|where||ΔZ||= impedance change [Ω/m³]|
|t0, t1||= time instants|
|Δσ||= conductivity change between the two time instants [S/m = 1/Ω·m]|
|LE||= lead field of the voltage measurement electrodes for unit reciprocal current [1/m2]|
|LI||= lead field of the current feeding electrodes for unit current [1/m2]|
|v||= volume [m3]|
In Equation 25.1, the region v consists of an inhomogeneous volume conductor whose conductivity (as a function of position) at time t0 is σ(t0). At t1, this has changed to σ(t1), and it is this change (t1) - (t0) = Δσ which is responsible for the measured impedance change ΔZ. Thus Equation 25.1 describes how the changes in volume conductor conductivity are converted into the impedance change evaluated from a measured voltage (at the voltage electrode pair) divided by applied current (at the current electrode pair). Note that the 4-electrode impedance method underlies Equation 25.1.
A special case of Equation 25.1 is one where we consider σ(t1) = εσ(t0), where ε is very small:
where all variables are evaluated at t0. Equation 25.2 describes how the macroscopic resistivity Z (impedance per unit volume) is derived from the spatial distribution of conductivity σ weighted by the dot product of the lead fields of the current and voltage electrodes. Note the similarity between Equation 25.2 and the fundamental equation of the lead field theory, Equation 11.30 (or 11.52), which describes the electric signal in the lead produced by a volume source formed by a distribution of the impressed current i. In these equations the corresponding variables are the measured signals: VLE and Z (= measured voltage per applied current), the distributions of sensitivity: LE in both of them, as well as the source distributions: i and LI.
If the introduction of the current is done with the same electrodes as the voltage measurement is made, the sensitivity distribution, that is the lead field LE is the same as the distribution of the applied current LI. This technique is, however, seldom used because of the artifact due to the electrode impedance. If the current-feeding electrodes are different from those of the voltage measurement electrodes, the sensitivity distribution is the dot product of the lead fields of the voltage electrodes LE and the current electrodes LI. Thus, any previous discussion in this book on the electric and magnetic lead fields in general (Chapters 11 and 12), in the head (Chapters 13 and 14) or in the thorax (Chapters 15 ... 18 and 20) may readily be applied to impedance plethysmography. Just as in the study of electrocardiography, one can design electrode systems for impedance measurement to give special emphasis to particular regions (the aorta, the ventricles, etc.). One can even have situations where the dot product is negative in a particular region so that if the conductivity increases in that region, the impedance Z will also increase. Some examples can be found in Plonsey and Collin (1977) and Penney (1986).
While Equation 25.1 is a suitable theoretical basis for impedance plethysmography, we are still left with considerable uncertainty how varies throughout the heart and torso or in what way the circulation modifies the thorax structure and conductivity as a function of time throughout the cardiac cycle. Further research is required to develop a physiologically adequate circulation model. Note, however, that Equation 25.1 may be more readily applied over a longer time frame (t1 - t0) to, say, the growth of a localized tumor in the thorax (other regions remaining the same).
|where||Zf||= impedance (as a function of frequency f )|
|R0||= resistance at f = 0|
|R||= resistance at f =|
|τ||= time constant (R2C)|
The Cole-Cole plot is a semicircle with radius (R 0 - R )/2 which intercepts the real axis at R 0 and R, a conclusion that can be verified by noting that the real (Re) and imaginary (Im) parts of Equation 25.3 satisfy
The right-hand side of Equation 25.4 is a constant where one recognizes the equation to be that of a circle whose center is at Im = 0, Re = (R 0 - R )/2 with a radius of (R 0 - R )/2, as stated. In the three-element circuit of Figure 25.1A, R0 = R1 + R2, R = R 1 , and τ = R2C.
In practice, the center of the semicircle is not necessarily on the real axis, but is located beneath it. The equation representing practical measurements may be described by Equation 25.5 (Schwan, 1957):
In the corresponding Cole-Cole plot, shown in Figure 25.1C, the depression angle is φ = (1 - α)π/2. Figure 25.2 shows the depression of the semicircle in the Cole-Cole plots for the transverse and longitudinal impedances of skeletal muscle as measured by Epstein and Foster (1983).
The reactive component of human blood has been studied, for example, by Tanaka et al. (1970) and Zhao (1992). The reactive component of tissue impedance seems to have an important role in impedance plethysmography, as will be discussed later in this Chapter in connection with determining body composition..
Fig. 25.5 Simplified cylindrical model of the average thorax containing a uniform blood and tissue compartment for determining the net torso impedance.
|where||Z||= longitudinal impedance of the model|
|Zb||= impedance of the blood volume|
|Zt||= impedance of the tissue volume|
The relationship between the impedance change of the thorax and the impedance change of the blood volume is found by differentiating Equation 25.6 with respect to Zb:
The impedance of the blood volume with blood resistivity ρb based on the cylindrical geometry of Figure 25.5, is:
|where||ρb||= blood resistivity|
|Ab||= cross-section of the blood area|
|l||= length of the thorax model|
The relationship between changes in blood volume vb and the blood volume impedance is found by solving for the blood volume in Equation 25.8 and differentiating:
|where||vb||= blood volume|
We finally derive the dependence of the change in blood volume on the change in thoracic impedance by solving for dZb in Equation 25.7 and substituting it into Equation 25.9:
Assuming that Δt equals the ejection time te, ΔZ can be determined from equation
With the above assumptions, the impedance change ΔZ can be determined by multiplying the ejection time by the minimum value of the first derivative of the impedance curve (that is, the maximum slope magnitude; the reader must remember that the slope is negative).
Finally, the formula for determining the stroke volume is obtained by substituting Equation 25.12 into Equation 25.10, which gives:
|where||SV||= stroke volume [ml]|
|ρb||= resistivity of the blood [Ω·cm]|
|l||= mean distance between the inner electrodes [cm]|
|Z||= mean impedance of the thorax [Ω]|
|= absolute value of the maximum deviation of the first derivative signal during systole [Ω/s]|
|te||= ejection time [s]|
The ejection time can be determined from the first-derivative impedance curve with the help of the phonocardiogram or carotid pulse. Then, the impedance curve itself is used only for control purposes (e.g., checking the breathing).
The resistivity of the blood is of the order of 160 Ωcm. Its value depends on hematocrit, as discussed in Section 7.4.
|Vena cava and right atrium||+20%|
|Pulmonary artery and lungs||+60%|
|Pulmonary vein and left atrium||+20%|
|Aorta and thoracic musculature||+60%|
|Source: Penney (1986)|
Mohapatra (1981) conducted a critical analysis of a number of hypotheses concerning the origin of the cardiac impedance signal. He concluded that it was due to cardiac hemodynamics only. Furthermore, the signal reflects both a change in the blood velocity as well as change in blood volume. The changing speed of ejection has its primary effect on the systolic behavior of ΔZ whereas the changing volume (mainly of the atria and great veins) affects the diastolic portion of the impedance curve.
These facts point out that the weakest feature of impedance plethysmography is that the source of the signal is not accurately known. Additional critical comments may be found in Mohapatra (1988).
|Event in the cardiac cycle||Notch|
|Closure of tricuspid valve||B|
|Closure of aortic valve||X|
|Closure of pulmonic valve||Y|
|Opening snap of mitral valve||O|
|Third heart sound||Z|
|Source: Lababidi et al., (1970)|
The first-derivative impedance curve can be used with some accuracy in timing various events in the cardiac cycle. The ejection time can be determined as the time between where the dZ/dt curve crosses the zero line after the B point, and the X point. However, in general, the determination of ejection time from the dZ/dt curve is more complicated. Thus, the need of the phonocardiogram in determining the ejection time depends on the quality and clarity of the dZ/dt curve. Though the timing of the various notches of the dZ/dt curve is well known, the origins of the main deflections are not well understood.
|where||ΔZ||= change of the impedance of the thorax|
|Z||= mean value of the impedance of the thorax|
|vtx||= volume of the thorax between the inner electrode pair|
They used the Fick principle as a reference for evaluating stroke volume. (The Fick principle determines the cardiac output from the oxygen consumption and the oxygen contents of the atrial and venous bloods.) In a study of six subjects at various exercise levels, the correlation between the impedance and Fick cardiac outputs was r = 0.962, with an estimated standard error of 12% of the average value of the cardiac output.
Harley and Greenfield (1968) performed two series of experiments with simultaneous dye dilution and impedance techniques. They estimated ΔZ from the impedance curve itself, instead of using the first-derivative technique. In the first experiment, 13 healthy male subjects were examined before and after an intravenous infusion of isoproterenol. The mean indicator dilution cardiac output was 6.3 /min before and 9.5 /min after infusion. The ratios of the cardiac outputs measured with impedance plethysmography and indicator dilution were 1.34 and 1.23, respectively. This difference (p > .2) was not significant. The second experiment included 24 patients with heart disease, including aortic and mitral insufficiencies. A correlation coefficient of r = .26 was obtained for this data. The poor correlation was caused in those cases with aortic and mitral insufficiency.
Bache, Harley, and Greenfield (1969) performed an experiment with eight patients with various types of heart disease excluding valvular insufficiencies. As a reference they used the pressure gradient technique. Individual correlation coefficients ranged from .58 to .96 with an overall correlation coefficient as low as .28.
Baker et al. (1971) compared the impedance and radioisotope dilution values of cardiac output for 17 normal male subjects before and after exercise. The regression function for this data was COZ = 0.80·COI + 4.3 with a correlation coefficient r = .58. The comparison between the paired values before and after exercise showed better correlation for the impedance technique. Baker examined another group of 10 normal male subjects by both impedance and dye techniques. In 21 measurements the regression function was COZ = 1.06·COD + 0.52, with correlation coefficient r = .68. In addition to this set of data, the impedance cardiac output was determined by using individual resistivity values determined from the hematocrit. The relation between resistivity and Hct was, however, not mentioned.
In this case, the regression function was COZ = 0.96·COD + 0.56 with correlation coefficient r = .66. A set of measurements was performed also on 11 dogs using electromagnetic flowmeters and the impedance technique. A comparison of 214 paired data points was made with intravenous injections of epinephrine, norepinephrine, acetylcholine, and isoproterenol. Values of the correlation coefficients from each animal ranged from 0.58 to 0.98 with a mean value of 0.92. The first two experiments of this paper are also presented in Judy et al. (1969).
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