Fabrizio Martelli

Diffuse optics measurements performed in the time domain provide the highest information level, but are plagued by a low spatial resolution and significant sensitivity to superficial regions (e.g., extracerebral changes). Here we compare the two main analysis methods (namely, statistical moments and time-windowing of the photon time-of-flight distribution) with the recently introduced dual-slope method. Both in-silico and phantom studies are performed. Our results show that time-windowing, although not previously applied in this context, maximizes the advantages of a dual-slope approach while also maximizing the depth selectivity. Furthermore, for time-windowing, a triple-slope approach is proposed, demonstrating superior rejection of superficial changes.

For over thirty years, the definition of the diffusion coefficient in light transport has been the subject of persistent debate. Its canonical expression includes an explicit dependence on absorption, violating a fundamental scaling property of the radiative transfer equation (RTE). In the time domain (TD), evidence shows that the correct definition is independent of absorption, yet absorption-dependent formulations unfortunately remain common in practical data analysis. In the continuous-wave regime, the radial decay of the RTE can be reproduced by an absorption-dependent coefficient, but this does not imply that steady-state transport is genuinely diffusive. Here, this controversy is resolved through an exact consistency test based solely on the requirement that diffusion converges to the RTE in the limit of infinite propagation. In the TD, we prove that the absorption-independent coefficient is both necessary and sufficient for exact long-time convergence. In the continuous-wave regime, we prove analytically that no scalar diffusion coefficient can make diffusion asymptotically equivalent to the RTE when absorption is nonzero: matching the dominant attenuation length fixes one coefficient, whereas matching the residue requires a different one. We further show that this failure is not caused by any finite set of ballistic or few-scattering contributions, and quantify the intrinsic error introduced by incorrect parameterizations in idealized absorption-retrieval scenarios. These results establish the time-domain formulation as the only self-consistent reference and call for a re-examination of how diffusion is applied in optical transport studies

Exact identities for radiative transfer are derived in scattering and absorbing media that relate emitted, received, and absorbed power (in the full domain or in arbitrary sub-volumes) to two trajectory statistics: the mean optical pathlength ⟨ ℓ ⟩ traveled by photons and the mean number of scattering events ⟨ k ⟩. For the received signal, the logarithmic derivatives with respect to absorption and scattering take particularly simple forms. In the continuous-wave case, ∂ ln ⁡( P _R )/ ∂ μ _a =−⟨ ℓ ⟩ _R and ∂ ln ⁡( P _R )/ ∂ μ _s =⟨ k ⟩ _R / μ _s −⟨ ℓ ⟩ _R , with analogous relations for time-resolved measurements. The second identity implies a counter-intuitive and yet highly consequential fact: the difference ⟨ k ⟩ _R / μ _s −⟨ ℓ ⟩ _R is generally nonzero, which is required to ensure that the detected signal is sensitive to scattering variations. Motivated by this observation, we formalize an “adding scattering” protocol, analogous to established adding-absorption approaches, to estimate scattering sensitivity via finite-difference changes in μ _s in both Monte Carlo simulations and experiments. The predicted relations are validated via independent Monte Carlo implementations and with measurements in calibrated liquid phantoms. Finally, the role of these identities as exact benchmarks for forward solvers in photon migration studies is discussed.

An analytical heuristic model has been developed to describe the time-domain reflectance Raman signal in a two-layer diffusive medium. The model is based on the solution of two coupled diffusion equations at the excitation and emission wavelengths, respectively. The accuracy of the model is verified by comparison with the results of Monte Carlo simulations. This solver correctly accounts for the Raman emitters within the layers and the full wavelength dependence of the optical properties of the layers. The computational efficiency of the solver is compatible with the requirements of rapid reconstructions of Raman spectra from time-domain Raman measurements acquired for a large number of different wavelengths.

We present a rigorous analytical model, based on the diffusion equation, to calculate the average generation depth of Raman photons in a semi-infinite, diffusive, homogeneous medium, taking into account variations in optical properties between the wavelength of the excitation photons and that of the Raman-scattered photons. The model quantifies how the generation depth depends on photon time-of-flight for time domain diffuse Raman spectroscopy (TD-DIRS) or source-detector distance for spatially offset Raman spectroscopy (SORS). Monte Carlo simulations demonstrate the model’s reliability. We applied this model to simulate representative cases of in vivo scenarios for which the probed depth is evaluated by the proposed theoretical model. This model provides the basic knowledge to significantly improve the interpretation of SORS and TD-DIRS measurements.

Ultrasound optical tomography enables noninvasive characterization of tissue and lesions up to 5 cm below the skin surface, by enhancing diffuse optical tomography with ultrasound focusing resolution and an increased number of measurement points. For optimal clinical imaging, it is essential to be able to reconstruct tissue optical properties in real time during in vivo measurements, and therefore, it is necessary to have fast analytical models for ultrasound-modulated light fluence calculation. We derive a simple analytical model for tagged light fluence calculation based on mean photon pathlengths. The continuous-wave diffusion equation and the perturbation approach are used for modeling light propagation through a diffusive medium with inhomogeneous absorption. We introduce, forwhat we believe to be the first time, a concept of conditional mean photon pathlengths and the partial pathlengths approach with its derivation. The proposed analytical model is validated against a previously developed tagged light Monte Carlo simulator. The tagged light transmittance from the analytical model differs 3 times faster (∼30 s instead of ∼20 h). This substantial improvement in computational efficiency will enable real-time inverse problem solving. The results also show potential for further refinement of the proposed analytical model and for the development of in vivo clinical applications, e.g., including the characterization of suspicious breast lesions or the measurement of oxygen saturation in tissues.

The term diffusive media refers to all the media for which the photon diffusion equation provides an accurate description of light propagation. Indeed, this is the case for a plethora of natural media, such as biological tissues and agricultural products, when illuminated by red and near-infrared light. Diffuse Optics (DO) is the branch of Optics that studies how absorption and scattering phenomena affect light propagation in diffusive media. In this review paper, we present an introduction to time domain (TD) DO, a specific implementation of DO that employs picosecond light pulses, fast and sensitive photodetectors and timing electronics to record the distribution of photon time-of-flight (or photon path lengths) in diffusive media. By interpreting the TD DO signals with the physical model provided by the photon diffusion theory, it is possible to estimate the absorption and scattering properties of the medium that in case of biological tissues can be related to physiological and pathological conditions. We focus on the physical principles of TD DO, the building blocks of TD DO instrumentation, and the applications of TD DO targeting human biological tissues (such as the brain, breast, muscle, and others).

Monte Carlo (MC) simulations can adequately describe photon migration in layered media; however, storing and querying the resulting dataset may be computationally prohibitive when detailed path data are needed for each photon trajectory. A heuristic approach that significantly reduces the stored information to the average path length traveled by the photons detected in each layer is proposed. Its accuracy is evaluated by comparing it with the exact time point spread function (TPSF) for a bilayered medium. This method, almost exact for small variations in absorption, is potentially useful to provide a small dataset for lookup tables to be used in inverse problems.

Monte Carlo (MC) simulations are the gold standard for describing various transport phenomena and have largely contributed to the understanding of these processes. However, while their implementation for classical transport governed by exponential step‐length distributions is well‐established, widely accepted approaches are still lacking for the more general class of anomalous transport phenomena. In this work, a set of rules for performing MC simulations in anomalous diffusion media is identified, which is also applicable in the case of finite‐size geometries and/or heterogeneous inclusions. The results are presented in the context of radiative transfer, however their implications extend to all types of anomalous transport. The proposed set of rules exhibits full compatibility with the pathlength invariance property for random trajectories, and with the important radiometric concept of fluence. Additionally, it reveals the counter‐intuitive possibility of introducing interfaces between independent subdomains with identical properties, which arise from the fact that non‐exponential step‐length distributions have a “memory” that can in principle be reset when traversing a boundary. These results have far‐reaching consequences not just for the physical interpretation of the corrections required to handle these discontinuities, but also for their experimental verification, due to their expected effects on the observable pathlength distributions.

The limits of applicability of scaling relations to generate new simulations of photon migration in scattering media by re-scaling an existing Monte Carlo simulation are investigated both for the continuous wave and the time domain case. We analyzed the convergence properties in various scenarios by numerical methods, trying to derive practical guidelines for the judicious use of this approach, as well as a deeper understanding of the physics behind such relations. In the case of scaling of the absorption coefficient, the convergence is always rigorous both for the forward and inverse problems, relying on the derivatives with respect to the absorption coefficient. Also, the regenerated simulation inherits the very same noise of the original Monte Carlo simulation. In the case of scaling of the scattering coefficient, the situation is more critical. For forward problems, even for just a 10% uniform increase in scattering, appreciable deviations are observed whenever a high number of scattering interactions is involved. We tested a practical criterion based on the number of scattering events in the original simulation to judge the convergence of the scaling factors. For inverse problems, the scaling relations provide accurate regenerated simulations apart from the noise level that is increased with respect to the initial simulation, although anyway lower than the noise level obtained by implementing the direct calculation. The results of this study are important whenever an increase of Monte Carlo code throughput is mandatory, e.g., for fast data analysis of diffuse data, or in machine-learning scenarios, when generating huge datasets is needed.