MCCL: an open-source software application for Monte Carlo simulations of radiative transport

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

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

Quantifying tissue properties and absolute hemodynamics using coherent spatial imaging

Significance

Measuring hemodynamic function is crucial for health assessment. Optical signals provide relative hemoglobin concentration changes, but absolute measurements require costly, bulky technology. Speckleplethysmography (SPG) uses coherent light to detect speckle fluctuations. Combining SPG with multispectral measurements may provide important physiological information on blood flow and absolute hemoglobin concentration.

Aim

To develop an affordable optical technology to measure tissue absorption, scattering, hemoglobin concentrations, tissue oxygen saturation (StO 2 ), and blood flow.

Approach

We integrated reflectance spectroscopy and laser speckle imaging to create coherent spatial imaging (CSI). CSI was validated against spatial frequency domain imaging (SFDI) using phantom-based measurements. In vivo arterial and venous occlusion experiments compared CSI with diffuse optical spectroscopy/diffuse correlation spectroscopy (DOS/DCS) measurements.

Results

CSI and SFDI agreed on tissue absorption and scattering in phantom tests. CSI and DOS/DCS showed similar trends and agreement in arterial occlusion results. During venous occlusion, both uncorrected and corrected blood flow decreased with increasing pressure, with an ∼ 200 % difference in overall blood flow decrease between the methods. CSI and DOS/DCS data showed expected hemoglobin concentrations, StO 2 , and blood flow trends.

Conclusions

CSI provides affordable and comprehensive hemodynamic information. It can potentially detect dysfunction and improve measurements, such as blood pressure, S p O 2 , and metabolism.

scatterBrains: an open database of human head models and companion optode locations for realistic Monte Carlo photon simulations

Significance

Monte Carlo (MC) simulations are currently the gold standard in the near-infrared and diffuse correlation spectroscopy (NIRS/DCS) communities for generating light transport paths through tissue. However, realistic and diverse models that capture complex tissue layers are not easily available to all; moreover, manually placing optodes on such models can be tedious and time consuming. Such limitations may hinder the adoption of representative models for basic simulations and the use of these models for large-scale simulations, e.g., for training machine learning algorithms.

Aim

We aim to provide the NIRS/DCS communities with an open-source, user-friendly database of morphologically and optically realistic head models, as well as a succinct software pipeline to prepare these models for mesh-based Monte Carlo simulations of light transport.

Approach

Sixteen anatomical models were created from segmented T1-weighted magnetic resonance imaging (MRI) head scans and converted to tetrahedral mesh volumes. Approximately 800 companion scalp surface locations were distributed on each model, comprising full head coverage. A pipeline was created to place custom source and optical detectors at each location, and guidance is provided on how to use these parameters to set up MC simulations.

Results

The models, head surface locations, and all associated code are freely available under the scatterBrains project on Github.

Conclusions

The NIRS/DCS community benefits from having shared resources for conducting MC simulations on realistic head geometries. We hope this will make MRI-based head models and virtual optode placement easily accessible to all. Contributions to the database are welcome and encouraged.

Designing and simulating realistic spatial frequency domain imaging systems using open-source 3D rendering software

Spatial frequency domain imaging (SFDI) is a low-cost imaging technique that maps absorption and reduced scattering coefficients, offering improved contrast for important tissue structures such as tumours. Practical SFDI systems must cope with various imaging geometries including imaging planar samples ex vivo, imaging inside tubular lumen in vivo e.g. for endoscopy, and measuring tumours or polyps of varying morphology. There is a need for a design and simulation tool to accelerate design of new SFDI systems and simulate realistic performance under these scenarios. We present such a system implemented using open-source 3D design and ray-tracing software Blender that simulates media with realistic absorption and scattering in a wide range of geometries. By using Blender's Cycles ray-tracing engine, our system simulates effects such as varying lighting, refractive index changes, non-normal incidence, specular reflections and shadows, enabling realistic evaluation of new designs. We first demonstrate quantitative agreement between Monte-Carlo simulated absorption and reduced scattering coefficients with those simulated from our Blender system, achieving 16 % discrepancy in absorption coefficient and 18 % in reduced scattering coefficient. However, we then show that using an empirically derived look-up table the errors reduce to 1 % and 0.7 % respectively. Next, we simulate SFDI mapping of absorption, scattering and shape for simulated tumour spheroids, demonstrating enhanced contrast. Finally we demonstrate SFDI mapping inside a tubular lumen, which highlighted a important design insight: custom look-up tables must be generated for different longitudinal sections of the lumen. With this approach we achieved 2 % absorption error and 2 % scattering error. We anticipate our simulation system will aid in the design of novel SFDI systems for key biomedical applications.

Analysis of relative error in perturbation Monte Carlo simulations of radiative transport

Significance

Perturbation and differential Monte Carlo (pMC/dMC) methods, used in conjunction with nonlinear optimization methods, have been successfully applied to solve inverse problems in diffuse optics. Application of pMC to systems over a large range of optical properties requires optimal "placement" of baseline conventional Monte Carlo (cMC) simulations to minimize the pMC variance. The inability to predict the growth in pMC solution uncertainty with perturbation size limits the application of pMC, especially for multispectral datasets where the variation of optical properties can be substantial.

Aim

We aim to predict the variation of pMC variance with perturbation size without explicit computation of perturbed photon weights. Our proposed method can be used to determine the range of optical properties over which pMC predictions provide sufficient accuracy. This method can be used to specify the optical properties for the reference cMC simulations that pMC utilizes to provide accurate predictions over a desired optical property range.

Approach

We utilize a conventional error propagation methodology to calculate changes in pMC relative error for Monte Carlo simulations. We demonstrate this methodology for spatially resolved diffuse reflectance measurements with ±20% scattering perturbations. We examine the performance of our method for reference simulations spanning a broad range of optical properties relevant for diffuse optical imaging of biological tissues. Our predictions are computed using the variance, covariance, and skewness of the photon weight, path length, and collision distributions generated by the reference simulation.

Results

We find that our methodology performs best when used in conjunction with reference cMC simulations that utilize Russian Roulette (RR) method. Specifically, we demonstrate that for a proximal detector placed immediately adjacent to the source, we can estimate the pMC relative error within 5% of the true value for scattering perturbations in the range of [ - 15 % , + 20 % ] . For a distal detector placed at ∼ 3 transport mean free paths relative to the source, our method provides relative error estimates within 20% for scattering perturbations in the range of [ - 8 % , + 15 % ] . Moreover, reference simulations performed at lower ( μ s ' / μ a ) values showed better performance for both proximal and distal detectors.

Conclusions

These findings indicate that reference simulations utilizing continuous absorption weighting (CAW) with the Russian Roulette method and executed using optical properties with a low ( μ s ' / μ a ) ratio spanning the desired range of μ s values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.

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.

Special Section Guest Editorial: Introduction to the Special Section Celebrating 30 years of Open Source Monte Carlo Codes in Biomedical Optics

The editorial introduces the JBO Special Section Celebrating 30 Years of Open Source Monte Carlo Codes in Biomedical Optics for Volume 27, Issue 8.