yckim
☆

2007-11-27 02:46

Posting: # 1334
Views: 20,561

## Calculation of AUCt [NCA / SHAM]

Hi, its very nice to meet you.
The informations from this forum are very helpful. I very appreciate it.

I have a question.
how do you calculate the AUCt and do statistical analysis if there are concentrations below LOQ (at early or late sampling point)? As I know, the 't' time in AUCt should be the same time. Someone recommend to make it zero in the early time. And they also recommend to delete it in the late time to calculate the AUCinf. Is it meaningful to do statistical analysis of AUCt with different 't' value? Please let me know how to proceed the study in such a case.

Sorry for the poor english. ^^
Thank you.

Yu Chul Kim

Edit: Category changed. [Helmut]
Helmut
★★★

Vienna, Austria,
2007-11-27 22:02

@ yckim
Posting: # 1337
Views: 19,605

## Calculation of AUCt, AUCall, AUCinf...

Hi Yu!

» Hi, its very nice to meet you.
» The informations from this forum are very helpful. I very appreciate it.

» how do you calculate the AUCt and do statistical analysis if there are concentrations below LOQ (at early or late sampling point)? As I know, the't' time in AUCt should be the same time. Someone recommend to make it zero in the early time. And they also recommend to delete it in the late time to calculate the AUCinf. Is it meaningful to do statistical analysis of AUCt with different 't' value? Please let me know how to proceed the study in such a case.

This topic comes up from time to time – not only on this forum, but also at the PKPD-list.

Let’s start with an example – a picture may tell more than a thousand words:
I generated data for a simple 1-compartment model, where ka = 1, kel = 0.2, and F×V/CL = 100 (reference) / 95 (test).
Since in BE we assume V and CL to be constant, and set Dtest = Dreference (in most regulations)
Frel = AUCtest / AUCreference.
AUC based on the parameters (calculating the definite integral) is 100/kel-100/ka for the reference and 95/kel-95/ka or AUCreference = 400 and AUCtest = 380. Therefore Frel = 380/400 = 95%. Fine.
Unfortunately we don’t have theoretical curves, but discrete sampling intervals, a lot of noise (biological and – to a much (!) lesser extent – analytical), and an artificial cut-off called LLOQ…
I used this algorithm (t1 0.5, tn 24, 10 sampling points in-between) to establish a sampling schedule. Sampling times were rounded to 2 decimal places, as well as the simulated concentrations; LLOQ was set to 0.8.
Obviously the concentration at the 24h sampling point was just above LLOQ for the reference (0.82), and BLQ (0.78) for the test.
┌──────┬──────┬──────┐ │  t   │ ref  │ test | ├──────┼──────┼──────┤ │ 0.00 │ 0.00 │ 0.00 | │ 0.50 │29.83 │28.34 | │ 0.71 │37.60 │35.72 | │ 1.01 │45.29 │43.02 | │ 1.44 │51.28 │48.72 | │ 2.05 │53.49 │50.82 | │ 2.91 │50.43 │47.91 | │ 4.14 │42.10 │39.99 | │ 5.89 │30.51 │28.99 | │ 8.37 │18.73 │17.79 | │11.90 │ 9.25 │ 8.79 | │16.92 │ 3.39 │ 3.22 | │24.00 │ 0.82 │(0.78)| └──────┴──────┴──────┘

We can calculate the AUC by numerical integration up to 24h (tlast) for the reference, and only up to 16.92h (tlast) for the test…

An enlarged view of the the last section of the profile is given below:

Now let’s start with the calculations (based on the linear trapezoidal rule for simplicity). We will assess the methods based on the bias (deviation from the theoretical value of T/R = 95%).

Method 1:
Calculation up to the last measured concentration for each formulation (AUCt)
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 401.88595 │ 367.64200 │ 91.48% │ -3.71% │ └───────────┴───────────┴────────┴────────┘

Method 2
Calculation up to the last time point where both formulations show concentrations above LLOQ (AUCp)
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 386.98255 │ 367.64200 │ 95.00% │  0.00% │ └───────────┴───────────┴────────┴────────┘

Method 3
Like Method #1, but a concentration of zero is assumed at the time point after tlast, i.e., a small triangle is added to AUCt. The method is available in WinNonlin; this metric was named by Pharsight ‘AUCall’ (you don’t find it in any textbook on PK – it’s Pharsight’s ‘invention’).
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 401.88595 │ 379.04080 │ 94.32% │ -0.72% │ └───────────┴───────────┴────────┴────────┘

Method 4
Extrapolation from tlast to t=∞ based on the observed concentration Clast (AUCinf, AUCinf-obs)
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 405.98047 │ 383.74829 │ 94.52% │ -0.50% │ └───────────┴───────────┴────────┴────────┘

Method 5
Extrapolation from tlast to t=∞ based on the predicted concentration Clast (AUCinf, AUCinf-pred)
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 405.93053 │ 383.74829 │ 94.54% │ -0.49% | └───────────┴───────────┴────────┴────────┘
(erroneous; see this post for corrections)

Now let’s look at the Pros and Cons.

Method #1 showed the largest bias, but has its merits. The method is available in all standard software (both commercial and open-source); the bias seen in the example most likely diminishes in ‘real world’ data sets. Pragmatically speaking: this method has already been applied in tenthousands of BE studies…

Although in this simple example Method #2 showed the smallest bias,1 I would not recommend it because:
• You would just throw away valuable information (all concentrations of the ‘longer’ profile after tlast of the ‘shorter’ profile will not be used).
• The method is not standard in any software I know. It would be rather tricky to get the right numbers out from a list of partial areas.
• You would make the life of inspectors miserable.
Method #3 is based on a flawed assumption. There’s only one concentration we simply know that it does not exist after a dose: Zero. Although it’s reasonable to start with a zero-concentration before the dose (the first triangle which is used in all methods), in some situations AUCall > AUCinf – which lead to a lot of questions in the past. This happens if a next sampling interval is quite large, and the estimated half life is comparable small. See also this post.

Extrapolation based on Method #4 is recommended by the FDA, but considered suboptimal (nicely speaking) by PK experts. Using this method we rely on the concentration with the largest variability (both inaccuracy and precision).
It’s definitely better to go with Method #5,2 which is standard according to the WHO, European, and many other countries’ guidelines.

Conclusion:
It’s like already discussed in this thread on sampling times; whatever method you choose, lay it down in the protocol and calculate all AUCs in the same manner. AUC is a quite robust metric, unless you have a relatively high LLOQ and large differences between formulations deviations simply mean out.
My personal recommendation would be AUCt calculated by the lin-up/log-down method followed by extrapolation by Method #5.

» Sorry for the poor english.

Oh, there are just a few native speakers here; we do all our best to communicate – somehow.

1. Edit (years later). Why not?
Fisher D, Kramer W, Burmeister Getz E. Evaluation of a Scenario in Which Estimates of Bioequivalence Are Biased and a Proposed Solution: tlast (Common). J Clin Pharm. 2016;56(7):794–800. doi:10.1002/jcph.663. free resource.
2. Method #5 is the default in EquivTest/PK (though #4 is available as well).

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
yckim
☆

2007-11-28 02:44

@ Helmut
Posting: # 1338
Views: 17,786

## Calculation of AUCt, AUCall, AUCinf...

Hi HS

Thank you for your detailed explanation.

Could you clarify "observed" and "predicted" concentration Clast?
Doesn't "predicted" conc. mean extrapolated Clast(24 h) of test drug? I wonder why the AUC values of the test are same in method 4 and 5.

Best regards,
Yu Chul Kim
Helmut
★★★

Vienna, Austria,
2007-11-28 03:49

@ yckim
Posting: # 1339
Views: 17,396

## Calculation of AUCt, AUCall, AUCinf...

Dear Yu!

» Thank you for your detailed explanation.

Welcome!

» Could you clarify "observed" and "predicted" concentration Clast? Doesn't "predicted" conc. mean extrapolated Clast(24 h) of test drug?

The last observed/measured concentrations are 0.82 at 24h (reference) and 3.22 at 16.92h (test). The 24h value of the test (0.78) is just below the LLOQ and is given in the table for demonstrational purposes only.
If we estimate kel from the last three values above the LLOQ for each formulation (linear regression of t vs. ln C), we obtain (interval of reference 11.90h – 24.00h, and test 8.37h – 16.92h):
┌───────────┬──────────┬──────────┐ │ parameter │    ref   │   test   │ ├───────────┼──────────┼──────────┤ │ slope     │ -0.20027 │ -0.19992 │ │ intercept │  4.60838 │  4.55224 │ └───────────┴──────────┴──────────┘
Whereas kel = |slope|, and C0 = ℯintercept.
Therefore we can estimate the concentration (Cpred) at tlast by calculating
Cpred = (intercept+kel·tlast).

» I wonder why the AUC values of the test are same in method 4 and 5.

Because in my simple example there was no noise in the data; only due to rounding to two decimal places small differences are observed. The observed and predicted concentrations show a bias of 0.04% (reference) and 0.02% (test) only. This will not be the case with ‘real world’ data. Since we already have agreed in using a couple of data points (≥3) in the estimation of kel, it’s justified to use the estimated concentration – instead of the observed one – in extrapolating beyond tlast.

Edit: If you use WinNonlin, check "Output intermediate calculations" in the Model Options. You will get the Intercept in "NCA Text", but not in the result worksheet. Example:
Intermediate Output -------------------   Value for Lambda_z:      0.2003, and intercept:     4.6084   Value for Lambda_z:      0.1999, and intercept:     4.5522

Edit: Gabrielsson and Weiner (see here) give a formula without requiring the intercept:
C24 = Cobs · ℯ-lambdaz·(24-tobs) or C24 = 3.22 · ℯ-0.19992·(24-16.92) = 0.78…

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
jmlee
●

2008-10-10 08:29
(edited by Ohlbe on 2008-10-10 09:59)

@ Helmut
Posting: # 2503
Views: 17,429

## Calculation of AUCt, AUCall, AUCinf...

Dear HS.
I always appreciate informations on this web site.

By the way, I have some questions.

I have calculated AUCinf-pred (Method #5) according to your explanations.
But, I couldn't get the same result as you did.

So, I present the calculation process at a lower column.

1. Concentration raw data (you have already presented)
┌──────┬──────┬──────┐ │  t   │ ref  │ test | ├──────┼──────┼──────┤ │ 0.00 │ 0.00 │ 0.00 | │ 0.50 │29.83 │28.34 | │ 0.71 │37.60 │35.72 | │ 1.01 │45.29 │43.02 | │ 1.44 │51.28 │48.72 | │ 2.05 │53.49 │50.82 | │ 2.91 │50.43 │47.91 | │ 4.14 │42.10 │39.99 | │ 5.89 │30.51 │28.99 | │ 8.37 │18.73 │17.79 | │11.90 │ 9.25 │ 8.79 | │16.92 │ 3.39 │ 3.22 | │24.00 │ 0.82 │ 0.78 | └──────┴──────┴──────┘

2. AUCt calculation (by Method #1)
┌─────────────────┬──────────┬──────────┐ │      AUC        │  ref     │  test    | ├─────────────────┼──────────┼──────────┤ │ AUC(0.00~0.50)  │  7.45750 │  7.08500 | │ AUC(0.50~0.71)  │  7.08015 │  6.72630 | │ AUC(0.71~1.01)  │ 12.43350 │ 11.81100 | │ AUC(1.01~1.44)  │ 20.76255 │ 19.72410 | │ AUC(1.44~2.05)  │ 31.95485 │ 30.35970 | │ AUC(2.05~2.91)  │ 44.68560 │ 42.45390 | │ AUC(2.91~4.14)  │ 56.90595 │ 54.05850 | │ AUC(4.14~5.89)  │ 63.53375 │ 60.35750 | │ AUC(5.89~8.37)  │ 61.05760 │ 58.00720 | │ AUC(8.37~11.90) │ 49.38470 │ 46.91370 | │ AUC(11.90~16.92)│ 31.72640 │ 30.14510 | │ AUC(16.92~24.00)│ 14.90340 │   -      | │ AUCt            │401.88595 │367.64200 | └─────────────────┴──────────┴──────────┘

3. Clast calculation
┌───────────┬──────────┬──────────┐ │ parameter │    ref   │   test   │ ├───────────┼──────────┼──────────┤ │ slope     │ -0.20027 │ -0.19992 │ │ intercept │  4.60838 │  4.55224 │ └───────────┴──────────┴──────────┘

2.1. Calculation of Clast
- ref (last time = 24.00)
: 0.82 × (-0.20027)+4.60838 = -0.19804
--> e(-0.19804) = 0.82033

- test (last time = 16.92)
: 3.22 × (-0.19992)+4.55224 = 1.16956
--> e(1.16956) = 3.2206

3. AUCinf-pred calculation
Formula : AUCt + Clast/Kel

1) AUCinf-pred of ref
= 401.99585 + 0.82033/0.20027
= 405.98213

2) AUCinf-pred of test
= 367.64200 + 3.22059/0.19992
= 383.75213

These values are not same your values.
- AUCinf-pred of ref : 405.98213(I) vs 405.93053 (yours)
- AUCinf-pred of test : 383.75213(I) vs 383.74829 (yours)

There's some severe mistakes in my calculation? (please check..)

And, could you please explain why AUCinf-pred is not a same value between us?

best regards,
Jungmin Lee

--
Edit: Full quote removed. Please see this post! [Ohlbe]
Helmut
★★★

Vienna, Austria,
2008-10-10 17:48

@ jmlee
Posting: # 2507
Views: 17,247

## Calculation of AUCt, AUCall, AUCinf...

Dear Jungmin!

» I always appreciate informations on this web site.

Thanks!

» I have calculated AUCinf-pred (Method #5) according to your explanations.
» But, I couldn't get the same result as you did.

Oops!

» These values are not same your values.
» - AUCinf-pred of ref : 405.98213(I) vs 405.93053 (yours)
» - AUCinf-pred of test : 383.75213(I) vs 383.74829 (yours)
»
» There's some severe mistakes in my calculation? (please check..)

You are absolutely correct; your values are right, whereas mine are wrong. Updated table:
Method 5
Extrapolation from tlast to t=∞ based on the predicted concentration Clast (AUCinf, AUCinf-pred)
┌───────────┬───────────┬────────┬────────┐ │   ref     │   test    │  T/R   │ % Bias │ ├───────────┼───────────┼────────┼────────┤ │ 405.98213 │ 383.75123 │ 94.52% │ -0.50% | └───────────┴───────────┴────────┴────────┘

» And, could you please explain why AUCinf-pred is not a same value between us?

Stupidity?
I’m greateful for your correction – sometimes I’m hit by the devil of copy & paste…

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
mittyri
★★

Russia,
2014-03-19 19:15

@ jmlee
Posting: # 12672
Views: 11,021

## Calculation of AUCt, AUCall, AUCinf...

Dear Jungmin, Helmut and All,

I cannot undestand these formulas:
» 2.1. Calculation of Clast
» - ref (last time = 24.00)
» : 0.82 X (-0.20027)+4.60838 = -0.19804
0.82 X (-0.20027)+4.60838 = 4.4442
<...>
» - test (last time = 16.92)
» : 3.22 X (-0.19992)+4.55224 = 1.16956
3.22 X (-0.19992)+4.55224 = 3.9085

Kind regards,
Mittyri
d_labes
★★★

Berlin, Germany,
2014-03-20 08:26

@ mittyri
Posting: # 12674
Views: 10,820

## Errors happen

Dear mittyri,

» I cannot undestand these formulas:
» » 2.1. Calculation of Clast
» » - ref (last time = 24.00)
» » : 0.82 X (-0.20027)+4.60838 = -0.19804
» 0.82 X (-0.20027)+4.60838 = 4.4442
» <...>
» » - test (last time = 16.92)
» » : 3.22 X (-0.19992)+4.55224 = 1.16956
» 3.22 X (-0.19992)+4.55224 = 3.9085

Me too !
You are absolutely correct in questioning these formulas. What here would be correct is:
24*(-0.20027)+4.60838 = -0.1981
and
16.92*(-0.19992)+4.55224=1.169594

Regards,

Detlew
fno
☆

Belgium,
2014-03-20 11:03

@ d_labes
Posting: # 12676
Views: 10,804

## Errors happen

» Me too !
» You are absolutely correct in questioning these formulas. What here would be correct is:
» 24*(-0.20027)+4.60838 = -0.1981
» and
» 16.92*(-0.19992)+4.55224=1.169594

And then (from Jungmin's AUCt),

For ref:
Clast_pred=exp(-0.1981)=0.82029
AUCinf_pred=401.88595+0.82029/0.20027=405.98186

For test:
Clast_pred=exp(1.169594)=3.22068
AUCinf_pred=367.642+3.22068/0.19992=383.75186

T/R=94.5244%
Bias=-0.5006%

Kind regards,
Fabrice
Ohlbe
★★★

France,
2007-11-28 10:37

@ Helmut
Posting: # 1340
Views: 17,143

## Calculation of AUCt, AUCall, AUCinf...

Dear HS,

Hey, that's a nice post and a nice demonstration. By the way, nice new home page, too.

» It's definitely better to go with Method #5, which is standard according to the WHO, European any many other countries' guidelines.

I couldn't find any recommendation on this in the EU BE guideline, which only states that the method used to calculate AUC should be specified, or in the old PK in man guideline. Anyway the EU BE guideline recommends the use of AUCt as the "most reliable estimate of the extent of absorption" anyway. I couldn't find it either in the WHO guideline, which mentions "Clast" and defines it as "last measurable concentration". I couldn't say about BA trials (where there would be more modelling anyway) but I think I've only seen once a BE trial using Method #5.

Is there any recommendation I have missed in EU or WHO guideline, or made at a conference ?

Regards
Ohlbe
Helmut
★★★

Vienna, Austria,
2007-11-28 12:28

@ Ohlbe
Posting: # 1341
Views: 17,330

## Calculation of AUCt, AUCall, AUCinf...

Dear Ohlbe!

» Hey, that's a nice post and a nice demonstration.

Thanks!

Oh, I just refurbished the CSS; that's the wonderful thing with valid (X)HTML – if you want to change the layout of an entire website, only a single file has to be modified. Theoretically.

» » It’s definitely better to go with Method #5, which is standard according to the WHO, European any many other countries’ guidelines.

» […] Anyway the EU BE guideline recommends the use of AUCt as the "most reliable estimate of the extent of absorption" anyway.

Without going into my files - the main parameter for extent of BA was changed from the 1992 version in 1998; and I’m still happy with it. There are a lot of papers published demonstrating for partial AUCs that beyond 2–4× tmax point estimates do not essentially change any more; only variability increases. The worst metric in terms of variability is AUC0-∞.

» I couldn't find it either in the WHO guideline, which mentions "Clast" and defines it as "last measurable concentration".

Area under the plasma/serum/blood concentration–time curve from time zero to time t (AUC0–t), where t is the last sampling time point with a measurable concentration of the API in the indi­vi­du­al formulation tested. The method of calculating AUC-values should be specified. In general AUC should be calculated using the linear/log trapezoidal integration method. The exclusive use of compartmental-based parameters is not recommended.

So the recommendation goes with Method #1 (not #2!).

Further down:

AUC0–t and Cmax are considered to be the most relevant parameters for assessment of bio­equi­va­lence. In addition it is recommended that the following parameters be estimated:
- area under the plasma/serum/blood concentration–time curve from time zero to time infinity (AUC0-∞) representing total exposure, where AUC0-∞ = AUC0–t + Clast/β; Clast is the last mea­sur­able drug concentration and β is the terminal or elimination rate constant calculated according to an appropriate method;

It’s nice to see AUC0–t as the recommended metric – but it’s a pity to go with Method #4 for extrapolation.

» Is there any recommendation I have missed in EU or WHO guideline, or made at a conference ?

Not in the guidelines; in papers, workshops & conferences – yes.
Perhaps I will post a little collection the next days.

» I couldn't say about BA trials (where there would be more modelling anyway) but I think I've only seen once a BE trial using Method #5.

Really? Was it one of my hundreds?
OK, to be more serious – I used Method #4 in only 3 out of >500 studies, because I failed in convincing these sponsors in the reasonability of applying Method #5.
Method #5 is my standard and also in a couple of CROs which are rather more scientifically than “cook­book”-driven in their evaluations. On the other hand I never got any deficiency letter claiming that I should recalculate my results obtained with Method #5 by Method #4.
Just the fact that Method #5 is implemented in commercial software packages (WinNonlin, Kinetica) gives you a hint that it must be used by somebody – except myself.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Helmut
★★★

Vienna, Austria,
2007-11-28 17:58

@ Helmut
Posting: # 1342
Views: 17,335

## References

Dear all!

» Hopefully I will post a little collection the next days.

As promised…

IMHO everybody dealing with NCA and BE-statistics should get the pioneering paper by Sauter et al. (1992)!

Sauter R, Steinijans VW, Diletti E, Böhm E, Schulz H-U. Presentation of results from bioequivalence studies. Int J Clin Pharm Ther Toxicol. 1992; 30(Suppl1): S7–S30.

Method #1 considered a better metric than Method #5 according to simulations:

Tozer TN, Bois FY. Metrics of Absorption; Simulation Approach. In: Blume HH, Midha KK (Eds.), Bio-International 2 - Bioavailability, Bioequivalence and Pharmacokinetic Studies. Stuttgart: medpharm; 1995. p. 51–60.

A nice comparison of Methods #4 and #5 in Chapter 3.7 (Non-Compartmental Analysis); lin-up/log-down trapezoidal recommended variant of Method #1:

Gabrielsson J, Weiner D. Pharmacokinetic an Pharmacodynamic Data Analysis: Concepts and Applications. Stockholm: Swedish Pharmaceutical Press; 3rd ed. 2000. p. 141–153.

Pabst G. Area under the concentration-time curve. In: Cawello W (Ed.), Parameters for Compartment-Free Pharmacokinetics. Aachen: Shaker-Verlag; 2003. p. 65–80.

Method #4 is given in (F5.9), and Method #5 as an alternative in (F5.9a).

Chapter 6 (Presentation of bioequivalence studies) from a recent book:

Hauschke D, Steinijans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007. p. 123–155.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
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