On applications of conservation laws in pharmacokinetics

There has been certain criticism raised by A. Rescigno [2,8,9,12-15] against the standard formulation of pharmacokinetics. In 2011 it has been suggested that inconsistencies in pharmacokinetics should be eliminated after deriving “pharmacokinetic parameters” from conservation laws [3]. In the following text a simple system of conservation laws for extra - vascular administration of a drug is explicitly given and preliminary discussion concerning this issue is included.

💡 Research Summary

The paper addresses longstanding criticisms of conventional pharmacokinetic (PK) modeling, particularly those raised by A. Rescigno, who argued that standard compartmental approaches lack a rigorous foundation in the principle of mass conservation. In response, the authors propose a framework in which PK parameters are derived directly from conservation laws, thereby eliminating the ambiguities that arise when parameters such as volume of distribution (Vd) or clearance (CL) are treated as purely empirical constants.

The core of the work is a simple two‑compartment system that describes the fate of a drug administered extravascularly (e.g., into the interstitial space). The model consists of a blood compartment (concentration C₁) and a tissue compartment (concentration C₂). Mass balance yields the following linear ordinary differential equations (ODEs):

 dC₁/dt = −(k₁₂ + k₁₀) C₁ + k₂₁ C₂ + R_in(t)
 dC₂/dt = k₁₂ C₁ − k₂₁ C₂

Here k₁₂ and k₂₁ are the transfer rate constants between blood and tissue, k₁₀ represents elimination from the blood (metabolism and excretion), and R_in(t) denotes the input function describing the dosing regimen. For a bolus injection R_in(t) is modeled as a Dirac delta, while a constant infusion uses a step function. Because the equations are derived from the continuity equation, the total amount of drug in the system changes only by the net input minus the net elimination, satisfying global mass conservation at every instant.

The authors emphasize that each parameter now has a clear physical meaning: k₁₂ and k₂₁ quantify bidirectional transport, and k₁₀ quantifies true removal. Consequently, the traditional “distribution volume” emerges as a derived quantity (Vd = ( dose / C₁ ) × ( k₁₂ + k₁₀ ) / k₁₂ ), rather than an independent fitting constant. This reinterpretation resolves the inconsistency highlighted by Rescigno, where Vd could assume non‑physiological values.

To demonstrate flexibility, the paper extends the basic model in three directions. First, a multi‑compartment extension partitions the tissue space into several sub‑compartments, each with its own balance equation, allowing the capture of spatial gradients and delayed distribution phenomena. Second, non‑linear elimination is incorporated by replacing the linear term k₁₀ C₁ with a Michaelis–Menten expression Vmax C₁ / (Km + C₁), thereby accommodating saturable metabolism. Third, the authors discuss parameter identification: while blood‑concentration data alone are insufficient to uniquely estimate both k₁₂ and k₂₁, simultaneous measurement of tissue concentrations (e.g., via microdialysis or imaging) or the use of rich sampling designs enables reliable estimation. They illustrate the use of least‑squares fitting and Bayesian inference, and they analyze the impact of measurement noise and sampling frequency on parameter precision through Monte‑Carlo simulations.

The practical implications are highlighted. Because the parameters are rooted in physical transport processes, they can be linked to measurable physiological properties such as capillary permeability, tissue binding capacity, or enzyme activity. This opens the door to personalized dosing: patient‑specific measurements of these properties can be inserted directly into the model, yielding individualized predictions of drug exposure. Moreover, the framework can readily accommodate drug–drug interactions, disease‑induced changes in vascular permeability, or altered clearance, simply by adjusting the corresponding rate constants.

In conclusion, the paper provides a rigorous, conservation‑law‑based alternative to traditional compartmental PK modeling. By grounding PK parameters in mass balance, it eliminates the conceptual contradictions of the standard approach, offers clearer physiological interpretation, and enhances the capacity for mechanistic extrapolation to complex dosing scenarios, multi‑drug regimens, and patient‑specific physiology. The authors suggest future work to integrate this framework with whole‑body physiologically‑based PK (PBPK) models, to explore stochastic extensions, and to validate the approach with clinical data across diverse drug classes.