# A Multifractal Vision of 5-Fluorouracil Release from Chitosan-Based Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### Theoretical Model

_{D}is the differential velocity (scale resolution dt, independent of time), the function f(α) is the singularity spectrum, α is dependent, and √ρ is a function of the state’s amplitude.

_{r}is much greater than the value t

_{d}, characteristic solvent diffusion time) or, as non-Fickian-type diffusion (when t

_{r}≈ t

_{d}), with the same ease. If the medicament release is of the multifractal type and takes place in a perfect immersed state, the initial conditions and conditions at the boundary are accepted

_{t}is the drug quantity dissolved in time t and ${M}_{\infty}$ is the drug quantity dissolved in the total time, considering that the pharmaceutical graduation shape has been already exhausted.

_{1}and U

_{4}are classic saturation curves, while curves U

_{2}and U

_{3}are increasing with time (tend to infinite values).

## 3. Evaluation of Polarized Optical Microscopy Pictures by Fractal Analysis

_{1}and U

_{2}, in contrast with the hydrogel formulations of reduced crosslinking density (U

_{4}), about which several things are discussed. Thus, a birefringent comportment and a granular texture were declared; a specific feature of the crystal is its sub-micrometric dimensions, but it still falls below the detection limit of the analysis devices used [17].

_{1}; (b) U

_{2}; (c) U

_{3}) is 20 microns. The POM image in Figure 7 of experimentally produced formulations were processed according to fractal analysis standards, calculating the fractal dimension and lacunarity of each one. The processing method and the values obtained for fractal dimension and lacunarity, together with the voxel representation for each image separately are presented below, in paper continuation.

_{1}, the values of fractal dimension D = 1.7602, standard deviation $\mathrm{s}=\pm \sqrt{{\sigma}^{2}}=\pm 0.2026$ (see [24]) and lacunarity $\mathsf{\Lambda}=0.0215$ were obtained [24,25,26], as in Table 1.

_{1}[25,27].

_{1}image area with the software Harmonic and Fractal Image Analyser Demo version 5.5.30 [28] of the fractal dimension for various ruler scales r.

_{1}from the modified area.

_{2}, the values of fractal dimension D = 1.7523, standard deviation $\mathrm{s}=\pm \sqrt{{\sigma}^{2}}=\pm 0.1949$ (see [24]) and lacunarity $\mathsf{\Lambda}=0.0363$ were obtained [24,25,26], as in Table 2.

_{2}[25,27].

_{2}image area with the software Harmonic and Fractal Image Analyser Demo version 5.5.30 [28] of the fractal dimension for various ruler scales r.

_{2}from the modified area.

_{4}, the values of fractal dimension D=1.7352, standard deviation $\mathrm{s}=\pm \sqrt{{\sigma}^{2}}=\pm 0.1831$ (see [24]) and lacunarity $\mathsf{\Lambda}=0.0385$ were obtained [24,25,26], as in Table 3.

_{1}, U

_{2}and U

_{3}from the original Figure 7a, Figure 11a and Figure 15a are shown.

_{4,}[25,27].

_{4}image area with the software Harmonic and Fractal Image Analyser Demo version 5.5.30 [28] for the fractal dimension with various ruler scales r.

_{4}from the modified area.

_{1}, U

_{2}and U

_{4}from the modified area. The three coordinate axes are distributed as follows: the number of pixels on the ox axis, the number of pixels on the oy axis, and the gray level for each pixel on the oz axis. In 3D computer graphics, the voxel represents a numerical value associated with the regular grid of a three-dimensional coordinate space.

## 4. Conclusions

_{1}, U

_{2}and U

_{4}compounds, obtained by means of polarized light microscopy, were investigated according to the standards of fractal analysis. Thus, for image U

_{1}, we found the fractal dimension D = 1.7602 ± 0.2026 and lacunarity $\mathsf{\Lambda}=0.0215$; for image U

_{2}, we found the fractal dimension D = 1.7523±0.1949 and lacunarity $\mathsf{\Lambda}=0.0363$; and for Image U

_{4}, we found the fractal dimension D = 1.7352±0.1831 and lacunarity $\mathsf{\Lambda}=0.0385$.

## 5. Materials and Methods

#### 5.1. Materials

#### 5.2. Synthesis of the Hydrogel Formulation Preparations

_{1}, U

_{2}, U

_{3}, U

_{4}, where the attached number was in agreement with the amino/aldehyde classes’ molar ratio.

#### 5.3. Methods

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three-dimensional plot representations of $U\left(\xi ,\eta \right)$multifractal function for two distinct sets of variable values ξ and η: (

**a**) ξ from −40 to 40; η from 0 to 10; (

**b**) ξ from −5 to 25; η from 0 to 10.

**Figure 2.**Three-dimensional plot representations of $V\left(\xi ,\eta \right)$multifractal function for two distinct sets of variable values ξ and η: (

**a**) ξ from −5 to 25; η from 0 to 10; (

**b**) ξ from −40 to 40; η from 0 to 10.

**Figure 4.**Two-dimensional contour (

**a**) and three-dimensional plot (

**b**) representations of f multifractal function.

**Figure 5.**5-fluorouracil release of experimental presentation [17]; the formulations are accommodated in multifractal theoretical replica (solid-colored lines).

**Figure 7.**Primary processing of the selected image U

_{1}. (

**a**)—original image (the entire portion); (

**b**)—the grayscale version; (

**c**)—the gray scale version without luminance; (

**d**)—binarized version. A threshold of 14 was used for binarization.

**Figure 11.**Primary processing of the selected image U

_{2}. (

**a**)—original image (the entire portion); (

**b**)—the grayscale version; (

**c**)—the gray scale version without luminance; (

**d**)—binarized version. A threshold of 25 was used for binarization.

**Figure 15.**Primary processing of the selected image U

_{4}. (

**a**)—original image (the entire portion); (

**b**)—the grayscale version; (

**c**)—the gray scale version without luminance; (

**d**)—binarized version. A threshold of 35 was used for binarization.

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image U_{1} | 1.7602 | ±0.2026 | 0.0215 |

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image U_{2} | 1.7523 | ±0.1949 | 0.0363 |

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image U_{4} | 1.7352 | ±0.1831 | 0.0385 |

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**MDPI and ACS Style**

Paun, M.-A.; Paun, V.-A.; Paun, V.-P.
A Multifractal Vision of 5-Fluorouracil Release from Chitosan-Based Matrix. *Gels* **2022**, *8*, 661.
https://doi.org/10.3390/gels8100661

**AMA Style**

Paun M-A, Paun V-A, Paun V-P.
A Multifractal Vision of 5-Fluorouracil Release from Chitosan-Based Matrix. *Gels*. 2022; 8(10):661.
https://doi.org/10.3390/gels8100661

**Chicago/Turabian Style**

Paun, Maria-Alexandra, Vladimir-Alexandru Paun, and Viorel-Puiu Paun.
2022. "A Multifractal Vision of 5-Fluorouracil Release from Chitosan-Based Matrix" *Gels* 8, no. 10: 661.
https://doi.org/10.3390/gels8100661