Validity conditions for the radiative transfer equation

Light transport in refractive turbid media

Light scattering in refractive media can be used for visualization of caustics and singularities of wavefronts of the incident radiation. Its modeling requires solving the radiative transfer equation. Numerical solution of the radiative transfer equation in turbid media with a spatially varying refractive index is presented and discussed in this paper. The approach is based on the self-consistent approximation of the radiative transfer equation and results in a relatively inexpensive and robust numerical algorithm. Simulations are presented for several cases including a weakly scattering medium with varying refractive index and a refractive weakly scattering medium with embedded highly scattering objects.

Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects

We present a simplified spherical harmonics approximation for the time-domain radiative transfer equation including the source-divergence effect. This leads to a set of coupled partial differential equations (PDEs) of the parabolic type that model diffuse light propagation in biological-tissue-like media. We introduce a finite element approach for solving these PDEs, thereby obtaining the time-dependent spatial profile of the fluence. We compare the results with the diffusion equation and Monte Carlo simulations. The fluence obtained via our model is shown to reproduce well the Monte Carlo results in all cases and improves on the solution of the diffusion equation in homogeneous diffusive-defying media. Our solution also shows more sensitivity than the diffusion equation to changes in the absorption coefficient of small inclusions.

Radiative transfer equations with varying refractive index: a mathematical perspective

Established mathematical techniques to model the energy density of high-frequency waves in random media by radiative transfer equations and to model the small mean-free-path limit of radiative transfer solutions by diffusion equations are reviewed. These techniques are then applied to the derivation of radiative transfer and diffusion equations for the radiance, also known as specific intensity, of electromagnetic waves in situations where the refractive index of the underlying structure varies smoothly in space.

Refractive index of carcinogen-induced rat mammary tumours

Near-infrared optical techniques for clinical breast cancer screening in humans are rapidly advancing. Based on the computational inversion of the photon diffusion process through the breast, these techniques rely on optical tissue models for accurate image reconstruction. Recent interest has surfaced regarding the effect of refractive index variations on these reconstructions. Although many data exist regarding the scattering and absorption properties of normal and diseased tissue, no measurements of refractive index appear in the literature. In this paper, we present near-infrared refractive index data acquired from N-methyl-N-nitrosourea-induced rat mammary tumours, which are similar in pathology and disease progression to human ductal carcinoma. Eight animals, including one control, were employed in this study, yielding data from 32 tumours as well as adjacent adipose and connective tissues.

Linear and nonlinear reconstruction for optical tomography of phantoms with nonscattering regions

Most research in optical imaging incorrectly assumes that light transport in nonscattering regions in the head may be modeled by use of the diffusion approximation. The effect of this assumption is examined in a series of experiments on tissue-equivalent phantoms. Images from cylindrical and head-shaped phantoms with and without clear regions [simulating the cerebrospinal fluid (CSF) filled ventricles] and a clear layer (simulating the CSF layer surrounding the brain) are reconstructed with linear and nonlinear reconstruction techniques. The results suggest that absorbing and scattering perturbations can be identified reliably with nonlinear reconstruction methods when the clear regions are also present in the reference data but that the quality of the image degrades considerably if the reference data does not contain these features. Linear reconstruction performs similarly to nonlinear reconstruction, provided the clear regions are present in the reference data, but otherwise linear reconstruction fails. This study supports the use of linear reconstruction for dynamic imaging but suggests that, in all cases, image quality is likely to improve if the clear regions are modeled correctly.

Recent advances in diffuse optical imaging

We review the current state-of-the-art of diffuse optical imaging, which is an emerging technique for functional imaging of biological tissue. It involves generating images using measurements of visible or near-infrared light scattered across large (greater than several centimetres) thicknesses of tissue. We discuss recent advances in experimental methods and instrumentation, and examine new theoretical techniques applied to modelling and image reconstruction. We review recent work on in vivo applications including imaging the breast and brain, and examine future challenges.

P(N) approximation for frequency-domain measurements in scattering media

Presented here are expressions for the P(N) approximation for light propagation in scattering media in the frequency domain. To elucidate parametric dependencies, the derivation uses normalization of the resulting expressions to either the total interaction coefficient or the reduced total interaction coefficient. For the latter case, a set of reduced phase function coefficients are introduced. Expression of the P(N) approximation as a conventional eigenvalue problem facilitates computation of the eigenvalues or attenuation coefficients. This approach is used to determine the attenuation coefficients in the asymptotic regime over the full values of the scattering albedo and reduced scattering albedo (0 to 1) and all positive values of the asymmetry factor (0 to 1). Frequency-domain measurements yield a sensitivity to turbid media optical properties for reduced scattering albedos as small as 0.2. P(N) calculations are used to assess the magnitude of errors associated with the P1 and P3 approximations over a range of scattering albedo, phase function, and modulation frequency.

Radiative transfer in a turbid medium with a varying refractive index: comment

In the literature one can encounter at least two different radiative transfer equations for media with spatially varying refractive indices. These are the results of Ferwerda [J. Opt. A Pure Appl. Opt. 1, L1 (1999)] and Tualle and Tinet [Opt. Commun. 228, 33 (2003)]. Accordingly, two different diffusion approximations are derived from these two radiative transfer equations. I reconsider the derivation of the radiative transfer equation in a medium with an inhomogeneous refractive index and confirm the result of Tualle and Tinet. In the diffusion approximation, a simple analytical solution has been found for the steady-state illumination of a non-absorbing turbid medium with a varying refractive index.