Image Segmentation And Tumor Biomedical Modeling Study Based On Partial Differential Equation

Partial differential equation is a reflection of the restricting relation between the time derivatives and the spatial derivatives of the unknown variables. The mathematical models in many fields could be described by partial differential equations, which are just the basic formula in many important areas, such as physics and mechanics. As early as the theory of calculus was introduced, partial differential equations were used to describe, explain and predict various natural phenomena, based on which, the relative research methods and research results were applied to various science and engineering technology, showing the importance of partial differential equations on the understanding of basic nature laws. Gradually, with the background of many practical problems in the physical, mechanical and other branches of science, the research of partial differential equations becomes one of the most important contents in traditional applied mathematics. It is directly linked to many natural phenomena and practical problems. This study constantly presents and introduces new directions and new methods, and promotes the developments of many related branches of mathematics (such as functional analysis, differential geometry, and computational mathematics), and many powerful tools used to solve practical problems are consequentially introduced. Partial differential equation has become an important component in contemporary mathematics, and also become an important bridge between many branches of pure mathematics and many areas of natural science and engineering technology.This thesis mainly discusses the application of the partial differential equation, as a powerful mathematical tool, in the fields of image segmentation and biomedical modeling.In the areas of image processing and computer vision, the methods based on partial differential equations have become research hotspots. On one hand, as an important branch of basis mathematics, it has formed theoretical system and numerical methods, on the other hand, it also benefits from the traditional image processing technology experience. In the past10years, partial differential equations are widely used in different fields of image processing, including image segmentation, amplification, super-resolution analysis, image reparation, sharpening, denoising, and contour tracking of the moving object. In image segmentation field, the application of partial differential equations began in the eighty's of the last century, when Kass et al proposed the active contour model which used partial differential equations as the basic form. This model studied the curve evolution procedure from the point of view of dynamics. By solving the partial differential equations, the boundary of interest region would be obtained. In the next thirty years, with the rapid development of the active contour model, it had gradually become the most active and the most successful image segmentation technique. And then various models with different features were derived from it, such as geometric model, active surface, deformation balloon, deformation contour, and deformation surface. In all active contour models, we focused on the theory and improvements of two kinds of classical geometric models. One is edge information based geodesic active contour (GAC) model, the other one is region information based C-V model. These two models both produced tremendous influences on subsequent image segmentation methods. However, due to their respective shortcomings, the scope of their applications is restricted. Therefore, in this paper, the study in image segmentation focuses on the improvements of these two classical active contour models, aiming at wider application scopes and better segmentation results.Partial differential equation can not only play a role in the field of image segmentation, this powerful mathematical tool has also been applied to the field of biomedical engineering. Biomedical Engineering is developed by combining electronics, microelectronics, computer technology, chemistry, polymer chemistry, mechanics, modern physics, optics, X-ray technology and precision machinery, with medicine. It contains a lot of research directions, such as biological systems modeling and simulation, biomedical signal detection and analysis, biomedical imaging and image processing, biological effects of electromagnetic field, artificial organs and related medical equipment development. In this thesis, we mainly utilizes mathematical theory of the partial differential equations, and a variety of physics theory and computer technology, to study two biological systems modeling and simulation which are both closely related to tumor.Tumor is one of the most serious threats to human life. It grows exuberant, and often presents continued growth. Malignant tumor has more unrestricted outward invasion, violating the vital organs, causing exhaustion, and resulting in deaths. With the development of science and technology, there have been increased research and analysis tumor methods. At present, the tumor research focuses on (1) studies on tumor growth trend, and (2) studies on intratumoral drug delivery process and effect. These two aspects of tumor knowledge are of scientific and practical significance for tumor growth mechanism understanding, and for the diagnosis and treatment of tumor.Experimental data and theoretical studies show that, there are many microenvironment factors influencing the tumor growth, thus tumor growth is a complex process. Not only tumor cells and cells interaction, and cells and matrix interaction, but also tumor cells pressure, nutrient concentration distribution and some other factors will have direct impacts on tumor growth. Similarly, for intratumoral drug delivery process, tumor complicated microstructure and microenvironment, as well as the characteristics of the drug itself will affect drug delivery distribution and results. Therefore, it is not sufficient to fully explain the correlation between factors affecting tumor growth and tumor growth trend, and also the correlation between complex tumor microenvironment, drug characteristics and drug delivery process, only by traditional clinical and experimental methods. Consequentially, considering various factors influencing tumor invasion and intratumoral drug delivery, to construct rational tumor growth model and intratumoral drug delivery model, through integrating mathematical and physical theories with computer technology, it is important to understand the invasion procedure of cancerous tissue and help for optimizing clinical tumor therapy plan. Therefore, in this paper, the study in biomedical modeling focuses on using partial differential equations as mathematical physics method to construct two models:the tumor growth model and intratumoral drug delivery model.The main innovations in this paper include:1. Aiming at realizing hippocampus segmentation in human sagittal brain MR images, the edge information based GAC model is improved. Based on the analysis of curve evolution behavior of GAC model, prior region force and prior shape force are reasonably combined into the original GAC model, realizing effective constraint from prior knowledge on curve evolution, through which prior region force could drive the curve rapidly evolve into the segmentation target area, and prior shape force could drive the curve more accurately close to the segmentation target edge. In the simulation experiments, good hippocampus segmentations in human sagittal brain MR images are obtained through the improved model, showing the effectiveness of combining prior knowledge into the GAC model. 2. Using image region based C-V model as the original model, based on the analysis of the curve evolution of local binary fitting (LBF) model which is an improved C-V model, Gaussian mixture model (GMM) and particle swarm optimization (PSO) algorithm are incorporated to achieve local distribution fitting model which has more rapid and effective curve evolution. For both of the global information based C-V model and the local information based modified LBF model, fitting gray distribution values in the target area and background area need to be updated with the curve evolution, thus in the modified local distribution fitting model, GMM is used to pre-estimate the fitting gray distribution function in both image target area and background area, in this process, PSO algorithm is used to optimize the estimation results. The modifications lead that fitting gray distribution functions in target area and background area no longer depend on curve evolution, but can be obtained before curve evolution and keep fixed. Simulation results show that the curve evolution of our improved model is faster, and leads to more accurate results.3. As for tumor growth modeling, through comprehensive considering tumor cell adhesion energy, cell movement velocity, pressure between tumor cells, internal quality exchange, as well as nutrition concentration distribution and some other factors, convection-diffusion-reaction equations, which can describe the law of mass conservation, and convection-diffusion equation, in accordance with the fluid mechanics, are used as basic equations to construct a discrete-continuous mixture mathematical model, a mathematical description of tumor growth. The detailed numerical implementation scheme is also studied. Results of simulation experiments prove that our models for tumor growth can simulate the evolution of tumors growing outward expansion trend; and according to the nutrient concentration changes and other conditions alterations, the corresponding evolution results can be obtained; simulating tumor growth results appear the protuberant structures, which is consistent with the actual tumor growth.4. As for intratumoral drug delivery modeling, using the Navier-Stokes (N-S) equations, which describe the momentum conservation of fluid substances, and coupled with the continuity equation and convection-diffusion equation, the intratumoral drug delivery mathematical model is constructed. To highlight influence from the tumor environment characteristics on drug delivery, we regard the tumor environment as a porous tissue. Derived from the mass, momentum and the drug concentration conservation, the porous tumor environment can be expressed by a porosity parameter, which is then reasonably incorporated into the continuity equation, N-S equation and the convection-diffusion equation, in addition, the friction effect due to the porous tumor environment is also considered. To highlight the effects of drug own characteristics on drug delivery, based on the actual experimental data that showed drug will deposit in delivery procedure. Thus, the drug and fluid are considered separately, through using a relation function between fluid velocity and drug velocity to distinguish fluid and drug. Simulation and results analysis show that this model is helpful for analysis and prediction in intratumoral drug delivery distribution with influences of various factors.