Radiative transport in the delta-P1 approximation for semi-infinite turbid media

Multi-frequency spatial frequency domain imaging: a depth-resolved optical scattering model to isolate scattering contrast in thin layers of skin

Significance

Current methods for wound healing assessment rely on visual inspection, which gives qualitative information. Optical methods allow for quantitative non-invasive measurements of optical properties relevant to wound healing.

Aim

Spatial frequency domain imaging (SFDI) measures the absorption and reduced scattering coefficients of tissue. Typically, SFDI assumes homogeneous tissue; however, layered structures are present in skin. We evaluate a multi-frequency approach to process SFDI data that estimates depth-specific scattering over differing penetration depths.

Approach

Multi-layer phantoms were manufactured to mimic wound healing scattering contrast in depth. An SFDI device imaged these phantoms and data were processed according to our multi-frequency approach. The depth sensitive data were then compared with a two-layer scattering model based on light fluence.

Results

The measured scattering from the phantoms changed with spatial frequency as our two-layer model predicted. The performance of two δ - P 1 models solutions for SFDI was consistently better than the standard diffusion approximation.

Conclusions

We presented an approach to process SFDI data that returns depth-resolved scattering contrast. This method allows for the implementation of layered optical models that more accurately represent physiologic parameters in thin tissue structures as in wound healing.

Wide-field illumination diffuse optical tomography within a framework of single-pixel time-domain spatial frequency domain imaging

We present a wide-field illumination time-domain (TD) diffusion optical tomography (DOT) for three-dimensional (3-D) reconstruction within a shallow region under the illuminated surface of the turbid medium. The methodological foundation is laid on the single-pixel spatial frequency domain (SFD) imaging that facilitates the adoption of the well-established time-correlated single-photon counting (TCSPC)-based TD detection and generalized pulse spectrum techniques (GPST)-based reconstruction. To ameliorate the defects of the conventional diffusion equation (DE) in the forward modeling of TD-SFD-DOT, mainly the low accuracy in the near-field region and in profiling early-photon migration, we propose a modified model employing the time-dependent δ-P1 approximation and verify its improved accuracy in comparison with both the Monte Carlo and DE-based ones. For a simplified inversion process, a modified GPST approach is extended to TD-SFD-DOT that enables the effective separation of the absorption and scattering coefficients using a steady-state equivalent strategy. Furthermore, we set up a single-pixel TD-SFD-DOT system that employs the TCSPC-based TD detection in the SFD imaging framework. For assessments of the reconstruction approach and the system performance, phantom experiments are performed for a series of scenarios. The results show the effectiveness of the proposed methodology for rapid 3-D reconstruction of the absorption and scattering coefficients within a depth range of about 5 mean free pathlengths.

Solutions for single-scattered radiance in the semi-infinite medium based on radiative transport theory

In this paper, some explicit analytical solutions for single-scattered radiance in a half-space medium under consideration of a reflecting boundary are derived. We consider both a unidirectional beam source as well as an isotropic point source. In addition to direct applications within optical tomography and computer graphics, the obtained solutions are also needed when solving the radiative transport equation after the separation of the unscattered and single-scattered contribution. Comparisons between the derived analytical solutions and the Monte Carlo method display excellent agreement.

Evaluation of Fluence Correction Algorithms in Multispectral Photoacoustic Imaging

Multispectral photoacoustic imaging (MPAI) is a promising emerging diagnostic technology, but fluence artifacts can degrade device performance. Our goal was to develop well-validated phantom-based test methods for evaluating and comparing MPAI fluence correction algorithms, including a heuristic diffusion approximation, Monte Carlo simulations, and an algorithm we developed based on novel application of the diffusion dipole model (DDM). Phantoms simulated a range of breast-mimicking optical properties and contained channels filled with chromophore solutions (ink, hemoglobin, or copper sulfate) or connected to a previously developed blood flow circuit providing tunable oxygen saturation (SO₂). The DDM algorithm achieved similar spectral recovery and SO₂ measurement accuracy to Monte Carlo-based corrections with lower computational cost, potentially providing an accurate, real-time correction approach. Algorithms were sensitive to optical property uncertainty, but error was minimized by matching phantom albedo. The developed test methods may provide a foundation for standardized assessment of MPAI fluence correction algorithm performance.

Analytical model for diffuse reflectance in single fiber reflectance spectroscopy

Cancer progression leads to changing scattering properties of affected tissues. Single fiber reflectance (SFR) spectroscopy detects these changes at small spatial scales, making it a promising tool for early in situ detection. Despite its simplicity and versatility, SFR signal modeling is hugely complicated so that, presently, only approximate models exist. We use a classic approach from geometrical probability to derive accurate analytical expressions for diffuse reflectance in SFR that shows a strong improvement over existing models. We consider the case of limited collection efficiency and the presence of absorption. A Monte Carlo light transport study demonstrates that we adequately describe the contribution of diffuse reflectance to the SFR signal. Additional steps are required to include semi-ballistic, non-diffuse reflectance also present in the SFR measurement.

P3 solution for the total steady-state and time-resolved reflectance and transmittance from a turbid slab

In this paper, we derive some explicit analytical solutions to the P₃ equations for the slab geometry that is illuminated by a collimated plane source. The resulting expressions for the total reflectance and transmittance are compared with the corresponding transport theory solution predicted by the Monte Carlo method. Further, for the special case of a non-absorbing anisotropically scattering slab, simple and accurate expressions in the P₁ approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm's law. In view of the time domain, we present an alternative method to the classical frequency-domain approach avoiding the use of complex numbers.

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.

Simple empirical master-slave dual-source configuration within the diffusion approximation enhances modeling of spatially resolved diffuse reflectance at short-path and with low-scattering from a semi-infinite homogeneous medium: erratum

We correct one typographical error of three equations in Appl. Opt.56, 1447 (2017)APOPAI0003-693510.1364/AO.56.001447.

Accelerating nonlinear reconstruction in laminar optical tomography by use of recursive SVD inversion

Image reconstruction in the most model-based biophotonic imaging modalities essentially poses an ill-posed nonlinear inverse problem, which has been effectively tackled in the diffusion-approximation-satisfied scenarios such as diffuse optical tomography. Nevertheless, a nonlinear implementation in high-resolution laminar optical tomography (LOT) is normally computationally-costly due to its strong dependency on a dense source-detector configuration and a physically-rigorous photon-transport model. To circumvent the adversity, we herein propose a practical nonlinear LOT approach to the absorption reconstruction. The scheme takes advantage of the numerical stability of the singular value decomposition (SVD) for the ill-posed linear inversion, and is accelerated by adopting an explicitly recursive strategy for the time-consuming repeated SVD inversion, which is based on a scaled expression of the sensitivity matrix. Experiments demonstrate that the proposed methodology can perform as well as the traditional nonlinear one, while the computation time of the former is merely 26.27% of the later on average.

Depth-resolved measurements with elliptically polarized reflectance spectroscopy

The ability of elliptical polarized reflectance spectroscopy (EPRS) to detect spectroscopic alterations in tissue mimicking phantoms and in biological tissue in situ is demonstrated. It is shown that there is a linear relationship between light penetration depth and ellipticity. This dependence is used to demonstrate the feasibility of a depth-resolved spectroscopic imaging using EPRS. The advantages and drawbacks of EPRS in evaluation of biological tissue are analyzed and discussed.

Rapid multispectral endoscopic imaging system for near real-time mapping of the mucosa blood supply in the lung

We have developed a fast multispectral endoscopic imaging system that is capable of acquiring images in 18 optimized spectral bands spanning 400-760 nm by combining a customized light source with six triple-band filters and a standard color CCD camera. A method is developed to calibrate the spectral response of the CCD camera. Imaging speed of 15 spectral image cubes/second is achieved. A spectral analysis algorithm based on a linear matrix inversion approach is developed and implemented in a graphics processing unit (GPU) to map the mucosa blood supply in the lung in vivo. Clinical measurements on human lung patients are demonstrated.

Explicit solutions of the radiative transport equation in the P3 approximation

PURPOSE

Explicit solutions of the monoenergetic radiative transport equation in the P3 approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light.

METHODS

A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P3 equations in closed form.

RESULTS

The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P3 approximation is close to the exact solution of the radiative transport equation.

CONCLUSIONS

The authors derived exact analytical solutions of the P3 equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.

Convolution model of the diffuse reflectance for layered tissues

We present an explicit model for the diffuse reflectance due to a collimated beam of light incident normally on layered tissues. This model is derived using the corrected diffusion approximation applied to a layered medium, and it takes the form of a convolution with an explicit kernel and the incident beam profile. This model corrects the standard diffusion approximation over all source-detector separation distances provided the beam is sufficiently wide compared to the scattering mean free path. We validate this model through comparison with Monte Carlo simulations. Then we use this model to estimate the optical properties of an epithelial layer from Monte Carlo simulation data. Using measurements at small source-detector separations and this model, we are able to estimate the absorption coefficient, scattering coefficient, and anisotropy factor of epithelial tissues efficiently with reasonable accuracy.

Estimating nanoparticle optical absorption with magnetic resonance temperature imaging and bioheat transfer simulation

PURPOSE

Optically activated nanoparticle-mediated heating for thermal therapy applications is an area of intense research. The ability to characterise the spatio-temporal heating potential of these particles for use in modelling under various exposure conditions can aid in the exploration of new approaches for therapy as well as more quantitative prospective approaches to treatment planning. The purpose of this research was to investigate an inverse solution to the heat equation using magnetic resonance temperature imaging (MRTI) feedback, for providing optical characterisation of two types of nanoparticles (gold-silica nanoshells and gold nanorods).

METHODS

The optical absorption of homogeneous nanoparticle-agar mixtures was measured during exposure to an 808 nm laser using real-time MRTI. A coupled finite element solution of heat transfer was registered with the data and used to solve the inverse problem. The L2 norm of the difference between the temperature increase in the model and MRTI was minimised using a pattern search algorithm by varying the absorption coefficient of the mixture.

RESULTS

Absorption fractions were within 10% of literature values for similar nanoparticles. Comparison of temporal and spatial profiles demonstrated good qualitative agreement between the model and the MRTI. The weighted root mean square error was <1.5 σMRTI and the average Dice similarity coefficient for ΔT = 5 °C isotherms was >0.9 over the measured time interval.

CONCLUSION

This research demonstrates the feasibility of using an indirect method for making minimally invasive estimates of nanoparticle absorption that might be expanded to analyse a variety of geometries and particles of interest.

Retrieving the optical parameters of biological tissues using diffuse reflectance spectroscopy and Fourier series expansions. I. theory and application

The determination of optical parameters of biological tissues is essential for the application of optical techniques in the diagnosis and treatment of diseases. Diffuse Reflection Spectroscopy is a widely used technique to analyze the optical characteristics of biological tissues. In this paper we show that by using diffuse reflectance spectra and a new mathematical model we can retrieve the optical parameters by applying an adjustment of the data with nonlinear least squares. In our model we represent the spectra using a Fourier series expansion finding mathematical relations between the polynomial coefficients and the optical parameters. In this first paper we use spectra generated by the Monte Carlo Multilayered Technique to simulate the propagation of photons in turbid media. Using these spectra we determine the behavior of Fourier series coefficients when varying the optical parameters of the medium under study. With this procedure we find mathematical relations between Fourier series coefficients and optical parameters. Finally, the results show that our method can retrieve the optical parameters of biological tissues with accuracy that is adequate for medical applications.

Modeling the diffuse reflectance due to a narrow beam incident on a turbid medium

We present a model for the diffuse reflectance when a continuous beam is incident normally on a half space composed of a uniform scattering and absorbing medium. This model is the result of an asymptotic analysis of the radiative transport equation for strong scattering, weak absorption, and a narrow beam width. Through comparison with the diffuse reflectance computed using the numerical solution of the radiative transport equation, we show that this diffuse reflectance model gives results that are accurate for small source--detector separation distances.

Photon diffusion near the point-of-entry in anisotropically scattering turbid media

From astronomy to cell biology, the manner in which light propagates in turbid media has been of central importance for many decades. However, light propagation near the point-of-entry in turbid media has never been analytically described, until now. Here we report a straightforward and accurate method that overcomes this longstanding, unsolved problem in radiative transport. Our theory properly treats anisotropic photon scattering events and takes the specific form of the phase function into account. As a result, our method correctly predicts the spatially dependent diffuse reflectance of light near the point-of-entry for any arbitrary phase function. We demonstrate that the theory is in excellent agreement with both experimental results and Monte Carlo simulations for several commonly used phase functions.

Measurement of optical scattering properties with low-coherence enhanced backscattering spectroscopy

Low-coherence enhanced backscattering (LEBS) is a depth selective technique that allows noninvasive characterization of turbid media such as biological tissue. LEBS provides a spectral measurement of the tissue reflectance distribution as a function of distance between incident and reflected ray pairs through the use of partial spatial coherence broadband illumination. We present LEBS as a new depth-selective technique to measure optical properties of tissue in situ. Because LEBS enables measurements of reflectance due to initial scattering events, LEBS is sensitive to the shape of the phase function in addition to the reduced scattering coefficient (μ(s) (*)). We introduce a simulation of LEBS that implements a two parameter phase function based on the Whittle-Matérn refractive index correlation function model. We show that the LEBS enhancement factor (E) primarily depends on μ(s) (*), the normalized spectral dependence of E (S(n)) depends on one of the two parameters of the phase function that also defines the functional type of the refractive index correlation function (m), and the LEBS peak width depends on both the anisotropy factor (g) and m. Three inverse models for calculating these optical properties are described and the calculations are validated with an experimental measurement from a tissue phantom.

A predictive model of backscattering at subdiffusion length scales

We provide a methodology for accurately predicting elastic backscattering radial distributions from random media with two simple empirical models. We apply these models to predict the backscattering based on two classes of scattering phase functions: the Henyey-Greenstein phase function and a generalized two parameter phase function that is derived from the Whittle-Matérn correlation function. We demonstrate that the model has excellent agreement over all length scales and has less than 1% error for backscattering at subdiffusion length scales for tissue-relevant optical properties. The presented model is the first available approach for accurately predicting backscattering at length scales significantly smaller than the transport mean free path.

Quantification of mucosa oxygenation using three discrete spectral bands of visible light

Quantification of the mucosa oxygenation levels during Endoscopic imaging provides useful physiological/diagnostic information. In this work a method for non-contact quantification of the oxygen saturation index during Endoscopic imaging using three discrete spectral-band in the blue, the green, and the red parts of the spectrum (RGB bands) has been investigated. The oxygen saturation index (TOI_rgb) was calculated from the three discrete RGB spectral bands using diffusion approximation modeling and least-square analysis. A parametric study performed to identify the optimum band width for each of the three spectral bands. The quantification algorithm was applied to in vivo images of the endobronchial mucosa to calculate (TOI_rgb) from selected areas within the image view. The results were compared to that obtained from the full visible spectral (470-700 nm, 10 nm) measurements. The analysis showed that a band width of at least 20 nm in the blue and the green is required to obtain best results. The results showed that the method provides accurate estimation of the oxygenation levels with about 90% accuracy compared to that obtained using the full spectra. The results suggest the potential of quantifying the oxygen saturation levels from the three narrow RGB spectral bands/images.

Frequency domain measurements on turbid media with strong absorption using the PN approximation

We have applied the frequency-domain technique to measurement of the optical properties of turbid media with strong absorption in the infinite medium limit. Absorption coefficients up to 2.3 cm(-1) for a modified scattering coefficient of 4.3 cm(-1) are studied, which corresponds to a reduced scattering albedo of 0.65. Low phase noise and good phase stability are required for these low albedo conditions. As the degree of absorption increases, the phase changes are reduced while amplitude changes increase. For this reason, correction of amplitude-phase cross talk is essential to achieve accurate measurements with strong absorption. Careful control of stray reflections is required to properly measure amplitude-phase cross talk. Because the diffusion approximation becomes less accurate, measurements are compared to calculations performed in the PN approximation, which is essentially an exact solution for the infinite medium limit. Agreement between theory and experiment is only obtained when correction for amplitude-phase cross talk is performed. These measurements can provide a good method for testing amplitude-phase cross talk.