yckim ☆ 20071127 01:46 Posting: # 1334 Views: 21,126 

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, 20071127 21:02 @ yckim Posting: # 1337 Views: 20,189 

Hi Yu! » Hi, its very nice to meet you. » The informations from this forum are very helpful. I very appreciate it. Thanks, nice to read... » 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 PKPDlist. Let’s start with an example – a picture may tell more than a thousand words: I generated data for a simple 1compartment model, where k_{a} = 1, k_{el} = 0.2, and F×V/CL = 100 (reference) / 95 (test). Since in BE we assume V and CL to be constant, and set D_{test} = D_{reference} (in most regulations) F_{rel} = AUC_{test} / AUC_{reference}. AUC based on the parameters (calculating the definite integral) is 100/k_{el}100/k_{a} for the reference and 95/k_{el}95/k_{a} or AUC_{reference} = 400 and AUC_{test} = 380. Therefore F_{rel} = 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 cutoff called LLOQ… I used this algorithm (t_{1} 0.5, t_{n} 24, 10 sampling points inbetween) 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. ┌──────┬──────┬──────┐ We can calculate the AUC by numerical integration up to 24h (t_{last}) for the reference, and only up to 16.92h (t_{last}) 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 (AUC_{t}) ┌───────────┬───────────┬────────┬────────┐ Method 2 Calculation up to the last time point where both formulations show concentrations above LLOQ (AUC_{p}) ┌───────────┬───────────┬────────┬────────┐ Method 3 Like Method #1, but a concentration of zero is assumed at the time point after t_{last}, i.e., a small triangle is added to AUC_{t}. The method is available in WinNonlin; this metric was named by Pharsight ‘AUC_{all}’ (you don’t find it in any textbook on PK – it’s Pharsight’s ‘invention’). ┌───────────┬───────────┬────────┬────────┐ Method 4 Extrapolation from t_{last} to t=∞ based on the observed concentration C_{last} (AUC_{inf}, AUC_{infobs}) ┌───────────┬───────────┬────────┬────────┐ Method 5 Extrapolation from t_{last} to t=∞ based on the predicted concentration C_{last} (AUC_{inf}, AUC_{infpred}) ┌───────────┬───────────┬────────┬────────┐ (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 opensource); 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:
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 AUC_{t} calculated by the linup/logdown 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.
— Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
yckim ☆ 20071128 01:44 @ Helmut Posting: # 1338 Views: 18,305 

Hi HS Thank you for your detailed explanation. Could you clarify "observed" and "predicted" concentration C_{last}? Doesn't "predicted" conc. mean extrapolated C_{last}(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, 20071128 02:49 @ yckim Posting: # 1339 Views: 17,915 

Dear Yu! » Thank you for your detailed explanation. Welcome! » Could you clarify "observed" and "predicted" concentration C_{last}? Doesn't "predicted" conc. mean extrapolated C_{last}(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 k_{el} 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): ┌───────────┬──────────┬──────────┐ Whereas k_{el} = slope, and C_{0} = ℯ^{intercept}. Therefore we can estimate the concentration (C_{pred}) at t_{last} by calculating C_{pred} = ℯ^{(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 k_{el}, it’s justified to use the estimated concentration – instead of the observed one – in extrapolating beyond t_{last}. 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 Edit: Gabrielsson and Weiner (see here) give a formula without requiring the intercept: C_{24} = C_{obs} · ℯ^{lambdaz·(24tobs)} or C_{24} = 3.22 · ℯ^{0.19992·(2416.92)} = 0.78… — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
jmlee ● 20081010 06:29 (edited by Ohlbe on 20081010 07:59) @ Helmut Posting: # 2503 Views: 17,944 

Dear HS. I always appreciate informations on this web site. By the way, I have some questions. I have calculated AUCinfpred (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. Please check my calculation 1. Concentration raw data (you have already presented) ┌──────┬──────┬──────┐ 2. AUCt calculation (by Method #1) ┌─────────────────┬──────────┬──────────┐ 3. Clast calculation ┌───────────┬──────────┬──────────┐ 2.1. Calculation of C_{last}  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. AUCinfpred calculation Formula : AUCt + Clast/Kel 1) AUCinfpred of ref = 401.99585 + 0.82033/0.20027 = 405.98213 2) AUCinfpred of test = 367.64200 + 3.22059/0.19992 = 383.75213 These values are not same your values.  AUCinfpred of ref : 405.98213(I) vs 405.93053 (yours)  AUCinfpred 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 AUCinfpred is not a same value between us? best regards, Jungmin Lee  Edit: Full quote removed. Please see this post! [Ohlbe] 
Helmut ★★★ Vienna, Austria, 20081010 15:48 @ jmlee Posting: # 2507 Views: 17,786 

Dear Jungmin! » I always appreciate informations on this web site. Thanks! » I have calculated AUCinfpred (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. »  AUCinfpred of ref : 405.98213(I) vs 405.93053 (yours) »  AUCinfpred 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 t_{last} to t=∞ based on the predicted concentration C_{last} (AUC_{inf}, AUC_{infpred}) ┌───────────┬───────────┬────────┬────────┐ » And, could you please explain why AUCinfpred 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, 20140319 18:15 @ jmlee Posting: # 12672 Views: 11,534 

Dear Jungmin, Helmut and All, I cannot undestand these formulas: » 2.1. Calculation of C_{last} »  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 Please correct me — Kind regards, Mittyri 
d_labes ★★★ Berlin, Germany, 20140320 07:26 @ mittyri Posting: # 12674 Views: 11,334 

Dear mittyri, » I cannot undestand these formulas: » » 2.1. Calculation of C_{last} » »  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 
Ohlbe ★★★ France, 20071128 09:37 @ Helmut Posting: # 1340 Views: 17,652 

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, 20071128 11:28 @ Ohlbe Posting: # 1341 Views: 17,845 

Dear Ohlbe! » Hey, that's a nice post and a nice demonstration. Thanks! » By the way, nice new home page, too. 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 AUC_{t} 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× t_{max} point estimates do not essentially change any more; only variability increases. The worst metric in terms of variability is AUC_{0∞}. » I couldn't find it either in the WHO guideline, which mentions "Clast" and defines it as "last measurable concentration". OK, the entire paragraph reads: Area under the plasma/serum/blood concentration–time curve from time zero to time t (AUC_{0–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 AUCvalues should be specified. In general AUC should be calculated using the linear/log trapezoidal integration method. The exclusive use of compartmentalbased parameters is not recommended. So the recommendation goes with Method #1 (not #2!).Further down: AUC_{0–t} and C_{max} are considered to be the most relevant parameters for assessment of bioequivalence. In addition it is recommended that the following parameters be estimated: It’s nice to see AUC_{0–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, 20071128 16:58 @ Helmut Posting: # 1342 Views: 17,850 

Dear all! » Hopefully I will post a little collection the next days. As promised… IMHO everybody dealing with NCA and BEstatistics should get the pioneering paper by Sauter et al. (1992)! Sauter R, Steinijans VW, Diletti E, Böhm E, Schulz HU. 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.), BioInternational 2  Bioavailability, Bioequivalence and Pharmacokinetic Studies. Stuttgart: medpharm; 1995. p. 51–60. A nice comparison of Methods #4 and #5 in Chapter 3.7 (NonCompartmental Analysis); linup/logdown trapezoidal recommended variant of Method #1: Gabrielsson J, Weiner D. Pharmacokinetic an Pharmacodynamic Data Analysis: Concepts and Applications. Stockholm: Swedish Pharmaceutical Press; 3^{rd} ed. 2000. p. 141–153. For a dicussion of different methods see also: Pabst G. Area under the concentrationtime curve. In: Cawello W (Ed.), Parameters for CompartmentFree Pharmacokinetics. Aachen: ShakerVerlag; 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 