Controlled drug delivery systems are of utmost importance for the improvement of drug bioavailability while limiting the side effects. For the improvement of their performances, drug release modeling is a significant tool for the further optimization of the drug delivery systems to cross the barrier to practical application. We report here on the modeling of the diclofenac sodium salt (DCF) release from a hydrogel matrix based on PEGylated chitosan in the context of Multifractal Theory of Motion, by means of a fundamental spinor set given by 2 Â 2 matrices with real elements, which can describe the drug-release dynamics at global and local scales. The drug delivery systems were prepared by in situ hydrogenation of PEGylated chitosan with citral in the presence of the DCF, by varying the hydrophilic/hydrophobic ratio of the components. They demonstrated a good dispersion of the drug into the matrix by forming matrix-drug entities which enabled a prolonged drug delivery behavior correlated with the hydrophilicity degree of the matrix. The application of the Multifractal Theory of Motion fitted very well on these findings, the fractality degree accurately describing the changes in hydrophilicity of the polymer. The validation of the model on this series of formulations encourages its further use for other systems, as an easy tool for estimating the drug release toward the design improvement. The present paper is a continuation of the work 'A theoretical mathematical model for assessing diclofenac release from chitosan-based formulations,' published in Drug Delivery Journal, 27(1), 2020, that focused on the consequences induced by the invariance groups of Multifractal Diffusion Equations in correlation with the drug release dynamics.

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.

A case study was performed, concerning the behavior and degradation of a polymeric biocomposite material – “liquid wood”. This material is biodegradable and it is obtained from renewable resources. Three presentation forms – Arbofill Fichte, Arboform F45 and Arboblend V2, were subjected to the action of external factors present in a marine environment. The results pertaining to the change in the physico – chemical properties of “liquid wood” when subjected to the action of seawater and seawater microorganisms, with significant – but nevertheless positive – consequences upon the environment. The material exhibits good performance after the surface and mass stabilization – due to water, C, Na, Cl and O absorption. As such – due to the emergence of a protective organic biofilm – growth of microorganisms significantly decreases and electric conductivity increases. This case study may be viewed as a starting point for subsequent studies of “liquid wood”.

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

In a multifractal paradigm of motion, Shannon’s information functionality of a minimization principle induces multifractal–type Newtonian behaviors. The analysis of these behaviors through motion geodesics shows the fact that the center of the Newtonian-type multifractal force is different from the center of the multifractal trajectory. The measure of this difference is given by the eccentricity, which depends on the initial conditions. In such a context, the eccentricities’ geometry becomes, through the Cayley–Klein metric principle, the Lobachevsky plane geometry. Then, harmonic mappings between the usual space and the Lobachevsky plane in a Poincaré metric can become operational, a situation in which the Ernst potential of general relativity acquires a classical nature. Moreover, the Newtonian-type multifractal dynamics, perceived and described in a multifractal paradigm of motion, becomes a local manifestation of the gravitational field of general relativity.

By assimilating shape memory alloys with mathematical multifractal-type objects, a theoretical model based on Scale Relativity Theory in the form of The Multifractal Theory of Motion, in order to explain the mechanical behavior of such material, is proposed. The model is validated by analyzing the mechanical behavior of Cu–Al–Zn shape memory alloy with various chemical compositions. More precisely, the multifractal tunnel effect can “mime” the mechanical hysteresis of such a material, a situation in which a direct correspondence for several mechanical properties of Cu–Al–Zn is highlighted (the chemical composition can be correlated with the shapes of the curves controlled through the multifractality degree, while the areas delimited by the same curves can be correlated with the multifractal specific potential, as a measure of the mechanical memory degree).

The existence of many forms of “liquid wood”, even in the same subgroup, is explained by the fact that the main matrix is made of lignin and lignin can be found in nature in over 60 presentation forms. The lignin molecule is a complex macro – molecule made of three molecules which link together in various shapes. At the same time, structurally speaking, the lignin molecule is dependent on the type of plant species from which it is sourced. It results that the type and the structure of the lignin molecule – and implicitly the “liquid wood” biocomposite matrix – has a major role in the forming and the structuring of every type of “liquid wood”. In the current article, a comparative study of the properties of “liquid wood” pertaining to all three subgroups is presented. The chosen types are: Arboform F45, Arbofill Fichte and Arboblend V2 (which is the subject of the entire study).

“Liquid wood” is a biopolymer composite exhibiting a lignin matrix. Lignin is separated as a by-product of paper. Following research by German scientists from a production company, an invention was made that determines a new use of lignin, the capitalization of said product in this case being superior. This material combines lignin with other natural substances: resins, waxes, vegetable fibres etc. It results a polymer that exhibits certain physic-chemical properties which recommend it for large scale use, being able to substitute plastic materials. Some of the properties of “liquid wood” are: it is a thermo-injectable polymer, it is obtained from renewable resources and it is biodegradable. Existing injection or extruding machines, used for synthetic polymers such as polyethylene and polypropylene, can be employed to obtain various “liquid wood” parts. It has been observed that injection process parameters influence mechanical and physic – chemical properties of products obtained from “liquid wood”. In order to find the optimal injection parameters for an object that exhibits certain mechanical and physic – chemical properties, it is necessary to determine the rheological properties of “liquid wood”. In this paper, several “liquid wood” rheological properties have been determined, in order to assist the optimization of the injection process.