Despite major scientific, medical and technological advances over the last few decades,
a cure for cancer remains elusive. The disease initiation is complex, and including
initiation and avascular growth, onset of hypoxia and acidosis due to accumulation
of cells beyond normal physiological conditions, inducement of angiogenesis from the
surrounding vasculature, tumour vascularization and further growth, and invasion of
surrounding tissue and metastasis. Although the focus historically has been to study
these events through experimental and clinical observations, mathematical modelling
and simulation that enable analysis at multiple time and spatial scales have also
complemented these efforts. Here, we provide an overview of this multiscale modelling
focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis.
While we briefly review discrete modelling, our focus is on the continuum approach.
We limit the scope further by considering models of tumour progression that do not
distinguish tumour cells by their age. We also do not consider immune system interactions
nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where
the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale)
elements, thus connecting the micrometre to the centimetre tumour scale. We review
recent examples that incorporate experimental data into model parameters. We show
that recent mathematical modelling predicts that transport limitations of cell nutrients,
oxygen and growth factors may result in cell death that leads to morphological instability,
providing a mechanism for invasion via tumour fingering and fragmentation. These conditions
induce selection pressure for cell survivability, and may lead to additional genetic
mutations. Mathematical modelling further shows that parameters that control the tumour
mass shape also control its ability to invade. Thus, tumour morphology may serve as
a predictor of invasiveness and treatment prognosis.