Chapter 1 Phase I design

mPTI

mTPI design is short for modified toxicity probability interval design (Ji et al. 2010).

Suppose the probability of toxicity at dose $d$ is $p_d$ and $n_d$ is the number of patient treated at dose $d$. The number of DLTs (dose limiting toxicity), $x_d$ has a binomial distribution $x_d\mid p_d \sim Bin(n_d, p_d)$.

Denote the target toxicity probability as $p_T$. There are 3 possible decisions by comparing $p_d$ and $p_T$:

• $M_E: p_d \in (0, p_T - \epsilon_1)$, escalate to the next level dose;
• $M_S: p_d \in (p_T - \epsilon_1, p_T + \epsilon_2)$, stay at the current dose;
• $M_D: p_d \in (p_T + \epsilon_2, 1)$, de-escalate to the lower level dose.

The mTPI design assumes the prior of $p_d$ is a Beta distribution, $p_d \sim Beta(\alpha, \beta)$. Usually $\alpha = \beta = 1$. The dose-finding decision rule is given by $$\mathcal{D}_{mTPI} = \arg\max_{i\in \{E, S, D\}} UPM(i, d)$$ where $UPM(i, d) = Pr(p_d \in M_i \mid x_d)/Pr(p_d\in M_i)$ is called the unit probability mass. In this case, $$UPM(i, d) = \frac{IBeta(c_{i2}; x+1, n-x+1) - IBeta(c_{i1}; x+1, n-x+1)}{c_{i2} - c_{i1}}$$ where $IBeta(q; a, b) = \frac{1}{B(a, b)}\int_0^q \theta^{a-1}(1-\theta)^{b-1}d\theta$.

knitr::include_app("https://eugenehao.shinyapps.io/mtpi/")

mTPI-2

mTPI-2 design is an extension of mTPI by cutting the escalation interval and de-escalation interval into subintervals with the same length equal to $(\epsilon_1 + \epsilon_2)$ (Guo et al. 2017). The equivalence interval (EI) is still $(p_T - \epsilon_1, p_T+ \epsilon_2)$. However, the set of intervals below EI (LI) are $\{(p_T - 2\epsilon_1 - \epsilon_2, p_T - \epsilon_1), \ldots, (p_T - k\epsilon_1 - (k-1)\epsilon_2, 0) \}$ and the set of intervals above EI (HI) are $(p_T + \epsilon_2, p_T + \epsilon_1 + 2\epsilon_2), \ldots, (p_T + l\epsilon_1 + (l+1)\epsilon_2, 1) \}$.

Let $A_i$ denote the set of intervals under the decision $i \in \{E, S, D\}$. The decision function of mTPI-2 design is given by $$\mathcal{D}_{mTPI2} = \arg\max_{i\in \{E, S, D\}} \max_{j \in A_i} UPM(j, d)$$

knitr::include_app("https://eugenehao.shinyapps.io/mtpi-2/")

BOIN

BOIN is short for Bayesian optimal interval design. Suppose we have $J$ prespecified doses. At the current dose level $j$, $n_j$ patients are treated and $m_j$ of them have experienced a toxicity. Let $\lambda_{1j}(n_j, \phi)$ and $\lambda_{2j}(n_j, \phi)$ be the prespecified lower and upper boundaries of the interval with $0 \leq \lambda_{1j}(n_j, \phi) \leq \lambda_2(n_j, \phi) \leq 1$ where $\phi$ is the target toxicity level. (Liu and Yuan 2015)

The next cohort dose assignment will be decided by the following steps:

• if $m_j/n_j \in [0, \lambda_{1j}]$: escalate to the next dose level $j+1$;
• if $m_j/n_j \in (\lambda_{1j}, \lambda_{2j})$: stay at the dose level $j$;
• if $m_j/n_j \in [\lambda_{2j}, 1]$: de-escalate to the dose level $j-1$.

Local BOIN

Define the three point hypothesis as: $$\begin{aligned} H_{0j}: p_j & = \phi \\ H_{1j}: p_j & = \phi_1\\ H_{2j}: p_j & = \phi_2 \end{aligned}$$

The correct decisions under $H_0$, $H_1$ and $H_2$ are retainment (R), escalation (E) and de-escalation (D). Suppose $Bin(x; n, p)$ as the CDF of a binomial distribution. The probability of making incorrect decision is given by

$$\begin{aligned} \alpha(\lambda_1, \lambda_2) &= Pr(H_0)Pr(\bar R\mid H_0) + Pr(H_1)Pr(\bar E\mid H_1) + Pr(H_2)Pr(\bar D\mid H_2) \\ & = \pi_0\left\{Bin(n\lambda_1; n, \phi) + 1 - Bin(n\lambda_2; n, \phi) \right\} + \pi_1(1 - Bin(n\lambda_1; n, \phi_1)) + \pi_2 Bin(n\lambda_2; n, \phi_2) \\ & = \alpha_1(\lambda_1) + \alpha_2(\lambda_2) + \pi_0 + \pi_1 \end{aligned}$$ where $$\begin{aligned} \alpha_1(\lambda_1) & = \pi_0Bin(n\lambda_1; n, \phi) - \pi_1Bin(n\lambda_1; n, \phi) \\ & = \sum_{y=0}^{b_{j}} \pi_{1}\left(\begin{array}{c}n \\ y\end{array}\right) \phi_{1}^{y}\left(1-\phi_{1}\right)^{n-y}\left\{\frac{\pi_{0 }}{\pi_{1}}\left(\frac{\phi}{\phi_{1}}\right)^{y}\left(\frac{1-\phi}{1-\phi_{1}}\right)^{n-y}-1\right\} \end{aligned}$$ and $$\alpha_2(\lambda_2) = \pi_2Bin(n\lambda_2; n, \phi_2) - \pi_0Bin(n\lambda_2; n, \phi).$$

$\alpha_1(\lambda_1)$ is minimized when $$n\lambda_1 = \max\left\{y: \frac{\pi_{0 }}{\pi_{1}}\left(\frac{\phi}{\phi_{1}}\right)^{y}\left(\frac{1-\phi}{1-\phi_{1}}\right)^{n-y} \leq 1 \right\} = \max \left\{y: \frac{Pr(H_0\mid y)}{Pr(H_1\mid y)} \leq 1\right\}$$ which leads to the solution $$\lambda_{1}=\frac{\log \left(\frac{1-\phi_{1}}{1-\phi}\right)+n_{j}^{-1} \log \left(\frac{\pi_{1}}{\pi_{0}}\right)}{\log \left(\frac{\phi\left(1-\phi_{1}\right)}{\phi_{1}(1-\phi)}\right)}.$$ Similarly, $$n\lambda_2 = \max \left\{y: \frac{Pr(H_2\mid y)}{Pr(H_1\mid y)} \leq 1\right\}$$ with solution $$\lambda_{2}=\frac{\log \left(\frac{1-\phi}{1-\phi_2}\right)+n_{j}^{-1} \log \left(\frac{\pi_0}{\pi_{2 }}\right)}{\log \left(\frac{\phi_2\left(1-\phi\right)}{\phi(1-\phi_2)}\right)}.$$

Global BOIN

Define the three composite hypothesis as: $$\begin{aligned} H_{0j} &: \phi_1 < p_j < \phi_2 \\ H_{1j} &: 0 \leq p_j \leq \phi_1\\ H_{2j} &: \phi_2 \leq p_j \leq 1 \end{aligned}$$ In this case, $$\begin{aligned} Pr(\bar R\mid H_0) & = \int f(p)Pr(\bar R\mid p, H_0)dp \\ Pr(\bar E\mid H_1) & = \int f(p)Pr(\bar E\mid p, H_1)dp \\ Pr(\bar D\mid H_2) & = \int f(p)Pr(\bar D\mid p, H_2)dp \end{aligned}$$ Therefore, $$\alpha(\lambda_1, \lambda_2) = Pr(H_{0}) + Pr(H_{1}) + \underbrace{\sum_{y = 0}^{b_{1}}f(y)\{Pr(H_0\mid y) - Pr(H_1\mid y) \}}_{\alpha_{g1}(\lambda_1)} + \underbrace{\sum_{y=0}^{b_2 - 1}\{Pr(H_2\mid y) - Pr(H_0\mid y) \}}_{\alpha_{g2}(\lambda_2)}$$ where $b_1 = \lfloor n\lambda_1\rfloor$ and $b_2 = \lfloor n\lambda_2 \rfloor$. Stilly, we can minimize $\alpha_{g1}(\lambda_1)$ and $\alpha_{g2}(\lambda_2)$ when $$n\lambda_1 = \max \left\{y: \frac{Pr(H_0\mid y)}{Pr(H_1\mid y)} \leq 1\right\}, \quad n\lambda_2 = \max \left\{y: \frac{Pr(H_2\mid y)}{Pr(H_0\mid y)} \leq 1\right\}.$$

Therefore, no matter we use local BOIN or global BOIN, the decision function can be defined as $$\mathcal{D}_{BOIN} = \max_{i \in 0, 1, 2} Pr(H_i\mid y)$$

References

Guo, Wentian, Sue-Jane Wang, Shengjie Yang, Henry Lynn, and Yuan Ji. 2017. “A Bayesian Interval Dose-Finding Design addressingOckham’s Razor: mTPI-2.” Contemporary Clinical Trials 58: 23–33.

Ji, Yuan, Ping Liu, Yisheng Li, and B Nebiyou Bekele. 2010. “A Modified Toxicity Probability Interval Method for Dose-Finding Trials.” Clinical Trials 7 (6): 653–63.

Liu, Suyu, and Ying Yuan. 2015. “Bayesian Optimal Interval Designs for Phase i Clinical Trials.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 64 (3): 507–23.