Clinical Pharmacokinetics & Biopharmaceutics
Oral Bateman PK: t_max, C_max & Bioavailability Formula
Calculate dose-normalized Absolute Bioavailability F, Harry Bateman’s oral peak time t_max = LN(k_a / k_el) / (k_a - k_el), peak plasma concentration C_max, and flip-flop kinetics flags.
Model extravascular oral absorption, peak exposure, and flip-flop kinetics!
When an oral drug of dose D_po absorbs from the gut at first-order rate k_a and clears from systemic volume V_d at rate k_el != k_a with Absolute Bioavailability F = (AUC_po * Dose_iv) / (AUC_iv * Dose_po), plasma concentration follows the Bateman equation C(t) = ((F * D_po * k_a) / (V_d * (k_a - k_el))) * (EXP(-k_el * t) - EXP(-k_a * t))! Setting dC/dt = 0 proves that Time to Peak Concentration t_max = LN(k_a / k_el) / (k_a - k_el) depends ONLY on rate constants k_a and k_el (completely independent of Dose and F!), while Peak Concentration simplifies at t_max to C_max = (F * D_po / V_d) * EXP(-k_el * t_max)!
Verify on Oral_Dose D_po = 400 mg (B2), Oral_AUC = 60 mg*hr/L (B3), Reference_IV_Dose D_iv = 200 mg (B4), Reference_IV_AUC = 40 mg*hr/L (B5), V_d = 30 L (B6), Absorption_Rate k_a = 1.00 hr^-1 (B7), and Elimination_Rate k_el = 0.25 hr^-1 (B8): Dose-Normalized Absolute Bioavailability is F = (60 * 200) / (40 * 400) = 0.7500 (75.00% — failing to normalize for 400 mg vs 200 mg would falsely claim F = 150%!); Time to Peak t_max = LN(1.00 / 0.25) / (1.00 - 0.25) = LN(4) / 0.75 = 1.8484 hr; and C_max = (0.75 * 400 / 30) * EXP(-0.25 * 1.84839) = 10 * 0.62996 = 6.2996 mg/L! Test sustained-release k_a = 0.10 < k_el = 0.25: terminal slope is governed by slow absorption ("Flip-Flop Kinetics (k_a < k_el)")!
| Extravascular Oral PK Metric (D_po=400, D_iv=200, k_a=1.0, k_el=0.25) | Formula | Verified Output |
|---|---|---|
| Absolute Bioavailability (F) | (AUC_po * Dose_iv) / (AUC_iv * Dose_po) | 0.7500 (75.00%) |
| Time to Peak Concentration (t_max) | LN(k_a / k_el) / (k_a - k_el) | 1.8484 hr |
| Peak Plasma Concentration (C_max) | (F * D_po / V_d) * EXP(-k_el * t_max) | 6.2996 mg/L |
| Kinetic Regime Check | IF(k_a < k_el, "Flip-Flop...", "Standard...") | Standard Absorption (k_a > k_el) |
Bateman Time-to-Peak t_max With Equal-Rate Limit (k_a = k_el) Guard Formula
When k_a = k_el, L’Hopital’s rule gives exact limit t_max = 1 / k_el and C_max = (F * D_po / V_d) * EXP(-1) — handling k_a = k_el explicitly prevents #DIV/0! in sensitivity tables!
=LET(ka_r, B7, ke_r, B8, IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, ka_r <= 0, ke_r <= 0), "Invalid Oral PK Input", ROUND(IF(ka_r = ke_r, 1 / ke_r, LN(ka_r / ke_r) / (ka_r - ke_r)), 4)))Peak Plasma Concentration C_max & Flip-Flop Regime Formula
Validate 0 < F <= 1.25 (bioequivalence outlier check) or positive PK inputs, and flag sustained-release flip-flop kinetics whenever k_a < k_el!
=LET(f_ba, (B3 * B4) / (B5 * B2), ka_r, B7, ke_r, B8, IF(ISTEXT(E3), E3, LET(t_mx, IF(ka_r = ke_r, 1 / ke_r, LN(ka_r / ke_r) / (ka_r - ke_r)), ROUND((f_ba * B2 / B6) * EXP(-ke_r * t_mx), 4))))Try the example
Use the original values or change them to test your understanding.