5-Fluorouracil Release from Chitosan-Based Matrix. Experimental and Theoretical Aspects

A series of four drug release formulations based on 5-fluorouracil encapsulated into a chitosan-based matrix were prepared by in situ hydrogelation with 3,7-dimethyl-2,6-octadienal. The formulations were investigated from structural and morphological aspects by FTIR spectroscopy, polarized light microscopy and scanning electron microscopy. It was established that 5-fluorouracil was anchored into the matrix as crystals, whose dimension varied as a function of the crosslinking density. The in vitro drug release simulated into a media mimicking the physiological environment revealed a progressive release of the 5-fluorouracil, in close interdependence with the crosslinking density. In the context of Pharmacokinetics behavioral analysis, a new mathematical procedure for describing drug release dynamics in polymer-drug complex system is proposed. Assuming that the dynamics of polymerdrug system’s structural units take place on continuous and nondifferentiable curves (multifractal curves), we show that in a one-dimensional hydrodynamic formalism of multifractal variables the drug release mechanism (Fickian diffusion, non-Fickian diffusion, etc) are given through synchronous dynamics at a differentiable and non-differentiable scale resolutions. Finally, the model is confirmed by the empirical data.


1.Introduction
Chitosan based formulations are of increasing interest in the drug delivery field, due to its intrinsic properties such as biocompatibility and biodegradability, which recommend it for in vivo applications [1][2][3]. In order to further improve the chitosan ability to anchor large amounts of drugs and to release them in a controlled manner, many attempts were pursued consisting mainly in chitosan crosslinking with various agents. In this line of thoughts, an eco-friendly method was developed by crosslinking chitosan with eco-friendly monoaldehydes [4][5][6][7][8][9][10]. The method proved to be a successful one, providing hydrogels whose properties can be simple controlled by the nature of the aldehyde. Thus, by using the natural aldehyde: 3,7-dimethyl-2,6-octadienal, also known under the commercial name: citral, hydrogels, with excellent biocompatibility and biodegradability which were further developed as matrix for drug delivery systems, were obtained.
On the other hand, the homogeneity assumption in its various forms (homogenous kinetic space, law of mass etc.) has become almost dogmatic in Pharmacokinetics (PK).The functionality of such a hypothesis allowed the development of a class of differentiable models in the description of dynamics of biological systems (i.e. "compartmental" analysis) and mainly, of drug release dynamics in such systems. However, biological systems are nowadays understood as inherently non-differential (fractal). Specifically, the microenvironments where any drug molecules with membrane interface, metabolic enzymes or pharmacological receptors are unanimously recognized as unstirred, space-restricted, heterogeneous and geometrically fractal. It is thus necessary to define a new class of models, this time non-differentiable, in describing biological system dynamics and particularly drug release dynamics in such systems.
Usually, such an approach, known as Fractal Pharmacokinetics, implies the use of fractional calculus, expanding on the notion of dimension etc. As such, it is possible in the context of "compartmental analysis" [11], to describe diffusion in dense objects [12], dynamics in polymeric networks [13,14], diffusion in porous and fractal media [4], kinetics in viscoelastic media [15] etc. More recently, "compartmental analysis" through PK allowed the modeling of processes such as drug dissolution [16], absorption [17], distribution [18], whole disposition [19], kinetics with bio-molecular reactions [20] etc.
In this paper, in the context of "compartmental analysis", a new method for describing drug release dynamics in complex systems (evidently discarding to fractional derivative and other standard "procedures" used in PK), considering that drug release dynamics can be described through continuous but non-differentiable curves (multifractal curves) is proposed. Then, instead of "working" with a single variable described by a strict, non-differentiable function, it is possible to "operate" only with approximations of these mathematical functions, obtained by averaging them on different scale resolutions. As a consequence, any variable purposed to describe drug release processes will still perform as the limit of a family of mathematical functions, this being non-differentiable for null scale resolutions and differentiable otherwise. Finally, the theoretical model is confirmed by the empirical date related to the 5-fluorouracil release from chitosan-based matrix.

Formulation preparation
The formulations were prepared by in situ hydrogelation of chitosan with 3,7-dimethyl-2,6octadienal in the presence of 5-fluorouracil, following an reported protocol [21]. Shortly, a 2% solution of 3,7-dimethyl-2,6-octadienal mixed with 5-fluorouracil was slowly dropped into a 3% solution of chitosan dissolved in aqueous acetic acid (1%), produced by Aldrich. The amount of drug and chitosan were kept constant, while the quantity of 3,7-dimethyl-2,6-octadienal was varied to reach different molar ratios of the amine/aldehyde groups, from 1/1 to 4/1, and thus to obtain hydrogels with different crosslinking density [7]. The hydrogelation time increased as the amount of aldehyde decreased. Thus, it instantly occurred for a 1/1 molar ratio of amine/ aldehyde functional groups and proceeded slowly, during 24 h for a 4/1 molar ratio. Finally, the obtained hydrogels were lyophilized and submitted to analysis. The formulations were coded U1, U2, U3, U4, the number corresponding to the molar ratio of amino/aldehyde groups.

Methods
The gelation time was determined when visually the reaction mixture was transformed from viscous to rubbery state. The xerogels have been obtained by lyophilization from the corresponding hydrogels, using a Labconco FreeZone Freeze Dry System, (FreeZoner2.5 Liter Freeze Dry Systems) equipment for 24 h at −50 o C and 0.04 mbar. The formulations were characterized by FTIR spectroscopy, on a FT-IR Bruker Vertex 70 Spectrophotometer. The xerogels morphology was investigated with a field emission Scanning Electron Microscope SEM EDAX -Quanta 200 at accelerated electron energy of 20 keV.
The release kinetics from the developed drug delivery systems has been monitored by registering the absorbance at 265 nm from the supernatant in which the release was done, after which the concentration was calculated using the Beer-Lambert law. The UV-vis spectra of the supernatant were registered on a Horiba Spectrophotometer, and the absorbance was fitted on a prior drawn calibration curve. The calibration curve for the 5-fluorouracil was traced using the absorption maximum from its spectrum, at 265 nm.

3.Results and discussions
The presence of the 5-fluorouracil into formulations was evidenced by polarized light microscopy (Figure 1), which revealed the clear segregation of the drug into the hydrogels with high crosslinking density (U1, U2), while for the hydrogel formulations with lower crosslinking density (U4) a birefringent, granular texture was observed, characteristic for submicrometric dimensions of the crystals, which fall under the detection limit of the equipment [22,23]. Further, the formulation morphology was assesed by scanning electron microscopy. As can be seen in Figure 2, they have a porous morphology, with evident drug crystals encapsulated into the pore walls ( Figure 2). The diameter of the drug crystals decreased as the crosslinking degree was diminished, in line with the polarized light microscopy observation, as also observed for other chitosan based formulations [24]. The in vitro drug release of the 5-fluorouracil showed a different trend, depending by the crosslinking density. Thus, the formulation U1 with the highest crosslinking density exhibited the fastest release rate, reaching almost 100% drug released in less than 24 h. The release rate slow down as the crosslinking degree decreased along with the total percent of the drug released, reaching in the case of the formulation U4 around 75 % drug release in less than 24 h (Figure 3). This correlated very well with the size of the drug into formulations; as the drug crystals size decreased, the release rate diminished, in agreement with the stronger anchoring of the drug into the chitosan based matrix [25].

3.1.Theoretical model
Let it be considered the one-dimensional multifractal hydrodynamic-type equations [26][27][28][29][30], in the form: In the abovewritten relations, x is the fractal spatial coordinate, is the nonfractal time having the role of an affine parameter of the motion curves, VD is the differential velocity independent on the scale resolution , , ( ) is the singularity spectrum of order and √ is the states function amplitude.
These equations for the initial and boundary conditions: Now taking out the quadratic term in between (11) and (13), it results that for = . the ratio is homographic dependent of by the form: From here, the condition (dynamical simultaneity): (i.e. the extension of the first principle of Newton to any scale resolution, or equivalently, "synchronizations" of drug release dynamics at differentiable scale with drug release dynamics at nondifferentiable scale), implies correlations between phase and amplitude of the shape function, by the form: where 0 and 0 are integration constants. Thus, it is stated that various "mechanisms" involved in the drug release process can be mimed through period doubling, quasi-periodicity, intermittences etc. (for details see [37]). Because through the restriction (15) given, for example, by = − , the multifractal type conservation laws (1) and (2) take the form of the multifractal type "diffusion" equation: it results that these "mechanisms" "manifest"/are "perceived" as diffusions at various scale resolutions in a multifractal space (fickian-type diffusion, non-fickian-type diffusion etc.) To expand on this hypothesis, we approach on investigating the following scenario: the one-dimensional drug diffusion of multifractal type from a controlled-release polymeric system with the form of a plane shut, of thickness . If drug release of multifractal type occurs under perfect sink condition, the following initial and boundary conditions can be assumed: where 0 is the initial drug states density of the multifractal type in the "device" of multifractal type and 1 is the drug states density at the "polymer-fluid" interface of multifractal type. This solution equation under these conditions can take the following form (for details in the classical case see [37]). In Figure 4 there are represented the In such a context, ∞ can be assimilated to the fraction of dissolved drug i.e.
where is the amount of drug dissolved in time and ∞ is the total amount of time dissolved when the pharmaceutical dosage form is exhausted [29,37]. A verification of our model is presented in Figure 5, for the drug release of 5-fluorouracil release from chitosan-based matrix. The empirical data was fitted with the mathematical function. The figure shows that the model is well equipped to predict the drugrelease dynamics [38].

Conclusions
A series of drug delivery systems were prepared by encapsulation of the 5-fluorouracil into a hydrogel formed by crosslinking chitosan with 3,7-dimethyl-2,6-octadienal. The hydrogel proved excellent ability to anchor the drug, assuring its prolonged release during 24 h. The release rate has been tuned by varying the crosslinking density, reaching 96 % for a high crosslinking density and a 75 % rate for a lower one. A theoretical model in a multi fractal paradigm was developed for understanding the drug release dynamics, considering that these behaviors are described by continuous but nondifferentiable curves. In such a context the irrotational type dynamic of the polymer drug structural units implies the functionality of a multifractal type hydrodynamic formalism. For the unidimensional case of this multifractal type hydrodynamic formalism, we can see that ratio between the differentiable velocity and the non-differentiable one for a certain distance depends in a homographic manner on time. The conditions for the simultaneous dynamics imply the synchronization of the drug release mechanisms at the two scale resolutions, expressed through diffusion functions of multifractal type (the diffusion process depends on the scale resolutions). The model is confirmed by experimental data .