We consider a non homogeneous Gompertz diffusion process whose parameters are modified by generally time-dependent exogenous factors included in the infinitesimal moments. The proposed model is able to describe tumor dynamics under the effect of anti-proliferative and/or cell death-induced therapies. We assume that such therapies can modify also the infinitesimal variance of the diffusion process. An estimation procedure, based on a control group and two treated groups, is proposed to infer the model by estimating the constant parameters and the time-dependent terms. Moreover, several concatenated hypothesis tests are considered in order to confirm or reject the need to include time-dependent functions in the infinitesimal moments. Simulations are provided to evaluate the efficiency of the suggested procedures and to validate the testing hypothesis. Finally, an application to real data is considered.