to that of the aortic pressure. The systolic time intervals can be derived solely from the esophageal elecFIG. 5. A line drawing made from a right lateral chest x ray of a dog Elecp. L-L!,a~~-ao~~-~rQ~t~~~r. 0 )--in&e. - trwr&egrx -MC! -the -~-I-E-I-s&al --z&s -&own -& --FJg.4. --Wikk!-SSophpIg9~LpEQb I-20, starting at distal end. Electrodes The preejection period for this dog was 80 ms and the trodes on probe are numbered 7, 8, 10, and 11 (heavy shaded) are used for EIEI measurement. left ventricular ejection time was 170 ms with a heart Electrocardiogram is sensed between electrodes 10 and 20, separate ---~-_--~ ~~-~~~ rata-&-g@ vr -&&&&$T~?,p G-Y-CIWW Qa_ rra- Jlinlc2__03?a - rvAr&alr l&VI zLr~~z& _ -refe-en~~-&Qrt~&~ &v&&-+rr-t~le-rr&;~e~a~~~n~~~p~~~~~~~~-~ro~e -v rrrithinm__n_rl\rmalm limits for the dog (19). and thoracic aorta (3), heart (4), lungs (5), and some of the pulmonary DISCUSSION
The electrode array employed in this study was positioned in that portion of the esophagus which is in close
arteries and veins (6) are shown. Also indicated are superior vena cava (7), sternum (8), spine (9), and tangential edge of the diaphragm (16). Most distal electrode is located at level at which esophagus traverses diaphragm and only appears to be subdiaphragmatic here because of curvature of diaphragm.
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- =~~
_
R172
M.
ElECTROCARDIOGRAM
AORTICPRESSURE
ESOPHAGEAL IMPEDANCE
0.2
0.4
0.6
0.8 1.0 SECUNCS
FIG, 6. Esophageal electrocardiogram (from lead 10 with reference electrode 20), aortic root pressure, and intrathoracic esophageal electrical impedance (from electrodes 7, 8, 10, and I I) recorded frGm a dog during apnea at functional residual capacity. Vertical lines mark onset of QRS complex and beginning and end of left ventricular ejection. Preejection period and left ventricular ejection time are determined to be 80 and 170 ms, respectively. Base-line intrathoracic esophageal change in electrical impedance 20, is 46.5 s1 and the peak-to-peak impedance with left ventricular ejection is 4.5 G2,
respect to the esophagus will, in general, be more important than diameter or volume changes in determining impedance effects. Thus, in general, timing information will be more readily accessible than volume information, no matter what the esophageal location of the electrodes. Figure 6 demonstrates how the preejection period and left ventricular ejection time can be determined solely from the esophageal electrocardiogram and the appropriate fiducial points of the EXE1 signal. The aortic root pressure tracing gives us an independent determination of the onset and end of left ventricular ejection and was used to validate the systolic time intervals as determined from the electrical measurements. In the dog, we found that the appropriate, easily identifiable, aortic signal could be obtained despite a 05cm variation in the longitudinal position of the electrode array within the esophagus. We can expect that in the human subject a similar impedance signal will be obtainable over a larger range of electrode array positions because the human descending thoracic aorta and esophagus are more nearly parallel over longer segments of their lengths and all relevant dimensions are greater. The fractional changes in impedance with cardiac function are much larger with esophageal electrodes than with surface electrodes. The fractional change in thoracic impedance with left ventricular ejection is generally on the order of 0.2% when measured from the surface (12). When the esophageal electrode configuration described here is used, the fractional change is 540%. This results in a much improved signal-to-noise ratio and makes the recording and processing of the impedance signal easier and more reliable. It appears that shunting of current
M.
MITCHELL
AND
R. S. NEWBOWER
fiow through the thoracic wall and other structures that are not of interest is minimized (al).’ In general, there are several practical reasons for exploring the placement of an electrode array on an esophageal probe for the measurement of variations in intrathoracic electrical impedance. The location is one that is generally available. The esophageal stethoscope is commonly employed during general anesthesia. It could be replaced by a simple multifunction probe capable of monitoring not only heart and breath sounds, but also esophageal temperature, electrocardiogram, and electrical impedance (21, 22). In the intensive-care setting, it would be feasible to instrument the esophagus of a crititally ill patient with such a probe in place of a routine nasogastric tube. We are hopeful that by extension of the modeling techniques described, an even better understanding of the nature of the origins of the intrathoracic electrical impedance signal can be obtained. Although it seems unlikely that an esophageal electrical impedance signal can be found whose major determinant is stroke volume or cardiac output, we are encouraged that at least systolic time intervals and possibly other useful physiological parameters can be reliably determined in such a simple fashion. APPENDIX Details
ofMethod
of Solution
of the Model
The exact solution of Maxwell’s equations for our model of the esophageal electrode array and the thoracic aorta can be obtained as follows. The simplified geometry assumed is shown in Fig. 1, where the cylinder is assumed to be infinite in the z dimension. We assume that the materials are isotropic, nondispersive, and independent of other parameters such as temperature and pressure (8). Furthermore, materials outside the aorta and inside the aorta are assumed to be identical except for their conductivities, which are CT]
and~ respectivelywhere~~~ o, 2,
We seek a stationary, quasi-ztatic solution (8). In practice, the measurements are made at 10 kHz, but this frequency is sufficiently low to allow treating this as a static problem. I3y conservation of charge
@ v-J=-z=o where J is the electric current density, charge density, in C/m’. Furthermore, constitutive relations become
in amperes; Maxwell’s
(Al) and p is the electric equations and the
VxE=O
M2)
VxH=J
L43)
v-D=0
(A4)
V*B = 0
b45)
J=uE where
--
e is the electric
field,
in V/m;
H is the magnetic
(A@ field,
in A/m;
~-
’ It is difficult to estimate what contribution the concepts described in this paper will make to understanding surface recordings from circumferential band electrodes. The small amplitude of the cardiac signal as observed from the surface implies substantial cancellation of opposing effects and significant shunting of currents. The physics modeled in this paper may, however, apply to a certain extent to arrays of spot electrodes on the surface, or to band electrodes viewed as a linear superposition of arrays of spot electrodes.
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ESOPHAGEAL
INTRATHORACIC
ELECTRICAL
D is the electric displacement, in C/m2; B is the magnetic in Wb/m2; and 0 is the conductivity, in 1/Q* m. Therefore, if
a, the scalar
electric
at the sources, V-J=
potential
function,
then
(A81
I [6(x
2 - B) - 8(x + A, y, z + B)]
n*(Jp is continuous
an impulse at the point (x’, y’, at the surface between regions
- J]) = 0 and there
: j
V
1
1 .
t-c
I
I B
/
of surface (All)
is, E tangential is continuous Clearly, this system is equivalent to an electrostatic problem of the same geometry if we replace 0 by E, J by D, and I by 4, where I is the current source magnitude, q is a charge, and l is the electrical permittivity. So we can use the method of images to solve this problem (15). The geometry in the (x, y, z) coordinate system is still intractable. This can be circumvented by employing a conformal transformation (31. Let W = x + iy, and T = u + iv, where i is the square root of -1; then a conformal transformation W = F(T) is a regular function which maps tu, u) + (x, y) in such a way that angles are preserved, and therefore so is the flux through corresponding areas and potentials at corresponding points (3). The conformal transformation of use in this application is of the bilinear form, generally
where, a, b, c, and d are complex employed is
+ d)
constants.
W2) The
specific
transform
W = (AZ/T)
W3)
of the near wall of the aortic lumen as shown in Fig. 7. This is equivalent + u2);
y = -A’v/(u”
from to
+ f));
v = -y/(A2($
l
-
+ v2)
7
I
-
FIG. 8, Method of images used to solve Maxwell’s equation for transformed geometry. Source I and -I are reflected with appropriate scale factor (02 - aJ/(oz + oJ, in plane perpendicular to u-axis through u = A’/2R.
frame. We need to derive an expression for the difference in potential between the two sensing points as a function of the geometry parameterized by A, R, B, C, and the current source magnitude, I. This requires establishing the two image sources shown in Fig, 8. The potential for a point source at the origin is (3)
the
and inversely u = x/(A2(;1c2
0;'
(AlO)
is no accumulation
that
W = (aT + b)/(cT
+C
Ir +
(A9
Ii x (E, - El) = 0
x = A2u/(u2
+B
.
+ A,y,
where A is the distance midline of the esophagus
I
l +
0;:
where
where A(#‘, y’, z’) is the Dirac function, z’) = (0, 0, 0). The boundary conditions of conductivity u1 and uz are
that is, J normal charge and
T
L47)
va=o except
7
flux density,
VQe-E defines
R173
IMPEDANCE
@ b, y, 21 = Therefore we find that region of interest is
I
(A141
4 7m Jx* + y2 + 2
by superposition,
the total
potential
for the
+ s;-‘,,
The transformation (x, y, z) t (u, v, z) is also conformal, and preserves angles, flux densities, and potentials. The interior of the aorta becomes the right half-space, u < A2/2R as shown in Fig. 7. We can then proceed with solution of the problem in the transformed
if, =-
The
1
I 4m
d(u
sensed
voltage
+ A)2 + v2 + (z - B)’
-1 + Jju+A)“-+rr’-q;-+B)’
is c
7. Bilinear conformal transformation, W = (AZ/Z’), which maps geometry (x, y) on Left into geometry (u, u) on right, and preserves angles of intersection and flux densities through corresponding areas.
v=-
FIG,
&dz
= a, (-A,
0, C) - 0 (-A,
I -c
Remembering
that
0, -C)
(Al61
B > C > 0, we find that
Downloaded from www.physiology.org/journal/ajpregu by ${individualUser.givenNames} ${individualUser.surname} (129.186.138.035) on January 11, 2019.
RI74
M. @ (-A,
0, C) = 4
(-A,
0, -C)
(A171
and therefore V= Because
2 = V/I,
the solution
2 if, (-A,
0, C)
(Af@
is
M.
MITCHELL
AND
R. S. NEWBOWER
Special thanks for help and guidance are given to Professor Jin Au Kong of the Massachusetts Institute of Technology Electrical Engineering Department and to Drs, Jeffrey B. Cooper, Michael C. Long, Richard J. Kitz, and James H. Philip, and to Mr. Josh Tolkoff of the Anesthesia Department at the Massachusetts General Hospital. This work was supported in part by Grant GM-15904 from the National Institute of General Medical Science, The MACSYMA facilities of the Massachusetts Institute of Technology Mathlab group (supported by the Defense Advanced Research Projects Agency work order 2095 under the Office of Naval Research Contract NOOO14-75-C 0661) were used to correct and verify the mathematics used herein. M. M. Mitchell acknowledges support, during the course of this work, by the Anesthesia Research Training Grant GM-01273. His present address: Dept. of Anesthesiology, University of California, San Diego, San Diego, California 92103. Received
19 December
1977; accepted
in final
form
29 September
1978.
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