There is Only One Valid Definition of Clearance: Critical Examination of Clearance Concepts Reveals the Potential for Errors in Clinical Drug Dosing Decisions

A commentary on "Are all measures of liver Kpuu a function of FH, as determined following oral dosing, or have we made a critical error in defining hepatic drug clearance?"

The clearance concept has been used in pharmacokinetics for over 50 years. However, there is still much debate regarding mathematical clearance models. A recent article discussed that there is a critical error in a basic assumption that leads to the mechanistic hepatic clearance models (Benet, L.Z., Sodhi, J.K., 2024. Are all measures of liver Kpuu a function of FH, as determined following oral dosing, or have we made a critical error in defining hepatic drug clearance? European Journal of Pharmaceutical Sciences 196, 106,753. https://doi.org/10.1016/j.ejps.2024.106753). This commentary discusses this point based on the extended clearance model (ECM), which is increasingly used in modern drug discovery and development. Confusion about clearance can be avoided by using clearly defined drug concentrations based on hierarchical body structures.

Are all measures of liver Kpuu a function of FH, as determined following oral dosing, or have we made a critical error in defining hepatic drug clearance?

Here we present, utilizing universally accepted relationships for hepatic clearance at steady state, that for all models of hepatic elimination the ratio of unbound liver drug concentration to unbound systemic blood concentration, Kpuu, is a function of or related to the hepatic bioavailability for that drug, FH. According to the derivation for the well-stirred model, Kpuu can never exceed unity, can frequently be a function of hepatic blood flow, and is equivalent to the value of FH as determined following oral dosing. For the parallel tube model, Kpuu will not equal FH but will be a function of FH and will also never be a value greater than 1. When hepatic clearance is rate limited by basolateral transporters, Kpuu will be less than 1, and less than FH. We believe that such outcomes are highly unlikely, and that the error arises from a basic assumption concerning hepatic clearance that leads to the mechanistic models of hepatic elimination, the well-stirred, parallel tube and dispersion models. That basic assumption is that the steady-state systemic concentration multiplied by the hepatic systemic clearance is equal to the product of the average unbound liver steady-state concentration and the intrinsic hepatic clearance (Css · CL = CH,u · CLint). Calculations of Kpuu and FH based on present methods of analysis provide a strong argument as to why this universally accepted relationship is not correct. Alternatively, we have shown in recent publications that hepatic clearance may be adequately determined based on Kirchhoff's Laws where no assumption of the above equality concerning hepatic intrinsic clearance is required, and where Kpuu is independent of hepatic extraction ratio and FH.

Introducing the Dynamic Well-Stirred Model for Predicting Hepatic Clearance and Extraction Ratio

The well-stirred model (WSM) incorporating the fraction of unbound drug (fu) to account for the effect of plasma binding on intrinsic clearance has been widely used for predicting hepatic clearance under the assumption that drug protein binding reaches equilibrium instantaneously. Our theoretical analysis reveals that the effect of protein binding on intrinsic clearance is better accounted for with the dynamic free fraction (fD), a measure of drug protein binding affinity, which leads to a putative dynamic well-stirred model (dWSM) without the instantaneous equilibrium assumption. Using recombinant CYP3A4 as the in vitro clearance system, we demonstrate that the binding effect of albumin on the intrinsic clearance of both highly bound midazolam and highly free verapamil is fully corrected by their corresponding fD values, respectively. On the other hand, fu only corrects the binding effect of albumin on the intrinsic clearance of verapamil, and yields severe over-correction of the intrinsic clearance of midazolam. The results suggest that the traditional WSM is suitable for highly free drugs like verapamil but not necessarily for highly bound drugs such as midazolam due to the violation of the instantaneous equilibrium assumption or under-estimating the true free drug concentration. In comparison, the dWSM incorporating fD holds true as long as drug elimination follows steady-state kinetics, and hence, it is more broadly applicable to drugs with different protein binding characteristics. Here we demonstrate with 36 diverse drugs, that the dWSM significantly improves the accuracy of predicting human hepatic clearance and liver extraction ratio from in vitro microsomal clearance data, highlighting the importance of drug plasma protein binding kinetics in addressing the under-prediction of hepatic clearance by the WSM.

An Explanation of Why Dose-Corrected Area Under the Curve for Alternate Administration Routes Can Be Greater than for Intravenous Dosing

It is generally believed that bioavailability (F) calculated based on systemic concentration area under the curve (AUC) measurements cannot exceed 1.0, yet some published studies report this inconsistency. We teach and believe, based on differential equation derivations, that rate of absorption has no influence on measured systemic clearance following an oral dose, i.e., determined as available dose divided by AUC. Previously, it was thought that any difference in calculating F from urine data versus that from systemic concentration AUC data was due to the inability to accurately measure urine data. A PubMed literature search for drugs exhibiting F > 1.0 and studies for which F was measured using both AUC and urinary excretion dose-corrected analyses yielded data for 35 drugs. We show and explain, using Kirchhoff's Laws, that these universally held concepts concerning bioavailability may not be valid in all situations. Bioavailability, determined using systemic concentration measurements, for many drugs may be overestimated since AUC reflects not only systemic elimination but also absorption rate characteristics, which is most easily seen for renal clearance measures. Clearance of drug from the absorption site must be significantly greater than clearance following an iv bolus dose for F(AUC) to correctly correspond with F(urine). The primary purpose of this paper is to demonstrate that studies resulting in F > 1.0 and/or greater systemic vs urine bioavailability predictions may be accurate. Importantly, these explications have no significant impact on current regulatory guidance for bioequivalence testing, nor on the use of exposure (AUC) measures in making drug dosing decisions.

The Uses and Advantages of Kirchhoff's Laws vs. Differential Equations in Pharmacology, Pharmacokinetics, and (Even) Chemistry

In chemistry, rate processes are defined in terms of rate constants, with units of time-1, and are derived by differential equations from amounts. In contrast, when considering drug concentrations in biological systems, particularly in humans, rate processes must be defined in terms of clearance, with units of volume/time, since biological volumes, which are highly dependent on drug partition into biological tissues, cannot be easily determined. In pharmacology, pharmacokinetics, and in making drug dosing decisions, drug clearance and changes in drug clearance are paramount. Clearance is defined as the amount of drug eliminated or moved divided by the exposure driving that elimination or movement. Historically, all clearance derivations in pharmacology and pharmacokinetics have been based on the use of differential equations in terms of rate constants and amounts, which are then converted into clearance equations when multiplied/divided by a hypothesized volume of distribution. Here, we show that except for iv bolus dosing, multiple volumes may be relevant. We have recently shown that clearance relationships, as well as rate constant relationships, may be derived independent of differential equations using Kirchhoff's Laws from physics. Kirchhoff's Laws may be simply translated to recognize that when two or more rate-defining processes operate in parallel, the total value of the overall reaction parameter is equal to the sum of those rate-defining processes. In contrast, when two or more rate-defining processes operate in series, the inverse of the total reaction parameter is equal to the sum of the inverse of those rate-defining steps.

Process and System Clearances in Pharmacokinetic Models: Our Basic Clearance Concepts Are Correct

Clearances are important parameters in pharmacokinetic (PK) models. All clearances in PK models are either process clearances that include diffusion, transport, and metabolism clearances or system clearances that include organ and systemic clearance. Clearance and volume of distribution are two independent parameters that characterize drug disposition in both individual compartments and systems of compartments. In this minireview, we show that systemic and organ clearances are net clearances that can be easily derived by partition analysis. When drugs are eliminated from the central compartment by first-order processes, systemic clearance is constant. When drugs are eliminated from a peripheral compartment, instantaneous systemic clearance will vary with time. However, average clearance and clearance at steady state will be constant and will equal dose divided by area under the curve. We show that peripheral elimination will not have a large impact on most pharmacokinetic analyses and that standard models of organ and systemic clearance are useful and appropriate. SIGNIFICANCE STATEMENT: There are two basic kinds of clearances used in pharmacokinetic models, process and system clearances. We show that organ and systemic clearances are net clearances with blood or plasma as the driving concentration. For linear pharmacokinetics, clearance is constant for elimination from the central compartment but varies with time for peripheral elimination. Despite the different kinds of clearance parameters and models, standard clearance models and concepts remain valid.

Plasma Protein-Mediated Uptake and Contradictions to the Free Drug Hypothesis: A Critical Review

According to the free drug hypothesis (FDH), only free, unbound drug is available to interact with biological targets. This hypothesis is the fundamental principle that continues to explain the vast majority of all pharmacokinetic and pharmacodynamic processes. Under the FDH, the free drug concentration at the target site is considered the driver of pharmacodynamic activity and pharmacokinetic processes. However, deviations from the FDH are observed in hepatic uptake and clearance predictions, where observed unbound intrinsic hepatic clearance (CL_int,u) is larger than expected. Such deviations are commonly observed when plasma proteins are present and form the basis of the so-called plasma protein-mediated uptake effect (PMUE). This review will discuss the basis of plasma protein binding as it pertains to hepatic clearance based on the FDH, as well as several hypotheses that may explain the underlying mechanisms of PMUE. Notably, some, but not all, potential mechanisms remained aligned with the FDH. Finally, we will outline possible experimental strategies to elucidate PMUE mechanisms. Understanding the mechanisms of PMUE and its potential contribution to clearance underprediction is vital to improving the drug development process.

Assessment of the Kochak-Benet Equation for Hepatic Clearance for the Parallel-Tube Model: A Response to Jusko and Li

This commentary seeks to clarify the relationships between conservation of mass, spatial continuity, advective mass transport, and first-order drug metabolism in the liver. It also provides additional insight into the meaning of clearance thereby proving that the Pang-Rowland equation and its extensions are incorrect. The Kochak-Benet equation has been verified and is not a rearrangement of the Pang-Rowland equation.

Review of the application of Kirchhoff's Laws of series and parallel flows to pharmacology: Defining organ clearance

Dosing rate decisions for drugs and changes in dosing in a patient due to disease states, drug interactions and pharmacogenomics are all based on clearance, a measure of the body's ability to eliminate drug. The primary organs of elimination are the liver and the kidney. Clearance for each of these organs is a summative composition of biologic processes. In 1857, Gustav Kirchhoff first developed his laws to describe the "motion of electricity in conductors... [and] ...in wires", recognizing that summative processes occur either in parallel or in series. Since then, Kirchhoff's Laws have also been applied to heat transfer, diffusion and drag force on falling objects, but not to pharmacology. Although not previously recognized, renal clearance always follow Kirchhoff's Laws, as does hepatic clearance for drugs where basolateral transporters are not clinically relevant. However, when basolateral transporters are clinically relevant, we demonstrate that the present accepted approach is inconsistent with recognized drug disposition processes. However, this clearance relationship can be easily corrected using Kirchhoff's Laws. The purpose of this review is to demonstrate that Kirchhoff's Laws, which define how to approach rate processes that occur in parallel versus processes that occur in series, can be applicable to pharmacology in addition to the over 160-year recognition of their use in physical sciences. We anticipate that the application to clearance will be only the first of many such pharmacological analyses.

Design of novel quinoline derivatives as antibreast cancer using 3D-QSAR, molecular docking and pharmacokinetic investigation

Breast cancer has been one of the most challenging women's cancers and leading cause of mortality for decades. There are several studies being conducted all the time to find a cure for breast cancer. Quinoline derivatives have shown their potential as antitumor agents in breast cancer therapy. In this work, three-dimensional quantitative structure-activity relationships (3D-QSAR) and molecular docking with aromatase enzyme (Protein Data Bank: 3S7S) studies were performed to suggest the current scenario of quinoline derivatives as antitumor agents and to refine the path of these derivatives to discover and develop new drugs against breast cancer. For developing the 3D-QSAR model, comparative molecular similarity indices analysis (CoMSIA) and comparative molecular field analysis (CoMFA) were included. To attain the high level of predictability, the best CoMSIA model was applied. External validation utilizing a test set has been used in order to validate the predictive capabilities of the built model. According to the findings, electrostatic, hydrophobic and hydrogen bond donor, and acceptor fields had a significant impact on antibreast cancer activity. Thus, we generated a variety of novel effective aromatase inhibitors based on prior findings and we predicted their inhibitory activity using the built model. In addition, absorption, distribution, metabolism, elimination and toxicity properties were employed to explore the effectiveness of new drug candidates.

Evidence of the Need for Modified Well-stirred Model in In Vitro to In Vivo Extrapolation

In vitro to in vivo extrapolation (IVIVE), an approach for hepatic clearance (CL_H) prediction used worldwide, remains controversial due to systematic underprediction. Among the various probable factors, the original assumption of the hepatic mathematical model (i.e., the well-stirred model, WSM) may become problematic, leading to the underestimation of drug CL_H. Having a similar prerequisite that the well-stirred conditions are homogenous with perfectly mixed reactants, but using a different driving concentration, the modified well-stirred model (MWSM) stands apart from the WSM. However, we believe that both models should coexist so that the entire well-stirred scenario can be completely illustrated. Consequently, we collected published data from the literature and employed a logistic regression method to differentiate the optimal timing of use between WSM and MWSM in drug CL_H prediction. Generally, variances adopted in the regression, including partition coefficient (logP), fraction unbound (fu), volumes of distribution at steady-state (Vss), and mean residence time (MRT), corresponded to our assumption when protein-facilitated uptake was considered. Furthermore, a new empirical approach was introduced to allow practical use of the MWSM. The results showed that this model could provide a more precise prediction compared to previous empirical approaches. Therefore, these preliminary results not only delineated a more detailed structure and mechanism of MWSM but also highlighted its necessity and potential.

Tissue Drug Concentration

Blood flow enables the delivery of oxygen and nutrients to the different tissues of the human body. Drugs follow the same route as oxygen and nutrients; thus, drug concentrations in tissues are highly dependent on the blood flow fraction delivered to each of these tissues. Although the free drug concentration in blood is considered to correlate with pharmacodynamics, the pharmacodynamics of a drug is actually primarily commanded by the concentrations of drug in the aqueous spaces of bodily tissues. However, the concentrations of drug are not homogeneous throughout the tissues, and they rarely reflect the free drug concentration in the blood. This heterogeneity is due to differences in the blood flow fraction delivered to the tissues and also due to membrane transporters, efflux pumps, and metabolic enzymes. The rate of drug elimination from the body (systemic elimination) depends more on the driving force of drug elimination than on the free concentration of drug at the site from which the drug is being eliminated. In fact, the actual free drug concentration in the tissues results from the balance between the input and output rates. In the present paper, we develop a theoretical concept regarding solute partition between intravascular and extravascular spaces; discuss experimental research on aqueous/non-aqueous solute partitioning and clinical research on microdialysis; and present hypotheses to predict in-vivo elimination using parameters of in-vitro metabolism.

Assessment of the Kochak-Benet Equation for Hepatic Clearance for the Parallel-Tube Model: Relevance of Classic Clearance Concepts in PK and PBPK

This report reviews concepts related to operation of the classic parallel-tube model (PTM) for hepatic disposition and examines two recent proposals of a newly derived equation to describe hepatic clearance (CL_H). It is demonstrated that the proposed equation is identical to a re-arrangement of an earlier relationship from Pang and Rowland and provides a means of calculation of intrinsic clearance (CL_int,PTM) rather than CL_H as posed. We further demonstrate how classic hepatic clearance models with an assumed CL_int, while subject to numerous limitations, remain highly useful and necessary in both traditional pharmacokinetics (PK) and physiologically based pharmacokinetic (PBPK) modeling.

Can In Vitro-In Vivo Extrapolation Be Successful? Recognizing the Incorrect Clearance Assumptions

For a number of years, our laboratory has been investigating the underlying reasons for the published poor in vitro-in vivo extrapolation (IVIVE) predictability of human clearance both from a theoretical and from an experimental perspective. Here, we critically examine clearance concepts and commonly employed IVIVE approaches, concluding that there is no theoretical reason that IVIVE should work, just as it does not. Our analysis, however, has identified 10 misconceptions and/or poorly understood aspects of clearance that are listed in the Conclusion section of this manuscript. Chief among these are that all published human drug clearance values are arterial clearances-clearance calculated as organ blood flow multiplied by the extraction ratio is the arterial clearance of the organ of elimination (and not the published drug clearance value)-and that the well-stirred model equation taught in all pharmacokinetic courses that relates organ blood flow, fraction unbound in blood, and intrinsic clearance has no validity. We further list 10 conclusions relating to the IVIVE process. The primary IVIVE-related conclusions are that the intrinsic clearance value determined from an in vitro incubation is an arterial intrinsic clearance, there is no theoretical basis upon which an arterial intrinsic clearance can be related to a whole-body arterial clearance to accomplish IVIVE, there are no published data demonstrating that in vitro intrinsic metabolic clearance can predict in vivo organ clearance as IVIVE assumes, and the scientific basis for the hypothesized albumin-mediated hepatic uptake phenomenon is invalid. We further propose three IVIVE process recommendations.