Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Porous systems are ubiquitous in natural world and in technology. Examples from nature include volcanic rocks such as pumice, oil sediments, soils, dry wood, leaves, bone, and the glomeruli of the kidney. The cell’s plasma membrane is studded with channel proteins that open and shut to admit certain ions and exclude others, and its nuclear membrane contains a lattice of pores that regulate traffic of macromoleculles between the nucleus and the cytoplasm. In the industrial world, porous systems are used for filtration, dialysis, reverse osmosis, adsorption, catalysis, flow control, mixing, and other applications where a large surface area/volume ratio is required. Paper is a porous medium that is designed to absorb and retain inks or graphite particles. Solid foams, cardboards, and aerogels are porous systems that are useful due to their mechanical and acoustic properties. Porosity may be an intentional property or it may be a side effect of manufacture, as in compacted powders.

Porosity plays a major role in many controlled release systems [1]. A typical controlled release system consists of a solid substrate, or matrix, into which drug is incorporated. Often, the matrix material is impermeable to the drug, and release occurs through a system of water filled pores. Pores are formed either as a result of phase separation or spinodal decomposition during preparation, by dissolution and leaching of pore forming soluble excipients, or by the drug itself which, upon release, leaves behind a void space. In recent years, advances in lithographic, electrochemical, and block polymer self assembly have produced highly reproducible, regular porous structures that can be used to precisely control release rate.

In this chapter, we first present examples of porous media relevant to controlled release. We then briefly review techniques for characterizing pore size distributions and pore structure. Finally, we discuss some mathematical approaches used to predict diffusional release from porous media.

There is no strict definition of a porous medium, and we prefer to work with the idea that the medium has one or more components that admit drug diffusion and others that serve as barriers to diffusion. In this respect, many of the concepts developed in this chapter can also be applied to other heterogeneous media, such as the stratum corneum of skin or the extracellular interstitial medium between living cells.

2 Example of Porous Media Relevant to Controlled Release

A very simple example of a porous system is the common laboratory filter membrane. Such membranes are composed of interwoven polymer fibers with gaps or pores between the fibers that permit passage of water and small molecules, but reject larger molecules. Placing a drug solution on one side of the membrane and a receiving medium on the other side, drug diffuses through the membrane at a rate that depends on the relative size of the drug molecule compared to a typical distance between the fibers. The fiber network is usually sufficiently dense to suppress convection, allowing diffusion control of drug transport.

Traditional filters and dialysis membranes are characterized by random porosity. Consequently, their selectivity by molecular size is not absolute, and size cutoff specifications are fuzzy. In fact, cutoffs are typically specified in terms of molecular weight, not diameter. For several decades, radiation track etched polycarbonate (Nuclepore®) membranes were regarded as the best alternative, since the etched micron sized pores were relatively monodisperse [2, 3]. However, these membranes, besides having pores that are too large to provide size selectivity, are characterized by randomly positioned pores, and their porosity (volume or area fraction of pores in the membrane) must be kept low in order to avoid pore overlap.

Since the 1990s, there have been major advances in fabrication of micro- and nanoporous arrays with high pore density. Solid state microporous arrays can be formed by a number of traditional chemical and plasma etching, and electron beam techniques, many of which were originally developed for the microelectronics industry but are now also used in microelectromechanical systems (MEMS) and microfluidics [4]. For example, a microporous array can be fabricated by placing a thin silicon wafer under a patterned mask that blocks reactive ions, except in an array of gaps introduced into the mask. The silicon under the gaps is etched away upon exposure to plasma, leaving behind an array of pores. Alternatively, an array of microposts can be formed by a mask/etch process. A microporous sheet can then be formed by pouring a thin layer of polymer, such as poly(dimethylsiloxane) (PDMS), to a level below the tips of the posts, curing the PDMS, and peeling off the cured sheet, which now contains a pore array. Under proper conditions, the silicon post array acts as a “master” from which identical microporous membranes can be cast repetitively.

Anopore® membranes are 60 μm thick aluminum oxide sheets containing a dense honeycomb array of near-regular cylindrical pores whose diameters can be as small as 20 nm. These regular structures are manufactured electrochemically under strong electric fields.

It has long been known that block polymers self-assemble into regular arrays whose structure and periodicity depend on the mutual compatibility of the polymer blocks and their lengths. Lamellar, hexagonal, cubic, and gyroid morphologies have been predicted and demonstrated in 3D block polymer materials [5]. When solutions of block polymers are spun onto a surface and the solvent is evaporated, they self-assemble into thin films that also exhibit 2D spatial periodicity. In one example [6], a polymer solution consisting of long polystyrene (PS) and short poly(lactic acid) (PLA) blocks, separated by an even shorter polyisoprene (PI) block, is spun onto a thin wafer. Following solvent evaporation these block polymers form hexagonal arrays, with PLA cylinders, lined by PI, dispersed in a PS continuum. Upon exposure to strong base the PLA cylinders are etched away, leaving behind an ultrathin (~100 nm thick), nanoporous (~40 nm diameter) array on top of the wafer. Using a combination of chemical and plasma etching techniques, an array of micropores is introduced into the underlying wafer. The result is an asymmetric membrane with a nanoporous carpet lying on top of a microporous substrate. The latter provides mechanical support to the former. If the nanopore diameters can be further reduced, then selectivity based on size is possible in this system. It is interesting to note that nuclear pores in the eukaryotic cell are of comparable diameters to the nanopores in this block polymer-based system.

In the past two decades, there has been extensive research into mesoporous silica nanoparticles (MSNs) [7, 8]. These nanoparticles, of diameter ~100 nm, are formed by condensation of silica around arrays of cylindrical micelle templates, followed by removal of the template. These structured particles contain arrays of rigid, parallel, cylindrical pores of diameter ~2–4 nm that extend from one end to the other. Pore diameter can be further reduced, if desired, by functionalizing the silica pore walls. Drug can be rapidly loaded into the pores by diffusion, and its partitioning into the MSN can be encouraged by functionalizing the pore walls with moieties that favor drug association. MSNs coated with lipid bilayer membranes, which hold hydrophilic drugs inside but release the drug when the membrane is destabilized (e.g., due to lowering of pH in an endosome), are under investigation [9].

In the forgoing examples, pores are of cylindrical shape. As is discussed below, a critical parameter affecting diffusion through narrow pores is the ratio of molecular diameter to pore diameter. The same holds true for narrow slit-like pores, which have been studied thoroughly in the past decade. In one example, very thin (~10 nm), sacrificial oxide layers are grown on micron sized walls of cavities etched into silicon wafers. The coated cavities are then backfilled with polysilicon, and the oxide is removed, leaving behind nanoscale, slit-like gaps that serve as channels for diffusion of drug [10].

We now turn to more traditional porous polymeric systems which, though more heterogeneous in their pore structure, are much cheaper to produce on a mass scale. A simple general procedure is to mix a polymer with an additive, which might be a gas, liquid, or solid under conditions, e.g., temperature or vapor pressure, where the components are compatible. If external conditions are changed so that the polymer and additive become incompatible, then the system will phase separate into polymer rich and additive rich domains. Upon removal of the additive, either by evaporation or liquid leaching, the domain structure becomes a randomly porous structure [11]. The resulting pore structure depends on whether coagulation of the polymer occurs by a nucleation/growth mechanism or by spinodal decomposition, and random, periodic, and cellular morphologies are possible, depending on processing conditions.

Another method is to prepare a multiphase mixture with interfaces stabilized by surfactants. By removing of one of the phases and “hardening” the other, a porous medium results. For example [12], drug-loaded porous microspheres can be formed by dissolving the drug in the internal aqueous phase of a water-in-oil-in-water (w/o/w) emulsion, in which the “oil” phase consists of droplets of an organic polymer solution and even smaller droplets of an internal aqueous phase, all suspended in a continuous aqueous medium. The phases are stabilized by emulsifying agents or surfactants. Under vigorous stirring, the organic solvent is removed by evaporation, and the resulting microspheres are then removed from the continuous phase, followed by drying of the internal aqueous phase, which leaves the drug behind in pores. This process can be adapted to spray systems, in which the evaporation steps are fast.

More direct methods for producing porosity are to blow a foam in a polymer solution or melt, followed by hardening of the polymer around the air bubbles or by suspending pore-forming agents, such as salts or incompatible polymers, in the initial polymeric preparation, followed by liquid leaching. Solid particles of the drug itself may constitute the pore-forming agent. Again, these procedures may be amenable to spray processing.

3 Characterization of Porous Materials

The definition of a pore is ambiguous, since it refers to void space within solid material, which also cannot be defined rigorously. The inferred porosity of a material and details regarding pore structure depend on the methods used to probe these properties. Furthermore, different probes provide different information about a porous medium [13].

Perhaps the most straightforward way to estimate void volume in a dry porous solid is to measure the amount of helium (He) that is introduced into it at a specified pressure. Since He is an inert gas, it interacts minimally with the solid component, and the pore volume can be calculated using either the ideal gas law or, more accurately, the van der Waals equation of state for He. This measure of porosity accounts for all pores except those whose diameters are less than that of a helium atom.

If instead of helium nitrogen gas (N2) is used, an estimate of the internal surface area of the porous media can be made. As a highly polarizable molecule, N2 adsorbs readily to most surfaces, so the first introduction of N2 coats all pore walls, except those that are not accessible to the gas because they are surrounded by nitrogen impermeable material. Using Brunauer–Emmet–Teller (BET) analysis, it is possible to ascertain both the internal surface area of a porous solid and the affinity of the solid for N2.

While He and N2 absorption isotherms are useful for determining pore volume and internal surface area of a porous solid, they cannot generally be used to determine pore sizes. Traditionally, mercury (Hg) intrusion measurements have been used for this purpose. As a liquid, Hg possesses a surface tension with air, γ, and a contact angle at the Hg/air/solid interface, θ. According to the Washburn equation, the pressure (excess of atmospheric pressure) required to drive Hg through a pore of diameter R p is given by

$$ \Delta P = 4\gamma \cos \theta /{R_{\rm{p}}}. $$
(9.1)

By plotting pressure versus the amount of Hg introduced, the distribution of pore diameters is determined. The Washburn equation predicts that large pores are filled before small pores. Unfortunately, this procedure cannot be accurate in general, since some large pores may be initially inaccessible, and can only be reached after initial penetration of surrounding smaller pores [1, 14, 15]. Moreover, the Washburn equation does not account for compressibility of the porous medium [16].

Various imaging and microscopy techniques can be used to characterize pore structure, including serial section microtomy, optical laser scanning confocal microscopy (LSCM), confocal Raman microscopy (CRM), scanning electron microscopy (SEM), transmission electron microscopy (TEM), and magnetic resonance imaging (MRI). Atomic force microscopy (AFM) can also be used to probe the surface terrain of a material and ascertain the location and sometimes the depth of nanopores. These techniques, together with present day data storage and computational capabilities, make it possible to reconstruct many of the details of pore structure, going beyond simple measures such as porosity and specific surface area. For example, many porous systems contain large pore bodies connected by relatively narrow throats. As is discussed below, these details are often crucial in determining transport processes in porous media.

4 Mathematical Models

The variety of pore morphologies and arrangements or topologies that are possible in porous media is such that no single mathematical description or model of diffusion inside the medium covers all possibilities. Hence, it is necessary to have at least an approximate idea as to how a medium is structured before modeling can be pursued. As indicated above, it is conceivable that a porous medium’s structure can be characterized with great accuracy by imaging/reconstruction algorithms, and this structure can be used with powerful and computationally intensive (e.g., finite element modeling) software packages to make predictions of release behavior. However, such a procedure may provide little insight into factors governing diffusional release. Here we discuss some relatively simple models.

4.1 Tubular Pores

The simplest model of a porous medium is a planar membrane containing a collection of circular cylindrical tubes passing from one face of a membrane, of thickness L, to the other, with longitudinal axes perpendicular to the membrane surfaces [2, 17–19]. Figure 9.1a is a rendering of such a membrane. Let A p be the area of a single pore. For a circular pore with radius R p, \( {A_{\rm{p}}} = \pi R_{\rm{p}}^{{2}} \). If there are, on average, n such pores per unit area of membrane, then the porosity, ε, of the membrane is given by \( \varepsilon = n{A_{\rm{p}}} = n\pi R_{\rm{p}}^2 \). Not all of the pore space is equally available, however, and we must be concerned with the freedom of a molecule to place itself fully inside the pore, considering steric or other interactions with the pore wall.

Fig. 9.1
figure 1_9

(a) Schematic of simple, straight, cylindrical pores crossing a membrane, with spherical molecules inside. Pore length and radius are L and R p, respectively, and diffusing molecule is of radius a centered at distance r from centerline (dashed) of pore. (b) Plot of normalized quantities ε s/ε (curve 1), D eff/D 0 (curve 2), and their product (ε s/ε)(D eff/D 0) (curve 3) as a function of ratio of molecular radius to pore radius (a/R p)

Full size image

For pores whose radii are in the nanometer scale, a critical parameter determining accessibility of the pore to a solute molecule is the ratio of the molecule’s radius, a, to the pore radius, \( \lambda = a{/}{R_{\rm{p}}} \). Since the center of the molecule cannot come any closer to the wall than a single molecular radius, the available porosity is

$$ {\varepsilon_{\rm{s}}} = n\pi {\left( {{R_{\rm{p}}} - a} \right)^2} = n\pi {a^2}{(1 - \lambda )^2} = \varepsilon {(1 - \lambda )^2}. $$
(9.2)

For drugs that are much smaller than the pore diameter (\( \lambda \ll 1 \)), the factor \( {(1 - \lambda )^2} \) is not significant, but it can be important for large molecules or narrow pores.

The parameter \( {\varepsilon_{\rm{s}}} \) is analogous to the partition coefficient, K, discussed in Chap. 6, and the two quantities are equal, provided drug is allowed only in the pores and all pore space is accessible to solute. As calculated above, \( {\varepsilon_{\rm{s}}} \) is affected only by steric interaction between drug molecule and pore wall. In fact, there are other interactions, such as van der Waals, hydrophobic, and dielectric and electrical forces, the latter being particularly important when both pore wall and drug are charged. Charged pore walls attract oppositely charged drug molecules and repel similarly charged molecules. The forces between pore wall and drug are attenuated by thermal excitation, and electrical forces on a charged solute are partially screened by neighboring ions of opposite charge.

Again assuming circular symmetry of the pore, these forces can be represented by a potential energy of interaction, U, between the molecule and the pore wall, which depends on the radial position, r, of the molecule’s center from the centerline of the pore, with \( 0\, <\, r \,<\, {R_{\rm{p}}}(1 - \lambda ) \). The “partition coefficient” of drug in the membrane is then given, according to statistical thermodynamics, by

$$ K = \frac{{2\varepsilon }}{{R_{\rm{p}}^2}}\int_0^{{{R_{\rm{p}}}(1 - \lambda )}} {r{{\hbox{e}}^{{ - U(r)/{k_{\rm{B}}}T}}}} {\hbox{d}}r, $$
(9.3)

which reduces to the earlier expression when there are no forces (U = 0). In this expression, k B is Boltzmann’s constant and T is temperature (°K). We shall not indulge in detailed calculations here, but note in the electrostatic case that positive U (similarly charged wall and drug) leads to decreased K while negative U (opposite charges) leads to increased K. The electrostatic range of influence of the wall on the drug molecule in aqueous solution is characterized, roughly, by the Debye length, \( {\ell_{\rm{D}}} = \sqrt {{RT{\varepsilon_{\rm{w}}}{\varepsilon_0}/1,000{F^2}(2I)}} \), where R is the gas constant, F is Faraday’s constant, \( {\varepsilon_0} \) is the dielectric permittivity of vacuum, \( {\varepsilon_{\rm{w}}} \) is the dielectric constant of water (~80) and I is the ionic strength of the solution. Under physiological conditions, I = 155 mM and \( {\ell_{\rm{D}}} \approx {8}\, \)Å, so electrostatic effects are confined to within a few nanometers of the pore wall.

The partitioning properties of nonspherical molecules into pores are also of interest. An extreme but illuminating case is a long, thin rod-like molecule that does not interact with the wall in any way other than sterically. Assume that the rod’s radius is much smaller than that of the pore, but that its length is larger than the pore diameter. The rod can fit easily into the pore by orienting itself closely parallel to the pore’s longitudinal axis, and therefore would seemingly be able to partition easily. However, this orientation is very particular. Outside the membrane, the rod is free to orient in any direction; hence, there is a high entropy cost associated with entering the pore. Similarly, linear polymer molecules that assume a random coil configuration whose radius of gyration is comparable to or larger than the pore radius could seemingly uncoil and form a “straight line” in a narrow pore, but doing so would come with considerable cost in conformational entropy. In both cases, the partition coefficient is greatly reduced [20, 21].

Finally, reversible or irreversible adsorption of molecules, such as proteins, may reduce \( {\varepsilon_{\rm{s}}} \) or K, especially when pore walls are hydrophobic. Globular proteins typically consist of a core containing hydrophobic residues surrounded by a surface containing polar residues, many of which are charged. Upon encountering a hydrophobic surface, the protein rearranges or unfolds such that its core residues adsorb onto the surface. The polar/charged residues project into the pore lumen, introducing extra steric and ionic interactions to other diffusing molecules, especially other proteins.

We now turn to wall effects on the diffusion coefficient. In dilute solution, the diffusion constant is given by the Stokes–Einstein relation, \( {D_0} = {k_{\rm{B}}}T/6\pi a\eta \), where \( \eta \) is the solvent viscosity. This equation is derived by balancing the thermal “force,” \( {k_{\rm{B}}}T \), against the viscous drag, \( 6\pi a\eta \), presented by the medium. Drag is due to a solvent fluid shear profile that extends away from the molecule, from the molecule’s surface to infinity. While the shear field decays away from the surface, it remains significant over a considerable distance. The presence of a rigid wall close to the moving molecule constrains the shear field, and provides extra resistance to the molecule’s motion. Calculations of this effect are complicated, and exact results are not available [22]. For spherical molecules that do not interact other than sterically with the wall, a useful expression due to Faxén [23],

$$ {D_{\rm{eff}}}{/}{D_0} = 1 - 2.104\,\lambda + 2.09\,{\lambda^3} - 0.95\,{\lambda^5}, $$
(9.4)

accounts reasonably well for cylindrical wall drag, provided \( \lambda\; <\;0.4 \). Here, \( {D_{\rm{eff}}} \) is the so called effective diffusion constant, which can be substituted for D in any of the expressions modeling drug release by diffusion given in Chap. 6. Similarly, the expression

$$ {\varepsilon_{\rm{s}}}{D_{\rm{eff}}} = {(1 - \lambda )^2}(1 - 2.104\,\lambda + 2.09\,{\lambda^3} - 0.95\,{\lambda^5})\varepsilon {D_0} $$
(9.5)

would replace the product KD in that chapter. Expressions for \( {\varepsilon_{\rm{s}}}{/}\varepsilon \), \( {D_{\rm{eff}}}{/}{D_0} \), and \( {\varepsilon_{\rm{s}}}{D_{\rm{eff}}}{/}\varepsilon {D_0} \) as a function λ, which reflect the effects the cylindrical pore wall, are plotted in Fig. 9.1b.

The Faxén expression was derived by considering drag effects on spherical molecules positioned at the centerline of the pore. It should be modified for nonspherical molecules or when there are nonsteric interactions between the pore wall and the solute, but precise calculations are difficult.

The discussion so far applies, strictly speaking, to dilute solutions, where solute molecules are rarely close enough to each other to interact. Many controlled release systems, however, are concentrated. While concentration affects diffusion and partition coefficients in general, the effects tend to be of extra significance in systems with nanoscale pores, where close proximity of pore walls may augment energetic and hydrodynamic interactions between solute molecules [24]. The normal formulation of Fick’s laws of diffusion, which assumes that solute molecule executes independent random walks, breaks down. In the limit where λ > 1/2, solute molecules must move single file through the pores. At high concentrations, entry of drug into cylindrical pores becomes difficult since the pores are already occupied by other molecules, and the rate of drug permeation tends to saturate.

It is interesting to note that saturation of transport at high solute concentrations has been also observed in microfabricated 2D “slit pores” (see above) when the slit width is of comparable dimension to that of a large solute (e.g. a protein), but the lateral dimension is much larger. In this case, single file diffusion cannot explain the behavior, but one can imagine that the energetic and viscous interactions between highly concentrated solute molecules can lead to highly correlated motions resembling single file diffusion [25–27].

4.2 Tortuous Pathways

Thus far, we have only considered the effects of pore width on partitioning and diffusion of drug. Another potential factor is pore length. In Fig. 9.1a, pores were drawn to be perpendicular to the membrane surfaces, with lengths the same as the membrane thickness, denoted by L in Chap. 6. In Fig. 9.2a, pores are still depicted as tubes, but they do not connect the two faces of the membrane by a perpendicular path. In this case, the path length is increased. We shall denote this increase in length by a “tortuosity factor,” τ, such that the “effective thickness” of the membrane becomes τL. This product can replace L or R in the equations for drug release presented inChap.6. For example, the expression (6.10) for release across a membrane becomes

Fig. 9.2
figure 2_9

(a) Various pore structures. From left: A straight pore as a 2D analog of the pores in Fig. 9.1; a “tilted” pore illustrating the simplest kind of increase in path length, with tortuosity factor 1/cos θ; a curved, tortuous single path from face to face; a tangle of curved pores with intersections and dead ends. (b) Brick and mortar representation of stratum corneum

Full size image
$$ {M_{{t}}} = \frac{{A{\varepsilon_{\rm{s}}}{D_{\rm{eff}}}{c_{\rm{s}}}}}{{\tau L}}\left( {t - \frac{{{\tau^2}{L^2}}}{{6{D_{\rm{eff}}}}}} \right) $$
(9.6)

and the Higuchi equation (6.24) becomes

$$ \frac{{{M_{\rm{t}}}}}{{{M_{\infty }}}} = 2\sqrt {{\left( {2 - \frac{{{\varepsilon_{\rm{s}}}{c_{\rm{s}}}}}{{{c_0}}}} \right)\left( {\frac{{{\varepsilon_{\rm{s}}}{c_{\rm{s}}}}}{{{c_0}}}} \right)\left( {\frac{{{D_{\rm{eff}}}t}}{{{\tau^2}{L^2}}}} \right)}}. $$
(9.7)

This expression combines the effects of porosity, pore width, and tortuosity.

In the past, tortuosity has referred to any feature of a porous network that slows down diffusion, and the effective diffusion constant was defined as \( {D_{\rm{eff}}} = \varepsilon D{/}\tau \) [28], where here the steric and hydrodynamic factors discussed above are not considered. For example, the Higuchi equation often appears with \( \tau \) in the denominator instead of \( {\tau^2} \). This definition is not tenable, however, since porosity by itself has no bearing on time lag while both \( D{/}\tau \) and \( D{/}{\tau^2} \) appear in (9.6). Neither of these combinations can account for both steady state and lag properties [1].

The basic idea behind tortuosity is that molecules pass through channels that do not direct them straight toward the release surface. For a straight pore making angle θ with the line directly connecting the two faces of a planar slab, \( \tau = 1{/}\cos \theta \). However, as shown in Fig. 9.2a, the direction of a pore may change with position. More generally, tubular pores can meet at junction points, where molecules switch direction as they move from one pore to another, and some tubular channels may lead to nowhere. Thus, tortuosity is often a statistical characteristic. For a membrane or monolithic system with well-connected pore network whose pores are uniformly distributed in diameter and direction, it can be shown that \( \tau = \sqrt {3} \) [1, 29].

Another example of a tortuous diffusion network is the stratum corneum, the outer epithelial layer of the skin. As described in Chap. 2, the stratum corneum consists of multiple layers of desiccated, proteinaceous cells surrounded by lipids through which lipophilic drugs diffuse, arranged in a “brick and mortar” fashion. Similar barrier structures with tortuous paths for diffusion have been created by dispersing clay platelets in polymer films [55, 56]. Figure 9.2b is a simplified rendering, in 2D, of such structures, in which the “bricks” are uniform and regularly spaced with successive layers in alternating register. The mortar separating any two bricks is assumed to be narrow compared to the brick dimensions. Let h and w be the vertical and lateral dimensions of a brick, respectively, and assume that each mortar channel crossing a layer begins and ends at the center of bricks of the previous and following layers. If there are m such layers, then the overall thickness of the “membrane” will be mh. However, the shortest “zig-zag” path that a diffusing drug molecule can take has length m(h + w/2). Because of the symmetry of this geometry, it can be shown that tortuosity is \( \tau = 1 + w{/}2h \). Clearly, very high tortuosities result given wide, thin bricks. More complicated results have been derived for similar structures with variable brick thicknesses and offsets between layers [57], and for 3D brick and mortar structures [58, 59].

4.3 Variations in Pore Diameter

In addition to tortuosity, there are other means by which pore structure can affect the rate of transport and release. Consider a medium containing relatively large, varicose pores connected by relatively narrow throats, as illustrated in Fig. 9.3a. Most of the time spent by a diffusing solute is in the large pores. To move from one large pore to the next, a molecule must find a throat and pass through it. However, the solute must do so by executing a random walk, and it may require considerable time to find a throat. Even after it enters that throat, the molecule may pass part way and then return, again at random, to the original pore, where it gets “lost” again. Thus, one may expect longer confinement in a pore body if its surface area is large compared to the total surface area of the exits to the throats from that pore and if the throats are long [30–33]. Under these circumstances, solute is “well mixed” in the pore, with nearly uniform concentration. This argument does not rely on any steric or hydrodynamic interactions between the solute and the pore wall, which were previously discussed.

Fig. 9.3
figure 3_9

(a) Schematic of Brownian motion (diffusion) of a single molecule in a porous medium with varicose pores and narrow throats. Upon entering the pore, the molecule makes numerous unsuccessful attempts to leave that pore, even occasionally venturing part way into one of its connecting throats but not making a full crossing. Eventually, the molecule crosses a throat into another pore. (b) Two dimensional rendering of model for retarded diffusion in porous media with varicose pore bodies and narrow throats. Pores of “volume” V p are each surrounded by n T throats of length L T and “area” A T. Pore centers are separated by distance d p

Full size image

A simple mathematical model of the effect of pore constrictions or throats is illustrated in Fig. 9.3b [31]. The pore’s volume is denoted by \( {V_{\rm{p}}} \), and \( {n_{\rm{T}}} \) throats of area \( {A_{\rm{T}}} \) and length \( {L_{\rm{T}}} \) emanate from the pore. Using simple dimensional analysis, we estimate the average time it takes for a molecule to leave the pore and reach one of its neighbors as being approximately \( {t_{\rm{P}}} = {V_{\rm{p}}}{/}{n_{\rm{T}}}({A_{\rm{T}}}D{/}{L_{\rm{T}}}) \). Now, let us assume that the pore centers are spaced, on average, at distance d P from their nearest neighbors. Again using dimensional analysis, the effective solute diffusion coefficient in this porous medium, D eff can be estimated according to \( {t_{\rm{P}}} = \omega d_{\rm{P}}^{{2}}{/}{D_{\rm{eff}}} \), where ω depends on the arrangement of the pores. Equating these two time estimates, we find that

$$ {D_{\rm{eff}}}{/}D = \frac{{\omega {n_{\rm{T}}}{A_{\rm{T}}}d_{\rm{P}}^2}}{{{L_{\rm{T}}}{V_{\rm{P}}}}}. $$
(9.8)

Further, if we define the total throat volume per pore as \( {V_{\rm{T}}} = {n_{\rm{T}}}{A_{\rm{T}}}{L_{\rm{T}}}{/}2 \) (denominator signifies that each throat connects two pores), then we obtain the relation

$$ {D_{\rm{eff}}}{/}D = \frac{\omega }{{2{{(\tau )}^2}}}\frac{{{V_{\rm{T}}}}}{{{V_{\rm{P}}}}}, $$
(9.9)

where \( \tau = {L_{\rm{T}}}{/}{d_{\rm{p}}} \) is an apparent tortuosity factor which may be significant if throats connecting pores are twisted. Because these relations were derived rather crudely, they cannot be exact, and more detailed computational tools would be needed for specific pore/throat configurations. Nevertheless, these relations suggest that the effective diffusion coefficient of a solute in a constricted porous medium can vary substantially due to pore and throat geometric factors. Of course, when throats are extremely narrow, steric and hydrodynamic factors also need to be accounted for.

The analysis to this point has dealt with pores that are accessible to the releasing surfaces of a monolithic device or to both sides of a membrane through which drug passes by diffusion. Clearly there are porous structures where this is not the case. Pores can be isolated from all channels leading to a device surface, and hence they become irrelevant for drug release, assuming that the matrix material is otherwise impermeable to drug. In the next section, we discuss concepts from percolation theory, in which the effects of random positioning of pores in a medium is shown to affect not only the accessibility of pore space, but also the rate of drug release by diffusion.

4.4 Percolation Theory

Percolation theory was developed initially to account for the connectedness of pores in rock, as a function of porosity [34, 60]. Later, it was extended to other phenomena such as electrical conduction in heterogeneous materials [46], mechanical strength of composite materials [61], tertiary oil recovery [62], groundwater flow [63], compaction of materials including pharmaceuticals [64, 65], and even forest fires and the spread of disease through random contacts [34, 60]. It also turns out that percolation phenomena are analogous to certain types of physical phase transitions [60]. The theory has been developed by mathematicians [35], physicists [36], and chemical engineers [37]. Because diffusion through random porous systems has elements in common with conduction, percolation concepts also apply to controlled release systems [31, 38–41]. In the following discussion, we restrict attention to site percolation theory.

The simplest predictions of percolation theory relate to pore connectedness. Figure 9.4 illustrates a sequence of equatorial cross sections of simulated spherical porous matrices of different porosities that are set up by random assignment of pores onto a 3D simple cubic (sc) lattice of sites that is embedded in the spherical geometry. The “radius,” N, of each sphere was taken to be 50 lattice sites. The porosity, ε, of a matrix determines the probability that any site is assigned as a pore. Yellow pores are connected to the surface of the sphere through a sequence of neighboring pores while black pores cannot make such a connection to the surface. Connectedness of pores to the surface is determined by the following algorithm. First, all pores at the surface are colored yellow. Next, pores that share a cubic face with any one of the surface pores are colored yellow. This second step is repeated over and over, connecting interior pores sharing cubic faces with already yellowed pores, until no new pores are available to be added by this process. The remaining pores, which do not share cubic faces with yellow pores, are colored black. (Since each cross section shown in Fig. 9.4 is a 2D slice, it should be kept in mind that pores that are not apparently connected by 2D paths might be connected by paths in 3D that go outside the slice.) At low porosities, say ε = 0.05, only pores near the surface are colored yellow. As ε increases, the yellow pores invade further into the center of the sphere, until nearly all of the pores are seen to be connected to the surface when ε = 0.4.

Fig. 9.4
figure 4_9

Model pore networks based on simple cubic lattice embedded in a sphere. Pictured are equatorial cross sections with gray lattice sites designating impermeable polymer matrix, yellow sites designating pores belonging to clusters connected to the surface, and black sites designating pores belonging to clusters that are isolated from the surface. Pore clusters calculated using site percolation procures, with body-centered cubic (bcc) connectivity

Full size image

The porous structures illustrated in Fig. 9.4 might be due to solid drug particles that are randomly deposited in a polymer matrix during formation of a spherical device. In this case, the yellow sites would refer to drug that is releasable from the device while the black sites would indicate drug that is trapped inside the device, assuming that the polymer is completely impermeable to drug. If the polymer is slightly permeable, then the yellow pores would indicate rapidly releasable drug while the black pores would signify drug that is released much more slowly. In the following, we assume that release from a pore is either all or none.

Figure 9.5a shows the predicted fraction of drug release, \( {F_{\infty }} \), from the model spherical systems, as a function of ε. In the limit of low porosity (ε → 0), \( {F_{\infty }} \to \varepsilon \times {\hbox{(surface fraction of lattice sites)}} \) while \( {F_{\infty }} \to 1 \) when ε → 1. Most interesting, however, is the rapid rise in \( {F_{\infty }}(\varepsilon ) \) over an intermediate range of porosities. This transition is due to the growth and coalescence of clusters of connected pores, until eventually one of these clusters extends, or “percolates” throughout the matrix. For values of ε below the transitional range, pores are mostly disconnected from each other and no percolating cluster exists. Moving through the transitional range, more pore clusters are recruited into the percolating cluster, until all belong. At any stage, contributions to \( {F_{\infty }}(\varepsilon ) \) include the percolating cluster, plus nonpercolating clusters containing at least one site on the surface.

Fig. 9.5
figure 5_9

(a) Calculation of fraction of releasable drug, \( {F_{\infty }}(\varepsilon ) \), for finite, simple cubic lattices embedded in spheres, as illustrated in Fig. 9.4. N denotes radius of sphere in lattice units. (b) Behaviors of \( {F_{\infty }}(\varepsilon ) \) and D eff/D 0 above the percolation threshold \( {\varepsilon_{\rm{c}}} \) and average cluster size, S(ε) below \( {\varepsilon_{\rm{c}}} \) for infinite 3D lattices. Precise value of \( {\varepsilon_{\rm{c}}} \) depends on lattice type, but general shapes of curves are universal across lattices of given dimension

Full size image

Figure 9.5a also shows predictions for spherical systems of smaller radius relative to pore size, i.e., smaller N. As this number decreases, the fraction of pores at or near the surface increases and the fraction of releasable drug increases. The transition range becomes less well defined.

For an infinitely extended lattice, it can be shown that a definite threshold value of ε, called the percolation threshold, exists at which a pore cluster of infinite extent appears. Below that threshold, denoted by \( {\varepsilon_{\rm{c}}} \), all pore clusters are isolated, so \( {F_{\infty }}(\varepsilon \leqslant {\varepsilon_{\rm{c}}}) \to 0 \) since the surface/volume ratio of the lattice becomes vanishingly small, and the fraction of finite pore clusters intersecting the surface must also vanish. For the finite size devices, with “lattice” site size determined by the ratio between device diameter and a typical pore diameter, \( {F_{\infty }}\, >\, 0 \) for all porosities and the threshold is diffuse, becoming sharper as device diameter increases or pore diameter related to drug particle size decreases. The threshold value, \( {\varepsilon_{\rm{c}}} \), is a useful concept even for finite size systems, since near that point behavior tends to change radically.

Close inspection of Fig. 9.4 reveals that near the percolation threshold the connected paths from internal pores to the surface are very tortuous. In this regime, an initial rapid burst release of drug from the surface is followed by much slower diffusion. In addition to the tortuous pathways, much of the pore space available to a diffusing molecule consists of “dead ends” into which the molecule may wander (see also the rightmost pore structure in Fig. 9.2a). These dead ends “distract” the molecule from its most direct path to the surface, and time is lost while the molecule finds its way back to the more direct path. This delay in finding the “proper way out” is similar to that which occurs in a large pore body when a molecule is trying to find a throat through which it can escape, as previously described. Delay due to dead ends also finds analogy in gel permeation chromatography, in which small molecules wander into and linger in gel interstices while larger molecules are sterically prohibited from doing so, the result being that the larger molecules are convected more rapidly through the column by the carrier solvent and are eluted more rapidly.

While the simple cubic lattice with a fraction of lattice sites containing pores is an idealization of true pore space, results can be generalized to other geometric partitionings of a matrix. A critical parameter for a given scheme is the average number of nearest neighbor “sites” that can potentially contain pores, which we denote by \( \bar{z} \). In the simple cubic lattice, each lattice site has exactly six nearest neighbors, \( \bar{z} = 6 \). Another regular structure, the hexagonal close packed (hcp) lattice, can be visualized as a stack of closely packed cannonballs, with cannonballs in each layer surrounded by six others forming a hexagon and each cannonball “resting” in a valley formed by a triangle of cannonballs in the underlying layer. For hcp, each site has 12 nearest neighbors, so \( \bar{z} = 12 \).

A well-studied nonregular geometric model is the Voronoi tessellation, in which “seed” points are distributed at random in a three dimensional medium, which is then partitioned into cells, each cell containing all points that are closest to a particular seed point. Cells are irregular polyhedra bounded by polygons with varying numbers of edges, and different cells have different number of nearest neighbors. It has been shown that for a completely random Voronoi tessellation in 3D, \( \bar{z} = 15.56 \) [42].

Percolation threshold values, \( {\varepsilon_{\rm{c}}} \), have been tabulated by computer for all regular 3D lattices and the Voronoi tesselation. For example, values of \( {\varepsilon_{\rm{c}}} \) for sc, hcp, and Voronoi lattices are, respectively, 0.312, 0.199, and 0.145. For values of \( \bar{z} \) ranging from 4 to 42, a useful empirical correlation is [40]

$$ {\varepsilon_{\rm{c}}} \approx \frac{1}{{1 + 0.356\bar{z}}}. $$
(9.10)

Models based on regular lattices or Voronoi tesselations do not exactly reproduce percolation thresholds observed in most porous systems, since real systems are not configured according to these models. Other “continuum” percolation models have been proposed [43, 44]. All such models still make assumptions regarding pore sizes, shapes, and configurations, and therefore can only make specialized predictions. One interesting observation is that the percolation threshold decreases as pores become more oblong at constant ε, since the chance of pore intersection increases [45, 66]. In the end, the percolation threshold \( {\varepsilon_{\rm{c}}} \) of a family of porous media characterized by certain rules of formation is particular to these rules, which are often not well understood. It is prudent, when dealing with complex real systems, to treat \( {\varepsilon_{\rm{c}}} \) as a free parameter. The power of percolation theory lies in its descriptions of behaviors near threshold, which are relatively insensitive to the particular value of \( {\varepsilon_{\rm{c}}} \), as is now discussed.

For infinite lattices, over a range or porosities just above \( {\varepsilon_{\rm{c}}} \), a power law relates fraction of pore space available for release to total porosity:

$$ {F_{\infty }}(\varepsilon ) = A{(\varepsilon - {\varepsilon_{\rm{c}}})^{\beta }}\quad {\varepsilon_{\rm{c}}} \leq \varepsilon \ll 1, $$
(9.11)

where β = 0.40 for all 3D cases, including Voronoi tessellations and other models. A different value β = applies to 2D systems, which do not concern us here. The fact that β only depends on dimension, but not the details of lattice structure or configuration of pores, indicates that a kind of “universality” exists in percolation phenomena. It should be noted, however, that \( {\varepsilon_{\rm{c}}} \), the prefactor A, and the range of validity of the power law depend on the particular lattice or other space partitioning structure. Universality of the power law exponent, but not the critical point or the prefactor, is related to universal behaviors that have been revealed in physics when comparing disparate critical phenomena, such as vaporization of liquids and ferromagnetism.

Just below the percolation threshold and onset of the infinite cluster, the average finite cluster size grows rapidly, and another power law behavior has been determined. Denoting average cluster size by S(ε), the universal expression, with ε close to \( {\varepsilon_{\rm{c}}} \), is

$$ S = {A_{\rm{S}}}{({\varepsilon_{\rm{c}}} - \varepsilon )^{{ - \gamma }}}\quad 0 \ll \varepsilon \,< \,{\varepsilon_{\rm{c}}}, $$
(9.12)

where γ = 1.8 for 3D (γ = 2.4 for 2D). The coefficient \( {A_{\rm{S}}} \) is system dependent. In finite-size systems below the percolation threshold, these clusters account for the releasable drug as determined by \( {F_{\infty }}(\varepsilon ) \). A similar power law accounts for rapid decrease in finite cluster size above \( {\varepsilon_{\rm{c}}} \) due to recruitment of finite clusters into the growing infinite cluster.

Another “universal” behavior of interest refers to the ratio of the effective diffusion coefficient of solute in the random pore structure and the diffusion coefficient of solute in water [46, 47]. Again, a power law relation is seen for \( {\varepsilon_{\rm{c}}} \leq \varepsilon \ll 1 \):

$$ \frac{{{D_{\rm{eff}}}(\varepsilon )}}{{{D_0}}} = {A_{\rm{D}}}{(\varepsilon - {\varepsilon_{\rm{c}}})^{\mu }}, $$
(9.13)

where μ = 2.0 for all 3D systems but \( {A_{\rm{D}}} \) is system dependent. (For 2D, μ = 1.3.) In addition to pore topology (e.g., the number of nearest neighbors), pore/throat geometry and pore wall hydrodynamic interactions (for narrow pores) are factors influencing \( A_{\rm D} \).

Plots comparing \( {F_{\infty }}(\varepsilon ) \), S(ε), and \( {D_{\rm{eff}}}(\varepsilon ){/}{D_0} \) for infinite lattices are shown in Fig. 9.5b. The sharp rise of \( {F_{\infty }}(\varepsilon ) \) and the much gentler rise of \( {D_{\rm{eff}}}(\varepsilon ){/}{D_0} \) reflect tortuosity of the percolating pore cluster and the predominance of dead-end pores near the percolation threshold.

Although power law behaviors apply, strictly speaking, only to infinite systems, they ultimately must be compared against finite sized systems. The larger the system with respect to pore diameter, the better the theory is supposed to hold. Several studies testing percolation theory have been carried out in the controlled release arena [31, 38–41, 67–69]. Here we summarize the results of two carefully constructed studies by Hastedt and Wright [40, 41]. These authors mixed various proportions of micronized solid benzoic acid (BzA) and poly(vinyl stearate) (PVS) particles, both with average diameter ~10 μm, in a die, and were compressed to form solid monolithic tablets of thickness 500 μm. Assuming random mixing, device thickness was approximately 50 times that of the particle, which is a suitably large ratio to apply power law predictions. PVS is water insoluble and BzA has a low solubility in water, so the Higuchi model was used to account for release. Porosity in the monolith was accounted for by the volume of the drug particles, plus a small void volume (0.02) that was measured by helium intrusion.

The authors found that release rate data fit well to (9.13), with the proper value of μ, above an apparent percolation threshold of \( {\varepsilon_{\rm{c}}} = 0.09 \). This threshold is much lower than would be predicted by the sc, hcp, or Voronoi models or any reasonable model that assumes random juxtaposition of BzA and PVS particles. Two possible explanations were offered. First, it was thought that the mixing of BzA and PVS particles was not completely random. As already discussed, nonrandom mixing, and nonspherical shapes of particle aggregates, may lower the percolation threshold. Second, it was noted that PVS is not completely impermeable to BzA, as the authors demonstrated by measuring the diffusion coefficient of BzA in a thin membrane containing pure PVS. Indeed, their experiments showed nearly complete release even at very low inclusion of BzA, with \( \varepsilon \;< \;{\varepsilon_{\rm{c}}} \). Thus, release was not exclusively through the BzA created pores. Instead, drug most likely was able to traverse from “isolated” BzA pores into other pores, albeit slowly, eventually reaching the surface, where drug was released.

It is usually desirable, when formulating a drug delivery system, to have all drug released, as was observed in Hastedt and Wright’s work, and it appears that the matrix must be extremely impermeable to drug in order for strict percolation theory to apply. For small but finite matrix permeabilities, “effective medium” theory [36, 46], which provides a means for averaging fast transport through pores and slow transport through the matrix material, can be used to estimate drug release kinetics.

One example, where “strict” percolation theory may apply, is in the release of paclitaxel (PTX) from coated stents that are used to treat cardiac angioplasty [48]. Here, the drug particles, along with a block polymer matrix material, polystyrene-b-isobutylene-b-polystyrene (SIBS), are dissolved in a common solvent and sprayed onto stent struts (see Chap. 14). Upon evaporation of solvent, the initial solution phase separates, with PTX nanodomains dispersed in SIBS. These domains are imaged using AFM. Below 25% PTX incorporation, release is incomplete while release is virtually complete above that loading. At low loadings, only drug at or close to the surface is released while drug that is deeper in the coating is trapped by surrounding SIBS. At the higher loadings, all drug appears to be available for release through a connected series of neighboring pores.

We conclude this section with a perspective. As already described, percolation theory makes interesting qualitative predictions regarding ultimate release fraction and release kinetics from porous systems. However, it is also pointed out that its utilization requires determination of the percolation threshold for a given family of porous systems. Pore sizes need to be much smaller than overall system size, and the matrix must be highly impermeable to drug in order for percolation effects to be fully manifested. When these conditions are fulfilled, there is a sharp rise in the fraction of drug that is releasable just above the percolation threshold. Leaving aside the observation that nonreleased drug is wasted drug, one should be aware of the quality control pitfalls that are likely to present themselves near the percolation threshold. Small changes in porosity may lead to large changes in fraction released. Thus, it might be recommended that once the percolation threshold is determined, one should strive to formulate with ε sufficiently larger than \( {\varepsilon_{\rm{c}}} \) that \( {F_{\infty }}(\varepsilon ) \) is close to 1. The gentler rise of \( {D_{\rm{eff}}}(\varepsilon ){/}{D_0} \) is such that one can still control the release rate by varying porosity over this “safe” range, although it should not be expected that effects will be as dramatic as they would be closer to \( {\varepsilon_{\rm{c}}} \).

When the matrix possesses a limited but finite permeability to drug, the percolation threshold gains a new significance. When individual drug particles/pores or clusters are isolated from each other, most drug must diffuse through the matrix material to be released, and one can use the solubility/diffusivity properties of drug in the matrix to make predictions. Drug that belongs to clusters intersecting with the surface is released in a burst, however. The strength of the burst depends on the cluster size distribution. Approaching \( {\varepsilon_{\rm{c}}} \) from below, average cluster size increases and the size distribution broadens, and one may again expect quality control problems when ε is too close to \( {\varepsilon_{\rm{c}}} \). Once \( {\varepsilon_{\rm{c}}} \) is determined, it is safest to stay away from it.

5 Concluding Remarks

The emphasis in the chapter has been on porous systems with complicated structural characteristics and randomness. Models have been qualitative in nature, and have primarily been developed to illustrate means by which pore structure can influence availability of drug for release, as well as release kinetics. As noted at the beginning, progress in methods for precisely characterizing the internal pore structure of a medium, and in computational power, may permit designers to obtain much more accurate predictions. We believe that qualitative modeling and quantitative characterization/computational approaches have complementary value. The former provides initial guidance while the latter enable fine tuning.

We have already noted that lessons learned from porous media can be applied to other drug delivery issues, such as transport across the skin and through tissue interstitium. These systems can all be regarded as structured media. In concluding this chapter, we briefly comment on diffusion through hydrogels, which constitute another important class of structured media encountered frequently in drug delivery applications.

As discussed elsewhere in this book, hydrogels are water-swollen polymer networks. Water soluble drugs passing through hydrogels diffuse more slowly than in bulk water because their pathways are obstructed by polymer chains and due to hydrodynamic interactions between the diffusing drug and the less mobile polymer. The analogy to diffusion in porous media is evident, but there are two more aspects that need to be considered in hydrogels. First, a substantial portion of hydrogel water is often “bound” to polymer chains and thus has different properties than bulk water. Second, and perhaps more importantly, hydrogel chains execute thermal “breathing” motions which may open and close water spaces available for drug diffusion. These fluctuations in time add a new statistical layer to analyzing transport. A large amount of data is now available for testing theories of partitioning and diffusion in hydrogels that take into account obstructions, hydrodynamic interactions, and chain fluctuations, as summarized in extensive reviews [49–52]. Such theories are most appropriate for drugs which have little affinity for the polymer.

There are of course cases in which drug binding to polymer in the hydrogel provides a means for controlling release rate, and building solute binding specificity is an active area of research, particularly for hydrogels intended for tissue engineering applications (see Chap. 17 ) [53].

Finally, we note that pore structure in controlled release systems may evolve with time. Bulk degrading polymers often feature growth of internal pores with time, prior to disintegration of the polymer mass. Similarly, porous systems have also been designed with osmotically active agents incorporated into pores that are isolated from the surface. As water enters the pores, the latter swell until the surrounding polymer bursts, opening new channels for drug release [54].

If solid drug is incorporated into a surface eroding polymer in order to achieve zero order release, then this should occur below the percolation threshold in order to guarantee erosion control. Otherwise, pathways for direct diffusional release are available.