Why a Strength Bridge Needs More Than Similar Exposure Means
When a drug is studied at one strength, a development program may need evidence to support additional strengths. The central question is whether exposure changes with dose in a predictable way over the studied range. If geometric-mean exposure is dose proportional, doubling dose is expected to double geometric-mean exposure. But apparent proportionality in a few geometric means is not enough: uncertainty, the study design, and the criteria specified before analysis all matter.
A strength bridge is not automatically a bioequivalence comparison. A proportionality analysis asks how exposure changes with dose; a bioequivalence analysis asks whether the exposure from specified test and reference treatments meets prespecified equivalence limits. A bridge may use both ideas: a model to assess proportionality and dose-normalized contrasts to compare particular strengths. The criteria and the endpoints needed for a bridge must be specified for the intended claim.
Model and Interpretation
For positive exposure measures such as AUC or Cmax, a common analysis models log exposure against log dose. In a crossover study, a model may include a subject effect and a separate period effect. Let \(Y_{ij}\) be the exposure for subject \(i\) after strength \(j\), let \(D_j\) be that strength’s dose, and let \(p(i,j)\) identify the period in which subject \(i\) received strength \(j\). One possible model is:
Here \(D_0\) is a chosen reference dose, \(\pi_{p(i,j)}\) is the effect for the actual period of that observation, and \(b_i\) is a subject effect. Thus, \(\beta\) describes the slope of expected log exposure and, for the geometric-mean target defined above, the slope of geometric-mean exposure. It does not generally determine the slope of arithmetic mean exposure when log-scale variance varies with dose. Treatment strength and period are distinct: their assignments can differ across subjects. The precise model depends on the design. In a parallel study, a within-subject term is not appropriate; the analysis must reflect the independent groups and the planned covariates.
The model assumes that the log-exposure relationship is adequately described by a straight line over the dose range, the log-scale residual distribution is adequately modeled, and the design and covariates account for relevant systematic effects. A crossover analysis also requires that the design and conduct support valid within-subject comparisons, including adequate washout and no material carryover affecting the results. If variability differs across doses, the variance model may need to allow for that rather than assume constant residual variance.
Under the power model, dividing geometric-mean exposure by dose gives \(G(D)/D=kD^{\beta-1}\). Thus, when \(\beta=1\), dose-normalized geometric-mean exposure does not change with dose. When \(\beta>1\), it tends to increase with dose; when \(\beta<1\), it tends to decrease. A slope estimate near 1 is descriptive, however. Evidence for a bridge should be judged against prespecified inferential criteria, not by whether the estimate merely looks close to 1.
From the Model to a Bridge Decision
A practical analysis begins by defining the intended claim. Specify the dose range, exposure endpoints, reference strength, comparison strengths, model, and decision criteria before examining results. For example, a protocol might require the confidence interval for \(\beta\) to lie within a prespecified interval and require each dose-normalized AUC contrast to lie within prespecified bounds. Those are example criteria, not general regulatory limits.
A model-based contrast compares geometric-mean exposures while accounting for the study design. For a strength \(D_j\) and reference strength \(D_0\), the dose-normalized geometric-mean exposure ratio is:
On the log scale, the contrast is \(\ln R_j\). Its confidence interval should be calculated using the fitted model and the covariance of its estimates, then transformed to the ratio scale. A confidence interval wholly within prespecified bounds supports that particular contrast under the specified analysis. An interval that crosses a bound does not establish equivalence under that criterion; it may indicate insufficient precision, a difference outside the criterion, or both.
If a bridge requires several conditions to hold together—for example, a slope criterion plus AUC and Cmax contrast criteria—the overall claim is supported only if every required condition passes. The endpoints and the logic for combining them must be prespecified. Selecting only the endpoint or strength that passes after seeing the results weakens the interpretation and can inflate error rates for selective claims.
Worked Example: Calculating Dose-Normalized Exposure Ratios
Suppose a hypothetical study reports geometric mean AUC values of 500, 1,010, and 1,980 ng·h/mL at doses of 50, 100, and 200 mg. Calculate the dose-normalized ratios for 50 mg and 200 mg relative to 100 mg.
At 50 mg, dose-normalized AUC is \(500/50=10.00\) ng·h/mL per mg. At 100 mg, it is \(1{,}010/100=10.10\) ng·h/mL per mg. Therefore:
At 200 mg, dose-normalized AUC is \(1{,}980/200=9.900\) ng·h/mL per mg, so:
These point estimates are close to 1, so the observed geometric means are descriptively consistent with proportional exposure. They do not, by themselves, show that either ratio meets an inferential bridge criterion: no uncertainty interval has been considered. Nor do these AUC results establish a conclusion for Cmax.
Worked Model-Based Decisions
Worked Example: Assessing Slope and Prespecified Contrasts
Consider a hypothetical three-strength study at 50, 100, and 200 mg. Suppose the protocol specifies that the 90% confidence interval for the log-dose slope must lie within 0.90 to 1.10, and that each dose-normalized AUC ratio relative to 100 mg must have its 90% confidence interval within 0.80 to 1.25. These criteria are illustrative and apply here only because they were stipulated for this example. The study model gives \(\hat{\beta}=0.993\), with a 90% confidence interval of (0.944, 1.042). The model-based dose-normalized AUC ratios and intervals are 0.990 (0.940, 1.043) for 50 mg and 0.980 (0.931, 1.032) for 200 mg.
Slope decision. The entire slope interval, 0.944 to 1.042, is within the prespecified 0.90 to 1.10 interval. The result supports the prespecified slope criterion over the studied dose range.
Contrast decisions. The 50 mg interval lies wholly within 0.80 to 1.25, as does the 200 mg interval. Both dose-normalized AUC contrasts meet their stipulated criteria. If those were all the required conditions for the AUC bridge, the evidence supports the 50 mg and 200 mg strengths relative to 100 mg for AUC under this analysis.
This conclusion is limited to the endpoint and criteria evaluated. If the protocol also requires Cmax, its results must be assessed before claiming the overall bridge. The intervals quantify uncertainty in the model-based estimates; they are not probabilities that the fixed ratios or slope lie inside the intervals.
Worked Example: A Strength That Does Not Meet the Bridge Criterion
Now consider a hypothetical study that includes 50, 100, 200, and 400 mg. Suppose the prespecified dose-normalized AUC criterion relative to 100 mg is that the 90% confidence interval must lie within 0.80 to 1.25. At 400 mg, the geometric mean AUC is 3,100 ng·h/mL, compared with 1,010 ng·h/mL at 100 mg. The point estimate of the dose-normalized ratio is:
Suppose the model-based 90% confidence interval for this ratio is (0.680, 0.880). Its upper limit is below 1, and the interval is not wholly within 0.80 to 1.25: the point estimate is below the lower bound, and the interval includes values below that bound. The 400 mg contrast therefore fails the specified criterion. It would be inaccurate to say that the 400 mg strength is supported by this criterion.
The conclusion is not that the true ratio has a stated probability of being below 0.80. Rather, this interval-based procedure did not meet its prespecified support rule, and the estimate indicates lower dose-normalized exposure at 400 mg in this study. Whether that result reflects genuine nonproportionality or remaining uncertainty requires interpretation of the full analysis and data. It does not invalidate evidence for other strengths that independently meet their required criteria.
Justification, Diagnostics, and Alternatives
The power-model interpretation follows directly from taking logs. If geometric-mean exposure follows \(G(D)=kD^\beta\), then:
For doses \(D_j\) and \(D_0\), subtracting the model-predicted mean log exposures with period and subject effects held fixed gives \(E[\ln Y\mid D_j]-E[\ln Y\mid D_0]=\beta\ln(D_j/D_0)\). Subtracting the log dose ratio gives:
Exponentiating both sides yields the dose-normalized ratio formula. This derivation also shows why a proportionality slope is not interchangeable with a set of pairwise bioequivalence comparisons: the slope summarizes a trend across doses, while each contrast concerns a particular comparison.
Diagnostics should look for patterns inconsistent with the model or design assumptions. Plot log exposure against log dose and inspect whether a straight-line trend is plausible; examine residuals for dose-related spread or systematic curvature; and review influential subjects and data-integrity concerns. In a crossover, assess whether period patterns or incomplete observations could affect the fitted contrasts. These checks do not replace the prespecified model, but they can reveal that its assumptions or the study’s conduct need further investigation.
A simple alternative is to calculate dose-normalized geometric means separately for each strength and compare each with the reference. This is transparent and directly addresses prespecified pairwise contrasts, but does not summarize the dose trend as efficiently and may be imprecise when several strengths are considered. Conversely, a single straight-line slope can conceal curvature: a slope near 1 across the full range does not guarantee that every local or pairwise contrast is acceptable. If the data show meaningful curvature, a linear log-dose model may be inadequate; interpretation should not be rescued by relying on an overall slope alone.
The analysis also cannot separate dose effects from strength-specific formulation effects when those factors change together. A conclusion is therefore about the tested strengths and product configurations, not a universal biological law about dose. Limited numbers of strengths, narrow dose ranges, small samples, missing observations, or high variability can also make proportionality difficult to assess. A confidence interval that fails a support rule is not automatically proof of nonproportionality; it can be too wide to resolve the question.
Common Mistakes
- Calling similar point estimates proof of proportionality. Point ratios do not show whether uncertainty is small enough to meet the prespecified criterion. Use the appropriate model-based confidence intervals.
- Equating proportionality with bioequivalence. A slope near 1 describes how exposure scales with dose; it does not by itself show that a test and reference formulation meet bioequivalence criteria.
- Using the same index for treatment and period. Strength and period are different factors in a crossover. Index the period separately, as \(p(i,j)\), to identify when subject \(i\) received strength \(j\).
- Generalizing beyond the study. Evidence over studied doses and endpoints does not establish proportionality at untested doses or for unmeasured exposure endpoints.
- Treating a failed criterion as proof of the opposite. A contrast that does not meet its support rule may be imprecise or may indicate a meaningful departure. Interpret the estimate, interval, assumptions, and prespecified rule together.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A power model gives \(\beta=1.08\). What does that imply about dose-normalized exposure as dose increases, and does it alone establish a bridge?
- At dose \(D_j=150\) mg, the geometric mean AUC is 1,260 ng·h/mL. At the 100 mg reference dose, it is 840 ng·h/mL. Calculate the dose-normalized ratio for 150 mg relative to 100 mg.
- A prespecified slope criterion is 0.90 to 1.10. The estimated slope is 1.02 and its 90% confidence interval is (0.88, 1.15). Does the slope criterion pass?
- Why must period and strength be represented as distinct factors in a crossover model?
Solutions. (1) Since \(\beta-1=0.08>0\), the model implies that dose-normalized exposure tends to increase with dose. The estimate alone does not establish a bridge; uncertainty and all required contrasts and endpoints must be assessed against prespecified criteria. (2) \(R_j=(1{,}260/840)(100/150)=1.5\times0.6667=1.000\). The point estimate is 1.000, but its uncertainty is still needed for an inferential conclusion. (3) No. Although the estimate is within the criterion, the confidence interval extends below 0.90 and above 1.10, so it is not wholly contained in the required interval. (4) Subjects can receive strengths in different periods. Strength identifies the treatment, while period identifies when it was administered; conflating the indices can misrepresent the design and the fitted effects.