Selected Topological Indices with Applications to the Identification of Anti-breast Cancer Agents
Simon Mukwembi1, Farai Nyabadza2, Opeyemi Dorcas Ogundeji3
1University of Witwatersrand, South Africa
2University of Johannesburg, South Africa
3African Institute for Mathematical Sciences (AIMS), Cape Town, South Africa
Abstract: – Breast Cancer is a global challenge that mostly affects women and its treatment is stage dependent. The treatment, among other methods, involves the use of medicines that shrink or kill the cancer cells. This type of treatment involves the use of chemical agents. In this research, the graph theoretic techniques to screen and identify chemical compounds with anti-breast cancer activities were used. In particular, the distance and graph based topological indices were used.
In this research, we developed a graph theory model that screens compounds for anti-breast cancer activity, and we also provide an assessment of the quality of model that was developed. Our newly developed graph theoretic model can predicts IC50 values of compounds for anti-breast cancer activity. We also recommended that normalized diameter is one of the key parameter in developing a model to predict some useful flavonoids that have anti-breast cancer activity.
We presented the upper bound on the known indices and explored their applications in the prediction of cytotoxic activity of flavonoids against breast cancer cells in drug design. A model that fits to data generated in the laboratories was formulated and analyzed. We did simulation on the correlations between parameters and IC50 values. Conclusions on relationship of activities and graphical structures will be made. The implications of relationship on the identification of compounds with the most activity against breast cancer are also discussed.
Keywords: Graphs • Diameter • Flavonoids • Distances • Eccentricity • IC50 values
I. Introduction
K¨onigsberg is an ancient city of Russia, which is currently known as Kalingrad, Russia. There are seven bridges in the city which was constructed on both sides of the Pregel River. The city was divided into two islands by the seven bridges. The seven bridges are: Honey bridge, Connecting bridge, Blacksmith bridge, Green bridge, Merchant?s bridge, and Wooden bridge. Some people in the city used to spend their Sunday afternoons walking around the city. Then, the people of the city made a game for themselves, and the goal was to devise a way to walk around the city while only crossing each of the seven bridges once. No one was able to devise a path that would allow them to cross each bridge once (Natalya, 2015). The story has captivated mathematicians? imaginations up to the present day. In an attempt to solve the bridge problem some centuries ago, L. Euler introduced the concept of graphs. The solving of the bridge problem led to an increase in the use of graphs in solving real life problems. Chemical graphs were introduced in 1895 and have received increased attention in the last two decades as a result of their usefulness in chemical engineering and pharmaceuticals (Arielle, 2017). Cancer is a disease in which some cells in the body develop uncontrollably and spread to other regions of the body. Cancer can begin practically at any place in the human body, which contains billions of cells (cancer institute, 2007). There are about 100 different types of cancers. Cancers are frequently named from the organs or tissues in which they develop. Lung cancer, for example, starts in the lungs, skin cancer begins in the skin while breast cancer begins in the breast. According to Judy (2022), skin cancer is the abnormal growth of skin cells which most often develops on skin exposed to the sun. On the other hand, breast cancer is a type of cancer that develops in the cells of the breasts. Breast cancer is the second most common cancer diagnosed in women after skin cancer. Breast cancer can affect both men and women, but it mostly affects women. Substantial support for breast cancer awareness and research funding has aided in the advancement of breast cancer diagnosis and treatment. Breast cancer survival rates have increased, and the number of deaths associated with the disease have steadily decreased, and to factors such as early detection, a new personalized approach to treatment, and a better understanding of the disease. Some analgesic activity agents are used to kill pain (Pain killers). Pain in breast is one of the common symptoms that occur which makes a lot of women seek medical care. A lump in the breast, bloody discharge from the nipple and changes in the form or texture of the nipple or breast are all symptoms of breast cancer (Salah, 2015).