Dual-slope method for enhanced depth sensitivity in diffuse optical spectroscopy

Abstract

Using diffusion theory, we show that a dual-slope method is more effective than single-slope methods or single-distance methods at enhancing sensitivity to deeper tissue. The dual-slope method requires a minimum of two sources and two detectors arranged in specially configured arrays. In particular, we present diffusion theory results for a symmetrical linear array of two sources (separated by 55 mm) that sandwich two detectors (separated by 15 mm), for which dual slopes achieve maximal sensitivity at a depth of about 5 mm for direct current (DC) intensity (as measured in continuous-wave spectroscopy) and 11 mm for phase (as measured in frequency-domain spectroscopy) under typical values of the tissue optical properties (absorption coefficient: ∼0.01mm^-1, reduced scattering coefficient: ∼1mm^-1). This result is a major advance over single-distance or single-slope data, which feature maximal sensitivity to shallow tissue (<2mm for the intensity, <5mm for the phase).

Fig. 1.

Top panel: linear source-detector arrangement for the dual-slope method with two sources (S₁ and S₂) and two detectors (A and B). Bottom panel: extended dual-slope method using two sources and seven detectors (A–G). In the bottom panel the distances are calculated with respect to the source S₁. Given the symmetric arrangement, these are also the distances between the detectors and S₂ but in the reverse order. Note that the x axis is the depth coordinate, while the y axis is along the line that contains all sources and detectors.

Fig. 2.

Sensitivity of raw data at a single-distance S_Y (top panel) and their slopes S_SlY (bottom panel) according to Eqs. (10) and (16), respectively, are obtained for the three data types (“Y”: DC intensity, AC amplitude, and φ phase; labeled as “DC”, “AC”, and “ϕ”, respectively) by scanning a layer of size 1 × 80 × 80 mm along the x axis (depth) at 1 mm steps. For layered changes in the absorption coefficient we remind that S_SSlY = S_DSlY. The optical properties of the medium are μ_a = 0.01 mm⁻¹, ${\mu }_s^{\prime }=1{\text{mm}}^{-1}$. The refractive index of the diffusive medium and the outer medium are n_i = 1.4, n_o = 1. The modulation frequency is 140 MHz. S_SlY were calculated by using the source-detector arrangement of Fig. 1 (top panel), while S_Y refer to the farthest distance (35 mm).

Fig. 3.

Ratio of estimated phasor D_E/O_E with DC intensity data and DC intensity slope labeled as “DC” (orange arrow, left panel) and “Sl_DC” (purple arrow, left panel), respectively. Phase data and phase slope data are labeled as “φ” (orange arrow, right panel) and “Sl_φ” (purple arrow, right panel), respectively. The true phasors ratio D_T/O_T for the top and bottom layers (red arrows) are indicated by the labels “T” and “B”, respectively. The layers are 1 mm thick and occupy the regions [0, 1] mm and [15, 16] mm, for “T” and “B”, respectively. We remind that for layered changes in absorption S_SSlY = S_DSlY; therefore, it is unnecessary to specify if we are considering single or dual slopes.

Fig. 4.

Ratio of estimated phasor D_E/O_E with DC intensity data and DC intensity slope indicated as “DC” (orange arrow, left panel) and “Sl_DC” (purple arrow, left panel), respectively. Phase data and phase slope data indicated as “φ” (orange arrow, right panel) and “Sl_φ” (purple arrow, right panel), respectively. The true phasors ratio D_T/O_T for the top and bottom layers (red arrows) are indicated by the labels “T” and “B,” respectively. The layers are 6 mm thick and occupy the regions [0, 6] mm [12, 18] mm for “S” and “B,” respectively. We remind that for layered changes in absorption S_SSlY = S_DSlY; therefore, it is unnecessary to specify if we are considering single or dual slopes.

Fig. 5.

Sensitivity maps for single-distance (35 mm) data (S_Y) for two data types (“Y”: DC intensity and φ phase) defined in Eq. (10), for DC intensity data (labeled as “DC,” top panel), and phase data (labeled as “ϕ,” bottom panel). The maps were obtained by scanning a 1 × 5 × 5 mm rectangular cuboid (in the x, y, and z) direction, along the depth (x axis) and horizontal directions (y and z axis) by steps of 1 mm. The optical properties of the background and modulation frequency were the same as those in Figs. 3 and 4 at 690 nm. The arrows indicate the position of the input source S₁ and detector B (see Fig. 1).

Fig. 6.

Cross-sectional plots along the horizontal direction (y) of the sensitivity (S_Y) maps of Fig. 5 for single-distance data and two data types (“Y”: DC intensity and φ phase). Each subplot includes both DC intensity and phase sensitivity (labeled as “DC” and “ϕ,” respectively) and it refers to a different depth (x) of the cuboid’s center in the medium (indicated on top of each subplot).

Fig. 7.

Sensitivity maps for single slopes (S_SSlY) for two data types (“Y”: DC intensity and φ phase) defined in Eq. (16), for DC intensity slope (labeled as “SSl_DC,” top panel) and phase slope (labeled as “SSl_ϕ,” bottom panel). They were obtained by scanning a 1 × 5 × 5 mm rectangular cuboid (in the x, y, and z) direction, along the depth (x axis) and horizontal directions (y and z axis) by steps of 1 mm. The optical properties of the background and modulation frequency are the same as those in Figs. 3 and 4 at 690 nm. The slopes were calculated by using the source detectors S₁A and S₁B of Fig. 1 (top panel) (distances: 20 and 35 mm), which are represented by arrows on top of the DC sensitivity map.

Fig. 8.

Cross-sectional plots along the horizontal direction (y) of the single-slope sensitivity (S_SSlY) maps of Fig. 7 for two data types (“Y”: DC intensity and φ phase). DC intensity single slope (labeled as “SSl_DC”) and phase single slopes (labeled as “SSl_ϕ”). Each subplot refers to a different depth (x) of the cuboid’s center in the medium (indicated on top of each subplot).

Fig. 9.

Sensitivity maps for dual-slope data (S_DSlY) for two data types (“Y”: DC intensity and φ phase) defined in Eq. (31), for DC intensity dual slope data (labeled as “DSl_DC,” top panel) and phase dual-slope data (labeled as “DSl_ϕ,” bottom panel). They were obtained by scanning a 1 × 5 × 5 mm rectangular cuboid (in the x, y, and z) direction, along the depth (x axis) and horizontal directions (y and z axis) by steps of 1 mm. The optical properties of the background were the same as those in Figs. 3 and 4 at 690 nm. The dual slopes were calculated by using the source detectors S₁A and S₁B for one slope and S₂BS₂A for the matched slope [Fig. 1 (top panel)], which are represented by arrows on top of the DC dual-slope sensitivity map.

Fig. 10.

Comparison of cross-sectional plots along the horizontal direction (y) of (1) the DC intensity dual-slope sensitivity map of Fig. 9 (black lines labeled as “DSl_DC”); (2) the DC intensity single-slope sensitivity map of Fig. 7 calculated with S₁A and S₁B (red lines labeled as “SSl_DC1”); (3) the sensitivity maps of the matched DC intensity single slope calculated with S₂B and S₂A (see Fig. 1 top panel) (green lines labeled as “SSl_DC2”).

Fig. 11.

Comparison of cross-sectional plots along the horizontal direction (y) of (1) the phase dual-slope sensitivity map of Fig. 9 (black lines labeled as “DSl_ϕ ”); (2) the phase single-slope sensitivity map pf Fig. 7 calculated with S₁A and S₁B (red lines labeled as SSl_ϕ1); (3) the sensitivity maps of the matched phase intensity single slope calculated with S₂B and S₂A (see Fig. 1 top panel) (green lines labeled as “SSl_ϕ2”).

Fig. 12.

Sensitivity maps for dual-slope data (S_DSlY) for two data types (“Y’: DC intensity and φ phase) defined in Eq. (31) for DC intensity dual-slope data (labeled as “DSl_DC,” top panel) and phase dual-slope data (labeled as “DSl_ϕ,” bottom panel). Obtained with two sources and seven detectors [Fig. 1 (bottom panel)]. The maps were obtained in the same situation of Fig. 9.

Fig. 13.

Schematic representation of a diffusive medium comprising three equal top regions with size (6,22,40) mm along (x, y, z) and one cubic deeper region of side 10 mm. The centers of the three top regions are (x, y, z) = (3, 5.5, 0) mm (region 1), (x, y, z) = (3, 27.5, 0) mm (region 2), (x, y, z) = (3, 49.5, 0) mm (region 3). The cubic region 4 (10 mm side) is centered at (x, y, z) = (15, 27.5, 0) mm. The source-detector arrangement is the same as in Fig. 1. The background optical properties are the same as those of Fig. 3.

Fig. 14.

First case of the ratio of estimated phasor D_E/O_E with single-distance DC intensity data and DC intensity dual slope, labeled “DC” (green arrow, left panel) and “DSl_DC” (blue arrow, left panel), respectively. Single-distance phase data and dual-slope phase, labeled “φ” (green arrow, right panel) and “DSl_φ” (blue arrow right panel), respectively (right panel). The true phasor ratio D_T/O_T for the three top layers and one deeper region (red arrows) are indicated by the numbers 1, 2, 3, and 4 (see Fig. 13 for the labeling of the four regions). The ratio of phasors obtained with the single slopes (not shown) are almost coincident with those obtained with the dual slopes.

Fig. 15.

Second case of the ratio of estimated phasor D_E/O_E with single-distance DC intensity data and DC intensity dual slope, labeled “DC” (green arrow, left panel) and “DSl_DC” (blue arrow, left panel), respectively. Single-distance phase data and dual-slope phase, labeled “φ” (green arrow, right panel) and “DSl_φ” (blue arrow right panel), respectively (right panel). The true phasor ratio D_T/O_T for the three top layers and one deeper region (red arrows) are indicated by the numbers 1, 2, 3, and 4 (see Fig. 13 for the labeling of the four regions). The ratio of phasors obtained with the single slopes (not shown) are almost coincident with those obtained with the dual slopes.

Fig. 16.

Maps of signal-to-noise ratio (SNR) for DC intensity data labeled “DC” (top panel), DC intensity single-slope data labeled “SSl_DC” (middle panel), and DC intensity dual-slope data labeled “DSl_DC” (bottom panel). Same source detector arrangement shown in Fig. 1 (top panel) with S₁ and S₂ as sources and A and B as detectors (top panel).

Fig. 17.

Maps of signal-to-noise ratio (SNR) for phase (φ) data labeled “ϕ” (top panel), phase single-slope data labeled “SSl_ϕ” (middle panel), and phase dual-slope data labeled “DSl_ϕ” (bottom panel). Same source detector arrangement shown in Fig. 1 (top panel) with S₁ and S₂ as sources and A and B as detectors (top panel).