Breast cancer is the second most common cancer in women, and incidences of it continue to rise. The cure rate is increasing for breast cancer that is diagnosed in its early stages. That is why the value of breast health care is so important. Malignant breast tumor detection is extremely important for the physician. Generally, this task has been performed based on the experience and knowledge of the radiologists by analyzing the visual characteristics of the structures that appear in mammograms. The irregularity that exists in the edge of the typical cell that forms part of the tumor and tumoral structures (angiogenesis) is one of the major characteristics. Particularly, some malignant tumors involve a more irregular boundary edge, as compared to the benign ones. Speculating this feature by means of evaluating the irregularities can help support the diagnosis of malignancy. Hence, study of chaotic time series (CTS) can afford the tools necessary to generate the procedures to speculate the irregularities in the edges, through the use of the concept of fractal dimension (FD), which can produce the parameters to describe and categorize the structures under study. To detect the tumors and generate the time series (TS) characterizing the edge, techniques of digital image processing are applied over breast thermal images. Ahmed concluded that tumor growth, argued as a dynamical system, is chaotic. Proposed chaotic models fit the observations well. Some of these models treat the tumor as a fractal.
In this chapter, Lyapunov exponents (LEs) are computed from CTS based on the Jacobian approach by using polynomial models. The chapter is organized as follows: the CTS is introduced in Section 9.2, the time-delay embedding (TDE) method is described in Section 9.3, followed by an explanation of LEs in 9.4. In Section 9.5, a method that is used to calculate the LEs is presented. The outline of the method to generate the TS of the edge is explained in Section 9.6, and Section 9.7 presents the results produced by using this process on the different types of tumors under study. Finally, the findings are concluded in Section 9.8.
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