# Graph Theory for Modeling and Analysis of the Human Lymphatic System

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## Abstract

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## 1. Introduction

## 2. Graph Models Based on Anatomical Data

#### 2.1. Reddy’s Directed Graph Model of the HLS

#### 2.2. Plastic Boy-Derived Graph Model of the HLS

#### 2.2.1. Undirected Graph of the HLS

- coordinates of all vertices in the graph are scaled to correspond to the basal 1750 mm height of a man;
- the right hand and the head right side vertices are attached to the major part of HLS graph by adding the edge from vertex 987 to the vertex 993;
- single output node (sink): (1) two vertices are added (numbered as 995 and 996) with vertex 996 being the output of the system, (2) three edges, i.e., between nodes 993–995, 994–995 and 995–996 were added;
- we iterate over graph vertices and search for irrelevant vertices that meet all the following conditions:
- (a)
- their degree is equal to two;
- (b)
- they are not lymph nodes;
- (c)
- their neighbor vertices ${v}_{a}$, ${v}_{b}$ are not connected with each other by an edge;

at each iteration, the irrelevant vertex was removed, and the edge ${v}_{a}-{v}_{b}$ was created.

#### 2.2.2. Lymph Flow Analysis for Specifying the Directed Graph

- there are no other sinks of the lymph except for the exit vertex number 996: ${V}_{out}=\left\{996\right\}$;
- all vertices of the graph that have only one connection with other vertices (except for the vertex 996) are the points of lymph entry (collecting lymphatics) into the lymphatic network;
- the pressure at all inflow vertices is adjusted to provide the lymph flow rate on the input edges to be equal to the output flow rate of the system divided by number of vertices with zero input edges;
- the radii of all the vessels are set to be same, equal to $1\phantom{\rule{3.33333pt}{0ex}}$mm;
- the constant viscosity is set in the lymphatic network.

## 3. Computational Algorithm for Generating a Random Rule-Based Directed Graph of the HLS

Algorithm 1: Generation of a rule-based directed graph of the HLS. |

## 4. Parameters of the Rule-Based Algorithm Providing Best Match to the Anatomy Data Graph

- The number of input nodes ${N}_{inp}$, i.e., the number of nodes with degree 1 and out-degree 0.
- Maximum degree of graph ${\Delta}_{G}$, i.e., the maximum degree of its vertices.
- Girth of the graph g, which is the length of the shortest (undirected) cycle in the graph.
- Diameter, i.e., the longest geodesic distance (in other terms, maximum eccentricity of any vertex),$$D=\underset{v\in V}{max}\u03f5\left(v\right)=\underset{v\in V}{max}\underset{u\in V}{max}d(u,v),$$
- Radius of the graph (minimum eccentricity of any vertex),$$r=\underset{v\in V}{min}\u03f5\left(v\right)=\underset{v\in V}{min}\underset{u\in V}{max}d(u,v).$$
- Average path length (mean geodesic distance),$${l}_{G}=\frac{1}{n(n-1)}\sum _{u,v\in V,\phantom{\rule{3.33333pt}{0ex}}u\ne v}d(u,v).$$
- The energy and the spectral radius of the graph are defined as follows,$$En\left(A\right)=\sum _{j=1}^{n}\mid {\lambda}_{j}\mid ,\phantom{\rule{1.em}{0ex}}\rho \left(A\right)=max\{|{\lambda}_{j}|\},$$
- Edge density of the graph, i.e., the number of edges divided by the number of all possible edges,$${\rho}_{d}=\frac{m}{n(n-1)}.$$
- The clustering coefficient C (transitivity) measures the probability that two neighbors of a vertex are connected. It can be computed as function of adjacency matrix A:$$C\left(A\right)=\frac{{\sum}_{i=1,j=1,k=1}^{n,n,n}{a}_{ij}\xb7{a}_{jk}\xb7{a}_{ki}}{{\sum}_{i=1}^{n}(\left({\sum}_{j=1}^{n}{a}_{ij}\right)\xb7(\left({\sum}_{j=1}^{n}{a}_{ij}\right)-1))}.$$
- Number of separators ${n}_{sep}$, i.e., the vertices removal of which disconnects the graph.
- The robustness R of the graph can be defined as the fraction of peripheral vertices that retained the connection with the output vertex after removing 5% of the vertices selected randomly, averaged over $k=1,\dots ,{N}_{r}$ removal realizations. Given adjacency matrix ${\tilde{A}}_{k}$ of the k-th realization of the graph, indices ${i}_{1},\dots ,{i}_{{n}_{inp}}$ of its input vertices and index o of the output vertex, the robustness is computed as$$R=\frac{1}{{N}_{r}}\sum _{k=1}^{{N}_{r}}{R}_{k},\phantom{\rule{1.em}{0ex}}{R}_{k}=\sum _{i=1}^{{n}_{inp}}\frac{F({\tilde{A}}_{k},{i}_{i},o,n)}{{n}_{inp}},\phantom{\rule{1.em}{0ex}}F({\tilde{A}}_{k},{i}_{i},o,n)=\left\{\begin{array}{c}1,\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}{\sum}_{j=1}^{n-1}|{({\tilde{A}}_{k}^{j})}_{{i}_{i},o}|>0,\hfill \\ 0,\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}{\sum}_{j=1}^{n-1}|{({\tilde{A}}_{k}^{j})}_{{i}_{i},o}|=0,\hfill \end{array}\right.$$
- Topological diversity of the vertices as a function of the Shannon entropy associated with flow rates through the incident edges,$${D}_{flow}\left({v}_{i}\right)=\frac{H\left({v}_{i}\right)}{log\left(k\right)}=\frac{-{\sum}_{j=1}^{k}{p}_{ij}log\left({p}_{ij}\right)}{log\left(k\right)},\phantom{\rule{1.em}{0ex}}{p}_{ij}=\frac{\left|{Q}_{ij}\right|}{{\sum}_{j=1}^{k}\left|{Q}_{ij}\right|},$$

## 5. Comparative Analysis of the HLS Graph Models

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HLS | Human Lymphatic System |

LNs | lymph nodes |

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**Figure 1.**Oriented graph of a simple network model of the human lymphatic system suggested by Reddy et al. [3] with 29 nodes and 28 edges. The black node with out-degree $de{g}^{+}=0$ corresponds to jugular vein (sink).

**Figure 2.**Graph of a human lymphatic system based on anatomical $Plasticboy$ data [5] with 996 vertices and 1117 edges (graph files are provided in Supplementary Materials). (

**A**) Face- and (

**B**) side views. Blue colored vertices represent 272 lymph nodes. The black vertex corresponds to the output node.

**Figure 3.**Statistical distributions of the anatomy data-based HLS graphs. (

**A**) Statistics of the data-based graph, (

**B**) of the Reddy’s graph. Statistics include the distributions of total degrees of the vertices, distributions of in- and out-degrees, as well as the edge-length distribution and degree distribution of the vertices that represent the lymph nodes (LNs) for data-based graph.

**Figure 4.**Steady-statedistribution of lymph flows in the HLS graph model. The output sink node is presented as black circle.

**Figure 5.**Oriented graph model of the HLS based on the analysis of the lymph flow directions in the network. (

**A**) Spatial view of the graph with numbered vertices (face-view projection of 3D coordinates). (

**B**) View of the graph on a plane with no-overlapping layout. The 2D layout was obtained using spring-based model algorithm [19] implemented in package igraph, followed by some manual tweaking to space out the vertices. The colored subgraphs correspond to different parts of the body, i.e., arms (dark and light blue), head/neck (magenta), torso (light green), legs (red and gray). LNs are represented by larger circles.

**Figure 6.**Adjacency matrix of the oriented data-based graph of human lymphatic system shown in Figure 5. Colors represent the lengths ${L}_{ij}$ of the edges ${e}_{ij}$.

**Figure 7.**Two examples (

**A**,

**B**) of algorithmically generated realizations of rule-based directed random graphs of the HLS. The output nodes are marked with black color. Algorithm parameters: ${N}_{v}=996$, ${N}_{inp}=357$, ${N}_{l}=41$, ${P}_{e}=0.035$, ${P}_{o}=0.21$. The metrics of topological fitness to anatomy-based graphs presented in figure are defined in Section 4.

**Figure 8.**Degree distribution of the rule-based graph presented in Figure 7A.

**Figure 9.**Directed graph adjacency matrix of the rule-based random graph presented in Figure 7A.

**Figure 10.**The landscape of topological fitness of rule-based graphs over different values of algorithm parameters. (

**A**) Image of ${log}_{10}\left({\Phi}_{mean}\right)$ values as function of varying parameters ${P}_{e}$ and ${P}_{o}$ and fixed ${N}_{l}=41$. The mean values are calculated in each image cell over 640 graphs realizations of rule-based graphs generated with corresponding parameter sets. (

**B**) Dependence of objective function on ${P}_{o}$ with fixed ${P}_{e}=0.035$, ${N}_{l}=41$. The statistics on values of objective function (14) are calculated over 10,000 realizations of the algorithm for each value of ${P}_{o}$.

**Figure 11.**Histograms of the robustness of (

**A**) Reddy’s graph, (

**B**) anatomical data-based graph and (

**C**) rule-based random graph presented in Figure 7A.

**Figure 12.**Histograms of the flow diversity of the inner vertices of anatomy-based graphs: (

**A**) for data-based graph, (

**B**) for Reddy’s graph.

**Table 1.**Summary statistics for Reddy’s, anatomy-based and rule-based graphs characterizing their topological properties. For algorithmically generated graphs, we present the statistics obtained over 10,000 graphs generated with algorithm parameters ${P}_{e}=0.035,\phantom{\rule{3.33333pt}{0ex}}{P}_{o}=0.21,\phantom{\rule{3.33333pt}{0ex}}{N}_{l}=41$. Robustness was calculated over 1000 algorithm realizations, with 1000 removal attempts for each graph.

Lymphatic Vascular System Graph Model | Reddy’s Model | Anatomy-Based Model | Rule-Based Model (Mean) | Rule-Based Model (SD) | Rule-Based m. (Min-Max Range) |
---|---|---|---|---|---|

$G(n,m)$ | $(29,28)$ | $(996,1117)$ | $(996,1029)$ | $(0,6.4)$ | (996, 1009–1056) |

${N}_{inp}$ | 16 | 357 | 357 | 0 | (357–357) |

Maximum degree, ${\Delta}_{G}$ | 4 | 8 | 16 | 1.58 | (8–21) |

Girth, g | 0 | 3 | 4 | 0.9 | (3–14) |

Diameter, D | 5 | 40 | $39.96$ | $0.22$ | (37–40) |

Radius, r | 4 | 30 | 28 | $2.3$ | (23–38) |

Average path length, ${l}_{G}$ | $2.46$ | $12.79$ | $15.3$ | $0.86$ | (13.6–18) |

Energy, $En$ | $32.1$ | $1224.5$ | 1190 | $4.9$ | (1173–1203) |

Spectral radius, $\rho $ | $2.58$ | $3.51$ | $4.18$ | $0.19$ | (3.28–4.72) |

Edge density, ${\rho}_{d}$ | $0.034$ | $0.001127$ | $0.001038$ | $6.4\times {10}^{-6}$ | (0.00102–0.00107) |

Clustering coefficient, C | 0 | $0.027$ | $0.0004$ | $0.0008$ | (0–0.0036) |

Number of separators, ${n}_{sep}$ | 13 | 401 | 496 | 20 | (437–548) |

Robustness, R | $0.7$ | $0.6$ | $0.66$ | $0.05$ | (0.45–0.77) |

Average flow diversity, $\overline{{D}_{flow}}$ | 0.908 | 0.996 | − | − | − |

**Table 2.**Summary statistics for the subgraphs of the anatomy-based graph which correspond to different parts of the body.

Subgraph | Left Arm | Right Arm | Head & Neck | Torso | Left Leg | Right Leg |
---|---|---|---|---|---|---|

$G(n,m)$ | $(149,181)$ | $(141,170)$ | $(198,208)$ | $(168,183)$ | $(163,176)$ | $(177,192)$ |

Number of input nodes | 46 | 45 | 64 | 68 | 66 | 71 |

Number of output nodes | 1 | 1 | 2 | 1 | 1 | 1 |

Maximum degree, ${\Delta}_{G}$ | 8 | 8 | 4 | 8 | 5 | 4 |

Girth, g | 3 | 3 | 4 | 3 | 3 | 3 |

Diameter, D | 23 | 22 | 21 | 18 | 22 | 22 |

Radius, r | 10 | 9 | 13 | 9 | 13 | 13 |

Average path length, ${l}_{G}$ | $8.77$ | $8.37$ | $7.24$ | $7.02$ | $8.95$ | $9.55$ |

Energy, $En$ | $192.22$ | $180.6$ | $236.24$ | $164.4$ | $196.57$ | $214.1$ |

Spectral radius, $\rho $ | $3.3$ | $3.32$ | $2.95$ | $3.51$ | $2.92$ | $3.08$ |

Edge density, ${\rho}_{d}$ | $0.0082$ | $0.0086$ | $0.0053$ | $0.0065$ | $0.0067$ | $0.0062$ |

Clustering coefficient, C | $0.049$ | $0.035$ | 0 | $0.055$ | $0.01$ | $0.009$ |

Number of separators, ${n}_{sep}$ | 41 | 41 | 46 (left), 46 (right) | 81 | 66 | 72 |

Number of LNs | 36 | 36 | 51 (left), 51 (right) | 59 | 21 | 18 |

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**MDPI and ACS Style**

Savinkov, R.; Grebennikov, D.; Puchkova, D.; Chereshnev, V.; Sazonov, I.; Bocharov, G.
Graph Theory for Modeling and Analysis of the Human Lymphatic System. *Mathematics* **2020**, *8*, 2236.
https://doi.org/10.3390/math8122236

**AMA Style**

Savinkov R, Grebennikov D, Puchkova D, Chereshnev V, Sazonov I, Bocharov G.
Graph Theory for Modeling and Analysis of the Human Lymphatic System. *Mathematics*. 2020; 8(12):2236.
https://doi.org/10.3390/math8122236

**Chicago/Turabian Style**

Savinkov, Rostislav, Dmitry Grebennikov, Darya Puchkova, Valery Chereshnev, Igor Sazonov, and Gennady Bocharov.
2020. "Graph Theory for Modeling and Analysis of the Human Lymphatic System" *Mathematics* 8, no. 12: 2236.
https://doi.org/10.3390/math8122236