This work presents a comprehensive high-performance 3D computer model simulating complex tissue growth processes. The model contains three major components: (a) a discrete algorithm simulating individual cell activities and cell-cell interactions, (b) transient partial differential equations (PDE's) describing the convection, diffusion, consumption and secretion of nutrients, growth factors or other important substances in tissue systems, and (c) rules describing how cell behavior is modulated by the local concentrations of nutrients and growth factors. The discrete model captures the essential features of important cellular activities including migration, proliferation and cell-cell contact. The diffusion-reaction PDE's are discretized with an implicit-explicit scheme and the resulting linear systems are solved with a highly efficient GMRES solver. The model is fully parallelized for execution on distributed-memory computer clusters. Simulation results reveal the intricate interplay of cell population and mass transfer dynamics during the growth of various tissues, such as bioartificial tissues and tumors.