On the assumption of cubic graphs of vascular networks

Sung Hyuk Cha, Sukmoon Chang, Michael L. Gargano

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationMedical Imaging 2006
Subtitle of host publicationPhysiology, Function, and Structure from Medical Images
Volume6143 II
DOIs
StatePublished - Jun 30 2006
EventMedical Imaging 2006: Physiology, Function, and Structure from Medical Images - San Diego, CA, United States
Duration: Feb 12 2006Feb 14 2006

Other

OtherMedical Imaging 2006: Physiology, Function, and Structure from Medical Images
CountryUnited States
CitySan Diego, CA
Period2/12/062/14/06

Fingerprint

Graph theory
Pattern matching
Blood vessels
Rats
Imaging techniques

All Science Journal Classification (ASJC) codes

  • Engineering(all)

Cite this

Cha, S. H., Chang, S., & Gargano, M. L. (2006). On the assumption of cubic graphs of vascular networks. In Medical Imaging 2006: Physiology, Function, and Structure from Medical Images (Vol. 6143 II). [61432M] https://doi.org/10.1117/12.648766
Cha, Sung Hyuk ; Chang, Sukmoon ; Gargano, Michael L. / On the assumption of cubic graphs of vascular networks. Medical Imaging 2006: Physiology, Function, and Structure from Medical Images. Vol. 6143 II 2006.
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Cha, SH, Chang, S & Gargano, ML 2006, On the assumption of cubic graphs of vascular networks. in 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.

Medical Imaging 2006: Physiology, Function, and Structure from Medical Images. Vol. 6143 II 2006. 61432M.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Cha SH, Chang S, Gargano ML. On the assumption of cubic graphs of vascular networks. In Medical Imaging 2006: Physiology, Function, and Structure from Medical Images. Vol. 6143 II. 2006. 61432M https://doi.org/10.1117/12.648766