Non invasive determination of the position of

obstruction in arterial system

K.B. Abdessalem ^{a, b}, W. Sahtout^b, 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.

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Abstract

In spite of the importance of the localization of a total occlusion (vascular pathology) in arterial system for any surgical operation, the literature shows a lack of research tasks which allow to an evaluation of the distance () between measurement site and pathological site. In this paper we present a theoretical model for deriving an expression of the distance (). Our method is based on the knowledge of instantaneous velocity and radius at only two sites. The simulated distances obtained by our method are in good agreement with theoretical values in different physical and geometrical condition.

Introduction

A traditional method based on the knowledge of the time lag between incident and retrograde waves and of the wave speed, provide to an evaluation of the distance between the measurement site and reflection site in temporal field:

(1)

Wherethe distance to the reflecting site, the wave velocity and the time lag between incident and retrograde waves. This method, required separation of wave on there forward and backward component, had been used by many authors [1]. Using the wave velocity and the frequency of the first minimum () of the input impedance, Milnor et al.(1989)[2] proposes an equation allowing a determination of the distance to the effective reflecting site () of the arterial trunk:

(2)

This formula had been used by Patrick Segers et al.(2000)[3] to calculate the distance to the reflecting site in vitro.

In the theory section of this work, we drive a formula of the distance of reflecting site in case of occluded artery. Then we studied, in first time, the effect of noise in the accurate of our method to simulate experimental condition. In second time, the position of reflecting site had been computed, respectively, for different attenuation values and tube length.

Theory:

Consider pulsate, laminar flow through a uniform, viscoelastic and impermeable vessels of instantaneous radius, of length L, terminated by an equivalent site of reflection, with reflection coefficient K=1(occlusion).

The fluid is assumed to be Newtonian, incompressible; the viscosity is taken to be constant, the effect of gravity was negligible. The velocity of the fluid enclosed within the cylindrical coordinates is denoted by , where r is the radial coordinate, x is the position along the vessel, t is time, u the radial velocity and v the axial velocity.

We assume that parietal deformations are small; the behaviour of the system is linear, for the range of requests applied, which is coherent with a fluid velocity small compared to wave speed.

The wall of the vessel undergoes radial motions only (wall is longitudinally tethered).

The flow is assumed to be axisymetric and No-slip condition is satisfied (the velocity of fluid at the wall equals the velocity of the wall.

The pressure does not vary much over the cross-sectional area of the vessel (Womersley [4]).

The radius is very small in front of the wave’s length .

Wave speed is large in front of the velocity of the fluid.

The model assumes the simultaneous measurements of velocity and radius at two different sites in the blood vessel of respectively coordinate and, where is the distance between measurements sites supposed to be known. Then we applied a Fourier analyse of each of the four measurements to obtain there harmonic components. From these data, the propagation coefficient can be computed accordance to a method first introduced by us [5].

Under these conditions we can treat each harmonic separately:

At the point of coordinate the velocity wave,, corresponding to nth harmonic, is the sum of an incident wave,, and a reflected wave,:

(3)

Equation (3) lead to:

(4)

The same is true at the point of coordinate:

(5)

The changes in forward or backward travelling wave are assumed to be exponential function of the distance travelled and the complex propagation coefficient, that is to say:

(6)

On the assumption of a total obstruction the velocity is null at the end of the tube:

(7)

is the distance between the reflecting site and the first measurement site.

From these assumptions and combining equations (3-7) to eliminate the forward and backward components, an expression of the unknown distance can be derived:

(8)

That lead to:

(9)

Equation (9) allows us to calculate the distance between the occlusion site and the measurement site. Its shows that the localisation of an occlusion is possible, starting from the measurement of instantaneous radius and velocities in two sections of an arterial tree or from only three measurement of velocity [7]. The method assume an known propagation coefficient that can be computed using two point method [8-9-10] or three point method [11]. Diameter of vessel can be obtained by echo-tracking [4] and the blood velocity can be also measured by ultrasound Doppler techniques [5-6].

Numerical procedure:

The numerical computation can be recapitulated in the following way (Tab.1).

The values of the instantaneous velocity and radius, at two sections, are given. The distance between measurement sections and the rheological parameters characterising the tube and the fluid are known. The validity of the numerical model has been checked for different position of the occlusion site.

Results and discussion

In order to study the effect of a variation of the physiological and geometrical conditions as well as change in the viscoelasticity of the wall and viscosity of fluid, we simulate in first time, an increase of the distance of reflecting site,, from the measurement site,, (fig.2-3) and we compute the simulated length,, for each theoretical value over the range of frequency investigated [1,10Hz]. Then, in second time, we simulate a variation of the attenuation (fig.4).

Top panels of fig.1 shows examples of radius and velocity waveforms used in simulation for computing the propagation coefficient in a vessel over the range of frequency investigated in this paper (between 1 and 10Hz). Bottom panels of fig.1 shows the harmonics content of the radius and velocity wave used in simulation.

Fig. 1 : (top panels) examples of radius and velocity signals used in simulation. (Bottom panels) harmonics content of these signals

We will test, first, the validity of our method for various physiological lengths. Typical results are presented in figure 2, corresponding to non noisy signals. This figure shows the simulated length,, computed for different theoretical location of the occlusion site, That had been varied from the measurement site by step of 5cm (=L0=0.1,0.15, 0.20, 0.25,0.3and0.35). the simulated length is plotted as a function of the frequency. Except the first harmonic, the values computed are in good agreement with theoretical values for all range of frequency investigated.

Fig.2: simulated values of distance from reflecting site, L, plotted as function of the frequency. L0=10, 15,20,25,30 and 35cm are the corresponding theoretical values. a=0.4m^-1 C=12.67m/s, x=0.05m and. d=0.04m.

Fig. 3: simulated length (see legend fig.2) corresponding to noisy signals of 5%.

Figure 3 shows in the same manner as fig.2 the results obtained in the case of noisy signals. The location of occlusion site had been varied from the measurement site by step of 5cm. The amplitude of noise is of 2% of the amplitude of signal. In this situation the simulated length shows an oscillatory behaviour. The simulated values oscillated slightly about the theoretical values. The amplitude of oscillation seems to be independent of the frequency, while the discrepancy is more graters for the first harmonic. The noise seem to affect slightly the determination of the true propagation coefficient.

Figure 4 (top panel) show three dimensional frequency patterns of Normalized length (ratio of simulated length,, and theoretical length) for several attenuation values which vary between 0.05 and 0.95 m^-1 by steep of 0.1m^-1 . The values of lengths obtained by simulation, in case of non noisy signals, are in good agreement with theoretical values. The values obtained by simulation are slightly over-estimed for the first harmonic and high attenuation, the error does not exceed 5%. The increase of the attenuation produces an error on the determination of the distance from the reflecting site, this error is 1% for an attenuation equal to (a=0.1 m^-1), and is equal to 4% for an attenuation equal to (a=0.9 m^-1). For high frequencies the distance from the site of reflection obtained by simulation becomes independent of the value of the attenuation.

Figure 4 (bottom panel) shows the effect of noise on the true determination of the distance of the reflecting site. The values of computed length oscillate slightly about the theoretical values. The discrepancy is more important at low frequency (F<4Hz) and high values of attenuation (a >0.6 m^-1 ).

Fig. 4: three dimensional view of frequency pattern of normalized length, for different attenuation (a vary between 0.05 m^-1 and 0.95 m^-1 with steep of 0.1m^-1). Tube length is L=0.443m, reflection coefficient K=1 and theoretical phase velocity C=12.76m/s, d=0.04m and x=0.05m.

Conclusion

In this work, we presented a new non invasive method which will allow an evaluation of the distance from the reflecting site by using ultrasonic velocimetric data. We have shown that for blood vessels assimilated to viscoelastic straight pipes and starting from the data of instantaneous velocity and radius at only two sections of an arterial trunk, we can localised the occlusion site. To validate this method we used simulated signals of these homodynamic sizes (figure 1). The quality of the determination is slightly affected by noise and the values of distance of reflecting sites obtained are in good agreement with the theoretical values except for some low frequency.

Neverless, the method cannot be applied in all physiological condition. For many artery the non-linear effects are no more negligible an no analytical expression of the distance to the reflecting site as a function of velocity is provided by the theory.

References:

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