## Abstract

Despite the important advancements in the stent technology for the treatment of diseased coronary arteries, major complications still affect the postoperative long-term outcome. The stent-induced flow disturbances, and especially the altered wall shear stress (WSS) profile at the strut level, play an important role in the pathophysiological mechanisms leading to stent thrombosis (ST) and in-stent restenosis (ISR). In this context, the analysis of the WSS topological skeleton is gaining more and more interest by extending the current understanding of the association between local hemodynamics and vascular diseases. This study aims to analyze the impact that a deployed coronary stent has on the WSS topological skeleton. Computational fluid dynamics (CFD) simulations were performed in three stented human coronary artery geometries reconstructed from clinical images. The selected cases presented stents with different designs (i.e., two contemporary drug-eluting stents and one bioresorbable scaffold) and included regions with stent malapposition or overlapping. A recently proposed Eulerian-based approach was applied to analyze the WSS topological skeleton features. The results highlighted that the presence of single or multiple stents within a coronary artery markedly impacts the WSS topological skeleton. In particular, repetitive patterns of WSS divergence were observed at the luminal surface, highlighting a WSS contraction action exerted proximal to the stent struts and a WSS expansion action distal to the stent struts. This WSS action pattern was independent from the stent design. In conclusion, these findings could contribute to a deeper understanding of the hemodynamics-driven processes underlying ST and ISR.

## Introduction

Percutaneous coronary intervention (PCI) with drug-eluting stent implantation is the gold standard endovascular treatment for patients suffering from coronary artery disease [1]. Contemporary stent platforms are made of cobalt–chromium or platinum–chromium, allowing for thinner stent struts (<100 μm) and more flexibility than old-generation devices, while maintaining adequate radial strength [2,3]. These modern devices have reduced the incidence rate of stent thrombosis (ST) below 1% after 1 yr of intervention [4,5]. Conversely, the incidence rate of in-stent restenosis (ISR) still remains at 5–10% [6]. Moreover, the period of ISR presentation is generally longer than that of the old-generation devices and often extends several years beyond the intervention [6]. To avoid late adverse events promoted by the persistence of the metallic platforms in the coronary vessel (including late ST and ISR), bioresorbable scaffolds, either based on a fully resorbable polymeric or metallic backbone, are currently under development and clinical testing [2]. These devices provide temporary mechanical support and drug delivery to the coronary vessel within 1 yr after implantation and completely resorb in the subsequent 1–2 yr, restoring the normal luminal diameter and vasomotor function [2,3]. Bioresorbable scaffolds present the notable advantage of reducing the need of the long-term dual antiplatelet therapy and allow for surgical revascularization, if needed [2]. As a counterpart, they present the disadvantage that thicker struts are necessary to provide radial forces similar to the contemporary drug-eluting stents [2], with the consequent exacerbation of the local flow disturbances. In this regard, in addition to systemic and biologic factors [6,7], numerous studies have identified local blood flow disturbances at the stent strut level as a key contributor for the ISR development [7,8]. In particular, a negative association between baseline wall shear stress (WSS) and neointimal thickness at follow-up has emerged in different clinical datasets [8]. However, the role of the altered WSS profile on the pathophysiological mechanisms leading to ISR is still under investigation. Moreover, the lack of endothelial coverage or delayed re-endothelialization observed in consequence of thicker stent struts, such as in bioresorbable scaffolds, increases the risk of thrombosis [9,10].

In this context, the analysis of the WSS topological skeleton, which is contributing to improve and extend the current understanding of the association between local hemodynamics and vascular diseases [1114], could allow to better identify the biomechanical stimuli involved in clinical adverse events eventually leading to failure of stenting procedures. The WSS topological skeleton is composed by fixed points, where the WSS vanishes, and by manifolds, connecting fixed points [11,12]. Recently, an approach has been proposed for (i) the identification of WSS manifolds [12,15] and (ii) the quantitative analysis of the contraction/expansion action exerted by the shear forces on the endothelium [1517], starting from the distribution of the WSS divergence on the arterial luminal surface. Using such a WSS divergence-based approach, evidences emerged that the variability of the WSS contraction/expansion action along the cardiac cycle is associated with wall degradation in the ascending thoracic aorta aneurysm [17] and is a predictor of the risk of long-term restenosis in the carotid bifurcation after endarterectomy [16]. Furthermore, the findings of a very recent longitudinal study focusing on coronary arteries indicated that the high variability in the WSS contraction/expansion action is a predictor of wall thickness change over time (a hallmark of atherosclerosis development), suggesting that both the WSS manifold dynamics along the cardiac cycle and the WSS magnitude concur to coronary atherosclerosis, acting as different hemodynamic stimuli [15]. Applied in a clinical setting, the high variability in the WSS contraction/expansion action added incremental predictive and discriminative capacity to area stenosis, virtual fractional flow reserve, and time-averaged WSS (TAWSS) to identify intermediate coronary lesion site of subsequent myocardial infarction at 5-yr follow-up [18].

Based on the above mentioned observations and on the direct link of the WSS topological skeleton features with near-wall flow stagnation, separation, and recirculation, which represent typical flow conditions of stented coronary arteries [11,12], in this study the impact that a deployed coronary stent has on the WSS topological skeleton is investigated. The possible involvement of WSS topological skeleton features in the processes leading to post-PCI complications is discussed. For this purpose, computational fluid dynamics (CFD) simulations are performed in stented human coronary artery models reconstructed from clinical images. The impact of three different stent designs (i.e., two contemporary drug-eluting stents and one bioresorbable scaffold) and of stent positioning (i.e., malapposition and overlapping) on the WSS topological skeleton features is analyzed.

## Methods

### Coronary Artery Geometries.

Three cases (A–C) of pathological left anterior descending coronary artery (LAD) of patients suffering from coronary artery disease who underwent percutaneous coronary stent implantation were investigated (Fig. 1). Cases A and B were treated with cobalt–chromium drug-eluting stents at University Hospital Doctor Peset (Valencia, Spain) [19,20]. In case A, a 3.0 × 28 mm Xience Prime everolimus-eluting stent (Abbott Laboratories, Abbott Park, IL) was implanted in the proximal portion of the LAD at the level of the first diagonal branch and a septal branch by means of the provisional side branch stenting technique concluded by proximal optimization. The Xience Prime stent is characterized by an open-cell peak-to-valley design and struts with square cross section and thickness of 81 μm (Fig. 2) [2]. In case B, two Endeavor Resolute zotarolimus-eluting stents (Medtronic, Dublin, Ireland) (sizes 2.75 × 15 mm and 3.0 × 15 mm) were deployed in the midportion of the LAD in correspondence of the first and second diagonal branches. The two stents were sequentially deployed using the provisional side branch technique, ensuring a stent overlapping region of ∼20 mm. The Endeavor Resolute stent presents an open-cell peak-to-peak design and struts with circular cross section and diameter of 91 μm (Fig. 2) [2]. Case C was treated with a drug-eluting bioresorbable scaffold at Rivoli Infermi Hospital (Turin, Italy). In particular, a 3.0 × 25 mm resorbable magnesium-based sirolimus-eluting scaffold Magmaris (Biotronik AG, Bülach, Switzerland) was implanted in the midportion of the LAD in correspondence of the second diagonal branch following a provisional stenting technique concluded by proximal optimization, without the need of final kissing balloon inflation. This bioresorbable scaffold is characterized by an open-cell peak-to-valley design with midstrut connector and struts with rectangular cross section and thickness of 150 μm (Fig. 2) [21]. The study was conducted in accordance with the principles of the Declaration of Helsinki and met the requirement of medical ethics. The study protocol was approved by the institutional review board of the involved hospitals. All patients gave written informed consent.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

Patient-specific geometries of the three stented coronary arteries were obtained from medical images (Fig. 1). Regarding cases A and B, pre-operative vessel geometries were reconstructed by combining conventional angiography and computed tomography angiography images. Finite element analyses replicating the entire stenting procedure were then performed to obtain the stented lumen configuration (i.e., post-PCI vessel configuration) to be used for CFD simulations. Details on vessel and stent geometry reconstruction, and virtual stenting procedure were extensively described elsewhere [19,20]. Regarding case C, the post-PCI vessel configuration was reconstructed through the fusion of conventional angiography and optical coherence tomography (OCT) images acquired immediately after scaffold implantation. More in detail, the following five-step procedure was applied [2224]: (i) semi-automatic detection of the lumen contours and stent struts on the OCT frames using an in-house developed algorithm; (ii) extraction of the vessel centerline from two angiographic views; (iii) placement of the lumen contours and stent struts orthogonal to the vessel centerline using the side branches as reference to properly orient the OCT frames; (iv) smooth connection of the lumen contours to obtain the lumen surface; and (v) creation of the three-dimensional stent model by means of a manual morphing procedure that adapts the stent skeleton in its straight free-expanded configuration toward the stent point cloud detected from OCT.

In addition to the three stented coronary artery geometries, the corresponding lumen geometries without stent (i.e., nonstented cases) were considered for comparison purposes. These vessel geometries were obtained by excluding the stents from the domain of interest and by smoothing the lumen surface using the open-source software vmtk (Orobix, Bergamo, Italy) to avoid abrupt local changes of the luminal surface after the stent removal.

### Computational Fluid Dynamics.

The coronary artery geometries were discretized into tetrahedral elements with five layers of prismatic elements at the luminal surface using the commercial software fluentmeshing (Ansys Inc., Canonsburg, PA). The element size was defined based on a mesh independence study, resulting in a mesh cardinality ranging from 5,068,182 to 13,948,023 elements and from 526,492 to 1,548,654 elements for the stented and nonstented vessel models, respectively. A smaller element size (0.02 mm) was defined at the stent struts.

The local hemodynamics of all cases under investigation was analyzed by performing transient CFD simulations. Specifically, the governing equations of the unsteady-state fluid motion were numerically solved using the finite volume-based commercial code fluent (Ansys Inc.). Details on boundary conditions imposed at inlet and outlet sections of each model were extensively described in Ref. [19]. Briefly, a pulsatile flow waveform, distinctive for the LAD [25], was applied as paraboloid-shaped velocity profile at the inlet cross section [26]. The pulsatile flow waveform amplitude was personalized to the specific patient according to an inflow section lumen diameter-based scaling law [27]. In this way, flow rate waveforms with anatomically derived, personalized average flow rate values (42.2 ml/min, 45.1 ml/min, and 34.0 ml/min for cases A, B, and C, respectively) were prescribed as inflow boundary conditions (Fig. S1 available in the Supplemental Materials on the ASME Digital Collection). A flow-split, estimated through a scaling law based on the lumen diameter of the daughter branches [27], was imposed at the outflow boundaries (Fig. S1 available in the Supplemental Materials on the ASME Digital Collection). The no-slip condition was applied at the vessel and stent walls, which were considered as rigid. The blood was modeled as an incompressible, homogeneous, non-Newtonian fluid with density of 1060 kg/m3 and viscosity described through the Carreau formulation [19]. The flow was considered laminar as the maximum Reynolds number at peak flow rate was 195, 260, and 140 for cases A, B, and C, respectively. Details on the numerical settings were exhaustively reported in a previous study [19].

### Wall Shear Stress Features.

The Eulerian-based method recently proposed for Newtonian fluids [12] and extended to the class of Reiner–Rivlin fluids [15] was applied to analyze the WSS topological skeleton features at the stented region, considered as the region of interest. More in detail, the WSS topological skeleton consists of a collection of fixed points (i.e., focal points where the WSS vanishes) and associated unstable/stable manifolds [11,12] (Fig. 3(a)). As reported in previous studies [12,15], WSS manifolds can be identified by computing the divergence of the normalized WSS vector field at the stented vessel surface, expressed as
$DIVWSS=∇⋅(τ∥τ∥2)=∇·τu$
(1)

where $τ$ and $τu$ are the WSS vector and its unit vector, respectively. From a physical perspective, vessel surfaces characterized by negative/positive values of $DIVWSS$ correspond to contraction/expansion regions approximating unstable/stable manifolds (Fig. 3(a)). To complete the WSS topological skeleton analysis, the WSS fixed points were identified at the vessel surface according to the procedure proposed elsewhere, based on the Poincaré index [11,12,28]. Then, the WSS fixed points were classified according to their nature (i.e., saddle point, node, or focus Fig. 3(a)), using the approach based on the eigenvalues of the WSS Jacobian matrix [1113,29].

Fig. 3
Fig. 3
Close modal
First, the cycle-average WSS vector field $τ¯$ was analyzed. Subsequently, the instantaneous WSS topological skeleton along the cardiac cycle was characterized. To do that, the amount of variation in the WSS contraction or expansion action along the cardiac cycle (Fig. 3(b)) was quantified by computing the topological shear variation index (TSVI) [1517], namely, the root-mean-square deviation of the divergence of the normalized WSS with respect to its average along the cardiac cycle
$TSVI= {1T∫0T[ ∇·(τu)−∇·(τu)¯ ]2 dt}1/2$
(2)
where T is the cardiac cycle period. Moreover, to characterize the unsteady nature of the WSS fixed points along the cardiac cycle, the WSS fixed point weighted residence time along the cardiac cycle was computed [12,15,16]
$RT∇xfp(e)=AavgAe1T∫0TIe(xfp,t)|(∇·τ)e|dt$
(3)

where $xfp(t)$ represents the location of a WSS fixed point at time $t∈[0,T]$, e is a generic element of the mesh of area $Ae$, $Aavg$ the average surface area of all elements of the mesh, $I$ is the indicator function, and $(∇·τ)e$ is the instantaneous WSS divergence. Equation (3) quantifies the fraction of cardiac cycle in which a generic mesh element e hosts a fixed point, weighting the residence time by the strength of the contraction/expansion action measured by the WSS divergence.

To provide a more complete picture of the near-wall hemodynamics, the well-known WSS-based descriptor TAWSS, namely, the average WSS magnitude value along the cardiac cycle, was computed in addition to the WSS topological skeleton features.

The exposure to large variations in the WSS contraction or expansion action was quantified by the relative surface area at the stented region exposed to high values of TSVI, considering as threshold value the 80th percentile of the TSVI distribution of each nonstented model. This variable was denoted as topological shear variation area (TSVA). Similarly, the exposure to the action of instantaneous WSS fixed points was quantified by the relative surface area exposed to non-null values of $RT∇xfp(e)$, namely, considering the luminal surface area at the stented region where fixed points occurred along the cardiac cycle. This variable was denoted as weighted fixed point area (wFPA). Finally, the exposure to low TAWSS values was quantified by the relative surface area at the stented region exposed to TAWSS values below a threshold value, corresponding to the 20th percentile of the TAWSS distribution of the nonstented model. This variable was denoted as low shear area (LSA).

## Results

### Hemodynamic Impact of Stenting.

Since the WSS topological skeleton features directly link to near-wall flow stagnation, separation, and recirculation areas [11,12], an overview of the intravascular hemodynamic features of the three cases under analysis is briefly presented in Fig. 4, comparing the stented with the nonstented models in terms of in-plane flow patterns on explanatory cross-sectional planes. As expected, stent implantation induced small flow recirculation regions close to the stent struts, whose extension was dictated by the peculiar design features of the stent. In these explanatory cases, the near-wall recirculation regions did not affect the large-scale secondary flow patterns, except for case C, where the stent struts broke the classical two-vortex Dean-like structure of the secondary flows (Fig. 4).

Fig. 4
Fig. 4
Close modal

The differences in the hemodynamics induced by the implantation of stents with different design were reflected by the WSS topological skeleton features. The cycle average WSS topological skeleton distribution at the luminal surface of the three cases under investigation, expressed in terms of divergence of the normalized WSS vector field, is displayed in Fig. 5. By visual inspection, in all three cases, the presence of the stent markedly impacted the WSS topological skeleton. In particular, the stented regions presented a repetitive pattern characterized by WSS contraction regions located immediately upstream from each stent ring, identified by negative DIVWSS values, and WSS expansion regions located immediately downstream from each stent ring, identified by positive DIVWSS values. This repetitive pattern was observed along the entire stented regions, except for the stent malapposed region (i.e., proximal stented segment of case A), where a reverse DIVWSS distribution was visible, and the stent overlapped region (i.e., midstented segment of case B), where a less repetitive pattern was present. The bifurcation regions of the stented cases showed a more complex WSS topological skeleton distribution than that of the nonstented cases, in which only a WSS expansion region was identified at the bifurcations' carina. In the stented cases, WSS fixed points were mainly located at the stent peaks and valleys, at the interface between the lumen and the stent struts. The number of fixed points was at least 2 orders of magnitude higher than that of the nonstented cases (Table 1), for which only a single (case A) or few saddle points (cases B and C) were present at the bifurcation regions. Consequently, a high percentage increase (>70%) in the wFPA was also observed in the stented cases with respect to the nonstented ones (Table 2).

Fig. 5
Fig. 5
Close modal
Table 1

Number of fixed points identified in the three investigated cases

Case ACase BCase C
Stented models
No. of foci/nodes533550211
Nonstented models
No. of foci/nodes000
Case ACase BCase C
Stented models
No. of foci/nodes533550211
Nonstented models
No. of foci/nodes000
Table 2

Percentage increase in wFPA, TSVA, and LSA at the stented region for the stented cases with respect to the nonstented ones

Case ACase BCase C
Percentage increase in wFPA165.47%440.16%71.07%
Percentage increase in TSVA680.87%603.23%593.85%
Percentage increase in LSA544.94%415.18%486.24%
Case ACase BCase C
Percentage increase in wFPA165.47%440.16%71.07%
Percentage increase in TSVA680.87%603.23%593.85%
Percentage increase in LSA544.94%415.18%486.24%

The distribution of TSVI along the luminal surface of the stented cases was different than that of the nonstented ones (Fig. 6). Specifically, in the stented cases, marked variations in the WSS contraction/expansion action along the cardiac cycle, as quantified by TSVI, were mainly located immediately downstream and upstream from the stent struts. The highest values of TSVI (>5000 m−1) were found at the stent peaks and valleys and at the links between the stent rings. Conversely, in the nonstented cases, high TSVI values were located only at the bifurcation regions and at the proximal segment of case A, where the coronary artery presented high curvature and tortuosity. The difference in the distributions of TSVI along the luminal surface between the stented and nonstented cases emerged also from the violin plots of Fig. 7. The stented cases were characterized by higher median values of TSVI as compared to the nonstented ones (862.79 m−1 versus 92.53 m−1, 802.78 m−1 versus 102.69 m−1, and 662.97 m−1 versus 119.18 m−1, for the stented versus the nonstented models of cases A, B, and C, respectively). Furthermore, the stent presence caused a marked percentage increase (>550%) in the TSVA (Table 2).

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

The distribution of TAWSS along the luminal surface of the stented cases was different than that of the nonstented ones, as visually emerged from the color maps of Fig. 8 and the corresponding violin plots of Fig. 9. In both the stented and nonstented cases, low TAWSS values were located at the bifurcation regions and at the proximal segment of case A. In the stented cases, low TAWSS was also present at the stent struts, as expected. The median values of TAWSS were lower in the case of the stented models as compared to the nonstented ones (0.54 Pa versus 1.02 Pa, 0.57 Pa versus 0.93 Pa, and 0.45 Pa versus 0.76 Pa, for the stented versus the nonstented models of cases A, B, and C, respectively). The LSA was markedly higher in the case of the stented models with respect to the nonstented ones (percentage increase > 400% in all cases, Table 2). Moreover, by comparing the color maps of TSVI and TAWSS of Figs. 5 and 7, a colocalization between high TSVI and low TAWSS values in the vicinity of the stent struts was observable.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

The distributions of DIVWSS, TSVI, and TAWSS on the stent surface are depicted in Fig. 10 for the three investigated stents. A continuous DIVWSS distribution can be observed between the luminal surface close to the stent struts and their side faces (i.e., negative and positive DIVWSS values for the proximal and distal lateral surfaces of the stent struts at the intersection with the luminal surface, respectively). A transition from positive to negative DIVWSS values (i.e., from WSS expansion to contraction regions) in the main flow direction was identified at the top faces of the stent struts. The highest values of TSVI colocalized with the lowest values of TAWSS at the stent peaks and valleys. Low values of TSVI and high values of TAWSS were present at the top faces of the stent struts.

Fig. 10
Fig. 10
Close modal

### Impact of Stent Design.

The repetitive pattern of the cycle average WSS topological skeleton distribution at the luminal surface, characterized by WSS contraction regions located upstream from the stent rings, and WSS expansion regions located downstream from the stent rings, was present in all the investigated cases (Fig. 5), suggesting that this pattern was independent from the stent design features. A different number of fixed points was found in the three investigated cases (Table 1). The number of fixed points depended on the number of peaks/valleys per stent ring and the stent length. The higher the number of peaks/valleys per stent ring and the longer the stent length, the higher the number of fixed points. Consequently, the highest number of fixed points was identified in case B, where two Endeavor Resolute stents were implanted (total length of ∼28 mm, ten peaks per stent ring), while the lowest number in case C, where the Magmaris stent was deployed (length of 25 mm, six peaks per stent ring). Accordingly, the highest percentage increase (440.16%) in the wFPA was observed in case B, while the lowest (71.07%) in case C (Table 2). The number of saddle points and foci/nodes was balanced only in case C (Table 1). Conversely, the number of saddle points was lower than that of the foci/nodes in case A and higher than that of the foci/nodes in case C (Table 1).

The distributions of TSVI along the luminal surfaces of the stented regions were similar for the three investigated cases (Figs. 6 and 7). Although the general patterns of the TAWSS along the stented region were similar for the different cases (i.e., low TAWSS close to the stent struts, gradually increasing toward the stent cell center, Fig. 8), the TAWSS median value for case C was lower than that of the other two cases (0.45 Pa versus 0.54 Pa and 0.57 Pa for case C versus cases A and B, respectively, Fig. 9), suggesting that the open-cell design of the Magmaris stent, characterized by thicker struts than the Xience Prime and Endevour Resolute stents, had a higher impact on the TAWSS distribution.

## Discussion

### Summary of Findings.

This study investigated the impact that the stenting of human coronary arteries has on the WSS topological skeleton, whose features highlight the complex nature of the interaction of fluid shear forces with the stent struts and the luminal surface. Transient CFD simulations were performed in three patient-specific coronary models featuring different implanted stents, namely, two contemporary drug-eluting stents and one bioresorbable scaffold. The WSS topological skeleton features were analyzed along the stented regions by applying a recently proposed Eulerian-based method [12,15]. The results of the work demonstrated that the presence of single or multiple stents within a coronary artery highly impacts the WSS vector field topological skeleton. More specifically, first a repetitive pattern of the WSS topological skeleton distribution at the luminal surface was observed in the stented regions of all investigated cases. As schematized in Fig. 11, this pattern was characterized by a WSS contraction action on the luminal surface (negative DIVWSS values) immediately proximal to the stent struts and by a WSS expansion action (positive DIVWSS values) immediately distal to the stent struts. Second, this pattern was independent of the stent design, stent strut cross section shape, and thickness. Third, the highest variations in the WSS contraction/expansion action exerted on luminal surface interacting with the streaming blood along the cardiac cycle, as captured by the TSVI, were located close to the stent struts. Last, high TSVI colocalized with low TAWSS values in the vicinity of the stent struts.

Fig. 11
Fig. 11
Close modal

### Wall Shear Stress Topological Skeleton and Post-Percutaneous Coronary Intervention Complications.

Despite the advancements in the stent technology, ST and ISR are major adverse events still affecting the post-PCI long-term outcome [8]. The stent-induced flow disturbances, and more specifically the altered WSS patterns at the stent strut level, play an important role in the pathophysiological mechanisms leading to ST and ISR [7,8]. In this context, this work extends the current knowledge on the impact of stent implantation on local hemodynamics, presenting for the first time the characterization of the WSS topological skeleton in human stented coronary arteries. Motivated by recent studies highlighting the influence that peculiar WSS topological skeleton features have on vascular adverse events (i.e., wall degradation in the ascending thoracic aorta aneurysm [17], long-term restenosis in the carotid bifurcation after endarterectomy [16], and wall thickness change over time in coronary arteries prone to atherosclerotic lesion development [15]), the analysis here proposed could represent an important starting point for better elucidating the role of altered WSS profiles in promoting ST and ISR in stented arteries. Moreover, we have recently demonstrated the feasibility of a comprehensive diagnostic approach that includes WSS topological skeleton analysis in a conventional clinical framework to stratify the risk of myocardial infarction in patients with mild coronary lesions [18], holding the promise of a future extension to other clinical applications such as stented coronary arteries.

Percutaneous coronary intervention with stenting causes severe vascular injury producing endothelial denudation [30]. A process of re-endothelialization begins immediately after stent deployment leading to the repopulation of the damaged endothelium in the weeks after intervention through the proliferation of the endothelial cells remaining within the stented portion, of those close to the treated lesion, and of circulating endothelial progenitor from the blood [3032]. This process can be incomplete or, even if endothelium is completely restored, it can be dysfunctional, thus resulting in impaired endothelial function [30]. Incomplete endothelium promotes the insurgence of ST by exposing potential thrombogenic material, such as the stent material, to the circulating prothrombotic factors [10,30]. Incomplete and dysfunctional endothelium represents a major factor promoting the development of ISR through the increase in permeability, in the expression of chemotactic molecules, in the recruitment and accumulation of monocytes and macrophages, in the smooth muscle cell proliferation and migration, and in the expression of procoagulatory molecules [3335]. Re-endothelialization rates are affected by the local hemodynamic environment of the stented arterial segment, in particular by the WSS patterns at the stent strut level [9,10]. Low WSS is known to (i) attenuate the endothelial production of nitric oxide, prostacyclin I2, and tissue plasminogen activator, and (ii) inhibit endothelial cell proliferation and delay re-endothelialization [7]. This study confirms evidence from literature [7,8,19,36]: luminal regions close to the stent struts exhibit low WSS, as emerged from the distributions of TAWSS of the three investigated cases (Fig. 8). Additionally, it shows that (i) the luminal regions upstream from the stent struts are characterized by WSS contraction while those downstream from the stent struts by WSS expansion (Fig. 5), and (ii) both these luminal regions are characterized by high TSVI (i.e., high variations in the WSS contraction or expansion action along the cardiac cycle) (Fig. 6). The presence of WSS contraction/expansion regions upstream/downstream from the stent struts may contribute to explain the preferential migration direction of endothelial cells, which were shown, at least in an in vitro experiment [37], to migrate in the direction of the flow upstream from the stent struts and to accumulate downstream from the stent struts, being entrapped at the recirculation zone behind the stent struts. Furthermore, the high variability of the WSS manifolds along the cardiac cycle may lead to the dysfunction of the cells in contact with blood immediately post-PCI, namely, the remaining endothelial cells and the smooth muscle cells (usually not in contact with the blood), by exerting a push/pull action on those cells.

Stent thrombosis can be induced by high WSS values, which can lead to platelet activation and triggering of the coagulation cascade [7,8]. High stent strut thickness (e.g., ≥150 μm as in the case of the current bioresorbable scaffolds, such as the Magmaris scaffold of case C) is associated with increased risk of ST [3,7,8,10]. This study confirms that high TAWSS is present at the stent strut top face (Fig. 10), in agreement with previous findings [7,8]. Moreover, it highlights that the stent strut top face exhibits high variability in the WSS contraction/expansion action (Fig. 10). Further investigation is recommended to elucidate the role of the WSS topological skeleton features on the mechanisms triggering ST.

Stent thrombosis and ISR seem to be favored by inadequate stent positioning, including the conditions of stent malapposition (e.g., proximal stented segment of case A) and stent overlapping (e.g., midstented segment of case B) [7,8,38]. Interestingly, a different DIVWSS luminal distribution was observed in these regions as compared to those featuring well-apposed stent struts (Fig. 5). Again, further investigation is required to better understand the role of the identified WSS topological skeleton features on the underlying mechanisms of ST and ISR.

### Impact of Stent Design.

Previous in silico, in vitro, and in vivo investigations have demonstrated that the stent design (e.g., closed-cell versus open-cell stent design, stent strut cross section shape and thickness, and strut connector length) has a strong impact on the local hemodynamics, which in turn may influence the vessel response to stenting, potentially triggering ST or ISR [7,8]. In this study, the impact of three stents with different design (peak-to-valley for the Xience Prime and Magmaris stents of cases A and C, respectively, and peak-to-peak for Endeavor Resolute stents of case B), strut cross section shape (square, circular, and rectangular cross section shape for the Xience Prime, Endeavor Resolute, and Magmaris stents, respectively), and thickness (81 μm, 91 μm, and 150 μm for the Xience Prime, Endeavor Resolute, and Magmaris stents, respectively) on the WSS topological skeleton features was analyzed. Despite the differences in the design of the three considered stent platforms, a similar repetitive pattern of the WSS topological skeleton distribution was observed at the luminal surface of the stented region in all cases (Fig. 5). However, the number of fixed points of each case was different and depended on (i) the number of peaks/valleys per stent ring and (ii) the stent length. Case B presented the highest number of fixed points, as the implanted Endeavor Resolute stents were characterized by the highest number of peaks per stent ring as compared to the other stent designs and by the longest total stent length. The TSVI distribution at the luminal surface was similar in all cases (Fig. 6), suggesting that the highest variations in the WSS manifolds along the cardiac cycle are located close to the stent struts independent of the stent design. From a quantitative viewpoint, case C, in which the Magmaris scaffold was deployed, presented lower median values of both TSVI and TAWSS than the other two cases. The lower median values of TAWSS could be explained by the square strut cross section of the Magmaris scaffold presenting with higher thickness than the other two stent designs [7,39,40]. In order to draw definitive conclusions, a dedicated comparative analysis should be conducted on an idealized vessel geometry with different implanted stents to systematically quantify the effect of the stent design parameters (e.g., stent strut thickness) on the WSS topological skeleton features, without considering the influence of the patient-specific vessel geometry and boundary conditions on the hemodynamic results.

### Limitations.

This study faces some limitations. Only three image-based stented coronary artery models were considered herein. A large number of cases, including different stent designs, stent malapposition, and stent overlapping regions, should be analyzed to provide more general conclusions. Regarding the CFD models, in the absence of patient-specific flow measurements, boundary conditions were derived from: a diameter-based scaling law applied at the outflow sections [27]; a characteristic pulsatile flow waveform available from the literature [25], whose amplitude was scaled to the diameter of the inflow section [27]. This allowed prescribing boundary conditions that were personalized with respect to the specific anatomical features of each case. The flow rate values at the inflow section were then imposed in terms of Dirichlet boundary condition by prescribing a parabolic velocity profile. We have recently demonstrated that the influence of the velocity profile shape imposed at the inflow section is limited to a very few diameter lengths in computational hemodynamic models of LADs [26]; thus, we expect here a small effect of the inflow velocity profile on the WSS distribution at stented regions. In general, the adopted boundary conditions might have some impact on the in-stent WSS topological skeleton features. However, at this stage of the investigation, the lack of measured patent-specific inflow rate waveforms does not entail the generality of the results, at least in terms of the localization of the WSS contraction and expansion regions, as by construction the WSS topological skeleton analysis is based on the normalized WSS. Moreover, the vessel and stent walls were assumed as rigid based on a previous fluid–structure interaction work highlighting that the rigid-wall assumption has marginal effect on the WSS distribution of a stented coronary artery model [41]. Finally, the coronary artery motion during the cardiac cycle was neglected. Previous computational findings suggested that the cardiac-induced motion in untreated coronary arteries has a moderate impact on the coronary flow and WSS distribution [42]. Nevertheless, further investigation is required to confirm these findings in the case of stented coronary arteries when analyzing the WSS topological skeleton features.

## Conclusions

In this study, a recently developed Eulerian-based method [12,15] was used to characterize the WSS topological skeleton of human stented coronary arteries. The findings of the study revealed that the presence of single or multiple stents within a coronary artery severely affects the WSS topological skeleton features. The high variability in the WSS contraction/expansion action exerted at the luminal surface close to the stent struts may have important implications in the pathophysiological mechanisms leading to ST and ISR. These findings contribute to a deeper understanding of the hemodynamics-driven processes underlying ST and ISR, stimulating further investigations in order to elucidate the link between the WSS topological skeleton features and post-PCI complications.

## Acknowledgment

The authors would like to thank Dr. Marco Bologna (Politecnico di Milano, Milan, Italy) for his contribution to the vessel reconstruction from OCT images.

## Funding Data

• This work has been supported by the Italian Ministry of Education, University and Research (FISR2019_03221, CECOMES).

## Conflict of Interest

The authors declare that they have no conflict of interest.

## Authors' Contributions

Conceptualization: C.C., V.M., D.G., U.M.; data curation: C.C., V.M., D.B., E.C.; 3D stented vessel reconstruction: C.C., M.L.R.; meshing: V.M., A.C., A.A.; simulations: V.M., A.A.; post-processing of the results: V.M.; interpretation of data: C.C., V.M., E.C., D.G., U.M.; writing—original draft preparation: C.C., V.M.; writing—review and editing: C.C., V.M., M.L.R., K.C., A.C., A.A., G.D.N., D.B., E.C., D.G., U.M. All authors discussed the results and reviewed the paper.

## Nomenclature

• CFD =

computational fluid dynamics

•
• $DIVWSS$ =

divergence of the normalized wall shear stress vector field

•
• ISR =

in-stent restenosis

•

left anterior descending coronary artery

•
• LSA =

low shear area

•
• OCT =

optical coherence tomography

•
• PCI =

percutaneous coronary intervention

•
• $RT∇xfp(e)$ =

WSS fixed point weighted residence time

•
• ST =

stent thrombosis

•
• TAWSS =

time-averaged wall shear stress

•
• TSVA =

topological shear variation area

•
• TSVI =

topological shear variation index

•
• wFPA =

weighted fixed point area

•
• WSS =

wall shear stress

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