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 (µs'/µa) 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 (µs'/µa) ratio spanning the desired range of µs values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.

Modeling nonlinear optical microscopy in scattering media, part I. Propagation from lens to focal volume: tutorial.

The development and application of nonlinear optical (NLO) microscopy methods in biomedical research have experienced rapid growth over the past three decades. Despite the compelling power of these methods, optical scattering limits their practical use in biological tissues. This tutorial offers a model-based approach illustrating how analytical methods from classical electromagnetism can be employed to comprehensively model NLO microscopy in scattering media. In Part I, we quantitatively model focused beam propagation in non-scattering and scattering media from the lens to focal volume. In Part II, we model signal generation, radiation, and far-field detection. Moreover, we detail modeling approaches for major optical microscopy modalities including classical fluorescence, multi-photon fluorescence, second harmonic generation, and coherent anti-Stokes Raman microscopy.

Modeling nonlinear optical microscopy in scattering media, part II. Radiation from focal volume to far-field: tutorial.

The development and application of nonlinear optical (NLO) microscopy methods in biomedical research has experienced rapid growth over the past three decades. Despite the compelling power of these methods, optical scattering limits their practical use in biological tissues. This tutorial offers a model-based approach illustrating how analytical methods from classical electromagnetism can be employed to comprehensively model NLO microscopy in scattering media. In Part I, we quantitatively model focused beam propagation in non-scattering and scattering media from the lens to focal volume. In Part II, we model signal generation, radiation, and far-field detection. Moreover, we detail modeling approaches for major optical microscopy modalities including classical fluorescence, multi-photon fluorescence, second harmonic generation, and coherent anti-Stokes Raman microscopy.

Radiance backscattered by a strongly scattering medium in the high spatial frequency limit.

We study the radiative transfer of a spatially modulated plane wave incident on a half-space composed of a uniformly scattering and absorbing medium. For spatial frequencies that are large compared to the scattering coefficient, we find that first-order scattering governs the leading behavior of the radiance backscattered by the medium. The first-order scattering approximation reveals a specific curve on the backscattered hemisphere where the radiance is concentrated. Along this curve, the radiance assumes a particularly simple expression that is directly proportional to the phase function. These results are inherent to the radiative transfer equation at large spatial frequency and do not have a strong dependence on any particular optical property. Consequently, these results provide the means by which spatial frequency domain imaging technologies can directly measure the phase function of a sample. Numerical simulations using the discrete ordinate method along with the source integration interpolation method validate these theoretical findings.

Corneal transparency and scleral opacity arises from the nanoarchitecture of the constituent collagen fibrils.

While human scleral and corneal tissues possess similar structural morphology of long parallel cylindrical collagen fibrils, their optical characteristics are markedly different. Using pseudospectral time-domain (PSTD) simulations of Maxwell's equations, we model light propagation through realistic representations of scleral and corneal nanoarchitecture and analyze the transmittance and spatial correlation in the near field. Our simulation results provide differing predictions for scleral opacity and corneal transparency across the vacuum ultraviolet to the mid-infrared spectral region in agreement with experimental data. The simulations reveal that the differences in optical transparency between these tissues arise through differences in light scattering emanating from the specific nanoscale arrangement and polydispersity of the constituent collagen fibrils.

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

The Monte Carlo Command Line application (MCCL) is an open-source software package that provides Monte Carlo simulations of radiative transport through heterogeneous turbid media. MCCL is available on GitHub through our virtualphotonics.org website, is actively supported, and carries extensive documentation. Here, we describe the main technical capabilities, the overall software architecture, and the operational details of MCCL.

Comparative analysis of discrete and continuous absorption weighting estimators used in Monte Carlo simulations of radiative transport in turbid media: erratum.

This erratum corrects the relative error plots and references in our paper [J. Opt. Soc. Am. A31, 301 (2014)JOAOD60740-323210.1364/JOSAA.31.000301].

Single-shot interferometric measurement of cavitation bubble dynamics.

We demonstrate an interferometric method to provide direct, single-shot measurements of cavitation bubble dynamics with nanoscale spatial and temporal resolution with results that closely match theoretical predictions. Implementation of this method reduces the need for expensive and complex ultra-high speed camera systems for the measurement of single cavitation events. This method can capture dynamics over large time intervals with sub-nanosecond temporal resolution and spatial precision surpassing the optical diffraction limit. We expect this method to have broad utility for examination of cavitation bubble dynamics, as well as for metrology applications such as optorheological materials characterization. This method provides an accurate approach for precise measurement of cavitation bubble dynamics suitable for metrology applications such as optorheological materials characterization.

Laser cavitation rheology for measurement of elastic moduli and failure strain within hydrogels.

We introduce laser cavitation rheology (LCR) as a minimally-invasive optical method to characterize mechanical properties within the interior of biological and synthetic aqueous soft materials at high strain-rates. We utilized time-resolved photography to measure cavitation bubble dynamics generated by the delivery of focused 500 ps duration laser radiation at λ = 532 nm within fibrin hydrogels at pulse energies of Ep = 12, 18 µJ and within polyethylene glycol (600) diacrylate (PEG (600) DA) hydrogels at Ep = 2, 5, 12 µJ. Elastic moduli and failure strains of fibrin and PEG (600) DA hydrogels were calculated from these measurements by determining parameter values which provide the best fit of the measured data to a theoretical model of cavitation bubble dynamics in a Neo-Hookean viscoelastic medium subject to material failure. We demonstrate the use of this method to retrieve the local, interior elastic modulus of these hydrogels and both the radial and circumferential failure strains.

Optical sampling depth in the spatial frequency domain.

We present a Monte Carlo (MC) method to determine depth-dependent probability distributions of photon visitation and detection for optical reflectance measurements performed in the spatial frequency domain (SFD). These distributions are formed using an MC simulation for radiative transport that utilizes a photon packet weighting procedure consistent with the two-dimensional spatial Fourier transform of the radiative transport equation. This method enables the development of quantitative metrics for SFD optical sampling depth in layered tissue and its dependence on both tissue optical properties and spatial frequency. We validate the computed depth-dependent probability distributions using SFD measurements in a layered phantom system with a highly scattering top layer of variable thickness supported by a highly absorbing base layer. We utilize our method to establish the spatial frequency-dependent optical sampling depth for a number of tissue types and also provide a general tool to determine such depths for tissues of arbitrary optical properties.

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.

Effect of scattering on coherent anti-Stokes Raman scattering (CARS) signals.

We develop a computational framework to examine the factors responsible for scattering-induced distortions of coherent anti-Stokes Raman scattering (CARS) signals in turbid samples. We apply the Huygens-Fresnel wave-based electric field superposition (HF-WEFS) method combined with the radiating dipole approximation to compute the effects of scattering-induced distortions of focal excitation fields on the far-field CARS signal. We analyze the effect of spherical scatterers, placed in the vicinity of the focal volume, on the CARS signal emitted by different objects (2µm diameter solid sphere, 2µm diameter myelin cylinder and 2µm diameter myelin tube). We find that distortions in the CARS signals arise not only from attenuation of the focal field but also from scattering-induced changes in the spatial phase that modifies the angular distribution of the CARS emission. Our simulations further show that CARS signal attenuation can be minimized by using a high numerical aperture condenser. Moreover, unlike the CARS intensity image, CARS images formed by taking the ratio of CARS signals obtained using x- and y-polarized input fields is relatively insensitive to the effects of spherical scatterers. Our computational framework provide a mechanistic approach to characterizing scattering-induced distortions in coherent imaging of turbid media and may inspire bottom-up approaches for adaptive optical methods for image correction.

Development of perturbation Monte Carlo methods for polarized light transport in a discrete particle scattering model.

We present a polarization-sensitive, transport-rigorous perturbation Monte Carlo (pMC) method to model the impact of optical property changes on reflectance measurements within a discrete particle scattering model. The model consists of three log-normally distributed populations of Mie scatterers that approximate biologically relevant cervical tissue properties. Our method provides reflectance estimates for perturbations across wavelength and/or scattering model parameters. We test our pMC model performance by perturbing across number densities and mean particle radii, and compare pMC reflectance estimates with those obtained from conventional Monte Carlo simulations. These tests allow us to explore different factors that control pMC performance and to evaluate the gains in computational efficiency that our pMC method provides.

Multi-scale silica structures for improved HIV-1 Capsid (p24) antigen detection.

Silica (SiO2) micro- and nanostructures fabricated with pre-stressed thermoplastic shrink wrap film have been shown to yield far-field fluorescence signal enhancements over their planar or wrinkled counterparts. The SiO2 structures have previously been used for improved detection of fluorescently labelled proteins and DNA. In this work, we probe the mechanism responsible for the dramatic increases in fluorescence signal intensity. Optical characterization studies attribute the fluorescence signal enhancements to increased surface density and light scattering from the rough SiO2 structures. Using this information, we come up with a theoretical approximation for the enhancement factor based off the scattering effects alone. We show that increased deposition thickness of SiO2 yields improved fluorescence signal enhancements, with an optimal SiO2 thin layer achieved at 20 nm. Finally, we show that the SiO2 substrates serve as a suitable platform for disease diagnostics, and show improved limits of detection (LOD) for the human immunodeficiency virus type 1 (HIV-1) p24 antigen.

Differential pathlength factor informs evoked stimulus response in a mouse model of Alzheimer's disease.

Baseline optical properties are typically assumed in calculating the differential pathlength factor (DPF) of mouse brains, a value used in the modified Beer-Lambert law to characterize an evoked stimulus response. We used spatial frequency domain imaging to measure in vivo baseline optical properties in 20-month-old control ([Formula: see text]) and triple transgenic APP/PS1/tau (3xTg-AD) ([Formula: see text]) mouse brains. Average [Formula: see text] for control and 3xTg-AD mice was [Formula: see text] and [Formula: see text], respectively, at 460 nm; and [Formula: see text] and [Formula: see text], respectively, at 530 nm. Average [Formula: see text] for control and 3xTg-AD mice was [Formula: see text] and [Formula: see text], respectively, at 460 nm; and [Formula: see text] and [Formula: see text], respectively, at 530 nm. The calculated DPF for control and 3xTg-AD mice was [Formula: see text] and [Formula: see text] OD mm, respectively, at 460 nm; and [Formula: see text] and [Formula: see text] OD mm, respectively, at 530 nm. In hindpaw stimulation experiments, the hemodynamic increase in brain tissue concentration of oxyhemoglobin was threefold larger and two times longer in the control mice compared to 3xTg-AD mice. Furthermore, the washout of deoxyhemoglobin from increased brain perfusion was seven times larger in controls compared to 3xTg-AD mice ([Formula: see text]).

Low-density plasma formation in aqueous biological media using sub-nanosecond laser pulses.

We demonstrate the formation of low- and high-density plasmas in aqueous media using sub-nanosecond laser pulses delivered at low numerical aperture (NA = 0.25). We observe two distinct regimes of plasma formation in deionized water, phosphate buffered saline, Minimum Essential Medium (MEM), and MEM supplemented with phenol red. Optical breakdown is first initiated in a low-energy regime and characterized by bubble formation without plasma luminescence with threshold pulse energies in the range of Ep ≈ 4-5 µJ, depending on media formulation. The onset of this regime occurs over a very narrow interval of pulse energies and produces small bubbles (Rmax = 2-20 µm) due to a tiny conversion (η < 0.01%) of laser energy to bubble energy EB. The lack of visible plasma luminescence, sharp energy onset, and low bubble energy conversion are all hallmarks of low-density plasma (LDP) formation. At higher pulse energies (Ep = 11-20 µJ), the process transitions to a second regime characterized by plasma luminescence and large bubble formation. Bubbles formed in this regime are 1-2 orders of magnitude larger in size [Formula: see text] due to a roughly two-order-of-magnitude increase in bubble energy conversion (η ≳ 3%). These characteristics are consistent with high-density plasma formation produced by avalanche ionization and thermal runaway. Additionally, we show that supplementation of MEM with fetal bovine serum (FBS) limits optical breakdown to this high-energy regime. The ability to produce LDPs using sub-nanosecond pulses focused at low NA in a variety of cell culture media formulations without FBS can provide for cellular manipulation at high throughput with precision approaching that of femtosecond pulses delivered at high NA.

High-throughput optical screening of cellular mechanotransduction.

We introduce an optical platform for rapid, high-throughput screening of exogenous molecules that affect cellular mechanotransduction. Our method initiates mechanotransduction in adherent cells using single laser-microbeam generated micro-cavitation bubbles (µCBs) without requiring flow chambers or microfluidics. These µCBs expose adherent cells to a microTsunami, a transient microscale burst of hydrodynamic shear stress, which stimulates cells over areas approaching 1mm2. We demonstrate microTsunami-initiated mechanosignalling in primary human endothelial cells. This observed signalling is consistent with G-protein-coupled receptor stimulation resulting in Ca2+ release by the endoplasmic reticulum. Moreover, we demonstrate the dose-dependent modulation of microTsunami-induced Ca2+ signalling by introducing a known inhibitor to this pathway. The imaging of Ca2+ signalling, and its modulation by exogenous molecules, demonstrates the capacity to initiate and assess cellular mechanosignalling in real-time. We utilize this capability to screen the effects of a set of small molecules on cellular mechanotransduction in 96-well plates using standard imaging cytometry.

Rapid computation of the amplitude and phase of tightly focused optical fields distorted by scattering particles.

We develop an efficient method for accurately calculating the electric field of tightly focused laser beams in the presence of specific configurations of microscopic scatterers. This Huygens-Fresnel wave-based electric field superposition (HF-WEFS) method computes the amplitude and phase of the scattered electric field in excellent agreement with finite difference time-domain (FDTD) solutions of Maxwell's equations. Our HF-WEFS implementation is 2-4 orders of magnitude faster than the FDTD method and enables systematic investigations of the effects of scatterer size and configuration on the focal field. We demonstrate the power of the new HF-WEFS approach by mapping several metrics of focal field distortion as a function of scatterer position. This analysis shows that the maximum focal field distortion occurs for single scatterers placed below the focal plane with an offset from the optical axis. The HF-WEFS method represents an important first step toward the development of a computational model of laser-scanning microscopy of thick cellular/tissue specimens.

Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media.

We present a coupled forward-adjoint Monte Carlo (cFAMC) method to determine the spatially resolved sensitivity distributions produced by optical interrogation of three-dimensional (3-D) tissue volumes. We develop a general computational framework that computes the spatial and angular distributions of the forward-adjoint light fields to provide accurate computations in mesoscopic tissue volumes. We provide full computational details of the cFAMC method and provide results for low- and high-scattering tissues probed using a single pair of optical fibers. We examine the effects of source-detector separation and orientation on the sensitivity distributions and consider how the degree of angular discretization used in the 3-D tissue model impacts the accuracy of the resulting absorption sensitivity profiles. We discuss the value of such computations for optical imaging and the design of optical measurements.

Comparative analysis of discrete and continuous absorption weighting estimators used in Monte Carlo simulations of radiative transport in turbid media.

We examine the relative error of Monte Carlo simulations of radiative transport that employ two commonly used estimators that account for absorption differently, either discretely, at interaction points, or continuously, between interaction points. We provide a rigorous derivation of these discrete and continuous absorption weighting estimators within a stochastic model that we show to be equivalent to an analytic model, based on the radiative transport equation (RTE). We establish that both absorption weighting estimators are unbiased and, therefore, converge to the solution of the RTE. An analysis of spatially resolved reflectance predictions provided by these two estimators reveals no advantage to either in cases of highly scattering and highly anisotropic media. However, for moderate to highly absorbing media or isotropically scattering media, the discrete estimator provides smaller errors at proximal source locations while the continuous estimator provides smaller errors at distal locations. The origin of these differing variance characteristics can be understood through examination of the distribution of exiting photon weights.