Clinical Pharmacokinetics & Biopharmaceutics
One-Compartment IV Bolus PK: k_el, V_d & Clearance Formula
Calculate first-order elimination rate constant k_el, biological half-life t_1/2, back-extrapolated C_0, apparent volume of distribution V_d, and systemic clearance CL from two plasma draws.
Derive full clinical pharmacokinetic parameters from two timed plasma concentrations!
Under a one-compartment intravenous (IV) bolus model with instantaneous distribution and first-order elimination C(t) = C_0 * EXP(-k_el * t), two post-distribution plasma samples (C_1 at time t_1 and C_2 < C_1 at time t_2 > t_1) determine the entire exposure profile! First-Order Elimination Rate Constant is k_el = LN(C_1 / C_2) / (t_2 - t_1), Biological Half-Life is t_1/2 = LN(2) / k_el, Back-Extrapolated Time-Zero Concentration is C_0 = C_1 * EXP(k_el * t_1), Apparent Volume of Distribution is V_d = Dose / C_0, Total Systemic Clearance is CL = k_el * V_d, and Total Exposure AUC_0_inf = Dose / CL = C_0 / k_el!
Verify on IV_Bolus_Dose = 500 mg (B2), Sample_1_Time t_1 = 3.0 hr (B3), Sample_1_Conc C_1 = 10.0 mg/L (B4), Sample_2_Time t_2 = 9.0 hr (B5), and Sample_2_Conc C_2 = 2.5 mg/L (B6): Elimination Rate Constant k_el = LN(10.0 / 2.5) / (9.0 - 3.0) = LN(4) / 6 = 0.2310 hr^-1; Biological Half-Life t_1/2 = LN(2) / (LN(4) / 6) = 3.0000 hours; Back-Extrapolated C_0 = 10.0 * EXP(0.231049 * 3) = 20.00 mg/L; Apparent Volume of Distribution V_d = 500 / 20.00 = 25.00 L; and Total Systemic Clearance CL = k_el * V_d = 5.7762 L/hr (yielding AUC_0_inf = 500 / 5.776227 = 86.56 mg*hr/L)!
| IV Bolus PK Parameter (Dose=500 mg, C(3h)=10, C(9h)=2.5 mg/L) | Formula | Verified Output |
|---|---|---|
| Elimination Rate Constant (k_el) | LN(C_1 / C_2) / (t_2 - t_1) | 0.2310 hr^-1 |
| Biological Half-Life (t_1/2) | LN(2) / k_el | 3.0000 hr |
| Extrapolated Initial Conc & V_d | C_0 = C_1 * EXP(k_el * t_1); V_d = Dose / C_0 | C_0 = 20.00 mg/L; V_d = 25.00 L |
| Total Systemic Clearance (CL) | k_el * V_d = Dose / AUC_0_inf | 5.7762 L/hr |
Exact Unrounded-Chain V_d & Systemic Clearance Formula
Keep unrounded k_el inside LET when back-extrapolating C_0 = C_1 * EXP(k_el * t_1): rounding k_el to 0.23 before exponentiating introduces avoidable drift in V_d and therapeutic drug monitoring (TDM) dose adjustments!
=LET(ds_mg, B2, t_on, B3, c_on, B4, t_tw, B5, c_tw, B6, IF(OR(ds_mg <= 0, t_on < 0, t_tw <= t_on, c_on <= 0, c_tw <= 0, c_tw >= c_on), "Invalid TDM Sample Pair", LET(k_el, LN(c_on / c_tw) / (t_tw - t_on), c_zr, c_on * EXP(k_el * t_on), v_ds, ds_mg / c_zr, ROUND(k_el * v_ds, 4))))Elimination Half-Life & Apparent Volume of Distribution Formula
Validate t_2 > t_1 >= 0 and C_1 > C_2 > 0 (post-distribution declining phase).
=LET(ds_mg, B2, t_on, B3, c_on, B4, t_tw, B5, c_tw, B6, IF(OR(ds_mg <= 0, t_on < 0, t_tw <= t_on, c_on <= 0, c_tw <= 0, c_tw >= c_on), "Invalid TDM Sample Pair", LET(k_el, LN(c_on / c_tw) / (t_tw - t_on), ROUND(ds_mg / (c_on * EXP(k_el * t_on)), 2))))Try the example
Use the original values or change them to test your understanding.