A numerical method for analysis of in vitro time-dependent inhibition data. Part 2. Application to experimental data

Enzyme-mediated drug-drug interactions: a review of in vivo and in vitro methodologies, regulatory guidance, and translation to the clinic

Enzyme-mediated pharmacokinetic drug-drug interactions can be caused by altered activity of drug metabolizing enzymes in the presence of a perpetrator drug, mostly via inhibition or induction. We identified a gap in the literature for a state-of-the art detailed overview assessing this type of DDI risk in the context of drug development. This manuscript discusses in vitro and in vivo methodologies employed during the drug discovery and development process to predict clinical enzyme-mediated DDIs, including the determination of clearance pathways, metabolic enzyme contribution, and the mechanisms and kinetics of enzyme inhibition and induction. We discuss regulatory guidance and highlight the utility of in silico physiologically-based pharmacokinetic modeling, an approach that continues to gain application and traction in support of regulatory filings. Looking to the future, we consider DDI risk assessment for targeted protein degraders, an emerging small molecule modality, which does not have recommended guidelines for DDI evaluation. Our goal in writing this report was to provide early-career researchers with a comprehensive view of the enzyme-mediated pharmacokinetic DDI landscape to aid their drug development efforts.

CYP2J2-mediated metabolism of arachidonic acid in heart: A review of its kinetics, inhibition and role in heart rhythm control

Cytochrome P450 2 J2 (CYP2J2) is primarily expressed extrahepatically and is the predominant epoxygenase in human cardiac tissues. This highlights its key role in the metabolism of endogenous substrates. Significant scientific interest lies in cardiac CYP2J2 metabolism of arachidonic acid (AA), an omega-6 polyunsaturated fatty acid, to regioisomeric bioactive epoxyeicosatrienoic acid (EET) metabolites that show cardioprotective effects including regulation of cardiac electrophysiology. From an in vitro perspective, the accurate characterization of the kinetics of CYP2J2 metabolism of AA including its inhibition and inactivation by drugs could be useful in facilitating in vitro-in vivo extrapolations to predict drug-AA interactions in drug discovery and development. In this review, background information on the structure, regulation and expression of CYP2J2 in human heart is presented alongside AA and EETs as its endogenous substrate and metabolites. The in vitro and in vivo implications of the kinetics of this endogenous metabolic pathway as well as its perturbation via inhibition and inactivation by drugs are elaborated. Additionally, the role of CYP2J2-mediated metabolism of AA to EETs in cardiac electrophysiology will be expounded.

Consideration of Nevirapine Analogs To Reduce Metabolically Linked Hepatotoxicity: A Cautionary Tale of the Deuteration Approach

Idiosyncratic drug reactions (IDRs) in their most deleterious form can lead to serious medical complications and potentially fatal events. Nevirapine (NVP), still widely used in developing countries for combinatorial antiretroviral and prophylactic therapies against HIV infection, represents a prototypical example of IDRs causing severe skin rashes and hepatotoxicity. Complex metabolic pathways accompanied by production of multiple reactive metabolites often complicate our understanding of IDR's origin. While assessment of NVP analogs has helped characterize the pathways involved in IDRs for NVP, which are largely driven by metabolism at the 12-methyl position, it has yet to be investigated if some of these analogs could be valuable replacement drugs with reduced reactive metabolite properties and drug-drug interaction (DDI) risks. Here, we evaluated a set of eight NVP analogs, including the deuterated 12-d3-NVP and two NVP metabolites, for their efficacy and inhibitory potencies against HIV reverse transcriptase (HIV-RT). A subset of three analogs, demonstrating >85% inhibition for HIV-RT, was further assessed for their hepatic CYP induction-driven DDI risks. This led to a closer investigation of the inactivation properties of 12-d3-NVP for hepatic CYP3A4 and a comparison of its propensity in generating reactive metabolite species. The metabolic shift triggered with 12-d3-NVP, increasing formation of the 2-hydroxy and glutathione metabolites, emphasized the importance of the dynamic balance between induction and metabolism-dependent inactivation of CYP3A4 and its impact on clearance of NVP during treatment. Unfortunately, the strategy of incorporating deuterium to reduce NVP metabolism and production of the electrophile species elicited opposite results, illustrating the great challenges involved in tackling IDRs through deuteration.

Complex Cytochrome P450 Kinetics Due to Multisubstrate Binding and Sequential Metabolism. Part 1. Theoretical Considerations

Complexities in P450-mediated metabolism kinetics include multisubstrate binding, multiple-product formation, and sequential metabolism. Saturation curves and intrinsic clearances were simulated for single-substrate and multisubstrate models using derived velocity equations and numerical solutions of ordinary differential equations (ODEs). Multisubstrate models focused on sigmoidal kinetics because of their dramatic impact on clearance predictions. These models were combined with multiple-product formation and sequential metabolism, and simulations were performed with random error. Use of single-substrate models to characterize multisubstrate data can result in inaccurate kinetic parameters and poor clearance predictions. Comparing results for use of standard velocity equations with ODEs clearly shows that ODEs are more versatile and provide better parameter estimates. It would be difficult to derive concentration-velocity relationships for complex models, but these relationships can be easily modeled using numerical methods and ODEs. SIGNIFICANCE STATEMENT: The impact of multisubstrate binding, multiple-product formation, and sequential metabolism on the P450 kinetics was investigated. Numerical methods are capable of characterizing complicated P450 kinetics.

Numerical Methods for Modeling Enzyme Kinetics

Differential equations are used to describe time-dependent changes in enzyme kinetics and pharmacokinetics. Analytical and numerical methods can be used to solve differential equations. This chapter describes the use of numerical methods in solving differential equations and its applications in characterizing the complexities observed in enzyme kinetics. A discussion is included on the use of numerical methods to overcome limitations of explicit equations in the analysis of metabolism kinetics, reversible inhibition kinetics, and inactivation kinetics. The chapter describes the advantages of using numerical methods when Michaelis-Menten assumptions do not hold.

Inhibition of CYP3A7 DHEA-S Oxidation by Lopinavir and Ritonavir: An Alternative Mechanism for Adrenal Impairment in HIV Antiretroviral-Treated Neonates

Prophylactic antiretroviral therapy (ART) in HIV infected pregnant mothers and their newborns can dramatically reduce mother-to-child viral transmission and seroconversion in the neonate. The ritonavir-boosted lopinavir regimen, known as Kaletra, has been associated with premature birth and transient adrenal insufficiency in newborns, accompanied by increases in plasma dehydroepiandrosterone 3-sulfate (DHEA-S). In the fetus and neonates, cytochrome P450 CYP3A7 is responsible for the metabolism of DHEA-S into 16α-hydroxy DHEA-S, which plays a critical role in growth and development. In order to determine if CYP3A7 inhibition could lead to the adverse outcomes associated with Kaletra therapy, we conducted in vitro metabolic studies to determine the extent and mechanism of CYP3A7 inhibition by both ritonavir and lopinavir and the relative intrinsic clearance of lopinavir with and without ritonavir in both neonatal and adult human liver microsomes (HLMs). We identified ritonavir as a potent inhibitor of CYP3A7 oxidation of DHEA-S (IC₅₀ = 0.0514 μM), while lopinavir is a much weaker inhibitor (IC₅₀ = 5.88 μM). Furthermore, ritonavir is a time-dependent inhibitor of CYP3A7 with a K_I of 0.392 μM and a k_inact of 0.119 min^–1, illustrating the potential for CYP3A mediated drug–drug interactions with Kaletra. The clearance rate of lopinavir in neonatal HLMs was much slower and comparable to the rate observed in adult HLMs in the presence of ritonavir, suggesting that the addition of ritonavir in the cocktail therapy may not be necessary to maintain effective concentrations of lopinavir in neonates. Our results suggest that several of the observed adverse outcomes of Kaletra therapy may be due to the direct inhibition of CYP3A7 by ritonavir and that the necessity for the inclusion of this drug in the therapy may be obviated by the lower rate of lopinavir clearance in the neonatal liver. These results may lead to a reconsideration of the use of ritonavir in neonatal antiretroviral therapy.

Inhibition and induction of CYP enzymes in humans: an update

The cytochrome P450 (CYP) enzyme family is the most important enzyme system catalyzing the phase 1 metabolism of pharmaceuticals and other xenobiotics such as herbal remedies and toxic compounds in the environment. The inhibition and induction of CYPs are major mechanisms causing pharmacokinetic drug–drug interactions. This review presents a comprehensive update on the inhibitors and inducers of the specific CYP enzymes in humans. The focus is on the more recent human in vitro and in vivo findings since the publication of our previous review on this topic in 2008. In addition to the general presentation of inhibitory drugs and inducers of human CYP enzymes by drugs, herbal remedies, and toxic compounds, an in-depth view on tyrosine-kinase inhibitors and antiretroviral HIV medications as victims and perpetrators of drug–drug interactions is provided as examples of the current trends in the field. Also, a concise overview of the mechanisms of CYP induction is presented to aid the understanding of the induction phenomena.

Mechanisms of Herb-Drug Interactions Involving Cinnamon and CYP2A6: Focus on Time-Dependent Inhibition by Cinnamaldehyde and 2-Methoxycinnamaldehyde

Information is scarce regarding pharmacokinetic-based herb-drug interactions (HDI) with trans-cinnamaldehyde (CA) and 2-methoxycinnamaldehyde (MCA), components of cinnamon. Given the presence of cinnamon in food and herbal treatments for various diseases, HDIs involving the CYP2A6 substrates nicotine and letrozole with MCA (KS = 1.58 µM; Hill slope = 1.16) and CA were investigated. The time-dependent inhibition (TDI) by MCA and CA of CYP2A6-mediated nicotine metabolism is a complex process involving multiple mechanisms. Molecular dynamic simulations showed that CYP2A6's active site accommodates two dynamic ligands. The preferred binding orientations for MCA and CA were consistent with the observed metabolism: epoxidation, O-demethylation, and aromatic hydroxylation of MCA and cinnamic acid formation from CA. The percent remaining activity plots for TDI by MCA and CA were curved, and they were analyzed with a numerical method using models of varying complexity. The best-fit models support multiple inactivator binding, inhibitor depletion, and partial inactivation. Deconvoluted mass spectra indicated that MCA and CA modified CYP2A6 apoprotein with mass additions of 156.79 (142.54-171.04) and 132.67 (123.37-141.98), respectively, and it was unaffected by glutathione. Heme degradation was observed in the presence of MCA (48.5% ± 13.4% loss; detected by liquid chromatography-tandem mass spectrometry). In the absence of clinical data, HDI predictions were made for nicotine and letrozole using inhibition parameters from the best-fit TDI models and parameters scaled from rats. Predicted area under the concentration-time curve fold changes were 4.29 (CA-nicotine), 4.92 (CA-letrozole), 4.35 (MCA-nicotine), and 5.00 (MCA-letrozole). These findings suggest that extensive exposure to cinnamon (corresponding to ≈ 275 mg CA) would lead to noteworthy interactions. SIGNIFICANCE STATEMENT: Human exposure to cinnamon is common because of its presence in food and cinnamon-based herbal treatments. Little is known about the risk for cinnamaldehyde and methoxycinnamaldehyde, two components of cinnamon, to interact with drugs that are eliminated by CYP2A6-mediated metabolism. The interactions with CYP2A6 are complex, involving multiple-ligand binding, time-dependent inhibition of nicotine metabolism, heme degradation, and apoprotein modification. An herb-drug interaction prediction suggests that extensive exposure to cinnamon would lead to noteworthy interactions with nicotine.

Numerical analysis of time-dependent inhibition kinetics: comparison between rat liver microsomes and rat hepatocyte data for mechanistic model fitting

Time-dependent inhibition (TDI) may confound drug interaction predictions. Recently, models were generated for an array of TDI kinetic schemes using numerical analysis of microsomal assays. Additionally, a distinct terminal inactivation step was identified for certain mechanism based inhibitors (MBI) following reversible metabolite intermediate complex (MIC) formation. Longer hepatocyte incubations potentially allow analysis of slow TDI and terminal inactivation. In the experiments presented here, we compared the quality of TDI parameterization by numerical analysis between hepatocyte and microsomal data. Rat liver microsomes (RLM), suspended rat hepatocytes (SRH) and sandwich-cultured rat hepatocytes (SCRH) were incubated with the prototypical CYP3A MBI troleandomycin and the substrate midazolam. Data from RLM provided a better model fit as compared to SRH. Increased CYP3A expression after dexamethasone (DEX) induction improved the fit for RLM and SRH. A novel sequential kinetic scheme, defining inhibitor metabolite production prior to MIC formation, improved the fit compared to direct MIC formation. Furthermore, terminal inactivation rate constants were parameterized for RLM and SRH samples with DEX-induced CYP3A. The low expression of CYP3A and experimental error in SCRH resulted in poor data for model fitting. Overall, RLM generated data better suited for elucidation of TDI mechanisms by numerical analysis.

Time-dependent enzyme inactivation: Numerical analyses of in vitro data and prediction of drug-drug interactions

Cytochrome P450 (CYP) enzyme kinetics often do not conform to Michaelis-Menten assumptions, and time-dependent inactivation (TDI) of CYPs displays complexities such as multiple substrate binding, partial inactivation, quasi-irreversible inactivation, and sequential metabolism. Additionally, in vitro experimental issues such as lipid partitioning, enzyme concentrations, and inactivator depletion can further complicate the parameterization of in vitro TDI. The traditional replot method used to analyze in vitro TDI datasets is unable to handle complexities in CYP kinetics, and numerical approaches using ordinary differential equations of the kinetic schemes offer several advantages. Improvement in the parameterization of CYP in vitro kinetics has the potential to improve prediction of clinical drug-drug interactions (DDIs). This manuscript discusses various complexities in TDI kinetics of CYPs, and numerical approaches to model these complexities. The extrapolation of CYP in vitro TDI parameters to predict in vivo DDIs with static and dynamic modeling is discussed, along with a discussion on current gaps in knowledge and future directions to improve the prediction of DDI with in vitro data for CYP catalyzed drug metabolism.

Numerical Analysis of Time-Dependent Inhibition by MDMA

Methylenedioxymethamphetamine (MDMA) is a known drug of abuse and schedule 1 narcotic under the Controlled Substances Act. Previous pharmacokinetic work on MDMA used classic linearization techniques to conclude irreversible mechanism-based inhibition of CYP2D6. The current work challenges this outcome by assessing the possibility of two alternative reversible kinetic inhibition mechanisms known as the quasi-irreversible (QI) model and equilibrium model (EM). In addition, progress curve experiments were used to investigate the residual metabolism of MDMA by liver microsomes and CYP2D6 baculosomes over incubation periods up to 30 minutes. These experiments revealed activity in a terminal linear phase at the fractional rates with respect to initial turnover of 0.0354 ± 0.0089 in human liver microsomes and 0.0114 ± 0.0025 in baculosomes. Numerical model fits to percentage of remaining activity (PRA) data were consistent with progress curve modeling results, wherein an irreversible inhibition pathway was found unnecessary for good fit scoring. Both QI and EM kinetic mechanisms fit the PRA data well, although in CYP2D6 baculosomes the inclusion of an irreversible inactivation pathway did not allow for convergence to a reasonable fit. The kinetic complexity accessible to numerical modeling has been used to determine that MDMA is not an irreversible inactivator of CYP2D6, and instead follows what can be generally referred to as slowly reversible inhibition. SIGNIFICANCE STATEMENT: The work herein describes the usage of computational models to delineate between irreversible and slowly reversible time-dependent inhibition. Such models are used in the paper to analyze MDMA and classify it as a reversible time-dependent inhibitor.

Impact of Lipid Partitioning on the Design, Analysis, and Interpretation of Microsomal Time-Dependent Inactivation

Nonspecific drug partitioning into microsomal membranes must be considered for in vitro-in vivo correlations. This work evaluated the effect of including lipid partitioning in the analysis of complex TDI kinetics with numerical methods. The covariance between lipid partitioning and multiple inhibitor binding was evaluated. Simulations were performed to test the impact of lipid partitioning on the interpretation of TDI kinetics, and experimental TDI datasets for paroxetine (PAR) and itraconazole (ITZ) were modeled. For most kinetic schemes, modeling lipid partitioning results in statistically better fits. For MM-IL simulations (KI,u = 0.1 µM, kinact = 0.1 minute-1), concurrent modeling of lipid partitioning for an fumic range (0.01, 0.1, and 0.5) resulted in better fits compared with post hoc correction (AICc: -526 vs. -496, -579 vs. -499, and -636 vs. -579, respectively). Similar results were obtained with EII-IL. Lipid partitioning may be misinterpreted as double binding, leading to incorrect parameter estimates. For the MM-IL datasets, when fumic = 0.02, MM-IL, and EII model fits were indistinguishable (δAICc = 3). For less partitioned datasets (fumic = 0.1 or 0.5), the inclusion of partitioning resulted in better models. The inclusion of lipid partitioning can lead to markedly different estimates of KI,u and kinact A reasonable alternate experimental design is nondilution TDI assays, with post hoc fumic incorporation. The best fit models for PAR (MIC-M-IL) and ITZ (MIC-EII-M-IL and MIC-EII-M-Seq-IL) were consistent with their reported mechanism and kinetics. Overall, experimental fumic values should be concurrently incorporated into TDI models with complex kinetics, when dilution protocols are used.

Kinetic mechanism of time-dependent inhibition of CYP2D6 by 3,4-methylenedioxymethamphetamine (MDMA): Functional heterogeneity of the enzyme and the reversibility of its inactivation

We investigate the mechanism of time-dependent inhibition (TDI) of human cytochrome P450 2D6 (CYP2D6) by 3,4-methylenedioxymethamphetamine (MDMA, ecstasy), one of the most widespread recreational drugs of abuse. In an effort to unravel the kinetic mechanism of the formation of metabolic inhibitory complex (MIC) of CYP2D6 with MDMA-derived carbene we carried out a series of spectrophotometric studies paralleled with registration of the kinetics of time-dependent inhibition (TDI) in CYP2D6-incorporated proteoliposomes. The high amplitude of spectral signal in this system allowed us to characterize the spectral properties of the formed MIC in details and obtain an accurate spectral signature of MIC formation. This information was then used in the studies with CYP2D6-containing microsomes of insect cells (CYP2D6 Supersomes™). Our results demonstrate that in both systems the formation of the ferrous carbene-derived MIC is relatively slow, reversible and is not associated with the accumulation of the ferric carbene intermediate, as takes place in the case of CYP3A4 and podophylotoxin. Furthermore, the limited amplitude of MIC formation suggests that only a fraction (∼50%) of spectrally detectable CYP2D6 in both proteoliposomes and Supersomes participates in the formation of MIC and is therefore involved in the MDMA metabolism. This observation reveals yet another example of a cytochrome P450 that exhibits persistent functional heterogeneity of its population in microsomal membranes. Our study provides a solid methodological background for further mechanistic studies of MIC formation in human liver microsomes and demonstrates that the potency and physiological relevance of MDMA-dependent TDI of CYP2D6 may be overestimated.

Improved Predictions of Drug-Drug Interactions Mediated by Time-Dependent Inhibition of CYP3A

Time-dependent inactivation (TDI) of cytochrome P450s (CYPs) is a leading cause of clinical drug-drug interactions (DDIs). Current methods tend to overpredict DDIs. In this study, a numerical approach was used to model complex CYP3A TDI in human-liver microsomes. The inhibitors evaluated included troleandomycin (TAO), erythromycin (ERY), verapamil (VER), and diltiazem (DTZ) along with the primary metabolites N-demethyl erythromycin (NDE), norverapamil (NV), and N-desmethyl diltiazem (NDD). The complexities incorporated into the models included multiple-binding kinetics, quasi-irreversible inactivation, sequential metabolism, inhibitor depletion, and membrane partitioning. The resulting inactivation parameters were incorporated into static in vitro-in vivo correlation (IVIVC) models to predict clinical DDIs. For 77 clinically observed DDIs, with a hepatic-CYP3A-synthesis-rate constant of 0.000 146 min-1, the average fold difference between the observed and predicted DDIs was 3.17 for the standard replot method and 1.45 for the numerical method. Similar results were obtained using a synthesis-rate constant of 0.000 32 min-1. These results suggest that numerical methods can successfully model complex in vitro TDI kinetics and that the resulting DDI predictions are more accurate than those obtained with the standard replot approach.

Role of Cytochrome P450 2C8 in Drug Metabolism and Interactions

During the last 10-15 years, cytochrome P450 (CYP) 2C8 has emerged as an important drug-metabolizing enzyme. CYP2C8 is highly expressed in human liver and is known to metabolize more than 100 drugs. CYP2C8 substrate drugs include amodiaquine, cerivastatin, dasabuvir, enzalutamide, imatinib, loperamide, montelukast, paclitaxel, pioglitazone, repaglinide, and rosiglitazone, and the number is increasing. Similarly, many drugs have been identified as CYP2C8 inhibitors or inducers. In vivo, already a small dose of gemfibrozil, i.e., 10% of its therapeutic dose, is a strong, irreversible inhibitor of CYP2C8. Interestingly, recent findings indicate that the acyl-β-glucuronides of gemfibrozil and clopidogrel cause metabolism-dependent inactivation of CYP2C8, leading to a strong potential for drug interactions. Also several other glucuronide metabolites interact with CYP2C8 as substrates or inhibitors, suggesting that an interplay between CYP2C8 and glucuronides is common. Lack of fully selective and safe probe substrates, inhibitors, and inducers challenges execution and interpretation of drug-drug interaction studies in humans. Apart from drug-drug interactions, some CYP2C8 genetic variants are associated with altered CYP2C8 activity and exhibit significant interethnic frequency differences. Herein, we review the current knowledge on substrates, inhibitors, inducers, and pharmacogenetics of CYP2C8, as well as its role in clinically relevant drug interactions. In addition, implications for selection of CYP2C8 marker and perpetrator drugs to investigate CYP2C8-mediated drug metabolism and interactions in preclinical and clinical studies are discussed.

Mechanism-Based Inhibition of CYP3A4 by Podophyllotoxin: Aging of an Intermediate Is Important for in Vitro/in Vivo Correlations

An in vitro observation of time-dependent inhibition (TDI) of metabolic enzymes often results in removing a potential drug from the drug pipeline. However, the accepted method for predicting TDIs of the important drug metabolizing cytochrome P450 enzymes often overestimates the drug interaction potential. Better models that take into account the complexities of the cytochrome P450 enzyme system will lead to better predictions. Herein we report the use of our previously described models for complex kinetics of podophyllotoxin. Spectral characterization of the kinetics indicates that an intermediate MI complex is formed, which slowly progresses to an essentially irreversible MI complex. The intermediate MI complex can release free enzyme during the time course of a typical 30 min TDI experiment. This slow rate of MI complex conversion results in an overprediction of the kinact value if this process is not included in the analysis of the activity versus time profile. In vitro kinetic experiments in rat liver microsomes predicted a lack of drug interaction between podophyllotoxin and midazolam. In vivo rat pharmacokinetic studies confirmed this lack of drug interaction.

Cytochrome P450 time-dependent inhibition and induction: advances in assays, risk analysis and modelling

INTRODUCTION

It is widely accepted that current practice of polypharmacy inevitably increases the incidence of drug-drug interactions (DDIs). Serious DDIs are a major liability for new molecular entities entering the pharmaceutical market. Various strategies are employed to avoid problematic compounds for clinical development. Progress made with reversible CYP DDIs has prompted a switch to study and model time-dependent inhibition and induction interactions.

AREAS COVERED

An overview of popular experimental practices is presented with discussion of techniques and algorithms used to analyse the clinical DDI risk. Emphasis is placed on the transition from early, simple static equations, via more complex net mechanistic, static models to dynamic approaches involving multiple perpetrators and metabolites, simultaneous inhibition and induction.

EXPERT OPINION

Inclusion of the more conservative terms for parameters required for DDI evaluation may eliminate promising chemical space, encourages poor practice and hampers innovation. Breakthroughs have originated from understanding of 'outliers' from such analyses where CYP enzyme-transporter interplay may be involved. The role of key transporters in drug disposition is firmly established as the chemistry required to address new targets deviates from traditional 'drug-like' space. Attempts to model more complex interactions for substrates of both CYP enzymes and drug transporters are still in their infancy and will benefit from dynamic modelling.

A numerical method for analysis of in vitro time-dependent inhibition data. Part 1. Theoretical considerations

Inhibition of cytochromes P450 by time-dependent inhibitors (TDI) is a major cause of clinical drug-drug interactions. It is often difficult to predict in vivo drug interactions based on in vitro TDI data. In part 1 of these manuscripts, we describe a numerical method that can directly estimate TDI parameters for a number of kinetic schemes. Datasets were simulated for Michaelis-Menten (MM) and several atypical kinetic schemes. Ordinary differential equations were solved directly to parameterize kinetic constants. For MM kinetics, much better estimates of KI can be obtained with the numerical method, and even IC50 shift data can provide meaningful estimates of TDI kinetic parameters. The standard replot method can be modified to fit non-MM data, but normal experimental error precludes this approach. Non-MM kinetic schemes can be easily incorporated into the numerical method, and the numerical method consistently predicts the correct model at errors of 10% or less. Quasi-irreversible inactivation and partial inactivation can be modeled easily with the numerical method. The utility of the numerical method for the analyses of experimental TDI data is provided in our companion manuscript in this issue of Drug Metabolism and Disposition (Korzekwa et al., 2014b).