Mathematical Modeling of Water-Soluble Astaxanthin Release from Binary Polysaccharide/Gelatin Blend Matrices

: Water-soluble AstaSana astaxanthin (AST) was loaded into 75/25 blend ﬁlms made of polysaccharides (carboxymethyl cellulose (CMC), gum Arabic (GAR), starch sodium octenyl succinate (OSA), water-soluble soy polysaccharides (WSSP)) and gelatin (GEL) at levels of 0.25, 0.5, and 1%, respectively. Due to the presence of starch granules in the AST formulation, the supplemented ﬁlms exhibited increased surface roughness as compared to the AST-free ﬁlms. Apart from the CMC/GEL carrier, the migration of AST to water (25 ◦ C, 32 h) was incomplete. Excluding the CMC-based carrier, the gradual rise in the AST concentration decreased the release rate. The Hopfenberg with time lag model provided the best ﬁt for all release series data. Based on the quarter-release times (t 25% ), the 0.25% AST-supplemented OSA/GEL ﬁlm (t 25% = 13.34 h) ensured a 1.9, 2.2, and 148.2 slower release compared to the GAR-, WSSP- and CMC-based carriers, respectively. According to the Korsmeyer– Peppas model, the CMC-based ﬁlms offered a quasi-Fickian release of AST ( n < 0.5) with the burst effect (t 100% = 0.5–1 h). In general, the release of AST from the other ﬁlms was multi-mechanistic ( n > 0.5), i.e., controlled at least by Fickian diffusion and the polymer relaxation (erosion) mechanism. The 1% AST-added WSSP/GEL system provided the most linear release proﬁle.


Introduction
Hydrocolloids are commonly used in almost all major dosage forms including tablets, capsules, suspensions, gels, films, and transdermal patches [1]. Currently, plant origin polysaccharides, due to their abundance, renewability, and biodegradability, are often considered as an alternative material for packaging production, including "biodegr-edible" controlled-release packaging, which is a new technology that relies on releasing active compounds (antimicrobials and antioxidants) at desirable rates to extend the shelf life of food [2]. Edible packaging can be used where the application of plastic packaging is limited, e.g., as coating, casings, capsules, micro-and nano-capsules, sachets for single serve products (e.g., coffee, spices, etc.) that dissolve after exposure to heat, layers separating components in complex food, etc.
Every biopolymer has its own characteristic [3,4]. For example, the easily soluble polysaccharides, such as carboxymethyl cellulose (CMC) and pullulan, offer rapid release (burst effect) of bioactive molecules upon contact with the aqueous medium, while various forms of starch carrier matrices can be fabricated for controlled-release purposes [4,5]. The knowledge of release rate is of utmost relevance, hence a rapid release causes fast consumption of the active compounds within a short time, after which the concentration of active compound required for the effective protective action is not maintained on the food

Film Preparation
The films were prepared according to the previous procedure [17]. Briefly, the aqueous 5% (w/w) polysaccharide/GEL solutions containing glycerol (1% w/w) and AST (0.25, 0.5, 1% w/w) were degassed and cast on leveled polytetrafluoroethylene-coated trays with an area of 4 cm 2 . The polysaccharide/GEL ratio was 75/25. A constant amount of total solids (0.0125 g/cm 2 ) was placed on the trays in order to maintain film thickness. The film-forming solutions were dried at 25 ± 2 • C and 50 ± 5% relative humidity (RH) for 24 h. The films were peeled from the trays and conditioned in a MLR-350 climatic test chamber (Sanyo Electric Biomedical Co. Ltd., Osaka, Japan) for 48 h under the conditions as above.

Film Thickness
The thickness of the film samples (4 cm 2 ) was determined using a 547-401 micrometer (Mitotuyo, Tokyo, Japan).

Scanning Electron Microscopy (SEM) and Light Microscopy
SEM was used in order to visualize the surface of the films, AST powder, and 1% AST aqueous solution. After drying in a Polaron CPD 7501 critical point dryer (Quorum Technologies Ltd., East Sussex, UK), the samples were sputter coated (Polaron SC7620, Quorum Technologies Ltd., East Sussex, UK) with 20-30 nm layer of gold palladium (Au/Pd) and examined using LEO 1430 VP scanning electron microscope (LEO Electron Microscopy Ltd., Cambridge, UK) at following presets: accelerating voltage of 15 kV, aperture size: 30 µm, beam current 30 µm, signal: secondary electron detector, chamber pressure: 10 −5 Pa, working distance: 10-13 mm, image resolution: 2048 × 1576 pixel, scanning mode: pixel noise reduction. Additionally, the AST aqueous solution was examined with the Olympus MCKX53 light microscope (Olympus, Tokyo, Japan).
The diameter of the objects was measured using AxioVision Rel. 4.8 software (Carl Zeiss Microscopy GmbH, Göttingen, Germany).

AST Release Study
The film discs (4 cm 2 ) were shaken with distilled water (25 mL, 25 ± 1 • C, 170 rpm, 32 h) in the shaking incubator (Incu-Shaker Mini, Benchmark Scientific, Inc., Sayreville, NJ, USA). 250 µL of the release media samples were taken at different time points and the absorbance was read at 464 nm using a microplate spectrophotometer (EPOCH 2 Microplate Spectrophotometer, BioTek, Winooski, VT, USA). The release tests were limited to 32 h because of a further progressive spoilage of film samples (the presence of a bad odour coming from the acceptor solution). The analyses were performed in triplicate.

Mathematical Modeling
DDSolver, i.e., add-in software for Microsoft Excel, was used for modeling the AST release kinetics. Eight mathematical models (Table 1) were chosen to fit the experimental data. The adjusted coefficient of determination (R 2 adjusted ) was used for the selection of the optimal mathematical models, which were used for the determination of the quarter-release times (t 25% ) [27]. Table 1. Mathematical models used to describe the dissolution curves [27].

Model Equation Parameters
Zero-order with T lag (Z-O T lag )

Model Equation Parameters
Hopfenberg with T lag (Hop T lag ) F is the fraction (%) of drug released in time t. k 0 is the zero-order release constant; k 1 is the first-order release constant; F max is the maximum fraction of the drug released at infinite time; T lag is the lag time prior to drug release. k HB is the combined constant in Hopfenberg model, , where k 0 is the erosion rate constant, C 0 is the initial concentration of drug in the matrix, and a 0 is the initial radius for a sphere or cylinder or the half thickness for a slab; n is 1, 2, and 3 for a slab, cylinder, and sphere, respectively; k KP is the release constant incorporating structural and geometric characteristics of the drug-dosage form; n is the diffusional exponent indicating the drug release mechanism; α is the scale factor in Logistic 1 and 2 models; β is the shape factor in Logistic 1 and 2 models. k MB , n, and k are empirical parameters in Makoid-Banakar model (k MB , n, k > 0); α is the scale parameter which defines the time scale of the process; β is the shape parameter which characterizes the curve as either exponential (β = 1; case 1), sigmoid, S-shaped, with upward curvature followed by a turning point (β > 1; case 2), or parabolic, with a higher initial slope and after that consistent with the exponential (β < 1; case 3); Ti is the location parameter which represents the lag time before the onset of the dissolution or release process and in most cases will be near zero.

Microstructure and Film Thickness
Consistent with the previous study [14], the surface of the control CMC/GEL film revealed the presence of GEL-rich microspheres with diameters ranging from 13.33 to 26.67 µm ( Figure 1). It confirms that CMC and GEL are two incompatible biopolymers [8] that cannot form the homogeneous blend films at the micro-level. In turn, good compatibility (lack of aggregates or undissolved particles) was found for the other polysaccharide/GEL blends. In accordance with the previous studies [7], the surfaces of WSSPcontaining film were rough and uneven, which could be attributed to the weak dissolution of the polysaccharide. The AST-added films were more heterogeneous as compared to the controls ( Figure 1). Since starch is the component of the AST formulation ( Figure 2), it is easy to deduce that the observed round-shaped inequalities were the starch granules. The size of the starch granules in the AST powder and the dehydrated AST solution ranged from 5.71 to 15.71 ( Figure 2), which corresponds to the size of corn starch granules [28].
It was found that the CMC75/GEL25 films with higher AST content (0.5-1%) exhibited higher thickness as compared to other carriers (p < 0.05, Table 2). This result may be explained by the fact that the combination of the phase-separated microspheres with the starch granules (from AST) was unable to form a compact film network ( Figure 1). However, regardless of the carrier type, no difference (p > 0.05) in thickness was observed between the AST-added and control films.

Figures 3 and 4
show the cumulative amount (mg/cm 2 ) and percentage of AST released from the films, respectively. It was found that during 32 h dissolution test, only the CMC/GEL carrier offered complete release. Depending on the AST concentration, the 100% release occurred after 0.5-1 h ( Figure 4). This result could be easily explained by the fact that CMC dissolves rapidly after coming into contact with water [8]. The t 25% values estimated for the CMC-based films ranged from 0.09 to 0.15 h ( Table 2). In the case of the other carriers,~14-89% of the initial dose of AST was released. The slowest release of AST was found for the OSA/GEL (t 25% = from 13.34 to >27.26 h) and the WSSP/GEL (t 25% = 6.16-50.95 h) carriers (Table 2). This result is likely associated with the partial solubility of these films at 25 • C [7]. It should be noted here that since the release of AST from the 1% AST-added OSA-based film was very slow and incomplete, it was not possible to predict t 25% for this system.       Regardless of the AST concentration, the WSSP/GEL film ensured the longest time without AST release (T lag = 50 min, Figure 4). It suggests a strong physicochemical entrapment of the carotenoid in this matrix. This result confirms our earlier observations, which showed that WSSP-based film offered longer T lag (thus slower release) than GARbased carrier [17]. As suggested before, the cause could be the highly branched structure of WSSP [29]. Apart from the 1% AST-added WSSP-based system, the release profiles exhibited a sudden burst of AST. This phenomenon can be attributed to the erosion of the films.
Apart from the CMC-based carrier, the gradual rise in the AST concentration decreased the release rate from the carriers. It seems possible that this result is due to increasing amounts of the starch in the AST-added films; i.e., the starch granules ( Figure 1) inhibited access of water molecules to the polymeric matrix, which limited dissolution of AST [17]. The highly erodible character of CMC probably meant that the presence of starch did not affect the dissolution behavior of the CMC/GEL carrier.
Eight mathematical equations (Table 1) were used to the quantitative interpretation of the data obtained from the ASTA release assay. The R 2 adjusted and the parameter values estimated for the models are presented in Tables 3-6. It was impossible to fit one optimal model to describe the migration of AST from the particular carrier types, however, based on R 2 adjusted averages the Hop T lag model provided the best fit for all release series data (R 2 adjusted mean = 0.9242). It was found that the Hig T lag model was the least useful for AST release prediction (R 2 adjusted mean = 0.6650). Among the tested models, the F-O F max , K-P T lag , Hop T lag , Log 2, and Wb F max were quite suitable for the description of AST release from the CMC-based carrier (R 2 adjusted = 0.9410-0.9890). The Wb F max model ensured the best fit (R 2 adjusted = 0.9610-0.9890). The shape parameter (β) obtained from the Wb F max model, showed that the AST release profile of the 0.25-0.5% AST-added CMC-based films was sigmoid (β > 1), while for the 1% AST-added carrier the release was more parabolic (β < 1). Nevertheless, the differences in the over mentioned shapes of the release curves were barely visible (Figures 4 and 5). model to describe the migration of AST from the particular carrier types, however, based on R 2 adjusted averages the Hop Tlag model provided the best fit for all release series data (R 2 adjusted mean = 0.9242). It was found that the Hig Tlag model was the least useful for AST release prediction (R 2 adjusted mean = 0.6650). Among the tested models, the F-O Fmax, K-P Tlag, Hop Tlag, Log 2, and Wb Fmax were quite suitable for the description of AST release from the CMC-based carrier (R 2 adjusted = 0.9410-0.9890). The Wb Fmax model ensured the best fit (R 2 adjusted = 0.9610-0.9890). The shape parameter (β) obtained from the Wb Fmax model, showed that the AST release profile of the 0.25-0.5% AST-added CMC-based films was sigmoid (β > 1), while for the 1% AST-added carrier the release was more parabolic (β < 1). Nevertheless, the differences in the over mentioned shapes of the release curves were barely visible (Figures 4 and 5).   The M-B Tlag model was the best to predict the release of AST from the GAR-based (R 2 adjusted = 0.9341-0.9380), OSA-based (R 2 adjusted = 0.8861-0.9461) and WSSP-based films (R 2 adjusted = 0.9187-0.9967). Interestingly, since the 1% AST-added WSSP/GEL film offered the most linear release profile (Figures 4 and 5), all mathematical models described fairly well the release from this system (R 2 adjusted = 0.9281-0.9967). The K-P model is often used to analyse the release of the active substance from polymeric dosage forms, when the release mechanism is not well known, or when more than one type of release phenomena could be involved [30]. In planar (thin films) geometry, the released mechanism is described as: (i) quasi-Fickian diffusion (n < 0.5), (ii) Fickian diffusion (n = 0.5), (iii) non-Fickian diffusion (0.5 < n < 1), (iv) case II transport (zero-order release) (n = 1), and (v) super case II transport (n > 1) [31]. It was found that in the case of all CMC-based films, the n values were below 0.5 (Table 5), which implies that release mechanism was   0.007-0.055) is characteristic for the burst release of AST ( Figure 5). This result may be explained by the fact that despite the CMC/GEL film has short initial ability to hold water (swelling ≈ 500%), it rapidly dissolves (due to high content of easily soluble polysaccharide fraction) [8], which results in a rapid dissolution of AST in the aqueous medium. This result supports evidence from previous observations [8], which showed that the release of potassium salts of iso-α-acids from the films based on CMC/GEL blends was beyond the limits of the K-P "power law" model (n < 0.5). The release of AST from the 0.25-0.5% AST-added WSSP-based carrier and was followed by a non-Fickian mechanism. It shows that the AST migration was governed by diffusion and controlled swelling, whose rate was similar. The rearrangement of the polymeric chains occurred slowly, and the diffusion simultaneously caused the time-depended anomalous release [33].
For the GAR-and OSA-based films (regardless of AST concentration), the n value was above 1, which is an indicator of the extreme form of transport (super case II model), The M-B T lag model was the best to predict the release of AST from the GAR-based (R 2 adjusted = 0.9341-0.9380), OSA-based (R 2 adjusted = 0.8861-0.9461) and WSSP-based films (R 2 adjusted = 0.9187-0.9967). Interestingly, since the 1% AST-added WSSP/GEL film offered the most linear release profile (Figures 4 and 5), all mathematical models described fairly well the release from this system (R 2 adjusted = 0.9281-0.9967). The K-P model is often used to analyse the release of the active substance from polymeric dosage forms, when the release mechanism is not well known, or when more than one type of release phenomena could be involved [30]. In planar (thin films) geometry, the released mechanism is described as: (i) quasi-Fickian diffusion (n < 0.5), (ii) Fickian diffusion (n = 0.5), (iii) non-Fickian diffusion (0.5 < n < 1), (iv) case II transport (zero-order release) (n = 1), and (v) super case II transport (n > 1) [31]. It was found that in the case of all CMC-based films, the n values were below 0.5 (Table 5), which implies that release mechanism was non-swellable matrix-diffusion [8,32]. It shows that the very small release exponent (n = 0.007-0.055) is characteristic for the burst release of AST ( Figure 5). This result may be explained by the fact that despite the CMC/GEL film has short initial ability to hold water (swelling ≈ 500%), it rapidly dissolves (due to high content of easily soluble polysaccharide fraction) [8], which results in a rapid dissolution of AST in the aqueous medium. This result supports evidence from previous observations [8], which showed that the release of potassium salts of iso-α-acids from the films based on CMC/GEL blends was beyond the limits of the K-P "power law" model (n < 0.5).
The release of AST from the 0.25-0.5% AST-added WSSP-based carrier and was followed by a non-Fickian mechanism. It shows that the AST migration was governed by diffusion and controlled swelling, whose rate was similar. The rearrangement of the polymeric chains occurred slowly, and the diffusion simultaneously caused the timedepended anomalous release [33].
For the GAR-and OSA-based films (regardless of AST concentration), the n value was above 1, which is an indicator of the extreme form of transport (super case II model), i.e., more than one mechanism (swelling, polymer chain disentanglement (relaxation), erosion) was involved in the AST release kinetics. Regarding the GAR-based carrier, the obtained result complies with the previous findings [34,35], which suggested that gelling properties of GAR (at high polymer concentrations) ensure the sustained drug release. Thus, it can be concluded that the water entrapped in the gel strongly affected the diffusional behavior of AST in the GAR-based films. In turn, limited release of AST from the OSA/GEL carrier (~19-46%, depending on the AST concentration) could be attributed to the encapsulation of ASX in the OSA/GEL complexes (coacervates) [36].
As for the 1% AST-added WSSP-based film, it is possible that the bulky amounts of starch granules (from AST) acted as a release modifier. It is known that native starch has multifunctional uses in the different physical forms of carriers serving as the binder, disintegrant, diluents, glidant, and lubricant [37]. Therefore, it is possible that the granules anchored in the film matrix (Figure 1), hindered the contact of dissolution media with WSSP/GEL matrix, which consequently limited erosion of the carrier, thus ASX release was predominately controlled by diffusion (n < 0.5, Table 5).

Conclusions
The CMC-based carrier offers a quasi-Fickian release of AST with the burst effect (100% release occurred after 0.5-1 h, depending on the AST concentration). According to the t 25% , at the lowest AST concentration (0.25%), the OSA/GEL film offered about 1.9, 2.2, and 148.2 times slower release compared to the GAR-, WSSP-and CMC-based carriers, respectively. The 1% AST-added WSSP-based system had the most linear AST release profile (close to the Z-O kinetics). It suggests that this formulation applied on the high moisture food will maintain the main dose of the carotenoid on the surface, where the oxidation reactions occur the most intensively. It was found that the Hopfenberg with T lag model provided the best fit for all release series data. The Korsmeyer-Peppas release exponent (n), for the GAR-, OSA-, and WSSP-based films was >0.5, confirming that these carriers ensured more multi-mechanistic AST release than the CMC-based film (n < 0.5).
In summary, this study showed that apart from the CMC/GEL carrier, the migration of AST from the films into the water at 25 • C (the highest predictable food/packaging contact temperature) was incomplete, although the compound was released for more than one day. The release tests were limited only to 32 h because of a further progressive spoilage of film samples (the presence of a bad odour coming from the acceptor solution). Therefore, the accelerated release testing at increased temperature (e.g., 30 • C) can be recommended to rapidly assess and predict "real-time" AST release profiles from the obtained films.
In order to facilitate the selection of suitable candidate material for a particular packaging application, further experimental investigations are needed to estimate the effect of increasing concentration of AST on the optical, mechanical, and barrier properties of the binary polysaccharide/GEL blend films.