Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 45 https://internationalpubls.com Border Detection of Skin Cancer Cells over Fractal Dimension Analysis and Image Processing Techniques 1P.Tharaniya, 2R.Ambrose Prabhu, 3V.Sumathi 4R.Rajalakshmi, 5R.Malarkodi, 6R.Geethanjaliyadav 1,2,3Department of Mathematics, Rajalakshmi Instititue of Technology, Chennai, India, 1tharamanisharmi@gmail.com, 2bennyamb0457@gmail.com, 4vsumathigeerthana83@gmail.com 4Assistant Professor, Department of Mathematics,Panimalar Engineering College, Chennai, rajimat2020@gmail.com, 5Department of Mathematics, St.Joseph’s college of Engineering and Technology, Thanjavur, India, r.malarkodi26@gmail.com 6Department of Mathematics, Rajalakshmi Instititue of Technology, Chennai, India, rvgeetha20@gmail.com Article History: Received: 19-09-2024 Revised: 03-11-2024 Accepted: 16-11-2024 Abstract: Fractal dimension analysis is a novel technique that uses the self-similarity qualities of fractals to identify irregular forms, such as those prevalent in diseased tissues, in order to detect the borders of skin cancer cells. Acquire detailed pictures of skin tissue samples that have cancer cells in them. A variety of imaging methods, including microscopy and medical imaging tools like MRIs and CT scans, can be used for this. Determine the image's fractal dimension by applying suitable methods, like the fractal signature method or box-counting. A geometric shape's complexity is measured by its fractal dimension, and because malignant cells have uneven edges, they typically show higher complexity. To increase the border recognition process' accuracy, clean the photos to get rid of noise and boost contrast. Here, methods such as morphological procedures, histogram equalization, and median filtering can be used. To increase the border detection system's accuracy and resilience, fine-tune the parameters and algorithms in light of the validation results. A reliable approach for identifying the borders of skin cancer cells can be created by fusing fractal dimension analysis with image processing methods. This will help with early detection and therapy planning. Based on the fractal dimension, choose an appropriate threshold value to divide the image into zones of interest. This stage aids in the malignant cells' separation from the surrounding tissue. Keywords: Fractal Dimension Analysis, Skin Cancer Cells, Border Detection, Image Processing, Medical Imaging MSC Classification Key: 28A80, 62P10, 65D18, 92C50, 68U10. 1. Introduction The largest organ in the body is frequently referred to as the skin. It keeps water and other bodily fluids inside the body, aids in controlling body temperature, produces vitamin D, and carries out a number of other intricate tasks that safeguard individuals. Of all cancers, skin cancer is the most prevalent. Fifty percent of all cancer cases are caused by it. It is a skin cancerous growth that has a variety of reasons. Usually, skin cancer starts in the epidermis, which is the skin's outermost layer. Therefore, Tumor is typically easily seen. Because of this, the majority of skin cancers can be found early on. The most common cause of this malignancy is prolonged skin exposure to UV (ultra violet) radiation. mailto:1tharamanisharmi@gmail.com mailto:2bennyamb0457@gmail.com mailto:4vsumathigeerthana83@gmail.com mailto:rajimat2020@gmail.com mailto:r.malarkodi26@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 46 https://internationalpubls.com A: Asymmetry B: Border irregularity C:Color D:Diameter: ΒΌ inch or 6mm Fig. 1.1: Diagnose Skin Cancer Certain physical characteristics, including as shape, edge, color, and surface texture, can be used to identify skin cancer. It found that the most important diagnostic criteria was the irregularity of the border of the pigmented skin lesions. According to research by Morris Smith, one of the main vocabulary words used in medical textbooks to characterize the border of malignant melanoma is irregularity. A stands for asymmetry, B for border, C for color, and D for diameter. The ABCD checklist (Fig. 1.1) can be used to diagnose skin cancer border irregularity. The Box-Counting Method (DB) can be used to determine the size of the cell, and the Sausage Method (DS) can be used to determine how invasive the malignancy is. Fractal geometry offers techniques for defining complexity, which can be utilized to qualify morphologies in image analysis content that are currently only qualitatively evaluable or that are thought to be random or irregular. Here, we have used the Sausage Method (DS) to highlight the invasiveness, and the Box Counting Method (DB), which is found for a small number of samples, can be used to determine the overall dimension. The cancer image is handled as if it were two-dimensional, with coordinates defined as (x, y). The picture pixel size in the tissue is represented by s, and the (x, y) coordinates are then divided into grids that measure. When the minimum and maximum binary image levels in the (𝑖, 𝑗)π‘‘β„Ž grid are (k-1) and l, respectively, and fall into the and boxes, the contribution of in the (𝑖, 𝑗)π‘‘β„Ž grid is defined as π‘›π‘Ÿ(𝑖, 𝑗) = 𝑙 βˆ’ π‘˜ + 1. The formula for calculating box counting dimension is calculated as ( )    log log limdim 0 FN FB βˆ’= β†’ , using least-squares linear regression to estimate the box-counting dimension yields the line's slope. If the least and highest binary image levels in the (𝑖, 𝑗)π‘‘β„Ž grid, respectively, are (k-1) and l, and they fit into the boxes, 𝑁(π‘Ÿ) is described as the total of the contributions made by each grid present in a window of the picture. ( ) ( )οƒ₯= ji r jinrN , , . The slope of the line that best fits points (log ( 1 π‘Ÿ ) , log 𝑁(π‘Ÿ)) can be used to estimate the fractal dimension if 𝑁(π‘Ÿ) is calculated for various amounts of scaling r. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 47 https://internationalpubls.com Step 1: Regular meshes with a mesh size of β€˜r’ are created from the image. Step 2: Determine 𝑁(π‘Ÿ) how many square boxes cross the image. Step 3: 𝑁(π‘Ÿ) depends on r Step 4: We count the corresponding number 𝑁(π‘Ÿ) after repeating for a number of size values. Step 5: Form the slope D by plotting log (𝑁(π‘Ÿ)) against log ( 1 π‘Ÿ ) The fractal's dimensions or level of complexity are indicated in Step 5. Finally, the least squares approach is used to fit a straight line to the plotted points in the diagram. The fractal dimension was estimated using a linear regression equation log(𝑁(π‘Ÿ)) = log π‘˜ + 𝐷 log ( 1 π‘Ÿ ), k is constant, Where D denotes the dimension of the fractal. Numerous patients have had the aforementioned technique used to determine the size of their cancer cells. The above algorithm has been programmed and run in MATLAB. Fig. 1.2: Twenty Original Images of skin lesions Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 48 https://internationalpubls.com Fig. 1.3: Dermatologist Image of skin lesions 1.1.1 Literature Review Ahmed [1] addresses the application of fractal geometry to the study and identification of cancer in a 1993 publication in the International Journal of Theoretical Physics. The study investigates the potential applications of fractal geometryβ€”which deals with intricate patterns that display self- similarity at various scalesβ€”to biological tissues. This idea could be applied to medical imaging to improve cancer detection and diagnosis, offering a non-invasive way to spot cancers. M. A. Aon and S.Cortassa [2] investigates the application of fractal analysis in comprehending and identifying cancer through cellular architecture in their 1994 paper published in FEBS Letters. Aon and Cortassa specifically look at how anomalies linked to cancer can be found by using the fractal character of cellular structures. Fractal analysis may be used to detect cancer since malignant cells frequently have distinct fractal dimensions from healthy cells. M. Battaglia-Parodi and D. D. Giusto [3] examines the use of fractal analysis in the diagnosis of eye- related diseases and makes comparisons with its application in the detection of skin cancer. They contrast the usefulness and usability of fractal analysis in skin cancer research with its application in ophthalmology diagnosis. Block. A [4] use of fractal geometry in biological systems is examined in this article, with a special emphasis on its uses in cancer research. Researchers may be able to detect and measure these anomalies in tissue samples by using fractal analysis, which could provide a novel method for the identification and diagnosis of cancer. Buckland-Wright. J. C., et. al [5] examines the application of fractal analysis to bone structure and its link with malignant alterations. The main focus of the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 49 https://internationalpubls.com research is on how to quantify the fractal dimension of bone formations and how variations in these dimensions may be related to cancer. Cross [6] uses of fractal dimension as a diagnostic technique to distinguish between tissues that are benign and those that are cancerous is covered in this article. In a study that they describe, the scientists used fractal analysis to analyze a variety of tissue samples and discovered that cancerous tissues frequently have larger fractal dimensions than benign ones. Dubuc. B [7] presents a brand-new method for studying cancer that makes use of fractal analysis. They investigate the use of fractal geometry in microscopic imaging and its relevance to the research of malignant tissues. The authors draw attention to how this strategy may further our knowledge of how cancer progresses and lead to better detection methods. Giusto, D. D [8] uses of fractal dimension analysis to the study of pathological disorders is the main topic of this article, which includes a case study specifically on skin cancer. The authors compare their findings to other clinical states and investigate the potential utility of the fractal dimension of skin tissue in identifying malignant alterations. Landini. G [9] have been applied to the study of diverse tissues, including cancer-affected tissues.An extensive summary of the use of fractal geometry in biological tissue analysis is given in this review paper. MacAulay, C [12] explain about how to fractal dimensions are used to quantitatively analyze cell architecture with an emphasis on cancer diagnosis. The authors explain how the complexity of cell structures can be quantified in cytology and histology by using fractal analysis. This approach's theoretical and practical elements are presented in the study, emphasizing its potential to increase the precision of cancer diagnosis. Post. U [13] investigates biological systems' fractal patterns and how they relate to cancer diagnosis. The writers talk on how fractals' erratic and self-similar patterns can be seen in a variety of biological tissues, including cancerous ones. The study offers a theoretical examination of these patterns along with suggestions for applying fractal geometry to the identification of malignant tissue alterations. Rogers. G. W [14] focuses on the cell-by-cell cancer detection utilizing fractal dimension analysis. The authors present a technique for evaluating the complexity of individual cells using fractal geometry and identifying malignant alterations. The study provides experimental data showing how well this method distinguishes between benign and malignant cells. Ryser, M. D [15] investigates the modeling of cancer cell invasion and proliferation using fractal dimension. The authors describe a mathematical model that mimics the behavior of cancer cells as they proliferate and infiltrate neighboring tissues by utilizing fractal geometry. The work emphasizes the diagnostic and therapeutic implications of fractal analysis, implying that a better knowledge of the fractal patterns of malignant growth may result in more potent therapy approaches. Sadana, A [16] use fractal geometry in conjunction with biosensing technology to improve cancer detection techniques is covered in this article. This strategy may result in earlier and more sensitive and accurate cancer detection, enhancing diagnostic capacity and treatment results. Traverso.S [18] use of fractal dimension analysis to identify malignant cells is presented in this article.. This suggests that fractal analysis may be a useful method for cellular cancer identification. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 50 https://internationalpubls.com The study also explores how this approach might be used in clinical settings to improve the precision and dependability of cancer diagnosis. 1.2 Research Gap La Brun [10] introduced in this article as a novel approach to cancer screening. The authors address the possibility of identifying and analyzing malignant tissues using fractal dimensions, and they suggest that this method could result in notable improvements in diagnostic precision. Liebovitch, L. L [11] tissues is examined the application of fractal analysis to the study of malignant in this article. With an emphasis on using fractal dimensions to distinguish between healthy and malignant tissues, the authors provide a mathematical framework for employing fractal geometry to analyze tissue architectures. We introduce the image fractal dimension method for distinguishing cancer cells. We apply new methods such as fractal signature method and box-counting method for determining the image’s fractal dimension. 1.3 Notations DB : Box-Counting Method DS : Sausage Method MM : Malignant Melanoma HMM : Hidden Markov Model Dim B F: box counting dimension p : Probability f(n) : function N : Natural numbers R : Real numbers d(i) : Radial distance 2 Percolation Model In the process of the disease spreading throughout the organ, a single percolation cluster (cancer cell) is produced. Each site in a square lattice represents a person who has a probability of (p) of infection and a probability of (1-p) of immunity. The person at the centre of the lattice (cell) is infected at initial time t = 0. We now assume that all non-immune nearby neighbour sites become infected by this infected site within a single unit of time. These infected sites will spread to all of their nearby, non-immune neighbour sites in the second unit of time, and so forth. This means that all of the non- immune spots in the square grid surrounding the cells are infected after t time steps, or the longest possible path between the infected sites and the cell is tl  . 2.1 Algorithm Step 1: Begin at the origin, which is the centre of the empty site (square lattice). Step 2: The closest neighbouring sites are either blocked with probability (p) or occupied with probability (p) from the origin. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 51 https://internationalpubls.com Step 3: The empty nearest neighbour sites are blocked with probability and proceed with probability (p) as in Step 2. The aforementioned technique is very helpful for researching the composition and characteristics of a single percolation cluster. Fig. 1.4: First four steps for the percolation model 3 Boundary Descriptors When representing an item with an irregular shape, the border direction is a better choice. Relative position or direction is indicated by consecutive points on a shape's boundary. Applying the Theorem of Maximum Modulus Let f(z) be analytic and non-constant in the interior of D and continuous in a closed, bounded area D. Then, and never within D, |f(z)| reaches its maximum value on D's perimeter. Cell growth is based on the aforementioned theorem. Inside the tissue, cell growth is erratic and non-constant, but it is continuous outside of a closed, bounded area. The cell can then reach its maximum on the tissue's periphery and never within. Thus, we may determine the maximal boundary of skin cancer (MM) based on the aforementioned theorem. Cell compactness in tissue and the Box Counting Method can both be used to demonstrate this. Compactness ( π‘ƒπ‘’π‘Ÿπ‘–π‘šπ‘’π‘‘π‘’π‘Ÿ2 π΄π‘Ÿπ‘’π‘Ž ) is a dimensionless number that is minimal for a region with an irregular shape. This straightforward test determines how aggressive the skin cancer is. 3.1 Radial Distance Measure Perimeter is an important feature of a cell. Contour based features which ignore the interior of a shape, depend on finding the perimeter or boundary points of the cell. This perimeter is used for the parametric boundary representation, 𝑇 = ∫ √π‘₯2(𝑑) + 𝑦2(𝑑) 𝑑𝑑. (1.1) Minkowski-Bouligand dimension is defined as if A is a bounded set of Euclidean space, then ( )ο₯A is the set of all points at a distance less than ο₯ from A, and the β€œthickened” set ( )ο₯A is also called Minkowski Sausage. It is the union of all balls of radius ο₯ centered on A. Denoting the volume by V, we get the Minkowski-Bouligand dimension is 𝐷𝑀 = lim βˆˆβ†’0 (3βˆ’ log 𝑉(𝐴(πœ–)) log πœ€ ). (1.2) By the aid of Sausage Method or Boundary Dilation Method which is very closely related to the Minkowski Dimension. The images were dilated with circles of increasing diameter. As an approximation for a circle boxes with pixel sizes of ,11ο‚΄ ,33ο‚΄ 1717,...,55 ο‚΄ο‚΄ were again used. 2 1 2 3 1 0 2 3 2 3 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 52 https://internationalpubls.com The corresponding approximated radius r in pixels was calculated by π‘Ÿ = ( 𝐴 πœ‹ ) 1/2 (1.3) where 𝐴 denotes the area in pixel. The slope kS of the regression line of the double logarithmic plot of the counted pixels with respect to the radii provides 𝐷𝑠 = 2βˆ’ π‘˜π‘  (1.4) The estimated fractal capacity dimension is DS. In this method, the diameter of the cell can be calculated. The quantitative parameters such as Area, Perimeter, Form factor and Invaslog can be found out using Sausage method (Fig. 1.2 & 1.3). πΉπ‘œπ‘Ÿπ‘šπ‘“π‘Žπ‘π‘‘π‘œπ‘Ÿ = 4πœ‹ π΄π‘Ÿπ‘’π‘Ž π‘ƒπ‘’π‘Ÿπ‘–π‘šπ‘’π‘‘π‘’π‘Ÿ2 (1.5) πΌπ‘›π‘£π‘Žπ‘  log = βˆ’log (π‘“π‘œπ‘Ÿπ‘šπ‘“π‘Žπ‘π‘‘π‘œπ‘Ÿ) (1.6) The value of πΌπ‘›π‘£π‘Žπ‘  π‘™π‘œπ‘” has been specially proven to be a strong quantitative measure of the invasiveness of skin cancer. Radial distance is the distance from the centre of the mass to the perimeter point (π‘₯𝑖, 𝑦𝑖) So the radial distance is defined as 𝑑(𝑖) = √(π‘₯𝑖 βˆ’ οΏ½Μ…οΏ½)2 + (𝑦𝑖 βˆ’ οΏ½Μ…οΏ½)2, 𝑖 = 0,1,2… . 𝑁 βˆ’ 1. (1.7) Here ( )id is a vector obtained by the distance measure of the boundary pixels. A normalized vector ( )ir is obtained by dividing ( )id by the maximum value of ( )id . 4. Statistical Analysis The reliability of statistical analysis results is influenced by the quantity of samples per statistical group, which supports the final conclusions. This number is constrained, on the one hand, by the experiment's real-time nature and the researchers' practical capacity to manage a high number of samples. However, since the variance is inversely related to the square root of the number of samples examined, the number must be suitably high. Therefore, for these trials and the statistical analysis of the data, more than 20 samples were selected. The probability distribution of the cells inside each tissue was used to calculate the fractal dimension for that tissue. The nonlinear regression equation y= a xb, where y is the number of cells within a square of the grid with a given radius x, an is the scaling coefficient, and b is the fractal dimension, was fitted to numerical experimental data. 4.1 Results Estimating the fractal dimension D with two different methods (DB & DS) for the twenty samples. The large set of image sequences yielded a large amount of information, if D is the dynamic behavior of the Cancer Invasion, Percolation Model. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 53 https://internationalpubls.com Table 1.1. Data analysis of a typical Image using Box-Counting Method (Original Image) Scaling Original Image I Area Perimeter Total Area Form Factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 4631 1945 1046 635 423 300 220 162 128 742 604 426 352.6 286 241 190 177 15.6 5373 2549 1472 987.6 709 541 410 339 278.6 0.106 0.067 0.072 0.064 0.065 0.65 0.77 0.065 0.071 0.975 1.174 1.143 1.194 1.187 1.187 1.114 1.187 1.149 1.33 Scaling Original Image II Area Perimeter Total Area Form Factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 6783 2923 1612 1009 683 490 376 283 229 488 389 293 231.6 194.83 170 137 128.89 107.3 7271 3312 1905 1240.6 877.83 660 513 411.8 336.3 0.358 0.243 0.235 0.236 0.226 0.213 0.252 0.214 0.250 0.446 0.614 0.629 0.627 0.646 0.672 0.599 0.670 0.602 1.21 Table 1.2. Data analysis of Atypical Image using Box-Counting Method (Dermatologist Image-1 ) Scaling Dermatologist Image I Area Perimeter Total Area Form Factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 1180 480 242 142 92 67 46 35 25 375 261 189 152 115 86 78 61 59 1555 741 431 294 207 153 124 96 84 0.105 0.089 0.085 0.077 0.087 0.114 0.095 0.118 0.090 0.979 1.051 1.071 1.114 1.060 0.944 1.022 0.928 1.046 1.67 Scaling Dermatologist Image II Area Perimeter Total Area Form Factor Invaslog Dimension Db 2 3 4 1757 738 400 277 197 140 2034 935 540 0.289 0.239 0.256 0.541 0.622 0.592 1.52 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 54 https://internationalpubls.com 5 6 7 8 9 10 245 165 118 88 68 53 108 88 71 59 46 40 353 253 189 147 114 93 0.264 0.268 0.294 0.318 0.404 0.416 0.578 0.572 0.532 0.498 0.394 0.381 Table 1.3: Data analysis of Atypical Image using Sausage Method for Image I & II Scaling Original Image I Dermatologist Image I Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 2 3 5 7 9 11 13 15 1945 635 300 162 103 69 47 36 24.882 14.217 9.772 7.181 5.726 4.687 3.868 3.385 0.67 1.33 480 142 67 35 21 11 7 7 12.361 6.723 4.618 3.338 2.585 1.871 1.493 1.493 0.33 1.67 Scaling Original Image II Dermatologist Image II Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 3 5 7 9 11 13 15 17 2923 1009 490 283 185 127 92 68 30.503 17.921 12.489 9.491 7.674 6.358 5.412 4.652 0.79 1.21 738 245 118 68 42 28 19 14 15.327 8.831 6.129 4.652 3.656 2.985 2.459 2.111 0.48 1.52 Benign Image :The given Table 1.1,1.2 and 1.3 shows the data produced by way of Box Counting Method and Sausage Method respectively. Table 1.4: Data analysis of Benign Image using Box-Counting Method (Original Image) Scaling Original Image I Area Perimeter Total Area Form factor Invaslog Dimension Db 2 3 4 5 6 7 4693 1813.7 940.4 563 373 262.4 1982 1442.33 961.75 702 513.17 396.86 6675 3256 1902.13 1265 886.17 659.29 0.015 0.011 0.013 0.014 0.018 0.021 1.824 1.959 1.886 1.854 1.745 1.678 1.34 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 55 https://internationalpubls.com 8 9 10 189.5 150 115 314.63 249.52 210.5 504.09 399.52 325.5 0.024 0.030 0.033 1.620 1.523 1.481 Scaling Original Image II Area Perimeter Total Area Form factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 6582 2730.67 1474 879.60 575.67 392.58 293.5 207.56 159.2 962 780.67 559 447.60 363.33 314.58 251 231.89 200.4 7544 3511 2033 1327.20 939 707.14 544.5 439.44 359.6 0.089 0.056 0.059 0.055 0.055 0.050 0.059 0.049 0.050 1.051 1.252 1.229 1.260 1.260 1.301 1.229 1.310 1.301 1.31 Table 1.5. Data analysis of Benign Image using Box-Counting Method (Dermatologist Image) Scaling Dermatologist Image I Area Perimeter Total area Form factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 1053 406 208 123 86 59 43 32 24 415 277 185 134 104 81 62 48 44 1468 683 393 257 190 140 105 80 68 0.077 0.066 0.076 0.086 0.100 0.113 0.141 0.175 0.156 1.114 1.181 1.119 1.066 1 0.947 0.851 0.757 0.807 1.67 Scaling Dermatologist Image II Area Perimeter Total area Form factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 1653 659 344 203 130 86 61 43 31 384 292 206 160 130 108 88 80 72 2037 951 550 363 260 194 149 123 103 0.141 0.097 0.102 0.100 0.097 0.093 0.099 0.084 0.075 0.851 1.013 0.991 1 1.013 1.032 1.004 1.076 1.125 1.67 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 56 https://internationalpubls.com Table 1.6. Data analysis of Benign Image using Sausage Method for Image I & II Scaling Original Image I Dermatologist Image I Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 3 5 7 9 11 13 15 17 1813.67 563 262.43 150 91.37 64 46 37 24.027 13.387 9.140 6.910 5.393 4.514 3.827 3.432 0.66 1.34 406 123 59 32 21 13 8 5 11.368 6.257 4.334 3.192 2.585 2.034 1.596 1.262 0.32 1.68 Scaling Original Image II Dermatologist Image II Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 3 5 7 9 11 13 15 17 2730.67 879.60 392.57 207.56 114.27 69.31 43.33 33 29.48 16.73 11.18 8.12 6.03 4.7 3.7 3.24 0.69 1.31 659 203 86 43 22 12 9 4 14.48 8.038 2.232 3.7 2.646 1.954 1.693 1.128 0.33 1.67 Malignant Melanoma Image The given Table 1.4, 1.5 & 1.6 shows the data produced by way of Box Counting Method and Sausage Method respectively. Table 1.7. Data analysis of Malignant Melanoma using Box-Counting Method (Original Image) Scaling Original Image I Area Perimeter Total area Formfactor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 6090 2379.33 1194 669 420 278 200 135 95 2200.5 1632.334 1186.5 910.200 717.500 580 458.25 400.1 337 8290.5 3999.67 2380.5 1579.20 1137.50 858 658.25 536 432 0.016 0.011 0.011 0.0101 0.0103 0.0104 0.012 0.011 0.011 1.80 1.96 1.96 1.996 1.987 1.983 1.921 1.959 1.959 1.8 Scaling Original Image II Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 57 https://internationalpubls.com Area Perimeter Total area Formfactor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 5065 2036.67 1084.62 658.04 430.83 297 228 162.07 122.20 1993.5 1478.67 1023.0 763.80 588.83 477.57 362.72 324.56 282.96 7058.5 3515.33 2107.63 1421.84 1019.67 774.57 590.72 486.63 405.16 0.016 0.012 0.013 0.014 0.015 0.016 0.022 0.019 0.019 1.80 1.92 108 1.85 1.82 1.80 1.66 1.72 1.72 1.788 Table 1.8. Data analysis of Malignant Melanoma using Box-Counting Method (Dermatologist Image) Scaling Dermatologist Image I Area Perimeter Totalarea Formfactor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 1635 647 311 181 105 68 49 32 27 589 414 321 239 203 164 134 118 93 2224 1061 634 420 308 232 183 150 120 0.059 0.047 0.038 0.040 0.032 0.032 0.034 0.029 0.039 1.229 1.328 1.420 1.398 1.495 1.495 1.469 1.538 1.409 1.8 Scaling Dermatologist Image II Area Perimeter Total area Form factor Invaslog Dimension Db 2 3 4 5 6 7 8 9 10 1130.5 461 234 137.2 85 56 44 27 20 545.5 350 252.75 189.6 150.94 123.16 95.38 89.35 72.28 1676 811 486.75 326.8 232.94 179.16 139.38 116.35 92.28 0.048 0.047 0.046 0.048 0.047 0.046 0.060 0.043 0.048 1.319 1.328 1.337 1.319 1.328 1.337 1.222 1.367 1.39 1.81 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 58 https://internationalpubls.com Table 1.9. Data analysis of Malignant Melanoma using Sausage Method for Image I & II Scaling Original Image I Dermatologist Image I Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 3 5 7 9 11 13 15 17 2379.33 669 278 135 73 35 23 14 27.520 14.593 9.407 6.555 4.820 3.338 2.706 2.111 0.57 1.43 647 181 68 32 16 10 4 2 14.351 7.590 4.652 3.192 2.257 1.784 0.128 0.798 0.25 1.75 Scaling Original Image II Dermatologist Image II Area Radius sK sD sKβˆ’= 2 Area Radius sK sD sKβˆ’= 2 3 5 7 9 11 13 15 17 2036.67 658.04 297 162.07 95 62 38.03 28 25.462 14.473 9.723 7.183 5.499 4.442 3.479 2.985 0.65 1.35 461 137.2 56 27 9 6 2 3 12.114 6.608 4.222 2.932 1.693 1.382 0.801 0.977 0.19 1.81 4. Conclusion Finally, there are a number of benefits and insights to be gained from using the fractal box counting method to the determination of the borders of malformed cells, especially in the context of skin cancer: Cell boundary complexity can be quantified via fractal box counting. This approach provides a mathematical foundation for characterizing irregular forms, which is very helpful for examining the complex and uneven boundaries that are frequently present in malignant cells. Cancer cells' uncontrollably fast proliferation and invasion frequently result in their asymmetrical, fractal-like shapes. Compared to conventional geometric methods, the fractal box counting method allows for a more accurate delineation of cell borders since it is sensitive to these imperfections. Based on the cell potential, a radial distance measure and the cells compactness of the irregularity boundary for pigmented skin lesions are suggested. It is suggested to use the Sausage Method and Box Counting Method to analyse both the dermatologist's provided image and the original. Protrusions and indentations were used in the current ways to explain the irregularity border. This will not get very accurate results [48]. The experimental results show unequivocally that every cell in the population has distinct irregularity borders in addition to having fractal dimensions. As a result, the suggested Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3s (2025) 59 https://internationalpubls.com Sausage Method (DS) and Box-Counting Method (DB) will produce extremely accurate results. We can determine the cancer's stages based on the cell's dimensions. The degree of invasiveness rises in tandem with the dimension. Conflict of Interest: The authors declare that they have no conflict of interest. References [1] Ahmed, E. (1993). Fractal dimension and its application in cancer detection. International Journal of Theoretical Physics, 32(3), 353-365. [2] Aon, M. A., & Cortassa, S. (1994). Fractal analysis of cellular structures: Implications for cancer detection. FEBS Letters, 344(1), 1-5. [3] Battaglia-Parodi, M., & Giusto, D. D. (1993). 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