# Network Modeling of Murine Lymphatic System
## 6 Conclusions  
In this work, we have developed a graph model of the LS network in mice. To define directions of edges in the original anatomy-based simple graph, we considered the mass balance of global lymph flow in the LS. The local interstitial pressure was not considered for the final definition of the oriented graph. It is the first study in which a geometric model of the murine LS has been developed and characterized in terms of its structural organization and the lymph transfer function. The study complements our previous analysis of the human lymphatic system [18]. The developed graph model of the LS in mice provides a computational tool for studying the spatial aspects of the immune system functioning. It goes in line with recent advances in experimental techniques to characterize the whole-body dynamics of systemic infections in experimental mice [34]. The predicted orientation of the edges in the graph deserves further biological verification. To this end, various intravital imaging techniques could be used, e.g., near-infrared fluorescence imaging of lymphatic drainage patterns in mice [35], indocyanine green lymphangiography, or Doppler optical coherence tomography [4], to name just few of them [6]. \begin{table}
\begin{tabular}{l r r r} \hline \hline
**Property** & **Whole Graph, \(g\)** & **Left Subgraph, \(g_{l}\)** & **Right Subgraph, \(g_{r}\)** \\ \hline \(G(n,m)\) & \(g(88,87)\) & \(g_{l}(61,60)\) & \(g_{r}(27,27)\) \\ Number of inlet vertices & 52 & 36 & 16 \\ Maximum degree & 5 & 5 & 5 \\ Girth & 3 & 0 & 3 \\ Diameter, oriented (simple) & 7 (11) & 7 (11) & 4 (7) \\ Radius & 4 & 6 & 4 \\ Average path length, dir. (undir.) & 2.5 (5.3) & 2.7 (5.5) & 1.9 (3.9) \\ Energy & 95.1 & 65.6 & 29.5 \\ Spectral radius & 2.93 & 2.9 & 2.93 \\ Edge density & 0.0114 & 0.0164 & 0.0385 \\ Clustering coefficient & 0.019 & 0 & 0.059 \\ Number of separators & 36 (in total) & 25 & 11 \\ Robustness & 0.917 & 0.863 & 0.825 \\ Average topological flow diversity, & & & \\ - scenario 1: & 0.8252 & 0.8028 & 0.9039 \\ - scenario 2: & 0.8242 & 0.8028 & 0.9015 \\ - scenario 3: & 0.8242 & 0.8028 & 0.9023 \\ Number of LNs & 27 & 19 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary statistics for the anatomy-based graph of murine lymphatic system. We have previously implemented a similar approach to the analysis and modeling of the network structure of the human LS. The mouse and human LS differ fundamentally in terms of their cardinality, i.e., the number of elements comprising the system. The LS of mice consists of 28-36 LNs, whereas their number in the human LS ranges from about 500 to 1000. As a direct consequence, the graph model of the human LS is characterized by a much larger variability and more prominent randomness in its structure. The estimates of the probabilities of a new edge creation \(P_{e}\) and the edge to connect nodes of different layers \(P_{o}\) in random graph approximation of the LS quantify a larger uncertainty in the structure of the human LS graph compared to the mouse LS graph, i.e., 0.035 vs. 0.851 and 0.21 vs. 0.66, respectively, [18; 36]. In addition, the diameter, radius, average path length, and energy features of the respective LS graphs differ substantially. However, the topological properties of the graph models, such as the girth, spectral radius, edge density, and clustering coefficients are close for both human and mouse LSs. The graph scheme of the mouse LS formulated to study the search time of antigen presenting cells by T cells in the LS has been recently presented in [36]. However, the essential details of the LS, such as the adjacency matrix, the orientation of the edges, and the lymph flow estimates are not provided. In our study, we present a systematic analysis of the available anatomical and physiological data to develop the network model of the mouse LS, quantify the topological properties of the LS graph, and calculate flow through the network after making a series of assumptions about vessel diameters and terminal pressures.