Mathematical models of tumour spheroids, proposed since the early seventies, have been
generally formulated in terms of a single diffusive nutrient which is critical for cell replication
and cell viability. Only recently, attempts have been made to incorporate in the models the cell
energy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).
By assuming glucose and lactate as the only fuel substrates, we propose a simple model for the
cell ATP production which takes into account the main reactions that occur in the glycolytic and
the oxidative pathway. Under the assumption that cell death occurs when ATP production falls
below a critical level, we have studied the free boundary problem for the concentration of glucose,
lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core is
guaranteed for a sufficiently large size of the spheroid.