CW sensitivity to absorption change — semi-infinite homogeneous medium
Sensitivity maps SY(r) for CW intensity measurements in a semi-infinite homogeneous medium under diffusion theory. Source–detector optodes are on the surface (z = 0); the map shows the x–z plane (y = 0).
Sensitivity \(S_Y(\mathbf{r})\) is defined as the ratio of the local Jacobian to the global Jacobian:
$$S_Y(\mathbf{r}) = \frac{\partial Y / \partial \mu_{a,\text{pert}}(\mathbf{r})} {\partial Y / \partial \mu_{a,\text{pert,homo}}}$$It is dimensionless and quantifies how a localised absorption perturbation \(\Delta\mu_{a,\text{pert}}(\mathbf{r})\) at position \(\mathbf{r}\) affects the measurement \(Y\) relative to a homogeneous perturbation throughout the medium. Equivalently,
$$S_Y(\mathbf{r}) = \frac{\Delta\mu_{a,Y}}{\Delta\mu_{a,\text{pert}}(\mathbf{r})}$$where \(\Delta\mu_{a,Y}\) is the recovered effective homogeneous \(\Delta\mu_a\) and \(\Delta\mu_{a,\text{pert}}(\mathbf{r})\) is the true local perturbation.
The medium is modelled as semi-infinite and homogeneous with the extrapolated boundary condition (same geometry as the Diffuse Reflectance tool). Optodes sit on the surface at \(z=0\); the map shows the \(xz\)-plane at \(y=0\).
For CW, the data type \(Y = \ln I\) is dimensionless since it always appears in differences. The generalized total path length is \(\langle L \rangle = -\partial Y/\partial\mu_{a,\text{homo}}\) (mm) and the generalized partial path length is \(\langle\ell(\mathbf{r})\rangle = -\partial Y/\partial\mu_{a,\text{pert}}(\mathbf{r})\) (mm). For the single-distance arrangement:
$$S(\mathbf{r}) = \frac{\langle\ell(\mathbf{r})\rangle}{\langle L \rangle}$$SD (Single Distance) uses one source and one detector. \(S(\mathbf{r}) = \langle\ell(\mathbf{r})\rangle/\langle L\rangle\).
SS (Single Slope) uses one source and two detectors at different separations \(\rho_0 < \rho_1\): \(S(\mathbf{r}) = (\langle\ell_1\rangle - \langle\ell_0\rangle)/(\langle L_1\rangle - \langle L_0\rangle)\).
DS (Dual Slope) uses two sources and two detectors, combining slopes measured in both directions. DS provides suppressed sensitivity to superficial layers:
$$S(\mathbf{r}) = \frac{(\langle\ell_1(\mathbf{r})\rangle - \langle\ell_0(\mathbf{r})\rangle) + (\langle\ell_3(\mathbf{r})\rangle - \langle\ell_2(\mathbf{r})\rangle)}{(\langle L_1\rangle - \langle L_0\rangle) + (\langle L_3\rangle - \langle L_2\rangle)}$$where measurements 0–3 follow the ordering S1→D1, S1→D2, S2→D2, S2→D1.
Giles Blaney, Angelo Sassaroli, and Sergio Fantini, Spatial sensitivity to absorption changes for various near-infrared spectroscopy methods: A compendium review, Journal of Innovative Optical Health Sciences, Vol. 17, No. 04, 2430001 (2024). https://doi.org/10.1142/S1793545824300015