# STOCHASTIC AND DETERMINISTIC COAGULATION MODELS, THEIR NUMERICAL APPROXIMATIONS AND APPLICATIONS TO CELL AGGREGATION

Open Access

- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 09, 2007
- Committee Members:
- Qiang Du, Committee Chair
- Cheng Dong, Committee Member
- Chun Liu, Committee Member
- Suzanne Michelle Shontz, Committee Member
- Ludmil Tomov Zikatanov, Committee Member

- Keywords:
- cell aggregation
- coagulation equation
- stochastic model
- time relaxed method

- Abstract:
- The work in the present thesis is to study the stochastic and deterministic coagulation models, their numerical approximations, and applications to polymorphonuclear neutrophil (PMN) and tumor cell aggregation in the parallel plate flow chamber. The work is motivated by some experimental and theoretical studies on tumor cell metastasis, in particular, the earlier studies that have been done in Prof. Dong's lab. Some previous work demonstrated that PMNs can facilitate tumor cell adhesion to the endothelium. Parallel plate flow chamber experiments have shown that under different flow conditions, tumor cells and PMNs adhere to each other and to the endothelium with different efficiencies. The comprehensive study of the cell aggregation and adhesion is proceeded from three aspects: experimental studies on the kinetics of cell aggregation, adhesion and deformation; computational fluid dynamical (CFD) modeling of individual PMN and tumor cell interaction and adhesion; statistical study of large populations of cell aggregation. The experimental study and CFD modeling work conducted by Prof. Dong in his lab provided us the basis of the present thesis work focusing on the statistical study of cell aggregation. The primary focus of this work is the development of mathematical models and numerical simulation methods for cell aggregation in populations and the exploration of how the flow condition changes the interactions of PMNs and tumor cells in the nonuniform shear flow. In this thesis, the coagulation equations (also called Smoluchowski or population balance equations) are used to model cell aggregation in the near wall region in nonuniform shear flow. These equations have been used widely to study the time evolution of the particle concentration in physics, biological and chemical engineering. This thesis is organized as follows. First, we study the population balance equations with bounded kernels and their numerical approximations. A comprehensive review of the coagulation equations is given first, such as the well-posedness problem and some other basic properties. For the numerical approximation of the coagulation equation, a new formulation motivated by the Wild sum is utilized to offer a convenient framework for the analysis of the population balance equations. It also leads naturally to a time relaxed method for the numerical solution of the coagulation equations. This time relaxed method is shown to be a high order convergent numerical scheme that preserves the non-negativity of the solution and the total volume conservation. Furthermore, we study several deterministic algorithms for the numerical approximations of the discrete Smoluchowski coagulation equations, such as the time relaxed marching method and the stabilized Euler method. Their stability and convergence properties are examined, and error estimates are derived. Particular attention is given to issues such as the high order accuracy, the preservation of the total volume, and the non-negativity of the population density. The numerical examples and comparison tests are presented to demonstrate the performance of the algorithms. We also check the consistency of the computational complexity estimation and the real CPU running time. The application of these numerical methods to coagulation-fragmentation is also taken into account. In order to bridge the deterministic coagulation equation and the stochastic coagulation dynamical system, we also develop a stochastic interpretation of the coagulation equation. We show that the density function of the stochastic coagulation process satisfies the coagulation equation. Based on this model, a backward Monte Carlo method is developed. We also present a formulation of the coagulation equation by using an energetic variation framework based on discussions with Prof. Liu. The energetic variation formulation model the coagulation equation by a deterministic dynamical system in which case the model can be described by the energy law. Overall, we provide both the stochastic and deterministic formulations for the coagulation equation. Furthermore, our studies on the relations of stochastic and deterministic formulations show the consistency of these two formulations. Several examples, for instance, the transport equation, the diffusion equation, and the coagulation equation, demonstrate that the deterministic model can be derived from the stochastic model and its consistency is verified. The applications of the coagulation model to cell aggregation is joint work with Prof. Dong and his research group. We develop a simple population balance model for cell aggregation and adhesion process in a nonuniform shear flow. Some Monte Carlo simulation results based on the model are presented for the heterotypic cell-cell collision and adhesion to a substrate under dynamic shear forces. In particular, we focus on leukocyte (PMN)-tumor cell emboli formation and subsequent tethering to the vascular endothelium (EC) as a result of cell-cell aggregation. The simulation results are compared with the results of experimental measurement. We also develop a modified population balance model to describe the cell aggregation in the near wall region. In this model, the tethering frequency, adhesion efficiency, and collision rate of deformable cells are also discussed. For the spatially inhomogeneous coagulation equation, we first introduced the Lagrangian model. By the deterministic formulation in Section 3.4, we derive the equation by considering the conservation law of a coupled system which involves spatial variable and particle size variable. By the conservation law formulation, first, we develop a model in which the coagulation kernel and fluid dynamics are given, then we consider a self-consistent model where the particle motion and coagulation are induced by a given transition probability. Finally, we present some issues for future studies.