Application of the photon average trajectories method to real-time reconstruction of tissue inhomogeneities in diffuse optical tomography of strongly scattering media

Abstract

The possibility of application of the photon average trajectories (PAT) method to real-time reconstruction of tissue inhomogeneities in diffuse optical tomography of strongly scattering media has been substantiated. By this method, the inverse problem is reduced to solution of the integral equation with integration along a conditional PAT. Such an approach allows the standard fast algebraic algorithms commonly used in projection computed tomography to be applied to diffuse optical image reconstruction. To demonstrate the capabilities of the PAT method, a numerical experiment on cross-sectional reconstruction of cylindrical strongly scattering objects with absorbing inhomogeneities has been done. Relative shadows caused by inhomogeneities are simulated via numerical solution of the non-stationary diffusion equation. To solve the inverse problem, the QR-factorization least-squares algorithm and the multiplicative algebraic reconstruction technique are used. The results are compared with those obtained by a well-known software package for temporal optical absorption and scattering tomography based on multiple solution of the diffusion equation. It is shown that the PAT method allows reconstruction of the optical structure of objects with comparable accuracy while saving reconstruction time considerably.

Methods for Postprocessing in Single-Step Diffuse Optical Tomography

“…At time moment τ = t this distribution forms a zone of the most probable trajectories of photons migrated from (r s ,0) to (r d ,t) . This zone is shaped as a banana (Lyubimov et al, 2002;Volkonskii et al, 1999) with vertices at the points of source and receiver localizations on the boundary of the scattering object. The effective width of this zone estimates the theoretical spatial resolution and is described by the standard rootmeansquare deviation of photon position from the PAT as follows (4) where R(τ ) is a radius-vector describing the PAT.…”

“…The total time is thus below 1 minute. The comparative analysis of computational speed presented in (Lyubimov et al, 2002) suggests that the use of the well-known package TOAST (Temporal Optical Absorption and Scattering Tomography, Schweiger & Arridge, 2008) which implements the Newton-Raphson algorithm will make the time of restoration several times longer. It should also be noted that there are good prospects for making the postprocessing procedure yet faster by using not MATLAB, but a faster programming environment and optimizing the measurement ratio.…”

“…the problem of radiation propagation through tissue is solved numerically many times and at each step of linearization it is necessary to adjust the matrix of the system of algebraic equations that describe the discrete reconstruction model. As shown in our earlier papers (Kalintsev et al, 2005;Konovalov et al, 2003;2006b;2007a;2007b;Lyubimov et al, 2002;, there exists a unique opportunity to make the reconstruction procedure much faster by changing over in equation (1) from volume integration to integration along an effective trajectory from source to receiver. The photon average trajectory method (PAT method) we have developed finds such a trajectory using a probabilistic interpretation of light energy transfer by photons from source to receiver.…”

“…The effective trajectory is a photon average trajectory (PAT) described by the mass center of the spatial distribution P over a time t . If we deal with absorbing inhomogeneities, then in the approximation of perturbation theory by Born or Rytov, integral (1) can be written as the fundamental equation of the PAT method (Kravtsenyuk & Lyubimov, 2000;Lyubimov et al, 2002; ( 2) where L is the PAT from the source point r s to the receiver point r d , l is distance along the PAT, and v(l) is a factor meaning the inverse relative velocity of the mass center of the distribution P along the PAT. The volume integral in the braces is the sought object function f (r) averaged over the spatial distribution of the photons that contribute to the signal recorded at time t .…”

“…For the second approach the single-step reconstruction means that the system of algebraic equations describing the discrete model is only once inverted and the matrix needs no adjustment. In our previous papers we provided some examples of 2D reconstruction from data simulated for the time-domain measurement technique to show that the PAT method can be implemented with integral algorithms 2007b;Lyubimov et al, 2003) as well as with algebraic ones (Konovalov et al, 2006b;2007a;Lyubimov et al, 2002). Compared with the multistep algorithms, the former give a terrific gain (a factor of about 100) in calculation time, but are too inaccurate in the reconstruction of optical inhomogeneities near the boundaries of the study object.…”