  In this paper we consider two diffuse interface models for tumor growth
coupling a Cahn-Hilliard type equation for the tumor phase parameter to a
reaction-diffusion type equation for the nutrient. The models are distinguished
by the presence of two different coupling source terms. For such problems, we
address the question of the limit, as the diffuse interface parameter tends to
zero, from diffuse interface models to sharp interface ones, justifying
rigorously what was deduced via formal asymptotics in [20]. The resulting
evolutions turn out to be varifold solutions to Mullins-Sekerka type flows for
the tumor region suitably coupled with the equation for the nutrient.
