I enjoyed reading a recent article by Kips et al. that described the circulatory determinants of the observed linear relationship between SBP and DBP using a computer simulation, as well as the corresponding editorial by Parati and Schillaci .
The wide interest in this relationship began in 2006, after the measure, 1 – (the DBP-on-SBP standard regression slope), called the ambulatory arterial stiffness index (AASI), was derived from ambulatory blood pressure (BP) recordings and showed that AASI has a prognostic significance and could be a surrogate measure for arterial stiffness (see references  and ). The latter statement stimulated debate regarding its validity. The computer model presented by Kips et al. elegantly demonstrates that AASI is strongly affected by heart rate (HR) and peripheral resistance, which seriously limit the use of AASI as a marker of stiffness. Furthermore, the pressure dependence of arterial stiffness increases AASI considerably. In view of the historical development of this field presented in the editorial , I find it appropriate to mention that the first identification of the SBP-on-DBP slope as an arterial property and its modelling were already reported in 2000 , and its derivation from ambulatory BP recordings and modification by interventions were reported in 2001  – a number of years before AASI was introduced.
In my opinion, the question of ‘what are the real determinants of AASI?’ should be preceded by a more fundamental one: what is the physiological origin of the linear relationship between SBP and DBP? In this letter, I present model-derived expressions for SBP-on-DBP slope and AASI that provide some answers to that question, as well as physiological rationale for Kips’ findings. The previously derived model [3,5] (see Fig. 1) shows that for any type of repeated BP measurements that displays a linear relationship, this relationship is equivalent to a linear relationship between arterial pressure P and the exponent exp(βV), where V is the arterial volume (or cross-sectional area, or diameter) of a tested arterial segment, and the exponent β is a constant expressing the ‘curving’ of the P–V curve . The parameter β has a known association with arterial wall structure and composition, and possesses diagnostic and prognostic significance. Defining the arterial stiffness G(P) as the steepness of the P–V curve, that is the derivative dP/dV, this exponential P–V relationship is equivalent to a linear relationship between stiffness and pressure that closely fits physiological data  (see Fig. 1). Thus, G(P) = G(0) + βP, where β=dG/dP expresses the increase of stiffness per unit of pressure. The model-derived SBP-on-DBP slope (‘S-D slope’) equals the average exp(βΔV), where ΔV is the pulse volume, that is the arterial volume change from its diastolic to systolic value . Furthermore, S-D slope expresses the relative increase of the arterial stiffness G(P) between its diastolic and the systolic value, that is the S-D slope also equals G(SBP)/G(DBP) [3,6]. Therefore, S-D slope on one hand expresses the arterial stiffening during the systole and on the other hand the quantity βΔV, where ΔV may be affected by a number of factors.
The statistical definition of AASI is important, as it is shown hereafter to affect both results and interpretation of the work of Kips et al.: when using experimental or simulated data, the statistically estimated SBP-on-DBP slope can be expressed, according to the principles of a preferred symmetric regression method, by the ratio between the SDs of the SBP and DBP, that is S-D slope(sym) = SD(SBP)/SD(DBP) . When determined by standard regression, AASI (standard) is given by 1 – R/ [S-D slope(sym)], where R is the SBP-DBP correlation coefficient. On the contrary, when using symmetric regression, AASI(sym) equals 1 – 1/[S-D slope(sym)], which is free of the bias by R[7,6]. Therefore, assuming that the model-derived S-D slope can be identified by S-D slope(sym), we find that AASI(sym) is expressed by 1 – G(DBP)/G(SBP) or 1 – exp(−βΔV) and AASI (standard) by 1 – R·G(DBP)/G(SBP) or 1 – R·exp(−βΔV). Thus, in case of a ‘stiffening artery’ for which G(SBP) > G(DBP), we expect S-D slope(sym) > 1 and AASI(sym) or AASI(standard) > 0. However, for an elastic artery for which β = 0, we find AASI(symmetric) = 0 and AASI(standard) = 1 − R that is positive for R less than 1. These results are compatible with the findings of Kips et al. that AASI increases for nonlinear arterial wall behaviour in comparison with elastic arterial wall for which AASI is small but positive. The latter result was explained by Kips et al. as a demonstrated discrepancy between the prediction of AASI equal to 0 [6,8] and the result AASI greater than 0. In fact, there is no discrepancy, as Kips et al. used AASI(standard), while the model predictions corresponded to AASI(sym)! More specifically, the value AASI(standard) equal to 0.18 for an elastic vessel shown in Fig. 5a [Kips] means that R is equal to 0.88, which makes sense when looking at that figure, but could be easily checked with the data. This problem is another manifestation of the statistical artifacts associated with AASI when determined using standard regression . It should be mentioned that the expression of S-D slope by exp(βΔV)  and the resulting AASI = 1 – exp(–βΔV) were rediscovered by Craiem et al., who determined AASI from a similar model by assuming an exponential P–V relationship and validated that expression using computer simulation and an animal model. Therefore,the results of Kips do not invalidate the view that AASI (and S-D slope) are markers for arterial stiffening (with pressure) but not for arterial stiffness.
The findings of Kips et al. that the increase in HR, peripheral resistance and arterial distensibility is associated with a decrease in AASI (and S-D slope) can be explained by a reduction in the pulse volume ΔV, as it expresses the decay of the arterial volume from its systolic to diastolic value; any factors that affect that decay are likely to affect ΔV. These include the net shortening of the heart period, that is increasing HR, which results in higher DBP and lower ΔV; an increase in peripheral resistance,which slows diastolic pressure decay, as it takes more time to release the blood stored by compliant arteries after the left ventricular ejection. As mentioned by Kips et al., the effects of HR and peripheral resistance on AASI were already explained by Westerhof et al.. These researchers derived an expression for AASI on the basis of concepts taken from the Windkessel model and described AASI as a ventriculo-arterial coupling factor by being a function of the quantity [heart period] [arterial stiffness]/ [peripheral resistance]. The interpretation of Westerhof et al. is similar to the present view regarding the role played by ΔV, as both relate to the diastolic decay of arterial pressure or volume. However, the Westerhof et al. model cannot explain all of the results of Kips et al., as it predicts that AASI is zero only for zero heart period, that is infinite HR, which is unrealistic. The problem may reflect the constancy of the arterial stiffness implicitly assumed by the Windkessel model. According to the present view, the lack of such constancy is essential for explaining S-D slope greater than 1 or AASI greater than 0 value (see Fig. 1).
Finally, by using the simulation method of Raymond et al. [ref. 16 in ], Kips et al. assumed that stiffness depends quadratically on pressure. This assumption may be problematic in simulating AASI (or S-D slope), as, according to the present view, the collinearily of SBP and DBP is associated with a linear relationship between stiffness and pressure. In view of these comments, I do believe that a deeper understanding of the physiological meaning of the SBP–DBP relationship could be achieved by focusing on the effect of different factors on the pulse volume and its relationship with the indices S-D slope(sym) or AASI(sym) and evaluating its model expression. Such investigation seems to combine the approaches of Craiem et al. and Kips et al.. However, an easily achievable step in this direction,based on the data of Kips et al., could be the transformation of AASI(standard) into AASI(sym) using the relationship AASI(sym) = 1 – (1 – AASI(standard))/R and reevaluate the validity of AASI(sym) = 0 for elastic arteries, as well as AASI(sym) distribution and its potential stiffness dependence. Similarly, S-D slope(sym) can be determined directly from the already simulated SBP and DBP data.
Conflicts of interest
The author has not declared any conflicts of interest.
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