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. MC simulations are considered a "gold standard" because they give exact solutions (in the statistical sense), however, in the case of statistical mixtures their implementation is often extremely complex. For this reason, the aim of the present contribution is to propose a new approach that should allow us in the future to simplify the MC approach. This is done through an explanatory example, i.e.; by deriving the 'exact' analytical expression for the probability density function of photons' random steps (single step function, SSF) propagating in a medium represented as a binary (isotropic-Poisson) statistical mixture. The use of the SSF reduces the problem to an 'equivalent' homogeneous medium behaving exactly as the original binary statistical mixture. This will reduce hundreds MC simulations, allowing to obtain one set of wanted parameters, to only one equivalent simple MC simulation. To the best of our knowledge the analytically 'exact' SSF for a binary (isotropic-Poisson) statistical mixture has never been derived before.

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. To give a better sense of the importance of these particular photon step lengths, we also show numerically that the described approach is essential in preserving the invariance property for light propagation. An interesting byproduct of the MC method for ART is that it allows us to simplify the structure of "classical" MC codes, utilized, for example, in biomedical optics.

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.

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. Due to the particular rules utilized to generate the MC photons' paths, the present results may have an interest not only for the biomedical optics community, but also from a pure mathematical point of view.

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.

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. The chosen anomalous photon transport model also includes the classic and the "generalized" Lambert-Beer's law (depending on the choice of parameters). A numerical algorithm allowing generation of the related anomalous random photon steps is also given. The clear influence of anomalous transport on the field autocorrelation function is shown and discussed for the proposed specific examples by comparing the general results with the classical case (Lambert-Beer's law).

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.

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. The comparisons were realized for two SP-LDF implementations: for "classical" and "fast" algorithms. "Classical" SP-LDF is not sensitive to small changes on the number of detected speckles and coherence length of the laser, usually described by a unique parameter "beta." This is a strong point when assessing blood flow index, e.g., in humans, where "beta" is particularly difficult to be determined in real time. The proposed theory may be utilized, e.g., to investigate other SP-LDF setups and optical/physiological parameter ranges or, generally, to optimize real SP-LDF instrumentation.

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. The obtained results show that the AC signal, for modulation frequencies < 200 MHz, has a depth sensitivity with similar characteristics to that of the continuous-wave (CW) domain (source modulation frequency of zero). Thus, the depth probed by such a signal can be estimated by using the formula of penetration depth for the CW domain (Sci. Rep.6, 27057 (2016)). However, the PH signal has a different behavior compared to the CW domain, showing a larger depth sensitivity at shallow depths and a less steep relative contrast as a function of depth. These results mark a clear difference in term of depth sensitivity between AC and PH signals, and highlight the complexity of the estimation of the actual depth probed in tissue spectroscopy.

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.

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. All the solvers have been obtained as solutions of the time dependent diffusion equation. The validation of the solvers has been performed by means of comparisons with the results of "gold standard" Monte Carlo simulations. These forward solvers provide an accurate tool to explore the information content encoded in the time-resolved Raman measurements.

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. Validation of the theory by means of comparisons with Monte Carlo simulations is also presented. The results are of interest for many research fields such as biomedical optics, advanced microscopy and disordered photonics.

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. It is shown that for the human tibia, the "Gaussian hypothesis" holds for interoptode distances ≥20 mm. This lower boundary seems to depend on the type of investigated tissue.

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 Levenberg­Marquardt method for a geometry and real experimental conditions typically used to reconstruct the optical properties of biological tissues such as muscle and brain. The absorption coefficient of the layers was varied in a range of values typical for biological tissues. The reconstructions performed demonstrate the substantial improvements achievable with the OE provided a priori information is available. We note the extreme reliability, robustness, and accuracy of the retrieved absorption coefficient of the second layer obtained with the OE that was found for up to six fit parameters, with an error in the retrieved values of less than 10%. A priori information on fit parameters and fixed forward model parameters clearly improves robustness and accuracy of the inversion procedure.

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. Soc. Am. A14, 486 (1997)]. In this paper, we extend these findings to the time-dependent DE and we discuss some implications to diffuse optical tomography.

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).10.1364/AO.54.002400APOPAI1559-128X] for diffuse correlation spectroscopy (DCS) may be successfully used to check the ability of a given DCS model to generate a reliable estimator of tissue blood flow. As an example, four different DCS models often found in the scientific literature are tested on a simulated tissue (semi-infinite geometry) with a Maxwell-Boltzmann probability distribution function for red blood cell speed. It is shown that the random model is the best model for the chosen speed distribution but that (1) some inaccuracies in the DCS model in taking into account red blood cell concentration and (2) some inaccuracies, probably due to a low-order approximation of the DCS model, are still observed. The method can be easily generalized for other speed/flow probability distribution functions of the red blood cells.

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. We hope that the present contribution will encourage the scientific community to put a greater attention on this approach.

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 (MCan) is proposed. The MCan approach allows to obtain noise exempt LDF spectra in a short time and with a simple personal laptop. The proposed MCan holds for a diffusive regime of light propagation and it is practically implemented for a semi-infinite geometry. Its validity is demonstrated by comparisons with the classical MC.

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. The SP-LDF has been validated by Monte Carlo simulations, as well as by experimental measurements on a bone tissue phantom for optical flowmeters and on human.