Annals of Biomedical Engineering, Vol. 22, pp. 392-397, 1994 0090-6964/94 $10.50 + .00 Printed in the USA. All rights reserved. Copyright 9 1994 Biomedical Engineering Society
Simple and Accurate Way for Estimating Total and Segmental Arterial Compliance: The Pulse Pressure Method
N. STERGIOPULOS,* J . - J . MEISTER,* and N . WESTERHOF-~
*Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, Ecublens, Switzerland and tLaboratory for Physiology, Institute for Cardiovascular Research, ICaR-VU, Free University, Amsterdam, The Netherlands
Abstract - -We derived and tested a new, simple, and accurate method to estimate the compliance of the entire arterial tree and parts thereof. The method requires the measurements of pressure and flow and is based on fitting the pulse pressure (systolic minus diastolic pressure) predicted by the two-element windkes- sel model to the measured pulse pressure. We show that the two-element windkessel model accurately describes the modulus of the input impedance at low harmonics (0-4th) of the heart rate so that the gross features of the arterial pressure wave, including pulse pressure, are accounted for. The method was tested using a distributed nonlinear model of the human systemic arterial tree. Pressure and flow were calculated in the ascending aorta, tho- racic aorta, common carotid, and iliac artery. In a linear version of the systemic model the estimated compliance was within 1% of the compliance at the first three locations. In the iliac artery an error of 7% was found. In a nonlinear version, we compared the estimates of compliance with the average compliance over the cardiac cycle and the compliance at the mean working pressure. At the first three locations we found the estimated and "actual" compliance to be within 12% of each other. In the iliac artery the error was larger. We also investigated an increase and decrease in heart rate, a decrease in wall elasticity and exercise condi- tions. In all cases the estimated total arterial compliance was within 10% of mean compliance. Thus, the errors result mainly from the nonlinearity of the arterial system. Segmental compli- ance can be obtained by subtraction of compliance determined at two locations.
Keywords--Windkessel model, Computer simulation, Nonlin- ear arterial system, Input impedance.
Peripheral resistance and total arterial compliance form the major part of the arterial load on the heart (1). Periph- eral resistance is commonly derived from the ratio of mean aortic pressure and flow. To obtain total arterial compli- ance many methods have been suggested, none of which has been universally accepted (4,5,6,7,9,10,13,14,15,
Acknowledgment--This work is supported by the Swiss National Sci- ence Foundation (grant number 21-32559.91).
Address correspondence to N. Stergiopulos, Ph.D., Biomedical En- gineering Laboratory, Swiss Federal Institute of Technology, Champs- Courbes 1, 1024 Ecublens, Switzerland.
(Received 260ct93, Revised 23Feb94, Revised t9Apr94, Accepted 19Apr94 )
16,17). The most well-known method is the diastolic de- cay method, which, on the basis of the two-element wind- kessel (2), fits a monoexponential curve to the time decay of the diastolic pressure (9,10,15). To avoid problems that arise when aortic pressure does not decay monoexponen- tially, the area method was suggested (7). Others have proposed taking full advantage of the information obtained in the entire waveforms of pressure and flow, and have, on this basis, suggested a parameter estimation technique us- ing the three element windkessel model (13,14).
Although the diastolic decay time and area methods are simple to use, they can only be applied when flow in diastole is zero, limiting them to the ascending aorta only. The parameter estimation technique based on the three element windkessel model is applicable to all locations where the distal tree can be modeled with a three element windkessel. However, input impedances of the femoral and carotid beds deviate from those of the three element windkessel model (8). This finding limits the use of these methods to proximal vessels.
In this study, a new method to derive arterial compli- ance distal to the site of a pressure and flow measurement is proposed that is based on the (linear) two element wind- kessel model. This simple method can be applied at any location of the systemic arterial tree and yields acceptable results when applied to an arterial system model with non- linear properties.
To test the proposed method, we used an extensive computer simulation of the human systemic circulation that provides an absolute reference point for the compli- ance estimates. The systemic arterial tree is modeled by fifty-five arterial segments accounting for all major arter- ies (11). The mathematical model is based on the one- dimensional ( I-D) flow equations and accounts for the nonlinearities due to convective effects. The arterial wall can be modeled as a linear or nonlinear elastic material. All terminal segments are ended with a three-element windkessel model to avoid unrealistic reflections. A de-
Evaluation of Compliance Estimation Methods 393
tailed description of the computer model, the governing equations, the physiologic parameters defining the geo- metric and elastic properties, as well as the numerical solution scheme, can be found in (11).
In the original computer model (11), the area-pressure relationship was taken to be quadratic, which in turn yielded a linear dependence of the area compliance on pressure. Several in vivo (12) and in vitro (3) studies, however, have shown that area compliance exhibits a strong nonlinear dependence on pressure, falling sharply from low- to mid-pressure values and tending to an as- ymptotic value toward high pressure values. For the pur- pose of the current model, where the nonlinear depen- dence of compliance on pressure is of primary interest, the area-pressure relationship, assumed to apply to all arterial segments, was given by
Ca=Ca,ref a+b PP-u " . (1)
The parameters were chosen as follows: C,,re f is the area compliance at the reference pressure of 100 mm Hg, Pa equals 20 mm Hg, Pb equals 30 mm Hg, and a and b are empirical constants taken as 0.4 and 5, respectively. Equation 1 is a modified form of the compliance-pressure relationship proposed by Langewouters et al. (3). As men- tioned in (3) the empirical constants do not have a partic- ular physical meaning, and are chosen here to fit well typical compliance-pressure curves observed in vivo and in vitro (3,12). A graphic representation of the above re- lationship is shown in Fig. 1.
The input to the computer simulation can be either a flow or a pressure pulse at the ascending aorta level. We have chosen flow as the input. The flow waveform used for the control case is depicted in Fig. 2. It represents a typical flow pulse from a healthy individual at rest and has
/ / 0.0 I I I I I I I I 60 90 120 150 180 210
Pressure [mm Hg]
FIGURE 1. Compliance as a function of pressure. Compliance is normalized to its value at 100 mm Hg.
120 (~ Fl~w [ \ -. - .... Pressure (distributed model I ~,," :,,~ " ' , - - - Pressure (2-element WK) 400
9 1" 9 9
I:: E 100 ~ "%, " . . Z o
' - ' i \ ' " " : # "li,, - - 9 , ..~.
80 ,: ,, "-:.-.=. ..' m e - , ,~ 0
0.5 Time [s]
FIGURE 2. Pressure and f low in the ascending aorta in the nonlinear distributed model of the human systemic arterial tree. Using aortic f low as the input to the two-e lement wind- kessel model with the known peripheral resistance and total arterial compliance, the dashed curve is found. The aortic pulse pressure (distributed model) and the one calculated us- ing the two-e lement windkessel are similar.
a period of 1 sec, peak systolic flow of 500 ml/sec, and mean flow of 88 ml/sec. Given this input flow, the com- puter simulation yields pressure and flow values at all locations of the systemic arterial tree. The resulting pres- sure pulse at the root of the aorta is shown in Fig. 2.
The simplest possible model of an arterial tree is the two-element windkessel model, originally proposed by Otto Frank in 1889 (2). In this model the whole arterial tree is modeled as an elastic chamber (windkessel) of con- stant compliance, C, and a peripheral resistance, R. The compliance is C = dV/dP, where V is the systemic arterial blood volume. The governing equation of the two-element windkessel is
d P(t) C -s P(t) + y = F(t) , (2)
where F denotes flow. In the frequency domain, the cor- responding input impedance is given by
R Zi. - (3)
1 + jcoRC "
The question posed was, how well does a two-element windkessel simulate the total arterial tree? A direct com- parison can be made by taking the aortic flow as the input to the two-element windkessel model, using the total R and mean C values of the nonlinear version of the systemic arterial model and comparing the computed and "actual" (as given by the nonlinear model) pressure waveforms. This comparison is shown in Fig. 2. We notice that all high frequency components of the pressure pulse are poorly represented by the two-element windkessel. More- over, the pressure in the windkessel shows a significant time delay. However, the pulse pressure predicted by the
394 N. STERGIOPULOS, J.-J. MEISTER, and N. WESTERHOF
two-element windkessel is very well matched in amplitude to the real pulse pressure.
To explain why the two-element windkessel model mimics the gross features of the arterial system so well, we have plotted the input impedance of the total arterial tree together with the input impedance of the two-element windkessel. Compliance and peripheral resistance were chosen equal to their arterial counterparts at mean pres- sure. The comparison between the moduli and phase an- gles is shown in Fig. 3, parts a and b, respectively. We notice that, up to about 5 Hz, the modulus of the input impedance of the arterial tree and that of the two-element windkessel are essentially identical. This quite striking
7_ E 0.02 n
I I I I
2 4 6 8 10 Frequency [Hz]
90 I i = I
(D .~. 0 (D
-90 0 2 4 6 8 0
FIGURE 3. Input impedance of the distributed model (solid line) and the two-element windkessel model (dashed line) when making use of the peripheral resistance and mean total arterial compliance of the systemic tree model. The moduli of the impedance, which determines the gross features of the pressure, are similar at low harmonics. The difference in phase angles mainly results in a phase shift in the pressure waveforms.
similarity has a very important consequence: if the gross features of the arterial pressure and flow pulse are con- tained within this frequency range, the magnitude of the pulses in the real system and the two-element windkessel will be essentially the same. Figure 3b shows that the phase angles of the input impedance of the two-element windkessel model differ from those of the actual input impedance. This explains the phase lag between the two pressure pulses. It also explains why fitting the two- element windkessel model to the entire arterial pulse would not yield satisfactory estimates.
The similarity of the moduli of the impedances suggests that the two-element windkessel models the low frequency characteristics of the arterial system correctly and can pre- dict the pulse pressure well for the right choice of com- pliance, On the basis of these observations we propose the following method to estimate compliance: given the pres- sure and flow at a certain point in the arterial tree, the ratio of their mean values gives peripheral resistance, and one can apply the two-element windkessel model to fit the predicted pressure pulse to the actual pulse by adjusting compliance. Compliance adjustment is done by a simple "trial and error" type of approach knowing, however, that lower compliance yields larger pulse pressures. Based on that, a few iterations always sufficed. The value of the compliance, C, that gives the closest fit of the pulse pres- sure is the best estimate of the compliance. The method is shown schematically in Fig. 4.
The method is tested as follows. Using the computer simulation of the entire human systemic arterial tree, the pressure and flow in the ascending aorta, descending tho- racic aorta and iliac and common carotid arteries are de- termined. The new method to derive compliance is applied at all these locations, both for the linearized version of the
Measured f low 2 -e le rnent WK Computed pressure
Measured pressure Ad jus t C . . . . .
FIGURE 4. Schematic diagram of the compliance estimation method. Simultaneous recordings of flow and pressure at a certain arterial location are required. Mean pressure over mean flow yields the value of peripheral resistance, R. PPa is the measured pulse pressure and PPd is the pulse pressure predicted by the two-element windkessel model using the known peripheral resistance value and an assumed compli- ance value, C. The iterative scheme looks for the value of C that yields the closest agreement between the two pulse pressures, This is the best estimate of the arterial compliance.
Evaluation of Compliance Estimation Methods 395
systemic simulation (constant compliance) and for the case where the wall material is nonlinear (compliance varying as in Fig. 1). Subsequently we have changed the heart rate and arterial system and determined total arterial compliance with the nonlinear version of the systemic ar- terial tree. The heart rate was both increased and de- creased (to 1.67 and 0.5 of control). To mimic systolic hypertension, compliance was decreased by a factor of two and resistance was increased by 30%. To mimic mild exercise conditions, local peripheral resistances were changed (muscle resistance decreased and intestinal resis- tances increased slightly) while heart rate was increased by 67%. In all of the above situations, total arterial com- pliance was determined and compared with the actual compliance of the model. In the linear version of the sys- temic arterial tree, the actual compliance is simply the sum of all (constant) compliances. In the nonlinear version, we have compared the derived compliance with (a) the aver- age compliance over the cardiac cycle, and (b) the com- pliance at mean pressure. Mean pressure is essentially constant throughout the arterial tree with only a small drop in the peripheral arteries which, in any case, have a min- imal contribution to the total compliance. Thus, we have calculated the volume compliance of each arterial segment at the local mean pressure and subsequently added all volume compliances to come up with the total arterial compliance at mean pressure.
Pressure and flow in the ascending aorta of the distrib- uted nonlinear model of the human systemic arterial tree are given in Fig. 2. Pressures in the thoracic aorta and iliac and common carotid arteries are given in Fig. 5. The magnitudes and wave shapes of the pressure and flow compare well with those found in the human arterial tree, but the wave shapes of pressure differ strongly from one
140 Common carotid
A , - . . . . . . . . Thoracic aorta l \ [ ', . . . . . Iliac
-~ I00 ~. "..-..,
0 0.5 1 Time Is]
FIGURE 5, Pressure waveforms in the thoracic aorta, iliac ar- tery, and common carotid artery, as given by the computer model of the human systemic arterial tree,
another. The estimates of arterial compliance at these lo- cations are given in Tables 1A and lB. In Table 1A, the estimates for a linear arterial tree (constant compliance) are given. We see that the estimates are very good. Errors are less than 1% except for the iliac artery where an error of 7% was found. When the arterial system is nonlinear, compliance depends on pressure. The pressure depen- dence of local compliance is known and we calculated both the compliance at mean pressure as well as the av- erage compliance over the cardiac cycle. The compliance derived on the basis of the pulse pressure method was compared with both mean compliance and the compliance at mean pressure (Table 1B). In general, the estimates were good. In the iliac artery the error was much larger than that found in the linear arterial tree. In Table 2 it is shown how estimated and derived total arterial compli- ances compare for changes in the nonlinear arterial system and heart. Again, the comparison is good. For systolic hypertension where pulse pressure is high, agreement with the compliance at mean pressure is poor (18% error), but the agreement with the mean compliance is acceptable (Table 2).
From the compliance determined at different locations, segmental compliance can be obtained by subtraction. Es- timated compliance in the thoracic aorta (Table 1B) when subtracted from the total arterial compliance yields the compliance of the upper part of the arterial tree of 12.25- 3.95 = 8.30 ml kPa- ]. This value compares well with the value of 12.38 - 4.05 = 8.33 ml kPa -1 for the actual compliance at mean pressure for this arterial section.
It should be noted that the present method is totally different from the classic PP/SV (pulse pressure over stroke volume) method. The PP/SV method is an empir- ical method that does not use the two-element windkessel model as the basic theoretical model for the arterial tree. The PP/SV method was tested using the data from the present study and, for all cases, it consistently overesti- mated the total arterial compliance by approximately 50%.
We proposed and tested a new simple technique to derive total and segmental arterial compliance from pres- sure and flow measurements. We tested the technique on the basis of an extensive distributed linear and nonlinear model of the human systemic arterial tree.
We used this distributed model for the tests because at present this is the only accurate way to perform quantita- tive comparisons. In the real human arterial tree, total arterial compliance cannot be obtained readily because determination of local compliances at all locations is not feasible and a pressure-volume relationship for the whole systemic tree is nearly impossible to obtain.
Errors in the estimation of compliance may arise from nonlinearities in the arterial system, or when the imped-
396 N. STERGIOPULOS, J.-J. MEISTER, and N. WESTERHOF
TABLE 1. Comparison of actual and derived compliances at different locations.
A. Linear arterial system. Ascending Aorta Thoracic Aorta lilac Artery Common Carotid
Pm (ram Hg) 92 90 90 92 Ps/Pd (m m Hg) 112/70 114/68 128/65 128/67 Cderive d (ml kPa -1) 11.04 3.65 0.61 0.11 Cactual (ml kPa- 1) 11.04 3.65 0.66 0.11 Accuracy (%) 0.0 0.0 -7.0 O.0
B. Nonlinear arterial system. Ascending Aorta Thoracic Aorta lilac Artery Common Carotid
Pm (mm Hg) 92 91 90 92 Ps/Pd (mm Hg) 112/74 112/74 129/70 132/73 Cderive d (ml kPa -1) 12.25 3.95 0.63 0.11 C@prn (ml kPa-1) a 12.38 4.05 0.74 0,12 Accuracy (%) -0 .3 -2 .5 - 16.0 -8 ,0 Cmean (ml kPa 1)b 12.65 4.15 0.78 0,12 Accuracy (%) -3 .2 -4 .8 - 19.0 - 11.5
aAt mean pressure, pro, bMean compliance over the heart cycle. Ps is systolic and Pd is diastolic pressure.
ance of the two-element windkessel model at low frequen- cies is not exactly equal to that of the real arterial bed under study. In this case the method falls short. The error in the estimated compliance in the iliac artery is still con- siderable, even in the linear system. This error is due to the fact that the actual input impedance deviates from the two-element windkessel already at the 3rd harmonic. This type of error is found in peripheral beds where the two- element windkessel falls short, especially at high rates.
To test the role of nonlinearities, we applied the method to a linearized version of the systemic arterial tree. We found much better estimates in the linear model than in the nonlinear version (compare Tables 1A and 1B). Thus, nonlinearities clearly contribute considerably to errors. We explain this on the basis of the results of Fig. 6 where the input impedance, calculated from the pressure and flow waves in the ascending aorta, is given for heart rates of 1 and 0.5 Hz. It may be seen that, in the linear system, the input impedance at both heart rates is exactly the same, while in the nonlinear system input impedances are not the same. At different heart rates the pulse pressure is differ-
ent, and therefore the mean compliance of the arterial system is somewhat different, even though mean pressure is the same. Thus, in the linear model, the unaffected input impedance would yield, for both heart rates, identi- cal and accurate estimates of the compliance. This is not the case in the nonlinear model.
The estimated compliance was compared with the com- pliance at mean pressure and to mean compliance over the heart cycle. If we calculate compliance at systolic and diastolic pressure for the simulation of systolic hy- pertension, we obtain values of 4.00 and 8.20 ml kPa -1, respectively. We see that the variations in com- pliance are large with respect to the value at mean pres- sure (5.25 ml kPa- l ) . It is clear that for a nonlinear compliance-pressure relationship the mean compliance over the heart beat is not equal to compliance at mean pressure.
We conclude that the pulse pressure method based on the two-element windkessel model yields good estimates of the mean arterial compliance, even in a nonlinear arte- rial system. The major advantage of this novel method is
TABLE 2. Comparison of actual and derived total arterial compliance: nonlinear arterial system.
Control High HR Low HR Decreased C Exercise
P= (mm Hg) 92 91 91 106 100 Ps/P d (mm Hg) 112/74 104/83 132/61 145/72 117/85 F m (ml/s) 87.6 87.6 87.6 78.9 125.5 Cderived (ml kPa -1) 12.25 11.50 13.00 6.20 10.50 C@p m (ml kPa- 1)a 12.38 12.21 12.43 5.25 11.21 Accuracy (%) - 0.3 - 5.7 4.6 17.4 - 6.0 Cmean (ml kPa-1) b 12.65 12.36 13.80 5.71 11.42 Accuracy (%) - 3.2 - 7.0 - 5.8 - 8.5 - 8.1
aAt mean pressure, Pro" bMean compliance over the heart cycle. Ps is systolic and Pd is diastolic pressure.
Evaluation of Compliance Estimation Methods 397
(r "1 "5 "0 0
I I I I
I t |
I I I I I I I
i t #
~o 9 .~- ~-Er ~,.  o~'o
I I I I
2 3 4 5 Frequency [Hz]
6 7 8
I I I I I I ! I I
I I I I I I I
t t ~]_
I J I I I I I
1 2 3 4 5 6 7 Frequency [Hz]
FIGURE 6. (a) Input impedance of the nonlinear model of the human systemic arterial tree. (b) input impedance of the linear version. Input impedances were determined at the heart rate of 1.0 (open circles) and 0.5 (open squares) Hz. In the linear model, impedance is the same at both heart rates, but this is not so in the nonlinear model.
that it can apply, equally well, in the aorta and in other locations of the arterial tree.
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a ,b =
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Ca,re f =
F F m = p =
Pa,Pb = Pd Pm Ps R t V z~. OJ
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empirical constants total volume compliance of systemic arterial tree area compliance area compliance at reference pressure of 100 mm Hg flow mean flow over the heart cycle pressure pressure constants
= diastolic pressure = mean pressure over the heart cycle = systolic pressure = peripheral resistance = time = systemic arterial blood volume = input impedance = basic angular frequency of the heart cycle