Monte carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain

A review of advances in imaging methodology in fluorescence molecular tomography

Objective. Fluorescence molecular tomography (FMT) is a promising non-invasive optical molecular imaging technology with strong specificity and sensitivity that has great potential for preclinical and clinical studies in tumor diagnosis, drug development and therapeutic evaluation. However, the strong scattering of photons and insufficient surface measurements make it very challenging to improve the quality of FMT image reconstruction and its practical application for early tumor detection. Therefore, continuous efforts have been made to explore more effective approaches or solutions in the pursuit of high-quality FMT reconstructions. Approach. This review takes a comprehensive overview of advances in imaging methodology for FMT, mainly focusing on two critical issues in FMT reconstructions: improving the accuracy of solving the forward physical model and mitigating the ill-posed nature of the inverse problem from a methodological point of view. More importantly, numerous impressive and practical strategies and methods for improving the quality of FMT reconstruction are summarized. Notably, deep learning methods are discussed in detail to illustrate their advantages in promoting the imaging performance of FMT thanks to large datasets, the emergence of optimized algorithms and the application of innovative networks. Main results. The results demonstrate that the imaging quality of FMT can be effectively promoted by improving the accuracy of optical parameter modeling, combined with prior knowledge, and reducing dimensionality. In addition, the traditional regularization-based methods and deep neural network-based methods, especially end-to-end deep networks, can enormously alleviate the ill-posedness of the inverse problem and improve the quality of FMT image reconstruction. Significance. This review aims to illustrate a variety of effective and practical methods for the reconstruction of FMT images that may benefit future research. Furthermore, it may provide some valuable research ideas and directions for FMT in the future, and could promote, to a certain extent, the development of FMT and other methods of optical tomography.

Recent methodology advances in fluorescence molecular tomography

Molecular imaging (MI) is a novel imaging discipline that has been continuously developed in recent years. It combines biochemistry, multimodal imaging, biomathematics, bioinformatics, cell & molecular physiology, biophysics, and pharmacology, and it provides a new technology platform for the early diagnosis and quantitative analysis of diseases, treatment monitoring and evaluation, and the development of comprehensive physiology. Fluorescence Molecular Tomography (FMT) is a type of optical imaging modality in MI that captures the three-dimensional distribution of fluorescence within a biological tissue generated by a specific molecule of fluorescent material within a biological tissue. Compared with other optical molecular imaging methods, FMT has the characteristics of high sensitivity, low cost, and safety and reliability. It has become the research frontier and research hotspot of optical molecular imaging technology. This paper took an overview of the recent methodology advances in FMT, mainly focused on the photon propagation model of FMT based on the radiative transfer equation (RTE), and the reconstruction problem solution consist of forward problem and inverse problem. We introduce the detailed technologies utilized in reconstruction of FMT. Finally, the challenges in FMT were discussed. This survey aims at summarizing current research hotspots in methodology of FMT, from which future research may benefit.

Optical architecture design for detection of absorbers embedded in visceral fat

Optically absorbing ducts embedded in scattering adipose tissue can be injured during laparoscopic surgery. Non-sequential simulations and theoretical analysis compare optical system configurations for detecting these absorbers. For absorbers in deep scattering volumes, trans-illumination is preferred instead of diffuse reflectance. For improved contrast, a scanning source with a large area detector is preferred instead of a large area source with a pixelated detector.

Light diffusion in a turbid cylinder. II. Layered case

This paper is the second of two dealing with light diffusion in a turbid cylinder. The diffusion equation was solved for an N-layered finite cylinder. Solutions are given in the steady-state, frequency, and time domains for a point beam incident at an arbitrary position of the first layer and for a circular flat beam incident at the middle of the cylinder top. For special cases the solutions were compared to other solutions of the diffusion equation showing excellent agreement. In addition, the derived solutions were validated by comparison with Monte Carlo simulations. In the time domain we also derived a fast solution ( approximately 10ms) for the case of equal reduced scattering coefficients and refractive indices in all layers.

Light diffusion in N-layered turbid media: steady-state domain

We deal with light diffusion in N-layered turbid media. The steady-state diffusion equation is solved for N-layered turbid media having a finite or an infinitely thick N'th layer. Different refractive indices are considered in the layers. The Fourier transform formalism is applied to derive analytical solutions of the fluence rate in Fourier space. The inverse Fourier transform is calculated using four different methods to test their performance and accuracy. Further, to avoid numerical errors, approximate formulas in Fourier space are derived. Fast solutions for calculation of the spatially resolved reflectance and transmittance from the N-layered turbid media ( approximately 10 ms) with small relative differences (<10(-7)) are found. Additionally, the solutions of the diffusion equation are compared to Monte Carlo simulations for turbid media having up to 20 layers.

Image reconstruction method for a two-layer tissue structure accounts for chest-wall effects in breast imaging

We develop a new tomographic imaging reconstruction algorithm for a two-layer tissue structure. Simulations and phantom experiments show more accurate reconstruction of target optical properties compared with those results obtained from a semi-infinite tissue model for layered structures. This improvement is mainly attributed to the more accurate estimation of background optical properties and more accurate estimation of weight matrix for imaging reconstruction by considering the light propagation effect in the second layer. Clinical results of breast lesions are also presented to demonstrate the utility of this new imaging algorithm.

Time-domain Green functions for diffuse light in two adjoining turbid half-spaces

Propagation of light emitted by an instantaneous source located above a plane interface between two semi-infinite turbid media is considered using the diffusion approximation. Green functions are derived for an instantaneous line source and an instantaneous point source by the method of Bellman et al. [Philos. Mag. 40, 297 (1949)], which is based on integral transforms. Both two-dimensional and three-dimensional Green functions for diffuse light have been obtained in the form of single integrals that allow for fast calculation of the specific intensity in the whole space. The influence of the optical parameters of the two media (diffusion coefficients, absorptions, and refractive indices) on the shapes of the contour lines of the specific intensity is analyzed.

Rapid simulation of steady-state spatially resolved reflectance and transmittance profiles of multilayered turbid materials

We present a technique for efficiently computing the reflection and transmission of light by arbitrary systems of turbid layers. To approximate the steady-state reflectance and transmittance without the need to solve difficult boundary conditions, we convolve the reflectance and transmittance profiles of individual layers. We extend single-slab boundary conditions to handle index-of-refraction mismatches between turbid slabs and account for interlayer scattering by applying methods similar to Kubelka-Munk theory in frequency space. We demonstrate good agreement between the reflectance and the transmittance predicted by our model and numerical Monte Carlo methods and show that the far-source reflectance and transmittance of multilayered turbid materials are dominated by interlayer scattering.

Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions

In this paper, a coupled radiative transfer equation and diffusion approximation model is extended for light propagation in turbid medium with low-scattering and non-scattering regions. The light propagation is modelled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The streamline diffusion modification is used to avoid the ray-effect problem in the finite element solution of the radiative transfer equation. The proposed method is tested with simulations. The results of the coupled model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation and with results of Monte Carlo simulation. The results show that the coupled model can be used to describe photon migration in turbid medium with low-scattering and non-scattering regions more accurately than the conventional diffusion model.

Optical characterization of thin female breast biopsies based on the reduced scattering coefficient

One of the main goals in optical characterization of biopsies is to discern between tissue types. Usually, the theory used for deriving the optical properties of such highly scattering media is based on the diffusion approximation. However, biopsies are usually small in size compared to the transport mean free path and thus cannot be treated with standard diffusion theory. To account for this, an improved theory was developed, by the authors, that can correctly describe light propagation in small geometries (Garofalakis et al 2004 J. Opt. A: Pure Appl. Opt. 6 725-35). The theory's limit was validated by both Monte Carlo simulations and experiments performed on tissue-like phantoms, and was found to be two transport mean free paths. With the aid of this theory, we have characterized 59 samples of breast tissue including cancerous samples by retrieving their reduced scattering coefficients from time-resolved transmission data. The mean values for the reduced scattering coefficients of the normal and the tumour tissue were measured to be 9.7 +/- 2.2 cm(-1) and 10.8 +/- 1.8 cm(-1), respectively. The correlation with age was also investigated.

Light transport in two-layer tissues

We study theoretically light backscattered by tissues using the radiative transport equation. In particular we consider a two-layered medium in which a finite slab is situated on top of a half space. We solve the one-dimensional problem in which a plane wave is incident normally on the top layer and is the only source of light. The solution to this problem is obtained formally by imposing continuity between the solutions for the upper and lower layers. However, we are interested solely in probing the top layer. Assuming that the optical properties in the lower layer are known, we remove it from the problem yielding a finite slab problem by prescribing an alternate boundary condition. This boundary condition is derived using the theory of Green's functions and is exact. Hence, one needs only to solve the transport equation in a finite slab using this alternate boundary condition. We derive an asymptotic solution for the case when the slab is optically thin. We extend these results to the three-dimensional problem using Fourier transforms. These results are validated by comparisons with numerical solutions for the entire two-layered problem.

Hybrid radiative-transfer-diffusion model for optical tomography

A hybrid radiative-transfer-diffusion model for optical tomography is proposed. The light propagation is modeled with the radiative-transfer equation in the vicinity of the laser sources, and the diffusion approximation is used elsewhere in the domain. The solution of the radiative-transfer equation is used to construct a Dirichlet boundary condition for the diffusion approximation on a fictitious interface within the object. This boundary condition constitutes an approximative distributed source model for the diffusion approximation in the remaining area. The results from the proposed approach are compared with finite-element solutions of the radiative-transfer equation and the diffusion approximation and Monte Carlo simulation. The results show that the method improves the accuracy of the forward model compared with the conventional diffusion model.

The use of spatially resolved fluorescence and reflectance to determine interface depth in layered fluorophore distributions

The possibility of using spatially resolved fluorescence and reflectance measurements to recover tissue optical properties, fluorophore concentration and the thickness of a superficial layer in a two-layer geometry was investigated. A diffusion theory model was used to fit reflectance and fluorescence data generated using Monte Carlo simulations or experimentally obtained using tissue-simulating phantoms. Initial analysis fitting diffusion theory generated data suggested that it should be possible to recover all parameters from a single set of spatially resolved fluorescence and reflectance measurements. However, when Monte Carlo or experimental data were fitted the results were less impressive. Overall, it was shown that there is a strong coupling between interface depth, fluorophore concentration and tissue absorption, especially at larger depths. The recovery of all input parameters from a single set of spatially resolved measurements was limited to interface depths less than 3 mm, which is a reasonable range for measuring fluorophore in skin. When the tissue optical properties and fluorophore concentrations were known, then the interface depth could be monitored with good accuracy in simulated serial measurements. These results may also point to deficiencies in the diffusion theory model that introduce significant errors in the fitted results.

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.

Asymptotic behavior and inverse problem in layered scattering media

The main challenge of noninvasive optical biopsy is to obtain an accurate value of the optical coefficients of an encapsulated organ (muscle, brain, etc.). The idea developed by us is that some interesting information could be deduced from the long-time behavior of the reflectance function. This asymptotic behavior is analyzed for layered media in the framework of the diffusion approximation. A new method is derived to obtain accurate values for the optical parameters of the deepest layers. This method is designed to work in a specific long-time regime that is still within the scope of standard time-of-flight experiments but far from being included in the mathematically defined asymptotic region. The limits of this method, linked to the cases where the asymptotic behavior is no longer governed by the deepest layer, are then discussed.

Determination of the optical properties of a two-layer tissue model by detecting photons migrating at progressively increasing depths

We have investigated a method for solving the inverse problem of determining the optical properties of a two-layer turbid model. The method is based on deducing the optical properties (OPs) of the top layer from the absolute spatially resolved reflectance that results from photon migration within only the top layer by use of a multivariate calibration model. Then the OPs of the bottom layer are deduced from relative frequency-domain (FD) reflectance measurements by use of the two-layer FD diffusion model. The method was validated with Monte Carlo FD reflectance profiles and experimental measurements of two-layer phantoms. The results showed that the method is useful for two-layer models with interface depths of >5 mm; the OPs were estimated, within a relatively short time (<1 min), with a mean error of <10% for the Monte Carlo reflectance profiles and with errors of <25% for the phantom measurements.

Transabdominal near infrared oximetry of hypoxic stress in fetal sheep brain in utero

The feasibility of transabdominal near-infrared (NIR) spectroscopy for detecting and quantifying fetal hypoxia in utero is demonstrated in a pregnant ewe model. A frequency domain NIR spectroscopy probe, consisting of two detectors and six sources operating at three wavelengths (675, 786, and 830 nm), was placed on the maternal abdomen directly above the fetal head. Fetal hypoxia was indirectly induced through occlusion of uterine blood flow for approximately 3 min. NIR photon diffusion measurements were made during a baseline period, during hypoxia of the fetus, and during recovery. Fetal blood samples were drawn from the fetal brachial artery and jugular veins at several time points during the cycle. Seven hypoxic cycles were induced in a total of five pregnant ewes. The NIR measurements were analyzed by using a two-layer diffusion model to deconvolve the fetal blood saturation from that of the pregnant ewe. Fetal hypoxia was detected. Good agreement was found between fetal blood saturation determined by the transabdominal NIR method and arterial and venous fetal blood saturation quantified from fetal blood samples by using a hemoximeter.

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.

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.

Experimental and numerical study of the colour appearance of tattoo models

The colour of tattooed skin has been predicted by a Monte Carlo method based on the optical coefficient spectra of the skin and tattoo dyes. Slices of pig skin, a tattoo phantom and skin phantoms with different thickness were prepared, and their reflectance and transmittance spectra were measured using an integrating sphere at wavelengths varying from 400nm to 700nm. The absorption and scattering coefficient spectra of skin phantoms, pig skins and the tattoo phantom were each calculated using the inverse Monte Carlo method. The skin phantoms and pig skins were overlaid on the tattoo phantom, and the reflectance spectra of the two-layered structures were measured. The reflectance spectra of the two-layered structures were calculated from the optical coefficient spectra using the Monte Carlo method. They agreed well with the measured spectra. The colour differences between the calculated and measured spectra were also evaluated by the L*a*b* colour space distances and showed good agreement, with 3.49 for the skin phantoms and 8.27 for the pig skins.

Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model

The general two-layer inverse problem in biomedical photon migration is to estimate the absorption and scattering coefficients of each layer as well as the top-layer thickness. We attempted to solve this problem, using experimental and simulated spatially resolved frequency-domain (FD) reflectance for optical properties typical of skin overlying muscle or skin overlying fat in the near infrared. Two forward models of light propagation were used: a two-layer diffusion solution [Appl. Opt. 37, 779 (1998)] and a hybrid Monte Carlo (MC) diffusion model [Appl. Opt. 37, 7401 (1998)]. MC-simulated FD reflectance data were fitted as relative measurements to the hybrid and the pure diffusion models. It was found that the hybrid model could determine all the optical properties of the two-layer media studied to ~5%. Also, the same accuracy could be achieved by means of fitting MC-simulated cw reflectance data as absolute measurements, but fitting them as relative ones is an ill-posed problem. In contrast, two-layer diffusion could not retrieve the top-layer optical properties as accurately for FD data and was ill-posed for both relative and absolute cw data. The hybrid and the pure diffusion models were also fitted to experimental FD reflectance measurements from two-layer tissue-simulating phantoms representative of skin-on-fat and skin-on-muscle baseline optical properties. Both the hybrid and the diffusion models could determine the optical properties of the lower layer. The hybrid model demonstrated its potential to retrieve quantitatively the transport scattering coefficient of skin (the upper layer), which was not possible with the pure diffusion model. Systematic discrepancies between model and experiment may compromise the accuracy of the deduced top-layer optical properties. Identifying and eliminating such discrepancies is critical to practical application of the method.

Recovery of optical parameters in multiple-layered diffusive media: theory and experiments

Diffuse photon density waves have lately been used both to characterize diffusive media and to locate and characterize hidden objects, such as tumors, in soft tissue. In practice, most biological media of medical interest consist of various layers with different optical properties, such as the fat layer in the breast or the different layers present in the skin. Also, most experimental setups consist of a multilayered system, where the medium to be characterized (i.e., the patient's organ) is usually bounded by optically diffusive plates. Incorrect modeling of interfaces may induce errors comparable to the weak signals obtained from tumors embedded deep in highly heterogeneous tissue and lead to significant reconstruction artifacts. To provide a means to analyze the data acquired in these configurations, the basic expressions for the reflection and transmission coefficients for diffusive-diffusive and diffusive-nondiffusive interfaces are presented. A comparison is made between a diffusive slab and an ordinary dielectric slab, thus establishing the limiting distance between the two interfaces of the slab for multiple reflections between them to be considered important. A rigorous formulation for multiple-layered (M-layered) diffusive media is put forward, and a method for solving any M-layered medium is shown. The theory presented is used to characterize a two-layered medium from transmission measurements, showing that the coefficients of scattering, mu'(s) , and absorption, mu(a) , are retrieved with great accuracy. Finally, we demonstrate the simultaneous retrieval of both mu;(s) and mu(a).

Non-invasive determination of muscle blood flow in the extremities from laser Doppler spectra

We investigate theoretically the non-invasive determination of blood flow in muscles of the extremities using laser Doppler measurements. Laser Doppler spectra are calculated using Monte Carlo simulations and solutions of the correlation diffusion equation. The extremities are modelled as a two-layered turbid medium. The first layer represents the skin and subcutaneous fat layer and the second layer the muscle. It is shown that the absolute root-mean-square velocity of the blood in the muscle layer can be accurately derived in many practical cases if the laser Doppler spectra are measured at a distance which is sufficiently far from the source, and if the optical properties of the muscle are simultaneously determined.