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THEORIES AND MECHANISMS OF DISSOLUTION TESTING By D.Narender M.pharmacy 1 st Semester DEPARTMENT OF PHARMACEUTICS UNIVERSITY COLLEGE OF PHARMACEUTICAL SCIENCES KAKATIYA UNIVERSITY OUT LINE Definitions Theories of Dissolution Mechanisms of drug release Wagner theory Zero order release First order release Hixon -Crowel model Higuchi model Peppas model Weibull model Conclusion Definitions: Dissolution: Dissolution is defined as a process in which a solid substance solubilizes in a given solvent i.e. mass transfer from solid surface to the liquid phase. Dissolution rate: Dissolution rate is defined as the amount of solute dissolved in a given solvent under standard conditions of temperature, pH, solvent composition and constant solid surface area. It is a dynamic process The rate of dissolution of drug substance is determined by the rate at which solvent-solute forces of attraction overcome the cohesive forces present in solid Drug Dissolution Process THEORIES OF DISSOLUTION: 3 Theories 1) Diffusion layer model / Film theory 2) Danckwert’s model (penetration or surface renewal theory) 3) Interfacial barrier model (double barrier or limited solvation theory) Diffusion layer model Assumes that there is a stagnant layer or diffusion layer which is saturated with the drug at the solid –liquid interface. From this stagnant layer, diffusion of soluble solute occurs to the bulk of the solution. Cs S Film boundary c bulk solution Stagnant layer Diffusion layer model Here ,the dissolution is diffusion controlled where the solvent-solute interaction is fast when compared with the transport of solute into bulk of solution Once the solute molecules pass the liquid film-bulk film interface rapid mixing occurs and concentration gradient is destroyed. The rate of solute movement and therefore the dissolution rate are determined entirely by the Brownian motion diffusion of molecules in liquid film. The rate of dissolution when the process is diffusion controlled is given by noyes-whitney equation Equation: dC/dt =D.A.Kw/o (Cs –Cb)\ v.h dC/dt = dissolution rate of the drug. D = diffusion coefficient of the drug. A = surface area of the dissolving solid Kw/o = water/oil partition coefficient of drug V = volume of dissolution medium h = thickness of stagnant layer Cs–Cb = concentration gradient of diffusion of drug Limitation : Assumes that surface area of the dissolving solid remains constant during dissolution which is practically not possible. To account for particle size ,Hixson and Crowell cube root law was developed Equation: w01/3 – w1/3 = k .t W=mass of drug remaining to be dissolvedat time t K=dissolution rate constant w0 =original mass of the drug. 2) Danckwert model: Did not approve the existence of stagnant layer as said by diffusion layer theory Instead, said that turbulence existed in dissolution medium near solid – liquid interface. Due to agitation, mass of eddies or packets reach the solid –liquid interface and absorb the solute and carry to bulk of solution since solvent molecules are exposed to new solid surface each time, the theory is called surface renewal theory Equation: V.dC/dT= dm/dt = A ( Cs-Cb). (ү.D)1/2 m=mass of solid dissolved Ү = rate of surface renewal. 3) Interfacial layer model Film boundary S Cs C Bulk solution Stagnant layer In this model it is assumed that the reaction at solid surface is not instantaneous i.e. the reaction at solid surface and its diffusion across the interface is slower than diffusion across liquid film. therefore the rate of solubility of solid in liquid film becomes the rate limiting than the diffusion of dissolved molecules equation : dm/dt = Ki (Cs – C ) K = effective interfacial transport rate constant Biopharmaceutical Classification System High Solubility (Dose Vol. NMT 250 mL) Low Solubility (Dose Vol. >250 mL) High Permeability (Fract. Abs. NLT 90%) CLASS І CLASS II e.g. Propranolol metoprolol e.g. piroxicam, naproxen Low Permeability (Fract. Abs. <90%) CLASS III CLASS IV e.g. ranitidine cimetidine e.g. furosemide hydrochlorothiazide Mechanisms of dissolution Wagner theory Wagner interpreted the percent dissolved time plots derived from the in vitro testing of regular tablets and capsules. this concept relates to the apparent first order kinetics under sink conditions to the fact that a percent dissolved value at time t may be equivalent to the percent surface area generated at same time. Wagner utilized the following mathematical method to desribe his theory for the dissolution kinetics of conventional tablets and capsules assuming that surface area available for dissolution decreases exponentially with time according to the equation; S = S0 e-ks ( t-to) --------------------------------> Where So is the surface area at time to. 1 But we know that dW/dt = K.S.Cs ------------------------(2) Substitution for S from equation (1) ,we get dW/dt = K.Cs.So.e-ks(t-to) ------------------------(3) Integration of above equation gives w= w0+K/ks Cs So [1-e-ks(t-to)] ------------------------(4) If it is assumed that W∞ is the amount in solution at infinite time and M= K/ks.Cs.So,then W∞ = Wo+M and W∞-W = M e-ks(t-to) -------------------(5) Applying log to both sides ,we get, log (W∞ - W) = log M – ks/2.303( t – to) ------------(6) Where W∞ - w is the amount of undissolved drug. Zero order release: Zero order refers to the process of constant drug release from a drug delivery device such as oral osmotic tablets,transdermal systems,matrix tablets with low soluble drugs constant refers to the same amount of drug is released per unit time. drug release from pharmaceutical dosage forms that donot disaggregate and release the drug slowly can be represented by the following equation W0 – Wt = K .t ------------------- 1 W0 = initial amount of drug in the dosage form. Wt = amount of drug in the pharmaceutical dosage form at time t K = proportionality constant. Dividing this equation by W0 and simplifying ft = K0 .t where ft = 1-(Wt/W0) Ft = fraction of drug dissolved in time t and Ko the zero order release constnat. A graphic of the drug dissolved fraction versus time will be linear. Applications: Zero order kinetic model can be used to describe the drug dissolution of several types of modified release pharmaceutical dosage forms, as in case of some trans dermal systems ,as well as matrix tablets with low soluble drugs, coated forms ,osmotic systems etc. First order release: If the amount of drug Q is decreasing at a rate that is proportional to he amount of drug Q remaining ,then the rate of release of drug Q is expressed as dQ/dt = -k.Q -----------------1 Where k is the first order rate constant. Integration of above equation gives, ln Q = -kt + ln Q0 ---------------- 2 The above equation is aslo expressed as Q = Q0 e-kt ------------------------ 3 Because ln=2.3 log, equation (2) becomes log Q = log Q0 + kt/2.303 ---------------------(4) This is the first order equation A graphic of the logarithm of released amount of drug versus time will be linear. Inference The pharmaceutical dosage forms following this model, such as those drugs containing water soluble drugs in porous matrices, release the drug in a way that is proportional to the amount of drug remaining in its interior. This model has been also used to describe absorption and elimination of drugs. Higuchi’s mechanism. Higuchi developed an equation for the release of drug from an ointment base and applied it to diffusion of solid drugs dispersed in homogenous and granular matrix devices. Higuchi pointed out that to develop mathematical relationship for the release of drugs from matrix tablets, two systems are considered. a) first, when the drug particles are dispersed in homogeneous uniform matrix, which acts as diffusional mechanism b) When the drug particles are incorporated in granular matrix and released by leaching action of penetrating solvent. Higuchi demonstrated that during the initial release phase from a spherical system until approximately 50% of drug content in vehicle has been released,the square root of time behaviour is dominating and then it depends on design of sustaine release system. From Ficks first law, dM/S.dt = dQ/dt = D.Cs/ h --------------------------(1) As the drug passes out of a homogeneous matrix. the boundary of drug( represented by the dashed vertical line), moves to the left by an infinitesimal distance, dh. The infinitesimal amount ,dQ, of the drug released because of this shift is given by dQ = A.dh – ½ Cs dh -----------------------(2) Substituting (2) in (1),we get D .Cs /h = (A – ½ Cs) dh/dt --------------------(3) The steps for derivation as given by higuchi are , 2A – Cs/2DCs ∫ h dh = ∫ dt ------------------ (4) t = (2A –Cs) h2/4DCs +C -------------------------(5) The integration constant C,can be evaluated at t=0 at which h=0.giving. t = (2A – Cs)h2/4DCs --------------------------------(6) h = ( 4.D.Cs t / 2A – Cs)1/2 ------------------------------(7) The amount of drug depleted per unit area of matrix .Q at time t is obtained by integrating the equation (2) to yield, Q = h.A -1/2 h.Cs ---------------------------- (8) Substituting Q = (D.Cs.t / 2A – Cs)1/2 . (2A – Cs) or Q = [D(2A-Cs)Cs.t]1/2 ------------------------------------- (9) This is known as higuchi equation. When the porosity and tortuosity of the matrix is concerned, the equation is modified as ; ( for heterogeneous type matrix) Q = [D€/t( 2A - € Cs)Cs.t]1/2 -------------------------------- (10) The instantaneous rate of release of a drug at time t is obtained by differentiating equation (10 ) to yield, dQ / dt = ½ [ D(2 A – Cs)Cs/t]1/2 ------------------------ (11) Ordinarily A is much greater that Cs and hence equation ( 9 ) reduces to Q = (2.A.D.Cs.t)1/2 --------------------------- (12) And hence equation ( 11) becomes . dQ/dt = (A.D.Cs/2t)1/2 ---------------------------- (13) Equation (12), indicates that the amount of drug released is proportional to square root of A , the total amount of drug in unit volume of matrix; D. the diffusion coefficient of the drug in matrix; Cs is the solubility of drug in polymeric matrix and t the time. Graph : graph is plotted between % drug release and square root of time. Applications: Higuchi describes the drug release as a diffusion process based on Ficks law, square root time dependent . This model is useful for studying the release of water soluble and poorly soluble drugs from variety of matrices ,including solids and semi solids. Hixon-crowell cube root law Hixon Crowell cube root equation for dissolution kinetics is based on assumption that: a) Dissolution occurs normal to the surface of the solute particles b) Agitation is uniform all over the exposed surfaces and there is no stagnation. c) The particle of solute retains its geometric shape The particle (sphere) has a radius r and surface area 4Π r2 Through dissolution the radius is reduced by dr and the infinitesimal volume of section lost is dV = 4Π r2 . dr ------------------(1) For N such particles,the volume loss is dV = 4N Π r2 dr ----------------------------(2) The surface of N particles is S = 4 N Π r2 -----------------------------(3) Now ,the infinitesimal weight change as represented by he noyes –whitney law ,equation is dW = k.S.Cs.dt ---------------------------(4) The drugs density is multiplied by the infinitesimal volume change ρ.dV, can be setequal to dW, ρ.dV = k.S.Cs.dt --------------------------- (5) Equations (2) and (3) are substituted into equation (5) , to yield -4 ρ N Π r2 . dr = 4 N Π r2 . K .Cs .dt -------------(6) Equation 6 is divided through by 4 N Π r2 to give - ρ . Dr = k Cs.dt -------------------------(7) Integration with r = ro at t= 0produces the expression r = ro – kCs .t/ ρ -----------------------------(8) The radius of spherical particles can be replaced by the weight of N particles by using the relationship of volume of sphere W = N ρ(Π/6)d3 ----------------------------(9) Taking cube root of the equation (9) yield, W 1/3 = [ N ρ(Π/6)]1/3. d. ----------------------------(10) The diameter d from equation (10) ,is substituted for 2r into equation 8 to give W01/3 - W1/3 =k t ------------------(11) Where k = [ N ρ(Π/6)]1/3.2 k Cs/ρ. Wo is the original weight of drug particles . Equation (11) is known as Hixson- Crowell cube root law ,and k is the cube root dissolution rate constant. Futher dividing euation (11) by w01/3 and simplifying,we get ( 1 – ft )1/3 = k t Where ft = 1-(w/w0) and it represents the drug dissolved fraction at time t And k is release constant. Korsmeyer and peppas model Also called as power law To understand the mechanism of drug release and to compare the release profile differences among these matrix formulations ,the percent drug released time versus time were fitted using this equation Mt / M∞ = k. tn Mt / M∞ = percent drug released at time t K= constant incorporating structural and geometrical characteristics of the sustained release device. n =exponential which characterizes mechanism of drug release Exponent n of the power law and drug release mechanism from polymeric controlled delivery systems of different geometry Exponent n slab Cylinder Sphere DR mechanism 0.5 0.45 0.43 Fickian diffusion Anomalous transport 0.5 < n < 1.0 0.45 < n < 0.89 1.0 0.89 0.43 < n < 0.85 0.85 Case-II transport Applications: 1. This equation can be applied to any kind of delivery system 2. This model is generally used to analyze the release of pharmaceutical dosage forms, when the release mechanisms is not well known or when more than one type of release phenomena could be involved. Weibull Model It expresses the accumulated fraction of the drug in solution at time by following equation: m = 1- exp [-(t –Ti )b /a ] m = accumulated fraction of the drug at time t a = scale parameter Ti = location parameter ( represents lag time before the onset of dissolution or release process and in most cases will be zero ) b = shape parameter. The equation may be rearranged into: log[ -ln(1-m)]= b log ( t-Ti )- log a graph: -ln(1-m) vs t gives linear relation and the slope is equal to shape parameter CONCLUSION The Quantitative interpretation of the values obtained in dissolution assays is easier using mathematical equations which describe the release profile in function of some parameters related with the pharmaceutical dosage forms. The release models with the major appliance and the best describing drug release phenomena are in general ,the Higuchi model, Zero order model and Korsmeyer- Peppas model. the Higuchi and Zero order models represent two limit cases in the transport and drug release phenomena and the Korsmeyer-Peppas model can be a decision parameter between these two models while the Higuchi model has a larger application in polymeric systems, the zero order model becomes ideal to describe coated dosage forms or membrane controlled dosage forms. References 1) Remington's “The science and practice of pharmacy” 21st edition page no 672-685. 2) “A Text book of Applied Bio pharmaceutics and pharmacokinetics”, by Leon Shargel,andrew , 4 th edition ,page no 131-195. 3) “Text book of Bio pharmaceutics and pharmacokinetics” ,by V.Venkateshwarlu page no.32-55. 4) Text book of Bio pharmaceutics and pharmacokinetics, by Brahmankar.page no.15-48. 5) Text book of Dissolution ,Bio availability and Bio equivalence, by hammed m.abdoue.page no 337-354. 6) Pharmaceutical Dissolution Testing ,by Umesh .V. Banakar, pg.no 1100,pg no 200- 350 7) Text book of Martins, physical pharmacy and pharmaceutical sciences. page no 337-354. 8)Encyclopedia of pharmaceutical technology, by James Swarbrick, James C.Boylan volume 4 page no 121-126 9) European Journal of Pharmaceutical sciences 13 (2001) page no.123 – 133.