Likelihoods for Free, Total, and Target Measurements

Why the Observation Model Matters in TMDD

A TMDD model predicts concentrations of free drug, free target, and drug–target complex. An assay does not necessarily measure each state separately: a total-drug assay may include drug bound in complex, while a target assay may detect free target, total target, or a particular mixture. Even when the predicted analyte is correctly specified, measurement variability can differ substantially across assay types and concentration ranges. The likelihood must represent both the assay’s mapping from model states to a measured quantity and its residual-error distribution.

The distinction matters because the likelihood weights discrepancies between predictions and observations. A noisy target assay contributes less precise evidence about target-related parameters than a precise assay with the same predicted response. Conversely, adding target measurements may provide information about parameters that free-drug observations alone constrain weakly—but only if the target assay actually measures the intended quantity and its error is modeled credibly.

From TMDD States to Assay Predictions

Let \(D(t)\) denote free-drug concentration, \(R(t)\) free-target concentration, and \(C(t)\) drug–target complex concentration at time \(t\). For a one-to-one drug–target complex, total drug expressed in drug-equivalent concentration units is \(D(t)+C(t)\), and total target expressed in target-equivalent concentration units is \(R(t)+C(t)\). These sums presume compatible molar units and one molecule of drug per molecule of complex. If the assay detects only free analyte, the complex is not included in its prediction.

Definition: An assay prediction \(m_a(t;\theta)\) is the model-implied mean or location of the measured response for assay type \(a\), at time \(t\), given model parameters \(\theta\). The prediction is formed by mapping the TMDD states to the molecular quantity that assay measures. For example, \(m_{\mathrm{free\,drug}}=D\), \(m_{\mathrm{total\,drug}}=D+C\), \(m_{\mathrm{free\,target}}=R\), and \(m_{\mathrm{total\,target}}=R+C\), when those are the respective assay definitions.

These mappings are not interchangeable. An assay called “target” does not automatically measure \(R\): depending on its reagents and protocol, it may detect total target or respond differently to target bound in complex. The assay definition, units, and calibration therefore need to be aligned with the model prediction before fitting. The earlier material on mapping TMDD states to measured concentrations establishes the mechanistic state definitions; here the focus is how those predictions enter the observation likelihood.

Residual-Error Models and Likelihoods

For observation \(y_{ak}\) from assay type \(a\) at record \(k\), write its prediction as \(m_{ak}=m_a(t_{ak};\theta)\). A common starting assumption is that, conditional on the model parameters and the predicted state, the residuals are independent and normally distributed across records and assay types. Independence is an assumption, not a consequence of using different assay labels; paired measurements from the same sample may have correlated errors.

Definition: Under an additive-plus-proportional normal error model, \(Y_{ak}\mid\theta\sim N(m_{ak},v_{ak})\), where \(v_{ak}=\sigma_{a,\mathrm{add}}^2+(\sigma_{a,\mathrm{prop}}m_{ak})^2\). The additive error scale \(\sigma_{a,\mathrm{add}}\) has the units of assay \(a\); the proportional scale \(\sigma_{a,\mathrm{prop}}\) is dimensionless. Setting one component to zero gives a purely proportional or purely additive normal model.

The normal density for one observation is \(f(y_{ak}\mid\theta)=(2\pi v_{ak})^{-1/2}\exp\{-(y_{ak}-m_{ak})^2/(2v_{ak})\}\). If observations are conditionally independent, their joint likelihood is the product of these densities, and the log-likelihood is the sum:

L(\theta)=\prod_{a,k}\frac{1}{\sqrt{2\pi v_{ak}}} \exp\left[-\frac{(y_{ak}-m_{ak})^2}{2v_{ak}}\right], \qquad \ell(\theta)=-\frac{1}{2}\sum_{a,k} \left[\log(2\pi v_{ak})+\frac{(y_{ak}-m_{ak})^2}{v_{ak}}\right].

The normalization term involving \(v_{ak}\) is part of the likelihood. It cannot be discarded when the variance depends on the prediction, as it does under proportional and combined error. Otherwise, a parameter change that increases the variance could appear to improve fit merely by making residuals easier to accommodate, without paying the corresponding density penalty.

Worked Example: A Combined-Error Free-Drug Observation

Suppose a hypothetical free-drug assay observes \(y=2.50\) nM when the TMDD model predicts \(m=2.00\) nM. Let the additive residual-error scale be \(0.20\) nM and the proportional scale be \(0.15\). Calculate the variance, standardized residual, and log-density contribution.

The combined variance is \(v=(0.20\ \mathrm{nM})^2+(0.15\times2.00\ \mathrm{nM})^2 =0.0400+0.0900=0.1300\ \mathrm{nM}^2\). Thus the residual standard deviation is \(\sqrt{0.1300}=0.3606\) nM.

The standardized residual is \(z=(2.50-2.00)/0.3606=1.387\). The log-density contribution, retaining the density’s units of inverse nM, is \(-\log(0.3606)-0.918939-\tfrac12(1.387)^2 =1.0201-0.9189-0.9615=-0.8604\). This is one observation’s contribution; it is not by itself a model-comparison result. The residual is about 1.39 residual standard deviations above the prediction.

Likelihoods for Different Assays

Each assay can have its own prediction mapping and residual-error scales. A free-drug observation can be compared with \(D(t)\), while a total-drug observation can be compared with \(D(t)+C(t)\). Similarly, a free-target assay may use \(R(t)\), and a total-target assay may use \(R(t)+C(t)\). In a joint analysis, every record contributes a density using its own assay prediction and error model. Assay-specific scales should be estimated or fixed on the basis of appropriate information; a single shared residual scale is not justified merely because all measurements came from one study.

Worked Example: A Total-Target Assay Contribution

At a hypothetical time point, suppose the model states are \(D=0.80\) nM, \(C=0.40\) nM, and \(R=0.60\) nM. A validated total-target assay measures free target plus complex, so its prediction is \(m=R+C=1.00\) nM. The observed total-target concentration is \(1.20\) nM, with an additive normal error scale of \(0.25\) nM.

The standardized residual is \(z=(1.20-1.00)/0.25=0.800\). The log-density is \(-\log(0.25)-0.918939-\tfrac12(0.800)^2 =1.386294-0.918939-0.320000=0.1474\). The positive value is permissible: a density has units of inverse nM and is not a probability. The likelihood contribution is based on the total-target prediction of \(1.00\) nM, not the free-target state \(R=0.60\) nM. If the assay instead measured only free target, the prediction and hence the likelihood contribution would be different.

A proportional normal model sets \(\sigma_{a,\mathrm{add}}=0\), so the residual standard deviation is \(\sigma_{a,\mathrm{prop}}m_{ak}\). This can represent variability that grows with concentration, but a normal distribution permits negative observations; that may be implausible near zero. A distinct alternative is a lognormal model. For positive \(Y\), one parameterization is \(\log Y\mid\theta\sim N(\log m,\omega^2)\), in which \(m\) is the conditional median. Its density on the original measurement scale includes a Jacobian factor \(1/y\):

f(y\mid m,\omega)= \frac{1}{y\,\omega\sqrt{2\pi}} \exp\left[-\frac{(\log y-\log m)^2}{2\omega^2}\right], \qquad y>0.

A lognormal model is not the same as a proportional normal model: it implies a different distribution and a multiplicative error structure. If \(m\) is intended to be the arithmetic mean rather than the median, the log-scale location must be adjusted to \(\log m-\omega^2/2\). The chosen interpretation should match the scientific meaning assigned to the model prediction.

Worked Example: Combining Free- and Total-Drug Contributions

Consider two hypothetical, conditionally independent observations at the same time. The free-drug observation is \(2.50\) nM with prediction \(2.00\) nM and proportional scale \(0.20\). The total-drug observation is \(2.00\) nM with prediction \(2.40\) nM and proportional scale \(0.10\). Assume purely proportional normal errors for both assays, with their own scales.

For free drug, the standard deviation is \(0.20(2.00)=0.400\) nM and \(z=(2.50-2.00)/0.400=1.250\). Its log-density is \(-\log(0.400)-0.918939-\tfrac12(1.250)^2 =0.916291-0.918939-0.781250=-0.7839\).

For total drug, the standard deviation is \(0.10(2.40)=0.240\) nM and \(z=(2.00-2.40)/0.240=-1.667\). Its log-density is \(-\log(0.240)-0.918939-\tfrac12(1.667)^2 =1.427116-0.918939-1.388889=-0.8807\). Under the stated independence assumption, the joint log-likelihood contribution is the sum, \(-0.7839-0.8807=-1.6646\). The two discrepancies are weighted using separate assay scales and predictions; treating them as one observation with a pooled error scale would describe a different model.

How Assay Noise Changes Parameter Information

The role of assay precision can be made explicit locally. Suppose an observation has a normal distribution with mean \(m(\theta)\), variance \(v(\theta)\), and differentiable dependence on parameter vector \(\theta\). Its expected information matrix is

\mathcal I(\theta)= \frac{1}{v(\theta)} \frac{\partial m}{\partial\theta} \left(\frac{\partial m}{\partial\theta}\right)^{\!T} + \frac{1}{2v(\theta)^2} \frac{\partial v}{\partial\theta} \left(\frac{\partial v}{\partial\theta}\right)^{\!T}.

To justify the expression, differentiate the normal log-density. Its score is \(\frac{y-m}{v}\frac{\partial m}{\partial\theta} +\frac12[\frac{(y-m)^2}{v^2}-\frac1v]\frac{\partial v}{\partial\theta}\). Taking the expected outer product under the model uses \(E(y-m)=0\), \(E[(y-m)^2]=v\), and the zero covariance between the centered first and second normal moments. The resulting two terms are the information from changes in the mean and in the variance, respectively. If the variance is fixed with respect to \(\theta\), only the first term remains.

The mean-information term shows why assay type matters: the sensitivity vector \(\partial m/\partial\theta\) depends on which TMDD state combination is measured. Free-drug data may respond to binding through depletion of \(D\); total-drug data also reflect drug in complex; target data can respond to target and complex dynamics. These are potential sources of complementary information, not guarantees that parameters are separately identifiable. Greater residual variance reduces the information from a given mean sensitivity, while overlapping sensitivity patterns can still leave parameter combinations difficult to distinguish.

Assumptions, Diagnostics, and Limitations

The likelihood above assumes the observation model is correctly calibrated, the assay prediction corresponds to the actual analyte, and residuals follow the specified distribution conditional on the model. It also assumes the stated independence structure. Check residuals by assay type rather than pooling them: standardized residuals should not show systematic trends with time or predicted concentration, and their spread should be broadly compatible with the stated error model. Persistent curvature or bias may indicate a wrong state-to-assay mapping or model misspecification; increasing spread with concentration may indicate that additive error alone is inadequate.

Check units and analyte definitions. Do not add concentrations expressed on incompatible molar or mass bases, or treat a total assay as a free-analyte assay.
Check residual scale by assay. A single scale can over-weight a noisy assay and under-weight a precise one.
Check dependence. If paired assays share sample handling or calibration errors, conditional independence may be inadequate; a product of marginal densities then overstates the information in the joint data.
Check the support of the model. A normal error model can assign probability to negative concentrations. This may be tolerable as an approximation well above zero, but can be unsuitable near the assay’s lower range.
Check assay behavior. Saturation, cross-reactivity, or failure to detect complex as assumed can invalidate the prediction mapping. Measurements below a quantification limit also require a likelihood for the censoring event rather than treating the limit as an exact observed concentration.

These observation models do not by themselves resolve structural or practical identifiability problems. They specify how measurements contribute evidence conditional on the mechanistic predictions. Parameter information still depends on the observed dose and time range, the sensitivity of predictions to parameters, and whether different parameter effects can be distinguished in the data.

Common Mistake: Using the Wrong Quantity or Dropping the Variance Term

A common error is to compare a total-drug observation with \(D(t)\) because free drug is the primary state of interest, or to omit \(\log(v)\) from the likelihood because the residual discrepancy term seems to be the main component. The first error gives the assay the wrong expected value; the second changes the likelihood whenever variance depends on parameters or predictions. The correction is to establish the assay’s molecular target, construct its prediction from the appropriate states, and retain the full density—including its normalization term. Another frequent error is to call a proportional normal model “lognormal”; these distributions are not interchangeable.

Key takeaway: A TMDD likelihood is built assay by assay: map the mechanistic states to the measured analyte, specify that assay’s residual distribution and scale, and sum the resulting log-density contributions only under a justified dependence assumption.

Check Your Understanding

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

For a one-to-one complex, what are the model predictions for total drug and total target if the assays detect both free analyte and its complex?
A free-drug prediction is \(3.00\) nM, with additive scale \(0.10\) nM and proportional scale \(0.20\). What is the combined residual variance?
Why must a proportional normal likelihood retain the term involving \(\log(v)\)?
State one reason a total-target assay may add information beyond a free-drug assay, and one reason it may fail to do so.

Solutions. (1) Total drug is \(D+C\), and total target is \(R+C\), when the assays count complex in the respective analyte-equivalent units. (2) \(v=(0.10)^2+(0.20\times3.00)^2=0.0100+0.3600=0.3700\ \mathrm{nM}^2\). (3) The variance depends on the prediction, so omitting its normalization term lets parameter values alter the variance without the corresponding density penalty. (4) Total target can respond to the target-plus-complex pool and thus have a different sensitivity pattern; it may add little if it is very noisy or its sensitivity duplicates information already present in other observations.