**1. Introduction**

Cardiovascular diseases, including atherosclerosis, are some of the main causes of mortality in developed countries [1]. They consist of the development of atheroma plaque in the arterial wall, which is caused by lipid deposition. This leads to an increase in the thickness of the arterial wall and, therefore, to a decrease in the area where blood flows, called the lumen. In addition, plaque can break and cause a blood clot that can travel through the circulatory system. For these reasons, it can lead to several consequences that are dependent on the artery affected, e.g., myocardial attacks, ischaemia, or ictus, among others.

The process of atheroma plaque formation begins with the deposition of low-density lipoproteins (LDLs) in the arterial wall and, once there, oxidise. Then, monocytes are deposited from the lumen into the arterial wall and differentiate into macrophages, which ingest oxidised LDL. Once they cannot ingest more oxidised LDL, they become foam cells. These foam cells compose the lipid core of the atheroma plaque. Concurrently, at the beginning of the inflammatory process, all muscular cells in the arterial wall are contractile,

**Citation:** Hernández-López, P.; Martínez, M.A.; Peña, E.; Cilla, M. Understanding the Parameter Influence on Lesion Growth for a Mechanobiology Model of Atherosclerosis. *Mathematics* **2023**, *11*, 829. https://doi.org/10.3390/ math11040829

Academic Editor: Fernando Simões

Received: 22 December 2022 Revised: 31 January 2023 Accepted: 2 February 2023 Published: 6 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

i.e., they cannot move or react with any other substance. However, due to the presence of cytokines in the arterial wall (segregated by macrophages), these muscle cells change their phenotype and become synthetic smooth muscle cells, which can move and interact with other substances in the arterial wall. Then synthetic smooth muscle cells surround foam cells, macrophages, and oxidised LDL, and segregate collagen fibre to isolate the lipid core, forming the fibrous layer of the plaque [2–4].

Therefore, atheroma plaques are developed as a consequence of an increase in endothelial permeability. This variation in permeability was accepted to be due to a change in the shape of endothelial cells, depending on some mechanical stimuli caused by blood flow [5]. These mechanical stimuli could be, among others, the wall shear stress of blood with the endothelium (*WSS*), time-averaged wall shear stress (*TAWSS*), and oscillatory shear index (*OSI*) in a cardiac cycle [6,7], or a combination of some of them [8].

There are some studies in the literature related to the influence of blood hemodynamics on the location of atheroma plaque for patient-specific geometries. Malvè et al. [9] modelled a carotid artery bifurcation and analysed the influence of *TAWSS* and *OSI* on the shape index (*SI*) of endothelial cells. Sáez et al. [6] analysed *WSS* on eight different geometries of coronary bifurcations. Alimohammadi et al. [7] analysed *TAWSS*, *OSI*, and a new index that they previously proposed [10], Highly Oscillatory Low Magnitude Shear (HOLMES), in the bifurcation of the abdominal aorta.

Some continuum mathematical models of the development of atherosclerosis have been developed. There is a huge quantity of these models developed as two-dimensional axisymmetric models. Olgac et al. [11] consider the LDL flow from the lumen to the arterial wall with the three-pore model, depending on the *WSS*, while Tomaso et al. [12] considered the flow of monocytes as well as LDL flow. Calvez et al. [13] used a two-dimensional model which, in addition to LDL and monocytes, also considered macrophages, cytokines, and foam cells. Finally, they modelled the fluid–structure interaction between the mesh displacement due to plaque formation and blood flow. Cilla et al. [4] modelled a two-dimensional axisymmetric geometry with LDL and monocyte flows into the arterial wall depending on *WSS* and the three-pore model, and the interactions between all the substances commented on before, as well as smooth muscle cells and collagen fibre. In addition, Shahzad et al. [14] studied the influence of hemodynamics in a two-dimensional geometry of an artery bifurcation with an obstacle plaque that disturbs blood flow. They used the fluid–structure interaction (FSI) with an elastic wall and computational fluid dynamics (CFD) considering a rigid wall.

There are also mechanobiological models with patient-specific geometries, such as by Siogkas et al. [15], who consider LDL that becomes oxidised, monocytes that differentiate into macrophages, and cytokines in the arterial wall. Filipovic et al. [16] modelled oxidised LDL, macrophages, and cytokines, depending on the *WSS* on coronary arteries, and adjusted the parameters of their model in order to obtain results based on experimental data. Hernández-López et al. [8] included also foam cells, smooth muscle cells, and collagen fibre, and analysed the effect of mechanical stimuli on the growth of atheroma plaque in carotid arteries. Kenjereš and De Loor [17] developed a computational model to simulate the flow of LDL through the arterial wall in a realistic geometry of a carotid bifurcation with a multilayered wall.

On the other hand, there are some agent-based models that study plaque development in the arteries [18–20]. The main advantage of continuum models against agent-based models is that they allow the simulation of plaque development in real geometries, whereas the advantage of agent-based models is that they can take into account the random behaviour of cells, which is not possible with continuum models. Olivares et al. [18] developed a model for early-stage atherosclerosis, in which they considered substances such as LDL, oxidised LDL, macrophage, foam cells, smooth muscle cells, endothelial cells, and autoantibodies. Bhui and Hayenga [19] developed a three-dimensional model of transendothelial migration during atherogenesis, in which they modelled endothelial cells, monocyte, macrophage, lymphocyte, neutrophil, and foam cells. In addition, Corti et al. [20] modelled atheroma

plaque growth depending on *WSS* in three-dimensional geometry. They considered a multilayered wall, composed of the intima, media, and adventitia layers, modelling also the internal and external elastic laminae.

There is a large number of substances and parameters involved in the process of formation of atheroma plaque, so it is important to understand how each substance and parameter influences plaque growth.

There are also some studies focused on parameter influence in previous related models, such as Tomaso et al. [12], who analysed the influence of different LDL concentrations in blood to determine how it affects plaque growth. Olivares et al. [18] analysed, in 8 cases, the influence in the plaque of the oxidation rate of LDL, migration speed of agents, and auto-antibodies maximum concentration. Cilla et al. [21] analysed the effect of transmural transport properties on atheroma plaque formation, attending to LDL and oxidised LDL diffusion coefficient anisotropy. Finally, Corti et al. [20] made both, mono-parametric and multi-parametric sensitivity analyses to evaluate the changes in their results caused by changes in the parameters of an agent-based model of plaque growth.

The aim of this study is to analyse the influence of the parameters of a previous computational model on the formation of atheroma plaque in the arteries [8], to determine the effect of these parameters on the growth and composition of atheroma plaque.

The mathematical model used has a total of 52 parameters, which come from different studies and can be related to experimental or computational analysis. Among the experimental ones, there are some differences between the analysis conditions. For example, some of the parameters have been determined from in vitro tests, while others are from in vivo tests. Moreover, they come from studies of different species and arteries (coronary, carotid, or aorta). Finally, some of them have been estimated based on computational results. Therefore, the values of the parameters can have a large variation, so it is relevant to perform a sensitivity analysis of the parameters of the model.

Although the study of parameter variation was performed in a geometry of a carotid artery, the influence of the parameters in the model would be the same in the case of other arteries or geometries.
