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Table Of Integrals Series And Products

Photoacoustic tomography (PAT) can create multiscale multicontrast images of living biological structures ranging from organelles to organs. This emerging technology overcomes the high degree of scattering of optical photons in biological tissue by making use of the photoacoustic effect. Light absorption by molecules creates a thermally induced pressure jump that launches ultrasonic waves, which are received by acoustic detectors to form images. Different implementations of PAT allow the spatial resolution to be scaled with the desired imaging depth in tissue while a high depth-to-resolution ratio is maintained. As a rule of thumb, the achievable spatial resolution is on the order of 1/200 of the desired imaging depth, which can reach up to 7 centimeters. PAT provides anatomical, functional, metabolic, molecular, and genetic contrasts of vasculature, hemodynamics, oxygen metabolism, biomarkers, and gene expression. We review the state of the art of PAT for both biological and clinical studies and discuss future prospects.

Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs

Photoacoustic imaging (also called optoacoustic or thermoacoustic imaging) has the potential to image animal or human organs, such as the breast and the brain, with simultaneous high contrast and high spatial resolution. This article provides an overview of the rapidly expanding field of photoacoustic imaging for biomedical applications. Imaging techniques, including depth profiling in layered media, scanning tomography with focused ultrasonic transducers, image forming with an acoustic lens, and computed tomography with unfocused transducers, are introduced. Special emphasis is placed on computed tomography, including reconstruction algorithms, spatial resolution, and related recent experiments. Promising biomedical applications are discussed throughout the text, including (1) tomographic imaging of the skin and other superficial organs by laser-induced photoacoustic microscopy, which offers the critical advantages, over current high-resolution optical imaging modalities, of deeper imaging depth and higher absorptioncontrasts, (2) breast cancerdetection by near-infrared light or radio-frequency–wave-induced photoacoustic imaging, which has important potential for early detection, and (3) small animal imaging by laser-induced photoacoustic imaging, which measures unique optical absorptioncontrasts related to important biochemical information and provides better resolution in deep tissues than optical imaging.

/pdf/photoacoustic-imaging-in-biomedicine-5c6h0us7a9.pdf

Photoacoustic imaging in biomedicine

Photoacoustic (PA) imaging, also called optoacoustic imaging, is a new biomedical imaging modality based on the use of laser-generated ultrasound that has emerged over the last decade. It is a hybrid modality, combining the high-contrast and spectroscopic-based specificity of optical imaging with the high spatial resolution of ultrasound imaging. In essence, a PA image can be regarded as an ultrasound image in which the contrast depends not on the mechanical and elastic properties of the tissue, but its optical properties, specifically optical absorption. As a consequence, it offers greater specificity than conventional ultrasound imaging with the ability to detect haemoglobin, lipids, water and other light-absorbing chomophores, but with greater penetration depth than purely optical imaging modalities that rely on ballistic photons. As well as visualizing anatomical structures such as the microvasculature, it can also provide functional information in the form of blood oxygenation, blood flow and temperature. All of this can be achieved over a wide range of length scales from micrometres to centimetres with scalable spatial resolution. These attributes lend PA imaging to a wide variety of applications in clinical medicine, preclinical research and basic biology for studying cancer, cardiovascular disease, abnormalities of the microcirculation and other conditions. With the emergence of a variety of truly compelling in vivo images obtained by a number of groups around the world in the last 2–3 years, the technique has come of age and the promise of PA imaging is now beginning to be realized. Recent highlights include the demonstration of whole-body small-animal imaging, the first demonstrations of molecular imaging, the introduction of new microscopy modes and the first steps towards clinical breast imaging being taken as well as a myriad of in vivo preclinical imaging studies. In this article, the underlying physical principles of the technique, its practical implementation, and a range of clinical and preclinical applications are reviewed.

/pdf/biomedical-photoacoustic-imaging-4s982wv96j.pdf

Biomedical photoacoustic imaging

Photoacoustic tomography (PAT) is probably the fastest-growing area of biomedical imaging technology, owing to its capacity for high-resolution sensing of rich optical contrast in vivo at depths beyond the optical transport mean free path (~1 mm in human skin). Existing high-resolution optical imaging technologies, such as confocal microscopy and two-photon microscopy, have had a fundamental impact on biomedicine but cannot reach the penetration depths of PAT. By utilizing low ultrasonic scattering, PAT indirectly improves tissue transparency up to 1000-fold and consequently enables deeply penetrating functional and molecular imaging at high spatial resolution. Furthermore, PAT promises in vivo imaging at multiple length-scales; it can image subcellular organelles to organs with the same contrast origin — an important application in multiscale systems biology research.

/pdf/multiscale-photoacoustic-microscopy-and-computed-tomography-4r9zgikqyo.pdf

Multiscale photoacoustic microscopy and computed tomography.

The limited-view problem is studied for thermoacoustic tomography, which is also referred to as photoacoustic or optoacoustic tomography depending on the type of radiation for the induction of acoustic waves. We define a “detection region,” within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any objects inside this region can be recovered stably. Otherwise some sharp details become blurred. One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. If the detector scans along a circle in a two-dimensional case, acquiring a sufficient view might require covering more than a π-, or less than a π-arc of the trajectory depending on the position of the object. Similar results hold in a three-dimensional case. In order to support our theoretical conclusions, three types of reconstruction methods are utilized: a filtered backprojection (FBP) approximate inversion, which is shown to work well for limited-view data, a local-tomography-type reconstruction that emphasizes sharp details (e.g., the boundaries of inclusions), and an iterative algebraic truncated conjugate gradient algorithm used in conjunction with FBP. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions.

/pdf/reconstructions-in-limited-view-thermoacoustic-tomography-1wx17z6vao.pdf

Reconstructions in limited-view thermoacoustic tomography

The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such a circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As is common for transforms of Radon type, its range has infinite codimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography—for instance, when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. Range conditions include the recently found set of moment‐type conditions, which happens to be incomplete, as well as other conditions that have less standard form. In order to explain the procedure better, a similar (nonstandard) treatment of the range conditions is described first for the usual Radon transform on the plane.

A range description for the planar circular radon transform

The circular Radon transform integrates a function over the set of all spheres with a given set of centres. The problem of injectivity of this transform (as well as inversion formulae, range descriptions, etc) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs and recently to newly developing types of tomography. A major breakthrough in the 2D case was made several years ago in a work by Agranovsky and Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, especially those based on simple PDE techniques, which would be less restrictive in more general situations. This paper provides some new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.

https://iopscience.iop.org/article/10.1088/0266-5611/21/2/004/pdf

On the injectivity of the circular Radon transform

The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As it is common for transforms of Radon type, its range has infinite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. Range conditions include the recently found set of moment type conditions, which happens to be incomplete, as well as the rest of conditions that have less standard form. In order to explain the procedure better, a similar (non-standard) treatment of the range conditions is described first for the usual Radon transform on the plane.

A range description for the planar circular Radon transform

Several novel imaging modalities proposed during the last couple of years are based on a mathematical model, which uses the V-line Radon transform (VRT). This transform, sometimes called broken-ray Radon transform, integrates a function along V-shaped piecewise linear trajectories composed of two intervals in the plane with a common endpoint. Image reconstruction problems in these modalities require inversion of the VRT. While there are ample results about inversion of the regular Radon transform integrating along straight lines, very little is known for the case of the V-line Radon transform. In this paper, we derive an exact inversion formula for the VRT of functions supported in a disc of arbitrary radius. The formula uses a two-dimensional restriction of VRT data, namely the incident ray is normal to the boundary of the disc, and the breaking angle is fixed. Our method is based on the classical filtered back-projection inversion formula of the Radon transform, and has similar features in terms of stability, speed, and accuracy.

Gaik Ambartsoumian

Papers

Reconstructions in limited-view thermoacoustic tomography

A range description for the planar circular radon transform

On the injectivity of the circular Radon transform

A range description for the planar circular Radon transform

Inversion of the V-line Radon transform in a disc and its applications in imaging