Investigation of two-layered turbid media with time-resolved reflectance

A comprehensive overview of diffuse correlation spectroscopy: Theoretical framework, recent advances in hardware, analysis, and applications

Diffuse correlation spectroscopy (DCS) is a powerful tool for assessing microvascular hemodynamic in deep tissues. Recent advances in sensors, lasers, and deep learning have further boosted the development of new DCS methods. However, newcomers might feel overwhelmed, not only by the already-complex DCS theoretical framework but also by the broad range of component options and system architectures. To facilitate new entry to this exciting field, we present a comprehensive review of DCS hardware architectures (continuous-wave, frequency-domain, and time-domain) and summarize corresponding theoretical models. Further, we discuss new applications of highly integrated silicon single-photon avalanche diode (SPAD) sensors in DCS, compare SPADs with existing sensors, and review other components (lasers, sensors, and correlators), as well as data analysis tools, including deep learning. Potential applications in medical diagnosis are discussed and an outlook for the future directions is provided, to offer effective guidance to embark on DCS research.

Statistics of maximum photon penetration depth in a two-layer diffusive medium

We present numerical results for the probability density function f(z) and for the mean value of photon maximum penetration depth ‹zmax› in a two-layer diffusive medium. Both time domain and continuous wave regime are considered with several combinations of the optical properties (absorption coefficient, reduced scattering coefficient) of the two layers, and with different geometrical configurations (source detector distance, thickness of the upper layer). Practical considerations on the design of time domain and continuous wave systems are derived. The methods and the results are of interest for many research fields such as biomedical optics and advanced microscopy.

Fast estimation of adult cerebral blood content and oxygenation with hyperspectral time-resolved near-infrared spectroscopy

Near-infrared spectroscopy (NIRS) can measure tissue blood content and oxygenation; however, its use for adult neuromonitoring is challenging due to significant contamination from their thick extracerebral layers (ECL; primarily scalp and skull). This report presents a fast method for accurate estimation of adult cerebral blood content and oxygenation from hyperspectral time resolved NIRS (trNIRS) data. A two-phase fitting method, based on a two-layer head model (ECL and brain), was developed. Phase 1 uses spectral constraints to accurately estimate the baseline blood content and oxygenation in both layers, which are then used by Phase 2 to correct for the ECL contamination of the late-arriving photons. The method was validated with in silico data from Monte-Carlo simulations of hyperspectral trNIRS in a realistic model of the adult head obtained from a high-resolution MRI. Phase 1 recovered cerebral blood oxygenation and total hemoglobin with an accuracy of 2.7 ± 2.5 and 2.8 ± 1.8%, respectively, with unknown ECL thickness, and 1.5 ± 1.4 and 1.7 ± 1.1% when the ECL thickness was known. Phase 2 recovered these parameters with an accuracy of 1.5 ± 1.5 and 3.1 ± 0.9%, respectively. Future work will include further validation in tissue-mimicking phantoms with various top layer thicknesses and in a pig model of the adult head before human applications.

Efficient computation of the steady-state and time-domain solutions of the photon diffusion equation in layered turbid media

Accurate and efficient forward models of photon migration in heterogeneous geometries are important for many applications of light in medicine because many biological tissues exhibit a layered structure of independent optical properties and thickness. However, closed form analytical solutions are not readily available for layered tissue-models, and often are modeled using computationally expensive numerical techniques or theoretical approximations that limit accuracy and real-time analysis. Here, we develop an open-source accurate, efficient, and stable numerical routine to solve the diffusion equation in the steady-state and time-domain for a layered cylinder tissue model with an arbitrary number of layers and specified thickness and optical coefficients. We show that the steady-state ([Formula: see text] ms) and time-domain ([Formula: see text] ms) fluence (for an 8-layer medium) can be calculated with absolute numerical errors approaching machine precision. The numerical implementation increased computation speed by 3 to 4 orders of magnitude compared to previously reported theoretical solutions in layered media. We verify our solutions asymptotically to homogeneous tissue geometries using closed form analytical solutions to assess convergence and numerical accuracy. Approximate solutions to compute the reflected intensity are presented which can decrease the computation time by an additional 2-3 orders of magnitude. We also compare our solutions for 2, 3, and 5 layered media to gold-standard Monte Carlo simulations in layered tissue models of high interest in biomedical optics (e.g. skin/fat/muscle and brain). The presented routine could enable more robust real-time data analysis tools in heterogeneous tissues that are important in many clinical applications such as functional brain imaging and diffuse optical spectroscopy.

Diffused reflectance measurements to detect tattoo ink location in skin using the crossover point

Tattoos are highly trendy in western culture, but many people regret their tattoos for many reasons. It is essential to be aware of the ink location in advance to reduce the long and short-term side effects. In this study, diffuse reflectance (DR) experiments were conducted on two-layer (2L) tissue-mimicking phantoms, where ink was sandwiched between the layers. An appreciable difference in the DR profile was found between the 2L phantom with and without the tattoo ink using the crossover point (Cp) method. Our technique was applied to ex vivo porcine skin. A point of intersection was found, between the skin and the tattooed skin. In the shorter wavelengths (500-600 nm), a distinguishable 2L behavior was found, and in longer wavelengths (600-850 nm), a single layer behavior was found between the tattooed skin before and after the intersection. In biological tissue, this Cp indeed finds the tattoo ink without harm to the surrounding skin.

A Monte Carlo study of near infrared light propagation in the human head with lesions-a time-resolved approach

Several clinical conditions leading to traumatic brain injury can cause hematomas or edemas inside the cerebral tissue. If these are not properly treated in time, they are prone to produce long-term neurological disabilities, or even death. Low-cost, portable and easy-to-handle devices are desired for continuous monitoring of these conditions and Near Infrared Spectroscopy (NIRS) techniques represent an appropriate choice. In this work, we use Time-Resolved (TR) Monte Carlo simulations to present a study of NIR light propagation over a digital MRI phantom. Healthy and injured (hematoma/edema) situations are considered. TR Diffuse Reflectance simulations for different lesion volumes and interoptode distances are performed in the frontal area and the left parietal area. Results show that mean partial pathlengths, photon measurement density functions and time dependent contrasts are sensitive to the presence of lesions, allowing their detection mainly for intermediate optodes separations, which proves that these metrics represent robust means of diagnose and monitoring. Conventional Continuous Wave (CW) contrasts are also presented as a particular case of the time dependent ones, but they result less sensitive to the lesions, and have higher associated uncertainties.

Bottom layer absorption coefficients extraction from two-layer phantoms based on crossover point in diffuse reflectance

SIGNIFICANCE

Numerous optical imaging and spectroscopy techniques are used to study the tissue-optical properties; the majority of them are limited in information regarding the penetration depth. A simple, safe, easily applicable diagnostic technique is required to get deeper tissue information in a multilayer structure.

AIM

A fiber-based diffuse reflectance (DR) technique is used to extract and quantify the bottom layer absorption coefficients in two-layer (2L) tissue-mimicking solid phantoms. We determine the Indian black ink concentrations in a deep-hidden layer that is sandwiched between agar and silicone-based phantom layers.

APPROACH

A fiber-based DR experiment was performed to study the optical properties of the tissue at higher penetration depth, with different fiber core diameters and a constant numerical aperture (0.5 NA). The optimal core diameter of the fiber was chosen by measuring solid phantoms. In 2L phantoms, the thickness of the top layer was kept 5.5 mm with a constant absorption and reduced scattering coefficients (μa  =  0.045  mm  -  1 and μs  '    =  2.622  mm  -  1), whereas the absorption coefficients of the bottom layers were varied from 0.014 to 0.037  mm  -  1 keeping the μs  '   the same as the top layer. A unique crossover point (Cp) was found in the DR intensity profile against distance. We examined the slope before and after the Cp. These two slopes indicate the difference between the optical properties of the top and bottom layers. Our technique got further verification, as we successfully determined the Cp with different Indian black ink concentrations, placed at the junction between the agar and silicone-based phantom layers.

RESULTS

The DR measurements were applied to 2L phantoms. Two different slopes were found in 2L phantoms compared to the one-layer (optical properties equal to the top layer of 2L). We extracted the slopes before and after the Cp in the 2L phantoms. The calculated absorption coefficients before the Cp were 0.014  ±  0.0004, 0.022  ±  0.0003, 0.028  ±  0.0003, and 0.036  ±  0.0014  mm  -  1, and the absorption coefficients after the Cp were 0.019  ±  0.0013, 0.013  ±  0.0004, 0.014  ±  0.0006, and 0.031  ±  0.0001  mm  -  1, respectively. The calculated absorption coefficients before the Cp were in good agreement with the optical properties of the bottom layer. The calculated absorption coefficients after the Cp were not the same as the top layer. Our DR system successfully determines the crossover points 12.14  ±  0.11 and 11.73  ±  0.15  mm for 70% and 100% ink concentrations placed at the junction of the agar and silicone layers.

CONCLUSIONS

In a 2L tissue structure, the Cp depends on the absorption coefficients of top and bottom layers and the thickness of the top layer. With the help of the Cp and the absorption coefficients, one can determine the thickness of the top layer or vice versa. The slope value before the Cp in the DR profile allowed us to determine the absorption properties of the bottom layer instead of having the average behavior of the 2L phantom in the far detection range (11.0 to 17.0 mm).

Cerebral blood volume increment after resuscitation measured by near-infrared time-resolved spectroscopy can estimate degree of hypoxic-ischemic insult in newborn piglets

Neonatal hypoxic–ischemic encephalopathy is a notable cause of neonatal death and developmental disabilities. To achieve better outcomes, it is important in treatment strategy selection to categorize the degree of hypoxia ischemia and evaluate dose response. In an asphyxia piglet model with histopathological brain injuries that we previously developed, animals survived 5 days after insult and showed changes in cerebral blood volume (CBV) that reflected the severity of injuries. However, little is known about the relationship between changes in CBV during and after insult. In this study, an HI event was induced by varying the amount and timing of inspired oxygen in 20 anesthetized piglets. CBV was measured using near-infrared time-resolved spectroscopy before, during, and 6 h after insult. Change in CBV was calculated as the difference between the peak CBV value during insult and the value at the end of insult. The decrease in CBV during insult was found to correlate with the increase in CBV within 6 h after insult. Heart rate exhibited a similar tendency to CBV, but blood pressure did not. Because the decrement in CBV was larger in severe HI, the CBV increment immediately after insult is considered useful for assessing degree of HI insult.

Space-enhanced time-domain diffuse optics for determination of tissue optical properties in two-layered structures

A novel methodology for solving the inverse problem of diffuse optics for two-layered structures is proposed to retrieve the absolute quantities of optical absorption and reduced scattering coefficients of the layers simultaneously. A liquid phantom with various optical absorption properties in the deep layer is prepared and experimentally investigated using the space-enhanced time-domain method. Monte-Carlo simulations are applied to analyze the different measurements in time domain, space domain, and by the new methodology. The deviations of retrieved values from nominal values of both layers' optical properties are simultaneously reduced to a very low extent compared to the single-domain methods. The reliability and uncertainty of the retrieval performance are also considerably improved by the new methodology. It is observed in time-domain analyses that for the deep layer the retrieval of absorption coefficient is almost not affected by the scattering properties and this kind of "deep scattering neutrality" is investigated and overcome as well.

A comparison between plausible models in layered turbid media with geometrical variations applying a Bayesian selection criterion

One possible application of Near Infrared techniques is to analyze human brain metabolic activity. Currently used models take into account the layered structure of the human head but, usually, they do not consider the non-planar surface of some of the boundaries, i.e. gray matter, which results in a much more complex structure, thus leading to more sophisticated models and longer calculation times. The main objective of this work is to determine if it is worth to replace a planar layered structure by a non-planar one. To this end we implement a Bayesian-based quantitative methodology for choosing between two competitive models describing light propagation in layered turbid media. Experiments of time-resolved diffuse reflectance measurements are performed in layered phantoms and complemented with numerical calculations. The resulting Distributions of Time of Flight of both models are compared using Bayesian model selection analysis. The non-planar interface was introduced in the simulations by a simple surface parametrization. Results suggest that, under certain conditions, a multilayer model with planar boundaries is good enough.

Time-Domain Near-Infrared Spectroscopy and Imaging: A Review

This article reviews the past and current statuses of time-domain near-infrared spectroscopy (TD-NIRS) and imaging. Although time-domain technology is not yet widely employed due to its drawbacks of being cumbersome, bulky, and very expensive compared to commercial continuous wave (CW) and frequency-domain (FD) fNIRS systems, TD-NIRS has great advantages over CW and FD systems because time-resolved data measured by TD systems contain the richest information about optical properties inside measured objects. This article focuses on reviewing the theoretical background, advanced theories and methods, instruments, and studies on clinical applications for TD-NIRS including some clinical studies which used TD-NIRS systems. Major events in the development of TD-NIRS and imaging are identified and summarized in chronological tables and figures. Finally, prospects for TD-NIRS in the near future are briefly described.

Recovery of layered tissue optical properties from spatial frequency-domain spectroscopy and a deterministic radiative transport solver

We present a method to recover absorption and reduced scattering spectra for each layer of a two-layer turbid media from spatial frequency-domain spectroscopy data. We focus on systems in which the thickness of the top layer is less than the transport mean free path   (  0.1  -  0.8l  *    )  . We utilize an analytic forward solver, based upon the N'th-order spherical harmonic expansion with Fourier decomposition   (  SHEFN  )   method in conjunction with a multistage inverse solver. We test our method with data obtained using spatial frequency-domain spectroscopy with 32 evenly spaced wavelengths within λ  =  450 to 1000 nm on six-layered tissue phantoms with distinct optical properties. We demonstrate that this approach can recover absorption and reduced scattering coefficient spectra for both layers with accuracy comparable with current Monte Carlo methods but with lower computational cost and potential flexibility to easily handle variations in parameters such as the scattering phase function or material refractive index. To our knowledge, this approach utilizes the most accurate deterministic forward solver used in such problems and can successfully recover properties from a two-layer media with superficial layer thicknesses.

Time-resolved near infrared light propagation using frequency domain superposition

Time-resolved temporal point spread function (TPSF) measurement of near infrared spectroscopic (NIRS) data allows the estimation of absorption and reduced scattering properties of biological tissues. Such analysis requires an iterative calculation of the theoretical TPSF curve using mathematical and computational models of the domain being imaged which are computationally complex and expensive. In this work, an efficient methodology for representing the TPSF data using a superposition of cosines calculated in frequency domain is presented. The proposed method is outlined and tested on finite element realistic models of the human neck and head. Using an adult head model containing ~140k nodes, the TPSF calculation at each node for one source is accelerated from 3.11 s to 1.29 s within an error limit of ± 5% related to the time domain calculation method.

Development of thin skin mimicking bilayer solid tissue phantoms for optical spectroscopic studies

In vivo spectroscopic measurements have the proven potential to provide important insight about the changes in tissue during the development of malignancies and thus help to diagnose tissue pathologies. Extraction of intrinsic data in the presence of varying amounts of scatterers and absorbers offers great challenges in the development of such techniques to the clinical level. Fabrication of optical phantoms, tailored to the biochemical as well as morphological features of the target tissue, can help to generate a spectral database for a given optical spectral measurement system. Such databases, along with appropriate pattern matching algorithms, could be integrated with in vivo measurements for any desired quantitative analysis of the target tissue. This paper addresses the fabrication of such soft, photo stable, thin bilayer phantoms, mimicking skin tissue in layer dimensions and optical properties. The performance evaluation of the fabricated set of phantoms is carried out using a portable fluorescence spectral measurement system. The alterations in flavin adenine dinucleotide (FAD)-a tissue fluorophore that provides important information about dysplastic progressions in tissues associated with cancer development based on changes in emission spectra-fluorescence with varied concentrations of absorbers and scatterers present in the phantom are analyzed and the results are presented. Alterations in the emission intensity, shift in emission wavelength and broadening of the emission spectrum were found to be potential markers in the assessment of biochemical changes that occur during the progression of dysplasia.

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. 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.

Towards the next generation of near-infrared spectroscopy

Although near-infrared spectroscopy (NIRS) was originally designed for clinical monitoring of tissue oxygenation, it has also been developing into a useful tool for neuroimaging studies (functional NIRS). Over the past 30 years, technology has developed and NIRS has found a wide range of applications. However, the accuracy and reliability of NIRS have not yet been widely accepted, mainly because of the difficulties in selective and quantitative detection of signals arising in cerebral tissue, which subject the use of NIRS to a number of practical restrictions. This review summarizes the strengths and advantages of NIRS over other neuroimaging modalities and demonstrates specific examples. The issues of selective quantitative measurement of cerebral haemoglobin during brain activation are also discussed, together with the problems of applying the methods of functional magnetic resonance imaging data analysis to NIRS data analysis. Finally, near-infrared optical tomography--the next generation of NIRS--is described as a potential technique to overcome the limitations of NIRS.

Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain

This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.

Retrieving skin properties from in vivo spectral reflectance measurements

A previously developed inverse method was applied to in vivo normal-hemispherical spectral reflectance measurements taken on the inner and outer forearm as well as the forehead of healthy white Caucasian and black African subjects. The inverse method was used to determine the thickness and melanin concentration in the epidermis, dermal blood volume fraction and oxygen saturation, and skin's spectral scattering coefficient. It was established that changes in melanin concentration due to racial difference and tanning, and differences in epidermal thickness and blood volume with anatomical location were detectable. The retrieved values were also consistent with independent measurements reported in the literature. The same method could be used for optical diagnosis of pathologies affecting the structure and pigmentation of human skin.

Light diffusion in a turbid cylinder. II. Layered case

This paper is the second of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for an N-layered finite cylinder. Solutions are given in the steady-state, frequency, and time domains for a point beam incident at an arbitrary position of the first layer and for a circular flat beam incident at the middle of the cylinder top. For special cases the solutions were compared to other solutions of the diffusion equation showing excellent agreement. In addition, the derived solutions were validated by comparison with Monte Carlo simulations. In the time domain we also derived a fast solution ( approximately 10ms) for the case of equal reduced scattering coefficients and refractive indices in all layers.

Light diffusion in N-layered turbid media: steady-state domain

We deal with light diffusion in N-layered turbid media. The steady-state diffusion equation is solved for N-layered turbid media having a finite or an infinitely thick N'th layer. Different refractive indices are considered in the layers. The Fourier transform formalism is applied to derive analytical solutions of the fluence rate in Fourier space. The inverse Fourier transform is calculated using four different methods to test their performance and accuracy. Further, to avoid numerical errors, approximate formulas in Fourier space are derived. Fast solutions for calculation of the spatially resolved reflectance and transmittance from the N-layered turbid media ( approximately 10 ms) with small relative differences (<10(-7)) are found. Additionally, the solutions of the diffusion equation are compared to Monte Carlo simulations for turbid media having up to 20 layers.

Estimation of the absorption coefficients of two-layered media by a simple method using spatially and time-resolved reflectances

Our newly developed method using spatially and time-resolved reflectances can easily estimate the absorption coefficients of each layer in a two-layered medium if the thickness of the upper layer and the reduced scattering coefficients of the two layers are known a priori. We experimentally validated this method using phantoms and examined its possibility of estimating the absorption coefficients of the tissues in human heads. In the case of a homogeneous plastic phantom (polyacetal block), the absorption coefficient estimated by our method agreed well with that obtained by a conventional method. Also, in the case of two-layered phantoms, our method successfully estimated the absorption coefficients of the two layers. Furthermore, the absorption coefficients of the extracerebral and cerebral tissue inside human foreheads were estimated under the assumption that the human heads were two-layered media. It was found that the absorption coefficients of the cerebral tissues were larger than those of the extracerebral tissues.

Optical measurement of adipose tissue thickness and comparison with ultrasound, magnetic resonance imging, and callipers

Near-infrared spectroscopy is used to quantify the subcutaneous adipose tissue thickness (ATT) over five muscle groups (vastus medialis, vastus lateralis, gastrocnemius, ventral forearm and biceps brachii muscle) of healthy volunteers (n=20). The optical lipid signal (OLS) was obtained from the second derivative of broad band attenuation spectra and the lipid absorption peak (lambda=930 nm). Ultrasound and MR imaging as well as mechanical calliper readings were taken as reference methods. The data show that the OLS is a good predictor for ATT (<16 mm) with absolute and relative errors of <0.8 mm and <24%, respectively. The optical method compares favourably with calliper reading. The finding of a non-linear relationship of optical signal vs. ultrasound is explained by a theoretical two-layer model based on the diffusion approximation for the transport of photons. The crosstalk between the OLS and tissue hemoglobin concentration changes during an incremental cycling exercise was found to be small, indicating the robustness of OLS. Furthermore, the effect of ATT on spatially-resolved spectroscopy measurements is shown to decrease the calculated muscle hemoglobin concentration and to increase oxygen saturation.

Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries

We apply a previously proposed perturbation theory of the diffusion equation for studying light propagation through heterogeneous media in the presence of absorbing defects. The theory is based on the knowledge of (a) the geometric characteristics of a focal inclusion, (b) the mean optical path length inside the inclusion, and (c) the optical properties of the inclusion. The potential of this method is shown in the layered and slab geometries, where calculations are carried out up to the fourth order. The relative changes of intensity with respect to the unperturbed (heterogeneous) medium are predicted by the theory to within 10% for a wide range of contrasts dDeltamu(a) (up to dDeltamu(a) approximately 0.4-0.8), where d is the effective diameter of the defect and Deltamu(a) the absorption contrast between defect and local background. We also show how the method of Padé approximants can be used to extend the validity of the theory for a larger range of absorption contrasts. Finally, we study the possibility of using the proposed method for calculating the effect of a colocalized scattering and absorbing perturbation.

Investigation of diffuse correlation spectroscopy in multi-layered media including the human head

In this work, diffuse correlation spectroscopy (DCS) is explored in multi-layered geometries. A quantitative comparison of an homogeneous versus a two-layered model efficiencies to recover flow changes is presented. By simulating a realistic human head with MRI anatomical data, we show that the two-layered model allows distinction between changes in superficial layers and brain. We also show that the two-layered model provides a better estimate of the flow change than the homogeneous one. Experimental measurements with a two-layered dynamical phantom confirm the ability of the two-layered analytical model to distinguish flow increase in each layer.

Double-layer estimation of intra- and extracerebral hemoglobin concentration with a time-resolved system

We present in vivo measurements of baseline physiology from five subjects with a four-wavelength (690, 750, 800, and 850 nm) time-resolved optical system. The measurements were taken at four distances: 10, 15, 25, and 30 mm. All distances were fit simultaneously with a two-layered analytical model for the absorption and reduced scattering coefficient of both layers. The thickness of the first layer, comprising the skin, scalp, and cerebrospinal fluid, was obtained from anatomical magnetic resonance images. The fitting procedure was first tested with simulations before being applied to in vivo measurements and verified that this procedure permits accurate characterization of the hemoglobin concentrations in the extra- and intracerebral tissues. Baseline oxyhemoglobin, deoxyhemoglobin, and total hemoglobin concentrations and oxygen saturation were recovered from in vivo measurements and compared to the literature. We observed a noticeable intersubject variability of the hemoglobin concentrations, but constant values for the cerebral hemoglobin oxygen saturation.

Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model

The use of perturbation and differential Monte Carlo (pMC/dMC) methods in conjunction with nonlinear optimization algorithms were proposed recently as a means to solve inverse photon migration problems in regionwise heterogeneous turbid media. We demonstrate the application of pMC/dMC methods for the recovery of optical properties in a two-layer extended epithelial tissue model from experimental measurements of spatially resolved diffuse reflectance. The results demonstrate that pMC/dMC methods provide a rapid and accurate approach to solve two-region inverse photon migration problems in the transport regime, that is, on spatial scales smaller than a transport mean free path and in media where optical scattering need not dominate absorption. The pMC/dMC approach is found to be effective over a broad range of absorption (50 to 400%) and scattering (70 to 130%) perturbations. The recovery of optical properties from spatially resolved diffuse reflectance measurements is examined for different sets of source-detector separation. These results provide some guidance for the design of compact fiber-based probes to determine and isolate optical properties from both epithelial and stromal layers of superficial tissues.

Rapid simulation of steady-state spatially resolved reflectance and transmittance profiles of multilayered turbid materials

We present a technique for efficiently computing the reflection and transmission of light by arbitrary systems of turbid layers. To approximate the steady-state reflectance and transmittance without the need to solve difficult boundary conditions, we convolve the reflectance and transmittance profiles of individual layers. We extend single-slab boundary conditions to handle index-of-refraction mismatches between turbid slabs and account for interlayer scattering by applying methods similar to Kubelka-Munk theory in frequency space. We demonstrate good agreement between the reflectance and the transmittance predicted by our model and numerical Monte Carlo methods and show that the far-source reflectance and transmittance of multilayered turbid materials are dominated by interlayer scattering.

Simple algorithm for the measurement of absorption coefficients of a two-layered medium by spatially resolved and time-resolved reflectance

An inversion procedure for the recovery of absorption coefficients of a two-layered semi-infinite diffusive medium by use of time-resolved reflectance measured at two different source-detector distances is proposed. The inversion procedure is based on the property of the photon diffusion equation; i.e., the solution of the diffusion equation for the time-resolved reflectance measured at a longer source--detector distance coincides with that measured at a shorter one by a proper temporal, spatial, and intensity transformation. This inversion procedure, used together with the results of one set of Monte Carlo simulations, is validated as working well when the values of the scattering coefficients of the two layers and the thickness of the first layer are within a range of interest in tissue optics.

Extraction of depth-dependent signals from time-resolved reflectance in layered turbid media

We try a new approach with near-IR time-resolved spectroscopy, to separate optical signals originated in the upper layer from those in the lower layer and to selectively determine the absorption coefficient (mu(a)) of each layer in a two-layered turbid medium. The difference curve in the temporal profiles of light attenuation between a target and a reference medium is divided into segments along the time axis, and a slope of each segment is calculated to determine the depth-dependent mu(a). The depth-dependent mu(a) values are estimated under various conditions in which mu(a) and the reduced scattering coefficient (mu(s)') of each layer are changed with a Monte Carlo simulation and in phantom experiments. Temporal variation of them represents the difference in mu(a) between two layers when mu(s)' of a reference is the same as that of the upper layer of the target. The discrepancies between calculated mu(a) and the real mu(a) depend on the ratio of the real mu(a) of the upper layer to that of the lower layer, and our approach enables us to estimate the ratio of mu(a) between the two layers. These results suggest the potential that mu(a) of the lower layer can be determined by our procedure.

Two-dimensional phased arrays of sources and detectors for depth discrimination in diffuse optical imaging

We present a multisource, multidetector phased-array approach to diffuse optical imaging that is based on postprocessing continuous-wave data. We previously showed that this approach enhances the spatial resolution of diffuse optical imaging. We now demonstrate the depth discrimination capabilities of this approach and its potential to perform tomographic sectioning of turbid media. The depth discrimination results from the dependence of the sensitivity function on the depth coordinate z. To demonstrate the potential of this approach, we perform an experimental study of a turbid medium containing cylindrical inhomogeneities that are placed 2.0, 3.0, and 4.0 cm from a seven-element, 2-D source array. A single detector element is placed at a distance of 6.0 cm from the source array, and the measurement is repeated after switching the positions of the detector and the source array to simulate the case where both sources and detectors consist of a 2-D array of elements. We find that the proposed phased-array method is able to separate cylinders at different depths, thus showing cross-sectioning capabilities.

Quantification of cerebral hemoglobin as a function of oxygenation using near-infrared time-resolved spectroscopy in a piglet model of hypoxia

Near-infrared spectroscopy (NIRS) has been used for measurement of cerebral hemoglobin (Hb) concentrations in neonates to study cerebral oxygenation and hemodynamics. We perform measurements by portable three-wavelength NIR time-resolved spectroscopy (TRS) in a piglet hypoxia model with various degrees of oxygenation to estimate the absorption coefficient (mu(a)) and reduced scattering coefficient (mu(s)') of the head. Measurements of absolute values of mu(a) at three wavelengths enable estimation of Hb concentration and Hb oxygen saturation in the head (SO2). However, there is a problem concerning which background absorption should be used to estimate Hb concentration in the head derived from mu(a) at three wavelengths because it is different from a simple in vitro model. Therefore, we use two different background absorption values with the assumption that background absorption is due only to 85% (by volume) water or that background absorption is equal to absorption of the piglet head with blood exchange transfusion by fluorocarbon (FC), and we compared SO2 measured by TRS with arterial Hb oxygen saturation (SaO2) and sagittal sinus venous Hb oxygen saturation (SvO2) measured by a co-oximeter at several inspired fractional O2(FI(O2)) concentrations. We find that SO2 values using the absorption (abs) of the piglet head with blood exchange transfusion (BET) by FC are not significantly different from SO2 values using the water-only background at FI(O2) in the range of 15 to 100%, but that the values using abs of the head with BET by FC are lower than the values using the water-only background at FI(O2) in the range of 12 to 4%. The SO2 values calculated from the water-only background are higher than those of SaO2 at FI(O2) in the range of 10 to 4%. However, SO2 values using the abs of the head with BET by FC are between those of SaO2 and SvO2 over the whole range of FI(O2). Therefore, abs of the head with BET by FC is more useful for estimation of the absolute values of oxyHb and deoxyHb of the piglet head.

Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements

An experimental validation of an inversion procedure for retrieving the optical properties of layered media from multidistance time-resolved reflectance measurements is presented. The results cover a wide range of optical properties, showing excellent effectiveness and reliability of the procedure in reconstructing the optical properties of a two-layered medium. The optical properties of the first layer and the absorption of the second layer could be retrieved with excellent precision, whereas the reduced scattering coefficient of the second layer was reconstructed with a large error. The first layer thickness could be retrieved with an error less than 50%. An example of retrieval of the optical properties of muscle from in vivo measurements during an arterial occlusion is shown.

Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons

We report on multidistance time-resolved diffuse reflectance spectroscopy of the head of a healthy adult after intravenous administration of a bolus of indocyanine green. Intracerebral and extracerebral changes in absorption are deduced from moments (integral, mean time of flight, and variance) of the distributions of times of flight of photons (DTOFs), recorded simultaneously at four different source-detector separations. We calculate the sensitivity factors converting depth-dependent changes in absorption into changes of moments of DTOFs by Monte Carlo simulations by using a layered model of the head. We validate our method by analyzing moments of DTOFs simulated for the assumed changes in absorption in different layers of the head model.

Asymptotic behavior and inverse problem in layered scattering media

The main challenge of noninvasive optical biopsy is to obtain an accurate value of the optical coefficients of an encapsulated organ (muscle, brain, etc.). The idea developed by us is that some interesting information could be deduced from the long-time behavior of the reflectance function. This asymptotic behavior is analyzed for layered media in the framework of the diffusion approximation. A new method is derived to obtain accurate values for the optical parameters of the deepest layers. This method is designed to work in a specific long-time regime that is still within the scope of standard time-of-flight experiments but far from being included in the mathematically defined asymptotic region. The limits of this method, linked to the cases where the asymptotic behavior is no longer governed by the deepest layer, are then discussed.

Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measurements

A procedure for retrieving the optical properties of a two-layered diffusive medium based on an exact analytical solution of the diffusion equation and on relative multidistance time-resolved reflectance measurements is presented. The method overcomes some limitations of previously developed procedures. Five parameters of the medium have been fitted: the absorption and the reduced scattering coefficients of both layers and the thickness of the first layer. The actual values of the parameters are correctly retrieved by the procedure. The inversion procedure does not require an initial guess for the unknown optical properties, but the starting value for the thickness of the first layer needs to be estimated with an error smaller than 50%.

Robust inference of baseline optical properties of the human head with three-dimensional segmentation from magnetic resonance imaging

We model the capability of a small (6-optode) time-resolved diffuse optical tomography (DOT) system to infer baseline absorption and reduced scattering coefficients of the tissues of the human head (scalp, skull, and brain). Our heterogeneous three-dimensional diffusion forward model uses tissue geometry from segmented magnetic resonance (MR) data. Handling the inverse problem by use of Bayesian inference and introducing a realistic noise model, we predict coefficient error bars in terms of detected photon number and assumed model error. We demonstrate the large improvement that a MR-segmented model can provide: 2-10% error in brain coefficients (for 2 x 10(6) photons, 5% model error). We sample from the exact posterior and show robustness to numerical model error. This opens up the possibility of simultaneous DOT and MR for quantitative cortically constrained functional neuroimaging.

Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method

An exact solution of the time-dependent diffusion equation for the case of a two- and a three-layered finite diffusive medium is proposed. The method is based on the decomposition of the fluence rate in a series of eigenfunctions and upon the solution of the consequent transcendental equation for the eigenvalues obtained from the boundary conditions. Comparisons among the solution of the diffusion equation and the results of Monte Carlo simulations show the correctness of the proposed model.

Analytical approximate solutions of the time-domain diffusion equation in layered slabs

Time-domain analytical solutions of the diffusion equation for photon migration through highly scattering two- and three-layered slabs have been obtained. The effect of the refractive-index mismatch with the external medium is taken into account, and approximate boundary conditions at the interface between the diffusive layers have been considered. A Monte Carlo code for photon migration through a layered slab has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the mean path length followed by photons inside each diffusive layer and the shape of the temporal profile of received photons, while discrepancies are observed for the continuous-wave reflectance or transmittance.

Recovery of optical parameters in multiple-layered diffusive media: theory and experiments

Diffuse photon density waves have lately been used both to characterize diffusive media and to locate and characterize hidden objects, such as tumors, in soft tissue. In practice, most biological media of medical interest consist of various layers with different optical properties, such as the fat layer in the breast or the different layers present in the skin. Also, most experimental setups consist of a multilayered system, where the medium to be characterized (i.e., the patient's organ) is usually bounded by optically diffusive plates. Incorrect modeling of interfaces may induce errors comparable to the weak signals obtained from tumors embedded deep in highly heterogeneous tissue and lead to significant reconstruction artifacts. To provide a means to analyze the data acquired in these configurations, the basic expressions for the reflection and transmission coefficients for diffusive-diffusive and diffusive-nondiffusive interfaces are presented. A comparison is made between a diffusive slab and an ordinary dielectric slab, thus establishing the limiting distance between the two interfaces of the slab for multiple reflections between them to be considered important. A rigorous formulation for multiple-layered (M-layered) diffusive media is put forward, and a method for solving any M-layered medium is shown. The theory presented is used to characterize a two-layered medium from transmission measurements, showing that the coefficients of scattering, mu'(s) , and absorption, mu(a) , are retrieved with great accuracy. Finally, we demonstrate the simultaneous retrieval of both mu;(s) and mu(a).

Determining changes in NIR absorption using a layered model of the human head

A theoretical approach is presented to determine absorption changes in different compartments of a layered structure from distributions of times of flight of photons. In addition resulting changes in spatial profiles of time-integrated intensity and mean time of flight are calculated. The capability of a single-distance, time-domain method to determine absorption changes with depth resolution is tested on a layered phantom. We apply this method to in vivo measurements on the human head (motor stimulation, Valsalva manoeuvre) and introduce a small-sized time-domain experimental set-up suitable for bedside monitoring.

Non-invasive and quantitative near-infrared haemoglobin spectrometry in the piglet brain during hypoxic stress, using a frequency-domain multidistance instrument

The frequency-domain multiple-distance (FDMD) method is capable of measuring the absolute absorption and reduced scattering coefficients of optically turbid media. Absolute measurement of absorption at two near-infrared (NIR) wavelengths makes possible the quantitation of tissue haemoglobin concentration and tissue haemoglobin oxygen-saturation (StO2). However, errors are introduced by the uncertainties of background absorption and the dissimilarities between real tissues and the simplified mathematical model on which these measurements are based. An FDMD-based tissue instrument has been used for the monitoring of tissue haemoglobin concentration and oxygenation in the brain of newborn piglets during periods of hypoxia and hyperoxia. These tissue haemoglobin saturation values were compared with arterial saturation (SaO2) and venous saturation (SvO2) measured by blood gas analyses. A linear correlation was observed between StO2 and the average of SaO2 and SvO2. However, StO2 is not equal to any fixed weighted average of SaO2 and SvO2 unless we introduce an effective background tissue absorption. The magnitude of the background absorption was about 0.08 cm(-1) at 758 nm and 0.06 cm(-1) at 830 nm, and it was nearly consistent between piglets. The origin of this 'effective' background absorption may be real, an artefact caused by the application of a simplified model to a complex sample, or a combination of factors.

Real-space Green's function calculation for the solution of the diffusion equation in stratified turbid media

We have derived the space-time Green's function for the diffusion equation in layered turbid media, starting from the case of a planar interface between two random scattering media. This new approach for working directly in real space permits highly efficient numerical processing, which is a decisive criterion for the feasibility of the inverse problem in biomedical optics. The results obtained by this method in the case of a two-layered medium are compared with Monte Carlo simulations.

Method for measuring the diffusion coefficient of homogeneous and layered media

A method for measuring the diffusion coefficient of homogeneous and layered media, based on multidistance measurements of time-resolved reflectance, is proposed. The diffusion coefficient is retrieved from the logarithm between two measurements of reflectance at two different distances. The proposed procedure is simpler than others usually employed and also provides a reliable criterion for retrieval of information on the layered structure of a diffusive medium.

Monte carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain

We propose a hybrid Monte Carlo (MC) diffusion model for calculating the spatially resolved reflectance amplitude and phase delay resulting from an intensity-modulated pencil beam vertically incident on a two-layer turbid medium. The model combines the accuracy of MC at radial distances near the incident beam with the computational efficiency afforded by a diffusion calculation at further distances. This results in a single forward calculation several hundred times faster than pure MC, depending primarily on model parameters. Model predictions are compared with MC data for two cases that span the extremes of physiologically relevant optical properties: skin overlying fat and skin overlying muscle, both in the presence of an exogenous absorber. It is shown that good agreement can be achieved for radial distances from 0.5 to 20 mm in both cases. However, in the skin-on-muscle case the choice of model parameters and the definition of the diffusion coefficient can lead to some interesting discrepancies.

Scattering delay time of Mie scatterers determined from steady-state and time-resolved optical spectroscopy

We have determined the scattering delay time of Mie scatterers (r = 255 nm quartz spheres in polyester resin) from a combination of steady-state (integrating-sphere) and time-resolved (frequency-domain) measurements performed in the multiple-scattering regime. The effective transport velocity of light was derived from intensity and phase measurements at four different wavelengths by using the time-integrated microscopic Beer-Lambert law. We could demonstrate a systematic underestimation of the effective transport velocity compared with the phase velocity in the medium. Assuming that this discrepancy was caused entirely by the transient nature of a single-scattering process, the data presented resulted in time delays of between 18 fs (lambda = 678 nm) and 177 fs (lambda = 1,064 nm) per scattering event. For three out of four wavelengths investigated, the measured values are in excellent agreement with values predicted by a theoretical model for the scattering delay time based on Mie theory.

In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model

We have investigated the possibility of determining the optical coefficients of muscle in the extremities with in vivo time-resolved reflectance measurements using a layered model. A solution of the diffusion equation for two layers was fitted to three-layered Monte Carlo calculations simulating the skin, the subcutaneous fat and the muscle. Relative time-resolved reflectance data at two distances were used to derive the optical coefficients of the layers. We found for skin and subcutaneous fat layer thicknesses (l2) of up to 10 mm that the estimated absorption coefficients of the second layer of the diffusion model have differences of less than 20% compared with those of the muscle layer of the Monte Carlo simulations if the thickness of the first layer of the diffusion model is also fitted. If l2 is known, the differences are less than 5%, whereas the use of a semi-infinite model delivers differences of up to 55%. Even if l2 is only approximately known the absorption coefficient of the muscle can be determined accurately. Experimentally, the time-resolved reflectance was measured on the forearms of volunteers at two distances from the incident beam by means of a streak camera. The thicknesses of the tissues involved were determined by ultrasound. The optical coefficients were derived from these measurements by applying the two-layered diffusion model, and results in accordance with the theoretical studies were observed.