A Bayesian method for individualized dose-finding based on efficacy and toxicity is presented. Dose-covariate interactions are included in the marginal outcome probabilities. Implementation requires an informative prior on covariate main effects. Elicited covariate-specific limits on the probabilities of efficacy and toxicity are used to construct bounding functions to determine the acceptability of each dose for each possible covariate vector. Each patient's dose is chosen to optimize the efficacy-toxicity trade-off for his/her covariates, based on elicited target outcome probability pairs. Because the selected doses are covariate-specific and the method is sequentially outcome-adaptive, different patients may receive different doses at the same point in the trial, and patient eligibility may change during the trial. The method is illustrated by a trial in acute leukemia.