# Mathematical Modeling in Drug Metabolism and Pharmacokinetics: Correct In Vitro, Not Always Valid In Vivo

## TL;DR

This paper introduces a new method for modeling drug metabolism using Kirchhoff’s Laws, which avoids traditional differential equations and provides more accurate in vivo predictions.

## Contribution

A novel, model-independent pharmacokinetic framework using Kirchhoff’s Laws that avoids assumptions about organ mechanisms.

## Key findings

- The Kirchhoff-based approach produced model-independent clearance equations that include blood flow and transport effects.
- Traditional differential equation methods may misestimate in vivo clearance when drug input is slow or volumes differ.
- The framework was successfully applied to a hypothetical drug case study (KL25A).

## Abstract

Background/Objectives: Chemical and metabolic kinetics have historically been derived from mass balance differential equations expressed in terms of amounts, and this framework was later extended to pharmacokinetics by converting amount-based equations to concentration-based clearance relationships. That conversion is valid for fixed-volume in vitro experiments, but may be unreliable in vivo, where input, distribution, and elimination can occur in different volumes of distribution. The objective of this study is to present an alternate, mechanistically agnostic framework for deriving pharmacokinetic relationships by adapting Kirchhoff’s Laws to treat pharmacokinetic systems as networks of parallel and in-series rate-defining processes, and to identify where differential equation approaches fail in vivo. Methods: Clearance and rate constant equations were derived using the adapted Kirchhoff’s Laws by summing parallel rate-defining processes and summing inverses for in-series processes, explicitly incorporating organ blood flow, net transporter, and delivery site effects. The resulting expressions were compared with differential equation hepatic disposition elimination models (well-stirred, parallel tube, dispersion) and the Extended Clearance Concept (ECC). Mean residence time concepts were used to extend the framework to oral input, and the full approach was applied to a case study of a hypothetical drug (KL25A). Results: The adapted Kirchhoff-based approach reproduced standard pharmacokinetic analyses without mechanistic organ assumptions and yielded model-independent hepatic and renal clearance equations that include blood flow, net transport, and delivery kinetics. Inconsistencies with the traditional differential-based derivations were highlighted, including the interpretation of pharmacokinetics associated with slow absorption site clearance, as illustrated by KL25A. Conclusions: For linear drug metabolism and pharmacokinetics, clearance and rate constant relationships can be derived by summing parallel and in-series rate-defining processes, without differential equations. Differential equation methods may misestimate in vivo clearance and bioavailability when drug input is slow or when volumes of distribution differ across processes. The adapted Kirchhoff framework offers a simpler, model-independent basis for interpreting clinical data.