# Fractal Dimension Analyses to Detect Alzheimer’s and Parkinson’s Diseases Using Their Thin Brain Tissue Samples via Transmission Optical Microscopy

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Mathematical Methods

#### 2.2.1. Fractals and Fractal Dimensions

_{box}(r) [1]. As such, for two different length scales, we can find the fractal dimension (D

_{f}) by the equation.

_{f}being the line’s slope.

#### 2.2.2. Fractal Dimension Calculation from Microscopic Images

#### 2.3. Experimental Setup

_{t}(x, y) is the transmission intensity at a spatial point (x, y), n(x, y) is the refractive index and ρ(x, y) is the mass density at that point.

#### 2.4. Analysis of Image Data

## 3. Results

#### 3.1. Change in Fractal Dimension (D_{f}) in Alzheimer’s Disease

#### 3.1.1. Change in D_{f} in Hippocampus

#### 3.1.2. Change in D_{f} in Precentral Gyrus

#### 3.1.3. Change in D_{f} in Postcentral Gyrus

#### 3.1.4. Change in D_{f} in Occipital Lobe

#### 3.1.5. Change in D_{f} in Cerebellum

#### 3.2. Change in Fractal Dimension (D_{f}) in Parkinson’s Disease

#### 3.2.1. Change in D_{f} Substantia Nigra

#### 3.2.2. Change in D_{f} in Hippocampus

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**AD (Hippocampus): (

**a**,

**b**) represents the bright field images of normal and severe brain tissue of AD respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) is the bar graph representation of the mean fractal values taken over several spots. The result shows that the fractal value for Low AD increases by 10.5%, intermediate by 13.18% and severe by 16.3% w.r.t the control, with actual fractal dimensions of 1.4731, 1.6283, 1.6758, 1.7133 for Control, Low, Intermediate and Severe AD, respectively. (p-values < 0.05 of low, intermediate, and severe AD cases w.r.t the control; averaged over n = 5 different samples and 10–15 spots for each sample.) (

**d**,

**e**) show the ln(1/r) plots for brain tissues for Control and Severe AD patients, respectively. It can be emphasized here that the ln(N(r)) vs. ln(r) follow nice straight lines, slope is the fractal dimension D

_{f}.

**Figure 2.**Alzheimer’s Disease (Precentral Gyrus): (

**a**,

**b**) represents the bright field images of normal brain tissue and severe AD tissue, respectively. (

**a’**,

**b’**) represent the corresponding binary image (

**c**) is the bar graph representation of the mean fractal values taken over several spots. As shown, there is a marginal increase of 3.4% in the fractal value for the disordered brain tissue compared to normal, with actual fractal dimensions of 1.687 and 1.744 for Control and AD, respectively. (p-values < 0.05 of AD w.r.t the control. Samples n = 5, ~10–15 spots from each sample).

**Figure 3.**Alzheimer’s Disease (Postcentral Gyrus): (

**a**,

**b**) represents the bright field images of normal brain tissue and brain tissue of severe AD, respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) The bar graph represents the mean fractal values taken over several 10–15 spots. As shown, a marginal increase of 5.63% in the fractal value for the disordered brain tissue compared to expected, with actual fractal dimensions of 1.662 and 1.756 for Control and AD, respectively (p-values < 0.05 of AD w.r.t the control; Samples n = 5, ~10–15 spots from each sample).

**Figure 4.**Alzheimer’s Disease (Occipital lobe): (

**a**,

**b**) represent the bright field images of AD’s normal and severe brain tissue, respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) is the bar graph representation of the mean fractal values taken over several spots. As shown, a marginal increase of 4.60% in the fractal value for the disordered brain tissue compared to normal, with actual fractal dimensions of 1.653 and 1.729 for Control and AD, respectively. (p-values < 0.05 AD w.r.t the control. Samples n = 5, ~10–15 spots per sample).

**Figure 5.**Alzheimer’s Disease (Cerebellum). (

**a**,

**b**) represents the bright field images of normal and severe brain tissue of AD, respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) is the bar graph representation of the mean fractal values taken over several spots. As shown, a marginal increase of 4.25% in the fractal value for the disordered brain tissue compared to normal, with actual fractal dimensions of 1.67 and 1.74 for Control and AD, respectively. (p-values < 0.05 of AD w.r.t the control; Samples n = 5, ~10–15 spots from each sample).

**Figure 6.**Parkinson’s Disease (Subtantia Nigra). (

**a**,

**b**) represent the bright field images of normal and severe brain tissue of PD, respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) is the bar graph representation of the mean fractal values taken over several spots. As shown, a marginal increase of 10.78% in the fractal dimension value for the disordered brain tissue compared to normal, with actual fractal dimensions of 1.507 and 1.67 for Control and PD, respectively. (p-values < 0.05 of PD w.r.t the control; n = 5, ~10–15 spots from each sample).

**Figure 7.**Parkinson’s Disease (Hippocampus). (

**a**,

**b**) represent the bright field images of normal and severe brain tissue of AD, respectively. (

**a’**,

**b’**) represent the corresponding binary images. (

**c**) is the bar graph representation of the mean fractal values taken over several spots. As shown, a marginal increase of 7.86% in the fractal value for the disordered brain tissue compared to normal, with actual fractal dimensions of 1.60 and 1.73 for Control and PD, respectively. (p-values < 0.05 of PD w.r.t the control; n = 5, ~10–15 spots from each sample).

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**MDPI and ACS Style**

Apachigawo, I.; Solanki, D.; Tate, R.; Singh, H.; Khan, M.M.; Pradhan, P.
Fractal Dimension Analyses to Detect Alzheimer’s and Parkinson’s Diseases Using Their Thin Brain Tissue Samples via Transmission Optical Microscopy. *Biophysica* **2023**, *3*, 569-581.
https://doi.org/10.3390/biophysica3040039

**AMA Style**

Apachigawo I, Solanki D, Tate R, Singh H, Khan MM, Pradhan P.
Fractal Dimension Analyses to Detect Alzheimer’s and Parkinson’s Diseases Using Their Thin Brain Tissue Samples via Transmission Optical Microscopy. *Biophysica*. 2023; 3(4):569-581.
https://doi.org/10.3390/biophysica3040039

**Chicago/Turabian Style**

Apachigawo, Ishmael, Dhruvil Solanki, Ruth Tate, Himanshi Singh, Mohammad Moshahid Khan, and Prabhakar Pradhan.
2023. "Fractal Dimension Analyses to Detect Alzheimer’s and Parkinson’s Diseases Using Their Thin Brain Tissue Samples via Transmission Optical Microscopy" *Biophysica* 3, no. 4: 569-581.
https://doi.org/10.3390/biophysica3040039