Continuous Dose Escalation With Model-Based Risk Curves

Why Use a Continuous Risk Curve?

Dose-escalation trials often administer a limited set of dose levels, but the dose that corresponds to a chosen toxicity risk may lie between those levels. A continuous dose-toxicity model makes that interpolation explicit: it estimates risk as a function of dose, then allows the dose associated with a target risk to be calculated. This is useful only when the model is a reasonable description of the dose-response relationship and when the resulting dose is compatible with the trial’s allowable dose choices.

The distinction between a continuous recommendation and a grid recommendation matters. A calculated dose such as 30.59 mg is not automatically a dose that should be administered. A protocol may permit only specified dose levels, and a safety rule may rule out some of them. The continuous estimate helps locate a risk boundary; the dose grid and trial rules determine which choices are available.

Model and Assumptions

Definition: Let $Y_i=1$ indicate a dose-limiting toxicity (DLT) for participant $i$ during the specified assessment window, and $Y_i=0$ otherwise. For a positive dose $d_i$, the two-parameter logistic risk model is $$ P(Y_i=1\mid d_i)=p(d_i)=\frac{1}{1+\exp[-\{\alpha+\beta\log(d_i/d_{\mathrm{ref}})\}]}, $$ where $d_{\mathrm{ref}}>0$ is a chosen reference dose, $\alpha$ is the log-odds of DLT at $d_{\mathrm{ref}}$, and $\beta$ is the change in log-odds per one-unit increase in log dose. When $\beta>0$, the modeled DLT risk increases with dose.

The reference dose sets the origin of the log-dose scale; changing it changes the intercept but not the fitted risk curve, provided the intercept is transformed consistently. The model requires positive doses because $\log(d_i/d_{\mathrm{ref}})$ is defined only when both doses are positive.

For the basic fit below, assume participants’ DLT outcomes are conditionally independent given dose, the assessment window and DLT definition are the same across participants, and participants at a given dose share the same risk under the model. The logistic form is assumed to describe risk over the dose range being used. A monotone increasing curve additionally requires $\beta>0$; a fitted negative slope would contradict the intended escalation interpretation and should prompt investigation rather than automatic use.

With $y_j$ DLTs among $n_j$ participants at dose $d_j$, the grouped data model is $y_j\sim\operatorname{Binomial}(n_j,p(d_j))$. The maximum-likelihood fit chooses $\alpha$ and $\beta$ to maximize the binomial likelihood. With more than two distinct dose levels, this is generally done numerically. The fitted coefficients describe the curve estimated from the observed outcomes; they do not remove uncertainty or establish that the logistic shape is correct.

Fitting the Curve and Finding a Target Dose

Worked Example: Fit a Curve From Two Dose Levels

Suppose a hypothetical trial has completed DLT assessment for all participants and observes 2 DLTs among 20 participants at 20 mg, and 8 among 20 at 40 mg. For a transparent calculation, use $d_{\mathrm{ref}}=20$ mg. The model is fitted to grouped binomial outcomes at these two dose levels.

At the two observed doses, the sample proportions are $2/20=0.10$ and $8/20=0.40$. With two parameters and two distinct dose levels, the model can reproduce both observed proportions exactly. The corresponding fitted logits are

\operatorname{logit}(0.10)=\log(0.10/0.90)=-2.197225,\qquad \operatorname{logit}(0.40)=\log(0.40/0.60)=-0.405465.

At 20 mg, $\log(20/20)=0$, so $\hat{\alpha}=-2.197225$. At 40 mg, $\log(40/20)=\log 2$. Therefore,

\hat{\beta}= \frac{-0.405465-(-2.197225)}{\log 2} =\frac{1.791760}{0.693147} =2.584963.

The fitted curve is thus

\hat{p}(d)= \frac{1}{1+\exp[-\{-2.197225+2.584963\log(d/20)\}]}.

This slope is positive, so the fitted curve increases with dose as assumed. The exact match at both observed doses follows from the two-level, two-parameter setup; it is not evidence that the logistic curve has been validated between those doses.

Derivation: To find the dose with target risk $q$, set $p(d)=q$. Taking the logit gives $\log\{q/(1-q)\}=\alpha+\beta\log(d/d_{\mathrm{ref}})$. For $\beta\ne0$, isolate the log dose and exponentiate: $$ d(q)=d_{\mathrm{ref}}\exp\left[ \frac{\log\{q/(1-q)\}-\alpha}{\beta} \right]. $$ When $\beta>0$, risks below $q$ occur at doses below $d(q)$, and risks above $q$ occur at doses above it.

Worked Example: Calculate the Dose at a Target Risk

Using the hypothetical fitted curve above, calculate the dose associated with a 25% DLT risk. Here $q=0.25$, so $\log\{q/(1-q)\}=\log(0.25/0.75)=-1.098612$. Substitution into the inverse curve gives

\begin{aligned} d(0.25) &=20\exp\left[ \frac{-1.098612-(-2.197225)}{2.584963} \right]\ \text{mg}\\ &=20\exp(0.42498)\ \text{mg}\\ &\approx 30.59\ \text{mg}. \end{aligned}

Thus, under this fitted model, the estimated DLT risk at approximately 30.59 mg is 25%. This is a model-based point estimate, not a guarantee of the risk at that dose or a statement that the parameter is known exactly. The data were observed only at 20 and 40 mg, so this recommendation interpolates between observed dose levels.

Risk Constraints and Discrete Dose Grids

A dose constraint can be written directly in terms of the risk curve. For an increasing curve, an upper-risk rule $p(d)\leq q_U$ is equivalent to $d\leq d(q_U)$. A target risk band $q_L\leq p(d)\leq q_U$, with $0<q_L<q_U<1$, corresponds to $d(q_L)\leq d\leq d(q_U)$. These equivalences depend on $\beta>0$; if the curve is not increasing, the inequality directions must be reconsidered.

A discrete dose grid is a separate decision layer. Given allowable doses $G=\{g_1,\ldots,g_K\}$, calculate or otherwise assess risk at each allowable dose, then apply the trial’s stated constraints. If the rule is to choose the highest allowable dose with fitted risk no greater than $q_U$, the recommendation is the largest $g_k\in G$ satisfying $\hat{p}(g_k)\leq q_U$. It is not necessarily the nearest grid dose to the continuous target.

Worked Example: Apply an Upper-Risk Rule on a Dose Grid

Retain the fitted curve and suppose the allowable doses are 20, 25, 30, 35, and 40 mg. The rule is to choose the highest grid dose with estimated DLT risk at most 25%. At 30 mg, the linear predictor is

\hat{\eta}(30)=-2.197225+2.584963\log(30/20) =-2.197225+2.584963(0.405465) \approx -1.1491.

Therefore,

\hat{p}(30)=\frac{1}{1+\exp(1.1491)}\approx 0.2408.

At 35 mg, the linear predictor is $-2.197225+2.584963\log(1.75)\approx-0.7506$, giving $\hat{p}(35)\approx0.3206$. Since the model is increasing, grid doses above 35 mg also exceed the 25% limit. Of the listed doses, 30 mg is the highest with fitted risk at most 25%; it is below the continuous boundary of approximately 30.59 mg.

A nearby grid dose is not automatically safe merely because it is close to the continuous estimate. For example, 31 mg is above the boundary; the fitted increasing curve therefore assigns it a risk greater than 25%. The exact available-dose rule must be applied to the actual grid, not to a rounded continuous result.

Worked Example: Find a Target-Risk Band and Its Grid Choice

Suppose the hypothetical design defines an acceptable fitted-risk band from 10% to 30%. The lower boundary is attained at 20 mg because the observed proportion there is 10% and the fitted curve passes through that point. For the upper boundary, use $q_U=0.30$, whose logit is $\log(0.30/0.70)=-0.847298$:

\begin{aligned} d(0.30) &=20\exp\left[ \frac{-0.847298-(-2.197225)}{2.584963} \right]\ \text{mg}\\ &=20\exp(0.52222)\ \text{mg}\\ &\approx 33.72\ \text{mg}. \end{aligned}

Because the curve is increasing, the continuous doses with fitted risk between 10% and 30% run from 20 mg through approximately 33.72 mg. On the grid $\{20,25,30,35,40\}$ mg, 20, 25, and 30 mg lie in this band, while 35 and 40 mg exceed its upper risk. The highest grid dose within the band is 30 mg. The continuous upper boundary and the discrete choice answer related but different questions.

Diagnostics, Alternatives, and Limitations

The two-point example is useful for showing the algebra, but a two-parameter curve fitted to exactly two distinct dose levels has no spare information for assessing whether its shape fits. With data at several dose levels, compare observed DLT proportions with fitted risks and inspect residual or deviance summaries for systematic departures. A pattern in which observed risks repeatedly bend away from the fitted curve may indicate that the assumed log-dose logistic form is inadequate. Check that the estimated slope has the intended sign, that the inverse target dose lies within or near the observed dose range, and that the fitted curve does not imply implausible risks over doses relevant to the decision.

Sparse data can also make a numerical fit unstable. If a dose group has no DLTs or all participants have DLTs, the observed group logit is not finite; with sparse groups, the estimate may be highly sensitive to a small number of outcomes. A very steep curve can put a target dose in a narrow interval, so rounding or grid conversion becomes especially consequential. These are reasons to inspect the data and fitting behavior, not reasons to treat a computed root as inherently reliable.

A discrete analysis that uses only observed grid risks avoids assuming a smooth curve between dose levels, but it may not provide a direct risk estimate at an unobserved dose and can be unstable with few participants per level. A continuous logistic curve offers interpolation and a compact, monotone summary when that shape is credible. It can, however, conceal lack of fit and make unsupported extrapolation appear precise. The choice between these approaches should reflect the dose range, data quantity, model diagnostics, and the protocol’s decision rule.

These calculations use a fitted risk curve, not a full uncertainty-aware safety decision. In particular, a point estimate below a risk limit does not establish that the true risk is below that limit. Model-based escalation procedures may incorporate parameter uncertainty and other safety rules; those decision criteria should not be replaced by the point-estimate calculation shown here.

Common Mistake

Common pitfall: A learner calculates a continuous target dose of 30.59 mg, rounds it to 31 mg, and assumes it still satisfies a 25% risk limit because it is close to the boundary. The fitted curve is increasing, so every dose above 30.59 mg has fitted risk above 25%; proximity does not preserve the constraint. Calculate risk at allowable doses or compare each dose with the exact boundary, and follow the protocol’s rounding and grid rules.

A second frequent error is to treat the observed proportions at two doses as confirmation of the entire continuous curve. With two groups and two parameters, the fit reproduces those points by construction. Evidence about the curve’s shape between them requires additional dose information and careful model checking.

Check Your Understanding

Work through each question, then compare with the solutions.

In the model $p(d)=\{1+\exp[-(\alpha+\beta\log(d/d_{\mathrm{ref}}))]\}^{-1}$, what condition on $\beta$ makes risk increase with dose?
For the fitted hypothetical model above, is 35 mg within the 25% upper-risk constraint? Give the fitted risk to three significant figures.
Why can a two-parameter logistic curve pass exactly through two observed dose-group proportions without providing a meaningful goodness-of-fit check?
If a protocol permits only 20, 25, 30, 35, and 40 mg and requires fitted risk at most 30%, what is the highest allowable dose under this fitted curve?

Solutions. 1. The condition is $\beta>0$, since increasing dose increases $\log(d/d_{\mathrm{ref}})$, and the logistic function is increasing in its linear predictor. 2. No. The fitted risk at 35 mg is approximately $0.3206$, or 32.1%, which exceeds 25%. 3. Two distinct dose levels provide two fitted proportions, while the curve has two parameters; the model is saturated at those levels, leaving no independent dose-level information to assess the assumed shape. 4. The 30% continuous boundary is approximately 33.72 mg. Of the listed grid doses, 30 mg is the highest dose not exceeding that boundary, so it is the highest grid dose with fitted risk at most 30%.