Hybrid Monte Carlo-diffusion method for light propagation in tissue with a low-scattering region

The heterogeneity of the tissues in a head, especially the low-scattering cerebrospinal fluid (CSF) layer surrounding the brain has previously been shown to strongly affect light propagation in the brain. The radiosity-diffusion method, in which the light propagation in the CSF layer is assumed to obey the radiosity theory, has been employed to predict the light propagation in head models. Although the CSF layer is assumed to be a nonscattering region in the radiosity-diffusion method, fine arachnoid trabeculae cause faint scattering in the CSF layer in real heads. A novel approach, the hybrid Monte Carlo-diffusion method, is proposed to calculate the head models, including the low-scattering region in which the light propagation does not obey neither the diffusion approximation nor the radiosity theory. The light propagation in the high-scattering region is calculated by means of the diffusion approximation solved by the finite-element method and that in the low-scattering region is predicted by the Monte Carlo method. The intensity and mean time of flight of the detected light for the head model with a low-scattering CSF layer calculated by the hybrid method agreed well with those by the Monte Carlo method, whereas the results calculated by means of the diffusion approximation included considerable error caused by the effect of the CSF layer. In the hybrid method, the time-consuming Monte Carlo calculation is employed only for the thin CSF layer, and hence, the computation time of the hybrid method is dramatically shorter than that of the Monte Carlo method.     

Near-infrared light propagation in an adult head model. I. Modeling of low-level scattering in the cerebrospinal fluid layer

Adequate modeling of light propagation in a human head is important for quantitative near-infrared spectroscopy and optical imaging. The presence of a nonscattering cerebrospinal fluid (CSF) that surrounds the brain has been previously shown to have a strong effect on light propagation in the head. However, in reality, a small amount of scattering is caused by the arachnoid trabeculae in the CSF layer. In this study, light propagation in an adult head model with discrete scatterers distributed within the CSF layer has been predicted by Monte Carlo simulation to investigate the effect of the small amount of scattering caused by the arachnoid trabeculae in the CSF layer. This low scattering in the CSF layer is found to have little effect on the mean optical path length, a parameter that can be directly measured by a time-resolved experiment. However, the partial optical path length in brain tissue that relates the sensitivity of the detected signal to absorption changes in the brain is strongly affected by the presence of scattering within the CSF layer. The sensitivity of the near-infrared signal to hemoglobin changes induced by brain activation is improved by the effect of a low-scattering CSF layer.     

Modeling of diffuse-diffuse photon coupling via a nonscattering region: a comparative study

It is well established that diffusion approximation is valid for light propagation in highly scattering media, but it breaks down in nonscattering regions. The previous methods that manipulate nonscattering regions are essentially boundary-to-boundary coupling (BBC) methods through a nonscattering void region based on the radiosity theory. We present a boundary-to-interior coupling (BIC) method. BIC is based on the fact that the collimated pencil beam incident on the medium can be replaced by an isotropic point source positioned at one reduced scattering length inside the medium from an illuminated point. A similar replacement is possible for the nondiffuse lights that enter the diffuse medium through the void, and it is formulated as the BIC method. We implemented both coupling methods using the finite element method (FEM) and tested for the circle with a void gap and for a four-layer adult head model. For mean time of flight, the BIC shows better agreement with Monte Carlo (MC) simulation results than BBC. For intensity, BIC shows a comparable match with MC data compared with that of BBC. The effect of absorption of the clear layer in the adult head model was investigated. Both mean time and intensity decrease as absorption of the clear layer increases.     

Effective scattering coefficient of the cerebral spinal fluid in adult head models for diffuse optical imaging

An efficient computation of the time-dependent forward solution for photon transport in a head model is a key capability for performing accurate inversion for functional diffuse optical imaging of the brain. The diffusion approximation to photon transport is much faster to simulate than the physically correct radiative transport equation (RTE); however, it is commonly assumed that scattering lengths must be much smaller than all system dimensions and all absorption lengths for the approximation to be accurate. Neither of these conditions is satisfied in the cerebrospinal fluid (CSF). Since line-of-sight distances in the CSF are small, of the order of a few millimeters, we explore the idea that the CSF scattering coefficient may be modeled by any value from zero up to the order of the typical inverse line-of-sight distance, or approximately 0.3 mm(-1), without significantly altering the calculated detector signals or the partial path lengths relevant for functional measurements. We demonstrate this in detail by using a Monte Carlo simulation of the RTE in a three-dimensional head model based on clinical magnetic resonance imaging data, with realistic optode geometries. Our findings lead us to expect that the diffusion approximation will be valid even in the presence of the CSF, with consequences for faster solution of the inverse problem.     

Finite-difference time-domain analysis of time-resolved reflectance from an adult head model composed of multilayered slabs with a nonscattering layer

Finite-difference time-domain (FDTD) analysis has been used to predict the time-resolved reflectance from multilayered slabs with a nonscattering layer. Light propagation across the nonscattering layer was calculated based on the light intensity characteristics along a ray in free space. Additional equivalent source functions due to light from scattering regions across the nonscattering region were introduced into the diffusion equation and an additional set of the diffusion equation was solved by FDTD analysis by employing new boundary conditions. The formulation was used to calculate time-resolved reflectances of three- and four-layered slabs containing a nonscattering layer. The received light intensity and the mean time of flight estimated from the time-resolved reflectance are in reasonable agreement with previously reported experimental data and Monte Carlo simulations.     

Monte Carlo prediction of near-infrared light propagation in realistic adult and neonatal head models

In near-infrared spectroscopy and imaging, the sensitivity of the detected signal to brain activation and the volume of interrogated tissue are clinically important. Light propagation in adult and neonatal heads is strongly affected by the presence of a low-scattering cerebrospinal fluid layer. The effect of the heterogeneous structure of the head on light propagation in the adult brain is likely to be different from that in the neonatal brain because the thickness of the superficial tissues and the optical properties of the brain of the neonatal head are quite different from those of the adult head. In this study, light propagation in the two-dimensional realistic adult and neonatal head models, whose geometries are generated from a magnetic resonance imaging scan of the human heads, is predicted by Monte Carlo simulation. The sandwich structure, which is a low-scattering cerebrospinal fluid layer held between the high-scattering skull and gray matter, strongly affects light propagation in the brain of the adult head. The sensitivity of the absorption change in the gray matter is improved; however, the intensely sensitive region is confined to the shallow region of the gray matter. The high absorption of the neonatal brain causes a similar effect on light propagation in the head. The intensely sensitive region in the neonatal brain is confined to the gray matter; however, the spatial sensitivity profile penetrates into the deeper region of the white matter.     

Linear and nonlinear reconstruction for optical tomography of phantoms with nonscattering regions

Most research in optical imaging incorrectly assumes that light transport in nonscattering regions in the head may be modeled by use of the diffusion approximation. The effect of this assumption is examined in a series of experiments on tissue-equivalent phantoms. Images from cylindrical and head-shaped phantoms with and without clear regions [simulating the cerebrospinal fluid (CSF) filled ventricles] and a clear layer (simulating the CSF layer surrounding the brain) are reconstructed with linear and nonlinear reconstruction techniques. The results suggest that absorbing and scattering perturbations can be identified reliably with nonlinear reconstruction methods when the clear regions are also present in the reference data but that the quality of the image degrades considerably if the reference data does not contain these features. Linear reconstruction performs similarly to nonlinear reconstruction, provided the clear regions are present in the reference data, but otherwise linear reconstruction fails. This study supports the use of linear reconstruction for dynamic imaging but suggests that, in all cases, image quality is likely to improve if the clear regions are modeled correctly.     

Optical tomography in the presence of void regions

There is a growing interest in the use of near-infrared spectroscopy for the noninvasive determination of the oxygenation level within biological tissue. Stemming from this application, there has been further research in the use of this technique for obtaining tomographic images of the neonatal head, with the view of determining the levels of oxygenated and deoxygenated blood within the brain. Owing to computational complexity, methods used for numerical modeling of photon transfer within tissue have usually been limited to the diffusion approximation of the Boltzmann transport equation. The diffusion approximation, however, is not valid in regions of low scatter, such as the cerebrospinal fluid. Methods have been proposed for dealing with nonscattering regions within diffusing materials through the use of a radiosity-diffusion model. Currently, this new model assumes prior knowledge of the void region location; therefore it is instructive to examine the errors introduced in applying a simple diffusion-based reconstruction scheme in cases in which there exists a nonscattering region. We present reconstructed images of objects that contain a nonscattering region within a diffusive material. Here the forward data is calculated with the radiosity-diffusion model, and the inverse problem is solved with either the radiosity-diffusion model or the diffusion-only model. The reconstructed images show that even in the presence of only a thin nonscattering layer, a diffusion-only reconstruction will fail. When a radiosity-diffusion model is used for image reconstruction, together with a priori information about the position of the nonscattering region, the quality of the reconstructed image is considerably improved. The accuracy of the reconstructed images depends largely on the position of the anomaly with respect to the nonscattering region as well as the thickness of the nonscattering region.     

Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head

Near-infrared light propagation in various models of the adult head is analyzed by both time-of-flight measurements and mathematical prediction. The models consist of three- or four-layered slabs, the latter incorporating a clear cerebrospinal fluid (CSF) layer. The most sophisticated model also incorporates slots that imitate sulci on the brain surface. For each model, the experimentally measured mean optical path length as a function of source-detector spacing agrees well with predictions from either a Monte Carlo model or a finite-element method based on diffusion theory or a hybrid radiosity-diffusion theory. Light propagation in the adult head is shown to be highly affected by the presence of the clear CSF layer, and both the optical path length and the spatial sensitivity profile of the models with a CSF layer are quite different from those without the CSF layer. However, the geometry of the sulci and the boundary between the gray and the white matter have little effect on the detected light distribution.     

Boundary conditions for light propagation in diffusive media with nonscattering regions

The diffusion approximation proves to be valid for light propagation in highly scattering media, but it breaks down in the presence of nonscattering regions. We present a compact expression of the boundary conditions for diffusive media with nonscattering regions, taking into account small-index mismatch. Results from an integral method based on the extinction theorem boundary condition are contrasted with both Monte Carlo and finite-element-method simulations, and a study of its limit of validity is presented. These procedures are illustrated by considering the case of the cerebro-spinal fluid in the brain, for which we demonstrate that for practical situations in light diffusion, these boundary conditions yield accurate results.     

Near-infrared spectroscopy of the adult head: effect of scattering and absorbing obstructions in the cerebrospinal fluid layer on light distribution in the tissue

Previous modeling of near-infrared (NIR) light distribution in models of the adult head incorporating a clear nonscattering cerebrospinal fluid (CSF) layer have shown the latter to have a profound effect on the resulting photon measurement density function (PMDF). In particular, the presence of the CSF limits the PMDF largely to the outer cortical gray matter with little signal contribution from the deeper white matter. In practice, the CSF is not a simple unobstructed clear layer but contains light-scattering membranes and is crossed by various blood vessels. Using a radiosity-diffusion finite-element model, we investigated the effect on the PMDF of introducing intrusions within the clear layer. The results show that the presence of such obstructions does not significantly increase the light penetration into the brain tissue, except immediately adjacent to the obstruction and that its presence also increases the light sampling of the adjacent skull tissues, which would lead to additional contamination of the NIR spectroscopy signal by the surface tissue layers.     

Linear single-step image reconstruction in the presence of nonscattering regions

There is growing interest in the use of near-infrared spectroscopy for the noninvasive determination of the oxygenation level within biological tissue. Stemming from this application, there has been further research in using this technique for obtaining tomographic images of the neonatal head, with the view of determining the level of oxygenated and deoxygenated blood within the brain. Because of computational complexity, methods used for numerical modeling of photon transfer within tissue have usually been limited to the diffusion approximation of the Boltzmann transport equation. The diffusion approximation, however, is not valid in regions of low scatter, such as the cerebrospinal fluid. Methods have been proposed for dealing with nonscattering regions within diffusing materials through the use of a radiosity-diffusion model. Currently, this new model assumes prior knowledge of the void region; therefore it is instructive to examine the errors introduced in applying a simple diffusion-based reconstruction scheme in cases where a nonscattering region exists. We present reconstructed images, using linear algorithms, of models that contain a nonscattering region within a diffusing material. The forward data are calculated by using the radiosity-diffusion model, and the inverse problem is solved by using either the radiosity-diffusion model or the diffusion-only model. When using data from a model containing a clear layer and reconstructing with the correct model, one can reconstruct the anomaly, but the qualitative accuracy and the position of the reconstructed anomaly depend on the size and the position of the clear regions. If the inverse model has no information about the clear regions (i.e., it is a purely diffusing model), an anomaly can be reconstructed, but the resulting image has very poor qualitative accuracy and poor localization of the anomaly. The errors in quantitative and localization accuracies depend on the size and location of the clear regions.     

Generalized diffusion model in optical tomography with clear layers

We introduce a generalized diffusion equation that models the propagation of photons in highly scattering domains with thin nonscattering clear layers. Classical diffusion models break down in the presence of clear layers. The model that we propose accurately accounts for the clear-layer effects and has a computational cost comparable to that of classical diffusion. It is based on modeling the propagation in the clear layer as a local tangential diffusion process. It can be justified mathematically in the limit of small mean free paths and is shown numerically to be very accurate in two- and three-dimensional idealized cases. We believe that this model can be used as an accurate forward model in optical tomography.     

Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model

The general two-layer inverse problem in biomedical photon migration is to estimate the absorption and scattering coefficients of each layer as well as the top-layer thickness. We attempted to solve this problem, using experimental and simulated spatially resolved frequency-domain (FD) reflectance for optical properties typical of skin overlying muscle or skin overlying fat in the near infrared. Two forward models of light propagation were used: a two-layer diffusion solution [Appl. Opt. 37, 779 (1998)] and a hybrid Monte Carlo (MC) diffusion model [Appl. Opt. 37, 7401 (1998)]. MC-simulated FD reflectance data were fitted as relative measurements to the hybrid and the pure diffusion models. It was found that the hybrid model could determine all the optical properties of the two-layer media studied to ~5%. Also, the same accuracy could be achieved by means of fitting MC-simulated cw reflectance data as absolute measurements, but fitting them as relative ones is an ill-posed problem. In contrast, two-layer diffusion could not retrieve the top-layer optical properties as accurately for FD data and was ill-posed for both relative and absolute cw data. The hybrid and the pure diffusion models were also fitted to experimental FD reflectance measurements from two-layer tissue-simulating phantoms representative of skin-on-fat and skin-on-muscle baseline optical properties. Both the hybrid and the diffusion models could determine the optical properties of the lower layer. The hybrid model demonstrated its potential to retrieve quantitatively the transport scattering coefficient of skin (the upper layer), which was not possible with the pure diffusion model. Systematic discrepancies between model and experiment may compromise the accuracy of the deduced top-layer optical properties. Identifying and eliminating such discrepancies is critical to practical application of the method.     

Hybrid diffusion and two-flux approximation for multilayered tissue light propagation modeling

Accurate and rapid estimation of fluence, reflectance, and absorbance in multilayered biological media has been essential in many biophotonics applications that aim to diagnose, cure, or model in vivo tissue. The radiative transfer equation (RTE) rigorously models light transfer in absorbing and scattering media. However, analytical solutions to the RTE are limited even in simple homogeneous or plane media. Monte Carlo simulation has been used extensively to solve the RTE. However, Monte Carlo simulation is computationally intensive and may not be practical for applications that demand real-time results. Instead, the diffusion approximation has been shown to provide accurate estimates of light transport in strongly scattering tissue. The diffusion approximation is a greatly simplified model and produces analytical solutions for the reflectance and absorbance in tissue. However, the diffusion approximation breaks down if tissue is strongly absorbing, which is common in the visible part of the spectrum or in applications that involve darkly pigmented skin and/or high local volumes of blood such as port-wine stain therapy or reconstructive flap monitoring. In these cases, a model of light transfer that can accommodate both strongly and weakly absorbing regimes is required. Here we present a model of light transfer through layered biological media that represents skin with two strongly scattering and one strongly absorbing layer.     

Effect of the scattering delay on time-dependent photon migration in turbid media

We modified the diffusion approximation of the time-dependent radiative transfer equation to account for a finite scattering delay time. Under the usual assumptions of the diffusion approximation, the effect of the scattering delay leads to a simple renormalization of the light velocity that appears in the diffusion equation. Accuracy of the model was evaluated by comparison with Monte Carlo simulations in the frequency domain for a semi-infinite geometry. A good agreement is demonstrated for both matched and mismatched boundary conditions when the distance from the source is sufficiently large. The modified diffusion model predicts that the neglect of the scattering delay when the optical properties of the turbid material are derived from normalized frequency- or time-domain measurements should result in an underestimation of the absorption coefficient and an overestimation of the transport coefficient. These observations are consistent with the published experimental data.     

Finite element model for the coupled radiative transfer equation and diffusion approximation

In this paper a coupled radiative transfer equation and diffusion approximation model for light propagation in tissues is proposed. The light propagation is modelled with the radiative transfer equation in sub‐domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The method is tested with simulations. The results of the proposed approach are compared with finite element solutions of the radiative transfer equation, the diffusion approximation and a previously proposed hybrid model. The results show that the method improves the accuracy of the forward solution for diffuse optical tomography compared to the conventional diffusion model. Copyright © 2005 John Wiley & Sons, Ltd.     

Photon diffusion coefficient in scattering and absorbing media

We present a unified derivation of the photon diffusion coefficient for both steady-state and time-dependent transport in disordered absorbing media. The derivation is based on a modal analysis of the time-dependent radiative transfer equation. This approach confirms that the dynamic diffusion coefficient is given by the random-walk result D = cl(*)/3, where l(*) is the transport mean free path and c is the energy velocity, independent of the level of absorption. It also shows that the diffusion coefficient for steady-state transport, often used in biomedical optics, depends on absorption, in agreement with recent theoretical and experimental works. These two results resolve a recurrent controversy in light propagation and imaging in scattering media.     

Experimental determination of photon propagation in highly absorbing and scattering media

Optical imaging and tomography in tissues can facilitate the quantitative study of several important chromophores and fluorophores. Several theoretical models have been validated for diffuse photon propagation in highly scattering and low-absorbing media that describe the optical appearance of tissues in the near-infrared (NIR) region. However, these models are not generally applicable to quantitative optical investigations in the visible because of the significantly higher tissue absorption in this spectral region compared with that in the NIR. We performed photon measurements through highly scattering and absorbing media for ratios of the absorption coefficient to the reduced scattering coefficient ranging approximately from zero to one. We examined experimentally the performance of the absorption-dependent diffusion coefficient defined by Aronson and Corngold [J. Opt. Soc. Am. A 16, 1066 (1999)] for quantitative estimations of photon propagation in the low- and high-absorption regimes. Through steady-state measurements we verified that the transmitted intensity is well described by the diffusion equation by considering a modified diffusion coefficient with a nonlinear dependence on the absorption. This study confirms that simple analytical solutions based on the diffusion approximation are suitable even for high-absorption regimes and shows that diffusion-approximation-based models are valid for quantitative measurements and tomographic imaging of tissues in the visible.     

Light propagation in tissues with forward-peaked and large-angle scattering

We study light propagation in tissues using the theory of radiative transport. In particular, we study the case in which there is both forward-peaked and large-angle scattering. Because this combination of the forward-peaked and large-angle scattering makes it difficult to solve the radiative transport equation, we present a method to construct approximations to study this problem. The delta-Eddington and Fokker-Planck approximations are special cases of this general framework. Using this approximation method, we derive two new approximations: the Fokker-Planck-Eddington approximation and the generalized Fokker-Planck-Eddington approximation. By computing the transmittance and reflectance of light by a slab we study the performance of these approximations.