Model-Assisted vs. Model-Based Dose-Finding Designs

Introduction

Early-phase oncology trials must answer a difficult question: which dose provides an acceptable balance between toxicity and the need to treat patients at sufficiently high exposure?

Traditional dose-escalation approaches often use simple rules such as the 3+3 design. More modern approaches fall into two broad statistical families: model-assisted and model-based dose-finding designs.

The distinction is important because both families use statistical thinking, but they use it in fundamentally different ways. A model-assisted design typically converts statistical calculations into a prespecified decision algorithm or dose-toxicity table. A model-based design explicitly fits a dose-toxicity model to the accumulating data and uses the fitted model to guide dose selection.

The Dose-Finding Problem

Suppose a Phase I oncology trial evaluates several dose levels:

• \(d_1\): lowest dose
• \(d_2\): second dose
• \(d_3\): third dose
• \(\ldots\)
• \(d_K\): highest prespecified dose

Let \(p_j\) denote the true probability of a dose-limiting toxicity (DLT) at dose \(d_j\).

The objective is usually to identify a dose whose toxicity probability is close to a prespecified target \(p_T\). For example, an oncology study might define:

The target is not necessarily the dose with the lowest observed DLT rate. Rather, the trial seeks a dose whose underlying toxicity probability is acceptably close to the target while maintaining appropriate safety constraints.

What Is a Model-Assisted Design?

A model-assisted design uses statistical modeling to construct a set of prespecified dose-escalation and de-escalation rules, but the operational decision can often be made from a simple table or algorithm based on the number of patients treated and the number of DLTs observed.

Examples include:

• Bayesian Optimal Interval (BOIN)
• modified Toxicity Probability Interval (mTPI and related interval designs)
• Keyboard designs
• other interval-based or model-assisted dose-finding procedures

The design is statistically derived, but investigators do not necessarily have to refit a dose-toxicity curve after every cohort.

What Is a Model-Based Design?

A model-based design explicitly specifies a mathematical relationship between dose and toxicity and estimates that relationship as patient data accumulate.

A generic dose-toxicity model can be written as:

where \(\theta\) represents unknown model parameters.

At each decision point, the accumulating DLT data are used to update information about \(\theta\). The fitted model is then used to estimate the toxicity probability at each candidate dose.

The next dose is selected according to a prespecified rule based on those model-based estimates, often choosing the dose whose estimated toxicity probability is closest to \(p_T\), subject to safety restrictions.

The classic example is the Continual Reassessment Method (CRM). Other model-based approaches include time-to-event extensions such as TITE-CRM and Bayesian escalation methods such as EWOC.

Model-Assisted vs. Model-Based: The Core Difference

The easiest way to understand the distinction is to ask: what is used directly to make the next dose decision?

Why the Distinction Matters

The choice affects how investigators design, execute, monitor, simulate, and communicate a Phase I study.

A model-assisted design may be attractive when the sponsor wants a statistically efficient design that is relatively simple to implement and explain. A model-based design may be attractive when the sponsor wants the dose-toxicity model itself to drive dose selection and is willing to accept greater statistical and operational complexity.

Defining the Target Toxicity Probability

Both design families require a clinically meaningful target. Let:

For illustration, suppose:

The trial is then attempting to identify the dose whose true DLT probability is near 25%.

The target should be established from clinical and pharmacologic considerations, not selected simply because it makes the statistical design convenient. Different therapeutic settings may justify different target toxicity probabilities.

The Three Fundamental Dose Decisions

Most dose-finding designs can be understood in terms of three possible actions:

The major difference is how the design determines whether the toxicity is “too low,” “close,” or “too high.”

• Model-assisted: use a prespecified interval or decision algorithm.
• Model-based: estimate the toxicity probability from the fitted dose-toxicity model.

Model-Assisted Designs: The Interval Concept

Many model-assisted designs define an interval around the target toxicity probability. For illustration, suppose the target is:

A design might use a statistically calibrated interval such as:

The operating rule is conceptually:

The actual boundaries used by a particular design should be calculated from the design methodology and target, rather than arbitrarily selected.

BOIN as a Model-Assisted Example

The Bayesian Optimal Interval design, commonly abbreviated BOIN, is a widely used model-assisted design.

Conceptually, BOIN creates escalation and de-escalation boundaries around the target toxicity probability. Let the target be \(p_T\), and let \(\lambda_e\) and \(\lambda_d\) denote the calibrated escalation and de-escalation boundaries.

The boundaries are not chosen merely as arbitrary percentages. They are derived from the target toxicity probability and the statistical decision criterion of the design.

Keyboard Designs

Keyboard designs use a similar interval-based concept, but organize the decision process into a sequence of probability intervals represented as “keys.”

The observed number of DLTs and treated patients determines which toxicity interval is most compatible with the data. The design then recommends escalation, staying, or de-escalation according to the prespecified decision rule.

The important conceptual point is that the operational procedure is table- or interval-driven, rather than requiring the trial team to fit and interpret a nonlinear dose-toxicity curve at each decision.

mTPI and Related Interval Designs

The modified Toxicity Probability Interval framework, or mTPI, also divides the toxicity probability space into regions around a target.

The statistical calculations determine which interval has the greatest posterior support, after which the design maps that result to an escalation, stay, or de-escalation action.

mTPI and its descendants are therefore often grouped with model-assisted rather than model-based designs in the practical taxonomy used for Phase I dose finding.

Model-Based Designs: The Dose-Toxicity Curve

A model-based design instead specifies a functional relationship between dose and DLT probability.

A generic logistic formulation is:

where \(x(d)\) is a transformed dose metric and \(\alpha,\beta\) are unknown parameters.

A Bayesian model-based design assigns prior distributions to the unknown parameters:

After observing DLT outcomes, the prior is updated to a posterior distribution:

The posterior distribution is then used to estimate toxicity probabilities at candidate doses.

The Continual Reassessment Method (CRM)

The Continual Reassessment Method is the canonical example of a model-based dose-finding design.

Suppose the candidate doses are represented by standardized scores \(x_1,\ldots,x_K\), and the model is:

The CRM updates the model as patient outcomes become available. The next dose is then selected using the updated estimate of the dose-toxicity relationship, typically with safeguards against overly aggressive escalation.

A simplified selection criterion is:

The actual CRM implementation can include overdose control, dose-skipping restrictions, cohort-size rules, and other safeguards. These details are essential parts of the design rather than optional additions.

Worked Example: Same Data, Different Design Logic

Consider five dose levels:

Suppose the target DLT probability is:

A model-assisted interval design evaluates the observed DLT rate at the currently tested dose against its prespecified escalation, stay, and de-escalation boundaries.

A model-based design instead considers all accumulated information and fits a dose-toxicity model. The estimated toxicity at 20, 30, 40, and 50 mg is therefore influenced by the assumed dose-response structure and by the data from neighboring dose levels.

Step 1: Define the Dose Levels

The first step is to specify the candidate doses and the maximum dose that may be administered.

The dose grid should be justified clinically and pharmacologically. The statistical design should not be used to compensate for a poorly chosen starting dose or implausible dose levels.

Step 2: Define the Target

For this example:

The target represents the toxicity probability that the design is attempting to identify.

It is useful to distinguish three quantities:

• Target toxicity probability: the design target.
• Observed DLT rate: the sample proportion observed so far.
• Estimated toxicity probability: a model-based or model-assisted statistical estimate used for decision making.

Step 3: Apply a Model-Assisted Rule

Suppose the selected model-assisted design has already been calibrated to produce an escalation boundary and a de-escalation boundary.

At a dose with \(n\) treated patients and \(y\) DLTs:

The decision is then made from the prespecified rule:

For an actual study, the boundaries must be taken from the chosen design's calibration rather than from an informal approximation.

Step 4: Fit the Model in a Model-Based Design

For a model-based design, the same DLT observations are incorporated into the dose-toxicity model.

Let \(Y_i=1\) if patient \(i\) experiences a DLT and \(Y_i=0\) otherwise. Then:

The likelihood is:

In a Bayesian implementation, the likelihood is combined with the prior to obtain the posterior distribution.

The fitted model can then produce posterior estimates for each candidate dose.

Step 5: Select the Next Dose

Suppose the fitted model produces:

With \(p_T=0.30\), the closest model-estimated toxicity probability is at 40 mg:

whereas at 30 mg:

A model-based design could therefore consider 40 mg the dose closest to the target, subject to its prespecified overdose-control and escalation rules.

Local Information vs. Borrowing Across Doses

One of the most important conceptual differences is how information from neighboring doses influences a decision.

A simple interval rule is strongly tied to the observed outcomes at the current dose. A model-based approach explicitly links doses through the dose-toxicity model.

For example, if a patient experiences a DLT at 30 mg, a fitted monotonic dose-toxicity model may update the estimated toxicity probabilities at 20, 30, 40, and 50 mg simultaneously.

This borrowing of information is one reason model-based designs can be statistically efficient when the dose-toxicity model is appropriate.

Why Model Misspecification Matters

The major tradeoff is that a model-based design depends on the assumed dose-toxicity model.

If the model is poorly specified, the fitted curve can provide misleading toxicity estimates. For example, a model that imposes a particular shape or monotonic relationship may not adequately describe the actual biological relationship.

Why Model-Assisted Designs Can Be Attractive

Model-assisted designs attempt to capture much of the statistical efficiency of principled dose-finding while simplifying implementation.

• Decision rules can be transparent.
• Operational implementation can be straightforward.
• The design can be simulated before the study.
• Statistical calibration can control escalation and de-escalation behavior.
• The design does not necessarily require real-time nonlinear model fitting.
• Investigators can often explain the decision process using a table.

This can be particularly useful when the clinical team wants a sophisticated design without making every dose decision dependent on a model-fitting workflow.

Why Model-Based Designs Can Be Attractive

Model-based designs can use the accumulating data more directly and can borrow information across dose levels through the assumed dose-toxicity relationship.

• They directly estimate the dose-toxicity curve.
• They can use information across multiple dose levels.
• They can accommodate flexible Bayesian or likelihood-based frameworks.
• They can be extended to time-to-event outcomes.
• They can incorporate formal overdose-control criteria.
• They can be highly efficient when the model is well specified.

CRM vs. BOIN: A Useful Comparison

3+3 Is Not the Same as Model-Assisted

A frequent source of confusion is the statement that the 3+3 design is “model-assisted” because it uses a dose-escalation rule.

That terminology should be used carefully.

The traditional 3+3 design is generally considered a rule-based algorithmic design, not a model-assisted design in the modern statistical taxonomy.

The important point is that a design can be algorithmic without being statistically classified as model-assisted.

Safety Constraints Apply to Both Families

Neither model-assisted nor model-based classification eliminates the need for explicit safety controls.

A dose-finding protocol may include rules concerning:

• Maximum tolerated dose or maximum administered dose
• Maximum allowable overdose probability
• Restrictions on skipping dose levels
• Maximum dose-escalation magnitude
• Stopping for excessive toxicity
• Stopping for insufficient information
• Requirements for evaluable patients
• Handling of pending DLT observations

Handling Pending Toxicity Data

Conventional cohort-based designs often wait for the required DLT assessment window before making the next decision. That can be operationally slow when the DLT window is long.

Model-based approaches can be extended to incorporate partial follow-up. The best-known example is TITE-CRM, which incorporates time-to-event information into dose-finding.

Conceptually:

The exact weighting or likelihood contribution depends on the selected time-to-event model and must be prespecified.

Model-Assisted vs. Model-Based: Operational Complexity

Statistical Operating Characteristics

Both design families should be evaluated through simulation before being used in a clinical trial.

Important operating characteristics include:

• Probability of selecting the true target dose
• Probability of selecting an overdose
• Probability of selecting an underdose
• Probability of excessive toxicity
• Average number of patients treated
• Probability of early stopping
• Number of dose escalations
• Number of dose de-escalations
• Trial duration
• Patient exposure at each dose

Let \(D^*\) denote the true target dose. Then one important metric is:

where \(\widehat D\) is the dose ultimately selected by the design.

Probability of Selecting the Target Dose

Suppose there are five dose levels and dose 3 is the true target dose. A useful simulation metric is the proportion of simulated trials that select dose 3 at the end of the study.

The target-selection probability is therefore:

The exact values are design-specific and should be obtained from simulation rather than assumed.

Overdose Control

Target-dose selection alone is not enough. A design may have a high probability of selecting the target dose but still expose too many patients to doses whose toxicity is unacceptably high.

Let \(p_{\mathrm{OD}}\) represent an overdose threshold. For example:

An overdose-control rule can be expressed conceptually as:

where \(q\) is a prespecified maximum acceptable probability of overdose.

The exact formulation depends on the selected design. Bayesian model-based designs can naturally express such posterior overdose probabilities, while model-assisted designs can incorporate safety boundaries into their decision rules.

Why Simulation Is Essential

A dose-finding design is a sequential decision process. Its behavior depends on the entire path of DLT outcomes, not merely on the final DLT rate.

For a candidate design, investigators should simulate many possible trials under a range of true dose-toxicity scenarios.

Example Simulation Scenarios

Suppose the target DLT probability is 30%. Three possible underlying dose-toxicity scenarios might be:

Scenario B has dose 3 as the target dose. Scenario A has dose 4 as the target dose. Scenario C creates a more difficult separation between doses 3 and 4.

A useful design comparison asks whether BOIN, CRM, or another candidate design behaves appropriately under all three scenarios rather than only under the scenario that is most favorable to one method.

Common Misconception: “Model-Based Is Always Better”

A more sophisticated statistical model is not automatically a better clinical trial design.

Model-based methods can be highly efficient when their assumptions are reasonable, but they can also be more sensitive to model specification and implementation details.

Model-assisted designs can offer an attractive compromise: statistically calibrated decision rules with less dependence on real-time model fitting.

Common Misconception: “Model-Assisted Means No Model”

This is also incorrect.

A model-assisted design may rely heavily on statistical modeling during its development. The distinction is that the final operational decision is expressed as a prespecified rule rather than requiring the trial team to fit the underlying model continuously.

Thus:

Instead:

Common Misconception: “Observed DLT Rate Is the Toxicity Probability”

If 1 of 3 patients experiences a DLT, the observed DLT rate is:

But the true toxicity probability is unknown.

The observed proportion is an estimate with substantial uncertainty. Model-assisted and model-based designs differ partly in how they translate that limited information into a dose decision.

Model-Assisted vs. Model-Based: Strengths and Limitations

When a Model-Assisted Design May Be Preferable

A model-assisted approach may be attractive when:

• The sponsor wants a modern statistical design with straightforward operational rules.
• The clinical team values transparent escalation and de-escalation decisions.
• Real-time model fitting would create unnecessary operational complexity.
• The trial has relatively standard discrete dose levels.
• The sponsor wants a design that can be implemented with a clear decision table.
• Strong safeguards against unsafe escalation are important.

When a Model-Based Design May Be Preferable

A model-based approach may be particularly attractive when:

• The dose-toxicity relationship is expected to contain useful information across neighboring dose levels.
• The sponsor has strong statistical programming and Bayesian modeling capabilities.
• Time-to-event information should be incorporated during escalation.
• Complex overdose-control rules are required.
• The trial requires substantial adaptation to accumulating evidence.
• Extensive simulation and model diagnostics are feasible.

Protocol Considerations

Regardless of the design family, the protocol should define the dose-finding procedure before enrollment begins.

At minimum, specify:

• Starting dose
• Candidate dose levels
• Cohort size
• Target toxicity probability
• DLT definition
• DLT observation window
• Escalation rule
• De-escalation rule
• Rules for staying at a dose
• Dose-skipping restrictions
• Overdose-control criteria
• Maximum administered dose
• Stopping rules
• Rules for pending DLT assessments
• Rules for unevaluable patients
• Final dose-selection rule

Statistical Analysis Plan Considerations

The statistical documentation should make the dose-finding procedure reproducible.

For a model-assisted design, document:

• Target toxicity probability
• Exact decision boundaries
• Decision table or algorithm
• Cohort rules
• Safety restrictions
• Final dose-selection procedure

For a model-based design, additionally document:

• Model form
• Dose transformation
• Prior distributions, if Bayesian
• Parameter constraints
• Posterior updating method
• Dose-selection criterion
• Overdose-control criterion
• Restrictions on escalation and dose skipping
• Handling of pending outcomes
• Final posterior or model-based dose-selection rule

Practical Workflow for Choosing Between the Two

R Implementation: A Simple Model-Assisted Decision Rule

The following example illustrates the logic of an interval-style model-assisted rule. It is intentionally generic; actual BOIN, Keyboard, or mTPI boundaries should be generated using the corresponding design methodology.

target <- 0.30
lower  <- 0.20
upper  <- 0.40

decision <- function(dlt, n) {
  p_hat <- dlt / n

  if (p_hat < lower) {
    "Escalate"
  } else if (p_hat > upper) {
    "De-escalate"
  } else {
    "Stay"
  }
}

decision(dlt = 0, n = 3)
decision(dlt = 1, n = 3)
decision(dlt = 2, n = 3)

The output illustrates the structure of an interval rule:

# 0/3  -> Escalate
# 1/3  -> Stay
# 2/3  -> De-escalate

R Implementation: A Generic Model-Based Framework

A simple Bayesian model-based workflow can be represented conceptually as:

# Candidate doses
dose <- c(10, 20, 30, 40, 50)

# Observed DLT data
n_dose   <- c(3, 3, 3, 3, 3)
dlt_dose <- c(0, 0, 1, 1, 2)

# Fit a prespecified dose-toxicity model
# and obtain estimated toxicity probabilities
#
# p_hat <- fitted_model(dose, n_dose, dlt_dose)

# Select the admissible dose closest to target
target <- 0.30
#
# next_dose <- dose[
#   which.min(abs(p_hat - target))
# ]

In an actual CRM implementation, the model-fitting and dose-selection functions would be defined explicitly, along with prior distributions, safety constraints, and rules for pending data.

Comparing Designs Through Simulation

A simple simulation framework repeatedly generates DLT outcomes under a specified true toxicity vector and applies the entire dose-finding algorithm.

true_p <- c(0.05, 0.10, 0.20, 0.30, 0.50)

simulate_trial <- function(true_p, nsim = 10000) {
  # Initialize counters
  selected <- integer(length(true_p))

  # Apply the complete prespecified
  # dose-finding algorithm in each simulation.
  #
  # The actual implementation depends
  # on the selected design.

  selected
}

The purpose of the simulation is not to prove that one design is universally better. It is to determine how each candidate design behaves under the specific scenarios relevant to the clinical program.

What a Good Simulation Report Should Show

Final Dose Selection Is Different From Next-Dose Selection

A subtle but important point is that the dose given to the next cohort is not necessarily the dose ultimately recommended for later development.

During escalation, the design repeatedly asks:

At the end, the study asks:

The final selection may incorporate all accumulated information and may use a different rule from the operational next-dose decision.

Relationship to the MTD Concept

Historically, Phase I oncology studies often focused on the maximum tolerated dose, or MTD.

Modern dose-finding programs increasingly recognize that the biologically and clinically appropriate dose may not be the highest tolerable dose. Exposure-response, pharmacodynamics, efficacy, safety, and the overall benefit-risk profile may all contribute to dose selection.

Consequently, a design may target a toxicity probability while the final clinical dose recommendation incorporates additional information.

This distinction becomes particularly important for targeted therapies, immunotherapies, molecularly driven treatments, and drugs with delayed or cumulative toxicity.

Common Mistakes

1. Calling every rule-based design model-assisted. The traditional 3+3 design is generally classified as rule-based rather than model-assisted.
2. Assuming model-assisted means model-free. Model-assisted designs are statistically derived; they simply operationalize the statistical information through prespecified rules.
3. Assuming model-based designs are automatically superior. Model-based efficiency depends on appropriate model specification and careful implementation.
4. Ignoring overdose control. Selecting the dose closest to the target without a safety constraint can expose patients to unacceptable toxicity.
5. Ignoring pending DLT information. A design must prespecify how incomplete follow-up affects dose decisions.
6. Failing to simulate the design. Small-cohort dose finding is highly variable, so analytical intuition alone is not sufficient.
7. Changing the algorithm after observing data. Unplanned changes can invalidate the original operating-characteristic evaluation.
8. Assuming the observed DLT rate equals the true toxicity probability. Small cohorts produce highly uncertain observed rates.
9. Using the same design for every program. The appropriate method depends on the drug, dose range, toxicity timing, clinical objective, and operational environment.
10. Assuming the MTD is necessarily the recommended Phase II dose. The final dose recommendation may require pharmacokinetic, pharmacodynamic, efficacy, and broader safety information.

Model-Assisted vs. Model-Based: Decision Framework

The Most Important Concept

The most important distinction is not whether a design uses mathematics. Both model-assisted and model-based designs are statistical methods.

The key question is: does the trial directly fit and update a dose-toxicity model to make the dose decision, or does it use a prespecified statistically calibrated decision rule?

Model-assisted designs such as BOIN, Keyboard, and mTPI-type methods provide a statistically principled middle ground between simple algorithmic escalation and fully model-based dose finding. Model-based methods such as CRM explicitly estimate the dose-toxicity relationship and can use information across dose levels more directly.

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