Comparing Turnover and Transit Models With Biomarker Data

Why Compare Delay Models?

A delayed biomarker response does not, by itself, reveal where the delay arises. The drug may alter formation of a precursor that subsequently supplies the biomarker, or the delay may be represented by a sequence of transit compartments describing maturation. Both explanations can produce a gradual response and a gradual recovery. The practical question is therefore not only which curve follows the observations more closely, but also whether its advantage is meaningful, whether its parameters are well determined, and whether the models make materially different predictions under a new dosing schedule.

This comparison uses a normalized biomarker whose untreated baseline is 1. A lower value indicates a drug-associated decrease. The examples are hypothetical and illustrate the reasoning, not a real study. The models assume that drug input is known, the biomarker is measured without a systematic assay bias, and the drug effect on the modeled pathway is represented by a constant inhibition magnitude. These assumptions should be checked against the study design and data before interpreting a fitted model mechanistically.

Two Candidate Models

Let $u(t)$ denote normalized drug input, with $u(t)=1$ during the stated dosing period and $u(t)=0$ otherwise. Let $I$ be the maximum fractional inhibition of the pathway, so $0\leq I\leq 1$. In both models, the normalized biomarker is $b(t)=1-q_b(t)$, where $q_b(t)$ is the biomarker deficit from baseline.

Definition: In the explicit precursor-pool model, $q_p(t)$ is the deficit in precursor availability and $q_b(t)$ is the biomarker deficit: $$ \frac{dq_p}{dt}=k_p\{Iu(t)-q_p(t)\},\qquad \frac{dq_b}{dt}=k_b\{q_p(t)-q_b(t)\},\qquad b(t)=1-q_b(t). $$ Here $k_p$ and $k_b$ are positive rate constants, in inverse time. Drug input inhibits precursor formation; the precursor deficit then drives the biomarker deficit. The two rates can differ.

The alternative represents delay by two sequential transit stages. It does not posit an explicitly measured or biologically distinct precursor pool.

Definition: In the two-stage transit model, $x_1(t)$ and $x_2(t)$ are deficits in successive transit stages: $$ \frac{dx_1}{dt}=k_{\mathrm{tr}}\{Iu(t)-x_1(t)\},\qquad \frac{dx_2}{dt}=k_{\mathrm{tr}}\{x_1(t)-x_2(t)\},\qquad b(t)=1-x_2(t). $$ The common rate $k_{\mathrm{tr}}>0$ governs both stages. The model has one fewer kinetic parameter than the explicit precursor model.

Both models start at baseline before dosing: their deficit states are zero. They also assume a linear, time-invariant response to the input, no feedback, and no direct drug effect on the measured biomarker apart from the modeled pathway. For fitting, suppose measurement errors are independent, normally distributed with mean zero and a common variance, conditional on the model and its parameters. If repeated measurements from one participant are correlated, this error assumption is not appropriate without an added correlation structure.

Step Responses and Their Meaning

For a unit step in input beginning at time zero, each model has a step-response function $F(t)$: the output deficit is $I F(t)$. In the explicit precursor model, solving the first equation gives $q_p(t)=I(1-e^{-k_p t})$. Substituting this into the biomarker equation and solving the resulting linear differential equation gives the response below, for $k_p\ne k_b$.

F_{\mathrm{pool}}(t)=1-\frac{k_b e^{-k_p t}-k_p e^{-k_b t}}{k_b-k_p}.

The expression starts at zero and tends to one. If $k_p=k_b=k$, the limiting response is $F_{\mathrm{pool}}(t)=1-(1+kt)e^{-kt}$; the apparent singularity in the unequal-rate expression is removable. For two equal-rate transit stages, solving the stage equations gives $x_2(t)=I[1-(1+k_{\mathrm{tr}}t)e^{-k_{\mathrm{tr}}t}]$, so

F_{\mathrm{tr}}(t)=1-(1+k_{\mathrm{tr}}t)e^{-k_{\mathrm{tr}}t}.

Thus, the two-stage transit response is the equal-rate special case of the two sequential linear states in the explicit model. The distinction is that the explicit model allows two different rates and assigns a precursor interpretation to the first state; a good fit cannot establish that biological interpretation on its own.

For a dose that starts at time zero and stops at $t_d$, linearity means the input is the difference between a step at zero and a step at $t_d$. Therefore, at $t\geq t_d$, the predicted deficit is $I\{F(t)-F(t-t_d)\}$. This relation makes the models' predictions under dosing and washout directly comparable.

Worked Example: Comparing Fits to a Hypothetical Biomarker Series

Suppose a hypothetical study records a normalized biomarker during dosing from 0 to 6 hours and during washout. The model fits use the observations below. The fitted parameter values are rounded for display; fitted predictions and residual summaries are also rounded. Residual means observed minus predicted.

Time (h)	Observed $b$	Explicit-pool prediction	Pool residual	Transit prediction	Transit residual
0	1.000	1.0000	0.0000	1.0000	0.0000
1	0.982	0.9801	0.0019	0.9799	0.0021
2	0.936	0.9393	-0.0033	0.9383	-0.0023
4	0.858	0.8532	0.0048	0.8499	0.0081
6	0.786	0.7898	-0.0038	0.7850	0.0010
8	0.816	0.8110	0.0050	0.8071	0.0089
10	0.870	0.8743	-0.0043	0.8736	-0.0036
12	0.928	0.9251	0.0029	0.9270	0.0010

For the explicit-pool fit, the approximate estimates are $I=0.300$, $k_p=0.55\ \mathrm{h}^{-1}$, and $k_b=0.32\ \mathrm{h}^{-1}$. For the transit fit, they are $I=0.300$ and $k_{\mathrm{tr}}=0.42\ \mathrm{h}^{-1}$. For example, the explicit-pool prediction at 4 hours is $1-I F_{\mathrm{pool}}(4)$. Substitution gives

F_{\mathrm{pool}}(4) =1-\frac{0.32e^{-0.55(4)}-0.55e^{-0.32(4)}}{0.32-0.55} \approx 0.4893,\qquad b(4)=1-0.300(0.4893)\approx 0.8532.

Squaring the seven nonzero residuals in the table and summing gives approximate residual sums of squares $RSS_{\mathrm{pool}}=0.0001039$ and $RSS_{\mathrm{tr}}=0.0001695$, in squared normalized-biomarker units. Under the independent normal-error assumption, a convenient AIC comparison is $n\ln(RSS/n)+2p$, up to a constant shared by both fits, where $p$ counts fitted mean-model parameters and the common variance parameter. Here $n=8$, $p_{\mathrm{pool}}=4$, and $p_{\mathrm{tr}}=3$. The difference is

AIC_{\mathrm{tr}}-AIC_{\mathrm{pool}} =8\ln\left(\frac{0.0001695}{0.0001039}\right)+2(3-4) \approx 1.9.

The explicit-pool fit has the lower AIC, but a difference of about 1.9 is modest evidence, not decisive separation. The fit with more kinetic flexibility also has one more parameter, and AIC penalizes that additional flexibility. This comparison does not establish which biological pathway is true.

Uncertainty Is Part of the Comparison

A parameter estimate is incomplete without information about its uncertainty. Under the assumed error model, a nonlinear fit commonly estimates parameter covariance from the local curvature of the likelihood or residual sum of squares. That approximation can be unreliable when the likelihood is asymmetric, parameters approach bounds, or different parameter combinations produce nearly the same curve. Profile likelihoods or simulation-based intervals can then be more informative than symmetric standard errors.

Worked Example: Reading Approximate Parameter Intervals

Suppose the hypothetical fitting output reports the following standard errors: for the explicit-pool model, $SE(I)=0.0045$, $SE(k_p)=0.040\ \mathrm{h}^{-1}$, and $SE(k_b)=0.025\ \mathrm{h}^{-1}$; for the transit model, $SE(I)=0.0050$ and $SE(k_{\mathrm{tr}})=0.012\ \mathrm{h}^{-1}$. As a quick large-sample approximation, form a 95% Wald interval as estimate $\pm 1.96SE$. For example, for $k_p$,

0.55\ \mathrm{h}^{-1}\ \pm 1.96(0.040\ \mathrm{h}^{-1}) =0.55\ \mathrm{h}^{-1}\ \pm 0.0784\ \mathrm{h}^{-1} \approx (0.472,\ 0.628)\ \mathrm{h}^{-1}.

The corresponding approximate intervals are $I: (0.291,\ 0.309)$ for the pool model, $k_b: (0.271,\ 0.369)\ \mathrm{h}^{-1}$, $I: (0.290,\ 0.310)$ for the transit model, and $k_{\mathrm{tr}}: (0.396,\ 0.444)\ \mathrm{h}^{-1}$. These are approximate intervals under the fitted model and covariance approximation, not probabilities assigned to fixed parameters. In this example, the intervals describe local parameter precision; they do not account for uncertainty about which model structure is appropriate.

The rates also have different meanings. $k_p$ and $k_b$ represent distinct processes only under the explicit-pool model; $k_{\mathrm{tr}}$ is a common transit rate. Comparing their numerical values directly as though they were the same biological quantity is not justified. A further warning sign would be a strong correlation between $k_p$ and $k_b$: if changing both rates together leaves predictions nearly unchanged, their separate values may be weakly identified even when the predicted biomarker curve is precise.

Residual Checks and Model Failure

A smaller residual sum of squares is useful only alongside checks of how residuals behave over time and across the response range. Plot residuals against time and fitted biomarker values; inspect their signs and sizes, and check whether observations from the same participant cluster. Under the assumed error model, residuals should show no systematic time pattern, no major change in spread, and no unexplained serial correlation. In the example, both fits have small residuals, but the transit residuals at 4 and 8 hours are positive and larger than neighboring residuals. With only eight observations, that is a clue to examine, not sufficient evidence by itself to reject the transit model.

A smooth sequence of positive residuals during onset followed by negative residuals during recovery can indicate that the model's delay shape is wrong. Increasing residual spread as the biomarker falls can suggest that constant additive error is inappropriate. A consistent offset in baseline can indicate that fixing $b(0)=1$ is inadequate. These patterns can arise from an incorrect structural model, a misspecified observation-error model, or data problems; residuals do not identify the cause automatically. Visual predictive checks and residual signatures are useful complements to a single fit statistic.

Do not infer the mechanism from the curve alone. Similar output trajectories can arise from different internal state interpretations.
Do not compare raw residual sums of squares without considering complexity. A more flexible model can reduce them simply by fitting more parameters.
Do not trust narrow local intervals blindly. Check parameter correlations, boundary estimates, and whether multiple parameter sets give similar predictions.

Predictions Under a Changed Dosing Schedule

A model comparison becomes more consequential when the models imply different responses to a schedule not used for fitting. Predictions should be reported with uncertainty, and their sensitivity to parameter and structural uncertainty should be assessed. For a single 4-hour dose followed by washout, the step-response difference gives the prediction at 8 hours without numerically integrating the differential equations again.

Worked Example: Predicting After a Four-Hour Dose

Use the fitted parameter values above, but suppose the new hypothetical schedule gives drug input from 0 to 4 hours and then stops. For the explicit-pool model, the step-response values are $F_{\mathrm{pool}}(4)\approx0.4893$ and $F_{\mathrm{pool}}(8)\approx0.8322$. At 4 hours, the deficit and biomarker are

q_b(4)=0.300F_{\mathrm{pool}}(4)\approx0.1468,\qquad b(4)=1-0.1468\approx0.8532.

At 8 hours, four hours after stopping, subtract the response to the step that ended at hour 4:

q_b(8)=0.300\{F_{\mathrm{pool}}(8)-F_{\mathrm{pool}}(4)\} =0.300(0.8322-0.4893)\approx0.1029,\qquad b(8)\approx0.8971.

For the transit model, $F_{\mathrm{tr}}(4)\approx0.5005$ and $F_{\mathrm{tr}}(8)\approx0.8486$. Thus $b(4)=1-0.300(0.5005)\approx0.8499$, while, using the unrounded step responses, $b(8)\approx0.8956$. The point predictions are close for this schedule. That agreement is valuable: it shows that the modest fit difference need not imply a large difference in this particular extrapolation. It does not guarantee agreement for every schedule, nor does it remove the need to propagate parameter uncertainty into prediction intervals.

How to Choose Between the Models

The explicit-pool model is appropriate when a precursor state has a defensible biological interpretation and the data can inform its separate formation and conversion rates. The transit model is a more compact representation when the main goal is to represent maturation delay without asserting a measured precursor pool. The earlier material on transit-stage choice and identifiability is relevant when deciding whether the available sampling times can distinguish these structures.

Neither model is universally preferable. The explicit model can be over-parameterized for sparse data; the transit model may impose an overly restrictive equal-rate chain. Both can fail if inhibition varies with concentration, if baseline drifts, if feedback changes the response, or if the observation errors are correlated or heteroscedastic. External measurements of precursor concentrations, denser sampling around onset and recovery, and prediction performance under a distinct schedule can help discriminate mechanisms more effectively than fitting the same biomarker curve alone.

Key takeaway: Compare the models on fit, complexity, parameter uncertainty, residual behavior, and predictions relevant to the intended use. A better numerical fit is evidence about predictive adequacy under the assumed error model; it is not, by itself, proof of a biological pathway.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In the explicit-pool model, what does $k_p$ describe, and why is it not interchangeable with $k_{\mathrm{tr}}$?
Why does a dose stopped at time $t_d$ produce the response $I\{F(t)-F(t-t_d)\}$ for $t\geq t_d$?
If two models have the same residual sum of squares but one has an additional fitted parameter, which has lower AIC under the stated comparison?
What does a run of positive residuals during onset followed by negative residuals during recovery suggest?
If two models give similar predictions for a new schedule, does that establish which internal mechanism is correct?

Solutions. (1) $k_p$ is the rate constant for the explicit precursor-deficit state; $k_{\mathrm{tr}}$ is the common rate for each transit stage, so they have different model roles. (2) A finite pulse is the difference of a step starting at zero and an equal step starting at $t_d$; linearity makes the responses subtract. (3) The model with fewer parameters has lower AIC, because the fit terms are equal and the more complex model receives a larger penalty. (4) It can indicate a systematic mismatch in the modeled delay or response shape, although other model or data problems are possible. (5) No. Similar predictions support agreement for that schedule, but do not prove either model's internal biological interpretation.