Theories and mechanisms of dissolution testing

D.Narender 1 st Semester
 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
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
3 Theories
1) Diffusion layer model / Film theory
Danckwert’s model (penetration or surface renewal theory)
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
Film boundary
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
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
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
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
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
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
Biopharmaceutical Classification System
High Solubility
(Dose Vol. NMT
250 mL)
Low Solubility
(Dose Vol. >250
High Permeability
(Fract. Abs. NLT
e.g. Propranolol
e.g. piroxicam,
Low Permeability
(Fract. Abs. <90%)
e.g. ranitidine
e.g. furosemide
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
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.
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.
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.
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
t = (2A – Cs)h2/4DCs --------------------------------(6)
h = ( 4.D.Cs t / 2A – Cs)1/2
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)
Q = (D.Cs.t / 2A – Cs)1/2 . (2A – Cs)
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.
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
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,
= [ 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
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
0.5 < n < 1.0 0.45 < n <
0.43 < n <
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
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
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.
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 no.15-48.
5) Text book of Dissolution ,Bio availability and Bio equivalence, by
hammed no 337-354.
6) Pharmaceutical Dissolution Testing ,by Umesh .V. Banakar, 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 –

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