**7. Computational modelling of mass transport from drug eluting stents**

The following computational models were created to illustrate how changes in stenting deployment conditions can influence drug concentrations within the artery wall, analysed after 30 and 60 minutes for each model. Figure 4 describes the five 2-D axis-symmetric computational models that were analysed. Model 1 in figure 4 depicts the locations of drug concentration measurement through the depth of the artery wall and axially down the artery at a depth of 25% and 50% of the wall thickness (WT) beneath the strut. The models are described as follows:

**Model 1.** In the absence of a lumen and subsequent BSMT, this model only considered WSMT from a single DES stent strut (150*µ*m) deployed flush against a single layer artery wall (200*µ*m). In this instance WSMT is purely diffusive.

**Model 2.** Similar to Model 1 but with the inclusion of a steady blood flow profile (mean velocity = 0.1m/s) through the arterial lumen. Here WSMT is purely diffusive and BSMT is modelled using the convection-diffusion equation.

**Model 3.** Similar to Model 2 but with the inclusion of a 20*µ*m thick layer of plaque along the artery wall.

**Model 4.** Similar to Model 2 only the stent strut becomes embedded in the artery wall as it compresses it by 25% of the wall thickness.

**Model 5.** Similar to Model 4 except upon compression of the wall the stent strut doesn't become embedded.

A hypothetical drug was used in the analysis with effective diffusivity values in each respective media defined in table 1. It is the combination of both the drug used and the characteristics of the media within which transport takes place that determines the effective diffusivity value. This fact becomes evident as the effective diffusivity of the drug in the compressed wall is determined. The drug remains the same but, as described in equation 12, changes to the tortuosity (*τ*) and porosity (*ε*) of the wall alters the effective diffusivity within.

From equation 15 the pore path (*L*) remains the same length but the distance (*X*) has reduced due to the 25% compression of the artery wall. Therefore the tortuosity of the compressed wall (*τCW*) can be described as a function of the tortuosity of the wall in its original state (*τW*).

$$
\tau\_{CW} = \frac{L}{0.75X} = 1.333 \frac{L}{X} = 1.333 \tau\_W
$$

Similarly, as the wall is compressed the total volume and pore volume of the wall under compression reduces but the fibre volume remains the same. To this end *εCW*= 0.787*εW* and the effective diffusivity of the compressed wall (D*CW*) can be described as follows:

$$D\_{\rm CV} = \frac{\varepsilon\_{\rm CW}}{\tau\_{\rm CV}} D\_{\rm free} = \frac{0.787 \,\varepsilon\_w}{1.333 \tau\_W} D\_{\rm free} = 0.59 D\_{\rm W} = 0.59 \times 10^{-12} m^2 \,/\,\text{s}$$


Table 1. Effective diffusivity values of the different layers of arterial DES models.

Fig. 4. Illustration of 2-D axis-symmetric computational models used in the DES mass transport analysis and their resulting drug concentration contours after 30 minutes.

Fig. 4. Illustration of 2-D axis-symmetric computational models used in the DES mass transport analysis and their resulting drug concentration contours after 30 minutes.

### **7.1 Computational predictions of mass transport in the artery wall**

Even though BSMT doesn't significantly contribute to the distal absorption of drugs into the artery wall, it is a necessary element of the modelling process as it provides a more realistic representation of how the drugs in the DES dissipate in the vasculature. Failure to model BSMT (Model 1) culminates in the entire reservoir of antirestenotic drug, from both the luminal and abluminal side of the stent, having no choice but to eventually transport into the wall. This as we know is not the case as a considerable amount of drug is lost to the blood stream. Figures 5-7 demonstrate that for both time points and at each location the drug concentration for Model 1 is greater than that of Model 2 due to the absence of BSMT. Model 3 examined the influence of arterial plaque, which was given a drug diffusivity of 1×10-13m2/s. In reality plaque size and composition will vary from patient to patient and the implication of this is a study in itself where a range of plaque types would need to be modelled in order to quantitatively predict its influence on WSMT. For the purpose of demonstrating the influence that the presence of any plaque may have on mass transport, a drug diffusivity was chosen that is an order of magnitude between the diffusivities in the DES coating and the uncompressed artery wall respectively. What Model 3 demonstrates is that even a 20*µ*m thick layer of plaque between the stent and the artery wall can significantly reduce uptake within the artery wall, even more so than arterial compression.

Fig. 5. Drug concentrations through the depth of the artery wall, as illustrated in Fig 4 Model 1. Concentrations are measured after 30 minutes (solid line) and 60 minutes (dashed line) respectively.

Fig. 6. Axial drug concentrations at 25% of the artery wall depth, illustrated in Fig 4 Model 1 by the 25% WT concentration line. Concentrations are measured after 30 minutes (solid line) and 60 minutes (dashed line) respectively.

Fig. 7. Axial drug concentrations at 50% of the artery wall depth, illustrated in Fig 4 Model 1 by the 50% WT concentration line. Concentrations are measured after 30 minutes (solid line) and 60 minutes (dashed line) respectively.

The influence of compression on a porous wall in relation to mass transport has been demonstrated experimentally by O'Connell and Walsh (2010) and is evident in the results from Models 4 and 5. Histological evidence demonstrated that upon stent implantation some struts almost cut into the wall (Model 4) while others merely compress it providing gradual recovery either side of the stent strut (Model 5). Contrasting concentration profiles from these models elucidate to the inclusion of compression alone may not necessarily be adequate enough because the surrounding tissue orientation will also play a role in WSMT. Intuitively the greater the DES surface area in contact with arterial tissue the greater its ability to transport drugs into the wall. This holds true when comparing the concentration profiles of Models 4 and 5. The depth concentrations at the early time point for both models are very similar (Figure 5) but there is a notable difference in the axial concentration plots at both 25% and 50% of the wall thickness, whereby the 60% greater wall contact surface area of model 4 results in an increased concentration profile.
