Bioequivalence and Bioavailability Forum

Dear Ratnakar!

❝ […] which method should be used/ generally recommended by regulatory authorities?

No specific recommendations except by the WHO (see below). State one in the protocol and stick to it.
BTW: see here.

❝ As one of our sponsor wants AUC to be calculated by Linear/Log Trapezoidal rule instead of linear trapezoidal by which we generally do the analysis.

In most situations the linear trapezoidal rule is positively biased if compared to the lin-up/log-down trapezoidal. The log-trapezoidal is negatively biased in the absorption phase. On the other hand if sampling points are properly chosen, in BE all formulations are expected to show the same bias – which should mean out in the comparison. For that reason I would guess most people use the linear trapezoidal rule for simplicity. I have never heard about a deficiency letter issued by a regulatory authority in that respect (e.g., if the linear rule was applied and the assessor is in a love-affair with another method). Most people use actual time points if deviating from scheduled ones. I had two deficiencies on my desk many years ago, where the scheduled ones were used (although stated in the protocol). An updated calculation changed the BE assessment (CI) in the second decimal place… See also this thread.

❝ After referencing it was found that there are following 4 methods […]

❝ by which we can calculate AUC:

❝ 1. Linear Trapezoidal

❝ 2. Linear/Log Trapezoidal

❝ 3. Linear Trapezoidal with Linear/Log Interpolation and

❝ 4. Linear Up and Log down Method

Seemingly your reference was WinNonlin’s manual?

❝ […] may be more?

Sure. For references see this post.

❝ Which of the method is more accurate […]

Most likely lin-up/log-down. If you want to get a nice estimate in the area of C_max, one of Purves’ methods should be considered.

❝ […] recommended by regulators or any regulatory guideline specifying about this?

The only one I know comes from the WHO (2006):

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 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.

❝ Also what is the difference between all these methods?

• Linear trapezoidal
  AUCi = AUCi-1 + ½(Ci + Ci-1) × (ti - ti-1), where i is the index of sampling time points t from ti=1=0 to ti=z (time point of the last concentration Cz ≥LLOQ). Most people set concentrations between time of administration and the first concentration ≥LLOQ = 0.
  
• Linear/log trapezoidal
  • If one concentration in a section is <LLOQ the linear trapezoidal rule is applied
    
  • If both concentrations in a section are ≥LLOQ and Ci ≠ Ci-1
AUCi = AUCi-1 + (ti - ti-1) × (Ci - Ci-1)/ln(Ci/Ci-1)
• Linear to t_max/log-down
  Guess
  
• Linear-up/log-down
  In every section concentrations are compared. If Ci>=Ci-1 the linear rule is applied and the linear/log otherwise

If you are using WinNonlin 5.2.1 and data are missing in the profile, beware! One might guess, that for missing values an interpolation is performed - which is not what you might expect…
Model Options > NCA Settings > Calculation Method > Linear Trapezoidal (Linear/Log Interpolation) gives the impression that a missing value where Ci>Ci-1 is linear interpolated and logarithmical if Ci<Ci-1. This is not the case. Let’s have a look at Clayton’s example data set, formulation t, subject 5:

t    C
0    0
0.5  0.12
1    1.85
1.5  2.92
2    1.85
3    0.99
4    0.73
6    0.42
8    0.19

AUC0-8 6.9475 (linear rule), 6.8176 (linear/log rule). So far, so good.
Actually WinNonlin is calculating AUC up to t_max according to the linear rule and log afterwards. Applying the linear/log-rule would give 6.6210!
Now we crack a virtual sample vial and set the three hour’s value to Missing.
Setting WinNonlin’s NCA-Options to Linear Trapezoidal (Linear Interpolation) we get AUC0-8 7.2475. OK, this is what we expect, since drawing a straight line between 2 and 4 hours should overestimate the AUC. Now we set Linear Trapezoidal (Linear/Log Interpolation) and get again 7.2475. Surprised?
My pocket calculator comes up 7.2475 for the linear rule, with 6.8010 for the logarithmic rule, and with 6.9976 for the linear/log rule.
Digging deeper into WinNonlin’s manual we read:

Linear Trapezoidal rule (Linear Interpolation). If a partial area is selected that has an endpoint that is not in the data set, then the linear interpolation rule, given below, is used to inject a concentration value for that endpoint. […]
Linear Trapezoidal Method with Linear/Log Interpolation. As above except when a partial area is selected that has an endpoint that is not in the data set. In that case, WinNonlin inserts a final concentration value using: 1) the linear interpolation rule, or 2) logarithmic interpola­tion if the endpoint is after Cmax, or after C0 for bolus models (if C0 > Cmax). If Cmax is not unique, then the first maximum is used.

In other words logarithmic interpolation is only used for partial areas; WinNonlin draws always a straight line between sampling time points enclosing missing values…

That’s why I have an SOP for data imputation in place.

PK-software vendors are members of the dark side of the Force.

• EquivTest/PK
  states in the manual:

  Under Missing Values, the user can select Linear Interpolation or No interpolation. If you select Linear Interpolation, then the system will linearly interpolate between any missing concentration values when calculating the AUC. If you select No Interpolation then any missing values will be excluded from the calculation of the AUC.

  Funny, leaving us the freedom to choose between Scylla and Charybdis. You can try it with your pocket calculator. Linear interpolation or omitting a time point give the same AUC. So, the user gets the impression he/she may select between two options – which actually give the same result.
  
• Kinetica 5.0 (and version 4.4.1)
  interpolates logarithmically and comes up with the correct result 6.9976!
  I used to write Thermo_Scientific until today, in this case I would switch to Thermo^Scientific.