Exact PK Equivalence
for a bridging study
Steven Novick, Harry Yang (MedImmune)
and Xiang Zhang (NC State)
NCB, October 2015
In submission
Journal of Biopharmaceutical Statistics
• Written with human-subject bridging study in mind.
• Easily adapted to animal PK study comparison.
Pharmacokinetics: the study of the time course of drug
absorption, distribution, metabolism, and excretion.
Reference Drug
Test Drug
Regulatory Requirement: In Vivo Bioequivalence
• Change rate and extent of absorption of drug
product for which the manufacture has changed.
– FDA 1995: Guidance for Industry Immediate Release
Solid Oral Dosage Forms, Scale-Up and Postapproval
Changes: […] In Vivo Bioequivalence Documentation
• A sponsor proposes manufacturing a generic
version of an approved off-patent product.
– FDA 2006: Guidance for Industry: Bioequivalence
Guidance
Regulatory Requirement: In Vivo Bioequivalence
• […] Demonstrate equivalence […] between the
generic medicinal product and a reference
medicinal product.
– EMEA 2010: Guideline on the Investigation of
Bioequivalence
Absorption rate=elimination rate
Cmax
Basic pharmacokinetic
(PK) considerations
Let the curve be: f( , t )
AUC
Tmax
T1/2
Bioequivalence definition
• EMEA 2010: The plasma drug concentration time curve is […]
used to assess bioequivalence between two formulations.
• FDA 2006: AUC and CMAX [are considered] as the pivotal
parameters for bioequivalence determination.
• Area under the curve (AUC): reflects the extent of exposure.
• The maximum plasma concentration or peak exposure (CMax)
• Statistical evaluations of Tmax and T1/2 are not required.
Testing for bioequivalence
Show:
𝑙𝑜𝑔 𝐴𝑈𝐶
𝑅
𝑅
/𝐴𝑈𝐶
𝑇
𝑙𝑜𝑔 𝐶𝑀𝑎𝑥 /𝐶𝑀𝑎𝑥
We add:
𝑅
𝑇
𝑙𝑜𝑔 𝑇1/2 /𝑇1/2
𝑇
< log 1.25
< log 1.25
< log(1.25)
Mice, Rats, and Dogs, oh my
• I was often asked to compare PK profiles of
mice, rats, and dogs on two different
formulations of the same drug compound via
AUC, Cmax, Tmax, and T1/2.
• Crossover studies
• Show statistical equivalence
• Sometimes shows statistical superiority
Some thoughts
• Westlake 1988: Equivalence of the PK metrics
does not necessarily imply equivalence in the
concentration time profile.
• EMEA 2010: The plasma drug concentration
time curve is […] used to assess bioequivalence
between two formulations.
Comparing the mean PK profiles
• Chen and Huang (2009)
𝑚𝑎𝑥
𝑡∈(0,𝜏)
log( 𝑓(𝜽𝑅 , 𝑡) / 𝑓(𝜽 𝑇 , 𝑡) ) < log(1.25)
• We agree!
We show: closeness in PK profiles is a more stringent
measure than both AUC and CMax.
𝑚𝑎𝑥
𝑡∈(0,𝜏)
==>
log( 𝑓(𝜽𝑅 , 𝑡) / 𝑓(𝜽𝑇 , 𝑡) ) < log(𝛿)
𝑙𝑜𝑔 𝐴𝑈𝐶
𝑅
𝑅
/𝐴𝑈𝐶
𝑇
𝑙𝑜𝑔 𝐶𝑀𝑎𝑥 /𝐶𝑀𝑎𝑥
𝑇
< 𝑙𝑜𝑔 𝛿
< 𝑙𝑜𝑔 𝛿
Does not imply anything about Tmax or T1/2.
Oral, first-order, single-compartment PK model
𝑒𝑥𝑝(−𝐾𝑒 𝑡 ) − 𝑒𝑥𝑝(−𝐾𝑎 𝑡 )
𝑓(𝜽, 𝑡) = 𝛼𝐾𝑎
𝐾𝑎 − 𝐾𝑒
Ka = absorption rate constant
Ke = elimination rate constant
 = (Dose)x(Fraction absorbed)/(Volume)
A quick counter-example
• Reference: 𝐾𝑒 = 0.331, 𝐾𝑎 = 0.357, 𝛼 = 300
• Test:
𝐾𝑒 = 0.277, 𝐾𝑎 = 0.301, 𝛼 = 300
𝑙𝑜𝑔 𝐴𝑈𝐶
𝑅
/𝐴𝑈𝐶
𝑅
𝑇
𝑙𝑜𝑔 𝐶𝑀𝑎𝑥 / 𝐶𝑀𝑎𝑥
𝑅
𝑇
𝑙𝑜𝑔 𝑇1/2 /𝑇1/2
𝑇
= log 1.18 
= log 1.00 
= log(1.19) 
Ratio of PK profiles
Bioequivalence
declared through
AUC, CMax, and T1/2
Bioequivalence not
declared through
PK profile comparison
Hypothesis testing
• Chen and Huang (2009): parametric bootstrap (probably ok)
• We propose Bayesian posterior probability
𝑚𝑎𝑥
𝑝 𝛿, 𝜏 = 𝑃𝑟( 𝑡∈(0,𝜏)
log( 𝑓(𝜽𝑅 , 𝑡) / 𝑓(𝜽𝑇 , 𝑡) )
< log(𝛿) | 𝑑𝑎𝑡𝑎 )
• If 𝑝 𝛿, 𝜏 > 0.95 (say), the PK profiles are
declared bioequivalent
• JAGS (Plummer, 2003), OpenBUGS (Lunn, Thomas, Best, and
Spiegelhalter, 2000) or STAN (Stan, 2014)
A small simulation
• Examine characteristics of p(, ) vs.
traditional bioequivalence testing.
• Two arms: Reference (R) and Test (T)
– 20 subjects per arm
– 11 time points: 0.5 – 16 hours
• Hierarchical model
Data Generation for Simulation
log10 ( 𝐶𝑖𝑗𝑡 ) = log10 {𝑓(𝜽𝑖𝑗 , 𝑡)} + 𝜀𝑖𝑗𝑡
– Between-subject: log e (𝜽𝑖𝑗 )~𝑁 log 𝑒 𝜽𝑖 , 𝑽
– Within-subject: ijt ~ N(0, 2)
0.01
𝑽=
0
0
0
0
0.01
0
0
0.01
and =0.022
Priors
• Centered at their true values
𝜽𝑖 ~ normal with SD = 0.5
– In practice, 𝜽𝑅 may be known with greater
precision.
V-1 ~ Wishart with 20 degrees of freedom
– We assumed knowledge of V.
 ~ U( 0, 1 )
Mean Reference PK curve
Showing time points
Example of 20 subjects
on Reference drug
Testing
Traditional Test:
𝑙𝑜𝑔 𝐴𝑈𝐶
𝑅
𝑅
/𝐴𝑈𝐶
< log 1.25
𝑇
< log 1.25
𝑇
< log(1.25)
𝑙𝑜𝑔 𝐶𝑀𝑎𝑥 /𝐶𝑀𝑎𝑥
𝑅
𝑇
𝑙𝑜𝑔 𝑇1/2 /𝑇1/2
Proposed Test:
𝑚𝑎𝑥
𝑡∈(0,𝜏)
log( 𝑓(𝜽𝑅 , 𝑡) / 𝑓(𝜽𝑇 , 𝑡) ) < log(1.35)
Six Scenarios
(=300 for all cases)
Ka
Ke
Max Ratio
Reference
0.257
0.294
Test 1
0.257
0.294
log10(1) = 0
Equal
Test 2
0.248
0.294
log10(1.1)
Equiv
Test 3
0.257
0.316
log10(1.1)
Equiv
Test 4
0.228
0.294
log10(1.35)
Borderline
Test 5
0.257
0.370
log10(1.35)
Borderline
Test 6
0.198
0.294
log10(1.88)
Not Equiv
Bayesian model fitting
• 4 independent MCMC chains via JAGS
• 200K posterior samples (total)
– Burn in = 10,000
– Thinning = 200
• Effective sample sizes > 10,000
• Simulation ran for four weeks!
Simulation Results
Alternative #1
Close…most of the time (q=70%)
100q%
Curves are close together for 100q% time
Let 𝑄 𝑡 = log( 𝑓(𝜽𝑅 , 𝑡) / 𝑓(𝜽 𝑇 , 𝑡) )
τ
𝐼{
0
𝑄 𝑡 < log 𝛿 } 𝑑𝑡
= proportion of t  (0, ) such that 𝑄 𝑡 < log 𝛿
Metric:
τ
𝑝 𝛿, 𝜏, 𝑞 = 𝑃𝑟(
𝐼{ 𝑄 𝑡 < log 𝛿 } 𝑑𝑡 ≥ 𝑞 | 𝑑𝑎𝑡𝑎 )
0
q = 80%, 90%, 95%, 100%
Simulation Results
Alternative #2
Subset of time points
60% of points <
1
𝑅
𝐶𝑚𝑎𝑥
10
1
10
𝑅
𝐶𝑚𝑎𝑥
are not equivalent
Subset of time points
• Let C be a subset of time points  (0, ).
• For example C = set of time points such that
𝑓 𝜃𝑅 , 𝑡 ≥
1
10
𝑅
𝐶𝑀𝑎𝑥
• A claim of |log( f(R, t) / f(T, t) )| <  for all t  C
𝑅
𝑇
implies | log(𝐶𝑚𝑎𝑥
/𝐶𝑚𝑎𝑥
) | < log() and implies
closeness in the AUC results on the set C.
Alternative #3: Differences
• The difference of the two curves are close together;
i.e., not on the log scale.
𝑚𝑎𝑥
𝑡∈(0,𝜏)
0.2
𝑓(𝜽𝑹 , 𝑡 ) − 𝑓(𝜽 𝑇 , 𝑡 ) <
𝐴𝑈𝐶
𝜏
𝑅
==>
𝑙𝑜𝑔 𝐴𝑈𝐶
𝑅
𝑅
/𝐴𝑈𝐶
𝑇
𝑙𝑜𝑔 𝐶𝑀𝑎𝑥 /𝐶𝑀𝑎𝑥
𝑇
< log 1.25
< log 1.25
• Posterior probability that log-ratio of two PK curves are close
together for all time points in a time range.
• Method provides better control over consumer’s risk from a
compliance point of view.
• Discussed three related, alternative metrics.
Thank you!
Questions?