Bioequivalence and Bioavailability Forum

Dear Vijay

❝ ❝ [...] the second one only looks like one.

❝ ❝ I wouldn't recommend it."

❝ Why does the second split only "look like" a 2x2x2 cross-over? I guess I am missing something when you write this. It seems like a conventional 2x2x2 cross-over for me.

No it isn't. Just compare the layout of William's design (top) to Atish's design with the two splits. In William's design every treatment has the same chance of being administered in periods 1-4. In Atish's design R₁ and T₁ have an equal chance of being administered to a treatment naïve subject (in period 1), whereas the second split is different: period 1 is really period 3 from the complete layout. So R₂ and T₂ are always administered to pre-treated subjects. Even assuming the design is planned to avoid sequence effects, regulatory questions are almost unavoidable. Statistically it's no problem, but how will you explain that you defined an inclusion criterion of e.g. four weeks drug-free prior to period 1, and then present results from period 3/4 as a conventional cross-over? If the wash-out is one week, will you present for the second part screening data which are 3+ weeks old? Etc, etc...

❝ [...] While planning a pilot BE study for choosing an optimal formulation, we are comparing 3 different formulations (T1, T2 & T3) given nasally to a rectal formulation (R). To ensure all subjects complete the 4 period study, we plan to administer the rectal formulation which is the reference to all subjects in the first period and randomize the three test formulations across periods 2, 3 and 4. We are doing this to avoid a situation with subjects who receive nasal drug in the first few periods not to return to the rectal administration in the later period due to compliance issues or acceptability.

IMHO you should avoid running into compliance issues by a good ICF. I don't know what type of formulation you are using, but for e.g., a DPI you may consider preparing a placebo device containg only lactose and perform a run-in-phase or a training session to familiarize volunteers with the device. Since you are interested in a comparison of the nasal formulations with the rectal one, what do you gain from your design if a subject drops out in period 2 due to problems in administration? You may not use data from period 1 anyhow.

❝ GP | P1 P2 P3 P4
❝ ---+-------------
❝ G1 | R  T1 T2 T3
❝ G2 | R  T2 T3 T1
❝ G3 | R  T3 T1 T2

❝

❝ P1-P4 = Period 1-4
❝ G1-G3 = Group 1-3

❝ Each group will have 4 subjects.

What is this? A 3×3 latin square with an extra period?

❝ I would like to mention that this will be a pilot study intended to obtain initial estimates of relative BA of the test formulations with respect to the reference and plans to do some exploratory BE analysis which would provide us with sufficient data and initial estimates to design an adequately powered BE study with the best chosen formulation.

❝

❝ Any comments on the design issue would be highly appreciated, both in terms of complexity of determining BE, or with regards to drawbacks of such a design. I understand that lack of balance is one of them. How valid will be the inclusion of Period as a fixed effect?.

Your design is lacking one main property which is neccessary in the evaluation of any cross-over design: randomization. In a randomized cross-over design the treatment effect can be separated from the period effect. Just give it a try: take the data from any study, multiple all values of the second period by 5 and run the BE-calculation. Both PE and the CI will be exactly the same.
Now for a Gedankenexperiment: a 2×2 cross-over (RT/TR), test and reference of equal true bioavailability of 1 giving a PK response of 100 units (left pannel). Due to some external influence (let's call it the 'phase of the moon') the measured PK responses in the second period are just 50 units (right pannel).
                 P1   P2                       P1   P2
sequence 1: RT  100  100      sequence 1: RT  100   50
sequence 2: TR  100  100      sequence 2: TR  100   50
period mean     100  100      period mean     100   50
reference mean  100           reference mean   75
test mean       100           test mean        75
PE = T/R =  100/100 =  1      PE = T/R =    75/75 =  1
The treatment effect is not influenced by the period effect, and an unbiased PE is obtained.

Now let's see what will happen if we don't randomize (OK, I still used the term 'sequence', but actually there are none):
                 P1   P2                       P1   P2
sequence 1: RT  100  100      sequence 1: RT  100   50
sequence 2: RT  100  100      sequence 2: RT  100   50
period mean     100  100      period mean     100   50
reference mean  100           reference mean  100
test mean       100           test mean        50
PE = T/R =  100/100 =  1      PE = T/R =   50/100 =0.5
We are dead.

Such a period effect may be extreme, but remember:
Without randomization any period effect will bias the treatment effect.

Your design is not a cross-over study at all. Just give it a try, feed your preferred software with random numbers and you most likely will end up with some kind of error. Technically speaking the variance matrix is ill-conditioned. The only way of evaluating such is study is to trust in absence of sequence effects and run a paired test. But still, period effects will bias the treatment effect. Since you wrote this is a pilot study things get even worse. Imagine the situation where the period effect affects the treatment effect in such a way, that you come up with a nice PE of 95% for one of your candidate formulations. You power the pivotal cross-over study for this PE, and - surprise - the PE turns out to 115%. Bad luck.

Another possibility comes into my mind - if you want to reduce the number of periods you may opt for a balanced incomplete block design. The basic layouts for four treatments with three and two periods are:
S/P |   I     II   III      S/P |   I     II 
----+-----------------      ----+-----------
  1 |   T1    T2    T3        1 |    R    T1
  2 |   T2    T3     R        2 |   T1    T2
  3 |   T3     R    T1        3 |   T2    T3
  4 |    R    T1    T2        4 |   T3     R
                              5 |    R    T2
                              6 |   T1    T3
                              7 |   T3    T1
                              8 |   T2     R
                              9 |    R    T3
                             10 |   T3    T2
                             11 |   T2    T1
                             12 |   T1     R
But the price to be paid for the reduction in periods is high. Only in 75% of sequences in the three-period design and in 50% of sequences in the two-period design subjects receive the reference formulation. Any kind of imbalance due to drop-outs is uncomfortable enough in a conventional design, in an IBD it may become a real problem.

❝ I did a literature search for such designs, but could not really come with the correct key words to find a good reference.

Me not either.