Helmut ★★★ Vienna, Austria, 20091221 13:08 Posting: # 4517 Views: 11,982 

Dear colleagues, since I saw a couple of deficiency letters concerning C_{min} in the recent past, I want to start a little survey and discussion. The BEDraft (2008) states (lines 558559): For studies to determine bioequivalence at steady state AUC_{tau}, C_{max,ss}, and C_{min,ss} should be analysed using the same acceptance interval as stated above. and in lines 551552:For these parameters the 90% confidence interval for the ratio of the test and reference products should be contained within the acceptance interval of 80125%. Lines 561562 state:[…] for highly variable drugs the acceptance interval for C_{max} 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 C_{max} is justifiable for C_{min}.
Edit 1: In the final BE Guideline C_{min} 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 commentdocument (see this post) states at page 89: By C_{min,ss} we mean the concentration at the end of the dosage interval, i.e. C_{trough}. However, in bioequivalence studies for immediate release formulations there is no need to report C_{trough} and fluctuation. The guideline has been revised. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Belgium?, 20091222 14:17 @ Helmut Posting: # 4518 Views: 10,127 

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 predose 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 C_{trough}, with trough being defined as predose. My 0.02 euros. EM. 
Helmut ★★★ Vienna, Austria, 20091222 14:30 @ ElMaestro Posting: # 4519 Views: 10,378 

Dear ElMaestro! » Clearly I would use option 1b; if predose 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 C_{trough}, with trough being defined as predose. Agree with C_{trough}; but I would opt for 1d, because if a lagtime comes into play, concentrations will still decrease after time of administration. In such a case C_{trough} ('trough' is a minimum in the English language) is not to be expected at predose. Weimann* states: 3.5 Trough value PKsoftware 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 1bd). 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 t_{min} we would end up with a nonsensical location of τ/2 with high variance (141% ≥ CV% ≥100%). In other words, comparing the predose concentration to the concentration at the end of the dosing interval within subjects leads to some kind of ‘ApplesandOrangesStatistics’. The expectation of such an A&O comparison in any sample is 50% (and 25% for both predose and end of dose). On the other hand, the true minimum (regardless the location) is required in the calculation of %PTF or Swing.
— Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Belgium?, 20091227 00:47 @ Helmut Posting: # 4522 Views: 10,137 

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 C_{max} is generally based on the assumption that the residuals are normal. If we subsequently also want to qualify Log C_{min} with some kind of statistic can we then conduct such an analysis with parametric statistics? Once we identify Log C_{max} 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 C_{min} 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, 20091227 02:01 @ ElMaestro Posting: # 4523 Views: 10,075 

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 C_{max} is generally based on the assumption that the residuals are normal. Yep. » If we subsequently also want to qualify Log C_{min} with some kind of statistic can we then conduct such an analysis with parametric statistics? Once we identify Log C_{max} 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 C_{max}. » Hence, there are constraints on the magnitudes of the residuals in the analysis of Log C_{min} 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 logtransformed have limits of [>0, +∞], but in both cases the probability equals 1 (in other words, for C_{min} and C_{max} the logtransformation 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 The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Belgium?, 20091227 13:28 @ Helmut Posting: # 4525 Views: 10,085 

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 logtransformed have limits of [0, +∞], but in both cases the probability equals 1 (in other words, for C_{min} and C_{max} the logtransformation lead to the same lower limit – I wouldn’t call it a constraint – namely zero). Hmmmm, this I would express differently. C_{max} etc varies from 0 and upwards whilst the transformed (log) C_{max} 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 C_{max} in the dataset (log or not is not the important issue here). Now we do a parametric analysis of (log) C_{min} and we consider the (log) C_{min} 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) C_{max} is higher than average (log) C_{min} so average (log) C_{min} plus e will be lower than (log) C_{max} 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 meaningdisturbing typo. EM. 
Helmut ★★★ Vienna, Austria, 20091227 15:07 @ ElMaestro Posting: # 4526 Views: 10,102 

Dear ElMaestro, I agree with: » Hmmmm And also with: » C_{max} 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 logtransfom, the probability of a negative value is infinitesimaly small (nitpicking: it’s not defined, because log(0)=?, but if x→∞, ℯ^{x}→0). » Let's for simplicity consider the subject with the highest C_{max} in the dataset (log or not is not the important issue here). Now we do a parametric analysis of (log) C_{min} and we consider the (log) C_{min} 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) C_{max} is higher than average (log) C_{min} so average (log) C_{min} plus e will be lower than (log) C_{max} 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 C_{max}>C_{min} (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 t_{max}, not caring about the continous one of C_{max}. » […] I meant nonparametric. Sorry about the meaningdisturbing typo. Nonparametrics never disturb me. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Belgium?, 20091228 02:46 @ Helmut Posting: # 4528 Views: 9,978 

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 C_{max}. Rids me of any issue. Case closed. Was an elmaestrolophystic stray of thoughts. EM. Edit: [Helmut] 
d_labes ★★★ Berlin, Germany, 20100104 10:45 @ Helmut Posting: # 4555 Views: 10,151 

Dear Helmut, Ad 1) » How do you define C_{min} in steady state? 1.a) as standard, 1.a) or 1.c) as option Ad 1e) » Additional question: your rationale for either of the above 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 Ad 2) » Do you only report values for the formulations (geometric mean, sd) ... It depends (not at least on sponsor's wish ). Ad 3) » If the latter, do you use the metric in a confirmatory analysis (e.g., state an acceptance range)? ... See ad 2) Never used 3.c) Acceptance range usually 0.81.25, but also widening. Ad 4) » 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 ... ". Ad 5) » Do you see a change in point of views by European regulators ... None due to rare use of steady state studies. Ad 6) » Do you consider widening of the acceptance range in a replicate design in steady state (i.e., TTRRRRTT) ^^^^ See ad 5) Best wishes for the New Year Regards D. Labes 
Helmut ★★★ Vienna, Austria, 20100104 13:53 @ d_labes Posting: # 4556 Views: 10,100 

Dear D. Labes, thanks for participating! » » replicate design in steady state (i.e., TTRRRRTT) » ^^^^ Something like this:
saturation phase T T switch over R R This is my interpretation of EMA’s MRNfG (Section 4.1.2): The interindividual 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 intraindividual 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. 🚮 Science Quotes 
beman ☆ 20100104 15:43 @ Helmut Posting: # 4557 Views: 9,943 

Hi, I define the CminValue 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?
beman Edit: Linked to another thread. See the Policy. [Helmut] 
Helmut ★★★ Vienna, Austria, 20100104 16:34 @ beman Posting: # 4559 Views: 10,041 

Dear beman (reads BEman?), » I define the CminValue 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 C_{min}. 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 PKsoftware only for AUC_{τ}; see this post). You would have to set up something on your own. The second one IMHO is ‘applesandoranges statistics’. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 