Soluble and Membrane Target Pools in Disposition Models

Why Target Location Belongs in the Disposition Model

A drug can encounter a soluble target in plasma and a membrane-associated target in tissue, but those targets need not be equally accessible. The free drug concentration at a tissue surface may differ from the plasma concentration, and binding can lead to different complex fates in the two locations. Treating both pools as one target may therefore obscure which binding events remove drug from circulation and which chiefly alter local exposure.

The full TMDD framework distinguishes free drug, free target, and drug–target complex states, with binding, dissociation, target turnover, and complex loss represented as rate processes. Here the same bookkeeping is applied to two locations: a plasma-accessible soluble pool and a tissue-accessible membrane pool. The model is mechanistic only to the extent that its locations and pathways correspond to the system being studied.

A Two-Location State Model

Let \(C_p\) be free drug concentration in plasma and \(C_t\) free drug concentration in the tissue region containing the membrane target. Let \(S\) and \(R\) be, respectively, free soluble-target concentration in plasma and free membrane-target concentration in tissue. Let \(C_s\) and \(C_m\) denote soluble and membrane drug–target complex concentrations. Concentrations of \(S\) and \(C_s\) use the plasma volume \(V_p\); concentrations of \(R\) and \(C_m\) use the tissue volume \(V_t\). Drug input \(R_{\mathrm{in}}\) is an amount per time, \(CL_p\) and \(CL_t\) are free-drug clearances, and \(Q\) is an intercompartmental exchange clearance.

For location \(j\), \(k_{\mathrm{on},j}^{\mathrm{eff}}\) is the effective association-rate constant, \(k_{\mathrm{off},j}\) is the dissociation-rate constant, and \(k_{\mathrm{int},j}\) is the rate constant for complex internalization or another complex-loss process that removes drug from the modeled free and bound states. Target synthesis and free-target turnover are represented by \(k_{\mathrm{syn},j}\) and \(k_{\mathrm{deg},j}\). A fraction \(\rho_j\) of target removed with complex loss may return promptly to the free-target pool; use \(\rho_j=0\) when no such recycling is represented. The effective association rate can include an accessibility factor, \(k_{\mathrm{on},j}^{\mathrm{eff}}=a_j k_{\mathrm{on},j}\), where \(0\leq a_j\leq1\). If local drug concentration already captures limited access, set \(a_j=1\) rather than accounting for the same restriction twice.

Each association term removes free drug and free target while forming complex in the same location. Dissociation restores both. Complex loss removes the complex; only the fraction \(\rho_j\) of the target is returned directly to the free-target state. The equations do not imply that all complex loss is literally internalization: the chosen mechanism must match the available biological evidence.

The drug exchange term \(Q(C_p-C_t)\) moves free drug between locations and cancels when the two drug amounts are summed. This is a simplified representation of distribution. A spatially heterogeneous tissue, intracellular transport, or explicit surface-binding region may require additional states rather than one effective tissue concentration.

How Complex Formation Becomes Drug Loss

Binding by itself is not necessarily irreversible clearance. A drug molecule in a complex can dissociate and return to the free pool. If complex loss competes with dissociation, only some binding events result in drug removal through that route. This distinction can be made precise using the complex equation.

Here \(D_j\) is the local free-drug concentration and \(T_j\) the corresponding free-target concentration: \((D_s,T_s)=(C_p,S)\) and \((D_m,T_m)=(C_t,R)\). If \(D_j\) and \(T_j\) are approximately constant over the interval of interest and the complex rapidly reaches a quasi-steady value, set \(dC_j/dt\approx0\). Rearranging gives \(C_j\approx k_{\mathrm{on},j}^{\mathrm{eff}}D_jT_j/(k_{\mathrm{off},j}+k_{\mathrm{int},j})\). The drug-loss rate per unit volume is then \(k_{\mathrm{int},j}C_j\), or equivalently \(k_{\mathrm{on},j}^{\mathrm{eff}}D_jT_j [k_{\mathrm{int},j}/(k_{\mathrm{off},j}+k_{\mathrm{int},j})]\).

The ratio \(k_{\mathrm{int},j}/(k_{\mathrm{off},j}+k_{\mathrm{int},j})\) is the probability that complex loss occurs before dissociation when those two competing events have constant rates. The quasi-steady expression is not a general steady-state solution for the entire TMDD system: it holds only when the complex adjusts quickly enough and the local free-drug and target concentrations change slowly on that timescale.

Accessibility Can Change the Apparent Binding Contribution

A membrane target may be abundant but poorly exposed to drug. One way to represent this is an explicit tissue drug concentration determined by distribution. Another is an effective association rate that summarizes limited access not otherwise represented. These approaches describe different levels of detail: if tissue distribution already lowers \(C_t\), applying a second accessibility factor for that same limitation exaggerates the reduction in binding.

Model Choice, Diagnostics, and Limitations

A single lumped target pool can be a useful simpler alternative when the two locations have similar exposure and kinetics, or when the data cannot distinguish them. It uses fewer parameters and states, but cannot separately attribute binding and complex loss to soluble and membrane targets. A two-location model can express different local concentrations, target turnover, and complex fates, but requires enough information to estimate or constrain those additional quantities.

Useful checks follow directly from the model structure. Confirm that each concentration is paired with the correct volume and that \(k_{\mathrm{on}}DT\) has units of concentration per time. Check that all rate constants and clearances are nonnegative and that the chosen initial conditions do not imply negative states. Compare predicted tissue exposure and soluble-target measurements with relevant observations where available. If the model requires an extreme accessibility factor, a large unexplained target pool, or compensating changes in association and internalization rates, the assumed location structure may be unsupported or insufficiently informed.

A common mistake is to count complex formation itself as irreversible clearance. That reasoning treats every binding event as drug loss, even though dissociation can restore free drug. The correction is to track complex as a state and count drug removal only through the modeled complex-loss pathway (or another explicitly defined removal process). Another frequent error is to apply both a low tissue concentration and an accessibility multiplier for the same barrier. Use one representation for each limitation unless the factors describe distinct mechanisms.

Separate pools do not automatically establish that a target is physically accessible, that internalization destroys the drug, or that a recycling fraction is known. Those are mechanistic assumptions, not consequences of fitting equations. If the data do not distinguish soluble from membrane pathways, the more detailed model may produce uncertain parameter estimates even when its aggregate predictions appear adequate.