Baroreflex sensitivity: An algebraic dilemma

The baroreflex system is a complex mechanism for short-term regulation of arterial blood pressure, involving the heart rate (HR), the heart contractility and the vascular tone in its efferent branches. The study of arterial baroreflexes relies on two different experimental approaches. On the one hand, the closed-loop approach analyses the mutual relationship between HR and arterial blood pressure, both in steady state and in dynamic conditions. The most classical method for the study of arterial baroreflexes with a closed-loop approach is the sequence method.1 The typical representation uses the R-to-R interval (RR) as dependent variable and provides positive linear relationships between RR and systolic arterial pressure (SAP). The slope of these lines is considered indicative of the baroreflex sensitivity (BRS) around the operating point.1 On the contrary, negative linear relationships between HR and SAP are reported if HR is used instead. Both representations have been simultaneously applied to the same data.2, 3 On the other hand, the open-loop approach investigates the HR or pressure response to an external perturbation, whether pharmacological or mechanical, which is suddenly imposed starting from a steady-state condition. The most classical representation of an open-loop investigation of arterial baroreflexes provides the saturating negative relationship between arterial pressure and carotid sinus pressure, which Kent et al4 treated with a logistic model. In more recent representations, HR instead of pressure was retained as the dependent variable, to represent the cardiac response.5 These curves are also accurately described by the logistic model of Kent.4 When RR is used instead of HR, inverted logistic curves, with positive slopes, are obtained. This Editorial concerns a theoretical analysis, validated with literature data, of the effects of using either the RR or HR as dependent variables. Its aim is to demonstrate that this choice is not equivalent in baroreflex analysis, because the slope of a RR vs pressure line is a conceptually different parameter from the slope of a HR vs pressure line. We firstly present and discuss the results of the analysis performed on closed-loop baroreflex sequences; then, we report the results obtained with the open-loop logistic baroreflex model of Kent. If we use HR instead of RR, as one is the reciprocal of the other, we obtain an equilateral translated hyperbola of curvature a−1 and vertical asymptote equal to b a-1 (see Appendix for calculations), which means that the relationship between HR and P cannot be treated as linear. Figure 1B was constructed on the basis of this mathematical transformation, using the same data as for Figure 1A, except for HR instead of RR. Thus, the prevailing HR at the operating point were 88.6 ± 15.5 bpm and 116.1 ± 17.5 bpm, at rest and exercise, respectively. The slope of the hyperbolic relationship between HR and P (i) is negative and increases (becomes flatter, thus less negative) when P is increased, (ii) is not equivalent to the given definition of BRS, which is rather the reciprocal of the curvature of the hyperbola, but (iii) includes it (see Appendix, Equation A2), quod erat demonstrandum. Some may argue that, as the sequence method investigates the BRS around the operating point, thus over a very narrow range of the entire baroreflex curve, the HR-vs-P relationship, when treated as linear instead of nonlinear (hyperbolic), introduces negligible error. This is true from the statistical viewpoint. Yet the key issue here is that, if we admit that the slope of an RR-vs-P relationship is equal to the BRS, then it necessarily follows that the slope of a HR-vs-P relationship is not the BRS. Of course, the reverse would be the case if we assume that, opposite to Bertinieri et al1 the linear relationship is that between HR and P. If we substitute RR for HR, we still obtain a logistic function (see Appendix for calculations). However, the maximum baroreflex gain is equal to . Thus, with respect to the HR-vs-P relationship represented by Equation 2, the maximum baroreflex gain (i) is positive instead of negative, (ii) depends also on δ and (iii) is lower at exercise than at rest (in fact 3 times lower for the data of Potts et al5). Moreover, the sigmoid midpoint corresponds to a centring pressure of , thus higher than γ. This means that if we use RR instead of HR, the steepest slope and the centring pressure change quantitatively and qualitatively, quod erat demonstrandum. The results of Potts et al5 on humans at rest and during steady-state exercise at 50% of peak oxygen consumption were used to graphically illustrate Kent's model. Kent's variables of Equation 2 resulted: α = 12.6 ± 4.8 bpm, β = 0.15 ± 0.21 per mm Hg, the centring pressure γ = 99.7 ± 7.9 mm Hg, and δ = 50.4 ± 9.3 bpm, at rest; at exercise, we had α = 16.4 ± 7.9 bpm, β = 0.13 ± 0.16 per mm Hg, γ = 126.0 ± 19.0 mm Hg and δ = 97.4 ± 18.0 bpm (Figure 1C). When the RR-vs-P relationship was computed on the same data, the centring pressure resulted equal to 101.2 mm Hg at rest and 127.2 mm Hg at exercise (Figure 1D). The above mathematical analysis implies that, when the gain of arterial baroreflexes is to be calculated, substituting RR for HR is not trivial, as long as the gain is not merely the negative slope of an otherwise positive relationship of the same type. Different concepts are at stake. In closed-loop context, in which the sequence method is used, different definitions of baroreflex sensitivity provide contradictory results and thus conceptually different representations. Therefore, physiologists should find an agreement on the definition of BRS, not only for conceptual purposes, but also and mostly for its operational consequences. If we axiomatically assume the RR-vs-P relationship as a valid representation of arterial baroreflexes,1 and thus, we admit that its slope is equal to the BRS, then the reciprocal HR-vs-P representation would not provide a baroreflex sensitivity in terms of slope of a linear relationship, but in terms of curvature of a hyperbolic relationship. The reverse would be the case if we axiomatically assume a linear relationship between HR and P as a correct representation of arterial baroreflexes. On the other hand, when in open-loop condition we apply Kent's model, the maximum baroreflex gain is the steepest slope of a logistic curve. This slope is negative if we use HR, positive if we use RR, but it is also independent of the minimum HR in the former case, and dependent on the maximum RR in the latter case. In open-loop context, there is agreement on the concept that the HR-vs-P relationship provides a correct representation of arterial baroreflexes.7 If we accept this principle, the maximal gain provided by the logistic model of Kent is independent of the minimal HR, but we would also have to accept that the sequence method would provide a conceptually different representation of baroreflex sensitivity. To sum up, there is an axiomatic issue, which should be someway resolved. Opening a debate on this issue is, in our vision, the best way to proceed towards its resolution. The authors declare no commercial or financial relationships that represent a potential conflict of interest. Analysis of the relationship between HR and P starting from Equation 1: Analysis of the relationship between RR and P starting from Equation 2: The steepest slope and the sigmoid midpoint derive from: when Thus identifies the steepest slope of g(P), that is situated at the sigmoid midpoint, whose coordinates are .