# Theoretical Importance of PVP-Alginate Hydrogels Structure on Drug Release Kinetics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

_{e}). Accordingly, the dependence of the elastic (G′) and viscous (G″) moduli on pulsation, ω = 2πf, is given by the following expressions:

_{R}is the number of Maxwell elements considered, g

_{i}, η

_{i}, and λ

_{i}represent, respectively, the elastic modulus, the viscosity, and the relaxation time of the ith Maxwell element. The equilibrium modulus, g

_{e}, measures the contribution of the purely elastic element. The simultaneous fitting of Equations (1) and (2) to experimental G′ and G″ data was performed assuming that relaxation times (λ

_{i}) were scaled by a factor 10 (λ

_{i+1}= 10λ

_{i}) [1]. Hence, the parameters of the model were 2 + n

_{R}(i.e., λ

_{1}, g

_{e}+ g

_{i}(1 ≤ i ≤ n

_{R})). Based on a statistical procedure [21], n

_{R}was selected in order to minimize the product, χ

^{2}*(2 + n

_{R}), where χ

^{2}is the sum of the squared errors. Three to four Maxwell elements plus the purely elastic element guarantee a statistically reliable fitting (positive F-test). Table 1 reports the fitting parameters values.

_{i}plus g

_{e}[22], Flory theory [23] allows the determination of the polymeric network crosslink density (ρ

_{x}), defined as the moles of junction points between different polymeric chains per hydrogel unit volume:

_{x}and the average mesh size of the polymeric network (ξ) is provided by the equivalent network theory [24]. As it is very difficult to account for all the irregularities of a real polymeric network, this theory replaces the real network structure by an idealized one constituted by a perfect three-dimensional cubic arrangement of crosslinks. The common aspect of the real architecture and idealized one relies on the same average, ρ

_{x}. By labeling with ξ the space occurring between two consecutive crosslinks and using the crosslink density definition, it is possible to establish a simple connection between ξ and ρ

_{x}. Indeed, the volume ((4/3)π(ξ/2)

^{3}) associated to each crosslink in both architectures has to be equal to 1/(N

_{A}ρ

_{x}), with N

_{A}the Avogadro number. Consequently, ρ

_{x}and ξ are connected through the following relation:

_{s}) of the xy component of the global magnetization vector. On the contrary, at fixed irradiation times, the effect of the concentration seems insignificant for hydrogels B and C, while, hydrogel 30A is characterized by a slower relaxation time with respect to hydrogel 20A. In order to determine the relaxation times’ distribution (A

_{i}, T

_{2i}), the free induction decay (FID) time decay (I

_{s}) (see Figure 2a,b), related to the extinction of the x–y component of the magnetization vector, was fitted according to its theoretical estimation, I(t) [25]:

_{i}is the pre-exponential factor (dimensionless), proportional to the number of protons relaxing with the relaxation time, T

_{2i}, and A

_{i%}is the percentage value of A

_{i}. The average relaxation time of protons, T

_{2m}, and its inverse value, (1/T

_{2})

_{m}, are defined as:

_{2i}) constituting the relaxation time distribution (A

_{i}, T

_{2i}). m was determined by minimizing the product, χ

^{2}*(2m), where χ

^{2}is the sum of the squared errors and 2m represents the number of fitting parameters of Equation (5) [21]. An inspection of Table 3, showing the values of the fitting parameters from Equation (5) makes it clear that two relaxation times (T

_{21}and T

_{22}) are always necessary to describe the hydrogen relaxation of the six specimen studied.

_{2m}) and the polymeric network mesh size (ξ). Indeed, it is well known [25,26,27] that the average value of the inverse relaxation time (1/T

_{2})

_{m}(see Equation (6)) referring to hydrogens belonging to water molecules trapped in the polymeric network is related to the polymer volume fraction, φ, and ξ, by the following equation:

_{2H2O}is the relaxation time of the free water hydrogens (≈ 3008 ms at 25 °C, 20 MHz [28]), i.e., the relaxation time of water hydrogens in the absence of polymer network. $\mathcal{M}$ is a parameter, called relaxivity, representing the ratio between the thickness and the relaxation time of the water layer close to polymeric chains (bound water layer) while C

_{0}and C

_{1}are two constants that, for a cubical network, are equal to 1 and 3π, respectively. In addition, ξ is related to φ and to the polymeric chains radius, R

_{f}, by the following equation:

_{2})

_{m}can be evaluated by fitting Equation (5) to the experimental relaxation data shown in Figure 2a,b (see Table 3), φ is unknown. Indeed, not only does hydrogel realization imply a washing step, aimed to remove un-crosslinked polymer, but alginate ionotropic crosslinking induces hydrogel syneresis. Accordingly, the φ real value can be very different from its nominal value, as evaluated from the % polymer mass percentage. Thus, φ is evaluated on the basis of Equation (7), assuming that the $\mathcal{M}$ value referring to our hydrogels, composed by PVP and alginate, is the same as the pure PVP (7.7 × 10

^{−3}nm/ms) [22]. This assumption is reasonable since the amount of alginate is very small compared to the PVP one (1/40 or 1/60). Then, Equation (8) allows the evaluation of R

_{f}. The relations between T

_{2m}, ξ, and φ descending from this analysis can be seen by looking at Figure 3.

_{2m}, ξ, and φ values is Gaussian (as demonstrated by the Kolmogorov-Smirnov test), the analysis of their inter-correlations was performed according to the Pearson correlation coefficient, r. Interestingly, Figure 3 shows a strong inverse linear correlation between T

_{2m}and φ (r = −0.976, t

_{r}(4, 0.95) < 8.5; correlation coefficient ρ = 0.92, t

_{ρ}(4, 0.95) < 4.6, F(4, 3, 0.95) > 2). On the contrary, no correlation exists between ξ and φ and between T

_{2m}and ξ. The inverse linear correlation between T

_{2m}and φ is theoretically correct as the φ increase implies the increase of the solid surface, represented by the polymeric chains’ surface, in contact with the water molecules. In turn, the solid surface is responsible for faster hydrogen relaxation [27]. On the contrary, the absence of a correlation between ξ and φ and between T

_{2m}and ξ simply states that, whatever φ, the mesh size is almost constant. This is the result of the counteracting phenomena connected to polymer removal (due to hydrogel washing, see Section 4.3) and hydrogel syneresis caused by alginate ionotropic crosslinking. As a matter of fact, these phenomena imply a reduction of φ with respect to the theoretical value for hydrogels 30A, 20A, and 30B and a φ increase for the others (20B, 30C, and 20C).

_{21}, and relative abundance, A

_{1%}, and the second by the relaxation time, T

_{22}, and relative abundance, A

_{2%}(see Table 3). In other words, assuming that $\mathcal{M}$ does not depend on the mesh size [25], Equation (7) holds also for each single population:

_{i}and (1/T

_{2})

_{i}represent the mesh size and the inverse of the relaxation time pertaining to the population or class ith, respectively. Thus, Equation (9) allows an evaluation of ξ

_{1}and ξ

_{2}, whose relative abundances are, respectively, A

_{1%}and A

_{2%}as reported in Table 4.

_{0}is the drug diffusion coefficient in pure water, r

_{s}is the radius of the sphere embodying the drug molecule, and Y is a parameter that can be set to be equal to 1 for correlation purposes [30] even if, in general, it can take higher values [29,31]. Assuming Y = 1 in Equation (10), Figure 5 shows the variation of the ratio of D/D

_{0}assuming drugs of different dimension (theophylline, r

_{s}= 0.39 nm; vitamin B

_{12}, r

_{s}= 0.86 nm; myoglobin, r

_{s}= 1.89 nm; bovine serum albumin (BSA) r

_{s}= 3.6 nm; immunoglobulin G r

_{s}= 5.63 nm [32]) and the ξ–φ couples competing with our six hydrogels (see Figure 3 and Table 4).

_{0}ratio, while a reduction of the r

_{s}/ξ ratio causes an increase of D/D

_{0}at fixed φ. It is important to underline that both φ and ξ variation, within the experimental field of this study, causes considerable variations of D/D

_{0}(up to 40%) that, in turn, affect the drug release kinetics. The same holds for r

_{s}, whose accuracy in estimation is fundamental to achieve a reliable evaluation of D. Experimentally, the Stokes radius, r

_{s}, is commonly obtained by hydrodynamics techniques. Centrifugation/sedimentation, size exclusion chromatography (SEC), and electrophoresis are routinely used [33,34,35]. Dynamic light scattering (DLS) and NMR spectroscopy-based techniques are other analytical methods generally employed to measure diffusion coefficients and derive the corresponding hydrodynamic radius from the Stokes-Einstein equation [36,37,38]. Alternatively, theoretical and molecular models, alone [39,40] or in combination with experimental data (e.g., small-angle X-ray scattering (SAXS), Small-angle neutron scattering (SANS), high-resolution NMR, X-ray crystallography, cryogenic electron microscopy (cryo-EM)) [41], can be chosen to predict the hydrodynamic properties of proteins, macromolecules, drugs, or nanoparticles. The HYDRO suite of algorithms [42] deserves special mention, which allows the calculation of hydrodynamic properties simply from medium or high resolution models (e.g., a coordinate file) with substantial accuracy. We report here, for comparison, the values of r

_{s}calculated for some of the substances considered in Figure 5 using the HYDROPRO module, starting from atomic coordinates: Theophylline, r

_{s}= 0.37 nm, vitamin B

_{12}, r

_{s}= 0.84 nm; myoglobin, r

_{s}= 1.86 nm; BSA r

_{s}= 3.5 nm [43,44]. Thus, this tool is particularly suited to the design and screening of molecules when experimental information is missing or scarce.

_{i}(1 ≤ i ≤ n, relative abundance, A

_{i%}/100), for each mesh class, Equation (10) reads:

_{i}value corresponding to the mesh size ξ

_{i}is equal to A

_{i%}/100. A simpler way to proceed is to assume that the mass flux, J, is the sum of n contributes, each one weighting for A

_{i%}/100:

^{3}) was suspended by a web in a stirred release environment containing 10 cm

^{3}of distilled water at 37 °C. At fixed times, the myoglobin concentration was spectrophotemetrically detected (see Section 4.5 for more details) so that the release profile shown in Figure 7 was obtained. The experimental release kinetics was fitted by a classical model relying on Fick’s law in the presence of a finite volume environment and assuming that drug release occurs only in the axial direction (radial diffusion was retained as negligible due to the small lateral surface) [32]:

_{t}

^{+}is the ratio between the amount of drug released at time t and the amount released after an infinite time, L is the gel semi-thickness, D is the myoglobin diffusion coefficient in the hydrogel while q

_{i}is the non-zero positive roots of:

_{r}is the release environment volume. Equation (15) was solved according to the bisection method fixing the tolerance to 10

^{−5}. Equation (14) best fitting (see solid line in Figure 7), assuming 100 terms in the summation (further terms revealed to be unnecessary), was statistically satisfactory (F(1, 18, 0,95) < 318) and yielded to D = (2.3 ± 0.13) × 10

^{−9}cm

^{2}/s. Remembering that the myoglobin diffusion coefficient in pure water at 37 °C is equal to 1.16 × 10

^{−7}cm

^{2}/s [32], it turns out that the ratio of D/D

_{0}for myoglobin is equal to 0.0195. Inserting this ratio in Equation (10), considering the myoglobin radius (1.91 nm) and the ξ–φ couple pertaining to hydrogel 20C (ξ = 18.6 nm, φ = 0.36), Y has to be equal to 6.57. As this value lies in between typical values found in the literature [29,31] (2.8 < Y < 30) for similar drugs, we can conclude that the approach used to characterize the hydrogel network (mesh size and polymer volume fraction) is reliable.

## 3. Conclusions

_{s}(solute radius) on drug release. Finally, in order to prove the reliability of the entire approach, experimental release data regarding myoglobin release from one of the studied hydrogels (20C) were fitted by means of Fick’s second law accounting for a finite release environment. Accordingly, the myoglobin diffusion coefficient was estimated for inside the hydrogel and, thus, the ratio of D/D

_{0}(D

_{0}is the myoglobin diffusion coefficient in water). It was verified that the Lustig-Peppas equation (Equation (10)) yields the same D/D

_{0}ratio (0.0195) provided that Y = 6.57, thus it was concluded that the entire approach is reliable as, for similar drugs, Y ranges between 2.8 and 30 [29].

## 4. Materials and Methods

^{5}Da), hydrogen peroxide (30% wt) and sodium alginate (~10

^{6}Da; high α-l-guluronic acid content ~70%) were provided by Sigma-Aldrich (Saint Louis, MO, USA). All the materials were used as received from the supplier and no further purifications were performed.

#### 4.1. Interpenetrating Polymeric Network (IPN) Preparation

_{2}O

_{2}(2% v/v) was added under 250 rpm, stirring at 50 °C, until a homogeneous mixing was achieved. H

_{2}O

_{2}plays the role of the photoinitiator. The UV radiation splits the H

_{2}O

_{2}molecule into OH radicals which attack the polymer chains generating macroradicals. These latter propagate the reaction, creating covalent bonds (cross-linking points), which build up the polymer network. Solutions were prepared fixing the PVP concentration (% mass fraction) at 20% and 30% while the alginate concentration (% mass fraction) was always set at 0.5%. After preparation, polymer solutions were poured into syringes and left for 24 h at room temperature for the removal of bubbles.

^{2}, samples A, B, and C received, respectively, an irradiated energy equal to 67 J, 100 J, and 134 J. Subsequently, hydrogel samples were removed from the molds with a spatula and placed in a Petri dish, where fixed volumes of a CaCl

_{2}aqueous solution ([Ca

^{++}] = 9 g/L) were sprayed onto their surface to get alginate ionotropic gelation. Alginate crosslinking was allowed to last for 5 min. After crosslinking, hydrogels were dipped in deionized water under stirring to extract un-crosslinked polymer and photoinitiator (washing).

#### 4.2. Rheology

#### 4.3. LF-NMR

_{2m}) was performed according to the CPMG (Carr–Purcell–Meiboom–Gill) [46] sequence {90°[−τ−180°−τ(echo)]n−T

_{R}} with a 8.36 μs wide 90° pulse, echo time τ = 250 μs, and T

_{R}(sequences repetition rate) equal to 5 s. The criterion adopted to choose n consisted in ensuring that the final FID (free induction decay) intensity corresponded to approximately 1% of the initial FID intensity. In the light of this acquisition strategy, n spanned between 235 and 540. Finally, each FID decay, composed by n points, was repeated 36 times (number of scans).

#### 4.4. TEM

#### 4.5. Release Test

_{r}) containing 10 mL of distilled water at 37 °C. Stirring was ensured by a magnetic stirrer at 100 rpm. At established time intervals, 3 μL of dissolution medium were collected to measure myoglobin concentration by a UV spectrophotometer (408 nm, path-lengths of 20 mm). Release experiments were performed in duplicate.

## Author Contributions

## Funding

## Conflicts of Interest

## References and Note

- Lapasin, R.; Pricl, S. Rheology of Industrial Polysaccharides: Theory and Applications; Blackie Academic & Professional: London, UK, 1995; Chapter 1. [Google Scholar]
- Lin, C.-C.; Metters, A.T. Hydrogels in controlled release formulations: Network design and mathematical modeling. Adv. Drug Deliv. Rev.
**2006**, 58, 1379–1408. [Google Scholar] [CrossRef] - Ahmed, E.M. Hydrogel: Preparation, characterization, and applications: A review. J. Adv. Res.
**2015**, 6, 105–121. [Google Scholar] [CrossRef] - Posocco, B.; Dreussi, E.; de Santa, J.; Toffoli, G.; Abrami, M.; Musiani, F.; Grassi, M.; Farra, R.; Tonon, F.; Grassi, G.; et al. Polysaccharides for the delivery of antitumor drugs. Materials
**2015**, 8, 2569–2615. [Google Scholar] [CrossRef] - Matricardi, P.; Alhaique, F.; Coviello, T. Introduction. In Polysaccharide Hydrogels: Characterization and Biomedical Applications; Matricardi, P., Alhaique, F., Coviello, T., Eds.; Pan Standford Publishing: Singapore, 2015; Chapter 1. [Google Scholar]
- Matricardi, M.; di Meo, C.; Coviello, C.; Hennink, W.E.; Alhaique, F. Interpenetrating polymer networks polysaccharide hydrogels for drug delivery and tissue engineering. Adv. Drug Deliv. Rev.
**2013**, 65, 1172–1187. [Google Scholar] [CrossRef] - Grassi, M.; Grassi, G. Application of mathematical modelling in sustained release delivery systems. Expert Opin. Drug Deliv.
**2014**, 11, 1299–1321. [Google Scholar] [CrossRef] - Grassi, M.; Lapasin, R.; Pricl, S. Modeling of drug release from a swellable matrix. Chem. Eng. Commun.
**1998**, 169, 79–109. [Google Scholar] [CrossRef] - Miller-Chou, A.A.; Koening, J.K. A review of polymer dissolution. Prog. Polym. Sci.
**2003**, 28, 1223–1270. [Google Scholar] [CrossRef] - Hasa, D.; Voinovich, D.; Perissutti, B.; Grassi, M.; Bonifacio, A.; Sergo, V.; Cepek, C.; Chierotti, M.R.; Gobetto, R.; Dall’Acqua, S.; et al. Enhanced oral bioavailability of vinpocetine through mechanochemical salt formation: Physico-chemical characterization and in vivo studies. Pharm. Res.
**2011**, 28, 1870–1880. [Google Scholar] [CrossRef] - Grassi, M.; Colombo, I.; Lapasin, R. Drug release from an ensemble of swellable crosslinked polymer particles. J. Control. Release
**2000**, 68, 97–113. [Google Scholar] [CrossRef] - Coceani, N.; Magarotto, L.; Ceschia, D.; Colombo, I.; Grassi, M. Theoretical and experimental analysis of drug release from an ensemble of polymeric particles containing amorphous and nano-crystalline drug. Chem. Eng. Sci.
**2012**, 71, 345–355. [Google Scholar] [CrossRef] - Grassi, G.; Hasa, H.; Voinovich, D.; Perissutti, B.; Dapas, B.; Farra, R.; Franceschinis, E.; Grassi, M. Simultaneous release and ADME processes of poorly water-soluble drugs: Mathematical modeling. Mol. Pharm.
**2010**, 7, 1488–1497. [Google Scholar] [CrossRef] - Singh, M.; Lumpkin, J.; Rosenblat, J. Mathematical modeling of drug release from hydrogel matrices via a diffusion coupled with desorption mechanism. J. Control. Release
**1994**, 32, 17–25. [Google Scholar] [CrossRef] - Singh, M.; Lumpkin, J.; Rosenblat, J. Effect of electrostatic interactions on polylysine release rates from collagen matrices and comparison with model predictions. J. Control. Release
**1995**, 35, 165–179. [Google Scholar] [CrossRef] - Huang, J.; Wigent, R.J.; Schwartz, J.B. Drug-polymer interaction and its significance on the physical stability of nifedipine amorphous dispersion in microparticles of an ammonio methacrylate copolymer and ethylcellulose binary blend. J. Pharm. Sci.
**2008**, 97, 251–262. [Google Scholar] [CrossRef] - Paulsson, M.; Edsman, K. Controlled drug release from gels using lipophilic interactions of charged substances with surfactants and polymers. J. Colloid Interface Sci.
**2002**, 248, 194–200. [Google Scholar] [CrossRef] [PubMed] - Byrne, M.E.; Park, K.; Peppas, N.A. Molecular imprinting within hydrogels. Adv. Drug Deliv. Rev.
**2002**, 54, 149–161. [Google Scholar] [CrossRef] - Milcovich, G.; Lettieri, S.; Antunes, F.E.; Medronho, B.; Fonseca, A.C.; Coelho, J.F.G.; Marizza, P.; Perrone, F.; Farra, R.; Dapas, B.; et al. Recent advances in smart biotechnology: Hydrogels and nanocarriers for tailored bioactive molecules depot. Adv. Colloid Interface Sci.
**2017**, 249, 163–180. [Google Scholar] [CrossRef] - Fanesi, G.; Abrami, M.; Zecchin, F.; Giassi, I.; Dal Ferro, E.; Boisen, A.; Grassi, G.; Bertoncin, P.; Grassi, M.; Marizza, P. Combined Used of Rheology and LF-NMR for the Characterization of PVP-Alginates Gels Containing Liposomes. Pharm. Res.
**2018**, 35, 171–182. [Google Scholar] [CrossRef] - Draper, N.R.; Smith, H. Applied Regression Analysis; John Wiley & Sons Inc.: New York, NY, USA, 1966. [Google Scholar]
- Marizza, P.; Abrami, M.; Keller, S.S.; Posocco, P.; Laurini, E.; Goswami, K.; Skov, A.L.; Boisen, A.; Larobina, D.; Grassi, G.; et al. Synthesis and characterization of UV photocrosslinkable hydrogels with poly(N-vinyl-2-pyrrolidone): Determination of the network mesh size distribution. J. Polym. Mater. Polym. Biomater.
**2016**, 65, 516–525. [Google Scholar] [CrossRef] - Flory, P.J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, USA, 1953. [Google Scholar]
- Schurz, J. Rheology of polymer solutions of the network type. Prog. Polym. Sci.
**1991**, 16, 1–53. [Google Scholar] [CrossRef] - Chui, M.; Phillips, R. Measurement of the porous microstructure of hydrogels by nuclear magnetic resonance. J. Colloid Interface Sci.
**1995**, 174, 336–344. [Google Scholar] [CrossRef] - Scherer, G.W. Hydraulic radius and mesh size of gels. J. Sol Gel Sci. Technol.
**1994**, 1, 285–291. [Google Scholar] [CrossRef] - Abrami, M.; Chiarappa, G.; Farra, R.; Grassi, G.; Marizza, P.; Grassi, M. Use of low field NMR for the characterization of gels and biological tissues. Admet Dmpk
**2018**, 6, 34–46. [Google Scholar] [CrossRef] - Coviello, T.; Matricardi, P.; Alhaique, F.; Farra, R.; Tesei, G.; Fiorentino, S.; Asaro, F.; Milcovich, G.; Grassi, M. Guar gum/borax hydrogel: Rheological, low field NMR and release characterizations. Express Polym. Lett.
**2013**, 7, 733–746. [Google Scholar] [CrossRef] - Amsden, B. Solute Diffusion within Hydrogels. Mechanisms and Models. Macromolecules
**1998**, 31, 8382–8395. [Google Scholar] [CrossRef] - Lustig, S.R.; Peppas, N.A. Solute diffusion in swollen membranes. IX. Scaling laws for solute diffusion in gels. J. Appl. Polym. Sci.
**1988**, 36, 735–747. [Google Scholar] [CrossRef] - Dalmoro, A.; Barba, A.A.; Lamberti, G.; Grassi, M.; d’Amore, M. Pharmaceutical Applications of Biocompatible Polymer Blends Containing Sodium Alginate. Adv. Polym. Technol.
**2012**, 31, 219–230. [Google Scholar] [CrossRef] - Grassi, M.; Grassi, G.; Lapasin, R.; Colombo, I. Understanding Drug Release and Absorption Mechanisms a Physical and Mathematical Approach; CRC Press Imprint of Taylor & Francis: Boca Raton, FL, USA, 2007; Chapter 4. [Google Scholar]
- Luo, Z.; Zhang, G. Scaling for sedimentation and diffusion of poly(ethylene glycol) in water. J. Phys. Chem. B
**2009**, 113, 12462–12465. [Google Scholar] [CrossRef] - Hong, P.; Koza, S.; Bouvier, E.S.P. Size-exclusion chromatography for the analysis of protein biotherapeutics and their aggregates. J. Liq. Chromatogr. Release Technol.
**2012**, 35, 2923–2950. [Google Scholar] - Stellwagen, N.C.; Magnusdottir, S.; Gelfi, C.; Righetti, P.G. Measuring the translational diffusion coefficients of small DNA molecules by capillary electrophoresis. Biopolymers
**2001**, 58, 390–397. [Google Scholar] [CrossRef] - Patil, S.M.; Keire, D.A.; Chen, K. Comparison of NMR and dynamic light scattering for measuring diffusion coefficients of formulated insulin: Implications for particle size distribution measurements in drug products. AAPS J.
**2017**, 19, 1760–1766. [Google Scholar] [CrossRef] [PubMed] - Stetefeld, J.; McKenna, S.A.; Patel, T.R. Dynamic light scattering: A practical guide and applications in biomedical sciences. Biophys. Rev.
**2016**, 8, 409–427. [Google Scholar] [CrossRef] - Macchioni, A.; Ciancaleoni, G.; Zuccaccia, C.; Zuccaccia, D. Determining accurate molecular sizes in solution through NMR diffusion spectroscopy. Chem. Soc. Rev.
**2008**, 37, 479–489. [Google Scholar] [CrossRef] [PubMed] - Wang, J.; Hou, T. Application of molecular dynamics simulations in molecular property prediction ii: Diffusion coefficient. J. Comput. Chem.
**2011**, 32, 3505–3519. [Google Scholar] [CrossRef] - Weiss, L.B.; Dahirel, V.; Marry, V.; Jardat, M. Computation of the hydrodynamic radius of charged nanoparticles from nonequilibrium molecular dynamics. J. Phys. Chem. B
**2018**, 122, 5940–5950. [Google Scholar] [CrossRef] [PubMed] - Durchschlag, H.; Zipper, P. Modeling the hydration of proteins: Prediction of structural and hydrodynamic parameters from X-ray diffraction and scattering data. Eur. Biophys. J.
**2003**, 32, 487–502. [Google Scholar] [CrossRef] - De la Torre, J.G. The HYDRO software suite for the prediction of solution properties of rigid and flexible macromolecules and nanoparticles. In Analytical Ultracentrifugation: Instrumentation, Software, and Applications; Uchiyama, S., Arisaka, F., Stafford, W.F., Laue, T., Eds.; Springer: Tokyo, Japan, 2016; pp. 195–217. [Google Scholar]
- Atomic coordinates: myoglobin, PDB ID: 1VXB; BSA, PDB ID: 3V03. Vitamin B12 and theophylline molecular models were built and optimized according to the computational procedure described in Reference [44].
- Marson, D.; Laurini, E.; Posocco, P.; Fermeglia, M.; Pricl, S. Cationic carbosilane dendrimers and oligonucleotide binding: An energetic affair. Nanoscale
**2015**, 7, 3876–3887. [Google Scholar] [CrossRef] - Kuijpers, A.J.; Engbers, G.H.M.; Feijen, J.; de Smed, S.C.; Meyvis, T.K.L.; Demeester, J.; Krijgsveld, J.; Zaat, S.A.J.; Dankert, J. Characterization of the network structure of carbodiimide cross-linked gelatin gels. Macromolecules
**1999**, 32, 3325–3334. [Google Scholar] [CrossRef] - Meiboom, S.; Gill, D. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. Instrum.
**1958**, 29, 688–691. [Google Scholar] [CrossRef] - Desmon, K. Techniques for Electron Microscopy; Blackwell Scientific Publication: Oxford, UK; Edinburg, TX, USA, 1965. [Google Scholar]
- Sabatini, D.D.; Bensch, K.; Barnett, R.J. Cytochemistry and electron microscopy. The preservation of cellular ultrastructure and enzymatic activity by aldehyde fixation. J. Cell. Biol.
**1963**, 17, 19–58. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Mechanical spectra referring to hydrogels, 20A (polymer mass percentage = 20.5% and 22 min irradiation), 20B (% polymer mass fraction = 20.5% and 33 min irradiation), and 20C (% polymer mass fraction = 20.5% and 44 min irradiation). (

**b**) Mechanical spectra referring to hydrogels, 30A (% polymer mass percentage = 30.5% and 22 min irradiation), 30B (% polymer mass percentage = 30.5% and 33 min irradiation), and 30C (% polymer mass percentage = 30.5% and 44 min irradiation). Filled symbols represent the storage modulus, G′, while open symbols indicate the loss modulus, G″. Solid lines are the best fitting of the generalized Maxwell model (Equations (1)–(2)).

**Figure 2.**Experimental relaxation behavior of the modulus (I

_{s}) of the xy component of the magnetization vector referring to (

**a**) hydrogels 20A (% polymer mass percentage = 20.5% and 22 min irradiation), 20B (% polymer mass percentage = 20.5% and 33 min irradiation), and 20C (% polymer mass percentage = 20.5% and 44 min irradiation); and (

**b**) hydrogels 30A (% polymer mass percentage = 30.5% and 22 min irradiation), 30B (% polymer mass percentage = 30.5% and 33 min irradiation), and 30C (% polymer mass percentage = 30.5% and 44 min irradiation). Equation (5) best fitting is not visible as it practically coincides with the experimental data. t is time.

**Figure 3.**Dependence of the mean relaxation time of water hydrogens (T

_{2m}) and mesh size (ξ) on the polymer volume fraction, φ, referring to the different hydrogels considered (20A, 20B, 20C, 30A, 30B, and 30C). Solid lines indicate the linear interpolation of the experimental data.

**Figure 4.**Transmission electron microscope (TEM) picture of hydrogel 20B. Gray ribbons represent polyvinylpyrrolidone (PVP) chains (the contrast agent weakly colors PVP chains), while dark zones represent crosslinked alginate strongly colored by the contrast agent.

**Figure 6.**Time evolution of the dimensionless amount of bovine serum albumin (BSA) released from a spherical matrix (radius = 0.5 mm; M

_{t}

^{+}= ratio between the amount of drug released at time t and the amount released after an infinite time). For BSA, D

_{0}(37 °C, water) = 6.35 × 10

^{−8}cm

^{2}/s [32].

**Figure 7.**Time evolution of the dimensionless amount of myoglobin released from a cylindrical 20C hydrogel (M

_{t}

^{+}is the ratio between the amount of drug released at time t and the amount released after an infinite time). Open circles indicate experimental data while the solid line is Equation (14) best fitting. Vertical bars represent datum standard error.

**Table 1.**Generalized Maxwell model (Equations (1)−(2)) fitting parameters referring to the data (G′ and G″) shown in Figure 1a,b. While λ

_{1}indicates the first Maxwell element relaxation time, g

_{i}represents the spring constant value of the ith Maxwell element. Finally, g

_{e}represents the purely elastic element.

System | λ_{1} (s) | g_{1} (Pa) | g_{2} (Pa) | g_{3} (Pa) | g_{4} (Pa) | g_{e} (Pa) |
---|---|---|---|---|---|---|

20A | (9.2 ± 2) × 10^{−2} | 8.2 ± 1.2 | 7.7 ± 1.4 | 15.1 ± 1.7 | - | 364 ± 18 |

20B | (5.9 ± 0.9) × 10^{−2} | 10.4 ± 0.8 | 11.1 ± 0.7 | 4.9 ± 0.9 | 17.3 ± 3.0 | 279 ± 9 |

20C | (10.0 ± 1) × 10^{−2} | 39.3 ± 1.3 | 25.5±1.5 | 19.5 ± 4.0 | 178 ± 23.0 | 854 ± 25 |

30A | (6.7 ± 3) × 10^{−2} | 13.0 ± 2 | 10.6 ± 2.0 | 9.1 ± 3.0 | 8.2 ± 8.0 | 337 ± 21 |

30B | (2.9 ± 2.6) × 10^{−2} | 200.6 ± 79 | 48.6 ± 26 | 16.0 ± 7.0 | 19.4 ± 7.6 | 453 ± 30 |

30C | (9.1 ± 1.1) × 10^{−2} | 136.6 ± 6 | 30.0 ± 5.0 | 33.0 ± 7.0 | 95.5 ± 25 | 1041 ± 32 |

**Table 2.**Shear modulus (G), crosslink density (ρ

_{x}), and average mesh size (ξ) referring to the six hydrogel considered in this paper.

System | G (Pa) | ρ_{x} (mol/cm^{3}) | ξ (nm) |
---|---|---|---|

20A | 395 ± 28 | (1.7 ± 0.10) × 10^{−7} | 26.0 ± 0.6 |

20B | 434 ± 8 | (1.9 ± 0.03) × 10^{−7} | 25.0 ± 0.1 |

20C | 1117 ± 35 | (4.9 ± 0.15) × 10^{−7} | 18.6 ± 0.2 |

30A | 432 ± 23 | (1.9 ± 0.10) × 10^{−7} | 25.5 ± 0.5 |

30B | 737 ± 89 | (3.2 ± 0.40) × 10^{−7} | 21.4 ± 0.9 |

30C | 1335 ± 42 | (5.8 ± 0.20) × 10^{−7} | 17.5 ± 0.2 |

**Table 3.**Equation (5) fitting parameters (A

_{i%}, T

_{2i}), average relaxation time, T

_{2m}, and average inverse relaxation time (1/T

_{2})

_{m}(Equation (6)).

System | A_{1%} | T_{21} (ms) | A_{2%} | T_{22} (ms) | T_{2m} (ms) | (1/T_{2})_{m} (ms^{−1}) |
---|---|---|---|---|---|---|

20A | 36 ± 1.5 | 1162 ± 14 | 64 ± 1.5 | 782 ± 5 | 920 ± 22 | (11.3 ± 0.2) × 10^{−4} |

20B | 80 ± 1.1 | 771 ± 3 | 20 ± 1.1 | 449 ± 10 | 708 ± 11 | (14.8 ± 0.3) × 10^{−4} |

20C | 58 ± 1.1 | 595 ± 2 | 42 ± 1.1 | 393 ± 7 | 509 ± 8 | (20.0 ± 0.4) × 10^{−4} |

30A | 68 ± 17.0 | 1349 ± 36 | 32 ± 17 | 1141 ± 79 | 1283 ± 310 | (7.8 ± 2.0) × 10^{−4} |

30B | 26 ± 1.2 | 1029 ± 36 | 74 ± 1.2 | 587 ± 79 | 703 ± 16 | (15.1 ± 0.3) × 10^{−4} |

30C | 79 ± 2.0 | 571 ± 5 | 21 ± 2 | 321 ± 159 | 518 ± 15 | (20.4 ± 0.9) × 10^{−4} |

System | A_{1%} | ξ_{1} (nm) | A_{2%} | ξ_{2} (nm) | ξ (nm) | R_{f} (nm) |
---|---|---|---|---|---|---|

20A | 36 ± 1.5 | 39.6 | 64 ± 1.5 | 22.1 | 26.0 ± 0.6 | 3.8 |

20B | 80 ± 1.1 | 30.3 | 20 ± 1.1 | 15.4 | 25.0 ± 0.1 | 5.1 |

20C | 58 ± 1.1 | 23.7 | 42 ± 1.1 | 14.4 | 18.6 ± 0.2 | 4.1 |

30A | 68 ± 17.0 | 28.2 | 32 ± 17 | 21.2 | 25.5 ± 0.5 | 2.0 |

30B | 26 ± 1.2 | 39.4 | 74 ± 1.2 | 18.4 | 21.4 ± 0.9 | 3.7 |

30C | 79 ± 2.0 | 21.1 | 21 ± 2 | 10.8 | 17.5 ± 0.2 | 3.6 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abrami, M.; Marizza, P.; Zecchin, F.; Bertoncin, P.; Marson, D.; Lapasin, R.; de Riso, F.; Posocco, P.; Grassi, G.; Grassi, M.
Theoretical Importance of PVP-Alginate Hydrogels Structure on Drug Release Kinetics. *Gels* **2019**, *5*, 22.
https://doi.org/10.3390/gels5020022

**AMA Style**

Abrami M, Marizza P, Zecchin F, Bertoncin P, Marson D, Lapasin R, de Riso F, Posocco P, Grassi G, Grassi M.
Theoretical Importance of PVP-Alginate Hydrogels Structure on Drug Release Kinetics. *Gels*. 2019; 5(2):22.
https://doi.org/10.3390/gels5020022

**Chicago/Turabian Style**

Abrami, Michela, Paolo Marizza, Francesca Zecchin, Paolo Bertoncin, Domenico Marson, Romano Lapasin, Filomena de Riso, Paola Posocco, Gabriele Grassi, and Mario Grassi.
2019. "Theoretical Importance of PVP-Alginate Hydrogels Structure on Drug Release Kinetics" *Gels* 5, no. 2: 22.
https://doi.org/10.3390/gels5020022