Image reconstruction with uncertainty quantification in 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.

Advanced data analysis in inertial confinement fusion and high energy density physics

Bayesian analysis enables flexible and rigorous definition of statistical model assumptions with well-characterized propagation of uncertainties and resulting inferences for single-shot, repeated, or even cross-platform data. This approach has a strong history of application to a variety of problems in physical sciences ranging from inference of particle mass from multi-source high-energy particle data to analysis of black-hole characteristics from gravitational wave observations. The recent adoption of Bayesian statistics for analysis and design of high-energy density physics (HEDP) and inertial confinement fusion (ICF) experiments has provided invaluable gains in expert understanding and experiment performance. In this Review, we discuss the basic theory and practical application of the Bayesian statistics framework. We highlight a variety of studies from the HEDP and ICF literature, demonstrating the power of this technique. Due to the computational complexity of multi-physics models needed to analyze HEDP and ICF experiments, Bayesian inference is often not computationally tractable. Two sections are devoted to a review of statistical approximations, efficient inference algorithms, and data-driven methods, such as deep-learning and dimensionality reduction, which play a significant role in enabling use of the Bayesian framework. We provide additional discussion of various applications of Bayesian and machine learning methods that appear to be sparse in the HEDP and ICF literature constituting possible next steps for the community. We conclude by highlighting community needs, the resolution of which will improve trust in data-driven methods that have proven critical for accelerating the design and discovery cycle in many application areas.

On the robustness of model-based algorithms for photoacoustic tomography: Comparison between time and frequency domains

For photoacoustic image reconstruction, certain parameters such as sensor positions and speed of sound have a major impact on the reconstruction process and must be carefully determined before data acquisition. Uncertainties in these parameters can lead to errors produced by a modeling mismatch, hindering the reconstruction process and severely affecting the resulting image quality. Therefore, in this work, we study how modeling errors arising from uncertainty in sensor locations affect the images obtained by matrix model-based reconstruction algorithms based on time domain and frequency domain models of the photoacoustic problem. The effects on the reconstruction performance with respect to the uncertainty in the knowledge of the sensors location are compared and analyzed both in a qualitative and quantitative fashion for both time and frequency models. Ultimately, our study shows that the frequency domain approach is more sensitive to this kind of modeling errors. These conclusions are supported by numerical experiments and a theoretical sensitivity analysis of the mathematical operator for the direct problem.

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.

Computationally Efficient Forward Operator for Photoacoustic Tomography Based on Coordinate Transformations

Photoacoustic tomography (PAT) is an imaging modality that utilizes the photoacoustic effect. In PAT, a photoacoustic image is computed from measured data by modeling ultrasound propagation in the imaged domain and solving an inverse problem utilizing a discrete forward operator. However, in realistic measurement geometries with several ultrasound transducers and relatively large imaging volume, an explicit formation and use of the forward operator can be computationally prohibitively expensive. In this work, we propose a transformation-based approach for efficient modeling of photoacoustic signals and reconstruction of photoacoustic images. In the approach, the forward operator is constructed for a reference ultrasound transducer and expanded into a general measurement geometry using transformations that map the formulated forward operator in local coordinates to the global coordinates of the measurement geometry. The inverse problem is solved using a Bayesian framework. The approach is evaluated with numerical simulations and experimental data. The results show that the proposed approach produces accurate 3-D photoacoustic images with a significantly reduced computational cost both in memory requirements and time. In the studied cases, depending on the computational factors, such as discretization, over the 30-fold reduction in memory consumption was achieved without a reduction in image quality compared to a conventional approach.

Deep learning for biomedical photoacoustic imaging: A review

Photoacoustic imaging (PAI) is a promising emerging imaging modality that enables spatially resolved imaging of optical tissue properties up to several centimeters deep in tissue, creating the potential for numerous exciting clinical applications. However, extraction of relevant tissue parameters from the raw data requires the solving of inverse image reconstruction problems, which have proven extremely difficult to solve. The application of deep learning methods has recently exploded in popularity, leading to impressive successes in the context of medical imaging and also finding first use in the field of PAI. Deep learning methods possess unique advantages that can facilitate the clinical translation of PAI, such as extremely fast computation times and the fact that they can be adapted to any given problem. In this review, we examine the current state of the art regarding deep learning in PAI and identify potential directions of research that will help to reach the goal of clinical applicability.

R-fMRI reconstruction from k-t undersampled data using a subject-invariant dictionary model and VB-EM with nested minorization

Higher spatial resolution in resting-state functional magnetic resonance imaging (R-fMRI) can give reliable information about the functional networks in the cerebral cortex. Typical methods can achieve higher spatial or temporal resolution by speeding up scans using either (i) complex pulse-sequence designs or (ii) k-space undersampling coupled with priors on the signal. We propose to undersample the R-fMRI acquisition in k-space and time to speedup scans in order to improve spatial resolution. We propose a novel model-based R-fMRI reconstruction framework using a robust, subject-invariant, spatially regularized dictionary prior on the signal. Furthermore, we propose a novel inference framework based on variational Bayesian expectation maximization with nested minorization (VB-EM-NM). Our inference framework allows us to provide an estimate of uncertainty of the reconstruction, unlike typical reconstruction methods. Empirical evaluation of (i) simulated R-fMRI reconstruction and (ii) functional-network estimates from brain R-fMRI reconstructions demonstrate that our framework improves over the state of the art, and, additionally, enables significantly higher spatial resolution.

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.

Temperature estimation of inflamed bowel by the photoacoustic inverse approach

Local temperature increase is one of the five classical signs of regions with inflammations. This work is focused on the application of the photoacoustic technique for the estimation of the temperature field in the colon, as the solution of an inverse problem, for the detection of inflamed regions. Two-dimensional cases are examined here involving a cross section of the bowel, which characterize either the inflammation of the whole mucosa layer, or three small inflamed regions. The inverse problem is solved for a rotating laser inside the intestine lumen, which imposes pulses for the generation of the acoustic waves. One single ultrasound detector, also located at the laser rotating shaft, provides the simulated measurements for the inverse analysis. The inverse problem is solved here with the minimization of the maximum a posteriori objective function. Results show that the proposed technique can be applied for accurate estimations of the temperature distribution in the region of interest, which might be used for the diagnosis of inflammatory bowel diseases (IBD).

Black Mesoporous Silicon as a Contrast Agent for LED-Based 3D Photoacoustic Tomography

Mesoporous silicon (PSi) nanoparticles have been widely studied in different biomedical imaging modalities due to their several beneficial material properties. However, they have not been found to be suitable for photoacoustic imaging due to their poor photothermal conversion performance. In the present study, biodegradable black mesoporous silicon (BPSi) nanoparticles with strong light absorbance were developed as superior image contrast agents for photoacoustic tomography (PAT), which was realized with a light-emitting diode (LED) instead of the commonly used laser. LED-based PAT offers the advantages of low cost, compactness, good mobility, and easy operation as compared to the traditional laser-based PAT modality. Nevertheless, the poor imaging sensitivity of the LED-PAT systems has been the main barrier to prevent their wide biomedical application because the LED light has low optical energy. The present study demonstrated that the imaging sensitivity of the LED-PAT system was significantly enhanced with the PEGylated BPSi (PEG-BPSi) nanoparticles. The PEG-BPSi nanoparticles were clearly detectable with a low concentration of 0.05 mg/mL in vitro and with an LED radiation energy of 5.2 μJ. The required concentration of the PEG-BPSi nanoparticles was 10 times lesser than that of the reference gold nanoparticles to reach the corresponding level of the imaging contrast. The ex vivo studies demonstrated that the submillimeter BPSi nanoparticle-based absorbers were distinguishable in chicken breast tissues. The strong contrast provided by the BPSi particles indicated that these particles can be utilized as novel contrast agents in PAT, especially in LED-based systems with low light intensity.

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.

Confidence Estimation for Machine Learning-Based Quantitative Photoacoustics

In medical applications, the accuracy and robustness of imaging methods are of crucial importance to ensure optimal patient care. While photoacoustic imaging (PAI) is an emerging modality with promising clinical applicability, state-of-the-art approaches to quantitative photoacoustic imaging (qPAI), which aim to solve the ill-posed inverse problem of recovering optical absorption from the measurements obtained, currently cannot comply with these high standards. This can be attributed to the fact that existing methods often rely on several simplifying a priori assumptions of the underlying physical tissue properties or cannot deal with realistic noise levels. In this manuscript, we address this issue with a new method for estimating an indicator of the uncertainty of an estimated optical property. Specifically, our method uses a deep learning model to compute error estimates for optical parameter estimations of a qPAI algorithm. Functional tissue parameters, such as blood oxygen saturation, are usually derived by averaging over entire signal intensity-based regions of interest (ROIs). Therefore, we propose to reduce the systematic error of the ROI samples by additionally discarding those pixels for which our method estimates a high error and thus a low confidence. In silico experiments show an improvement in the accuracy of optical absorption quantification when applying our method to refine the ROI, and it might thus become a valuable tool for increasing the robustness of qPAI methods.

Three dimensional photoacoustic tomography in Bayesian framework

The image reconstruction problem (or inverse problem) in photoacoustic tomography is to resolve the initial pressure distribution from detected ultrasound waves generated within an object due to an illumination by a short light pulse. Recently, a Bayesian approach to photoacoustic image reconstruction with uncertainty quantification was proposed and studied with two dimensional numerical simulations. In this paper, the approach is extended to three spatial dimensions and, in addition to numerical simulations, experimental data are considered. The solution of the inverse problem is obtained by computing point estimates, i.e., maximum a posteriori estimate and posterior covariance. These are computed iteratively in a matrix-free form using a biconjugate gradient stabilized method utilizing the adjoint of the acoustic forward operator. The results show that the Bayesian approach can produce accurate estimates of the initial pressure distribution in realistic measurement geometries and that the reliability of these estimates can be assessed.

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].

A photoacoustic imaging reconstruction method based on directional total variation with adaptive directivity

BACKGROUND

In photoacoustic tomography (PAT), total variation (TV) based iteration algorithm is reported to have a good performance in PAT image reconstruction. However, classical TV based algorithm fails to preserve the edges and texture details of the image because it is not sensitive to the direction of the image. Therefore, it is of great significance to develop a new PAT reconstruction algorithm to effectively solve the drawback of TV.

METHODS

In this paper, a directional total variation with adaptive directivity (DDTV) model-based PAT image reconstruction algorithm, which weightedly sums the image gradients based on the spatially varying directivity pattern of the image is proposed to overcome the shortcomings of TV. The orientation field of the image is adaptively estimated through a gradient-based approach. The image gradients are weighted at every pixel based on both its anisotropic direction and another parameter, which evaluates the estimated orientation field reliability. An efficient algorithm is derived to solve the iteration problem associated with DDTV and possessing directivity of the image adaptively updated for each iteration step.

RESULTS AND CONCLUSION

Several texture images with various directivity patterns are chosen as the phantoms for the numerical simulations. The 180-, 90- and 30-view circular scans are conducted. Results obtained show that the DDTV-based PAT reconstructed algorithm outperforms the filtered back-projection method (FBP) and TV algorithms in the quality of reconstructed images with the peak signal-to-noise rations (PSNR) exceeding those of TV and FBP by about 10 and 18 dB, respectively, for all cases. The Shepp-Logan phantom is studied with further discussion of multimode scanning, convergence speed, robustness and universality aspects. In-vitro experiments are performed for both the sparse-view circular scanning and linear scanning. The results further prove the effectiveness of the DDTV, which shows better results than that of the TV with sharper image edges and clearer texture details. Both numerical simulation and in vitro experiments confirm that the DDTV provides a significant quality improvement of PAT reconstructed images for various directivity patterns.