Pharmacokinetic Simulations Involving Convolution Approaches

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Abdul Hakim Abdullah Ahmed Khaled

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Mathematics

May 2014

Department of Mathematics The Islamia University of Bahawalpur

Bahawalpur 63100, Pakistan

Pharmacokinetic Simulations Involving Convolution Approaches

Abdul Hakim Abdullah Ahmed khaled

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in

Mathematics

May 2014

Supervised by

Dr. Khalid Pervaiz Akhter Dr. Ghulam Murtaza

Department of Mathematics The Islamia University of Bahawalpur, Pakistan

Bahawalpur 63100, Pakistan

In The Name Of Allah, The Most Beneficent, The Most Merciful

Dedicated To

My respectful mother, my lovely wife, sons, and my daughter, who always pray, love, support and encourage me.

Approval

It is hereby certified that the work presented by Abdul Hakim Abdullah Ahmed Khaled S/O

Abdullah Ahmed in the dissertation entitled “Pharmacokinetic Simulations Involving

Convolution Approaches” has been successfully carried out under our supervision in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, at the

Department of Mathematics, Faculty of Science, The Islamia University of Bahawalpur,

Pakistan.

[email protected] Mobile: +92-3468783832

[email protected] Mobile: + 92-3142082826; fax: + 92-62925556

III

Acknowledgement

All praises and gratitude to my Lord Allah, the most gracious, the most merciful for guiding me out of darkness, helping in difficulties and bestowing upon me with the courage to accomplish this dissertation. All respect to the holy prophet Muhammad peace be upon him for showing the right path to humanity.

I wish to express my profound gratitude to my supervisors, Dr. Khalid Pervaiz Akhter,

Associate Professor, and Dr. Ghulam Murtaza, Assistant Professor, for their professional guidance, help, and scientific advice.

I would like to express my sincere thanks to Prof. Dr. Tahir Mahmood, Chairman Department of Mathematics, The Islamia University of Bahawalpur, whose guidance, advice, and great help during the different phases in my study. Hearty appreciations to my teachers, and class fellows,

I would like to thank my dearest friends, Dr. Abdul Gaffar, Dr. Aqeel Khan, Saddam Al-Adwar, and Amir Ahmed who never failed in helping me whenever I needed their assistance. My last few words devoted to my family: Mother, brothers and sisters, to my wife I would like to say that

I owe my success to her moral support. My deepest gratitude and thanks to the two expert evaluators: Prof. Dr. Muhammad Saleem, Department of Mathematics, San Jose State

University, U.S.A. and Prof. Dr. Doron Levy, Department of Mathematics, University of

Maryland, U.S.A. who have accepted to evaluate my thesis. I am very thankful to external examiners Prof. Dr. Muhammad Ozair Ahmed, Chairman Department of Mathematic, UET

Lahore, and Dr. Muhammad Nawaz Naeem, Associate Professor, Department of Mathematics, the University of Agriculture, Faisalabad , all of your notes and highlights will be really helpful and worth to be added. Finally, thanks to the Amran University which sent me to Pakistan in order to study for a Ph. D. the peace and the safe to my beloved homeland Yemen.

List of symbols and abbreviations

PK Pharmacokinetic

ADME Absorption, Distribution, Metabolism, Excretion

BCS Bio-pharmaceutics classification system

FDA Food and drug administration

USP United States Pharmacopeia

AUC Area under the concentration time curve

퐴푈퐶0−∞ Total area under the curve

퐴푈퐶0−푡 The area under the plasma drug concentration-time curve from zero to t

퐶푚푎푥 Maximum concentration

IVIVC In vitro in vivo correlation

IVIVP In vitro to in vivo profiling

∗ Symbol of convolution

∕∕ Symbol of deconvolution

푘푎 First – order absorption rate constant

푘푒푙 First – order elimination rate constant

푘12 Distribution rate constant for transfer of drug from compartment one to two

푘21 Distribution Rate Constant for transfer of drug from Compartment two to one

UIR Unit impulse response (퐶훿)

푇푚푎푥 Time needed to reach maximum blood drug concentration

PH Potential of Hydrogen

SD Standard deviation

SPSS Statistical Product and Service Solution

ANOVA Analysis of variance

IRF Input response function

퐹푇 Fraction of absorption at any time T

푓2 Similarity factor

(퐶푃)0 Concentration at time zero

퐾퐻 Higuchi model

Γ(. ) Gamma function

VII

Abstract

This thesis is a part of a research project on pharmacokinetic modeling of enteric coated microparticulate formulations of Metoprolol tartrate. First part of this study dealt with pharmaceutical aspects i.e. formulation development, while second part of study dealt with mathematical modeling of pharmacokinetics. Firstly, this study was aimed to develop in vitro in vivo correlation (IVIVC) level A, B and C for encapsulated Metoprolol tartrate (T1, T2 and

T3 having Metoprolol tartrate/polymer ratio of 1:1, 1:1.5 and 1:2 by weight/weight). The in vitro data was correlated with in vivo data. For IVIVC level A, drug absorption data was calculated using Wagner-Nelson method. In addition, convolution approach was used to approximate plasma drug levels from in vitro dissolution data. The coefficient of determination (R2) for level

A was 0.720, 0.905, 0.928 and 0.878 for Mepressor®, T, T2 and T3 formulations, respectively with acceptable percent error (<15%). The value of (R2) for level B and C was 0.231 and 0.714, respectively. It is also concluded that IVIVC level A is a proficient mathematical model for biowaiver studies involving study parameters as those implemented for T1S (T1 formulation tested for dissolution in the presence of sodium lauryl sulphate) revealing that IVIVC level A is dosage form specific, rather than to be drug specific. Secondly, the aimed of this study was to assess and apply the in vitro to in vivo profiling (IVIVP), a new biowaiver approach, in designing a product with specific release pattern. The IVIVC was established by plotting the observed and predicted plasma drug concentrations. For IVIVC, convolution approach was employed to estimate plasma drug concentrations from in vitro dissolution profiles. The IVIVC for T1S exhibited a good correlation coefficient (R2 = 0.963) followed by the T2 (R2 = 0.682),

T3 (R2 = 0.665), T1 (R2 = 0.616), and Mepressor® (R2 = 0.345). Establishing an IVIVP, based on

VIII the convolution approach, can be more useful and practicable in the biowaiver studies, rather than present complicated practice of IVIVC estimated via deconvolution approach. This study also elaborates that there is good correlation between the IVIV profiles of the developed

Metoprolol tartrate formulations, particularly for T1S.

Keywords: Metoprolol tartrate, Eudragit® FS, Microparticles, Convolution, Deconvolution,

IVIVC, IVIVP.

List of content

Serial No. Title Page No. Title page Bismillah Dedication I Declaration II Approval III Certificate V Acknowledgement IV List of abbreviations VI Abstracts VIII List of contents X List of tables XII List of figures XII Chapter 1 Introduction 1 1.1 Introduction 2 1.2 Basic definitions 5 1.3 Convolution /Deconvolution 10 1.4 Pharmacokinetic 18 1.4.1 Absorption 21 1.4.2 Distribution 22 1.4.3 Metabolism 22 1.4.4 Excretion 23 1.5 Simulation 23 1.6 Administration drugs and pharmacokinetic process 24 1.7 Classifications of drugs 25 1.7.1 BCS class I drugs 25

1.8 Metoprolol 26 Chapter 2 Literature review 27 2.1 Introduction 28 2.2 Pharmacokinetics 29 2.2.1 Pharmacokinetic modeling 32 2.2.2 Compartment models 37 2.2.2.1 One - compartment model 40 2.2.2.1.1 One - compartment model intravascular administration 40 2.2.2.1.2 One – compartment model extravascular administration 42 2.2.2.1.3 Application of one compartment model (Wagner – Nelson method) 49 2.2.3 Two – compartment model 51 2.2.3.1 Analytical solution for two compartments 54 2.2.3.2 Application of two - compartment model (Loo-Riegelman method) 57 2.2.4 푛- compartment Analysis 68 Chapter 3 Convolution/Deconvution and their applications 63 3.1 Introduction 64 3.2 Linear system analysis 65 3.2.1 Definition of convolution 66 3.2.1.1 Properties of convolution 67 3.2.1.2 Definition of Dirac Delta function 68

3.2.1.3 Properties of the Dirac Delta function 68

3.2.1.4 Properties of convolution of derivative 69 3.2.1.5 Convolution of Properties of integration 70 3.3.2 Definition Delta function (Kronecker function) 70

3.3.2.1 Properties of Delta function 훿 70 3.4 Convolution modeling and application 71 3.4.3 Plasma input by using convolution 76 3.6 Deconvolution modeling 77 3.6.1 Methods of deconvolution 80

Chapter 4 Materials and methods 90 4.1 Preparation of investigational tablets 91

4.1.1 Compatibility analysis 92

4.2 Drug release kinetics 94

4.3 Convolution of in vitro dissolution data to approximate plasma 96

4.4 Experimental protocols for in vivo study 103

4.5 Deconvolution of in vivo data IVIVC development 103 4.5.1 Computation of absorption data and IVIVC development 106 4.6 Mathematical and statistical analysis 106 Chapter 5 Results and discussion 107 Results and discussion 108 5.1 Compatibility analysis 108 5.2 Drug Release kinetics 110 5.3 In vivo study 112 5.4 Development of in vitro in vivo correlation 113 5.5 Development of in vitro in vivo profiling 120 Conclusion 122 References 125 Original publications from thesis 138

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List of tables

Serial No. Title Page No. 3.1 Convolution integral for different functions 88 4.1 Convolution of dissolution data Mepressor® of formulation 99 4.2 Convolution of dissolution data of formulation T1 100 4.3 Convolution of dissolution data of formulation TIS 101 4.4 Convolution of dissolution data of formulation T2 102 u 4.5 Convolution of dissolution data of formulation T3 103 5.1 Pharmacokinetic parameters for all tablets obtained from in vivo 115 experiments and convolution method 5.2 In vitro in vivo correlation data 120 5.3 In vitro to in vivo profiling data 120

List of figures

Serial No. Title Page No. 2. a Drug removed from the body 32 2. b Scheme of one –compartment model 45 2. c Scheme of one-compartment model 47 2. d Scheme of one –compartment model 48 2.2.a Scheme of two-compartment model 54 5.1 Predicted plasma drug concentration (in ng/ml) and dissolution 112 profiles (in percentage) 5.2 In vitro in vivo correlation level A for formulations Mepressor® 117 (A), T1 (B), T2 (C), T3 (D) and T1S(E) 5.3 In vitro in vivo correlation level B 118 5.4 In vitro in vivo correlation level C 119 5.5 In vitro to in vivo profiling for formulations T1 (A), T1S (B), T2 124 (C), T3 (D) and Mepressor®(E)

XIII

Chapter 1 IInnttrroodduuccttiioonn

1.1. Introduction

This work is focused on model-independent methods (convolution∕deconvolution) as well as model-dependent methods. For computation processes of pharmacokinetic in linear systems the convolution operation calculates response functions using input and output functions.

The convolution operation and its inverse deconvolution can achieve algebraically using Laplace transforms [12]. They studied time courses of drug metabolism and concentration in biological fluids and tissues. Convolution and deconvolution are the classical in vitro in vivo correlation tools in vitro in vivo correlation to describe the relationship between input responses in a linear system, where input represents the in vitro drug release. Convolution and deconvolution are normal mathematical tools for the analysis of linear systems, based on the validity of the superposition principle. In dependent method based on one compartment model with first order absorption is the application of regular residual methods (Gibadiet et al.1982, 1987; Garrett,

1993). While for two compartments method discussed by Loo-Riegelman, 1968. In this method we can compute fraction absorption. In general convolution is defined as the integral of the product of two functions, (Bracewell, 1986). According to the convolution theorem, it is mapped as multiplication in the frequency domain [40]. Absorption kinetics was studied by (Vaughan,

Dennis, 1978, Pedersen, 1980), while pharmacokinetic is dedicated to the study of the time course of substances and their relationship with an organism and describes the concentration of such substances into the organism over the time. The primary goal of pharmacokinetic is to generate parameters that are mathematical abstraction physiological processes and can be used as an aid for the best understanding drug disposition. Mathematical modeling generates parameters that may vary as the physiology varies as results of concentration in the body [50]. Usually in

vivo absorption or dissolution a mathematical model based on the convolution integral is used, a general procedure for assessing absorption kinetics of a drug is the Wagner-Nelson method. It can be used when a one-compartment model of the body applies, but a modified form of it is also useful for models with more than one compartment. It is based on the concept that the observation after an extra vascular dose must be a convolution of both the input and disposition of a drug. If the disposition function is known (obtained after an intravenous bolus dose), then by deconvolution the input function can be obtained. A possible solution is to use an estimate of elimination rate constant in the subject, obtained following administration of a rapid-release dosage form on a separate occasion. The drug in the body and the cumulative amount of the drug eliminated used Wagner-Nelson,1964 assuming one compartment. Loo-Riegelman1968, introduced a method fordetermining the fraction of absorption of the drug by usingatwo compartment model. There are three important differences compared to the previous Wegner-

Nelson method. First, the very important peripheral compartment(s) may be included. Second, the model is forced to distinguish between the rate constant for the drug disappearance from the body and the specific elimination constant. Third, the volume constant used to convert the amount to concentration is the volume of the central compartment rather that the apparent volume of distribution, as long as the absorption and elimination are taking place in the central compartment. For instance, Wagner, 1983 extended the Loo-Riegelman equation to more than two-compartment open model with first order absorption and elimination via the central compartment. The absorption kinetic based on a deconvolution method as pharmacokinetic modeling refers to one or more compartments, the operational principle of the convolution is based on molecular stochastic independence that exhibits the linearity between the kinetic response variables with linear superposition. It is a mathematical operation on two functions,

an input function and a unit impulse response function, producing a response function.

In principle the deconvolution through a convolution iterative procedure consists of three steps.

These steps are repeated until the objective function is optimized [21]. This dissertation is distributed as follows:

In Chapter 1, a brief introduction is given to convolution and deconvolution terms, basics definition, simulation, classification of drugs in BCS, the routes of administration of drugs as well as the processes of pharmacokinetics. In Chapter 2, a brief pharmacokinetic history, and discuss two pharmacokinetic one compartment and two compartments models like Wagner-

Nelson & Loo-Riegelman, as well as non-compartment models. The aim of this chapter is to introduce the concept of pharmacokinetic modeling and to bring in the typical structure of pharmacokinetics compartment models. Essentially focused on two compartment models based on linear differential equations with either intravenous injection or oral absorption of a drug, in this approach, one compartment describes the blood and the other is associated with tissue.

In Chapter 3, we give a brief introduction of convolution history, properties of convolution.

In Chapter 4, constitutes the material and methods used, in Chapter 5, included the results discussion, and conclusion.

1.2. Basic definitions

ADME: ADME is an acronym in pharmacology for four simultaneously occurring processes; absorption, distribution, metabolism, and excretion. The four processes, working together, are responsible for blood concentrations of a drug within the body.

Absorption: Absorption is the process of releasing a drug from its dosage formulation and transferring into the bloodstream.

Distribution: Distribution is the movement of a drug in the blood stream it enters the circulation and is carried throughout the body.

Metabolism: Metabolism is a term that is used to describe all chemical reactions involved in maintenance of living state of the cells and the organisms.

Excretion: Excretion is the process of eliminating a drug or its metabolites from the body.

The major organ of excretion is the kidney.

Bioavailability: The term bioavailability can be defined as the rate and extent to which the active drug component or curative is absorbed from a drug product and becomes obtainable at the site of drug action.

Bioequivalence: The term bioequivalence can be defined as the absence of a significant difference in the rate and extent to which the active ingredient or active pharmaceutical equivalents becomes available at the site of drug action when administered at the same molar dose under similar conditions in an appropriately designed study.

Biowaiver: The term biowaiver refers to the regulatory acceptance of an in vitro dissolution testing as a credible replacement for an in vivo bioequivalence study.

Dissolution: The term dissolution refers to the process of dissolving a solid substance into a solvent to make a solution.

In vitro: The term in vitro refers to experimentation outside a living organism.

In vivo: The term in vivo refers to experimentation using a whole living organism.

In vitro in vivo correlation (IVIVC): The IVIVC can be defined as a predictive mathematical model describing the relationship between an in vitro property of an extended release dosage form and a relevant in vivo response [116].

Level A IVIVC: A predictive mathematical model for the relationship between the entire in vitro dissolution time course and the entire in vivo response time course, and represents a point- to-point relationship between in vitro dissolution rate and in vivo input rate of the drug from the dosage form. Percent of drug absorbed calculated by model dependent techniques such as

Wagner-Nelson procedure or Loo-Riegelman.

Level B IVIVC: A level B IVIVC utilizes the principles of statistical moment analysis.

In this level of correlation, the mean in vitro dissolution time of the product is compared to either mean in vivo residence time or the mean in vivo dissolution time.

Level C IVIVC: In this level of correlation, one dissolution time point (t50%, t90%, etc.) is compared to one pharmacokinetic parameter such as AUC, 푡푚푎푥 or 퐶푚푎푥 [65].

Similarity factor풇ퟐ: We can define a similarity factor as a "logarithmic reciprocal square root transformation of one plus the mean squared (the average sum of squares) differences of drug percent dissolved between the test and the reference products" [88].

Impulse response It is the response of linear time invariance system to a unit impulse function applied at time zero.

Superposition: The superposition is a term that can refer to the superposition integral or sum, which is identical to the convolution integral or sum except that it refers to the response of a linear time invariance system, or can be applied to the property of superposition that defines a linear system. These two rules taken together are often referred to as the principle of superposition. Mathematically, the principle of superposition is expressed as:

푙 훼푥 + 훽푦 = 훼푙 푥 + 훽푙(푦). Homogeneity is a special case in which one of the signals is absent. Additively is a special case in which 훼 = 훽 = 1

Linear operator: A linear operator is a mathematical operator with the following property

푙 휇1푥1 + 휇2푥2 = 휇1푙 푥1 + 휇2푙(푥2), where 푥1 and 푥2 are two arbitrary variables where

휇1 and 휇2 are two arbitrary constants. A linear combination of linear operation is simplified as a linear operator, 휇1푙1 푥 + 휇2푙2 푥 + … + 휇푛 푙푛 푥 = 푙 푥 , for all values of linear constants,

휇1 , 휇2 , … ., 휇푛

Linear drug disposition: A linear drug disposition is defined as a linear relationship between the input of a drug into the systemic circulation and the resulting measured concentration of drug in the systemic circulation.

Mean residence time (MRT): The mean residence time (MRT) of drug molecules in a kinetic space is the mean total time which spent in the kinetic space. If all the molecules enter the kinetic space at the same time (푡 = 0) and the drug leaves irreversibly from the kinetic space, then mean total time (MTT) and MRT are equal and are given by

∞ 푀푅푇 = 푀푇푇 = 퐹 푡 푑푡 (1.1) 0 where 퐹(푡) is the fraction of the molecules remaining in the kinetic space at time 푡. If the molecules can return to the kinetic space, then MRT is still obtained by the same formula but 7

MTT will not be equal to MRT, if f (t) denotes an arbitrary rate of elimination of the drug molecules from a kinetic space where the drug molecules are all introduced at 푡 = 0 then:

∞ ∞ ∞ 푡 ∞ ∞ ∞ 푀푅푇 = 퐹 푡 푑푡 = 푓 푡 푑푡 − 푓 푡 푑푡 푑푡/ 푓 푡 푑푡 = 푡푓 푡 푑푡/ 푓 푡 푑푡 0 0 0 0 0 0 0

If the rate of elimination, 푓(푡) is proportional to the measured concentration,

∞ where 퐴푈푀퐶 = 푡퐶(푡)푑푡 0

The MRT of the kinetic is equal to the sum of the MRT of the mutually exclusive kinetic [56].

Mean arrival time (MAT): The mean arrival time of drug molecules entering a kinetic space is the average time taken by the molecules to arrive in the kinetic space. The mean arrival time for the absorption MAT (absorption), is the average time it takes the bio-available drug molecules after entering the absorption space to enter the disposition space. The MAT (absorption) can generally be calculated from the absorption rate 푓(푡), according to the formula

∞ ∞

푀퐴푇푎푏푠표푟푝푡푖표푛 = 푡푓 푡 푑푡 ∕ 푓 푡 푑푡 0 0

If 푓푖푛 (푡), denotes the rate input of drug kinetic space and 푓표푢푡 (t) denotes the rate of irreversible elimination from that space, then the MRT of the drug in the kinetic space is

∞ ∞ ∞ ∞

푀푅푇 = 푡푓푖푛 푡 푑푡 ∕ 푓푖푛 푡 푑푡 − 푡 푓표푢푡 푡 푑푡 ∕ 푓표푢푡 푡 푑푡 0 0 0 0

MRT relationships are also readily derived from linear input response relationships from the fundamental convolution relationship involving the measured concentration 퐶(푡), the rate absorption 푓(푡) and the unit impulse response: 퐶(푡) = 푓(푡) ∗ 푈퐼푅(푡) using the LapLace transform technique it can be shown that:

∞ ∞ ∞ ∞ ∞ ∞ 푡퐶 푡 푑푡 ∕ 퐶 푡 푑푡 = 푡푓 푡 푑푡 ∕ 푓 푡 푑푡 + 푡푈퐼푅 푡 푑푡 ∕ 푈퐼푅 푡 푑푡 0 0 0 0 0 0

1.3. Convolution/Deconvolution

The convolution of two functions represents the amount of overlap between the two functions where one function is input and other is called kernel of convolution, which are often used for filtering [90].

Convolution plays an important role for computation of pharmacokinetics processes.

This technique has been employed to prognosticate drug level profiles from the in vitro data, the convolution derived from the numerical method and then applied to the dissolution data; Huag’s paper (1982) used the convolution simulation [48]. Moreover, convolution techniques which allow the temporal delay to be characterized by poly-exponential [7]. The convolution and deconvolution is partially based on the application of software [39] that is designated for numerical convolution [41]. Convolution and deconvolution are the classical IVIVC tools to describe the relationship between input and unit impulse response in a linear system, where input represents the in vitro drug release [41]. The convolution method described by [38], was elaborated by several authors, and first published their method in 1980. Convolution is often used to find the output of a linear time-invariant system to an input signal as well as to use it in pharmacokinetics processes. Convolution is a mathematical time integration operation that gives the result of an input function combining with a system’s impulse response to obtain its output.

The convolution is the process to predict drug input and parameters describing drug dissolution and∕or elimination are combined in to get predicted time courses [69]. Recently convolution methods have been established which convolute the in vitro dissolution profiles without implementing the correlation of the in vivo absorption/dissolution profile with the in vitro dissolution profile. A major advantage of convolution methods for IVIVC is that no additional in vivo data such as intravenous injections or oral solutions are required. The typical process of developing IVIVC involves the in vivo plasma profile to estimate the in vivo release, followed by comparison of the in vivo fraction of drug absorbed to the in vitro fraction of drug dissolved.

The weighting function represented by the average cumulative urinary excretion profile observed after oral administration of oral solution. The characteristic function, 훼(푡) which describes the disposition after intravenous bolus administration, and the output function 퐶(푡), which is describing drug input as a function of time after administration drug this technique is also called superposition, integration and uses the observed profile and the characteristic profile to determine the input profile using the relations 퐶 푡 = 훼 푡 ∗ 훽훿 푡 (Convolution technique) while if using the relation 훽훿 푡 = 퐶(푡) ∕∕ 훼(푡) ( Deconvolution ). For a linear system, the system input-output relationship can be described by the convolution integral. We can describe this scheme by the figure.

훼 푡 퐶 푡 = 훼 푡 ∗ 훽훿(푡)

Weighting function (훽훿(푡)) Input function (Linear system) Output function

The weighting function may be a physical system that transforms an input into an output, the input and the output are given by continuous or sectional continuous functions. The disposition

function can be obtained from an intravenous administration and the input function corresponding to oral administration or topically is obtained following an oral administration.

Convolution plays a role when the tracer is administered over an extended time and the measured uptake is considered as a sum of time shifted exponential functions. The convolution method introduced by [92], fits the immediate release plasma concentration time profile. Gaynor, Dunne et al., 2007 demonstrated the unbiased prediction of the summary measures AUC and 퐶푚푎푥 for the two-stage convolution method. Convolution method is a mathematical approach which is very efficiently employed to extract drug concentration in blood from in vitro dissolution profiles. A typical application of poly-exponential convolution is the prediction of in vivo response for hypothetic product formulations, where the response to an oral solution is expressed as a poly-exponential, the same way as all in vitro release profiles [9, 12]. Deconvolution is a process of solving convolution integral for either input rate of drug 훼(푡) or impulse response

훽훿(푡) when one is known and the response to input 퐶(푡) is also known. Usually, deconvolution is used to solve for input rate in order to characterize the rate of drug absorption, especially when oral control release dosage forms are involved. This approach allows comparison of the absorption rate portion of bioavailability for formulations given by the same route of administration. In brief deconvolution is the mathematical process employed to find the in vivo drug input function that balances an observed response to drug input with a characteristic degradation or elimination function unique to a drug pair [63, 69]. In general the convolution approach is a simple, accurate, efficient and scientifically valid method for predicting concentration-time profiles for the development and evaluation of pharmacokinetic [72].

The best approach to establish the process of drug absorption is to analyze the plasma concentration time profile obtained after oral administration using a deconvolution technique.

Convolution and deconvolution are powerful tools to optimize drug formulation [81, 82].

Impulse response function that describes the cumulative effects of response changes, following all drugs input at time푡. Deconvolution is a process of solving the convolution integral for either input rate of drug or impulse response when one or the other is known and the response to input is also a known. Input rate function which describes the time-course of in vivo absorption rate of drug from the gastrointestinal tract. For controlled release formulations, we assume that absorption is a dissolution rate limited. Impulse response was approximated by exponential function with a literature-derived elimination rate constant as the exponential factor. Integration of in vivo absorption rate functions obtained from deconvolution give cumulative amounts of drug absorbed, which is in vivo dissolution when absorption is limited by dissolution rate. In brief the convolution is the mathematical process employed to find the in vivo drug input function that balances an observed response to drug input with a characteristic elimination function [78]. The system can be described by static and dynamic characteristic functions; the static characteristic function expresses the relationship between the system input and output in steady state. The elimination of an oral solution or immediate release formulation may be determined by Chan (1987). The drug can be described by three functions: Input function, impulse unit response function and output function, by knowing two of these three functions, the respective third can be calculated. Concentration time profile 퐶푝(푡) can be determined by taking an oral formulation, as follows:

푡

퐶푝 푡 = 훼 푡 ∗ 훽훿 푡 = 훼 푡 − 휏 훽훿 휏 푑휏 (1.2) 0

They are known in transform theory as convolution integrals and in classical mathematics as

Duhamel integrals, where ∗ is symbol of convolution 퐶푝 푡 is output function plasma 12

concentration of drug, 훼(푡) is input function and 훽훿 (푡) is impulse response drug elimination.

The intravenous administration is most appropriate because this type of drug delivery does not release or absorption of the action substance receives in the response administration [59].

푡 푡 퐷 퐷 −푘푒푙 (푡−휏) −푘푎 휏 −푘푒푙 푡 (푘푒푙 −푘푎 )휏 퐶푝 푡 = 푒 푘푎 푒 푑휏 = 푒 푒 푑휏 푉푑 푉푑 0 0

퐷푘푎 1 퐷푘푎 = 푒−푘푒푙 푡 푒(푘푒푙 −푘푎 ) − 1 = (푒−푘푎 푡 − 푒−푘푒푙 푡 ) (1.3) 푉푑 푘푒푙 − 푘푎 푉푑 푘푒푙 − 푘푎

where 퐷 is dose, 푉푑 is volume of distribution, 푘푎 is absorption rate constant, and 푘푒푙 is eliminate rate constant. This equation is called Batman function. The plasma concentration considered to be the convolution of a mono-exponential or bi-exponential absorption function with a bi- exponential unit disposition function,

−훼푖푡 푈푅퐹 = 퐵푖푒 (1.4) 푖=1

where 푈푅퐹 is unite response function. The input rate function 퐼푟 for the mono-exponential

−푘푎 푡 absorption model is given by Ir = D. F. 퐾푎 푒 , where 퐷 is the dose, 퐹 is the fraction absorbed

(bioavailability), and 푘푎 is the first-order absorption rate constant; the input rate for the bio- exponential absorption model is given by

−푘푎1푡 −푘푎2 퐼푟 = 퐷. 퐹 푏퐾푎1푒 + (1 − 푏)푘푎2푒 .

where 푘푎1 and 푘푎2 are first order absorption rate constants, 푏 is the proportion of the bioavailable dose absorbed by 푘푎1 and (1 − 푏) is the proportion of the bio-available dose absorbed by 푘푎2.

The convolution of these input functions with bi-exponential disposition function yielded

equations which describe the plasma concentrations over time after IM injection for the mono-exponential and bi-exponential absorption models, respectively.

2 퐵푖 −훼푖푡 −푘푎 푡 퐶푝 푡 = 퐷. 퐹. 푘푎 . 푒 − 푒 (1.5) 푘푎 − 훼푖 푖=1

2 푘푎1. 푏. 퐵푖 푘푎1. 1 − 푏 . 퐵푖 −훼푖푡 −푘푎1푡 −훼푖푡 −푘푎2푡 퐶푝 푡 = 퐷. 퐹. ( 푒 − 푒 ) + 푒 − 푒 (1.6) 푘푎1 − 훼푖 푘푎2 − 훼푖 푖=1

The parameter F is not identifiable and is incorporated into the models as 퐹. 퐵1 and 퐹. 퐵2.

The simplest is one-compartment open model with drug being directly introduced in the circulatory system by an intravenous administration and was first published in 1967, by Smith.

In the numerical deconvolution approach the drug unit impulse response function describes distribution and elimination phases respectively. In a convolution approach the parameters estimated would be those related to the characteristic response [20]. Deconvolution method used a numerical stepwise integration this method was first described by Turner and colleagues in

1971, where they used their technique to investigate first-phase insulin delivery to the systemic circulation. Mathematically, change can be expressed as a differential equation [12] therefore, the concentration-time profile can be described by a differential equation. The tools for numerical deconvolution have the possibility of obtaining and calculative absorption curves of plasma level curves in the mean time of tool IVIVC studies [40], other possible procedures are based on transformation techniques, like Laplace transforms. It has been shown that it is possible to determine the impulse/response function of individual parts of flow-injection systems by a mathematical deconvolution [40] including Bayesian approaches to assist the estimation parameter [13]. Numerical deconvolution of the simulated serum level curves was accomplished

using a program. Independent of the deconvolution approach under adoption the two main pieces of information required to estimate the system input via deconvolution are the samples of the output and the system kernel [8]. The deconvolution technique is a powerful mathematical operation that can be used to extract the impulse response function of a linear time-invariant system. Turner used a deconvolution technique requiring known half –life for insulin removal, this concepts studied by [9]. Also several deconvolution techniques have been considered for biomedical systems, Cutler, 1978; Pedersen, 1980; Hovorka, 1998 and this method can be used successfully to provide point estimates for immeasurable quantities in patients. Furthermore, they assumed a known functional form for the input function (e.g. polynomial Cutler, 1978 or quadratic Hovorka, 1998) which may be appropriate for the drug synthesis but is not applicable for the in vivo synthesis of antibody proteins. This method was used to estimate the insulin secretion in fasting rats, enabling a more improved marker of insulin secretion to be defined

Watson, 2007. The discrete deconvolution of the piecewise function approximation used by

Sparacino, 2001, the deconvolution problem has been drawing the attention of mathematicians, physicists and engineers since the early sixties. Model-independent deconvolution type requires in vivo plasma data from an oral solution or intravenous impulse function for the application.

Deconvolution requires data obtained after both oral and intravenous administration in the same subject and assumes no differences in the pharmacokinetics of drug distribution and elimination from one study to the other [10]. The deconvolution technique can be applied to estimate systemic bioavailability in linear pharmacokinetic systems given the time-concentration profile following an intravenous bolus dose, and the venous time-concentration profile following the oral dose [11], and it is estimation of the time course of drug input, usually in vivo absorption or dissolution using a mathematical model based on the convolution integral. A typical application

of poly-exponential convolution is the prediction of in vivo response for hypothetic product formulations, where the response to an oral solution is expressed as a poly-exponential, and the same way as all in vitro release profiles. In deconvolution approach hypothetical drug release profiles were calculated by numerical deconvolution from the urinary excretion data obtained after per-oral administration [12, 43,51]. Deconvolution approach represents a valuable tool for the identification of drug products in vivo dissolution kinetics (Hanano, 1967; Langenbucher,

1982; Gillespies and Veng Pedersen, 1985; Niclasson et al. 1987), to separate drug input from drug distribution and elimination [10, 11]. Deconvolution of the subcutaneous data using prescribed input functions , namely rectangular pulse functions of variable duration mono and bi- exponential input functions Cutler, 1978, indicated a mono-exponential input function best described absorption from the injection site judged by the criterion of Imbimbo 1989 [12,44], however, Moate used deconvolution method to estimate C-peptide secretion in humans to improve the performance of flow injection analysis by deconvolution of experimental data.

Deconvolution methods provide bioavailability data from drug concentrations measured only over the time of drug absorption. These methods have also been employed to assess the in vivo dissolution of oral dosage forms and to determine the rate and extent of absorption in vivo dissolution. A wide range of functional forms has been used in deconvolution procedures to estimate input functions for a large variety of biological systems [38, 39, 46]. However, no universal solution is possible for deconvolution in its most general formulation. In that sense, the mathematical formulation of the deconvolution problem, and the equation used in convolution is,

푛 푡푛 −푡푖−1 1 푑푋 퐶 = 퐶푑푡 (1.7) 표푟 퐷 푑푡 푖=1 푡푛 −푡푖

푑푋 at= 0, 푋 푡 = 0 표푟 = 0 푑푡 16

푑푋 where 퐶 concentration for oral administration predicted by simulation 퐷 is bolus dose , is 표푟 푑푡 rate of absorption, and 퐶 is concentration after bolus administration. The typical process of developing a Level A IVIVC involves deconvolution of the in vivo plasma profile to estimate the in vivo release, followed by comparison of the in vivo fraction of drug absorbed to the in vitro fraction of drug dissolved. The point-area numerical deconvolution method has been implemented and tested using data from oral absorption and to improve the performance of flow injection analysis, [17] computed in vitro-in vivo correlation based on convolution and deconvolution techniques, the IVIVC uses deconvolution followed by convolution to carry out predictions of plasma concentration [45,71]. The convolution approach assures direct correlation of dissolution data and pharmacokinetic parameters (퐶푚푎푥 , 푇푚푎푥 ). Predicting plasma concentration time course in a single step using convolution method is relatively simple when compared to the two stage methodology in deconvolution model [116].

1.4. Pharmacokinetics

Pharmacokinetics is not an easy field and it requires knowledge of mathematics, statistics, and computer programming. There are four major areas in pharmacokinetic: absorption, distribution, metabolism and elimination [87]. The term pharmacokinetics was first introduced by the German

Podiatrist F. H. Dost, 1953, who published and discussed it in 1961. So, Dost is considered the founder of term pharmacokinetics. He analyzed drug manners, and applied linear one compartment models to describe different drugs and derived several physiological characteristics. Later at the School of Pharmacy, United State of New York he made a number of important contributions to pharmacokinetics [12].

During (1973-1979) articles of pharmacokinetics literature grew at a very rapid rate, by the end of 1972, there were several journals which published pharmacokinetics articles.

In pharmacokinetic the data is analyzed by using a mathematical representation of a part or whole of an organism Wagner (1968). In pharmacokinetics, mathematical modeling is used to clarify the relationship between drug concentration in plasma and the drug concentration in affected site [13, 14]. The process in which pharmacokinetic parameter that describe the constant drug absorption, distribution and elimination that are not changed with the change in drug dose is known as linear pharmacokinetics. The elimination and distribution processes are called disposition. If the drug is well absorbed the high hepatic clearance leads to the low oral bioavailability [19]. Prediction of pharmacokinetic profile in humans based on in vivo or in vitro, which is a useful tool for drug discovery and development to identity compounds with attractive pharmacokinetic properties. In the second half of the 20th century biomathematics has emerged as a new tool to rationalize the complex biological methods based on their mechanism of action.

Presently bio-mathematicians are becoming an authentic in pharmacology specially those who are working on vehicle based drug delivery and release system [17, 18]. The application of mathematical modeling for some problems arising in medicine and biology is considered in this dissertation. However in the last forty years mathematical biology has become very popular so, it is not wrong to say that mathematics is a universal language. The assessment of bioavailability using convolution techniques and comparisons with conventional AUC methods has been presented [19, 20]. For many drugs, it has been observed that the rate of absorption, after extravascular administration is approximately proportional to the amount of drug remaining to be unabsorbed at the absorption site. A process in which the rate of reaction is proportional to the amount present is known a first order and represented by linear differential equation [21].

There is limited amount of enzyme present in the liver, where drug metabolism mainly takes place; therefore an increase in drug concentration has autonomic disposition kinetics, the rate of change of drug concentration [23, 24]. The process of drug transport is divided into the following stages which are connected with each other, transport through and out of the dosage form, transport from the gastric liquid to the blood and transport from the blood to the surrounding.

The Wagner-Nelson method is the most frequently used for estimation of absorption kinetics.

This method can be applied if the pharmacokinetic parameters for administration are known.

The researcher computes the drug concentration in plasma or blood and absorption of drug as well as disposition processes in the body at any time by using convolution techniques for simulation data, because the plasma concentration of the drug is the main focus in pharmacokinetics. In 2000, the FDA introduced regulatory guidance for BCS biowaivers; however fewer biowaiver-based new generic oral drug applications have been received. The

World Health Organization (WHO, 2006) has actively utilized BCS, biowaivers; the working group on the BCS of the International Pharmaceutical Federation (FIP) has produced a series of publications in the Journal of pharmaceutical Sciences U.S.A. The two-stage convolution-based method was compared with the novel one-stage convolution based method [30]. Once the compartments and interactions are defined it is possible through mass balance principles, to generate a system of differential equations mathematically describing the compartmental quantities over time. There are many approaches that have been opted for pharmacokinetic modeling, compartmental, physiological and model-independent method. Model-independent calculations represent pure statistical analysis that operates under no assumptions about compound compartments or the distribution of the drug in these compartments.

1.4.1. Absorption

Absorption is the process of drug movement from the site of drug administration to blood circulation. However drug absorption from the site of administration permits entry of the therapeutic agent into plasma. The absorption site or port of entry is determined by the route of drug administration. We know that the routes of administration are either extravascular or intravascular. Absorption of drug occurs along the gastrointestinal tract including the mouth, stomach, intestine and rectum. Also absorption depends on the physical characteristics of the drug. Almost 50 % of the drug that is absorbed from GI passes by the liver, the absorption through the skin depends on the characteristics of the drug and the condition of the skin. The rate of absorption is limited by the size and solution of the drug in the interstitial fluid. If the blood flow is more at the administration site, absorption will be high [50, 52]. Drug absorption from extravascular sites often occurs via first-order rate, which can be described by an absorption rate constant (ka) as long as the amount of drug in solution. In the case of oral drug dosing absorption is complicated by transit time through the gastrointestinal tract. The fraction absorbed or bioavailability (퐹) is determined by comparison with intravenous dosing where 0 < 퐹 ≤ 1.

One method to estimate the fraction absorbed using an appropriate technique such as Wagner-

Nelson procedure, Loo-Riegelman method or convolution techniques for each formulation and subject [34, 37].

1.4.2. Distribution

Distribution can be defined as the partitioning of drugs among various body compartments.

In other words drug distribution of drug refers to the movement of a drug from the blood and is often described in terms of perfusion or diffusion-limited tissue uptake. Drug movement into tissues depends on the movement of drug molecules from the blood into the tissue with factors of

facilitated transport and tissue properties effecting the rate and extent of drug uptake.

The important parameters of distribution are the rate and extent of distribution [35, 36], the extent of distribution depends on the physicochemical properties of the drug and physiological factors; the rate of distribution of a drug depends on blood flow and the rate of diffusion across cell membranes of various tissues and organs.

1.4.3. Metabolism

Metabolism is a term that is used to describe all chemical reactions involved in maintenance of living state of the cells and the organisms. The drug may be bio-transformed by metabolism in the liver or other tissues and is the enzymatic biotransformation of drugs. In metabolism some substances are broken down to yield energy for vital processes while other substances are formed to support the life of the cell. It is the chemical and physical processes in an organism by which protoplasm is produced and then decomposed to make energy available and it also refer to the entire chemical reactions by which complex molecules are taken into an organism are broken down to produce energy [3, 6].

1.4.4. Excretion

The drug and its metabolites are eliminated from the body in urine and bile. When drugs enter in the general circulation after it is distributed throughout the body. Drugs are excreted or eliminated from the body as metabolites. Lungs are important for the elimination; the most important excretory organ is kidney. Excretion means how the drug substance leaves the body, where the main elimination path is via kidneys for most of the compounds. But elimination can also occur through lungs, skin, and other body secretions like sweat, serum and milk [50, 52].

1.5. Simulation

Historically simulations were used in different fields which were developed largely independently, but 20th century studies of system theory with spreading use of computers across all those fields have led to some unification and more systematic view of the idea [5, 6].

In 1990, the first ride to be done entirely with computer graphics and based on random variables along with many mathematical, statistical, and graphical functions. Simulations began by implementing numerical approximations of the differential equations under study with the target of solving the computation of more complex models in a fast and precise way, [31- 33].

The simulations are two kinds. Stochastic is the first simulation, and deterministic is this simulation. A simulation is the imitation of the operation of a real-world process or system over time, whether done by hand or on a computer to verify analytic solutions. Pharmacokinetic modeling and simulation are integral components of the drug development process with potential impact on the regulatory approval process. A pharmacokinetic simulation model may be a useful tool to combine information from different sources [1]. One aim of mathematical modeling is to differentiate the information or systematic component in the system. For an example, the equation for one compartment model after bolus administration is

퐶퐿 퐷 − 푡 퐶 푡 = 푒 푉 + 휀 (1.8) 푉

Where 퐷 and 푡 areindependent variables, where 퐶 is concentration, 퐷 is dose, 푉 is volume of distribution, 퐶퐿 is clearance, and 휀 is error, which represents the deviation between model predicted concentrations and observed concentration. The first conference was in 1989, at the

National Institutes of Health, and the one conclusion from the conference was biomedical research can progress and develop more efficiently by application of mathematics in developing

models for research. The use of modeling in drug absorption, for which many of the examples,

Aarons et al. 200, used simulation modeling in development of drug , the blood samples used for pharmacokinetic endpoint that are measured would then represent the set of output. Both the input and the output cannot be measured perfectly [54]. Modeling and simulation are often spoken interchangeably, but actually they are not the same, and there are important differences between them. Modeling understands the system in its current state, while simulation can be used to understand the system in alternative states. The simulation model is defined as computational and/or mathematical tool that interprets drug kinetics in living environment under specific conditions [28]. Computer simulations were undertaken to evaluate the performance of the estimation procedure as a function of the input tissue curves [10, 12], the simulation results a good agreement with both the mathematical model and in the in vitro data, and it is useful to develop pharmacokinetic models for the prediction of pharmacokinetic parameters.

The convolution method is a simulation method used to predict the blood concentration using percent absorbed data when a drug is administered orally [29, 30]. Beal in 1983, made a simulation study to investigate the effect of the addition of single measurement in one- compartment model with intravascular administration. Currently simulation is most widely used in categories of research, design, analysis, training and education. Simulation is a tool to evaluate the performance of a system, existing or proposed under different configurations of interest and over long periods of time [31, 73].

1.6. Administration drugs and pharmacokinetic processes

The routes of administration drugs are either extravascular or intravascular; and can be illustrated by the following diagram.

Drug Administration

Oral Intravenous

Absorption Distribution Enteric transport and metabolism Intravascular and Extravascular

Metabolism

Hepatic influx transport

Biliary Excretion Intestinal Excretion Renal Excretion

Efflux transport Efflux transport

1.7. Classifications of drugs

The drugs can be classified as class I drugs have high absorption number and high dissolution rate. Class II drugs have high absorption number but low dissolution number. Class III drugs exhibit a high variable in the rate and extent of absorption. Class IV drugs exhibit poor and variation bioavailability [61, 65]. There is another classification of drugs according to the

Bio-pharmaceutics classification system where class I high permeability, high solubility, like

Metoprolol class II high permeability, low solubility, like amide class III low permeability, high solubility, like Atenolol, class IV low permeability, low solubility like Hydrochlorothiazide [9].

1.7.1 BCS Class I drugs

The Bio-Pharmaceutics classification system is the result of continuous efforts in mathematical analysis for the explanation of the kinetics and dynamics of drug process in the gestrointesental tract for new drug application. For drug development process, IVIVC has been defined by the

FDA as a predictive mathematical model describing the relationship between an in vitro property of a dosage form and an in vivo response. Generally the in vitro property is the rate or extent of drug dissolution or release while the in vivo response is the plasma drug concentration or amount of drug absorbed [8, 26]. The Bio-pharmaceutics classification system is a drug development tool that allows estimation of the contribution of three major factors, dissolution, solubility, and intestinal permeability that effect oral drug absorption from immediate release solid oral products, in 1995, Amidon and co-workers introduced the BCS for orally administered drugs.

Solubility, dissolution and permeability are the main factors that affect the rate and extent of oral absorption [4, 11].

1.8. Metoprolol

Metoprolol is a BCS class I drug having high solubility, high permeability; and is used in the treatment of several diseases of the cardiovascular system, especially hypertension. Metoprolol is also prescribed for off-label use in performance anxiety and social anxiety disorder. Metoprolol may worsen the symptoms of heart failure in some patients, who may experience chest pain or discomfort; dilated neck veins; extreme fatigue; irregular breathing; an irregular heartbeat,

shortness of breath; swelling of the face, fingers, feet, or lower legs; weight gain; or wheezing.

This medicine may cause changes in blood sugar levels or cover up signs of low blood sugar, such as a rapid pulse rate. This medicine may cause some people to become less alert than they are normally, making it dangerous for them to drive, use machines, or do other activities.

Excessive doses of Metoprolol can cause severe hypotension, metabolic acidosis, seizures and cardio respiratory arrest [9, 10, 14]. Because of this, Metoprolol is always manufactured in a salt- based solution as drugs with low melting points are difficult to work with in a manufacturing environment.

Chapter 2 LLiitteerraattuurree RReevviieeww

Literature review 2.1. Introduction

Pharmacokinetics can be defined as relationship between drug dose and the response of the drug in the body. In this work we have discussed pharmacokinetics models based on convolution approach supported with applications. Pharmacokinetic analysis is performed by compartment or non-compartment model. Compartment can be classified into mechanistic models and physiological models. Compartmental models often occur in the course of describing the kinetics of a tracer injected into a physiological system. A physiologic system is often described by decomposition into a number of interacting subsystems, called compartments [75].

These models based on mathematical equations for deriving various pharmacokinetic parameters from experimental data, such as area under the curve; absorption; volume of distribution.

Pharmacokinetics is divided into several areas including the extent and rate of absorption, distribution, metabolism and excretion. These four factors will eventually determine the concentration of a drug at its goal site and the magnitude of its pharmacological effect.

The Swedish physiologist Teorell published first paper on the compartment models [79], to describe the time course of drug in biological system as well as he introduced a systematic study of pharmacokinetics. He also extended the idea of compartment to include the transformation of a drug substance to another chemical form without changing its spatial localization. Then in

1960, Bellman and Kabala derived mathematical expression for drug distribution in a tissue.

The convolution method in the Pharmacokinetics, allow the development of Laplace equation, for the amount of drug in the central compartment, by simple multiplication of the input function and the disposition function. Using the algebraic approach differential and integral equations were developed for the two compartment model. The input function describes the route of

administration, while the disposition function describes a first order distribution and elimination processes. Linear system analysis is a pharmacokinetic mode which depends on convolution and deconvolution rather than ordinary differential equations. The model was introduced to the field of pharmacokinetics by Roberts and Rowland who used it for liver perfusion experiments [63].

The implementation of mathematics into biology, physiology, pharmacology, and medicine is not new, but it had grown in the last five decades. If a model is logically suitable, we can analyze it by mathematical equations and can get meaningful results. A general linear pharmacokinetics model where 푘푒푙 is elimination through urine, 푘푚 is metabolism, 푘푒푥 is excretion, and 푘표푡푕푒푟 other processes, 푘푒푙 = 푘푒푥 + 푘푚 + 푘표푡푕푒푟 . Every pharmacokinetic model needs to be represented by a formula or an equation. Understanding these equations and parameters in these equations is important. The weighting function was represented by the average cumulative urinary excretion profile observed after oral administration. There are two models for these processes, first is model-dependent (compartment models) as Wagner-Nelson and Loo-Riegelman, second is model-independent as convolution based on theory of linear system analysis, while deconvolution is model-dependent and model independent as well [14].

2.2. Pharmacokinetic

Pharmacokinetics involves the application of mathematical and biochemical techniques in a physiologic and pharmacologic context. We can say that pharmacokinetics is multidisciplinary.

Pharmacokinetics aims to describe all relevant processes that influence absorption, distribution, metabolism and excretion. In general the purposes of pharmacokinetics are to reduce data to a number of meaningful parameter values, and to reduce data predict either the results of future experiments or the results of studies which would be too costly and time-consuming to complete, a similar definition has been given by other authors [74].

푉 = Τ Ω 2.1

∞ 푡. 퐶 푡 푑푡 Τ = 0 (2.2) ∞ 0 퐶 푡 푑푡

Equation (2.1) has been written in pharmacokinetics articles published by Galeazei 1979, where

푉 is the volume of the system, Τ is the mean transit time, Ω is the blood flow, and 퐶 is the concentration in plasma at time 푡. The numerator and denominator in equation (2.2) are the area under the first moment of concentration-time curve, and the area under the concentration-time curve respectively. Dominguez, 1987 was the first to derive and apply equation (2.3) to estimate the rate of absorption as a function of time, by using ordinary differential equation.

푑푋 푑퐶 = 푉 + 푉 푘 퐶 (2.3) 푑푡 푑 푑푡 푑 푒푙

푑푋 where is the rate of absorption at time t, 푉 is the volume of distribution, 퐶 is the plasma drug 푑푡 푑 concentration-time, and 푘푒푙 is the first-order elimination rate constant. Bioavailability theory has become an important topic in pharmacokinetics. This concept was introduced by Oser in 1945, use of the computer for fitting and simulating pharmacokinetics data in model building was introduced by Garrett [31]. The first conference on pharmacokinetics was held in USA, in 1972, and then many theoretical articles were published which later became part of classical pharmacokinetics. These articles included a new method of estimating drug bioavailability.

Pederson later published another treatment for input and elimination into one or more compartments. Two approaches were described; one depends on Laplace transformation and other that avoids transformation of the input function(s) and the use of convolution integrals.

The latter approach is important when complex input functions do not have a simple Laplace transform.

Pharmacokinetics modeled with linear systems, by which plasma concentrations following bolus

푖푣 intravenous injection 휒푝 will be described by a poly-exponential equation as follows:

푛

푖푣 −훼푖푡 휒푝 = 퐶푖푒 (2.4) 푖=1

푡푕 where, 퐶푖 , 훼푖, 푖 = 1, 2, … , 푛 are the coefficients of 푖 exponential term of the poly-exponential equation and t is the time of these numerical values obtained from a given set of data depending on the specific linear pharmacokinetic model and the numerical values of the rate constants.

The poly-exponential equation describing the plasma concentration after either intravenous or oral administration can readily vanish a few minutes after administration. The drug amount 푋푒 at time t is given by the equation,

퐷 퐴푈퐶0→푡 푋푒 = (2.5) (퐴푈퐶0→∞)

And the amount of drug in the body at time 푡, 푋푏 is given by

퐷 퐴푈퐶푡→∞ 푋푏 = (2.6) (퐴푈퐶0→∞)

Also the amount of drug 푋푝 in the plasma compartment at time 푡, is given by equation

푖푣 푋푝 = 푉푝 휒푝 (2.7)

The amount of drug 푋0 in other compartment in the plasma compartment at time 푡, is given by

푋0=푋푏 − 푋푝 2.8

Equation (2.8) describes the differences between amount of drug in the body and the amount drug in the plasma compartment at time 푡 [12].

We can describe this process of pharmacokinetics by the following scheme:

Tablet

Absorption

Drug in Drug in Drug in system tissues kidneys circulation Distribution Disposition

Elimination

Drug removed from the body

Figure (2.a)

2.2.1. Pharmacokinetic modeling

Different mathematical models can be devised to simulate and use for the quantitative description of drug processes. These mathematical models consist of equations that purport to describe drug concentration in the various organs of the body. In the physiological approach, the body is divided into compartments based on anatomical regions like, heart, liver, kidney, and stomach. The differential equation of the one compartment model can be solved analytically.

Physiological models are based on well-defined and structured compartments interconnected by

blood flow. Some typical pharmacokinetics parameters are defined by mathematical equations.

Compartment mode and non-compartment mode (Veng-Pedersen, 2001) are types of modeling that benefit from the quantitative structure-pharmacokinetic relationships that are described by experimental mathematical algorithms. They can be used to estimate the motion of a compound based on numerical format [77]. Compartmental modeling is used to describe systems that differ in time but not in space, which are often described by a single ordinary differential equation.

Pharmacokinetics parameters allow the clinician to design and optimize treatment regimens, including decisions as to the route of administration for a specific drug, the amount and frequency of each dose and duration of treatment [55]. After administration of an intravenous bolus injection the plasma concentration against time profile has two phases: Intravenous bolus

−푘푡 administration and plasma concentration 퐶푝 for a one compartment model is 퐶푝 = (퐶푝)0푒 and for two-compartment model is

−훼푡 −훽푡 퐶푝 = 푎1푒 + 푎2푒 , 푎1 , 푎2 are intercept constants while 훼 , 훽 are hybrid rate constants.

In general there is dependent variable 푌 expressed as a function of independent variable 푥 with various constant and/or parameters 푦 = 푓(푥, 푝), where 푦 −dependent variable, 푥 −independent variable (often time) & 푝 −parameter.

Models of data usually referred to as empirical models require a few assumptions about the data.

Models of system or mechanistic models are based on physical and physiological principles; these models usually take the form of ordinary differential equations or partial differential equations based on, mass-balance, or mass-action principles. It is typically assumed that a drug pharmacokinetics is stationary or variant over time so that the principle of superposition applies.

Models are also classified whether they are deterministic or stochastic if involving a chance or probability. However, deterministic models are useful to understand the properties of system,

a pharmacokinetic model which relates dose and dosing frequency to drug concentration, usually plasma or serum after the administration of dose. The model parameters from the two-term exponential equation also directly translate to pharmacokinetic parameters, such as volume of distribution expanded the number as stage by [34, 37]. Pharmacokinetics could be characterized by two-compartment model with first-order elimination, which produces similar pharmacokinetic estimates. Wagner, 1975 has discussed formulating pharmacokinetic models; one compartment model is given by the equation

퐷 퐶 푡 = 푒−훼푡 (2.9) 푉

where 퐶(푡) is concentration, 퐷 is dose, 푉푑 is volume of distribution. This model has three estimable parameters, 퐷, 푉푑 , and 훼, to estimate volume of distribution, take log-transformation concentrations, we give

퐷 퐷 ℓ푛퐶 푡 = ℓ푛 − 훼푡, at time 푡 = 0, thus 퐶0 = ℓ푛 , if the dose were known, then this equation 푉푑 푉 can easily be solved. One-compartment model after extravascular administration assuming complete bioavailability, is given by

퐷 푘푎 퐶 푡 = 푒−푘10푡 − 푒−푘푎 푡 (2.10) 푉푑 푘푎 − 푘10

where 푘푎 , 푎푛푑 푘10 are the absorption and elimination rate constants respectively, this model is can be written as

−휑1 푡 −휑2 푡 퐶 푡 = 휑0 푒 − 푒

퐷 푘푎 where, 휑0 = , 휑1 = 푘10 , 휑2 = 푘푎 푉푑 푘푎 − 푘10

퐷 휑2 푠표 푉푑 = , 휑0 휑2 − 휑1

−휑2 푡 −휑1 푡 퐶 푡 = 휑0 푒 − 푒 (2.11)

where the solution for 푉푑 is

퐷 휑2 푉푑 = (2.12) 휑0 휑2 − 휑1 hence, the two solutions

퐷 휑2 푘10, 푘푎 , 푉푑 = [휑1, 휑2, ] 휑0 휑2 − 휑1

퐷 휑2 {푘10, 푘푎 , 푉푑} = 휑1, 휑2, − 휑0 휑1−휑2

Provide identical concentration–time profile for a drug, 푘푎 → 푘10, implies 푘푎 − 푘10 → 0 ,

푘푎 implies → ∞, taking limit as 푘푎 → 푘10 , we get 푘푎 −푘10

퐷푘 ′푡 ′ 퐶 푡 = 푒−푘 푡 , where 푘′ is a hybrid estimate of 푘 , 푘 traditionally for linear compartment 푉 푎 10 model. This involves Laplace transforms. For one compartment model after first order absorption with complete bioavailability, the model can be written as:

푑푌0 = −푘푎 푌0, 푑푡 푑푌 1 = 푘 푌 − 푘 푌 , 푑푡 푎 0 10 1 푌 퐶 = 0 , 푌 0 = 퐷, 푌 0 = 0 푉 0 1

where 푌0 represents the absorption compartment and 푌1 is the central compartment, in Laplace transform, we have

푆푌 − 퐷 = −푘 푌 , 0 푎 0 (2.13) 푆푌 1 = 푘푎 푌 0 − 푘10푌 1

Solving for 푌 0, and 푌1 and then substituting 푌 0 into 푌1 , gives

푘푎 퐷 푌 1 = 푠 + 푘푎 (푠 + 푘10)

1 If the Laplace transform of the absorption function is scaled by then Ψ 푠 is defined as 푉푑 follows:

ℒ[푦(푡, 푝) Ψ 푠 = ℒ[푢(푡, 푝) where ℒ 푦 푡, 푝 denotes the Laplace transform function, and ℒ 푢 푡, 푝 is the input function in

Laplace transforms.

Let ℒ 푢 푡, 푝 = 퐷,

푘 퐷 ℒ 푦 푡, 푝 = 푎 푠 + 푘푎 (푠 + 푘10)

ℒ[푦(푡, 푝) 푘푎 Ψ 푠 = = ℒ[푢(푡, 푝) 푉 푠 + 푘푎 (푠 + 푘10)

푏 which is equivalent to Ψ 푠 = 2 (2.14) 푠 + 푎1푠 + 푎2

푘 where 푏 = 푎 & 푎 = 푘 + 푘 & 푎 = 푘 푘 푉 1 푎 10 2 0 10

The coefficients of the power of s in Eq. (2.14) are called moment invariants, when F is unknown, then

퐹퐷 푘푎 퐶 푡 = 푒−푘10푡 − 푒−푘푎 푡 (2.15) 푉푑 푘푎 − 푘10

ℒ[푦(푡, 푝) 퐹푘 Ψ 푠 = = 푎 ℒ[푢(푡, 푝) 푉 푠 + 푘푎 (푠 + 푘10)

푏 퐹푘 푎 Ψ 푠 = 2 , and 푏 = 푠 + 푎1푠 + 푎2 푉

First recognized as a problem by Wagner [12], and formalized by Godfrey and colleagues [54], pharmacokinetic models utilize mathematical equations to describe drug concentrations measured in the body as a function of time. These mathematical models can be used to generate pharmacokinetic parameters that describe the processes of absorption, distribution, and elimination of a substance in the body [27]. The convolution integral of Stephenson, was useful in linear systems analysis. Some model-independent prediction methods for use in pharmacokinetics were discussed by [41]. One of these methods to estimate the asymptote of a curve according to first-order kinetics; these methods are most useful for estimating the area under a concentration-time curve from zero to infinite time or the cumulative amount of drug after a single dose of drug. The method was later extended to bi-exponential processes [12].

2.2.2. Compartmental models

Compartmental models depend upon linear or non-linear differential equations. Until now essentially ordinary differential equations are used to construct pharmacokinetics models.

Therefore we remark that the application of ordinary differential equation is not new in pharmacokinetics. In 1982, presented model for pharmacokinetic based on ordinary equations.

One compartment describes the blood and tissue while two compartments describe the time course for most drugs. Main assumption in pharmacokinetics is that the drugs after metabolism and excretion from the body are through the blood compartments. The process of the metabolism takes place in the liver and the excretion by kidneys [76]. Compartmental modeling is performed on pharmacokinetics data sets. Compartmental analysis has applications in clinical medicine, pharmacokinetics and chemical reaction. It can be described as the analysis of a system in which the system is separated into a finite number of component parts which are called compartments.

Compartment model is used to study drug kinetics. It is based on dividing the body into finite number of compartments. It is not physiological or anatomical region but a mathematical concept. All the compartments may be assumed as various organs or tissues that altogether handle a drug like homogenous compartments. Input and output in these compartments are defined in from of 1st order kinetics each compartment is considered to be rapidly and uniformly distributed. The rate constants are used to represent the overall rate processes of the drug entry into and exit from the compartments. Two compartment models are also known as delayed distribution models. The body is considered to be comprised of two compartments, one compartment is called central compartment, and second compartment is called tissue compartment or peripheral compartment. The two compartments can be categorized into three types, two compartment models with elimination from central compartment, two compartment models with elimination from peripheral compartment [46], two compartment models with elimination from both the compartments. We will describe two elements of the required solution, the function 푒−푘푡 and convolution. The function 푓 푡 = 푒푡 , in mathematical has a very important property, that the rate of change of this functions equal to the function at any time, so it plays a

푑푋 very important role in solution. The rate of drug excretion actually has two components, the 푑푡 rate of the process and the amount of compound available for

푑푋 = 푘푋푛 (2.16) 푑푡 where 푘 is the fractional rate constant compartment, 푋 is amount of drug in compartment, 푛 is the order of the process.

First-order rates, for a first order process, let 푛 = 1, in Eq. (2.16) becomes

푑푋 = 푘푋 푑푡

푑푋 By definition first-order or linear processes, the rate of process varies in a direct proportion 푑푡

푑푋 to 푋 if we move the system per unit time, when 푋 increases, also increases in the direct 푑푡 proportion. This distinction is very important for linear models, the rate constant is fixed.

Compounds that undergo absorption, distribution, and elimination in direct proportion to a concentration gradient are by definition first–order rate processes.

Zero-order rates, for zero-order process, let 푛 = 0, in Eq. (2.16), the rate equation becomes

푑푋 푑푋 = 푘 , in case of to describe the rate of process, differential equation express rate in terms 푑푡 표 푑푡 of the change (푑푋), over small interval of time (푑푡), defining rate with differentiation is analogous to taking slopes, while the inverse process of integration produces parameters that may often be expressed and numerically estimated area. In case the rate of compound decays

푑푋 푑푋 (− ), we can write this equation such that as = −푘푡, and integrate the equation at time 푑푡 푑푡 zero (푥0) through 푋 at time 푡 , (푋푡 ) to obtain a formula for the mass of drug at any time푡, we

−푘푡 get the equation 푋푡 = 푥0푒 . This differential equation describing the rate generates the exponential term found in most linear pharmacokinetic models. In reality any method of pharmacokinetic analysis using exponential functions can be used to describe physiological processes [52].

2.2.2.1. One-compartment model

The drug is administrated into the compartment and distributed throughout the body instantaneously either extravascular or intravascular route [50]. Likewise the drug is eliminated directly from the one compartment so the one-compartment model is the simplest pharmacokinetic model compared with two or more compartments model.

2.2.2.1.1. One-compartment model intravascular administration

Compartment models are a special class of linear models where the response variable is described by an ordinary differential equation. These differential equations describe the change of concentration of drug in the compartment over time, a process that is usually of first order kinetics. First order kinetics means that the rate of change of drug concentration in a compartment at time t is directly proportional to the drug concentration in that compartment at that time. Therefore, the first kinetics can be described by the following differential equation,

푑퐶(푡) = −푘 퐶 푡 (2.17) 푑푡 푒푙

where 퐶(푡) is the drug concentraed at time t, and 푘푒푙 is rate constant. In all the models presented here we consider first order kinetics for the rate constants and that, the drug is administered by a single dose with intravascular or extravascular administration. In an intravascular administration the drug is directly placed into the bloodstream usually considered as the central compartment,

and therefore we assume that the drug is rapidly mixed in the blood. If a one-compartment model is used, then elimination for extravascular administration, the drug must be absorbed by the central compartment. One compartment model of intravascular administration has just one rate constant that is the elimination rate constant. The pharmacokinetic model is obtained by direct

−푘푒푙 푡 integration of previous equation and is given by 퐶 푡 = 퐶0푒

In this model 푘푒푙 is the elimination rate constant and 퐶0 is the drug concentration at time zero.

Compartmental modeling is the mainstay of pharmacokinetic modeling and these models can be used to describe concentration data. The most basic mathematical description of drug distribution and elimination is the one-compartment model. So, we have the one-compartment model with bolus intravenous injection and multiple doses administrated at uniform time intervals or one- compartment open model with constant rate intravenous infusion. The mathematical of the rate of drug accumulation in the one-compartment linear system was discovered by Wagner -Nelson,

1967. Then a year later, exact solution was given for the number of doses for the one and two- compartment model with first–order absorption [15]. The rate of drug elimination from the central compartment is a first-order process. The mathematical description of linear decay of a drug from the compartment is a function of time. In this method we use one compartment model with first order absorption. Application of regular residual methods and Wagner-Nelson method have been shown to give parameter estimates where, 0 ≤ 푘푒푙 ≤ 1

2.2.2.1.2. One-Compartment model extravascular administration

In this model we have absorption and elimination rate constants and the pharmacokinetic model

−푘푒푙 푡 −푘푎 푡 is given by 퐶 푡 = 퐴푒 − 퐵푒 where, 푘푎 is the absorption rate constant, the coefficients 퐴 and 퐵 are equal to,

퐷퐹퐾 퐴 = 퐵 = 푎 푉(퐾푎 − 푘푒푙 ) where F is that fraction of administered dose which is absorbed, we can write this model as,

퐷퐹푘푎 퐶 푡 = 푒−푘푎 푡 − 푒−푘푒푙 . 푉(푘푎 − 푘푒푙 )

The first–order elimination rate constant characterizing the overall elimination of a drug from a one compartment model is usually written as 푘푒푙 and also represents the sum of two or more rate constants of elimination processes. In oral administration of a drug the distribution in the body is according to a one compartment model and is eliminated via apparent first-order kinetics, the rate of loss of drug from the body is given by

푑푌 = −푘 푌 (2.18) 푑푡 푒푙

This differential equation is derived from one compartment which can be presented by the following scheme:

Drug in the

body ( Y ) 푘푒푙

Figure (2. b). Scheme of one compartment model

where Y is the amount of drug in the body at time 푡 after oral administration, 푘푒푙 is the apparent first-order elimination rate constant for the drug. The negative sign indicate that drug is being lost from the body, by taking the Laplace transforms for Eq. (2.18) becomes

푠푌 푠 − 푌0 = −푘푒푙 푌 푠 (2.19)

where 푌0 is the dose and 푠 is a variable, rearrangement of Eq.(2.19), gives

푌0 푌 푠 = . (2.20) 푠 + 푘푒푙

By solving Eq. (2.20) using inverse Laplace transforms we get,

−푘푒푙 푡 푌 푡 = 푌0푒 (2.21)

Taking the natural logarithm of both sides of Eq. (2.21) gives

ℓ푛푌 = ℓ푛푌0 − 푘푒푙 푡 (2.22)

ℓ푛푎 We know that, 2.303 ℓog 푎=ℓn 푎, this means, ℓog푎 = so, Eq. (2.22) becomes 2.303

푘 푡 ℓog 푌 = ℓ표푔푌 − 푒푙 (2.23) 0 2.303

The relationship between drug concentration in plasma 퐶 and the amount of drug in the body is

푌 = 푉. 퐶 푡 (2.24) where 푉 is the volume of distribution apparent.

푘 푡 ℓ표푔퐶 = ℓ표푔퐶 − 푒푙 (2.25) 0 2.303

푘 where 퐶 is the drug concentration in plasma immediately after oral administration an (− 푒푙 ) 0 2.303 is slope resulting from a plot of ℓog 퐶 versus time. If a drug is administered intravenously at constant rate, the following differential equation may be written for the change in amount of drug in the body with time.

푘푎

Drug in the body (Y) 푘푒푙

Figure (2. c). Scheme of one-compartment model

We can derive first order ordinary differential equation from above scheme as follows:

푑푌 = 푘 − 푘 푌 (2.26) 푑푡 푎 푒푙

where, 푘푎 is the rate of drug infusion, expressed in amount per unit time, the Laplace transform of Eq. (2.26) is

푘 푠푌 푠 = 푎 − 푘 푌 푠 (2 .27) 푠 푒푙

After rearranging we have

푘 푌 푠 = 푎 (2.28) 푠(푠 + 푘푒푙 )

Solving Eq. (2.28) by using inverse Laplace transforms, gives the relationship between the amounts of drug in the body given by

푘푎 푌 푡 = 1 − 푒−푘푒푙 푡 (2.29) 푘푒푙

From Eq. (2.29), let 퐶(푡) = 푌 푡 /푉 we get,

푘푎 퐶(푡) = 1 − 푒−푘푒푙 푡 (2.30) 푉푘푒푙

푘푎 where, 퐶(푡) represents concentration in plasma, as 푡 → ∞,푒−푘푒푙 푡 → 0 and 퐶 → , this drug 푉푘푒푙 concentration called infusion equilibrium. So the steady-state concentration in plasma 퐶푠푠 is given by

푘푎 퐶푠푠 = . (2.31) 푉푘푒푙

From Eq. (2.30) and Eq. (2.31), we get

−푘푒푙 푡 퐶 = 퐶푠푠 1 − 푒 (2.32)

A very large number of plasma concentration-time curves can be obtained after extravascular dosing e.g. oral administration of drugs can be described by one compartment model with first- order absorption, elimination, and distributes in the body, the following differential equation, which can be derived from the following scheme.

푌푎

푘푎

Y 푘푒푙

Figure (2. d). Scheme of one-compartment model

푑푌 = 푘 푌 − 푘 푌 (2.33) 푑푡 푎 푎 푒푙 dY = rate of inputs – rate of output, this differential equation is derived from one compartment dt where, 푌 and 푘푒푙 have been already defined, 푘푎 is the apparent first-order absorption rate constant and 푌푎 is the amount of drug at the absorption site. For analytical solution of one compartment model, we use the Laplace transform of Eq. (2.33) which gives

푠푌 푠 = 푘푎 푌푎 푠 − 푘푒푙 푌 푠 (2.34)

The rate of loss of drug from the absorption site is

푑푌 푎 = −푘 푌 (2.35) 푑푡 푎 푎

The Laplace transform of which is

푠푌푎 푠 − 퐹푌0 = −푘푎 푌푎 푠 (2.36)

where F is the fraction of the dose administrated 푌0 that is amount of drug absorbed following extravascular administration, solving Eq.(2.36) for 푌푎 (푠), substituting this value for 푌푎 (푠) and solving for 푌(푠) we get

푘 퐹푌 푌 푠 = 푎 0 (2.37) 푠 + 푘푒푙 (푠 + 푘푎 )

Solving Eq. (2.37) by using inverse Laplace transforms, we get a relationship between the amount of drug in the body and time

푘 퐹푌 ℒ−1 푌 푠 = ℒ−1 푎 0 푠 + 푘푒푙 (푠 + 푘푎 )

푘푎 퐹푌0 푌 푡 = 푒−퐾푡 − 푒−퐾푎 푡 (2.38) 푘푎 − 푘푒푙

Dividing both sides of Eq. (2.38) by 푉푑 we get drug concentration which is given by

푘푎 퐹푌0 퐶(푡) = 푒−푘푡 − 푒−푘푎 푡 (2.39) 푉푑(푘푎 − 푘푒푙 ) where 퐶(푡) is the plasma concentration of drug at any time 푡 following the administration of dose 푋0, 푉푑 is the apparent volume of distribution, F is the fraction of the orally administered dose which is absorbed, 푘푎 and 푘푒푙 are the apparent first-order absorption and elimination rate

−푘푎 푡 constants respectively. At the time after administration the term 푒 → 0, when 푘푎 ≫ 푘푒푙 and

Eq. (2.39) becomes,

푘푎 퐹푌0 퐶(푡) = 푒−푘푒푙 푡 (2.40) 푉(푘푎 − 푘푒푙 )

When a time is evident, the appropriate equation to describe the course of drug concentrations in plasma is

푘푎 퐹푌0 퐶(푡) = 푒−푘푒푙 (푡−푡0) − 푒−퐾푎 (푡−푡0) (2.41) 푉(푘푎 − 푘푒푙 )

where 푡0 is the drug concentration in plasma versus time data after oral administration integration Eq. (2. 41) from zero to infinity, we get

∞ ∞ 푘푎 퐹푌0 퐶 푡 푑푡 = 푒−퐾푡 − 푒−퐾푎 푡 푑푡 푉(푘푎 − 푘푒푙 ) 0 0

푘푎 퐹푌0 1 1 퐴푈퐶 = − (2.42) 푉(푘푎 − 푘푒푙 ) 푘푒푙 퐾푎

Rearrangement of Eq. (2.42) gives

퐹푌 퐴푈퐶 = 0 (2.43) 푉푘푒푙

where, 퐹. 푌0 the amount of drug is absorbed, in general, the fraction absorbed or percentage of drug absorbed can be calculated by model-dependent techniques, such as the Wagner-Nelson procedure and the Loo–Riegelman method or by model-independent numerical deconvolution.

According to the Wagner-Nelson procedure, the cumulative fraction of drug absorbed F can be calculated, while the absorption half-life can be calculated using expression 0.693/푘푎 .

More details about the derivation of equations for both procedures have been widely discussed elsewhere. However, it is necessary to mention that both of these methods have been widely employed and included in a variety of software for pharmacokinetic model.

2.2.2.1.3. Application of one-compartment model (Wagner–Nelson method)

The fraction absorbed (퐹abs) was determined from the plasma concentration-time data by deconvolution using the Nelson-Wagner method. The Wagner-Nelson method is applied to one- compartment open model data with first order and zero order input to the central compartment to provide of amount of drug absorbed per unit volume of distribution versus time and plots of percent of drug absorbed versus time. Extravascular drug delivery is complicated by variables at the absorption site including possible drug degradation and significant patient differences in the rate and extent of absorption. Drug first-order absorption kinetics occurs if the amount absorbed is dependent on dose. Most drugs exhibit first order absorption kinetics. The consideration of physicochemical laws is essential for complete understanding of drug kinetics. The amount of drug that has been absorbed into the systemic circulation, 푋푎 at any time after administration will

be equal to the sum of the amount of drug in the body 푋, and the cumulative amount of drug eliminated 푋푒푙 , by urinary excretion by metabolism and by all other routes at that time, thus

푋푎 = 푋 + 푋푒푙 which when differentiate with respect to time becomes

푑푋푎 푑푋 푑푋푒푙 = + 푑푡 푑푡 푑푡

The term 푑푋푒푙 /푑푡 (elimination rate of drug) is by definition equal to the product of the amount 푋 of drug in the body and the apparent first-order elimination rate constant of drug from the body

푑푋푒푙 = 푘 푋 푑푡 푒푙

Substitution of 푘푒푙 푋 for 푑푋푒푙 /푑푡 where 푘푒푙 is the first-order elimination rate constant from the central compartment, we get

푑푋 푑푋 푎 = + 푘 푋 푑푡 푑푡 푒푙

We know that 푋=푉퐶, where 푉 and 퐶(푡) are the apparent volume of distribution and plasma concentration of drug respectively, the above equation can be written as

푑푋 푑퐶 푎 = 푉 + 푘 푉퐶 푑푡 푑푡 푒푙

Integrating this equation from zero to 푇, we get

푇

(푋푎 )푇 = 푉퐶푇 + 푘푒푙 푉 퐶푑푡 0 where 퐶푇 is the plasma concentration of drug at time T, again integrating the above differential equation from time zero to infinity and recognizing that 퐶푇 equals zero at both times zero and infinity, we get the total amount of drug absorbed as following:

∞

(푋푎 )∞ = 푘푒푙 푉 퐶푑푡 0

푇 ∞ where 퐶푑푡 푎푛푑 퐶푑푡 0 0

Are the area and the total area under the plasma concentration versus time curve respectively, the traction of drug absorbed at any time is,

(푋푎 )푇 = 퐹푇 (푋푎 )∞

This relates the cumulative amount of drug absorbed after a certain time, to the amount of drug ultimately absorbed, rather than to the dose administrated. According to Wagner-Nelson method, the cumulative fraction of drug absorbed at time 푡 is calculated from equation as follows:

푇 퐶푇 + 푘푒푙 0 퐶푑푡 퐹푇 = ∞ 푘푒푙 0 퐶푑푡

퐹푇 is the cumulative fraction absorbed at any time 푇 [60]. An important characteristic of the

Wagner-Nelson method for evaluating absorption data fraction at any time 푇 is given by

푇 (푋푎)푇 퐶푇 + 푘푒푙 [퐴푈퐶]0 1 − 퐹푇 = 1 − = 1 − ∞ (푋푎 )∞ 푘푒푙 [퐴푈퐶]0

푇 where [퐴푈퐶]0 is the area under the plasma concentration versus time curve from time zero to

∞ 푇 and [퐴푈퐶]0 is the total area under the concentration versus time curve from zero to infinity. 2.2.3. Two-compartment model

Two-compartment models are composed of a central compartment and a peripheral compartment they are often preferred to more complicated models in terms of fewer parameters. The central compartment represents the circulatory system, where the drug is exchanged rapidly with other parts of the body [61, 66].

We can derive the differential equations of two compartment model from the following scheme.

푘푎 푘12 Central Peripheral 푞 Compartment (1) 3 Compartment (2) 푘 푞1 21 푞2

푘푒푙

Figure (2.2a) Scheme of two-compartment model

A two-compartment model consists of two physiological meaningful parts. The first is 푞1

identified with the blood and organs like liver or kidney. The second is called peripheral

compartment and 푞2 describes tissue or more generally, 푞3 dose of drug. These compartments are

connected with each other in both directions and therefore, an exchange of drug between the first

and the second compartment. The drug when administrated bolus intravenous injection into the

blood or the drug indirectly administrated as orally either tablets or solution. We will discuss two

cases of drug administration in two compartments either intravenous or oral. In case of

intravenous administration the conditions are

푖푣 0 푘푒푙 , 푘12, 푘21, 푘푎 > 0 & 푞 0 = (푞1 , 0,0)

푝표 0 In case of oral administration, the conditions are, 푘푒푙 , 푘12 , 푘21 ,푘푎 > 0 & 푞 0 = (0,0, 푞3 ).

The form of a two-compartment model describing either intravenous or oral drug administration

is as follows:

푑푞 푡 1 = − 푘 + 푘 푞 푡 + (푘 )푞 푡 + 푘 푞 & 푞 0 = 푞0 ≥ 0 (2.44) 푑푡 12 푒푙 1 21 2 푎 3 1 1

푑푞2 푡 = 푘 푞 푡 − 푘 푞 푡 , 푞 0 = 0 (2.45) 푑푡 12 1 21 2 2 푑푞 (푡) 3 = −푘 푞 푡 , 푞 0 = 퐹. 푞0 ≥ 0 푤푕푒푟푒 0 < 퐹 ≤ 1 (2.46) 푑푡 푎 3 3 3 where 퐹 is a fraction parameter (bioavailability), when there is no loss, then 퐹 = 1. For our mathematical consideration between central and peripheral compartment Eq. (2.44) describes the blood compartment, Eq. (2.45) describes the peripheral compartment and Eq. (2.46) describes the absorption of administration drug where 푘푒푙 describes the elimination rate constant and

푘12, 푎푛푑 푘21 are the rates transit from central compartment to peripheral compartment and vice versa, and 푘푎 is the absorption rate constant, equations (2.44-2.46), can be writing in matrix notation as

−푘푒푙 − 푘12 푘21 푘푎 ′ 푞 푡 = 푘12 −푘21 0 (2.47) 0 0 푘푎

We can write the above matrix as 푄′ 푡 = 퐴. 푄 푡 , 푞 0 = 푏, the eigen values of the sub matrix

−푘 − 푘 푘 1 0 푀 = 푒푙 12 21 are real because 퐵−1푀퐵 = 퐶 is symmetric with 퐵 = . 푘12 −푘12 0 푘12푘21

We know that the Eq. (2.47) is a linear homogeneous differential equation. The amount of drug in the first compartment 푞1(푡) has to be evaluated in each iteration and the different measurement at many or different time points. If a slope based optimization method is used, then the gradient 푞1(푡) has to be calculated. For each compartment, the rate of change of drug concentration is described by ordinary differential equations.

2.2.3.1. Analytical solution for two compartments

We can compute the analytical solution of the blood compartment 푞1(푡) of Eq. (2.47) by the

Laplace transform ℒ, the advantage of this transformation is that differentiation and integration in the time domain correspond to simple algebraic operations, so take Laplace transforms of equation (2.47),

ℒ 푞′ 푡 = ℒ 퐴. 푞 푡 = 푠푄 푠 − 푞 0 = 퐴. 푄 푠 = 푠퐼 − 퐴 푄 푠 = 푞(0), then

0 푠 + 푘12 + 푘푒푙 −푘21 −푘푎 푄1(푠) 푞1 −푘12 푠 + 푘21 0 푄2(푠) = 0 0 0 0 푠 + 푘푎 푄3(푠) 푞3

We can write this system as

퐿 푠 푄 푠 = 퐵 (2.48)

Solving the system (2.48) by Cramer’s rule and determinant of 퐿(푠), we get

퐷et 퐿 푠 = Det[푠퐼 − 퐴] = (푠 + 푘푎 )[ 푠 + 푘12 + 푘푒푙 푠 + 푘21 − 푘21푘12]

2 = (푠 + 푘푎 )[푠 + 푠 푘21 + 푘12 + 푘푒푙 + 푘푒푙 푘21] = 푠 + 푘푎 푠 + 훼 (푠 + 훽)

1 1 where 훼, 훽 = { 푘 + 푘 + 푘 ± ( [푘 + 푘 + 푘 ]2 − 4푘 푘 )2 } 2 12 21 푒푙 12 21 푒푙 21 푒푙

(훼 + 훽) ± (훼 + 훽)2 − 4훼훽 휉 = 2

where 훼훽 = 푘21푘푒푙 푎푛푑 훼 + 훽 = 푘12 + 푘21 + 푘푒푙

훼, 훽 ∈ ℝ, therefore 퐷et 퐿 푠 = 푠 + 푘푎 푠 + 훼 (푠 + 훽) for all 푠 ≥ zero, to compute the solution of the central compartment 푞1we substitute the vector 푏 into the first column of matrix

퐿 푠 denoted by 퐿1(s) so,

det[퐿1 푠 ] 푄 푠 = (2.49) 1 det[퐿 푠 ]

Now we consider the drug administrated case (intravenous injection), we know that

0 퐷et[퐿푖푣 푠 ] = 푠 + 푘푎 푠 + 훼 (푠 + 훽), 퐷et 퐿1,푖푣 푠 = 푞1 (푠 + 푘21), consequently we taking inverse Laplace transform of Eq. (2.49) we have

푞0(푠 + 푘 ) 푠 1 ℒ−1 푄 푠 = ℒ−1 1 21 = 푞0ℒ−1 + 푞0푘 ℒ−1 1 푠 + 훼 (푠 + 훽) 1 푠 + 훼 (푠 + 훽) 1 21 푠 + 훼 (푠 + 훽)

Hence we obtain the solution the first compartment Eq. (2.49)

훼 훽 1 1 ℒ−1 푄 푠 = 푞 푡 = 푞0 푒−훼푡 + 푒−훽푡 + 푞0푘 푒−훼푡 + 푒−훽푡 1 1,푖푣 1 훽 − 훼 훼 − 훽 1 21 훽 − 훼 훼 − 훽

푞0(푘 − 훼) 푞0(푘 − 훽) hence 푞 푡 = 1 21 푒−훼푡 + 1 21 푒−훽푡 (2.50) 1,푖푣 훽 − 훼 훼 − 훽

We can use the same technique in case of oral drug administration to get solution of 푞1,푝표 (푡) is

0 0 0 푞3 푘푎 (푘21 − 훼) 푞3 푘푎 (푘21 − 훽) 푞3 푘푎 (푘21 − 푘푎 ) −훼푡 −훽푡 −푘푎 푡 푞1,표푟 푡 = 푒 + 푒 + 푒 (2.51) (푘푎 − 훼)(훽 − 훼) (푘푎 − 훽)(훼 − 훽) 푘푎 − 훽 (푘푎 − 훼)

In practice, the drug concentration is measured in blood. Therefore, the volume of distribution

푉푑 > 0 for the central compartment 푞1 푡 is introduced to obtain the drug concentration which is a proportionality factor between the amount of drug and the drug concentration [76].

푞 (푡) 퐶 푡 = 1 (2.52) 푉푑

In this work, 퐶(푡) will always denote the drug concentration in blood, the model parameters of the two-compartment model are obtained Eq. (2.48)

Ψmic ,iv = 푘푒푙 , 푘12 , 푘21 , 푉푑 or Ψ푚푖푐 ,표푟 = (푘푒푙 , 푘12, 푘21 , 푉푑 , 푘푎 )

0 0 which are called the micro constant parameterization, the initial values 푞1 or 푞3 are symbols for dose when speaking of concentration terms, based on equations (2.50-2.52), they are defined as

푘21 − 훼 푘21 − 훽 푎푖푛푡푒푟푣푒푛표푢푠 = , 푏푖.푣 = 푉푑 (훽 − 훼) 푉푑 (훼 − 훽)

푘푎 푘푎 푎푂푟푎푙 = 푎푖푣 & 푏푂푟 = 푏푖푣 푘푎 − 훼 푘푎 − 훽 where Ψ푚푎푐 ,푖푣 = 푎푖푣, 푏푖푣, 훼, 훽 or Ψ푚푎푐 ,표푟 = (푎표푟 , 푏표푟 ,훼, 훽, 푘푎 ), so

−훼푡 −훽푡 −훼푡 −훽푡 −푘푎 푡 퐶푖푣 푡 = 푎푖푣푒 + 푏푖푣푒 and 퐶표푟 푡 = 푎표푟 푒 + 푏표푟 푒 − (푎표푟 + 푏표푟 )푒 , and

푘 푘 − 훼 푘 푘 − 훽 푘 (푘 − 푘 ) 푎 21 + 푎 21 = − 푎 21 푎 푘푎 − 훼 훽 − 훼 (푘푎 − 훼) 푘푎 − 훽 훼 − 훽 (푘푎 − 훽) 훼 − 푘푎 (훽 − 푘푎 )

2 푎푖푣훽 + 푏푖푣훼 훼훽 훼훽(푎푖푣 + 푏푖푣) 푎푖푣푏푖푣(훽 − 훼) 푘21 = , 푘푒푙 = = , 푘12 = , 푎푖푣 + 푏푖푣 푘21 푎푖푣훽 + 훼푏푖푣 푎푖푣 + 푏푖푣 (푎푖푣훽 + 푏푖푣훼)

퐷 푉푑 = 푎푖.푣 + 푏푖.푣

2.2.3.2. Application of two compartment model (Loo-Riegelman method)

We can use Loo-Riegelman method for intravenous and oral administration. It can be applied generally to linear compartment pharmacokinetic models. The derivation that follows is based on a drug with two-compartment characteristics. The amount of drug absorbed into the systemic circulation at any time is given by 푋푎 = 푋푐 + 푋푒푙 + 푋푃 , where 푋푒푙 is the cumulated amount of drug eliminated by all pathways and 푋푐 and 푋푝 are the amounts of drug in the central and peripheral compartments respectively. Differentiating this equation with respect to time, we get

푑푋 푑푋 푑푋 푑푋 푎 = 푐 + 푒푙 + 푝 . 푑푡 푑푡 푑푡 푑푡

The rate of elimination of drug 푑푋푒푙 /푑푡 assuming first-order kinetics, is by definition

푑푋 푒푙 = 푘 푋 푑푡 푒푙 푐 where 푘푒푙 is the apparent first-order rate constant of drug from the central compartment. By substituting 푘푒푙 푋푐 for 푑푋푒푙 /푑푡 and dividing both sides of the above equation by the apparent volume of the central compartment푉푐equation written as

1 푑푋푎 1 푑푋푐 1 1 푑푋푃 = + 푘푒푙푋푐 + 푉푐 푑푡 푉푐 푑푡 푉푐 푉푐 푑푡

Integrating from zero to 푇, the amount of drug absorbed in time 푇 is obtained as

푇 (푋푎 )푇 (푋푃 )푇 = 퐶푇 + 퐾푒푙 퐶푑푡 + 푉푐 푉푐 0 where, 퐶푇 & ( 푋푃)푇 are the plasma concentration and amount of drug in the peripheral compartment at time 푇 respectively, the expression for the amount of drug ultimately absorbed,

(푋푎 )∞ is obtained by integrating from zero to infinity

∞ (Xa)∞ = 푘푒푙 퐶푑푡 푉퐶 0

The apparent absorption rate constant 푘푎 could be obtained from the least square fitted linear plot of the percent unabsorbed versus time [49, 65]. The Loo-Riegelman method requires drug concentration time data after both oral and intravenous administration of the drug to the same subject and the fraction absorbed at any time 푇 is given by:

푇 ∞ (푋푃)푇 퐹푇 = [퐶푇 + 푘푒푙 퐶푑푡 + ]/푘푒푙 퐶푑푡 푉푐 0 0

(푋푎 )푇 where 퐹푇 = , the fraction absorbed at any time 푇, (푋푝 )푇 is the amount of drug in the (푋푎 )∞ peripheral compartment as a function of time after oral administration

(푋푝)푇 (푋푝 )0 푘12 퐶0 ∆퐶 ∆푡 −푘21∆푡 −푘21∆푡 2 = 푒 + 1 − 푒 + 푘12 ( ) 푉푐 푉푐 푘21 ∆푡 2 where, 푘12 is distribution rate constant from compartment one to two and 푘21is distribution rate constant from compartment two to one and ∆푡, ∆퐶 are time and concentration intervals respectively. In addition to symbols defined previously, (푋푝)푇 is the amount of drug in the peripheral compartment as a function of time after oral administration and 푉푐 is the apparent volume of the central compartment, 푘푒푙 is the apparent first order elimination rate constant of drug from the central compartment. Subsequent intravenous study of the same subject (푋푝 )푇/푉푐 can be estimated by a more complicated approximation procedure requiring both oral and intravenous.

2.2.4. 풏-compartment analysis

Suppose that the drug is distributed among 푛-compartment, but only one can be sampled.

Also suppose that the concentration 퐶 푡 , in the same compartment after a bolus intravenous administration at time 푡 = 0 can be approximated reasonably well by a sum of exponential functions,

−훼1 푡 −훼2푡 −훼푛 푡 퐶 푡 = 퐴1푒 + 퐴2푒 + ⋯ + 퐴푛 푒

By using Laplace transform the equation take the form

퐴 퐴 퐴 퐶 푠 = 1 + 2 + ⋯ + 푛 푠 + 훼1 푠 + 훼2 푠 + 훼푛

푛−1 푛−2 푏0푠 + 푏1푠 + ⋯ + 푏푛−1 퐶(푠) = 푛 푛−1 푛−2 푠 + 푞1푠 + 푞2푠 + ⋯ + 푞푛

All pharmacokinetic models are derived from a set of basic differential equations. Basic differential equations can be integrated because it is most efficient to fit the data to the integrated

equations. But it is also possible to fit data to the differential equations by numerical integration.

A set of 푛-compartment represented by a set of 푛-differential equations as follows:

푑푥 1 = −퐾 푥 푡 + 푘 푥 푡 + ⋯ 푘 푥 푡 푑푡 1 1 21 21 푛1 푛 푑푥 2 = 푘 푥 푡 − 퐾 푥 푡 + ⋯ + 푘 푥 푡 (2.53) 푑푡 12 1 2 2 푛2 푛

… …

푑푥 푛 = 푘 푥 + 푘 푥 + ⋯ − 퐾 푥 (푡) 푑푡 1푛 1 2푛 2 푛 푛

These equations can be written in matrix form as:

푑푥 푑푥 푑푥 1 2 … . . 푛 = − 푥 (푡) 푥 (푡) … 푥 (푡) . 푑푡 푑푡 푑푡 1 2 푛

We can write the above equation as

푑푋 = −푋 푡 퐾 푑푡

The matrix K multiplied by the column vector 푋, and then the element of K in the 푖푡푕 row and

푗푡푕column is the transfer rate to 푖 from 푗 , the drug can be moved from one compartment to another in many different ways, but it never leaves the system. Remember that 푥푖 푡 is amount of drug in compartment 푖, 퐾푖 is fractional rate which leaves a compartment푖,

푘푖푗 , is fractional rate of transfer from compartment 푖 to compartment 푗 where

1….푛

퐾푖 ≥ 푘푖푗 푖≠푗

To transform the above equations from amount of drug to concentration, we divide each term by the corresponding volume 푉푖 , 푖 = 1,2, … , 푛, thus

푑퐶1 푉2푘21 푉푛 푘푛1 = −퐾1퐶1 푡 + 퐶2 푡 + ⋯ + 퐶푛1 푡 , 푑푡 푉1 푉1

푑퐶2 푉1푘12 푉푛 푘푛2 = + 퐶1 푡 −퐾2퐶2 푡 + ⋯ + 퐶푛1 푡 , (2.54) 푑푡 푉2 푉2

………,

푑퐶푛 푉1푘1푛 푉2푘2푛 = + 퐶1 푡 + 퐶2 푡 + ⋯ − 퐾푛 퐶푛 푡 푑푡 푉푛 푉푛

By using Laplace transform on equations (2.53), become as:

푠 푥1 − 푥1 0 = −퐾1 푥1 + 푘21 푥2 + ⋯ + 푘푛1 푥푛 ,

푠 푥2 − 푥2 0 = 푘12 푥1 − 퐾2 푥2 + ⋯ + 푘푛2 푥푛 , (2.55)

… … … .,

푠 푥푛 − 푥푛 0 = 푘1푛 푥1 + 푘2푛 푥2 + ⋯ − 퐾푛 푥푛 .

Re-arranging all terms, we get

푠 + 퐾1 푥1 − 푘21 푥2 − ⋯ −푘푛1 푥푛 = 푥1 0 ,

−푘12 푥1 + 푠 + 퐾2 푥2 − ⋯ − 푘푛2 푥푛 = 푥2 0 , (2.56)

… … …,

−푘1푛 푥1 − 푘2푛 푥2 − ⋯ + 푠 + 퐾푛 푥푛 = 푥푛 0

In general the dose is administrated in one compartment and there is no loss of generality if it is taken to be compartment one. With this condition, the terms at the right hand side of equations

(2.56) 푥1 0 ≠ 0, otherwise zero. The solution for these equations is

[푥 푡 ] ∆ (푠) 푖 = 푖,1 , 푖 = 1, 2, 3, … , 푛 , where , (2.57) 푥1(0) ∆(푠)

The determinant ∆ is formed by the coefficients on the left hand side of equation, and ∆푖,1(s) is the determinant obtained from ∆(푠) by suppressing row 푖 and column 푖. By expanding ∆(푠) we

푛 푛−1 푛−2 get a polynomial in 푠 of degree 푛 is obtaineds ∆ 푠 = 푠 + 푏1푠 + 푏2푠 + ⋯ + 푏푛 where 푏푖 is the sum of all products of the constants 퐾 taken 푖 by 푖, minus all products of the constants 푘 forming rings of length 푖.

By evaluating ∆푖,1(푠) we get a polynomial in s of degree, 푛 − 1 if 푖 ≠ 1,we have the polynomial in 푠 as

푛−1 푛−2 ∆푖,1 푠 = 푞0푠 + 푞1푠 + ⋯ + 푞푛−1.

If −휇1 , −휇2, … , −휇푛 , are the roots of equation ∆ 푠 = 0, they can be called eigen values of system of compartments, or characteristic values or the exponents of the sum of exponential functions. The Eigen values are typical model parameters.

푏1 = 휇푖 , 푏2 = 휇푖휇푗 , 푏3 = 휇푖 휇푗 휇푙 , … 푖 푖,푗 푖,푗,푙

Equation (2.57) can be written as:

푛−1 푛−2 푛 [푥푖 푡 ] ∆푖,1(푠) 푞0푠 + 푞1푠 + ⋯ + 푞푛−1 퐴푖푗 = = 푛 푛−1 푛−2 = , 푖 = 1 , 2 , … , 푛 푥1(0) ∆(푠) 푠 + 푏1푠 + 푏2푠 + ⋯ + 푏푛 푠 + 휇푗 푗 =1

By using inverse Laplace transform, we get

푛 푥푖 푡 −휇 푗 = 퐴푖푗 푒 푡. (2.58) 푥1(0) 푗 =1

It is equal to function which measures the amount of drug in compartment 푖 when a unit dose is given to compartment 1 as a bolus at time zero. The real eigen values of ∆ 푠 are non-positive, and the complex eigen values have the real part non-positive. For the time being we assume that if one of the Eigen values is zero,

푛

퐾푖 = 푘푖푗 , 푖 = 1,2,3, … , 푛 (2.59) 푗 ≠푖

Formula (2.58) converges to zero when 푡 → ∞ and when formula (2.59) holds, the equation

(2.58) contains at least one exponential term with non-negative coefficient [83].

Chapter 3 CCoonnvvoolluuttiioonn//DDeeccoonnvvoolluuttiioonn aanndd TThheeiirr AApppplliiccaattiioonnss

Convolution/Deconvultion and their applications 3.1. Introduction

Convolution is an important and useful concept. According to foundation and history of convolution, most likely the real convolution integral first appeared in the year 1754 when the mathematician D’Alembert derived Taylor's expansion theorem. Also another expression of this kind was by Sylvester Francois in his book entitled “Treatise on Differences and Series”, which is the last of volumes of the series Paris, 1797-1800. Soon after that, convolution operations appeared in the work of Simon Laplace, Joseph Fourier, and others. The term itself did not come into wide use until the 1950s, before which it was occasionally known as folding, superposition integral; Carson's integral appeared as early as, 1903. A particular case of composition products disseminated by the Italian mathematician, Volterra, in 1913, the idea to use convolution of two functions has spread widely and it has many applications in several sciences such as electrical engineering, Physics, Biology, Mathematics, Statistics and Medical Science. Borel’s theorem describes convolution as the action of an observing mechanism when it takesaweighted mean of some physical quantities over a narrow range of some variables. It was shown that the appearnce of convolution with linearity of time or space invariance, and also with sinusoidal response.

The concept of convolution was first introduced for finding the solution of a first order differential equation by the techniques of the variation of parameters. Convolution is the process that depicts combined effect of dissolution and elimination of drug from the body, so it reflects blood drug concentration-time profile. The in vitro dissolution profile to a plasma concentration can take place via convolution (input to output). Recently convolution methods have been established, which convolutes the in vitro dissolution profiles [26]. A major advantage of convolution methods for IVIVC is that no additional in vivo data such as intravenous injections

or oral solutions are required. Even if the results are mathematically correct, it may not be applicable in pharmacokinetic or physiological models. Convolution is defined for pairs of functions in which at least one is linear. In the context of pharmacokinetics, the relation between these functions and the processes from which they are abstracted should be kept in mind.

In particular, assumptions concerning linearity and time invariance may not be perfectly valid

[25]. Properties of the convolution operation given in result from the defining integral, convolution term can be classified in two types, either convolution of continuous type or convolution of discrete type: The convolution method uses in vitro dissolution data to derive blood drug levels [68]. However, in mathematics, the convolution power is the 푛-fold iteration of the convolution with itself then the convolution power is defined by 훼 ∗ 훼 ∗ 훼 ∗ … ∗ 훼 = 훼∗푛 is the power convolution of order n or is called the nth fold convolution of 훼 where

훼∗1 = 훼 and 훼∗0 = 훿 which is called the delta function.

3.2. Linear systems analysis

The linear system analysis is a modeling approach in pharmacokinetics which applies general linear Principles such as convolution and deconvolution to simplify and generalize linear pharmacokinetic relationships. The concept of linearity is defined in different context in pharmacokinetics. For example, in regression analysis, linearity often refers to linear regression equations which have the following general form, linear in the parameters,

푦 = 푏1푓1(푥) + 푏2푓2(푥) + … + 푏푛 푓푛 (푥). The terms 푓푖(푥) are arbitrary functions of the independent variable(s), linear pharmacokinetics, or linear drug disposition as kinetics described by linear compartmental models or first order processes. The linear compartmental model

involves first order compartmental transfer; we can say general linear systems are useful to define input-output relationships for continuous and discrete function [56, 63].

3.2.1. Definition of convolution

Convolution is a mathematical operator which takes two functions 훼(푡) and 훽(푡) to produce third function 푦.

Let 훼 푡 , 훽 푡 ∈ ℝ, then (훼 ∗ 훽)(푡) ∈ ℝ, is defined as convolution integral where ∗ is the convolution symbol. For two functions 훼 푡 , 훽(푡), it is written as 훼 푡 ∗ 훽(푡) and is defined as

푡 푦 푡 = 훼 푡 ∗ 훽 푡 = 훼(휏)훽 푡 − 휏 푑휏 (3.1) 0 where 훼 푡 , 훽(푡) as continuous functions in case of discrete functions, we write summation instead of integration.

3.2.1.2. Properties of convolution

The convolution, whether continuous case or discrete case has the same properties, the most important of which are as follows

I. Convolution has commutative property

If the order of functions is changed, then the result of the convolution operation remains the same, i.e. 훼 푡 ∗ 훽 푡 = 훽 푡 ∗ 훼 푡 , which can be written as

푡 푡 훼 푡 ∗ 훽 푡 = 훽 푡 ∗ 훼 푡 = 훽 푡 − 휏 훼 휏 푑휏 = 훽 휏 훼 푡 − 휏 푑휏 휏=0 휏=0

II. Convolution has associative property

Convolution for given three functions 훼 푡 , 훽 푡 , 훾(푡), the associative property of convolution means the changing the order of functions does not change value of convolution integral [83], this means 훼 푡 ∗ 훽 푡 ∗ 훾 푡 = 훼 푡 ∗ [훽 푡 ∗ 훾 푡 ]

III. Convolution is distributive over addition

Convolution is distributive over addition as follow:

훼 + 훽 (푡) ∗ 훾(푡) = 훼 ∗ 훾 (푡) + 훽 ∗ 훾 (푡), 푎푛푑 훼 ∗ 훽 + 훾 (푡) = 훼 ∗ 훽 (푡) + 훼 ∗ 훾 (푡)

IV. Linear convolution property

푡 Consider 훼(푡) ∗ 훽 푡 = 훼 휏 훽 푡 − 휏 푑푡 ∀푡 휏=0

It is a linear operation, i. e. 훼(푡) ∗ 푘1훽 + 푘2훾 푡 = 푘1훼(푡) ∗ 훽 푡 + 푘2훼(푡) ∗ 훾(푡), where 푘1 and 푘2 are arbitrary constants.

V. Shift convolution property

If 훼 푡 ∗ 훽 푡 = 휆1 푡 then 훼 푡 ∗ 훽 푡 − 휏 = 휆1 푡 − 휏 , and훼 푡 − 휏 ∗ 훽 푡 = 휆1(푡 − 휏).

If α t , 훽(푡) have duration of λ1, and λ2 respectively, then the duration of α t ∗ β t is λ1 + 휆2.

If 푦 푡 = 훼 푡 ∗ 훽(푡), then 훼 푡 − 휃 ∗ 훽 푡 − 휆 = 푦(푡 − 휃 − 휆)

3.2.1.3. Definition of Dirac Delta function

If 휙 푡 : ℝ → ℝ is a continuous function, such that 휙 푡 is an input function, then an impulse on the input produces an identical impulse on the output, this means all functions are passed through the system without change, any function convolved with delta function is exactly the same function, this means, there some properties as follow :

휙 푡 ∗ 훿 푡 = 휙(푡), and 휙 푡 ∗ 훿 푡 − 훽0 = 휙(푡 − 훽0)

푡 휙(푡) ∗ 훿 푡 = 휙( 휏)훿 푡 − 휏 푑휏 = 휙 푡 , 휏=0

푡

휙 푡 ∗ 훿 푡 − 훽0 = 휙 휏 훿 푡 − 훽0 − 휏 푑휏 = 휙 푡 − 훽0 0

1 푓표푟 푡 = 0 where 훿 푡 = 0 푓표푟 푡 ≠ 0

3.2.1.4. Properties of the Dirac Delta function with convolution

∞ 휖 1) 훿 푡 푑푡 = 1 , and ℒ 훿 푡 = 훿 푡 푒−푠푡 푑푡 = 1 −휖 0

2) 훼 푡 ∗ 훿 푡 − 휏 = 훼(휏)

∞ 3) 훼 휏 훿 푡 − 휏 푑휏 = 훼 푡 −∞

1 4) 훿 푎푡 = 훿(푡) 푎

5) 훿 푡 = 훿(−푡) 67

∞

6) 훼 푡 훿 푡 − 푡0 푑푡 = 훼 푡0 , 훼 푡 is continuous a푡 푡 = 푡0 −∞

푡2 훼 푡 , 푡 < 푡 < 푡 7) 훼 푡 훿 푡 − 푡 푑푡 = 0 1 0 2 0 0 , otherwise 푡1

∞

8) 훼 푡 − 푡0 훿 푡 푑푡 = 훼 −푡0 , 훼 푡 is continuous at 푡 = 푡0 −∞

9) 훼 푡 훿 푡 − 푡0 = 훼 푡0 훿 푡 − 푡0 , 훼 푡 is continuous at 푡 = 푡0

푡 1 , 푡 > 푡0 10) 푢 푡 − 푡0 = 훿 휏 − 푡0 푑휏 = 0, 푡 < 푡0 −∞

1 훽 11) 훿 훼푡 − 훽 = 훿(푡 − ) 훼 훼

푡2 푡2 1 훽 훽 훿 푡 − 푑푡 , 푡 < < 푡 12) 훿 훼푡 − 훽 푑푡 = 훼 훼 1 훼 2 푡 푡 1 1 0, otherwise

3.2.1.5. Properties of convolution derivative

If there are two functions 훼(푡) and 훽(푡) then, the first derivative is written as,

푑 I. 훼 푡 ∗ 훽 푡 = 훼 푡 ∗ 훽′ 푡 = 훼′ 푡 ∗ 훽(푡) 푑푡

∞ ∞ 푑 푑 훼 푡 ∗ 훽 푡 = 훼 휃 훽 푡 − 휃 푑휃 = 훼 푡 ∗ 훽′ (푡 − 휃)푑휃 = 훼 푡 ∗ 훽’(푡) 푑푡 푑푡 −∞ −∞

∞ 푑 푑 = 훽 휃 훼 푡 − 휃 푑휃 = 훽 푡 ∗ 훼 푡 = 훽 푡 ∗ 훼′ 푡 = 훼′ 푡 ∗ 훽(푡) 푑푡 푑푡 −∞

푑2 II. 훼 ∗ 훽 푡 = 훼′ 푡 ∗ 훽′ (푡) 푑2

3.2.1.6. Convolution properties of integration

If 푦 푡 = 훼(푡) ∗ 훽(푡), then algebraic meaning of convolution in analysis can be seen from the following formula:

∞ ∞ ∞ I. 훼 푡 ∗ 훽 푡 = 훼 푡 푑푡 훽 푡 푑푡 0 퐨 0

∞ ∞ ∞ ∞ ∞ II. 푡 훼 ∗ 훽 푡 푑푡 = [ 푡훼 푡 푑푡][ 훽 푡 푑푡] + [ 훼 푡 푑푡][ 푡훽 푡 푑푡] 0 0 0 0 0

푡 푡 푡 III. 훼 ∗ 훽 휏 푑휏 = 훼 푡 ∗ 훽 휏 푑휏 = 훽 푡 ∗ 훼 휏 푑휏 −∞ −∞ −∞

3.3.2. Definition Delta function (Kronecker function)

Convolution of any function with a delta function is the same function. Mathematically let 훼(푛) ∈ ℤ ⟶ ℝ , where 훿푖,푗 is a Kronecker function defined as:

1, 푖 = 푗 훿 = 푖,푗 0, 푖 ≠ 푗

3.3.2.1. Properties of Kronecker function 휹풊,풋

(1) 훼 푛 ∗ 훿 푛 = 훼(푛)

(2) 훼 푛 ∗ 푘훿 푛 = 푘훼(푛)

(3) 훼 푛 ∗ 훿 푛 + 휏 = 훼(푛 + 휏)

(4) 훼 푛 ∗ 훿(푛 − 휏) = 훼 푛 − 휏

(5) 훿 푛 ∗ 훿[푛] = 훿[푛]

∞ 6 f 푛 = 푓 푗 훿[푛 − 푗] 푗 =−∞

푛 1 , 푛 ≥ 0 7 훿 푗 = 0, 푛 < 0 푗 =−∞

∞ 8 훼 ∗ 훿 푛 = 훼 푖 훿 푛 − 푖 = 훼 푛 , ∀ n 푖=0

3.4. Convolution modeling and applications

Convolution is the mathematical operation which describes many physical processes in the system or in the body. So, most of the applications are based on mathematical models.

The convolution integral is the main mathematical tool for analyzing a system in series and it is represented by Laplace transform. In the convolution approach, one needs to start with dissolution results or profiles and develop the in vivo or drug concentration-time profiles [68].

In the development of convolution model the drug concentration time profiles obtained from

dissolution results may be evaluated using criteria for in vivo bioavailability assessment based on

퐶푚푎푥 and AUC parameters. In mathematical terminology, dissolution results become an input function and concentrations become a weighting factor or function resulting in an output function. Software has been developed which implements a differential equation based convolution approach. But it is a task that can be time consuming and complex [62].

Mathematically we can write the convolution as

푡

퐶 푡 = 퐶훿 푡 ∗ 휂 푡 = 퐶훿 휏 휂 푡 − 휏 푑휏 0 where 퐶 푡 is the plasma drug of concentrations after oral dose 퐶훿(푡) is plasma concentration after an intravenous dose or after oral solution. The relationship between measured quantities (in vitro release and plasma drug concentrations) is modeled directly in a single stage rather than via an indirect two stage approach. As a result the modeling focuses on the ability to predict measured quantities. The results are more readily interpreted in terms of the effect of in vitro release on conventional bioequivalence metrics [62]. Concentration after I.V bolus dose, 퐷푖푣 can be after approximation by a poly-exponential to express about these processes,

푛

−훼푖푡 퐶푖푣 = 퐴푖푒 훼푖 > 0 푖=1

If the system is linear, the characteristic response function is given by

푛 1 −훼푖푡 퐶훿 푡 = 퐴푖 푒 퐷푖푣 푖=1

퐶푖푣 푡 is the concentration between the unit impulse response 퐶훿 (푡). It is obvious that most of drugs are administrated orally, so the dissolution and absorption both take place in gastrointestinal tract. The elimination an orally administered solution or immediate release formulation may be determined the body is considered as a linear system in convolution.

The input function describes the inclusion of the substance from the dosage form into dissolution medium, which in the normal case consenting in vivo dissolution. By means of the input response function or unit impulse response is the disposition phase of the drug. Convolution is a method used to estimate drug concentration in the body using a mathematical model based on the integral. In pharmacokinetic output function 퐶(푡) describes the plasma concentration- time curve and 훼(푡) input function describes drug absorption during the time and 훽(푡) is called weight function or unit impulse response. Assume that 훼 (푡) is input function (absorption function), and

β t is the unit impulse response function, where,

1 −퐾푎 푡 −푘푒 푡 훼 푡 = 퐹. 퐷. 푘푎 푒 and 훽(푡) = 푒 . 푉푑

Solving 퐶(푡) by use convolution technique, given 훼(푡) and 훽(t),

푡 1 퐹. 퐷. 퐾푎 −퐾푎 푡 −푘푒 푡 −푘푎 (푡−휏) −푘푒 휏 퐶 푡 = 퐹. 퐷. 푘푎 푒 ∗ 푒 = 푒 푒 푑휏 푉푑 푉푑 0

퐹. 퐷. 퐾푎 then 퐶 푡 = 푒−푘푒 푡 − 푒−푘푎 푡 푉푑

This is the equation for single oral dose where 퐶(푡) is the output function expressing drug concentration in blood/plasma, 퐷 is the dose of drug, 퐹 is the fraction of administrated dose

푘푎 is first-order absorption rate constant, 푘푒푙 is the elimination rate constant, 푉푑 the apparent volume of distribution. The linear system analysis derivations can be done simply by elementary convolution operations in conjunction with the input-response convolution relationship between concentration, 퐶(푡) and the rate of input 훼(푡). Most drugs with a linear disposition have a poly- exponential response to a system bolus input. Accordingly by definition the unit impulse response is well described by

푛 −푘훼 푡 푈퐼푅 푡 = 퐴푖 푒 푖 (3.2) 푖=1

−ka Unit impulse response 훼 t = F. D. Kae is first- order input. If 퐶 푡 = 푈 푡 ∗ 훼 푡 , then

푛

−푘푎 푡 −훼 푡 퐶 푡 = (퐹. 퐷. 퐾푎 푒 ) ∗ 퐴푖 푒 푖 푖=1

푛 퐴푖 −훼푖푡 −푘푎 푡 퐶 푡 = 퐹. 퐷푘푎 푒 − 푒 3.3 푘푎 − 훼푖 푖=1

−푘푎 푡 −훼1 푡 −푘푎 푡 −훼2 푡 −푘푎 푡 −훼푛 푡 퐶 푡 = 퐹. 퐷. 퐾푎 푒 ∗ 퐴1푒 + 퐹. 퐷. 퐾푎 푒 ∗ 퐴2푒 + ⋯ + 퐹. 퐷. 퐾푎 푒 ∗ 퐴푛 푒

퐴1 퐴2 −훼1푡 −푘푎 푡 −훼2푡 −푘푎 푡 퐶 푡 = 퐹. 퐷. 푘푎 푒 − 푒 + 퐹. 퐷. 푘푎 푒 − 푒 + ⋯ 푘푎 − 훼1 푘푎 − 훼2

퐴푛 −훼푛 푡 −푘푎 푡 + 퐹. 퐷. 푘푎 푒 − 푒 푘푎 − 훼푛

퐶 푡 = 퐶1 푡 + 퐶2 푡 + 퐶3 푡 + ⋯ 퐶푛 푡 (3.4)

Equation (3.4) is the total plasma concentration; a linear drug disposition is defined as a linear relationship between the input of drug into the systemic circulation and the resulting measured concentration of drug in the systemic circulation. Such that 퐴1 + 퐴2 + 퐴3 + ⋯ + 퐴푛 ≥ 0 and

훼1 < 훼2 < ⋯ < 훼푛 , and 푡 ≥ 0 , where C t is the total plasma concentration and 퐴푖 is the coefficient of the 푖푡푕 exponential term and αi is the exponent of the 푖푡푕 exponential term, that is always positive for this disposition, where typically n =2 or n = 3 for most of drugs. For n = 2, from Eq. (3.3), it is obtained:

퐴 퐴 1 −훼1푡 −푘푎 푡 2 −훼2푡 −푘푎 푡 퐶 푡 = 퐹. 퐷. 퐾푎 푒 − 푒 + 퐹. 퐷. 퐾푎 (푒 − 푒 ), 푘푎 −훼1 푘푎 −훼2

For n = 3, from Eq. (3.3), it resulted in:

퐴1 퐴2 −훼1푡 −푘푎 푡 −훼2 푡 −푘푎 푡 퐶 푡 = 퐹. 퐷. 퐾푎 푒 − 푒 + 퐹. 퐷. 퐾푎 푒 − 푒 푘푎 − 훼1 푘푎 − 훼2

퐴3 + 퐹. 퐷. 퐾푎 푘푎 − 훼3

Convolution and Laplace transforms

The relationship between convolution and Laplace/inverse Laplace transforms represented by the following diagram,

푔(푡) 푓 푡 푦 푡 = 푔 푡 ∗ 푓(푡)

ℒ 푓 푡 ℒ{푔 푡 ℒ−1{퐺 푠 . 퐹 푠 }

퐺(푠) 퐹(푠) 푌 푠 = 퐺 푠 . 퐹(푠)

which is equivalent to the formula, ℒ−1 퐹 푠 퐺 푠 = 푓 ∗ 푔 (푡) & ℒ 푓 ∗ 푔 (푡) = 퐹 푠 퐺 푠 , where ℒ , ℒ−1Laplace and inverse Laplace transform respectively.

3.5. Plasma input by using convolution

The convolution of the impulse response function with the plasma input function is given by

푛

−휑푖푡 퐶푡 푡 = 휓푖 푒 ∗ 퐶푝 (3.5) 푖=1

In the linear time invariant system, it is possible to derive the general equation for plasma input compartmental in the systems impulse response function (IRF) which is a sum of exponentials,

푛

−휑푖푡 퐼푅퐹 = 휓푖 푒 (3.6) 푖=1

When 푛 is the total number of tissue compartments, 휓푖 and 휑푖 are constants which are functions

, −푘1푡 of the individual rate constants (푘 푠). When 푛 = 1, 휓1 = 푘푡, 휑1 = 푘1, then, 퐼푅퐹 = 푘푡 푒 .

The convolution of the impulse response function with the plasma input function, is given by

푛

−휑푖푡 퐶푡 = 휓푖푒 ∗ 퐶푝 (3.7) 푖=1 where 퐶푡 and 퐶푝 are the tracer concentrations in tissue and arterial plasma respectively [68,114].

The delivery of the tracer to the tissue is given by

푛

푘푡 = 휓푖 (3.8) 푖=1

Suppose that 푉푑 is the volume of distribution which is equal to the integral of the impulse response function is given by

∞ 푛 휓푖 푉푑 = 퐼푅퐹푑푡 = (3.9) 휑푖 0 푖=1

The estimation of the total volume of distribution 푉푑 is such that

푡 푡 0 퐶푡 푑푡 퐶푡 푑푡/퐶푡 = 푉푑 (3.10) 퐶푡 0

Substituting equation (3.10) in the left hand with Eq. (3.8), we get

휓 푡 푡 푛 푛 푖 −휑푖푡 휓 푒−휑푖푡 ∗ 퐶 푑푡 푖=1 1 − 푒 ∗ 퐶푡 0 푖=1 푖 푝 휑푖 퐶푡 푑푡/퐶푡 = = 퐶푡 퐶푡 0

휓푖 휓푖 푡 휓푖 푛 ∗ 퐶 푛 푒−휑푖푡 ∗ 퐶 푛 푛 푒−휑푖푡 ∗ 퐶 푖=1 휑 푝 푖=푖 휑 푝 휓 0 퐶푝푑푡 푖=1 휑 푝 = 푖 − 푖 = 푖 − 푖 퐶푡 퐶푡 휑푖 퐶푡 퐶푡 푖=1

휓 푡 푛 푖 푒−휑푖 푡 ∗ 퐶 푉푑 퐶푡 푑푡 푖=1 휑 푝 = 0 − 푖 (3.11) 푛 −휑푖푡 퐶푡 푖=1 휓푖 푒 ∗ 퐶푝 for large t, equation is

푡 푡 퐶푡 푑푡 푉푑 퐶푝 푑푡 1 0 ≅ 0 − (3.12 ) 퐶푡 퐶푡 휑푛

The linear addition of the separate response is the key to understand the convolution principle.

By considering the input to a compartment as a series of separate impulses, the convolution can be broken down into its constituent components. Using the tools of the unit response function and the convolution, the special condition of bolus injection and constant function can be examined. Exponential convolution, which is a more physiological description, the estimation of the irreversible uptake rate constant from plasma (푘푖 ) and is given by

푡 퐶 푘푖 퐶푝푑푡 푡 ≅ 0 (3.13) 퐶푝 퐶푡

From the general expression for irreversible plasma input system the equation for the target tissue is given by,

푛−1

−휑푖푡 퐶푡 = 휓푖 푒 + 휓푖 ∗ 퐶푝 (3.14) 푖=1

And the irreversible uptake rate constant from plasma, 푘푖 = 휓푛 , substituting equation (3.13) into the left hand side equation (3. 12), equation becomes;

퐶 푛−1 휓 푒−휑푖푡 + 휓 ∗ 퐶 푡 = 푖=1 푖 푛 푝 (3.15) 퐶푝 퐶푝

푡 푛 −1 −휑푖푡 퐶푡 휓푛 0 퐶푝푑푡 푖=1 휓푖 푒 ∗ 퐶푝 = + ; as 푘푖 = 휓푛 , so 퐶푝 퐶푝 퐶푝

푡 푛−1 −휑 푡 퐶 푘푖 퐶푝(푡) 푑푡 휓 푒 푖 ∗ 퐶 푡 = 0 + 푖=1 푖 푝 3.16 퐶푝 퐶푝 (푡) 퐶푝

푡 푛−1 퐶 푘푖 퐶푝푑푡 휓 for large t, equation becomes 푡 ≅ 0 + 푖 (3.17) 퐶푝 퐶푝 휑푖 푖=1

This formula describes the ratio of tracer concentrations between tissues and arterial plasma [57].

3.6. Deconvolution modeling

Deconvolution is a numerical method used to estimate the time course of drug input using a mathematical function and is based on a convolution integral. In a pharmacokinetics context deconvolution allows to give the disposition function and data obtained after extravascular administration. Extravascular administration is dependent on a rate of input 푘푎 and rate of output

푘푒푙 since 푘푎 often cannot be measured directly, pharmacokinetic analysis must frequently be

performed with 푘푒푙 . Deconvolution in pharmacokinetics has been widely used for almost 40 years. It is an algorithm-based process employing the reverse of the effects of convolution by trying to solve a convolution equation 훼(푡) ∗ 훽(푡) = 퐶(푡). This can be applied in typical pharmacokinetics as 퐶 being the plasma concentration measure at different time points.

Therefore, deconvolution is necessary to solve for 훽(푡) in order to estimate the pharmacokinetic parameters. It must be kept in mind that pharmacokinetics is a dynamic system in which instances of simultaneous absorption and distribution, absorption and elimination are encountered. Deconvolution is used to assess drug interactions by taking the intravenous data as the weighing function and the test formulation data, following administration, as the response function. This characteristic allows the determination of the absorption process with no requirement of first-order input or other pharmacokinetic processes. The deconvolution technique requires the comparison of in vivo dissolution profile which can be obtained from the blood profiles with in vitro dissolution profiles [62, 68]. When in vivo dissolution curves are obtained there is no parameter available with associated statistical confidence and physiological relevance which would be used to establish the similarity or dissimilarity of the curves. A more serious limitation of this approach is that it often requires multiple products having potentially different in vivo release characteristics (slow, medium, fast). These products are used to define experimental conditions for an appropriate dissolution test to reflect their in vivo behavior.

This approach is more suited for method development as release characteristics of test products are to be known rather product evaluation. To separate drug input from drug distribution and elimination, model-dependent approaches, such as Wagner-Nelson and Loo-Riegelman, or model-independent procedures, based on numerical deconvolution may be utilized. In the model dependent approaches, the distribution and elimination rate constants describe pharmacokinetics

after absorption, while in the numerical deconvolution approach the drug unit impulse response function describes distribution and elimination phases, respectively [62]. This method uses numerical stepwise integration and was first described by Turner and colleagues in 1971, they used this technique to investigate first-phase insulin delivery to the systemic circulation, the functional form of the input function is assumed so that the deconvolution problem becomes one of parameter for estimation and this can be performed in numerical methods. A wide range of functional forms has been used in deconvolution procedures to estimate input functions for a large variety of biological systems. When the serum level curves are known, the cumulative inputs corresponding to different routes of administration and dosage forms may be determined by deconvolution [8, 9]. Deconvolution is a powerful mathematical operation that can be used to extract the impulse response function of a linear time-invariant system for kinetic analysis of data which may be appropriate for the drug synthesis but is not applicable for the in vivo synthesis of antibody proteins. The deconvolution method was used to estimate the insulin secretion in fasting rats, enabling a more improved marker of insulin secretion to be defined.

The deconvolution problem is concerned with recovering the unknown input of a convolution system given measurements of the output. The in vivo profile to the in vivo absorption profile is known as deconvolution (output to input). Model-independent numerical deconvolution requires in vivo plasma data from an oral solution or intravenous as impulse function for the application.

Deconvolution requires data obtained after both oral and intravenous administration in the same subject and assumes no differences in the pharmacokinetics of drug distribution and elimination from one study to the other. Given the corresponding input-response function and the impulse response function for the system can be applied to estimate systemic bioavailability in linear pharmacokinetic systems given the time-concentration profile following an intravenous bolus.

Deconvolution of the subcutaneous data using prescribed input functions, namely rectangular pulse functions of variable duration mono and bi-exponential input functions, indicated that a mono-exponential input function best described absorption from the injection site by [15]. Eaton first published their method in 1980, to estimate insulin secretion; deconvolution approach involves the use of simulation procedure by Jens Anderson 2001 improves the performance of flow injection analysis by deconvolution of experimental data [16].

3.6.1. Methods of deconvolution

There are several methods of pharmacokinetic deconvolution we can describe them as follow:

(1) Deconvolution through convolution method

The inverse of convolution is not as simple as what it seems to be. Indeed, the mathematical inverses of multiplication and addition are division and subtraction, respectively. Deconvolution is the inverse operation of convolution and is mainly applied to determine the input function.

The formula of convolution is 퐶(푡) = 훼 ∗ 훽 in deconvolution for solving 훼(푡), given 퐶(푡) and

훽(푡) or solving 훽(푡), given 훼(푡) and 퐶 푡 . So, the deconvolution deals with either of two problems. We can explain the relation between a convolution and deconvolution by the following scheme.

훼 푡 Convolution 퐶 푡 Deconvolution 훼 푡

훽 푡 훽(푡)

Mainly, given 퐶(푡) and 훼(푡) to determine 훽(푡) or given 퐶 푡 and 훽(푡) to determine 훼(푡).

Several deconvolution methods exist for determining the input to solve deconvolution problem.

Basically these can all be classified in two categories. Consequently, the deconvolution operation to obtain the input function can be solved algebraically as a quotient using the transformed functions and is denoted by the symbol ∕∕ e.g. 훽 푡 = 퐶(푡) ∕ 훼 푡 , or

훼 푡 = 퐶(푡) ∕∕ 훽(푡). The mathematical processes of convolution and deconvolution must be done numerically. Therefore, the input function 훼(푡) and the output function 퐶(푡) have to be sampled at discrete intervals 훼 푖 , and 훽 푖 respectively; the most straightforward procedure for convolution is given by

푘=∞ 퐶 푘 = 훼 푖 훽 푘 − 푖 푖=−∞

From this formula, 훽 푖 , can be calculated as

퐶 1 퐶 2 − 훼 2 훽 0 훽 0 = & 훽 1 = 훼 1 훼 1

퐶 3 − 훼 2 훽 1 − 훼 3 훽 0 훽 2 = 푒푡푐 . Similarly we can find 훼 푖 in same way. 훼 1

(2) The direct deconvolution method

In this method, the function Ψ(t) is unit impulse response. Furthermore a suitable function is fitted to the input- response data from the unknown input to determine Φ 푡 . The input function y(푡) is then determined directly from the functionsΦ(t) and Ψ(t) using the following deconvolution formula that is applicable when Ψ(t) ≠ 0:

Ψ′ 0 Φ′ 푡 − Φ 푡 − 푕 푡 ∗ Φ 푡 y 푡 = Ψ 0 Ψ (0)

The auxiliary function 푕(푡) in this formula is called the distribution function and is obtained from the following Laplace transform inversion

Ψ 0 ′ 푠 − ℒ Ψ(t) − Ψ 0

푕 푡 = ℒ−1 Ψ 0 ℒ(Ψ(t))

Consider a simple case where Ψ(t) =퐴푒−훼푡 , here 푕 (푡) becomes zero, leading to the following one exponential deconvolution formula: y(푡) = Φ′ 푡 + 훼Φ 푡 /퐴

Consider another example, the more common case where

−훼1푡 −훼2푡 Ψ(t) =퐴1푒 + 퐴2푒

The distribution function is then given by

퐴1훼1+퐴2 훼2 퐴1 퐴2 푠 + + − 퐴1 + 퐴2 퐴1+퐴2 푠+훼1 푠+훼2 푕 푡 = 퐴 퐴 1 + 2 푠+훼1 푠+훼2

Resulting, 푕 푡 = 휙푒휆푡 , where

훼1 +훼2 2 휙 = 퐴1퐴2( ) 푎푛푑 휆 = −(퐴1훼1 + 퐴2훼2)/(퐴1+퐴2), thus, the input rate is 퐴1 +퐴2

퐴 훼 +퐴 훼 Φ′ 푡 + 1 1 2 2 Φ 푡 + 휙푒휆푡 ∗ Φ 푡 퐴 +퐴 푖 푡 = 1 2 퐴1 + 퐴2

푛

−훼푖푡 and Ψ 푡 = 퐴푖푒 is given by 푖=1

푛 푛−1 푛 푖=1 퐴푖훼푖 휆푖푡 푖 푡 = {Φ′ 푡 + 푛 Φ 푡 + Φ(푡) ∗ 휙푖푒 }/ 퐴푖 퐴푖 푖=1 푖=1 푖=1

The 휆푖 are obtained as the (푛 − 1) roots of the following polynomial:

푛 푛

푝 푥 = 퐴푖 (훼푗 + 푥) 푖=1 푗 =1 푗 ≠푖

Each corresponding 휙푗 parameter is obtained from the 휆푖 parameter as

푛 푛 푛 퐴 1 푗 −1 휙푖 = −( 퐴푙 ){ + 훼푘 } 휆푖 + 훼푗 휆푖 + 훼푘 푙=1 푗 =1 푘=1 푘≠푗

The methodically exact direct deconvolution does not apply when unit impulse response is equal to zero at time zero, this mean Ψ 0 = 0. So, the method is easily extended to include Ψ(t) = 0.

Differentiation of Φ with respect to time in this case gives

Φ'(t) = Ψ(0)푦(푡) + Ψ′ (t)∗ y (t) = Ψ′ (t)∗ y(푡), [56]. So, it becomes:

Ψ′′ 0 Φ′′ 푡 − Φ′ 푡 − 푕 푡 ∗ Φ′ 푡 Ψ(t) ′ 0 y(푡) = Ψ′ (0)

Ψ′′ 0 where 푕 푡 = ℒ−1 푠 − ℒ Ψ′ − Ψ′ 0 / ℒ(Ψ′) Ψ′ 0

Ψ′′ 0 = ℒ−1 푠 − 푠ℒ Ψ − Ψ′ 0 / 푠ℒ(Ψ) Ψ′ 0

−훼1푡 −훼2푡 If Ψ (t) = A (푒 − 푒 ), 훼2 > 훼1, where, Ψ is an oral solution and the input response is the

′ concentration of the drug in the general systemic circulation Ψ (0) = A (훼2 − 훼1) and

2 2 Ψ''(0) = A(훼1 − 훼2 ), for 푕 above gives 푕= −훼1훼2, then

퐴 Φ 푡 + 훼 + 훼 Φ′ 푡 + 훼 훼 Φ 푡 y 푡 = 1 2 1 2 , (푎2 − 푎1) thus, if the same subject was given an oral test formulation resulting in a drug concentration

−훽1 푡 −훽2 푡 −훽3 푡 response, Φ 푡 =푎1푒 + 푎2푒 + 푎3푒 , such that 푎1 + 푎2 + 푎3 = 0 then the input function y(푡), which in this case is the rate of release of dissolved drug into the gestrointesental tract, is obtained from the above expression for y(푡) with:

′ −훽1 푡 −훽2 푡 −훽3 푡 Φ (푡) = −훽1푎1푒 − 훽2푎2푒 − 훽3푎3푒 ,

′′ 2 −훽 1 푡 2 −훽2 푡 2 −훽3 푡 Φ 푡 = 훽1 푎1푒 + 훽2 푎2푒 + 훽3 푎3푒 .

The deconvolution through convolution method is particularly suitable to use when the functional forms of the prescribed input function 휉(푡) and unit impulse response function Ψ(푡) are such that an analytical expression can be found for their convolution. In such cases numerical convolution which is less exact than analytical convolution, is avoided [56]. For example, let

−훼1푡 −훼2푡 Ψ 푡 = 퐴1푒 + 퐴2푒

−훽1 −훽2 Consider an exponential input function as 휉 푡 = 퐵1푒 + 퐵2푒

2 2 퐴푖 퐵푗 휑 푡 = (휉 ∗ Ψ) 푡 = 푒−훽푗푡 − 푒−훼푖푡 훼푖 − 훽푗 푖=1 푗 =1

(3) Point area deconvolution method (Numerical method)

The rate of input in a small time interval (푡푖 − 푡푖−1) is assumed to be 푓(푡푖). Then, the poly- exponential unit impulse response is writing as

푛

−훼푗 푡 퐶훿 푡 = 퐶푗 푒 푗 =1

The input response is given by

푘 푡푖

퐶 푡 = 휓(푡푖) 퐶훿 푡푖 − 휏 푑휏 ,

푖=1 푡푖−1 and the rate of input can be achieved from the following formula:

푛 퐶푗 훼푗 푡1 휓(푡1) = 퐶(푡1)/( 1 − 푒 훼푗 푗 =1

퐶 푘−1 푛 푗 −훼푗 푡푘 −푡푖 −훼푗 푡푘 −푡푖−1 퐶(푡푖) − 푖=1 휓 푡푖 푗 =1 푒 − 푒 훼푗 휓 푡푖 = 퐶 푛 푗 훼푗 (푡푘 −푡푘−1 ) 푗 =1 ( 1 − 푒 ) 훼푗

The point-area numerical deconvolution method has been implemented and tested using data from oral absorption [57].

Table (3.1) Convolution integral for different functions

No. ϕ(t) ψ(t) (ϕ ∗ ψ) t

1 e−αt e−βt (e−αt − e−βt)/(β − α) for α ≠ β

2 e−αt e−αt te−αt for α = β

−αt −αt 3 휇1 e 휇1(1 − e )/α

4 휇1 휇2 휇1휇2푡

5 휇 tn 휇tn+1 , n = 0,1,2, … n + 1

6 푡 e−αt 1 − e−αt t − /α α

7 t2 e−αt 2t 2 1 − e−αt t2 − + /α α α2

8 t3 e−αt 3t2 6t 6 1 − e−αt t3 − + − /α α t2 α3

9 φ(t)) δ(t − τ) φ(t − τ)

10 eαtφ(t) φ(t) 1 eαt − 1 φ(t) α

11 φ(t) φ(t) tφ(t)

No. ϕ(t) ψ(t) (ϕ ∗ ψ) t

12 eαtφ(t) eβtφ(t) 1 eβt − eαt φ t , α ≠ β β − α

13 eαtφ(t) eαtφ(t) teαtφ t

14 teαtφ(t) eαtφ(t) 1 t2eαtφ t 2 n n! n! eαtφ t − tn−iφ(t) 15 tnφ(t) eαtφ(t) αn+1 αi+1 n−i ! i=0 n! m! tn+m+1φ(t) 16 tnφ(t) tmφ(t) n + m + 1 !

1 βt αt αt 2 e − e + α − β te φ(t) 17 teαtφ(t) eβtφ(t) (α − β) n! m! tn+m+1eαtφ(t) 18 tn eαtφ(t) tm eαtφ(t) n + m + 1 ! n −1 in! n + i ! tn−ieαtφ t i! n − i ! (α − β)m+i+1 i=0 19 tn eαtφ(t) tm eβtφ(t) m −1 jm! n + j ! + tm−jeβtφ t , j! m − j ! (β − α)n+j+1 j=0 α ≠ β cos θ − ϕ eγt − e−αt cos βt + θ − ϕ φ t , 20 e−αt cos βt eγtφ(t ) (α + γ)2 + β2

+ θ φ(t) −1 β ϕ = tan [ − (α + γ)]

αt αt 1 21 e φ(t) e φ(−t ) eαtφ t + eβtφ(−t) , Re β > 푅푒훼 β−α

No. ϕ(t) ψ(t) (ϕ ∗ ψ) t

22 eαtφ(−t) eβtφ(−t) 1 eαt − eβt φ(−t ) β − α

23 푡푛 푡푚 n! m! tn+m+1 n + m + 1 ! 24 푡푛 푡−푚 푛! Γ(1 − m) 푡푛−푚+1 Γ(푛 − 푚 + 2)

where, 휇, 휇1, 휇2 are constants, 훿 is Delta unit function, and Γ(. ) is called Gamma function such that Γ n = n − 1 ! for any positive integer푛.

CChhaapptteerr 44 MMaatteerriiaallss aanndd MMeetthhooddss

Materials and methods

Metoprolol tartrate was gifted by Novartis Pharma, Pakistan. Eudragit® was purchased from

Rohm Pharma, Germany. Metoprolol tartrate tablets (Mepressor® 200 mg, Novartis Pharma-

Pakistan) was purchased from open market. Liquid paraffin, methanol, petroleum ether, and other chemicals of analytical grade were procured from Merck, Germany and were used as received.

4.1. Preparation of investigational Tablets

Metoprolol tartrate loaded Eudragit® FS microparticles were prepared by solvent evaporation technique. Metoprolol tartrate (1g) and Eudragit® FS (1, 1.5 or 2 g) were dissolved in acetone

(20 ml) using magnetic stirrer (stirring speed 450 rpm) to prepare drug-polymer solution. Light liquid paraffin (40 ml) solution containing span 80 (0.2 g) was prepared and added to drug- polymer solution with continuous stirring for four hours at room temperature. After complete removal of acetone, the resultant microparticles were harvested by vacuum filtration.

The microparticles were washed three times with (100 ml) and dried in an oven at 30 °C.

The microparticles (drug/polymer ratios of 1:1, 1:1.5 and 1:2 (w/w)) were directly compressed into tablets (T1, T2 and T3 named as investigational tablets/tabletted microparticles or test formulations containing 1, 1.5 and 2g polymer, respectively) so that each tablet contained metoprolol tartrate equivalent to 200 mg of metoprolol. The hardness of test tablets was achieved at desired level by varying compression force. Each tablet contained 200 mg metoprolol. Based on in vitro and in vivo studies, these investigational tablets were compared with reference commercial sustained release tablet, Mepressor® 200 mg SR (Batch No. 457X, Novartis Pharma,

Pakistan) [94].

4.1.1. Compatibility analysis

To analyze any possible chemical interaction between drug and polymer during the process of microencapsulation, Fourier transform infra-red (FTIR) (MIDAC M 2000, USA) spectroscopy, x-ray diffractometry (X-RD) (D8 Discover, Bruker, Germany) and differential scanning calorimetry (DSC) (TA Instruments, USA) analytical tests were performed for pure metoprolol,

Eudragit® FS and microparticles (T2 microparticles before compression into tablets) [94].

Morphology study and particle size determination

Scanning electron microscopy (Philips-XL-20, Netherlands) was used for the morphological study of microparticles. After mounting the microparticles directly onto the sample stub, a thin

(200 nm) coating of gold was applied to assess their morphology under reduced pressure (0.133 Pa). Light microscope (XSZ-150A, Ningbo, China) was used for the determination of microparticle size. The light microscope consisted of a microscope stage, a digital camera and a computer. The suspension of microparticles was placed on a glass slide and its photomicrographs were taken with digital camera. The microparticulate diameter was determined from image analysis [94].

Tablet evaluation

Weight variation, tablet hardness, friability, disintegration and dissolution tests for reference and test tablets were performed according to USP (6). Six test tablets were tested for disintegration in

0.1 N HCl solution for 2 h by using USP basket rack assembly and then in phosphate buffer pH

6.8. The disintegration tests for reference tablets were also performed [94].

High-performance liquid chromatography analysis of Metoprolol samples

The quantitative estimation of Metoprolol tartrate in the dissolution and biosamples was performed by HPLC method (95-2). The HPLC (Isocratic HPLC, Agilent, California, USA) was connected with UV/Vd etector (Agilent, USA) operated at 273 nm and Hypersil ODS-C18 column (250 mm × 4.6 mm internal diameter, particle size 5 mm; Agilent, USA) operated at 27

°C. The HPLC system was operated with the Chem Station software. A degassed mixture of acetonitrile and triple distilled water containing 0.4% of triethylamine (pH adjusted to 3.6 with

5% ortho-phosphoric acid) in the ratio of 15:85 was employed as mobile phase and was eluted at a flow rate 1.0 mL/min. Total run time for each sample was set at 15 minutes. Tramadol hydrochloride, as the internal standard, showed no interference with the peaks of Metoprolol.

The validation parameters [96] for the proposed method were also determined. A calibration curve (푛=7) was constructed for the estimation of Metoprolol tartrate in a concentration of 20–

200 ng/ml. The coefficient of determination (R2) of this curve was 0.9986.

Encapsulation efficiency and yield

An accurately weighed quantity of drug loaded microparticles containing drug equivalent to 100 mg of Metoprolol tartrate was dissolved in 10 mL of Methanol by vortexing for about 24 h for complete extraction of Metoprolol tartrate. After the filtration of solution, the filtrate was analyzed by HPLC as described previously. Encapsulation efficiency (%) were determined by dividing actual drug loading with theoretical drug loading and multiplied by 100. While drug loading (%) was calculated by dividing amount of drug with the amount of microparticles and multiplied by 100. The product yield was determined by dividing amount of microparticles with the total initial amount of drug plus polymer. Each determination was made in triplicate [94].

Stability studies

The optimum formulation was filled in 10 amber colored glass bottles and stored at accelerated stability conditions (40 °C ± 2 °C / 75% RH ± 5% RH) for 6 months. One bottle was used each month for the investigation of dissolution behavior and drug contents of stored formulation [94].

Dissolution test by sequential pH change method

The reference and test formulations were passed through the dissolution test by sequential pH change method. To simulate gastrointestinal transit conditions, dissolution conditions were: USP dissolution apparatus II at temperature 37 ± 5°C and 50 rpm in three dissolution media [pH 1.2

(0.1 N hydrochloric acid) for 2 h, pH 4.5 (phosphate buffer) for 2 h and pH 7.0 (phosphate buffer) for 8 h] with a final volume up to 900 ml. Dissolution of T1 formulation (labeled as T1S) was also performed using same dissolution conditions with 0.5% sodium lauryl sulfate.

Dissolution samples (5 ml) were withdrawn at 0, 1, 2, 3, 4, 6, 8, 10 and 12 h, filtered through a 0.45μm filter and analyzed as described previously [94]. The formulations were observed visually to assess any physical changes to the particles occurred during the dissolution testing process. All dissolution studies were performed in triplicate.

4.2. Drug release kinetics

To assess the mode of in vitro drug release from the formulations, the release profiles were analyzed using zero-order (equation 4.1), first-order (equation 4.2), Higuchi’s model (equation

4.3) and Korsmeyer–Peppas (equation 4.4) [96-98]. The best-fit model was identified by calculating the regression coefficients (R2), where the highest R2 value elaborates the best fit.

푀푡 = 푀0 − 푘0푡 (4.1)

푙푛푀푡 = 푙푛푀0 − 푘1푡 (4.2)

1/2 푄푡 = 퐾퐻푡 (4.3)

where 푀푡 = quantity of drug remaining undissolved at time t, 푀0= quantity of drug remaining undissolved at time 푡 = 0, 푄푡 = quantity of drug remaining undissolved at time 푡 where 푘0, 푘1, and 퐾퐻 are the release rate constants for zero-order kinetic, first-order kinetic and Higuchi’s model, respectively.

For the evaluation of mechanism of drug release, drug release data was put in Korsmeyer-Peppas equation (equation 4.4) [99].

푀 ℓ표푔 푡 = ℓ표푔 푘 + 푛ℓ표푔 푡 (4.4) 푀푓 where 푀푡 = quantity of drug release at time 푡, 푀푓 = quantity of drug release at infinite time, 푘 is the release rate constant and 푛= release exponent that indicates the drug release mechanism.

For a cylindrical shape matrix, the value of 푛 ≤ 0.45 represents Fickian release; 푛 greater than

0.45 but lesser than 0.89 indicates anomalous (non-Fickian) release; and 푛 greater than 0.89 represents super case-II type release. Model independent analysis of dissolution data was also conducted by calculating similarity factor 푓2.

Finally all the formulations were compared with the marketed formulation based on similarity factor (푓2) using equation (4.5)

푛 −1/2 1 푓 = 50 × ℓ표푔 1 + ( 푅 − 푇 )2 × 100 (4.5) 2 푛 푡 푡 푡=1 where 푛 is sampling number, Rt is percent of drug dissolved of the reference product and Tt is percent of drug dissolved of test products. For similar release profiles, similarity factor should be in the range of 50-100 (more closed to 100) [13,100].

4.3. Convolution of In Vitro dissolution data to approximate plasma drug levels

Biowaiver study is, actually, the comparative dissolution analysis approximating the in vivo absorption of a drug product. IVIVC, estimated via deconvolution approach), a predictive mathematical model, is a vital tool that is used in developing and evaluating the drug products.

The IVIVC represents an association between the in vitro dissolution profiles and in vivo performance of drug. The in vivo performance is computed from plasma drug concentration-time data. Then a plot is drawn between these in vitro and in vivo profiles to get a straight line

[13,100,101].

Based on FDA guidelines, level A IVIVC is expected for modified release formulations of BCS class I drugs (like Metoprolol tartrate) where dissolution is the rate limiting step. The main advantage of IVIVC is the prediction capability of in vivo performance of an alternative formulation of predefined nature from specific dissolution characteristics and IVIVC function

[100,102].

No such literature exists which has employed IVIVC (straight line equation) to approximate the relevant in vivo profiles, since it is mathematical impossibility. Thus, it should be observed that establishing an IVIVC cannot be employed for the development and evaluation of drug products aside from its complex procedure [100-102].

The requirement for the development and assessment of drug product should be the approximation of plasma drug profiles, thus one should possibly make use of the suggested terminology of IVIVP, rather than IVIVC.

This study represents a part, IVIVP, of our project that was planned for the development and characterization (in vitro as well as in vivo) of tableted microparticles of Eudragit® FS loaded with Metoprolol tartrate followed by the development of IVIVP.

Convolution of in vitro dissolution data was done to get the 퐶 푡 (predicted plasma drug concentration) from the dissolution data utilizing 퐶훿 (Unit impulse response which is found from the intravenous bolus dose data or standard oral solution data) and 푋푣푖푡푟표 (drug input rate in vitro from oral solid dosage form) as follows (equation 4.6):

푡

퐶 푡 = 퐶훿 푡 − 휏 푋푣푖푡푟표 휏 푑휏 (4.6) 0

To predict plasma drug concentration from the in vitro dissolution profiles, the distinct drug concentrations, obtained from the percentage in vitro dissolution data during each sampling interval, were converted into the bio-available drug concentrations utilizing the published bioavailability data of drug. Then the calculation of reducing levels of plasma drug concentrations during each interval utilizing the published elimination data of drug carried out.

All the determined drug concentrations for each time point were added, and finally the predicted plasma drug level at each time point was determined (Table 4.1-4.5) using the published values of volume of distribution of drug as well as the adult body weight (70 Kg in average) [104].

® Table 4.1.Convolution of dissolution data of formulation Mepressor

Half life (h) 3.5 Time (h) 0 0.08 0.17 0.25 0.5 0.75 1 1.5 2 3 4 6 8 10 12 Percent release Ke per hour 0.198 1 2 4 6 8 10 15 21 27 33 43 55 64 71 (Cumulative) 0 Percent release (Within F 0.5 1 1 2 2 2 2 5 6 6 6 10 12 9 7 sampling interval) Amount (mg) release Vd (L/Kg) 5.6 2 2 4 4 4 4 10 12 12 12 20 24 18 14 (Within sampling interval) Total blood Conc. Body weight 70 Te after absorption (h) amount (mg) (ng/ml) at (Kg) after absorption times 200 Dose 0 mg 0.08 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2.55102

0.17 1.96 2 3.965 5.057

0.25 1.93 1.96 4 7.902 10.08

0.5 1.84 1.87 3.80 4 11.52 14.69

0.75 1.75 1.78 3.62 3.80 4 14.96 19.09

1 1.66 1.69 3.44 3.62 3.80 4 18.24 23.27

1.5 1.50 1.53 3.12 3.28 3.44 3.62 10 26.52 33.83

2 1.36 1.39 2.82 2.97 3.12 3.28 9.05 12 36.02 45.95

3 1.12 1.14 2.32 2.43 2.56 2.69 7.43 9.84 12 41.55 53

4 0.92 0.93 1.90 2.00 2.10 2.20 6.09 8.07 9.84 12 46.09 58.79

5 0.75 0.76 1.56 1.64 1.72 1.81 5.00 6.62 8.07 9.84 20 57.81 73.74

6 0.61 0.63 1.28 1.34 1.41 1.48 4.10 5.43 6.62 8.07 16.40 24 71.42 91.1

7 0.50 0.51 1.05 1.10 1.16 1.21 3.36 4.45 5.43 6.62 13.46 19.68 18 76.59 97.7

8 0.41 0.42 0.86 0.90 0.95 1.00 2.76 3.65 4.45 5.43 11.04 16.15 14.7 14 76.84 98.01

9 0.34 0.34 0.70 0.74 0.78 0.82 2.26 3.00 3.65 4.45 9.05 13.25 12.11 11.48 63.03 80.4

10 0.28 0.28 0.58 0.60 0.64 0.67 1.85 2.46 3.00 3.65 7.43 10.87 9.93 9.42 51.71 65.96

11 0.23 0.23 0.47 0.50 0.52 0.55 1.52 2.01 2.46 3.00 6.09 8.91 8.15 7.72 42.42 54.11

12 0.18 0.19 0.39 0.41 0.43 0.45 1.25 1.65 2.01 2.46 5.00 7.31 6.68 6.34 34.8 44.39

13 0.15 0.15 0.32 0.33 0.35 0.37 1.02 1.35 1.65 2.01 4.10 6.00 5.48 5.20 28.55 36.42

14 0.12 0.12 0.26 0.27 0.29 0.30 0.84 1.11 1.35 1.65 3.36 4.92 4.50 4.26 23.42 29.87

15 0.10 0.10 0.21 0.22 0.23 0.25 0.69 0.91 1.11 1.35 2.76 4.03 3.69 3.50 19.21 24.51

16 0.08 0.08 0.17 0.18 0.19 0.20 0.56 0.75 0.91 1.11 2.26 3.31 3.02 2.87 15.76 20.11

17 0.07 0.07 0.14 0.15 0.16 0.16 0.46 0.61 0.75 0.91 1.87 2.71 2.48 2.35 12.93 16.49

18 0.05 0.05 0.11 0.12 0.13 0.13 0.38 0.50 0.61 0.75 1.52 2.23 2.03 1.93 10.61 13.53

19 0.04 0.04 0.09 0.10 0.10 0.11 0.31 0.41 0.50 0.61 1.25 1.82 1.67 1.58 8.703 11.1

20 0.03 0.03 0.08 0.08 0.08 0.09 0.25 0.33 0.41 0.50 1.02 1.50 1.37 1.30 7.14 9.107

21 0.03 0.03 0.06 0.06 0.07 0.07 0.21 0.27 0.33 0.41 0.84 1.23 1.12 1.06 5.857 7.471

22 0.02 0.02 0.05 0.05 0.05 0.06 0.17 0.22 0.27 0.33 0.69 1.01 0.92 0.87 4.805 6.129

23 0.02 0.02 0.04 0.04 0.04 0.05 0.14 0.18 0.22 0.27 0.56 0.82 0.75 0.71 3.942 5.028

24 0.01 0.01 0.03 0.03 0.04 0.042 0.11 0.15 0.18 0.22 0.46 0.67 0.621 0.58 3.234 4.125

Table 4.2.Convolution of dissolution data of formulation T1

Half life (h) 3.5 Time (h) 0 0.08 0.17 0.25 0.5 0.75 1 1.5 2 3 4 6 8 10 12 Ke per hour 0.198 Percent release (Cumulative) 0 0 0 0 0 0 0 0 0 3 10 39 50 65 71 Percent release (Within F 0.5 sampling interval) 0 0 0 0 0 0 0 0 3 7 29 11 15 6 Amount (mg) release (Within Vd (L/Kg) 5.6 sampling interval) 0 0 0 0 0 0 0 0 6 14 58 22 30 12 Body weight Total blood amount Conc. (ng/ml) (Kg) 70 Time after absorption (h) (mg) after absorption at times 200 Dose mg 0 0.08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.17 0 0 0 0 0.25 0 0 0 0 0 0.5 0 0 0 0 0 0 0.75 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1.5 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 6 6 7.65 4 0 0 0 0 0 0 0 0 4.92 14 18.92 24.14 5 0 0 0 0 0 0 0 0 4.04 11.50 58 73.52 93.78 6 0 0 0 0 0 0 0 0 3.31 9.42 47.60 22 82.32 105.00 7 0 0 0 0 0 0 0 0 2.72 7.73 39.00 18.0 30 97.53 124.40 8 0 0 0 0 0 0 0 0 2.23 6.34 32.00 15.0 24.60 12 92.01 117.40 9 0 0 0 0 0 0 0 0 1.83 5.20 26.30 12.0 20.20 9.84 75.48 96.28 10 0 0 0 0 0 0 0 0 1.50 4.27 21.60 10.0 16.60 8.07 61.92 78.98 11 0 0 0 0 0 0 0 0 1.23 3.50 17.70 8.20 13.60 6.62 50.80 64.80 12 0 0 0 0 0 0 0 0 1.01 2.87 14.50 6.70 11.10 5.43 41.68 53.16 13 0 0 0 0 0 0 0 0 0.83 2.36 11.90 5.50 9.14 4.45 34.19 43.61 14 0 0 0 0 0 0 0 0 0.68 1.93 9.76 4.50 7.50 3.65 28.05 35.77 15 0 0 0 0 0 0 0 0 0.56 1.59 8.01 3.70 6.15 3.00 23.01 29.35 16 0 0 0 0 0 0 0 0 0.46 1.30 6.57 3.00 5.05 2.46 18.88 24.08 17 0 0 0 0 0 0 0 0 0.38 1.07 5.39 2.50 4.14 2.02 15.49 19.75 18 0 0 0 0 0 0 0 0 0.31 0.88 4.42 2.00 3.40 1.65 12.70 16.20 19 0 0 0 0 0 0 0 0 0.25 0.72 3.63 1.70 2.79 1.35 10.42 13.29 20 0 0 0 0 0 0 0 0 0.21 0.59 2.98 1.40 2.29 1.11 8.55 10.91 21 0 0 0 0 0 0 0 0 0.17 0.48 2.44 1.10 1.88 0.91 7.01 8.94 22 0 0 0 0 0 0 0 0 0.14 0.40 2.00 0.90 1.54 0.75 5.75 7.33 23 0 0 0 0 0 0 0 0 0.11 0.33 1.64 0.80 1.26 0.61 4.72 6.02 24 0 0 0 0 0 0 0 0 0.09 0.27 1.35 0.60 1.04 0.50 3.87 4.93

Table 4.3.Convolution of dissolution data of formulation T1S

Half life (h) 3.5 Time (h) 0 0.08 0.17 0.25 0.5 0.75 1 1.5 2 3 4 6 8 10 12 Percent release Ke per hour 0.198 0 0 0 0 1 2 3 7 9 17 44 58 75 78 79 (Cumulative) Percent release (Within F 0.5 0 0 0 1 1 1 4 2 8 27 14 17 3 1 sampling interval) Amount (mg) release Vd (L/Kg) 5.6 0 0 0 2 2 2 8 4 16 54 28 34 6 2 (Within sampling interval) Body weight Total blood amount Conc. (ng/ml) 70 Time after absorption (h) (Kg) (mg) after absorption at times Dose (mg) 200 0 0

0.08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.17 0 0 0 0

0.25 0 0 0 0 0

0.5 0 0 0 2 2 2.55

0.75 0 0 0 1.91 2 3.903 4.98

1 0 0 0 1.83 1.9 2 5.715 7.29

1.5 0 0 0 1.61 1.72 1.82 8 13.18 16.8

2 0 0 0 1.55 1.56 1.64 7.25 4 15.93 20.3

3 0 0 0 1.23 1.28 1.31 5.94 3.28 16 29.07 37.1

4 0 0 0 1.01 1.05 1.14 4.88 2.69 13.11 54 77.85 99.3

5 0 0 0 0.83 0.86 0.912 4.02 2.21 10.80 44.29 28 91.87 117

6 0 0 0 0.75 0.71 0.74 3.28 1.81 8.83 36.34 22.97 34 109.4 139

7 0 0 0 0.61 0.58 0.62 2.69 1.49 7.25 29.81 18.80 27.89 6 95.72 122

8 0 0 0 0.52 0.48 0.53 2.21 1.22 5.95 24.46 15.50 22.77 4.92 2 80.52 103

9 0 0 0 0.45 0.39 0.41 1.81 1.04 4.88 20.07 12.71 18.79 4.04 1.64 66.06 84.3

10 0 0 0 0.34 0.32 0.34 1.49 0.82 4.01 16.46 10.42 15.42 3.31 1.35 54.19 69.1

11 0 0 0 0.31 0.26 0.35 1.22 0.67 3.28 13.51 8.54 12.63 2.72 1.1 44.46 56.7

12 0 0 0 0.22 0.22 0.23 1.01 0.55 2.69 11.08 7.01 10.38 2.23 0.91 36.47 46.5

13 0 0 0 0.24 0.18 0.22 0.82 0.45 2.21 9.09 5.74 8.51 1.83 0.74 29.92 38.2

14 0 0 0 0.15 0.15 0.21 0.67 0.37 1.81 7.46 4.71 7.03 1.5 0.61 24.55 31.3

15 0 0 0 0.13 0.12 0.10 0.55 0.3 1.49 6.12 3.87 5.69 1.23 0.5 20.14 25.7

16 0 0 0 0.12 0.10 0.12 0.45 0.25 1.22 5.02 3.17 4.64 1.01 0.41 16.52 21.1

17 0 0 0 0.11 0.08 0.14 0.37 0.21 1.02 4.12 2.59 3.87 0.83 0.34 13.55 17.3

18 0 0 0 0.10 0.07 0.11 0.31 0.17 0.82 3.38 2.13 3.19 0.68 0.28 11.12 14.2

19 0 0 0 0.12 0.05 0.10 0.25 0.14 0.67 2.77 1.75 2.58 0.56 0.23 9.121 11.6

20 0 0 0 0.04 0.04 0.09 0.21 0.11 0.55 2.27 1.44 2.11 0.46 0.19 7.482 9.54

21 0 0 0 0.04 0.04 0.07 0.17 0.09 0.45 1.86 1.18 1.72 0.38 0.15 6.138 7.83

22 0 0 0 0.03 0.03 0.05 0.14 0.08 0.37 1.53 0.97 1.43 0.31 0.13 5.036 6.42

23 0 0 0 0.02 0.02 0.04 0.11 0.06 0.31 1.26 0.79 1.21 0.25 0.1 4.131 5.27

24 0 0 0 0.02 0.02 0.03 0.09 0.05 0.25 1.03 0.65 1.04 0.21 0.08 3.389 4.32

Table 4.4.Convolution of dissolution data of formulation T2

Half life (h) 3.5 Time (h) 0 0.08 0.17 0.25 0.5 0.75 1 1.5 2 3 4 6 8 10 12 Percent release Ke per hour 0.198 0 0 0 0 0 0 0 0 2 5 29 36 53 58 (Cumulative) 0 Percent release (Within sampling F 0.5 0 0 0 0 0 0 0 0 2 3 24 7 17 5 interval) Amount (mg) release (Within Vd (L/Kg) 5.6 0 0 0 0 0 0 0 0 4 6 48 14 34 10 sampling interval) Body weight Total blood amount Conc. (ng/ml) 70 Time after absorption (h) (Kg) (mg) after absorption at times 200 Dose 0 mg 0.08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.17 0 0 0 0

0.25 0 0 0 0 0

0.5 0 0 0 0 0 0

0.75 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

1.5 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 4 4 5.10

4 0 0 0 0 0 0 0 0 3.28 6 9.281 11.84

5 0 0 0 0 0 0 0 0 2.69 4.92 48 55.61 70.94

6 0 0 0 0 0 0 0 0 2.21 4.04 39.38 14 59.62 76.05

7 0 0 0 0 0 0 0 0 1.81 3.31 32.30 11.5 34 82.91 105.80

8 0 0 0 0 0 0 0 0 1.49 2.72 26.5 9.42 27.89 10 78.02 99.52

9 0 0 0 0 0 0 0 0 1.22 2.23 21.74 7.73 22.88 8.20 64.01 81.64

10 0 0 0 0 0 0 0 0 1.00 1.83 17.84 6.34 18.77 6.73 52.51 66.97

11 0 0 0 0 0 0 0 0 0.82 1.50 14.63 5.20 15.40 5.52 43.08 54.94

12 0 0 0 0 0 0 0 0 0.67 1.23 12.00 4.27 12.63 4.53 35.34 45.07

13 0 0 0 0 0 0 0 0 0.55 1.01 9.84 3.50 10.36 3.72 28.99 36.98

14 0 0 0 0 0 0 0 0 0.45 0.83 8.07 2.87 8.50 3.05 23.78 30.34

15 0 0 0 0 0 0 0 0 0.37 0.68 6.62 2.36 6.97 2.50 19.51 24.89

16 0 0 0 0 0 0 0 0 0.30 0.56 5.43 1.93 5.72 2.05 16.01 20.42

17 0 0 0 0 0 0 0 0 0.25 0.46 4.46 1.59 4.69 1.68 13.13 16.75

18 0 0 0 0 0 0 0 0 0.21 0.38 3.65 1.30 3.85 1.38 10.77 13.74

19 0 0 0 0 0 0 0 0 0.17 0.31 3.00 1.07 3.15 1.13 8.83 11.27

20 0 0 0 0 0 0 0 0 0.14 0.25 2.46 0.88 2.59 0.93 7.25 9.24

21 0 0 0 0 0 0 0 0 0.11 0.21 2.02 0.72 2.12 0.76 5.94 7.58

22 0 0 0 0 0 0 0 0 0.09 0.17 1.65 0.59 1.74 0.63 4.87 6.22

23 0 0 0 0 0 0 0 0 0.08 0.14 1.36 0.48 1.43 0.51 4.00 5.10

24 0 0 0 0 0 0 0 0 0.06 0.11 1.11 0.40 1.17 0.42 3.28 4.18

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Table 4.5.Convolution of dissolution data of formulation T3

Half life (h) 3.5 Time (h) 0 0.08 0.17 0.25 0.5 0.75 1 1.5 2 3 4 6 8 10 12 Percent release Ke per hour 0 0 0 0 0 0 0 0 1 3 25 30 48 56 0.198 (Cumulative) 0 Percent release (Within F 0 0 0 0 0 0 0 0 1 2 22 5 18 8 0.5 sampling interval) Amount (mg) release Vd (L/Kg) 0 0 0 0 0 0 0 0 2 4 44 10 36 16 5.6 (Within sampling interval) Total blood Conc. Body weight Time after absorption (h) amount (mg) (ng/ml) at (Kg) 70 after absorption times 200 Dose 0 mg 0.08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.17 0 0 0 0

0.25 0 0 0 0 0

0.5 0 0 0 0 0 0

0.75 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

1.5 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0 0

3 0 0 0 0 0 0 0 0 2 2 2.55

4 0 0 0 0 0 0 0 0 1.64 4 5.64 7.19

5 0 0 0 0 0 0 0 0 1.34 3.28 44 48.62 62.02

6 0 0 0 0 0 0 0 0 1.10 2.69 36.09 10 49.89 63.63

7 0 0 0 0 0 0 0 0 0.90 2.20 29.61 8.20 36 76.93 98.12

8 0 0 0 0 0 0 0 0 0.74 1.81 24.29 6.73 29.53 16 79.11 100.90

9 0 0 0 0 0 0 0 0 0.60 1.48 19.92 5.52 24.22 13.12 64.90 82.781

10 0 0 0 0 0 0 0 0 0.50 1.21 16.34 4.52 19.87 10.76 53.24 67.91

11 0 0 0 0 0 0 0 0 0.41 1.00 13.41 3.71 16.30 8.83 43.67 55.71

12 0 0 0 0 0 0 0 0 0.33 0.82 11.00 3.04 13.37 7.24 35.83 45.70

13 0 0 0 0 0 0 0 0 0.27 0.67 9.02 2.50 10.97 5.94 29.39 37.49

14 0 0 0 0 0 0 0 0 0.22 0.55 7.40 2.05 9.00 4.87 24.11 30.75

15 0 0 0 0 0 0 0 0 0.18 0.45 6.07 1.68 7.38 4.00 19.78 25.23

16 0 0 0 0 0 0 0 0 0.15 0.37 4.98 1.38 6.05 3.28 16.22 20.70

17 0 0 0 0 0 0 0 0 0.12 0.30 4.08 1.13 4.97 2.69 13.31 16.98

18 0 0 0 0 0 0 0 0 0.10 0.25 3.35 0.92 4.07 2.20 10.92 13.93

19 0 0 0 0 0 0 0 0 0.08 0.20 2.75 0.76 3.34 1.81 8.96 11.42

20 0 0 0 0 0 0 0 0 0.06 0.16 2.25 0.62 2.74 1.48 7.35 9.37

21 0 0 0 0 0 0 0 0 0.05 0.13 1.85 0.51 2.25 1.21 6.03 7.69

22 0 0 0 0 0 0 0 0 0.04 0.11 1.51 0.42 1.84 1.00 4.94 6.31

23 0 0 0 0 0 0 0 0 0.03 0.09 1.24 0.34 1.51 0.82 4.05 5.17

24 0 0 0 0 0 0 0 0 0.03 0.07 1.02 0.28 1.24 0.67 3.329 4.24

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4.4. Experimental protocols for In Vivo study

This in vivo study was carried out at the Centre for Bioequivalence Studies, the Faculty of

Pharmacy and Alternative Medicines, the Islamia University of Bahawalpur, Bahawalpur, under the supervision of an expert clinical team including clinicians and nurses for monitoring the tolerability of volunteers. The principles of Helsinki Declaration and Good Clinical Practice were observed for the conductance of this study. Based on biochemical and pathological tests, twenty eight young healthy human (age 23±4 years, weight 70±12.5 kg) were randomly recruited for this four way, four periods, single dose, randomized cross over study. The volunteers were advised to avoid the use of any drug 14 days before the start of study. They were informed about the possible undesired effects of drug and signed consents were taken. The volunteers received the investigational and reference tablets (containing approximately 200 mg Metoprolol) with 200 ml of water in a randomized order with a washout period of 10 days. The volunteers were provided with a standard lunch after 12 hour pre-dose and 4 hour post-dose fasting.

After oral administration of tablets, venous blood samples (5 ml) from antecubital vein were collected via an intravenous cannula (20 gauge) in heparinized-glass tubes at predetermined time points (0, 1, 2, 3, 4, 6, 8, 10 and 12 h) for each protocol. The samples were immediately centrifuged at 4000 rpm for 10 min to collect plasma and stored in tabelled eppendr of tube in a freezer at 20°C until quantitative bio-analysis by HPLC to establish the Metoprolol tartrate concentration.

4.5. Deconvolution of In vivo data IVIVC development

Indeed, IVIVC has remained a topic of constant discussion with several dosage forms, especially solid orals, since the publication of IVIVC guidance by the FDA in 1997. The goal of an IVIVC is to establish a relationship between the in vitro dissolution behavior and in vivo performances

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of a drug product. An IVIVC is generally described by a linear relationship between parameters derived from the in vitro and in vivo experiments as quantified by the Pearson correlation.

Once established, an IVIVC can be used to guide formulation and/or process development changes in the various stages of drug development and also simplify any scale-up or post- approval changes. Additionally, an IVIVC allows setting of clinically relevant in vitro dissolution specifications to ensure product quality. A particular benefit of an IVIVC is that it can be used to support the use of dissolution testing as a surrogate for human bioequivalence studies, which would reduce the number of human studies needed for drug applications. A well- described in vitro in vivo relationship could also be used to set clinically meaningful dissolution specifications for monitoring drug manufacture [115]. Based on FDA guidelines, level A IVIVC is expected for modified release formulations of BCS class I drugs (like Metoprolol tartrate) where dissolution is the rate limiting step. IVIVC Level A is the highest correlation for the submission of New Drug Application (NDA) and Abbreviated New Drug Application (ANDA)

[102]. IVIVC Level B and C are also of remarkable importance in biowaiver studies.

An increasing trend of IVIVC development has been observed in recent research. The main advantage of IVIVC is the prediction capability of in vivo performance of an alternative formulation of predefined nature from specific dissolution characteristics and IVIVC function

[102].According to Food and Drug Administration (FDA) guidelines for the establishment of

IVIVC, three formulations of the subject drug with different release rates are required followed by the internal or external validation of IVIVC [103]. Previously, many studies have been conducted for the formulation of Metoprolol tartrate sustained release solid oral dosage form

[101-105]. However, present polymer for enteric delivery of drug is not available in literature except this work [106,107].

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This study is a part of work that was designed to develop Metoprolol tartrate-Eudragit® FS modified release pH-dependent formulations i.e. tabletted microparticles using various concentrations of polymer. Then the in vitro and in vivo evaluation of the prepared and reference formulations (Mepressor® 200 mg, Novartis Pharmaceuticals, Karachi, Pakistan) was conducted followed by the development of in vitro in vivo correlation.

4.5.1. Computation of absorption data and IVIVC development

In order to establish an IVIVC level A, in vivo absorption (%) data was calculated using

Wagner-Nelson equation. Firstly, the area under the plasma concentration-time curve from zero to time 푡 (퐴푈퐶0−푡) was evaluated from plasma drug concentration (퐶푡) data using the trapezoidal rule, and then the area under the plasma concentration-time curve from zero to time infinity

( 퐴푈퐶0−∞) was calculated by adding 퐴푈퐶0−푡to the last 푙표푔-linear concentration divided by the terminal disposition rate constant. Secondly, elimination rate constant (푘푒) was multiplied with

퐴푈퐶0−푡 (resulting in 푘푒 × 퐴푈퐶0−푡 ) as well as with 퐴푈퐶0−∞(resulting in 푘푒 × 퐴푈퐶0−∞) and then the product of 푘푒 × AUC0-t was added to the respective Ct at each time point [resulting in

퐶푡+ (푘푒 × (퐴푈퐶0−푡 )]. Finally, each 퐶푡 + (푘푒 × 퐴푈퐶0−푡 ) was divided by the product of

푘푒 × 퐴푈퐶0−∞×100 to calculate the percentage of drug absorbed (퐹) at each time point using following equation (4.7) [108]:

퐶 + 푘 × 퐴푈퐶 퐹 = 푡 푒 0−푡 × 100 (4.7) 푘푒 × 퐴푈퐶0−∞

IVIVC level A was developed by drawing a plot between the percentage drug absorbed (along x- axis) of a formulation and its percentage drug dissolved (along y-axis) followed by the regression analysis of each curve to evaluate the strength of correlation determining whether the curve is linear or non-linear. Closer the value of determination coefficient to 1, stronger is the correlation

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and linear is the curve. IVIVC level B is developed by plotting the values of MDT (along x-axis) against MRT (along y-axis) of a formulation followed by the regression analysis of curve.

IVIVC level C is a single point correlation which is developed by plotting 푡50% (along x-axis) and pharmacokinetic parameter like AUC (along y-axis) followed by the regression analysis of curve [108].

Predictability of IVIVC

Predictability of IVIVC was determined using following equation (4.8) [105].

퐶푚푎푥 푂푏푠푒푟푣푒푑 − 퐶푚푎푥 푃푟푒푑푖푐푡푒푑 Prediction error (%)퐶푚푎푥 = × 100 (4.8) 퐶푚푎푥 푂푏푠푒푟푣푒푑

4.6. Mathematical and statistical analysis

Experimental results were expressed as mean ± SD.

The values of 퐶푚푎푥 (maximum drug concentration), 푡푚푎푥 (time to reach peak concentration),

AUC (area of curve of drug), 푘푒 (elimination rate constant), 푡1/2 (half life) and MRT (mean residence time) were measures from plasma drug concentration versus time profiles for each volunteers, using Microsoft excel, 2007.

The significance of difference between various pharmacokinetic parameters was evaluated by

Analysis of Variance (ANOVA) using software, SPSS version 13.0. The level of significance was set at 0.05. As same doses of reference and test formulations were given, the relative bioavailability (퐹%) was calculated by dividing 퐶푚푎푥 , 푙표푔 transformed 퐴푈퐶0−푡 and 퐴푈퐶0−∞ of test formulation with the 퐶푚푎푥 푙표푔 transformed 퐴푈퐶0−푡 and 퐴푈퐶0−∞ of reference formulation, respectively. Two compared formulations are considered bioequivalent if 90% class interval for these ratios lie between 80% and 125%[109].

The in vivo absorption (%) was calculated was by Wagner-Nelson equation to establish IVIVC.

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Chapter 5

RReessuullttss aanndd DDiissccuussssiioonn

106

Results and discussion

Introduction

Eudragit® FS microparticles loaded with Metoprolol tartrate were prepared by solvent evaporation [111]. Since eudragit® FS and metoprolol tartrate were both soluble in acetone, an emulsion of acetone in light liquid paraffin was used. Microparticles were formulated with Span

80 as surfactant, which is extensively used. Other than dissolution and in vivo testing are not presented in this thesis used in the formulation of microspheres by solvent evaporation, the pharmaceutical additionally the influence of the concentration of eudragit® FS on the release profile was probed to establish IVIVC [112]. The drug release data and the encapsulation efficiency (%) for Metoprolol tartrate of the resulting microparticles were compared.

5.1. Compatibility analysis

Differential scanning calorimetry was used to characterize the physical state of Metoprolol tartrate within the microparticles. A sharp endotherm was observed for metoprolol tartrate at approximately 135°C, indicating its melting transition point. This peak was observed also in the thermogram of the microparticles, indicating that the nature of metoprolol tartrate remained intact in the form of microparticles. The DSC results were further verified by X-RD tests. XRD patterns indicate that the crystalline nature of Metoprolol tartrate in the microparticles was not different from its original crystal, however there was significant reduction in peak intensities which suggest that the extent of Metoprolol tartrate crystallinity was reduced by Eudragit® FS .

In addition, FTIR spectra elaborated that the principal FTIR peaks observed in the spectra of

Metoprolol tartrate were in close resemblance to those in the spectra of the microparticles.

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Morphology study and particle size determination

Surface morphology of scanning electron microscopic photo of the T2 microparticles (before compression into tablets) showed that the prepared microparticles were spherical and discrete

(except few agglomerated microparticles) with plugged porous surface. Average size was 56±4.5

µm, 59±6.3 µm and 102±7.9 µm for T1, T2 and T3 microparticulate formulations, respectively.

Tablet evaluation

The weight variation, hardness, friability, disintegration, and dissolution for both formulations were within allowed limits of USP [112].

Encapsulation efficiency and yield

The encapsulation efficiency of eudragit® FS microparticles loaded with metoprolol tartrate was, determined as described above, 78.6±6.7%, 78.1±9.5% and 69.7±7.3% for T1, T2 and T3 microparticulate formulations, respectively. The product yield was approximately 87% for all microparticulate formulations.

Stability studies

The percentage residual drug content of tabletted microparticles (T2) stored at accelerated stability condition was 99.1, 98.9, 99.0, 98.2, 98.3 and 97.7 % at 1st to 6th month, respectively indicating non-significant (p > 0.05) decrease in drug contents. No significant (푃 > 0.05,

푓2> 99.0 for all comparisons) difference was found in the release behavior of tabletted microparticles, which indicates the reliability and reproducibility of the manufacturing process employed. Also, the release kinetics remained unaltered for up to three months of storage, and there were no changes in the tablet characteristics suggesting that metoprolol tartrate is stable in the tabletted microparticles for the above mentioned period.

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5.2. Drug release kinetics

The influence of Eudragit® FS ratio in the encapsulated formulation on the release profiles of metoprolol tartrate microparticles is shown in Figure 5.1. Then in vitro release results were also evaluated by various model dependent and independent approaches. The eudragit® FS concentration played an important role in regulating the release behavior of metoprolol tartrate microparticles. Metoprolol tartrate was released approximately 1.0% in pH 1.2 dissolution medium until 2 h for T1, T2 and T3. After changing to fresh medium to pH 4.5 at 2 h, drug release was about 3, 5 and 10% and more than 56, 58 and 71% in pH 7.0 at 8 h after changed to fresh medium from T1, T2 and T3, respectively. The in vitro release data for T1, T2 and T3 were compared by 푓2 test to investigate the influence of polymer concentration. According to results,

T1 versus T2 and T2 versus T3 are similar to each other while T1 versus T3 are different from each other. As the polymer ratio was increased from T1 to T3, the acid-resistant property of eudragit® FS matrix increased and eventually the release of metoprolol tartrate from formulations decreased. This can be attributed to the fact that the higher polymer concentration produced larger particles with proportionately less drug. So that the polymer quantity was changed and thus release was reduced. Almost none of the drug released from formulations in pH 1.2 and pH

4.5 dissolution media. In contrast, the cumulative drug release was about 56-71% in pH 7.0, regardless of the polymer concentration. This could be due to the fact that eudragit® FS was pH sensitive copolymers that started to be dissolved from pH 6.0. In pH 7.0 medium, eudragit® FS could to be dissolved and the channels were then quickly created in the coating membrane and thus allowing for the higher drug release [96, 113]. Based on similarity factor 푓2(a model independent approach), release profile of T1 was similar (푓2=55) to that of reference formulation.

By contrast, dissolution data of T2 and T3 were dissimilar (푓2<50) from that of reference

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formulation. On the other hand, best fit kinetic model to the dissolution data of test formulations was zero order followed by Higuchi’s and then first order model.

T1 (ng/ml) T2 (ng/ml) T3 (ng/ml) Mepressor (ng/ml) T1S (ng/ml) T1 (%) T2 (%) T3 (%) Mepressor (%) T1S (%) 150

100

50 Drug concentration Drug

0 0 6 12 18 24 Time (h)

Figure 5.1 Predicted plasma drug concentration (in ng/ml) and dissolution profiles (in percentage)

110

Zero order models illustrated the concentration independent release of drug. The release of drug from reference tablets occurred by Higuchi’s model followed by zero and first order models.

The mode of drug release from reference, T1 and T2 was case II relaxation transport while from

T3 was anomalous type.

5.3. In vivo study

This probe involves pharmacokinetic, bioavailability, bioequivalence and simulation studies of 4

Metoprolol tartrate oral formulations; a reference (Mepressor®) tablet formulation and 3 test

(investigational) tabletted microparticulate formulations. All tablet formulations met USP criteria for in vitro analysis.

The in vivo evaluations involved 28 healthy human volunteers. In this probe, 퐶푚푎푥 for all test formulations was non-significantly (p > 0.05) different from each other but significantly

(p < 0.05) lesser than that of reference. The 푡푚푎푥 for the test formulations was significantly higher as compared to the reference which certifies increased transit or stay time of tabletted microparticles in GIT. This probe is further proved by higher MRT for test formulations as compared to that reference. MRT indicates stay time of drug in body. Significant (p < 0.05) reduction in 퐶푚푎푥 , 퐴푈퐶0−푡 and 퐴푈퐶0−∞ are observed in case of test formulations which can be attributed to the enteric nature of these formulations.

Either one Metoprolol tartrate 200 mg tablets (reference) or three Metoprolol tartrate 200 mg enteric tablets (test) was orally administered in the 4×4 crossover experimental design in human.

The pharmacokinetic parameters are evaluated and given in Table 4.1.

In the pharmacokinetic analysis, 퐶푚푎푥 (Mean ± SD, μg/ml) for the reference, T1, T2, and T3 was

190, 144, 133 and 129 (P < 0.05), respectively. The 퐴푈퐶0−푡 (Mean ± SD, μg.h/ml) for the reference, T1, T2, and T3 was 1423, 849, 857 and 823 (P <0.05), respectively. The 퐴푈퐶0−∞

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(Mean ± SD, μg.h/ml) for the reference, T1, T2, and T3 was 1675, 927, 1019 and 999 (P <0.05), respectively. The 푇푚푎푥 (Mean ± SD, h) for the reference and all test formulations was 4 and 6,

(P < 0.05), respectively. The statistical analysis of data by ANOVA showed no influence of sequence and period on the pharmacokinetic parameters.

To elaborate bioequivalence, the 90% for the ratios of the 푙푛 (natural log) transformed 퐶푚푎푥 ,

퐴푈퐶0−푡 and 퐴푈퐶0−∞ for the reference, T1, T2, and T3 were calculated and these results lied in the bioequivalence range (80% to 125%) indicating bioequivalence between the compared formulations.

Clinical observations

During in vivo study, the physical observation by clinician and biochemical and hematological tests encountered no serious undesired effect with either formulation except mild gastrointestinal disturbance with the reference formulation in one volunteer in second period of study.

The literature has also showed gastrointestinal disturbance as an undesired effect with

Metoprolol tartrate.

5.4. Development of in vitro in vivo correlation

In simulation studies, the coefficient of determination (R2) between in vivo absorption (%) and in vitro release (%) was 0.720, 0.905, 0.928 and 0.878 for reference, T1, T2 and T3 formulations, respectively with acceptable percent error (<10%). These results showed good correlation between in vitro and in vivo performance of formulations.

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Table 5.1 Pharmacokinetic parameters for all tablets obtained from in vivo experiments and convolution method

Parameters Approach Mepressor® T1 T1S T2 T3 6 ± 6 ± In vivo 4 ± 0.45 6 ± 0.38 6 ± 0.50 0.38 0.43 푡 (h) 푚푎푥 6 ± 8 ± Convolution 8 ± 0.34* 7 ± 0.13* 7 ± 0.38* 0.09 0.28* 190.23 ± 144.92 ± 144.92 133.57 ± 129.54 In vivo 11.42 9.58 ± 9.58 6.96 ± 7.72 퐶 (ng/ml) 100.91 푚푎푥 124.40 ± 139.49 105.75 ± Convolution 98.05 ± 3.27* ± 5.01* ± 4.23 6.37* 3.58* 999.41 1675.27 ± 927.19 ± 927.19 1028.37 ± In vivo ± 19.87 18.56 ± 18.56 17.04 17.09 퐴푈퐶0−∞(ng.h/ml) 1114.74 797.77 990.70 ± 1095.88 ± 828.72 ± Convolution ± ± 7.63* 15.97* 11.92* 13.71* 8.53* *p < 0.05 vs. in vivo value

In this study, the evaluation and application methods of IVIVC in setting in vitro release specification for a product are given. For biowaiver studies, comparative dissolution analysis is employed in addition to routine quality control tests, and subsequently the obtained dissolution data is evaluated. Biowaiver study is usually carried out for formulations with different strengths

(thus different release rates) [105]. To approximate in vivo activity of a formulation, the application of dissolution testing as a quality control tool has considerably increased after developing an IVIVC. There are many applications of IVIVC such as the selection of the biorelevant in vitro dissolution medium in bioequivalence studies, use of validated IVIVC for providing details for a biowaiver in scale-up or post approval changes. Indeed, a biowaiver is only granted if the prediction of in vivo performance of the formulation with the modified in vitro release rate remains bioequivalent with that of the originally tested formulation [105].

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Figure 5.3-5.5 and Table 5.2 exhibit an effort at a level A IVIVC for encapsulated Metoprolol tartrate formulation. The IVIVC is considered as the most valuable tool for the approximation of in vivo activity from dissolution profiles. It is also remarkable to identify the subsistence of a superb relationship between drug absorbed (%) in vivo and drug dissolved (%) in vitro in this study, which also involves the prediction of plasma drug concentration and absorption kinetics from dissolution data as well as the drug release kinetics. It is apparent from data that the developed formulations have revealed insensitivity to the hydrodynamic conditions which is a climatic feature of gastrointestinal tract; this characteristic helps in the prediction of in vivo plasma drug levels.

The IVIVC level A for T2 revealed a good correlation coefficient (R2 = 0.928) followed by the

T1 (R2 = 0.905) and T3 (R2 = 0.878) (Table 5.2). It clearly indicates the enteric nature of formulated products which is further confirmed from a weaker IVIVC (R2 = 0.720) for

Mepressor® (Figure 5.3) as it a non-enteric formulation performing unlikely in sequential pH change dissolution test. In addition, the value of R2 = 0.973 is significantly (P <0.05) higher for

T1S which involved the dissolution in the presence of 0.1% sodium dodecyl sulphate. This use of surfactant enhanced the rate of dissolution which resulted in the close resemblance of dissolution conditions to that of normal physiology. In addition, percentage prediction error was found to be 14% exhibiting convolution technique as a proficient procedure for predicting plasma drug levels.

There was a very weak correlation coefficients (R2 = 0.231) for level B IVIVC, while 0.714 was the correlation coefficient in case of level C IVIVC (Figure 5.4 and 5.5).

In short, Wagner-Nelson equation for the development of an IVIVC was revealed using a targeted release formulation as a model system. The technique possesses the benefit of

114

tolerating the data characteristically obtainable from a formulation development program to be used for establishing an IVIVC.

Figure 5.2.IVIVC Level A for formulations Mepressor® (A), T1 (B), T2 (C), T3 (D) and T1S (E)

115

7 MRT (h) MRT y = 0.805x + 5.180 R² = 0.231

0 0 4 8

MDT (h)

Figure 5.3.IVIVC Level B

116

200

100

% (h) %

50 T y = -12.41x + 272.3 R² = 0.714

0 0 7 14

Cmax (μg/ml)

Figure 5.4. IVIVC Level C

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Table 5.2. In vitro In vivo correlation data

Level of in vitro in vivo correlation Factor Result R2 for Mepressor® 0.720 2 R for T1 0.928 2 IVIVC level A R for T2 0.905 2 R for T3 0.878 R2 for T1S 0.973 Prediction error (%) 14 IVIVC level B R2 0.231 IVIVC level C R2 0.714

Table 5.3.In vitro to in vivo profiling data

Formulation R2 values

Mepressor® 0.345

T1 0.616

T2 0.682

T3 0.665

T1S 0.963

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5.5. Development of IVIV profiling

After developing formulations with specific release behavior (T1, T2 and T3), the formulations were analyzed for in vitro dissolution behavior. Thus the key purpose in performing the dissolution analysis is to predict in vivo drug release in human physiology. Finally, predicted data for all four products was used to calculate pharmacokinetic parameters like 퐴푈퐶, 퐶푚푎푥 , and

푇푚푎푥 (Table 5.1) to evaluate pharmacokinetic behavior of formulations. The in vitro dissolution data and predicted blood drug levels of all formulations are given in Figure5.1. Though, the values for all pharmacokinetic parameters for all formulations (except T1S) from convolution approach was significantly (p <0.05) different from that of obtained via in vivo experimentation, however the values of derived pharmacokinetic parameters from the predicted drug concentration in blood were amazingly comparable to that calculated from the corresponding human in vivo data. In addition, some variation in results exhibits dissimilarity between the actual physiological conditions of gastrointestinal tract and the used dissolution medium. When dissolution testing of T1 was conducted after adding surfactant (sodium lauryl sulphate), amazingly different results (labeled as T1S in Table 5.2) were observed. The convoluted values of 푇푚푎푥 and 퐶푚푎푥 for T1S were significantly (p < 0.05) similar to that of calculated via in vivo study. However, the values of AUC0-∞ obtained from both methods were significantly (p < 0.05) different, that could be due to the lack of normal physiological conditions in vitro testing. In normal human physiology, there are many factors responsible for the removal of drug substance from systemic circulation, which is responsible for the lower 퐴푈퐶0−∞ obtained from in vivo study.

The predicted blood drug levels also indicate the accurate prediction capability of convolution procedure in comparison to the results narrated in published data. It is noteworthy that the

119

퐶푚푎푥 values are higher than those achieved from in vitro dissolution values for T1 tablets compared to T2 and T3 tablets, which could be attributed to the low polymeric contents in T1 compared to T2 and T3 tablets. Similarly, 퐴푈퐶0−∞ (ng.h/ml) values are higher those achieved from in vitro dissolution values for T1 tablets compared to T2 and T3 tablets. It can also be attributed to the low polymeric contents in T1 compared to T2 and T3 tablets. Similar trend in results are presented in previous publications [97, 99]. Thus, it is not dangerous to extract that undeniably the approach of achieving blood levels, as presented here, looks validated and applicable.

® On the other hand, pharmacokinetic parameters (except 푇푚푎푥 ) for Mepressor from in vivo experiments were approximately double to that calculated via convolution method. However, such drastic variation is not observed in case of the designed formulations (T1, T2 and T3) which could be due to the colonic nature of these formulations in contrast to Mepressor® is a sustained release formulation, rather than a colonic delivery system. Mepressor® also does not contain such recipients which could encourage its drug release in the colonic pH range. These results also elaborate the use of sequential pH change method as biorelevant dissolution medium.

This study describes the assessment and application of IVIVP in designing a product with specific release pattern. Since the approximation of plasma drug concentration data is needed from dissolution profiles for the fabrication of drug products, such data can be acquired by amalgamating the in vitro dissolution profiles with the pharmacokinetic parameters of the drug.

This amalgamating and independent step is termed as the convolution approach [99].

In this context, present study involves the description and application of this easy and realistic approach in approximating the plasma drug concentration-time profiles on the basis of convolution methodology.

120

The IVIVP for T1S exhibited a good correlation coefficient (R2 = 0.963) followed by the T2

(R2 = 0.682), T3 (R2 = 0.665), T1 (R2 = 0.616), and Mepressor® (R2 = 0.345) (table 4.8, Figure

4.6). Significantly high (P <0.05) values for correlation coefficient for T1S suggests that the nature of used dissolution medium closely resembles the human physiology conditions. Similar results were observed when IVIVC for T1S was developed using deconvolution approach where the highest value of R2 i.e. 0.973 was found for T1S [99,100]. The employed dissolution medium was 0.1 N HCl, pH 6.8 phosphate buffer, and 0.1% sodium dodecyl sulphate as an attempt to simulate the biochemistry of gastrointestinal tract environment. In addition, the results also support the hypothesis for employing Eudragit® FS in formulation development for metoprolol which is efficiently absorbed in intestine compared to that in stomach. The use of surfactant perhaps enhances the dissolution rate resulting in the decrease in dissimilarity between the dissolution conditions of in vitro and in vivo environment [100].

In brief, convolution approach is successfully applied for the development of an IVIVP a targeted release formulation as a model system. This approach exhibits the benefit of treating the data typically available from a formulation development program to be used for developing an IVIVP.

121

Figure 5.5.In vitro to in vivo profiling for formulations T1 (A), T1S (B), T2 (C), T3 (D) and Mepressor® (E)

122

CCoonncclluussiioonn

This study elaborates that there is good correlation between the in vitro and in vivo profiles of the developed Metoprolol tartrate formulations, particularly for T1S, and study corroborates that there is an excellent in vitro in vivo correlation for Metoprolol tartrate formulations encapsulated into Eudragit® FS, principally for T1S. It is also concluded that level AIVIVC is a proficient mathematical model for biowaiver studies involving study parameters as those implemented for

T1S revealing that IVIVC level A is dosage form specific, rather than to be drug specific.

Convolution is a promising technique to compute blood drug levels using the in vitro dissolution profiles avoiding the involvement of physiological variability and refraining from the complex procedure of in vivo studies. However, the use of biosimilar dissolution conditions is necessary for accurate prediction of blood drug concentration. On the other hand, establishing an in vitro in vivo profiling, based on the convolution approach, can be more productive and practicable in the biowaiver studies, rather than present in adequate practice of in vitro in vivo correlation estimated via deconvolution approach.

123

RReeffeerreenncceess

[1] D. T. Sturrock, G.W. Evans, introduction to discrete–even system simulation-part one,

(1999).

[2] J. A. Sokolowski, Principle of modeling and simulation A Multidisciplinary Approach,

University of California at Los Angeles, Volume 31, Book Review 3 (2009).

[3] G. L. Amidon, H. Lennenas, A theoretical basis for a bio-pharmaceutics drug classification:

The correlation of in vitro drug product dissolution and in vivo bioavailability, (1995).

[4] A. S. Hussain, L. J. Lesko, R. L. Williams, The bio-pharmaceutics classification system:

Highlights of the FDA guidance dissolution, (2011).

[5] C. Hacket, L.P Isbster, The use of high-dose insulin – glucose euglycemia in beta-blocker

overdose: a case report, J. Med. Tox. 5: (2009), 139-143.

[6] R. Seema, R. Ankur, N. Arun, Biopharmaceutics classification system: A strategic tool for

classifying drug substances IRJP 2:7, (2011), 53-59.

[7] S. F. Marshall, application of nonlinear mixed effects modeling in the early phases of drug

development, Ph.D. thesis, university of Glasgow, (1999).

[8] R. H. Eyzagurre Perez, Analysis and design of pharmacokinetic models university of

technology (Mathematics and Computer Science) (2006).

[9] J. U. Fluckiger, Absolute quantification of pharmacokinetic parameters in contrast enhanced

magnetic resonance imaging, Ph.D. thesis (Department of Bioengineering) university of

Utah, (2011).

[10] H. Kortejavi, Modeling and simulation approaches for waiving in vivo pharmacokinetic

formulation studies, Ph.D. thesis university of Helsinki Finland, (2008).

124

[11] S. Pilari, Novel approaches to mechanistic pharmacokinetic/pharmacodynamic

modeling, Ph.D. thesis (Department of mathematics) university of Berlin, (2011).

[12] G. J. Wagner, History of pharmacokinetics, Pharmacy Vol. 12, (1982), 537-562.

[13] C. Mircioiu, S. A. Borisova and V. A. Voicu, Biopharmaceutic Metrics Applied in

Comparison of Clusters of time Courses of Effect in Clinical Trials, Journal of Applied

Biopharmaceutics and Pharmacokinetics, 1, (2013), 37-44.

[14] S. E. Tent, C. Day, Bioavailability of hydroxychorquine tablets using deconvolution

technique, J. pharm Sci. 81: (1992), 155-159.

[15] D. J. Tett, R.O Day, Absorption and in vivo dissolution of hydroxycholorquine in fed

subjects assessed using deconvolutin techniques, Clin pharmacy, 36:(1993), 405-410.

[16] M. Nicklasson, K. Ellstrom, Linear systems analysis and moment analysis in the

determination, J. pharmacokinetic Biopharm, 12 :( 1984), 467-478.

[17] K. Anders, Computational pharmacokinetics, Chapman and Hall/CRC, (2008).

[18] I. H. Mohammad, A.K. Moharram, Mathematical modeling as a tool for improving the

acting of drugs, Journal of Basic and applied Sciences Vol. 4, No. 2 (2008), 81-88.

[19] S. J. Chapman, Multi-scale mathematical modeling in medical and biology, 18th world

IMACS/MODSIM Congress, Australia, (2009), 13-17.

[20] E. M. Ouriemchi, J. Bouzon, J.M. Vergnaud, Modeling the process of controlled release

of drug in vitro and in vivo test, (2005), 54-67.

[21] K. Younggil, Handbook of essential pharmacokinetics and drug metabolism for industrial

scientists, Plenum publisher, 2nd edition (2001).

[22] R. Panchagnula, N. Sunil Thomas Bio-pharmaceutics and pharmacokinetics in drug

research, International Journal of Pharmaceutics 201, (2000), 131–150.

125

[23] P. L. Madan, J. Jayme, Bio-pharmaceutics and pharmacokinetic 2nd edition, (2004), 72-

76.

[24] M. Rowland, T. N. Tozer, Clinical pharmacokinetics and pharmacodynamics concepts

and applications, 4th edition, (2011).

[25] J. B. Dressman, oral drug absorption prediction and assessment, by Marcel Dekker, Inc.

(2000).

[26] T. Jacobs. S. Rossenu, A. Dunne, Combined model for data from in vitro in vivo

correlation experiments, Journal Biopharm. Stat. 18: (2008), 1197-1211.

[27] X. Hubert, Feasibility study estimating non-invasive the arterial input function for PET

imaging in human, Ph.D. thesis, Paris laboratory of applied mathematics and system, (2009).

[28] G. Murtaza, M. Ahmed, Development of in vitro in vivo correlation for pharmacokinetic

simulation, African journal of pharmacy and pharmacology, Vol.6 (4) (2012), 257- 263.

[29] H. Kwon, pharmacokinetic/pharmacodynamic modeling/simulation and novel gastric

retention formulation, Ph.D. thesis Oregon State University, (2003).

[30] T. Jacobs, Non-linear mixed-effects modeling for complex biopharmaceutical data, Ph.D.

thesis, Hasselt University (2009).

[31] N. Sivadas, In vivo animal models for drug delivery across the lung mucosal barrier,

Advanced Drug Delivery Reviews, Advanced Drug Delivery Reviews 60 (2008).

[32] D. M. Chilukuri, Pharmaceutical product development in vitro in vivo correlation, drug

and the pharmaceutical sciences,Vol.165 Informa Healthcare USA, Inc.(2007).

[33] R. Gomeni, In Silico prediction of optimal in vivo delivery properties using convolution-

based model and clinical trial simulation, Pharmaceutical Research, 19(1), (202), 99-103.

126

[34] A. D. Rodrigues, Drug –Drug interactions, drugs and the pharmaceutical

sciences,Vol.179, 2nd edition, by Informa Healthcare USA, Inc. (2008).

[35] A. M. Segrave, An investigation of the pharmacokinetics and lymphatic transport of

recombinant human leukaemia inhibitory factors, Ph.D. thesis, Monash University (2004).

[36] S. B. Karch, Pharmacokinetics and pharmacodynamics of abused drug, by Taylor&

Francis Group, LLC (2008).

[37] S. Haruta, Prediction of concentration-time curve of orally administrated theophyline

based on a scintigraphic monitoring of gastrointestinal tract transit in human volunteers,

International journal of pharmaceutics 233, (2002), 179-190.

[38] G. Sparacino, A stochastic deconvolution interactive programme for physiological and

pharmacokinetic system, Computer methods and programme in biomedical, Vol. 67, (2002),

67-77

[39] M. L. Johnson, A review and appraisal of deconvolution methods to evaluate in vivo,

Journal of Neuroendocrinology,Vol.2, No. 6, Oxford University,(1990).

[40] M. Gibaldi, Pharmacokinetics, drugs and the pharmaceutical sciences, Vol.15, 2 nd

edition, Informa Healthcare USA, Inc. (2007).

[41] F. Langenbucher, Handling of computational in vitro in vivo correlation problems by

Microsoft Excel: III Convolution and deconvolutio, European Journal of pharmaceutics and

Bio-pharmaceutics 56, (2003), 429-437.

[42] G. Sanchez, Mathematical toolbox for modeling Bioknetic systems, Mathematical

Education and Research Vol.10 No.2 (2010).

127

[43] O. I. Corriganb, An investigation into the usefulness of generalized regression neural

network analysis in the development of level A in vitro–in vivo correlation, European Journal

of Pharmaceutical Sciences 30, (2007), 264-272.

[44] W. Weber, M. Looby, The pharmacokinetics and pharmacodynamics of recombinant

human erythropoietin in haemodialysis patients, Br. J. clin. Pharmac, 34: (1992), 499-508.

[45] D. J. Cutler, Absorption and in vivo dissolution of hydroxycholoroquine in fed subjects

assessed using deconvolution techniques, Br J clin Pharmac;36: (1993),405-411.

[46] R. C. Boston, A numerical deconvolution method to estimate C-peptide secretion in

humans after an intravenous glucose tolerance test, Metabolism Clinical and Experimental

58: (2009),891–900.

[47] R. Liu, D. Sun, Convolution method to predict drug concentration profiles of

tetramethylpyrazine following transdermal application, International Journal of

Pharmaceutics, (2003), 39–45.

[48] L. Hovgaard, H.G. Kristensen, L. Knutson, A. Lindahl, A comparison between direct

determination of in vivo dissolution and the deconvolution technique in humans, European

Journal of Pharmaceutical Sciences, 8 :(1999),19–27.

[49] A. Royce, L. S. Weaver, U. Shah, In vitro and in vivo evaluation of three controlled

release principles of 6-N-cyclohexyl-2'-Omethyladenosine. J. Contrl. Rel. 97(1): (2004),79-

90.

[50] S. B. Karch, Pharmacokinetics and pharmacodynamics of Abused Drugs, by Taylor &

Francis Group, LLC. (2008).

128

[51] P. Macheras, A. Iliadis, Modeling in bio-pharmaceutics/Pharmacokinetics homogeneous

and heterogeneous approaches, Interdisciplinary Applied Mathematics, Vol. 30, Springer

Science-Business Media, Inc., (2006).

[52] J. E. Riviera, Comparative Pharmacokinetics Principles, Techniques, and Applications,

2nd edition, by John Wiley & Sons, Inc. ISBN: 978-0-813-82993-7, (2011).

[53] Z. D. Argenio, Advanced Methods of Pharmacokinetic and Pharmacodynamic systems

Analysis, the Kluwer International Series in engineering and computer science, University of

Southern California, Vol.3 ,(2004).

[54] P. L. Bonate, Pharmacokinetic-Pharmacodynamic Modeling and Simulation, 2nd edition,

Springer Science-Business Media, LLC. (2011).

[55] R. A.Harvey, pharmacology,5th edition, Lippincott’s Williams, (2012).

[56] P. Veng-Pedersen, Non-compartmentally-based pharmacokinetic modeling, Advanced

Drug Delivery Reviews 48, (2001), 265–300.

[57] R. P. Maguire, K.L. Leenders , PET Pharmacokinetic course, Osaka, Japan, (2007).

[58] L. Dedik, M. Durisova, CXT-MAIN: a software package for determination of the

analytical form of the pharmacokinetic system weighting function, computer methods and

programs in Biomedicine, 15, ,(1996),183-192.

[59] M. L.Pfefferle, Possibilities of simultaneous investigation of the release and heavy-

water absorption drugs from solid and dosage forms by bio-relevant in vitro models, PhD

thesis of the faculty of mathematics natural science of university of Tubingen German,

(2010).

129

[60] N. Sirisuth, N. D. Eddington, The influence of first pass metabolism on the development

and validation of an IVIVC for Metoprolol extended release tablets, European Journal of

Pharmaceutics and Bio-pharmaceutics 53, (2002), 301–309.

[61] S. Nainar, K. Rajiah, R. Kasibhatta, Biopharmaceutical Classification System in In-vitro/

In-vivo correlation: Concept and Development Strategies in Drug Delivery, Tropical Journal

of Pharmaceutical Research April, 11 (2): (2012), 319-329.

[62] S. Sakor, B. Chakraborty, In Vitro–In Vivo Correlation (IVIVC): A Strategic tool in drug

development, J. BioeqivAvailab, (2011).

[63] M .H. M. El-Komy, A pharmacokinetic receptor-based recirculation model for target-

mediated disposition drugs, Ph.D. thesis, University of Iowa, (2012).

[64] J. Popovic, Derivation of Laplace transform for the general disposition deconvolution

equation in drug metabolism kinetics, Exp Toxic Pathol 51: (1999), 409-411.

[65] J. Emami, In vitro - in vivo Correlation: from Theory to Applications, J Pharm

Pharmaceut Sci. 9 (2): (2006), 169-189.

[66] Yu. Jihnhee, E. Thomas, An approach to the residence time distribution for stochastic

multi-compartment models, Mathematical Biosciences 191 (2004) 185–205.

[67] R. Bhandari, Pharmacokinetics, tissue distribution and relative bioavailability of

Isoniazid-solid lipid nanoparticles, International Journal of Pharmaceutics 441, (2013), 202-

212.

[68] S. A. Qureshi, In Vitro-In Vivo Correlation (IVIVC) and Determining Drug

Concentrations in Blood from Dissolution Testing-A Simple and Practical Approach, The

Open Drug Delivery Journal, 4, (2010),38-47.

130

[69] K.E. Holt, Bioequivalence Studies of Ketoprofen: Product Formulation,

Pharmacolcinetics, Deconvolution, and In Vitro In Vivo Correlations, Ph.D. thesis, Oregon

State University (1997).

[70] C. F. Minto, C. howe, S. Wishart, A. J. Conway, D. J. Handelsman, Pharmacokinetics

and Pharmacodynamics of Nandrolone Esters in Oil Vehicle: Effects of Ester, Injection Site

and Injection Volume1, Journal of Pharmacology and experimental therapeutics the

American Society for Pharmacology and Experimental Therapeutics U.S.A. JPET 281; 199

(1997),93–102.

[71] V. Kumar, L. K. Khurana, S. Ahmad, R. B. Singh, Application of Assumed IVIVC in

Product Life Cycle Management: A case Study of TrimetazidineDihydrochloride Extended

Release Tablet, Varinder Kumar, Ranbaxy Laboratories Limited, Gurgaon, Haryana, India,

(2013).

[72] S. A. Qureshi, Prediction of plasma drug levels from dissolution results for OROS-based

nifedipine products, (www.drug-dissolution-testing.com), (2013).

[73] V.M. Nguyen, Mathematical modeling and simulation, (2008).

[74] A. F. S. Mohamed, Pharmacokinetic and Pharmacodynamic Modeling Antibiotics

Bacteria, Ph.D. thesis, Uppsala University (2013).

[75] A. Almusharrf, Development of fractional trigonometry and an application of fractional

calculus to pharmacokinetic model, Msc thesis Western Kentucky University, (2011).

[76] G. Koch, Modeling of pharmacokinetics and pharmacodynamics with application to

cancer and arthritis, Ph.D. thesis (Mathematics and Natural Sciences Section-Department of

Mathematics and Statistics), University of Konstanz German, (2012).

131

[77] H. Pang, A review on pharmacokinetic modeling and the effects of environmental

stressors on pharmacokinetics for operational medicine: Operational pharmacokinetics,

(2009).

[78] K. E. Holt, Bioequivalence Studies of Ketoprofen: Product Formulation,

Pharmacokinetics, Deconvolution, and In Vitro In Vivo Correlations, Ph.D. thesis Oregon

State University, (1998).

[79] L.M. Pereira, Fractal pharmacokinetics, Computational and Mathematical Methods in

Medicine Vol. 11, No. 2: (2010), 161–184.

[80] L. Lunney, Pharmacokinetics: A Look at Different Approaches, Biomathematics North

Carolina State University, (1992).

[81] T. Kind, I. Houtzager , J. C. Faes, B. M. Hofman, Evaluation of model-independent

deconvolution techniques to estimate lung parenchymal perfusion, ConfProc IEEE Eng Med

Biol Soc. ( 2010),2602-2607.

[82] P. L. Toutain, A. Bousquet, Bioavailability and its assessment, J. vet. Pharmacol. Therap:

455–466, (2004).

[83] A. Rescigno, Foundations of pharmacokinetics, Kluwer Academic/Plenum Publishers,

(2004).

[84] M. J. Roberts, Web-AppendixD-Derivations of convolution Properties (2007)

[85] D. Rafaja, Deconvolution versus convolution-a comparison for materials with

concentration gradient, Materials Structure, Vol.7, (2000).

1 [86] N. J. Kasdin, Discrete Simulation of Colored Noise and Stochastic Processes and 푓훼

Power Law Noise Generation, IEEE, VOL. 83, NO.5 (1995).

132

[87] B. K. A. Rasool, S. A. Fahmy, Development of coated beads for oral controlled delivery

of In vitro evaluation, Pharm. 63, (2013), 31-44.

[88] R. H. Shawahna, Investigation and Revision of Current BCS Based Biowaiver Criteria:

in vivo correlation (ISIVC), PhD. thesis, The Islamia University of Bahawalpur Pakistan,

(2011).

[89] J. Emami, Comparative in vitro and in vivo evaluation of three tablet formulations of

amiodarone in healthy subjects, DARU Vol. 18, No. 3 (2010).

[90] D. Heeger, Signals, Linear Systems, and Convolution, (2000).

[91] F. Keller, Computational foundations of convolution and kernel, (2010).

[92] P. Veng-Pedersen, Mean time parameters in pharmacokinetic, definition computation

and clinical implications (Part II), Clin. Pharmacokinetic 17, (1989), 424–440.

[93] P. Veng-Pedersen, Mean time parameters in pharmacokinetic, definition computation

and clinical implications (Part II) Part I), Clin. Pharmacokinetic 17,(1989),345–366.

[94] F. Rasool, M. Ahmad, G. Murtaza, H. M. S. Khan and S. Ali Khan. Eudragit FS based

colonic microparticls of metoprolol tartrate. ActaPoloniaPharmaceutica-Drug Research, 69:

(2012), 347-353.

[95] F. Rasool, M. Ahmad, G. Murtaza, HMS Khan, SA Khan. Pharmacokinetic Studies on

MetoprololEudragit Matrix Tablets and Bioequivalence Consideration with Mepressor®.

Tropical J Pharm Res, 11: (2012), 281-287.

[96] S. A. Qureshi, Choice of rotation speed (rpm) for biorelevant drug dissolution testing

using a crescent-shaped spindle. Eur. J. Pharm. Sci. 23, (2004), 271–275.

133

[97] M .C. Meyer, A. B. Straughn, R.M. Mhatre; V. P. Shah, R.L Williams, L. J. L. Lesko,

Lack of in vivo/in vitro correlations for 50 mg and 250 mg primidone tablets. Pharm Res. 15,

(1998), 1085-1089.

[98] S. Riegelman, P. Collie, The application of statistical moment theory to the evaluation of

in vivo dissolution time an absorption time. Journal of Pharmacokinetics and

Biopharmaceutics. 8(5): (1980), 509-534.

[99] USP 2007. The United States Pharmacopoeial Convention, Inc., Rockville, MD, p. 323.

[100] S. A. Khan, M. Ahmad, G. Murtaza, H. M. Shoaib, M. N. Aamir, R. Kousar, F. Rasool,

A. Madni, Development of In-Vitro In-Vivo Correlation for Nimesulide Loaded

EthylcelluloseMicroparticles. Latin American Journal of Pharmacy, 29 (6): (2010), 1029-

1034.

[101] M. N. Aamir, M. Ahmad, N. Akhtar, G. Murtaza, S. Ali Khan, Shahiq-uz-Zaman, Ali

Nokhodchi.Development and in vitro–in vivo relationship of controlled-release

microparticles loaded with tramadol hydrochloride. International Journal of Pharmaceutics,

407: (2011), 38-43.

[102] S .Khiljee, M. Ahmad, G. Murtaza, A. Madni, N. Akhtar, M. Akhtar, Bioequivalence

evaluation of norfloxacin (Noroxin® AND Norocin®) tablets based on in vitro – in vivo

correlation. Pak. J. Pharm. Sci. 24 (4): (2011), 421-426.

[103] G. Murtaza, M. Ahmad, N. Akhtar, and Biowaiver study of oral tablettedethylcellulose

microcapsules of a BCS class I drug. Bull. Chem. Soc. Ethiop., 22(2): (2000),1-16.

[104] F. Rasool, M. Ahmad, G. Murtaza, H. M. S. Khan and S. Ali Khan. Eudragit FS® based

colonic microparticles of metoprolol tartrate. ActaPoloniacPharmaceutica-Drug Research, 69

(2): (2012), 347-353.

134

[105] A. A. A. Khaled, K. Pervaiz, K. Farzana, G. Murtaza. Metoprolol-Eudragit

Microcapsules: Pharmacokinetic Study using Convolution Approach. Latin American

Journal of Pharmacy, 31 (6): (2012), 914-917.

[106] F. Rasool, M. Ahmad, G. Murtaza, HMS Khan, SA Khan. Pharmacokinetic Studies on

Metoprolol – Eudragit Matrix Tablets and Bioequivalence Consideration with Mepressor®.

Tropical Journal of Pharmaceutical Research, 11 (2): (2012), 281-287.

[107] G. Murtaza, S. Azhar, A. Khalid, B. Nasir, M. Ubaid, M. K. Shahzad, F. Saqib, I. Afzal,

S. Noreen, M. Tariq, T. A. Chohan, Abdul Malik. Development of in vitro-in vivo correlation

for pharmacokinetic simulation. AJPP, 6 (4): (2012), 257-263.

[108] A. A. A. Khaled, Khalid Pervaiz, Sonia Khiljee, Sabiha Karim, Qurat-ul-AinShoaib, G.

Murtaza. In vitro to in vivo profiling: An easy idea for biowaiver study.

ActaPoloniaePharmaceutica-Drug Research, 70 (5): (2013), 873-875.

[109] A. A. A. Khaled, Khalid Pervaiz, Sabiha Karim, KalsoomFarzana, G. Murtaza.

Development of in vitro in vivo correlation for encapsulated metoprolol tartrate.

ActaPoloniaePharmaceutica-Drug Research, 70 (4): (2013), 743-747.

[110] G. Murtaza, M. Ahmad, Microencapsulation of tramadol hydrochloride and

physicochemical evaluation of formulations. Pak. J. Chem. Soc., 31 (4), (2009), 511-519.

[111] S.A. Khan, M. Ahmad, G. Murtaza, M. N. Aamir, N. Akhtar, R. Kousar, Dual coated

microparticles for intestinal delivery of nimesulide. Latin. Am. J. Pharm. 29 (1), (2011), 739-

745.

[112] F. Rasool, M. Ahmad, G. Murtaza, H. M. S. Khan, S. A. Khan, Metoprolol tartrate-

EthylcelluloseTablettedMicroparticles: Formulation and in vitro evaluation. Lat. Am. J.

Pharm. 9 (3), (2010), 984-990.

135

[113] A. Sharif, M. Rabbani, M. F. Akhtar, B. Akhtar, A. Saleem, K. Farzana, A. Usman, G.

Murtaza, Design and evaluation of modified release bilayer tablets of flurbiprofen. Adv. Clin.

Exp. Med., 20 (2), (2012), 343-349.

[114] J. Jiao, G. E. Searle, A. C. Tziortzi, C. A. Salinas, R. N. Gunn, J. A. Schnabel, Spatio-

temporal pharmacokinetic model based registration of 4D PETneuroimaging

data,NeuroImage 84 (2014) 225–235.

[115] S. D'Souza, J. A. Faraj, S. Giovagnoli, P. P. DeLuc, IVIVC from Long Acting

Olanzapine Microspheres, Hindawi Publishing Corporation, International Journal of

Biomaterials, (2014), Article ID 407065, pp 1-11.

[116] S. Chakraborty, K. Pandya, D. Aggarwal, Establishing Prospective IVIVC for Generic

Pharmaceuticals: Methodologies Assessment, the Open Drug Delivery Journal, 2014, 5, 1-7.

136

Original Publications from thesis

1. Metoprolol Eudragit Microcapsules: Pharmacokinetic study using convolution

approach. Latin American Journal of Pharmacy, 31 (6): Abdul Hakim Abdullah,

Khalid Pervaiz Akhter and Ghulam Murtaza (2012), 914-917.

2. Development of In Vitro In Vivo correlation for encapsulated Metoprolol

tartrate, Acta Poloniae Pharmaceutica-Drug Research, 70 (4): Abdul Hakim

Abdullah, Khalid Pervaiz Akhter and Ghulam Murtaza (2013), 743-747.

3. In Vitro to In Vivo Profiling: An easy idea for Biowaiver study, Acta Poloniae

Pharmaceutica-Drug Research, 70 (5): Abdul Hakim Abdullah, Khalid Pervaiz

Akhter and Ghulam Murtaza (2013), 873-875.

4. Pharmacokinetic based on convolution/deconvolution and Laplace transforms,

Bulletin of the Malaysian Mathematical Sciences Society (submitted).

137