# The Role of Macrophage Dynamics in Atherosclerosis Analyzed Using a Petri Net-Based Model
###### Abstract
For example, one of the ODE models focused on the development of atherosclerotic plaques integrates the macrophage response with the dynamics of lipids, examining factors such as macrophage migration and lipid influx in plaque progression [18]. Another extends a lipid framework to account for macrophage responses to lipid levels, specifically their apoptosis, emigration, and proliferation, showing that lipid-influenced macrophage behaviors significantly affect plaque development by modifying lipid patterns [13, 19]. Unfortunately, in the models mentioned above, simplification involves consolidating lipoproteins into a single group without distinguishing between their types, and also macrophage polarization is not considered. To explore macrophage polarization's role in plaque progression, a dynamic stochastic model was developed, capturing changes in plaque components through PDEs and macrophage phenotype shifts with a Bernoulli distribution [16]. Simulations aligned with experimental data suggest that manipulating macrophage polarization can alter the plaque environment, potentially reversing plaque formation, particularly in the early stages, which is consistent with the results of our analysis. An agent-based model was also proposed that simulates macrophage dynamics and polarization using bifurcation analysis and incorporates the diffusion of cytokines through ODEs [15]. Other mathematical and computational models have been developed to investigate the mechanobiological mechanisms involved in plaque advancement. These include a dynamic interaction between blood flow and arterial walls, together with a system of PDEs and ODEs to capture the multiscale dynamics of solute concentrations in plaque growth [14, 22]. Although many models have been previously proposed, differing in the range of details and complexity, most focus on specific and limited aspects of the role of macrophages in atherosclerosis. A comprehensive mathematical model of atherosclerosis that encompasses the fundamental and complex processes associated with macrophage behavior and cholesterol balance has yet to be developed. In our previous article [23], using the Petri net theory, we created a model of macrophage differentiation and analyzed the impact of selected micro-environmental signals. In addition, we focused on the relationships between different macrophage subpopulations. This allowed us, among other things, to explore the interaction of macrophages in relation to the activation of T lymphocytes. The current model is a continuation of the research on this topic. Here, to better understand the associations between macrophages and atherosclerosis, a system approach was utilized that involved using Petri nets [24; 25; 26], as explained in this paper. This approach examining the complex link between cholesterol balance, macrophage behavior, and atherosclerosis enhances knowledge organization. It deepens the understanding of the biological system by constructing a mathematical model that depicts a complex network of interactions. Following this approach, the study aimed to identify molecules, factors, or processes within the examined system that, when blocked, could effectively stop or diminish the progression of atherosclerosis. This was performed using a Petri net-based model illustrating the interaction pathways between cholesterol homeostasis, macrophage dynamics, and atherosclerosis. It is worth noting that Petri net-based models allow for a detailed description of the structure of the analyzed biological system, even in the case where it is composed of dozens or hundreds of elementary components. The relationships between all of them can be precisely described in the form of a bipartite graph and then simulated and analyzed using formal methods. The graph-based structure of the model helps us to easily understand the structure of the system and to perform a simulation in an intuitive way (i.e., the flow of signals in the system is shown in the graph as a flow of tokens between vertices).