Optical sampling depth in the spatial frequency domain

Abstract

We present a Monte Carlo (MC) method to determine depth-dependent probability distributions of photon visitation and detection for optical reflectance measurements performed in the spatial frequency domain (SFD). These distributions are formed using an MC simulation for radiative transport that utilizes a photon packet weighting procedure consistent with the two-dimensional spatial Fourier transform of the radiative transport equation. This method enables the development of quantitative metrics for SFD optical sampling depth in layered tissue and its dependence on both tissue optical properties and spatial frequency. We validate the computed depth-dependent probability distributions using SFD measurements in a layered phantom system with a highly scattering top layer of variable thickness supported by a highly absorbing base layer. We utilize our method to establish the spatial frequency-dependent optical sampling depth for a number of tissue types and also provide a general tool to determine such depths for tissues of arbitrary optical properties.

Keywords: Monte Carlo simulation; diffuse optical spectroscopy; diffuse optics; photon migration; spatial frequency domain.

Fig. 1

Schematic of an SFD $P_{\mathrm{V}\cap \mathrm{D}}$ MC simulation within a tissue subdivided into layered surfaces.

Fig. 2

Experimental setup and schematic of two-layer phantom with top layer thickness $d$ varying from $[0-7.5]l^{*}$ where $l^{*}=2\mathrm{mm}$. Theta ($\theta$) is 15 deg. This experimental setup was used to acquire reflectance as a function of spatial frequency $(f_x)$ and layer thickness $d$. All measurements were performed at $\lambda =731\mathrm{nm}$.

Fig. 3

(a) $P_{\mathrm{V}\cap \mathrm{D}}(z)$ generated for media with optical properties ${\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{a}}=100$, $l^{*}=1\mathrm{mm}$ using $f_x=0,0.025,0.05,0.075,0.1,0.125,0.15,0.175,0.2,0.250,0.3,0.5{\mathrm{mm}}^{-1}$ with $1-\sigma$ error bars. Data were obtained at a $\mathrm{\Delta }z=0.01\mathrm{mm}$. Data symbols are shown on this plot at $z$-intervals of 1 mm. (b) $P_{z_{\max }}(z)$ derived from $P_{\mathrm{V}\cap \mathrm{D}}(z)$ distributions shown in (a).

Fig. 4

(a) Experimentally measured and calibrated $R_d$ (solid lines) and $P_{z_{\max }}(z\le d/l^{*})$ from simulation (dashed lines) and (b) their difference.

Fig. 5

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ for media with optical properties ${\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{a}}=100$ and $l^{*}=1\mathrm{mm}$. Table of plot values are provided in Table 3.

Fig. 6

(a) $d_{50}$, (b) $d_{75}$, (c) $d_{90}$ sampling depths for human brain, mouse skin, human skin and human breast tissues at $\lambda =851\mathrm{nm}$ (solid lines) and 731 nm (dashed lines).

Fig. 7

Median sampling depth $d_{50}$ normalized by the transport mean-free path $l^{*}$ as a function of ${\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{a}}$ at spatial frequencies $f_x{l}^{*}=0$, 0.05, 0.1, 0.2, and 0.3. For clarity, results for only a subset of $f_x$ values are plotted here. Results for additional $f_x$ values are provided in the supplemental data.

Fig. 8

Relative difference between the median depth determined by the lookup table of general optical properties and the median depth determined by running an MC simulation using the real tissue optical properties.

Fig. 9

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human breast at 731 nm.

Fig. 10

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human breast at 851 nm.

Fig. 11

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human brain at 731 nm.

Fig. 12

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human brain at 851 nm.

Fig. 13

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human skin at 731 nm.

Fig. 14

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for human skin at 851 nm.

Fig. 15

Median sampling depth ($d_{50}$) with [25 to 5]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for mouse skin at 731 nm.

Fig. 16

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ and table of plot values for mouse skin at 851 nm.

Fig. 17

Median sampling depth ($d_{50}$) with [25 to 75]% (gray rectangle) and [10 to 90]% (vertical-capped line) intervals versus $f_x$ for general tissue properties. Values of ${\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{a}}$ plotted are a subset of those in the supplemental material.