21st European Symposium on Computer Aided Process Engineering – ESCAPE 21 E.N. Pistikopoulos, M.C. Georgiadis and A.C. Kokossis (Editors) c 2011 Elsevier B.V. All rights reserved.
Optimal design of chitosan-based scaffolds for controlled drug release using dynamic optimization Belmiro P.M. Duartea,b∗ , Nuno M.C. Oliveirab, Maria J.C. Mouraa a Dep. Chemical and Biological Engineering; Polytechnic Institute of Coimbra; R. Pedro Nunes; 3030–190 Coimbra; Portugal. b GEPSI-PSE Group; CIEPQPF; Dep. Chemical Engineering; University of Coimbra; R. Sílvio Lima — Pólo II; 3030-790 Coimbra; Portugal.
Abstract This paper addresses the optimal design of drug delivery matrices employing Dynamic Optimization (DO), using the scaffold matrix dimensions, the concentration of the crosslinking agent and initial drug concentration as decision variables to reach a target profile within a given time horizon. The conceptual approach is demonstrated via the design of chitosan based matrices previously cross-linked with genipin in immobilized liposomes, used to encapsulate a drug. The kinetic model employed was derived from data obtained in lab assays. The PDE model describing the diffusion mechanism is discretized with Finite Differences, and the resulting ODEs together with a description of the degradation rate are handled via a simultaneous approach, with a discretization scheme based on orthogonal collocation in Finite Elements in time. The results obtained illustrate the adequacy of this type of methodologies for drug delivery design. Keywords: Product design, Mathematical Programming, Drug release, Diffusion processes.
1. Introduction The delivery of bioactive molecules employing polymeric materials derived from natural polysaccharides has attracted considerable attention in recent years, and is nowadays gaining more and more relevance in the controlled drug release field (Lewis, 1990). The trend of using biodegradable polymers as excipients is expanding, since it allows avoiding complex follow-up surgical interventions after the drug is depleted. Among the most promising biopolymers for drug release purposes is chitosan. Chitosan can be used to form hydrogels with the property of swelling but not dissolving in contact with water (Ganji and Vasheghani-Farahani, 2009). Chitosan-based hydrogel scaffolds are commonly employed either for drug release purposes or cell encapsulation vehicles for tissue engineering (Ahmadi and de Bruijn, 2007). Another relevant aspect regarding chitosan-based hydrogels is that they possess thermosensitive properties, thus making them suitable to be used as injectable in situ, forming scaffolds. However, chitosan-based matrices have poor mechanical properties and exhibit uncontrollable degradation rates in lysozyme environments. To handle these drawbacks, cross-linking agents are employed with the basic aim of modulating the mechanical properties, namely the viscosity, the ∗
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injectability and also the diffusivity of the drug in the hydrogel network. Here, natural cross-linking agents are also preferred due to their biocompatibility with living cells and the ability of organisms to assimilate the residuals, and genipin is among this group. The use of hydrogel scaffolds for drug release suffers from one additional difficulty resulting from the higher diffusion rate, particularly for low-molecular-weight hydrophilic drugs (< 1000 g/mol). In order to handle this problem, the drug is firstly loaded into liposomes, which are then incorporated into hydrogel matrices subsequently injected in situ, forming scaffolds (Ruel-Gariépy et al., 2002). The application of PSE tools to the design of this type of drug release matrices can be highly beneficial, not only from an economical perspective but also to improve and systematize this task.
2. Model development Data from laboratorial trials was initially used to develop a kinetic model for the degradation rate. To guarantee the structural stability of the crosslinked matrices, the degradation rate of samples prepared with different concentrations of genipin was monitored during 28 days, employing gravimetric analysis. Samples consisting in hydrogel tablets with cylindrical form previously cured for 2 h (∼ 0.4 g) were used. The weight loss of initially weighed samples was monitored as a function of incubation time, in 10 mL of phosphate buffer saline (PBS) at 37 ◦ C, with gentle agitation in an orbital shaker. At specified time intervals, samples were removed from the medium, the surface water blotted gently with a filter paper and weighed. The extent of in vitro degradation is expressed as percentage of weight loss. Laboratorial assays were carried for four different crosslinking concentrations gel matrices (0.05, 0.10, 0.15 and 0.20% (w/w) of genipin). An empirical kinetic model of the form dα = −k1 (1 − α )k2 , dt
α (0) = 1
(1)
with α representing the weight loss extent, k1 = P21 (Cr ) and k2 = P22 (Cr ), was fitted to experimental data. Here P21 ( · ) and P22 ( · ) stand for general 2nd order polynomials and Cr is the cross-linking concentration. The assumption of independency of the parameters was firstly considered to fit different models to each data set. A Least Squares based algorithm was employed, with the results proving the initial assumption regarding the independence of both parameters. All models fitted revealed parametric correlations between k1 and k2 well below 0.9, and the χ 2 test showed that in every case good models could be produced. The kinetic parameters determined were subsequently polynomially fitted to Cr . The model adequacy is demonstrated in Figure 1. A paddle method dissolution apparatus rotating at 100 rev/min was used in the drug release experiments, with a low-molecular-weight hydrophilic anti-carcinogenic agent. All experiments were carried out in 250 mL PBS (pH=7.0) at 37 ± 0.5 ◦ C, where hydrogel samples previously cured and produced with different cross-linking formulations (0.05, 0.10, 0.15 and 0.20% (w/w) of genipin) were immersed. Test samples of 1 mL were extracted at specific time intervals and analysed spectrophotometrically. To guarantee sink conditions requirements, each sample removed was replaced by fresh buffer solution. Drug release kinetics may be affected by many factors such as polymer swelling, polymer erosion, drug dissolution/diffusion characteristics, drug distribution inside the matrix, drug/polymer ratio, system geometry and, in this particular system, by crosslinking concentration and liposomes characteristics. Here we consider that the release
100 90 80
Weight loss (%)
70 60 50 40 obs. (C =0.05 %) r
30
obs. (Cr=0.10 %) obs. (C =0.15 %)
20
r
obs. (C =0.20 %) r
10
pred. (C =0.05 %) r
pred. (C =0.10 %)
0
r
0
pred. 5 (C =0.15 %) r
10
pred. (C =0.20 %) r
15 Time (days)
20
25
Figure 1. Degradation profiles and model predictions.
is diffusion-driven, with a Fickian law representing the dynamics of the concentration within the matrix (Grassi et al., 2007). Additionally, we also consider that the tablets have a cylindrical form, with thickness much smaller than diameter. Therefore, the axial dimension can be considered dominant in the diffusion mechanism, leading to the conceptualization of the tablet as a slab with cylindrical shape, with an x coordinate representing the spatial domain [0, L/2], assuming symmetric concentration profiles. Another aspect considered is that the erosion of the matrix is negligible during the drug diffusion mechanism. Thus, the matrix geometrical stability is maintained during the drug release interval, with the degradation rate modulated via the cross-linking concentration. The drug concentrations in the hydrogel matrix and in the buffer solution are described by the equations (Crank, 1975):
∂c ∂ 2c = De 2 , c(x, 0) = cin , c(L/2,t) = k cen , ∂t ∂x dcen S De ∂ c(L/2,t) =− , cen (0) = 0 dt V ∂x
∂ c(0,t) =0 ∂x
(2) (3)
Here c is the drug concentration in hydrogel matrix, De the diffusion coefficient, cin the initial drug concentration (assumed as homogeneous), cen the drug concentration in the buffer medium, S the release area, V the medium volume, k the partition coefficient matrix-medium, and cen is the variable observed within time. The same approach described for model fitting of weight loss was employed here, adjusting the parameters De and k. All models fitted corresponding to the four cross-linking concentrations tested denoted good statistical quality, with parametric non-collinearity holding. The correlations obtained are presented as following: De = −3.854 × 10−1 Cr2 − 1.599 × 10−2 Cr + 1.870 × 10−2 [cm2 /day] k = 9.554 × 10
4
Cr2 − 1.297 × 104 Cr + 6.466 × 102
3
3
(4)
[cm .matrix/cm .buffer] (5)
3. Optimal design of the drug release matrix Here we address the design of hydrogel tablets employed for drug release in vitro or in vivo environments, using as sizing parameters the thickness and diameter (the cylindrical form used in laboratory tests is maintained to minimize extrapolation of parameters values), the cross-linking concentration employed to modulate diffusion and degradation features, and the initial drug concentration in the matrix. The objective function is the minimization of the squared difference between the drug concentration forecast by the model and a target profile within a given time horizon T , here denoted as c∗en (t). Significantly different reference profiles can be selected, to adapt to the particular characteristics of the subject or the drug being administered. This DO problem is formulated as follows: min
L,D,cin ,Cr
s.t.
Z T 0
(c∗en (t) − cen(t))2 dt
(6)
Equations (1–5) α (T ) ≤ αl
(7) (8)
L/D ≤ 0.20
(9)
Equation (8) represents a terminal condition for the weight loss required to guarantee the structural stability of the matrix, and avoid variations in the model parameters. The maximum limit of weight loss allowed is αl =0.65. Equation (9) expresses a constraint required to assume the axial diffusion as dominant, similarly to the phase of model development. Here T is set to 12 days, and the parameter correlations fitted in Section 2 are used. The volume of medium was considered equal to 250 mL, assuming a one-compartment based model. Additionally, we assumed that no relevant drug degradation occurs during this period, and that the diffusion for the other contiguous compartments is negligible. This simplistic conceptual basis can be improved by introducing drug consumption and intercompartmental flow terms, both requiring additional laboratorial assays to estimate the corresponding diffusion parameters. The target profile considered in this example is represented by c∗en (t) = 1 − exp(−0.8t), an instance of a general type of curves commonly used in therapeutics based on controlled drug release. Equation (2) was discretized with respect to axial dimension using Centered Finite Differences (11 nodes). Since L is a decision variable, the nodes were fixed in a unitary domain with the real positions determined by the rule xi = L (i − 1)/N, 2 ≤ i ≤ N + 1, and x1 = 0 where xi is the position of ith node. The integral term involved in Equation (6) is computed with a Gaussian Quadrature rule based in 4 points per finite element, corresponding to the zeros of Gauss-Legendre 3rd order polynomials intercepting the finite element limits. The time domain was discretized with orthogonal collocation in finite elements, each element having the duration of 0.5 days, employing the same space of polynomial approximations used for integral computation. Figure 2 illustrates the agreement between target and optimal drug release profile. Table 1 lists the optimal design parameters. Table 1. Optimal design parameters. L [cm] 0.1813
D [cm] 1.0072
Cr [%] 0.1941
cin [µg/cm3.matrix] 11.358
100 Drug released 90 80
Weight loss
Percentage (%)
70 60 50 40 30 20 Target profile (drug release) Optimal profile (drug release) Weight loss
10 0
0
2
4
6 Time (days)
8
10
12
Figure 2. Target and optimally designed drug release profiles.
4. Conclusions The optimal design of drug release matrices based on thermosensitive hydrogel systems modulated employing cross-linking agents with the drug previously encapsulated in immobilized liposomes was addressed. The methodology uses a simultaneous DO approach to determine the optimal structural dimensions and features of the hydrogel matrix minimizing the squared error norm of drug release dynamics with respect to a target profile. The models employed to describe drug release within the tablets and matrix degradation were initially fitted to laboratorial data and later used to predict the dynamics of the system. The strategy proposed is successfully demonstrated via the design of chitosan based matrices employing genipin as cross-linking agent for the purpose of releasing a low-molecular weight anti-carcinogenic drug.
References Ahmadi, R., de Bruijn, J. D., 2007. Biocompatibility and gelation of chitosan-glycerol phosphate hydrogels. Journal of Biomedical Materials Research Part A 86A, 824–832. Crank, J., 1975. The Mathematics of Diffusion, 2nd Edition. Oxford University Press, New York. Ganji, F., Vasheghani-Farahani, E., 2009. Hydrogels in controlled drug delivery systems. Iranian Polymer Journal 18, 63–88. Grassi, M., Grassi, G., Lapasin, R., Colombo, I., 2007. Understanding Drug Release and Absorption Mechanisms. CRC Press, Boca Raton. Lewis, D. H., 1990. Control release of bioactive agents from lactide/glycolide polymers. Marcel Dekker, Inc., pp. 1–41. Ruel-Gariépy, E., Leclair, G., Hildgen, P., Gupta, A., Leroux, J. C., 2002. Thermosensitive chitosanbased hydrogel containing liposomes for the delivery of hydrophilic molecules. Journal of Control Release 82, 373–383.
