  We consider a model describing the evolution of a tumor inside a host tissue
in terms of the parameters $\varphi_p$, $\varphi_d$ (proliferating and dead
cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The
variables $\varphi_p$, $\varphi_d$ satisfy a Cahn-Hilliard type system with
nonzero forcing term (implying that their spatial means are not conserved in
time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a
quasistatic diffusion equation. The main novelty of the present work stands in
the fact that we are able to consider a configuration potential of singular
type implying that the concentration vector $(\varphi_p,\varphi_d)$ is
constrained to remain in the range of physically admissible values. On the
other hand, in view of the presence of nonzero forcing terms, this choice gives
rise to a number of mathematical difficulties, especially related to the
control of the mean values of $\varphi_p$ and $\varphi_d$. For the resulting
mathematical problem, by imposing suitable initial-boundary conditions, our
main result concerns the existence of weak solutions in a proper regularity
class.
