skip to main content


Title: Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation
Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.  more » « less
Award ID(s):
1750488 1740707
NSF-PAR ID:
10158022
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Entropy
Volume:
21
Issue:
3
ISSN:
1099-4300
Page Range / eLocation ID:
291
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime. 
    more » « less
  2. Optical coherence tomography (OCT) leverages light scattering by biological tissues as endogenous contrast to form structural images. Light scattering behavior is dictated by the optical properties of the tissue, which depend on microstructural details at the cellular or sub-cellular level. Methods to measure these properties from OCT intensity data have been explored in the context of a number of biomedical applications seeking to access this sub-resolution tissue microstructure and thereby increase the diagnostic impact of OCT. Most commonly, the optical attenuation coefficient, an analogue of the scattering coefficient, has been used as a surrogate metric linking OCT intensity to subcellular particle characteristics. To record attenuation coefficient data that is accurately representative of the underlying physical properties of a given sample, it is necessary to account for the impact of the OCT imaging system itself on the distribution of light intensity in the sample, including the numerical aperture (NA) of the system and the location of the focal plane with respect to the sample surface, as well as the potential contribution of multiple scattering to the reconstructed intensity signal. Although these considerations complicate attenuation coefficient measurement and interpretation, a suitably calibrated system may potentiate a powerful strategy for gaining additional information about the scattering behavior and microstructure of samples. In this work, we experimentally show that altering the OCT system geometry minimally impacts measured attenuation coefficients in samples presumed to be singly scattering, but changes these measurements in more highly scattering samples. Using both depth-resolved attenuation coefficient data and layer-resolved backscattering coefficients, we demonstrate the retrieval of scattering particle diameter and concentration in tissue-mimicking phantoms, and the impact of presumed multiple scattering on these calculations. We further extend our approach to characterize a murine brain tissue sample and highlight a tumor-bearing region based on increased scattering particle density. Through these methods, we not only enhance conventional OCT attenuation coefficient analysis by decoupling the independent effects of particle size and concentration, but also discriminate areas of strong multiple scattering through minor changes to system topology to provide a framework for assessing the accuracy of these measurements.

     
    more » « less
  3. McCulloch, R. (Ed.)
    Varying coefficient models (VCMs) are widely used for estimating nonlinear regression functions for functional data. Their Bayesian variants using Gaussian process priors on the functional coefficients, however, have received limited attention in massive data applications, mainly due to the prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We address this problem using a divide-and-conquer Bayesian approach. We first create a large number of data subsamples with much smaller sizes. Then, we formulate the VCM as a linear mixed-effects model and develop a data augmentation algorithm for obtaining MCMC draws on all the subsets in parallel. Finally, we aggregate the MCMC-based estimates of subset posteriors into a single Aggregated Monte Carlo (AMC) posterior, which is used as a computationally efficient alternative to the true posterior distribution. Theoretically, we derive minimax optimal posterior convergence rates for the AMC posteriors of both the varying coefficients and the mean regression function. We provide quantification on the orders of subset sample sizes and the number of subsets. The empirical results show that the combination schemes that satisfy our theoretical assumptions, including the AMC posterior, have better estimation performance than their main competitors across diverse simulations and in a real data analysis. 
    more » « less
  4. Structural optical coherence tomography (OCT) images of tissue stand to benefit from greater functionalization and quantitative interpretation. The OCT attenuation coefficientµ, an analogue of the imaged sample’s scattering coefficient, offers potential functional contrast based on the relationship ofµto sub-resolution physical properties of the sample. Attenuation coefficients are computed either by fitting a representativeµover several depth-wise pixels of a sample’s intensity decay, or by using previously-developed depth-resolved attenuation algorithms by Girardet al.[Invest. Ophthalmol. Vis. Sci.52,7738(2011).10.1167/iovs.10-6925] and Vermeeret al.[Biomed. Opt. Express5,322(2014).10.1364/BOE.5.000322]. However, the former method sacrifices axial information in the tomogram, while the latter relies on the stringent assumption that the sample’s backscattering fraction, another optical property, does not vary along depth. This assumption may be violated by layered tissues commonly observed in clinical imaging applications. Our approach preserves the full depth resolution of the attenuation map but removes its dependence on backscattering fraction by performing signal analysis inside individual discrete layers over which the scattering properties (e.g., attenuation and backscattering fraction) vary minimally. Although this approach necessitates the detection of these layers, it removes the constant-backscattering-fraction assumption that has constrained quantitative attenuation coefficient analysis in the past, and additionally yields a layer-resolved backscattering fraction, providing complementary scattering information to the attenuation coefficient. We validate our approach using automated layer detection in layered phantoms, for which the measured optical properties were in good agreement with theoretical values calculated with Mie theory, and show preliminary results in tissue alongside corresponding histological analysis. Together, accurate backscattering fraction and attenuation coefficient measurements enable the estimation of both particle density and size, which is not possible from attenuation measurements alone. We hope that this improvement to depth-resolved attenuation coefficient measurement, augmented by a layer-resolved backscattering fraction, will increase the diagnostic power of quantitative OCT imaging.

     
    more » « less
  5. Abstract

    We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography.

     
    more » « less