Tissue engineering involves growing cells within supporting scaffolds to obtain structures for in vivo implantation with adequate functionality. Obtaining a proper oxygen supply, high cell density, and a uniform cell distribution in a three-dimensional (3D) growth support are important challenges. Both experiments and quantitative mathematical models are needed to better understand the physical, mechanical, and biochemical phenomena and for the rational design of suitable reactor geometries and operating protocols for the production of functional engineered artificial grafts. In this work, a comprehensive mathematical model of convection and diffusion in a perfusion bioreactor, combined with cell growth kinetics, is developed. The model describes the spatial-temporal evolution of oxygen concentration and cell density within a 3D polymeric scaffold. The fluid dynamics of the medium flow inside the bioreactor is described through the Navier-Stokes equations for incompressible fluids while convection through the scaffold is modeled using Brinkman's extension of Darcy's law for porous media. The species balances are written in general form, but only that for oxygen, considered the limiting factor for cell growth, is used here. The oxygen uptake rate is described by Michaelis-Menten kinetics, and cell growth is modeled as a function of oxygen concentration through the Contois equation, accounting for contact inhibition. Convection, diffusion, and cell growth interact strongly through the geometric properties of the scaffold, such as porosity and permeability, which change in time and space. The model is used to simulate two conditions: total flow perfusion and partial flow perfusion. The former highlights the significant effects of various operating conditions on culture performance; the latter demonstrates the adverse effects of flow channelling at the walls of the perfusion bioreactor, which is frequently observed in experiments. Results are presented for a specific application involving the culture of immortalized rat cells C2C12 on a collagen scaffold.