Existence and uniqueness of solutions for a class of models for stress-modulated growth is proven in one spatial dimension. The model features the multiplicative decomposition of the deformation gradient F into an elastic part Fe and a growth-related part G. After the transformation due to the growth process, governed by G, an elastic deformation described by Fe is applied in order to restore the Dirichlet boundary conditions and, therefore, the current configuration might be stressed with a stress tensor S. The growth of the material at each point in the reference configuration is given by an ordinary differential equation for which the right-hand side may depend on the stress S and the pull-back of a nutrient concentration in the current configuration, leading to a coupled system of ordinary differential equations.