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Fluid–structure interaction modelling of the venous valve with elastic leaflets

The paper presents the results of a numerical study of the flow in a symmetric model of a healthy popliteal venous valve with different leaflet elasticity. The focus is on the analysis of the stagnant area under the valve leaflet. The stagnant area under the valve leaflet, as well as behind the valve, is of great practical interest to phlebologists in terms of the possibility of blood clots. The flow in the constructed simplified model of the venous valve gave good qualitative agreement with the clinical ultrasound data for a leaflet with Young modulus of 1.2 MPa.


Introduction
Bicuspid valves in veins prevent retrograde flow away from the heart [1]. They are able to regulate the flow and keep it unidirectional preventing the backflow when the pressure above the valves is greater than the pressure below them acting as a check valve in a pumping pipe system. Under load conditions, a healthy valve fully opens and closes on a periodic basis [2][3][4]. Under resting conditions, valve remains open and the amplitude of its leaflet's oscillation is approximately 3-5% of the vein radius.
A venous valve eliminates any retrograde flow of blood in the vein circulation unless there are any disturbances such as clots [5]. Ageing makes venous valve leaflets more rigid preventing the normal functioning of the valve. Blood clots form in the stagnant zone, which is located behind the valve leaflets in the vein expansion behind the valve (sinus). The size of this zone is of great interest to phlebologists.
Research [6] provides useful insights for better diagnosis and understanding of the vein blockage induced by deep venous thrombosis and the occurrence of reverse flow in human veins, allowing for a proper detection of serious diseases related to deep venous insufficiency. The purpose of this simulation was to explore the effects of clot formation on the flow in a vein segment comprising several valves. Different periods of time were analyzed here according to the vein wall contraction and expansion, showing how each venous valve behaves. Previously, in our study [7] a two-dimensional computational model of the popliteal venous valve was constructed and the effect of the gap width between the valve leaflets (venous valve incompetence) on the valve reverse flow (reflux) was investigated. Paper [8] presents the results of a numerical study of the flow in this model with different leaflet elasticity. The focus is on the analysis of the velocity field and the stagnant region. Meanwhile, The purpose of the work is to study the effect of leaflet elasticity on the stagnant zone in the model of a geometrically healthy venous popliteal venous valve (figure 1a) under resting conditions, and also to compare the numerical results with the results of ultrasound measurements.

Venous valve geometry
In this study, a two-dimensional symmetric model of the venous valve was selected [4]. As the vein radius is R = 5 mm, this model corresponds to the popliteal vein valve (figure 1b).   2a). The cycle time is T = 1.9 s. A flat velocity profile is set at the inlet boundary. Constant pressure P = 0 is set at the outlet boundary. The symmetry condition is applied at the upper boundary, and the no-slip condition is applied at the bottom boundary. The condition of interface between solid and fluid regions was applied on the leaflet surface: Г = Г ; Г = Г ; Г = − Г , where d is displacement, u is the velocity, and F is the force. The approximation of rigid walls was used due to the insignificance of their displacements for the popliteal vein.

Computation aspects
The hydrodynamic mesh consisted of triangular elements and was refined on the valve leaflet. A mesh independence study was conducted. The mesh with 18100 cells is enough to obtain a practically converged solution, the difference from the coarse mesh with 9050 cells is 15%, and the maximum difference with the fine mesh (with 36200 cells) is 5%. The mechanical mesh consisted of 800 rectangular elements.
A smoothing algorithm was used to set up a dynamic hydrodynamic meshes, it smoothes the mesh and avoids negative cell volumes in the calculation process, which may stop computation. Figure 3 shows axial velocity fields with streamlines for a leaflet with Young's modulus of 1.2 MPa at different time instances of the cycle. It can be seen that a jet forms between the valve leaflets. A recirculation zone is formed behind the valve leaflet. A zone with low velocity (stagnant zone) is observed under the valve leaflet.  When the leaflet elasticity decreases, the length of the stagnant zone increases ( figure 5). This is due to the fact that with an increase in Young's modulus, the amplitude of oscillations (∆R) of the leaflets decreases and the fluid below it moves at a lower velocity. With an increase in Young's modulus from 0.6 MPa to 20 MPa, the length of the stagnant zone increases from 0.4L to L.

Conclusions
The dependence was obtained between the stagnant zone length and the amplitude of the leaflet oscillations, and Young's modulus. The length of the stagnant zone increases as the amplitude of leaflet oscillations decreases. The calculation results obtained in a symmetric two-dimensional model of the healthy venous valve of the popliteal vein gave good qualitative agreement with the clinical ultrasound data on the position and size of the stagnant zone behind the valve.