Drug-Drug Interactions Pharmacokinetic Models with Extravascular Administration: Estimation of Elimination and Absorption Rate Constants

One and two-compartment pharmacokinetic models with drug-drug interactions are proposed. Two drugs are given orally simultaneously, so that their interaction affects the drug absorption process and subsequently the elimination process. The aim of this paper is to estimate the elimination and absorption rate constants by evaluating the data set of time and drug concentration. This data set was divided into two time phases: large-time elimination phase to estimate the elimination rate constant, and small-time absorption phase to estimate the absorption rate constant. Since the models are nonlinear, the Taylor expansion is employed to so that the Wagner-Nelson and the Loo-Riegelman methods can be used for estimation. Finally, simulations were performed using the generated arbitrary data set of time and concentration, instead of an actual data set, to derive the solution of drug concentration concerning time numerically. In these simulations we compared the original parameter values with their estimates for the one and two-compartment models, and we concluded that the two-compartment model produced better estimates than the one-compartment model. Qualitatively, the two-compartment model gives smaller drug concentration curve deviations between the original and the estimated curve compared with the one-compartment model

Integration-Based Method as an Alternative Way to Estimate Parameters in the IV Bolus Compartment Model

An alternative method of integration-based parameter estimation applied in pharmacokinetics problems is proposed here. The method, introduced by Holder and Rodrigo, is used to estimate the rate of drug elimination and distribution when it enters the body via intravenous bolus. The estimation results are then compared with the classical method, the least squares method for the one-compartment model, and the residual method for the two-compartment model. Graphical simulations of drug concentration versus time are also performed in this article to view not only the dynamics of drug delivery in the body, but also the comparisons between the approximate solutions and the arbitrarily generated data points. Comparisons are also presented when the data points take into account noise in the form of random values. Based on the estimation and simulation results, the integration-based method gives good results and even better than the classical method although when noise is applied to the data points

Pharmacokinetic Consideration to Formulate Sustained Release Drugs: Understanding the Controlled Drug Diffusion through the Body Compartment of the Systemic Circulation and Tissue Medium-A Caputo Model

الهدف من هذه الدراسة هو تقديم لمحة عامة عن النماذج المختلفة لدراسة انتشار الدواء لفترة طويلة في جسم الإنسان وداخله. تم التأكيد على نماذج المقصورة الرياضية باستخدام نهج المشتقة الجزئية (نموذج كابوتو) للتحقيق في التغير في تركيز الدواء المستدام في أجزاء مختلفة من نظام جسم الإنسان من خلال الطريق الفموي أو الطريق الوريدي. و تم استخدام قانون العمل الجماعي ، وحركية الدرجة الأولى ، ومبدأ الإرواء لفيك لتطوير نماذج المقصورة الرياضية التي تمثل انتشارا مستداما للأدوية في جميع أنحاء جسم الإنسان. للتنبؤ بشكل كافٍ بانتشار الدواء المستمر في أجزاء مختلفة من جسم الإنسان، وضعنا في الاعتبار(نموذج كابوتو (للتحقيق في معدل تغير التركيز اعتمادًا على التغيير في ترتيب التمايز الجزئي في جميع الأجزاء الممكنة من الجسم، أي الدوران الجهازي وحجرات الأنسجة. أيضا ، تم تعيين قيمة معلمة عددية لمعدل تدفق الدواء في مقصورات مختلفة لتقدير تركيز الدواء. تم حساب النتائج وتصوير الأرقام باستخدام برنامج MATLAB (الإصدار R2020a). التأثيرات الرسومية الموضحة للتغير في معدل التركيز بافتراض قيم وسيطة مختلفة وفقا للمشتقة الكسرية (نموذج كابوتو ). التأثيرات الرسومية الموضحة للتغير في معدل التركيز بافتراض قيم وسيطة مختلفة وفقا للمشتقة الكسرية (نموذج كابوتو). يخلص التمثيل البياني الناتج إلى أنه بالنظر إلى ترتيب قيم المعادلات التفاضلية ، يختلف تركيز الدواء اعتمادا على معدل الثوابت في المقصورات المتعلقة بالوقت.   النظر في الحالة الأولية للتقدير التقريبي حيث يشير الجسم كحجرة كاملة، بعد تقسيم الجسم إلى مقصورتين نموذجيتين. في حين أن النموذج الأول يمثل المعدة والكبد والدم الجهازي ؛ والنموذج الثاني يأخذ في الاعتبار الدم الشرياني وأنسجة الكبد والدم الوريدي.The aim of this study is to provide an overview of various models to study drug diffusion for a sustained period into and within the human body. Emphasized the mathematical compartment models using fractional derivative (Caputo model) approach to investigate the change in sustained drug concentration in different compartments of the human body system through the oral route or the intravenous route. Law of mass action, first-order kinetics, and Fick's perfusion principle were used to develop mathematical compartment models representing sustained drug diffusion throughout the human body. To adequately predict the sustained drug diffusion into various compartments of the human body, consider fractional derivative (Caputo model) to investigate the rate of concentration changing depending upon the change in the order of fractional differentiation in all the possible compartments of the body, i.e., systemic circulation and tissue compartments. Also, assigned a numerical parameter value to the rate of drug flow in different compartments to estimate the drug concentration. Results were calculated and figures were depicted by using MATLAB software (version R2020a). Illustrated graphical effects of change in concentration rate by assuming various intermediate values according to the fractional derivative (Caputo model). The resultant graphical representation concludes that considering the order of the differential equation values, the drug concentration varies depending upon its rate of constants in compartments concerning time. Considering the initial case for rough estimation where the body is indicated as a whole compartment, following division of the body into two model compartments. Whereas, the model I represents stomach, liver, and systemic blood, and model II consider arterial blood, liver tissue, and venous blood

Fractional order compartment models

Compartment models have been used to describe the time evolution of a system undergoing reactions between populations in different compartments. The governing equations are a set of coupled ordinary differential equations. In recent years fractional order derivatives have been introduced in compartment models in an ad hoc way, replacing ordinary derivatives with fractional derivatives. This has been motivated by the utility of fractional derivatives in incorporating history effects, but the ad hoc inclusion can be problematic for flux balance. To overcome these problems we have derived fractional order compartment models from an underlying physical stochastic process. In general, our fractional compartment models differ from ad hoc fractional models and our derivation ensures that the fractional derivatives have a physical basis in our models. Some illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided

A general framework for fractional order compartment models

Compartment models are a widely used class of models that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments. Classic examples include SIR models in epidemiology and many pharmacokinetic models used in pharmacology. These models are formulated as sets of coupled ordinary differential equations, but in recent years there has been increasing interest in generalizations involving fractional differential equations. The majority of such generalizations have been performed in an ad hoc manner by replacing integer order derivatives with fractional derivatives. Such an approach does allow for the incorporation of history effects into the models, but may be problematic in a number of ways, such as breaking conservation of matter. To overcome these problems we have developed a systematic approach for the inclusion of fractional derivatives into compartment models by deriving the deterministic governing equations from an underlying physical stochastic process. This derivation also reveals the connection between these fractional order models and age-structured models. Unlike the ad hoc addition of fractional derivatives, our approach ensures that the model remains physically reasonable at all times and provides for an easy interpretation of all the parameters in the model. Illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided

An explicit numerical scheme for solving fractional order compartment models from the master equations of a stochastic process

We derive the generalized master equations for a stochastic process representing populations entering, leaving, or waiting in compartments at discrete times. This discrete time compartment model limits to a fractional order continuous time compartment model for a particular choice of waiting time distribution and the appropriate limiting process. We demonstrate that the discrete time master equations can be used to provide an explicit numerical method that can be employed for solving the fractional order compartment equations. The advantage of this approach is that the numerical scheme has a physical interpretation, it is stable, and it is easy to implement. The method can be applied to a wide class of fractional order compartment model equations that arise in a broad range of applications