Bioequivalence and Bioavailability Forum

Dear Yung-jin & Detlew,

❝ ❝ But as a tool to restrict the upper number of points to consider in the terminal phase it may be also used for other shapes. Hopefully the fit criterion you use (adjR2, AIC or whatever you prefer) will pick the linear part. In that sense I have used the TTT rule and found it giving reasonable results.

❝

❝ […] Yes, it should be picking linear part only. Since when it approaches "nonlinear" part, adjR² will start decreasing or AIC will increase in that case (theoretically?).

Maybe. Maybe not.
Scheerans et al. recommend as a starting point the intersection of the last two log/linear lines (“[…] the crossing of the imaginary first and second disposition phase lines in the (semi-logarithmic) plasma profile post C_max is generally used as a ‘visual marker’ for the beginning of the monoexponential terminal phase.”)¹ Unfortunately this point might be difficult to spot if (1) there is more than moderate noise and (2) distribution/elimination don’t differ too much.

I’m currently exploring this workflow:

1. Find TTT by the usual algo (see here).
   
2. Generate a subset of data in the interval [TTT, t_z].
   
3. Perform a 2-part linear stick-regression or MARS with a common intercept on log(C), where the initial estimate of the breakpoint δ = 2^nd time point after TTT (3 in my example below).
   Y = β₀ + β₁t + ifelse(t≥δ, β₂(t-δ), 0)
   
4. Set the starting point of the last phase similar to TTT: Use the first time point which is ≥ the estimated breakpoint δ.
   
5. Visual inspection still mandatory.

I used a simple two-compartment model without lagtime, parameterized in macro-constants C(t) = Ae^–αt + Be^–βt + Ce^–k_at, where C = –(A+B). No clear story yet…

If you want to play:
A 75, B 25, α 0.5, β 0.1, k_a 2, t {0, 0.25, 0.5, 1, 1.5, 2, 2.5, 3, 4, 6, 9, 12, 16, 24}
After adding some noise / rounding I got:
  0.25  29.63
  0.5   46.07
  1     53.68
  1.5   52.75
  2     45.99
  2.5   40.24
  3     34.87
  4     27.19
  6     17.59
  9     11.83
 12      7.39
 16      4.08
 24      2.75
t_max 1, TTT 2, time-interval for stick-regression {2, 24}, δ 7.73 → starting-point for λ_z 9.
   t { }    n   λz     %RE   R²adj
  1.5 – 24 10 0.1384 +38.36 0.9391 ← max. R²adj, min. AIC
  2   – 24  9 0.1342 +34.19 0.9382 ← TTT/max. R²adj, TTT/min. AIC
  2.5 – 24  8 0.1295 +29.47 0.9362
  3   – 24  7 0.1236 +23.65 0.9333
  4   – 24  6 0.1157 +15.68 0.9290
  6   – 24  5 0.1051 + 5.07 0.9206
  9   – 24  4 0.0950 – 5.04 0.8800 ← ‘breakpoint method’
 12   – 24  3 0.0777 –22.35 0.8183
The common algos would pick nine to ten values – resulting in extremely biased λ_z.² This demonstrates how important visual inspection of the fit is (aka don’t naïvely trust in automatic methods). The algo based on the breakpoint would pick four values, which is also the ‘best’ in terms of bias (–5.04%). By eye-balling likely I would choose the last three (bias –22.35%).

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1. Scheerans C, Derendorf H, Kloft C. Proposal for a Standardised Identification of the Mono-Exponential Terminal Phase for Orally Administered Drugs. Biopharm Drug Dispos. 2008;29(3):145–57. doi 10.1002/bdd.596
   
2. Phoenix by default and bear restrict the first value to the one following t_max. Classical WinNonlin’s algo is even more greedy and would include even t_max (!): R²_adj 0.9431, λ_z 0.1408, RE +40.80%… See also this post.

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P.S.: I would not use function lee() in package PK – smells already too much of modeling to me (sorry, Martin; see here). All methods detect two phases if the starting point is set to TTT, but…
lee(conc[6:14], time[6:14], method="ols"): last 3, λ_z 0.0777 (–22.35%)
lee(conc[6:14], time[6:14], method="lad"): last 4, λ_z 0.0824 (-17.62%)
lee(conc[6:14], time[6:14], method="hub"): last 3, λ_z 0.0777 (–22.35%)
lee(conc[6:14], time[6:14], method="npr"): last 4, λ_z 0.0973 (-2.73%)
Maybe a way forward would be to extract only the breakpoint?