Tissue engineering is a novel and promising approach to repair articular cartilage defects. This procedure requires that autologous articular chondrocytes are isolated from a small biopsy, expanded in vitro on two-dimensional culture plates to augment their original number and then cultured onto specific three-dimensional synthetic biomaterials to generate implantable cartilaginous constructs. The phase of cell expansion is critical, since for such technique large number of chondrocytes is needed. We thus aimed our research at devising a mathematical model to deeply understand the growth dynamics of chondrocytes. Experimentally, it was observed that when the cells in culture reach a critical level, they spontaneously tend to align along some common axis of orientation. Our intention in this book is to study the parameter regimes which lead to the formation of one or more patterns, and to compare them to the experiments. With this goal in mind we implemented simulations of the three dimensional model by a combination of Tchebychef method, finite differences and trapezoidal rule and perform experiments to approximate some of the parameters.