Bioequivalence and Bioavailability Forum

Hi John,

after administration of an IR formulation (if we have no IV data in the same subject) we assume that k_a > k_el (absorption is faster than elimination). In a controlled release formulation we try to slow down the absorption process. We reach flip-flop PK if k_a = k_el. Note that in the common formula we have FDk_a in the numerator and V(k_a–k_el) in the denominator. FDk_a divided by zero? Oops! Therefore, we use the given modification. Note, that in Phoenix/WinNonlin for the same reason we have Model #5 (and #6 with a lag-time).

❝ How did he arrive to k=1/Tmax?

Calculus followed by curve sketching and a little algebra. Start with the first derivative of the con­cen­tra­tion-time curve. Its root is at t_max (concentrations increasing before t_max = positive slope, decreasing after t_max = negative slope). Step-by-step:
The flip-flop models is

(1) C = kFD/Vtℯ^–kt

The first derivative is

(2) dC/dt = kFD/Vℯ^–kt – k²tFD/Vℯ^–kt

We know that at t_max dC/dt = 0. Therefore, we can write

(3) kFD/Vℯ^–kt_max = k²t_maxFD/Vℯ^–kt_max

Solving for t_max (shortening some stuff and dividing by k), we get

(4) k = k²t_max
(5) 1 = kt_max and finally
(6) t_max = 1/k

Rearranging (6) gives the k = 1/t_max you quoted above.

Example: k 0.25, F 1, D 100, V 10

k = 1/4 = 0.25, Q.E.D.
BTW, like in the usual one-compartment model the inflection point – where the tangent chances the “side” – is at the minimum of slopes (or the root of the 2^nd derivative) = 2×t_max.
If you go for a semilog-plot you will notice that we have no linear phase. Therefore, in PK modeling we have a problem. If we aim for a one-compartment model and get similar estimates of k_a and k_el (generally with large variances and overlapping CIs) it’s time to try a flip-flop model.

❝ I don't have the Gibaldi PK book on hand.

An overrated book, IMHO. Gibaldi/Perrier work most of the time with decadic logarithms and therefore have to introduce this stupid t_½ = 2.303/k… Not quite elegant didactics.