A mathematical model to aid in the interpretation of drug effects on cell kinetics

A bilinear model is introduced to represent the time varying effects of anti-tumor drugs on the kinetics of the cell cycle. Optimal control theory is applied for the estimation of these time-varying effects. This methodology will enable developers of new cancer drugs to quantify the effects of a drug on each phase of the cycle. Initially, a compartmental model with constant parameters, in which each of the three compartments represents the number of cells in the three phases of the cycle is formulated. The constant parameters are determined utilizing the Gauss-Newton algorithm. Knowing the constant parameters, we then introduce the control function to represent the drug effects on the various phases of the cycle. The model is then formulated as a bilinear control process. The problem is one of minimizing the performance function subject to the constraints of the dynamics of the model. With the use of the discrete minimum principle, the Hamiltonian is determined and a steepest descent algorithm is then applied to obtain a minimum. The results, which were consistent with current concepts on the pharmacology of the drug, suggest that this method is reliable for determining drug effects on the cell cycle.