# Critical Issues in Modelling Lymph Node Physiology

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## Abstract

**:**

## 1. Introduction

## 2. Major Structural Elements of a Paradigmatic Lymph Node

## 3. Computational Models of FRC- and Blood Vascular Networks of Lymph Nodes

#### 3.1. Cellular Potts Modelling of the FRC Network

#### 3.2. Modelling Blood Vascular Network

#### 3.2.1. Initial Data

#### 3.3. Algorithm of Network Graph Generation

- Step 1.
- Graph topology organisation. In this step we generate the basis points and edges of connections.
- Step 2.
- Local edge length optimization. In this step we use the algorithm from [7] just for a local (i.e., for neighbouring nodes) adjustment of the mismatch of the model and target graph edges lengths. In this and the next steps, the following parameter from Step 1 is used:
**blos**(length of segmentation of the vessels). It’s the canonical length of segments of the vessels graph. - Step 3.
- Global network structure optimization. In this step we use a modified algorithm from [7] for (i) minimization of the edge length deviation from the real data for all neighbouring nodes; (ii) pushing apart disconnected nodes from each other to prevent merger of the vessels; and (iii) shifting the nodes away from the prohibited domains associated with other LN structures.

// Initialise the data arrays |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\u201clength\phantom{\rule{3.33333pt}{0ex}}distribution\u201d\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}\mathbf{ld};$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\{5\phantom{\rule{3.33333pt}{0ex}}5\phantom{\rule{3.33333pt}{0ex}}5\phantom{\rule{3.33333pt}{0ex}}6\}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}integer\phantom{\rule{3.33333pt}{0ex}}array\phantom{\rule{3.33333pt}{0ex}}\mathbf{bf};$ |

// Specify the segmentation accuracy, vessels radius and decreasing, processing zone size |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\u201cinitial\phantom{\rule{3.33333pt}{0ex}}input/output\phantom{\rule{3.33333pt}{0ex}}vessels\phantom{\rule{3.33333pt}{0ex}}diameter\u201d\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{vd};$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}4\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}4.0\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{blos};\phantom{\rule{3.33333pt}{0ex}}//\phantom{\rule{4.pt}{0ex}}\mathrm{length}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{segmentation}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{vessels},\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}5\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}{2}^{\frac{1}{2}}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{rf};\phantom{\rule{3.33333pt}{0ex}}//\phantom{\rule{4.pt}{0ex}}\mathrm{coefficient}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{vessels}\phantom{\rule{4.pt}{0ex}}\mathrm{radius}\phantom{\rule{4.pt}{0ex}}\mathrm{decreasing}$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}6\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\u201cwork\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}radius\u201d\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}\mathbf{R};\phantom{\rule{3.33333pt}{0ex}}//\phantom{\rule{4.pt}{0ex}}\mathrm{simplify}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{constructing}$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}7\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}graph\phantom{\rule{3.33333pt}{0ex}}structure\phantom{\rule{3.33333pt}{0ex}}\mathbf{t};$ |

// Attach the input and output vessels |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}8\phantom{\rule{3.33333pt}{0ex}}insert\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\u201cinput\phantom{\rule{3.33333pt}{0ex}}vessel\u201d\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\u201coutput\phantom{\rule{3.33333pt}{0ex}}vessel\u201d\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{t};$ // simple lines, splitted into 100 segments |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}9\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}(integer\phantom{\rule{3.33333pt}{0ex}}\mathbf{j}=1\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}NC)\phantom{\rule{3.33333pt}{0ex}}begin$ |

// In this loop, we create new vessels, growing from input and output vessels |

$10\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{vdl}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{vd};$ |

$11\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{ld};$ |

$12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}/\phantom{\rule{3.33333pt}{0ex}}\mathbf{blos};$ |

$13\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}integer\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl};$ |

$14\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc};$ |

$15\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}\mathbf{sl}\phantom{\rule{3.33333pt}{0ex}}>\phantom{\rule{3.33333pt}{0ex}}0.5\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}1;$ |

// sc defines the number of segments for current generating line |

$16\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pin}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}input\phantom{\rule{3.33333pt}{0ex}}vessel;$ |

$17\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pout}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}output\phantom{\rule{3.33333pt}{0ex}}vessel;$ |

$18\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{11}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$19\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{12}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$20\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{21}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$21\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{22}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

// we used points pmXX to avoid helical structures while the second and |

// third parts of graph construction. |

$22\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}init\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}that\phantom{\rule{3.33333pt}{0ex}}goes\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{pin}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{12}\phantom{\rule{3.33333pt}{0ex}}through\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{11}\phantom{\rule{3.33333pt}{0ex}}splitted\phantom{\rule{3.33333pt}{0ex}}into\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}segments;$ |

$23\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}assign\phantom{\rule{3.33333pt}{0ex}}\mathbf{vdl}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}diameter;$ |

$24\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}insert\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{t};$ |

$25\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}init\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{2}\phantom{\rule{3.33333pt}{0ex}}that\phantom{\rule{3.33333pt}{0ex}}goes\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{pout}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{22}\phantom{\rule{3.33333pt}{0ex}}through\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{21}\phantom{\rule{3.33333pt}{0ex}}splitted\phantom{\rule{3.33333pt}{0ex}}into\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}segments;$ |

$26\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}assign\phantom{\rule{3.33333pt}{0ex}}\mathbf{vdl}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{2}\phantom{\rule{3.33333pt}{0ex}}diameter;$ |

$27\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}insert\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{2}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{t};$ |

$28\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}execute\phantom{\rule{3.33333pt}{0ex}}code\phantom{\rule{3.33333pt}{0ex}}lines\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}18\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}27;$ |

$29\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}integer\phantom{\rule{3.33333pt}{0ex}}\mathbf{nbf}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{bf};$ |

$30\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}(integer\phantom{\rule{3.33333pt}{0ex}}\mathbf{i}=1\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{nbf})\phantom{\rule{3.33333pt}{0ex}}begin$ |

// In this cycle in each loop we create two sub-vessels for each |

// couple [lx1, lx2], created while previous loop |

$31\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}real\phantom{\rule{3.33333pt}{0ex}}\mathbf{vdl}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{\mathbf{vd}}{{\mathbf{rf}}^{i}};$ |

$32\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{12}\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pin};$ |

$33\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{22}\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{2}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pout};$ |

$34\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}if(\mathbf{i}\phantom{\rule{3.33333pt}{0ex}}<\phantom{\rule{3.33333pt}{0ex}}\mathbf{nbf})\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}execute\phantom{\rule{3.33333pt}{0ex}}code\phantom{\rule{3.33333pt}{0ex}}lines\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}15,\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}18\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}26;$ |

$35\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}if(\mathbf{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\mathbf{nbf})\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}begin$ |

// here we connect the inner and outer parts of vessel |

$36\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}execute\phantom{\rule{3.33333pt}{0ex}}code\phantom{\rule{3.33333pt}{0ex}}lines\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}15;$ |

$37\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$38\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{2}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$39\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}set\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{3}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}random\phantom{\rule{3.33333pt}{0ex}}point\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}sphere\phantom{\rule{3.33333pt}{0ex}}with\phantom{\rule{3.33333pt}{0ex}}radius\phantom{\rule{3.33333pt}{0ex}}R;$ |

$40\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}init\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}that\phantom{\rule{3.33333pt}{0ex}}goes\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}\mathbf{pin}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{pout}\phantom{\rule{3.33333pt}{0ex}}through\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{1},\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{2},\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{3}\phantom{\rule{3.33333pt}{0ex}}splitted\phantom{\rule{3.33333pt}{0ex}}into\phantom{\rule{3.33333pt}{0ex}}\mathbf{sc}\phantom{\rule{3.33333pt}{0ex}}segments;$ |

$41\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}assign\phantom{\rule{3.33333pt}{0ex}}\mathbf{vdl}\phantom{\rule{3.33333pt}{0ex}}as\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}diameter;$ |

$42\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}insert\phantom{\rule{3.33333pt}{0ex}}line\phantom{\rule{3.33333pt}{0ex}}\mathbf{lx}\mathbf{1}\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}\mathbf{t};$ |

$43\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}end$ |

$44\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}execute\phantom{\rule{3.33333pt}{0ex}}code\phantom{\rule{3.33333pt}{0ex}}lines\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}34\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}43;$ |

$\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}//\phantom{\rule{3.33333pt}{0ex}}Note:\phantom{\rule{3.33333pt}{0ex}}inside\phantom{\rule{3.33333pt}{0ex}}cycle\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}all\phantom{\rule{3.33333pt}{0ex}}\mathbf{i}>1\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{12},\phantom{\rule{3.33333pt}{0ex}}\mathbf{pm}\mathbf{22}\phantom{\rule{3.33333pt}{0ex}}should\phantom{\rule{3.33333pt}{0ex}}be\phantom{\rule{3.33333pt}{0ex}}used\phantom{\rule{3.33333pt}{0ex}}from\phantom{\rule{3.33333pt}{0ex}}previous\phantom{\rule{3.33333pt}{0ex}}loop\phantom{\rule{3.33333pt}{0ex}}of\phantom{\rule{3.33333pt}{0ex}}cycle.$ |

$45\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}end$ |

$46\phantom{\rule{3.33333pt}{0ex}}end$ |

#### 3.4. Integrative Geometric Model of Vascular Networks

## 4. Lymph Dynamics in Conduit Elements of FRC Network

#### 4.1. Transport Through a Single Conduit

#### 4.2. Diffusive Transport in an FRC Conduit

^{®}and a cuboid block of a generated FRC network consisting of 6927 edges and 3374 nodes was subjected to analysis. Contiguous nodes were made co-planar to give a domain of known dimensions, $151\times 193\times 187\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m. Pressure or concentration gradients were placed across the block, a linear system formed and solved for the fluid permeability or diffusive flux using a direct solver. One species often considered in lymphatics is diphtheria toxin with a diffusivity $D=6.2\times {10}^{-7}$ cm/s [36] and a molecular weight of 58 kDa [37]. For the network under consideration a 2 ng/g imposed gradient across the first axis of the network gave a diffusive flux of $2.68\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$g/s. Whilst for an imposed pressure gradient of 6 cm H${}_{2}$O gives a fluid flux of $0.0214\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{3}$/s. The fluid flux would have to be ∼10 orders higher to have a similar contribution to mass transport as diffusion.

## 5. Modelling Lymph Flow in Conduit System of FRC Network in Idealized LN

#### 5.1. Normal FRC Network

#### 5.2. Disrupted FRC Network

## 6. Percolation Robustness of the FRC Network

#### 6.1. Graph Measures

#### 6.2. Percolation Threshold

## 7. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FRC | Fibroblastic reticular cell |

SCS | Subcapsular sinus |

LN | lymph node |

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**Figure 1.**(

**a**) The initialized reticular network as the system of conduits of given topology. State of the simulation at MCS = 0; (

**b**) The reticular network reached its target volume. State of the simulation at MCS = 20.

**Figure 2.**Blood vessels length distribution summarized from [21].

**Figure 3.**(

**left**) An artificially generated 3D blood vessel network (for a sphere with a diameter of about 200 $\mathsf{\mu}$m); (

**right**) Vessels length distribution of the computationally constructed blood microvascular model.

**Figure 5.**FRC network. (

**left**) Network graph; (

**center**) Node degree distribution; (

**right**) Edges lengths distribution.

**Figure 6.**Integrated model of vascular vessels and FRC network (for a sphere of about 200 $\mathsf{\mu}$m).

**Figure 7.**Diagram showing the packaged bed model of an FRC conduit. Reticular fibres are hexagonally close packed and encased in FRCs. ${r}_{f}$ is the radius of the fibres.

**Figure 8.**(

**left**) Table of area independent parameter χ against degrees of freedom in the FEA model; (

**right**) Numerically computed velocity profile of flow through a curvilinear triangle with area ${A}_{p}$.

**Figure 9.**Schematic view of the SCS pressure distribution for two types of pressure boundary conditions (1) gradient and (2) constant. Branches emanating from the inner SCS circle represent the input and output edges of the network.

**Figure 10.**Pressure values depending on coordinate. (

**left**) for gradient boundary conditions; (

**right**) for constant boundary conditions. Boundary points are highlighted in red.

**Figure 11.**Scheme of network edges deletion from graph. (

**left**) Normal network structure; (

**right**) Randomly removed (rm) edges are marked in pink and non-functional ones are indicated as (nf).

**Figure 12.**Pressure gradient for (

**a**–

**f**) 0%–50% edges removed (constant boundary conditions). Boundary points are highlighted in red.

**Figure 13.**Pressure gradient for (

**a**–

**i**) 0%–80% edges removed (gradient boundary conditions). Boundary points are highlighted in red.

**Table 1.**Parameters of Cellular Potts Model used to generate the reticular network (a.u.e. stands for arbitrary units of energy).

Parameter | Description | Value |
---|---|---|

${l}_{px}$ | length of the voxel | 0.3 $\mathsf{\mu}$m |

${L}_{x}\times {L}_{y}\times {L}_{z}$ | sizes of computational domain | $590\phantom{\rule{4.pt}{0ex}}\mathrm{px}\times 658\phantom{\rule{4.pt}{0ex}}\mathrm{px}\times 670\phantom{\rule{4.pt}{0ex}}\mathrm{px}$ |

${d}_{conduit}$ | diameter of the conduits | 1.0 $\mathsf{\mu}$m |

${N}_{FRCs}$ | number of FRCs in domain | 3374 |

${N}_{FRCs}{V}_{target}(FRC)/{L}^{3}$ | volume fraction of reticular network | 4% |

${J}_{i,j}$ | adhesion energies | 0 a.u.e. |

${\lambda}_{FRC}$ | spring modulus of FRCs | 10 a.u.e./px${}^{6}$ |

${T}_{max}$ | amplitude of intrinsic motility | 2 a.u.e. |

$2s$ | characteristic diameter of FRC body | 12 px |

FRCn | Vascular Network | |
---|---|---|

Surface area | 1,131,209 $\mathsf{\mu}$m${}^{2}$ | 61,264 $\mathsf{\mu}$m${}^{2}$ |

Relative volume | 7.98% | 1.71% |

Damage (%) | Nodes | Edges | n-f Edges | Inputs | Outputs | Relative Outflow | Relative Sum. Flow |
---|---|---|---|---|---|---|---|

0 | 3694 | 7253 | 0 | 164 | 156 | 1.0 | 1.0 |

10 | 3671 | 6528 | 29 | 156 | 147 | 0.82 | 0.84 |

20 | 3608 | 5802 | 108 | 143 | 132 | 0.615 | 0.653 |

30 | 3454 | 5077 | 1792 | 117 | 105 | 0.414 | 0.386 |

40 | 3135 | 4352 | 2138 | 94 | 78 | 0.16 | 0.149 |

50 | 2713 | 3626 | 2365 | 55 | 89 | 0.044 | 0.019 |

60 | 2202 | 2901 | 2443 | 16 | 85 | 0.0 | 0.0 |

Damage (%) | Nodes | Edges | n-f Edges | Inputs | Outputs | Relative Outflow | Relative Sum. Flow |
---|---|---|---|---|---|---|---|

0 | 3694 | 7253 | 0 | 169 | 151 | 1.0 | 1.0 |

10 | 3671 | 6528 | 27 | 157 | 146 | 0.854 | 0.845 |

20 | 3608 | 5802 | 110 | 140 | 135 | 0.678 | 0.66 |

30 | 3454 | 5077 | 226 | 109 | 113 | 0.49 | 0.398 |

40 | 3135 | 4352 | 346 | 85 | 87 | 0.33 | 0.23 |

50 | 2713 | 3626 | 454 | 68 | 76 | 0.26 | 0.149 |

60 | 2202 | 2901 | 599 | 46 | 55 | 0.185 | 0.095 |

70 | 1623 | 2176 | 512 | 34 | 41 | 0.135 | 0.074 |

80 | 1062 | 1451 | 254 | 13 | 25 | 0.072 | 0.039 |

90 | 513 | 725 | 76 | 5 | 13 | 0.028 | 0.013 |

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**MDPI and ACS Style**

Grebennikov, D.; Van Loon, R.; Novkovic, M.; Onder, L.; Savinkov, R.; Sazonov, I.; Tretyakova, R.; Watson, D.J.; Bocharov, G.
Critical Issues in Modelling Lymph Node Physiology. *Computation* **2017**, *5*, 3.
https://doi.org/10.3390/computation5010003

**AMA Style**

Grebennikov D, Van Loon R, Novkovic M, Onder L, Savinkov R, Sazonov I, Tretyakova R, Watson DJ, Bocharov G.
Critical Issues in Modelling Lymph Node Physiology. *Computation*. 2017; 5(1):3.
https://doi.org/10.3390/computation5010003

**Chicago/Turabian Style**

Grebennikov, Dmitry, Raoul Van Loon, Mario Novkovic, Lucas Onder, Rostislav Savinkov, Igor Sazonov, Rufina Tretyakova, Daniel J. Watson, and Gennady Bocharov.
2017. "Critical Issues in Modelling Lymph Node Physiology" *Computation* 5, no. 1: 3.
https://doi.org/10.3390/computation5010003