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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., (© 2023. The Author(s).)

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.

In this paper, we derive some explicit analytical solutions to the P 3 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 1 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.

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.

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.

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.

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.

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.

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.

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.

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. 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.

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.

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.

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 an ) is proposed. The MC an approach allows to obtain noise exempt LDF spectra in a short time and with a simple personal laptop. The proposed MC an 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.

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.

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. An experimental demonstration of a possible use of the optical phantom, utilizing a laser-Doppler flowmeter, is also presented.

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. In some specific cases, this phenomenon might invalidate laser Doppler analytical models.

The interest in, and the need for effective measures to be used in screening, diagnosis, and the follow-up of skeletal pathologies is growing markedly. This paper proposes a completely new and non-invasive technique allowing the study of the human tibia bone marrow (BM) haemodynamics with a time resolution of 1 s. The technique, based on near infrared spectroscopy, is sensitive enough to allow the detection of BM blood volume and/or oxygen saturation changes during orthostatic variations imposed by a tilt bed. An increase in the slope of the bed of 15 degrees is sufficient to detect this phenomenon. The ability to study the possible presence of a neural control of BM haemodynamics is also discussed. No other existing technique currently allows one to obtain the proposed results and this approach might open up a new field of study related to human BM physiology.

It is demonstrated, that the bioenergetic model combined with the mathematical constraints determined by the experimental knowledge of the aerobic metabolism and the Lohmann reaction dictates the exact lactate (La)-time relationship during exercise. The theory predicts that La is necessarily produced (above the resting baseline), even during extremely low work loads, where the metabolism was usually considered in the past to be "pure" aerobic. The La rate of production increases linearly as a function of the work load. The anaerobic threshold is strictly determined by the saturation of the La clearance mechanisms of the body different from the "La shuttle" and not by the involvement of a sudden increased La production at the cellular level. These results imply that the half time of the PCr breakdown kinetics at the onset of a constant load exercise can be expressed as a function of the onset speed of the aerobic and of the anaerobic metabolism, even in the case of a very low mechanical power. The PCr half-time does not depend on the workload and represents a physiological invariant. The bioenergetic model was created during a long historical period, when it was believed that the La production was not present at all for very low exercise levels but, actually, the bioenergetic model predicts exactly the opposite result!

Background: A new method of thermoablation with hot water vapour based on a new type of microtube was developed. This approach allows tumours, with volume and anatomical positions not accessible to other techniques (cryoablation, radiofrequency ablation, laser ablation) to be treated., Materials and Methods: The method was tested on a human colon carcinoma grafted subcutaneously in Swiss nude mice and the experiment monitored under magnetic resonance imaging., Results: It was found that 2.52 cal s(-1) per cm3 of tumour were necessary to reduce tumour size. The microtube is built to withstand a large range of temperatures and pressures and is biocompatible., Conclusion: A specific feature of this technique is that, besides hot vapour, several types of drugs can be delivered through the same microtube depending of the location, type or size of the tumour. These properties make it a unique device for multi-therapeutic treatments.

A human model allowing the non-invasive study of bone marrow haemodynamics has been developed. A decrease in postischaemic tissue reperfusion capability (postischaemic hyperaemia) as a function of age (range 25-72 years) was observed both in the human tibia and tibialis anterior muscle. In the tibia bone marrow the reperfusion capability started to decrease after 50 years and was lower than for muscle for all the age range. Mean basal muscle O(2) saturation (80.8% at 25 years) decreases as a function of age (-0.35%+/-0.13% per year) whereas it remains constant for bone marrow (84.8+/-2.8%). A Monte Carlo simulation has been performed to evaluate the accuracy of the derived O(2) saturation measurements and has shown that this parameter is robust even in the presence of substantial noise. It has also been demonstrated that it is necessary to use a multi wavelength NIR spectrometer and a second derivative based fitting algorithm to obtain reliable measurements from the bone marrow, and that the tissue scattering changes occurring during the protocol do not allow the use of the standard near infrared spectroscopy algorithms. The human tibia bone marrow model presented here and the related measurement technique should enable access to new areas of physiological research.

The age dependence of human gastrocnemius Mg2+ concentration is demonstrated. To quantitate Mg2+ concentration, an original and accurate fitting algorithm using quantum mechanics-based prior knowledge is detailed. In a group of 28 volunteers (14 females) in the age range 5-80 years, pH, PCr/ATP and Pi/ATP values in the gastrocnemius were 7.02 +/- 0.02 pH, 4.16 +/- 0.33 and 0.13 +/- 0.02, respectively and independent of age and sex. By contrast, intracellular Mg2+ concentration (mM) decreased linearly (p < 0.05) with age (Mg2+ = 0.7803 +/- 0.0247-0.0027 +/- 0.0005 * age). No difference was found between sexes. From these results, it follows that care must be taken when comparing muscle Mg2+ data from subjects of different age. The hypothesis can be put forward that during aging insufficient intake and/or increased depletion of Mg2+ (e.g., intestinal hypoabsorption or urinary leakage) may affect the musculoskeletal system.

The aim of the present study was to quantify changes in human skeletal muscle pennation angle (F theta) values during growth and adult life. The human gastrocnemius medialis muscle of 162 subjects (96 males and 66 females) in the age range 0-70 years was scanned with ultrasonography. The subjects were laying prone, at rest, with the ankle maintained at 90 degrees with all muscles relaxed. F theta increased monotonically starting from birth (0 years) and reached a stable value after the adolescent growth spurt. There was a significant (p < 0.05) linear relationship between F theta and muscle thickness (TK). F theta = 0.84 (+/- 0.09) * TK + 3.15 (+/- 1.13). Human gastrocnemius medialis F theta and TK data found in the literature seem to fit the F theta-TK plot in a coherent manner, independent of the physiological or anatomical characteristics of the subject. The present findings indicate that F theta is not a constant parameter but evolves, as is the case for bone length and height, as a function of age.

The aim of this review is to describe the effects induced by local temperature changes on human skeletal muscle metabolism. More specifically, we will consider the influence of temperature on the mechanical properties of muscle contraction, on aerobic metabolism, anaerobic metabolism and on the Lohmann reaction. The text has been voluntarily organized on the basis of a simple bioenergetic model describing the different energy fluxes appearing in the muscle system. This approach should better highlight some of the points that still need to be investigated. Although it was not always possible to restrict the discussion to human muscle, the references report mainly data obtained directly on humans or on isolated human fibres. A short comment on skeletal muscle temperature measurement techniques, on humans, is also included.

Oxidative metabolism is the dominant source of energy for skeletal muscle. Near-infrared spectroscopy allows the non-invasive measurement of local oxygenation, blood flow and oxygen consumption. Although several muscle studies have been made using various near-infrared optical techniques, it is still difficult to interpret the local muscle metabolism properly. The main findings of near-infrared spectroscopy muscle studies in human physiology and clinical medicine are summarized. The advantages and problems of near-infrared spectroscopy measurements, in resting and exercising skeletal muscles studies, are discussed through some representative examples.

Rationale and Objectives: The authors investigated the usefulness of dynamic phosphorus-31 magnetic resonance (MR) spectroscopy in the assessment of hepatic function by studying the effect of a fructose load on a rat model of liver cirrhosis., Methods: In vivo P-31 MR liver spectra of eight rats with bile duct ligature and 10 control rats were obtained every 4.6 minutes before and after intraperitoneal fructose load (10 mmol per kilogram of body weight)., Results: In the basal spectra of the experimental group, the phosphomonoester peak was higher than in the control group (P = .026). After the fructose load, the phosphomonoester peak increase and the inorganic phosphate peak decrease were significantly less marked in the experimental group (P = .003). There was a linear correlation between the serum level of bilirubin and the phosphomonoester increase (r = .61, P < .001)., Conclusion: Dynamic P-31 MR spectroscopy may be useful in the assessment of hepatic function in chronic liver disease.