A Novel Differential Pathlength Factor Model for Near-Infrared Diffuse Optical Imaging

TL;DR

This work addresses the sensitivity of CW-NIR chromophore quantification to how the differential pathlength factor is defined. By performing extensive Monte Carlo photon transport simulations, the authors derive a physically grounded true DPF and introduce a practical inverse-distance DPF that closely tracks the true pathlength across a broad range of tissue-like optical properties. They demonstrate that these models yield sub-$10\%$ errors in absorption estimates, outperforming conventional DPF formulations (which can exceed $100\%$ error), with independent phantom experiments validating the simulations. The results offer a robust, computationally efficient framework to improve the quantitative reliability of CW-NIR imaging in varied geometries and tissue conditions, with potential extensions to heterogeneous tissues and other NIR modalities.

Abstract

Near infrared diffuse optical imaging can be performed in reflectance and transmission mode and relies on physical models along with measurements to extract information on changes in chromophore concentration. Continuous-wave near-infrared diffuse optical imaging relies on accurate differential pathlength factors (DPFs) for quantitative chromophore estimation. Existing DPF definitions inherit formulation-dependent limitations that can introduce large errors in modified Beer--Lambert law analyses. These errors are significantly higher at smaller source-detector separations in a reflectance mode of measurement. This minimizes their applicability in situations where large area detection is used and also when signal depth is varying. Using Monte Carlo simulations, we derive two distance- and property-dependent DPF models one ideal and one experimentally practical and benchmark them against standard formulations. The proposed models achieve errors below 10 percent across broad optical conditions, whereas conventional DPFs can exceed 100 percent error. The theoretical predictions are further validated using controlled phantom experiments, demonstrating improved quantitative accuracy in CW-NIR imaging.

Figures (6)

• Figure 1: (a) Schematic of the Monte Carlo simulation. One million photons were launched into a homogeneous tissue slab using a pencil-beam source (red arrow), and their trajectories (orange) were recorded. The trajectory data were used to determine each photon's pathlength $s$, source--detector separation $d$, and corresponding detected intensity (orange arrow). (b–e) Representative photon trajectories for different combinations of optical properties: (b) low $\mu_a$, low $\mu_s$; (c) low $\mu_a$, high $\mu_s$; (d) high $\mu_a$, low $\mu_s$; and (e) high $\mu_a$, high $\mu_s$. Red, orange, and yellow traces denote photons that exited the top surface near the source ($<1\,\mathrm{cm}$), exited farther from the source ($>1\,\mathrm{cm}$), or were absorbed within the medium, respectively. The thickness of each trajectory bundle is proportional to the number of photons following that path.
• Figure 2: (a) Optical density as a function of source--detector separation $d$ for representative combinations of absorption and scattering coefficients $(\mu_a,\mu_s)$. Solid curves show Monte Carlo--derived OD values. Dashed curves represent fits of the same data to the mixed linear model given in Eq. \ref{['eq:OD_linear_fit']}, demonstrating that the simulated OD profiles are well approximated by a linear combination of $d$ and $\log_{10}(d)$ with a nonzero intercept. (b) Comparison of the corresponding true DPF values (solid curves), computed directly from photon pathlength statistics, with the inverse-distance DPF estimates (dashed curves) obtained using Eq. \ref{['eq:DPF_inv']}.
• Figure 3: Experimental setup for validation using a homogeneous tissue-mimicking phantom. The phantom consists of a water-based super-absorbent polymer hydrogel matrix mixed with titanium dioxide particles and cast in a $15 \times 15 \times 6$ cm$^3$ acrylic mold. A fixed 750 nm light-emitting diode source illuminates the phantom surface at the center of a $13 \times 13$ measurement grid with a 1 cm pitch, while a movable silicon photodiode detector samples diffuse reflectance at varying source--detector separations. The detected signal is conditioned by readout electronics, digitized by a microcontroller unit, and recorded on a host computer.
• Figure 4: Comparison of DPF models and their corresponding absorption coefficient estimation errors for a representative optical property pair ($\mu_a = 0.15$ cm$^{-1}$, $\mu_s = 50$ cm$^{-1}$). (Left) Variation of the DPF as a function of source--detector separation for the constant, semi-infinite, mean-pathlength, slope-based, true, and inverse-distance models. (Right) Relative error in estimating the absorption coefficient using each DPF formulation as a function of source--detector separation.
• Figure 5: (a) Average absolute relative error in estimating the absorption coefficient using the constant DPF model as a function of absorption and scattering coefficients. Errors are averaged over source--detector separations between 1 and 8 cm. (b) Average absolute relative error obtained using the true DPF under the same conditions. (c--e) Variation of the inverse-distance DPF parameters $A$ (panel c), $B$ (panel d), and $C$ (panel e) across the $\mu_a$--$\mu_s$ space, comparing empirically fitted values with regression-derived values. (f) Average absolute relative error obtained using the fully empirical inverse-distance DPF model.