## Abstract

The aim of our study was twofold: *1*) to establish a mathematical link between mean aortic pressure (MAP) and systolic (SAP) and diastolic aortic pressures (DAP) by testing the hypothesis that either the geometric mean or the harmonic mean of SAP and DAP were reliable MAP estimates; and *2*) to critically evaluate three empirical formulas recently proposed to estimate MAP. High-fidelity pressures were recorded at rest at the aortic root level in controls (*n* = 31) and in subjects with various forms of cardiovascular diseases (*n* = 108). The time-averaged MAP and the pulse pressure (PP = SAP − DAP) were calculated. The MAP ranged from 66 to 160 mmHg [mean = 107.9 mmHg (SD 18.2)]. The geometric mean, i.e., the square root of the product of SAP and DAP, furnished a reliable estimate of MAP [mean bias = 0.3 mmHg (SD 2.7)]. The harmonic mean was inaccurate. The following MAP formulas were also tested: DAP + 0.412 PP (Meaney E, Alva F, Meaney A, Alva J, and Webel R. *Heart* 84: 64, 2000), DAP + 0.33 PP + 5 mmHg [Chemla D, Hébert JL, Aptecar E, Mazoit JX, Zamani K, Frank R, Fontaine G, Nitenberg A, and Lecarpentier Y. *Clin Sci (Lond)* 103: 7–13, 2002], and DAP + [0.33 + (heart rate × 0.0012)] PP (Razminia M, Trivedi A, Molnar J, Elbzour M, Guerrero M, Salem Y, Ahmed A, Khosla S, Lubell DL. *Catheter Cardiovasc Interv* 63: 419–425, 2004). They all provided accurate and precise estimates of MAP [mean bias = −0.2 (SD 2.9), −0.3 (SD 2.7), and 0.1 mmHg (SD 2.9), respectively]. The implications of the geometric mean pressure strictly pertained to the central (not peripheral) level. It was demonstrated that the fractional systolic (SAP/MAP) and diastolic (DAP/MAP) pressures were reciprocal estimates of aortic pulsatility and that the SAP times DAP product matched the total peripheral resistance times cardiac power product. In conclusion, although the previously described thumb-rules applied, the “geometric MAP” appears more valuable as it established a simple mathematical link between the steady and pulsatile component of aortic pressure.

- pulse pressure
- cardiac power
- total peripheral resistance

an improved description of central aortic pressure is a key issue for cardiovascular physiologists and clinicians, because central aortic pressure is a major determinant of myocardial oxygen consumption, coronary perfusion pressure, and the pressure for arterial baroreflex. Aortic pressure is currently analyzed according to both its steady component [mean aortic pressure (MAP)] and pulsatile component [systolic aortic pressure (SAP), diastolic aortic pressure (DAP), and pulse pressure (PP) = SAP − DAP]. The MAP accounts for >80% of the hydraulic load put on the left ventricle and significantly contributes to vascular load (9, 21, 34). MAP is adequately described by the cardiac output (CO) times total peripheral resistance product, and MAP is considered as the perfusion pressure through each tissue bed. The overall strategy of the cardiovascular system is to provide all organs with constant perfusion pressure, and MAP is closely monitored via central and peripheral control mechanisms (2, 31).

MAP is the time-averaged aortic pressure throughout cardiac cycle length. Previous studies have established a link between the steady and pulsatile component of aortic pressure by estimating MAP from SAP and DAP values using various empirical formulas that rely on different physiological aspects of the human circulation (5, 20, 30). The characterization of MAP by using empirical formulas is focused on the amount of information contained in the pressure database, and this may be of interest for several reasons. First, mathematical solution for any system may be a step for improving the rational modeling of the system (1), and a mathematical solution relating the steady (MAP) and pulsatile (SAP, DAP) aortic pressures may be especially valuable. Second, empirical formulas may help reduce the number of independent variables. A properly defined collection of biological data must minimize redundancy, because redundancy may hamper the understanding of pathophysiological processes and the relevance of statistical analysis (15, 17, 22). Third, this approach may be utilitarian with regard to precisely predicting one of the variables under study.

Although valuable, the previously proposed thumb-rules (5, 20, 30) do not belong to the so-called mathematical means [arithmetic (*A*), geometric (*G*), harmonic means (*H*)]. While it is admitted that MAP is lower than the *A* of SAP and DAP (10, 39), the potential value of the *G* and *H* remains to be established. The aim of our study was to test the hypothesis that MAP could be accurately estimated by using either the *G* or the *H* of SAP and DAP in the aorta of resting humans. As we observed that the *G* (i.e., the square root of the product of SAP and DAP) was an accurate and precise estimate of MAP, this new formula was compared with the three empirical formulas recently proposed for estimating MAP (5, 20, 30). Finally, because indexes of aortic pressure pulsatility, such as fractional systolic pressure (FSP) (SAP/MAP) and fractional diastolic pressure (FDP) (DAP/MAP), could relate to the risk of coronary heart diseases (16, 23, 26), the clinical implications of our results were also discussed.

## MATERIALS AND METHODS

#### Patients.

Our study involved 139 patients [109 men/30 women; 49 yr (SD 12)]. The subjects were referred to our laboratory for diagnostic left heart (*n* = 66) and left and right heart (*n* = 73) catheterization for symptoms of chest pain, heart failure, or other cardiovascular symptoms. The final diagnosis was as follows: subjects with normal cardiac function and coronary angiograms (*n* = 31), hypertensive patients (*n* = 46), grafted hearts (*n* = 18), idiopathic dilated cardiomyopathy (*n* = 14), and miscellaneous cardiac diseases, mainly coronary artery disease (*n* = 30). Part of the study population has been described elsewhere (4, 5). All investigations were approved by our institution, and informed consent was obtained for all patients.

#### Pressure recordings.

Catheterization procedures and data analysis and calculations have been previously described (4, 5). In brief, pressures were recorded in resting subjects using micromanometer-tipped catheter (Sentron, Cordis Laboratory, Roden, The Netherlands). Unlike fluid-filled catheters, micromanometer-tipped catheters provide high-fidelity pressure values without the errors stemming from differences in the reference zero level, as observed with external pressure transducers (13). Pressure measured by a micromanometer catheter tip instrument depends on the vertical position of the catheter tip in the chest, given the hydrostatic pressure component. This was not a significant problem in our study, as pressure was always recorded at the same site in the ascending aorta in all supine patients, namely above the aortic cusp. Mean pressure was automatically calculated as the area under the pressure curve divided by the cardiac cycle length. Systolic, diastolic, and mean pressures were averaged out over nine beats in 66 patients (4) and over a 15-s period in 73 patients (5). Stroke volume was calculated from monoplane angiograms using the area-length method (*n* = 66) or by the triplicate thermodilution method (*n* = 73). Cardiac index and total peripheral resistance were calculated by using standard formulas. Cardiac power (W′) was calculated as the MAP times cardiac index product (6, 7, 12).

#### Calculations of the mathematical means of SAP and DAP.

The *A*, *G*, and *H* are traditionally thought to have been discovered by Pythagoras of Samos and coworkers, from the Greek school of mathematicians around 500 BC, and these means formed the basis of classical Greek theories of architecture, perspective, music, and processes of nature (38). Mathematical means have been largely used in medicine and physiology, as illustrated by the following examples in the cardiovascular field. The *A* of systolic and diastolic brachial artery pressure has demonstrated great value in predicting mortality risk (18). Geometric progression has been documented in the distribution of coronary artery lesions in the human heart (11). If a circuit has two resistors connected in parallel, the average resistance is one-half of their *H* (2, 21, 31).

The following mathematical means of SAPs and DAPs were calculated as follows From a mathematical point of view As far as *G* is concerned, Pythagoras of Samos and coworkers first noted that, if a rectangle is formed with side lengths *x* and *y*, the *G* of *x* and *y* gives the side length of a square with the same area. This appears to be a simple way to memorize the mathematical formula of the *G*, which is often called mean proportional. Finally, it must be noted that other mathematical means have been described (e.g., quadratic mean). The quadratic mean [*Q* = was not tested because *Q* > *A*.

#### Empirical formulas for estimating MAP.

In peripheral systemic arteries, MAP can be reasonably estimated by adding one-third (0.33) of PP to diastolic pressure. This very popular formula appears in all physiological and medical textbooks and is currently used in clinical hypertension trials (2, 8, 9, 10, 31, 33, 39). In the aorta, three MAP empirical formulas (E1, E2, and E3) have been recently proposed (5, 20, 30). Given the sine-wave-like pattern of aortic pressure, it has been suggested nearly one century ago that the fraction of PP that must be added to DAP may be between 0.33 and 0.50 (10, 39). Consistently, Meaney et al. (20) have recently documented the following empirical formula in the ascending aorta of 150 patients (86 men/64 women) with various forms of cardiac diseases On the other hand, in 73 patients (65 men/8 women), we have observed that the classic empirical formula currently used at the peripheral level underestimates central MAP by 5 mmHg such that the following formula applied (5) It was proposed (5) that the E2 formula is consistent with the classic empirical formula currently used at the peripheral level, if one considers that both MAP and DAP remain nearly constant from central to peripheral arteries, while there is a physiological amplification of SAP from aorta to periphery (2, 25, 31) that amounts to 15 mmHg on average (28, 35) (5 mmHg = 0.33 × 15 mmHg).

Finally, when the classic empirical formula currently used at the peripheral level was corrected for the increasing time dominance of systole with increasing heart rates, Razminia et al. (30) have reported that the following heart rate-corrected MAP accurately applied in 12 patients at increasing paced rates In the second part of our study, the accuracy and precision of E1, E2, and E3 were tested.

#### FSP and FDP.

The aortic FSP (SAP/MAP) and FDP (DAP/MAP) were also calculated. FSP and FDP have been recently proposed as valuable estimates of aortic pulsatility and could relate to the risk of coronary heart diseases (16, 23, 26). We tested the hypothesis that FSP and 1/FDP were reciprocal estimates of aortic pulsatility.

#### Comparisons with studies using fluid-filled catheters.

In the clinical setting, fluid-filled catheter-manometer systems are used, not micromanometer-tipped catheters. To document how our analysis compares with fluid-filled catheter studies, we performed a retrospective analysis of the main recent papers relating cardiovascular risk and aortic pressures recorded by using fluid-filled catheters (for a review, see Ref. 26). Average values of *G*, FSP, and 1/FDP were calculated from published values of SAP, DAP, and MAP. Only studies in which the time-averaged MAP was documented entered the final analysis (14, 19, 24, 29, 37), while papers using the classic empirical formula did not enter the final analysis. Indeed, as previously discussed, the DAP + 0.33 (SAP − DAP) formula underestimates the time-averaged aortic MAP by 5 mmHg (5).

#### Statistical analysis.

Results are expressed as means (SD). For each estimate of mean pressure, the mean bias (estimate minus the time-averaged mean pressure) and SD of the bias were calculated. The mean bias and SD reflect the accuracy and the precision of the estimate, respectively. Comparisons were performed by using analysis of variance. The mean pressure bias was plotted against the time-averaged mean pressure and was also plotted against the average of the time-averaged mean pressure and mean pressure estimate, as previously recommended by Bland and Altman (3). The same statistical analysis was performed for comparisons between FSP and 1/FDP and for comparisons of hemodynamic formulas involving W′ (see results). Between-groups differences in pressures were tested by using analysis of variance followed by paired *t*-test with Bonferroni correction. Linear regressions were obtained by using the least squares method. A *P* value < 0.05 was considered significant.

## RESULTS

Data were obtained over a 66- to 160-mmHg MAP range. Characteristics of the study population are listed in Table 1. Among the three mathematical means of SAP and DAP under study, the *A* significantly overestimated the time-averaged MAP, whereas the *H* significantly underestimated MAP (Table 2). Only the *G*, i.e., the square root of the product of SAP and DAP, furnished an accurate and precise estimate of the time-averaged MAP (Table 2 and Figs. 1 and 2).

The three empirical formulas (E1, E2, and E3) were accurate and precise estimates of the time-averaged MAP (Table 3). The E1 formula proposed by Meany et al. (20) applied in our subjects. The E2 formula proposed by our group (5) was also confirmed in the present study on a larger number of subjects. The E3 formula proposed by Razminia et al. (30) at increasing pacing rates also applied in our study in patients with markedly different cardiac cycle duration at rest (from 506 to 1,267 ms).

There was a significant influence of MAP on the biases for E2 and E3 and for the *A* and *H*, but not for the *G* and E1 (Tables 2 and 3). Similar results were obtained when the influence of the pressure level on the bias was tested using Bland and Altman plots (*x*-axis = the average between MAP and the pressure estimate). Finally, resting heart rate was not related to the biases for *G*, *H*, E2, and E3, whereas there was a weak, negative relationship between heart rate and the biases for *A* (*R*^{2} = 0.032; *P* = 0.03) and E1 (*R*^{2} = 0.044; *P* = 0.01).

The practical implications of our results were studied. It was found that MAP, SAP, and DAP were related as follows (1) and thus that (2) Put differently, our findings implied that (3) (4) Given potential implications for risk stratification, this was tested in the overall population as well as in each subgroup. On average, the FSP and 1/FDP values were similar in the overall population [1.35 (SD 0.12) vs. 1.34 (SD 0.12)]. The bias appeared clinically moderate and was not influenced by the mean (Figs. 3 and 4). The FSP and 1/FDP values were also similar within each subgroup, and the (1/FDP − FSP) bias appeared small enough to be negligible (Table 4). Both FSP and 1/FDP were higher in hypertensive subjects than in controls. Compared with the values documented in controls, both FSP and 1/FDP were lower in patients with grafted heart, idiopathic dilated cardiomyopathy, and miscellaneous cardiac diseases.

The applicability of our results to clinical conditions where fluid-filled catheters are used was tested. Five studies having recently related FSP, FDP, and cardiovascular risk were selected (see materials and methods) and involved an overall population of 503 individuals (Table 5). Our retrospective calculations confirmed that the geometric MAP matched the time-averaged MAP in four studies (bias ≤ 1 mmHg) (Refs. 14, 19, 24, 37). In one study (Ref. 29), *G* slightly overestimated MAP by 3–4 mmHg. The FSP vs. 1/FDP matching was essentially confirmed (Table 5).

The implications of our results for new developments of current hemodynamic formulas were studied. The MAP can be precisely expressed as follows (5) where R is systemic vascular resistance, and Pra is right atrial pressure (which is often neglected). This equality, in conjunction with *Eq. 1*, implies that (6) We further attempted to simplify *Eq*. *6* because the handling of square roots in physiological equations is not intuitive. Assuming that Pra is small enough vs. MAP as to be negligible, *Eq. 5* reduces as follows (7) where *R* is total peripheral resistance. Multiplying both sides of *Eq. 7* by MAP, we obtain (8) where W′ is mean cardiac power (with W′ = MAP × CO). From *Eqs. 2* and *8* we obtain (9) Thus one physiological implication of our study is that the (SAP × DAP) product [12,052 mmHg^{2} (SD 4,075)] matched the product of total peripheral resistance times mean cardiac power [12,016 mmHg^{2} (SD 3,992), *P* = not significant].

## DISCUSSION

The square root of the product of SAP and DAP (*G*) matched the time-averaged MAP in the aorta of resting humans. Although the previously described empirical formulas (5, 20, 30) demonstrated essentially equivalent accuracy and precision for MAP estimation, the “geometric MAP” appears more valuable as it established a simple mathematical link between the steady and pulsatile component of aortic pressure.

#### Physiological implications.

While MAP is accurately described by CO, R, and Pra, the determinants of SAP and DAP are not quantitatively known (36). For a given ejection pattern, SAP and DAP are mainly determined by peripheral vascular resistance and arterial stiffness (8, 9, 25, 27, 32, 36, 40). In patients with normally compliant arteries (younger individuals, normotensive subjects), SAP and DAP mainly depend on R value (and, therefore, on MAP), with increasing R resulting in both SAP and DAP increases (9). Conversely, in patients with stiff arteries (older individuals, hypertensive subjects), the decreased buffering capacities of the proximal aorta and abnormalities in the traveling arterial pressure/flow waves are responsible for the increased PP, with opposite effects on SAP (increased) and DAP (decreased) (9, 27, 32, 36). Indeed, ejection into a stiff proximal aorta generates a wider PP than in a high-compliance system. Furthermore, it is generally admitted that the increased backward-traveling wave (increased wave reflection) results in a higher SAP by adding to the incident pressure wave in systole, while it tends to decrease DAP given the loss of the physiological boosting pressure effect in early diastole (27, 32). The opposite contribution of arterial stiffness to SAP and DAP may have been canceled when one multiplies SAP by DAP, thus unmasking the remaining influence of R. Overall, the major role of R on SAP and DAP values in patients with highly compliant arteries, and the opposite effects of arterial stiffness on SAP and DAP in patients with stiff arteries, may furnish the physiological basis of *Eq. 1*.

*Equation 9* was the axiomatic consequence of *Eq. 1*. Although essentially phenomenological, *Eq. 9* established a simple relationship between the pulsatile (SAP, DAP) and steady (R, W′) hemodynamic variables. Resistance is the only component capable of extracting energy from a circuit by transforming it into heat (frictional loss) (21). The time-averaged mean power of the instantaneous pressure-flow product is the total hydraulic power. Mean cardiac power (W′ = MAP × CO) represents the mean rate of energy input that the systemic vasculature receives from the heart at the level of the aortic root, i.e., the hydraulic power that would produce the same average flow in a steady stream without pulsation (6, 7, 12, 21). The total hydraulic power minus mean power difference is the oscillatory power, which entails pulsatile pressure wave without resulting in net forward flow. The pulsatile power proportionally increases in stiff arterial systems, but it is not calculated in practice because it requires precise recordings of instantaneous pressure and flow. Recent clinical works have demonstrated the interest in studying W′ and the W′/R relationship in the diagnosis, follow-up, and prognosis of heart failure (6) and cardiogenic shock (7).

#### Clinical implications.

Aortic pressure determines the hemodynamic burden put on the heart, coronary arteries, and carotid arteries (2, 31), and this may have an impact on the risk of heart failure, myocardial infarction, and stroke (26, 33). Indexes that measure central pressure are more directly related to cardiovascular events than measures of pressure in peripheral arteries (26, 27, 32). Given that the higher the MAP, the higher the fluctuations around the MAP (25, 32), aortic FSP (SAP/MAP) and FDP (DAP/MAP) may be valuable indexes of aortic pressure pulsatility (16, 23). After multivariable adjustments in 445 patients, Jankowski et al. (16) have recently demonstrated that both FSP and FDP are related to the risk of coronary heart disease. One implication of the matching between MAP and the *G* was that FSP and FDP were essentially reciprocal estimates of aortic pulsatility, both in the overall population and in the various diseased subgroups. This may have implications for simplifying the battery of indexes used for risk stratification.

#### Limitations and strengths of the study.

Our results strictly pertain to pressures recorded at the central aortic level and do not apply to sphygmomanometer recordings. Peripheral DAP measured by sphygmomanometer may be higher than pressures determined invasively at the aortic root, and there is a physiological amplification of SAP from the aorta to periphery. Overall, this could introduce some errors in estimating peripheral pressure by means of any formula obtained at the central level. Thus the applications of our results to noninvasive peripheral blood pressure measurements must be avoided.

Data were obtained over a 60- to 160-mmHg MAP range and in a large high-fidelity pressure data set (*n* = 139), and consistent results were documented in the various subgroups. Because micromanometer-tipped catheters are not used in the clinical setting, one may argue that the frequency-dependent amplitude distortion caused by fluid-filled catheter may obscure the relationship we reported here. Although our results remain to be confirmed with aortic pressure recorded by using fluid-filled catheter, the retrospective analysis of previously published pressure data involving 503 patients (14, 19, 24, 29, 37) confirmed that our results remain essentially valid with fluid-filled catheters.

In conclusion, in resting humans, the time-averaged MAP can be reliably estimated using the square root of the product of SAP and DAP. The “geometric MAP” strictly applies to aortic root pressure and not to peripheral blood pressure. This simple formula may offer a valuable mathematical link between the steady and pulsatile component of central pressure.

## Acknowledgments

Part of this study has been presented at the 24th Meeting of the French Hypertension Society and consequently appears as an abstract (*J Hypertens* 23: A2–A3, 2005).

## Footnotes

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 © 2005 the American Physiological Society