J Electr Bioimp, vol. 1, pp. 2–17, 2010 Received 25 Nov 2009, published: 3 Dec 2009 URN:NBN:no-23978 Review Article Impedance cardiography: Pulsatile blood flow and the biophysical and electrodynamic basis for the stroke volume equations
Donald P. Bernstein
Palomar Medical Center, Escondido, CA, USA E–mail: [email protected]
Abstract Keywords: Impedance cardiography, stroke volume, cardiac Impedance cardiography (ICG) is a branch of bioimpedance output, dZ/dtmax, acceleration, volume conductor, extravascular primarily concerned with the determination of left ventricular lung water 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 Pulsatile Blood Flow magnitude, high frequency, low amplitude alternating current (AC). By Ohm’s Law, the voltage difference measured within the Stroke volume (SV) is defined as that quantity of current field is proportional to the electrical impedance Z (Ω). blood ejected from the cardiac ventricles for each heart Without ventilatory or cardiac activity, Z is known as the beat. More specifically, it relates to the volume of blood transthoracic, static base impedance Z0. Upon ventricular ejection, ejected from opening to closure of the semilunar valves, a characteristic time dependent cardiac-synchronous pulsatile and in the case of the left ventricle, the aortic valve. The impedance change is obtained, ∆Z(t), which, when placed time interval over which left ventricular ejection occurs is electrically in parallel with Z0, constitutes the time-variable total transthoracic impedance Z(t). ∆Z(t) represents a dual-element known as left ventricular ejection time (TLVE, s). For composite waveform, which comprises both the radially-oriented purposes of simplicity, and considering this is not an in- volumetric expansion of and axially-directed forward blood flow depth analytical treatise on pulsatile blood flow, per se, the within both great thoracic arteries. In its majority, however, ∆Z(t) following simplistic model is proposed as an operational is known to primarily emanate from the ascending aorta. tool for understanding the opposing hypotheses concerning Conceptually, commonly implemented methods assume a SV determination by means of the transthoracic electrical volumetric origin for the peak systolic upslope of ∆Z(t), (i.e. –1 bioimpedance technique (TEB). dZ/dtmax), with the presumed units of Ω·s . A recently introduced In geometric terms, consider stroke volume V (mL) a method assumes the rapid ejection of forward flowing blood in cylinder of length S (cm) and cross-sectional area earliest systole causes significant changes in the velocity-induced –1 π 2 2 blood resistivity variation (∆ρb(t), Ωcm·s ), and it is the peak rate r (CSA, cm ). In this model it is assumed that SV is –2 of change of the blood resistivity variation dρb(t)/dtmax (Ωcm·s ) surrounded by a thin-walled viscoelastic aorta, equal in that is the origin of dZ/dtmax. As a consequence of dZ/dtmax length S to the SV contained within at end-systole. peaking in the time domain of peak aortic blood acceleration, Hemodynamically, as a function of ventricular ejection, let –2 dv/dtmax (cm·s ), it is suggested that dZ/dtmax is an ohmic mean –2 S (cm) also represent the time velocity integral, also known acceleration analog (Ω·s ) and not a mean flow or velocity as stroke distance (cm). surrogate as generally assumed. As conceptualized, the normalized value, dZ/dtmax/Z0, is a dimensionless ohmic mean 2 acceleration equivalent (s–2), and more precisely, the electro- Vr= π S (mL) (1) dynamic equivalent of peak aortic reduced average blood –2 acceleration (PARABA, d
Bernstein: Impedance cardiography. J Electr Bioimp, 1, 2-17, 2010
i –1 derivative 1 of equation 2, if 2/π rdr dt equals dA/ dt QA= × v (mL·s ) (3) ISA Wall (time-rate of change of aortic area, cm2·s–1), then Q(t) for
derivative 1 can be given as, 2 For derivative 2, if πr is the CSA of the aorta at a discreet point of measurement and dS(t)/dt the axial blood i dA dV() t –1 velocity (v, cm·s–1) measured at that point, then axial Q can QS= = (mL·s ) (7) dt dt be given as,
i –1 QA=CSA × v Blood (mL·s ) (4) 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 Inasmuch as ACSA changes continuously over the –1 cardiac cycle as a function of aortic valve radius dr, the dP(t)/dt (mmHg·s ), respectively, the following pertains following is an animated representation of derivative 2 of [3]: equation 2 and equation 4 [1]. dV() t dP () t –1 = C (mL·s ) (8) i RRdA dt dt Q= 22π rv= π rdrv (mL·s–1) (5) ∫∫00dr where the right hand side (rhs) of equation 8 defines the At an operational level, derivative 2 of equation 2 and magnitude and shape of the flow wave, and C is the aortic equations 4 and 5 represent the basis for SV determination compliance. by means of both the Doppler/echo method and If aortic dP(t) is a function of force F and area A, both electromagnetic flowmetry. To find SV simply requires changing with time over the ejection interval, dP(t) –1 integration of the velocity profile over the ejection period at (mmHg) and dP(t)/dt (mmHg ⋅ s ) are given as, a discreet point in the aortic root and multiplying this by the aortic root CSA at the point of velocity measurement. dF() t dP () taorta 1 dF () t F dA()t dP() t =⇒=− (9) aorta dA() t dt A dt A2 dt Common sites for measurement include the left ventricular outflow tract and aortic valve leaflets. Thus, respectively, and compliance C is given as, t = π 2 1 π 2 ×× =π 2 SV r∫ v() t dt = r v tLVE rS (mL) (6) t0 dA dV CS= = (mL·mmHg–1) (10) dP dP where v is the mean velocity during ejection and the interval t0 to t1, TLVE. Extrapolating derivative 1 of equation where dA/dP is a non-linear function of Pulse pressure (∆P) 2 and equation 3 for determination of SV is something very and mean distending pressure Pm. Thus, much more complex, for it represents the geometric foundation of the windkessel model for SV determination dV() t dA ( P ) dP () t dA ( P )⎛⎞ 1 dF () t F dA () t = = − [1]. For this model of pulsatile blood flow, it is assumed SS⎜⎟2 = dt dP dt dP⎝⎠ A dt A dt that the proximal aorta acts as a compliance chamber C and C dF() t CF dA () t the distal vasculature a static resistance, Rs. Upon the force − –1 2 (mL·s ) (11) of ventricular ejection in early systole, and because of distal A dtA dt vascular hindrance, the proximal aorta expands, storing a part of the inflow (Q(t)in) as potential energy (Q(t)stored). where the mechanical expression for dV/dt of equation 11 Simultaneously, due to open outflow, blood is expelled is equivalent to the geometrical definition for dV/dt of axially (Q(t)out) to the periphery, where Q(t)out is commonly derivative 1 in equation 2. As an analytical expansion of referred to as “runoff”. When aortic pressure exceeds left equation 8, the rhs of equation 11 more precisely defines ventricular pressure from mid-systole to aortic valve the temporal landmarks, shape and magnitude of the closure, the recoiling aorta converts potential energy to radially oriented flow wave. kinetic energy, expelling the stored blood to the periphery. Figure 1 shows the pressure curve P(t), the rate of From aortic valve closure (end-systole) to end-diastole, change of aortic pressure, dP/dt, aortic blood flow (Q(t)) non-pulsatile flow dissipates along a quasi-mono- and the rate of change of flow, which is acceleration of exponential pressure decay. Pulsatile blood flow is flow, dQ(t)/dt. It is clear from equation 11 that dF/dtmax, the therefore buffered and converted from periodic oscillating major derivative, will occur early in systole, when dA/dt is flow to near-continuous flow for each cardiac cycle [2]. It is trivial. Likewise, dA/dtmax will occur when dF/dt = 0. When this aforementioned physical phenomenon that is the basis dP/dt is modulated by C, dA/dtmax will occur at the peak for conceptualization of the plethysmographic-based ICG- magnitude of the flow wave dV/dtmax (i.e. Qmax) and derived SV equations, and the mechanics underlying the dF/dtmax will occur on the steepest portion of the flow wave, windkessel model for pulsatile blood flow. Referring to
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immediately following aortic valve opening in early Rs. Rs is defined as the quotient of mean pressure to mean systole. flow Pm/Qm. Q(t)total is thus given as [4],
i dPt() Pt () –1 QCin− total = + (mL·s ) (13) dt Rs
Integration of Q(t)strored yields stored volume inflow, SVstored:
t1 dP() t = = Δ SVstored C∫ dt CP (mL) (14) t0 dt
where ∆P is the pulse pressure, which is aortic peak systolic
pressure Ps minus aortic diastolic pressure Pd (Ps–Pd) [5]. Since SV = C∆P, 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 [6].
i dP() t dCP () t –1 QCin = + P (mL·s ) (15) dt dt
where derivatives 1 and 2 represent Q(t)stored and Q(t)out, respectively. Differentiating dCp(t)/dt results in,
Fig. 1. ECG, aortic pressure, aortic blood flow Q, aortic rate of change of ⎛⎞ΔV pressure dP/dt, and rate of change of flow, which is acceleration dQ/dt. d ⎜⎟ dCp () t ΔP 1 dV V dP From a canine model In: Li J-K. The arterial circulation: physical =⎝⎠ = − 2 (16) principles and clinical applications. Totowa, NJ: Humana Press; 2000, p. dt dt P dtP dt 78. In terms of total flow into segment S, equations 12 and 15 When flow is differentiated with respect to time are equivalent. Integration of equation 13 over an ejection dQ/dtmax will occur when dA/dt and dV/dt are minimal. interval yields SV and is thus given as [2], Also apparent is that when flow acceleration = 0, dV/dt is at maximum. Therefore, dQ(t)/dtmax and dV/dtmax are i tt111 dP() t 1 t monotonically restricted to their respective time-domains, SV=∫∫ Q() t dt = C dt+ ∫ P() t dt (17) tt00 t 0 dt Rs 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 where equation 17 represents a two-element windkessel flow into the segment would be the total input flow from model. Derivatives 1 and 2 on the rhs equation 17 represent left ventricular ejection. If, however, segment S is an open Vstored and Vout, respectively. outflow conduit, equation 8 would represent a net flow. For Although a theoretically correct model for SV, a simple 2-element windkessel model, net flow into the equation 17 is not easily solved. Clearly, P(t), dP(t)/dt, and segment (Q(t)in) is given as [4], Pm can be computed by direct measurement and signal processing, but C and Rs must be obtained independently by alternative means [2]. This is clear from equation 10 for the i dPt() Pt () –1 QC= − (mL·s ) (12) determination of C and the relationship of Rs to Q(t)mean in in− net dt R s equation 17. From equations 15 and 16, it is also clear that C is not a constant as equation 17 implies. Pressure where derivatives 1 and 2 on the rhs of (12) represent dependent C and the rate of change of C with respect to Q(t)stored and Q(t)out, respectively. time, dC/dt, are not accounted for in equation 17. To more To obtain total flow Q(t)total through the segment over accurately predict SV from the windkessel model, it is a single systolic interval (TLVE, s), Q(t)out must be measured. necessary to incorporate the transverse frequency- With regard to a two-element windkessel model, Q(t)out is dependent pulsatile resistance, also known as mechanical or obtained by dividing the pressure change over the ejection characteristic aortic input impedance, Zc. Zc is approxi- interval P(t) by the distal, or systemic vascular resistance, mated by aortic pulse pressure divided by peak aortic blood
flow (∆P/Qmax). Incorporating characteristic aortic impe-
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dance into the windkessel model, which constitutes the part (XC), both collectively known as the reactance. With three-element model, is now considered the “gold-standard” regard to the magnitude of X, XL is proportional and XC [4]. Thus, as an approximate unifying statement for SV inversely proportional (1/f) to the frequency of the applied from equations 5, 6 and 17: AC. │Z│ is determined by the geometric addition of the resistive and reactive components.
Rt1`t dP() t 1 t1 SV=∫∫2ππ rdr v () t dt =≅+ r2 S C ∫ dt ∫ P( t) dt 0 tt00t0 2 2 dt Rs Z= R +−() XX (20) LC (mL) (18) where the reactance X is, Impedance Cardiography X = (XX− ) (21) Modeling the transthoracic impedance Z(t) L C
Thus, Impedance cardiography (ICG) is a branch of bioimpedance primarily concerned with the determination Z =RX22 + (22) of left ventricular stroke volume (SV) [7–10]. It was the first and still the only truly noninvasive, continuous, where the magnitudes of R and X on Cartesian coordinates operator-independent, hands-off, beat-to-beat method used are determined by the phase angle φ, and thus, R tan φ = X. in clinical practice. Despite these desirable attributes, For the remainder of the discussion, let Z = Z. general acceptance of the method as a replacement for invasive reference standards in critically-ill humans has not For cylindrical electrical conductors, and as a been realized. As summarized by Raaijmakers et al. and companion equation to equation 19 [13], Moshkovitz et al., both correlation and agreement against ρLLρ 2 standard reference methods have been far too variable to Z = ≡ (Ω) (23) reliably apply the method clinically for modulation of A V patient therapy [7,12]. As emphasized by Kauppinen et al. [11], the major problem in cardiovascular impedance where ρ = the resistivity (Ω⋅ cm), L = the length of the measurements is the inability to accurately correlate the conductor, and A and V are the CSA (cm2) and volume magnitude of the impedance waveforms to the (mL) of the conductor, respectively. Thus, substituting hemodynamic variables they pretend to mimic. The lack of ρL2/V in equation 23 for Z in equation (19) and consistently high correlation and agreement between ICG rearranging,2 measurements and reference standards would suggest that, the equation models describing SV in ICG are not ρL2 sufficiently robust and the physical acquisition of the I()tU⋅= ()t (U) (24) V impedance signals not sensitive and specific enough for clinical application in the critically ill [11]. If the thorax is considered a cylindrical bulk electrical While measurements can be obtained by the whole conductor of length L between the voltage sensing body technique, the method is usually implemented by electrodes, with ρT the transthoracic specific resistance and means of the transthoracic approach (i.e. transthoracic VT the volume of thorax, then, without ventilatory or electrical bioimpedance cardiography, TEB, ICG) [7]. The cardiac activity, Z in equations 19 and 23 represents the latter technique, and subject of this review, involves adynamic or static transthoracic base impedance Z0. When applying a current field longitudinally across a segment L cardiac and ventilatory activity are superimposed on Z0, a (cm) of thorax by means of a constant magnitude, high time-variable transthoracic impedance is registered, Z(t). frequency (50–100 kHz), low amplitude AC (I(t)) (1–4 By eliminating the oscillating cardiac-asynchronous mA). By Ohm’s Law, the voltage difference U(t) (volt, U) ventilatory component ∆Zvent, Z(t) comprises, in parallel, a measured within the current field is proportional to the static DC component Z0 (22Ω–45Ω) and a dynamic AC transthoracic electrical impedance Z. component ∆Z(t) (0.1Ω–0.2Ω) [14,15] (Fig.2). It should be
noted that the magnitude of Z0 not only varies between Ut() = Z (Ω) (19) individuals and the frequency of the applied AC, but also It() with the electrode configuration used for signal acquisition. For four common electrode configurations a computer where Z is the frequency-dependent AC analog of the static model predicts a Z0 range of 26.3Ω–34.3Ω [11]. DC resistance R. The magnitude of the impedance, or modulus, │Z│, comprises a resistive component, also 2 known as the real part, R, and an imaginary part X. X is For the remainder of the discussion, the character “U” is used in lieu of further comprised of an inductive part (XL) and capacitive “v” for volt, so as not to confuse v with velocity “v” or volume “V”.
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When electrical resistances or impedances are added in Ω·cm·s–1), where, in end-diastole, the state of highest blood parallel, they are summed as their reciprocals. resistivity, red blood cells are randomly oriented. By contrast, during peak aortic blood acceleration and the rapid 11 1 ejection phase of early systole, red blood cells become =Δ= =+ Zt() Z0 Zt () (25) deformed and assume a well defined state of parallel Z()t Z0 Δ Zt () orientation along their long axis of symmetry [19,22,23]
(fig. 3). As a composite impedance with tissues of widely varying specific resistances, Z0 comprises, in parallel, a static, relatively non-conductive multi-compartmental tissue 20 impedance Zt (400Ω·cm–10 Ω·cm) [11,16], a highly conductive blood impedance Zb (or resistance Rb 100Ω·cm– 180Ω·cm) varying with hematocrit [17], and a very highly conductive interstitial extra-vascular lung water (EVLW) impedance Ze (60Ω·cm–70Ω·cm).
Fig. 3. Behavior of AC as applied to pulsatile blood flow. → = AC flow. v
= red cell velocity; ∆ρb(t) = changing specific resistance of flowing blood.;
Qin = flow into aortic segment; Qout = simultaneous flow out of aortic segment; dA(t) = time-dependent change in aortic CSA. From reference Fig. 2. Schematic of a multi-compartmental parallel conduction model of 13. the thorax. The transthoracic electrical impedance Z(t) to an applied AC field represents the parallel connection of a quasi-static base impedance Z0 Parallel alignment opens clear current pathways and a time-dependent component of the blood impedance, ∆Z (t). Z b 0 through the highly conductive plasma, causing a decrease in represents the parallel connection of all static tissue impedances, Zt, and the static component of the blood resistance, Zb. Ze represents the quasi- blood resistivity and transthoracic resistivity [19,22] static EVLW compartment. A voltmeter (U) and AC generator (~) are (i.e.↑Conductivity). The rapidity of and degree to which shown. From reference 37. attainment of complete parallel orientation is achieved
determines the magnitude of the resistivity change ∆ρ (t) Z is thus an intensity weighted mean of the reciprocal sum b max 0 (Ω·cm·s–1) [23]. The maximum resistivity change occurs of all thoracic tissue impedances, the sum of the reciprocals when the long axis of the red cell is oriented within 200 to being less than the lowest tissue impedance of the thorax. the direction of blood flow [23]. Figure 4 shows that, in early ejection, maximum acceleration of red cell reduced 1 111 average velocity (i.e. mean spatial velocity) parallels the Z0 = ZZZtbe = =++ (26) Z ZZZ change in blood resistivity (i.e. conductivity) (r = 0.99). By 0 tbe comparison, upon negative acceleration, the resistivity change does not parallel the reduced average velocity, with In this model, the blood resistance Rb is considered a cylindrical tube of constant length L, surrounded by the subsequent delay in reaching baseline [19,23]. This disparity is, no doubt, a consequence of the red cells’ highly conductive extravascular lung water impedance Ze and both surrounded by a non-conductive thoracic inability to achieve complete randomization at end-systole. 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.
∆Zb(t): The Transthoracic Cardiogenic Impedance Pulse Variation
The velocity and volume components
The systolic portion of the cardiogenically-induced impedance pulse variation ∆Z(t), hereafter known as ∆Zb(t), Fig. 4. Absolute spatial average velocity (i.e. reduced average velocity comprises two components, arguably of equal magnitude
Bernstein: Impedance cardiography. J Electr Bioimp, 1, 2-17, 2010
The second component generating a change in Stroke Volume Equations transthoracic specific resistance is the transversely or laterally-oriented volume displacement of non-conductive Nyboer equations: Foundation and Rationale for the alveolar gas (ρ =1020 Ω·cm) by stroke volume-induced Plethysmographic Hypothesis and Parallel Conduction expansion of the ascending aorta, principally [24,25], with Model in Impedance Cardiography highly conductive blood (ρb=100–180 Ω·cm) ∆Vb(t) (∆Ω·cm(t)). As discussed under pulsatile blood flow, in The original Nyboer equation [27] was specifically addition to distal vascular hindrance Rs, the volumetric proposed for determination of segmental blood volume expansion of the aorta is due to pressure-induced, changes in the upper and lower extremities. It is based upon compliance modulated changes in cross sectional area, πr2 the assumption that the arteries of the extremities are rigidly (cm2), and, over segment L, a change in internal surface encased in muscle and connective tissue, and that both the area, 2π·(dA/dr)L, (cm2) [1,26]. For example, assume that arteries and surrounding tissue can be approximated as equation 23 represents the impedance Z of a segment of cylindrical conductors placed in parallel alignment. Nyboer very thin walled aorta embedded within the thorax. Assume found that, by placing spaced-apart circumferential current that the aortic segment is of length L with open outflow, the injecting electrodes on a limb segment and voltage sensing content of which is blood of static specific resistance ρb and electrodes within the current field proximate the current volume Vb. Thus, an increase in aortic volume ∆Vb(t) injectors, impedance changes proportional to strain gauge would result in a corresponding decrease in vessel and determined blood volume changes could be measured. He transthoracic impedance ∆Zb(t)volume. also determined that, in order to compensate for simultaneous volume outflow from the limb segment during ρ L2 arterial inflow, venous outflow occlusion was necessary. ↓ΔZt() = b (∆Ω(t)) b volume Thus, the ∆Zb(t) waveform represents the sum of two ↑ΔVtb () signals: one caused by inflow and the other caused by ρ 2 b L outflow (runoff) between the voltage sensing electrodes ⇒ ↑ΔVtb () = (∆volume(t)) (27) ↓ΔZb ()t [28]. Using venous outflow obstruction, absolute volume changes during each pressure pulse were accurately where the rhs of equation 27 is the ohmic equivalent of approximated. Nyboer referred to this method, and quite equation 14, wherein ∆Zb(t) and ∆Vb(t) represent net accurately as, electrical impedance plethysmography. The changes. The combined velocity and volume effects cause a foundation of Nyboer’s equation, and for that matter the steep drop in transthoracic impedance. earliest SV equations, originate from equation 27, where, For the velocity-induced change in blood resistivity, when solving for ∆Vb(t)net, the decrease in vessel and transthoracic impedance is given as, ρ L2 Δ=Vt() b ( mL) (31) b net ΔZb()t net Δρ 2 b ()tL –1 Δ=Ztb() velocity (Ω·s ) (28) Vb However, since a large extremity artery (i.e. brachial or femoral) is encased in muscle and connective tissue, where ∆ρb(t) is of diminishing value upon ejection. ∆Zb(t) had to be further refined, reflecting a parallel Combining equations 27 and 28, connection of the static tissue and dynamic blood impedances. Since the magnitude of Z(t) of an extremity is 22 larger than extremity Z0 by a trivial factor of extremity Δρρbb()tL L ΔZtb () = + =ΔZtb() velocity +Δ Zt b () volume (29) ∆Zb(t), the approximation that extremity Z(t) and Z0 are VΔ Vt() bb virtually identical is a plausible assumption. Without
assumption, it also follows that Z(t)–Z0 = ∆Zb(t). Thus, if Combining equations 24, 26, and 29, the following equation 25 is solved for ∆Zb(t), using the reciprocal rule impedance model describes the static and dynamic for impedances added in parallel: impedances in parallel, when an AC field is applied longitudinally across the thorax (fig 2). 2 Z0 ⋅ Zt() Z0 Δ=Ztb () ≅ (Ω(t)) (32) Z0 −Zt() −Δ Zb () t It()⎡⎤ ZZZΔ Zt() +Δ Zt () + U +Δ U()t (30) ⎣⎦⎢⎥()t b e() b velocity b volume 0 b Substituting the right hand side of equation 32 for ∆Zb(t) into equation 31 results in the operational model for measurement of blood volume changes in an extremity. For the maximum volume change ∆Vb(max) with venous outflow occlusion [28],
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ρ L2 ρ L2 ΔV =−Δb Z (mL) (33) SV =−Δb Z (Down-slope extrapolation) (34) b(max) Z 2 max Nyboer Z 2 max 0 0
where ∆Zmax is the peak magnitude of the waveform and where Z0 is the static transthoracic base impedance and 2 2 ρbL /Z0 is a constant. For an arterial conduit with closed ∆Zmax is the maximum backward extrapolated value of the outflow, equation 33 is the bioelectric equivalent of the thoracic cardiogenic impedance pulse variation, which hemodynamic expression given for SV in equation 14 with includes runoff. While credible SV values could be outflow obstruction. For an extremity vessel with open obtained with equation 34, it was not widely accepted, this outflow, equation 33 is analogous to the integrated value being due to difficulty in manually determining the true for equation 12. Clearly, as applied to SV calculations, maximum downslope. equation 33 is inadequate, because complete aortic outflow obstruction for calculation of total SV is impossible. Kubicek’s Method: Maximum Systolic Upslope Forward
Extrapolation of ∆Zb(t) Stroke Volume Calculation As a consequence of the deficiencies of the backward Nyboer’s method: Maximum Systolic Down-slope Back- extrapolation procedure, Kubicek et al. proposed a ward Extrapolation of ∆Zb(t). maximum systolic forward extrapolation of ∆Zb(t), which is the basis for all subsequent plethysmographic conceptually- As applied to thoracic measurements of SV, Nyboer based SV equations using the transthoracic approach professed to have solved the outflow problem by manually [29,30]. Kubicek et al. made the assumption that, if the –1 –1 determining the maximum systolic down-slope of ∆Zb(t) maximum systolic upslope (i.e. ∆Z′ = ∆Z·s , Ω·s ) is held and extrapolating it backward to the beginning of ejection constant throughout ejection, then compensation for
[28]. Thus, the maximum impedance change resulting from outflow, before and after attainment of Qmax is achieved. the backward extrapolation method was believed to be Theory underlying Kubicek’s method assumes that little equivalent to the maximum impedance change attained as if arterial runoff occurs during the inertial phase of rapid no arterial runoff occurred during ejection. systolic ejection. Unlike Nyboer’s direct measurement of
∆Zmax, Kubicek’s method requires multiplying ∆Z′ (forward extrapolation) by left ventricular ejection time TLVE (i.e. ∆Z′× TLVE = ∆Zmax). In the original description of the technique, TLVE was determined by means of phono- cardiography. As opposed to manual extrapolation of ∆Z′,
electronic differentiation of ∆Zb(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, ∆Zmax = dZ/dtmax × TLVE = Ω. Kubicek’s equation is thus obtained by substituting dZ/dtmax × TLVE for ∆Zmax in equation 34. The Kubicek equation is given below and purportedly equivalent to equation 17. Thus [29,30],
2 ρb L dZ() t SVKubicek = TLVE (mL) (35) Z 2 dt Fig. 5. ECG, ∆Z(t) and dZ/dt waveforms from a human subject. For ∆Z(t) 0 max note Nyboer’s maximum down-slope backward extrapolation to find ∆Zmax. Also note Kubicek’s maximal systolic up-slope forward where ρ is the static specific resistance of blood, which extrapolation of ∆Z(t). For dZ/dt, point B = aortic valve opening; point X b = aortic valve closure; Y = pulmonic valve closure; O = rapid ventricular was initially fixed at 150 Ω·cm [29]. As concerns the filling wave; Q–B interval = pre-ejection period (s); B–C interval = time- appropriate value for ρ, Quail et al [31] rearranged the to-peak (TTP, s) of dZ/dtmax; B–X interval; left ventricular ejection time Kubicek equation (assuming it correct), solved for ρ as the (TLVE, s). dZ/dt waveform to right shows the square wave integration dependent variable, and measured SV by EMF. By means (shaded area), dZ/dtmax remaining constant over the ejection interval, which represents outflow compensation. Modified from reference 13. of normovolemic hemodilution exchange transfusion, they showed that, over a wide range of hematocrit from 26%– Thus, SV, or the maximum volume change ∆Vb 66%, the value of ρ remained virtually constant about a measured between the voltage sensing electrodes is given mean of 135Ω·cm at hematocrit 40%. L (cm) is the as, 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- 8
Bernstein: Impedance cardiography. J Electr Bioimp, 1, 2-17, 2010
static base impedance measured between the voltage 9. Outflow, or runoff during ventricular ejection, can be sensing electrodes. In an in vitro expansible tube model, compensated for by extrapolating the peak rate of when equation 35 is compared to equation 33 without change of ∆Zb(t) over the ejection interval (i.e. dZ/dtmax outflow occlusion, equation 33 systematically × TLVE) to obtain ∆Zmax [29–30]. underestimates SV obtained from 35 [26]. By elimination 10. The specific resistance (resistivity) of blood ρ is of a Z term, the volume conductor V in equation 35 is b 0 C constant during ejection [29] given as, 2 11. The transthoracic specific resistance, ρT, is constant ρb L VC = (mL) (36) [31]. Z0
where, when Z0 varies with alterations in extravascular lung Sramek-Bernstein Method: A Constant-Magnitude Volume water, the VC is non-constant. Equation 35 is, in theory, of Electrically Participating Thoracic Tissue VEPT modeled as the ohmic equivalent of windkessel equation 17, where outflow correction is explicitly defined. Thus, by The assumptions of the Sramek-Bernstein equation are analogy and inspection of equation 17, Kubicek’s equation virtually identical to those of the Kubicek method, except is conceptually and explicitly dependent on aortic pressure, for the physical definition and magnitude of the VC [32]. In the rate of change of aortic pressure and mean arterial Sramek’s interpretation of Kubicek’s VC, ρ and a Z0 pressure. However, by comparison with equation 17, and variable are eliminated by mathematical substitution. This without theoretical or mathematical explanation, Kubicek’s simplification assumes, as per the work of Quail et al. [29] equation implicitly integrates the unknowns of compliance that, the ρT equivalent, Z0A/L can replace ρT. By this
C, characteristic (mechanical) aortic input impedance (Zc) substitution, VC is rendered a personal constant for each and systemic vascular resistance Rs into the overall individual. Sramek [32] named the modification of theoretical gestalt. Kubicek’s VC as the volume of electrically participating thoracic tissue, VEPT. Conceptually, as opposed to Assumptions of the Nyboer/Kubicek method [14,15]: Kubicek’s cylindrical VC, Sramek’s VEPT is geometrically a frustum, or truncated cone. Full mathematical derivation 1. The transthoracic impedance Z(t) is considered the and justification for Sramek’s model is provided elsewhere parallel connection of an aggregate of static cylindrical [32]. The original Sramek equation is given as,
tissue impedances, Zt, considered as one, and, encased 3 within, a dynamic blood resistance, Rb, otherwise dZ// dt L dZ dt SV= V max T = max T known as the blood impedance, Zb. Sramek EPT ZZLVE 4.25 LVE 00 2. The blood resistance is considered a homogeneously (mL) (37) conducting blood-filled cylinder of constant length L, or a parallel connection of an aggregate of cylinders where L is the measured distance (cm) between the voltage considered as one. sensing electrodes and 4.25 an experimentally-derived 3. The current distribution in the blood resistance is constant. The Bernstein modification of VEPT assumes that uniform. SV is not only a function of thoracic length, but also body weight and blood volume. Correcting for the magnitude of 4. All current flows through the blood resistance. the blood resistance (i.e. intrathoracic blood volume, ITBV, mL) and using the work of Feldschuh and Enson [33], a 5. The volume conductor VC is homogeneously perfused factor δ (delta) was appended to the Sramek equation. The with blood of specific resistance ρb. Sramek-Bernstein equation is given as [32], 6. The magnitude of stroke volume is directly proportional to power functions of measured distance L L3 dZ/ dt between the voltage sensing electrodes [29–30], or to SV = δ max T (mL) (38) SB− 4.25 Z LVE height-based thoracic length equivalents [32]. 0
7. All pulsatile impedance changes ∆Zb(t) are due to where L is a thoracic length equivalent, equal to 17% of vessel volume changes ∆V (t), and, in the context of b body height (i.e. 0.17⋅H) (cm), and delta δ is the weight assumption 2, ∆Z (t) is due exclusively to changes in b correction for blood volume. δ is a dimensionless vessel radius dr(t) and CSA dA(t). parameter, which corrects for deviation from ideal body
8. In the context of assumption 7, dZ/dtmax is the bioelec- weight (kg) at any given height and further modified for the –1 i indexed blood volume (mL·kg ) at that weight deviation tric equivalent of dV/dt , Q max max [32,33]. dA() t dV() t (i.e. ×=L ). See equation 7. dtmax dtmax
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Validity of the Plethysmographic Hypothesis and its 7. Origin of dZ/dtmax in the hemodynamic time domain Assumptions: [37].
In order for equations 35, 37 and 38 to yield Despite the objections to the plethysmographic equivalent results to equation 17, it must be demonstrated hypothesis, correlation with reference standards is that dZ/dt is explicitly dependent on aortic systolic max considered good (r2 = 0.67, range = 0.52–0.81 and, r = 0.82, pressure P(t), the rate of change of aortic pressure dP/dt, range = 0.70–0.90 [7,12]. However, high correlation and mean arterial pressure P , systemic vascular resistance R mean s reasonable agreement with invasive reference standards and aortic compliance C. In an early, well designed does not verify the plethysmographic hypothesis as correct. experimental study, Yamakoshi et al. [1] demonstrated that, It may simply mean that the product of dZ/dt , T , and in an in vitro distensible tube model, the magnitude of max LVE a best-fit volume conductor V , yield results that mimic impedance-derived SV is not related to any one of the C reference method SV [14,15]. above windkessel parameters. Despite their results, which clearly invalidate the plethysmographic hypothesis, the Origin of dZ/dt literature has ignored these findings. Data provided by max
Djordjevich et al. [34], though purporting to show a close Resolution of Origin by Differential Time-Domain Analysis relationship between dZ/dtmax and blood pressure levels, actually demonstrated that the correlations are weak, at If equation 23 is differentiated by parts with respect to best. Clinically, Brown et al. [35] demonstrated that, time, the following results: despite advancing age (20–80 years) and progressive stiffening of the aorta, ICG CO is nonetheless highly dZ() t dZlength () t dZ() t dZ () t correlated with thermodilution CO (TDCO). It is also = +−vel vol (39) dt dt dt dt interesting to note that, as an alleged analog of dV/dtmax, dZ/dtmax supposedly provides an ohmic mean velocity analog necessary for SV calculation (vide infra). As Furthermore, if all variables in equation 23 are continuously concerns the volume conductor (VEPT) of the Kubicek differentiable functions of time and are expressed within equation (equation 36), it seems quite improbable that the the respective dZ/dt derivatives of equation 39 [37], transthoracic specific resistance, ρT, remains constant in the 2 2 face of changing values of Z0. By inspection of equation 23, dZ() tρ 2 L dL () t L d ρρ() t L dV () t =bb +−b b (40) L and V remaining constant, it is apparent that this is dt V1 dt V dt V 2 dt b bb impossibility, because, ρT must vary directionally with its dependent variable Z0. Other unanswered questions Since dL and dL/dt are of trivial magnitude, dZ/dt concerning the validity of the plethysmographic hypotheses comprises derivatives 2 and 3, the units of which are Ω·s–2 include the following: and Ω·s–1, respectively. Using older plethysmographic techniques, which are 1. The proper value, physiologic definition, and conceptually analogous to the simple two-element theoretical basis for ρ: i.e. does ρ vary with hematocrit windkessel model, simultaneous outflow during inflow over (ρb), or is it a constant based on thoracic resistivity aortic segment L is compensated for by using the maximal (ρT)? systolic upslope extrapolation of ∆Zb(t). The peak slope is 2. The physiological relevance of measured L or thoracic conceptually believed to be the peak slope of ∆Zvol(t), and, length equivalents in the Kubicek and Sramek- according to theoretical assumptions, represents the ohmic Bernstein equations regarding SV. equivalent of the peak rate of change of aortic volume –1 (peak flow, dV/dtmax, mL·s ), at and before which little 3. The validity of the outflow extrapolation procedure outflow is thought to occur [36]. When extrapolated over [36]. the ejection interval, this convention is believed to be 4. A coherent physiologic basis and correlate for the proportional to total SV [29,30]. To better define the peak empirically-derived volume conductors and their slope, ∆Zb(t) is electronically differentiated to dZ/dt (i.e. relevance to SV. d[∆Zb(t)]/dt), the peak magnitude of which is dZ/dtmax and conceptually analogous to the peak value of derivative 3 of 5. Lack of consideration for the changing transthoracic equation 40 (i.e. d[∆Z (t)]/dt ). That is, base impedance (∆Z ) in critical illness, typified by vol max 0 increased thoracic liquids (pulmonary edema) causing dZ() tρ L2 dV () t aberrant electrical conduction [37]. vol = bb (Ω⋅ s–1) (41) dt V 2 dt 6. Lack of regard for the effect of the blood velocity- max b max
induced change in the transthoracic specific resistance A recently introduced ICG method conceptualizes the ∆ρb(t) [37]. peak value of derivative 2 of equation 40 to represent
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dZ/dtmax, which is the peak rate of change of the red cell anesthesia. With profound myocardial depression, it is velocity-induced blood resistivity variation dρb(t)/dtmax easily demonstrated that dZ/dtmax peaks synchronously with –2 –2 (Ω⋅ cm·s ) (i.e d[∆Zvel(t)]/dtmax, Ω·s ) [37]. Thus, dv/dtmax and appreciably before Q (t)max.
dZ() tL2 dρ () t vel = b (Ω·s–2) (42) dtmax Vb dtmax
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 (dv/dtmax, dQ/dtmax).
Resolution of Origin by Comparative Time-Domain Analysis: Evidence for a New Paradigm
Comparative time domain analysis confirms that peak flow velocity occurs at ∼ 100±20ms after opening of the aortic valve [38], whereas dZ/dt, (as extrapolated from the data of Matsuda et al. [39] and Lozano et al. [40]) and acceleration dv/dt peak at a mean value of 50 (Lozano)→60 Fig. 6. Relationship between esophageal ECG (A), aortic pressure (B), (Matsuda)±20ms. As testaments to the validity of these rise aortic expansion (C), aortic blood flow (D), pulmonic expansion (E), ∆Z(t) times, which is the temporal interval between aortic valve (F), and dZ/dt (G). Note that dZ/dtmax precedes peak aortic blood flow opening, designated as point B on the dZ/dt tracing to (Qmax) during the period of peak flow acceleration in early systole. Modified from reference 13. dZ/dt , data provided by Matsuda et al. [39] show that, max when the time interval from the Q wave of the ECG to Figure 6 shows that dZ/dtmax intersects the flow curve LVdP/dtmax is subtracted from the time interval between the during peak acceleration and appreciably before Q(t)max. Q wave and dZ/dt , TTP of dZ/dt = 60±15ms (i.e. max max Simultaneously obtained waveforms comparing TTP of [Q→dZ/dtmax(ms)] – [Q→LVdP/dtmax (ms)] = 60±15 ms). peak flow velocity and dZ/dtmax, showing the latter This analysis is valid, because, in the absence of aortic temporally preceding the former, can be found elsewhere valve disease, LVdP/dt almost invariably occurs within max [30,45,46,47]. 2–5 ms just prior to aortic valve opening. Corroborative evidence from Lozano et al., obtained by subtracting the R(ECG)→B(dZ/dt) interval (mean 68 ms, range 47–84 ms) from the R→dZ/dtmax interval (mean 120ms, range 80– 160ms), shows that the mean rise time (TTP) from point B to dZ/dtmax = 52±20 ms (i.e. [R→dZ/dtmax ms] – [R→B ms] = 52±20 ms). These calculated rise times are corroborated by actual measurements obtained by Debski et al. [41]. It can also be shown that TTP of both dv/dt and dZ/dt occur in the first 10–20% of systole, whereas peak flow velocity occurs at the end of the first third of systole [13]. Matsuda et al. have shown that, while peaking out of phase, the maximum upslope and TTP of LVdP/dtmax and dZ/dtmax are identical. Regression equations by Adler et al. [42] and graphic evidence interpolated from Lyssegen et al. [43] Fig. 7. Ascending aortic dP/dt, ascending aortic pressure P, and dZ/dt from show that, at normocardia (70–90 bpm) TTP of LVdP/dt a human. Note that dZ/dtmax peaks precisely with ascending aortic dP/dtmax. max Note computer artifact. Courtesy of Kirk L. Peterson, M.D.; From the is 50–60 ms, which is equivalent to the rise time of cardiac catheterization laboratory at the University of California School of dZ/dtmax. These data suggest that the inotropic forces Medicine, San Diego. governing the potential energy generated during isovolumic contraction time (IVCT) are transferred, unabated, Figure 7 shows that dZ/dtmax peak synchronously with converting potential energy to equivalent kinetic energy (as aortic dP/dtmax, where, as extrapolated from equation 9 and expressed as dv/dtmax or equivalently dZ/dtmax) during the its following discussion (vide supra) dP/dtmax corresponds earliest phase of ejection. Welham et al. [44] have in time with dF/dtmax (major derivative) when dA/dtmax definitively shown that, during halothane-induced (minor derivative) is trivial. This indicates that the peak rate myocardial depression LVdP/dtmax, dv/dtmax, and dZ/dtmax of change of force of ventricular ejection is manifest as decrease and recover proportionately with the depth of dZ/dtmax in impedance cardiography. As equations 9 and 11 11
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predict and figure 1 indicates, Q(t)max will occur Square Root Acceleration Step-down Transformation: synchronously with dA/dtmax, when aortic dP/dt, dF/dt and Ohmic Mean Velocity from Ohmic Mean Acceleration. dQ/dt=0.
Figure 6 indicates that, at Q(t)max, dZ/dt = 0. These As discussed by Gaw [23] and Visser [19], the relative observations infer, if not prove that dZ/dtmax occurs in change of blood resistivity (or equivalently blood earliest systole, contemporaneously with aortic dP/dtmax and conductivity) due to aortic blood flow is related, peak blood flow acceleration (dv/dtmax, dQ/dtmax). Further hemodynamically, to an exponential power (n) of the proof that dZ/dtmax peaks with and is the electrical analog of reduced average blood velocity (i.e. mean spatial velocity) n –1 n –2 dv/dtmax is verified experimentally by their corresponding [(
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.
L2 dρ () t1 dZ / dt b = max (s–1) (46) Vb dtmax Z 0 Z0
It naturally follows that,
m ⎛⎞d〈〉 v/ dtmax dZ/ dtmax –1 ⎜⎟≡ (s ) (47) ⎝⎠RZ0 Fig. 8. ECG, ultra-low frequency ballistocardiogram (BCg), dZ/dt, and ∆Z(t) from a human. Note that the “I” wave of the HJ interval peaks precisely with dZ/dtmax. The unlabeled noisy signal above ECG is a Inasmuch as dZ/dtmax is the electrodynamic equivalent of phonocardiogram. From reference 49. mean aortic blood acceleration, and SV is obtained from a mean velocity calculation, it is suggested that equations 35, Gaw et al. [23] showed that the relative blood 37, and 38 produce a mean acceleration surrogate of SV, –1 conductivity change (∆σ/σ (%), i.e. [∆ρb(t)/ρb] (%)) which is impossible according to equation 6. Thus, parallels the peak acceleration of the red cell reduced average velocity (mean spatial velocity,
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t Effect of Hematocrit on dZ/dt and ICG-Derived SV 221 dZ/ dtmax max SV=ππ r v() t dt = r ⋅⋅ v TLVE = V EPT TLVE (49) ∫t0 and CO Z0
In view of this discussion, the outflow correction As inferred, the static specific resistance of blood, ρ , factor, dZ/dt x T , is rendered moot, because dZ/dt b max LVE max nor the static specific resistance of the thorax, ρ , are represents axial blood acceleration and not radially-oriented T included in the ohmic acceleration-based SV method. rate of change of volume according to windkessel theory Despite Visser’s [19] and Hoetink’s [22] in vitro [36]. Close correspondence between the Kubicek or experimental findings, showing that the relative blood Sramek-Bernstein equations with reference method CO is resistivity change is proportional to hematocrit, Visser et al. simply due to the fact that peak velocity and mean [14] also showed that, although the peak magnitude of acceleration are highly correlated (r=0.75) and both peak ∆Z (t) (i.e ∆Z (t) ) is hemocrit-dependent, its maximum and mean aortic blood acceleration are highly correlated b b max upslope, dZ/dt , is not (figure 3 in Visser et al.). In a with SV (r = 0.75) as well as with the systolic velocity max canine model, Quail et al. [31] showed that normovolemic integral (stroke distance) (r = 0.75) [53–55]. Thus over a hemodilution over a hematocrit range of 66%–26%, selected range of ohmic acceleration (i.e. dZ/dt ), dZ/dt max max produced an appropriate increase in SV, which was not × TLVE provides acceleration facsimiles of ohmic mean different from electromagnetic flow-meter-derived (EMF) velocity. SV. Corroboratively, an in vivo ICG study by Wallace et As per equation 46, the relationship between ohmic al. [57] showed that, over a wide range of hematocrit mean velocity and ohmic mean acceleration is parabolic (20%–35%) and blood conductivity (ρ=80–160Ω·cm) (ohmic mean acceleration = (ohmic mean velocity)2 or y = induced by normovolemic hemodilution, ICG CO was in x2. This relationship predicts that, over wide ranges of agreement with transit time flow probe CO. dZ/dt , ohmic mean velocity will be overestimated and max underestimated at its upper and lower extremes, The Volume Conductor (V , V ) respectively. EPT C
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
Fig. 9. Square root acceleration step-down transformation. On the y-axis, total blood volume (TBV), SV and CO than patient height the ohmic equivalent of peak aortic reduced average blood acceleration [62–66]. By magnitude, the ITBV represents approximately –1 (PARABA), which is dZ/dtmax/Z0., and the linear extrapolation of the 25% of TBV, or about 17.5 mL⋅ kg , which results in square root transformation, ohmic mean velocity on the x-axis (x = √y and 1.02 VITBV = 17.5 x Wkg, or equivalently 16Wkg [37,67] By y = x2). From reference 13. comparison, existing volume conductors associated with the Indeed, Yamakoshi et al. [1] showed that ICG-derived SV plethysmographic hypothesis are modeled as simple overestimates reference method SV in healthy canines by as geometric abstractions, which are firmly rooted in basic much as 70%, and correspondingly underestimates electrical theory. By virtue of their “best-fit” mathematical reference SV by as much as 25% when myocardial failure construction, they bare little relevance to, and have virtually is induced. Similarly, Ehlert and Schmidt [56] could not no biophysical relationship with other commonly accepted find a linear relationship between ICG and EMF-derived physiologic, anatomic, or hemodynamic parameters. SV over a wide range of hemodynamic perturbations. Determinants of the Magnitude of dZ/dtmax
Evidence suggesting that dZ/dtmax varies inversely with aortic valve CSA, or radius r, as a function of
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PARABA, is inferentially demonstrated through the work Expanding equation 30, the following is a useful of Sageman [68]. He clearly showed that an inversely analytical tool for studying the effect of excess EVLW on proportional and highly negatively correlated (r = –0.75) the various compartments of the transthoracic impedance relationship exists between dZ/dtmax/Z0 and body mass in with the AC field. healthy humans. Aortic valve CSA has been shown to correlate highly with body mass (kg) and body surface area 2 (BSA, m ). But by contrast with dZ/dtmax/Z0, Doppler and EMF peak velocities and systolic velocity integrals are totally independent of body mass [69]. 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,
Fig. 10. Box plots showing that, as oleic acid-induced pulmonary edema dZ/ dtmax –1 Area×= v Volume × (mL·s ) (50) worsened, as reflected by decreasing Z0, the systematic underestimation of Z0 flow-probe CO by ICG-derived CO increased. From reference 74.
⎡⎤ Because of the absolute dependency of dZ/dtmax/Z0 ⎛⎞ρρρL222 L L⎛⎞Δ ρ() tL 2 ρ L 2 I()tU⎢⎥⎜⎟tbe⎜⎟ b+=Δ b U ()t upon PARABA, this means that, for any given value of ⎢⎥⎜⎟V V V⎜⎟ VΔ Vt() 0 b ⎢⎥⎝⎠tbe(1) (2) (3) ⎝⎠ b b(4) mean acceleration, the magnitude of dZ/dtmax is related to ⎣⎦ and explicitly dependent on aortic root CSA by its (52) dependency on R. Since aortic root CSA is a function of body mass, age, and gender, dZ/dtmax will vary accordingly. In this model it is assumed that AC (I(t)) flows
Thus, the magnitude of dZ/dtmax is multi-factorial and not exclusively through the blood resistance Zb (100Ω·cm– wholly dependent on the respective levels of myocardial 180Ω·cm) (element 2), despite the fact that Ze (element 3) is contractility and Z0. As a first order approximation, of lower specific resistance (60Ω·cm–70Ω·cm). This assumption is probably not accurate, because the blood 2 resistance R is considered a cylindrical conductor ⎡ m ⎤ b π 3 ⎛⎞〈〉 dZ() t⎢ r ⎜⎟⎛⎞d v/ dtmax ⎥ –2 surrounded by a more highly conductive EVLW impedance = Z0 ⎜⎟(Ω·s ) (51) dt⎢ V⎜⎟⎝⎠ R ⎥ Z . By the reciprocal rule for parallel impedances, the max ⎣ C ⎝⎠⎦ e resultant impedance (i.e. Zb||Ze) would be lower than either
impedance alone. If Ze is of variable magnitude and Zb is Index of Transthoracic Aberrant Conduction: Genesis held constant, then Z0 should vary with Ze. This is precisely of the Three-Compartment Parallel Conduction Model what is observed clinically. When Ze reaches a critical
volume (Ve(CRIT)), Ze becomes the lowest impedance as One of the major drawbacks of the impedance relates to the components of Z0. This causes electrical technique has been its inability to correctly predict SV in shunting, and in the extreme case (Z0 =10Ω–12Ω), a the presence of excess EVLW [70–72]. Critchley et al. [73] complete short circuiting of current away from the blood tested the hypothesis that the poor agreement between resistance. This results in preferential flow through Ze at the invasive reference standards and ICG is due to excess expense of Zb, with the result being a decrease in the EVLW. Typified by Sepsis, they were able to show that magnitude of Z0 and the amplitude of ∆Zb(t) [1]. At this ICG CO underestimated its thermodilution counterpart and critical level of Z0 (i.e. Z0(CRIT), ZC), which in humans is that the degree of underestimation was related to the degree 20Ω±3Ω, excess EVLW causes spuriously reduced values of EVLW excess. They also noted that, in general, excess of dZ/dtmax. Thus, dZ/dtmax no longer parallels its EVLW was associated with values of Z0<20–22Ω. In a hemodynamic equivalent, PARABA, resulting in subsequent experimental model in canines, where systematic underestimation of ICG SV/CO. In order to pulmonary edema was induced by oleic acid infusion, they compensate for pathologic conduction through excess found a systematic progressive bias between flow probe EVLW, an index of transthoracic aberrant conduction has and ICG CO [74]. As pulmonary edema progressively been derived (ζ, zeta), which, as its magnitude decreases, worsened, ICG CO progressively underestimated the flow creates a larger VEPT. probe estimate. In concert with the progressive divergence of the two methods, Z0 progressively decreased (fig. 10). 14
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In patients without excess EVLW, VEPT is equivalent blood acceleration. This can only occur, when, by volume, to VITBV. For all values of Z0<20 Ω, 0<ζ<1 and for Z0≥20 the blood resistance is the compartment with the lowest Ω, ζ=1. For all values of Z0<20 Ω, the following pertains: impedance. Thus, when excess extravascular lung water exceeds some critical volume, current is diverted to the V compartment of lowest impedance, which is now that of the VV=ITBV ≡+V (mL) (53) EPT ζ 2 ITBV EVLW extravascular lung water. With progressive diversion of AC where ζ is given as [37], away from the blood resistance, the magnitude of ∆Zb(t) and dZ/dtmax diminish and therefore are not reflective of or proportional to the hemodynamic state. Clearly, a more Z2 −+ ZZ K ζ = CC0 (Dimensionless) (54) robust mathematical solution for excess extravascular lung 22+− + 23ZCCZ00 ZZ K water is desirable; that is, if is at all possible. where ZC = critical level of base impedance Z0(CRIT) = 20Ω, References corresponding to Ve(critical). Z0 = measured transthoracic 1. Yamakoshi K, Togawa T, Ito H. Evaluation of the theory of impedance ≤ ZC, and K = a trivial constant →0. cardiac-output computation from transthoracic impedance plethysmogram. Med Biol Eng Comput 1977;15:479–88. Stroke Volume Equation for Impedance Cardiography 2. Bernstein DP. Pressure pulse contour-derived stroke volume and cardiac output in the morbidly obese patient. Obes Surg 2008;18:1015–21. V dZ( t )/ dt SV = ITBV max T (mL) (55) 3. Quick CM, Berger DS, Noordergraaf A. Apparent arterial ICG ζ 2 LVE Z0 compliance. Am J. Physiol (Heart Circ Physiol 43) 1998;274:H1393–1403. 1.02 where VITBV = 16W(kg) . Equation 55 [37] has been 4. Fogliardo R, Di Donfrancesco M, Burattini R. Comparison prospectively tested in critical and non-critically-ill patients of linear and nonlinear formulations of the three-element and been shown to provide SV and CO values comparable winkdessel model. Am J Physiol (Heart Circ Physiol 40) to standard reference methods [37,75–78]. As reviewed and 1996;271:H2661–68. compiled by Moshkovitz et al. [7] there are other whole 5. Chemla D, Hebert J-L, Coirault C, et al. Total arterial body and transthoracic equations, but they have not been compliance estimated by stroke volume-to-aortic pulse effectively prospectively tested. pressure ratio in humans. Am J Physiol (Heart Circ Physiol 43)1998;274:H500–05.
Conclusions 6. Olufsen MS, Ottesen JT, Tran HT, et al. Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. J Appl Physiol From the results of this review, it is strongly suggested 2005;99:1523–37. that the conceptually-based plethysmographic hypothesis 7. Moshkovitz Y, Kaluski E, Milo O, et al. Recent for ICG-derived CO is flawed. The incontrovertible developments in cardiac output determination by bio- evidence provided herein shows quite conclusively that impedance: comparison with invasive cardiac output and dZ/dtmax is a mean acceleration analog and not that of the potential cardiovascular applications. Curr Opin Cardiol peak rate of change of aortic volume. Therefore, to obtain 2004;19:229–37. ohmic mean velocity from dZ/dtmax/Z0, square root 8. Summers RL, Shoemaker WC, Peacock DF, et al. Bench to transformation is obligatory. Because dZ/dtmax represents bedside: electrophysiologic and clinical principles of non- axial blood acceleration, the “outflow compensation” for invasive hemodynamic monitoring using impedance runoff is entirely irrelevant as a theoretical problem to be cardiography. Acad Emerg Med 2003;10:669–80. proved or disproved, as per the work of Faes et al. (36). The 9. Newman DG, Callister R. The non-invasive assessment of fact that dZ/dtmax is a mean acceleration analog permits the stroke volume and cardiac output by impedance square wave integration, dZ/dtmax x TLVE, as a valid cardiography: a review. Aviat Space Environ Med. convention similar to equations 6 and 49 for the 1999;70:780–9. Doppler/EMF method of SV determination. As for the 10. Woltjer HH, Bogaard HJ, deVries PM. The technique of volume conductors implemented by both the impedance cardiography. Eur Heart J 1997;18:1396–403. Nyboer/Kubicek and Sramek/Bernstein models, there 11. Kauppinen PK, Hyttinen JA, Malmivuo JA. Senstitivity seems little physiologic justification for either approach. distributions of impedance cardiography using band and The volume of electrically participating thoracic tissue spot electrodes analyzed by a three-dimensional computer involved in dynamic conduction is clearly the blood model. Ann Biomed Eng 1998;26:694–702. resistance. By magnitude, the blood resistance translates 12. Raaijmakers E, Faes TJ, Scholten RJ, et al. A meta-analysis directly into the intrathoracic blood volume. While basing of three decades of validating thoracic impedance the volume conductor on a fixed volume appears to be cardiography. Crit Care Med 1999;27:1203–13. concordant with the Sramek/Bernstein model, this can only 13. Bernstein DP. Impedance cardiography: development of the be true when dynamic conduction through the blood stroke volume equations and their electrodynamic and biophysical foundations. In: Leondes CT, editor. Bio- resistance is in parallel with peak aortic reduced average 15
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