New late-intensification schedules for cancer treatments.

The nonlinear Gompertz equation that describes the evolutions of tumors under different treatments was investigated. It was shown that late logarithmic intensification asymptotically reduces the tumor cell population to zero. Using the same total amount of therapy, the schedule following a logarithmic function produces a larger reduction of tumor cell population than the schedule using a therapy with constant intensity. On the other hand, a logarithmic intensification should be more tolerable by the patients than a faster temporal growth therapy. We have shown that during the treatment the dose intensity should not be decreased at any time while the therapy is applied, because this will allow the tumor to relapse. The therapy intensity should be continuously increased as possible. We have solved an optimization problem for several late intensification treatments by the constraints of the total dose and the maximum individual (or daily) dose. Based on our results, we have designed new chemotherapy and radiotherapy treatments.