IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i21p2726-d666140.html
   My bibliography  Save this article

Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics

Author

Listed:
  • John R. Jungck

    (Departments of Biological Sciences and Mathematical Sciences, Delaware Biotechnology Institute, University of Delaware, 15 Innovation Way, Newark, DE 19716, USA)

  • Michael J. Pelsmajer

    (Department of Applied Mathematics, Illinois Institute of Technology Engineering 1, Room 206 10 West 32nd Street, Chicago, IL 60616, USA)

  • Camron Chappel

    (Department of Statistics, University of Delaware, Newark, DE 19716, USA)

  • Dylan Taylor

    (Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA)

Abstract

Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi polygons; Delaunay triangulations; perpendicular bisectors; circumcenters; convex hulls; minimal spanning trees; Ulam trees; Pitteway violations; circularity; Clark-Evans spatial statistics; variance to mean ratios; Gabriel graphs; and, minimal spanning trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara’s Law (daughter cell areas should be roughly half that of their parent cells’ areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.

Suggested Citation

  • John R. Jungck & Michael J. Pelsmajer & Camron Chappel & Dylan Taylor, 2021. "Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics," Mathematics, MDPI, vol. 9(21), pages 1-24, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2726-:d:666140
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/21/2726/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/21/2726/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Matthew C. Gibson & Ankit B. Patel & Radhika Nagpal & Norbert Perrimon, 2006. "The emergence of geometric order in proliferating metazoan epithelia," Nature, Nature, vol. 442(7106), pages 1038-1041, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Patrik Sahlin & Henrik Jönsson, 2010. "A Modeling Study on How Cell Division Affects Properties of Epithelial Tissues Under Isotropic Growth," PLOS ONE, Public Library of Science, vol. 5(7), pages 1-9, July.
    2. Sebastian A Sandersius & Manli Chuai & Cornelis J Weijer & Timothy J Newman, 2011. "Correlating Cell Behavior with Tissue Topology in Embryonic Epithelia," PLOS ONE, Public Library of Science, vol. 6(4), pages 1-11, April.
    3. Y. Chélin & J. Averseng & P. Cañadas & B. Maurin, 2013. "Divided media-based simulations of tissue morphogenesis," Computer Methods in Biomechanics and Biomedical Engineering, Taylor & Francis Journals, vol. 16(S1), pages 2-3, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2726-:d:666140. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.