Explore the vast range of titles available.

To quantify fluorescence imaging of biological tissues, we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissues. We begin by giving a concise mathematical argument to derive coupled diffusion equations with the Robin boundary condition as an approximation of the radiative transfer system. Then by using this coupled system of equations as a model for the fluorescence imaging, we have a nonlinear inverse problem to identify the absorption coefficient in this system. The associated linearized inverse problem is to ignore the absorbing effect on the excitation field. We firstly establish the estimates of errors on the excitation field and the solution to the inverse problem, which ensures the reasonability of the model approximation quantitatively. Some numerical verification is presented to show the validity of such a linearizing process quantitatively. Then, based on the analytic expressions of excitation and emission fields, the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for the absorption coefficient in the special form, revealing the physical difficulty of the 3-dimensional imaging model by the back scattering diffusive system.

Neonatal hypoxic-ischemic encephalopathy (HIE) is a significant cause of neonatal mortality and developmental disabilities. It has been revealed that the temporal behavior of the cerebral blood volume (CBV) carries information on the degree of hypoxia-ischemia. CBV can be estimated by means of near-infrared spectroscopy. The change of CBV after the insult is related to the change of CBV during the insult. In this paper, we consider a mathematical model which governs the time evolution of CBV after the insult. We show that the temporal behavior of CBV can be predicted with the Kalman filter which is based on the mathematical model. Finding a governing equation opens up possibilities for a more quantitative diagnosis of HIE.

Mass spectrometry imaging (MSI) allows us to visualize the spatial distribution of molecular components in a sample. A large amount of mass spectrometry data comprehensively provides molecular distributions. In this study, we focus on the information in the obtained data and use the Shannon entropy as a quantity to analyze MSI data. By calculating the Shannon entropy at each pixel on a sample, the spatial distribution of the Shannon entropy is obtained from MSI data. We found that low-entropy pixels in entropy heat maps for kidneys of mice had different structures between two ages (3 months and 31 months). Such changes cannot be visualized by conventional imaging techniques. We further propose a method to find informative molecules. As a demonstration of the proposed scheme, we identified two molecules by setting a region of interest which contained low-entropy pixels and by exploring changes of peaks in the region.

The use of the linear Boltzmann equation is proposed for transport in porous media in a column. By column experiments, we show that the breakthrough curve is reproduced by the linear Boltzmann equation. The advection–diffusion equation is derived from the linear Boltzmann equation in the asymptotic limit of large propagation distance and long time.

Near-infrared spectroscopy (NIRS) including diffuse optical tomography is an imaging modality which makes use of diffuse light propagation in random media. When optical properties of a random medium are investigated from boundary measurements of reflected or transmitted light, iterative inversion schemes such as the Levenberg–Marquardt algorithm are known to fail when initial guesses are not close enough to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome using Markov chain Monte Carlo. Using time-resolved measurements performed against a polyurethane-based phantom, we present a case that the Levenberg–Marquardt algorithm fails to work but the proposed hybrid method works well. Then, with a toy model of diffuse optical tomography we illustrate that the Levenberg–Marquardt method fails when it is trapped by a local minimum but the hybrid method can escape from local minima by using the Metropolis–Hastings Markov chain Monte Carlo algorithm until it reaches the valley of the global minimum. The proposed hybrid scheme can be applied to different inverse problems in NIRS which are solved iteratively. We find that for both numerical and phantom experiments, optical properties such as the absorption and reduced scattering coefficients can be retrieved without being trapped by a local minimum when Monte Carlo simulation is run only about 100 steps before switching to an iterative method. The hybrid method is compared with simulated annealing. Although the Metropolis–Hastings MCMC arrives at the steady state at about 10,000 Monte Carlo steps, in the hybrid method the Monte Carlo simulation can be stopped way before the burn-in time.

The diffusion approximation has been one of the central topics in near-infrared spectroscopy (NIRS). When NIRS measurements are analyzed by the diffusion theory, the measurements must be performed in the diffusive regime. However, since most of past researches have focused on theoretical or qualitative nature of the diffusion approximation, it is not easy to know if each measurement is designed in the diffusive regime. In this paper, we consider the diffusion approximation quantitatively and propose indicators that quantify the degree of validness of the diffusion approximation. The difference between the measurement and diffusion theory can be evaluated with the χ 2 value, ℓ 1 and ℓ 2 norms, and Kullback-Leibler divergence. We conduct a liquid phantom experiment to test the proposed χ 2 value. Moreover, the χ 2 value is further investigated by Monte Carlo simulations. We find the χ 2 value becomes significantly large when measurements are performed in the nondiffusive or transport regime. The proposed indicators similarly work. In particular, the χ 2 value is shown to work as an indicator which evaluates the degree of validness of the diffusion approximation. These indicators are general and can be used for different numerical, experimental, and clinical measurements in NIRS.

Albeit the past intensive research, the governing equation of anomalous diffusion which is observed for the transport of particles underground is still an open problem. In this paper, as a governing equation, the advection-diffusion equation with a time-fractional derivative term is derived from the radiative transport equation with an integral term for waiting time.

Mass spectrometry imaging (MSI) allows us to visualize the spatial distribution of molecular components in a sample. A large amount of mass spectrometry data comprehensively provides molecular distributions. In this study, we focus on the information in the obtained data and use the Shannon entropy as a quantity to analyze MSI data. By calculating the Shannon entropy at each pixel on a sample, the spatial distribution of the Shannon entropy is obtained from MSI data. We found that low-entropy pixels in entropy heat maps for kidneys of mice had different structures between two ages (3 months and 31 months). Such changes cannot be visualized by conventional imaging techniques. We further propose a method to find informative molecules. As a demonstration of the proposed scheme, we identified two molecules by setting a region of interest which contained low-entropy pixels and by exploring changes of peaks in the region.

The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly observed. Examples include medical imaging techniques such as X-ray CT and optical tomography. Indeed, the mathematics of inverse problems has often originated from challenges posed by other fields. Inverse problems are often ill-posed and solutions are unstable. In this lecture, we will explore methods to solve such inverse problems.

Rotated reference frames offer fast algorithms for the radiative transport equation (RTE). We review the singular-eigenfunction approach and related numerical methods for the multi-dimensional RTE with rotated reference frames.