Statistical & Mathematical Formulas
Gompertz Growth Curve: Double-Exponential Volume & K/e Inflection
Model decelerating tumor spheroid, microbial, and SaaS adoption curves via Laird’s Gompertz equation N(t) = K * EXP(-LN(K / N_0) * EXP(-α * t)) and locate the exact peak-growth inflection time t_inf = LN(LN(K / N_0)) / α.
Why biological tumors inflect at K / e (36.8%) instead of logistic K / 2 (50%)!
Verhulst’s symmetric logistic S-curve assumes per-capita growth slows linearly with population size and peaks symmetrically at 50% of carrying capacity K. Real solid tumors, bacterial colonies, and mature market curves lose proliferative fraction exponentially over time (Anna K. Laird, British Journal of Cancer 1964), obeying Benjamin Gompertz’s double-exponential law: N(t) = K * EXP(-LN(K / N_0) * EXP(-α * t))!
Because the inner growth rate decays as EXP(-α * t), maximum absolute growth speed dN/dt occurs strictly EARLY at N_inf = K / e = 0.3679 * K, reached at time t_inf = LN(LN(K / N_0)) / α! Track a N_0 = 10 mm^3 tumor spheroid approaching vascular plateau K = 1,000 mm^3 with decay constant α = 0.15 day^-1: peak growth strikes at t_inf = 10.18 days (367.88 mm^3), reaches 568.97 mm^3 by day 14, and plateaus at 950.13 mm^3 by day 30!
| Growth Time Horizon (N0 = 10, K = 1,000, α = 0.15/d) | Gompertz Size N(t) | % of Plateau K | Dynamic Growth Phase |
|---|---|---|---|
| Asymmetric Inflection Peak (t_inf = 10.18 d) | 367.88 mm^3 | 36.79% (= 1 / e!) | Maximum Absolute dN/dt Rate |
| Two-Week Checkpoint (t = 14.00 d) | 568.97 mm^3 | 56.90% | Post-Inflection Deceleration |
| One-Month Plateau (t = 30.00 d) | 950.13 mm^3 | 95.01% | Asymptotic Carrying Capacity |
Gompertz Double-Exponential Growth Trajectory N(t)
Evaluates K * EXP(-LN(K / N_0) * EXP(-α * t)) for initial size 0 < N_0 < K and decay rate α > 0.
=IF(OR(B2 <= 0, B3 <= B2, B4 <= 0, B5 < 0), "Invalid Gompertz Inputs", ROUND(B3 * EXP(-LN(B3 / B2) * EXP(-B4 * B5)), 2))Exact Inflection Time t_inf = LN(LN(K / N_0)) / α
Returns the time when N(t) crosses K / e (if N_0 < K / e; negative values indicate initial N_0 already started past inflection).
=IF(OR(B2 <= 0, B3 <= B2, B4 <= 0), "Invalid Gompertz Inputs", ROUND(LN(LN(B3 / B2)) / B4, 2))Try the example
Use the original values or change them to test your understanding.