  The growth of many solid tumors has been found to be driven by chemo- and
radiotherapy-resistant cancer stem cells (CSCs). A suitable therapeutic avenue
in these cases may involve the use of a differentiating agent (DA) to force the
differentiation of the CSCs and of conventional therapies to eliminate the
remaining differentiated cancer cells (DCCs). To describe the effects of a DA
that reprograms CSCs into DCCs, we adapt a differential equation model
developed to investigate tumorspheres, which are assumed to consist of jointly
evolving CSC and DCC populations. We analyze the mathematical properties of the
model, finding the equilibria and their stability. We also present numerical
solutions and phase diagrams to describe the system evolution and the therapy
effects, denoting the DA strength by a parameter \(a_{dif}\).To obtain
realistic predictions, we choose the other model parameters to be those
determined previously from fits to various experimental datasets. These
datasets characterize the progression of the tumor under various culture
conditions. Typically, for small values of \(a_{dif}\) the tumor evolves
towards a final state that contains a CSC fraction, but a strong therapy leads
to the suppression of this phenotype. Nonetheless, different external
conditions lead to very diverse behaviors. For some environmental conditions,
the model predicts a threshold not only in the therapy strength, but also in
its starting time, an early beginning being potentially crucial. In summary,
our model shows how the effects of a DA depend critically not only on the
dosage and timing of the drug application, but also on the tumor nature and its
environment.
