André Liemert

Exact and efficient solution of the radiative transport equation for the semi-infinite medium

An accurate and efficient solution of the radiative transport equation is proposed for modeling the propagation of photons in the three-dimensional anisotropically scattering half-space medium. The exact refractive index mismatched boundary condition is considered and arbitrary rotationally invariant scattering functions can be applied. The obtained equations are verified with Monte Carlo simulations in the steady-state, temporal frequency, and time domains resulting in an excellent agreement.

Light transport in three-dimensional semi-infinite scattering media.

The three-dimensional radiative transfer equation is solved for modeling the light propagation in anisotropically scattering semi-infinite media such as biological tissue, considering the effect of internal reflection at the interfaces. The two-dimensional Fourier transform and the modified spherical harmonics method are applied to derive the general solution to the associated homogeneous problem in terms of analytical functions. The obtained solution is used for solving boundary-value problems, which are important for applications in the biomedical optics field. The derived equations are successfully verified by comparisons with Monte Carlo simulations.

Light diffusion in a turbid cylinder. II. Layered case

This paper is the second of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for an N-layered finite cylinder. Solutions are given in the steady-state, frequency, and time domains for a point beam incident at an arbitrary position of the first layer and for a circular flat beam incident at the middle of the cylinder top. For special cases the solutions were compared to other solutions of the diffusion equation showing excellent agreement. In addition, the derived solutions were validated by comparison with Monte Carlo simulations. In the time domain we also derived a fast solution ( approximately 10ms) for the case of equal reduced scattering coefficients and refractive indices in all layers.

Infinite space Green’s function of the time-dependent radiative transfer equation

This study contains the derivation of an infinite space Green’s function of the time-dependent radiative transfer equation in an anisotropically scattering medium based on analytical approaches. The final solutions are analytical regarding the time variable and given by a superposition of real and complex exponential functions. The obtained expressions were successfully validated with Monte Carlo simulations.

Sources of errors in spatial frequency domain imaging of scattering media

Abstract. Knowledge of the impact of potential sources of error in spatial frequency domain imaging (SFDI) is essential for the quantitative characterization of absorption and scattering in tissue and other turbid media. We theoretically investigate the error in the derived absorption and scattering parameter, subject to typical experimental and theoretical sources of errors. This provides a guideline to properly assess the significance of various parameters related to the measurement and the theoretical evaluation of spatial frequency domain reflectance data. At the same time, this study serves as a reference to estimate the overall precision of derived optical parameters of semi-infinite scattering media using SFDI.

Light diffusion in a turbid cylinder. I. Homogeneous case.

This paper is the first of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for a homogeneous finite cylinder that is illuminated at an arbitrary location. Three solutions were derived for an incident delta -light source in the steady-state, frequency, and time domains, respectively, applying different integral transformations. The performance of these solutions was compared with respect to accuracy and speed. Excellent agreement between the solutions, of which some are very fast (< 10ms), was found. Six of the nine solutions were extended to a circular flat beam which is incident onto the top side. Furthermore, the validity of the solutions was tested against Monte Carlo simulations.

Model-based analysis on the influence of spatial frequency selection in spatial frequency domain imaging

Frequency variation in spatial frequency domain imaging is a powerful tool for adjusting the penetration depth of the imaging signal and the parameter sensitivity toward absorption and diffusive and subdiffusive scattering. Through our computational analysis, using an analytical solution of the radiative transfer equation, we add quantitation to this tool by linking the different spatial frequency regimes to their relative information content and to their absolute depth sensitivity. Special focus is placed on high spatial frequencies by analysis of the phase function parameter γ and its significance and ambiguity in describing subdiffusive scattering.

Analytical solution of the radiative transfer equation for infinite-space fluence

This Brief Report presents the derivation of analytical expressions for the fluence of the steady state radiative transfer equation in an infinitely extended and anisotropically scattering medium in arbitrary dimensions for different source types. The fluence, which is composed of an infinite sum of diffusion-like Greens functions, was compared to the Monte Carlo method. Within the stochastic nature of the Monte Carlo simulations, an exact agreement was found in the steady state and time domains. It is shown that the use of low-order approximations is sufficient for many relevant cases.

Quantifying phase function influence in subdiffusively backscattered light

Abstract. Light backscattering at short source–detector separations is considerably influenced by the scattering phase function of a turbid medium. We seek to more precisely relate a medium’s subdiffusive backscattering to the angular scattering characteristics of its microstructure. First, we demonstrate the inability of the scattering asymmetry g1= to predict phase function influence on backscattering and reveal ambiguities related to the established phase function parameter γ. Through the use of high-order similarity relations, we introduce a new parameter that more accurately relates a scattering phase function to its subdiffusive backscattering intensity. Using extensive analytical forward calculations based on solutions to the radiative transfer equation in the spatial domain and spatial frequency domain, we demonstrate the superiority of our empirically derived quantifier σ over the established parameter γ.

Radiative transfer in two-dimensional infinitely extended scattering media

In this study, Green’s function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media.