Prostate cancer bone metastasis poses significant health challenges, affecting countless individuals. While treatment with the radioactive isotope radium-223 ( ${}^{223}$Ra) has shown promising results, there remains room for therapy optimization. In vivo studies are crucial for optimizing radium therapy; however, they face several roadblocks that limit their effectiveness. By integrating in vivo studies with in silico models, these obstacles can be potentially overcome. Existing computational models of tumor response to ${}^{223}$Ra are often computationally intensive. Accordingly, we here present a versatile and computationally efficient alternative solution. We developed a PDE mathematical model to simulate the effects of ${}^{223}$Ra on prostate cancer bone metastasis, analyzing mitosis and apoptosis rates based on experimental data from both control and treated groups. To build a robust and validated model, our research explored three therapeutic scenarios: no treatment, constant ${}^{223}$Ra exposure, and decay-accounting therapy, with tumor growth simulations for each case. Our findings align well with experimental evidence, demonstrating that our model effectively captures the therapeutic potential of ${}^{223}$Ra, yielding promising results that support our model as a powerful infrastructure to optimize bone metastasis treatment.

In this study, we introduced a mathematical model mimicking as much as possible the evolutions and interactions between glioma and lactate in the brain, in order to test different therapies and administration protocols. We simulated both glioma cell density evolution and lactate concentration, and considered two therapies: chemotherapy and a treatment targeting lactate production. Three different protocols for administrating the therapies were tested. We compared the efficiency of the combined therapies, depending on the administration protocols and the dosage of the drugs, in order to evaluate the importance of controlling lactate production. Results show that the use of an agent to reduce lactate concentration permits one to significantly reduce the dose of the chemotherapeutic drug.