Large Eddy Simulation of Pulsating Flow Before and After CoA Repair - CFD for Intervention Planning

Large Eddy Simulation of Pulsating Flow

Before and After CoA Repair - CFD for

Intervention Planning

Roland Gårdhagen, Fredrik Carlsson and Matts Karlsson

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

Roland Gårdhagen, Fredrik Carlsson and Matts Karlsson, Large Eddy Simulation of Pulsating

Flow Before and After CoA Repair - CFD for Intervention Planning, 2015, Advances in

Mechanical Engineering, (7), 2.

http://dx.doi.org/10.1155/2014/971418

Copyright: Hindawi Publishing Corporation / SAGE Publications

http://www.hindawi.com/

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-100917

Research Article

Large Eddy Simulation of Pulsating Flow before and after CoA

Repair: CFD for Intervention Planning

R. Gårdhagen,

F. Carlsson,

and M. Karlsson

1_{Department of Management and Engineering, Link¨oping University, 581 83 Link¨oping, Sweden} 2_{FS Dynamics Sweden AB, M¨olndalsv¨agen 24, 412 63 G¨oteborg, Sweden}

Correspondence should be addressed to R. G˚ardhagen; roland.gardhagen@liu.se Received 14 April 2014; Accepted 19 September 2014

Academic Editor: Yosry S. Morsi

Copyright © R. G˚ardhagen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Large eddy simulation was applied to investigate hemodynamics in a model with coarctation of the aorta (CoA) and poststenotic dilatation. Special focus was put on the role of hemodynamics for success of CoA repair. Several parameters previously identified as related to cardiovascular disease were studied. Known risk factors were observed both with CoA and after repair, and the restoration of the anatomy seems to be crucial for a successful result.

1. Introduction

Coarctation of the aorta (CoA), a blocking at the end of the arch of the major artery of the body, constitutes 5–12% of all congenital heart defects [1–3]. Typically the condition is diag-nosed and treated during childhood; however, the disease may also remain undiscovered until the person is grownup [1]. In addition, restenosis may occur after treatment [1,4,

5], and the remaining hypertension [6], after a seemingly successful treatment, is still an elusive issue. CoA patients therefore have a worse long-term prognosis compared to the general population. Abnormal hemodynamics was early suggested as a key-factor in the genesis of comorbidities [7,8]; however, as pointed out by LaDisa et al. [9], this issue has since then been relatively sparsely addressed.

When diagnosed, CoA is typically corrected either with surgical repair (first practiced during the 1940s), with isolated balloon angioplasty (introduced during the 1980s), or since the 1990s with implantation of a stent. The latter two options would be preferable due to less hospitalization and less incon-venience for the patient as well as significantly lower costs for the health care. However, endovascular treatments are unfortunately associated with higher prevalence of restenosis [4–6] and the remaining hypertension [4,5], which is not the case with surgical repair [1].

Hemodynamics and its relation to vascular diseases have been a constantly growing field of research during the last decades [10]. Its role in the genesis of atherosclerotic plaques has been studied extensively, and several studies contributing to increased understanding of the pathogenesis have been presented, for example, [11–17]. Wall shear stress (WSS) has been identified as a key parameter, as regions with increased intima-media thickness often correlate with relatively low and/or oscillating WSS. Atherosclerosis differs from CoA in that atherosclerosis is a thickening of the vessel wall, whereas CoA is a narrowing of the vessel. However, WSS might also be a possible determinant for development of comorbidities as well as for success of CoA repair.

WSS is obtained as the shear rate (SR) at the wall multi-plied by the blood viscosity. Apart from its close relation to WSS, SR per se has also been associated with cardiovascular disease. In an experimental study by Merino et al. [18] the role of SR in thrombus formation at the site of a wall injury, that is, a damaged endothelial cell layer, was investigated. The work demonstrated increased risk for thrombus formation if the SR was greater than 5 000 s−1. Such conditions are unlikely to occur to a greater extent in a healthy cardiovascular system with normal flow conditions. However, in the presence of a constriction the vessel wall might be damaged and the SR very high; the damage might even appear as a result of

Outlet Pre Inlet x y z 72 mm 240 mm (a) Post Inlet Outlet x y z 72 mm 240 mm (b)

Figure 1: Vessel model with CoA and poststenotic dilatation (a). Corresponding model after CoA repair, with a remaining reduction of cross-sectional area of 21% (b).

high SR. Several previous works have addressed the topic; in a summarizing publication Sakariassen et al. [19] present results from a number of their experimental studies. These suggest that high SR induces platelet activation, and the longer the exposure time for high shear the more the activa-tion. High SR in their studies was 10 500 s−1; however no SR in the range between 2 600 and 10 500 s−1 was investigated. Platelet activation was found to initiate platelet microparticle formation, which in turn resulted in a markedly enhanced degree of platelet thrombus formation.

Last but not least flow after a stenosis might change from laminar to turbulent, manifested as more or less high-frequent fluctuations of the flow field variables (pressure, velocity, etc.). Such disturbances do not occur under normal conditions [20] (p. 58) and have been identified to influence pathological events in blood vessels in several ways. Already in the 1960s it was proposed to weaken the arterial wall and thereby to cause poststenotic dilatation [7]. In an experimen-tal study Stein and Sabbah [21] found turbulence (in terms of fluctuations in the velocity field) to constitute increased risk for thrombus formation, and yet another experimental study by Davies et al. [22] revealed endothelial cell turnover to be considerably more sensitive to turbulent shear stresses than laminar stresses regardless of magnitude. Reynolds stresses, which are related to the fluctuating velocity components of a turbulent flow, and their relation to red blood cell hemolysis were investigated by Sallam and Hwang [23] by studying the content of free hemoglobin in the blood for various levels of Reynolds stresses. No signs of hemolysis were found for low values, while continuously increased hemolysis occurred above a certain limit. All together this calls for a thorough investigation of the hemodynamics at the site of a CoA, including additional parameters than WSS, with particular focus on how the parameters are affected by CoA repair.

Computational fluid dynamics (CFD) has been the pri-mary tool for many hemodynamic studies, especially those investigating the WSS pattern and its role in the pathogenesis, for example, [16,17]. Thanks to increased computer capacity and sophisticated computational models, CFD has shown to be a promising and potential tool for subject specific vessel modeling and intervention planning [9,24,25]. The predictive functionality is unique; subject specific vessel

models are easily created, modified as by different treatments, and used for flow simulations. Secondly, CFD offers a yet superior combination of spatially and temporally resolved flow field data to evaluate and assess the simulated flow field. A major challenge for CFD in general and biofluid CFD in particular has been appropriate modeling of turbulent flows. The simplest option is to apply a relatively fast time-average model based on the Reynolds-Averaged Navier-Stokes (RANS) equations. With this approach all turbulence is modeled in one way or another, and no fluctuating motion is resolved. On the other side of the spectrum is direct numer-ical simulation (DNS), with which all turbulent motion is temporally as well as spatially resolved, unfortunately to a usually unaffordable prize in terms of computational power. In between these techniques, large eddy simulation (LES) combines the benefits of RANS and DNS; larger, anisotropic turbulent scales are resolved, whereas smaller, less energy containing, and more likely isotropic scales are modeled. LES is more time consuming than RANS simulations, but yet affordable for the model size and Reynolds number range of cardiovascular problems. Its usefulness for stenotic flows has been demonstrated previously [26–28].

During recent years some CoA related simulation studies have been presented. Quite naturally, these use previous knowledge and apply simulations to evaluate and contrast the outcome of surgical repair versus stent implantation [9,29] or to investigate a particular treatment [30,31]. However, until now, very few have addressed the turbulent nature of the flow as a part of the analysis [32].

In this work we apply LES to simulate the pulsating flow through a generic vessel with CoA and poststenotic dilatation before and after CoA repair. We extend the number of parameters investigated in CoA models with some previously linked to vascular dysfunction and discuss how they are affected by an intervention.

2. Method

2.1. Geometrical Model and Computational Mesh. Generic

models of a vessel with and without CoA, both with post-stenotic dilatation, were constructed based on dimensions from a middle aged person with a CoA that had restenosed

RMS of axial velocity component (m/s) 1 1 1 0.5 0.5 0.5 0.2 0.4 0 0 0 0 −0.5 −1 1 0.5 0 −0.5 −1 1 0.5 0 −0.5 −1 x = 24 mm x = 48 mm x = 72 mm y/ R (—) 8.9 MC 12 MC 27 MC

Figure 2: Profiles of the RMS-values of the axial velocity at three positions in the dilatation along the𝑦-axis for the baseline mesh containing 8.9 million cells (MC) and two refined meshes containing 12 MC and 27 MC. It was decided that the baseline mesh was sufficient to capture the fluctuations.

multiple times after treatments, Figure 1. The model with CoA is referred to as pre, while the model without CoA (i.e., after treatment) is referred to as post. The arch had a diameter of 18 mm and radius of curvature of 30 mm for the inner side. After the arch the pipe narrows according to a cosine formula to the minimum diameter of 6 mm at the stenosis throat, after which another cosine expression widens the pipe to the maximum diameter 36 mm in the dilatation. The converging part of the dilatation is obtained by a third cosine expression and ends 60 mm after the stenosis with 12 mm diameter. After this the pipe continued for another 180 mm (15 diameters) before the outlet to ensure that this would not affect the flow in the diseased region. A hexahedral mesh was constructed in Ansys ICEM CFD 13.0 (ANSYS Inc., Canonsburg, PA, USA). Steady state flow simulations were used to reach desired mesh characteristics including a target value of the nondi-mensional wall distance𝑦+less than unity. This was achieved with smaller cells at the wall. Cells along the arch and through the CoA were gradually refined to ensure appropriately cap-tured separation and flow through and after the constriction. After the dilatation, the cells were gradually stretched in the axial direction. The mesh contained 8.9 million cells (MC). Two-point correlations and comparison with finer meshes, containing 12 MC and 27 MC, ensured sufficient resolution. Good agreement was also found for the RMS-values of the fluctuation of the axial velocity component, Figure 2. The velocity field agreed well with the finest mesh; the difference was caused by a somewhat earlier jet breakdown for the

coarser mesh. Thus, the predicted flow fields were essentially identical disregarding a small spatial displacement relative to each other. For the purpose of the present study, this was considered sufficient. In the beginning of the pipe after the dilatationΔ𝑥+andΔ𝑧+ were around 25 and 8, respectively. This vouches for sufficiently resolved LES.

The treated vessel, Figure 1(b), was constructed analo-gously. A small stenosis often remains after treatment [30]; in this case 21% reduction of cross-sectional area (11% reduction of cross-sectional diameter) was chosen. All mesh parameters were directly transferred from the pre case.

2.2. CFD Model. Flow simulations under the assumption of

an incompressible (density𝜌 = 1060 kg/m3) and Newtonian (viscosity𝜇 = 0.004 kg/(m⋅s)) fluid [33] and rigid walls were conducted in Ansys Fluent 13.0 (ANSYS Inc., Canonsburg, PA, USA) using an inlet velocity pulse profile based on MRI measurements,Figure 3. The numerics have previously been validated against experimental results [28]. The flow was uniformly distributed on a circle of radius 6 mm (the radius of the inlet was 9 mm) in the middle of the inlet, with zero flow outside this circle, to represent the flow through the aortic valve and provide a reasonably disturbed flow in the descending part of the arch [20] (p. 58). No additional perturbations were added. All simulations were run with the same pulse profile. At the outlet, located in the thoracic region, the gage pressure was prescribed and all other parameters were extrapolated from the interior.

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Time (s) (m/s) Inlet velocity

Figure 3: Pulse profile of the inlet velocity.

LES is a time resolved numerical technique, controlled by the Courant Friedrichs Levy (CFL) number, which should be less than unity. In this case a time step of 5 ⋅ 10−5s fulfilled the criterion. Time advancement was done using a second order implicit scheme. Central differencing was used for the convective and diffusive flux terms, and the pressure at the cell sides was computed using a discrete continuity balance for a staggered control volume. Subgrid scales, that is, turbulent eddies smaller than the cells of the mesh, were modeled using the dynamic Smagorinsky-Lilly model. The simulations were run on the Linux cluster Kappa provided by the National Supercomputer Centre, Link¨oping, Sweden (https://www.nsc.liu.se/).

2.3. Turbulence Statistics and Postprocessing. Since LES, in

contrast to, for example, RANS simulations, is time resolved (the RANS equations are solved directly for time averaged quantities) all statistical calculations must be carried out as part of the postprocessing of the results. Fifteen pulses were run before collection of data for statistical analysis was started and continued for 50 consecutive pulses. Axial velocity was monitored in points along the centerline from the stenosis, every diameter (outlet size) throughout the dilatation until one diameter after the dilatation. Axial and circumferential WSS were monitored along four lines on the surface (i.e., the vessel wall) where the𝑥-𝑦 plane and 𝑥-𝑧 cut the wall,Figure 1. The lines in the 𝑥-𝑦 plane are referred to as 𝑦 = 𝑅 and 𝑦 = −𝑅, respectively, depending on whether the 𝑦 coordinate is positive or negative (𝑅 is the local radius of the vessel wall), and analogously the lines in the𝑥-𝑧 plane are referred to as 𝑧 = 𝑅 and 𝑧 = −𝑅, respectively. As expected the flow on 𝑧 = ±𝑅 was symmetric on average; hence only 𝑧 = −𝑅 was considered in the analysis. Phase average WSS (PAWSS) of the WSS components was computed according to (1), in which 𝑁 is the number of cardiac cycles (i.e., 50) and 𝑇 the time of each cycle, which in this work was 0.8 s. Time averaged WSS (TAWSS), based on PAWSS, was calculated according

to (2), and oscillatory shear index (OSI) during one pulse was computed using (3) PAWSS= 1 𝑁 + 1 𝑁−1 ∑ 𝑛=0𝜏𝑤(𝑡 + 𝑛 ⋅ 𝑇) , (1) TAWSS= 1 𝑇∫ 𝑇 0 |PAWSS| 𝑑𝑡, (2) OSI= 1 2[ [ 1 − 󵄨󵄨󵄨󵄨󵄨󵄨∫ 𝑇 0 𝜏𝑤𝑑𝑡󵄨󵄨󵄨󵄨󵄨󵄨 ∫₀𝑇󵄨󵄨󵄨󵄨𝜏_𝑤󵄨󵄨󵄨󵄨𝑑𝑡]_]. (3)

3. Result and Discussion

In this work CFD was used as a tool for cardiovascular intervention planning. Previously identified hemodynamic risk factors were studied in a generic model of an aorta before and after a possible repair of a coarctation. A coexisting post-stenotic dilatation was left untreated. Established knowledge was thus applied for better understanding of a severe disease with still unsatisfactory low long-term success when treated. Whether the flow becomes turbulent or not is a key issue from a biological as well a computational perspective. Previous studies have shown how turbulence may lead to poststenotic dilatation, thrombus formation, and a very oscillating WSS pattern [7,21,34]. Furthermore it has been shown that appropriate calculation of WSS requires a scale resolving technique [34]. In this work LES was used to handle the turbulence.

3.1. Velocity and Pressure. Figure 4(a)shows monitor plots of instantaneous axial velocity at three points located 24 mm, 48 mm, and 72 mm downstream of the constriction, respec-tively (i.e., two in the dilatation and one in the downstream pipe). Analogous to [27] considerable differences between the pulses were also observed in this study. As expected, turbulent fluctuations dominated the systolic flow; the instabilities were first seen at the most upstream point and propagate quickly downstream, even in the pipe after the dilatation. Thus, a patient with a CoA would also be exposed to increased risk for thrombus formation and endothelial cell turnover due to turbulence [21,22].

Distal to the CoA the flow was dominated by a jet and its breakdown, resulting in high-shear regions and consid-erable fluctuations in the velocity field. Reynolds stresses based on phase average velocity were computed. The normal components reached values up to 600 Pa, whereas the shear components typically peaked between 200 and 400 Pa. In the study of Sallam and Hwang a corresponding threshold value for hemolysis was found to be 400 Pa [23]. Nevertheless, the observed levels were far from normal aortic flow conditions. Hence, endovascular or surgical repair of CoA should elimi-nate these hemodynamic abnormalities.

After intervention the remaining reduction of cross-sectional area of 21% was assumed, while the poststenotic dilatation was left untreated. Velocity signals from this model are shown in Figure 4(b); the points were the same as in

0 0.2 0.4 0.6 0.8 0 2 4 t (s) (m/s)

Axial velocity, pre

x = 24 mm x = 48 mm x = 72 mm (a) 2 0 0.2 0.4 0.6 0.8 0 4 t (s) (m/s)

Axial velocity, post

x = 24 mm x = 48 mm x = 72 mm

(b)

Figure 4: Monitor of axial velocity during one pulse. Three locations on the centerline of the vessel model are shown: 24 mm, 48 mm (in the divergent and convergent part of the dilatation, resp.), and 72 mm (one diameter into the pipe) downstream of the constriction for (a) pre and (b) post.

some instabilities in the flow field at the later part of the diastolic phase. These were a consequence of flow separation in the divergent part of the dilatation as the velocity increased. Thus, the entire dilatation was now a region of unstable and oscillating low-speed flow that previously only dominated the region surrounding the jet. Hence, the risk for thrombus formation remained elevated.

Compared to the results from the rabbit aorta presented by Menon et al. [30] one can note that both models featured a jet; however, the present work showed a jet going unstable and resulting in massive turbulence, whereas the jet in the rabbit aorta hits the vessel wall with seemingly less turbulence as a consequence. This suggests that several mechanisms may be involved in the development of the poststenotic dilatation: turbulence [7], direct jet-impingement, or a combination. O’Rourke and Cartmill [8] state that the aim of surgical cor-rection of coarctation should be to restore normal anatomical structure of the aorta, which supports the previously dis-cussed need of a restored flow field. With the results from the present work it seems probable that, seemingly, minor details like a modest remaining diameter reduction and the dilatation might affect the outcome crucially. This is also supported by the study of Roach [7], in which the poststenotic dilatation developed within the first ten days after induction of the CoA and then remained. Thus, even the moderate remaining flow disturbances probably prevent regression of the dilatation.

The pressure drop across the coarctation along the centerline peaked around 50 mmHg at peak systole, which agrees with the average pressure gradient reported by Carr [6], and was well above the limit of 20 mmHg typically indicating treatment of CoA [35]. After treatment the peak systolic pressure drop was reduced below 3 mmHg across the coarctation; however the contraction of the flow at the end of the dilatation now caused a drop of about 15 mmHg, which in turn was greater than the 10 mmHg for which reintervention

after balloon dilatation in adults should be considered due to the increased risk for restenosis [1]. Firstly, this demonstrated the influence of the dilatation on the flow field, and secondly it called for careful pressure measurements in order to detect the drop over the entire diseased region, not only the stenosis. At the wall the turbulence and recirculation induced pressure oscillations on the order of a few mmHg in terms of root mean square values around the phase average, and after treatment the corresponding values were only a few tenths of a mmHg.

3.2. Shear Rate. Constriction of a blood vessel is known to

increase the shear rate significantly; according to Wootton and Ku [10] levels well above 10 000 s−1 are possible. Other studies have found SR levels above certain threshold values to increase the risk for thrombus formation at the site of a wall injury. Merino et al. [18] identified SR≥ 5 000 s−1to constitute such a risk.Figure 5shows the shear rate in the CoA and dilatation, in (a) pre and (b) post, along𝑦 = 𝑅 during systole. For the CoA case the pattern was relatively symmetric around the pipe. However, in contrast to the description in [10] of the dilatation as a low-shear region, our results showed that high SR prevailed at the wall along a large part of the dilatation and that dangerous levels also occurred at the end of the dilatation during the ejection phase of the heart. In addition longer residence times associated with recirculation were a complicating factor according to the findings by [19]. In the constriction SR also reached levels above 10 500 s−1, which was the critical level observed by [19]. After CoA repair the SR at that site and in most of the dilatation was reduced significantly,Figure 5(b). Only on𝑦 = 𝑅 the SR exceeded 5 000 s−1 at the end of the dilatation around peak systole (0.1 s).

3.3. Wall Shear Stress. WSS is the friction load exerted on the

(mm) (s) 0 20 40 60 0 0.1 0.2

0.3 Shear rate, y = R, pre

(a) (mm) (s) 0 20 40 60 0 0.1 0.2

0.3 Shear rate, y = R, post

(b)

Figure 5: Regions with SR> 5000 s−1, which have been identified to increase the risk for thrombus formation at the site of an injured vessel wall [18]. The image shows the diseased region (0 ≤ 𝑥 ≤ 72 mm) during systole (0 ≤ 𝑡 ≤ 0.3 s) for (a) pre and (b) post along the line 𝑦 = 𝑅.

20 40 60 0 0.2 0.4 0.6 0.8 0 50 100 150 (mm) (s) (P a) −50

Axial PAWSS, z = −R, pre

(a) 20 40 60 0 0.2 0.4 0.6 0.8 (mm) (s) 0 50 100 150 (P a) −50

Axial PAWSS, z = −R, post

(b) 20 40 60 0 0.2 0.4 0.6 0.8 0 20 (P a) (mm) (s) −20

Circumferential PAWSS, z = −R, pre

(c) 20 40 60 0 0.2 0.4 0.6 0.8 (mm) (s) 0 20 (P a) −20

Circumferential PAWSS, z = −R, post

(d)

Figure 6: Phase average wall shear stress (PAWSS) computed from 50 consecutive pulses in the axial direction (a, b) and circumferential direction (c, d) before CoA repair (a, c) and after (b, d).

and acts along the surface in a certain direction. Axial and circumferential PAWSS along𝑧 = −𝑅 from the CoA (𝑥 = 0) until 𝑥 = 0.072 m are shown in Figure 6. (a) and (c) show axial and circumferential PAWSS for pre, while (b) and (d) show the same for post. As for the axial PAWSS component, its pattern of maximums and minimums was the same on all lines, whereas differences were observed for

the circumferential PAWSS. On a certain spatial location axial PAWSS showed either a maximum or a minimum during the pulse, reflecting the jet, its breakdown, flow separation, and turbulence. Circumferential PAWSS on𝑧 = −𝑅 showed a maximum and minimum on the same location in the dilatation but at different instants of time. It is also almost an order of magnitude lower than the axial PAWSS.

0 20 40 60 0

20 40

60 Axial TAWSS pre

(P a) −20 −40 −60 x (mm) y = R y = −R z = −R (a) 0 20 40 60 0

Axial TAWSS post

20 40 60 (P a) −20 −40 −60 x (mm) y = R y = −R z = −R (b)

Figure 7: Time average wall shear stress (TAWSS) in the axial direction during a pulse. The values are based on the phase averaged wall shear stress. In the CoA model (a) and after removal of CoA (b). The dashed lines show maximum and minimum WSS during a pulse.

The circumferential variation of the circumferential compo-nent was more pronounced in both the constriction and the dilatation.

Low time average WSS (TAWSS) during a pulse has been correlated with increased medial thickness proximal to the constriction [30], and of course too high TAWSS will eventually harm the endothelial cells as well.Figure 7(a)

shows TAWSS (solid lines) along the three previously defined surface lines, together with maximum and minimum WSS (dashed lines) during a pulse. TAWSS was relatively low except in the constriction and in the convergent part of the dilatation, and most regions had a not too high maximum load in either the forward or backward direction. How-ever, some regions in the dilatation were distinguished, for example, in that both TAWSS and the extreme values were small or that large maximum and minimum values appear on the same location. Previously discussed similarities and differences with [30] were again reflected here. The most striking difference was the high TAWSS where the jet hits the wall of the rabbit aorta, which caused a strong circumferential variation.

PAWSS in the repaired model is presented in Figures6(b)

and6(d). Since the small remaining diameter reduction was assumed, a small PAWSS peak still existed at𝑥 = 0. A jet did no longer develop though (as discussed previously), and the separation was very limited; forward flow prevailed until maximum flow was reached. Maximum WSS now occurred in the convergent part of the dilatation, and asymmetry concerning the magnitude appeared relatively early and

increased throughout the entire pulse. At maximum flow a recirculation zone had developed, and the maximum PAWSS at the end of the dilatation was as high as in the CoA-case. Thus, the endothelial cells at this site would sense no relief in shear when the constriction was removed.

After repair axial TAWSS essentially maintained the same pattern but with somewhat smaller maximums and minimums. The highest values were now those at the end of the dilatation accompanied by high maximum WSS. The minimum values in the dilatation were strongly affected by the irregular flow in the dilatation and showed a more oscillatory pattern. Circumferential TAWSS was generally rather small. The highest values were found in the middle of the dilatation.

Since WSS is a vector, it has a magnitude as well as a direction. Results discussed so far have mainly focused on the magnitude, although the direction was included in terms of forward or backward WSS. A more detailed assessment of the direction is provided by the directional (angular) distribution of WSS during the pulse, that is, a plot showing the part of the pulse that the WSS vector points in a certain direction. This was studied at four spatial locations, three in the dilatation and one after it (𝑥 = 0.072 m) on the lines on the surface.

Figure 8shows the results on𝑦 = 𝑅, 𝑦 = −𝑅, and 𝑧 = −𝑅 for pre.Figure 9shows the same for post.

The pattern seen in PAWSS was also reflected here; however a more pronounced circumferential asymmetry could now be discerned. Two things should be noted: first that WSS in the 𝑥-𝑧 plane altered between two opposite

2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 10000 20000 30 210 60 240 90 270 120 300 150 330 180 0 1000 2000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 1000 2000 30 210 60 240 90 270 120 300 150 330 180 0 1000 2000 30 210 60 240 90 270 120 300 150 330 180 0 5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 x = 15 mm x = 30 mm x = 45 mm x = 72 mm y = R y = −R z = −R

Figure 8: Directional distribution of WSS during a pulse on the lines𝑦 = 𝑅, 𝑦 = −𝑅, and 𝑧 = −𝑅. The locations are 15 mm, 30 mm, 45 mm, and 72 mm downstream of the constriction. Zero corresponds to WSS in the main flow direction, whereas 180∘corresponds to flow in the opposite direction (i.e., backward flow). All diagrams are for the pre case.

directions during the entire pulse and second that up to the middle of the dilatation WSS pointed essentially in opposite directions on𝑦 = 𝑅 and 𝑦 = −𝑅. This means that EC in the dilatation were exposed to different kinds of shearing loads, and consequently the shape and orientation of the endothelial cells will vary [36,37]. This supported the need for detailed local WSS results from CFD studies, emphasized by Menon et al. [30]. The asymmetry of WSS was also reflected in OSI,

Figure 10. As expected the values were high in the entire dilatation (the dashed vertical line marks the axial location of the last peak).

Average OSI for the entire dilatation was 0.29, and when the coarctation was removed the corresponding value was 0.31, that is, essentially the same or even higher, while the location of the last peak came earlier in the dilatation (which ended at 𝑥 = 0.06 m), thus, indicating a smaller region with higher OSI. The tendency in the plots of directional distribution, Figure 9, was towards flow (and hence WSS) in opposite directions even in the𝑥-𝑦 plane, although not as pronounced as in the𝑥-𝑧. The results also indicated that WSS on𝑦 = 𝑅 and 𝑦 = −𝑅 now rather pointed in opposite directions in the convergent part of the dilatation. In addition, WSS in the dilatation was, even when the CoA was removed, very oscillating. Based on OSI, WSS was in fact oscillating

even more than before removal. Together, PAWSS and the directional behavior suggest that WSS has become lower and more oscillating in a large part of the dilatation during a large part of the pulse. This change was even more pronounced in [30], where OSI of the CoA aorta was very low in the dilatation, implying moderate back flow as well as turbulence, since the jet hit the wall before it broke down into turbulence. After repair, on the other hand, OSI increased significantly.

4. Concluding Remarks: Clinical Implications

Intervention planning using CFD has potential to be a useful complement to experience in the future. In this study the predictive functionality of CFD has been demonstrated for a generic model of an aorta with coarctation and poststenotic dilatation. Previously identified risk factors for cardiovas-cular disease were investigated before and after CoA repair in order to identify potential indicators of a successful treatment.

Even the moderate remaining reduction of cross-sectional area and the dilatation were found to cause instabilities in the flow of the same type as is believed to cause the dilatation itself. This calls for special caution concerning the details of the anatomical restoration. Comparison

5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 5000 10000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 2500 5000 30 210 60 240 90 270 120 300 150 330 180 0 x = 15 mm x = 30 mm x = 45 mm x = 72 mm y = R y = −R z = −R

Figure 9: Directional distribution of WSS during a pulse on the lines𝑦 = 𝑅, 𝑦 = −𝑅, and 𝑧 = −𝑅. The locations are 15 mm, 30 mm, 45 mm, and 72 mm downstream of the constriction. Zero corresponds to WSS in the main flow direction, whereas 180∘corresponds to flow in the opposite direction (i.e., backward flow). All diagrams are for the case after CoA repair.

0 0.02 0.04 0.06 0 0.1 0.2 0.3 0.4 0.5 OSI, pre x (m) y = R y = −R z = −R (—) (a) 0 0.02 0.04 0.06 0 0.1 0.2 0.3 0.4 0.5 OSI, post y = R y = −R z = −R x (m) (—) (b)

with other studies shows that development of poststenotic dilatation might be caused by several mechanisms, either disturbances in the flow field or direct jet impingement on the vessel wall.

Based on the previous statement, instabilities in the flow, for example, in terms of fluctuations of the velocity field should be evaluated. In addition, SR has been shown to reach levels well above a threshold for thrombus formation and hence should be included when the results are assessed. WSS in the diseased region shows the same behavior as has been identified to cause growth of the intima-media layer that might clearly be a risk in this case too. A suitable continuation of this work would be to apply the procedure to subject specific models before and after treatment to verify the findings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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