Optimal estimation reconstruction of the optical properties of a two-layered tissue phantom from time-resolved single-distance measurements.

In this work, we have tested the optimal estimation (OE) algorithm for the reconstruction of the optical properties of a two-layered liquid tissue phantom from time-resolved single-distance measurements. The OE allows a priori information, in particular on the range of variation of fit parameters, to be included. The purpose of the present investigations was to compare the performance of OE with the Levenberg­Marquardt method for a geometry and real experimental conditions typically used to reconstruct the optical properties of biological tissues such as muscle and brain. The absorption coefficient of the layers was varied in a range of values typical for biological tissues. The reconstructions performed demonstrate the substantial improvements achievable with the OE provided a priori information is available. We note the extreme reliability, robustness, and accuracy of the retrieved absorption coefficient of the second layer obtained with the OE that was found for up to six fit parameters, with an error in the retrieved values of less than 10%. A priori information on fit parameters and fixed forward model parameters clearly improves robustness and accuracy of the inversion procedure.

Modified reciprocity relation for the time-dependent diffusion equation.

The classical reciprocity relation of radiative transfer fails for two points placed in regions having different indices of refraction. A modified reciprocity relation that involves the relative refractive index between the two points considered was previously derived for the continuous wave (cw) radiative transfer equation and for the cw diffusion equation (DE) [J. Opt. Soc. Am. A14, 486 (1997)]. In this paper, we extend these findings to the time-dependent DE and we discuss some implications to diffuse optical tomography.

Retrieval procedure for time-resolved near-infrared tissue spectroscopy based on the optimal estimation method.

We propose the use of a retrieval procedure for time-resolved near-infrared tissue spectroscopy based on the 'optimal estimation' method. The aim of this retrieval method is to obtain an improved estimate of the target parameters compared with standard nonlinear least-squares routines, since the inverse problem dedicated to retrieve the optical properties of tissue is ill posed. A priori information on target and forward model parameters is used, so that a larger number of target parameters can be retrieved, and/or a better accuracy and precision can be achieved on the retrieved target parameters. The procedure has been tested on time-resolved simulated experiments generated, using solutions of the diffusion equation and with solutions of the radiative transfer equation reconstructed with Monte Carlo simulations. The results obtained show that, by using a priori information on target parameters, we have a smaller difference between retrieved values and true values, and lower retrieved error bars. Similarly, a more correct estimate of the errors of the forward model parameters improves the retrieval of the target parameters.

Sun-induced leaf fluorescence retrieval in the O2-B atmospheric absorption band.

Sun-induced leaf fluorescence was inferred by using high resolution (0.5 cm(-1)) radiance measurements and simulated spectra of the solar irradiance at the ground level, in the region of the O(2)-B absorption band. The minimization of a cost function was performed in the Fourier transform domain in order to make an accurate fit of the Instrumental Line- Shape that convoluted the simulated spectrum. Second- order polynomials were used to fit the leaf fluorescence and reflectance in the 100-cm(-1)-wide spectral window. The scale and the instrumental conversion factor were also fitted in order to obtain an accuracy that could not be attained by using the radiance measurements alone.

Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model.

A diffusion-based model for photon migration through a three-layer medium is described. The main purpose of this work is to investigate the performance of a diffusion equation (DE)-based forward model for studying photon migration through a diffusive layered medium having a low scattering layer. This geometrical model can be used as a simple model of the adult head. Numerical results are shown for a set of values of the optical properties typical of the adult human head, where scalp and skull are lumped in the first layer while the second and third layer are associated with the cerebrospinal fluid (CSF) and the brain, respectively. Due to the presence of the CSF, which is a relatively clear layer, the diffusion-based model yields an approximate solution of photon migration. Nevertheless, comparisons with MC simulations show that the model can predict the total and the partial mean path length in the different layers with an error less than 20%. In particular, the partial mean path length in the third layer, representative of the brain, is calculated with an error less than 10% if the reduced scattering coefficient of the second layer, representative of the CSF, is assumed 0.25 mm(-1).

Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation.

We characterize the capability of time-resolved reflectance measurements at small source-detector separation (less than 5 mm) to localize small inhomogeneities embedded in an otherwise homogeneous or layered diffusive medium. By considering both absorption and scattering inhomogeneities, we demonstrate the improvement of this approach in terms of contrast and spatial resolution, as compared to more typical set-ups involving larger source-detection separations (few centimeters). Simulations are performed exploiting an analytical perturbation approach to diffusion theory and a four-layer heterogeneous time-resolved Monte Carlo code, considering realistic tissue geometries. Exhaustive investigation in the parameters space is reported.

Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging.

We propose a novel approach to imaging in diffusive media based on time-resolved reflectance measurements at null source-detector separation. This approach yields better spatial resolution and contrast as compared to the classical approach, which typically employs a separation of 20-40 mm. Results are obtained by an analytical perturbation approach to diffusion theory and on Monte Carlo simulations. Practical implementation with state-of-the-art technology and performance of a complementary approach based on the use of small but not null source-detector separation are also discussed.

Perturbation model for light propagation through diffusive layered media.

A fast and novel perturbation approach is proposed to account for the effects of absorbing inhomogeneities on light propagation through layered media. The calculation has been implemented with the Born approximation. Examples of results are reported both for a two- and for a three-layered medium. The method presented has been validated with the results of Monte Carlo simulations. The forward solver presented in this paper can be of significant use in investigating the feasibility of real experiments, e.g. for functional imaging studies.

Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements.

An experimental validation of an inversion procedure for retrieving the optical properties of layered media from multidistance time-resolved reflectance measurements is presented. The results cover a wide range of optical properties, showing excellent effectiveness and reliability of the procedure in reconstructing the optical properties of a two-layered medium. The optical properties of the first layer and the absorption of the second layer could be retrieved with excellent precision, whereas the reduced scattering coefficient of the second layer was reconstructed with a large error. The first layer thickness could be retrieved with an error less than 50%. An example of retrieval of the optical properties of muscle from in vivo measurements during an arterial occlusion is shown.

Effect of the refractive index mismatch on light propagation through diffusive layered media.

The effect of the refractive index mismatch on light propagation through diffusive layers has been investigated. The refractive index mismatch changes the balance of energy inside the medium determining a temporal and spatial redistribution of light. Light penetration through the medium is obstructed (facilitated) by a negative (positive) refractive index step variation. An analytical solution of the time-dependent diffusion equation that accounts for this effect has been obtained. The solution has been validated by comparisons with the results of Monte Carlo simulations. An excellent description of light propagation is given even for a high refractive index mismatch.

Effect of a clear layer at the surface of a diffusive medium on measurements of transmittance and reflectance.

The effect of a clear layer at the surface of a diffusive medium on measurements of reflectance and transmittance has been investigated with Monte Carlo simulations. To quantify the effect of the clear layer Monte Carlo results have been fitted with the solution of the diffusion equation for the homogeneous medium in order to reconstruct the optical properties of the diffusive medium. The results showed that the clear layer has a small effect on measurements of transmittance. On the contrary measurements of reflectance are greatly perturbed and the accurate reconstruction of the optical properties of the diffusive medium becomes almost impossible.

Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measurements.

A procedure for retrieving the optical properties of a two-layered diffusive medium based on an exact analytical solution of the diffusion equation and on relative multidistance time-resolved reflectance measurements is presented. The method overcomes some limitations of previously developed procedures. Five parameters of the medium have been fitted: the absorption and the reduced scattering coefficients of both layers and the thickness of the first layer. The actual values of the parameters are correctly retrieved by the procedure. The inversion procedure does not require an initial guess for the unknown optical properties, but the starting value for the thickness of the first layer needs to be estimated with an error smaller than 50%.

Measurements of optical properties of high-density media.

Measurements of optical properties (scattering coefficient mu(s), absorption coefficient mu(a), reduced scattering coefficient mu(s)', and asymmetry factor g) have been carried out up to a volume particle concentration of rho = 0.227. The results for mu(s) and mu(s)' show significant deviations from the linear dependence on rho as expected when the independent scattering assumption is fulfilled. The asymmetry factor also changed significantly. In contrast, the dependence of mu(a) remained linear even at the largest concentration investigated. The simple linear dependence of absorption on the chromophore concentration expected from the independent scattering assumption is thus applicable also to spectroscopic measurements of dense media. A comparison with an approximate theoretical model based on the Foldy-Twersky equation is also reported. The model provides a good description of the dependence of mu(s) on particle concentration.

Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method.

An exact solution of the time-dependent diffusion equation for the case of a two- and a three-layered finite diffusive medium is proposed. The method is based on the decomposition of the fluence rate in a series of eigenfunctions and upon the solution of the consequent transcendental equation for the eigenvalues obtained from the boundary conditions. Comparisons among the solution of the diffusion equation and the results of Monte Carlo simulations show the correctness of the proposed model.

Method to measure the optical properties of small volumes of diffusive media.

The method consists of measuring the perturbation provoked by a small volume of the diffusive medium on light propagating through a medium of known optical properties. The absorption and the reduced scattering coefficients of the medium are retrieved from multidistance continuous-wave measurements of transmittance. The inversion procedure is based on the solution of the diffusion equation obtained with a perturbative approach. The method has been validated with Monte Carlo results. Examples of experimental results are reported.

Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation.

The depth at which photons penetrate into a diffusive medium before being re-emitted has been investigated with reference to a semi-infinite homogeneous medium illuminated by a pencil beam. By using the diffusion equation analytical expressions have been obtained for the probability that photons penetrate at a certain depth before being detected, and for the mean path length they travel inside each layer of the medium. Expressions have been obtained both for the cw and the time domain, and simple approximate scaling relationships describing the dependence on the scattering properties of the medium have been found. For time-resolved measurements both the probability and the mean path length are expected to be independent of the distance from the light beam at which the detector is placed and of the absorption coefficient of the medium. The penetration depth increases as the time of flight increases. In contrast, for cw measurements both the probability and the mean path length strongly depend on the distance and absorption. The penetration depth increases as the distance increases or absorption decreases. The accuracy of the analytical expressions has been demonstrated by comparisons with cw experimental results. The penetration depth and the mean path length provide useful information, for instance, for measurements of tissue oxygenation and for functional imaging of muscle and brain. In particular, the depth reached by received photons provides overall information on the volume of the tissue actually investigated, while the mean path is strictly related to the sensitivity to local variations of absorption.