Flow-tissue interaction with compliance mismatch in a model stented artery

https://doi.org/10.1016/S0021-9290(03)00259-8Get rights and content

Abstract

The insertion of an endovascular prosthesis is known to have a thrombogenic effect that is also a consequence of the interaction between the flowing blood and the stented arterial segment; in fact the prosthesis induces a compliance mismatch and a possible small expansion along the vessel that eventually gives rise to an anomalous distribution of wall shear stresses.

The fluid dynamics inside a rectilinear elastic vessel with compliance and section variation is studied here numerically. A recently introduced perturbative approach is employed to model the interaction between the fluid and the elastic tissue; this approximate technique is first validated by comparison with a complete solution within a simple one-dimensional model of the same system. Then it is applied to an axisymmetric model in order to evaluate the flow dynamics and the distribution of wall shear stress in the stented vessel.

Compliance mismatch is shown to produce more intense negative wall shear stresses in the stented segment while rapid variations of wall shear stress are found at the stent ends. These effects are enhanced when the prosthesis is accompanied by a small increase of the vessel lumen.

Introduction

The unsteady flow in conduits with elastic walls is a topic relevant to many different fields. It is of particular interest for cardiovascular studies in order to understand the dynamics corresponding to a specific physiology, predict the evolution of pathologies due to vessel deformation and the hydraulic behaviour after the insertion of a prosthesis.

The birth and development of arteriosclerosis is known to be closely related with the presence of a locally irregular flow, as a consequence of the boundary-layer separation and the vorticity layers formation that cause an unnatural distribution of wall shear stress. The recirculating flow, which is associated with low shear values (also reversed with respect to the undisturbed case) and their irregularity along the vessel are directly correlated with the presence of arteriosclerosis plaques, i.e. an alteration of the arterial wall, characterised by an accumulation of lipidic material and a progressive thickening of vascular walls. This process eventually causes stenosis, a narrowing of the vascular lumen whose entity represents a determining factor for the ictus risk. The normal surgical treatment for arterial pathologies consists of a cut on the vessel neck to remove the nidificated fat that reduces the blood flow lumen. The endovascular technique is another less invasive option. It consists of a prosthesis insertion, a stent generally made of a stainless steel mesh, usually inserted in the narrowed artery through the inguinal canal (groin), by using a catheter. When the catheter reaches the artery narrowed section, the compressed stent is released and expanded, by means of a balloon or special materials, commonly producing a slight enlargement of the previously occluded vessel.

The stent itself may present a thrombogenic power because it also induces irregularities in the flow at the prosthesis place caused by the slight geometrical modifications and by the compliance mismatch with the surrounding vessel. The experimental works (Campbel et al., 1999; Back et al., 1994) studied the response of the arterial wall to the insertion of an endovascular prosthesis, some aspects of the biomechanics interaction between a stent and a stenotic artery have been analysed (Auricchio et al., 2001; Fabregeus et al., 1997; Dumoulin and Cochelin, 2000) and possible modified typologies have been proposed in order to limit the restenosis process; an estimation of the wall shear stress distribution in a stented artery has been reported in Wentzel et al. (2000) as a factor connected to the cellular growth and thrombi increase.

In this work we study a pulsatile flow inside a circular vessel with variable elastic properties, either with a constant or a variable section, as a basic model for a stented elastic artery. The objective is to verify the main flow modifications that occur in presence of a compliance mismatch which is possibly also associated with a slight change in the vessel section.

The analysis is performed by computational techniques. The modelling of flow over elastic boundaries deserves several difficulties because the fluid and the wall equations are, in general, coupled in a strong interaction. Their unsteady coupled solution is so time consuming that the studies about fluid-structure interactions are often accompanied by the presentation of modified techniques adapted for the specific situation (Cancelli and Pedley, 1985; Wang and Tarbell, 1992; Luo and Pedley, 1996; Davies and Carpenter, 1997; Lucey et al., 1997; Pedrizzetti, 1998; Wiplier and Ehrenstein, 2000). The first numerical approaches to a complete solution of the three-dimensional flow into compliant vessels (Anayiotos et al., 1994; Perktold and Rapitsch, 1995) have been performed by alternating the solution of the fluid and tissue equations by separated numerical packages and require a relevant computational and tuning effort.

The perturbative approach, first introduced in Pedrizzetti et al. (2002) for the solution of pulsatile flow inside moderately elastic arteries, is either tested and employed here. It is an approximate method that appears appropriate for the study of spatially limited district in large artery flow. In fact, as partially explained in Pedrizzetti et al. (2002): (i) the pulsation period is very large when compared with the convective time scale (e.g. the Strouhal number is small, order 10−2), (ii) the pressure variation along the vessel is small when compared to the physiologic (systolic–diastolic) pressure changes in time, therefore, locally, propagation phenomena are negligible; and finally, (iii) the artery stiffness is large enough that wall deformation is of relatively small entity, below 10% of the vessel diameter, and a perturbation parameter can be defined.

A preliminary evaluation of the flow alteration in a stented rectilinear vessel is done here by using a complete coupled solution under a one-dimensional approximation. The new perturbative technique is then applied to the same one-dimensional problem in order to verify the validity and limits of the perturbative approach. The one-dimensional approximation is advantageous to grossly describe the system behaviour, to preliminarily estimate the most important parameters, and to verify the confidence of the perturbative approach. Afterward the flow details are worked out using an axisymmetric approximation. Through this model we analyse the fluid-wall interaction, the dynamics of the boundary layer, and the space-time pattern of the wall shear stress.

The physical problem is formulated in Section 2; the one-dimensional approximation with the complete numerical method and the perturbative method are reported in 3 One-dimensional formulation, 4 One-dimensional perturbative formulation, respectively. The application of the perturbative approach to the axisymmetric formulation is given in Section 5. The major results are reported in Section 6 where also the role of the free parameters is discussed. The conclusions with a summary of the most important issues are drawn in Section 7.

Section snippets

Definition of the physical problem

We model the pulsatile flow inside a vessel after the positioning of a stent. The physical problem is that of a deformable vessel, with a rectilinear axis and a circular cross section. We consider both the case with a constant section and that of a slightly increase of the radius in correspondence of the stiffness increase. These geometries model the insertion of a stent which is perfectly adherent to the vessel walls and the variation of elasticity due to the stiffer prosthesis. Let us fix the

One-dimensional formulation

The vessel axis is the x-axis of a Cartesian system of coordinates. In the one-dimensional approximation we assume that pressure p(t,x) and longitudinal velocity u(t,x) are uniformly distributed across the vessel section. Under these conditions the Navier–Stokes equations are reduced to the one-dimensional Euler equation:u∂u∂x+St∂u∂t+∂p∂x=0,in addition to the continuity equationSt∂Ω∂t+∂Q∂x=0,where Ω(x,t)=πR2(x,t). When dealing with an elastic boundary an additional equation, the law of the

One-dimensional perturbative formulation

The wall motion in the main arteries is characterised by relatively small amplitudes. The deformation is commonly smaller than the 10% of the vessel diameter in response to the typical systolic–diastolic pressure fluctuations; typical values of the dimensionless elasticity coefficient ε given by (11) are contained in a range between 10−4 and 10−2. We therefore propose the use of a perturbative approach, for the interacting wall-fluid system, based on the presence of the small parameter ε.

The

Perturbative axisymmetric formulation

We consider an incompressible Newtonian viscous fluid, with density ρ and kinematic viscosity ν, flowing with the pulsatile flow volume rate (4) and pressure (5). Assume the duct axis as the x-axis of a cylindrical system (x,r,θ), the approximation of axial symmetry makes the flow independent of the θ-coordinate. The governing equations are the axisymmetric form of the Navier–Stokes equations, which are written in the vorticity-streamfunction formulation asSt∂ω∂t+Jψ,ωr=1Re∇2ω,ω(x,r,t) is the

Results

The value of the Reynolds number Re=300 has been chosen in addition to the dimensionless parameters already specified in the preliminary one-dimensional analysis of Section 3.2.

The flow dynamics is first outlined in the rigid vessel case, which also constitutes the zeroth-order term of the perturbative solution. The instantaneous vorticity fields and the corresponding streamlines are reported in Fig. 6 for ΔR=0.1 during systole and diastole. The dynamics when ΔR=0 is a well-known one and can be

Conclusion

The fluid dynamics in a rectilinear vessel after the insertion of an endovascular prosthesis is here studied numerically. The influence of compliance mismatch, and possibly area variation, to the distribution of wall shear stress is analysed.

The interacting dynamics between the fluid and the elastic tissue is formulated by a perturbative approach, introduced previously, that is well suited for the flow into large arteries. This method transforms the coupled fluid-tissue problem into a cascade

Acknowledgements

The partial support of the National Research Council (CNR), grant Agenzia2000, is acknowledged.

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