Limits of high-order perturbation theory in time-domain optical mammography

Higher order corrections to the Born approximation in perturbation theory are derived in order to improve its performance with the experiments in slablike geometry. A general expression of the $n\text{th}$ order correction to absorption is developed. The cross talking between absorption and scattering is given. The convergence for higher orders of perturbation analysis for absorbing inclusions was studied. Second order absorption and scattering contributions to the transmitted flux are discussed by analyzing the data from forward simulations. The validity of the results is proven in the experiments with phantoms simulating breast tumors. The significant improvement for the fitted values of the absorption is observed. The alternative application of developed formalism as the first order theory to treat the multiple inclusions is suggested.

Figure 1

(a) Geometry of the second order perturbation theory for a spherical inclusion with radius $R_{\mathit{imp}}$ embedded in a homogeneous slab of thickness $d$; $z_e$ is the distance to the extended boundary; the source is at distance $z_0$ below the origin 0; (b) same picture in the case of the $n\text{th}$ order theory. The wavy lines are connecting points of different scattering events. Cases with $m_p\ne 0$ relating to the mirror imaged spheres are not shown in this picture.

Figure 2

Higher order perturbation calculations on spherical inhomogeneity with $R=10\mathrm{mm}$, placed in depth $z$ of a $d=60\mathrm{mm}$ thick slab with optical coefficients ${\mu }_{s0}^{\prime }=10.75{\mathrm{cm}}^{-1}$ and ${\mu }_{a0}=0.045{\mathrm{cm}}^{-1}$. (a) Comparison of the first and second order absorption contributions to transmission, case $\delta {\mu }_a=0.046{\mathrm{cm}}^{-1}$, $z=30\mathrm{mm}$; (b) first and second order scattering contributions to transmission, case reduced scattering $\delta {\mu }_s^{\prime }=-5.4{\mathrm{cm}}^{-1}$, $z=30\mathrm{mm}$; (c) cross-talk shape functions, $J_{\mathit{abs}\text{−}\mathit{scat}}$ and $J_{\mathit{scat}\text{−}\mathit{abs}}$, calculated from (A3, A4), for the middle of the slab, $z=30\mathrm{mm}$. No significant difference can be detected; (d) Same as (c) but for depths $z=15\mathrm{mm}$ and $z=45\mathrm{mm}$, reflecting symmetric displacements from the slab midline. (e) Schematic illustration of the time reversibility of perturbation solution to diffusion equation (equivalence to geometrical optics). $S$ stands for the scattering, and $A$ for the absorbing event.

Figure 3

Depth difference sensitivity of the perturbation method after Eq. (31) [case 4 in Fig. 2e] applied to the coaxially placed absorber and scatterer. Solid transmission curve corresponds to the absorber-scatterer depth difference $\Delta z=20\mathrm{mm}$; dotted curve to the depth difference of $\Delta z=30\mathrm{mm}$.

Figure 4

(a) Absorption coefficients ${\mu }_a$ of agarose spheres with the same ink concentration, obtained by fitting with the help of different approximations of perturbation theory and photon density wave method (PDW) as a function of sphere radius ($D_0=0.034{\mathrm{cm}}^{-1}$, ${\mu }_{a0}=0.04{\mathrm{cm}}^{-1}$, $\delta {\mu }_a=0.045{\mathrm{cm}}^{-1}$); (b) deviation of absorption coefficients ${\mu }_a$ of phantoms from the values ${\mu }_a^{\mathit{ref}}$ of the reference slab as a function of growing absorption ${\mu }_a^{\mathit{ref}}$ (sphere with a frequent tumor radius $R_{imp}=10\mathrm{mm}$); (c) same for the case $R_{imp}=15\mathrm{mm}$. Value for the reference slab, ${\mu }_a=0.087{\mathrm{cm}}^{-1}$, is shown with a dashed line.