  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 final 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.
