Helmut
★★★

Vienna, Austria,
2009-12-21 13:08

Posting: # 4517
Views: 11,575

## Cmin (EU) [Surveys]

Dear colleagues,

since I saw a couple of deficiency letters concerning Cmin in the recent past, I want to start a little survey and discussion. The BE-Draft (2008) states (lines 558-559):

For studies to determine bioequivalence at steady state AUCtau, Cmax,ss, and Cmin,ss should be analysed using the same acceptance interval as stated above.

and in lines 551-552:

For these parameters the 90% confidence interval for the ratio of the test and reference products should be contained within the acceptance interval of 80-125%.

Lines 561-562 state:

[…] for highly variable drugs the acceptance interval for Cmax may in certain cases be widened (see section 4.1.10).

I would say the same rationale for widening the acceptance range (clinical justification, high variability) for Cmax is justifiable for Cmin.
1. How do you define Cmin in steady state?
1. Minimum concentration within the profile
2. Pre-dose concentration
3. Last concentration within the dosing interval
4. Minimum concentration within time of administration and tmax
2. Do you only report values for the formulations (geometric mean, sd) or do you calculate and report a confidence interval?
3. If the latter, do you use the metric in a confirmatory analysis (e.g., state an acceptance range)?
1. Yes, always
2. Yes, but only if clinical concerns, e.g. for long-term use analgetics, antibiotics, …
3. Yes, but one-sided (i.e., not inferior to reference)
4. No
If yes, which acceptance range?
5. 0.80-1.25
6. 0.75-1.33
7. 0.70-1.43
8. other
How do you deal with the inherent variability (low power)?
4. Did you have any problems with your approach?
5. Do you see a change in point of views by European regulators within the last years – especially after publication of the BE-Draft?
6. Do you consider widening of the acceptance range in a replicate design in steady state (i.e., TTRR-RRTT), which would need measuring four profiles?
P.S.: Cmin is a quite nasty metric. In a recent study (n=40) with ≈100% PTF (Cmin ≈25% of Cmax) I saw a CVintra of 15.6% for Cmax, but 62% (!) for Cmin

Edit 1: In the final BE Guideline Cmin is not given any more as a steady state metric (for IR formulations). So the question stays valid for MR formulations – would you dare to go with scaling?

Edit 2: The comment-document (see this post) states at page 89:

By Cmin,ss we mean the concentration at the end of the dosage interval, i.e. Ctrough. However, in bioequivalence studies for immediate release formulations there is no need to report Ctrough and fluctuation. The guideline has been revised.

Cheers,
Helmut Schütz

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ElMaestro
★★★

Belgium?,
2009-12-22 14:17

@ Helmut
Posting: # 4518
Views: 9,773

## Cmin (EU)

Hi HS,

I am not the one to answer this interesting question, and I hope there will be some input from more qualified people.

Clearly I would use option 1b; if pre-dose is not the lowest in practice then we are looking at data that do not reflect reality but random fluctuations or mixups (unless, of course, the body is capable of synthesizing the drug or metabolite under investigation which is generally not the case for the majority of substances). And in fact Cmin,ss is to some people like me a misnomer; should be called Ctrough, with trough being defined as predose.

My 0.02 euros.
EM.
Helmut
★★★

Vienna, Austria,
2009-12-22 14:30

@ ElMaestro
Posting: # 4519
Views: 10,012

## Terminology issues

Dear ElMaestro!

» Clearly I would use option 1b; if pre-dose is not the lowest in practice then we are looking at data that do not reflect reality but random fluctuations or mixups (unless, of course, the body is capable of synthesizing the drug or metabolite under investigation which is generally not the case for the majority of substances). And in fact Cmin,ss is to some people like me a misnomer; should be called Ctrough, with trough being defined as predose.

Agree with Ctrough; but I would opt for 1d, because if a lag-time comes into play, concentrations will still decrease after time of administration. In such a case Ctrough ('trough' is a minimum in the English language) is not to be expected at pre-dose.

Weimann* states:

3.5 Trough value
The trough value is defined as the drug concentration measured at the end of dose interval at steady-state. […] the trough value is defined as the concentration measured immediately before the next drug-intake […]
3.6 Minimum concentration
The minimum concentration Cmin is defined as the lowest concentration on a concentration-time curve at steady-state and within one dosing interval. The Cmin value can – but need not necessarily – be identical to the trough value. Drugs for which Cmin and trough values differ from each other are products with sustained release properties. After administration of such a formulation, the drug concentration continues to decrease until substance absorption from the administered drug product commences at the end of the lag time. […] Like Cmax, the accuracy of the Cmin value depends on the blood sampling pattern. This is in contrast to the trough value, which is simply determined as the predose concentration.

I leave it to you to find the contradictions.

PK-software in their default configuration (Phoenix/WinNonlin, Kinetica) come up with 1a (it needs some experience to tweak the software to come up with either one of 1b-d). IMHO 1a does not make sense, since in true steady state we would expect 50% of values at the start and end of the dosing interval due to random variability. If we look at the time point tmin we would end up with a nonsensical location of τ/2 with high variance (141% ≥ CV% ≥100%). In other words, comparing the pre-dose concentration to the concentration at the end of the dosing interval within subjects leads to some kind of ‘Apples-and-Oranges-Statistics’. The expectation of such an A&O comparison in any sample is 50% (and 25% for both pre-dose and end of dose). On the other hand, the true minimum (regardless the location) is required in the calculation of %PTF or Swing.

• HJ Weimann
Drug concentrations and directly derived parameters
In: W Cawello (Ed.), Parameters for Compartment-Free Pharmacokinetics
Shaker-Verlag, Aachen pp 31–4 (2003)

Cheers,
Helmut Schütz

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ElMaestro
★★★

Belgium?,
2009-12-27 00:47

@ Helmut
Posting: # 4522
Views: 9,782

## Philostics

Dear HS,

Thank for this and for the reference. I can learn a lot, and there is a loooooong way to go for me.

I wonder about a issue that is on the border between philosophy and statistics... Analysis of Log Cmax is generally based on the assumption that the residuals are normal. If we subsequently also want to qualify Log Cmin with some kind of statistic can we then conduct such an analysis with parametric statistics? Once we identify Log Cmax we know that the rest will be smaller than that value. Hence, there are constraints on the magnitudes of the residuals in the analysis of Log Cmin and therefore I would be inclined to think we violate an important assumption in parametric statistics?

(I shall abstain from proposing parametric statistics as a potential alternative, notch notch wink wink!)

Best regards,
EM.
Helmut
★★★

Vienna, Austria,
2009-12-27 02:01

@ ElMaestro
Posting: # 4523
Views: 9,723

## Philostics

Howdy ElMaestro!

» I can learn a lot, and there is a loooooong way to go for me.

For all of us…

» I wonder about a issue that is on the border between philosophy and statistics... Analysis of Log Cmax is generally based on the assumption that the residuals are normal.

Yep.

» If we subsequently also want to qualify Log Cmin with some kind of statistic can we then conduct such an analysis with parametric statistics? Once we identify Log Cmax we know that the rest will be smaller than that value.

We also know, that the residuals of AUC will always be lower than the ones of Cmax.

» Hence, there are constraints on the magnitudes of the residuals in the analysis of Log Cmin and therefore I would be inclined to think we violate an important assumption in parametric statistics?

Not quite. The assumptions in the parametric model of any given metric are independent from each other,
Untransformed metrics have limits of [-∞, +∞], whilst log-transformed have limits of [>0, +∞], but in both cases the probability equals 1 (in other words, for Cmin and Cmax the log-transformation lead to the same lower limit – I wouldn’t call it a constraint – namely zero).

» (I shall abstain from proposing parametric statistics as a potential alternative, notch notch wink wink!)

You mean nonparametric statistics? Well, cough…

Cheers,
Helmut Schütz

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ElMaestro
★★★

Belgium?,
2009-12-27 13:28

@ Helmut
Posting: # 4525
Views: 9,738

## Philostics

Dear HS,

» Not quite. The assumptions in the parametric model of any given metric are independent from each other,
» Untransformed metrics have limits of [-∞, +∞], whilst log-transformed have limits of [0, +∞], but in both cases the probability equals 1 (in other words, for Cmin and Cmax the log-transformation lead to the same lower limit – I wouldn’t call it a constraint – namely zero).

Hmmmm, this I would express differently. Cmax etc varies from 0 and upwards whilst the transformed (log) Cmax etc is from -inf to +inf. Log transformed parameters (C, AUC) go from -inf to +inf.

I meant something else though: Let's for simplicity consider the subject with the highest Cmax in the dataset (log or not is not the important issue here). Now we do a parametric analysis of (log) Cmin and we consider the (log) Cmin residual e for the same subject and we disregard other factors for now (makes no difference but makes it less easy to grasp). We know something about e; the (log) Cmax is higher than average (log) Cmin so average (log) Cmin plus e will be lower than (log) Cmax for that subject.

» » (I shall abstain from proposing parametric statistics as a potential alternative, notch notch wink wink!)
»
» You mean nonparametric statistics? Well, cough…

Yes, I meant nonparametric. Sorry about the meaning-disturbing typo.
EM.
Helmut
★★★

Vienna, Austria,
2009-12-27 15:07

@ ElMaestro
Posting: # 4526
Views: 9,749

## Philostics

Dear ElMaestro,

I agree with:

» Hmmmm

And also with:

» Cmax etc varies from 0 and upwards …

… in ‘reality’ – since there is nothing like a negative concentration. But based on normal theory (parameters: mean, variance), we must accept the (small) probability of values <0 in the population. Example: two samples (1, 3), mean 2, variance 2, probability of a value of C≤0 in the population = 0.23%. If we log-transfom, the probability of a negative value is infinitesimaly small (nitpicking: it’s not defined, because log(0)=?, but if x→-∞, ℯx→0).

» Let's for simplicity consider the subject with the highest Cmax in the dataset (log or not is not the important issue here). Now we do a parametric analysis of (log) Cmin and we consider the (log) Cmin residual e for the same subject and we disregard other factors for now (makes no difference but makes it less easy to grasp). We know something about e; the (log) Cmax is higher than average (log) Cmin so average (log) Cmin plus e will be lower than (log) Cmax for that subject.

Can you reword your simple statement for my even more simple mind? I did not get your point. From a PK point of view we are sure that Cmax>Cmin (by definition), but from a statistical POV I don’t see why this a priori knowledge should influence the distributional assumptions of either metric. If we build a statistical model for a metric we base it on distributional assumptions for that particular metric – and don't peek across the fence for another metric; e.g. we rely on a discrete distribution of tmax, not caring about the continous one of Cmax.

» […] I meant nonparametric. Sorry about the meaning-disturbing typo.

Nonparametrics never disturb me.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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ElMaestro
★★★

Belgium?,
2009-12-28 02:46

@ Helmut
Posting: # 4528
Views: 9,633

## Philostics

Dear HS,

» Can you reword your simple statement for my even more simple mind?

I think I should rather contradict myself. Let's say we do not analyse Cmax. Rids me of any issue. Case closed. Was an elmaestrolophystic stray of thoughts.

EM.

Edit: [Helmut]
d_labes
★★★

Berlin, Germany,
2010-01-04 10:45

@ Helmut
Posting: # 4555
Views: 9,794

## Cmin (EU)

Dear Helmut,

» How do you define Cmin in steady state?
1.a) as standard, 1.a) or 1.c) as option
for 1.a) Common sense (what is a minimum?), ease of calculation and consistency (same value used for other parameters like swing or PTF)
for 1.a) or 1.c) Sponsor's wish

» Do you only report values for the formulations (geometric mean, sd) ...
It depends (not at least on sponsor's wish ).

» If the latter, do you use the metric in a confirmatory analysis (e.g., state an acceptance range)? ...
Never used 3.c)
Acceptance range usually 0.8-1.25, but also widening.

» Did you have any problems with your approach?
No problems so far for the definition of Cmin (regardless of which).
Usual problems in case of widening acceptance range:
"The applicant should justify the widening ... ".

» Do you see a change in point of views by European regulators ...
None due to rare use of steady state studies.

» Do you consider widening of the acceptance range in a replicate design in steady state (i.e., TTRR-RRTT)
         ^^^^

Best wishes for the New Year
Regards

D. Labes
Helmut
★★★

Vienna, Austria,
2010-01-04 13:53

@ d_labes
Posting: # 4556
Views: 9,737

## Repeated profiles

Dear D. Labes,

thanks for participating!

»                                             ^^^^

Something like this:
 saturation phase     T       T     switch over         R       R saturation phase     R       R     switch over         T       T pre dose values   profile profile  pre dose values  profile profile

This is my interpretation of EMA’s MR-NfG (Section 4.1.2):

The inter-individual variability of the pharmacokinetic parameters of interest should be compared between the modified and immediate release formulation and the variability of the modified release formulation should not exceed that of the immediate release formulation. It may be valuable to assess the intra-individual variability. This could be achieved by either repeated measurements of the concentration profile at steady state or by performing a single dose study with replicate design.

(my emphasis)

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
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beman
☆

2010-01-04 15:43

@ Helmut
Posting: # 4557
Views: 9,598

## Cmin-value <LLOQ

Hi,

I define the Cmin-Value in multiple dose bioequivalence studies as the 'Last concentration within the dosing interval'.

If the last value in the dosing interval is <LLOQ, which value should i use?
• If i set the value to zero, the ratio T/R can't be calculated.
• I can calculate a value <LLOQ (with the average half life time of the corresponding single dose study or the literature).
• I can use the Last concentration value >=LLOQ (for both the reference and the test formulation)
• I can use the Last concentration value >=LLOQ (for only one formulation, for the other i use the last concentration value).
thank you

beman

Helmut
★★★

Vienna, Austria,
2010-01-04 16:34

@ beman
Posting: # 4559
Views: 9,691

## Cmin-value <LLOQ

» I define the Cmin-Value in multiple dose bioequivalence studies as the 'Last concentration within the dosing interval'.

Well, that’s option 1.c and the ‘Trough concentration’ defined by Weimann and WHO (2006).

» If the last value in the dosing interval is <LLOQ, which value should i use ?

Hhm, this problem applies to all definitions of Cmin. Nasty for a low accumulation index…

» - If i set the value to zero, the ratio T/R can't be calculated.

Sure. Though I’m not a fan of setting anything to LOQ/2, I used that approach in a recent study accepted by the German BfArM (low accumulation, concentrations ranging more than three orders of magnitute).

» - I can calculate a value <LLOQ (with the average half life time of the corresponding single dose study or the literature).

Well, why not use an estimate (from the half life of the particular subject in the multiple dose study)? We had a little discussion there.
Using the (average!) half life from another study is unacceptable IMHO. Even more from the literature.

» - I can use the Last concentration value >=LLOQ (for both the reference and the test formulation)
» - I can use the Last concentration value >=LLOQ (for only one formulation, for the other i use the last concentration value).

This problem is similar to the ‘missing 72 h value’ for truncated AUC. See this post for an example. Your first option would give an unbiased T/R (but is available in standard PK-software only for AUCτ; see this post). You would have to set up something on your own. The second one IMHO is ‘apples-and-oranges statistics’.

Cheers,
Helmut Schütz

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