數值分析與其在光聲影像之應用

Numerical analysis and its applications in photoacoustic imaging

許曄琳(Yae-Lin Sheu)

指導教授 ： 李百祺

國立臺灣大學/電機資訊學院/電機工程學研究所/博士(2010年)

https://doi.org/10.6342/NTU.2010.02382

摘要

本論文主體有二：其一研討光聲波在液體中之生成與傳播方程的數值解；其二乃探究傳統有限可視角影像重建方法在血管內光聲影像中之應用。
醫用光聲相關技術所採之簡易物理模型無法應用於任意光能照射時間與非線性且耗散之傳播。本論文第一部分由熱傳導、狀態、連續、推導與Navier-Stokes方程歸納出一般液態物經受熱轉形聲波的機制。此系統方程對任意初始光能分佈之解析解不易求得，故以數值偏微分之技巧，諸如時域有限差分和時域擬譜方式來計算光聲波生成與傳播。模擬架構為一柱對稱座標系，斷面之邊界則以理想匹配層(PMLs)吸收散射之光聲波。細微結構須隨不同差分技巧而調整。兩方法所得之數值結果符合系統方程之線性解析解。結果顯示擬譜方法之數值解的誤差優於有限差分方法之解。其中擬譜方法中只用了四個取樣格點，而有限差分則須至少15取樣格點方可免於數值頻散現象。
傳統光聲影像取得架構有三，為環型、柱型與平面掃描方式。今血管內光聲造影為一新興技術，其中所得掃得之光聲訊號唯以座標轉換重建，致使所建之影像與真實大有出入。論文第二部分將有限可視角影像重建方法中之濾波反投影(FBP)法、局部重建法與期望最大化(EM)演算法應用於血管內光聲影像架構。模擬與研究結果顯示濾波反投影與局部重建之法雖非此影像反問題之解析解，但可揭示部分物體形態；而期望最大化演算法藉由迭代計算物體影像所投射之光聲訊號與真實訊號之間的最大相似度來改善重建影像品質。

關鍵字

光聲影像 ； 有限差分 ； 擬譜方法 ； 濾波反投影 ； 局部重建法 ； 局部重建法與期望最大化

並列摘要

Several photoacoustic (PA) techniques, such as photoacoustic imaging, spectroscopy, and parameter sensing, measure quantities that are closely related to optical absorption, position detection, and laser irradiation parameters. The photoacoustic waves in biomedical applications are usually generated by elastic thermal expansion, which has advantages of nondestructiveness and relatively high conversion efficiency from optical to acoustic energy. Most investigations describe this process using a heuristic approximation, which is invalid when the underlying assumptions are not met. This thesis developed a numerical solution of the general photoacoustic generation equations involving the heat conduction theory and the state, continuity, and Navier-Stokes equations in 2.5D axisymmetric cylindrical coordinates using a finite-difference time-domain (FDTD) and a pseudospectral time-domain (PSTD) scheme.
Simulation using the FDTD method included staggered grids and Berenger’s perfectly matched layers (PMLs). The spatial derivatives in the system of equations were approximated by the second-order-accurate central difference, and the time evolution was advanced by the method of lines. The numerical results were validated using the linear-perturbation analytical solutions and a heuristic model for generation and propagation of photoacoustic waves. In addition, as a free-space boundary, the 10-layer PML produced the lowest reflection error for commonly used boundary conditions.
Computation efficiency and accuracy can be further improved by the PSTD method. Derivatives in spatial domain were calculated using the differentiation theorem for Fourier transforms. Infinite order of accuracy is achieved with at least two sampling grids per minimum wavelength, as compared with the FDTD method, in which 15 or more sampling grids are necessary to reduce numerical dispersion. We developed an extended computation domain to avoid Gibbs phenomenon and wraparound effect associated with the periodicity inherent in the FFT operation by applying the concepts of PMLs and symmetry properties of boundary conditions in our simulation model. The accuracy of using four sampling grids was shown superior to those using 15 grids in FDTD method in the photoacoustic wave simulation. Furthermore, a nonlinear propagation of photoacoustic wave was demonstrated.
In the second part of the thesis, we explored feasibility and efficacy of applying limited-view reconstruction methods to intravascular photoacoustic (IVPA) imaging as a new type of scanning geometry in photoacoustic tomography. A very limited view angle can attribute to an enclosed scanning trajectory within the phantom. According to the data sufficiency condition, only a small portion of object interfaces can be reconstructed stably in such imaging configuration. To the present, intravascular images are obtained using only scan conversion, resulting in image distortion if a point-like detector is used. Analytical formulas such as the filtered backprojection (FBP) and the lambda-tomography method were employed to suggest object geometries. The reconstruction quality was further improved by the expectation maximization (EM) method which could minimize the error between the measured signals and those generated by a reconstructed image. Computer simulations and experiments were carried out to validate the methods.

並列關鍵字

photoacoustic imaging ； finite difference and time-domain method ； pseudospectral and time-domain method ； filtered backprojection method ； local tomography method ； expectation maximization method

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