Cumulant solution of the elastic boltzmann transport equation in an infinite uniform medium

Extended angular-spectrum modeling (EASM) of light energy transport in scattering media

The exact modeling of light transport in scattering media is critical in biological imaging, free-space communication, and phosphor-converted lighting. Angular spectrum is proved to be a fast and effective approach to reconstructing the wavefront dynamics during the propagation in scattering media, however, finding it difficult in acquiring the wavefront and energy change simultaneously. Besides, conventional methods for energy tracing, such as the Monte Carlo method, are inefficient in speed and hard to simulate the wavefront change. Here, we propose an extended angular-spectrum modeling (EASM) approach using tenuous scattering approximate solutions to obtain a time-efficient and accurate method for reconstruction of energy and wavefront dynamics in various scattering media. The generality of our method is numerically simulated and experimentally verified with a set of scattering media with different properties. EASM has a time advantage under the guarantee of calculation accuracy, especially when calculating several thickness changes after the calculation model is established. Furthermore, multi-layered media can also be simulated by EASM with a good precision. The results suggest that EASM performs certain computations more efficiently than the conventional method and thus provides an effective and flexible calculation tool for scattering media.

Merging Mie solutions and the radiative transport equation to measure optical properties of scattering particles in optical phantoms

We present a new method to calculate the complex refractive index of spherical scatterers in a novel optical phantom developed by using homemade monodisperse silica nanospheres embedded into a polyester resin matrix and an ethanol-water mixture for applications in diffuse imaging. The spherical geometry of these nanoparticles makes them suitable for direct comparison between the values of the absorption and reduced scattering coefficients (μ_a and μs', respectively) obtained by the diffusion approximation solution to the transport equation from scattering measurements and those obtained by the Mie solution to Maxwell's equations. The values of the optical properties can be obtained by measuring, using an ultrafast detector, the time-resolved intensity distribution profiles of diffuse light transmitted through a thick slab of the silica nanosphere phantom, and by fitting them to the time-dependent diffusion approximation solution to the transport equation. These values can also be obtained by Mie solutions for spherical particles when their physical properties and size are known. By using scanning electron microscopy, we measured the size of these nanospheres, and the numerical results of μ_a and μs' can then be inferred by calculating the absorption and scattering efficiencies. Then we propose a numerical interval for the imaginary part of the complex refractive index of SiO₂ nanospheres, n_s, which is estimated by fixing the fitted values of μ_a and μs', using the known value of the real part of n_s, and finding the corresponding value of Im(n_s) that matches the optical parameters obtained by both methods finding values close to those reported for silica glass. This opens the possibility of producing optical phantoms with scattering and absorption properties that can be predicted and designed from precise knowledge of the physical characteristics of their constituents from a microscopic point of view.

Comparison of spatially and temporally resolved diffuse transillumination measurement systems for extraction of optical properties of scattering media

This paper discusses the main differences between two different methods for determining the optical properties of tissue optical phantoms by fitting the spatial and temporal intensity distribution functions to the diffusion approximation theory. The consistency in the values of the optical properties is verified by changing the width of the recipient containing the turbid medium; as the optical properties are an intrinsic value of the scattering medium, independently of the recipient width, the stability in these values for different widths implies a better measurement system for the acquisition of the optical properties. It is shown that the temporal fitting method presents higher stability than the spatial fitting method; this is probably due to the addition of the time of flight parameter into the diffusion theory.

Numerical investigation of polarization filtering for direct optical imaging within scattering media

We investigate the ability of polarization filtering to improve direct imaging of absorbing objects which are buried within scattering environments. We extend on previous empirical investigations by exploiting an efficient perturbation-based formalism, which is applicable to arbitrarily arranged sources and detectors with arbitrary polarizations. From this approach, we are able in some cases to find certain non-trivial linear combinations of polarization measurement channels that maximize the object resolution and visibility.

Stochastic model for the differential Mueller matrix of stationary and nonstationary turbid media

We show the existence of different regimes in spatial evolution of depolarization in turbid media characterized by a diagonal Mueller matrix (pure depolarizer). Experimental results previously published already established the existence of a first regime, where the depolarization follows a parabolic law with the thickness of stationary medium traveled by light. New experiments first confirm the existence of a second regime, which we have previously demonstrated, where the depolarization follows a linear law on a large scale. They also confirm the existence of much more complex evolution laws even under small-scale approximation. A stochastic approach is proposed to model the phenomenon. It perfectly describes all these different experimental results and allows us to analyze the behavior of the polarization in the case of solid or liquid scattering media. The influence of the measurement setup is also analyzed.

Green's function of the time-dependent radiative transport equation in terms of rotated spherical harmonics

The time-dependent radiative transport equation is solved for the three-dimensional spatially uniform infinite medium which is illuminated by a point unidirectional source using a spherical harmonics transform under rotation. Apart from the numerical evaluation of a spherical Hankel transform which connects the spatial distance with the radial distance in Fourier space, the dependence on all variables is found analytically. For the special case of a harmonically modulated source, even the spherical Hankel transform can be carried out analytically. Additionally, a special solution for the isotropically scattering infinite medium is given. The Monte Carlo method is used for a successful verification of the derived solution.

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.

Computational Laser Spectroscopy in a Biological Tissue

We present a numerical spectroscopic study of visible and infrared laser radiation in a biological tissue. We derive a solution of a general two-dimensional time dependent radiative transfer equation in a tissue-like medium. The used model is suitable for many situations especially when the external source is time-dependent or continuous. We use a control volume-discrete ordinate method associated with an implicit three-level second-order time differencing scheme. We consider a very thin rectangular biological-tissue-like medium submitted to a visible or a near infrared light sources. The RTE is solved for a set of different wavelength source. All sources are assumed to be monochromatic and collimated. The energetic fluence rate is computed at a set of detector points on the boundaries. According to the source type, we investigate either the steady-state or transient response of the medium. The used model is validated in the case of a heterogeneous tissue-like medium using referencing experimental results from the literature. Also, the developed model is used to study changes on transmitted light in a rat-liver tissue-like medium. Optical properties depend on the source wavelength and they are taken from the literature. In particular, light-transmission in the medium is studied for continuous wave and for short pulse.

Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo

Imaging of targeted fluorescent probes offers significant advantages for investigating disease and tissue function in animal models in vivo. Conversely, macroscopic tomographic imaging is challenging because of the high scatter of light in biological tissue and the ill-posed nature of the reconstruction mathematics. In this work, we use the earliest-transmitted photons through Lewis Lung Carcinoma bearing mice, thereby dramatically reducing the effect of tissue scattering. By using a fluorescent probe sensitive to cysteine proteases, the method yielded outstanding imaging performance compared with conventional approaches. Accurate visualization of biochemical abnormalities was achieved, not only in the primary tumor, but also in the surrounding tissue related to cancer progression and inflammatory response at the organ level. These findings were confirmed histologically and with ex vivo fluorescence microscopy. The imaging fidelity demonstrated underscores a method that can use a wide range of fluorescent probes to accurately visualize cellular- and molecular-level events in whole animals in vivo.

Inversion with early photons

Optical tomography using early photons can improve resolution and reduce the ill-posed nature of the inversion problem. In this work we use 360 degrees projection experimental data to investigate the inversion performance of three commonly used numerical inversion methods: the random algebraic reconstruction technique (rART), singular value decomposition (SVD), and the conjugate-gradient-type method LSQR. Results are contrasted to each other and the effects of different photon propagation models are also investigated. We find that all methods perform adequately given appropriate regularization parameters, and that an experimentally measured photon weight function yields superior results over two approximate weights that have been previously used.

Analytical cumulant solution of the vector radiative transfer equation investigates backscattering of circularly polarized light from turbid media

The backscattering of circularly polarized light pulses from an infinite uniform scattering medium is studied as a function of helicity of the incident light and size of scatterers in the medium. The approach considers a polarized short pulse of light incident on the scattering medium, and uses an analytical cumulant solution of the vector radiative transfer equation with the phase matrix obtained from the Mie theory to calculate the temporal profile of scattered polarized photons for any position and any angle of detection. The general expression for the scattered photon distribution function is an expansion in spatial cumulants up to an arbitrary high order. Truncating the expansion at the second-order cumulant, a Gaussian analytical approximate expression for the temporal profile of scattered polarized photons is obtained, whose average center position and half width are always exact. The components of scattered light copolarized and cross polarized with that of the incident light can be calculated and used for determining the degree of polarization of the scattered light. The results show that circularly polarized light of the same helicity dominates the backscattered signal when scatterer size is larger than the wavelength of light. For the scatterers smaller than the wavelength, the light of opposite helicity makes the dominant contribution to the backscattered signal. The theoretical estimates are in good agreement with our experimental results.

Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation

An analytical expression of the particle distribution based on an analytical cumulant solution of the time-dependent elastic Boltzmann transport equation (BTE) is presented. This expression improves upon the previous second order cumulant solution of the BTE described by a Gaussian distribution in two aspects: (1) separating the ballistic component from the scattered component to ensure that the summation in expressions is convergent; and (2) enforcing the causality condition to ensure that no particle travels faster than the free speed of the particles. Time-resolved profiles obtained using the analytical form are compared with those obtained by the Monte Carlo simulation, for both transmission and backscattering. The calculating time using our analytical form is much faster than that using the Monte Carlo approach.

First photon detection in time-resolved transillumination imaging: a theoretical evaluation

First photon detection, as a special case of time-resolved transillumination imaging, is studied through the derivation of the temporal probability density function (pdf) for the first arriving photon. The pdf for different laser intensities, media and second and later arriving photons were generated. The arrival time of the first detected photon reduced as the laser power increased and also when the scattering and absorption coefficients decreased. The pdf for an imbedded totally absorbing 3 mm inhomogeneity may be distinguished from the pdf of a homogeneous turbid medium similar to that of human breast in dimensions and optical properties.

Multiple passages of light through an absorption inhomogeneity in optical imaging of turbid media

Multiple passages of light through an absorption inhomogeneity of finite size deep within a turbid medium are analyzed for optical imaging by use of the self-energy diagram. The nonlinear correction becomes more important for an inhomogeneity of a larger size and with greater contrast in absorption with respect to the host background. The nonlinear correction factor agrees well with that from Monte Carlo simulations for cw light. The correction is approximately 50%-75% in the near infrared for an absorption inhomogeneity with the typical optical properties found in tissues and five times the size of the transport mean free path.

Photon transport parameters of diffusive media with highly anisotropic scattering

The dependence of the photon transport parameters on the optical characteristics of diffusive media such as biological tissue with strongly forward biased scattering is examined with respect to the influence of the large angle scattering component and higher moments of the phase function. The latter are particularly significant for the temporal evolution of the angular intensity. The P3 approximation gives clear physical insight into the influence of boundaries on the radiative flux and is applied here as an analytic method of evaluating certain phase functions reported in the literature, while higher order P(N) approximations are used to calculate accurate time-dependent angular intensity distributions of the scattered light.

Quantitative fluorescence lifetime spectroscopy in turbid media: comparison of theoretical, experimental and computational methods

A Monte Carlo model developed to simulate time-resolved fluorescence propagation in a semi-infinite turbid medium was validated against previously reported theoretical and computational results. Model simulations were compared to experimental measurements of fluorescence spectra and lifetimes on tissue-simulating phantoms for single and dual fibre-optic probe geometries. Experiments and simulations using a single probe revealed that scattering-induced artefacts appeared in fluorescence emission spectra, while fluorescence lifetimes were unchanged. Although fluorescence lifetime measurements are generally more robust to scattering artefacts than are measurements of fluorescence spectra, in the dual-probe geometry scattering-induced changes in apparent lifetime were predicted both from diffusion theory and via Monte Carlo simulation, as well as measured experimentally. In all cases, the recovered apparent lifetime increased with increasing scattering and increasing source-detector separation. Diffusion theory consistently underestimated the magnitude of these increases in apparent lifetime (predicting a maximum increase of approximately 15%), while Monte Carlo simulations and experiment were closely matched (showing increases as large as 30%). These results indicate that quantitative simulations of time-resolved fluorescence propagation in turbid media will be important for accurate recovery of fluorophore lifetimes in biological spectroscopy and imaging applications.

Photon migration in turbid media using a cumulant approximation to radiative transfer

A photon transport model for light migration in turbid media based on a cumulant approximation to radiative transfer is presented for image reconstruction inside an infinite medium or a bounded medium with a planar geometry. This model treats weak inhomogeneities through a Born approximation of the Boltzmann radiative transfer equation and uses the second-order cumulant solution of photon density to the Boltzmann equation as the Green's function for the uniform background. It provides the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation. At early times, it agrees much better with the result from the Monte Carlo simulation than the diffusion approximation. Both approximations agree well with the Monte Carlo simulation at later times. The weight function for image reconstruction under this proposed model is shown to have a strong dependence at both early and later times on absorption and/or scattering inhomogeneities located in the propagation direction of and close to the source, or in the field of view of and close to the detector. This effect originates from the initial ballistic motion of incident photons, which is substantially underestimated by the diffusion approximation.

Diffusion coefficient depends on time, not on absorption

The recent controversy over whether the photon diffusion coefficient depends on absorption is addressed by use of the analytical solution of the photon transport equation in an infinite homogeneous scattering medium. The diffusion coefficient is found to be independent of absorption but temporally dependent. After a long period of time, the photon diffusion coefficient approaches D=1/3mu(s)(?) , which supports a claim made by Furutsu and Yamada [Phys. Rev. E 50, 3634 (1994)]. At early times, the diffusion coefficient is smaller than D=1/3mu(s)(?) , but this reduction cannot be expressed as D=1/3(mu(s)(?)+mu(a)) , since the time-dependent diffusion coefficient is found to be unrelated to absorption.

Photon-transport forward model for imaging in turbid media

A photon-transport forward model for image reconstruction in turbid media is derived that treats weak inhomogeneities through a Born approximation of the Boltzmann radiative transfer equation. This model can conveniently replace the commonly used diffusion approximation in optical tomography. An analytical expression of the background Green's function is obtained from the cumulant solution of the Boltzmann equation. Our model provides the correct behavior of photon migration at early times and reduces at long times to the center-moved diffusion approximation. Numerical comparisons between this model and the standard and center-moved diffusion models are presented.

Analytical solution of the polarized photon transport equation in an infinite uniform medium using cumulant expansion

An analytical solution for time-dependent polarized photon transport equation in an infinite uniform isotropic medium is studied using a circular representation of the polarized light and expansion in the generalized spherical functions. We extend our cumulant approach for solving the scalar (unpolarized) photon transport equation to the vector (polarized) case. As before, an exact angular distribution is obtained and a cumulant expansion is derived for the polarized photon distribution function. By a cutoff at the second cumulant order, a Gaussian analytical approximate expression of the polarized photon spatial distribution is obtained as a function of the direction of light and time, whose average center position and half-width are always exact. The central limit theorem claims that this spatial distribution approaches accuracy in detail when the number of collisions or time becomes large. The analytical expression of cumulants up to an arbitrary high order is also derived, which can be used for calculating a more accurate polarized photon distribution through a numerical Fourier transform. Contrary to what occurs in other approximation techniques, truncation of the cumulant expansion at order n is exact at that order and cumulants up to and including order n remain unchanged when higher orders are added, at least as applied in our photon transport equation.