1. Exercise 5 Monte Carlo simulations, Bioequivalence and Withdrawal time

2. Objectives of the exercise
To understand regulatory definitions of Bioequivalence and Withdrawal time
To understand the principles and goals of Monte Carlo simulation (MCS)
To simulate a data set using MCS with Crystal Ball (CB) to show that two formulations of a drug (pioneer and generic) can be bioequivalent while having different withdrawal times.
To compute a Withdrawal time using the EMEA software (WT 1.4 by P Heckman)
To compute a Bioequivalence using WNL (crossover design)

3. Origin of the question

4. A regulatory decision
The new EMEA guideline on bioequivalence (draft 2010) states that it is possible to get a marketing authorization for a new generic without having to consolidate the withdrawal time associated with the pioneer formulation except if there are tissular residues at the site of injection.

5. The EMA guideline

6. Comments on guideline by EMA

7. The question
As an expert, you have to express your opinion on this regulatory decision.
As a kineticist you know that the statistical definitions of Withdrawal time (WT) and bioequivalence (BE) are fundamentally different
So, you decide to demonstrate, with a counterexample, it is not true and for that you have to build a data set corresponding to a virtual trial for which BE exist while WT are different
For that you will use MCS

8. EMA definition of BE (2010)
For two products, pharmacokinetic equivalence (i.e. bioequivalence) is established if the rate and extent of absorption of the active substance investigated under identical and appropriate experimental conditions only differ within acceptable predefined limits.
Rate and extent of absorption are estimated by Cmax (peak concentration) and AUC (total exposure over time), respectively, in plasma.

9. EMA definition of BE
The EMA consider the ratio of the population geometric means (test/reference) for the parameters (Cmax or AUC) under consideration.
For AUC, the 90% confidence intervals for the ratio should be entirely contained within the a priori regulatory limits 80% to 125%.
For Cmax, the a priori regulatory limits 70% to 143% could in rare cases be acceptable

10. Decision procedures in bioequivalence trials
BE not accepted
BE not accepted 1 2
the 90 % CI of f the ratio
- 20%
+20%
BE accepted µt
Mean of ref formulation

12. The a priori BE interval: why 80-125%
The reference value is 100 thus
After a logarithmic transformation:
This interval is no longer symmetric around the reference value

13. The a priori BE interval: why 80-125%
After a logarithmic transformation:
This interval is no longer symmetric around the reference value
This interval is now symmetric around the reference value in the Ln domain

14. Different types of bioequivalence
Average (ABE) : mean
Population (PBE) : prescriptability

15. Average bioequivalence
reference
test1
test2
AUC: Same mean but different distributions

16. Population bioequivalence Population dosage regimenNo
Yes
Pigs that eat less:
Possible underexposure
Pigs that eat more

17. Bioequivalence and withdrawal time
Formulation A
Formulation B
AUCA = AUCB
A and B are BE
Mean curve
Mean curve
Concentration
Individuals
individuals
Time

18. Bioequivalence and withdrawal time
Formulation A
Formulation B
WTA < WTB
MRL
Concentration
WTA
Time
WTB

19. Bioequivalence and the problem of drug residues
Bioequivalence studies in food-producing animals are not acceptable in lieu of residues data: why?

20. Definitions ans statistics associated to (average) bioequivalence and withdrawal time are fundamentally different

21. EMA guidance for WT

22. Definition of WT by EMA

23. The EU statistical definition of the Withdrawal Time
"WT is the time when the upper one- sided 95% tolerance limit for residue is below the MRL with 95% confidence"
Tolerance limits: Limits for a percentage of a population

24. WT definition is related to a tolerance interval (limits)
Tolerance interval is a statistical interval within which a specified proportion of a population falls (here 95%) with some confidence (here 95%)
To compute a WT you have to specify two different percentages.
The first expresses what fraction (percentage) of the values (animals) the interval will contain.
The second expresses how sure you want to be
If you set the second value (how sure) to 50%, then a tolerance interval is the same as a prediction interval (see our first exercise).

25. Tolerance limits vs Confidence limits
How sure we are (risk fixed to 95%)
95% for EMA and 99% for FDA

26. Tolerance limits vs Confidence limits
When sample size increase, the confidence interval converge toward the parameter and the width of the confidence interval shows convergence toward 0
When sample size increase tolerance limits converge toward two limiting parameters (constant) namely those corresponding to the quantiles of the population between which lies the percentage of the population to which the tolerance limits relate.

27. Tolerance limits Without confidence probability
When very many samples of the same size are taken from the same stable population and the tolerance limits calculated each time, then these limits will enclose on the average 95% of the population
With confidence probability
When very many samples of the same size are taken from the same stable population and the tolerance limits calculated each time, then these limits will enclose at least 95% of the population in an average of 95% of cases

28. Tolerance interval
The tolerance interval estimates the range which should contain a certain percentage of each individual measurement in the population.
Because tolerance intervals are based upon only a sample of the entire population, we cannot be 100% confident that that interval will contain the specified proportion. Thus there are two different proportions associated with the tolerance interval: a degree of confidence, and a percent coverage. For instance, we may be 95% confident that 95% of the population will fall within the range specified by the tolerance interval.

29. Confidence interval
A Confidence interval is a range of values which span from the Lower Confidence Limit to the Upper Confidence Limit.
We expect this range to encompass the population parameter of interest, such as the population mean, with a degree of certainty which we specify up fron

30. Bioequivalence and withdrawal time
Bioequivalence is related to a confidence interval for a parameter (e.g. mean AUC-ratio for 2 formulations)
Withdrawal time is related to a tolerance limit (quantile 95% EU or 99% in US) and it is define as the time when the upper one- sided 95% tolerance limit for residue is below the MRL with 95% confidence“
The fact to guarantee that the 90% confidence interval for the AUC-ratio of the two formulations lie within an acceptance interval of do not guarantee that the upper one- sided 95% tolerance limit for residue is below the MRL with 95% confidence for both formulations“

31. Who is affected by an inadequate statistical risk associated to a WT
It is not a consumer safety issue
It is the farmer that is protected by the statistical risk associated to a WT
It is the risk, for a farmer, to be controlled positive while he actually observe the WT.
When the WT is actually observed, at least 95% of the farmers in an average of 95% of cases should be negative

32. How to build a counterexample to show that it is possible to have two formulations complying with BE requirements while their WTs largely differ.

33. Counterexample
In mathematics, counterexamples are often used to show that certain conjectures are false,

34. How to build a counter example
Considering that BE is demonstrated using plasma concentration over a rather short period of time (e.g. 24 or 48h) but that WT is generally much longer (e.g. 12 days), you can expect that two bioequivalent formulations exhibiting a so-called very late terminal phase could have different WT.

35. Bioequivalence and withdrawal time
Withdrawal time are generally much more longer than the time for which plasma concentration were measured for BE demonstration
Pionner
WT for the generic
Generic
WT for the pionner BE
LOQ WT

36. Use of Monte Carlo simulations

37. What is the origin of the word Monte Carlo?
Toulouse
Monte-Carlo
(Monaco)

38. Selecting a model to simulate our data set

39. Equation describing our model

40. Step1: Implementation of the selected model in an Excel sheet
We have to write this equation in Excel and to solve it for:
times ranging from 0 to 144h for plasma concentrations for BE
from 144 to 1440 h for tissular concentrations for WT.

41. Implementation of the selected model in an Excel sheet

42. Step2: Monte Carlo simulation to establish a data set using CB
For the BE trial, we need 12 animals to carry out a 2x2 crossover design i.e. 24 vectors of plasma concentrations from time 0 to 144h (12 vectors for formulation1 and 12 vectors for formulation 2).
For the WT we are planning a trial with 4 different slaughter times (14, 21, 28 and 45 days) and for each slaughter time, we need 6 samples i.e. 24 animals per formulation; thus the total number of animals to simulate for the WT is 48.
The total number of vectors to simulate is of n=72 i.e. 36 for formulation1 and 36 for formulation2.

43. Step2: Monte Carlo simulation to establish a data set using CB
Only variability is introduced in F1;
for the first formulation F1 is normally distributed with a mean F1=0.7 associated with a relatively low inter-animal variability of 10%.
For the second formulation we also fixed F1=0.7 but with an associated CV of 30% meaning more variability between animals for this second formulation but the same average PK parameters.

44. Step2: Monte Carlo simulation to establish a data set using CB

45. Simulated concentrations at 120h post dosing

46. Plasma concentration profiles for formulation A and B

47. NCA of the data set Computation of AUC using the NCA by WNL
Computation of some statistics using the statistical tool of WNL.
It appeared that the ratio of the AUC (geometric means) was 0.83 (659.2/788.8).
This point estimate is to close to the a priori lower bound of the a priori confidence interval for a BE trial (lower bound is 0.8) thus I know a priori that it will be impossible to conclude to a BE with this data set

48. New simulation
So, I decided to re-run my simulations but with a lower variances for the second formulation
Second simulation CV=10 and 20% for formulations A and B respectively; same mean F=0.70.

49. Results of the second simulation
Descriptive statistics
The point estimate of the AUCs ratio is now of 0.89 and I can expect to demonstrate a BE.

50. Assessment of bioequivalence of the two formulations

51. Bioequivalence in WNL

53. Bioequivalence output
Cmax:
the regulatory 90% CI was from 73.9-:98.3
AUC
the regulatory CI 90% was :
I permuted the 2 CI in the document?)

54. Computation of withdrawal time for the two formulations

55. How to obtain raw data
Simulate the model between 144 and 1440 h (i.e. from 6 to 60 days)
You need 6 tissular samples for each of the four sampling times
Sampling times: 14, 21 ,28 and 45 days
Tissular concentrations are 100 times plasma concentrations
Thus for the 6 first animals, you select forecasts at day 14, then forecasts at day 21 for animals 7 to 12 and so on.

56. MRL=30

57. Computation of a WT

60. Conclusion
This example shows that we are in position to demonstrate that an average bioequivalence between two formulations is not a proof to guarantee that the formulations have identical withdrawal times.