Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

A. Ferrarini; A. Finotello; G. Salsano; F. Auricchio; D. Palombo; G. Spinella; B. Pane; M. Conti

doi:10.1007/s10409-021-01066-2

Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Ferrarini, A.; Finotello, A.; Salsano, G.; Auricchio, F.; Palombo, D.; Spinella, G.; Pane, B.; Conti, M. 2021-03-30 00:00:00 Endovascular treatment of the femoro-popliteal artery has recently become a valuable therapeutic option for popliteal arterial aneurysms. However, its efﬁcacy remains controversial due to the relatively high rate of complications, such as stent occlusion as result of intra-stent thrombosis. The elucidation of the interplay among vessel geometrical features, local hemodynamics, and leg bending seems crucial to understand onset and progression of popliteal intra-stent thrombosis. To this aim, patient- speciﬁc computational ﬂuid dynamic simulations were performed in order to assess the intra-stent hemodynamics of two patients endovascularly treated for popliteal arterial aneurysm by stent-grafts and experiencing intra-stent thrombosis. Both Newtonian and non-Newtonian blood rheological models were considered. Results were presented in terms of tortuosity, luminal area exposed to low (< 0.4 Pa) and high (> 1.5 Pa) time-averaged wall shear stress (TAWSS), area exposed to high (> 0.3) oscillatory shear index (OSI), and ﬂow helicity. Study outcomes demonstrated that leg bending induced signiﬁcant hemodynamic differences (> 50% increase) in both patients for all the considered variables, except for OSI in one of the two considered patients. In both leg conﬁgurations, stent-graft overlapping induced a severe discontinuity of the lumen diameter where the proximal stented zone is characterized by low tortuosity, low velocity, low helicity, low TAWSS, and high OSI; while the distal part has higher tortuosity, velocity, helicity, TAWSS, and lower OSI. Sensitivity study on applied boundary conditions showed that the different inlet velocity proﬁles for a given inlet waveform affect slightly the numerical solution; conversely, the shape and magnitude of the prescribed inlet waveform is determinant. Focusing on the comparison between the Newtonian and non-Newtonian blood models, the area with low TAWSS is greater in the Newtonian model for both patients, while no signiﬁcant difference occurs between the surfaces with high TAWSS. Keywords Popliteal artery aneurysm · Peripheral stenting · Endovascular treatment · Femoropopliteal segment · Medical image analysis 1 Introduction Executive Editor: Jizeng Wang Popliteal arterial aneurysms (PAA) are common peripheral aneurysms. Although in the last few years endovascular treat- Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s10409- ment of the femoro-popliteal artery (FPA) has become a 021-01066-2. valuable therapeutic option, its efﬁcacy remains controver- sial due to the relatively high rate of complications, such B M. Conti as stent occlusion, intra-stent thrombosis or even stent frac- michele.conti@unipv.it ture [1]. All these drawbacks could be related to the intrinsic Department of Civil Engineering and Architecture, University morphology of the FPA segment that presents unique char- of Pavia, Pavia, Italy acteristics in terms of extreme mobility and biomechanical Department of Surgical and Integrated Diagnostic Sciences, forces and severe loading conditions due to repetitive leg ﬂex- University of Genoa, Genoa, Italy ion during daily activities [2]. If on the one hand the stent Department of Radiology, IRCCS Ospedale Policlinico San fracture can be traced back almost exclusively to repeated Martino, Genoa, Italy bending of the stented leg, on the other the mechanisms that Vascular and Endovascular Surgery Unit, IRCCS Ospedale lead to intra-stent thrombosis are not fully understood even Policlinico San Martino, Genoa, Italy 123 280 A. Ferrarini, et al. if hemodynamics is suspected of playing an important role Carreau model to describe the non-Newtonian viscosity of in this process [3]. blood. Computational ﬂuid dynamics (CFD) analyses are increas- According to the literature, CFD provides a useful tool ingly exploited to quantify the blood ﬂow inside the FPA and for understanding and predicting disease progression and evaluate the changes on hemodynamic patterns due to the hemodynamic-related post-stent complications. However, combination of endovascular stenting and leg movements. literature studies about patient speciﬁc CFD of popliteal Blood ﬂow patterns, and in particular low shear stress, promi- stenting are scarce; in particular, many of them involve ide- nent secondary ﬂows or huge variations of arterial wall shear alized geometry [3,9] and literature boundary conditions stress (WSS) are indeed known to correlate with pathological [3,10], without any information regarding the follow-up conditions [4–6], as brieﬂy resumed in the following. intra-stent thrombosis. Moreover, morphological variations First studies investigating ﬂow patterns in patient-speciﬁc during knee ﬂexion in the FPA could signiﬁcantly inﬂuence superﬁcial femoral arteries date back to 2006 by Wood et al. the local hemodynamic [9,10]. To the best of our knowledge, [6], who combined magnetic resonance imaging and CFD a complete computational study including all these aspects to assess the relationship between curvature and tortuosity is still missing. of superﬁcial femoral arteries and ﬂow patterns as function Based on such considerations, we performed patient- of sex and age. More recently, the study of Desyatova et speciﬁc CFD simulations in order to assess the impact of al. [7], who investigated the effects of aging on mechani- leg bending and the interplay among geometrical features on cal stresses, deformations, and hemodynamics, has identiﬁed the local hemodynamic of two patients treated with endovas- the popliteal artery as the location with greatest intramu- cular stent-graft placement for PAA, experiencing intra-stent ral stresses along the leg arteries. Moreover, the association thrombosis. Moreover, we deepened and improved our pre- of vessel restenosis with hemodynamical markers derived vious study [10] by investigating the impact of the different from blood ﬂow has been investigated by Gogkol et al. [3], inlet boundary conditions on the solution (in a similar way in patients undergoing endovascular treatment for periph- to Hua et al. [17]), and assessing the hypothesis of non- eral artery diseases (PAD). However, the proposed work was Newtonian behavior of blood (using the Carreau model), in based on vessel geometries reconstructed from 2D angio- comparison with the usual approximation of blood as a New- graphic images thus idealizing the lumen cross-sections. tonian ﬂuid. This limitation was overcome by Colombo et al. [8], who presented a fully patient-speciﬁc computational framework based on geometric reconstructions from Computed Tomog- 2 Materials and methods raphy (CT) images and boundary conditions taken from Doppler ultrasound images. However, despite the proposed Signed informed consent was obtained from the patients innovations, only the straight-leg conﬁguration has been and all procedures were performed in accordance with the studied, thus neglecting the analysis of the effects of leg bend- Declaration of Helsinki and submitted to the local institu- ing on the geometry and hemodynamics of the stented area. tional medical ethics committee. Two patients with PAA and In a more recent study led by Colombo et al. [9], knee ﬂexion endovascularly treated at Vascular and Endovascular Surgery and complete movement of walking have been assessed in an Unit of University Hospital of Genoa were enrolled for an idealized model of FPA. Finally, the impact of leg bending imaging study with double CT acquisition at straight- and on geometrical and hemodynamic features have been inves- bent-leg. More details of the imaging acquisition protocol tigated in our previous work, where patient-speciﬁc CFD were provided in a previous study [18]. simulations have been performed on a single patient, using A 78-year old man (Patient 1) with PAA in the left leg was literature boundary conditions [10] and Newtonian model successfully treated with two Viabahn devices (W.L. Gore for blood rheology. Although it is known that blood is a non- & Associates, Flagstaff, AZ, USA) sized 9 mm ×150 mm Newtonian ﬂuid [11], literature review about CFD modeling (proximal stent) and 7 mm ×150 mm (distal stent). At 12 of the actual rheology of blood is controversial. While the months follow-up, post-operative CT showed partial stent assumption of treating blood as a Newtonian ﬂuid is widely thrombosis in the transition zone between the two partially accepted [12], it still represents a pivotal issue in case of overlapped devices. The second enrolled patient (Patient 2) small or mid arteries. Moreover, while some studies high- was a 68 year-old man treated in the left leg for PAA with lighted the importance of the non-Newtonian rheology [13], two Viabahn devices sized 10 mm×100 mm (proximal stent) others found that the use of a Newtonian blood model repre- and 9 mm ×150 mm (distal stent). In this patient, intra-stent sents a good approximation [14,15]. In particular, focusing thrombosis was revealed already by 1 month follow-up CT on modeling of the FPA, most of the studies [3,6,16] assumed scan. a constant viscosity, even if Colombo et al. [8] adopted the Postoperative CT images were anonymized and trans- ferred to a workstation for image processing. Segmentation 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 281 of the vessel lumen from the femoral artery bifurcation – A3: equivalent to the boundary condition A2 with a to the popliteal artery bifurcation, intra-stent thrombosis, parabolic velocity proﬁle set at the inlet of the ﬂow exten- leg bones, and the implanted stent-graft(s) was performed sion. by means of Vascular Modeling ToolKit (VMTK) libraries – B1: inﬂow waveform taken from Nichols et al. [22](see [19]. A surrogate pre-thrombotic model of the lumen was Fig. 1) with a ﬂat velocity proﬁle. derived by virtually removing the thrombosis during the – B2: equivalent to the boundary condition B1 but with the image segmentation in order to correlate hemodynamic with ﬂat velocity proﬁle set at the inlet of the ﬂow extension. thrombosis onset. Rigid registration of bent-leg structures – B3: equivalent to the boundary condition B2 but with on their corresponding straight counterparts was automat- a parabolic velocity proﬁle set at the inlet of the ﬂow ically performed by means of the Iterative Closest Point extension. algorithm implemented in VMTK. Centerline vessel was automatically extracted, smoothed and resampled by 0.5 mm by means of VMTK libraries [20]. Centerline tortuosity was Hence, we are imposing the same velocity waveform when then computed: it represents an important factor in different using the boundary conditions A1, A2, and A3, while the cardiovascular diseases, i.e., atherosclerosis, abdominal aor- same inﬂow waveform when adopting the boundary con- tic aneurysm, and, in particular, in thrombus initiation [21]. ditions B1, B2, and B3, thus implying slightly different Tortuosity (T ) is measured as follows: given the centerline inlet velocity waveforms according to the inlet radius of length (L) and the shortest distance between the two center- our computational domain. For example, in Fig. 1 the veloc- line endpoints (ED), T = L/ED−1; therefore, with this ity waveforms corresponding to the boundary conditions A1 deﬁnition, the tortuosity of a straight line is 0. (equivalent to A2, A3) and B1 for Patient 1 in straight-leg Transient CFD analyses were performed using Intel Xeon conﬁguration are represented. In particular, we extracted the W-2123 computing workstation (3.6 GHz, 32 GB RAM) with values of the velocity waveform, taken from Ref. [6] (and the commercial software FLUENT (v.19.2, ANSYS Aca- analogously for the inﬂow waveform [22]), using the soft- demic Research). We considered the two patients in both ware WebPlotDigitizer 4.4 (WebPlotDigitizer). Then, the straight- and bent-leg conﬁgurations in order to assess the data obtained by the literature waveform were interpolated effects of leg bending and the impact of inlet boundary with an 8th order Fourier series by using the Curve Fitting condition on the ﬂow solution. Uniform meshes were gen- App, given by the software Matlab R2018a (The Mathworks erated using VMTK, ranging from 648879 and 1471373 Inc.). The transient inlet velocity waveform was deﬁned in number of tetrahedral elements, according to the previ- FLUENT by the meaning of a user deﬁned function (UDF). ously performed mesh sensitivity analysis. In particular, Therefore, the inlet boundary conditions with a ﬂat velocity the mesh was reﬁned until the difference in the luminal proﬁle (i.e., scenarios A1, A2, B1, B2) were given by area exposed to TAWSS < 0.62 Pa between successive grids was < 1%. Geometry and boundary conditions are the main features affecting the CFD simulations in hemo- dynamics. We evaluated the impact of the inlet boundary conditions on the numerical solution by considering two literature waveforms (boundary conditions A and B) with three different scenarios, i.e., with or without ﬂow exten- sion and varying the velocity proﬁle (ﬂat or parabolic). In particular, the ﬂow extensions that we used in the simu- lations have been chosen in order to reduce the effect of the arbitrary choice of the velocity proﬁle. They were mod- eled using VMTK, with a length corresponding to 3.5 times the dimension of the inlet diameter, according to Colombo et al. [8]. The following inlet boundary conditions were tested: – A1: velocity inlet waveform taken from Wood et al. [6] Fig. 1 Inlet velocity waveforms in m/s colored according to: the liter- (see Fig. 1) with a ﬂat velocity proﬁle. ature inlet velocity taken from Wood et al. [6] and imposed at the inlet – A2: equivalent to the boundary condition A1 with the ﬂat boundary in scenario A1 (and analogously in A2, A3); the inlet velocity velocity proﬁle set at the ﬂow extension of the inlet of computed from the literature inﬂow taken from Nichols et al. [15]and used in scenario B1 (and analogously in B2, B3). the patient-speciﬁc models (see Fig. 2). 123 282 A. Ferrarini, et al. 2 2 (n−1)/2 η = η + (η − η )(1 + λ γ˙ ) , (3) ∞ 0 ∞ where η is the effective viscosity, η the inﬁnite shear rate viscosity, η the zero shear rate viscosity, λ the natural time, γ˙ the shear rate, and n the power law index. The parameter values were set according to Quanyu et al. [23] and listed in Table 1. In each simulation we prescribed the no-slip con- dition on the wall of the artery. Regarding the outlets, the following ﬂow splits were assigned as percentages of the FPA output, according to Crawford et al. [24]: the anterior tibial artery 20%, posterior tibial artery 40%, and peroneal artery 40%. The ﬂow was assumed in laminar regime since the maximum Reynolds number among all the simulations was 1328 at systolic peak (occurring with A1, A2, and A3 con- ditions). Semi-implicit method for pressure linked equations Fig. 2 Femoro-popliteal artery of the two patients considered in the (SIMPLE) was used to solve the Navier–Stokes equations. CFD simulations in the straight-leg conﬁguration. Both are colored Second order scheme for both pressure and momentum spa- according to the ﬂow extension, added to our computational domains tial discretization was adopted. After a sensitivity analysis, a in scenarios A2, A3, B2, B3, and to the three zones under investiga- tion (proximal artery, proximal stent, and distal stent). Moreover, the constant time-step size was set to 0.001 s and three cardiac sections considered in the post processing S ,S ,…,S are represented. 0 1 4 cycles were performed for each simulation to guarantee the The region marked with asterisk denotes the overlapping zone of the repeatability of solution. two stents in both the patients In order to evaluate the impact leg bending on the local hemodynamics of FPA, with a focus on the stented and throm- botic regions, the FPA segments of both the patients were u = a + [a cos(iωt ) + b sin(iωt )], (1) ﬂat 0 i i divided into 3 zones (see Fig. 2): (1) proximal artery, i.e., i =1 lumen of the artery above the proximal end of the proxi- mal stent (excluding the ﬂow extension); (2) proximal stent, where ω is the fundamental frequency (see Table 1), t the i.e., the lumen of the proximal stent, excluding the overlap- simulation time, and a , a , b for i = 1, 2, ··· , 8, the values 0 i i ping zone; (3) distal stent, i.e., the lumen of the distal stent of the Fourier parameters given by the Curve Fitting App. The including the overlapping zone. We performed both a quali- inlet boundary conditions with a parabolic velocity proﬁle tative and quantitative analysis comparing the results of the (i.e., scenarios A3 and B3) were prescribed as two patients in straight- and bent-leg conﬁgurations obtained from the CFD simulations. Firstly, to evaluate the impact of boundary conditions, we focused on the velocity streamlines, u = 2u 1 − , (2) parabolic ﬂat the vectors of velocity magnitude, and the velocity proﬁles at the following cross sections corresponding to: the ﬂow where r denotes the distance between a point on the con- extension inlet, S ; FPA inlet, S ; proximal stent inlet, S ; 0 1 2 strained surface and the center of the surface, and R is the distal stent inlet, S ; distal stent outlet, S (see Fig. 2). The 3 4 radius of the constrained surface. velocity streamlines, the vectors of velocity magnitude, and The proposed six boundary conditions were imposed the velocity proﬁles were reported at the systolic peak. on the patient-speciﬁc model of the two patients for both We computed the time-averaged wall shear stress (TAWSS) straight- and bent-leg conﬁgurations, therefore we performed and oscillatory shear index (OSI), regarding the near wall 24 simulations (six boundary conditions for two patients for ﬂow features, and local normalized helicity (LNH) and helic- two leg conﬁgurations). ity intensity (h index), relating to the bulk ﬂow. TAWSS and Firstly, blood was assumed as an incompressible and OSI were computed as follow: Newtonian ﬂuid, with 1060 kg/m density and 0.0035 Pa s viscosity [16]. Then, in order to evaluate the impact of the non-Newtonian behavior of blood, we chose the boundary TAWSS = |WSS|dt , (4) conditions A1 and B1, i.e., two velocity waveforms with a ﬂat velocity proﬁle, running 8 simulations (two boundary | WSSdt | OSI = 0.5 1 − , (5) conditions per two patients per two leg conﬁgurations). The WSSdt viscosity was modeled using the Carreau model described in the following equation: 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 283 Table 1 Fundamental frequency used in UDF inlet waveform—Eq. (1); parameters of the Carreau model—Eq. (3) −1 Frequency ω(s ) Inﬁnite shear rate Zero shear rate Time constant λ (s) Power low index n (B1, B2, B3) (A1, A2, A3) viscosity η (kg/(m·s)) viscosity η (kg/(m·s)) ∞ 0 Value 6.981 7.854 0.0035 0.056 3.313 0.3568 Fig. 3 Streamlines, contours, and velocity vectors colored according to velocity magnitude at systolic peak, corresponding to the scenarios A1, A2 and A3, in both straight- and bent-leg conﬁgurations of the two patients where T is the cardiac period and |WSS| the norm of the of the WSS. In particular, high values of OSI denote sites WSS vector. WSS is deﬁned as follows where the WSS deviates from the main ﬂow direction in a large fraction of the cardiac cycle [26]. According to Gok- gol et al. [3], luminal area exposed to high OSI (> 0.3) was WSS = σ n −[(σ n) · n]n, (6) computed. Regarding the bulk ﬂow, qualitatively, we com- puted the LNH, which corresponds to the cosine of the angle where σ is the Cauchy stress tensor and n the normal vector formed between the vorticity vector and the velocity vector to the surface. In particular, in an incompressible ﬂuid, the Cauchy stress tensor is deﬁned as follow (∇× u) · u LNH = = cos α, (8) σ = η(∇u +∇u ) − pI , (7) |∇ × u|·|u| where u is the velocity vector, p the pressure, and I the iden- where α is the angle formed between the vorticity vector tity matrix. TAWSS plays a pivotal role in the development of (∇× u) and velocity vector u. It is a measure of the align- arterial stenosis and in prediction of the risk of wall rupture ment/misalignment of the local velocity and vorticity vectors. and thrombus deposition. According to Malek et al. [25], we LNH ranges from − 1 to 1, and its sign indicates the direc- calculated the luminal surface exposed to low and high val- tion of helical structures. Quantitatively, we computed the h ues of TAWSS, i.e., ranging between 0 and 0.4 Pa and above helicity, that is an index regarding the bulk ﬂow: it is given 1.5 Pa, respectively. OSI measures the temporal oscillations by time-averaging the absolute value of the helicity [27]: 123 284 A. Ferrarini, et al. Fig. 4 Streamlines, contours, and velocity vectors colored according to velocity magnitude at systolic peak, corresponding to the scenarios B1, B2, and B3 in both straight- and bent-leg conﬁgurations of the two patients cross-sections of the stented regions, by ﬁxing the inlet wave- h = |u · (∇× u)|dV dt , (9) TV T V form. Therefore, from now on we consider only the results relating to the scenarios A1 and B1 for both the patients in where V is the arterial volume. The h helicity index straight- and bent-leg conﬁgurations. However, the ﬁgures expresses the helicity intensity in the ﬂuid domain, irrespec- including all the scenarios relating to the Newtonian model tive of direction. Recalling that the helicity is deﬁned by are contained in the Supplementary Materials and Methods the spatial integral of the scalar product of the velocity and section. vorticity, we assume h index has higher values in the ﬂuid 2 Figure 5 highlights the arterial lumen colored according domain in which velocity and vorticity vectors are aligned. to low (ranging from 0 to 0.4 Pa) and high TAWSS (higher than 1.5 Pa). The results suggest that the distal part of the artery is exposed to high TAWSS in both straight- and bent- leg conﬁguration with a limited inﬂuence of inﬂow boundary 3 Results conditions; such a result is particularly evident in the case of Patient 1, while for Patient 2 the B1 scenario is resulting in Firstly, we reported the results obtained by the 24 simula- tions performed with constant viscosity. Figures 3 and 4 show physiological TAWSS in most of the whole artery for both conﬁgurations. Figure 6 shows the arterial lumen colored the results of CFD simulations for the straight- and bent-leg according to high OSI (> 0.3) is represented. High OSI are conﬁgurations of both the patients in the six scenarios that located for all the cases under considerations in the proximal have been tested (A1, A2, A3 and B1, B2, B3 in Figs. 3 and part of artery irrespective to the adopted boundary conditions. 4, respectively), reporting streamlines, velocity proﬁles, and Helical blood ﬂow structures developing into the endo- velocity vectors colored according to the velocity magnitude prostheses are represented in Fig. 7 using iso-surfaces of at the systolic peak. These two ﬁgures prove that only the LNH at the systolic peak with a threshold of ± 0.25, accord- imposed waveform at the inlet (taken from Wood et al. [6]or ing to Colombo et al. [8], for both the patients in straight-leg Nichols et al. [22]) signiﬁcantly affects the solution. Indeed, each scenario has similar velocity proﬁles and contours in the conﬁguration, relating to the scenario A1. The results show 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 285 Fig. 7 Femoro-popliteal artery and three zoom views of the lumen (rotating clockwise) of both the patients in straight-leg conﬁguration: the area where the thrombosis is localized is highlighted by a black box. Moreover, blood ﬂow helicity is represented: in blue the ﬂow with negative LNH and in red the ﬂow with positive LNH Fig. 5 Arterial lumen colored according to low (< 0.4 Pa) and high (> that the bulk ﬂow in the artery for both patients is char- 1.5 Pa) TAWSS in both straight- and bent-leg conﬁgurations of the two acterized by two counter-rotating helical structures and in patients particular the helical shape of the thrombosis seems to ﬂow the path of the negative LNH region. Figure 8 reports the bar-plots of the value of h index, tortuosity, and the percentage of luminal surface exposed to low and high TAWSS, and high OSI corresponding to each zone (proximal artery, proximal stent, and distal stent) for both the patients in straight- and bent-leg conﬁgurations. The results show that leg bending induces a difference of the computed hemodynamics indices for Patient 1 with both A1 and B1 boundary conditions. Indeed, a percentage difference above 50% between the two conﬁgurations is present for each hemodynamic quantity that we computed in all the tested scenarios, except for the percentage difference relating to the luminal area exposed to high OSI in Patient 2 (with a maximum percentage difference of 24% in the distal stent region). In particular, our results show a signiﬁcant variation of tortuosity between the two conﬁgurations, accentuated in the distal stent zone, where the tortuosity is greater in the bent-leg conﬁguration. Finally, we treated the blood as a non-Newtonian ﬂuid and we assessed the results, comparing them with the previ- ous analyses, obtained using the Newtonian model. Figure 9 shows the arterial lumen of both patients colored according to TAWSS magnitude, low (ranging from 0 to 0.4 Pa) and high TAWSS (higher than 1.5 Pa), based on both the Carreau Fig. 6 Arterial lumen colored according to high OSI (> 0.3) in both (non-Newtonian) and Newtonian models. Figure 9 represents straight- and bent-leg conﬁgurations of the two patients only the results relating to the scenario B1, which provides greater differences between the two models under considera- 123 286 A. Ferrarini, et al. Fig. 8 Bar plot of tortuosity, helicity (h index), and percentage of luminal area exposed to both low (< 0.4 Pa) and high (> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in both leg conﬁgurations, corresponding to scenarios A1 and B1 tion and allows us a wider discussion, as we will introduce in thrombosis during follow-up. In particular, the role of leg the next section. Figures 10 and 11 represent the bar-plots of bending on the local hemodynamic was elucidated by mod- the value of h index, and the percentage of luminal surface eling both straight- and bent-leg conﬁgurations. exposed to low and high TAWSS, and high OSI correspond- Focusing on the velocity magnitude, Figs. 3 and 4 show ing to each zone (proximal artery, proximal stent, and distal a higher ﬂow velocity in the distal stent region than to the stent) for both the patients. In particular, Fig. 10 refers to the proximal one, due to the luminal narrowing given by the over- straight-leg conﬁguration, while Fig. 11 to the bent-leg one. lapping of the two stents-grafts. As we already pointed out, the results suggest that the different inlet velocity proﬁles used in the simulations slightly affect the numerical solu- tion, conversely to the determinant role of the prescribed inlet 4 Discussions waveform. In order to obtain reliable results of clinical sig- niﬁcance, patient-speciﬁc inﬂow waveforms would be very In this study, we have evaluated the local hemodynamic useful in understanding the hemodynamic behavior. How- and the interplay among geometric features in two patients ever, our geometrical study shows velocity sensitivity, i.e., endovascularly treated for PAA, who experienced intra-stent 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 287 velocity magnitude variations between the two patients occur along the two FPAs by ﬁxing a velocity inlet (i.e., scenario A or B). Although the behavior of stented FPAs has already been investigated in the literature [8], to date there is still no information on the different response between the various portions of the stented artery itself. Figure 8 suggests that the overlapping of the stent grafts seems to induce a severe discontinuity of lumen diameter, dividing the region treated with endovascular stent-graft in two zones: (1) the proximal part, where thrombosis is located, it is characterized by low tortuosity, low velocity, low helicity, low TAWSS, and high OSI; (2) the distal part that presents higher tortuosity, pro- moting higher velocity, higher helicity, higher TAWSS, and lower OSI. In particular, by focusing on the tortuosity of the vessel (see Fig. 8 at the bottom), the stented FPA respects the behavior that we would have expected, when considered in its entirety, i.e., increased tortuosity values with leg bend- ing. Analyzing the stented area by portions, we have found that in both patients the tortuosity increases from the prox- imal artery region to the distal one; this result matches the Fig. 9 Arterial lumen colored according to the TAWSS magnitude, low ﬁndings of Wood et al. [6], who performed CFD simula- (< 0.4 Pa) and high TAWSS (> 1.5 Pa) in both straight- and bent-leg tions in the superﬁcial femoral artery of 9 healthy men and conﬁgurations of the two patients. The TAWSS values represented refers 9 healthy women, showing that tortuosity was signiﬁcantly to the scenario B1 greater for men than women, but the highest values were found in the most distal segment, regardless of sex. Then, when considering the comparison between straight- and bent- an inverse relationship between helical ﬂow and OSI evalu- leg conﬁguration, we observed that in both patients proximal ating four bypass geometries in ascending aorta, according vessel and distal stent segments tortuosity increases with leg to our results. Moreover, our results denote that the spiral bending. However, the proximal stent, characterized by its shape of thrombosis matches the path of the negative LNH larger diameter, slow velocity, low TAWSS, and low helicity, region; this is more evident in Patient 1 (Fig. 7). Figure 7 straightens with leg ﬂexion. This area is also the one in which refers only to scenario A1, but an analogous pattern of the thrombosis was found in both patients, conﬁrming that the LNH was found using the boundary condition B1. It is hard formation of thrombosis is linked to a combination of both to formulate a conclusion to explain this result; given the lim- hemodynamic and geometric factors. Hence the importance ited number of analysed patients, further analysis involving of conducting the analyses by investigating the stented FPA a cohort of patients should be investigated in order to provide not only in its entirety but by dividing it into the various por- more information to elucidate this observation. tions, in order to be able to identify areas more at risk of From our results we found that the study in both straight- thrombosis. and bent-leg conﬁgurations is crucial in understanding and The role of low TAWSS in thrombotic regions has been assessing the numerical results in stented arteries, given the previously corroborated in literature. Boyd et al. [28]showed increase of the tortuosity of the distal part of the artery due to a correlation between regions of low WSS, where ﬂow the leg bending. Our ﬁndings match with Wensing et al. [32], recirculation predominated, and thrombus deposition, by per- who highlighted the importance of considering the impact forming CFD simulations in 7 abdominal aortic aneurysms. of knee ﬂexion in femoral and popliteal arteries, showing The luminal area exposed to low TAWSS and high OSI in increasing tortuosity in bent-leg conﬁguration of 22 healthy the proximal zone is greater in Patient 1 than in Patient 2 (see volunteers. Moreover, the increase of tortuosity in leg bend- Fig. 8), suggesting that patient-speciﬁc geometrical features ing implies a reduction of the blood velocity in each scenario also affect the near wall ﬂow features. that we assumed for both patients (see velocity streamlines Regarding the bulk ﬂow, our results suggest that intra- and contours represented in Figs. 3 and 4). stent thrombosis is located in the region where the intensity The alternate bending of the legs is known to inﬂuence of helicity is low (see Fig. 8). The fundamental role of helical the mechanical solicitation of the stent [33], the shape of the (or swirling ﬂow) in the prevention of thrombosis and disease artery [34], and the local hemodynamics [9] but its role in the progression has been conﬁrmed in many literature studies thrombosis onset and progression is still unknown. From our [29–31]. In particular, Morbiducci et al. [30] also presented results, it is evident that leg bending increases the tortuosity 123 288 A. Ferrarini, et al. Fig. 10 Comparison between results considering Newtonian and non-Newtonian behavior: bar plot of helicity (h index), and percentage of luminal area exposed to both low (< 0.4Pa) andhigh(> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in the straight-leg conﬁguration, corresponding to scenariosA1 and B1 of the distal stent segment, which combined with an overall the magnitude of WSS is relatively small (< 1N/m ). Anal- blood ﬂow velocity, exacerbate the difference between the ogously, we found similar results using the scenario A1, but distal and proximal part of the stented region, with the latter with less marked differences, since the surface exposed to more exposed to the risk of thrombosis (i.e., lower velocity, low TAWSS is very small even in the Newtonian model. For wider area of low wall shear stress, higher oscillatory shear this reason we chose to omit the qualitative analysis given by stress, and lower helicity). Such considerations are however the scenario A1. hardly generalizable with data proposed by the present paper, Figures 10 and 11 allow deepening the comparison which deals with only two patients, but, at the same time, call between the Newtonian and non-Newtonian models. Sig- for future developments focused on such hypotheses. niﬁcant differences based on the luminal surfaces exposed Focusing on the qualitative comparison between the New- to low TAWSS are highlighted (with a maximum decrease tonian and non-Newtonian model, Fig. 9 shows an optimal in the proximal artery zone, compared to the non-Newtonian agreement on the distribution of the TAWSS magnitude model, of 8.7% and 8.5% for Patients 1 and 2, respectively, in between the two models. These results reproduce the assump- the straight-leg conﬁguration and in the scenario B1), a good tions discussed by Liu et al. [35], who introduced that the agreement occurs for the other analyzed outcomes (with a blood viscosity properties do not affect the spatial pattern maximum OSI decrease in the proximal artery zone, com- of the TAWSS qualitatively. However, looking at the lumi- pared to the non-Newtonian model, of 3.2% for patient 2 in nal surface exposed to low and high TAWSS, the area with the straight-leg conﬁguration and in the scenario B1). In par- low TAWSS is greater in the Newtonian model for both the ticular, as we mentioned before, the scenario B1, in which the patients, while no signiﬁcant difference occurs between the inlet average velocity is lower and also low TAWSS values surfaces with high TAWSS. Our ﬁndings are in agreement occur, provides major differences. As the velocity increases with Soulis et al. [15] and Liu et al. [35], who proved an (see the results given by the scenario A1 in Fig. 10), the underestimated WSS given by the Newtonian model, when 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 289 Fig. 11 Comparison between results considering Newtonian and non-Newtonian behavior: bar plot of tortuosity, helicity (h index), and percentage of luminal area exposed to both low (< 0.4 Pa) and high (> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in the bent-leg conﬁguration, correspondingto scenarios A1 and B1 Newtonian and non-Newtonian models become more simi- different time instants, from early post-operative to annual lar, according to Liu et al. [35]. follow-up exams. In the present study we dealt with thrombosis only from a ﬂuid dynamic point of view. However, further analysis should include the role of hemodynamic stress in the platelet activation [36], recently proved to be associated with aortic 5 Limitations thrombus formation [37]. Finally, according to previous studies [3,8], we did not take The present work, based on the analysis of only two into account the stent struts; however, further developments cases, represents a proof-of-concept study, aimed at link- will include this aspect, since Al-Hakim et al. [38]showed ing post-stent geometry, hemodynamics, and thrombosis in that stent struts have an effect on WSS. endovascular repair of popliteal aneurysms. Further analyzes should be performed in order to obtain statistically and clin- ically relevant conclusions. We have already discussed the importance of considering patient-speciﬁc inlet boundary 6 Conclusions condition; therefore, in future studies inﬂow data elaborated by echo doppler measurements will be set at the inlet of the The present study suggests that the overlapping of the stent- computational domain. grafts seems to induce a severe discontinuity of lumen The computational domains considered in the simula- diameter, dividing the region treated with endovascular stent- tions represent surrogate geometrical models of the lumen graft into two zones: the proximal part, where thrombosis is of each patient prior to thrombosis by virtually removing the located, it is characterized by low tortuosity, low velocity, thrombus during the segmentation process. Such a limitation low helicity, low TAWSS, and high OSI; the distal part that could be overcome by analyzing the CT scans performed at presents higher tortuosity, promoting higher velocity, higher 123 290 A. Ferrarini, et al. helicity, higher TAWSS, and lower OSI. Since this analy- 3. Gökgöl, C., Diehm, N., Räber, L., et al.: Prediction of restenosis based on hemodynamical markers in revascularized sis is limited to two cases, a further study with a cohort of femoro-popliteal arteries during leg ﬂexion. Biomech. Model. patients should be investigated in order to generalize and Mechanobiol. 18, 1883–1893 (2019) validate our results. Boundary conditions affect the solu- 4. Glagov, S., Zarins, C., Giddens, D.P., et al.: Hemodynamics and tion only when considering different velocity waveforms, atherosclerosis. Insights and perspectives gained from studies of human arteries. Arch. Pathol. Lab. Med. 112(10), 1018–1031 dependent on time; different inlet velocity proﬁles and the (1988) use of ﬂow extension do not provide signiﬁcant variations. 5. Casa, L.D., Deaton, D.H., Ku, D.N.: Role of high shear rate in Accounting for actual ﬂow rate is essential for accurate and thrombosis. J. Vasc. Surg. 61(4), 1068–1080 (2015) reliable results. The Newtonian and non-Newtonian blood 6. Wood, N.B., Zhao, S.Z., Zambanini, A., et al.: Curvature and tor- tuosity of the superﬁcial femoral artery: a possible risk factor for treatments provide similar results in both the patients, except peripheral arterial disease. J. Appl. Physiol. 101(5), 1412–1418 when the magnitude of the TAWSS is relatively small (< 0.4 (2006) Pa). In this latter case the Newtonian model gives lower val- 7. Desyatova, A., Poulson, W., Deegan, P., et al.: Limb ﬂexion ues of TAWSS than the non-Newtonian one. However, the induced twist and associated intramural stresses in the human femoropopliteal artery. J. R. Soc. Interface 14, 20170025 (2017) Newtonian blood treatment should be a good choice in all 8. Colombo, M., Bologna, M., Garbey, M., et al.: Computing patient- cases in which the analysis of WSS is not necessary. Leg speciﬁc hemodynamics in stented femoral artery models obtained bending induces signiﬁcant hemodynamic differences com- from computed tomography using a validated 3D reconstruction pared to the straight leg conﬁguration in each of the scenarios method. Med. Eng. Phys. 75, 23–35 (2020) 9. Colombo, M., Luraghi, G., Cestariolo, L., et al.: Impact of lower we studied for both patients. The helical form of intra-stent limb movement on the hemodynamics of femoropopliteal arteries: thrombosis suggests an implication of ﬂow helicity in the a computational study. Med. Eng. Phys. 81, 105–117 (2020) onset and progression of thrombosis. However, further stud- 10. Conti, M., Ferrarini, A., Finotello, A., et al.: Patient-speciﬁc com- ies should be considered to investigate this aspect. putational ﬂuid dynamics of femoro-popliteal stent-graft thrombo- sis. Med. Eng. Phys. 86, 57–64 (2020) 11. Merrill, E.W.: Rheology of blood. Physiol. Rev. 49, 863–888 (1969) Acknowledgements This work was partially supported by the “Pro- 12. Wootton, D.M., Ku, D.N.: Fluid mechanics of vascular systems, gramma Operativo Por FSE Regione Liguria 2014-2020” (RLOF18A- diseases, and thrombosis. Annu. Rev. Biomed. Eng. 1, 299–329 SSRIC/38/1). (1999) 13. Cho, Y.I., Kensey, K.R.: Effects of the non-Newtonian viscosity of Funding Open access funding provided by Università degli Studi di blood on ﬂows in a diseased arterial vessel. Part 1: Steady ﬂows. Pavia within the CRUI-CARE Agreement. Biorheology 28, 241–262 (1991) 14. Johnston, B.M., Johnston, P.R., Corney, S., et al.: Non-Newtonian blood ﬂow in human right coronary arteries: transient simulations. Compliance with ethical standards J. Biomech. 39, 1116–1128 (2005) 15. Soulis, J.V., Giannoglou, G.D., Chatzizisis, Y.S., et al.: Spatial and Ethical approval The study was approved by the Liguria Regional phasic oscillation of non-Newtonian wall shear stress in human Ethics Committee (Comitato Etico Regionale Liguria) on 15/07/2019 left coronary artery bifurcation: an insight to atherogenesis. Coron. (Ref. gr-2018-12368376; internal Amendment No: 4587). Artery Dis. 17, 351–358 (2006) 16. Desyatova, A., MacTaggart, J., Romarowski, R., et al.: Effect of Open Access This article is licensed under a Creative Commons aging on mechanical stresses, deformations, and hemodynamics Attribution 4.0 International License, which permits use, sharing, adap- in human femoropopliteal artery due to limb ﬂexion. Biomech. tation, distribution and reproduction in any medium or format, as Model. Mechanobiol. 17, 181–189 (2018) long as you give appropriate credit to the original author(s) and the 17. Hua, Y., Oh, J.H., Kim, Y.B.: Inﬂuence of parent artery segmenta- source, provide a link to the Creative Commons licence, and indi- tion and boundary conditions on hemodynamic characteristics of cate if changes were made. The images or other third party material intracranial aneurysms. Yonsei Med. J. 56, 1328–1337 (2015) in this article are included in the article’s Creative Commons licence, 18. Spinella, G., Finotello, A., Pane, B., et al.: In vivo morphological unless indicated otherwise in a credit line to the material. If material changes of the femoropopliteal arteries due to knee ﬂexion after is not included in the article’s Creative Commons licence and your endovascular treatment of popliteal aneurysm. J. Endovasc. Ther. intended use is not permitted by statutory regulation or exceeds the 26(4), 496–504 (2019) permitted use, you will need to obtain permission directly from the copy- 19. Antiga, L., Piccinelli, M., Botti, L., et al.: An image-based model- right holder. 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Qiu, Y., Yuan, D., Wang, Y., et al.: Hemodynamic investigation Publisher’s Note Springer Nature remains neutral with regard to juris- of a patient-speciﬁc abdominal aortic aneurysm with iliac artery dictional claims in published maps and institutional afﬁliations. tortuosity. Comput. Methods Biomech. Biomed. Eng. 21, 824–833 (2018) 32. Wensing, P.J.W., Scholten, F.G., Buijs, P.C., et al.: Arterial tortuos- ity in the femoropopliteal region during knee ﬂexion: a magnetic resonance angiographic study. J. Anat. 186, 133–139 (1995) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Acta Mechanica Sinica" Springer Journals http://www.deepdyve.com/lp/springer-journals/impact-of-leg-bending-in-the-patient-specific-computational-fluid-ooxDc9SX9M

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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2021
ISSN: 0567-7718
eISSN: 1614-3116
DOI: 10.1007/s10409-021-01066-2
Publisher site: See Article on Publisher Site

Abstract

Endovascular treatment of the femoro-popliteal artery has recently become a valuable therapeutic option for popliteal arterial aneurysms. However, its efﬁcacy remains controversial due to the relatively high rate of complications, such as stent occlusion as result of intra-stent thrombosis. The elucidation of the interplay among vessel geometrical features, local hemodynamics, and leg bending seems crucial to understand onset and progression of popliteal intra-stent thrombosis. To this aim, patient- speciﬁc computational ﬂuid dynamic simulations were performed in order to assess the intra-stent hemodynamics of two patients endovascularly treated for popliteal arterial aneurysm by stent-grafts and experiencing intra-stent thrombosis. Both Newtonian and non-Newtonian blood rheological models were considered. Results were presented in terms of tortuosity, luminal area exposed to low (< 0.4 Pa) and high (> 1.5 Pa) time-averaged wall shear stress (TAWSS), area exposed to high (> 0.3) oscillatory shear index (OSI), and ﬂow helicity. Study outcomes demonstrated that leg bending induced signiﬁcant hemodynamic differences (> 50% increase) in both patients for all the considered variables, except for OSI in one of the two considered patients. In both leg conﬁgurations, stent-graft overlapping induced a severe discontinuity of the lumen diameter where the proximal stented zone is characterized by low tortuosity, low velocity, low helicity, low TAWSS, and high OSI; while the distal part has higher tortuosity, velocity, helicity, TAWSS, and lower OSI. Sensitivity study on applied boundary conditions showed that the different inlet velocity proﬁles for a given inlet waveform affect slightly the numerical solution; conversely, the shape and magnitude of the prescribed inlet waveform is determinant. Focusing on the comparison between the Newtonian and non-Newtonian blood models, the area with low TAWSS is greater in the Newtonian model for both patients, while no signiﬁcant difference occurs between the surfaces with high TAWSS. Keywords Popliteal artery aneurysm · Peripheral stenting · Endovascular treatment · Femoropopliteal segment · Medical image analysis 1 Introduction Executive Editor: Jizeng Wang Popliteal arterial aneurysms (PAA) are common peripheral aneurysms. Although in the last few years endovascular treat- Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s10409- ment of the femoro-popliteal artery (FPA) has become a 021-01066-2. valuable therapeutic option, its efﬁcacy remains controver- sial due to the relatively high rate of complications, such B M. Conti as stent occlusion, intra-stent thrombosis or even stent frac- michele.conti@unipv.it ture [1]. All these drawbacks could be related to the intrinsic Department of Civil Engineering and Architecture, University morphology of the FPA segment that presents unique char- of Pavia, Pavia, Italy acteristics in terms of extreme mobility and biomechanical Department of Surgical and Integrated Diagnostic Sciences, forces and severe loading conditions due to repetitive leg ﬂex- University of Genoa, Genoa, Italy ion during daily activities [2]. If on the one hand the stent Department of Radiology, IRCCS Ospedale Policlinico San fracture can be traced back almost exclusively to repeated Martino, Genoa, Italy bending of the stented leg, on the other the mechanisms that Vascular and Endovascular Surgery Unit, IRCCS Ospedale lead to intra-stent thrombosis are not fully understood even Policlinico San Martino, Genoa, Italy 123 280 A. Ferrarini, et al. if hemodynamics is suspected of playing an important role Carreau model to describe the non-Newtonian viscosity of in this process [3]. blood. Computational ﬂuid dynamics (CFD) analyses are increas- According to the literature, CFD provides a useful tool ingly exploited to quantify the blood ﬂow inside the FPA and for understanding and predicting disease progression and evaluate the changes on hemodynamic patterns due to the hemodynamic-related post-stent complications. However, combination of endovascular stenting and leg movements. literature studies about patient speciﬁc CFD of popliteal Blood ﬂow patterns, and in particular low shear stress, promi- stenting are scarce; in particular, many of them involve ide- nent secondary ﬂows or huge variations of arterial wall shear alized geometry [3,9] and literature boundary conditions stress (WSS) are indeed known to correlate with pathological [3,10], without any information regarding the follow-up conditions [4–6], as brieﬂy resumed in the following. intra-stent thrombosis. Moreover, morphological variations First studies investigating ﬂow patterns in patient-speciﬁc during knee ﬂexion in the FPA could signiﬁcantly inﬂuence superﬁcial femoral arteries date back to 2006 by Wood et al. the local hemodynamic [9,10]. To the best of our knowledge, [6], who combined magnetic resonance imaging and CFD a complete computational study including all these aspects to assess the relationship between curvature and tortuosity is still missing. of superﬁcial femoral arteries and ﬂow patterns as function Based on such considerations, we performed patient- of sex and age. More recently, the study of Desyatova et speciﬁc CFD simulations in order to assess the impact of al. [7], who investigated the effects of aging on mechani- leg bending and the interplay among geometrical features on cal stresses, deformations, and hemodynamics, has identiﬁed the local hemodynamic of two patients treated with endovas- the popliteal artery as the location with greatest intramu- cular stent-graft placement for PAA, experiencing intra-stent ral stresses along the leg arteries. Moreover, the association thrombosis. Moreover, we deepened and improved our pre- of vessel restenosis with hemodynamical markers derived vious study [10] by investigating the impact of the different from blood ﬂow has been investigated by Gogkol et al. [3], inlet boundary conditions on the solution (in a similar way in patients undergoing endovascular treatment for periph- to Hua et al. [17]), and assessing the hypothesis of non- eral artery diseases (PAD). However, the proposed work was Newtonian behavior of blood (using the Carreau model), in based on vessel geometries reconstructed from 2D angio- comparison with the usual approximation of blood as a New- graphic images thus idealizing the lumen cross-sections. tonian ﬂuid. This limitation was overcome by Colombo et al. [8], who presented a fully patient-speciﬁc computational framework based on geometric reconstructions from Computed Tomog- 2 Materials and methods raphy (CT) images and boundary conditions taken from Doppler ultrasound images. However, despite the proposed Signed informed consent was obtained from the patients innovations, only the straight-leg conﬁguration has been and all procedures were performed in accordance with the studied, thus neglecting the analysis of the effects of leg bend- Declaration of Helsinki and submitted to the local institu- ing on the geometry and hemodynamics of the stented area. tional medical ethics committee. Two patients with PAA and In a more recent study led by Colombo et al. [9], knee ﬂexion endovascularly treated at Vascular and Endovascular Surgery and complete movement of walking have been assessed in an Unit of University Hospital of Genoa were enrolled for an idealized model of FPA. Finally, the impact of leg bending imaging study with double CT acquisition at straight- and on geometrical and hemodynamic features have been inves- bent-leg. More details of the imaging acquisition protocol tigated in our previous work, where patient-speciﬁc CFD were provided in a previous study [18]. simulations have been performed on a single patient, using A 78-year old man (Patient 1) with PAA in the left leg was literature boundary conditions [10] and Newtonian model successfully treated with two Viabahn devices (W.L. Gore for blood rheology. Although it is known that blood is a non- & Associates, Flagstaff, AZ, USA) sized 9 mm ×150 mm Newtonian ﬂuid [11], literature review about CFD modeling (proximal stent) and 7 mm ×150 mm (distal stent). At 12 of the actual rheology of blood is controversial. While the months follow-up, post-operative CT showed partial stent assumption of treating blood as a Newtonian ﬂuid is widely thrombosis in the transition zone between the two partially accepted [12], it still represents a pivotal issue in case of overlapped devices. The second enrolled patient (Patient 2) small or mid arteries. Moreover, while some studies high- was a 68 year-old man treated in the left leg for PAA with lighted the importance of the non-Newtonian rheology [13], two Viabahn devices sized 10 mm×100 mm (proximal stent) others found that the use of a Newtonian blood model repre- and 9 mm ×150 mm (distal stent). In this patient, intra-stent sents a good approximation [14,15]. In particular, focusing thrombosis was revealed already by 1 month follow-up CT on modeling of the FPA, most of the studies [3,6,16] assumed scan. a constant viscosity, even if Colombo et al. [8] adopted the Postoperative CT images were anonymized and trans- ferred to a workstation for image processing. Segmentation 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 281 of the vessel lumen from the femoral artery bifurcation – A3: equivalent to the boundary condition A2 with a to the popliteal artery bifurcation, intra-stent thrombosis, parabolic velocity proﬁle set at the inlet of the ﬂow exten- leg bones, and the implanted stent-graft(s) was performed sion. by means of Vascular Modeling ToolKit (VMTK) libraries – B1: inﬂow waveform taken from Nichols et al. [22](see [19]. A surrogate pre-thrombotic model of the lumen was Fig. 1) with a ﬂat velocity proﬁle. derived by virtually removing the thrombosis during the – B2: equivalent to the boundary condition B1 but with the image segmentation in order to correlate hemodynamic with ﬂat velocity proﬁle set at the inlet of the ﬂow extension. thrombosis onset. Rigid registration of bent-leg structures – B3: equivalent to the boundary condition B2 but with on their corresponding straight counterparts was automat- a parabolic velocity proﬁle set at the inlet of the ﬂow ically performed by means of the Iterative Closest Point extension. algorithm implemented in VMTK. Centerline vessel was automatically extracted, smoothed and resampled by 0.5 mm by means of VMTK libraries [20]. Centerline tortuosity was Hence, we are imposing the same velocity waveform when then computed: it represents an important factor in different using the boundary conditions A1, A2, and A3, while the cardiovascular diseases, i.e., atherosclerosis, abdominal aor- same inﬂow waveform when adopting the boundary con- tic aneurysm, and, in particular, in thrombus initiation [21]. ditions B1, B2, and B3, thus implying slightly different Tortuosity (T ) is measured as follows: given the centerline inlet velocity waveforms according to the inlet radius of length (L) and the shortest distance between the two center- our computational domain. For example, in Fig. 1 the veloc- line endpoints (ED), T = L/ED−1; therefore, with this ity waveforms corresponding to the boundary conditions A1 deﬁnition, the tortuosity of a straight line is 0. (equivalent to A2, A3) and B1 for Patient 1 in straight-leg Transient CFD analyses were performed using Intel Xeon conﬁguration are represented. In particular, we extracted the W-2123 computing workstation (3.6 GHz, 32 GB RAM) with values of the velocity waveform, taken from Ref. [6] (and the commercial software FLUENT (v.19.2, ANSYS Aca- analogously for the inﬂow waveform [22]), using the soft- demic Research). We considered the two patients in both ware WebPlotDigitizer 4.4 (WebPlotDigitizer). Then, the straight- and bent-leg conﬁgurations in order to assess the data obtained by the literature waveform were interpolated effects of leg bending and the impact of inlet boundary with an 8th order Fourier series by using the Curve Fitting condition on the ﬂow solution. Uniform meshes were gen- App, given by the software Matlab R2018a (The Mathworks erated using VMTK, ranging from 648879 and 1471373 Inc.). The transient inlet velocity waveform was deﬁned in number of tetrahedral elements, according to the previ- FLUENT by the meaning of a user deﬁned function (UDF). ously performed mesh sensitivity analysis. In particular, Therefore, the inlet boundary conditions with a ﬂat velocity the mesh was reﬁned until the difference in the luminal proﬁle (i.e., scenarios A1, A2, B1, B2) were given by area exposed to TAWSS < 0.62 Pa between successive grids was < 1%. Geometry and boundary conditions are the main features affecting the CFD simulations in hemo- dynamics. We evaluated the impact of the inlet boundary conditions on the numerical solution by considering two literature waveforms (boundary conditions A and B) with three different scenarios, i.e., with or without ﬂow exten- sion and varying the velocity proﬁle (ﬂat or parabolic). In particular, the ﬂow extensions that we used in the simu- lations have been chosen in order to reduce the effect of the arbitrary choice of the velocity proﬁle. They were mod- eled using VMTK, with a length corresponding to 3.5 times the dimension of the inlet diameter, according to Colombo et al. [8]. The following inlet boundary conditions were tested: – A1: velocity inlet waveform taken from Wood et al. [6] Fig. 1 Inlet velocity waveforms in m/s colored according to: the liter- (see Fig. 1) with a ﬂat velocity proﬁle. ature inlet velocity taken from Wood et al. [6] and imposed at the inlet – A2: equivalent to the boundary condition A1 with the ﬂat boundary in scenario A1 (and analogously in A2, A3); the inlet velocity velocity proﬁle set at the ﬂow extension of the inlet of computed from the literature inﬂow taken from Nichols et al. [15]and used in scenario B1 (and analogously in B2, B3). the patient-speciﬁc models (see Fig. 2). 123 282 A. Ferrarini, et al. 2 2 (n−1)/2 η = η + (η − η )(1 + λ γ˙ ) , (3) ∞ 0 ∞ where η is the effective viscosity, η the inﬁnite shear rate viscosity, η the zero shear rate viscosity, λ the natural time, γ˙ the shear rate, and n the power law index. The parameter values were set according to Quanyu et al. [23] and listed in Table 1. In each simulation we prescribed the no-slip con- dition on the wall of the artery. Regarding the outlets, the following ﬂow splits were assigned as percentages of the FPA output, according to Crawford et al. [24]: the anterior tibial artery 20%, posterior tibial artery 40%, and peroneal artery 40%. The ﬂow was assumed in laminar regime since the maximum Reynolds number among all the simulations was 1328 at systolic peak (occurring with A1, A2, and A3 con- ditions). Semi-implicit method for pressure linked equations Fig. 2 Femoro-popliteal artery of the two patients considered in the (SIMPLE) was used to solve the Navier–Stokes equations. CFD simulations in the straight-leg conﬁguration. Both are colored Second order scheme for both pressure and momentum spa- according to the ﬂow extension, added to our computational domains tial discretization was adopted. After a sensitivity analysis, a in scenarios A2, A3, B2, B3, and to the three zones under investiga- tion (proximal artery, proximal stent, and distal stent). Moreover, the constant time-step size was set to 0.001 s and three cardiac sections considered in the post processing S ,S ,…,S are represented. 0 1 4 cycles were performed for each simulation to guarantee the The region marked with asterisk denotes the overlapping zone of the repeatability of solution. two stents in both the patients In order to evaluate the impact leg bending on the local hemodynamics of FPA, with a focus on the stented and throm- botic regions, the FPA segments of both the patients were u = a + [a cos(iωt ) + b sin(iωt )], (1) ﬂat 0 i i divided into 3 zones (see Fig. 2): (1) proximal artery, i.e., i =1 lumen of the artery above the proximal end of the proxi- mal stent (excluding the ﬂow extension); (2) proximal stent, where ω is the fundamental frequency (see Table 1), t the i.e., the lumen of the proximal stent, excluding the overlap- simulation time, and a , a , b for i = 1, 2, ··· , 8, the values 0 i i ping zone; (3) distal stent, i.e., the lumen of the distal stent of the Fourier parameters given by the Curve Fitting App. The including the overlapping zone. We performed both a quali- inlet boundary conditions with a parabolic velocity proﬁle tative and quantitative analysis comparing the results of the (i.e., scenarios A3 and B3) were prescribed as two patients in straight- and bent-leg conﬁgurations obtained from the CFD simulations. Firstly, to evaluate the impact of boundary conditions, we focused on the velocity streamlines, u = 2u 1 − , (2) parabolic ﬂat the vectors of velocity magnitude, and the velocity proﬁles at the following cross sections corresponding to: the ﬂow where r denotes the distance between a point on the con- extension inlet, S ; FPA inlet, S ; proximal stent inlet, S ; 0 1 2 strained surface and the center of the surface, and R is the distal stent inlet, S ; distal stent outlet, S (see Fig. 2). The 3 4 radius of the constrained surface. velocity streamlines, the vectors of velocity magnitude, and The proposed six boundary conditions were imposed the velocity proﬁles were reported at the systolic peak. on the patient-speciﬁc model of the two patients for both We computed the time-averaged wall shear stress (TAWSS) straight- and bent-leg conﬁgurations, therefore we performed and oscillatory shear index (OSI), regarding the near wall 24 simulations (six boundary conditions for two patients for ﬂow features, and local normalized helicity (LNH) and helic- two leg conﬁgurations). ity intensity (h index), relating to the bulk ﬂow. TAWSS and Firstly, blood was assumed as an incompressible and OSI were computed as follow: Newtonian ﬂuid, with 1060 kg/m density and 0.0035 Pa s viscosity [16]. Then, in order to evaluate the impact of the non-Newtonian behavior of blood, we chose the boundary TAWSS = |WSS|dt , (4) conditions A1 and B1, i.e., two velocity waveforms with a ﬂat velocity proﬁle, running 8 simulations (two boundary | WSSdt | OSI = 0.5 1 − , (5) conditions per two patients per two leg conﬁgurations). The WSSdt viscosity was modeled using the Carreau model described in the following equation: 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 283 Table 1 Fundamental frequency used in UDF inlet waveform—Eq. (1); parameters of the Carreau model—Eq. (3) −1 Frequency ω(s ) Inﬁnite shear rate Zero shear rate Time constant λ (s) Power low index n (B1, B2, B3) (A1, A2, A3) viscosity η (kg/(m·s)) viscosity η (kg/(m·s)) ∞ 0 Value 6.981 7.854 0.0035 0.056 3.313 0.3568 Fig. 3 Streamlines, contours, and velocity vectors colored according to velocity magnitude at systolic peak, corresponding to the scenarios A1, A2 and A3, in both straight- and bent-leg conﬁgurations of the two patients where T is the cardiac period and |WSS| the norm of the of the WSS. In particular, high values of OSI denote sites WSS vector. WSS is deﬁned as follows where the WSS deviates from the main ﬂow direction in a large fraction of the cardiac cycle [26]. According to Gok- gol et al. [3], luminal area exposed to high OSI (> 0.3) was WSS = σ n −[(σ n) · n]n, (6) computed. Regarding the bulk ﬂow, qualitatively, we com- puted the LNH, which corresponds to the cosine of the angle where σ is the Cauchy stress tensor and n the normal vector formed between the vorticity vector and the velocity vector to the surface. In particular, in an incompressible ﬂuid, the Cauchy stress tensor is deﬁned as follow (∇× u) · u LNH = = cos α, (8) σ = η(∇u +∇u ) − pI , (7) |∇ × u|·|u| where u is the velocity vector, p the pressure, and I the iden- where α is the angle formed between the vorticity vector tity matrix. TAWSS plays a pivotal role in the development of (∇× u) and velocity vector u. It is a measure of the align- arterial stenosis and in prediction of the risk of wall rupture ment/misalignment of the local velocity and vorticity vectors. and thrombus deposition. According to Malek et al. [25], we LNH ranges from − 1 to 1, and its sign indicates the direc- calculated the luminal surface exposed to low and high val- tion of helical structures. Quantitatively, we computed the h ues of TAWSS, i.e., ranging between 0 and 0.4 Pa and above helicity, that is an index regarding the bulk ﬂow: it is given 1.5 Pa, respectively. OSI measures the temporal oscillations by time-averaging the absolute value of the helicity [27]: 123 284 A. Ferrarini, et al. Fig. 4 Streamlines, contours, and velocity vectors colored according to velocity magnitude at systolic peak, corresponding to the scenarios B1, B2, and B3 in both straight- and bent-leg conﬁgurations of the two patients cross-sections of the stented regions, by ﬁxing the inlet wave- h = |u · (∇× u)|dV dt , (9) TV T V form. Therefore, from now on we consider only the results relating to the scenarios A1 and B1 for both the patients in where V is the arterial volume. The h helicity index straight- and bent-leg conﬁgurations. However, the ﬁgures expresses the helicity intensity in the ﬂuid domain, irrespec- including all the scenarios relating to the Newtonian model tive of direction. Recalling that the helicity is deﬁned by are contained in the Supplementary Materials and Methods the spatial integral of the scalar product of the velocity and section. vorticity, we assume h index has higher values in the ﬂuid 2 Figure 5 highlights the arterial lumen colored according domain in which velocity and vorticity vectors are aligned. to low (ranging from 0 to 0.4 Pa) and high TAWSS (higher than 1.5 Pa). The results suggest that the distal part of the artery is exposed to high TAWSS in both straight- and bent- leg conﬁguration with a limited inﬂuence of inﬂow boundary 3 Results conditions; such a result is particularly evident in the case of Patient 1, while for Patient 2 the B1 scenario is resulting in Firstly, we reported the results obtained by the 24 simula- tions performed with constant viscosity. Figures 3 and 4 show physiological TAWSS in most of the whole artery for both conﬁgurations. Figure 6 shows the arterial lumen colored the results of CFD simulations for the straight- and bent-leg according to high OSI (> 0.3) is represented. High OSI are conﬁgurations of both the patients in the six scenarios that located for all the cases under considerations in the proximal have been tested (A1, A2, A3 and B1, B2, B3 in Figs. 3 and part of artery irrespective to the adopted boundary conditions. 4, respectively), reporting streamlines, velocity proﬁles, and Helical blood ﬂow structures developing into the endo- velocity vectors colored according to the velocity magnitude prostheses are represented in Fig. 7 using iso-surfaces of at the systolic peak. These two ﬁgures prove that only the LNH at the systolic peak with a threshold of ± 0.25, accord- imposed waveform at the inlet (taken from Wood et al. [6]or ing to Colombo et al. [8], for both the patients in straight-leg Nichols et al. [22]) signiﬁcantly affects the solution. Indeed, each scenario has similar velocity proﬁles and contours in the conﬁguration, relating to the scenario A1. The results show 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 285 Fig. 7 Femoro-popliteal artery and three zoom views of the lumen (rotating clockwise) of both the patients in straight-leg conﬁguration: the area where the thrombosis is localized is highlighted by a black box. Moreover, blood ﬂow helicity is represented: in blue the ﬂow with negative LNH and in red the ﬂow with positive LNH Fig. 5 Arterial lumen colored according to low (< 0.4 Pa) and high (> that the bulk ﬂow in the artery for both patients is char- 1.5 Pa) TAWSS in both straight- and bent-leg conﬁgurations of the two acterized by two counter-rotating helical structures and in patients particular the helical shape of the thrombosis seems to ﬂow the path of the negative LNH region. Figure 8 reports the bar-plots of the value of h index, tortuosity, and the percentage of luminal surface exposed to low and high TAWSS, and high OSI corresponding to each zone (proximal artery, proximal stent, and distal stent) for both the patients in straight- and bent-leg conﬁgurations. The results show that leg bending induces a difference of the computed hemodynamics indices for Patient 1 with both A1 and B1 boundary conditions. Indeed, a percentage difference above 50% between the two conﬁgurations is present for each hemodynamic quantity that we computed in all the tested scenarios, except for the percentage difference relating to the luminal area exposed to high OSI in Patient 2 (with a maximum percentage difference of 24% in the distal stent region). In particular, our results show a signiﬁcant variation of tortuosity between the two conﬁgurations, accentuated in the distal stent zone, where the tortuosity is greater in the bent-leg conﬁguration. Finally, we treated the blood as a non-Newtonian ﬂuid and we assessed the results, comparing them with the previ- ous analyses, obtained using the Newtonian model. Figure 9 shows the arterial lumen of both patients colored according to TAWSS magnitude, low (ranging from 0 to 0.4 Pa) and high TAWSS (higher than 1.5 Pa), based on both the Carreau Fig. 6 Arterial lumen colored according to high OSI (> 0.3) in both (non-Newtonian) and Newtonian models. Figure 9 represents straight- and bent-leg conﬁgurations of the two patients only the results relating to the scenario B1, which provides greater differences between the two models under considera- 123 286 A. Ferrarini, et al. Fig. 8 Bar plot of tortuosity, helicity (h index), and percentage of luminal area exposed to both low (< 0.4 Pa) and high (> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in both leg conﬁgurations, corresponding to scenarios A1 and B1 tion and allows us a wider discussion, as we will introduce in thrombosis during follow-up. In particular, the role of leg the next section. Figures 10 and 11 represent the bar-plots of bending on the local hemodynamic was elucidated by mod- the value of h index, and the percentage of luminal surface eling both straight- and bent-leg conﬁgurations. exposed to low and high TAWSS, and high OSI correspond- Focusing on the velocity magnitude, Figs. 3 and 4 show ing to each zone (proximal artery, proximal stent, and distal a higher ﬂow velocity in the distal stent region than to the stent) for both the patients. In particular, Fig. 10 refers to the proximal one, due to the luminal narrowing given by the over- straight-leg conﬁguration, while Fig. 11 to the bent-leg one. lapping of the two stents-grafts. As we already pointed out, the results suggest that the different inlet velocity proﬁles used in the simulations slightly affect the numerical solu- tion, conversely to the determinant role of the prescribed inlet 4 Discussions waveform. In order to obtain reliable results of clinical sig- niﬁcance, patient-speciﬁc inﬂow waveforms would be very In this study, we have evaluated the local hemodynamic useful in understanding the hemodynamic behavior. How- and the interplay among geometric features in two patients ever, our geometrical study shows velocity sensitivity, i.e., endovascularly treated for PAA, who experienced intra-stent 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 287 velocity magnitude variations between the two patients occur along the two FPAs by ﬁxing a velocity inlet (i.e., scenario A or B). Although the behavior of stented FPAs has already been investigated in the literature [8], to date there is still no information on the different response between the various portions of the stented artery itself. Figure 8 suggests that the overlapping of the stent grafts seems to induce a severe discontinuity of lumen diameter, dividing the region treated with endovascular stent-graft in two zones: (1) the proximal part, where thrombosis is located, it is characterized by low tortuosity, low velocity, low helicity, low TAWSS, and high OSI; (2) the distal part that presents higher tortuosity, pro- moting higher velocity, higher helicity, higher TAWSS, and lower OSI. In particular, by focusing on the tortuosity of the vessel (see Fig. 8 at the bottom), the stented FPA respects the behavior that we would have expected, when considered in its entirety, i.e., increased tortuosity values with leg bend- ing. Analyzing the stented area by portions, we have found that in both patients the tortuosity increases from the prox- imal artery region to the distal one; this result matches the Fig. 9 Arterial lumen colored according to the TAWSS magnitude, low ﬁndings of Wood et al. [6], who performed CFD simula- (< 0.4 Pa) and high TAWSS (> 1.5 Pa) in both straight- and bent-leg tions in the superﬁcial femoral artery of 9 healthy men and conﬁgurations of the two patients. The TAWSS values represented refers 9 healthy women, showing that tortuosity was signiﬁcantly to the scenario B1 greater for men than women, but the highest values were found in the most distal segment, regardless of sex. Then, when considering the comparison between straight- and bent- an inverse relationship between helical ﬂow and OSI evalu- leg conﬁguration, we observed that in both patients proximal ating four bypass geometries in ascending aorta, according vessel and distal stent segments tortuosity increases with leg to our results. Moreover, our results denote that the spiral bending. However, the proximal stent, characterized by its shape of thrombosis matches the path of the negative LNH larger diameter, slow velocity, low TAWSS, and low helicity, region; this is more evident in Patient 1 (Fig. 7). Figure 7 straightens with leg ﬂexion. This area is also the one in which refers only to scenario A1, but an analogous pattern of the thrombosis was found in both patients, conﬁrming that the LNH was found using the boundary condition B1. It is hard formation of thrombosis is linked to a combination of both to formulate a conclusion to explain this result; given the lim- hemodynamic and geometric factors. Hence the importance ited number of analysed patients, further analysis involving of conducting the analyses by investigating the stented FPA a cohort of patients should be investigated in order to provide not only in its entirety but by dividing it into the various por- more information to elucidate this observation. tions, in order to be able to identify areas more at risk of From our results we found that the study in both straight- thrombosis. and bent-leg conﬁgurations is crucial in understanding and The role of low TAWSS in thrombotic regions has been assessing the numerical results in stented arteries, given the previously corroborated in literature. Boyd et al. [28]showed increase of the tortuosity of the distal part of the artery due to a correlation between regions of low WSS, where ﬂow the leg bending. Our ﬁndings match with Wensing et al. [32], recirculation predominated, and thrombus deposition, by per- who highlighted the importance of considering the impact forming CFD simulations in 7 abdominal aortic aneurysms. of knee ﬂexion in femoral and popliteal arteries, showing The luminal area exposed to low TAWSS and high OSI in increasing tortuosity in bent-leg conﬁguration of 22 healthy the proximal zone is greater in Patient 1 than in Patient 2 (see volunteers. Moreover, the increase of tortuosity in leg bend- Fig. 8), suggesting that patient-speciﬁc geometrical features ing implies a reduction of the blood velocity in each scenario also affect the near wall ﬂow features. that we assumed for both patients (see velocity streamlines Regarding the bulk ﬂow, our results suggest that intra- and contours represented in Figs. 3 and 4). stent thrombosis is located in the region where the intensity The alternate bending of the legs is known to inﬂuence of helicity is low (see Fig. 8). The fundamental role of helical the mechanical solicitation of the stent [33], the shape of the (or swirling ﬂow) in the prevention of thrombosis and disease artery [34], and the local hemodynamics [9] but its role in the progression has been conﬁrmed in many literature studies thrombosis onset and progression is still unknown. From our [29–31]. In particular, Morbiducci et al. [30] also presented results, it is evident that leg bending increases the tortuosity 123 288 A. Ferrarini, et al. Fig. 10 Comparison between results considering Newtonian and non-Newtonian behavior: bar plot of helicity (h index), and percentage of luminal area exposed to both low (< 0.4Pa) andhigh(> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in the straight-leg conﬁguration, corresponding to scenariosA1 and B1 of the distal stent segment, which combined with an overall the magnitude of WSS is relatively small (< 1N/m ). Anal- blood ﬂow velocity, exacerbate the difference between the ogously, we found similar results using the scenario A1, but distal and proximal part of the stented region, with the latter with less marked differences, since the surface exposed to more exposed to the risk of thrombosis (i.e., lower velocity, low TAWSS is very small even in the Newtonian model. For wider area of low wall shear stress, higher oscillatory shear this reason we chose to omit the qualitative analysis given by stress, and lower helicity). Such considerations are however the scenario A1. hardly generalizable with data proposed by the present paper, Figures 10 and 11 allow deepening the comparison which deals with only two patients, but, at the same time, call between the Newtonian and non-Newtonian models. Sig- for future developments focused on such hypotheses. niﬁcant differences based on the luminal surfaces exposed Focusing on the qualitative comparison between the New- to low TAWSS are highlighted (with a maximum decrease tonian and non-Newtonian model, Fig. 9 shows an optimal in the proximal artery zone, compared to the non-Newtonian agreement on the distribution of the TAWSS magnitude model, of 8.7% and 8.5% for Patients 1 and 2, respectively, in between the two models. These results reproduce the assump- the straight-leg conﬁguration and in the scenario B1), a good tions discussed by Liu et al. [35], who introduced that the agreement occurs for the other analyzed outcomes (with a blood viscosity properties do not affect the spatial pattern maximum OSI decrease in the proximal artery zone, com- of the TAWSS qualitatively. However, looking at the lumi- pared to the non-Newtonian model, of 3.2% for patient 2 in nal surface exposed to low and high TAWSS, the area with the straight-leg conﬁguration and in the scenario B1). In par- low TAWSS is greater in the Newtonian model for both the ticular, as we mentioned before, the scenario B1, in which the patients, while no signiﬁcant difference occurs between the inlet average velocity is lower and also low TAWSS values surfaces with high TAWSS. Our ﬁndings are in agreement occur, provides major differences. As the velocity increases with Soulis et al. [15] and Liu et al. [35], who proved an (see the results given by the scenario A1 in Fig. 10), the underestimated WSS given by the Newtonian model, when 123 Impact of leg bending in the patient-speciﬁc computational ﬂuid dynamics of popliteal stenting 289 Fig. 11 Comparison between results considering Newtonian and non-Newtonian behavior: bar plot of tortuosity, helicity (h index), and percentage of luminal area exposed to both low (< 0.4 Pa) and high (> 1.5 Pa) TAWSS, respectively, and high OSI (> 0.3). The data are reported for the three zones under investigation (proximal artery, proximal stent, and distal stent) of the two patients in the bent-leg conﬁguration, correspondingto scenarios A1 and B1 Newtonian and non-Newtonian models become more simi- different time instants, from early post-operative to annual lar, according to Liu et al. [35]. follow-up exams. In the present study we dealt with thrombosis only from a ﬂuid dynamic point of view. However, further analysis should include the role of hemodynamic stress in the platelet activation [36], recently proved to be associated with aortic 5 Limitations thrombus formation [37]. Finally, according to previous studies [3,8], we did not take The present work, based on the analysis of only two into account the stent struts; however, further developments cases, represents a proof-of-concept study, aimed at link- will include this aspect, since Al-Hakim et al. [38]showed ing post-stent geometry, hemodynamics, and thrombosis in that stent struts have an effect on WSS. endovascular repair of popliteal aneurysms. Further analyzes should be performed in order to obtain statistically and clin- ically relevant conclusions. We have already discussed the importance of considering patient-speciﬁc inlet boundary 6 Conclusions condition; therefore, in future studies inﬂow data elaborated by echo doppler measurements will be set at the inlet of the The present study suggests that the overlapping of the stent- computational domain. grafts seems to induce a severe discontinuity of lumen The computational domains considered in the simula- diameter, dividing the region treated with endovascular stent- tions represent surrogate geometrical models of the lumen graft into two zones: the proximal part, where thrombosis is of each patient prior to thrombosis by virtually removing the located, it is characterized by low tortuosity, low velocity, thrombus during the segmentation process. Such a limitation low helicity, low TAWSS, and high OSI; the distal part that could be overcome by analyzing the CT scans performed at presents higher tortuosity, promoting higher velocity, higher 123 290 A. 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Surg. 61(4), 1068–1080 (2015) reliable results. The Newtonian and non-Newtonian blood 6. Wood, N.B., Zhao, S.Z., Zambanini, A., et al.: Curvature and tor- tuosity of the superﬁcial femoral artery: a possible risk factor for treatments provide similar results in both the patients, except peripheral arterial disease. J. Appl. Physiol. 101(5), 1412–1418 when the magnitude of the TAWSS is relatively small (< 0.4 (2006) Pa). In this latter case the Newtonian model gives lower val- 7. Desyatova, A., Poulson, W., Deegan, P., et al.: Limb ﬂexion ues of TAWSS than the non-Newtonian one. However, the induced twist and associated intramural stresses in the human femoropopliteal artery. J. R. Soc. Interface 14, 20170025 (2017) Newtonian blood treatment should be a good choice in all 8. Colombo, M., Bologna, M., Garbey, M., et al.: Computing patient- cases in which the analysis of WSS is not necessary. 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Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

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Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

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