Bioequivalence and Bioavailability Forum

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:

1. 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).
   
2. Clearances: As above, but instead of k₁₂, k₂₁, and k₀₁ clearances from the central and peripheral compartments and V₂/F are fitted.
   
3. 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 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…

❝ 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.