Photon Diffusion Approximation Research Articles

Tutorial on the Use of the Photon Diffusion Approximation for Fast Calculation of Tissue Optical Properties.

Computer simulations, which are performed at a single wavelength at a time, have been traditionally used to estimate the optical properties of tissues. The results of these simulations need to be interpolated. For a broadband estimation of tissue optical properties, the use of computer simulations becomes time consuming and computer demanding. When spectral measurements are available for a tissue, the use of the photon diffusion approximation can be done to perform simple and direct calculations to obtain the broadband spectra of some optical properties. The additional estimation of the reduced scattering coefficient at a small number of discrete wavelengths allows to perform further calculations to obtain the spectra of other optical properties. This study used spectral measurements from the heart muscle to explain the calculation pipeline to obtain a complete set of the spectral optical properties and to show its versatility for use with other tissues for various biophotonics applications.

Toward Cloud Tomography from Space Using MISR and MODIS: Locating the “Veiled Core” in Opaque Convective Clouds

Abstract For passive satellite imagers, current retrievals of cloud optical thickness and effective particle size fail for convective clouds with 3D morphology. Indeed, being based on 1D radiative transfer (RT) theory, they work well only for horizontally homogeneous clouds. A promising approach for treating clouds as fully 3D objects is cloud tomography, which has been demonstrated for airborne observations. However, more efficient forward 3D RT solvers are required for cloud tomography from space. Here, we present a path forward by acknowledging that optically thick clouds have “veiled cores” (VCs). Sunlight scattered into and out of this deep region does not contribute significant information about the inner structure of the cloud to the spatially detailed imagery. We investigate the VC location for the MISR and MODIS imagers. While MISR provides multiangle imagery in the visible and near-infrared (IR), MODIS includes channels in the shortwave IR, albeit at a single view angle. This combination will enable future 3D retrievals to disentangle the cloud’s effective particle size and extinction fields. We find that, in practice, the VC is located at an optical distance of ~5, starting from the cloud boundary along the line of sight. For MODIS’s absorbing wavelengths the VC covers a larger volume, starting at smaller optical distances. This concept will not only lead to a reduction in the number of unknowns for the tomographic reconstruction but also significantly increase the speed and efficiency of the 3D RT solver at the heart of the algorithm by applying, say, the photon diffusion approximation inside the VC.

Photon path length distribution in random media from spectral speckle intensity correlations

We show that the spectral speckle intensity correlation (SSIC) technique can be profitably exploited to recover the path length distribution of photons scattered in a random turbid medium. We applied SSIC to the study of Teflon slabs of different thicknesses and were able to recover, via the use of the photon diffusion approximation theory, the characteristic transport mean free path l∗ and absorption length s a of the medium. These results were compared and validated by means of complementary measurements performed on the same samples with standard pulsed laser time of flight techniques.

Transport-based quantitative photoacoustic tomography: simulations and experiments

In this paper we describe a method for quantitative photoacoustic tomography (qPAT) based on the photon radiative transfer equation (RTE) coupled with the Helmholtz photoacoustic wave equation. Considerable simulations and tissue-like phantom experiments are conducted to evaluate transport-based qPAT in comparison with diffusion-based qPAT. In these comparative simulations and experiments, we systematically examine the effects of the ratio of μa/μ′s (absorption/reduced scattering coefficient), the anisotropy factor (g) and the imaging domain size on the transport- and diffusion-based photoacoustic image reconstruction. The results obtained show that transport-based qPAT allows for clearly more accurate recovery of the absolute absorption coefficient images of heterogeneous media over diffusion-based qPAT for all the cases examined and provides considerably improved image quality for cases where the photon diffusion approximation (DA) is invalid.

Quantitative photoacoustic tomography based on the radiative transfer equation

We describe a method for quantitative photoacoustic tomography (PAT) based on the radiative transfer equation (RTE) coupled with the Helmholtz photoacoustic wave equation. This RTE-based quantitative PAT allows for accurate recovery of absolute absorption coefficient images of heterogeneous media and provides significantly improved image reconstruction for the cases where the photon diffusion approximation may fail. The method and associated finite element reconstruction algorithm are validated using a series of tissuelike phantom experiments.

Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy

Near-infrared (NIR) optical properties of turbid media, e.g., tissue, can be accurately quantified noninvasively using methods based on diffuse reflectance or transmittance, such as frequency domain photon migration (FDPM). Factors which govern the accuracy and sensitivity of FDPM-measured optical properties include instrument performance, the light propagation model, and fitting algorithms used to calculate optical properties from measured data. In this article, we characterize instrument, model, and fitting uncertaintics of an FDPM system designed for clinical use and investigate how each of these factors affects the quantification of NIR absorption (μa) and reduced scattering (μs′) parameters in tissue phantoms. The instrument is based on a 500 MHz, multiwavelength platform that sweeps through 201 discrete frequencies in as little as 675 ms. Phase and amplitude of intensity modulated light launched into tissue, i.e., diffuse photon density waves (PDW), are measured with an accuracy of ±0.30° and ±3.5%, while phase and amplitude precision are ±0.025° and ±0.20%, respectively. At this level of instrument uncertainty, simultaneous fitting of frequency-dependent phase and amplitude nonlinear model functions derived from a photon diffusion approximation provides an accurate and robust strategy for determining optical properties from FDPM data, especially for media with high absorption. In an optical property range that is characteristic of most human tissues in the NIR (5×10−3<μa<5×10−2 mm−1, 0.5<μs′<2 mm−1), we theoretically and experimentally demonstrate that the multifrequency, simultaneous-fit approach allows μa and μs′ to be quantified with an accuracy of ±5% and ±3%, respectively. Although exceptionally high levels of precision can be obtained using this approach (<1% of the estimated absorption and scattering values), we show that the absolute accuracy of optical property measurements is highly dependent on specific factors associated with instrument performance, model function relevance, and details of the fitting strategy used to calculate μa and μs′.

Nonparametric characterization of human breast tissue by the Laguerre expansion of the kernels technique applied on propagating femtosecond laser pulses through biopsy samples

Ultrafast laser pulses transmitted through excised human breast tissue have been detected by a streak camera. Experimental data of the temporal spread of the ultrafast pulse during the transmission through the tissue have been analyzed using the Laguerre expansion technique. This method treats the medium of propagation as a “black box system” and using data sets of incident–transmitted pulse, it relates to this system a set of coefficients as well the first-order system kernels. This analysis could present an alternative method of tissue characterization when, due to the limited optical thickness of small biopsy samples, the photon diffusion approximation cannot be used successfully.

Anisotropic scattering of light in random media: incoherent backscattered spotlight

We discuss the anisotropic scattering of unpolarized light in optically dense random media and the flux analysis of an incoherent backscattered spotlight. We present a classic statistical approach based on the photon-diffusion approximation and Monte Carlo simulations to describe the anisotropic propagation of ballistic and long-path photons in a semi-infinite random medium with internal reflections. An imagery technique with high gray-level resolution is used to measure the surface flux density in the incoherent backscattered spotlight. We investigated light scattering from homogeneous suspensions of nonspherical alumina particles in water. We analyzed the particle volume fraction and the particle-size dependence of the surface flux density to determine the transport mean free path and the optical properties of scatterers from scaling laws that account for short-path photons and internal reflections.

Modeling of Ultrashort Light Pulse Propagation in Light Scattering Media

Light pulses have found increasing use in medicine and biology. Biological tissues, however, scatter light very strongly, and the modeling of light pulse propagation in biological tissues has not been established because of scattering. We compare the results of three modeling methods, the photon diffusion approximation, the diffusive wave approximation, and the two-flux method with those of the discrete ordinate method which provides the exact solution of the transient equation of radiative transfer for the case of 1 picosecond pulse incidence on scattering and absorbing slab or semi-infinite media simulating biological tissues. The early behavior of the light pulses predicted by the photon diffusion approximation is quite different from that by the diffusive wave approximation. In the later period both the photon diffusion and diffusive wave approximations are able to give reasonable results, but the two-flux method is not. Because of much shorter computation time, the diffusive wave approximation is advantageous to the photon diffusion approximation.

Angular dependence of enhanced backscattering of light by Brownian dynamics in the diffusive transport theory of photons

The normalized autocorrelation function of time-varying fluctuations of light scattered multiply from a dense disordered medium is theoretically derived within the framework of a photon-diffusion approximation. The interference effect between the scattered lights traveling along the time-forward and the time-reversed paths in the medium is renormalized to a path-length distribution function. The resultant theory enables us to investigate the characteristic dependence of the relaxation time of fluctuations of multiply scattered light on the backscattering angle within the framework of a diffusion approximation. The theory derived here is confirmed fairly well by experiments.

Fiber‐optic diffusing‐wave spectroscopy on concentrated dispersions of large polymer latex spheres

AbstractFiber‐optic diffusing‐wave spectroscopy (FODWS) measurements were carried out on polymer latex dispersions (10 wt%) with large particle diameters (1.07, 3.2 and 5.6 μm). Autocorrelation functions (ACFs) were measured from light multiply scattered between two single‐mode optical fibers immersed in the dispersion. As expected from a photon diffusion approximation the logarithms of the ACFs were linear functions of the fiber separation and the square root of the correlation delay time. A small downward curvature of the curves at short delay times could be attributed to a slight absorption by the samples. From the ACFs a parameter κ was derived which strongly depended on particle size. Reasonable agreement was found between the experimental κ‐values and predictions by a hard‐sphere model. Potential applications of FODWS are the on‐line process control of emulsions, viscous pigment or ceramic dispersions.

Diffusing-wave spectroscopy in concentrated latex dispersions: An investigation using single-mode fibers

It is shown that diffusing-wave spectroscopy (DWS) of multiple light scattering from concentrated dispersions can be performed using immersed single-mode fibers to emit and collect the light. The intensity autocorrelation functions (ACFs) were measured from a 270 nm poly(styrene) latex dispersion (weight fractions 5, 10 and 30%) as a function of the separation between the source and detector fiber faces. Different orientations between the fiber axes (0, 10, 45 and 180°) were studied. As expected from a photon diffusion approximation the logarithms of the ACFs of the multiply scattered light at the larger fiber separations were linear functions of the square root of time. As further predicted by the theory, the slopes of these curves were linear with fiber separation and were independent of fiber orientation. The slopes are proportional to the square root of the particle diffusion coefficient and depend on the static scattering properties of the dispersion.

Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit

Dynamic light scattering is extended to optically thick (opaque) media which exhibit a very high degree of multiple scattering. This new technique, called diffusing-wave spectroscopy (DWS), exploits the diffusive nature of the transport of light in strongly scattering media to relate the temporal fluctuations of the multiply scattered light to the motion of the scatterers. A simple theory of DWS, based on the diffusion approximation for the transport of light, is developed to calculate the temporal electric field autocorrelation functions of the multiply scattered light. Two important scattering geometries are treated : transmission and backscattering. The theory is compared to experimental measurements of Brownian motion of submicron-diameter polystyrene spheres in aqueous suspension. The agreement between theory and experiment is excellent. The limitations of the photon diffusion approximation and the polarization dependence of the autocorrelation functions are discussed for the backscattering measurements. The effects of absorption of light and particle polydispersity are also incorporated into the theory and verified experimentally. It is also shown how DWS can be used to obtain information about the mean size of the particles which scatter light.

Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation.

We find experimentally that in the forward direction the length scale of quasiballistic photons dominates the dynamic intensity-intensity correlation function of light multiply scattered from randomly diffusing spheres. This implies a breakdown of the photon-diffusion approximation, which is characterized by a single length, the transport mean free path. We suggest why this breakdown occurs, introduce phenomenologically a dynamic correlation length as an additional length scale, and with this extension reconcile the theory with experiment.

Diffuse reflectance of whole blood: model for a diverging light beam.

Previous theories describing the optical properties of whole blood have assumed a collimated light source in the derivation of reflectance and transmittance equations. However, many practical instruments which measure blood oxygen rely on noncollimated LED's and optical fibers as light sources. In addition, in the case of blood-perfused cuvettes, the light source is not in direct contact with the blood. Thus, many theories have limited applicability to practical instruments. Therefore, we present here an existing photon diffusion theory that we have generalized to account for the effects of a diverging light source and a thickness of transparent material between the light source and the blood. Comparisons to experimental data are made for both the new equation and the previous one. Better results are obtained with the new equation when applied to a cuvette geometry which is more accurately described by the generalized parameters. Limitations of both the new equation and the photon diffusion approximation in general are discussed.