Solving a Tumor Growth and Kill Model by Hand

Why Solve the Model by Hand?

A hand solution makes the assumptions and the effect of each treatment interval visible. It provides a direct check on whether a predicted tumor burden is consistent with the growth and killing rates, and it can expose errors in units, signs, or treatment-switch calculations before those errors are carried into a more elaborate analysis. The calculations here use an exponential model with rates that are constant within each interval.

Model and Assumptions

Let \(V(t)\) denote tumor volume at time \(t\), measured, for example, in \(\mathrm{cm}^3\). During an untreated interval, assume tumor volume grows exponentially at a constant per-capita rate \(r\). During a treated interval, assume treatment adds a constant per-capita killing rate \(k\). Both rates have units of inverse time, such as \(\mathrm{day}^{-1}\). The model treats growth and killing as proportional to the current volume, so their net effect is the difference between the two rates.

The hand solution assumes that volume is positive at the start of each interval, the stated rates remain constant over that interval, and no separate process causes an instantaneous change in volume. In a multi-interval example, the untreated growth rate may be specified separately for different intervals; this represents a model assumption, not an effect generated automatically by the treatment-kill term. The simple model also assumes that the same net rate applies at every tumor volume within a given interval.

Deriving the Solution on One Interval

For a constant net rate \(g\), separate the variables and integrate from the start of an interval, \(t_a\), to any time \(t\) within it. The starting volume is \(V(t_a)\), and the solution follows by exponentiating both sides:

The exponent must be dimensionless: \(g\) has units of inverse time and \(t-t_a\) has units of time. If \(g>0\), the volume increases; if \(g<0\), it decreases; and if \(g=0\), it remains constant. These statements concern the model's trajectory, not a guarantee that real tumors follow exponential behavior indefinitely.

For several consecutive intervals, use the volume at the end of one interval as the starting volume for the next. If the rate is \(g_i\) on the interval from \(t_{i-1}\) to \(t_i\), then after \(m\) intervals the solution is

This expression is obtained by multiplying the exponential factors from the individual intervals. It is often more accurate to add the exponents first and round only the final volume.

Switch Times, Continuity, and Interpretation

Suppose the rate changes at time \(t_s\), from \(g_1\) before the switch to \(g_2\) after it. With no instantaneous volume change, the value at the switch is the end of the first interval and the start of the second:

The derivative can change at \(t_s\), because the rate changes, even though the volume remains continuous. This distinction is important: changing the slope of the trajectory is not the same as imposing a jump in tumor volume. If a model includes an instantaneous procedure or other discrete volume change, that jump must be stated and calculated separately.

Thresholds and Limiting Behavior

On treatment, the sign of \(r-k\) determines the direction of change. If \(k>r\), the model predicts shrinkage; if \(k=r\), it predicts a constant volume; and if \(k<r\), it predicts continued growth despite treatment. This comparison concerns the two model rates, not a direct guarantee of a clinical response.

More generally, under a constant net rate, a positive rate leads to unbounded exponential growth in the mathematical model, a zero rate gives a constant volume, and a negative rate leads asymptotically toward zero. These limiting behaviors help identify what the equations imply if their assumptions are extended indefinitely. They also highlight why a constant-rate model may be suitable over a specified interval but inadequate for long-term prediction.

Diagnostics, Alternatives, and Limitations

A hand solution is most useful when its inputs and intermediate steps can be checked. Confirm that time and rate units match; a rate in \(\mathrm{day}^{-1}\) must be paired with an interval in days. Confirm the sign convention: killing is subtracted from growth in \(r-k\). At each switch, verify that the ending volume of the preceding interval is the starting volume of the next. Finally, check that the direction of each predicted change matches the net-rate sign.

The hand solution is exact for the stated piecewise-constant-rate model. If rates vary continuously with time, the corresponding exponential solution depends on the accumulated rate over time rather than one constant rate times an interval duration. Numerical methods can be useful when a model has time-varying rates, interacting compartments, or other equations without a simple closed-form solution. They do not remove the need to check units, assumptions, and behavior at treatment changes.

A common mistake is to round each interval's tumor volume and then use that rounded value as the next interval's starting value. Repeated rounding can shift the final result. The correction is to carry the unrounded value, or equivalently add all rate-times-duration contributions in the exponent and round only the reported final volume. Another error is to treat a treatment switch as an instantaneous reduction in volume even though the model changes only the rate. A rate switch changes the subsequent slope; a volume jump requires a separate, explicit model rule.

The model's limitations are structural: constant per-capita rates imply exponential trajectories within intervals; treatment killing is represented as a fixed rate rather than a changing drug effect; and the equation does not itself represent measurement error, tumor-size constraints, or other biological mechanisms. A good fit over one interval does not establish that those assumptions remain reasonable over another.