The starting point of the theoretical model is the Scale Relativity Theory [
16,
17] that, in recent years, has been used successfully in describing the dynamics of complex systems, such as nanofluids [
17] and plasma [
18,
19,
20,
21,
22,
23]. The hypothesis underlying this theory is that the entities of any complex system move on continuous and non-differential curves named fractal curves, i.e., three dimensional fractured lines, whose non-linearity is dependent and proportional with the number of interactions within the system. In this context, the fractalization degree will be defined as a measure of the system complexity, and the physical quantities, characterizing the system evolution, will be fractal functions dependent both on the spatio-temporal coordinates and resolution scales [
24].
Further, we will use this hypothesis in the case of our system, which will be considered as a complex system consisting of a hydrogel, drug and release medium. Previous studies showed that the theoretical models built on this hypothesis allowed insights concerning drug release at the microscopic level, which has been unaddressed thus far due to the complexity of the phenomena involved. For example, this revealed that, in the case of polymeric particles based on chitosan and gelatin, the trajectories of drug molecules follow cnoidal oscillation modes [
25], while, for polysaccharide-based hydrogels, their trajectories evolved from a normal period doubling state towards damped oscillating via strong modulated dynamics [
26].
For chitosan-gelatin and chitosan-poly(vinyl alcohol) hydrogels, analyzing the release kinetics at long time scales (i.e., up to 25 days), when the hydrogel films degrade, it was found that the release efficiency was dependent on the time and system nonlinearity, i.e., the fractalization degree [
27]. Moreover, it was found that, along with the evolution through all the phases of the release process (i.e., burst, swelling, equilibrium and degradation phases), the fractalization degree increased, thus, reflecting the increasing system complexity [
28].
In our opinion, the two representations describing the dynamics of release are not excluded; on the contrary, they are complementary. As the problem of compatibility of the two representations in the description of the release dynamics has not been analyzed thus far, next we will analyze such a problem and its implications.
4.2.1. Two Models of Describing the Release Dynamics through the Scale Relativity Theory
In all the above assumptions, the motion equation of any of the system component (either drug and either release medium molecules) can be written in the form:
where
where
is the multifractal spatial coordinate,
is the non-fractal time with the role of an affine parameter of the motion curves,
is the complex velocity,
is the differential velocity independent of scale resolution,
is the non-differential/fractal velocity dependent of scale resolution,
is the scale resolution,
is the singularity spectrum of order
,
is the singularity index and a function of fractal dimension
,
is the constant tensor associated with the differentiable–non-differentiable transition,
is the constant vector associated with the backward differentiable-non-differentiable dynamic processes,
is the constant vector associated with the backward differentiable-non-differentiable dynamic processes, and
is a multifractal function.
In the case of our drug release system, multifractalization will be considered by means of Markovian stochastic processes, i.e., processes for which the probability of possible “events” in system evolution depends only on the state attained in the previous “event”. From a mathematical point of view, the following restrictions must be fulfilled:
where
is a constant associated with the differentiable–non-differentiable transition, which, in the case of release dynamics, is actually the transition from Fickian to non-Fickian diffusion, and
is Kronecker’s pseudo-tensor [
16,
17,
24].
Imposing the restrictions from (7), Equation (5) becomes:
where
represents the local multifractal acceleration,
the multifractal convection and
the multifractal dissipation. Equality with zero from Equation (8) reflects the fact that the local multifractal acceleration, the local multifractal convection and the multifractal dissipation are in equilibrium at all points of the motion curve. Moreover, the existence of the complex dissipation term confirms that the multifractal fluid with which the drug release complex system can be assimilated exhibits memory according to the assumption of Markovian stochastic processes.
Considering that, in the release flow, each element of the moving fluid undergoes no net rotation (i.e., rotational movements of system structural units), the complex velocity field from (6) becomes
where
is the complex scalar potential of the complex velocity fields from (6), and
is the state function that describes the equilibrium state of the system. Substituting (9) in (8) and using the mathematical procedure from [
24], Equation (8) takes the form of the multifractal Schrödinger type equation:
Therefore, for the complex velocity field (9), the dynamics of any complex system entities are described through multifractal Schrödinger “regimes”, i.e., Schrödinger equations at various scale resolutions.
From such a perspective, the meaning of
can be given based on a conservation law. Indeed, multiplying Equation (11) with
(complex conjugate of
) and the complex conjugate of Equation (11) with
, followed by their difference, the multifractal law of conservation for the state density is obtained in the form [
29]:
where
In the relations (12) and (13),
is the multifractal state density, while
is the multifractal current density. In the case of the existence of a constraint, such as the hydrogel network-drug interactions, in the form
, the usual derivative from the Scale Relativity Theory is substituted with the covariant derivative,
:
where
is a vector potential and
is a coupling constant between
and
. In such a context, (12) maintains its validity, with the difference that
from (13) can be expressed as
Moreover, if
is chosen of the form (Mandelung’s choice):
where
is the amplitude of the state quantities and
the phase, then the complex velocity field (9) takes the explicit form:
which leads to the determination of velocity fields:
By (17) and (18) and using the mathematical approaches from [
24,
29], the Equation (8) reduces to the multifractal Mandelung’s equations:
where
denotes the multifractal specific potential:
Equation (19) corresponds to the multifractal specific momentum conservation law, while Equation (20) corresponds to the multifractal density conservation law [
24]. The specific multifractal potential
expressed by (21) implies the existence of a specific multifractal force:
that quantifies the multifractality degrees of the motion trajectories.
Therefore, for the complex velocity fields (18), the dynamics of any complex system are described through multifractal Madelung “regimes” (i.e., Madelung equations at various scale resolutions). In this last context, the following consequences result:
- (i)
Any complex system entities are in permanent interaction with a multifractal medium through the multifractal specific force (18) [
29].
- (ii)
Any complex system can be identified with a multifractal fluid, the dynamics of which is described by the multifractal Madelung model (Equations (19) and (20)).
- (iii)
The velocity field does not represent the contemporary dynamics; since is missing from (20), this velocity field contributes to the transfer of the multifractal specific momentum and to the multifractal energy.
- (iv)
Any analysis of should consider the “self” nature of the multifractal specific momentum transfer; then, the conservation of the multifractal energy and the multifractal momentum ensure the reversibility and the existence of the multifractal eigenstates.
- (v)
If a multifractal tensor is considered:
the equation defining the multifractal “forces” that derive from
can be written in the form of a multifractal equilibrium equation:
Since
can be also written in the form:
with
a multifractal linear constitutive equation for a multifractal “viscous fluid” can be highlighted. In such a context, the coefficient
can be interpreted as multifractal dynamic viscosity coefficient of multifractal fluid.
4.2.2. The Compatibility of the Two Models of Describing the Release Dynamics through the Scale Relativity Theory
As anticipated, the two multifractal descriptions of the dynamics of complex systems, the multifractal Schrödinger description and the multifractal Madelung one, are not mutually exclusive but, on the contrary, are complementary. Let us explain this on the basis of the following hypotheses:
- (a)
The dynamics of any complex system, independent of the two scale resolutions (differentiable and non-differentiable scale resolutions), are one-dimensional dynamics. It thus results that the conservation law for the multifractal states density (12) becomes
where
In particular, for the state function in the form (16), Equation (28) becomes
- (b)
The synchronization of the dynamics of any complex system at the two scale resolutions is achieved by “compensating” the speed fields
and
given by the restriction:
Through (18) and (19), a dependence results between the amplitude
and phase
of the state function
in the form:
with
an integration constant.
- (c)
The vectorial field
is uniform, a situation in which the potential vector
has the expression:
In particular, for one-dimensional dynamics of any complex system,
can be chosen in the form:
Since, on the basis of the above hypotheses (a)–(c), Equation (29) takes the form:
Equation (27) becomes the multifractal equation Fokker–Planck:
Therefore, the correlations between the two descriptions, i.e., the multifractal Schrödinger and the multifractal Mandelung one, imply multifractal Fokker–Planck description of complex system dynamics.
With the notations:
where
can be correlated with the strength of the hydrogel network and
is the fractalization degree, as previously defined, the solution of Equation (35) has the expression:
signifying that the density of states is a multifractal Gaussian whose average multifractal value decreases exponentially to zero and whose multifractal variance tends asymptotically towards
.