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Analysis of a multiphase free boundary problem

Abdelouahab Ahlem, Bensid Sabri

Opuscula Mathematica

2024

Vol. 44, no. 5

631--649

In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution.

Stochastic Models of Progression of Cancer and Their Use in Controlling Cancer-Related Mortality

Kimmel M., Gorlova O. Y.

International Journal of Applied Mathematics and Computer Science

2003

Vol. 13, no 3

279-287

A construction of a realistic statistical model of lung cancer risk and progression is proposed. The essential elements of the model are genetic and behavioral determinants of susceptibility, progression of the disease from precursor lesions through early (localized) tumors to disseminated disease, detection by various modalities, and medical intervention. Using model estimates as a foundation, mortality reduction caused by early-detection and intervention programs can be predicted under different scenarios. Genetic indicators of susceptibility to lung cancer should be used to define the highest-risk subgroups of the high-risk behavior population (smokers). The calibration and validation of the model requires applying our techniques to a variety of data sets available, including public registry data of the SEER type, data from the NCI lung cancer chest X-ray screening studies, and the recent ELCAP CT-scan screening study.

Dynamics of the Tumor-Immune System Competition-the Effect of Time Delay

Gałach M.

International Journal of Applied Mathematics and Computer Science

2003

Vol. 13, no 3

395-406

The model analyzed in this paper is based on the model set forth by V.A. Kuznetsov and M.A. Taylor, which describes a competition between the tumor and immune cells. Kuznetsov and Taylor assumed that tumor-immune interactions can be described by a Michaelis-Menten function. In the present paper a simplified version of the Kuznetsov-Taylor model (where immune reactions are described by a bilinear term) is studied. On the other hand, the effect of time delay is taken into account in order to achieve a better compatibility with reality.