Department of Mathematics
http://hdl.handle.net/10211.2/1914
Mathematics Master's Thesis Collection | Faculty Publications and Research2017-03-31T00:39:08ZComputational Optical Biopsy
http://hdl.handle.net/10211.3/177802
Computational Optical Biopsy
Li, Yi; Jiang, Ming; Wang, Ge
Optical molecular imaging is based on fluorescence or bioluminescence, and hindered by photon scattering in the tissue, especially in patient studies. Here we propose a computational optical biopsy (COB) approach to localize and quantify a light source deep inside a subject. In contrast to existing optical biopsy techniques, our scheme is to collect optical signals directly from a region of interest along one or multiple biopsy paths in a subject, and then compute features of an underlying light source distribution. In this paper, we formulate this inverse problem in the framework of diffusion approximation, demonstrate the solution uniqueness properties in two representative configurations, and obtain analytic solutions for reconstruction of both optical properties and source parameters.
2005-01-01T00:00:00ZQuantitative Interpretation of a Genetic Model of Carcinogenesis Using Computer Simulations
http://hdl.handle.net/10211.3/177803
Quantitative Interpretation of a Genetic Model of Carcinogenesis Using Computer Simulations
Dai, Donghai; Beck, Brandon; Wang, Xiaofang; Howk, Cory; Li, Yi
The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor heterogeneity. Our model describes normal tissue regeneration, malignant transformation, cancer incidence including dormant/transient tumors, and tumor evolution. Further, a novel mechanism for the initiation of metastasis resulting from substantial cell death is proposed. Finally, model simulations suggest two different mechanisms of metastatic inefficiency for aggressive and less aggressive cancer cells. Our work suggests that cellular de-differentiation is one major oncogenic pathway, a hypothesis based on a numerical description of a cell's differentiation status that can effectively and mathematically interpret some major concepts in V/W theories such as progressive transformation of normal cells, tumor evolution, and cancer hallmarks. Our model is a mathematical interpretation of cancer phenotypes that complements the well developed V/W theories based upon description of causal biological and molecular events. It is possible that further developments incorporating patient- and tissue-specific variables may build an even more comprehensive model to explain clinical observations and provide some novel insights for understanding cancer.
2011-01-01T00:00:00ZSingularity of Super-Brownian Local Time at a Point Catalyst
http://hdl.handle.net/10211.3/177804
Singularity of Super-Brownian Local Time at a Point Catalyst
Dawson, Donald; Fleishmann, Klaus; Li, Yi; Mueller, Carl
In a one-dimensional single point-catalytic continuous super-Brownian motion studied by Dawson and Fleischmann, the occupation density measure λc at the catalyst's position Cis shown to be a singular (diffuse) random measure. The source of this qualitative new effect is the irregularity of the varying medium δc describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters χ appearing in the Levy-Khintchine representation of λc in a historical process setting and the fact that these χ have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.
1995-01-01T00:00:00ZPerturbation of Global Solution Curves for Semilinear Problems
http://hdl.handle.net/10211.3/177798
Perturbation of Global Solution Curves for Semilinear Problems
Korman, Philip; Li, Yi; Ouyang, Tiancheng
We revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in 161. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approach, and study the evolution of turning points. We give conditions under which the critical (turning) points continue on smooth curves, which allows us to reduce the problem to the easier case of f (0) = 0. We show that the smallest root of f (u) does not have to be restricted.
2003-01-01T00:00:00Z