## Abstract

Late systolic augmentation of the ascending aortic pressure waveform is believed to be caused by particular impedance patterns but also could be caused by particular left ventricular outflow patterns. Using a linear mathematical model of the entire human arterial tree, we derived realistic impedance patterns by altering *1*) Young’s modulus of the arterial wall of the individual branches,*2*) peripheral reflection coefficients, and *3*) distal compliances at the terminations. These calculated impedance patterns were then coupled to realistic left ventricular outflow patterns determined by unique *1*) end-diastolic and end-systolic pressure-volume relationships,*2*) preload-recruitable stroke work relationships, and *3*) shortening paths simulated by altered aortic flow contours. As determined by the ratio of the individual parameter coefficient of determination (*r*
^{2}) to the overall model *r*
^{2}, late systolic pressure augmentation was more strongly determined by left ventricular outflow patterns than by arterial impedance parameters (*r*
^{2} ratio: 53% vs. 33%). Thus left ventricular outflow patterns are at least as important as impedance parameters in determining late systolic pressure augmentation in this model.

- arterial model
- hemodynamics
- ventricular function
- hypertension

one of the prominent features of the ascending aortic pressure waveform in elderly humans is the presence of a late systolic peak (11, 15). This late systolic peak is regarded as detrimental to left ventricular energetics because it increases the myocardial oxygen demand-to-supply ratio via increased late systolic load and decreased pressure during diastole (18). It is considered also as a major factor in the development and progression of left ventricular hypertrophy, an independent risk factor for cardiovascular mortality (8).

The relationship between the late systolic peak and ascending aortic impedance patterns was first demonstrated by Murgo et al. (15). It was suggested that increased reflected pressure waves from the periphery, as evidenced from altered ascending aortic impedance patterns, summed with the forward-going pressure waves during late systole (15, 16). Subsequent studies showed that various drugs alter the modulus of zero and first harmonic components of the arterial impedance while simultaneously altering the late systolic pressure peak amplitude (10,19, 25). Further studies also found that the late systolic pressure peak increased with age (11), shorter body length (14), and increased arterial stiffness (19). Because pulse wave velocity, characteristic impedance, and zero crossing of the impedance phase were associated with the timing of reflected waves and distal arterial compliance and resistance were associated with the intensity of reflected waves, these observations on the amplitude of the late systolic peak were explained as manifestations of altered timing and intensity of reflected pressure waves (17).

By definition, however, the aortic pressure waveform is a function not only of the load (i.e., arterial impedance) but also of the pump, the left ventricle. One would anticipate, therefore, that the contractile behavior of the left ventricle might be an important factor in explaining pressure wave augmentation. Pioneering studies that formed the basis for establishing the link between the late systolic pressure peak and reflected waves, however, assumed that the contractile behavior of the left ventricle remained unchanged across individuals (11, 15, 18). As a consequence of this assumption, alterations in outflow patterns of left ventricles were not analyzed to determine their relationship to the late systolic pressure peak.

The purpose of the present study was to quantify the relative importance of the left ventricular outflow patterns and arterial impedance in determining late systolic pressure augmentation. We used a mathematical model of the entire human arterial tree (9) to compute a range of realistic arterial input impedance patterns. As input to these ascending aortic impedances, we synthesized different left ventricular outflow patterns by altering the point, integral, and path descriptors of the left ventricular contraction behavior. From the computed aortic pressure waveforms, late systolic pressure peaks were identified and analyzed.

## METHODS

#### Synthesis of ascending aortic impedance patterns.

Multibranched models that use realistic anatomic and physical parameters can reproduce all the features of the human arterial impedance patterns. Simpler arterial models based on a few elements, such as asymmetric T tubes and modified three- or four-element windkessels, cannot achieve the same sophistication. In this study, we used a previously described multibranched mathematical model of the human arterial system (2) that was subsequently modified to simulate increased wave velocity with aging and the presence of distal arterial compliances (9). These modifications yielded realistic impedance patterns in good agreement with the previously published data. Both the age-related changes (rightward shift of first zero crossing of the phase angle, increased first minimum of impedance moduli, and increased characteristic impedance) and altered distal compliances and reflected waves were reproduced (9, 17) (Fig.1
*B*).

These changes included the nonuniform increase of the Young’s modulus (*E*) of the individual branches to simulate age-related increases in wave velocity from 20 to 60 yr and the precise description of the frequency-domain characteristics of terminal mismatches: the resistive component of the terminal mismatches (Γ_{0}), and the windkessel time constant (τ). These changes in *E*, τ, and Γ_{0} modified the moduli of impedance spectrum at higher (>3 Hz), middle (1–3 Hz), and lower (<1 Hz) harmonics, respectively. Table1 documents the simulated ranges. As explained earlier, changes made to *E*were nonuniform because of variation in the distribution of pulse wave velocities (a measure of stiffening) of large arteries with aging (3): the ascending aortic wall elasticity increased from 5.28 to 11.30 × 10^{6} dyn/cm^{2}, while brachial artery wall elastance increased from 9.56 to 13.62 × 10^{6} dyn/cm^{2}. To simulate conditions ranging from total vasodilation to total vasoconstriction, we altered Γ_{0}from 0.0 to 1.00 in steps of 0.1. Increased distal compliance was simulated by altering τ from 0.0 to 0.5 s in steps of 0.1 s.

Because the mathematical assumptions used in this model become less valid at the capillary level, the multibranched model could not be extended to compute peripheral resistances. Nevertheless, it is possible with the existing mathematical model to set the total peripheral resistance (*R*
_{p}) as a constant function of age, where*R*
_{p}= 9.7409 × age + 853.97 dyn ⋅ s ⋅ cm^{−5}(17) and then compute the entire arterial input impedance spectrum.

#### Synthesis of left ventricular outflow patterns.

For each synthesized arterial impedance, we defined a unique left ventricular outflow pattern. For this purpose, the diastolic behavior of the left ventricle was defined by a constant end-diastolic pressure-volume (EDPV) relationship, according to the equation
Equation 1where P is pressure, V is volume, and *a*(intercept) and *K* (volume constant) are given as 0.72 and 0.02, respectively (1). The work output of the left ventricle was also constrained by the preload-recruitable stroke work (PRSW) relationship, described by the equation
Equation 2where SW and V_{ed} are the stroke work and end-diastolic volume of the left ventricle, respectively, and*M*
_{w} and V_{w} represent the slope and volume-axis intercept of the relationship, respectively (6). In this study, we have set V_{w} as constant and altered *M*
_{w}within a realistic range (Ref. 5; Table 1). The end-systolic point reached by the left ventricle was defined by the end-systolic pressure volume (ESPV) relationship, described by the equation
Equation 3where P_{es} and V_{es} are the end-systolic pressure and volume, respectively, and*E*
_{es} and V_{0} are the slope and volume-axis intercept of the relationship, respectively (21). We kept V_{0} constant but altered*E*
_{es} within a realistic range (Ref. 22; Table 1).

These three descriptors of the systolic and diastolic function of the heart do not provide a complete definition of a unique outflow pattern of the left ventricle. There is also a need to describe the pattern of outflow in a closed form. Unfortunately, the point (ESPV relationship) and integral (PRSW relationship) descriptions of left ventricular systolic function are necessary but not sufficient to describe the path of shortening under different loads. Although, at first sight, the time-varying ventricular elastance concept [*E*(*t*)] could be seen to be useful for this purpose, the lack of a closed-form description of a unique*E*(*t*) makes it impossible to define particular*E*(*t*) curves during ejection. Also, there is little empirical evidence describing the precise contour of the*E*(*t*) curve during ejection in vivo. In fact, the available evidence indicates that the*E*(*t*) curve is most highly variable and unpredictable during the ejection period (22), rendering the choice of an*E*(*t*) trajectory during ejection arbitrary.

The alternative we chose was to select specific patterns of the aortic flow waveform within the constraints provided by the three descriptors of left ventricular systolic and diastolic function described above. This method constrains the choice of left ventricular outflow patterns as a function of independently quantifiable physiological parameters.

For this purpose, in a template aortic flow waveform representing a heart rate of 72 beats/min and an ejection duration of 310 ms, we defined several feature points in relation to the foot of the wave (Fig. 1
*A*). These were the time to peak flow velocity (*T*
_{1}), time to a shoulder defined on the flow deceleration phase (*T*
_{2}), and the amplitude of this shoulder normalized to the peak flow velocity [flow shoulder index (FSI)]. With the exception of FSI (see*Data analysis and statistics*), we altered these parameters over the entire range that was reported previously (Refs. 11, 17; Table 1).

Once all these model constraints (i.e., the arterial impedance, the EDPV, ESPV, and PRSW relationships, and a specific pattern of the aortic flow waveform) were established, we used the following computational procedure. *1*) For a given end-diastolic volume, stroke volume, and flow pattern, a calibrated aortic flow waveform was calculated.*2*) The calibrated aortic flow waveform was given as input to the arterial impedance, and aortic pressure waveforms were synthesized.*3*) The end-systolic elastance (slope of ESPV relationship) was calculated using *Eq.3
*. *4*) If this end-systolic elastance was found to be different from the preset elastance, the end-diastolic volume was allowed to change.*5*) If the stroke work calculated for this new end-diastolic volume was different from stroke work derived from *Eq. 2
*, the stroke volume was allowed to change. *6*) Finally,*processes 1–5* were repeated until all the external constraints were met. We used a generic Newton-Ralphson method (see Ref. 20) to find this point in the parameter space, with convergence to this point usually achieved within 10 iterations (Fig. 2).

#### Data analysis and statistics.

Initially, we subdivided the parameter space of the left ventricular outflow patterns linearly. We encountered difficulties, however, with this approach. First, the linear steps had to be coarse for realistic allocation of computational time. This precluded detailed examination of the parameter space. Second, this method implied that these parameters are uniformly distributed across the range. This assumption, in contrast to observed situations in vivo, puts undue weight on the extremes of each parameter range. We defined each parameter, therefore, as a Gaussian function with mean and standard deviation (Table 1), with the exception of FSI. Because we could not find any published data for the FSI range, we employed a uniform distribution. To enable comparisons with the measurements done in vivo, the features of the derived left ventricular outflow patterns, peak flow acceleration, peak aortic flow (PF_{T}
_{1}) and aortic flow at shoulder point (PF_{T}
_{2}) were analyzed.

To quantify the pressure wave augmentation, we extracted the features of the synthesized ascending aortic pressure waveform corresponding to the timing of the peak flow (P_{T}
_{1}) and to the timing of reflected waves (P_{T}
_{2}) (Fig. 1
*C*) using techniques described previously (24). These feature points and the pressures measured at the wave foot (P_{T}
_{f}) then were related to define the shoulder index (SI) as a measure of the late systolic pressure augmentation: SI = PP_{T}
_{2}/PP_{T}
_{1}, where PP_{T}
_{2} = P_{T}
_{2} − P_{T}
_{f} and PP_{T}
_{1} = P_{T}
_{1} − P_{T}
_{f}. This index is similar to the widely accepted augmentation index, defined as AP/PP, where AP is the augmented pressure (P_{T}
_{2} − P_{T}
_{1}) and PP is the pulse pressure (11). By definition, the augmentation index can be negative whenever the peak pressure occurs early in systole. In contrast, the SI is always positive. Thus SI values of 0, 1, and 2 correspond with augmentation indexes of −100, 0, and 100%, respectively.

Analyses were performed using a commercially available statistical package (SPSS for Windows version 7.5, SPSS). For each dependent parameter, univariate regression was first performed and the predictors that had significant correlations were included in the final multiple-regression model. The variables with a high degree of multicollinearity, quantified as a variance inflation factor > 2.0, were excluded (23). For example, V_{ed} was found to be collinear with the flow waveform features (PF_{T}
_{1}, PF_{T}
_{2}, and*T*
_{2}) and therefore was not utilized in the final model. The stepwise multiple-regression analysis was then performed, and the standardized regression coefficient (β) for each parameter was calculated. The importance of each independent model parameter was assessed by the ratio of part *r*
^{2}(defined as the square of the correlation coefficient after controlling the effects of additional variables) to model-adjusted*r*
^{2} (RPM). Data are expressed as means ± SD, and the*P* < 0.05 level was accepted as significant.

## RESULTS

The simulations carried out in the parameter space represented 330 different arterial impedance patterns coupled to 10,000 different left ventricular outflow patterns. The hemodynamic parameters calculated from these simulations yielded results that were comparable to those observed in vivo (Table 2). Figure3
*A*illustrates changes in the pressure contour accompanying changes with aging, simulated by increasing Young’s modulus*E* of the arterial tree nonuniformly. The arterial pulse pressure and the amplitude of the first and second pressure peaks were increased, but the arterial pressure contour was little affected, with no change in the ratio of the peaks, the SI (Fig.3
*A*,*left*,*top*). This suggests that early return of reflected waves caused by increased wave velocity does not affect the SI. Increased distal vascular compliance, which decreases the arterial impedance moduli at middle frequencies (1–3 Hz), reduced the second pressure peak and AP but had no effect on the SI (Table3).

This contrasts with the effects of increasing the resistive component of the terminal mismatches (Γ_{0}), which increases the arterial impedance moduli at lower frequencies (<1 Hz) (Fig.3
*B*,*left*,*top*) and the FSI (Fig.3
*C*,*left*,*top*), where the entire pressure contour was altered. The second pressure peak, AP, and SI were all increased. In each of these simulations, V_{ed}, P_{es}, and stroke work increased (Fig. 3, *right*).

Table 3 summarizes the main predictors of various multiple-regression models. The model-adjusted regression coefficients were high, ranging between 0.77 and 0.93. As in the case of increased arterial wall elastance, increased early shortening of the left ventricle [indexed as aortic flow at*T*
_{1}(PF_{T}
_{1})] also increased the first aortic pressure peak. Because there was no corresponding increase in the second pressure peak, this decreased the SI. This contrasts with the effect of increased late-systolic shortening, which did not influence the first pressure peak and had a minor effect on the second pressure peak but otherwise dominated the other left ventricular contraction trajectory parameters. It increased the AP and SI to an extent similar to that caused by the increased wave reflection coefficient.

Analysis of the multiple-regression coefficients indicated that the combination of the impedance-related parameters (*E*, Γ_{0}, and τ) explained the variability of pulse pressure at the flow shoulder point (*T*
_{2}) better (RPM = 0.90) than did the combination of all parameters of the left ventricular outflow patterns (RPM = 0.09). In contrast, parameters of the left ventricular outflow patterns were stronger than impedance-related parameters as determinants of the amplitude of the first pressure peak (RPM = 0.56 vs. 0.40), SI (RPM = 0.53 vs. 0.33), and AP (RPM = 0.48 vs. 0.38).

Among the descriptors of left ventricular systolic function (i.e., PRSW and ESPV relationships), only the slope of the PRSW relationship (*M*
_{w}) had an impact on the aortic pressure waveform: there was a small but positive association between*M*
_{w} and the amplitude of the second pressure peak.

## DISCUSSION

In this study, with the exception of FSI, we used previously published data as input to our model simulations (Table 1) to synthesize arterial impedances and left ventricular outflow patterns. These simulations yielded hemodynamic parameters that were similar to those observed in vivo (Table 2; Refs. 5, 22).

Using arterial impulse response functions derived from arterial impedances, Latson and colleagues (12) found that the contour of the input aortic flow waveform influenced the calculated aortic pressure waveform. Unfortunately, that study did not systematically explore the extent of this influence. It has been suggested that the observed increase in the augmentation index with aging could be caused by two factors, the early return of reflected waves, a result of increased arterial stiffness with aging, and the increased intensity of reflected waves caused by an increase in the reflection coefficient (11, 17). The former explanation is inconsistent with our results, which show that increased wave velocity influences not only the late systolic peak but also the pulse pressure and the first peak, so that the SI is little altered (Table 3).

Our results indicate that the wave reflection coefficient has a definitive impact on the pressure waveform. It is equally possible, however, that the left ventricular outflow patterns, independent of the arterial impedance patterns, can also affect the wave reflection indexes derived from the pressure waveform. This finding suggests that the increased intensity of reflected waves may not be the sole factor in the formation of the late systolic pressure augmentation.

#### Limitations.

In this study, ascending aortic impedances were derived from a mathematical model of the human arterial system. Although this mathematical model reproduces most of the features of the human arterial system, there might still be differences caused by some of the physiological parameters used as constants in the model. Although the model uses realistic parameters such as physical dimensions, wall viscosities, and anatomic organization of the arteries, together with blood viscosity and density, the specific values of these parameters might be different in individuals. Also important to note is that the model employed is linear, and, although small, there are certain nonlinearities in the arterial system. The nonlinear pressure-diameter relationship of the arterial wall and the convective terms in the governing equations might have influenced our results.

Also not considered in the determination of the left ventricular outflow patterns were possible nonlinearities of the ESPV relationship (13), the positive and/or negative inotropic effects of the ejection (4, 7), and the possible curvilinearities of the early and late systolic portions of the outflow pattern. In this study, we did not attempt to model these effects. Nevertheless, the current simulations yielded hemodynamic parameters that were in close agreement with their counterparts in vivo (Table 2).

In conclusion, our results indicate that left ventricular outflow patterns can be a major determinant of late systolic pressure augmentation. The relevance of this finding to the observed late systolic pressure augmentation in vivo in human subjects would depend, however, on the relative range of variation in the left ventricular outflow patterns and aortic input impedance in human subjects, which has yet to be determined. This study indicates a need to examine alterations in the left ventricular outflow patterns due to age, sex, coronary artery disease, left ventricular contractility changes, and administration of vasoactive substances in vivo.

Finally, because pressure wave augmentation observed in the central aortic pressure waveform is a function of both the arterial impedance and the left ventricular outflow pattern, “wave reflection” indexes derived from the aortic pressure waveform alone may not be accurate measures of the true extent of wave reflection.

## Acknowledgments

This study was supported by a grant from the National Health and Medical Research Council of Australia (NH&MRC 0980225).

## Footnotes

Address for reprint requests and other correspondence: M. Karamanoglu, Cardiology Dept., St. Vincent’s Hospital, Victoria St., Sydney, Australia 2010 [E-mail: m.karamanoglu{at}unsw.edu.au].

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 1999 the American Physiological Society