by Yongzhi Xu
My current research is focused on inverse scattering problems and mathematical model of breast cancer.
Inverse scattering problems
Scattering theory is concerned with the effect an inhomogeneity has on an incident wave or particle. The inverse scattering problem is to determine the unknown inhomogeneity from the knowledge of the incident wave and the measured scattered wave. Its applications include medical imaging, underwater imaging and underground imaging.
Currently I am cooperating with Professor Jun Zou of Chinese University of Hong Kong and Professor Dinghua Xu of Zhejiang University of Technology on developing effective numerical algorithms for computer simulations. We developed a parallel radial bisection algorithm and a direct sampling method for the inverse scattering problems, which greatly improved the existing methods.
Mathematical modeling of cancer
I continue my research on cancer modeling, analysis and computation. Developing mathematical models of tumor growth is an emerging field and has attracted much research in recent years. Some models have been studied by biomathematicians. Assuming that a tumor consists of a continuum of live and dead cells, we can describe local volume changes and cell movement due to cell growth and death by a system of partial differential equations. Hence we are able to capture the early growth and developing spatial composition of the tumor.
I have been studying a special kind of cancer, ductal carcinoma in situ (DCIS), in order to investigate possible procedures to connect free boundary model of DCIS with clinical data. The study focuses on a class of free boundary value problem models of DCIS and their comparisons with clinical data. In particular, we formulate a number of inverse problems for the well-posed free boundary valued problem related to clinical diagnose of cancer. To the best of my knowledge, there is no publication on this kind of inverse free boundary valued problems.
This research focuses on a central issue of computational bio-mathematics, which meshes well with UofL’s efforts in cancer research. A computational model of tumor growth enables scientists to use computers to study the effects of different factors related to tumor growth. It can help to find new strategies to enhance inhibitors of tumor growth.