# Developing Computational Geometry and Network Graph Models of Human Lymphatic System

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## Abstract

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## 1. Introduction

- the network can be modeled as a graph,
- no long-distance edges are allowed,
- multiple edges can enter a node,
- the nodes’ connections are locally acyclic,
- the lymphatic system network is a circular graph.

## 2. Anatomy of the Human Lymphatic System

#### 2.1. General Characteristics of the Lymphatic System: Invariant Features and Variable Characteristics

- inguinal lymph nodes, 4-20,
- axillary lymph nodes, 12–45,
- mesenteric lymph nodes, 64–404,
- lumbar lymph nodes, 1–17,
- cardiac gastric lymph nodes, 1–11,
- pyloric lymph nodes, 2–16,
- bronchopulmonary lymph nodes, 4–25,
- hepatic lymph nodes, 1–10,
- parasternal lymph nodes, 2–20.

#### 2.2. 3D Anatomy Model of the Human Male Lymphatic System

## 3. Computational Algorithm for Generating a Graph from a 3D Polygonal Model

- Construct a voxel-based approximation of the polygonal model of the vascular system. To do this, each polygon of the model to be approximated is covered by a regular grid with a quarter-size voxel step. After that, the coordinates of the grid points are converted to the coordinates of the voxel, and it is entered in the list of voxels approximating the model.
- Complete the void filling in the voxel-based approximation of the model under study, as shown in Figure 6.
- Divide the model into layers, and within the layers, isolate the connected groups of voxels.
- Identify the intersections of the voxel groups on adjacent layers and connect (with edges) their centers (nodes) to the branch of the emerging graph, as shown in Figure 8.

## 4. Graph Representation of the Human Lymphatic System

#### 4.1. Lymphatic System Network Graph Refinement

- each edge corresponds to a certain single vessel,
- each vertex belongs to one or several vessels,
- each lymph node from the 3D model is represented by one vertex in the 1D model,
- each lymphatic vessel is represented by at least one chain of vertices and edges.

#### 4.2. Topological Analysis of the Lymphatic System Network Graph

## 5. Towards a Large-Scale 3D Modeling of the Lymphatic System

#### 5.1. Anatomically-Distinct Parts of the Lymphatic System Model

#### 5.2. Lymphangion-Oriented Transformations of the Lymphatics Network Graph

- Take the skeleton of the basis network graph. Its vertices are invariant.
- Replace each edge of the skeleton by a parabolic spline (see Figure 16) with vertices of the basis graph specified as knots.
- Place equidistant points on each spline curve (the distance is defined separately for each anatomically-specific group of vessels) and link them into a chain by consequent edges.
- Replace each edge of the skeleton with a chain of equidistant vertices and edges.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Various types of connections characterizing lymphatic vessel networks (from [18]).

**Figure 2.**Schematic view of a lymph node with afferent and efferent lymphatics (from [18]).

**Figure 3.**Global view of the 3D polygonal surface model of the lymphatic system consisting of 270 lymph nodes, 443 lymph vessels, spleen and cisterna chyli. (

**Left**) Original PlasticBoy [20] model; (

**Right**) the model after processing.

**Figure 4.**Some mismatches of the lymphatics system model of PlasticBoy [20]. (

**Left**) Disconnected lymph node; (

**Right**) disconnected vessels.

**Figure 5.**3D polygonal surface model of the lymphatic system after removal of pendant vessels and differentiation of vessels. (

**Left**) Spleen and vessels draining to cisterna chyli; (

**Right**) head, neck, chest and armpits from PlasticBoy project [20].

**Figure 6.**Filling a closed area. Gray areas denote the voxels of the surface; green ones refer to voxels filling the internal voids of the approximated model.

**Figure 7.**Filling an open area. Gray areas denote voxels of the surface; green denotes voxels filling the internal voids of the approximating model; and red zones indicate contact with the boundary of the region to be approximated.

**Figure 8.**Searching the intersections in the voxel layers. Gray areas denote voxels of neighboring layers of the approximating model; red zones are the intersections of the voxel groups, which are used to construct the graph.

**Figure 9.**Three-dimensional objects in the lymphatic system model from the PlasticBoy project [20]. (

**Top**) The lymph node is an ellipsoid surface object, which has a spatial intersection with at least one lymphatic vessel. (

**Bottom**) The lymphatic vessel is a series of cylinder tubes. Polygonal 3D objects are colored in gray; 1D model vertices are colored in red; edges are colored in scarlet.

**Figure 10.**The basis graph of the lymphatics system network. Red color refers to graph vertices; black color refers to graph edges. The physical scale is in centimeters (cm).

**Figure 11.**The body-wide distribution of the vertices degrees in the basis network graph. (

**Left**) $deg=1$ for the inputs and outputs of the system, $deg=2$ for the inner point of the vessel, $deg\ge 3$ for vessel junctions; (

**right**) vertices of the basis graph representing the lymph nodes are marked in red.

**Figure 12.**The skeleton graph of the lymphatic system network. Red color stands for vertices, and black color designates the edges.

**Figure 13.**Topological characteristics of the skeleton network graph of the lymphatic system. (

**Left**) Distribution of vertices degrees; (

**Right**) distribution of edges lengths (note that these distributions refer to an intermediate model, and the final network graph model properties are presented in Section 5.2).

**Figure 14.**Anatomically-clustered vessel groups. Lymphatic vessel groups’ color code numbering refers to the following: head (0), neck (1), arms (2), chest (3), trunk (4), waist (5), legs (6). (

**Top**) Basis network graph; (

**Bottom**) skeleton network graph.

**Figure 15.**(

**Top**) The basis graph model in 3D color code specifies the equal edge lengths; (

**Middle**) degree distribution of vertices; (

**Bottom**) edge lengths’ distribution (note that these characteristics are for the intermediate model; the final uniform network graph model properties are presented in Section 5.2).

**Figure 16.**Parabolic spline interpolation. The curve representing lymphatic vessels is interpolated using given knots (

**top**) and then divided into equal parts by evenly-spaced points (

**bottom**).

**Figure 18.**(

**Top**) The uniform edge length graph model; (

**Middle**) degree distribution of vertices; (

**Bottom**) edge lengths’ distribution.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tretyakova, R.; Savinkov, R.; Lobov, G.; Bocharov, G.
Developing Computational Geometry and Network Graph Models of Human Lymphatic System. *Computation* **2018**, *6*, 1.
https://doi.org/10.3390/computation6010001

**AMA Style**

Tretyakova R, Savinkov R, Lobov G, Bocharov G.
Developing Computational Geometry and Network Graph Models of Human Lymphatic System. *Computation*. 2018; 6(1):1.
https://doi.org/10.3390/computation6010001

**Chicago/Turabian Style**

Tretyakova, Rufina, Rostislav Savinkov, Gennady Lobov, and Gennady Bocharov.
2018. "Developing Computational Geometry and Network Graph Models of Human Lymphatic System" *Computation* 6, no. 1: 1.
https://doi.org/10.3390/computation6010001