Tumour growth
Mathematical tumour-growth models describe how tumour cell number, mass or volume changes through time. Their purpose is not to give one universal law for cancer, but to provide simplified growth rules that can be compared with data and biological mechanisms.
Choose the state variable first
Let \(N(t)\) denote a measure of tumour size. Depending on the study, \(N\) might represent cell number, volume, mass or another measurable quantity.
The basic form is
\[\boxed{\frac{dN}{dt}=\text{net tumour growth rate}.}\]The right-hand side combines proliferation and loss into one net growth law.
Exponential growth
The simplest model assumes that every tumour cell contributes the same constant net growth rate:
\[\boxed{\frac{dN}{dt}=rN.}\]Its solution is
\[\boxed{N(t)=N_0e^{rt}.}\]Here \(r>0\) is the net per-capita growth rate.
Why exponential growth accelerates
If the tumour doubles in size, there are twice as many cells contributing to future growth. Therefore the absolute growth rate \(rN\) also doubles.
This creates accelerating growth and a constant doubling time
\[\boxed{t_d=\frac{\ln2}{r}.}\]Where exponential growth is useful
It can be a useful approximation over an early interval when resource limitation and spatial constraints are not yet strongly affecting growth.
However, indefinite exponential growth is biologically unrealistic because it predicts unlimited acceleration.
Logistic growth
The logistic model introduces a carrying-capacity scale \(K\):
\[\boxed{\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).}\]When \(N\ll K\), growth is approximately exponential. As \(N\) approaches \(K\), net growth slows.
Logistic solution
\[\boxed{N(t)=\frac{K}{1+\left(\frac{K-N_0}{N_0}\right)e^{-rt}}.}\]The curve is sigmoidal and the absolute growth rate is largest at \(N=K/2\).
Gompertz growth
\[\boxed{\frac{dN}{dt}=rN\ln\left(\frac{K}{N}\right).}\]Its solution is
\[\boxed{N(t)=K\exp\left[\ln\left(\frac{N_0}{K}\right)e^{-rt}\right].}\]The per-capita growth rate \(r\ln(K/N)\) decreases continuously as \(N\) increases.
Compare the three models
The following curves use the same initial size \(N_0=1\). For the bounded models, \(K=100\). Parameters are chosen only to illustrate qualitative differences.
Absolute growth versus per-capita growth
| Model | Per-capita growth rate |
|---|---|
| Exponential | \(r\) |
| Logistic | \(r(1-N/K)\) |
| Gompertz | \(r\ln(K/N)\) |
Why growth may slow biologically
Possible mechanisms include nutrient or oxygen limitation, spatial crowding, necrosis, immune interactions, mechanical constraints and changes in the proliferating fraction. A simple growth law compresses many such mechanisms into a few parameters.
Carrying capacity is a model parameter
In logistic and Gompertz models, \(K\) is a limiting scale in the equation. It should not automatically be interpreted as a fixed physical maximum imposed by one specific mechanism.
Volume and radius
If a tumour is approximated as a sphere of radius \(R\),
\[V=\frac{4}{3}\pi R^3.\]A growth law written for radius is therefore not mathematically identical to the same-looking law written for volume.
Treatment as an additional loss term
\[\boxed{\frac{dN}{dt}=G(N)-T(N,t).}\]For example, \(T(N,t)=u(t)N\) gives a simple proportional time-dependent treatment effect. This is a modelling structure rather than a clinical dosing rule.
Extensions
More detailed models may distinguish sensitive and resistant populations, represent spatial tumour density with reaction–diffusion equations, or include stochastic birth, death and mutation events.
Parameter estimation and uncertainty
Growth parameters are estimated from longitudinal measurements. Different models can fit limited data similarly well, measurement error can be substantial, and parameters such as \(K\) may be poorly identifiable if observations cover only the early growth phase.
A practical modelling workflow
Define the measured tumour-size variable. Plot the data. Start with a simple growth law. Fit parameters and inspect residuals. Compare alternative models. Quantify uncertainty. Add treatment, heterogeneity, spatial structure or stochasticity only when the biological question and data support the extra complexity.