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New non-invasive method of determination of wave speed and attenuation in arterial system K.B. Abdessalem a, b, W. Sahtoutb, P. Flaud a, M.H. Gazzah c, Z. Fakhfakh b a Laboratoire Matière et Systèmes Complexes, , CNRS URA 343, Université Paris VII, 2, Place Jussieu, 75005 Paris, France b Département de Physique, Faculté des sciences de Sfax, Tunisia c Département de Physique, Faculté des sciences de Monastir, Tunisia Khaled Ben Abedessalem : Faculté des sciences de Sfax, 3018, Tunisia. E-mail address: khaledabdessalem@yahoo.fr Abstract The propagation coefficient of pulse wave’s harmonics in an artery can be determined by measuring pulsating blood velocity and radius at each of two points along the vessel. In this work an expression of propagation coefficient in pulsatile flow through a viscoelastic vessel is derived and used to study the effects produced by changing in physical tube parameters. We consider in particular, the effects of increasing the reflection coefficient, on the determination of the true propagation coefficient. The method developed here is based on the knowledge of instantaneous velocity and radius values at only two sites. It takes into account, a reflection site of unknown reflection coefficient, localised in the distal end of the vessel. The results of wave speed and attenuation obtained from simulation of this method are in good agreement with theoretical values. Keywords: Pulse wave propagation, Wave reflection, phase velocity, attenuation. 1. Introduction: The determination of propagation coefficient is of a obtained by some authors [5] using the three points great importance. To describe the contribution of method, measuring pressures at three sites, is not in arterial properties on pressure, velocities and radius good agreement with the theory. Phase velocity found wave propagation, the propagation coefficient should by these investigators was 20% lower as the attempted be calculated. Attenuation, a, is related to the energy value, and the attenuation coefficient was about eight dissipation due to the blood viscosity, viscoelasticity of times higher. In other researches, an agreement was the wall, reflection in bifurcations and disease sites as found between theoretical and experimental atherosclerosis or occlusion [1]. The imaginary part of propagation coefficient values, for uniform tube [6]. propagation coefficient is inversely proportional to the A second approach in case of unknown reflection phase velocity that represents a parameter of clinical coefficient, using four transducers, requiring a pair of interest [2]. The determination of the propagation pressure and flow measurement at two sections of the coefficient in arterial system has been the subject of vessel investigated, has been described by Milnor and many researches in the last few decades. In absence of Nichols [7], and used by Milnor and Bertram [8] to significant reflection or when the reflection coefficient measure the propagation coefficient in the femoral and of the distal site is known (equal to unity), a two-point carotid artery. Bertram et al. [9] described, an iterative method based on the measurement of pressure or flow general method, for calculating propagation coefficient in two sites [3] can be used on the derivation of the true using two pressure measurements, one flow rate and propagation coefficient. Investigators using two-point one vessel diameter. method [4] found values in agreement with the All the methods found in literature, used for Womersley theory. The accuracy of this two-point derivation of propagation coefficient are based on method is acceptable, but it has a limited application : invasive measurement. We solve this problem by using It assumes a known reflection coefficient produced by a non-invasive method based on ultrasound a total occlusion, which is not the case in most measurements of velocities and radius at two sites hemodynamic conditions, when the lumen of artery is separated with a known distance (d). This method partially or not occluded. doesn't require the knowledge of the reflection Without any restrictive assumption about the coefficient as it is generally the case in major literature reflections from local vascular sites, Taylor developed works using two-point method. We will critically re- a method known as three-point method. It is based on examine the exactitude of the method of wave speed the measurements of pressure or flow in three and attenuation determination, for different equidistant sites of the vessel of interest. The results hemodynamic conditions, by using numerical 1 simulations. In particularly we will study, the effects of At point x (2) the expression of forward velocity is increasing the reflection coefficient. 1 V V2fn = V2 n + R2 n 1an (3) 2. Mathematical model 2 R1an Consider pulsate, laminar flow through a uniform, viscoelastic and impermeable vessels of Where Vin the running velocity waves at location f instantaneous radius R ( x, t ) , of length L, and i, Vin the forward running velocity waves at location terminated by an equivalent site of reflection, with i, f Rin the running radius waves at location i, Rin the reflection coefficient K. The fluid is assumed to be Newtonian, and forward running radius waves at location i, V1an and incompressible; the viscosity is taken to be constant, and the effect of gravity is negligible. The velocity is R1an are respectively the component of running axial denoted by V = [u ( r , x, t ), v ( r , x, t )] , where r is the velocity and radius waves function of Womersley radial coordinate, x is the position along the vessel, t is parameter, at location 1, U 1an component of running time, u the radial velocity and v the axial velocity. We radial velocity waves, dependant of variable r, at assume that parietal deformations are small; the location 1. behaviour of the system is linear, which is coherent Since the forward wave at site (1) can been expressed with a fluid velocity small compared to wave speed. as a function of the forward wave at site (2), we can The signal is assumed to be periodic. write for sites (1-2): V2 fn = V1n .e −γ n .d (4) Let us consider a vessel, with a distal reflection site. f We will take two measurements of radius and velocities at two sites separated by the distance (d). At With a development limited, of order 3, in the vicinity each point of the vessel, the velocity or radius which is − γ .d associated to the nth harmonic is the sum of two of zero of e and using equation (2) and (3), we components, a forward and backward travelling wave. can obtain an equation of fourth degree with complex The forward component for each harmonic at point x coefficients: the unknown complex coefficient is can be expressed as a function of total radius and propagation coefficient, and H is a function of the velocity using an expression derived by Rogova [10]. Womersley parameter α=R(ω/ν)1/2. At point x (1) the expression of forward velocity is 1 V Expressions (4), shows that the determination of V1nf = V1n + R1n 1an (2) propagation coefficient is possible, starting from the 2 R1an measurement of instantaneous radius and velocities in two sections of an arterial tree. The diameter of vessel can be obtained by echo-tracking; blood velocity can be also measured by ultrasound Doppler techniques. d 3V2n 3 d ²V d 3 HR2n 2 d ² HR2n γn 4 + γ n ( 2n + ) + γ n dV2n + + γ n (V2n − V1n + R2n dH ) + (R2n − R1n )H = 0 (4) 6 2 6 2 3. Results: Fig. 1: examples of signals used in simulation and there discrete spectrum at x=5cm of the origin of the tube (non noisy signals). 2 The number of harmonic of velocity and radius signal method are in good agreement with theoretical values is n=15, the theoretical wave speed is C=8.78m/s, the for low and high reflection coefficient tube length is L=44.3cm and the attenuation a=0.4m-1. Fig. 4a shows three dimensional frequency pattern of Figure 2 shows the frequency pattern of normalized wave speed, between 1 and 10Hz for different values of phase velocity between 1 and 15Hz, in a tube of length reflection coefficient, which increases from 0.06 to L, for low reflection coefficient (top panels) and high 0.96. The phase velocity values computed by RV-two reflection coefficient (bottom panel), obtained using a point method are in close agreement with theoretical small distance (d=3cm, left) or a large values over the investigated range of frequency. The distance(d=8cm,) between transducers based on augmentation of reflection coefficient has no effect on sections 1-2. In each panel we plotted the apparent the determination of the true phase velocity. phase velocity derived from the following equation γ n = (1 / d ) ln( φ1n − φ 2 n ) where φ in is the Figure 4.b shows three dimensional frequency pattern of Normalised attenuation, between 1 and 10Hz for phase of the nth harmonic. The phase velocity different reflection coefficient values, which increase computed by our method (RV-two point method), for from 0.06 to 0.96. Attenuation computed on all the same physical and geometrical state in the case of sections, is also in a good agreement with theoretical non noisy signals. values. Increase of reflection coefficient has no effect Figure 3 shows the results of Normalized attenuation on the determination of wave speed computed by RV- over the range of frequency in a similar manner as in method. The deviations of attenuation values computed fig.2. Figures 2 and 3 show a good agreement between by RV-method over the range of frequency are very simulated and theoretical wave speed and attenuation. small; it varies between 1.001 and 0.999 times of Values obtained are close to the theoretical one at low theoretical values for all values of reflection coefficient and high values of the reflection coefficient. While used over the investigate range of frequency. The apparent wave speed and attenuation show an deviations of phase velocity and attenuation, are small oscillatory behaviour over the range of frequency compared to theoretical values, the maximum of wave investigated. Moreover the amplitude of apparent wave speed obtained by RV-method over the range of speed and attenuation increases with the augmentation frequency investigated is 1.001 times the theoretical of reflection coefficient, as shown by fig. 2 (bottom). values; the minimum simulated wave speed was about The phase’s velocity and attenuation, derived from our 0.999 times the theoretical phase velocity. Fig. 2: Normalized phase velocity RV-method: without noise (∆), normalized apparent phase velocity (o), for uniform viscoelastic tube, of length L=43.3cm at low reflection coefficient K=0.36 (top panel) and high reflection coefficient K=0.86 (bottom panel). The panel in the left column show data obtained for (d=3cm), the panel in the right column show data obtained for (d=8cm).C=8.67m/s 3 Fig. 3: Normalized attenuation RV-two point method: without noise (∆), normalized apparent attenuation (o), for uniform viscoelastic tube, of length L=43.3cm. The legend panels are the same as in (fig.2). Fig. 4: three dimensional view of frequency pattern of normalized phase velocity (left) and normalised attenuation (right) for different reflection coefficient values (K=0.06: step=0.1:0.96) 5. Conclusions: tube wall behaviour can be solved in the frequency domain. The aim of our study is to present a theoretical linear The simulation of the mathematical formula of this method, for estimating propagation coefficient in method can be summarized as follows: arterial vessels. We have validated the method by In all case studied, both attenuation and phase velocity simulation and investigated the effect produced by the computed by our method are in good agreement with increase of reflection coefficient on the determination theoretical values. Small deviations are shown at high of propagation coefficient. Our method based on four frequency when increasing reflection coefficient. The arterial wave form measurements, (radius and centre accuracy of the method increases when decreasing line velocities at two arterial sites) can be applied non- frequency. The discrepancies between theoretical and invasively using Doppler techniques. Our investigation computed values of the attenuation are larger than based on Womersley and transmission lines theories, those of the phase velocity for the same physical and takes into account the viscoelastic behaviour of wall geometrical conditions, but still very small. The effect vessel. Linear analysis, has the advantage that the of noise can be studded in a next time to evaluate the system of equations taking into account the viscoelastic applicability of the method in vivo. 4 References [1]. Alderson H, Zamir M. Smaller, stiffer coronary bypass can moderate or reverse the adverse effects of wave reflections. Journal of Biomechanics 2001; 34:1455-1462. [2]. Sawabe M, Takaahashi R, Matsushita S, Ozawa T, Arai T. Aortic pulse wave velocity and the degree of atherosclerosis in the elderly: a pathological study based on 304 autopsy cases. Atherosclerosis 2005; 179:345-351. [3]. Ursino M, Artioli E, Gallerani M. An Experimental comparison of different methods of measuring wave propagation in viscoelastic tubes. Journal of Biomechanics 1994; 27:979- 990. [4]. Li J, Melbin KJ, Riffle J, Noordergraaf A. Pulse wave propagation coefficient. Circulation Res. 1981; 49:442-452. [5]. McDonald, D.A. Blood flow in Arteries. Arnolds, London (1974). [6]. Milnor, W.R.: Hemodynamics; Williams & Wilkins, Baltimore (1982). [7]. Milnor WR, Nichols WW. A new Method of measuring propagation coefficients and characteristic Impedance in blood vessels. Circulation Res. 1975; 36:631-639. [8]. Milnor WR, Bertram CD. The relation between arterial viscoelasticity and wave propagation in the canine femoral artery in vivo. Circulation Res. 1987; 43:870-879. [9]. Bertram CB, Greenwald SE. A General method of determining the frequency-dependent propagation coefficient and characteristic impedance of an artery in the presence of reflections. Journal of Biomechanical Engineering 1992; 114:2-10. [10]. Rogova Irina : Propagation d’ondes en hémodynamique artérielle : Application à l’évaluation indirecte des paramètres physiopathologiques (Thesis Paris 1998). 5