Two-layered blood-lipid phantom and method to determine absorption and oxygenation employing changes in moments of DTOFs.

Near-infrared spectroscopy (NIRS) is an established technique for measuring tissue oxygen saturation (StO2), which is of high clinical value. For tissues that have layered structures, it is challenging but clinically relevant to obtain StO2 of the different layers, e.g. brain and scalp. For this aim, we present a new method of data analysis for time-domain NIRS (TD-NIRS) and a new two-layered blood-lipid phantom. The new analysis method enables accurate determination of even large changes of the absorption coefficient (Δµa) in multiple layers. By adding Δµa to the baseline µa, this method provides absolute µa and hence StO2 in multiple layers. The method utilizes (i) changes in statistical moments of the distributions of times of flight of photons (DTOFs), (ii) an analytical solution of the diffusion equation for an N-layered medium, (iii) and the Levenberg-Marquardt algorithm (LMA) to determine Δµa in multiple layers from the changes in moments. The method is suitable for NIRS tissue oximetry (relying on µa) as well as functional NIRS (fNIRS) applications (relying on Δµa). Experiments were conducted on a new phantom, which enabled us to simulate dynamic StO2 changes in two layers for the first time. Two separate compartments, which mimic superficial and deep layers, hold blood-lipid mixtures that can be deoxygenated (using yeast) and oxygenated (by bubbling oxygen) independently. Simultaneous NIRS measurements can be performed on the two-layered medium (variable superficial layer thickness, L), the deep (homogeneous), and/or the superficial (homogeneous). In two experiments involving ink, we increased the nominal µa in one of two compartments from 0.05 to 0.25 cm-1, L set to 14.5 mm. In three experiments involving blood (L set to 12, 15, or 17 mm), we used a protocol consisting of six deoxygenation cycles. A state-of-the-art multi-wavelength TD-NIRS system measured simultaneously on the two-layered medium, as well as on the deep compartment for a reference. The new method accurately determined µa (and hence StO2) in both compartments. The method is a significant progress in overcoming the contamination from the superficial layer, which is beneficial for NIRS and fNIRS applications, and may improve the determination of StO2 in the brain from measurements on the head. The advanced phantom may assist in the ongoing effort towards more realistic standardized performance tests in NIRS tissue oximetry. Data and MATLAB codes used in this study were made publicly available.

Single scattering models for radiative transfer of isotropic and cone-shaped light sources in fog.

The simulation of rare edge cases such as adverse weather conditions is the enabler for the deployment of the next generation of autonomous drones and vehicles into conditions where human operation is error-prone. Therefore, such settings must be simulated as accurately as possible and be computationally efficient, so to allow the training of deep learning algorithms for scene understanding, which require large-scale datasets disallowing extensive Monte Carlo simulations. One computationally-expensive step is the simulation of light sources in scattering media, which can be tackled by the radiative transfer equation and approximated by analytical solutions in the following. Traditionally, a single scattering event is assumed for fog rendering, since it is the dominant effect for relatively low scattering media. This assumption allows us to present an improved solution to calculate the so called air-light integral that can be evaluated fast and robustly for an isotropic point source in homogeneous media. Additionally, the solution is extended for a cone-shaped source and implemented in a computer vision rendering pipeline fulfilling computational restrictions for deep learning uses. All solutions can handle arbitrary azimuthally symmetric phase functions and were tested with the Henyey-Greenstein phase function and an advection fog phase function calculated from a particle distribution using Mie's theory. The used approximations are validated through extensive Monte Carlo simulations and the solutions are used to augment good weather images towards inclement conditions with focus on visible light sources, so to provide additional data in such hard-to-collect settings.

Analytical solution of the vector radiative transfer equation for single scattered radiance.

In this paper, derivation of the analytical solution of the vector radiative transfer equation for the single scattered radiance of three-dimensional semi-infinite media with a refractive index mismatch at the boundary is presented. In particular, the solution is obtained in the spatial domain and spatial frequency domain. Besides the general derivation, determination of the amplitude scattering matrix, which is required for the analytical solution, is given in detail. Furthermore, the incorporation of Fresnel equations due to a refractive index mismatch at the boundary is presented. Finally, verification of the derived formulas is performed using a self-implemented electrical field Monte Carlo method based on Jones formalism. For this purpose, the solution based on Jones formalism is converted to Stokes-Mueller formalism. For the verification, spherical particles are assumed as scatterers, whereby arbitrary size distributions can be considered.

Analytical solution of the vector radiative transfer equation for single scattered radiance: erratum.

We have discovered a small error in our recently published work [J. Opt. Soc. Am. A39, 2045 (2022)JOSAAH0030-394110.1364/JOSAA.467890], which we correct in this erratum. The error is located in Section 2.D and affects the solution for polydisperse distributions.

Analytical solution of the radiative transfer theory for the coherent backscattering from two-dimensional semi-infinite media.

An analytical solution for coherent backscattering (CBS) in two dimensions was derived by solving the radiative transfer equation. Particularly, the single scattered radiance from a semi-infinite medium containing perpendicularly illuminated cylinders was obtained. At the boundary, a refractive index mismatch was taken into account. Furthermore, the link between the radiance and the CBS was shown in the small angle approximation. An excellent agreement was found between Monte Carlo simulations and the analytical solution. Additionally, it was shown that the often applied solution in the spatial frequency domain for quantifying the CBS delivered significantly different results compared to the derived exact analytical solution.

Two-step verification method for Monte Carlo codes in biomedical optics applications.

SIGNIFICANCE: Code verification is an unavoidable step prior to using a Monte Carlo (MC) code. Indeed, in biomedical optics, a widespread verification procedure for MC codes is still missing. Analytical benchmarks that can be easily used for the verification of different MC routines offer an important resource. AIM: We aim to provide a two-step verification procedure for MC codes enabling the two main tasks of an MC simulator: (1) the generation of photons' trajectories and (2) the intersections of trajectories with boundaries separating the regions with different optical properties. The proposed method is purely based on elementary analytical benchmarks, therefore, the correctness of an MC code can be assessed with a one-sample t-test. APPROACH: The two-step verification is based on the following two analytical benchmarks: (1) the exact analytical formulas for the statistical moments of the spatial coordinates where the scattering events occur in an infinite medium and (2) the exact invariant solutions of the radiative transfer equation for radiance, fluence rate, and mean path length in media subjected to a Lambertian illumination. RESULTS: We carried out a wide set of comparisons between MC results and the two analytical benchmarks for a wide range of optical properties (from non-scattering to highly scattering media, with different types of scattering functions) in an infinite non-absorbing medium (step 1) and in a non-absorbing slab (step 2). The deviations between MC results and exact analytical values are usually within two standard errors (i.e., t-tests not rejected at a 5% level of significance). The comparisons show that the accuracy of the verification increases with the number of simulated trajectories so that, in principle, an arbitrary accuracy can be obtained. CONCLUSIONS: Given the simplicity of the verification method proposed, we envision that it can be widely used in the field of biomedical optics.

Semianalytical solution for the transient temperature in a scattering and absorbing slab consisting of three layers heated by a light source.

We derived a semianalytical solution for the time-dependent temperature distribution in a three-layered laterally infinite scattering and absorbing slab illuminated by an obliquely incident collimated beam of light. The light propagation was modeled by the low-order [Formula: see text] and [Formula: see text] approximations to the radiative transfer equation with closed form expressions for eigenvalues and eigenvectors, yielding a quickly computable solution, while the heat conduction was modeled by the Fourier equation. The solution was compared to a numerical solution using a Monte Carlo simulation for the light propagation and an FEM method for the heat conduction. The results showed that using the [Formula: see text] solution for the light propagation offers a large advantage in accuracy with only a moderate increase in calculation time compared to the [Formula: see text] solution. Also, while the [Formula: see text] solution is not a very good approximation for the spatially resolved absorbance itself, its application as a source term for the heat conduction equation does yield a very good approximation for the time-dependent temperature.

Solutions for single-scattered radiance in the semi-infinite medium based on radiative transport theory.

In this paper, some explicit analytical solutions for single-scattered radiance in a half-space medium under consideration of a reflecting boundary are derived. We consider both a unidirectional beam source as well as an isotropic point source. In addition to direct applications within optical tomography and computer graphics, the obtained solutions are also needed when solving the radiative transport equation after the separation of the unscattered and single-scattered contribution. Comparisons between the derived analytical solutions and the Monte Carlo method display excellent agreement.

P3 solution for the total steady-state and time-resolved reflectance and transmittance from a turbid slab.

In this paper, we derive some explicit analytical solutions to the P3 equations for the slab geometry that is illuminated by a collimated plane source. The resulting expressions for the total reflectance and transmittance are compared with the corresponding transport theory solution predicted by the Monte Carlo method. Further, for the special case of a non-absorbing anisotropically scattering slab, simple and accurate expressions in the P1 approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm's law. In view of the time domain, we present an alternative method to the classical frequency-domain approach avoiding the use of complex numbers.

Spatial frequency domain imaging using an analytical model for separation of surface and volume scattering.

A method to correct for surface scattering in spatial frequency domain imaging (SFDI) is presented. The use of a modified analytical solution of the radiative transfer equation allows calculation of the reflectance and the phase of a rough semi-infinite geometry so that both spatial frequency domain reflectance and phase can be applied for precise retrieval of the bulk optical properties and the surface scattering. For validation of the method, phantoms with different surface roughness were produced. Contrarily, with the modified theory, it was possible to dramatically reduce systematic errors due to surface scattering. The evaluation of these measurements with the state-of-the-art theory and measuring modality, i.e., using crossed linear polarizers, reveals large errors in the determined optical properties, depending on the surface roughness, of up to ≈100 % . These results were confirmed with SFDI measurements on a phantom that has a structured rough surface.

Analytical solutions of the radiative transport equation for turbid and fluorescent layered media.

Accurate and efficient solutions of the three dimensional radiative transport equation were derived in all domains for the case of layered scattering media. Index mismatched boundary conditions based on Fresnel's equations were implemented. Arbitrary rotationally symmetric phase functions can be applied to characterize the scattering in the turbid media. Solutions were derived for an obliquely incident beam having arbitrary spatial profiles. The derived solutions were successfully validated with Monte Carlo simulations and partly compared with analytical solutions of the diffusion equation.

Derivation of the correlation diffusion equation with static background and analytical solutions.

A new correlation diffusion equation has been derived from a correlation transport equation allowing one to take into account the presence of moving scatterers and static background. Solutions for the reflectance from a semi-infinite medium have been obtained (point-like and ring detectors). The solutions have been tested by comparisons with "gold standard" Monte Carlo (MC) simulations. These formulas suitably describe the electric field autocorrelation function, for Brownian or random movement of the scatterers, even in the case where the probability for a photon to interact with a moving scatterer is very low. The proposed analytical models and the MC simulations show that the "classical" model, often used in diffuse correlation spectroscopy, underestimates the normalized field autocorrelation function for increasing correlation times.

Analytical solution of the correlation transport equation with static background: beyond diffuse correlation spectroscopy.

The correlation transport equation (CTE) is the natural generalization of the theory for diffusion correlation spectroscopy and represents a more precise model when dealing with measurements of particle movement in fluids or red blood cell flow in biological tissues. Unfortunately, the CTE is not methodically used due to the complexity of finding solutions. It is shown that actually a very simple modification of the theory/software for the solution of the radiative transport equation allows one to obtain exact solutions of the CTE. The presence of a static background is also taken into account and its influence on the CTE solutions is discussed. The proposed approach permits one to easily work beyond the diffusion regime and potentially for any optical and/or physiological value. The validity of the approach is demonstrated by using "gold standard" Monte Carlo simulations.

Quantifying phase function influence in subdiffusively backscattered light.

Light backscattering at short source-detector separations is considerably influenced by the scattering phase function of a turbid medium. We seek to more precisely relate a medium's subdiffusive backscattering to the angular scattering characteristics of its microstructure. First, we demonstrate the inability of the scattering asymmetry g1 = < cos θ > to predict phase function influence on backscattering and reveal ambiguities related to the established phase function parameter γ. Through the use of high-order similarity relations, we introduce a new parameter that more accurately relates a scattering phase function to its subdiffusive backscattering intensity. Using extensive analytical forward calculations based on solutions to the radiative transfer equation in the spatial domain and spatial frequency domain, we demonstrate the superiority of our empirically derived quantifier σ over the established parameter γ.

Detecting structural information of scatterers using spatial frequency domain imaging.

We demonstrate optical phantom experiments on the phase function parameter γ using spatial frequency domain imaging. The incorporation of two different types of scattering particles allows for control of the optical phantoms' microscopic scattering properties. By laterally structuring areas with either TiO2 or Al2O3 scattering particles, we were able to obtain almost pure subdiffusive scattering contrast in a single optical phantom. Optical parameter mapping was then achieved using an analytical radiative transfer model revealing the microscopic structural contrast on a macroscopic field of view. As part of our study, we explain several correction and referencing techniques for high spatial frequency analysis and experimentally study the sampling depth of the subdiffusive parameter γ.

Model-based analysis on the influence of spatial frequency selection in spatial frequency domain imaging.

Frequency variation in spatial frequency domain imaging is a powerful tool for adjusting the penetration depth of the imaging signal and the parameter sensitivity toward absorption and diffusive and subdiffusive scattering. Through our computational analysis, using an analytical solution of the radiative transfer equation, we add quantitation to this tool by linking the different spatial frequency regimes to their relative information content and to their absolute depth sensitivity. Special focus is placed on high spatial frequencies by analysis of the phase function parameter γ and its significance and ambiguity in describing subdiffusive scattering.

Novel analytical solution for the radiance in an anisotropically scattering medium.

We report on a novel analytical solution of the three-dimensional radiative transport equation for the case of an infinitely extended anisotropically scattering medium that is illuminated by an isotropic point light source. The resulting expression for the radiance can be evaluated efficiently and accurately and exhibits significant improvements with respect to the convergence and the numerical stability compared to the solutions found in the literature so far. The equations obtained were successfully verified by comparisons with the Monte Carlo method.

Surface layering properties of Intralipid phantoms.

Intralipid has become an extensively studied and widely used reference and calibration phantom for diffuse optical imaging technologies. In this study we call attention to the layering properties of Intralipid emulsions, which are commonly assumed to have homogeneous optical properties. By measurement of spatial frequency domain reflectance in combination with an analytical solution of the radiative transfer equation for two-layered media, we make quantitative investigations on the formation of a surface layer on different dilutions of Intralipid. Our findings are verified by an independent spatially resolved reflectance setup giving evidence of a time dependent, thin and highly scattering surface layer on top of Intralipid-water emulsions. This layer should be considered when using Intralipid as an optical calibration or reference phantom.

Explicit solutions of the radiative transport equation in the P3 approximation.

PURPOSE: Explicit solutions of the monoenergetic radiative transport equation in the P3 approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. METHODS: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P3 equations in closed form. RESULTS: The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P3 approximation is close to the exact solution of the radiative transport equation. CONCLUSIONS: The authors derived exact analytical solutions of the P3 equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.