Steady-State Accumulation R_ac & Loading Dose Formula

Predict drug accumulation and hit steady-state target exposure on Dose 1!

When a maintenance dose D_maint is administered repeatedly every tau hours for a drug with elimination half-life t_1/2 (so k_el = LN(2) / t_1/2), each dose builds upon residual plasma drug until elimination per interval equals D_maint (reaching ~97% of steady state after 5 half-lives)! By geometric-series summation 1 + r + r^2 + ... with residual fraction r = EXP(-k_el * tau), the Exact Accumulation Index is R_ac = 1 / (1 - EXP(-k_el * tau))! Consequently, Steady-State Peak is C_max,ss = (D_maint / V_d) * R_ac, Steady-State Trough is C_min,ss = C_max,ss * EXP(-k_el * tau), and administering Immediate Loading Dose D_load = D_maint * R_ac achieves steady-state C_max,ss immediately without waiting 4–5 half-lives!

Verify on Maintenance_IV_Dose D_maint = 250 mg (B2), Elimination_Half_Life t_1/2 = 8.0 hr (B3 -> k_el = LN(2)/8 = 0.08664 hr^-1), Dosing_Interval tau = 8.0 hr (B4, dosing once every half-life!), and Volume_of_Distribution V_d = 25.0 L (B5 -> Single-Dose C_0 = 10.00 mg/L): Residual Fraction per interval is EXP(-LN(2) * 8 / 8) = 0.5000, Accumulation Index R_ac = 1 / (1 - 0.5000) = 2.0000x, Steady-State Peak C_max,ss = 10.00 * 2.0000 = 20.00 mg/L, Steady-State Trough C_min,ss = 20.00 * 0.5000 = 10.00 mg/L, and Immediate Loading Dose D_load = 250 * 2.0000 = 500.00 mg! Widen interval to tau = 16.0 hr (2 half-lives): residual fraction drops to 0.25 and R_ac falls to 1.3333x!

Accumulation Index R_ac Formula

Expressing the exponent directly as -LN(2) * (tau / t_1/2) eliminates intermediate k_el rounding error!

=LET(dm_mg, B2, th_hr, B3, tu_hr, B4, vd_lt, B5, IF(OR(dm_mg <= 0, th_hr <= 0, tu_hr <= 0, vd_lt <= 0), "Invalid Dosing Regimen", ROUND(1 / (1 - EXP(-LN(2) * (tu_hr / th_hr))), 4)))

Steady-State Trough C_min,ss & Exact Loading Dose Formula

Verify therapeutic window compliance: C_max,ss below toxicity ceiling and C_min,ss above minimum inhibitory/effective concentration (MIC/MEC)!

=LET(dm_mg, B2, th_hr, B3, tu_hr, B4, vd_lt, B5, IF(ISTEXT(E2), E2, LET(f_rs, EXP(-LN(2) * (tu_hr / th_hr)), r_ac, 1 / (1 - f_rs), ROUND((dm_mg / vd_lt) * r_ac * f_rs, 2))))

Further reading

• Arte Sheets Lesson — One-Compartment IV Bolus PK: k_el, V_d & Clearance Formula
• Arte Sheets Lesson — Mosteller BSA & Cockcroft-Gault Creatinine Clearance Formula