Mapping TMDD States to Measured Concentrations

From TMDD States to Assay Readouts

A TMDD model describes biological quantities, but an assay reports a measurement produced by a particular laboratory method. Those are not automatically the same thing. For example, a model may track unbound drug and drug bound to target separately, while an assay may report only unbound drug, or may dissociate the complex and report both forms together. The observation equation must connect the states to what that assay actually measures.

Use \(D(t)\) for free-drug concentration, \(R(t)\) for free-target concentration, and \(C(t)\) for drug–target-complex concentration at time \(t\). For the ideal equations below, assume concentrations are expressed in the same sampled compartment and in molar units, and that each complex contains one drug molecule and one target molecule. These equations map states to assay predictions; they do not replace the TMDD state equations that describe how the concentrations change over time.

Definition: An observation equation specifies the assay-scale prediction from the model states. Under ideal, selective assays and one-to-one binding, free-drug and free-target predictions are \(D(t)\) and \(R(t)\). Total-drug and total-target predictions are \(D(t)+C(t)\) and \(R(t)+C(t)\), respectively, because each complex contains one molecule of each.

Why the Complex Appears in Both Total Measurements

“Total drug” and “total target” refer to different molecular totals. A total-drug assay counts drug molecules whether they are free or bound in complex. A total-target assay counts target molecules whether free or bound. In the assumed one-to-one complex, each complex contributes one drug molecule to the drug total and one target molecule to the target total. Thus the same \(C(t)\) appears in both equations, although each total is the total of a different analyte.

\begin{aligned} \widehat{Y}_{D,\mathrm{free}}(t)&=D(t),\\ \widehat{Y}_{D,\mathrm{total}}(t)&=D(t)+C(t),\\ \widehat{Y}_{R,\mathrm{free}}(t)&=R(t),\\ \widehat{Y}_{R,\mathrm{total}}(t)&=R(t)+C(t). \end{aligned}

The hat denotes the model-predicted assay result, not an additional TMDD state. The equations assume the assay is selective for the named analyte, recovers the relevant free and bound forms as intended, and reports a result proportional to the molar amount of analyte. A free assay is modeled as excluding complex-bound analyte. An ideal total assay is modeled as making the complex-bound analyte measurable, for example by dissociating the complex during sample processing.

The totals are not generally constant over time. Drug and target can enter or leave the modeled compartment through dosing, synthesis, distribution, or elimination. The equations are accounting identities at a given time, not claims that either total is conserved.

Deriving the Total-Concentration Equations

The total-drug equation follows by counting drug molecules in the sampled compartment. Free drug contributes \(D(t)\) moles per unit volume. Each mole of complex contains one mole of drug, so the complex contributes \(C(t)\) moles of drug per unit volume. Adding those contributions gives \(D(t)+C(t)\). Counting target molecules gives \(R(t)+C(t)\) by the same argument.

\begin{aligned} \text{drug molecules per unit volume} &=\text{free drug}+\text{drug in complex}\\ &=D(t)+1\cdot C(t),\\ \text{target molecules per unit volume} &=\text{free target}+\text{target in complex}\\ &=R(t)+1\cdot C(t). \end{aligned}

This bookkeeping also shows why a total assay should not be predicted as \(D(t)+R(t)+C(t)\): target molecules are not drug molecules, and drug molecules are not target molecules. The two analyte totals are formed separately. If the modeled complex has a different stoichiometry, or the assay reports a signal rather than a molar analyte concentration, the observation equation must reflect that different definition.

Worked Example: Four Ideal Assay Predictions

Suppose a hypothetical model predicts \(D=2.4\) nM, \(R=0.8\) nM, and \(C=1.6\) nM at a sampling time. Assume all three states refer to the same compartment, the complex is one-to-one, and the assays have the ideal selectivity and recovery just described.

Free drug: \(\widehat{Y}_{D,\mathrm{free}}=D=2.4\) nM.

Total drug: \(\widehat{Y}_{D,\mathrm{total}}=D+C=2.4+1.6=4.0\) nM.

Free target: \(\widehat{Y}_{R,\mathrm{free}}=R=0.8\) nM.

Total target: \(\widehat{Y}_{R,\mathrm{total}}=R+C=0.8+1.6=2.4\) nM.

The two total predictions differ because the free-drug and free-target concentrations differ; the shared complex contribution is 1.6 nM in each. The result would be incorrect if the complex were added to a free-only assay prediction without evidence that the assay detects bound analyte.

Assay Response, Recovery, and Units

Real assays need not behave like the ideal equations. One useful general representation is a calibrated linear observation map:

\widehat{Y}_j(t) =a_{jD}D(t)+a_{jR}R(t)+a_{jC}C(t).

Here \(j\) labels an assay, and the coefficients describe how its reported scale responds to each state. Their interpretation depends on the output units and calibration. For an ideal free-drug assay reported in molar drug units, the coefficients are \(a_{jD}=1\), \(a_{jR}=0\), and \(a_{jC}=0\). For an ideal total-drug assay in the same units, they are \(1,0,1\). A target assay has the corresponding target-specific pattern. These coefficients are a way to describe the observation map, not a substitute for checking the assay’s documented analyte and calibration.

For instance, an assay that responds to free drug and only a fraction \(\rho\) of complex-bound drug could have the prediction \(D(t)+\rho C(t)\), where \(0\leq\rho\leq1\) on a calibrated molar scale. When \(\rho=0\), the bound drug is not detected; when \(\rho=1\), it is recovered as fully as free drug. A nonzero response to free target or target-containing complex may also matter for an assay with cross-reactivity. Do not assume that such terms are present or absent without an assay definition.

Units require the same care. If states are in mol/L but the assay reports mass concentration, convert each molecular analyte using its own molecular weight. For total drug, the complex term counts the drug moiety and is converted using the drug molecular weight. For total target, that same complex concentration counts the target moiety and is converted using the target molecular weight. The complex is not assigned one shared mass conversion for both analyte totals.

Worked Example: Converting Total Drug to Mass Concentration

Suppose, hypothetically, \(D=3.0\) nM and \(C=2.0\) nM. An ideal total-drug assay reports mass concentration, and the drug molecular weight is \(150{,}000\) g/mol. Assume the assay reports drug-moiety mass, including drug released from complex during processing.

First calculate the total drug in molar units:

\widehat{Y}_{D,\mathrm{total}}=D+C=3.0+2.0=5.0\ \mathrm{nM}.

Since \(1\) nM is \(10^{-9}\) mol/L, the mass concentration is

5.0\times10^{-9}\ \mathrm{mol/L} \times150{,}000\ \mathrm{g/mol} =7.5\times10^{-4}\ \mathrm{g/L} =0.75\ \mathrm{mg/L}.

Thus the predicted total-drug result is \(0.75\) mg/L. The calculation converts the drug concentration represented by both \(D\) and \(C\); it does not add the target’s mass to the drug assay result.

Worked Example: A Partially Complex-Recovering Assay

Suppose a hypothetical assay is calibrated in nM drug equivalents and responds fully to free drug but recovers only 25% of complex-bound drug. At a sampling time, \(D=3.0\) nM and \(C=2.0\) nM. Assume no response to target alone.

The appropriate prediction is \(\widehat{Y}=D+0.25C\), not the ideal total \(D+C\). Substituting gives

\widehat{Y}=3.0+0.25(2.0)=3.0+0.50=3.50\ \mathrm{nM\ drug\ equivalents}.

The ideal free-drug prediction would be \(3.0\) nM, while the ideal total-drug prediction would be \(5.0\) nM. The assay’s documented partial recovery puts its prediction between these values for these nonnegative states. That comparison is a check on this particular observation map, not a universal rule for assays with other response scales or cross-reactivity.

Practical Checks and Common Failure Modes

Before using an observation equation, verify what molecule the assay quantifies, whether it measures free or total analyte, and whether sample processing preserves or dissociates complexes. Confirm that the model states describe the same compartment and that the concentration and reporting units are compatible. These checks concern the mapping from states to assay outputs; they do not determine how the state trajectories themselves should be modeled.

Adding the complex to every assay. This incorrectly treats a free-specific assay as a total assay. Include \(C(t)\) only to the extent that the assay detects complex-bound analyte.
Using one total equation for both analytes. Total drug is \(D+C\); total target is \(R+C\). The free term must match the analyte being measured.
Adding drug and target concentrations. \(D+R+C\) does not represent either analyte total. Form a separate molecular count for each analyte.
Mixing units or compartments. Adding a concentration in one compartment to a concentration from another, or adding mass concentration directly to molar concentration, is not valid without a justified conversion and model link.
Assuming a reported “total” guarantees full complex recovery. The assay name alone does not establish recovery, cross-reactivity, or the analyte’s reporting scale. Use the assay definition and validation information.

A basic numerical check is that, for nonnegative states and ideal molar equations, total drug is at least free drug and total target is at least free target. If measured values appear to violate those comparisons, possible explanations include assay error, unmatched measurement times or compartments, incompatible units, or an incorrect observation map. Observed values need not obey exact inequalities in every noisy sample; the checks apply directly to the model predictions under the stated ideal assumptions.

Choosing an Observation Map

The main choice is not between different TMDD state equations, but between ways of connecting those states to the available assay data. If an assay measures free drug selectively, predict \(D(t)\); if it measures total drug with full complex recovery, predict \(D(t)+C(t)\). Use the corresponding target equations for target assays. If assay recovery is partial or there is cross-reactivity, represent that response explicitly when it is known and supported.

A convenient total assay may be easier to interpret as a combined analyte measurement, but it does not reveal free and bound concentrations separately. A free assay may directly inform the unbound state, but it does not count the bound fraction. Paired free and total measurements can, under matching units, time, compartment, and ideal one-to-one assumptions, provide two distinct views of the states. Their differences are not automatically exact measurements of complex: measurement error and assay-specific recovery can affect that calculation.

These observation equations are limited by the definitions and performance of the assays. They cannot correct for an assay that measures an unmodeled species, for uncertain sample handling, or for a state defined in a different compartment from the sample. In those situations, refine the observation map or the modeled states using appropriate assay information rather than forcing the ideal equations onto the data.

Check Your Understanding

Answer each question using the one-to-one complex assumption unless another condition is stated. Solutions follow.

If \(D=1.2\) nM, \(R=0.7\) nM, and \(C=0.5\) nM, what are the ideal total-drug and total-target predictions?
An ideal free-target assay is reported in the same molar units as \(R\) and \(C\). Should its prediction include \(C\)? Explain.
An assay responds to \(D\) and 40% of complex-bound drug, with \(D=2.0\) nM and \(C=1.5\) nM. What is its predicted result in drug-equivalent units?
Why is it incorrect to convert a total-drug prediction using the target molecular weight?

Solutions. 1. Total drug is \(D+C=1.2+0.5=1.7\) nM; total target is \(R+C=0.7+0.5=1.2\) nM. 2. No. An ideal free-target assay predicts \(R=0.7\) nM and excludes complex-bound target. 3. The prediction is \(D+0.40C=2.0+0.40(1.5)=2.6\) nM drug equivalents. 4. A total-drug assay counts drug moieties, including those in complex, so its mass conversion uses the drug molecular weight; the target molecular weight applies to target-moiety measurements.