Target-Mediated Drug Disposition and PK/PD · Tutorial 253 of 598

Linking Target Occupancy to Pharmacodynamic Response

Build a direct occupancy–response link and determine when dynamic TMDD states, rather than a concentration-based equilibrium formula, should define occupancy.

Advanced 12 min read

What You'll Learn

Define fractional occupancy from free-target and drug–target-complex states.
Derive the equilibrium occupancy formula and identify its assumptions.
Link occupancy to a direct Emax-type pharmacodynamic response.
Calculate occupancy and response from hypothetical TMDD states.
Diagnose when equilibrium-based occupancy is inconsistent with dynamic binding.
Distinguish an occupancy-driven response from a concentration-driven or delayed response.

Why Link Response to Occupancy?

A drug concentration is an exposure measure; target occupancy is a measure of how much target is bound. When a drug acts by binding a target, occupancy can provide a mechanistic intermediate between the TMDD model and the pharmacodynamic (PD) response. The link is useful only if the occupancy quantity is defined consistently with the model: a fraction calculated from circulating free drug is not automatically the same as the fraction bound in a dynamic target-binding model.

Here, “direct response” means that the modeled response at a given time depends on occupancy at that same time, without a separate effect-compartment delay or a response-turnover process. This is a modeling assumption, not a claim that every biological response changes instantaneously when binding changes.

Definition: Let $R(t)$ be free target concentration and $RC(t)$ be drug–target-complex concentration. The dynamic fractional occupancy is $$ \theta_{\mathrm{dyn}}(t)=\frac{RC(t)}{R(t)+RC(t)}. $$ It is the fraction of the modeled target pool that is bound at time $t$. When both states are nonnegative and their sum is positive, $0\leq\theta_{\mathrm{dyn}}(t)\leq1$. If the total target pool is zero, this fraction is undefined.

The definition uses the target states, not total drug concentration. In a TMDD model, free drug concentration $C(t)$, free target, and complex can change on different timescales. Binding, dissociation, target synthesis and turnover, and complex internalization can therefore make the complex fraction differ from a value inferred from $C(t)$ alone.

Dynamic Occupancy in a TMDD Model

For clarity, consider a single-compartment full TMDD model with concentration states $C$, $R$, and $RC$. Let $u(t)$ be drug input expressed as a concentration rate, $k_{\mathrm{el}}$ the first-order nonspecific elimination rate constant, $k_{\mathrm{on}}$ and $k_{\mathrm{off}}$ the binding and dissociation rate constants, $k_{\mathrm{syn}}$ the target synthesis rate, $k_{\mathrm{deg}}$ the free-target degradation rate constant, and $k_{\mathrm{int}}$ the complex internalization rate constant. A simple set of equations is $$ \frac{dC}{dt}=u(t)-k_{\mathrm{el}}C-k_{\mathrm{on}}CR+k_{\mathrm{off}}RC, $$ $$ \frac{dR}{dt}=k_{\mathrm{syn}}-k_{\mathrm{deg}}R-k_{\mathrm{on}}CR+k_{\mathrm{off}}RC, $$ $$ \frac{dRC}{dt}=k_{\mathrm{on}}CR-(k_{\mathrm{off}}+k_{\mathrm{int}})RC. $$

These equations assume that drug and target concentrations are expressed on compatible concentration scales, binding is represented by mass action, and the displayed processes are the relevant ones for this simplified model. More detailed models may represent additional compartments or different target and complex fates. The occupancy definition remains a ratio of the relevant bound target to the total target pool represented by the model.

Once the states are obtained by solving the differential equations, $\theta_{\mathrm{dyn}}(t)$ is calculated from $R(t)$ and $RC(t)$ at each time. It reflects the binding history encoded in those states. For example, free drug may decline while complex remains present, or complex may be removed by internalization. In either case, occupancy need not be determined by the current free drug concentration through an equilibrium formula.

Worked Example: Calculating Occupancy and Response From Dynamic States

Suppose a hypothetical TMDD model predicts $R=8.0$ nM and $RC=12.0$ nM at a particular time. Let the direct stimulatory response model be $E=E_0+E_{\max}\theta^\gamma/(\theta_{50}^\gamma+\theta^\gamma)$, with $E_0=10$ response units, $E_{\max}=50$ response units, $\theta_{50}=0.40$, and $\gamma=1.5$. Calculate occupancy and response.

The total modeled target concentration is $8.0+12.0=20.0$ nM, so $$ \theta_{\mathrm{dyn}}=\frac{12.0}{8.0+12.0}=0.600. $$ Occupancy is dimensionless; the concentration units cancel.

Substitute this occupancy into the response model: $$ E=10+50\frac{0.600^{1.5}}{0.40^{1.5}+0.600^{1.5}}. $$ Here, $0.600^{1.5}=0.4648$ and $0.40^{1.5}=0.2530$. Thus, $$ E=10+50\frac{0.4648}{0.2530+0.4648} =10+32.38 =42.38\ \text{response units}. $$ Under this model, the predicted response is 42.38 units at the specified time. This calculation does not establish that the response is biologically instantaneous; it applies the stated direct-link assumption.

Equilibrium Occupancy From Free Drug

A familiar alternative calculates occupancy from free drug concentration and the binding dissociation constant. The formula follows from a particular equilibrium assumption; it is not a general definition of occupancy.

Derivation: For reversible binding $C+R\rightleftharpoons RC$, with free drug concentration $C$, define $K_D=k_{\mathrm{off}}/k_{\mathrm{on}}$. If binding is at equilibrium, then $k_{\mathrm{on}}CR=k_{\mathrm{off}}RC$, so $RC/R=C/K_D$. Therefore, $$ \theta_{\mathrm{eq}}=\frac{RC}{R+RC} =\frac{C/K_D}{1+C/K_D} =\frac{C}{K_D+C}. $$ This expression assumes equilibrium for the reversible binding reaction, uses free drug concentration at the target site, and treats complex loss and changing target states as negligible over the binding-equilibration timescale. It is not generally valid as an exact dynamic occupancy formula when binding is out of equilibrium or complex is being removed.

The dynamic and equilibrium quantities answer related but distinct questions. $\theta_{\mathrm{dyn}}$ is calculated from the bound and total target states generated by the TMDD model. $\theta_{\mathrm{eq}}$ is an estimate implied by the current free concentration under an equilibrium binding assumption. Under the assumptions in the derivation, they agree. Outside those conditions, the difference can be meaningful rather than an arithmetic error.

Worked Example: Comparing Dynamic and Equilibrium Occupancy

At a hypothetical time point, suppose the free drug concentration is $C=2.0$ nM and $K_D=1.0$ nM. A dynamic TMDD model at the same time predicts $R=6.0$ nM and $RC=6.0$ nM. Calculate both occupancies and interpret the difference.

The equilibrium estimate from free drug is $$ \theta_{\mathrm{eq}}=\frac{2.0}{1.0+2.0}=0.667. $$ The dynamic occupancy is $$ \theta_{\mathrm{dyn}}=\frac{6.0}{6.0+6.0}=0.500. $$ The equilibrium formula implies 66.7% occupancy, whereas the modeled target states imply 50.0% occupancy. The discrepancy indicates that the equilibrium formula and dynamic states do not give the same occupancy at this time. It does not, by itself, identify the cause: binding may not have equilibrated, complex may be internalized, or the concentration used in the equilibrium calculation may not represent free drug at the target site. The model states should be used for a response link intended to reflect the dynamic TMDD model.

Choosing and Interpreting the Response Link

A common direct link is a Hill Emax model using occupancy as its input: $$ E(t)=E_0+E_{\max}\frac{\theta(t)^\gamma}{\theta_{50}^\gamma+\theta(t)^\gamma}. $$ Here $E_0$ is baseline response, $E_{\max}$ is the maximum change from baseline in the modeled direction, $\theta_{50}$ is the occupancy producing half of that maximum change, and $\gamma>0$ controls the steepness of the relationship. If the response decreases as occupancy increases, an inhibitory form can be written $E(t)=E_0-I_{\max}\theta(t)^\gamma/(\theta_{50}^\gamma+\theta(t)^\gamma)$, where $I_{\max}$ is the maximum decrease. The response units are those of $E_0$ and $E_{\max}$ (or $I_{\max}$); occupancy and $\theta_{50}$ are fractions.

The occupancy-response parameter $\theta_{50}$ is not generally the binding constant $K_D$. $K_D$ describes the concentration relation for equilibrium binding under its assumptions; $\theta_{50}$ describes the occupancy at which the specified response link reaches half its maximum change. They have different meanings and units: $K_D$ is a concentration, whereas $\theta_{50}$ is dimensionless.

A linear link, $E(t)=E_0+\beta\theta(t)$, may be suitable over a restricted occupancy range, but it does not have the same saturation behavior as the Hill Emax model. A concentration-driven Emax model instead uses drug concentration directly as its input. That can describe concentration-response data, but it does not explicitly make target occupancy the mechanistic mediator. Selecting an occupancy-driven link is most informative when occupancy is measured or credibly predicted and the scientific question concerns the connection from target binding to response.

Worked Example: Calculating the Local Rate of Occupancy Change

Consider a hypothetical time point with $C=1.0$ nM, $R=4.0$ nM, and $RC=2.0$ nM. Let $k_{\mathrm{on}}=0.20$ nM$^{-1}$h$^{-1}$, $k_{\mathrm{off}}=0.10$ h$^{-1}$, and $k_{\mathrm{int}}=0.05$ h$^{-1}$. For the target states, take $k_{\mathrm{syn}}=0.40$ nM/h and $k_{\mathrm{deg}}=0.10$ h$^{-1}$. Use the model equations to find the instantaneous rate of change of dynamic occupancy.

First, the complex rate is $$ \frac{dRC}{dt}=(0.20)(1.0)(4.0)-(0.10+0.05)(2.0) =0.80-0.30=0.50\ \text{nM/h}. $$ The free-target rate is $$ \frac{dR}{dt}=0.40-(0.10)(4.0)-(0.20)(1.0)(4.0)+(0.10)(2.0) =-0.60\ \text{nM/h}. $$ Let $T=R+RC$, so $dT/dt=dR/dt+dRC/dt=-0.10$ nM/h. Differentiating $\theta_{\mathrm{dyn}}=RC/T$ gives $$ \frac{d\theta_{\mathrm{dyn}}}{dt} =\frac{(dRC/dt)T-RC(dT/dt)}{T^2} =\frac{(0.50)(6.0)-(2.0)(-0.10)}{6.0^2} =0.08889\ \text{h}^{-1}. $$ The current occupancy is $2.0/6.0=0.333$, and it is increasing at this instant. The calculation shows why current free drug alone does not specify the trajectory: occupancy depends on the rates of change in both bound and total target.

Assumptions, Diagnostics, and Common Pitfalls

A direct occupancy-response model assumes that the selected occupancy state represents the target engagement relevant to the measured response, that the response link has the stated form, and that any important delay between binding and response is negligible or intentionally omitted. The TMDD states must also be defined consistently: the denominator should include the target pools that belong to the modeled occupancy definition. If the response is driven by occupancy in a tissue but only plasma states are available, equating the two requires an additional biological and distribution assumption.

Useful checks include verifying that $R$ and $RC$ remain nonnegative, that their sum is positive when occupancy is calculated, and that the computed fraction stays between zero and one. Check concentration units in the binding terms: $k_{\mathrm{on}}CR$ must have concentration-per-time units. Examine whether the fitted model predicts plausible occupancy trajectories and whether response changes track those trajectories as the direct-link assumption requires. Systematic response lags or hysteresis—different responses at the same occupancy on rising and falling phases—suggest that a direct instantaneous link may be inadequate. A delayed or indirect response model may then be more appropriate, though it represents a different mechanism.

Common mistake: “The free concentration is $2K_D$, so occupancy must be 67%; I will use that value even though the TMDD model predicts a different complex fraction.” The correction is to distinguish the equilibrium estimate from model-based dynamic occupancy. The value $C/(K_D+C)$ follows only under the equilibrium assumptions; if the intended response model is linked to the dynamic TMDD states, calculate $RC/(R+RC)$ from those states instead.

An occupancy-response relationship also has limitations. A good fit does not prove that occupancy causes the response, and estimated occupancy-response parameters can be sensitive to errors in binding parameters or target measurements. Sparse observations may not distinguish a steep occupancy link from a delayed response or a concentration-driven link. Compare these alternatives using the scientific mechanism, data quality, and predictive adequacy rather than treating one parameterization as universally preferable.

Check Your Understanding

Answer each question, then compare with the solutions below.

A model predicts $R=3.0$ nM and $RC=9.0$ nM. What is dynamic occupancy?
For $C=3.0$ nM and $K_D=2.0$ nM, calculate equilibrium occupancy. Name one assumption needed to interpret it as actual occupancy.
Why is $\theta_{50}$ in an occupancy-response model not interchangeable with $K_D$?
A response at the same modeled occupancy differs substantially on the rising and falling phases. What assumption should be reconsidered?

Solutions. (1) $\theta_{\mathrm{dyn}}=9.0/(3.0+9.0)=0.750$, or 75.0%. (2) $\theta_{\mathrm{eq}}=3.0/(2.0+3.0)=0.600$, or 60.0%; interpreting this as actual occupancy requires binding to be at equilibrium, among other conditions. (3) $K_D$ is a concentration describing equilibrium binding, whereas $\theta_{50}$ is a dimensionless occupancy at half the maximum response change in the chosen PD link. (4) Reconsider the direct instantaneous response assumption; the pattern may indicate a delay or other response dynamics not represented by the direct link.

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