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Effects of friction and nonlinearities on the separation of arterial waves into their forward and backward components
VW
Pergamon
PII: SOO21-9290(%)00071-l
J. Biomechmics, Vol. 29, No. 11, pp. 141’%1423, 1996 Cowriaht . . _ 0 1996 Elwier Science Ltd. All riahts reserved Printed in &eat Britain 0021-9290/96 $15.00 + .OO
EFFECTS OF FRICTION AND NONLINEARITIES ON THE SEPARATION OF ARTERIAL WAVES INTO THEIR FORWARD AND BACKWARD COMPONENTS F. Pythoud,* N. Stergiopulos,* C. D. Bertramt and J.-J. Meister* *Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, Lausanne, Switzerland 1015; and tCentre of Biomedical Engineering, University of New South Wales, Kensington, Sydney, Australia 2033 Abstract-In this paper we examine the importance of fluid friction and nonlinearities due to the area-pressure relationship and to the convective acceleration on the separation of arterial pressure and flow waves into their forward and backward components. Experiments were run in straight uniform nonlinearly elastic tubes. Different degrees of fluid friction and nonlinearities, covering the physiological range, have been tested. We predicted the forward and backward running pressure components using two wave separation methods: the classical linear method (Westerhof et al., Cardiouasc. Res. 6,648656,1972) and the first order correction (FOC) method (Pythoud et al., Trans ASME J. Biomech. Engng, in press) which takes nonlinearities and fluid friction into account. We found that the two methods yield somewhat different predictions. The differences tend to increase with the degree of fluid friction and nonlinearities and are typically of the order of 4-8%. We further compared the transmission ratio of forward and backward waves predicted by both methods. The transmission ratio was found to be overestimated by 10% by the classical linear method. The nonlinear method gave more accurate estimates, consistent with theory. We conclude that, for in oioo applications, the classicallinear method should be the method of choice becauseit is simpler to use and the errors involved (4-8%) are comparable to measurement errors. Copy right 0 1996 Elsevier Science Ltd. Keywords:
Separation; Attenuation; Transmission; Nonlinear; Wall friction. NOTATION
pi p: %X
Q Rt,R, u ZC Greek F
P
wave speed wave speed at mean pressure P,,, shear stress inertia coefficient: c, = 1 + Im[iGlo(a)] shear stress viscous coefficient: c, = Re[a’iG,,(a)/8] fluid friction term: f = 8xpc,/(pA,c,) cross-sectional area cross-sectional area at mean pressure P, compliance compliance at mean pressure P, monotonic function used to calculate the Riemann invariants - 2J,(ai3/‘)) (see Womersley) 2J,(cri3/2)/(cci 3’2 J,(ai3’*) pressure mean pressure forward pressure wave component forward pressure wave component calculated taking nonlinearities into account forward pressure wave component calculated taking fluid friction into account forward pressure wave component calculated by the classical linear method forward pressure wave component calculated by FOC method flow Riemann invariants velocity averaged over the cross-section U = Q/A characteristic impedance letters
dynamic viscosity specific weight
in final form 20 March 1996. Address correspondence to: N. Stergiopulos Ph.D., Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland. Received
Y K cp co
elastic nonlinearities parameter convective acceleration parameter fluid friction parameter frequency INTRODUCTION
Separation of arterial pressure and flow waves into their forward and backward running components is generally used to quantify reflections in the arterial system as well as to analyze heart-arterial system interaction (Murgo et al., 1981; Van den Bos et al., 1982). It may also be used to determine the reflection and transmission properties of areas of special physiological interest or of pathological section of the arterial tree such as stenoses, aneurysms, bifurcations, etc. (Stergiopulos et al., 1996). The first and most popular separation method was proposed by Westerhof et al. (1972). The method assumes linear wave propagation theory, and requires pressure and flow measurements at the same arterial location, as well as an estimate of the local characteristic impedance from these measurements. However, the one-dimensional flow equations, describing the propogation of arterial pressure and flow waves, are nonlinear and dissipative. Convective acceleration as well as nonlinearity of the area-pressure relationship of the artery contributes to nonlinearity (Langewouters et al., 1984,198s; Tardy et al., 1991) which was neglected in most of the studies on wave transmission (Bertram and Greenwald, 1992; Busse et al., 1979; Cox, 1971; Milnor and Nichols, 1975). Fluid friction, due to the blood viscosity, and wall friction, due to the viscoelastic properties of the wall, introduce dissipation into the system.
1419
1420
F. Pythoud et al.
A general wave separation theory, taking into account nonlinearity as well as fluid friction, has been recently developed and applied on computer simulation data (Pythoud et al., in press). In the present study we will compare the predictions of this general wave separation theory with the classical linear theory using data from in vitro experiments. The experiments were performed on straight tubes in conditions that permit important elastic nonlinearities and fluid friction to be present. The goal is to assess the influence of fluid friction and elastic nonlinearities on the prediction of the forward and backward running waves. METHODS
Experimental
setup
The experimentalsetupis shownin Fig. 1. A Harvard Apparatus” pump generatedphysiologicalflow waves. The working fluid was water for low viscosity experimentsand a glycerine/water mixture for high viscosity experiments(p = 0.023Pas). PVC tubesconducted the fluid from the pump to the elastictube simulatingthe artery via a small Windkesselor air chamberusedto attenuate high-frequencyoscillationsproduced by the pump. The elastictube was made out of a Sylgard 184 (Dow Corning)elastomerwhich wasshownto reproduce the nonlinear elastic properties of conduit arteries (Pythoud et al., 1994).The tube had an unstretched diameterof 8 mm, wall thickness0.105mm, and an unstretchedlength of 300mm. It was stretchedlongitudinally by approximately 50% betweenPlexiglas(perspex) connectionsto avoid buckling at high transmuralpressures.The tube wassubmersed in a bath containing the samefluid as the circulating one, primarily to compensate gravitational effects,while allowing for good flow rate and diametermeasurements usingultrasonic devices.The distal load of the tube consistedof a Windkessel (air reservoir)and a resistorwhich wasmadeof a pieceof tube filled up with smallspheresof lead.The linearity or not of the resistor does not affect the results. Mean pressurewithin the elastictubewascontrolled by a distal fluid column of adjustableheight. Pressureand flow rate were measuredat two sites alongthe tube 310mm apart. Pressurewasmeasuredby two catheter-tip pressuretransducers(Millar@ 2.5F)
introduced into the tube from the proximal and distal ends,respectively.The flow rate wasmeasuredby transittime ultrasonicdevices(Triton@Technology,SanDiego) of sufficientinsidediameter(14 and 17mm) so that the tube at maximumdistensiondid not touch the surrounding cuff. The pressurecatheterswereplacedabout 12mm from the center of the flow probes.Both pressureand flow signalswereamplifiedand sampledat 500 samples Hz by an analog-to-digitalacquisitionboard (National InstrumentsaLab-NB). They werestoredin a Macintosh II-Ci. The internal diameterof the tube, and thereforeits cross-sectional area,wasmeasuredseparatelybeforeand after experimentsby an echo-trackingultrasonic device (Nius@,OmegaElectronics,Switzerland)usinga 4 MHz ultrasonic probe. In each experiment, pressuretransducerswerecalibratedagainsta static fluid columnin the absenceof flow, and flow meterswerecalibratedagainst constantflow rates.Linear regressionof thesedata lead to the calibration factors and showeda potential error of 0.5% on pressuremeasurements, and a potential error of 3-4% on flow rate measurements. We assumedstatic calibration curvesto be valid with the sameaccuracy in pulsatileflow conditions.Diameter measurement errors, given by the manufacturer,are 0.1%. Weperformedfour experimentsat a room temperature of 27°C. In the first one, the working fluid was water (p = 0.00086Pas, p = 1000kg m- 3, to reducethe frictional effects.The pump rate wasset to 0.7 Hz and the pump stroke volume to 6ml. With this stroke volume, the pressurerangedfrom 14 to 19kPa. The area-pressure relationship,despitebeing not strictly linear, was approximatedaslinear over this limited pressurerange. In the secondexperimentthe strokevolume of the pump wasincreasedto 20 ml to extend the pressurerangeand thusto obtain asmucharea-pressure nonlinearity (differencein the area-pressure curve slopeat the two extremes of the pressurerange)as possiblewithout risking tube rupture. The third and fourth experimentswere as the first two except that fluid viscosity was increasedto about four times blood viscosity (to amplify viscous losses)thereforeincreasingdissipationand wave attenuation. This wasobtainedby usinga mixture of glycerine and water (volume ratio 7:3) giving a viscosity of p = 0.023Pas and a density of p = 1192kgme3. Data analysis
The digitizedflow rate data werefirst correctedfor the phase-shifteffectsintroduced by the output filter in the flow meters,that consistsof two cascaded2-polefilters of breakpoint 100Hz and Q-factor 0.7. Thesefilesproduced a time shift of the order of 5 msfor frequenciesranging between1 and 100Hz. The other phaseshift of multiplexed analogto digital conversionwassmaller(0.5ms) and thus considerednegligible.Furthermore, pressure and flow rate signalswere not measuredexactly at the samelocation, and the phaseshift resulting from the distanceof about 12mm betweenthe two transducers Fig. 1. Schematic diagramof theexperimental setup.Twocatheter tip transducers measure the pressures PI and Pz. Theflow was also corrected. These steps have been taken to rate QI and Q, are given by two transit-time ultrasonic devices guaranteean accuracyof the order of 3% on the forward placed around the elastic tube. and backward running pressurecomponents.
1421
Effects of friction and nonlinearities on the separation of arterial waves Linear forward and backward running pressure wave components were obtained from a classical linear separation method (Westerhof et al., 1972) which neglects both friction and nonlinearities:
P: = fP + ZcQX
(1)
PP = t(f' - GQ),
(2)
where P is the pressure, Q the flow, and Z, = ,,/m) the characteristicimpedance.p is the fluid density,A, and C, are cross-sectional area and complianceat meanpressure. Resultsof the linear analysiswerethen comparedwith thoseof nonlinear analysisdescribedby Pythoud et al. (in press).The first-order correction (FOC) separation method is obtained as a combination of two methods: a first one taking nonlinearitiesonly into account and secondone taking dissipation(fluid friction) only into account. The first solution is obtained by defining a smooth monotonic function F(P) and the Riemann . . mvanants RI Rz as F(P) =
s
p dP
(3)
RExJLm
Figure 2 showsthe area-pressurerelationshipmeasured, first in small pressureamplitude experiment (14 to 19kPa, compliancerange 3.8x 10e9m2Pa-‘-2.6 x 10e9mzPa-‘), and then duringlargepressureamplitude experiment (P varying from 11 to 21 kPa, compliance range4.6 x 10m9m* Pa-‘-2.1 x 10e9mzPa-‘). The areapressurerelationshipwasapproximatedby a second-order polynomial.Typical pressureand flow rate measurementsat the entrance(fully drawn) and exit (dashed)are shownin Fig. 3. Figure 4 showsthe forward waves obtained at the proximal location after separationin forward and backward components.The fully drawn linesare the waves derivedfrom the linear, classical,theory and the dashed linesare the wavesobtainedusingthe nonlineartheory. As seenin Fig. 4, the forward running pressurecomponentspredictedby the two methodsare very similar in shapebut differ significantlyin amplitude.This difference increases with friction (increasedfluid viscosity,compare Fig. 4(a)with (c) and (b) with (d) as well as with nonlinearity (large pressurerange, compareFig. 4(a) with (b)
P,pco'
RIO’, U) = (F(P) + W/Z
(4)
R,(P, U) = (F(P) - W2,
(5)
1.2
wherePO is the diastolicpressure,c = ,/m the wave speed,and U = Q/A blood velocity averagedover the cross-section.The forward and backward running pressurecomponents,PiI and P,“1, are found by solving the following equations: W:,)
= RI
(6)
F(Pj,)
= R2.
(7)
The solutionto the dissipationproblemis obtainedby defininga frequency-dependent characteristicimpedance -
-
l-
I 0.8
“'I IO
I 14
I
I 18
Pressure[kPa]
I
I 22
Fig.2. Thearea-pressure relationship at proximallocationduring smallandlargepressure amplitudeexperiments. with f = 8npz,/(pA,c,). The viscousand inertia coefficientsof shearstress,c, and c,, are definedas c, = 1 + ImCiG&)l, c, = Re[u2iGio(a)/8]
(9) (10)
and dependon the Womersleyparametertl (Womersley, 1957;Stergiopuloset al., 1992).To accountfor dissipation effects, forward and backward running pressure components,Pi and Pi, are calculatedin the frequency domainaccordingto equations(1) and (2).FOC method combinesthe two stepsgiven above: Got = Pi, + Pi - P:.
(11)
Similar expressionis obtained for backward pressure components.All of thesemethodsrequire simultaneous Fig.3. Proximal(continuous line)anddistal(dashed line)presmeasurements of pressureand flow rate at the same sureand flow rate signalsrecordedduringa largepressure location aswell asknowledgeof the local area-pressure amplitude experiment in whichthecirculatingfluidwasaglycerinemixture. relationship.
F. Pythoud et al.
1422
a)
b)
18
0
1
Time [s]
Time [s]
1
Time [s]
1
Fig. 4. Comparison between forward running proximal pressure components calculated with the classical linear separation method (continuous line), and the FOC method (dashed line). The panels (a)-(d) refer to the four experiments performed (a) low-pressure amplitude with water, (b) large pressure amplitude with water, (c) low-pressure amplitude with glycerine mixture and (d) large pressure amplitude with glycerine mixture.
and (c) with (d)). In order to measurethe effect of fluid friction and nonlinearitieson the separation method, three dimensionless parameterswere used to quantify fluid friction (cp),elasticnonlinearities(y), and convective acceleration(K). Theseparametersquantify the relative importance of the terms of fluid friction, elastic nonlinearities, and convective acceleration in the one dimensionalflow equations(Pythoud et al., in press). They typically vary between0 and 1, and are definedas
y = IAwCtP) - AmICmlmax A&m ’
(13) (14)
wheresubscriptmax refersto be maximumvalue within a pulsecycle.The differenceEbetweenthe classicallinear and FOC separationmethodsis definedby the maximum deviation in the predictedforward runningpressurewave and is expressedin percent of the total pressureamplitude (maximum-minimum).The valuesof parameters4, y, and K for the four experimentsare listed in Table 1.
Table 1. Percent difference, E, in pressure wave amplitude between classical linear and FOC separation methods as function of the friction parameter q, convective acceleration parameter K, and elastic nonlinearity parameter y Experiment
(4 (4
cp
K
0.11 0.12
0.10 0.13
0.60
1.15
1 8
0.88 0.84
0.08 0.13
0.38 0.99
4 4
Y
&[%]
Figure 5 comparesthe transmissionof forward and backwardrunningwavespredictedby the classicallinear methodwith thosepredictedby FOC method.We define the forward transmissionratio asthe ratio betweendistal and proximal forward running pressurewave amplitudes (maximum-minimum).The backwardtransmissionratio is defined similarly. Being a global number, it correspondsto an averageof the frequency dependenttransmissionratio over the two of threefirst harmonics.In the caseof linear separation,wefound a transmissionratio of 0.95 and 0.91 for the forward and backward running
Effects of friction and nonlinearities on the separation of arterial waves
1423
forward and backward running pressurecomponent amplitudes. Acknowledgements-This work is supported by the Swiss National Science Foundation (Grant Number 21-32559.91) REFERENCES
Fig. 5. Transmission of forward and backward running waves estimated with (a) the classical linear separation method, (b) with FOC method that accounts for both nonlinear effectsas well as for dissipation. Thick and thin drawn lines give the proximal forward and backward running pressure waves, respectively. Thick and thin dashed lines give the forward and backward waves at the distal end of the tube.
waves,respectively.Using FOC method, we found 0.84 and 0.81. DISCUSSION
We have tested the classicallinear and FOC wave separationmethodsandcomparedtheir predictions.The differencestend to increasewith the degreeof fluid friction and nonlinearitiesand are typically of the order of 4-8%.
The transmissionratio of forward and backward waves was found to be larger when estimatedby the classicallinear method.The attenuationof wavesis principally due to frictional effectsrather than to nonlinear effects,sincenonlineareffectstend, at leastin the caseof pressureand flow waves in straight tubes, to steepen wave fronts without changingtheir amplitude(Pythoud et al., in press). Assuminga linear but dissipativesystem, we determine the transmissionper wavelength in an elastictube usingWomersley’stheory (1957)as
(
exp -2x
Y +owx x-owx
Bertram, C. D. and Greenwald, S. E. (1992) A general method of determining the frequency-dependent propagation coefficient and characteristics impedance of an artery in the presence of reflections. Trans. ASME J. Biomeck. Engng 114, 2-9. Busse, R., Bauer, R. D., Schabert, A., Summa, Y. and Wetterer, E. (1979) An improved method for the determination of the pulse transmission characteristics of arteries in oiuo. Circ. Res. 44,630-636.
Cox, R. (1971) Determination of the true phase velocity of arterial pressure waves in ho. Circ. Res. 29,407418. Langewouters, G. J., Wessehng, K. H. and Goedhard, W. J. A. (1984) The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model. J. Biomeckanics 17, 425435. Langewouters, G. J., Wesseling, K. H. and Goedhard, W. J. A. (1985) The pressure dependent dynamic elasticity of 35 thoracic and 16 abdominal human aortas in vitro described by a five component model. J. Biomeckanics l&613-620. Milnor, W. R. (1989) Hemodynamics, 2nd Edn. Williams and Wilkins, Baltimore. Milnor, W. R. and Nichols, W. W. (1975) A new method of measuring propagation coefficients and characteristic impedance in blood vessels.Circ. Res. 36, 631-635. Murgo, J. P., Westerhof, N., Giolma, J. P. and Altobelli, S. A. (1981) Manipulation of ascending aortic pressure and flow wave reflections with the Valsalva maneuver: relationship to input impedance. Circulation 63, 122-132. Pythoud, F., Stergiopulos, N. and Meister, J.-J. (1994) Modeling of the wave transmission properties of large arteries using nonlinear elastic tubes. J. Biomeckanics 27, 1379-1381. Pythoud, F., Stergiopulos, N. and Meister, J.-J. Separation of arterial pressure and tlow waves in forward and backward components. Trans. ASME, J. Biomeck. Engng (in press). Stergiopulos, N., Young, D. F. and Rogge, T. R. (1992) Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomeckanics 25, 1477-1488. Stergiopulos, N., Pythoud, F. and Meister, J.-J. The use of forward running waves in the analysis of wave propagation. Annals of Biomedical. Engng (in review). Stergiopulos, N., Spiridon, M., Pythoud, F. and Meister, J.-J. (1996) On the wave transmission and reflection properties of stenoses. 1. Biomeckanics 29, 31-38. Tardy, Y., Meister, J.-J., Perret, F., Brunner, H. R. and Arditi, M. (1991) Non-invasive estimate of the mechanical properties of peripheral arteries from ultrasonic and photoplethysmographic measurements. Clin. Pkys. Pkysiol. Meas. 12, 39-54. Van den Bos, G. C., Westerhof, N. and Randall, 0. S. (1982) Pulse wave reflection: can it explain the differences between systemic and pulmonary pressure and flow waves? Circ. Res. $1, 479-485. - Westerhof, N., Sipkema, P., Van den Bos, C. G. and Elzinga, G. (1972) Forward and backward waves in the arterial svstem.
1’ whereX and Y dependon the wall constraint K(K = 0 for a freely moving tube), on the Poissonratio 0 (estimated to be 0.35 for Sylgard) and on the Womersley parameterct.o W is a constantquantifying the viscoelasticity of the material which is approximately equal to l/2 tan@, @being the phaseshift betweenpressureand diameter, estimatedto be 0.13 in this experiment.Applying equation(15),we found that the theoreticaltransmissionratio over a distanced = 0.31m was ranging from 0.87 to 0.82 for frequenciesbetweenthe first and third harmonics(0.7 to 2.1Hz). This result is insensitive to the wall constraintK (2% changesbetweenK = 0 and hard&ax. Res. 6, 648-656. K = - CO)and compareswell with the resultsof the Womersley, J. R. (1957) An elastic tube theory of pulse transmiswave separationanalysiswherenonlinearitiesand fluid sion and oscillatory flow in mammalian arteries. WADC friction are taken into account. Neglectingfriction and Technical Report TR 56-614, Write Air Development Center nonlinear termsmay thus yield wrong estimatesof the Wright Patterson Air Force Base, Ohio.
Pergamon
PII: SOO21-9290(%)00071-l
J. Biomechmics, Vol. 29, No. 11, pp. 141’%1423, 1996 Cowriaht . . _ 0 1996 Elwier Science Ltd. All riahts reserved Printed in &eat Britain 0021-9290/96 $15.00 + .OO
EFFECTS OF FRICTION AND NONLINEARITIES ON THE SEPARATION OF ARTERIAL WAVES INTO THEIR FORWARD AND BACKWARD COMPONENTS F. Pythoud,* N. Stergiopulos,* C. D. Bertramt and J.-J. Meister* *Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, Lausanne, Switzerland 1015; and tCentre of Biomedical Engineering, University of New South Wales, Kensington, Sydney, Australia 2033 Abstract-In this paper we examine the importance of fluid friction and nonlinearities due to the area-pressure relationship and to the convective acceleration on the separation of arterial pressure and flow waves into their forward and backward components. Experiments were run in straight uniform nonlinearly elastic tubes. Different degrees of fluid friction and nonlinearities, covering the physiological range, have been tested. We predicted the forward and backward running pressure components using two wave separation methods: the classical linear method (Westerhof et al., Cardiouasc. Res. 6,648656,1972) and the first order correction (FOC) method (Pythoud et al., Trans ASME J. Biomech. Engng, in press) which takes nonlinearities and fluid friction into account. We found that the two methods yield somewhat different predictions. The differences tend to increase with the degree of fluid friction and nonlinearities and are typically of the order of 4-8%. We further compared the transmission ratio of forward and backward waves predicted by both methods. The transmission ratio was found to be overestimated by 10% by the classical linear method. The nonlinear method gave more accurate estimates, consistent with theory. We conclude that, for in oioo applications, the classicallinear method should be the method of choice becauseit is simpler to use and the errors involved (4-8%) are comparable to measurement errors. Copy right 0 1996 Elsevier Science Ltd. Keywords:
Separation; Attenuation; Transmission; Nonlinear; Wall friction. NOTATION
pi p: %X
Q Rt,R, u ZC Greek F
P
wave speed wave speed at mean pressure P,,, shear stress inertia coefficient: c, = 1 + Im[iGlo(a)] shear stress viscous coefficient: c, = Re[a’iG,,(a)/8] fluid friction term: f = 8xpc,/(pA,c,) cross-sectional area cross-sectional area at mean pressure P, compliance compliance at mean pressure P, monotonic function used to calculate the Riemann invariants - 2J,(ai3/‘)) (see Womersley) 2J,(cri3/2)/(cci 3’2 J,(ai3’*) pressure mean pressure forward pressure wave component forward pressure wave component calculated taking nonlinearities into account forward pressure wave component calculated taking fluid friction into account forward pressure wave component calculated by the classical linear method forward pressure wave component calculated by FOC method flow Riemann invariants velocity averaged over the cross-section U = Q/A characteristic impedance letters
dynamic viscosity specific weight
in final form 20 March 1996. Address correspondence to: N. Stergiopulos Ph.D., Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland. Received
Y K cp co
elastic nonlinearities parameter convective acceleration parameter fluid friction parameter frequency INTRODUCTION
Separation of arterial pressure and flow waves into their forward and backward running components is generally used to quantify reflections in the arterial system as well as to analyze heart-arterial system interaction (Murgo et al., 1981; Van den Bos et al., 1982). It may also be used to determine the reflection and transmission properties of areas of special physiological interest or of pathological section of the arterial tree such as stenoses, aneurysms, bifurcations, etc. (Stergiopulos et al., 1996). The first and most popular separation method was proposed by Westerhof et al. (1972). The method assumes linear wave propagation theory, and requires pressure and flow measurements at the same arterial location, as well as an estimate of the local characteristic impedance from these measurements. However, the one-dimensional flow equations, describing the propogation of arterial pressure and flow waves, are nonlinear and dissipative. Convective acceleration as well as nonlinearity of the area-pressure relationship of the artery contributes to nonlinearity (Langewouters et al., 1984,198s; Tardy et al., 1991) which was neglected in most of the studies on wave transmission (Bertram and Greenwald, 1992; Busse et al., 1979; Cox, 1971; Milnor and Nichols, 1975). Fluid friction, due to the blood viscosity, and wall friction, due to the viscoelastic properties of the wall, introduce dissipation into the system.
1419
1420
F. Pythoud et al.
A general wave separation theory, taking into account nonlinearity as well as fluid friction, has been recently developed and applied on computer simulation data (Pythoud et al., in press). In the present study we will compare the predictions of this general wave separation theory with the classical linear theory using data from in vitro experiments. The experiments were performed on straight tubes in conditions that permit important elastic nonlinearities and fluid friction to be present. The goal is to assess the influence of fluid friction and elastic nonlinearities on the prediction of the forward and backward running waves. METHODS
Experimental
setup
The experimentalsetupis shownin Fig. 1. A Harvard Apparatus” pump generatedphysiologicalflow waves. The working fluid was water for low viscosity experimentsand a glycerine/water mixture for high viscosity experiments(p = 0.023Pas). PVC tubesconducted the fluid from the pump to the elastictube simulatingthe artery via a small Windkesselor air chamberusedto attenuate high-frequencyoscillationsproduced by the pump. The elastictube was made out of a Sylgard 184 (Dow Corning)elastomerwhich wasshownto reproduce the nonlinear elastic properties of conduit arteries (Pythoud et al., 1994).The tube had an unstretched diameterof 8 mm, wall thickness0.105mm, and an unstretchedlength of 300mm. It was stretchedlongitudinally by approximately 50% betweenPlexiglas(perspex) connectionsto avoid buckling at high transmuralpressures.The tube wassubmersed in a bath containing the samefluid as the circulating one, primarily to compensate gravitational effects,while allowing for good flow rate and diametermeasurements usingultrasonic devices.The distal load of the tube consistedof a Windkessel (air reservoir)and a resistorwhich wasmadeof a pieceof tube filled up with smallspheresof lead.The linearity or not of the resistor does not affect the results. Mean pressurewithin the elastictubewascontrolled by a distal fluid column of adjustableheight. Pressureand flow rate were measuredat two sites alongthe tube 310mm apart. Pressurewasmeasuredby two catheter-tip pressuretransducers(Millar@ 2.5F)
introduced into the tube from the proximal and distal ends,respectively.The flow rate wasmeasuredby transittime ultrasonicdevices(Triton@Technology,SanDiego) of sufficientinsidediameter(14 and 17mm) so that the tube at maximumdistensiondid not touch the surrounding cuff. The pressurecatheterswereplacedabout 12mm from the center of the flow probes.Both pressureand flow signalswereamplifiedand sampledat 500 samples Hz by an analog-to-digitalacquisitionboard (National InstrumentsaLab-NB). They werestoredin a Macintosh II-Ci. The internal diameterof the tube, and thereforeits cross-sectional area,wasmeasuredseparatelybeforeand after experimentsby an echo-trackingultrasonic device (Nius@,OmegaElectronics,Switzerland)usinga 4 MHz ultrasonic probe. In each experiment, pressuretransducerswerecalibratedagainsta static fluid columnin the absenceof flow, and flow meterswerecalibratedagainst constantflow rates.Linear regressionof thesedata lead to the calibration factors and showeda potential error of 0.5% on pressuremeasurements, and a potential error of 3-4% on flow rate measurements. We assumedstatic calibration curvesto be valid with the sameaccuracy in pulsatileflow conditions.Diameter measurement errors, given by the manufacturer,are 0.1%. Weperformedfour experimentsat a room temperature of 27°C. In the first one, the working fluid was water (p = 0.00086Pas, p = 1000kg m- 3, to reducethe frictional effects.The pump rate wasset to 0.7 Hz and the pump stroke volume to 6ml. With this stroke volume, the pressurerangedfrom 14 to 19kPa. The area-pressure relationship,despitebeing not strictly linear, was approximatedaslinear over this limited pressurerange. In the secondexperimentthe strokevolume of the pump wasincreasedto 20 ml to extend the pressurerangeand thusto obtain asmucharea-pressure nonlinearity (differencein the area-pressure curve slopeat the two extremes of the pressurerange)as possiblewithout risking tube rupture. The third and fourth experimentswere as the first two except that fluid viscosity was increasedto about four times blood viscosity (to amplify viscous losses)thereforeincreasingdissipationand wave attenuation. This wasobtainedby usinga mixture of glycerine and water (volume ratio 7:3) giving a viscosity of p = 0.023Pas and a density of p = 1192kgme3. Data analysis
The digitizedflow rate data werefirst correctedfor the phase-shifteffectsintroduced by the output filter in the flow meters,that consistsof two cascaded2-polefilters of breakpoint 100Hz and Q-factor 0.7. Thesefilesproduced a time shift of the order of 5 msfor frequenciesranging between1 and 100Hz. The other phaseshift of multiplexed analogto digital conversionwassmaller(0.5ms) and thus considerednegligible.Furthermore, pressure and flow rate signalswere not measuredexactly at the samelocation, and the phaseshift resulting from the distanceof about 12mm betweenthe two transducers Fig. 1. Schematic diagramof theexperimental setup.Twocatheter tip transducers measure the pressures PI and Pz. Theflow was also corrected. These steps have been taken to rate QI and Q, are given by two transit-time ultrasonic devices guaranteean accuracyof the order of 3% on the forward placed around the elastic tube. and backward running pressurecomponents.
1421
Effects of friction and nonlinearities on the separation of arterial waves Linear forward and backward running pressure wave components were obtained from a classical linear separation method (Westerhof et al., 1972) which neglects both friction and nonlinearities:
P: = fP + ZcQX
(1)
PP = t(f' - GQ),
(2)
where P is the pressure, Q the flow, and Z, = ,,/m) the characteristicimpedance.p is the fluid density,A, and C, are cross-sectional area and complianceat meanpressure. Resultsof the linear analysiswerethen comparedwith thoseof nonlinear analysisdescribedby Pythoud et al. (in press).The first-order correction (FOC) separation method is obtained as a combination of two methods: a first one taking nonlinearitiesonly into account and secondone taking dissipation(fluid friction) only into account. The first solution is obtained by defining a smooth monotonic function F(P) and the Riemann . . mvanants RI Rz as F(P) =
s
p dP
(3)
RExJLm
Figure 2 showsthe area-pressurerelationshipmeasured, first in small pressureamplitude experiment (14 to 19kPa, compliancerange 3.8x 10e9m2Pa-‘-2.6 x 10e9mzPa-‘), and then duringlargepressureamplitude experiment (P varying from 11 to 21 kPa, compliance range4.6 x 10m9m* Pa-‘-2.1 x 10e9mzPa-‘). The areapressurerelationshipwasapproximatedby a second-order polynomial.Typical pressureand flow rate measurementsat the entrance(fully drawn) and exit (dashed)are shownin Fig. 3. Figure 4 showsthe forward waves obtained at the proximal location after separationin forward and backward components.The fully drawn linesare the waves derivedfrom the linear, classical,theory and the dashed linesare the wavesobtainedusingthe nonlineartheory. As seenin Fig. 4, the forward running pressurecomponentspredictedby the two methodsare very similar in shapebut differ significantlyin amplitude.This difference increases with friction (increasedfluid viscosity,compare Fig. 4(a)with (c) and (b) with (d) as well as with nonlinearity (large pressurerange, compareFig. 4(a) with (b)
P,pco'
RIO’, U) = (F(P) + W/Z
(4)
R,(P, U) = (F(P) - W2,
(5)
1.2
wherePO is the diastolicpressure,c = ,/m the wave speed,and U = Q/A blood velocity averagedover the cross-section.The forward and backward running pressurecomponents,PiI and P,“1, are found by solving the following equations: W:,)
= RI
(6)
F(Pj,)
= R2.
(7)
The solutionto the dissipationproblemis obtainedby defininga frequency-dependent characteristicimpedance -
-
l-
I 0.8
“'I IO
I 14
I
I 18
Pressure[kPa]
I
I 22
Fig.2. Thearea-pressure relationship at proximallocationduring smallandlargepressure amplitudeexperiments. with f = 8npz,/(pA,c,). The viscousand inertia coefficientsof shearstress,c, and c,, are definedas c, = 1 + ImCiG&)l, c, = Re[u2iGio(a)/8]
(9) (10)
and dependon the Womersleyparametertl (Womersley, 1957;Stergiopuloset al., 1992).To accountfor dissipation effects, forward and backward running pressure components,Pi and Pi, are calculatedin the frequency domainaccordingto equations(1) and (2).FOC method combinesthe two stepsgiven above: Got = Pi, + Pi - P:.
(11)
Similar expressionis obtained for backward pressure components.All of thesemethodsrequire simultaneous Fig.3. Proximal(continuous line)anddistal(dashed line)presmeasurements of pressureand flow rate at the same sureand flow rate signalsrecordedduringa largepressure location aswell asknowledgeof the local area-pressure amplitude experiment in whichthecirculatingfluidwasaglycerinemixture. relationship.
F. Pythoud et al.
1422
a)
b)
18
0
1
Time [s]
Time [s]
1
Time [s]
1
Fig. 4. Comparison between forward running proximal pressure components calculated with the classical linear separation method (continuous line), and the FOC method (dashed line). The panels (a)-(d) refer to the four experiments performed (a) low-pressure amplitude with water, (b) large pressure amplitude with water, (c) low-pressure amplitude with glycerine mixture and (d) large pressure amplitude with glycerine mixture.
and (c) with (d)). In order to measurethe effect of fluid friction and nonlinearitieson the separation method, three dimensionless parameterswere used to quantify fluid friction (cp),elasticnonlinearities(y), and convective acceleration(K). Theseparametersquantify the relative importance of the terms of fluid friction, elastic nonlinearities, and convective acceleration in the one dimensionalflow equations(Pythoud et al., in press). They typically vary between0 and 1, and are definedas
y = IAwCtP) - AmICmlmax A&m ’
(13) (14)
wheresubscriptmax refersto be maximumvalue within a pulsecycle.The differenceEbetweenthe classicallinear and FOC separationmethodsis definedby the maximum deviation in the predictedforward runningpressurewave and is expressedin percent of the total pressureamplitude (maximum-minimum).The valuesof parameters4, y, and K for the four experimentsare listed in Table 1.
Table 1. Percent difference, E, in pressure wave amplitude between classical linear and FOC separation methods as function of the friction parameter q, convective acceleration parameter K, and elastic nonlinearity parameter y Experiment
(4 (4
cp
K
0.11 0.12
0.10 0.13
0.60
1.15
1 8
0.88 0.84
0.08 0.13
0.38 0.99
4 4
Y
&[%]
Figure 5 comparesthe transmissionof forward and backwardrunningwavespredictedby the classicallinear methodwith thosepredictedby FOC method.We define the forward transmissionratio asthe ratio betweendistal and proximal forward running pressurewave amplitudes (maximum-minimum).The backwardtransmissionratio is defined similarly. Being a global number, it correspondsto an averageof the frequency dependenttransmissionratio over the two of threefirst harmonics.In the caseof linear separation,wefound a transmissionratio of 0.95 and 0.91 for the forward and backward running
Effects of friction and nonlinearities on the separation of arterial waves
1423
forward and backward running pressurecomponent amplitudes. Acknowledgements-This work is supported by the Swiss National Science Foundation (Grant Number 21-32559.91) REFERENCES
Fig. 5. Transmission of forward and backward running waves estimated with (a) the classical linear separation method, (b) with FOC method that accounts for both nonlinear effectsas well as for dissipation. Thick and thin drawn lines give the proximal forward and backward running pressure waves, respectively. Thick and thin dashed lines give the forward and backward waves at the distal end of the tube.
waves,respectively.Using FOC method, we found 0.84 and 0.81. DISCUSSION
We have tested the classicallinear and FOC wave separationmethodsandcomparedtheir predictions.The differencestend to increasewith the degreeof fluid friction and nonlinearitiesand are typically of the order of 4-8%.
The transmissionratio of forward and backward waves was found to be larger when estimatedby the classicallinear method.The attenuationof wavesis principally due to frictional effectsrather than to nonlinear effects,sincenonlineareffectstend, at leastin the caseof pressureand flow waves in straight tubes, to steepen wave fronts without changingtheir amplitude(Pythoud et al., in press). Assuminga linear but dissipativesystem, we determine the transmissionper wavelength in an elastictube usingWomersley’stheory (1957)as
(
exp -2x
Y +owx x-owx
Bertram, C. D. and Greenwald, S. E. (1992) A general method of determining the frequency-dependent propagation coefficient and characteristics impedance of an artery in the presence of reflections. Trans. ASME J. Biomeck. Engng 114, 2-9. Busse, R., Bauer, R. D., Schabert, A., Summa, Y. and Wetterer, E. (1979) An improved method for the determination of the pulse transmission characteristics of arteries in oiuo. Circ. Res. 44,630-636.
Cox, R. (1971) Determination of the true phase velocity of arterial pressure waves in ho. Circ. Res. 29,407418. Langewouters, G. J., Wessehng, K. H. and Goedhard, W. J. A. (1984) The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model. J. Biomeckanics 17, 425435. Langewouters, G. J., Wesseling, K. H. and Goedhard, W. J. A. (1985) The pressure dependent dynamic elasticity of 35 thoracic and 16 abdominal human aortas in vitro described by a five component model. J. Biomeckanics l&613-620. Milnor, W. R. (1989) Hemodynamics, 2nd Edn. Williams and Wilkins, Baltimore. Milnor, W. R. and Nichols, W. W. (1975) A new method of measuring propagation coefficients and characteristic impedance in blood vessels.Circ. Res. 36, 631-635. Murgo, J. P., Westerhof, N., Giolma, J. P. and Altobelli, S. A. (1981) Manipulation of ascending aortic pressure and flow wave reflections with the Valsalva maneuver: relationship to input impedance. Circulation 63, 122-132. Pythoud, F., Stergiopulos, N. and Meister, J.-J. (1994) Modeling of the wave transmission properties of large arteries using nonlinear elastic tubes. J. Biomeckanics 27, 1379-1381. Pythoud, F., Stergiopulos, N. and Meister, J.-J. Separation of arterial pressure and tlow waves in forward and backward components. Trans. ASME, J. Biomeck. Engng (in press). Stergiopulos, N., Young, D. F. and Rogge, T. R. (1992) Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomeckanics 25, 1477-1488. Stergiopulos, N., Pythoud, F. and Meister, J.-J. The use of forward running waves in the analysis of wave propagation. Annals of Biomedical. Engng (in review). Stergiopulos, N., Spiridon, M., Pythoud, F. and Meister, J.-J. (1996) On the wave transmission and reflection properties of stenoses. 1. Biomeckanics 29, 31-38. Tardy, Y., Meister, J.-J., Perret, F., Brunner, H. R. and Arditi, M. (1991) Non-invasive estimate of the mechanical properties of peripheral arteries from ultrasonic and photoplethysmographic measurements. Clin. Pkys. Pkysiol. Meas. 12, 39-54. Van den Bos, G. C., Westerhof, N. and Randall, 0. S. (1982) Pulse wave reflection: can it explain the differences between systemic and pulmonary pressure and flow waves? Circ. Res. $1, 479-485. - Westerhof, N., Sipkema, P., Van den Bos, C. G. and Elzinga, G. (1972) Forward and backward waves in the arterial svstem.
1’ whereX and Y dependon the wall constraint K(K = 0 for a freely moving tube), on the Poissonratio 0 (estimated to be 0.35 for Sylgard) and on the Womersley parameterct.o W is a constantquantifying the viscoelasticity of the material which is approximately equal to l/2 tan@, @being the phaseshift betweenpressureand diameter, estimatedto be 0.13 in this experiment.Applying equation(15),we found that the theoreticaltransmissionratio over a distanced = 0.31m was ranging from 0.87 to 0.82 for frequenciesbetweenthe first and third harmonics(0.7 to 2.1Hz). This result is insensitive to the wall constraintK (2% changesbetweenK = 0 and hard&ax. Res. 6, 648-656. K = - CO)and compareswell with the resultsof the Womersley, J. R. (1957) An elastic tube theory of pulse transmiswave separationanalysiswherenonlinearitiesand fluid sion and oscillatory flow in mammalian arteries. WADC friction are taken into account. Neglectingfriction and Technical Report TR 56-614, Write Air Development Center nonlinear termsmay thus yield wrong estimatesof the Wright Patterson Air Force Base, Ohio.