Impedance cardiography (ICG) is a branch of bioimpedance primarily concerned with the determination of left ventricular stroke volume (SV). As implemented, using the transthoracic approach, the technique involves applying a current field longitudinally across a segment of thorax by means of a constant magnitude, high frequency, low amplitude alternating current (AC). By Ohm’s Law, the voltage difference measured within the current field is proportional to the electrical impedance Z (Ω). Without ventilatory or cardiac activity, Z is known as the transthoracic, static base impedance Z0. Upon ventricular ejection, a characteristic time dependent cardiac-synchronous pulsatile impedance change is obtained, ΔZ(t), which, when placed electrically in parallel with Z0, constitutes the time-variable total transthoracic impedance Z(t). ΔZ(t) represents a dual-element composite waveform, which comprises both the radially-oriented volumetric expansion of and axially-directed forward blood flow within both great thoracic arteries. In its majority, however, ΔZ(t) is known to primarily emanate from the ascending aorta. Conceptually, commonly implemented methods assume a volumetric origin for the peak systolic upslope of ΔZ(t), (i.e. dZ/dtmax), with the presumed units of Ω·s–1. A recently introduced method assumes the rapid ejection of forward flowing blood in earliest systole causes significant changes in the velocity-induced blood resistivity variation (Δρb(t), Ωcm·s–1), and it is the peak rate of change of the blood resistivity variation dρb(t)/dtmax (Ωcm·s–2) that is the origin of dZ/dtmax. As a consequence of dZ/dtmax peaking in the time domain of peak aortic blood acceleration, d
- Impedance cardiography
- stroke volume
- cardiac output
- volume conductor
- extravascular lung water
Stroke volume (SV) is defined as that quantity of blood ejected from the cardiac ventricles for each heart beat. More specifically, it relates to the volume of blood ejected from opening to closure of the semilunar valves, and in the case of the left ventricle, the aortic valve. The time interval over which left ventricular ejection occurs is known as left ventricular ejection time (TLVE, s). For purposes of simplicity, and considering this is not an in-depth analytical treatise on pulsatile blood flow, per se, the following simplistic model is proposed as an operational tool for understanding the opposing hypotheses concerning SV determination by means of the transthoracic electrical bioimpedance technique (TEB).
In geometric terms, consider stroke volume V (mL) a cylinder of length S (cm) and cross-sectional area π
If the dependent variable, V, and independent variables r and S are continuously differentiable functions of time t, the rate of change of aortic volume dV/dt or flow Q(t) (mL·s–1) is given as,
If 2π NB: Q(t) in lieu of
NB: Q(t) in lieu of
For derivative 2, if π
At an operational level, derivative 2 of equation 2 and equations 4 and 5 represent the basis for SV determination by means of both the Doppler/echo method and electromagnetic flowmetry. To find SV simply requires integration of the velocity profile over the ejection period at a discreet point in the aortic root and multiplying this by the aortic root CSA at the point of velocity measurement. Common sites for measurement include the left ventricular outflow tract and aortic valve leaflets. Thus,
In terms of mechanics, since dA(t) and dA(t)/dt are non-linear functions of vessel material composition, pressure P(t) (mmHg) and rate of change of aortic pressure dP(t)/dt (mmHg·s–1), respectively, the following pertains :
where the right hand side (rhs) of equation 8 defines the magnitude and shape of the flow wave, and C is the aortic compliance.
If aortic dP(t) is a function of force F and area A, both changing with time over the ejection interval, dP(t) (mmHg) and dP(t)/dt (mmHg ⋅ s–1) are given as,
respectively, and compliance C is given as,
where dA/dP is a non-linear function of Pulse pressure (ΔP) and mean distending pressure Pm. Thus,
where the mechanical expression for dV/dt of equation 11 is equivalent to the geometrical definition for dV/dt of derivative 1 in equation 2. As an analytical expansion of equation 8, the rhs of equation 11 more precisely defines the temporal landmarks, shape and magnitude of the radially oriented flow wave.
Figure 1 shows the pressure curve P(t), the rate of change of aortic pressure, dP/dt, aortic blood flow (Q(t)) and the rate of change of flow, which is acceleration of flow, dQ(t)/dt. It is clear from equation 11 that dF/dtmax, the major derivative, will occur early in systole, when dA/dt is trivial. Likewise, dA/dtmax will occur when dF/dt = 0. When dP/dt is modulated by C, dA/dtmax will occur at the peak magnitude of the flow wave dV/dtmax (i.e. Qmax) and dF/dtmax will occur on the steepest portion of the flow wave, immediately following aortic valve opening in early systole.
When flow is differentiated with respect to time dQ/dtmax will occur when dA/dt and dV/dt are minimal. Also apparent is that when flow acceleration = 0, dV/dt is at maximum. Therefore, dQ(t)/dtmax and dV/dtmax are monotonically restricted to their respective time-domains, with acceleration of blood flow, dQ/dtmax peaking, on the mean, approximately 50ms before peak flow, dV/dtmax.
If S is a segment of aorta with closed outflow, then flow into the segment would be the total input flow from left ventricular ejection. If, however, segment S is an open outflow conduit, equation 8 would represent a net flow. For a simple 2-element
where derivatives 1 and 2 on the rhs of (12) represent Q(t)stored and Q(t)out, respectively.
To obtain total flow Q(t)total through the segment over a single systolic interval (TLVE, s), Q(t)out must be measured. With regard to a two-element
Integration of Q(t)strored yields stored volume inflow, SVstored:
where ΔP is the pulse pressure, which is aortic peak systolic pressure Ps minus aortic diastolic pressure Pd (Ps–Pd) .
Since SV = CΔP, where C is a non-constant time variable function of pressure P, the following expression also represents total flow into a segment of aorta of length S and internal surface area wall velocity dA/dt .
where derivatives 1 and 2 represent Q(t)stored and Q(t)out, respectively. Differentiating dCp(t)/dt results in,
Although a theoretically correct model for SV, equation 17 is not easily solved. Clearly, P(t), dP(t)/dt, and Pm can be computed by direct measurement and signal processing, but C and Rs must be obtained independently by alternative means . This is clear from equation 10 for the determination of C and the relationship of Rs to Q(t)mean in equation 17. From equations 15 and 16, it is also clear that C is not a constant as equation 17 implies. Pressure dependent C and the rate of change of C with respect to time, dC/dt, are not accounted for in equation 17. To more accurately predict SV from the
Impedance cardiography (ICG) is a branch of bioimpedance primarily concerned with the determination of left ventricular stroke volume (SV) [7–10]. It was the first and still the only truly noninvasive, continuous, operator-independent, hands-off, beat-to-beat method used in clinical practice. Despite these desirable attributes, general acceptance of the method as a replacement for invasive reference standards in critically-ill humans has not been realized. As summarized by Raaijmakers et al. and Moshkovitz et al., both correlation and agreement against standard reference methods have been far too variable to reliably apply the method clinically for modulation of patient therapy [7,12]. As emphasized by Kauppinen et al. , the major problem in cardiovascular impedance measurements is the inability to accurately correlate the magnitude of the impedance waveforms to the hemodynamic variables they pretend to mimic. The lack of consistently high correlation and agreement between ICG measurements and reference standards would suggest that, the equation models describing SV in ICG are not sufficiently robust and the physical acquisition of the impedance signals not sensitive and specific enough for clinical application in the critically ill .
While measurements can be obtained by the whole body technique, the method is usually implemented by means of the transthoracic approach (i.e. transthoracic electrical bioimpedance cardiography, TEB, ICG) . The latter technique, and subject of this review, involves applying a current field longitudinally across a segment L (cm) of thorax by means of a constant magnitude, high frequency (50–100 kHz), low amplitude AC (I(t)) (1–4 mA). By Ohm’s Law, the voltage difference U(t) (volt, U) measured within the current field is proportional to the transthoracic electrical impedance Z.
where Z is the frequency-dependent AC analog of the static DC resistance R. The magnitude of the impedance, or modulus, │Z│, comprises a resistive component, also known as the real part, R, and an imaginary part X. X is further comprised of an inductive part (XL) and capacitive part (XC), both collectively known as the reactance. With regard to the magnitude of X, XL is proportional and XC inversely proportional (1/f) to the frequency of the applied AC. │Z│ is determined by the geometric addition of the resistive and reactive components.
where the reactance X is,
where the magnitudes of R and X on Cartesian coordinates are determined by the phase angle φ, and thus, R
For the remainder of the discussion, let
where ρ = the resistivity (Ω ⋅ cm), L = the length of the conductor, and A and V are the CSA (cm2) and volume (mL) of the conductor, respectively. Thus, substituting ρL2/V in equation 23 for Z in equation (19) and rearranging, For the remainder of the discussion, the character “U” is used in
For the remainder of the discussion, the character “U” is used in
If the thorax is considered a cylindrical bulk electrical conductor of length L between the voltage sensing electrodes, with ρT the transthoracic specific resistance and VT the volume of thorax, then, without ventilatory or cardiac activity, Z in equations 19 and 23 represents the adynamic or static transthoracic base impedance Z0. When cardiac and ventilatory activity are superimposed on Z0, a time-variable transthoracic impedance is registered, Z(t). By eliminating the oscillating cardiac-asynchronous ventilatory component ΔZvent, Z(t) comprises, in parallel, a static DC component Z0 (22Ω–45Ω) and a dynamic AC component ΔZ(t) (0.1Ω–0.2Ω) [14,15] (Fig.2). It should be noted that the magnitude of Z0 not only varies between individuals and the frequency of the applied AC, but also with the electrode configuration used for signal acquisition. For four common electrode configurations a computer model predicts a Z0 range of 26.3Ω–34.3Ω .
When electrical resistances or impedances are added in parallel, they are summed as their reciprocals.
As a composite impedance with tissues of widely varying specific resistances, Z0 comprises, in parallel, a static, relatively non-conductive multi-compartmental tissue impedance Zt (400Ω·cm–1020Ω·cm) [11,16], a highly conductive blood impedance Zb (or resistance Rb 100Ω·cm–180Ω·cm) varying with hematocrit , and a very highly conductive interstitial extra-vascular lung water (EVLW) impedance Ze (60Ω·cm–70Ω·cm).
Z0 is thus an intensity weighted mean of the reciprocal sum of all thoracic tissue impedances, the sum of the reciprocals being less than the lowest tissue impedance of the thorax.
In this model, the blood resistance Rb is considered a cylindrical tube of constant length L, surrounded by the highly conductive extravascular lung water impedance Ze and both surrounded by a non-conductive thoracic encompassing cylinder Zt. It should be noted that, at the frequencies used in ICG, blood is almost purely resistive with a trivial reactive component. Therefore, the term blood resistance, Rb, is justifiably used in lieu of blood impedance Zb.
The systolic portion of the cardiogenically-induced impedance pulse variation ΔZ(t), hereafter known as ΔZb(t), comprises two components, arguably of equal magnitude [14,18–21]. The first is a velocity-induced change in the specific resistance of axially directed flowing blood (Δρb(t), Ω·cm·s–1), where, in end-diastole, the state of highest blood resistivity, red blood cells are randomly oriented. By contrast, during peak aortic blood acceleration and the rapid ejection phase of early systole, red blood cells become deformed and assume a well defined state of parallel orientation along their long axis of symmetry [19,22,23] (fig. 3).
Parallel alignment opens clear current pathways through the highly conductive plasma, causing a decrease in blood resistivity and transthoracic resistivity [19,22] (i.e.↑Conductivity). The rapidity of and degree to which attainment of complete parallel orientation is achieved determines the magnitude of the resistivity change Δρb(t)max (Ω·cm·s–1) . The maximum resistivity change occurs when the long axis of the red cell is oriented within 200 to the direction of blood flow . Figure 4 shows that, in early ejection, maximum acceleration of red cell reduced average velocity (i.e. mean spatial velocity) parallels the change in blood resistivity (i.e. conductivity) (r = 0.99). By comparison, upon negative acceleration, the resistivity change does not parallel the reduced average velocity, with subsequent delay in reaching baseline [19,23]. This disparity is, no doubt, a consequence of the red cells’ inability to achieve complete randomization at end-systole.
The second component generating a change in transthoracic specific resistance is the transversely or laterally-oriented volume displacement of non-conductive alveolar gas (ρ =1020 Ω·cm) by stroke volume-induced expansion of the ascending aorta, principally [24,25], with highly conductive blood (ρb=100–180 Ω·cm) ΔVb(t) (ΔΩ·cm(t)). As discussed under pulsatile blood flow, in addition to distal vascular hindrance Rs, the volumetric expansion of the aorta is due to pressure-induced, compliance modulated changes in cross sectional area, πr2 (cm2), and, over segment L, a change in internal surface area, 2π·(dA/dr)L, (cm2) [1,26]. For example, assume that equation 23 represents the impedance Z of a segment of very thin walled aorta embedded within the thorax. Assume that the aortic segment is of length L with open outflow, the content of which is blood of static specific resistance ρb and volume Vb. Thus, an increase in aortic volume ΔVb(t) would result in a corresponding decrease in vessel and transthoracic impedance ΔZb(t)volume.
where the rhs of equation 27 is the ohmic equivalent of equation 14, wherein ΔZb(t) and ΔVb(t) represent net changes. The combined velocity and volume effects cause a steep drop in transthoracic impedance.
For the velocity-induced change in blood resistivity, the decrease in vessel and transthoracic impedance is given as,
The original Nyboer equation  was specifically proposed for determination of segmental blood volume changes in the upper and lower extremities. It is based upon the assumption that the arteries of the extremities are rigidly encased in muscle and connective tissue, and that both the arteries and surrounding tissue can be approximated as cylindrical conductors placed in parallel alignment. Nyboer found that, by placing spaced-apart circumferential current injecting electrodes on a limb segment and voltage sensing electrodes within the current field proximate the current injectors, impedance changes proportional to strain gauge determined blood volume changes could be measured. He also determined that, in order to compensate for simultaneous volume outflow from the limb segment during arterial inflow, venous outflow occlusion was necessary. Thus, the ΔZb(t) waveform represents the sum of two signals: one caused by inflow and the other caused by outflow (runoff) between the voltage sensing electrodes . Using venous outflow obstruction, absolute volume changes during each pressure pulse were accurately approximated. Nyboer referred to this method, and quite accurately as, electrical impedance plethysmography. The foundation of Nyboer’s equation, and for that matter the earliest SV equations, originate from equation 27, where, when solving for ΔVb(t)net,
However, since a large extremity artery (i.e. brachial or femoral) is encased in muscle and connective tissue, ΔZb(t) had to be further refined, reflecting a parallel connection of the static tissue and dynamic blood impedances. Since the magnitude of Z(t) of an extremity is larger than extremity Z0 by a trivial factor of extremity ΔZb(t), the approximation that extremity Z(t) and Z0 are virtually identical is a plausible assumption. Without assumption, it also follows that Z(t)–Z0 = ΔZb(t). Thus, if equation 25 is solved for ΔZb(t), using the reciprocal rule for impedances added in parallel:
Substituting the right hand side of equation 32 for ΔZb(t) into equation 31 results in the operational model for measurement of blood volume changes in an extremity. For the maximum volume change ΔVb(max) with venous outflow occlusion ,
where ΔZmax is the peak magnitude of the waveform and ρbL2/Z02 is a constant. For an arterial conduit with closed outflow, equation 33 is the bioelectric equivalent of the hemodynamic expression given for SV in equation 14 with outflow obstruction. For an extremity vessel with open outflow, equation 33 is analogous to the integrated value for equation 12. Clearly, as applied to SV calculations, equation 33 is inadequate, because complete aortic outflow obstruction for calculation of total SV is impossible.
As applied to thoracic measurements of SV, Nyboer professed to have solved the outflow problem by manually determining the maximum systolic down-slope of ΔZb(t) and extrapolating it backward to the beginning of ejection . Thus, the maximum impedance change resulting from the backward extrapolation method was believed to be equivalent to the maximum impedance change attained as if no arterial runoff occurred during ejection.
Thus, SV, or the maximum volume change ΔVb measured between the voltage sensing electrodes is given as,
where Z0 is the static transthoracic base impedance and ΔZmax is the maximum backward extrapolated value of the thoracic cardiogenic impedance pulse variation, which includes runoff. While credible SV values could be obtained with equation 34, it was not widely accepted, this being due to difficulty in manually determining the true maximum downslope.
As a consequence of the deficiencies of the backward extrapolation procedure, Kubicek et al. proposed a maximum systolic forward extrapolation of ΔZb(t), which is the basis for all subsequent plethysmographic conceptually-based SV equations using the transthoracic approach [29,30]. Kubicek et al. made the assumption that, if the maximum systolic upslope (i.e. ΔZ′ = ΔZ·s–1, Ω·s–1) is held constant throughout ejection, then compensation for outflow, before and after attainment of Qmax is achieved. Theory underlying Kubicek’s method assumes that little arterial runoff occurs during the inertial phase of rapid systolic ejection. Unlike Nyboer’s direct measurement of ΔZmax, Kubicek’s method requires multiplying ΔZ′ (forward extrapolation) by left ventricular ejection time TLVE (i.e. ΔZ′ × TLVE = ΔZmax). In the original description of the technique, TLVE was determined by means of phono-cardiography. As opposed to manual extrapolation of ΔZ′, electronic differentiation of ΔZb(t) was implemented. The first time-derivative clearly defines dZ/dtmax, with the presumed units of Ω·s–1, as a distinct point C and left ventricular ejection time (s) as the interval between points B (aortic valve opening) and X (aortic valve closure) (fig. 5). Thus, ΔZmax = dZ/dtmax × TLVE = Ω. Kubicek’s equation is thus obtained by substituting dZ/dtmax × TLVE for ΔZmax in equation 34. The Kubicek equation is given below and purportedly equivalent to equation 17. Thus [29,30],
where ρb is the static specific resistance of blood, which was initially fixed at 150 Ω·cm . As concerns the appropriate value for ρ, Quail et al  rearranged the Kubicek equation (assuming it correct), solved for ρ as the dependent variable, and measured SV by EMF. By means of normovolemic hemodilution exchange transfusion, they showed that, over a wide range of hematocrit from 26%–66%, the value of ρ remained virtually constant about a mean of 135Ω·cm at hematocrit 40%. L (cm) is the longitudinal distance between the voltage sensing electrodes on the base of the neck and those on the lower thorax at the level of the xiphoid process. Z0 is the quasi-static base impedance measured between the voltage sensing electrodes. In an in vitro expansible tube model, when equation 35 is compared to equation 33 without outflow occlusion, equation 33 systematically underestimates SV obtained from 35 . By elimination of a Z0 term, the volume conductor VC in equation 35 is given as,
where, when Z0 varies with alterations in extravascular lung water, the VC is non-constant. Equation 35 is, in theory, modeled as the ohmic equivalent of
The transthoracic impedance Z(t) is considered the parallel connection of an aggregate of static cylindrical tissue impedances, Zt, considered as one, and, encased within, a dynamic blood resistance, Rb, otherwise known as the blood impedance, Zb.
The blood resistance is considered a homogeneously conducting blood-filled cylinder of constant length L, or a parallel connection of an aggregate of cylinders considered as one.
The current distribution in the blood resistance is uniform.
All current flows through the blood resistance.
The volume conductor VC is homogeneously perfused with blood of specific resistance ρb.
All pulsatile impedance changes ΔZb(t) are due to vessel volume changes ΔVb(t), and, in the context of assumption 2, ΔZb(t) is due exclusively to changes in vessel radius dr(t) and CSA dA(t).
In the context of assumption 7, dZ/dtmax is the bioelectric equivalent of dV/dtmax,
The specific resistance (resistivity) of blood ρb is constant during ejection 
The transthoracic specific resistance, ρT, is constant .
The assumptions of the Sramek-Bernstein equation are virtually identical to those of the Kubicek method, except for the physical definition and magnitude of the VC . In Sramek’s interpretation of Kubicek’s VC, ρ and a Z0 variable are eliminated by mathematical substitution. This simplification assumes, as per the work of Quail et al.  that, the ρT equivalent, Z0A/L can replace ρT. By this substitution, VC is rendered a personal constant for each individual. Sramek  named the modification of Kubicek’s VC as the volume of electrically participating thoracic tissue, VEPT. Conceptually, as opposed to Kubicek’s cylindrical VC, Sramek’s VEPT is geometrically a frustum, or truncated cone. Full mathematical derivation and justification for Sramek’s model is provided elsewhere . The original Sramek equation is given as,
where L is the measured distance (cm) between the voltage sensing electrodes and 4.25 an experimentally-derived constant. The Bernstein modification of VEPT assumes that SV is not only a function of thoracic length, but also body weight and blood volume. Correcting for the magnitude of the blood resistance (i.e. intrathoracic blood volume, ITBV, mL) and using the work of Feldschuh and Enson , a factor δ (delta) was appended to the Sramek equation. The Sramek-Bernstein equation is given as ,
where L is a thoracic length equivalent, equal to 17% of body height (i.e. 0.17⋅H) (cm), and delta δ is the weight correction for blood volume. δ is a dimensionless parameter, which corrects for deviation from ideal body weight (kg) at any given height and further modified for the indexed blood volume (mL·kg–1) at that weight deviation [32,33].
In order for equations 35, 37 and 38 to yield equivalent results to equation 17, it must be demonstrated that dZ/dtmax is explicitly dependent on aortic systolic pressure P(t), the rate of change of aortic pressure dP/dt, mean arterial pressure Pmean, systemic vascular resistance Rs and aortic compliance C. In an early, well designed experimental study, Yamakoshi et al.  demonstrated that, in an
The proper value, physiologic definition, and theoretical basis for ρ: i.e. does ρ vary with hematocrit (ρb), or is it a constant based on thoracic resistivity (ρT)?
The physiological relevance of measured L or thoracic length equivalents in the Kubicek and Sramek-Bernstein equations regarding SV.
The validity of the outflow extrapolation procedure .
A coherent physiologic basis and correlate for the empirically-derived volume conductors and their relevance to SV.
Lack of consideration for the changing transthoracic base impedance (ΔZ0) in critical illness, typified by increased thoracic liquids (pulmonary edema) causing aberrant electrical conduction .
Lack of regard for the effect of the blood velocity-induced change in the transthoracic specific resistance Δρb(t) .
Origin of dZ/dtmax in the hemodynamic time domain .
Despite the objections to the plethysmographic hypothesis, correlation with reference standards is considered good (r2 = 0.67, range = 0.52–0.81 and, r = 0.82, range = 0.70–0.90 [7,12]. However, high correlation and reasonable agreement with invasive reference standards does not verify the plethysmographic hypothesis as correct. It may simply mean that the product of dZ/dtmax, TLVE, and a best-fit volume conductor VC, yield results that mimic reference method SV [14,15].
If equation 23 is differentiated by parts with respect to time, the following results:
Since dL and dL/dt are of trivial magnitude, dZ/dt comprises derivatives 2 and 3, the units of which are Ω·s–2 and Ω·s–1, respectively.
Using older plethysmographic techniques, which are conceptually analogous to the simple two-element
A recently introduced ICG method conceptualizes the peak value of derivative 2 of equation 40 to represent dZ/dtmax, which is the peak rate of change of the red cell velocity-induced blood resistivity variation dρb(t)/dtmax (Ω⋅cm·s–2) (i.e d[ΔZvel(t)]/dtmax, Ω·s–2) . Thus,
While differential time domain analysis provides the two possible origins of dZ/dtmax, the definitive origin requires temporal correspondence of dZ/dtmax with either peak ascending aortic blood flow (Q(t)max, dV/dtmax) or peak aortic blood acceleration (d
Comparative time domain analysis confirms that peak flow velocity occurs at ∼ 100±20ms after opening of the aortic valve , whereas dZ/dt, (as extrapolated from the data of Matsuda et al.  and Lozano et al. ) and acceleration d
Figure 6 shows that dZ/dtmax intersects the flow curve during peak acceleration and appreciably before Q(t)max. Simultaneously obtained waveforms comparing TTP of peak flow velocity and dZ/dtmax, showing the latter temporally preceding the former, can be found elsewhere [30,45,46,47].
Figure 7 shows that dZ/dtmax peak synchronously with aortic dP/dtmax, where, as extrapolated from equation 9 and its following discussion (
Figure 6 indicates that, at Q(t)max, dZ/dt = 0. These observations infer, if not prove that dZ/dtmax occurs in earliest systole, contemporaneously with aortic dP/dtmax and peak blood flow acceleration (d
Gaw et al.  showed that the relative blood conductivity change (Δσ/σ (%), i.e. [Δρb(t)/ρb]–1(%)) parallels the peak acceleration of the red cell reduced average velocity (mean spatial velocity, <
As discussed by Gaw  and Visser , the relative change of blood resistivity (or equivalently blood conductivity) due to aortic blood flow is related, hemodynamically, to an exponential power (n) of the reduced average blood velocity (i.e. mean spatial velocity) [(<
Through the relationship for mean flow velocity given by Visser , PARABA can be reduced to mean blood flow velocity by square root transformation,
Preliminary evidence suggests that the exponent m is in the range of 1.15–1.25.
To obtain ohmic mean velocity (s–1) from the mean acceleration analog, dZ/dtmax/Z0 must also undergo square root transformation.
It naturally follows that,
Inasmuch as dZ/dtmax is the electrodynamic equivalent of mean aortic blood acceleration, and SV is obtained from a mean velocity calculation, it is suggested that equations 35, 37, and 38 produce a mean acceleration surrogate of SV, which is impossible according to equation 6. Thus,
Based on the above discussion, this leads to the inevitable conclusion that,
In view of this discussion, the outflow correction factor, dZ/dtmax x TLVE, is rendered moot, because dZ/dtmax represents axial blood acceleration and not radially-oriented rate of change of volume according to
As per equation 46, the relationship between ohmic mean velocity and ohmic mean acceleration is parabolic (ohmic mean acceleration = (ohmic mean velocity)2 or y = x2. This relationship predicts that, over wide ranges of dZ/dtmax, ohmic mean velocity will be overestimated and underestimated at its upper and lower extremes, respectively.
Indeed, Yamakoshi et al.  showed that ICG-derived SV overestimates reference method SV in healthy canines by as much as 70%, and correspondingly underestimates reference SV by as much as 25% when myocardial failure is induced. Similarly, Ehlert and Schmidt  could not find a linear relationship between ICG and EMF-derived SV over a wide range of hemodynamic perturbations.
As inferred, the static specific resistance of blood, ρb, nor the static specific resistance of the thorax, ρT, are included in the ohmic acceleration-based SV method. Despite Visser’s  and Hoetink’s 
In contrast to the earlier methods, where the swept volume of the thorax (or portion thereof) is considered the appropriate VC, the VEPT corresponding to the new method is conceptualized physiologically and by magnitude as the intrathoracic blood volume (ITBV, VITBV). As opposed to linear-based volume conductors, using thoracic length or a height-based equivalent, ITBV is biophysically assumption-free, inherently unambiguous in physiologic meaning and intuitively understood as the physical embodiment of the blood resistance, Rb. Physiologically, ITBV has been shown to be highly correlated with left ventricular preload, as expressed by left ventricular end-diastolic volume (LVEDV), and thus with absolute values for SV and directional changes thereof [58–61]. Computationally, VITBV is found through linear allometric equivalents of body mass (kg). Supporting this relationship, studies show that body mass correlates linearly, and much more closely with total blood volume (TBV), SV and CO than patient height [62–66]. By magnitude, the ITBV represents approximately 25% of TBV, or about 17.5 mL⋅ kg–1, which results in VITBV = 17.5 x Wkg, or equivalently 16Wkg1.02 [37,67] By comparison, existing volume conductors associated with the plethysmographic hypothesis are modeled as simple geometric abstractions, which are firmly rooted in basic electrical theory. By virtue of their “best-fit” mathematical construction, they bare little relevance to, and have virtually no biophysical relationship with other commonly accepted physiologic, anatomic, or hemodynamic parameters.
Evidence suggesting that dZ/dtmax varies inversely with aortic valve CSA, or radius r, as a function of PARABA, is inferentially demonstrated through the work of Sageman . He clearly showed that an inversely proportional and highly negatively correlated (r = –0.75) relationship exists between dZ/dtmax/Z0 and body mass in healthy humans. Aortic valve CSA has been shown to correlate highly with body mass (kg) and body surface area (BSA, m2). But by contrast with dZ/dtmax/Z0, Doppler and EMF peak velocities and systolic velocity integrals are totally independent of body mass . Thus, Newtonian-based peak velocities of equal magnitude, measured between age-matched individuals of different body mass and aortic valve CSAs, will produce correspondingly disparate values of dZ/dtmax/Z0. It follows that, while there is no direct proportionality between hemodynamically-based and impedance-derived systolic velocity integrals within or between individuals, a linear equivalence exists through their respective mean flow values. Specifically, as a convoluted abstraction of the equation of continuity,
Because of the absolute dependency of dZ/dtmax/Z0 upon PARABA, this means that, for any given value of mean acceleration, the magnitude of dZ/dtmax is related to and explicitly dependent on aortic root CSA by its dependency on R. Since aortic root CSA is a function of body mass, age, and gender, dZ/dtmax will vary accordingly. Thus, the magnitude of dZ/dtmax is multi-factorial and not wholly dependent on the respective levels of myocardial contractility and Z0. As a first order approximation,
One of the major drawbacks of the impedance technique has been its inability to correctly predict SV in the presence of excess EVLW [70–72]. Critchley et al.  tested the hypothesis that the poor agreement between invasive reference standards and ICG is due to excess EVLW. Typified by Sepsis, they were able to show that ICG CO underestimated its thermodilution counterpart and that the degree of underestimation was related to the degree of EVLW excess. They also noted that, in general, excess EVLW was associated with values of Z0<20–22Ω. In a subsequent experimental model in canines, where pulmonary edema was induced by oleic acid infusion, they found a systematic progressive bias between flow probe and ICG CO . As pulmonary edema progressively worsened, ICG CO progressively underestimated the flow probe estimate. In concert with the progressive divergence of the two methods, Z0 progressively decreased (fig. 10).
Expanding equation 30, the following is a useful analytical tool for studying the effect of excess EVLW on the various compartments of the transthoracic impedance with the AC field.
In this model it is assumed that AC (I(t)) flows exclusively through the blood resistance Zb (100Ω·cm–180Ω·cm) (element 2), despite the fact that Ze (element 3) is of lower specific resistance (60Ω·cm–70Ω·cm). This assumption is probably not accurate, because the blood resistance Rb is considered a cylindrical conductor surrounded by a more highly conductive EVLW impedance Ze. By the reciprocal rule for parallel impedances, the resultant impedance (i.e. Zb||Ze) would be lower than either impedance alone. If Ze is of variable magnitude and Zb is held constant, then Z0 should vary with Ze. This is precisely what is observed clinically. When Ze reaches a critical volume (Ve(CRIT)), Ze becomes the lowest impedance as relates to the components of Z0. This causes electrical shunting, and in the extreme case (Z0 =10Ω–12Ω), a complete short circuiting of current away from the blood resistance. This results in preferential flow through Ze at the expense of Zb, with the result being a decrease in the magnitude of Z0 and the amplitude of ΔZb(t) . At this critical level of Z0 (i.e. Z0(CRIT), ZC), which in humans is 20Ω±3Ω, excess EVLW causes spuriously reduced values of dZ/dtmax. Thus, dZ/dtmax no longer parallels its hemodynamic equivalent, PARABA, resulting in systematic underestimation of ICG SV/CO. In order to compensate for pathologic conduction through excess EVLW, an index of transthoracic aberrant conduction has been derived (ζ, zeta), which, as its magnitude decreases, creates a larger VEPT.
In patients without excess EVLW, VEPT is equivalent to VITBV. For all values of Z0<20 Ω, 0<ζ<1 and for Z0≥20 Ω, ζ=1. For all values of Z0<20 Ω, the following pertains:
where ζ is given as ,
where ZC = critical level of base impedance Z0(CRIT) = 20Ω, corresponding to Ve(critical). Z0 = measured transthoracic impedance ≤ ZC, and K = a trivial constant →0.
where VITBV = 16W(kg)1.02. Equation 55  has been prospectively tested in critical and non-critically-ill patients and been shown to provide SV and CO values comparable to standard reference methods [37,75–78]. As reviewed and compiled by Moshkovitz et al.  there are other whole body and transthoracic equations, but they have not been effectively prospectively tested.
From the results of this review, it is strongly suggested that the conceptually-based plethysmographic hypothesis for ICG-derived CO is flawed. The incontrovertible evidence provided herein shows quite conclusively that dZ/dtmax is a mean acceleration analog and not that of the peak rate of change of aortic volume. Therefore, to obtain ohmic mean velocity from dZ/dtmax/Z0, square root transformation is obligatory. Because dZ/dtmax represents axial blood acceleration, the “outflow compensation” for runoff is entirely irrelevant as a theoretical problem to be proved or disproved, as per the work of Faes et al. (36). The fact that dZ/dtmax is a mean acceleration analog permits the square wave integration, dZ/dtmax x TLVE, as a valid convention similar to equations 6 and 49 for the Doppler/EMF method of SV determination. As for the volume conductors implemented by both the Nyboer/Kubicek and Sramek/Bernstein models, there seems little physiologic justification for either approach. The volume of electrically participating thoracic tissue involved in dynamic conduction is clearly the blood resistance. By magnitude, the blood resistance translates directly into the intrathoracic blood volume. While basing the volume conductor on a fixed volume appears to be concordant with the Sramek/Bernstein model, this can only be true when dynamic conduction through the blood resistance is in parallel with peak aortic reduced average blood acceleration. This can only occur, when, by volume, the blood resistance is the compartment with the lowest impedance. Thus, when excess extravascular lung water exceeds some critical volume, current is diverted to the compartment of lowest impedance, which is now that of the extravascular lung water. With progressive diversion of AC away from the blood resistance, the magnitude of ΔZb(t) and dZ/dtmax diminish and therefore are not reflective of or proportional to the hemodynamic state. Clearly, a more robust mathematical solution for excess extravascular lung water is desirable; that is, if is at all possible.