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 \[ \mathcal{D}_{BOIN} = \max_{i \in 0, 1, 2} Pr(H_i\mid y) \]