Transactions on Biomedicine and Health vol 2, 1995 WIT Press, ISSN

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1 Indirect instantaneous velocitiy profiles and wall shear rate measurements in arteries: a centre line velocity method applied to non newtonian fluids P.Flaud,*A.Bensalahk "Laboratoire de Biorheologie et d'hydrodynamique Physicochimique, CNRS URA 343 and Universite Paris VII, 2 Place Jussieu, Paris, France ^Laboratoire d'informatique et de Mathematiques Appliquees, C.N.C.P.R.S.T., B.P. 1346, Rabat, Morocco Abstract Noninvasive measurement of the arterial wall shear rate has been studied in many works, in relation with the various mechanical factors involved in atherosclerosis genesis (Nerem [1], Noller [2]). In a previous work (Bensalah, [3]), it has been shown how to compute reliably this factor in the case of a linear newtonian flow. This method required the only measurement of the arterial centre line velocity, by means of the ultrasonic Doppler velocimetry. Furthermore it avoided the use of curve fitting methods (Lou, [4]), inefficient because of the rather bad quality of the velocity determination near the wall (convolution effects and sensitivity of the apparatus). The present work deals with the extent of this method when accounting for the non newtonian behaviour of the blood. In that case, the numerical solution of the Navier-Stokes equations requires the approximation of the longitudinal non linear terms and the modelling of the shear thinning rheological behaviour of the blood. Thus, it can be shown that the best way for determining in vivo the arterial wall shear rate is to measure the instantaneous axial centre line velocity (Flaud, [5]), and the instantaneous radius, thus accounting for the linear or non linear elastic properties of the wall (mainly in the greatest arteries). The non newtonian rheological behaviour of the blood can be easily measured and modelised. It is significant when: i) the wall shear rate is less than 100 s^, ii).the rheological behaviour of the blood is abnormal and highly non newtonian, iii) back flow occurs.

2 192 Computer Simulations in Biomedicirie 1 Introduction In spite of recent progress in velocimetric technics, blood velocity profiles in human arteries still remain difficult to be obtained near the arterial walls. In this region the shape of the velocity is strongly affected by different artefacts such as the moving walls, or the convolution effects. Efficient deconvolution procedures have been proposed (Bensalah, [6]), which improve the quality of the measurement. Nevertheless they give unsatisfactory results when considering the evaluation of the wall shear rate. The use of curve fitting, as attempted by different authors, is an alternative method which is very difficult to be used with convoluted and noised velocity profiles (Lou, [4]). An other way consists to use a linear mathematical model (Bensalah, [3]) and the measured centre line velocity. Nevertheless, these theories neither account for the non linear and high amplitude wall displacement, nor for the non newtonian behaviour of the blood. In the present work, a theory is presented and discussed. It assumes the knowledge of both the instantaneous diameter and the instantaneous velocity measured near the center of the vessel, where noise level and convolution effects are minimized. The additional knowledge of the blood and wall rheological properties allows to reconstruct the whole instantaneous velocity profile, the wall shear rate or wall shear stress, and the pressure gradient generating the flow. 2 Mathematical model The description of the non newtonian fluid flow in an elastic tube requires the knowledge of the mechanical behaviour of both the fluid and the wall. a/ mechanical properties of the wall The mechanical properties of the vessel wall have been extensively studied during the last twenty years. Numerous mechanical models have been proposed to describe the non linear Pressure P(t) Radius R(t) relationship. Assuming a viscoelastic behaviour with a static E and a dynamic E^ Young's modulus, the radius pressure relationship can be written as (1): where ^»o"ov^s ^d ; (1) P0 is the mean value of the transmural pressure, RQ and hg the radius and thickness of the tube at zero transmural pressure. This relationship can be generalised for non linear materials with E^(k) and E^(X) (X = R/RQ), or for sake of simplicity in the following way (1):

3 Computer Simulations in Biomedicine 193 /, * * _p_) where R is the mean value of the radius, and C is the wave speed defined as: _ \1 2, 4 R h08qe b Rheological behaviour of the fluid ^~ti 0,00 0,00 1, Shear rate (Log) 3,00 Fig. 1: comparison between the model (solid line) and the experimental data (human blood). Numerous models have been proposed to describe the rheological behaviour of the blood viscosity. A non newtonian fluid like blood exhibits two viscous newtonian plateaux for very low shear rate (p^), or high shear rate (m^), and a transition zone, the position and the range of which can be characterised by two parameters, (i^andp). Among these models, the Quemada's one, based on a theory of minimisation of dissipated energy can be considered as a reference one. But a simple phenomenological one, such as Cross model defined by (2): gives also results allowing a good description of the experimental data (Fig. 1). c/ Fluid dynamic We assume a fully developed laminar axisymmetrical flow in an elastic tube filled with a non newtonian fluid. In such a case, the Navier Stokes equations can be expressed as (3): d\v dw chv 1 dp 1 d ( <3w d\\ + u + w = - + rji( ) (3) di dr dz. p dz pr dr \ dr or (2) fe ^L = o with: u(r) = and w(r) = 0 dr r dz di

4 194 Computer Simulations in Biomedicine where u and w are respectively the radial and the longitudinal velocity, r and z the radial and longitudinal co-ordinate, and JA( ) the non newtonian viscosity dr of the fluid as expressed by (2). Introducing a reduced variable for the radial coordinate and using the approximation suggested by Ling and Attabek (Ling [8]) for the longitudinal velocity gradient, this set of equation can be written as (4): aw _ iap i ar aw i aw S(^) w w at' p az' R at' ^ arj r arj Sm(l) Sm( 1) J ±8P8R(2. R az' dp(r\ ar ^ Sm(l) )) ' Sm(l) 1 R 2 j_aw rj ar 1 p,q- i<x) I 1 a\v aw Icomp p(l+comp)[ T] dr\ drf 1 + comp dr? where z'=z, r = r/r(t), t'=t, S(r() = J^Tiwdrj, Sm(rj) - J^TI w dr, and comp = - R This equation appears mostly as a generalisation for a non newtonian fluid of the Ling Attabek model which allows, assuming the knowledge of R(t) and (t), the computation of the resultant velocity profile, as soon as the a% rheological behaviour of the fluid (v>p<h^lo and the wall (via the wave speed C) are known. The numerical computation is stable and convergent if: where dr\ = dr / R is the reduced increment in radial displacement, 9 = dt / T is the reduced increment in time, and T a characteristic time, for instance the period for a periodic flow Nevertheless, for biomedical applications, the pressure gradient is quite difficult to measure in an atraumatic way and this method has to be rejected. In an other hand, the measurement of the centre line velocity is precise, and fully atraumatic. Moreover, the convolution effects are minimised in the core of the vessel (Bensalah, [6]). The echographic technics allows a precise measurement of the instantaneous diameter, and the wave speed can also been evaluated in an atraumatic way. For these reasons, we present an alternative method using the measured instantaneous centre line velocity w^(t) instead of the pressure gradient in order to evaluate the wall shear rate, the wall shear stress, the whole velocity profile, and the pressure gradient governing the flow. This approach is similar to

5 Computer Simulations in Biomedicine 195 a previous work (Flaud, [7]), but account for non linearity both of the fluid and the vessel wall, and doesn't assume a periodic flow. This method is based on the following equation (5), obtained from (3) at the center of the vessel (n=0): r 2 (5) R ^at' Sm(l) Sm(l) dz' cp) R* p drf which allows, at a given value of the time to compute the pressure gradient if the instantaneous centre line velocity w^(t) and radius R(t) are known. 3 Numerical model The numerical computation can be summarised in the following way (see Tab 1): The values of the instantaneous centre line velocity and radius are given, and the rheological parameters characterising the tube and the fluid are known. An arbitrary velocity profile is used to initialise the computation. Read Wcl(t) and R(t) Read rheological parameters Arbitrary initial velocity profile /\ Computation of the pressure gradient at t - Computation of the velocity profile at t+dt I Computation of the radial velocity Wall shear rate, wall shear stress. Tab. 1: Simplified representation of the numerical procedure. At each time t, the associated pressure gradient is computed by (5), which allows, using (3) the computation of the velocity profile at the next time step t+dt. The radial velocity 11(1,2%,t + dt), the wall shear rate y(r.z^.t + dt), and wall shear stress i(r.zq,t + dt) can then be computed. In a first approach we used simulated centre line velocity w (t) and radius R(t) to get evidence of the effects of the non newtonian behaviour of the fluid, as presented in the next section. The validity of the numerical model has been checked both with Womersley analytical solution (for unstationnary flow of newtonian fluids in

6 196 Computer Simulations in Biomedicine rigid tubes) and with non newtonian static flows in rigid tubes. In these cases, the numerical solution matches exactly the analytical one. 4 Results and discussion In order to get evidence of the non newtonian effects, the newtonian viscosity was taken equal to the high shear rate viscosity of the non newtonian fluid, as commonly admitted. Typical results are presented which correspond to the flow conditions in middle or large size arteries, where the non newtonian effects are generally neglected. The use of the centre line velocity method implies that the velocity is given at the vessel axis, and known (zero) at the wall. The knowledge of these two values of the velocity independently of the rheological behaviour of the fluid implies that this non newtonian behaviour moderately affects the shape of the computed velocity profiles. (Fig 2). Reduced radial position Reduced radial position ig. 2: Comparison between an non newtonian effects on computed velocity profiles, for large size arteries (R=6mm., on the left) or middle size arteries (R=4mm, on the right), and at t\vo values of the reduced time: t/t Nevertheless, it appears that when back flow occurs, the non newtonian effects cannot be neglected and give significant differences in the shape of the velocity profile. This observation has to be related to the small value of the shear rate after the systolic peak. When the shear rate is less than 100 s-\ the non newtonian effects induces a noticeable enhancement of the viscosity of the fluid (see Fig. 1), which cannot be neglected. The incidence on the computed wall shear rate is more complex to interpret when comparing the results corresponding to large or middle size arteries (Fig. 3). The main point to be underlined is the decrease of the wall shear when the size of the arteries is decreasing. This induces an obvious increase of the non newtonian effects.

7 Computer Simulations in Biomedicine ,J 7 Non newtonian Reduced time t/t --/OO - Non newtonian Reduced time t/t 0,J 7 A Non newtonian Reduced time t/t Fig. 3: Comparison between newtonian and newtonian effects on computed wall shear rate for large size arteries (R=6mm. top left), or middle size arteries (R=4mm. top right and R=2mm.) On an other hand, the computation of the pressure gradient seems to be more or less unaffected by the non newtonian behaviour of the fluid (Fig. 4) VOOO ^ 2000 t "# (3 i bo 0 M ^ a o g V «& tf* \lawt A Non newtonian ^ JOOO - A Non newtonian Reduced time t/t gradien Non newtonian reduced time Fig. 4: Comparison between newtonian and newtonian effects on the computed pressure gradient for large size arteries (R=6mm. top left), or middle size arteries (R=4mm. top right and R=2mm.)

8 198 Computer Simulations in Biomedicine This can be related to the order of magnitude of both the viscous terms in equation (5) which can be neglected for a blunted velocity profile, and the non linear one which appear also as corrective terms quasi independent of the non newtonian effects. 5 Conclusion In this work, our interest was focused on the noninvasive evaluation of the arterial wall shear stress by using ultrasonic velocimetric data We have shown that for blood vessels assimilated to viscoelastic straight pipes, the centre-line velocity method is the most efficient way for the in vivo determination of the arterial wall shear rate, the wall shear stress, the pressure gradient or the flow rate. The quality of the determination is improved when measuring the instantaneous radius and thus taking into account the linear or non linear elastic properties of the wall (mainly in the great arteries). The non newtonian rheological behaviour which is generally neglected and which is shown to have little influence on the computation of the pressure gradient, has to be taken into account when: i/ the wall shear rate is less than 100 s^ \i/ the rheological behaviour of the blood is abnormal iii/ back flow occurs. Nevertheless, the centre-line velocity method cannot be applied everywhere in the arterial systemic tree For singular sites (bifurcation, bending), non-linear effects are no more negligible and no analytical expression of the wall shear rate as a function of the centre-line velocity is provided by the theory. The unique approach remains the curve fitting of velocity profiles which is not practicable for convoluted profiles In this case, one has to attempt a deconvolution of the arterial velocity profiles 6 References I/ Nerem R M. Vascular Fluid Mechanics, the arterial wall, and Atherosclerosis. Journal of Biomechanical Engineering Transactions of the AS ME Vol. 114, n 3, pp , Noller, M. U, Hall, E R., Eskin, S. G. and Mclntire L V The effect of the shear stress on the uptake and metabolism of arachidonic acid by human endothelial cells. Biochimica et Biophysica Acta, 1005 : 72-78, / Bensalah, A. and Flaud, P. Non-traumatic study of the arterial system by indirect determination of clinical parameters. A theoretical linear model. Computer Method in Water Resources II, Vol 3, Computer Aided Engineering in Water Resources, Eds Brebbia C.A., Ouazar, D, Bensari, D. Computational Mechanics Publications, pp , Rabat, Morocco, Feb / Lou, Z, Yang, W.J. and Stein, P.D Errors in the estimation of arterial wall shear rates that result from curvefittingof velocity profiles. J. Biomechanics, 26 : , 1993

9 Computer Simulations in Biomedicine 199 5/ Flaud, P., Bensalah, A., Counord, J.L., Levenson, J and Simon, A. A new geometric procedure for in vivo pulsed Doppler evaluation of velocity distribution inside the diametrical section of large arteries in humans. Annals of Biomedical Engineering, Vol. 18:1-13,1990 6/ Bensalah A. and Flaud P.: Ultrasonic indirect arterial wall shear rate measurement: comparison between a centre-line velocity method and curve fitting of velocity profiles. Effects of the convolution. Computational in Biomedicine (II), Eds: Brebbia, Hart, Power, Ciskowski,. Computational mechanics publications, , 1993 II Flaud, P., Bensalah, A. and De Jouvenel, F. Experimental and theoretical determination of the wall shear stress in an unsteady periodic flow. Euromech 272, Response of shear flows to imposed unsteadiness, pp , Aussois, France, Jan / Ling, S., Atabek, H, A non linear analysis of pulsatile flow in arteries. J Fluid Meek Vol. 55 part 3 pp , 1972.