Choosing Full TMDD, QSS, or Michaelis-Menten Models

Why Model Choice Is More Than a Fit Statistic

Full target-mediated drug disposition (TMDD) models describe drug binding, complex formation, and target turnover as dynamic processes. Quasi-steady-state (QSS) and Michaelis–Menten (MM) models simplify those processes, which can make estimation more stable and reduce the number of parameters. But a simpler model is useful only if its assumptions are compatible with the dose, time-course, measurements, and predictions of interest.

The practical choice is therefore not “which model has the largest likelihood?” A more informative question is whether a reduced model preserves the features of the data that matter, while its assumptions and parameter estimates remain credible. Likelihood-based criteria, predictive residuals, and mechanistic checks each answer a different part of that question.

What the Three Models Represent

A minimal full TMDD model tracks free drug concentration \(C\), free target concentration \(R\), and drug–target complex concentration \(RC\). In a one-compartment description with linear drug elimination rate constant \(k_e\), association rate \(k_{\mathrm{on}}\), dissociation rate \(k_{\mathrm{off}}\), and complex internalization rate \(k_{\mathrm{int}}\), the binding and turnover terms can be written as:

\begin{aligned} \frac{dC}{dt} &= \text{input} - k_e C - k_{\mathrm{on}}CR + k_{\mathrm{off}}RC,\\ \frac{dR}{dt} &= k_{\mathrm{syn}} - k_{\mathrm{deg}}R - k_{\mathrm{on}}CR + k_{\mathrm{off}}RC,\\ \frac{dRC}{dt} &= k_{\mathrm{on}}CR - (k_{\mathrm{off}}+k_{\mathrm{int}})RC. \end{aligned}

The equations are schematic: the input term must use units consistent with concentration, and the observation model must correspond to the measured quantity, such as free drug or total drug. The full model lets the complex change dynamically rather than imposing an instantaneous binding relationship.

Definition: The QSS approximation replaces the dynamic complex equation with the algebraic relationship \(RC \approx R_{\mathrm{tot}}C/(K_{\mathrm{ss}}+C)\), where \(R_{\mathrm{tot}}=R+RC\) and \(K_{\mathrm{ss}}=(k_{\mathrm{off}}+k_{\mathrm{int}})/k_{\mathrm{on}}\). It still allows target amount to change through synthesis, degradation, and internalization. An MM reduction further treats target availability as approximately constant, yielding saturable elimination with \(V_{\max}\approx k_{\mathrm{int}}R_{\mathrm{tot}}\) and \(K_m\approx K_{\mathrm{ss}}\) under that reduction.

QSS is motivated when complex formation and loss adjust rapidly relative to changes in free drug and target. MM adds a stronger simplification: target availability must remain sufficiently stable that a fixed-capacity saturable-elimination term is adequate. The rapid-binding equilibrium reduction uses \(K_d=k_{\mathrm{off}}/k_{\mathrm{on}}\); it is not interchangeable with QSS, whose constant includes complex internalization.

A Practical Comparison Strategy

Begin by fitting candidates to the same observations, using the same observation model and a consistent likelihood definition. In NLME, likelihood-based comparisons should be based on maximum likelihood, not a mixture of likelihoods computed by different approximations or conditioning conventions. Record the number of estimated parameters \(k\), the maximized log-likelihood \(\ell\), and the number of observations \(N\) used for the information criterion.

\mathrm{AIC}=-2\ell+2k,\qquad \mathrm{BIC}=-2\ell+k\ln(N).

Lower AIC or BIC indicates a better balance of fit and complexity among the candidates being compared; neither proves that a model is biologically correct. AIC penalizes each additional parameter by 2 units. BIC applies a penalty of \(\ln(N)\) per parameter, which is larger than 2 when \(N>e^2\). For hierarchical data, report the convention used for \(N\) and apply it consistently.

A likelihood-ratio test is not automatically appropriate just because one model is reduced. Its familiar reference distribution requires nested models and regularity conditions, including parameters that are identifiable under the null and not on problematic boundaries. Approximation-based reductions may be limiting or differently parameterized rather than regular nested alternatives. If those conditions are not defensible, compare information criteria and predictive performance instead of reporting a routine chi-square likelihood-ratio p-value.

Predictive residuals add information that a single likelihood number cannot. A standardized residual can be expressed as observed minus predicted concentration, divided by the predictive standard deviation. Positive values indicate underprediction; negative values indicate overprediction. Examine residuals against time, dose, and predicted concentration, and distinguish population predictions from predictions conditioned on individual random-effect estimates. Conditional predictions can look reassuring partly because they use information from that individual’s observations. Simulation-based predictive checks can help assess whether the fitted model reproduces the observed patterns and variability.

Worked Comparisons

Worked Example: Comparing AIC and BIC

Suppose three candidate models are fitted to the same hypothetical data set of \(N=120\) concentration observations, with the same likelihood construction. The full TMDD, QSS, and MM fits have \(-2\ell=240, 242,\) and \(252\), respectively, and use \(k=8, 6,\) and \(4\) estimated parameters.

Their AIC values are:

\begin{aligned} \mathrm{AIC}_{\mathrm{full}} &= 240+2(8)=256,\\ \mathrm{AIC}_{\mathrm{QSS}} &= 242+2(6)=254,\\ \mathrm{AIC}_{\mathrm{MM}} &= 252+2(4)=260. \end{aligned}

AIC favors QSS among these candidates. For BIC, \(\ln(120)\approx4.787\), giving:

\begin{aligned} \mathrm{BIC}_{\mathrm{full}} &= 240+8(4.787)=278.3,\\ \mathrm{BIC}_{\mathrm{QSS}} &= 242+6(4.787)=270.7,\\ \mathrm{BIC}_{\mathrm{MM}} &= 252+4(4.787)=271.1. \end{aligned}

BIC also favors QSS, though its value is close to MM. This comparison does not establish that QSS is mechanistically true. It identifies QSS as a good fit–complexity compromise for these data; predictive diagnostics and the approximation assumptions still need review.

Worked Example: Using Residual Patterns to Challenge a Simpler Fit

Consider a hypothetical MM fit to early concentration measurements. At one time point, the observed concentration is \(2.40\ \mathrm{nmol/L}\), the population prediction is \(2.00\ \mathrm{nmol/L}\), and the predictive standard deviation is \(0.25\ \mathrm{nmol/L}\). The standardized residual is:

\frac{2.40-2.00}{0.25}=1.60.

This positive residual means the model underpredicts this observation by \(0.40\ \mathrm{nmol/L}\), or 1.60 predictive standard deviations. Suppose, in the same hypothetical analysis, residuals across early observations average \(+1.1\), while residuals across late observations average \(-0.9\). That systematic change in sign suggests a time-dependent mismatch, not merely one unusually high measurement. If QSS residuals are instead centered near zero in both periods and show no clear trend, QSS has a predictive advantage worth investigating.

The averages alone do not prove that QSS is adequate: inspect plots and the spread of residuals as well. Confirm that predictions are calculated consistently and that the pattern persists across subjects rather than being driven by one individual. The evidence favors QSS only if its assumptions and parameter estimates are also plausible.

Worked Example: Checking Whether the MM Assumption Is Credible

Suppose a hypothetical QSS fit estimates \(k_{\mathrm{off}}=0.04\ \mathrm{h^{-1}}\), \(k_{\mathrm{int}}=0.01\ \mathrm{h^{-1}}\), and \(k_{\mathrm{on}}=0.50\ \mathrm{L\,nmol^{-1}\,h^{-1}}\). Then:

K_{\mathrm{ss}}= \frac{0.04+0.01}{0.50} =0.10\ \mathrm{nmol/L}.

Suppose the fitted target amount is \(R_{\mathrm{tot}}=0.50\ \mathrm{nmol/L}\) at baseline, but the model’s predicted target trajectory falls to \(0.20\ \mathrm{nmol/L}\) during high exposure. That is a reduction of \(0.30\ \mathrm{nmol/L}\), or \(0.30/0.50=60\%\) of baseline. A fixed-capacity MM approximation based on the baseline target would then use a capacity that is poorly representative of the predicted target state. Even if MM has a competitive AIC, this large predicted depletion is a reason to prefer a model that allows target amount to change, such as QSS, or to investigate whether the assumptions or fit are wrong.

Justifying the Reduced Relationship

The QSS relationship follows by setting the complex derivative close to zero relative to the other changing quantities. From the full complex equation, this gives:

0\approx k_{\mathrm{on}}CR-(k_{\mathrm{off}}+k_{\mathrm{int}})RC, \qquad RC\approx\frac{CR}{K_{\mathrm{ss}}}.

Since \(R_{\mathrm{tot}}=R+RC\), substitute \(R=R_{\mathrm{tot}}-RC\) into the last relationship:

RC\approx\frac{C(R_{\mathrm{tot}}-RC)}{K_{\mathrm{ss}}} \quad\Longrightarrow\quad RC(K_{\mathrm{ss}}+C)\approx CR_{\mathrm{tot}} \quad\Longrightarrow\quad RC\approx\frac{R_{\mathrm{tot}}C}{K_{\mathrm{ss}}+C}.

This algebra explains the saturation-shaped complex relationship; it does not establish that complex adjustment is fast enough in a particular data set. That is a timescale assumption to assess against the model’s rates and observed concentration changes. MM adds the further approximation that \(R_{\mathrm{tot}}\) can be treated as nearly constant. If it changes substantially, the fixed \(V_{\max}\) interpretation becomes questionable.

Making the Choice and Recognizing Failure

Confirm comparability.
Fit the same data with compatible observation models and likelihood definitions. Check convergence and ensure the reported likelihoods are genuinely comparable.

Compare fit and complexity.
Use AIC or BIC as summaries, not as a substitute for scientific judgment. Do not apply an ordinary likelihood-ratio test unless nesting and regularity conditions are justified.

Inspect prediction errors.
Look for systematic residual trends by time, dose, concentration, and subject. Patterns can expose missing dynamics even when an aggregate criterion is favorable.

Check assumptions and parameter plausibility.
Ask whether complex adjustment is plausibly fast enough for QSS and whether target availability is stable enough for MM. Check units, boundary estimates, large uncertainty, and implausible parameter combinations.

Choose for the intended use.
Prefer the simplest candidate that gives adequate predictions for the relevant dose and time range and has credible assumptions. Retain the full model when reduced-model failures matter for the intended predictions.

Common warning signs include optimizer failure, estimates pressed against bounds, large changes in estimates from different initial values, highly uncertain parameters, and residual patterns that persist across subjects. A low information criterion cannot repair a failed fit or validate an approximation outside the observed dose and time range. Conversely, the full model is not automatically preferable: if the data do not support its extra parameters, estimates may be unstable and prediction may not improve.

The main alternative to selecting one model is to report sensitivity across plausible candidates, especially when conclusions change with the approximation. A full model can be useful as a mechanistic benchmark; QSS can retain evolving target behavior with fewer dynamic states; MM can be practical when target availability is effectively stable. None is universally best. The data may not contain enough information to distinguish them, so conclusions should reflect that uncertainty.

Common pitfall: Choosing MM solely because it has fewer parameters and a similar AIC, then interpreting its fixed \(V_{\max}\) as a measured constant target capacity. The correction is to examine target depletion and predictive residuals, and to verify that target availability is sufficiently stable over the exposure range. A simpler fit can be statistically competitive while its mechanistic approximation is inappropriate.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

What additional approximation does MM make beyond QSS?
For the hypothetical comparison above, which model has the smallest AIC, and what does that conclusion mean?
A standardized residual is \(-2.0\). Is the observation above or below its prediction?
Why might a routine likelihood-ratio test be invalid when comparing a full TMDD model with a reduced approximation?
If target amount is predicted to decline substantially during dosing, which approximation is most directly challenged, and why?

Solutions. (1) MM treats target availability as approximately constant, creating a fixed saturable capacity. (2) QSS has the smallest AIC, 254; among the candidates it gives the best AIC fit–complexity compromise, not proof of mechanistic truth. (3) The observation is below its prediction because the residual is observed minus predicted. (4) The reduced model may not be a regular nested special case, or identifiability and boundary conditions required for the usual reference distribution may fail. (5) MM is most directly challenged because a fixed target-based capacity assumes target availability is nearly stable; a changing target may require a dynamic representation such as QSS or full TMDD.