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Archives of cardiovascular diseases
Volume 110, n° 12
pages 659-666 (décembre 2017)
Doi : 10.1016/j.acvd.2017.03.008
Received : 1 December 2016 ;  accepted : 23 Mars 2017
Influence of critical closing pressure on systemic vascular resistance and total arterial compliance: A clinical invasive study
Étude clinique invasive de l’influence de la pression critique de fermeture sur la résistance vasculaire systémique et la compliance artérielle totale

Denis Chemla a, b, c, , Edmund M.T. Lau d, Philippe Hervé e, Sandrine Millasseau f, Mabrouk Brahimi a, b, Kaixian Zhu a, b, Caroline Sattler a, b, c, Gilles Garcia a, b, c, Pierre Attal g, Alain Nitenberg a, b
a Service de physiologie, hôpital de Bicêtre, hôpitaux universitaires Paris-Sud, 94275 Le Kremlin-Bicêtre, France 
b Université Paris-Sud, 94275 Le Kremlin-Bicêtre, France 
c Inserm UMR_S999, LabEx Lermit, centre chirurgical Marie-Lannelongue, 92350 Le Plessis Robinson, France 
d Sydney medical school, university of Sydney, Camperdown, Australia 
e Departement de chirurgie thoracique, vasculaire et de transplantation pulmonaire, hôpital Marie-Lannelongue, 92350 Le Plessis Robinson, France 
f Pulse wave consulting, 95320 Saint-Leu-la-Forêt, France 
g Department of otolaryngology, head and neck surgery, Shaare-Zedek medical centre, Hebrew university medical school, Jerusalem, Israel 

Corresponding author. Service des explorations fonctionnelles cardiovasculaires, Broca 4, hôpital de Bicêtre, 78, rue du Général-Leclerc, 94275 Le Kremlin-Bicêtre, France.

Systemic vascular resistance (SVR) and total arterial compliance (TAC) modulate systemic arterial load, and their product is the time constant (Tau) of the Windkessel. Previous studies have assumed that aortic pressure decays towards a pressure asymptote (P∞) close to 0mmHg, as right atrial pressure is considered the outflow pressure. Using these assumptions, aortic Tau values of ∼1.5seconds have been documented. However, a zero P∞ may not be physiological because of the high critical closing pressure previously documented in vivo.


To calculate precisely the Tau and P∞ of the Windkessel, and to determine the implications for the indices of systemic arterial load.


Aortic pressure decay was analysed using high-fidelity recordings in 16 subjects. Tau was calculated assuming P∞=0mmHg, and by two methods that make no assumptions regarding P∞ (the derivative and best-fit methods).


Assuming P∞=0mmHg, we documented a Tau value of 1372±308ms, with only 29% of Windkessel function manifested by end-diastole. In contrast, Tau values of 306±109 and 353±106ms were found from the derivative and best-fit methods, with P∞ values of 75±12 and 71±12mmHg, and with ∼80% completion of Windkessel function. The “effective” resistance and compliance were ∼70% and ∼40% less than SVR and TAC (area method), respectively.


We did not challenge the Windkessel model, but rather the estimation technique of model variables (Tau, SVR, TAC) that assumes P∞=0. The study favoured a shorter Tau of the Windkessel and a higher P∞ compared with previous studies. This calls for a reappraisal of the quantification of systemic arterial load.

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La résistance vasculaire systémique (RVS) et la compliance artérielle totale (CAT) modulent la charge artérielle. Leur produit est la constante de temps (Tau) de la décroissance de la pression aortique diastolique selon le modèle du Windkessel. Les travaux antérieurs font l’hypothèse d’une asymptote de pression en débit nul (P∞) égale à 0mmHg, aboutissant à des valeurs de Tau d’environ 1,5 secondes. Une valeur nulle de P∞ n’est cependant pas physiologique.


Calculer précisément Tau et P∞ à l’aide de nouvelles méthodes numériques.


La pression aortique a été acquise par sondes haute-fidélité chez 16 patients. Tau a été d’abord calculée par la méthode semi-logarithmique en faisant l’hypothèse que P∞=0mmHg, puis par deux méthodes ne faisant pas d’hypothèse sur P∞ : la méthode des dérivées et la méthode du meilleur ajustement.


En faisant l'hypotthèse P∞=0mmHg, la valeur de Tau est de 1372±308ms et l’utilisation de la fonction Windkessel en télé-diastole est très incomplète (29 %). Avec les méthodes des dérivées et du meilleur ajustement, Tau=306±109 et 353±106ms, respectivement, P∞=75±12 et 71±12mmHg, et la fonction Windkessel est utilisée à ∼80 % en télédiastole. Avec ces deux dernières méthodes, la résistance « effective » était ∼70 % plus basse que la RVS et la compliance ∼40 % plus basse que CAT (méthode des surfaces).


L’existence d’une pression critique de fermeture élevée pourrait être responsable d’une surestimation franche de la RVS (et à un moindre degré de CAT) par les méthodes classiques qui font l’hypothèse d’une pression d’aval négligeable.

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Keywords : Arterial Windkessel, Systemic vascular resistance, Total arterial compliance, Exponential aortic pressure decay

Mots clés : Windkessel artériel, Résistance vasculaire systémique, Compliance artérielle totale, Pression aortique diastolique

Abbreviations : Pd, Pes, Pm, P∞, SVR, TAC, Tau


Systemic vascular resistance (SVR) and total arterial compliance (TAC) modulate the arterial load faced by the left ventricle. While SVR modulates the steady component of arterial load, as reflected in mean aortic pressure (Pm), TAC modulates the pulsatile component, as reflected in aortic pulse pressure, which is also influenced by wave reflections [1, 2, 3]. SVR and TAC are mostly dependent on the distal portion (resistive arterioles) and proximal portion (elastic large arteries) of the vascular tree, respectively. Right atrial pressure is considered as the downstream pressure of the systemic circulation, and is usually small enough to be negligible. As a result, SVR is commonly calculated by the Pm/cardiac output ratio, thus allowing estimation of vasomotor tone, and helping haemodynamic decision making [1, 2, 3]. The reference method used to estimate TAC, namely the area method, also assumes a zero downstream pressure [4].

According to the widely-used Windkessel model, the SVR×TAC product is the time constant (Tau) of the exponential diastolic aortic pressure (Pd) decay [5, 6, 7]. The Windkessel function is undoubtedly of major physiological importance in dampening blood pressure fluctuations, thus decreasing cardiac work, and for maintaining organ perfusion during diastole. During systole, the major part of stroke volume is stored in the compliant aorta, which reduces left ventricular afterload, while a smaller part of the stroke volume runs off through SVR. During diastole, the blood stored in the compliant aorta is released, resulting in improvement in coronary blood flow and left ventricular relaxation, and generating essentially steady flow at the capillary level. This beneficial systolic-diastolic interplay represents the Windkessel function [5, 6, 7]. Previous studies have assumed that the pressure asymptote (P∞) of Pd decay is close to 0mmHg, resulting in a Tau of the Windkessel in the 1–2 second range, on average [4, 8, 9, 10, 11, 12, 13, 14] (Table 1).

This classic view may be questioned because systemic blood flow ceases at an arterial pressure well above central venous pressure, called the critical closing pressure [15, 16, 17, 18, 19]. A high critical closing pressure will mathematically decrease Tau and SVR [4, 16, 17, 20], while the effects on TAC remain to be documented. The aim of our clinical study was to calculate precisely the Tau of the Windkessel using high-frequency digitization of high-fidelity aortic pressure and mathematical tools that make no assumption about the level of P∞, and to determine the implications for SVR and TAC values. Preliminary results have been presented elsewhere [21].


Our retrospective study included patients with symptoms of chest pain or other cardiovascular symptoms who were referred for diagnostic right and left heart catheterization (n =16). Part of the study population has been described elsewhere [22]. Patients with end-stage heart failure, rhythm disturbances, aortic stenosis or mitral valve insufficiency were excluded from the study. The final diagnoses were as follows: subjects with normal LV function and coronary angiograms (n =5); patients undergoing coronary artery bypass grafts (n =6); and patients with miscellaneous cardiac diseases (n =5). In patients receiving vasoactive drugs, treatment was discontinued 24hours before the investigation. All patients fasted for at least 12hours before the investigation. All patients gave their informed consent, and the study conformed to the standards set by the latest revision of the Declaration of Helsinki. The procedures were approved by the ethics committee of our institution (IRB Bicêtre Hospital, Le Kremlin-Bicêtre, France).

High-fidelity aortic pressure study

Heart catheterization was performed according to our routine protocol [22]. In brief, using micromanometer angiographic catheters (Sentron; Cordis Europa N. V., Roden, The Netherlands), aortic root pressures were obtained after a 5-minute equilibrium period. The data were computed on a personal computer with customized software (sampling rate 500Hz). Cardiac output was measured in triplicate using the thermodilution technique, and stroke volume was calculated by dividing cardiac output by heart rate.

Data analysis

The following indices were computed: central aortic systolic pressure; Pd; Pm; aortic pulse pressure; heart period in ms; heart rate (60,000/heart period); left ventricular ejection time (from the foot of aortic pressure to the incisura); and left ventricular diastolic time (heart period–left ventricular ejection time). We used a three-element Windkessel model of the circulation. Characteristic impedance was calculated in the time domain, as described previously [23, 24]. The resistance and capacitance/compliance elements of the Windkessel were calculated, assuming that Pd decays exponentially with time (t), according to Equation 1 [4, 16]: P(t)=(Pes – P∞)×e (–t/Tau)+P∞, where Pes is the end-systolic aortic pressure at the onset of exponential analysis (t=0). Tau (i.e. the resistance×compliance product) represents the time necessary for aortic pressure to exponentially decrease to 36.8% (1/e=0.368) of the Pes minus P∞ pressure difference. In theory, it will take three Tau values to complete ∼95% of this pressure drop (Figure 1).

Figure 1

Figure 1. 

The time constant of diastolic aortic pressure decay according to the classic model, which assumes a fixed pressure asymptote=0mmHg in all subjects. Typical exponential decline is superimposed over a schematic aortic pressure curve. The time constant Tau (τ) represents the time necessary for aortic pressure to decrease in value to 36.8% (1/e=0.368) of end-systolic pressure (100%). Assuming a classic Tau value of ∼1.5seconds, it will take theoretically three Tau values (∼4.5seconds) to complete 95% of the pressure change from end-systolic pressure to 0mmHg. Diastolic time at rest is in the <1 second range, i.e. <1 Tau. Thus, the classic model implies a poor use of Windkessel (WK) function at the end of diastole, which is counterintuitive.


As previously recommended, the onset of analysis (t=0) was determined by the operator on the P(t) vs dP/dt phase-plane, as the point when dP/dt linearly increases from negative values towards zero [25, 26]. Thus, the dicrotic notch and the subsequent transient plateau phase or the transient pressure oscillation phase were excluded from the monoexponential analysis. Overall, Pd decay was analysed over the last three-quarters of the diastolic period. Three numerical approaches were used over the same time period [4, 25, 26, 27]: the semilogarithmic method where P∞=0 in Equation 1; the derivative method in which the slope of the negative linear relationship between P(t) and dP/dt was calculated (Tau=–1/slope and P∞ is the extrapolated pressure intercept of the relationship when dP/dt=0); and the best-fit method in which Equation 1 was computed (semilogarithmic method) by imposing 50 P∞ values from 0 mmHg up to 95% Pd (the P∞ value corresponding to the best fit [highest r2 value of the monoexponential fit] was retained for analysis as well as the corresponding Tau).

We calculated the resistance corresponding to each of the three different levels of P∞ obtained with the semilogarithmic method, the derivative method and the best-fit method using Equation 2: resistance=(Pm–P∞)/cardiac output, where resistance is the part of systemic vascular resistance corresponding to flow resistance upstream of the site where the aortic pressure equals P∞ [15, 16, 17, 18, 19, 28]. Classic estimates of TAC were computed using the area method [4] or the stroke volume/aortic pulse pressure ratio [9, 22]. We also estimated TAC using systolic pulse contour analysis, as described previously [24, 29].

Following exponential curve fitting and Tau calculation with the semilogarithmic method, the derivative method and the best-fit method, the compliance element of the Windkessel model was calculated using Equation 3: compliance=Tau/resistance.

Finally, for each corresponding curve-fitting method, the percentage of Windkessel function completed by end-diastole was calculated using Equation 4: % of Windkessel function completed by end-diastole=100×(1–[(Pd–P∞)/(Pes–P∞)]).

Statistical analysis

Data are expressed as means±standard deviations. Comparisons were performed using the Wilcoxon test (paired samples). MedCalc software, version (MedCalc Software, Mariakerke, Belgium), was used for statistical analysis. Statistical significance was assumed for two-sided P <0.05.


Baseline characteristics of the study population are shown in Table 2.

Zero P∞ model

Using the semilogarithmic method with fixed P∞=0mmHg, Tau values of 1372±308ms were documented (Table 3). The median r2 value of the fit was 0.993 (range 0.987–0.998). Tau was markedly longer than diastolic duration (578±128ms), resulting in a calculated percentage of Windkessel function completed by end-diastole of 29±7% only (Table 3).

Derivative and best-fit method with variable P∞

The median r2 value of the derivative method fit was 0.947 (range 0.618–0.995). Derivation is known to increase the signal noise [4, 25, 26], and this explained the slightly lower r2 values compared with the semilogarithmic method. The median r2 value of the best-fit method was 0.998 (range 0.994–0.999). Using the two independent methods that make no assumption regarding P∞, we documented Tau values of 306 ±109ms (derivative method) and 353 ±106ms (best-fit method), which were ∼4×shorter than Tau calculated with the semilogarithmic method with zero P∞ (both P <0.05) (Table 3). The P∞ was 75±12mmHg (derivative method) and 71±12mmHg (best-fit method), thus reaching 91% and 87% of Pd, respectively.

The percentage of Windkessel function completed by end-diastole was 84±13% for the derivative method and 77±13% for the best-fit method, each significantly higher than the percentage calculated with the semilogarithmic method with zero P∞ (both P <0.05) (Table 3).

Implications for the resistance and compliance elements of the three-element Windkessel

Assuming a zero P∞, resistance was 2013±813dyn×s×cm−5. Resistance was 72% and 69% less using the derivative and best-fit methods, respectively (both P <0.05) (Table 4).

Assuming a zero P∞, compliance was 1.02±0.40mL/mmHg (Table 4). When no assumption was made regarding P∞ (derivative and best-fit methods), or when P∞ did not enter the model (systolic pulse contour analysis), there was no significant difference between the estimates of arterial compliance (Table 4). These three estimates of compliance were ∼40% less than TAC (area method or the stroke volume/pulse pressure ratio) (both P <0.05).


This high-fidelity pressure study did not challenge the Windkessel model, but rather the different estimation techniques of the model variables (P∞, Tau, SVR and TAC). A zero outflow pressure is commonly assumed while, on the contrary, a much higher critical closing pressure has been documented in systemic circulation. Using two exponential fitting methods of Pd decay that make no assumptions regarding P∞, we documented a higher P∞, a shorter Tau and a markedly decreased SVR and TAC compared with previous studies. This calls for a reappraisal of the quantification of systemic arterial load.

The Windkessel function is of major importance for cardiovascular homeostasis. It is generally assumed that Pd decays exponentially towards a P∞ close to 0mmHg, resulting in a Tau of the Windkessel=SVR×TAC that is in the 1–2 second range, on average [4, 8, 9, 10, 11, 12, 13, 14] (Table 1). Given the high critical closing pressure recently documented in vivo [15, 16, 17, 18, 19], our study hypothesis was that Tau must be calculated using numerical methods that make no assumptions regarding P∞. Using two exponential fitting methods (the derivative and best-fit methods), we documented a P∞ value closer to Pd than to 0mmHg. We also documented Tau values of 300–350ms on average (i.e. Tau was 4×shorter than its empiric estimate). It is reassuring that these two independent methods gave essentially similar results. Interestingly, the derivative and best-fit methods have been used extensively in the cardiovascular field to calculate the Tau and P∞ of left ventricular isovolumic pressure fall, while using a fixed zero P∞ is not recommended [25, 26, 27, 30]. Langer et al. recently concluded that “The estimation of the time constant of isovolumic pressure fall with a preset zero asymptote is heavily biased…” [30], and here we suggest that the same conclusion may apply in the setting of aortic pressure decay.

To our knowledge, only two clinical studies have used the derivative method [4, 31]. Liu et al. did not succeed in calculating Tau with the derivative method, and this may be explained by the low sampling rate they used [4]. Conversely, Chung et al. [31] successfully reported Tau values using the derivative method in their study involving 13 subjects (12 men), of whom one was hypertensive and one had diabetes (sampling rate 200Hz). The authors reported Tau values of 366 ±126ms [31], and our findings (sampling rate 500Hz) appear consistent with theirs.

Assuming in the first instance a fixed P∞ value of 0mmHg for all subjects, we found that only 29% of the Windkessel function had been completed by end-diastole. This is counterintuitive, given the physiological importance of this function. Furthermore, left ventricular ejection would occur at a time when emptying of large arteries is not completed, thus leading to acute vascular overload after a few beats. Conversely, with the derivative and best-fit methods, a higher P∞ and a shorter Tau were documented, such that the majority of the Windkessel function (78–84%) was completed by end-diastole, which seems more physiological [32]. The early diastolic portion of the aortic pressure decay was not included in the analysis, as it deviates from exponentiality, and this may explain why we did not document full completion of Windkessel function by end-diastole. Other hypotheses may be that P∞ is the theoretical, never-reached, zero flow pressure, or that the aortic pressure decay may slightly deviate from a single exponential function under near zero flow conditions.

A zero P∞ may not be physiological (Table 5). Under prolonged arrested heart conditions, pressure equilibrates at a similar level in the systemic and venous circulation. This so-called mean systemic filling pressure (∼7–15mmHg) is often believed to reflect the effective downstream pressure of the systemic circulation, acting as a Starling resistor located at the systemic venous level [33, 34, 35]. However, cardiac arrest is far from physiological conditions. Other studies performed with the heart beating have suggested that systemic blood flow may cease at a “critical closing pressure”, well above the mean systemic filling pressure. Experimental studies with Pm plotted at various aortic flows have documented 40–100mmHg zero flow pressure in animals whose autonomous system was kept intact [28, 36]. Using a monoexponential function of time, Kottenberg-Assenmacher et al. [16] reported that P∞ was 56.8±16.2mmHg in the systemic circulation (naturally-beating heart) of 10 anaesthetized patients scheduled for defibrillator implantation. Using inspiratory-hold manoeuvres in 10 mechanically-ventilated postcardiac surgery patients, Maas et al. [17] documented cardiac output and Pm at various plateau pressure values, thus allowing the estimation of critical closing pressure (45.5±11.1mmHg). In the latter studies [16, 17], patients were anaesthetized/sedated, and this may have reduced their vascular tone and sympathetic drive. The higher sympathetic drive in our awake patients may well have resulted in a higher P∞[28].

The high P∞ we documented may reflect the mean weighted pressure of the various vascular beds below which the arterioles collapse. This mechanism and its consequences have been previously termed critical closing pressure or waterfall effect or Starling resistor [33, 34, 35, 36, 37, 38]. At each organ level, the critical closing pressure is generated by the vasomotor tone of arterioles, the precapillary vascular smooth muscle state and local metabolic conditions modulating tissue pressure surrounding the vasculature. Based on previous clinical studies and the present study, it may be concluded that, on average, the critical closing pressure of the systemic circulation as a lump sum of organ-specific critical closing pressures is in the 45–75mmHg range in the human systemic circulation with a naturally-beating heart. Thus, right atrial pressure should not be considered as the systemic outflow pressure.

Both SVR and TAC are invaluable indices of arterial load [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. However, their ability to offer reliable metrics of the resistance and capacitance elements of the Windkessel was challenged in our study. By assuming a non-zero P∞, the resistive element of the Windkessel was ∼70% less than the empiric estimate of SVR, and this is consistent with recent findings [16, 17]. Another result of our study was that compliance appeared proportionally less influenced by P∞, as the compliance element of the Windkessel was ∼40% less than empiric estimates of TAC.

So defined, resistance may be called either the “effective” resistance of the peripheral systemic circulation or the arterial vascular resistance [15, 17, 20, 28]. A critical closing pressure may present advantages over a purely resistive pressure-flow arrangement [17, 19]. First, it could disconnect systemic arteries from the influence of venous pressure changes, and thus help maintain the stability of arterial flow to the organs (vascular waterfall). Second, for identical pressure and flow, resistance is smaller because of the decreased pressure gradient, and equal variations of cardiac output will result in smaller pressure changes, thus requiring less active pressure control. Third, because diastolic pressure decays toward the high critical pressure, this may prevent low Pd values, thus preserving coronary perfusion and decreasing oscillatory energy loss.

Study limitations

There were only five controls, and 11 patients had cardiovascular diseases; these small samples prevent any between-group comparisons of Tau or P∞, a point that deserves further study. Although the three-element Windkessel model is used widely in practice because of simplicity and physiological relevance, the model has limitations [1, 2, 3, 4, 5, 6, 7, 29]. The model is zero dimensional and does not take into account the finite pulse wave velocity and the phenomena of blood propagation and wave reflection. However, when distributed models are used, an increased SVR or a decreased TAC strongly affects the forward and backward pressure waves, thus reinforcing the interest in a detailed understanding of their meaning and determinants. We did not use the still-debated reservoir-wave approach, but it must be noted that the fundamental assumption of this approach is that P∞ is greater than zero (normally 30–40mmHg) [28]; furthermore, P∞ may increase up to 100mmHg following sympathetic stimulation [28]. Aortic pressure decay may deviate from a single exponential function under the near zero flow conditions encountered by end-diastole. This cannot account for the observed high P∞ value, as applying a dual exponential function to fit aortic pressure decay in the naturally-beating heart results in a mild decrease in P∞ (–7%) [16]. It was not the aim of our study to calculate Tau and P∞ during cardiac pauses. Interestingly, using the best-fit method in subjects with irregular cardiac rhythm, a recent preliminary study indicated that P∞ is 70% Pd in cases where heart beat duration is nearly doubled [39].

Clinical implications

The main implication of our study was that the empiric estimates of SVR and TAC, which assume zero outflow pressure, may be irrelevant because of high effective outflow pressure. Taking into account precisely the P∞ level in the evaluation of arterial load variables may have major clinical implications, because the critical closing pressure may vary depending on physiological (e.g. age), pathological (e.g. hypertension) or pharmacological (e.g. vasoactive agents) states. The implications for clinical monitoring (e.g. cardiac output estimation from pulse contour analysis) also deserve further study.


In conclusion, the present study has challenged the classic estimation of Windkessel model variables (P∞, Tau, SVR and TAC). While a zero outflow pressure is commonly assumed, we documented a high P∞ of the arterial Windkessel by using two independent exponential fitting methods of Pd decay (derivative and best-fit methods). This is consistent with the high critical closing pressure recently documented in the human systemic circulation (naturally-beating heart). We documented a shorter Tau and a markedly decreased SVR (–70%) and TAC (–40%) compared with classic estimates. This calls for a reappraisal of the quantification of systemic arterial load.

Sources of funding

Université Paris-Sud.

Disclosure of interest

The authors declare that they have no competing interest.


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