**6. Results**

Tables 6 and 7 show the results obtained by increasing and decreasing the selected parameters by 25% and 10%, respectively.

**Table 6.** Results of foam cells, synthetic smooth muscle cells, and collagen volume variations (second, third, and fourth columns, respectively), and stenosis ratio variation (fifth column), by reducing and increasing the values of the parameters of the first column by 25%.



**Table 7.** Results of the foam cells, synthetic smooth muscle cells, and collagen volume variations (second, third, and fourth columns, respectively), and variation in the stenosis ratio (fifth column), reducing and increasing the values of the parameters of the first column by 10%.

The first column of Tables 6 and 7 represents the parameter whose influence is being analysed. The second, third, and fourth double columns are, respectively, changes in the volume of foam cells, synthetic smooth muscle cells, and collagen fibre in the plaque, caused by the considered parameter variation. Finally, the last double column is the change in the stenosis ratio of the artery due to the change in the considered parameter.

As can be seen in Tables 6 and 7, the trend of the results is the same in the cases of variation of 25% and 10% parameters variation. Therefore, the results will be discussed only with reference to the 10% variation table (Table 7), and can be extrapolated for the 25% variation table (Table 6).

As can be seen in Tables 6 and 7, the variation of the substance was limited to a maximum of 100%. Therefore, in cases where the variation in the percentage of a substance was greater than 100%, the stenosis ratio was not calculated.

As can be seen in Table 7, an increase in the diffusion parameters, *md*, *LDLox*,*r*, *nFC*, *dc*, *<sup>c</sup>thc*,*w*, and *rapop* causes a decrease in the stenosis ratio produced by the plaque. In contrast, an increase in *dLDL*, *dm*, *Cr*, *Sr*, *pss*, *Gr*, and *mr* induces an increase in it. In addition, there are some parameters of the model that have more influence on the results than others, and their variation causes a change greater than 100% in the volume of any of the substances in the plaque.

When considering the parameters that influence the change in the volume of foam cells in the plaque (Table 7), these are, in order of influence, *nFC*, *dLDL*, *DLDL*,*<sup>w</sup>* = *DLDLox*,*<sup>w</sup>* and *Dm*,*<sup>w</sup>* = *DM*,*w*, which are related to foam cells, LDL, and the diffusion properties of substances in the arterial wall, respectively. As can be seen, none of the variations produces a change in the volume of foam cells greater than 100%.

The parameters that cause a higher change in the volume of synthetic smooth muscle cells are, in order of influence: *dLDL*, *Cr*, *pss*, *mr*, *rapop*, *dc*, *<sup>C</sup>thc*,*w*, and *LDLox*,*r*, when increased (the first four produce a change greater than 100%). When their values decrease, the most influential are, in order: *DLDL*,*<sup>w</sup>* = *DLDLox*,*w*, *Dm*,*<sup>w</sup>* = *DM*,*w*, *LDLox*,*r*, *dc*, *<sup>C</sup>thc*,*w*, *rapop*, *pss*, *Cr*, and *dLDL*, the first six of which cause changes greater than 100%. Therefore, for the change in the volume of synthetic smooth muscle cells volume in the plaque, the parameters *dLDL*, *Cr*, *pss*, *rapop*, *dc*, *<sup>C</sup>thc*,*w*, and *LDLox*,*<sup>r</sup>* have huge influences, regardless if their values are increased or decreased.

For the case of the influence on volume change due to SSMC, *dLDL*, *Cr*, *pss*, *rapop*, *dc*, *<sup>C</sup>thc*,*<sup>w</sup>* and *LDLox*,*<sup>r</sup>* have a grea<sup>t</sup> influence on the results obtained.

For the variation of collagen volume in the plaque, the most influential parameters are, when increased: *dLDL*, *Cr*, *pss*, *rapop*, *dc*, *<sup>C</sup>thc*,*<sup>w</sup>* and *LDLox*,*r*, while when decreased: *DLDL*,*<sup>w</sup>* = *DLDLox*,*w*, *Dm*,*<sup>w</sup>* = *DM*,*w*, *LDLox*,*r*, *dc*, *<sup>C</sup>thc*,*w*, and *rapop*.

In Figure 4, the variation in the volume of foam cells, synthetic smooth muscle cells, and collagen fibre is represented in a graph of parallel bars for variations of ±10%. In cases of parameters that cause a volume variation higher than 100% in any of the considered substances, the bar of this substance is represented in red, i.e., the cases of the diffusion coefficients *dLDL*, *LDLox*,*r*, *Cr*, *dc*, *pss*, *<sup>c</sup>thc*,*w*, *mr*, and *rapop*.

**Figure 4.** Variation of the volume of FC (blue color), SSMC (yellow color), and collagen (green color) when increasing and decreasing the parameters by 10% (solid and striped colors, respectively). Bars in red represent a variation of one of the substances higher than 100%.

Figure 5 represents the change in the stenosis ratio due to the variation of the considered parameters when varied ±10%. The red bars refer to cases in which at least one of the substances that adds volume to the plaque has a volume variation greater than 100%.

**Figure 5.** Variation of the stenosis ratio (blue color) when increasing and decreasing the parameters by 10% (solid and striped colors, respectively). Bars in red represent a variation of one of the substances higher than 100%.

### **7. Discussion**

In this work, an analysis of the influence of some parameters of a previous mathematical model of atherosclerosis development in arteries was performed [8]. The mathematical model has a large number of parameters that can affect the growth of the plaque. However, some of them are considered well-known due to, for example, corresponding to geometrical properties of arteries or substances. Therefore, the parameters whose influence on plaque growth was analysed are related to the reactive terms of the equations referred to substances in the arterial wall. These parameters have been modified by increasing and decreasing their value in different cases by 10% and, to determine how they affect plaque growth, variations in the volume of substances that add volume to the plaque have been analysed (foam cells, synthetic smooth muscle cells, and collagen fibre). In addition, the variation in the plaque stenosis ratio was analysed.

As can be seen in the results, on the one hand, the parameters whose variations are directly proportional to the stenosis ratio are the degradation rate of LDL (*dLDL*), the monocyte differentiation rate (*dm*), the parameters referring to the production and degradation of cytokines ( *Cr* and *Sr*), the proliferation rate of synthetic smooth muscle cells (*pss*), the segregation rate of collagen ( *Gr*) and the parameter related to the amount of monocyte recruited by the endothelium ( *mr*). On the other hand, some of the analysed parameters are inversely proportional to the growth of the plaque, and an increase in their values will cause a decrease in the volume of the plaque and, therefore, in the stenosis ratio. It is the case of the diffusion parameters of substances on the arterial wall (*DLDL*,*<sup>w</sup>* = *DLDLox*,*<sup>w</sup>* and *Dm*,*<sup>w</sup>* = *D <sup>M</sup>*,*<sup>w</sup>*), the rate of death of monocytes ( *md*), the rate of oxidised LDL that is uptaken by macrophages, and the maximum amount of oxidised LDL that a macrophage can ingest (*LDLox*,*<sup>r</sup>* and *nFC*), the cytokine degradation rate, its threshold in the arterial wall (*dc* and *Cth <sup>c</sup>*,*<sup>w</sup>*), and the rate of apoptosis of synthetic smooth muscle cells (*rapop*).

As can be noticed, the parameters that influence the change in the volume of synthetic smooth muscle cells also influence the change in the volume of collagen. It is due to the segregation of collagen fibre by SSMC, so collagen depends directly on them.

The less influential parameters in the volume change of substances in the plaque are: *dm*, *md*, *Sr*, and *Gr*. It should be noted that *dm* and *md* are parameters referring to monocytes, which act at the beginning of the process. Therefore, a grea<sup>t</sup> influence on them could be expected. However, monocytes highly affect the results with the parameter *mr*, which is the monocytes recruitment from the lumen, and its variation has a huge influence on the volume of FC in the plaque and, therefore, in the stenosis ratio.

As can be seen in Figure 4, *rapop* has a large influence on the variation of volume of synthetic smooth muscle cells and collagen for both cases, when increasing and decreasing its value by 10%. The smaller the *rapop* value, the more plaque is generated, as it is an apoptosis factor of synthetic smooth muscle cells. However, its influence on the variation of the stenosis ratio is greater in the case of increasing its value than in the case of decreasing it. As it is a parameter related to the apoptosis ratio of synthetic smooth muscle cells, its change does not cause variation in the results of foam cells (Figure 5).

*mr* has more influence in both concentrations and stenosis ratio variation when increased (Figures 4 and 5). This is because, when its value is decreased, the amount of monocytes deposited in the arterial wall is reduced.

*Cth c*,*w* has a large influence in the variations of the results, having more influence when increased (the change of volume and stenosis ratio are greater than 100% in this case, as can be seen in Figures 4 and 5). It is a parameter involved in the differentiation of contractile smooth muscle cells into synthetic ones due to the presence of cytokines in the arterial wall. Thus, it does not influence the volume of foam cells.

As *Gr* is a parameter of collagen fibre segregation, it only has influence on the change of the volume of collagen in the plaque (Figure 4). Therefore, its influence on the stenosis ratio is limited (Figure 5).

*pss* is related to the proliferation of synthetic smooth muscle cells, so it does not influence the results of foam cells. On the contrary, as can be seen in Figure 4, increasing it by 10% produces a change greater than 100% in the variation of synthetic smooth muscle cells and collagen fibre.

*Sr* changes do not cause a large variation in the volume of any substance or the plaque stenosis ratio (Figure 5).

*dc* is a parameter that also has a large influence on the volume variations (Figure 4). This parameter represents the cytokine degradation; thus, the higher it is, the more cytokines are degraded and, thus, the volume and the ratio of stenosis are lower (Figure 5). The same occurs with *Cr*, which represents the cytokine production.

*nFC* represents the maximum amount of oxidised LDL that a macrophage can ingest before becoming a foam cell. Therefore, an increase in its value produces a reduction in the volume of foam cells and the stenosis ratio (Figures 4 and 5). Its influence on the variation of the volume of the substances is not very large, but it produces an important variation in the stenosis ratio. When these results are contrasted with those of a substance that produces a large variation in synthetic smooth muscle cells and collagen volumes (for example, *dc*), it can be observed that a smaller change in the volume of foam cells produces a larger change in the stenosis ratio. It can be explained by attending to Equation (45): The volume of a foam cell is equal to 1.489 · 10−<sup>14</sup> *m*3, while the volume of a synthetic smooth muscle cell is equal to 6.774 · 10−<sup>15</sup> *m*3. Therefore, due to their size difference, less change in foam cell volume is needed to produce a similar stenosis ratio variation.

*LDLox*,*<sup>r</sup>* is related to the oxidised LDL uptaken by macrophages, so it affects the volume of each of the considered substances. An increase in its value produces a reduction in the volume of substances (Figure 4) and therefore of the stenosis ratio (Figure 5).

*dm* and *md* are both parameters referring to monocytes. The first one is related to their differentiation, while the second one refers to their apoptosis. Therefore, their influences are opposite. Their influence is more notable for synthetic smooth muscle cells and collagen volumes (Figure 4).

*dLDL* is the degradation rate of LDL, so it has an influence on foam cells, synthetic smooth muscle cells, and collagen fibre and, therefore, in the stenosis ratio of the plaque (Figures 4 and 5). So, it is one of the most influential parameters of the model and the most influential in the stenosis ratio when it is reduced.

*DLDL*,*<sup>w</sup>* = *DLDLox*,*<sup>w</sup>* and *Dm*,*<sup>w</sup>* = *D M*,*<sup>w</sup>* are related to the diffusion of substances in the arterial wall, so they affect all the processes. Therefore, they influence the results of all the substances, and are some of the most important parameters in the model (Figures 4 and 5).

With all of this information, knowing the influence of all the parameters, they could be adjusted to achieve more or less vulnerable atheroma plaque, according to the percentage volume of foam cells, synthetic smooth muscle cells, and collagen fibre [67,68]. The vulnerability of a plaque is dependent on multiple factors, such as its size or stresses caused by blood flow in it, but it is also dependent on its composition. There is a high risk of rupture of plaque with a large lipidic nucleus and a thin fibrous cap [67,69]. Therefore, a high quantity of foam cells and a small amount of synthetic smooth muscle cells and collagen fibre will be indicators of plaque with a high risk of rupture (and, therefore, less stable) than one with a large quantity of synthetic smooth muscle cells and collagen [67–70]. Therefore, reducing the maximum amount of oxidised LDL that a macrophage can ingest and the ratio of oxidation of LDL (*nFC*) will cause plaque with bigger lipid nuclei, which can develop into more unstable plaque. However, as can be seen, it has no influence on SSMC and collagen volumes in the plaque. Conversely, increasing the apoptosis ratio of SSMCs, the cytokine threshold in the arterial wall and its degradation rate, and the rate of oxidised LDL uptaken by macrophages (*rapop*, *cth c*,*w*, *dc* and *LDLox*,*r*, respectively), and reducing SSMC proliferation, cytokine production, and the oxidation LDL ratio (*pss*, *Cr*, and *dLDL*, respectively) will produce plaque with less fibrotic layer and, thus, a high risk of rupture.

Some studies in the literature focus on determining the most influential parameters in the development of atheroma plaque in different mathematical models. It is the case of Cilla et al. [21], in which the effect of the anisotropy of the arterial wall on the diffusion coefficients of LDL was analysed. Their results have been considered to implement the anisotropy of the diffusion coefficients in the present model. There are also some studies on parameter influence in agent-based models. It is the case of Olivares et al. [18], who focus on determining how the migration of agents, the velocity of oxidation of LDL, and the maximum amount of autoantibodies can affect the plaque. In addition Corti et al. [20] have mono-parametric and multi-parametric studies to determine the influence of the parameters on their model. However, it is not possible to contrast their results with the ones obtained in this article, as each of the models considers different substances in the process of atheroma plaque formation and, therefore, their parameters are referred to other substance values.

The findings of this study should be interpreted in the context of its limitations. For example, the study of the influence of parameters was done only in the geometry of the carotid artery. However, the behaviour of the mathematical model would be the same for other geometrical configurations and arteries, for example, coronary or aorta arteries, adapting the values of the corresponding parameters if necessary. Another limitation of the study is that it was done with a 2D-axisymmetric model instead of a real geometry. This produces a higher plaque growth than in real cases as diffusion in the circumferential direction is not allowed and therefore causes a higher accumulation of substances in the area of the plaque. However, it should not influence the development of the plaque according to the variations in the parameters that were analysed here. In addition, blood flow and the inflammatory process are not coupled, which could influence the shape and stenosis ratio of the developed plaque. In this study, we also do not consider the influence of the mechanics of the arterial wall in the development of the plaque (such as tortuosity or changes in the permeability of the arterial wall due to the thickness variation of it with the cardiac cycle).
