Download 2.4. Assessment of intrinsic contractility. Referring to figure 2.2

2.4. Assessment of intrinsic contractility.
Referring to figure 2.2, representing the cardiac cycle in the pressure-volume plane,
indices of contractility can be classified according to the phase of the phase the cardiac
cycle during which they are obtained [95, 254] :
1. the phase of isovolumic contraction,
2. the phase of ventricular ejection,
3. the end-systolic pressure-volume relationship (ESPVR),
4. the relationship between stroke work and end-diastolic volume (i.e. the
preload recruitable stroke work) [85].
2.4.1. Load-dependent indices of contractility.
The paradigm of an isovolumic contraction phase index of contractility is the maximum
rate of increase of left ventricular pressure [dP/dt]max [146].
It requires continuous sampling of left ventricular pressure with a high fidelity
micromanometer tipped catheter and on-line electronic or off-line digital differentiation.
Values of [dP/dt]max are :
-
relatively independent of afterload (because the aortic valve is closed), but
dependent on arterial diastolic pressure [146],
-
easy to interpret if no distortion of the signal by wall properties and no valve
dysfunction [95],
-
sensitive enough to detect acute alterations in contractility,
-
but substantially preload dependent (figure 2.29) [214].
90
LVP (mmHg)
60
50
40
30
dLVP/dt (mmHg/sec)
20
10
0.4
0.2
0.0
-0.2
-0.4
time (sec)
Figure 2.29
Tracings of left ventricular pressure and its time derivative during a “rapid” caval inflow
reduction, illustrating the preload dependence of [dP/dt] max.
To compensate for preload dependence the following corrections can be applied [95,
214, 254]:
-
the ratio of dP/dt to developed pressure P and calculation of [(dP/dt)/P]max,
-
dP/dt related to a fixed pressure (e.g. 50 mmHg) [189],
-
or the ratio of [dP/dt]max to EDV (end-diastolic volume).
The standard ejection phase index of contractility is the ejection fraction EF :
EF =
SV
EDV " ESV
ESV
=
= 1"
EDV
EDV
EDV
with ESV : end-systolic volume.
!
91
(Eq. 2.23)
Ejection fraction is described as being an afterload-sensitive measure of ventricular
function, but according equation 2.23 an hyperbolic relation exists between EF and
EDV, implicating a substantial sensitivity of EF on EDV at low EDV [118, 207, 214].
As a consequence the usefulness of EF as a measure of ventricular function is limited,
but according to Robotham the ejection fraction should be considered as a measure of
the integrated cardiovascular system’s ability to maintain homeostasis [207]. However
EF remains a popular measurement for clinicians, because:
-
EF is easy to determine with the availability of noninvasive cardiac imaging
techniques, e.g. echocardiography,
-
an association has demonstrated between EF and prognosis of patients with
coronary artery disease [207, 254] .
2.4.2. The end-systolic pressure-volume relationship.
The idea of assessing ventricular contractility by the end-systolic pressure-volume
relationship originates from Otto Frank representation of the cardiac cycle in the
pressure-volume plane (figure 2.2) and his observations concerning the relationship
between peak (systolic) pressure and end-diastolic volume during isovolumic
contractions in the frog ventricle in the late 19th century [212-214, 247]. Subsequent
investigations in dogs revealed that this relationship was reasonably linear over the
physiologic range of EDV and that the slope of peak isovolumic (systolic) pressure –
EDV relation is directly related with the contractile state of the ventricle and heart rate
(figure 2.30) [18, 160, 241]. By definition, this relationship is independent of afterload
: isovolumic contraction or non-ejecting beats) [247].
92
Peak isovolumic pressures are difficult to obtain in animals and impossible to obtain in
humans. Investigations on isolated canine hearts demonstrated that the line connecting
the left upper corners of the pressure-volume loops, the so-called “end-systolic pressure
volume points” [117] of ejecting contractions closely approximates the EDV – peak
isovolumic line (figure 2.30) [160, 212, 241, 267]. Furthermore the end-systolic
pressure-volume relationship i.e. the line connecting the end-systolic pressure volume
points obtained at various preloads and constant inotropic state, appeared to be linear
and independent of loading conditions (figure 2.31) [243]. As a consequence the slope
of this end-systolic pressure-volume relationship (ESPVR), called the end-systolic
elastance Ees, has been put forward as load independent index of ventricular
contractility. However there is an influence of heart rate on the ESPVR (Bowditch
phenomenon) as demonstrated by Maughan in the isolated canine heart and Freeman in
conscious dogs [76, 148], and other observations suggest that the ESPVR may be quite
variable, nonlinear, and shifted by alterations in arterial loading conditions [8, 31, 140,
264].
93
Figure 2.30
Assessment of contractility using a ventricular pressure-volume loop. The loop A’B’C’D’E’F’ is the
normal curve. At the same normal state of contractility, the loop ABCDEF is generated by
decreasing end-diastolic volume, and the loop A”B”C”D”E”F” is generated by increasing enddiastolic volume. The slope of the line through the points at end-systole (F, F’ and F”) represents
the end-systolic pressure-volume relation (ESPVR).
From : Boelpaep, E. L. (2003). The Heart as a Pump. In W. F. Boron & E. L. Boelpaep (Eds.),
Medical Physiology (First ed., pp. 508-533). Philadelphia: Saunders.
94
Figure 2.31.
Left ventricular pressure-volume loops of a denervated heart. Abscissa : left ventricular volume
(milliliter). Ordinate : left ventricular pressure (mmHg) Mean arterial pressure was fixed at three
different levels while cardiac output was kept constant during both the control (solid loops) and the
enhanced (2 µg/kg/min epinephrine infusion) contractile state (broken loops).
From : Suga, H., Sagawa, K., & Shoukas, A. A. (1973). Load independence of the instantaneous
pressure-volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the
ratio. Circ Res, 32(3), 314-322
95
2.4.2.1. The time-varying elastance model of the ventricle.
The concept of elastance was first introduced in analog computer models of the
circulation by Defares and Beneken and led to Suga’s experiments in the canine left
ventricle on the instantaneous ratio of intraventricular pressure to volume as a function
of time during one cardiac cycle (figure 2.32) [211, 238, 241, 243].
Figure 2.32
Ventricular pressure-volume diagram (left) and pressure/volume ratio E(t) (right) defined as
P(t)/V(t).
From : Suga, H. (1990). Cardiac mechanics and energetics - from Emax to PVA. Front Med Biol
Eng, 2(1), 3-22. Originally first published in : Suga, H (1969) : Analysis of left ventricular pumping
by its pressure-volume coefficient. Jpn J Med Electron Biol Eng , 7,406-415
The left panel of figure 2.32 depicts three pressure-volume loops in the left-ventricle of
a dog. The right panel of figure 2.32 depicts the instantaneous intraventricular pressurevolume P(t)/V(t) ratio, called the time-varying elastance E(t), for the three P-V loops.
E(t) corresponds to the slope, as a function of time, of the instantaneous P-V relation
line connecting instantaneous P-V data points at a specified time of the cardiac cycle on
multiple P-V loops under varied loading conditions in a given contractile state [238].
96
The right panel indicates that, despite the difference in preloaded volume or afterloaded
pressure between these three contractions, the instantaneous ratios E(t) are
approximately the same [211]. E(t) reached a maximum value called Emax near the end
of ejection, and the instant at which E(t) equals Emax was called end systole [214, 238].
Referring to figure 2.32, Emax is the slope of the end-systolic P-V relation line. The time
to Emax from the onset of systole was called Tmax [241].
Suga demonstrated that the E(t) curve, Emax and Tmax was markedly affected by
inotropic interventions [215, 241, 243]. Further experiments by Suga and Sagawa
however revealed a unique basic shape of the E(t) curve, particularly the systolic
portion : the basic shape is invariant with dogs, loading conditions, heart rates and
contractile state (!). Therefore the systolic portion of any given E(t) curve can be fully
represented by Emax and Tmax [215, 241, 243]. Emax was obtained as the maximal value
of E(t), determined as the time-dependent increase in the slope of regression lines
applied to multiple isochronous sets of instantaneous pressure-volume data points
(figure 2.33) [214, 215].
97
Figure 2.33
Ventricular pressure and volume data and instantaneous pressure-volume relations. These
pressure and volume data were obtained from six isovolumic and six ejecting contractions of a left
ventricle under a constant contractile state. Solid circles are the pressure-volume data points
obtained at 100 milliseconds after the onset of systole. Open circles are data obtained at 160
milliseconds. The solid rectilinear lines are the linear regression lines of pressure on volumes
obtained at those specified times as shown in the figure. Except for the very early part of systole,
the linear regression lines are a good approximation to the instantaneous pressure-volume relation.
From : Sagawa, K., Suga, H., Shoukas, A. A., & Bakalar, K. M. (1977). End-systolic
pressure/volume ratio: a new index of ventricular contractility. Am J Cardiol, 40(5), 748-753.
98
The regression analysis indicated that the instantaneous pressure-volume relationship at
any specified time ti, can be specified by the following equation [215] :
[
P(t i ) = E(t i ) V (t i ) " Vd ( i)
]
(Eq. 2.24)
where P(ti) : ventricular pressure at time ti
!
V(ti) : ventricular volume at time ti
E(ti) : the slope of the pressure-volume regression line at ti
Vd(i) : the volume-axis intercept of the afore mentioned regression at ti.
At end-systole (define as an instant of time in the ejection phase at which E(t) reaches
its maximum) equation 2.24 becomes [214]:
(Eq. 2.25)
Pes = E max [Ves " Vd (es) ]
Stroke volume equals end-diastolic volume Ved minus end-systolic volume Ves
!
(figure 2.2) :
(Eq. 2.26)
SV = Ved " Ves
Rearranging equation 2.26 and substituting the result in equation 2.25 leads to :
!
(Eq. 2.27)
Pes = E max [Ved " SV " Vd (es) ]
!
99
The foregoing account emphasize the fact that the ESPVR, indexed by Ees, and Emax are
different concepts and are obtained in a different way, although in isolated hearts with
isovolumetric or ejecting beats determined over a range of preloads, Emax and Ees are
nearly identical [117]. However, when the afterload is significantly altered among the
cardiac cycles used for Ees or Emax determination, the two can very different (figure
2.34) [52, 117].
Figure 2.34
Disparity between Emax and Ees in situ. Data are from a closed-chest dog, with volumes measured by
conductance catheter. The E(t) relations are shown at four time points : 10, 40, 60 (dotted lines),
and 260 milliseconds after the onset of systole (dashed line), the last being the time of maximal E(t)
or Emax. The solid line shows the ESPVR, with a slope significantly less than Emax.
From : Kass, D. A., & Maughan, W. L. (1988). From 'Emax' to pressure-volume relations: a
broader view. Circulation, 77(6), 1203-1212.
In the experimental and clinical setting Ees is usually determined rather than Emax, Ees is
considered the more useful measure of systolic properties for the purpose of assessing
pump function, while Emax is a useful concept for ventricular modeling [117]. In respect
of the latter, Suga also showed a close relationship between the empirically obtained
E(t) curve and the known force-velocity relation of heart muscle [211, 242].
100
2.4.2.2. Practical aspects concerning the determination of Ees :
single-beat estimation.
Determination of the ESPVR requires the simultaneous measurement of instantaneous
intraventricular pressure and volume during multiple P-V loops generated by altering
preload or afterload [117]. Accurate measurement of intra-ventricular pressure is
accomplished by the use of a high fidelity micromanometer catheter positioned in the
left ventricle. Since the introduction of the volume conductance catheter by Baan
measurement of instantaneous intraventricular volume is possible with reasonable
accuracy in the intact subject [7, 9, 119, 214]. Although the method is well validated in
animals and humans, its wide spread application is hampered by technical difficulties
and especially by its limited availability due to the high costs [214].
Pressure-volume loops are obtained during a transient change in preload or afterload,
but such maneuvers may affect sympathetic tone and myocardial contractility and may
either not be acceptable in patients or may not be desired when studying the effects of
anesthetics [27, 31, 35, 108, 207, 250].
In order to circumvent the afore mentioned problems, methods were developed to
estimate Ees from a single beat: the peak systolic pressure Pmax of an isovolumic beat is
estimated by curve fitting from the isovolumic portions of the LV pressure curve of a
single ejecting beat (figures 2.35 and 2.36) [27, 247, 249]. The ESPVR line was drawn
from Pmax tangential to the left upper corner of the P-V loop of the ejection contraction
(i.e. the actual ejecting beat ‘end systole’) [27, 249]. The slope of this ESPVR line is
considered to represent Ees.
101
The P-V loop (limited to isovolumic contraction, ejection and isovolumic relaxation)
was reconstructed from (measured) instantaneous LV pressure and aortic flow [27,
108]. The instantaneous left ventricular volume of the ejecting contraction was
calculated by substracting the time integral of aortic flow from an arbitrary EDV [108].
By choosing an arbitrary EDV (cfr. upper pannel of figure 2.36) and taking account of
the relation between the change in left ventricular volume and aortic flow (cfr. equation
2.30), the single beat estimation avoids absolute (ventricular) volume measurements
[108, 164].
Figure 2.35
Determination of ventricular end-systolic elastance (Ees) and effective arterial elastance (Ea).
Left : end-systolic pressure of an isovolumic beat is computed by sine wave extrapolation from the
ejecting beat by using pressure values recorded before maximal first derivative of pressure
development over time (dP/dt) and after minimal dP/dt.
Right : this maximal vetricular pressure of isovolumic beats (Pmax) value is drawn on the pressurevolume diagram. The ESPVR line is drawn from Pmax down and tangent to the pressure-volume
curve, i.e., from predicted isovolumic beat end systole to actual ejecting beat end systole (defined by
the contact point of pressure-volume curve and ESPVR line). The effective arterial elastance line is
drawn from end systole to end diastole. Ees is the slope of the ESPVR line, and Ea is the absolute
slope of the arterial elastance line.
From : Brimioulle, S., Wauthy, P., Ewalenko, P., Rondelet, B., Vermeulen, F., Kerbaul, F., et al.
(2003). Single-beat estimation of right ventricular end-systolic pressure-volume relationship. Am J
Physiol Heart Circ Physiol, 284(5), H1625-1630.
102
A single beat estimation of Ees can be considered as a direct application of Otto Frank’s
relationship between peak (systolic) pressure and EDV during an isovolumic
contraction (figures 2.30 and 2.36) [164, 247].
Figure 2.36
Panel A : Schematic of the conventional end-systolic pressure-volume relation (ESPVR) line
determined by a set of three pressure-volume loops. Pmax is the peak isovolumic pressure at enddiastolic volume. Ees is the slope of ESPVR.
Panel B : Schematic of a method for determining the ESPVR line from a single ejecting beat.
Pmax(E) at end-diastolic volume was estimated by a curve-fitting technique. ESPVR (estimated
ESPVR) line was drawn from the Pmax(E) - volume point tangential to the left upper corner of the
pressure-volume loop. Slope of this line is the estimated Ees [Ees(E)], and the volume axis intercept
of the estimated ESPVR line is the estimated V0 [V0(E)].
From : Takeuchi, M., Igarashi, Y., Tomimoto, S., Odake, M., Hayashi, T., Tsukamoto, T., et al.
(1991). Single-beat estimation of the slope of the end-systolic pressure-volume relation in the
human left ventricle. Circulation, 83(1), 202-212.
103
2.4.3. Preload recruitable stroke work (PRSW).
The preload recruitable stroke work represent a linearization of the Frank-Starling
relationship [85]. To measure PRSW, venous return to the intact heart is acutely
decreased and the area of each P-V loop, which represents the external work performed
during the beat, is plotted as a function of EDV (figure 2.37) [95].
Figure 2.37
An example of global PRSW relationships derived from a series of LV pressure-volume loops
recorded during a single vena caval occlusion. The global PRSW relationship is the relationship
between stroke work, calculated as the pressure-volume loop area (shaded for one beat) and the
corresponding end-diastolic volume for the same beat. M w and Vw are the slope and volume-axis
intercept, respectively, of the global PRSW relationship.
From : Karunanithi, M. K., & Feneley, M. P. (2000). Single-beat determination of preload
recruitable stroke work relationship: derivation and evaluation in conscious dogs. J Am Coll
Cardiol, 35(2), 502-513.
The slope of this relationship determines how much work the heart is capable of at any
given preload. When contractility increases or decreases, the slope rises or falls as the
heart performs more or less work for a given preload [95].
104
2.5. Ventricular-arterial interaction and
mechanoenergetic efficiency.
2.5.1. Ventricular work and power.
Because the heart ejects an amount of blood at a given pressure against the systemic
arterial impedance, characterization of the pump performance of the heart by its cardiac
output is unsatisfactory in mechanical terms. In technical fluid mechanics pump
performance of a fluid pump is given by the “head-capacity curve” (figure 2.38 and
2.39) [65, 66]. Such a graph relates, for a given setting of the pump, the amount of fluid
it can handle to the pressure head it has to overcome. This is a inverse relationship,
because for a certain head the pressure opposing flow is so high that output will be zero,
whereas at zero pressure, flow is at its maximum [65, 268].
Figure 2.38
Pump function of an ordinary roller pump measured at three settings of the pump :
 control,  increased roller speed,  increased pressure on the tubing at control roller speed
(more occlusive).
From : Elzinga, G., & Westerhof, N. (1979). How to quantify pump function of the heart. The value
of variables derived from measurements on isolated muscle. Circ Res, 44(3), 303-308.
105
Figure 2.39
Pump function graphs of an isolated ejecting feline heart for :  control,  increased end-diastolic
volume,  increased inotropic state.
From : Elzinga, G., & Westerhof, N. (1979). How to quantify pump function of the heart. The value
of variables derived from measurements on isolated muscle. Circ Res, 44(3), 303-308.
Implicit in this approach is that performance of a pump can be characterized by the
work performed by the pump or the power output of the pump.
Otto Frank’s experiments on the frog ventricle, also led to the realization that the area
enclosed by the pressure-volume trajectory loop represents external mechanical work,
or stroke work, performed by the ventricle that contributes to ejection of blood into the
arterial system during one cardiac cycle (figure 2.2) [155, 214]. Mathematically:
106
EEV
SW =
" P (t) dV
(Eq. 2.28)
v
EDV
with : SW : stroke work, i.e. the mechanical energy the ventricle has to perform in
!
order to eject the stroke volume (unit: Joule),
Pv(t) : instantaneous ventricular pressure,
dV : ventricular volume change,
EDV : end-diastolic volume,
EEV : end-ejection volume.
Considering figures 2.2 and 2.4 and taking Pee ≈ Pes ≈ AoP and Ped ≈ PAOP, equation
2.28 can be approximated by :
!
SW = SV " (AoP # PAOP)
with
!
(Eq. 2.29)
Pee : ventricular pressure at end ejection,
Pes : ventricular pressure at end systole,
Ped : ventricular pressure at end diastole,
AoP : mean aortic pressure,
PAOP : pulmonary arterial occluded pressure.
!
A decrement in ventricular volume is manifested as aortic flow Qao(t) :
Qao (t) =
dV (t)
dt
(Eq. 2.30)
note : as for the ventricle in systole dV(t) < 0, a minus sign should be written on the
!
right hand side of equation 2.30.
107
A change of integration variable in equation 2.28 results in :
Te
W =
"P
ao
(t)
Tb
dV (t)
dt
dt
(Eq. 2.31)
Te
W =
"P
ao
(t)Qao (t)dt
Tb
with
!
W : total hydraulic work of the ventricle, i.e. the work (energy) the ventricle
has to perform in order to insure flow in the arterial system, given a
certain aortic impedance,
Pao(t): instantaneous aortic pressure,
Qao(t) : instantaneous aortic pressure,
Tb : time of beginning of the cardiac cycle,
Te : time of end of cardiac cycle.
Due to this change of integration variable, comparing equation 2.28 to equation 2.31 a
few changes will be noted :
-
the integration limits has been changed from the period of ejection to a
cardiac cycle,
-
Pv(t) has been substituted by Pao(t) : refering to figure 2.4 , a small difference
is noted between Pv and Pao during ejection, but for the purpose of
integration over the cardiac cycle (: no forward flow during diastole) this
difference can be neglected [155, 214],
-
stroke work SW, has been substituted by total hydraulic work W.
Both SW and W represents pressure-volume work external to the “ventricle”, i.e. work
performed by the ventricle to the outside system (: the arterial system) during one
cardiac cycle [169, 214]. SW emphasize the energy expended by the ventricle during
ejection of the stroke volume [154], while W emphasize the energy imparted to blood
by the ventricle but which is lost as blood flows through the circulation [169].
108
The product Pao(t).Qao(t) in equation 2.31 represents the instantaneous hydraulic power,
i.e. external work per unit time (unit : Watt = Joule/s), and is depicted for one cardiac
cycle in the upper panel of figure 2.40.
Figure 2.40
Pulsatile pressure, flow and their product, hydraulic power, in the main pulmonary artery of an
unanesthetized dog. Power was recorded by multiplication pressure and flow at every instant with
an electronic multiplier. Abscissa, time in seconds.
Above, power supplied by the right ventricle (omitting kinetic energy).
Below, pressure (scale on right in 104 dyn/cm2) and flow (scale on left in milliliters/second).
Averages of the three curves over one cardiac cycle are shown by dashed lines.
Mean pressure ( P ) multiplied by mean flow ( Q ) is indicated by solid line in upper panel.
Difference between P " Q product and the true average power is the extra energy entailed in
pulsations.
!from : Milnor, W. R. (1972).!Pulsatile blood flow. N Engl J Med, 287(1), 27-34.
Figure
!
Text from : Milnor, W. R. (1982). Hemodynamics. Baltimore: Williams & Wilkins.
109
Time averaged hydraulic power, called total (external) hydraulic power W˙ tot , indicated
by the dotted line in the upper panel of figure 2.40, equals :
!
Te
W˙ tot =
"P
ao
(t)Qao (t)dt
(Eq. 2.32)
Tb
T
with T: heart period, Tb: time of begin and Te: time of end of cardiac cycle.
!
It is important to realize that this time averaged hydraulic power exceeds the product of
mean pressure aortic pressure AoP and mean aortic flow AoQ , indicated by the solid
line in the upper panel of figure 2.40. The product of the means is the hydraulic power
that would be involved in producing the same average flow in a steady stream, without
!
!
pulsations. The difference between this product of the means and the true mean of the
power curve is an extra energy entailed in the pulsatile wave form [154-156].
Consequently the total hydraulic power can be divided into two components [154-156]
:
•
1. steady flow power W st is expended principally in moving blood through
arterioles, and hence depends to a great extent on peripheral resistance,
and is calculated as:
!
•
(Eq. 2.33)
W st = AoP " AoQ
If!input resistance is defined as : Rin = AoP
AoQ
!
110
then :
AoP
W st = AoQ " Rin =
Rin
•
2
2
(Eq. 2.34)
•
2. oscillatory
flow power W osc is exerted in pulsations and is largely
!
determined by the visco-elastic properties of the large arteries. This
oscillatory flow power is conveniently calculated in terms of the aortic input
!
impedance [157]:
•
W osc =
1 N
(Qn ) 2 Z n cos" n
#
2 n=1
(Eq. 2.35)
with Qn : the amplitude of the n-th flow harmonic,
!
Zn : the modulus of the n-th harmonic of the aortic input impedance,
θn : the phase angle of the n-th harmonic of the aortic input
impedance,
N : the total number of harmonics computed.
The total hydraulic power is the combination of steady flow and oscillatory flow
components:
•
•
•
(Eq. 2.36)
W tot = W st + W osc
!
111
Equations 2.34 and 2.35 demonstrates that hydraulic power is calculated from the same
measurements as needed for the calculation of the aortic input impedance spectrum:
pulsatile (instantaneous) aortic pressure and flow. This fact indicates that arterial
resistance and impedance are important in determining external ventricular work, which
is to be expected because resistance and impedance express the opposition to the
ejection of blood that is imposed by the physical properties of the arterial tree [155].
In the systemic circulation W˙ osc amounts to 15% of W˙ tot , while in the pulmonary
!
circulation W˙ osc amounts to 30% of W˙ tot [154, 155]. The ratio W˙ osc /W˙ tot is used as
inverse index of arterial efficiency, with the implication that steady flow power is
!
!
physiologically useful in pumping blood forward whereas pulsatile power is wasted in
!
!
!
arterial pulsations [169].
Remark : in the foregoing and in the remainder of this work the kinetic component of
external ventricular work is omitted, because the contribution of the kinetic energy
component is low as compared to the contribution of the potential energy component
[154, 155, 169, 214].
2.5.2. Cardiac energetics and efficiency.
Suga hypothesized that in the context of the time varying elastance model of the
ventricle, an increase in the elastance accompanied by pressure development would be a
manifestation of increased mechanical potential energy as in stretched spring [238,
239]. Figures 2.2, 2.30 and 2.41 indicates that the pressure-volume trajectories of
ejecting beats are enveloped by the end-systolic pressure-volume line and the enddiastolic pressure-volume curve [214]. Considering Otto Frank’s acknowledgment that
the area within the pressure-volume trajectory represents external mechanical work,
Suga proposed that the specific area in the pressure-volume plane circumscribed by the
end-systolic pressure-volume line, the systolic segment of the pressure-volume
112
trajectory (IC and EJ in figure 2.2) and the end-diastolic pressure-volume curve
represents the total mechanical energy that the ventricle generates in one cardiac cycle
(figure 2.41) [214, 238, 239]. This area is called the systolic pressure-volume area and
abbreviated as PVA. The PVA consists of two areas (figure 2.41) [214, 238, 239]:
-
a rectangular area within the P-V trajectory, representing external
mechanical work (EW) during systole,
-
a triangular area between the end-systolic pressure-volume line and the enddiastolic curve on the left side of the P-V trajectory, representing endsystolic elastic potential energy (PE) presumed to be stored in the
myocardium at end systole.
Figure 2.41
Pressure-volume area, consisting of external work and end-systolic elastic potential energy.
From : Sagawa, K., Maughan, L., Suga, H., & Sunagawa, K. (1988). Cardiac Contraction and the
Pressure-Volume Relationship. New York: Oxford University Press.
113
Suga also demonstrated that total left ventricular oxygen consumption per beat (Vo2) is
linearly correlated to the PVA [214, 239], with a regression line in the form :
Vo2 = a.PVA + b
(Eq. 2.37)
The regression constant b represents that part of Vo2 responsible for basal metabolism
and excitation-contraction coupling, which is not reflected by PVA [214, 227].
The efficiency of energy conversion in a system is generally defined as the ratio of the
amount of useful energy delivered by the system to the amount of energy supplied to it
[214, 240]. In the heart mechanical work is considered to be the useful energy. The
conventional definition of the cardiac mechanical efficiency (CME) is the ratio of
external work done by the heart to the total energy supplied to and used by the heart.
Under aerobic conditions, the energy used by the heart is equivalent to cardiac oxygen
consumption [214].
Cardiac MechanicalEfficiency =
EW SW
=
VO2
VO2
(Eq. 2.38)
Cardiac mechanical efficiency has been reported to range between 0 and 35%, with a
!
normal functioning value of 10% - 25% [178, 240].
Mathematical manipulation of equation 2.38 leads to:
SW
SW PVA
=
"
VO2 PVA VO2
(Eq. 2.39)
!
114
The ratio PVA/Vo2 is called myocardial efficiency conversion and represents the
efficiency of conversion of the total energy input into the total mechanical energy
output [214]. The ratio SW/PVA is called left ventricular mechanical efficiency and
reflects the mechanical efficiency of converting total mechanical energy (PVA)
available to the left ventricle to external work (i.e. SW) [227]. Equation 2.39 indicates
that CME is determined by the product of SW/PVA and PVA/Vo2. As a consequence
CME increases or decreases depending on the relative changes of SW/PVA and
PVA/Vo2 [214]. It can be demonstrated that SW/PVA approaches 100% as the endsystolic pressure Pes becomes very small or Ees becomes very large [214].
The quantity SW appears as the numerator in equations 2.38 and 2.39, and reflects
the fact that SW represents the useful fraction of ventricular energy transferred to
the arterial system [32]. From the point of view of the body tissues, SW is the important
measure of the service that the heart provides in transporting oxygen [32].
2.5.3. Ventricular-Arterial Coupling and
Effective Arterial Elastance.
The following issue concerns the conditions under which the left ventricle is coupled to
its arterial load in order to attain the highest CME, or to produce maximal SW. In
conceptual terms, the left ventricle and arterial circulation represent two mechanical
units that are joined together to form a coupled biologic system [169]. The overall
mechanical behavior of the coupled system, therefore, is a function of the mechanical
properties of each unit, as is the operating point of coupled equilibrium between the two
units [169].
115
One already demonstrated that the concept of ventricular elastance (i.e. a pressurevolume relation) is an appropriate tool to describe the mechanical behavior of the left
ventricle. From equation 2.27 one can infer that Pes varies inversely with SV in linear
way (when Ees,Ved and Vd constant) : the larger the ejected stroke volume the lower left
ventricular end-systolic pressure Pes. This relationship is implicit in figure 2.42,
representing two PV trajectories with the same EDV but different Pes, but explicit in
figure 2.43. The Pes-SV relation is a representation of the function of the left ventricle as
a mechanical pump [169]. One should note the similarity between figure 2.43 and
figures 2.28 and 2.39 (: the head capacity curves).
Figure 2.42
Pressure-volume relationships with increased afterload. SV1 : stroke volume (normal); SV2 : stroke
volume with increased afterload; ESV : end-systolic volume; ESV2 : end-systolic volume with
increased afterload; EDV : end-diastolic volume
From : Power, I., & Kam, P. (2001). Principles of Physiology for the Anaesthetist. London: Arnold.
116
Figure 2.43
(A) Left ventricular end-systolic pressure (Pes) – stroke volume (SV) relation shifts rightward in a
parallel fashion with an increase in the end-diastolic volume (EDV). (B) Augmenting left
ventricular contractility by an increase in Ees rotates the ventricular Pes –SV relation clockwise
around the SV-axis intercept. Data were obtained from an isolated canine heart preparation.
From : Nichols, W. W., & O'Rourke, M. F. (1990). McDonald's Blood Flow in Arteries : theoretic,
experimental and clinical principles (Third ed.). London: Edward Arnold
If the mechanical properties of the arterial system could be characterized by a similar
relationship, i.e. a relation between arterial end-systolic pressure and stroke volume,
investigation of the coupling between the left ventricle and the arterial system would be
straightforward in a mathematical and a graphical way.
Sunagawa observed that for any bolus of blood (i.e. the SV) ejected into the arterial
system, a pressure is generated: the larger the SV the higher the pressure generated.
If arterial system characteristics (e.g. peripheral resistance, characteristic impedance,
arterial compliance) and duration of arterial diastole and ejection time are kept constant,
plotting various values of SV against the generated end-systolic pressure a linear
relation is obtained (figure 2.44) [244]. The slope of this Pes -SV relationship is called
effective arterial elastance Ea [214, 244].
117
Figure 2.44
End-systolic pressure versus stroke volume relationship of the arterial system.
Panel A : slope of the line markedly increased with increases in arterial resistance.
Panel B : slope of the relationship line only mildly increased with decreases in arterial compliance.
From : Sunagawa, K., Maughan, W. L., Burkhoff, D., & Sagawa, K. (1983). Left ventricular
interaction with arterial load studied in isolated canine ventricle. Am J Physiol, 245(5 Pt 1), H773780
The slope of the ventricular (figure 2.43) and arterial (figure 2.44) end-systolic pressure
–stroke volume relationships have both the dimension of volume elastance (mmHg/ml).
As a consequence the arterial system is treated as if it was an elastic chamber with a
volume elastance Ea, just as the left ventricle was treated as an elastic chamber with the
end-systolic elastance Ees (Sunagawa 1983). Referring to figure 2.19 ejection of blood
from the left ventricle into the arterial system can be considered as a shift of blood from
one elastic chamber into another elastic chamber. The distribution of the volume
between the two chambers will be determined by the ratio of their volume elastance
values [244].The slope of the arterial Pes -SV relationship represents not the physical
elastance (or compliance) of the arterial system but an “effective” arterial elastance
[244]. Effective arterial elastance changes more with changes physical arterial
resistance than with changes in physical arterial compliance (figure 2.44) [214, 244].
118
Effective arterial elastance is an index of arterial vascular load that describes a
mechanical characteristic of the arterial system and that incorporates non-pulsatile and
pulsatile components of this load [125, 254].
Figure 2.43 and 2.44 can be combined into one single graph: the intersection of the
ventricular and arterial end-systolic pressure – stroke volume relationship determines
the equilibrium stroke volume (figure 2.45).
Figure 2.45
Panel A : graphical determination of equilibrium stroke volume when ventricle with a volume
elastance Ees is coupled to the arterial system with a volume elastance Ea. Stroke volume is obtained
as the abscissa of the intersection of these two lines.
Panel B : physical meaning of the proposed ventriculo-arterial coupling model. Both ventricle and
arterial system are considered as elastic chambers with known volume elastances (Ees and Ea
respectively). Stroke volume is determined as the amount of fluid transferred from the ventricle to
the arterial system when the two are connected.
From : Sunagawa, K., Maughan, W. L., Burkhoff, D., & Sagawa, K. (1983). Left ventricular
interaction with arterial load studied in isolated canine ventricle. Am J Physiol, 245(5 Pt 1), H773780
119
The fundamental concept of ventricular-arterial coupling is clearly demonstrated in the
Pes-SV plane: how much volume (i.e. SV) will be transferred if one couples an elastic
chamber (LV) with a know Ees, Ved and Vdes to a second elastic chamber (arterial
system) with a known Ea (figure 2.46) [244] ?
Figure 2.46
Panel A : effects of changes in end-diastolic volume on equilibrium stroke volume (SV). Ventricular
Ps-SV relationship line (VPSVR) will shift rightward (tick arrow) with increases in end-diastolic
volume. Arterial Ps-SV relationship line (APSVR) will not change with changes in ventricular
preload. Equilibrium stroke volume will increase (thin arrow) from control (open circle) to a new
steady state (closed circle) with an increase in end-diastolic volume.
Panel B : effect of changes in contractility on equilibrium SV. The ventricular Ps-SV relationship
line becomes steeper (thick arrow) with increases in contractility. Equilibrium SV will increase as
indicated by the shift in intersections (thin arrow) from control (open circle) to the new steady state
(closed circle).
Panel C : effects of an increase in arterial resistance on SV. Arterial Ps-SV relationship line is
steeper (thick arrow) with increases in arterial resistance, while the ventricular Ps-SV is
unchanged. Equilibrium SV decreases as shown by the shift in intersections (thin arrow) from
control (open circle) to the new steady state (closed circle).
From : Sunagawa, K., Maughan, W. L., Burkhoff, D., & Sagawa, K. (1983). Left ventricular
interaction with arterial load studied in isolated canine ventricle. Am J Physiol, 245(5 Pt 1), H773780
Due to the introduction of describing the ventricular properties by Ees in the pressurevolume plane, the practical graphical analysis of ventricular-arterial coupling is usually
performed in the pressure-volume plane and not in the Pes-SV plane. Therefore the
arterial Pes-SV graph has to be reflected and translated so that its origin lies at the
120
appropriate end-diastolic volume (figure 2.47). End-systolic volume is given as the
intersection between the ventricular and arterial ESPVR line. Stroke volume is the
difference between the end-diastolic volume (the abscissal coordinate of the lower right
corner of the PV loop) and end-systolic volume (the abscissal coordinate of the upper
left corner of the PV loop) [214].
Figure 2.47
The slope of the arterial ESPVR (line A) is Ea with the volume axis intercept of Ved (end-diastolic
volume). Line B represents the ventricular ESPVR with the volume axis intercept of V0. Endsystolic volume is given as the intersection between these two lines. Stroke volume is the difference
between the end-diastolic volume (the abscissal ordinate of the lower right corner of the pressure
volume loop) and end-systolic volume (the abscissal ordinate of the upper left corner of the
pressure volume loop).
From : Sagawa, K., Maughan, L., Suga, H., & Sunagawa, K. (1988). Cardiac Contraction and the
Pressure-Volume Relationship. New York: Oxford University Press
Figure 2.48 illustrates the effects on stroke volume of changing :
-
afterload
-
end-diastolic volume or preload (Frank-Starling mechanism)
-
contractility.
121
Figure 2.48
Effects on stroke volume of changes in arterial resistance (left panel), end-diastolic volume (middle
panel) and contractility (right panel). The closed circle represents the equilibrium end-systolic
pressure-volume point under control conditions.
From : Sunagawa, K., Sagawa, K., & Maughan, W. L. (1987). Ventricular Interaction with the
Vascular System in Terms of Pressure-Volume Relationships. In F. C. Yin (Ed.),
Ventricular/Vascular Coupling : Clinical, Physiological, and Engineering Aspects (pp. 210-239). New
York: Springer Verlag.
Referring to figure 2.45 and figure 2.47 it is clear that many combinations of ventricular
and arterial end-systolic pressure – stroke volume lines can result in the same
equilibrium point (i.e. stroke volume). Consequently it is reasonable to ask if a
particular combination is better or optimal than others. In order to answer this question,
one has to establish criteria with which to judge what is “optimal” [169].
The are three optimization criteria [169] :
-
maximal power transfer between the energy source and its load, i.e. maximal
SW
-
maximal ventricular mechanical efficiency (SW/Vo2)
-
minimal relative oscillatory power (cfr. 2.5.1).
Sunagawa demonstrated in the isolated canine left ventricle, loaded with a 3-element
Windkessel arterial model, that the left ventricle does maximal external work model to
the arterial load when the ventricular and arterial elastance are equalized [246]. Based
upon this experimental work of Sunagawa, Burkhoff and Sagawa demonstrated through
122
model-based analytical expressions for stroke work and mechanical efficiency, that
(figure 2.49) [32] :
1. stroke work is always maximal when Ea = Ees
2. mechanical efficiency is always maximal when Ea = 0.5 x Ees.
Figure 2.49 (Ees = 7)
Relation between afterload impedance as quantified by effective arterial elastance (Ea) and stroke
work SW (panel A), myocardial oxygen consumption MVO2 (panel B), and ventricular efficiency
(panel C).
Dashed lines, Ea value that produces greatest SW; dotted lines, Ea value that produces greatest
efficiency.
Ees equals 7 mmHg/milliliter.
From : Burkhoff, D., & Sagawa, K. (1986). Ventricular efficiency predicted by an analytical model.
Am J Physiol, 250(6 Pt 2), R1021-1027
123
In addition figure 2.49 demonstrates that efficiency is more sensitive than the external
work to changes in Ea [214]. Burkhoff and Sagawa also demonstrated that SW and
CME of a heart with impaired contractility (i.e. a low Ees value) are more sensitive to
changes in Ea (: afterload) than that of a heart with increased contractility (i.e. a high Ees
value) (figure 2.50) [32].
Figure 2.50
Relation between afterload impedance as quantified by effective arterial elastance (Ea) and stroke
work SW (panel A), myocardial oxygen consumption MVO2 (panel B), and ventricular efficiency
(panel C) for different values of end-systolic elastance (Ees) as indicated in panel A. Arrow heads in
panel A : Ea value equal to set Ees value. This is point at which SW is maximum.
From : Burkhoff, D., & Sagawa, K. (1986). Ventricular efficiency predicted by an analytical model.
Am J Physiol, 250(6 Pt 2), R1021-1027
124
2.5.4 Computational aspects for determining ventricular-arterial
coupling and CME.
2.5.4.1. Effective arterial elastance Ea can be calculated using the formula [244] :
Ea =
Pes
SV
(Eq. 2.40)
However Ea can be calculated during the single beat estimation of Ees, as the slope of
! connecting the end-systolic to end-diastolic point (figure 2.35).
the line
2.5.4.2. Cardiac mechanical efficiency was defined as the ratio of stroke work to
oxygen consumption per beat (equation 2.38). Rooke and Feigl devised the pressure
work index (PWI) as an empirical index for estimating left ventricular oxygen
consumption in the dog [208] :
# (0.8 " P + 0.2 " P ) " HR " SV &
sys
dias
( + 1.43
PWI = K1 ( Psys " HR) + K 2 %%
(
BW
$
'
!
with
(Eq. 2.41)
PWI : pressure work index (ml O2/min/100 g)
Psys : systolic arterial blood pressure (mmHg)
Pdias : diastolic arterial blood pressure (mmHg)
SV : stroke volume (ml)
BW : body weight (kg)
K1 = 4.08 x 10-4
K2 = 3.25 x 10-4
It is important to realize that the PWI is an empirical index, and it should be emphasized
that all indexes are only correlations and that experimental data may be satisfactorily fit
by a number of indices [208]. Correlation quantifies the strength of an association
between variables [82], but association does not mean causation. As such there is no
125
physiological explanation for the term Pdias in equation 2.41 (arterial diastolic variables
are considered to have no effect on left ventricular oxygen consumption). The objective
of Rooke and Feigl was dual [208]:
-
To evaluate the importance of left ventricular stroke volume on left
ventricular oxygen consumption,
-
To device a index estimating left ventricular oxygen consumption, from
easy to perform cardiovascular measurements, that incorporates the
(systolic) rate-pressure product (= Psyst x HR) and left ventricular external
work (equation 2.28) [208]. Referring to equation 2.28, Walker et al. found
that (Dias + 0.8 x PP) is a close estimate of mean pressure during ejection
[265].
Because PWI is expressed in ml O2 “per minute” per 100 g, and stroke work as work
per beat (i.e. Joule per beat) (equations 2.28 and 2.29) caution should be applied with
regard to the units used for calculating CME [214]. There are several equivalent ways to
proceed, e.g. :
-
on a “per beat” basis and in terms of energy (Joule) :
1. calculate (left ventricular) stroke work (in Joule) by equation 2.29,
2. divide PWI by the heart rate per minute,
3. convert milliliters O2 to Joule by means of the caloric equivalent
[214, 239, 275] :
1 milliliter O2 ≈ 20 Joules.
-
(Eq. 2.42)
on a “per minute” basis and in terms of oxygen consumption:
1. calculate steady power W˙ st (in Watts) (equation 2.33),
2. converts watts to milliliters O2 per minute by means of the caloric
equivalent [275] :
!
1 milliliter O2/minute ≈ 0.33 Watt.
(Eq. 2.43)
Although these calculations are mathematically equivalent, but when interpreting the
CME results, one should take into account the potentially confounding effect of the
heart rate.
126