Biomechanics and Modeling in Mechanobiology

, Volume 14, Issue 2, pp 413–425 | Cite as

Vertex dynamics simulations of viscosity-dependent deformation during tissue morphogenesis

  • Satoru OkudaEmail author
  • Yasuhiro Inoue
  • Mototsugu Eiraku
  • Taiji Adachi
  • Yoshiki Sasai
Original Paper


In biological development, multiple cells cooperate to form tissue morphologies based on their mechanical interactions; namely active force generation and passive viscoelastic response. In particular, the dynamic processes of tissue deformations are governed by the viscous properties of the tissues. These properties are spatially inhomogeneous because they depend on the tissue constituents, such as cytoplasm, cytoskeleton, basement membrane and extracellular matrix. The multicellular mechanics of tissue morphogenesis have been investigated in vertex dynamics models. However, conventional models are applicable only to quasi-static deformation processes, which do not account for tissue viscosities. We propose a vertex dynamics model that simulates the viscosity-dependent dynamic deformation processes during tissue morphogenesis. By incorporating local velocity fields into the governing equation of vertex movements, the model turns Galilean invariant. In addition, the viscous properties of tissue components are newly expressed by formulating friction forces on vertices as functions of the relative velocities among the vertices. The advantages of the proposed model are examined by epithelial growth simulations under the employed condition for quasi-static processes. As a result, the epithelial vesicle simulated by the proposed model is linearly elongated with nearly free stress, while that simulated by the conventional model is undulated with compressive residual stress. Therefore, the proposed model is able to reflect the timescale of deformations by satisfying Galilean invariance. Next, the applicability of the proposed model is assessed in epithelial growth simulations of viscous extracellular materials. In this test, the epithelial vesicles are deformed into tubular shapes by oriented cell divisions, and their morphologies are extremely sensitive to extracellular viscosity. Therefore, the dynamic deformations in the proposed model depend on the viscous properties of tissue components. The proposed model will be useful for simulating dynamic deformation processes of tissue morphogenesis depending on viscous properties of various tissue components.


Multicellular morphogenesis Dynamic deformation process Viscous property Vertex dynamics model Computational biomechanics Developmental biomechanics 



The authors appreciate valuable comments from Dr. Yoshihiro Morishita, Katsuhiko Sato and Yukitaka Ishimoto at the RIKEN Center for Developmental Biology, Japan. This study was supported by JSPS KAKENHI Grant Numbers 25889070.

Supplementary material

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Satoru Okuda
    • 1
    Email author
  • Yasuhiro Inoue
    • 2
  • Mototsugu Eiraku
    • 3
  • Taiji Adachi
    • 2
  • Yoshiki Sasai
    • 1
  1. 1.Organogenesis and Neurogenesis Group, Center for Developmental BiologyRIKENKobeJapan
  2. 2.Department of Biomechanics, Institute for Frontier Medical SciencesKyoto UniversityKyotoJapan
  3. 3.Four-Dimensional Tissue Analysis Unit, Center for Developmental BiologyRIKENKobeJapan

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