Rule-based dose-escalation methods include the Storer up-and-down design  and the frequently applied 3 + 3 design (Fig. 2). Although several variants of this long-established approach have been published, the earliest source is a book chapter arising from the proceedings of a clinical pharmacology course held in Brussels in May 1972 . The main principle of algorithmic designs is that a small group of patients is treated at a given dose and, dependent on the observed number of toxicities, a decision is made on to whether to study a further group of patients at the next dose up the scale, to study more patients at the same dose or to stop the trial.
The 3 + 3 design, for example, enters three patients into the trial one at a time at intervals of seven days or more and treats them at the chosen starting dose. The core features of this design, as outlined by Storer , are:
If no dose-limiting toxicity (DLT) occurs in the first cycle of any of the initial group of 3 patients treated at a dose, the dose should be escalated for the next group of patients.
If two or more of the 3 patients treated at a given dose experience a DLT, the trial should stop.
If one patient of the 3 treated at a given dose level experiences a DLT, a further 3 patients are treated at the same dose level. If a DLT has occurred in exactly one of these 6 patients, escalation may continue as in (a); otherwise, the trial should stop.
A frequent variation allows the dose to be reduced if more than one DLT is observed within the first three patients studied at a particular dose.
Following completion of the trial according to this design, the MTD is defined as either the actual dose at which the trial was stopped or the next lower dose, possibly depending on the frequency and severity of toxicity observed in the patients evaluated in the final group . A number of adaptations of this design attempt to reduce the number of patients treated at the lowest dose levels . Such accelerated titration designs [18, 21] include single-patient dose levels and more rapid dose escalation (e.g., dose doubling) until the first DLT is observed, after which the dose-escalation scheme would then revert to the modified Fibonacci sequence.
Model-based Bayesian methods
Bayesian modeling combines prior knowledge of the drug with the observed data from the current trial to provide updated information about the distribution of the trial outcome of interest. Bayesian methods have been incorporated into a number of early phase clinical trial designs, the best known being the continual reassessment method (CRM) [4, 5]. In contrast to the algorithmic approach of the 3 + 3 design, the continual reassessment method aims to identify the dose at which the proportion of patients experiencing a DLT reaches a specific target level (e.g., 25 %). Furthermore, the investigator performs repeated analyses on all of the data gathered to date—hence the term continual reassessment—rather than simply observing the data recorded at the current dose, as is the case for the 3 + 3 design.
The CRM uses a one-parameter Bayesian model which assumes that the probability of toxic response increases with dose: as we move from one dose up to the next higher dose, the toxicity risk at the higher dose is greater than or equal to that at the previous dose. Before the trial has commenced, Bayesian modeling requires a prior distribution for the dose–toxicity curve parameters to be specified , representing the expected shape of the dose–toxicity relationship. In the original version of the CRM, a simple one-parameter prior distribution is recommended; subsequent developments allow the prior to be informed by data (such as information from pre-clinical studies) or expert opinion (see for example  for alternatives). The model may be updated as soon as new data become available on previously included patients. Following this the next patient, or group of patients, is treated at the dose, based on the evidence to date, that has an estimated probability of toxicity closest to the target level.
Once all patients have been treated and followed up, the MTD is taken to be the dose at which the estimated toxicity probability is closest to the target level. Although the originally proposed CRM method allows for the initial dose to be above the lowest available dose level, in practice, a modified version that enforces dose escalation to start from the lowest dose  is usually employed. Stopping rules have been developed for the CRM  to allow early discontinuation of the trial in the situation where all doses are found to be excessively toxic.
Alternative Bayesian model-based approaches include escalation with overdose control (EWOC) designs  and the TITE-CRM extension of the CRM which allows the trial to be completed more quickly by incorporating time to toxicity event data .
Curve-free Bayesian approaches
This class of escalation strategies includes work by Gasparini and Eisele  and Whitehead et al.  and aims to minimize the number of assumptions within the dose–toxicity modeling. No assumption is made about the form of the relationship between dose and toxicity except that, as in the CRM, the probability of toxic response increases with dose. The risk of toxicity is modeled directly, resulting in an easy to interpret table of probabilities for each risk level.
The possible levels of toxicity risk at a dose are described qualitatively as very safe, safe, ideal, risky or toxic. The numerical probability value that corresponds to each of these descriptors depends on the target toxicity level. The “ideal” risk category has exactly the target toxicity probability, while the “safe” and “very safe” categories have progressively lower risks of toxicity and the “risky” and “toxic” categories have progressively higher risks. For example, in a study aiming to identify the dose which has toxicity rate 25 %, the descriptors very safe, safe, ideal, risky and toxic might be assigned the probability values 0.05, 0.15, 0.25, 0.4 and 0.65, respectively.
The prior distribution summarizing previous knowledge of the expected risk at each dose level, required under the Bayesian framework, is informed by investigator opinion. As data accumulate during the trial, the updated probabilities of each risk level being associated with each dose are calculated. The dose to be allocated to the next patient enrolled in the trial is selected to avoid doses that have anything other than a small chance of being “toxic” and to target the dose that has the greatest chance of being “ideal”. Table 1 illustrates the probabilities of each level of toxicity during a hypothetical trial. Here, 25 % is the target toxicity rate, and dose level 7 is the one with the highest probability of having that toxicity rate.
At the end of the trial, one of three approaches may be used to determine which dose to take forward for further study. The first method bases the decision on the final table of updated risk level probabilities. If the dose with a toxicity risk of 25 % was being sought, then the dose with the highest probability of having a toxicity risk of 25 % would be recommended. In the second, perhaps more realistic, strategy, the table of updated risk level probabilities would form just part of the information being considered by investigators when deciding what dose to recommend. The complete study data set will contain far more information than DLT occurrences: pharmacokinetic and clinical data and the expert opinion of investigators could also inform the recommendation. The third approach would be to take the data set and apply any appropriate method of statistical analysis, independently of the dose allocation method used in the trial.