K-space approach to biomedical imaging with diffusive photon density waves

In recent years, a variety of techniques for imaging turbid media such as tissue with diffuse light have been explored.' Most of these methods employ direct matrix inversion (e.g., singular value decomposition) or iterative techniques (e.g., ART, SIRT and conjugate gradient) for image reconstruction, and they are in general computationally intensive. In this paper, we introduce a new imaging methodology which is computationally much faster. Our approach is essentially a near-field wave technique* that relies on a series of twodimensional fast Fourier transforms (FFTs). While the method (including the angular spectrum representation) has attracted the attention of a few researchers in the photon migration field,3 in this contribution we present a rigorous account of the theory, and the first experimental images of absorbing and scatteringobjects in turbid media using the approach. In addition to producing information about the position and shape of the hidden object(,), we have found that under some circumstances it is possible to utilize projection images to deduce the optical properties within a thin slice without the need for a complex reconstruction procedure such as matrix inversion. It should be possible to obtain clinical projection images in real time with this fast FFT approach. Light propagation in highly scattering turbid media can be well described as diffuse photon density wave (DPDW).4 In heterogeneous turbid media, the photon density wave is scattered by the heterogeneity and the total DPDW in this case is a superposition of incident wave (corresponding to the DPDW in homogeneous media) and scattered wave. We adopt the frequency domain picture in our discussion here. To the first order in the variation of optical absorption and reduced scattering coefficients, the scattered wave is Ul(rd , rs, w) = J,T[r, U,,(r, rs, w)] G( lrd rl)d3r where U, is the incident wave and G is the Green's function solution for the DPDW in homogeneous media. The heterogeneity function Tabs[r, U,(r, rs, w)] = [S1*.,(r)v/D0] U,(r, rs, w) for absorbing objects, and Tscatr[r, U,(r, r,, w) ] = [8p:(r) 3DOkO'b) V In (pi,, + Spi)] U,(r, rs, w) for scattering objects. We consider a planar geometry, which is applicable to the case of compressed breast configuration (see Fig. la). Two-dimensional measurements of the scattered wave are made by scanning the detector in a x-y plane. We then take the spatial two-dimensional Fourier transform of the data Ul(rd, r,, w) in terms of spatial frequencies ( p , q) and utilize the Weyl expansion of Green's f ~ n c t i o n , ~ to obtain the following relation between the heterogeneity function T, and the scattered wave in K-space at the detection plane, fhl