# The Role of Macrophage Dynamics in Atherosclerosis Analyzed Using a Petri Net-Based Model
## 2 Materials and Methods  
### Petri Nets  
The basis of the system's approach is a mathematical model of the analyzed phenomenon. Such a model can be created using methods from some branches of mathematics. One of them is the Petri net theory. Models based on such nets are quantitative, which means that they describe the structure of the investigated system, and its quantitative aspects are taken into account only in a very limited manner. This can be seen as a limitation of such models, but in the case of biological systems, it is rather an advantage because the availability of quantitative data describing the properties of biological phenomena is usually very limited. In other words, collecting a complete set of quantitative data necessary for constructing a quantitative model of a biological system is usually difficult or even impossible. Hence, Petri nets are an interesting alternative for differential equations, for example, where such a data set is necessary. Moreover, many extensions of Petri netsexist that allow one to consider quantitative data of various types. So, it is possible to extend a qualitative model when quantitative data are easily available [25; 27]. Petri nets have a structure of directed weighted bipartite graphs. Graphs of this type are composed of two disjoint sets of vertices such that each arc is an ordered pair of vertices, where one of them belongs to one of these sets and the other is an element of the second set. In the case of Petri nets, vertices belonging to one of these sets are called places, and elements of the other set are transitions. Places correspond to passive elements of the modeled biological system, e.g., chemical compounds, while transitions are counterparts of elementary processes, e.g., chemical reactions. Arcs connecting places with transitions and transitions with places describe casual relations between passive and active components (elementary processes) of the modeled system. The bipartite graphs describe the structure of the system. A descriptive power of Petri nets follows from the fact that, by using them, a kind of system dynamic can also be modeled. This is due to the existence of components of another type, i.e., tokens. They are located in different places and can move from one place to another via transitions. The movement of tokens corresponds to the flow of substances, signals, information, etc., through the modeled system. This is governed by the transition firing rule, according to which a transition is active if in every place directly preceding it (such a place is called a preplace of the transition), the number of tokens is equal to at least the weight of the arc connecting the place with the transition. An active transition can be fired, which means that tokens flow from their preplaces to their postplaces, i.e., the ones that directly follow the transition. The number of flowing tokens is equal to the weight of the respective arcs [25; 28]. One of the great advantages of Petri nets is their intuitive graphical representation. This helps us to understand the structure of the modeled system and its behavior, especially when software for simulating the nets is used. It is also very helpful at the stage of constructing the model. In this representation, places are depicted as circles, transitions as rectangles, arcs as arrows, and tokens as dots or numbers located inside places. Although the graphical representation is very useful, it is not very well suited for an algorithmic model analysis. For this purpose, another representation, i.e., incidence matrix \(A\), is used. In this matrix, rows correspond to places, and columns correspond to transitions. A number in entry \(a_{ij}\) of matrix \(A\) is equal to a change in the number of tokens residing in place \(p_{i}\) before and after firing transition \(t_{j}\). As was mentioned before, transitions correspond to some elementary processes, but in the context of biological systems, an analysis of some more complex processes is also very important.