Adjusting Doses When Exposure Is Nonproportional

Why an AUC Ratio May Not Give the Right Dose

A dose change based on a DDI exposure ratio is straightforward only when exposure is proportional to dose over the relevant range. Under that condition, if a perpetrator doubles AUC at the prescribed dose, halving the dose under the interacting condition restores the original AUC. With nonlinear pharmacokinetics, however, the dose–AUC relationship can bend: halving the dose may reduce AUC by more or less than half. The ratio measured at one dose then does not, by itself, specify the replacement dose.

The practical alternative is to define the AUC to be matched and use the simulated dose–exposure relationship under the interacting condition to find a dose that produces it. The task is an inversion of a model-predicted curve, not an automatic conversion of one ratio into one dose.

Definition: Let $AUC_{\mathrm{ref}}(D)$ denote the AUC under the reference condition at dose $D$, and let $AUC_{\mathrm{DDI}}(D)$ denote the AUC under the interacting condition at dose $D$. For a reference dose $D_0$, an AUC-matching replacement dose $D^*$ under the interacting condition satisfies $$ AUC_{\mathrm{DDI}}(D^*)=AUC_{\mathrm{ref}}(D_0). $$ The dose $D^*$ is a model-based dose estimate for matching that specified AUC; it is not automatically a recommendation to prescribe that dose.

The target is commonly the reference-condition AUC at the current dose. It must be stated explicitly: a different target, such as an exposure within a predefined range, leads to a different dose-selection problem. Use the same AUC definition, dosing interval, route, and relevant population assumptions when comparing the reference and interacting predictions. Matching AUC alone does not ensure that peak or trough concentrations, safety, efficacy, or clinical outcomes are also matched.

When Inverse Scaling Is Valid

At the reference dose $D_0$, define the AUC ratio as $R=AUC_{\mathrm{DDI}}(D_0)/AUC_{\mathrm{ref}}(D_0)$. Inverse scaling proposes $D_{\mathrm{inv}}=D_0/R$. This formula follows only under a proportionality assumption for the interacting-condition curve over the relevant dose range: $AUC_{\mathrm{DDI}}(D)=kD$, with a constant $k$. Under that assumption,

AUC_{\mathrm{DDI}}(D_0/R) = kD_0/R = AUC_{\mathrm{DDI}}(D_0)/R = AUC_{\mathrm{ref}}(D_0).

The last equality follows from the definition of $R$. If the curve is not proportional, the first equality need not hold: changing the dose changes the local dose-to-AUC relationship. A ratio evaluated at $D_0$ summarizes the comparison at that dose; it does not describe the entire curve.

Worked Example: A Supraproportional DDI Curve

Suppose a simulated reference-condition AUC at $100$ mg is $30$ mg·h/L. Under the interacting condition, simulated AUC values are $9$, $24$, $39$, and $60$ mg·h/L at doses of $25$, $50$, $75$, and $100$ mg, respectively. These are hypothetical model outputs. The target is $30$ mg·h/L.

At $100$ mg, the ratio is $R=60/30=2.00$, so inverse scaling proposes $100/2=50$ mg. But the interacting-condition curve predicts only $24$ mg·h/L at $50$ mg, below the target. The ratio-based dose therefore does not restore the target AUC.

The target lies between the simulated values at $50$ and $75$ mg. Using linear interpolation between those two points, $$ D^*=50+\frac{30-24}{39-24}(75-50) =50+\frac{6}{15}(25) =60\text{ mg}. $$ At $60$ mg, the interpolated AUC is $24+(60-50)(15/25)=30$ mg·h/L. In this example the replacement dose is $60$ mg, not the $50$ mg obtained by inverse scaling.

The curve’s changing increments show why the ratio at $100$ mg is insufficient: the AUC change per milligram is not constant across the simulated range. Linear interpolation is an approximation between simulated points, so a final model run at $60$ mg should confirm the estimate.

Finding a Replacement Dose from Simulated Values

For a monotone increasing dose–AUC curve, a target between two simulated AUC values can be estimated by local interpolation. If the target $AUC_{\mathrm{target}}$ lies between predictions at $D_1$ and $D_2$, linear interpolation gives

D^*=D_1+ \frac{AUC_{\mathrm{target}}-AUC_{\mathrm{DDI}}(D_1)} {AUC_{\mathrm{DDI}}(D_2)-AUC_{\mathrm{DDI}}(D_1)} (D_2-D_1).

This estimate assumes a locally linear segment between the two simulated points. It is not a claim that the full pharmacokinetic relationship is linear. If the model can be evaluated repeatedly, a numerical root-finding method can instead seek a dose for which $AUC_{\mathrm{DDI}}(D)-AUC_{\mathrm{target}}=0$. In either case, the resulting dose should be run through the model and its AUC checked directly.

Set the target.
Specify the reference dose and the exposure quantity to match, such as reference-condition AUC at that dose.

Obtain the interacting-condition curve.
Simulate doses that cover the plausible replacement-dose range using the relevant regimen and model assumptions.

Locate a bracketing interval.
Find adjacent doses whose predicted AUC values lie on opposite sides of the target. Interpolate locally or solve numerically within the interval.

Verify the candidate.
Simulate the estimated dose directly, inspect other relevant exposure measures, and check whether dose rounding changes the predicted exposure materially.

Worked Example: A Less Dramatic Ratio Still Gives a Different Dose

In a second hypothetical case, the reference-condition AUC is $80$ mg·h/L at $200$ mg. The interacting-condition simulations predict AUCs of $75$, $100$, and $120$ mg·h/L at $100$, $150$, and $200$ mg, respectively. The target is $80$ mg·h/L.

At $200$ mg, $R=120/80=1.50$. Inverse scaling gives $200/1.50=133.3$ mg. Yet the target lies between $75$ mg·h/L at $100$ mg and $100$ mg·h/L at $150$ mg. Interpolating gives $$ D^*=100+\frac{80-75}{100-75}(150-100) =100+\frac{5}{25}(50) =110\text{ mg}. $$

The interpolated AUC at $110$ mg is $75+(110-100)(25/50)=80$ mg·h/L. By contrast, the dose from inverse scaling, $133.3$ mg, lies on the $100$-to-$150$ mg segment and gives an interpolated AUC of $75+(133.3-100)(25/50)$, or approximately $91.7$ mg·h/L. Thus even a ratio of $1.50$ does not guarantee that inverse scaling matches the intended AUC when the curve is nonproportional.

Interpolation, Verification, and Dose Resolution

Simulated curves are often available only at selected doses. Local interpolation provides a transparent estimate when the target is bracketed, the curve behaves smoothly in that interval, and the dose spacing is sufficiently fine for the intended decision. It becomes less dependable across a wide interval or where the curve bends sharply. More simulations near the candidate dose can reveal that curvature; a direct model run at the proposed dose is the simplest verification.

Worked Example: Rounding a Model-Based Estimate

Consider a hypothetical target AUC of $32$ mg·h/L. The interacting-condition simulations predict $24$ mg·h/L at $40$ mg and $42$ mg·h/L at $60$ mg. Linear interpolation estimates $$ D^*=40+\frac{32-24}{42-24}(60-40) =40+\frac{8}{18}(20) \approx48.89\text{ mg}. $$

Suppose the available dose increments are $10$ mg. Rounding to $50$ mg is a practical candidate, but the interpolated $48.89$ mg is not itself a verification. Assume a direct model simulation at $50$ mg predicts $33$ mg·h/L, and that the hypothetical analysis had specified an acceptable matching band of $30$ to $34$ mg·h/L in advance. The $50$ mg candidate falls within that band. If the direct prediction had fallen outside the band, the rounded dose would require reconsideration rather than being justified by the unrounded interpolation.

This example illustrates why the permitted target range and the available dose increments should be considered explicitly. Rounding is a practical constraint, not a property of the pharmacokinetic model.

Assumptions, Diagnostics, and Alternatives

Interpreting an inverted curve requires checking the conditions behind the calculation. The reference and interacting simulations should represent comparable conditions apart from the interaction and dose change of interest. The target must be defined using a consistent AUC metric and regimen. The model should be applicable to the population and dose range being considered, and the candidate dose should be within the range supported by simulations or other evidence.

A monotone curve makes a single dose estimate easier to interpret: increasing dose consistently increases AUC, so a target within the simulated AUC range has a corresponding dose. If the curve is nonmonotone, a target may have multiple dose solutions or none. If the target lies above or below the simulated AUC range, interpolation cannot identify it; extrapolating a dose from sparse points is not a substitute for additional justified simulations. Irregular changes in a simulated curve should prompt checks for model behavior, numerical error, and the spacing or setup of the simulations.

Inverse AUC-ratio scaling is a useful shortcut when proportionality is justified across the dose change. Direct model-based inversion is preferable when the curve is visibly nonlinear or proportionality is uncertain. Interpolation is convenient for a bracketed target and a smooth local curve; denser simulation or numerical root finding is preferable when curvature is substantial or precision matters. These approaches estimate an AUC-matching dose; none, by itself, establishes that matching AUC is sufficient for a clinical decision.

Wrong reasoning: “The AUC ratio is 2, so halving the dose must halve AUC.” Correction: that conclusion requires proportional dose–AUC behavior over the relevant range. Check the interacting-condition curve or justify proportionality.
Wrong reasoning: “The interpolated dose matches AUC, so all exposure measures match.” Correction: AUC matching does not imply matching $C_{\max}$, trough concentrations, or the concentration-time profile. Check other measures relevant to the question.
Wrong reasoning: “A target outside the simulated range can be handled by the same interpolation formula.” Correction: interpolation requires a bracketing interval. Extend the simulations where justified, and avoid treating unsupported extrapolation as a reliable dose estimate.

Key takeaway: When exposure is nonproportional, choose a replacement dose by solving for the dose under the interacting condition that reaches a clearly specified AUC target. Treat inverse scaling as valid only when proportionality is justified, and verify any interpolated or rounded candidate with the model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A reference AUC at $80$ mg is $24$ mg·h/L. Under the interacting condition, AUC is $48$ mg·h/L at $80$ mg, $20$ mg·h/L at $40$ mg, and $32$ mg·h/L at $60$ mg. What dose matches the reference AUC by linear interpolation between $40$ and $60$ mg?
Why does an AUC ratio of $1.8$ at a single dose not generally prove that dividing the dose by $1.8$ will match the reference AUC?
A target AUC is $50$ mg·h/L, but simulated values range only from $10$ to $42$ mg·h/L. What does this imply about interpolation?
State two checks to make after estimating a replacement dose from a dose–AUC curve.

Solutions. (1) The target is $24$ mg·h/L. Interpolation gives $40+(24-20)/(32-20)(60-40)=40+(4/12)(20)\approx46.67$ mg. A direct simulation at approximately $46.67$ mg should verify the estimate. (2) The ratio describes exposure at one dose. Dividing by it matches the target only if interacting-condition AUC is proportional to dose over the changed range. (3) The target is not bracketed by the simulated AUC values, so interpolation cannot estimate a matching dose; additional justified simulations are needed rather than unsupported extrapolation. (4) Examples include directly simulating the candidate dose to verify its AUC, checking that the target is bracketed and the curve is suitable for inversion, assessing the effect of practical dose rounding, and checking other relevant exposure measures.