  Ductal carcinoma {\em in situ} (DCIS) lesions are non-invasive tumours of the
breast which are thought to precede most invasive breast cancers (IBC). As
individual DCIS lesions are initiated, grow and invade (i.e. become IBC) the
size distribution of the DCIS lesions present in a given human population will
evolve. We derive a differential equation governing this evolution and show,
for given assumptions about growth and invasion, that there is a unique
distribution which does not vary with time. Further, we show that any initial
distribution converges to this stationary distribution exponentially quickly.
It is therefore reasonable to assume that the stationary distribution is equal
to the true DCIS size distribution, at least for human populations which are
relatively stable with respect to the determinants of breast cancer. Based on
this assumption and the size data of 110 DCIS lesions detected in a
mammographic screening program between 1993 and 2000, we produce maximum
likelihood estimates for certain growth and invasion parameters. Assuming that
DCIS size is proportional to a positive power $p$ of the time since tumour
initiation we estimate $p$ to be 0.50 with a 95% confidence interval of $(0.35,
0.71)$. Therefore we estimate that DCIS lesions follow a square-root growth law
and hence that they grow rapidly when small and relatively slowly when large.
Our approach and results should be useful for other mathematical studies of
cancer, especially those investigating biological mechanisms of invasion.
