Compartment Pharmacokinetic Model Research Articles

Statistical Analysis of Two-Compartment Pharmacokinetic Models with Drug Non-adherence.

Poor drug adherence is considered one of major barriers to achieving the clinical and public health benefits of many pharmacotherapies. In the current paper, we aim to investigate the impact of dose omission on the plasma concentrations of two-compartment pharmacokinetic models with two typical routes of drug administration, namely the intravenous bolus and extravascular first-order absorption. First, we reformulate the classical two-compartment pharmacokinetic models with a new stochastic feature, where a binomial random model of dose intake is integrated. Then, we formalize the explicit expressions of expectation and variance for trough concentrations and limit concentrations, with the latter proved of the existence and uniqueness for steady-state distribution. Moreover, we mathematically demonstrate the strict stationarity and ergodicity of trough concentrations as a Markov chain. In addition, we numerically simulate the impact of drug non-adherence to different extents on the variability and regularity of drug concentration and compare the drug pharmacokinetic preference between one and two compartment pharmacokinetic models. The results of sensitivity analysis also suggest the drug non-adherence as one of the most sensitive model parameters to the expectation of limit concentration. Our modelling and analytical approach can be integrated into the chronic disease models to estimate or quantitatively predict the therapy efficacy with drug pharmacokinetics presumably affected by random dose omissions.

Pharmacokinetics with intravenous infusion of two-compartment model based on Liu process

By describing the absorption, distribution, metabolism, and excretion of drugs, pharmacokinetics helps to find optimal therapies for patients and speed up drug development. In pharmacokinetics, dynamic uncertainties, such as fluctuations in hormone levels and external environment factors, are ubiquitous. Due to sparse clinical data and patient disease specificity, these uncertainties are mainly epistemic uncertainties rather than aleatory uncertainties, which cannot be handled well by probability theory based methods. Therefore, several pharmacokinetics based on uncertain differential equations under the framework of uncertainty theory were investigated, which all considered one compartment models. Noting that many drugs follow two compartment kinetics, this article proposes a two compartment pharmacokinetic model based on uncertain differential equations. Based on the proposed model, several essential pharmacokinetic parameters are investigated, which are important information required by regulatory bodies to approve drugs for public use. Estimation for unknown parameters in the model are given. Finally, a real data analysis using lidocaine drug concentration illustrates our methodology in details.

Pharmacokinetic parameter identification using a meshless method approach for transdermal drug delivery

The Localized Collocation Meshless Method (LCMM) is applied to the solution of a set of partial differential equations governing the diffusive transport of a dermally injected compound as described by the Compartmental Pharmacokinetic (CPK) model. This efficient and accurate numerical technique then provides the framework for the estimation of pharmacokinetic (PK) parameters by the comparison of two models, both containing information derived from experimental results, using the iterative Golden Section search algorithm. This comparison is quantified using a norm between the two solution spaces. While this method does provide an estimate of the effective-diffusion coefficient, it ultimately demonstrates that a simple, free-diffusion model is inadequate for quantifying the spatial and temporal distribution of a dermally-injected compound if elimination effects are not accounted for properly.

IMPORTANCE OF SUFFICIENT TIME POINTS FOR EFFICIENT PHARMACOKINETIC (PK) COMPARTMENTAL MODELING

Objective: Modeling and simulation are the two widely used terms, usually simultaneously mentioned in most PK discussions. There are several modeling strategies to model pharmacokinetic (PK) profiles. Compartmental modeling divides the body into different compartments based on the observed C-t profile and model comparison functions. Most C-t profiles are efficiently modeled using at max three compartments model (one, two, or three compartments). While there are many important applications of classical compartmental models, it emphasizes the importance of selecting the best model to explain the observed data. Therefore, initial data generation is very important. In many instances, insufficient data collection might not lead to the best model, which can be proved later costly by underpredicting or overpredicting PK parameters. This paper illustrates that adequate data collection can lead to correct model selection. Methods: Data was generated using the three-compartmental model's explicit equation for twenty-five simulated patients with 15% random variability. Generated data were fitted to different compartmental models using sufficient time points (case a) and without enough time points (case b). Results: In the case of a, generated data from three compartmental models was explained best by three compartmental models. In the case of b, the same data was presented better by two compartmental models. Finally, in the case of b, with sufficient time points, data generated from three compartmental models could be explained better by three compartmental models. Conclusion: With sufficient time points, the compartmental PK model can converge to an accurate one. Although almost all pharmacometricians know the importance of time points, there is no paper with a mathematical explanation of this incident. This paper will help the current and future pharmacometricians to help design efficient in vivo works.

Pharmacokinetic Modeling of the Second-Wave Phenomenon in Nanobubble-Based Contrast-Enhanced Ultrasound.

With a typical 100-500 nm diameter, nanobubbles are a promising new-generation ultrasound contrast agent that paves ways for several applications, such as efficient drug delivery, molecular imaging, and assessment of vascular permeability. Due to their unique physical properties, nanobubbles exhibit distinct in vivo pharmacokinetics. We have shown that the first pass of the nanobubble bolus is usually accompanied by the appearance of a second bolus (wave) within a time range of about 15 minutes. Such phenomenon, to the best of our knowledge, has never been observed with conventional microbubbles and smaller molecular contrast agents used in MRI and CT. In a previous study, we showed the potential of this phenomenon in supporting cancer diagnosis. This study focuses on developing a new compartmental pharmacokinetic model that can be used to interpret the second-wave phenomenon. With this model, we can analyze more in-depth the roles of several physiological factors affecting the characteristics of the second-wave phenomenon.

Model-Informed drug development of gastroretentive release systems for sildenafil citrate

Gastroretentive drug delivery systems (GRDDS) are modified-release dosage forms designed to prolong their residence time in the upper gastrointestinal tract, where some drugs are preferentially absorbed, and increase the drug bioavailability. This work aimed the development of a novel GRDDS containing 60 mg of sildenafil citrate, and the evaluation of the feasibility of the proposed formulation for use in the treatment of pulmonary arterial hypertension (PAH), for once a day administration, by using in silico pharmacokinetic (PK) modeling and simulations using GastroPlusTM. The Model-Informed Drug Development (MIDD) approach was used in formulation design and pharmacokinetic exposure prospecting. A 22 factorial design with a central point was used for optimization of the formulation, which was produced by direct compression and characterized by some tests, including buoyancy test, assay, impurities, and in vitro dissolution. A compartmental PK model was built using the GatroPlusTM software for virtual bioequivalence of the proposed formulations in comparison with the defined target release profile provided by an immediate release (IR) tablet formulation containing 20 mg of sildenafil administered three times a day (TID). The results of the factorial design showed a direct correlation between the dissolution rate and the amount of hydroxypropyl methyl cellulose (HPMC) in the formulations. By comparing the PK parameters predicted by the virtual bioequivalence, the formulations F1, F2, F3 and F5 failed on bioequivalence. The F4 showed bioequivalence to the reference and was considered the viable formulation to substitute the IR. Thus, GRDDS could be a promising alternative for controlling the release of drugs with a pH-dependent solubility and narrow absorption window, specifically in the gastric environment, and an interesting way to reduce dose frequency and increase the drug bioavailability. The MIDD approach increases the level of information about the pharmaceutical product and guide the drug development for more assertive ways.

A compatibility evaluation between the physiologically based pharmacokinetic (PBPK) model and the compartmental PK model using the lumping method with real cases.

Pharmacokinetic (PK) modeling is a useful method for investigating drug absorption, distribution, metabolism, and excretion. The most commonly used mathematical models in PK modeling are the compartment model and physiologically based pharmacokinetic (PBPK) model. Although the theoretical characteristics of each model are well known, there have been few comparative studies of the compatibility of the models. Therefore, we evaluated the compatibility of PBPK and compartment models using the lumping method with 20 model compounds. The PBPK model was theoretically reduced to the lumped model using the principle of grouping tissues and organs that show similar kinetic behaviors. The area under the concentration–time curve (AUC) based on the simulated concentration and PK parameters (drug clearance [CL], central volume of distribution [Vc], peripheral volume of distribution [Vp]) in each model were compared, assuming administration to humans. The AUC and PK parameters in the PBPK model were similar to those in the lumped model within the 2-fold range for 17 of 20 model compounds (85%). In addition, the relationship of the calculated Vd/fu (volume of distribution [Vd], drug-unbound fraction [fu]) and the accuracy of AUC between the lumped model and compartment model confirmed their compatibility. Accordingly, the compatibility between PBPK and compartment models was confirmed by the lumping method. This method can be applied depending on the requirement of compatibility between the two models.

Nonlinear Elimination of Pharmaceutical Substances in Single Compartment Pharmacokinetic Models

Pharmacokinetic modeling is a mathematical representation of the behavior of a pharmaceutical substance in the presence or anatomical and topographical area, created to describe pharmacological or refined kinetic characteristics. Pharmacokinetics is the science of basic processes observed over time and approaching the action of clinical manifestations. These nutritional patterns model biological processes, refer to the kinetic behavior of a drug after it has been introduced into the body, leading to a better understanding of its natural effects. Mathematical modeling is currently preferred, used in the study of research and biochemical systems. The spectrum of its use is from non-compartmental to large-scale multichamber models. In the case of compartmental models ("block model", "compartment model"), mass balance equations are used to represent each compartment. The number of compartments in the models depends on the rate of drug distribution over different topographic areas of the body. Most studies use one- or two-chamber models. When a substance is excreted from the body, its concentration in the systemic circulation and in all tissues decreases at the same rate due to the rapid distribution of distribution. This article is spent on the components of a one-component pharmacokinetic model specifically designed for various drug elimination models. The single-compartment open model is a simple way to describe the distribution and excretion of drug compounds from the body. This model assumes that drugs can enter or leave the body (an "open" model), and the whole body appears as a whole.

Natural and commercial antibiotic comparison with drugs modeling cell integrity cell stability of Bio-Kinetics changes under morphological topographies cells with lower toxicological characteristics for multidrug resistances problem

Antibacterial drug-resistant strains are a serious problem of bacterial treatments nowadays and have a concern. The plant exacts of Adhatoda vasica and Calotropis procera are well-known for their role as antibiotic agents. The extraction of novel antibiotic compounds was done by HPLC-DAD, their yield is quantified by numerous solvents. The complete biological activity with antioxidants, bio-kinematicof four compounds of B-Sitosteryl linoleate, Myristyl diglucoside, D-Triglucopyranoside, and S- allylcysteine acids were studied. The supercritical fluid extraction techniques were the best strategies for higher yield, accuracy clarity, and inter, intra process of all four compounds. A. vasica and C. procera samples and investigated in six different solvents. D-Triglucopyranoside (13.81 ± 0.48%), Myristyl diglucoside (11.81 ± 0.41%), B- Sitosteryl linoleate (12.81 ± 0.48%), and s-allylcysteine acids (14.81 ± 0.31%) were higher. The design and action of compounds were applied to proper compartmental pharmacokinetic modelling for in-depth design understanding. The morphology and structure of bacterial cells with the extracted compounds upheld the permeability of cell membranes, membrane integrity, and membrane potential and lower the bacterial binding capacity the infectious index was measured in transmission electron microscopy (TEM) and their alteration process. Plants have well upheld the cellular permeability The toxicity test was performed on both extracted samples with concentrations (1, 0.4, and 0.8%). The areas under plasma half-life of compounds with their solubility, abortion level were higher in four compounds showed the potential of novel antibiotics. The novel medicinal plants used as antibiotics could be the best sources of infection control as a source of future medicines with antibacterial potential solving multidrug issues of bacteria in the world.

Dose-dependent bioavailability, absorption-rate limited elimination, and tissue distribution of the ATR inhibitor BAY-1895344 (elimusertib) in mice.

Ataxia Telangiectasia and Rad3-related (ATR) is a pivotal component of the DNA damage response and repair pathways that is activated in responses to cytotoxic cancer treatments. Several ATR inhibitors (ATRi) are in development that block the ATR mediated DNA repair and enhance the damage associated with cytotoxic therapy. BAY-1895344 (elimusertib) is an orally available ATRi with preclinical efficacy that is in clinical development. Little is known about the pharmacokinetics (PK) which is of interest, because tissue exposure and ATR inhibition may relate to toxicities or responses. To evaluate BAY-1895344 PK, a sensitive LC-MS/MS method was utilized for quantitation in mouse plasma and tissues. PK studies in mice were first conducted to determine dose linearity. In vivo metabolites were identified and analyzed semi-quantitatively. A compartmental PK model was developed to describe PK behavior. An extensive PK study was then conducted in tumor-bearing mice to quantitate tissue distribution for relevant tissues. Dose linearity was observed from 1 to 10mg/kg PO, while at 40mg/kg PO bioavailability increased approximately fourfold due to saturation of first-pass metabolism, as suggested by metabolite analyses and a developed compartmental model. Longer half-lives in PO treated mice compared to IV treated mice indicated absorption-rate limited elimination. Tissue distribution varied but showed extensive distribution to bone marrow, brain, and spinal cord. Complex PK behavior was limited to absorption processes which may not be recapitulated clinically. Tissue partition coefficients may be used to contrast ATR inhibitors with respect to their efficacy and toxicity.

Real-Time Analytical Solutions as Series Formulas and Heaviside off/on Switch Functions for Multiple Intermittent Intravenous Infusions in One- and Two-Compartment Models

Pharmacokinetic compartment models are the only models that can extract pharmacokinetic parameters from data collected in clinical studies but their estimates lack accuracy, explanations and physiological significance. The objective of this work was to develop particular solutions to drug concentration and AUC in the form of mathematical series and Heaviside functions for repetitive intermittent infusions in the one- and two-compartment models, as a function of dose number and total time using differential calculus. It was demonstrated that the central and peripheral compartment volumes determined from regression analysis of the aminoglycoside antibiotic Sisomicin concentration in plasma represent the actual physiological body fluid volumes accessible by the drug. The drug peak time and peak concentration in the peripheral compartment were also calculated as a function of dose number. It is also shown that the time of intercompartmental momentary distribution equilibrium can be used to determine the drug’s apparent volume of distribution within any dosing interval in multi-compartment models. These estimates were used to carry out simulations of plasma drug concentration with time in the one-compartment model. In conclusion, the two-compartment open mammillary pharmacokinetic model was fully explained for the aminoglycoside antibiotic sisomicin through the new concept of the apparent volume of distribution.

Designing Drug Regimens that Mitigate Nonadherence.

Medication adherence is a well-known problem for pharmaceutical treatment of chronic diseases. Understanding how nonadherence affects treatment efficacy is made difficult by the ethics of clinical trials that force patients to skip doses of the medication being tested, the unpredictable timing of missed doses by actual patients, and the many competing variables that can either mitigate or magnify the deleterious effects of nonadherence, such as pharmacokinetic absorption and elimination rates, dosing intervals, dose sizes, and adherence rates. In this paper, we formulate and analyze a mathematical model of the drug concentration in an imperfectly adherent patient. Our model takes the form of the standard single compartment pharmacokinetic model with first-order absorption and elimination, except that the patient takes medication only at a given proportion of the prescribed dosing times. Doses are missed randomly, and we use stochastic analysis to study the resulting random drug level in the body. We then use our mathematical results to propose principles for designing drug regimens that are robust to nonadherence. In particular, we quantify the resilience of extended release drugs to nonadherence, which is quite significant in some circumstances, and we show the benefit of taking a double dose following a missed dose if the drug absorption or elimination rate is slow compared to the dosing interval. We further use our results to compare some antiepileptic and antipsychotic drug regimens.

Novel Quantification of Extracellular Vesicles with Unaltered Surface Membranes Using an Internalized Oligonucleotide Tracer and Applied Pharmacokinetic Multiple Compartment Modeling

PurposeWe developed an accessible method for labeling small extracellular vesicles (sEVs) without disrupting endogenous ligands. Using labeled sEVs administered to conscious rats, we developed a multiple compartment pharmacokinetic model to identify potential differences in the disposition of sEVs from three different cell types.MethodsCrude sEVs were labeled with a non-homologous oligonucleotide and isolated from cell culture media using a commercial reagent. Jugular vein catheters were used to introduce EVs to conscious rats (n = 30) and to collect blood samples. Digital PCR was leveraged to allow for quantification over a wide dynamic range. Non-linear mixed effects analysis with first order conditional estimation – extended least squares (FOCE ELS) was used to estimate population-level parameters with associated intra-animal variability.Results86.5% ± 1.5% (mean ± S.E.) of EV particles were in the 45–195 nm size range and demonstrated protein and lipid markers of endosomal origin. Incorporated oligonucleotide was stable in blood and detectable over five half-lives. Data were best described by a three-compartment model with one elimination from the central compartment. We performed an observation-based simulated posterior predictive evaluation with prediction-corrected visual predictive check. Covariate and bootstrap analyses identified cell type having an influence on peripheral volumes (V2 and V3) and clearance (Cl3).ConclusionsOur method relies upon established laboratory techniques, can be tailored to a variety of biological questions regarding the pharmacokinetic disposition of extracellular vesicles, and will provide a complementary approach for the of study EV ligand-receptor interactions in the context of EV uptake and targeted therapeutics.

Development of PBPK model for intra-articular injection in human: methotrexate solution and rheumatoid arthritis case study

A physiologically based model describing the dissolution, diffusion, and transfer of drug from the intra-articular (IA) space to the plasma, was developed for GastroPlus® v9.8. The model is subdivided into compartments representing the synovial fluid, synovium, and cartilage. The synovium is broken up into two sublayers. The intimal layer acts as a diffusion barrier between the synovial fluid and the subintimal layer. The subintimal layer of the synovium has fenestrated capillaries that allow the free drug to be transported into systemic circulation. The articular cartilage is broken up into 10 diffusion sublayers as it is much thicker than the synovium. The cartilage acts as a depot tissue for the drug to diffuse into from synovial fluid. At later times, the drug will diffuse from the cartilage back into synovial fluid once a portion of the dose enters systemic circulation. In this study, a listing of all relevant details and equations for the model is presented. Methotrexate was chosen as a case study to show the application and utility of the model, based on the availability of intravenous (IV), oral (PO) and IA administration data in patients presenting rheumatoid arthritis (RA) symptoms. Systemic disposition of methotrexate in RA patients was described by compartmental pharmacokinetic (PK) model with PK parameters extracted using the PKPlus™ module in GastroPlus®. The systemic PK parameters were validated by simulating PO administration of methotrexate before being used for simulation of IA administration. For methotrexate, the concentrations of drug in the synovial fluid and plasma were well described after adjustments of physiological parameters to account for RA disease state, and with certain assumptions about binding and diffusion. The results indicate that the model can correctly describe PK profiles resulting from administration in the IA space, however, additional cases studies will be required to evaluate ability of the model to scale between species and/or doses.

Development of a Compartmental Pharmacokinetic Model for Molecular Radiotherapy with 131I-CLR1404.

Pharmacokinetic modeling of the radiopharmaceuticals used in molecular radiotherapy is an important step towards accurate radiation dosimetry of such therapies. In this paper, we present a pharmacokinetic model for CLR1404, a phospholipid ether analog that, labeled with 124I/131I, has emerged as a promising theranostic agent. We follow a systematic approach for the model construction based on a decoupling process applied to previously published experimental data, and using the goodness-of-fit, Sobol’s sensitivity analysis, and the Akaike Information Criterion to construct the optimal form of the model, investigate potential simplifications, and study factor prioritization. This methodology was applied to previously published experimental human time-activity curves for 9 organs. The resulting model consists of 17 compartments involved in the CLR1404 metabolism. Activity dynamics in most tissues are well described by a blood contribution plus a two-compartment system, describing fast and slow uptakes. The model can fit both clinical and pre-clinical kinetic data of 124I/131I. In addition, we have investigated how simple fits (exponential and biexponential) differ from the complete model. Such fits, despite providing a less accurate description of time-activity curves, may be a viable alternative when limited data is available in a practical case.

Topical finasteride dose evaluation for treatment of androgenetic alopecia using computer simulations

Our objective was to evaluate the absorption of finasteride administered by oral and topical routes for treatment of androgenetic alopecia, by means of computer simulations using the GastroPlus ® software. In vivo plasma concentration profile of oral administration of finasteride 1 mg tablets from the literature was used in the software to build a compartmental pharmacokinetic model that was extrapolated to simulate topical administration of finasteride 0.25% solution in the scalp. Results were compared to literature and other drug concentrations (0.1% and 1%) were also predicted. Compared to literature data, predictive plasma curve from oral administration of finasteride 1 mg showed good correlation, R2=0.992, and Cmax=4.6769 ng/mL. Simulations of topical administration in the scalp for finasteride 0.25% solution showed good correlation, R2=0.908, and Cmax=0.04325 ng/mL when compared to literature data. Topical administration was also simulated at concentrations 0.1% and 1%, both with low plasma concentrations. The results obtained in this study suggest that topical finasteride is a potential treatment for androgenetic alopecia in the concentrations analyzed using computer simulations in GastroPlus ®.

A Mathematical Treatment of Multiple Intermittent Intravenous Infusions in a One-Compartment Model

Compartment models in pharmacokinetics are generally described for a single dose and can only simulate drug concentration as a function of the first dosing interval. Our objective was to create a one-compartment model for constant rate repetitive intravenous intermittent infusions where drug concentration can be simulated as a function of real time. The analytical solutions to differential equations that were set for the time periods of drug infusion and elimination after an intermittent intravenous infusion were used to derive sequences patterns and determine the partial sums of mathematical series for multiple intermittent infusion doses. The model's original theory was supplemented with explicit solutions to the concentration and AUC, at non steady state conditions, with and without a loading dose, for both the infusion and elimination time periods after repetitive intermittent intravenous infusions. The validity and accuracy of these formulas in calculating drug concentration and AUC was verified by mathematical proofs, numerical methods and the principle of superposition. Drug concentration and AUC can be simulated using the newly described one-compartment pharmacokinetic model in the entire time-domain of therapy after multiple and repetitive intermittent intravenous infusions and not just within the first dosing interval.

Back Cover: Preclinical pharmacokinetics of trigonelline using ultra‐performance liquid chromatography–tandem mass spectrometry and pharmacological studies targeting type 2 diabetes

DOI: 10.1002/sscp.202000118 The cover picture shows, Trigonelline (Tr) a quaternary base alkaloid, which acts by affecting β-cell regeneration and insulin secretion. Tr lowers the blood glucose and lipid levels due to which it is used in the treatment of co-morbid diseases. Herein, we have aimed to develop a bioanalytical method for estimation of Tr using UPLC-MS/MS and explore the pharmacokinetic profile. The anti-diabetic, antilipidemic efficacy studies were also conducted in the high-fat diet-induced streptozotocin-diabetic rat. The experimental results suggest that Tr follow the two compartmental pharmacokinetic model. In addition, Tr exhibited a significant anti-hyperglycemic and antihyperlipidemic efficacy in high-fat diet-induced streptozotocin induced diabetic rats.

Preclinical pharmacokinetics of trigonelline using ultra‐performance liquid chromatography–tandem mass spectrometry and pharmacological studies targeting type 2 diabetes

AbstractTrigonelline is a quaternary base alkaloid and zwitterionic complex that acts by affecting β‐cell regeneration and insulin secretion. Tr inhibits enzymatic activities, lowering the blood glucose and lipid levels due to which it is used in the treatment of co‐morbid diseases such as diabetes, Alzheimer's, etc. Herein, it was aimed to develop a bioanalytical method for estimation of Tr using ultra‐performance liquid chromatography–tandem mass spectrometry and explore the pharmacokinetic profile. The anti‐diabetic, antilipidemic efficacy studies of Tr in the high‐fat diet‐induced streptozotocin‐diabetic rat model was also explored. The separation of analyte was achieved with acetonitrile and 0.1% formic acid (20:80) with a flow of 0.4 mL/min. The ionization of the analyte was achieved in positive electrospray ionization mode with the precursor to product ion transitions of Tr (138.14 > 94), and Trigonelline D3 (143.19 > 97.13). The validated assay was effectively applied for the estimation of compartmental pharmacokinetic by using Phoenix WinNolin8.0 (Certera™, USA) and it was observed that the Tr follow two compartmental pharmacokinetic model. The experimental results also suggest that Tr distributed through the central compartment to the peripheral compartment and redistributed to the central compartment. In addition, Tr exhibited significant anti‐hyperglycemic and antihyperlipidemic efficacy against high‐fat diet‐induced streptozotocin‐induced type 2 diabetic rats.

Constant infusion case of one compartment pharmacokinetic model with simultaneous first-order and Michaelis-Menten elimination: analytical solution and drug exposure formula.

The main objective of this article is to propose the closed-form solution of one-compartment pharmacokinetic model with simultaneous first-order and Michaelis-Menten elimination for the case of constant infusion. For the case of bolus administration, we have previously established a closed-form solution of the model through introducing a transcendent X function. In the same vein, we found here a closed-form solution of constant infusion could be realized through introducing another transcendent Y function. For the general case of constant infusion of limited duration, the closed-form solution is then fully expressed using both X and Y functions. As direct results, several important pharmacokinetic surrogates, such as peak concentration [Formula: see text] and total drug exposure AUC[Formula: see text], are found the closed-form expressions and ready to be analyzed. The new pharmacokinetic knowledge we have gained on these parameters, which largely exhibits in a nonlinear feature, is in clear contrast to that of the linear case. Finally, with a pharmacokinetic model adapted from that formerly reported on phenytoin, we numerically analyzed and illustrated the roles of different model parameters and discussed their influence on drug exposure. To conclude, the present findings elucidate the intrinsic quantitative structural properties of such pharmacokinetic model and provide a new avenue for future modelling and rational drug designs.