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Drug release is a complex phenomenon due to the large number of interdependent side effects that occur simultaneously, involving strong nonlinear dynamics. Therefore, since their theoretical description is difficult in the classical mathematics modelling, we have built a theoretical model based on logistic type laws, validated by the correlations with the experimental data, in a special case of drug release from hydrogels. The novelty of our approach is the implementation of multifractality in logistic type laws, situation in which any chaotic system, characterized by a small number of nonlinear interactions, gets memory and, implicitly, characterization through a large number of nonlinear interactions. In other words, the complex system polymer-drug matrix becomes “pseudo-intelligent.”

Hydrogels are an important class of materials with a large range of applications in key domains as biomedicine [

Another important aspect related to the developing of drug delivery systems is related to the efficient use of the drugs, limiting thus their side effects on (i) the human body and (ii) the environment. As an example, many antitumor drugs, e.g., 5-fluorouracil, which are extensively used in the treatment of a wide variety of tumor lines [

In this context, targeting the preparation of hydrogels for the local therapy of cancer, chitosan has been reacted with nitrosalicylaldehyde, when biocompatible hydrogels with high cytotoxicity against the HeLa cells were obtained [

The novelty of our approach is the implementation of multifractality in logistic type laws for the description of the drug release processes. In such a situation, the complex system polymer-drug matrix gets memory, becoming “pseudo-intelligent.”

Chitosan of low-molecular weight, nitrosalicylaldehyde of 98% purity, diclofenac sodium salt (DCF) of purity higher of 99%, and phosphate buffer solution were purchased from Aldrich and used as received.

The drug delivery systems which are the subject of the present paper were obtained according to a procedure already reported [

Graphical representation of the reagents weight used for hydrogel preparation.

In order to monitor the drug delivery profile of the studied formulations, equal amounts of the synthetized A2D–A5D systems were prepared as pills by pressing with a hydraulic press (2 N/m^{2}). The pills were introduced into sealed vials containing 10 ml of PBS at 37°C. At predetermined moments, 2 mL of supernatant was withdrawn and replaced with 2 mL of PBS. The supernatant samples were collected and analyzed by quantitative absorption spectroscopy, by measuring the absorbance of the DCF drug. The values of the absorbance obtained at different investigation times were fitted on the DCF calibration curve [

In vitro drug release investigation.

The release kinetics of the studied formulations was first investigated by fitting the resulted release data on the equations of 5 different models:

Zero-order model:

First-order model:

Higuchi model:

Korsmeyer–Peppas model:

Hixson–Crowell model:

Due to the empirical character of these equations, the release kinetics was further investigated through nonlinear dynamics in the form of logistic type laws.

The polymeric matrix loaded with drug is, both structurally and functionally, a complex system, hereafter referred to as polymer-drug matrix. In this case, the complexity refers both to the collective behaviour of the polymer-drug matrix (dictated by the extremely high number of nonlinear interactions between its structural units) and to the constraints that the polymeric-drug matrix system supports in relation to the “environment” generated by various biological structures. From such a perspective, the polymer-drug matrix evolves away from the state of equilibrium (at the “edge of chaos” between determinism and randomness), in a critical state built from an “archeology”/history of “unpredictable and unexpected events” through feedback cycles, self-organized structures, etc., thus, the archeology/history is grounded as the main feature of the polymer-drug matrix.

The absence of archeology/history is specific to chaotic systems. Here, the behaviour of any such system is dictated by the relatively small number of nonlinear interactions between its structural units, and it has the fundamental property the “chaotic order.” Thus, the chaotic systems can be assimilated with a subset of complex systems without history. In such context, the following question arises: is it possible like a chaotic polymer-drug matrix, without history, to mimic the complex polymer-drug matrix, a system with history? The answer is affirmative only to the extent that the chaotic system (chaotic polymer-drug matrix) is “assigned” a type of archeology/history. In this regard, let us first notice that the polymer-drug matrix as a chaotic system can “support” a release mechanism based on the nondimensional logistic type law [

Release curves of the same

Release curves of the same

Release curves of the same

Release curves of the same

A possible procedure for assigning archeology/history to polymer-drug matrix is through the multifractality of the release curves (continuous curves with varying degrees of nondifferentiability, hereinafter referred to as multifractal release curves). Consequently [

In such a context, the indiscernibility of the release curves is a result of multifractalisation by multistocasticization—simultaneously, any monofractalization from the singularity spectrum of the release dynamics being compatible with a stochastic process. However, the discernibility of the release curves involves a selection process (obviously at scale resolution of finite multifractal type) of multifractal space geodesics based on a measurement process, from the multitude of multifractals, manifested at a given moment in the singularity spectrum of the release dynamics; only one confers globality through the type of the stochastic process.

In such a context was analyzed the dynamics of release from different types of drug delivery systems, such as microparticles [

In these conditions, the release dynamics of the polymer-drug matrix, considered now as complex system, can be described by the generalized nondimensional logistic type law:

In order to solve equation (

Next, let us introduce a new substitution given by

This equation admits the following solution:

The above solution has, in the variable

While in the variable

The stationary points can be calculated as above. They are

The derivative of function

As

The inflexion point of the solution is given by means of condition:

Applying the procedure of in situ hydrogelation described in Section

SEM microimages of the studied drug delivery systems: (a) A2D; (b) A3D; (c) A4D; (d) A5D.

The monitoring of the DCF release from the under study systems is represented in Figure

Monitoring of the DCF release from the A2D–A5D formulations.

The release data were fitted on the equations of five different standard (empirical/semiempirical) models [

As can be seen in Figure

Linear forms of the zero order (a); first order (b); Higuchi (c); Korsmeyer–Peppas (d); and Hixson–Crowell (e) models applied for the release of DCF from the A2D-A5D formulations.

Taking into account our theoretical model presented in Section

(a) 3D dependence of the amount of drug released,

In the present paper, release dynamics are analyzed, both theoretical and experimental, for a special polymer-drug matrix, based on standard models (empirical/semiempirical), zero order, first order, Higuchi, Korsmeyer–Peppas, and Hixson–Crowell, and on the basis of a new model given by a logistic type law.

Our approach is based on the following: (i) first, the logistic type law (1) is proposed in the description of the release dynamics. In this relation, the term

A mathematical model that intends to describe the dynamics of controlled drug release systems should allow (a) to predict the drug release kinetics and the phenomena involved, avoiding thus repetitive time and money expensive experiments; (b) to optimize the drug release kinetics; (c) to estimate the effect of the polymer matrix design parameters, i.e., shape, size, and composition on the drug release kinetics; (d) to predict the global therapeutic efficiency and drug safety; and (e) overall, to design a new drug delivery system depending on the release kinetics imposed by the therapeutic requirements.

So far, successful mathematical models designed to describe the dynamics of the drug controlled release system have been developed based on either an almost intuitive selection of dominant phenomena relative to the polymer matrix configuration and geometry or just by analysing the experimental results, resulting in a wide variety of empirical/semiempirical models such as the diffusion model, the zero-kinetic model, the Higuchi model, the Korsemeyer–Peppas model, the Hixon–Crowell model, the Weibull model, the Baker Lonsdale model, the Hopfenberg model, the Gompertz model, the sequential layer model, the Peppas–Sahlin model, etc.

In our opinion, this diversity of models demonstrates the inability of differential and integral mathematical procedures to describe the complexity of controlled drug release dynamics. Thus, the success of the abovementioned models should be understood only sequentially/progressively in domains where differentiability and integrity are still valid. However, the differential and integral mathematical procedures “suffer” when one wants to describe the dynamics of drug controlled release systems, dynamics involving both nonlinearity and chaos.

The model developed in this paper is precisely intended to substitute the standard mathematical procedures, i.e., of differential and integral types, with nondifferential and nonintegrative mathematical procedures, applied in describing the dynamics of controlled drug release systems. In such a conjecture, we first developed the logistic model of dynamics of drug controlled release systems (where the system is only chaotic, without memory, behaviour dictated by a relatively small number of nonlinear interactions between its entities and still subject to differential and integral mathematical procedures). Then, by extension, we developed the multifractal logistic model for the dynamics of controlled drug release systems (where the system is complex, with memory, its behaviour being imposed by a large number of nonlinear interactions between its entities, subjected only to nondifferential and nonintegrative mathematical procedures); such a description proves to be reducible to the dynamics of drug controlled release systems at various scale resolutions.

The model was finally validated on the basis of the experimental results on diclofenac sodium salt release from nitrosalicyl-imine-chitosan hydrogels, confirming thus that complex systems, i.e., polymer-drug matrices, are pseudo-intelligent systems, with memory. In such an approach, phenomena as drug diffusion, swelling, and degradation of the polymeric matrix can be described as dynamics of pseudo-intelligent materials at various scale resolutions.

Previously reported experimental data were used to support this study and are available at

The authors declare that they have no conflicts of interest.

This work was supported by the European Union’s Horizon 2020 Research and Innovation Programme (grant no. 667387) and Romanian National Authority for Scientific Research MEN—UEFISCDI (grant nos. PN-III-P1-1.2-PCCDI2017-0569 and 10PCCDI/2018).

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