Abstract The Rytov approximation is known in near-infrared spectroscopy including diffuse optical tomography. In diffuse optical tomography, the Rytov approximation often gives better reconstructed images than the Born approximation. Although related inverse problems are nonlinear, the Rytov approximation is almost always accompanied by the linearization of nonlinear inverse problems. In this paper, we will develop nonlinear reconstruction with the inverse Rytov series. By this, linearization is not necessary and higher order terms in the Rytov series can be used for reconstruction. The convergence and stability are discussed. We find that the inverse Rytov series has a recursive structure similar to the inverse Born series.

Light propagation through diffusive media can be described by the diffusion equation in a space–time domain. Furthermore, fluorescence can be described by a system of coupled diffusion equations. This paper analyzes time-domain measurements. In particular, the temporal point-spread function is measured at the boundary of a diffusive medium. Moreover, the temporal profile of fluorescence is considered. In both cases, we refer to the maximum temporal position of measured light as the peak time. In this paper, we provide proofs of the existence and uniqueness of the peak time and give explicit expressions of the peak time. The relationship between the peak time and the object position in a medium is clarified.

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.

Diffuse optical tomography (DOT) is an imaging modality that uses near-infrared light. Although iterative numerical schemes are commonly used for its inverse problem, correct solutions are not obtained unless good initial guesses are chosen. We propose a numerical scheme of DOT, which works even when good initial guesses of optical parameters are not available. We use simulated annealing (SA), which is a method of the Markov-chain Monte Carlo. To implement SA for DOT, a spin Hamiltonian is introduced in the cost function, and the Metropolis algorithm or single-component Metropolis–Hastings algorithm is used. By numerical experiments, it is shown that an initial random spin configuration is brought to a converged configuration by SA, and targets in the medium are reconstructed. The proposed numerical method solves the inverse problem for DOT by finding the ground state of a spin Hamiltonian with SA.

Abstract 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 χ² value, ℓ¹ and ℓ² norms, and Kullback-Leibler divergence. We conduct a liquid phantom experiment to test the proposed χ² value. Moreover, the χ² value is further investigated by Monte Carlo simulations. We find the χ² value becomes significantly large when measurements are performed in the nondiffusive or transport regime. The proposed indicators similarly work. In particular, the χ² 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.

The decay behavior of specific intensity is studied for spatial-frequency domain imaging (SFDI). It is shown using the radiative transport equation that the decay is given by a superposition of different decay modes, and the decay rates of these modes are determined by spatial frequencies and Case’s eigenvalues. This explains why SFDI can focus on shallow regions. The fact that light with nonzero spatial frequency rapidly decays makes it possible to exclusively extract optical properties of the top layer of a layered medium. We determine optical properties of the top layer of a solid phantom. This measurement is verified with different layered media of numerical phantoms.

A fast algorithm for fluorescence diffuse optical tomography is proposed. The algorithm is robust against the choice of initial guesses. We estimate the position of a fluorescent target by assuming a cuboid (rectangular parallelepiped) for the fluorophore target. The proposed numerical algorithm is verified by a numerical experiment and an experiment with a meat phantom. The target position is reconstructed with a cuboid from measurements in the time domain. Moreover, the long-time behavior of the emission light is investigated making use of the analytical solution to the diffusion equation.

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case, the Born series converges regardless the value of the impedance term in the Robin boundary condition.

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.

We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.

The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.

光輸送方程式について

近赤外イメージングは被曝の心配なく簡便な装置で計測できる技術として、医学はもちろん理学・工学の様々な分野での活躍が期待されています。しかし、そこに現れるのは非線形で非常に非適切な逆問題です。本研究ではこの逆問題に対し、輻射輸送方程式を用いた高精度化と逆級数を用いた高速化で挑みます。さらに継続計測における計測間の時系列に着目し、情報科学と融合した新たなデータ駆動型アプローチを構築します。

令和3年度は、令和2年度に開発した基本解の数値計算法を基に初期値をゼロとした境界値問題の数値解法を実装し、主に誤差の収束特性について数値実験により検討した。その結果、誤差の収束について、以下のことがわかった。(i) 空間変数については電荷点の個数に対し、指数関数的である。なお収束の速度は電荷点と拘束点の距離を大きくとった方が高い、(ii) 時間変数については時間刻み幅に対し、線型オーダーである。この中で、(i)の特性はLaplace方程式に対する代用電荷法が持っている性質と類似のものであり、好ましいものである。一方で、(ii) の特性は不満足ではあるものの、ある程度、予想されたものであった。これらの研究成果は、日本応用数理学会年会（2021年10月）および環瀬戸内応用数理研究部会第27回シンポジウムにおいて口頭発表を行った。 その後、層状領域への拡張を検討していたが、数値実験を進めていくうちに、時間刻み幅を小さくした場合に不安定性が生じることが判明した。線型安定解析に基づく分析を行った結果、電荷点と拘束点の距離を小さくすれば多少改善されるものの、この不安定性は今回用いている手法、すなわち時間方向について単純な線形和を取る手法が本質的に持っている性質であり、そのままの形での解決は困難であるものと考えられた。そのため、時間方向の近似手法を根本から考え直すことが必要となった。 年度末になり、時間依存問題に対する境界要素法の手法について調べた結果、Lubichが開発したConvolution Quadrature Method(CQM) を適用することで解決できるのではないかとの着想を得た。現在、CQMを適用した代用電荷法の数値計算法の開発を行い、予備的数値実験を行っている段階であるが、良好な結果を得つつある。

多孔質媒体を水とともに流れる分子の輸送が輻射輸送方程式に支配されることをさらに検証した。以前の研究では半空間について輻射輸送方程式の解を求めたが、今回は実験を反映して両端の境界を考慮した。実験では、砂などを詰めたシリンダー状の容器にターゲット分子をまぜた水を流す。実験状況から、1次元の空間における輸送を扱った。輻射輸送方程式の解は固有モードの重ね合わせとして求める。斉次方程式は解析的に解くことができ、固有モードは数値計算せずに得られる。このとき、輻射輸送方程式の積分項を離散方位法によって離散化しておくと、固有モードに超関数が現れない。時間に依存する輻射輸送方程式を解いたが、ラプラス変換によって時間に依存しない方程式に帰着させた後、逆ラプラス変換で時間に依存する解を得た。逆ラプラス変換には二重指数関数型数値積分公式を利用した。吸収係数・散乱係数の異なる様々な実験の結果と比較したところ、いずれの場合にも輻射輸送方程式の解とよく合うことがわかった。
光トモグラフィーの逆問題は非線型であり反復法におけるコスト関数は非常に複雑な多谷構造を持つ。反復法にかわる逆問題解法として、シミュレーテッドアニーリングの利用を検討した。この方法では、乱数を用いることで局所解から抜け出すことができる。最初の試みとして、輻射輸送方程式に対する拡散近似を用いた。拡散方程式による光トモグラフィーについてシミュレーテッドアニーリングの方法を試すことにより、この手法が有望であることがわかった。

拡散光トモグラフィ（DOT）は、生体機能と構造情報を可視化できる生体光計測技術の中で最も高度な技術の一つである。DOT画像再構成アルゴリズムは、基本的に順問題解析と逆問題解析によって構成される。生体の光学特性値（吸収係数と換算散乱係数）は、順問題の条件設定や逆問題解析の初期値として必要であるが、その真の値は未だ不明である。本研究は、脳組織を灰白質と白質に分けて、それぞれの光学特性値をin situ計測することを目的とした。そのために、フェムト秒時間分解計測システムを構築して生きているラットとサルの脳を計測し、モンテカルロ法で作成したルックアップテーブルを用いて光学特性値を推定した。

メゾスコピック系の電気伝導において、メゾスコピック系内の電子間相互作用や、メゾスコピック系に接続されている導線がどのような効果を及ぼすかを議論した。前者が多体の束縛状態を、後者が共鳴状態を生み出すことを明らかにした。