Dose-Finding and Phase I/II Designs
Bayesian Phase I dose-finding using the modified Toxicity Probability Interval design of Ji, Liu, Li & Bekele (2010). Enter the target toxicity probability, equivalence interval, and observed DLTs at the current dose to calculate the posterior unit probability masses and the corresponding escalate, stay, or de-escalate decision. Runs entirely in your browser.
The modified Toxicity Probability Interval (mTPI) design is a Bayesian interval-based Phase I dose-finding method introduced by Ji, Liu, Li, and Bekele in 2010. It uses a beta-binomial model and replaces the calibration parameters used in the earlier TPI method with a prespecified equivalence interval around the target toxicity probability.
Let x be the number of DLTs among n patients treated at the current dose. The original mTPI method uses a uniform Beta(1,1) prior for the dose-specific toxicity probability p. The resulting posterior distribution is:
The calculator uses this continuous beta posterior directly. No normal approximation or rounding of the posterior distribution is used.
Given target toxicity probability φ, the mTPI method defines an equivalence interval using ε1 and ε2:
The original paper notes that values such as 0.05 for both ε1 and ε2 are commonly used, although the two quantities can be different when the clinical setting calls for an asymmetric acceptable toxicity interval.
For an interval (a,b), its UPM is its posterior probability divided by the interval width. Thus the three quantities are:
The largest UPM determines the dose-assignment action: the underdosing interval corresponds to escalation, the proper-dosing interval corresponds to staying at the current dose, and the overdosing interval corresponds to de-escalation. This is the central mTPI decision rule described by Ji et al.
The practical mTPI safety rule evaluates the posterior probability that the current dose exceeds the target toxicity probability. If Pr(p > φ | data) exceeds a prespecified cutoff ξ, conventionally 0.95 in the original method, the current dose and higher doses are excluded. If the lowest dose is found excessively toxic, the trial is stopped for safety.
Target φ = 0.30; ε1 = 0.05; ε2 = 0.05; n = 3; x = 1.
Posterior: Beta(2, 3).
UPME = 1.046875, UPMS = 1.753000, UPMD = 0.866125.
Expected decision: Stay at the current dose.
This matches the published mTPI decision table for a target DLT probability of 30%, where 1 DLT among 3 patients produces an S (stay) decision.
Ji, Y., Liu, P., Li, Y., & Bekele, B. N. (2010). A modified toxicity probability interval method for dose-finding trials. Clinical Trials, 7(6), 653–663. DOI: 10.1177/1740774510382799 .
Ji, Y., & Wang, S.-J. (2013). Modified Toxicity Probability Interval Design: A Safer and More Reliable Method Than the 3 + 3 Design for Practical Phase I Trials. Journal of Clinical Oncology, 31(14), 1785–1791. DOI: 10.1200/JCO.2012.45.7903 .
The primary methodological source defines the mTPI decision through the beta-binomial posterior and unit probability mass, while the later JCO paper provides published examples and decision tables for practical Phase I implementation.