Why use Gompertz instead of logistic for tumors?

Empirical tumor growth curves often show rapid early expansion and a prolonged slowdown near maximum size without the symmetric inflection of logistic growth. The Gompertz logarithmic factor r ln(K/V) enforces that slowdown because growth rate vanishes as V→K, similar to logistic, but the time course is skewed—useful as a teaching contrast on the same plot.

What does the chemotherapy term −kV mean?

It assumes the drug instantaneously removes a fraction of cells proportional to current volume—linear or first-order kill. Larger tumors lose more cells per unit time, a simple stand-in for cytotoxic therapy without modeling drug concentration, resistance, or cell-cycle phases.

Why integrate two trajectories at once?

The dashed gray curve is the same growth law with identical r, K, and V(0) but k=0. The green curve includes chemo after t_start. Their separation isolates the effect of treatment on the same biological parameters.

Can V exceed K or go negative?

The integrator clamps V to a tiny positive floor after each RK4 step. Mathematically both laws push V toward K when k=0; strong chemo can drive V well below K and in extreme slider settings toward the numerical floor—interpret that as near-eradication in this toy model, not a clinical prediction.

Tumor Growth (Gompertz / Logistic)

This simulator models tumor burden as a scalar volume V(t) approaching a maximum plateau K. Two classical growth laws are available. Logistic growth follows dV/dt = rV(1 − V/K), producing a symmetric S-shaped curve familiar from ecology. The Gompertz law dV/dt = rV ln(K/V) is widely used in oncology because it captures decelerating growth with an asymmetric, long-tailed approach to K—tumor cells proliferate quickly when small but slow markedly as they approach carrying capacity. Chemotherapy is represented in the simplest pharmacodynamic caricature as a proportional sterilizer: an additional term −kV added to dV/dt when treatment is active, modeling first-order cell kill by cytotoxic drugs. You can delay treatment with a start time t_start so the green treated curve initially tracks the dashed untreated reference, then diverges once k>0. The untreated trajectory is integrated in parallel with k=0 for direct comparison. The ODEs are advanced with fourth-order Runge–Kutta; a horizontal line marks K and a vertical marker shows when chemo begins. The model omits spatial heterogeneity, angiogenesis, immune response, resistance, and dose scheduling—extensions common in clinical modeling.

Who it's for: Students of mathematical biology, pharmacokinetics, or introductory oncology modeling who want to compare Gompertz versus logistic saturation and see how linear kill terms shrink tumor volume.

Key terms

Gompertz growth
Logistic growth
Tumor volume
Carrying capacity
Chemotherapy kill rate
First-order kinetics
Plateau
Runge–Kutta integration

How it works

Tumor volume V(t) with logistic or Gompertz growth toward a plateau K. Optional chemotherapy modeled as first-order kill −kV (a simple sterilizing drug). Compare treated trajectory (green) with an untreated reference (dashed) integrated in parallel.

Key equations

Logistic: V' = rV(1−V/K) − kV · Gompertz: V' = rV ln(K/V) − kV · k=0 or t<t_start → no chemo

Frequently asked questions

Why use Gompertz instead of logistic for tumors?: Empirical tumor growth curves often show rapid early expansion and a prolonged slowdown near maximum size without the symmetric inflection of logistic growth. The Gompertz logarithmic factor r ln(K/V) enforces that slowdown because growth rate vanishes as V→K, similar to logistic, but the time course is skewed—useful as a teaching contrast on the same plot.
What does the chemotherapy term −kV mean?: It assumes the drug instantaneously removes a fraction of cells proportional to current volume—linear or first-order kill. Larger tumors lose more cells per unit time, a simple stand-in for cytotoxic therapy without modeling drug concentration, resistance, or cell-cycle phases.
Why integrate two trajectories at once?: The dashed gray curve is the same growth law with identical r, K, and V(0) but k=0. The green curve includes chemo after t_start. Their separation isolates the effect of treatment on the same biological parameters.
Can V exceed K or go negative?: The integrator clamps V to a tiny positive floor after each RK4 step. Mathematically both laws push V toward K when k=0; strong chemo can drive V well below K and in extreme slider settings toward the numerical floor—interpret that as near-eradication in this toy model, not a clinical prediction.

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Key equations

Frequently asked questions

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