Drug release kinetics

Drug release kinetics and
mathematical models
Mr. sagar Kishor savale
Department of Pharmaceutics
avengersagar16@gmail.com
2015-2016
Department of Pharmacy (Pharmaceutics) | Sagar savale

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.
 Peppas & Sahlin equation.
 wiebull equation.
 Baker and Lansdale model.
 Hopfenberg Model
 Drug release from slab, sphere and cylinder.
 Drug release from erosion controlled matrix.
 Drug release from lipophilic matrix.
24/19/2016 sagar kishor savale

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?
o “Drug release kinetics is application of
mathematical models to drug release process.”

Drug
release
form
matrix
Drug diffusion from the
non-degraded polymer
(diffusion-controlled
system).
Enhanced drug
diffusion due to
polymer swelling
(swelling-controlled
system)
Drug release due
to polymer
degradation and
erosion (erosion-
controlled
system).

Non-Biodegradable
polymer matrix
Concentration
gradient
Diffusion
Matrix swelling
Biodegradable
polymer
Hydrolytic
cleavage of
polymer chains
Matrix erosion
Diffusion

MATHEMATICAL MODELS-
1. Zero order release model
2. First order release model
3. Hixson-crowell release model
4. Higuchi release model
5. Korsmeyer – peppas release model.
6. Korsmeyer-peppas equation.
7. Peppas & Sahlin equation.
8. wiebull equation.
9. Baker and Lansdale model.
10. Hopfenberg Model

FICK’S FIRST LAW
J = dM
s dt
dM = change in mass of material.
S = barrier surface area.
dm/dt = rate of mass transfer.
J = flux.
Deals with the mass diffusing across a unit
area of barrier in unit time .
Fig. diffusion cell .donor compartment
contains diffusant at conc. c

FICK’S SECOND LAW
 An equation for mass transport that emphasizes the
change in concentration with time at a definite
location.
 It states “the change in conc. with time in particular
region is proportional to change in conc. gradient at
that point in the system.”
dC = D d²c
dt dx²

ZERO ORDER RELEASE MODEL
The equation for zero order release is
Qt = Q0 + K0 t
Q0 = initial amount of drug
Qt = cumulative amount of drug
release at time “t”.
(released occurs rapidly
after drug dissolves.)
K0 = zero order release constant.
t = time in hours
• INDEPENDENT OF
CONCENTRATION.DRUG RELEASE RATE
• % CDR VS TIME.
• STRAIGHT LINE OBTAIN.
GRAPHICAL
REPRESENTATION 94/19/2016 sagar kishor savale

Examples
Zero
order
Transdermal
Implantable
depot
Oral control
release
Matrix tablet
with low
solubility
drug
Suspension
Oral osmotic
pressure
“CONSTANT RELEASE” same amount of
drug release per unit time.

First order release kinetics
The first order release equation is-
Log Qt = Log Q0+ Kt /2.303
or Qt =Q0e-Kt
Q0 = initial amount of drug.
Qt = cumulative amount of drug
release at time “t”.
K = first order release constant.
t = time in hours.
• Depends on the
concentration.
DRUG RELEASE
RATE
• log of % drug remaining vs
time
GRAPHICAL
REPRESENTATION

First
order
release
system
Matrix
dissolution
controlled
release
Matrix
diffusion
controlled
release
Sustained
release
Solution

HIXSON – CROWELL RELEASE EQUATION.
The Hixson - Crowell release equation is
Q0 = Initial amount of drug.
Qt = Cumulative amount of drug release
at time “t”.
KHC = Hixson Crowell release constant.
t = Time in hours.
Drug releases by dissolution
Changes in surface area
Changes diameter of the particles
release is not by diffusion
What makes this
equation different from
noyes –Whitney's

For spherical particle
Wt
1/3 = W0 – k1/3t
Wt = particle weight at time t
W0 = initial particle weight
k1/3 = dissolution rate constant
Under sink condition
k1/3 = (4П/3p2)1/3 .DCs/h
For multiparticulate system
wt
½ = w0
1/2 – k1/2t
Under sink condition
k1/2 = (3П/2p)1/2. DCs/k
K = proportionality constant betweendiffusion layer
thickness and particle size
• cube root of % remaining vs
time (hrs)
GRAPHICAL
REPRESENTATION

HIGUCHI’S EQUATION / MODEL
The equation for Higuchi’s
For drug release through matrix
Q = KH t1/2
0r
Mt / M0 = kt
1/2
Q = cumulative amount of drug
release at time “t”
KH = Higuchi constant
t = time in hours
• Amount of drug released
Vs square root of time. .
Graphical
representation

water soluble drug
Poorly water soluble
drug
semisolids
Solids.
some Transdermal
patches
• pentoxyphylline
• morphine.
• Diclofenac gel.
What is the significance HIGUCHI’S
EQUATION / MODEL …….
It describes the drug release as a diffusion process
based on the Fick’s law which is square root time
dependent.

KORSMEYER-PEPPAS EQUATION
Korsmeyer – peppas equation is
F = (Mt /M ) = Km tn
F = Fraction of drug released at time ‘t’
Mt = Amount of drug released at time ‘t’
M = Total amount of drug in dosage form
Km = Kinetic constant
n = Diffusion or release exponent
t = Time in hours

• log % CDR vs log time taken .Graphical representation
‘n’ is estimated from linear regression of log ( Mt/M ) versus log t
n INDICATION
Less than 0.45 Quasi fickian
0.45 fickian diffusion
0.45<n<0.89 anomalous
diffusion or non-
fickian diffusion.
0.89-1
Zero order.
case-2 relaxation
or non fickian
case 2.
>1 Non fickian super
case 2 Fig:This plot explains diffusion
mechanism by which drug
diffuses from dosage form.

• Diffusion release
• Square root of time depended.
• Fickian release.
n= 0.5
• Swelling controlled
• Release rate independent of time.
• Zero order/case II .
n= 1
• Diffusion and swelling
• Release is time depended.
• First order/anamolus /non-
fickian.
n= between 0.5-1
19
fig:- graph of drug release kinetics

Erosion controlled
release rate
Diffusion
Release
Rate
Anomalous diffusion
or
non-fickian diffusion
What is the significance KORSMEYER-
PEPPAS EQUATION….
Release behavior of drug from hydrophilic
matrix.
Case-2
relaxation
or
super case
transport-2
Erosion of polymeric chain.
stresses and state-transition in
hydrophilic glassy polymers
which swell in water or
biological fluids
Diffusion and
dissolution controlled
dosage form
Swellable & erodible
polymer.

This equation can be used to analyze the first
60% of a release curve, regardless of geometric
shape.
Fickian diffusional release and a Case-II
relaxation release, are the limits of this
phenomenon.
Hence modification was need for above stated
condition

PEPPAS & SAHLIN EQUATION
Mt / M0 = Kdt
m
+ Krt
2m
Kd = Diffusion constant
Kr = Relaxation constant
m = Purely fickian diffusion exponent for
device of any geometrical shape, which exhibit
controlled release.
when release of drug depends
upon it’s diffusion as well as on
fickian relaxation of polymer.
Case 2
transport
diffusional
contribution
Used
for???

Diffusion exponent (m) Mechanism
Film Cylinder Sphere
0.50 0.45 0.43 Fickian
diffusion
0.50 < m< 1.00 0.45 < m<
0.89
0.43 < m< 0.85 Anomalous
transport
1.00 0.89 0.85 Case-II
transport
Table :- Describes the limits of this analysis.

WIEBULL DISTRIBUTION
• Almost all systemsAPPLICABLE FOR
• Log of dissolved amt Vs log timeGRAPHICAL
REPRESENTATION
m - accumulated fraction,
a - time scale process,
Ti -lag time ( generally zero),
b - shape factor.
24
b = 1 indicates exponential curve,
b= 2 indicates sigmoid curve,
b= 3 indicates parabolic curve.
What the need of this equation?
It can be widely used for the analysis and characterization of
DRUG DISSOLUTION process from different dosage form.

Baker and Lansdale model
(Mt /M∞) 2/3 - Mt /M∞ = (3DmCms t)/ (r0
2C0)
where Mt is the amt of drug released at time t and M∞ is amt of
drug released at time ∞
 For Controlled release from a spherical matrix.
Hopfenberg Model
Mt /M∞ = 1 – [1 – (kot)/ (C0a0)] n
Mt is the amt of drug dissolved in time t.
Ko is erosion rate constant.
 drug release from slabs, spheres and infinite cylinder displaying
heterogeneous erosion
• Mt/M∞ Vs Time.Graphical representation
• Mt/M∞ Vs Time.Graphical representation

HOW DO WE
USE THESE
MODELS
CAN ONE FORMULATION
FOLLOW DIFFERENT
EQUATIONS AT A TIME
HOW DO WE COME TO
KNOW WHICH MODEL IS
FIT.
 The kinetic model that best fits the dissolution data is
evaluated by comparing the correlation coefficeint ( r ) values
obtained in various models.
 The model that gave higher ‘r’ value is considered as best fit
model.
 yes ,one model can follow two types of release systems.

Sr No. Model Graph
1. Zero Order Cummulative Release Vs
Time
2. First Order Log Cummulative Release
Vs Time
3. Weibull Model Log Cummulative Release
Vs Log Time
4. Higuchi Model % Cummulative Release Vs
(Time) ½
5. Korsemeyer Peppas Model % CRt/CR∞ Vs (Time)1/2
6. Hixson Crowell Model (Unreleased fraction)1/3 Vs
Time
7. Baker-Lonsdale Model Mt/M∞ Vs Time

Article review
 Formulation, and Evaluation of Pentoxifylline-
Loaded Poly (ἑ-caprolactone) Microspheres :-
 Table 1:-yield , drug entrapment and avg. particle
size of pentoxyphylline loaded poly(ἑ-
caprolactone) microspheres.
Formulation
code
Drug :
polymer
% yield Drug
entrapped
Avg .
Particle
size
F1 1 : 3 79.63 73.14 59.3
F2 1 : 4 80.97 76.92 65.6
F3 1 : 4 83.34 74.69 78.52
F4 1: 5 81.28 71.96 86.22

MODEL F1 F2 F3 F4
ZERO
ORDER
0.853 0.865 0.883 0.877
FIRST
ORDER
0.994 0.983 0.965 0.954
HIGUCHI 0.987 0.985 0.978 0.979
TABLE 2:- VALUE OF R² FROM RELEASE DATA OF
VARIOUS FORMULATIONS FOR DIFFERENT MODELS.
THE VALUES LISTED ARE THE VALUES OF COEFFICIENT
RELATED (R²) OBTAINED FROM RELEASE DATA OF VARIOUS
FORMULATIOS FOR DIFFERENT MODELS OF MECHANISM OF
DRUG RELEASE.

ZERO ORDER FIRST ORDER
HIGUCHI

 By applying In vitro release data to various kinetics models
the drug release mechanism was found to be—
 Microspheres were diffusion controlled as plots of the
amount released versus square root of time was found to be
linear.
 The correlation coefficient (r2) was in the range of 0.978-
0.987 for various formulations.
 When log percentage of drug remaining to be released vs.
time was plotted in accordance with first order equation,
straight lines were obtained (r2 >0.95) indicated that drug
release followed first order kinetics.

Mathematical models for diffusion-controlled
systems
RESERVOIR SYSTEM.
Mt = 4∏ RRi Dci
t (R-Ri)
R-Ri = layer of thickness.
Cr = drug concentration in the reservoir.
Mt/t = Cumulative amount of drug release.
MATRIX SYSTEM.
Mt = 6 Dt ½ 3Dt for Mt < 0.4
M∞ ∏r² r² M∞
Mt/M ∞ = Cumulative amount of drug release.
D = diffusion coefficient.
Mt = 1 6 - ∏²Dt
M∞ ∏²exp r²
For Mt > 0.6
M ∞
DISSOLVED
DRUG (Co<Cs)

Mt = 1- r’ ³
M∞ R
FOR DISPERSED SYSTEM
(Co > Cs)

Imbibing water
Polymer concentration decrease
Polymer disentanglement
Swelling/rubbery gel layer
Disentanglement + dissolution
Non fickian release
DRUG RELEASE FROM SWELLING CONTROLLED SYSTEM .

LEE AND PEPPAS MODEL.
Swelling Thermodynamic
equilibrium
Glassy –rubbery
Front (R)
Rubbery- solvent
Front (S)
Water penetration
Initial thickness of
the carrier
• R moves inward.
• S moves outward
Early-time
swelling
• Thermodynamic
attend at S .
• S starts dissolves.
• S moves inwards
(shrinks).
late-time swelling
• Both fronts
moves inwards.
• R diminishes as
the glassy core
disappears
• Rubber region is
only present.
final dissolution
process
35
Fig: one dimensional swelling process–lee fig.

water
No polymer
Relaxation
Fickian
diffusion
Glassy matrix
water
Non fickian
Case ii
transport
Zero order
Anomalous
Diffusion +
Dissolution
First order

MATHEMATICAL MODELS FOR EROSION-
CONTROLLED SYSTEMS
Empirical
No
physiochemical
phenomenon.
Surface erosion.
that exhibits zero
order.
Mechanistic
physiochemical
phenomenon.
Surface erosion.
Involving mass
transfer +
chemical reaction

EMPIRICAL MODEL -
 Weibull equation.
 Hofenbergs equation.
 Hixson-crowell cube root law.
MECHANISTIC MODEL -
Mechanistic
Diffusion and reaction
model
Diffusion + chemical
erosion model
Cellular automata Random erosion
 For diffusion and reaction model.
∂C = De d²c + 2dc k(ε cs-c)
∂t dr² rdr
C - drug concentration .
De - effective diffusivity in liquid-filled pores .
k - drug dissolution rate constant
ε - porosity of the polymer matrix
Cs - equivalent drug saturation concentration in the solution in pores.

 Modified higuchi equation.
dmt = s 2pco ½
dt 2 t
S -surface area on both sides of the planar film,
C0 - initial drug concentration(loading) in the polymer,
P - permeability of the drug inside the polymer matrix.
Mt = (1-b) xa(t)фa,o + xb(t) фb,o p +B
M ∞ фa,o + фb,o p
xa = release mass fraction for fast eroding matrix
xb = release mass fraction for slow eroding matrix
фa,o = surface fraction for fast eroding matrix
фb,o = surface fraction for slow eroding matrix
p = ratio of drug conc in matrix
B = bust release

Cellular automata
ξmax
Mt =∫0 ε(τ, ξ )dξ
M∞
ξ- dispersion parameter .
ε- porosity of the device.
τ- pore size parameter, exact geometry and the encapsulated
drug .

DRUG RELEASE FROM LIPOPHILIC MATRIX
DEPENDING ON DRUG : POLYMER RATIO IT CAN
FOLLOW
 Zero order release
 First order release
 Higuchi square root time model = Mt/M∞ = kt 1/2
 Hixson and Crowell cube-root equation
 Weibull distribution

Factor
affecting
drug release
mechanism
Diluents
Form / state
of drug
Salt
formation
Temp
Viscosity
particle
size
wetting
Binders
Lubricants
solubility
HPMCK4M
HPMC K100M
GELATIN
EC
LACTOSE
STARCH
MCC
griseofulvin

Release profile comparison
 Statistical methods-ANOVA, MANOVA
 Model independent method- AUC, etc.
 Model dependent methods- all stated models.
Similarity factor(f2)
 It Calculates similarity in the % dissolution between two
curves & it is a logarithmic reciprocal square root
transformation of the sum of squared error, value between 50 to
100.

Dissimilarity factor(f1)
 Calculates % difference between two curves at each time
point & is measurement of the relative error between two
curves, value between 0 to 15.
R & T = Dissolution measurements at n time points of the
reference and test, respectively.
n = no. of time points.
t = Time
 Rescigno Index may also be used for comparison.

PROCEDURE SET BY FDA FOR DISSOLUTION
PROFILE COMPARISON:
• At least 12 units of reference & test product used.
• Use mean dissolution values from both curves at each
time interval to calculate f1 & f2 .
• Measurement should be carried out under same test
conditions.
• f1 & f2 values are sensitive to number of dissolution time
points.
• For rapidly dissolving products comparison is not
necessary.
• For curves to be considered similar, f1 values should be
close to 0 & f2 close to 100.

Drug release from thin polymer film.
Mt = 4 Dt ½
M∞ ∏ L²
Mt = mass of drug release at time t.
M ∞ = mass of drug release at ∞.
D = diffusion coefficient.
L = thickness of the film.
 Fickian diffusional release from a thin film is characterize by an
initial t ½ time dependence of the drug released.
 The short-time approximation is valid for the first 60% of the
total released drug (Mt/ M∞ ≤ 0.06)
Assumptions
•Perfect sink
condition.
•One dimensional
isothermal solute
release.

Semi-empirical equation for drug
release from thin polymer slabs.
Mt / M∞ = k’ t
Mt/M ∞ = 1 - exp( - kt) if release order is first
k’ = constant.
t = square root of time.
if release order is zero
For slab ,
spheres,cylinder
Fickian diffusion
may be defined by
an initial t ½ time
THEORETICAL
STANDPOINT
Practical
stand point
Fickian diffusion dependence on t ½
time valid only for first 15% release for
sphere and cylinder

RELEASE FROM CYLINDERS/ SPHERES
By coupling of fickian and non fickian mechanism-
Mt = ktⁿ
M∞
n = diffusional exponent.
t = square root of time.
k = constant (characteristics of macromolecule
specifically geometry).
 Above equ. Shows that the relationship between the diffusional
exponent n and the corresponding release mechanism is clearly
dependent upon the geometry.

References
1. Davis Yohanes Arifin , Lai Yeng Lee , Chi-hwa Wang ; Mathematical Modeling
And Simulation Of Drug Release From Microspheres: Implications To Drug
Delivery Systems ; Advanced Drug Delivery Reviews 58 (2006) 1274–1325.
2. Philip L. Ritger And Nikolaos A. Peppas ; A Simple Equation For Description
Of Solute Release Fickian And Non-fickian Release From Non-swellable
Devices In The Form Of Slabs, Spheres, Cylinders Or Discs ; Journal Of
Controlled Release, 5 (1987) ;elsevier Science Publishers B.V. ;Amsterdam; 23-
36.
3. Nikolaos A. Peppas and Jennifer J. Sahlin; A simple equation for the
description of solute release.Iii. Coupling of diffusion and relaxation;
International Journal of Pharmaceutics, 57 (1989) 169-172.
4. Tamizharasi S, Rathi JC, Rathi V. Formulation and evaluation of
Pentoxifylline-loaded Poly(ἑ-caprolactone) microspheres . Ind J of Pharm Sci,
2008may; 70(3):333-5.

5. Edith Mathiowitz. Encyclopedia of controlled drug delivery. 1stedition, Volume.II, pg: 698-29.
6. Martin’s physical pharmacy &pharmaceutical sciences, fifth edition, Patrick J. sinko ,lippincott
Williams & Wilkins publication ,p.337-349.
7. Compaction properties, drug release kinetics and fronts movement studies from matrices
combining mixtures of swellable and inert polymers, International Journal of Pharmaceutics,
September 2007, 61–73.
8. Desai S.J, Sing P, Simonelli A.P, Higuchi W.I, Investigation of Factors Influencing Releaese of
Solid Drug Dispersed In Inert Matrices, III, Quantitative Studies involving the Polyethylene
Plastics Matrix, Journal of Pharmaceutical Sciences, 1966a, 55, 1230-1234.
9. Higuchi W.I, Analysis of Data on the Medicament Release from Ointments, J. Pharm. Sci,
1962, 51, 802 – 804.
10. Judit Dredin*, Istvin Antal, Istvin R/lcz .Evaluation of mathematical models describing drug
release from lipophilic matrices ,International,Journal of Pharmaceutics.

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