Cardiovascular disease is the leading cause of death in the US and is attributed to one in every three deaths (Benjamin et al., 2017). Stenting is the most common treatment for stenosed arteries, with over one million procedures performed annually in the US alone (Benjamin et al., 2019). However in-stent restenosis occurs in 5–10% of cases (Byrne et al., 2015). Interventions such as stenting can change the arrangement of collagen fibers within the vessel. A study carried out on sheep found that after implanting self-expanding heart valve stents, regional differences were found in the collagen organization with collagen fibers near the strut aligning in the direction of the strut and random collagen orientation observed between struts (Ghazanfari et al., 2016). These changes in the collagen alignment alter the cell environment and the structural cues experienced by the cells. Previous research has demonstrated how mechanical and structural cues regulate the alignment of cells and collagen in vitro (Dickinson et al., 1994; Lee et al., 2008; Melvin et al., 2011; Rouillard and Holmes, 2012), and there is a need to better understand how cells integrate these cues as they remodel the extracellular matrix in biological processes like in-stent restenosis (Nolan and Lally, 2018), vascular graft repopulation (Zahedmanesh and Lally, 2012), and wound healing (Rouillard and Holmes, 2012). To date, research has shown that the orientation of cells greatly influences the in vivo mechanical properties and functionality of soft tissues (Ristori et al., 2018). In particular, cells excrete collagen along their primary direction (Sawhney and Howard, 2002; Wang et al., 2003; Matsugaki et al., 2013). As collagen is the main load-bearing component (Ristori et al., 2018), an understanding of cellular orientation is of huge importance in understanding the response of soft tissues to mechanical stimuli or injury. Previous experimental studies have demonstrated that cells align to topographical cues provided by collagen (Guido and Tranquillo, 1993) with numerous studies demonstrating that cells seeded in differentially constrained collagenous constructs align in the direction of the constraints (Huang et al., 1993; Thomopoulos et al., 2005; Henshaw et al., 2006; Foolen et al., 2012, 2018; Ristori et al., 2018). However, in the presence of uniaxial or cyclic strain, cells prefer to align themselves perpendicularly to the mechanical stimulus, known as “strain avoidance” (Foolen et al., 2012, 2014; de Jonge et al., 2013). Further to this, the ability of cells to reorient in response to mechanical stimuli is dependent on the density of the collagen fibers (Foolen et al., 2012, 2014). This interplay between collagen fibers and cells is not yet fully understood with further research required (Foolen et al., 2012; Ristori et al., 2018). A study by Rouillard and Holmes (Rouillard and Holmes, 2012) developed an agent-based model (ABM) to examine healing in infarcts that included structural, mechanical and chemical cues. This study, however, did not include strain avoidance demonstrated by cells under cyclic loading conditions.

Recent experimental studies in our lab examined the strain response of Rat Aortic Smooth Muscle Cells (RASMC) to varying amplitudes of cyclic strain on both unstructured dimethylpolysiloxane (PDMS) (Figure 1) substrates (Mathieu et al., 2020) and on decellularized collagenous tissue substrates (Figure 2) which presented topographical cues to the cells (Mathieu, 2020). It was found that RASMC seeded on PDMS exhibited a strain avoidance response that increased as the strain amplitude increased (additional experimental data can be found in the Supplementary Material). It was also found that RASMC strain avoid to a greater degree after 72 h. Furthermore, a strain-induced decrease in cell number was observed for the RASMC (Mathieu et al., 2020). When seeded on decellularized collagenous tissue substrates RASMC aligned with the topographical cues in two of the four samples while two samples exhibited strain avoidance behavior. The cells that remained aligned parallel to the strain direction showed a greatly reduced cell number in compared to the samples in which cells reoriented perpendicular to the strain direction. Figure 2 demonstrates that cells aligned with the direction of collagen fibers irrespective of the direction of strain (Mathieu, 2020).

The objective of this study is to investigate the combined response of cells to both structural and mechanical cues using a numerical framework and verify this framework using this aforementioned experimental data. Combining these agent-based modeling techniques with the existing experimental data allows us to generate a greater understanding of cellular behavior. This work aims to develop a modeling framework that can be used to investigate this response in a manner which cannot be assessed in vitro/in vivo.

A 2D ABM was developed in MATLAB (R2018b, MathWorks) to simulate the collective response of RASMC to varying amplitudes of cyclic strain (0–10%, 2–8%, 4–6%) when seeded on an unstructured (PDMS) and a structured (decellularized collagenous tissue) substrate. It is known that smooth muscle cells are spindle-like in shape (Zhou et al., 2018). However, in this model, each individual vascular smooth muscle cell (VSMC) was represented as a circle agent with a radius, R_c and the cell direction is defined by a vector with angle, θ_cell,t. In vitro and in vivo cells can move and change their shape to accommodate cells around them in highly dense populations. Cells in this model will only proliferate if there is space to do so. It was found that using a cell radius of 12.96 μm (Zahedmanesh and Lally, 2012; Zahedmanesh et al., 2014) lead to artificial over confluence at high cell densities. Therefore, to accurately represent the cell turnover at higher cell densities a value of 0.3888 μm was chosen as R_c. Further information on this can be found in the Supplementary Material.

VSMC agents were seeded randomly as a monolayer on 2D constructs representing PDMS and decellularized collagenous tissue based on (Mathieu et al., 2020) and (Mathieu, 2020) with seeding densities of 5.5 × 10³ cells/cm² and 1.33 × 10⁴ cells/cm² on each of the respective constructs. Each VSMC agent was then randomly assigned an angle between π2 and -π2 radians.

Model calibration found the following parameters (Table 5) to best fit the experimental data.

In this study, a 2D ABM is presented to investigate the influence of cell behavior in response to cyclic strain and structure. This model was calibrated to experimental data of cell behavior to cyclic strain and used to predict cell growth when subjected to the combined influence of strain and structure. While the model was able to predict the trends in fold change in response to the combined response of strain and structure, it was somewhat limited by the lack of experimental data necessary to calibrate certain parameters in the model. By combining further experimental work with this model, a more robust design tool could be developed. This could be implemented to better understand cellular behavior and also as an efficient means to design and optimize the compliance and topological structure of implantable devices. Furthermore, a model such as this with further experimental calibration could be used to aid the design of next-generation vascular grafts and stents. Below is a comprehensive discussion of each of the model parameters, how they individually influence cellular response and suggestions for further experimentation that could lead to a more accurate representation of cellular behavior.

The first assumption of the model was a reduction in cell radius that is found for RASMC in literature. Using the RASMC cell radius value from literature it was found that the model could not predict the high cell densities seen in the experimental data. This is due to the fact that in vitro cells can change their shape to conform to cells around them. However, in the model cells were a strict circular shape and could not conform. It was observed at higher cells densities even though the total free area on the construct would allow for further proliferation, due to the rigidity of the cell shape in the model there was not adequate space for cells to double. Therefore, the model would achieve confluence earlier than the experimental data. Performing a sensitivity analysis, it was found that a cell radius of 0.3888 μm allowed proliferation to higher cell densities. Further information on this can be found in the Supplementary Material.

Another assumption was C_prolif was chosen based on the experimental study (Endlich et al., 2000). However, doubling time is calculated as the time it takes cells to double in number and consequently the balance between B_apop and C_prolif needs to be carefully considered. This ratio of cell death to cell growth will ultimately dictate the final cell density. Although understanding the ratio of cell death to growth can give an insight to cell behavior, further studies are required to understand the influence of these two parameters individually. Using stains like Ki67 as in Mathieu et al. (2020) could give further insight into the degree of cell proliferation occurring, however, Ki67 stains for a section of the mitosis cycle, whereas in the ABM mitosis is modeled as instant cell division making markers like this difficult to use for calibration. Future studies into the degree of proliferation and cell death occurring at hourly time points may give further insight into the individual influence of these two parameters on the collective behavior of this cell population. Furthermore, the mechanics of PDMS and collagenous tissue are different and this must be noted as substrate stiffness can influence cell properties leading to increased proliferation, migration and decreased cell contractility (Dieffenbach et al., 2017; Yi et al., 2019). However, the levels of strain induced are comparable between the two materials and as most variables were calibrated to the change in strain, the only assumption made between the two different substrates is that the cells had a similar doubling time and rate of apoptosis when left unstrained. Given the cells are the same cell type and this parameter was based on values taken from literature the authors feel this is a reasonable assumption.

Once the ABM performance had been calibrated by comparison to experimental data of cells without any strain or structural stimulus the model was used to investigate how the cells responded to cyclic strain. The rate of response of cells to cyclic strain (k_rot,c) is a factor that the model has shown to be key in representing the correct reorientation of the cells. The model suggests that although cell reorientation appears to influence the overall cell alignment there is also a bias of cells proliferating in areas of low cyclic strain and cell death occurring in areas of high cyclic strain. Therefore, more apoptosis and less proliferation inherently occur in areas where cells are aligned with strain and conversely, less apoptosis and more proliferation occur where cells are aligned perpendicular to strain. One of the most interesting findings of this model is that even in the absence of cell reorientation (i.e., k_rot,c0) we can still see that when predicting the same fold change as seen in the experimental data we observe similar reorientation plots due to the effective strain-based bias of apoptosis and proliferation. This finding leads the authors to two hypotheses (1) that cell reorientation is not occurring to any significant level in these experiments and that the “reorientation” observed is due to a greater extent to the bias of effective strain, or (2) that the rate of cell reorientation does not dominate the final cell position and it is the bias of effective strain and cells slowly reorienting combined that give us this overall result of cells aligned perpendicular to strain. Further experimental studies such as live-cell images of cell reorientation in response to cyclic strain could answer these questions concerning the rate of cell reorientation more definitively. Understanding the rate of cellular response and how cells respond to cyclic strain can have a huge impact on phenomena such as in-stent restenosis and other cascading events. Further insight into whether cell reorientation itself or whether the effective strain bias is dominating the beginning of this cellular response could provide further insight into these phenomena.

Additionally,k_rot,f could be investigated using a similar live-cell imaging approach to assess the rate of response of cells to structure. Currently, in this model depending on the strain rate the cells reorient at 0.007° per hour in response to strain and 2.8° per hour in response to structure, however, this would need to be verified through further experimentation examining more regular time points. Furthermore, the influence of fiber dispersion can be observed in the model, it was predicted that with an increase in fiber alignment, fold change increases when fibers are perpendicular to strain and decreases when fibers are aligned parallel to strain. This can be explained due to the fact that an increase in overall fiber alignment would lead to a greater number of cells aligning in the predominant fiber direction meaning in the case of perpendicular collagen alignment a greater proportion of cells would experience less strain leading to decreased apoptosis and increased proliferation generating an increase in foldchange. The opposite would be seen when collagen is aligned with strain, that is, a greater proportion of cells would experience more strain and therefore increased apoptosis and decreased proliferation would occur leading to the lower fold change as predicted in the model. The alignment and dispersion of the construct prior to cell seeding and cell alignment should be examined in future experiments. In the results shown here, a dispersion of κ between 2 and 5 seems most likely for the experimental constructs, however, this would need to be quantified.

In this model some terms were included that could not be calibrated from the existing experimental data and would require further experimental research to be able to accurately quantified. Firstly, the model contains a threshold fiber density parameter, ϕ_Thres, and M, the slope, that dictates the density at which fibers begin to influence cellular behavior over other cues like cyclic strain. This may vary from cell type to cell type depending on the size of the cells relative to the space between fibers on a structured construct. In the model, ϕ_Thres and M are used to represent the point at which fibers begin to dominate the influence of cell behavior based on the normalized cell density (0–1) where 0 means no fibers are present and 1 represents a dense collagenous tissue. In the model, it was assumed PDMS has a fiber density of zero and collagen tissue has a fiber density of 1. Therefore, in the experiments used to calibrate the model ϕ_Thres and M cannot be quantified and remain undefined. These parameters were included to allow for the improvement of the model with further experimentation. Experiments with different structured constructs of different density are necessary to calibrate these parameters. Furthermore, the maximum strain that can influence cell behavior, ε_Max, was briefly examined in the paper and assigned a value of 0.4, however, experiments examining higher levels of strain must be carried out to quantify this parameter. Similarly, to correctly identify the threshold strain, ε_Thres, lower levels of cyclic strain must be examined experimentally.

Finally, the influence of A_prolif,A_apop, and B_prolif are parameters that are challenging to quantify experimentally as separating the influence of proliferation and apoptosis in cellular experiments is very difficult. The calibration shown does not capture the experimental data at all three levels of cyclic strain. However, having a better understanding of other parameters in the model would lead to increased accuracy in the prediction of these parameters and a better understanding of cell behavior in relation to cyclic strain.

In conclusion, this paper presents an ABM that can be used to predict cell behavior in response to cyclic strain and structural cues. The model was calibrated against existing experimental data and used to predict the fold change in cell number after 10 days of 0–10% cyclic strain when seeded on structured constructs aligned perpendicular and parallel to the direction of cyclic strain. The model successfully captured the trends in cell growth seen experimentally. This model gives insight into the role of different factors influencing cells that cannot easily be examined experimentally, such as highlighting the role of effective strain bias in influencing cell alignment. Tools such as this ABM can be used in conjunction with in vitro experimental data to further enhance the understanding of cellular behavior in response to intravascular medical devices.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

OM was the primary researcher and developed the model. DN contributed to the model and its development. PM provided the experimental data referenced here and used in the manuscript. CL was the principal investigator who conceived the study and provided insight into the model. All authors contributed to writing and editing the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.