To define an optimal radiation therapy strategy, the dependence of the probability of cure and of significant complications, on the parameters controlled by the radiotherapist must be determined. The recent success of the Linear Quadratic (LQ) model in constructing isoeffect relations for normal tissue damage and in describing in vitro cell survival curves, indicate that this model could be used to determine this dependence. The problem of tumor control is addressed. Using LQ model parameters obtained from human tumor cell lines, the sigmoid dose-response curves for controlling tumors of fixed size but of several histologies, with a fractionated course of radiotherapy is obtained. Except for squamous cell carcinoma, the calculated average tumor control doses (TCD37 or TCD50) are unrealistically low, but the model can be made more realistic by including inhomogeneities in the spatial dose distribution and heterogeneous tumor cell populations. The slope of the dose response curves are determined and the significance of the "relative slope" parameter rho as a measure of the number of cells in a tumor's most radioresistant clone is noted. The relation of the model's predictions to qualitative features of the experimental animal data for both tumor control and for normal tissue damage is discussed. Experiments to test the validity of this type of model are suggested.