On aortic pressure waveforms and a happy or unhappy marriage between wave propagation and Windkessel models

Several well respected research groups have an already long-standing history and tradition in reporting, with varying degrees of success, mathematical modeling of physiologically important aspects of arterial hemodynamics. Some of the best elaborated and most rewording models are Windkessel-based (two and three element models) and wave propagation-based models. Windkessel models incorporate vascular resistance, compliance and characteristic impedance, whereas wave propagation models are suited to deal with concepts such as pulse wave velocity and wave reflection. More recently, attempts have been made to create models combining elements of both. For instance as an alternative to the widely accepted frequency-domain model of arterial hemodynamics; Wang et al.[1] proposed about 15 years ago a functional, time-domain, arterial model that combined Windkessel with wave propagation; the so-called reservoir-wave model. In their model, measured pressure can be resolved into two components, the volume-related and the wave-related component, respectively, termed as ‘reservoir’ and the ‘excess’ pressures. The ‘wave-independent’ reservoir pressure should be causally related to filling and emptying of the arteries over the cardiac cycle, whereas the excess pressures should tell the comprehensive story related to the propagation and reflection of waves. The idea covers the elementary notion that changes in pressure should not be ascribed to passages of waves only but should contain a term interrogating volume changes of an elastic artery and changes in chamber elasticity (ventricular contraction and relaxation). Then, the pressure waveform is conceptually and mathematically modeled as the sum of a rather uniform reservoir pressure throughout the arteries that is ‘delayed’ by a wave propagation time dependent on location. The history, the exact definitions and the key steps in the development of the concept are summarized in a long series of papers by the initiators of the concept. Because of space limitations the editorial commentary can only refer to a few of these landmark papers [1–6]. However, more than a decade of intense research on the topic also proved how tough it is to create a reservoir pressure that subtracted from the measured pressure yields an excess pressure uniquely dependent on wave motion. Therefore and for reasons summarized below, it is not surprising that the reservoir-wave paradigm has been criticized by other research groups [7,8] who consider the reservoir-wave concept as an ultimately unhappy marriage between a Windkessel and a wave propagation model. Applied to selected conditions the reservoir-wave concept even seems to violate and contradict generally accepted basic principles of physics and wave-based physiology. Under such conditions rather simple mathematical modeling (1-D, elastic uniform tubes, and terminal Windkessels) already lead to conflicting data with the concept [7,8] (See also below). One of the believers’ contraarguments to the published criticisms has always been that the nonbelievers’ mathematical modeling is far too simplistic (missing the homogenizing effect of highly dispersed reflection sites even after introduction of multiple bifurcations into the models) which may lead to nonphysiological and even erratic conclusions. Another contra-argument is that the so-called conceptual criticisms essentially arise from misinterpretation and misunderstanding of the underlying principles of the concept [6]. Moreover, believers continuously argue that the concept should be viewed as a heuristic hypothesis and not as a theoretical physiology-based concept and that the only valid way of founded criticism should come from judgment of the clinically applicability as a gold standard for the theoretical evaluation procedure is still lacking [3–4]. Applying the reservoir waveform model to a clinical situation is precisely what Narayan et al.[9] did in this issue of the Journal. They analyzed high-fidelity aortic pressure waveforms obtained at five anatomical sites (ascending aorta, transverse aortic arch, diaphragm, renal arteries, aortic bifurcation) from 40 individuals referred for a diagnostic catheterization procedure. As predicted the magnitude of the calculated reservoir pressure was nearly constant throughout the aorta, whereas the excess pressure progressively increased along the aorta in concert with the concept of increase of pulse pressure from the proximal to the distal aorta. Thus the reservoir pressure (assumed to be determined mainly by global aortic compliance) plays an important role in the ascending aorta and the excess pressure (local wave transmission) plays a more important role in the more distal parts of the aorta. In the authors’ opinion, their heuristic model proved its utility in a specific clinical condition to get better mechanistic insights. In sharp contrast to the nearly constant reservoir pressure, the systolic rate constant for reservoir filling (ks) sharply decreased by more than a third from the aortic root to the bifurcation whereas the diastolic rate constant (kd) somewhat increased. How to correctly interpret the rate constants given that one of them (ks) even proved prognostic information in an earlier study by the same authors [10]? A straightforward interpretation might be decreasing aortic compliance and increasing impedance along the trajectory. Other saving graces of the concept despite more or less well founded objections against underlying physiological principles comprise the correlations that have been described between reservoir pressure and the amplitude of backward travelling pressure waveform derived from a classical wave separation analysis and more importantly the outcome data in selected patient groups predicted by reservoir-wave related parameters. For instance, Hametner et al.[11] studied 694 patients undergoing coronary angiography with an excellent correlation between reservoir pressure and the amplitude of the backward travelling pressure waveform and with both pressures predicting cardiovascular outcome (after a median follow-up of 1395 days) in appropriately adjusted Cox proportional hazard models: reservoir pressure (hazard ratio per 1 SD = 1.37) and the amplitude of the backward travelling pressure waveform (hazard ratio per 1 SD = 1.42). In the same line Narayan et al.[10] showed that ks independently predicted outcome in the Second Australian National Blood Pressure Study (n = 838). More recently, Cheng et al.[12] confirmed this finding in two long-term follow-up studies (The Kinmen and Two-Township Studies) on about 3500 individuals and even showed prognostic value for the diastolic rate constant of the reservoir pressure. These three studies as well as other recent work [13] contribute to the idea that the added value of the reservoir waveform model should be appreciated within is applicability in a clinical context. However, excellent predictive capacity of a heuristic model in a clinical context does not necessarily mean scientific and physiological accuracy. Several research groups [7,8] conceptually question the existence of a happy marriage between Windkessel (not compatible with an interpretation in terms of waves) and wave transmission models. One of the ongoing conceptual issues is whether or not the reservoir pressure is a wave in the original classical definition of the word. The current viewpoint (as described in the Supplemental readings accompanying the manuscript) of the initiators of the concept is that the reservoir pressure is made up of forward and backward ‘wave fronts’ but limited to those wave fronts ‘that have been in the system long enough to have visited an extensive part of the arterial system and have been influenced by the global properties of the system’ (e.g., total compliance). Wave fronts generated by the most recent left ventricular contraction and reflections from nearby locations constitute the excess pressure. It is a matter of debate whether all classically trained physiologists would agree on that indeed rather heuristic approach by ‘wave fronts’. Although considered by the initiators of the reservoir waveform model as a rather simplistic and disputable approach one-dimensional computational modeling [8,14] (a vascular network with three bifurcations and an anatomically based model of the systemic circulation) indicated that the reservoir-wave approach underestimated and even eliminated reflected compression waves, overestimated and even introduced (as artifacts) forward and backward expansion waves, and displayed interactions between distal reflection sites and early systolic waves classically considered as out of physiological range. Moreover from energetic point of view the model (if applied as a nonheuristic strictly mathematical model) can easily lead to violations of basic principles of physics [15]. In a three-dimensional computational model of the aorta [16] in simulated clinical cases (normal, aortic and stented aortic coarctation) reservoir pressure did not show an unequivocal relationship with aortic volume as predicted by the reservoir wave model and thus violating one of the key assumptions of that model. In pressure–volume diagrams hysteresis loops were looped in a clockwise way. For a given volume pressure would be lower on the upslope than on the downslope, thus ‘work done on the fluid by the aorta’; an implausible characteristic of reservoir pressure! Consequently, the intrinsic assumption that excess pressure should be attributed to wave travel and reflection will fail to a large degree. An argument possibly favoring the use of the reservoir waveform concept is the so-called anomalous behavior of classical wave separation analysis in the canine aorta [17]. Opposite of what should have been observed the foot of the backward wave arrived first at the most proximal measuring site and progressively later at the more distal sites. The authors’ findings in essence tackle as well the principle that a higher augmentation index should be the result of a mid-systolic reflected and pressure-augmenting wave. Westerhof et al.[18] argued against that interpretation. The awkward anomalous finding of counter-intuitively traveling waves results more likely from an oversimplification of the model. A simple single uniform tube with a single discrete terminal reflection site should be replaced by a model of a nearly continuum of reflection sites because of branching and tapering and the compound central backward wave should be made up of all individual waves. Application of that more complex model (an analogue condition to echo's in a gallery with arcades) will finally lead to a more physiological conclusion not obviously violating elementary principles. Are the conflicts between the three-element Windkessel and reservoir-wave models both unavoidable and insurmountable? Vermeersch et al.[19] analyzed noninvasively (tonometry) recorded scaled carotid pressure waveforms in 2019 study participants from the large population-based Asklepios cohort. In spite of its ‘outward appearance’ the wave-separation model seems to have more in common with a traditional three-element Windkessel model than generally perceived. The three-element Windkessel (frequency domain) adds a third component that disregards wave phenomena, whereas wave separation models the fast-acting phenomena by accounting for wave-propagation effects in the time domain. The Resistance-Compliance element and the reservoir functionally act as a filter separating low and high-frequency wave phenomena. The highest degree of similarity between the time domain reservoir pressure and frequency domain Windkessel approach was observed for study participants with a high reflection magnitude and what the authors called a ‘high Windkesselness’ of the arterial system. However, the authors [19] also had to tackle serious problems and intriguing observations while comparing both concepts. First, a fitting problem for venous pressure has led to nonphysiological values in many cases. A simplified model with zero venous pressure still led to unrealistic fittings in about 4 to 15% of the cases dependent on introducing some fitting modifications. Second, ‘low Windkesselness’ C-type waves without exponential pressure decay in diastole are particularly prone to erratic fittings for the reservoir component. Third, the relation between excess pressure and flow linearized with increasing reflection magnitude. Thus, the more important reflection magnitude the better the reservoir-wave concept will hold. However, the conclusion that the reservoir-wave concept would hold when the arterial system would display ‘Windkesselness’ to a large degree indeed looks counter intuitive at first glance. In conclusion, Narayan et al.[9] in this issue of the Journal confirm in an invasive clinical context one of the cornerstones of the reservoir-wave concept: a nearly constant magnitude of the calculated reservoir pressure waveform along the aorta notwithstanding its structural inhomogeneity. But are we facing a happy marriage between a Windkessel and a wave progation model? The authors’ heuristic approach in this and earlier published material and the clinical outcome data are pleading in favor of their case. On the other hand, both simple and more complex computational models and partial failure of the model after confrontation with some basic principles of physics and physiology are pleading in favor of the nonbelievers’ opinion. Whether Windkessel and wave propagation models will ever peacefully go hand in hand and harmonize will strongly depend on the ability of all investigators working in the field to tolerate some violations of first principles which will be unavoidably created by considering mixed models of that kind and to hope that some of these ‘imperfect parameters’ derived from mixed models will continuously show prognostic information and thus constitute valuable additions to the clinician's armamentarium. ACKNOWLEDGEMENTS Conflicts of interest There are no conflicts of interest.