Pfli.igers Archiv
Pfltigers Arch. 380, 221-226 (1979)
EuropeanJoun'el of Physiology 9 by Springer-Verlag 1979
Separate Determination of the Pulsatile Elastic and Viscous Forces Developed in the Arterial Wall in vivo R. D. Bauer, R. Busse, A. Schabert, Y. Summa, and E. Wetterer
Institut ftir Physiologicund Kardiologieder UniversitfitErlangen-Niirnberg,Waldstrasse6, D-8520 Erlangen, Federal Republic of Germany
Abstract. The viscoelastic behaviour of arteries in vivo is analyzed by separate representation of the purely elastic and the purely viscous properties, using natural pressure and diameter pulses of various dog arteries recorded under steady-state conditions. The circumferential wall stress (a) and the radius (r) of the mean wall layer are calculated as functions of time and the hysteresis of the ~r-r diagram is represented. The stress is regarded as the sum of an elastic stress (ael) which is a function of r, and a viscous stress (o-vj~) which is a function of dr/dt. Thus ael = o- - avis. Since the a~1-r diagram must be free from hysteresis, the disappearance of the loop is the criterion that indicates that a~l has been found. O-vl~ is formulated as a second degree polynomial of dr/dt whose coefficients are determined using that criterion. The a~-r curve is always nonlinear and the elastic modulus increases with increasing radius. The avis-dr/dt curve, too, is nonlinear. Its slope decreases with increasing dr/dt. The same applies to the wall viscosity (pseudoplastic behaviour). The nonlinear properties can be represented adequately by processing the experimental data in the time domain. Problems inherent in investigations based on the frequency domain, as reported in the literature, are pointed out.
Key words: Arterial elasticity - Arterial wall viscosity - Arterial stress-strain relationship - Models of the arterial wall.
Introduction
The mechanical behaviour of the arterial wall can be characterized by elasticity, viscosity, and plasticity. Under steady-state pulsatile conditions, only elasticity and viscosity need to be considered. The viscosity gives
rise to a time lag of the pulsations of the arterial diameter with respect to the pressure or, expressed in other terms, of the circumferential walt strain with respect to the tensile stress. For this reason, the pressure-diameter diagram exhibits a hysteresis loop, from which the purely elastic and the purely viscous stress-strain relationships cannot be obtained directly. However, the separate representation of these two functions would be of considerable interest both with respect to their fundamental significance, and for a number of practical reasons. For example, a linear elastic behaviour in the range of the given pulse amplitude is a prerequisite for the application of Fourier analysis to natural arterial pulses [13]. This also applies to the viscous behaviour. Only the separation of the two functions enables one to establish whether, and to what extent, each deviates from linearity. Previous investigators determined the elastic and viscous properties of arteries either in vitro with the aid of small sinusoidal changes of pressure and volume at various frequencies [2, 4, 12, 14], or in vivo using the Fourier components of natural pulses under the assumption of linearity [7, 13]. If nonlinearities of the elastic and viscous properties are to be taken into account, the evaluation of natural pulses carried out in the frequency domain involves serious problems. For this reason, we have developed a procedure which permits the separate representation of elasticity and viscosity applying a calculation carried out in the time domain. It is the aim of this study to describe the principles of the method and to present, as examples, the results obtained on two types of arteries of the dog. An extensive application will be left to future work.
Theoretical Basis of the Method In preliminary reports [3, 24] we used the relation between the pulsatile transmural pressure and the
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external diameter for the separate determination of the elastic and viscous behaviour of the arterial wall. However, from the theoretical point o f view it is more meaningful to use the pulsatile circumferential wall stress instead of the pressure and the pulsatile length of the m e a n circumferential wall layer or the length of its radius instead of the external diameter. In practice, the shapes of the pressure-diameter and the stress-radius diagrams are very similar. The average circumferential wall stress (a) is: a =p"
ri/h
(1)
where p = transmural pressure, r i = internal vessel radius, and h = wall thickness. The radius of the mean wall layer (r) is calculated as the geometric mean: r = ]/Tii" re
(2)
where ~ = external radius. We are using the geometric rather than the arithmetic mean because the former has a more general significance. If the wall is homogeneous, eq. (2) represents the radius of that wall layer whose circumferential stress equals the stress given by eq. (1) irrespective of whether the tube is thin- or thickwalled [231. By definition, the purely elastic forces depend on the circumferential stretch itself, while the viscous forces are related to the rate of change of the circumferential stretch. We can, therefore, subdivide the pulsatile stress into two components, the "elastic stress" (a~l) which is a function of the pulsatile radius (r), and the "viscous stress" (avis) which is a function of the rate of change of the radius (dr/dt): a = a~, + a~, = f~l (r) + f~is (dr/dt)
(3)
where f~l and f~is are symbols of functions. In order to find the purely elastic behaviour of the artery, i.e. the ael-r relationship, avis must be subtracted from a. As general formulation of avis, we write: a,,~, = F . dr/dt
(4)
For a first approximation, the factor Fis assumed to be constant throughout a pulse cycle. This constant factor is called F~. Thus we have: ~1 = a - F j " dr/dt.
(5)
The C%l-r diagram must be free from hysteresis. Therefore the value of the factor F~ can be found by means of a procedure based on the disappearance of the hysteresis loop. This is done by trial and error in the following way. For each point of a pulse cyde, the time derivative of the radius is multiplied by an arbitrarily chosen value of F t . These products are subtracted, according to eq. (5), from the corresponding values of the stress. The difference is plotted against the radius
and the value of F~ is varied systematically until the loop area is minimized. In principle, total disappearance of the hysteresis loop is the criterion which indicates that the optimal value of the factor F1 and the true aol-r curve have been found. As will be shown below, a constant factor F~ generally permits a noticeable reduction of the loop area. However, it is not possible to reduce this area to an acceptable minimum. For this reason, a further approximation of the viscous term proved to be necessary in which the factor F of eq.(4) comprises a constant part and another part which is proportional to dr/dt: F = F t + F2 [ dr/dt[.
(6)
Since F is assumed to be independent of the sign of dr/dt, the absolute value of dr/dt is used. If eq. (6) is inserted into eq. (4), we obtain: av~s = (Fa + g e l dr/dt I) dr/dt.
(7)
Inserting eq. (7) into eqs, (3) and (4) leads to: a~1 = a -- (Fa + F2 l dr/dt 1)dr/dt.
(8)
N o w the factors Fa and F 2 have to be varied until the hysteresis loop reaches a minimum. As will be shown, the two-factor method generalIy leads to a virtuaIIy complete, or at least almost complete, elimination of the loop. In all cases examined so far, F2 has turned out to be negative. The physical significance of the term F 2 [dr/dt[ will be discussed below. The second-degree polynomial of eq. (7) represents the first two terms of a modified Taylor expansion of the viscous stress with respect to dr/dt. The addition of terms of a still higher order might be envisagedfrom a theoretical point of view. In practice, however,such an addition did not yield any clear improvementwith respect to the elimination of the loop, as will be discussedbelow. This also appliesto second and higher order time derivatives of the radius. Another question is whether in eq. (3) the functionsf~l and fvis should be related to the strain and the rate of change of strain rather than to the stretch and the rate of change of stretch. With respect to the functionf~, the strain would then be defined as r/ro where ro is a fixed radius, e.g. the enddiastolic radius. Replacing r by r / r o in the functionf~ would make only a formal differencewithout affectingthe shape of the respectivediagram for cry. With respect to the function fv~, the rate of change of strain would then have to be formulated as (dr/dt)/r where r is the actual radius which changes during the pulse cycle. However, the percentage changes of r are mostly not great enough to provide a means of deciding whether the use of the rate of change of stretch or of the rate of change of strain would be more appropriate. Once the factors F~ and F 2 have been determined in the way described, relationships other than that between ae~ and r can also be obtained. F r o m the a~-r curve, the circumferential elastic modulus of the arterial wall can be calculated for each point of the pulse cycle. Furthermore, the avis-dr/dt relationship is of special interest [eq. (7)] and can be used to demonstrate the dependence of the wall viscosity on the rate of
R. D. Bauer et al. : Pulsatile Elastic and Viscous Forces in the Arterial Wall c h a n g e o f the w a l l strain. F i n a l l y , the h a e m o d y n a m i cally i m p o r t a n t r e l a t i o n s h i p b e t w e e n the t r a n s m u r a l p r e s s u r e a n d the i n t e r n a l c r o s s - s e c t i o n a l a r e a o f t h e a r t e r y c a n be c a l c u l a t e d for e a c h p o i n t o f the p u l s e cycle a n d r e s o l v e d i n t o its e l a s t i c a n d v i s c o u s c o m p o n e n t s . If we return to eq. (3), we see that two general functionsfel andfv~s are first assumed. Then the function f ~ is formulated in a suitable way by eqs. (4) and (6) and the functionf~ 1determined experimentally using the criterion of loop elimination which gives us the two factors of the function f~i~- No special assumptions are made regarding the function f~i. The following approach is also possible. First, the function foj is formulated in a suitable way, for example as a polynomial [19], and its coefficients determined by the criterion of the elimination of the a-dr~dr loop. In principle, this approach would be equivalent to the procedure described above.
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[rnm Hg]
1V
/
130 -J
[dyn/cm2]
lcm] 1.44-
0.329
0.70 r
0.5
0.360
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Ib)
2.35 -
0
r [cmI
1
2.34-
Oel 2 [ dyn/cm ]
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[dynlcm ]
Methods Recordings of intraarterial pressure and outside diameter were performed in the common carotid artery, the descending thoracic aorta, the abdominal aorta, the iliac, and the femoral arteries of 6 mongrel dogs weighing 20-35 kg. The animals were anaesthetized with sodium pentobarbital (30 mg/kg i.v.). The artery under investigation was exposed and carefully freed from surrounding tissue over a length of about 3 cm. Pressure was recorded using a cathetertip manometer (Millar PC 350A) introduced through a side branch. The pressure-sensitive membrane of this manometer type is located in such a way that lateral pressure is measured. The bridge circuit of the manometer was fed from a high-stabilized voltage source and the pressure signal amplified by an instrumentation amplifier (AD 605 L). A contact-free photoelectric device [25] was employed for the recording of the pulsatile outside diameter of the artery simultaneously with the pressure. The above-mentioned amplifier type was also used for the photoelectric device whose resolving power was about 1 ~tm. The calibration was carried out in situ during the measurements. The pressure-sensitive membrane of the manometer was located exactly at the site of diameter measurement. The frequency responses of the manometer system and of the photoelectric device were mainly determined by the amplifiers and were, therefore, almost identical. The amplitude response was virtually fiat up to 200 Hz. In this range, the time delay was about 0.2 ms and independent of frequency. The cut-off frequency was higher than 2 kHz. At the end of each experiment, the artery was excised and the wall thickness determined in vitro. Assuming wall volume constancy, the wall thickness was calculated for the in-situ conditions. The recorded data were stored on analog magnetic tape (Bell & Howell VR 3200) and processed, after analog-digital coIwersion (Hewlett-Packard 5610A), by a digital computer (HP 2100A). The sampling interval was 2.5 ms in the case of short pulse cycles and 5 ms for long cycles. From a series of steady-state pressure and diameter pulses, one cycle was selected for further analysis. An appropriate computer program allowed the systematic variation of the factors F 1 and F2 and, for each pair of factors, the calculation according to eq. (8) for the whole pulse cycle. Each result was visually inspected on the display (HP 1300A) and the respective diagrams were plotted (HP 7210A). The loop was considered to have been optimally eliminated when its ascending and descending limbs coincided at a maximum number of points. We also used the method of the least squares of the differences between the two limbs of the loop. However, the values of the factors F t and F2 determined in this way did not noticeably differ from those found by visual control. In any case, the accuracy of the loop elimination is limited by noise, as will be discussed below.
1.440.329
1.45-
r Icm]
(c)
0.360
0.329
r[cml
0.360
(d)
Fig. 1 a--d. Canine abdominal aorta, a Recorded intraarterial pressure (p) and outside diameter (D) as functions of time. Vertical arrows indicate the limits of the section used for evaluation, b Stress
(~r)-radius (r) diagram exhibiting a hysteresis loop. c Result of application of eq.(5) using F~ : 7.0 910" dyn . s/cm 3. Loop area somewhat reduced, d G r r diagram obtained by applying eq. (8) using F 1 = 13.2 - 10~ dyn - s/cm 3 and Fz : -3.8 ' 104 dyn. s2/cm *
Results F i g u r e s l a a n d b s h o w a n e x a m p l e o f the r e c o r d e d p r e s s u r e a n d d i a m e t e r o f a c a n i n e a b d o m i n a l a o r t a as f u n c t i o n s o f t i m e a n d the c o r r e s p o n d i n g s t r e s s - r a d i u s d i a g r a m e x h i b i t i n g a h y s t e r e s i s - l o o p . I n Fig. 1 c a n d d, the results o f t w o d i f f e r e n t steps in the e l i m i n a t i o n o f the l o o p are d e m o n s t r a t e d . I n Fig. 1 c, eq. (5) is a p p l i e d a n d t h a t v a l u e o f the f a c t o r F 1 is u s e d w h i c h l e a d s to the b e s t p o s s i b l e r e d u c t i o n o f the l o o p a t t a i n a b l e w i t h a c o n s t a n t factor. H o w e v e r m a r k e d r e m n a n t s o f the l o o p are still to be seen. T h e d i a g r a m n o w has the s h a p e o f a s p i n d l y " 8 " w h o s e i n t e r s e c t i o n p o i n t c a n be s h i f t e d u p a n d d o w n by slightly v a r y i n g t h e f a c t o r F 1 w i t h o u t i m p r o v i n g the result. By a p p l y i n g e q . ( 8 ) w i t h t w o f a c t o r s , F1 a n d F2, the d i a g r a m s h o w n in Fig. I d, w h i c h is v i r t u a l l y free o f a n y signs o f a l o o p , is o b t a i n e d . It is i m p o r t a n t t h a t the f a c t o r F 2 p r o v e d to be n e g a t i v e . A s a l r e a d y m e n t i o n e d , this was f o u n d to be t r u e for all o t h e r l o o p e l i m i n a t i o n s we effected. D u e to the n e g a tivity o f the f a c t o r ?2, the t w o - f a c t o r m e t h o d a l w a y s gives a g r e a t e r v a l u e o f F 1 t h a n the o n e - f a c t o r m e t h o d (cf. L e g e n d o f Fig. 1 c a n d d). T h e r e s u l t i n g G~-r c u r v e o f Fig. l d is m a r k e d l y n o n l i n e a r . T h e p u l s a t i l e r a d i u s c h a n g e is a b o u t 9 % o f the e n d d i a s t o l i c radius. A s a s e c o n d e x a m p l e , F i g . 2 a a n d b s h o w the recorded pressure and diameter of a canine femoral
X/ ;/
224
Pflfigers Arch. 380 (1979)
1.31
[rnrn Hg] / 106 J
O [d yn/cmz]
GeL
z
[dyn/cm ]
0.940.197
T I
0.5
(a)
0.94 r [crn]
0.201
0.197
r[cm]
0.201
I
sec
(c)
(b)
Fig. 2 a--c. Canine femoral artery, a Recorded intraarterial pressure and outside diameter as functions of time. Vertical arrows as before9 b Stressradius diagram exhibiting a hysteresis loop. c cr~l-r diagram obtained by application of eq. (8) using F 1 = 5.0 9 10 s dyn- s/cm ~ and F: = - 2 . 3 9106 dyn. s2/cm4
artery as functions of time, and the corresponding stress-radius loop. In Fig. 2c, elimination of the loop is effected using the two-factor method [eq. (8)]. In this case, there is only a slight nonlinearity of the %-r curve. This can be understood from the small relative pulsatile radius change which is only 2 ~. The procedure demonstrated in Figs. 1 and 2 was also applied with equal success to the descending thoracic aorta and the carotid and iliac arteries of the dog. As may be mentioned, the values of the two factors determined on a series of consecutive steady-state pulses showed only negligible variations. The viscous behaviour of the arterial wall can be demonstrated by means of the avj~-dr/dt diagram derived from eq. (7). An example is shown in Fig. 3 for the same pulse cycle of the femoral artery which was evaluated in Fig. 2, and using the same factors as noted for Fig.2c. The a~i~-dr/dt curve is S-shaped. Its two limbs are concave towards the dr/dr axis, which is due to the negative sign of the factor F 2. Usually the a~j~-dr/dt diagram exhibits a shorter limb in the third than in the first quadrant because the ascent of the natural pulse is commonly steeper than its descent and, therefore, higher values of dr/dt are attained in the positive than in the negative direction.
Discussion
The practical application of the procedure described above is impaired by the noise on the recordings which is largely caused by the magnetic tape. The necessity of using the differential quotient of the radius and the square of this quotient [eq. (8)] increases the effect of the noise since differentiation of a signal with respect to the time decreases the signal-to-noise ratio. Thus the ~x-r curve shows more noise than the original a-r loop. For this reason, the accuracy of the determination of the
0.25- x 10-
~ -
/
Gvis [ d y n / c m 2] /
dr/dr [cm/sec]
0 / -
0
.
0,05
0,I
-0.25Fig. 3. Canine femoral artery, avid-dr~dr diagram calculated by means of eq. (7). Factors F1 and FE as in Fig. 2c
factors F 1 and F 2 is limited and it is difficult to decide whether the addition of further terms of third and higher orders in dr/dt in eqs. (7) and (8) would yield any significant improvement. The crvis-dr/dt curve (Fig. 3), however, is smooth because O-vis is calculated according to eq. (7) using fixed values of the factors F~ and F 2 determined before. Now our results obtained on arteries in vivo are compared, with the help of an example, with those obtained by previous authors using small artificial sinusoidal pressure and diameter changes of arteries in vitro. We restrict the comparison to experiments carried out without any mechanical contact of the arterial wall and the diameter measuring device. A quantity appropriate for such a comparison is the elastic modulus. Ranke [18], Hardung [14], McDonald and Taylor [17], and Bergel [4] have defined a complex elastic modulus composed of a real part they called dynamic modulus (Eay,), and an imaginary part they called loss modulus (Elos~ = oat/), where co is the angular frequency of the sinusoidal oscillations and t/ the coefficient of viscosity of the arterial wall [4, 14]. The modulus of elasticity calculated from our ael-r curve is comparable to Edyn and the coefficient of viscosity
R. D. Bauer et al. : Pulsatile Elastic and Viscous Forces in the Arterial Wail calculated from o u r (Tvis-dF/dt curves is comparable to t/ contained in E~o~. However, such a comparison involves a number of difficulties. Eay. and Eloss were determined assuming linear elastic and viscous behaviour of the arterial wall within the range of the small artificial sinusoidal oscillations at given frequencies. In contrast, our determinations were carried out on natural pulses whose pressure and diameter amplitudes are far greater, so that elastic nonlinearity becomes apparent and the elastic modulus varies with stretch. In addition, there is a viscous nonlinearity since, as is shown in Fig. 3, the viscous stress and also the wall viscosity depend on the rate of change of stretch. We calculate the circumferential elastic modulus (Et) from the smoothed oel-r curve using the formula:
Er = (A%,/Ar)r (1 - bt2).
(9)
In essence, this formula corresponds to that used by Bergel [4], in which the longitudinal constraint of the artery is also taken into account by the factor (1 - la2), where bt = Poisson's ratio = 0.5. (As to the problem of definition of E t, see [23]). For the canine femoral artery (Fig. 2c), we obtain values o f E t which increase from 1.4 910 v to 1.9- 107 dyn/cm 2 from the lowest to the highest part of the curve. The corresponding pressures are 110 and 140 mm Hg. The value OfEdy n found by Bergel ([4], Table 1) as the mean of the values of 5 canine femoral arteries in vitro is (1.2 _+0.08)- 10 v dyn/cm 2 at frequencies of 2 and 5 Hz and at a mean pressure of 100 mm Hg. Our value of 1.4 9 107 dyn/cm 2 determined at 110 mm Hg is in fair agreement with this figure. Furthermore, we calculate the viscosity t/ of the arterial wall as the ratio of the circumferential viscous stress to the strain velocity, i.e. the rate of change of the circumferential wall strain: Crvi~'r (1 -- ~t2).
(10)
t ~ - dr/dt The longitudinal constraint of the artery is again taken into account by the factor (1 - la2). For the canine femoral artery, the values of O-visand dr/dt can be taken from the diagram of Fig.3. One can see from the marked nonlinearity that t/ decreases with increasing strain velocity. The radius itself undergoes changes of only + 1 ~ and can, therefore, be taken as approximately constant (0.2 cm in this example). At very low values of dr/dr, we obtain t / = 0.75.10 s dyn. s/cm 2 and at the highest value ofdr/dt we find t / = 0.36 910 s dyns/cm 2. For a comparison with Bergel's data ([4] Fig. 3) we refer to his value of the loss modulus cot/ of the femoral artery noted for the lowest frequency used (2 Hz). This value is 1.35.106 dyn/cm 2. Hence t / = 1.07 9 l0 s dyn 9 s/cm 2 which is not far above our value oft/ determined at very low dr/dt. Although the strain amplitudes applied in Bergel's experiments are not
225
mentioned in his paper, one may assume that, at 2 Hz, the strain velocity did not exceed that of the almost linear part of the curve in our Fig. 3. At higher frequencies, Bergel presumably had greater strain velocities. Indeed, the values oft/calculated from his loss moduli decrease when the frequency rises. A decrease in arterial wall viscosity with increasing frequency of strain has also been found by other authors [5, 10, 12, 15, 16, 20]. This property of viscoelastic materials is commonly attributed to "pseudoplasticity" or also to "thixotropy". The diagram of our Fig. 3 offers an explanation of the dependence of t/ on frequency. At a constant amplitude of sinusoidal strain, the peak of the strain velocity rises in proportion to the frequency. Thus with increasing frequency the range covered by the sinusoidal strain velocity will exceed the initial linear part of the diagram to an ever greater extent, and project into the flatter parts of the S-shaped characteristic9 The viscous stress is now no longer sinusoidal and it seems difficult to judge to what point of the avis-dr~& curve the value of viscosity calculated from frequency, amplitudes of stress and strain, and phase angle is related. Although this value of viscosity, irrespective of how it may be determined, is found to decrease when the frequency rises, the representation in the frequency domain can hardly give a true picture of the dependence of wall viscosity on strain velocity. This dependence can only be represented in the time domain as described above. Finally, a consideration of the model underlying the theoretical principles applied in this study is given. Basically, the separate representation of the purely elastic and the purely viscous behaviour of the arterial wall is in conformity with the Voigt model which consists of a spring and a dashpot arranged in parallel. The spring is characterized by a linear relationship between elastic force and length, and the dashpot by a linear relationship between viscous force and velocity of stretch. In contrast to this original model, our calculation does not presuppose a linear behaviour of the elastic and viscous components. This can be seen from eq. (3) which was formulated without postulating linearity of the functionsfol andfvis. The model to which these two functions and the following equations can be related, is, therefore, a modified Voigt model. Indeed, our results have shown that both the elastic and the viscous components of the arterial wall possess nonlinear properties. The nonlinear Voigt model meets the requirements for the analysis of the viscoelastic behaviour of arteries under the condition of steady-state pulsations, which is a prerequisite for the procedure of loop elimination. This model cannot be replaced by any arrangement of multiple linear elements discussed in the literature [1, 6, 8 - 1 1 , 18, 21-23].
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References 1. Apter, J. T.: Correlation of viscoelastic properties of large arteries with microscopic structure. I, III. Circ. Res. 19, 104-121 (1966) 2. Bauer, R. D., Pasch, Th. : The quasistatic and dynamic circumferential elastic modulus of the rat tail artery studied at various wall stresses and tones of the vascular smooth muscle. Pfliigers Arch. 330, 335-346 (1971) 3. Bauer, R. D., Busse, R., Schabert, A., Summa, Y., Wetterer, E. : The determination of the nonlinear elastic behaviour of viscoelastic solids demonstrated on arteries in vivo. Pfliigers Arch. 368, R7 (1977) 4. Bergel, D. H. : The dynamic elastic properties of the arterial wall. J. Physiol. (Lond.) 156, 458-469 (1961) 5. Bergel, D. H., Schultz, D. L. : Arterial elasticity and fluid dynamics. Progr. Biophys. Mol. Biol. 22, 1 - 3 6 (1971) 6. Cox, R. H. : A model for the dynamic mechanical properties of arteries. J. Biomech. 5, 135-152 (1972) 7. Cox, R. H. : Pressure dependence of the mechanical properties of arteries in vivo. Am. J. Physiol. 229, 1371-1375 (1975) 8. Fliigge, W. : Viscoelasticity, 2rid ed. Berlin, Heidelberg, New York: Springer 1975 9. Fung, Y. C. : Comparison of different models of the heart muscle. J. Biomech. 4, 289-295 (1971) 10. Fung, Y. C.: Stress-strain-history relati0ns of soft tissues in simple elongation. In: Biomechanics - Its Foundations and Objectives (Y. C. Fung, N. Perrone, and M. Anliker, eds.) pp. 181-208. New Jersey: Prentice-Hall 1972 11. Goedhard, W. J. A., Knoop, A. A. : A model of the arterial wall. J. Biomech. 6, 281-288 (1973) 12. Gow, B. S. : The influence of vascular smooth muscle on the viscoelastic properties of blood vessels. In: Cardiovascular fluid dynamics, Vol. II (D. H. Bergel, ed.), pp. 65-110. London, New York: Academic Press 1972 13. Gow, B. S., Taylor, M. G. : Measurement of viscoelastic properties of arteries in the living dog. Circ. Res. 23, 111-122 (1968) 14. Hardung, V. : Vergleichende Messungen der dynamischen Elastizitfit nnd ViskositM yon Blutgef/iBen, Kautschuk und synthetischen Elastomeren. Helv. Physiol. Pharmacol. Acta 11, 1 9 4 211 (1953)
Pfliigers Arch. 380 (1979) 15. Hardung, V. : Dynamische Elastizit/it und innere Reibung muskul/irer BlutgefgBe bei verschiedener durch Dehnung und tonische Kontraktion hervorgerufener Wandspannung. Arch. Kreisl.-Forsch. 61, 8 3 - 1 0 0 (1970) 16. Learoyd, B. M., Taylor, M. G.: Alterations with age in the viscoelastic properties of human arterial walls. Circ. Res. 18, 278-292 (1966) 17. McDonald, D. A., Taylor, M. G.: The hydrodynamics of the arterial circulation. Progr. Biophys. Biophys. Chem. 9, 105 - 173
(1959) 18. Ranke, O. F.: Die D~impfung der Pulswelle und die innere Reibung der Arterienwand. Z. Biol. 95, 179-204 (1934) 19. Summa, Y. : Determination of the tangential elastic modulus of human arteries in vivo. In: The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 95-100. Berlin, Heidelberg, New York: Springer 1978 20. Taylor, M. G. : Hemodynamics. Annu. Rev. Physiol. 35, 8 7 - 1 1 6 (1973) 21. Wesseling, K. H., Weber, H., de Wit, B. : Estimated five component viscoelastic model parameters for human arterial walls. J. Biomech. 6, 1 3 - 2 4 (1973) 22. Westerhof, N., Noordergraaf, A. : Arterial viscoelasticity: a generalized model. J. Biomech. 3, 357-379 (1970) 23. Wetterer, E., Kenner, Th. : Grnndlagen der Dynamik des Arterienpulses. Berlin, Heidelberg, New York: Springer 1968 24. Wetterer, E., Bauer, R. D., Busse, R. : New ways of determining the propagation coefficient and the visco-elastic behaviour of arteries in situ. In : The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 35-47. Berlin, Heidelberg, New York: Springer 1978 25. Wetterer, E., Busse, R., Bauer, R. D., Schabert, A., Summa, Y. : Photoelectric device for contact-free recording of the diameters of exposed arteries in situ. Pflfigers Arch. 368, 149-152 (1977)
Received December 11, 1978
Pfltigers Arch. 380, 221-226 (1979)
EuropeanJoun'el of Physiology 9 by Springer-Verlag 1979
Separate Determination of the Pulsatile Elastic and Viscous Forces Developed in the Arterial Wall in vivo R. D. Bauer, R. Busse, A. Schabert, Y. Summa, and E. Wetterer
Institut ftir Physiologicund Kardiologieder UniversitfitErlangen-Niirnberg,Waldstrasse6, D-8520 Erlangen, Federal Republic of Germany
Abstract. The viscoelastic behaviour of arteries in vivo is analyzed by separate representation of the purely elastic and the purely viscous properties, using natural pressure and diameter pulses of various dog arteries recorded under steady-state conditions. The circumferential wall stress (a) and the radius (r) of the mean wall layer are calculated as functions of time and the hysteresis of the ~r-r diagram is represented. The stress is regarded as the sum of an elastic stress (ael) which is a function of r, and a viscous stress (o-vj~) which is a function of dr/dt. Thus ael = o- - avis. Since the a~1-r diagram must be free from hysteresis, the disappearance of the loop is the criterion that indicates that a~l has been found. O-vl~ is formulated as a second degree polynomial of dr/dt whose coefficients are determined using that criterion. The a~-r curve is always nonlinear and the elastic modulus increases with increasing radius. The avis-dr/dt curve, too, is nonlinear. Its slope decreases with increasing dr/dt. The same applies to the wall viscosity (pseudoplastic behaviour). The nonlinear properties can be represented adequately by processing the experimental data in the time domain. Problems inherent in investigations based on the frequency domain, as reported in the literature, are pointed out.
Key words: Arterial elasticity - Arterial wall viscosity - Arterial stress-strain relationship - Models of the arterial wall.
Introduction
The mechanical behaviour of the arterial wall can be characterized by elasticity, viscosity, and plasticity. Under steady-state pulsatile conditions, only elasticity and viscosity need to be considered. The viscosity gives
rise to a time lag of the pulsations of the arterial diameter with respect to the pressure or, expressed in other terms, of the circumferential walt strain with respect to the tensile stress. For this reason, the pressure-diameter diagram exhibits a hysteresis loop, from which the purely elastic and the purely viscous stress-strain relationships cannot be obtained directly. However, the separate representation of these two functions would be of considerable interest both with respect to their fundamental significance, and for a number of practical reasons. For example, a linear elastic behaviour in the range of the given pulse amplitude is a prerequisite for the application of Fourier analysis to natural arterial pulses [13]. This also applies to the viscous behaviour. Only the separation of the two functions enables one to establish whether, and to what extent, each deviates from linearity. Previous investigators determined the elastic and viscous properties of arteries either in vitro with the aid of small sinusoidal changes of pressure and volume at various frequencies [2, 4, 12, 14], or in vivo using the Fourier components of natural pulses under the assumption of linearity [7, 13]. If nonlinearities of the elastic and viscous properties are to be taken into account, the evaluation of natural pulses carried out in the frequency domain involves serious problems. For this reason, we have developed a procedure which permits the separate representation of elasticity and viscosity applying a calculation carried out in the time domain. It is the aim of this study to describe the principles of the method and to present, as examples, the results obtained on two types of arteries of the dog. An extensive application will be left to future work.
Theoretical Basis of the Method In preliminary reports [3, 24] we used the relation between the pulsatile transmural pressure and the
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external diameter for the separate determination of the elastic and viscous behaviour of the arterial wall. However, from the theoretical point o f view it is more meaningful to use the pulsatile circumferential wall stress instead of the pressure and the pulsatile length of the m e a n circumferential wall layer or the length of its radius instead of the external diameter. In practice, the shapes of the pressure-diameter and the stress-radius diagrams are very similar. The average circumferential wall stress (a) is: a =p"
ri/h
(1)
where p = transmural pressure, r i = internal vessel radius, and h = wall thickness. The radius of the mean wall layer (r) is calculated as the geometric mean: r = ]/Tii" re
(2)
where ~ = external radius. We are using the geometric rather than the arithmetic mean because the former has a more general significance. If the wall is homogeneous, eq. (2) represents the radius of that wall layer whose circumferential stress equals the stress given by eq. (1) irrespective of whether the tube is thin- or thickwalled [231. By definition, the purely elastic forces depend on the circumferential stretch itself, while the viscous forces are related to the rate of change of the circumferential stretch. We can, therefore, subdivide the pulsatile stress into two components, the "elastic stress" (a~l) which is a function of the pulsatile radius (r), and the "viscous stress" (avis) which is a function of the rate of change of the radius (dr/dt): a = a~, + a~, = f~l (r) + f~is (dr/dt)
(3)
where f~l and f~is are symbols of functions. In order to find the purely elastic behaviour of the artery, i.e. the ael-r relationship, avis must be subtracted from a. As general formulation of avis, we write: a,,~, = F . dr/dt
(4)
For a first approximation, the factor Fis assumed to be constant throughout a pulse cycle. This constant factor is called F~. Thus we have: ~1 = a - F j " dr/dt.
(5)
The C%l-r diagram must be free from hysteresis. Therefore the value of the factor F~ can be found by means of a procedure based on the disappearance of the hysteresis loop. This is done by trial and error in the following way. For each point of a pulse cyde, the time derivative of the radius is multiplied by an arbitrarily chosen value of F t . These products are subtracted, according to eq. (5), from the corresponding values of the stress. The difference is plotted against the radius
and the value of F~ is varied systematically until the loop area is minimized. In principle, total disappearance of the hysteresis loop is the criterion which indicates that the optimal value of the factor F1 and the true aol-r curve have been found. As will be shown below, a constant factor F~ generally permits a noticeable reduction of the loop area. However, it is not possible to reduce this area to an acceptable minimum. For this reason, a further approximation of the viscous term proved to be necessary in which the factor F of eq.(4) comprises a constant part and another part which is proportional to dr/dt: F = F t + F2 [ dr/dt[.
(6)
Since F is assumed to be independent of the sign of dr/dt, the absolute value of dr/dt is used. If eq. (6) is inserted into eq. (4), we obtain: av~s = (Fa + g e l dr/dt I) dr/dt.
(7)
Inserting eq. (7) into eqs, (3) and (4) leads to: a~1 = a -- (Fa + F2 l dr/dt 1)dr/dt.
(8)
N o w the factors Fa and F 2 have to be varied until the hysteresis loop reaches a minimum. As will be shown, the two-factor method generalIy leads to a virtuaIIy complete, or at least almost complete, elimination of the loop. In all cases examined so far, F2 has turned out to be negative. The physical significance of the term F 2 [dr/dt[ will be discussed below. The second-degree polynomial of eq. (7) represents the first two terms of a modified Taylor expansion of the viscous stress with respect to dr/dt. The addition of terms of a still higher order might be envisagedfrom a theoretical point of view. In practice, however,such an addition did not yield any clear improvementwith respect to the elimination of the loop, as will be discussedbelow. This also appliesto second and higher order time derivatives of the radius. Another question is whether in eq. (3) the functionsf~l and fvis should be related to the strain and the rate of change of strain rather than to the stretch and the rate of change of stretch. With respect to the functionf~, the strain would then be defined as r/ro where ro is a fixed radius, e.g. the enddiastolic radius. Replacing r by r / r o in the functionf~ would make only a formal differencewithout affectingthe shape of the respectivediagram for cry. With respect to the function fv~, the rate of change of strain would then have to be formulated as (dr/dt)/r where r is the actual radius which changes during the pulse cycle. However, the percentage changes of r are mostly not great enough to provide a means of deciding whether the use of the rate of change of stretch or of the rate of change of strain would be more appropriate. Once the factors F~ and F 2 have been determined in the way described, relationships other than that between ae~ and r can also be obtained. F r o m the a~-r curve, the circumferential elastic modulus of the arterial wall can be calculated for each point of the pulse cycle. Furthermore, the avis-dr/dt relationship is of special interest [eq. (7)] and can be used to demonstrate the dependence of the wall viscosity on the rate of
R. D. Bauer et al. : Pulsatile Elastic and Viscous Forces in the Arterial Wall c h a n g e o f the w a l l strain. F i n a l l y , the h a e m o d y n a m i cally i m p o r t a n t r e l a t i o n s h i p b e t w e e n the t r a n s m u r a l p r e s s u r e a n d the i n t e r n a l c r o s s - s e c t i o n a l a r e a o f t h e a r t e r y c a n be c a l c u l a t e d for e a c h p o i n t o f the p u l s e cycle a n d r e s o l v e d i n t o its e l a s t i c a n d v i s c o u s c o m p o n e n t s . If we return to eq. (3), we see that two general functionsfel andfv~s are first assumed. Then the function f ~ is formulated in a suitable way by eqs. (4) and (6) and the functionf~ 1determined experimentally using the criterion of loop elimination which gives us the two factors of the function f~i~- No special assumptions are made regarding the function f~i. The following approach is also possible. First, the function foj is formulated in a suitable way, for example as a polynomial [19], and its coefficients determined by the criterion of the elimination of the a-dr~dr loop. In principle, this approach would be equivalent to the procedure described above.
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Methods Recordings of intraarterial pressure and outside diameter were performed in the common carotid artery, the descending thoracic aorta, the abdominal aorta, the iliac, and the femoral arteries of 6 mongrel dogs weighing 20-35 kg. The animals were anaesthetized with sodium pentobarbital (30 mg/kg i.v.). The artery under investigation was exposed and carefully freed from surrounding tissue over a length of about 3 cm. Pressure was recorded using a cathetertip manometer (Millar PC 350A) introduced through a side branch. The pressure-sensitive membrane of this manometer type is located in such a way that lateral pressure is measured. The bridge circuit of the manometer was fed from a high-stabilized voltage source and the pressure signal amplified by an instrumentation amplifier (AD 605 L). A contact-free photoelectric device [25] was employed for the recording of the pulsatile outside diameter of the artery simultaneously with the pressure. The above-mentioned amplifier type was also used for the photoelectric device whose resolving power was about 1 ~tm. The calibration was carried out in situ during the measurements. The pressure-sensitive membrane of the manometer was located exactly at the site of diameter measurement. The frequency responses of the manometer system and of the photoelectric device were mainly determined by the amplifiers and were, therefore, almost identical. The amplitude response was virtually fiat up to 200 Hz. In this range, the time delay was about 0.2 ms and independent of frequency. The cut-off frequency was higher than 2 kHz. At the end of each experiment, the artery was excised and the wall thickness determined in vitro. Assuming wall volume constancy, the wall thickness was calculated for the in-situ conditions. The recorded data were stored on analog magnetic tape (Bell & Howell VR 3200) and processed, after analog-digital coIwersion (Hewlett-Packard 5610A), by a digital computer (HP 2100A). The sampling interval was 2.5 ms in the case of short pulse cycles and 5 ms for long cycles. From a series of steady-state pressure and diameter pulses, one cycle was selected for further analysis. An appropriate computer program allowed the systematic variation of the factors F 1 and F2 and, for each pair of factors, the calculation according to eq. (8) for the whole pulse cycle. Each result was visually inspected on the display (HP 1300A) and the respective diagrams were plotted (HP 7210A). The loop was considered to have been optimally eliminated when its ascending and descending limbs coincided at a maximum number of points. We also used the method of the least squares of the differences between the two limbs of the loop. However, the values of the factors F t and F2 determined in this way did not noticeably differ from those found by visual control. In any case, the accuracy of the loop elimination is limited by noise, as will be discussed below.
1.440.329
1.45-
r Icm]
(c)
0.360
0.329
r[cml
0.360
(d)
Fig. 1 a--d. Canine abdominal aorta, a Recorded intraarterial pressure (p) and outside diameter (D) as functions of time. Vertical arrows indicate the limits of the section used for evaluation, b Stress
(~r)-radius (r) diagram exhibiting a hysteresis loop. c Result of application of eq.(5) using F~ : 7.0 910" dyn . s/cm 3. Loop area somewhat reduced, d G r r diagram obtained by applying eq. (8) using F 1 = 13.2 - 10~ dyn - s/cm 3 and Fz : -3.8 ' 104 dyn. s2/cm *
Results F i g u r e s l a a n d b s h o w a n e x a m p l e o f the r e c o r d e d p r e s s u r e a n d d i a m e t e r o f a c a n i n e a b d o m i n a l a o r t a as f u n c t i o n s o f t i m e a n d the c o r r e s p o n d i n g s t r e s s - r a d i u s d i a g r a m e x h i b i t i n g a h y s t e r e s i s - l o o p . I n Fig. 1 c a n d d, the results o f t w o d i f f e r e n t steps in the e l i m i n a t i o n o f the l o o p are d e m o n s t r a t e d . I n Fig. 1 c, eq. (5) is a p p l i e d a n d t h a t v a l u e o f the f a c t o r F 1 is u s e d w h i c h l e a d s to the b e s t p o s s i b l e r e d u c t i o n o f the l o o p a t t a i n a b l e w i t h a c o n s t a n t factor. H o w e v e r m a r k e d r e m n a n t s o f the l o o p are still to be seen. T h e d i a g r a m n o w has the s h a p e o f a s p i n d l y " 8 " w h o s e i n t e r s e c t i o n p o i n t c a n be s h i f t e d u p a n d d o w n by slightly v a r y i n g t h e f a c t o r F 1 w i t h o u t i m p r o v i n g the result. By a p p l y i n g e q . ( 8 ) w i t h t w o f a c t o r s , F1 a n d F2, the d i a g r a m s h o w n in Fig. I d, w h i c h is v i r t u a l l y free o f a n y signs o f a l o o p , is o b t a i n e d . It is i m p o r t a n t t h a t the f a c t o r F 2 p r o v e d to be n e g a t i v e . A s a l r e a d y m e n t i o n e d , this was f o u n d to be t r u e for all o t h e r l o o p e l i m i n a t i o n s we effected. D u e to the n e g a tivity o f the f a c t o r ?2, the t w o - f a c t o r m e t h o d a l w a y s gives a g r e a t e r v a l u e o f F 1 t h a n the o n e - f a c t o r m e t h o d (cf. L e g e n d o f Fig. 1 c a n d d). T h e r e s u l t i n g G~-r c u r v e o f Fig. l d is m a r k e d l y n o n l i n e a r . T h e p u l s a t i l e r a d i u s c h a n g e is a b o u t 9 % o f the e n d d i a s t o l i c radius. A s a s e c o n d e x a m p l e , F i g . 2 a a n d b s h o w the recorded pressure and diameter of a canine femoral
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Pflfigers Arch. 380 (1979)
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[rnrn Hg] / 106 J
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GeL
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[dyn/cm ]
0.940.197
T I
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(a)
0.94 r [crn]
0.201
0.197
r[cm]
0.201
I
sec
(c)
(b)
Fig. 2 a--c. Canine femoral artery, a Recorded intraarterial pressure and outside diameter as functions of time. Vertical arrows as before9 b Stressradius diagram exhibiting a hysteresis loop. c cr~l-r diagram obtained by application of eq. (8) using F 1 = 5.0 9 10 s dyn- s/cm ~ and F: = - 2 . 3 9106 dyn. s2/cm4
artery as functions of time, and the corresponding stress-radius loop. In Fig. 2c, elimination of the loop is effected using the two-factor method [eq. (8)]. In this case, there is only a slight nonlinearity of the %-r curve. This can be understood from the small relative pulsatile radius change which is only 2 ~. The procedure demonstrated in Figs. 1 and 2 was also applied with equal success to the descending thoracic aorta and the carotid and iliac arteries of the dog. As may be mentioned, the values of the two factors determined on a series of consecutive steady-state pulses showed only negligible variations. The viscous behaviour of the arterial wall can be demonstrated by means of the avj~-dr/dt diagram derived from eq. (7). An example is shown in Fig. 3 for the same pulse cycle of the femoral artery which was evaluated in Fig. 2, and using the same factors as noted for Fig.2c. The a~i~-dr/dt curve is S-shaped. Its two limbs are concave towards the dr/dr axis, which is due to the negative sign of the factor F 2. Usually the a~j~-dr/dt diagram exhibits a shorter limb in the third than in the first quadrant because the ascent of the natural pulse is commonly steeper than its descent and, therefore, higher values of dr/dt are attained in the positive than in the negative direction.
Discussion
The practical application of the procedure described above is impaired by the noise on the recordings which is largely caused by the magnetic tape. The necessity of using the differential quotient of the radius and the square of this quotient [eq. (8)] increases the effect of the noise since differentiation of a signal with respect to the time decreases the signal-to-noise ratio. Thus the ~x-r curve shows more noise than the original a-r loop. For this reason, the accuracy of the determination of the
0.25- x 10-
~ -
/
Gvis [ d y n / c m 2] /
dr/dr [cm/sec]
0 / -
0
.
0,05
0,I
-0.25Fig. 3. Canine femoral artery, avid-dr~dr diagram calculated by means of eq. (7). Factors F1 and FE as in Fig. 2c
factors F 1 and F 2 is limited and it is difficult to decide whether the addition of further terms of third and higher orders in dr/dt in eqs. (7) and (8) would yield any significant improvement. The crvis-dr/dt curve (Fig. 3), however, is smooth because O-vis is calculated according to eq. (7) using fixed values of the factors F~ and F 2 determined before. Now our results obtained on arteries in vivo are compared, with the help of an example, with those obtained by previous authors using small artificial sinusoidal pressure and diameter changes of arteries in vitro. We restrict the comparison to experiments carried out without any mechanical contact of the arterial wall and the diameter measuring device. A quantity appropriate for such a comparison is the elastic modulus. Ranke [18], Hardung [14], McDonald and Taylor [17], and Bergel [4] have defined a complex elastic modulus composed of a real part they called dynamic modulus (Eay,), and an imaginary part they called loss modulus (Elos~ = oat/), where co is the angular frequency of the sinusoidal oscillations and t/ the coefficient of viscosity of the arterial wall [4, 14]. The modulus of elasticity calculated from our ael-r curve is comparable to Edyn and the coefficient of viscosity
R. D. Bauer et al. : Pulsatile Elastic and Viscous Forces in the Arterial Wail calculated from o u r (Tvis-dF/dt curves is comparable to t/ contained in E~o~. However, such a comparison involves a number of difficulties. Eay. and Eloss were determined assuming linear elastic and viscous behaviour of the arterial wall within the range of the small artificial sinusoidal oscillations at given frequencies. In contrast, our determinations were carried out on natural pulses whose pressure and diameter amplitudes are far greater, so that elastic nonlinearity becomes apparent and the elastic modulus varies with stretch. In addition, there is a viscous nonlinearity since, as is shown in Fig. 3, the viscous stress and also the wall viscosity depend on the rate of change of stretch. We calculate the circumferential elastic modulus (Et) from the smoothed oel-r curve using the formula:
Er = (A%,/Ar)r (1 - bt2).
(9)
In essence, this formula corresponds to that used by Bergel [4], in which the longitudinal constraint of the artery is also taken into account by the factor (1 - la2), where bt = Poisson's ratio = 0.5. (As to the problem of definition of E t, see [23]). For the canine femoral artery (Fig. 2c), we obtain values o f E t which increase from 1.4 910 v to 1.9- 107 dyn/cm 2 from the lowest to the highest part of the curve. The corresponding pressures are 110 and 140 mm Hg. The value OfEdy n found by Bergel ([4], Table 1) as the mean of the values of 5 canine femoral arteries in vitro is (1.2 _+0.08)- 10 v dyn/cm 2 at frequencies of 2 and 5 Hz and at a mean pressure of 100 mm Hg. Our value of 1.4 9 107 dyn/cm 2 determined at 110 mm Hg is in fair agreement with this figure. Furthermore, we calculate the viscosity t/ of the arterial wall as the ratio of the circumferential viscous stress to the strain velocity, i.e. the rate of change of the circumferential wall strain: Crvi~'r (1 -- ~t2).
(10)
t ~ - dr/dt The longitudinal constraint of the artery is again taken into account by the factor (1 - la2). For the canine femoral artery, the values of O-visand dr/dt can be taken from the diagram of Fig.3. One can see from the marked nonlinearity that t/ decreases with increasing strain velocity. The radius itself undergoes changes of only + 1 ~ and can, therefore, be taken as approximately constant (0.2 cm in this example). At very low values of dr/dr, we obtain t / = 0.75.10 s dyn. s/cm 2 and at the highest value ofdr/dt we find t / = 0.36 910 s dyns/cm 2. For a comparison with Bergel's data ([4] Fig. 3) we refer to his value of the loss modulus cot/ of the femoral artery noted for the lowest frequency used (2 Hz). This value is 1.35.106 dyn/cm 2. Hence t / = 1.07 9 l0 s dyn 9 s/cm 2 which is not far above our value oft/ determined at very low dr/dt. Although the strain amplitudes applied in Bergel's experiments are not
225
mentioned in his paper, one may assume that, at 2 Hz, the strain velocity did not exceed that of the almost linear part of the curve in our Fig. 3. At higher frequencies, Bergel presumably had greater strain velocities. Indeed, the values oft/calculated from his loss moduli decrease when the frequency rises. A decrease in arterial wall viscosity with increasing frequency of strain has also been found by other authors [5, 10, 12, 15, 16, 20]. This property of viscoelastic materials is commonly attributed to "pseudoplasticity" or also to "thixotropy". The diagram of our Fig. 3 offers an explanation of the dependence of t/ on frequency. At a constant amplitude of sinusoidal strain, the peak of the strain velocity rises in proportion to the frequency. Thus with increasing frequency the range covered by the sinusoidal strain velocity will exceed the initial linear part of the diagram to an ever greater extent, and project into the flatter parts of the S-shaped characteristic9 The viscous stress is now no longer sinusoidal and it seems difficult to judge to what point of the avis-dr~& curve the value of viscosity calculated from frequency, amplitudes of stress and strain, and phase angle is related. Although this value of viscosity, irrespective of how it may be determined, is found to decrease when the frequency rises, the representation in the frequency domain can hardly give a true picture of the dependence of wall viscosity on strain velocity. This dependence can only be represented in the time domain as described above. Finally, a consideration of the model underlying the theoretical principles applied in this study is given. Basically, the separate representation of the purely elastic and the purely viscous behaviour of the arterial wall is in conformity with the Voigt model which consists of a spring and a dashpot arranged in parallel. The spring is characterized by a linear relationship between elastic force and length, and the dashpot by a linear relationship between viscous force and velocity of stretch. In contrast to this original model, our calculation does not presuppose a linear behaviour of the elastic and viscous components. This can be seen from eq. (3) which was formulated without postulating linearity of the functionsfol andfvis. The model to which these two functions and the following equations can be related, is, therefore, a modified Voigt model. Indeed, our results have shown that both the elastic and the viscous components of the arterial wall possess nonlinear properties. The nonlinear Voigt model meets the requirements for the analysis of the viscoelastic behaviour of arteries under the condition of steady-state pulsations, which is a prerequisite for the procedure of loop elimination. This model cannot be replaced by any arrangement of multiple linear elements discussed in the literature [1, 6, 8 - 1 1 , 18, 21-23].
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References 1. Apter, J. T.: Correlation of viscoelastic properties of large arteries with microscopic structure. I, III. Circ. Res. 19, 104-121 (1966) 2. Bauer, R. D., Pasch, Th. : The quasistatic and dynamic circumferential elastic modulus of the rat tail artery studied at various wall stresses and tones of the vascular smooth muscle. Pfliigers Arch. 330, 335-346 (1971) 3. Bauer, R. D., Busse, R., Schabert, A., Summa, Y., Wetterer, E. : The determination of the nonlinear elastic behaviour of viscoelastic solids demonstrated on arteries in vivo. Pfliigers Arch. 368, R7 (1977) 4. Bergel, D. H. : The dynamic elastic properties of the arterial wall. J. Physiol. (Lond.) 156, 458-469 (1961) 5. Bergel, D. H., Schultz, D. L. : Arterial elasticity and fluid dynamics. Progr. Biophys. Mol. Biol. 22, 1 - 3 6 (1971) 6. Cox, R. H. : A model for the dynamic mechanical properties of arteries. J. Biomech. 5, 135-152 (1972) 7. Cox, R. H. : Pressure dependence of the mechanical properties of arteries in vivo. Am. J. Physiol. 229, 1371-1375 (1975) 8. Fliigge, W. : Viscoelasticity, 2rid ed. Berlin, Heidelberg, New York: Springer 1975 9. Fung, Y. C. : Comparison of different models of the heart muscle. J. Biomech. 4, 289-295 (1971) 10. Fung, Y. C.: Stress-strain-history relati0ns of soft tissues in simple elongation. In: Biomechanics - Its Foundations and Objectives (Y. C. Fung, N. Perrone, and M. Anliker, eds.) pp. 181-208. New Jersey: Prentice-Hall 1972 11. Goedhard, W. J. A., Knoop, A. A. : A model of the arterial wall. J. Biomech. 6, 281-288 (1973) 12. Gow, B. S. : The influence of vascular smooth muscle on the viscoelastic properties of blood vessels. In: Cardiovascular fluid dynamics, Vol. II (D. H. Bergel, ed.), pp. 65-110. London, New York: Academic Press 1972 13. Gow, B. S., Taylor, M. G. : Measurement of viscoelastic properties of arteries in the living dog. Circ. Res. 23, 111-122 (1968) 14. Hardung, V. : Vergleichende Messungen der dynamischen Elastizitfit nnd ViskositM yon Blutgef/iBen, Kautschuk und synthetischen Elastomeren. Helv. Physiol. Pharmacol. Acta 11, 1 9 4 211 (1953)
Pfliigers Arch. 380 (1979) 15. Hardung, V. : Dynamische Elastizit/it und innere Reibung muskul/irer BlutgefgBe bei verschiedener durch Dehnung und tonische Kontraktion hervorgerufener Wandspannung. Arch. Kreisl.-Forsch. 61, 8 3 - 1 0 0 (1970) 16. Learoyd, B. M., Taylor, M. G.: Alterations with age in the viscoelastic properties of human arterial walls. Circ. Res. 18, 278-292 (1966) 17. McDonald, D. A., Taylor, M. G.: The hydrodynamics of the arterial circulation. Progr. Biophys. Biophys. Chem. 9, 105 - 173
(1959) 18. Ranke, O. F.: Die D~impfung der Pulswelle und die innere Reibung der Arterienwand. Z. Biol. 95, 179-204 (1934) 19. Summa, Y. : Determination of the tangential elastic modulus of human arteries in vivo. In: The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 95-100. Berlin, Heidelberg, New York: Springer 1978 20. Taylor, M. G. : Hemodynamics. Annu. Rev. Physiol. 35, 8 7 - 1 1 6 (1973) 21. Wesseling, K. H., Weber, H., de Wit, B. : Estimated five component viscoelastic model parameters for human arterial walls. J. Biomech. 6, 1 3 - 2 4 (1973) 22. Westerhof, N., Noordergraaf, A. : Arterial viscoelasticity: a generalized model. J. Biomech. 3, 357-379 (1970) 23. Wetterer, E., Kenner, Th. : Grnndlagen der Dynamik des Arterienpulses. Berlin, Heidelberg, New York: Springer 1968 24. Wetterer, E., Bauer, R. D., Busse, R. : New ways of determining the propagation coefficient and the visco-elastic behaviour of arteries in situ. In : The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 35-47. Berlin, Heidelberg, New York: Springer 1978 25. Wetterer, E., Busse, R., Bauer, R. D., Schabert, A., Summa, Y. : Photoelectric device for contact-free recording of the diameters of exposed arteries in situ. Pflfigers Arch. 368, 149-152 (1977)
Received December 11, 1978
