Hairy roots are plants genetically transformed by Agrobacterium rhizogenes, which do not produce shoots and are composed mainly by roots. Hairy roots of Ophiorrhiza mungos Linn. are currently gaining interest of pharmacologists, since a secondary product of their metabolism, camptothecin, is used in chemotherapy. To optimize the production of valuable secondary metabolites it is necessary to understand the metabolism and growth of these roots systems. In this work, a mathematical model for description of apical growth of a dense root network (e.g. hairy roots) is derived. A continuous approach is used to define densities of root tips and root volume. Equations are posed to describe the evolution of these and are coupled to the distribution of nutrient concentration in the medium and inside the network. Following the principles of irreversible thermodynamics, growth velocity is defined as the sum over three different driving forces: nutrient concentration gradients, space gradients and root tip diffusion. A finite volume scheme was used for the simulation and parameters were chosen to fit experimental data from O. mungos Linn. hairy roots. Internal nutrient concentration determines short-term growth. Long-term behavior is limited by the total nutrient amount in the medium. Therefore, mass yield could be increased by guaranteeing a constant supply of nutrients. Increasing the initial mass of inoculation did not result in higher mass yields, since nutrient consumption due to metabolism also rose. Four different growth strategies are compared and their properties discussed. This allowed to understand which strategy might be the best to increase mass production optimally. The model is able to describe very well the temporal evolution of mass increase and nutrient uptake. Our results provide further understanding of growth and density distribution of hairy root network and therefore it is a sound base for future applications to describe, e.g., secondary metabolite production.