Multi-fractal modeling of curcumin release mechanism from polymeric nanomicelles

Abstract

The physicochemical properties of “smart” or stimuli-sensitive amphiphilic copolymers can be modeled as a function of their environment. In special, pH-sensitive copolymers have practical applications in the biomedical field as drug delivery systems. Interactions between the structural units of any polymer-drug system imply mutual constraints at various scale resolutions and the nonlinearity is accepted as one of the most fundamental properties. The release kinetics, as a function of pH, of a model active principle, i.e., Curcumin, from nanomicelles obtained from amphiphilic pH-sensitive poly(2-vinylpyridine)-b-poly(ethylene oxide) (P2VP-b-PEO) tailor-made diblock copolymers was firstly studied by using the Rietger-Peppas equation. The value of the exponential coefficient, n, is around 0.5, generally suggesting a diffusion process, slightly disturbed in some cases. Moreover, the evaluation of the polymer-drug system’s nonstationary dynamics was caried out through harmonic mapping from the usual space to the hyperbolic one. The kinetic model we developed, based on fractal theory, fits very well with the experimental data obtained for the release of Curcumin from the amphiphilic copolymer micelles in which it was encapsulated. This model is a variant of the classical kinetic models based on the formal kinetics of the process.

A Holographic-Type Model in the Description of Polymer–Drug Delivery Processes

“…Since these curves display the self-similarity property (a single unit represents the entire system, and the whole reflects the part, miming holographic-type behaviors), we can state that, through holographic-type models, we can describe various drug release dynamics. Recent papers support our approach [ 82 , 83 , 84 ].…”

“…In Equation (1), are the multifractal space coordinates, is a non-multifractal time coordinate, is the state function, is a constant correlated to the multifractal-non-multifractal scale transition, is the scale resolution, is the singularity spectrum of order with , and the fractal dimension of the motion curves [ 82 , 83 ]. For other details referring to the values of the previously mentioned factors, please see [ 85 , 86 , 87 ].…”

“…Now, for in the form (the Madelung substitution): where is the amplitude, and is the phase, the complex velocity field [ 31 , 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 ] takes the explicit form: Relation (4) implies the real velocity fields: where is the differentiable velocity field, i.e., the velocity at differentiable resolution scale, and is the non-differentiable velocity field, i.e., the velocity at non-differentiable resolution scale.…”