Residual Patterns From Incorrect Indirect Mechanisms

Why the Order of Residuals Matters

A residual is not merely a measure of how far a prediction misses an observation. In an indirect-response model, the order and timing of the misses can matter: errors that cluster during onset, peak effect, or washout may indicate that the fitted model is not representing the response dynamics adequately. A residual pattern can raise a specific question about mechanism or turnover, but it does not, by itself, establish the cause.

This tutorial considers two diagnostic situations: production inhibition fitted to data generated by loss inhibition, and a fitted turnover rate that is too large or too small. The interpretation depends on the direction of response change, the phase of the profile, the residual definition, and what other parts of the model were estimated.

Models and Assumptions Behind the Diagnostic

Let \(R(t)\) denote a response, with baseline \(R_0\). In the usual turnover formulation, baseline equilibrium satisfies \(k_{\mathrm{in}}=k_{\mathrm{out}}R_0\), where \(k_{\mathrm{in}}\) is the production rate and \(k_{\mathrm{out}}\) is the first-order loss-rate constant. For a concentration-driven inhibitory effect \(I(C)\), with \(0\leq I(C)\leq1\), the two mechanisms are:

These equations assume that production and loss follow the stated forms, the response is described by one turnover pool, and drug concentration \(C(t)\) is supplied by an adequate PK model. They also assume that observation error is represented appropriately when residuals are assessed. The concentration-effect function might, for example, be an inhibitory \(E_{\max}\) function, but residuals alone do not identify that function.

At baseline, when \(I(C)=0\), both equations give zero net change at \(R_0\). During drug effect, however, the two mechanisms alter different terms. If \(I(C)>0\) and the response starts at baseline, production inhibition initially lowers the response, whereas loss inhibition initially raises it. A fitted model may partly absorb the discrepancy by changing baseline, effect magnitude, turnover, or other parameters. Consequently, the resulting residuals need not retain a simple sign throughout the profile.

Worked Example: Reading Residuals Separately by Phase

Mechanism Mismatch: Production Versus Loss Inhibition

The initial direction of change provides a basic mechanism check. Suppose concentration is approximately constant over a short interval, the response begins at \(R_0\), and the effect is \(I>0\). Using \(k_{\mathrm{in}}=k_{\mathrm{out}}R_0\), the initial slopes are:

Thus, when the observed response rises after dosing but a production-inhibition fit predicts a fall, early residuals are likely to be positive as the observation pulls above the prediction. If the fitted production-inhibition model is adjusted to produce a rise through other parameter choices, the residual sequence can be more complicated; the initial-slope argument is a local comparison under the stated baseline and constant-effect conditions, not a guarantee about every fitted profile.

Turnover-Rate Error and Washout Residuals

When drug effect has ended, a first-order turnover response returns toward baseline exponentially. If the response at washout start is \(R_s\), then \(R(t)=R_0+(R_s-R_0)e^{-k_{\mathrm{out}}t}\), where \(t\) is time since washout. This relation makes the direction of turnover-rate error explicit. A larger \(k_{\mathrm{out}}\) predicts faster approach to baseline; a smaller value predicts slower approach.

For a response below baseline, an assumed rate that is too large predicts values closer to baseline than the true values. Its residuals \(Y-\hat{Y}\) tend to be negative during washout. For a response above baseline, the same too-large rate predicts a faster decline toward baseline, so the residual direction is reversed. Therefore, “positive residual means turnover is too fast” is not a general rule; the sign depends on whether the response is above or below baseline.

A Short Justification for the Washout Pattern

The turnover diagnostic follows from the residual expected when the true and fitted washout profiles share \(R_0\) and \(R_s\). Write \(k_T\) for the true rate and \(k_F\) for the fitted rate. Then:

If the response starts below baseline, \(R_s-R_0<0\). If \(k_F>k_T\) and \(t>0\), then \(e^{-k_Tt}>e^{-k_Ft}\), so the parenthesized term is positive and \(e(t)<0\). This proves the negative expected residual direction in the preceding example. If the response starts above baseline, \(R_s-R_0>0\), and the same rate ordering yields \(e(t)>0\). If the fitted rate is too small, these signs reverse. The result assumes drug effect is absent, both profiles share the same starting response and baseline, and the only difference is the turnover rate.

Diagnostics, Comparisons, and Limitations

A useful diagnostic begins with residuals plotted against actual time, with dosing, concentration, and response phases marked. Also inspect residuals against predictions and concentration. Use raw residuals to understand discrepancies in response units; use standardized residuals when the assumed error model gives different variances across observations. For example, under proportional error, a raw deviation of 5 units may be relatively small at a prediction of 200 and large at a prediction of 10.

Look for runs of residuals with the same sign, systematic curvature, or a change in pattern around peak concentration or washout. These can signal temporal misspecification, but sparse sampling may miss a phase entirely. Repeated observations from the same individual may also be correlated, so a sequence of residuals should not automatically be treated as independent evidence. Residuals calculated after parameter estimation are constrained by the fitting process; they are not independent tests of each model feature.

Residual inspection complements, rather than replaces, likelihood-based estimation and model assessment. A likelihood comparison evaluates how well candidate models account for the data under their specified error distributions and parameter constraints. Residual plots show where and when discrepancies occur, which can guide model checking. Neither a visually appealing residual plot nor a better fit alone proves a biological mechanism. A wrong PK input, baseline drift, incorrect observation variance, delayed effect, or omitted response process can mimic some patterns attributed to turnover or mechanism.

When production inhibition is fitted to loss-inhibition data, a systematic pattern may arise because the models act on different terms and can initially predict opposite directions. When turnover is wrong, washout residuals may show a phase-specific trend. But if parameters such as baseline, effect magnitude, and turnover are estimated together, they can redistribute error across time. Residual patterns are therefore most informative when interpreted alongside the model equations, concentration-time data, sampling schedule, and plausible sources of variability.

The practical objective is not to diagnose a mechanism from one residual. It is to identify reproducible, time-structured discrepancies that motivate checking assumptions or estimating a scientifically plausible alternative. Conclusions should remain tentative unless supported by the complete model comparison and relevant design information.