Mathematical Modeling and Numerical Simulation for Application of DCE-MRI in Early Detection of Chronic Liver Disease

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姓名

簡正軒(Cheng-Hsuan Jain) 查詢紙本館藏

畢業系所

數學系

論文名稱

(Mathematical Modeling and Numerical Simulation for Application of DCE-MRI in Early Detection of Chronic Liver Disease)

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摘要(中)

肝臟疾病在台灣排名為前十大死亡原因之一。一般來說，肝臟疾病可以分為三個階段，分別為肝纖維化，肝硬化及肝癌。此研究中我們專注的臨床應用是透過DCE-MRI所測得在肝臟內不同時間的顯影劑濃度然後建立一個肝臟血流數學模型，藉此找到一個有效的參數可以去區別不同期的肝纖維化程度。我們假設肝臟裡的構造是種多孔介性界質且血液是穩態的牛頓流體，在數學方程式方面，我們使用Darcy equation 搭配 unsteady convective-diffusive equation。在離散化方面，對於空間上的離散使用stabilized finite element method，而時間上的離散則使用 implicit backward Euler finite difference method。最後我們發現 porosity 是個有效的指標來區別不同的期的肝纖維化程度。

摘要(英)

Liver diseases are always on the list of the top 10 causes of death in Taiwan. Generally speaking, the progression of liver disease can be classified into three stages, including liver fibrosis, liver cirrhosis, and liver cancer. Recently, using the noninvasive Dynamic Contrast Enhanced MRI (DCE-MRI) technique for the early detection of chronic liver disease is quite promising. The research focus for clinical application is to define some index related to the relative signal enhancement in a liver to identify the degree of liver fibrosis. In reach the goal, we build both of the mathematical models for blood flows through the liver and the relative signal enhancement scanned by MRI varied with respect to time. Under assumptions that liver is a kind of porous medium, and the blood flow is Newtonian, Laminar, in steady state, the governing equations consist of the Darcy equation weakly coupled with unsteady convective-diffusive equation. The stabilized finite element Darcy solver together with time-dependent convective-diffusive solver are verified by a case with analytical solution and the mathematical models are validated by the experiment of fluid flow through the sponge. In addition, our numerical result is consistent with the clinical data. Finally, we find the porosity is potentially to be a good index to identify the degree of liver fibrosis.

關鍵字(中)

★ 肝臟纖維化
★ 有限元素法

關鍵字(英)

論文目次

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Darcy equation weakly coupled with unsteady convective-diffusive equation 5
2.1.1 Darcy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Convection diffusion equation . . . . . . . . . . . . . . . . . . . 6
2.2 The classical weak formulation . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The stabilized weak formulations . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Discretization of Darcy’s Law by using FEM . . . . . . . . . . . 9
2.4.2 Discretization of convection diffusion equation by using FEM . . 10
3 Code verification and model validation . . . . . . . . . . . . . . . . . . . . . 12
3.1 Matlab code verification . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 Darcy’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2 Convection diffusion equation . . . . . . . . . . . . . . . . . . . 15
3.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Clinical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Clinical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
viii

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指導教授

黃楓南(Feng-Nan Hwang)

審核日期

2015-8-18

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