Deriving Ionized-Drug Contributions in Rodgers-Rowland

Why Derive the Ionized Contributions?

A tissue-to-plasma partition coefficient is not determined by lipophilicity alone when a drug ionizes. Acid–base equilibrium sets the relative amounts of neutral and charged drug in each aqueous phase, while the phases differ in their ability to accommodate each species. The Rodgers–Rowland approach accounts for those contributions by combining tissue composition with phase-specific partitioning.

The key bookkeeping step is to choose a reference concentration. For a monoprotic base, use the neutral species \(B\) in the aqueous phase. Acid–base equilibrium then gives the concentration of \(BH^+\) at each pH relative to \(B\). The tissue terms can be written using that common reference, summed, and compared with the plasma concentration on the desired basis.

Acid–Base Equilibrium Gives the Species Terms

For a monoprotic base, the conjugate acid dissociates as \(BH^+ \rightleftharpoons B+H^+\). The Henderson–Hasselbalch relationship gives the ionized-to-neutral ratio in an aqueous phase at pH \(pH_j\):

The ratio is phase-specific because it depends on pH. A lower pH increases the relative concentration of \(BH^+\) for a base. In the model derivation below, the neutral species is assumed to equilibrate between the aqueous phases, so \(C_{0,j}=C_{0,p}\). This is an equilibrium assumption about neutral drug, not a claim that the ionized concentrations are identical across phases.

Let \(V_{EW}\) and \(V_{IW}\) denote extracellular-water and intracellular-water fractions of tissue. Let \(V_{NL}\) and \(V_P\) denote neutral-lipid and phospholipid fractions. Each fraction is expressed per unit tissue volume. Define \(K_{NL}\) as the neutral-lipid-to-aqueous partition coefficient for neutral drug; \(K_{P,0}\) and \(K_{P,+}\) as phospholipid-to-aqueous partition coefficients for neutral and ionized drug, respectively. They are phase-specific concentration ratios under the model conditions.

With extracellular pH \(pH_e\), intracellular pH \(pH_i\), and \(r_e=10^{pK_a-pH_e}\), \(r_i=10^{pK_a-pH_i}\), the aqueous tissue amounts per unit tissue volume are proportional to \(V_{EW}(1+r_e)\) and \(V_{IW}(1+r_i)\). Neutral drug in neutral lipid contributes \(V_{NL}K_{NL}\). Phospholipid contributes separately for the two species: \(V_P K_{P,0}\) for neutral drug and \(V_P K_{P,+}r_i\) for ionized drug, using intracellular aqueous concentration as the reference for the ionized phospholipid term.

Consequently, the tissue concentration relative to the neutral aqueous reference is a sum of terms. The plasma unbound concentration has its own species sum, \(C_{u,p}=C_{0,p}(1+r_p)\), where \(r_p=10^{pK_a-pH_p}\). Dividing by this plasma sum gives \(K_{pu}\); multiplying by \(f_{u,p}\) converts to total-plasma \(K_p\).

This expression makes the species contributions explicit. The \(1\) in each aqueous factor represents neutral drug; the \(r_j\) term represents ionized drug. In the phospholipid term, the ionized contribution is weighted by both its aqueous abundance and its phospholipid partition coefficient. In particular, ionization alone does not imply strong phospholipid association: that depends on \(K_{P,+}\) as well as \(r_i\).

Derivation and Conditions for Use

The decomposition follows directly from conservation of drug across the model phases. For example, the extracellular-water drug amount per unit tissue volume is \(V_{EW}C_{0,p}(1+r_e)\). The intracellular-water amount is \(V_{IW}C_{0,p}(1+r_i)\). The neutral-lipid amount is \(V_{NL}K_{NL}C_{0,p}\). The phospholipid amount is \(V_PC_{0,p}(K_{P,0}+K_{P,+}r_i)\). Adding these amounts and dividing by tissue volume gives the numerator multiplied by \(C_{0,p}\). The plasma unbound concentration is \(C_{0,p}(1+r_p)\), so the common reference concentration cancels in the ratio.

The cancellation is the useful result: the partition coefficient depends on pH ratios, composition fractions, and phase-specific coefficients rather than on the absolute concentration of drug, provided the model is linear and unsaturated. The conversion to total-plasma Kp follows from \(C_{u,p}=f_{u,p}C_{\mathrm{plasma}}\), hence \(C_{\mathrm{tissue}}/C_{\mathrm{plasma}}=f_{u,p}(C_{\mathrm{tissue}}/C_{u,p})\).

This derivation assumes rapid acid–base equilibrium within each aqueous phase; neutral-drug equilibrium across phases; specified, constant pH values; linear partitioning with no concentration-dependent saturation; and phase fractions and partition coefficients that represent the tissue under consideration. The equation is a composition-based prediction, not a guarantee that all tissues reach equilibrium in vivo. If neutral drug does not equilibrate, or if transport, binding, or kinetics violate the assumptions, this calculation may not describe the observed distribution.

Diagnostics, Alternatives, and Limitations

A reliable calculation is auditable term by term. Check that each pH ratio uses \(pK_a-pH\) for a base, that phase fractions are dimensionless and use a consistent tissue-volume basis, and that the listed fractions and coefficients match the intended model. Verify that the plasma denominator uses plasma pH, not intracellular pH. Finally, check whether the requested output is \(K_{pu}\) or total-plasma \(K_p\); the latter includes \(f_{u,p}\).

Useful failure signals include a negative contribution, a phase fraction entered as a percentage rather than a fraction, or an unexpectedly large result caused by applying the same ionization factor twice. The ionized amount in a particular aqueous phase is already included through \(1+r_j\); the phospholipid ionized term is a separate phase contribution and should be included only with its own coefficient. A coefficient defined relative to the ionized species must not be applied to the neutral concentration without the \(r_j\) multiplier.

The composition-based derivation differs from a neutral-only partition estimate, which cannot represent the explicit \(r_j\)-weighted ionized terms. It also differs from a direct empirical Kp estimate, which may predict a ratio from observed data without resolving its contributions by phase. The Rodgers–Rowland decomposition is useful for interpreting how pH and composition enter a prediction, but it requires more inputs and inherits their uncertainty. Uncertain pKa, pH, phase fractions, or phospholipid coefficients can change the result materially; the calculation alone cannot establish whether equilibrium assumptions hold in a particular tissue.