A predictive multiscale model of in-stent restenosis in femoral arteries: linking haemodynamics and gene expression with an agent-based model of cellular dynamics

Abstract

In-stent restenosis (ISR) is a maladaptive inflammatory-driven response of femoral arteries to percutaneous transluminal angioplasty and stent deployment, leading to lumen re-narrowing as consequence of excessive cellular proliferative and synthetic activities. A thorough understanding of the underlying mechanobiological factors contributing to ISR is still lacking. Computational multiscale models integrating both continuous- and agent-based approaches have been identified as promising tools to capture key aspects of the complex network of events encompassing molecular, cellular and tissue response to the intervention. In this regard, this work presents a multiscale framework integrating the effects of local haemodynamics and monocyte gene expression data on cellular dynamics to simulate ISR mechanobiological processes in a patient-specific model of stented superficial femoral artery. The framework is based on the coupling of computational fluid dynamics simulations (haemodynamics module) with an agent-based model (ABM) of cellular activities (tissue remodelling module). Sensitivity analysis and surrogate modelling combined with genetic algorithm optimization were adopted to explore the model behaviour and calibrate the ABM parameters. The proposed framework successfully described the patient lumen area reduction from baseline to one-month follow-up, demonstrating the potential capabilities of this approach in predicting the short-term arterial response to the endovascular procedure.

1. Introduction

Peripheral artery disease (PAD) is an atherosclerosis-related disease affecting more than 230 million people worldwide [1]. PAD mainly occurs in lower limb arteries, predominantly in the superficial femoral artery (SFA) [2]. Percutaneous transluminal angioplasty (PTA) with self-expanding stent implantation is a common endovascular treatment of atherosclerotic SFAs [3]. However, its long-term success is often compromised by the occurrence of in-stent restenosis (ISR), characterized by exacerbated smooth muscle cell (SMC) activity leading to excessive neointimal growth and subsequent lumen re-narrowing [4]. In the SFA, the ISR incidence rate ranges between 15% and 32% [5].

ISR and vascular adaptation processes are driven by complex, multifactorial and multiscale networks of events involving feedback mechanisms, cause–effect relationships and mutual interactions of components at different spatio-temporal scales, from molecular (e.g. gene pattern alteration) to cellular and tissue/organ ones [6]. Among the heterogeneous factors favouring ISR, the vessel wall damage-related inflammatory response induced by stent implantation and the stent-related haemodynamic alteration play significant roles [4]. The lack of a thorough understanding of ISR mechanobiological mechanisms has fostered extensive in vitro, in vivo and in silico research aimed at providing insights into the process. In this context, in silico multiscale models inspired by systems biology principles have been recently proposed to investigate the underlying mechanisms of ISR [6]. Different modelling strategies, based on continuum (e.g. [7,8]) and/or discrete (e.g. [9]) approaches, have been adopted [6]. Among these, multiscale agent-based modelling frameworks, integrating both continuous- and agent-based approaches, has emerged as promising tools being able to (i) include components across molecular, cellular and tissue scales, and (ii) capture the dynamic interplay of mechanobiological events responding to stent implantation [6].

The available multiscale frameworks have mainly described the cellular events and the subsequent arterial wall remodelling triggered by the intervention-induced damage and/or haemodynamic alteration in idealized vessel geometries [6]. However, patient-specific applications are lacking. Furthermore, multi-omics data (e.g. gene expression data, protein networks) have not been integrated in these frameworks yet, although deemed to be the future target of cardiovascular medicine [10]. Recently, monocyte gene expression analysis has enabled the identification of markers of systemic inflammation (target genes) associated with the clinical success/failure of PTA and stenting of femoral arteries [11]. Its integration into an in silico framework will elucidate the ISR mechanisms and contribute to improve the management of PAD.

Accordingly, this work presents a novel multiscale framework that integrates the effects of local haemodynamics and monocyte gene expression on cellular dynamics to simulate ISR mechanobiological processes in a patient-specific model of stented SFA. The framework consists of a computational fluid dynamics (CFD) simulation (haemodynamics module) coupled with an agent-based model (ABM) of cellular dynamics (tissue remodelling module). The model replicates cellular behaviours and arterial wall remodelling in response to the CFD-derived haemodynamic input and the inflammatory cues (monocyte gene expression data) triggered by stenting. Sensitivity analysis and surrogate modelling were adopted to explore the model behaviour and calibrate the ABM parameters, respectively. To assess framework feasibility and verify its capability to describe the short-term arterial response to the endovascular procedure, the framework was applied to one patient-specific case and the simulated lumen area reduction at one month was compared with the patient's follow-up data.

2. Methods

2.1. Patient-specific multiscale framework

Figure 1 schematizes the patient-specific multiscale framework of ISR. The framework receives as input the patient-specific stented SFA three-dimensional (3D) geometry (i.e. post-intervention condition), the blood velocity waveform and the longitudinal data of the patient's monocyte gene expression. The framework returns the one-month follow-up 3D lumen geometry. The framework sees two coupled modules: (i) a haemodynamics and (ii) a tissue remodelling module. Within the first, the fluid domain of the 3D stented SFA is meshed and a steady-state CFD simulation is performed. The wall shear stress (WSS) profiles are extracted at several cross-sectional planes in the stented portion and given as input to the tissue remodelling module, which simulates the arterial adaptation in response to the intervention-induced local haemodynamic and the systemic inflammatory cues. Within the tissue remodelling module, a bidimensional (2D) ABM, implemented for each plane, replicates the cellular and extracellular matrix (ECM) dynamics leading to the overall arterial wall remodelling. At the end of the ABM simulations, corresponding to a one-month period, the 3D lumen geometry is reconstructed from the ABM lumen contour of the simulated planes. The framework is fully automated and executed through an external subroutine in Matlab (MathWorks, Natick, MA, USA).

Figure 1.

Multiscale computational framework. Starting from the patient-specific inputs, the framework (dashed red box) simulates arterial wall remodelling and in-stent restenosis (ISR) along one simulated month and generates as output a three-dimensional (3D) model of arterial lumen geometry at one month of follow-up. The framework consists of two modules, namely (i) the haemodynamics module at the tissue-seconds scale, in which the 3D artery model is meshed and the computational fluid dynamics (CFD) simulation is performed to compute the wall shear stress (WSS), and (ii) the tissue remodelling module at the cell-days scale, in which an agent-based model (ABM) simulates cellular dynamics and arterial wall remodelling, and an updated 3D arterial lumen geometry is reconstructed from the ABM outputs.

2.1.1. Clinical data

Fourteen patients who underwent PTA with stenting at the Malcom Randall VAMC (Gainesville, FL, USA) between 2007 and 2012 were selected to measure the monocyte gene expression. All patients were treated with self-expanding EverFlex stent (EV3, Medtronic, Dublin, Ireland). Blood samples were collected at 1 h pre-intervention, and 2 h and 1, 7 and 28 days post-intervention to perform monocyte gene expression analysis. Framework feasibility was assessed with a one-patient pilot study. The patient was a 57-year-old man, treated with a 40 mm long stent, and presented both computer tomography (CT) and Doppler ultrasound images at one-week (baseline condition) and one-month follow-up. The study was approved by the Institutional Review Board at the University of Florida (Gainesville, FL, USA) and conformed to the Helsinki Declaration on human research of 1975, as revised in 2000.

2.1.2. Monocyte gene expression analysis

Monocytes were isolated using a RosetteSep negative selection kit (StemCell Technologies, Vancouver, British Columbia, Canada). RNA was isolated using RNeasy MiniKit and the quality assessed with an Agilent Bioanalyzer. cDNA was created using Ovation Pico WT kit (NuGEN, San Carlos, CA, USA) and labelled with GeneChip WT Terminal Labelling (Affymetrix, Santa Clara, CA, USA). The Glue Grant Human Transcriptome Array (GGH2, Affymetrix) [12] was used to measure the expression level of 34 834 genes and the raw expression data were normalized with Partek Genomics Suite (Partek, St. Louis, MO, USA) for statistical analysis. Two-way analysis of variance (ANOVA) test was applied to identify the time-dependent genes (1263 genes, p < 0.05). A custom clustering algorithm [13] was adopted to group genes with similar dynamics. Accordingly, the 1263 time-dependent genes were divided into 34 clusters. Moreover, a selection process was applied to detect the clusters that were most likely associated with the procedure outcome, namely the clinical success/failure at one year. Specifically, the clinical failure was defined as angiographic or Doppler ultrasound evaluation demonstrating occlusion or high-grade stenosis (greater than a 3.5-fold increase in peak systolic velocity) at the site of intervention, an associated interval decrease in ankle-brachial index (ABI) > 15% with return of clinical symptoms, or the need for a secondary intervention of the index lesion [11]. The selection process was based on four criteria: (i) significant (p < 0.05) outcome difference identified by the clustering algorithm (24 clusters), (ii) significant (p < 0.05) outcome difference identified by ANOVA (10 clusters), (iii) normalized Cohen d effect size at any time point between outcome groups > 0.8 (10 clusters), and (iv) expression fold change at any time point from pre-operative condition < 0.7 or greater than 1.4 (three clusters). As a result of the clustering and selection processes, three clusters containing 22 genes were found to be differentially expressed between the outcome groups, and thus potentially able to discriminate between one-year clinical success/failure (figure 2). Specifically, the failure group was associated with a downregulation of cluster 5 genes and an upregulation of clusters 8 and 24 genes. The patient's gene expression profile of the three clusters (figure 2, red curves) was considered to represent the patient's systemic inflammation level and was assumed to contribute to the overall restenosis process, by influencing cellular dynamics in the ABM of tissue remodelling (detailed in §2.1.5).

Figure 2.

Monocyte gene expression data of the 14 patients. The gene expression (GE) of three out of 34 clusters that were found to be significantly differentially expressed between success and failure groups are shown. Each grey curve represents a patient's gene profile of the specific cluster (average of the patient's genes belonging to the cluster), with the red curves representing to the patient selected for the application of the multiscale framework. The black curves are the average curves of all the patient's curves. All the gene expression curves were log2-transformed, namely the displayed value of gene expression was computed as GE(t) = log₂(v(t)/v(t0)), where v(t) is the measured gene expression level at time t and v(t0) is the pre-operative gene expression level.

2.1.3. Three-dimensional superficial femoral artery model

A previously developed semi-automatic method [14] was applied to reconstruct the patient-specific 3D vessel lumen geometry at baseline and one-month follow-up. Since the CT images presented poor resolution and metallic artefacts, the stent struts were invisible, and a direct reconstruction of the stent was impossible. Hence, a 40-mm long EverFlex stent model was virtually implanted in the baseline model through a morphing procedure [15] using the mesh manipulation tool HyperMorph, available in HyperMesh (Altair Engineering, Troy, MI, USA) (figure 3a). The correct position of the stent model within the vessel was determined by the stent edges visible on CT. Additional morphing procedure details are reported in electronic supplementary material. The baseline stented SFA model was used to provide the framework initial condition, while the one-month SFA model was used for its calibration.

Figure 3.

(a) Patient-specific model of a superficial femoral artery (SFA) treated with the EverFlex stent at the Malcom Randall VA Medical Center (Gainesville, FL, USA). The model includes a small portion of the common femoral artery (CFA) and the profunda femoris artery (PFA). (b) Detail of the polyhedral computational fluid dynamics (CFD) mesh of the inlet and stented portion. (c) CFD model, with the applied boundary conditions and the resulting wall shear stress (WSS) contour map. In the stented portion, cross-sectional planes (1–9) were selected (one every two stent rings) for the subsequent agent-based analysis. (d) Agent-based model (ABM) of plane 5 and detail on the hexagonal grid.

2.1.4. Haemodynamics module

The baseline 3D stented SFA model was meshed using polyhedral elements and five boundary layers of prism elements near the wall (figure 3b). Curvature- and proximity-based refinement criteria were adopted to capture the stent-lumen intersection with a level of accuracy required for the generation of the ABM planes. Accordingly, a mesh sensitivity analysis, based on both the haemodynamic output and the ABM generation was performed, and the resulting mesh (approx. 3 000 000 elements) was used in the framework. A steady-state CFD simulation was performed in the baseline stented SFA model (figure 3c). The boundary conditions were based on our previous studies [14,16,17]. Briefly, a parabolic velocity profile, with mean velocity derived from patient's Doppler ultrasound images (0.05 m s⁻¹ corresponding to a flow-rate of 246 ml min⁻¹), was applied at the inlet; a flow-split of 0.67:0.33 was prescribed at the SFA and profunda femoris artery outlets, respectively; a no-slip condition was specified at the walls, considered as rigid. Blood was modelled as a non-Newtonian Carreau fluid with a density of 1060 kg m⁻³ [14]. Details about the solver settings are reported in the electronic supplementary material. The software Fluent (Ansys Inc., Canonsburg, PA, USA) was employed for both the meshing and CFD simulation. To perform the subsequent ABM analysis, WSS profiles were extracted at nine cross-sectional planes (one plane every two stent rings) within the stented portion (the region of interest). This choice ensured the best compromise between computational costs and accuracy of the 3D surface reconstruction from the ABM outputs.

2.1.5. Tissue remodelling module

A 2D ABM, based on our work on atherosclerosis [18,19], was developed in Matlab to simulate, for each of the nine planes, the one-month follow-up arterial wall remodelling in response to patient-specific local haemodynamic and systemic gene expression stimuli.

The 2D ABM rests on a 300 × 300 site hexagonal grid (figure 3d). The initial configuration was generated by reconstructing the lumen wall and the stent struts from the 3D coordinates, after proper scaling, rotation and translation operations, needed to fit the ABM of the arterial wall within the computational grid. Specifically, a 0.0375 mm site⁻¹ scale factor was adopted, so that each ABM site corresponds to approximately 1.5 cells (SMC diameter approx. 25 µm [20]). Intima, media and adventitia were generated according to the literature thickness values of human femoral arteries [21,22] (table 1). The internal elastic lamina (IEL) shape was assumed to mirror the lumen wall contour, while the external elastic lamina (EEL) and the vessel border were assumed as circular. The ABM was seeded with stent strut agents, SMCs and ECM (collagen and elastin) in the intima and in the media, and fibroblasts and collagen in the adventitia, with previously adopted densities [18].

Table 1.

Arterial wall dimensions adopted for the agent-based model (ABM) construction.

dimension	value	artery
wall thickness/lumen diameter [21]	0.22	human popliteal artery
intima thickness/wall thickness [22]	0.18	human common femoral artery
media thickness/wall thickness [22]	0.52	human common femoral artery
adventitia thickness/wall thickness [22]	0.30	human common femoral artery

The ABM was initialized with WSS- and gene expression-based inputs (WSSinput and GEinput, respectively). WSSinput was set to reflect the emerging evidence of inverse relationship between the WSS and ISR, according to which low WSS promotes ISR [23]. Indeed, low WSS enhances SMC proliferative and synthetic activity both directly and through endothelial-mediated mechanisms, namely by contributing to a condition known as endothelial dysfunction characterized, among all, by altered nitric oxide release affecting SMC activity. Accordingly, each lumen wall ABM site was initialized with the corresponding CFD-derived WSS, from which a variable representing the level of endothelial dysfunction D was computed according to a sigmoid shaped curve defined as follows (electronic supplementary material, figure S1):

$$D{(\mathrm{WSS})}^i=\hspace{0.17em}-{\displaystyle \frac{1}{1+{\mathrm{e}}^{L_1(WS{S}^i-L_2)}}+1,}$$ 2.1

where WSSⁱ is the WSS value at the i-^th site of the lumen wall, and L₁ and L₂ are parameters defining the slope and the WSS value at which D = 0.5, respectively. D can assume values between 0 and 1, with higher levels of endothelial dysfunction associated with low WSS. As in [18], the primary endothelial lesion D, computed at the lumen wall sites, triggers a perturbation state that diffuses throughout the entire intima layer influencing intimal agent dynamics. WSSinput is thus constituted by (i) lumen wall sites endothelial dysfunction D, and (ii) resulting intimal sites perturbation level, and is expressed as

$$\begin{array}{rl}& WSSinput=\\ & \{\begin{array}{cc}{D}^i& \hspace{0.17em}\mathrm{lumen}\hspace{0.17em}\mathrm{wall}\hspace{0.17em}\mathrm{sites}\\ \sum _{i}D{(\mathrm{WSS})}^i\times \mathrm{Amp}\times \left(1+\cos \left(\pi {\displaystyle \frac{x}{\mathrm{dist}}}\right)\right)& \hspace{0.17em}\mathrm{intima}\hspace{0.17em}\mathrm{sites}\end{array},(2.2)\end{array}$$ 2.2

with WSSinput∈[0 1]. Specifically, WSSinput in the intima, at a distance x < dist from the lumen wall (i^th sites), was computed as the sum of the decays of D, described as the cosine curve portion with negative slope and defined by the parameter Amp, amplitude of the cosine function.

Regarding GEinput, patient's monocyte gene expression curves (red curves in figure 2) and the average success curves of the three selected clusters (black curves in figure 2, success group) were considered to formulate a patient-specific weight of the systemic inflammatory level. First, a generic inflammatory curve, inspired from the literature [24], was generated (figure 4a), presenting a rapid increase in the first post-operative days (peak around day 2 [25]), followed by a rapid and then slow decay, vanishing after one month (in accordance with the monocyte gene expression curves). Second, for each cluster, the difference between the patient-specific curve and the average success curve was computed. Third, it was assumed that the more the patient-specific curve of the cluster deviates from the average success curve (towards the failure direction), the greater the inflammatory level (electronic supplementary material, figure S2A). Accordingly, for each cluster a weight w was computed as a sigmoid-shaped function of the above defined difference (electronic supplementary material, figure S2B)

$$w_5=-{\displaystyle \frac{1}{1+{\mathrm{e}}^{\hspace{0.17em}{p}_{ge}\ast {\mathrm{\Delta }}_5}}+1,}$$ 2.3

$$w_8={\displaystyle \frac{1}{1+{\mathrm{e}}^{\hspace{0.17em}{p}_{ge}\ast {\mathrm{\Delta }}_8}}}$$ 2.4

$$\mathrm{and}\hspace{0.17em}{w}_{24}={\displaystyle \frac{1}{1+{\mathrm{e}}^{\hspace{0.17em}{p}_{ge}\ast {\mathrm{\Delta }}_{24}}},}$$ 2.5

where Δ₅, Δ₈ and Δ₂₄ are the differences between the patient-specific curve and average success curve for the clusters 5, 8 and 24, respectively, and p_ge is a coefficient defining the slope of the sigmoid function. According to equations (2.3)–(2.5), the more negative Δ₅ is, and the more positive Δ₈ and Δ₂₄ are, the greater the inflammatory weights, since the patient-specific curves approach the failure one (electronic supplementary material, figure S2). The GEinput was thus computed as:

$$GEinput(t)=\hspace{0.17em}{w}_{\mathrm{tot}}\times \mathrm{Inflammation}\hspace{0.17em}(t-\mathrm{delay}),$$ 2.6

where w_tot is the product of the three cluster-specific weights, Inflammation(t) is the generic inflammatory curve in figure 4a and delay is the time period that was assumed for the activation of the cellular and ECM response to the inflammatory stimulus. In accordance with previous animal findings [26], GEinput was translated of 3 days to reproduce this delayed effect (figure 4b). To summarize, differently from WSSinput, which locally influences cellular activities, GEinput is associated with the systemic inflammatory response, thus constituting a trigger to all the intimal agents (i.e. all the cells/ECM intimal components are equally affected by GEinput).

Figure 4.

(a) Generic inflammatory curve, inspired from the literature [24]. (b) Example of gene expression-based input (GEinput) of the agent-based model (ABM). This curve was obtained with a value of p_ge = −3.854.

Each agent was initialized with a random even number, representing the internal clock (i.e. the biological time), between 0 and T_agent (T_cell = 24 h [27] and T_ECM = 4 h [28], representing the agent biological cycle), ensuring the desynchronization of cellular activities. After the initialization phase, the ABM performs the cellular activities, followed by proper geometrical regularization, with a 2 h time step over a one-month period. At each time step, the agents reaching the end of their biological cycle (i.e. internal clock = T_agent) are identified as potentially active agents (i.e. ready for a biological event). The potentially active agents are randomly interrogated and the occurrence of a biological event is assessed. Specifically, for each potentially active agent, a random number between 0 and 1 is generated and compared with the agent probability of mitosis/apoptosis (for cell agents only), or ECM production/degradation (for cell and ECM agents) [18]. If the agent probability associated with the biological event (mitosis/apoptosis or ECM production/degradation) is greater than the randomly generated number, then the potentially active agent is upgraded to active agent and the specific biological event occurs. The baseline probabilities for cell mitosis/apoptosis and ECM production/degradation were set as in [18] and applied to the media and adventitia dynamics. In the intima, cell mitosis and ECM production were perturbed depending on WSSinput and GEinput of each agent h, as expressed in equations (2.7) and (2.8):

$$p_{\mathrm{division}}^h=\hspace{0.17em}{\alpha }_1+{\alpha }_{2}WSSinpu{t}^h+\hspace{0.17em}{\alpha }_{3}GEinpu{t}^h$$ 2.7

and

$$p_{\mathrm{ECM}\hspace{0.17em}\mathrm{production}}^h=\hspace{0.17em}{\alpha }_4+{\alpha }_{5}WSSinpu{t}^h+\hspace{0.17em}{\alpha }_{6}GEinpu{t}^h.$$ 2.8

α₁ and α₄ were set to 0.05 and 0.008, respectively [18]. α₂, α₃, α₅ and α₆, weighing the effects of haemodynamics and systemic inflammation on cell mitosis or ECM production, were calibrated as explained in §2.2. Once the active agents are determined, they are randomly accessed, the specific biological event is replicated and the internal clock of both the newly formed agent (in case of mitosis or ECM production) and the active one is reset.

Cell mitosis and ECM production imply the addition of a new agent next to the active one, while cell apoptosis and ECM degradation the removal of the active agent. Agent dynamics in the intima are inward-oriented while those in the media and adventitia are outward-oriented. The details of tissue reorganization at event occurrence are reported in [18]. Finally, regularization algorithms were applied at each time step to guarantee the preservation of smooth contours. Specifically, (i) at the lumen wall, agents were locally redistributed to maximize their contact, (ii) the IEL was reinitialized to its initial configuration, and (iii) the EEL and the external border were redefined as circles with their current average radii.

Due to the ABM stochasticity, three simulations were performed for each plane starting from the same initial condition. The lumen coordinates obtained from the three simulations were collected and the lumen contour minimizing the root mean square deviation from the average one was selected for each plane [18]. Finally, the one-month follow-up 3D arterial lumen geometry was obtained by lofting the selected lumen contour of the ABM planes using Rhinoceros (Robert McNeel & Associates, Seattle, WA, USA).

2.2. Sensitivity analysis, surrogate modelling and calibration

A preliminary sensitivity analysis was performed on a single ABM plane to identify the parameters that mainly govern the ABM outputs, reducing the total number of parameters to be calibrated and thus the computational efforts. Then, surrogate modelling of tissue remodelling module was adopted to define an approximate relationship between selected ABM parameters and the global outputs of interest referred to the resulting follow-up arterial configuration in the stented portion, namely the one-month lumen area and the normalized intimal ECM/SMC ratio (ECM/SMC_{ratio_int}). The surrogate models were (i) used to explore the relation between the ABM parameters and the global outputs, and (ii) employed in the calibration process, in which the selected ABM parameters were calibrated through genetic algorithm optimization.

2.2.1. Preliminary sensitivity analysis

A set of j = 8 ABM parameters }α₂, α₃, α₅, α₆, L₁, L₂, Amp and p_ge}, with ranges reported in table 2, was considered. In particular, the range of L₁, L₂ was set to satisfy a 10% tolerance on the value of D, meaning that D(WSS = 0) > 0.9 and D(WSS = 1) < 0.1. Similarly, a 10% tolerance criterion was considered to set the range of p_ge, so that w₅(Δ = 1) < 0.1, w₅(Δ = −1) > 0.9, w_8,24(Δ = 1) > 0.9 and w_8,24(Δ = −1) < 0.1, with Δ = ±1 being extreme values for the patient-specific case. The ranges of α₂, α₃, α₅, α₆ and Amp were defined by running extreme case scenarios (i.e. WSSinput = 1 and GEinput = 0.7, which represented the maximum possible values for the investigated patient) and choosing upper limits that guaranteed successful one-month ABM simulations. Latin hypercube sampling (LHS) was adopted to sample the admissible range of the j parameters in k = 1000 equal intervals, needed to achieve statistical significance in the sensitivity analysis. A LHS matrix (k × j) was obtained, identifying k parameter combinations [18,29]. Since all the planes presented similar WSSinput (i.e. WSS < 0.4 Pa) and identical GEinput the analysis was performed on a single ABM plane, being the aim of the analysis that of obtaining a generic relationship between the ABM parameters and the output. Plane 2 was selected for the analysis because it presented the lowest WSS, thus allowing exploring the extreme case scenario. Partial rank correlation coefficients (PRCCs) were computed in Matlab to measure the correlation between each ABM parameter and the ABM outputs of interest (i.e. the one-month ABM lumen area, and the intimal cell and ECM content, normalized by the initial value) [18,29]. Statistically significant correlations were assumed for p-value < 0.05.

Table 2.

Agent-based model (ABM) parameters. WSSinput, wall shear stress input; GEinput, gene expression input; SMC, smooth muscle cell; ECM, extracellular matrix; D, variable representing the level of endothelial dysfunction.

parameter	meaning	range
α2	multiply WSSinput in SMC probability	[0; 0.1]
α3	multiply GEinput in SMC probability	[0; 1.7]
α5	multiply WSSinput in ECM probability	[0; 0.1]
α6	multiply GEinput in ECM probability	[0; 1.77]
L1	slope of D	[−25; −5.5]
L2	half decay of D	[0.4; 0.6]
Amp	amplitude of the cosine function representing the diffusion of D in the intima	[0.013; 0.13]
pge	slope of the weights of gene expression	[−6; −2.5]

2.2.2. Surrogate modelling

A Gauss process surrogate model of the tissue remodelling module was developed in Matlab for each global outputs, namely (i) the median lumen area of the stented portion, and (ii) the normalized ECM/SMC_{ratio_int}, both referred to the resulting follow-up arterial configuration, and computed as detailed in the electronic supplementary material. Only the n (≤ j) ABM parameters that were found to be significantly correlated with the ABM outputs in the preliminary sensitivity analysis were considered to build the two surrogate models. To construct them, s = 10 × n samples were generated through LHS (training dataset). Specifically, a LHS matrix (s × n) was obtained by dividing the range of the n = 6 {α₂, α₃, α₅, α₆, Amp and p_ge} ABM parameters (see §3.1 for the results of the preliminary sensitivity analysis, identifying six significantly correlated parameters) in s = 60 equal intervals, and s parameter combinations were obtained. Parameters L₁, L₂ were maintained constant at their half range values (i.e. L₁ = −15.25 and L₂ = 0.5). For each parameter combination s, the tissue remodelling module (consisting of three ABM simulations of the nine planes and the one-month 3D arterial lumen geometry reconstruction) was run, and the median lumen area of the stented portion and the normalized ECM/SMC_{ratio_int} were computed. The surrogate models were validated by: (i) applying the leave-one-out method, and (ii) using a validation set of 10 samples, chosen to span over the possible entire range of lumen area (considering the parameter range reported in table 2) [30]. The consistency between the predicted and simulated outputs was assessed in the leave-one-out process by plotting the predicted outputs as a function of the simulated ones and then evaluating the coefficient of determination $R_{\mathrm{pred}}^2$, and by computing the standardized cross-validated residual (SCVR) values [30]. The $R_{\mathrm{pred}}^2$ between the predicted and simulated outputs was also quantified in the case of the validation set [30].

2.2.3. Calibration

To calibrate the ABM parameters, an optimization problem was defined so that the optimal n parameters within their admissible range could: (i) minimize the difference between the median lumen area of the stented portion of the model and that measured in vivo at one month, and (ii) guarantee a normalized ECM/SMC_{ratio_int}∈[0.5 1.5]. Based on the available clinical data, the patient's lumen area at one month was the only comparative information usable in the calibration process. Moreover, a constraint on the normalized ECM/SMC_{ratio_int} was set to prevent excessive unbalanced neointima composition, with respect to the baseline condition. Indeed, both increased SMC and/or ECM content may be observed in neointimal ISR tissue. However, the predominant content of cells or ECM depends on the subject and the post-procedural time [31,32]. Accordingly, as a conservative hypothesis, no extreme temporal changes in the ECM/SMC_{ratio_int} were assumed as detailed below. Mathematically, the objective and constraint functions were defined as

$$f_{\mathrm{LA}}\hspace{0.17em}(\mathit{x})=|L{A}_{\mathrm{SM}}-\hspace{0.17em}L{A}_{\mathrm{patient}}|$$ 2.9

and

$$\begin{array}{r}{f}_{\mathrm{ECM}/\mathrm{cell}}(\mathit{x})=\hspace{0.17em}{{\displaystyle \frac{{\mathrm{ECM}/\mathrm{cell}|}_{1\hspace{0.17em}\mathrm{month}}}{{\mathrm{ECM}/\mathrm{cell}|}_{\mathrm{day}\hspace{0.17em}0\hspace{0.17em}}}}|}_{\mathrm{SM}},\end{array}$$ 2.10

where x is the vector of the ABM parameters; $f_{LA}\hspace{0.17em}(\mathit{x})$ is the optimization objective, with LA_SM and LA_patient being the one-month median lumen area of the stented portion predicted by the surrogate model and measured on the patient, respectively; and f_ECM/cell(x) is the constraint, computed as the one-month normalized ECM/SMC_{ratio_int} predicted by the surrogate model. Within the optimization process, the two surrogate models were thus interrogated through the definition of the objective and constraint functions. The single, constrained optimization problem was written as

$$\{\begin{array}{lll}& & \stackrel{\min }{x\in A}{f}_{LA}\hspace{0.17em}(\mathit{x})\\ & & 0.5<f_{\mathrm{ECM}/\mathrm{cell}}(\mathit{x})<1.5\\ s.t:& \{A=\left\{\begin{array}{l}x=[{\alpha }_2,{\alpha }_3,{\alpha }_5,{\alpha }_6,\mathrm{Amp},p_{ge}]:\hspace{0.17em}\hspace{0.17em}{\alpha }_2\in [0,0.1],{\alpha }_3\in [0,1.7],{\alpha }_5\in [0,1.77],\\ \hspace{0.17em}{\alpha }_6\in [0,0.1],\mathrm{Amp}\in [0.013,0.13],\hspace{0.17em}{p}_{ge}\in [-6,\hspace{0.17em}-2.5]\end{array}\right\}\end{array}$$ 2.11

The non-dominated sorting genetic algorithm (NSGA-II) was adopted in Matlab to find the optimal solution. The settings of the algorithm are reported in the electronic supplementary material. Finally, the resulting optimum parameters were set in the ABM and the tissue remodelling module was run to assess the goodness of the calibration procedure.

3. Results

3.1. Preliminary sensitivity analysis

The PRCCs between the j = 8 ABM parameters and the three ABM outputs of interest, normalized by their initial values, are illustrated in figure 5. Six out of eight parameters (i.e. α₂, α₃, α₅, α₆, Amp and p_ge) presented significant PRCCs with all investigated outputs and hence were identified as the driving ABM parameters. Since the remaining two parameters (L₁ and L₂) were not correlated with any output, they were excluded from the subsequent global sensitivity analysis (i.e. parameter exploration through the surrogate models referred to the follow-up arterial configuration) and calibration process, reducing the associated computational costs without affecting the output validity.

Figure 5.

Results of the preliminary sensitivity analysis. Partial rank correlation coefficients (PRCCs) between the ABM parameters and (a) the one-month lumen area, (b) extracellular matrix (ECM) intimal content and (c) smooth muscle cell (SMC) intimal content. To compute the PRCCs, the average ABM outputs obtained from the three repetitions (run for each parameter combination to account for stochasticity) were considered. PRCCs range from −1 to +1, with negative PRCCs corresponding to a negative correlation and positive PRCCs to a positive correlation. (*) Significant PRCC, p-value < 0.05.

The six parameters exhibiting significant PRCCs were all negatively correlated with the final lumen area, with Amp presenting the highest negative correlation. Moreover, among the agent probability parameters (α₂, α₃, α₅, α₆), (i) those associated with the WSSinput (α₂, α₅) were more strongly correlated with the final lumen area than those multiplying the GEinput in the same probability equation (α₃, α₆), and (ii) the parameters governing the ECM production (α₅, α₆) affected more the final lumen area than the dual ones in the SMC proliferation probability equation (α₂, α₃). The same considerations held for the final intimal ECM content, but with positive PRCCs. As expected, higher α₅, α₆, Amp and p_ge lead to increased ECM content, being α₅ and α₆ the direct driver of ECM production in the intima, and Amp and p_ge associated with the WSSinput and GEinput, respectively. Moreover, α₂ and α₃, by directly driving SMC proliferation, had a role in the ECM production since ECM deposition is mediated by the cell agents in the ABM. Accordingly, the higher the cellular content, the more ECM production is promoted. Finally, regarding the SMC content, positive correlations were found for α₂, α₃, Amp and p_ge, as expected. Conversely, negative correlations were found for α₅ and α₆, highlighting a counterintuitive behaviour of the ABM for which an increase in α₅ and α₆ (promoting ECM production) leads to reduced cellularity.

3.2. Surrogate modelling: validation and parameter exploration

Sixty samples (i.e. derived from 60 combinations of the six previously identified driving ABM parameters), corresponding to 60 one-month arterial configurations (stented portion), were defined through LHS to build the surrogate models (training set). The 60 samples in terms of distributions of one-month lumen area of the stented portion and normalized ECM/SMC_{ratio_int} are shown in figure 6. The median values of these distributions were considered for the generation of the surrogate models, as detailed in §2.2.2.

Figure 6.

Samples used for the construction of the surrogate models. The grey box plots represent the distribution of the patient's lumen area of the stented portion at time 0 (T0) and at one month (1M-p). The red box plots represent the distribution of the one-month lumen area of the stented portion obtained from 60 simulated cases and ordered according to the absolute difference from the patient one-month lumen area (1M-p), considering the median values. The blue box plots represent the distributions of the one-month normalized intimal extracellular matrix/smooth muscle cell ratio (ECM/SMC_{ratio_int}) obtained from the 60 simulated cases.

The outputs of the surrogate model validation procedure, based on the leave-one-out principle and 10 additional simulations, are presented in figure 7 and electronic supplementary material, figure S3. In both analyses, the surrogate model predictions and the simulated values of the one-month lumen area and normalized ECM/SMC_{ratio_int} exhibited excellent agreement, as supported by the high values of $R_{\mathrm{pred}}^2$ (figure 7). Furthermore, in the leave-one-out principle, the SCVR values of the surrogate model predictions fell within the required interval [−3, 3] (electronic supplementary material, figure S3) [30], thus confirming the validity of the surrogate models.

Figure 7.

Surrogate model validation. (a) Leave-one-out predicted values of the one-month lumen area and normalized intimal extracellular matrix/smooth muscle cell ratio (ECM/SMC_{ratio_int}) in function of the corresponding simulated values. (b) Predicted values of the one-month lumen area and normalized ECM/SMC_{ratio_int} as a function of the corresponding simulated values for 10 additional cases (validation dataset).

The validated surrogate models were used to investigate the influence of each parameter on the two outputs of interest (i.e. one-month lumen area and normalized ECM/SMC_{ratio_int}) by varying three parameters at a time while maintaining the others fixed at their half-range values (figure 8). The parameters were grouped based upon the association with WSSinput (i.e. α₂, α₅, Amp) and GEinput (i.e. α₃, α₆, p_ge). Overall, the WSSinput-associated parameters had more impact on the model outputs than the GEinput-associated ones (figure 8). Regarding the final lumen area, as expected, increases in α₂ and α₅ led to a decrease in the lumen area by enhancing cell mitosis and ECM production, respectively, and their effect was more pronounced at higher levels of Amp (figure 8a). Similar considerations held for the effect of α₃, α₆ and p_ge on the final lumen area (figure 8b). Regarding the normalized ECM/SMC_{ratio_int}, the highest values were observed for concurrent decrease of α₂ and increase of α₅ and Amp, thus confirming the major impact of the haemodynamic-related parameters (figure 8c). Moreover, as also confirmed by the observed relation of α₂ and α₅, and of α₃ and α₆, the normalized ECM/SMC_{ratio_int} raised with enhanced ECM production and reduced cell proliferation (figure 8c,d). Finally, similarly to the effects on the lumen area, Amp amplified the impact of α₂ and α₅, and p_ge amplified that of α₃ and α₆, on the normalized ECM/SMC_{ratio_int} (figure 8c,d).

Figure 8.

Parameter exploration. Predicted values of the one-month lumen area and normalized intimal extracellular matrix/smooth muscle cell ratio (ECM/SMC_{ratio_int}) by varying three parameters at the time while maintaining the others at their half-range values. (a) Predicted one-month lumen area (coloured variable) at the variation of the parameters associated with the haemodynamic input WSSinput (α₂, α₅, Amp). (b) Predicted one-month lumen area (coloured variable) at the variation of the parameters associated with the gene expression input GEinput (α₃, α₆, p_ge). (c) Predicted normalized ECM/SMC_{ratio_int} (coloured variable) at the variation of the parameters associated with the haemodynamic input WSSinput (α₂, α₅, Amp). (d) Predicted normalized ECM/SMC_{ratio_int} (coloured variable) at the variation of the parameters associated with the gene expression input GEinput (α₃, α₆, p_ge).

3.3. Calibrated patient-specific model

The calibration procedure (§2.2.3) provided the optimum parameters x = [α₂, α₃, α₅, α₆, Amp, p_ge] (table 3) that minimized the objective function $f_{\mathrm{LA}}\hspace{0.17em}(\mathit{x})=4.3\times {10}^{-11}$, guaranteeing an acceptable normalized ECM/SMC_{ratio_int}, $f_{\mathrm{ECM}/\mathrm{cell}}\hspace{0.17em}(\mathit{x})=1.3$. The ISR framework was run with the computed optimum ABM parameters. The model was verified in terms of consistency with the rules, robustness and repeatability. Furthermore, the lumen geometry obtained at one month was compared with the patient's lumen geometry reconstructed at the same time point to evaluate the framework capability to capture the short-term arterial response after the calibration process.

Table 3.

Calibrated agent-based model (ABM) parameters.

α2	α3	α5	α6	Amp	pge
0.06	1.614	0.019	0.928	0.065	−3.854

The WSS distribution in the stented portion and the temporal evolution of three explanatory ABM planes (i.e. planes 1, 5 and 9) at day 0, 10, 20 and 30 are shown in figure 9a and electronic supplementary material, video. The temporal trends of the normalized lumen, intima, media and adventitia areas, intimal SMC and ECM content, and normalized ECM/SMC_{ratio_int}, resulting from the three repeated simulations of the planes 1, 5 and 9, are reported in figure 9b,c,d,e, respectively. The planes experienced a similar lumen area reduction (approx. 25%) and presented a homogeneous distribution of intimal growth. The absence of focal restenosis was due to the small variability of the WSS, both circumferentially within the plane and longitudinally among the three considered planes. The temporal trends of figure 9 elucidate the presence of four phases in the ABM dynamics that reflected the GEinput (dashed black line in figure 9b), namely: (i) a primary slight activation attributable to the sole WSSinput (silent GEinput), (ii) intense response starting at day 3 (corresponding to the activation of the GEinput), (iii) growth attenuation (corresponding to the GEinput decay), and (iv) linear growth (corresponding to the constant GEinput phase). As expected, the intima was the only layer involved in the restenosis process, undergoing a 2.5-fold area growth, while the media and adventitia preserved their initial areas. Within the intima, both the ECM and SMC agents contributed to ISR, with the ECM increasing more than the SMC, as also demonstrated by the rise of the normalized ECM/SMC_{ratio_int} over time (consistent with the one-month normalized ECM/SMC_{ratio_int} obtained from the calibration process, $f_{\mathrm{ECM}/\mathrm{cell}}\hspace{0.17em}(\mathit{x})=1.3$). Finally, the repeatability of the model is evincible from the small interquartile range (figure 9b) and the high similarity of the outputs resulting from the three repetitions of each plane (electronic supplementary material, figure S4). Similar considerations held for all the nine planes, whose selected one-month ABM outputs are depicted in figure 10a.

Figure 9.

Results of the calibrated framework for three explanatory planes of the stented region of the patient-specific superficial femoral artery model. (a) Wall shear stress (WSS) contour of the stented portion (left) and temporal evolution of the agent-based models (ABM) of three explanatory planes (planes 1, 5 and 9) along one simulated month (days 0, 10, 20 and 30). For each ABM plane, the results were retrieved from one out of three ABM simulation, namely the one presenting the lumen configuration minimizing the root mean square deviation from the average, as detailed in §2.1.5. (b) Normalized lumen area over time for planes 1 (green), 5 (yellow) and 9 (violet). The dashed black line represents the (not scaled) gene expression-based input (GEinput), also reported in figure 4b. (c) Normalized area over time of the intima (solid line), media (dashed line) and adventitia (dotted line) layers for the planes 1 (green), 5 (yellow) and 9 (violet). (d) Normalized intimal content of smooth muscle cells (SMC) (solid line) and extracellular matrix (ECM) (dashed line) over time for the planes 1 (green), 5 (yellow) and 9 (violet). (e) Normalized intimal ECM/SMC ratio (ECM/SMC_{ratio_int}) over time for the planes 1 (green), 5 (yellow) and 9 (violet).

Figure 10.

Geometry reconstruction of the stented region of the superficial femoral artery model at one month. (a) One-month ABM outputs of the selected simulations for the nine planes within the stented region. (b) Lumen contours extracted from the one-month ABM outputs of the nine planes. (c) Lumen surface reconstruction.

The one-month lumen contour of the nine planes were used to reconstruct the one-month lumen surface of the stented portion (figure 10b,c). The obtained lumen geometry was compared with the patient-specific lumen geometries reconstructed from CT immediately after stenting and at one month (figure 11a). No significant differences were observed between the patient's lumen area at one month and the model prediction (figure 11b). In both the simulated and the patient's cases, a significant lumen area reduction was found at one month with respect to the condition immediately after stenting (p < 0.05). Overall, the calibrated model provided an excellent estimation of the one-month lumen area of the stented portion (model lumen area of 23.85 [22.29 24.47] mm² and patient's lumen area of 23.61 [22.68 25.79] mm²). However, the model was not fully able to capture the local lumen geometrical variability, especially at the proximal region of the stented portion where focal narrowing was detected at one month in the patient's case (figure 11a, dashed box).

Figure 11.

Evaluation of the model prediction. (a) Left: lumen surface of the stented portion of the patient-specific superficial femoral artery model immediately after the intervention (T0) reconstructed from computed tomography (CT) images; centre: lumen surface of the stented portion of the patient-specific model at one month, reconstructed from CT (1M-patient); right: lumen surface of the stented portion at one month, obtained from the framework (1M-model). (b) Box plots of the lumen area of the stented portion immediately after the intervention (T0), at one month of the patient's case (1M-patient) and at one month of the calibrated model case (1M-model). Kruskal–Wallis test with multiple comparison was used to compare the three groups. *p-Value < 0.05 (p-value correction through Tukey–Kramer method).

4. Discussion

Previous cellular-scale ABMs were successfully integrated with molecular/tissue scale continuum-models to investigate the arterial response to haemodynamic and mechanical stimuli post-stenting [33–43]. However, applications to patient-specific scenarios are lacking and multi-omics data have never been included within these models [6]. The present study constitutes the first attempt to include patient gene expression data in a multiscale agent-based modelling framework of vascular adaptation. Specifically, a patient-specific multiscale framework of ISR integrating inputs at different spatio-temporal levels, namely the haemodynamics at the tissue level and monocyte gene expression accounting for systemic inflammation after stenting, at the molecular level, was developed. The framework was used to model the short-term response after self-expanding stent placement in a human SFA. Following a calibration procedure, the framework successfully described the lumen area change from baseline to one-month follow-up.

The gene expression data used herein embed information on the patient's early activation state of the systemic inflammatory system in response to PTA with stenting and, similarly to [11], were found to be associated with the one-year clinical success or failure of the procedure (as defined in §2.1.2). These data were assumed, in combination with the haemodynamic stimuli, as key drivers of cellular dynamics and arterial adaptation after the endovascular intervention. However, given the lack of deeper ontological analysis, a direct association between specific genes and cellular activity was not possible. Accordingly, the main model assumption was that a pronounced and persistent inflammatory response (represented by GEinput) triggers a more intense and long-lasting cellular activity, resulting in intimal hyperplasia and ISR. This hypothesis agrees with the numerous studies demonstrating the relevant role of inflammation in restenosis and identifying biomarkers of systemic inflammation associated with restenosis (e.g. [11,25,44,45]). Furthermore, although the considered gene expression-based stimulus potentially encompasses several pro-inflammatory factors related to the overall intervention, it was supposed to be mainly associated with the arterial wall damage induced by PTA. Indeed, the PTA-provoked injury, consisting of endothelial denudation and potential arterial wall dissection, likely represents the major intervention-related trauma triggering the early inflammatory response [46]. Given the systemic nature of the gene expression data and being unable (due to the available data) to identify regions subjected to higher PTA-induced trauma, the GEinput was assumed to equally influence intimal cellular activities, thus constituting a global trigger to ISR. Consequently, the simulated process of ISR was overall governed by the systemic inflammatory response and locally modulated by the haemodynamics.

Previous multiscale agent-based modelling frameworks of ISR in idealized vessel geometries focused on the effect of local arterial wall damage induced by stenting on the ABM-simulated cellular behaviour and ISR process [35,40–43]. In these studies, the authors (i) performed a finite element analysis of stent deployment and computed the arterial wall damage as function of the von Mises stress, (ii) modelled the damage-induced temporal variation of inflammatory variables (e.g. growth and matrix degrading factors) through a set of ordinary differential equations, and (iii) implemented an ABM of cellular behaviours driven by the damage-induced inflammatory cues. In the present study, the pre-operative patient-specific vessel was completely obstructed, preventing us from performing a finite element analysis of PTA and stent deployment, and obtaining an estimation of local damage. Furthermore, in the previous idealized models [35,40,42,43] a key role was attributed to the re-endothelialization process, through the explicit modelling of endothelial cells and the definition of rules accounting for the inhibitory effect of the endothelial released nitric oxide on SMC activity. However, the haemodynamics was not computed and the WSS influence on endothelial nitric oxide production was not modelled. In the studies by Hoekstra's research group [36–39], to reproduce the endothelium-mediated haemodynamic influence on SMCs, the probability of the presence of functional endothelium over time was modelled without explicitly including endothelial cells and a literature-derived relation between nitric oxide endothelial production and WSS was used to drive SMC dynamics. In the present framework, the endothelial cells were not explicitly modelled. Instead, a phenomenological representation of the direct and indirect (endothelial-mediated) haemodynamic effects on SMC activity was proposed. Each lumen wall agent was initialized with a variable accounting for the endothelial dysfunction, computed as function of the WSS, from which each intimal agent was then initialized with a variable (WSSinput) representing the overall haemodynamic-related trigger to SMC and ECM activity. Accordingly, WSSinput embedded, among all, the nitric oxide mediated mechanisms.

In this study, a feasible and effective workflow for the sensitivity analysis and calibration of the ABM of ISR was also presented, constituting fundamental phases of the modelling process towards the assessment of the model reliability. Besides Nikishova et al. [47], in which a robust sensitivity analysis and uncertainty quantification were performed, in the previously cited multiscale ISR models [33–43], the model response was solely analysed by tuning few clinically relevant parameters (e.g. stent strut thickness, re-endothelialization rate) to assess the model capability to identify intervention-related factors affecting the procedural outcome. Here, the preliminary sensitivity analysis performed on the ABM, based on PRCC evaluation, enabled the driving ABM parameters to be identified. This, besides providing insights into the working mechanisms of the ABM, was useful for the calibration process. Indeed, the calibration of all the ABM parameters in a single-step process through the evaluation of the lumen area and the normalized ECM/SMC_{ratio_int} as outputs may not only be computationally expensive, but also ineffective. Hence, focusing the calibration problem only on those parameters that mainly drive the ABM outputs allowed the computational costs to be reduced while maintaining the high accuracy of the results.

The combination of LHS with the Gauss process surrogate model was successfully applied, enabling the formulation of metamodels of the tissue remodelling module, which provided an estimate of the one-month lumen area and normalized ECM/SMC_{ratio_int}. The surrogate models were used to explore the interrelation among the parameters and the ABM outputs. Moreover, through their interrogation within the genetic algorithm optimization, the ABM parameters were successfully calibrated. Overall, the calibrated multiscale framework of ISR was able to replicate the patient's lumen area reduction at one month. However, some local geometrical features of the vessel were not captured by the simulation, suggesting that additional factors may play a role in the remodelling process.

This study has some limitations. As a pilot study, only one patient-specific stented SFA lesion was modelled. This enabled assessment of the feasibility and potentiality of the developed method. However, additional cases are required to validate the proposed computational framework and subsequently evaluate its applicability in the real-world clinical practice. As described above, the model was unable to fully capture the local heterogeneous response of the artery to stenting. This could be partially attributed to two aspects. First, GEinput, being related to the systemic inflammation, was spatially uniform within the whole domain. The haemodynamic-related factor (WSSinput) was the only local input that could determine heterogeneous growth patterns. Nevertheless, due to the little variability of the WSS (both circumferentially and longitudinally), a uniform intimal growth was obtained within all the considered planes. Even increasing the number of planes would not result in a markedly more heterogeneous arterial response. This suggests that additional inputs (e.g. the local arterial wall damage) should be taken into account. Second, given the unavailability of patient-specific data of arterial tissue composition, a uniform distribution of cells and ECM was assumed to model the ABM intima, media and adventitia layers. However, the real atherosclerotic vessel was probably composed by non-uniform tissue with plaque components (e.g. calcifications) that could likely lead to heterogeneous growth patterns. In this context, intravascular imaging techniques detecting the plaque composition (e.g. virtual-histology intravascular ultrasound or the combination of optical coherence tomography with near-infrared spectroscopy [48]) would allow additional lesion-specific features to be determined, besides improving the 3D vessel reconstruction. In the present study, only the short-term (i.e. one-month) arterial response to stenting was simulated. Although the first post-operative month was found to be the most dynamic, being also affected by the early inflammatory response (as represented by the available monocyte gene expression data), the peak of restenosis was observed at one year in a study on stented SFAs [49]. Moreover, the dynamics of vessel remodelling during the first year after SFA stenting seems to be characterized by three main phases, namely: (i) fast lumen area reduction in the first post-operative month, (ii) slow lumen area change until six months, and (iii) potential focal restenosis after six months [16]. Given the clinical relevance of the long-term (e.g. one year) arterial response, in the future, the proposed computational framework will be adapted to investigate the vessel remodelling processes for longer post-intervention periods. In particular, a bi-directional coupling between the haemodynamics and the tissue remodelling modules will be performed to compute the haemodynamics in the current remodelled vessel geometry and consequently update the WSSinput to the ABM cellular activities, as previously done in [18]. In the long-term scenario, which can be characterized by focal restenosis and thus by larger intra- and inter-patient heterogeneity than the short-term period, the inclusion of patient comorbidities (e.g. diabetes and coronary artery disease) and lesion characteristics within the framework might be even more impactful in its predictive power.

5. Conclusion

The developed patient-specific multiscale framework of ISR simulated arterial wall remodelling following stent deployment intervention by integrating the effects of local haemodynamics and monocyte gene expression within an ABM of cellular dynamics. The multiscale framework was applied to a human stented SFA lesion. After proper calibration, the computational framework was able to describe the patient's lumen area reduction from baseline to one-month follow-up. The overall results highlighted the potential capability of the mechanobiological model in predicting the short-term arterial response to the endovascular procedure. In the future, the model will be validated on the other available patient-specific cases and its capability to predict the long-term outcome will be assessed.

Ethics

This study was approved by the Institutional Review Board at the University of Florida and conformed to the Helsinki Declaration on human research of 1975, as revised in 2000. Written informed consent was obtained from the patients. No animal studies were carried out by the authors for this article.

Data accessibility

The input data used to set up the computational framework are available from previous studies. In particular, gene expression data were derived from the study of DeSart et al. J. Vasc. Surg. 2016 (doi:10.1016/j.jvs.2015.04.399) [11]. Gene expression analysis was based on the study of Wang et al. Brief. Bioinform. 2012 (doi:10.1093/bib/bbr032) [13]. The patient-specific vessel geometry was reconstructed from the study of Colombo et al. Ann. Biomed. Eng. 2021 (doi:10.1007/s10439-021-02776-1) [16]. The computational framework is exhaustively described in the methods of the manuscript and the electronic supplementary materials [50]. The description enables the reproducibility of the framework.

Authors' contributions

A.C.: conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft, writing—review and editing; M.C.: data curation, methodology, writing—review and editing; J.M.R.: data curation, formal analysis, writing—review and editing; S.C.: conceptualization, writing—review and editing; Y.H.: writing—review and editing; D.C.: formal analysis, writing—review and editing; F.M.: resources, supervision, writing—review and editing; J.F.R.M.: resources, supervision, writing—review and editing; S.A.B.: resources, supervision, writing—review and editing; C.C.: conceptualization, funding acquisition, resources, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

The authors declare that they have no competing interests.

Funding

This work has been supported by Fondazione Cariplo, Italy (grant no. 2017-0792, TIME). F.M. and C.C. have been also partially supported by the Italian Ministry of Education, University and Research (FISR2019_03221, CECOMES).