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  • Several limitations of this study should be acknowledged Fir


    Several limitations of this study should be acknowledged. Firstly, all data came from the TCGA database. There was no verification of miRNA expression in tissue samples and no exploration of synergistic molecular mechanisms by experiment. Secondly, the OS of EC patients also came from the TCGA database, with no prospective follow-up data from other databases or clinical trials. Finally, for the 4-miRNA model, the AUC for the ROC curves in the entire set was 0.704 at 5-year OS. Herein, further studies are needed to be performed to obtain a more accurate prognostic model. However, this study proved the potential of the 4-miRNA model as a biomarker of EC prognosis, providing new insights into the prognosis and individualized treatment of EC.
    In conclusion, a 4-miRNA prognostic model was established and had the potential to predict the survival prognosis of EC patients. Through the expression levels of these four miRNAs, the patient's risk score can be calculated. By combining the risk score with the clinical factors, the long-term survival rate of EC patients can be estimated. This conclusion needs further clinical trials to be verified. 
    Conflicts of interest
    The authors declare that they have no conflict of interest.
    Appendix A. Supplementary data
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    multifaceted regulator in organogenesis, homeostasis, and disease. Dev. Dyn. 246, 285–290.
    Torres, A., Torres, K., Pesci, A., Ceccaroni, M., Paszkowski, T., Cassandrini, P., Zamboni, G., Maciejewski, R., 2013. Diagnostic and prognostic significance of miRNA sig-natures in tissues and plasma of endometrioid endometrial carcinoma patients. Int. J. Cancer 132, 1633–1645.
    Microenvironment-induced PTEN loss by exosomal microRNA primes CFTRinh-172 metas-tasis outgrowth. Nature 527, 100–104.
    Contents lists available at ScienceDirect
    Chaos, Solitons and Fractals
    Nonlinear Science, and Nonequilibrium and Complex Phenomena
    journal homepage:
    A fractional mathematical model of breast cancer competition model
    J.E. Solís-Pérez a, J.F. Gómez-Aguilar b,∗, A. Atangana c a Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México b CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México c Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
    Article history:
    Fractional derivatives and integrals
    Integral transforms
    Laplace transform
    Breast cancer competition model 
    In this paper, a mathematical model which considers population dynamics among cancer stem cells, tu-mor cells, healthy cells, the effects of excess estrogen and the body’s natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville–Caputo sense were considered. Special solutions using an iterative scheme via Laplace trans-form were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana–Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful in-formation about the complexity of the dynamics of the breast cancer competition model.
    1. Introduction
    Cancer is a leading cause of death in many countries around the world. Cancer development is a stepwise process through which normal somatic cells acquire mutations which enable them to es-cape their normal function in the tissue and become self-su cient in survival. Breast cancer is the most common cancer among women in worldwide. Rates of breast cancer are increasing world-wide, with a particular increase in postmenopausal and estrogen receptor-positive cases [1]. The natural history of breast cancer has been di cult to study. However, mathematical models and com-putation simulations can help offering the ability to monitor tu-mor growth, cellular distribution and to observe the genetic mu-tations that lead to aggressive growth and metastasis. Mufudza in [2], examined the effects of excess estrogen on breast cancer dy-namics with the inclusion of an immune cell population to model the body’s natural response to tumor growth. In [3], developed a mathematical model for cancer cells in a post-pubertal breast from healthy or pre-cancerous cells.The model consider four differential equations describing the stepwise mutations from a normal breast stem cell to a tumour cell. In [4], the authors studied the role of excess estrogen in the development of breast cancer and its impact on the body’s natural immune response, the interactions among cancer stem cells, non-proliferative tumor cells, and healthy ep-ithelial cells in the breast tissue were considered. Another inter-esting works can be found in [5–11].