Worked Analysis Of A Two-Formulation Crossover

From Crossover Concentrations to Formulation Ratios

A crossover study gives each participant both formulations, so the comparison can use within-participant information while modeling the entire concentration-time profile. A nonlinear mixed-effects analysis connects those measurements to pharmacokinetic parameters and then to endpoints such as area under the concentration-time curve (AUC) and maximum concentration (Cmax). The key is to make the estimand explicit: a ratio derived from a fitted concentration model is conditional on that model, its assumptions, and the parameters being compared.

The worked analysis below uses a hypothetical, equal-dose, two-period crossover. It illustrates one-compartment disposition with first-order absorption. It is not a real study, and its model-based confidence intervals are not a substitute for checking whether the design, model, and inferential procedure are appropriate for a particular application.

Model and Assumptions

Let \(C_{ij}(t)\) be the concentration for participant \(i\) in period \(j\), measured \(t\) hours after dosing. Let \(T_{ij}=1\) for the test formulation and \(T_{ij}=0\) for the reference. For an equal dose \(D\), use the one-compartment oral model

C_{ij}(t)= \exp(a_i+\pi_j+\theta_F T_{ij}) \frac{k_i^{\,0} k_{a,ij}}{k_{a,ij}-k_i} \left(e^{-k_i t}-e^{-k_{a,ij}t}\right), \qquad k_{a,ij}=k_{a,R}\exp(\theta_a T_{ij}).

Here \(k_i\) is participant \(i\)'s elimination-rate constant, \(k_{a,R}\) is the reference absorption-rate constant, and \(\theta_a\) is the test-versus-reference log absorption-rate contrast. The scale \(a_i\) includes the participant-specific dose-to-volume and reference bioavailability contribution; \(\theta_F\) is the test-versus-reference log relative-bioavailability effect. The fixed period term \(\pi_j\) allows a common period shift, with one period set as the reference. In practice, the random-effects structure can include correlated participant-level deviations in scale, elimination, and absorption parameters. The displayed form emphasizes the treatment contrasts rather than prescribing a single random-effects structure.

A typical fit assumes that the chosen concentration model specifies the structural (typical) concentration-time curve; doses and sampling times are correctly recorded; observations are conditionally independent given the random effects and model, or have a specified residual covariance; and the residual distribution is appropriate. For example, a proportional lognormal residual model represents positive observations by \(\log C_{ij}(t)=\log \mu_{ij}(t)+\epsilon_{ij}(t)\), with \(\epsilon_{ij}(t)\) centered at zero and a modeled variance. Random effects require a specified distribution and covariance structure. The crossover must also be analyzed consistently with its randomization and period structure. The random-effects and residual-model diagnostics considered earlier are part of the analysis, not optional refinements.

Definition: A model-based formulation ratio is the ratio of a specified endpoint calculated from the fitted concentration model under test and reference conditions. An AUC or Cmax ratio is not defined until the dose, time horizon, endpoint calculation, and treatment contrast have been specified.

For the illustrative comparison, use equal doses, complete absorption over the modeled time course, linear one-compartment disposition, and no carryover. The principal target is the ratio for a typical participant at the fitted population parameters, with period held at the same value for both formulations. This is not automatically the ratio of arithmetic mean endpoints across a population: with nonlinear endpoints and random effects, those quantities can differ. A marginal population estimand would require integrating the endpoint over the random-effects distribution.

Worked Analysis

Worked Example: Translating a Hypothetical Fit Into Model Parameters

Suppose 24 participants are randomized equally to the two treatment sequences and receive both formulations in separate periods. Concentrations are collected at prespecified times from 0.25 to 24 hours after the same dose. A converged nonlinear mixed-effects fit gives \(k_{a,R}=1.00\ \mathrm{h}^{-1}\), typical \(k=0.150\ \mathrm{h}^{-1}\), \(\theta_F=0.03922\), and \(\theta_a=0.2231\). Assume the fitted residual model is proportional lognormal with residual standard deviation 0.12 on the log-concentration scale. These are hypothetical fitted estimates, not raw observations.

The test-to-reference relative bioavailability estimate is obtained by exponentiating the log contrast:

\widehat{F_T/F_R}=\exp(\widehat{\theta}_F) =\exp(0.03922)=1.040.

The estimated absorption-rate ratio is \(\exp(0.2231)=1.250\). The fitted typical test absorption rate is therefore \(1.00\times1.250=1.250\ \mathrm{h}^{-1}\), compared with \(1.00\ \mathrm{h}^{-1}\) for reference. These estimates alone do not establish equivalence: uncertainty in the endpoint ratios must also be evaluated.

AUC Ratio and Its Uncertainty

For the model above, integrating the concentration curve from zero to infinity gives the AUC; dividing by dose gives the dose-normalized AUC. The two exponential terms integrate to \(1/k_i\) and \(1/k_{a,ij}\), respectively, so

\begin{aligned} AUC_{ij}(0,\infty) &=\exp(a_i+\pi_j+\theta_F T_{ij}) \frac{k_i^{\,0} k_{a,ij}}{k_{a,ij}-k_i} \left(\frac{1}{k_i}-\frac{1}{k_{a,ij}}\right)\\ &=\exp(a_i+\pi_j+\theta_F T_{ij})\frac{k_i^{\,0}}{k_i} =\frac{\exp(a_i+\pi_j+\theta_F T_{ij})}{k_i}. \end{aligned}

The cancellation follows because \(1/k_i-1/k_a=(k_a-k_i)/(k_i k_a)\). Thus, under this model and equal doses, with period held at the same value for both formulations, the same participant's test-to-reference AUC ratio is \(\exp(\theta_F)\): the absorption-rate contrast does not affect total AUC when absorption is complete and disposition is linear and unchanged by formulation. For observations from different periods \(j_T\) and \(j_R\), the ratio is \(\exp(\theta_F+\pi_{j_T}-\pi_{j_R})\). This is a model result, not a general claim that absorption rate never matters for exposure measured over a finite interval.

Worked Example: AUC Ratio and 90% Confidence Interval

Suppose the fitted standard error of \(\widehat{\theta}_F\) is 0.0350, and its large-sample sampling distribution is adequately approximated by a normal distribution. For a two-sided 90% confidence interval, use 1.645 standard errors on the log-ratio scale.

\begin{aligned} \widehat{AUC_T/AUC_R}&=\exp(0.03922)=1.040,\\ CI_{\log}&=0.03922\pm1.645(0.0350) =0.03922\pm0.05758\\ &=(-0.01836,\ 0.09680),\\ CI_{\mathrm{ratio}}&=(\exp(-0.01836),\exp(0.09680))\\ &=(0.9818,\ 1.102). \end{aligned}

The estimated typical-subject AUC is 4.00% higher for test than reference. The 90% confidence interval for the model-based ratio is 0.9818 to 1.102. In repeated applications of the stated procedure under its assumptions, 90% of intervals constructed this way would cover the fixed target parameter; this is not a 90% probability that this particular interval contains the parameter.

For this example, the confidence interval uses the estimated covariance matrix from the nonlinear mixed-effects fit and a normal approximation. If the sample is small, the likelihood is weakly identified, or the estimate is near a boundary, that approximation may be unreliable. A bootstrap or another inferential approach may be considered, with the resampling scheme reflecting the crossover design and the uncertainty in model estimation.

Cmax From the Fitted Curve

For positive \(k\) and \(k_a\) with \(k_a\ne k\), the concentration curve reaches its interior maximum where its time derivative is zero. Differentiating the time-dependent part gives \(-k e^{-kt}+k_a e^{-k_a t}=0\). Therefore,

t_{\max}=\frac{\log(k_a/k)}{k_a-k}, \qquad C_{\max}= \exp(a_i+\pi_j+\theta_F T) \frac{k_a}{k_a-k} \left(e^{-k t_{\max}}-e^{-k_a t_{\max}}\right).

When \(k_a=k_i\), the continuous-limit concentration curve is \(C_{ij}(t)=\exp(a_i+\pi_j+\theta_F T_{ij})\,k_i t e^{-k_i t}\), and its maximum occurs at \(t_{\max}=1/k_i\).

The fitted Cmax ratio is consequently not just the bioavailability ratio: changing \(k_a\) changes the peak factor as well as the time of the peak. Parameter uncertainty must be propagated through both effects. One convenient approach is the delta method. For a smooth function \(g(\boldsymbol{\beta})\) of fitted parameters with covariance matrix \(\widehat{\Sigma}\), the estimated variance is approximately \(\nabla g^\mathsf{T}\widehat{\Sigma}\nabla g\). Here it is applied to the log Cmax ratio, then the resulting interval is exponentiated.

Justification: At \(t_{\max}\), the derivative of log concentration with respect to time is zero. Consequently, when differentiating the log peak with respect to a model parameter, the term involving the derivative of \(t_{\max}\) vanishes by the chain rule. The derivative of the peak height can therefore be calculated by differentiating log concentration with respect to that parameter while holding \(t=t_{\max}\).

Worked Example: Cmax Ratio With Delta-Method Uncertainty

Use the fitted values from the first example: \(k=0.150\ \mathrm{h}^{-1}\), reference \(k_a=1.00\ \mathrm{h}^{-1}\), test \(k_a=1.250\ \mathrm{h}^{-1}\), and \(\theta_F=0.03922\). The reference and test peak times are

t_{\max,R}=\frac{\log(1.00/0.150)}{1.00-0.150} =2.232\ \mathrm{h}, \qquad t_{\max,T}=\frac{\log(1.250/0.150)}{1.250-0.150} =1.928\ \mathrm{h}.

Substitution in the peak-height formula gives reference peak factor 0.7156 and test peak factor 0.7489, in the same concentration scale before applying the common participant scale. Thus,

\widehat{C_{\max,T}/C_{\max,R}} =\exp(0.03922)\frac{0.7489}{0.7156} =1.088.

The log Cmax ratio is \(0.03922+\log(0.7489/0.7156)=0.0847\). Suppose the fitted covariance estimates are \(SE(\widehat{\theta}_F)=0.0350\), \(SE(\widehat{\theta}_a)=0.120\), and \(\mathrm{Cov}(\widehat{\theta}_F,\widehat{\theta}_a)=-0.000500\). For this calculation, treat \(k\) and \(k_{a,R}\) as fixed; the interval is conditional on those values and includes uncertainty only in \(\theta_F\) and \(\theta_a\). Differentiating the log peak factor with respect to \(\log k_a\) at the test parameters gives approximately 0.192. The AUC scale contrast has derivative 1 with respect to \(\theta_F\) and zero with respect to \(\theta_a\), so the conditional delta-method variance for the log Cmax ratio is

\begin{aligned} \widehat{\mathrm{Var}}(\log\widehat{C_{\max,T}/C_{\max,R}}) &\approx 0.0350^2+(0.192)^2(0.120)^2 +2(1)(0.192)(-0.000500)\\ &=0.0015638416,\\ SE&=\sqrt{0.0015638416}=0.03955. \end{aligned}

The 90% interval is \(0.0847\pm1.645(0.039545)\) on the log scale, or approximately \((0.01965,0.14975)\). Exponentiating gives a Cmax ratio interval of approximately \((1.020,1.162)\). The estimated typical-subject Cmax is 8.8% higher for test; the interval quantifies uncertainty in that model-derived ratio under the fit and approximation.

Equivalence Assessment, Diagnostics, and Limitations

Under conventional average bioequivalence limits of 0.80 to 1.25, the assessment asks whether the entire 90% confidence interval for each specified endpoint ratio lies within those bounds. In the hypothetical analysis, the AUC interval is \((0.9818,1.102)\), and the Cmax interval is \((1.020,1.162)\). Both lie within 0.80 to 1.25, so both modeled endpoints meet this interval criterion in this example. This conclusion is conditional on the model, estimand, study design, and uncertainty calculation. A model-based decision requires a justified analysis plan and suitable validation; these calculations alone do not establish regulatory acceptability.

Several checks are essential before interpreting the intervals. Confirm convergence, stable parameter estimates across reasonable starting values, and an estimable covariance matrix. Examine observed and model-predicted concentration-time profiles by formulation and period, residual patterns over time and predicted concentration, and participant-level fits. Check whether the random-effects distribution and covariance structure are credible, whether residual variability changes with concentration or time, and whether influential participants drive the treatment contrast. Assess whether sampling times identify the absorption phase and terminal decline; sparse or poorly placed samples can leave \(k_a\), \(k\), or the peak poorly determined even when the optimizer reports convergence.

The simple model also has substantive limits. It assumes first-order absorption, a single disposition compartment, and formulation differences represented by relative scale and absorption rate. It may not represent lag time, multiple absorption phases, nonlinear elimination, or time-dependent input. AUC to infinity relies on the model's complete absorption and elimination assumptions; a finite-interval estimand requires its own calculation. Random-effects heterogeneity means the typical-participant ratio need not equal the ratio of population arithmetic means. Model uncertainty and sensitivity to absorption assumptions should be considered rather than hidden by a precise-looking interval.

A conventional analysis of individual log-transformed AUC and Cmax summaries is a familiar alternative when those endpoints are reliably observed and the design and assumptions support it. It focuses directly on endpoint contrasts, while the nonlinear model uses the full concentration-time profile and can provide model-defined endpoints, including peak estimates between sample times. The model-based route is more dependent on structural and residual assumptions; the summary-endpoint route can be sensitive to endpoint construction and sampling. Neither approach is automatically superior. They answer related but not identical questions, and disagreements should prompt investigation of the estimands, data, and model assumptions.

Common pitfall: It is incorrect to call a point estimate inside 0.80 to 1.25 evidence of equivalence, or to conclude equivalence because the interval includes 1. The required check is that the full prespecified confidence interval is inside both limits. Here both hypothetical intervals satisfy that check; a wider interval extending beyond either limit would not.

Check Your Understanding

Use the model and hypothetical estimates in this tutorial. Solutions follow each question.

If \(\theta_F=0\), what is the model-based AUC ratio under the stated equal-dose, complete-absorption assumptions? Solution: \(\exp(0)=1\), so the modeled AUCs are equal.
Why can the Cmax ratio differ from the AUC ratio? Solution: Cmax depends on both relative scale and the peak factor, which changes when the absorption rate changes; total AUC in this model depends on relative scale alone.
Using \(SE(\widehat{\theta}_F)=0.0350\), would an AUC ratio estimate of 1.040 alone establish equivalence? Solution: No. Equivalence requires the entire prespecified confidence interval to lie within the limits, not merely an acceptable point estimate.
What does a 90% confidence interval for the Cmax ratio describe? Solution: It gives a range produced by a procedure with 90% repeated-sampling coverage for the fixed model-based ratio, under the model and inferential assumptions; it is not a 90% probability statement about the parameter given this interval.
Name one reason the Cmax result could be unreliable despite optimizer convergence. Solution: Poorly timed or sparse samples may not identify the absorption phase, making the estimated absorption rate and model-derived peak uncertain or biased.