Modeling subdiffusive light scattering by incorporating the tissue phase function and detector numerical aperture

Two-term scattering phase function for photon transport to model subdiffuse reflectance in superficial tissues

For Monte Carlo simulations of light transport in a variety of diffuse scattering applications, a single-scattering two-term phase function with five adjustable parameters is sufficiently flexible to separately control the forward and backward components of scattering. The forward component dominates light penetration into a tissue and the resulting diffuse reflectance. The backward component controls early subdiffuse scatter from superficial tissues. The phase function consists of a linear combination of two phase functions [Reynolds and McCormick, J. Opt. Soc. Am.70, 1206 (1980)10.1364/JOSA.70.001206] that were derived from the generating function for Gegenbauer polynomials. The two-term phase function (TT) accommodates strongly-forward anisotropic scattering with enhanced backscattering and is a generalization of the two-term, three-parameter Henyey-Greenstein phase function. An analytical inverse of the cumulative distribution function for scattering is provided for implementation in Monte Carlo simulations. Explicit TT equations are given for the single-scattering metrics g 1, g 2, γ, and δ. Scattering data from previously published bio-optical data are shown to fit better with the TT than other phase function models. Example Monte Carlo simulations illustrate the use of the TT and its independent control of subdiffuse scatter.

Subdiffuse scattering and absorption model for single fiber reflectance spectroscopy

Single fiber reflectance (SFR) spectroscopy is a technique that is sensitive to small-scale changes in tissue. An additional benefit is that SFR measurements can be performed through endoscopes or biopsy needles. In SFR spectroscopy, a single fiber emits and collects light. Tissue optical properties can be extracted from SFR spectra and related to the disease state of tissue. However, the model currently used to extract optical properties was derived for tissues with modified Henyey-Greenstein phase functions only and is inadequate for other tissue phase functions. Here, we will present a model for SFR spectroscopy that provides accurate results for a large range of tissue phase functions, reduced scattering coefficients, and absorption coefficients. Our model predicts the reflectance with a median error of 5.6% compared to 19.3% for the currently used model. For two simulated tissue spectra, our model fit provides accurate results.

Analytical model for diffuse reflectance in single fiber reflectance spectroscopy

Cancer progression leads to changing scattering properties of affected tissues. Single fiber reflectance (SFR) spectroscopy detects these changes at small spatial scales, making it a promising tool for early in situ detection. Despite its simplicity and versatility, SFR signal modeling is hugely complicated so that, presently, only approximate models exist. We use a classic approach from geometrical probability to derive accurate analytical expressions for diffuse reflectance in SFR that shows a strong improvement over existing models. We consider the case of limited collection efficiency and the presence of absorption. A Monte Carlo light transport study demonstrates that we adequately describe the contribution of diffuse reflectance to the SFR signal. Additional steps are required to include semi-ballistic, non-diffuse reflectance also present in the SFR measurement.

Subdiffuse scattering model for single fiber reflectance spectroscopy

To detect small-scale changes in tissue with optical techniques, small sampling volumes are required. Single fiber reflectance (SFR) spectroscopy has a sampling depth of a few hundred micrometers. SFR spectroscopy uses a single fiber to emit and collect light. The only available model to determine optical properties with SFR spectroscopy was derived for tissues with modified Henyey-Greenstein phase functions. Previously, we demonstrated that this model is inadequate for other tissue phase functions. We develop a model to relate SFR measurements to scattering properties for a range of phase functions, in the absence of absorption. Since the source and detector overlap, the reflectance cannot be accurately described by diffusion theory alone: SFR measurements are subdiffuse. Therefore, we describe the reflectance as a combination of a diffuse and a semiballistic component. We use the model of Farrell et al. for the diffuse component, solved for an overlapping source and detector fiber. For the semiballistic component, we derive a new parameter, p_sb, which incorporates the integrals of the phase function over 1 deg in the backward direction and 23 deg in the forward direction. Our model predicts the reflectance with a median error of 2.1%, compared to 9.0% for the currently available model.

Multidiameter single-fiber reflectance spectroscopy of heavily pigmented skin: modeling the inhomogeneous distribution of melanin

When analyzing multidiameter single-fiber reflectance (MDSFR) spectra, the inhomogeneous distribution of melanin pigments in skin tissue is usually not accounted for. Especially in heavily pigmented skins, this can result in bad fits and biased estimation of tissue optical properties. A model is introduced to account for the inhomogeneous distribution of melanin pigments in skin tissue. In vivo visible MDSFR measurements were performed on heavily pigmented skin of type IV to VI. Skin tissue optical properties and related physiological properties were extracted from the measured spectra using the introduced model. The absorption of melanin pigments described by the introduced model demonstrates a good correlation with the co-localized measurement of the well-known melanin index.

Simple analytical total diffuse reflectance over a reduced-scattering-pathlength scaled dimension of [10-5, 10-1] from a medium with HG scattering anisotropy

Model approximation is necessary for reflectance assessment of tissue at sub-diffusive to non-diffusive scale. For tissue probing over a sub-diffusive circular area centered on the point of incidence, we demonstrate simple analytical steady-state total diffuse reflectance from a semi-infinite medium with the Henyey-Greenstein (HG) scattering anisotropy (factor $g$g). Two physical constraints are abided to: (1) the total diffuse reflectance is the integration of the radial diffuse reflectance; (2) the radial and total diffuse reflectance at $g \gt {0}$g>0 analytically must resort to their respective forms corresponding to isotropic scattering as $g$g becomes zero. Steady-state radial diffuse reflectance near the point of incidence from a semi-infinite medium of $g \approx 0$g≈0 is developed based on the radiative transfer for isotropic scattering, then integrated to find the total diffuse reflectance for $g \approx 0$g≈0. The radial diffuse reflectance for $g \ge 0.5$g≥0.5 is semi-empirically formulated by comparing to Monte Carlo simulation results and abiding to the second constraint. Its integration leads to a total diffuse reflectance for $g \ge 0.5$g≥0.5 that is also bounded by the second constraint. Over a collection diameter of the reduced-scattering pathlength ($1/\mu _s^{ \prime}$1/μs') scaled size of [${{10}^{ - 5}}$10^-5, ${{10}^{ - 1}}$10^-1] for $g = [{0.5},{0.95}]$g=[0.5,0.95] and the absorption coefficient as strong as the reduced scattering coefficient, the simple analytical total diffuse reflectance is found to be accurate, with an average error of 16.1%.

Goniometer for determination of the spectrally resolved scattering phase function of suspended particles

An apparatus for direct determination of the spectral resolved scattering phase function of suspensions and emulsions is described. The system consists of a polychromatic xenon lamp as a light source and a spectrometer as a detector. Together with a stepper motor, the system enables spectrally and angularly resolved measurements in the range of 450 nm-950 nm and 10°-170°, respectively. A model for light propagation inside the cuvette in the regime of single scattering, which also takes the spectral dimension into account, was developed. A postprocessing algorithm applying the model and an extrapolation to the measured angular data allows a direct determination of the complete scattering phase function. By comparing measurements on polystyrene microspheres with Mie theory, the concept of the presented instrument was validated. Finally, the method was used to determine the scattering phase function of different types and brands of soybean oil emulsions such as Intralipid or Lipovenös. The measured scattering phase functions were then used to calculate the corresponding Legendre moments to an order of 20 and for wavelengths between 450 nm and 750 nm (available online), which besides the anisotropy factor also allow the determination of higher order factors such as gamma.

Refractive index measurement using single fiber reflectance spectroscopy

A method using single fiber reflectance spectroscopy to measure the refractive indices of transparent and turbid media over a broad wavelength range is presented and tested. For transparent liquid samples, the accuracy is within 0.2%, and the accuracy increases with increasing wavelength. For liquid turbid media, the accuracy is within 0.3% and increases with decreasing wavelength. For solid turbid samples, such as human skin, the accuracy critically depends on the optical contact between the fiber and sample surface. It is demonstrated that this technique has the potential to measure refractive indices of biological tissue in vivo.

Efficient estimation of subdiffusive optical parameters in real time from spatially resolved reflectance by artificial neural networks

Subdiffusive reflectance captured at short source-detector separations provides increased sensitivity to the scattering phase function and hence allows superficial probing of the tissue ultrastructure. Consequently, estimation of subdiffusive optical parameters has been the subject of many recent studies focusing on lookup-table-based (LUT) inverse models. Since an adequate description of the subdiffusive reflectance requires additional scattering phase function related optical parameters, the LUT inverse models, which grow exponentially with the number of estimated parameters, become excessively large and computationally inefficient. Herein, we propose, to the best of our knowledge, the first artificial-neural-network-based inverse Monte Carlo model that overcomes the limitations of the LUT inverse models and thus allows efficient real-time estimation of optical parameters from subdiffusive spatially resolved reflectance. The proposed inverse model retains the accuracy, is about four orders of magnitude faster than the LUT inverse models, grows only linearly with the number of estimated optical parameters, and can be easily extended to estimate additional optical parameters.

Heuristic model for ballistic photon detection in collimated transmittance measurements

An heuristic model for ballistic photon detection in continuous-wave measurements of collimated transmittance through a slab is presented. The model is based on the small angle approximation and the diffusion equation and covers all the ranges of optical thicknesses of the slab from the ballistic to the diffusive regime. The performances of the model have been studied by means of comparisons with the results of gold standard Monte Carlo simulations for a wide range of optical thicknesses and two types of scattering functions. For a non-absorbing slab and field of view of the receiver less than 3° the model shows errors less than 15% for any value of the optical thickness. Even for an albedo value of 0.9, and field of view of the receiver less than 3° the model shows errors less than 20%. These results have been verified for a large set of scattering functions based on the Henyey-Greenstein model and Mie theory for spherical scatterers. The latter has also been used to simulate the scattering function of Intralipid, a diffusive material widely used as reference standard for tissue simulating phantoms. The proposed model represents an effective improvement compared to the existing literature.

Single fiber reflectance spectroscopy calibration

To accurately determine sample optical properties using single fiber reflectance spectroscopy (SFR), an absolute calibration of the reflectance is required. We investigated two SFR calibration methods, using a calibrated mirror and using the Fresnel reflection at the fiber tip as a reference. We compared these to commonly used calibration methods, using either Intralipid-20% in combination with Monte Carlo simulations or Spectralon as a reference. The Fresnel reflection method demonstrated the best reproducibility and yielded the most reliable result. We therefore recommend the Fresnel reflection method for the measured absolute reflectance calibration of SFR.