DRUG RELEASE KINETICS AND MATHEMATICAL MODELLING.pptx

Drug Release Kinetics

1. CONTENTS Introduction Mathematical models Fick’s first law Fick’s second law Zero order release model First order release kinetics Hixson - Crowell release equation Korsmeyer-peppas equation 1

2. DRUG RELEASE KINETICS AND MATHEMATICAL MODELLING

3. 3 WHAT DO MEAN BY DRUG RELEASE ? “ It is a process by which a drug leaves a drug product & is subjected to ADME & eventually becoming available for pharmacological action.” It involves the study of drug release rate, dissolution /diffusion/erosion studies and the study of factors affecting release rate of the drug. WHAT DO MEAN BY DRUG RELEASE KINETICS? “Drug release kinetics is application of mathematical models to drug release process.

4. Laws in the kinetics of drug release Noyes-Whitney law : Noyes-Whitney developed the fundamental principle of evaluation of drug release, which explains as: dC /dt=K (C s -C t ) Where dC /dt is dissolution rate of drug K is dissolution rate constant, C s is the concentration of drug in the stagnant layer (also called as the saturation or maximum drug solubility) C t is the concentration of the drug in the bulk of the solution at time t 4

5. Ficks law of diffusion : According to this law drug molecules diffuse from a region of higher concentration to one of lower concentration until equilibrium is attained and that the rate of diffusion is directly proportional to concentration gradient across the membrane. This can be expressed mathematically as dQ /dt= DAK m /w (C GIT - C)/h Where dQ /dt is rate of drug diffusion, amount per time (rate of appearance of drug in blood), D is the diffusion coefficient A is the surface area for absorbing membrane for drug diffusion (area), Km/w Partition coefficient of drug between lipoidal membrane and aqueous GI fluid (no unit), C GIT - C is the difference in the concentration of drug in the GI fluid and the plasma called concentration gradient (amount per volume), h is the thickness of the membrane 5

6. There are number of kinetic models, which described the overall release of drug from the dosage forms. Because qualitative and quantitative changes in a formulation may alter drug release and in vivo performance, developing tools that facilitate product development by reducing the necessity of bio-studies is always desirable. In this regard, the use of in vitro drug dissolution data to predict in vivo bio-performance can be considered as the rational development of controlled release formulations . 6 Release kinetic modeling

7. The methods of approach to investigate the kinetics of drug release from controlled release formulation can be classified into three categories: ● Statistical methods (exploratory data analysis method, repeated measures design, multivariate approach [MANOVA: multivariate analysis of variance] ● Model dependent methods (zero order, first order, Higuchi, Korsmeyer-Peppas model, Hixson Crowell, Baker-Lonsdale model, Weibull model, etc. ● Model independent methods [ difference factor (f1), similarity factor (f2)] 7

8. ZERO-ORDER MODEL The equation for zero order release is Q t = Q + K t; Q = initial amount of drug Q t = cumulative amount of drug release at time “t”. K = zero order release constant. t = time in hours 8

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10. First Order Model This model has also been used to describe absorption and/or elimination of some drugs. The release of the drug which followed first order kinetics can be expressed by the equation: where K is first order rate constant Equation can be expressed as: where C is the initial concentration of drug, k is the first order rate constant, and t is the time. The data obtained are plotted as log cumulative percentage of drug remaining vs. time which would yield a straight line with a slope of K/2.303. 10 Application: • This relationship can be used to describe the drug dissolution in pharmaceutical dosage forms such as those containing water-soluble drugs in porous matrices.

11. A mathematical model to describe the drug dissolution from matrix systems was not developed until the 1960s. From 1961 to 1963, there was a great advance in the development of mathematical models to understand drug release. In 1961, Higuchi published probably the most famous and most often used mathematical equation to describe the release rate of drugs from matrix systems. The rate of drug release from ointment bases (planar systems) containing drugs in suspension was the object of investigation (Higuchi, 1961). Development of many theoretical models to describe the release of active agents that are less soluble as well as very soluble, contained in solid and semi-solid matrices. 11 Higuchi model

12. This model is based on the hypotheses that: ( i ) initial drug concentration in the matrix is much higher than drug solubility (ii) drug diffusion takes place only in one dimension (iii) drug particles are much smaller than system thickness (iv) matrix swelling and dissolution are negligible (v) drug diffusivity is constant; and (vi) perfect sink conditions are always attained in the release environment 12

13. Accordingly, model expression is given by the equation: 13 where Q is the amount of drug released in time t per unit area A, C is the drug initial concentration, C s is the drug solubility in the matrix media and D is the diffusivity of the drug molecules (diffusion coefficient) in the matrix substance This relationship is valid over the time of dissolution, except when the drug-released levels tend to saturate the liquid medium contained in the matrix

14. To study the dissolution from a planar heterogeneous matrix system, where the drug concentration in the matrix is lower than its solubility and the release occurs through pores in the matrix, the expression is given by equation: 14 where D is the diffusion coefficient of the drug molecule in the solvent, δ is the porosity of the matrix, τ is the tortuosity of the matrix . Tortuisity is defined as the dimensions of radius and branching of the pores and canals in the matrix After simplifying the above equation, Higuchi equation can be represented in the simplified form Q=K H × t 1/2 where, K H is the Higuchi dissolution constant. The data obtained is plotted as cumulative percentage drug release versus square root of time if the drug release follows H iguchi model- high correlation coefficient- it implies the process of release is diffusion controlled)

15. Hixson-Crowell cube root law The cube root law was first proposed by Hixson and Crowell (1931a) as a means of representing dissolution rate that is normalized for the decrease in solid surface area as a function of time. Hixson- Crowell cube root law describes the release from systems where there is a change in surface area and diameter of particles or tablets. Provided there is no change in shape as a suspended solid dissolves, its surface decreases as the two-thirds power of its weight. This relation has been used by Hixson and Crowell in the derivation of the cube root law. For a drug powder consisting of uniformly sized particles, it is possible to derive an equation that expresses the rate of dissolution based on the cube root of the particles . 15 They discovered that a group of particles’ regular area is proportional to the cube root of its volume.

16. When sink conditions are applied, the cube root law can be written 16 Q = Initial amount of drug. Q t = Cumulative amount of drug release at time “t”. K HC = Hixson Crowell release constant. t = Time in hours Drug releases by dissolution Changes diameter of the particles Changes in surface area release is not by diffusion  

17. The assumptions made for the validity of the law by Hixson and Crowell can be summarized as follows: The law is claimed to be more suitable for monodispersed, predominantly spheroidal, materials, i.e., the solid is in the form of a single unit or all units having identical properties regarding size, shape, surface and volume characteristics The dissolution takes place normal to the surface. The difference in rates at different crystal faces is considerably less and the effect of agitation of the liquid against all parts of the surface remains same. The liquid is agitated intensely to prevent stagnation in the nearest places of the dissolving particle thus resulting in a slow rate of diffusion. 17

18. KORSMEYER-PEPPAS EQUATION Korsmeyer – Peppas equation is F = (M t /M ) = K m t n F = Fraction of drug released at time ‘t’ M t = Amount of drug released at time ‘t’ M = Total amount of drug in dosage form K m = Kinetic constant of incorporation of structural modifications and geometrical characteristics of the system n = Diffusion or release exponent indicative of the mechanism of transport of drug through the polymer. The n value is used to characterize different release mechanisms t = Time in hours This model was developed specifically for the release of a drug molecule from a polymeric delivery systems (both swellable and non swellable) such as a hydrogel . 18

19. When the drug release process is characterized by an abrupt increase of initial drug release (burst effect), the following equation was proposed : 19 The power law model is useful for the study of drug release from polymeric systems when the release mechanism is not known or when more than one type of phenomenon of drug release is involved. M ∞ is the amount of drug at the equilibrium state M i is the amount of drug released over time t

20. Over the last many years, many scientists have employed a modified form of this equation that contains the latency time (l), whichmarks the beginning of drug release from the system ( El-Arini&Leuenberger , 1998;Ford et al., 1991; Kim & Fassihi , 1997; Pillay & Fassihi , 1999):   or the logarithmical version:        l is the latency time.  

21. Depending on the value of n that better adjusts to the release profile of an active agent in a matrix system, it is possible to establish a classification, according to the type of observed behavior : 21

22. In the Fickian model (Case I), n=0.5 and the drug release are governed by diffusion. When n = 1, the model is non- Fickian (Case II), the drug release rate corresponds to zero-order release kinetics and the mechanism driving the drug release is the swelling or relaxation of polymeric chains. At the end of Case II transport, a fast increase of absorption rate of solvent may sometimes be observed. In this situation, the transport Case II evolved to transport Super Case II, due to the expansion of forces exercised by swollen gel in the vitreous nucleus. • When 0.5<n < 1, constituting an extreme form of transport. During the sorption process, tension and breaking of the polymer (solvent crazing- interaction b/w polymer and surface active liquid environment ) occurs. To determine the exponent n, it is recommended to use the first 60% of drug release data. Applications: • Release data from microcapsules and microspheres 22 Fick’s diffusion equation - dQ /dt= DAK m /w (C GIT - C)/h

23. Between the three classes of non- Fickian diffusion (Case II, Anomalous Case and Super Case II transport), the main difference is the velocity of solvent diffusion. In Case II, the velocity of solvent diffusion is less than the polymeric relaxation process. In Anomalous transport, the velocity of solvent diffusion and the polymeric relaxation possess similar magnitudes. In Super Case II, the velocity of solvent diffusion is much higher, causing an acceleration of solvent penetration. Transport Case II is observed when the solvent has a high affinity by matrix.