Quantification by random walk of the optical parameters of nonlocalized abnormalities embedded within tissuelike phantoms

Efficient algorithm to calculate the optical properties of breast tumors by high-order perturbation theory

An efficient algorithm to obtain the solutions for n-th order terms of perturbation expansions in absorption, scattering, and cross-coupling for light propagating in human tissue is presented. The proposed solution is free of any approximations and makes possible fast and efficient estimates of mammographic, optical tomographic, and fluorescent images, applying a perturbation order of 30 and more. The presented analysis sets the general limits for the applicability of the perturbation approach as a function of tumor size and optical properties of the human tissue. The convergence tests of the efficient calculations for large absorbing objects show excellent agreement with the reference data from finite element method calculations. The applicability of the theory is demonstrated in experiments on breast-like phantoms with high absorbing and low-scattering lesions.

Trapping of diffusing particles by small absorbers localized in a spherical region

We study trapping of particles diffusing in a spherical cavity with an absorbing wall containing small static spherical absorbers localized in a spherical region in the center of the cavity. The focus is on the competition between the absorbers and the cavity wall for diffusing particles. Assuming that the absorbers and, initially, the particles are uniformly distributed in the central region, we derive an expression for the particle trapping probability by the cavity wall. The expression gives this probability as a function of two dimensionless parameters: the transparency parameter, characterizing the efficiency of the particle trapping by the absorbers, and the ratio of the absorber-containing region radius to that of the cavity. This work is a generalization of a recent study by Krapivsky and Redner [J. Chem. Phys. 147, 214903 (2017)] who considered the case where the absorber-containing region occupies the entire cavity. The expression for the particle trapping probability is derived in the framework of a steady-state approach which, in our opinion, is much simpler than the time-dependent approach used in the above-mentioned study.

Closed-form solution of the steady-state photon diffusion equation in the presence of absorbing inclusions

We have developed a theoretical model for photon migration through scattering media in the presence of an absorbing inhomogeneity. A closed-form solution for the average diffuse intensity has been obtained through an iterative approximation scheme of the steady-state diffusion equation. The model describes absorbing defects in a wide range of values. Comparisons with the results of Monte Carlo simulations show that the error of the model is lower than 3% for size inclusion lower than 4 mm and absorption contrast up to the threshold value of the "black defect." The proposed model provides a tractable mathematical basis for diffuse optical and photoacoustic tomographic reconstruction techniques.

Higher-order perturbation theory for the diffusion equation in heterogeneous media: application to layered and slab geometries

We apply a previously proposed perturbation theory of the diffusion equation for studying light propagation through heterogeneous media in the presence of absorbing defects. The theory is based on the knowledge of (a) the geometric characteristics of a focal inclusion, (b) the mean optical path length inside the inclusion, and (c) the optical properties of the inclusion. The potential of this method is shown in the layered and slab geometries, where calculations are carried out up to the fourth order. The relative changes of intensity with respect to the unperturbed (heterogeneous) medium are predicted by the theory to within 10% for a wide range of contrasts dDeltamu(a) (up to dDeltamu(a) approximately 0.4-0.8), where d is the effective diameter of the defect and Deltamu(a) the absorption contrast between defect and local background. We also show how the method of Padé approximants can be used to extend the validity of the theory for a larger range of absorption contrasts. Finally, we study the possibility of using the proposed method for calculating the effect of a colocalized scattering and absorbing perturbation.

Limits of high-order perturbation theory in time-domain optical mammography

Higher order corrections to the Born approximation in perturbation theory are derived in order to improve its performance with the experiments in slablike geometry. A general expression of the nth order correction to absorption is developed. The cross talking between absorption and scattering is given. The convergence for higher orders of perturbation analysis for absorbing inclusions was studied. Second order absorption and scattering contributions to the transmitted flux are discussed by analyzing the data from forward simulations. The validity of the results is proven in the experiments with phantoms simulating breast tumors. The significant improvement for the fitted values of the absorption is observed. The alternative application of developed formalism as the first order theory to treat the multiple inclusions is suggested.

Advances in optical spectroscopy and imaging of breast lesions

A review is presented of recent advances in optical imaging and spectroscopy and the use of light for addressing breast cancer issues. Spectroscopic techniques offer the means to characterize tissue components and obtain functional information in real time. Three-dimensional optical imaging of the breast using various illumination and signal collection schemes in combination with image reconstruction algorithms may provide a new tool for cancer detection and treatment monitoring.

Monte Carlo simulations of increased/decreased scattering inclusions inside a turbid slab

We analyse the effect on scattered photons of anomalous optical inclusions in a turbid slab with otherwise uniform properties. Our motivation for doing so is that inclusions affect scattering contrast used to quantify optical properties found from transmitted light intensity measured in transillumination experiments. The analysis is based on a lattice random walk formalism which takes into account effects of both positive and negative deviations of the scattering coefficient from that of the bulk. Our simulations indicate the existence of a qualitative difference between the effects of these two types of perturbations. In the case of positive perturbations the time delay is found to be proportional to the square of the size of the inclusion while for negative perturbations the time delay is a linear function of its volume.

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.

Recent advances in diffuse optical imaging

We review the current state-of-the-art of diffuse optical imaging, which is an emerging technique for functional imaging of biological tissue. It involves generating images using measurements of visible or near-infrared light scattered across large (greater than several centimetres) thicknesses of tissue. We discuss recent advances in experimental methods and instrumentation, and examine new theoretical techniques applied to modelling and image reconstruction. We review recent work on in vivo applications including imaging the breast and brain, and examine future challenges.

Three-dimensional localization and optical imaging of objects in turbid media with independent component analysis

A new approach for optical imaging and localization of objects in turbid media that makes use of the independent component analysis (ICA) from information theory is demonstrated. Experimental arrangement realizes a multisource illumination of a turbid medium with embedded objects and a multidetector acquisition of transmitted light on the medium boundary. The resulting spatial diversity and multiple angular observations provide robust data for three-dimensional localization and characterization of absorbing and scattering inhomogeneities embedded in a turbid medium. ICA of the perturbations in the spatial intensity distribution on the medium boundary sorts out the embedded objects, and their locations are obtained from Green's function analysis based on any appropriate light propagation model. Imaging experiments were carried out on two highly scattering samples of thickness approximately 50 times the transport mean-free path of the respective medium. One turbid medium had two embedded absorptive objects, and the other had four scattering objects. An independent component separation of the signal, in conjunction with diffusive photon migration theory, was used to locate the embedded inhomogeneities. In both cases, improved lateral and axial localizations of the objects over the result obtained by use of common photon migration reconstruction algorithms were achieved. The approach is applicable to different medium geometries, can be used with any suitable photon propagation model, and is amenable to near-real-time imaging applications.

Tissue characterization by quantitative optical imaging methods

Optical methods have a long history in the field of medical diagnosis. The biomolecular specificity possible with optical methods has been particularly valuable in microscopy and histopathology while in vivo imaging of deep structures has traditionally been the domain of X-ray and MRI. The use of optical methods in deep tissue has been limited by multiple-scattering which blurs or distorts the optical signal. New stochastic methods which account for multiple scattering have been developed that are extending the usefulness of optical methods deep into tissue. In optical mammography, photons may travel through 10 cm of tissue before arriving at the detector. We have developed a method for quantifying parameters of anomalous sites in breast tissue that may be used for functional characterization of tumors. In other work presented here, we are developing fluorescence based methods to detect and monitor tumor status. The immune response to a tumor is a target for fluorescently labeled specific antibodies. We have developed a method to localize the tumor site using CW fluorescence. Additionally, we have developed a method which uses time-resolved data and capitalizes on probe lifetime sensitivity to metabolic parameters such as pH and temperature to obtain functional information from the tumor site.

Optical characterization of mammalian tissues by laser reflectometry and Monte Carlo simulation

The optical characterization of various goat organs/tissues, by measurement of the spatial variation of the diffuse reflectance from the surface by laser multi-probe reflectometer, is carried out. For determination of the optical parameters, these profiles are matched by iterative procedures with that obtained by Monte Carlo simulation by best-fit procedure with chi value 0.99. The first set of measurements is carried out with milk phantom. Thereafter, the absorption and scattering coefficients and anisotropy parameter of goat's heart, lungs, kidney, liver, spleen, skeletal muscle, brain and adipose tissues are measured. These parameters vary over a wide range, which is in agreement with results reported by others. Based on these data, their laser scattering profiles along the depth in terms of depth of penetration (DP) and maximum scattered beam width (MSBW) are determined. These are maximum (0.030 and 0.038 m) for kidney and minimum (0.007 and 0.006 m) for spleen, respectively. The backscattered intensity measured 0.002, 0.004 and 0.006 m away from the beam entry point shows the maximum contribution from the respective depths, irrespective of the nature of the organs/tissues.

Characterization and imaging of compositional variation in tissues

The diffuse surface reflectance profiles of the goat's isolated heart, spleen, and adipose tissues by multiprobe laser reflectometer are measured. The normalized backscattered intensity values for adipose, heart, and spleen tissues at source-detector separation 0.2 cm, are 0.060, 0.021, and 0.003, respectively. The optical parameters of these tissues are determined by the best fit (chi2(0.99)) of their spatial profiles with that as obtained by Monte Carlo simulation by iterative procedure. As the optical parameters of these vary over a wide range, adipose and spleen tissues are treated as inhomogeneity of diameter 0.1, 0.2, or 0.3 cm, and placed inside the control (heart) tissue at different depths. Anisotropic simulation of light backscattering or photon depth distribution is significantly different for various tissues. The surface intensity profiles vary depending on the changes in tissue composition. From the horizontal scans of the subtracted images, the photon backscattering simulated images of control and combination of tissues are obtained. By analysis of peak intensity and full-width at half maximum, the type, location, and size of the tissue compositional variation are determined.

Experimental test of a perturbation model for time-resolved imaging in diffusive media

We report what to our knowledge is a novel perturbation approach for time-resolved transmittance imaging in diffusive media, based on the diffusion approximation with extrapolated boundary conditions. The model relies on the method of Padé approximants and consists of a nonlinear approximation of time-resolved transmittance curves in the presence of an inclusion. The proposed model is intended to extend the range of applicability of perturbation models when applied to inclusions that are non-point-like. We test the model on different tissue phantoms with scattering only, absorbing only, and both scattering and absorbing inclusions. Maps of the optical properties are displayed, and the results are compared with those obtained by means of the usual linear approximation of time-resolved transmittance curves. We found that the nonlinear approach gives a better prediction for absolute values of the scattering and absorption coefficients of inclusions, when the inclusion optical properties are higher than the surrounding background. Furthermore, better-resolved spots and a reduced cross talk between the two parameters are found in the reconstructed maps. Because the range of the optical properties spanned by the considered phantoms covers the values expected for optical mammography, the application of the reported reconstruction method to in vivo images of a breast appears promising from a diagnostic viewpoint.

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.

Number of distinct sites visited by a random walker trapped by an absorbing boundary

The number of distinct sites visited by a lattice random walker is a subject of continuing interest in both mathematics and physics. All previous investigations have used the assumption that the lattice is unbounded. An assessment of the amount of tissue interrogated by a photon in reflectance measurements for diagnostic purposes suggests analyzing properties of the average number of distinct sites visited by a random walker trapped by an absorbing plane at time t. We show that for sufficiently large t this number is the same as the average number of distinct sites visited for this time when the surface is not present. A more complete analysis is possible for a random walk on a line terminated by an absorbing point.

Analytical calculation of the mean time spent by photons inside an absorptive inclusion embedded in a highly scattering medium

The mean time spent by photons inside a nonlocalized optically abnormal embedded inclusion has been derived analytically. The accuracy of the results has been tested against Monte Carlo and experimental data. We show that for quantification of the absorption coefficient of absorptive inclusions, a corrective factor that takes into account the size of the inclusion is needed. This finding suggests that perturbation methods derived for very small inclusions which are used in inverse algorithms require a corrective factor to adequately quantify the differential absorption coefficient of nonlocalized targets embedded in optically turbid media.

Quantitative oximetry of breast tumors: a near-infrared method that identifies two optimal wavelengths for each tumor

We present a noninvasive optical method to measure the oxygen saturation of hemoglobin in breast lesions. This method introduces the novel concept that the best choice of near-infrared wavelengths for noninvasive tumor oximetry consists of a wavelength pair (lambda1, lambda2) within the range 680-880 nm, where the specific values of lambda1 and lambda2 depend on the optical properties of the specific tumor under examination. Our method involves two steps: (1) identify the optimal wavelength pair for each tumor and (2) measure the tumor oxygenation using the optical data at the two selected wavelengths. We have tested our method by acquiring experimental optical data from turbid media containing cylindrical or irregularly shaped inhomogeneities and by computing theoretical data for the case of spherical lesions embedded in a highly scattering medium. We have found that our optical method can provide accurate and quantitative measurements of the oxygenation of embedded lesions without requiring knowledge of their size, shape, and depth.

Quantification of optical properties of a breast tumor using random walk theory

For the first time we use a random walk methodology based on time-dependent contrast functions to quantify the optical properties of breast tumors (invasive ductal carcinoma) of two patients. Previously this theoretical approach was successfully applied for analysis of embedded objects in several phantoms. Data analysis was performed on distributions of times of flight for photons transmitted through the breast which were recorded in vivo using a time-domain scanning mammograph at 670 and 785 nm. The size of the tumors, their optical properties, and those of the surrounding tissue were reconstructed at both wavelengths. The tumors showed increased absorption and scattering. From the absorption coefficients at both wavelengths blood oxygen saturation was estimated for the tumors and the surrounding tissue.

Accuracy of a perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration

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.