A contemporary systematic review on deterministic numerical simulations of light propagation in head tissues

The attenuation of near-infrared (NIR) light intensity as it propagates in a turbid medium like biological tissue is described by modified the Beer–Lambert law (MBLL). The MBLL is generally used to quantify the changes in tissue chromophore concentrations for NIR spectroscopic data analysis. Even though MBLL is effective in terms of providing qualitative comparison, it suffers from its applicability across tissue types and tissue dimensions. In this work, we introduce Lambert-W function-based modeling for light propagation in biological tissues, which is a generalized version of the Beer–Lambert model. The proposed modeling provides parametrization of tissue properties, which includes two attenuation coefficients μ0 and η. We validated our model against the Monte Carlo simulation, which is the gold standard for modeling NIR light propagation in biological tissue. We included numerous human and animal tissues to validate the proposed empirical model, including an inhomogeneous adult human head model. The proposed model, which has a closed form (analytical), is first of its kind in providing accurate modeling of NIR light propagation in biological tissues.

Monte Carlo (MC) simulation is the most frequently used method to numerically model the light propagation in biological tissues because of its high flexibility and precision. Although MC simulation is assumed to be capable of achieving any desired precision, larger number of photons are always necessary for more precise simulation, leading to its major limitation of intensive computation. In this work, the authors present a way to adapt generative adversarial networks (GAN) to accelerate MC simulation. The pix2pix network, a variant of GAN, was investigated to reconstruct precise MC simulation results from the results roughly modeled by small amount of photons, thus the computation time was expected to be significantly saved. The proposed method was tested on single-layer embedded tumor models to derive the absorption distribution maps. The results demonstrate that the absorption distribution maps reconstructed from the simulation of only 10000 photons were very similar to those modeled by using 1000000 photons, based on the criterion of peak signal to noise ratio (PSNR) and percentage difference of power coupling efficiencies, and the simulation process was proved to be accelerated by approximately 102 times. For the first time, GAN was adapted to save computation time of MC simulation of light propagation. By achieving MC simulation with acceptable quality, the proposed method can speed up the computation by hundreds of times.

In this work we introduce the finite volume (FV) approximation to the simplified spherical harmonics (SP_N) equations for modeling light propagation in tissue. The SP_N equations, with partly reflective boundary conditions, are discretized on unstructured grids. The resulting system of linear equations is solved with a Krylov subspace iterative method called the generalized minimal residual (GMRES) algorithm. The accuracy of the FV-SP_N algorithm is validated through numerical simulations of light propagation in a numerical phantom with embedded inhomogeneities. We use a FV implementation of the equation of radiative transfer (ERT) as the benchmark algorithm. Solutions obtained using the FV-SP_N (N > 1) algorithm are compared to solutions obtained with the ERT and the diffusion equation (SP₁). Compared to the SP₁, the SP₃ solutions obtained using the FV-SP_N algorithm can better approximate ERT solutions near boundary sources and in the vicinity of void-like regions. Solutions using the SP₃ algorithm are obtained 9.95 times faster than solutions with the ERT-based algorithm.

In brain function measurement using near‐infrared light, it is very important to make a theoretical analysis of light propagation in the head tissue, since it is impossible to measure the effect of optical fiber arrangement on the region of measurement or its sensitivity. The Monte Carlo method is widely used in the analysis of near‐infrared light propagation in biological tissue, since it allows light propagation analysis of heterogeneous tissue by a relatively simple algorithm. In this study, a model simulating the cross‐sectional structure of the neonatal head is constructed by using square elements, and the light propagation is analyzed. First, the variance‐reduction (VR) method and the delta‐scattering (DS) method, which are typical algorithms of the Monte Carlo method, are applied to the model composed of square elements, and the results of the analysis and the computation time for light propagation are compared and discussed. It is shown that the DS method is useful for a model composed of small square elements, since the computation time depends little on the size of the elements. As a result of analysis of light propagation in the neonatal head model using the DS method, we see that light propagation is greatly affected by the heterogeneity of the head tissue. © 2004 Wiley Periodicals, Inc. Syst Comp Jpn, 35(9): 60–69, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10246

The light propagation in biological tissue having anisotropic optical properties is investigated. Monte Carlo simulations employing the phase function of infinitely long cylindrical scatterers and the Henyey-Greenstein function are performed and compared to spatially-resolved reflectance measurements of semi-infinite turbid media. In addition, simulations are shown for the spatially-resolved reflectance and transmittance from the surfaces of cubic turbid media. It is found that the light propagation in anisotropic turbid media is very different compared to turbid media that have isotropic optical properties. For example, it is observed that the light which is incident perpendicular to the top surface of the cube may be transmitted mainly from a single lateral side.

Numerical modeling was used for the theoretical analysis of the propagation of optical radiation in the tissues of the human head, generated by a single source placed on the surface of the scalp. Of special interest and importance is the propagation of radiation within the layer of cerebrospinal fluid contained in the subarachnoid space (SAS), which is the only low absorption/high transmittance medium whose width can vary rapidly. Qualitative and quantitative assessment of changes in propagation of radiation within the SAS could become a source of information on changes in the geometry of this anatomical compartment playing a crucial role in cranio-spinal physiology and pathology. Essential for the idea of the possible noninvasive assessment of changes in width of the SAS by an optical method is the dependence of intensity of radiation reaching a photodetector located at a certain distance from the source on changes in the width of this fluid layer, which acts like a biological optical waveguide. Monte Carlo modeling and numerical analysis confirmed the feasibility of assessing changes in the width of the subarachnoid space optically. Presented here are details of the Monte Carlo simulation of light propagation in the tissues of human head and the results of such simulation as a function of the width of the subarachnoid space, calculated for different distances between the source and detector and for a few selected values of bone thickness. Results of numerical modeling were then compared with those of experiments on a mechanical-optical model.

Precise knowledge about light propagation in biological tissues is necessary for accurate diagnostics and effective therapies utilizing optical technologies. In the current paper, the Monte Carlo simulation is applied to study light dispersion in normal and cancerous breast after irradiating to different laser beam shapes. Two distinct laser wavelengths (800-1100nm) with planar and Gaussian shapes were employed. The spatially resolved steady-state diffuse reflectance of normal tissue and tumor was investigated using Monte Carlo simulation method via MCML and MCXLAB computations. The diffusion equation was solved to simulate the fluence rate at the tissue surface based on the optical parameter values (i.e., scattering and absorption coefficients). The results confirm differences in diffuse reflectance and optical fluence distribution between the normal and tumor tissues at each wavelength. Tissue optical parameters and the utilized laser beam shape control the distribution of the fluence rate within tissues. Therefore, offering visual representations of these distributions can provide a secure visual route for biological diagnostics.

Some of our experimental and theoretical results on optical properties of atmospheric turbulence and light propagation through atmosphere are introduced as follows. 1) A mixed model for the scaling property of high-order structure functions of temperature is proposed to study the turbulence intermittency effect in numerical simulation of light propagation through turbulence. 2) The difference of refractive index structure constant C²_n of atmosphere and light beam wander measured in horizontal and vertical directions suggest that the atmospheric turbulence is not isotopc in many cases. 3) The fluctuation of the scaling exponent of optical turbulence spectrum and light intensity fluctuation spectrum indicate taht the atmospheric turbulence is not a Kolmogorov one in many cases. 4) A general description of turbulence strength is proposed using the variance of the refractive index σ²_n, the outer scale L₀, the scaling power γ and the inner scale l₀. 5) Characteristics of the power spectrum of laser scintillation in atmosphere depend on the laesr beam property, diameter of the receiving aperture, turbulence spectrum in the dissipation range. 6) The intermittency of ligh scintillation is rather stable at different scintillation index for light propagation through turbulence over both land and sea surfaces. 7) The number density of phaser branch points of light wave field in turbulence generally increases with turbulence strength but present large diversity.

Studying light propagation in biological tissues is critical for developing biophotonics techniques and their applications. Monte Carlo (MC) method, a stochastic solver for radiative transfer equation, has been recognized as the gold standard for modeling light propagation in turbid media. However, due to the stochastic nature of the MC method, millions even billions of photons are usually required to achieve accurate results using the MC method, leading to a long computational time even with the acceleration using graphical processing units (GPU). Furthermore, due to the rapid advances in multi-scale optical imaging techniques such as optical coherence tomography (OCT) and multiphoton microscopy (MPM), there is an increasing need to model light propagation in extremely complex tissues such as vessel networks. The mesh-based Monte Carlo (MMC) is usually superior to the voxel-based MC method for such modeling since, unlike grid-like voxels, tetrahedral meshes can represent arbitrary shapes with curved boundaries. However, the mesh density can be excessively high when the tissue structure is extremely complex, resulting in high computational costs and memory demand. The goal of this proposal is to focus on solving the challenges mentioned above. To tackle the first challenge, we came up with a filtering approach with GPU acceleration to improve the signal-to-noise ratio (SNR) of the results while keeping the simulated photons low. The adaptive non-local means (ANLM) filter is selected to suppress the stochastic noise in MC results because 1) the filtering process on each voxel is mutually independent, making it possible for parallel computing; 2) it has high performance in denoising and a strong capacity in edge-preserving. For the second problem, a novel method, implicit mesh-based Monte Carlo (iMMC), was proposed to significantly reduce the mesh density. The iMMC utilizes the edge, node, and face of the tetrahedral mesh to model tissue structures with shapes of the cylinder, sphere, and thin layer. The typical applications for an edge, node, and face-based iMMC are vessel networks, porous media, and membranes, respectively. Lastly, we applied MC simulations and the aforementioned filter on segmented brain models derived from MRI neurodevelopmental atlas to estimate the light dosage for transcranial photobiomodulation (t-PBM), a technique for treating major depressive disorder using near-infrared, across the lifespan. The MMC simulation was also applied to evaluate the impact of human hair on brain sensitivity for functional near-infrared spectroscopy (fNIRS). Furthermore, a new approach that can improve the penetration depth in optical brain imaging, as well as PBM, is proposed. In this approach, the possibility of placing light sources in head cavities is investigated using MC simulations. The preliminary results demonstrate better performance in deep brain monitoring compared to the standard transcranial approach using 10-20 EEG positioning system.--Author's abstract

Monte Carlo (MC) simulation for light propagation in tissue is the gold standard for studying the light propagation in biological tissue and has been used for years. Interaction of photons with a medium is simulated based on its optical properties. New simulation geometries, tissue-light interaction methods, and recording techniques recently have been designed. Applications, such as whole mouse body simulations for fluorescence imaging, eye modeling for blood vessel imaging, skin modeling for terahertz imaging, and human head modeling for sinus imaging, have emerged. Here, we review the technical advances and recent applications of MC simulation.

Monte Carlo (MC) numerical simulation of light propagation in living tissue is widely used for characterization of optical coherence tomography (OCT) signals. Using MC simulation we obtained OCT images of skin with dysplastic nevus. Two various positions of skin nevus in depth were considered. Comparing these OCT simulation results with image of skin without nevus, we showed that OCT medical approach allows to detect dysplastic nevus at different stages of its life.

An accurate knowledge of tissue optical properties (absorption coefficients, μa, and reduced scattering coefficients, μs’) is critical for precise modeling of light propagation in biological tissue, essential for developing diagnostic and therapeutic optical techniques that utilize diffusive photons. A great number of studies have explored the optical properties of various tissue, and these values are not known in detail due to difficulties in the experimental determination and significant variations in tissue constitution. Especially, in situ estimates of the optical properties of brain tissue, a common measurement target in optical imaging, is a challenge because of its layer structure (where the thin gray matter covers the white matter). Here, we report an approach to in situ estimates of the μa and μs’ of the gray and white matter in living rat and monkey brains by using femtosecond time-resolved measurements and Monte Carlo simulation. The results demonstrate that the μa of the gray matter is larger than that of the white matter, while there was no significant difference in the μs’ between the gray and white matter. The optical properties of the rat brain were very similar to those of the monkey brain except for the μa of the gray matter here.

Diffusion equations (DEs) or simplified spherical harmonic equations are commonly used forward models in bioluminescence tomography (BLT), which are usually numerically calculated by the finite element method to construct the system matrix for reconstruction. However, the numerical solver is not accurate enough. The Monte Carlo (MC) method is regarded as the golden standard for modeling light propagation in biological tissue. In this paper, we proposed a GPU-accelerated inverse MC method for BLT reconstruction. The main feature is that the system matrix for BLT reconstruction is calculated by the MC method instead of the model-based numerical approximation. We evaluated the performance of the proposed method with both phantom-based simulation and animal-based in vivo experiment. The results show that, compared with the DE-based method, the proposed GPU-accelerated inverse MC method is more accurate and effective in BLT reconstruction.

Spatially resolved spectroscopy provides a means for measuring the optical properties of biological tissues, based on analytical solutions to diffusion approximation for semi-infinite media under the normal illumination of an infinitely small light beam. The method is, however, prone to error in measurement because the actual boundary condition and light beam often deviate from that used in deriving the analytical solutions. It is therefore important to quantify the effect of different boundary conditions and light beams on spatially resolved diffuse reflectance in order to improve the measurement accuracy of the technique. This research was aimed at using finite element method (FEM) to model light propagation in turbid media, subjected to normal illumination by a continuous-wave beam of infinitely small or finite size. Three types of boundary conditions [i.e., partial current (PCBC), extrapolated (EBC), and zero (ZBC)] were evaluated and compared against Monte Carlo (MC) simulations, since MC could provide accurate fluence rate and diffuse reflectance. The effect of beam size was also investigated. Overall results showed that FEM provided results as accurate as those of the analytical method when an appropriate boundary condition was applied. ZBC did not give satisfactory results in most cases. FEM-PCBC yielded a better fluence rate at the boundary than did FEM-EBC, while they were almost identical in predicting diffuse reflectance. Results further showed that FEM coupled with EBC effectively simulated spatially resolved diffuse reflectance under the illumination of a finite size beam. A large beam introduced more error, especially within the region of illumination. Research also confirmed an earlier finding that a light beam of less than 1 mm diameter should be used for estimation of optical parameters. FEM is effective for modeling light propagation in biological tissues and can be used for improving the optical property measurement by the spatially resolved technique.

This work presents a deterministic and a stochastic model for the simulation of industrial-size deionized water and water for injection (DI/WFI) systems. The objective of the simulations is to determine if additional DI/WFI demand from future production processes can be supported by an existing DI/WFI system. The models utilize discrete event simulation to compute the demand profile from the distribution system; they also use a continuous simulation to calculate the variation of the water level in the storage tank. Whereas the deterministic model ignores uncertainties, the stochastic model allows for both volume and schedule uncertainties. The Monte Carlo method is applied to solve the stochastic method. This paper compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications and that the stochastic model should only be used if found necessary by the deterministic simulation. The models are programmed within Excel 2003 and are available for download as open public domain software (1), allowing for public modifications and improvements of the model. The proposed models may also be utilized to determine size or analyze the performance of other utilities, such as heat transfer media, drinking water, etc. Water for injection (WFI) and other pharmaceutical water distribution systems are notoriously difficult to analyze analytically due to the highly dynamic variable demand that is drawn from these systems. Discrete event simulation may provide an answer where the typical engineering approach of utilizing a diversity factor fails. This paper develops an Excel based deterministic and stochastic model for a WFI system with the latter allowing for the modeling of offtake volume and schedule uncertainty. The paper also compares the deterministic and stochastic models and shows that the deterministic model may be suitable for most applications while the stochastic model should only be used if found necessary. The models are available for download as open public domain software allowing for modifications and improvements of the model.