Optimal strategy in chemotherapy for Malthusian model of cancer growth.

PURPOSE The problem of optimal strategy in cancer chemotherapy is reconsidered. Two incompatible goals should be completed: the number of cancer cells in the patient's body should be reduced and the toxic effect of the therapy should be minimized. Such problem may be formulated in optimal control. The control function is the amount of the drug administered in the time unit. METHODS The Malthusian model of cell population growth is employed where the rate of increase of the number of cancer cells is proportional to the number of cells in population and an intrinsic rate that usually is assumed to be constant. The performance index is the amount of the drug cumulated in the patient's body and it is minimized. A non-standard method of optimal control is used - method of Miele. RESULTS The optimal solutions are obtained for three cases: constant intrinsic rate, monotonically increasing/decreasing intrinsic rate and for periodic intrinsic rate. The optimal control is ununique for the first case - the result is irrespective of the strategy. Such result has been known earlier. The optimal control is unique for other cases and it is of bang-bang type. CONCLUSIONS The ununique solution for constant intrinsic rate is surprising, therefore a mechanical analogy is given. The optimal strategy is in accordance with clinical experience for decreasing intrinsic rate. The optimal control is a periodic function of time for the intrinsic rate of sin/cos type - the drug should be administered, as its value is relatively high.