Bulletin of Mathematical Biology

, Volume 70, Issue 3, pp 677–712 | Cite as

A Computational Study of the Development of Epithelial Acini: I. Sufficient Conditions for the Formation of a Hollow Structure

  • Katarzyna A. Rejniak
  • Alexander R. A. Anderson
Original Article


Normal hollow epithelial acini are 3-dimensional culture structures that resemble the architecture and functions of normal breast glands and lobules. This experimental model enables in vitro investigations of genotypic and molecular abnormalities associated with epithelial cancers. However, the way in which the acinar structure is formed is not yet completely understood. Gaining more information about consecutive stages of acini development—starting from a single cell that gives rise to a cluster of randomly oriented cells, followed by cell differentiation that leads to a layer of polarised cells enclosing the hollow lumen—will provide insight into the transformations of eukaryotic cells that are necessary for their successful arrangement into an epithelium. In this paper, we introduce a two-dimensional single-cell-based model representing the cross section of a typical acinus. Using this model, we investigate mechanisms that lead to the unpolarised cell growth, cell polarisation, stabilisation of the acinar structure and maintenance of the hollow lumen and discuss the sufficient conditions for each stage of acinar formation. In the follow-up paper (Rejniak and Anderson, A computational study of the development of epithelial acini. II. Necessary conditions for structure and lumen stability), we investigate what morphological changes are observable in the growing acini when some assumptions of this model are relaxed.


Development of epithelial acini Cell polarisation Cell apoptosis Single-cell-based model Immersed boundary method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P., 2002. Molecular biology of the cell. Garland Science, 4th edn. Google Scholar
  2. Anderson, A.R.A., 2005. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22, 163–186. zbMATHCrossRefGoogle Scholar
  3. Anderson, A.R.A., Weaver, A.M., Cummings, P.T., Quaranta, V., 2006. Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127, 905–915. CrossRefGoogle Scholar
  4. Ardens, M.J., Wyllie, A.H., 1991. Apoptosis: mechanisms and roles in pathology. Int. Rev. Exp. Pathol. 32, 223–254. Google Scholar
  5. Bowen, I.D., Lockshin, R.A., 1981. Cell Death in Biology and Pathology. Kluwer Academic, Dordercht. Google Scholar
  6. Debnath, J., Brugge, J.S., 2005. Modelling glandular epithelial cancers in three-dimensional cultures. Nat. Rev. Cancer 5, 675–688. CrossRefGoogle Scholar
  7. Debnath, J., Mills, K.R., Collins, N.L., Reginato, M.J., Muthuswamy, S.K., Brugge, J.S., 2002. The role of apoptosis in creating and maintaining luminal space within normal and oncogene-expressing mammary acini. Cell 111, 29–40. CrossRefGoogle Scholar
  8. Debnath, J., Muthuswamy, S.K., Brugge, J.S., 2003. Morphogenesis and oncogenesis on MCF-10A mammary epithelial acini grown in three-dimensional basement membrane cultures. Methods 30, 256–268. CrossRefGoogle Scholar
  9. Dembo, M., Harlow, F., 1986. Cell motion, contractile networks, and the physics of interpenetrating reactive flow. Biophys. J. 50, 109–121. CrossRefGoogle Scholar
  10. Dillon, R., Othmer, H.G., 1999. A mathematical model for outgrowth and spatial pattering of the vertebrate limb bud. J. Theor. Biol. 197, 295–330. CrossRefGoogle Scholar
  11. Ferguson, D.J.P., 1985. Ultrastructural characterisation of the proliferative (stem?) cells within the parenchyma of the normal “resting” breast. Virchows Arch. Pathol. Anat. 407, 379–385. CrossRefGoogle Scholar
  12. Ferguson, D.J.P., 1988. An ultrastructural study of mitosis and cytokinesis in normal “resting” human breast. Cell Tissue Res. 252, 581–587. CrossRefGoogle Scholar
  13. Ferguson, D.J.P., Anderson, T.J., 1981. Ultrastructural observations on cell death by apoptosis in the “resting” human breast. Virchows Arch. Pathol. Anat. 393, 193–203. Google Scholar
  14. Laurent, V.M., Planus, E., Fodil, R., Isabey, D., 2003. Mechanical assessment by magnetocytometry of the cytosolic and cortical cytoskeletal compartments in adherent epithelial cells. Biorheology 40, 235–240. Google Scholar
  15. Mills Shaw, K.R., Wrobel, C.N., Brugge, J.S., 2004. Use of three-dimensional basement membrane cultures to model oncogene-induced changes in mammary epithelial morphogenesis. J. Mammary Gland Biol. Neoplasia 9, 297–310. CrossRefGoogle Scholar
  16. Nelson, C.M., Bissell, M.J., 2005. Modeling dynamic reciprocity: engineering three-dimensional culture models of breast architecture, function, and neoplastic transformation. Semin. Cancer Biol. 15, 342–352. CrossRefGoogle Scholar
  17. Nelson, W.J., 2003. Adaptation of core mechanisms to generate cell polarity. Nature 422, 766–774. CrossRefGoogle Scholar
  18. O’Brien, L.E., Zegers, M.M.P., Mostov, K.E., 2002. Building epithelial architecture: insights from three-dimensional culture models. Nat. Rev. Mol. Cell Biol. 3, 531–537. CrossRefGoogle Scholar
  19. Peskin, C.S., 1972. Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271. zbMATHCrossRefMathSciNetGoogle Scholar
  20. Peskin, C.S., 1977. Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252. zbMATHCrossRefMathSciNetGoogle Scholar
  21. Peskin, C.S., 2002. The immersed boundary method. Acta Numer. 11, 479–517. zbMATHCrossRefMathSciNetGoogle Scholar
  22. Rejniak, K.A., 2002. A computational model of the mechanics of growth of a trophoblast tissue. PhD thesis, Tulane University. Google Scholar
  23. Rejniak, K.A., 2005. A single-cell approach in modeling the dynamics of tumor microregions. Math. Biosci. Eng. 2, 643–655. zbMATHMathSciNetGoogle Scholar
  24. Rejniak, K.A., 2007a. An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development. J. Theor. Biology 247, 186–204. CrossRefMathSciNetGoogle Scholar
  25. Rejniak, K.A., 2007b. Modelling the development of complex tissues using individual viscoelastic cells. In: A.R.A. Anderson, M.A.J. Chaplain, K.A. Rejniak (Eds.), Single Cell Based Models in Biology and Medicine. Birkhäuser, Basel Google Scholar
  26. Rejniak, K.A., Anderson, A.R.A., 2007. A computational study of the development of epithelial acini. II. Necessary conditions for structure and lumen stability, in preparation. Google Scholar
  27. Rejniak, K.A., Dillon, R.H., 2007. A single cell based model of the ductal tumor microarchitecture. Comput. Math. Methods Med. 8(1), 51–69. zbMATHCrossRefMathSciNetGoogle Scholar
  28. Rejniak, K.A., Kliman, H.J., Fauci, L.J., 2004. A computational model of the mechanics of growth of the villous trophoblast bilayer. Bull. Math. Biol. 66, 199–232. CrossRefMathSciNetGoogle Scholar
  29. Wang, A.Z., Ojakian, G.K., Nelson, W.J., 1990a. Steps in the morphogenesis of a polarised epithelium I. Uncoupling the roles of cell–cell and cell-substratum contact in establishing plasma membrane polarity in multicellular epithelial (MDCK) cysts. J. Cell Sci. 95, 137–151. Google Scholar
  30. Wang, A.Z., Ojakian, G.K., Nelson, W.J., 1990b. Steps in the morphogenesis of a polarised epithelium II. Disassembly and assembly of plasma membrane domains during reversal of epithelial cell polarity in multicellular epithelial (MDCK) cysts. J. Cell Sci. 95, 153–165. Google Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • Katarzyna A. Rejniak
    • 1
  • Alexander R. A. Anderson
    • 1
  1. 1.Division of MathematicsUniversity of DundeeDundeeUK

Personalised recommendations