POPULATION DYNAMICS MODELING OF HETEROTYPIC CELL AGGREGATIONS AND RELATED PARAMETER IDENTIFICATION PROBLEMS

Open Access
Author:
Ma, Yanping
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 05, 2011
Committee Members:
  • Qiang Du, Dissertation Advisor/Co-Advisor
  • Cheng Dong, Dissertation Advisor/Co-Advisor
  • Qiang Du, Committee Chair/Co-Chair
  • Cheng Dong, Committee Chair/Co-Chair
  • Xiantao Li, Committee Member
  • John Fricks, Committee Member
  • Qiang Du; Cheng Dong, Dissertation Advisor/Co-Advisor
Keywords:
  • Population dynamics
  • Bayesian framework
  • inverse problem
  • cell coagulation
  • cancer metastasis
Abstract:
Cancer has been one of the leading causes of death around the world for decades. Metastasis, the spread of the tumor cells from the primary site to other locations in the body via the lymphatic system or through the blood stream, is responsible for most of the cancer deaths. Massive experimental studies have been done in these areas. The work of this thesis brings together the experimental, numerical, and mathematical studies on the step of polymorphonuclear neutrophils (PMNs) tethering to the vascular endothelial cells (EC), and the subsequent melanoma cell emboli formation in a nonlinear shear flow, both of which are important in tumor cell extravasations from the circulation during metastasis. The primary focus of this work was the development of mathematical models of heterotypic aggregation between PMNs and melanoma cells in the near-wall region of an in vitro parallel-plate flow chamber. A population balance model based on the Smoluchowski coagulation equation was applied to study the process, and then in vivo cell-substrate adhesion from the vasculatures were simulated by combining mathematical modeling and numerical implementations with experimental observations. This work proposed a new coagulation kernel, and also, developed a multiscale near-wall aggregation model, which to the best of our knowledge, was the first one that could incorporate the effects of both cell deformation and general ratios of heterotypic cells on the cell aggregation process. Quantitative agreement was found between numerical predictions and in vitro experiments. The effects of factors, including: intrinsic binding molecule properties, near-wall heterotypic cell concentrations, and cell deformations on the coagulation process, were discussed. Sensitivity analysis has been done, and we concluded that the reaction coefficient along with the critical bond number on the aggregation process should be recommended as the most critical variables. The conclusions from this thesis work also managed to propose more experimental time-frame design by the end of chapter two. The success of mathematical modeling not only depends on the theoretical model development, but also crucially relies on the accuracy of parameter estimations. Following the comparisons between numerical simulations and experimental results, it was proposed that the parameters in our model, instead of being constants over time, could be modeled by functions with stochastic uncertainties. The other part of this thesis work, chapter three, focused on the application of the Bayesian framework to the parameter identification problems for our models, and hence, provided quantitative instead of purely qualitative analyses for the inverse problems. We adopted the Bayesian framework studied in literatures, and theoretically verified the assumptions for our models. Moreover, we successfully found the maximum a posteriori estimator (MAP estimator) for the parameters following variational methods, and estimated the sensitivity of the parameters with respect to the data using Monte Carlo methods. This work thus offered a systematic parameter identification tool specially tailored to our models.