• Phase 1 trials are deceptively complicated
• 3+3 not easily modifiable, except on ad-hoc basis

• Occurrence of DLT usually not instantaneous
• What to do if next patient enrolls?

1. …calculate next dose assignment excluding patients with incomplete follow-up (wastes information)
2. OR assume no DLT (\(y_i=0\)) and calculate next dose assignment (DLT may still occur)
3. OR wait until observation window is complete (refuse treatment to patient; extend trial duration)
4. OR?

• (day 1)
• S1 enrolled on \(d_1\)

• (day 15; no DLT yet for S1)
• S2 enrolls on \(d_1\)

• (day 45; no DLT yet for S1 [45 days]; no DLT yet for S2 [30 days])
• S3 enrolls on \(d_1\)

• (day 55; no DLT yet for S1 [55 days]; no DLT yet for S2 [40 days]; S3 had DLT after 5 days)
• S4 enrolls on \(d_1\) based on S3 data

• (day 95; no DLT for S1 [90 days]; no DLT yet for S2 [80 days]; S3 had DLT after 5 days; no DLT yet for S4 [40 days])
• S3 will enroll to either \(d_1\) or \(d_2\) using S1, S3 data

• Information wasted. S2 nearly complete follow-up at day 95
• Escalation is slow

• Time-to-event CRM (TITE-CRM) (Cheung and Chappell, 2000)
• When dose assignment needed:

1. Record current status of all patients (DLT / no DLT yet)
2. Calculate weights \(w_i\in(0,1]\) for those with no DLT and are still being observed
3. Use weighted binomial likelihood:

\[ L_w({ y}|\theta) = \prod_{i=1}^n x_i^{\theta y_i}(1-w_ix_i^\theta)^{1-y_i}, \]

What is the likelihood contribution of \((1-w_ix_i^\theta)\) relative to \((1-x_i^\theta)\)?

• \(w_i\) is function of time, increases to 1 as follow-up completes

• Function form based on on expected timing of DLT

• e.g. if DLT is early onset, then 80/90 days of completed follow-up should imply \(w_i\approx 1\).

• titesim
• Different choices for \(w\)
• Simulate arrival process of patients
• Calculate estimated trial duration

#June 5, 2017
#3 subjects for >1 cycle each at dose level 3 free of DLT; (001-003)
#1 subject at dose level 4 with DLT; (004)
#1 subject at dose level 4 without DLT (005)
#1 subject at dose level 4 without DLT for 9/21 days (dropped due to progression) (006)
#1 subject enrolled to dose level 5, never received drug (007)
#1 sujbect enrolled to dose level 5 without DLT for 7/21 days (dropped due to progression) (008)
#1 subject enrolled to dose level 5 without DLT ; (009)
#Data
tox_dat = c(0,0,0,1,0,0,0,0,0);
level = c(3,3,3,4,4,4,5,5,5);
n = length(tox_dat);
weights = c(1,1,1,1,1,9/21,0,7/21,1);

titecrm(prior = c(2,5,10,15,20,30,40,45)/100,
        target = 0.30,
        tox = tox_dat,
        level = level,
        weights = weights,
        method = "bayes",
        model = "logistic",
        intcpt = 0,
        scale = sqrt(1.34));

## Today:  Fri Jan 26 12:21:44 2018 
## DATA SUMMARY (TITE-CRM) 
## PID   Level   Toxicity    f/u     Weight      Included 
## 1     3   0       N/A     1       1 
## 2     3   0       N/A     1       1 
## 3     3   0       N/A     1       1 
## 4     4   1       N/A     1       1 
## 5     4   0       N/A     1       1 
## 6     4   0       N/A     0.429       1 
## 7     5   0       N/A     0       1 
## 8     5   0       N/A     0.333       1 
## 9     5   0       N/A     1       1 
## 
## Toxicity probability update (with 90 percent probability interval): 
## Level     Prior   n   total.wts   total.tox   Ptox    LoLmt   UpLmt 
## 1     0.02    0   0       0       0.048   0   0.274 
## 2     0.05    0   0       0       0.095   0.001   0.324 
## 3     0.1     3   3       0       0.156   0.006   0.366 
## 4     0.15    3   2.43        1       0.209   0.017   0.393 
## 5     0.2     3   1.33        0       0.257   0.037   0.414 
## 6     0.3     0   0       0       0.343   0.12    0.447 
## 7     0.4     0   0       0       0.423   0.278   0.475 
## 8     0.45    0   0       0       0.462   0.384   0.487 
## Next recommended dose level: 6 
## Recommendation is based on a target toxicity probability of 0.3 
## 
## Estimation details:
## Logistic dose-toxicity model: p = {1 + exp(-a-exp(beta)*dose)}^{-1} with a = 0 
## dose = -3.89 -2.94 -2.2 -1.74 -1.39 -0.847 -0.405 -0.201 
## Normal prior on beta with mean 0 and variance 1.34 
## Posterior mean of beta: -0.265 
## Posterior variance of beta: 0.464

If \(s_j = 0.3\), \[ \begin{aligned} \bar{\theta}_A(y) =\bar{\theta}_B(y)&\Rightarrow s_j^{\bar{\theta}_A(y)}=s_j^{\bar{\theta}_B(y)}\\ {\Pr}_A(\theta>1|y) > {\Pr}_B(\theta>1|y)&\Rightarrow {\Pr}_A(p_j < p_T) < {\Pr}_B(p_j < p_T) \end{aligned} \]

• \(p_T\) = target rate of DLT (as before)
• Model: \(p_j = s_j^\theta\)
• Calculate posterior probability that dose \(j\) constitutes an "overdose", given data on first \(i\) patients: \[ \begin{aligned} \gamma_{j} &\equiv \Pr(p_j > p_T|{y})\\ &= \Pr(s_j^\theta > p_T|{ y})\\ &= \Pr(\theta <\log p_T/ \log s_j|{ y})\\ &= \int_{0}^{c_j} \pi(\theta|{ y})d\theta,\qquad (c_j = \log p_T/ \log s_j)\\ &= \dfrac{\int_{0}^{c_j} L(y|\theta)\pi(\theta) d\theta}{f(y)}, \end{aligned} \]

What is the ordering of the \(\gamma_j\)s? \[ \begin{aligned} \gamma_{j-1} \quad ? \quad \gamma_{j} \end{aligned} \]

• At each patient, identify [smallest / largest ] \(j\) such that \(\gamma_{j} \quad [> / <]\quad \gamma_T\), i.e. \(j\) such that probability of overdose is tolerable.

• \(\gamma_T\) is a pre-specified acceptable overdose probability

• Replaces \(\arg\min_j |p_j-p_T|\) (from original CRM) with \(\max_j \{j:\gamma_{j} < \gamma_T\}\) (Babb, Rogatko and Zacks, 1998)
• Reminiscent of MTPI overdose control
• What are differences?
• Implemented in bcrm package (Sweeting, Mander and Sabin, 2013) (requires additional Bayesian software packages)

• How do the designs compare?

• Sample size
• Conduct
• Adaptability / extensibility

• Ethics