Application limits of the scaling relations for Monte Carlo simulations in diffuse optics. Part 1: theory.

Monte Carlo (MC) is a powerful tool to study photon migration in scattering media, yet quite time-consuming to solve inverse problems. To speed up MC-simulations, scaling relations can be applied to an existing initial MC-simulation to generate a new data-set with different optical properties. We named this approach trajectory-based since it uses the knowledge of the detected photon trajectories of the initial MC-simulation, in opposition to the slower photon-based approach, where a novel MC-simulation is rerun with new optical properties. We investigated the convergence and applicability limits of the scaling relations, both related to the likelihood that the sample of trajectories considered is representative also for the new optical properties. For absorption, the scaling relation contains smoothly converging Lambert-Beer factors, whereas for scattering it is the product of two quickly diverging factors, whose ratio, for NIRS cases, can easily reach ten orders of magnitude. We investigated such instability by studying the probability-distribution for the number of scattering events in trajectories of given length. We propose a convergence test of the scattering scaling relation based on the minimum-maximum number of scattering events in recorded trajectories. We also studied the dependence of MC-simulations on optical properties, most critical in inverse problems, finding that scattering derivatives are ascribed to small deviations in the distribution of scattering events from a Poisson distribution. This paper, which can also serve as a tutorial, helps to understand the physics of the scaling relations with the causes of their limitations and devise new strategies to deal with them.

Broadband absorption spectroscopy of heterogeneous biological tissue.

Absorption spectra (∼600 to 1064 nm) of six tissues in three healthy volunteers were measured by combining dual-slope continuous-wave broadband spectroscopy with self-calibrated frequency-domain measurements of scattering at two wavelengths (690 and 830 nm). The spectral fit with a linear combination of oxy- and deoxyhemoglobin, water, and lipids extinction spectra is improved by a wavelength-independent absorption background. The need to introduce this background is assigned to the inhomogeneous distribution of absorbers in tissue. By using a two-layer model, the relationship between recovered concentrations and their two-layer values was investigated, and the implications for non-invasive tissue spectroscopy are discussed.

Dual-slope imaging in highly scattering media with frequency-domain near-infrared spectroscopy.

We present theoretical and experimental demonstrations of a novel, to the best of our knowledge, diffuse optical imaging method that is based on the concept of dual slopes (DS) in frequency-domain near-infrared spectroscopy. We consider a special array of sources and detectors that collects intensity (I) and phase (ϕ) data with multiple DS sets. We have recently shown that DSϕ reflectance data features a deeper sensitivity with respect to DSI reflectance data. Here, for the first time, we describe a DS imaging approach based on the Moore-Penrose inverse of the sensitivity matrix for multiple DS data sets. Using a circular 8-source/9-detector array that generates 16 DS data sets at source-detector distances in the range 20-40 mm, we show that DSI images are more sensitive to superficial (<5mm) perturbations, whereas DSϕ images are more sensitive to deeper (>10mm) perturbations in highly scattering media.

Dual-slope method for enhanced depth sensitivity in diffuse optical spectroscopy.

Using diffusion theory, we show that a dual-slope method is more effective than single-slope methods or single-distance methods at enhancing sensitivity to deeper tissue. The dual-slope method requires a minimum of two sources and two detectors arranged in specially configured arrays. In particular, we present diffusion theory results for a symmetrical linear array of two sources (separated by 55 mm) that sandwich two detectors (separated by 15 mm), for which dual slopes achieve maximal sensitivity at a depth of about 5 mm for direct current (DC) intensity (as measured in continuous-wave spectroscopy) and 11 mm for phase (as measured in frequency-domain spectroscopy) under typical values of the tissue optical properties (absorption coefficient: ∼0.01mm-1, reduced scattering coefficient: ∼1mm-1). This result is a major advance over single-distance or single-slope data, which feature maximal sensitivity to shallow tissue (<2mm for the intensity, <5mm for the phase).

Nonlinear extension of a hemodynamic linear model for coherent hemodynamics spectroscopy.

In this work, we are proposing an extension of a recent hemodynamic model (Fantini, 2014a), which was developed within the framework of a novel approach to the study of tissue hemodynamics, named coherent hemodynamics spectroscopy (CHS). The previous hemodynamic model, from a signal processing viewpoint, treats the tissue microvasculature as a linear time-invariant system, and considers changes of blood volume, capillary blood flow velocity and the rate of oxygen diffusion as inputs, and the changes of oxy-, deoxy-, and total hemoglobin concentrations (measured in near infrared spectroscopy) as outputs. The model has been used also as a forward solver in an inversion procedure to retrieve quantitative parameters that assess physiological and biological processes such as microcirculation, cerebral autoregulation, tissue metabolic rate of oxygen, and oxygen extraction fraction. Within the assumption of "small" capillary blood flow velocity oscillations the model showed that the capillary and venous compartments "respond" to this input as low pass filters, characterized by two distinct impulse response functions. In this work, we do not make the assumption of "small" perturbations of capillary blood flow velocity by solving without approximations the partial differential equation that governs the spatio-temporal behavior of hemoglobin saturation in capillary and venous blood. Preliminary comparison between the linear time-invariant model and the extended model (here identified as nonlinear model) are shown for the relevant parameters measured in CHS as a function of the oscillation frequency (CHS spectra). We have found that for capillary blood flow velocity oscillations with amplitudes up to 10% of the baseline value (which reflect typical scenarios in CHS), the discrepancies between CHS spectra obtained with the linear and nonlinear models are negligible. For larger oscillations (~50%) the linear and nonlinear models yield CHS spectra with differences within typical experimental errors, but further investigation is needed to assess the effect of these differences. Flow oscillations larger than 10-20% are not typically induced in CHS; therefore, the results presented in this work indicate that a linear hemodynamic model, combined with a method to elicit controlled hemodynamic oscillations (as done for CHS), is appropriate for the quantitative assessment of cerebral microcirculation.

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.

Implantable, multifunctional, bioresorbable optics.

Advances in personalized medicine are symbiotic with the development of novel technologies for biomedical devices. We present an approach that combines enhanced imaging of malignancies, therapeutics, and feedback about therapeutics in a single implantable, biocompatible, and resorbable device. This confluence of form and function is accomplished by capitalizing on the unique properties of silk proteins as a mechanically robust, biocompatible, optically clear biomaterial matrix that can house, stabilize, and retain the function of therapeutic components. By developing a form of high-quality microstructured optical elements, improved imaging of malignancies and of treatment monitoring can be achieved. The results demonstrate a unique family of devices for in vitro and in vivo use that provide functional biomaterials with built-in optical signal and contrast enhancement, demonstrated here with simultaneous drug delivery and feedback about drug delivery with no adverse biological effects, all while slowly degrading to regenerate native tissue.

Validation of a novel hemodynamic model for coherent hemodynamics spectroscopy (CHS) and functional brain studies with fNIRS and fMRI.

We report an experimental validation and applications of the new hemodynamic model presented in the companion article (Fantini, 2014-this issue) both in the frequency domain and in the time domain. In the frequency domain, we have performed diffuse optical measurements for coherent hemodynamics spectroscopy (CHS) on the brain and calf muscle of human subjects, showing that the hemodynamic model predictions (both in terms of spectral shapes and absolute spectral values) are confirmed experimentally. We show how the quantitative analysis based on the new model allows for autoregulation measurements from brain data, and provides an analytical description of near-infrared spiroximetry from muscle data. In the time domain, we have used data from the literature to perform a comparison between brain activation signals measured with functional near-infrared spectroscopy (fNIRS) or with blood oxygenation level dependent (BOLD) fMRI, and the corresponding signals predicted by the new model. This comparison shows an excellent agreement between the model predictions and the reported fNIRS and BOLD fMRI signals. This new hemodynamic model provides a valuable tool for brain studies with hemodynamic-based techniques.

Forward solvers for photon migration in the presence of highly and totally absorbing objects embedded inside diffusive media.

In this paper, after a critical review of the literature, we present two forward solvers and a new methodology for description of photon migration in the presence of totally absorbing inclusions embedded in diffusive media in both time and CW domains. The first forward solver is a heuristic approach based on a higher order perturbation theory applied to the diffusion equation (DE) [denoted eighth-order perturbation theory (EOPT)]. The second forward solver [denoted eighth-order perturbation theory with the equivalence relation (EOPTER) ] is obtained by combining the EOPT solver with the adoption of the equivalence relation (ER) [J. Biomed. Opt.18, 066014 (2013)]. These forward solvers can possibly overcome some evident limitations of previous approaches like the theory behind the so-called banana-shape regions or exact analytical solutions of the DE in the presence of highly or totally absorbing inclusions. We also propose the ER to reformulate the problem of a totally absorbing inclusion in terms of another inclusion having a finite absorption contrast and a re-scaled volume. For instance, we have shown how this approach can indeed be used to simulate black inclusions with the Born approximation. By means of comparisons with the results of Monte Carlo simulations, we have shown that the EOPTER solver can model totally absorbing inclusions with an error smaller than about 10%, whereas the EOPT solver shows an error smaller than about 20%, showing a performance largely better than that observed with solvers proposed previously.

Novel data types for frequency-domain diffuse optical spectroscopy and imaging of tissues: characterization of sensitivity and contrast-to-noise ratio for absorption perturbations.

In frequency-domain (FD) diffuse optics it is known that the phase of photon-density waves (ϕ) has a stronger deep-to-superficial sensitivity ratio to absorption perturbations than the alternate current (AC) amplitude, or the direct current intensity (DC). This work is an attempt to find FD data types that feature similar or even better sensitivity and/or contrast-to-noise for deeper absorption perturbations than phase. One way is to start from the definition of characteristic function (Xt(ω)) of the photon's arrival time (t) and combining the real (ℜ(Xt(ω))=ACDCcos(ϕ)) and imaginary parts (ℑ[Xt(ω)]=ACDCsin(ϕ)) with phase to yield new data types. These new data types enhance the role of higher order moments of the probability distribution of the photon's arrival time t. We study the contrast-to-noise and sensitivity features of these new data types not only in the single-distance arrangement (traditionally used in diffuse optics), but we also consider the spatial gradients, which we named dual-slope arrangements. We have identified six data types that for typical values of the optical properties of tissues and depths of interest, have better sensitivity or contrast-to-noise features than phase data and that can be used to enhance the limits of imaging of tissue in FD near infrared spectroscopy (NIRS). For example, one promising data type is ϕ-ℑ[Xt(ω)] which shows, in the single-distance source-detector arrangement, an increase of deep-to-superficial sensitivity ratio with respect to phase by 41% and 27% at a source-detector separation of 25 and 35 mm, respectively. The same data type also shows an increase of contrast-to noise up to 35% with respect to phase when the spatial gradients of the data are considered.

Dual-slope imaging of cerebral hemodynamics with frequency-domain near-infrared spectroscopy.

Significance: This work targets the contamination of optical signals by superficial hemodynamics, which is one of the chief hurdles in non-invasive optical measurements of the human brain. Aim: To identify optimal source-detector distances for dual-slope (DS) measurements in frequency-domain (FD) near-infrared spectroscopy (NIRS) and demonstrate preferential sensitivity of DS imaging to deeper tissue (brain) versus superficial tissue (scalp). Approach: Theoretical studies (in-silico) based on diffusion theory in two-layered and in homogeneous scattering media. In-vivo demonstrations of DS imaging of the human brain during visual stimulation and during systemic blood pressure oscillations. Results: The mean distance (between the two source-detector distances needed for DS) is the key factor for depth sensitivity. In-vivo imaging of the human occipital lobe with FD NIRS and a mean distance of 31 mm indicated: (1) greater hemodynamic response to visual stimulation from FD phase versus intensity, and from DS versus single-distance (SD); (2) hemodynamics from FD phase and DS mainly driven by blood flow, and hemodynamics from SD intensity mainly driven by blood volume. Conclusions: DS imaging with FD NIRS may suppress confounding contributions from superficial hemodynamics without relying on data at short source-detector distances. This capability can have significant implications for non-invasive optical measurements of the human brain.

Fluence rate directly derived from photon pathlengths: a tool for Monte Carlo simulations in biomedical optics.

In biomedical optics, the mean fluence rate of photons, assessed in a sub-volume of a propagating medium, is classically obtained in Monte Carlo simulations by taking into account the power deposited by the absorbed photons in the sub-volume. In the present contribution, we propose and analytically demonstrate an alternative method based on the assessment of the mean pathlength traveled by all the photons inside the sub-volume. Few practical examples of its applications are given. This method has the advantage of improving, in many cases, the statistics and the convergence of the Monte Carlo simulations. Further, it also works when the absorption coefficient is nil and for a non-constant spatial distribution of the absorption coefficient inside the sub-volume. The proposed approach is a re-visitation of a well-known method applied in radiation and nuclear physics in the context of radiative transfer, where it can be derived in a more natural manner.

Dual-ratio approach for detection of point fluorophores in biological tissue.

Significance: Diffuse in vivo flow cytometry (DiFC) is an emerging fluorescence sensing method to non-invasively detect labeled circulating cells in vivo. However, due to signal-to-noise ratio (SNR) constraints largely attributed to background tissue autofluorescence (AF), DiFC's measurement depth is limited. Aim: The dual ratio (DR)/dual slope is an optical measurement method that aims to suppress noise and enhance SNR to deep tissue regions. We aim to investigate the combination of DR and near-infrared (NIR) DiFC to improve circulating cells' maximum detectable depth and SNR. Approach: Phantom experiments were used to estimate the key parameters in a diffuse fluorescence excitation and emission model. This model and parameters were implemented in Monte Carlo to simulate DR DiFC while varying noise and AF parameters to identify the advantages and limitations of the proposed technique. Results: Two key factors must be true to give DR DiFC an advantage over traditional DiFC: first, the fraction of noise that DR methods cannot cancel cannot be above the order of 10% for acceptable SNR. Second, DR DiFC has an advantage, in terms of SNR, if the distribution of tissue AF contributors is surface-weighted. Conclusions: DR cancelable noise may be designed (e.g., through the use of source multiplexing), and indications point to the AF contributors' distribution being truly surface-weighted in vivo. Successful and worthwhile implementation of DR DiFC depends on these considerations, but results point to DR DiFC having possible advantages over traditional DiFC.

Single-distance and dual-slope frequency-domain near-infrared spectroscopy to assess skeletal muscle hemodynamics.

Significance: Non-invasive optical measurements of deep tissue (e.g., muscle) need to take into account confounding contributions from baseline and dynamic optical properties of superficial tissue (adipose tissue). Aim: Discriminate superficial and deep tissue hemodynamics using data collected with frequency-domain (FD) near-infrared spectroscopy (NIRS) in a dual-slope (DS) configuration. Approach: Experimental data were collected in vivo on the forearm of three human subjects during a 3-min arterial occlusion or 1-min venous occlusion. Theoretical data were generated using diffusion theory for two-layered media with varying values of the reduced scattering coefficient (µs') (range: 0.5 to 1.1 mm-1) and absorption coefficient (µa) (range: 0.005-0.015 mm-1) of the two layers, and top layer thickness (range: 2 to 8 mm). Data were analyzed using diffusion theory for a homogeneous semi-infinite medium. Results: Experimental data in vivo were consistent with simulated data for a two-layered medium with a larger µs' in the top layer, comparable absorption changes in the top and bottom layers during venous occlusion, and smaller absorption changes in the top vs. bottom layers during arterial occlusion. Conclusions: The dataset generated by DS FD-NIRS may allow for discrimination of superficial and deep absorption changes in two-layered media, thus lending itself to individual measurements of hemodynamics in adipose and muscle tissue.

Phase-amplitude investigation of spontaneous low-frequency oscillations of cerebral hemodynamics with near-infrared spectroscopy: a sleep study in human subjects.

We have investigated the amplitude and phase of spontaneous low-frequency oscillations (LFOs) of the cerebral deoxy- and oxy-hemoglobin concentrations ([Hb] and [HbO]) in a human sleep study using near-infrared spectroscopy (NIRS). Amplitude and phase analysis was based on the analytic signal method, and phasor algebra was used to decompose measured [Hb] and [HbO] oscillations into cerebral blood volume (CBV) and flow velocity (CBFV) oscillations. We have found a greater phase lead of [Hb] vs. [HbO] LFOs during non-REM sleep with respect to the awake and REM sleep states (maximum increase in [Hb] phase lead: ~π/2). Furthermore, during non-REM sleep, the amplitudes of [Hb] and [HbO] LFOs are suppressed with respect to the awake and REM sleep states (maximum amplitude decrease: 87%). The associated cerebral blood volume and flow velocity oscillations are found to maintain their relative phase difference during sleep, whereas their amplitudes are attenuated during non-REM sleep. These results show the potential of phase-amplitude analysis of [Hb] and [HbO] oscillations measured by NIRS in the investigation of hemodynamics associated with cerebral physiology, activation, and pathological conditions.

Equivalence of four Monte Carlo methods for photon migration in turbid media.

In the field of photon migration in turbid media, different Monte Carlo methods are usually employed to solve the radiative transfer equation. We consider four different Monte Carlo methods, widely used in the field of tissue optics, that are based on four different ways to build photons' trajectories. We provide both theoretical arguments and numerical results showing the statistical equivalence of the four methods. In the numerical results we compare the temporal point spread functions calculated by the four methods for a wide range of the optical properties in the slab and semi-infinite medium geometry. The convergence of the methods is also briefly discussed.

Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette.

Many applications seek to measure a sample's absorption coefficient spectrum to retrieve the chemical makeup. Many real-world samples are optically turbid, causing scattering confounds which many commercial spectrometers cannot address. Using diffusion theory and considering absorption and reduced scattering coefficients on the order of 0.01 mm-1 and 1 mm-1, respectively, we develop a method which utilizes frequency-domain to measure absolute optical properties of turbid samples in a standard cuvette (45 mm × 10 mm × 10 mm). Inspired by the self-calibrating method, which removes instrumental confounds, the method uses measurements of the diffuse complex transmittance at two sets of two different source-detector distances. We find: this works best for highly scattering samples (reduced scattering coefficient above 1 mm-1); higher relative error in the absorption coefficient compared to the reduced scattering coefficient; accuracy is tied to knowledge of the sample's index of refraction. Noise simulations with 0.1 % amplitude and 0.1° = 1.7 mrad phase uncertainty find errors in absorption and reduced scattering coefficients of 4 % and 1 %, respectively. We expect that higher error in the absorption coefficient can be alleviated with highly scattering samples and that boundary condition confounds may be suppressed by designing a cuvette with high index of refraction. Further work will investigate implementation and reproducibility.

Broadband diffuse optical spectroscopy of two-layered scattering media containing oxyhemoglobin, deoxyhemoglobin, water, and lipids.

We investigated the relationship between chromophore concentrations in two-layered scattering media and the apparent chromophore concentrations measured with broadband optical spectroscopy in conjunction with commonly used homogeneous medium inverse models. We used diffusion theory togenerate optical data from a two-layered distribution of relevant tissue absorbers, namely, oxyhemoglobin, deoxyhemoglobin, water, and lipids, with a top-layer thickness in the range 1-15 mm. The generated data consisted of broadband continuous-wave (CW) diffuse reflectance in the wavelength range 650-1024 nm, and frequency-domain (FD) diffuse reflectance at 690 and 830 nm; two source-detector distances of 25 and 35 mm were used to simulate a dual-slope technique. The data were inverted using diffusion theory for a semi-infinite homogeneous medium to generate reduced scattering coefficients at 690 and 830 nm (from FD data) and effective absorption spectra in the range 650-1024 nm (from CW data). The absorption spectra were then converted into effective total concentration and oxygen saturation of hemoglobin, as well as water and lipid concentrations. For absolute values, it was found that the effective hemoglobin parameters are typically representative of the bottom layer, whereas water and lipid represent some average of the respective concentrations in the two layers. For concentration changes, lipid showed a significant cross-talk with other absorber concentrations, thus indicating that lipid dynamics obtained in these conditions may not be reliable. These systematic simulations of broadband spectroscopy of two-layered media provide guidance on how to interpret effective optical properties measured with similar instrumental setups under the assumption of medium homogeneity.

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.

Optical imaging and spectroscopy for the study of the human brain: status report.

This report is the second part of a comprehensive two-part series aimed at reviewing an extensive and diverse toolkit of novel methods to explore brain health and function. While the first report focused on neurophotonic tools mostly applicable to animal studies, here, we highlight optical spectroscopy and imaging methods relevant to noninvasive human brain studies. We outline current state-of-the-art technologies and software advances, explore the most recent impact of these technologies on neuroscience and clinical applications, identify the areas where innovation is needed, and provide an outlook for the future directions.