Between-Subject Target Abundance and TMDD Exposure

Why Between-Subject Target Abundance Matters

At the same administered dose, two subjects need not have the same free-drug exposure. In a target-mediated drug disposition (TMDD) model, drug binding to target can create an additional route of drug elimination, particularly when drug-target complex is internalized. If baseline target abundance differs between subjects, the capacity for that route differs as well. The resulting exposure difference can carry through to target occupancy and pharmacodynamic response.

The key is to distinguish a subject’s baseline target abundance from the amount of target that is free or bound at a particular time. Here, baseline abundance is represented by a subject-specific target capacity. A quasi-steady-state model will make the exposure consequence explicit; it is an approximation, not a replacement for a turnover model when target abundance changes substantially during treatment.

Definition: Let $R_i$ be the effective target abundance for subject $i$, expressed as a concentration of binding sites. Between-subject variability in baseline target abundance means that $R_i$ differs across subjects before drug exposure. A common population model is $R_i=R_{\mathrm{pop}}\exp(\eta_{R,i})$, where $R_{\mathrm{pop}}$ is the population median and $\eta_{R,i}\sim N(0,\omega_R^2)$. This lognormal model ensures positive abundance; its arithmetic mean is $R_{\mathrm{pop}}\exp(\omega_R^2/2)$.

A Fixed-Pool Quasi-Steady-State Exposure Model

Consider constant intravenous infusion at rate $R_{\mathrm{inf}}$ into a central volume $V$. Let $C$ be free-drug concentration, $X$ be drug-target complex concentration, $k_0=CL/V$ be the first-order nonspecific elimination rate constant, $k_{\mathrm{on}}$ and $k_{\mathrm{off}}$ be binding and dissociation rate constants, and $k_{\mathrm{int}}$ be the complex internalization rate constant. Drug is eliminated nonspecifically from the free pool and through internalization of complex.

For this reduced model, assume a fixed total target pool $R_i$, rapid equilibration of binding relative to the timescale of exposure change, and constant parameters within each subject. Under the quasi-steady-state approximation, complex concentration is $X=R_iC/(K_{\mathrm{ss}}+C)$, where $K_{\mathrm{ss}}=(k_{\mathrm{off}}+k_{\mathrm{int}})/k_{\mathrm{on}}$. This $K_{\mathrm{ss}}$ is not necessarily the equilibrium dissociation constant $K_D=k_{\mathrm{off}}/k_{\mathrm{on}}$, because complex internalization also removes complex.

\frac{R_{\mathrm{inf}}}{V} = k_0C+k_{\mathrm{int}}X = k_0C+k_{\mathrm{int}}R_i\frac{C}{K_{\mathrm{ss}}+C}

The left side is the drug input rate per unit volume; both terms on the right have units of concentration per time. The first is nonspecific elimination, while the second is target-mediated elimination. Thus, even with the same infusion rate and nonspecific clearance, a larger $R_i$ can support more target-mediated elimination and lower the steady-state free concentration.

Assumptions and scope: This fixed-pool approximation is useful for isolating the effect of target abundance. It assumes that the effective target pool remains constant across the exposure range being compared. If drug binding, internalization, or synthesis and degradation appreciably alter target abundance during treatment, a dynamic target-turnover model is needed; $R_i$ should then be interpreted as a baseline parameter, not as the instantaneous target amount.

How Target Abundance Changes Exposure

For positive input and positive model parameters, the steady-state equation has one positive solution for $C$. Its right side is zero at $C=0$, increases continuously with $C$, and approaches an unbounded value because of the $k_0C$ term. The response to abundance can be established by implicit differentiation.

Derivation: Write $u=R_{\mathrm{inf}}/V$ and $f(C,R_i)=k_0C+k_{\mathrm{int}}R_iC/(K_{\mathrm{ss}}+C)-u=0$. Differentiating with respect to $R_i$, while holding the infusion and other parameters fixed, gives $$ \frac{\partial C}{\partial R_i} = -\frac{k_{\mathrm{int}}C/(K_{\mathrm{ss}}+C)} {k_0+k_{\mathrm{int}}R_iK_{\mathrm{ss}}/(K_{\mathrm{ss}}+C)^2} <0. $$ The numerator and denominator before the minus sign are positive. Therefore, within this model, increasing target abundance decreases steady-state free-drug concentration. The denominator is the slope of total elimination with respect to free concentration, so the formula also shows why the exposure change depends on the operating concentration and model parameters.

The result is conditional on holding dose, nonspecific clearance, binding parameters, and the effective target pool structure fixed. It does not imply that every subject with a larger measured baseline biomarker must have lower exposure: other subject-level differences, measurement error, or treatment-induced target changes may alter the observed relationship.

Worked Example: Comparing Two Target Abundances at One Infusion Rate

Suppose, hypothetically, that $u=1.00$ concentration unit per hour, $k_0=0.100$ per hour, $k_{\mathrm{int}}=0.500$ per hour, and $K_{\mathrm{ss}}=2.00$ concentration units. Compare subjects with $R_i=1.00$ and $R_i=3.00$ concentration units. The values are hypothetical and illustrate the reduced model.

For $R_i=1.00$, solve the steady-state equation:

1=0.100C+0.500\frac{C}{2.00+C} \quad\Longrightarrow\quad 0.100C^2-0.300C-2.00=0

The positive root is $C=(3+\sqrt{89})/2=6.217$ concentration units. The negative root is not a physically meaningful concentration. At this concentration, nonspecific elimination is $0.100(6.217)=0.6217$ concentration units per hour, and target-mediated elimination is $0.500(1.00)(6.217/8.217)=0.3783$ concentration units per hour. Their sum is $1.000$, matching the input.

For $R_i=3.00$, the equation becomes:

1=0.100C+1.50\frac{C}{2.00+C} \quad\Longrightarrow\quad 0.100C^2+0.700C-2.00=0

The positive root is $C=(-7+\sqrt{129})/2=2.179$ concentration units. Nonspecific elimination is $0.100(2.179)=0.2179$, while target-mediated elimination is $0.500(3.00)(2.179/4.179)=0.7821$, in concentration units per hour. Again, the sum is $1.000$. In this example, tripling target abundance lowers steady-state free concentration from $6.217$ to $2.179$ concentration units because the target-mediated route accounts for a larger share of elimination.

Propagating Exposure Differences to Response

Exposure is not itself a pharmacodynamic response. To translate the concentration difference, specify a response model. For example, if response is proportional to target occupancy, occupancy may be written $\theta=C/(K_{\mathrm{occ}}+C)$, where $K_{\mathrm{occ}}$ is the concentration producing half-maximal occupancy under the chosen occupancy model. A direct effect model might instead use a baseline plus an $E_{\max}$ function of concentration. The response consequences therefore depend on both the exposure change and the concentration-response relationship.

Worked Example: From Free Concentration to Occupancy

Use the two steady-state concentrations from the preceding example and suppose, hypothetically, that $K_{\mathrm{occ}}=1.00$ concentration unit. Under the occupancy model, the subject with $R_i=1.00$ has $\theta=6.217/(1.00+6.217)=0.8615$, or 86.15% occupancy. The subject with $R_i=3.00$ has $\theta=2.179/(1.00+2.179)=0.6854$, or 68.54% occupancy.

Thus, for this specified response model and fixed dose, the higher-abundance subject has lower predicted occupancy because free concentration is lower. This is a model-based prediction, not evidence that abundance causally changes response in an observed population. If response depends on a different concentration, a delayed effect compartment, or a changing target pool, this calculation alone is insufficient.

Between-subject variability also means that a population-average response generally should not be calculated by inserting an average target abundance into the model and treating the result as the average response. Nonlinear transformations do not usually commute with averaging. The following small hypothetical population makes that distinction visible.

Worked Example: Average of Individual Predictions Versus a Typical Subject

Suppose a hypothetical population has equal numbers of two subject types, with target abundances $R_i=1.00$ and $R_i=3.00$. Their geometric-mean abundance is $\sqrt{1.00\times3.00}=1.732$ concentration units, while their arithmetic mean abundance is $2.00$. Use the same infusion and parameters as before.

For the geometric-mean abundance, the steady-state equation reduces to $0.100C^2+(0.500(1.732)-0.800)C-2.00=0$, giving the positive solution $C=4.154$ concentration units. With $K_{\mathrm{occ}}=1.00$, the occupancy for this typical subject is $4.154/5.154=0.8060$, or 80.60%.

The mean of the two individual free concentrations is $(6.217+2.179)/2=4.198$ concentration units, not $4.154$. The mean of their individual occupancies is $(0.8615+0.6854)/2=0.7735$, or 77.35%, not 80.60%. The differences are modest in this example, but they arise because the concentration and occupancy mappings are nonlinear. Population predictions should therefore be obtained by evaluating individual predictions across the target-abundance distribution and then summarizing those predictions.

Quantifying Sensitivity and Representing Variability

A useful local summary is the elasticity of concentration with respect to target abundance: the percent change in $C$ associated approximately with a 1% change in $R_i$, near a specified parameter set and exposure. It is obtained by multiplying the derivative by $R_i/C$. The elasticity is local, not a constant effect across the full range of abundance.

Worked Example: Local Exposure Sensitivity to Target Abundance

At $R_i=2.00$, with the parameters used above, the steady-state equation is $0.100C^2+0.200C-2.00=0$, whose positive solution is $C=3.583$ concentration units. The concentration elasticity is:

\frac{\partial \log C}{\partial \log R_i} = -\frac{k_{\mathrm{int}}R_i/(K_{\mathrm{ss}}+C)} {k_0+k_{\mathrm{int}}R_iK_{\mathrm{ss}}/(K_{\mathrm{ss}}+C)^2} = -\frac{1.00/5.583}{0.100+2.00/(5.583)^2} \approx -1.091

Near this abundance and concentration, a 1% increase in $R_i$ predicts approximately a 1.09% decrease in free-drug concentration, with the other quantities held fixed. For occupancy with $K_{\mathrm{occ}}=1.00$, the local elasticity of occupancy to concentration is $K_{\mathrm{occ}}/(K_{\mathrm{occ}}+C)=1.00/4.583=0.2182$. The corresponding local occupancy elasticity to abundance is approximately $(-1.091)(0.2182)=-0.238$. Thus, near this point, a 1% abundance increase predicts approximately a 0.238% decrease in occupancy. This local approximation should not be extrapolated far from the evaluated conditions.

In a population model, the lognormal random effect $R_i=R_{\mathrm{pop}}\exp(\eta_{R,i})$ provides a parsimonious way to represent positive abundance values. If the coefficient of variation (CV) is specified, the corresponding log-scale variance is $\omega_R^2=\log(1+\mathrm{CV}^2)$. For example, a 50% CV gives $\omega_R^2=\log(1.25)=0.2231$ and $\omega_R=0.4723$. The parameter $R_{\mathrm{pop}}$ is then the median, not the arithmetic mean, of the lognormal abundance distribution.

A random effect is appropriate when the scientific model treats abundance as varying among subjects around a population distribution. A measured baseline target biomarker can instead enter as a covariate, with an estimated relationship between biomarker value and target abundance. The two approaches answer different modeling questions: a random effect describes unexplained heterogeneity, whereas a covariate represents a measured predictor. They can be combined when justified, but data may not support separate estimates of measurement error, covariate effects, and residual between-subject variability.

Diagnostics, Alternatives, and Limitations

A target-abundance variability model should be assessed against the information that can actually identify it. Useful checks include comparing observed baseline biomarker distributions with the assumed abundance distribution, examining whether exposure or response residuals show systematic trends across measured baseline abundance, and verifying that individual predictions satisfy the drug mass balance. If concentrations are consistently underpredicted in one abundance range and overpredicted in another, the assumed abundance-exposure relationship may be inadequate or another parameter may vary with abundance.

Such patterns are not proof that target abundance is the cause. Biomarker assay error, correlation between target abundance and nonspecific clearance, dose differences, or treatment-related changes in target expression can produce similar patterns. Estimates of individual random effects can also be imprecise when each subject has sparse concentration data. A weak observed association therefore does not establish that target abundance is constant; it may indicate limited information.

Approach	What it represents	Main caution
One fixed target abundance for all subjects	A common effective target capacity	Can miss systematic exposure variation if abundance truly differs.
Measured baseline abundance as a covariate	An estimated relationship between a recorded biomarker and target abundance	Biomarker error and imperfect correspondence to the model target can weaken or bias the relationship.
Random effect on target abundance	Unexplained positive between-subject variability	May be difficult to distinguish from variability in clearance, binding, or other parameters.
Individual sensitivity analysis over plausible abundances	Predicted exposure range under specified abundance scenarios	Shows model implications but does not estimate a population distribution by itself.

The fixed-pool equation is most informative when target abundance is approximately stable over the exposure conditions of interest and rapid binding equilibration is reasonable. It can be misleading if target synthesis, turnover, or complex internalization changes the pool materially, or if binding kinetics are not fast relative to the concentration changes. In those cases, fit or simulate a model with explicit free target and complex dynamics, while retaining baseline abundance as a subject-level parameter. The reduced model also assumes that parameters other than target abundance are identical across the compared subjects; real populations may require additional sources of variability.

Common Mistake

A common mistake is to conclude that a 50% increase in target abundance must cause a 50% decrease in free-drug exposure. The fixed-dose relationship is nonlinear: its slope depends on nonspecific clearance, internalization, binding capacity, and the current concentration. The derivative and elasticity above quantify a local effect; they do not imply a proportional inverse relationship. Another mistake is to calculate response at the population-average abundance and call it the population-average response. When the exposure and response functions are nonlinear, predictions must be calculated across individual abundance values before averaging.

Key takeaway: In a fixed-dose TMDD model, greater effective target abundance can increase target-mediated elimination and lower free-drug exposure. The magnitude of the effect depends on the kinetic regime, and its translation to response depends on the response model. Represent subject variability explicitly and distinguish an individual prediction from a population average.

Check Your Understanding

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

Under the fixed-pool model, why does increasing $R_i$ lower steady-state free concentration at a fixed infusion rate?
With $k_{\mathrm{off}}=0.20$ per hour, $k_{\mathrm{int}}=0.30$ per hour, and $k_{\mathrm{on}}=0.25$ per concentration unit per hour, calculate $K_{\mathrm{ss}}$.
Why is $R_{\mathrm{pop}}$ the median rather than the arithmetic mean under the specified lognormal random-effects model?
What does a negative local elasticity of concentration with respect to abundance mean, and why should it not be treated as a constant effect?
Name two reasons a measured baseline biomarker might not show the predicted association with free-drug exposure.

Solutions. (1) At a fixed positive concentration, the target-mediated elimination term $k_{\mathrm{int}}R_iC/(K_{\mathrm{ss}}+C)$ increases with $R_i$; the balance with fixed input is restored at a lower concentration. (2) $K_{\mathrm{ss}}=(0.20+0.30)/0.25=2.00$ concentration units. (3) Since $\eta_R$ has mean zero, the median of $\exp(\eta_R)$ is 1, so the median abundance is $R_{\mathrm{pop}}$; the arithmetic mean is $R_{\mathrm{pop}}\exp(\omega_R^2/2)$. (4) It indicates that a small proportional increase in abundance predicts a proportional decrease in concentration near the evaluated conditions, holding other parameters fixed; the derivative changes with concentration and abundance. (5) Examples include biomarker measurement error, correlation with clearance or other model parameters, and treatment-related changes in target abundance.