Minimal Morphoelastic Models of Solid Tumour Spheroids: a Tutorial

Benjamin J Walker, Giulia Celora, Alain Goriely, Derek Moulton, Helen Byrne

Research output: Contribution to journalArticlepeer-review

3 Citations (SciVal)


Tumour spheroids have been the focus of a variety of mathematical models, ranging from Greenspan’s classical study of the 1970s through to contemporary agent-based models. Of the many factors that regulate spheroid growth, mechanical effects are perhaps some of the least studied, both theoretically and experimentally, though experimental enquiry has established their significance to tumour growth dynamics. In this tutorial, we formulate a hierarchy of mathematical models of increasing complexity to explore the role of mechanics in spheroid growth, all the while seeking to retain desirable simplicity and analytical tractability. Beginning with the theory of morphoelasticity, which combines solid mechanics and growth, we successively refine our assumptions to develop a somewhat minimal model of mechanically regulated spheroid growth that is free from many unphysical and undesirable behaviours. In doing so, we will see how iterating upon simple models can provide rigorous guarantees of emergent behaviour, which are often precluded by existing, more complex modelling approaches. Perhaps surprisingly, we also demonstrate that the f inal model considered in this tutorial agrees favourably with classical experimental results, highlighting the potential for simple models to provide mechanistic insight whilst also serving as mathematical examples.
Original languageEnglish
Article number38
Number of pages35
JournalBulletin of Mathematical Biology
Issue number5
Early online date29 Mar 2023
Publication statusPublished - 31 Mar 2023

Bibliographical note

Funding BJW is supported by the Royal Commission for the Exhibition of 1851. GLC is supported by EPSRC and MRC Centre for Doctoral Training in Systems Approaches to Biomedical Science (grant number EP/L016044/1) and Cancer Research UK. The work of AG was supported by the Engineering and Physical Sciences Research Council grant EP/R020205/1.


  • Mathematical modelling
  • Morphoelasticity
  • Stress-dependent growth
  • Tumour dynamics

ASJC Scopus subject areas

  • General Neuroscience
  • Immunology
  • General Mathematics
  • General Biochemistry,Genetics and Molecular Biology
  • General Environmental Science
  • Pharmacology
  • General Agricultural and Biological Sciences
  • Computational Theory and Mathematics


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