# Chapter 1: Overview

### 1.1 What’s New in U-Design Version 1.1.5

1. Chinese version. You could switch between English and Chinese using the selection button at the right corner of the global top menu.
2. i3+3. A new single agent cohort-based dose finding design, i3+3, was added.
3. Updated simulation results report. Simulation results are embedded in the clinical trial protocol template now.
4. Restore function. Added a restore button to restore simulation settings

### 1.2 What’s New in U-Design Version 1.1.4

1. A new UI and more informative content for U-Design. U-Design Version 1.1.4 has a new home page. Users will find everything they need to know about U-Design, such as overview, quick demo, tutorials, pricing, FAQ, etc.
2. Tutorial Videos. Tutorial videos were added to U-Design HOW IT WORKS section.
3. Random seed in single-agent modules. Users can now input random seed values in both single-agent cohort-based and rolling based designs. See Chapters 3 and 4 for more details.
4. BLRM decision table generation in Single Agent - Cohort-Based Designs. This version of U-Design will generate the empirical BLRM decision table if BLRM was included in the simulation studies. See Section 3.4.1 for details.
5. Simulation results embedded in the clinical trial protocol template. The generated downloadable simulation report zip file includes a word file that incorporates simulations results, including all line plots, tables and decision tables of CRM and BLRM, in addition to the clinical trial protocol template. See Chapter 2 for details.

### 1.3 What’s New in U-Design Version 1.1

1. Dual-agent cohort-based dose finding designs. This version of U-Design added a new module for dual-agent dose-finding designs, including the Bayesian logistic regression model (BLRM; Neuenschwander et al. (2015)), and the method of the Product of Independent beta Probabilities dose Escalation (PIPE; Mander and Sweeting (2015)). See Chapter 6 for more details.

### 1.4 Overview of U-Design Version 1.0

U-Design is a web-based next-generation statistical tool that aims to speed up drug development and improve patient care by providing reliable, cutting-edge, and user-friendly designs and methods for clinical studies.

U-Design was ﬁrst released in April 2017. The latest ofﬁcial version of U-Design, Version 1.0, is released in June 2018 . In this version, an exclusive module of Rolling-Based Designs is introduced with the ability to compare several rolling-based and cohort-based designs in terms of trial duration. Currently, U-Design focuses on phase I dose- ﬁnding designs for single agent and offers eight of them. They are the 3+3 design (Storer, 1989), the modiﬁed toxicity probability interval (mTPI) design (Ji et al., 2010, Ji and Wang, 2013, Yang et al., 2015), the mTPI-2 design (Guo et al., 2017), the rolling six design (Skolnik et al., 2008), the R-TPI design (Guo et al., sion), the modiﬁed cumulative cohort design (mCCD; the original CCD design was introduced in (Ivanova et al., 2007)), the continual reassessment method (CRM) (O’Quigley et al., 1990), and the Bayesian logistic regression method (BLRM) (Neuenschwander et al., 2008).

U-Design is an extremely user-friendly web-based tool that generates efﬁcient and powerful designs and a sophisticated protocol template including a detailed statistical section in a few minutes. To our knowledge, U-Design is the only web-based design tool that 1) does not require software download or maintenance, 2) only requires internet access and an internet browser (Chrome, Firefox, etc), 3) can be accessed on any computers regardless of operating system, and 4) can be personalized for individual accounts.

The U-Design web-tool contains three modules to fulﬁll the need to design a phase I dose-ﬁnding clinical trial. The ﬁrst module is Cohort-Based Designs (Chapter 3), which allows users to conduct computer-simulated trials and compare up to four different cohort-based designs using these trials. The second module is Rolling-Based Designs (Chapter 4), which allows users to simulate the timeline settings in real-life trials, such as the speed of patient enrollment and the maximum follow-up time . This module of rolling-based designs is the only tool on the market that incorporates the comparison of trial duration among different designs, including rolling-based designs (rolling six and R-TPI) that aim to accelerate phase 1 trials, and cohort-based designs (3+3 and mTPI-2). Besides the trial duration comparison, users can also compare operating characteristics of different designs which are also provided in the module of cohort-based designs. The third module is Decision & MTD (Chapter 5), which allows users 1) to generate and examine dose-ﬁnding decision tables for four designs, mTPI, mTPI-2, mCCD and 3+3, 2) to make the mTPI-2 decision of MTD selection based on accumulated data for a real trial.

Besides, U-Design provides draft of the statistical section for the trial protocol (Chapter 2), thereby eliminating the need for manual and non-reproducible word processing. The free draft, which can be downloaded by U-Design subscribers for free, contains the template of a clinical trial protocol. Moreover, a full version of the draft contains the full details of the statistical section of the protocol, which would be ready for FDA submission. The full version can be purchased via U-Design and provided upon the users' request.

# Chapter 2: Clinical Trial Protocols

U-Design offers a free clinical trial design protocol template with a draft statistical section, that consists of two parts:

1. A summary of recommended dose-finding design used in the main protocol.
2. An appendix section with detailed technical description of designs and simulation studies.

Two versions of the free clinical trial protocol are available on U-Design, based on either of Cohort-Based Designs or Rolling-Based Designs:

1. The template for single drug cohort-based dose-finding designs described in Chapter 3, including mTPI-2, mTPI, 3+3, CRM, BLRM, etc.
2. The template for single drug rolling-based dose-finding designs described in Chapter4, including R-TPI, rolling six, mTPI-2, and 3+3.

They can be downloaded for free on U-Design website together with the detailed simulation results after launching the simulation. See Sections 3.4.3, 4.4.3, and 6.4.3 for details.

Based on the free template, users can modify the template contents with their choice of design and the corresponding simulation results, and finalize an official protocol for their own designs by themselves. In addition, upon users’ request, U-Design can also provide a more sophisticated protocol including a simulation full report and detailed description of designs, tailored for users’ needs. The simulation full report is built on top of simulation results of our recommended dose-finding designs for the specifics of your clinical trial, which will be ready to be submitted to FDA. Contact us to request full report info@laiyaconsulting.com.

# Chapter 3: Single Agent – Cohort-Based Designs

This module of Cohort-Based Designs performs trial simulation to examine the operating characteristics of six de- signs, including mTPI-2 (Guo et al., 2017), CRM (O’Quigley et al., 1990), 3+3 (Storer, 1989), mTPI (Ji et al., 2010, Ji and Wang, 2013, Yang et al., 2015), mCCD (Ivanova et al., 2007) and BLRM (Neuenschwander et al., 2008). In the Cohort-Based Designs page, there are two main tabs: Simulation Settings and Simulation Results.

In the Simulation Settings tab, there are three panels: 1) Review design & scenario, 2) Add design, and 3) Add scenario (See Figure 3.1). Users need to first add the selected designs for comparison in the panel of Add design, and then add scenarios in the panel of Add scenarios to carry out the simulation. All the added designs and scenarios will be reviewed in the panel of Review design & scenario, and users can make further modification on the added designs and scenarios per their discretion in this panel. After the simulation is launched, the results of simulations will be shown in the Simulation Results tab. Specifically, the simulation process can be monitored in real time in the panel of Running Simulations and the simulation results will be stored and displayed in the Simulation History panel. Detailed steps of using this module are elaborated in Section 3.1 - 3.4.

Figure 3.1: Construction of designs and scenarios in the Cohort-Based Designs

### 3.1 Construct Designs

In the module of Cohort-Based Designs, U-Design provides six designs for simulation. Users can choose up to 4 of them for simultaneous comparison in the Simulation Settings tab each time.

Instructions:

In the Add Design panel, there are six designs: mTPI-2, CRM, 3+3, mTPI, mCCD and BLRM (Figure 3.2). Click a XX button to add the XX design. Enter the required parameter values for the selected design (Table 3.1). Then click Submit button to submit the candidate designs to the panel of Review design & scenario. Click Delete button to remove each design (Figure 3.2). Figure 3.2 is an example if one click the buttons for mTPI-2, CRM and 3+3. Notice that the required parameters for different designs are different, and the parameter input for each design is summarized in Table 3.1.

Figure 3.2: Add designs in the Cohort-Based Designs.

Output:

If one clicks Submit button in Figure 3.2, the candidate designs will be submitted to the panel of Review design & scenario (see Figure 3.3). One can also modify the designs parameters in the panel of Review design & scenario following the instructions in Chapter 3.1.2.

##### 3.1.2 Edit Designs

Designs can be modified in the panel of Review design & scenario. This allows for flexible modifications in practice.

Table 3.1: Input arguments for designs in the Cohort-Based Designs.
Notation Arguments Description
$n$
(all designs)
Sample size The maximum number of patients to be treated in the trial. Here, the hard limit is set at 100 since the number of patients that are enrolled in phase I clinical trial is typically small. For most cases, users can input a number smaller than 100, e.g., 30.
$n_{cohort}$
(all designs)
Cohort size The number of patients in each cohort (e.g., 3).
$ε_1, ε_2$
(mTPI, mTPI-2, mCCD, BLRM)
$ε_1, ε_2$ Two small fractions used to define the equivalence/target interval of the MTD (default is 0.05 in all interval-based designs). Any doses with a toxicity probability falling into the interval $(p_T - ε_1, p_T + ε_2)$ will be considered an acceptable dose level as MTD.
$p_{EWOC}$
(BLRM)
Cutoff probability of escalation with over-dose control The threshold of controlling the probability of excessive or unacceptable toxicity. (Only for BLRM)
$δ$
(CRM)
Half-width The starting dose level in the simulation trials.
$d_{start}$
(all designs)
Starting dose level The starting dose level in the simulation trials.
Figure 3.3: Design list in the panel of Review Design & Scenario of Cohort-Based Designs.

Instructions:

Click Delete All button at the bottom of design list (Figure 3.3) to remove all the existing scenarios. Click Edit button to edit the parameter of each design or remove the selected design. After clicking Edit button in Figure 3.3, the edit mode is enabled (Figure 3.4). All the values in the input-cell can be changed. Click the Delete button to delete a specific design. When all the changes are made, click Save button to confirm and reload the design list.

Figure 3.4: Edit designs in the Cohort-Based Designs.

### 3.2 Construct Scenarios

Simulated trials are generated under scenarios in which the true toxicity probabilities for the experimental doses are given. In the panel of Add scenario, there are two parts of settings: Basic setting and Scenario setting. In the Scenario setting, users can select to add Default scenarios or add their own scenarios of interest manually. Figure 3.5 provides the user interface for an illustration. The detailed descriptions of input parameters in Figure 3.5 are defined in Table 3.2.

Figure 3.5: Add scenarios in the Cohort-Based Designs.
Table 3.2: Input arguments for scenarios in the Cohort-Based Designs.
Notation Arguments Description
$p_T$ Target toxicity probability The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability closest to or lower than the target probability.
$n_{sim}$ The number of simulated trials The maximum number of simulated trials is 5000.
$n_{dose}$ The number of dose levels The maximum number of prespecified dose levels is 10.

Instructions

In the Basic setting, specify the target toxicity probability pT and the number of simulated trials nsim. In the Scenario setting, U-Design provide two ways of adding scenarios:

• Click Default scenarios. Then by clicking numeric button (right to $n_{dose} =$), U-Design will automatically generate a series of default scenarios with diverse patterns.
• Click Manual scenarios button to unfold the manually input interface. Scenarios can be constructed manually using two different means: Add a single scenario for adding a single scenario and Add a batch of scenarios for adding multiple scenarios in a batch.
• In Add a single scenario section, select a number of doses ($n_{dose}$) and there will subsequently appear a list of dose level input boxes under the Add a single scenario section; then type in the true toxicity probability for each dose level. Then click Submit button to submit the single scenario.
• In Add a batch of scenarios section, users can submit multiple scenarios in batches and then click Submit button to submit them. The format of input must comply with following instructions.
• Scenarios should be separated by line break;
• The parameters in one scenario should be ordered in accordance with this sequence:
• Target toxicity probability, Number of simulated trials, True toxicity probabilities of all the dose levels; Each parameter must be separated by a WHITE SPACE or COMMA.

In addition, U-Design provides a handy function to automatically generate the 42 scenarios in (Ji and Wang, 2013). Click the The 42 Scenarios in Ji and Wang (2013, JCO) button to insert these 42 scenarios into the Add a batch of scenarios panel for simulation. We don’t suggest to run these 42 scenarios using CRM or BLRM unless users can wait for a long time, say hours, because these two methods, especially BLRM, are slow.

Output:

The submitted scenarios are displayed as a scenario list (Figure 3.6). The list will appear in the top region of the web page. See Figure 3.1 for an example.

Figure 3.6: Scenario list in the Cohort-Based Designs.
##### 3.2.2 Edit Scenarios

Scenarios can be modified in the panel of Review design & scenario. This allows for flexible modifications in practice.

Instructions

Click Delete All button at the bottom of scenario list (Figure 3.6) to remove all the existing scenarios. Click Edit button to edit parameter in the scenario settings or remove the selected scenario.

Figure 3.7: Edit scenario in the Cohort-Based Designs.

After clicking Edit the button, the edit mode is enabled (Figure 3.7). All the values in the input-box can be changed. Click the Delete button to delete a specific scenario. When all the changes are made, click Save button to confirm the edit and reload the scenario list.

### 3.3 Launch Simulation

Once the designs and scenarios are constructed, users can conduct simulated clinical trials to examine the operating characteristics of these designs using these scenarios, by clicking the Launch Simulation button (Figure 3.1). A green "Launch Successful" message will then be displayed on the website as in Figure 3.8 to indicate that the simulation has been successfully launched. Also, users can click the here hyperlinks to either track the simulation processing status (Figure 3.9) and simulation results in the Simulation Results tab or reset the Simulation Settings page.

Figure 3.8: The displayed "Launch Successful" message after launching simulations in the Cohort-Based Designs.

In addition, U-Design allows users to change the random seed for the simulation studies (the box right to $R_{seed} =$ located at the upper-right of the Review design & scenario panel, see Figure 3.7).

### 3.4 View Simulation Results

The results of submitted simulations can be viewed in the Simulation Results tab. In the Simulation Results tab, the Running Simulation panel exhibits the process of running simulation (see Figure 3.9 for an illustration). Once the simulations are completed, click the Refresh button on the top right corner of the Simulation History panel (the blue box in Figure 3.9) to load the latest simulation results. All completed simulations are listed in the Simulation History panel. Click the View button (the green box in Figure 3.9) to unfold the simulation results. Click the Restore button (the black box in Figure 3.9) will restore the simulation settings that produced this simulation result to the Simulation Settings tab.

The results of simulation are shown in two ways: Simulation Result Plots and Simulation Result Tables.

Figure 3.9: Simulation progress in the Cohort-Based Designs.
##### 3.4.1 Simulation Result Plots

There are three sections in the Simulation Result Plots:

1. Line plots reflecting four summary statistics of the simulation results for all the designs (Figure 3.10).
2. A table of mean and standard deviation for four summary statistics of the simulation results for all the designs across all simulated scenarios (Figure 3.11).
3. [Optional] An empirical CRM decision table if CRM is selected in the simulation (Figure 3.12).
Figure 3.10: Simulation result plots in the Cohort-Based Designs.
Figure 3.11: Simulation summary in the Cohort-Based Designs.

Note for simulation result plots:

• The four summary statistics are
1. Prob. of Select MTD: The probability of selecting the true MTD in the end. For interval-based designs (mTPI, mTPI-2, BLRM, & mCCD), the true MTDs is defined as the dose levels of which the true toxicity probabilities fall into the equivalence interval; if none of the doses has a toxicity probability that falls in the equivalence interval, the true MTD is defined as the dose with the highest toxicity probability that is below $p_T$ . But for those non-interval-based designs, e.g., 3+3 and CRM, the true MTDs is defined as the dose levels with the highest toxicity probabilities lower than or equal to $p_T$ . To fairly compare the operating characteristics of multiple designs submitted in batch in one simulation study, the definition of MTD should be unified. If any of interval-based designs (mTPI, mTPI-2, BLRM, & mCCD) are used in the simulation, dose levels of which the true toxicity probabilities fall into the widest equivalence interval are defined as the true MTDs (if none of doses has a toxicity probability that falls in, the dose with highest toxicity probability that is below $p_T$ is the true MTD). For example, consider a case that one wants to compare four designs,mTPI,mTPI-2,CRM and 3+3,in a simulation study targeting $p_T = 0.3$,and $ε_1$ and $ε_2$ are 0.02 and 0.05 for mTPI, but 0.05 and 0.03 for mTPI-2. In this case, the true MTD is the dose levels with toxicity probabilities falling in (0.3-0.05, 0.3+0.05); if none of the dose has a toxicity probability in (0.3-0.05, 0.3+0.05), the dose with the highest toxicity probability lower than 0.3 is the true MTD.
2. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulated trials.
3. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
4. Prob. of No Selection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
• For each plot, the x axis is the index of scenario and the y axis is the value of summary statistics (Prob. of No Selection, Prob. of Toxicity, Prob. of Select Dose-over-MTD, Prob. of Select MTD). Lines with different colors represent different designs.
• The plots are interactive for better visualization.
• Hover the mouse on the plots and a box will display the value of each design at corresponding scenario (bottom left plot in Figure 3.10: Prob. of Select Dose-over-MTD).
• Hover the mouse on the design label to highlight the corresponding line and fade others (top right plot in Figure 3.10: Prob. of Toxicity ).
• Click the design label to hide the corresponding line and click again to change it back (bottom right plot in Figure 3.10: Prob. of No Selection).

Note for CRM decision table: An empirical CRM decision table will be given in the simulation results if CRM is included in the simulation (Figure 3.12). This table summarizes the frequency of decisions made by CRM across all the simulated trials.

• The lengths of three colored bars in one cell represent the frequencies of dose-finding decisions under the corresponding combination of number of patients and number of DLTs (Figure 3.12). The longer the bar, the higher the frequency. For example, in the cell above pointed by the mouse, it shows that CRM de-escalates the dose 59.6% of the times when 2 out of 3 patients experienced DLTs.
Figure 3.12: CRM decision table in the Cohort-Based Designs.

Note for BLRM decision table: An empirical CRM decision table will be given in the simulation results if BLRM is included in the simulation (Figure 3.13). This table summarizes the frequency of decisions made by BLRM across all the simulated trials.

• The lengths of three colored bars in one cell represent the frequencies of dose-finding decisions under the corresponding combination of number of patients and number of DLTs (Figure 3.13). The longer the bar, the higher the frequency. For example, in the cell above pointed by the mouse, it shows that BLRM stay at the current dose 75% of the times when 1 out of 3 patients experienced DLT.
Figure 3.13: BLRM decision table in the Cohort-Based Designs.
##### 3.4.2 Simulation Result Tables

In Figure 3.13, one table represents one scenario, consisting of two parts. In the upper part, the first two columns summarize the scenario, with dose levels and their true toxicity probabilities; the remaining columns report three summary statistics, selection probability, number of patients treated and number of toxicities. The true MTD(s) of the scenario is(are) highlighted by yellow. In the lower part, four additional rows report four more summary statistics.

Output:

Figure 3.14: Simulation result tables in the Cohort-Based Designs.
• The seven summary statistics are
1. Selection Prob.: The probability of selecting each dose level as the MTD.
2. # of Patients Treated: The average number of patients treated at each dose level.
3. # of Toxicities: The average number of patients experienced DLT at each dose level.
4. Prob. of Select MTD: The probability of selecting the true MTD. For the definition of the true MTD, please refer to Section 3.4.1.
5. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulation trial.
6. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
7. Prob. of No Selection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
• The design setting will be kept at the top of page while scrolling down the page to review the simulation results.

There will be a Download U-Design Simulation Report button at the upper left corner of each simulation results panel. Click it to download a zip file, which includes two word files and four images of line plots of summary statistic shown in Figure 3.10. The two word files are a template for the statistical section in a protocol and a simulation report with complete simulation results under the designs and scenarios users added in the simulation settings tab, respectively. Users could modify the protocol template and update the simulation settings and results tailored for your trial by themselves or contact us via email info@laiyaconsulting.com for consulting service to get the full version of the statistical section for protocol and the simulation results.

### 3.5 Methods Review

In the module of Cohort-Based Designs we provide the 3+3, mTPI, mTPI2, mCCD, CRM and BLRM designs. Assume that $I$ doses $\{d_1 , d_2 , … , d_I \}$ are considered in the trial and let $p_i$ be the true toxicity probability for dose $d_i$ . Let $n_i$ and $y_i$ be the number of patients treated at $d_i$ and the number of patients with DLTs, respectively. The method description of each design is given in the following subsections.

##### 3.5.1 The 3+3 Design

The 3+3 design is an algorithmic approach. The full algorithm for 3+3 is provided in Figure 3.15 (Yang et al., 2015).

Assuming dose $i$ is the dose currently being used in the trial, we provide the above schema to illustrate the 3+3 design. Notation $n_i+1$ refers to the number of patients that have been treated at dose level $i + 1$, the next higher dose.

Figure 3.15: Schema of the 3+3 design.
##### 3.5.2 The Modified Toxicity Probability Interval Design (mTPI)

The mTPI design(Ji et al., 2010) is an extension of the toxicity probability interval method (Ji et al., 2007) and employs a simple independent beta-binomial model. Decision rules are based on calculating the normalized probability (i.e. the unit probability mass (UPM) ) of three intervals corresponding to under, proper, and over dosing. Specifically, the under-dosing interval is defined as $(0, p_T - ε_1 )$, the over-dosing interval as $(p_T + ε_2 , 1)$, and the equivalence interval as $(p_T - ε_1, p_T + ε_2)$ for proper dosing, where $ε_1$ and $ε_2$ are small fractions, such as 0.05, to account for the uncertainty around the true target toxicity. The three dosing intervals are associated with three different dose-escalation decisions. The under-dosing interval corresponds to a dose escalation (E), the over-dosing interval corresponds to a dose de-escalation (D), and the equivalence interval corresponds to staying at the current dose (S). Given an interval and a probability distribution, define the unit probability mass (UPM) of that interval as the probability of the interval divided by the length of the interval. The mTPI design calculates the UPMs for the three dosing intervals, and the one with the largest UPM implies the corresponding dose-finding decision. That decision provides the dose level to be used for future patients. For example, if the under-dosing interval has the largest UPM, decision E will be executed and the next cohort of patients will be treated at the next higher dose level. At the end of the trial, isotonic transformation is applied and the posterior mean of $p_i$ is calculated, denoted as $p_i↖{\^}$. The mTPI design selects the dose with a $p_i↖{\^}$ that is closest to $p_T$ and in the meanwhile less than $pT + ε_2$ as the MTD.

In addition, two safety rules are applied to protect patients from being exposed to overly toxic doses.

• Safety rule I: if the lowest dose (by default the 1st dose) has a high chance of being above the MTD, defined as the probability $Pr\{p_1 > p_T |data\} > 0.95$, and in the meanwhile at least two DLTs have been observed at the lowest dose, the trial will be terminated before the maximum the sample size is reached.
• Safety rule II: if for any dose i, $Pr{p_i > p_T |data} > 0.95$, and in the meanwhile at least two DLTs have been observed at dose $i$, dose $i$ and all the doses higher than $i$ will be excluded from the trial.

These rules are applied to the mTPI design, the mTPI-2 design (Chapter 3.5.3), the mCCD design (Chapter 3.5.4), CRM (Chapter 3.5.5) and BLRM (Chapter 3.5.6).

##### 3.5.3 The Modified Toxicity Probability Interval-2 Design (mTPI-2)

The mTPI-2 design (Guo et al., 2017) is an extension of mTPI (Ji et al., 2010). Following the idea of the mTPI design, decisions of mTPI-2 are based on the posterior probabilities that the toxicity rate $p_i$ of dose level $i$ falls into different intervals.

The mTPI-2 design improves over the mTPI design by blunting the Ockhams razor that leads to some statistically sound but practically debatable decisions in mTPI. For example, for a target toxicity probability of $p_T =0.3$,mTPI will recommend no dose change if 3 out of 6 patients experienced DLTs at a dose, which may be considered too aggressive by physicians. However, mTPI-2 resolves the problem fundamentally by dividing the intervals $(0, p_T - ε_1 )$ and $(p_T + ε_2, 1)$ into shorter subintervals with length $(ε_1 + ε_2)$, which is the same as the length of the equivalence interval $(p_T - ε_1, p_T + ε_2)$. As a result, for the same settings in the example above, mTPI-2 will recommend de-escalation to the lower dose, which is compatible with physicians opinion. In particular, each subinterval corresponds to one of the three decisions for dose finding, E – escalate the dose by one level, S – stay at the current dose, and D – de-escalate the dose by one level. The subintervals below $p_T$ correspond to E, subintervals above $p_T$ correspond to D, and the interval $(p_T - ε_1, p_T + ε_2)$ corresponds to S. The mTPI-2 design selects the interval with the maximum posterior probability as the winning interval in which the toxicity rate pi falls, and takes the decision (E, S, or D) corresponding to the winning interval. It is shown that the mTPI-2 design is optimal in that the decision rule above minimizes the Bayesian risk of wrong dose allocations (Guo et al., 2017). The two safety rules of the mTPI design described in Chapter 3.5.2 are also applied to mTPI-2. In addition, the dose selection rule of mTPI-2 is the same as that of the mTPI design.

##### 3.5.4 The Modified Cumulative Cohort Design (mCCD)

The original CCD design (Ivanova et al., 2007) is based on a simple algorithm which can be described below.

Assuming dose $i$ is the dose currently being used in the trial:

1. Escalate to dose $(i+1)$ if $y_i\/n_i < p_T - ε_1$
2. Stay at dose $i$ if $y_i\/n_i ∊ (p_T - ε_1, p_T + ε_2)$
3. De-escalate to dose $(i-1)$ if $y_i\/n_i > p_T + ε_2$

We set $ε_1$ and $ε_2$ as user provided input values, just as the mTPI and mTPI-2 designs in which $(p_T - ε_1, p_T + ε_2)$ is considered as the equivalence interval. With this modification and the additional safety rule I and II in Chapter 3.5.2, we construct a modified CCD (mCCD) design. The mCCD design performs similar to the mTPI-2 design and is essentially the same as the $BOIN_λ$ design described in (Guo et al., 2017), which has been shown to exhibit desirable operating characteristics. At the end of the trial, isotonic transformation is applied and the posterior mean of the toxicity rate of dose $i$, $p_i$, is calculated, denoted as $p_i$. The mCCD design selects the dose with a $p_i$ that is closest to $p_T$ as the MTD.

##### 3.5.5 The Continuous Reassessment Method (CRM)

CRM is a Bayesian model-based dose-finding design first introduced in (O’Quigley et al., 1990). Its model delineates the stochastic relationship between dose and probability of toxicity, and borrows strength across all the dose levels in the trial. In the original CRM by (O’Quigley et al., 1990), it is possible to escalate the current dose by more than one dose levels, which may assign patients to fairly high doses quite early. (Goodman et al., 1995) proposed several practical rules on the original CRM to remedy this drawback.

As part of the CRM design, one should first specify the one-parameter dose-response curve, e.g. power model, one-parameter logistic model or hyperbolic tangent model (Cheung, 2011). U-Design uses a simple one-parameter power model $$p_i = c_i^θ, θ>0,$$ where $θ$ is the parameter of the model and $c_i$'s are pre-specified prior toxicity probability ('skeleton') values, monotonically increasing with $i$. The skeleton $c_i$'s reflect the initial guesses of toxicity probability at each dose. Lee and Cheung (2011) proposed a fast and systematic approach for selecting the skeleton based on indifference intervals. The approach is imbedded in U-Design by default, and users need to specify the half-width ($λ$) of the indifference interval manually to estimate the skeleton.

The posterior distribution of $θ$ is given by $$p(θ | n, y)∝∏↙{i=1}↖{I}(c_i^θ)^{y_{i}}(1 − c_i^θ)^{n_i −y_i} π(θ),$$ where $n = \{n_1,… , n_I \}$ and $y = {y_1,… , y_I }$, and $π(θ)$ is the prior density of $θ$. In U-Design $π(θ)$ follows the log-normal distribution with mean $0$ and variance 1.34. Using the posterior distribution of $π(θ)$, the posterior mean of $π(θ)$, $θ↖{\^}$ can be calculated, which will be used as the point estimate of $θ$ . The standard dose recommendation rule in the originally CRM is based on the point estimate of toxicity rate at each dose, $p_i↖{\^} = c_i^{θ↖{\^}}$. In particular, the next cohort will be assigned to the dose with $p_i↖{\^}$ closest to $p_T$ , i.e. $argmin_{i=1,…,I}{|p_i↖{\^}-p_T|}$. In U-Design an additional no-skipping escalation rule is added to the standard dose recommendation rule in order to ensure the safety of the trial conduct (Goodman et al., 1995). That is, restricting the escalation to one level for doses that have not been used. In addition, the safety rules I and II of mTPI are also applied in our CRM design. See Chapter 3.5.2 for more details. Lastly, no escalation is permitted if the empirical DLT rate of the most recent cohort is higher than $p_T$. At the end of the trial, the mTPI design selects the dose with a $p_i↖{\^}$ that is closest to $p_T$ as the MTD.

##### 3.5.6 The Bayesian Logistic Regression Method (BLRM)

BLRM is a Bayesian model-based method proposed by (Neuenschwander et al., 2008). BLRM assumes a two-parameter logistic model between $d_i$ and $p_i$, which is given by $$logit(p_i)=log(α)+βlog(d_{i}\/d_{ref})$$ where $d_{ref}$ is the reference dose level. Model parameters $α$ and $β$ follow a multivariate log-normal prior, given by $$(\table α; β)~lognormal(\table (\table μ_1; μ_2), ∑), \text"where" ∑=(\table σ_1^2, ρσ_1σ_2; ρσ_1σ_2, σ_2^2)$$ Let $θ = \{μ_1, μ_2, σ_1, σ_2, ρ\}$ be the hyperparameter set of the model. U-Design uses a default set of doses, $d_i = 5 × i$, and the default reference dose level is the ceiling of $(I + 1)\/2$, where $I$ is the number of doses. As a result users don’t have to input doses and reference dose manually on U-Design; however, we provide service of customized input of these values upon users requests. In U-Design we use the quantile-based non-informative prior calculator proposed by Neuenschwander et al. (2008) to obtain the values of $θ$, and use the same prior guess for the lowest and highest doses as described in Appendix A.1 in Neuenschwander et al. (2008). The posterior distribution of $(α, β)$ is given by $$p(α, β | n, y)∝∏↙{i=1}↖{I}(p_i)^{y_i}(1 − p_i)^{n_i −y_i} π(α, β),$$ where $n = \{n_1,… , n_I \}$ and $y = \{y_1,… , y_I \}$. Using Markov chain Monte Carlo (MCMC) simulation, the posterior sample could be drawn for $(α, β)$ and posterior inference can be made.

The standard dose recommendation of BLRM relies on maximizing the probability of targeted toxicity interval $(p_T + ε_1, p_T + ε_2)$. In particular, the next cohort will be assigned to the dose whose posterior probability in the targeted toxicity interval is the largest, i.e. $argmax_{i=1,…,I} Pr{p_i ∊ (p_T + ε_1, p_T + ε_2) | y, n}$. An additional safety rule (Escalation with Overdose Control, EWOC) is added to the standard dose recommendation. That is, the probability of excessive toxicity of the recommended dose should be less than a given threshold $p_{EWOC}$, i.e. $Pr{p_i > pT + ε_2 | y,n} < p_{EWOC}$. Safety rules I and II of mTPI are also applied to BLRM. Besides safety rule I, if all doses violate the EWOC rule, the trial will be terminated before the maximum the sample size is reached. We found that BLRM is the slowest in the simulation due to its use of MCMC.

##### 3.5.7 The i3+3 Design

The i3+3 design is a rule-based dose finding design proposed by (Liu et al., 2019), where the letter i represents the word interval. The i3+3 design is based on a short set of simple rules that accounts for the variabilities in the observed data, a main reason why model-based designs perform well. The i3+3 design has been demonstrated to perform as good as major model-based designs due to the improved rules. Similar to the model-based designs, it can accommodate different target toxic probability and different cohort sizes. Assume the target toxicity probability is pT and the equivalence interval is (pT - ε1; pT + ε2). Suppose dose d is the dose currently used to treat patients, n is the number of patients who have been treated at dose d, and x is the number of patients who have experienced DLTs at dose d. Assume patients are enrolled and assigned to doses in cohorts. Given the observed patients data, the task is to identify an appropriate dose for the next cohort of patients. The i3+3 design can be summarized as following rules:

At the current dose d, calculate two ratios x / n and (x - 1) / n
• If x / n is below the EI, escalate (“E”) and enroll patients at the next higher dose (d + 1);
• – else, if x=n is inside the EI, stay (“S”) and continue to enroll patients at the current dose d;
• * else, if x / n is above the EI, there are two cases:
1. if (x - 1) / n is below the EI, stay (“S”) and continue to enroll patients at the current dose d,
2. else, de-escalate (“D”) and enroll patients at the next lower dose (d - 1).

# Chapter 4: Single Agent – Rolling-Based Designs

This module of Rolling-Based Designs performs trial simulation to examine the operating characteristics of four designs, including R-TPI (Guo et al., 2018) and rolling six (Skolnik et al., 2008), in which patient is are enrolled to the trial in a rolling fashion to accelerate the trial conduct, and modified mTPI-2 (Guo et al., 2017) and 3+3 (Storer, 1989), in which an additional "decision-in-advance" rule is applied to further mimic the real-life trials (details of the rule are described in Section 4.5).

In the page of Rolling-Based Designs, there are two main tabs: Simulation Settings and Simulation Results. In the Simulation Settings, there are three panels: 1) Review design & scenario, 2) Add design; 3) Add scenario (See Figure 4.1). Users need to first add the selected designs for comparison in the panel of Add scenario, and then add scenarios in the panel of Add scenario to carry out the simulation. All the added designs and scenarios will be reviewed in the panel of Review design & scenario, and users can make further modification on the added designs and scenarios per their discretion in this panel. After the simulation is launched, the results of simulations will be shown in the Simulation Results tab. Specifically, the simulation process can be monitored in real time in the panel of Running Simulations and the simulation results will be stored and displayed in the Simulation History panel. Besides the operating characteristics provided in the module of Cohort-Based Designs, the comparison of trial duration and sample sizes of different designs are also provided.

Figure 4.1: Construction of designs and scenarios in the Rolling-Based Designs

### 4.1 Construct Designs

In the module of Rolling-Based Designs, U-Design provides four designs for simulation. Users can select up to 4 of them with different design settings for simultaneous comparison in the Simulation Settings tab each time.

Instructions:

In the Add Design panel, there are four designs: mTPI-2, 3+3, rolling six and R-TPI (Figure 4.2). Click a XX button to add the XX design and input the required parameter values for the selected design. Notice that the required parameters for different designs are different, and the parameter input for each design is summarized in Table 4.1.Then click Submit button to submit the candidate designs to the panel of Review design & scenario. Click Delete button to remove each design (Figure 4.2).

In addition, for the sample size n in mTPI-2 and R-TPI, two options are provided: 1) match with 3+3, if 3+3 is selected; 2) manually input. Figure 4.2 is an example if one selects mTPI-2, 3+3, rolling six and R-TPI, with the sample size of mTPI-2 matching with 3+3 and the sample size of R-TPI being a manually input value, 21.

Figure 4.2: Add designs in the Rolling-Based Designs.

Output:

If one clicks Submit button in Figure 4.2, the candidate designs will be submitted to the panel of Review design & scenario (see Figure 4.3). One can also modify the designs parameters in the panel of Review design & scenario following the instructions in Chapter 4.1.2.

Table 4.1: Input arguments for designs in the Rolling-Based Designs.
Notation Arguments Description
$n$
(all designs)
Sample size The maximum number of patients to be treated in the trial. Here, the hard limit is set at 100 since the number of patients that are enrolled in phase I clinical trial is typically small. For most cases, users can input a number smaller than 100, e.g., 30.
$n_{cohort}$
(mTPI-2)
Cohort size The number of patients in each cohort (e.g., 3).
$ε_1, ε_2$
(mTPI-2, R-TPI)
$ε_1, ε_2$ Two small fractions used to define the equivalence/target interval of the MTD (default is 0.05 in all interval-based designs). Any doses with a toxicity probability falling into the interval $(p_T - ε_1, p_T + ε_2)$ will be considered an acceptable dose level as MTD.
$d_{start}$
(all designs)
Starting dose level The starting dose level in the simulation trials.
$C$
(R-TPI)
The maximum number of pending patients allowed in the trial The maximum number of pending patients without observed out- comes allowed in the trial. It can be provided by users to control the enrollment speed.
Figure 4.3: Design list in the Rolling-Based Designs.
##### 4.1.2 Edit Designs

Designs can be modified in the panel of Review design & scenario. This allows for flexible modifications in practice.

Instructions

Click Delete All button at the bottom of design list (Figure 4.3) to remove all the existing scenarios. Click Edit button to edit the parameter of each design or remove the selected design.

After clicking Edit button in Figure 4.3, the edit mode is enabled (Figure 4.4). All the values in the input-cell can be changed. Click the Delete button to delete a specific design. When all the changes are made, click to confirm and reload the design list.

Figure 4.4: Edit designs in the Rolling-Based Designs.

### 4.2 Construct Scenarios

Simulated trials are generated under scenarios in which the true toxicity probabilities for the experimental doses are given. In the panel of Add scenario, there are three parts of settings: Enrollment setting, Basic setting and Scenario setting. In the Scenario setting, users can select Default scenarios or Manual scenarios. Figure 4.5 provides the user interface for an illustration. The detailed descriptions of input parameters in Figure 4.5 are defined in Table 4.2.

Figure 4.5: Add scenarios in the Rolling-Based Designs.
Table 4.2: Input arguments for scenarios in the Rolling-Based Designs.
Notation Arguments Description
$T_{follow-up}$ Follow-up time (days) The maximum follow-up time for each patient in the trial.
$MIAT$ Mean interpatient arrival time (days) The mean chronologic time for a patient to arrive in the clinic and be eligible for study.
$IR$ Inevaluable rate The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability closest to or lower than the target probabil- ity.
$p_T$ Target toxicity probability The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability closest to or lower than the target probability.
$n_{sim}$ The number of simulated trials The maximum number of simulated trials is 5000.
$n_{dose}$ The number of dose levels The maximum number of prespecified dose levels is 10.

Instructions:

1. In the Enrollment setting, enter the follow-up time $T_{follow-up}$, the mean interpatient arrival time $MIAT$ and the inevaluable rate $IR$;
2. In the Basic setting, specify the target toxicity probability $p_T$ and the number of simulated trials $n_{sim}$;
3. In the Scenario setting, U-Design provide two ways of adding scenarios:
• Click Default scenarios. Then by clicking numeric button (right to ndose =), U-Design will automati- cally generate a series of default scenarios with diverse
• Click Manual scenarios button to unfold the manually input interface. Scenarios can be constructed manually using two different means: Add a single scenario for adding a single scenario and Add a batch of scenarios for adding multiple scenarios in a batch.
• In Add a single scenario section, select a number of doses ($n_{dose}$) and there will subsequently appear a list of dose level input boxes under the Add a single scenario section; then type in the true toxicity probability for each dose level. Then click Submit button to submit the single scenario.
• In Add a batch of scenarios section, users can submit multiple scenarios in batches and then click Submit button to submit them. The format of input must comply with following instructions.
• Scenarios should be separated by line break;
• The parameters in one scenario should be ordered in accordance with this sequence: Target toxicity probability, Number of simulated trials, True toxicity probabilities of all the dose levels;
• Each parameter must be separated by a WHITE SPACE or COMMA.

In addition, U-Design provides a handy function to automatically generate the 42 scenarios in (Ji and Wang, 2013). Click the The 42 Scenarios in Ji and Wang (2013, JCO) button to insert these 42 scenarios into the Add a batch of scenarios panel for simulation.

Output:

The submitted scenarios are displayed as a scenario list (Figure 4.6). The list will appear in the top region of the web page. See Figure 4.1 for an example.

Figure 4.6: Scenario list in the Rolling-Based Designs.
##### 4.2.2 Edit Scenarios

Scenarios can be modified after being added. This allows for flexible modifications in practice.

Instructions

Clicking Delete All button at the bottom of scenario list (Figure 4.6) to remove all the existing scenarios. Click Edit button to edit parameter in the scenario settings or remove selected scenarios.

After clicking the Edit button, the edit mode is enabled (Figure 4.7). All the values in the input-box can be changed. Click the Delete button to delete a specific scenario. When all the changes are made, click Save button to confirm the edit and reload the scenario list.

Figure 4.7: Edit scenario in the Rolling-Based Designs.

### 4.3 Launch Simulation

Once the designs and scenarios are constructed, users can conduct simulated clinical trials to examine the operating characteristics of these designs using these scenarios, by clicking the Launch Simulation button. A green "Launch Successful" message will then be displayed on the website as in Figure 4.8 to indicate that the simulations have been successfully launched. Also, users can click the here hyperlinks to either track the simulation processing status (Figure 4.9) and simulation results in the Simulation Results tab or reset the Simulation Settings page.

In addition, U-Design allows users to change the random seed for the simulation studies (the box right to $R_{seed} =$ located at the upper-right of the Review design & scenario panel, see Figure 4.7).

Figure 4.8: The displayed "Launch Successful" message after launching simulations in the Rolling-Based Designs.
Figure 4.9: Simulation progress in the Rolling-Based Designs.

### 4.4 View Simulation Results

The results of submitted simulations can be viewed in the Simulation Results tab. In the Simulation Results tab, the Running Simulation panel exhibits the process of running simulation (see Figure 4.9 for an illustration). Once the simulations are completed, click the Refresh button on the top right corner of the Simulation History panel (the blue box in Figure 4.9) to load the latest simulation results. All completed simulations are listed in the Simulation History panel. Click the View button (the green box in Figure 4.9) to unfold the simulation results. Click the Restore button (the black box in Figure 3.9) will restore the simulation settings that produced this simulation result to the Simulation Settings tab.

The results of simulation are shown in two ways: Simulation Result Plots and Simulation Result Tables.

##### 4.4.1 Simulation Result Plots

There are two sections in the Simulation Result Plots:

1. Line plots reflecting six summary statistics of the simulation results for all the designs (Figure 4.10).
2. A table of mean and standard deviation for six summary statistics of the simulation results for all the designs across all simulated scenarios (Figure 4.11).

Note for simulation result plots:

• The six summary statistics are
1. Prob. of Select MTD: The probability of selecting the true MTD in the end. For interval-based designs (mTPI-2 & R-TPI), the true MTDs is defined as the dose levels of which the true toxicity probabilities fall into the equivalence interval; if none of the doses has a toxicity probability that falls in the equivalence interval, the true MTD is defined as the dose with the highest toxicity probability that is below pT . But for those non-interval-based designs, e.g., 3+3 and rolling six, the true MTDs is defined as the dose levels with the highest toxicity probabilities lower than or equal to $p_T$ . To fairly compare the operating characteristics of multiple designs submitted in batch in one simulation study, the definition of MTD should be unified. If any of interval-based designs are used in the simulation, dose levels of which the true toxicity probabilities fall into the widest equivalence interval are defined as the true MTDs (if none of doses has a toxicity probability that falls in, the dose with highest toxicity probability that is below $p_T$ is the true MTD). For example, consider a case that one wants to compare four designs, R-TPI, mTPI-2, rolling six and 3+3, in a simulation study targeting $p_T = 0.3$, and $ε_1$ and $ε_2$ are 0.02 and 0.05 for R-TPI, but 0.05 and 0.03 for mTPI-2. In this case, the true MTD is the dose levels with toxicity probabilities falling in $(0.3-0.05, 0.3+0.05)$; if none of the dose has a toxicity probability in $(0.3-0.05, 0.3+0.05)$, the dose with the highest toxicity probability lower than 0.3 is the true MTD.
Figure 4.10: Simulation result plots in the Rolling-Based Designs.
Figure 4.11: Simulation summary in the Rolling-Based Designs.
2. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulation trial.
3. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
4. Prob. of No Selection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
5. Trial Duration: The average time duration for trial conduct.
6. Number of Enrolled Patients: The averaged number of patients enrolled in the trial, including the patients with outcomes observed and patients who drop out and therefore become inevaluable.
• For each plot, the x axis is the index of scenario and the y axis is the value of summary statistics (Prob. of No Selection, Prob. of Toxicity, Prob. of Select Dose-over-MTD, Prob. of Select MTD, Trial Duration, and Number of Enrolled Patients). Lines with different colors represent different designs.
• The plots are interactive for better visualization.
• Hover the mouse on the plots and a box will display the value of each design at corresponding scenario (upper left plot in Figure 4.10: Prob. of Select MTD).
• Hover the mouse on the design label to highlight the corresponding line and fade others (upper right plot in Figure 4.10: Prob. of Toxicity).
• Click the design label to hide the corresponding line and click again to change it back (middle left plot in Figure 4.10: Prob. of Select Dose-over-MTD).
##### 4.4.2 Simulation Result Tables

One table represents one scenario, consisting of two parts. In the upper part, the first two columns summarize the scenario, with dose levels and their true toxicity probabilities; the remaining columns report three summary statistics, selection probability, number of patients treated and number of toxicities. The true MTD(s) of the scenario is(are) highlighted by yellow. In the lower part, six additional rows report four more summary statistics.

Output:

• The nine summary statistics are
1. Selection Prob.: The probability of selecting each dose level as the MTD.
2. # of Patients Treated: The average number of patients treated at each dose level.
3. # of Toxicities: The average number of patients experienced DLT at each dose level.
4. Prob. of Select MTD: The probability of selecting the true MTD. For the definition of the true MTD, please refer to Section 3.4.1.
5. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulation trial.
6. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
7. Prob. of No Selection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
8. Trial Duration: The average time duration for the trial conduct.
9. Number of Enrolled Patients: The average number of patients enrolled in the trial, including the patients with outcomes observed and patients who drop out and therefore become inevaluable.
• The design setting will be kept at the top of page while scrolling down the page to review the simulation results.
Figure 4.12: Simulation result tables in the Rolling-Based Designs.

There will be a Download U-Design Simulation Report button at the upper left corner of each simulation results panel. Click it to download a zip file, which includes two word files and six images of line plots of summary statistic shown in Figure 4.10. The two word files are a template for the statistical section in a protocol and a simulation report with complete simulation results under the designs and scenarios users added in the simulation settings tab, respectively. Users could modify the protocol template and update the simulation settings and results tailored for your trial by themselves or contact us via email info@laiyaconsulting.com for consulting service to get the full version of the statistical section for protocol and the simulation results.

### 4.5 Methods Review

The module for Rolling-Based Designs includes rolling six (Skolnik et al., 2008), R-TPI (Guo et al., 2018), mTPI-2 (Guo et al., 2017) and 3+3 (Storer, 1989) designs. Besides the operating characteristics reported in the module of Cohort-Based Designs, this module enables users to compare the trial duration and sample size of different designs based on real-life settings, which are characterized as three user-input parameters, the mean inter-patient arrival time ($MIAT$), the maximum follow-up time ($T_{follow-up}$), and the inevaluable rate of enrolled patients ($IR$).

##### 4.5.1 The Rolling Six Design (RSD)

The rolling six design (Skolnik et al., 2008) extends 3+3 with the aim to reduce the occurrence of accrual suspension. In 3+3, patient accrual is suspended after enrolling 3 patients until the data of these 3 patients are completely observed. During the period of suspension, new eligible patients have to be turned away, resulting in a waste of patient resources. The rolling six design reduces this type of suspension by allowing up to 6 patients to be enrolled without breaks at a dose level, i.e. no suspension after the enrollment of the first 3 patients at a dose. Dosing decisions for a new patient is based on the number of patients enrolled at the current dose, the number of DLTs, and the number of patients still being followed without definitive DLT outcomes. RSD is a rule-based design and all dose assignment rules for the six patients can be prespecified.

##### 4.5.2 The Rolling Toxicity Probability Interval Design (R-TPI)

R-TPI is a rolling enrollment design that combines the features of model-based designs such as mTPI-2 (Guo et al., 2017) and rule-based designs such as rolling six (Skolnik et al., 2008). The R-TPI design aims to shorten the duration but still maintain safety and desirability of phase I dose-finding clinical trials. Unlike traditional cohort-based designs, the R-TPI design does not require to enroll patients according to a pre-planned cohort size, and thus avoids undesirable enrollment suspension. On the other hand, R-TPI overcomes weakness of the rolling six design by borrowing features from model-based inference in mTPI-2. Therefore, R-TPI reduces the trial suspension between two cohorts, targets different $p_T$ values with wider applications, and at the same time maintains simplicity and transparency in the implementation. In R-TPI, dosing decisions for a new patient are based on integrating the mTPI-2 decision of the observed data and the mTPI2 dosing decisions of the most extreme cases of the pending data (i.e. all pending outcomes are DLTs or non-DLTs) at the current dose. Safety is strongly enforced across the dosing decisions by stringent rules. An important threshold $C$, the maximum number of pending patients without observed outcomes allowed in the trial, can be provided by users to control the enrollment speed. The larger value of C the more rapid the enrollment is. Besides, the safety rules applied in mTPI-2 (Chapter 3.5.3), are also applied to R-TPI.

##### 4.5.3 The Modified Toxicity Probability Interval-2 Design (mTPI-2)

The mTPI-2 design (Guo et al., 2017) is a model-based design that enrolls patients according to a pre-planned cohort size. For the detailed description of mTPI-2, please refer to Chapter 3.5.3. We also applied an additional "decision-in-advance" rule to the mTPI-2 design in the module of Rolling-Based Designs in order to further mimic the real-life trials. That is, if a cohort for a current dose is not fully enrolled or completely observed, a decision can be made in advance if and only if the decision would not be changed by the pending data for the cohort of patients, either enrolled but still being followed or yet to be enrolled. For example, under $p_T = 0.3$, if 2 patients have been enrolled to a newly-used dose d and both of them experience DLTs, stop enrolling the third patient to $d$ and de-escalate to $d - 1$ immediately. This is ethical and sensible, as the decision would still be de-escalation if a third patient is enrolled to dose $d$ and experiences non-DLT eventually. This rule of “decision-in-advance” can accelerate the trial and make the trial conduct more ethical.

##### 4.5.4 The 3+3 Design

The 3+3 design is an algorithm-driven design that enrolls patients in a cohort of three. The detailed description is in Chapter 3.5.1. Also, the rule of “decision-in-advance” mentioned in Chapter 4.5.3 is also applied to 3+3.

# Chapter 5: Single Agent – Decision & MTD

The Decision & MTD module consists of two main tabs to achieves two goals: 1) the Generate Decision Table tab for generating a decision table before the trial is started and 2) the Estimate the MTD tab for estimating the MTD after the trial is completed.

### 5.1 Generate Dose-Finding Decision Table

This function generates a decision table based on the mTPI, mTPI-2, mCCD and 3+3 designs which can be used to conduct a phase I dose-finding trial. The CRM and BLRM designs do not provide a decision table before the trial is started. However, for both designs, we provide empirical decision tables after launching simulations (See Section 3.4.1).

Instructions

Figure 5.1 shows the interface with which users provide the required input as explained in Table 5.2.

Figure 5.1: Input arguments in the Decision & MTD.

Output:

Once users provide the input values and click the Generate button, four decision tables will be generated (Figure 5.2). Users can click the tabs to switch between the tables for the mTPI-2, mTPI, mCCD and 3+3 designs. For each decision table, the column represents the number of patients treated at the current dose, which is the dose currently being used to treat patients in the trial, and the row represents the number of patients among those treated at the current dose who have experienced dose-limiting toxicity (DLT) events. Note that these are the counts of patients, not DLT events. For example, column 3 and row 1 means that 3 patients have been treated at the current dose and 1 of them experienced DLT.

Table 5.1: Input arguments in the Decision & MTD.
Notation Arguments Description
$n$ Sample size The maximum number of patients to be treated in the trial. Here, the hard limit is set at 50 since the number of patients that are enrolled in phase I clinical trial is typically small. For most cases, users can input a number smaller than 100, e.g., 10, which would be sufficient to monitor a phase I trial.
$p_T$ Target toxicity probability The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability no higher than or close to the target toxicity probability.
$ε_1, ε_2$ $ε_1, ε_2$ Two small fractions used to define the equivalence interval of the MTD (de- fault is 0.05). Any doses with a toxicity probability falling into the interval $(p_T - ε_1, p_T + ε_2)$ will be considered an acceptable dose level as MTD.

Each cell in the table provides the dose-assignment decision based on the readouts from the corresponding row and column. For example, for column 3 and row 1, i.e., 1 out of 3 patients experiencing DLTs, the decision is S. The letters in the decision table represent different dose-assignment decisions as shown below:

• E stands for escalating to the next higher dose,
• S stands for staying at the current dose,
• D stands for de-escalating to the previous lower dose,
• DU stands for de-escalating to the previous lower dose and marking the current dose as unacceptable so that it will never be used again in the remainder of the trial.
Figure 5.2: Decision tables in the Decision & MTD.

Notes

• The 3+3 decision table is fixed regardless of different trial parameters.
• For CRM (or BLRM), the decision table cannot be easily summarized since the dose-assignment decision under CRM (or BLRM) for a given outcome (say, 1 DLT out of 3 patients) and a given dose are random, depending on existing data in the entire trial including those at other doses. In other words, CRM (or BLRM) could stay, escalate or de-escalate when 1 out of 3 patients having DLT at a dose, which makes it impossible to provide a fixed decision table. Nevertheless, U-Design provides empirical CRM (or BLRM) decision table in the simulation section when CRM (or BLRM) is implemented in simulation trials (See Chapter 3.4.1).
• A pop-up box will show the row and column numbers when hovering the mouse over each cell in the decision table (Figure 5.2). It is useful for looking over a large decision table.
• Click the Download Design Decision Table button to save the above decision table as a word file (.docx).

### 5.2 Estimate the MTD

Based on the Pool Adjacent Violators Algorithm (PAVA), U-Design estimates the MTD when the trial is completed and the data is collected. This approach is described in (Ji et al., 2010).

Instructions

• Enter $p_T , ε_1, ε_2$ and $n_{dose}$ of the trial. Then an editable table will be shown in the page (Figure 5.3).
• Provide the required input in Figure 5.3 as explained in Table 5.2.
• Click the Estimate button to estimate the MTD.
Figure 5.3: Input arguments in the Decision & MTD.

Output:

The estimated MTD is highlighted as shown in Figure 5.4.

Figure 5.4: Deciding MTD report in the Decision & MTD.
Table 5.2: Input arguments in the Decision & MTD.
Notation Arguments Description
$p_T$ Target toxicity probability The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability no higher than or close to the target toxicity probability.
$ε_1, ε_2$ $ε_1, ε_2$ Two small fractions used to define the equivalence interval of the MTD (de- fault is 0.05). Any doses with a toxicity probability falling into the interval $(p_T - ε_1, p_T + ε_2)$ will be considered an acceptable dose level as MTD.
$n_{dose}$ The number of dose levels The number of dose levels
# of DLTs The number of patients with DLTs at each dose level A non-negative integer number of patients with DLT at each dose level
# of patients The number of patients treated at each dose level A positive integer number of patients treated at each dose level.

# Chapter 6: Dual Agents – Cohort-Based Designs

This module of Dual Agents – Cohort-Based Designs performs trial simulation to examine the operating characteristics of two dual agent dose-finding designs, including the Bayesian logistic regression model (BLRM; Neuenschwander et al. (2015)), and the method of the Product of Independent beta Probabilities dose Escalation (PIPE; Mander and Sweeting (2015)). In the Dual Agents – Cohort-Based Designs page, there are two main tabs: Simulation Settings and Simulation Results.

In the Simulation Settings tab, there are three panels: 1) Review design & scenario, 2) Add design, and 3) Add scenario (See Figure 6.1). Users need to first add the selected designs for comparison in the panel of Add design, and then add scenarios in the panel of Add design to carry out the simulation. All the added designs and scenarios will be reviewed in the panel of Review design & scenario, and users can make further modification on the added designs and scenarios per their discretion in this panel. After the simulation is launched, the results of simulations will be shown in the Simulation Results tab. Specifically, the simulation process can be monitored in real time in the panel of Running Simulations and the simulation results will be stored and displayed in the Simulation History panel. Detailed steps of using this module are elaborated in Section 6.1 - 6.4.

Figure 6.1: Construction of designs and scenarios in the Dual Agents – Cohort-Based Designs.

### 6.1 Construct Designs

In the module of Dual Agents – Cohort-Based Designs, U-Design provides two designs for simulation.

Output:

In the Add Design panel, there are two designs: BLRM and PIPE (Figure 6.2). Click a XX design button to add the design. Enter the required parameter values for the selected design (Table 6.1). Then click Submit button to submit the candidate designs to the panel of Review design & scenario. Click Delete button to remove each design (Figure 6.2). Figure 6.2 is an example if one click the buttons for BLRM and PIPE. Notice that the required parameters for different designs are different, and the parameter input for each design is summarized in Table 6.1.

Figure 6.2: Add designs in the Dual Agents – Cohort-Based Designs.

Output:

If one clicks Submit button in Figure 6.2, the candidate designs will be submitted to the panel of Review design & scenario (see Figure 6.3). One can also modify the designs parameters in the panel of Review design & scenario following the instructions in Chapter 6.1.2.

##### 6.1.2 Edit Designs

Designs can be modified in the panel of Review design & scenario. This allows for flexible modifications in practice.

Table 6.1: Input arguments for designs in the Dual Agents – Cohort-Based Designs.
Notation Arguments Description
$n$
(all designs)
Sample size The maximum number of patients to be treated in the trial. Here, the hard limit is set at 100 since the number of patients that are enrolled in phase I clinical trial is typically small. For most cases, users can input a number smaller than 100, e.g., 30.
$n_{cohort}$
(all designs)
Cohort size The number of patients in each cohort (e.g., 3).
$ε_1, ε_2$
(BLRM)
$ε_1, ε_2$ The maximum number of patients to be treated in the trial. Here, the hard limit is set at 100 since the number of patients that are enrolled in phase I clinical trial is typically small. For most cases, users can input a number smaller than 100, e.g., 30.
$p_{EWOC}$
(BLRM)
Cutoff probability of escalation with over-dose control The threshold of controlling the probability of excessive or unacceptable toxicity. (Only for BLRM)
$d_{start1}$
(all designs)
Starting dose level for agent 1 The starting dose level for agent 1 in the simulation trials.
$d_{start2}$
(all designs)
Starting dose level for agent 2 The starting dose level for agent 2 in the simulation trials.
Figure 6.3: Design list in the panel of Review Design & Scenario of Dual Agents – Cohort-Based Designs.

Instructions:

Click Delete All button at the bottom of design list (Figure 6.3) to remove all the existing scenarios. Click Edit button to edit the parameter of each design or remove the selected design.

After clicking Edit button in Figure 6.3, the edit mode is enabled (Figure 6.4). All the values in the input-cell can be changed. Click the Delete button to delete a specific design. When all the changes are made, click Save to confirm and reload the design list.

Figure 6.4: Edit designs in the Dual Agents – Cohort-Based Designs.
Table 6.2: Input arguments for scenarios in the Dual Agents Cohort-Based Designs.
Notation Arguments Description
$p_T$ Target toxicity probability The target toxicity probability of the maximum tolerated dose (MTD). The main objective of phase I clinical trials is to find the highest dose with a toxicity probability closest to or lower than the target probability.
$n_{sim}$ The number of simulated trials The maximum number of simulated trials is 5000.
$n_{dose}$ The number of dose levels The maximum number of prespecified dose levels is 10.

### 6.2 Construct Scenarios

Simulated trials are generated under scenarios in which the true toxicity probabilities for the experimental doses are given. In the panel of Add scenario, there are three parts of settings: Basic setting, Dose levels and Toxicity profiles. In the Toxicity profiles, users can select to add Default scenarios or generate their own scenarios of interest by three options, Logistic regression, Marginals & interaction or Manually type in. Figure 6.5 provides the user interface for an illustration. The detailed descriptions of input parameters in Figure 6.5 are defined in Table 6.2.

Figure 6.5: Add scenarios in the Dual Agents – Cohort-Based Designs.

Instructions:

In the Basic setting, specify the target toxicity probability pT and the number of simulated trials $n_{sim}$. In the Dose levels, specify the number of doses and the dose levels for each agent. In the Toxicity profiles, U-Design provide four ways of adding scenarios:

• Default scenarios: By clicking Default scenarios button, U-Design will automatically generate a series of default scenarios with diverse patterns.
• Logistic regression: Click Logistic regression button to unfold the corresponding input interface. Scenarios can be constructed by specifying the coefficients of logistic function. Click Generate button to generate the toxicity probabilities corresponding to the coefficients input, which will be displayed at the bottom of the panel of Add scenario. Then click Submit button to submit them.
• Marginals & interactions: Click Marginals & interactions Scenarios can be constructed by specifying the marginal toxicity probabilities of two agents and the interaction between them. Click Generate button to generate the toxicity probabilities corresponding to the input marginal toxicity probabilities of two agents and the interaction between them, which will be displayed at the bottom of the panel of Add scenario. Then click Submit button to submit them.
• Manually type in: At the bottom of the panel of Add scenario, users can manually type in the true toxicity probability for each combination of two agents. Then click Submit button to submit them.

Output:

The submitted scenarios are displayed in the top region of the panel of Review design & scenario (Figure 6.6).

##### 6.2.2 Edit Scenarios

Scenarios can be modified in the panel of Review design & scenario. This allows for flexible modifications in practice.

Instructions:

Click Delete All button at the bottom of scenario list (Figure 6.6) to remove all the existing scenarios. Click Edit button to edit parameter in the scenario settings or remove the selected scenario.

After clicking the Edit button, the edit mode is enabled (Figure 6.7). All the values in the input-box can be changed. Click the Delete button to delete a specific scenario. When all the changes are made, click Save button to confirm the edit and reload the scenario list.

Figure 6.6: Scenario list in the Dual Agents – Cohort-Based Designs.
Figure 6.7: Edit scenario in the Dual Agents – Cohort-Based Designs.

### 6.3 Launch Simulation

Once the designs and scenarios are constructed, users can conduct simulated clinical trials to examine the operating characteristics of these designs using these scenarios, by clicking the Launch Successful button (Figure 6.1). A green "Launch Successful" message will then be displayed on the website as in Figure 6.8 to indicate that the simulation has been successfully launched. Also, users can click the here hyperlinks to either track the simulation processing status (Figure 6.9) and simulation results in the Simulation Results tab or reset the Simulation Settings page.

In addition, U-Design allows users to change the random seed for the simulation studies (the box right to $R_{seed} =$ located at the upper-right of the Review design & scenario panel, see Figure 6.7).

Figure 6.8: The displayed "Launch Successful" message after launching simulations in the Dual Agents –- Cohort-Based Designs.
Figure 6.9: Simulation progress in the Dual Agents -– Cohort-Based Designs.

### 6.4 View Simulation Results

The results of submitted simulations can be viewed in the Simulation Results tab. In the Simulation Results tab, the Running Simulation panel exhibits the process of running simulation (see Figure 6.9 for an illustration). Once the simulations are completed, click the Refresh button on the top right corner of the Simulation History panel to load the latest simulation results. All completed simulations are listed in the Simulation History panel. Click the View button to unfold the simulation results. Click the Restore button will restore the simulation settings that produced this simulation result to the Simulation Settings tab.

The results of simulation are shown in two ways: Simulation Result Plots and Simulation Result Tables.

##### 6.4.1 Simulation Result Plots

There are three sections in the Simulation Result Plots:

1. Line plots reflecting four summary statistics of the simulation results for all the designs (Figure 6.10).
2. Bar plots reflecting the selection probabilities, number of patients treated and number of DLTs for each dose combination (Figure 6.11).
3. A table of mean and standard deviation for four summary statistics of the simulation results for all the designs across all simulated scenarios (Figure 6.12).

Note for simulation result plots:

• The four summary statistics are
1. Prob. of Select MTD: The probability of selecting the true MTD in the end. For BLRM, the true MTDs is defined as the dose combinations of which the true toxicity probabilities fall into the equivalence interval; if none of the dose combinations has a toxicity probability that falls in the equivalence interval, the true MTD is defined as the dose with the highest toxicity probability that is below pT . But for PIPE, the true MTDs is originally defined as the dose combinations with the highest toxicity probabilities lower than or equal to pT . To fairly compare the operating characteristics of multiple designs submitted in batch in one simulation study, the definition of MTD should be unified. If BLRM and PIPE are both used in the simulation, the definition of true MTDs for BLRM is also applied to PIPE
2. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulated trials.
3. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
4. Prob. of No Selection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
• For each plot, the x axis is the index of scenario and the y axis is the value of summary statistics (Prob. of No Selection, Prob. of Toxicity, Prob. of Select Dose-over-MTD, Prob. of Select MTD). Lines with different colors represent different designs.
• The plots are interactive for better visualization.
• Hover the mouse on the plots and a box will display the value of each design at corresponding scenario (bottom left plot in Figure 6.10: Prob. of Select Dose-over-MTD).
• Hover the mouse on the design label to highlight the corresponding line and fade others (lower left plot in Figure 6.10: Prob. of Select Does-over-MTD).
• Click the design label to hide the corresponding line and click again to change it back (lower right plot in Figure 6.10: Prob. of Select MTD).
Figure 6.10: Line plots in the Dual Agents Cohort-Based Designs.
Figure 6.11: Bar plots in the Dual Agents Cohort-Based Designs.
Figure 6.12: Simulation summary in the Dual Agents Cohort-Based Designs.
##### 6.4.2 Simulation Result Tables

In Figure 6.13, one table represents one scenario, consisting of two parts. In the upper part, the first two columns summarize the scenario, with dose levels and their true toxicity probabilities; the remaining columns report three summary statistics, selection probability, number of patients treated and number of toxicities. The true MTD(s) of the scenario is(are) highlighted by yellow. In the lower part, four additional rows report four more summary statistics.

Output:

Figure 6.13: Simulation result tables in the Dual Agents Cohort-Based Designs.
• The seven summary statistics are
1. Selection Prob.: The probability of selecting each dose level as the MTD.
2. # of Patients Treated: The average number of patients treated at each dose level.
3. # of Toxicities: The average number of patients experienced DLT at each dose level.
4. Prob. of Select MTD: The probability of selecting the true MTD. For the definition of the true MTD, please refer to Section 6.4.1.
5. Prob. of Toxicity: The proportion of patients who have experienced DLT across all the simulation trial.
6. Prob. of Select Does-over-MTD: The probability of selecting the dose levels above the true MTD.
7. Prob. of NoSelection: The proportion of the simulation trials in which none of the dose levels are selected as the MTD.
• The design setting will be kept at the top of page while scrolling down the page to review the simulation results.

### 6.5 Methods Review

In the module of Dual Agents – Cohort-Based Designs we provide the BLRM (Neuenschwander et al., 2015) and PIPE (Mander and Sweeting, 2015) designs. Assume that $I$ doses $\{d_{11}, d_{12}, … , d_{1I} \}$ for agent 1 and $J$ doses $\{d_{21}, d_{22}, … , d_{2J} \}$ for agent 2 are considered in the trial and let $p_{ij}$ be the true toxicity probability for combination of dose $d_{1i}$ and dose $d_{2j}$ for drugs 1 and 2, respectively. Let $n_{ij}$ and $y_{ij}$ be the number of patients treated at the combination of dose $d_{1i}$ and dose $d_{2j}$ for drugs 1 and 2, and the number of patients with DLTs, respectively. The method description of each design is given in the following subsections.

##### 6.5.1 BLRM

The BLRM method (Neuenschwander et al., 2015) assumes a logistic model between the marginal toxicity probability of each agent and the dose levels, and the toxicity of probability of the dual agent combination is constructed by the marginal toxicity probability of each agent and the interaction between them. The relationship of the marginal toxicity probability of each agent and the dose levels is given by: $$logit(p_i^{(1)})=logit(odds_i^{(1)})=log(α_1)+β_1*log(d_{1i}\/d_{ref1})$$ $$logit(p_j^{(2)})=logit(odds_j^{(2)})=log(α_2)+β_2*log(d_{2j}\/d_{ref2})$$ where $p_i^{(1)}$ is the marginal toxicity probability of agent 1 dose $i$, $p_j^{(2)}$ is the marginal toxicity probability of agent 2 dose $j$, $odds_i^{(1)}=p_i^{(1)}\/(1 − p_i^{(1)})$ and $odds_j^{(2)}=p_j^{(2)}\/(1 − p_j^{(2)})$ are the odds of toxicity events for agent 1 dose $i$ and agent 2 dose $j$, $d_{ref1}$ and $d_{ref2}$ are the reference dose levels for each agent respectively, and $α_1, β_1, α_2$, and $β_2$ are the unknown parameters modeling the relationship of the marginal toxicity probability of each agent and the dose levels. In the special case of no interaction, $α_1, β_1, α_2$, and $β_2$ fully determine the toxicity probability for a drug combination. For dose combination $(i, j)$ the probability of having no DLT is $(1 − p_i^{(1)})(1 − p_j^{(2)})$. Hence, the probability of DLT under no interaction is $$p_{ij}^0=1-(1-p_i^{(1)})(1-p_j^{(2)})=p_i^{(1)}+p_j^{(2)}-p_i^{(1)}p_j^{(2)}$$ On the odds scale this is equivalent to $$odds_{ij}^0=odds_i^{(1)}+odds_j^{(2)}-odds_i^{(1)}odds_j^{(2)}$$ where $odds_{ij}^0$ is the odds of DLT of dose combination $(i,j)$ under no interaction. Interaction parameter $η$ has the interpretation of an odds-multiplier as follows: $$odds_{ij}=odds_{ij}^0*exp^η$$ where $odds_{ij}$ is the odds of DLT of dose combination $(i, j)$. Hence, the the probability of DLT of dose combination $(i, j)$ is given by $$p_{ij}=odds_{ij}\/(1+odds_{ij})$$ The value of $η$ has the interpretation as follows:

• $η = 0$: No interaction, i.e. the drug combination produces a toxic effect whose magnitude is equal to that obtained if the drugs act independently in the body.
• $η < 0$: Protective, i.e. the drug combination produces a toxic effect less than that obtained if the drugs act independently in the body.
• $η > 0$: Synergistic, i.e. the drug combination produces a toxic effect greater than that obtained if the drugs act independently in the body.

Model parameters $α_k$ and $β_k$ ($k=1$ or 2, denoting different agents) follow a multivariate log-normal prior, given by $$(\table α_k; β_k)~lognormal(\table (\table μ_{k1}; μ_{k2}), ∑), \text"where" ∑=(\table σ_{k1}^2, ρ_kσ_{k1}σ_{k2}; ρ_kσ_{k1}σ_{k2}, σ_{k2}^2)$$ The interaction parameter $η$ follows a normal distribution as follows $η ∼ N (μ_η , σ_η^2 )$.

The posterior distribution of $θ = (α_1, β_1, α_2, β_2, η)$ is given by $$p(θ | n, y)∝∏↙{i,j}(p_{ij})^{y_{ij}}(1 − p_{ij})^{n_{ij} −y_{ij}} π(θ)$$ where $n=\{n_{ij} :1≤i≤I,1≤j≤J\}$ and $y=\{y_{ij} :1≤i≤I,1≤j≤J\}$.UsingMarkovchainMonteCarlo (MCMC) simulation, the posterior sample could be drawn for θ and posterior inference can be made.

The standard dose recommendation of BLRM relies on maximizing the probability of targeted toxicity interval $(p_T − ε_1, p_T + ε_2)$. In particular, the next cohort will be assigned to the dose combination whose posterior probability in the targeted toxicity interval is the largest, i.e. $argmax_{i=1,...,I,j=1,...,J} Pr\{p_{ij} ∈ (p_T − ε_1, p_T + ε_2) | y, n\}$. Notice that the dose level change can not be larger than 1 for a single agent, and the diagonal escalation for both agent is not allowed. For example, if the current dose combination is $(i, j)$, the candidate dose combinations for the next cohort are $\{(i,j),(i + 1,j),(i,j + 1),(i − 1,j),(i,j − 1),(i − 1,j − 1),(i + 1,j − 1),(i − 1,j + 1)\}$. An additional safety rule (Escalation with Overdose Control, EWOC) is added to the standard dose recommendation. That is, the probability of excessive toxicity of the recommended dose combination should be less than a given threshold $p_{EWOC}$ , i.e. $Pr{p_{ij} > p_T + ε_2 | y, n} < p_{EWOC}$. Safety rules I and II of mTPI (Section 3.5.2) are also applied to BLRM. Besides safety rule I, if the dose combination of the lowest doses for both agents violates the EWOC rule, the trial will be terminated before the maximum the sample size is reached. At the end of the trial, the MTD is selected as the explored dose combination that maximizes the probability of targeted toxicity interval.

##### 6.5.2 The Product of Independent beta Probabilities dose Escalation (PIPE)

The Product of Independent beta Probabilities dose Escalation design (PIPE) was proposed by Mander and Sweeting (2015). This method assumes independent beta prior for the toxicity rates of all dose combinations and uses posterior probabilities from all proposed dose combinations for dose escalation. It assumes a weak "ordering" constraint over the posterior probabilities from all proposed dose combinations, and aims to target a MTD contour (defined as $MTC_θ$ hereinafter, where $θ = p_T$ is the target toxicity probability) such the risk of toxicity for dose combinations on this contour is a prespecified $p_T$.

Assume that the probabilities of toxicity at all dose combinations follow an independent Beta distribution, i.e. $p_{ij} ∼ Beta(a_{ij},b_{ij})$. Under the assumption of monotonicity, toxicity risk increases with increasing dose, that is, $p_{ij} ≤ p_{(i+1)j}$ and $p_{ij} ≤ p_{i(j+1)}$. Therefore, the MTCθ should satisfy this monotonicity assumption. Mander and Sweeting (2015) has a detailed introduction to the $MTC_θ$ . A candidate contour for $MTC_θ$ can be defined by a matrix $C_S = \{C_S[i,j] : i = 1,...,I,j = 1,...,J\}$, where $[i,j]$ refers to the element of a matrix in row $i$, column $j$, and $C_S[i,j]$ is 0 or 1 representing the corresponding drug combination $(i,j)$ being under or over the $MTC_θ$. Assume a gain function for combination $(i, j)$ and contour $C_S$ given by $$G(C_S[i,j],p_{ij}) = {\{\table C_S[i,j], p_{ij} ≥θ, ; 1-C_S[i,j], p_{ij} < θ,} (6.1)$$ Thus, the expected posterior gain for $C_S$ is given by $$P_{C_S}(θ | data)=∏↙{i,j}\{1-P_{ij}(θ | data)\}^{C_S{[ij]}}P_{ij}(θ | data)^{1-C_S{[ij]}}$$ where $P_{ij} (θ | data) = Pr(p_{ij} < θ | data)$ is the posterior probability that $p_{ij}$ is smaller than $θ$. The PIPE algorithm aims to find the $C_S$ that maximize the $P_{C_S} (θ | data)$, denoted as {MTC_θ} . In U-Design, the admissible dose set in the dose escalation is defined as the set of the "closest" doses combination near the {MTC_θ}. The closest doses are defined as those adjacent doses below/above the contour that cannot move up (for below) or down (for above) by one dose level without crossing the contour. The admissible dose with smallest effective sample size (defined as the sample size plus $a_{ij} + b_{ij}$ , the effective sample size of the Beta prior) will be assigned to the next cohort of patients. Besides, we also apply the safety rules I and II of mTPI (Section 3.5.2) as well as the no dose skipping rule.

At the end of the trial, all the "closest" dose combinations from below the estimated {MTC_θ} are considered as the selected dose combinations. As a result, the sum of probabilities of dose selection for PIPE across all dose combinations may be larger than 1.

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