The structure of the bileaflet heart valves in this study was based on an experimental system used by Hutchinson et al. [
57]. This specific valve was selected because of the detailed schematics available, which allowed for a more accurate computational representation, and because of the imaging method used to obtain the experimental measurements. Their group used particle image velocimetry (PIV), one of the most commonly used flow imaging techniques, which has been found to have a greater resolution than other imaging methods like magnetic resonance imaging (MRI) and Doppler [
58]. Their experimental system included an artificial heart valve modeled after the Carbomedics No. 25 aortic bileaflet mechanical heart valve. A virtual representation of Hutchinson’s bileaflet valve system for this research was created using ANSYS Workbench DesignModeler 18.1 (see
Figure 1). The structure with axial lengths consisted of a 400 mm inlet, a 7.7 mm valve frame, a 27.8 mm sinus of varying diameter, and a 270 mm outlet. Both the inlet and outlet regions had diameters of 26.8 mm.
Flow Simulation
CFD simulations for turbulent flow were carried out with both
k-ε and
k-
ω models for the fully open, functioning valve. The selection of specific CFD simulation results to be used for eddy analysis was based on comparison of the different model results with velocity profiles in the literature for this system [
59,
60].
In the
k-ε turbulence model, equations are applied using a finite volume solver to find the turbulence kinetic energy (
) and dissipation rate of turbulent kinetic energy (
) for the mean fluctuating strain
. These are determined with:
where
ui’ and
uj’ are the velocity fluctuations in the
i direction and
j directions respectively, the overbar indicates average,
is density,
µ is viscosity,
ν is kinematic viscosity,
µt is turbulent viscosity,
Gk is the generation of turbulent kinetic energy due to the mean velocity gradients, and
Sij is the mean strain rate. Model parameters have the standard values of
C2 = 1.9 and
Cµ = 0.09 and the turbulent Prandtl numbers for
k and
ε are
σk = 1.0,
σε = 1.2 [
61].
In the
k-
ω SST method, the model transport equations are solved for turbulent kinetic energy,
k and
ω with the latter defined as
. Both
k-
ε and
k-
ω approaches use the same equation for k while they differ when solving the second variable. Transport equations of the
k-
ω SST model are as follows:
where
Gω is the generation of
ω,
Yk and
Yω are dissipation of k and
ω due to turbulence,
Dω is the cross-diffusion term. The above terms can be found from:
where
y is the distance to the next surface and
is the positive portion of the cross-diffusion term.
The model constants are:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
Prior to running the flow simulation, boundary conditions were set for all of the surfaces in the model. The inlet was treated as a mass flow inlet, the outlet as outflow (a setting used when the conditions of the outlet are not known), the planes of symmetry as symmetry, and all other surfaces were treated as no-slip walls. For methods, the SIMPLE pressure-velocity coupling scheme, Green–Gauss cell based gradient, and standard pressure were used. Finally, the fluid properties of the volumetric region were defined.
For initial literature comparisons and results, properties in the simulation were set to match the values of the test fluid used in the Hutchinson et al. experimental system [
50]: a kinematic viscosity of 1.57 × 10
−6 m
2/s and a density of 1796 kg/m
3. Based on the density, viscosity, and diameter of the system, the inlet velocity was set to 0.445 m/s to obtain a Reynolds number of 7600 (as was used by Hutchinson et al.) [
60]. The purpose of these runs was to develop validated computational methods before carrying out simulations of blood flow through the valve. Results were also compared to those of Blackmore et al., who ran computations with large eddy simulation (LES) techniques for the same physical experiment [
59]. The comments in Blackmore’s paper indicate that they trusted the detailed velocity profile from the LES model over that from the physical experiment, because of issues with PIV, including the scarcity and settling of flow tracking seed particles in the fluid, preventing the detection of flow separation around the leaflets [
59].
A mesh independence analysis was done using the region adaptation function in Fluent to refine the mesh until the difference in mesh size did not produce results with significant differences.
Table 2 is a presentation of a comparison between the coarse, medium, and fine mesh sizes. For the different mesh sizes, variables were compared along the centerline of the computational domain and along cuts orthogonal to the centerline at various axial locations within the flow field. Examples are shown in
Figure 4 and
Figure 5, which present total pressure with axial position along the centerline and
KLS values at an orthogonal cut 405 mm downstream of the inlet, respectively. The medium mesh was selected to conduct the computational analysis, because the distinct difference in the results between the coarse and medium mesh is not seen between the medium and fine meshes. With fewer computational mesh cells (meaning a decreased calculation time) in the medium mesh, the iterations converged to lower residual levels faster without a high percentage difference from those of the fine mesh.
Turbulence simulations require the selection of a closure model to account for the fluctuations in the flow. An error analysis based on the velocity profile presented by Blackmore et al. was conducted to determine which turbulence closure approach to use [
59]. Simulations were run at both first and second order convergence using the
k-
ε turbulence model with enhanced wall functions and the
k-
ω SST turbulence model with curvature and low-Re corrections. The velocity profile in Blackmore et al. was divided into four increments to create analytical polynomial equations to be used for error analysis. The predicted values from these equations were compared to the computational values from each turbulence model as seen in
Figure 6. Error analysis showed that the
k-
ω SST first order model gave the best predictions for this geometry (
Table 3). The computation of the error was done as follows: First, the velocity profile provided in Blackmore et al. was assumed to be the correct solution relative to which the error was calculated. This assumption was justified because the results provided by Blackmore et al. [
59] were obtained using a large eddy simulation approach that is more accurate than Reynolds-Averaged Navier–Stokes models. Using 98 points across half of the profile (as the results are symmetric), the data plot in [
59] was split into four lines, and each line was fit to a polynomial equation with the lowest root mean square error prior to overfitting (determined as a plateau in an error vs. degree of polynomial plot). From these equations, literature results were calculated based on the location of points from our Fluent model. The mean value of the absolute difference between each point in our model results and the data interpolated from [
59] was calculated, as well as the mean square of this difference, and reported in
Table 3.
After selecting the appropriate mesh size and turbulence model based on agreement with Hutchinson and Blackmore’s work, simulations of blood flow through the valve were run treating blood as a Newtonian fluid with a viscosity of 0.002 Pa·s and a density of 1050 kg/m
3 [
52]. These simulations for blood were conducted to obtain
KLS distributions in the flow through the valve to use for eddy analysis and hemolysis predictions. The flowrates in this case were set to peak systolic flow, when turbulence is at its highest and blood is most likely to be damaged. Peak velocity of blood through heart valves in systolic flow has been cited as anywhere from 1.0–1.8 m/s, and in some cases has been found to be higher through artificial valves [
44,
62,
63,
64]. General values of 1.25 m/s and 1.5 m/s were selected for these simulations.
For eddy analysis of blood damage, cross-sectional surfaces were created 0.5 mm apart axially along the centerline. The surfaces were created along the length of the flow field in regions that contained eddies of 10 μm or smaller. The KLS values were calculated on each surface by using a user-defined equation (Equation (2)) in Fluent. Using this information, the total surface areas of all spheres with diameters equal to a specific KLS value can be calculated in 1-unit intervals (a unit being 1 μm) starting from the smallest diameter identified in the flow up to a diameter of 9 or 10 μm depending on which prediction model was applied (Equations (3) and (4)). To do this, the area of a particular KLS size range was determined on two consecutive surfaces and multiplied by the distance between the two surfaces, yielding the volume of the fluid containing eddies of that diameter.
The number of eddies (
Neddy) of each size can be found by dividing the volume within the region made up by eddies of that specific size by the volume of one eddy (
Veddy)
when the assumption that the eddies are spherical is made:
Finally, the total surface area of eddies for each
KLS value (
Aeddy) was calculated by multiplying the surface area of one eddy with that specific
KLS value by the total number of eddies of that size as follows:
The total surface areas for each size range were then normalized by dividing by the total volume of the region where hemolysis may occur (i.e., the region containing eddies with a KLS of either 9 or 10 μm or less, depending on the equation). This normalization process yields EAKLS(D1–D2) and allows for the calculation of hemolysis expected per volumetric unit. Using this basis facilitates calculations of hemolysis values that are device-independent, allowing the comparison of damage from different types of devices, in this case extending to heart valves.