Simulating Treatment Schedules in a Tumor Growth Model

From a Treatment Schedule to a Tumor Trajectory

A treatment schedule is an input to a tumor model, not a tumor outcome by itself. To predict a trajectory, we need to specify how the schedule determines drug exposure, how exposure changes the tumor growth or kill rate, and how the tumor state evolves over time. This tutorial uses a deliberately simple model with a prescribed concentration profile and fixed parameters. It isolates the schedule-to-response calculation; it does not estimate parameters or generate drug concentrations from doses through a pharmacokinetic model.

The distinction matters when comparing regimens. Equal total exposure, or equal average concentration, does not necessarily mean equal average drug effect. If the exposure-effect relationship is nonlinear, the timing and magnitude of concentration can change the resulting tumor trajectory.

When concentration is constant over an interval, the net rate \(r=k_g-k_{\max}q(C)\) is constant there. The tumor volume over an interval of length \(\Delta t\) can then be calculated exactly as \(V(t+\Delta t)=V(t)e^{r\Delta t}\). For a schedule with several constant-concentration intervals, apply this relationship consecutively. A numerical solver is useful when the input varies continuously or the model is more complicated.

Fixed Parameter Set and Simulation Choices

For the worked comparisons, use the following hypothetical parameter values: \(V(0)=100\) mm³, \(k_g=0.080\) day\(^{-1}\), \(k_{\max}=0.12\) day\(^{-1}\), and \(EC_{50}=1\) concentration unit. These values are fixed across schedules. The comparison therefore asks how the specified concentration profiles behave under this model; it does not establish which schedule would be better in a patient.

We compare two 10-day profiles. The continuous profile holds \(C(t)=1\) throughout. The intermittent profile alternates between \(C(t)=2\) for the first 0.5 day of each day and \(C(t)=0\) for the next 0.5 day. Both profiles have an average concentration of 1 concentration unit over each day. In this example, concentration is prescribed directly; equal average concentration is not a claim of equal administered dose or equal clinically achievable exposure.

Implementing a Schedule

For a piecewise-constant concentration profile, a reliable implementation divides time at every concentration change. At each interval, evaluate \(q(C)\), calculate the net rate, and update the volume. The exact update avoids discretization error within such intervals. If using a numerical method for more general models, the time step should be short enough to resolve both the concentration changes and the tumor response.

Comparing Continuous and Intermittent Exposure

In the intermittent profile, \(q(2)=2/3\) during each 0.5-day high interval, while \(q(0)=0\) during each 0.5-day drug-free interval. The net rate during a high interval is \(0\) day\(^{-1}\), as calculated above. During a drug-free interval it is \(0.080\) day\(^{-1}\). Across 10 days, there are ten intervals of each type: five total days at each concentration. The exact endpoint is therefore $$ V(10)=100e^{(0)(5)+(0.080)(5)}=100e^{0.4}=149.2\ \text{mm}^3. $$ The intermittent profile produces a larger modeled tumor volume than the continuous profile at day 10: 149.2 versus 122.1 mm³. The result follows from the chosen model and parameters, not from a general rule that intermittent treatment is inferior.

The two profiles have the same average concentration, but not the same average fractional effect. The continuous profile has \(q=0.5\) throughout. The intermittent profile has average effect $$ \bar q=\frac{(0.5)(2/3)+(0.5)(0)}{1}=\frac{1}{3}. $$ Since the effect function saturates, doubling concentration from 1 to 2 does not double the effect: it rises from 0.5 to 0.6667. The time spent at zero exposure then lowers the intermittent profile’s average effect. This explains the difference in the net tumor growth accumulated over time.

Why the Calculation Works—and When It Does Not

The exact interval update follows by dividing the model equation by \(V(t)\), assuming \(V(t)>0\), and integrating over an interval with constant concentration. Since \(q(C)\) and hence \(r\) are constant on that interval, $$ \frac{1}{V}\frac{dV}{dt}=r,\qquad \int_{V(t)}^{V(t+\Delta t)}\frac{dV}{V}=\int_t^{t+\Delta t}r\,dt. $$ Therefore $$ \ln\left(\frac{V(t+\Delta t)}{V(t)}\right)=r\Delta t,\qquad V(t+\Delta t)=V(t)e^{r\Delta t}. $$ For multiple intervals, the exponents add. Thus, in this model, the endpoint depends on the time-integrated net rate, not on the order of intervals when their durations and concentrations are unchanged. That property need not hold in models with delays, changing sensitivity, or other state variables.

A useful diagnostic is to inspect intermediate values, not only the final tumor volume. Check that concentration switches at the intended times, that zero concentration gives zero drug effect, that the net rate at \(C=2\) is zero for these parameters, and that the model predicts growth in drug-free intervals. A numerical result that contradicts any of these checks suggests an implementation, units, or schedule-boundary problem.

There are three common implementation choices. Exact interval updates are simplest and avoid numerical integration error when concentration and the net rate are piecewise constant. Euler steps are transparent and easy to code, but their error depends on step size. Adaptive ODE solvers adjust step size to control local numerical error and are often convenient for more complex models, but they do not correct an incorrect model or input schedule. For any numerical method, repeat the simulation with a smaller maximum step and confirm that the predicted trajectory is stable to a practically acceptable precision.

The comparison here has important limits. The concentration profiles are prescribed rather than generated from dose, absorption, distribution, and elimination. The model does not represent delayed response, tumor carrying capacity, measurement error, parameter uncertainty, or changing sensitive and resistant subpopulations. Its fixed effect relationship also assumes that the same concentration has the same modeled effect throughout treatment. Consequently, these simulations are conditional model predictions, not evidence of clinical benefit or a dosing recommendation.

Common Pitfall

A frequent mistake is to conclude that equal average concentration guarantees equal tumor response. That reasoning treats concentration and effect as proportional. In this example, \(q(C)=C/(1+C)\) is saturable, so the average effect for alternating concentrations 2 and 0 is \(1/3\), not the continuous-profile effect of \(1/2\). The correction is to apply the prescribed exposure-effect function at the actual concentrations over time, then integrate the resulting net tumor rate. Equal average concentration alone is insufficient.