Joint Reconstruction of Absorbed Optical Energy Density and Sound Speed Distributions in Photoacoustic Computed Tomography: A Numerical Investigation

Optimizing quantitative photoacoustic imaging systems: the Bayesian Cramér-Rao bound approach

Quantitative photoacoustic computed tomography (qPACT) is an emerging medical imaging modality that carries the promise of high-contrast, fine-resolution imaging of clinically relevant quantities like hemoglobin concentration and blood-oxygen saturation. However, qPACT image reconstruction is governed by a multiphysics, partial differential equation (PDE) based inverse problem that is highly non-linear and severely ill-posed. Compounding the difficulty of the problem is the lack of established design standards for qPACT imaging systems, as there is currently a proliferation of qPACT system designs for various applications and it is unknown which ones are optimal or how to best modify the systems under various design constraints. This work introduces a novel computational approach for the optimal experimental design of qPACT imaging systems based on the Bayesian Cramér-Rao bound (CRB). Our approach incorporates several techniques to address challenges associated with forming the bound in the infinite-dimensional function space setting of qPACT, including priors with trace-class covariance operators and the use of the variational adjoint method to compute derivatives of the log-likelihood function needed in the bound computation. The resulting Bayesian CRB based design metric is computationally efficient and independent of the choice of estimator used to solve the inverse problem. The efficacy of the bound in guiding experimental design was demonstrated in a numerical study of qPACT design schemes under a stylized two-dimensional imaging geometry. To the best of our knowledge, this is the first work to propose Bayesian CRB based design for systems governed by PDEs.

Learning-based sound speed estimation and aberration correction for linear-array photoacoustic imaging

Photoacoustic (PA) image reconstruction involves acoustic inversion that necessitates the specification of the speed of sound (SoS) within the medium of propagation. Due to the lack of information on the spatial distribution of the SoS within heterogeneous soft tissue, a homogeneous SoS distribution (such as 1540 m/s) is typically assumed in PA image reconstruction, similar to that of ultrasound (US) imaging. Failure to compensate for the SoS variations leads to aberration artefacts, deteriorating the image quality. Various methods have been proposed to address this issue, but they usually involve complex hardware and/or time-consuming algorithms, hindering clinical translation. In this work, we introduce a deep learning framework for SoS estimation and subsequent aberration correction in a dual-modal PA/US imaging system exploiting a clinical US probe. As the acquired PA and US images were inherently co-registered, the estimated SoS distribution from US channel data using a deep neural network was incorporated for accurate PA image reconstruction. The framework comprised an initial pre-training stage based on digital phantoms, which was further enhanced through transfer learning using physical phantom data and associated SoS maps obtained from measurements. This framework achieved a root mean square error of 10.2 m/s and 15.2 m/s for SoS estimation on digital and physical phantoms, respectively and structural similarity index measures of up to 0.88 for PA reconstructions compared to the conventional approach of 0.69. A maximum of 1.2 times improvement in the signal-to-noise ratio of PA images was further demonstrated with a human volunteer study. Our results show that the proposed framework could be valuable in various clinical and preclinical applications to enhance PA image reconstruction.

Challenges and advances in two-dimensional photoacoustic computed tomography: a review

Significance

Photoacoustic computed tomography (PACT), a hybrid imaging modality combining optical excitation with acoustic detection, has rapidly emerged as a prominent biomedical imaging technique.

Aim

We review the challenges and advances of PACT, including (1) limited view, (2) anisotropy resolution, (3) spatial aliasing, (4) acoustic heterogeneity (speed of sound mismatch), and (5) fluence correction of spectral unmixing.

Approach

We performed a comprehensive literature review to summarize the key challenges in PACT toward practical applications and discuss various solutions.

Results

There is a wide range of contributions from both industry and academic spaces. Various approaches, including emerging deep learning methods, are proposed to improve the performance of PACT further.

Conclusions

We outline contemporary technologies aimed at tackling the challenges in PACT applications.

Piecewise Acoustic Source Imaging with Unknown Speed of Sound Using a Level-Set Method

We investigate the following inverse problem: starting from the acoustic wave equation, reconstruct a piecewise constant passive acoustic source from a single boundary temporal measurement without knowing the speed of sound. When the amplitudes of the source are known a priori, we prove a unique determination result of the shape and propose a level set algorithm to reconstruct the singularities. When the singularities of the source are known a priori, we show unique determination of the source amplitudes and propose a least-squares fitting algorithm to recover the source amplitudes. The analysis bridges the low-frequency source inversion problem and the inverse problem of gravimetry. The proposed algorithms are validated and quantitatively evaluated with numerical experiments in 2D and 3D.

Quantitative Photoacoustic Tomography Using Iteratively Refined Wavefield Reconstruction Inversion: A Simulation Study

The ultimate goal of photoacoustic tomography is to accurately map the absorption coefficient throughout the imaged tissue. Most studies either assume that acoustic properties of biological tissues such as speed of sound (SOS) and acoustic attenuation are homogeneous or fluence is uniform throughout the entire tissue. These assumptions reduce the accuracy of estimations of derived absorption coefficients (DeACs). Our quantitative photoacoustic tomography (qPAT) method estimates DeACs using iteratively refined wavefield reconstruction inversion (IR-WRI) which incorporates the alternating direction method of multipliers to solve the cycle skipping challenge associated with full wave inversion algorithms. Our method compensates for SOS inhomogeneity, fluence decay, and acoustic attenuation. We evaluate the performance of our method on a neonatal head digital phantom.

Fast iterative reconstruction for photoacoustic tomography using learned physical model: Theoretical validation

Iterative reconstruction has demonstrated superior performance in medical imaging under compressed, sparse, and limited-view sensing scenarios. However, iterative reconstruction algorithms are slow to converge and rely heavily on hand-crafted parameters to achieve good performance. Many iterations are usually required to reconstruct a high-quality image, which is computationally expensive due to repeated evaluations of the physical model. While learned iterative reconstruction approaches such as model-based learning (MBLr) can reduce the number of iterations through convolutional neural networks, it still requires repeated evaluations of the physical models at each iteration. Therefore, the goal of this study is to develop a Fast Iterative Reconstruction (FIRe) algorithm that incorporates a learned physical model into the learned iterative reconstruction scheme to further reduce the reconstruction time while maintaining robust reconstruction performance. We also propose an efficient training scheme for FIRe, which releases the enormous memory footprint required by learned iterative reconstruction methods through the concept of recursive training. The results of our proposed method demonstrate comparable reconstruction performance to learned iterative reconstruction methods with a 9x reduction in computation time and a 620x reduction in computation time compared to variational reconstruction.

Multi-segmented feature coupling for jointly reconstructing initial pressure and speed of sound in photoacoustic computed tomography

SIGNIFICANCE

Photoacoustic computed tomography (PACT) is a fast-growing imaging modality. In PACT, the image quality is degraded due to the unknown distribution of the speed of sound (SoS). Emerging initial pressure (IP) and SoS joint-reconstruction methods promise reduced artifacts in PACT. However, previous joint-reconstruction methods have some deficiencies. A more effective method has promising prospects in preclinical applications.

AIM

We propose a multi-segmented feature coupling (MSFC) method for SoS-IP joint reconstruction in PACT.

APPROACH

In the proposed method, the ultrasound detectors were divided into multiple sub-arrays with each sub-array and its opposite counterpart considered to be a pair. The delay and sum algorithm was then used to reconstruct two images based on a subarray pair and estimated a direction-specific SoS, based on image correlation and the orientation of the subarrays. Once the data generated by all pairs of subarrays were processed, an image that was optimized in terms of minimal feature splitting in all directions was generated. Further, based on the direction-specific SoS, a model-based method was used to directly reconstruct the SoS distribution.

RESULTS

Both phantom and animal experiments demonstrated feasibility and showed promising results compared with conventional methods, with less splitting and blurring and fewer distortions.

CONCLUSIONS

The developed MSFC method shows promising results for both IP and SoS reconstruction. The MSFC method will help to optimize the image quality of PACT in clinical applications.

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.

Adaptive photoacoustic computed tomography

For many optical imaging modalities, image qualities are inevitably degraded by wavefront distortions caused by varying light speed. In optical microscopy and astronomy, adaptive optics (AO) has long been applied to compensate for such unwanted aberrations. Photoacoustic computed tomography (PACT), despite relying on the ultrasonic wave for image formation, suffers from the acoustic version of the same problem. However, this problem has traditionally been regarded as an inverse problem of jointly reconstructing both the initial pressure and the sound speed distributions. In this work, we proposed a method similar to indirect wavefront sensing in AO. We argued that wavefront distortions can be extracted and corrected by a frequency domain analysis of local images. In addition to an adaptively reconstructed aberration-free image, the speed of sound map can be subsequently estimated. We demonstrated the method by in silico, phantom, and in vivo experiments.

Iterative image reconstruction in transcranial photoacoustic tomography based on the elastic wave equation

Photoacoustic computed tomography (PACT) is an emerging computed imaging modality that exploits optical contrast and ultrasonic detection principles to form images of the photoacoustically induced initial pressure distribution within tissue. The PACT reconstruction problem corresponds to a time-domain inverse source problem, where the initial pressure distribution is recovered from the measurements recorded on an aperture outside the support of the source. A major challenge in transcranial PACT of the brain is to compensate for aberrations and attenuation in the measured data due to the propagation of the photoacoustic wavefields through the skull. To properly account for these effects, a wave equation-based inversion method can be employed that can model the heterogeneous elastic properties of the medium. In this study, an optimization-based image reconstruction method for 3D transcranial PACT is developed based on the elastic wave equation. To accomplish this, a forward-adjoint operator pair based on a finite-difference time-domain discretization of the 3D elastic wave equation is utilized to compute penalized least squares estimates of the initial pressure distribution. Computer-simulation and experimental studies are conducted to investigate the robustness of the reconstruction method to model mismatch and its ability to effectively resolve cortical and superficial brain structures.

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.

A survey of computational frameworks for solving the acoustic inverse problem in three-dimensional photoacoustic computed tomography

Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging technique that holds great promise for biomedical imaging. PACT is a hybrid imaging method that can exploit the strong endogenous contrast of optical methods along with the high spatial resolution of ultrasound methods. In its canonical form that is addressed in this article, PACT seeks to estimate the photoacoustically-induced initial pressure distribution within the object. Image reconstruction methods are employed to solve the acoustic inverse problem associated with the image formation process. When an idealized imaging scenario is considered, analytic solutions to the PACT inverse problem are available; however, in practice, numerous challenges exist that are more readily addressed within an optimization-based, or iterative, image reconstruction framework. In this article, the PACT image reconstruction problem is reviewed within the context of modern optimization-based image reconstruction methodologies. Imaging models that relate the measured photoacoustic wavefields to the sought-after object function are described in their continuous and discrete forms. The basic principles of optimization-based image reconstruction from discrete PACT measurement data are presented, which includes a review of methods for modeling the PACT measurement system response and other important physical factors. Non-conventional formulations of the PACT image reconstruction problem, in which acoustic parameters of the medium are concurrently estimated along with the PACT image, are also introduced and reviewed.

Feature coupling photoacoustic computed tomography for joint reconstruction of initial pressure and sound speed in vivo

Photoacoustic imaging relies on diffused photons for optical contrast and diffracted ultrasound for high resolution. As a tomographic imaging modality, often an inverse problem of acoustic diffraction needs to be solved to reconstruct a photoacoustic image. The inverse problem is complicated by the fact that the acoustic properties, including the speed of sound distribution, in the image field of view are unknown. During reconstruction, subtle changes of the speed of sound in the acoustic ray path may accumulate and give rise to noticeable blurring in the image. Thus, in addition to the ultrasound detection bandwidth, inaccurate acoustic modeling, especially the unawareness of the speed of sound, defines the image resolution and influences image quantification. Here, we proposed a method termed feature coupling to jointly reconstruct the speed of sound distribution and a photoacoustic image with improved sharpness, at no additional hardware cost. Simulations, phantom studies, and in vivo experiments demonstrated the effectiveness and reliability of our method.

An image reconstruction method for endoscopic photoacoustic tomography in tissues with heterogeneous sound speed

The idealized assumption of a constant speed of sound (SOS) in acoustically inhomogeneous biological tissues usually results in blurred details, acoustic distortion and artifacts in in vivo endoscopic photoacoustic tomographic (EPAT) images. In this paper, we propose an image reconstruction method to improve EPAT imaging for luminal structures with the variable SOS. In our method, an optimal SOS providing the maximal local focusing of a measuring location within the imaging region is firstly determined. The deviation in the ultrasonic propagation time caused by the variable SOS is then compensated. The grayscale images of the optical absorption distribution on the cross-sections of the luminal structures are finally reconstructed with a filtered back-projection (FBP) algorithm based on the corrected propagation time. Any prior knowledge of the SOS distribution in the imaged tissues is not required. The results of numerical simulation experiments demonstrated that the proposed method can effectively improve the image quality by reducing the misalignment of tissues, acoustic distortion and artifacts caused by the variable SOS.

Optoacoustic image reconstruction: the full inverse problem with variable bases

Optoacoustic imaging was for a long time concerned with the reconstruction of energy density or optical properties. In this work, we present the full inverse problem with respect to optical absorption and diffusion as well as speed of sound and mass density. The inverse problem is solved by an iterative gradient-based optimization procedure. Since the ill-conditioning increases with the number of sought parameters, we propose two approaches to improve the conditioning. The first approach is based on the reduction of the size of the basis for the parameter spaces, that evolves according to the particular characteristics of the solution, while maintaining the flexibility of element-wise parameter selection. The second approach is a material identification technique that incorporates prior knowledge of expected material types and uses the acoustical gradients to identify materials uniquely. We present numerical studies to illustrate the properties and functional principle of the proposed methods. Significant convergence speed-ups are gained by the two approaches countering ill-conditioning. Additionally, we show results for the reconstruction of a mouse brain from in vivo measurements.

Parameterized joint reconstruction of the initial pressure and sound speed distributions for photoacoustic computed tomography

Accurate estimation of the initial pressure distribution in photoacoustic computed tomography (PACT) depends on knowledge of the sound speed distribution. However, the sound speed distribution is typically unknown. Further, the initial pressure and sound speed distributions cannot both, in general, be stably recovered from PACT measurements alone. In this work, a joint reconstruction (JR) method for the initial pressure distribution and a low-dimensional parameterized model of the sound speed distribution is proposed. By employing a priori information about the structure of the sound speed distribution, both the initial pressure and sound speed can be accurately recovered. The JR problem is solved by use of a proximal optimization method that allows constraints and non-smooth regularization functions for the initial pressure distribution. The gradients of the cost function with respect to the initial pressure and sound speed distributions are calculated by use of an adjoint state method that has the same per-iteration computational cost as calculating the gradient with respect to the initial pressure distribution alone. This approach is evaluated through 2D computer-simulation studies for a small animal imaging model and by application to experimental in vivo measurements of a mouse.

Parameterized joint reconstruction of the initial pressure and sound speed distributions for photoacoustic computed tomography

Accurate estimation of the initial pressure distribution in photoacoustic computed tomography (PACT) depends on knowledge of the sound speed distribution. However, the sound speed distribution is typically unknown. Further, the initial pressure and sound speed distributions cannot both, in general, be stably recovered from PACT measurements alone. In this work, a joint reconstruction (JR) method for the initial pressure distribution and a low-dimensional parameterized model of the sound speed distribution is proposed. By employing a priori information about the structure of the sound speed distribution, both the initial pressure and sound speed can be accurately recovered. The JR problem is solved by use of a proximal optimization method that allows constraints and non-smooth regularization functions for the initial pressure distribution. The gradients of the cost function with respect to the initial pressure and sound speed distributions are calculated by use of an adjoint state method that has the same per-iteration computational cost as calculating the gradient with respect to the initial pressure distribution alone. This approach is evaluated through 2D computer-simulation studies for a small animal imaging model and by application to experimental in vivo measurements of a mouse.

Joint reconstruction of the initial pressure and speed of sound distributions from combined photoacoustic and ultrasound tomography measurements

The initial pressure and speed of sound (SOS) distributions cannot both be stably recovered from photoacoustic computed tomography (PACT) measurements alone. Adjunct ultrasound computed tomography (USCT) measurements can be employed to estimate the SOS distribution. Under the conventional image reconstruction approach for combined PACT/USCT systems, the SOS is estimated from the USCT measurements alone and the initial pressure is estimated from the PACT measurements by use of the previously estimated SOS. This approach ignores the acoustic information in the PACT measurements and may require many USCT measurements to accurately reconstruct the SOS. In this work, a joint reconstruction method where the SOS and initial pressure distributions are simultaneously estimated from combined PACT/USCT measurements is proposed. This approach allows accurate estimation of both the initial pressure distribution and the SOS distribution while requiring few USCT measurements.

Generation of anatomically realistic numerical phantoms for photoacoustic and ultrasonic breast imaging

Photoacoustic computed tomography (PACT) and ultrasound computed tomography (USCT) are emerging modalities for breast imaging. As in all emerging imaging technologies, computer-simulation studies play a critically important role in developing and optimizing the designs of hardware and image reconstruction methods for PACT and USCT. Using computer-simulations, the parameters of an imaging system can be systematically and comprehensively explored in a way that is generally not possible through experimentation. When conducting such studies, numerical phantoms are employed to represent the physical properties of the patient or object to-be-imaged that influence the measured image data. It is highly desirable to utilize numerical phantoms that are realistic, especially when task-based measures of image quality are to be utilized to guide system design. However, most reported computer-simulation studies of PACT and USCT breast imaging employ simple numerical phantoms that oversimplify the complex anatomical structures in the human female breast. We develop and implement a methodology for generating anatomically realistic numerical breast phantoms from clinical contrast-enhanced magnetic resonance imaging data. The phantoms will depict vascular structures and the volumetric distribution of different tissue types in the breast. By assigning optical and acoustic parameters to different tissue structures, both optical and acoustic breast phantoms will be established for use in PACT and USCT studies.

Nonconfocal all-optical laser-ultrasound and photoacoustic imaging system for angle-dependent deep tissue imaging

Biomedical imaging systems incorporating both photoacoustic (PA) and ultrasound capabilities are of interest for obtaining optical and acoustic properties deep in tissue. While most dual-modality systems utilize piezoelectric transducers, all-optical systems can obtain broadband high-resolution data with hands-free operation. Previously described reflection-mode all-optical laser-ultrasound (LUS) systems use a confocal source and detector; however, angle-dependent raypaths are lost in this configuration. As a result, the overall imaging aperture is reduced, which becomes increasingly problematic with depth. We present a reflection-mode nonconfocal LUS and PA imaging system that uses signals recorded on all-optical hardware to create angle-dependent images. We use reverse-time migration and time reversal to reconstruct the LUS and PA images. We demonstrate this methodology with both a numerical model and tissue phantom experiment to image a steep-curvature vessel with a limited aperture 2-cm beneath the surface. Nonconfocal imaging demonstrates improved focusing by 30% and 15% compared to images acquired with a single LUS source in the numerical and experimental LUS images, respectively. The appearance of artifacts is also reduced. Complementary PA images are straightforward to acquire with the nonconfocal system by tuning the source wavelength and can be further developed for quantitative multiview PA imaging.