Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain

This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.     

A method for determination of the absorption and scattering properties interstitially in turbid media

We have developed a method to quickly determine tissue optical properties (absorption coefficient mu(a) and transport scattering coefficient mu'(s)) by measuring the ratio of light fluence rate to source power along a linear channel at a fixed distance (5 mm) from an isotropic point source. Diffuse light is collected by an isotropic detector whose position is determined by a computer-controlled step motor, with a positioning accuracy of better than 0.1 mm. The system automatically records and plots the light fluence rate per unit source power as a function of position. The result is fitted with a diffusion equation to determine mu(a) and mu'(s). We use an integrating sphere to calibrate each source-detector pair, thus reducing uncertainty of individual calibrations. To test the ability of this algorithm to accurately recover the optical properties of the tissue, we made measurements in tissue simulating phantoms consisting of Liposyn at concentrations of 0.23, 0.53 and 1.14% (mu'(s) = 1.7-9.1 cm(-1)) in the presence of Higgins black India ink at concentrations of 0.002, 0.012 and 0.023% (mu(a) = 0.1-1 cm(-1)). For comparison, the optical properties of each phantom are determined independently using broad-beam illumination. We find that mu(a) and mu'(s) can be determined by this method with a standard (maximum) deviation of 8% (15%) and 18% (32%) for mu(a) and mu'(s), respectively. The current method is effective for samples whose optical properties satisfy the requirement of the diffusion approximation. The error caused by the air cavity introduced by the catheter is small, except when mu(a) is large (mu(a) > 1 cm(-1)). We presented in vivo data measured in human prostate using this method.     

Determination of optical properties in heterogeneous turbid media using a cylindrical diffusing fiber

For interstitial photodynamic therapy (PDT), cylindrical diffusing fibers (CDFs) are often used to deliver light. This study examines the feasibility and accuracy of using CDFs to characterize the absorption (μ(a)) and reduced scattering (μ'(s)) coefficients of heterogeneous turbid media. Measurements were performed in tissue-simulating phantoms with μ(a)?between 0.1 and 1?cm(-1)?and μ'(s) between 3 and 10?cm(-1)?with CDFs 2 to 4?cm in length. Optical properties were determined by fitting the measured light fluence rate profiles at a fixed distance from the CDF axis using a heterogeneous kernel model in which the cylindrical diffusing fiber is treated as a series of point sources. The resulting optical properties were compared with independent measurement using a point source method. In a homogenous medium, we are able to determine the absorption coefficient μ(a)?using a value of μ'(s) determined a priori (uniform fit) or μ'(s) obtained by fitting (variable fit) with standard (maximum) deviations of 6% (18%) and 18% (44%), respectively. However, the CDF method is found to be insensitive to variations in μ'(s), thus requiring a complementary method such as using a point source for determination of μ'(s). The error for determining μ(a)?decreases in very heterogeneous turbid media because of the local absorption extremes. The data acquisition time for obtaining the one-dimensional optical properties distribution is less than 8?s. This method can result in dramatically improved accuracy of light fluence rate calculation for CDFs for prostate PDT in vivo when the same model and geometry is used for forward calculations using the extrapolated tissue optical properties.     

Determination of optical properties of turbid media using pulsed photothermal radiometry.

Pulsed photothermal radiometry (PPTR) measures blackbody radiation emitted by a sample after absorption of an optical pulse. Three techniques for obtaining the absorption coefficient of absorbing-only, semi-infinite samples are examined and shown to give comparable results. An analytic theory for the time dependence of the PPTR signal in semi-infinite scattering and absorbing media has been derived and tested in a series of controlled gel phantoms. This theory, based on the diffusion approximation of the radiative transport equation, is shown to model the time course of the detected signal accurately. Furthermore, when the incident fluence is known, the theory can be used in a non-linear, two-parameter fitting algorithm to determine the absorption and reduced scattering coefficients of a turbid sample with an accuracy of 10-15% for transport albedos ranging from 0.42-0.88.     

Determination of optical scattering properties in turbid media using Mueller matrix imaging

A need exists for the continued development of diagnostic tools and methods capable of distinguishing and characterizing slight differences in the optical properties of tissues. We present a method to estimate the scattering coefficient contribution as a function of particle size in complex mixtures of polystyrene spheres. The experimental method we used is a Mueller matrix imaging approach. The Mueller matrix encodes the polarization-dependent properties of the sample and describes how a given sample will transform an incident light polarization state. A partial least-squares approach is used to form a model around a set of Mueller matrix image-based measurements to accurately predict the individual scattering coefficient contributions in phantoms containing 0.2, 0.5, 1, and 2 microm-diameter polystyrene spheres. The results show individual scattering coefficient contribution errors as low as 0.1585 cm(-1) can be achieved. In addition, it is shown how the scattering type (i.e., Rayleigh and Mie) is encoded within the Mueller matrix. Such methods may eventually lead to the development of improved diagnostic tools capable of characterizing and distinguishing between tissue abnormalities, such as superficial cancerous lesions from their benign counterparts.     

Image reconstruction through turbid media under a transmission-mode microscope

In this paper, image enhancement and reconstruction through a turbid medium by utilizing polarization gating and mathematical image reconstruction methods in a microscopic imaging system are investigated. A Monte Carlo simulation model based on Mie theory and the concept of the effective point spread function (EPSF) is adopted to study image formation under a transmission-mode microscope. The results show that polarization gating methods, and particularly the differential polarization gating method, can be efficient in suppressing highly scattered light, which leads to a significant enhancement of image quality. An image reconstruction method based on the concept of the EPSF is demonstrated to be efficient in further improving image quality.     

Quantitative fluorescence lifetime spectroscopy in turbid media: comparison of theoretical,experimental and computational methods

A Monte Carlo model developed to simulate time-resolved fluorescence propagation in a semi-infinite turbid medium was validated against previously reported theoretical and computational results. Model simulations were compared to experimental measurements of fluorescence spectra and lifetimes on tissue-simulating phantoms for single and dual fibre-optic probe geometries. Experiments and simulations using a single probe revealed that scattering-induced artefacts appeared in fluorescence emission spectra, while fluorescence lifetimes were unchanged. Although fluorescence lifetime measurements are generally more robust to scattering artefacts than are measurements of fluorescence spectra, in the dual-probe geometry scattering-induced changes in apparent lifetime were predicted both from diffusion theory and via Monte Carlo simulation, as well as measured experimentally. In all cases, the recovered apparent lifetime increased with increasing scattering and increasing source-detector separation. Diffusion theory consistently underestimated the magnitude of these increases in apparent lifetime (predicting a maximum increase of approximately 15%), while Monte Carlo simulations and experiment were closely matched (showing increases as large as 30%). These results indicate that quantitative simulations of time-resolved fluorescence propagation in turbid media will be important for accurate recovery of fluorophore lifetimes in biological spectroscopy and imaging applications.     

Optical imaging of turbid media using independent component analysis: theory and simulation

A new imaging approach for 3-D localization and characterization of objects in a turbid medium using independent component analysis (ICA) from information theory is developed and demonstrated using simulated data. This approach uses a multisource and multidetector signal acquisition scheme. ICA of the perturbations in the spatial intensity distribution measured on the medium boundary sorts out the embedded objects. The locations and optical characteristics of the embedded objects are obtained from a Green's function analysis based on any appropriate model for light propagation in the background medium. This approach is shown to locate and characterize absorptive and scattering inhomogeneities within highly scattering medium to a high degree of accuracy. In particular, we show this approach can discriminate between absorptive and scattering inhomogeneities, and can locate and characterize complex inhomogeneities, which are both absorptive and scattering. The influence of noise and uncertainty in background absorption or scattering on the performance of this approach is investigated.     

Angular measurements of light scattered by turbid chiral media using linear Stokes polarimeter

The effects of turbid chiral media on light polarization are studied in different directions around the scattering samples using a refined linear Stokes polarimeter, which simplifies the signal analysis, and allows for the detailed investigations of scattered light. Because no moving parts are involved in a measurement at a specific detection direction, the determination accuracy of polarization states is increased. The results show that light depolarization increases with both turbidity and detection angle for low and moderately turbid samples; however, the angular dependence decreases with increasing turbidity. When the turbidity is increased to approximately 100 cm(-1), the depolarization becomes higher in the forward than in the backward direction. Polarization sensitive Monte Carlo simulations are used to verify some experimental observations. The results also demonstrate that surviving linear polarization fraction and overall intensity are more sensitive to the increase of glucose concentration in backward than in the forward direction in highly turbid media, indicating that backward geometry may be preferable for potential glucose detection in a biomedical context. Comparison measurements with optically inactive glycerol suggest that the refractive index matching effect, and not the chiral nature of the solute, dominates the observed optical rotation engendered by glucose in highly turbid media.     

Error estimation of measuring total interaction coefficients of turbid media using collimated light transmission

The error of measuring the total interaction coefficients of turbid media using collimated light transmission was estimated with an analytical expression, which was verified with accurate Monte Carlo simulations. The expression is based on the Henyey-Greenstein phase function of scattering and the probabilities of non-scattered and singly scattered photons transmitted through a tissue slab with a unit anisotropy factor.     

Experimental spectro-angular mapping of light distribution in turbid media

We present a new approach to the analysis of radiance in turbid media. The approach combines data from spectral, angular and spatial domains in a form of spectro-angular maps. Mapping provides a unique way to visualize details of light distribution in turbid media and allows tracking changes with distance. Information content of experimental spectro-angular maps is verified by a direct comparison with simulated data when an analytical solution of the radiative transfer equation is used. The findings deepen our understanding of the light distribution in a homogenous turbid medium and provide a first step toward applying the spectro-angular mapping as a diagnostic tool for tissue characterization.     

Propagation of polarized light in birefringent turbid media: a Monte Carlo study

A detailed study, based on a Monte Carlo algorithm, of polarized light propagation in birefringent turbid media is presented in this paper. Linear birefringence, which results from the fibrous structures, changes the single scattering matrix and alters the polarization states of photons propagating in biological tissues. Some Mueller matrix elements of light backscattered from birefringent anisotropic turbid media present unusual intensity patterns compared with those for nonbirefringent isotropic turbid media. This result is in good agreement with the analytic results based on the double-scattering model. Degree of polarization, Stokes parameters, and diffuse reflectance as functions of linearly birefringent parameters based on numerical results and theoretical analysis are discussed and compared in an effort to understand the essential physical processes of polarized light propagation in fibrous tissues.     

Path-length-resolved measurements of multiple scattered photons in static and dynamic turbid media using phase-modulated low-coherence interferometry

In optical Doppler measurements, the path length of the light is unknown. To facilitate quantitative measurements, we develop a phase-modulated Mach-Zehnder interferometer with separate fibers for illumination and detection. With this setup, path-length-resolved dynamic light scattering measurements of multiple scattered light in static and dynamic turbid media are performed. Optical path length distributions spanning a range from 0 to 11 mm are measured from the area under the phase modulation peak around the modulation frequency in the power spectrum. A Doppler-broadened phase modulation interference peak is observed that shows an increase in the average Doppler shift with optical path length, independent of absorption. Validation of the estimated path length distributions is done by measuring their deformation for increasing absorption and comparing these observations with predictions based on Lambert-Beer's law.     

State-of-the art of acousto-optic sensing and imaging of turbid media

Acousto-optic (AO) is an emerging hybrid technique for measuring optical contrast in turbid media using coherent light and ultrasound (US). A turbid object is illuminated with a coherent light source leading to speckle formation in the remitted light. With the use of US, a small volume is selected,which is commonly referred to as the "tagging" volume. This volume acts as a source of modulated light, where modulation might involve phase and intensity change. The tagging volume is created by focusing ultrasound for good lateral resolution; the axial resolution is accomplished by making either the US frequency, amplitude, or phase time-dependent. Typical resolutions are in the order of 1 mm. We will concentrate on the progress in the field since 2003. Different schemes will be discussed to detect the modulated photons based on speckle detection, heterodyne detection, photorefractive crystal (PRC) assisted detection, and spectral hole burning (SHB) as well as Fabry-Perot interferometers. The SHB and Fabry-Perot interferometer techniques are insensitive to speckle decorrelation and therefore suitable for in vivo imaging. However, heterodyne and PRC methods also have potential for in vivo measurements. Besides measuring optical properties such as scattering and absorption, AO can be applied in fluorescence and elastography applications.     

Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory

The goal of frequency-domain optical absorption spectroscopy is the non-invasive determination of the absorption coefficient of a specific tissue volume. Since this allows the concentration of endogenous and exogenous chromophores to be calculated, there is considerable potential for clinical application. The technique relies on the measurement of the phase and modulation of light, which is diffusely reflected or transmitted by the tissue when it is illuminated by an intensity-modulated source. A model of light propagation must then be used to deduce the absorption coefficient. For simplicity, it is usual to assume the tissue is either infinite in extent (for transmission measurements) or semi-infinite (for reflectance measurements). The goal of this paper is to examine the errors introduced by these assumptions when measurements are actually performed on finite volumes. Diffusion-theory calculations and experimental measurements were performed for slabs, cylinders and spheres with optical properties characteristic of soft tissues in the near infrared. The error in absorption coefficient is presented as a function of object size as a guideline to when the simple models may be used. For transmission measurements, the error is almost independent of the true absorption coefficient, which allows absolute changes in absorption to be measured accurately. The implications of these errors in absorption coefficient for two clinical problems--quantitation of an exogenous photosensitizer and measurement of haemoglobin oxygenation--are presented and discussed.     

Drug quantification in turbid media by fluorescence imaging combined with light-absorption correction using white Monte Carlo simulations

Accurate quantification of photosensitizers is in many cases a critical issue in photodynamic therapy. As a noninvasive and sensitive tool, fluorescence imaging has attracted particular interest for quantification in pre-clinical research. However, due to the absorption of excitation and emission light by turbid media, such as biological tissue, the detected fluorescence signal does not have a simple and unique dependence on the fluorophore concentration for different tissues, but depends in a complex way on other parameters as well. For this reason, little has been done on drug quantification in vivo by the fluorescence imaging technique. In this paper we present a novel approach to compensate for the light absorption in homogeneous turbid media both for the excitation and emission light, utilizing time-resolved fluorescence white Monte Carlo simulations combined with the Beer-Lambert law. This method shows that the corrected fluorescence intensity is almost proportional to the absolute fluorophore concentration. The results on controllable tissue phantoms and murine tissues are presented and show good correlations between the evaluated fluorescence intensities after the light-absorption correction and absolute fluorophore concentrations. These results suggest that the technique potentially provides the means to quantify the fluorophore concentration from fluorescence images.     

Simulation study of second-harmonic microscopic imaging signals through tissue-like turbid media

We establish, for the first time, a simulation model for dealing with the second-harmonic signals under a microscope through a tissue-like turbid medium, based on the Monte Carlo method. With this model, the angle-resolved distribution and the signal level eta of second-harmonic light through a slab of the turbid medium are demonstrated and the effects of the thickness (d) of the turbid medium, the numerical aperture (NA) of the objective as well as the size (rho) of the scatterers forming the turbid medium are explored. Simulation results reveal that the use of a small objective NA results in a narrow angle distribution but strong second-harmonic signals. A turbid medium consisting of large scattering particles has a strong influence on the angle distribution and the signal level eta, which results in a low penetration limit for second-harmonic signals made up of ballistic photons. It is approximately 30 microm in our situation.     

Monte-Carlo-based model for the extraction of intrinsic fluorescence from turbid media

A Monte-Carlo-based model of fluorescence is developed that is capable of extracting the intrinsic fluorescence properties of tissue, which are independent of the absorption and scattering properties of tissue. This model is flexible in its applicability to different illumination-collection geometries and is also valid for a wide range of optical properties, representative of tissue in the UV-visible spectrum. This is potentially useful in a variety of biomedical applications, including cancer diagnostics and monitoring the physiological response to therapy. The model is validated using phantoms composed of hemoglobin (absorber), polystyrene spheres (scatterer), and furan-2 (fluorophore). It is found that this model is able to retrieve the intrinsic fluorescence spectra of the tissue phantoms and recover the intrinsic fluorescence intensity of furan within the phantoms to within a mean error of less than 10%.     

Extraction of depth-dependent signals from time-resolved reflectance in layered turbid media

We try a new approach with near-IR time-resolved spectroscopy, to separate optical signals originated in the upper layer from those in the lower layer and to selectively determine the absorption coefficient (mu(a)) of each layer in a two-layered turbid medium. The difference curve in the temporal profiles of light attenuation between a target and a reference medium is divided into segments along the time axis, and a slope of each segment is calculated to determine the depth-dependent mu(a). The depth-dependent mu(a) values are estimated under various conditions in which mu(a) and the reduced scattering coefficient (mu(s)') of each layer are changed with a Monte Carlo simulation and in phantom experiments. Temporal variation of them represents the difference in mu(a) between two layers when mu(s)' of a reference is the same as that of the upper layer of the target. The discrepancies between calculated mu(a) and the real mu(a) depend on the ratio of the real mu(a) of the upper layer to that of the lower layer, and our approach enables us to estimate the ratio of mu(a) between the two layers. These results suggest the potential that mu(a) of the lower layer can be determined by our procedure.