3.3.5. Cytokines

Due to their sizes, cytokines do not have convection. In addition, they are enclosed by macrophages, so they do not have diffusion either. Cytokines in the arterial wall are segregated by macrophages and also experience degradation, phenomena that can be seen in their reactive terms:

$$\frac{\partial \mathbb{C}\_{\mathfrak{c},w}}{\partial t} = \mathbb{C}\_{r} \mathbb{C}\_{\alpha x LDL\_{w}} \mathbb{C}\_{M,w} - d\_{\mathfrak{c}} \mathbb{C}\_{\mathfrak{c},w\prime} \tag{40}$$

where *Cc*,*<sup>w</sup>* is cytokine concentration in the arterial wall. *Cr* and *dc* are, respectively, the ratios of production and degradation of cytokines.

### 3.3.6. Contractile Smooth Muscle Cells

They have neither convection nor diffusion because of their large size. At the beginning of the process, all smooth muscle cells of the arterial wall have a contractile phenotype. Then, due to the presence of cytokines, they experience a change of phenotype into synthetic smooth muscle cells, which is represented in their reactive term:

$$\frac{\partial \mathbb{C}\_{\text{CSMC},w}}{\partial t} = -\mathbb{C}\_{\text{CSMC},w} \cdot \mathbb{S}\_r \cdot \left(\frac{\mathbb{C}\_{\text{c,w}}}{k\_\text{c} \cdot \mathbb{C}\_{\text{c,w}}^{th} + \mathbb{C}\_{\text{c,w}}}\right) \tag{41}$$

*CCSMC*,*<sup>w</sup>* is CSMC concentration in the arterial wall. *Sr* is the CSMC differentiation rate into SSMC and *Cth c*,*w* is the maximum cytokine concentration allowed in the arterial wall. Finally, *kc* is a constant for the saturation equation.

### 3.3.7. Synthetic Smooth Muscle Cells

Equal to CSMC and because of their sizes, SSMCs neither have convection nor diffusion. CSMCs change their phenotypes into SSMCs due to the presence of cytokines in the

arterial wall. They also experience proliferation and apoptosis. These three phenomena are represented in their reactive terms: 

$$\frac{\partial \mathbb{C}\_{\text{SSMC},w}}{\partial t} = \mathbb{C}\_{\text{CSMC},w} \cdot \mathbf{S}\_r \cdot \left(\frac{\mathbb{C}\_{\text{c,w}}}{k\_\text{c} \cdot \mathbb{C}\_{\text{c,w}}^{th} + \mathbb{C}\_{\text{c,w}}}\right) +$$

$$\left(\frac{p\_{s\text{s}}\mathbb{C}\_{\text{c,w}}}{\mathbb{C}\_{\text{c,w}/2}^{th} + \mathbb{C}\_{\text{c,w}}}\right) \mathbb{C}\_{\text{SSMC},w} - r\_{Apwp} \cdot \mathbb{C}\_{\text{SSMC},w} \tag{42}$$

*CSSMC*,*<sup>w</sup>* represents the SSMC concentration in the arterial wall. In addition, *pss* is the SSMC proliferation rate and *rApop* is the SSMC apoptosis rate.

### 3.3.8. Foam Cells

Due to their large sizes, foam cells neither have convection nor diffusion. Once a macrophage cannot ingest a greater quantity of oxidised LDL, it becomes a foam cell, which is represented in the reactive term of foam cells:

$$\frac{\partial \mathbb{C}\_{\text{FC},w}}{\partial t} = \frac{LDL\_{\text{ox},r}}{n\_{\text{FC}}} \cdot \mathbb{C}\_{LDL\_{w}} \mathbb{C}\_{M,w} \tag{43}$$

### 3.3.9. Collagen Fibre

Collagen fibre do not experience convection or diffusion because they are composed of many molecules. Collagen fibre experience segregation by SSMC, which can be seen in its reactive term.

$$\frac{\partial \mathbb{C}\_{\mathbb{C},w}}{\partial t} = G\_r \cdot \mathbb{C}\_{\mathbb{S}SM\mathbb{C},w\prime} \tag{44}$$

where *CCg*,*<sup>w</sup>* is collagen concentration in the arterial wall, and *Gr* is its secretion rate due to plaque formation. Natural segregation and degradation of collagen in the arterial wall were not considered, as they occur in healthy areas of the arterial wall, not related to plaque generation.

Table 3 contains all the parameters to calculate the inflammatory process in the arterial wall.

**Table 3.** Parameters to calculate the inflammatory process in the arterial wall.



### **Table 3.** *Cont.*

### *3.4. Plaque Growth*

Finally, with Equation (45), it is possible to calculate the growth of plaque in the arterial wall. We consider plaque to be composed of a lipid nucleus of foam cells and a fibrous layer of synthetic smooth muscle cells and collagen fibre. Therefore, considering the isotropic growth of plaque, it is possible to determine the change in volume in the arterial wall due to plaque appearance:

$$\nabla \cdot \boldsymbol{v} = \frac{\partial \mathbb{C}\_{\text{FC},\text{w}}}{\partial t} \cdot \text{Vol}\_{\text{FC}} + \frac{\partial \Delta \mathbb{C}\_{\text{SSMC,w}}}{\partial t} \cdot \text{Vol}\_{\text{SSMC}} + \frac{\partial \mathbb{C}\_{\text{Cg},\text{w}}}{\partial t} \cdot \frac{1}{\rho\_{\text{Cg}}},\tag{45}$$

where *<sup>∂</sup>Ci*,*<sup>w</sup> ∂t* is the variation of concentration with respect to the initial concentration of the considered substance. *VolFC* and *VolSSMC* are the volumes of a foam cell and a synthetic smooth muscle cell, respectively. Finally, *ρCg* is the collagen density.

To calculate the volume of foam cells, they have been approximated as spherical geometries, while synthetic smooth muscle cells are modelled as ellipsoids, so their volumes can be calculated with Equations (46) and (47).

$$Vol\_{\rm FC} = \frac{4}{3} \pi R\_{\rm FC}^{\rm 3} \tag{46}$$

$$Vol\_{SSMC} = \frac{4}{3} \pi R\_{SSMC}{}^2 \cdot l\_{SSMC} \tag{47}$$

with *RFC* and *RSSMC* being foam cells and the synthetic smooth muscle cell radius, and *lSSMC* being the lengths of synthetic smooth muscle cells. Parameters for calculating plaque growth in the arterial wall are given in Table 4.


**Table 4.** Parameters to calculate plaque growth in the arterial wall.

### **4. Numerical Methods**

The software COMSOL Multiphysics (COMSOL AB, Burlington, MA, USA) was used to computationally solve the model. It was modelled using four consecutive steps, which can be seen in Figure 3. In the first step, three cardiac cycles and hemodynamic stimuli were calculated in a transient step, determining the values of *TAWSS* and *OSI* for the third cardiac cycle. Then, in the second step, which is stationary, the plasma flow through the endothelium is calculated. In the third step, the inflammatory process is calculated in a transient mode, determining the concentrations of all substances in the arterial wall for a total of 30 years. Finally, a last stationary step was developed to calculate the growth of the plaque, knowing the concentrations of all substances for a period of 30 years.

**Figure 3.** Pathline of the computational model.

A direct solver (multi-frontal massively parallel sparse direct solver, MUMPS) was used to calculate transient blood flow along the lumen, as well as plasma flow across the endothelium, the inflammatory process in the arterial wall, and plaque growth. The inflammatory process was calculated iteratively, using groups of segregated steps for the different substances.

### **5. Sensitivity Analysis**

As can be seen in Section 3, the computational model has a large number of parameters that can affect the composition and growth of the plaque. There is a wide range of values of these parameters in the literature that also come from studies under different conditions, such as in vivo or in vitro experiments, or different species or arteries. Therefore, a sensitivity analysis can help to understand the role of every parameter in the generated plaque.

The objective of this study is to analyse plaque growth and its composition. To do so, a previous selection of the parameters to analyse was made. Therefore, geometric parameters, initial concentrations, material properties, flow properties, and other factors that are well known have not been considered in the analysis because their influence on the model seems to be clear (for example, an increase in the radius of foam cells would result in an increase in the stenosis ratio due to an increase in the volume of the plaque). Thus, in this study, only the parameters related to the reactive terms of the convection–diffusion–reaction equations of the inflammatory process are analysed, as well as the diffusion coefficients of substances in the arterial wall.

Table 5 contains all the parameters whose variation was analysed and their descriptions. The analysed parameters have been increased and reduced by 25 % and 10 % in 61

different simulations in a mono-variant sensitivity analysis. The values of the parameters for the case of ± 10% variation are included in Table 5, as an example:


**Table 5.** Analysed parameters and values.

The percentage change in the volume of the plaque due to each of the substances involved in its growth (foam cells that compose the lipidic core of the plaque, and synthetic smooth muscle cells and collagen fibre, which correspond to the fibrous layer of the plaque) was analysed, as well as the variation in the stenosis ratio, which is defined as:

$$SR(\%) = \left(1 - \frac{Area\ of\ human\ with\ plane}{Area\ of\ hallny\ lumen}\right) \cdot 100\tag{48}$$
