A mathematical model of the tumour growth along a blood vessel is proposed. Themodel employs the mixture theory approach to describe a tissue which consists of cells, extracellularmatrix and liquid. The growing tumour tissue is supposed to be surrounded by the hosttissue. Tumours where complete oxydation of glucose prevails are considered. Special attention ispaid to consistent description of oxygen consumption and growth processes based on the energybalance. A finite difference numerical method is proposed. The level set method is used to trackan interface between the tissues. The simulations show localization of the tumour within a limiteddistance from the vessels and constant expansion velocity along the vessels.