Mathematical Modelling of Dissolution Kinetics in Dosage forms

 

Mohamed Rizwan I1, Damodharan N2*

1Department of Pharmaceutics, SRM College of Pharmacy, SRM Institute of Science and Technology,

Kattankulathur-603203

2Department of Pharmaceutics, SRM College Of Pharmacy, SRM Institute of Science and Technology,

Kattankulathur-603203.

*Corresponding Author E-mail: damodhan@srmist.edu.in

 

ABSTRACT:

Drug delivery is a technique by which the drug substances are administered to attain the therapeutic responses in humans or animals. It is crucial to study about the drug release or dissolution for a new solid dosage form. The prediction of correct data of discharge from the drug in the dissolution using mathematical model, which can help to understand the innovation in drug design and formulation with desired therapeutic efficacy and safety. Generally, the mathematical models are used to determine the release of drug profiles are zero-order kinetic model, first order kinetic model, Korsmeyer-peppas model, Higuchi model, Hopfenberg model, Baker Lonsdale model, Hixson Crowell model and Weibull model. Zero-order model explains the constant releases from the formulation such as transdermal, implantable depot, oral control release, matrix tablet with low solubility and suspension. First-order have explained about the release from the formulation like matrix dissolution controlled release, matrix diffusion controlled release, sustained release. Higuchi model explained the drug has released from matrix type. Hixson Crowell cube root law described the drug release has occurred from the variation in particles diameter of system and in surface area. Korsmeyer-peppas model created the experimental equation to evaluate the both fickian and non-fickian release of drug. From the Higuchi model, the Baker and Lonsdale has improved which describes the drug discharge from the matrices of spherical. The Weibull model defined the drug release from the aggregated fraction of drug ‘m’ at time ‘t’ in solution. Hopfenberg model compares the drug release from surface eroding polymers.

 

KEYWORDS: Dissolution, Release kinetics, Release models, Mathematical modelling.

 

 


INTRODUCTION:

Dissolution and release of drugs are the most important phenomenon for all kind of dosage forms like solid dosage form such as capsules, tablets and semi-solid dosage forms such as creams, ointments and implants which deliver the drugs over the intended period time ranging from hours, weeks and years. It is also applicable in the design and optimisation of all kinds of modified release dosage forms like sustained, controlled and delayed dosage forms[1].

 

In vitro dissolution is crucial constituent in the drug development. From the modified and immediate release dosage forms are explained the delivery of drug by various concepts of models related to release kinetics. The dissolution profiles are denoted by many kinetic models where ft is a function at time t connected from the pharmaceutical dosage form to the bulk of drug dissolved[2] . The common equations are converting the dissolution curve into some function of parameters mathematically according to certain parameters associated with pharmaceutical dosage forms. So that the dissolution test values are acquired easily through the equation. In certain cases, the theoretical analysis of the process can deduce this equation, such as in zero order kinetics. An empirical equation was used sufficiently when the theoretical basis did not occur on capsule, tablet, prolonged release form and coated forms. The release kinetics are affected by solubility, crystallinity, polymorphic form and drug quantity in the pharmaceutical dosage form. Dissolution is a technique in which the drug substances are dissolved in an appropriate solvent[3] . The dissolution means the mass is transferred from a solid substance to a liquid phase. This process is composed of two consecutive stages which are as follows. The solid phase liberates the solute molecules due to the interfacial reaction and the transport of solute away from the interfacial boundary under the influence of diffusion/ convection[4] . A few release kinetic models explained about the drug dissolution from dosage forms in that the volume of drug dissolved represents Q is the time function t is expressed as Q= f(t). The Noyes Whitney equation which describes the rate of dissolution is proportional to the surface area (S) of solid dosage and concentration gradient.

 

The dissolution rate =

 

Where, ‘Cs’ represents the concentration of boundary layer to solid surface

 

‘C’ means the concentration of the medium and ‘K’ represents the constant of dissolution rate [4].

 

If dissolution parameter has to takes place the drug particle should cross a membrane known as the diffusion layer. Therefore, the solid particles should meet with GI fluids that will create the saturated layer of drug dissolution in solid surface and immediately surrounding them is called as diffusion layer[5]. The major two possible cases for dissolution of the drug is i) Absorption occurs when the solid particles dissolved in solvent at a faster dissolution rate and in this case, the rate of diffusion controls the rate of absorption. ii) Absorption takes place from the solution after slow dissolution rate of solids and the dissolution rate controls the absorption rate in this case[6]. In statistic studies, model independent methods and model dependent methods are used to correlate the dissolution profiles between the two drug products[3] . Model dependent techniques depends on different mathematical functions that explained the dissolution profiles are zero-order, first-order, Korsmeyer-peppas model, Higuchi model, Hixson Crowell method, Weibull model and Baker Lonsdale model[7]. One or more mechanisms are involved in the discharge of drug kinetics depending upon the composition of matrix, method of preparation, geometry and the dissolution media of the drug release [8]. The mathematical model is used to get the release performance and has a crucial capability to ease the product advancement[9].

 

 

Fundamentals of drug release kinetics:

Noyes Whitney rule:

The basic proposition of the kinetics for evaluation of drug delivery was offered by Noyes Whitney. The equation is expressed as follows,

 

where, ‘M’ represents the mass

‘S’ represents the surface area of solid particles

‘Cs – Cb’ is concentration gradient

‘Cb means the bulk drug concentration in the solution and ‘Cs means the drug concentration in stagnant layer is known as saturation or maximum drug solubility [10] .

 

Nernst and Brunner theory:

To reveal the inter relationship between the constant in Noyes Whitney equation dM/dt = KS (Cs – Ct) and a coefficient of the diffusion in solute using Nernst and Brunner theory according to Fick’s law of diffusion. The Nernst and Brunner experimental equation is expressed as,

 

 Where, ‘D’ represents the diffusion coefficient

‘S’ is surface area

‘γ’ represents volume of the solution

‘h’ is the diffusion layer

 

The process gets much faster at the surface than the transport process simulated by Nernst and Brunner and concentration gradient of linearity is produced confining within the layer of adhering solution to the solid surface particles. According to Noyes Whitney equation, the dissolution process inherits the first order reaction [11] .

According to Fick’s law, this can be mathematically expressed by the following equation

 

The rate of drug diffusion is denoted by dQ/dt

‘D’ represents a diffusion coefficient

‘A’ represents the area of absorbing membrane surface

Km/w is partition coefficient of the drug between aqueous GI fluid and lipoidal membrane

CGIT is drug concentration in the GI fluid

‘V’ represents the volume of media in dissolution

‘h’ means the membrane thickness

(CGIT – C) represents the concentration gradient [6] .

 

Model dependent methods:

Kinetics in dosage form:

The dissolution profiles are explained by the model dependent methods based on various mathematical roles. The model dependent methods included zero-order, first-order, Higuchi model, Hixson model, Korsmeyer-peppas model, Baker Lonsdale model, Weibull model and Hopfenberg model [10]

 

Zero-order model:

The zero-order kinetics is a method which release the substances constantly from the dosage form which are independent to the concentration. A dissolution of the drug from various dosage form which does not segregate, causes slow drug release occurs. it can be described by the equation

 

‘Q’ means quantity of drug discharged or solubilized

‘Qo’ means quantity of drug in solution at the initial stage

‘K’ represents zero-order release constant [12].

 

The linearity will be the fraction of drug solubilized versus time in graphical representation. A value of K gets from the slope of the curve in zero order release kinetics [13]. The graphical representation was plotted as cumulative amount of drug released versus time [14] [15]. The zero-order kinetics used to explain about the dissolution of many kind of formulations such as transdermal and matrix tablet from the modified release dosage form and osmotic type as well [16].

 

First order release model:

Gibaldi and Feldman was first proposed the first order release kinetic models in 1967 [17] followed by Wagner in 1969 [18] [19]. The first order kinetic is a process which release the drug from the system where the rate of release is dependent on concentration. Absorption and elimination of few drugs are illustrated by the first order model and it is very hard to conceptualise the mechanism by theoretical manner [20] [21]. The first order equation is expressed by an equation,

 

Where ‘K’ is first order rate constant

Changing the above equation into logarithmic form is expressed as,

 

Where, ‘Co’ is the initial concentration of drug

‘K’ is first order rate constant

‘t’ represents the time [22].

This equation predicts a first order dependence on the concentration gradient (Co – C) between the static liquid layer and the bulk liquid. The graphical representation was plotted as the logarithm cumulative percentage release versus time[23]. The drug particle dissolution rate is limited to the drug release rate while using this model and not by diffusion[3]. This first order kinetics of drug release is used to illustrate the drug dissolution in pharmaceutical dosage form such as drug soluble in water[24] [25].

 

Higuchi model:

Higuchi developed a first mathematical model to explain the release of drug from a matrix system in dosage forms in 1961[10] [21]. The Higuchi model was considered that the drug release is takes place due to the diffusion, at the beginning this Higuchi model is recommend for planar systems which is prolonged to the various geometrics and porous systems [26]. Higuchi formulated a several methods to study the release of hydrophilic and hydrophobic drugs are combined in semisolid or solid matrixes. This method was constructed the propositions which are a) Solubility of the drug is much lower than the initial drug concentration in the matrix. b) In one dimension, the diffusion of drug takes place including the edge outcome should be avoided. c) Thickness of the system is higher than the drug particles. d) Dissolution and swelling of matrices are inconsequential. e) Drug diffusivity is stable. f) perfect sink condition is constantly reached in the release environment [3]. This Higuchi model is expressed by the equation as,

 

where ‘Q’ is quantity of drug discharged

‘C’ is initial concentration of the drug

‘Cs’ is drug solubility in media

‘D’ is drug diffusivity[3].

 

The planar heterogenous matrix for the dissolution studies implies that when the drug solubility is higher than the concentration of drug in the matrix, the drug release takes place through pores of the matrix, Higuchi model is expressed by the equation as,[27] [28]

 

 where ‘D’ is the diffusion coefficient

δ is matrix porosity

τ is matrix tortuosity

Tortuosity means the radius, bulk and dividing of the canals and holes in the matrix. Simplified the Higuchi model can be written as

 

 

where,

‘KH’ is the dissolution constant and t is the time [10].

 

The dissolution data is collected from the in vitro dissolution study and plotted as the cumulative percentage released versus square root of time[29][30]. This Higuchi model usually describes drug dissolution studies from different modified release kind of dosage forms, for instance, used in some matrix tablets contains water soluble property and transdermal systems[31] [32].

 

Hixson Crowell model:

Hixson and Crowell were developed a model in 1931 [33]. When there is an alteration in the size (diameter) of the particles and surface area of the tablets there occurs the delivery of the drug from the system. Hixson and Crowell cube root law explains the release of drugs from the system where there is change in diameter of the particles and surface area of the particles [26]. The Hixson Crowell model states that the drug particles and dissolution rate are assumed as the rate of drug release is limited and not by the diffusion [34]. Hence, this model results in proportion between the cube root of its volume and surface area of particle. It is expressed by an equation,

 

Where

‘W0 represents the drug quantity at initial stage

‘Wt’ is the remaining of drug presents at time t

‘κ’ (kappa) is a constant

 

This equation is used for exposition of dissolution data of immediate release dosage form like dispersible tablets, conventional tablets and from other dosage forms[34]. To study the drug release kinetics the dissolution data gets from the in vitro drug dissolution and the data was plotted as the cube root of percentage drug remaining versus time[35] [36]. The Hixson Crowell model was used to illustrate the dissolution plan and in the pharmaceutical dosage form such as tablets and capsules, if a tablet reduces the proportionality the dissolution in planes which are collateral to the drug surface[37]

 

Korsmeyer Peppas model:

This model was first proposed by Korsmeyer Peppas in 1983, obtained an easy relationship which processes are illustrated the release of drug from a polymeric form[38] [39]. This model formulated a simple model and semi-empiric model. The Korsmeyer-peppas model is expressed by an equation,

 

Where,

‘Mt is drug quantity at time t

‘M’ is total amount of drug released after time t

‘n’ is drug release exponent

‘K’ is Korsmeyer release rate constant[6].

 

Korsmeyer and Peppas developed a factual equation to examine both fickian release which obey the fickian laws and non-fickian release of drug which does not obey the fickian law from swelling polymeric and non-swelling polymeric delivery system as well[40]. Fickian diffusion outcome is when n = 0.5 and when n = 1, where the drug release occurred and the release obeyed to zero order kinetics[41]. Korsmeyer-peppas model accept mostly 60% of the drug release data which helps to find out the mechanism of drug release. The graphical representation of the model was studied by acquires release data from in vitro drug release and it is plotted as log cumulative percentage drug release versus log time [42][43]. The Korsmeyer peppas model used in several formulation of microsphere and microcapsules with the linearization of release data[21].

 

Weibull model:

The Weibull model proposed by Weibull in 1951 [44] followed by langenbucher in 1972[45]. The drug release curves are explained using Weibull equation. The Weibull equation mostly fits to all kind of dissolution curves [46] [47]. The Weibull equation builds up the fraction of the drug (m) when applied the equation to release of drugs

 

 

Where M0 is quantity of the drug dissolved, the scale parameter, ‘T’ describes the lag time before the onset of dissolution or release process and it will be zero in most cases [48]. The scale parameter ‘a’ is process time, while ‘b’ describes the advancement in the shape of dissolution curve. This equation may be rearranged into logarithmic form as follows

 

 

Where the shape parameter ‘b’ defines when b=1, a correlation of dissolution curve shape and an exponential profile with the constant, the curve shape becomes sigmoid with a turning point when the ‘b’ value more than 1, whereas when the ‘b’ value is lower than 1 the shape of the curve reveals the parabolic form [3]. To study the drug release kinetics the dissolution data acquired from the in vitro drug release studies and it is plotted as logarithm of the dissolved amount of drug versus the logarithm of time [49] [50]. The drug release profile in matrix form of drug delivery is usually compared using Weibull model [51-53].

 

Baker Lonsdale model:

From the Higuchi model, Baker and Lonsdale was developed in 1974 which illustrates the drug discharge from spherical matrix in a controlled manner, it is expressed as the following equation [26],

 

where

‘Mt’ is volume of drug released at time t

‘M’ represents drug released quantity at the time infinity

‘Dm is diffusion coefficient

‘Cms represents drug solubilization

‘r0’ is the radius

‘C0’ is drug initial concentration

 

The graphical representation of drug release kinetics, the release data were collected from the in vitro drug release studies and it is plotted as [d (Mt / M)]/ dt versus the root of time inverse [54]. The baker Lonsdale model is mainly useful for providing the linearity report of release data from various dosage form such as microspheres and microcapsules [55].

 

Hopfenberg model:

Hopfenberg in 1976 and katzhendler et al in 1997 developed a common analytical equation reporting the release from spheres, slabs and infinite cylinders exhibits heterogenous erosion. On the degradation process, the surface area remains constant when discharge of the drug rectified from the surface eroding polymers using these mathematical models. [56] [57]. It can be expressed by an equation,

 

Where

‘Mt’ represents the quantity of drug solubilized at time ‘t’

‘M’ represents quantity of drug solubilized at infinite time

‘Mt / M is drug dissolving fraction

‘K0’ represents the constant of erosion rate

‘C0’ represents the drug initial concentration

‘a0’ represents the sphere radius at beginning

‘n’ means exponent that differs with geometry which values are 1, 2 and 3 for sphere, cylinder and slab correspondingly[58]. At the initial stage of drug discharge from the pharmaceutical dosage form is to attain the reflection of action associated with the absorption phase and drug migration to the absorption phase is known as lag time [59].

 

Where kl is equivalent to k0/C0a0. The assumption of this model relates to the rate limiting step by the disintegration of matrix which is dependent upon the time diffusion opposition either internal or extrinsic towards the erosion of matric does not influence it[60] [61]. This model identifies the release mechanism, the data obtained from the optimized oily-sphere through the complex profile, which results the discharge of drug at two different rates or two different time is known as biphasic release in the specified site[62] [63].

 

CONCLUSION:

This review focuses on the mathematical models for the dissolution kinetics study. The following models are used to evaluate the discharge of drug from the formulations are zero-order kinetic, first-order kinetic model, Higuchi model, Korsmeyer Peppas model, Hixson Crowell model, Weibull model, Baker Lonsdale model, and Hopfenberg model. Zero-order model explains the constant releases from the formulation such as transdermal, implantable depot, oral control release, matrix tablet with low solubility, suspension and osmotic pressure type. First-order have explained about the release from the formulation like matrix dissolution controlled release, matrix diffusion controlled release, sustained release. Hixson Crowell model describes the releases from the systems where the change in surface area and changes diameter of the particles. Korsmeyer-Peppas model have described about the discharge of the drug from both the fickian and non-fickian diffusion system. Baker Lonsdale model explains the release from the spherical matrices. Weibull model mostly applicable to all systems but more complicate to evaluate. Hopfenberg model have explained about the drug releases from the eroding polymers on surface. From this review, it is evident that there is no single mathematical model which is widely accepted to determine the dissolution profile of the drug if dissolution profiles are similar. It’s very crucial to study the drug release mechanism, which can ultimately derives the drug release mathematically. Moreover, the mathematical models are used to determine the drug release and it will help to develop the design of new drug delivery system with appropriate therapeutic efficacy and safety.

 

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Received on 14.09.2019            Modified on 11.10.2019

Accepted on 16.10.2019           © RJPT All right reserved

Research J. Pharm. and Tech 2020; 13(3):1339-1345.

DOI: 10.5958/0974-360X.2020.00247.4