Living donors are a significant and increasing source of livers for transplantation, mainly due to the insufficient supply of cadaveric organs. We consider the problem of optimally timing a living-donor liver transplant in order to maximize the patient's total reward such as quality-adjusted life expectancy. We formulate a Markov decision process (MDP) model in which the state of the process is described by patient health. We derive structural properties of the MDP model, including a set of

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... itive conditions that ensure the existence of a control-limit optimal policy. We use clinical data in our computational experiments, which show that the optimal policy is typically of control-limit type.