Analytical Solution of the Elastic Boltzmann Transport Equation in an Infinite Uniform Medium Using Cumulant Expansion†

Clean image synthesis and target numerical marching for optical imaging with backscattering light

Scanning backscattering imaging and independent component analysis (ICA) are used to probe targets hidden in the subsurface of a turbid medium. A new correction procedure is proposed and used to synthesize a "clean" image of a homogeneous host medium numerically from a set of raster-scanned "dirty" backscattering images of the medium with embedded targets. The independent intensity distributions on the surface of the medium corresponding to individual targets are then unmixed using ICA of the difference between the set of dirty images and the clean image. The target positions are localized by a novel analytical method, which marches the target to the surface of the turbid medium until a match with the retrieved independent component is accomplished. The unknown surface property of the turbid medium is automatically accounted for by this method. Employing clean image synthesis and target numerical marching, three-dimensional (3D) localization of objects embedded inside a turbid medium using independent component analysis in a backscattering geometry is demonstrated for the first time, using as an example, imaging a small piece of cancerous prostate tissue embedded in a host consisting of normal prostate tissue.

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.

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.

Optical imaging of turbid media using independent component analysis: theory and simulation

A new imaging approach for 3-D localization and characterization of objects in a turbid medium using independent component analysis (ICA) from information theory is developed and demonstrated using simulated data. This approach uses a multisource and multidetector signal acquisition scheme. ICA of the perturbations in the spatial intensity distribution measured on the medium boundary sorts out the embedded objects. The locations and optical characteristics of the embedded objects are obtained from a Green's function analysis based on any appropriate model for light propagation in the background medium. This approach is shown to locate and characterize absorptive and scattering inhomogeneities within highly scattering medium to a high degree of accuracy. In particular, we show this approach can discriminate between absorptive and scattering inhomogeneities, and can locate and characterize complex inhomogeneities, which are both absorptive and scattering. The influence of noise and uncertainty in background absorption or scattering on the performance of this approach is investigated.

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.

Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis

A new approach for optical imaging and localization of objects in turbid media that makes use of the independent component analysis (ICA) from information theory is demonstrated. Experimental arrangement realizes a multisource illumination of a turbid medium with embedded objects and a multidetector acquisition of transmitted light on the medium boundary. The resulting spatial diversity and multiple angular observations provide robust data for three-dimensional localization and characterization of absorbing and scattering inhomogeneities embedded in a turbid medium. ICA of the perturbations in the spatial intensity distribution on the medium boundary sorts out the embedded objects, and their locations are obtained from Green's function analysis based on any appropriate light propagation model. Imaging experiments were carried out on two highly scattering samples of thickness approximately 50 times the transport mean-free path of the respective medium. One turbid medium had two embedded absorptive objects, and the other had four scattering objects. An independent component separation of the signal, in conjunction with diffusive photon migration theory, was used to locate the embedded inhomogeneities. In both cases, improved lateral and axial localizations of the objects over the result obtained by use of common photon migration reconstruction algorithms were achieved. The approach is applicable to different medium geometries, can be used with any suitable photon propagation model, and is amenable to near-real-time 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.