2.1.

Monte Carlo Simulations

MC simulations were used to evaluate how the optical contrast of a thin ($<1000\text{-}\mu \mathrm{m}$ thick) superficial layer with respect to the subsurface main medium would affect the spatially resolved diffuse reflectance measured on the surface of the superficial layer. The medium geometry of the MC simulation is shown in Fig. 2. A planar two-layer medium geometry was implemented for evaluating the diffuse reflectance $R(\rho )$ measured at a lateral distance $\rho$ from the point of light injection into the medium. A set of optical properties including the refractive index $n$, the anisotropy factor $g$, the absorption coefficient ${\mu }_a$, and the reduced scattering coefficient ${\mu }_s^{\prime }$ were assigned independently to the superficial layer and the main medium. These four optical properties were used as the primary parameters for each of the two layers. The parameters associated with the main medium are marked by a subscript of “0,” and those with the superficial layer by a subscript of “1.”

Fig. 2

The two-layer medium geometry in the MC simulation for evaluating the diffuse reflectance $R(\rho )$ at a distance $\rho$ from the position of photon injection to the medium. The superficial layer or the main medium is characterized by four independent optical properties including refractive index $n$, anisotropy factor $g$, absorption coefficient ${\mu }_a$, and reduced scattering coefficient ${\mu }_s^{\prime }$. The optical properties of the main medium are marked with a subscript of “0,” and those of the superficial layer “1.” The thickness of the superficial layer is $\tau$.

The MC simulation was performed using the “MC solver panel” of Virtual Photonics General-Purpose ATK 2.2.0 Beta,³³ by custom defining the positions of the detector and the superficial-layer thickness in the off-line editable Input File and changing the medium optical properties using the online graphical user interface. A total of 1000 detector points were placed on the surface of the superficial layer, corresponding to a distance of 0.01 to 10 mm from the point of photon injection at an interval of $10\mu \mathrm{m}$. The 10-mm span of the SDS in the MC simulations covers the 3-mm separation between the source and the detector of the single source–detector pair on the applicator probe used for diffuse reflectance measurements from the liver and kidney as will be detailed in the corresponding section. Each MC simulation was executed with a total of 100,000 photons, for a total tissue thickness of 10 cm. The MC simulations were executed with only the Henyey–Greenstein (HG) phase function as it was the one accessible on the virtual photonics MC solver.

Each MC simulation corresponded to the superficial layer differing from the main medium in only one of the four primary parameters including $n_1$, $g_1$, ${\mu }_{a1}$, and ${\mu }_{s1}^{\prime }$. Two stages of MC simulations were performed. The first stage of MC simulations evaluated the effect of each of the four primary parameters of the superficial layer at a fixed setting of the main medium parameters on the diffuse reflectance for a superficial layer thickness ranging from 10 to $1000\mu \mathrm{m}$. The individual patterns revealed by the first-stage MC simulations, which constituted 94.9% of the total number of simulations, regarding how each of the four capsular contrasts over the main medium affected the diffuse reflectance were used to gauge the analytical model development. As will be shown from the first-stage of the MC simulations conducted for a fixed setting of the main medium properties, the diffuse reflectance from the two-layer geometry with a thin superficial layer ($\le 1000\text{-}\mu \mathrm{m}$ thick) is not sensitive to $n_1$ and $g_1$ (or equivalently the contrast of each over its main medium counterpart) over their respective ranges covering more than what a biological tissue will present. These outcomes were based upon to constrain the parameter setting in the next stage of MC simulation that was intended for evaluating the effect of the change of main medium properties on diffuse reflectance in a more pathologically relevant parameter setting of the superficial layer. The subsequent stage of MC simulations thus was performed at the absence of a superficial layer contrast on the refractive index or anisotropy factor over the main medium. In consideration of the high scattering of organ capsule due to collagen content²⁵ and a liver capsule being $\sim 10\text{-}\mu \mathrm{m}$ thick and a kidney capsule being $\sim 200\text{-}\mu \mathrm{m}$ thick, the superficial layer in the second stage of MC simulations was thus fixed at a high scattering property with the thicknesses of 10 and $200\mu \mathrm{m}$ only, whereas the ${\mu }_{a0}$ and ${\mu }_{s0}^{\prime }$ of the main medium were changed to evaluate how much the main medium property would affect the spatially resolved surface diffuse reflectance at a fixed setting of a highly scattering superficial layer.

Table 1 shows the parameter configuration of the first-stage MC simulations. In these MC simulations, the main medium was kept fixed at the following baseline set of parameters: $n_0=1.40$, $g_0=0.90$, ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$. To evaluate the effect of the refractive index of the superficial layer to the diffuse reflectance, the $n_1$ was changed from 1.30 to 1.50 at a step increment of 0.01, whereas keeping other parameters constant as $g_1=0.90$, ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s1}^{\prime }=1.0{\mathrm{mm}}^{-1}$. To evaluate the effect of the anisotropy factor of the superficial layer to the diffuse reflectance, the $g_1$ was changed from 0.80 to 0.99 at a step increment of 0.01, whereas keeping other parameters constant as $n_1=1.40$, ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s1}^{\prime }=1.0{\mathrm{mm}}^{-1}$. To evaluate the effect of the absorption properties of the superficial layer to the diffuse reflectance, the ${\mu }_{a1}$ was changed from 0.001 to $0.01{\mathrm{mm}}^{-1}$ at a step increment of $0.001{\mathrm{mm}}^{-1}$, then from 0.01 to $0.1{\mathrm{mm}}^{-1}$ at a step increment of $0.01{\mathrm{mm}}^{-1}$, and thereafter from 0.1 to $1.0{\mathrm{mm}}^{-1}$ at a step increment of $0.1{\mathrm{mm}}^{-1}$, whereas keeping other parameters constant as $n_1=1.40$, $g_1=0.90$, and ${\mu }_{s1}^{\prime }=1.0{\mathrm{mm}}^{-1}$. To evaluate the effect of the reduced scattering properties of the superficial layer to the diffuse reflectance, the ${\mu }_{s1}^{\prime }$ was changed from 0.1 to $1.0{\mathrm{mm}}^{-1}$ at a step increment of $0.1{\mathrm{mm}}^{-1}$, and then from 1.0 to $10.0{\mathrm{mm}}^{-1}$ at a step increment of $1.0{\mathrm{mm}}^{-1}$, whereas keeping other parameters constant as $n_1=1.40$, $g_1=0.90$, and ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$. The diffuse reflectance at any of the aforementioned setting of the optical parameters of the superficial layer was evaluated at the superficial layer thickness ranging from 10 to $100\mu \mathrm{m}$ at a step increment of $10\mu \mathrm{m}$, and then from 100 to $1000\mu \mathrm{m}$ at a step increment of $100\mu \mathrm{m}$. These configurations resulted in $\sim 1670$ MC simulations corresponding to the superficial layer that differs from the main medium in one of four primary optical properties, not including the duplicated baseline sets.

Table 1

The properties of the superficial layer at fixed main medium properties.

Table 2 shows the parameter configuration of the second-stage MC simulations specific to superficial layer with fixed optical properties including a highly reduced scattering coefficient and an underlying main medium at various absorptions and reduced scattering coefficients. In these MC simulations, the properties of the superficial layer were kept fixed at the following parameters: $n_1=1.40$, $g_1=0.90$, ${\mu }_{a1}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s1}^{\prime }=10.0{\mathrm{mm}}^{-1}$. To evaluate how the absorption of the main medium affects the diffuse reflectance, the ${\mu }_{a0}$ was changed from 0.001 to $0.01{\mathrm{mm}}^{-1}$ at a step increment of $0.001{\mathrm{mm}}^{-1}$, then from 0.01 to $0.1{\mathrm{mm}}^{-1}$ at a step increment of $0.01{\mathrm{mm}}^{-1}$, and thereafter from 0.1 to $1.0{\mathrm{mm}}^{-1}$ at a step increment of $0.1{\mathrm{mm}}^{-1}$, with the other parameters of the main medium kept constant as $n_0=1.40$, $g_0=0.90$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$. To evaluate how the reduced scattering of the main medium affects the diffuse reflectance, the ${\mu }_{s0}^{\prime }$ was changed from 0.1 to $1.0{\mathrm{mm}}^{-1}$ at a step increment of $0.1{\mathrm{mm}}^{-1}$, and then from 1.0 to $10.0{\mathrm{mm}}^{-1}$ at a step increment of $1.0{\mathrm{mm}}^{-1}$, with the other parameters of the main medium kept constant as $n_0=1.40$, $g_0=0.90$, and ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$. The range of the absorption coefficient of the main medium spanning three decades from 0.001 to $1.0{\mathrm{mm}}^{-1}$ and the range of the reduced scattering coefficient of the main medium spanning two decades from 0.1 to $10.0{\mathrm{mm}}^{-1}$ cover the optical properties typical to soft biological tissues including the liver and kidney.³⁴ The diffuse reflectance at any of the aforementioned setting of the optical parameters of the main medium was evaluated at the superficial layer thicknesses of 10 and $200\mu \mathrm{m}$ only as these two numbers represent, respectively, the thickness of the liver capsule and kidney capsule. These configurations resulted in over 90 MC simulations corresponding to the main medium that differs from the superficial layer in one of the four primary optical properties. Tables 1 and 2 when combined amount to $\sim 1760$ MC simulations corresponding to the semi-infinite tissue geometry containing a superficial layer that is optically heterogeneous comparing with the underlying main medium. These MC simulations were executed at random sequences on multiple computers having various system configurations.

Table 2

The properties of the main medium at fixed superficial layer properties.

2.2.

Analytical Model of Surface Diffuse Reflectance From a Semi-Infinite Medium in the Presence of a Thin Superficial Layer

The effect of capsule on the surface diffuse reflectance can be appreciated by identifying how the surface diffuse reflectance of the subcapsular parenchyma at the presence of the capsule may differ from that at the absence of the capsule. At the absence of the capsule, the parenchyma, which is referred to as the main or baseline medium, approximates a homogeneous domain of photon propagation for surface diffuse reflectance. The presence of the capsule introduces a thin superficial layer that may also be optically heterogeneous in comparison with the main medium. How much this capsular layer changes the surface diffuse reflectance from the subsurface main medium when compared with the baseline will be affected by not only the extent of the contrasts of the optical properties of the capsular layer over the main medium but also the thickness of the capsule. Apparently, if the superficial layer presents no contrasts of the optical properties over the main medium, the presence of the superficial layer shall have no effect to the surface diffuse reflectance of the composite media containing the superficial layer and the main medium, regardless of the thickness of the capsular layer. In contrast, if the capsular layer is optically heterogeneous comparing with the main medium, the effect of this capsular layer on the surface diffuse reflectance will decrease and eventually vanish as the capsular layer becomes thinner and disappears. An analytical model addressing the effect of the superficial layer on the surface diffuse reflectance from a layered medium thus shall converge to the case of surface diffuse reflectance from a homogeneous medium when either the contrast of the optical properties of the superficial layer over the main medium vanishes or the thickness of the superficial layer zeros. Conversely, the effect of the superficial layer to the surface diffuse reflectance of the multilayer medium may be accounted for by a differential change of the surface diffuse reflectance from a baseline homogeneous medium, which shall elevate as the contrast of the optical properties of the superficial layer over the main medium increases or the superficial layer thickens. These considerations, heretofore, substantiate the analytical model development as detailed in the following.

Recently, we have demonstrated a simple, convenient, and transparent model of diffuse reflectance from a homogeneous semi-infinite medium³³ that has been validated over an SDS ranging from 0.01 to 10 mm and tissue diffusing conditions including the absorption coefficient as strong in magnitude as the scattering coefficient $({\mu }_a\sim {\mu }_s^{\prime })$ at a medium reduced scattering coefficient of ${\mu }_s^{\prime }=1.0{\mathrm{mm}}^{-1}$. The central scheme of this model is a master–slave dual-source configuration that sets two virtual sources, with one being the primary or the master source and the other being the secondary or the slave source, whose position and strength are dependent upon the master source. Both the master source and the slave source are placed along the ballistic line of photon injection for the apparent geometric symmetry, as shown in Fig. 3(a). For a homogeneous medium confined in a semi-infinite geometry that is illuminated by a directional point-beam, the master source is set at a depth of one reduced scattering pathlength or $1/{\mu }_s^{\prime }$, and the slave source is set to be much closer to the medium boundary at a depth less than one scattering step-size [i.e., the depth $<1/{\mu }_s$ or $(1-g)/{\mu }_s^{\prime }$]. The master source is similar, albeit not identical, to the commonly implemented isotropic source in delivering the far-field pattern of surface photon irradiation for diffuse reflectance. The slave source that is set to be much closer to the medium boundary than the master source is conveniently produced the aggressive peaking of the diffuse reflectance toward the point-of-entry as applying to the common fiber-based contact irradiation geometry involving a point-injection of photons into the medium.

Fig. 3

The medium geometry used for the analytical model development. (a) The geometry of the medium in the absence of the superficial layer, when only the master source and the slave source are considered. The master–source $S$ locates at a depth of $z_a=1/{\mu }_s^{\prime }$ from the point of entry. The slave-source $S^{*}$ locates at a depth of $z_a^{*}$ that is proximal to the point of entry than the position of a depth of $1/{\mu }_s$. The image source of S is at $z_a+2z_b$ away from the medium surface, and the image source of $S^{*}$ is at $z_a^{*}+2z_b$ away from the medium surface. (b) The geometry of the medium in the presence of a superficial layer that does not change the surface diffuse reflectance significantly when compared with that in the absence of the superficial layer. The master-source $S^{\text{master}}$ locates at a depth $z_a^{\text{master}}$ from the point of entry. The slave-source $S^{\text{slave}}$ locates at a depth of $z_a^{\text{slave}}$ that is much proximal to the point of entry than the position of the master source. A differential source $S^{\mathrm{diff}}$ appears with its position and strength dependent upon the significance of the superficial layer in terms of the optical contrast over the main medium and the layer thickness. The image source of $S^{\text{master}}$ is at $z_a^{\text{master}}+2z_b^{\text{master}}$ away from the medium surface. The image source of $S^{\text{slave}}$ is at $z_a^{\text{slave}}+2z_b^{\text{slave}}$ away from the medium surface. The image source of $S^{\mathrm{diff}}$ is at $z_a^{\mathrm{diff}}+2z_b^{\mathrm{diff}}$ away from the medium surface. (c) The geometry of the medium in the presence of a superficial layer that changes the surface diffuse reflectance significantly when compared with that in the absence of the superficial layer. The master-source $S^{\text{master}}$ locates at a depth $z_a^{\text{master}}$ from the point of entry. A weak slave-source $S^{\text{slave}}$ locates at a depth of $z_a^{\text{slave}}$ that is much proximal to the point of entry than the position of the master source. A strong differential source $S^{\mathrm{diff}}$ appears with its position and strength dependent upon the significance of the superficial layer in terms of the optical contrast over the main medium and the layer thickness. The image source of $S^{\text{master}}$ is at $z_a^{\text{master}}+2z_b^{\text{master}}$ away from the medium surface. The image source of $S^{\text{slave}}$ is at $z_a^{\text{slave}}+2z_b^{\text{slave}}$ away from the medium surface. The image source of $S^{\mathrm{diff}}$ is at $z_a^{\mathrm{diff}}+2z_b^{\mathrm{diff}}$ away from the medium surface.

We modify this master–slave dual-source approach to model the spatially resolved diffuse reflectance from an otherwise homogeneous semi-infinite medium in the presence of a thin superficial layer, as shown in Fig. 3(b). The modification introduces a differential change to the master–slave dual-source configuration to account for the change to surface diffuse reflectance from the composite media due to the presence of the thin superficial layer when compared with that from the homogeneous main medium only—referred to hereafter as the baseline. When either the contrast of the optical properties of the superficial layer with a fixed thickness over the main medium or the thickness of the superficial layer with a fixed contrast of the optical properties over the main medium increases, the change to the diffuse reflectance with respect to the baseline shall increase. The increased change to the diffuse reflectance at a stronger superficial layer as shown in Fig. 3(c) (corresponding to either a greater contrast of the optical properties of the superficial layer over the main medium for a given thickness of the superficial layer or a thicker superficial layer at a given contrast of the optical properties of the superficial layer over the main medium) at the same main medium could cause the master–slave sources to deviate more from their respective baseline settings. As the superficial layer becomes more significant in terms of the contrast of its optical properties over the main medium or the layer thickens, the effective medium properties associated with each source could also vary as to affect the photon irradiation contributed by the individual source. In regard to the changes in the diffuse reflectance from the baseline, a differential source is thus introduced. The contribution to the diffuse reflectance by the differential source is set to increase as the superficial layer becomes stronger in either its contrast of optical properties over the main medium or layer thickness and conversely to disappear as the superficial layer vanishes due to no contrast of optical properties over the main medium or too thin to be accounted for.

The following analytical development treats the diffuse reflectance from a semi-infinite medium containing a thin superficial layer overlaying an otherwise homogeneous main medium as the composition of three photon irradiation components: a master component, a slave component, and a differential component. When the superficial layer becomes stronger due to either increasing of the contrast of optical properties over the main medium or thickening of the layer, the differential component is set to increase, whereas the master and the slave components are set to deviate from their corresponding baseline values. The decomposition of the surface photon irradiation to three components must also warrant that when the superficial layer vanishes at the cases of either the superficial layer presenting no contrast of optical properties over the main medium or the thickness of the superficial layer with an accountable contrast of optical properties over the main medium approaching zero, the differential component must vanish, whereas the master component and the slave component shall return to their respective baseline values associated with a homogeneous main medium.

In implementing the master–slave dual-source configuration for the diffuse reflectance from a semi-infinite homogeneous medium,³³ the contribution to the surface diffuse photon irradiance by the master source and the slave source was computed using the same set of basic formula, with only the variables differing according to whether it is associated with the master source or the slave source. The basic set of formula for the master–slave dual-source configuration was established according to the image-source approach commonly associated with the implementation of an extrapolated zero-boundary condition^35,36 that sets zero the photon fluence rate on a virtual boundary slightly away from the physical boundary. The position of the extrapolated boundary on which the photon fluence rate is set at zero is determined by the refractive index mismatch between the two media forming the physical boundary.^35,36 Similarly in this work of modeling the diffuse reflectance from a semi-infinite composite medium containing a thin superficial layer overlying an otherwise homogeneous medium, the master, slave, and differential components of the surface diffuse photon irradiance are computed also using the same set of basic formula governed by the image-source approach associated with the extrapolated zero-boundary condition but with the variables differing among the three components, according to the contrast of the optical properties of the superficial layer over the main medium and the thickness of the superficial layer. In the following, the basic set of formula governed by the image–source approach associated with the extrapolated zero-boundary condition and applying equally to the master, slave, and differential components is presented first. The principle of formulating the variables associated, respectively, with the master, slave, and differential components for the variables to depend upon the contrast of the optical properties of the superficial layer over the main medium and the thickness of the superficial layer is described afterward. The actual forms of all variables used for the master, slave, and differential components are listed in Appendix A.

In the analytics that follows, any variable or entity that is associated with the master component is marked by a superscript of “master,” the slave component a superscript of “slave,” and the differential component a superscript of “diff.” To be consistent with the parameter notations used for the MC simulations, any variable that is uniquely defined by the properties of only the main medium is characterized by an ending subscript of “0,” and its superficial layer counterpart by an ending subscript of “1.” The four parameters including $n$, $g$, ${\mu }_a$, and ${\mu }_s^{\prime }$ are considered as the primary parameters as in MC simulations for either the main medium or the superficial layer, inferring that all other parameters used for deriving the diffuse reflectance are secondary parameters to be determined by these four primary parameters.

2.3.

Diffuse Reflectance Spectroscopy of Liver and Kidney Using a Lab-on-a-Crate Device

DRS of liver and kidney specimens was acquired using a device shown in Fig. 4. The DRS device was developed onto a wheeled crate for use in a clinical pathology laboratory. The output of a Halogen light source (Cuda I-150, Jacksonville, Florida) through a fiber-optical light-conduit was coupled by a $40\times$ achromatic objective lens (Olympus RMS40X, Thorlabs Inc, Newton, New Jersey) to a $400\text{-}\mu \mathrm{m}$ fiber that was directed to a custom applicator probe as the source channel for tissue illumination. The light diffusely propagated through the tissue was collected by a second $400\text{-}\mu \mathrm{m}$ fiber on the same applicator probe and at a 3-mm distance from the source fiber. The detector fiber was coupled to a compact visible/near-infrared spectrometer (NT58-303; Edmund Optics Inc., Barrington, New Jersey) with a 16-bit data resolution (0-65535).

Fig. 4

(Upper panel): Schematic diagram of the DRS system. A light source directs visible/near-infrared light to the tissue via a fiber on the tissue surface. Another fiber at a 3-mm distance from the source fiber collects the diffusely propagated light to a compact spectrometer. (Lower panel) the device built on a wheeled crate (left), the applicator probe (middle), and the hands-free configuration of the applicator probe for measuring liver specimen.

The applicator probe consisted of a 12.5-mm diameter inner probe and a 2-in.-diameter outer adaptor, both made of black plastic materials. The inner probe contained two $400\text{-}\mu \mathrm{m}$ fibers separated at 3 mm as the source–detector pair for DRS. The 12.5-mm diameter inner probe was tightly plugged into the outer adaptor with an outer diameter of 2 in. (50.4 mm). Once plugged in, the inner probe and the outer adaptor formed an integral handheld probe of 2 in. in diameter. The outer adaptor of 2 in. in diameter served for two purposes: (1) to suppress the ambient light affecting the volume of tissue sampled by the 3-mm separation source–detector pair at the center region of the inner probe having a diameter of 12.5 mm and (2) to align easily and reliably with a solid phantom of known properties for acquisition of a reference DRS profile. The handheld probe of 2 in. in diameter was also positioned on a platform for the probe to descend by its own weight onto a tissue specimen for hands-free contact with the specimen.

A reference DRS spectral profile, denoted as $I_{\text{phantom}}(\lambda )$, was acquired from a solid phantom (PB0335, INO, Québec, Canada) having an absorption coefficient of $0.021{\mathrm{mm}}^{-1}$ and a reduced scattering coefficient of $1.08{\mathrm{mm}}^{-1}$ at 800 nm. A noise spectral profile, denoted as $I_{\text{noise}}(\lambda )$, was acquired by placing the 2 in. handheld probe on a dark cloth under normal ambient light. The raw DRS spectral profile acquired from a tissue as denoted by $I_{\text{raw}}(\lambda )$ was postprocessed according to the following equation

The DRS study on the liver specimens was approved by the University of Oklahoma Health Sciences Center (OUHSC, IRB #8155). Within 24 h of cross clamp, livers not meeting the criteria for transplantation based on information including patient clinical history, warm ischemia time, or gross appearance were measured using the hands-free DRS surface probe in the clinical pathology lab one floor below the Surgery Floor of the University of Oklahoma Medical Center of OUHSC. The DRS measurements were obtained from a random site on the anterior surface of the right lobe and a cut section of the right lobe. The probe was topically cleaned using a swab saturated with 70% isopropyl alcohol before measuring on a different surface of a specimen. The DRS measurement from each site was the result of averaging over five data acquisitions of 100-ms exposure time each. A total of 10 livers were measured, resulting in 50 combined DRS measurements from the anterior surfaces of the right lobes that were averaged to generate one DRS profile representing DRS measured on the surface of an intact liver. Another set of 50 DRS measurements from the cut-sections of the right lobes of the same 10 livers were averaged to generate one DRS profile specific to measuring on the liver parenchyma in the absence of the liver capsule. Direct DRS measurement on the liver parenchyma immediately adjacent to the capsule was not attempted due to the difficulty of removing only the extremely thin liver capsule.

The kidney specimens were measured by DRS under the same protocol (IRB #8155). Within 24 h of cross clamp, kidneys not used for transplantation based on information including patient clinical history, warm ischemia time, gross appearance, and duration of preservation were measured. The DRS measurements were obtained from a random site on the anterior surface of the upper pole of a kidney, and the surface of the upper cortex after the capsule was removed. The DRS measurement from each site was the result of averaging over five data acquisitions of 100-ms exposure time each. A total of nine kidneys were measured, resulting in 45 combined DRS measurements from the anterior surfaces of the upper poles that were averaged to generate a DRS profile corresponding to measuring on the surface of an intact kidney. Another set of 45 DRS measurements from the capsule-removed upper cortex of the same nine kidneys were averaged to generate a DRS profile specific to measuring directly on the kidney parenchyma in the absence of the capsule.

3.1.

Effects of the Contrasts of Refractive Index and Anisotropy Factor of a Thin Superficial Layer Over the Main Medium on Spatially Resolved Diffuse Reflectance

Figure 6 shows the diffuse reflectance when the superficial layer presents various contrast of refractive index or anisotropy factor over the main medium with all other parameters kept at the corresponding baseline values. The results corresponding to varying the refractive index of the superficial layer are presented at the left column, and their counterparts for anisotropy factor at the right column. Any diffuse reflectance profile is normalized with respect to the baseline value corresponding to a homogeneous medium of $n_0=1.40$, $g_0=0.90$, ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$.

Fig. 6

The diffuse reflectance when the superficial layer presents various contrast of refractive index or anisotropy factor over the main medium with all other parameters kept at the corresponding baseline values. The results corresponding to varying the refractive index of the superficial layer are presented in (a) and (c), and their counterparts for anisotropy factor in (b) and (d). Any diffuse reflectance profile is normalized with respect to the baseline value. (a) Corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has a refractive index of 1.30 or 1.50, as the superficial layer thickness increases from $10\mu \mathrm{m}$ to 1 mm. (c) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $10\text{-}\mu \mathrm{m}$ thick or 1-mm thick while the refractive index of it increases from 1.30 to 1.50. (b) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has an isotropy factor of 0.80 or 0.99, as the superficial layer thickness increases from $10\mu \mathrm{m}$ to 1 mm. (d) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $10\text{-}\mu \mathrm{m}$ thick or 1-mm thick while the anisotropy factor of it increases from 0.80 to 0.99.

The Fig. 6(a) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has a refractive index of 1.30 (circles for MC and the lower solid blue line for model prediction) or 1.50 (diamonds for MC and the upper solid red line for model prediction), versus a main medium refractive index of 1.40, as the superficial layer thickness increases from 10 to $1000\mu \mathrm{m}$. The pattern of change of the diffuse reflectance estimated by the MC simulations over the range of the superficial layer thickness at a given refractive index of the superficial layer is not clear. The relative magnitude of the diffuse reflectance estimated by the MC simulations between a refractive index of 1.30 and a refractive index of 1.50 of the superficial layer is not clear either. The model-prediction, however, projects a slight reduction ($<5\%$) of the diffuse reflectance as the superficial layer thickens at either of the two refractive indices shown, and a slightly higher ($<10\%$) diffuse reflectance at superficial layer refractive index of 1.50 than that at 1.30 over the entire range of the superficial layer thickness. The observed changes associated with the refractive index of the superficial layer agree with what can be expected from changing the refractive index as it will cause a slight shift of the positions of the image sources through small modifications to the positions of the respective extrapolated zero-boundary lines. Figure 6(c) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $10\text{-}\mu \mathrm{m}$ thick (circles) or $1000\text{-}\mu \mathrm{m}$ thick (diamonds), whereas the refractive index of it increases from 1.30 to 1.50. The pattern of change of the diffuse reflectance estimated by the MC simulations over the range of the superficial layer refractive index is not clear. The model-prediction (solid line for $1000\text{-}\mu \mathrm{m}$-thick superficial layer), however, projects a slight elevation ($<10\%$) of the diffuse reflectance as the superficial layer refractive index increases from 1.30 to 1.50 at the $1000\text{-}\mu \mathrm{m}$ superficial-layer thickness.

Figure 6(b) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has an isotropy factor of 0.80 (circles for MC and the overlapping solid line for model prediction) or 0.99 (diamonds for MC and the overlapping solid line for model prediction), versus the main medium anisotropy factor of 0.90, as the superficial layer thickness increases from 10 to $1000\mu \mathrm{m}$. The pattern of change of the diffuse reflectance estimated by the MC simulations over the range of the superficial layer thickness is not pronounced. The relative magnitude of the diffuse reflectance estimated by the MC simulations between an isotropy factor of 0.80 and an anisotropy factor of 0.99 of the superficial layer is not pronounced either. The observed patterns that are indifferent between an isotropy factor of 0.80 and an anisotropy factor of 0.99 are expected at the 3-mm SDS as the difference in anisotropy for the same reduced scattering property will be exhausted at a pathlength several times of the reduced scattering pathlength. The model-prediction, however, projects a slight reduction ($<5\%$) of the diffuse reflectance as the superficial layer thickens at either of the two isotropic factors shown, and indistinguishable diffuse reflectance values between the superficial layer isotropy factor of 0.99 and 0.80 over the entire range of the superficial layer thickness. Figure 6(d) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $10\text{-}\mu \mathrm{m}$ thick (circles) or $1000\text{-}\mu \mathrm{m}$ thick (diamonds), whereas the anisotropy factor of it increases from 0.80 to 0.99. The pattern of change of the diffuse reflectance estimated by the MC simulations over the range of the superficial layer anisotropy factor is not pronounced. The model-prediction (solid line for $1000\text{-}\mu \mathrm{m}$-thick superficial layer) projects unrecognizable change of the diffuse reflectance as the superficial layer anisotropy factor increases from 0.80 to 0.99 at the $1000\text{-}\mu \mathrm{m}$-superficial-layer thickness.

3.2.

Effects of the Contrasts of Absorption Coefficient of a Thin Superficial Layer Over the Main Medium on Spatially Resolved Diffuse Reflectance

Figure 7 shows the diffuse reflectance when the superficial layer presents various contrast of absorption coefficient over the main medium with all other parameters kept at the corresponding baseline values. The results corresponding to increasing the thickness of the superficial layer when fixing the absorption coefficient of the superficial layer at $1.0{\mathrm{mm}}^{-1}$ are presented at the left column. The results corresponding to increasing the absorption coefficient of the superficial layer at a fixed superficial layer thickness of $1000\mu \mathrm{m}$ are presented at the right column. All values of the diffuse reflectance are normalized with respect to their respective baseline values corresponding to a homogeneous medium of $n_0=1.40$, $g_1=0.90$, ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$.

Fig. 7

The diffuse reflectance when the superficial layer presents various contrast of absorption coefficient over the main medium with all other parameters kept at the corresponding baseline values. All values of the diffuse reflectance are normalized with respect to their respective baseline values. (a) Corresponds to the diffuse reflectance at SDSs ranging from $10\mu \mathrm{m}$ to 10 mm when the superficial layer has a high absorption coefficient of $1.0{\mathrm{mm}}^{-1}$ and the following layer thicknesses: 10, 100, 500, and $1000\mu \mathrm{m}$. (c) Corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has an absorption coefficient of $1.0{\mathrm{mm}}^{-1}$ while the thickness of it increases from $10\mu \mathrm{m}$ to 1 mm. (b) Corresponds to the diffuse reflectance at SDSs ranging from $10\mu \mathrm{m}$ to 10 mm when the superficial layer of 1-mm thick has the following absorption coefficients: 0.01, 0.1, 0.5, and $1.0{\mathrm{mm}}^{-1}$. (d) Corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is 1-mm thick while the absorption coefficient of it increases from 0.001 to $1.0{\mathrm{mm}}^{-1}$.

Figure 7(a) corresponds to the diffuse reflectance at SDSs ranging from 0.01 to 10 mm when the superficial layer presents a high absorption coefficient of $1.0{\mathrm{mm}}^{-1}$ and the following thicknesses: 10, 100, 500, and $1000\mu \mathrm{m}$. The ${\mu }_{a1}=1.0{\mathrm{mm}}^{-1}$ represents a superficial layer having a contrast of 100 of the absorption coefficient over the main medium and the absorption as strong in magnitude as the reduced scattering (${\mu }_{a1}={\mu }_{s1}^{\prime }$). The following patterns are identified from (A): (1) for the superficial layer that is more absorbing than the main medium, the deviation of the diffuse reflectance with respect to the baseline value increases as the SDS increases. (2) For the superficial layer that is more absorbing than the main medium, the deviation of the diffuse reflectance with respect to the baseline value increases as the layer thickens. These patterns are all attributed to the increased attenuation of the photons by the superficial layer as it becomes more absorbing or thicker. Figure 7(c) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has an absorption coefficient of $1.0{\mathrm{mm}}^{-1}$, whereas the thickness of it increases from 10 to $1000\mu \mathrm{m}$. The pattern of the reduction of the diffuse reflectance as projected by the model (solid line) agrees well with that estimated by the MC simulations (diamonds) over the range of the superficial layer thickness. For a superficial layer with an absorption coefficient of $1.0{\mathrm{mm}}^{-1}$ representing 100 times more absorbing than the main medium, a $10\text{-}\mu \mathrm{m}$-thick layer will reduce the diffuse reflectance to $\sim 90\%$, a $100\text{-}\mu \mathrm{m}$-thick layer to $\sim 50\%$, and a $1000\text{-}\mu \mathrm{m}$ layer to $<3\%$ of the baseline value.

Figure 7(b) corresponds to the diffuse reflectance at SDSs ranging from 0.01 to 10 mm when the superficial layer of $1000\text{-}\mu \mathrm{m}$ thick has the following absorption coefficients: 0.001, 0.1, 0.5, $1.0{\mathrm{mm}}^{-1}$, corresponding to 1/10, 10, 50, and 100 times, respectively, of the absorption coefficient of the main medium. The following patterns are identified from (B): (1) for a $1000\text{-}\mu \mathrm{m}$-thick superficial layer, the diffuse reflectance with respect to the baseline value decreases as the absorption coefficient of the superficial layer increases. Figure 7(d) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $1000\text{-}\mu \mathrm{m}$ thick while the absorption coefficient of it increases from 0.001 to $1.0{\mathrm{mm}}^{-1}$. The MC simulations (diamonds) revealed the pattern of the diffuse reflectance that decreases as the superficial layer becomes more absorbing than the main medium. The pattern projected by the model (solid line) agrees with that estimated by MC (diamonds) when the superficial layer is at least three times as absorbing as the main medium. For a $1000\text{-}\mu \mathrm{m}$ superficial layer, an absorption coefficient of $0.03{\mathrm{mm}}^{-1}$ of the superficial layer will reduce the diffuse reflectance to $\sim 90\%$, a $0.2{\mathrm{mm}}^{-1}$ to $<50\%$, and a $1.0{\mathrm{mm}}^{-1}$ to $<3\%$ of the baseline value. For the superficial layer that is less than three times as absorbing as the main medium, the pattern projected by the model (solid line) oscillates with respect to the relatively smooth pattern estimated by MC (diamonds), indicating rooms for the model improvement.

3.3.

Effects of the Contrasts of Reduced Scattering Coefficient of a Thin Superficial Layer Over the Main Medium on Spatially Resolved Diffuse Reflectance

Figure 8 shows the diffuse reflectance when the superficial layer presents various contrast of reduced scattering coefficient over the main medium with all other parameters kept at the corresponding baseline values. The results corresponding to increasing the thickness of the superficial layer when fixing the reduced scattering coefficient of the superficial layer at $10.0{\mathrm{mm}}^{-1}$ are presented at the left column. The results corresponding to increasing the reduced scattering coefficient of the superficial layer at a fixed superficial layer thickness of $1000\mu \mathrm{m}$ are presented at the right column. All values of the diffuse reflectance are normalized with respect to their respective baseline values corresponding to a homogeneous medium of $n_0=1.40$, $g_1=0.90$, ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$.

Fig. 8

The diffuse reflectance when the superficial layer presents various contrast of reduced scattering coefficient over the main medium with all other parameters kept at the corresponding baseline values. All values of the diffuse reflectance are normalized with respect to their respective baseline values. (a) Corresponds to the diffuse reflectance at SDSs ranging from $10\mu \mathrm{m}$ to 10 mm when the superficial layer with a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ has the following layer thicknesses: 10, 100, and $1000\mu \mathrm{m}$. (c) Corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ while the thickness of it increases from $10\mu \mathrm{m}$ to 1 mm. (b) corresponds to the diffuse reflectance at SDSs ranging from $10\mu \mathrm{m}$ to 10 mm when the superficial layer of 1-mm thick has the following reduced scattering coefficients: 0.1, 2.0, 5.0, and $10.0{\mathrm{mm}}^{-1}$. (d) Corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is 1-mm thick while the reduced scattering coefficient of it increases from 0.1 to $10.0{\mathrm{mm}}^{-1}$.

Figure 8(a) corresponds to the diffuse reflectance at SDSs ranging from 0.01 to 10 mm when the superficial layer with a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ has the following thicknesses: 10, 100, and $1000\mu \mathrm{m}$. The ${\mu }_{s1}^{\prime }=10{\mathrm{mm}}^{-1}$ represents a superficial layer having a contrast of 10 of the reduced scattering coefficient over the main medium. The following patterns are remarkable: (1) for a superficial layer that is more scattering than the main medium, the diffuse reflectance with respect to the baseline value increases at near-field ($<0.1\mathrm{mm}$) but decreases at far-field ($>1\mathrm{mm}$). (2) For a superficial layer that is more scattering than the main medium, the deviation of the diffuse reflectance from the baseline value increases as the layer thickens. The second pattern is straightforward to appreciate. The first pattern can be expected qualitatively as the superficial layer becomes more scattering. As the superficial layer picks up scattering, the near-field diffuse reflectance associated with less photon pathlength thus more contribution from the superficial layer will increase, however, the far-field diffuse reflectance associated with deeper photon path will experience a reduction of the photon remission due to more photons escaping the path between the light injection and light detection after the photons are scattered more by the superficial layer. Figure 8(c) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer has a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ while the thickness of it increases from 10 to $1000\mu \mathrm{m}$. The pattern of the reduction of the diffuse reflectance projected by the model (solid line) agrees globally with that estimated by the MC simulations (diamonds) over the range of the superficial layer thickness. However, the model prediction seems to have under-estimated the deviation from the baseline values when the superficial layer is thinner than $60\mu \mathrm{m}$ and over-estimated the deviation from the baseline values when the superficial layer is thicker than $300\mu \mathrm{m}$. We note that, the model is not developed as a direct fit to the MC results; therefore, deviations of the model predictions from the MC results are expected when the medium properties or geometries do not strongly support the model basis. For a superficial layer with a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ representing 10 times more scattering than the main medium, a $10\text{-}\mu \mathrm{m}$-thick layer has essentially no effect to the diffuse reflectance at the 3-mm SDS, but a $100\text{-}\mu \mathrm{m}$ layer will reduce the diffuse reflectance at the 3-mm SDS to $\sim 90\%$, and a $1000\text{-}\mu \mathrm{m}$ layer to $<25\%$ of the baseline value, according to the smooth model-prediction.

Figure 8(b) corresponds to the diffuse reflectance at SDSs ranging from 0.01 to 10 mm when the superficial layer of $1000\text{-}\mu \mathrm{m}$ thick has the following reduced scattering coefficients: 0.1, 2.0, 5.0, and $10.0{\mathrm{mm}}^{-1}$, corresponding to 1/10, 2, 5, and 10 times of the reduced scattering coefficient of the main medium. The following patterns are remarkable: (1) For a $1000\text{-}\mu \mathrm{m}$-thick superficial layer, the deviation of the diffuse reflectance with respect to the baseline value increases as the reduced scattering coefficient of the superficial layer increases from the baseline value. The subfigure (D) corresponds to the diffuse reflectance at a 3-mm SDS when the superficial layer is $1000\text{-}\mu \mathrm{m}$ thick, whereas the reduced scattering coefficient of it increases from 0.1 to $10.0{\mathrm{mm}}^{-1}$. The MC simulations (diamonds) revealed a pattern of the reduction of the diffuse reflectance as the scattering contrast of the superficial layer over the main medium increases but with a lesser extent at a negative contrast of the reduced scattering of the superficial layer over the main medium. The pattern projected by the model (solid line) agrees qualitatively with that estimated by MC (diamonds), but the model prediction seems to have over-estimated the deviation from the baseline values when the superficial layer is less scattering than the main medium and under-estimated the deviation from the baseline values when the superficial layer is more scattering than the main medium. For a $1000\text{-}\mu \mathrm{m}$-thick superficial layer, a reduced scattering coefficient of $0.1{\mathrm{mm}}^{-1}$ of the superficial layer will reduce the diffuse reflectance to $\sim 70\%$, and a $10.0{\mathrm{mm}}^{-1}$ to $<25\%$ of the baseline value according to the smoother model predictions.

3.4.

Effects of the Absorption and Reduced Scattering Coefficients of the Main Medium on Spatially Resolved Diffuse Reflectance from a Highly Scattering Superficial Layer

Figure 9 shows the diffuse reflectance at a 3-mm SDS when the superficial layer of 10- or $200\text{-}\mu \mathrm{m}$ thick is set at a high reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ and the main medium presents various contrast of absorption and reduced scattering coefficient over the superficial layer with other optical parameters kept at the corresponding baseline values. All values of the diffuse reflectance are normalized with respect to the baseline values at a homogeneous medium of $n_0=1.40$, $g_1=0.90$, ${\mu }_{a0}=0.01{\mathrm{mm}}^{-1}$, and ${\mu }_{s0}^{\prime }=1.0{\mathrm{mm}}^{-1}$.

Fig. 9

The diffuse reflectance at a 3-mm SDS when the superficial layer of 10- or $200\text{-}\mu \mathrm{m}$ thick is set at a high reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ and the main medium presents various contrast of absorption and reduced scattering coefficient over the superficial layer with other parameters kept at the corresponding baseline values. All values of the diffuse reflectance are normalized with respect to the baseline values. (a) Corresponds to changing the absorption coefficient of the main medium from 0.001 to $1.0{\mathrm{mm}}^{-1}$, when keeping the superficial layer at an absorption coefficient of $0.01{\mathrm{mm}}^{-1}$ and reduced scattering coefficient of $10{\mathrm{mm}}^{-1}$. (b) Corresponds to changing the reduced scattering coefficient of the main medium from 0.1 to $10.0{\mathrm{mm}}^{-1}$ when keeping the superficial layer at an absorption coefficient of $0.01{\mathrm{mm}}^{-1}$ and reduced scattering coefficient of $10{\mathrm{mm}}^{-1}$.

The results in Fig. 9(a) correspond to changing the absorption coefficient of the main medium from 0.001 to $1.0{\mathrm{mm}}^{-1}$ when keeping the superficial layer at an absorption coefficient of $0.01{\mathrm{mm}}^{-1}$ and a reduced scattering coefficient of $10{\mathrm{mm}}^{-1}$. As the absorption of the main medium increases, the diffuse reflectance at the 3-mm SDS reduces monotonically. When the superficial layer having a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ is $10\text{-}\mu \mathrm{m}$ thick, the diffuse reflectance at the 3-mm SDS is $\sim 115\%$ of the baseline value for a main medium absorption coefficient of $0.001{\mathrm{mm}}^{-1}$, and reduces monotonically to $<3\%$ of the baseline value for a main medium absorption coefficient of $1.0{\mathrm{mm}}^{-1}$. When the superficial layer having a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ is $200\text{-}\mu \mathrm{m}$ thick, the diffuse reflectance at the 3-mm SDS is $\sim 75\%$ of the baseline value for a main medium absorption coefficient of $0.001{\mathrm{mm}}^{-1}$, and reduces monotonically to $<3\%$ of the baseline value when the main medium absorption coefficient reaches $1.0{\mathrm{mm}}^{-1}$.

The results in Fig. 9(b) correspond to changing the reduced scattering coefficient of the main medium from 0.1 to $10.0{\mathrm{mm}}^{-1}$ when keeping the superficial layer at an absorption coefficient of $0.01{\mathrm{mm}}^{-1}$ and a reduced scattering coefficient of $10{\mathrm{mm}}^{-1}$. As the reduced scattering coefficient of the main medium increases, the diffuse reflectance at the 3-mm SDS reveals a biphasic pattern of increasing initially and decreasing later. This biphasic pattern may be appreciated as the following: as the main medium picks up scattering, more photons reaching the main medium from the superficial layer will be scattered back into the superficial layer after multiple scattering events to cause an initial increase in the diffuse reflectance at the surface of the superficial layer; however, as the main medium continues to pick up scattering, more photons reaching the main medium from the superficial layer will be lost in the main medium due to multiple scattering to cause less photons to be returned to the superficial layer and subsequently a decrease in the diffuse reflectance at the surface of the superficial layer. When the superficial layer having a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ is $10\text{-}\mu \mathrm{m}$ thick, the diffuse reflectance at the 3-mm SDS is $\sim 20\%$ of the baseline value for a main medium reduced scattering coefficient of $0.1{\mathrm{mm}}^{-1}$, peaks at $\sim 99\%$ of the baseline value for a main medium reduced scattering coefficient of $1.0{\mathrm{mm}}^{-1}$, then decreases to $\sim 20\%$ of the baseline value for a main medium reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$. When the superficial layer having a reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$ is $200\text{-}\mu \mathrm{m}$ thick, the diffuse reflectance at the 3-mm SDS is $\sim 5\%$ of the baseline value for a main medium reduced scattering coefficient of $0.1{\mathrm{mm}}^{-1}$, peaks at $\sim 60\%$ of the baseline value for a main medium reduced scattering coefficient of $2.0{\mathrm{mm}}^{-1}$, then decreases to $\sim 20\%$ of the baseline value for a main medium reduced scattering coefficient of $10.0{\mathrm{mm}}^{-1}$.

3.5.

Diffuse Reflectance Spectroscopy of Liver and Kidney with and Without the Capsule

The device-specific DRS profiles corresponding to measuring on the capsular surface (thicker solid violet line) and subcapsular surface (thicker dashed red line) of a kidney appear below the black line of unity in Fig. 10(a). The device-specific DRS profiles corresponding to measuring on the capsular surface (thinner solid light blue line) and cross section (thinner dashed blue line) of a liver appear globally below the lines corresponding to the kidney in Fig. 10(a). The diffuse reflectance from the liver cross section is lower than that from the subcapsular kidney over 400 to 940 nm, indicating stronger pigment attenuation in the liver than in the kidney if the tissue scattering in the liver can be assumed to be similar to that in the kidney. Figure 10(b) plotted the ratio between the device-specific DRS measured on the capsular surface over that on the subcapsular surface of the kidney (thick solid red line) and the ratio between the device-specific DRS measured on the capsular surface over that on the cross-section of the liver (thin solid blue line), both appearing above the black line of unity. The ratio corresponding to the kidney varies between 1.01 (or 101%) around 800 nm and 1.7 (or 170%) that occurs at $\sim 500\mathrm{nm}$. The ratio corresponding to the liver varies between 1.05 (or 105%) around 800 nm and 1.2 (or 120%) that occurs at shorter ($<400\mathrm{nm}$) or longer ($>1000\mathrm{nm}$) wavelength ranges.

Fig. 10

(a) The DRS profile corresponding to measuring on the capsular surface of a kidney is plotted as the thicker solid violet line. The DRS profile corresponding to measuring on the subcapsular surface of a kidney is plotted as the thicker dashed red line. The DRS profile corresponding to measuring on the capsular surface of a liver is plotted as the thinner solid light blue line. The DRS profile corresponding to measuring on the cross section of a liver is plotted as the thinner dashed blue line. (b) The ratio between the DRS measured on the capsular surface over that on the subcapsular surface of a kidney is plotted as thicker solid red line. The ratio between the DRS measured on the capsular surface over that on the cross-section of a liver is plotted as the thinner solid blue line. (c) The model implementation to calculate the ratio between DRS measured on the surface of a kidney and on the subcapsular surface of a kidney, assuming a kidney capsule thickness of $200\mu \mathrm{m}$. The model implementation to calculate the ratio between DRS measured on the surface of a liver and on the subcapsular parenchyma of a liver, assuming a liver capsule thickness of $10\mu \mathrm{m}$. (d) The model-predicted ratio of the diffuse reflectance from the kidney in the presence of the realistic $200\text{-}\mu \mathrm{m}$ capsule over that in the absence of the capsule is displayed as the thick solid red line. The model-predicted ratio of the diffuse reflectance from the liver in the presence of the realistic $10\text{-}\mu \mathrm{m}$ capsule over that in the absence of the capsule is displayed as the thinner solid blue line.

To evaluate that if the much greater difference of the capsular versus subcapsular device-specific DRS in the kidney than in the liver is potentially caused by the much thicker capsule of the kidney than that of the liver, the device-specific DRS for the kidney or liver as plotted in Fig. 10(b) is used to estimate the main medium absorption property, which is then implemented in the analytical model presented earlier to estimate the difference between capsular and subcapsular DRS from a liver or a kidney for a $10\text{-}\mu \mathrm{m}$-thick liver capsule and a $200\text{-}\mu \mathrm{m}$-thick kidney capsule, as schematically shown in Fig. 10(c). The estimation is conducted according to the following procedures: (1) the spectrum of tissue absorption coefficient was calculated as ${\mu }_a(\lambda )=\{\ln {[R_{\mathrm{tiss}}(\lambda )]}^{-1}\}/9+{\mu }_{a\_\text{phantom}}$, where the number “9” is the product of the SDS in mm “3” and a differential pathlength factor of “3” estimated for the 3-mm SDS.³⁷ The ${\mu }_{a\_\text{phantom}}=0.021{\mathrm{mm}}^{-1}$ is the absorption coefficient of the reference solid phantom. The extracted absorption coefficient spectrum was then assigned as the absorption coefficient spectrum of the main medium. (2) The absorption coefficient spectrum of the capsular layer is estimated as ${\mu }_{a1}(\lambda )=0.3*{\lambda }^{-0.1}$, where $\lambda$ is the wavelength in $\mu \mathrm{m}$, based on the absorption properties suggested by experimental reports.³⁸ (3) The reduced scattering coefficient of the main medium is set as $1.0{\mathrm{mm}}^{-1}$. (4) The reduced scattering coefficient spectrum of the superficial layer is estimated as ${\mu }_{s1}^{\prime }(\lambda )=3.2*{\lambda }^{-1}$, where $\lambda$ is the wavelength in $\mu \mathrm{m}$, based on the reduced scattering properties suggested by experimental reports.³⁸ (5) The afore-estimated spectra were implemented in the analytical model. The ratio of device-specific DRSs between measuring on the kidney parenchyma in the presence of a realistic $200\text{-}\mu \mathrm{m}$-thick capsule and in the absence of the capsule is estimated as the ratio between the model-outputs of the diffuse reflectance at a superficial layer thickness of 200 and $0.1\mu \mathrm{m}$ for a main medium absorption spectrum derived using the kidney measurements. Similarly, the ratio of device-specific DRSs between measuring on the liver parenchyma in the presence of a realistic $10\text{-}\mu \mathrm{m}$-thick capsule and in the absence of the capsule is estimated as the ratio between the model outputs of the diffuse reflectance at a superficial layer thickness of 10 and $0.1\mu \mathrm{m}$ for a main medium absorption spectrum derived using the liver measurements.

The model-prediction of the ratio of the diffuse reflectance from the kidney in the presence of the realistic $200\text{-}\mu \mathrm{m}$ capsule over that in the absence of the capsule is shown in Fig. 10(d) as the thick solid red line, after aligning the peak magnitude around the 500 nm to that of the experimentally measured profile shown in Fig. 10(b). The DRS profile for the kidney shown in (D) based on a realistic $200\text{-}\mu \mathrm{m}$ capsule resembles closely that shown in (B), indicating that the ratio between the surface DRS and subcapsular DRS varies significantly over the spectrum of measurements. The ratio corresponding to the kidney as shown in (D) estimated for a realistic $200\text{-}\mu \mathrm{m}$ capsule varies between 0.81 (or 81%) around 300 nm and peaks at 1.7 (or 170%) that occurs at $\sim 500\mathrm{nm}$. The profile corresponding to the kidney estimated with a realistic capsule thickness of $200\mu \mathrm{m}$ reproduces the experimentally measured profile at $>420\mathrm{nm}$ but deviates significantly from the experimentally measured profile at $<420\mathrm{nm}$. The model-prediction of the ratio of the diffuse reflectance from the liver in the presence of the realistic $10\text{-}\mu \mathrm{m}$ capsule over that in the absence of the capsule is shown in Fig. 10(d) as the thinner solid blue line, after aligning the plateau at $<400\mathrm{nm}$ with that of the experimentally measured profile shown in Fig. 10(b). The DRS profile for the liver shown in (D) based on a realistic $10\text{-}\mu \mathrm{m}$ capsule resembles globally that shown in (B), indicating that the ratio between the surface DRS and subcapsular DRS varies insignificantly over the spectrum of measurements. The ratio corresponding to the liver as shown in (D) estimated for a realistic $10\text{-}\mu \mathrm{m}$ capsule varies between 1.17 (or 117%) over 500 to 600 nm and 1.2 (or 120%) that occurs at shorter ($<400\mathrm{nm}$) or longer ($>1000\mathrm{nm}$) wavelength ranges.