Hybrid Method for Solving the Radiative Transport Equation ^†

Abstract

The spherical harmonics method ($P_N$ method) is often used for solving the radiative transport equation in terms of analytical functions. A severe and unsolved problem in this context was the evaluation of the angle-resolved radiance near sources and boundaries, which is a serious limitation of this method in view of concrete applications, e.g., in biomedical optics for investigating the different types of optical microscopy, within NIR spectroscopy, such as for the determination of ingredients in foods or in pharmaceuticals, and within physics-based rendering. In this article, we report on a hybrid method that enables accurate evaluation of the angle-resolved radiance directly at the boundary of an anisotropically scattering medium, avoiding the problems of the traditional $P_N$ methods. The derived integral equation needed for the realization of the hybrid $P_N$ method is formally valid for an arbitrary convex bounded medium. The proposed approach can be evaluated with practically the same computational effort as the traditional $P_N$ method while being far more accurate.

1. Introduction

The radiative transport equation (RTE) is the fundamental equation to describe light propagation within many applications. It is involved in different areas of science, such as astrophysics, neutron transport, climate research, heat transfer, biomedical optics, and computer graphics [1,2,3,4,5,6]. It is considered to be the gold standard for the prediction of light propagation in random media, for example biological tissue [7], because it provides a valid approximation of Maxwell’s equations in many cases [8,9], avoiding the high computational cost of solving Maxwell’s equations numerically, which, in general, restricts their application to microscopic volumes or to the microscopic scale. For quantitative investigations of the photon transport in biological media, the RTE is used for all scales, i.e., for macroscopic, mesoscopic, and microscopic applications [10], where interference effects can be neglected. For example, it is applied for non-invasive investigations of the light propagation in the human brain, in small animals, or for quantitative microscopy [11,12,13,14,15]. However, the derivation of simple analytical solutions to the RTE, e.g., in the form of closed-form expressions, is still challenging. Therefore, the RTE is mainly solved numerically by means of Monte Carlo methods [16], the finite element method [17], the boundary element method [18], the discrete ordinate method [19], or the finite volume method [20]. Additionally, the RTE is approximated by less complicated equations, such as the diffusion equation (DE) [5,21,22,23], which can be solved analytically in the steady state and time domain for several relevant geometries [5,24]. The DE is a second-order equation of the parabolic type, which typically fails in describing the correct behavior of the particle motion at early times. By contrast, the telegrapher’s equation (TE), which can be derived from the time-dependent $P_1$ equations, is a hyperbolic equation that overcomes the unrealistic feature of infinite speed propagation. However, the resulting particle velocity deviates from the correct value by a factor of $\sqrt{3}$. In general, the differences between an exact RTE solution and that predicted by low-order approximations such as the DE or the TE are too large for many situations of high practical importance [21,25,26]. In particular, the angle-resolved radiance predicted by diffusion-based models is not applicable because it consists (in view of the angular dependence) only of a cosine plus a constant. As a consequence, higher-order approximations such as the $P_N$ equations [27] (for $N>1$) must be applied. For more details concerning the theory of the classical $P_N$ method, we refer the readers to the textbooks in [1,2,3]. In recent years, a modified $P_N$ method that uses generalized spherical functions was developed for solving the system of $P_N$ equations in a more convenient way [28,29,30,31,32,33,34]. However, both the classical and modified $P_N$ methods lead to radiance in the form of an infinite spherical harmonics series. It is well known from the harmonic analysis that the convergence of a Fourier series strongly depends on the smoothness of the underlying function. In view of the transport theory, the radiance at the boundary is typically not smooth, so a large number of spherical harmonics would be necessary for an accurate representation. However, the computational effort for the determination of the corresponding expansion coefficients within $P_N$ methods increases rapidly with the approximation order N. In addition, the resulting linear equations arising from the boundary conditions become difficult to solve due to the increasing condition number. This means that small perturbations, e.g., due to unavoidable rounding errors, will lead to large deviations from the correct solution. Therefore, accurate and stable evaluation of angular quantities (especially near sources and boundaries) under the use of $P_N$ methods is still an open problem. In the past, there were several attempts to improve the numerical stability and convergence of $P_N$ methods. For example, the radiance has been decomposed into two parts, namely the unscattered component and the diffuse part, where the latter one is obtained via $P_N$ methods [32,33,34]. In addition to this separation procedure, the delta-Eddington approximation [35] was proposed to achieve further improvements, especially when highly anisotropic phase functions are taken into account. Nevertheless, the evaluation of the surface radiance without unphysical oscillations still remains impossible.

In this work, we report on an alternative approach, which we term the hybrid $P_N$ method [36]. It enables the computation of radiance more precisely compared to the known $P_N$ methods, with practically no additional computational effort. Concerning the implementation, the first step is the conversion of the RTE (given in integro-differential form) into a purely integral equation, considering the exact boundary and interface conditions at the same time. To derive the associated integral kernel, it is necessary to solve the RTE for unscattered light due to an anisotropic point source. For media bounded by planes such as a half-space, a slab, or a layered medium, the kernel function can be given in closed form. Secondly, to solve the derived integral equation, we incorporate the radiance predicted by the classical $P_N$ method. The resulting radiance obtained by the hybrid method is then given as a combination of elementary functions, such as exponentials and a spherical harmonics series, which exhibit a higher order of convergence due to the additional separation of the single-scattered contribution. The comparisons with Monte Carlo simulations and an exact analytical solution demonstrate the improvements achieved under the use of the hybrid $P_N$ method.

2. Hybrid ${\mathbf{P}}_{\mathbf{N}}$ Method

In this section, we describe the main steps for obtaining the hybrid $P_N$ method. An overview of this process is shown in Figure 1. We start with the RTE in the steady-state domain that is given by the integro-differential equation

$$\Omega ·\nabla I(\mathbf{x},\Omega )+{\mu }_t\left(\mathbf{x}\right)I(\mathbf{x},\Omega )={\mu }_s\left(\mathbf{x}\right){\int }_{{\mathbb{S}}^2}f(\mathbf{x},\Omega ,{\Omega }^{\prime })I(\mathbf{x},{\Omega }^{\prime })d{\Omega }^{\prime }+S(\mathbf{x},\Omega ),$$

(1)

with I being the radiance, $\mathbf{x}\in V\subseteq {\mathbb{R}}^3$, and the unit vector $\Omega \in {\mathbb{S}}^2$ is given by

$$\Omega =\left(\begin{array}{c}{\Omega }_1\\ {\Omega }_2\\ {\Omega }_3\end{array}\right)=\left(\begin{array}{c}sin\theta cos\varphi \\ sin\theta sin\varphi \\ cos\theta \end{array}\right)=\left(\begin{array}{c}\sqrt{1-{\mu }^2}cos\varphi \\ \sqrt{1-{\mu }^2}sin\varphi \\ \mu \end{array}\right).$$

We have S as the internal source distribution; ${\mu }_t={\mu }_a+{\mu }_s$ is the total attenuation coefficient, with ${\mu }_a$ and ${\mu }_s$ being, respectively, the absorption and scattering coefficients, and f is the scattering phase function. The outward normal vector $\widehat{\mathbf{n}}\left(\mathbf{x}\right)$, which depends on the position $\mathbf{x}\in \partial V$, is simply written as $\widehat{\mathbf{n}}$. The radiance at a point $\mathbf{y}\in \partial V$ evaluated for inward directions $\Omega ·\widehat{\mathbf{n}}<0$ must satisfy the reflecting boundary condition (BC)

$$I(\mathbf{y},\Omega )=R_f(-\Omega ·\widehat{\mathbf{n}})I(\mathbf{y},\overline{\Omega }),$$

(2)

where $\overline{\Omega }=\Omega -2(\widehat{\mathbf{n}}·\Omega )\widehat{\mathbf{n}}$ is the reflection of the vector $\Omega$ on the tangent plane $\widehat{\mathbf{n}}\left(\mathbf{y}\right)·(\mathbf{x}-\mathbf{y})=0$ through the boundary point $\mathbf{y}\in \partial V$. Furthermore, $R_f$ denotes the Fresnel reflection coefficient that is defined as

$$R_f\left(\mu \right)=\{\begin{array}{c}{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mu -n{\mu }_0}{\mu +n{\mu }_0}}\right)}^2+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\mu }_0-n\mu }{{\mu }_0+n\mu }}\right)}^2,\mu >{\mu }_c,\hfill \\ 1,\mu \le {\mu }_c,\hfill \end{array}$$

where ${\mu }_c=\sqrt{n^2-1}/n$ equals the cosine of the critical angle, ${\mu }_0=\sqrt{1-n^2(1-{\mu }^2)}$ is the cosine of the angle of refraction, and n is the relative refractive index. In view of the hybrid method, it is necessary to convert the RTE (1) into an integral equation by inverting the differential operator $\mathcal{L}I:=\Omega ·\nabla I+{\mu }_{t}I$ under consideration of (2). Thus, we write the scattering term in the form

$${\mu }_s\left(\mathbf{x}\right){\int }_{{\mathbb{S}}^2}f(\mathbf{x},\Omega ,{\Omega }^{\prime })I(\mathbf{x},{\Omega }^{\prime })d{\Omega }^{\prime }={\int }_V{\int }_{{\mathbb{S}}^2}{\mu }_s\left({\mathbf{x}}^{\prime }\right)f({\mathbf{x}}^{\prime },\Omega ,{\Omega }^{\prime })\delta (\mathbf{x}-{\mathbf{x}}^{\prime })I({\mathbf{x}}^{\prime },{\Omega }^{\prime })d{\Omega }^{\prime }d{\mathbf{x}}^{\prime }.$$

Next, we determine a function $K=K(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime })$ as solution of the transport equation

$$\mathcal{L}K={\mu }_s\left({\mathbf{x}}^{\prime }\right)f({\mathbf{x}}^{\prime },\Omega ,{\Omega }^{\prime })\delta (\mathbf{x}-{\mathbf{x}}^{\prime }),$$

(3)

where ${\mathbf{x}}^{\prime }\in V$ and ${\Omega }^{\prime }\in {\mathbb{S}}^2$, under consideration of the BC

$$K(\mathbf{y},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime })=R_f(-\Omega ·\widehat{\mathbf{n}})K(\mathbf{y},\overline{\Omega },{\mathbf{x}}^{\prime },{\Omega }^{\prime }),$$

(4)

where $\mathbf{y}\in \partial V$ and $\Omega ·\widehat{\mathbf{n}}\left(\mathbf{y}\right)<0$. The resulting function K, which becomes our integral kernel, depends on the domain V. For example, in the case of an arbitrary convex region with varying scattering and absorption coefficients, we have the following under matched conditions:

$$K(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime })={\mu }_s\left({\mathbf{x}}^{\prime }\right){\displaystyle \frac{f({\mathbf{x}}^{\prime },\Omega ,{\Omega }^{\prime })}{|\mathbf{x}-{\mathbf{x}}^{\prime }{|}^2}}exp\left(-{\int }_0^{|\mathbf{x}-{\mathbf{x}}^{\prime }|}{\mu }_t(\mathbf{x}-\ell \Omega )d\ell \right)\delta \left({\displaystyle \frac{\mathbf{x}-{\mathbf{x}}^{\prime }}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}-\Omega \right).$$

(5)

For the important case of an inhomogeneous reflecting half-space $V=\{\mathbf{x}\in {\mathbb{R}}^3|z>0\}$, where $\widehat{\mathbf{n}}=-\widehat{\mathbf{z}}$, with depth-dependent optical properties and a rotationally invariant scattering phase function of the form $f(\mathbf{x},\Omega ,{\Omega }^{\prime })=f(z,\Omega ·{\Omega }^{\prime })$, we have the kernel

$$\begin{array}{cc}\hfill & K(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime })={\mu }_s\left(z^{\prime }\right)f(z^{\prime },\Omega ·{\Omega }^{\prime }){\displaystyle \frac{exp\left(-\frac{1}{\mu }{\int }_{z^{\prime }}^z{\mu }_t(\ell )d\ell \right)}{|\mathbf{x}-{\mathbf{x}}_1{|}^2}}\delta \left({\displaystyle \frac{\mathbf{x}-{\mathbf{x}}_1}{|\mathbf{x}-{\mathbf{x}}_1|}}-\Omega \right)\hfill \\ \hfill & +{\mu }_s\left(z^{\prime }\right)f(z^{\prime },\tilde{\Omega }·{\Omega }^{\prime })R_f\left(\mu \right)\Theta \left(\mu \right){\displaystyle \frac{exp\left(-\frac{1}{\mu }{\int }_0^z{\mu }_t(\ell )d\ell -\frac{1}{\mu }{\int }_0^{z^{\prime }}{\mu }_t(\ell )d\ell \right)}{|\mathbf{x}-{\mathbf{x}}_2{|}^2}}\delta \left({\displaystyle \frac{\mathbf{x}-{\mathbf{x}}_2}{|\mathbf{x}-{\mathbf{x}}_2|}}-\Omega \right),\hfill \end{array}$$

where ${\mathbf{x}}_1={\mathbf{x}}^{\prime }$, ${\mathbf{x}}_2={(x^{\prime },y^{\prime },-z^{\prime })}^T$, $\tilde{\Omega }={({\Omega }_1,{\Omega }_2,-\mu )}^T$, and $\Theta (·)$ being the Heaviside step function. For evaluation of the backscattered radiance under the use of the hybrid $P_N$ method, see at the end of this section, we need the kernel at the boundary $z=0$ and for directions $\mu <0$. In this case, the restriction $K_s{:=K|}_{(z=0,\mu <0)}$ becomes

$$K_s={\displaystyle \frac{{\mu }_s\left(z^{\prime }\right)}{\left|\mu \right|}}f(z^{\prime },\Omega ·{\Omega }^{\prime })\delta (x-x^{\prime }+z^{\prime }tan\theta cos\varphi )\delta (y-y^{\prime }+z^{\prime }tan\theta sin\varphi )exp\left({\displaystyle \frac{1}{\mu }}{\int }_0^{z^{\prime }}{\mu }_t(\ell )d\ell \right).$$

(6)

In the field of medical physics and biophotonics, layered media are frequently used because many parts of the human body can be approximately modeled by layers, e.g., the brain, the arm, the leg, or the skin. In these cases, the optical properties are described by piecewise constant functions. Assume an inhomogeneous half-space consisting of $M\ge 1$ layers, with $s_j$$(1\le j\le M)$ denoting the thickness of the jth layer, where $s_M=\infty$. Then, the remaining integral in (6) can be taken analytically using

$$z^{\prime }\in [L_{k-1},L_k]:{\int }_0^{z^{\prime }}{\mu }_t(\ell )d\ell ={\mu }_{tk}{z}^{\prime }+\sum _{j=1}^{k-1}({\mu }_{tj}-{\mu }_{tk})s_j,$$

where $L_k:={\sum }_{j=1}^k{s}_j$, and ${\mu }_{tj}$ denotes the total attenuation coefficient in the jth layer. For applications in the spatial frequency domain, we also note, on an anisotropically scattering slab of thickness L that is characterized by the absorption ${\mu }_a$, the scattering coefficient ${\mu }_s$ and a relative refractive index n. It is illuminated by a spatially modulated light source of the form

$$S(\mathbf{x},\Omega )=exp(i\mathbf{q}·\rho )\delta \left(z\right)\delta (\Omega -{\Omega }_0),$$

with $\mathbf{q}={(q_1,q_2)}^T$ being the real-valued modulation wave vector, and ${\Omega }_0$ denotes the direction of the incident light beam with ${\mu }_0>0$. In that case, the corresponding integral kernel needed for the hybrid model is given by

$$\begin{array}{cc}K(z,\Omega ,z^{\prime },{\Omega }^{\prime })=\hfill & {\mu }_s{\displaystyle \frac{f(\Omega ·{\Omega }^{\prime })}{\left|\mu \right|}}exp\left(-\sigma {\displaystyle \frac{z-z^{\prime }}{\mu }}\right)\Theta \left({\displaystyle \frac{z-z^{\prime }}{\mu }}\right)+{\mu }_s{\displaystyle \frac{R_f\left(\right|\mu \left|\right)}{\left|\mu \right|}}{\displaystyle \frac{e^{-\sigma z/\mu }}{1-R_f^2\left(\right|\mu \left|\right)e^{-2\sigma L/\left|\mu \right|}}}\hfill \\ & \times \{\begin{array}{c}f(\tilde{\Omega }·{\Omega }^{\prime })exp\left(-{\displaystyle \frac{\sigma z^{\prime }}{\mu }}\right)+R_f\left(\mu \right)f(\Omega ·{\Omega }^{\prime })exp\left(\sigma {\displaystyle \frac{z^{\prime }-2L}{\mu }}\right),\mu >0,\hfill \\ R_f\left(\right|\mu \left|\right)f(\Omega ·{\Omega }^{\prime })exp\left(\sigma {\displaystyle \frac{z^{\prime }+2L}{\mu }}\right)+f(\tilde{\Omega }·{\Omega }^{\prime })exp\left(\sigma {\displaystyle \frac{2L-z^{\prime }}{\mu }}\right),\mu <0,\end{array}\end{array}$$

where $\sigma :={\mu }_t+iq\sqrt{1-{\mu }^2}cos\varphi$. Suppose that a specific kernel has been obtained.

Then, acting on both sides of (1) by the inverse operator ${\mathcal{L}}^{-1}$ under the use of

$$\begin{array}{cc}\hfill & {\mathcal{L}}^{-1}\left\{{\mu }_s\left({\mathbf{x}}^{\prime }\right)f({\mathbf{x}}^{\prime },\Omega ,{\Omega }^{\prime })\delta (\mathbf{x}-{\mathbf{x}}^{\prime })\right\}=K(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime }),\hfill \\ \hfill & {\mathcal{L}}^{-1}\left\{S({\mathbf{x}}^{\prime },\Omega )\delta (\mathbf{x}-{\mathbf{x}}^{\prime })\right\}=K_S(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime }),\hfill \end{array}$$

leads to the integral equation

$$I(\mathbf{x},\Omega )=\left(TI\right)(\mathbf{x},\Omega )+I_0(\mathbf{x},\Omega ),(\mathbf{x},\Omega )\in V\times {\mathbb{S}}^2,$$

(7)

with T being a linear integral operator that is defined as

$$\left(TI\right)(\mathbf{x},\Omega ):={\int }_V{\int }_{{\mathbb{S}}^2}K(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime },{\Omega }^{\prime })I({\mathbf{x}}^{\prime },{\Omega }^{\prime })d{\Omega }^{\prime }d{\mathbf{x}}^{\prime }$$

(8)

and $I_0$ denotes the unscattered radiance having the representation

$$I_0(\mathbf{x},\Omega )={\int }_V{K}_S(\mathbf{x},\Omega ,{\mathbf{x}}^{\prime })d{\mathbf{x}}^{\prime }.$$

(9)

Here, $K_S$ is a kernel with regard to the internal source distribution that satisfies

$$\mathcal{L}{K}_S=S({\mathbf{x}}^{\prime },\Omega )\delta (\mathbf{x}-{\mathbf{x}}^{\prime })$$

and the BC $K_S(\mathbf{y},\Omega ,{\mathbf{x}}^{\prime })=R_f(-\Omega ·\widehat{\mathbf{n}})K_S(\mathbf{y},\overline{\Omega },{\mathbf{x}}^{\prime })$. We note that $K_S$ is, apart from a pre-factor, the same as the integral kernel K defined in (3). A formal solution to (7) can be given in form of the Neumann series $I(\mathbf{x},\Omega )={\sum }_{n\ge 0}{I}_n(\mathbf{x},\Omega )$, where the sequence of functions ${\left(I_n\right)}_{n\in {\mathbb{N}}_0}$ are defined recursively by

$$I_{n+1}(\mathbf{x},\Omega )=\left(T{I}_n\right)(\mathbf{x},\Omega ),n\in {\mathbb{N}}_0,$$

with the unscattered radiance (9) as initial value. Suppose that I is a solution of (7); then, it also satisfies the original Equation (1) due to

$$\begin{array}{cc}\hfill & \mathcal{L}I(\mathbf{x},\Omega )=\mathcal{L}\left(TI\right)(\mathbf{x},\Omega )+\mathcal{L}{I}_0(\mathbf{x},\Omega )\hfill \\ \hfill & ={\mu }_s\left(\mathbf{x}\right){\int }_{{\mathbb{S}}^2}f(\mathbf{x},\Omega ,{\Omega }^{\prime })I(\mathbf{x},{\Omega }^{\prime })d{\Omega }^{\prime }+S(\mathbf{x},\Omega ).\hfill \end{array}$$

Further, to verify the BC (2), we consider (7) for an arbitrary $\mathbf{y}\in \partial V$ with $\Omega ·\widehat{\mathbf{n}}\left(\mathbf{y}\right)<0$, yielding

$$\begin{array}{cc}I(\mathbf{y},\Omega )\hfill & =(TI+I_0)(\mathbf{y},\Omega )=R_f(-\Omega ·\widehat{\mathbf{n}})(TI+I_0)(\mathbf{y},\overline{\Omega })=R_f(-\Omega ·\widehat{\mathbf{n}})I(\mathbf{y},\overline{\Omega }).\hfill \end{array}$$

Instead of solving the integral Equation (7) via discretizations or subspace methods, we use the radiance $L_N$ predicted by the classical $P_N$ method. In terms of a truncated series, it has the following representation

$$L_N(\mathbf{x},\Omega )\approx I_0(\mathbf{x},\Omega )+L_d(\mathbf{x},\Omega ),$$

(10)

with $L_d(\mathbf{x},\Omega )={\sum }_{\ell m}{\psi }_{\ell }^m\left(\mathbf{x}\right)Y_{\ell }^m(\Omega )$ being the diffuse part, ${\psi }_{\ell }^m$ are the corresponding moments, and the unscattered component is the same as defined in (9). Of course, the incorporated radiance $L_N$ must also satisfy the reflecting BC (2). Then, the solution predicted by the hybrid $P_N$ method is obtained after replacing the argument of the integral operator in (7) by expression (10), yielding

$$I(\mathbf{x},\Omega )=I_0(\mathbf{x},\Omega )+I_1(\mathbf{x},\Omega )+\left(T{L}_d\right)(\mathbf{x},\Omega ),$$

where we have used $\left(T{I}_0\right)=I_1$. As a byproduct, in addition to the unscattered part, we have separated the single-scattered radiance $I_1$ from the complete solution. In the case of the layered medium (piecewise constant coefficients), the backscattered surface radiance $I_s{:=I|}_{(z=0,\mu <0)}$ can be given more explicitly by making use of the kernel (6). The corresponding expression is

$$\begin{array}{cc}{I}_s(\rho ,\Omega )=\hfill & I_{0,s}(\rho ,\Omega )+I_{1,s}(\rho ,\Omega )+{\displaystyle \frac{1}{\left|\mu \right|}}{\int }_0^{\infty }{\mu }_s\left(\xi \right)exp\left({\displaystyle \frac{1}{\mu }}{\int }_0^{\xi }{\mu }_t(\ell )d\ell \right)\sum _{\ell m}{f}_{\ell }\left(\xi \right)\hfill \\ \hfill & \times Y_{\ell }^m(\Omega ){\psi }_{\ell }^m(x+\xi tan\theta cos\varphi ,y+\xi tan\theta sin\varphi ,\xi )d\xi ,\hfill \end{array}$$

where $f_{\ell }$ denotes the piecewise constant anisotropy factor with property $f_{\ell }\left(z\right)=f_{\ell ,k}$ for $z\in [L_{k-1},L_k]$. This factor positively influences the convergence of the spherical harmonics series due to $|f_{\ell }|\le 1$. In the case of the frequently used Henyey–Greenstein model, we have in the kth layer $f_{\ell ,k}=g_k^{\ell }$.

3. Numerical Experiments

In this section, numerical experiments are carried out in order to illustrate the improvements achieved under the proposed hybrid $P_N$ method. For this, solutions were compared with Monte Carlo simulations [37] and a simple exact analytical solution for the special case of a non-absorbing medium that is illuminated by a Lambertian light source [38]. The Monte Carlo method simulates the propagation of photons through the scattering medium using appropriate probability distributions and a random number generator. In the limit of an infinitely large number of photons used in the simulations, the Monte Carlo method converges to the exact RTE solution. In many applications such as in biomedical optics, usually the angular reflectance is calculated and compared to experimental data. Thus, we also take into account the angular reflectance according to the definition

$$I_r(\rho ,\Omega ):={\displaystyle \frac{1-R_f(-\mu )}{n^2}}{I}_s\left(\rho ,\pi -arcsin{\displaystyle \frac{\sqrt{1-{\mu }^2}}{n}},\varphi \right),\mu <0.$$

In Figure 2, we simulated the angle-resolved reflectance from a semi-infinite medium caused by a perpendicularly incident plane source. In order to emphasize the effect of Fresnel’s reflection, two relative refractive indices, namely $n=1.4$ and $n=2.0$, were considered. The remaining optical properties are assumed to be ${\mu }_a=0.01{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$, and $g=0.8$, where the Henyey–Greenstein phase function (HG) is used. We note that these are typical optical properties of biological tissue in the red and infrared wavelength ranges. The solution obtained by the hybrid $P_N$ method for the order $N=19$ is denoted by smooth lines, whereas the filled dots represent the data predicted by the Monte Carlo method. We observe good agreement for both refractive indices, which can be seen by the relative differences shown in the inset of Figure 2.

Next, we consider the half-space medium from the first numerical experiment with $n=1.0$, which is now illuminated by a uniform Lambertian light source instead of a collimated incident beam. The corresponding radiance at the surface is computed with the classical $P_N$ method for orders $N=29$ and $N=401$ as well as with the hybrid $P_N$ method under the use of $N=29$. Figure 3 shows that the radiance predicted by the classical $P_N$ method suffers from unphysical oscillations, including the Gibbs phenomenon, even for $N=401$. These oscillations, which are always present when using the classical $P_N$ method, cannot be removed even for $N\to \infty$. The agreement with the MC method is indicated by the inset.

In general, closed-form solutions to the RTE for bounded media are not available. An exceptional case is the non-absorbing medium when it is illuminated by a Lambertian light source. In this situation, it can by proved via inserting that the exact solution to the RTE corresponds with a constant radiance, namely $I=1{\mathrm{sr}}^{-1}$ [38]. For the third comparison, we took advantage of this exact solution in order to verify the proposed hybrid method. Figure 4 displays the radiance at the boundary of a non-absorbing half-space with properties ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$, $g=0.8$ (HG) and a relative refractive index of $n=1.4$. While the hybrid $P_N$ for $N=29$ is already very close to the exact solution, the classical $P_N$ method again suffers from strong unphysical oscillations. The relative differences of the hybrid solution to the exact analytical solution are shown in the inset of Figure 4. While the conventional $P_{29}$ approximation has errors up to $20\%$, those of the hybrid $P_{29}$ method are for all angles more than 100 times smaller. Figure 5 shows the convergence of both methods to the exact solution with the approximation order N, where the root mean square and the maximum relative errors of the angle-resolved radiance were plotted. The weakness of the classical $P_N$ method due to the Gibbs phenomenon is particularly evident in the maximum relative errors, whereas the hybrid method converges to the correct solution as expected.

In Figure 6, we considered the angle-resolved reflectance due to a perpendicular plane source backscattered from a two-layered medium with optical properties ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$, ${\mu }_{s1}^{\prime }=0.9{\mathrm{mm}}^{-1}$, ${\mu }_{a2}=0.001{\mathrm{mm}}^{-1}$, and ${\mu }_{s2}^{\prime }=1.5{\mathrm{mm}}^{-1}$. The refractive indices of the two layers are $n_1=n_2=1.4$, and that of the external medium was set to $n_0=1.0$. Furthermore, we took into account the HG phase function for both layers with anisotropy factors $g_1=g_2=0.8$. The solid line is the result obtained with the hybrid $P_N$ method for $N=19$, and the noisy curve corresponds with the data predicted by the Monte Carlo simulation. We additionally computed the relative differences and included them in the inset.

Modeling of light that is obliquely projected onto the boundary of a scattering medium is also of interest in the biomedical optics field. In Figure 7, we have considered the radiance backscattered from a semi-infinite medium due to an external incident beam coming from direction $({\theta }_0,{\varphi }_0)=(\pi /4,0)$. The radiance is evaluated for the azimuthal angle $\varphi =0$ as function of $\mu =cos\theta$. The optical properties of the half-space are assumed to be ${\mu }_a=0.01{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$, and $g=0.8$ (HG). The relative refractive index is set to $n=1.4$. We note that the relative differences show different behavior compared to the statistical errors presented above. The reason for this is that we have used in the present case a recently developed integral-equation-based variance reduction method for the Monte Carlo simulations [39].

For the last numerical experiment, we have reconsidered the semi-infinite medium from the first comparison that is now illuminated by a spatially modulated light source of the form $S(\mathbf{x},\Omega )=exp\left(iqx\right)\delta \left(z\right)\delta (\Omega -\widehat{\mathbf{z}})$ with angular modulation frequency $q=1{\mathrm{mm}}^{-1}$. This light source is usually applied in spatial frequency domain imaging, a method with which the absolute concentration of chromophores and the microstructure of biological tissue can be determined [40,41]. The resulting absolute value of the radiance at the surface is shown in Figure 8. The solid line is the result obtained with the hybrid $P_N$ method for $N=31$, and the orange curve corresponds with the data predicted by the Monte Carlo simulation. Again, we have used the recently developed Monte Carlo approach [39] for verification purposes.

In addition to the absolute value of the backscattered radiance, we also performed a comparison for the shift of the reflected light pattern relative to the incident modulated light source. Figure 9 and the corresponding inset display the comparison between the hybrid $P_N$ method (blue curve) for $N=31$ and the result obtained from the Monte Carlo simulation (orange curve).

4. Concluding Remarks

The classical $P_N$ method enables formally the exact solution of the RTE under the use of spherical harmonics. A serious problem of this method is the slow/bad convergence of the radiance in series form, leading in the case of the surface radiance to unacceptable oscillations. To overcome this shortcoming and to improve the traditional $P_N$ methods, we have proposed a hybrid model for approximating the exact solution of the associated integral equation by incorporating the radiance of the classical $P_N$ method. It is also mentionable that smooth curves (without unphysical oscillations) are important in situations when the derivative of the solution is needed. In the article, we have described solutions for the plane source problem and for spatially modulated light sources. The corresponding solutions in the space domain can be reconstructed by means of the Fourier transform [5]. It is worth mentioning that the derived integral equation is also applicable in the context of other solution approaches. This means that, in addition to using the radiance predicted by $P_N$ methods, one can alternatively incorporate radiance data generated by other deterministic or stochastic approaches, such as the Monte Carlo method [39]. Of course, the accuracy of the resulting hybrid expression depends heavily on that of the inserted approximation, although it will never be worse.