Most of the optical-physiological diagnoses are based on the illumination of light, with known parameters, onto a tested tissue, followed by the measurement of the reflected or transmitted light. Changes in the optical properties of this light such as its spectrum, polarization and intensity, compared to the injected light, result due to the interactions of the irradiated light with the tissue’s components [1], and hence is used for diagnostic purposes. Due to the important information which can be inferred from the changes in the re-emitted or transmitted light, photon migration in the irradiated tissue was, and still is, intensively investigated [2], [3], [4].

The basis for the development of optical-based diagnosis methods lies in the theory of light transfer in biological tissues. The optical regime is highly desired for biomedical applications as it is a nonionizing radiation, abundantly available, and inexpensive. Several approaches have been developed in order to best describe the photon transfer in biological tissues. The main equation used for the description of the photon movement is the radiative transport equation (RTE) [5], [6]:

$$\frac{1}{c}\frac{\partial {L}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}}{\partial t}=-\widehat{k}\nabla {L}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}-{\mu}_{t}{L}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}+{\mu}_{s}\underset{0}{\overset{4\pi}{{\displaystyle \int}}}{L}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}{P}_{\mathrm{(}{\widehat{k}}^{\prime}*\widehat{k}\mathrm{)}}d\Omega +{S}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}$$(1)

where

$${\mu}_{t}={\mu}_{a}+{\mu}_{s}$$(2)

*c* is the speed of light, *L* is the radiance, *P* is the phase function, representing the probability of light to be scattered toward the *k* direction, ${S}_{\mathrm{(}r,\widehat{k},t\mathrm{)}}$ is the source function [7], and *μ*_{a} and *μ*_{s} are the absorption and scattering coefficients of the tissue, respectively.

Equation (1) has two main different solving approaches, the numerical and the analytical approaches. In the numerical field, the Monte Carlo (MC) simulation is the most popular and provides accurate results that best correlate between the optical properties of the tissue and the reflected or transmitted photons [7]. This method simulates random trajectories for the photons within the tissue, in which each photon can be randomly absorbed or scattered in different directions, and determined according to the optical properties of the tissue:

$${r}_{n+1}={r}_{n}+s{\widehat{k}}_{n}$$(3)

$$s=-\frac{ln\mathrm{(}\u03f5\mathrm{)}}{{\mu}_{t}}$$(4)

*r*_{n} is the photon location in the tissue in the *n*th step, *s* is the step length of the photon, *ε* is a computational random number, and ${\widehat{k}}_{n}$ is the vector location of the photon in step *n*:

$$\begin{array}{c}{k}_{x}=\text{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}cos}\text{\hspace{0.17em}}\phi \\ \text{\hspace{0.17em}}{k}_{y}=\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\phi \\ \hspace{0.17em}{k}_{z}=\mathrm{cos}\text{\hspace{0.17em}}\theta \end{array}$$(5)

where *θ* and *φ* are randomly determined, based on the scattering properties of the tissue.

The MC simulations usually result in the collection of the diffuse transmitted and reflected light (*T*_{d} and *R*_{d}, respectively). In this article, we focus on the simulations and measurements of the light reflected from the tissue, measured at the tissue’s surface. *R*_{d} can be schematically described as shown in Figure 1, which illustrates the symmetry in the diffusion measurements, resulting in the same intensity values and shape for a given radius *ρ* (light source-detector separation). Therefore, it is common to describe the diffusive reflectance by circles around the light injecting point (Figure 1B). In the MC simulation, the collected *R*_{d}(*ρ*) is normalized by these circles area [7].

Figure 1: A schematic description of the diffusion reflection measurements.

(A) A pencil beam illumination on a semi-infinite lattice with an absorption coefficient *μ*_{a} and a scattering coefficient *μ*_{s}. The diffusive reflectance is collected at a specific distance *ρ* from the light source. (B) The symmetry in the reflectance from the tissue’s surface in a given *ρ* is illustrated as circles around light irradiation point, with given radii *ρ*.

The MC method was, and still is, successfully applied for biomedical imaging purposes [8], [9], [10], [11], such as for brain activity [12], [13], cardiovascular investigation [14], [15], X-ray imaging [16], oxygen saturation measurements [17], etc. [18], [19]. In the DR-GNRs method, the MC simulation was used to adjust between the tissue’s optical properties and the DR profile, as will be further described in this review.

The main approach studied for solving the RTE is the analytical, diffusion approach [20], [21]. By this approach, the spherical harmonics [22], [23] describe the radiance propagation in the tissue according to the following expressions:

$${Y}_{n,m}\mathrm{(}\theta ,\text{\hspace{0.17em}}\phi \mathrm{)}={\mathrm{(}-1\mathrm{)}}^{m}\sqrt{\frac{\mathrm{(}2n+1\mathrm{)}\mathrm{(}n-\left|m\right|\mathrm{)}!}{4\pi \mathrm{(}n+\left|m\right|\mathrm{)}!}}\text{exp(}jm\phi \text{)}{P}_{n,m}\mathrm{(}\mathrm{cos}\theta \mathrm{)}$$(6)

where *P*_{n,m} (cos*θ*) is the Legendre polynomial, given by:

$${P}_{n,m}\mathrm{(}\mathrm{cos}\theta \mathrm{)}=\frac{{\mathrm{(}1-co{s}^{2}\hspace{0.17em}\theta \mathrm{)}}^{\left|m\right|/2}}{{2}^{n}n!}\frac{{d}^{n+\left|m\right|}}{d{\mathrm{(}\mathrm{cos}\theta \mathrm{)}}^{n+\left|m\right|}}{\mathrm{(}co{s}^{2}\hspace{0.17em}\theta -1\mathrm{)}}^{n}$$(7)

The spherical harmonics enables to represent a radiance, which does not depend on the spatial angles *φ* and *θ*, by the arithmetic progression:

$$L\mathrm{(}r,\text{\hspace{0.17em}}\widehat{k},\text{\hspace{0.17em}}t\mathrm{)}\approx {\displaystyle \sum}_{n=0}^{N}{\displaystyle \sum}_{m=n}^{n}{L}_{n,m}\mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}{Y}_{n,m}\mathrm{(}\widehat{k}\mathrm{)}$$(8)

where *L* is the radiance penetrating the tissue.

Using this solution, one arrives at the RTE which depends only on the radial distance *r* and the time *t*:

$$\frac{1}{c}\frac{\partial}{\partial t}\varnothing \mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}-D{\nabla}^{2}\varnothing \mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}+{\mu}_{a}\varnothing \mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}=S\mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}$$(9)

where:

$$\varnothing \mathrm{(}r,\text{\hspace{0.17em}}t\mathrm{)}=\underset{0}{\overset{2\pi}{{\displaystyle \int}}}L\mathrm{(}r,\text{\hspace{0.17em}}\widehat{k},\text{\hspace{0.17em}}t\mathrm{)}\text{d}\widehat{k}$$(10)

$$D=\frac{1}{3{{\mu}^{\prime}}_{t}}=\frac{1}{3\mathrm{(}{\mu}_{a}+{\mu}_{s}\mathrm{(}1-g\mathrm{)}\mathrm{)}}$$(11)

Equation (9) is called the optical diffusion equation [24]. It has several optional solutions, depending on the given boundary conditions. In this review, we will focus on the DR-GNRs method developed in our lab, presenting boundary conditions of semi-infinite turbid medium and a continuous laser source (resulting with a time independent equation) placed on a single point on the tissue surface. Thus, the result for ∅(*r*) under these conditions is (using a point spread function or the Green’s function [25]):

$$\varnothing \mathrm{(}r\mathrm{)}=P\frac{\text{exp}\mathrm{(}-\rho /c\mathrm{)}}{4\pi D\rho}$$(12)

According to Fick’s first law, the DR from the semi-infinite scattering medium is, approximately, the current density reflected onto the surface normal, given by:

$${R}_{d}\mathrm{(}\rho \mathrm{)}=D\frac{\partial \varnothing}{\partial z}{|}_{z=0}$$(13)

Thus, by inserting Eq. (13) into Eq. (12), we get:

$$\begin{array}{l}{R}_{d}\mathrm{(}\rho \mathrm{)}={a}^{\prime}\frac{{z}^{\prime}\mathrm{(}1+{\mu}_{\text{eff}}{\rho}_{1}\mathrm{)}\mathrm{exp}\mathrm{(}-{\mu}_{\text{eff}}{\rho}_{1}\mathrm{)}}{4\pi {\rho}_{1}{}^{3}}\\ \text{\hspace{1em}}+\text{\hspace{0.17em}}{\alpha}^{\prime}\frac{\mathrm{(}{z}^{\prime}+4D\mathrm{)}\mathrm{(}1+{\mu}_{\text{eff}}{\rho}_{2}\mathrm{)}\mathrm{exp}\mathrm{(}-{\mu}_{\text{eff}}{\rho}_{2}\mathrm{)}}{4\pi {\rho}_{2}{}^{3}}\hspace{0.17em}\end{array}$$(14)

Here, *α*′ is the transport albedo, *z*′ is equal to ${l}_{t}\prime =1/\mathrm{(}{\mu}_{a}+{{\mu}^{\prime}}_{s}\mathrm{)},$ and *ρ*_{1} and *ρ*_{2} are the distances between the observation point (r, 0, 0) and the light source injection point (0, 0, *l**t*_{′}) and between the observation point and the image source point (0, 0, −*l**t*_{′}+4D), respectively (see in Ref. [7]). μ*s*′ is the reduced scattering coefficient, as described by:

$$\hspace{0.17em}{{\mu}^{\prime}}_{s}={\mu}_{s}\mathrm{(}1-g\mathrm{)}$$(15)

Patterson et al. [26] have presented a slightly different result for *R*_{d}, under the condition of ${\mu}_{a}\ll {\mu}_{s\prime}$ and an image source point at ${z}_{0}=1/{\mu}_{s\prime}$:

$${R}_{d}\mathrm{(}\rho \mathrm{)}=\frac{1}{2\pi}\text{exp}\mathrm{(}\frac{-{\mu}_{\text{eff}}\sqrt{{\rho}^{2}+\frac{1}{{\mu}_{s\prime}{}^{2}}}}{\frac{1}{{\mu}_{s\prime}{}^{2}}+{\rho}^{2}}\mathrm{)}\mathrm{(}{\mu}_{\text{eff}}+\frac{1}{\sqrt{{\rho}^{2}+\frac{1}{{\mu}_{s\prime}{}^{2}}}}\mathrm{)}$$(16)

The equation given suggests that *R*_{d}(*ρ*) depends on *ρ*^{2}, rather than on *ρ*. Under restricted ranges of *ρ*, Farrell et al. have simplified the *R*_{d}(*ρ*) expression from Eq. (16) into a simple term [27], which enables the simple extraction of the tissue’s optical properties from the DR profile:

$${R}_{d}\mathrm{(}\rho \mathrm{)}=\frac{{C}_{1}}{{\mathrm{(}\rho \mathrm{)}}^{m}}\text{exp(}-\mu \rho \text{)}$$(17)

This simplified equation highly depends on the distance from the light source, as well as on the optical aperture of the setup. *C*_{1} is a constant, depending on the optical properties of the medium and the sizes of the source and detector apertures. *μ* is the effective attenuation coefficient given by:

$$\mu =\sqrt{3\text{\hspace{0.17em}}{\mu}_{a}{{\mu}^{\prime}}_{s}}$$(18)

as long as μ_{a}≪μ_{s}′ [25], [27]. *m* is the power of *ρ*, which depends on *ρ*’s range and on the scattering and absorption properties of the tissue [27].

For the analyses of the DR-GNRs measurements presented in this article, Farrell’s simple equation was found to highly fit the experimental results [28], [29], as discussed hereinafter.

## 2.1 Calculating the tissue’s optical properties

As mentioned earlier, the DR model enables to calculate the absorption coefficient (*μ*_{a}) and the reduced scattering coefficient (*μ*_{s}′) from *R*_{d}(*ρ*) of the measured tissue, as long as the power *m* in the diffusion equation is known. Therefore, determining the power *m* was the first required step in order to establish a quantitative approach for our DR measurements. The DR computerized MC simulations and experimental data analyses were performed, as described in our article written in 2012 [29], in order to find the model that gives the best agreement between *R*_{d}(*ρ*) and the measured optical properties. Results indicated that *m=*2, suggesting that the best correlation between the DR profile and the tissue’s optical properties is achieved when the DR profile is presented in the logarithmic form ln(*ρ*^{2}*R*(*ρ*)). This result is consistent with Eq. (17) for *R*_{d}(*ρ*), which was developed for turbid medium presenting *μ*_{a}≪*μ*_{s}′, as simulated and measured using our DR experiments [30]. In addition, in order to provide a mathematical base for our experiments, we have experimentally tested whether measurements were performed in the diffusive regime. The DR measurements presented in this article were performed in relatively small values of *ρ* (1<*ρ*<6 mm), and also a low value of scattering coefficient (~1.6 mm^{−1}). Thus, the diffusive regime, which is determined by the mean free path: ${l}_{t\prime}=\frac{1}{{\mu}_{{s}^{\prime}+}{\mu}_{a}},$ was ~0.6 mm, resulting in low optical path lengths and small light source-detector separations [7].

In the case of *m*=2, the reflectance profile is highly sensitive to the optical properties of the tissue and, as a result, better distinguishes between absorption coefficients that only slightly differ from each other. By inserting *m*=2 into Eq. (17), it can be rewritten as:

$$\mathrm{ln}\mathrm{(}{\rho}^{2}R\mathrm{(}\rho \mathrm{)}\mathrm{)}={C}_{2}-\mu \rho $$(19)

Equation (19) presents a simple, linear correlation between ln(*ρ*^{2}*R*(*ρ*)) and *μ*. The square slope of the linear curve depends on the product of the absorption and the reduced scattering coefficients of the tissue. This equation was the basis for the analyses of the DR measurements presented in this article.

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