# Network Modeling of Murine Lymphatic System
## 2 Anatomy and Physiology of Murine Lymphatic System  
Available anatomical and physiological data provide the empirical basis for specifying the network structure of the lymphatic system in mice [23; 24] using the notion of a simple graph. A simple graph \(G=(V,E)\) is a pair of sets \(V\) and \(E\), with elements of \(V\) being vertices or nodes and \(E\) being edges [25]. The simple graph with edges oriented in only one direction is called an oriented graph. There are some variations in the number of lymph nodes, i.e., ranging from 22 to 36 as indicated in Table 1. A generalized graph of the murine LS, consisting of 88 nodes and 87 edges is shown in Figure 1. It was developed using anatomical descriptions from [23; 24]. The vertices of the graph refer to either the lymph nodes, outlet vertices with out-degree \(deg^{+}=0\) corresponding to the sink into jugular veins, the confluences of lymphatic vessels, or inlet vertices with in-degree \(deg^{-}=0\) corresponding to the collecting lymphatics of various body tissues. The geometric characteristics of the lymphatic vessels and the baseline parameters of the lymph flow through various parts of the LS network are detailed in Table 1. To set the pressure at the sink nodes \(p_{out}\), we used the estimate of the murine central venous pressure: 7.4 (5.9-8.9) cm H\({}_{2}\)O [26; 27]. Lymph viscosity is taken to be equal to 1.81 mPa\(\cdot\)s.  
The anatomy data enable specifying a simple graph of the murine LS. The adjacency matrix \(A\) of the LS graph is shown in Figure 2B. As the LS functions to transport the lymph from the drained tissues to the venous part of cardiovascular system, additional analysis is required to transfer the simple graph representation to a physiologically meaningful oriented graph of the LS. To generate an oriented graph mode of the LS, we used a combination of experimental stud ies on fluid dynamics in various parts of the LS in mice [28; 29; 30; 31] and computation of the lymph flow through the system in accordance to an overall mass balance using the Hagen-Poiseuille equations. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Property** & **Characteristic Value/Range** & **Commentaty/Source** \\ \hline Lymph nodes: & & \\ Number & 22 & (BALB/cAnNcr) [24] \\  & 28–36 & (DD/NIH) [23] \\ Diameter & 1–2.3 mm & (C57 Bl/6J, Nude, CB-17 SCID) [31; 32] \\  & 1–17.3 mm & (DD/NIH) [23] \\ \hline Thoracic duct: & & \\ Radius & 300 μm & [28] \\ Flow & 417–1250 \(\mu\)L/h & (10 mL/day for immobilized mice, 30 mL/day after movements) & [28] \\ Velocity & 410–1228 μm/s & [28] \\ \hline Vessels afferent to popliteal nodes: & & \\ Radius & 20–40 μm & [28; 30] \\ Flow & 0.3–3.4 \(\mu\)L/h & (mean flow = 0.3 \(\mu\)L/h [30]) [28; 29; 30] \\ Velocity & 37–186 μm/s & (mean velocity = 53 \(\pm\)16 μm/s [30]) [28; 29; 30] \\ \hline Collecting lymphatics in hind limbs: & & \\ Velocity & 50–100 μm/s & [31] \\ \hline Collecting lymphatics in ears: & & \\ Velocity & 0–400 μm/s & [30] \\ \hline Collecting lymphatics in the tail: & & \\ Velocity & 4.2 μm/s & [29] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physiological and anatomical properties of the murine lymphatic system. Figure 1: Oriented graph of the murine lymphatic system based on the anatomic data with 88 vertices and 87 edges. The vertices of the graph belong to four groups as detailed in the legend box: (1) lymph nodes (large blue), (2) outlet vertices with out-degree \(deg^{+}=0\) corresponding to the sink into jugular veins (red), (3) connectors, i.e., the confluences of lymphatic vessels (light blue), (4) inlet vertices with in-degree \(deg^{-}=0\) corresponding to the collecting lymphatics of various body tissues (orange). The vertex IDs and edge IDs are enumerated arbitrarily for correspondence with the matrix representation of the graph in Figure 2. The presented graph is specified in the CSV files containing the vertex and edge lists in Supplementary Materials.