Bioequivalence and Bioavailability Forum

Hi Divyen & all,

I simulated IR ibuprofen.

One-compartment model, D = 400 mg, V = 7 L (lognormal distribution, CV 40%), ƒ = 0.9 (uniform distribution 0.8 – 1.0), t_½ = 2 h. Associated k₁₀-values (lognormal distribution, CV 25%). Seven formulations with t_max 1.25 h (Reference), at the lower (1.000 h) and upper (1.562 h) ‘limits’, fast (1.125 h, 1.188 h), and slow (1.316 h, 1.389 h). Associated k₀₁-values (lognormal distribution, CV 35%), analytical error (normal distribution, CV 7.5%), LLOQ set to 5% of the reference’s error-free model C_max. Concentrations <LLOQ before t_max set to zero, and after to NA. Lots of samples…
16 subjects in order to achieve ≥80% power for C_max (CV 18%, T/R 0.95).

Lengthy -script (302 LOC) upon request. I got:

Simulation settings: 2,500 studies with 16 subjects
  Sampling every five minutes up to 2 × tmax of R (2.50 h),
  exponentially increasing intervals to tlast (16 h) = 38 samples.
    0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70,
    75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135,
    140, 145, 150 min, 3.5, 4, 5.5, 7, 9.5, 12.5, 16 h
  Seven formulations
  L  = lower limit: tmax = 1.000 h, ka = 2.190 / h, t½,a = 19.0 min
  T1 = fastest    : tmax = 1.125 h, ka = 1.822 / h, t½,a = 22.8 min
  T2 = fast       : tmax = 1.188 h, ka = 1.672 / h, t½,a = 24.9 min
  R  = Reference  : tmax = 1.250 h, ka = 1.539 / h, t½,a = 27.0 min
  T3 = slow       : tmax = 1.316 h, ka = 1.417 / h, t½,a = 29.4 min
  T4 = slowest    : tmax = 1.389 h, ka = 1.296 / h, t½,a = 32.1 min
  U  = upper limit: tmax = 1.562 h, ka = 1.065 / h, t½,a = 39.0 min

Simulation results:
  L = lower limit
    Median  : 1.0833 h (Range: 0.7500 - 1.4583 h)
    Skewness: +0.4452  (Bias: +0.0833)
  T1 = fastest
    Median  : 1.2083 h (Range: 0.8750 - 1.6250 h)
    Skewness: +0.4093  (Bias: +0.0833)
  T2 = fast
    Median  : 1.2917 h (Range: 0.9167 - 1.7917 h)
    Skewness: +0.3858  (Bias: +0.1042)
  R  = Reference
    Median  : 1.3333 h (Range: 1.0000 - 1.7917 h)
    Skewness: +0.3505  (Bias: +0.0833)
  T3 = slow
    Median  : 1.4167 h (Range: 1.0000 - 1.8750 h)
    Skewness: +0.2960  (Bias: +0.1009)
  T4 = slowest
    Median  : 1.5000 h (Range: 1.1250 - 2.0000 h)
    Skewness: +0.2594  (Bias: +0.1111)
  U = upper limit
    Median  : 1.6667 h (Range: 1.2500 - 2.2500 h)
    Skewness: +0.0878  (Bias: +0.1042)

Comparisons:
  passed ‘±20% median criterion’ (80.00-125.00%)
    L  = lower limit: 52.1%
    T1 = fastest    : 80.9%
    T2 = fast       : 90.4%
    T3 = slow       : 91.2%
    T3 = slowest    : 84.8%
    U  = upper limit: 57.0%

The positive skewness of t_max-values confirmed the theoretical considerations of the two Lászlós.¹ Interesting that the skewness decreased with increasing t_max. All medians were positively biased when compared to the models’ true values.

What changed to the simulations I presented in the comments to the guidance?

• The chance to pass is higher due to the upper limit of 125% (instead of 120%).
  As mentioned by other stakeholders, the approach favors slower formulations – even more than with the draft’s 120% limit. Surprisingly we read on page 16:
  • It is not agreed that slower formulations will be developed to use a wider acceptance range since quicker formulations are desired from a clinical and marketing point of view.
  Whatever the development’s target might be, slower formulations will more likely pass. Full stop. How does marketing enter the game in bioequivalence? Is it OK nowadays to claim in the SmPC: “The product releases quicker than the originator”?
  
• If a study is powered for C_max and the median t_max of a formulation does not deviate more than ~8 (‼) minutes from the reference, it will likely pass.
  However, is a Δ of ±8 mi­nutes really clinically relevant for a pain-killer? I doubt it.
  
• If a formulation’s t_max is expected to lie close to the limits, one has to increase the sample size substantially. On page 17 we read:
  • The sample size required for the demonstration of bioequivalence for C_max and AUC should be sufficient to obtain a reliable and representative median T_max.
  Really? Probably it’s cheaper to toss a coin…
  
• On page 13 about the sampling schedule:
  • […] if T_max is expected 30 minutes after administration, sampling should be at e.g. 10, 20, 25, 30, 35, 40, etc., which would allow to conclude that the difference is less than 5 minutes if the T_max is observed in the same or an adjacent sampling time.
  I doubt it. My simulations at the end. Power compromised.
  
• A picky comment on page 25:
  • ‘T’ is the SI symbol for the absolute temperature. Use the correct SI symbol ‘t’ for time, at least for consistency with the overarching guideline.
  Was accepted but ignored at the end of the day.
  
• On page 26 about the comparison of ranges:
  • The assessment of the range is more subjective. If all the values except one are the same, the ranges would be considered acceptable. Therefore, only if differences are evident and worse for the test product, the range could be used for a regulatory decision.
  I see. What might by an ‘evident difference’ for an assessor? Is that better than the ‘apparent difference’ we have in the IR and MR GLs?
  
• On page 28:
  • The comparison of the medians does not intend to preserve the type 1 error [sic] but to exclude formulations with different onset of action.
  Aha! Patient’s risk not important.
  
• On page 29 about the proposed CI-interval inclusion approach based on nonparametric methods:
  • […] the power of a statistical test (usually be performed using a confidence interval), and consequently the sample size needed, will depend on the requested equivalence range and significance level (the allowed type-1 error rate). Equivalence range could be wider than the range that is applied for point estimate. Also, the allowed type-1 error rate (or equivalently, the coverage probability of the confidence interval) may be less strict than for AUC and C_max. This would allow for assessing the consumers risk for T_max but on a different level than for AUC and C_max. Still an agreement on both, equivalence range and significance level to be used, may be difficult to achieve.
  IMHO, that’s funky. Instead of taking the effort of defining an equivalence range based on a clinically relevant Δ, the EMA has chosen the easiest way. “The allowed type-1 error rate […] may be less strict than for AUC and C_max.” I beg your pardon? If t_max is considered less important than AUC and C_max, specify wider limits! Don’t even think about fiddling with α.
  • It is considered that while the Hodges-Lehmann estimator is an adequate estimator to compare T_max of Test (generic) and Reference (innovator) products […] the present revision of the product specific guideline concerns […] not introducing a new method particularly one for which EMA experience in regulatory submissions is limited.
  C’mon, “limited experience”! What about my 600+ studies? A nonparametric method was even recommended for 19 years^2,3 (till the 2010 GL). How many studies dwell in the gloomy dungeons of the agency? Exhume them and perform retrospective assessments.

Simulation settings: 2,500 studies with 16 subjects
  Sampling every five minutes up to 2 × tmax of R (1.00 h),
  exponentially increasing intervals to tlast (16 h) = 18 samples.
    0, 10, 20, 25, 30, 35, 40, 45, 50, 55, 60 min, 1.5, 2, 3.5,
    5, 7, 11, 16 h
  Seven formulations
  L  = R –5 min   : tmax = 25 min, ka = 7.828 / h, t½,a = 5.31 min
  T1 = pretty fast: tmax = 27 min, ka = 7.037 / h, t½,a = 5.91 min
  T2 = fast       : tmax = 28 min, ka = 6.527 / h, t½,a = 6.37 min
  R  = Reference  : tmax = 30 min, ka = 6.074 / h, t½,a = 6.85 min
  T3 = slow       : tmax = 32 min, ka = 5.650 / h, t½,a = 7.36 min
  T4 = pretty slow: tmax = 33 min, ka = 5.233 / h, t½,a = 7.95 min
  U  = R +5 min   : tmax = 35 min, ka = 4.881 / h, t½,a = 8.52 min

Simulation results:
  L = R –5 min
    Median  : 0.5000 h (Range: 0.3333 - 0.6667 h)
    Skewness: +0.4994  (Bias: +0.0833)
  T1 = pretty fast
    Median  : 0.5417 h (Range: 0.3750 - 0.7500 h)
    Skewness: +0.4606  (Bias: +0.0917)
  T2 = fast
    Median  : 0.5417 h (Range: 0.3750 - 0.7500 h)
    Skewness: +0.4086  (Bias: +0.0667)
  R  = Reference
    Median  : 0.5833 h (Range: 0.4167 - 0.8333 h)
    Skewness: +0.3538  (Bias: +0.0833)
  T3 = slow
    Median  : 0.5833 h (Range: 0.4167 - 0.8333 h)
    Skewness: +0.2857  (Bias: +0.0570)
  T4 = pretty slow
    Median  : 0.6250 h (Range: 0.4167 - 0.8333 h)
    Skewness: +0.2113  (Bias: +0.0694)
  U = R +5 min
    Median  : 0.6667 h (Range: 0.4583 - 0.8750 h)
    Skewness: +0.1550  (Bias: +0.0833)

Comparisons:
  passed ‘±20% median criterion’ (80.00-125.00%)
    L  = R –5 min   : 73.7%
    T1 = pretty fast: 83.2%
    T2 = fast       : 87.1%
    T3 = slow       : 88.3%
    T3 = pretty slow: 82.6%
    U  = R +5 min   : 76.8%

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1. Tóthfálusi L, Endrényi L. Estimation of C_max and T_max in Populations After Single and Multiple Drug Ad­mi­ni­stra­tion. J Pharma­co­kin Pharma­codyn. 2003; 30(5): 363–85. doi:10.1023/b:jopa.0000008159.97748.09.
   
2. Commission of the EC. Note for Guidance. Inves­ti­ga­tion of Bioavailability and Bioequivalence. Appendix III: Tech­ni­cal Aspects of Bioequivalence Statistics. Brussels. December 1991. Online.
   
3. EMEA, CPMP. Note for Guidance on the Inves­ti­ga­tion of Bioavailability and Bioequivalence. London. 26 July 2001. Online.