# Time-Domain Near-Infrared Spectroscopy and Imaging: A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background of TD-NIRS

#### 2.1. Radiative Transfer Equation (RTE)

**r**and at time t through the unit area oriented in the direction of the unit vector ŝ and through the unit solid angle along ŝ in a medium as shown in Figure 1. The most fundamental equation describing light propagation in biological tissue, which is accepted in this field, is the radiative transfer equation in time-domain (TD-RTE) (or the Boltzmann transport equation) for radiance [5],

_{a}(

**r**) and μ

_{s}(

**r**) are the absorption and scattering coefficients, respectively, $p(\widehat{s},\widehat{s}\prime )$ is the scattering phase (angular) function describing the probability of scattering from direction ŝ′ into direction ŝ, dΩ′ is the solid angle for integration, and $q(r,\widehat{s},t)$ is the light source. Here we assume that the radiance is for a specific wavelength and that the velocity of light is constant throughout the medium. The RTE is an energy conservation equation, and each term has physical meanings; the total temporal change in the radiance, the energy inflow due to the gradient of the radiance (or the diffusion of the radiance), the energy gain by absorption and scattering, the energy inflow to direction ŝ by scattering from direction ŝ′ over the entire solid angle, and the energy gain by light sources. Here we note: if the radiance, $I(r,\widehat{s},t)$, having a dimension of W/(m

^{2}sr) is divided by the velocity of light, c, it has a dimension of J/(m

^{3}sr) and is often called as the photon energy density, $u(r,\widehat{s},t)=I(r,\widehat{s},t)/c$. Equation (1) is sometimes expressed as $u(r,\widehat{s},t)$ instead of $I(r,\widehat{s},t)$.

#### 2.2. Expansion of the RTE by Spherical Harmonics and the P_{N} Approximations

_{l}

^{m}(ŝ) (l = 0, 1, 2, …, m = −l, −l + 1, …, l − 1, l), to separate the angular dependences of $I(r,\widehat{s},t)$ and $q(r,\widehat{s},t)$ on ŝ from the dependences on

**r**and t [5,6] [7] (pp. 282–288). After some mathematical manipulations, the RTE is rewritten in terms of a series of spherical harmonics for the expansion coefficients i

_{lm}(

**r**, t) of $I(r,\widehat{s},t)$,

_{lm}(

**r**, t) and p

_{l}are the expansion coefficients which are known. For a particular combination of (l, m) = (L, M), Equation (2) is transformed to an infinite number of coupled partial-differential equations for i

_{LM}(

**r**, t) with L ranging from 0 to ∞ and M ranging from −L to +L [7] (pp. 282–288).

^{2}, the same as the number of unknown functions, i

_{LM}(

**r**, t). The system of these (N + 1)

^{2}equations for (N + 1)

^{2}unknowns is the P

_{N}approximation. For example, for the P

_{1}approximation, there are four unknowns of i

_{LM}(

**r**, t), and for the P

_{3}approximation, there are 16 unknowns of i

_{LM}(

**r**, t). Even-order P

_{N}approximations are not useful and only odd-order P

_{N}approximations are considered.

#### 2.3. The P_{1} Approximation

_{1}approximation, four unknowns, i

_{00}(

**r**, t), i

_{1−1}(

**r**, t), i

_{10}(

**r**, t), and i

_{11}(

**r**, t) are related to the fluence rate, ϕ(

**r**, t), and to the flux vector of the fluence rate,

**F**(

**r**, t), expressed as,

_{1}approximation is expressed by two coupled equations for ϕ(

**r**, t) and

**F**(

**r**, t), including the reduced scattering coefficient, μ

_{s}′(

**r**) = [1 − g]μ

_{s}(

**r**), and the anisotropy parameter, g(

**r**), defined as the average cosine of the phase function as,

_{4π}p(ŝ, ŝ′)dΩ = 1.

**r**, t) is derived as,

**r**) = 1/[3(μ

_{s}′(

**r**) + μ

_{a}(

**r**))] is the diffusion coefficient, q

_{0}(

**r**, t) and

**q**

_{1}(

**r**, t) are related to q

_{lm}(

**r**, t) in Equation (2) describing the isotropic and anisotropic components of the light source, $q(r,\widehat{s},t)$. Equation (5) is the equation for ϕ(

**r**, t) in the P

_{1}approximation having a form of the telegraph equation (TE) which is an elliptic type of partial differential equation including the second derivatives with respect to both time and space indicating a phenomenon of wave propagation in the medium.

#### 2.4. Diffusion Approximation and Diffusion Equation (DE)

_{1}approximation is further simplified to the time-domain diffusion equation (TD-DE) by adding conditions of (i) strong scattering meaning μ

_{s}′ » μ

_{a}or Dμ

_{a}« 1, (ii) slow temporal changes in the fluence rate and the light source leading to,

**q**

_{1}(

**r**, t) = 0. Then, Equation (5) reduces to the TD-DE,

_{s}′c) ≈ 10(3D/c) under the condition of μ

_{s}′ » μ

_{a}. Here, 3D/c is the characteristic time of interaction. For a case of μ

_{s}′ = 1.0 mm

^{−1}typical for biological tissue, the DE fails for light propagation within times shorter than 0.05 ns (50 ps), and the RTE is required in this period of times.

**F**(

**r**, t), can be measured by time-resolved (TR) detectors and the measured fluxes such as TR reflectance and TR transmittance are often expressed as the time-of-flight distribution (TOF-distribution) in this article.

#### 2.5. Diffusion Coefficient Independent of the Absorption Coefficient in TD-DE

**r**) = 1/[3(μ

_{s}′(

**r**) + μ

_{a}(

**r**))] in the process of deriving the DE in the framework of the P

_{1}approximation stated above. However, there was a long controversy about the expression of the diffusion coefficient and whether it depends on μ

_{a}or not. Furutsu and Yamada [8] first discussed that in time-domain, D(

**r**) is independent of μ

_{a}(

**r**), i.e., D(

**r**) = 1/[3μ

_{s}′(

**r**)], while in the CW-domain D(

**r**) may depend on μ

_{a}(

**r**). For optically homogeneous media, the radiance in the RTE for an impulse source can be written as,

_{0}(

**r**, ŝ, t) is the radiance for non-absorbing medium without absorption, Equation (9),

_{1}approximation described above can be applied to Equation (9), then the diffusion coefficient is clearly given as D = 1/(3μ

_{s}′) independent of μ

_{a}. For inhomogeneous media, the derivation of D(

**r**) = 1/[3μ

_{s}′(

**r**)] is not straightforward like for homogenous media, but Furutsu and Yamada [8] proved it mathematically.

_{a}or not [10,11,12]. But experimental and numerical studies supported D(

**r**) = 1/[3μ

_{s}′(

**r**)] [13,14,15], and mathematical studies using processes different from the P

_{1}approximation derived D(

**r**) independent of μ

_{a}for the TD-DE and D(

**r**) dependent on μ

_{a}for the CW-DE [16,17]. Finally, the following expressions are likely to be accepted in this field,

_{a}, of biological tissue is much smaller than μ

_{s}′ in the NIR range (typically μ

_{a}~0.01 mm

^{−1}and μ

_{s}′ ~1.0 mm

^{−1}), and the change in the magnitude of D upon including μ

_{a}is very small. Therefore, D(

**r**) = 1/[3μ

_{s}′(

**r**)] is a good choice for all cases.

#### 2.6. Boundary Condition for DE

**r**

_{b}is a position on the interface, n indicates the direction outward normal to the interface, and A is a reflection parameter given by A = (1 + R

_{in})/(1 − R

_{in}), with R

_{in}denoting the internal diffusive reflectivity estimated by the Fresnel reflection coefficient or other empirical models. In the process of obtaining analytical solutions of the DE, extrapolated boundary conditions are often employed with the mirror image method to satisfy Equation (12). For simplicity, the zero-boundary condition where the fluence rate at the interface is given as zero is sometimes used, and this is the case for A = 0 in Equation (12).

#### 2.7. Analytical Solutions for TD-DE

_{1}= [(z − z

_{0})

^{2}+ ρ

^{2}]

^{1/2}and r

_{2}= [(z + z

_{0}+ 2z

_{b})

^{2}+ ρ

^{2}]

^{1/2}are the distances from the position of interest, P(ρ, z), to the positive and negative point impulse sources, respectively, The reflectance at the surface, P(ρ, 0) with the source-detector (SD) distance of ρ, is given by Fick’s law as Equation (15),

_{10}= (z

_{0}

^{2}+ ρ

^{2})

^{1/2}and r

_{20}= [(z

_{0}

^{2}+ 2z

_{b})

^{2}+ ρ

^{2}]

^{1/2}. Equation (15) is the Green’s function for semi-infinite homogeneous media and is often used in the derivation of analytical solutions using the perturbation theory.

#### 2.8. Monte Carlo Simulations

_{i}is scattered with a scattering path length of L to the direction of the polar and azimuthal angles of θ and φ, respectively, and the energy is attenuated to ${E}_{i+1}={E}_{i}\left[{\mu}_{s}/({\mu}_{s}+{\mu}_{a})\right]$. L, θ, and φ are determined using uniformly distributed random numbers in the range of (0, 1), R

_{1}, R

_{2}, and R

_{3}by the following equations,

_{2}(θ) is the accumulated phase function of p(θ), f

_{2}

^{−1}means the inverse of f

_{2}(θ). When p(θ) is constant, meaning that g = 0 and the diffusion approximation is applicable, f

_{2}(θ) becomes as simple as f

_{2}(θ) = (1 − cosθ)/2, and resultantly θ is simply determined by θ = cos

^{−1}(1 − R

_{2}).

#### 2.9. Time-Domain Sensitivity Functions of Optical Signals

_{0}(

**r**, t), is a solution of the TD-DE in a homogeneous medium with the absorption and diffusion coefficients, μ

_{a}

_{0}and D

_{0}, for the impulse point source at the origin and time zero. If μ

_{a}

_{0}and D

_{0}in a small volume of the medium, V

_{p}, are perturbed to μ

_{a}= μ

_{a}

_{0}+ δμ

_{a}and D = D

_{0}+ δD, the perturbations in the fluence rate due to the perturbations of δμ

_{a}and δD, δϕ

_{a}(

**r**, t), and δϕ

_{D}(

**r**, t), respectively, are derived by the first order perturbation theory (explained in detail in a later section), and the TD sensitivities (weights or Jacobians) of the fluence rate to the changes in μ

_{a}and D, defined as the limits of δμ

_{a}→ 0 and δD → 0 of the absolute values of δϕ

_{a}(

**r**, t)/δμ

_{a}and δϕ

_{D}(

**r**, t)/δD, respectively, are given as,

**r**,

**r**′, t − t′) is the Green’s function which is the solution of ϕ(

**r**, t) in the TD-DE for the impulse point source at position

**r**′ and time t′. The right-hand sides of Equations (17) and (18) are calculated analytically using the analytical solutions of TD-DE for simple geometries or numerically for complex geometries. Equations (17) and (18) are the TD sensitivities of the fluence rate to the changes in μ

_{a}and D, respectively, but the TD sensitivities of any optical signals such as the reflectance and transmittance can be defined.

**r**

_{d},

**r**

_{s}, t) (

**r**

_{d}and

**r**

_{s}are the detector and source positions, respectively) is considered, the TD sensitivity of the reflectance to the change in μ

_{a}is expressed approximately as,

^{(R)}(

**r**

_{d},

**r**′, t) is the Green’s function of the reflectance measured at

**r**

_{d}for the impulse point source at position

**r**′ and time t′, and G(

**r**′,

**r**

_{s}, t − t′) is the Green’s function of the fluence rate measured at position

**r**′ and time t − t′ for the impulse point source at position

**r**

_{s}and time t′.

_{p}= δ(

**r**′ −

**r**) around

**r**, and after some mathematical manipulations, a three-point function of

**r**

_{d},

**r**

_{s}, and

**r**can be defined as Equation (20),

**r**,

**r**

_{s}, t′) means the fluence rate at position

**r**and at time t′ generated by an impulse point source at position

**r**

_{s}and time 0, and G(

**r**

_{d},

**r**, t − t′) in the first equality means the fluence rate at position

**r**

_{d}at time t − t′ generated by an impulse point source at position

**r**and time t′ as indicated in Figure 5a. The function H(

**r**

_{d},

**r**

_{s},

**r**, t) indicates the probability of photons that exist at position

**r**after being injected from the source at position

**r**

_{s}at time 0 and finally being detected by the detector at position

**r**

_{d}at time t, and when illustrated in 2D or 3D images at varying times t, they show the temporal evolution of probabilistic light paths in the medium between the source and detector. In case of reflectance from semi-infinite media, the probabilistic light paths exhibit so-called banana shapes, and the banana shapes grow as time evolves. In case of transmittance through slabs, similar formulations are derived as in the case of reflectance, and the probabilistic light paths exhibit so-called spindle-like shapes.

**r**

_{d},

**r**

_{s},

**r**, t) is called various terminology such as the sensitivity to absorption, photon probability distribution [26], photon hitting density [27], photon measurement density function [28], relative photon visit probability [29], photon visiting probability [30], photon sampling function, photon weight function, Jacobian with respect to absorption, etc., with variation in the proportionality constant, α. The sensitivity to scattering changes can also be derived from Equation (18), but its physical meaning is not as clear as H(

**r**

_{d},

**r**

_{s},

**r**, t). The sensitivities play important roles not only in understanding the propagating path of scattered light but also in the inversion process of reconstructing absorption and scattering images in DOT as explained later.

**r**

_{d},

**r**

_{s},

**r**, t) by time-gated measurements of the fluence rates through a homogeneous semi-infinite medium. The second equality in Equation (20) means that H(

**r**

_{d},

**r**

_{s},

**r**, t) is given by the convolution between the Green’s function for the input position at the source,

**r**

_{s}, and that for the input position at the detector,

**r**

_{d}, as shown in Figure 5b. Sawosz et al. [30] used this characteristic of H(

**r**

_{d},

**r**

_{s},

**r**, t) to acquire the images in Figure 6 showing the banana shapes growing bigger as time evolved. Continuous wave sensitivity is obtained by extending the upper time limit in Equation (20) to infinity.

#### 2.10. Time-Resolved (TR) Mean Depth of Light Propagation

**r**

_{d},

**r**

_{s},

**r**, t), it is possible to calculate the TR mean depth of light propagation, <z>(

**r**

_{d},

**r**

_{s}, t), in a semi-infinite homogeneous medium by,

**r**

_{d},

**r**

_{s},

**r**, t).

_{eff}= (3μ

_{s}′μ

_{a})

^{1/2}, as <z>(ρ) = (1/2)(ρ/μ

_{eff})

^{1/2}[31].

#### 2.11. Time-Resolved (TR) Pathlength

_{i}is time when the light is propagating inside the i-th layer. For CW light, the (partial) pathlengths are obtained by evolving the upper limit of time t to infinity, ∞.

_{i}, ρ)exp(−μ

_{ai}ct) as seen from the TR reflectance of Equation (15), and Equation (23) is modified as follows,

_{eff}/(2μ

_{a})][1+1/(ρμ

_{eff})]

^{−1}for your reference [31].

#### 2.12. Physiological Information and Optical Properties

_{i}is the extinction coefficient or the molar absorption coefficient of the component i, C

_{i}is the molar concentration or the molarity of the component i, and N

_{C}is the number of components. Note that ε

_{i}is often based on common logarithm while μ

_{a}is based on natural logarithm, necessitating correction between the two logarithms. If ε

_{i}(λ) are known for all the components, measurements of μ

_{a}(λ) at N

_{C}(or more than N

_{C}) wavelengths will determine C

_{i}of all the components. Oxygenated hemoglobin (oxy-Hb) and deoxygenated hemoglobin (deoxy-Hb) are two dominant chromophores in NIRS with known spectra of ε

_{oxy-Hb}(λ) and ε

_{deoxy-Hb}(λ). Then C

_{oxy-Hb}and C

_{deoxy-Hb}are determined and total hemoglobin (total-Hb) concentration, C

_{total-Hb}= C

_{oxy-Hb}+ C

_{deoxy-Hb}, and oxygen saturation, S

_{O2}= C

_{oxy-Hb}/C

_{total-Hb}, can be calculated to show hemodynamic statuses in brain, muscle, breast, skin, etc.

_{a}, the scattering properties, μ

_{s}and μ

_{s}′, indicate the statuses of microstructures and sub-cellular components, because scattering is caused by gradients and discontinuities of refractive indexes in tissue.

## 3. Instruments for TD-NIRS

#### 3.1. Overview of TD-NIRS Instruments

#### 3.2. Time-Correlated Single Photon Counting (TCSPC) Technique

#### 3.2.1. Principle, Components, Characteristics, and Operation of the TCSPC Technique

_{true}= I

_{meas}/(1 − αI

_{meas}) where I

_{true}and I

_{meas}[s

^{−1}] are the corrected and integrated intensities, respectively, and α is the dead time [s] when the measuring systems are non-paralyzed models.

#### 3.2.2. Single- and Dual-Channel TD-NIRS Systems Based on the TCSPC Technique

_{a}and μ

_{s}′ of the media are estimated under the assumption that the media are homogeneous and semi-infinite. The TR reflectance from a homogeneous semi-infinite medium is given by the analytical solution of the TD-DE as Equation (15). For the zero-boundary condition, Equation (15) is slightly simplified as the following,

_{a}and μ

_{s}′ of the medium using a non-linear least-squared technique based on the Levenberg–Marquardt method. Applying Equation (25) to the case of the two components of oxy-Hb and deoxy-Hb with background absorption gives the following equation with the wavelength dependence written explicitly,

_{a}(λ) = ε

_{oxy-Hb}(λ)C

_{oxy-Hb}+ ε

_{deoxy-Hb}(λ)C

_{deoxy-Hb}+ μ

_{a,BG}(λ)

_{a,BG}is the absorption coefficient of the background medium. Solving the simultaneous equations of Equation (29) for the three wavelengths used in the instrument provides C

_{oxy-Hb}and C

_{deoxy-Hb}, and C

_{total-Hb}, and S

_{O2}= C

_{oxy-Hb}/C

_{total-Hb}is calculated.

_{O2}from 0% to 100% [44]. The values of C

_{oxy-Hb}, C

_{deoxy-Hb}, and C

_{total-Hb}measured by the TRS-10 agreed well with the values calculated from the measurements of pH, P

_{O2}, and P

_{CO2}in the phantom by conventional methods.

#### 3.2.3. Multi-Channel TD-NIRS Systems Based on TCSPC for DOT

#### 3.3. Other New TD-NIRS Systems

#### 3.3.1. MONSTIR II

#### 3.3.2. TD-DOT and TD-NIRS Systems for Optical Mammography

#### 3.3.3. Compact TD-NIRS Systems Using MPPCs (or SiPMs)

^{2}for the detector, and a home-made TDC with a temporal resolution of 10 ps for TR measurements. For TD-NIRS, the same group also developed a compact detector probe integrating a fast SiPM and its electronics, which can be directly put in contact with the skin [57]. Many compact detector probes can be installed into a head-cap without the need for optical fibers for collecting light.

#### 3.3.4. TD-NIRS Systems Using SPADs

#### 3.3.5. TD-NIRS Systems Using Pseudo-Random Bit Sequences

#### 3.3.6. TD-NIRS Systems Using ICCD

#### 3.3.7. Compact TD-NIRS System Incorporating Devices Used in Telecommunications

#### 3.3.8. TD-NIRS Systems for Measurement of Water, Lipid, and Collagen Contents

#### 3.4. Future Trend of TD-NIRS Instruments

## 4. Advanced Theories and Methods for TD-NIRS

#### 4.1. Solving the TD-RTE and TD-DE Numerically

#### 4.2. Analytical Solutions for the TD-RTE

_{N}approximation and by expressing the expansion coefficients analytically [78]. They also derived a Green’s function of the TD-RTE for the radiance in a 3D anisotropically-scattering medium with an impulse point unidirectional source [79]. The solution involves a spherical Hankel transform necessitating numerical calculation, but the dependences on all the variables were found analytically.

#### 4.3. The TD-RTE with Spatially Varying Refractive Index

#### 4.4. Solutions of the Telegraph Equation (TE)

_{1}approximation, Equation (5) [85]. They obtained analytical solutions for infinite slabs and semi-infinite media as the forms of Laplace transform, and the solutions obtained by inverse Laplace transform cannot be expressed explicitly. Their analytical solutions agreed well with the results of MC simulations, although small differences were observed in very early times.

#### 4.5. Perturbation Theory

#### 4.5.1. Formulation of the TD-Perturbation Based on the TD-DE

_{0}(

**r**, t), under the unperturbed μ

_{a}

_{0}(

**r**) and D

_{0}(

**r**) = 1/3μ

_{s}

_{0′}(

**r**), and the TD-DE for the perturbed fluence rate, ϕ

_{p}(

**r**, t) = ϕ

_{0}(

**r**, t) + δϕ(

**r**, t), under the perturbed coefficients, μ

_{ap}(

**r**) = μ

_{a}

_{0}(

**r**) + δμ

_{a}(

**r**) and D

_{p}(

**r**) = D

_{0}(

**r**) + δD(

**r**). By subtracting the unperturbed equation from the perturbed equation, the TD-DE for δϕ(

**r**, t) is obtained as,

_{v}(

**r**,t) = {−δμ

_{a}(

**r**) + ∇•[δD(

**r**)∇]}ϕ

_{p}(

**r**,t), and Equation (30) has the same form as the unperturbed equation. Then, if the Green’s function of the unperturbed equation is known as G(

**r**,

**r**′, t − t′), the solution of Equation (33) for the virtual source of Q

_{v}(

**r**, t) is given by the Green’s function method as,

_{p}indicates the volume occupied by the perturbations. Substitution of Q

_{v}(

**r**,t) into Equation (31) and some mathematical operations leads to the following equation,

_{p}(

**r**, t) (note: the integral equation composed of Equations (32), (33), and (34) is called the Fredholm equation of the second kind), and various assumptions and approximations have been employed. Usually, the perturbations are assumed constant inside the inclusions, i.e., δμ

_{a}(

**r**) = δμ

_{a}= const and δD(

**r**) = δD = const, and the simplest approximation is to substitute ϕ

_{0}(

**r**, t) for ϕ

_{p}(

**r**, t) by assuming ϕ

_{p}(

**r**, t) ≈ ϕ

_{0}(

**r**, t) resulting in the followings,

_{p}(

**r**, t) ≈ ϕ

_{0}(

**r**, t) + δϕ

_{a}

^{(1)}+ δϕ

_{D}

^{(1)}resulting in the followings,

_{a}and δD begin to play a role.

_{a}and D from those of the background. For large inclusions with large changes in μ

_{a}and D such as in the case of breast cancers, the second or third order perturbation may be necessary. For this purpose, many studies of higher order perturbation theory have been reported [92,93,94,95].

#### 4.5.2. First Order TD-Perturbation Using the TD-DE

#### 4.5.3. Higher Order TD Perturbation Using the TD-DE

_{a}|R

_{inc}

^{2}/D

_{0}< 3 where R

_{inc}is the radius of a spherical inclusion. The paper provided calculation results of the second and third perturbations of the TR transmittance from a 60-mm thick slab having an absorbing inclusion with a diameter of 10 mm and a contrast of δμ

_{a}/μ

_{a}

_{0}about unity, and it summarized that the second perturbation is sufficient for accurate predictions.

#### 4.5.4. TD-Perturbation Using the TD-RTE

**I**= {I, Q, U, V}, where I, Q, U, and V are the Stokes parameters describing the polarization state [86]. For simplicity of discussions, they first defined the operator of the RTE,

**L**, for

**I**(t,

**r**,

**n**) at point

**r**in the direction

**n**at time t, and its adjoint operator,

**L**

^{*}, for the Stokes vector of the adjoint radiance,

**I**

^{*}(t,

**r**,

**n**), and then expressed an optical measurement of E (such as reflectance or transmittance) as the scalar product of a response function (equivalent to the Green’s function) of

**R**and the radiance of

**I**, E = <

**R**

^{+},

**I**> (superscript + denotes the transpose of the vector). Finally, they derived two basic equations for the perturbation approach as,

_{a}and μ

_{s}′. It is very difficult to explicitly give analytical solutions of the TD-RTE for the response function of

**R**, because Equation (39) is expressed implicitly, but the perturbations of Equation (39) are given by the direct and adjoint solutions of the RTE,

**I**

_{b}and

**I**

^{*}

_{b}, and for the background medium.

#### 4.6. Multi-Layered Media

_{a}and μ

_{s}′ of the layers. But the analytical solutions derived using the Fourier-transform approach were not explicitly given in TD. Tualle et al. [99] reported another analytical solution of TR reflectance for two-layered semi-infinite media based on the TD-DE by applying the mirror image method for semi-infinite homogeneous media to two-layered media using distributed sources. Further, Tualle et al. extended Kienle et al. study to multi-layered media and obtained explicitly the asymptotic solution of TR reflectance in the very late time when the asymptotic solution is dependent on μ

_{a}and μ

_{s}′ of the deepest layer [98,100]. Liemert and Kienle extended the method for two-layered media to general N-layered media although explicit analytical solutions were not given [98,101]. The TD solutions were obtained using a fast Fourier transform from the FD solutions for N-layered media at many (512 for example) frequencies.

**r**) and time (t) based on the microscopic Beer–Lambert law, the analytical solution of the TD-DE for a three-layered finite cylinder shown in Figure 14 are obtained as [103],

_{0}is the 0th order Bessel function, K

_{l}are the roots of J

_{0}(K

_{l}L) = 0, and for K

_{lni}, γ

_{lni}, and N

_{ln}refer to the reference [103]. The TR reflectance, R(ρ, t), is obtained from Equation (40) by use of Fick’s law at z = 0, and the TR partial mean pathlength inside each layer, <l

_{i}(ρ, t)>, is given analytically by the following equation,

#### 4.7. Advanced Monte Carlo Simulations

_{a}and μ

_{s}by correcting the result of one MC simulation for the reference combination of μ

_{a}= 0 and μ

_{s}= μ

_{s}

_{0}[105]. When the TR reflectance from a semi-infinite homogeneous medium with μ

_{a}= 0 and μ

_{s}= μ

_{s}

_{0}is R

_{0}(ρ, t), the TR reflectance, R(ρ, t), for a medium with μ

_{a}and μ

_{s}is given by,

_{a}and μ

_{s}′ from the measured TR reflectance. A similar MC method was employed by Alterstam et al. [106] where they called the method white Monte Carlo.

_{a}= 0 and μ

_{s}= μ

_{s}

_{0}, and all the coordinates of scattering positions only for detected photons are stored. The weights of the detected photons are unity at this step. At the second step, another medium with an inhomogeneity is considered with μ

_{ai}and μ

_{si}for the inhomogeneity and with μ

_{a}

_{0}and μ

_{s}

_{0}for the surroundings. Then the corrections of the weights of the detected photons are given by the following two scaling relationships, w

_{a}for absorption and w

_{s}for scattering,

_{i}and l

_{0}are the pathlength of the photons inside and outside of the inhomogeneity, respectively, and K

_{i}is the number of scattering events inside the inhomogeneity. The two scaling relationships were combined into one equation by Hayakawa et al. [108]. By this method the computation time is significantly reduced. Later, Sassaroli proposed a faster perturbation MC method by storing the seed values of random number series only for the detected photons [109]. This can speed up computation by 1000 times under some conditions.

#### 4.8. Hybrid RTE and DE Models

#### 4.9. Anisotropic Light Propagation in TD

## 5. Studies toward Clinical Applications of TD-NIRS

#### 5.1. Features of TD-Light Propagation Including Penetration Depth, Optical Pathlength, etc.

#### 5.1.1. Light Propagation Based on the Microscopic Beer–Lambert Law

#### 5.1.2. Mean-TOF, Partial Pathlength, and Sensitivity for the Head Model

_{i}>, and the sensitivity distribution for CW on the SD distance, and to see the effect of the CSF layer on <t> and <l

_{i}>. More realistic head models were used to extend the study [127].

_{R}(t) and H

_{<t>}(t), and <l

_{i}(t)>. They found that H

_{<t>}(t) has larger values than H

_{R}(t) in deeper regions.

_{i}(t) > calculated by MC simulations. Their results were verified by experiments using two- and three-layered phantoms and by in vivo experiments.

#### 5.1.3. Use of Null SD Distance

_{i}(t)> and penetration depth, <z(ρ, t)>, for semi-infinite two-layered media using the analytical solution of the TD-DE. Based on this study, Torricelli et al. [131] proposed the usage of null SD distance for improving contrast and resolution of diffuse optical imaging from TR reflectance measurements. They concluded that the null SD distance for TR reflectance measurement provides four advantages over the conventional SD distances of a few centimeters, i.e., (i) stronger optical signals, (ii) deeper penetration depths, (iii) larger contrast for absorbing inclusions, and (iv) higher localization of the inclusions, but also discussed a problem of too strong signals of early photons. They argued several ways for solving the problem: (i) gating the detector to measure photons arriving at 200 ps and later, (ii) using SPADs which are not damaged by the burst of initial photons, and (iii) the use of non-null but small SD distance, while keeping the advantages of a null SD distance. The method of the null SD distance for TR reflectance measurements was further studied for two-layered media by Spinnelli et al. [132] using MC simulations and the TD-DE. Later, proof-of principle tests of the null SD distance technique were reported using fast-gated SPAD as well as non-contact probes showing good agreements of the depth sensitivity and spatial resolution between the phantom experiments and MC simulations [133]. These results indicated the feasibility and potentiality of the null SD distance technique for applications to non-contact and high-density diffuse TR reflectance measurements.

#### 5.1.4. Measurement of Mean Pathlength

#### 5.2. Measuring Optical Properties

#### 5.2.1. Homogenous Semi-Infinite Media or Infinite Slab

_{a}and μ

_{s}′ in homogeneous semi-infinite media by TD techniques is essentially based on fitting analytical or numerical solutions of the TD-DE to measured TR reflectances. The TR-reflectance, R(ρ, t), is expressed by Equation (15) for the TD-DE under the extrapolated boundary condition, and for simplicity the zero-boundary condition is sometimes employed as Equation (28). For the zero-boundary condition, the following two equations are derived [18],

_{max}is the time of maximum R(ρ, t). From these equations, it is possible to determine μ

_{a}and μ

_{s}′ of semi-infinite homogeneous media from the measured R(ρ, t). Especially, it should be noted that μ

_{a}is determined from the slope of lnR(ρ, t) at very late time.

_{max}due to the IRF. Therefore, using Equation (43) is not appropriate for accurate determination of μ

_{a}and μ

_{s}′, and fittings of the IRF convoluted analytical solutions of Equation (15) or Equation (28) to measured R(ρ, t) are often used to recover μ

_{a}and μ

_{s}′. Note that deconvolution of measured R(ρ, t) by the IRF enhances measurement noises and instead the analytical solutions are convoluted by the IRF.

_{a}and μ

_{s}′ in homogeneous infinite slabs by TR transmittance measurements is conducted using the analytical equation for TR transmittance, which is derived similarly to that for TR reflectance using the mirror image source method as Equation (15) [18,19].

_{a}and μ

_{s}′ of a piglet brain using a TD system at three wavelengths (759, 794, 824 nm) by the method stated above. To separate the contributions of different head layers, the measured R(ρ, t) were acquired at the surfaces of skin, skull, dura mater and brain, step by step. The values of ρ were chosen to assure the mean penetration depth within each layer. Measured μ

_{a}and μ

_{s}′ were compared with the other in vivo results reported in literatures, and the differences were discussed.

_{a}and μ

_{s}′ of human foreheads in the wavelength range from 700 nm to 1000 nm. They found that the estimated μ

_{a}and μ

_{s}′ were close to those of the superficial (scalp and skull) layers by additional MC simulations for four-layered media simulating the structure of the human head.

_{a}from the background at the seven wavelengths, Δμ

_{a}(λ), using a time-gated perturbation analysis based on the microscopic Beer–Lambert law under the assumption that the compressed breasts were homogeneous slabs [54,139]. The Δμ

_{a}(λ) images were converted to the changes in the contents of oxy-Hb, deoxy-Hb, water, lipid, and collagen using Equation (25), and from statistical analyses they concluded that the collagen content was the most important parameter for discriminating malignant and benign lesions. Ohmae et al. evaluated the performance of the six-wavelength TD-NIRS system using phantoms varying the contents of water, lipid, and an absorber [68]. The performance was confirmed with the measurements using a magnetic resonance imaging system.

_{a}(λ) and μ

_{s}′(λ) of irregularly-shaped homogeneous media from TD-NIRS measurements. The analytical solutions of the TD-DE are usually not applicable to irregularly shaped media, and the finite element method (FEM) was employed to solve the TD-DE. The FEM solutions for non-absorbing medium were multiplied by exp(-μ

_{a}ct) to incorporate the attenuation by absorption.

#### 5.2.2. Multi-Layered Media

_{a}

_{1}, μ

_{a}

_{2}, μ

_{s}

_{1}′, and μ

_{s}

_{2}′ with known thicknesses of the top layer (denoted by s

_{1}) [98]. They concluded that μ

_{a}

_{2}can be determined accurately even if s

_{1}is known only approximately. Gagnon et al. used the same analytical solutions for two-layered media as Kienle and Glanzman to estimate the intra- and extra-cerebral hemoglobin concentrations [141,142]. The upper layer included the scalp, skull, and CSF, and the lower layer was the brain. They observed noticeable inter-subject variations in the hemoglobin concentrations and constant oxygen saturation of the cerebral hemoglobin.

_{a}

_{1}, μ

_{a}

_{2}, μ

_{s}

_{1}′, μ

_{s}

_{2}′, and s

_{1}accurately [143,144]. The errors of the estimated μ

_{a}

_{1}, μ

_{a}

_{2}, and μ

_{s}

_{1}′ were small while those of μ

_{s}

_{2}′ were large.

_{a}

_{2}in two-layered semi-infinite media based on the microscopic Beer–Lambert law. Extending this method, a simple algorithm was developed to recover μ

_{a}

_{1}and μ

_{a}

_{2}in a two-layered medium from the TR reflectances measured at two values of ρ and was verified by numerical simulations and in vivo experiments using human foreheads [146,147].

_{a}and μ

_{s}′ of the four layers to fit the simulated TR reflectances to the measured ones and estimated the partial pathlengths and sensitivities of the four layers. As of the results, they found the followings: (i) the total pathlength ranging from 100 mm to 250 mm was closely related to the thickness of the scalp ranging from 2 mm to 10 mm; (ii) the partial pathlength of the brain ranged from 3% to 13% of the total pathlength; (iii) the brain tissue deeper than 25 mm from the head surface were hardly detected by the near-infrared light; (iv) the scalp tissue at the depth of 4 mm had the highest sensitivity inside the heads; and (v) most of the signals attributed to the brain came from the superficial layer with the thickness of 1 to 2 mm from the brain surface. These findings are important for functional NIRS and could not have been obtained without TD-measurements.

_{a}of the brain (μ

_{a,brain}) from TR reflectances in the case of five-layered media. Training data of the TR reflectances with a single value of ρ were generated by the analytical solutions of the TD-DE for five-layered media as well as MC simulations which can model the CSF layer more accurately than the TD-DE [101]. One-hundred to 500 training data with different noises were input into the NN, and the NN estimated μ

_{a,brain}under the condition that the optical properties of the five layers except μ

_{a,brain}and the thicknesses of the upper four layers are known with some uncertainties. Resultantly, μ

_{a,brain}was estimated with errors less than 5% even if there are uncertainties of 20% in the other optical properties and the thicknesses.

_{a}

_{1}, μ

_{a}

_{2}, μ

_{s}

_{1}′, and μ

_{s}

_{2}′ from the TR reflectances provided by MC simulations and phantom experiments with better performances than the standard non-linear least-squares methods. The thickness of the top layer, s

_{1}, and the time origin of the TR reflectance, t

_{0}, were also included in the unknowns. The values of μ

_{a}

_{2}were very accurately estimated because the TR partial pathlength of the bottom layer, l

_{2}(t), becomes much longer than that of the top layer, l

_{1}(t), after 2 ns in the decaying period. Even if the thickness of the top layer, s

_{1}, is unknown the values of μ

_{a}

_{2}are obtained with errors less than 10%, and this method will be well suited for applications to human heads where the effect of the CSF is mainly to decrease μ

_{s}

_{2}′.

_{a}

_{2}[32] (pp. 109–125) [152,153]. When the spectroscopic TR reflectance, R(t, λ), is divided into many time gates, the ratio of the perturbed and unperturbed R(t, λ) averaged over the g-th time-gate is derived from Equation (24) as,

_{0g}(λ) and R

_{g}(λ) are the unperturbed and perturbed R(t, λ) averaged over the g-th time-gate, respectively, Δμ

_{a,i}(λ) is the perturbation of μ

_{a,i}in the i-th layer, N is the number of layers, and l

_{g,i}(λ) is the partial pathlength of the i-th layer averaged over the g-th time-gate, which is given by the analytical solution of the TD-DE in case of two-layered media (N = 2). Then a system of linear equations of Equation (44) with the number of the time-gates is constructed and Δμ

_{a,i}(λ) is obtained by inverting the system of the linear equations.

_{a}

_{1}, μ

_{a}

_{2}, μ

_{s}

_{1}′, μ

_{s}

_{2}′, and s

_{1}were estimated. Although the inter-laboratory differences are rather large, a definite trend was found after comparisons with the results using R(t, λ) generated by MC simulations.

_{a}of the brain and superficial layers after the intravenous administration of indocyanine-green from multi-distance TR reflectance measurements and MC simulations for a two-layered model [155]. In the estimation process, the information of the partial pathlengths of the two layers was important for calculating the sensitivities to absorption changes in the layers.

#### 5.3. Time-Domain Diffuse Optical Tomography (TD-DOT)

#### 5.3.1. General Concept of TD-DOT

_{a}and μ

_{s}′ images. The software developed in this framework was named as “TOAST” and its revised version “TOAST++” was presented as an open-source software [165]. A general algorithm of MOBIIR is summarized below.

_{a}= 0 and T

_{b}= ∞ corresponds to the CW intensity, and multiple time-gated intensities with a small time-step of T

_{b}− T

_{a}= ΔT dividing the whole TOF distributions are equivalent to the use of full TOF distributions. The n-th temporal moment with n = 0 is also identical to the CW intensity, and those with n = 1, 2, 3 are the mean, M

_{1}= <t>, variance, M

_{2}, and skew, M

_{3}, of the TOF distribution, respectively. The Mellin–Laplace transform with n = 0 or p = 0 corresponds to the simple Laplace or Mellin transform, respectively. Equation (48) describes the transformation of the TD data to the FD data at a frequency of ω.

**Γ**

_{cal}and

**Γ**

_{mes}are the vectors of the measured and calculated (featured) TD data with the components of

**Γ**

_{cal}(

**r**

_{sm},

**r**

_{dm}:μ

_{a}(

**r**

_{n}), D(

**r**

_{n})), and

**Γ**

_{mes}(

**r**

_{sm},

**r**

_{dm}), respectively, the index m (=1, …, M) specifies the TD data number with M being the number of the TD data, the index n (=1, …, N) specifies the voxel number with N being the number of voxels in the medium, respectively, x = [μ

_{a}(

**r**

_{n}), D(

**r**

_{n})]

^{T}is a vector consisting of the optical properties at the n-th voxel, and λ is the regularization parameter. Here, the measurement errors are neglected for simplicity. The Levenberg–Marquardt algorithm, which is a standard method in the Newton-type minimization process, leads to the following inversion matrix equation for the updates at the k-th iteration,

**δx**

^{(k)}[69,166],

**W**is the sensitivity (weight or Jacobian) matrix of the featured TD data to the changes in the optical properties,

**δx**. If the time-gated intensity is chosen for the featured TD data,

**W**will be given by equations similar to Equations (17) and (18). Then the optical properties are updated by use of the Tikhonv regularization as follows,

_{DC}

_{(0,∞)}= E, first temporal moment M

_{1}= <t>, 3rd central moment c

_{3}, normalized Laplace transform M

_{ML}(0,0.001)/E, normalized Mellin–Laplace transform set M

_{ML}(1,0.01)/E + M

_{ML}(3,0.001)/E, and combination of c

_{3}+ M

_{ML}(0,0.001)/E [169]. Here, ${c}_{n}={E}^{-1}{\displaystyle {\int}_{0}^{\infty}{(t-<t>)}^{n}\Gamma (t)dt}$. Among these six featured TD data, the combination of c

_{3}+ M

_{ML}(0,0.001)/E provided the best reconstruction results.

_{2}, and <t> + c

_{2}+ c

_{3}even for 2D reconstruction [170]. Figure 16 illustrates that the μ

_{a}image reconstructed using the full TOF distributions shows qualities better than those using the featured TD data of E and <t>.

#### 5.3.2. Modified Generalized Pulse Spectrum Technique for TD-DOT

_{1}and p

_{2}), R = Γ(p

_{2})/Γ(p

_{1}), as the featured TD data. Then, with some assumptions, the sensitivities of R to the changes both in μ

_{a}and in μ

_{s}′ are expressed using the product of the Green’s functions of the Laplace transformed fluence rate and flux as follows,

_{s}′, W

_{s}

^{(R)}(

**r**

_{d},

**r**

_{s}, t), is easily calculated while the sensitivities of other featured TD data to the change in μ

_{s}′ are usually expressed by the spatial gradients of both the Green’s functions of the fluence rate and flux as Equation (18), which makes computation ineffective and leads to crosstalk between μ

_{a}and μ

_{s}′ images. This difference in calculating the sensitivity to the change in μ

_{s}′ is the biggest advantage of the modified GPST over the other featured TD data approaches. In addition, taking positive and negative Laplace parameters for p

_{1}and p

_{2}is interpreted as weighting the early and late times in the TOF distributions, thus covering the key features of the TOF distributions. With these advantages, the modified GPST makes the simultaneous reconstruction of μ

_{a}and μ

_{s}′ images possible with less crosstalk between them. μ

_{s}′ images provide anatomical information while μ

_{a}images provide physiological information. The modified GPST was successfully extended from a 2D case to a semi-3D case with numerical simulations and phantom experiments [171,172,173].

#### 5.3.3. Other Techniques for TD-DOT

#### 5.3.4. Brain Imaging

_{total-HB}and S

_{O2}indicated the existence of hemorrhage with physiologically reasonable values. Hebden et al. also obtained DOT images of ventilated infant brains to show the hemodynamic responses to changes in the oxygen and carbon dioxide partial pressures in the ventilating air [184]. The changes in C

_{total-HB}and S

_{O2}revealed by the DOT images were in qualitative agreement with physiologically expected changes.

_{a}are easily reconstructed. Two sets of optodes were attached onto the right and left of the forehead of a subject performing a video game task. The DOT images of ΔC

_{oxy-HB}, ΔC

_{deoxy-HB}, and ΔC

_{total-HB}overlaid on the MR images of his brain clearly showed the localized activity of the prefrontal cortex as shown in Figure 17, although the hemodynamic changes appeared in the dura mater due to localization errors.

#### 5.3.5. Breast Imaging

_{a}and μ

_{s}′ than other tissues so that transmittance measurements are available. Ntziachristos et al. [188] developed a transmittance TD-DOT system to image the optical properties of breast phantoms and reported the potentiality of TD-DOT for breast tumor diagnosis. The same group used the TD system with an exogenous agent, indocyanine green (ICG), for image enhancement in in vivo measurements on patients [189]. They obtained better localization of target tumors, observed the differences in the ICG distributions among malignant tumors, benign tumors, and healthy tissues, and compared the DOT images with the concurrently acquired gadolinium-enhanced MRI images. However, the use of exogenous agents for frequent measurements may cause complications for patients and may thus be avoided.

_{oxy-HB}, C

_{deoxy-HB}, water content, and lipid content in compressed breasts from 49 patients, and differences in the blood content and water fraction between malignant and benign tumors were found.

_{a}of the tumor after chemotherapy drastically decreased by 34% from 0.0076 mm

^{−1}before chemotherapy, indicating the effectiveness of quantitative evaluation of chemotherapy. Yoshimoto et al. [53] used a 12-channel TD-NIRS system with a hand-held probe for use at bed side and indicated the usefulness of the TD-DOT system for evaluation of the effectiveness of chemotherapy to breast cancers.

_{total-HB}strongly correlated with the clinical assessment of response to hormone treatment of breast cancers.

_{a}, μ

_{s}′) and (y

_{f}, τ

_{f}) where y

_{f}and τ

_{f}are the fluorescence yield and lifetime, respectively. The images of (μ

_{a}, μ

_{s}′) and (y

_{f}, τ

_{f}) were reconstructed independently with a reasonable quantitativeness, but when the images of (y

_{f}, τ

_{f}) were used to provide the locations of targets as prior information for reconstructing (μ

_{a}, μ

_{s}′) images (fluorescence guided DOT), the quantitativeness of the (μ

_{a}, μ

_{s}′) images was highly improved; for example, the ratios of the reconstructed to correct values of (μ

_{a}, μ

_{s}′) were (23.5%, 31.0%) for separate reconstruction and (76.6%, 86.2%) for fluorescence guided reconstruction.

#### 5.3.6. Muscle Imaging

_{a}and μ

_{s}′ inside a human adult forearm during hand grip exercises with transmittance measurements using MONSTIR [49,193]. The DOT images at wavelengths of 790 and 820 nm showed the responses of μ

_{a}and μ

_{s}′ to the exercises.

_{a}and μ

_{s}′ in human lower legs and forearms [47,194]. For the case of the forearms, DOT images of μ

_{a}, μ

_{s}′, ΔC

_{oxy-HB}, ΔC

_{deoxy-HB}, and ΔC

_{total-HB}in human forearms during excises were obtained as shown in Figure 18.

#### 5.4. Time-Domain Fluorescence Diffuse Optical Spectroscopy (TD-FS) and Tomography (TD-FT)

#### 5.4.1. Fundamental Equations for TD-FS and TD-FT

**r**), γ(

**r**), and τ(

**r**) are the spatial distributions of the fluorescence properties, i.e., concentration, quantum efficiency, and lifetime of the fluorophore, respectively, and ε is the extinction coefficient of the fluorophore at the excitation wavelength. The goal of FT is to reconstruct the distributions of fluorescence properties, and N(

**r**) and τ(

**r**) are the main targets for molecular imaging. In general, the image reconstruction algorithm for TD-DOT is extended for TD-FT, which uses the emitted fluorescence light intensities measured at the object surface as the input data in addition to the measured reemitted excitation light intensities. The optical and fluorescence properties are assumed first, and Equations (55) and (56) are solved as the forward problem to obtain the calculated excitation and fluorescence light intensities, which are compared with the measured ones. If they do not agree, the optical and fluorescence properties are upgraded by an optimization procedure introduced in Section 5.3.1, and Equations (55) and (56) are solved again. This process is repeated until convergence is reached. Many studies have been conducted on TD-FS and TD-FT, and some studies are reviewed in the following.

#### 5.4.2. Analytical Solutions of the Equations for TD-FS

#### 5.4.3. Clinical Applications of TD-FS

#### 5.4.4. TD-FT Using Full TOF-Distributions and Effects of Featured Data Types

#### 5.4.5. TD-FT Using Early Arriving Photons

_{a}and μ

_{s}′ images inside small animals such as mice for the purpose of improvement of TD-FT images [207]. They used high-speed time-gated ICCD with a time gate less than 300 ps to obtain TR transmission data which were converted to FD data for image reconstruction of μ

_{a}and μ

_{s}′ using a non-linear iterative inversion algorithm.

^{2}ϕ(

**r**, t)/∂t

^{2}, because the TD-DE was derived by assuming ∂

^{2}ϕ(

**r**, t)/∂t

^{2}<< ∂ϕ(

**r**, t)/∂t which may breakdown for early photons.

#### 5.4.6. TD-FT Using the GPST Algorithm

_{a}, μ

_{s}′, N, and τ, and hemoglobin images were obtained from μ

_{a}images at two wavelengths of 780 nm and 830 nm [192].

#### 5.4.7. Total Light Approach in TD-FS and TD-FT

_{t}= ϕ

_{x}+ ϕ

_{m}

^{*}/γ, where ϕ

_{m}

^{*}is the fluence rate of the emission light when the fluorescence life time is zero, i.e., ϕ

_{m}(

**r**, t) = ϕ

_{m}

^{*}(

**r**, t)$\otimes $(1/τ)exp(−t/τ) with $\otimes $ denoting the convolution operator. With additional assumption of μ

_{ax}= μ

_{am}= μ

_{a}and D

_{x}= D

_{m}= D, the DE for ϕ

_{t}is given as,

_{a}and εN(

**r**) in a 2D circular medium with a single and double fluorophore targets using <t> as a featured TD-data [213]. Nishimura et al. applied the total light approach to estimate the life-time function, (1/τ)exp(−t/τ), in heterogeneous scattering media with a more general scheme using the TD-RTE [214].

#### 5.4.8. Transformation of TD-FT to FD-FT

#### 5.4.9. Application of MC Method for TD-FT

## 6. Clinical Applications of Commercially Available TD-NIRS Systems by Japanese Researchers

#### 6.1. Group from Kagawa Medical University

_{s}′, DPF, cerebral blood volume (CBV), etc., in 22 neonate heads, and found a significant correlation between the postconceptional age, μ

_{s}′, and CBV [220,221,222]. Ogawa et al. measured the changes in C

_{oxy-Hb}, C

_{deoxy-Hb}, C

_{total-Hb}, and So

_{2}of breast tissue during breastfeeding and found that all four parameters decreased during breastfeeding [223]. Koyano et al. studied the effect of blood transfusion on cerebral hemodynamics in preterm infants and found that cerebral So

_{2}decreased after transfusion while CBV increased [224]. Nakamura et al. tried to use the TRS-10 to evaluate the seriousness of asphyxiated neonates and found CBV and cerebral So

_{2}were significantly higher for neonates with adverse outcomes of hypothermic therapy [225]. Kusaka et al. reviewed the usefulness of TD-NIRS for cerebral hemodynamic treatments in neonates [226].

#### 6.2. Group from Kagoshima University Hospital

_{oxy-Hb}, C

_{deoxy-Hb}, C

_{total-Hb}and S

_{O2}) of patients during cardiopulmonary bypass surgery [227]. They found that C

_{total-Hb}correlated well with hematocrit measured by a blood gas analyzer. Then, Kakihana et al. evaluated the occurrence of postoperative cognitive dysfunction (POCD) after cardiopulmonary bypass surgery with hypothermic treatment by S

_{O2}measured by TRS-10 and internal jugular vein oxygen saturation (S

_{jvO2}) measured by a conventional method [228]. In the patients with POCD, S

_{jvO2}were found to be significantly larger than S

_{O2}. The same group also tried to monitor hepatic oxygenation by TRS-10 and found that abdominal C

_{total-Hb}were significantly higher in the liver area than other area in healthy people and that hepatic oxygenation measured by TRS-10 may work for early detection of intestinal ischemia [229]. Recently, monitoring of post-resuscitation encephalopathy by TRS-10 was experimentally tried for pigs with cardiac arrest induced by electrical stimuli for preliminary study toward clinical applications [230].

#### 6.3. Group from Nihon University School of Medicine

_{oxy-Hb}and C

_{total-Hb}increased while C

_{deoxy-Hb}decreased. On the other hand, C

_{oxy-Hb}and C

_{total-Hb}decreased while C

_{deoxy-Hb}increased during the driving simulation task. The mean optical pathlengths did not change by the tasks. Yokose et al used the TRS-10 to detect vasospasm of the patients with subarachnoid hemorrhage by measuring the cortical oxygen saturation [232]. Tanida et al. used the TRS-10 to conclude that the change in C

_{oxy-Hb}at the lateral prefrontal cortex during a working memory task correlated with working memory performance [233]. Sakatani et al. examined the effect of the extract of Ginko biloba leaves on the performance of working memory with TRS-10 probes attached on the prefrontal cortex and found that the right laterality score of C

_{oxy-Hb}increased by administration of Ginko biloba leaves [234]. Machida et al. showed that cosmetic therapy to elderly women with mild cognitive impairment was effective to improve the activities of the prefrontal cortex with increased C

_{oxy-Hb}and C

_{total-Hb}measured by the TRS-20 [235]. Tanida et al. also used TRS-20 to evaluate the difference in the women’s emotion between pleasure and displeasure induced by the difference in lipsticks and found that the difference in the lipsticks resulted in different activities in the left and right prefrontal cortex [236]. Murayama et al. studied the relationship between cognitive function and cerebral blood oxygenation measured with the TRS-20 in the prefrontal cortexes of 113 elderly people and found strong correlations between working memory function and C

_{oxy-Hb}, C

_{total-Hb}, and S

_{O2}[237].

#### 6.4. Group of Professor Hamaoka (Tokyo Medical University)

_{oxy-Hb}, C

_{deoxy-Hb}, C

_{total-Hb}, and S

_{O2}in radial digitorum extensor muscles with arterial occlusion and found reasonable agreements of S

_{O2}measured by TD-NIRS and a blood gas analyzer [238]. Later, the group used TD-NIRS to examine the activity of brown adipose tissue (BAT), which can be a counter-measure of obesity and obesity-induced metabolic disorders. Brown adipose tissue has more capillary and mitochondria than other tissues, and the densities of capillaries and mitochondria were found to be strongly correlated with μ

_{a}and μ

_{s}′, respectively, which are measurable with TD-NIRS [239]. Using TD-NIRS, Nirengi et al. showed that daily ingestion of capsinoids (thermogenic capsisin analog) for eight weeks increased the BAT density by 46%, resulting in the same results with the conventional method using

^{18}F-fluoro-deoy-glucose positron emission tomography (PET) combined with X-ray CT [240]. Fuse et al. performed a study with 423 Japanese subjects to confirm the relationship between the BAT density and C

_{total-Hb}measured with TD-NIRS [241].

#### 6.5. Other Groups in Japan

_{O2}measured by the TRS-20 showed clear decreases from 67% to 54%, while Sp

_{O2}measured by pulse oximetry did not change. The TD-NIRS systems were found useful for monitoring bleeding during caesarean section. Ueda et al. evaluated the usefulness of TD-DOS imaging of primary breast cancers using the TRS-20 [245]. Lesions with C

_{total-Hb}20% higher than normal tissue exhibited more advanced cancer stage, higher mitotic counts, and higher

^{18}Fluoro-deoxy-glucose uptake in PET, and TD-DOS imaging of C

_{total-Hb}was found useful for prediction of patient prognosis and potential response to treatment.

## 7. Summary

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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**Figure 5.**(

**a**) Light propagation from the source position,

**r**

_{s}, to the detector position,

**r**

_{d}, through an arbitrary position,

**r**. (

**b**) Light propagation from the source position,

**r**

_{s}, to the position,

**r**, and from the detector position,

**r**

_{d}, to the position,

**r**.

**Figure 6.**Distributions of the time-resolved sensitivity of the reflectance from a homogeneous semi-infinite medium with the source-detector (SD) distance, ρ, of 30 mm, μ

_{a}= 0.01 mm

^{−1}, and μ

_{s}′ = 0.5 mm

^{−1}for different delays of the time windows of 0.86 ns, 1.66 ns, and 2.46 ns from left to right. Reproduced from Reference [30].

**Figure 8.**Photograph and block diagram of the TRS-20 system. The picosecond light pulser emit lights with wavelengths of 760, 800, and 830 nm; the photo detector unit consists of a variable optical attenuator, the photomultiplier tube (PMT), and a fast amplifier; the TCSPC unit consists of a CFD/TAC, an A/D converter, a histogram memory, etc.

**Figure 9.**Photo and block diagram of the 64-channel TD-NIRS system developed by the group in Japan. Modified from Reference [45].

**Figure 11.**Photos of (

**a**) the 48-channel TD-DOT for optical mammography [52]; (

**b**) the 12-channel TD-NIRS system for optical mammography with a hand-held probe [53], and (

**c**) the transmittance TD-NIRS imaging system for optical mammography with breast compression [54]. (

**b**) Reproduced from Reference [53]. (

**c**) Reproduced from Reference [54].

**Figure 13.**Photo of the compact dual-wavelength TD-NIRS system developed by the group in Italy. Modified from Reference [56].

**Figure 14.**Geometry of a three-layered finite cylinder simulating the human head. Modified from Reference [103].

**Figure 16.**Effects of the type of featured data on the quality of the reconstructed DOT images of μ

_{a}. (

**a**) Geometry of a 2D object; reconstructed DOT images using (

**b**) E (CW), (

**c**) <t>, and (

**d**) full TOF-distributions normalized by E. The 2D object has a diameter of 80 mm and contains two small absorbers with a diameter of 10 mm and a separation of 16 mm. Modified from Reference [170].

**Figure 17.**Hemodynamic changes in the prefrontal cortex observed by TD-DOT images during a video game task. Modified from Reference [50]. Copyright (c) 2005 The Japan Society of Applied Physics.

**Figure 18.**TD-DOT images of the human forearm. (

**Left**) MRI of the forearm showing the ulna and radius. (

**Center Two**) Reconstructed μ

_{a}and μ

_{s}′ images for 759 nm at rest state. μ

_{s}′ image reflects the positions of the two bones. (

**Right**) Change in C

_{oxy-HB}from rest to task states. Red areas indicate the positions of thick blood vessels while deep-blue areas indicate muscles. Modified from Reference [47].

**Figure 19.**Schematic view of the chronological development of the TD-NIRS instruments classifying the sources, detectors, and systems, indicating incorporation of the sources and detectors into the systems by vertical arrows (abbreviation: pLD—pulsed laser diode).

**Figure 20.**Schematic view of the chronological development of TD-NIRS theories, methods, and applications, indicating their relationship with vertical arrows.

**Table 1.**Chronology of major events in instruments of TD-NIRS (Abbreviations: “Ch” for channel and “pLD” for pulsed laser diode).

Year | Event | Ref. |
---|---|---|

1988 | TD-NIRS system using a streak camera or TCSPC | [34,36] |

1999 | 2-wavelength multi-Ch TD-NIRS oximeter using pLDs and multi-anode PMTs | [39] |

Commercial 1-Ch TD-NIRS system using TCSPC for research use: “TRS-10” | [40] | |

1-wavelength 1-Ch TD optical mammography using a pLD and PMT-TCSPC | [41] | |

64-channel TD-DOT system using pLDs and PMT-TCSPCs | [45] | |

2000 | 32-channel TD-DOT system (MONSTIR) | [49] |

2003 | TD-NIRS system using the spread spectrum technique and pseudo-random bit sequences | [63] |

2005 | TD-NIRS system using time-gated ICCD | [65] |

16-channel TD-DOT system using pLDs and PMT-TCSPCs | [50] | |

2009 | Commercial 2-Ch TD-NIRS system using TCSPC for research use: “TRS-20” | [43] |

2011 | 48-Ch 3-wavelength TD-NIRS optical mammography | [52] |

2013 | TD-DOT system incorporating SPADs into TCSPC | [58] |

2014 | Commercial 2-Ch TD-NIRS system using MPPCs for medical use: “tNIRS-1” | [55] |

MONSTIR II employing an SC laser with an AOTF for 4 wavelengths | [51] | |

2016 | TD-NIRS mammography for imaging the contents of water, lipid, collagen, oxy-Hb and deoxy-Hb using 7 wavelengths | [54] |

Compact 2-wavelength TD-NIRS system and detector probe using SiPM | [56,57] | |

TD-NIRS system using an SC laser and SPADs for non-contact measurements | [60] | |

2017 | 12-Ch TD-NIRS mammography with a hand-held probe | [53] |

2018 | TD-DOT system using an SC laser and SPAD camera | [62] |

Compact TD-NIRS system for measuring the contents of water, lipid, oxy-Hb, and deoxy-Hb using 6 wavelengths | [68] | |

Compact 1-Ch TD-NIRS system using telecommunication devices | [3] |

Year | Event | Ref. |
---|---|---|

1983 | Monte Carlo method applied to photon migration | [22] |

1988 | TD measurement of optical pathlength | [34] |

TD-NIRS of hemoglobin and myoglobin in muscle | [36] | |

1989 | Analytical solutions of the TD-DE for semi-infinite and slab media | [18] |

1991~1995 | TD sensitivity functions | [26,27,28,29] |

1991, 1993 | Method of TD-DOT image reconstruction including forward and inverse models | [69,164] |

1992 | Analytical solutions of the TD-DE for various simple geometries | [19] |

Monte Carlo code for multi-layered tissue, MCML | [23] | |

1993, 1995 | TOF and absorbance imaging of biological media and neonates | [156,181] |

1994 | Mathematical model for TD-FT | [197] |

1994, 2006 | Diffusion coefficient independent of the absorption coefficient | [8,17] |

1996 | TD imaging based on the perturbation model | [87] |

1996, 1998 | Perturbation Monte Carlo simulation | [105,107] |

1997 | Light propagation in a model of the adult head | [126] |

TD-FT using early-arriving photons | [205] | |

1998 | TR reflectance from two-layered media | [98] |

Simultaneous MR and TD-NIRS mammography | [191 | |

2000, 2014 | Open source software for TD-DOT: TOAST and TOAST++ | [49,165] |

2001 | TD-DOT of human forearm | [193] |

Photon path distribution based on the microscopic Beer–Lambert law | [9] | |

2002 | GPST and full TR algorithms for TD-DOT | [170,171] |

3D TD-DOT of premature infant brain | [185] | |

2005 | Perturbation model for layered media | [91] |

TR reflectance at null space SD distance | [131] | |

Measurements of optical properties in neonates using a commercial TD-NIRS system: TRS-10 | [222] | |

2010 | (Monograph) Light propagation through biological tissue | [32] |

2014 | Hybrid TD-RTE and TD-DE | [119,120] |

2014, 2016 | Estimate of tissue composition in breasts using TR reflectance at 7 wavelengths | [54,139] |

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**MDPI and ACS Style**

Yamada, Y.; Suzuki, H.; Yamashita, Y.
Time-Domain Near-Infrared Spectroscopy and Imaging: A Review. *Appl. Sci.* **2019**, *9*, 1127.
https://doi.org/10.3390/app9061127

**AMA Style**

Yamada Y, Suzuki H, Yamashita Y.
Time-Domain Near-Infrared Spectroscopy and Imaging: A Review. *Applied Sciences*. 2019; 9(6):1127.
https://doi.org/10.3390/app9061127

**Chicago/Turabian Style**

Yamada, Yukio, Hiroaki Suzuki, and Yutaka Yamashita.
2019. "Time-Domain Near-Infrared Spectroscopy and Imaging: A Review" *Applied Sciences* 9, no. 6: 1127.
https://doi.org/10.3390/app9061127