Cancer treatment models

Cancer treatment models extend tumour-growth equations by adding therapy as a process that changes tumour growth, survival or composition through time.

Core intuition. Untreated tumour cells tend to proliferate. Treatment pushes in the opposite direction by reducing growth or increasing cell loss. The mathematical outcome depends on the balance between those effects and on how treatment changes through time.

Start with untreated growth

Let \(N(t)\) be tumour size. An untreated model might be

\[\frac{dN}{dt}=G(N),\]

where \(G(N)\) could be exponential, logistic, Gompertz or another growth law.

Treatment adds an additional term:

\[\boxed{\frac{dN}{dt}=G(N)-T(N,t).}\]

The term \(T(N,t)\) represents treatment-induced reduction of viable tumour burden.

The simplest proportional treatment effect

A simple model assumes that treatment removes tumour cells at a rate proportional to tumour size:

\[\boxed{T(N,t)=u(t)N.}\]

Then

\[\boxed{\frac{dN}{dt}=G(N)-u(t)N.}\]

The function \(u(t)\) represents treatment intensity or an effective cell-kill rate in the model.

Constant treatment against exponential growth

If untreated growth is exponential,

\[G(N)=rN,\]

and treatment is constant, \(u(t)=u\), then

\[\frac{dN}{dt}=(r-u)N.\]

The solution is

\[\boxed{N(t)=N_0e^{(r-u)t}.}\]

This immediately gives a threshold:

\[\boxed{u>r\quad\Longrightarrow\quad N(t)\text{ declines}.}\]

If \(u

Treatment threshold graph

Each curve is generated from \(N(t)=N_0e^{(r-u)t}\) for the same tumour growth rate \(r\) but different treatment intensities \(u\). The change from growth to decline occurs at \(u=r\).

Why this simple model is useful

It shows the fundamental competition clearly:

\[\text{net growth rate}=\text{tumour growth rate}-\text{treatment kill rate}.\]

But real therapies rarely maintain a constant effective kill rate forever, so more realistic models make treatment time-dependent.

Link treatment to drug concentration

Suppose a pharmacokinetic model provides a drug concentration \(C(t)\). Then the treatment effect can depend on that concentration:

\[\boxed{\frac{dN}{dt}=G(N)-k(C(t))N.}\]

This creates a PK–tumour model:

\[\text{dose}\longrightarrow C(t)\longrightarrow k(C(t))\longrightarrow N(t).\]

A saturating drug-kill function

A common effective form is

\[\boxed{k(C)=k_{\max}\frac{C}{EC_{50}+C}.}\]

At low concentration, increasing \(C\) strongly increases treatment effect. At high concentration, the killing rate approaches the maximum \(k_{\max}\).

This prevents the model from assuming that treatment effect can increase without bound.

Concentration-dependent treatment graph

Generated directly from \(k(C)=k_{max}C/(EC_{50}+C)\). The marked point is the exact half-maximal treatment effect.

Why dose and effect are not the same

Dose determines input into the pharmacokinetic system. Concentration then changes through absorption, distribution and elimination. The tumour effect depends on concentration or exposure through a pharmacodynamic relationship.

Therefore

\[\boxed{\text{dose}\neq\text{concentration}\neq\text{tumour effect}.}\]

Pulsed treatment

Many treatments are given intermittently. A simple treatment schedule can be represented as a time-dependent function \(u(t)\) that is high during treatment windows and low or zero between them.

The tumour can then shrink during treatment periods and regrow between them.

A model-generated treatment schedule

The following example uses logistic tumour growth with a periodic treatment term:

\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-u(t)N.\]

The treatment is switched on for short intervals every 10 time units.

The curve is the numerical solution of the displayed logistic-treatment ODE with the stated periodic treatment schedule. Shrinkage and regrowth are generated by the model, not hand-drawn.

Why tumour size can rebound between treatments

When treatment is active, the net growth rate may become negative. When treatment stops, the treatment term falls and the remaining tumour cells can resume growth.

This produces cycles of reduction and regrowth in some model structures.

Fractional kill

If a treatment removes a fixed fraction of the current tumour population, then larger tumours lose more cells in absolute number than smaller tumours.

A discrete treatment event can be represented as

\[\boxed{N^+=(1-f)N^-,}\]

where \(f\) is the killed fraction, \(N^-\) is tumour size immediately before treatment and \(N^+\) immediately after.

This creates a hybrid model: continuous growth between treatment events, with discrete jumps at treatment times.

Continuous versus discrete treatment models

Continuous treatment term	Discrete treatment event
effect acts continuously through time	effect occurs at selected treatment times
represented inside the ODE	represented by a jump rule
useful for infusions or effective continuous exposure	useful for idealised pulses or fractions removed per treatment

Radiotherapy-style survival models

A common idealised radiobiological description uses a surviving fraction after dose \(d\):

\[\boxed{S(d)=e^{-\alpha d-\beta d^2}.}\]

Then an idealised treatment event maps

\[N^+=S(d)N^-.\]

The parameters \(\alpha\) and \(\beta\) describe linear and quadratic components of the modelled radiation response.

Important. This is a modelling relation used to study radiation-response principles. It is not a clinical prescription rule.

Fractionation

Radiotherapy is commonly divided into fractions. Mathematically, fractionation changes the sequence of survival events and regrowth periods.

A model can therefore compare how changing the timing and number of treatment fractions alters the theoretical tumour trajectory under specified assumptions.

Sensitive and resistant tumour cells

A single tumour population assumes all cells respond similarly. A more realistic model may separate

\[S(t)=\text{sensitive cells},\qquad R(t)=\text{resistant cells}.\]

One simple structure is

\[\frac{dS}{dt}=G_S(S,R)-k_S(C)S,\]\[\frac{dR}{dt}=G_R(S,R)-k_R(C)R,\]

with

\[k_R(C)for a treatment to which the resistant population responds less strongly.

Why resistance can emerge even if resistant cells grow more slowly

Treatment changes the competitive environment. Sensitive cells may be removed strongly while resistant cells survive. The resistant fraction can therefore increase even if resistant cells had no advantage before treatment.

This is an example of treatment-driven selection.

A simple resistant-fraction example

If sensitive and resistant cells grow exponentially but have different treatment sensitivities,

\[\frac{dS}{dt}=(r_S-u_S)S,\qquad \frac{dR}{dt}=(r_R-u_R)R.\]

The ratio \(R/S\) changes according to the difference between their net growth rates.

This makes it clear why total tumour size and tumour composition are separate modelling questions.

Healthy tissue and toxicity

Treatment can affect normal tissue as well as tumour cells. A model may include a healthy-cell variable \(H(t)\):

\[\frac{dH}{dt}=F(H)-T_H(C)H.\]

Now treatment design involves competing objectives: suppressing tumour while limiting damage to healthy tissue.

Why maximum tumour kill is not automatically the best model objective

A treatment schedule that produces the strongest tumour reduction may also create unacceptable toxicity in a model that includes healthy tissue.

Mathematical optimisation therefore often introduces both tumour-control and toxicity terms.

Optimal-control formulation

A schematic objective might be

\[J=\int_0^T\left[w_NN(t)+w_uu(t)^2\right]dt,\]

subject to tumour and treatment equations.

The first term penalises tumour burden. The second penalises aggressive treatment intensity.

The weighting parameters encode the mathematical trade-off being studied.

Adaptive treatment

Instead of fixing the entire schedule in advance, treatment can be modelled as responding to tumour measurements:

\[u(t)=\Phi(N(t),R(t),\ldots).\]

For example, treatment may be switched on or off according to modelled tumour thresholds.

This produces a feedback-control system rather than a predetermined schedule.

Immune-mediated treatment

Immunotherapy models may include immune cells \(I(t)\) interacting with tumour cells:

\[\frac{dN}{dt}=G(N)-\kappa IN,\]

together with an equation for immune activation, recruitment or exhaustion.

Treatment may then act indirectly by changing the immune response rather than killing tumour cells directly.

Treatment delay and damaged cells

Observed tumour volume may not fall immediately after therapy. A model can distinguish viable cells from damaged or dying cells:

\[V(t),\qquad D(t).\]

Treatment transfers cells from viable to damaged states, while damaged cells are removed later.

This can explain delayed changes in measured tumour volume without assuming that treatment had no biological effect.

Spatial treatment models

Drug concentration, oxygen and treatment response can vary across a tumour. Spatial models may therefore use

\[N=N(x,t),\qquad C=C(x,t),\]

with diffusion, transport and spatially varying killing terms.

Such models can represent poorly penetrated regions or local microenvironmental differences.

Stochastic treatment models

When resistant clones are rare, mutation and extinction events can be probabilistic. A stochastic model can estimate the probability that resistance is already present or emerges during treatment.

This is especially relevant when rare-cell populations can determine long-term outcome.

Parameter estimation and validation

Treatment-response parameters should be estimated from suitable longitudinal data. A model should then be checked against observations that were not used to construct or fit it whenever possible.

Uncertainty in growth, pharmacokinetics, pharmacodynamics and measurement should be propagated into model predictions.

Prediction versus scenario exploration

A model can be useful even when it is not accurate enough for patient-specific prediction. It may clarify mechanisms, identify important parameters or compare hypothetical scenarios.

Clinical prediction requires substantially stronger validation than conceptual or theoretical modelling.

A practical modelling workflow

Choose an untreated tumour-growth law. Specify how treatment enters the system. Link treatment to concentration when pharmacokinetics matters. Decide whether treatment acts continuously or as discrete events. Add resistance, healthy tissue or immune components only when relevant. Estimate parameters from data. Analyse tumour size and composition separately. Quantify uncertainty and validate before interpreting the model predictively.

Key idea. Cancer treatment models combine tumour growth with a treatment mechanism. The essential mathematical question is whether treatment changes the net dynamics enough to produce control, decline or selection for other tumour subpopulations—and how this depends on timing, exposure and biological heterogeneity.