Bioequivalence and Bioavailability Forum • which time points to use for calculation of kel

Samir Malhotra
☆

India,
2012-06-12 12:59
(4685 d 07:50 ago)

Posting: # 8698
Views: 13,679

which time points to use for calculation of kel [NCA / SHAM]

Read with interest the discussion and posts of Detlew Labes about the calculation of terminal half life, especially of choosing the linear part.

Please help us in applying that to our real data from 16 volunteers who received a single oral dose of a new (numbered) compound. Which method (ARS, TTT, any other) should be used? And which time points?

Time    Mean Concentration

 0.5         18.63

 0.75        41.24

 1.0         78.27

 1.5        112.04

 2.0        121.33

 4.0         85.60

 6.0         41.57

 8.0         26.64

12.0         15.23

18.0          6.38

24.0          4.67

48.0          2.07

72.0          0.64

ElMaestro ★★★ Denmark, 2012-06-12 13:16 (4685 d 07:33 ago) @ Samir Malhotra Posting: # 8701 Views: 11,874	which time points to use for calculation of kel Post reply
	Hello Samir, this is to some extent a subjective matter. When I plot your points with logs applied then I think the last 4 points would be OK for this subject in this period. — Pass or fail! ElMaestro

Samir Malhotra ☆ India, 2012-06-12 13:28 (4685 d 07:21 ago) @ ElMaestro Posting: # 8702 Views: 11,967	which time points to use for calculation of kel Post reply
	Thanks ElMaestro That seems reasonable but please explaine why last 4 and not 3 or 5 or 6 Samir ❝ this is to some extent a subjective matter. ❝ When I plot your points with logs applied then I think the last 4 points would be OK for this subject in this period.

ElMaestro
★★★

Denmark,
2012-06-12 13:35
(4685 d 07:14 ago)

@ Samir Malhotra
Posting: # 8703
Views: 11,924

which time points to use for calculation of kel

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Hi Samir,

❝ That seems reasonable but please explaine why last 4 and not 3 or 5 or 6

This is exactly why I say it is a subjective matter - it is a matter of judgement, opinion, gut feeling, instinct, etc. If you think the last 3, 4, 5 or 6 points all give you a linear piece then by all means go ahead and use the number of points that you find appropriate. You have freedom to do so because no guideline dictates how it should be done. At the end of the day you need to be reasonable to the extent that your judgement pleases the regulator.

—
Pass or fail!
ElMaestro

Samir Malhotra
☆

India,
2012-06-12 13:59
(4685 d 06:50 ago)

@ Samir Malhotra
Posting: # 8704
Views: 11,894

which time points to use for calculation of kel

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Then another point, El (if I can say) --
this data is mean of 16 volunteers, do I need to check the plot for each volunteer or the plot of mean concentration. In the earlier posts it has been mentioned that looking at individual plots is not correct. What would be your opinion?
Thanks
samir

Edit: Full quote removed. Please delete anything from the text of the original poster which is not necessary in understanding your answer; see also this post! [Helmut]

ElMaestro
★★★

Denmark,
2012-06-12 14:39
(4685 d 06:10 ago)

@ Samir Malhotra
Posting: # 8705
Views: 11,842

which time points to use for calculation of kel

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Hi Samir,

❝ this data is mean of 16 volunteers, do I need to check the plot for each volunteer or the plot of mean concentration. In the earlier posts it has been mentioned that looking at individual plots is not correct. What would be your opinion?

I am sorry, I might have misunderstood you then. In BE you generally to go through all individual plots and derive an elimination constant. But if you are working with the overall mean plot here then you might not be doing BE but rather characterising a new chemical entity?
If so, then I guess you can consider the grand mean, pop PK as well as individual plots.

Perhaps you can briefly explain the overall purpose of the trial?

—
Pass or fail!
ElMaestro

Helmut
★★★

Vienna, Austria,
2012-06-12 17:59
(4685 d 02:50 ago)

@ ElMaestro
Posting: # 8710
Views: 12,005

Wetware required

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Hi Samir & ElMaestro!

❝ ❝ this data is mean of 16 volunteers,

Hopefully not the arithmetic means? Geometric means would make more sense.

❝ ❝ Which method (ARS, TTT, any other) should be used? And which time points?

As ElMaestro pointed out there is subjectivity in it. There’s no ‘one-size-fits-all’ (aka automatic) method which will work in all situations. For a one-compartment model both ARS and TTT generally work reasonably well (though I would prefer the latter). TTT is not suitable for 2+ compartments (see the original paper; authors recommend as a starting point the inflection of the curve). ARS might fail as well. No automatic method can handle multiple peaks.

❝ ❝ do I need to check the plot for each volunteer or the plot of mean concentration.

The former.

❝ ❝ In the earlier posts it has been mentioned that looking at individual plots is not correct. What would be your opinion?

Which post? Mean plots are nice but unsuitable to derive any PK conclusions. Imagine an extreme situation: You have two subjects with exactly the same PK, except a large difference in lag-times. The mean plot is nonsense.

[image]

❝ I am sorry, I might have misunderstood you then. In BE you generally to go through all individual plots and derive an elimination constant. But if you are working with the overall mean plot here then you might not be doing BE but rather characterising a new chemical entity?

❝ If so, then I guess you can consider the grand mean, pop PK as well as individual plots.

❝

❝ Perhaps you can briefly explain the overall purpose of the trial?

Yes, what are you trying to achieve? In NCA you estimate λ_z individually. If you want to present mean values, hard-core pharmacokineticists would present the harmonic mean and a jacknife standard deviation. Next comes the geometric mean. Arithmetic means are the worst, IMHO.

Concerning what’s the best: If I ignore the fact that this are means I can fit a PK model. Best was a two-compartment with lag-time, weights C_pred^-2. I got an elimination of 16.80456 h.

[image]

In NCA you must not used weights anyhow. I got (almost) the same values, both if the last 3 or 4 values are fitted. 3 values are slightly better, if I assume the model’s as the ‘true’ value. Bias -0.38% (3) and -0.49% (4).

If you want to do PK modeling you essentially have two options:

Classical 2-step: Fit subjects and calculate mean values of PK parameters. There’s a lot of literature which mean to use (harmonic means for rate constants/half lives, geometric means for volumes and clearances). This method works only if all subjects follow the same PK-model!
Population PK: Preferred nowadays. You may include covariates (e.g., sex, body weight, creatinine clearance,…) in the model in order to ‘explain’ part of the variance. Needs experience + patience.

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d_labes ★★★ Berlin, Germany, 2012-06-12 18:28 (4685 d 02:21 ago) @ Helmut Posting: # 8711 Views: 11,870	Geometric, arithmetic, median ... ? Post reply
	Dear Helmut, ❝ Hopefully not the arithmetic means? Geometric means would make more sense. Seems some members of this forum don't share your opinion . — Regards, Detlew

Helmut
★★★

Vienna, Austria,
2012-06-12 19:51
(4685 d 00:58 ago)

@ d_labes
Posting: # 8715
Views: 12,217

Geometric, arithmetic, median ... ?

Post reply

Dear Detlew!

❝ ❝ Hopefully not the arithmetic means? Geometric means would make more sense.

❝

❝ Seems some members of this forum don't share your opinion :cool: .

Ooh-ooh; asymptotically unbiased estimator. ;-)

Don’t understand what Martin meant in referring to Källén’s Figure 3.4 (a plot of geometric means of infusion data of two drugs).
But see the same author (pp48–49; THX to OCR):

3.5 Population average vs. subject-specific approach
When describing plasma profiles, mean values can be used. However, values below LOQ cannot simply be considered missing when computing these mean values, since they carry information. It is therefore important that trailing missing values, due to values below LOQ, are replaced with estimates, based on mono-exponentials using the estimated terminal elimination rate discussed in the previous section. Also other values below LOQ need to be filled in with an appropriate algorithm.
When computing mean values, it is preferable to compute geometric means. Much of the variability resides in dosing, at least from an extravascular site,^* which is a multiplicative factor to the plasma concentration, and by taking the geometric mean, we get the product of the geometric mean of doses and the geometric mean of dose one response curves (assuming linear kinetics).
It is important to know what a mean value curve represents. Its value at time t represents what the plasma concentration is expected to be, if we take one sample at that time for a randomly chosen individual after having dosed according to the schedule used. There is a potentially important difference between this mean curve and an individual curve. In order to illustrate the difference, assume a situation with a first order absorption profile and a one-compartment model such that all individuals in the world have the same plasma concentration curve, except that there is a time-lag until absorption starts. But assume that this time-lag varies substantially between individuals, so that the absorption starts much later for some individuals as compared to for others.
This is illustrated in Figure 3.2, in which we have shown five typical subject profiles, the middle of which is thicker than the others. This is a mean parameter curve, in this case the subject profile you get if you take as lag-time the mean value of all individual lag-times. It looks like the other curves and lies in the middle of the family.

[image]

The other thick curve is the mean curve, the curve obtained by taking the means of all the other (not only the five shown, but from the whole population). It looks quite different, and does not resemble an individual curve. But it tells us what the mean plasma concentration should be, if we fix a time point and sample many individuals at that time point.
This curve is the population average curve, and describing it the population average approach to data analysis. The mean parameter curve represents the subject specific approach to data analysis.
Strictly speaking the difference between the mean parameter curve and the mean curve makes most sense if we discuss a NONMEM application, in which there is a clear meaning of the former. But we can think of a mean parameter curve as “the most typical individual curve”, obtained from “an average individual”.

Concerning your question in the subject line – and in the spirit of Källén: I don’t know (“other values below LOQ need to be filled in with an appropriate algorithm”). Lag-times & values <LLOQ are nasty. ;-)

But on the other hand: Is this really important? We don’t base any conclusions on curves; they are simple illustrations. Experienced pharmacokineticists want to see spaghetti plots anyway. Who never was confronted with a question like “I measured the C_max of test and reference from the plot on page 4 as 65 and 70; so why do you state on page 3 the PE is only 88%?”

Wouldn’t say so. More likely F, V, CL. Content uniformity should protect us from substantial variability in D.

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d_labes
★★★

Berlin, Germany,
2012-06-13 10:53
(4684 d 09:55 ago)

@ Helmut
Posting: # 8722
Views: 11,777

Geometric, arithmetic, median ... ?

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Dear Helmut!

❝ But on the other hand: Is this really important? We don’t base any conclusions on curves; they are simple illustrations ...

That's "des Pudels Kern" (J.W. Goethe, "Faust, part I").
Beside curves with "nasty Lag-times & values <LLOQ" it seldom makes a difference.

❝ Who never was confronted with a question like “I measured the Cmax of test and reference from the plot on page 4 as 65 and 70; so why do you state on page 3 the PE is only 88%?” ...

I definitely was. Frequently from 'experienced ...' (fill in what fits your need) :-D

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2012-06-13 10:58
(4684 d 09:51 ago)

@ d_labes
Posting: # 8723
Views: 11,698

Geometric, arithmetic, median ... ?

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Dear Detlew!

❝ ❝ But on the other hand: Is this really important? We don’t base any conclusions on curves; they are simple illustrations ...

❝

❝ Beside curves with "nasty Lag-times & values <LLOQ" it seldom makes a difference.

Agree.

❝ ❝ Who never was confronted with a question like “I measured the Cmax of test and reference from the plot on page 4 as 65 and 70; so why do you state on page 3 the PE is only 88%?” ...

❝

❝ I definitely was. Frequently from 'experienced ...' (fill in what fits your need) :-D .

Will meet ElMaestro today. Will discuss and come up with a list to fill in the gap. ;-)

Edit: Sorry, cannot publish that. :-D

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jag009 ★★★ NJ, 2012-06-12 19:38 (4685 d 01:11 ago) @ Helmut Posting: # 8713 Views: 11,924	Wetware required Post reply
	Hi Helmut, ❝ ❝ ❝ this data is mean of 16 volunteers, ❝ ❝ Hopefully not the arithmetic means? Geometric means would make more sense. Really? I think it's more appropriate to present mean curve with arithmetic means. Can you elaborate? Thanks John

Helmut
★★★

Vienna, Austria,
2012-06-12 21:50
(4684 d 22:59 ago)

@ jag009
Posting: # 8718
Views: 11,921

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Hi John!

❝ Really? I think it's more appropriate to present mean curve with arithmetic means. Can you elaborate?

No, no – like in GxP: Guilty until proven innocent! Why do you think arithmetic means are more suitable?
I myself have great difficulties accepting negative concentrations in pharmacokinetics.^* What do you prefer: negative volume or negative mass? [image]

Perhaps this is the first glimpse of the existence of exotic matter and in the near future we will be able to construct a wormhole from a test tube containing plasma.

A consequence of using the arithmetic mean as the minimum-variance unbiased estimator of normal distributed data.

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jag009
★★★

NJ,
2012-06-12 19:40
(4685 d 01:09 ago)

@ Helmut
Posting: # 8714
Views: 11,824

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Hi Helmut,

❝ If you want to do PK modeling you essentially have two options:

❝ • Classical 2-step: Fit subjects and calculate mean values of PK parameters. There’s a lot of literature which mean to use (harmonic means for rate constants/half lives, geometric means for volumes and clearances). This method works only if all subjects follow the same PK-model!

What if some subjects do not fit with the same PK model?

Thanks

John

Helmut
★★★

Vienna, Austria,
2012-06-12 20:23
(4685 d 00:26 ago)

@ jag009
Posting: # 8716
Views: 12,156

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Hi John!

❝ What if some subjects do not fit with the same PK model?

Then I would consider the 2-step method nonsense. Average subjects with similar PK only. Imagine you get a long apparent (!) half life in a 1-compartment and a 2-compartment. In the former case this is linked to elimination from the central compartment¹ but in the latter case there are two possibilities

Slow elimination from V_c – like in the one-compartment case; fast exchange between central and peripheral (k₁₂ & k₂₁ >> k₁₀).
Fast elimination from V_c, but a slow distribution phase (either k₁₂ or k₂₁).

Now we end up with parameter identifiability problems.²
In comparing subjects following 1- & 2-compartment PK it would be possible to average the ‘central’ compartment and the elimination – but only in case #1, IMHO.

Nitpicking there is no ‘central’ compartment because we don’t have peripheral(s) in this model. The whole body is seen as a big bucket (yeah: our bones are assumed to be made out of the same stuff as the vitreous body. Funny idea). One-compartment PK doesn’t exist – sometimes the analytical method is not able to catch the slower phase.
Bonate PL. Pharmacokinetic-Pharmacodynami Modeling and Simulation.
‘Identifiability of Compartmental Models’, New York: Springer; 2006. p. 29–37.

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Samir Malhotra
☆

India,
2012-06-13 11:39
(4684 d 09:10 ago)

@ Helmut
Posting: # 8725
Views: 11,742

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A few clarifications: this is data from healthy volunteers aged 18-45y who received a single oral dose. The values are arithmetic mean. We wish to estimate PK parameters. The LLOQ is 1.96
Queries:
1. It was recommended in the Forum to use the last 4 time points for kel. Do we take the last four time points for each or look at the graph of each volunteer and decide (4 points for some and 3 or 5 for others)?
2. The last time point is 72 hours and there are several volunteers in whom no drug was detected. Do we write zero and include that into calculations or we do not take that time point at all for that particular volunteer ?
thanks

Helmut
★★★

Vienna, Austria,
2012-06-13 22:33
(4683 d 22:16 ago)

@ Samir Malhotra
Posting: # 8731
Views: 12,013

Fit individuals; ignore BQLs

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Dear Samir!

❝ A few clarifications: this is data from healthy volunteers aged 18-45y who received a single oral dose. The values are arithmetic mean. We wish to estimate PK parameters. The LLOQ is 1.96

THX for the information. If you want to estimate PK parameters, you have to fit a PK model. Unless you go with Population PK, the LLOQ is irrelevant (only measured concentrations are used). Assuming that the mean curve reasonably reflects all subjects go with a two-compartment extravascular model with a lag-time. There are three possible parameterizations:

Micro-constants: V₁/F (volume of distribution of the central compartment / bioavailability), k₀₁ (absorption rate constant), k₁₂ & k₂₁ (intercompartmental rate constants; central ↔ peripheral), k₀₁ (elimination rate constant), and t_lag (lag-time).
Clearances: As above, but instead of k₁₂, k₂₁, and k₀₁ clearances from the central and peripheral compartments and V₂/F are fitted.
Macro-constants: Data are fitted to a sum of three exponential terms; C_t=Aℯ^–αt+Bℯ^–βt+Cℯ^–k₀₁t

Whether #1 or #2 should be used is an almost religious debate of epic dimensions (see here). If you want to program everything from scratch, #3 is the way to go. Regardless which method you use for fitting, the respective other parameters can be calculated from the primary estimates – but the formulas are quite complicated (get a textbook). Here are all of them for your mean curve (assuming a dose of 100):

V1/F     0.5254

k01      0.9780

K10      0.2328

K12      0.09487

K21      0.06168

Tlag     0.3982

A      275.9

Alpha    0.3481

B       13.23

Beta     0.04125

CL/F     0.1223

V2/F     0.8082

CL×D/F   0.04984

Cmax   110.2

Tmax     2.097

AUC∞   817.6

But here is the bioequivalence forum. In BE we may only use NCA (Noncompartmental Analysis). No model (PK parameters); only PK metrics (aka PK variables) derived by one of the variants of the trapezoidal rule to calculate AUC_t and estimate λ_z. With my preferred method (the lin-up/log-down method and extrapolation based of the predicted C_t) I got:

Cmax   121.33

Tmax     2

λz       0.04145

Ct       0.64

Cpred    0.6796

AUCt   808.4

AUCinf 824.9

Vz/F     2.928

CL/F     0.1212

Note that λ_z from NCA (0.04145) is similar to β from the PK model (0.04125). This is not elimination (k₀₁ 0.2328), but a composite.

❝ 1. It was recommended in the Forum to use the last 4 time points for kel. Do we take the last four time points for each or look at the graph of each volunteer and decide (4 points for some and 3 or 5 for others)?

Where were 4 recommended? Doesn’t make sense. If I recall it correctly package bear for [image]

sets a limit of 4 points, but if this is true it’s time for an update.
You should fit as many time points as possible; the further ‘up the curve’ you can go (without leaving the linear range of log-transformed data) the better. Higher values are more reliable (analytical accuracy / precision). But: don’t go too far. The elimination must not be influenced by absorption (one-compartment) or distribution (two-compartments). That’s the reason behind the TTT-method and the inflection point. In your example 3 points are better than 4 (if compared to the model’s k₁₀). Here the subjectivity comes in. My gut feeling suggested 3 and ElMaestro’s 4… :-D

❝ 2. The last time point is 72 hours and there are several volunteers in whom no drug was detected. Do we write zero and include that into calculations or we do not take that time point at all for that particular volunteer ?

Fit subjects separately. Even if you get best fits from different numbers of samples after different formulations in the same subject, fine. In a 2×2 cross-over you get two fits/subject (e.g., test: 24-48-72, reference: 12-18-24-48). Write ‘BQL’ in the report. Zero will not work. Try it with C:\WINDOWS\system32\calc.exe: Set view to ‘scientific’, enter 0, click ln and see yourself what happens.

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