一类反问题—生物体发光断层成像—的理论分析和数值模拟
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摘要
近年来,随着科学技术的发展,生物医学成像得到了长足的进步。分子成像是在细胞和分子水平上研究活体生理和病理过程的一个快速发展着的生物医学成像方法。它的目标是非入侵地准确描述活体细胞和分子过程。分子成像主要基于三种技术:核素成像技术,核磁共振成像技术和光学成像技术。与传统的成像方法,如计算机层析成像(CT)、正电子发射断层成像(PET)、核磁共振成像(MRI)以及它们的合成等,相比,作为光学成像模型,生物放光成像(BLI)有它独特的优点。对于活体人体研究,由于组织的吸收与发散特性,光学成像主要局限于表层位置,此时MRI或PET是比较合适的选择。然而,对于小动物来说,由于较短的路径长度,一大部分光子可以到达动物的外表,从而能以比MRI和PET低得多的成本探测到生物光和荧光信号。对于活体应用来说,用生物光源进行成像特别具有吸引力,因为它并不需要外部的刺激光源,从而背景噪音低而灵敏性高。
生物发光断层成像(BLT)是一个非常有前途的BLI,因为他能实时地揭示生物体的分子和细胞活动。几年来,人们就BLT问题做了大量的理论分析和数值仿真。BLT的主要任务就是利用边界测量的光信号来决定小动物体内或一些大器官的表层的光子密度分布。利用它,我们能在三维空间中对小动物内部的光源进行定位并定量计算出生物光源的强度密度分布。进行BLT的第一步是确定组织的光学特性,这正是光学断层摄影(DOT)问题的的主要任务。因为光在任何实体中的传送都受到吸收和散射的影响,所以我们需要知道光子在生物组织中传送的精确表达式。一般来说,生物光光子在组织中的传播过程能被漫射转移方程(RTE)或蒙特卡罗模型(MCM)准确描述。然而,就当前而言,两者在计算上都是不可行的。因此,当光的波长在600 nm附近或更大时,人们通常用扩散近似方程来代替RTE。从数学上来讲,BLT是一个反问题,通常需要对它进行正则化处理。在这篇论文中,我们讨论了BLT和多谱BLT的理论分析和数值求解等问题.全文共分五章。
在第一章里,我们首先简单介绍一下BLT问题的生物医学背景。然后,在1.2节中,我们给出了BLT问题的数学形式。在1.3节,我们给出了一些主要符号,它们将会在接下来的章节里被反复地用到.
第二章主要是关于经典BLT的理论和数值方法的研究。2.1节陈述了经典BLT的形式。然后,在2.2节里,在一些假定下,利用共轭方法,我们将一个正则化的最小化问题转化为一个偏微分方程组。这为快速重构光源函数提供了可能。在第2.3节中,利用正则化方法,我们研究了BLT问题的一个更为广义框架。在这个框架下,我们建立了问题的适定性结果,考察了它与以前别人文章之间的关系。在第2.4节中,我们发展了BLT问题的一个新的变分形式。利用该变分形式,我们可以从理论上证明光源函数并不连续依赖于测量数据。在这一节里,我们还给出了正则化有限元解最优收敛阶的证明。另外,利用伴随方程,我们构造了一个简单而有效的迭代格式。
第三章研究的是多谱BLT问题。首先,在3.2节,基于一个惩罚策略,我们提出了关于多谱BLT的全新的方法。这个新的数学框架的特点是其数值测量的使用涉及到两个相关但并不相同的边值问题(BVPs)。该数学框架包含了通常的多谱BLT模型。然后,在第3.2节,我们改进[1]中的误差阶。接着,在第3.3.2小节里,我们用线性而不是通常的常数有限元来离散光源函数空间.利用有效集策略,我们获得更高阶的有限元误差估计.
第四章我们继续深入讨论BLT问题。首先,在4.2节,我们研究了具有随空间改变折射率的BLT问题。为此,利用Tikhonov正则化方法,我们介绍一个广义的框架,证明了它的适定性,并利用有限元方法(FEMs),建立了数值解的误差估计式。在第4.3节,我们提出了一个数学模型,它将BLT和DOT结合在一起,也就是说,同时进行两种类型的重构而不是一前一后。因为在实际中,光学参数通常被假定是常数或分片光滑函数,因此我们设定光学参数的求解范围是有界变差(BV)函数空间。我们证明了解的存在性和数值解的收敛性并给出了一个数值求解格式。
最后,在第五章,我们作了总结并展望未来进一步要做的工作。
With the development of science and technology, these years have witnessed the rapid progress in biomedical imaging. Molecular imaging is a rapidly developing biomedical imaging technique for studying physiological and pathological processes in vivo at the cellular and molecular levels. The goal of molecular imaging is to depict non-invasively cellular and molecular processes in vivo sensitively. Molecular imaging is broadly based on three technologies: nuclear imaging、magnetic resonance imaging and optical imaging. Bioluminescent imaging (BLI), as one of the optical imaging modalities, has its own advantages over traditional imaging methods such as computed tomography (CT), positron emission tomography (PET), magnetic resonance imaging (MRI) as well as their combinations. For in vivo studies in humans, optical imaging is largely limited to superficial sites owing to the absorption and scattering properties of tissue, and MRI or PET are preferred modalities. However, in small animals, due to shorter path lenghts, a large fraction of photons reaches the surface of the animal and detection of bioluminescent and fluorescent signals is possible at significantly lower cost compared to MRI and PET. Imaging of bioluminescent sources is particularly attractive for in vivo applications because no external excitation source is needed, and, in turn, background noise is low and sensitivity is high.
Bioluminescence tomography (BLT) is a promising BLI because of the possibility of revealing molecular and cellular activities in real time. Over the past several years, numerous work has been devoted to theoretical analysis and numerical simulations of BLT. The main object of BLT is to determine the photon density distribution within small animals or on the superficial of some big organs from the light measurement on the boundary. With BLT, a bioluminescent source distribution inside a living small animal can be localized and quantified in 3D. The first step of BLT is to determine the optical properties of tissue and this is the issue of a diffuse optical tomography (DOT) problem. Because the transport of light in any entity is subject to both absorption and scattering, accurate representation of the photon transport in biological tissue is required. In general, the bioluminescent photon propagation in a tissue can be described accurately by either the radiative transfer equation (RTE) or the Monte Carlo model (MCM). However, at the moment, neither of them is computationally feasible. Usually a diffusion approximation equation of the RTE is employed when the wavelength of light is in the range of around or bigger than 600 nm. Mathematically, BLT is an ill-posed inverse source problem, usually studied through a regularization technique. In this thesis, we discuss theoretical analysis and numerical solution of BLT and its multispectral version. This thesis includes five chapters.
In Chapter 1, we at first give a brief introduction of biomedical background of BLT. Then in Section 1.2, its mathematical formulation is presented. At last, in Section 1.3, we give some principle symbols which will be used repeatedly in the following chapters.
Chapter 2 is devoted to some theoretical study and numerical experiments for the classic BLT. Section 2.1 gives a presentation of the classic BLT. Then in Section 2.2, under some assumptions, we convert a regularized minimal problem into a couple of partial differential equations (PDEs) by using an adjoint strategy. This makes it possible for reconstructing light source function fast. In Section 2.3, we study the BLT problem through a general framework together with a Tikhonov regularization. For the proposed formulation, we establish a well-posedness result and explore its relation to the formulation studied previously in other papers. In Section 2.4, a new variational formulation for the BLT problem is developed. It is used to explain rigorously the reason behind the loss of the continuous dependence of the light source function solution on the measurements. We also prove an optimal error order in finite element solution. Additionally, by using adjoint equations, a simple but efficient iterative scheme is also explored.
In the third chapter, we focus on multispectral BLT. In Section 3.2, based on a penalization strategy, we propose a novel approach for the multispectral BLT. The new feature of the mathematical framework is to use numerical prediction results based on two related but distinct boundary value problems. This mathematical framework includes the conventional framework in the study of multispectral BLT. Then in Section 3.3, we first improve finite element error order compared to [1]. In Subsection 3.3.2, a linear finite element is used for the discretization of the space of light source. Using active set strategy, a higher finite element error order is obtained.
We continue to discuss BLT extensively in Chapter 4. We study the BLT problem for media with spatially varying refractive index in Section 4.2. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish error bounds for its numerical solutions by the finite element methods (FEMs). Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index. Then in Section 4.3, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. We seek the optical parameters in the space of functions with bounded variation. We show the solution existence, prove convergence of the numerical solution and introduce numerical schemes.
At last, some conclusions are given in Chapter 5. We also list some problems which may deserve future effort.
生物发光断层成像(BLT)是一个非常有前途的BLI,因为他能实时地揭示生物体的分子和细胞活动。几年来,人们就BLT问题做了大量的理论分析和数值仿真。BLT的主要任务就是利用边界测量的光信号来决定小动物体内或一些大器官的表层的光子密度分布。利用它,我们能在三维空间中对小动物内部的光源进行定位并定量计算出生物光源的强度密度分布。进行BLT的第一步是确定组织的光学特性,这正是光学断层摄影(DOT)问题的的主要任务。因为光在任何实体中的传送都受到吸收和散射的影响,所以我们需要知道光子在生物组织中传送的精确表达式。一般来说,生物光光子在组织中的传播过程能被漫射转移方程(RTE)或蒙特卡罗模型(MCM)准确描述。然而,就当前而言,两者在计算上都是不可行的。因此,当光的波长在600 nm附近或更大时,人们通常用扩散近似方程来代替RTE。从数学上来讲,BLT是一个反问题,通常需要对它进行正则化处理。在这篇论文中,我们讨论了BLT和多谱BLT的理论分析和数值求解等问题.全文共分五章。
在第一章里,我们首先简单介绍一下BLT问题的生物医学背景。然后,在1.2节中,我们给出了BLT问题的数学形式。在1.3节,我们给出了一些主要符号,它们将会在接下来的章节里被反复地用到.
第二章主要是关于经典BLT的理论和数值方法的研究。2.1节陈述了经典BLT的形式。然后,在2.2节里,在一些假定下,利用共轭方法,我们将一个正则化的最小化问题转化为一个偏微分方程组。这为快速重构光源函数提供了可能。在第2.3节中,利用正则化方法,我们研究了BLT问题的一个更为广义框架。在这个框架下,我们建立了问题的适定性结果,考察了它与以前别人文章之间的关系。在第2.4节中,我们发展了BLT问题的一个新的变分形式。利用该变分形式,我们可以从理论上证明光源函数并不连续依赖于测量数据。在这一节里,我们还给出了正则化有限元解最优收敛阶的证明。另外,利用伴随方程,我们构造了一个简单而有效的迭代格式。
第三章研究的是多谱BLT问题。首先,在3.2节,基于一个惩罚策略,我们提出了关于多谱BLT的全新的方法。这个新的数学框架的特点是其数值测量的使用涉及到两个相关但并不相同的边值问题(BVPs)。该数学框架包含了通常的多谱BLT模型。然后,在第3.2节,我们改进[1]中的误差阶。接着,在第3.3.2小节里,我们用线性而不是通常的常数有限元来离散光源函数空间.利用有效集策略,我们获得更高阶的有限元误差估计.
第四章我们继续深入讨论BLT问题。首先,在4.2节,我们研究了具有随空间改变折射率的BLT问题。为此,利用Tikhonov正则化方法,我们介绍一个广义的框架,证明了它的适定性,并利用有限元方法(FEMs),建立了数值解的误差估计式。在第4.3节,我们提出了一个数学模型,它将BLT和DOT结合在一起,也就是说,同时进行两种类型的重构而不是一前一后。因为在实际中,光学参数通常被假定是常数或分片光滑函数,因此我们设定光学参数的求解范围是有界变差(BV)函数空间。我们证明了解的存在性和数值解的收敛性并给出了一个数值求解格式。
最后,在第五章,我们作了总结并展望未来进一步要做的工作。
With the development of science and technology, these years have witnessed the rapid progress in biomedical imaging. Molecular imaging is a rapidly developing biomedical imaging technique for studying physiological and pathological processes in vivo at the cellular and molecular levels. The goal of molecular imaging is to depict non-invasively cellular and molecular processes in vivo sensitively. Molecular imaging is broadly based on three technologies: nuclear imaging、magnetic resonance imaging and optical imaging. Bioluminescent imaging (BLI), as one of the optical imaging modalities, has its own advantages over traditional imaging methods such as computed tomography (CT), positron emission tomography (PET), magnetic resonance imaging (MRI) as well as their combinations. For in vivo studies in humans, optical imaging is largely limited to superficial sites owing to the absorption and scattering properties of tissue, and MRI or PET are preferred modalities. However, in small animals, due to shorter path lenghts, a large fraction of photons reaches the surface of the animal and detection of bioluminescent and fluorescent signals is possible at significantly lower cost compared to MRI and PET. Imaging of bioluminescent sources is particularly attractive for in vivo applications because no external excitation source is needed, and, in turn, background noise is low and sensitivity is high.
Bioluminescence tomography (BLT) is a promising BLI because of the possibility of revealing molecular and cellular activities in real time. Over the past several years, numerous work has been devoted to theoretical analysis and numerical simulations of BLT. The main object of BLT is to determine the photon density distribution within small animals or on the superficial of some big organs from the light measurement on the boundary. With BLT, a bioluminescent source distribution inside a living small animal can be localized and quantified in 3D. The first step of BLT is to determine the optical properties of tissue and this is the issue of a diffuse optical tomography (DOT) problem. Because the transport of light in any entity is subject to both absorption and scattering, accurate representation of the photon transport in biological tissue is required. In general, the bioluminescent photon propagation in a tissue can be described accurately by either the radiative transfer equation (RTE) or the Monte Carlo model (MCM). However, at the moment, neither of them is computationally feasible. Usually a diffusion approximation equation of the RTE is employed when the wavelength of light is in the range of around or bigger than 600 nm. Mathematically, BLT is an ill-posed inverse source problem, usually studied through a regularization technique. In this thesis, we discuss theoretical analysis and numerical solution of BLT and its multispectral version. This thesis includes five chapters.
In Chapter 1, we at first give a brief introduction of biomedical background of BLT. Then in Section 1.2, its mathematical formulation is presented. At last, in Section 1.3, we give some principle symbols which will be used repeatedly in the following chapters.
Chapter 2 is devoted to some theoretical study and numerical experiments for the classic BLT. Section 2.1 gives a presentation of the classic BLT. Then in Section 2.2, under some assumptions, we convert a regularized minimal problem into a couple of partial differential equations (PDEs) by using an adjoint strategy. This makes it possible for reconstructing light source function fast. In Section 2.3, we study the BLT problem through a general framework together with a Tikhonov regularization. For the proposed formulation, we establish a well-posedness result and explore its relation to the formulation studied previously in other papers. In Section 2.4, a new variational formulation for the BLT problem is developed. It is used to explain rigorously the reason behind the loss of the continuous dependence of the light source function solution on the measurements. We also prove an optimal error order in finite element solution. Additionally, by using adjoint equations, a simple but efficient iterative scheme is also explored.
In the third chapter, we focus on multispectral BLT. In Section 3.2, based on a penalization strategy, we propose a novel approach for the multispectral BLT. The new feature of the mathematical framework is to use numerical prediction results based on two related but distinct boundary value problems. This mathematical framework includes the conventional framework in the study of multispectral BLT. Then in Section 3.3, we first improve finite element error order compared to [1]. In Subsection 3.3.2, a linear finite element is used for the discretization of the space of light source. Using active set strategy, a higher finite element error order is obtained.
We continue to discuss BLT extensively in Chapter 4. We study the BLT problem for media with spatially varying refractive index in Section 4.2. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish error bounds for its numerical solutions by the finite element methods (FEMs). Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index. Then in Section 4.3, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. We seek the optical parameters in the space of functions with bounded variation. We show the solution existence, prove convergence of the numerical solution and introduce numerical schemes.
At last, some conclusions are given in Chapter 5. We also list some problems which may deserve future effort.
引文
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