Probability density functions for photon propagation in a binary (isotropic-Poisson) statistical mixture with unmatched positive and negative refractive indexes.

The exact homogenized probability density function, for a photon making a step of length s has been analytically derived for a binary (isotropic-Poisson) statistical mixture with unmatched refractive indexes. The companions, exact, homogenized probability density functions for a photon to change direction ("scatter"), with polar ϑ and azimuthal φ angles, and the homogenized albedo, have also been obtained analytically. These functions also apply to negative refractive indexes and can reduce the number of Monte Carlo simulations needed for photon propagation in complex binary (isotropic-Poisson) statistical mixtures from hundreds to just one, for an equivalent homogeneous medium.

Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture.

Monte Carlo (MC) simulations allowing to describe photons propagation in statistical mixtures represent an interest that goes way beyond the domain of optics, and can cover, e.g., nuclear reactor physics, image analysis or life science just to name a few.

Monte Carlo simulations in anomalous radiative transfer: tutorial.

Anomalous radiative transfer (ART) theory represents a generalization of classical radiative transfer theory. The present tutorial aims to show how Monte Carlo (MC) codes describing the transport of photons in anomalous media can be implemented. We show that the heart of the method involves suitably describing, in a "non-classical" manner, photon steps starting from fixed light sources or from boundaries separating regions of the medium with different optical properties.

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.

Topological complexity of photons' paths in biological tissues.

In the present contribution, three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe, and the minimal crossing number of photons' paths generated by Monte Carlo (MC) simulations, for different sets of optical parameters. It is observed that the complexity of the photons' paths increases for increasing light source/detector spacing, and that highly "knotted" paths are formed.

P 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 P 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 P approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm's law.

Generalized time-independent correlation transport equation with static background: influence of anomalous transport on the field autocorrelation function.

A generalized time-independent correlation transport equation (GCTE) is proposed for the field autocorrelation function. The GCTE generalizes various models for anomalous transport of photons and takes into account the possible presence of a static background. In a tutorial example, the GCTE is solved for a homogeneous semi-infinite medium in reflectance configuration through Monte Carlo simulations.

Time domain diffuse Raman spectrometer based on a TCSPC camera for the depth analysis of diffusive media.

We present a time domain diffuse Raman spectrometer for depth probing of highly scattering media. The system is based on, to the best of our knowledge, a novel time-correlated single-photon counting (TCSPC) camera that simultaneously acquires both spectral and temporal information of Raman photons. A dedicated non-contact probe was built, and time domain Raman measurements were performed on a tissue mimicking bilayer phantom.

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.

Study on the mathematical relationship existing between single-photon laser-Doppler flowmetry and diffuse correlation spectroscopy with static background.

In the present contribution, the theoretical relationship existing between the blood flow index measured by diffuse correlation spectroscopy and single-photon laser-Doppler flowmetry (SP-LDF) is investigated. A specific mathematical description that accounts for the properties of single-photon detectors for SP-LDP was developed. Static background has also been considered and, to the best of our knowledge, this has never been included before in SP-LDF analytical theories.

Depth sensitivity of frequency domain optical measurements in diffusive media.

The depth sensitivity functions for AC amplitude, phase (PH) and DC intensity signals have been obtained in the frequency domain (where the source amplitude is modulated at radio-frequencies) by making use of analytical solutions of the photon diffusion equation in an infinite slab geometry. Furthermore, solutions for the relative contrast of AC, PH and DC signals when a totally absorbing plane is placed at a fixed depth of the slab have also been obtained. The solutions have been validated by comparisons with gold standard Monte Carlo simulations.

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.

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.

Time-domain Raman analytical forward solvers.

A set of time-domain analytical forward solvers for Raman signals detected from homogeneous diffusive media is presented. The time-domain solvers have been developed for two geometries: the parallelepiped and the finite cylinder. The potential presence of a background fluorescence emission, contaminating the Raman signal, has also been taken into account.

There's plenty of light at the bottom: statistics of photon penetration depth in random media.

We propose a comprehensive statistical approach describing the penetration depth of light in random media. The presented theory exploits the concept of probability density function f(z|ρ, t) for the maximum depth reached by the photons that are eventually re-emitted from the surface of the medium at distance ρ and time t. Analytical formulas for f, for the mean maximum depth 〈zmax〉 and for the mean average depth reached by the detected photons at the surface of a diffusive slab are derived within the framework of the diffusion approximation to the radiative transfer equation, both in the time domain and the continuous wave domain.

Probability density function of the electric field in diffuse correlation spectroscopy of human bone in vivo.

Diffuse correlation spectroscopy (DCS) is the technique of choice for non-invasive assessments of human bone blood flow. However, DCS classical algorithms are based on the fundamental assumption that the electric field of the light reaching the DCS photodetector is a zero-mean complex Gaussian variable. The non-validity of this hypothesis might produce inaccurate blood flow estimations.

Optimal estimation reconstruction of the optical properties of a two-layered tissue phantom from time-resolved single-distance measurements.

In this work, we have tested the optimal estimation (OE) algorithm for the reconstruction of the optical properties of a two-layered liquid tissue phantom from time-resolved single-distance measurements. The OE allows a priori information, in particular on the range of variation of fit parameters, to be included. The purpose of the present investigations was to compare the performance of OE with the Levenberg–Marquardt method for a geometry and real experimental conditions typically used to reconstruct the optical properties of biological tissues such as muscle and brain.

Modified reciprocity relation for the time-dependent diffusion equation.

The classical reciprocity relation of radiative transfer fails for two points placed in regions having different indices of refraction. A modified reciprocity relation that involves the relative refractive index between the two points considered was previously derived for the continuous wave (cw) radiative transfer equation and for the cw diffusion equation (DE) [J. Opt.

Near-infrared photons: a non-invasive probe for studying bone blood flow regulation in humans.

The study of bone blood flow regulation in humans has always represented a difficult task for the clinician and the researcher. Classical measurement techniques imply the presence of ionizing radiation or contrast agents, or they are slow or cannot be repeated too often in time. In the present review, we would like to give a perspective on how the optical approach might overcome some of these problems and give unique solutions to the study of bone blood flow regulation.

Assessing the reliability of diffuse correlation spectroscopy models on noise-free analytical Monte Carlo data.

It is shown that an analytical noise-free implementation of Monte Carlo simulations [Appl. Opt.54, 2400 (2015).

Random numbers free analytical implementation of Monte Carlo for laser-Doppler flowmetry at large interoptode spacing: application to human bone tissue.

Classical Monte Carlo (MC) simulations for laser-Doppler flowmetry (LDF) often necessitate too long computation times and specialized hardware. This is particularly true for LDF at large interoptode spacing with low absorption coefficients and large anisotropic factors representing real biological tissues. For this reason, a random numbers free "analytical" implementation of the classical MC (MC) is proposed.

Time-domain algorithm for single-photon laser-Doppler flowmetry at large interoptode spacing in human bone.

A new laser-Doppler flowmeter at large interoptode spacing, based on single-photon counting (single-photon laser-Doppler flowmetry [SP-LDF]) and allowing assessment of blood flow deep in bone tissue, is proposed and implemented. To exploit the advantages of the new SP-LDF hardware, a dedicated simple and efficient time-domain algorithm has been developed. The new algorithm is based on the zero-order moment of the power density spectrum of the ad hoc prefiltered photoelectric current.

Bone tissue phantoms for optical flowmeters at large interoptode spacing generated by 3D-stereolithography.

A bone tissue phantom prototype allowing to test, in general, optical flowmeters at large interoptode spacings, such as laser-Doppler flowmetry or diffuse correlation spectroscopy, has been developed by 3D-stereolithography technique. It has been demonstrated that complex tissue vascular systems of any geometrical shape can be conceived. Absorption coefficient, reduced scattering coefficient and refractive index of the optical phantom have been measured to ensure that the optical parameters reasonably reproduce real human bone tissue in vivo.

Influence of light source-detector spacing on shape of probability density functions of scattering angles in laser Doppler flowmetry.

Analytical models, describing laser Doppler flowmetry and its derived applications, are based on fundamental assumptions of photon scattering angles. It is shown by means of Monte Carlo simulations that, even in the case these assumptions are correct, the presence of a specific source-detector configuration may bias the shape of the probability density functions describing scattering angle behavior. It is found that these biased shapes are generated by selective filtering of photons induced by a particular source-detector configuration.

Pulsatile and steady-state hemodynamics of the human patella bone by diffuse optical spectroscopy.

The cardiac cycle related pulsatile behavior of the absorption and scattering coefficients of diffuse light and the corresponding alterations in hemoglobin concentrations in the human patella was studied. The pulsations in scattering is considerably smaller than absorption. The difference in amplitude of absorption coefficient pulsations for different wavelengths was translated to pulsations in oxygenated and deoxygenated hemoglobin, which leads to strong pulsations in the total hemoglobin concentration and oxygen saturation.