Research ArticleA Bioequivalence Test by the DirectComparison of Concentration-versus-Time CurvesUsing Local Polynomial Smoothers

Suyan Tian123 Howard H Chang4 Dana Orange56

Jingkai Gu78 and Mayte Suaacuterez-Farintildeas2910

1 Division of Clinical Research First Hospital of Jilin University 71 Xinmin Street Changchun Jilin 130021 China2 Center for Clinical and Translational Science The Rockefeller University 1230 York Avenue New York NY 10065 USA3 School of Mathematics Jilin University 2699 Qianjin Street Changchun Jilin 130012 China4 Department of Biostatistics and Bioinformatics Rollins School of Public Health Emory University 1518 Clifton Road NEAtlanta GA 30322 USA

5 Laboratory of Molecular Neurooncology The Rockefeller University 1230 York Avenue New York NY 10065 USA6 Division of Rheumatology Hospital for Special Surgery 535 East 70th Street New York NY 10021 USA7 Clinical Pharmacology Center Research Institute of Translational Medicine First Hospital of Jilin UniversityDongminzhu Street Changchun 130021 China

8 College of Life Science Jilin University 2699 Qianjin Street Changchun 130012 China9 Center for Biostatistics Department of Population Health Science and Policy Icahn School of Medicine at Mount SinaiNew York NY 10029 USA

10Department of Genetics and Genomics Science Icahn School of Medicine at Mount Sinai New York NY 10029 USA

Correspondence should be addressed to Jingkai Gu gujkjlueducn and Mayte Suarez-Farinas maytesuarezfarinasmssmedu

Received 23 June 2016 Revised 29 August 2016 Accepted 7 September 2016

Academic Editor Andrzej Kloczkowski

Copyright copy 2016 Suyan Tian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to test if two chemically or pharmaceutically equivalent products have the same efficacy andor toxicity a bioequivalence(BE) study is conductedThe 80125 rule is the most commonly used criteria for BE and states that BE cannot be claimed unlessthe 90 CIs for the ratio of selected pharmacokinetics (PK) parameters of the tested to the reference drug are within 08 to 125Considering that estimates of these PK parameters are derived from the concentration-versus-time curves a direct comparisonbetween these curves motivates an alternative and more flexible approach to test BE Here we propose to frame the BE test interms of an equivalence of concentration-versus-time curves which are constructed using local polynomial smoother (LPS) Ametric is presented to quantify the distance between the curves and its 90 CIs are calculated via bootstrapping Then we appliedthe proposed procedures to data from an animal study and found that BE between a generic drug and its brand name cannotbe concluded which was consistent with the results by applying the 80125 rule However the proposed procedure has theadvantage of testing only on a single metric instead of all PK parameters

1 Introduction

In order to test if two chemically or pharmaceutically equiv-alent products for example a generic drug and its brandname have the same efficacy andor toxicity a bioequivalence(BE) study is usually conducted [1ndash3] The objective of

a BE trial is to determine whether the test (119879) and thereference (119877) formulation of a pharmaceutical product areldquoequivalentrdquo with respect to blood concentration times timeprofiles In contrast to a difference test the null hypothesisin an equivalence test states that two agents differ in terms ofthe endpoint under consideration by at least the minimum

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 4680642 6 pageshttpdxdoiorg10115520164680642

2 Computational and Mathematical Methods in Medicine

tolerable amount called the equivalence margin Δ whereasthe alternative hypothesis states that such difference is lessthan the equivalence margin Δ

1198670 10038161003816100381610038161205831003816100381610038161003816 ge Δversus 1198671 10038161003816100381610038161205831003816100381610038161003816 lt Δ (1)

In aBE test someparameters derived from the concentra-tion-versus-time curves are evaluated Those parameterstraditionally include the area under plasma concentration-versus-time curves (AUC1199050) peak plasma concentrations(119862max) and its corresponding time (119879max) Among themAUC is the most acceptedmeasure of absorption rate119862max isalso of importance because for some drugs a certain level ofconcentration needs to be reached to guarantee the desiredtherapeutic effect 119879max is a relevant measure for drugs suchas antibiotics that must reach the peak concentration quicklybut it may not be an appropriate measure for drugs requiringmultiple dosage before a therapeutic effect is observed Sincenone of these three parameters are universally superior to theothers a BE test usually considers them together For exam-ple under FDA regulations BE can be claimed only when the90 confidence intervals (CIs) for the ratio of119862max AUC(0minus119905)and AUC(0minusinfin) of the tested (eg generic) to the referencedrug (eg brand name) are within 80 to 125 [4] Thisis referred to as 80125 rule corresponding to a plusmn0223rule on the logarithmic scale (of the 119879119877 ratio) Hence thevalue of Δ is usually set to 0223 Of note the 90 CIs of theBE-endpoint represent a 005 significance level on the equiv-alence test since the hypothesis-testing problem in an equiva-lence test is divided into two one-sided hypothesis tests [25 6] one where the null hypothesis states that the differencebetween two agents is less than minusΔ whereas the other oneassumes such difference is larger than Δ

11986701 120583 lt minusΔversus 1198671198861 120583 ge minusΔ

11986702 120583 gt Δversus 1198671198862 120583 le Δ

(2)

Although the BE parameters can be easily obtained fromeither one of the concentration-versus-time curves usinga suitable pharmacokinetic (PK) model such as a one-compartment model or a nonparametric method [7] thereare many parameters to be tested which inflates the typeI error rate requiring the adjustment for multiple compar-isons Because these parameters may be highly correlatedsuch an adjustment is challenging

On the other hand some researchers had pointed outthat the requirement of all confidence intervals falling withinthe equivalence bounds might lead to a conservative resultdepending on the correlations among the PK parameters andthe study power [8 9] To address this simultaneous testing ofall PK parameters had been explored and developed see forexample the semiparametric Bayesian approach proposed byGhosh and Gonen [10] However such multivariate methodsare less popular than the univariate approach of testing PK

parameters partially because of the modeling complexityassociated with multivariate methods as compared with theirunivariate counterparts

More importantly when the confidence intervals of119862max119879max and AUC between two drugs all fall within the equiva-lence boundary (such that BE is concluded) it does not implythat the drugs are clinically equivalent since the overall shapesof the concentration-versus-time curves may in fact differ[11] A falsely determined BE between two drugs when theyare not clinically equivalent may be very harmful to the pub-lic To alleviate these limitations and drawbacks we proposeto make a direct comparison of the concentration-versus-time curves that accounts for the differences between the pro-file shapes for BE testingThemain objective of this paper is topresent a strategy to test BE with the aid of local polynomialsmoother (LPS) which is used to construct concentration-versus-time curves Then a summary statistic whose stand-ard error is estimated by bootstrapping is definedThis allowsthe calculation of CIs upon which decisions over equivalencebetween such two curves are made

LPS is a flexible nonparametric regression method tomodel curves or surfaces Specifically the fitted regressionfunction at a covariate value 119909 is based only on observationswithin a prespecified neighborhood of 119909 LPS can be tracedback to the late 19th century in actuarial sciences where itwas used to estimate the gradation of mortality rates and intime series modeling [12] In contrast another popular non-parametric regression method locally weighted scatterplotsmoothing (LOWESS) obtained smooth fitted values by aweighted linear least squares regression over the prespecifiedspans In this study however we do not emphasize theirdifferences and instead treat them as synonymous

With advances in computing the potential applicationsof LPS continue to expand For example it has been used fordye normalization of two-color microarray experiments [13]Recently Anders andHuber [14] used LPS in Rrsquos DESeq pack-age to estimate the mean-variance relationship in a negativebinomial model a technique now commonly used to modelRNA-Seq data The popularity of LPS in statistical applica-tions may be due to its several advantages including satisfac-tory boundary behavior and straightforward interpretabilityof the nonlinear relationships [15 16]

2 Methods and Materials

21 Experimental Data The study included 24 beagle dogsweighting 9ndash11 kg and those dogs were randomly assigned toreceive either single ormultiple (every four weeks for 3 times)administrations of either 14mgkg Sandostatin or a genericdrug developed by GenSci There were 6 dogs including3 females and 3 males per treatment-dosage group Bloodsamples for PK were collected into tubes containing heparinsodium and centrifuged at room temperature with 3500 rpmfor 20min to obtain plasma samples This animal studywas conducted in accordance with the rules and regulationsof Chinese Pharmacopoeia 2000 EditionVersion 2 andapproved by the Committee on the Ethics of Animal Exper-iments of Jilin University The plasma concentrations were

Computational and Mathematical Methods in Medicine 3

determined using an Agilent 1100 liquid chromatography-tandemmass spectrometry In the single-dose regimen bloodsamples were collected at the initial time corresponding tohour 0 and at the hours of 025 05 1 15 2 4 8 24 48 96 144192 240 336 432 528 624 720 816 912 1008 1104 and 1200

Based on [7] PK parameters in this study were calculatedas follows For each dog themaximumplasma concentration(119862max) and its corresponding time (119879max) were determined byvisual inspection of the profilesThe apparent terminal elimi-nation rate constant (120582) was calculated by linear regression ofthe natural logarithms of the terminal plasma concentrationsThe terminal half-life (11990512) was derived as 0693120582 The areaunder the curve (AUC) to the last measured point (AUC1199050)was calculated using the trapezoidal rule The AUC for theplasma concentration-versus-time function from 0 hours toinfinity (AUC0minusinfin) was calculated as the sum of AUC1199050 and119862119905120582 while 119862119905 was the last quantifiable concentration22 The LPS Model Let (1198831 1198841) (119883119899 119884119899) be covariate-response pairs for 119899 observations To simplify the notationsuppose there is only one covariate Then the response 119884119894 isrelated to the covariate119883119894 through the following model

119884119894 = 119892 (119883119894) + 120576119894 119894 = 1 119899 (3)

where the error term 120576119894 is assumed to be independentlydistributed with zero mean and variance equal to 1205902 Thegoal of a local polynomial smoother is to estimate the smoothfunction 119892(119909) = 119864(119884 | 119883 = 119909) using an approximationprovided by a local polynomial of low order in the neigh-borhood of the point 119909 Usually the model includes onlyfirst- and second-degree polynomials that is either locallylinear or locally quadratic This is because any function canbe well approximated in a small neighborhood by low-orderpolynomials and the inclusion of higher order terms oftendoes not improve the model fit dramatically and even maylead to overfitting instead Specifically for 119909 minus ℎ le 119909119894 le 119909 + ℎ

119892 (119909119894) asymp 119901sum119895=0

120573119895 (119909119894 minus 119909)119895 (4)

The abovementioned approximation is fitted by localweighted least squares where the estimated coefficients arethose that minimize the following function

119899sum119894=1

119882((119909119894 minus 119909)ℎ )(119910119894 minus 119901sum119895=0

120573119895 (119909119894 minus 119909)119895)2

(5)

where the weighted function 119882 gives greater weight to 119909119894in the vicinity of 119909 and is usually specified as a symmetricbounded function such as a normal kernel function

There are two commonly used approaches for the selec-tion of bandwidth ℎ in the equation above [12] One choice issimply to make the bandwidth a constant which is adequateif the regression function behaves smoothly The other one isto allow the bandwidth to change as a function of 119909 In ourapplications we chose the latter by using a nearest-neighborbandwidth selection method [17] Briefly in the nearest-neighbor bandwidth selection method a new parameter 120572

was included which was multiplied by the sample size 119899and rounded up to an integer 119896 Then the bandwidth ℎ(119909)was defined as the distance from 119909 to the 119896th closest 119909119894In this study measurements were taken at nonuniformlyspaced time points with more frequent measurements at thebeginning where a larger variation of the PK dynamic wasexpected among subjects Under this condition the nearest-neighbor bandwidth selection approach will provide betterfit than the constant bandwidth The nearest-neighbor band-width parameter 120572was determined by a leave-one-out (LOO)cross-validation due to the small sample size Specificallyfor a grid of possible 120572 values cross-validated mean squareerror (MSE) was calculated and the 120572 corresponding to theminimumMSE was selected

23 Statistical Language The statistical analysis was carriedout in theR language version 215 (httpwwwr-projectorg)The R-code is available upon request

3 Results and Discussion

31 Testing the Equivalence of Two Curves by CombiningLPS and Bootstrapping The Proposed Procedure Here wepropose to evaluate bioequivalence between drugs by directlycomparing the plasma concentration curves PC(119905) Supposewe have two treatments where 1198991 subjects were randomlyassigned to receive the test drug119879 and 1198992 to the reference drug119877 We proposed to reframe the hypothesis represented in (1)as

1198670 119889 (119892119879 (119905) 119892119877 (119905)) gt Δ (6)

versus 1198671 119889 (119892119879 (119905) 119892119877 (119905)) le Δ (7)

with 119889 representing the difference between the PK curves for119879 and 119877 drugs For the study subjects the plasma concen-trations of these drugs were measured for those at a grid oftime points Using LPS (details in Section 2) concentration-versus-time curves for both treatments can be estimated as119892119877(119909) and 119892119879(119909) To evaluate if the differences between twocurves are within the tolerable range we define the followingestimator of the average difference between the two curves ata grid of prespecified time points for example the time pointswhere the concentrations were taken in this study

ln () = sum119870119896=1 10038161003816100381610038161003816ln (119879119905119896119877119905119896)10038161003816100381610038161003816119870 (8)

where 119879119905119896 and 119877119905119896 are the fitted values of plasma concentra-tion using LPS at time 119905119896 for the test drug 119879 and the referencedrug 119877 respectively Here 119896 = 1 2 119870 and 119870 is thenumber of time points at which the difference betweentwo curves is evaluated Note that ln(119903) is a measure of thedifference between the curves and when the two curves areidentical and they should be identical at all time points andthus the value of ln(119903) should be 0

The standard error for ln() can be estimated using boot-strapping [18] For each bootstrapped replicate LPS curve

4 Computational and Mathematical Methods in Medicine

0 200 400 600 800 1000 1200Hour

GenSciSandostatin

Con

centratio

n

4

3

2

1

0

minus1

Figure 1 The fitted local polynomial smoother (LPS) curvesfor both GenSci and Sandostatin Black triangle the observedconcentrations for dogs using Sandostatin red circle the observedconcentrations for dogs using GenSci

was fitted for 119879 and 119877 and the difference between them iscalculated using (8) The 90 CIs of 119903 can be obtained as

(119890ln()minus1645times119904119890(ln()) 119890ln()+1645timesse(ln())) (9)

If the 90 CIs is within 08sim125 then the equivalencebetween two curves is claimed

32 Case Study Octreotide is an octapeptide that mimicsnatural somatostatin pharmacologically even though it is amore potent inhibitor of growth hormone glucagon andinsulin than the natural hormone Brand name Sandostatin(Novartis Pharmaceuticals) of octreotide had been approvedfor the treatment of several diseases such as acromegaly andgigantism To evaluate the BE between a generic octreotidedeveloped by GenSci Pharmaceuticals (Changchun China)and Brand name Sandostatin a pilot study using 24 beagledogs was conductedThe generic drug is referred to as GenScihereafter Using this study as an example we showcase how toincorporate LPS seamlessly with other statistical methods totest BE Once concentration-versus-time curves are obtainedvia LPS testing for BE will be akin to testing if the two LPS-derived curves are the same

Firstly we conducted an equivalence test using the proce-dure described in Methods The fitted plasma concentrationPC(119905) curves are shown in Figure 1 Interestingly the curvesestimated by LPS suggested some PK differences betweenthe two drugs Specifically we observed a surge in plasmaconcentration and then a steep drop in the GenSci group Inaddition the GenSci group entered the plateau stage fasterand remained in this stage longer than the Sandostatin groupdropping down at a sharper scope Based on these obser-vations we suspected that GenSci might have a better PKbehavior than Sandostatin

The 90 CIs for the proposed metric of GenSci againstSandostatin were calculated as 11902 sim16699 With itsupper bound above 125 the equivalence between the two

Table 1 The 90 CIs of PK parameters

90 CI Single dose Combined data119873 = 6 119873 = 12119862max (7223 16071) (8293 16088)11987903 ngnL (994 38263) (951 14460)AUC0ndash1200 (7198 17608) mdashAUC0ndash720 (6174 16985) (9144 20850)

drugs cannot be established based on this method As acomparison we also estimated the 90 CI of the ratio ofthe traditionally considered PK parameters (ie (AUC1199050)119862max and 119879max) based on the FDA guidelines [4] The resultssummarized in Table 1 show that all 90 CIs of the ratioof (AUC1199050) 119862max and 119879max of GenSci against Sandostatinwere outside 80 to 125 the FDA-sanctioned BE criteriaIn summary BE cannot be established using either theFDA-criteria or our proposed procedure Nevertheless theproposed procedure has the advantage of testing only a singlestatistic and hence avoiding multiple hypotheses issues

The sample size of 6 dogs per group raisesmajor concernsin light of the failure to establish BE between GenSci andSandostatin To explore the impact of the sample size inthis pilot study we combined the data from this single-dosecohort with additional data from a multiple-dose cohort butincluding only data collected in the first 720 hours where noboosts were given and the experimental methodsconditionswere exactly the same as in the single-dose cohorts Interestedreaders may see Section 2 for more information on howthe experiments were conducted Analysis on the combinedcohorts including 12 dogs per group led to similar resultsGiven that this application represents a pilot study as pre-liminary work for larger studies no confirmative conclusionswere drawn here By presenting this study our intent isto illustrate the LPS-based BE test as an alternative to theclassical univariate tests on PK parameters

33 Simulation Study We conducted a simulation study tocharacterize our BE-testing procedure Here we used the PKmodel presented in [11] where the concentration function isrepresented by

119888 (119905) prop 119870119886119870119886 minus 119870119890 (119890minus119896119890119905 minus 119890minus119896119886119905) (10)

where 119896119886 and 119896119890 are the absorption and the eliminationrate constants The concentration-versus-time curves for thereference drug and test drug were simulated as

log (119884119877) = log (119888 (119905)) + 120576log (119884119879) = log (119888 (119905)) + nabla + 120576

120576 sim MVN (0119901 Σ119901times119901) (11)

where the correlation between two measures on the log-arithm scale from the same subject 119896 (119896 = 1 2 2119899)was specified as a positive constant that is corr(120576119894119896 120576119895119896) =

Computational and Mathematical Methods in Medicine 5

Table 2 The exact coverage of the 90 CIs for ln(119903) (the proposed measure of the difference between two curves) using simulations

119899 = 12 119899 = 18 119899 = 24 119899 = 30 119899 = 50nabla = 0 119896119886119879119896119886119877 = 1 (two curves are identical)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 999 100 100 100 100120588 = 08 978 996 100 100 100nabla = 0223 119896119886119879119896119886119877 = 1 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 987 997 100 100 100120588 = 08 955 972 991 997 100nabla = 1 119896119886119879119896119886119877 = 1 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 33 07 03 0 0120588 = 05 116 66 37 32 07120588 = 08 18 112 79 58 26119896119886119879119896119886119877 = 125 nabla = 0 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 999 100 100 100 100120588 = 05 977 991 997 100 100120588 = 08 95 975 988 993 100119896119886119879119896119886119877 = 3 nabla = 0 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 5 15 06 04 0120588 = 05 132 82 54 38 07120588 = 08 177 158 102 65 27Note 119896119886119879 represents the absorption rate for the generic drug and 119896119886119877 represents the absorption rate for the reference drug 119899 represents the sample size of eachgroup nabla represents a constant difference between the concentrations of two drugs on the logarithm scale 120588 is the correlation coefficient between measuresover time from the same subject its influence on the estimation is expected to be bigger when its value is larger

120588 for 119894 = 119895 The curves above differ only by a constant nabla onthe logarithm scale In the second set of simulations the twoconcentration curves for the test drug and the reference drugwere set to have different absorption rates (ie 119896119886119879 and 119896119886119877)instead of nabla = 0

By varying the values for 120588nabla and 119896119886119879119896119886119877 and the samplesize we simulated 100 different scenarios in total The empir-ical coverage of the 90 CIs for each scenario was calculatedvia simulating 1000 bootstrap replicates The simulationresults are presented in Table 2 from which we found thata sample size of 24 per group is large enough to provide anadequate statistical power for the proposed testing procedureFurthermore the empirical coverage of the 90 CIs whentwo curves are equivalent is approximately 95or above sug-gesting the 80125 rule is applicable for the proposedmetric

34 Our Extension to the Superiority Test between Curves Itis worth pointing out that by virtue of being coupled with apermutation test [19 20] to calculate an overall 119901 value LPScan also be used to evaluate if two curves are different In thiscase the hypothesis tested is

1198670 119889 (119892119860 (119909) 119892119861 (119909)) = 0versus 1198671 119889 (119892119860 (119909) 119892119861 (119909)) = 0 (12)

with 119889 representing the difference between the two curves Asshown on the simulation studies (in Supplementary Materialavailable online at httpdxdoiorg10115520164680642)the permutation-based test provides a valid means for testingthe difference of two curves achieving the desired type I errorrate We applied this procedure to longitudinal data collectedby Orange et al [21] and showed that the results obtainedwere consistent with those obtained by using a generalizedestimating equation model [22 23] Therefore the proposedprocedure to test the difference between curves using LPS canbe employed in a wide range of applications in longitudinaldata analysis

4 Conclusions

In this study we presented a framework which directlycompares the concentration-versus-time curves constructedusing LPS to test BE This approach avoids multiple testingby evaluating the equivalence between curves with one singlemetric instead of multiple PK parameters Furthermoresince LPS can be viewed as an extension of linear weightedregression and can be easily implemented in any standardstatistical software our proposed procedure can be easilyutilized in other applications

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

tolerable amount called the equivalence margin Δ whereasthe alternative hypothesis states that such difference is lessthan the equivalence margin Δ

1198670 10038161003816100381610038161205831003816100381610038161003816 ge Δversus 1198671 10038161003816100381610038161205831003816100381610038161003816 lt Δ (1)

In aBE test someparameters derived from the concentra-tion-versus-time curves are evaluated Those parameterstraditionally include the area under plasma concentration-versus-time curves (AUC1199050) peak plasma concentrations(119862max) and its corresponding time (119879max) Among themAUC is the most acceptedmeasure of absorption rate119862max isalso of importance because for some drugs a certain level ofconcentration needs to be reached to guarantee the desiredtherapeutic effect 119879max is a relevant measure for drugs suchas antibiotics that must reach the peak concentration quicklybut it may not be an appropriate measure for drugs requiringmultiple dosage before a therapeutic effect is observed Sincenone of these three parameters are universally superior to theothers a BE test usually considers them together For exam-ple under FDA regulations BE can be claimed only when the90 confidence intervals (CIs) for the ratio of119862max AUC(0minus119905)and AUC(0minusinfin) of the tested (eg generic) to the referencedrug (eg brand name) are within 80 to 125 [4] Thisis referred to as 80125 rule corresponding to a plusmn0223rule on the logarithmic scale (of the 119879119877 ratio) Hence thevalue of Δ is usually set to 0223 Of note the 90 CIs of theBE-endpoint represent a 005 significance level on the equiv-alence test since the hypothesis-testing problem in an equiva-lence test is divided into two one-sided hypothesis tests [25 6] one where the null hypothesis states that the differencebetween two agents is less than minusΔ whereas the other oneassumes such difference is larger than Δ

11986701 120583 lt minusΔversus 1198671198861 120583 ge minusΔ

11986702 120583 gt Δversus 1198671198862 120583 le Δ

(2)

Although the BE parameters can be easily obtained fromeither one of the concentration-versus-time curves usinga suitable pharmacokinetic (PK) model such as a one-compartment model or a nonparametric method [7] thereare many parameters to be tested which inflates the typeI error rate requiring the adjustment for multiple compar-isons Because these parameters may be highly correlatedsuch an adjustment is challenging

On the other hand some researchers had pointed outthat the requirement of all confidence intervals falling withinthe equivalence bounds might lead to a conservative resultdepending on the correlations among the PK parameters andthe study power [8 9] To address this simultaneous testing ofall PK parameters had been explored and developed see forexample the semiparametric Bayesian approach proposed byGhosh and Gonen [10] However such multivariate methodsare less popular than the univariate approach of testing PK

parameters partially because of the modeling complexityassociated with multivariate methods as compared with theirunivariate counterparts

More importantly when the confidence intervals of119862max119879max and AUC between two drugs all fall within the equiva-lence boundary (such that BE is concluded) it does not implythat the drugs are clinically equivalent since the overall shapesof the concentration-versus-time curves may in fact differ[11] A falsely determined BE between two drugs when theyare not clinically equivalent may be very harmful to the pub-lic To alleviate these limitations and drawbacks we proposeto make a direct comparison of the concentration-versus-time curves that accounts for the differences between the pro-file shapes for BE testingThemain objective of this paper is topresent a strategy to test BE with the aid of local polynomialsmoother (LPS) which is used to construct concentration-versus-time curves Then a summary statistic whose stand-ard error is estimated by bootstrapping is definedThis allowsthe calculation of CIs upon which decisions over equivalencebetween such two curves are made

LPS is a flexible nonparametric regression method tomodel curves or surfaces Specifically the fitted regressionfunction at a covariate value 119909 is based only on observationswithin a prespecified neighborhood of 119909 LPS can be tracedback to the late 19th century in actuarial sciences where itwas used to estimate the gradation of mortality rates and intime series modeling [12] In contrast another popular non-parametric regression method locally weighted scatterplotsmoothing (LOWESS) obtained smooth fitted values by aweighted linear least squares regression over the prespecifiedspans In this study however we do not emphasize theirdifferences and instead treat them as synonymous

With advances in computing the potential applicationsof LPS continue to expand For example it has been used fordye normalization of two-color microarray experiments [13]Recently Anders andHuber [14] used LPS in Rrsquos DESeq pack-age to estimate the mean-variance relationship in a negativebinomial model a technique now commonly used to modelRNA-Seq data The popularity of LPS in statistical applica-tions may be due to its several advantages including satisfac-tory boundary behavior and straightforward interpretabilityof the nonlinear relationships [15 16]

2 Methods and Materials

21 Experimental Data The study included 24 beagle dogsweighting 9ndash11 kg and those dogs were randomly assigned toreceive either single ormultiple (every four weeks for 3 times)administrations of either 14mgkg Sandostatin or a genericdrug developed by GenSci There were 6 dogs including3 females and 3 males per treatment-dosage group Bloodsamples for PK were collected into tubes containing heparinsodium and centrifuged at room temperature with 3500 rpmfor 20min to obtain plasma samples This animal studywas conducted in accordance with the rules and regulationsof Chinese Pharmacopoeia 2000 EditionVersion 2 andapproved by the Committee on the Ethics of Animal Exper-iments of Jilin University The plasma concentrations were

Computational and Mathematical Methods in Medicine 3

determined using an Agilent 1100 liquid chromatography-tandemmass spectrometry In the single-dose regimen bloodsamples were collected at the initial time corresponding tohour 0 and at the hours of 025 05 1 15 2 4 8 24 48 96 144192 240 336 432 528 624 720 816 912 1008 1104 and 1200

Based on [7] PK parameters in this study were calculatedas follows For each dog themaximumplasma concentration(119862max) and its corresponding time (119879max) were determined byvisual inspection of the profilesThe apparent terminal elimi-nation rate constant (120582) was calculated by linear regression ofthe natural logarithms of the terminal plasma concentrationsThe terminal half-life (11990512) was derived as 0693120582 The areaunder the curve (AUC) to the last measured point (AUC1199050)was calculated using the trapezoidal rule The AUC for theplasma concentration-versus-time function from 0 hours toinfinity (AUC0minusinfin) was calculated as the sum of AUC1199050 and119862119905120582 while 119862119905 was the last quantifiable concentration22 The LPS Model Let (1198831 1198841) (119883119899 119884119899) be covariate-response pairs for 119899 observations To simplify the notationsuppose there is only one covariate Then the response 119884119894 isrelated to the covariate119883119894 through the following model

119884119894 = 119892 (119883119894) + 120576119894 119894 = 1 119899 (3)

where the error term 120576119894 is assumed to be independentlydistributed with zero mean and variance equal to 1205902 Thegoal of a local polynomial smoother is to estimate the smoothfunction 119892(119909) = 119864(119884 | 119883 = 119909) using an approximationprovided by a local polynomial of low order in the neigh-borhood of the point 119909 Usually the model includes onlyfirst- and second-degree polynomials that is either locallylinear or locally quadratic This is because any function canbe well approximated in a small neighborhood by low-orderpolynomials and the inclusion of higher order terms oftendoes not improve the model fit dramatically and even maylead to overfitting instead Specifically for 119909 minus ℎ le 119909119894 le 119909 + ℎ

119892 (119909119894) asymp 119901sum119895=0

120573119895 (119909119894 minus 119909)119895 (4)

The abovementioned approximation is fitted by localweighted least squares where the estimated coefficients arethose that minimize the following function

119899sum119894=1

119882((119909119894 minus 119909)ℎ )(119910119894 minus 119901sum119895=0

120573119895 (119909119894 minus 119909)119895)2

(5)

where the weighted function 119882 gives greater weight to 119909119894in the vicinity of 119909 and is usually specified as a symmetricbounded function such as a normal kernel function

There are two commonly used approaches for the selec-tion of bandwidth ℎ in the equation above [12] One choice issimply to make the bandwidth a constant which is adequateif the regression function behaves smoothly The other one isto allow the bandwidth to change as a function of 119909 In ourapplications we chose the latter by using a nearest-neighborbandwidth selection method [17] Briefly in the nearest-neighbor bandwidth selection method a new parameter 120572

was included which was multiplied by the sample size 119899and rounded up to an integer 119896 Then the bandwidth ℎ(119909)was defined as the distance from 119909 to the 119896th closest 119909119894In this study measurements were taken at nonuniformlyspaced time points with more frequent measurements at thebeginning where a larger variation of the PK dynamic wasexpected among subjects Under this condition the nearest-neighbor bandwidth selection approach will provide betterfit than the constant bandwidth The nearest-neighbor band-width parameter 120572was determined by a leave-one-out (LOO)cross-validation due to the small sample size Specificallyfor a grid of possible 120572 values cross-validated mean squareerror (MSE) was calculated and the 120572 corresponding to theminimumMSE was selected

23 Statistical Language The statistical analysis was carriedout in theR language version 215 (httpwwwr-projectorg)The R-code is available upon request

3 Results and Discussion

31 Testing the Equivalence of Two Curves by CombiningLPS and Bootstrapping The Proposed Procedure Here wepropose to evaluate bioequivalence between drugs by directlycomparing the plasma concentration curves PC(119905) Supposewe have two treatments where 1198991 subjects were randomlyassigned to receive the test drug119879 and 1198992 to the reference drug119877 We proposed to reframe the hypothesis represented in (1)as

1198670 119889 (119892119879 (119905) 119892119877 (119905)) gt Δ (6)

versus 1198671 119889 (119892119879 (119905) 119892119877 (119905)) le Δ (7)

with 119889 representing the difference between the PK curves for119879 and 119877 drugs For the study subjects the plasma concen-trations of these drugs were measured for those at a grid oftime points Using LPS (details in Section 2) concentration-versus-time curves for both treatments can be estimated as119892119877(119909) and 119892119879(119909) To evaluate if the differences between twocurves are within the tolerable range we define the followingestimator of the average difference between the two curves ata grid of prespecified time points for example the time pointswhere the concentrations were taken in this study

ln () = sum119870119896=1 10038161003816100381610038161003816ln (119879119905119896119877119905119896)10038161003816100381610038161003816119870 (8)

where 119879119905119896 and 119877119905119896 are the fitted values of plasma concentra-tion using LPS at time 119905119896 for the test drug 119879 and the referencedrug 119877 respectively Here 119896 = 1 2 119870 and 119870 is thenumber of time points at which the difference betweentwo curves is evaluated Note that ln(119903) is a measure of thedifference between the curves and when the two curves areidentical and they should be identical at all time points andthus the value of ln(119903) should be 0

The standard error for ln() can be estimated using boot-strapping [18] For each bootstrapped replicate LPS curve

4 Computational and Mathematical Methods in Medicine

0 200 400 600 800 1000 1200Hour

GenSciSandostatin

Con

centratio

n

4

3

2

1

0

minus1

Figure 1 The fitted local polynomial smoother (LPS) curvesfor both GenSci and Sandostatin Black triangle the observedconcentrations for dogs using Sandostatin red circle the observedconcentrations for dogs using GenSci

was fitted for 119879 and 119877 and the difference between them iscalculated using (8) The 90 CIs of 119903 can be obtained as

(119890ln()minus1645times119904119890(ln()) 119890ln()+1645timesse(ln())) (9)

If the 90 CIs is within 08sim125 then the equivalencebetween two curves is claimed

32 Case Study Octreotide is an octapeptide that mimicsnatural somatostatin pharmacologically even though it is amore potent inhibitor of growth hormone glucagon andinsulin than the natural hormone Brand name Sandostatin(Novartis Pharmaceuticals) of octreotide had been approvedfor the treatment of several diseases such as acromegaly andgigantism To evaluate the BE between a generic octreotidedeveloped by GenSci Pharmaceuticals (Changchun China)and Brand name Sandostatin a pilot study using 24 beagledogs was conductedThe generic drug is referred to as GenScihereafter Using this study as an example we showcase how toincorporate LPS seamlessly with other statistical methods totest BE Once concentration-versus-time curves are obtainedvia LPS testing for BE will be akin to testing if the two LPS-derived curves are the same

Firstly we conducted an equivalence test using the proce-dure described in Methods The fitted plasma concentrationPC(119905) curves are shown in Figure 1 Interestingly the curvesestimated by LPS suggested some PK differences betweenthe two drugs Specifically we observed a surge in plasmaconcentration and then a steep drop in the GenSci group Inaddition the GenSci group entered the plateau stage fasterand remained in this stage longer than the Sandostatin groupdropping down at a sharper scope Based on these obser-vations we suspected that GenSci might have a better PKbehavior than Sandostatin

The 90 CIs for the proposed metric of GenSci againstSandostatin were calculated as 11902 sim16699 With itsupper bound above 125 the equivalence between the two

Table 1 The 90 CIs of PK parameters

90 CI Single dose Combined data119873 = 6 119873 = 12119862max (7223 16071) (8293 16088)11987903 ngnL (994 38263) (951 14460)AUC0ndash1200 (7198 17608) mdashAUC0ndash720 (6174 16985) (9144 20850)

drugs cannot be established based on this method As acomparison we also estimated the 90 CI of the ratio ofthe traditionally considered PK parameters (ie (AUC1199050)119862max and 119879max) based on the FDA guidelines [4] The resultssummarized in Table 1 show that all 90 CIs of the ratioof (AUC1199050) 119862max and 119879max of GenSci against Sandostatinwere outside 80 to 125 the FDA-sanctioned BE criteriaIn summary BE cannot be established using either theFDA-criteria or our proposed procedure Nevertheless theproposed procedure has the advantage of testing only a singlestatistic and hence avoiding multiple hypotheses issues

The sample size of 6 dogs per group raisesmajor concernsin light of the failure to establish BE between GenSci andSandostatin To explore the impact of the sample size inthis pilot study we combined the data from this single-dosecohort with additional data from a multiple-dose cohort butincluding only data collected in the first 720 hours where noboosts were given and the experimental methodsconditionswere exactly the same as in the single-dose cohorts Interestedreaders may see Section 2 for more information on howthe experiments were conducted Analysis on the combinedcohorts including 12 dogs per group led to similar resultsGiven that this application represents a pilot study as pre-liminary work for larger studies no confirmative conclusionswere drawn here By presenting this study our intent isto illustrate the LPS-based BE test as an alternative to theclassical univariate tests on PK parameters

33 Simulation Study We conducted a simulation study tocharacterize our BE-testing procedure Here we used the PKmodel presented in [11] where the concentration function isrepresented by

119888 (119905) prop 119870119886119870119886 minus 119870119890 (119890minus119896119890119905 minus 119890minus119896119886119905) (10)

where 119896119886 and 119896119890 are the absorption and the eliminationrate constants The concentration-versus-time curves for thereference drug and test drug were simulated as

log (119884119877) = log (119888 (119905)) + 120576log (119884119879) = log (119888 (119905)) + nabla + 120576

120576 sim MVN (0119901 Σ119901times119901) (11)

where the correlation between two measures on the log-arithm scale from the same subject 119896 (119896 = 1 2 2119899)was specified as a positive constant that is corr(120576119894119896 120576119895119896) =

Computational and Mathematical Methods in Medicine 5

Table 2 The exact coverage of the 90 CIs for ln(119903) (the proposed measure of the difference between two curves) using simulations

119899 = 12 119899 = 18 119899 = 24 119899 = 30 119899 = 50nabla = 0 119896119886119879119896119886119877 = 1 (two curves are identical)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 999 100 100 100 100120588 = 08 978 996 100 100 100nabla = 0223 119896119886119879119896119886119877 = 1 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 987 997 100 100 100120588 = 08 955 972 991 997 100nabla = 1 119896119886119879119896119886119877 = 1 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 33 07 03 0 0120588 = 05 116 66 37 32 07120588 = 08 18 112 79 58 26119896119886119879119896119886119877 = 125 nabla = 0 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 999 100 100 100 100120588 = 05 977 991 997 100 100120588 = 08 95 975 988 993 100119896119886119879119896119886119877 = 3 nabla = 0 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 5 15 06 04 0120588 = 05 132 82 54 38 07120588 = 08 177 158 102 65 27Note 119896119886119879 represents the absorption rate for the generic drug and 119896119886119877 represents the absorption rate for the reference drug 119899 represents the sample size of eachgroup nabla represents a constant difference between the concentrations of two drugs on the logarithm scale 120588 is the correlation coefficient between measuresover time from the same subject its influence on the estimation is expected to be bigger when its value is larger

120588 for 119894 = 119895 The curves above differ only by a constant nabla onthe logarithm scale In the second set of simulations the twoconcentration curves for the test drug and the reference drugwere set to have different absorption rates (ie 119896119886119879 and 119896119886119877)instead of nabla = 0

By varying the values for 120588nabla and 119896119886119879119896119886119877 and the samplesize we simulated 100 different scenarios in total The empir-ical coverage of the 90 CIs for each scenario was calculatedvia simulating 1000 bootstrap replicates The simulationresults are presented in Table 2 from which we found thata sample size of 24 per group is large enough to provide anadequate statistical power for the proposed testing procedureFurthermore the empirical coverage of the 90 CIs whentwo curves are equivalent is approximately 95or above sug-gesting the 80125 rule is applicable for the proposedmetric

34 Our Extension to the Superiority Test between Curves Itis worth pointing out that by virtue of being coupled with apermutation test [19 20] to calculate an overall 119901 value LPScan also be used to evaluate if two curves are different In thiscase the hypothesis tested is

1198670 119889 (119892119860 (119909) 119892119861 (119909)) = 0versus 1198671 119889 (119892119860 (119909) 119892119861 (119909)) = 0 (12)

with 119889 representing the difference between the two curves Asshown on the simulation studies (in Supplementary Materialavailable online at httpdxdoiorg10115520164680642)the permutation-based test provides a valid means for testingthe difference of two curves achieving the desired type I errorrate We applied this procedure to longitudinal data collectedby Orange et al [21] and showed that the results obtainedwere consistent with those obtained by using a generalizedestimating equation model [22 23] Therefore the proposedprocedure to test the difference between curves using LPS canbe employed in a wide range of applications in longitudinaldata analysis

4 Conclusions

In this study we presented a framework which directlycompares the concentration-versus-time curves constructedusing LPS to test BE This approach avoids multiple testingby evaluating the equivalence between curves with one singlemetric instead of multiple PK parameters Furthermoresince LPS can be viewed as an extension of linear weightedregression and can be easily implemented in any standardstatistical software our proposed procedure can be easilyutilized in other applications

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 3

determined using an Agilent 1100 liquid chromatography-tandemmass spectrometry In the single-dose regimen bloodsamples were collected at the initial time corresponding tohour 0 and at the hours of 025 05 1 15 2 4 8 24 48 96 144192 240 336 432 528 624 720 816 912 1008 1104 and 1200

Based on [7] PK parameters in this study were calculatedas follows For each dog themaximumplasma concentration(119862max) and its corresponding time (119879max) were determined byvisual inspection of the profilesThe apparent terminal elimi-nation rate constant (120582) was calculated by linear regression ofthe natural logarithms of the terminal plasma concentrationsThe terminal half-life (11990512) was derived as 0693120582 The areaunder the curve (AUC) to the last measured point (AUC1199050)was calculated using the trapezoidal rule The AUC for theplasma concentration-versus-time function from 0 hours toinfinity (AUC0minusinfin) was calculated as the sum of AUC1199050 and119862119905120582 while 119862119905 was the last quantifiable concentration22 The LPS Model Let (1198831 1198841) (119883119899 119884119899) be covariate-response pairs for 119899 observations To simplify the notationsuppose there is only one covariate Then the response 119884119894 isrelated to the covariate119883119894 through the following model

119884119894 = 119892 (119883119894) + 120576119894 119894 = 1 119899 (3)

where the error term 120576119894 is assumed to be independentlydistributed with zero mean and variance equal to 1205902 Thegoal of a local polynomial smoother is to estimate the smoothfunction 119892(119909) = 119864(119884 | 119883 = 119909) using an approximationprovided by a local polynomial of low order in the neigh-borhood of the point 119909 Usually the model includes onlyfirst- and second-degree polynomials that is either locallylinear or locally quadratic This is because any function canbe well approximated in a small neighborhood by low-orderpolynomials and the inclusion of higher order terms oftendoes not improve the model fit dramatically and even maylead to overfitting instead Specifically for 119909 minus ℎ le 119909119894 le 119909 + ℎ

119892 (119909119894) asymp 119901sum119895=0

120573119895 (119909119894 minus 119909)119895 (4)

The abovementioned approximation is fitted by localweighted least squares where the estimated coefficients arethose that minimize the following function

119899sum119894=1

119882((119909119894 minus 119909)ℎ )(119910119894 minus 119901sum119895=0

120573119895 (119909119894 minus 119909)119895)2

(5)

where the weighted function 119882 gives greater weight to 119909119894in the vicinity of 119909 and is usually specified as a symmetricbounded function such as a normal kernel function

There are two commonly used approaches for the selec-tion of bandwidth ℎ in the equation above [12] One choice issimply to make the bandwidth a constant which is adequateif the regression function behaves smoothly The other one isto allow the bandwidth to change as a function of 119909 In ourapplications we chose the latter by using a nearest-neighborbandwidth selection method [17] Briefly in the nearest-neighbor bandwidth selection method a new parameter 120572

was included which was multiplied by the sample size 119899and rounded up to an integer 119896 Then the bandwidth ℎ(119909)was defined as the distance from 119909 to the 119896th closest 119909119894In this study measurements were taken at nonuniformlyspaced time points with more frequent measurements at thebeginning where a larger variation of the PK dynamic wasexpected among subjects Under this condition the nearest-neighbor bandwidth selection approach will provide betterfit than the constant bandwidth The nearest-neighbor band-width parameter 120572was determined by a leave-one-out (LOO)cross-validation due to the small sample size Specificallyfor a grid of possible 120572 values cross-validated mean squareerror (MSE) was calculated and the 120572 corresponding to theminimumMSE was selected

23 Statistical Language The statistical analysis was carriedout in theR language version 215 (httpwwwr-projectorg)The R-code is available upon request

3 Results and Discussion

31 Testing the Equivalence of Two Curves by CombiningLPS and Bootstrapping The Proposed Procedure Here wepropose to evaluate bioequivalence between drugs by directlycomparing the plasma concentration curves PC(119905) Supposewe have two treatments where 1198991 subjects were randomlyassigned to receive the test drug119879 and 1198992 to the reference drug119877 We proposed to reframe the hypothesis represented in (1)as

1198670 119889 (119892119879 (119905) 119892119877 (119905)) gt Δ (6)

versus 1198671 119889 (119892119879 (119905) 119892119877 (119905)) le Δ (7)

with 119889 representing the difference between the PK curves for119879 and 119877 drugs For the study subjects the plasma concen-trations of these drugs were measured for those at a grid oftime points Using LPS (details in Section 2) concentration-versus-time curves for both treatments can be estimated as119892119877(119909) and 119892119879(119909) To evaluate if the differences between twocurves are within the tolerable range we define the followingestimator of the average difference between the two curves ata grid of prespecified time points for example the time pointswhere the concentrations were taken in this study

ln () = sum119870119896=1 10038161003816100381610038161003816ln (119879119905119896119877119905119896)10038161003816100381610038161003816119870 (8)

where 119879119905119896 and 119877119905119896 are the fitted values of plasma concentra-tion using LPS at time 119905119896 for the test drug 119879 and the referencedrug 119877 respectively Here 119896 = 1 2 119870 and 119870 is thenumber of time points at which the difference betweentwo curves is evaluated Note that ln(119903) is a measure of thedifference between the curves and when the two curves areidentical and they should be identical at all time points andthus the value of ln(119903) should be 0

The standard error for ln() can be estimated using boot-strapping [18] For each bootstrapped replicate LPS curve

4 Computational and Mathematical Methods in Medicine

0 200 400 600 800 1000 1200Hour

GenSciSandostatin

Con

centratio

n

4

3

2

1

0

minus1

Figure 1 The fitted local polynomial smoother (LPS) curvesfor both GenSci and Sandostatin Black triangle the observedconcentrations for dogs using Sandostatin red circle the observedconcentrations for dogs using GenSci

was fitted for 119879 and 119877 and the difference between them iscalculated using (8) The 90 CIs of 119903 can be obtained as

(119890ln()minus1645times119904119890(ln()) 119890ln()+1645timesse(ln())) (9)

If the 90 CIs is within 08sim125 then the equivalencebetween two curves is claimed

32 Case Study Octreotide is an octapeptide that mimicsnatural somatostatin pharmacologically even though it is amore potent inhibitor of growth hormone glucagon andinsulin than the natural hormone Brand name Sandostatin(Novartis Pharmaceuticals) of octreotide had been approvedfor the treatment of several diseases such as acromegaly andgigantism To evaluate the BE between a generic octreotidedeveloped by GenSci Pharmaceuticals (Changchun China)and Brand name Sandostatin a pilot study using 24 beagledogs was conductedThe generic drug is referred to as GenScihereafter Using this study as an example we showcase how toincorporate LPS seamlessly with other statistical methods totest BE Once concentration-versus-time curves are obtainedvia LPS testing for BE will be akin to testing if the two LPS-derived curves are the same

Firstly we conducted an equivalence test using the proce-dure described in Methods The fitted plasma concentrationPC(119905) curves are shown in Figure 1 Interestingly the curvesestimated by LPS suggested some PK differences betweenthe two drugs Specifically we observed a surge in plasmaconcentration and then a steep drop in the GenSci group Inaddition the GenSci group entered the plateau stage fasterand remained in this stage longer than the Sandostatin groupdropping down at a sharper scope Based on these obser-vations we suspected that GenSci might have a better PKbehavior than Sandostatin

The 90 CIs for the proposed metric of GenSci againstSandostatin were calculated as 11902 sim16699 With itsupper bound above 125 the equivalence between the two

Table 1 The 90 CIs of PK parameters

90 CI Single dose Combined data119873 = 6 119873 = 12119862max (7223 16071) (8293 16088)11987903 ngnL (994 38263) (951 14460)AUC0ndash1200 (7198 17608) mdashAUC0ndash720 (6174 16985) (9144 20850)

drugs cannot be established based on this method As acomparison we also estimated the 90 CI of the ratio ofthe traditionally considered PK parameters (ie (AUC1199050)119862max and 119879max) based on the FDA guidelines [4] The resultssummarized in Table 1 show that all 90 CIs of the ratioof (AUC1199050) 119862max and 119879max of GenSci against Sandostatinwere outside 80 to 125 the FDA-sanctioned BE criteriaIn summary BE cannot be established using either theFDA-criteria or our proposed procedure Nevertheless theproposed procedure has the advantage of testing only a singlestatistic and hence avoiding multiple hypotheses issues

The sample size of 6 dogs per group raisesmajor concernsin light of the failure to establish BE between GenSci andSandostatin To explore the impact of the sample size inthis pilot study we combined the data from this single-dosecohort with additional data from a multiple-dose cohort butincluding only data collected in the first 720 hours where noboosts were given and the experimental methodsconditionswere exactly the same as in the single-dose cohorts Interestedreaders may see Section 2 for more information on howthe experiments were conducted Analysis on the combinedcohorts including 12 dogs per group led to similar resultsGiven that this application represents a pilot study as pre-liminary work for larger studies no confirmative conclusionswere drawn here By presenting this study our intent isto illustrate the LPS-based BE test as an alternative to theclassical univariate tests on PK parameters

33 Simulation Study We conducted a simulation study tocharacterize our BE-testing procedure Here we used the PKmodel presented in [11] where the concentration function isrepresented by

119888 (119905) prop 119870119886119870119886 minus 119870119890 (119890minus119896119890119905 minus 119890minus119896119886119905) (10)

where 119896119886 and 119896119890 are the absorption and the eliminationrate constants The concentration-versus-time curves for thereference drug and test drug were simulated as

log (119884119877) = log (119888 (119905)) + 120576log (119884119879) = log (119888 (119905)) + nabla + 120576

120576 sim MVN (0119901 Σ119901times119901) (11)

where the correlation between two measures on the log-arithm scale from the same subject 119896 (119896 = 1 2 2119899)was specified as a positive constant that is corr(120576119894119896 120576119895119896) =

Computational and Mathematical Methods in Medicine 5

Table 2 The exact coverage of the 90 CIs for ln(119903) (the proposed measure of the difference between two curves) using simulations

119899 = 12 119899 = 18 119899 = 24 119899 = 30 119899 = 50nabla = 0 119896119886119879119896119886119877 = 1 (two curves are identical)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 999 100 100 100 100120588 = 08 978 996 100 100 100nabla = 0223 119896119886119879119896119886119877 = 1 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 987 997 100 100 100120588 = 08 955 972 991 997 100nabla = 1 119896119886119879119896119886119877 = 1 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 33 07 03 0 0120588 = 05 116 66 37 32 07120588 = 08 18 112 79 58 26119896119886119879119896119886119877 = 125 nabla = 0 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 999 100 100 100 100120588 = 05 977 991 997 100 100120588 = 08 95 975 988 993 100119896119886119879119896119886119877 = 3 nabla = 0 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 5 15 06 04 0120588 = 05 132 82 54 38 07120588 = 08 177 158 102 65 27Note 119896119886119879 represents the absorption rate for the generic drug and 119896119886119877 represents the absorption rate for the reference drug 119899 represents the sample size of eachgroup nabla represents a constant difference between the concentrations of two drugs on the logarithm scale 120588 is the correlation coefficient between measuresover time from the same subject its influence on the estimation is expected to be bigger when its value is larger

120588 for 119894 = 119895 The curves above differ only by a constant nabla onthe logarithm scale In the second set of simulations the twoconcentration curves for the test drug and the reference drugwere set to have different absorption rates (ie 119896119886119879 and 119896119886119877)instead of nabla = 0

By varying the values for 120588nabla and 119896119886119879119896119886119877 and the samplesize we simulated 100 different scenarios in total The empir-ical coverage of the 90 CIs for each scenario was calculatedvia simulating 1000 bootstrap replicates The simulationresults are presented in Table 2 from which we found thata sample size of 24 per group is large enough to provide anadequate statistical power for the proposed testing procedureFurthermore the empirical coverage of the 90 CIs whentwo curves are equivalent is approximately 95or above sug-gesting the 80125 rule is applicable for the proposedmetric

34 Our Extension to the Superiority Test between Curves Itis worth pointing out that by virtue of being coupled with apermutation test [19 20] to calculate an overall 119901 value LPScan also be used to evaluate if two curves are different In thiscase the hypothesis tested is

1198670 119889 (119892119860 (119909) 119892119861 (119909)) = 0versus 1198671 119889 (119892119860 (119909) 119892119861 (119909)) = 0 (12)

with 119889 representing the difference between the two curves Asshown on the simulation studies (in Supplementary Materialavailable online at httpdxdoiorg10115520164680642)the permutation-based test provides a valid means for testingthe difference of two curves achieving the desired type I errorrate We applied this procedure to longitudinal data collectedby Orange et al [21] and showed that the results obtainedwere consistent with those obtained by using a generalizedestimating equation model [22 23] Therefore the proposedprocedure to test the difference between curves using LPS canbe employed in a wide range of applications in longitudinaldata analysis

4 Conclusions

In this study we presented a framework which directlycompares the concentration-versus-time curves constructedusing LPS to test BE This approach avoids multiple testingby evaluating the equivalence between curves with one singlemetric instead of multiple PK parameters Furthermoresince LPS can be viewed as an extension of linear weightedregression and can be easily implemented in any standardstatistical software our proposed procedure can be easilyutilized in other applications

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

4 Computational and Mathematical Methods in Medicine

0 200 400 600 800 1000 1200Hour

GenSciSandostatin

Con

centratio

n

4

3

2

1

0

minus1

Figure 1 The fitted local polynomial smoother (LPS) curvesfor both GenSci and Sandostatin Black triangle the observedconcentrations for dogs using Sandostatin red circle the observedconcentrations for dogs using GenSci

was fitted for 119879 and 119877 and the difference between them iscalculated using (8) The 90 CIs of 119903 can be obtained as

(119890ln()minus1645times119904119890(ln()) 119890ln()+1645timesse(ln())) (9)

If the 90 CIs is within 08sim125 then the equivalencebetween two curves is claimed

32 Case Study Octreotide is an octapeptide that mimicsnatural somatostatin pharmacologically even though it is amore potent inhibitor of growth hormone glucagon andinsulin than the natural hormone Brand name Sandostatin(Novartis Pharmaceuticals) of octreotide had been approvedfor the treatment of several diseases such as acromegaly andgigantism To evaluate the BE between a generic octreotidedeveloped by GenSci Pharmaceuticals (Changchun China)and Brand name Sandostatin a pilot study using 24 beagledogs was conductedThe generic drug is referred to as GenScihereafter Using this study as an example we showcase how toincorporate LPS seamlessly with other statistical methods totest BE Once concentration-versus-time curves are obtainedvia LPS testing for BE will be akin to testing if the two LPS-derived curves are the same

Firstly we conducted an equivalence test using the proce-dure described in Methods The fitted plasma concentrationPC(119905) curves are shown in Figure 1 Interestingly the curvesestimated by LPS suggested some PK differences betweenthe two drugs Specifically we observed a surge in plasmaconcentration and then a steep drop in the GenSci group Inaddition the GenSci group entered the plateau stage fasterand remained in this stage longer than the Sandostatin groupdropping down at a sharper scope Based on these obser-vations we suspected that GenSci might have a better PKbehavior than Sandostatin

The 90 CIs for the proposed metric of GenSci againstSandostatin were calculated as 11902 sim16699 With itsupper bound above 125 the equivalence between the two

Table 1 The 90 CIs of PK parameters

90 CI Single dose Combined data119873 = 6 119873 = 12119862max (7223 16071) (8293 16088)11987903 ngnL (994 38263) (951 14460)AUC0ndash1200 (7198 17608) mdashAUC0ndash720 (6174 16985) (9144 20850)

drugs cannot be established based on this method As acomparison we also estimated the 90 CI of the ratio ofthe traditionally considered PK parameters (ie (AUC1199050)119862max and 119879max) based on the FDA guidelines [4] The resultssummarized in Table 1 show that all 90 CIs of the ratioof (AUC1199050) 119862max and 119879max of GenSci against Sandostatinwere outside 80 to 125 the FDA-sanctioned BE criteriaIn summary BE cannot be established using either theFDA-criteria or our proposed procedure Nevertheless theproposed procedure has the advantage of testing only a singlestatistic and hence avoiding multiple hypotheses issues

The sample size of 6 dogs per group raisesmajor concernsin light of the failure to establish BE between GenSci andSandostatin To explore the impact of the sample size inthis pilot study we combined the data from this single-dosecohort with additional data from a multiple-dose cohort butincluding only data collected in the first 720 hours where noboosts were given and the experimental methodsconditionswere exactly the same as in the single-dose cohorts Interestedreaders may see Section 2 for more information on howthe experiments were conducted Analysis on the combinedcohorts including 12 dogs per group led to similar resultsGiven that this application represents a pilot study as pre-liminary work for larger studies no confirmative conclusionswere drawn here By presenting this study our intent isto illustrate the LPS-based BE test as an alternative to theclassical univariate tests on PK parameters

33 Simulation Study We conducted a simulation study tocharacterize our BE-testing procedure Here we used the PKmodel presented in [11] where the concentration function isrepresented by

119888 (119905) prop 119870119886119870119886 minus 119870119890 (119890minus119896119890119905 minus 119890minus119896119886119905) (10)

where 119896119886 and 119896119890 are the absorption and the eliminationrate constants The concentration-versus-time curves for thereference drug and test drug were simulated as

log (119884119877) = log (119888 (119905)) + 120576log (119884119879) = log (119888 (119905)) + nabla + 120576

120576 sim MVN (0119901 Σ119901times119901) (11)

where the correlation between two measures on the log-arithm scale from the same subject 119896 (119896 = 1 2 2119899)was specified as a positive constant that is corr(120576119894119896 120576119895119896) =

Computational and Mathematical Methods in Medicine 5

Table 2 The exact coverage of the 90 CIs for ln(119903) (the proposed measure of the difference between two curves) using simulations

119899 = 12 119899 = 18 119899 = 24 119899 = 30 119899 = 50nabla = 0 119896119886119879119896119886119877 = 1 (two curves are identical)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 999 100 100 100 100120588 = 08 978 996 100 100 100nabla = 0223 119896119886119879119896119886119877 = 1 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 987 997 100 100 100120588 = 08 955 972 991 997 100nabla = 1 119896119886119879119896119886119877 = 1 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 33 07 03 0 0120588 = 05 116 66 37 32 07120588 = 08 18 112 79 58 26119896119886119879119896119886119877 = 125 nabla = 0 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 999 100 100 100 100120588 = 05 977 991 997 100 100120588 = 08 95 975 988 993 100119896119886119879119896119886119877 = 3 nabla = 0 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 5 15 06 04 0120588 = 05 132 82 54 38 07120588 = 08 177 158 102 65 27Note 119896119886119879 represents the absorption rate for the generic drug and 119896119886119877 represents the absorption rate for the reference drug 119899 represents the sample size of eachgroup nabla represents a constant difference between the concentrations of two drugs on the logarithm scale 120588 is the correlation coefficient between measuresover time from the same subject its influence on the estimation is expected to be bigger when its value is larger

120588 for 119894 = 119895 The curves above differ only by a constant nabla onthe logarithm scale In the second set of simulations the twoconcentration curves for the test drug and the reference drugwere set to have different absorption rates (ie 119896119886119879 and 119896119886119877)instead of nabla = 0

By varying the values for 120588nabla and 119896119886119879119896119886119877 and the samplesize we simulated 100 different scenarios in total The empir-ical coverage of the 90 CIs for each scenario was calculatedvia simulating 1000 bootstrap replicates The simulationresults are presented in Table 2 from which we found thata sample size of 24 per group is large enough to provide anadequate statistical power for the proposed testing procedureFurthermore the empirical coverage of the 90 CIs whentwo curves are equivalent is approximately 95or above sug-gesting the 80125 rule is applicable for the proposedmetric

34 Our Extension to the Superiority Test between Curves Itis worth pointing out that by virtue of being coupled with apermutation test [19 20] to calculate an overall 119901 value LPScan also be used to evaluate if two curves are different In thiscase the hypothesis tested is

1198670 119889 (119892119860 (119909) 119892119861 (119909)) = 0versus 1198671 119889 (119892119860 (119909) 119892119861 (119909)) = 0 (12)

with 119889 representing the difference between the two curves Asshown on the simulation studies (in Supplementary Materialavailable online at httpdxdoiorg10115520164680642)the permutation-based test provides a valid means for testingthe difference of two curves achieving the desired type I errorrate We applied this procedure to longitudinal data collectedby Orange et al [21] and showed that the results obtainedwere consistent with those obtained by using a generalizedestimating equation model [22 23] Therefore the proposedprocedure to test the difference between curves using LPS canbe employed in a wide range of applications in longitudinaldata analysis

4 Conclusions

In this study we presented a framework which directlycompares the concentration-versus-time curves constructedusing LPS to test BE This approach avoids multiple testingby evaluating the equivalence between curves with one singlemetric instead of multiple PK parameters Furthermoresince LPS can be viewed as an extension of linear weightedregression and can be easily implemented in any standardstatistical software our proposed procedure can be easilyutilized in other applications

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

Table 2 The exact coverage of the 90 CIs for ln(119903) (the proposed measure of the difference between two curves) using simulations

119899 = 12 119899 = 18 119899 = 24 119899 = 30 119899 = 50nabla = 0 119896119886119879119896119886119877 = 1 (two curves are identical)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 999 100 100 100 100120588 = 08 978 996 100 100 100nabla = 0223 119896119886119879119896119886119877 = 1 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 100 100 100 100 100120588 = 05 987 997 100 100 100120588 = 08 955 972 991 997 100nabla = 1 119896119886119879119896119886119877 = 1 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 33 07 03 0 0120588 = 05 116 66 37 32 07120588 = 08 18 112 79 58 26119896119886119879119896119886119877 = 125 nabla = 0 (the difference is within the equivalence bound)120588 = 0 100 100 100 100 100120588 = 02 999 100 100 100 100120588 = 05 977 991 997 100 100120588 = 08 95 975 988 993 100119896119886119879119896119886119877 = 3 nabla = 0 (the difference is beyond the equivalence bound)120588 = 0 0 0 0 0 0120588 = 02 5 15 06 04 0120588 = 05 132 82 54 38 07120588 = 08 177 158 102 65 27Note 119896119886119879 represents the absorption rate for the generic drug and 119896119886119877 represents the absorption rate for the reference drug 119899 represents the sample size of eachgroup nabla represents a constant difference between the concentrations of two drugs on the logarithm scale 120588 is the correlation coefficient between measuresover time from the same subject its influence on the estimation is expected to be bigger when its value is larger

120588 for 119894 = 119895 The curves above differ only by a constant nabla onthe logarithm scale In the second set of simulations the twoconcentration curves for the test drug and the reference drugwere set to have different absorption rates (ie 119896119886119879 and 119896119886119877)instead of nabla = 0

By varying the values for 120588nabla and 119896119886119879119896119886119877 and the samplesize we simulated 100 different scenarios in total The empir-ical coverage of the 90 CIs for each scenario was calculatedvia simulating 1000 bootstrap replicates The simulationresults are presented in Table 2 from which we found thata sample size of 24 per group is large enough to provide anadequate statistical power for the proposed testing procedureFurthermore the empirical coverage of the 90 CIs whentwo curves are equivalent is approximately 95or above sug-gesting the 80125 rule is applicable for the proposedmetric

34 Our Extension to the Superiority Test between Curves Itis worth pointing out that by virtue of being coupled with apermutation test [19 20] to calculate an overall 119901 value LPScan also be used to evaluate if two curves are different In thiscase the hypothesis tested is

1198670 119889 (119892119860 (119909) 119892119861 (119909)) = 0versus 1198671 119889 (119892119860 (119909) 119892119861 (119909)) = 0 (12)

with 119889 representing the difference between the two curves Asshown on the simulation studies (in Supplementary Materialavailable online at httpdxdoiorg10115520164680642)the permutation-based test provides a valid means for testingthe difference of two curves achieving the desired type I errorrate We applied this procedure to longitudinal data collectedby Orange et al [21] and showed that the results obtainedwere consistent with those obtained by using a generalizedestimating equation model [22 23] Therefore the proposedprocedure to test the difference between curves using LPS canbe employed in a wide range of applications in longitudinaldata analysis

4 Conclusions

In this study we presented a framework which directlycompares the concentration-versus-time curves constructedusing LPS to test BE This approach avoids multiple testingby evaluating the equivalence between curves with one singlemetric instead of multiple PK parameters Furthermoresince LPS can be viewed as an extension of linear weightedregression and can be easily implemented in any standardstatistical software our proposed procedure can be easilyutilized in other applications

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

6 Computational and Mathematical Methods in Medicine

van derMeersch et al [24] have argued that the objectivesand methodology of bioequivalence trials differ greatly fromthose of noninferiority and equivalence trials due to thedecision criteria and study design In our opinion theirclaims are not persuasive In terms of decision criteria it isdifficult to define an equivalence margin in a longitudinalsense because themeasures such as plasma concentrations arecollected over time It might represent one major reason whythe FDA and other regulatory agencies have defined BE byusing confidence limits for a grid of PK parameters insteadof a single hypothesis testing Regarding the study designvan der Meersch et al [24] deemed a crossover design to besuitable and appropriate In practice however there aremanyBE studies using a parallel design that the FDA reports [4]explicitly listed the cases where a parallel design is consideredmore suitable than a crossover design Given that a BE testis a longitudinal equivalence test by nature we believe thatthe use of nonparametric smoothers such as LPS as a basisto construct the concentration-versus-time curves for directcomparisons warrants additional attention and research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by a Clinical and TranslationalScience Award Grant UL1RR024143 Suyan Tian was alsosupported by the National Natural Science Foundation ofChina (no 31401123) Mayte Suarez-Farinas was also partiallysupported by the Irma T Hirschl Career Scientist AwardJingkai Gu was supported by the National Natural ScienceFoundation of China (no 81102383) the Science and Tech-nologyMajor Specialized Projects for ldquoSignificantNewDrugsCreationrdquo of the 12th five-year plan (2012ZX09303-015) andthe National Key Technology RampD Program of the Ministryof Science and Technology (2012BAI30B00)

References

[1] B Ewald ldquoMaking sense of equivalence and non-inferioritytrialsrdquo Australian Prescriber vol 36 no 5 pp 170ndash173 2013

[2] J Ocana M Sanchez A Sanchez and J L Carrasco ldquoOnequivalence and bioequivalence testingrdquo SORT Statistics andOperations Research Transactions vol 32 no 2 pp 151ndash1762008

[3] A S Kesselheim A S Misono J L Lee et al ldquoClinical equiva-lence of generic and brand-name drugs used in cardiovasculardisease a systematic review and meta-analysisrdquo Journal of theAmerican Medical Association vol 300 no 21 pp 2514ndash25262008

[4] FDA Guidance for Industry Bioequivalence Guidance 2006

[5] L E Barker E T Luman M M McCauley and S Y ChuldquoAssessing equivalence an alternative to the use of differencetests for measuring disparities in vaccination coveragerdquo Ameri-can Journal of Epidemiology vol 156 no 11 pp 1056ndash1061 2002

[6] E Walker and A S Nowacki ldquoUnderstanding equivalence andnoninferiority testingrdquo Journal ofGeneral InternalMedicine vol26 no 2 pp 192ndash196 2011

[7] S S Jambhekar and P J Breen Basic Pharmacokinetics Phar-maceutical Press 2009

[8] HQuan J Bolognese andWYuan ldquoAssessment of equivalenceon multiple endpointsrdquo Statistics in Medicine vol 20 no 21 pp3159ndash3173 2001

[9] W Wang J T G Hwang and A Dasgupta ldquoStatistical tests formultivariate bioequivalencerdquo Biometrika vol 86 no 2 pp 395ndash402 1999

[10] P Ghosh and M Gonen ldquoBayesian modeling of multivariateaverage bioequivalencerdquo Statistics inMedicine vol 27 no 13 pp2402ndash2419 2008

[11] H A Bayoud and A M Awad ldquoPerformance of severalbioequivalence metrics for assessing the rate and extent ofabsorptionrdquo Journal of Bioequivalence and Bioavailability vol3 no 7 pp 174ndash177 2011

[12] W S Cleveland and C Loader ldquoSmoothing by local regressionprinciples and methodsrdquo in Statistical Theory and Computa-tional Aspects of SmoothingWHardle andMG Schimek EdsContributions to Statistics pp 10ndash49 Springer New York NYUSA 1996

[13] J Quackenbush ldquoMicroarray data normalization and transfor-mationrdquo Nature Genetics vol 32 no 5 supplement pp 496ndash501 2002

[14] S Anders and W Huber ldquoDifferential expression analysis forsequence count datardquo Genome Biology vol 11 no 10 articleR106 2010

[15] W S Cleveland ldquoRobust locally weighted regression andsmoothing scatterplotsrdquo Journal of the American StatisticalAssociation vol 74 no 368 pp 829ndash836 1979

[16] J Fan and I Gijbels ldquoVariable bandwidth and local linearregression smoothersrdquoTheAnnals of Statistics vol 20 no 4 pp2008ndash2036 1992

[17] C R Loader ldquoBandwidth selection classical or plug-inrdquo TheAnnals of Statistics vol 27 no 2 pp 415ndash438 1999

[18] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[19] P I Good Permutation Parametric and Bootstrap Tests ofHypotheses Springer Berlin Germany 3rd edition 2005

[20] W J Welch ldquoConstruction of permutation testsrdquo Journal of theAmerican Statistical Association vol 85 no 411 pp 693ndash6981990

[21] D E Orange N E Blachere J Fak et al ldquoDendritic cellsloaded with FK506 kill T cells in an antigen-specific mannerand prevent autoimmunity in vivordquo eLife vol 2013 no 2 articlee00105 2013

[22] K Y Liang and S L Zeger ldquoLongitudinal data analysis usinggeneralized linear modelsrdquo Biometrika vol 73 no 1 pp 13ndash221986

[23] S L Zeger and K Y Liang ldquoLongitudinal data analysis fordiscrete and continuous outcomesrdquoBiometrics vol 42 no 1 pp121ndash130 1986

[24] A van der Meersch A Dechartres and P Ravaud ldquoQuality ofreporting of bioequivalence trials comparing generic to brandname drugs a methodological systematic reviewrdquo PLoS ONEvol 6 no 8 article e23611 2011

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom