First in Human trials investigate whether a potential new medicine is safe for humanuse. The aim of the trial is typically to find the maximum tolerated dose of the potentialnew medicine. The trial design must specify the dose escalation scheme, a rule thatstates which dose to give to which subjects at which point in the trial. It is importantto allocate doses in such a way that avoids exposing subjects in the trial to unacceptablerisk and also provides information on the relationship between dose and toxicity. If wedefine the aims of the trial using a loss function we can use dynamic programming toobtain the optimal dose escalation scheme for a First in Human trial with respect tothat loss function. Different loss functions lead to different schemes. Thus, this workprovides a flexible framework that can be used to compare different trial designs. Thisshows that some common First in Human trial designs involve dosing decisions thatput subjects at greater risk than necessary, or dosing an excess of subjects at doses thatdo not contribute information that improves the estimation of the maximum tolerateddose.Dynamic programming requires a set of calculations to be performed for every possibledata set at each stage of the trial. Even with a small trial this state space is large. Weconsider reformulating the state space as the space of posterior density functions for thedose-response model parameter and adapting the dynamic programming algorithm to asample of this space. This produces a dose escalation scheme that is an approximationto the optimal rule produced by performing dynamic programming on the space ofall possible data sets. With this approximate version of the algorithm, we extend themethodology to find an optimal dose escalation scheme for a First in Human trial witha binary efficacy endpoint as well as a binary safety endpoint.