  We investigate the possible blow-up of strong solutions to a biological
network formation model originally introduced by D. Cai and D. Hu \cite{HC}.
The model is represented by an initial boundary value problem for an
elliptic-parabolic system with cubic non linearity. We obtain an algebraic
equation for the possible blow-up time of strong solutions. The equation yields
information on how various given data may contribute to the blow-up of
solutions. As a by-product of our development, we establish a $W^{1,q}$
estimate for solutions to an elliptic equation which shows the explicit
dependence of the upper bound on the elliptic coefficients.
