Bioequivalence Tests Based on Individual Estimates Using Non-compartmental or Model-Based Analyses: Evaluation of Estimates of Sample Means and Type I Error for Different Designs

Abstract

Purpose

The main objective of this work is to compare the standard bioequivalence tests based on individual estimates of the area under the curve and the maximal concentration obtained by non-compartmental analysis (NCA) to those based on individual empirical Bayes estimates (EBE) obtained by nonlinear mixed effects models.

Methods

We evaluate by simulation the precision of sample means estimates and the type I error of bioequivalence tests for both approaches. Crossover trials are simulated under H ₀ using different numbers of subjects (N) and of samples per subject (n). We simulate concentration-time profiles with different variability settings for the between-subject and within-subject variabilities and for the variance of the residual error.

Results

Bioequivalence tests based on NCA show satisfactory properties with low and high variabilities, except when the residual error is high, which leads to a very poor type I error, or when n is small, which leads to biased estimates. Tests based on EBE lead to an increase of the type I error, when the shrinkage is above 20%, which occurs notably when NCA fails.

Conclusions

For small n or data with high residual error, tests based on a global data analysis should be considered instead of those based on individual estimates.

APPENDIX

Approximation of the Variance of log(C _max ) by the Delta Method

For a one-compartment model with first-order absorption and first-order elimination, C _max is defined in Eq. 6 as a function of the three PK parameters: k _a , CL/F and V/F. The variance of log(C _max ), ${{\omega }}_{{{\text{C}}_{{ \max }}}}^{\text{2}}$, is approximated by the delta method (33) as follows:

$$ \omega_{{C_{\max }}}^2 \approx \left( {\frac{{\partial { \log }\left( {{C_{\max }}} \right)}}{{\partial { \log }\left( {{k_a}} \right)}}} \right)_{{ \log }\left( \mu \right)}^2\omega_{{k_a}}^2 + \left( {\frac{{\partial { \log }\left( {{C_{\max }}} \right)}}{{\partial { \log }\left( {CL/F} \right)}}} \right)_{{ \log }\left( \mu \right)}^2\omega_{CL/F}^2 + \left( {\frac{{\partial { \log }\left( {{C_{\max }}} \right)}}{{\partial { \log }\left( {V/F} \right)}}} \right)_{{ \log }\left( \mu \right)}^2\omega_{V/F}^2 $$

(13)

where log(μ) = (log(μ _ka ), log(μ _CL/F), log(μ _V/F ))′. After computing the derivatives, ${{\omega }}_{{{\text{C}}_{{ \max }}}}^{\text{2}}$ can be approximated by the following:

$$ \begin{array}{*{20}{c}} {\omega_{{C_{\max }}}^2 \approx \,{\Delta^2}\left( {\omega_{{k_a}}^2 + \omega_{CL/F}^2} \right) + {{\left( {\Delta - 1} \right)}^2}\omega_{V/F}^2} \hfill \\ {{\text{with}}\;\Delta = \frac{{{\mu_{CL/F}}\left( {{\mu_{CL/F}} - {\mu_{{k_a}}}{\mu_{V/F}}} \right) + {\mu_{{k_a}}}{\mu_{CL/F}}{\mu_{V/F}}{ \log }\left( {\frac{{{\mu_{{k_a}}}{\mu_{V/F}}}}{{{\mu_{CL/F}}}}} \right)}}{{{{\left( {{\mu_{{k_a}}}{\mu_{V/F}} - {\mu_{CL/F}}} \right)}^2}}}} \hfill \\ \end{array} $$

(14)

In this simulation study, the general formula above is applied to approximate the variance of log(C _max ) for both treatment groups (Ref and Test). Given the treatment effect we simulate for the treatment Test, both approximations, $\omega_{C_{\max }^{\left( {{\rm Re} f} \right)}}^2$ and $\omega_{C_{\max }^{\left( {Test} \right)}}^2$, are equal.

To approximate the variance of log(C _max ) by the delta method, we use the true simulated values of μ^(Ref) and Ω^(Ref) described in “Estimation Based on Nonlinear Mixed Effects Model.” To evaluate the delta method, we also estimate the variance of log(C _max ), using the simulated parameter values of the rich design (N = 40, n = 10) for the reference treatment, under S _l,l and S _h,l . For both variability settings, $\omega_{C_{\max }^{\left( {Ref} \right)}}^2$ is estimated as the empirical variance of the 40000 true simulated values of log$\left( {{C_{\max }}_i^{\left( {{\rm Re} f} \right)}} \right)$. For S _l,l , the standard deviation of log(C _max ) for the reference treatment expressed in percent is 10.5% both by simulation and the delta method. For S _h,l , it is 46.3% and 46.7% by simulation and the delta method, respectively.

These results on the true simulated values validate the approximation of the variance of log(C _max ) by the delta method. Consequently, we apply it to the data of each treatment group for each simulated trial of the simulation study to approximate $\widehat\omega_{C_{_{\max }}^{\left( {{\rm Re} f} \right)}}^2$ ($\widehat\omega_{C_{_{\max }}^{\left( {Test} \right)}\;}^2$respectively) using ${\mu^{\left( {{\rm Re} f} \right)}}$ (${\widehat\mu^{\left( {Test} \right)}}$ respectively) and ${\widehat\Omega^{\left( {{\rm Re} f} \right)}}$ (${\widehat\Omega^{\left( {Test} \right)}}$ respectively).