In this paper, the application of a robust nonlinear model predictive control (NMPC) framework to mammalian cell cultures is proposed, dealing with possible large kinetic parameter uncertainties. Industrial constraints formulated in view of good manufacturing practice and quality-by-design approach are also considered, namely the assurance that all state trajectories are contained within a corridor defined by lower and upper safety bounds. The latter are assimilated to the well-known tube-based paradigm which is used to formulate the corresponding robust NMPC problem. Both classical and tube-based NMPC performances are assessed in numerical simulations where specific key-species are regulated while dealing with an uncertain plant model. The capability of the tube-based method to reduce the impact of the parameter variations on the state trajectories and the violation of the constraints is highlighted, suggesting the transfer of the method on a real pharmaceutical process.