Dual-slope imaging in highly scattering media with frequency-domain near-infrared spectroscopy

Abstract

We present theoretical and experimental demonstrations of a novel diffuse optical imaging (DOI) method that is based on the concept of dual slopes (DS) in frequency-domain near-infrared spectroscopy (FD-NIRS). We consider a special array of sources and detectors that collects intensity (I) and phase (ϕ) data with multiple DS sets. We have recently shown that DSϕ reflectance data features a deeper sensitivity with respect to DSI reflectance data. Here, for the first time, we describe a DS imaging approach based on the Moore-Penrose inverse of the sensitivity matrix for multiple DS data sets. Using a circular 8-source/9-detector array that generates 16 DS data sets at source-detector distances in the range 20-40 mm, we show that DSI images are more sensitive to superficial (<5 mm) perturbations, whereas DSϕ images are more sensitive to deeper (>10 mm) perturbations in highly scattering media.

Near-infrared spectroscopy (NIRS) and diffuse optical imaging (DOI) are measurement techniques that have found a variety of applications in the study of highly scattering media. A key application is the non-invasive measurement of biological tissues such as brain [1], breast [2], and skeletal muscle [3]. These studies of biological tissue are typically performed in the wavelength range 600 – 1000 nm. The primary optical property of interest is the absorption coefficient of tissue (μ_a), or its changes with respect to a reference or baseline condition (Δμ_a). By considering the Beer’s law relationship between absorption and chromophore concentrations, absorption measurements at two or more wavelengths yield concentrations (or concentration changes) of oxyhemoglobin ([HbO₂]) and deoxyhemoglobin ([Hb]) in tissue.

In diffuse reflectance measurements, signal contributions from superficial layers may obscure contributions from deeper portions of turbid media. In the study of brain, skeletal muscle, and breast tissue, such superficial contributions to the optical signals are associated with scalp/skull or skin/lipid layers. The importance of addressing confounding effects from superficial tissue has been especially recognized in the field of cerebral and functional NIRS (fNIRS) [4].

NIRS and DOI can be implemented in three domains: continuous-wave (CW) with steady state illumination and detection, frequency-domain (FD) with intensity-modulated illumination and phase-resolved detection, and time-domain (TD) with pulsed illumination and time-resolved detection [5]. In this work, we consider FD-NIRS, where the optical intensity (I), which is also measured in CW-NIRS, is complemented by the phase (ϕ) of the modulated optical signal.

Practical NIRS measurements can be performed using a single source and a single detector that are separated by a given distance, or by a combination of either a single source and multiple detectors, or a single detector and multiple sources to collect data at multiple source-detector distances. We refer to the first approach as a single-distance (SD) method, and the latter as a single-slope (SS) method, since the multi-distance data are used to measure a gradient or slope of optical signals versus source-detector distance. SS methods have been used to perform absolute measurement of optical properties [6] and to minimize the sensitivity to a thin superficial tissue layer [7,8]. A dual SS configuration was proposed as a more robust method for measurement of absolute optical properties (termed the self-calibrating approach) [9]. The main advantage of this approach was its insensitivity to coupling factors and instrumental drifts. This dual SS configurations was later developed as a more effective approach to reduce sensitivity to superficial portions of the medium [10]. The latter application was named the dual-slope (DS) method, and its implementation with FD-NIRS was particularly effective when applied to phase data [10-12]. A similar method was also proposed in TD-NIRS, where it was particularly effective when applied to the first and second moments of the photon time-of-flight distribution [13].

It is possible to translate intensity or phase measurements collected in SD, SS, or DS configurations into effective absorption changes in the medium ($\Delta {\mu }_a^{(\text{Meas})}$) by assuming that the absorption changes are homogeneous. In the case of SDI data collected in CW-NIRS, this is commonly done through the modified Beer-Lambert law, which can be formally extended to SDϕ data, as well as to SSI, SSϕ, DSI, and DSϕ data [10-12]. These six methods to obtain $\Delta {\mu }_a^{(\text{Meas})}$ feature different sensitivities to localized absorption changes in the medium. We have previously described these regions of sensitivity using diffusion theory [10]. In this Letter, we report, for the first time, theoretical and experimental results of a novel approach to imaging that is entirely based on the DS method.

When designing a DS imaging array, the sources and detectors must be arranged in a way that realizes multiple DS sets with overlapping regions of sensitivity. The requirements of DS sets (in the case of two sources and two detectors) are as follows [12]: 1) A DS set features two paired SS measurements (SS₁ and SS₂), each based on SD data collected at two source-detector distances (ρ₁ and ρ₂), 2) the difference in source-detector distances for SS₁ and SS₂ must be the same (∣ρ₂ – ρ₁ ∣_SS1 = ∣ρ₂ – ρ₁ ∣_SS2), 3) if source “1” and detector “A” comprise the shorter distance for SS₁ then source “2” and detector “A” must comprise the longer distance for SS₂ and vice versa. The first requirement simply describes the makeup of a DS set, while the second and third requirements are needed to cancel out a variable probe-tissue coupling and instrumental drifts [9]. Additional design considerations based on sensitivity, dynamic range, and signal-to-noise ratio set further constraints on the lower and upper limits of source-detector distances (typically taken as 20 mm and 40 mm, respectively), and on the difference between the distances for the SS measurements (typically taken as 10 mm and 20 mm, respectively). When designing a DS imaging array, the above criteria must be met by all DS sets, whose sensitivities must also spatially overlap to achieve image reconstruction. Each of the DS sets in the array yields four measurements (two SD for each of the two SS) for each datatype, which are converted into one DS-measured absorption change per datatype after analysis.

The starting point of the analysis is the translation of intensity and phase changes associated with an absorption perturbation (measured in DS configurations) into measured absorption changes ($\Delta {\mu }_a^{(\text{Meas})}$). $\Delta {\mu }_a^{(\text{Meas})}$ may be interpreted as effectively homogeneous absorption changes associated with the measured intensity of phase changes. This is done by applying diffusion theory for a semi-infinite, homogeneous diffusive medium [14], where the sources and detectors are placed at the surface of the medium. In the SS case, one considers changes in the slopes of suitable functions of intensity or phase that are linear with distance (ΔSSI, ΔSSϕ) [units of inverse length]. In the DS case, one considers the average of two paired single slopes (SS₁ and SS₂). The linear functions of intensity and phase derived from diffusion theory in semi-infinite media may be approximated by ln(ρ²I and ϕ, respectively. In the following discussion, we drop the distinction between intensity I or phase ϕ and consider datatype Y which can be either. In the limit of small perturbations, the measured absorption changes are proportional to the actual absorption perturbation. A key parameter to translate a measured slope change into a measured absorption change is the differential slope factor (DSF_Y) [unitless], which may be different for the two SSs in a DS set (if the DS set is asymmetric and consists of SSs with different distance sets). With the introduction of this factor, which is determined by the optical properties of the medium [11], one can express $\Delta {\mu }_a^{(\text{Meas})}$ for DS as follows:

$$\Delta {\mu }_a^{(Meas,DSY)}=-\frac{1}{2}\left(\frac{\Delta SS{Y}_1}{DS{F}_{Y1}}+\frac{\Delta SS{Y}_2}{DS{F}_{Y2}}\right)$$ (1)

$$DS{F}_Y=\frac{\langle L_Y\rangle ({\rho }_2)-\langle L_Y\rangle ({\rho }_1)}{{\rho }_2-{\rho }_1}$$ (2)

where, ⟨L_Y⟩(ρ) is the generalized total pathlength of datatype Y at distance ρ [units of length].

Now we express the sensitivity (S_DSY,ij) of a given DSY absorption measurement “i” to a localized absorption change in voxel “j” within the medium, as the average of the two SSY sensitivities (that make up measurement i) to voxel j (S_SSY1,j and S_SSY2,j) [unitless]. This can also be expressed as the ratio of the measured DSY absorption change i ($\Delta {\mu }_{a,i}^{(\text{Meas},\text{DS}Y)}$) to the actual absorption change in voxel j ($\Delta {\mu }_{a,j}^{(\text{Actual})}$):

$$S_{DSY,ij}=\frac{1}{2}{\left(S_{SSY1,j}+S_{SSY2,j}\right)}_i=\frac{\Delta {\mu }_{a,i}^{(Meas,DSY)}}{\Delta {\mu }_{a,j}^{(Actual)}}$$ (3)

$${\left(S_{SSY1,j}\right)}_i=\frac{\langle l_Y\rangle \left({\rho }_{2,i1},\overrightarrow{r_j}\right)-\langle l_Y\rangle \left({\rho }_{1,i1},\overrightarrow{r_j}\right)}{\langle L_Y\rangle \left({\rho }_{2,i1}\right)-\langle L_Y\rangle \left({\rho }_{1,i1}\right)}$$ (4)

The two SS sensitivities [(S_SSY1,j)_i and (S_SSY2,j)_i] to voxel j for the i^th dual-slope set are expressed in terms of a ratio of differences between generalized partial and total pathlengths. Equation (4) expresses the sensitivity of the first single-slope of dual-slope set i, where ⟨l_Y⟩(ρ_2,i1, $\overrightarrow{r_j}$) is the generalized partial pathlength for voxel j at position $\overrightarrow{r_j}$ associated with datatype Y measured at the long distance (ρ₂) with the first source-detector pair of dual-slope set i [units of length]. We have previously shown how to perform theoretical calculations of generalized partial and total pathlengths in terms of the complex fluence rate and reflectance for a semi-infinite medium [10].

To investigate the effectiveness of the novel imaging method based on dual-slopes, we performed experiments on highly scattering liquid phantoms containing solid optical inhomogeneities. Liquid phantoms were made from water, 2% reduced fat milk, and black India-ink (Higgins, Leeds, MA, USA). For optical inclusions, solid phantoms were constructed from Rubber Glass silicone (Smooth-On, Macungie, PA, USA), Calli India-ink (Daler-Rowney, Bracknell, England), and titanium dioxide powder (AEE, Upper Saddle River, NJ, USA). The absorption coefficient (μ_a) and the reduced scattering coefficient (${\mu }_s^{\prime }$) of the liquid and solid phantoms were measured using scanned multi-distance FD-NIRS. We used a frequency-domain tissue spectrometer (Imagent V2, ISS, Champaign, IL, USA) operating at a modulation frequency of 140.625 MHz, and at an optical wavelength of 690 nm. The optical probe consisted of 400 μm multimode fibers to deliver light, and 3 mm diameter fiber bundles to collect light.

For demonstration of the novel dual-slope diffuse optical imaging approach, we considered a circular array of 8 sources and 9 detectors that features 16 DS sets (Fig. 1(a)). The arrangement contains 16 DS sets since each set must meet the DS requirements described above, and these are the only sets within the array that do so. Eight DS sets are asymmetric linear sets that use source-detector separations of (20, 30) mm and (30, 40) mm for the two paired SSs. The other eight DS sets are trapezoidal sets that use source-detector separations of (20, 32) mm for both paired SSs. The individual sensitivity maps for these DS sets can be found in [12]. Fig. 1(c)-(d) shows the overall region of sensitivity associated with all 16 sets for intensity (Fig. 1(c)) and phase (Fig. 1(d)) at the depth of respective maximum sensitivity (z = 4.3 mm for intensity, z = 10.8 mm for phase). Equations (3) and (4) were summed over all the dual-slope measurements to obtain the overall sensitivity for DSI and DSϕ at each voxel “j”.

Fig. 1:

(a) Schematic of 8-source/9-detector imaging array that features 16 DS sets [DS1, orange; DS11, green]. (b) Schematic of imaging configuration showing the superficial [at (5$\widehat{x}$ + 1.5$\widehat{z}$) mm] and deep [at (−5$\widehat{x}$ + 11.5$\widehat{z}$) mm] 10 × 10 × 3 mm³ perturbations. Region of sensitivity for intensity (c) and phase (d) at plane of maximal sensitivity.

We considered two perturbations (size: 10 × 10 × 3 mm³); a superficial one centered at (5$\widehat{x}$ + 1.5$\widehat{z}$) mm, and a deep one centered at (−5$\widehat{x}$ + 11.5$\widehat{z}$) mm (see Fig. 1(b)). In this experiment, we used an optical wavelength of 690 nm, for which the optical properties of the liquid phantom were μ_a = 0.093 ± 0.003 cm⁻¹ and ${\mu }_s^{\prime }$ = 8.5 ± 0.5 cm⁻¹, and those of the solid phantom were μ_a = 0.130 ± 0.001 cm⁻¹ and ${\mu }_s^{\prime }$ = 8.18 ± 0.04 cm⁻¹. The optical properties of the liquid phantom resulted in DSF values for intensity and phase of DSF_I = 7.4 and DSF_ϕ = 1.4, respectively [11]. Three cases were considered: 1) the presence of only the superficial perturbation, 2) the presence of only the deep perturbation, and 3) the presence of both superficial and deep perturbations. For each of these cases the solid phantom perturbation was immersed in the liquid phantom and held on the tip of a glass pipette which was filled with liquid phantom. To achieve the measurement, the perturbation was first placed far from the array during baseline then brought into the desired location by a 3-axis computer numerical control (CNC) system.

Here, the goal is not to achieve a quantitative reconstruction of absolute changes in optical properties, but rather to identify and localize (in the lateral x-y plane) deeper perturbations with minimal interference from shallower perturbations. This goal is consistent with the objective of brain measurements with non-invasive cerebral NIRS. In addition to collecting experimental data, we also present theoretical calculations assuming no scattering perturbations or refractive index mismatch at the perturbation boundary. Additionally, we have investigated the effect of modeling a scattering perturbation so that we may comment on its effect.

For image reconstruction, we considered the following forward model:

$$\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(Meas)}=\mathit{J}\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(Actual)}\iff \left[\begin{array}{c}\hfill \Delta {\mu }_{a,1}^{(Meas)}\hfill \\ \hfill ⋮\hfill \\ \hfill \Delta {\mu }_{a,k}^{(Meas)}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill S_{11}\hfill & \hfill \cdots \hfill & \hfill S_{1n}\hfill \\ \hfill ⋮\hfill & \hfill S_{ij}\hfill & \hfill ⋮\hfill \\ \hfill S_{k1}\hfill & \hfill \cdots \hfill & \hfill S_{kn}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \Delta {\mu }_{a,1}^{(Actual)}\hfill \\ \hfill ⋮\hfill \\ \hfill \Delta {\mu }_{a,n}^{(Actual)}\hfill \end{array}\right]$$ (5)

where $\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(\text{Meas})}$ is a k × 1 vector of measured absorption changes (where k is the number of DS sets, 16 in this work), J is a k × n matrix of sensitivities (where n is the number of voxels considered, 61 × 61 × 2 = 7442 in this work), and $\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(Actual)}$ is a n × 1 vector of actual voxel absorption changes. To solve the inverse problem, we use the Moore-Penrose pseudoinverse of J (J⁺) with Tikhonov regularization [15]:

$$\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(Recon)}={\mathit{J}}^{+}\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(Meas)}$$ (6)

$${\mathit{J}}^{+}={\mathit{J}}^T({\mathit{JJ}}^T+\alpha \mathit{I}{)}^{-1}$$ (7)

where $\stackrel{\rightharpoonup }{\Delta \mu }{}_a^{(\text{Recon})}$ is the vector of reconstructed voxel absorption changes that is the least-squares solution of Equation (5), and α is the Tikhonov regularization parameter expressed as a × max(diag(JJ^T)). We have used a value a = 0.01, which we found to achieve a good compromise between signal-to-noise and smoothing of the image, and is consistent with values previously used [15]. We sought to reconstruct 2-dimensional (2D) images in the x-y plane that are mostly sensitive to deeper perturbations, while using the same voxel size in the analysis of intensity and phase data for a fair comparison of DSI and DSϕ images. With this goal in mind, we chose voxels of size (0.5, 0.5, 5) mm (superficial layer) and (0.5, 0.5, 10) mm (deeper layer) in (x, y, z), and we only show the x-y image corresponding to the deeper layer of voxels (z = 5-15 mm). This layer of voxels includes a portion of the medium that is sensed by both datatypes and that does not include the most superficial layer (z = 0-5 mm). This is in line with the overall goal of DS, which is to suppress signal contributions from superficial layers. Therefore, we do not wish to create images at different depths, we focus on 2D images representative of deep perturbations.

The results of the imaging experiments are shown in Fig. 2(a)-(l) for intensity (column 1: theory; column 2: experiment) and phase (column 3: theory; column 4: experiment). In the images of Fig. 2(a)-(l), the superficial perturbation is on the right, i.e. positive x (row 1), the deep perturbation is on the left, i.e. negative x (row 2), and both are present in row 3. Overall, there is a good agreement between theory and experiment. The images based on DSI (columns 1-2) show a stronger sensitivity to the superficial perturbation, whereas the images based on DSϕ show a stronger sensitivity to the deeper perturbation.

Fig. 2:

DSI and DSϕ images, as labeled with only the superficial perturbation [(a),(c): theory; (b),(d): experiment], only the deep perturbation [(e),(g): theory; (f),(h): experiment], and with both the superficial and the deep perturbation [(i),(k): theory; (j),(l): experiment]. Note: superficial perturbation at (5$\widehat{x}$ + 1.5$\widehat{z}$) mm and deep perturbation at (−5$\widehat{x}$ + 11.5$\widehat{z}$) mm.

In the presence of only the superficial perturbation, the theoretical DSI map (Fig. 2(a)) features a greater Δμ_a than the experimental DSI maps (Fig. 2(b)) (likely because of different boundary effects in the theoretical and experimental cases), whereas both theoretical and experimental DSϕ maps (Fig. 2(c)-(d)) show a small absorption decrease at the perturbation location, due to the negative superficial sensitivity of DSϕ [10]. In the presence of only the deep perturbation, the theoretical and experimental maps are in excellent agreement and show a smaller contrast with DSI (Fig. 2(e)-(f)) than with DSϕ (Fig. 2(g)-(h)). In this case, the experimental localization error in the lateral plane (x-y) is 2 mm for both DSI and DSϕ (reconstructed maximum and deep true location).

The most significant case is the one where both superficial and deep perturbations are present. This is the case that emulates the presence of superficial confounds (for example, scalp hemodynamics) in the non-invasive study of deeper optical dynamics (for example, cerebral hemodynamics). The absorption maps based on DSI (Fig. 2(i)-(j)) show that they are mostly representative of the superficial perturbation (lateral localization error of the deep object: 12 mm), whereas the maps based on DSϕ (Fig. 2(k)-(l)) are mostly representative of the deeper perturbation (lateral localization error of the deep object: 2 mm).

While there is general agreement between theory and experiment in the results of Fig. 2, we have observed some discrepancy between the absolute values of the reconstructed absorption changes in theoretical and experimental results. We assign these discrepancies to experimental conditions that deviate from the assumptions made in theoretical calculations. These assumptions include 1) lack of scattering contrast for the optical inclusion (in our experiments, the scattering contrast was in the range −9% to +2%), 2) refractive index matching at the inclusion/background interface (in our experiments, the absorbing inclusion had n ≈ 1.4, while the liquid phantom had n ≈ 1.35 [16]), and 3) extrapolated boundary conditions at the flat phantom/air interface (in the experiment, the presence of illumination and collection optical fibers, surface tension effects, and the proximity of the perturbation to the phantom/air boundary resulted in modified boundary effects).

To investigate the most likely source of the discrepancy between theory and experiment, we performed theoretical calculations that included scattering perturbations consistent with the experimental conditions. We found that the scattering mismatch alone cannot explain the difference between theory and experiment. Therefore, we expect that the discrepancy is likely due to points 2) and 3) above, which refer to the unknown refractive index mismatch and to the inevitable variability of geometric conditions that cannot be fully controlled within the scope of this experiment. We have also opted to present theoretical results without modeling a scattering perturbation to emphasize our focus on measurements of absorption perturbations. However, we stress that our goal here is not to achieve reconstruction of absolute optical properties but instead to 1) correctly localize perturbations in the x-y plane, 2) achieve a stronger relative sensitivity to deeper versus shallower perturbations, and 3) correctly identify the direction of the absorption change caused by the perturbation (increase or decrease).

The key message is that DSϕ features a deeper sensitivity and is less impacted by superficial perturbations than DSI. This feature is exploited for imaging in Fig. 2, which shows that DSϕ images selectively reproduce deeper perturbations (see Fig. 2(k)-(l)), whereas DSI images are more strongly affected by superficial perturbations (see Fig. 2(i)-(j)). Of course, one can take advantage of the complementary information of DSI and DSϕ images to extract information on both superficial and deep perturbations.

In summary, we have presented a novel imaging approach based on DS data that can advance the capabilities of non-invasive diffuse optical imaging. It can also provide a robust practical approach, since it inherits all the desirable features of the self-calibrating approach [9] in terms of insensitivity to optical coupling to tissue, instrumental drifts, and motion artifacts. One application that can especially benefit from DS imaging is functional NIRS (fNIRS), where scalp hemodynamics can confound, or even obscure, the cerebral hemodynamics of interest.