mathematical models for drug release studies

1. Assignment on
Mathematical models used in
drug release studies
By
Sajan Maharjan
M. Pharm (1st Sem.)
Submitted to:
Dr. Panna Thapa
Department of Pharmacy
Kathmandu University
Dhulikhel, Nepal

2. Introduction
Pharmaceutical Drug Delivery Systems (DDS) are the systems which deliver a drug into the
body. A DDS consists of one or more bio-active agents, and one or more excipients, which forms
the vehicle or medium for the administration of the active agents. The ultimate aim of a DDS is
to undergo dissolution in a biological medium (eg.: gastrointestinal tract for tablets, biological
tissues for implants etc.). In order to ensure a desired effect, there must be an appropriate rate of
drug release and then proper absorption of active agents.[2]
Different modes of drug release from a drug delivery system may be explained as follows [1] :
In many drug delivery applications, the release of the excipient (involving diffusion, dissolution,
erosion or other mechanisms) determines the release of the bio-active agents. Some modern
formulations are very complex, and the relationship between the formulation and process
variables and the release profiles are not entirely understood [2].
A large number of drug delivery system has been extensively developed in last few decades.
Thus, different mathematical models may be an instrumental tool for understanding the drug
release from different drug delivery systems. Mathematical modeling and numerical simulations
of drug transport inside the human vessels and inside the arterial wall can help to better
understand the efficacy of the treatment and guide the design of a better model. However,
modeling the local drug release remains a difficult and challenging task because of the
complexity and heterogeneity of the physical phenomena involved, both in the arterial wall and
in the delivery system. Generally, realistic numerical simulations cannot be performed without
taking into consideration phenomena that take place on different scales and are characterized by

3. different kinetics [3]. The development of mathematical models, which can predict delivery
performance, will facilitate the design of various delivery systems.
Mathematical models are generally designed to obtain following objectives:
 To design the new drug delivery systems based on general release expressions.
 To predict the effect of design parameters like shape, size and composition on the
resulting drug release rate.
 To optimize the release kinetics.
 To elucidate the underlying mass transport mechanisms.
 To accurately predict the drug release profile and improve the overall therapeutic efficacy
and safety of these drugs.
Mathematical Models for drug release studies
Mathematical models for the drug release studies plays a significant role as it establishes a
mechanism of drug release and provides more general guidelines for the development of other
systems. It is accepted that numerous successful drug delivery systems have been developed as a
result of an almost arbitrary selection of components, configurations and geometrics. Hence, we
survey some of mathematical models as given below.
A large number of mathematical models were developed to describe the release rate of drugs
from different drug delivery systems. Some of the important models are:
1. Diffusion Model
2. Zero order Kinetics Model
3. First order Kinetics Model
4. Higuchi Model
5. Korsmeyer–Peppas Model
6. Hixson–Crowell Model
7. Weibull Model
8. Baker–Lonsdale Model
9. Hopfenberg Model
10. Gompertz model
11. Sequential Layer Model
1. Diffusion Model
Diffusion refers to the process by which molecules intermingle as a result of their kinetic
energy of random motion.

4. It can be well understood by placing a drop of dye in a beaker of water. The dye molecules
tend to diffuse throughout the water and given uniform color. At equilibrium, when a uniform
color exists throughout the beaker of water, the dye molecules are uniformly distributed and
no further net movement is observed. The same case is seen in the diffusion of drug molecule
in our body which can be well explained by Fick’s laws of diffusion.
i. Fick’s first law of diffusion: Fick's first law relates the diffusive flux to the concentration under
the assumption of steady state. It postulates that the flux goes from regions of high
concentration to regions of low concentration, with a magnitude that is proportional to the
concentration gradient. In one dimension, the law is
J = - D (dC / dx)
Where,
J is the amount of substance passing perpendicularly through a unit surface area per time,
D is the diffusion coefficient and
dc / dx is the concentration gradient
The negative sign in the above equation signifies that diffusion occurs in the direction
opposite to the increasing concentration.
 This equation is a first order process because the rate depends on the concentration of one
chemical species. This law describes the transfer of molecules from an area of higher
concentration to an area of lower concentration.
ii. Fick’s second law of diffusion: It is derived from Fick’s first law for unsteady state situations,
where concentration is changing with time. Fick’s second law states that the rate of change in
concentration in a volume within the diffusional field is proportional to the rate of change in
spatial concentration gradient at that point in the field; the proportionality constant is the
diffusion coefficient and represented as:
=
Thus it states that the change in concentration with time in a particular region is proportional
to the change in concentration gradient at that point.
Applications of diffusion models


Fick’s first and second laws are applied to find flux and concentration across membranes.
Dissolution of a solid in a liquid is well explained by the modified Fick’s law, i.e NoyesWhitney equation:

5. =
Where,
dM/dt is the rate of dissolution
D= Diffusion coefficient of the solute in solution
S= Surface area of the exposed solid.
h= Thick ness of the diffusion layer.
Cs= concentration in a saturated liquid layer
Cb = Concentration of solute in the bulk solution at time t
Cs – Ct = Concentration gradient
2. Zero order kinetics model
Drug dissolution from dosage forms that do not disaggregate and release the drug slowly can
be represented by the equation:
Q0 - Qt = K0t
Rearrangement of above equation,
Qt = Q0 + K0t
Where,
Qt is the amount of drug dissolved in time t,
Q0 is the initial amount of drug in the solution (most times, Q0 = 0) and
K0 is the zero order release constant expressed in units of concentration/time.
To study the release kinetics, data obtained from in vitro drug release studies, are plotted as
cumulative amount of drug released versus time.
Application:
This relationship can be used to describe the drug dissolution of several types of modified
release pharmaceutical dosage forms, as in the case of some transdermal systems, as well as
matrix tablets with low soluble drugs in coated forms, osmotic systems, etc. [4]

6. Such model is particularly important in certain classes of medicines intended, for example,
for antibiotic delivery, heart and blood pressure maintenance, pain control and
antidepressants.
3. First order kinetics model
This model is used to describe absorption and elimination of some drugs, although it is
difficult to conceptualize this mechanism on a theoretical basis. The release of the drug
which follows first order kinetics can be expressed by the equation:
= -KC
Where, K is first order rate constant expressed in units of time-1
Above equation can be expressed as:
log C = log C0 - Kt / 2.303 (6)
where,
C0 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.
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.
4.
Higuchi model
Higuchi proposed this model in 1961 to describe the drug release from matrix system.
Higuchi 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 (edge effect must be negligible)
(iii) drug particles are much smaller than system thickness
(iv) matrix swelling and dissolution are negligible
(v) drug diffusivity is constant and

7. (vi) perfect sink conditions are always attained in the release environment.
Higuchi model is given by the equation:
ft = Q = A√
where,
Q is the amount of drug released in time t per unit area A,
C is the drug initial concentration,
Cs is the drug solubility in the matrix media and
D is the diffusivity of the drug molecules (diffusion coefficient) in the matrix substance.
This relation is valid during all the time, except when the total depletion of the drug in the
therapeutic system is achieved. 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:
ft = Q = √
where,
D is the diffusion coefficient of the drug molecule in the solvent,
δ is the porosity of the matrix,
τ is the tortuisity of the matrix and
Q, A, Cs and t have the meaning assigned above.
Tortuisity is defined as the dimensions of radius and branching of the pores and canals in
the matrix.
After simplification, Higuchi model can be written as:
ft = Q = KH . t1/2
where,
KH is the Higuchi dissolution constant
The data obtained were plotted as cumulative percentage drug release versus square root
of time.

8. Application:
This model can be used to describe the drug dissolution from several modified release
dosage forms like, some transdermal systems and matrix tablets with water soluble drugs.
5. Korsmeyer–Peppas Model (The power law)
A simple relationship which described drug release from a polymeric system equation was
derived by Korsmeyer et al. in 1983. To find out the mechanism of drug release, first 60%
drug release data were fitted in
Korsmeyer-Peppas model:
Mt / M∞ = Ktn
Where,
Mt / M∞ is a fraction of drug released at time t,
k is the release rate constant and
n is the release exponent.
The n value is used to characterize different release for cylindrical shaped matrices and the
value of n characterizes the release mechanism of drug as described in Table below:
Release exponent (n)
Drug transport mechanism
Rate as a function of time
0.5
Fickian diffusion
t-0.5
0.5<n<1
Non Fickian diffusion
t n-1
1
Case II transport
Zero order release
Higher than 1
Super Case II transport
tn-1
The following assumptions were made in this model:
i. The generic equation is applicable for small values of t or short times and the
portion of release curve where Mt/M ∞ < 0.6 should only be used to determine
the exponent n.
ii. Drug release occurs in a one dimensional way.
iii. The system’s length to thickness ratio should be at least 10.
To study the release kinetics, data obtained from in vitro drug release studies were plotted as log
cumulative percentage drug release versus log time.

9. Application:
This model has been used frequently to describe the drug release from several modified release
dosage forms.
6. Hixson–Crowell Model
According to Hixson-Crowell, regular area of particles is proportional to the cube root of its
volume. i.e,
Q0 1/3 – Qt 1/3 = Ks t
Where
Q0 = Initial amount of drug
Qt = Cumulative amount of drug release at time “t”
Ks = Hixson crowell release constant
t = Time in hours.
It describes the drug releases by dissolution, with the changes in surface area and diameter of the
particles or tablets.
A linear plot of the cube root of the initial concentration minus the cube root of percent
remaining versus time in hours for the dissolution data in accordance with the Hixson-crowell
equation is plotted to study the release kinetics.
The plotted graph will be linear if the following conditions are fulfilled:• The equilibrium conditions are not reached and
• The geometrical shape of the pharmaceutical dosage form diminishes proportionally over time.
Application:
This expression applies to pharmaceutical dosage form such as tablets, where the dissolution
occurs in planes that are parallel to the drug surface if the tablet dimensions diminish
proportionally, in such a manner that the initial geometrical form keeps constant all the time.
7. Weibull Model
Weibull model is generally applied to drug dissolution or release from pharmaceutical dosage
forms. The accumulated fraction of the drug M in solution at time t is given by Weibull equation:

10. M = M0 [1-
]
Where,
M is the amount of drug dissolved as a function of time t
M0 is total amount of drug being released
T accounts for the lag time measured as a result of the dissolution process
a denotes a scale parameter that describes the time dependence
b describes the shape of the dissolution curve progression.
This equation may be rearranged into:
Log [-In (1-m)] = b log (t – Tt) – log a
From this equation a linear relation can be obtained for a log-log plot of
– In(1 – m) versus time t.
Application:
The Weibull model is more useful for comparing the release profiles of matrix type drug
delivery.
8. Baker–Lonsdale Model
This model is the modification of the Higuchi model and is extensively used to the linearization
of the release data from several formulations. It is expressed as:
Where,
Mt is the drug release amount at time t,
M∞ is the amount of drug released at an infinite time and
K is the release constant which corresponds to the slope of the graph when plotted as [d(Mt /
M∞)] / dt with respect to the root of time inverse.
Application:
This equation has been used to the linearization of release data from several formulations of
microcapsules or microspheres.
9. Hopfenberg Model
Hopfenberg developed a mathematical model to correlate the drug release from surface eroding
polymers so long as the surface area remains constant during the degradation process. The
cumulative fraction of drug released at time t was described as:

11. where,
Mt is the amount of drug dissolved in time t,
M∞ is the total amount of drug dissolved when the pharmaceutical dosage form is exhausted,
ko is the erosion rate constant,
Co is the initial concentration of drug in the matrix and
a is the initial radius for a sphere or cylinder or the half thickness for a slab
The value of n is 1, 2 and 3 for a slab, cylinder and sphere respectively.
Application:
This model is used to identify the mechanism of release from the optimized oil spheres using
data derived from the composite profile, which essentially displayed site-specific biphasic
release kinetics.
10. Gompertz model
Gompertz model is a simple model which describes in-vitro dissolution profile by a simpler
exponential equation:
X(t) = Xmax e [-α eβ log t]
Where,
X(t) is percent dissolved at time t divided by 100;
Xmax is maximum dissolution;
α determines the undissolved proportion at time t = 1 and described as location or scale
parameter and
β is dissolution rate per unit of time described as shape parameter.
This model has a steep increase in the beginning and converges slowly to the asymptotic
maximal dissolution.
Application:
The Gompertz model is more useful for comparing the release profiles of drugs having good
solubility and intermediate release rate.
11. Sequential Layer Model
It is a more rigorous method for predicting molecule release from swelling-controlled system.
In sequential layer model, the tablet system is assumed to have a certain amount of single
layers penetrated by water and the model is performed in a computational grid and the
modified structure of the grid is required for numerical analysis. An advantage of using
computational grid is that it allows the modeling of inhomogeneous swelling. Here, swelling
is considered to take place layer by layer, in which the outermost layer swells first followed
by the neighboring inner layers. The model is able to capture the major feature of swellingcontrolled system, which is the substantial change in volume of the system in the outer layer.
This improvement is important since the substantial change in the system volume will change
the concentrations of all species as well as influence the mobility of the species (increasing
diffusion coefficients of water and drug). Here, only water and drug are considered in the

12. transport model, whereas the polymer undergoes a reduction in molecular weight
characterized by the constant dissolution rate constant, kdiss. This model has obvious
improvements in accounting for the volume change in the system that affects concentration of
species in the system as well as in the increasing mobility of species due to water penetration
[5].
A dissolution rate constant, kdiss was considered characterizing the polymer mass loss velocity
normalized to the actual surface of the system:
Mpt =Mpo -kdiss Att
Where,
Mpt and Mpo are the dry polymer matrix mass at time t, and t = 0, respectively;
At denotes the surface area of the device at time t
Water and drug diffusion were described using Fick’s second law of
diffusion equation:
= Dk (
Here, Ck and Dk are the concentration and diffusion coefficient of the diffusing species (k = 1
for water, k = 2 for the drug), respectively;
t represents time.
Application:
This model is useful in predicting the shape and dimensions of HPMC tablets needed to
acquire desired profiles.
Selection of the model for release study
Based on the statistical treatments, there are various criteria for the selection of the mathematical
models. The most widely used method employs the coefficient of determination, R2, to assess the
fit of the model equation. This method can be used when the parameters of the model equations
are similar. But when the parameters of the comparing equations increased; a modification is
incorporated in this technique where an adjusted coefficient of determination (R2 adjusted) given
by:
R2adjusted = 1-
R2 )
Where
n is the number of dissolution data points and
p is the number of parameters in the model.
Hence, the best model is the one with the highest adjusted coefficient of determination.
Similarly other statistical tools like correlation coefficient (R), Analysis of Variance (ANOVA)
and Multivariate analysis of variance (MANOVA) are used for the comparison and selection of
the models.

13. Conclusion
The main aim of formulation of any drug delivery system is to ensure safety and efficacy to
patients, for which study of drug release from the dosage form is of utmost importance.
Mathematic models are derived from the theoretical analysis of the occurring process. In most of
the cases the theoretical concept does not exist and some empirical equations have proved to be
more appropriate. Thus, the mathematical model is an important cursor for prediction and
elucidation of the exact behavior of drug or drug release profile from a specific drug delivery
system.
In general, the release models with major application and fitting drug release
phenomena are the Higuchi model, zero order model, Weibull model and Power law. The
Higuchi and zero order model represents two limit cases in the transport and drug release
phenomena, and Korsmeyer-Peppas model can be decision parameter between these two models.
The Higuchi model has a large application in polymeric matrix systems. The zero order model
becomes ideal to describe coated dosage forms or membrane controlled dosage forms [6].
It is difficult to identify and develop an appropriate mathematical theory
for a specific drug delivery system as
there is no general model covering all the possible
chemical and physical processes that can occur. Hence a single system can be explained by
various models with their comparative study and correlating outcomes of different models.
References
1) Mathematical modeling as a tool for improving action of drugs, journal of basic and
applied science, Vol. 4, No. 2, 81-88, 2008.
2) Probabilistic model for drug dissolution Part-1, Review of Monte carlo and Stochastic
cellular Automata Approaches.
3) Mathematical and numerical modeling of controlled drug release by Sara Minisini.
4) Kinetic modeling on drug release from controlled drug delivery systems by Suvakanta et.
al.
5) Mathematical modeling and simulation of drug release from microspheres: implications
to drug delivery systems: advanced drug delivery reviews: 58(2006)
6) Diffusional release of a dispersed solute from planar and spherical matrices into finite
external volume, journal of controlled release 43 (1997).