# Network Modeling of Murine Lymphatic System
## 1 Introduction  
Networks of various natures, e.g., structural, functional, spatial, underlie the dynamics and mechanisms of regulation in live systems ranging from cells to physiological organs and to whole organisms [1]. Network concepts are increasingly used to describe the structure and function of the immune system [2]. The immune system represents an example of a highly complex network of interacting and migration cell populations embedded into the spatially distributed components of the lymphatic system (LS) [3]. The LS is a body-wide network of lymphatic vessels and lymphoid organs with two major functions: (1) fluid transport from tissues to the blood system to maintain fluid homeostasis and (2) trafficking of antigens and immune cells to lymph nodes where the immune responses take place [4, 5]. Lymphoid organs include a large number of lymph nodes as well as the spleen, thymus, tonsils, and bone marrow [5]. The lymphatic vessels are the conduits that facilitate the directional lymph transport from peripheral tissues to secondary lymph nodes [4]. Understanding of the lymphatic structural and functional organization is essential to discern how the LS interacts with different tissues and organs within the body [6]. Animal models of diseases, particularly mice, are considered to be the cornerstone for translational research in immunology [7, 8]. Research with laboratory mice enabled invaluable insight into mammalian immune systems [9]. Despite numerous advances in understanding the immune system from mouse studies, there exist fundamental differences between mouse and human immune systems [8]. The structural organization ofthe lymphatic system represents a straightforward example. However, a comprehensive mathematical network-based characterization of the LS in mice is still unavailable. The primary function of the lymphatic system is the maintenance of the interstitial fluid homeostasis [10]. The structure and topology of the LS network is heterogeneous and remains to be systematically explored [5]. The existing mathematical models of the lymphatic system refer to specific parts of it, such as the lymphatic capillary network [11], collecting lymphatics [12], lymphangions [13], or branching networks of lymphatic vessels [14]. Computational models of the whole lymphatic system network are still rare [15; 16; 17; 18]. Comprehensive reviews of existing approaches to modeling the lymphatic system structure and function can be found in [10; 19]. One of the major bottlenecks in developing the computational models of the lymphatic system is due to the lack of comprehensive anatomical and physiological data [10]. The latest research activity has clearly stated, \"Thus, gross lymphatic anatomy has not been updated for more than a century.... our knowledge of macro-lymphatic anatomy remains rudimentary\" [6]. The existing gaps [20], require further systematic research [21] including mathematical modeling [22], which serves to integrate available knowledge and to identify critical issues amenable to further biological testing. The application of graph theory methods to describe the spatial organization of the human LS has been addressed in a number of recent studies (see for a review [18]). In [18], we developed a computational algorithm for representing the anatomy-based and rule-based graphs of the LS in humans. The graph models enabled the analysis of different metrics of complexity of the human LS such as spectral radius, clustering coefficient, average path length, and number of separators. A similar analysis for the mouse LS remains to be performed. The aim of the present study is to model and analyze the network structure of the murine lymphatic system. The algorithm for building the graph model makes use of anatomical data. To identify the edge directions of the graph model, a mass balance approach to lymph dynamics based on the Hagen-Poiseuille equation is applied. Various matrix forms for graph representation are specified. The lymph transfer times between various nodes are estimated.