Skip to content

Research at St Andrews

Mechanical models of pattern and form in biological tissues: the role of stress-strain constitutive equations

Research output: Contribution to journalArticlepeer-review

Author(s)

Chiara Villa, Mark A. J. Chaplain, Alf Gerisch, Tommaso Lorenzi

School/Research organisations

Abstract

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin–Voigt model of linear viscoelasticity to represent the stress–strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress–strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin–Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress–strain relation of the ECM, while no patterns are observed when the Kelvin–Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress–strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.
Close

Details

Original languageEnglish
Article number80
Number of pages38
JournalBulletin of Mathematical Biology
Volume83
DOIs
Publication statusPublished - 26 May 2021

    Research areas

  • Pattern formation, Mechanical models, Murray-Oster theory, Biological tissues, Stress-strain constitutive equations, Linear viscoelasticity

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Modeling the emergence of phenotypic heterogeneity in vascularized tumors

    Villa, C., Chaplain, M. A. & Lorenzi, T., 2021, In: SIAM Journal on Applied Mathematics. 81, 2, p. 434-453 20 p.

    Research output: Contribution to journalArticlepeer-review

  2. Evolutionary dynamics in vascularised tumours under chemotherapy: mathematical modelling, asymptotic analysis and numerical simulations

    Villa, C., Chaplain, M. A. J. & Lorenzi, T., 6 Oct 2020, (E-pub ahead of print) In: Vietnam Journal of Mathematics. First Online

    Research output: Contribution to journalArticlepeer-review

  3. A novel 3D atomistic-continuum cancer invasion model: in silico simulations of an in vitro organotypic invasion assay

    Franssen, L. C., Sfakianakis, N. & Chaplain, M. A. J., 7 Aug 2021, In: Journal of Theoretical Biology. 522, 14 p., 110677.

    Research output: Contribution to journalArticlepeer-review

  4. Calibrating models of cancer invasion: parameter estimation using approximate Bayesian computation and gradient matching

    Xiao, Y., Thomas, L. & Chaplain, M. A. J., 16 Jun 2021, In: Royal Society Open Science. 8, 6, 17 p., 202237.

    Research output: Contribution to journalArticlepeer-review

Related by journal

  1. A mathematical dissection of the adaptation of cell populations to fluctuating oxygen levels

    Ardaševa, A., Gatenby, R. A., Anderson, A. R. A., Byrne, H. M., Maini, P. K. & Lorenzi, T., 1 Jun 2020, In: Bulletin of Mathematical Biology. 82, 6, 81.

    Research output: Contribution to journalArticlepeer-review

  2. A mathematical framework for modelling the metastatic spread of cancer

    Franssen, L. C., Lorenzi, T., Burgess, A. & Chaplain, M. A. J., Jun 2019, In: Bulletin of Mathematical Biology. 81, 6, p. 1965-2010 46 p.

    Research output: Contribution to journalArticlepeer-review

  3. Stability, convergence, and sensitivity analysis of the FBLM and the corresponding FEM

    Sfakianakis, N. & Brunk, A., Nov 2018, In: Bulletin of Mathematical Biology. 80, 11, p. 2789-2827 39 p.

    Research output: Contribution to journalArticlepeer-review

ID: 274287410

Top