1. DRUG RELEASE FROM MATRIX SYSTEMS
1.3.5 Diffusion, Dissolution and Swelling controlled systems
Systems falling in this category are typically represented by polymeric matrices characterised by covalent crosslinks or strong physical crosslinks as it happens for alginates and crosslinked polyvinylpyrrolidone. This means that erosion and drug polymer interactions play a negligible role at least before drug release has been completed. Indeed, it can not be excluded (and sometimes it is desirable) that matrix undergoes erosion after a long time exposure to the release environment conditions. In addition, in vivo, the action of enzymes and other factors can lead to matrix erosion (surface or bulk). Finally, matrix/drug interactions are supposed negligible.
In order to give a better understanding, this section discusses separately they key factors of this category systems, namely swelling equilibrium, swelling kinetics and drug diffusion/dissolution.
1.3.5.1 Matrix swelling equilibrium
Swelling properties of polymeric matrices can affect both the drug release kinetics and the drug loading properties, hence, also the modes of preparation and use of the release system [156]. The swelling behaviour is strongly dependent on the number of intermolecular junctions per unit of volume, namely the crosslink density x, and on the state of the interactions between the solvent molecules and the polymeric chain segments. In many situations, the latter feature of the gel network can be simply characterized by the Flory interaction parameter . For neutral polymers, the amount of absorbable solvent depends on both the solvent chemical affinity for the polymer and the elastic properties of the swollen polymeric network which, in turn, mainly depend on the number of intermolecular bonds, namely the crosslink density. In the case of charged polymers, the swelling equilibrium of the polymeric matrix is more complicated [157,158] as it heavily depends also on the ionic strength. Indeed, crosslinked polyelectrolite chains tend to assume more extended conformations in pure water, while such an extension is usually hindered in an aqueous salt solution where electrostatic interactions occur between the polymer charges and the mobile ions present in solution [159,160].The swelling equilibrium conditions of a polymeric gel can be obtained from the Gibbs free energy by imposing the condition of zero osmotic pressure [157,161,162]:
(6.113)
where is the osmotic pressure, and are the chemical potentials of the solvent in thematrix (g) and the liquid (l) phases, respectively, and V1 is the molar volume of the solvent. Although other approaches can be followed [163-165], the osmotic pressure can be written according to the Flory-Rehner theory [157, 166]:
(6.112)
where is the mixing free energy term, is the elastic contribution, connected with the deformation of polymeric network, is the ionic contribution, due to the difference in ion concentration between the gel and the liquid phase, and is the electrostatic contribution deriving from the repulsive effects between equal charges present in the network.Even though more sophisticated approaches might be followed to express the mixing term [162, 167], the Flory-Huggins expression can be used:
(6.113)
where 2 is the polymer volume fraction, R is the universal gas constant, T is the temperature, and is the Flory interaction parameter, related to the difference between the free energies of a polymer segment-segment, and polymer-solvent interactions.In the most general case, the elastic term could be calculated by assuming that the real structural conditions of the polymeric network are somewhat intermediate between two opposite ideal limits, corresponding to the affine and the phantom networks, respectively [160,162,168]. According to the phantom theory, the intermolecular constraints can freely fluctuate within the polymeric network and such an assumption leads to the following expression:
(6.114)
where x is the crosslink density (mole/cm3), and 20 is the polymer volume fraction at the reference state. Assuming the dry polymer as reference state leads to 20 = 1. Conversely, the fluctuations are hindered in the affine polymeric network, and such a structural condition yields:
(6.115)
For a real system lying between the phantom and affine network condition, we may write:
(6.116)
where F ranges from 0 to 1, depending on the degree of interpenetration of the polymeric network and on its dilatation ratio (20/2)1/3 [162,168,169]. Obviously, the affine network condition (F = 1) which is more suitable to represent the structural conditions and the intermolecular constraints of a chemically crosslinked polymer network.Equations (6.114) and (6.115) are derived on condition that the Gaussian statistic distribution could be assumed valid for the distance between two consecutive crosslinks. Such an assumption seems to be reasonable only for limited chain extensions. Indeed, for very extended swelling conditions, the polymeric chains closely approach their fully stretched conformation and behave as nonlinear elastic connectors so that a more complex distribution should be considered, namely the Langevin one [170,171]. The resulting expression of the elastic term is, then:
(6.117)
where 2c is the polymer volume fraction in the reference state, 2min is the minimum polymer volume fraction attainable by the swollen gel (all the polymeric chains are completely stretched) and is defined by:
(6.118)
where is defined by:
(6.119)
being the inverse of the Langevin function whose approximate form, according to Warner [172], is:
(6.120)
The ionic contribution to the osmotic pressure is derived from the Donnan equilibrium theory [157,158,160,162]. In the presence of a solution, the concentration of mobile ions within the gel phase is larger than in the external phase because of the charges located on the polymer chains. This difference gives origin to the following ionic contribution:
(6.121)
where n is the number of ionizable groups present on the monomer, j is the ionization degree (fraction of monomers carrying the ionizable groups), Vu is the monomer molar volume cs* and cs are the salt concentrations in the external solution and in the gel phase, respectively, () is the sum of the positive and negative valences of the dissociating salt. In the case of monovalent ions present both in solution and on the polymeric chains, = 2 andeq.(6.121) becomes:
(6.122)
Finally, the electrostatic term deriving from the repulsive effects between equal charges present in the polymeric network could be calculated from the Ilavsky approach [170], but its contribution is usually negligible in comparison with the ionic term [162].In conclusion, for not highly swellable matrices, the equilibrium swelling conditions is:
(6.123)
For highly swellable matrices, the equilibrium condition reads:
(6.124)
If salt concentration in the swelling medium is zero, eq.(6.124) becomes:
(6.124’)
Coviello and co-workers [173], studying drug permeation through crosslinked (1,6-hexanedibromide crosslinker) sclerox (oxidized scleroglucan) membranes, gives an example of eq.(6.124) application.In particular, fixing polymer concentration to 1.6 % (w/v), two different crosslinking ratios r (r = (equivalents of reagent)/(equivalents of polymer)) (0.5, 1), two temperatures (37°C, 7°C) and two NaCl concentrations in the swelling agent (water) are considered (0, 0.154 M) (see table6.4).A profitable strategy applicable to calculate and x is to collect the swelling equilibrium data in presence () and in absence () of an external salt (same temperature), and then solving, with respect to and x, eq. (6.124) and eq. (6.124’):
(6.125)
where:
(6.126)
(6.127)
Although 2minexperimental determination is not an easy task, it can be demonstrated that an exact knowledge of its value may not be strictly necessary and only an approximate evaluation is needed. For this purpose, let us focus the attention on the swelling equilibrium characterizing test 1a and 1b (table 6.4). Eqs. (6.124) - (6.124’) systemsolution is performed considering different 2min values and assuming V1 = 18.1 cm3/mol, n = 2 and j = 1 (each monomer carries two univalent ions), calculating the monomer molar volume (Vu = 288.8 cm3/mol) by means of the solubility parameters [174] and knowing that 2w = (1.1 0.35) 10-2 and 2s = (2.86 0.4) 10-2 (see also table 6.4). As figure6.19 clearly reveals, the unknown and x simultaneously assume a physically consistent value only when 0.82*10-22min < 0.97*10-2. Moreover, while strongly depends on 2min, x is virtually independent and could be evaluated, for instance, by choosing the mean 2min value characterizing the above mentioned interval. Accordingly, on condition to renounce to get the value, x can be calculated choosing 2min = 0.895x10-2. Eqs. (6.124) - (6.124’) system solution repeated for the other experimental conditions, leads to similar conclusions, namely the virtually independence of x on 2min in the interval where both x and assume physically consistent values. Consequently, the mean 2min values can be selected (see table 6.4).It is worth mentioning that these 2min values accomplish a reasonable topological condition requiring that at higher r values correspond higher 2min values at both temperatures (37 and 7 °C) (see table 6.4). Moreover, given anr value, higher is the temperature, lower is 2min.On this basis, the x calculation is performed for each experimental condition examined and the results are reported in table 6.4. Notably, at each temperature, the calculated x value coming from the test characterized by a higher r value is, approximately, two times the value coming from the test characterized by a lower r value. Moreover, test 1a - 1b and test 3a - 3b give similar x values as it happens for test 2a - 2b and test 4a - 4b.
Mantovani and co-workers [156], dealing with different kinds of crosslinked sodium starch glycolate matrices (sodium salt of a poly-a-glucopyranose, prepared by crosslinking(sodium trimetaphosphate) and bycarboxymethylation of potato starch), makes use ofeq.(6.123) to get x and . In particular, the solution strategy implies to get experimental data relative to matrix – liquid water (2L) and matrix - vapour water (2v) equilibrium performed at the same temperature. In the case of matrix - vapour equilibrium, eq.(6.123) becomes [175-179]:
(6.123’)
whereP/P0 is the relative vapour pressure. The simultaneous solution of eq.(6.123) and (6.123’) yields problem solution:
(6.128)
where:
(6.129)
(6.130)
Table 6.5 shows the and x values for the four sodium starch glycolate kinds considered knowing thatP/P0 = 0.97,V1=18.1 cm3/mole, Vu= 376.4 cm3/mole,n*j= 0.75, = 10-7 mole/cm3and different values for cs(gel/vapour equilibrium) calculated on the basis of the residual NaCl mass (residual presence of the saltin the matrix is due to the crosslinking and carboxymethylation reactions).Interestingly, the x knowledge allows the determination of Mc, the molecular weightof the chain segment comprised between two consecutive crosslinks. Indeed, the following relation [158] holds:
(6.131)
where p is polymer density and Mn is polymer molecular weight before crosslinking.For all the four sodium starch glycolate kinds considered (CLV, LOW PH, V17 and EXPLOTAB), the Mn value reported in the technical bulletin provided by the manufacturing [180] is 106. Accordingly, Mcis equal to 6455, 5156, 23 180, 7910 for the CLV, LOW PH, V17 and EXPLOTAB samples, respectively. The corresponding number of monomeric units betweentwo consecutive crosslinks is 10.5, 8.2, 37.7, 12.9, for CLV, LOW PH, V17 and EXPLOTAB,respectively.According to a molecular simulation (DTMM software),it is possible to conclude that the end-to-end distance of the sodium starch glycolate monomer is equal to 0.892 nm. Assuming that (i) each branchbetween two consecutive crosslinks can be modeled as a freely jointed chain composed of monomerunits and (ii) all the monomers comprised in the branch are perfectly aligned so that the chainconfiguration is fully extended, mesh size values should be 10.8, 9.5, 47.5 and13.9 nm, for the CLV, LOW PH, V17 and EXPLOTAB, respectively. Obviously, this simplifiedcalculation leads to an overestimated mesh sizes since the actual configuration of the polymericbranches, resulting from the balance between the intrinsic constraints and the tension state producedby the network swelling, is likely less extended.
The above mentioned examples show how to estimate and xexclusivelyresorting to equilibrium data. However, a more traditional way is to use eqs.(6.123) or (6.124) to calculate once x has been determined by means of mechanical experiments. Indeed, resorting torubber elasticity and linear viscoelastic theory,it can bedemonstrated that the Young (E) and shear (G) modulus of a swollen polymeric matrix can be expressed by [157,181]:
(6.132)
where R is universal gas constant and T is temperature. While for swollen matrices behaving like a solid,E (or G = E/3) can be estimated by matrix compression or elongation (or shear deformation) experiments, for viscolesatic matrices (where matrix response to a solicitation is not instantaneous but it develops in time), E (or G) identifies with the pure matrix elastic characterE0(G0 = E0/3). In case of relaxation tests, it follows [182]:
(6.133)
where is normal tension, 0 is deformation, i an Ei represent generalised Maxwell model parameters (see chapter 3) and t1 is the time required to get the desired deformation 0. Obviously, when (instantaneous compression), eq.(6.133) becomes:
(6.133’)
where i/Eican be seen as relaxation time. Eqs.(6.133) and (6.133’) clearly evidence that the normal stress approaches 0E0 after a complete relaxation. Pasut and co-workers [39] apply eq.(6.133) on experimental data referring to the relaxation behaviour of different types of sodium alginates cylinders characterised by polymer concentration ranging from 1% t0 4% w/w. Correspondingly, he finds that x ranges between 3*10-8 and 2*10-6 mole/cm3.
1.3.5.2 Matrix swelling kinetics
Modeling the drug release from a swellable matrix implies formulating the relevant mass balance and solving the corresponding constitutive equation for the flux of both the swelling fluid (entering the matrix) and the drug (leaving it), respectively. Whereas a suitable choice can be the classical Fick’s law as the constitutive equation of the drug, in the case of the swelling fluid, recourse must be made to an equation which can account for the complex phenomena governing the flux of swelling fluid entering the matrix, and particularly the viscoelastic properties of the swellable matrix.
Several approaches have been proposed to interpret and model the swelling fluid diffusion in a glassy polymer matrix. For example, Joshi and Astarita assumed a time-dependent composition at the polymer/swelling fluid interface [183], whereas other authors imposed a swelling propagation law at the swelling front [184]. In an interesting series of papers, Cohen firstly generalized Crank’s idea [76] of a time-dependent swelling fluid diffusion coefficient [185], and then supposed that the swelling fluid flux could be properly described by coupling the concentration and stress gradients, being the stress dependent on concentration and time via a Maxwell-like viscoelastic relationship [186]. Later on, Cohen and coworkers developed and improved the above mentioned approach by modifying the stress dependence on time and concentration [187-189]. The existence of a stress-related convective contribution to the swelling fluid flux was postulated by Frisch [190] and taken into consideration also by other authors [191,192]. According to Adib and Neogi [193] and Camera-Roda and Sarti [194], the swelling fluid diffusion flux may depend on the history of the swelling fluid concentration gradient. Singh and Fan [195] developed a generalized mathematical model for the simultaneous transport of a drug and a solvent in a planar glassy polymer matrix. The drug diffuses out of the matrix which is undergoing macromolecular chain relaxation and volume expansion due to solvent absorption from the environment into the matrix. The swelling behaviour of the polymer is characterised by a stress-induced drift velocity term ‘v’. The change of volume due to the relaxation phenomenon is assumed instantaneous. The model incorporates convective transport of the two species induced by volume expansion and by stress gradient. However, it was developed for planar systems, not for cylindrical matrices. The most general theory about the swelling fluid diffusion in a viscoelastic polymer matrix was developed by Lustig [196] who assumed that the swelling fluid flux should depend on several driving forces such as temperature gradient, species’ inertial and body forces, chemical potentials and stress gradient. One of the main advantages of this theory is that all the material properties taken into account may be measured so that no phenomenological coefficient must be introduced [197]. The predictions provided by the Lustig model are in semiquantitative agreement with the experimental results [198].
Despite its lower generality, we believe that the Camera-Roda and Sarti equation [199] can be used to model non fickian diffusion. This model avoids the determination of a great number of experimental information normally requested by other, more general models. Obviously, it implies the use of some phenomenological parameters which are not, however, empirical in nature, since all the parameters of the Camera-Roda and Sarti model do possess a well defined physical meaning. Basically, these authors suppose that swelling fluid (SF) flux J may be expressed as the sum of two terms: Jf, characterized by a zero relaxation time and representing the Fickian contribution to the global flux, and Jr, characterized by a non-zero relaxation time and representing the non-Fickian contribution to the global flux, respectively. Accordingly, the global flux can be expressed as:
(6.134)
where:
(6.135)
(6.136)
in which C is the swelling fluid concentration, is the relaxation time of the given polymer/SF system (in so doing it is implicitly assumed that system viscoelastic behaviour is represented by the simple Maxwell model constituted by the series of a hookean spring and viscous dashpot; see chapter 3), Df is the diffusion coefficient relative to the Fickian flux, Dr is the diffusion coefficient relative to the non-Fickian flux and t is time. Under these assumptions, the balance equation governing the SF adsorption phenomenon is given by:
(6.137)
On the basis of the free volume theory, Camera-Roda and Sarti (CRS) [199] assume, for Df, Dr and , the following functional dependencies on C:
Df (C) = D0(6.138)
Dr (C) = Deq exp [g (C - Ceq)] - D0(6.139)
(C) = eq exp [k (Ceq - C)](6.140)
in which D0 is the diffusion coefficient of the swelling fluid in the dry matrix (C = 0),Deq is the diffusion coefficient of the swelling fluid in the swollen matrix (C = Ceq), Ceq is the swelling fluid concentration in the polymeric matrix at equilibrium, g and k are two adjustable parameters and eq is the matrix relaxation time at equilibrium. Assuming to deal with a one-dimension problem (SF diffusion takes place only in the X direction), eq.(6.137) has to be solved under the following initial conditions:
Jr = 0 0 XL0(6.141)
C = 0 0 XL0(6.142)
C = 0.8CeqX = L0(6.143)
and boundary conditions:
J = Jf = Jr = 0 X = 0(6.144)
(6.145)
where L0 is the dry matrix thickness, L(t) is the matrix thickness at time t(matrix thickness increases due to swelling) and is thematrix relaxation time defined by eq. (6.140). Although in most cases SF equilibrium concentration holds at the matrix / external environment interface, in order to infer a wider generality to the model, it is assumed that a relaxation process takes place even at that boundary. Eqs.(6.143) and (6.144) rule this relaxation supposing that a reasonable C starting value may be 0.8 Ceq and that its increase rate is proportional to the difference Ceq - C via 1/[199].