The Bayesian Optimal Interval (BOIN) design is a structured approach for dose-finding in phase I clinical trials. It aims to balance effective dose escalation with patient safety by using observed dose-limiting toxicity (DLT) rates to guide dose adjustments.

BOIN Design for Single-Agent Trials

In the BOIN design, a series of doses `d`

are tested, each with an associated DLT probability _{1} < d_{2} < ... < d_{J}`p`

. The target DLT rate for the Maximum Tolerated Dose (MTD) is denoted as _{j}`φ`

. The design proceeds as follows:

**Initial Dose Assignment:**The first cohort of patients is treated at the lowest dose level or a predetermined starting dose.**Dose Adjustment Rules:**At each dose level`j`

, the observed DLT rate`^p`

is calculated, where_{j}= y_{j}/ n_{j}`y`

is the number of DLTs and_{j}`n`

is the total number of patients at dose_{j}`d`

. Based on_{j}`^p`

, the dose adjustment decisions are made as follows:_{j}- If
`^p`

, escalate to the next higher dose_{j}<= λ_{e}`d`

._{j+1} - If
`^p`

, de-escalate to the next lower dose_{j}>= λ_{d}`d`

._{j-1} - Otherwise (i.e.,
`λ`

), maintain the current dose level._{e}< ^p_{j}< λ_{d}

- If
**Trial Termination and MTD Selection:**The trial continues until the maximum sample size`N`

is reached or the trial is terminated due to excessive toxicity. The MTD is identified as the dose where the isotonic estimate of the DLT probability is closest to the target_{max}`φ`

. If the trial is stopped due to excessive toxicity, no dose is selected as the MTD.**Safety Stopping Rule:**If the posterior probability`P(p`

and_{j}> φ | y_{j}, n_{j}) > 0.95`n`

, the current dose_{j}>= 3`d`

and higher doses are eliminated from the trial. The trial is terminated if the lowest dose is eliminated._{j}

The posterior probability `P(p`

is evaluated using a beta-binomial model, where _{j} > φ | y_{j}, n_{j})`y`

and _{j} | n_{j}, p_{j} ~ Binom(n_{j}, p_{j})`p`

. The posterior distribution of _{j} ~ Beta(1, 1)`p`

is given by:_{j}

p_{j} | y_{j}, n_{j} ~ Beta(y_{j} + 1, n_{j} - y_{j} + 1)

The escalation and de-escalation boundaries `λ`

and _{e}`λ`

are calculated to minimize the risk of incorrect dose decisions. The formulas for these boundaries are:_{d}

\[ \lambda_e = \frac{\log \left( \frac{1 - \phi_1}{1 - \phi} \right)}{\log \left( \frac{\phi (1 - \phi_1)}{\phi_1 (1 - \phi)} \right)}, \] \[ \lambda_d = \frac{\log \left( \frac{1 - \phi}{1 - \phi_2} \right)}{\log \left( \frac{\phi_2 (1 - \phi)}{\phi (1 - \phi_2)} \right)}, \]

In these formulas, `φ`

and _{1}`φ`

are thresholds for dose escalation and de-escalation, respectively. Default values are often _{2}`φ`

and _{1} = 0.6 φ`φ`

, but these can be adjusted to meet specific trial requirements._{2} = 1.4 φ

The BOIN design provides a clear framework for dose adjustments and safety monitoring, making it user-friendly for both clinicians and regulatory agents. By setting appropriate boundaries and calibrating the design parameters, the BOIN approach ensures that trials are conducted with a focus on patient safety and effective dose determination.

In the example below using R, we use the `get.boundary`

function with the following parameters:

` ````
```bound <- get.boundary(target=0.3, ncohort=10, cohortsize=3)
summary(bound)

` ````
```Escalate dose if the observed DLT rate at the current dose <= 0.2364907
Deescalate dose if the observed DLT rate at the current dose > 0.3585195
This is equivalent to the following decision boundaries
Number of patients treated 3 6 9 12 15 18 21 24 27 30
Escalate if # of DLT <= 0 1 2 2 3 4 4 5 6 7
Deescalate if # of DLT >= 2 3 4 5 6 7 8 9 10 11
Eliminate if # of DLT >= 3 4 5 7 8 9 10 11 12 14
A more completed version of the decision boundaries is given by
Number of patients treated 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Escalate if # of DLT <= 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5
Deescalate if # of DLT >= 1 1 2 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 7 8 8 8 9 9 9
Eliminate if # of DLT >= NA NA 3 3 4 4 5 5 5 6 6 7 7 8 8 8 9 9 9 10 10 11 11 11 12
Number of patients treated 26 27 28 29 30
Escalate if # of DLT <= 6 6 6 6 7
Deescalate if # of DLT >= 10 10 11 11 11
Eliminate if # of DLT >= 12 12 13 13 14
Default stopping rule: stop the trial if the lowest dose is eliminated.

In the BOIN design, the target toxicity rate, number of cohorts, and cohort size determine the decision boundaries for dose escalation, de-escalation, and elimination.

The function output helps in making decisions based on the observed Dose-Limiting Toxicity (DLT) rate at each dose level:

If the observed DLT rate at the current dose level is below the escalation threshold, the dose should be increased. If it is above the de-escalation threshold, the dose should be reduced. If the DLT rate is high enough, the dose level should be removed from consideration.

In general, the decision boundaries adapt based on the total number of patients treated:

**Escalation:**Increase the dose if the number of observed DLTs falls below a certain threshold.**De-escalation:**Decrease the dose if the number of observed DLTs exceeds a certain threshold.**Elimination:**Remove the dose if the number of observed DLTs is very high.

The boundaries become more detailed with increasing numbers of patients, allowing for finer adjustments in decision-making:

**Escalation:**Thresholds for increasing the dose are adjusted as the number of patients treated increases.**De-escalation:**Thresholds for decreasing the dose also change with the total number of patients treated.**Elimination:**Criteria for removing a dose are similarly refined based on patient numbers.

The trial typically stops if the lowest dose level is eliminated based on these predefined boundaries.