INTRODUCTION

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EFFECTIVE ARTERIAL ELASTANCE (E_a), commonly known as the ratio of left ventricular (LV) end-systolic pressure and stroke volume (SV) (31, 32), is a simple and convenient way to characterize the arterial load from pressure-volume data measured in the LV. As outlined by Sunagawa et al. (32), E_a can also be approximated from knowledge of total peripheral resistance (R), total arterial compliance (C), aortic characteristic impedance (Z₀), and systolic and diastolic time intervals, parameters that can be derived from pressure and flow measured in the ascending aorta. E_a thus incorporates both steady (R) and pulsatile (C, Z₀) components of the arterial load. It was later shown by Kelly and co-workers (15) that there is a good agreement between E_a calculated from pressure-volume data and E_a calculated from arterial impedance, E_a(Z), in normal and hypertensive human subjects. Provided that 1) end-systolic pressure can be approximated by mean arterial pressure, and 2) the time constant of the arterial system, RC, is large compared with the diastolic time interval, E_a further reduces to R/T, where T is the cardiac cycle length (15, 32).

E_a is, however, a parameter originating from studies considering mechanicoenergetic aspects of heart-arterial interaction (15, 31, 32), where it has been combined with E_max (i.e., the slope of the LV end-systolic pressure-volume relation; Ref. 30) to be used as the heart-arterial coupling parameter E_a/E_max (1, 2, 4, 6, 14, 19, 20, 22, 27, 31, 32). Analytical work based on the assumption E_a  R/T revealed that the heart delivers maximal stroke work (SW) when E_a/E_max = 1 (4, 32), whereas optimal efficiency (ratio of SW to myocardial oxygen consumption) is obtained when E_a/E_max = 0.5 (4). This theoretical relation has been confirmed in experimental work (4, 9, 31, 32), although it has also been observed that SW remains near maximal within a relatively wide range of E_a/E_max values (9).

E_a is increasingly being used as a means to quantify the properties of the arterial system (6-8, 10, 15, 20, 21). Although E_a incorporates steady and pulsatile features of arterial impedance, it is important to realize that it is not a surrogate of impedance (31), which can only be calculated from the ratio of measured aortic pressure and flow and which is expressed in terms of complex harmonics in the frequency domain (17, 18). E_a lumps the steady and pulsatile components of the arterial load into a single number, but it does not provide any information on their relative contribution. Additional information (e.g., R) is required for an unequivocal characterization of the arterial system. Furthermore, by itself, E_a is notdespite its dimensional unitsa measure of arterial stiffness, because R and heart rate (HR) also contribute to E_a.

The aim of this study was twofold. First, we wanted to illustrate that E_a, by itself, contains insufficient information to fully capture the arterial system. Second, we wanted to illustrate how the finite arterial compliance interferes with the theoretical relationship between E_a/E_max and LV SW generation. Using a previously validated heart-arterial interaction model (23, 24, 28), we calculated LV pressure-volume loops, aortic pressure and flow, and E_a for a set of chosen and fixed cardiac parameters but with values for arterial resistance and compliance covering the human pathophysiological range. This allowed us to demonstrate 1) the relation between R and C with E_a; 2) the nonspecific character of E_a and the impact on calculated aortic pressure and flow; 3) the relation between E_a, E_a(Z), and R/T; and 4) the impact on E_a/E_max as a determinant of LV SW and hydraulic power generation.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The heart-arterial interaction model. Aortic blood pressure is computed using a previously validated heart-arterial interaction model (Refs. 23, 24, 28; Fig. 1). LV function is described by a time-varying elastance model (30) and is coupled to a four-element lumped-parameter windkessel model representing the systemic arterial load (29). The arterial model parameters are R, C, total inertance (L), and Z₀. Time-varying elastance is calculated as E(t) = P_LV/(V_LV  V_d), where P_LV and V_LV are LV pressure and volume, respectively, and V_d is the intercept of the end-systolic pressure-volume relation. It has been shown that the shape of the normalized E(t) curve [E_N(t_N)], obtained after normalization of E(t) with respect to amplitude and time, remains constant under various pathophysiological conditions (25). E_N(t_N) is thus assumed to be constant and has been implemented in the model (23, 24). The actual E(t) is then characterized by a limited number of cardiac parameters: the slope (E_max) and intercept (V_d) of the end-systolic pressure-volume relation, LV end-diastolic volume (LVEDV), venous filling pressure (P_v), HR, and the time to reach maximal elastance (t_P). Cardiac valves are simulated as frictionless, perfectly closing devices, allowing forward flow only.

View larger version (21K): [in this window] [in a new window]   Fig. 1.   In the heart-arterial interaction model (A), the heart function is modeled as a time-varying elastance function E(t) (B). The arterial model is a lumped-parameter model consisting of total compliance (C), total peripheral resistance (R), characteristic impedance of the aorta (Z0), and the inertia of blood in the systemic arteries (L). The model directly yields left ventricular (LV) pressure and volume and aortic pressure and flow (C). Rmv, mitral valve resistance, which was assumed 0 for these simulations.

Relation between arterial parameters R and C and E_a. E_max and V_d are taken as 1.7 mmHg/ml and 15 ml (6), respectively. LVEDV is chosen as 120 ml, and P_v is set to 5 mmHg. HR is 75 beats/min, and t_P is 0.3 s (38% of cardiac cycle length). Control values for Z₀ (18) and L (29) are 0.033 mmHg · ml¹ · s and 0.005 mmHg · ml¹ · s², respectively. These parameters are kept constant during the computations. R is varied from 0.6 to 1.8 mmHg · ml¹ · s (0.12 mmHg · ml¹ · s increments), and C is varied from 0.5 to 2 ml/mmHg (0.15 ml/mmHg increments) giving 11 values for each parameter covering the normal to pathophysiological range. Model simulations have been done for the 121 possible combinations of R and C. For each of these simulations, E_a is calculated from the data as the ratio of end-systolic pressure (P_es) and stroke volume (SV). E_a is then presented as a function of R and 1/C (total arterial stiffness).

Ea/Emax as determinant of LV mechanical energetics. Hydraulic power () is calculated as the product of instantaneous aortic pressure (P_Ao) and flow (Q_Ao), and its maximum yields maximal power (_max). LVSW is calculated as the area contained within each of the P-V loops for the combinations of R and C studied. SW and _max are presented as a function of E_a/E_max.

	RESULTS

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Relation between arterial parameters R and C and E_a. As an illustration of our results, Fig. 2 shows P-V loops and aortic pressure and flow calculated for three different combinations of R and C, each yielding the same E_a of 1.7 mmHg/ml (R = 1.08, 1.2, and 1.32 mmHg · ml¹ · s and corresponding C = 0.8, 1.1, and 2 ml/mmHg, respectively). For all 121 simulations, E_a is plotted as a function of R and 1/C in Fig. 3. For a given compliance value, E_a practically linearly increases with resistance. Although the relation of E_a with compliance is nonlinear, it linearizes when E_a is expressed as a function of 1/C. The relations E_a(R) or E_a(1/C) shift with C and R, but their slopes are independent of C and R and are 1.28 s¹ and 0.31, respectively. Multiple-linear regression analysis with E_a as independent and R/T (with, in this case, constant T = 0.8 s) and 1/C as dependent variables yields E_a = 0.127 + 1.023R/T + 0.314/C (r² = 0.99). For a given resistance, E_a tends toward an asymptotic value (R/T) for high values of C (low values of 1/C). It may be seen that several possible combinations of R and C yield the same E_a.

View larger version (16K): [in this window] [in a new window]   Fig. 2.   LV pressure-volume loops (A), aortic pressure (B), and aortic flow (C) for 3 combinations of R and C, each giving effective arterial elastance (Ea) = 1.7 mmHg/ml (Ea/Emax = 1, where Emax is slope of LV end-systolic pressure-volume relation). Vd, intercept of end-systolic pressure-volume relation.

View larger version (17K): [in this window] [in a new window]   Fig. 3.   The heart-arterial model was loaded with values for R (0.6-1.8 mmHg · ml1 · s) and C (0.5-2 ml/mmHg) covering the human pathophysiological range. Ea was then calculated as the ratio of LV end-systolic pressure and stroke volume (SV). A: data are organized to show the variation of Ea with R for fixed values of C. B: the variation of Ea with 1/C for fixed values of R. C: the quasi-perfect agreement (solid line) between Ea calculated as the ratio of end-systolic pressure and SV and Ea predicted from R and 1/C with the multiple-linear regression equation. T, cardiac cycle length.

View larger version (21K): [in this window] [in a new window]   Fig. 4.   A: the difference between Ea(Z), calculated from arterial system properties, and Ea, calculated as Pes/SV, is given as a function of Pes/SV. The solid line is the mean difference; the dashed lines represent mean difference ± 2SD. B: the difference between Ea and R/T.

E_a/E_max as determinant of LV mechanical energetics. SW and _max are given as a function of E_a/E_max for constant compliance values (Fig. 5). For C = 2 ml/mmHg, SW is maximal when E_a/E_max equals 1. For all other values, the relation is less clear, but maximal SW is reduced and the maximum is found for E_a/E_max between 0.6 and 1.1. For a given E_a/E_max, maximal power increases with C. For a given C, _max first decays with E_a/E_max, reaches a minimum, and then increases with E_a/E_max. The value of E_a/E_max corresponding to this minimum is a function of C. 

View larger version (18K): [in this window] [in a new window]   Fig. 5.   Stroke work (A) and maximal power (B) as a function of Ea/Emax. For each curve, C is constant and R varies. For fixed Ea/Emax, higher C yields higher stroke work and maximal power.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Our results demonstrate that for a given condition of the heart (HR, contractility, and end-diastolic volume), E_a is linearly related to R and to 1/C. There is an excellent correlation and good agreement between E_a(Z) calculated from vascular system properties and E_a calculated as P_es/SV. The sensitivity of E_a to 1/C is three times lower than to R/T. For large compliance values (>2 ml/mmHg), E_a approximates R/T. E_a, by itself, cannot be used to quantify the arterial system, because there are different combinations of R and C, yielding the same E_a but representing totally different arterial loads. This limitation becomes obvious when SW is plotted as a function of E_a/E_max. For large C values, we find the theoretical relation with maximal SW at E_a/E_max = 1. For C values within the normal pathophysiological range, however, maximal SW is reduced, and this maximal SW value occurs within a wider range of E_a/E_max values.

We varied R and C over what we consider the pathophysiological range in the adult human. There is, however, a large variability in reported values for R and C, both in control and pathological conditions, because of different flow measuring techniques and different methods to estimate C. Aortic pulse wave velocity, independent of flow measuring techniques, can change by a factor of 2 in aging and in hypertension (3). C is proportional to the square of pulse wave velocity (17) and may thus change by a factor of 4. We varied C from 0.5 to 2 ml/mmHg, thereby covering the reported range of values in normal and pathological conditions in humans (5, 13, 16, 26). Changes in R were between 0.6 and 1.8 mmHg · ml¹ · s (12, 13, 26).

Because E_a depends both on R and C, it is clear that it cannot represent a unique arterial load. The question is whether different combinations of R and C, giving the same E_a, actually occur in humans, because in aging and in hypertension both R and arterial stiffness tend to increase. However, the data in the literature show that, within healthy or pathological populations, there is considerable biological diversity in both R and C (3, 5, 12, 13, 16, 20, 26). Figure 2 also illustrates that despite identical E_a, markedly different pressure and flow wave profiles are found, each with physiological values for blood pressure and SV. These simulations thus show that it is reasonable to assume that combinations of R and C presenting the same E_a actually occur. Figure 2 also shows that E_a is not necessarily related to indexes characterizing the arterial wave shape or wave reflection such as the augmentation index. This may explain why in a recent study, despite different values for E_a, the augmentation index was similar in two groups of hypertensive patients (20).

The agreement between E_a(Z) and P_es/SV was previously demonstrated in humans (15). Our computer simulation data confirm the excellent correlation between E_a(Z) and P_es/SV, but E_a(Z) is, on average, 0.13 mmHg/ml higher than P_es/SV [although Kelly et al. (15) found a small underestimation of E_a(Z) compared with P_es/SV]. We believe the discrepancy is because we calculated E_a(Z) by using the parameters of the four-element windkessel model that was used as arterial load, whereas the expression for E_a(Z) is based on a three-element windkessel model. It is known that the latter characterizes the impedance spectrum with higher values for C and lower values for Z₀ than the four-element windkessel model (29). Our multiple-linear regression analysis results indicate that the relation between P_es/SV and arterial system properties can be further simplified as a linear relation with R/T and 1/C. However, this relation requires further validation in vivo, where, besides arterial system properties, HR, t_s and t_d also vary.

It has been shown theoretically that, for a given preload (LVEDV) and inotropic state (E_max and V_d) of the heart, SW is determined only by E_a/E_max and SW is maximal when E_a/E_max = 1 (4). This relation was derived under the assumptions that E_a  R/T and that SW can be approximated by the product of SV and P_es. In experimental studies, the E_a/E_max value corresponding to maximal SW has been reported to be <1, with SW remaining close to maximal (>90% of optimal value) for a wide range of E_a/E_max values (0.3-1.3) (9). We found that a single value of E_a may correspond to different values for SW and (Fig. 4). E_a/E_max corresponding to maximal SW as well as the range over which SW remains maximal change with C. For large C values (C = 2 ml/mmHg), the relation between E_a/E_max and SW approximates the theoretical prediction, with SW being maximal for E_a/E_max = 1. For lower C values, maximal SW is reduced and is reached for E_a/E_max < 1 and maximal SW can be achieved for a wider E_a/E_max range. We also plotted the relation between E_a/E_max and _max, a parameter frequently used to characterize cardiac performance. Again, a single value for E_a/E_max corresponds to very distinct values of _max. Because E_max was constant for all simulations, this further demonstrates that E_a is not an unequivocal measure for arterial load and, therefore, E_a/E_max is not a specific measure for heart-arterial interaction.

The use of E_a and E_a/E_max has been promoted by theoretical and experimental studies linking E_a/E_max to LV mechanicoenergetics (1, 2, 4, 6, 14, 22, 27, 32). It has been shown in humans that E_a/E_max is ~1 in the normal heart and that the LV operates close to optimal efficiency or SW (2, 6). This optimal energetic coupling of the heart and arterial system seems to be preserved in normal aging (6, 8) and in hypertension (7). In contrast, in heart failure, cardiac contractility (E_max) is impaired, whereas E_a generally increases and E_a/E_max increases progressively (14, 22). Note, however, that E_a/E_max is mainly a parameter related to LV volumes. With E_a = P_es/SV and E_max = P_es/(LVEDV  SV  V_d) and assuming V_d to be small enough that it can be neglected, E_a/E_max = LVEDV/SV  1 or E_a/E_max = 1/EF  1, where EF is ejection fraction. In normal hearts, where EF is ~0.5, E_a/E_max is indeed 1. In failing, dilated hearts, EF decreases and E_a/E_max thus increases. Why LVEF is ~0.5 in the normal heart can be argued on mechanical-energetic grounds, but it has also been shown that this value is explicable on basis of evolutionary arguments (11). Also, the human body has no sensors or receptors sensitive to SW or power output. It is therefore unlikely that there are control mechanisms maintaining constant E_a/E_max to operate at maximal power or maximal efficiency.

In conclusion, we have shown that E_a is related to R/T and arterial elastance, i.e., 1/C, in a linear way, but the sensitivity of E_a to a change in R/T is about three times higher than to a similar change in arterial stiffness. E_a is a convenient parameter, lumping pulsatile and steady components of the arterial load in a concise way, but it does not unequivocally characterize arterial system properties. The nonspecific character of E_a and the fact that E_a can be approximated as R/T only for high C values contribute to the discrepancy between the observed and theoretical relationship between E_a/E_max and SW.