Bioequivalence and Bioavailability Forum

Dear all,

TTT is the time of the inflection (t_inpt) of the plasma profile (or the minimum of the first derivative = the root of the second derivative). If we have a lagtime, the inflection occurs not at 2×t_max but is shifted by t_lag (if you don’t trust me, do some calculus yourself).*

Simple example (noise-free data, parameters of this post differing only in lagtimes – 0 and 2):
  t       C1      C2
  0       BQL     BQL
  0.25  29.917    BQL
  0.5   45.403    BQL
  1     54.577    BQL
  1.5   51.966    BQL
  2     46.228    BQL  ← tinpt1
  2.5   40.284  45.403
  3     35.007  54.577
  4     26.875  46.228 ← tinpt2
  6     17.454  26.875 ← picked by TTT-algo: not the inflection point!
  9     10.997  14.679
 12      7.716   9.702
 16      5.073   6.233
 24      2.268   2.771
 48      0.206   2.251
tlag     0       2     ← simple definition: time point before the first C ≥LLOQ
tmax     1       3
TTT      2       6
λz-range 2–48    6–48
λz       0.115   0.107 ← different estimates
t_max in green, t_inpt in red.
Although true elimination is identical we get different estimates because in the second case t_inpt is not at 6 but already at 4 (=2×t_max–t_lag). C_inpt1=C_inpt2=46.228. Therefore, I suggest to introduce a »TTT’« which corrects for lagtimes and gives similar estimates:
TTT’     2       4
λz-range 2–48    4–48
λz       0.115   0.113 ← similar estimates

Theoretically AUC_∞ should be identical for both profiles (not affected by lagtimes), i.e., 75/0.5+25/0.1–100/2=350. Let’s compare estimated AUCs (by the lin-up/log-down trapezoidal rule):
based on TTT  AUC∞    367.19   371.15
              bias     +4.91%   +6.04%
              extrap    5.35%    6.98%
based on TTT’ AUC∞    367.19   369.88
              bias     +4.91%   +5.37%
              extrap    5.35%    6.66%
With the second algo bias is smaller and similar for both profiles.

This is not (only) academic nitpicking, but of practical interest. In the recent MR draft EMA considered the possibility of waiving the multiple dose study if the residual area in single dose (truncated at τ) is <10% of AUC_∞. For the lagtime-profile we would get 7.48% (TTT) and 7.14% (TTT’).

Does anybody have a clever idea how I can find the lagtime in R without a loop? Essentially I would search backwards from t_max until I find the first NA.

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• Before you go crazy in curve sketching: The equations can be solved only for a one-compartment model (w/wo lagtime) in closed form. In R optimize() and deriv() is the way to go. If you want to take a sledgehammer to crack a nut: Get package numDeriv.