Oncology Pharmacology & Biostatistics
Bliss Independence & Loewe Combination Index Formula
Calculate Chester Bliss’s Probabilistic Additive Inhibition E_Bliss = E_A + E_B - E_A * E_B, Excess Over Bliss (EOB), and Loewe / Chou-Talalay Isobole Combination Index CI = d_A / D_{x,A} + d_B / D_{x,B}.
Stop summing fractional inhibitions E_A + E_B and test drug synergy with Bliss & Loewe!
In combination oncology and antimicrobial screening, summing single-agent fractional cell-kill rates E_A + E_B is a classic probability error (two drugs each killing 60% of tumor cells cannot kill 120% of cells!). Under Chester Bliss’s (1939) Probabilistic Independence model—where surviving fraction (1 - E_AB) equals (1 - E_A) * (1 - E_B)—Expected Additive Inhibition is E_Bliss = E_A + E_B - E_A * E_B, and Excess Over Bliss EOB = E_AB - E_Bliss > 0 signals probabilistic synergy! Alongside Bliss, Siegfried Loewe (1928) and Ting-Chao Chou & Paul Talalay (1984) evaluate dose-sparing isoboles at observed effect level x = E_AB: if combination doses (d_A, d_B) achieve the same kill x that requires D_{x,A} of Drug A alone or D_{x,B} of Drug B alone, the Combination Index is CI = d_A / D_{x,A} + d_B / D_{x,B}, where CI < 1 proves Synergy, CI = 1 Additivity, and CI > 1 Antagonism!
Verify on Drug_A_Inhibition E_A = 0.40 (40%, B2), Drug_B_Inhibition E_B = 0.50 (50%, B3), Observed_Combo_Inhibition E_AB = 0.85 (85%, B4), Combo_Dose_dA = 20 nM (B5), Alone_Isoeffective_DxA = 80 nM (B6), Combo_Dose_dB = 15 nM (B7), and Alone_Isoeffective_DxB = 60 nM (B8): Bliss Expected Inhibition is 0.40 + 0.50 - 0.20 = 0.7000 (70.00% surviving 0.60 * 0.50 = 30.0%, E2); Excess Over Bliss is 0.85 - 0.70 = +0.1500 (+15.00% in E3); Loewe Combination Index is 20 / 80 + 15 / 60 = 0.25 + 0.25 = 0.5000 (requiring half the additive dose, E4); and Classification confirms "Synergistic (CI < 1)" (E5)! Test antagonistic interference (E_AB = 0.60, d_A = 60, d_B = 45): EOB = -0.1000 and CI = 1.5000 ("Antagonistic (CI > 1)")!
| Drug Combination Metric (EA=0.40, EB=0.50, EAB=0.85; dA=20/80, dB=15/60) | Formula | Verified Output |
|---|---|---|
| Bliss Expected Additive Effect | E_A + E_B - E_A * E_B | 0.7000 (70.00% Kill) |
| Excess Over Bliss (EOB) | E_AB - (E_A + E_B - E_A * E_B) | +0.1500 (+15.00% Lift) |
| Loewe / Chou-Talalay CI | d_A / D_{x,A} + d_B / D_{x,B} | 0.5000 (2x Dose-Sparing) |
| Pharmacodynamic Interaction | CI < 1 Synergy; CI > 1 Antagonism | Synergistic (CI < 1) |
Loewe / Chou-Talalay Combination Index (CI) Formula
Validate all fractional inhibitions in [0, 1] and positive concentrations d_A, D_{x,A}, d_B, D_{x,B} > 0!
=LET(e_a, B2, e_b, B3, e_ab, B4, d_a, B5, dx_a, B6, d_b, B7, dx_b, B8, IF(OR(e_a < 0, e_a > 1, e_b < 0, e_b > 1, e_ab < 0, e_ab > 1, d_a <= 0, dx_a <= 0, d_b <= 0, dx_b <= 0), "Invalid Synergy Assay", ROUND(d_a / dx_a + d_b / dx_b, 4)))Excess Over Bliss & Isobole Classification Formula
Use a 0.0001 tolerance around CI = 1.0000 to classify exact sham-combination additivity cleanly!
=IF(ISTEXT(E4), E4, IF(ABS(E4 - 1) < 0.0001, "Additive (CI = 1)", IF(E4 < 1, "Synergistic (CI < 1)", "Antagonistic (CI > 1)")))Try the example
Use the original values or change them to test your understanding.