Worked Example: Tumor Response Intervals Across Doses

From Uncertain Parameters to Tumor-Response Intervals

A QSP model can predict a tumor-size change for a specified dose and time, but uncertain growth and drug-effect parameters mean that prediction is not a single certain outcome. A dose-specific response interval summarizes the spread in predicted changes when the model is evaluated across a distribution of plausible parameter values or virtual patients. It is therefore important to state what uncertainty the interval represents and which population and treatment conditions it applies to.

The worked calculations below use a deliberately simple hypothetical model. It is not intended to describe any actual drug or cancer. The purpose is to show how a model prediction becomes a response interval, how intervals at two doses can overlap, and why the distribution of paired dose differences is often more informative for comparing doses.

A Hypothetical Tumor-Growth Model

Suppose tumor size follows exponential growth with a dose-dependent reduction in net growth rate. Let \(g\) be the tumor-growth rate without treatment, \(k\) the maximum drug-effect rate, and \(C(D)\) a dimensionless exposure index for dose \(D\). The drug effect is represented by a saturating function with dimensionless half-effect value \(EC_{50}\). The model is

Here \(g\) and \(k\) have units of week\(^{-1}\), while \(C(D)\) and \(EC_{50}\) are dimensionless. For the examples, set \(C(D)=D/(100\text{ mg})\), \(EC_{50}=1\), and evaluate the model at week 8. At a fixed parameter draw, the net rate is \(r(D)=g-kC(D)/(EC_{50}+C(D))\). The resulting response is \(100\{\exp[r(D)t]-1\}\), so it does not depend on the chosen value of \(T_0\) when expressed as a percent change.

To form a predictive distribution, draw parameter values from a specified joint distribution representing the intended patient population and parameter uncertainty, then evaluate the model at each draw. The draws should preserve relevant dependence among parameters: independently sampling correlated parameters can create implausible combinations and distort the tails. The examples use matched draws across doses, so each virtual patient retains the same growth and drug-effect parameters when evaluated at both doses.

Constructing and Reading a Percentile Interval

In practice, a simulation might use thousands of parameter draws. For each draw \(m\), calculate \(R_m(D,t)\), then sort the resulting values from smallest to largest. A central 80% interval uses the 10th and 90th percentiles. One transparent finite-sample convention is linear interpolation: for \(M\) sorted values \(x_{(1)},\ldots,x_{(M)}\), calculate \(h=1+(M-1)p\) for percentile \(p\). Interpolate between the sorted observations bracketing \(h\). This convention is used in the small demonstration below; software packages may use other documented quantile conventions.

A short justification for the percentile construction is that the interval endpoints are chosen to leave equal probability in the two tails of the predictive distribution: \(P(R<q_{0.10})\) is approximately 0.10 and \(P(R>q_{0.90})\) is approximately 0.10. Thus the central interval covers approximately 80% of that distribution. With a finite simulation, its endpoints are estimates of those quantiles and can vary across Monte Carlo runs.

Comparing Doses: Marginal Intervals and Paired Contrasts

Now suppose the same five matched virtual-patient draws are evaluated at 120 mg. Their hypothetical week-8 responses are \(-20.00\%\), \(-17.20\%\), \(-16.00\%\), \(-14.00\%\), and \(-13.00\%\), respectively. Applying the same interpolation rule gives a 10th percentile of \(-20.00+0.4(2.80)=-18.88\%\). The 90th percentile is \(-14.00+0.6(1.00)=-13.40\%\). Thus the central 80% interval at 120 mg is \([-18.88\%,-13.40\%]\).

The dose-specific intervals overlap from \(-16.24\%\) to \(-13.40\%\). That overlap describes shared portions of the two marginal predictive distributions. It is not, by itself, a test of whether the doses have the same effect, and it does not establish equivalence. Nor does the absence of overlap, in another analysis, automatically supply a formal test or decision rule. The intervals summarize outcomes at each dose separately; a direct dose comparison concerns differences between outcomes.

Because the draws are matched, calculate each virtual patient’s contrast as the 120 mg response minus the 100 mg response. The five contrasts are \(-2.84\), \(-2.34\), \(-1.80\), \(-1.507\), and \(-1.26\) percentage points. Their central 80% interval, using the stated interpolation rule, is \([-2.64,-1.3588]\), or approximately \([-2.64,-1.36]\) percentage points. For these illustrative draws, the higher dose predicts greater shrinkage for each matched virtual patient, with the central 80% of paired differences between about 1.36 and 2.64 percentage points more negative. The small number of draws makes this an arithmetic demonstration, not a reliable estimate of clinical uncertainty.

Diagnostics, Alternatives, and Limitations

A response interval is only useful if the simulation and its interpretation are credible. First check that every draw uses the same baseline definition, time horizon, dose regimen, and endpoint calculation. Verify units in the rate equation, and inspect predicted tumor sizes and percent changes for impossible values or numerical failures. In this simple model, tumor size remains positive because it is an exponential; a numerical implementation that produces negative size signals a problem.

Next assess Monte Carlo stability. Recompute quantiles using larger simulation batches or independent runs and check whether the endpoints move materially. Tail quantiles generally require more draws to estimate stably than central summaries. Inspect extreme draws and confirm that the parameter sampling preserves the intended joint distribution. A visually smooth dose-response band is not evidence that the tails are stable. If uncertainty in model discrepancy, adherence, or measurement is relevant to the prediction target, it must be represented or the interval should be described as excluding it.

Several alternatives answer different questions. A single median or mean response is a compact summary, but it conceals dispersion and tail behavior. A normal approximation, such as one based on the mean and standard deviation of simulated responses, can be efficient when the distribution is roughly symmetric and well behaved; percentile intervals are less dependent on that shape assumption but require adequate simulation in the tails. A delta-method approximation can be useful for local uncertainty propagation, but nonlinear transformations or skewed response distributions can make a percentile simulation more informative. These approaches do not change what the prediction distribution represents.

For comparing doses, report both marginal response intervals when the absolute predicted outcomes matter and a paired contrast distribution when the same virtual patients can be evaluated under each dose. The contrast retains within-patient dependence and directly expresses the modeled dose difference. It still inherits the assumptions and uncertainties of the model and parameter distribution; it does not show that changing the dose causes the predicted difference in real patients unless the model and its intended use support that interpretation.