Bayesian Image Reconstruction in Quantitative Photoacoustic Tomography

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustic tomography

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

Modeling toolchain for realistic simulation of photoacoustic data acquisition

SIGNIFICANCE

Physics-based simulations of photoacoustic (PA) signals are used to validate new methods, to characterize PA setups and to generate training datasets for machine learning. However, a thoroughly validated PA simulation toolchain that can simulate realistic images is still lacking.

AIM

A quantitative toolchain was developed to model PA image acquisition in complex tissues, by simulating both the optical fluence and the acoustic wave propagation.

APPROACH

Sampling techniques were developed to decrease artifacts in acoustic simulations. The performance of the simulations was analyzed by measuring the point spread function (PSF) and using a rotatable three-channel phantom, filled with cholesterol, a human carotid plaque sample, and porcine blood. Ex vivo human plaque samples were simulated to validate the methods in more complex tissues.

RESULTS

The sampling techniques could enhance the quality of the simulated PA images effectively. The resolution and intensity of the PSF in the turbid medium matched the experimental data well. Overall, the appearance, signal-to-noise ratio and speckle of the images could be simulated accurately.

CONCLUSIONS

A PA toolchain was developed and validated, and the results indicate a great potential of PA simulations in more complex and heterogeneous media.

Adaptive stochastic Gauss-Newton method with optical Monte Carlo for quantitative photoacoustic tomography

SIGNIFICANCE

The image reconstruction problem in quantitative photoacoustic tomography (QPAT) is an ill-posed inverse problem. Monte Carlo method for light transport can be utilized in solving this image reconstruction problem.

AIM

The aim was to develop an adaptive image reconstruction method where the number of photon packets in Monte Carlo simulation is varied to achieve a sufficient accuracy with reduced computational burden.

APPROACH

The image reconstruction problem was formulated as a minimization problem. An adaptive stochastic Gauss-Newton (A-SGN) method combined with Monte Carlo method for light transport was developed. In the algorithm, the number of photon packets used on Gauss-Newton (GN) iteration was varied utilizing a so-called norm test.

RESULTS

The approach was evaluated with numerical simulations. With the proposed approach, the number of photon packets needed for solving the inverse problem was significantly smaller than in a conventional approach where the number of photon packets was fixed for each GN iteration.

CONCLUSIONS

The A-SGN method with a norm test can be utilized in QPAT to provide accurate and computationally efficient solutions.

Photoacoustic computed tomography for functional human brain imaging [Invited]

The successes of magnetic resonance imaging and modern optical imaging of human brain function have stimulated the development of complementary modalities that offer molecular specificity, fine spatiotemporal resolution, and sufficient penetration simultaneously. By virtue of its rich optical contrast, acoustic resolution, and imaging depth far beyond the optical transport mean free path (∼1 mm in biological tissues), photoacoustic computed tomography (PACT) offers a promising complementary modality. In this article, PACT for functional human brain imaging is reviewed in its hardware, reconstruction algorithms, in vivo demonstration, and potential roadmap.

Shape-Driven EIT Reconstruction Using Fourier Representations

Shape-driven approaches have been proposed as an effective strategy for the electrical impedance tomography (EIT) reconstruction problem in recent years. In order to augment the shape-driven approaches, we propose a new method that transforms the shape to be reconstructed as basic primitives directly modeled by using Fourier representations. To allow automatic topological changes between the basic primitives and surrounding objects simultaneously, Boolean operations are employed. The Boolean operations with direct representation of primitives can be utilized for dimensionality and ill-posedness reduction, enabling feasible shape and topology optimization with shape-driven approaches. As a proof of principle, we leverage the proposed method for two dimensional shape reconstruction in EIT with various conductivity distributions. We demonstrate that our method is able to improve EIT reconstructions by enabling accurate shape and topology optimization.

Deep Learning-Based Spectral Unmixing for Optoacoustic Imaging of Tissue Oxygen Saturation

Label free imaging of oxygenation distribution in tissues is highly desired in numerous biomedical applications, but is still elusive, in particular in sub-epidermal measurements. Eigenspectra multispectral optoacoustic tomography (eMSOT) and its Bayesian-based implementation have been introduced to offer accurate label-free blood oxygen saturation (sO2) maps in tissues. The method uses the eigenspectra model of light fluence in tissue to account for the spectral changes due to the wavelength dependent attenuation of light with tissue depth. eMSOT relies on the solution of an inverse problem bounded by a number of ad hoc hand-engineered constraints. Despite the quantitative advantage offered by eMSOT, both the non-convex nature of the optimization problem and the possible sub-optimality of the constraints may lead to reduced accuracy. We present herein a neural network architecture that is able to learn how to solve the inverse problem of eMSOT by directly regressing from a set of input spectra to the desired fluence values. The architecture is composed of a combination of recurrent and convolutional layers and uses both spectral and spatial features for inference. We train an ensemble of such networks using solely simulated data and demonstrate how this approach can improve the accuracy of sO2 computation over the original eMSOT, not only in simulations but also in experimental datasets obtained from blood phantoms and small animals (mice) in vivo. The use of a deep-learning approach in optoacoustic sO2 imaging is confirmed herein for the first time on ground truth sO2 values experimentally obtained in vivo and ex vivo.

Perturbation Monte Carlo Method for Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography aims at estimating optical parameters from photoacoustic images that are formed utilizing the photoacoustic effect caused by the absorption of an externally introduced light pulse. This optical parameter estimation is an ill-posed inverse problem, and thus it is sensitive to measurement and modeling errors. In this work, we propose a novel way to solve the inverse problem of quantitative photoacoustic tomography based on the perturbation Monte Carlo method. Monte Carlo method for light propagation is a stochastic approach for simulating photon trajectories in a medium with scattering particles. It is widely accepted as an accurate method to simulate light propagation in tissues. Furthermore, it is numerically robust and easy to implement. Perturbation Monte Carlo maintains this robustness and enables forming gradients for the solution of the inverse problem. We validate the method and apply it in the framework of Bayesian inverse problems. The simulations show that the perturbation Monte Carlo method can be used to estimate spatial distributions of both absorption and scattering parameters simultaneously. These estimates are qualitatively good and quantitatively accurate also in parameter scales that are realistic for biological tissues.

Modeling of Errors Due to Uncertainties in Ultrasound Sensor Locations in Photoacoustic Tomography

Photoacoustic tomography is an imaging modality based on the photoacoustic effect caused by the absorption of an externally introduced light pulse. In the inverse problem of photoacoustic tomography, the initial pressure generated through the photoacoustic effect is estimated from a measured photoacoustic time-series utilizing a forward model for ultrasound propagation. Due to the ill-posedness of the inverse problem, errors in the forward model or measurements can result in significant errors in the solution of the inverse problem. In this work, we study modeling of errors caused by uncertainties in ultrasound sensor locations in photoacoustic tomography using a Bayesian framework. The approach is evaluated with simulated and experimental data. The results indicate that the inverse problem of photoacoustic tomography is sensitive even to small uncertainties in sensor locations. Furthermore, these uncertainties can lead to significant errors in the estimates and reduction of the quality of the photoacoustic images. In this work, we show that the errors due to uncertainties in ultrasound sensor locations can be modeled and compensated using Bayesian approximation error modeling.

Modelling of errors due to speed of sound variations in photoacoustic tomography using a Bayesian framework

Inverse problem of estimating initial pressure in photoacoustic tomography is ill-posed and thus sensitive to errors in modelling and measurements. In practical experiments, accurate knowledge of the speed of sound of the imaged target is commonly not available, and therefore an approximate speed of sound is used in the computational model. This can result in errors in the solution of the inverse problem that can appear as artefacts in the reconstructed images. In this paper, the inverse problem of photoacoustic tomography is approached in a Bayesian framework. Errors due to uncertainties in the speed of sound are modelled using Bayesian approximation error modelling. Estimation of the initial pressure distribution together with information on the reliability of these estimates are considered. The approach was studied using numerical simulations. The results show that uncertainties in the speed of sound can cause significant errors in the solution of the inverse problem. However, modelling of these uncertainties improves the accuracy of the solution.

Image Reconstruction with Reliability Assessment in Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography is a novel imaging method which aims to reconstruct optical parameters of an imaged target based on initial pressure distribution, which can be obtained from ultrasound measurements. In this paper, a method for reconstructing the optical parameters in a Bayesian framework is presented. In addition, evaluating the credibility of the estimates is studied. Furthermore, a Bayesian approximation error method is utilized to compensate the modeling errors caused by coarse discretization of the forward model. The reconstruction method and the reliability of the credibility estimates are investigated with two-dimensional numerical simulations. The results suggest that the Bayesian approach can be used to obtain accurate estimates of the optical parameters and the credibility estimates of these parameters. Furthermore, the Bayesian approximation error method can be used to compensate for the modeling errors caused by a coarse discretization, which can be used to reduce the computational costs of the reconstruction procedure. In addition, taking the modeling errors into account can increase the reliability of the credibility estimates.

Impact of depth-dependent optical attenuation on wavelength selection for spectroscopic photoacoustic imaging

An optical wavelength selection method based on the stability of the absorption cross-section matrix to improve spectroscopic photoacoustic (sPA) imaging was recently introduced. However, spatially-varying chromophore concentrations cause the wavelength- and depth-dependent variations of the optical fluence, which degrades the accuracy of quantitative sPA imaging. This study introduces a depth-optimized method that determines an optimal wavelength set minimizing an inverse of the multiplication of absorption cross-section matrix and fluence matrix to minimize the errors in concentration estimation. This method assumes that the optical fluence distribution is known or can be attained otherwise. We used a Monte Carlo simulation of light propagation in tissue with various depths and concentrations of deoxy-/oxy-hemoglobin. We quantitatively compared the developed and current approaches, indicating that the choice of wavelength is critical and our approach is effective especially when quantifying deeper imaging targets.

A Bayesian Approach to Eigenspectra Optoacoustic Tomography

The quantification of hemoglobin oxygen saturation (sO2) with multispectral optoacoustic (OA) (photoacoustic) tomography (MSOT) is a complex spectral unmixing problem, since the OA spectra of hemoglobin are modified with tissue depth due to depth (location) and wavelength dependencies of optical fluence in tissue. In a recent work, a method termed eigenspectra MSOT (eMSOT) was proposed for addressing the dependence of spectra on fluence and quantifying blood sO2 in deep tissue. While eMSOT offers enhanced sO2 quantification accuracy over conventional unmixing methods, its performance may be compromised by noise and image reconstruction artifacts. In this paper, we propose a novel Bayesian method to improve eMSOT performance in noisy environments. We introduce a spectral reliability map, i.e., a method that can estimate the level of noise superimposed onto the recorded OA spectra. Using this noise estimate, we formulate eMSOT as a Bayesian inverse problem where the inversion constraints are based on probabilistic graphical models. Results based on numerical simulations indicate that the proposed method offers improved accuracy and robustness under high noise levels due the adaptive nature of the Bayesian method.

Segmentation of vessel structures from photoacoustic images with reliability assessment

Photoacoustic imaging enables the imaging of soft biological tissue with combined optical contrast and ultrasound resolution. One of the targets of interest is tissue vasculature. However, the photoacoustic images may not directly provide the information on, for example, vasculature structure. Therefore, the images are improved by reducing noise and artefacts, and processed for better visualisation of the target of interest. In this work, we present a new segmentation method of photoacoustic images that also straightforwardly produces assessments of its reliability. The segmentation depends on parameters which have a natural tendency to increase the reliability as the parameter values monotonically change. The reliability is assessed by counting classifications of image voxels with different parameter values. The resulting segmentation with reliability offers new ways and tools to analyse photoacoustic images and new possibilities for utilising them as anatomical priors in further computations. Our MATLAB implementation of the method is available as an open-source software package [P. Raumonen, Matlab, 2018].

Context encoding enables machine learning-based quantitative photoacoustics

Real-time monitoring of functional tissue parameters, such as local blood oxygenation, based on optical imaging could provide groundbreaking advances in the diagnosis and interventional therapy of various diseases. Although photoacoustic (PA) imaging is a modality with great potential to measure optical absorption deep inside tissue, quantification of the measurements remains a major challenge. We introduce the first machine learning-based approach to quantitative PA imaging (qPAI), which relies on learning the fluence in a voxel to deduce the corresponding optical absorption. The method encodes relevant information of the measured signal and the characteristics of the imaging system in voxel-based feature vectors, which allow the generation of thousands of training samples from a single simulated PA image. Comprehensive in silico experiments suggest that context encoding-qPAI enables highly accurate and robust quantification of the local fluence and thereby the optical absorption from PA images.

Quantitative photoacoustic tomography augmented with surface light measurements

Quantitative photoacoustic tomography is an imaging modality in which distributions of optical parameters inside tissue are estimated from photoacoustic images. This optical parameter estimation is an ill-posed problem and it needs to be approached in the framework of inverse problems. In this work, utilising surface light measurements in quantitative photoacoustic tomography is studied. Estimation of absorption and scattering is formulated as a minimisation problem utilising both internal quantitative photoacoustic data and surface light data. The image reconstruction problem is studied with two-dimensional numerical simulations in various imaging situations using the diffusion approximation as the model for light propagation. The results show that quantitative photoacoustic tomography augmented with surface light data can improve both absorption and scattering estimates when compared to the conventional quantitative photoacoustic tomography.

A Multi-Grid Iterative Method for Photoacoustic Tomography

Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multi-grid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction.

Accelerated high-resolution photoacoustic tomography via compressed sensing

Current 3D photoacoustic tomography (PAT) systems offer either high image quality or high frame rates but are not able to deliver high spatial and temporal resolution simultaneously, which limits their ability to image dynamic processes in living tissue (4D PAT). A particular example is the planar Fabry-Pérot (FP) photoacoustic scanner, which yields high-resolution 3D images but takes several minutes to sequentially map the incident photoacoustic field on the 2D sensor plane, point-by-point. However, as the spatio-temporal complexity of many absorbing tissue structures is rather low, the data recorded in such a conventional, regularly sampled fashion is often highly redundant. We demonstrate that combining model-based, variational image reconstruction methods using spatial sparsity constraints with the development of novel PAT acquisition systems capable of sub-sampling the acoustic wave field can dramatically increase the acquisition speed while maintaining a good spatial resolution: first, we describe and model two general spatial sub-sampling schemes. Then, we discuss how to implement them using the FP interferometer and demonstrate the potential of these novel compressed sensing PAT devices through simulated data from a realistic numerical phantom and through measured data from a dynamic experimental phantom as well as from in vivo experiments. Our results show that images with good spatial resolution and contrast can be obtained from highly sub-sampled PAT data if variational image reconstruction techniques that describe the tissues structures with suitable sparsity-constraints are used. In particular, we examine the use of total variation (TV) regularization enhanced by Bregman iterations. These novel reconstruction strategies offer new opportunities to dramatically increase the acquisition speed of photoacoustic scanners that employ point-by-point sequential scanning as well as reducing the channel count of parallelized schemes that use detector arrays.

Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radiance

Forward and adjoint Monte Carlo (MC) models of radiance are proposed for use in model-based quantitative photoacoustic tomography. A two-dimensional (2-D) radiance MC model using a harmonic angular basis is introduced and validated against analytic solutions for the radiance in heterogeneous media. A gradient-based optimization scheme is then used to recover 2-D absorption and scattering coefficients distributions from simulated photoacoustic measurements. It is shown that the functional gradients, which are a challenge to compute efficiently using MC models, can be calculated directly from the coefficients of the harmonic angular basis used in the forward and adjoint models. This work establishes a framework for transport-based quantitative photoacoustic tomography that can fully exploit emerging highly parallel computing architectures.

Direct Estimation of Optical Parameters From Photoacoustic Time Series in Quantitative Photoacoustic Tomography

Estimation of optical absorption and scattering of a target is an inverse problem associated with quantitative photoacoustic tomography. Conventionally, the problem is expressed as two folded. First, images of initial pressure distribution created by absorption of a light pulse are formed based on acoustic boundary measurements. Then, the optical properties are determined based on these photoacoustic images. The optical stage of the inverse problem can thus suffer from, for example, artefacts caused by the acoustic stage. These could be caused by imperfections in the acoustic measurement setting, of which an example is a limited view acoustic measurement geometry. In this work, the forward model of quantitative photoacoustic tomography is treated as a coupled acoustic and optical model and the inverse problem is solved by using a Bayesian approach. Spatial distribution of the optical properties of the imaged target are estimated directly from the photoacoustic time series in varying acoustic detection and optical illumination configurations. It is numerically demonstrated, that estimation of optical properties of the imaged target is feasible in limited view acoustic detection setting.

Quantitative photoacoustic tomography by stochastic search: direct recovery of the optical absorption field

We present, perhaps for the first time, a stochastic search algorithm in quantitative photoacoustic tomography (QPAT) for a one-step recovery of the optical absorption map from time-resolved photoacoustic signals. Such a direct recovery is free of the numerical inaccuracies inherent in conventional two-step approaches that depend on an accurate estimation of the absorbed energy distribution. The absorption profile parameterized as a vector stochastic process is additively updated over time recursions so as to drive the measurement-prediction misfit to a zero-mean white noise. The derivative-free additive update is a welcome departure from the conventional gradient-based methods requiring evaluation of Jacobians at every recursion. The quantitative accuracy of the recovered absorption map from both numerical and experimental data is good with an overall error of less than 10%.

Image reconstruction with uncertainty quantification in photoacoustic tomography

Photoacoustic tomography is a hybrid imaging method that combines optical contrast and ultrasound resolution. The goal of photoacoustic tomography is to resolve an initial pressure distribution from detected ultrasound waves generated within an object due to an illumination of a short light pulse. In this work, a Bayesian approach to photoacoustic tomography is described. The solution of the inverse problem is derived and computation of the point estimates for image reconstruction and uncertainty quantification is described. The approach is investigated with simulations in different detector geometries, including limited view setup, and with different detector properties such as ideal point-like detectors, finite size detectors, and detectors with a finite bandwidth. The results show that the Bayesian approach can be used to provide accurate estimates of the initial pressure distribution, as well as information about the uncertainty of the estimates.

Quantitative photoacoustic tomography using illuminations from a single direction

Quantitative photoacoustic tomography is an emerging imaging technique aimed at estimating optical parameters inside tissues from photoacoustic images, which are formed by combining optical information and ultrasonic propagation. This optical parameter estimation problem is ill-posed and needs to be approached within the framework of inverse problems. It has been shown that, in general, estimating the spatial distribution of more than one optical parameter is a nonunique problem unless more than one illumination pattern is used. Generally, this is overcome by illuminating the target from various directions. However, in some cases, for example when thick samples are investigated, illuminating the target from different directions may not be possible. In this work, the use of spatially modulated illumination patterns at one side of the target is investigated with simulations. The results show that the spatially modulated illumination patterns from a single direction could be used to provide multiple illuminations for quantitative photoacoustic tomography. Furthermore, the results show that the approach can be used to distinguish absorption and scattering inclusions located near the surface of the target. However, when compared to a full multidirection illumination setup, the approach cannot be used to image as deep inside tissues.

Bayesian approximation error approach in full-wave ultrasound tomography

In ultrasound tomography, the spatial distribution of the speed of sound in a region of interest is reconstructed based on transient measurements made around the object. The computation of the forward problem (the full-wave solution) within the object is a computationally intensive task and can often be prohibitive for practical purposes. The purpose of this paper is to investigate the feasibility of using approximate forward solvers and the partial recovery from the related errors by employing the Bayesian approximation error approach. In addition to discretization error, we also investigate whether the approach can be used to reduce the reconstruction errors that are due to the adoption of approximate absorbing boundary models. We carry out two numerical studies in which the objective is to reduce the computational times to around 3% of the time that would be required by a numerically accurate forward solver. The results show that the Bayesian approximation error approach improves the reconstructions.