We examine the critical behavior of a lattice model of tumor growth where supplied nutrients are correlated with the distribution of tumor cells. Our results support the previous report [Ferreira et al., Phys. Rev. E 85, 010901(R) (2012)], which suggested that the critical behavior of the model differs from the expected directed percolation (DP) universality class. Surprisingly, only some of the critical exponents (β, α, ν([perpendicular]), and z) take non-DP values while some others (β', ν(||), and spreading-dynamics exponents Θ, δ, z') remain very close to their DP counterparts. The obtained exponents satisfy the scaling relations β=αν(||), β'=δν(||), and the generalized hyperscaling relation Θ+α+δ=d/z, where the dynamical exponent z is, however, used instead of the spreading exponent z'. Both in d=1 and d=2 versions of our model, the exponent β most likely takes the mean-field value β=1, and we speculate that it might be due to the roulette-wheel selection, which is used to choose the site to supply a nutrient.