Diffuse correlation tomography in the transport regime: A theoretical study of the sensitivity to Brownian motion

Diffuse correlation tomography (DCT) uses the electric-field temporal autocorrelation function to measure the mean-square displacement of light-scattering particles in a turbid medium over a given exposure time. The movement of blood particles is here estimated through a Brownian-motion-like model in contrast to ordered motion as in blood flow. The sensitivity kernel relating the measurable field correlation function to the mean-square displacement of the particles can be derived by applying a perturbative analysis to the correlation transport equation (CTE). We derive an analytical expression for the CTE sensitivity kernel in terms of the Green's function of the radiative transport equation, which describes the propagation of the intensity. We then evaluate the kernel numerically. The simulations demonstrate that, in the transport regime, the sensitivity kernel provides sharper spatial information about the medium as compared with the correlation diffusion approximation. Also, the use of the CTE allows one to explore some additional degrees of freedom in the data such as the collimation direction of sources and detectors. Our results can be used to improve the spatial resolution of DCT, in particular, with applications to blood flow imaging in regions where the Brownian motion is dominant.

Figure 1

Correlation function sensitivity kernel computed via Monte Carlo simulations in (a), (c) transmission geometry and (b), (d) reflection geometry. The solution of the CTE is shown in panels (a) and (b) and its $P_1$ approximation in panels (c) and (d). All distances are shown in units of ${\ell }^{*}$. The source-detector separation in panels (b) and (d) is $1.5{\ell }^{*}$.

Figure 2

Correlation function sensitivity kernel computed via Monte Carlo simulations in reflection geometry. The solution of the CTE is shown in panels (a)–(c) and its $P_1$ approximation in panel (d). The source-detector separation is $2.7{\ell }^{*}$. In panels (a) and (c) the source and detector are rotated.

Figure 3

Geometry used in simulated experiment for imaging a dynamic absorber (straight capillary). The source and detector have normal orientation with respect to the surface.

Figure 4

Correlation function sensitivity computed via Monte Carlo simulations. The images are obtained by scanning on the surface of the sample containing a tube filled with particles undergoing Brownian motion with $D_B=0.5\times {10}^{-8}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$. The images correspond to the shadow of the dynamically absorbing object (straight capillary) shown in panel (c). The solution of the CTE is shown in panels (a) and (b) for transmission and reflection geometries respectively. The result for the $P_1$ approximation is shown in panel (d) in transmission geometry (very similar result is obtained in reflection geometry). The slab thickness is $1{\ell }^{*}$.

Figure 5

(a), (b) Correlation function sensitivity kernel in transmission geometry. (c), (d) Correlation function sensitivity, where the images correspond to the shadow of the dynamically absorbing object. (a), (c) The CTE solution and (b), (d) its $P_1$ approximation. The slab thickness is $2.5{\ell }^{*}$.

Figure 6

Correlation function sensitivity kernel computed via Monte Carlo simulations in reflection geometry solving the CTE with dynamic absorption corresponding to different correlation times (a) $\tau =1\mu \mathrm{s}$, (b) $\tau =10\mu \mathrm{s}$, (c) $\tau =100\mu \mathrm{s}$, and (d) $\tau =300\mu \mathrm{s}$. The source-detector separation is $1.5{\ell }^{*}$.