  We consider a continuum mechanical model of cell invasion through thin
membranes. The model consists of a transmission problem for cell volume
fraction complemented with continuity of stresses and mass flux across the
surfaces of the membranes. We reduce the original problem to a limiting
transmission problem whereby each thin membrane is replaced by an effective
interface, and we develop a formal asymptotic method that enables the
derivation of a set of biophysically consistent transmission conditions to
close the limiting problem. The formal results obtained are validated via
numerical simulations showing that the relative error between the solutions to
the original transmission problem and the solutions to the limiting problem
vanishes when the thickness of the membranes tends to zero. In order to show
potential applications of our effective interface conditions, we employ the
limiting transmission problem to model cancer cell invasion through the
basement membrane and the metastatic spread of ovarian carcinoma.
