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1Cardiovascular Research Group, Departments of Cardiac Sciences, Physiology and Biophysics, and Civil Engineering, University of Calgary, Calgary, Canada; and 2Department of Bioengineering, Imperial College of Science, Technology and Medicine, London, United Kingdom
Submitted 13 May 2005 ; accepted in final form 16 August 2005
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ABSTRACT |
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62% of total systemic vascular resistance, whereas windkessel-related venous resistance was only 7%. Total venous compliance was found to be 21 times larger than arterial compliance (n = 3). Inferior vena caval compliance (0.32 ± 0.015 ml·mmHg–1·kg–1; mean ± SE) was 14 times the aortic compliance (0.023 ± 0.002 ml·mmHg–1·kg–1; n = 8). Despite greater venous compliance, the variation in venous windkessel volume (i.e., compliance x windkessel pulse pressure; 7.8 ± 1.1 ml) was only 32% of the variation in aortic windkessel volume (24.3 ± 2.9 ml) because of the larger arterial pressure variation. In addition, and contrary to previous understanding, waves generated by the right heart propagated upstream as far as the femoral vein, but excellent proportionality between the excess pressure and venous outflow suggests that no reflected waves returned to the right atrium. Thus the venous windkessel model not only successfully accounts for variations in the venous pressure and flow waveforms but also, in combination with the arterial windkessel, provides a coherent view of the systemic circulation.
systemic circulation
In 1992, Tyberg (33) proposed a steady-state hydraulic model in which the volumes of the arterial and venous reservoirs were functions of mean arterial and venous pressures, via their respective compliances; this model was used to describe the translocation of blood volume in terms of parallel shifts in venous pressure-volume relationships. Recently, the concept of a constant-pressure arterial reservoir was extended to the time-varying state in which the arterial windkessel discharges during diastole, to be recharged during the subsequent systole (35). We now propose that the venous system is characterized by an analogous windkessel behavior: during the later part of diastole, pressure rises exponentially to approach an asymptotic value as inflow continues while venous outflow is negligible. In addition, waves propagate on this time-varying reservoir, the venous windkessel.
We first calculated the pressure change due (via venous compliance) to the volume change in the venous reservoir. By integrating this rate of pressure change, we obtained the windkessel pressure (PWk) over the cardiac cycle. We defined the difference between pressure in the inferior vena cava (PIVC) and PWk as the excess pressure (Pex), the pressure due to waves (17, 35). We then compared Pex with the measured outflow (QIVC). Our finding that Pex is proportional to QIVC suggests that waves generated by the right atrium (RA) and right ventricle (RV) account for the temporal variation in QIVC.
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METHODS |
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VWk-a
VWk-IVC
Venous windkessel theory.
The change in venous reservoir pressure, PWk, was assumed proportional to the change in the venous reservoir volume (i.e., VWk), which in turn, is the difference between inflow and outflow,
 | (1) |
, toward which PIVC exponentially rises during diastole) and the venous reservoir pressure (PWk), across a resistance (R), or in mathematical form, Qin(t) = [P – PWk(t)]/R.
By substituting Qin in terms of PWk and P
, Eq. 1 can be rewritten in terms of the windkessel pressure, PWk, as
 | (2) |
 | (3) |
= R x C, characterizes the exponential rise of PIVC.
To solve Eq. 3, we had to determine R, C, and P using experimental data. Because QIVC is minimal during late diastole and because there is no significant wave motion, the observed exponential rise in PIVC can be attributed to the charging of the venous system by inflow. Therefore, PIVC approximates PWk during this time.
P was initially determined by fitting PIVC data during the later part of diastole (when venous outflow is minimal; see the vertical dashed lines, Fig. 1) using a three-parameter exponential-rise equation. With arbitrary initial guesses for R and C (5 mmHg·min·l–1 and 2 ml/mmHg, respectively), they can be determined using a nonlinear search algorithm to minimize the mean-squared error between the calculated PWk and the measured PIVC. The minimization was done using the Matlab (Mathworks, Natick, MA) routine "fminsearch," which uses the Nelder-Mead simplex (direct search) method. Many combinations of R and C yield the same , but only very narrow sets of values yield a PWk that relates to PIVC appropriately, throughout the entire cardiac cycle. The calculation is not restricted to one cardiac cycle; it can be extended to multiple cycles. The difference between PIVC and PWk is defined as excess pressure (i.e., the pressure due to waves): Pex = PIVC – PWk (35).
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VWk = C x PWk. Experimental preparation and protocol. The protocol for the animal experiments conformed to the "Guiding Principles of Research Involving Animals and Human Beings" of the American Physiological Society and was approved by the University of Calgary animal care committee.
Experiments were performed on eight healthy mongrel dogs weighing between 16 and 30 kg. Dogs were anesthetized with thiopental sodium (20 mg/kg) followed by fentanyl citrate (30 µg·kg–1·h–1) and ventilated with a 1:1 nitrous oxide-oxygen mixture. The rate of a constant-volume respirator (model 607; Harvard Apparatus, Natick, MA; tidal volume = 15 ml/kg) was adjusted to maintain normal blood gas tensions and pH. Body temperature was maintained at 37°C with the use of a circulating-water warming blanket and a heating lamp. A 10% pentastarch solution (Pentaspan; Bristol-Myers Squibb Canada, Montreal, Quebec, Canada) was infused to increase RV end-diastolic pressure (PRVED).
Pressures were measured using high-fidelity catheter-tip manometers (Millar Instruments, Houston, TX) referenced to pressures measured externally (Statham-Gould, Oxnard, CA) via liquid-filled lumens and zeroed to the height of the RA. Flows were measured using an ultrasonic flowmeter (Transonic Systems, Ithaca, NY). We measured RV pressure (PRV) by inserting a manometer through the RV apex and RA pressure (PRA) by inserting a micromanometer (2F; Millar) through the RA appendage. Simultaneous pressures and flows were measured in the IVC (no more than 2 cm from the RA) and in the aortic root. In three dogs, pressure and flow also were measured in the superior vena cava (SVC). In the other five dogs, two additional flow probes were positioned, one just downstream to the renal veins and the other at the bifurcation, to measure simultaneous pressure and flow during pull back (see below). The IVC manometer was inserted through a femoral vein and the SVC manometer through a jugular vein. The aortic manometer was inserted through a femoral artery. The second jugular vein was cannulated for volume loading. To control the heart rate during the pull-back measurements, we paced the RA via epicardial electrodes by using a laboratory stimulator (model S88; Grass Instrument, Quincy, MA). All data were recorded after the respirator had been turned off at end expiration.
After the control recordings were obtained (PRVED 3 mmHg), PRVED was raised to 5, 10, and then 15 mmHg. At PRVED 
10 mmHg, the heart was paced at the lowest possible rate [zatebradine (ULFS 49)], and the IVC manometer was pulled back by 2-cm increments to the femoral vein, with pressures recorded at each position with the ventilator turned off at end expiration. Three-dimensional plots of venous pressure versus time and distance from the RA were constructed, using the ECG as a temporal reference. Distal venous flows were related to simultaneously measured pressures. The IVC manometer was repositioned back to within 2 cm of the RA before PRVED was increased to 15 mmHg. At the end of the volume loading, a bolus of acetylcholine (10 mg) was injected into the RA to produce a long (4–10 s) diastolic pause.
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RESULTS |
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values are shown in Fig. 2, top. The measured outflow, QIVC, the calculated inflow, Qin, and the calculated excess pressure, Pex, are shown in Fig. 2, bottom. (To compare the contours of Pex and QIVC, we reversed the sign of Pex and adjusted the scale so that their peak values coincided.) When Qin equals QIVC (indicated by the vertical dashed lines), there is no change in PWk. During diastole, Qin > QIVC and PWk increases as the reservoir (windkessel) is charged by Qin from the arterial circulation. Qin is regulated by a venous peripheral resistance, RWk, and is driven by the gradient between P and PWk. The charging of the reservoir would cease when PIVC reached the level of P. The pressure due to waves, which was characterized by Pex, the difference between the measured PIVC and PWk, is precisely proportional to QIVC (bottom). The proportionality factor (Pex/QIVC), which we interpret as the resistance of the large veins, is numerically equal to the characteristic impedance of the IVC (35).
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, R, and C were determined for the IVC and SVC by using the long diastole of the last beat, and each set of parameters was applied to calculate PWk-IVC and PWk-SVC during the whole series of beats. Note that the variation in PIVC was slightly less than that of PSVC but that the variation in QIVC was approximately threefold greater than that in QSVC.
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10 mmHg. Waves, corresponding to the maxima and minima in PRA, propagated through the IVC as far as measurements were made (i.e., to the femoral vein). However, during the later part of diastole, no wave motion was apparent and pressure increased uniformly and simultaneously throughout the length of the venous system. This uniformity can be best appreciated from the contour plot (Fig. 4, bottom; isobar increment = 0.2 mmHg) between 0.85 and 1.05 s.
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18 cm to the RA), and further upstream at the bifurcation (30 cm from the RA). At sites progressively further upstream, the variation of flow diminished but the variation of pressure was relatively unchanged. To observe wave propagation, on each pair of pressure flow panels we marked the beginning of the c wave and the beginning of the y descent with vertical lines. It took 44 ms for these waves to travel from the RA to the renal veins and another 27 ms to reach the bifurcation, corresponding to averaged wave speeds of 4.1 m/s between the RA and renal veins and 4.4 m/s between the renal veins and the bifurcation.
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DISCUSSION |
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Overall, however, the study of venous physiology has been dominated by Guyton's paradigm (11). With the laudable intent of demonstrating an important role for peripheral circulatory factors in modulating cardiac output, he focused attention on "venous return," the flow of blood (from distributed arterial and venous reservoirs held at an equivalent mean circulatory pressure) that discharged through a "resistance to venous return" into the RA. Although this construct effectively emphasized the principle that the heart can only pump the blood that returns from the venous system, the narrow focus on this segment of the total circulation arguably overemphasized the resistance of the venules and veins, which together compose only 10% (7) of the total resistance between the aorta and the RA (Guyton related distributed total blood volume to cardiac output, which required that he weight the arterial and venous resistances according to their respective capacitances), and underemphasized those elements of the circulation that continually refill the reservoirs that are the source of venous return. Thus, in our view, his interpretation did not provide complete balance with respect to the circulation as a whole. This paradigm reduced to a system in which PRA, "the back pressure to venous return," determined venous return and became the independent variable.
In 1979, Levy (16) reported the results of an experiment that was fundamentally similar to Guyton's and proposed an alternative paradigm in which cardiac output was the independent variable. Both Guyton and Levy emphasized the inverse relationship between cardiac output (equal to venous return in the steady state) and venous or RA pressure. However, from Levy's perspective, increasing the cardiac output augmented the arterial reservoir and depleted the venous reservoir and, therefore, caused venous pressure to decrease. In 1992, on the basis of the pressure-volume relationships of the arteries and the veins, Tyberg extended Levy's model and represented the peripheral systemic circulation as a high-pressure, low-compliance arterial reservoir functionally separated from a low-pressure, high-compliance venous reservoir by a lumped systemic vascular resistance (33). Our present conception of the arterial and the venous systems is, in fact, an extension of that steady-state, mean-pressure hydraulic reservoir model to a dynamic model in which arterial and venous windkessel pressures vary during the cardiac cycle in precise proportion to the volumes that they contain. During later diastole, the arterial reservoir discharges and the venous reservoir charges; during systole, the arterial reservoir charges and, during early diastole, the venous reservoir discharges. [A century ago, Otto Frank used the term "windkessel," a term first appearing in a German translation of Hales's Haemostatics (22), to describe the change in pressure associated with the dynamic variations in reservoir volume (9, 29).]
We propose that the venous system behaves as waves propagating on a time-varying reservoir, the venous windkessel, which is the exact analog of our study of the arterial windkessel (35) but with reversed functionality. The analysis is simple and straightforward in principle. We first quantify the windkessel pressure by observing that, after the diastolic E wave and before the A wave (see Figs. 1 and 2), QIVC virtually stops and the gradient among the RV, RA, and IVC is negligible. Therefore, the calculated pressure associated with venous reservoir volume, PWk, should equal PIVC during this period. The exponential rise of PIVC is due to the charging of the venous reservoir by blood flowing into it from the arterial system. According to Lighthill (17; see also Ref. 35), the pressure associated with waves is the excess pressure, Pex, which is the difference between the observed pressure (PIVC) and the reservoir pressure (PWk). Under all experimental circumstances, Pex was proportional to QIVC, which suggests that the variations in Pex and QIVC could be completely accounted for by backward (i.e., upstream)-going waves and that no significant reflections were propagated back to the root of the IVC (35).
The strict proportionality between Pex and QIVC over a wide range of PIVC does imply that the irregular relationship between the venous pressure and flow can be simply explained by two "first-order" mechanisms: waves propagated upstream and volume variations due to windkessel charging and discharging. In the past, the irregular venous pressure and flow contours have only been considered to be the results of nonlinear effects, such as the effects of venous valves, muscle pumps (32), and/or partially collapsed vessels (27). However, the effects of venous valves and muscle pumps should have been minimal, because the animals in those studies were anesthetized in the supine position. There is a general perception that veins collapse to some degree, even with positive transmural pressure and with the animals in the supine position, where gravity plays a minor role (13). The collapsible tube phenomenon was first quantitatively described by Holt's bench-top experiment (14), which became the paradigm and was extensively studied using experimental (8, 15), mathematical (4, 15), and computational analysis (10, 25). However, all these previous studies were focused on large negative transmural pressures, and the results showed that nonlinear effects on pressure-flow relationship were insignificant if transmural pressure was positive or only slightly negative (4, 10, 25).
Our approach incorporates a three-element windkessel, but there are distinct differences between our time-domain approach and the impedance-analysis approach, and there may be theoretical and practical reasons why previous applications of impedance analysis failed to demonstrate three-element windkessel characteristics. In our view, the most important reason is theoretical and pertains to the fundamental conceptual difficulty associated with applying the logic of the arterial windkessel to the venous system. The venous windkessel is an upside-down version of the arterial windkessel, but the logic of this inversion is subtle. In addition, although the exponential behavior of the arterial windkessel may have been recognized, to our knowledge, a non-zero asymptotic pressure was never identified. For example, in developing their method to calculate arterial compliance, Yin et al. (18) used equations that are identical to ours except for the fact that they implicitly assumed that flow continued until arterial pressure equaled zero. Similarly, Guyton (12) and Sunagawa (34) devised circuits composed of a series of capacitors, all discharging to earth/ground level. According to our venous windkessel, venous pressure rises exponentially toward an asymptotic level and the flow into the venous windkessel is related to the inverse of the difference between this asymptotic pressure and venous (IVC) pressure. Zero pressure is completely irrelevant.
This problem is compounded by the fact that, with the conventional impedance analysis, mean values are subtracted out before the remaining periodic variations of pressure/flow are resolved into time-shifted sinusoidal components of variable magnitudes. Thus, even though the exponential nature of the equations and the electrical circuit might have been apparent, all absolute orientation (e.g., with respect to pressure in general and P in particular) is immediately lost.
However, all these problems notwithstanding, it is possible in retrospect to use Fourier analysis to define Pex and PWk. Noting that our ratio Pex/QIVC is equal to characteristic impedance (Z0), one can multiply Z0 by QIVC to get Pex (26). Pex then can be subtracted from PIVC to yield PWk. Thus, once one appreciates that PIVC is equal to the sum of PWk and Pex, one can use Z0 to duplicate our time-domain analysis.
An important practical reason for the difficulty in using impedance analysis is because it is severely limited in the absence of a steady state; the waveforms are not then truly periodic in the sense of being exactly reproducible and, if impedance analysis on single non-steady beats is attempted, spurious additional components have to be generated in each harmonic to accommodate the discontinuity. On the other hand, our approach is facilitated by the occasional appearance of longer diastolic periods during which QIVC is negligible.
Changing mean PIVC (with volume loading) between 2 and 13 mmHg does not seem to change venous compliance significantly or consistently (Fig. 6). Although Magder (21) found that compliance decreased as pressure increased, at pressures less than 20 mmHg, no trend was clearly defined. Under control condition, IVC compliance averaged among eight dogs was 0.32 ± 0.02 ml·mmHg–1·kg–1, which is 14 times of the arterial compliance (0.023 ± 0.002 ml·mmHg–1·kg–1; see Table 1). In the past, venous compliance has been studied using the two-port analysis: venous compliance ranged from 0.34 ± 0.11 to 1.25 ± 0.62 ml·mmHg–1·kg–1, depending on the manipulation of the baroreflex and infusion of norepinephrine (27).
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Although the comparison of compliances is informative, a better index of their relative "size" may be a comparison of the volumes of blood that charge and discharge the windkessels during each cardiac cycle. This was obtained for the aorta and the IVC by taking the product of the respective windkessel pressure and compliance to obtain a "windkessel pulse volume" (i.e., the difference between the maximum and minimum windkessel volume). This analysis showed that the change in volume of the aortic windkessel was 3.33 ± 0.32 times the change in volume of the IVC windkessel. Even though the IVC compliance is much greater, the larger pressure variation in the aortic windkessel makes the aortic volume change substantially larger and, in a sense, more important physiologically.
Our studies of the arterial and venous windkessels have led us to modify our view of the peripheral systemic circulation. Consistent with previous ideas, in the language of an electrical analog, we suggest that a continuous resistor, the systemic vascular resistance (SVR), separates the potential levels corresponding to mean Pa and PIVC. The extreme ends of this resistor represent large-artery and large-vein resistances and are mathematically equal to the respective ratios of Pex to flow (i.e., Pex/Qa and Pex/QIVC). [In turn, these ratios are quantitatively equal to arterial and venous characteristic impedance (35). It is interesting to note that Quick et al. (26) showed that the product of flow times characteristic impedance was equal to "reflectionless" pressure; we would say "windkessel-less" pressure, i.e., Pex.] Segments of the resistor next to the large-artery and large-vein resistances are the resistances that define the time constants ( = R x C) of the windkessels. (Compliances are well-defined physical vessel properties, but, to produce the observed time constants, they must be coupled with resistances.)
These concepts can be illustrated by considering order-of-magnitude values (Table 2 and Fig. 7; only the flow that returns to the heart via the IVC, 60% of aortic flow, is considered so that arterial parameters are scaled appropriately.) The value of the large-artery resistance is 3% of the total resistance, and the resistance of the arterial windkessel is 59%. The value of the large-vein resistance is 1%, and the resistance of the venous windkessel is 6%. By difference, this implies that the value of the remaining intermediate segment of the total systemic vascular resistance is 31%. This implies that the arterial windkessel extends downstream 60% of the extent of the equivalent series resistor (i.e., SVR) and that the venous windkessel extends upstream 7%.
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39%) for the microcirculation (i.e., arterioles, capillaries, and venules) is essentially equal to the residual resistance not accounted for by the arterial and venous windkessels (32%; purple segments in Fig. 7). Their values for large artery resistance (7%) include arterial segments as small as 250 µm and, for large-vein resistance (6%), venous segments as small as 60 µm. Thus our respectively smaller values for large artery and large-vein resistances (1.2 and 0.6%) would seem to correspond to the resistances of the larger of the "large" vessels, as they must if they represent the resistances of vessels interposed between the heart and the respective windkessels.
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-a, so the difference between mean PWk-a and P-a should govern windkessel outflow during the cycle. The same is true of the venous system. The fact that mean IVC pressure is measurably lower than mean venous windkessel pressure (the difference being related to Pex-v) is consistent with blood flowing through a large-vein resistance. Also, according to the theory given above, instantaneous flow into the venous windkessel is proportional to the difference between P-v and PWk-v, and so the difference between mean P-v and PWk-v should govern venous windkessel inflow during the cycle. Finally, we hold that the resistance between the arterial and venous windkessels (i.e., the microvascular resistance) should be a function of the difference between P-a and P-v (purple crosshatching in Fig. 8, left). As shown in Fig. 8, right, we compared these pressure differences to incremental resistances calculated from this same beat and found an approximately linear relation, as expected (r2 = 0.998).
In addition, Fig. 8 suggests a practical expedient that may prove useful. The fraction of total SVR accounted for by the microcirculatory resistance may vary widely, from 40% (e.g., methoxamine) to 15% (e.g., sodium nitroprusside) (unpublished data). As a first approximation, the resistance related to the venous windkessel (blue hatching in Fig. 8, left) might be ignored. The arterial diastolic data then could be fit to a simple, three-parameter exponential equation to estimate the value of P-a, which would indicate how much of the SVR is related to the arterial windkessel (red hatching) and how much to microcirculatory resistance (purple crosshatching). [This estimation might be difficult, practically, if the heart rate is high (i.e., >80 beats/min) but not with data from a longer cardiac cycle, as in the presence of normal sinus arrhythmia.]
The extents of the arterial and venous windkessels are defined by the values of their respective P values. For the arterial system, these values correspond to "waterfall" (6, 19), Starling resistor (30, 31), or critical closing pressure (5) phenomena. Under our normal experimental conditions, P-a is 35 mmHg (35), but under the influence of methoxamine, this value can approach 100 mmHg (unpublished observations). Also, under the influence of an acetylcholine injection that can produce a diastolic pause lasting 15–20 s, aortic pressure declines to 15 mmHg, consistent with observations designed to estimate mean circulatory filling pressure (28). Finally, after death, the arterial-venous pressure gradient undoubtedly disappears. All these observations suggest that the value of P-a is a function of the local biochemical, humoral, and mechanical milieu and that these factors can vary.
Limitations.
In general, it is more difficult to determine windkessel parameters if the heart rate is fast (>70 beats/min). We view this as a practical but not a theoretical limitation. This difficulty can be minimized by selecting a sequence of beats that includes a longer diastole. (Since completion of the present experiments, we have found that momentary stimulation of the right vagus nerve prolongs diastole conveniently.) Because baroreceptor activity is known to affect the venous compliance (27), we allowed time between volume-loading interventions to achieve reequilibration.
In conclusion, we represent the venous circulation as a system in which waves propagate on a time-varying reservoir. This new model not only explains the details of the seemingly irregular IVC and SVC pressure and flow waveforms but also, in relation to systemic vascular resistance, provides a coherent view of the systemic circulation as a whole.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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a), arterial windkessel (