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1Center for Autonomic and Peripheral Nerve Disorders, Department of Neurology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, Massachusetts 02215; and 2Council of Research and Development, CID-001-99, Favaloro University, Buenos Aires, Argentina
Submitted 9 October 2002 ; accepted in final form 6 February 2003
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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plethysmography; modeling; vascular capacitance
70% of the total blood volume
and plays a critical role in hemodynamic control. Veins influence cardiac
output by active and passive participation in cardiac filling
(20,
21). Knowledge of the filling
and emptying characteristics of the venous system is therefore essential to
understanding the role of this system in cardiovascular homeostasis. Several recent studies have examined venous compliance and its effect on cardiovascular control in health and disease. Common to these studies is the determination of the pressure-volume relationship of the venous system in response to a physiological perturbation; however, linear and nonlinear analytic techniques have been used to characterize the pressure-volume relationship (4, 10, 15, 17, 19, 25). Because there is evidence that venous compliance is not linear within the physiological range of pressures, it is likely that linear modeling of the venous pressure-volume relationship may not describe venous physiology adequately. Furthermore, previous studies have shown that analysis of the nonlinear behavior in arteries can be used to separate the contribution of the different components of the arterial wall to arterial compliance (1-3, 7). Similar structure-function analyses may be applicable to the venous system as well.
We hypothesized that limb venous compliance could be described by using a biphasic model with two linear phases: a low-pressure phase that represents the steep gradient of the low pressures of the venous compliance curve and a high-pressure phase that represents the shallow gradient and plateau of the mid to high pressures. This model is consonant with the two physiological functions of the venous system: to enable large translocations of blood from the venous system in response to small changes in pressure at low venous pressures and to minimize venous pooling at high venous pressures (20, 21). To test this hypothesis, we obtained pressure-volume data from a group of healthy subjects and compared the performance of the biphasic model to linear and exponential models of venous compliance in the calf and forearm.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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Data analysis. The data recorded during each trial were averaged
into 1-mmHg segments from 60 to 10 mmHg. The data were reduced to 50 pressure
and volume data points. These data was used to generate pressure-volume curves
(see Fig. 1A). To
avoid any assumption on the pressure-volume relationship, compliance was
calculated as the numerical derivative (3-point derivative) of each data point
by use of the following equation:
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Results of the numerical derivative of the pressure-volume relationship from a representative control subject are shown in Fig. 2A.
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Compliance models. We examined the pressure-compliance relationship with biphasic, linear, and a family of exponential models. All models use the compliance determined by the numerical derivative of the pressure-volume relationship as raw data.
The biphasic model assumes that the pressure-compliance relationship is comprised of two linear phases, one at low pressures (LP) and the other at high pressures (HP) (see Fig. 2A). The breakpoint pressure, which separates the two linear phases of this model, is determined by using the coefficient of determination (r2) of a linear regression that starts at 60 mmHg. The algorithm to calculate this model begins at 60 mmHg and sequentially adds pressure-volume pairs throughout the data range to 10 mmHg. For each pressure-volume pair added, a new r2 is calculated (Fig. 1B shows the r2-vs.-pressure relation). The r2 increases as additional pressure-volume pairs are added, until a maximum is reached and the r2 begins decreasing. The breakpoint, defined as the pressure at 99% of the maximum r2, indicates the end of the high-pressure, linear phase of the pressure-volume relationship (see Fig. 1). The high-pressure phase is thus defined as the region from 60 mmHg to the breakpoint, and the low-pressure phase is defined as the region from the breakpoint to 10 mmHg. A linear function was then used to fit both the high- and low-pressure phases of the compliance-vs.-pressure relationship (Fig. 2A). The biphasic model yields an intercept (y0 LP and y0 HP) and slope (mLP and mHP) for each phase of the pressure-compliance relationship.
The linear model is a linear fit of the pressure-compliance relationship. This fit yields the intercept (y0) and the slope (m) of the fitted line (see Fig. 2B).
A family of exponential models was used: 1) single-exponential
model (2 parameters) [Compliance(Pressure) = a ·
e-b·Pressure]
(see Single exponential model under RESULTS and
Fig. 2C); 2)
single-exponential plus intercept model (3 parameters) [Compliance(Pressure) =
y0 + a ·
e-b·Pressure];
3) two-exponential model (4 parameters) [Compliance(Pressure) =
a ·
e-b·Pressure
+ c ·
e-d·Pressure];
4) two-exponential plus intercept model (5 parameters)
[Compliance(Pressure) = y0 + a ·
e-b·Pressure
+ c ·
e-d·Pressure].
Additional exponential models included a three-exponential plus intercept
model (7 parameters) [Compliance(Pressure) = y0 +
a ·
e-b·Pressure
+ c ·
e-d·Pressure
+ g ·
e-h·Pressure]
and a single-exponential with rational power model (3 parameters)
[
].
Data for these models are not shown.
Statistical analysis. The parameters of all models were examined for statistical significance; the criterion for goodness of fit for the model can be tested if P < 0.05 for any parameter estimation. The performance of those models with statistically significant parameters was then further characterized by calculating the coefficient of determination between compliance (the numerical derivative of the pressure-volume relation) and each model for the calf and forearm of each subject.
The Shapiro-Wilk test was then used to estimate whether the residuals for each subject and model follow a normal distribution (22, 23); a P value of <0.05 indicates a nonnormal distribution of the residuals, therefore a systematic error of the model under study. Friedman's test, a nonparametric (distribution-free) test, was used to compare the residuals between models for each subject. The breakpoint pressure between the calf and forearm was compared with a paired t-test; the coefficient of determination for the three models was compared with a paired t-test. The analyses of the effects of models and pressure and their interactions were investigated by using a three-way ANOVA for repeated measures; this ANOVA was implemented by using multiple-regression analysis. Data not normally distributed were logarithmically transformed. Data are reported as means ± SD unless otherwise indicated. All analyses were done by using R language (12) and Systat 10.0 (SPSS, Chicago, IL).
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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Characterization of models. The parameters of the models are shown in Tables 1, 2, 3, 4, 5, 6. The estimation of all parameters for the linear, biphasic, and single-exponential models was statistically significant. In most cases, other exponential models did not yield statistically significant estimation of the parameters, and some subject data could not be modeled (see Tables 4, 5, 6). The linear, biphasic, and single-exponential models were considered most representative of the data. Additional analyses performed on these models are reported below.
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Biphasic model. The biphasic model results for the calf and forearm are shown in Table 1. The low-pressure slope for the calf was -0.0029 ± 0.002 ml · dl-1 · mmHg-2 and the high-pressure slope was -0.0014 ± 0.0003 ml · dl-1 · mmHg-2 (P = 0.045). The low-pressure slope for the forearm was -0.0037 ± 0.004 ml · dl-1 · mmHg-2, and the high-pressure slope -0.0013 ± 0.0006 ml · dl-1 · mmHg-2 (P = 0.05). The low-pressure slopes in the calf and forearm and high-pressure slopes in the calf and forearm did not differ significantly. The breakpoint pressure between the low- and high-pressure phases was significantly higher in the calf than the forearm, 34.4 ± 3.9 vs. 29.1 ± 4.5 mmHg, P = 0.003.
The coefficient of determination for the calf biphasic model ranged from 0.86 to 0.98, r2 = 0.94 ± 0.04. The coefficient of determination for the forearm biphasic model ranged from 0.49 to 0.98, r2 = 0.83 ± 0.17. The residuals for the calf biphasic model, analyzed with the Shapiro-Wilk test, had a normal distribution in eight of nine subjects (Table 1, subjects 1, 2, 3, 4, 6, 7, 8, and 9). The residuals for the forearm biphasic model had a normal distribution in five of nine subjects (Table 1, subjects 1, 3, 6, 7, and 9).
Linear model. The linear model results are shown in Table 2. The slope for the calf was -0.0034 ± 0.0011 ml · dl-1 · mmHg-2. The slope for the forearm was -0.0030 ± 0.0015 ml · dl-1 · mmHg-2. There were no significant differences between the calf and forearm. The coefficient of determination for the calf ranged from 0.45 to 0.61, r2 = 0.54 ± 0.05. The coefficient of determination for the forearm ranged from 0.37 to 0.55, r2 = 0.51 ± 0.05. The residuals for both the calf and forearm linear models, analyzed with the Shapiro-Wilk test, were not normally distributed in any subjects, suggesting a systematic error in the model (Table 2).
Single-exponential model (2 parameters). The single-exponential model results are shown in Table 3, where a and b are the parameters of the exponential fit and where parameter b governs the decay of the exponential function. The parameter a of the exponential model was 0.1990 ± 0.1088 ml · dl-1 · mmHg-1 in the calf and 0.2199 ± 0.1593 ml · dl-1 · mmHg-1 in the forearm. The coefficient b of the exponential model was 0.0467 ± 0.0148 mmHg-1 in the calf and 0.0521 ± 0.0206 mmHg-1 in the forearm. There was no significant difference between the calf and the forearm. The coefficient of determination for calf ranged from 0.42 to 0.92, r2 = 0.81 ± 0.16. The coefficient of determination for the forearm ranged from 0.54 to 0.96, r2 = 0.79 ± 0.15. The residuals for the calf exponential model, analyzed with the Shapiro-Wilk test, were normally distributed in four of nine subjects (Table 3, subjects 2, 5, 7, and 9). The residuals for the forearm exponential model were normally distributed in four of nine subjects (Table 3, subjects 3, 5, 8, and 9).
Comparison of the models. The coefficient of determination for the biphasic model was greater than that of the linear and exponential models in the calf (biphasic model 0.94 ± 0.04, linear model 0.54 ± 0.05, and exponential model 0.81 ± 0.16; P < 0.001 between biphasic and linear models, P = 0.059 between biphasic and exponential models). Similar results were present in the forearm (biphasic model 0.83 ± 0.17, exponential model 0.79 ± 0.15, and linear model 0.51 ± 0.05; P < 0.001 between biphasic and linear models, P = 0.42 between biphasic and exponential models).
Figure 3 shows the residuals for the biphasic (Fig. 3A), linear (Fig. 3B), and exponential (Fig. 3C) models in a representative subject (subject 1, the calf). In the biphasic model, the distribution of residuals is uniform around zero, with slightly higher values at low pressures. The linear model underestimates compliance at low (below 25 mmHg) and high (above 45 mmHg) pressures, with a uniform distribution of residuals at midpressure ranges. The exponential model in the same subject shows similar results to the biphasic model.
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In the calf, the residuals of the biphasic model were significantly lower than the residuals of the linear model in six subjects (Table 7, subjects 4, 5, 6, 7, 8, and 9), and significantly lower than the residuals of the exponential model in three subjects (Table 7, subjects 7, 8, and 9). In the forearm, the residuals of the biphasic model were significantly lower than the linear model in all subjects (Table 7) and significantly lower than the residuals of the exponential model in two subjects (Table 7, subjects 1 and 5).
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The three-way ANOVA of the log-transformed residuals for the calf was significant for pressure (P < 0.001) and models (P < 0.001) and the interaction of models and pressure (P < 0.001, Fig. 4A). The same analysis for the forearm showed that the log-transformed residuals were not significantly different for pressure [P = not significant (NS)]. They were significantly different for models (P < 0.001) and the interaction of models and pressure (P < 0.001, Fig. 4B).
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The mean absolute values of the residuals derived from a multiple regression analysis for the ANOVA in the calf were as follows: 1) biphasic model, 0.0034 ml · dl-1 · mmHg-1 (P < 0.001); 2) linear model, 0.0128 ml · dl-1 · mmHg-1 (P < 0.001); and 3) exponential model, 0.0077 ml · dl-1 · mmHg-1 (P < 0.001). The mean absolute values of the residuals in the forearm were 1) biphasic model, 0.0089 ml · dl-1 · mmHg-1 (P < 0.001); 2) linear model, 0.0265 ml · dl-1 · mmHg-1 (P < 0.001); and 3) exponential model, 0.0080 ml · dl-1 · mmHg-1 (P = NS). The log transformed group mean ± SE residuals are displayed in Fig. 4. The residuals-vs.- pressure plot for the biphasic model in the calf (Fig. 4A) remains constant across a range of pressures. The linear model plot is constant for pressures of <35 mmHg, then grows exponentially (note that the plot appears quasi-linear because the residuals are log-transformed). The biphasic and exponential plot is constant across the range of pressures.
At low pressures in the calf (
35 mmHg, low-pressure phase) the
log-transformed residuals were lower for the biphasic model compared with
linear model, -5.839 ± 0.086 log(ml · dl-1
· mmHg-1) vs. -5.306 ± 0.099 log(ml
· dl-1 · mmHg-1),
P < 0.001; and the log-transformed residuals were lower for the
biphasic model compared with single-exponential model, -5.079 ± 0.079
log(ml · dl-1 ·
mmHg-1), P < 0.001 (see
Fig. 4A).
At high pressures in the calf (>35 mmHg, high-pressure phase) the log-transformed residuals were lower for the biphasic model compared with linear model, -5.574 ± 0.114 vs. -3.405 ± 0.067 log(ml · dl-1 · mmHg-1), P < 0.001; and the log-transformed residuals were similar for the biphasic model compared with the single-exponential model, -5.578 ± 0.075 log(ml · dl-1 · mmHg-1), P = NS (see Fig. 4A).
At low pressures in the forearm (30 mmHg, low-pressure phase), the
log-transformed residuals were similar for the biphasic model compared with
linear model, -5.616 ± 0.095 vs. -5.314 ± 0.113 log(ml ·
dl-1 · mmHg-1), P
= 0.098; and the log-transformed residuals were lower for the biphasic model
compared with the single-exponential model, -4.927 ± 0.083 log(ml
· dl-1 · mmHg-1),
P < 0.001 (see Fig.
4B).
At high pressures in the forearm (>30 mmHg, high-pressure phase), the log-transformed residuals were lower for the biphasic model compared with linear model, -5.542 ± 0.086 vs. -3.958 ± 0.090 log(ml · dl-1 · mmHg-1), P < 0.001, and the log-transformed residuals were similar for the biphasic model compared with the single-exponential model, -5.535 ± 0.072 log(ml · dl-1 · mmHg-1), P = NS (see Fig. 4B).
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DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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The biphasic model is congruent with the well-established, nonlinear features of venous compliance (20, 21). Specifically, the venous system has high compliance at low pressures and low compliance at high pressures. This physiology allows for large translocations of blood from the venous system in response to small changes in pressure to maintain cardiovascular homeostasis when central blood volume is compromised and central venous pressures are low. In contrast, the physiology minimizes venous pooling at high venous pressures, such as those attained when bipedal humans assume the upright position, thereby preserving central blood volume to maintain cerebral perfusion and orthostatic tolerance (see Fig. 1A) (20, 21). The biphasic model provides separate compliance data and slopes for the high- and low-pressure phases, which represent different physiological functions of the venous system, thereby allowing for individual comparisons between these two pressure regions in the upper and lower extremities in health and disease.
The results obtained from biphasic modeling of venous compliance illustrate differences between the operating ranges of the high- and low-pressure phases of the calf and forearm. Inherent to the biphasic model is the hypothesis that venous compliance is best described by two linear phases separated by a break-point. The breakpoint pressure, which separates the two phases of the model, is defined as the point where the pressure volume correlation (moving from high to low pressures) falls below the 99% of the maximum r2, denoting the end of the high-pressure phase and the onset of the low-pressure phase. Our data indicate that the low-pressure phase in the calf extends to higher pressures compared with the veins in the forearm, i.e., the forearm has a narrower operating range at low pressures.
This finding is consistent with the functional requirements of the dependent venous system because the calf veins support higher pressures than the forearm veins, particularly during standing. Furthermore, by extending the steep low-pressure phase of the compliance curve, the range of pressures over which hemodynamic regulation by the calf venous circulation occurs is extended. Although there is controversy as to the extent to which hemodynamic regulation occurs actively (owing to autonomic innervation of the venous system) or passively (owing to elastic recoil and calf muscle contraction) (21, 24), it is within this region of the compliance curve that large changes in volume occur with only small changes in pressure. The biphasic model thus demonstrates the manner in which the larger capacity, dependent venous circulation is adapted to maintain cardiovascular homeostasis. These data are consonant with the recent report of Halliwill and colleagues (9) that emphasized the relative hemodynamic importance of vascular pooling in the legs compared with the splanchnic and pelvic vascular beds. In contrast, the flat portion of the curve, i.e., the region in which gravity-induced venous pooling is resisted, begins at higher pressures in the lower than upper extremities. These data are consistent with the higher hydrostatic pressures promoted by gravity in the upright posture in the dependent venous system of the lower extremities.
The biphasic analysis did not reveal differences between venous compliance in the upper and lower extremities (although we cannot exclude a type 2 error). This finding is consistent with our previous report (8). Furthermore, others have observed increased venous compliance in the lower extremity (10, 13). We have previously noted that this observation is counterintuitive, because one might anticipate reduced venous compliance in the dependent venous system to minimize venous pooling in response to the increase in lower extremity hydrostatic pressures generated on assumption of the upright posture (8). The biphasic analysis allows this observation to be viewed from a different perspective, that of different operating ranges for the high- and low-pressure phases in the upper and lower extremity. The analysis suggests that the buffering function of the dependent venous system, i.e., the ability to translocate large amounts of blood from the lower extremities to the central vascular compartment in response to small changes in pressure and thereby increase cardiac filling and cardiac output, extends to higher pressures in the lower extremity than in the upper extremity. This characteristic of the dependent venous system promotes cardiovascular homeostasis in response to hemodynamic stress but may, by reducing the range in which gravity-induced venous pooling is resisted (the flat portion of the venous compliance curve), promote orthostatic intolerance. The shape of the compliance curve also highlights the functional differences between the upper and lower extremities. Although there was a significant difference between the slopes of the high- and low-pressure phases in the lower extremity, the slopes of the two phases in the upper extremity merely approached statistical significance.
Biphasic analysis may yield additional information on the structural components of veins and soft tissues that play a role in compliance. The occlusion plethysmographic technique, although assessing the limb volume change in response to venous filling, measures whole limb compliance and does not discriminate between the venous and soft tissue elements involved in limb compliance. The inability to isolate the specific components involved in compliance may be viewed as a limitation of the strain-gauge plethysmography technique; however, because the veins and supporting soft tissues are both involved in limb compliance [indeed compliance may be altered significantly by changes in the supporting tissues (5, 6, 14)], the whole limb occlusion plethysmographic technique provides an appropriate functional measure of compliance.
Reports of arterial compliance have revealed that elastin and smooth muscle
determine the compliance at low pressures whereas the stiffer collagen fibers,
which limit vascular distension, determine compliance at higher pressures
(1-3,
7). Structurally, veins have
thinner walls that have proportionally more collagen and less elastin and
smooth muscle than arteries
(16,
20). For example, the aortic
wall is composed of 48% elastin, 20% smooth muscle, and 30% collagen; in
contrast, the wall of large veins is composed of 12% elastin, 25% smooth
muscle, and 60% collagen (16,
20). Although the greater
proportion of elastin in arteries leads to a higher breakpoint (the breakpoint
in arteries is 90 mmHg compared with 30 mmHg in this study), the
relatively higher proportion of collagen provides a fibrous network that
resists high distending pressures so that veins at higher pressures are
stiffer than arteries at the same pressure. One might speculate that the
supporting soft tissue complements the role played by collagen, limiting
vascular distension. Although venous compliance is 30 times greater than
arterial compliance (18), the
biphasic model proposed in this work suggests that a structure-function
relation, similar to that reported in arteries
(1-3),
exists in veins.
An important limitation of the occlusion plethysmographic method is that there is no direct measure of venous pressure. It is thus essential to elevate the limb under study to ensure adequate limb emptying. We and others have observed that data below 10 mmHg are not reliable (10). This may in part be related to inadequate limb emptying and the presence of un-stressed intravascular volume. An additional limitation to this method is that a steady state is not attained during cuff deflation. The time-dependent aspects of venous compliance therefore may not be incorporated into the assessment of venous compliance at individual pressures. The importance of this limitation is, to an extent, diminished in the biphasic model because compliance is modeled over a range of pressures, thereby incorporating some time-dependent changes in the model.
Although the single-exponential model describes the data adequately, more complex exponential models did not add further precision. Nevertheless, it is possible that a more complex, multiexponent exponential model may describe the pressure-volume relationship even more precisely. However, the relative simplicity of the biphasic model, its concordance with the known physiological functions of the venous system, and the possible structure-function relationship all lend support to the value of this model.
In summary, we have shown that a biphasic model can describe limb venous compliance and delineate differences in venous physiology at high and low pressures. This model, compared with a linear and single-exponential model, most consistently maintains accuracy across the entire range of pressures. The slopes of the high- and low-pressure phases are significantly different in the calf, and the breakpoint between the high- and low-pressure phases is higher in the calf than the forearm. The findings based on this analysis are consistent with the physiological role of the dependent venous system.
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DISCLOSURES |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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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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REFERENCES |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONDISCLOSURESREFERENCES |
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