In this work, synthetic time-domain data are generated as if it were collected with a state-of-the-art multi-view experimental optical scanner developed in our group for small animal imaging, and used in a tomographic image reconstruction algorithm. The collected data comprises full time-dependent optical signals leaving the biological medium and acquired all around the medium. The diffuse optical tomography (DOT) algorithm relies on the time dependent parabolic simplified spherical harmonics (TD-pSPN) equations as the forward model to recover the 3D absorption and diffusion coefficient maps of the medium. The inverse problem is casted and solved as an iterative constrained optimization problem where an objective function determines the accuracy of the forward model predictions at each iteration. Time-dependent adjoint variables are introduced to accelerate the calculation of the gradient of the objective function. A three-dimensional case involving an absorption heterogeneity in a homogeneous medium is presented, reproducing practical situations encountered in our lab. The results support our hypothesis that accurate quantitative 3D maps of optical properties of biological tissues can be retrieved using intrinsic measurements obtained with our experimental scanner along with our DOT algorithm.
We investigate the problem of retrieving the optical properties (absorption and scattering) of biological tissue from a set of optical measurements. A diffuse optical tomography (DOT) algorithm that incorporates constrained optimization methods is implemented. To improve image quality, the DOT algorithm exploits full time-domain data. The time-dependent parabolic simplified spherical harmonics equations (TD-pSP N ) are used as the forward model. Time-dependent adjoint variables are resorted to in the calculation of the gradient of the objective function. Several numerical experiments for small geometric media with embedded inclusions that mimic small animal imaging are performed. In the experiments, optical coefficient values are varied in the range of realistic values for the near-infrared spectrum, including high absorption values. Single and multiparameter reconstructions are performed with the diffusion equation and higher orders of the TD-pSP N equations. The results suggest the DOT algorithm based on the TD-pSP N model outperforms the DE, and accurately reconstructs optical parameter distributions of biological media both spatially and quantitatively.
We present a diffuse optical tomography (DOT) algorithm for imaging the absorption distribution in a biological
tissue using time-domain optical measurements. The time-dependent parabolic simplified spherical polynomials
approximation of the radiative transfer equation (the TD-pSPN model) serves as the forward model. The DOT
algorithm is implemented using a nested analysis and design (NAND) method developed for solving constrained
optimization problems. Numerical experiments are provided for small geometry media to mimic small animal
imaging. In these experiments, the optical absorption coefficient value is varied within typical values found in
the near infrared range for biological tissues, including high absorption values. The results show good spatial
and quantitative reconstructions and support our TD-pSPN-based DOT algorithm as an accurate approach to
image absorption in biological media.
The equation of radiative transfer (ERT) is generally accepted as the most accurate model for light propagation in biological tissues. The ERT is notoriously expensive to solve numerically. Recently, Klose and Larsen have approximated the time-independent ERT using the simplified spherical harmonics equations ( SPN approximation). In this work, we outline how to derive the SPN approximation of the time-dependent ERT and obtain the associated integro- partial differential equations involving temporal convolution integrals. No approximation is made as regards the time variable in our derivation. To simplify the numerical solution of these equations, we introduce a "memory function". We discuss the numerical solution for N = 1 in the 2D and homogeneous case. We provide time evolution maps of the solution and compare it with the diffusion approximation of the ERT. The findings presented here straightforwardly extend to 3D inhomogeneous media and for higher values of N. These more complicated cases along with further details will be reported elsewhere.
The radiative transfer equation (RTE) is an important theoretical tool in biomedical optics for describing light propagation in tissues. The solutions to its derived diffusion equation (DE) are used, for example, for dose calculation in photodynamic therapy and for optical tomography. The RTE is valid for constant refractive index and zero ray divergence. These conditions limit its applicability in biomedical optics. To eliminate these drawbacks three new RTEs have been proposed. In this paper we test the standard RTE and the new RTEs by solving them for the irradiance of rays propagating in an infinite medium with no scattering, no absorption and no amplification. The solutions to this problem must coincide with the irradiance laws of geometrical optics. We show that only one of those equations gives solutions, which are consistent with irradiance laws of geometrical optics due to its ability to model, the effect of spatially varying refractive index and non-negligible ray divergence. Consequently that equation gives a better description of light propagation in scattering media with spatially varying refractive index and near sources, a physical situation occurring frequently in biomedical optics.
The diffusion equation (DE) is widely used in biomedical optics for describing light propagation in tissue. However, the DE yields inaccurate results near sources. This drawback is important in practical situations, when it is of primary interest to calculate the dose of light applied or to retrieve the optical properties of the tissue near the light source, e.g., the distal end of an optical fiber. To study this problem we derived a diffusion equation for constant refractive index and rays of arbitrary divergence (DErad) from a modified radiative transfer equation for spatially varying refractive index. We solve the DErad for a time-independent point source in near field and far field, which are defined by a parameter Rcrit. The far-field solution is the solution to the time-independent DE, the near-field solution agrees well with Monte Carlo simulation results and the Rcrit coincides with the reported radius of inaccuracy of the DE. These results suggest that the inaccuracy of the time-independent DE near a point source is due to a non-negligible ray divergence.
The radiative transfer equation (RTE) is the fundamental equation of the radiative transfer theory and one of more important theoretical tools in biomedical optics for describing light propagation in biological tissues. The RTE assumes that the refractive index of the medium is constant and the ray divergence is zero. These assumptions limit its range of applicability. To eliminate this drawback three new RTE have been proposed recently. Obviously, those equations must be carefully studied and compared. With that aim we solve the standard RTE and the new radiative transfer equations for the specific case of a time-independent isotropic point source in an infinite non-absorbing non-amplifying non-scattering linear medium with constant refractive index. The solution to this problem is the well-known inverse square law of geometrical optics. We show that only one of those equations gives solutions consistent with the inverse square law for the irradiance, due to its ability to model non-negligible ray divergence near a point source.
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