**Non-linear mixed-effects models for tests of interaction or of lack of interaction in cross-over and parallel pharmacokinetic studies: application to the test of interaction between protease inhibitors and nucleoside analogs in HIV patients**

Xavière Panhard and France Mentré

INSERM U738, CHU Bichat-Claude Bernard, Paris, France

**Introduction: **Pharmacokinetic (PK) interaction, or lack of interaction, of a drug B on a drug A, is usually tested by comparing the log AUC and the log C_{max} of A given alone (or with placebo) versus A given with B. The standard method, recommended in guidelines, is the comparison of log AUC and log C_{max} of A, using a Student test or a Schuirmann’s two-one sided test (TOST) to test interaction or absence of interaction, respectively. In order to take benefit of the knowledge accumulated on the PK of the studied drugs and to decrease the number of samples per patient, tests based on non-linear mixed effects models (NLMEM) are a good approach. Our objectives were to propose and evaluate tests based on NLMEM for evaluation of both PK interaction or absence of interaction, first in cross-over PK trials, which are designed to test such interactions, and second during usual population PK analyses, where interactions are usually tested a posteriori.

**Cross-over PK trials***a) Methods*

We focused on tests based on log AUC, but the proposed methods can easily be extended to log C_{max}. Both for evaluation of interaction or lack of interaction, several tests can be proposed based on NLMEM, to compare the log AUC of drug A taken with or without drug B. First, concentration data of A can be analysed to derive individual Empirical Bayes (EB) estimates, separately in the two groups of data with and without comedication with B. Second, they can be analyzed globally, with or without the estimation of the effect of drug B. Four tests for testing interaction on the log AUC can therefore be derived: paired parametric and non parametric tests comparing the EB estimates, the likelihood ratio test (LRT) between a model with or without interaction effect and a Wald test on the interaction effect. We adapted these approaches to the test of lack of interaction, i.e. equivalence, except the LRT, which does not have any simple extension. More precisely, we extended the method of the TOST to the EB tests and to the Wald test, using the standard error (SE) of the interaction effect.

*b) Evaluation by simulation without modelling inter-occasion variability (IOV)*

We evaluated by simulation the type I error and the power of both the proposed interaction or lack of interaction tests in cross-over trials [1]. Trials mimicking the theophyllin data set were simulated under H0 and several H1 and analysed with the nlme function, assuming a one-compartment model with first order absorption and elimination parametrized in k_{a}, AUC and V/F. Different configurations of the number of patients (N = 12, 24 and 40) and of the number of samples per patient (n = 10, 5 and 3) were evaluated. The type I errors of the Wald interaction tests in 5000 trial replications were 22% in the original design (N = 12, n = 10), 14% in the intermediate design (N = 24, n = 5) and 7.7% for the sparse design (N=40, n=3). The LRT achieved very similar results. Power was satisfactory for both tests, after Monte-Carlo correction of the significance threshold. Similar results were obtained for lack of interaction or equivalence tests: the type I errors of the Wald test in the 5000 replications were 25%, 16% and 7.4% in the original, intermediate and sparse designs, respectively. In both types of tests, the Wald test presented the best power after correction of the significance threshold. EB tests achieved satisfactory type I error and power in the interaction case, but were not of a great value for testing equivalence.

*c) Evaluation by simulation when modelling IOV*

In the preceding simulation study, we did not model IOV since the simulated IOV was small: 5% for V/F and 10% for ka and AUC. As we found that global tests (LRT and Wald) suffered from an important inflation of the type I error, we evaluated the impact of modelling IOV on the properties of those tests. For interaction tests when IOV was estimated on each parameter, the type I errors of the Wald test on the 5000 replications were 7.5%, 6.4% and 3.5% in the original, intermediate and sparse designs, respectively, which were much closer to 5% than when IOV was not taken into account. The LRT achieved very similar results. Power was satisfactory for both tests and for the three considered designs. For tests of absence of interaction when IOV was estimated, the type I errors were found to be 6.4%, 5.7% and 4.2% for the original, intermediate and sparse designs, respectively. NLMEM is therefore confirmed as a good and useful approach to test interaction or lack of interaction in cross-over trials. Global tests like LRT or Wald seem to achieve adequate type I error only if IOV, even small, is taken into account.

*d) Application to the interaction of tenofovir (TFV) on the PK of atazanavir (ATZ)*

The Wald test was applied to compare the PK of ATZ, an HIV protease inhibitor, administered with and without TFV, a nucleoside analog, in a cross-over study which was a PK substudy of the ANRS 107 – Puzzle 2 trial conducted in 11 HIV-infected patients. Six blood samples (1, 2, 3, 5, 8 and 24h after dosing) were taken at two periods: first without and then with co-administration of TFV. Concentration data of both periods were analysed assuming a one-compartment model with zero order absorption and first order elimination parametrized in absorption duration (T_{abs}), AUC and V/F. For each PK parameter, inter-individual variability (IIV), IOV and a treatment effect were estimated. Significant effects of comedication with TFV were found on ATZ AUC (p<10^{-4}) and T_{abs} (p<10^{-3}), which were decreased by decreased by 1.46 fold and increased by 1.45 fold, respectively, when patients received TFV.

**Test of interaction during usual population PK analyses***a) Methods*

Several authors showed an inflation of the type I error for the test of binary covariates in population analyses which might lead to false inclusion of effects. This inflation can be important for limited number of patients, for sparse sampling designs, for large residual error or when the structural model is complex. One method to correct that inflation is the use of randomization tests. This method can be applied to a posteriori test of PK interaction. We extended it to the test of lack of interaction using, as for cross-over trials, a TOST based on the estimate of the treatment effect and its SE.

*b) Application to the interaction of zidovudine (ZDV) on the PK of nelfinavir (NFV) and its metabolite*

We applied the randomization test to the effect of ZDV in the simultaneous population PK analysis of NFV and its major metabolite M8 [2]. Concentration data were obtained from 46 patients enrolled in the Cophar 1 – ANRS 102 study. Seven blood samples were taken per patient, five at a first visit (before and 0.5, 1, 3, 6 and 12 h after dosing) and two at a second visit 3 months later (before and 3 h after dosing). A one-compartment model with first order absorption and elimination was used to describe NFV concentrations, with an additional compartment with a first order rate-constant k_{m} describing metabolization into M8. The identifiable parameters were V/F, Cl/F and k_{a} for NFV, and V_{m}/Fk_{m} and Cl_{m}/Fk_{m} for M8. Concentrations of NFV and M8 from both visits were modelled simultaneously in all patients with the nlme function. IIV was estimated on V/F, Cl/F and Cl_{m}/Fk_{m}, and IOV was estimated on Cl/F. Interaction effects with nucleoside analogs were tested using the Wald test. The only significant interaction effect was that of ZDV, received by 27 patients, on Cl_{m}/Fk_{m} (p=0.011), which decreased by 1.8 fold in patients receiving ZDV. We then evaluated the real p-value of this effect by a randomization test. We performed R=1000 random permutations of comedication with ZDV, and analysed the corresponding data sets using the previous PK model. The type I error for the Wald test was found to be 6% for Cl_{m}/Fk_{m}, which led to a corrected p-value of 0.016 for the original data set. This result, obtained with nlme, did not confirm the necessity of correcting the type I error of interaction tests based on NLMEM even in a complex PK model with limited number of patients and large variability.

**Conclusion: **Tests based on NLMEM allow both to test PK interaction or lack of interaction while greatly decreasing the number of samples per patient. This point is of great interest when performing such trials in patients, for instance in HIV patients as illustrated here, or in special populations (children, older patients). The necessity of using or not a correction method for the type I error should be further evaluated and depends on the estimation method or algorithm. Our next step is the design of such PK interaction studies. Since the expected standard error of the interaction effect can be estimated with IOV using an extension of PFIM [3], the power of the interaction or lack of interaction test can be derived and thus the sample size for a given power.

**References:**

[1] Panhard, Mentré. Evaluation by simulation of tests based on NLMEM models in PK interaction and bioequivalence cross-over trials. *Stat Med*, 2005 [Epub ahead of print].

[2] Panhard, Goujard, Legrand, Taburet, Diquet, Mentré. Population PK of NFV and M8 in HIV patients on HAART. Abstract 6.13, *4th International Workshop on Clinical Pharmacology of HIV Therapy*, March 2003, Cannes, France.

[3] Retout, Mentré. Further developments of the Fisher information matrix in NLMEM with evaluation in population PK. *J Biopharm Stat*, 2003; 30:417-43.

**Acknowledgements:**

Puzzle2 - ANRS 107 study team (investigator: Dr Piketti, pharmacology: Dr Taburet, methodology: Dr Aboulker)

Cophar1 - ANRS 102 study team (investigator: Dr Goujard, pharmacology: Dr Taburet, methodology: Pr Mentré)