2.1.

Spatial Frequency Domain Imaging

SFDI is a widefield DOI method that provides pixel-level extractions of optical properties. SFDI instrumentation, data acquisition, processing, and analysis procedures have been described in detail in the previous publications.^11,12 Briefly, SFDI utilizes projections of spatially modulated light onto a phantom or tissue surface, and the remitted diffuse light is collected with a CCD camera. Images are typically collected at multiple wavelength bands and at least two spatial frequencies. A demodulation algorithm is used to extract the AC tissue response from each spatial frequency. Demodulated image data are then calibrated to remove the instrument response using measurements from a tissue-simulating phantom with known optical properties. Corrections for height and angle of the object surface may also be employed for nonflat surfaces.²⁴ Diffuse reflectance ($R_d$) maps acquired at a minimum of at least two spatial frequencies ($f_x$) are then used as inputs to an inverse model to extract tissue absorption (${\mu }_a$) and reduced scattering (${\mu }_s^{\prime }$) as a function of wavelength.^12,25

In this study, the OxImager RS SFDI system (Modulated Imaging, Inc., Irvine, California) was used for all measurements. The system has an imaging field of view of $15\times 20\mathrm{cm}$, and for this study illumination wavelengths of 659, 691, 731, and 851 nm and spatial frequencies of 0 and $0.1{\mathrm{mm}}^{-1}$ were utilized. The sample was illuminated with one-dimensional sinusoidal spatial patterns, and each $f_x$ was projected at three offset spatial phases (0, 120, and 240 deg). Demodulation was performed using previously described methods, specifically Eq. (20) in Cuccia et al.¹² An MC-based two-frequency LUT inversion algorithm was used to map $R_d$ values to optical properties (described in more detail in Sec. 2.2). For tissue measurements, the four ${\mu }_a$ values extracted at the four illumination wavelengths were used to compute tissue-level chromophore concentrations using Beer’s Law. Oxyhemoglobin (${\mathrm{HbO}}_2$) ($\mu \mathrm{M}$), deoxyhemoglobin (Hb) ($\mu \mathrm{M}$), total hemoglobin (THb) ($\mu \mathrm{M}$), and oxygen saturation [${\mathrm{StO}}_2$ (%): ${\mathrm{HbO}}_2/\mathrm{THb}\times 100$] were extracted from mouse tissue measurements.^5,11 Scattering amplitude and scattering slope were also computed by fitting the wavelength-dependent ${\mu }_s^{\prime }$ values to a power law on a pixel by pixel basis.¹¹

2.2.

Monte Carlo Simulations for the Generation of LUT-Based Inverse Models

2.2.2.

Monte Carlo simulation parameters for generating homogeneous and two-layer LUTs

The Martinelli method was used to construct a homogeneous LUT, and the Gardner method was used to construct both a homogeneous LUT and a two-layer LUT. The geometries of the MC simulations used to produce these LUTs are shown in Figs. 1(a) and 1(b). Both models were constructed for the purpose of extracting optical properties from a subcutaneously implanted tumor on the mouse flank. For the homogeneous case, the tumor was modeled as a semi-infinite geometry, and the effects of the superficial skin layer were ignored. For the two-layer case, the top layer represents the skin layer, with fixed (i.e., known) ${\mu }_a$, ${\mu }_s^{\prime }$, and thickness ($d$), and the bottom layer represents a semi-infinite tumor layer. For all LUTs, the tumor optical properties are the free parameters of the inversion algorithm. We describe the properties of each layer in more detail below.

Fig. 1

(a) Schematic of tissue model for the homogeneous case, (b) schematic of the tissue model for the two-layer case, and (c) an example of a custom-made two-layer silicone phantom used to validate the accuracy of the resulting two-layer inverse algorithm.

Skin layer: Optical properties of the upper (skin) layer of the MC simulations were adapted from Sabino et al.,³¹ who recently reported ${\mu }_a$ and ${\mu }_s^{\prime }$ of skin from BALB/c male mice using a Kubelka–Munk model of photon transport and spectrophotometric measurements for the wavelength range of 400 to 1400 nm. The authors used skin from the mouse dorsal region and shaved any excess hair before measurements. We utilized the average ${\mu }_a$ and ${\mu }_s^{\prime }$ values of the reported skin properties calculated over the four SFDI wavelengths (659, 691, 731, and 851 nm) for the upper layer properties in our two-layer MC simulations: ${\mu }_a=0.096{\mathrm{mm}}^{-1}$ (SD: $0.0075{\mathrm{mm}}^{-1}$) and ${\mu }_s^{\prime }=0.78{\mathrm{mm}}^{-1}$ (SD: $0.12{\mathrm{mm}}^{-1}$).

The thickness of the skin layer was estimated using caliper measurements of eight excised tumor skin samples from C57BL/6N female mice. The average skin thickness, which included the epidermis, dermis, and hypodermis, was $312.5\mu \mathrm{m}$. H&E staining of representative tumor skin cross sections was conducted to validate the skin thicknesses (agreement was within 3.1% for $n=2$ samples). The C57BL/6N mouse strain was used to estimate thickness since it is commonly used for mouse tumor imaging,³² and we plan to utilize this strain in future studies of mammary carcinoma. Our thickness measurements were similar to past reports of C57BL/6N skin thickness, where female mice were found to have a skin thickness of 371,³³ 300,³⁴ and $364\mu \mathrm{m}$.³¹ We note that data collected from male severe combined immunodeficient (SCID) mice are used in Sec. 3.5 of this work as well as in our prior study.¹¹ The thickness of male SCID mouse skin was determined by measuring 18 skin samples taken prior to treatment ($n=6$), during treatment with either DC101 or cyclophosphamide (CPA) ($n=6$), and after treatment ($n=6$) using brightfield microscopy of frozen tissue sections. The average skin thickness was found to be $326.9\mu \mathrm{m}$, which is within 5% of the thickness used in the two-layer model ($312.5\mu \mathrm{m}$). The difference between the thickness of samples taken before and during or after treatment was within 11%. These results suggest that the two-layer model is appropriate for subcutaneous tumor models in both female C57BL/6N and male SCID mice. Other mouse strains have somewhat similar skin thickness, including the commonly used BALB/c mouse strain ($336\mu \mathrm{m}$ for female and $393\mu \mathrm{m}$ for male),³¹ immunocompetent albino mice ($441\mu \mathrm{m}$),³⁵ immunocompromised athymic nude mice ($420\mu \mathrm{m}$),³⁴ and female SCID mice ($220\mu \mathrm{m}$).³⁶

Tumor layer: In all three LUTs, the bottom (tumor) layer contains 400 ${\mu }_a$ values, ranging from 0.0005 to $0.2{\mathrm{mm}}^{-1}$ ($\mathrm{\Delta }{\mu }_a=0.0005{\mathrm{mm}}^{-1}$) and 192 ${\mu }_s^{\prime }$ values ranging from 0.1 to $3.58{\mathrm{mm}}^{-1}$ ($\mathrm{\Delta }{\mu }_s^{\prime }=0.018{\mathrm{mm}}^{-1}$). A separate MC simulation was run for each ${\mu }_s^{\prime }$ value for both the homogeneous and two-layer Gardner LUTs. The tumor layer depth in all simulations was set to be at least 20 times the maximum $l^{*}$ [$=1/({\mu }_a+{\mu }_s^{\prime })$] to mimic a semi-infinite tissue geometry, where $l^{*}$ was determined using the lowest ${\mu }_a$ value ($0.0005{\mathrm{mm}}^{-1}$). Based on this criterion, a tumor layer thickness of 100 mm was used for all MC simulations except for the lowest 16 ${\mu }_s^{\prime }$ values (0.1 to $0.37{\mathrm{mm}}^{-1}$), for which the thickness was increased to 200 mm.

2.3.

Two-Layer Tissue-Simulating Optical Phantoms

A set of two-layer solid silicone phantoms was fabricated to optically simulate subcutaneous tumors in a mouse with a range of optical properties. These phantoms were used to test the accuracy of the Gardner LUT inversion algorithms. The phantoms consisted of a thin skin layer above a tumor layer. The top layer thickness and optical properties were fabricated to closely match the parameters described for MC simulations in Sec. 2.2.2. Four different two-layer phantoms were fabricated, all of which used the same skin layer. In all phantoms, silicone was used as the base solvent, nigrosin as the absorber, and titanium dioxide as the scatterer. The optical properties of the phantoms were adjusted by varying the amount of absorber and scatterer during fabrication as previously described.³⁹

The thin upper layer phantom was made by adapting a previously described technique.⁴⁰ First, an aluminum phantom mold was fabricated by machining a well that was $330\mu \mathrm{m}$ in depth and ${1.5}^{″}\times {1.5}^{″}$ in the lateral dimensions using a computer-controlled milling machine (SV-2414S-M, Sharp Industries). After the phantom ingredients were mixed together, the liquid mixture was poured into the aluminum mold. A microtome blade was used to draw and spread the mixture evenly across the well, and the edges of the blade remained in contact with the top surface of the mold at all times. The phantom was then left to cure, uncovered, overnight. During curing, the silicone layer was observed to shrink in the center of the well. Once cured, the thin silicone layer was removed from the mold and cut to the size of $1^{″}\times 1^{″}$ to remove the uneven and thicker edge. The thickness of the phantom was confirmed using caliper measurements by confining the thin layer between two microscope slide coverslips for stability and consistency. Because the top layer phantom was too thin for accurate optical property measurements with diffuse imaging techniques, a much larger, 2.5-cm thick homogeneous phantom was made from the same batch of material and SFDI was used to extract the optical properties.

Similarly, for the bottom (tumor) layer, four homogeneous phantoms were fabricated in collaboration with Dr. Muldoon’s group⁴¹ and measured with SFDI. The thickness of each phantom was 2.5 cm, and the ${\mu }_a$ and ${\mu }_s^{\prime }$ values of each phantom were targeted to span known mouse tumor optical properties. The skin layer and tumor layer phantoms were stacked to create the two-layer phantoms. First, the thin skin layer phantom was cleaned using an alcohol wipe. Then a small amount of ethanol was poured on a thick tumor layer phantom, and the thin layer was directly placed on top of the tumor layer, making sure that no visible air bubbles remained. The two-layer phantom was left under the chemical hood overnight to allow the ethanol to evaporate without leaving any air pockets between the layers. An example of one of the two-layer phantoms is shown in Fig. 1(c). The procedure was repeated 4 times to generate the four two-layer phantoms.

2.4.

Longitudinal Monitoring of a Mouse Tumor Xenograft During Cancer Treatment

In our prior work, we conducted SFDI longitudinal monitoring of the PC3/2G7 prostate tumor xenograft model during treatment with anticancer agents.^11,42 We have reprocessed a portion ($n=2$ mice) of this longitudinal data using the LUT inversion algorithms presented in this work in order to visualize the effect of using the multilayer model in a relevant physiologic system. Key details of the longitudinal study are presented here: SCID hairless outbred mice (SHO Mouse, Crl:SHOPrkdcscidHrhr), age 5 to 6 weeks old (21 to 23 grams), were purchased from Charles River Laboratories. When the average tumor volume reached $\sim 500{\mathrm{mm}}^3$, mice were treated with either the cytotoxic anticancer drug CPA (every 6 days for 3 cycles) or the antiangiogenic agent DC101 (every 3 days for 6 cycles), both given with intraperitoneal injections. During SFDI measurements, mice were anesthetized using isoflurane by inhalation. Mice were monitored longitudinally with SFDI for a total of 45 days. SFDI measurements were taken 5 times during 17 days of tumor growth, every day during 18 days of treatment, and every 2 days following the treatment period. Additional details including methods for tumor cell preparation, tumor cell inoculation, animal handling and care, SFDI measurement repeatability, angle and height corrections, etc., are described in Tabassum et al.²⁵ All animal procedures and measurements were conducted under an institutionally approved protocol.

3.1.

Comparison between MC Simulation Results and LUT Inversion Algorithms: Diffuse Reflectance

We first compared the results of the three different MC simulations methods (i.e., Martinelli homogeneous, Gardner homogeneous, and Gardner two-layer) and then compared how the LUT inversion algorithms based on these simulations differently map $R_d$ values to optical property values.

Figure 2 shows an illustrative example of the differences and similarities in the MC simulation results. $R_d$ is shown as a function of $f_x$ for varying $l^{*}$ values at a constant ratio of ${\mu }_s^{\prime }/{\mu }_a=100$. The optical properties corresponding to each $l^{*}$ value are as follows: for $l^{*}=0.5\mathrm{mm}$: ${\mu }_a=0.0198{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=1.98{\mathrm{mm}}^{-1}$; $l^{*}=1\mathrm{mm}$: ${\mu }_a=0.0099{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=0.99{\mathrm{mm}}^{-1}$; $l^{*}=2\mathrm{mm}$: ${\mu }_a=0.005{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=0.5{\mathrm{mm}}^{-1}$; and $l^{*}=4\mathrm{mm}$: ${\mu }_a=0.0025{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=0.25{\mathrm{mm}}^{-1}$. For the two-layer simulations, the skin layer properties were as previously described (${\mu }_a=0.096{\mathrm{mm}}^{-1}$, ${\mu }_s^{\prime }=0.78{\mathrm{mm}}^{-1}$, $d=312.5\mu \mathrm{m}$). From Fig. 2, it is evident that there is very little difference in MC simulation results between the Martinelli homogeneous and the Gardner homogeneous methods. However, the incorporation of the skin layer introduces a significant alteration in these results, as expected. Note that the effect of the skin layer is to sometimes shift $R_d$ values higher than the homogeneous results and to sometimes shift them lower. This effect is dependent on spatial frequency as well as the specific optical properties of the tumor and skin layers. Because the Martinelli and Gardner homogeneous results are nearly identical, the remainder of the analysis will focus on the Gardner homogeneous and Gardner two-layer MC results and the LUT inversion algorithms based on these results.

Fig. 2

Comparison of MC simulation results from the Martinelli homogeneous, the Gardner homogeneous, and the Gardner two-layer methods. Diffuse reflectance ($R_d$) is shown as a function of spatial frequency ($f_x$) for varying values of $l^{*}$ at a constant ratio of ${\mu }_s^{\prime }/{\mu }_a=100$. The Martinelli homogeneous and Gardner homogeneous results are nearly identical, while the introduction of the top (skin) layer introduces significant shifts in $R_d$ that are dependent on $f_x$ and the $l^{*}$ of the bottom (tumor) layer.

Figure 3 shows differences between the Gardner homogeneous and Gardner two-layer LUT inversion algorithms for two spatial frequencies: $f_x=0{\mathrm{mm}}^{-1}$ (DC $R_d$) and $0.1{\mathrm{mm}}^{-1}$ (AC $R_d$). Here, the $x$-axis displays ${\mu }_a$, the $y$-axis displays ${\mu }_s^{\prime }$, and the color axis displays $R_d$. Isolines of constant $R_d$ are displayed as an aid to visual comparisons between the subplots. As in Fig. 2, these plots visually show the substantial impact that the skin layer has on the mapping between diffuse reflectance and optical properties.

Fig. 3

Comparison of LUT inversion algorithms based on Gardner homogeneous and Gardner two-layer methods. $R_d$ values are shown both in the color dimension and as labeled isolines for the entire range of simulated ${\mu }_a$ and ${\mu }_s^{\prime }$ values. (a) and (b) Optical properties versus DC $R_d$ for the homogeneous and two-layer LUTs, respectively. (c) and (d) Optical properties versus AC $R_d$ for the homogeneous and two-layer LUTs, respectively.

3.2.

Comparison between LUT Inversion Algorithms: Optical Property Extraction

In practice, the LUT inversion algorithms accept experimentally measured $R_d$ values as inputs and provide optical property extractions as outputs. We investigated the extent to which the LUTs provide different optical property extractions for the same input measurements (i.e., $R_d$ values). To do this, we first choose 30 DC $R_d$ values evenly spaced between 0.0419 and 0.749, and 30 AC $R_d$ values, evenly spaced between 0.0289 and 0.4514. This range of $R_d$ values was chosen as they are present in both the homogeneous and two-layer LUTs. For all combinations of the chosen DC and AC $R_d$ values, ${\mu }_a$ and ${\mu }_s^{\prime }$ were extracted using both LUTs, and the difference in ${\mu }_a$ and ${\mu }_s^{\prime }$ extractions was computed. Absolute differences in optical properties between the LUTs are shown in Figs. 4(a) and 4(c) for ${\mu }_a$ and ${\mu }_s^{\prime }$, respectively, and the percent differences (relative to the homogeneous LUT values) are shown in Figs. 4(b) and 4(d). For some DC and AC $R_d$ combinations, the differences in optical property extractions are substantial. For example, from Fig. 4(b), we see that percent differences for ${\mu }_a$ are larger ($\sim 60\%$ to 80%) at higher DC $R_d$ values. In Fig. 4(d), larger differences in ${\mu }_s^{\prime }$ extractions occur at higher AC $R_d$ values. This analysis shows that the two LUTs differently map optical properties from $R_d$ inputs. In the following section, we demonstrate that the two-layer LUT improves the accuracy of tumor layer extractions.

Fig. 4

The impact on optical property extractions of the Gardner two-layer LUT inversion algorithm shown as absolute and % differences compared to the Gardner homogeneous case. (a) Absolute and (b) percent differences in ${\mu }_a$ extractions. (c) Absolute and (d) percent difference in ${\mu }_s^{\prime }$ extractions.

3.3.

Two-Layer LUT Improves the Accuracy of Tumor Layer Optical Property Extractions Using SFDI

Experimental measurements were conducted to determine if the Gardner two-layer LUT inversion algorithm improves the accuracy of tumor layer optical property extractions compared to the Gardner homogeneous LUT. This accuracy test utilized the four two-layer phantoms described in Sec. 2.3. Each of the two-layer phantoms used the same top (skin) layer. The measured thickness of the skin layer was $310\mu \mathrm{m}$ at its center, which is within 0.8% of the skin layer thickness defined in the MC simulations used to generate the Gardner two-layer LUT. Absorption of the skin layer was $0.0936{\mathrm{mm}}^{-1}$ at 659 nm, which is within 2.52% of the MC absorption parameter, and the ${\mu }_s^{\prime }$ value was $0.780{\mathrm{mm}}^{-1}$ at 659 nm, which is within 0.063% of the MC value. For the tumor layer, four different pairs of optical properties were utilized, spanning a range of optical properties observed in our prior work, in which we monitored PC3/2G7 mouse xenografts over 45 days using SFDI.¹¹ These pairs are labeled as tumor 1 through tumor 4 in Fig. 5. The optical property pairs, reported at 659 nm, are as follows: for tumor 1: ${\mu }_a=0.0244{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=2.054{\mathrm{mm}}^{-1}$; tumor 2: ${\mu }_a=0.002{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=2.314{\mathrm{mm}}^{-1}$; tumor 3: ${\mu }_a=0.0039{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=0.714{\mathrm{mm}}^{-1}$; and tumor 4: ${\mu }_a=0.0301{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=0.676{\mathrm{mm}}^{-1}$.

Fig. 5

Comparisons in bottom (tumor) layer optical property extraction errors for the Gardner homogeneous and Gardner two-layer LUT inversion algorithms. Diffuse reflectance measurements of four two-layer tissue-simulating optical phantoms were made with SFDI, and both inversion models were used to extract the bottom (tumor) layer optical properties (labeled as tumors 1 to 4). (a) The absolute extraction error compared with the known tumor layer ${\mu }_a$. (b) The same data but with a zoomed-in $y$-axis so that small extraction errors can be visualized. (c) Absolute errors in tumor layer ${\mu }_s^{\prime }$ extractions and (d) the same data with a zoomed-in $y$-axis. Optical properties were measured at 659 nm.

Each two-layer phantom was measured with SFDI, and the bottom (tumor) layer optical properties were extracted using both the Gardner homogeneous and Gardner two-layer LUTs. Since these phantoms have flat surfaces, no corrections for height or angle were implemented. The absolute differences between the measured and true ${\mu }_a$ for the tumor layer are shown in Figs. 5(a) and 5(b). The absolute differences between the measured and true ${\mu }_s^{\prime }$ for the tumor layer are shown in Figs. 5(c) and 5(d). In both cases, Figs. 5(b) and 5(d) recapitulates the data in Figs. 5(a) and 5(c) but with a zoomed-in $y$-axis to allow visualization of the small error values obtained for some phantoms. Table 1 shows the percent error in tumor layer optical property extractions for both the homogeneous and two-layer LUTs. In the worst-case, the ${\mu }_a$ and ${\mu }_s^{\prime }$ extraction errors were 20.33% and 10.87% for the two-layer LUT.

Table 1

Accuracy of optical property extractions in four two-layer tissue-simulating optical phantoms. Each two-layer phantom is designated as tumors 1 to 4.

In all cases, the error in tumor layer optical property extraction is substantially lower for the two-layer LUT versus the homogeneous LUT. This effect is not as pronounced in ${\mu }_s^{\prime }$ for tumors 3 and 4, as ${\mu }_s^{\prime }$ values in these tumors are very similar to that of the skin layer (${\mu }_s^{\prime }=0.78{\mathrm{mm}}^{-1}$). Note that the decrease in error by the two-layer LUT is between 7 and 256 times for ${\mu }_a$ and between 2 and 24 times for ${\mu }_s^{\prime }$. Taken together, these results confirm that the two-layer LUT provides a better estimate of the true tumor layer optical properties than the homogeneous LUT.

3.4.

Sensitivity Analysis of the Two-Layer LUT

We conducted a sensitivity analysis to characterize how mismatches in the skin layer optical properties and thickness affect the results of the Gardner two-layer LUT. Stated another way, the sensitivity analysis provides an indication of how well one must know the true skin layer properties in order to obtain accurate tumor layer optical property extractions.

Additional MC simulations were conducted of a two-layer medium, in which the properties of the skin layer (${\mu }_a$, ${\mu }_s^{\prime }$, and $d$) were varied to introduce a mismatch to those used to generate the Gardner two-layer LUT. The skin layer ${\mu }_a$ value was varied by up to $\pm 40\%$ of the original Gardner two-layer value, ${\mu }_s^{\prime }$ was varied by up to $\pm 40\%$, and $d$ was varied from $-40\%$ to $+80\%$. Unlike optical properties, $d$ was varied by up to $+80\%$ in order to simulate large skin thickness values, such as $d=530\mu \mathrm{m}$, reported previously in athymic nude mouse.³⁴ For this analysis, four tumor layer optical property pairs were chosen to span a physiologically relevant range, and extraction errors are reported for each pair. The tumor layer optical property pairs at 659 nm are listed here: For tumor 1: ${\mu }_a=0.031{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=2.025{\mathrm{mm}}^{-1}$; tumor 2: ${\mu }_a=0.004{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=2.2{\mathrm{mm}}^{-1}$; tumor 3: ${\mu }_a=0.005{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=0.676{\mathrm{mm}}^{-1}$; and tumor 4: ${\mu }_a=0.033{\mathrm{mm}}^{-1}$ and ${\mu }_s^{\prime }=0.645{\mathrm{mm}}^{-1}$.

In order to compute extraction errors induced by the mismatched skin layer, the $R_d$ values at the DC and AC spatial frequencies produced from each of the new MC simulations (which each had mismatched skin layer properties) were fed to the original Gardner two-layer LUT and resulting optical property extractions were recorded. The error was then computed, defined as the difference between the known tumor layer optical properties and the recorded optical property extractions. These errors are shown in Figs. 6(a) and 6(f). As expected, the errors in ${\mu }_a$ and ${\mu }_s^{\prime }$ extractions increase as the mismatch in skin layer properties increases. For example, as described in Sec. 2.2.2, Dodig et al.³⁶ reported that female SCID mice had a skin thickness of $\sim 220\mu \mathrm{m}$. This is an approximately $-30\%$ mismatch with the skin thickness used for our model ($312.5\mu \mathrm{m}$). As shown in Figs. 6(e) and 6(f), assuming the skin optical properties match, this $-30\%$ mismatch in skin layer thickness may induce an error in ${\mu }_a$ of as much as $-0.0023{\mathrm{mm}}^{-1}$ and a ${\mu }_s^{\prime }$ error of as much as $0.29{\mathrm{mm}}^{-1}$ (these are the worst-case errors observed from this analysis, substantially smaller errors were observed for some tumor optical property combinations).

Fig. 6

Results from a sensitivity analysis for the two-layer Gardner LUT inversion algorithm. Errors in tumor layer ${\mu }_{\mathrm{a}}$ and ${\mu }_s^{\prime }$ extractions are shown for various skin layer property mismatches (a)–(f). In (f), the inset image has a zoomed-in $y$-axis. Optical properties were measured at 659 nm.

Additional MC simulations were conducted, in which the properties of the skin layer (${\mu }_a$, ${\mu }_s^{\prime }$, and $d$) were varied simultaneously to explore the effect of combining these mismatches. The maximum errors observed when all three parameters were mismatched in the negative direction (i.e., all three parameters decreased by 40%) was $-0.0078{\mathrm{mm}}^{-1}$ for ${\mu }_a$ and $0.44{\mathrm{mm}}^{-1}$ for ${\mu }_s^{\prime }$. The maximum errors observed when all three parameters were mismatched in the positive direction (i.e., ${\mu }_a$ and ${\mu }_s^{\prime }$ increased by 40%, $d$ increased by 80%) was $0.021{\mathrm{mm}}^{-1}$ for ${\mu }_a$ and $-0.51{\mathrm{mm}}^{-1}$ for ${\mu }_s^{\prime }$. Whether errors of this magnitude are tolerable depending on the specific application and biological questions posed.

3.5.

Gardner Two-Layer LUT Reveals Larger Therapy-Induced Optical Scattering Dynamics and a More Hypoxic Tumor Environment During Longitudinal Monitoring of Tumor Xenografts

The homogeneous and two-layer LUT inversion algorithms were used to reanalyze a prior data set, in which SFDI was used to monitor mice longitudinally during the course of anticancer therapy.¹¹ The details of the data acquisition and analysis, including the methods for region of interest (ROI) selection, are described in detail in Tabassum et al.¹¹ Briefly, each SFDI measurement was repeated thrice and averaged to minimize breathing artifacts. The demodulated images were corrected for height and angle.²⁴ Two-by-two binning of the CCD was applied to improve the SNR. Mice were monitored longitudinally for a total of 45 days. $R_d$ data acquired with SFDI were analyzed with both the Gardner homogeneous and Gardner two-layer LUT inversion algorithms for comparison.

Figure 7(a) shows changes in tumor ${\mu }_s^{\prime }$ at 659 nm from a DC101-treated tumor over the course of 45 days. Injection dates are indicated by vertical dashed lines. The mean and standard deviation of ${\mu }_s^{\prime }$ values extracted over a manually chosen ROI are shown. The two-layer LUT reveals higher ${\mu }_s^{\prime }$ values throughout the study compared to the homogeneous LUT, and the changes in ${\mu }_s^{\prime }$ over time are also larger. Figure 7(b) shows tumor ${\mu }_s^{\prime }$ colormaps overlaid on a planar mouse image at day 0 and day 30 for both LUTs. The increase in ${\mu }_s^{\prime }$ is apparent throughout the tumor region at these time points.

Fig. 7

An example of the Gardner homogeneous and the Gardner two-layer LUT inversion algorithms applied to SFDI data collected during a longitudinal treatment monitoring study of PC3/2G7 prostate tumor xenografts. (a) ${\mu }_s^{\prime }$ extractions from both LUTs are shown during and after DC101 treatment. (b) ${\mu }_s^{\prime }$ colormaps overlaid on DC101-treated planar mouse images at day 0 and day 30. (c) ${\mathrm{StO}}_2$ values determined using four wavelength optical property extractions from both LUTs for a CPA-treated tumor. (d) ${\mathrm{StO}}_2$ colormaps overlaid on CPA-treated planar mouse images at day 0 and day 30.

Figure 7(c) shows changes in tumor ${\mathrm{StO}}_2$ from a CPA-treated tumor on a metronomic schedule, followed by a rebound period. Mean and standard deviation of ${\mathrm{StO}}_2$ values extracted over a manually chosen ROI are shown. The two-layer LUT reveals lower ${\mathrm{StO}}_2$ values over the entire study period compared to the homogeneous LUT; the values decrease by roughly the same extent throughout the study. Figure 7(d) shows tumor ${\mathrm{StO}}_2$ colormaps overlaid on planar mouse images at day 0 and day 30 for both LUTs. A decrease in ${\mathrm{StO}}_2$ is apparent throughout the tumor region at these time points.