Investigating the steady state of multicellular spheroids by revisiting the two-fluid model

In this paper we examine the steady state of tumour spheroids considering a structure in which the central necrotic region contains an inner liquid core surrounded by dead cells that keep some mechanical integrity. This partition is a consequence of assuming that a finite delay is required for the degradation of dead cells into liquid. The phenomenological assumption of constant local volume fraction of cells is also made. The above structure is coupled with a simple mechanical model that views the cell component as a viscous fluid and the extracellular liquid as an inviscid fluid. By imposing the continuity of the normal stress throughout the whole spheroid, we show that a steady state can exist only if the forces on cells at the outer boundary (provided e.g. by a surface tension) are intense enough, and in such a case we can compute the stationary radius. By giving reasonable values to the parameters, the model predicts that the stationary radius decreases with the external oxygen concentration, as expected from experimental observations.