Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 2, 2023



Article Number  39  
Number of page(s)  13  
DOI  https://doi.org/10.1051/jeos/2023037  
Published online  29 September 2023 
Research Article
Determination of scattering coefficient and scattering anisotropy factor of tissuemimicking phantoms using linefield confocal optical coherence tomography (LCOCT)
^{1}
Université ParisSaclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, Palaiseau 91127, France
^{2}
DAMAE Medical, Paris 75013, France
^{*} Corresponding author: lena.waszczuk@universiteparissaclay.fr
Received:
30
May
2023
Accepted:
29
August
2023
Linefield Confocal Optical Coherence Tomography (LCOCT) is an imaging modality based on a combination of timedomain optical coherence tomography and reflectance confocal microscopy. LCOCT provides threedimensional images of semitransparent samples with a spatial resolution of ∼1 μm. The technique is primarily applied to in vivo skin imaging. The image contrast in LCOCT arises from the backscattering of incident light by the sample microstructures, which is determined by the optical scattering properties of the sample, characterized by the scattering coefficient μ_{s} and the scattering anisotropy factor g. In biological tissues, the scattering properties are determined by the organization, structure and refractive indexes of the sample. The measurement of these properties using LCOCT would therefore allow a quantitative characterization of tissues in vivo. We present a method for extracting the two scattering properties μ_{s} and g of tissuemimicking phantoms from 3D LCOCT images. The method provides the mean values of μ_{s} and g over a lateral field of view of 1.2 mm × 0.5 mm (x × y). It can be applied to monolayered and bilayered samples, where it allows extraction of μ_{s} and g of each layer. Our approach is based on a calibration using a phantom with known optical scattering properties and on the application of a theoretical model to the intensity depth profiles acquired by LCOCT. It was experimentally tested against integrating spheres and collimated transmission measurements for a set of monolayered and bilayered scattering phantoms.
Key words: Linefield confocal optical coherence tomography / Optical coherence tomography / Reflectance confocal microscopy / Optical properties / Scattering
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Linefield Confocal Optical Coherence Tomography (LCOCT) is a recently introduced optical imaging technique combining the principles of timedomain OCT and reflectance confocal microscopy (RCM) with line illumination and line detection using broadband near infrared light. LCOCT performs two dimensional (2D) and threedimensional (3D) imaging with a quasiisotropic resolution of ∼1 μm [1–4]. Up to now, LCOCT has been applied to skin imaging for healthy skin characterization [5, 6] and early detection and diagnosis of various skin diseases including skin cancers [7–10]. LCOCT provides images with cellular resolution similar to histology, while being noninvasive.
As in OCT and RCM, the contrast of LCOCT images originates from the backscattering of light, due to refractive index inhomogeneities within the sample. The amount of detected backscattered light is determined by three main optical parameters specific to the imaged sample [11]: the absorption coefficient μ_{a}, the scattering coefficient μ_{s}, and the scattering anisotropy factor g. μ_{s} and μ_{a} correspond to the probability per unit of distance of absorption or scattering events respectively and thus reflect the amount of absorbed and scattered light. g is defined as the average of the scattering angles cosines along the photon path and hence characterizes the directivity of the scattered light. These optical parameters rule the decay of the OCT image intensity with depth I(z), commonly described by a BeerLambert law in the singlescattering regime [12]:(1)where μ = μ_{a} + μ_{s} is the attenuation coefficient, characterizing the attenuation of the incident light during indepth propagation, due to absorption and scattering. The factor 2 accounts for roundtrip propagation of light in the sample.
Beyond morphological information, LCOCT images contain information on optical properties of the sample. Those optical properties are determined by the size, shape and density of tissue light scatterers (nucleus, mitochondria, cytoskeleton, extracellular fiber such as collagen) at cellular and subcellular levels. The extraction of optical properties would thus allow access to quantitative information complementary to the morphological information provided by the LCOCT images. Optical properties could be used as a tool for tissue characterization and quantification of tissue structural changes, which could for instance occur during the development of a pathology.
Quantification of the attenuation coefficient μ using conventional OCT has been used for several biomedical applications, such as atherosclerosis plaque characterization [13], human brain cancer infiltration detection [14], glaucoma diagnosis [15], urothelial carcinoma staging and grading [16] or basal cell carcinoma characterization and subtypes differentiation [17]. Several methods have been developed to measure the attenuation coefficient μ, as reviewed by [12, 18]. The most common approach is to fit an exponential decay curve on the OCT image intensity as a function of depth to extract the attenuation coefficient μ. More recently, Vermeer et al. [19] introduced a new method for depthresolved measurement of the attenuation coefficient. Based on two assumptions (the incident light is completely attenuated within the OCT imaging depth range and the backscattered light is a fixed fraction of the attenuated light), their approach provides a pixelbypixel estimation of μ, thereby producing attenuation maps with the spatial resolution of the OCT image.
In contrast, fewer work has been reported on the separate measurement of the optical parameters μ_{a}, μ_{s} and g using OCT. However, the interest in measuring the optical properties, and especially the scattering parameters μ_{s} and g, is that each of them provides different and complementary information on tissue structure. μ_{s} is determined by several factors, including the density, size and shape of the scattering particles, while g is defined by the size and shape of these particles [20]. The measurement of both parameters μ_{s} and g could therefore provide a comprehensive insight into tissue structural changes occurring in pathologies. The depthresolved method is limited to the measurement of the attenuation coefficient μ due to its assumption that the backscattered light is a fixed fraction of the attenuated light.
A few methods based on curvefitting allow the measurement of the optical properties μ_{a}, μ_{s} and g. Thrane et al. [21] introduced a model to extract the optical parameters μ_{a}, μ_{s} and g from the OCT image intensity with depth I(z). This approach consists in fitting a theoretical model on I(z), based on the extended Huygens–Fresnel (EHF) principle. This model was later applied to the skin to distinguish between biopsies of melanoma and benign nevi [22]. In this study, the different layers of skin have not been dissociated and the skin was considered as a monolayered tissue. Studies have been carried out to show the applicability to bilayered samples, however the model becomes cumbersome to implement [23]. Another simpler approach for extracting the optical properties from OCT images was introduced by Jacques et al. [24]. The authors proposed to extract the scattering properties μ_{s} and g from OCT images using a linear fit. The model, based on MonteCarlo simulations, was designed for OCT in focustracking mode and RCM. It was validated on phantoms with known optical properties [25] and later applied to the extraction of scattering properties of various biological tissues including the skin [26, 27]. This model was also applied on bilayered tissues [28].
As a combination of RCM and OCT (in focustracking mode as will be described later), LCOCT is particularly suitable for the application of the model developed by Jacques et al. [24]. Based on this model, we present a method to determine both μ_{s} and g from LCOCT images of tissuemimicking phantoms. Our approach provides the mean values of μ_{s} and g over a lateral field of view of 1.2 mm × 0.5 mm (x × y). It can be applied to monolayered and bilayered samples, where it allows extraction of μ_{s} and g of each layer. The method was experimentally validated against integrating spheres and collimated transmission measurements on a set of nine monolayered phantoms and two bilayered with distinct optical properties in terms of both scattering properties μ_{s} and g. The phantoms were made of a polydimethylsiloxane (PDMS) polymer including different types of scattering particles and aim at mimicking tissue scattering properties in the near infrared range. The developed method is simple to implement since it relies on a linear fit and a calibration using one of the fabricated phantoms whose scattering properties were determined by integrating spheres and collimated transmission. The method provides additional quantitative bulk information (i.e., at a macroscopic scale), complementary to LCOCT images providing information at a microscopic scale.
2 Materials and methods
2.1 Scattering phantoms
In order to mimic optical properties of biological tissues such as skin [29–31], phantoms were designed with a scattering coefficient μ_{s} ranging from ∼2 to 20 mm^{−1} and an anisotropy factor g from ∼0.6 to 1. Since most biological tissues have negligible absorption compared to scattering (scattering is 10–100 times higher than absorption in the near infrared spectrum [29]), we chose μ_{a} ≪ μ_{s}. In the following, we focus on the measurement of the scattering properties μ_{s} and g, neglecting the role of μ_{a}.
The phantoms are composed of a PDMS matrix (Sylgard 184 silicone Elastomer Kit, Neyco, France) with a refractive index of 1.42 at the wavelength of 800 nm [32, 33]. To create phantoms with various optical scattering properties, different micro and nanopowders were added into the PDMS matrix: titanium dioxide (TiO_{2}) powder (SigmaAldrich Corporation, USA), zinc oxide (ZnO) powder (US Research Nanomaterials, Inc., USA) and silicon dioxide (SiO_{2}) powder (US Research Nanomaterials, Inc., USA). All powders have negligible absorption compared to scattering in the near infrared.
In order to select the appropriate scatterers to mimic optical properties of the skin, Mie theory (assuming spherical particles) was used to simulate μ_{s} and g values of the phantoms as a function of the refractive index, size and density of the particles embedded in PDMS. However, all the parameters necessary to predict the optical properties by Mie theory were not precisely known. For example, only the mean particle size was sometimes given by the manufacturer, without information on the size distribution. The refractive index of the nanoparticles was not provided either. For ZnO and SiO_{2} particles, we approximated the refractive index by optical index values extracted from the literature at the 800 nm wavelength [34, 35]. For TiO_{2} particles, a wide range of refractive indices was found [34, 36, 37], so an average of the different refractive indices reported in the literature was chosen. Ten different phantoms were fabricated. The detailed description of the composition of the phantoms is given in Table 1, with target values of μ_{s} and g computed using Mie theory when possible. The target values of μ_{s} and g are not given for phantoms 0 and 3 due to the lack of precise information on the size of the particles used (<5 μm).
Summary of monolayered phantoms composition. All phantoms are made of a PDMS matrix including scattering particles of different materials, sizes and concentrations.
The PDMS matrix was fabricated using a ratio of 10:1 by weight of PDMS prepolymer and curing agent. All phantoms were prepared by first mixing the curing agent with a variable amount of scattering powder. The mixture was placed in an ultrasonic bath for 30 min to prevent particles aggregation, and then mixed with the PDMS prepolymer. The obtained mixture was poured on a 40mm diameter petri dish. The volume poured was chosen in order to obtain 1–2.5 mm thick phantoms, a thickness allowing their characterization by integrating spheres. Air bubbles were removed using a vacuum pump for 1 h, and the phantom mixture was finally cured for h at 80 °C.
We also designed two bilayered phantoms with distinct optical properties in each layer, as listed in Table 2. To create the top layer, a small volume of the PDMS/scattering powder mixture was poured in a petri dish. The phantom was left for 24 h in a vacuum chamber at room temperature, allowing the mixture to spread into a thin layer before curing. This process allows to achieve a thickness of . Since this phantom is very fragile, another thicker phantom was fabricated using the same batch mixture, facilitating the measurement of the top layer scattering properties using integrating spheres and collimated transmission setup. Once cured, the obtained top layer was stacked on a thicker phantom with different optical properties to create a bilayered phantom, using a small drop of paraffin oil (refractive index of ) between the two layers for optical contact. A representative LCOCT image corresponding to a vertical image (i.e., crosssectional view) of one of the bilayered phantoms is shown in Figure 1.
Figure 1 LCOCT vertical image (i.e., crosssectional view) of a bilayered phantom. The LCOCT image is displayed in a logarithmic intensity scale (arbitrary unit). 
Summary of bilayered phantoms composition, given by the material, size and weight concentration of the scattering particles embedded in PDMS.
2.2 Integrating spheres and collimated transmission measurements
Mie theory allows to define target values for μ_{s} and g. However, as previously mentioned, the characterization of the optical properties by Mie theory is not very reliable due to uncertainties on the size distribution and the refractive indices of the particles used in our phantoms. Therefore, these target values are indicative values but cannot be used as reference values to compare with values obtained from LCOCT images. Thus, the optical properties of our samples were characterized experimentally, using integrating spheres and collimated transmission devices.
Integrating spheres measurement is a common method to measure optical properties of samples [38–40]. It consists in measuring the diffuse reflectance and diffuse transmittance of a sample using a setup with one or two hollow spheres with a diffuse reflective internal coating. The absorption coefficient μ_{a} and the reduced scattering coefficient are then derived from these two measurements using theoretical models [41]. In order to separate the reduced scattering coefficient into the scattering coefficient μ_{s} and the scattering anisotropy factor g, a collimated transmission measurement is required. This measurement is obtained by collecting the ballistic light from a laser beam propagating perpendicularly to the phantom. A more detailed description of integrating spheres and collimated transmission measurements can be found in [42].
We used a setup with two integrating spheres (2″ Integrating Sphere IS2004, Thorlabs, USA). The sample was placed between the two spheres. Light from a highpower broadband Xenon light source (HPX 2000, Ocean Insight, Dunedin, Florida) was guided to one of the two spheres using a 50/125 optical fiber and collimated into a 6mm diameter beam. Diffuse reflected and transmitted light were collected using an optical fiber (plastic fiber, PMMA Toray, 50cm long, 1mm diameter, 0.22 NA) coupled to a spectrometer (AvaSpec2048, Avantes, Netherlands). Collimated transmission measurement was carried out with a monochromatic fiber laser diode at 785 nm (Micron Cheetah, Sacher LaserTechnik, Germany). The laser beam was collimated at the fiber output and its diameter was reduced using a diaphragm. The phantom was placed a few cm behind the diaphragm, perpendicular to the laser beam. A collection fiber (105 μm core diameter, 0.1 NA), connected to the spectrometer, was placed one meter behind the phantom in order to collect the ballistic light. Due to its small diameter, low numerical aperture and large distance to the sample, the fiber acts as a pinhole rejecting scattered photons.
All measurements were corrected for dark noise and normalized with respect to blank measurements without sample. The optical properties were computed using the Inverse AddingDoubling (IAD) algorithm developed by Prahl [43]. μ_{s} and g were determined at the wavelength of 785 nm. Despite the slight difference in wavelength from that of the LCOCT device centered at 800 nm, the difference in μ_{s} and g values between these two wavelengths can be considered negligible compared to the accuracy of our measurements [39], allowing for comparison of integrating spheres and collimated transmission measurements with measurements obtained from LCOCT images.
In practice, the collimated transmission measurement could not be performed on some of the highly concentrated phantoms (3, 5, 7 and 9) since the ballistic intensity was too weak. However, phantoms 3, 5, 7 and 9 are made of the same scattering particles as phantoms 1, 4, 6 and 8 respectively. Since g is ruled by particle size and shape [20], the values of g of phantoms 3, 5, 7 and 9 were considered identical to those of phantoms 1, 4, 6 and 8 respectively.
For bilayered phantoms, the top and bottom layers were analyzed separately since integrating spheres and collimated transmission do not allow to separate the scattering properties of two stacked layers. As previously mentioned, since the 100 μmthick top layer was very fragile, the scattering properties of the top layer were measured on a thicker phantom fabricated with the same batch mixture as the thin top layer.
2.3 LCOCT measurements
2.3.1 LCOCT device
The LCOCT device used in this work is described in [1–4]. The device is based on a Linnik interferometer with a 0.5 NA waterimmersion microscope objective in each arm. A supercontinuum light source centered around 800 nm is used to generate spatially coherent ultrabroadband light. Line illumination of the sample is achieved using a cylindrical lens, and backscattered light is detected with a line camera. LCOCT is capable of generating 2D vertical section images (i.e., crosssectional views, x × z) and horizontal section images (i.e., enface views, x × y). By combining the depth and lateral scans used to generate 2D images, 3D images can also be obtained, with a quasiisotropic resolution of over a field of view of 1.2 mm × 0.5 mm × 0.5 mm (x × y × z). During imaging, samples are placed in contact with a glass plate to stabilize them and maintain them at the position corresponding to the working distance of the microscope objective. Paraffin oil is placed between the phantoms and the glass plate to minimize parasitic reflections at the interface.
2.3.2 Theoretical model for measuring μ_{s} and g
Within the singlescattering regime and in focustracking mode, the reflectance R(z) measured along an Ascan acquired by OCT or LCOCT image (i.e., the depthdependent fraction of incident light backscattered and collected by the device) can be described by an exponential decay with depth [29]:(2)ρ is the fraction of light backscattered at the focus into the collection solid angle of the LCOCT microscope objective; μ = μ_{s} + μ_{a} is the attenuation coefficient previously introduced, and the factor 2 accounts for roundtrip attenuation inside the sample. Since the factor represents the average distance between two scattering events [44], the single scattering regime is valid as long as μ_{s}z ≤ 1. This model of the LCOCT signal is therefore valid for media with low scattering properties or/and at shallow penetration depths.In order to be able to measure the optical properties of more scattering media, Jacques et al. [29] proposed a model, also based on an exponential decay law, allowing to take into account forward directed multiple scattering:(3)where(4)where G(NA,g), a(g) and b(g,NA) are model parameters described in [24] and Δz the axial resolution of the imaging device. Note that in the case of RCM, the authors mention that a corrective factor may be necessary to account for a difference between the theoretical axial resolution of the RCM system and the effective axial resolution to be taken into account in the model [28]. In the case of LCOCT, the axial resolution was computed as and no corrective factor was required based on results presented later in this paper.
G(NA,g) describes the average optical path length traveled by the photons collected by the OCT system, taking into account the numerical aperture NA and the scattering anisotropy factor g of the imaged sample. This term takes into account the fact that photons emitted at the edge of the beam travel a longer optical path than photons emitted on the optical axis. For NA = 0.5 (numerical aperture of the LCOCT device), G is close to 1 [24].
a(g) is called the scattering efficiency factor. It reflects the contribution of forward scattered photons, i.e., serpentile photons. Serpentile photons can contribute to the backscattered intensity because of their trajectory similar to that of ballistic photons. Then, if they follow a serpentile trajectory on the return path, photons can contribute to the collected intensity. Thus, serpentile photons slow down the extinction of the incident intensity by scattering in the sample. This effect has been formalized in the form of a function a(g):(5)with u, v, and w parameters determined by MonteCarlo simulations [24, 26]. For small values of g (isotropic scattering medium), a(g) is close to 1, which implies that μ_{eff} = μ as in the singlescattering regime. However, for values of g tending towards 1 (highly forward scattering medium), a(g) tends towards 0, leading μ_{eff} < μ, meaning that attenuation of light is slowed down by the contribution of serpentile photons. The authors have shown that this modeling of the contribution of multiple forward scattered photons is valid as long as μ_{s}z ≤ 3 [25]. However, this modeling does not take into account the possible contribution of photons multiply scattered along a nonballistic optical path. This typically allows the measurement of scattering coefficients on the order of 10 mm^{−1} down to a depth of 400 μm, and down to 200 μm for the most scattering phantoms (up to 20 mm^{−1}). Jacques’ model thus appears to be very well adapted to the measurement of optical properties of biological tissues imaged with LCOCT.
b(g,NA) is qualified as collection efficiency and represents the fraction of light scattered from probed volume and collected by the microscope objective of the LCOCT device. b(g,NA) is ruled by the phase function of the sample, described by the Henyey–Greenstein function in this model [45], and the collection angle of the microscope objective, given by the value of NA.
In this modeling of R(z), ρ and μ_{eff} can be considered as experimental observables and can be extracted from an LCOCT image as described in the following section.
2.3.3 Processing of LCOCT images for extracting μ_{s} and g on monolayered phantoms
The principle of our method consists in measuring μ_{eff} and ρ parameters from an LCOCT image, and to convert these parameters into the optical properties μ_{s} and g using the previously described model proposed by Jacques et al. To do so, a 3D LCOCT image is acquired for each phantom. The 3D image results from the acquisition of a stack of 2D LCOCT horizontal images (i.e., enface sections). Each image of the stack is obtained by demodulating the interferometric lines acquired successively in the horizontal plane during the lateral scan, using a phaseshifting algorithm [4]. No processing is applied to the LCOCT images in order not to distort the measurement of the scattering properties. The first step is to compute the reflectance R(z). Since phantoms have a homogeneous distribution of scattering particles, the mean reflectance R(z) in the 3D image is considered. All Ascans within the 3D LCOCT image are averaged to obtain a mean depthdependent intensity I(z). Since the phantoms are composed of discrete scattering particles, averaging all Ascans within the 3D image provides the smoothest possible intensity profile I(z), which will then help maximize the quality of the linear regression that will be applied as described in the following. The mean reflectance R(z) is then obtained by converting I(z) into dimensionless units of reflectance using a calibration constant f: R(z) = fI(z). This simple calibration using a proportionality factor is made possible since LCOCT operates in focustracking mode and thus does not require confocal functions to be taken into account as is sometimes the case with conventional OCT [12, 18].
ρ and μ_{eff} are extracted by applying a least square linear regression on log R(z) (Fig. 2), from the surface of the sample down to a depth of about 150 μm. ρ is obtained from log ρ, which corresponds to the intercept with depth z = 0 representing the interface between the glass plate and the phantom (intensity peak visible at z = 0 in Fig. 2); μ_{eff} is given by half the slope of the linear regression.
Figure 2 (a) 3D LCOCT image (horizontal or en face view and vertical or crosssectional view) of a phantom made of PDMS and TiO_{2} particles, in logarithmic intensity scale and (b) averaged intensity profile I(z) in the 3D LCOCT image as a function of depth, in logarithmic scale. A linear regression (red line) is applied on the intensity profile to obtain the measurement of the pair of observables μ_{eff} and ρ. 
Finally, the resulting parameters ρ and μ_{eff} extracted from the 3D LCOCT image are mapped to the optical properties μ_{s} and g using the model by Jacques et al. previously described. Several ways can be considered to recover μ_{s} and g from μ_{eff} and ρ using equations (4) of the Jacques’ model. In this paper, the following relationship was used considering μ_{a} ≪ μ_{s}:(6)The expression of , proposed by Jacques et al. in [24], is a function of λ, NA and g and is independent from μ_{s}. The curve as a function of g (for λ = 800 nm and NA = 0.5), shown in Figure 3, can thus be used to compute the value of g from the two observables ρ and μ_{eff}. μ_{s} can then be recovered from the value of μ_{eff} and g using equations (4). The resulting μ_{s} and g values correspond to the mean optical properties over the LCOCT lateral field of view of 1.2 mm × 0.5 mm (x × y).
Figure 3 Ratio of μ_{eff}/ρ as a function of g, calculated for λ = 800 nm, NA = 0.5 and Δz = 1.2 μm. 
2.3.4 Measurements on bilayered phantoms
For bilayered phantoms, two distinct areas of linear decay can be observed on the averaged intensity profile plotted in Figure 4, corresponding to the two layers of the sample. Since the two layers have different optical properties, they exhibit different slopes. As for the monolayered phantoms, the intensity peak at depth z = 0 indicates the interface between the glass slide and the phantom. In this example, the intensity peak around 110 μm depth shows the interface between the two layers. To extract the values of μ_{eff} and ρ in each layer (noted ρ_{top}, μ_{efftop} and ρ_{bottom}, μ_{effbottom}), the two layers are segmented by manually delineating the two areas of linear decay. A linear fit is then applied separately on each layer, as depicted in Figure 4. In the bottom layer, the intercept with the interface between the two layers (at depth z = 110 μm) corresponds to log due to attenuation in the top layer of thickness δz. ρ_{bottom} is retrieved from by multiplying it with a correction factor to compensate from attenuation in the top layer.
Figure 4 Averaged intensity profile R(z) in the 3D LCOCT image of a bilayered phantom as a function of depth (in logarithmic scale). For each layer, a linear regression (red line) is applied on the intensity profile to obtain the measurement of the pair of observables (ρ_{top}, μ_{top}) and (ρ_{bottom}, μ_{bottom}). The value of ρ_{bottom} is retrieved from the intercept with the interface between the two layers (z = 110 μm) corrected from attenuation in the top layer. 
The resulting parameters ρ_{top}, μ_{efftop} and ρ_{bottom}, μ_{effbottom} are then mapped to the optical properties μ_{s} and g in each layer using the model by Jacques et al. as described in the previous section.
2.3.5 Calibration
As introduced in the previous section, the mean intensity I(z) (in gray levels) in an LCOCT image can be converted into dimensionless units of reflectance R(z) using a calibration constant f such that R(z) = fI(z). To determine f, one of the phantoms was used as a calibration phantom. On the one hand, μ_{s} and g of the calibration phantom were obtained from integrating spheres and collimated transmission measurements. Using the model of Jacques et al. [24], the ratio was determined from the g value measured for this phantom. On the other hand, a 3D image of the calibration phantom was acquired. The values of I_{0}ρ_{calib} and μ_{calib} were extracted by applying a linear regression on the mean intensity I(z) in the image. Knowing the value of the expected r_{calib} ratio, the value of ρ_{calib} can be obtained from μ_{effcalib}. A calibration factor f can then be calculated as the ratio of the expected value for ρ_{calib} and the value of I_{0}ρ_{calib} measured from the 3D LCOCT image:(7)Other calibration approaches can be considered, but this approach has the advantage that it does not involve the value of μ_{s} measured using integrating spheres and collimated transmission. This is an interesting feature: while the value of g is independent from particle density, this is not the case of μ_{s}. Since integrating spheres and collimated transmission measurements are performed on a field of 6 mm diameter and LCOCT measurements on a field of the order of 1 mm, it is preferable in case of particle density heterogeneities in the calibration phantom to use only the g value obtained using integrating spheres and collimated transmission to calibrate reflectance in LCOCT images.
2.4 Evaluation of uncertainties
For each phantom, integrating spheres and collimated transmission measurements were performed three times. Mean values of μ_{s} and g obtained with integrating spheres and collimated transmission were calculated from these three measurements.
Since LCOCT measurements are based on a calibration performed with a calibration phantom whose optical properties are obtained by integrating spheres and collimated transmission measurements, the uncertainties of the integrating spheres and collimated transmission method have a significant impact on the accuracy of LCOCT measurements of μ_{s} and g. Values of μ_{s} and g were computed from mean values of μ_{s} and g obtained from three 3D LCOCT image, and uncertainties on LCOCT measurements were estimated as follows: first, the uncertainty on the calibration factor f was estimated from uncertainties obtained on μ_{s} and g values of the calibration phantom measured using integrating spheres and collimated transmission measurements. From the resulting uncertainty on f, the uncertainty on ρ, due to calibration, was obtained for each fabricated phantom. Additional uncertainties on ρ and μ_{eff} were experimentally observed due both to least square linear regression on the intensity depth profile extracted from the 3D image and to slight variations of ρ and μ_{eff} from one image to another of the same phantom. Finally, uncertainties on μ_{s} and g were evaluated for each fabricated phantom by propagating total uncertainties on ρ and μ_{eff} through the model of Jacques [24] previously described.
3 Results and discussion
3.1 Monolayered phantoms
Comparisons of μ_{s} and g values of monolayered phantoms obtained from LCOCT mean intensity depth profiles and integrating spheres and collimated transmission measurements are given in Figures 5 and 6. Additionally, values of μ_{a} obtained from integrating spheres and collimated transmission measurements are given in Figure A1 of Appendix A.
Figure 5 Comparison of μ_{s} values obtained by LCOCT and combined integrating spheres and collimated transmission measurements on monolayered phantoms. Mean values and error bars were determined as explained in detail in Section 2.4. 
Figure 6 Comparison of g values obtained by LCOCT and combined integrating spheres and collimated transmission measurements on monolayered phantoms. Mean values and error bars were determined as explained in detail in Section 2.4. 
First of all, one can observe that the optical scattering properties obtained correspond to the range of properties targeted, with μ_{s} values between 1 and 25 mm^{−1} and g values between 0.68 and 0.94. The obtained results confirm that μ_{a} ≪ μ_{s} for all phantoms. For phantoms made of the same particles (in terms of refractive index and size), one can notice that the value of μ_{s} increases with the particle concentration, in agreement with Mie theory. The measured values of μ_{s} do not correspond exactly to the theoretical values, but the order of magnitude is similar. As stated earlier, the theoretical values are indicative due to uncertainties in the size and refractive index of the scattering particles. Moreover, it should be noted that during the fabrication process, particle losses may have occurred when pouring the particle/hardener mixture into the prepolymer, which may also contribute to some deviations in μ_{s}. Concerning the g parameter, we can note that here again the values of g for the phantoms 1, 4 and 8 have the same order of magnitude compared to Mie theory. However, the value of g obtained for phantom 6, composed of 400 nm SiO_{2} particles, is largely higher (0.94) than the target value (0.7) and is equivalent to the value of g obtained for phantom 8, composed of SiO_{2} particles of larger size (1 μm). A possible explanation could be the aggregation of 400 nm particles into larger particles, of the order of 1 μm.
When looking at the comparison of μ_{s} and g values between the two measurement methods, results show that, for phantoms 1–6 and 8, LCOCT is capable of providing measurements of optical properties consistent with those obtained by integrating spheres and collimated transmission measurements, with overlapping error bars (nearly overlapping for the μ_{s} measurement for phantom 2). For phantoms 7 and 9, one observes a good agreement of g values between the two methods. On the other hand, we observe a significant discrepancy between the value of μ_{s} measured by integrating spheres and collimated transmission and that measured by LCOCT. To date, we have no definitive explanation for this discrepancy, but a potential explanation could be the presence of aggregates in phantoms. As shown by highresolution bright field microscopy images (acquired with a 1.35 NA objective) given in Figure 7, aggregates could be observed in some areas of our phantoms, while other areas were well homogeneous. We also observed a greater number of aggregates as we penetrated deeper into the phantoms. LCOCT images covered a 1.2 mm × 0.5 mm × 0.5 mm (x × y × z) field of view, and were acquired in regions excluding aggregates (or at least minimizing their number). In contrast, the integrating spheres and collimated transmission measurements are based on transmission through the entire thickness of the phantom and over a lateral diameter of 6 mm, which potentially takes into account much more aggregates than LCOCT. This issue shows the limitation of our fabrication process. In the future, it would be necessary to revise the manufacturing process, for instance with better mechanical mixing to improve homogeneity.
Figure 7 Highresolution microscopic images of phantom 9 acquired with a 1.35 NA objective. (a) Image acquired in a homogeneous region without aggregates and (b) image acquired in a region with particle aggregates. 
The measurements of μ_{s} and g have fairly high uncertainties, represented by the error bars in Figures 5 and 6. The error bars associated with the integrating spheres and collimated transmission measurements reflect the difficulty to perform these measurements. Indeed, although being a reference method, the collimated transmission measurement is an experimental measurement that was difficult to perform in a repeatable way and that required improvements to the initial setup. In particular, since the phantoms were not perfectly flat, the ballistic beam could be deflected. It was therefore necessary to readjust the injection in the collecting fiber for each phantom. Another potential source of error in the collimated transmission measurement is the contribution of multiple scattered photons to the ballistic intensity when the value of g is high. The proportion of scattered photons collected compared to ballistic photons depends on the scattering properties and on the acceptance angle of the collection system. In this paper, the contribution of multiple scattering to collimated transmission was minimized at best using a fiber with a small core, low numerical aperture and placed far from the sample allowing to minimize the collection angle. Experimental difficulties lead to uncertainties not only on μ_{s} and g values determined with integrating spheres and collimated transmission, but also on LCOCT measurements since the calibration is performed using a phantom with optical properties determined by this method, as explained in Section 2.4. When propagated through the model, these errors are amplified for high values of μ_{s} and g. This is why the error bars are strongly dependent on the optical properties of the phantoms. Thus, they are the widest for phantoms 6–9 which have a high value of g.
In summary, despite the issues encountered with phantoms 7 and 9, results obtained show that our measurement method is well suited to samples with medium to high scattering anisotropy (0.7 < g < 0.9) and scattering coefficients μ_{s} up to 12 mm^{−1}, which already allows for application to biological tissues.
3.2 Bilayered phantoms
The results obtained on the bilayered phantoms are shown in Figure 8. The two layers of the bilayered phantoms have very different values of μ_{s} and g. The results show that for each layer, LCOCT provides optical properties that are consistent with those obtained by the integrating spheres and collimated transmission method, with overlapping error bars. A very good agreement of g values can be observed between the two measurement methods, both for the top and bottom layers of the two bilayer phantoms.
Figure 8 Comparison of μ_{s} and g values (mean ± standard deviation) obtained by LCOCT and combined integrating spheres and collimated transmission measurements on two different bilayered phantoms. 
Concerning μ_{s} values, one can also observe a good agreement for all layers. Overall, the difference between the two measurement methods on μ_{s} remains of the same order as the difference obtained on the monolayered phantoms. One can notice that the value of μ_{s} in the bottom layer of the bilayer phantom 2 is slightly lower than the value of μ_{s} obtained with the monolayer phantom 5 alone, a difference that can possibly be explained by a more important contribution of multiple scattering than with the monolayer phantom 5 since the linear regression on the bottom layer is applied deeper (from 100 to 200 μm). The model of Jacques et al. takes into account the multiple scattering in certain extent and when scattering is directed forward. In order to better account for multiple scattering in its entirety, multiple scattering models such as the EHF model proposed by Thrane et al. [21] could be used. However, the investigations of the EHF model carried out so far have not been conclusive and have led to unrealistic values of optical properties, an issue that was previously mentioned in [46]. It was suggested in [18] that the model needs a priori knowledge of μ_{s} or g since they are codependent parameters in the model. Moreover, this model is not yet suitable for the application to multilayered samples due to its complexity.
Nevertheless, our results obtained on bilayered phantoms demonstrate the potential of LCOCT to provide layerbased measurements of distinct μ_{s} and g values from a single 3D image, unlike the integrating spheres and collimated transmission method which requires the two layers to be analyzed separately. Compared to the current state of the art, the application of our method on bilayered phantoms with different values of μ_{s} and g in each layer is innovative since previously reported multilayered phantoms have identical g values in all layers. However, our fabrication process would need to be improved for obtaining a more homogeneous thin top layer. Indeed, as for monolayered phantoms, aggregate deposition in the bottom of the thin layers has sometimes been observed. One solution could be to cut a thin layer from a thicker sample that would be more homogeneous.
4 Conclusion and perspectives
We have developed a method for measuring the scattering coefficient μ_{s} and the scattering anisotropy factor g of a sample from a single 3D image of that sample acquired by LCOCT. Our approach is based on the extraction of two observables ρ and μ_{eff} from the LCOCT image, by applying a linear fit to the mean intensity depthprofile within the 3D image (in logarithmic scale). Using a calibration with a phantom of known optical properties and a model previously introduced in the literature [24], the values of ρ and μ_{eff} can be mapped to the values of the scattering parameters μ_{s} and g. We tested our method on PDMS phantoms containing nano and microscattering particles mimicking optical properties of typical biological tissues, and compared it to another commonly used method based on integrating spheres and collimated transmission measurements. The values obtained measured by LCOCT are consistent with those obtained from the integrating spheres and collimated transmission measurements, suggesting that LCOCT is a relevant tool to measure the optical scattering properties of semitransparent samples.
Compared to integrating spheres and collimated transmission measurements that require manipulating the sample (place it between the spheres, then move it on the collimated transmission bench) and involves constraints on the sample physical characteristics (minimum size corresponding to the beam diameter, minimum thickness to be placed between the two spheres without risk of damaging the sample, maximum thickness for the collimated transmission to be possible), LCOCT does not require specific sample handling and constraints, which simplifies the measurement procedure and would facilitate in vivo measurements of μ_{s} and g. Moreover, LCOCT allows measurements on multilayered samples with distinct optical properties in each layer, using a single 3D image, unlike integrating spheres and collimated transmission which require analyzing the different layers separately. In this paper, we demonstrate the applicability of our method to bilayered samples, with the ultimate aim of applying the approach to the two main layers of the skin, the epidermis and the dermis. However, our method is not limited to two layers, but could be applied to multilayered samples (>2) following the same approach as for bilayered samples, as described in Section 2.3.4.
Our layerbased approach is interesting since few papers have reported measurement by OCT of 1 distinct μ_{s} and g values on multilayered samples [47, 48]. Compared to these previous works, our developed method has the advantage of being very simple to implement since it is based on a linear fit and a scattering model, and requires only a preliminary calibration step, using a phantom with known optical properties. Since LCOCT is based on timedomain OCT with dynamic focusing, LCOCT measurements do not require corrections accounting for the confocal function of the focusing lens [12]; this is another reason why LCOCT is particularly appropriate for the application of Jacques’ model.
The model of Jacques et al. takes into account multiple scattering when scattering is strongly directed forward, but it does not extensively account for multiple scattering as the extended Huygens–Fresnel (EHF) model does [21]. However, investigations of the EHF model have not yet given conclusive results on our phantoms. Moreover, Jacques’ model does not account for absorption, which restricts application to purely scattering samples (excluding for instance biological tissues with a high concentration of blood vessels or melanin). Our approach is also limited by the calibration step using a phantom with optical properties determined by integrating spheres and collimated transmission measurements. In particular, the collimated transmission measurement is complex to implement, leading to significant uncertainties of the measurement of μ_{s} and g, especially for high values of μ_{s} and g. Other techniques for measuring g (e.g., goniometry) could be investigated to reduce the uncertainties of collimated transmission measurements. However, goniometry, as collimated transmission, is complex to perform and requires constraints on the geometry of the samples in order to be able to collect scattered intensity from all angles [49].
Since LCOCT provides morphological images, a future prospect could be to perform spatially resolved measurements of μ_{s} and g in a horizontal plane, in addition to the layerbased measurements obtained on the bilayered phantoms. Such spatiallyresolved measurements could be obtained by dividing the 3D LCOCT image into macrovoxels, in which values of μ_{s} and g could be extracted using the mean intensity depth profile in each macrovoxel. A macroscopic spatial distribution of μ_{s} and g could then be obtained. Further investigation is required to apply our method to biological tissues. LCOCT is an imaging modality designed for biological tissue imaging, and in particular for skin imaging in vivo. Applied to biological tissue, the presented method could provide LCOCT with the ability to quantify the scattering properties of tissue imaged in vivo, complementary to the qualitative information provided by the highresolution of images. Potentially, the method could be used in vivo to characterize biological tissues and monitor structural changes that occur during the development of a pathology.
Acknowledgments
The authors would like to thank Pr. Audrey K. Bowden for the valuable and relevant discussion on the models allowing the measurement of optical properties in OCT. The authors also wish to thank PhD student Marc Chammas for his help in performing measurements using integrating spheres and collimated transmission setup. They are also grateful to the whole team of engineers at DAMAE Medical for technical support.
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A. Absorption coefficient measurement
See Figure A1.
Figure A1 Values of μ_{a} obtained at 785 nm from integrating spheres measurements for monolayered phantoms. 
All Tables
Summary of monolayered phantoms composition. All phantoms are made of a PDMS matrix including scattering particles of different materials, sizes and concentrations.
Summary of bilayered phantoms composition, given by the material, size and weight concentration of the scattering particles embedded in PDMS.
All Figures
Figure 1 LCOCT vertical image (i.e., crosssectional view) of a bilayered phantom. The LCOCT image is displayed in a logarithmic intensity scale (arbitrary unit). 

In the text 
Figure 2 (a) 3D LCOCT image (horizontal or en face view and vertical or crosssectional view) of a phantom made of PDMS and TiO_{2} particles, in logarithmic intensity scale and (b) averaged intensity profile I(z) in the 3D LCOCT image as a function of depth, in logarithmic scale. A linear regression (red line) is applied on the intensity profile to obtain the measurement of the pair of observables μ_{eff} and ρ. 

In the text 
Figure 3 Ratio of μ_{eff}/ρ as a function of g, calculated for λ = 800 nm, NA = 0.5 and Δz = 1.2 μm. 

In the text 
Figure 4 Averaged intensity profile R(z) in the 3D LCOCT image of a bilayered phantom as a function of depth (in logarithmic scale). For each layer, a linear regression (red line) is applied on the intensity profile to obtain the measurement of the pair of observables (ρ_{top}, μ_{top}) and (ρ_{bottom}, μ_{bottom}). The value of ρ_{bottom} is retrieved from the intercept with the interface between the two layers (z = 110 μm) corrected from attenuation in the top layer. 

In the text 
Figure 5 Comparison of μ_{s} values obtained by LCOCT and combined integrating spheres and collimated transmission measurements on monolayered phantoms. Mean values and error bars were determined as explained in detail in Section 2.4. 

In the text 
Figure 6 Comparison of g values obtained by LCOCT and combined integrating spheres and collimated transmission measurements on monolayered phantoms. Mean values and error bars were determined as explained in detail in Section 2.4. 

In the text 
Figure 7 Highresolution microscopic images of phantom 9 acquired with a 1.35 NA objective. (a) Image acquired in a homogeneous region without aggregates and (b) image acquired in a region with particle aggregates. 

In the text 
Figure 8 Comparison of μ_{s} and g values (mean ± standard deviation) obtained by LCOCT and combined integrating spheres and collimated transmission measurements on two different bilayered phantoms. 

In the text 
Figure A1 Values of μ_{a} obtained at 785 nm from integrating spheres measurements for monolayered phantoms. 

In the text 
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