Some of the key factors that regulate growth and remodeling of tissues are fundamentally mechanical. However, it is important to take into account the role of bioavailability together with the stresses and strains in the processes of normal or pathological growth. In this sense, the model presented in this work is oriented to describe the growth of soft biological tissue under "stress driven growth" and depending on the biological availability of the organism. The general theoretical framework is given by a kinematic formulation in large strain combined with the thermodynamic basis of open systems. The formulation uses a multiplicative decomposition of deformation gradient, splitting it in a growth part and viscoelastic part. The strains due to growth are incompatible and are controlled by an unbalanced stresses related to a homeostatic state. Growth implies a volume change with an increase of mass maintaining constant the density. One of the most interesting features of the proposed model is the generation of new tissue taking into account the contribution of mass to the system controlled through biological availability. Because soft biological tissues in general have a hierarchical structure with several components (usually a soft matrix reinforced with collagen fibers), the developed growth model is suitable for the characterization of the growth of each component. This allows considering a different behavior for each of them in the context of a generalized theory of mixtures. Finally, we illustrate the response of the model in case of growth and atrophy with an application example. ; Peer Reviewed ; Postprint (published version)