Detecting Turnover and Disease-Progression Misspecification

Why the Time Pattern Matters

A dynamic pharmacodynamic (PD) model can fit the average response reasonably well and still represent its time course incorrectly. A response may approach its treatment level too quickly or too slowly; meanwhile, the untreated response may drift because of disease progression. These errors can produce structured residuals even when residuals at an overall level appear centered near zero.

The key diagnostic question is not merely whether residuals vary with time, but which time scale and pattern they follow. A transient pattern after treatment begins can point toward response turnover. A persistent trend with calendar time, including in untreated or placebo observations, can point toward an incorrect untreated trajectory. Neither pattern uniquely identifies a model error: residuals are clues to investigate against the design, exposure, and other model components.

Definition: For an observation $Y_{ij}$ from subject $i$ at time $t_{ij}$, define the raw residual as $e_{ij}=Y_{ij}-\hat{Y}_{ij}$, where $\hat{Y}_{ij}$ is the model prediction on the response scale. Thus, a positive residual means the observed response is higher than predicted, and a negative residual means it is lower. A standardized residual divides this difference by an appropriate estimate of its standard deviation. Interpret patterns using the residual type and conditioning scheme actually plotted.

Conditional residuals compare observations with predictions that use estimated subject-specific effects; population-level residuals compare them with population predictions. Those views can differ, so a pattern should be checked in the relevant prediction view rather than presumed to be present in both. If the response has a nonconstant or transformed error model, use residuals appropriate to that scale when assessing signs and spread.

Turnover Models and Their Time Signature

A simple response-turnover model describes a response moving toward a target at a rate determined by $k_{\text{out}}$. In the absence of treatment, suppose the target is $R_0$, and let a treatment change the target to $R_1$ at time zero. The response then follows a first-order approach to the new target:

\frac{dR}{dt}=k_{\text{out}}(R_1-R), \qquad R(t)=R_1+(R_0-R_1)e^{-k_{\text{out}}t}.

This form assumes the target changes at the stated time, $k_{\text{out}}>0$ is constant over the interval, the response starts at $R_0$, and no other time-varying process changes the response. It is a useful diagnostic reference, not a claim that every PD response follows a single exponential.

For a suppressive treatment, $R_1<R_0$. If the fitted turnover rate is too high, its predicted response falls toward $R_1$ faster than the true response. Early after treatment, the prediction is therefore too low and $e=Y-\hat{Y}$ tends to be positive. As the response approaches the target, this discrepancy shrinks. If the fitted rate is too low, the prediction lags above the actual response and early residuals tend to be negative. For a stimulatory treatment, $R_1>R_0$, the sign pattern reverses. The pattern should be considered relative to the direction of effect, not interpreted from residual sign alone.

Derivation: Suppose the true rate is $k$, while the fitted rate is $\hat{k}$, with the same starting value and target. Subtracting the fitted response from the true response gives $$ R(t)-\hat{R}(t)=(R_0-R_1)\left(e^{-kt}-e^{-\hat{k}t}\right). $$ For suppression, $R_0-R_1>0$. If $\hat{k}>k$, then $e^{-kt}>e^{-\hat{k}t}$ for $t>0$, so the residual is positive. It approaches zero as both exponentials approach zero. This establishes the expected transient sign under these assumptions; it does not establish that turnover is the only possible cause of such a pattern.

In practice, inspect residuals against time since treatment initiation or a relevant treatment change, and compare early, intermediate, and late durations. A model with an overly rapid response may show positive residuals early that diminish later for a suppressive effect. A model with an overly slow response may show negative early residuals that diminish. Persistent late residuals, or patterns that recur when exposure is absent, are reasons to consider explanations beyond turnover rate alone.

Worked Example: A Turnover Rate That Is Too Fast

Suppose a hypothetical suppressive treatment moves the response from $R_0=100$ units toward $R_1=80$ units. The true turnover rate is $k=0.20$ per hour, but the fitted model uses $\hat{k}=0.50$ per hour. At two hours, calculate the true response, fitted prediction, and expected residual.

Using $R(t)=80+(100-80)e^{-kt}$, the true response is $80+20e^{-0.20(2)}=80+20e^{-0.4}=93.41$ units.

The fitted prediction is $80+20e^{-0.50(2)}=80+20e^{-1}=87.36$ units.

Thus the expected residual is $93.41-87.36=+6.05$ units. The model predicts too much suppression at this early time. Under the stated step-change assumptions, this positive residual is consistent with a fitted turnover rate that is too high. It is not, by itself, proof of that explanation.

Untreated Trajectories and Disease Progression

An untreated response need not remain constant. A disease-related process can cause its expected level to rise or fall over calendar time, whether or not treatment is being given. A simple way to express this in a turnover model is to let the untreated target change with time:

\frac{dY}{dt}=k_{\text{out}}\{B_0+gt-Y(t)\}.

Here $B_0$ is the untreated target at time zero and $g$ is the rate of change in that target, in response units per unit time. This model assumes a linear target over the interval and constant turnover. It is only one possible representation of the untreated trajectory; another scientifically justified trajectory may be needed if a linear change is inadequate.

If the fitted model instead assumes a constant target $B_0$, its expected response may systematically miss observations as time passes. For $g>0$, the actual response rises above the fixed-target prediction; for $g<0$, it falls below. The resulting residual trend may appear in untreated, placebo, and treated observations, although treatment dynamics can modify its size and timing. A trend against time since first dose alone does not distinguish disease progression from turnover if all subjects start treatment at the same calendar time.

Derivation: Let $Y(0)=B_0$. For the changing-target model, write $Z(t)=Y(t)-B_0$. Then $dZ/dt=k_{\text{out}}(gt-Z)$, with $Z(0)=0$. Solving this linear differential equation gives $$ Y(t)=B_0+g\left[t-\frac{1-e^{-k_{\text{out}}t}}{k_{\text{out}}}\right]. $$ The bracketed term is zero at time zero and becomes approximately $t-1/k_{\text{out}}$ at later times. Thus, a model that predicts $B_0$ throughout will increasingly underpredict when $g>0$, after allowing for the response’s finite turnover.

Worked Example: A Rising Untreated Trajectory

Consider a hypothetical untreated response with $B_0=100$ units, $g=1.2$ units per day, and $k_{\text{out}}=0.25$ per day. The fitted model incorrectly holds the response at 100 units. Calculate the expected residual at day 4 under the changing-target model.

Using the derived expression, $Y(4)=100+1.2[4-(1-e^{-0.25(4)})/0.25]$. Since $e^{-1}=0.3679$, the bracket is $4-(1-0.3679)/0.25=4-2.5284=1.4716$ days.

Therefore $Y(4)=100+1.2(1.4716)=101.77$ units. The fixed-target model predicts 100 units, so the expected residual is $101.77-100=+1.77$ units. A positive trend among observations not receiving active treatment would be consistent with an omitted rising untreated trajectory, subject to checks for other time-related structure and uncertainty.

Use Duration, Calendar Time, and Exposure Together

Treatment duration and calendar time are not interchangeable. Duration is elapsed time since treatment began or changed; calendar time is elapsed time from a common study or disease reference point. If everyone begins treatment on the same day, these variables may be nearly indistinguishable. Then a residual trend cannot reliably be assigned to treatment-related turnover rather than disease progression from the plot alone.

Exposure provides another diagnostic dimension. A residual pattern that appears after treatment starts but tracks concentration or predicted effect may reflect misspecification in the exposure-response relationship rather than turnover or untreated progression. Conversely, residual drift that is also present during untreated periods makes a purely treatment-driven explanation less persuasive. These are comparisons of evidence, not definitive decision rules.

Worked Example: Comparing Residuals Across Treatment Durations

Suppose a hypothetical study has four subjects in each of three treatment-duration groups. Their mean raw residuals are as follows. Positive residuals mean observed responses exceed predictions.

Time since treatment start	Residuals (units)	Mean residual (units)
2 hours	5, 3, 4, 4	4.00
8 hours	2, 1, 3, 2	2.00
24 hours	0, 1, -1, 0	0.00

At 2 hours, the mean is $(5+3+4+4)/4=16/4=4.00$ units. At 8 hours it is $(2+1+3+2)/4=8/4=2.00$ units, and at 24 hours it is $(0+1-1+0)/4=0/4=0.00$ units. The positive residuals that diminish with duration are compatible with an overly fast fitted suppression process under the simple turnover example.

This summary is descriptive, not a formal test. Before attributing the pattern to turnover, check whether the groups differ in concentration, baseline, calendar time, or sampling schedule, and examine individual trajectories. If untreated observations also show an increasing positive residual trend, an omitted untreated trajectory may contribute instead.

Diagnostic Workflow and Competing Explanations

A useful assessment combines plots with model and study-design knowledge. The following sequence keeps the residual sign, time scale, and plausible alternatives explicit.

Establish what is plotted.
Identify raw or standardized residuals, the prediction type, response scale, and the meaning of positive and negative values.

Separate time scales.
Compare residuals against time since treatment start or change, and against calendar time or time since disease onset when those are available.

Check treatment context.
Compare treated and untreated or placebo periods, and inspect residuals against concentration and predicted effect to assess exposure-related alternatives.

Review trajectories and design.
Look for the same pattern within subjects, account for sampling imbalance, and consider whether treatment start times provide information to separate duration from calendar time.

Evaluate a targeted alternative.
Fit a scientifically plausible turnover or untreated-trajectory alternative and check whether it removes the relevant structure without creating new misfit elsewhere.

Residual-versus-time plots are the direct visual tool for a temporal pattern; individual fits and predictive checks help assess whether it recurs across subjects and whether the model reproduces the observed distribution. A structured residual trend is evidence that the current model may be incomplete, not a stand-alone criterion for selecting a replacement.

Common Mistakes and Limitations

Reading the sign without the residual convention. With $e=Y-\hat{Y}$, positive means underprediction. If a software diagnostic uses the opposite difference, the interpretation reverses.
Calling every duration trend a turnover error. Calendar-time progression, exposure misspecification, baseline changes, and sampling patterns can also create time structure. Compare relevant plots and design periods before assigning a cause.
Assuming a residual signature is unique. Different model errors can produce similar patterns, and multiple errors can offset one another. Treat the signature as a hypothesis to check.
Ignoring uncertainty and dependence. Individual residuals are noisy, and repeated observations from one subject are not independent evidence. Use subject-level patterns and appropriate residual diagnostics rather than treating every point as an independent replicate.
Extrapolating a simple trajectory too far. A linear untreated target or a single turnover rate may only be reasonable over a limited interval. A visually improved fit does not establish that the added structure is biologically correct.

The main alternatives are not interchangeable. A turnover-rate change targets the speed of response adjustment; an untreated-trajectory change targets the response expected without active treatment. A change to the exposure-response component targets how treatment exposure maps to effect. When treatment timing and calendar time are confounded, or untreated data are sparse, the available observations may not distinguish these explanations. That limitation should be reported rather than hidden by a more complex fit.

Key takeaway: For observed-minus-predicted residuals, a transient signed pattern after a treatment change can be consistent with a turnover-rate error, while a persistent trend that also appears without active treatment can be consistent with an incorrect untreated trajectory. Confirm the time scale, treatment context, and competing explanations before changing the model.

Check Your Understanding

Answer each question using the observed-minus-predicted residual convention. Brief solutions follow the questions.

For a suppressive treatment, the fitted turnover rate is too low. What early residual sign is expected under the simple step-change model, and why?
An untreated response has a rising trajectory, but the model predicts a constant level. What residual direction is expected as time passes?
All subjects begin treatment at the same calendar time, and residuals trend with time since treatment. Why is that insufficient to distinguish turnover error from disease progression?
Residuals are positive early after treatment and diminish later, but the same positive trend occurs during placebo periods. What should be investigated before attributing the pattern to turnover?

Solutions. 1. Negative: with a suppressive effect, a rate that is too low makes the fitted response lag above the faster-changing actual response, so observed minus predicted is below zero. 2. Positive: the actual response rises above the constant prediction, so $Y-\hat{Y}>0$. 3. Treatment duration and calendar time move together in that design, so either a treatment-linked dynamic error or an untreated time trend could produce the association; staggered treatment starts or informative untreated observations would help separate them. 4. Investigate an omitted rising untreated trajectory and other calendar-time structure, as well as baseline, exposure, and sampling differences; the pattern is not unique to turnover.