Testing for bioequivalence of highly variable drugs from TR-RT crossover designs with heterogeneous residual variances

Transcription

1 Testing for bioequivalence of highly variable drugs from T-T crossover designs with heterogeneous residual variances Christopher I. Vahl, PhD Department of Statistics Kansas State University Qing Kang, PhD Chief Statistical Scientist The Statistical Intelligence Group, LLC

2 Introduction Traditional bioavailability (BA) studies assess average bioequivalence (ABE) between the test (T) and reference () products under the T-T crossover design. With highly variable (HV) drugs whose within-subject coefficient of variation (CV) in pharmacokinetic measures is 30%, assertion of ABE becomes difficult. In 011, the FDA adopted a BE criterion with mixed-scaling. For HV drugs, the equivalence limits for the geometric mean ratio (GM) are scaled to within-subject variance of the reference product. This is known as reference scaled ABE (SABE). GM H 0 H H 0 : ( T ) kmax(0.5, ) vs. 1 : ( T ) kmax(0.5, ) H 1 T ln( CV 1); k ln(t: GM); [ln(1.5)] 0.5.

3 Introduction The FDA extended the statistical methods for assessing individual BE (IBE) to testing SABE. The recommended procedure operates exclusively under T-T-T and TT-TT designs. Testing IBE calls for separate estimation of subjectformulation variance and withinsubject variances, which could only be achieved by replicate crossover designs. In 003, the FDA discontinued the IBE criterion due to the lack of evidence confirming the existence of subjectformulation. Designs with more than periods are not always feasible. The volume of blood taken from each subject may exceed the acceptable limit. They tend to have a large amount of missing data and a high dropout rate. To avoid carryover effect, the washout period lasts for 5 half lives. A -period design is more practical for drugs with a long half life. Subject s physiological changes affect the variability of systemic drug concentration. A lengthy study could not be conducted on growing animals or those susceptible to stress under prolonged confinement and repeated dosing. Goal: To investigate how to evaluate HV drugs under the T-T design.

4 Model Consider a T-T design where the washout time between periods is sufficient to eliminate any carryover effect. Subject j in sequence i provides a vector of ln-transformed responses (Y ij1,y ij ). Alternative BE criteria under the T-T design were assessed using the model with heterogeneous residual variance. 1 1 T ijt i ij 1 1 i Y ij1 τ ij Y ij 1 1 ij i ij i 1 τ T ijt τij : subject effect, τij ~ N(, S ); : sequence effect; i : period effect; j T & : product effects; Treatment Sequence (i). of Subjects* Period 1 Period 1 n 1 T n T τ ijt & ij : residual errors, ijt ~ N(0, T ), ij ~ N(0, );, & are mutually indept. ij ijt ij *n n N 1

5 Model The FDA s model for replicate crossover designs allows ij to have a covariance matrix of any positive definite structure. τ 0 0 ~ N(, ), ~ N(, ) ST ST S S ST S 1j j 0 τ 0 ST S S ST S ST The variance of subjectformulation is ( ) (1 ) Int ST S ST S The T-T design confounds Int no subjectformulation, with within-subject variances. But when there is ~ (, ) ST S S Int τij N S It is then plausible to assess SABE from the T-T design.

6 Summary Statistics Convert (Y ij1, Y ij ) into within-subject sum and T- difference. Yij1Yij i 1 Yij Yij1 Yij Yij Yij Yij1 i Yij 4 S T T Var( ) Y ij T T ( ) ( ) T T T T Define the summary statistics with respect to (Y ij, Y ij ) as n n i i Y Y n Y Y n D ( Y Y ) i j1 ij i i j1 ij i 1 ni ni ni i 1 j 1 ij i i 1 j 1 ij i i 1 j 1 ij i ij i S ( Y Y ) S ( Y Y ) S ( Y Y )( Y Y ) B S S S S S S te : D and S /[(N-)] correspond the LS mean difference and MSE in the classical ANOVA model; B and S represent the slope and the SSE when regressing Y ij on Y ij with the same slope but different intercept for i=1,.

7 Summary Statistics By ao(1973), these summary statistics have the distributional properties of 4nn N S [ D ( )] Z ~ N(0,1) U ~ ( N ) 1 T T T 1 S ( B ) S Z ~ N (0,1) U ~ ( N 3) 4 (1 ) : conditional variance of Y given Y S T ij ij te: Z, Z, U, and U are mutually independent; B is not independent of S and its marginal distribution is not normal; The estimator for T, and T are D, B and S /(N-); The estimator for 0.5(1 ) is whose distribution does not follow any classic form. T 0.5(1 B) S ( N )

8 Testing Procedures The current test of SABE employs the modified large sample (MLS) method which is applicable under replicate crossover designs. The MLS method approximates the confidence limits (CLs) of a linear combination of parameters by restricting it to be exact when only one parameter is unknown. It calls for independent summary statistics with known distributions. Let b, s and d be the values of B, S and D observed from the T-T design. The estimate of is and the CLs for T is 1 (1 bs ) N 4 N d t0.95, N s 4 n n ( N ) 1 The MLS method is inappropriate under the T-T design because is not estimated from independent summary statistics with classic marginal distributions.

9 Testing Procedures The generalized pivotal quantity (GPQ) method is used to test SABE under the T- T design. The distribution of the GPQ produces a fiducial-type inference which, in many situations, meets frequentists standards. The distributional properties of the summary statistics suggest that the GPQ for T, T, and are 4nn N s N s T1 d ( D) d Z N n n S n n U 1 T T T 1 T s S T s U 1 s s T3 b ( B ) S b Z S s s U The GPQ for ( T ) k ( T ) 0.5 k T(1 ) T 0.5 kt (1 T ) 1 3 is then assembled as

10 Testing Procedures The distribution of this GPQ is obtained via a resampling algorithm. Step 1. Independently sample z, u, z and u and N(0,1) and (N-3). from N(0,1), (N-), Step. eplace Z, U, Z, and U in T 1, T and T 3 with z, u, z and u. This yields t 1, t and t 3. Step 3. Calculate t 1 0.5kt (1-t 3 ) and accumulate its value by repeating steps1 and many times. The 95% upper CL of ( T ) k is given by the 95 th percentile of the resampling distribution. The test product passes SABE when this percentile is 0.

11 Analysis of Example Dataset Source: Dataset #7 in FDA s databank for BA studies. Data: Cmax measured in the first two periods of a TT-TT-TT-TT design. d s s s b s T ( n1 1) T ( n 10) Parameter Point Estimate (PE) Conf. Level Conf. Interval (CI) exp( T ) % (0.9148, ) ( T ) k % (-, )

12 Analysis of Example Dataset The FDA s decision rule for testing SABE under replicate crossover designs applies TOST when To ensure public confidence and harmonize with various regulatory agencies, the FDA requires PE of the estimated T: geometric mean ratio to lie within 0.8~1.5. TOST-MLS Procedure W/ PE constraint Data from the replicate crossover design TOST-GPQ Procedure W/ PE constraint Data from the T-T design Test SABE via the MLS method 0.94 Test SABE via the GPQ method % UCL of ( ) k 0 T Test ABE via TOST 90% CI of exp( T ) [0.8,1.5] 95% UCL of ( ) k 0 T PE constaint: 0.8 ln( d ) 1.5 Fail PE constaint: 0.8 ln( d ) 1.5 Pass Pass

13 Simulation Study Simulation parameters were chosen based on historical publications on BA studies. Total Sample size T-T designs: N =4,, 7 with n 1 =n =N/; T-T-T designs: N=18,,48 with n 1 =n =n 3 =N/3; Variation of the reference product High: CV = 40%,,80%; Borderline high: CV = 5% (the changeover point) and 30%; egular: CV = 0%. Within-subject variance Heterogeneous: T / =0.5 and ; Homogeneous: T / =1. S =0.,,1. Zero sequence and period effects. 10,000 simulations at each sample size and parameter setting. 5, 000 resamples within each simulated dataset.

14 Simulation Study Type I error rate at ( T ) =kmax(0.5, ). Each schematic boxplot summarizes results of 75 combinations of simulation setting.

15 Simulation Study te: The TOST-GPQ procedure was moderately liberal at CV =5% but held a desirable type I error rate at all other examined values of CV. The TOST-MLS procedure preserved its nominal level when CV 5%. In comparison to the TOST-GPQ procedure, it had a slightly lower inflation of type I error at CV =5%. The PE constraint led to ultra conservativeness when CV 40% but had little impact when CV 5%.

16 Power Simulation Study N 4 N 48 N 7 CV CV CV S Power for the TOST-GPQ procedure with PE constraint at T =0 & T / =1. te: Power increased as S decreased and as N increased. There was no monotonic relationship between power and CV. For a given S & N, power was always the highest when CV is around 0% and dipped to the bottom when CV is around 50%.

17 Passing ate Simulation Study CV =CV T =30% CV =CV T =60% exp( ) T exp( ) T S Passing rate of the TOST-GPQ procedure under the T-T design of size 54 ( ) and the TOST-MLS procedure under the T-T-T design of size 36 (- - -). te: These hybrid procedures performed similarly well at S =0. & 0.6. The TOST-GPQ procedure underperforms at S =1.

18 Sample Size Estimation Total sample size needed to establish BE with mixed scaling at T =0. 80% power CV T =CV T-TT-T Design S =0.4 T-T Design S =0.6 S =0.8 30% 1 8 (1.3 ) 3 (1.5) 38 (1.8) 40% 34 (1.5) 44 ( ) 56 (.5) 50% 34 (1.5) 44 ( ) 56 (.5) 60% 3 34 (1.5) 4 (1.8) 5 (.3) 70% 5 36 (1.4) 4 (1.7) 50 ( ) Provided by Tothfalusi & Endrenyi 01. The ratio of sample sizes. te: The two designs have a trivial difference at S =0.4. The T-T design is more expensive at S =0.6 & 0.8.

19 Discussion So far, BA studies on HV drugs have all been carried out under replicate crossover designs with 3 or 4 periods. In cases that these designs are infeasible, parallel designs have been considered as a backup choice. The setbacks include prohibitive sample sizes; difficulty in translating SABE when the between- and within-subject variability are confounded. The proposed approach utilizes the fact that subjectformulation is negligible; operates under the traditional T-T design; allows continued exercise of the existing BE criterion. When S 0.4, the resulting inference is at least as good as that derived from replicate crossover designs. Larger S weakens the inference. ecommendation: eplicate crossover designs are preferred for BA studies on HV drugs. Under practical restrictions, producers are encouraged to implement the T- T design and the proposed TOST-GPQ procedure.

20 Discussion The FDA requires producers to determine, a priori, whether the reference drug is HV. Variability classification is a known problem for human and animal drugs. Dispute arises when producers classify the same reference drug differently. The protocol specifies to use SABE for a HV drug Data from the replicate crossover design Data from the T-T design Test SABE via the MLS method 0.94 Test ABE via TOST 95% UCL of ( ) k 0 T PE constaint: 0.8 exp( d ) 1.5 Fail 90% CI of exp( T ) [0.8,1.5] Pass

21 Discussion The proposed approach promotes coherence in regulatory evaluation. The protocol specifies to test BE with mixed scaling Data from a crossover design Test SABE via the MLS or GPQ method 0.94 Test ABE via TOST 95% UCL of ( ) k 0 T PE constaint: 0.8 exp( d ) 1.5 Fail 90% CI of exp( T ) [0.8,1.5] Pass

22 eferences Claxton et al. Estimating product bioequivalence for highly variable veterinary drugs. J. Vet. Pharmaco. Ther. 01; 35 (Suppl. 1): Davit BM et al. Implementation of a reference-scaled average bioequivalence approach for highly variable generic drug products by the US Food and Drug Administration. AAPS J. 01; 14: Draft guidance on progesterone, GuidanceComplianceegulatoryInformation/Guidances/UCM0994.pdf. Guilbaud O. Exact inference about the within-subject variability in crossover trials. JASA 1993; 88: Hannig J et al. Fiducial generalized confidence intervals. JASA 006; 101: Haidar et al. Evaluation of a scaling approach for the bioequivalence of highly Variable drugs. AAPS J. 008; 10: ao C. Linear statistical inference and its applications (nd ed.). Wiley: New York; Tothfalusi L & Endrenyi L. Sample sizes for designing bioequivalence studies for highly variable drugs. J. Pharm. Pharm. Sci. 01; 15: Weerahandi S. Generalized confidence Intervals. JASA 1993; 88: