Modified Toxicity Probability Interval (mTPI) Design: A Worked Example

The target-region probability is:

and the over-dosing probability is:

These quantities make the mTPI calculation straightforward to implement computationally.

R Implementation: Basic Setup

The following R code specifies the core design parameters.

pT <- 0.25

epsilon1 <- 0.05
epsilon2 <- 0.05

pL <- pT - epsilon1
pU <- pT + epsilon2

prior_alpha <- 1
prior_beta  <- 1

The resulting equivalence interval is:

pL
# 0.20

pU
# 0.30

R Function for the mTPI Decision

mtpi_decision <- function(
  x,
  n,
  pT = 0.25,
  epsilon1 = 0.05,
  epsilon2 = 0.05,
  prior_alpha = 1,
  prior_beta = 1
) {

  pL <- pT - epsilon1
  pU <- pT + epsilon2

  post_alpha <- prior_alpha + x
  post_beta <- prior_beta + n - x

  prob_under <- pbeta(
    pL,
    post_alpha,
    post_beta
  )

  prob_target <- pbeta(
    pU,
    post_alpha,
    post_beta
  ) - prob_under

  prob_over <- 1 - pbeta(
    pU,
    post_alpha,
    post_beta
  )

  upm_under <-
    prob_under / pL

  upm_target <-
    prob_target / (pU - pL)

  upm_over <-
    prob_over / (1 - pU)

  upms <- c(
    under = upm_under,
    target = upm_target,
    over = upm_over
  )

  action <- names(which.max(upms))

  if (action == "under") {
    decision <- "Escalate"
  } else if (action == "target") {
    decision <- "Stay"
  } else {
    decision <- "De-escalate"
  }

  list(
    posterior_alpha = post_alpha,
    posterior_beta = post_beta,
    prob_under = prob_under,
    prob_target = prob_target,
    prob_over = prob_over,
    upm_under = upm_under,
    upm_target = upm_target,
    upm_over = upm_over,
    decision = decision
  )
}

Apply the Function to the Worked Example

For one DLT among three patients:

result <- mtpi_decision(
  x = 1,
  n = 3
)

result

The key values are approximately:

result$prob_under
# 0.1808

result$prob_target
# 0.1675

result$prob_over
# 0.6517

result$upm_under
# 0.904

result$upm_target
# 1.675

result$upm_over
# 0.931

result$decision
# "Stay"

The largest UPM is the target-region UPM.

Therefore:

Generate the Complete Decision Table in R

The mTPI decision can be calculated for every possible number of DLTs in a three-patient cohort.

results <- lapply(
  0:3,
  function(x) {

    z <- mtpi_decision(
      x = x,
      n = 3
    )

    data.frame(
      DLTs = x,
      Posterior_Alpha = z$posterior_alpha,
      Posterior_Beta = z$posterior_beta,
      UPM_Under = z$upm_under,
      UPM_Target = z$upm_target,
      UPM_Over = z$upm_over,
      Decision = z$decision
    )
  }
)

do.call(rbind, results)

mTPI Does Not Require a Parametric Dose-Toxicity Curve

One important feature of mTPI is that it does not require investigators to specify a particular mathematical relationship between dose and toxicity.

For example, the design does not require a logistic model such as:

Instead, mTPI evaluates the toxicity probability at the current dose using the accumulated data and makes a local decision.

This is one reason mTPI is classified as a model-assisted rather than a fully model-based dose-finding design.

mTPI Versus the CRM

The continual reassessment method (CRM) and mTPI both use Bayesian reasoning, but they solve the dose-finding problem differently.

The distinction is important.

CRM attempts to estimate a dose-toxicity relationship across dose levels.

mTPI instead uses the posterior distribution of toxicity probability at the current dose and compares its concentration across predefined probability regions.

mTPI Versus BOIN

The Bayesian optimal interval (BOIN) design is another model-assisted dose-finding method.

Both designs are considerably more formal than the traditional 3+3 design, but their decision rules are derived differently.

mTPI, mTPI-2, and Related Designs

The original mTPI design motivated subsequent developments in interval-based dose-finding.

In particular, the mTPI-2 design was developed to address some practical and interpretational issues associated with the original UPM formulation.

mTPI-2 uses a series of equal-length intervals around the target toxicity probability and assigns decisions according to posterior probability mass rather than relying on the unequal-width outer regions of the original mTPI construction.

Why the Original mTPI Uses UPM

Consider the three regions:

The target interval is only 0.10 wide, while the over-dosing region is 0.70 wide.

Simply comparing posterior probabilities would therefore give a structural advantage to broad regions.

For example, a broad over-dosing region can contain a large amount of probability simply because it covers most of the parameter space.

Dividing by interval length removes this direct dependence on region width.

Interpreting the mTPI Decision Geometrically

A useful conceptual picture is to imagine the posterior distribution spread across the possible toxicity probabilities from 0 to 1.

The mTPI algorithm asks:

Where is the posterior probability most concentrated per unit of toxicity-probability space?

If concentration is greatest below the target interval, the dose is considered too safe and escalation is appropriate.

If concentration is greatest inside the target interval, the current dose is appropriate and the design stays.

If concentration is greatest above the target interval, the dose appears too toxic and the design de-escalates.

The Role of Sample Size

The amount of accumulated information at the current dose strongly affects the posterior distribution.

With very few patients, the posterior is relatively diffuse.

As more patients are treated, the posterior generally becomes more concentrated around the observed toxicity rate.

For example, with:

the posterior is:

whereas with:

the posterior becomes:

The second posterior contains more information because it is based on five times as many patients.

Posterior Uncertainty Is Essential

A common mistake is to use only the observed toxicity proportion:

For example:

This number alone does not communicate how uncertain the estimate is.

mTPI instead uses:

which incorporates both the observed toxicity count and the uncertainty associated with the finite sample.

Decision Boundaries Are Not the Same as Confidence Intervals

The mTPI equivalence interval should not be confused with a confidence interval for the toxicity probability.

The equivalence interval is a design-defined clinical target region.

For the worked example:

is determined before the trial based on what investigators consider to be an acceptable toxicity range around the target.

The posterior distribution then determines how much probability lies in each decision region.

Safety Constraints Must Be Added to the Statistical Rule

The basic mTPI decision rule should not be implemented without considering clinical safety constraints.

For example, a protocol may specify:

• Never escalate by more than one dose level.
• Never skip dose levels during escalation.
• Do not escalate beyond the highest planned dose.
• Do not de-escalate below the lowest dose.
• Stop the trial if excessive toxicity is observed.
• Stop the trial if all available doses appear excessively toxic.

No Escalation Beyond the Highest Dose

Suppose the trial is currently at the highest planned dose.

Even if the mTPI calculation recommends escalation, escalation is impossible.

The protocol should therefore specify:

The statistical recommendation and the operational action are therefore not always identical at the boundaries of the dose range.

No De-Escalation Below the Lowest Dose

Similarly, if the current dose is the lowest available dose and the mTPI algorithm recommends de-escalation:

The protocol must specify what happens next.

Depending on the trial, this could involve stopping enrollment, declaring that no dose is acceptable, or applying another prespecified rule.

Complete Dose-Finding Algorithm

A practical mTPI implementation can be summarized as follows.

What Does "Stay" Mean?

A stay decision means that the current evidence is most consistent with the toxicity probability lying within the target region.

It does not mean that the toxicity probability is known to equal \(p_T\).

For example:

and:

A stay decision means that the target interval has the largest UPM, not that:

is large.

For a continuous parameter, the posterior probability of exactly one point is zero.

What Does "Escalate" Mean?

An escalation decision indicates that the under-dosing region has the largest UPM.

In other words, the posterior evidence is most concentrated below the target toxicity interval.

The statistical interpretation is:

The next cohort is therefore treated at a higher dose, subject to protocol constraints.

What Does "De-Escalate" Mean?

A de-escalation decision occurs when:

The posterior evidence is most concentrated in the region above the target toxicity interval.

The next cohort is therefore treated at a lower dose, subject to protocol constraints.

What Is the Final MTD Selection Rule?

The dose-finding rule and the final MTD selection rule should not be assumed to be identical.

During the trial, mTPI determines whether to:

• Escalate
• Stay
• De-escalate

At the end of the trial, a separate prespecified rule may be used to identify the recommended dose.

For example, investigators may select the dose whose posterior toxicity probability is closest to the target:

Other selection criteria may be used.

Probability of Correct Dose Selection

One important operating characteristic of a dose-finding design is the probability of correctly selecting the target dose.

If the true dose-toxicity probabilities are known in simulation, one can calculate:

This quantity is generally evaluated through simulation because the full adaptive path of a dose-finding trial can be complex.

Patient Allocation Across Doses

Another important operating characteristic is the percentage of patients treated at each dose.

For dose \(d\), define:

A good dose-finding design should generally allocate a substantial proportion of patients near the true target dose while limiting exposure to excessively toxic doses.

Simulation is commonly used to evaluate this behavior.

Overdose Control

An especially important safety metric is the probability that the trial selects a dose whose true toxicity probability is above an unacceptable threshold.

Suppose an unacceptable toxicity threshold is \(p_U\).

An overdose-selection probability can be defined as:

This should be evaluated under multiple dose-toxicity scenarios.

Simulation Is Central to Design Evaluation

Unlike a simple fixed-sample hypothesis test, an adaptive dose-finding design has a large number of possible patient-by-patient paths.

Simulation can reproduce these paths.

A typical simulation proceeds as follows:

1. Specify a true toxicity probability for each dose.
2. Start at the prespecified starting dose.
3. Generate patient toxicity outcomes.
4. Apply the mTPI decision rule.
5. Move to the next dose according to the decision.
6. Continue until the stopping rule is reached.
7. Select the final recommended dose.
8. Repeat thousands of times.

R Simulation Skeleton

The following code illustrates the structure of an mTPI simulation.

set.seed(123)

true_tox <- c(
  0.05,
  0.10,
  0.18,
  0.25,
  0.40,
  0.60
)

n_sim <- 5000
cohort_size <- 3

simulate_trial <- function(true_tox) {

  n_dose <- length(true_tox)

  current_dose <- 3

  n_treated <- rep(0, n_dose)
  n_dlt <- rep(0, n_dose)

  for (cohort in 1:10) {

    outcomes <- rbinom(
      1,
      cohort_size,
      true_tox[current_dose]
    )

    n_treated[current_dose] <-
      n_treated[current_dose] +
      cohort_size

    n_dlt[current_dose] <-
      n_dlt[current_dose] +
      outcomes

    decision <- mtpi_decision(
      x = n_dlt[current_dose],
      n = n_treated[current_dose]
    )$decision

    if (decision == "Escalate") {

      current_dose <-
        min(current_dose + 1, n_dose)

    } else if (decision == "De-escalate") {

      current_dose <-
        max(current_dose - 1, 1)
    }
  }

  list(
    final_dose = current_dose,
    n_treated = n_treated,
    n_dlt = n_dlt
  )
}

This is only a simulation framework. A production implementation should include all protocol-specific stopping, dose-skipping, overdose-control, and final-dose-selection rules.

Evaluating Multiple Scenarios

Suppose the true toxicity probabilities are:

Dose 4 is the true target dose because:

The simulation can then estimate:

• Probability of selecting dose 4
• Probability of selecting an overly toxic dose
• Average number of patients treated at each dose
• Probability of reaching each dose
• Average number of DLTs
• Probability of early stopping

Scenario-Based Design Evaluation

A comprehensive simulation study should consider several possible dose-toxicity relationships.

mTPI Does Not Automatically Guarantee Patient Safety

A statistical dose-finding algorithm cannot replace clinical safety oversight.

A complete Phase I design may require separate rules for:

• Severe or unexpected toxicity
• Non-DLT serious adverse events
• Organ toxicity
• Treatment-related deaths
• Stopping for excessive toxicity
• Stopping because no dose is sufficiently safe
• Stopping because the maximum dose has been reached

These rules must be incorporated into the protocol independently of the basic mTPI UPM calculation.

Delayed Toxicities

The basic mTPI formulation assumes that toxicity outcomes are sufficiently observed before the next dose decision is made.

This assumption can be problematic when DLTs occur after a long observation period.

For example, suppose DLTs are evaluated over a 28-day window.

If a new cohort is enrolled before all members of the previous cohort have completed the window, the toxicity data are incomplete.

A standard mTPI design may therefore not be appropriate without modification.

Specialized designs such as time-to-event extensions can be considered when toxicity outcomes are delayed.

Handling Partial Follow-Up

A protocol should specify:

• The DLT observation window
• When a patient is considered evaluable
• How incomplete follow-up is handled
• Whether enrollment pauses while toxicity data mature
• Whether a time-to-event extension is used

These details can materially affect the operating characteristics.

Prior Sensitivity

Because mTPI is Bayesian, the prior distribution should be considered during design development.

The common Beta(1,1) prior is:

But other priors are possible.

For example:

has prior mean:

whereas:

also has prior mean 20% but contains more prior information.

The prior concentration therefore matters, not just the prior mean.

Prior Sensitivity Should Be Evaluated

A useful sensitivity analysis can compare:

• Beta(1,1)
• Other weakly informative beta priors
• Clinically informed priors
• Alternative prior concentrations

The objective is to determine whether reasonable changes in the prior materially alter dose decisions or operating characteristics.

What mTPI Does Not Do

The mTPI design does not directly estimate treatment efficacy.

It does not determine the recommended dose based on:

• Response rate
• Progression-free survival
• Overall survival
• Pharmacokinetic exposure
• Pharmacodynamic response

unless these quantities are incorporated into a separate dose-selection framework.

The primary statistical target of the basic mTPI design is toxicity.

MTD Does Not Necessarily Mean Best Dose

The dose with the target toxicity probability is not necessarily the dose that will ultimately be selected for later-phase development.

A Phase I trial may identify a dose that has an acceptable toxicity profile, but additional information may be required to determine the recommended Phase II dose.

That information may include:

• Pharmacokinetics
• Pharmacodynamics
• Efficacy signals
• Exposure-response relationships
• Safety beyond the DLT window
• Chronic tolerability

Common Mistakes

1. Confusing mTPI with CRM. Both are Bayesian dose-finding methods, but mTPI does not require the same global dose-toxicity model used by CRM.
2. Using the posterior mean alone. mTPI decisions are based on UPM, not simply whether the posterior mean is above or below the target.
3. Ignoring the equivalence interval. The target is represented by an interval, not merely a point.
4. Comparing raw posterior probabilities without accounting for interval width. The UPM calculation divides posterior probability by interval length.
5. Confusing mTPI with mTPI-2. They are related but have different decision constructions.
6. Ignoring accumulated data. The posterior should incorporate all relevant toxicity observations at the current dose according to the protocol.
7. Allowing dose skipping without prespecified rules. Dose movement restrictions should be specified before the trial begins.
8. Ignoring delayed toxicity. A standard mTPI calculation assumes sufficiently mature toxicity information.
9. Failing to specify the final dose-selection rule. The dose-finding algorithm and final selection rule should be explicitly defined.
10. Failing to evaluate operating characteristics through simulation. Dose-finding designs should be evaluated across clinically plausible dose-toxicity scenarios.
11. Treating the target toxicity probability as a purely statistical parameter. The target should be justified clinically.
12. Assuming the MTD is automatically the optimal Phase II dose. The final clinical dose may require additional safety, PK, PD, and efficacy considerations.

mTPI Design Workflow

What Should Be Reported in the Protocol?

A complete statistical section for an mTPI study should clearly document the dose-finding algorithm.

At minimum, specify:

• Definition of DLT
• DLT observation window
• Target toxicity probability
• Lower equivalence limit
• Upper equivalence limit
• Prior distribution
• Starting dose
• Available dose levels
• Cohort size
• Dose-escalation rules
• Dose-de-escalation rules
• Dose-skipping restrictions
• Maximum dose
• Safety stopping rules
• Rules for incomplete toxicity information
• Final MTD or recommended-dose selection rule
• Simulation scenarios used for design evaluation
• Operating characteristics

Worked Example Summary

The Most Important Concept

The most important conceptual point is that mTPI does not simply compare the observed toxicity rate with a target.

Instead, it asks where the posterior distribution of the unknown toxicity probability is most concentrated after accounting for the amount of information available.

The process is:

The three possible decisions are:

• Escalate when the under-dosing region has the largest UPM.
• Stay when the target region has the largest UPM.
• De-escalate when the over-dosing region has the largest UPM.

This creates a transparent bridge between Bayesian inference and a practical dose-escalation rule.

References

Ji, Y., Li, Y. & Bekele, B.N. (2007). Dose-finding in phase I clinical trials based on toxicity probability intervals. Clinical Trials, 4(3), 235–244.
Ji, Y. & Wang, S.J. (2013). Modified toxicity probability interval design: a safer and more reliable method than the 3+3 design for dose-finding trials. Clinical Trials, 10(1), 89–98.
Ji, Y., Liu, P., Li, Y. & Bekele, B.N. (2010). A modified toxicity probability interval method for dose-finding trials. Clinical Trials.
Guo, W., Wang, S.J. & Ye, H. (2017). More efficient dose-finding designs based on toxicity probability intervals. Statistics in Medicine.
Neuenschwander, B., Branson, M. & Gsponer, T. (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in Medicine, 27, 2420–2439.
Yuan, Y., Hess, K.R., Hilsenbeck, S.G. & Gilbert, M.R. (2016). Bayesian optimal interval designs for phase I clinical trials. Journal of the Royal Statistical Society: Series C, 65(4), 507–523.