  We study the spatial evolutionary dynamics of solid tumors as they obtain
additional driver mutations. We start with a cancer clone that expands
uniformly in three dimensions giving rise to a spherical shape. We assume that
cell division occurs on the surface of the growing tumor. Each cell division
has a chance to give rise to a mutation that activates an additional driver
gene. The resulting clone has an enhanced growth rate, which generates a local
ensemble of faster growing cells, thereby distorting the spherical shape of the
tumor. We derive analytic formulas for the geometric boundary that separates
the original cancer clone from the new mutant as well as the expanding frontier
of the new mutant. The total number of original cancer cells converges to a
constant as time goes to infinity, because this clone becomes enveloped by
mutants. We derive formulas for the abundance and diversity of additional
driver mutations as function of time. Our model is semi-deterministic: the
spatial growth of the various cancer clones follows deterministic equations,
but the arrival of a new mutant is a stochastic event.
