Balancing Efficacy and Safety Across Candidate Doses

From Two Exposure-Response Models to One Dose Decision

A dose that produces more efficacy may also produce more adverse events. Comparing candidate doses therefore requires more than identifying the dose with the largest predicted treatment effect: the decision must put the efficacy and safety predictions against explicit clinical criteria. In this tutorial, a hypothetical Phase II case uses separate exposure-response models for a clinical benefit and an adverse event. The models are evaluated at exposures predicted for each candidate dose, then interpreted together.

The comparison is only meaningful when both models refer to a clearly defined treatment regimen, population, exposure measure, and time period. For example, the efficacy prediction might be a mean improvement over placebo at week 12, while the safety prediction is the probability of at least one specified adverse event during the same 12-week treatment period. These are distinct outcomes and need not have the same units.

Definition: A model-based dose comparison evaluates each candidate dose by first predicting its relevant exposure and then using separate exposure-response models to predict clinical benefit and harm. A dose satisfies a benefit-harm criterion only if its predictions meet the prespecified clinical requirements for both outcomes. This rule does not imply that the two outcomes are independent or can be collapsed into a single score.

Case Models and Decision Criteria

Suppose a hypothetical 12-week Phase II study is being used to compare daily doses of 25, 50, and 100 mg. A population PK analysis provides the steady-state average concentration, \(x\), expected at each dose. The efficacy endpoint is improvement over placebo on a symptom scale, in points; larger positive values indicate greater benefit. A saturable model is used for the mean improvement:

E(x)=\frac{E_{\max}x}{EC_{50}+x}

Here, \(E_{\max}\) is the maximum model-predicted improvement and \(EC_{50}\) is the exposure at which the predicted improvement is half of \(E_{\max}\). For this illustrative case, the fitted estimates are \(E_{\max}=4.8\) points and \(EC_{50}=1.5\) mg/L.

The safety endpoint is whether a patient experiences at least one specified adverse event during the 12-week period. Its probability is modeled with a logistic exposure-response relationship:

\operatorname{logit}\{p_{\mathrm{AE}}(x)\} = \log\left(\frac{p_{\mathrm{AE}}(x)}{1-p_{\mathrm{AE}}(x)}\right) = \alpha+\beta x

The fitted values are \(\alpha=-2.4\) and \(\beta=0.75\) per mg/L. The coefficient is positive, so the model predicts higher adverse-event probability as exposure increases. For the worked case, the clinical team has specified two criteria: predicted mean improvement must be at least 2.0 points, and predicted adverse-event probability must be no more than 0.25 over 12 weeks. These thresholds are decision inputs, not universal standards.

Assumptions for this comparison: The chosen exposure measure is relevant to both endpoints; the PK predictions represent the regimen and population being considered; the efficacy relationship is adequately described by a saturable curve over the candidate exposure range; and the adverse-event log odds are adequately described as linear in exposure over that range. Predictions also assume the study’s endpoint definitions and follow-up period apply to the intended decision. If patient characteristics alter either response, predictions should be standardized to a common target population rather than compared using different covariate distributions.

Worked Example: Predicting Benefit and Harm at Each Candidate Dose

Assume the PK model predicts steady-state average concentrations of 0.8, 1.6, and 3.2 mg/L at 25, 50, and 100 mg daily, respectively. Calculate both model predictions at each exposure, then apply the stated criteria.

At 25 mg, \(x=0.8\) mg/L. The predicted improvement is:

E(0.8)=\frac{4.8(0.8)}{1.5+0.8} =\frac{3.84}{2.3} =1.670\text{ points}

The adverse-event log odds are \(-2.4+0.75(0.8)=-1.8\). Thus:

p_{\mathrm{AE}}(0.8) = \frac{\exp(-1.8)}{1+\exp(-1.8)} =0.1419

The 25 mg dose meets the harm criterion but falls short of the 2.0-point benefit threshold.

At 50 mg, \(x=1.6\) mg/L:

E(1.6)=\frac{4.8(1.6)}{1.5+1.6} =\frac{7.68}{3.1} =2.477\text{ points}

The adverse-event log odds are \(-2.4+0.75(1.6)=-1.2\), giving:

p_{\mathrm{AE}}(1.6) = \frac{\exp(-1.2)}{1+\exp(-1.2)} =0.2315

The 50 mg dose meets both point-prediction criteria: its predicted improvement is at least 2.0 points, and its predicted adverse-event probability is below 0.25.

At 100 mg, \(x=3.2\) mg/L:

E(3.2)=\frac{4.8(3.2)}{1.5+3.2} =\frac{15.36}{4.7} =3.268\text{ points}

The log odds are \(-2.4+0.75(3.2)=0\), so \(p_{\mathrm{AE}}(3.2)=\exp(0)/(1+\exp(0))=0.5000\). The predicted benefit is greater than at 50 mg, but the predicted adverse-event probability exceeds the maximum allowed by the criterion. On point predictions alone, 50 mg is the only candidate that meets both requirements.

Deriving the Exposure Range That Meets Both Criteria

The candidate-dose calculation can be checked by deriving the exposures at which each criterion begins or ceases to hold. The benefit condition is \(E(x)\geq 2.0\) points. For positive exposure, the efficacy model is increasing, so multiplying by the positive denominator gives:

\frac{4.8x}{1.5+x}\geq 2.0 \quad\Longleftrightarrow\quad 4.8x\geq 3.0+2.0x \quad\Longleftrightarrow\quad x\geq 1.071\text{ mg/L}

For the safety condition, the logistic function is increasing, so \(p_{\mathrm{AE}}(x)\leq 0.25\) is equivalent to requiring its log odds to be no greater than the log odds of 0.25:

-2.4+0.75x \leq \log\left(\frac{0.25}{1-0.25}\right) = -1.099 \quad\Longleftrightarrow\quad x\leq 1.735\text{ mg/L}

Thus, under these fitted models and thresholds, both point-prediction criteria are met for exposures from approximately 1.071 to 1.735 mg/L. Among the three candidate doses, only the 50 mg dose has a predicted exposure in that range. This derivation is useful because it shows how a dose can be disqualified at either end: exposure may be too low to reach the benefit threshold or too high to remain below the harm threshold.

Worked Example: Checking Whether a New Candidate Dose Is in Range

Suppose a 40 mg daily dose is predicted to produce an average concentration of 1.3 mg/L. Determine whether its point predictions meet both criteria.

The exposure is above the benefit boundary of 1.071 mg/L and below the safety boundary of 1.735 mg/L. Direct substitution confirms the result:

E(1.3)=\frac{4.8(1.3)}{1.5+1.3} =\frac{6.24}{2.8} =2.229\text{ points}

The adverse-event log odds are \(-2.4+0.75(1.3)=-1.425\), so:

p_{\mathrm{AE}}(1.3) = \frac{\exp(-1.425)}{1+\exp(-1.425)} =0.1939

The 40 mg dose meets both point-prediction thresholds. The prediction does not establish that 40 mg is a clinically optimal dose: it establishes only that, under the assumed models, exposure prediction, and chosen thresholds, its point predictions pass these two criteria.

Accounting for Model Uncertainty

A point prediction is not the same as a precise prediction. Uncertainty in the PK exposure estimate and in the efficacy and safety model parameters can move a candidate dose across either threshold. A practical approach is to resample study participants, refit the PK and both exposure-response models in each replicate, and calculate the two predictions for every candidate dose. Resampling participants jointly is important when the same participants contribute information to both endpoint models: it can preserve dependence between the estimated efficacy and safety predictions.

For a prespecified uncertainty rule, the team could require that both criteria be met in at least 80% of model-based bootstrap replicates. This is a decision rule about the stability of predictions under the resampling procedure; it is not a probability that a model parameter is true, nor does it remove uncertainty from the decision.

Worked Example: A Point-Prediction Pass That Is Uncertain

Suppose 1,000 joint bootstrap replicates are generated for the 50 mg candidate. In 640 replicates, predicted improvement is at least 2.0 points. In 580, predicted adverse-event probability is at most 0.25. In 420, both conditions are met. The relevant fraction for the joint criterion is:

\widehat{P}_{\mathrm{boot}}(\text{both criteria met}) = \frac{420}{1000} =0.420

Although the fitted point predictions at 50 mg meet both criteria, only 42.0% of these hypothetical replicates meet both together. The 50 mg candidate therefore fails a rule requiring at least 80% of joint replicates to meet both criteria. The marginal counts, 640 and 580, cannot simply be multiplied to obtain the joint fraction: that would require independence of the two replicate-level events, which has not been established. This result calls for examining the sources of uncertainty and the decision’s tolerance for it; it does not by itself identify another dose as acceptable.

Diagnostics, Alternatives, and Limitations

Before using the comparison, check whether the model predictions are credible where the decision depends on them. Plot observed efficacy and adverse-event data against exposure, alongside the fitted relationships and uncertainty intervals. For efficacy, inspect whether the apparent plateau and curvature are supported by observations rather than imposed by extrapolation. For safety, compare observed event rates across exposure ranges with fitted probabilities and look for lack of fit, influential observations, or a pattern inconsistent with a linear exposure effect on log odds. Check the number of patients and events near the exposure boundaries that determine whether the thresholds are met.

Also check the PK link in the decision. If the 100 mg exposure estimate is outside the observed exposure range, the resulting efficacy and safety predictions are extrapolations. If exposure varies substantially among patients at the same dose, a prediction evaluated only at the mean exposure may not equal the average outcome probability across patients. In that case, integrate predictions over the exposure distribution, ideally accounting for relevant patient characteristics. A model based on average exposure should not be presented as an individual patient’s risk.

The threshold approach differs from selecting the dose with the greatest predicted efficacy, which ignores whether its harm prediction is acceptable. It also differs from a safety-only rule, which could select a dose that does not achieve meaningful benefit. Explicit benefit and harm criteria make the trade-off visible without combining endpoints measured in different units into one score. A simple comparison of observed dose-group averages is another alternative, but it may be less informative when exposure varies within dose groups or when candidate doses were not directly studied; model-based predictions, in turn, depend on model form and extrapolation. Neither approach substitutes for checking whether the underlying evidence supports the decision.

The main limitations are that clinical thresholds may be uncertain or population-specific, the two models may be misspecified, and separate endpoint models do not describe all patient-level relationships between efficacy and harm. A dose passing both criteria is not automatically proven safe or effective, and a dose failing a point-prediction threshold is not necessarily unsuitable under every possible clinical objective. The model-based result is conditional on the endpoint definitions, population, regimen, exposure predictions, assumptions, and criteria used.

Common Pitfall

Wrong reasoning: “The 100 mg dose has the largest predicted improvement, so it is the best dose.”
Correction: The 100 mg dose also has a predicted adverse-event probability of 0.5000, above the prespecified maximum of 0.25. Compare both endpoints against their criteria before drawing a dose conclusion.
Wrong reasoning: “The 50 mg point predictions pass, so the dose is established as acceptable.”
Correction: Passing point-prediction thresholds does not show that predictions are sufficiently certain or that the model assumptions are adequate. Evaluate uncertainty and diagnostics, and state the conclusion conditionally.

Key takeaway: For each candidate dose, connect the dose to predicted exposure, calculate efficacy and adverse-event predictions from their respective models, and apply explicit clinical criteria to both. Report uncertainty and the conditions under which the comparison holds.

Check Your Understanding

Use the case models and thresholds above. The solutions follow the questions.

At what exposure does the efficacy model predict exactly 2.0 points of improvement?
What adverse-event probability does the model predict at \(x=2.0\) mg/L, and does this meet the harm criterion?
Why is it not justified to multiply 0.640 by 0.580 to obtain the joint bootstrap fraction in the worked uncertainty example?
If a candidate dose has predicted exposure of 1.0 mg/L, which criterion does it fail based on point predictions?

Solutions. (1) Set \(4.8x/(1.5+x)=2.0\). Then \(4.8x=3.0+2.0x\), so \(x=3.0/2.8=1.071\) mg/L. (2) The log odds are \(-2.4+0.75(2.0)=-0.900\). Therefore, \(p_{\mathrm{AE}}(2.0)=\exp(-0.900)/(1+\exp(-0.900))=0.2891\), which exceeds 0.25 and fails the harm criterion. (3) Multiplication assumes the bootstrap events “benefit criterion met” and “harm criterion met” are independent. The separate marginal fractions do not establish that assumption; the joint fraction must be calculated from paired replicates. (4) At 1.0 mg/L, predicted improvement is \(4.8(1.0)/(1.5+1.0)=1.920\) points, below 2.0. Its adverse-event probability is below 0.25, so it fails the benefit criterion only.