3. the end-systolic pressure-volume relationship (ESPVR),

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 ]

[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]

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].

60 50 40 30 20 10

LVP (mmHg)

time (sec) 0.4

0.2 0.0 -0.2 -0.4

dLVP/dt (mmHg/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]

, - dP/dt related to a fixed pressure (e.g. 50 mmHg) [189],

- or the ratio of [dP/dt]

to EDV (end-diastolic volume).

The standard ejection phase index of contractility is the ejection fraction EF :

EF = SV

EDV = EDV " ESV

EDV = 1" ESV

EDV (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 19

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].

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 E

, 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].

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 end- diastolic 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.

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

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 pressure- volume 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].

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 E

near the end of ejection, and the instant at which E(t) equals E

was called end systole [214, 238].

Referring to figure 2.32, E

is the slope of the end-systolic P-V relation line. The time to E

from the onset of systole was called T

[241].

Suga demonstrated that the E(t) curve, E

and T

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 E

and T

[215, 241, 243]. E

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].

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.

The regression analysis indicated that the instantaneous pressure-volume relationship at any specified time t

, can be specified by the following equation [215] :

!

P(t

) = E (t

) [ V (t

) " V

] (Eq. 2.24)

where P(t

) : ventricular pressure at time t

V(t

) : ventricular volume at time t

E(t

) : the slope of the pressure-volume regression line at t

V

: the volume-axis intercept of the afore mentioned regression at t

.

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]:

!

P

= E

[ V

" V

] (Eq. 2.25)

Stroke volume equals end-diastolic volume V

minus end-systolic volume V

(figure 2.2) :

!

SV = V

" V

(Eq. 2.26)

Rearranging equation 2.26 and substituting the result in equation 2.25 leads to :

!

P

= E

[ V

" SV " V

] (Eq. 2.27)

The foregoing account emphasize the fact that the ESPVR, indexed by E

, and E

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, E

and E

are nearly identical [117]. However, when the afterload is significantly altered among the cardiac cycles used for E

or E

determination, the two can very different (figure 2.34) [52, 117].

Figure 2.34

Disparity between E_max and E_es 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 E

is usually determined rather than E

, E

is

considered the more useful measure of systolic properties for the purpose of assessing

pump function, while E

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].

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 E

from a single beat: the peak systolic pressure P

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 P

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 E

.

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 (E_es) and effective arterial elastance (E_a).

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 pressure- volume diagram. The ESPVR line is drawn from P_max 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. E_es is the slope of the ESPVR line, and E_a 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.

A single beat estimation of E

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 end- diastolic volume. E_es is the slope of ESPVR.

Panel B : Schematic of a method for determining the ESPVR line from a single ejecting beat.

P_max(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 E_es [E_es(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.

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 V_w 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].

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

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:

!

SW = P

(t) dV

EDV EEV

" (Eq. 2.28)

with : SW : stroke work, i.e. the mechanical energy the ventricle has to perform in order to eject the stroke volume (unit: Joule),

P

(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 P

≈ P

≈

!

AoP and P

≈ PAOP, equation 2.28 can be approximated by :

!

SW = SV " (AoP # PAOP) (Eq. 2.29)

with P

: ventricular pressure at end ejection, P

: ventricular pressure at end systole, P

: ventricular pressure at end diastole,

!

AoP : mean aortic pressure,

PAOP : pulmonary arterial occluded pressure.

A decrement in ventricular volume is manifested as aortic flow Q

(t) :

!

Q

(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.

A change of integration variable in equation 2.28 results in :

!

W = P

(t) dV (t) dt dt

Tb Te

"

W = P

(t)Q

(t)dt

Tb Te

"

(Eq. 2.31)

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,

P

(t): instantaneous aortic pressure, Q

(t) : instantaneous aortic pressure, T

: time of beginning of the cardiac cycle, T

: 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,

- P

(t) has been substituted by P

(t) : refering to figure 2.4 , a small difference is noted between P

and P

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].

The product P

(t).Q

(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

!

Q

Difference between

!

P " Q

Figure from : Milnor, W. R. (1972). Pulsatile blood flow. N Engl J Med, 287(1), 27-34.

Text from : Milnor, W. R. (1982). Hemodynamics. Baltimore: Williams & Wilkins.

Time averaged hydraulic power, called total (external) hydraulic power

!

W ˙

, indicated by the dotted line in the upper panel of figure 2.40, equals :

!

W ˙

=

P

(t)Q

(t)dt

Tb Te

"

T (Eq. 2.32)

with T: heart period, T

: time of begin and T

: 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

is expended principally in moving blood through arterioles, and hence depends to a great extent on peripheral resistance, and is calculated as:

!

W

= AoP " AoQ (Eq. 2.33)

If input resistance is defined as :

!

R

= AoP

AoQ

then :

!

W

= AoQ

" R

= AoP R

2

(Eq. 2.34)

2. oscillatory flow power

!

W

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

= 1

2 (Q

)

Z

cos"

n=1 N

# (Eq. 2.35)

with Q

: the amplitude of the n-th flow harmonic,

Z

: the modulus of the n-th harmonic of the aortic input impedance, θ

: 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:

!

W

= W

+ W

(Eq. 2.36)

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 ˙

amounts to 15% of

!

W ˙

, while in the pulmonary circulation

!

W ˙

amounts to 30% of

!

W ˙

[154, 155]. The ratio

!

W ˙

/ ˙ W

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 end-

diastolic 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

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 end- diastolic curve on the left side of the P-V trajectory, representing end- systolic 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.

Suga also demonstrated that total left ventricular oxygen consumption per beat (Vo

) is linearly correlated to the PVA [214, 239], with a regression line in the form :

Vo

= a.PVA + b (Eq. 2.37)

The regression constant b represents that part of Vo

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 V

2

= SW V

2

(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 V

2

= SW

PVA " PVA V

2

(Eq. 2.39)

The ratio PVA/Vo

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/Vo

. As a consequence CME increases or decreases depending on the relative changes of SW/PVA and PVA/Vo

[214]. It can be demonstrated that SW/PVA approaches 100% as the end- systolic pressure P

becomes very small or E

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].

One already demonstrated that the concept of ventricular elastance (i.e. a pressure- volume relation) is an appropriate tool to describe the mechanical behavior of the left ventricle. From equation 2.27 one can infer that P

varies inversely with SV in linear way (when E

,V

and V

constant) : the larger the ejected stroke volume the lower left ventricular end-systolic pressure P

. This relationship is implicit in figure 2.42,

representing two PV trajectories with the same EDV but different P

, but explicit in figure 2.43. The P

-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.

Figure 2.43

(A) Left ventricular end-systolic pressure (P_es) – 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 E_es rotates the ventricular P_es –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 P

-SV relationship is called

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), H773- 780

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 E

, just as the left ventricle was treated as an elastic chamber with the end-systolic elastance E

(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 P

-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].

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 E_es is coupled to the arterial system with a volume elastance E_a. 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 (E_es and E_a 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), H773- 780

The fundamental concept of ventricular-arterial coupling is clearly demonstrated in the P

-SV plane: how much volume (i.e. SV) will be transferred if one couples an elastic chamber (LV) with a know E

, V

and Vd

to a second elastic chamber (arterial system) with a known E

(figure 2.46) [244] ?

Figure 2.46

Panel A : effects of changes in end-diastolic volume on equilibrium stroke volume (SV). Ventricular P_s-SV relationship line (VPSVR) will shift rightward (tick arrow) with increases in end-diastolic volume. Arterial P_s-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 P_s-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), H773- 780

Due to the introduction of describing the ventricular properties by E

in the pressure-

volume plane, the practical graphical analysis of ventricular-arterial coupling is usually

performed in the pressure-volume plane and not in the P

-SV plane. Therefore the

arterial Pes-SV graph has to be reflected and translated so that its origin lies at the

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. End- systolic 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.

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/Vo

) - 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

model-based analytical expressions for stroke work and mechanical efficiency, that (figure 2.49) [32] :

1. stroke work is always maximal when E

= E

2. mechanical efficiency is always maximal when E

= 0.5 x E

.

Figure 2.49 (E

= 7)

Relation between afterload impedance as quantified by effective arterial elastance (E_a) and stroke work SW (panel A), myocardial oxygen consumption MVO2 (panel B), and ventricular efficiency (panel C).

Dashed lines, E_a value that produces greatest SW; dotted lines, E_a value that produces greatest efficiency.

In addition figure 2.49 demonstrates that efficiency is more sensitive than the external work to changes in E

[214]. Burkhoff and Sagawa also demonstrated that SW and CME of a heart with impaired contractility (i.e. a low E

value) are more sensitive to changes in E

(: afterload) than that of a heart with increased contractility (i.e. a high E

value) (figure 2.50) [32].

Figure 2.50

Relation between afterload impedance as quantified by effective arterial elastance (E_a) and stroke work SW (panel A), myocardial oxygen consumption MVO₂ (panel B), and ventricular efficiency (panel C) for different values of end-systolic elastance (E_es) as indicated in panel A. Arrow heads in panel A : E_a value equal to set E_es 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

2.5.4 Computational aspects for determining ventricular-arterial coupling and CME.

2.5.4.1. Effective arterial elastance E

can be calculated using the formula [244] :

!

E

= P

SV (Eq. 2.40)

However E

can be calculated during the single beat estimation of E

, as the slope of the line connecting the end-systolic to end-diastolic point (figure 2.35).

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] :

!

PWI = K

( P

" HR ) ^{+ K}

( ^{0.8 " P}

^{+ 0.2 " P}

) ^{" HR " SV}

BW

#

$

% %

&

'

( ( + 1.43 (Eq. 2.41)

with PWI : pressure work index (ml O

/min/100 g) P

: systolic arterial blood pressure (mmHg) P

: diastolic arterial blood pressure (mmHg) SV : stroke volume (ml)

BW : body weight (kg) K

= 4.08 x 10

K

= 3.25 x 10

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

physiological explanation for the term P

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 (= P

x HR) and left ventricular external work (equation 2.28) [208]. Referring to equation 2.28, Walker et al. found that (D

+ 0.8 x PP) is a close estimate of mean pressure during ejection [265].

Because PWI is expressed in ml O

“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 O

to Joule by means of the caloric equivalent [214, 239, 275] :

1 milliliter O

≈ 20 Joules. (Eq. 2.42) - on a “per minute” basis and in terms of oxygen consumption:

1. calculate steady power

!

W ˙

(in Watts) (equation 2.33),

2. converts watts to milliliters O

per minute by means of the caloric equivalent [275] :

1 milliliter O

/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.