Gompertz Growth Calculator

The Gompertz growth model describes how a population, tumor, or microbial colony grows over time when growth slows as the system approaches a fixed upper limit. Benjamin Gompertz published the underlying function in 1825 to describe human mortality, but it has since become the workhorse model for any biological system that follows an asymmetric S-shaped trajectory.

The formula (parameterized form):

N(t) = K · exp(-b · exp(-c · t))

Where K is the carrying capacity (asymptote), b sets the initial displacement from K, c is the growth rate constant, and t is time. The model produces a sigmoid curve that rises slowly at first, accelerates through an inflection point, and then decelerates as it approaches K.

Why Gompertz and not logistic? The logistic curve N(t) = K / (1 + a·exp(-r·t)) is symmetric about its inflection point. Real biological growth is rarely symmetric. Tumors, bacterial colonies, and organ growth all decelerate over a longer interval than they accelerate. Gompertz captures that asymmetry. The inflection point of a Gompertz curve sits at N = K/e ≈ 0.368·K, not at K/2 like logistic. That single number is the most useful diagnostic when fitting growth data: if the curve flattens out somewhere around 37% of the final value, Gompertz is the better choice.

Tumor growth in oncology: The Norton-Simon hypothesis, which underlies a large fraction of modern chemotherapy scheduling, assumes Gompertzian growth. A tumor that follows this model has a maximum growth rate when its volume is around K/e, and treatment that lands at that peak rate is far more effective than treatment given when the tumor is small (still accelerating) or near K (already decelerating). This is why some chemo protocols cycle aggressive doses around predicted peak-growth windows.

Bacterial growth in food safety: Predictive microbiology uses Gompertz to estimate how long a refrigerated food remains safe. Given K (the spoilage threshold), b (related to initial cell count), and c (temperature-dependent growth rate), the model predicts time-to-spoilage. The lag phase appears naturally: at small t, the inner exp term is large, so the outer exp(-b·exp(-c·t)) is essentially exp(of a large negative number) which is essentially zero, and the curve stays flat. As t grows, the inner term shrinks, and growth kicks in.

Worked example, bacterial culture: A culture starts with N₀ = 10³ cells, will saturate at K = 10⁹ cells, and has a growth rate c = 0.5 per hour. Find N at t = 12 hours.

First compute b from the initial condition: at t = 0, N(0) = K · exp(-b · 1) = K · exp(-b), so b = ln(K/N₀) = ln(10⁶) ≈ 13.82.

At t = 12: inner = b · exp(-c·t) = 13.82 · exp(-6) = 13.82 · 0.00248 = 0.0342.

N(12) = 10⁹ · exp(-0.0342) = 10⁹ · 0.9664 ≈ 9.66 × 10⁸ cells.

The culture is now at 97% of carrying capacity, near saturation.

Worked example, tumor doubling time: A 1 cm³ tumor grows toward K = 100 cm³ at rate c = 0.05/day. With b such that N(0) = 1, b = ln(100) = 4.605. At t = 30 days, inner = 4.605 · exp(-1.5) = 1.027, and N(30) = 100 · exp(-1.027) = 35.8 cm³. The tumor more than tripled in 30 days. Doubling time is not constant in Gompertz; it lengthens as the tumor grows, which is exactly the clinical observation.

Inflection point and peak growth: The maximum growth rate dN/dt occurs at N = K/e. At that point dN/dt = c·K/e ≈ 0.368·c·K. The time at which this happens, given b, is t_inflection = ln(b)/c.

Parameter intuition:

• K is the easy one. It is the saturation value.
• c controls how fast you reach K. Doubling c halves the time to any given fraction of K.
• b controls where you start. b = ln(K/N₀) means small N₀ gives large b (long lag); N₀ near K gives small b (essentially no growth phase left). When b < 1, the curve is already past inflection at t = 0.

When the model fails: Gompertz assumes a single growth phase with no external perturbation. It does not capture treatment events (a chemo cycle that knocks the tumor back), competition between subpopulations (multiple cell lines with different growth rates), or oscillatory behavior. For multi-phase systems, segmented Gompertz fits or full Lotka-Volterra dynamics are needed.