Posterior Distribution of the Maximum Tolerated Dose

From a Dose-Toxicity Curve to an MTD Distribution

A dose-toxicity model describes uncertainty about toxicity risk across doses. A single fitted curve may suggest one dose at which the risk reaches the target, but the fitted model is uncertain. The corresponding maximum tolerated dose (MTD) is therefore uncertain too. Its posterior distribution represents that uncertainty directly, rather than reducing it immediately to one recommended dose.

The MTD in this tutorial is a dose on a continuous scale at which the model's toxicity probability equals a prespecified target rate. This definition requires a model that gives a unique target crossing. A trial may offer only a finite set of dose levels; converting the continuous MTD distribution to that grid is a separate step, considered below.

Logistic Model and Posterior Transformation

For a clear derivation, consider a one-parameter logistic dose-toxicity curve with a fixed, positive slope \(b\) and an uncertain intercept \(a\):

Here \(d\) is dose on the scale used by the model, and \(\operatorname{logit}(u)=\log\{u/(1-u)\}\). The positive slope makes toxicity probability increase with dose. More generally, a two-parameter logistic model can assign a posterior distribution to both \(a\) and \(b\); requiring \(b>0\) throughout the relevant posterior support ensures monotonicity. In either case, the model and prior must give proper posterior uncertainty for the parameters.

For \(n_i\) patients treated at dose \(d_i\), let \(Y_i\) be the number with toxicity. A standard likelihood assumes independent patient outcomes conditional on the doses and model parameters, with \(Y_i\) binomial with size \(n_i\) and toxicity probability \(p(d_i\mid\theta)\). A prior distribution for \(\theta\), together with this likelihood, gives the posterior distribution. The posterior MTD is obtained by applying the function \(d^*(\theta)\) to that posterior; no separate likelihood for the MTD is needed.

Write \(q=\operatorname{logit}(\tau)\). Solving the logistic model equation for the dose gives:

Thus each posterior draw of \(a\) and \(b\) yields one MTD draw. The transformation also gives a useful expression for the posterior cumulative distribution function (CDF). For \(b>0\), the event \(d^*\le d\) is equivalent to \(a+bd\ge q\). Because the logistic function is increasing, that is the event that the modeled toxicity probability at dose \(d\) is at least the target:

This identity is a direct way to calculate the MTD CDF from posterior draws: at each dose \(d\), count the posterior draws whose toxicity probability at \(d\) meets or exceeds \(\tau\). Notice the direction: this is the probability that the MTD is at or below \(d\), not the probability that it is above \(d\).

Why the Transformation Gives the Distribution

The general principle is that a function of a posterior random parameter is itself a posterior random quantity. If \(\theta\) has posterior density \(\pi(\theta\mid\text{data})\), then for any set \(A\) of doses,

where \(I\{\cdot\}\) equals 1 when the condition inside braces holds and 0 otherwise. This expression is the posterior distribution induced by the MTD function. For the logistic model with \(b>0\), the CDF identity follows because:

If a joint posterior density for \((a,b)\) is available and \(b>0\), a density for the MTD can also be written by changing variables from \(a\) to \(d^*\), holding \(b\) fixed. Since \(a=q-bd^*\), the absolute Jacobian is \(b\), so:

In practice, posterior simulation is often simpler than evaluating this integral. For each posterior draw, calculate \(d^*=(q-a)/b\); then use the resulting draws to estimate probabilities, quantiles, and intervals. With equal-weight draws, a posterior probability is estimated by the fraction satisfying the event. With weighted draws, use the sum of the weights for those draws.

Continuous Dose Scale and a Discrete Trial Grid

A model can define \(d^*\) continuously even when a trial can administer only doses from a finite set. The continuous posterior answers questions about the target crossing itself. A dose-grid summary requires a rule for translating the crossing to an available dose. One common descriptive rule maps each posterior parameter draw to the candidate dose whose modeled toxicity probability is closest to \(\tau\). Another rule may choose the nearest dose to \(d^*\) on the dose scale. These need not agree when doses are unevenly spaced or the curve is steep.

The following example uses the closest-to-target-probability rule. It demonstrates how a posterior distribution over candidate dose labels is induced by the model; it does not make that grid distribution identical to the continuous MTD distribution.

Summaries, Diagnostics, and Limits

Useful summaries include posterior quantiles of \(d^*\), posterior probabilities such as \(P(d^*\le d\mid\text{data})\), and the probability that the MTD falls within the studied dose range. A central posterior interval gives a range containing the specified posterior probability under the model and prior. It is not a frequentist confidence interval, and its coverage interpretation depends on the Bayesian model and prior.

A continuous logistic model has a finite root for every finite parameter draw with \(b>0\), but that root may lie outside the doses studied or available. Report the probability that \(d^*\) lies within the clinically allowable range; do not hide extrapolation by reporting only a posterior mean. If the estimand is deliberately restricted to the allowable range, state how roots outside it are handled. Capping roots at the boundaries, for instance, creates point mass at those boundaries and changes the distribution being summarized.

Diagnostics should examine more than whether posterior computations converged. Check that the model enforces the intended direction of dose-toxicity, that posterior draws yield a unique target crossing, and that the inferred MTD is supported by data rather than remote extrapolation. A slope posterior that allows \(b\le0\) does not satisfy the monotonicity condition used in the derivation; with a negative slope the CDF inequality reverses, while a zero slope may yield no unique crossing. If the model averages over several dose-toxicity skeletons, the MTD distribution must also include uncertainty over the model index. If model \(k\) has posterior weight \(w_k\) and CDF \(F_k(d)\), the averaged CDF is \(\sum_k w_kF_k(d)\).

The posterior mean of the MTD is generally not the MTD calculated from posterior mean parameters. In the logistic model, \(d^*=(q-a)/b\) is nonlinear in the parameters, so in general \(E\{(q-a)/b\mid\text{data}\}\ne(q-E(a\mid\text{data}))/E(b\mid\text{data})\). Transforming each posterior draw first preserves the induced distribution and avoids treating a plug-in value as though it summarized all posterior uncertainty.

The posterior distribution also answers a different question from selecting the dose with posterior mean toxicity closest to target. The latter compares dose-level summaries; the MTD distribution tracks uncertainty in the target crossing under each parameter draw. A discrete-grid rule can be useful operationally, but its output depends on the mapping rule and is not interchangeable with continuous MTD quantiles. The interpretation of the posterior remains conditional on the dose-toxicity model, the prior, and the observed trial data.