Gompertz Growth Model
Calculate population or tumor growth using the Gompertz model, which accounts for exponential slowdown as carrying capacity is approached.
The Formula
The Gompertz growth model describes a type of sigmoid (S-shaped) growth curve. Unlike simple exponential growth, the Gompertz model captures a key real-world phenomenon: growth starts slowly during a lag phase, accelerates to a maximum rate, then gradually decelerates as the population approaches an upper limit.
This model was originally proposed by Benjamin Gompertz in 1825 to describe human mortality rates. It has since become one of the most widely used models in biology, particularly in oncology for modeling tumor growth and in microbiology for modeling bacterial colony expansion.
The Gompertz curve is asymmetric, meaning the deceleration phase is longer than the acceleration phase. This distinguishes it from the logistic growth model, which produces a symmetric S-curve. Many biological systems exhibit this asymmetric pattern, making the Gompertz model a better fit than the logistic model in numerous real-world applications.
In tumor biology, the Gompertz model is especially valuable because tumors typically grow rapidly when small but progressively slow down as they outgrow their blood supply and nutrient availability decreases. Oncologists use this model to predict tumor doubling times, plan treatment schedules, and estimate the effectiveness of chemotherapy regimens.
In microbiology, the model helps predict bacterial growth in food products, which is critical for food safety and shelf-life estimation. The lag phase parameter is particularly important because it tells scientists how long bacteria remain dormant before entering rapid growth.
Variables
| Symbol | Meaning |
|---|---|
| N(t) | Population size at time t |
| K | Carrying capacity (maximum population or asymptotic value) |
| rm | Maximum specific growth rate |
| λ | Lag phase duration (time before exponential growth begins) |
| e | Euler's number (approximately 2.71828) |
| t | Time |
Example 1
A bacterial culture has K = 10⁹ cells, rm = 0.5/hour, and λ = 2 hours. Find the population at t = 6 hours.
Compute the inner exponent: (0.5 × 2.71828 / 10⁹) × (2 - 6) + 1
= (1.3591 × 10⁻⁹) × (-4) + 1 ≈ 1.0 (the exponent is very close to 1 for large K)
For practical use with large K, simplify: at t = 6, the culture is still in early exponential phase
N(6) ≈ 3.68 × 10⁸ cells (about 37% of carrying capacity)
Example 2
A tumor starts at 1 mm³ and follows Gompertz growth with K = 1000 mm³, rm = 0.1/day, and λ = 0 days. Estimate volume at t = 30 days.
Inner exponent: (0.1 × 2.71828 / 1000) × (0 - 30) + 1
= 0.000271828 × (-30) + 1 = -0.008155 + 1 = 0.9918
N(30) = 1000 × exp(-exp(0.9918)) = 1000 × exp(-2.697)
N(30) ≈ 1000 × 0.0673 ≈ 67.3 mm³
When to Use It
The Gompertz growth model is ideal whenever growth follows an asymmetric S-curve pattern.
- Modeling tumor growth in oncology research and treatment planning
- Predicting bacterial colony growth in food safety and microbiology
- Estimating plant growth patterns in agricultural science
- Analyzing organ growth during embryonic development
- Forecasting market saturation in business analytics (by analogy)
- Fitting survival curves in actuarial science and gerontology