  Cancer cell mutations occur when cells undergo multiple cell divisions, and
these mutations can be spontaneous or environmentally-induced. The mechanisms
that promote and sustain these mutations are still not fully understood. This
study deals with the identification (or reconstruction) of the usually unknown
cancer cell mutation law, which lead to the transformation of a primary tumour
cell population into a secondary, more aggressive cell population. We focus on
local and nonlocal mathematical models for cell dynamics and movement, and
identify these mutation laws from macroscopic tumour snapshot data collected at
some later stage in the tumour evolution. In a local cancer invasion model, we
first reconstruct the mutation law when we assume that the mutations depend
only on the surrounding cancer cells (i.e., the ECM plays no role in
mutations). Second, we assume that the mutations depend on the ECM only, and we
reconstruct the mutation law in this case. Third, we reconstruct the mutation
when we assume that there is no prior knowledge about the mutations. Finally,
for the nonlocal cancer invasion model, we reconstruct the mutation law that
depends on the cancer cells and on the ECM. For these numerical
reconstructions, our approximations are based on the finite difference method
combined with the finite elements method. As the inverse problem is ill-posed,
we use the Tikhonov regularisation technique in order to regularise the
solution. Stability of the solution is examined by adding additive noise into
the measurements.
