Choosing Transit-Stage Number From Response Shapes

Why Response Shape Can Inform Stage Count

A delayed biomarker response can begin gradually, peak after a lag, and remain spread over time. In a transit-chain model, the number of sequential stages helps determine that shape: more stages produce a more concentrated distribution of transit times when the mean delay is held fixed. Thus response shape can help distinguish candidate stage counts, but only if delays due to drug input and biomarker turnover are also considered.

The target is not simply to find the first time at which a measured value differs from baseline. That apparent onset depends on measurement precision, sampling times, and the threshold used to call a change. A useful stage-count assessment instead compares the timing and dispersion predicted by each candidate chain with the response information supported by the data.

The Timing Distribution and Its Shape

Consider a linear chain in which each stage passes material onward at rate \(k_{tr}\). Let \(u(t)\) be the input to the first stage, and let \(x_j(t)\) denote the amount in stage \(j\). For \(N\) stages, the chain is

For an impulse entering the first stage, the fraction of the signal emerging from the final stage per unit time is the transit-time density

This is a density: its integral over nonnegative time is one. Equivalently, the total transit time is the sum of \(N\) independent exponential waiting times, each with rate \(k_{tr}\). The mean and variance are therefore

For \(N>1\), the density reaches its mode at \((N-1)/k_{tr}\); for \(N=1\), it decreases from its maximum at time zero. These summaries show how to compare chains. If mean transit time is held at a fixed value \(\tau\), set \(k_{tr}=N/\tau\). The variance is then \(\tau^2/N\), so increasing \(N\) narrows the timing distribution and moves its mode closer to the mean. A larger \(N\) does not, by itself, imply a longer mean delay.

Using Moments to Narrow Candidate Counts

If the mean and variance of the transit-time distribution can be estimated, the equal-rate model implies

The stage count must be a positive integer. A non-integer result is not a literal number of stages; it indicates that the moment estimates do not identify an exact integer count, or that the equal-rate chain is an imperfect description. In practice, moments are estimated with uncertainty, so this calculation is best used to identify a small set of candidate counts for comparison.

Compare Predicted Shapes at a Common Mean

A fair shape comparison holds the mean delay constant across candidate values of \(N\), adjusting \(k_{tr}\) for each. Otherwise, a difference in response timing could be attributed either to the stage count or simply to a changed mean transit time. At a fixed mean of 8 hours, \(N=1,2,\) and \(4\) imply rates \(0.125,\ 0.250,\) and \(0.500\ \text{h}^{-1}\), respectively.

A pharmacodynamic response is often not the transit density itself. The drug-driven input can have its own time course, and biomarker turnover can smooth or extend the output. In a linear setting, sequential delays act by convolution: the observed timing distribution combines the transit-chain timing with the timing of the input and any downstream response process. For independent timing components, means and variances add. If those other components are known or separately characterized, their contributions can be accounted for before using moments to assess \(N\). If they are unknown, the observed response dispersion cannot be assigned to the transit chain alone.

Why the Moment Relationship Holds

The moment formula follows directly from the sum-of-waiting-times representation. If \(T_i\) is the waiting time in stage \(i\), then each \(T_i\) has mean \(1/k_{tr}\) and variance \(1/k_{tr}^2\). The stage waiting times are independent under the model, and \(T=\sum_{i=1}^N T_i\). Linearity of expectation gives \(E(T)=N/k_{tr}\); independence gives

Dividing the squared mean by the variance yields

This derivation also shows the assumptions behind the result: a fixed number of stages, independent exponential waiting times, and one common rate. If stage rates differ, the sum is not described by this equal-rate gamma model, and the moment ratio need not equal an integer stage count.

Onset, Diagnostics, and Limits

The mathematical transit density has support from time zero onward; it does not impose a positive minimum delay. For \(N>1\), its density begins at zero and rises, but it is positive at any \(t>0\). Observed onset is usually defined by a detectable departure from baseline, not by the first instant the model response becomes nonzero. Consequently, an apparent onset time depends on assay precision, baseline variability, sampling schedule, and the size of the input. It can be useful for rejecting a grossly incompatible model, but it is not a direct estimate of \(N\).

Useful diagnostics include checking whether candidate chains reproduce the observed rise, peak timing, and spread, not just the mean delay. Inspect predictions against observations over the entire response period. A systematic early or late rise, an overly sharp or broad peak, or persistent timing discrepancies can indicate a poor stage-count choice or a failure of model assumptions. Sparse sampling around the rising phase and peak makes these distinctions weak. Uncertainty in input timing or turnover can also make different stage counts predict nearly indistinguishable responses.

The equal-rate chain is a parsimonious shape model, not a claim that each biological maturation step has been directly observed. Candidate integer chains can be fitted and compared while accounting for the rest of the response model; a flexible distributed-delay description can instead represent dispersion without requiring an integer stage interpretation. More flexible approaches may fit shape better but can be harder to interpret and estimate. In either case, apparent fit improvements should be weighed against the information in the data and uncertainty in the timing parameters.