A multi-center study has been set up to accurately characterize the optical properties of diffusive liquid phantoms based on Intralipid and India ink at near-infrared (NIR) wavelengths. Nine research laboratories from six countries adopting different measurement techniques, instrumental set-ups, and data analysis methods determined at their best the optical properties and relative uncertainties of diffusive dilutions prepared with common samples of the two compounds. By exploiting a suitable statistical model, comprehensive reference values at three NIR wavelengths for the intrinsic absorption coefficient of India ink and the intrinsic reduced scattering coefficient of Intralipid-20% were determined with an uncertainty of about 2% or better, depending on the wavelength considered, and 1%, respectively. Even if in this study we focused on particular batches of India ink and Intralipid, the reference values determined here represent a solid and useful starting point for preparing diffusive liquid phantoms with accurately defined optical properties. Furthermore, due to the ready availability, low cost, long-term stability and batch-to-batch reproducibility of these compounds, they provide a unique fundamental tool for the calibration and performance assessment of diffuse optical spectroscopy instrumentation intended to be used in laboratory or clinical environment. Finally, the collaborative work presented here demonstrates that the accuracy level attained in this work for optical properties of diffusive phantoms is reliable.
A liquid phantom for investigating light propagation through layered diffusive media is described. The diffusive medium is an aqueous suspension of calibrated scatterers and absorbers. A thin membrane separates layers with different optical properties. Experiments showed that a material with scattering properties should be used for the membrane to avoid the perturbation due to the guided propagation that occurs through a transparent layer. Examples of measurements on a three-layered medium are reported both in the cw and in the time domain.
By use of the solution of the diffusion equation for cylindrical and spherical geometry, two fitting procedures for retrieval of the optical properties from time-resolved measurements have been implemented. The fitting procedures are based on the Levenberg-Marquardt algorithm, in which the fitting parameters are the absorption coefficient, the reduced scattering coefficient, and an amplitude factor. Monte Carlo data generated for cylindrical and spherical geometry were fitted by these fitting procedures, and the retrieved optical properties were compared with those obtained from the inversion procedure with a mismatched geometry of a semi-infinite medium. The effects of refractive-index mismatch and of different boundary conditions of the diffusion equation were also studied, together with the effects of several sources of error that are typically found in time-resolved measurements. The advantages and drawbacks of these fitting procedures, including many details in several situations of interest in the field of tissue optics, are discussed. The results also offer a guideline to understanding the effects of mismatching in curved geometry as functions of source-detector distance and radii of cylinders or spheres.
The accuracy of the perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration has been investigated by comparisons with experimental and numerical results. Comparisons for scattering inhomogeneities showed that the model gives satisfactory results both for the intensity and for the temporal profile of the perturbation over a large range of values for the scattering properties of the defect. As for absorbing inhomogeneities, the model provides an excellent description for the temporal profile, but the results for the intensity are accurate only when the perturbation is small. For absorbing inhomogeneities an empirical model that has a significantly more extended application range has been proposed. The model is based on an expression for the time-resolved mean path length that detected photons have followed inside the inhomogeneity. The application range of the proposed model covers the values expected for the optical properties and for the volumes of inhomogeneities of practical interest for optical mammography.
The accuracy of results obtained from the diffusion equation (DE) has been investigated for the case of an isotropic point source in a homogeneous, weakly absorbing, infinite medium. The results from the DE have been compared both with numerical solutions of the radiative transfer equation obtained with Monte Carlo (MC) simulations and with cw experimental results. Comparisons showed that for the cw fluence rate, discrepancies are of the same order as statistical fluctuations on MC results (within 1%) when the distance r from the source is > 2/mu(s)', (mu(s)' is the reduced scattering coefficient). For these values of r, discrepancies for the time-resolved fluence rate are of the same order of statistical fluctuations (within 5%) when the time of flight is t > 4t0 with to time of flight for unscattered photons. For shorter times the DE overestimates the fluence discrepancies are larger for larger values of the asymmetry factor. As to the specific intensity, for small values of r the MC results are more forward peaked than expected from the DE, and the forward peak is stronger for photons arriving at short times. We assumed r > 2/mu(s)' and t > 4t0 for the domain of validity of the DE and we determined the requirements for which the simplifying assumptions necessary to obtain the DE, expressed by two inequalities, are fulfilled. Comparisons with cw experimental results showed a good agreement with MC results both at high and at small values of r mu(s)', while the comparison with the DE showed significant discrepancies for small values of r mu(s)'. Using MC results we also investigated the error made on the optical properties of the medium when they are retrieved using the solution of the DE. To obtain accuracy better than 1% from fitting procedures on time-resolved fluence rate data it is necessary to disregard photons with time of flight < 4t0. Also from cw data it is possible to retrieve the optical properties with good accuracy: by using the added absorber technique discrepancies are < 1%, both on mu(s)' and on mu(a), if the absorption coefficient is small (mu(a)/mu(s)' < 0.005).
Photons , Biophysical Phenomena , Biophysics , Models, Theoretical , Monte Carlo Method , Optics and Photonics , Scattering, Radiation
A method for measuring the diffusion coefficient of homogeneous and layered media, based on multidistance measurements of time-resolved reflectance, is proposed. The diffusion coefficient is retrieved from the logarithm between two measurements of reflectance at two different distances. The proposed procedure is simpler than others usually employed and also provides a reliable criterion for retrieval of information on the layered structure of a diffusive medium.
The method of images is employed to insert the effects of the presence of a single lateral boundary on contrast functions previously derived for an infinite slab using a random walk model of photon transport. The predictions of the model for zero and extrapolated boundary conditions are compared with Monte Carlo (MC) simulations and with experimental results obtained using a homogenous phantom with tissue-like optical properties. As expected, the extrapolated boundary condition yields better agreement between the theoretical predictions and results obtained from MC and experiments. This indicates that the random walk model has potential as a forward model in iterative imaging schemes developed for optical tomography.
Optics and Photonics , Biophysical Phenomena , Biophysics , Humans , Image Processing, Computer-Assisted , Models, Theoretical , Monte Carlo Method , Photons , Scattering, Radiation , Spectroscopy, Near-Infrared
By using the diffusion approximation of the radiative transfer equation and the partial-current boundary condition, an analytical expression for the angular dependence of the specific intensity emerging from a diffusive medium has been obtained. The analytical expression for the angular distribution has been validated by comparisons with the results of Monte Carlo simulations. By using the diffusion equation and the extrapolated boundary condition, an heuristic analytical expression for the diffuse time-resolved reflectance has also been obtained by assuming that the photon flux is simply proportional to the fluence rate. For the case of the semi-infinite medium, comparisons with Monte Carlo results are presented and time-resolved reflectance data are fitted with the simple fluence rate formula. The results obtained show that the simple expression correctly describes the time-resolved reflectance giving an error in the retrieved optical parameters smaller than that of other commonly used expressions.
Light , Biophysical Phenomena , Biophysics , Models, Theoretical , Monte Carlo Method , Nonlinear Dynamics , Optics and Photonics , Photons
A method of measuring the mean time of flight, ?t(i)? , spent by photons inside a generic volume element of a highly diffusing medium is presented. The method comes from a general property of the radiative transfer equation and is based on relative measurements of cw attenuation that correspond to small variations of the absorption coefficient inside the volume element. By use of a liquid phantom and small gels with known optical properties it was possible to measure ?t(i)? with good accuracy, even when it was only a few picoseconds long. The results were in good agreement with Monte Carlo results.
A Monte Carlo procedure has been developed to study photon migration through highly scattering nonhomogeneous media. When two scaling relationships are used, the temporal response when scattering or absorbing inhomogeneities are introduced can be evaluated in a short time from the results of only one simulation carried out for the homogeneous medium. Examples of applications to the imaging of defects embedded into a diffusing slab, a model usually used for optical mammography, are given. Comparisons with experimental results show the correctness of the results obtained.
The diffusion approximation of the radiative transfer equation is a model used widely to describe photon migration in highly diffusing media and is an important matter in biological tissue optics. An analysis of the time-dependent diffusion equation together with its solutions for the slab geometry and for a semi-infinite diffusing medium are reported. These solutions, presented for both the time-dependent and the continuous wave source, account for the refractive index mismatch between the turbid medium and the surrounding medium. The results have been compared with those obtained when different boundary conditions were assumed. The comparison has shown that the effect of the refractive index mismatch cannot be disregarded. This effect is particularly important for the transmittance. The discussion of results also provides an analysis of the role of the absorption coefficient in the expression of the diffusion coefficient.
In our companion paper we presented a model to describe photon migration through a diffusing slab. The model, developed for a homogeneous slab, is based on the diffusion approximation and is able to take into account reflection at the boundaries resulting from the refractive index mismatch. In this paper the predictions of the model are compared with solutions of the radiative transfer equation obtained by Monte Carlo simulations in order to determine the applicability limits of the approximated theory in different physical conditions. A fitting procedure, carried out with the optical properties as fitting parameters, is used to check the application of the model to the inverse problem. The results show that significant errors can be made if the effect of the refractive index mismatch is not properly taken into account. Errors are more important when measurements of transmittance are used. The effects of using a receiver with a limited angular field of view and the angular distribution of the radiation that emerges from the slab have also been investigated.
We investigated the dependence of the diffusion coefficient on the absorption coefficient by studying the propagation of light emitted by an isotropic source in an infinitely extended medium. Comparisons with both experimental and numerical results showed that the diffusion equation gives a better description of photon migration when the diffusion coefficient is assumed to be independent of absorption.
The results of a study on imaging that is based on the absorption method are presented. This method is based on attenuation measurements carried out in the presence of a sufficiently high absorption coefficient by the use of a continuous-wave source. The benefit of absorption on image quality comes from the strong attenuation of photons traveling along long trajectories. When the absorption coefficient is increased, the received energy decreases, but the mean path length of received photons decreases. The effect of increasing the absorption coefficient is similar to that of decreasing the gating time when the time-gating technique is used. Experimental results showed that the spatial resolution obtained with the absorption technique is similar to that obtained with the time-gating technique.
The results of lidar measurements on laboratory-scaled cloud models are presented. The lidar system was based on a picosecond laser source and a streak camera. The cloud was simulated by a homogeneous aqueous suspension of calibrated microspheres. Measurements were repeated for different concentrations of diffusers and for different values of the receiver angular field of view. The geometric situation was similar to one of an actual lidar sounding a 300-m-thick cloud at a distance of 1200 or 7800 m. The results show how the effect of multiple scattering depends on the extinction coefficient of the sounded medium and on the geometric parameters. The depolarization introduced by multiple scattering was also investigated. Measurements were carried out in well-controlled conditions. The results can thus be useful to validate the accuracy of numerical or analytical procedures that have been developed to study multiple-scattering contribution in lidar returns.
Numerical simulations based on a semianalytic Monte Carlo procedure have been developed to investigate the feasibility of a time-gated transillumination imaging system that could be useful for breast cancer screening. Numerical results showed that attenuation of earlier received photons strongly increases when the gating time decreases and strongly depends on single-scattering properties of the medium. For gating times shorter than ~ 5 ps, attenuation of scattered received energy approaches the exponential attenuation expected for unscattered photons. For typical optical properties of a healthy breast the use of gating times shorter than 100 ps seems to be questionable because of the low level of received energy. The image resolution expected with gating times longer than lOOps is of the order of 10mm. A comparison with predictions or the diffusion equation showed the inadequacy of this theory to describe the dependence of earlier photons on the single-scattering properties of the medium. However, thepredictions of the diffusion equation are within the range of values obtained from numerical simulations for the different scattering properties investigated.
The experimental results of laboratory simulations of lidar returns from clouds are presented. Measurements were carried out on laboratory-scaled cloud models by using a picosecond laser and a streak-camera system. The turbid structures simulating clouds were suspensions of polystyrene spheres in water. The geometrical situation was similar to that of an actual lidar sounding a cloud 1000 m distant and with a thickness of 300 m. Measurements were repeated for different concentrations and different sizes of spheres. The results show how the effect of multiple scattering depends on the scattering coefficient and on the phase function of the diffusers. The depolarization introduced by multiple scattering was also investigated. The results were also compared with numerical results obtained by Monte Carlo simulations. Substantially good agreement between numerical and experimental results was found. The measurements showed the adequacy of modern electro-optical systems to study the features of multiple-scattering effects on lidar echoes from atmosphere or ocean by means of experiments on well-controlled laboratory-scaled models. This adequacy provides the possibility of studying the influence of different effects in the laboratory in well-controlled situations.
Using a Monte Carlo method, we investigate the effect of a turbid medium on image transmission by means of the modulation transfer function approach. We present results that refer to a medium that consists of a random distribution of water spherical particles in air. We analyze the effect of geometric conditions (medium width and position) and source characteristics (Lambertian, beam emission). We present results for small spheres (Rayleigh scattering) and spheres (1.0-microm diameter) that are not small in comparison with the wavelength lambda = 0.6328 microm. Numerical data show a large modulation transfer function dependence on the source emission aperture and a substantial independence of the medium width for a fixed value of the optical depth. In accordance with reciprocity principles, we test an inverse scheme of Monte Carlo calculation, the advantage of this scheme being a substantial reduction in calculation time.
