### Abstract

A vascular network is often represented by a Reeb graph, which is a topological skeleton, and graph theory has been widely applied to analyze properties of a vascular network. A Reeb graph model for a vascular network is obtained by assigning the branch points of the network to be the vertices of the graph and the vessels between branch points to be the edges of the graph. Vascular networks develop by way of angiogenesis, a growth process that involves the biological mechanisms of vessel sprouting (budding) and splitting (intussusception). Vascular networks develop by way of two biological mechanisms of vessel sprouting (budding) and splitting (intussusception). According to a graph theory modeling of two vascular network growth mechanisms, all nodes in the Reeb graph must be cubic in degree except for two special nodes: the afferent (A) and efferent (E) nodes. We define that a vascular network is cubic if all internal nodes are cubic in degree. We consider six normal adult rat renal glomerular networks and use their reeb graphs already constructed and published in the literature. We observe that five of them contain internal vertices of degree higher than three. Branch points in vascular networks may appear to be of a higher degree if the imaging resolution cannot differentiate between blood vessels that are very close in proximity. Here, we propose a random graph theory model that edits a non-cubic vascular network into a cubic graph. We observe that the edited cubic graph from a non-cubic vascular network has the similar size and order as the one cubic vascular network. Renal glomerular network, random graph process, graph degree, graph invariants, pattern matching.

Original language | English (US) |
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Title of host publication | Medical Imaging 2006 |

Subtitle of host publication | Physiology, Function, and Structure from Medical Images |

Volume | 6143 II |

DOIs | |

State | Published - Jun 30 2006 |

Event | Medical Imaging 2006: Physiology, Function, and Structure from Medical Images - San Diego, CA, United States Duration: Feb 12 2006 → Feb 14 2006 |

### Other

Other | Medical Imaging 2006: Physiology, Function, and Structure from Medical Images |
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Country | United States |

City | San Diego, CA |

Period | 2/12/06 → 2/14/06 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Engineering(all)

### Cite this

*Medical Imaging 2006: Physiology, Function, and Structure from Medical Images*(Vol. 6143 II). [61432M] https://doi.org/10.1117/12.648766

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*Medical Imaging 2006: Physiology, Function, and Structure from Medical Images.*vol. 6143 II, 61432M, Medical Imaging 2006: Physiology, Function, and Structure from Medical Images, San Diego, CA, United States, 2/12/06. https://doi.org/10.1117/12.648766

**On the assumption of cubic graphs of vascular networks.** / Cha, Sung Hyuk; Chang, Sukmoon; Gargano, Michael L.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - On the assumption of cubic graphs of vascular networks

AU - Cha, Sung Hyuk

AU - Chang, Sukmoon

AU - Gargano, Michael L.

PY - 2006/6/30

Y1 - 2006/6/30

N2 - A vascular network is often represented by a Reeb graph, which is a topological skeleton, and graph theory has been widely applied to analyze properties of a vascular network. A Reeb graph model for a vascular network is obtained by assigning the branch points of the network to be the vertices of the graph and the vessels between branch points to be the edges of the graph. Vascular networks develop by way of angiogenesis, a growth process that involves the biological mechanisms of vessel sprouting (budding) and splitting (intussusception). Vascular networks develop by way of two biological mechanisms of vessel sprouting (budding) and splitting (intussusception). According to a graph theory modeling of two vascular network growth mechanisms, all nodes in the Reeb graph must be cubic in degree except for two special nodes: the afferent (A) and efferent (E) nodes. We define that a vascular network is cubic if all internal nodes are cubic in degree. We consider six normal adult rat renal glomerular networks and use their reeb graphs already constructed and published in the literature. We observe that five of them contain internal vertices of degree higher than three. Branch points in vascular networks may appear to be of a higher degree if the imaging resolution cannot differentiate between blood vessels that are very close in proximity. Here, we propose a random graph theory model that edits a non-cubic vascular network into a cubic graph. We observe that the edited cubic graph from a non-cubic vascular network has the similar size and order as the one cubic vascular network. Renal glomerular network, random graph process, graph degree, graph invariants, pattern matching.

AB - A vascular network is often represented by a Reeb graph, which is a topological skeleton, and graph theory has been widely applied to analyze properties of a vascular network. A Reeb graph model for a vascular network is obtained by assigning the branch points of the network to be the vertices of the graph and the vessels between branch points to be the edges of the graph. Vascular networks develop by way of angiogenesis, a growth process that involves the biological mechanisms of vessel sprouting (budding) and splitting (intussusception). Vascular networks develop by way of two biological mechanisms of vessel sprouting (budding) and splitting (intussusception). According to a graph theory modeling of two vascular network growth mechanisms, all nodes in the Reeb graph must be cubic in degree except for two special nodes: the afferent (A) and efferent (E) nodes. We define that a vascular network is cubic if all internal nodes are cubic in degree. We consider six normal adult rat renal glomerular networks and use their reeb graphs already constructed and published in the literature. We observe that five of them contain internal vertices of degree higher than three. Branch points in vascular networks may appear to be of a higher degree if the imaging resolution cannot differentiate between blood vessels that are very close in proximity. Here, we propose a random graph theory model that edits a non-cubic vascular network into a cubic graph. We observe that the edited cubic graph from a non-cubic vascular network has the similar size and order as the one cubic vascular network. Renal glomerular network, random graph process, graph degree, graph invariants, pattern matching.

UR - http://www.scopus.com/inward/record.url?scp=33745388571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745388571&partnerID=8YFLogxK

U2 - 10.1117/12.648766

DO - 10.1117/12.648766

M3 - Conference contribution

AN - SCOPUS:33745388571

SN - 0819461865

SN - 9780819461865

VL - 6143 II

BT - Medical Imaging 2006

ER -