Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation

We characterize the capability of time-resolved reflectance measurements at small source-detector separation (less than $5\mathrm{mm}$) to localize small inhomogeneities embedded in an otherwise homogeneous or layered diffusive medium. By considering both absorption and scattering inhomogeneities, we demonstrate the improvement of this approach in terms of contrast and spatial resolution, as compared to more typical set-ups involving larger source-detection separations (few centimeters). Simulations are performed exploiting an analytical perturbation approach to diffusion theory and a four-layer heterogeneous time-resolved Monte Carlo code, considering realistic tissue geometries. Exhaustive investigation in the parameters space is reported.

Figure 1

Scheme of the reflectance measurement geometry in the case of an inclusion of optical properties ${\mu }_{si}^{\prime },{\mu }_{ai}$ embedded in a homogeneous medium of optical properties ${\mu }_s^{\prime },{\mu }_a$.

Figure 2

Maximum (upper row) and FWHM (lower row) of the contrast line scans as a function of the inclusion depth $z$, for different interfiber distances $\rho$ and times $t$ in the case of a pointlike absorption inclusion embedded in a homogeneous medium. Optical parameters are: ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$, ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ for background and ${\mu }_{si}^{\prime }=1{\mathrm{mm}}^{-1}$, ${\mu }_{ai}=0.02{\mathrm{mm}}^{-1}$ for the inclusion.

Figure 3

The same as Fig. 1, with a less scattering background. Optical parameters are: ${\mu }_s^{\prime }=0.5{\mathrm{mm}}^{-1}$, ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ for background and ${\mu }_{si}^{\prime }=0.5{\mathrm{mm}}^{-1}$, ${\mu }_{ai}=0.02{\mathrm{mm}}^{-1}$ for the inclusion.

Figure 4

The same as Fig. 1, with a more scattering background. Optical parameters are: ${\mu }_s^{\prime }=1.5{\mathrm{mm}}^{-1}$, ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ for background and ${\mu }_{si}^{\prime }=1.5{\mathrm{mm}}^{-1}$, ${\mu }_{ai}=0.02{\mathrm{mm}}^{-1}$ for the inclusion.

Figure 5

Line scans of the contrast along the $x$ axis, for different interfiber distance $\rho$, at various times $t$ (different columns) and depths $z$ (different rows), for pointlike less scattering inclusion embedded in a homogeneous medium. Optical parameters are: ${\mu }_s^{\prime }=1{\mathrm{mm}}^{-1}$, ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ for background and ${\mu }_{si}^{\prime }=0.5{\mathrm{mm}}^{-1}$, ${\mu }_{ai}=0.005{\mathrm{mm}}^{-1}$ for the inclusion. In subplots, solid lines correspond to the case $\rho =0$, dashed lines to $\rho =5\mathrm{mm}$, dotted lines to $\rho =15\mathrm{mm}$ and dashed-dotted lines to $\rho =30\mathrm{mm}$.

Figure 6

Scheme for the three-layer medium ($s_1=8\mathrm{mm}$, ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$, ${\mu }_{s1}^{\prime }=1.5{\mathrm{mm}}^{-1}$; $s_2=4\mathrm{mm}$, ${\mu }_{a2}=0.005{\mathrm{mm}}^{-1}$, ${\mu }_{s2}^{\prime }=0.5{\mathrm{mm}}^{-1}$; $s_3=100\mathrm{mm}$, ${\mu }_{a3}=0.005{\mathrm{mm}}^{-1}$, ${\mu }_{s3}^{\prime }=2.5{\mathrm{mm}}^{-1}$) with an inclusion in the third layer (depth $z_i=15\mathrm{mm}$, volume $1{\mathrm{mm}}^3$, ${\mu }_{si}^{\prime }=1{\mathrm{mm}}^{-1}$, ${\mu }_{ai}=0.02{\mathrm{mm}}^{-1}$).

Figure 7

(a) Scan of the contrast along the $x$ axis at the interfiber distances $\rho =0$ (closed symbols) and $\rho =30\mathrm{mm}$ (open symbols), for $z_i=15\mathrm{mm}$ and times 1, 2, 3, and $4\mathrm{ns}$, in the case of the three-layer medium. Maximum (b) and FWHM (c) of the contrast line scans as a function of time for $z_i=15\mathrm{mm}$. Optical parameters are the same as in Fig. 5.

Figure 8

(a) Time-resolved reflectance curve for interfiber distance $\rho =0$ and $\rho =30\mathrm{mm}$ in a four-layer medium. (b) Maximum contrast as a function of time for the two approaches.

Figure 9

Effect of the refractive index mismatch on the contrast for the two approaches ($\rho =0$ and $\rho =30\mathrm{mm}$). The contrast is reported as a function of time for a two-layer medium ($s_1=8\mathrm{mm}$, ${\mu }_{s1}^{\prime }=1{\mathrm{mm}}^{-1}$, ${\mu }_{a1}=0.004{\mathrm{mm}}^{-1}$, $s_2=100\mathrm{mm}$, ${\mu }_{s2}^{\prime }=0.5{\mathrm{mm}}^{-1}$, ${\mu }_{a2}=0.03{\mathrm{mm}}^{-1}$) both for $n_1=1.44$, $n_2=1.37$, and for $n_1=n_2=1.40$.