Advanced Drug Delivery Reviews 48 (2001) 139–157
`
`www.elsevier.com/ locate/drugdeliv
`
`Modeling of drug release from delivery systems based on
`hydroxypropyl methylcellulose (HPMC)
`*
`J. Siepmann
`, N.A. Peppas
`
`a ,
`
`b
`
`a
`
`´
`College of Pharmacy,Universite d’Angers,16 Boulevard Daviers,49100 Angers, France
`bSchool of Chemical Engineering, Purdue University,West Lafayette, IN 47907,USA
`
`Received 20 October 2000; received in revised form 7 December 2000; accepted 15 December 2000
`
`Abstract
`
`The objective of this article is to review the spectrum of mathematical models that have been developed to describe drug
`release from hydroxypropyl methylcellulose (HPMC)-based pharmaceutical devices. The major advantages of these models
`are: (i) the elucidation of the underlying mass transport mechanisms; and (ii) the possibility to predict the effect of the
`device design parameters (e.g., shape, size and composition of HPMC-based matrix tablets) on the resulting drug release rate,
`thus facilitating the development of new pharmaceutical products. Simple empirical or semi-empirical models such as the
`classical Higuchi equation and the so-called power law, as well as more complex mechanistic theories that consider
`diffusion, swelling and dissolution processes simultaneously are presented, and their advantages and limitations are
`discussed. Various examples of practical applications to experimental drug release data are given. The choice of the
`appropriate mathematical model when developing new pharmaceutical products or elucidating drug release mechanisms
`strongly depends on the desired or required predictive ability and accuracy of the model. In many cases, the use of a simple
`empirical or semi-empirical model is fully sufficient. However, when reliable, detailed information are required, more
`complex, mechanistic theories must be applied. The present article is a comprehensive review of the current state of the art
`of mathematical modeling drug release from HPMC-based delivery systems and discusses the crucial points of the most
`important theories.
`2001 Elsevier Science B.V. All rights reserved.
`
`Keywords: Controlled drug delivery; HPMC; Hydrophilic matrices; Hydroxypropyl methylcellulose; Modeling; Release mechanism
`
`Contents
`
`1. Introduction ............................................................................................................................................................................
`2. Physicochemical characterization of HPMC ..............................................................................................................................
`3. Overall drug release mechanism from HPMC-based systems ......................................................................................................
`4. Empirical and semi-empirical mathematical models ...................................................................................................................
`4.1. Higuchi equation..............................................................................................................................................................
`4.2. Power law .......................................................................................................................................................................
`4.3. Other empirical and semi-empirical models........................................................................................................................
`
`140
`141
`143
`144
`144
`145
`147
`
`*Corresponding author. Tel.: 133-241-735-846; fax: 133-241-735-853.
`E-mail address: siepmann@zedat.fu-berlin.de (J. Siepmann).
`
`0169-409X/ 01/ $ – see front matter
`PII: S0169-409X( 01 ) 00112-0
`
`2001 Elsevier Science B.V. All rights reserved.
`
`(cid:211)
`(cid:211)
`

`
`140
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`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`5. Comprehensive mechanistic theories.........................................................................................................................................
`6. Conclusions ............................................................................................................................................................................
`References ..................................................................................................................................................................................
`
`147
`154
`154
`
`1. Introduction
`
`Hydroxypropyl methylcellulose (HPMC) is the
`most important hydrophilic carrier material used for
`the preparation of oral controlled drug delivery
`systems [1,2]. One of its most important characteris-
`tics is the high swellability, which has a significant
`effect on the release kinetics of an incorporated drug.
`Upon contact with water or biological fluid the latter
`diffuses into the device, resulting in polymer chain
`relaxation with volume expansion [3,4]. Then, the
`incorporated drug diffuses out of the system.
`For the design of new controlled drug delivery
`systems which are based on HPMC and aimed at
`providing particular, pre-determined release profiles,
`it is highly desirable: (i) to know the exact mass
`transport mechanisms involved in drug release; and
`(ii) to be able to predict quantitatively the resulting
`drug release kinetics. The practical benefit of an
`adequate mathematical model is the possibility to
`simulate the effect of
`the design parameters of
`HPMC-based drug delivery systems on the release
`profiles [5]. In the ideal case,
`the required com-
`position (type and amount of drug, polymer and
`additives) and geometry (size and shape) of the new
`controlled drug delivery system designed to achieve
`a certain drug release profile can be predicted
`theoretically. Thus, the number of necessary experi-
`ments can be minimized and the development of new
`pharmaceutical products significantly facilitated.
`Diffusion, swelling and erosion are the most
`important rate-controlling mechanisms of commer-
`cially available controlled release products [6]. Dif-
`fusion can be described using Fick’s second law
`[7–9]. There are various ways to apply the respective
`equations [10]. First of all, the considered geometry
`is important. Assuming one-dimensional transport in
`thin films results in rather simple mathematical
`expressions, but this approach is only valid for flat,
`planar devices. In the case of HPMC-based drug
`delivery
`systems
`three-dimensional,
`cylindrical
`geometries (tablets) are more relevant, but mathe-
`matically more difficult to treat. In addition, it is
`
`necessary to decide whether to assume constant or
`non-constant diffusivities. The mathematical
`treat-
`ment of constant diffusivity problems is much sim-
`pler, but only valid in the case of polymers that do
`not significantly swell upon contact with water (e.g.,
`ethylcellulose). For HPMC tablets, the drug diffusion
`coefficients are strongly dependent on the water
`content of the system [11]. Here, the assumption of
`constant diffusivities leads to less realistic mathe-
`matical models. Depending on the degree of substitu-
`tion and chain length of the HPMC type used,
`polymer dissolution might be observed during drug
`release. This will complicate the solution of Fick’s
`second law of diffusion, leading to moving boundary
`conditions. In addition to the physicochemical prop-
`erties of the polymer also the characteristics of the
`drug have to be considered. For example, drug
`dissolution has to be taken into account in case of
`poorly water-soluble drugs.
`Depending on the complexity of the resulting
`system of mathematical equations describing diffu-
`sion, swelling and /or dissolution processes, analyti-
`cal and/or numerical solutions can be derived.
`Analytical solutions have the major advantage of
`being more informative. The involved design and
`physicochemical parameters still appear in the equa-
`tions.
`In the case of explicit analytical solutions, we can
`obtain direct relationships between the dependent
`and independent variables. In the case of implicit
`analytical solutions, this dependence is not as obvi-
`ous. However, compared to numerical solutions, it is
`still much easier to get an idea of the effect of certain
`independent variables on particular dependent vari-
`ables. Thus, it is highly desirable to derive explicit
`analytical solutions. Unfortunately, this is only pos-
`sible in the case of rather simple forms of the
`diffusion equations, e.g., assuming constant dif-
`fusivities.
`In general, physically more realistic
`models are mathematically more complex and very
`often it is difficult to find analytical solutions of the
`respective set of equations [12]. Three important
`methods to derive exact mathematical solutions can
`
`

`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`141
`
`be distinguished: (i) the method of reflection and
`superposition; (ii) the method of separation of vari-
`ables; and (iii) the method of the Laplace transform.
`For a discussion of the advantages and disadvantages
`of these methods the reader is referred to other
`literature (e.g., [7,13,14]). Also a description of the
`principles of numerical analysis is beyond the scope
`of this review, but excellent textbooks are available
`(e.g., [7,13–15]). In contrast to analytical solutions,
`only approximate solutions are derived. The resulting
`error can be controlled using various different meth-
`ods. Generally, cumbersome mathematical calcula-
`tions are required to reduce the approximation error
`to an acceptable level (e.g. , 0.1%). The develop-
`ment of digital computers dramatically decreased the
`time necessary to perform the calculations, so that
`nowadays numerical methods have become econ-
`omic even for routine use.
`It is the scope of this article to review the most
`important mathematical models which have been
`developed to describe drug release from HPMC-
`based pharmaceutical systems. Simple and very
`comprehensive theories are presented and their ad-
`vantages and limitations are discussed. For a better
`understanding of the described theories, first
`the
`most relevant physicochemical properties of HPMC
`and the major principles of the overall drug release
`mechanisms from HPMC-based delivery systems are
`presented. Due to the substantially high number of
`variables, no effort was made in this review to
`present a uniform picture of the different systems of
`notation defined by the respective authors. The
`original nomenclatures are used and only some cases
`are modified by using more common abbreviations to
`avoid misunderstandings.
`
`2. Physicochemical characterization of HPMC
`
`HPMC is a propylene glycol ether of methyl-
`cellulose; its chemical structure is illustrated in Fig.
`1. The substituent R represents either a –CH , or a
`3
`–CH CH(CH )OH group, or a hydrogen atom. The
`2
`3
`physicochemical properties of
`this polymer are
`strongly affected by: (i) the methoxy group content;
`(ii) the hydroxypropoxy group content; and (iii) the
`molecular weight. The USP distinguishes four differ-
`ent
`types of HPMC, classified according to their
`
`Fig. 1. Chemical structure of HPMC. The substituent R represents
`either a –CH , or a –CH CH(CH )OH group, or a hydrogen
`3
`2
`3
`atom.
`
`relative –OCH and –OCH CH(CH )OH content:
`3
`2
`3
`HPMC 1828, HPMC 2208, HPMC 2906 and HPMC
`2910. The first two numbers indicate the percentage
`of methoxy-groups, the last two numbers the per-
`centage of hydroxypropoxy-groups, determined after
`drying at 1058C for 2 h. The exact limits for the
`degree of substitution defining the respective HPMC
`types are given in Table 1. In addition, the USP
`describes a method to determine the apparent vis-
`cosity of an aqueous 2% solution of the polymer
`using a suitable viscosimeter of the Ubbelohde type.
`This apparent viscosity serves as a measure for the
`average chain length of the polymer. The measured
`value must lie within the 80.0 to 120.0% range of the
`viscosity stated on the label for HPMC types of 100
`mPa s or less, and within the 75.0 to 140.0% range
`for HPMC types of higher viscosity.
`reported broad
`Interestingly, Dahl et al.
`[16]
`variations concerning important characteristics of
`seven batches HPMC 2208 with a labeled viscosity
`of 15,000 mPa s, provided by two different manufac-
`turers. All samples had similar viscosities, except
`one batch which was outside the USP specifications.
`The methoxy-group content was uniformly high and
`
`Table 1
`USP specifications for different types of HPMC, classified accord-
`ing to their degree of methoxy- and hydroxypropoxy-substitution
`
`Substitution
`type
`
`1828
`2208
`2906
`2910
`
`Methoxy (%)
`
`Hydroxypropoxy (%)
`
`Min.
`
`16.5
`19.0
`27.0
`28.0
`
`Max.
`
`20.0
`24.0
`30.0
`30.0
`
`Min.
`
`23.0
`4.0
`4.0
`7.0
`
`Max.
`
`32.0
`12.0
`7.5
`12.0
`
`

`
`142
`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`three batches fell outside the USP limits of 19.0 to
`24.0%. The hydroxypropoxy-group content (although
`within the USP specifications of 4.0 to 12.0%),
`varied relatively more than the methoxy group
`content. These variations lead to significant differ-
`ences concerning the resulting release rate of naprox-
`en from compressed matrix tablets in vitro. The
`authors concluded that each batch (even from the
`same manufacturer) should be carefully controlled
`and that
`the specifications of the USP and other
`pharmacopoeas might have to be reinforced.
`The glass transition temperature, T , of a polymer
`g
`is an important characteristic constant, in particular
`with respect to applications in the field of controlled
`drug delivery. Below the T the mobility of the
`g
`macromolecules is very low. The material is in its
`glassy state resulting in extremely small diffusion
`rates [10]. In contrast, above the glass transition
`temperature the mobility of the polymer chains is
`markedly increased (rubbery state), leading to much
`higher mass transfer rates of water and drug. Thus,
`we must know the T of the polymer when modeling
`g
`drug release from controlled delivery systems. A
`good summary of work that has been done to
`determine the T of HPMC has been provided by
`g
`Doelker [17]. He compares the results of various
`researchers [18–26] and lists values ranging from
`154 to 1848C (Table 2). Various techniques have
`been used to determine the glass transition tempera-
`ture: differential scanning calorimetry (DSC), dif-
`ferential thermal analysis (DTA), thermomechanical
`analysis (TMA), torsional braid analysis (TBA) and
`dynamic mechanical analysis
`(DMA). Different
`methods often lead to different T -values, and usual-
`g
`ly only the results achieved with one special method
`can be compared directly. In addition, the variation
`of
`the degree of substitution and the molecular
`weight plays a role in the observed variance of the
`T . Furthermore, a 578C-value was reported by
`g
`Conte et al. [26], which seems to correspond to a
`low-energy secondary transition. The relevance of
`this low-temperature transition is yet unknown, but
`could be of significance in the diffusion of oxygen
`and water.
`Numerous studies have been reported in the
`literature investigating the drug release kinetics from
`HPMC-based delivery systems
`[27–31]. Various
`techniques have been used to elucidate the physical
`
`Table 2
`Reported glass transition temperatures for HPMC (adapted from
`Doelker [17], with permission from Springer–Verlag)
`
`Material
`
`Method
`
`T (8C)
`g
`
`HPMC Type 2910
`Methocel E15
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 606
`Pharmacoat 603
`Pharmacoat 606
`Pharmacoat 615
`Pharmacoat 606
`HPMC Type 2208
`Methocel K4M
`Methocel K4M
`
`TMA
`DSC
`DSC
`DSC
`DTA
`TBA
`DSC
`TMA
`DMA
`DMA
`DMA
`DMA
`
`DSC
`DSC
`
`a
`
`172–175
`177
`155
`180
`169–174
`153.5, 158.5
`155.8
`163.8, 174.4
`160
`170
`175
`154
`
`184
`(57)
`
`Ref.
`
`[18]
`[19]
`[20]
`[21]
`[21]
`[21]
`[22]
`[22]
`[23]
`[23]
`[23]
`[24]
`
`[25]
`[26]
`
`a The values obtained by TMA in the penetration mode have
`been reported by the authors as softening temperatures.
`
`processes involved. For example, Melia and co-
`workers [32–35] characterized the water mobility in
`the gel
`layer of hydrating HPMC matrices using
`NMR imaging. It has been shown that there is a
`diffusivity gradient across this layer and that it is
`affected by the degree of substitution of the polymer.
`Also Fyfe and Blazek [36] investigated the HPMC
`hydrogel formation by NMR spectroscopy pointing
`out the complications due to the presence of trapped
`gas [37]. Recently, they studied the release behavior
`of two model drugs,
`triflupromazine–HCl and 5-
`fluorouracil from HPMC tablets [38]. The tablet
`swelling was restricted to one dimension and dis-
`tributions of the water and model drugs were ob-
`1
`19
`tained by H and
`F imaging. The distributions of
`triflupromazine–HCl and HPMC paralleled each
`other and the drug was only released at the eroding
`edge of the tablet where the HPMC concentration
`dropped below 10%. In contrast, 5-fluorouracil was
`released much more rapidly from the tablet and
`appeared to escape by diffusion from regions as high
`as 30% HPMC. They also developed a system for
`performing NMR imaging experiments on drug
`delivery devices within a flow-through dissolution
`apparatus, USP Apparatus 4 [39].
`Ford and coworkers [40–42] used DSC techniques
`to study the distribution and amount of water in
`
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`(cid:210)
`

`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`143
`
`HPMC gels. Water loosely bound to the polymer was
`detected as one or more events appearing at
`the
`low-temperature side of the main endotherm for the
`melting of free water in the DSC scans. The HPMC
`molecular weight, HPMC substitution type, gel
`storage time, and added drug influenced the appear-
`ance of these melting events [43,44]. Pham and Lee
`[28] designed a new flow-through cell to provide
`well-defined hydrodynamic conditions during the
`experimental studies and to allow precise measure-
`ments of dissolution and swelling front positions
`versus time. The rate of polymer swelling and
`dissolution as well as the corresponding rate of drug
`release were found to increase with either higher
`levels of drug loading or lower viscosity grades of
`HPMC. Gao and Meury [45] developed an optical
`image analysis method to examine the dynamic
`swelling behavior of HPMC-based matrix tablets in
`situ. In addition to providing precise determinations
`of the apparent gel layer thickness and the tablet
`dimensions, this method is also capable of estimating
`the HPMC concentration profile across the gel layer.
`They used this technique to characterize the effect of
`the HPMC/lactose ratio and HPMC viscosity grade
`(molecular weight) on the swelling of the matrix
`[46]. For all formulations tested it was found that (i)
`swelling is anisotropic with a preferential expansion
`in the axial direction; and (ii) swelling is isotropic
`with respect
`to the gel
`layer thickness and com-
`position in both, axial and radial directions.
`The modification of the surface area of HPMC
`tablets in order to achieve a desired release rate has
`been studied by Colombo et al. [47,48]. Different
`surface portions of an HPMC matrix tablet were
`covered with an impermeable coating. They investi-
`gated the drug release mechanisms and studied the
`influence of the type of coating on the resulting
`release rate.
`In order
`to facilitate the industrial
`production, the manual film-coating process can be
`avoided using press-coating techniques [49].
`
`3. Overall drug release mechanism from
`HPMC-based systems
`
`The overall drug release mechanism from HPMC-
`based pharmaceutical devices strongly depends on
`the design (composition and geometry) of the par-
`
`ticular delivery system. The following phenomena
`are involved:
`(i) At the beginning of the process, steep water
`concentration gradients are formed at the polymer /
`water interface resulting in water imbibition into the
`matrix. To describe this process adequately,
`it
`is
`important to consider (i) the exact geometry of the
`device; (ii) in case of cylinders, both, axial and radial
`direction of the mass transport; and (iii) the signifi-
`cant dependence of the water diffusion coefficient on
`the matrix swelling ratio [50,51]. In dry systems the
`diffusion coefficient is very low, whereas in highly
`swollen gels it is of the same order of magnitude as
`in pure water. Water acts as a plasticizer and reduces
`the glass transition temperature of the system. Once
`the T equals the temperature of the system,
`the
`g
`polymer chains undergo the transition from the
`glassy to the rubbery state.
`(ii) Due to the imbibition of water HPMC swells,
`resulting in dramatic changes of polymer and drug
`concentrations, and increasing dimensions of
`the
`system.
`(iii) Upon contact with water the drug dissolves
`and (due to concentration gradients) diffuses out of
`the device.
`(iv) With increasing water content the diffusion
`coefficient of the drug increases substantially.
`(v) In the case of poor water-solubility, dissolved
`and non-dissolved drug coexist within the polymer
`matrix. Non-dissolved drug is not available for
`diffusion.
`(vi) In the case of high initial drug loadings, the
`inner structure of the matrix changes significantly
`during drug release, becoming more porous and less
`restrictive for diffusion upon drug depletion.
`(vii) Depending on the chain length and degree of
`substitution of the HPMC type used, the polymer
`itself dissolves more or less rapidly. In certain cases
`this phenomenon is negligible, for example if all
`drug has already been released before polymer
`dissolution becomes important.
`As a result of conditions (i), (ii), (iv), (vi), and
`(vii) the mathematical description of the diffusional
`processes requires strongly time-dependent terms.
`From the aforementioned possible phenomena it is
`obvious that
`there is no universal drug release
`mechanism that is valid for all kinds of HPMC-based
`systems. In contrast,
`there are many devices that
`
`

`
`144
`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`exhibit various mechanisms that control drug release,
`mechanisms such as polymer swelling, drug dissolu-
`tion, drug diffusion or combinations of the above.
`The physicochemical characteristics and geometry of
`each device determine the resulting governing pro-
`cesses. Concerning the mathematical modeling of
`drug release from HPMC-based systems, one must
`identify the most important transport phenomenon
`for the investigated device and neglect
`the other
`processes, otherwise the mathematical model be-
`comes too complex for facile use.
`
`4. Empirical and semi-empirical mathematical
`models
`
`4.1. Higuchi equation
`
`the design variables of the system. Thus, the fraction
`of drug released is proportional to the square root of
`time. Alternatively, the drug release rate is propor-
`tional to the reciprocal of the square root of time.
`An important advantage of these equations is their
`simplicity. However, when applying them to con-
`trolled drug delivery systems, the assumptions of the
`Higuchi derivation should carefully be kept in mind:
`(i) The initial drug concentration in the system is
`much higher than the solubility of the drug. This
`assumption is very important, because it provides the
`basis for the justification of the applied pseudo-
`steady state approach. The resulting concentration
`profiles of a drug initially suspended in an ointment
`are illustrated in Fig. 2. The solid line represents the
`drug concentration profile after exposure of
`the
`ointment to perfect sink for a certain time t.
`As can be seen there is a sharp discontinuity at
`distance h from the surface. For this distance h
`above the absorbing surface the concentration gra-
`dient is essentially constant, provided, the initial drug
`concentration within the system, c , is much greater
`0
`than the solubility of the drug (c . .c ). After an
`0
`s
`additional time interval, Dt, the new concentration
`profile of the drug is given by the broken line. Again,
`a sharp discontinuity and otherwise linear concen-
`tration profiles result. Under these particular con-
`ditions Higuchi derived the very simple relationship
`between the release rate of the drug and the square
`root of time.
`(ii) Mathematical analysis is based on one-dimen-
`
`A w
`
`In 1961, Higuchi [52] published the probably most
`famous and most often used mathematical equation
`to describe the release rate of drugs from matrix
`systems. Initially valid only for planar systems, it
`was later modified and extended to consider different
`geometries and matrix characteristics including po-
`rous structures [53–57]. We have pointed out in the
`past [9] that the classical Higuchi equation [52] was
`derived under pseudo-steady state assumptions and
`generally cannot be applied to ‘real’ controlled
`release systems.
`The basic equation of the Higuchi model is:
`M ]]]]
`t] 5 D(2c 2 c )c t
`ˇ
`0
`s
`s
`here M is the cumulative absolute amount of drug
`t
`released at
`time t, A is the surface area of the
`controlled release device exposed to the release
`medium, D is the drug diffusivity in the polymer
`carrier, and c
`and c
`are the initial drug con-
`0
`s
`centration, and the solubility of the drug in the
`polymer, respectively. Clearly, Eq. (1) can be ex-
`pressed as:
`
`for c . c
`0
`
`s
`
`(1)
`
`Mt
`]˛] 5 K t
`M‘
`where M is the absolute cumulative amount of drug
`‘
`released at infinite time (which should be equal to
`the absolute amount of drug incorporated within the
`system at time t50), and K is a constant reflecting
`
`(2)
`
`Fig. 2. Pseudo-steady state approach applied for the derivation of
`the classical Higuchi equation. Theoretical concentration profile
`existing in an ointment containing suspended drug and in contact
`with a perfect sink (adapted from [52] with permission from Wiley
`& Sons).
`
`

`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`145
`
`sional diffusion. Thus, edge effects must be negli-
`gible.
`(iii) The suspended drug is in a fine state such that
`the particles are much smaller in diameter than the
`thickness of the system.
`(iv) Swelling or dissolution of the polymer carrier
`is negligible.
`(v) The diffusivity of the drug is constant.
`(vi) Perfect sink conditions are maintained.
`It is evident that these assumptions are not valid
`for most controlled drug delivery systems based on
`HPMC. However, due to the extreme simplicity of
`the classical Higuchi equation (Eq. (1)), the latter is
`often used to analyze experimental drug release data
`to get a rough idea of the underlying release mecha-
`nism. But the information obtained should be viewed
`with caution. The superposition of various different
`effects, such as HPMC swelling, transition of the
`macromolecules from the glassy to the rubbery state,
`polymer dissolution, concentration-dependent water
`and drug diffusion etc. might also result
`in an
`apparent square root of time kinetics. In addition, a
`proportionality between the fractional amount of
`drug released and the square root of time can as well
`be derived from an exact solution of Fick’s second
`law of diffusion for thin films of thickness d under
`perfect sink conditions, uniform initial drug con-
`centration with c ,c
`(monolithic solutions) and
`0
`s
`assuming constant diffusivities ([7,58,59]):
`
`J]
`
`nd
`]]
`˛
`2 Dt
`
`S D H
`p 1 2O (21) ierfc
`
`M
`Dt
`t
`5 4
`] ]
`M
`2
`d
`‘
`
`1 / 2
`
`21 / 2
`
`‘
`
`n 51
`
`n
`
`based on these physical circumstances which are
`substantially different from those studied by Higuchi
`for the derivation of his classical equation (mono-
`lithic solutions versus monolithic dispersions). How-
`ever,
`in both cases diffusion is the dominating
`mechanism and hence a proportionality between the
`cumulative amount of drug released and the square
`root of time is commonly regarded as an indicator
`for diffusion-controlled drug release.
`Various researchers used the Higuchi equation to
`interpret their experimental drug release data. For
`example, Sung et al. [60] investigated the effect of
`formulation variables
`(HPMC /lactose ratio and
`HPMC viscosity grade) on the resulting Higuchi rate
`constants of a water-soluble model drug, adinazolam
`mesylate. For HPMC K15M and HPMC K100M
`they found similar values, whereas for HPMC K4M
`and HPMC K100LV and various HPMC/ lactose
`ratios significantly different constants were obtained.
`This study is a good example for the use of the
`Higuchi equation for HPMC-based drug delivery
`systems, because although good fittings were ob-
`tained, the authors make clear that additional mathe-
`matical analysis is needed to make definitive mech-
`anistic conclusions. Talukdar and Kinget [61] not
`only determined Higuchi constants, but used Eq. (1)
`to obtain the diffusivities of
`three model drugs
`(indometacin, indometacin sodium and caffeine) in
`hydrated HPMC and xanthan gum gels. In addition,
`they used a special apparatus for the experimental
`studies, restricting drug release to one surface only.
`
`(3)
`
`4.2. Power law
`
`Here, M and M are the absolute cumulative
`t
`‘
`amount of drug released at time t and infinite time,
`respectively; D represents the diffusivity of the drug
`within the polymeric system. As the second term in
`the second brackets vanishes at short
`times, a
`sufficiently accurate approximation of Eq. (3) for
`M /M ,0.60 can be written as follows:
`t
`‘
`
`S
`
`M
`Dt
`t
`5 4
`] ]]
`M
`2
`pd
`‘
`
`D
`
`1 / 2
`
`5 k9
`
`]˛
`t
`
`(4)
`
`where k9 is a constant.
`Thus, a proportionality between the fraction of
`drug released and the square root of time can also be
`
`A more comprehensive, but still very simple,
`semi-empirical equation to describe drug release
`from polymeric systems is the so-called power law:
`M
`t
`n] 5 kt
`M‘
`Here, M and M are the absolute cumulative
`t
`‘
`amount of drug released at time t and infinite time,
`respectively; k is a constant incorporating structural
`and geometric characteristics of the device, and n is
`the release exponent, indicative of the mechanism of
`drug release.
`As can be seen, the classical Higuchi equation
`(Eq. (1)) as well as the above described short time
`
`(5)
`
`

`
`146
`
`J. Siepmann, N.A. Peppas / Advanced Drug Delivery Reviews 48(2001)139–157
`
`approximation of the exact solution of Fick’s second
`law for thin films (Eq. (4)) represent the special case
`of the power law where n50.5. Peppas and co-
`workers [62,63] were the first to give an introduction
`into the use and the limitations of this equation. The
`power law can be seen as a generalization of the
`observation that superposition of
`two apparently
`independent mechanisms of drug transport, a Fickian
`diffusion and a case-II transport [64,65], describes in
`many cases dynamic swelling of and drug release
`from glassy polymers, regardless of the form of the
`constitutive equation and the type of coupling of
`relaxation and diffusion [66].
`It is clear from Eq. (5) that when the exponent n
`takes a value of 1.0, the drug release rate is in-
`dependent of time. This case corresponds to zero-
`order release kinetics. For slabs, the mechanism that
`creates the zero-order
`release is known among
`polymer scientists as case-II
`transport. Here the
`relaxation process of the macromolecules occurring
`upon water imbibition into the system is the rate-
`controlling step. Water acts as a plasticizer and
`decreases the glass transition temperature of the
`polymer. Once the T equals the temperature of the
`g
`system, the polymer chains undergo the transfer from
`the glassy to the rubbery state, with increasing
`mobility of the macromolecules and volume expan-
`sion.
`Thus, Eq. (5) has two distinct physical realistic
`meanings in the two special cases of n50.5 (indicat-
`ing diffusion-controlled drug release) and n51.0
`(indicating swelling-controlled drug release). Values
`of n between 0.5 and 1.0 can be regarded as an
`indicator for the superposition of both phenomena
`(anomalous transport). It has to be kept in mind that
`the two extreme values for the exponent n, 0.5 and
`1.0, are only valid for slab geometry. For spheres
`and cylinders different values have been derived
`[67,68], as listed in Table 3. Unfortunately, this fact
`
`is not always taken into account, leading to mis-
`interpretations of experimental results.
`In the case of HPMC-based systems it has to be
`pointed out that the application of the power law can
`only give limited insight
`into the exact
`release
`mechanism of
`the drug. Even if values of
`the
`exponent n are found that would indicate a diffusion-
`controlled drug release mechanism, this is not auto-
`matically valid for HPMC. The derivation of the
`Higuchi equation and the above described short time
`approximation of Fick’s second law for slab geome-
`try assume constant diffusivities and constant dimen-
`sions of the device during drug release. However,
`HPMC swells to a significant extend, the diffusion
`coefficients of water and incorporated drugs are
`strongly concentration dependent [69], and HPMC
`itself dissolves more or less rapidly. An apparent
`square root of time release kinetics can thus result
`from the superposition of various effects, and is not
`necessarily based on a simple drug diffusion-control.
`As in the case of the H