Evaluation of higher-order time-domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms

Time-domain perturbation theory of photon diffusion up to third order was evaluated for its accuracy in deducing optical properties of breast tumors using simulated and physical phantoms and by analyzing 141 projection mammograms of 87 patients with histology-validated tumors that had been recorded by scanning time-domain optical mammography. The slightly compressed breast was modeled as (partially) homogeneous diffusely scattering infinite slab containing a scattering and absorbing spherical heterogeneity representing the tumor. Photon flux densities were calculated from densities of transmitted photons, assuming extended boundary conditions. Explicit formulas are provided for second-order changes in transmitted photon density due to the presence of absorbers or scatterers. The results on phantoms obtained by perturbation theory carried up to third order were compared with measured temporal point spread functions, with numerical finite-element method (FEM) simulations of transmitted photon flux density, with results obtained from the diffraction of diffuse photon density waves, and from Padé approximants. The breakdown of first-, second-, and third-order perturbation theory is discussed for absorbers and a general expression was derived for the convergence of the Born series in this case. Taking tumor optical properties derived by the diffraction model as reference we conclude that estimates of tumor absorption coefficients by perturbation theory agree with reference values within $\pm 25\%$ in only 65% (first order), 66% (second order), and 77% (third order) of all mammograms analyzed. In the remaining cases tumor absorption is generally underestimated due to the breakdown of perturbation theory. On average the empirical Padé approximants yield tumor absorption coefficients similar to third-order perturbation theory, yet at noticeable lower computational efforts.

Figure 1

Simulated time-domain photon flux densities through a partially homogeneous slab bearing a spherical heterogeneity ($R_{\mathit{imp}}=1.0\mathrm{cm}$, $z_{\mathit{imp}}=3\mathrm{cm}$, $d=6\mathrm{cm}$, on-axis geometry, ${\mu }_{a0}=0.045{\mathrm{cm}}^{-1}$, ${\mu }_{s0}^{\prime }=9.4{\mathrm{cm}}^{-1}$, $\delta {\mu }_a=+0.07{\mathrm{cm}}^{-1}$, $\delta {\mu }_s^{\prime }=0{\mathrm{cm}}^{-1}$) using various forward models. All curves are normalized to the amplitude $T_0\left(t_{\max ,0}\right)$ of the photon flux density $T_0$ (dash-dotted line) transmitted through a homogeneous slab, shown for comparison.

Figure 2

Normalized changes of transmitted photon flux density $(T_{\mathit{\text{total}}}-T_0)∕T_0$ through partially homogeneous slab (same geometry and parameters as Fig. 1 unless stated otherwise) at time $t=t_{\max ,0}$ versus changes in absorption and scattering of spherical heterogeneity over background values. Transmitted flux densities simulated by various forward models assuming (a) radius $R_{\mathit{imp}}=1.0\mathrm{cm}$, $\delta {\mu }_s^{\prime }=0{\mathrm{cm}}^{-1}$ (pure absorber); (b) radius $R_{\mathit{imp}}=1.5\mathrm{cm}$, $\delta {\mu }_s^{\prime }=0{\mathrm{cm}}^{-1}$; (c) radius $R_{\mathit{imp}}=1.0\mathrm{cm}$, $\delta {\mu }_a=0{\mathrm{cm}}^{-1}$ (pure scatterer); (d) radius $R_{\mathit{imp}}=1.5\mathrm{cm}$, $\delta {\mu }_a=0{\mathrm{cm}}^{-1}$. Results obtained from third-order perturbation theory are given for pure absorbers only. Breakdown limits of first-, second-, and third-order perturbation theories are indicated by arrows in (a) and (b).

Figure 3

Reconstruction of optical properties $(\delta {\mu }_a,\delta {\mu }_s^{\prime })$ of spherical heterogeneities derived by fitting results obtained from perturbation theory up to second order in scattering changes $\left(\delta {\mu }_s^{\prime }\right)$ and third order in absorption changes $\left(\delta {\mu }_a\right)$ to simulated FEM data; (a), (c) changes in absorption coefficient $\left(\delta {\mu }_a\right)$ and (b), (d) associated changes in reduced scattering coefficient $\left(\delta {\mu }_s^{\prime }\right)$ obtained from fits versus “true” absorption changes $\delta {\mu }_a^{\mathit{FEM}}$ used in FEM simulations for pure absorber $(\delta {\mu }_s^{\prime \mathit{FEM}}=0)$. Breakdown limits [see Figs. 2a, 2b] of first-, second-, and third-order perturbation theory are indicated by vertical dashed lines labeled $L_1$, $L_2$, and $L_3$, respectively. Geometry and background optical properties same as for Figs. 1, 2.

Figure 4

Reconstructed absorption coefficients ${\mu }_a$ of pure absorbers of various sizes at center $(z_{\mathit{imp}}=3\mathrm{cm})$ of rectangular physical and virtual phantom ($d=6\mathrm{cm}$, ${\mu }_{a0}=0.043{\mathrm{cm}}^{-1}$, ${\mu }_{s0}^{\prime }=9.0{\mathrm{cm}}^{-1}$, on-axis geometry); absorption coefficients of spherical absorbers reconstructed by fitting results of various forward models to measured temporal point spread functions (symbols) and to simulated (FEM) transmitted photon flux densities (lines); vertical dashed lines labeled $C$ correspond to Cauchy limit $C_{\mathit{\text{Cauchy}}}=1$, vertical dashed lines labeled $L_1$, $L_2$, and $L_3$ to breakdown limits of first-, second-, and third-order perturbation theory [see Figs. 2a, 2b]; (a) reconstructed absorption coefficient of spheres ($R_{\mathit{imp}}=0.5\mathrm{cm}$, $1.0\mathrm{cm}$, $1.5\mathrm{cm}$) made from same batch of agarose material $({\mu }_a^{\mathit{ref}}=0.082{\mathrm{cm}}^{-1})$; absorption coefficients of agarose material and of background medium indicated by horizontal straight lines labeled ${\mu }_a^{\mathit{ref}}$, ${\mu }_{a0}$, respectively; (b) difference ${\mu }_a-{\mu }_a^{\mathit{ref}}$ between reconstructed absorption coefficient ${\mu }_a$ and true value of spherical absorbers $(R_{\mathit{imp}}=1.0\mathrm{cm})$; (c) same plot as (b) but for $R_{\mathit{imp}}=1.5\mathrm{cm}$; results of first-, second-, and third-order perturbation theory beyond their breakdown limits indicated as dashed lines (b), (c) and by open symbols (c).

Figure 5

Histograms (number of cases) of (a) histologically determined tumor size (isovolumetric radius) of 87 breast carcinomas and (b) of tumor absorption contrast ${\mu }_{a,T}^{\mathit{DPDW}}∕{\mu }_{a,\mathit{\text{local}}}$ at $785\mathrm{nm}$ obtained from analysis of 151 optical projection mammograms.

Figure 6

Scatter plots of tumor absorption coefficients from perturbation theory versus corresponding reference values obtained by the diffraction model (left column) and related histograms displaying (normalized) frequency of tumor absorption coefficients normalized to reference values (right column); (a), (e) first-order, (b), (f) second-order, (c), (g) third-order perturbation theory, (d), (h) Padé approximants. Data refer to 141 optical mammograms, open symbols to absorption coefficients (reference values) above Cauchy limit $C_{\mathit{\text{Cauchy}}}=1$ (first-, third-order perturbation theory, Padé approximants) or above limit $L_2$ (second-order perturbation theory). Error bars indicated in (a)–(d) for arbitrarily selected data points correspond to an estimated statistical uncertainty of 10%.