International Journal of Pharmaceutics 418 (2011) 54–77
Contents lists available at ScienceDirect
International Journal of
Pharmaceutics
j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /i j p h a r
m
Review
Mechanistic modelling of drug relea from polymer-coated and swelling and dissolving polymer matrix systems
Erik Kaunisto a ,Mariagrazia Marucci b ,Per Borgquist a ,Anders Axelsson a ,∗
a Department of Chemical Engineering,LTH,Lund University,P.O.Box 124,SE-22100Lund,Sweden b
AstraZeneca R&D Mölndal,SE-43183Mölndal,Sweden
a r t i c l e
i n f o
Article history:
Received 24November 2010
Received in revid form 5January 2011Accepted 12January 2011
Available online 21 January 2011
Keywords:
Controlled relea
Mathematical modelling Swelling
Osmotic pumping Coated pellets Matrix systems
a b s t r a c t
The time required for the design of a new delivery device can be nsibly reduced if the relea mechanism is understood and an appropriate mathematical model is ud to characterize the system.Once all the model parameters are obtained,in silico experiments can be performed,to provide estimates of the relea from devices with different geometries and compositions.In this revi
ew coated and matrix systems are considered.For coated formulations,models describing the diffusional drug relea,the osmotic pumping drug relea,and the lag pha of pellets undergoing cracking in the coating due to the build-up of a hydrostatic pressure are reviewed.For matrix systems,models describing pure polymer dissolution,diffusion in the polymer and drug relea from swelling and eroding polymer matrix formulations are reviewed.Importantly,the experiments ud to characterize the process occurring during the relea and to validate the models are prented and discusd.
© 2011 Elvier B.V. All rights rerved.
Contents 1.Introduction ..........................................................................................................................................562.
Coated formulations .................................................................................................................................572.1.Experimental characterization of the relea mechanism ...................................................................................572.2.Modelling of drug relea from polymer coated systems ...................................................................................
572.2.1.Empirical models ...................................................................................................................572.2.2.Diffusion models ....................................................................................................................582.2.3.Single unit ..........................................................................................................................582.2.4.Multiple unit systems ..............................................................................................................
592.3.Relea by osmotic pumping and diffusion ..................................................................................................612.4.Lag time for pellets developing cracks in the coating due to hydrostatic pressure-build-up ...............................................643.
Modelling of polymer dissolution and swelling and dissolving matrix formulations ..............................................................653.1.Dissolution of amorphous glassy polymers ..................................................................................................653.2.Dissolution of mi-crystalline polymers ....................................................................................................683.3.Diffusion models in polymers ................................................................................................................
703.3.1.Models bad on obstruction effects ...............................................................................................703.3.2.Hydrodynamic models ................
.............................................................................................703.3.3.Models bad on free
713.4.Drug relea from swelling and dissolving polymer matrix formulations ..................................................................713.5.Model parameters and discriminating experiments .........................................................................................754.
Conclusions and suggested future work .............................................................................................................75References ...........................................................................................................................................
75
∗Corresponding author.Tel.:+46462223423.
E-mail address:Anders.Axelsson@chemeng.lth. (A.Axelsson).
0378-5173/$–e front matter © 2011 Elvier B.V. All rights rerved.doi:10.1016/j.ijpharm.2011.01.021
E.Kaunisto et al./International Journal of Pharmaceutics418 (2011) 54–7755
Nomenclature
Symbols
c concentration(for type e subscript)(kg/m3)
t time(s)
x spatial coordinate×direction(m)
D diffusion coefficient(for type e subscript and
superscript)(m2/s)
f factor
F diffusion coefficient function
¯r dimensionless dissolution velocity
¯t dimensionless time
E Young’s modulus(Pa)
R gas constant(J/mol/K)
T temperature(K)
ˆV molar volume(m3/mol)
L half-thickness of the polymer(or e subscript)(m) U function
K function
G memory function
k coefficient/constant
H Heaviside function
r spatial coordinate radial direction(or radius)(m) z spatial coordinate axial direction(m)
jflux(kg/m2/s)
c numerically stable interval(kg/m3)
t numerically stable interval(s)
n normal vector
M molar mass of diffusant(kg/mol)
V volume(m3)
A area(m2)
m cumulative absolute amount relead(kg)
Z axial element length(m)
N volumetricflux between elements(m/s)
s axial expansion parameter
v velocity(m/s)
Y empirical constant
Re Reynolds number
Sc Schmidt number
J volumetric osmotic pumpingflow
Greek letters
tortuosity
stress(Pa)
Ánormalized spatial coordinate
viscosity/viscosity of the polymer solvent mixture (Pa s)
osmotic pressure(Pa)
numerical constant
ıe subscript for definition
density(kg/m3)
ˇconcentration dependence parameter
˝computational domain(m2)
Flory’s interaction parameter
(∂˝)bnum boundary(m)
solvent shape factor(ranging from1.5for rods to2 for spheres)
␣parameter
Poisson’s coefficient(or e subscript)
Flory’s exponent for excluded volume
dissolution term(m/s)
empirical constant
strain Subscript
1solvent/diffusant(lf-diffusion)
2polymer(lf-diffusion)
12mutual(solvent/polymer)
v volume fraction(v/v)
sol solvent(water)diffusion in the polymer
drug drug
pol polymer diffusion in liquid solution
gs gel–solvent interface(critical value/equilibrium value)
sg solid–gel interface(threshold value)
front front
init initial
coop cooperative diffusion coefficient
rep reptation
m mass transfer coefficient(m/s)
core core
crys crystal-unfolding rate(1/s)
a amorphous
c crystalline
dis dintanglement rate(m/s)
w mass fraction(w/w)
r radial direction
z axial direction
gel gel-layer
lag lag time
ALE ALE frame(spatial frame)
b bound to polymer
pure in pure solvent
h hydrodynamic radius of diffusing molecule
e effective cylindrical radius o
f afibre or diffusion
coefficient
n molar fraction(n/n)
o obstacle
V average free volume per molecule
VSOL solvent free volume in the polymer solution
sat solubility(saturation)
craz critical for crazing
x,xx x direction
g gel-layer thickness(m)
bnum boundary number(bnum=1,2,3,4)
screp screening hydrodynamic interaction parameter padp particle dependent parameter
scap scaling parameter
prop proportionality factor
tot total before dissolution
dir dirac delta function
surf surface area of device
p axial discretization index(finite element)/solvent permeability
q radial discretization index(finite element)
k index
an annular element thickness(m)
ON overall normalized drug concentration
N normalized drug concentration in the solvent pha fick Fickian contribution
caII ca II contribution
kin kinetic exponent
∞infinity(time)
t time
0initial(time)
ex external
in internal
56 E.Kaunisto et al./International Journal of Pharmaceutics418 (2011) 54–77
ffilm
part partition coefficient
coat coating thickness,(m)
reflreflection coefficient
rel relea medium
mixflow from relea medium into tank
outflow through pore
pore pore
poi Poisson’s ratio
diss dissolved pha
bf bulkflowing throughfilm
bpore bulkflowing through pore
d dissolution rate
f0interface between coatingfilm and internal solution fh interface between coatingfilm and external solution Tank tank
i uncoated pellet
Superscript
v volume-bad
es envelope surface(radial direction)
cs cross-ction(axial direction)
out external
in internal
B bulk
w solvent(water)
A drug component
D diffusiveflux
n empirical constant
1.Introduction
Drug relea models can be classified as empirical models or mechanistic models.An empirical approach is bad on the exper-imental behaviour of the system studied.No physical mechanisms are considered in the description of the problem.The kinds of models can often mimic the behaviour of the actual system very well,especially if an appropriate number of parameters are included in the model.However,such a model does not provide any information on the mechanisms that control the process.Con-quently,an empirical model cannot be ud to predict what effect a change in the a change infilm thickness of a rer-voir system,will have on the relea rate.Thus,the models rve the same purpo as any mathematical polynomial with sufficient properties tofit the experimental data.The u of empirical models for simulating drug relea profiles is therefore restricted to simple curve-fitting procedures.
Mechanistic drug relea models are bad on the physical mechanisms that influence the relea process.Thus,the model parameters have physical significance,and it is therefore possible to u a mechanistic model to make predictive simulations.How-ever,it is still necessary to confirm the validity of the model against experimental data.In this procedure it is important to validate not only the output of the a relea profile,but the values of all the parameters included in the model.This vali
dation is possi-ble becau the parameters have physical significance,in contrast to the empirical parameters.Another important issue is to restrict the model to an appropriate level of complexity.A general rule is to identify the rate-limiting process of the system.This is espe-cially important if the parameter values of the process are unknown and are to be determined from parameterfitting procedures.The more detailed the model description,the more detailed the exper-imental verification procedure must be.The prentation of the different models should be en in the light of this leading idea of modelling.
Since1961,when Higuchi(1961)prented his well-known equation for describing drug relea from solid drugs suspended in ointment bas,there have been numerous contributions to empirically and mechanistically model the drug relea process. Several review papers have been published to summarize the drug relea models for coated formulations(Siepmann and Siepmann, 2008;Grassi et al.,2007)and for matrices(Siepmann and Siepmann, 2008;Siepmann and Peppas,2001;Narasimhan,2001).However, for coated formulations,the discussion of the mathematical mod-els ud to describe the process that occur when the coating is a mi-permeable membrane is not prented.This implies,for example,that the models ud to describe the osmotic pumping relea from coated formulations or the lag pha for pellets who coating cra
cks due to the hydrostatic pressure build-up have not been summarized.Similarly,in the ca of matrix systems,the focus has been mostly on models in general and very often only focusing on models including drug relea.However,the funda-mental basis for understanding drug relea from polymer matrix systems is at least understanding drug properties and polymer properties.This means that the polymer matrix system should be studied both with and without drug in order to under-stand the drug relea process.Understanding polymer dissolution is therefore vital to be able to design polymer matrix formula-tions.In earlier reviews of polymer dissolution models and their conquences for drug delivery(Narasimhan,2001;Miller-Chou and Koenig,2003)important conclusions from the authors were that an extended knowledge of polymer dissolution is necessary to understand the full applicability of polymers,and that the gap between theoreticians and experimentalists need to be bridged. Several models for polymer dissolution have included parame-ters that cannot be determined from experiments and numerous experimental results cannot be explained by current theories,thus raising questions regarding their quality.
In this review paper,models for describing the drug relea from coated formulations and from swelling and dissolving matrix formulations are prented.Although empirical and mechanis-tic models are summarized,a special attention was paid to the mechanistic models.The novelty of the p
art devoted to the coated formulations consists in the description of the models ud to describe the lag pha for pellets developing cracks in the coat-ing due to the osmotic pressure build-up,and the drug relea by osmotic pumping.A special discussion was prented for the models applied to describe the relea from multiple-unit coated systems.Regarding the part devoted to polymer dissolu-tion and polymer matrix formulations,the novelty can be found in the comprehensive prentation of models within the two rearchfields,since a mechanistic understanding of polymer dis-solution plays an integral role in the understanding of polymer matrix formulations.In addition,the importance of experimental data that can help to discriminate between model parameters is discusd.
Moreover,a ction was devoted to the experimental charac-terization of the relea mechanism from coated formulations.In the ca of polymer dissolution and swelling and dissolving matrix formulations,experimental data for model verification as well as important mechanisticfindings are prented.
Due to the huge amount of details and similarities between some models,only relevant equations will be shown/discusd in the text.For specific model details the reader is referred to the original text.Further,the nomenclature in the prent work will remain consistent,implying that deviations from
the original ref-erence text may occur.Unless stated in the text,the meaning of every symbol occurring can be deduced from the“Nomenclature”ction.
E.Kaunisto et al./International Journal of Pharmaceutics418 (2011) 54–7757
2.Coated formulations
2.1.Experimental characterization of the relea mechanism
It is fundamental to understand the underlying mechanism of diffusion and/or osmotic pumping)in order to choo the right relea model.Drug relea occurs by diffusion when the coating is permeable towards the drug investigated. Drug relea occurs mainly by osmotic pumping when thefilm is mi-permeable towards the drug investigated.Unfortunately, models that describe the relea by diffusion are often ud to fit experimentally determined relea data withoutfirst ensuring experimentally that the coatingfilm is actually permeable to the drug and that the relea occurs by diffusion.This can of cour lead to rious errors in the calculations.In order to obtain a com-plete understanding of the relea process,each parate pha of the relea must be he lag pha,the zero-order relea pha and the decaying pha.A full understanding of the relea mechanisms during the whole relea process may be com-plicated an
d require veral different kinds of experiments.In this ction a short summary of the experiments that has been ud to characterize the mechanism of relea is prented.A more detailed description has been prented by Marucci(2009).
Specially designed do relea experiments can be performed to understand the relea process after the lag pha.Diffusional relea and osmotic pumping relea from coated formulations are often differentiated in an easy and convenient way by perform-ing do relea experiments at different osmotic pressures of the dissolution medium(Zentner et al.,1985;Lindstedt et al.,1989; Ozturk et al.,1990;Verma and Garg,2004;Marucci et al.,2010). The experiments have been performed also to calculate thefluid permeability of the coating and the diffusion coefficient of the drug in the coating(Zentner et al.,1985;Liu et al.,2007;Marucci et al., 2010).Do relea experiments have been performed at differ-ent temperatures to elucidate the state of the coating rubbery or glassy)and to explain the change in relea mechanism from transport through water-filled pores to transport through the polymerfilm when the glass transition temperature of thefilm is higher than the temperature at which relea takes place(Frohoff-Hulsmann et al.,1999a).
When the drug delivery device is formulated as a multiple-unit system,the properties of the populatio
n of he varia-tion of the relea-controlling properties,becomes important and will control the overall relea characteristics of the device.In some cas erroneous conclusions regarding the relea mechanism from multi-particulate formulations can be drawn if only do exper-iments are performed(Dappert and Thies,1978;Hoffman et al., 1986).Hence,in order to properly describe the relea characteris-tics of a multiple-unit system it is appropriate to take into account subunit-to-subunit variations.Single-pellet relea has been stud-ied in small vesls and cells(Benita et al.,1988;Jorgenn et al., 1997;Lippold et al.,1999),and inflow-through cells(USP appara-tus4)(Schultz and Kleinebudde,1997)as well as in an absorbance microplate reader(Folestad et al.,2000;Borgquist et al.,2002,2004; Marucci et al.,2009b,2010).
Swelling experiments have been ud to study the uptake of water,the mass accumulation inside the pellets and the related hydrostatic pressure build-up during the lag pha,as well as the zero-order relea pha and the decaying pha(Schultz and Kleinebudde,1997;Hjärtstam and Hjertberg,1998).The accumu-lation of mass indicates that the coating is mi-permeable to the drug being studied.However,it is not always easy to detect swelling due to the mechanical properties of the coating.Magnetic reso-nance imaging(Shapiro et al.,1995;Fyfe and Blazek-Welsh,2000), nuclear magnetic resonance(Ensslin et al.,2008)and electron para-magnetic resonance(Strubing et al.,2007)
have been ud to study the process of water uptake and drug dissolution,which can be ud to characterize thefirst step of the relea process.
Scanning electron microscopy(SEM)has been widely ud to characterize the surface of coated pellets(and in some cas also of freefilms)and the cross ction of the coating,and to crudely identify relea paths.Comparison of the coating before and after drug relea may help us to understand the effect of the solvent on the coating properties.A sponge-like structure can be obrved at the end of the relea process in the ca of coatings containing leachable substances(Zentner et al.,1985;Marucci et al.,2009b). For formulations coated with a mi-permeablefilm,the prence of small cracks in the coating at the end of relea testifies to the mechanical failure of the coating caud by the swelling of the sys-tem due to solvent accumulation(Schultz and Kleinebudde,1997; Nevsten et al.,2005).
Confocal lar scanning microscopy(CLSM)has been ud to measure thefilm thickness and uniformity of a coated pellet (Haddish-Berhane et al.,2006;Marucci et al.,2009b),and the migration of drug in the coating and explain the unexpected ini-tially high relea rate(Felton,2007).
The leaching of the water-soluble compound from freefilms and from coatingfilms has been studied t
o understand the change in the physico-chemical and transport properties of thefilm during the relea(Siepmann et al.,2007;Marucci et al.,2009a,b).
The knowledge of the coating transport he drug diffusion coefficient in the coating,the water permeability and the drug reflection coefficient,and mechanical properties is fundamen-tal for the description and prediction of the relea rate from coated formulation.The parameters can be measured easily in freefilms. Freefilms are often ud as a convenient model for coatingfilms to explore physico-chemical and mechanical properties and,impor-tantly,how the change from the dry to the wet state(Bodmeier and Paeratakul,1994;Frohoff-Hulsmann et al.,1999b;Siepmann et al.,2007;Marucci et al.,2009a).Electronic speckle pattern inter-ferometry was ud to characterize the nature of freefilms and made it possible to discriminate between a permeable and a mi-permeablefilm(Marucci et al.,2006).Special relea cells that mimic a coated formulation have been developed(Marucci et al., 2009a,2006;Okimoto et al.,1999)and freefilms have also been ud to better understand the mechanism of relea from coated formulations(Marucci et al.,2006,2009a)and how the change during the relea(Marucci et al.,2009a).Marucci et al.(2009a) developed a relea cell equipped with a manometer to measure the pressure build-up inside the cell.The combination of pressure and relea data made it possible to easily and accuratel
y charac-terize the relea mechanism from a formulation coated with an ethyl cellulo badfilm,and how the mechanism changes during the relea due to leaching of the water-soluble compound prent in the polymerfilm(Marucci et al.,2009a).
2.2.Modelling of drug relea from polymer coated systems
As already described in Section1,the models can be classi-fied into empirical and mechanistic.In this ction a review of the empirical models is prented.Among the mechanistic model,a review of the following models is prented:models that describe drug relea by diffusion,models that describe drug relea by osmotic pumping and diffusion,models that describe the lag pha for formulations coated with afilm undergoing cracking due to the hydrostatic pressure build-up inside the pellet.
2.2.1.Empirical models
Much work has historically been devoted tofitting empirical relations to drug relea data from rervoir as well as matrix sys-tems(Ritger and Peppas,1987;Jorgenn,1996;Narasimhan et al., 1999;Korsmeyer et al.,1983).The power law equation prented
58 E.Kaunisto et al./International Journal of Pharmaceutics418 (2011) 54–77 by Korsmeyer et al.and Ritger and Peppas was originally developed
to describe the relea from matrix systems,but it has also been
applied to curvefitting of rervoir systems(Ritger and Peppas,
1987;Narasimhan et al.,1999):
m t
m∞
=Y·t n,(1)
where m t is the amount relead and m∞is the amount of drug
relead over an infinite time.Y and n are empirical constants and
t is time.
Jorgenn and co-workers developed the“order model”,an
empirical expression for the relea from single,film-coated pellets
and enmbles offilm-coated pellets(Jorgenn and Christenn,
1996;Jorgenn,1996;Jorgenn et al.,1997):
m t
m∞
=1−[1−Y·(1−n)(t−U(t init))]1/(1−n),(2)
where the function,U,allows for exponential behaviour during the
lag pha:
U(t init)=t init
1−exp
−t
abs(t init)
(3)
The kinds of models,Eqs.(1)and(2),are very successful in fitting drug relea profiles–perhaps an obvious statement as the empirical constants n and Y are not bad on any physical mech-anism.It can be questioned whether the information gained from simple curvefitting increas our knowledge of the relea pro-cess.The values of the empirical constants can,of cour,help us e how relea occurs(zero-order,etc.),but gives us very limited information about why it occurs.
2.2.2.Diffusion models
Diffusional mass transfer through the polymer membrane may take place in the(pure)polymer pha and/or in solvent-filled pores or cracks in the membrane(Langer and Peppas,1983;Good and Lee,1984;Siegel,1989;Narasimhan et al.,1999).Therefore,a truly mechanistic description of the transport through a polymerfilm must include information on numerous importantfilm-properties, e.g.the solute diffusion coefficients in the polymer pha,the solute diffusion coefficient in the water pha,partitioning,porosity,and pore size distribution,e Fig.1.
Simple models have been derived bad on the identification of the transport through the polymerfilm as the rate-limiting step in the relea process.The models,although simple,are bad
on a solid mechanistic approach as oppod to the empirical models. Assuming that the rate limitation of the drug relea is transport through thefilm,and that the system is at steady state,results in the following expression for a spherical device(Langer and Peppas, 1983;Good and Lee,1984;Narasimhan et al.,1999):
m t=4 ·r ex·r in·D·k part· c
r ex−r in
·t
(4)
Fig.1.Schematic picture of drug relea from rervoir systems(in this ca afilm-coated pellet).where r ex is the external and r in is the internal radius of the rer-voir,D is the diffusion coefficient of the drug in the polymer,k part is the partition coefficient and c is the concentration difference over the membrane.The u of Eq.(4)is valid when the coating is homogeneous.For heterogeneous coatings,the overall relea is affected by the diffusion coefficient and partition coefficient of each pha prent in the coating,including obstruction and exclu-sion effects.Instead of determining the parameters for each pha and the geometry of the different phas,it is convenient to u an effective diffusion coefficient,D e,which is a lumped parameter that characterizes the transport properties across thefilm and the parti-tioning between the water pha and the coating.The driving force for the relea can still be written as the difference in the drug concentration across the coating.Eq.(4)only applies in the ca of a constant concentration as long as a saturated solution exists inside the device and if perfect sink conditions are fulfilled.Furthermore,the dissolution rate of the solid drug(if the device is loaded above the drug solubility)is assumed to be rapid, and the external mass transfer resistances are neglected.Therefore, simulations using Eq.(4)are only applicable for the zero-order part of the releas
e profile.The lag time of rervoir systems cannot be described with this model if a constant D e is assumed,as pointed out by veral rearchers(Langer and Peppas,1983;Good and Lee, 1984;Narasimhan et al.,1999).The lag time was explained as the unsteady-state behaviour of the device during the initial pha of the relea process,when the concentration gradient is developing. This may be true for the lag time obrved for devices with thick polymerfilms.However,rervoir systems with very thin polymer films can also have substantial lag times.
2.2.
3.Single unit
The steady-statefilm approach has also been ud in modelling the relea of fertilizers from latex-coated urea balls(Lu and Lee, 1992;Lu and Chen,1993;Lu and Yu,1994;Lu,1994).The mod-els were evaluated against experimental relea data from single film-coated urea balls.Thefirst model prented could simulate the relea from an initially saturated solution in the core to a less than saturated solution during the declining pha(Lu and Lee,1992). Dissolution was assumed to be rapid and external mass trans-fer resistances were neglected.Non-perfect sink conditions were modelled.The model was extended in a further study to include diffusion in the core for the ca of u
nsaturated cores(Lu and Chen, 1993),and to include fast initial relea rates due to solid drug in thefilm(Lu and Yu,1994;Lu,1994).Since the polymerfilm is fairly thick(about200–400m on a15mm urea core)it is reasonable to assume that the lag pha in this ca is due tofilm dynamics as discusd above.The effect of the initial conditions in thefilm on the relea profilag pha or fast initial relea,was studied later(Lu and Chen,1995).
The drug relea from chitosan-coated tablets,where thefilm itlf dissolves during the relea process,has also been modelled using a steady-statefilm approach(Koizumi et al.,2001).Perfect sink conditions and fast dissolution of the solid drug were assumed, and the external mass transfer hindrances were neglected.The model wasfitted to experimental relea data.The excellence of thefit is not surprising since the number offitted model parameters were as many asfive,including the lag time.
The relea fromfilm-coated matrix a combina-tion of a rervoir and a matrix system,has been modelled by Lee and co-workers(Lee and Liao,1995;Liao and Lee,1997;Chen and Lee,2001,2002).The main focus of their studies was the effect of deformations in the coating on the relea rate.Unsteady-state dif-fusion in the matrix core and in the coating was included in the model.The concentration in the matrix was assumed to be less than the saturation concentration,and
external mass transfer resis-tances were neglected.Furthermore,perfect sink conditions were
