COMPUTATIONAL FLUID DYNAMIC EVALUTATION OF AN AORTIC BENCH-TOP MODEL.
With minimally invasive catheter surgeries becoming prevalent for treating various heart diseases like aortic aneurysms or aortic valve stenosis, there is a need for bench-top models to assess catheter effectiveness and train surgeons for these types of surgical interventions. Aortic bench-top models have been utilized for years to develop an understanding of secondary flows, vortices, and wall-shear stresses (WSS) existing at the walls of the aortic arch that can lead to numerous arterial diseases. One of the first aortic bench-top models was developed by Yearwood and Chandran (1979), and took into account the complex geometry, tapering, and irregular cross-sections in the aorta. They found high WSS existent along the outer wall of the aortic arch with a prevalence of secondary flows developing on the inner wall during ventricular systole. Despite the lack of important geometrical factors like the aortic sinus region and branching arteries, they quantified important flow conditions generated during high flow rates in the aortic arch . A few years later Khalighi et al. (1983) developed an acrylic in situ casting of the aortic arch including the branching arteries and aortic sinus. This further developed an understanding of flows existing in the regions of interest and how they are influenced by the bifurcating branches at the mid-aortic arch region. This work also allowed for the evaluation of valve prostheses . Since then, Liepsch et al. (1992) developed a nonrigid silicone model of the aorta to understand fluid-structure interaction (FSI) and the impact of wall compliance on the flow patterns generated in the aortic arch . This work has subsequently lead to the use of non-rigid, patient-specific aortic models in the evaluation of endovascular techniques and the assessment of various medical devices as in the works by Sulaiman et al. (2008) and Biglino et al. (2013) [4,5].
Computational fluid dynamics (CFD) is often utilized within the aortic arch region to analyze the complex three-dimensional (3D) flow patterns generated during ventricular systole, and to measure parameters that are difficult to obtain in vivo. CFD in the cardiovascular system was first validated in the work by Olufsen et al. (2000) where the one-dimensional theory from the Navier-Stokes equation was utilized in a CFD of the entire arterial tree and generated blood flow and pressure measurements in agreement with values obtained from magnetic resonance imaging (MRI) . Since then, numerous works have validated the use of CFD analysis in the aorta, like the work done by Morris (2005). This work developed a CFD simulation of an aortic model including the aortic sinuses and branching arteries, and although making the assumption of rigid walls, still quantified values relevant to in vivo data . CFD simulations have continued to advance to point of being utilized to assess the efficacy of endovascular treatments. The work by Tokuda et al. (2008) displayed the effects on aortic flow from arterial cannulas during cardiopulmonary bypass surgery, and the work by Fung (2008) quantified factors of failure for stent-grafts within the aorta from high velocity blood flow [8,9].
This paper provides a computational fluid dynamic (CFD) model that simulates flows in an aortic bench-top apparatus in order to influence hardware settings on the bench through the quantification of fluid resistance at the aortic branches. Also, using the resistance values determined, perform time-variable flow simulations to verify the water glycerol mixture developed in the lab as a physiologically relevant blood analog fluid. The CFD simulation seeks to characterize the complex three-dimensional flows in an aortic arch bench-top model during physiological pulsatile flow. By quantifying the flow patterns generated, parameters affecting a catheter's path can be determined and utilized in the optimization of the device. The acrylic bench-top model must be robust and durable to accommodate for numerous engineering changes during prototyping and the repeated in vitro testing of preliminary catheter devices. This is reflected in the CFD model by allowing for the integration of design tables, various bench-top configurations, and patient-specific aortic arch geometry.
All three-dimensional models were created using the computer-aided design software SolidWorks (SolidWorks, Concord, MA, USA). The aortic arch was modeled from a set of DICOM files obtained from the Visible Human Project's CT scans of a middle-aged female. An STL model of the aortic arch was developed in image segmentation software called ITK-SNAP (open-source software, www.itk-snap.org). The STL was then imported into SolidWorks where it was sliced into numerous cross-sections orthogonal to the centerline of the aortic arch, and each cross-section was traced and lofted together to create a functional solid-body model as seen in Fig. 2(a).
Due to the non-planar geometry of the aortic arch and branching arteries, visualizing flow through the center of the geometry becomes difficult. In order to illustrate the flow profiles existent in the bench-top model, a series of lofted surfaces were constructed. Using a series of slices of the aortic arch and branching arteries, centroid points were sketched at each cross-section. These were used to construct centerlines for each branch and the arch, and a series of horizontal lines coinciding with the centerline were sketched at each cross-section. All lines were then lofted together generating the non-planar, centerline surfaces as seen in Fig. 4.
Computational Fluid Dynamics
Numerous computational fluid dynamic (CFD) simulations were performed in order to quantify the dependence of fluid resistance on nozzle angle. Once this relationship was determined, appropriate fluid resistance values were applied to each branching artery and descending aorta, and numerous time-variable CFD simulations were performed with both blood and water glycerol in order to verify water glycerol as a blood analog fluid for this bench-top model. "SolidWorks Flow Simulation" was the CFD software utilized in order to characterize the flows existent within the bench-top model.
Like other CFD software, "SolidWorks Flow Simulation" combines ease of use with high-level functionality. This was desired in order to accommodate for the numerous engineering changes to bench-top configuration and patient-specific aortic arch geometry utilized during prototyping. The CFD code is based on solving 3-D Navier-Stokes equations with a finite volume method (FVM). This method divides the computational domain into smaller volumes around each node in the grid; this ensures continuity of flow between each node. The continuity and momentum equations governing flow for blood and water glycerol can be found in Eq. 1 and 2 .
In order to ensure accurate convergence of the CFD simulations, numerous surface goals with flow controlling criteria were introduced into the simulation. Surface goals were placed on probes at areas of interest within the bench-top model--aortic branch cross-sections (Fig. 4) and the bench-top outlets. Each probe had a volumetric flow rate and bulk average total pressure surface goal, and in order for the CFD simulation to converge all surface goals had to be satisfied.
[mathematical expression not reproducible] (eq. 1)
[mathematical expression not reproducible] (eq. 2)
Flow Simulation calculates the minimum gap size using information about the faces where boundary conditions and goals are specified. Therefore, before generating the mesh, all conditions are set to their desired values. Minimum gap size is a certain number of cells per specified gap that can be generated, thus this is the primary parameter driving the computational mesh . Large gap sizes result in inadequate simulation results due to the low fidelity of the automatically generated mesh. In order to generate sufficient simulations results, and stay within the computational resources of the computer, gap size was incrementally decreased until an optimal gap size was determined. For all meshes, the minimum gap size was specified to be 4.2 cm.
Boundary conditions during this study were obtained through the experimental work done by Bazan and Ortiz (2016), which characterized the volume flow rate waveforms at the inlet of the aorta for various heart rates . Inflow conditions are initiated ~50mm upstream to the aortic valve. In order to reduce computational work load, a series of three steady flow rates were applied to determine the nozzle angle-fluid resistance relationship for both blood and water glycerol. The three steady flow rates simulated onset ventricular systole (5L/min), mid-systole (15L/min), and peak systole (25 L/min) . The inflow waveform employed during the time-variable CFD simulations can be seen in Fig. 5. Due to the small role in high-velocity hemodynamic conditions and the targeted region of the heart cycle for this bench-top model, the small portion of retrograde flow during ventricular diastole has been assumed zero (as displayed by the solid line in Fig. 5). The assumption of a flat or plug velocity profile at the aortic inlet is used and has been verified by various in vivo measurements [12,13]. A uniform, static pressure at the outlets of the model were applied for all simulations. Due to the nature of the bench-top model application, the approximation of rigid walls was also applied to the simulations. Previous works have also displayed how the assumption of rigid walls within the aorta generates results comparable to in vivo measurements .
Flow Simulation is capable of computing laminar flows of inelastic non-Newtonian fluids. Thus, blood is characterized as an incompressible, non-Newtonian liquid with a density of 1005 kg/[m.sup.3]. The viscous shear stress tensor (t) governing the fluid properties of blood are defined by Eq. 3 and 4. The dynamic viscosity (p) of blood was determined using the Power-Law model (Eq. 5), where the values of p are restricted to maximum and minimum values and K is the consistency coefficient. Water glycerol has been used in various other studies on hemodynamics, and has proven to display characteristics similar to blood [10,14]. The water glycerol mixture used for this bench-top is 60%, by volume, glycerin, and has a constant density of 1153 kg/[m.sup.3] and viscosity of 0.00936 Pa-s as seen in Table 1.
[mathematical expression not reproducible], (eq. 3)
where shear rate,
[mathematical expression not reproducible] (eq. 4)
[mathematical expression not reproducible] (eq. 5)
By quantifying the pressure drop from the aortic branches to the nozzle outlet and the volume flow rate at the exit of the model, fluid resistance could be calculated using eq. 6. Once the fluid resistance-nozzle angle relationship was defined, effective fluid resistances were applied at the branching arteries and descending aorta to perform the time-variable simulations of blood and water glycerol. The values applied to the simulation can be found in Table 2 .
[P.sub.2] - [P.sub.1]/[??] (eq. 6)
During the fluid resistance analysis, a series of parametric CFD simulations were performed with both blood and water glycerol. During each study, the cone nozzles applied to the outlet of the simulation were incrementally increased at each design point in order to determine fluid resistance dependence on nozzle angle. The results of the various steady flow rate simulations can be found in Fig. 6. Fluid resistance is expected to remain relatively the same regardless of changing flow rate, but at a nozzle angle of 163[degrees], the fluid resistance increased as the flow rate was increased. However, starting at a nozzle angle of 164[degrees], all flow rates displayed a similar fluid resistance--300 mmHg/L/min. The CFD simulations performed with water glycerol (dashed lines) generated fluid resistances significantly higher than blood (solid lines), although this is to be expected due to the shear-thinning nature of blood and the effect of viscous drag on pressure drop. Now, that fluid resistance is quantified with nozzle angles, fluid resistances are applied to each branch of the study in order to run time-variable simulations.
Blood v. Water Glycerol
Fluid resistance values, found in Table 2, where applied to the simulation and time-variable CFD simulations were performed with both water glycerol and blood using the flow rate waveform found in Fig. 5. The resulting velocity magnitude contours can be found in Fig. 7. Both contours displayed similar flow patterns in the regions of interest--aortic arch, aortic branches, and descending aorta. There are slight discontinuities near the aortic root region; however, due to the primary interest of flows downstream, this difference can be ignored. During the time-variable simulations, blood and water glycerol generated flow waveforms nearly identical to each other at all areas of interest. This can be seen in Fig. 8, where at the ascending aorta, mid-arch, and descending aorta water glycerol (blue) generates a flow rate slightly higher than blood (red); however these differences are negligible and it can be assumed that water glycerol will generate hemodynamic flows.
This study generated data that is capable of being utilized when sizing fluid resistance hardware settings during future experimentation on the bench. The cone nozzle at the outlets of the model are capable of producing fluid resistances within the range of physiological relevancy [14,15]. There are discontinuities at each of the steady flow rates for fluid resistance in the branching arteries when the nozzle angle is set to 163[degrees]. However, the fluid resistances generated at this design point are far too high to apply to a physiologically relevant experiment, so these differences can be ignored. Once the nozzle angle geometry begins to produce relevant fluid resistances (164[degrees]), the values are similar across all steady flow rates (5 L/min, 15 L/min, 25 L/min).
Water Glycerol v. Blood
This study computationally verifies water glycerol as an effective blood analog fluid for this bench-top during constant and pulsatile flow conditions. During the pulsatile flow CFD simulations, both blood and water glycerol were evaluated in order to determine the relevance of water glycerol to natural hemodynamics. Water glycerol produced results similar to blood in these CFD simulations, and also to previous published clinical data . The velocity magnitude surface contour comparison (Fig. 7) illustrates how the flow patterns generated at the regions of interest for both blood and water glycerol are comparable to each other. There are slight differences in flow along the wall of the aorta that influence flow downstream in the descending aortic region due to the high viscosity of blood. However, these discontinuities are distinguishable only during the high velocities generated during peak systole, and it can be assumed that the flow patterns generated by water glycerol correspond to blood performance. Flow rate waveforms generated from the CFD simulations in the regions of interest by water glycerol are nearly identical to those generated by blood (Fig. 8); further verifying water glycerol as a blood analog for this bench-top configuration.
A high-fidelity CFD simulation was avoided during this study in order to accommodate for the numerous design changes during prototyping and their timely integration into a computational model. Due to the preliminary catheter designs tested on this bench-top, the apparatus had to be robust and durable to accommodate for repeated testing under various conditions. This is partially reflected in the assumption of no wall motion in the aortic arch which reduces the physiological relevancy of the CFD simulation; however, wall motion has been determined to have a small effect on flow patterns generated in the aortic arch . Also, because CFD analysis of the aorta requires numerous complex processes, only one representative case of aortic geometry and pulsatile waveform was simulated during this study. Finally, the boundary conditions at the outlets of the model were assumed to be the same and constant.
Although, this is one of the most commonly used assumptions for outflow boundary conditions while simulating blood flow, ideally there should be a comprehensive representation of all downstream effects from vasculature [13,16]. However, this would greatly increase the complexity and solving time of the computational model.
Once effective catheter devices have been developed through utilizing the rigid-wall bench-top model, high fidelity bench-tops can be employed. This would mean a variety of patient-specific aortic bench-top models that include complaint wall motion and diseased aortic leaflets to further determine the flow characteristics existent within the aorta to optimize catheter devices. This would also be reflected in the computational modeling of the bench-top by implementing more advanced CFD software like CRIMSON or SimVascular, and incorporate aortic leaflet motion simulation model for verification and validation (V&V) of prosthetic aortic heart valves.
This study attempted to computationally characterize flow characteristics within an aortic bench-top model for the optimization of minimally invasive surgical strategies for endovascular repair. By computationally quantifying fluid resistance values in order to influence bench-top hardware setting, future testing and experimentation can be expedited, and optimal catheter design can be determined for a range of conditions. The CFD simulations of the bench-top model was found capable of producing fluid resistance values within a wide range of physiological relevance. Water glycerol also produced comparable results to hemodynamic performance, thus computationally verifying it as an effective blood analog for this bench-top apparatus.
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Jonathan Primeaux, Charles Taylor, Ph.D., Jacob King, Clint Bergeron
University of Louisiana at Lafayette, 104 East University Avenue, Lafayette, LA 70504
Department of Mechanical Engineering, 241 E. Lewis St., Lafayette, LA 70503
Caption: Figure 1. Flow chart outlining chronology of research.
Caption: Figure 2. Steps of bench-top fabrication (a) 3D SolidWorks model, (b) aortic arch printed in PVA
Caption: Figure 3. Illustration of nozzle geometry's dependence on nozzle angle
Caption: Figure 4. 3-D render of the aortic bench-top model with lofted plot surfaces (red) for (1) aortic branch cross-sections, (2) aortic branches, and (3) aortic arch
Caption: Figure 5. Pulsatile volume flow rate waveform aortic inlet (60 bpm) 
Caption: Figure 6. Fluid resistance plots for steady flow rates (a) 5 L/min, (b) 15 L/min, and (c) 25L/min. All fluid resistance values are in mmHg/L/min. Solid lines represent blood and dashed lines represent water glycerol.
Caption: Figure 7. Comparison of resulting velocity magnitude surface contours at 15 L/min between (a) blood and (b) water glycerol.
Caption: Figure 8. Volume flow rate waveform (60 BPM) for regions of interest - (a) ascending aorta, (b) mid-arch, and (c) descending aorta. Blood is represented in red and water glycerol is represented in blue.
Table 1. Fluid properties applied to CFD simulation. Properties Blood Water Glycerol Density [kg/[m.sup.3]] 1003 1153 Maximum Dynamic Viscosity [Pa-s] 0.012171 0.00936 Minimum Dynamic Viscosity' [Pa-s] 0.003038 -- [text unreadable in original source] -- Table 2. Fluid resistance values applied to time- variable simulation  Vessel Resistance (mmHg/L/min) Brachiocephalic 182.7 Left Common Carotid 230.7 Left Subclavian 878.6 219 Descending Aorta 25.85
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|Author:||Primeaux, Jonathan; Taylor, Charles; King, Jacob; Bergeron, Clint|
|Publication:||Journal of the Mississippi Academy of Sciences|
|Date:||Apr 1, 2018|
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