Mathematical models in drug delivery: how modeling has shaped the way we design new drug delivery systems.

COREL-07275; No of Pages 7 Journal of Controlled Release xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Journal of Controlled Release journal homepage: www.elsevier.com/locate/jconrel

Mathematical models in drug delivery: How modeling has shaped the way we design new drug delivery systems☆ Nicholas A. Peppas a,b,c,⁎, Balaji Narasimhan d a

Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA Department of Biomedical Engineering, The University of Texas at Austin, Austin, TX 78712, USA College of Pharmacy, The University of Texas at Austin, Austin, TX 78712, USA d Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 50011, USA b c

a r t i c l e

i n f o

Article history: Received 11 April 2014 Accepted 21 June 2014 Available online xxxx Keywords: Mathematical modeling Controlled release Diffusion Erosion Swelling Dissolution

a b s t r a c t In this review we present some of the seminal contributions that have established the mathematical foundations of controlled drug delivery and led to the modern models. Mathematical modeling is no longer just a dry exercise in generating more and more complex models or a parametric fitting process, but rather an advanced analysis that can lead to a priori examination of a release/delivery process or a series of design equations that help the practitioner achieve a better formulation. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Over the past 50 years mathematical modeling of diffusional and release processes has been used to design a number of simple and complex drug delivery systems and devices and to predict the overall release behavior. Such systems have been used to design pharmaceutical formulations, analyze drug release processes in vitro and in vivo and in general come up with the optimal design for new systems. Mathematical models have been used mainly to predict the temporal release of the encapsulated cargo molecule(s). These models add value in terms of ensuring optimal design of pharmaceutical formulation(s) as well as to understand release mechanism(s) through experimental verification [1]. Only combining precise experimental observations with models that capture the underlying physics will provide new insights into the release mechanism. Many existing models are predicated upon diffusion equations. Because diffusion of drugs is a strong function of the structure through which the diffusion takes place, models need to account for polymer morphology [1,2]. Classification of controlled release systems is predicated upon drug release mechanisms as follows: (i) diffusion-controlled; (ii) chemically controlled; (iii) osmotically controlled; and (iv) swelling-

☆ Paper for the 30th anniversary of the Journal of Controlled Release. ⁎ Corresponding author.

and/or dissolution-controlled [2]. Each of these mechanisms is individually discussed in this review. We begin by discussing the fundamentals of diffusion because of the central role it plays in controlled release systems. 2. Diffusion of therapeutic compounds Therapeutic compounds are released from a polymeric carrier by diffusion through the bulk of the carrier. Mass transfer at the carrier/water interface plays an important role in diffusion. Drug transport can be described by Fick's law of diffusion, which is shown by Eqs. (1) and (2) for one-dimensional diffusion in a planar geometry [3]: dci dx

ð1Þ

∂ci ∂2 c ¼ Dip 2i ∂t ∂x

ð2Þ

ji ¼ −Dip

Here, ci, and ji represent the concentration and mass flux of species i, respectively, Dip is the diffusion coefficient of species i in the polymer matrix, and x and t represent position and time, respectively. Equivalent equations are available to describe drug transport through thick slabs, cylinders, and spheres [3]. The experimental conditions that constrain the carrier are captured by initial and boundary conditions and

http://dx.doi.org/10.1016/j.jconrel.2014.06.041 0168-3659/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: N.A. Peppas, B. Narasimhan, Mathematical models in drug delivery: How modeling has shaped the way we design new drug delivery systems, J. Control. Release (2014), http://dx.doi.org/10.1016/j.jconrel.2014.06.041

2

N.A. Peppas, B. Narasimhan / Journal of Controlled Release xxx (2014) xxx–xxx

solutions to Eqs. (1) and (2) have been obtained for a wide variety of initial and boundary conditions [3]. Strictly speaking, the drug diffusion coefficient, Dip, is a function of drug concentration. Incorporation of a concentration dependent diffusion coefficient results in Eq. (3), which can be solved with appropriate initial and boundary conditions: ∂ci ∂ ∂c ¼ Dip ðci Þ i ∂t ∂x ∂z

ð3Þ

In Eq. (3), Dip(ci) is the concentration-dependent diffusion coefficient, whose form is a strong function of the polymer morphology. Table 1 shows some typically used forms of the diffusion coefficient. Eyring [4] presented a theory to estimate the diffusion coefficient through a polymer, in which diffusion occurs via a series of discontinuous jumps rather than continuously. In the Eyring model (Eq. (4) in Table 1), λ is the diffusional jump of the drug and v is the jump frequency. Free volume-based drug diffusion coefficients with an exponential dependence of the drug diffusion coefficient on the free volume, νf, were proposed by Fujita [5], as shown by Eq. (8) in Table 1. Yasuda and Lamaze [12] refined Fujita's theory and arrived at Eq. (9), to predict the drug diffusion coefficient through a polymer matrix. In the Peppas– Reinhart free volume model [9] that describes drug transport in highly swollen, nonporous hydrogels, the free volume of the polymer was approximated as the free volume of the solvent (Eq. (9) in Table 1). Models for moderately or poorly swollen [13] and semicrystalline [14] hydrogels are also available. Drug diffusion coefficients have also been estimated by drawing analogies with transport in porous rocks, resins, and catalysts [6]. In these approaches, diffusion occurs through water-filled pores. In such cases, an effective diffusive coefficient, Deff, (Eq. (5) in Table 1) can be used. In this equation, ε is the porosity, or void fraction, of the polymer, which is a measure of the available pore volume and τ is the tortuosity, which describes pore geometry. The term Kp is an equilibrium partition coefficient, which is the ratio between the drug concentrations inside and outside the pore. The term Kr is the fractional reduction in diffusivity within the pore when ds (solute diameter) ~dr (pore diameter). As another example, Faxén [7] proposed Eq. (6) in Table 1 to describe diffusion of spheres through porous media. Herein, λ is the ratio between rs (drug radius) and rp (pore average radius), D and Db are the diffusion coefficients of the sphere through the pore and in the bulk, respectively; and α, β, and γ are constants. As the value of λ decreases, the normalized term D/Db goes to 1. 3. Diffusion-controlled systems Reservoir diffusion-controlled systems contain a bioactive cargo within a core that is sequestered by a polymer membrane. The drug release rate from such systems using Eq. (1) is constant (i.e., zero-order), regardless of geometry. As an example, Eqs. (10) and (11) show the rate

of drug release and the total amount of drug released from spherical systems. 4πDip K dMt ðc −c Þ ¼ dt ðre −ri Þ=ðre ri Þ i2 i1

ð10Þ

4πDip Kðci2 −ci1 Þ t ðre −ri Þ=ðre ri Þ

ð11Þ

Mt ¼

Here, Dip is the concentration independent diffusion coefficient, Mt is the amount of drug released at time t, K is the drug partition coefficient, re and ri are the external and internal radii of the sphere, respectively, and ci1 and ci2 are the drug concentrations inside and outside the matrix, respectively. In these systems, the amount of drug released can be controlled by the membrane thickness, concentration differential across the membrane, drug partition coefficient, and the drug diffusion coefficient. In diffusion-controlled matrix systems, the drug can be dissolved into or dispersed throughout the system [10]. The drug release from these systems can be modeled using Eq. (3) with a concentration dependent diffusion coefficient (Eqs. (8) through (10)). Solutions to Eq. (3) show that the drug release is proportional to t1/2. However, one can obtain zero order release (i.e., release proportional to t and release rate independent of t) in diffusion-controlled systems by tailoring the geometry of the device as shown elegantly by Langer and coworkers by using a hemispherical device coated with an impermeable layer on all sides except a small opening on the lateral surface [15]. In this manner, as water penetrates the hydrogel from the uncoated surface, the gel diffusion layer gets larger, but is compensated for by increasingly large drug fronts. Higuchi [16] was the first one to postulate a widely used mathematical model to describe drug release from matrix systems in 1961. In the first few years after its publication this equation was simply used to fit release data. However later in that decade, the so-called Higuchi equation became the preferred model for drug transport in different geometries and within porous matrices [17–21]. It is important to note that the classical Higuchi equation (see Eq. (12)) was derived using pseudosteady state assumptions and is not applicable to many controlled release systems [22]. Mt ¼ A

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D cs ð2 c0 −cs Þt

ð12Þ

Here, Mt is the cumulative amount of drug released at time t, A is the surface area of the device, D is the drug diffusivity in the carrier, and c0 and cs are the initial drug concentration and the drug solubility within the polymer, respectively. The amount of drug released, Mt, is proportional to the square root of time and the drug release rate, dMt/dt, varies as the reciprocal of the square root of time.

Table 1 Various forms of the diffusion coefficient. Type of carrier

Eq.

Form of Dip

Porous

4

Dip ¼ λ6ν Deff ¼ Diw Kp Kr τε Dip 2 1 þ αλ þ βλ3 þ γλ5 Db ¼ ð1‐λÞ n o Dip ¼ Do exp − νkf h i D2;1;3 qs 1 D2;1 ¼ φðqs Þ exp −B v f;1 H −1 2 D2;1;3 Mc −Mc 2 rs exp − kQ−1 D2;1 ¼ k1

Porous

5

Microporous

6

Nonporous

7

Nonporous

8

Nonporous (highly swollen)

9

Ref.

2

Mn −Mc

4 5 7 6 8 4 Fig. 1. Conceptual framework of the Higuchi model.



While the Higuchi model is conceptually simple, its validity in describing controlled drug release systems is dependent upon the assumptions used in deriving it: (i) The initial drug concentration is much higher than the drug solubility, justifying the pseudo-steady state approach. The drug concentration profile initially suspended in a matrix (an ointment in the case of the original system) is shown in Fig. 1. The solid line indicates the drug concentration profile under perfect sink conditions. The broken line indicates the new concentration profile after a time interval, Δt, has elapsed. (ii) Drug diffusion is one-dimensional, making edge effects negligible. (iii) The suspended drug micro- or nanoparticles are much smaller than the thickness of the system. (iv) Swelling or dissolution of the polymer carrier can be neglected. (v) The drug diffusion coefficient is constant. (vi) Perfect sink conditions prevail and are maintained. Even though these assumptions do not work for many controlled drug release systems, the Higuchi equation is often used to analyze experimental release profiles, which may lead to erroneous conclusions about release mechanisms [22]. The combination of different phenomena, such as swelling, glassy/rubbery transitions, dissolution, and concentration-dependent diffusion may result in release data that display a square root dependence upon time. Indeed an exact solution of Eq. (3) for thin films of thickness δ with uniform initial drug concentration and constant diffusivity under perfect sink conditions results in a Mt~t0.5 relationship [3,23,24]: " # ∞ X Mt 8 D ð2n þ 1Þ2 π2 t ¼ 1− exp − 2 2 M∞ δ2 n¼0 ð2n þ 1Þ π

ð13Þ

Here, Mt and M∞ are the cumulative amounts of drug released at time t and infinite time, respectively and D is the drug diffusivity. As the second term in the summation diminishes at short times, an approximation of Eq. (13) for Mt/M∞ b 0.60 can be written as: Mt ¼4 M∞

rffiffiffiffiffiffiffiffiffi Dt π δ2

ð14Þ

It is useful to remember that regardless of the device geometry and release conditions, diffusion is the dominating mechanism. More recently, generalized models based on chemical potential gradients have been proposed to understand drug release from polymer matrices [25]. A semi-empirical power law equation has been widely used to describe drug release from polymeric systems [26]: Mt n ¼ kt M∞

Yet, one characteristic of the simple power law Eq. (15) remains. This equation is not simply a fitting expression but the result of a detailed mechanistic process that involved a pure diffusional (Fickian) mechanism with an additional relaxational or convection mechanism usually associated with a major state or phase change. Thus, Eq. (15) stems from a molecular and mechanistic analysis, while other approaches have a more fitting approach. It is also interesting that Eq. (15) has been accepted by most scientists in the field over other equations because of its simple form. Indeed, those interested in using a more sophisticated equation to determine the underlying mechanisms of a release process where the only available results are simple Mt/M∞ versus t data points can do so by a simple logarithmic plot of the data and identification of the characteristic exponent n, as long as the statistical analysis of the logarithmic plot has a significant number of data points so that the associated correlation coefficient is statistically important [11,26]. 4. Chemically controlled systems Chemically controlled systems can be classified into erodible and pendant chain systems. The drug release rate in erodible systems is controlled by polymer degradation. In pendant chain systems, the drug release is controlled by the hydrolytic or enzymatic degradation of a chemical bond between the drug and the polymer carrier. Erodible systems exhibit ordinary diffusion and the drug release mechanism could be diffusion- or erosion-controlled. If matrix erosion is much slower than drug diffusion through the polymer, the drug release kinetics is diffusion-controlled and can be dealt with as described in the previous section. In contrast, in erosion-controlled systems, the drug diffusion rate from the matrix is low and it remains within the matrix. There are two possible mechanisms of erosion, surface and bulk (Fig. 2). Water is rapidly transported into the matrix of less hydrophobic polymers and the polymer degradation rate everywhere within the matrix is the same, leading to erosion throughout the polymer matrix (i.e., bulk erosion). In contrast, water is excluded from the bulk of the matrix of hydrophobic polymers, leading to erosion only at the surface. In reality, many polymers erode by a combination of surface and bulk erosion; however, the conceptual picture provided by the two idealized extremes is valuable because it affords an excellent starting point to model these phenomena. Surface erodible matrices exhibit near zeroorder release kinetics, making them valuable in the pharmaceutical field. Broadly, erosion models can be classified as phenomenological, probabilistic, or empirical [30]. Phenomenological models are based on the transport models that govern reaction, diffusion, and dissolution.

ð15Þ

Here, Mt and M∞ are the cumulative amounts of drug released at time t and infinite time, respectively; k contains structural and geometric information about the device, and n is indicative of the drug release mechanism. The mathematical solutions of Eq. (1) for Fickian drug diffusion result in release kinetics described by Eq. (15) with n = 0.5. When the release kinetics is zero order, n = 1 in Eq. (15). The various limits of this equation are discussed by Ritger and Peppas [26] where it has been shown that the mechanistic limits of n are dependent on the geometry of the associated release device. For the record, Eq. (15) is also known as the Korsmeyer–Peppas or the Peppas equation [11,22]. Modifications of the power law model have been suggested, including the use of the Weibull function [27]. Macheras and co-workers have also used Monte Carlo simulations to model drug release in Euclidian and fractal spaces [28,29]. These very important mathematical methods have provided further understanding of the drug transport process and have often offered a more accurate representation of the release process.

3

Fig. 2. Mechanisms of bulk and surface erosion of polymers.


4


They can be broadly applied to various polymer chemistries, device geometries, and conditions. Reaction–diffusion models for drug release from bulk eroding polymers have been proposed by Thombre and Himmelstein [31,32], Antheunis and co-workers [33,34], and Prabhu and Hossainy [35] and account for autocatalytic effects due to the reaction and its effect thereof on drug release kinetics. Other phenomenological models for erosion have considered aspects such as crystallization [36,37], surface microstructure [38,39], and moving boundaries [40,41]. A phenomenological model to describe drug release from bulk eroding microspheres was proposed by Batycky et al. [42] by considering both random- and endchain scissions as well as molecular weight distributions that result upon erosion. Finally, Arosio and co-workers postulated two models to describe a cylindrical bulk eroding polymer and accounted for reaction-controlled and diffusion-controlled states [43,44]. Phenomenological models for drug release from surface erodible polymers have considered aspects such as copolymer phase behavior and drug partitioning into various micro-domains of the matrix [38,39]. Other phenomenological models have accounted explicitly for both surface and bulk erosion [45,46]. A model proposed by von Burkersroda et al. [47] defined a dimensionless parameter accounting for conditions under which polymers would undergo surface or bulk erosion: ε¼ 4Deff

2 hxi λπ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 ln ½hxi− ln Mn =NA ðN−1Þρ

ð16Þ

Here Deff is the effective water diffusivity, Mn is the polymer number average molecular weight, NA is Avogadro's number, N is the degree of polymerization, ρ is polymer density, λ is a degradation rate constant, and bxN is a mean diffusion distance of water. Such models can be effective in describing the more realistic situation of hybrid erosion and may be better able to model drug release from erodible polymers, as proposed by Rothstein et al.[48]. Probability-based models have also been used to describe polymer erosion. A lifetime is assigned to each element of the matrix and the probability of erosion of the element (or cell) is calculated using Monte Carlo methods by accounting for important physical phenomena occurring during erosion [49–54]. These models are the basis for extending the probabilistic approach to describe drug release [55–57]. In many of these cases, diffusional aspects of drug release are mostly ignored, which is an important limitation. In addition to the above approaches, several empirical models have been proposed to study drug release from eroding polymers [58–60]. Limitations of empirical models are that they are not broadly applicable to all systems and conditions and usually require several fitting parameters. Consequently, it becomes difficult to obtain mechanistic information from such models. 5. Osmotically-controlled systems When polymers are uniformly loaded with highly soluble drugs, the local osmotic pressure is very high and results in rupture of the system, leading to drug release. The OROS® oral drug delivery technology developed by ALZA for gastrointestinal transport utilizes osmotic pressure to control water permeation and drug release. The mathematical models for drug delivery from osmotic systems are predicated on irreversible thermodynamics and the Kedem–Katchalsky [61] analysis. The rate of drug release from these systems can be written as: dM A ¼ Lp σc Δπs dt δ

ð17Þ

Here A is the cross sectional area and δ is the thickness of the device, c is the drug concentration, Lp is a permeability coefficient, σ is a reflection coefficient, and Δπs is the osmotic pressure of water. As shown in

Eq. (17), zero order release can be obtained from these systems. These approaches have been used to achieve release of encapsulated payload in a single pulse from spherically shaped devices [62]. In this work, it was shown that the ratio of the polymer yield stress to the stiffness determined the radius at which the spheres burst and release the encapsulated drug. 6. Swelling- and dissolution controlled systems The models for swelling-controlled released systems based on polymers need to account for complex macromolecular changes during release. In these systems, water-soluble drugs are initially loaded within glassy polymers. When placed in water or buffer, the solvent diffuses into the polymer, leading to swelling and volume expansion of the polymer. The swelling behavior is typically characterized by two moving fronts (or interfaces): (i) the swelling interface which delineates the rubbery (i.e., swollen) state from the glassy state and which moves inward; and (ii) the polymer interface which contacts water and moves outward. It is known that swelling of glassy polymers is accompanied by chain relaxation at the swelling interface [63]. This relaxation affects the drug diffusion through the polymer, which can be Fickian or non-Fickian diffusion. This system is referred to as Stefan or Stefan–Neumann or moving boundary problem and is modeled by solving the Fickian diffusion Eq. (2) with concentration-dependent or independent drug diffusion coefficients and moving boundary conditions at the two fronts. Adding macromolecular relaxation to this system results in a realistic model to describe the transport [3,64,65]. The transport of the drug is determined by the rate of chain relaxation and/or the drug diffusion rate through the rubbery polymer and can be characterized by a dimensionless number called the Deborah number, De [66], De ¼

λ θ

ð18Þ

where λ is the relaxation time and θ is the diffusion time. When De ≫1 (relaxation-controlled) or when De ≪1 (diffusion-controlled), Fickian behavior is observed. When De ~ 1, the relaxation time is of the same order as the diffusion time and results in non-Fickian or anomalous diffusion. Two other dimensionless numbers have been proposed to model drug release from swellable systems. The first is the “swelling interface” number, Sw, defined as the ratio between the rates of convective transport of water and drug diffusion [67], and the second is the “swelling area” number, which accounts for the variable area during swelling and its effect on drug transport [68]. In particular, the swelling area number has been used to accurately design the commercially available Geomatrix® systems. More complex analyses of drug release from swellable systems are available that account for enthalpy balances and front synchronization [69,70], non-uniform drug distributions within the polymer [40, 71–74], geometric effects [75], multi-dimensional transport [76], timedependent [77] or concentration-dependent [78–80] drug diffusion coefficients, and free volume [81]. Based on these models, conditions to achieve zero order release of the drug have been identified. The most comprehensive model to describe drug release from swellable polymers accounts for polymer viscoelasticity and uses a rational thermodynamics framework to describe the concentration-dependent drug diffusion and three-dimensional swelling [82]. Typically, swellable polymers also respond to environmental cues such as temperature, pH, and ionic strength, which affect drug release from these systems. Models are available to account for all of these effects [83–87]. All of these models have provided valuable information to understand drug release from hydroxypropyl methylcellulose (HPMC), poly(2-hydroxyethyl methacrylate) (PHEMA), and other hydrogel-based systems. In addition to swelling and relaxation, when the polymer chains are uncrosslinked, they will begin to disentangle and dissolve when



brought into contact with a thermodynamically compatible solvent. Conceptually, the physics is similar to that of the swellable systems described above, with the important difference that the polymer interface first moves outward due to swelling, but starts to move inward when chain disentanglement and diffusion start to dominate swelling and eventually, the fronts collapse and the polymer is fully dissolved [88]. This has resulted in the design of soluble polymers for drug delivery and a number of modeling efforts are available to understand drug release mechanisms from dissolving polymers. Harland et al. [89] proposed a model for drug release from a dissolving polymer and showed that synchronization between the rates of movement of the two fronts led to zero-order drug release. Similar models that accounted for both swelling and dissolution and radial transport have been proposed by Nixon and co-workers [90,91]. The first one-dimensional model for drug release from soluble polymers that accounted for a molecular level understanding of chain disentanglement was proposed by Narasimhan and Peppas [92]. This allows for tailoring the design of drug delivery systems for specific applications. From this model, an expression for drug release was derived as: υd;eq þ υd pffiffiffiffiffiffiffiffiffiffiffi Md 2At þ Bt ¼ Md;∞ 2L

ð19Þ

Here, L is the half-thickness of the polymer and υ1,eq. and υd,eq. are the equilibrium concentrations of solvent and drug, respectively. A and B depend upon the diffusion coefficients, the chain disentanglement rate (calculated using reptation theory [93–95]), and the solvent and drug concentrations. Conditions for zero order release and the transition between Fickian and non-Fickian type behavior were identified and used to explain experimentally obtained drug release profiles from a number of systems. Siepmann and co-workers proposed the first multi-dimensional model to account for drug release from dissolving polymers [96–98]. These models accounted for both axial and radial transport, nonhomogeneous swelling, different drug loadings, and device geometry. These models have been able to explain a number of experimental observations associated with drug release from HPMC-based systems. A similar model was developed by Wu et al. [99] for drug release from polyethylene oxide-based systems. A more mechanistic analysis of this approach was postulated by Borgquist and co-workers [100], which resulted in a model that was tested against experimental data on front movement, drug release, and polymer dissolution. All of the above models are valid for drug release from dissolution of amorphous polymers. For semicrystalline polymers, dissolution is typically preceded by the crystal unfolding [101,102]. Drug release from such systems was modeled by Mallapragada and Peppas [103]. It was shown that the dynamics between the rates of crystal unfolding and drug diffusion could be used to obtain zero order release of the drug. These types of models have enhanced our understanding of using materials such as poly(vinyl alcohol) as drug delivery vehicles. 7. Outlook and perspectives Our understanding of the mechanism of drug release from polymeric devices has been advanced significantly by mathematical models. The models have ranged from being simple and empirical to phenomenological to probabilistic to molecular, and have helped guide the design of drug delivery devices. Starting from the use of steady state and transient descriptions of drug diffusion using Fick's law, the models have accounted for chain relaxation, polymer microstructure, glassy/rubbery transitions, chain disentanglement, polymer crystallinity, environmental effects, concentration effects, multi-dimensional effects, non-uniform drug distributions, and device geometry. Moving forward, one of the key gaps that continues to remain between experiment and theory in the field of drug delivery is the need for accurate molecular, thermodynamic, and transport parameters. In

5

this context, enhanced collaborations between experimentalists and theoreticians are critical to drive our understanding. The standard comparisons between experiments and models for drug delivery have predominantly used release kinetics data. As discussed in this article, the release kinetics data represents a cumulative effect of several phenomena, interactions between various components of the formulation, and the polymer microstructure. Thus, it appears to make sense that the models need to be validated against experimental observations that measure these phenomena, interactions, and microstructure. For example, techniques such as rheometry, small angle scattering, electron microscopy, and solid state NMR need to be used to measure aspects of the polymer microstructure for appropriate comparisons with the models. Increasingly, the next generation of drug delivery devices is focused on delivering payload to specific locations within cells. These include the need to deliver DNA to the nucleus for gene therapy, antigen to endolysosomal or cytosolic compartments for vaccine delivery, and drug to organelles such as mitochondria to slow down degenerative diseases. In all of these cases, there is a strong need to understand the role of both the extracellular as well as the intracellular environment on the drug release kinetics. Mathematical models may play an important role in helping us rationally design delivery vehicles that can release the right amount of payload at the right location and at the right time. Finally, many of the currently used drug delivery models are focused on the release of small molecular weight payloads. As we have transitioned to an era of protein-based drugs and/or antigens, accurate models of protein delivery are needed, which account for conformational changes of the protein and the alterations in the protein structure caused by the synthesis of the vehicle and during release. Such models can significantly reduce the large numbers of experiments needed to identify the optimal formulations for enhanced efficacy or bioavailability. Acknowledgments Over the years we have benefited from numerous discussions with others who have contributed in the field. We acknowledge with gratitude the contributions of our collaborators Jennifer Sinclair Curtis, Phillip Ritger, the late Gavin Sinclair, Jennifer Sahlin, Jürgen Siepmann, Surya Mallapragada, Ron Harland and Ping I. Lee. We are also indebted to John Petropoulos, Kenneth Himmelstein, John Cushman, Panos Macheras, Achim Goepferich and James Caruthers who provided interesting and challenging viewpoints on diffusion, anomalous transport and diffusion/reaction models. References [1] B. Narasimhan, N.A. Peppas, The role of modeling in the development of future controlled release devices, in: K. Park (Ed.), Controlled Drug Delivery, American Chemical Society, Washington, D.C., 1997, pp. 529–557. [2] R.S. Langer, N.A. Peppas, Chemical and physical structure of polymers as carriers for controlled release of bioactive agents: a review, J. Macromol. Sci., Rev. Macromol. Chem. Phys. C23 (1) (1983) 61–126. [3] J. Crank, The Mathematics of Diffusion, 2nd ed. Oxford University Press, 1975. [4] H. Eyring, Viscosity, plasticity and diffusion as examples of reaction rates, J. Chem. Phys. 4 (1936) 283. [5] H. Fujita, Diffusion in polymer-diluent systems, Fortschr. Hochpolym. Forsch. 3 (1961) 1–47. [6] E.N. Lightfoot, Transport Phenomena and Living Systems, Wiley, New York, 1974. [7] H. Faxen, Die Bewegung einer Einer Starren Kuegel laengs der Achse mit zaeher Fluessigkeit gefuellten Rohres, Ark. Mat. Astron. Fys. 17 (1923) 1. [8] H. Yasuda, A. Peterlin, C.K. Colton, K.A. Smith, E.W. Merrill, Permeability of solutes through hydrated polymer membranes III. Theoretical background for the selectivity of dialysis membranes, Macromol. Chem. Phys. 126 (1969) 177. [9] N.A. Peppas, C.T. Reinhart, Solute diffusion in swollen membranes, I. A new theory, J. Membr. Sci. 15 (1983) 275–287. [10] R.S. Langer, N.A. Peppas, Present and future applications of biomaterials in controlled drug delivery systems, Biomaterials 2 (1981) 201–214. [11] N.A. Peppas, Mathematical modeling of diffusion processes in drug delivery polymeric systems, in: V.F. Smolen, L.A. Ball (Eds.), Drug Product Design and Performance, Controlled Drug Bioavailability, vol. 1, John Wiley, New York, 1984, pp. 203–237.


6


[12] H. Yasuda, C.E. Lamaze, Permselectivity of solutes in homogeneous water-swollen polymer membranes, J. Macromol. Sci., Phys. B5 (1971) 111–120. [13] N.A. Peppas, H.J. Moynihan, Solute diffusion in swollen membranes. IV. Theories for moderately swollen networks, J. Appl. Polym. Sci. 30 (1985) 2589–2606. [14] R.S. Harland, N.A. Peppas, Solute diffusion in swollen networks. VII. Diffusion in semicrystalline networks, Colloid Polym. Sci. 267 (1989) 218–225. [15] W.D. Rhine, V. Sukhatme, D.S.T. Hseih, R. Langer, A new approach to achieve zero-order release kinetics from diffusion-controlled polymer matrix systems, in: R. Baker (Ed.), Controlled Release of Bioactive Materials, Academic Press, New York, NY, 1980, pp. 177–187. [16] T. Higuchi, Rate of release of medicaments from ointment bases containing drugs in suspensions, J. Pharm. Sci. 50 (1961) 874–875. [17] T. Higuchi, Mechanisms of sustained action mediation. Theoretical analysis of rate of release of solid drugs dispersed in solid matrices, J. Pharm. Sci. 52 (1963) 1145–1149. [18] S.J. Desai, A.P. Simonelli, W.I. Higuchi, Investigation of factors influencing release of solid drug dispersed in inert drug delivery devices matrices, J. Pharm. Sci. 54 (1965) 1459–1464. [19] S.J. Desai, P. Singh, A.P. Simonelli, W.I. Higuchi, Investigation of factors influencing release of solid drug dispersed in inert matrices II, J. Pharm. Sci. 55 (1966) 1224–1229. [20] H. Lapidus, N.G. Lordi, Some factors affecting the release of a water-soluble drug from a compressed hydrophilic matrix, J. Pharm. Sci. 55 (1966) 840–843. [21] H. Lapidus, N.G. Lordi, Drug release from compressed hydrophilic matrices, J. Pharm. Sci. 57 (1968) 1292–1301. [22] J. Siepmann, N.A. Peppas, Higuchi equation: derivation, applications, use and misuse, Int. J. Pharm. 418 (2011) 6–11. [23] H.S. Carslaw, J.C. Jaeger (Eds.), Conduction of Heat in Solids, Clarendon Press, Oxford, 1959. [24] R.W. Baker, H.K. Lonsdale, Controlled release: mechanisms and rates, in: A.C. Tanquary, R.E. Lacey (Eds.), Controlled Release of Biologically Active Agents, Plenum Press, New York, 1974, pp. 15–72. [25] J.H. Petropoulos, K.G. Papadokostaki, M. Sanopoulou, Higuchi's equation and beyond: overview of the formulation and application of a generalized model of drug release from polymeric matrices, Int. J. Pharm. 437 (2012) 178–191. [26] P.L. Ritger, N.A. Peppas, A simple equation for description of solute release. 1. Fickian and non-Fickian release from non-swellable devices in the form of slabs, spheres, cylinders or discs, J. Control. Release 5 (1987) 23–36. [27] V. Papadopoulou, K. Kosmidis, M. Vlachou, P. Macheras, On the use of the Weibull function for the discernment of drug release mechanisms, Int. J. Pharm. 309 (2006) 44–50. [28] K. Kosmidis, P. Argyrakis, P. Macheras, A reappraisal of drug release laws using Monte Carlo simulations: the prevalence of the Weibull function, Pharm. Res. 20 (2003) 988–995. [29] K. Kosmidis, P. Argyrakis, P. Macheras, Fractal kinetics in drug release from finite fractal matrices, J. Chem. Phys. 119 (2003) 6373–6377. [30] C.K. Sackett, B. Narasimhan, Mathematical models for polymer erosion: consequences for drug delivery, Int. J. Pharm. 418 (2011) 104–114. [31] A.G. Thombre, K.J. Himmelstein, Modeling of drug release kinetics from a laminated device having an erodible drug reservoir, Biomaterials 5 (1984) 250–254. [32] A.G. Thombre, K.J. Himmelstein, A simultaneous transport-reaction model for controlled drug delivery from catalyzed bioerodible polymer matrices, AIChE J. 31 (1985) 759–766. [33] H. Antheunis, J.C. van der Meer, M. de Geus, A. Heise, C.E. Koning, Autocatalytic equation describing the change in molecular weight during hydrolytic degradation of aliphatic polyesters, Biomacromolecules 11 (2010) 1118–1124. [34] H. Antheunis, J.C. van der Meer, M. de Geus, W. Kingma, C.E. Koning, Improved mathematical model for the hydrolytic degradation of aliphatic polyesters, Macromolecules 42 (2009) 2462–2471. [35] S. Prabhu, S. Hossainy, Modeling of degradation and drug release from a biodegradable stent coating, J. Biomed. Mater. Res. Part A 80A (2007) 732–741. [36] X.X. Han, J.Z. Pan, A model for simultaneous crystallisation and biodegradation of biodegradable polymers, Biomaterials 30 (2009) 423–430. [37] Y. Wang, J.Z. Pan, X.X. Han, C. Sinka, L.F. Ding, A phenomenological model for the degradation of biodegradable polymers, Biomaterials 29 (2008) 3393–3401. [38] D. Larobina, G. Mensitieri, M.J. Kipper, B. Narasimhan, Mechanistic understanding of degradation in bioerodible polymers for drug delivery, AIChE J. 48 (2002) 2960–2970. [39] M.J. Kipper, B. Narasimhan, Molecular description of erosion phenomena in biodegradable polymers, Macromolecules 38 (2005) 1989–1999. [40] P.I. Lee, Diffusional release of a solute from a polymeric matrix — approximate analytical solutions, J. Membr. Sci. 7 (1980) 255–275. [41] A. Charlier, B. Leclerc, G. Couarraze, Release of mifepristone from biodegradable matrices: experimental and theoretical evaluations, Int. J. Pharm. 200 (2000) 115–120. [42] R.P. Batycky, J. Hanes, R. Langer, D.A. Edwards, A theoretical model of erosion and macromolecular drug release from biodegrading microspheres, J. Pharm. Sci. 86 (1997) 1464–1477. [43] P. Arosio, V. Busini, G. Perale, D. Moscatelli, M. Masi, A new model of resorbable device degradation and drug release — part I: zero order model, Polym. Int. 57 (2008) 912–920. [44] G. Perale, P. Arosio, D. Moscatelli, V. Barri, M. Muller, S. Maccagnan, M. Masi, A new model of resorbable device degradation and drug release: transient 1-dimension diffusional model, J. Control. Release 136 (2009) 196–205. [45] M.P. Zhang, Z.C. Yang, L.L. Chow, C.H. Wang, Simulation of drug release from biodegradable polymeric microspheres with bulk and surface erosions, J. Pharm. Sci. 92 (2003) 2040–2056.

[46] J.S. Soares, P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks, Biomaterials 31 (2010) 3032–3042. [47] F. von Burkersroda, L. Schedl, A. Gopferich, Why degradable polymers undergo surface erosion or bulk erosion, Biomaterials 23 (2002) 4221–4231. [48] S.N. Rothstein, W.J. Federspiel, S.R. Little, A unified mathematical model for the prediction of controlled release from surface and bulk eroding polymer matrices, Biomaterials 30 (2009) 1657–1664. [49] A. Gopferich, R. Langer, Modeling of polymer erosion, Macromolecules 26 (1993) 4105–4112. [50] A. Gopferich, R. Langer, Modeling monomer release from bioerodible polymers, J. Control. Release 33 (1995) 55–69. [51] A. Gopferich, Polymer bulk erosion, Macromolecules 30 (1997) 2598–2604. [52] K. Zygourakis, Development and temporal evolution of erosion fronts in bioerodible controlled release devices, Chem. Eng. Sci. 45 (1990) 2359–2366. [53] K. Zygourakis, P.A. Markenscoff, Computer-aided design of bioerodible devices with optimal release characteristics: a cellular automata approach, Biomaterials 17 (1996) 125–135. [54] J. Siepmann, N. Faisant, J.P. Benoit, A new mathematical model quantifying drug release from bioerodible microparticles using Monte Carlo simulations, Pharm. Res. 19 (2002) 1885–1893. [55] N. Faisant, J. Siepmann, J. Richard, J.P. Benoit, Mathematical modeling of drug release from bioerodible microparticles: effect of gamma-irradiation, Eur. J. Pharm. Biopharm. 56 (2003) 271–279. [56] N. Bertrand, G. Leclair, P. Hildgen, Modeling drug release from bioerodible microspheres using a cellular automaton, Int. J. Pharm. 343 (2007) 196–207. [57] R.X. Yu, H.L. Chen, T.N. Chen, X.Y. Zhou, Modeling and simulation of drug release from multi-layer biodegradable polymer microstructure in three dimensions, Simul. Model. Pract. Theory 16 (2008) 15–25. [58] H.B. Hopfenberg, Controlled release from bioerodible slabs, cylinders, and spheres, ACS Symp. Ser, 1976, pp. 26–32. [59] L.L. Lao, S.S. Venkatraman, N.A. Peppas, Modeling of drug release from biodegradable polymer blends, Eur. J. Pharm. Biopharm. 70 (2008) 796–803. [60] X. Li, L.K. Petersen, S.R. Broderick, B. Narasimhan, K. Rajan, Identifying factors controlling protein release from combinatorial biomaterial libraries via hybrid data mining methods, ACS Comb. Chem. 13 (1) (2011) 50–58. [61] O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochem. Biophys. Acta 27 (1958) 229–246. [62] D.O. Kuethe, D.C. Augenstein, J.D. Gresser, D.L. Wise, Design of capsules that burst at predetermined times by dialysis, J. Control. Release 18 (1992) 159–164. [63] N.A. Peppas, R. Korsmeyer, Polymers for sustained release of macromolecules, Polym. News 6 (1980) 149–155. [64] J.C. Wu, N.A. Peppas, Modeling of penetrant diffusion in glassy polymers with an integral sorption Deborah number, J. Polym. Sci. Polym. Phys. Ed. 31 (1993) 1503–1518. [65] N.A. Peppas, J.C. Wu, E.D. von Meerwall, Mathematical modeling and experimental characterization of polymer dissolution, Macromolecules 27 (1994) 5626–5638. [66] J.M. Vrentas, C.M. Jarzebski, J.L. Duda, A Deborah number for diffusion in polymersolvent systems, AIChE J. 21 (1975) 894–900. [67] N.A. Peppas, N.M. Franson, The swelling interface number as a criterion for prediction of diffusional solute release mechanisms in swellable polymers, J. Polym. Sci. Polym. Phys. Ed. 21 (1983) 983–997. [68] P. Colombo, P.L. Catellani, N.A. Peppas, L. Maggi, U. Conte, Swelling characteristics of hydrophilic matrices for controlled release: new dimensionless number to describe the swelling and release behavior, Int. J. Pharm. 88 (1992) 99–109. [69] P.I. Lee, Effect of non-uniform initial drug concentration distribution on the kinetics of drug release from glassy hydrogel matrices, Polymer 25 (1984) 973–978. [70] R.S. Harland, C. Dubernet, J.P. Benoit, N.A. Peppas, A model of dissolutioncontrolled, diffusional drug release from non-swellable polymeric microspheres, J. Control. Release 7 (1988) 207–215. [71] P.I. Lee, Initial concentration distribution as a mechanism for regulating drug release from diffusion-controlled and surface erosion-controlled matrix systems, J. Control. Release 4 (1986) 1–11. [72] R. Bodmeier, O. Paeratakul, Drug release from laminated polymeric films prepared from aqueous latexes, J. Pharm. Sci. 79 (1990) 32–36. [73] U. Conte, L. Maggi, P. Colombo, A. La Manna, Multi-layered hydrophilic matrices as constant release devices (Geomatrix® systems), J. Control. Release 26 (1993) 39–47. [74] R.A. Fassihi, W.A. Ritschel, Multiple-layer, direct-compression, controlled-release system: in vitro and in vivo evaluation, J. Pharm. Sci. 82 (1993) 750–754. [75] S. Kiil, K. Dam-Johansen, Controlled drug delivery from swellable hydroxypropylmethylcellulose matrices: model-based analysis of observed radial front movements, J. Control. Release 90 (2003) 1–21. [76] J. Siepmann, K. Podual, M. Sriwongjanya, N.A. Peppas, R. Bodmeier, A new model describing the swelling and drug release kinetics from hydroxypropylmethylcellulose tablets, J. Pharm. Sci. 88 (1999) 65–72. [77] P.I. Lee, Interpretation of drug release kinetics from hydrogel matrices in terms of time-dependent diffusion coefficients, in: P.I. Lee, W. Good (Eds.), Controlled Release Technology, ACS Symposium SeriesAmerican Chemical Society, Washington, DC, 1987, pp. 71–83. [78] J.H. Kou, D. Fleisher, G.L. Amidon, Modeling drug release from dynamically swelling poly(hydroxyethyl methacrylate-co-methacrylic acid) hydrogels, J. Control. Release 12 (1990) 241–250. [79] D.S. Cohen, T. Erneux, Controlled drug release asymptotics, SIAM J. Appl. Math. 58 (4) (1998) 1193–1204.


N.A. Peppas, B. Narasimhan / Journal of Controlled Release xxx (2014) xxx–xxx [80] R.W. Korsmeyer, S.R. Lustig, N.A. Peppas, Solute and penetrant diffusion in swellable polymers. 1. Mathematical modeling, J. Polym. Sci. Polym. Phys. Ed. 24 (1986) 395–408. [81] S.R. Lustig, N.A. Peppas, Solute and penetrant diffusion in swellable polymers, J. Appl. Polym. Sci. 33 (1987) 533–549. [82] S.R. Lustig, J.M. Caruthers, N.A. Peppas, Continuum thermodynamics and transport theory for polymer-fluid mixtures, Chem. Eng. Sci. 47 (1992) 3037–3057. [83] N.A. Peppas, L. Brannon-Peppas, Hydrogels at critical conditions. 1. Thermodynamics and swelling behavior, J. Membr. Sci. 48 (1990) 281–290. [84] L. Brannon-Peppas, N.A. Peppas, Time-dependent response of ionic polymer networks to pH and ionic strength changes, Int. J. Pharm. 70 (1991) 53–57. [85] C.L. Bell, N.A. Peppas, Complexation in poly(ethylene glycol-methacrylic acid) hydrogels, Polym. Prepr. 34 (1) (1993) 831–832. [86] D. Hariharan, N.A. Peppas, Swelling of ionic and neutral polymer networks in ionic solutions, J. Membr. Sci. 78 (1993) 1–12. [87] D. Hariharan, N.A. Peppas, Modeling of water transport and solute release in physiologically sensitive gels, J. Control. Release 23 (1993) 123–136. [88] B. Narasimhan, N.A. Peppas, The physics of polymer dissolution: modeling approaches and experimental behavior, Adv. Polym. Sci. 128 (1997) 157–207. [89] R.S. Harland, A. Gazzaniga, M.E. Sangani, P. Colombo, N.A. Peppas, Drug/polymer matrix swelling and dissolution, Pharm. Res. 5 (1988) 488–494. [90] R.T.C. Ju, P.R. Nixon, M.V. Patel, Drug release from hydrophilic matrices. 1. New scaling laws for predicting polymer and drug release based on the polymer disentanglement concentration and the diffusion layer, J. Pharm. Sci. 84 (1995) 1455–1463. [91] R.T.C. Ju, P.R. Nixon, M.V. Patel, D.M. Tong, Drug release from hydrophilic matrices. 2. A mathematical model based on the polymer disentanglement concentration and the diffusion layer, J. Pharm. Sci. 84 (1995) 1464–1477. [92] B. Narasimhan, N.A. Peppas, Molecular analysis of drug delivery systems controlled by dissolution of the polymer carrier, J. Pharm. Sci. 86 (3) (1997) 297–304.

7

[93] P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. [94] B. Narasimhan, N.A. Peppas, Disentanglement and reptation during dissolution of rubbery polymers, J. Polym. Sci. Polym. Phys. Ed. 34 (1996) 947–961. [95] B. Narasimhan, N.A. Peppas, On the importance of chain reptation in models of dissolution of glassy polymers, Macromolecules 29 (1996) 3283–3291. [96] J. Siepmann, H. Kranz, R. Bodmeier, N.A. Peppas, HPMC-matrices for controlled drug delivery: a new model combining diffusion, swelling and dissolution mechanisms and predicting the release kinetics, Pharm. Res. 16 (1999) 1748–1756. [97] J. Siepmann, N.A. Peppas, Hydrophilic matrices for controlled drug delivery: an improved mathematical model to predict the resulting drug release kinetics (the “sequential layer” model), Pharm. Res. 17 (2000) 1290–1298. [98] J. Siepmann, A. Streubel, N.A. Peppas, Understanding and predicting drug delivery from hydrophilic matrix tablets using the “sequential layer” model, Pharm. Res. 19 (2002) 306–314. [99] N. Wu, L.S. Wang, D.C.-W. Tan, S.M. Moochhala, Y.Y. Yang, Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: polyethylene oxide with high molecular weights, J. Control. Release 102 (2005) 569–581. [100] P. Borgquist, A. Körner, L. Piculell, A. Larsson, A. Axelsson, A model for the drug release from a polymer matrix tablet—effects of swelling and dissolution, J. Control. Release 113 (2006) 216–225. [101] S.K. Mallapragada, N.A. Peppas, Mechanism of dissolution of semicrystalline poly(vinyl alcohol) in water, J. Polym. Sci. Polym. Phys. Ed. 34 (1996) 1339–1346. [102] S.K. Mallapragada, N.A. Peppas, Crystal unfolding and chain disentanglement during semicrystalline polymer dissolution, AIChE J. 43 (1997) 870–876. [103] S.K. Mallapragada, N.A. Peppas, Crystal dissolution-controlled release systems: I. Physical characteristics and modeling analysis, J. Control. Release 45 (1997) 87–94.


Drug delivery systems 5A. Oral drug delivery.

Drug delivery systems. 6. Transdermal drug delivery.

Radiation sterilization of new drug delivery systems.

Mathematical modeling analysis of intratumoral disposition of anticancer agents and drug delivery systems.

Implantable drug-delivery systems.

Molecular Communication Modeling of Antibody-Mediated Drug Delivery Systems.

Lipid-Based Drug Delivery Systems.

ATP-Responsive Drug Delivery Systems.

MEMS: Enabled Drug Delivery Systems.

Kidney-targeted drug delivery systems.

pH-responsive drug-delivery systems.

Implantable neurosurgical drug delivery systems.

Polycaprolactone thin-film drug delivery systems: Empirical and predictive models for device design.

Marine Origin Polysaccharides in Drug Delivery Systems.

Biomedical Imaging in Implantable Drug Delivery Systems.

Mucoadhesive systems in oral drug delivery.

In situ gelling systems for drug delivery.

Ocular drug delivery systems: An overview.

Supramolecular hydrogels as drug delivery systems.

Pulmonary drug delivery systems for tuberculosis treatment.

Drug delivery from gelatin-based systems.

Raman imaging of drug delivery systems.

Drug delivery systems versus dosage forms.

Advanced drug delivery systems for therapeutic applications.