Bayesian interval dose-finding designs: Methodology and Application Yuan Ji, PhD. May 29, 2018 Yuan Ji, PhD. Interval-Design
Conflict of Interests The U-Design webtool is developed and hosted by Laiya Consulting, Inc., a statistical consulting firm co-founded by Yuan Ji. Yuan Ji, PhD. Interval-Design
Contents 1. Standard and new dose-finding designs 2. Interval-based designs for dose finding 3. Application via the U-Design Webtool Yuan Ji, PhD. Interval-Design
Standard and new dose-finding designs Yuan Ji, PhD. Interval-Design
Phase I dose-finding (Oncology) Consider trials with fixed doses. Setup Climb up and down a sequence of D ordered doses of a new drug to determine the maximum tolerated dose (MTD). Data At each dose i , n i patients are tested, y i patients experienced toxicity outcome (DLT). Parameters Dose i has a toxicity probability of p i (unknown). Sampling Model Binomial y i | p i ∼ Bin ( n i , p i ) Assumption Toxicity Monotonicity : p i ≤ p i +1 . Hidden assumption Efficacy Monotonicity : q i ≤ q i +1 – if not, why escalate when the dose is safe? Goal to find the MTD, defined as the highest dose with toxicity rate lower (or close to) a target rate, p T , e.g., p T = 0 . 30 . Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
A hypothetical dose-finding trial Yuan Ji, PhD. Interval-Design
Existing Designs 3+3 • Storer (1989). Algorithmic design; simple and transparent • Insufficient “power” to find the MTD even with sufficient resources • Contradicting rules : Stay for 1/3; Escalate for 1/6 (MTD not exceeded) Yuan Ji, PhD. Interval-Design
Existing Designs 3+3 • Storer (1989). Algorithmic design; simple and transparent • Insufficient “power” to find the MTD even with sufficient resources • Contradicting rules : Stay for 1/3; Escalate for 1/6 (MTD not exceeded) CRM • The first model-based design. First publication in 1990. • Performs well but could be sensitive to prior models; • black-box to physicians • Model-based interval designs in an algorithmic mTPI and mTPI-2 presentation (Ji et al., 2010; Guo et al., 2017). • Optimal under a formal Bayesian decision framework • Cumulative cohort design (CCD, Ivanova et al. CCD and BOIN 2007) – a Markov process and a simple rule • BOIN (Liu and Ying) is an extension of CCD • Their inference is based on point estimates (not accounting for variabilities) of toxicity probabilities; BOIN’s asymptotic behavior is strange Yuan Ji, PhD. Interval-Design
The Industry-Standard 3+3 Design – Yang, Wang, and Ji (2015) An integrated dose-finding tool for phase I trials in oncology. Yuan Ji, PhD. Interval-Design
The 3+3 Design: should be a past history? Features • Simple; • Transparent; • FDA review – reward for being inferior and naive Problems • Target highest dose with no more than 1/6 DLT rate; but STAY when 1 out 3 patients has DLT • Cannot target p T values different from 1/6 • Cannot have large power (3+3 treats no more than 6 patients per dose) – what if one has more resources to spend? Yuan Ji, PhD. Interval-Design
The 3+3 Design: it should be a past history • Ji and Wang (2013, JCO ) showed that with matched sample size, 3+3 is less safe and reliable when compared to the mTPI design , a model-based design. • The 2015 FDA/AACR Dose-Finding Symposium concluded that (Nie et al., 2016, Clinical Cancer Research ) “ The MTD/3+3 approach is not optimal and may result in recommended doses that are unacceptably toxic for many patients and in dose reduction/interruptions that might have an impact on effectiveness. ” Yuan Ji, PhD. Interval-Design
The CRM Design – a specific model Perhaps the most popular version of the CRM is the power model: • The dose-response curve : p i = p exp( α ) , where p i 0 are fixed and i 0 prespecified constants, and α is a parameter that describes the dose response curve. • The prior for α is N (0 , 2) . • The p i 0 ’s are decided by solving E [ p exp( α ) ] = s i , where s i ’s are a set i of prior probabilities that one must determine (called ”skeletons” ). • A binomial likelihood: � d i =1 p y i i (1 − p i ) n i − y i . • Posterior of α is obtained by numerical integration. • The next dose is arg min i | ˆ p i − p T | , where ˆ p i is the posterior mean. Yuan Ji, PhD. Interval-Design
The CRM Design – trial conduct and decision tables • Challenging to implement in practice (logistic and effort support) • Coherence (some version is NOT coherence ) and Over-dose control (e.g., no skipping dose when escalation) • Team meetings are needed for every patient allocation – CRM decisions may be overruled Output CRM “Decision Tables” for assessment (U-Design Platform ( udesign.laiyaconsulting.com )) Yuan Ji, PhD. Interval-Design
Interval Designs Yuan Ji, PhD. Interval-Design
Hallmark of ”Interval Designs” The decision of dose finding is based on toxicity probability intervals. • Interval designs : up-and-down decisions based on intervals (mTPI, mTPI-2, CCD, BOIN) Stay Escalate De-escalate p i ∈ ( p T − ǫ 1 , p T + ǫ 2 ) p i ∈ (0 , p T − ǫ 1 ) p i ∈ ( p T + ǫ 2 , 1) • Non-interval designs : • CRM chooses the dose arg min | ˆ p i − p T | , i • 3+3 uses up-and-down decisions based on y i n i , Yuan Ji, PhD. Interval-Design
Interval-Based Decision Setting Given the target toxicity probability p T (e.g., p T = 0 . 16 or 0 . 3 ), and “effect size” ǫ 1 , ǫ 2 , the interval-based decision is based on the following framework. Idea Divide (0 , 1) into three intervals: (0 , p T − ǫ 1 ) ( p T − ǫ 1 , p T + ǫ 2 ) ( p T + ǫ 2 , 1) � �� � � �� � � �� � Under dosing interval Equivalence Interval Under dosing interval [Associate with actions] E, S, D Decision rule Use Bayes’ rule to decide which action (decision) to take for the next patient. Next: Let us use mTPI and mTPI-2 as examples. Yuan Ji, PhD. Interval-Design
Decision Theory 101 Data Y is used to define likelihood function p ( Y | θ ) . Dose finding : p ( Y | θ ) ∝ � i p y i i (1 − p i ) n i − y i . Parameters are quantities of interests. Dose finding : θ = { p 1 , · · · , p D } ; Prior p ( θ | η ) p ( η ) ( η are additional parameters) Actions a to be taken (e.g., estimators). Dose finding : a ∈ { D, S, E } Optimal Decision Making is based on the following steps: Loss (Utility) function Define loss ℓ ( a, θ ) as a function of a and θ Optimality criterion define what you want to optimize. � For Bayesian, maximize posterior expected loss : ℓ ( a, θ ) p ( θ | Y ) d θ Optimal decision rule determines the action that achieves the optimality criterion. For Bayesian, we use Bayes’ Rule � a ∗ = arg min ℓ ( a, θ ) p ( θ | Y ) d θ a Yuan Ji, PhD. Interval-Design
The mTPI and mTPI-2 Designs Use Bayes’ Rule Consider mTPI for now. Data Y mTPI : p ( Y | θ ) ∝ � i p y i i (1 − p i ) n i − y i . Parameter θ Tox probs { p i } = { p 1 , · · · , p D } ; In addition to { p i } , introduce another parameter m i ∈ { M D , M S , M E } and M D : p i ∈ ( p T + ǫ 2 , 1) M S : p i ∈ ( p T − ǫ 1 , p T + ǫ 2 ) M E : p i ∈ (0 , p T − ǫ 1 ) Then mTPI : θ = { p i } ∪ { m i } Actions a mTPI : a ∈ { D, S, E } . Loss function mTPI : � 1 , if i � = j ; ℓ ( a = i, m i = M j ) = for i, j ∈ { E, S, D } . 0 , if i = j, Optimal decision mTPI uses Bayes’ rule : For dose i , choose decision a ∗ ( p T , ǫ 1 , ǫ 2 ) = arg j ∈{ D,S,E } Pr ( m i = M j | Y ) max Yuan Ji, PhD. Interval-Design
The mTPI Design Given p T , ǫ 1 , ǫ 2 , all the decisions a ∗ are pre-calculated for all the possible data Y at dose i . – An mTPI decision table but in an algorithmic form Yuan Ji, PhD. Interval-Design
UPM and Bayes rule If we assume a simple “working” model (Ji et al., 2010), indep y i | p i ∼ Binom ( n i , p i ) , and p i ∼ Beta (1 , 1) then the working posterior is q ( p i | y ) = Beta (1 + y i , 1 + n i − y i ) Turns out the mTPI’s rule to max Pr ( m i = M j | Y ) is the equivalent tomax the unit probability mass under the working posterior . Pr ( m i = M j | Y ) = UPM ( M j ) = q ( p i ∈ M j | Y ) length ( M j ) , j ∈ { D, S, E } Yuan Ji, PhD. Interval-Design
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