Deriving the Poulin-Theil Tissue Partition Equation

From Phase Equilibrium to a Tissue-to-Plasma Ratio

A tissue-to-plasma partition coefficient, \(K_p\), is a ratio of total drug concentrations at distribution equilibrium. A composition-based prediction needs to account for the fact that total tissue drug is distributed among distinct tissue phases, while plasma binding determines how much total plasma drug corresponds to a given unbound concentration. The Poulin-Theil expression brings those pieces together for neutral compounds.

The derivation is a concentration and amount balance: specify the concentration in each tissue phase relative to the aqueous concentration, sum the amounts held by the phases, and divide by total plasma concentration. The result is not a fitted curve by itself. It follows from explicit assumptions about equilibrium and how the drug associates with tissue water and lipids.

The volume fractions refer to fractions of the tissue volume, so their associated volumes sum to the tissue volume when all phases are represented. The partition coefficients are ratios of concentrations in a lipid phase and water at equilibrium; they are dimensionless when both concentrations use the same units. The plasma unbound fraction is also dimensionless and lies between 0 and 1. Under the model’s idealization, free aqueous drug concentration in tissue water equals unbound drug concentration in plasma.

Assumptions and Derivation

The derivation applies to a neutral compound under passive equilibrium. It assumes linear, concentration-independent partitioning between water and each lipid phase; a homogeneous tissue represented by the stated phase fractions; and no active transport, metabolism, or other process that maintains unequal free concentrations. It treats tissue total concentration as the sum of drug in the represented phases. Plasma binding is represented by \(f_{u,p}\), and the model equates unbound plasma concentration with the free aqueous concentration available to tissue water. These are modeling assumptions, not consequences of the algebra.

Let \(C_w\) be the equilibrium concentration in tissue water. The two lipid-phase concentrations are \(P_{nl/w}C_w\) and \(P_{ph/w}C_w\). To obtain concentration per unit tissue volume, sum the drug amounts in the phases and divide by tissue volume. Since the phase volumes are given as fractions of tissue volume, that calculation gives:

The plasma term follows directly from the definition \(f_{u,p}=C_{u,p}/C_{p,\mathrm{total}}\). Substituting \(C_{u,p}=C_w\) gives \(C_{p,\mathrm{total}}=C_w/f_{u,p}\). Thus, stronger plasma binding (smaller \(f_{u,p}\)) increases total plasma concentration relative to the same free aqueous concentration and lowers the predicted total tissue-to-plasma ratio. The equation is dimensionless: the volume fractions, partition coefficients, and unbound fraction are all dimensionless.

Worked Predictions

Checking Inputs, Comparing Approaches, and Knowing the Limits

A useful diagnostic is to calculate the tissue capacity before applying plasma binding. This makes it easier to locate an implausible prediction: an unexpected value may arise from a tissue fraction, a phase partition coefficient, or \(f_{u,p}\). Check that the fractions use a consistent volume basis and represent the phases included in the model. If they are intended to exhaust tissue volume, they should sum to one. If only some phases are represented, the omitted volume must not silently be treated as an additional modeled phase.

Also check the scale and meaning of every partition coefficient. A coefficient for lipid-to-water concentration must multiply the matching lipid volume fraction; entering a reverse water-to-lipid ratio in its place can greatly distort the prediction. Confirm that \(f_{u,p}\) is the unbound fraction, not the bound fraction, and that it lies between zero and one. For nonnegative fractions and partition coefficients, the predicted \(K_p\) cannot be negative. A calculation that violates this or fails to change proportionally when \(f_{u,p}\) changes signals an input or arithmetic error.

The main alternative for a tissue with reliable measurements is to use an experimentally determined tissue-to-plasma ratio under conditions relevant to the intended application. A measured ratio reflects the actual tissue and drug under those conditions, but may not transfer to a different setting or concentration if equilibrium, linearity, or comparability of measurements is not established. The composition-based Poulin-Theil prediction instead makes its assumptions visible and can be used when a direct tissue value is unavailable, but it inherits uncertainty in tissue fractions, partition coefficients, and plasma binding.

The method is specifically for neutral compounds under its stated phase-equilibrium representation. It does not derive the contributions of charged species or account for pH-driven ionization and ion trapping. Those require a model that explicitly represents ionizable-drug behavior; applying this neutral equation unchanged to such a compound is not justified simply because some neutral fraction is present. Even for neutral compounds, a mismatch between observed and predicted \(K_p\) may indicate that the assumed phase description or equilibrium conditions are inadequate, rather than an arithmetic error alone.

A common mistake is to report the bracketed capacity as \(K_p\) and omit \(f_{u,p}\). The capacity describes tissue concentration relative to free aqueous concentration, not relative to total plasma concentration. The required conversion is \(C_w=f_{u,p}C_{p,\mathrm{total}}\), so the final ratio includes \(f_{u,p}\). Another frequent error is to add the phase volumes without weighting each lipid fraction by its partition coefficient; that would implicitly treat every phase concentration as equal to the aqueous concentration.