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.

Background-oriented schlieren sensitivity in terms of geometrical parameters of measurement setup.

Background-oriented schlieren imaging is a recently proposed noninvasive optical method for imaging of full ultrasound fields. In this work, the impact of uncertainty in geometrical parameters of a background-oriented schlieren measurement setup for imaging of full ultrasound fields is studied using numerical simulations. The studied parameters are focal length of the camera and positions and orientations of the camera, water tank, and ultrasound field. The results demonstrate that the most sensitive parameters affecting the accuracy of the reconstructed ultrasound fields are the orientations of the camera that change the direction of an effective optical axis. Other sensitive parameters are the focal length of the camera and the position of the ultrasound field in perpendicular directions of an optical axis. This synthetic study demonstrates the accuracy requirements for calibrating the geometrical parameters of a measurement setup that would be required to achieve accuracy comparable to that of hydrophone measurements using the background-oriented schlieren imaging. Explicitly, limits of the variation ranges of the geometrical parameters resulting in relative error ranges of 5% and 10% are given. The results of this study may contribute to help design future background-oriented schlieren measurement setups intended for measurement of full ultrasound fields.

Nonlinear estimation of pressure projection of ultrasound fields in background-oriented schlieren imaging.

Background-oriented schlieren imaging is a recently proposed method for measuring projections of ultrasound fields. The method is based on observing deflection of light in a heterogeneous refractive index field that is induced by ultrasound via an acousto-optic effect. The deflection of light manifests as apparent perturbations in an imaged target, forming a potential flow estimation problem. In this work, the potential flow approach is formulated as a nonlinear regularized least-squares approach to alleviate limitations of approaches that linearize the problem. The nonlinear approach is shown to outperform the linear one when estimating projections of medically relevant ultrasound fields.

Evaluation of temporal moments and Fourier transformed data in time-domain diffuse optical tomography.

Time-domain diffuse optical tomography (TD-DOT) uses near-infrared pulsed lasers as light sources to measure time-varying exitance on the boundary of the target. These are used to estimate optical properties of the imaged target. Several integral-transform-based moments of the time-resolved data have been utilized in TD-DOT, the most common being the mean time of flight and variance. Recently, it has been shown that Fourier transforming the time-domain data to frequency domain enables utilization of these data at one or several frequencies, producing equally as good estimates as the whole time-domain data. In this work, we present a systematic comparison of the usage of the temporal moments and Fourier transformed data in TD-DOT. Both absolute and difference imaging are evaluated using numerical simulations. The simulations show that utilizing temporal moments and Fourier transformed data in TD-DOT provides good quality reconstructions with a good estimation accuracy. These estimates are improved if more than one data type is used. Furthermore, the simulations show that the frequency-domain computations enable computationally cheaper and straightforward implementation of the inverse solver when compared to the temporal moments.

Time-domain diffuse optical tomography utilizing truncated Fourier series approximation.

Diffuse optical tomography (DOT) uses near infrared light for in vivo imaging of spatially varying optical parameters in biological tissues. It is known that time-resolved measurements provide the richest information on soft tissues, among other measurement types in DOT such as steady-state and intensity-modulated measurements. Therefore, several integral-transform-based moments of the time-resolved DOT measurements have been considered to estimate spatially distributed optical parameters. However, the use of such moments can result in low-contrast images and cross-talks between the reconstructed optical parameters, limiting their accuracy. In this work, we propose to utilize a truncated Fourier series approximation in time-resolved DOT. Using this approximation, we obtained optical parameter estimates with accuracy comparable to using whole time-resolved data that uses low computational time and resources. The truncated Fourier series approximation based estimates also displayed good contrast and minimal parameter cross-talk, and the estimates further improved in accuracy when multiple Fourier frequencies were used.

Acoustic pressure field estimation methods for synthetic schlieren tomography.

Synthetic schlieren tomography is a recently proposed three-dimensional (3D) optical imaging technique for studying ultrasound fields. The imaging setup is composed of an imaged target, a water tank, a camera, and a pulsed light source, which is stroboscopically synchronized with an ultrasound transducer to achieve tomographically stationary imaging of an ultrasound field. In this technique, ultrasound waves change the propagation of light rays by inducing a change in refractive index via the acousto-optic effect. The change manifests as optical flow in the imaged target. By performing the imaging in a tomographic fashion, the two-dimensional tomographic dataset of the optical flow can be transformed into a 3D ultrasound field. In this work, two approaches for acoustic pressure field estimation are introduced. The approaches are based on optical and potential flow regularized least square optimizations where regularization based on the Helmholtz equation is introduced. The methods are validated via simulations in a telecentric setup and are compared quantitatively and qualitatively to a previously introduced method. Cases of a focused, an obliquely propagating, and a standing wave ultrasound field are considered. The simulations demonstrate the efficiency of the introduced methods also in situations in which the previously applied method has weaknesses.

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.

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.

Compensation of modeling errors due to unknown domain boundary in diffuse optical tomography.

Diffuse optical tomography is a highly unstable problem with respect to modeling and measurement errors. During clinical measurements, the body shape is not always known, and an approximate model domain has to be employed. The use of an incorrect model domain can, however, lead to significant artifacts in the reconstructed images. Recently, the Bayesian approximation error theory has been proposed to handle model-based errors. In this work, the feasibility of the Bayesian approximation error approach to compensate for modeling errors due to unknown body shape is investigated. The approach is tested with simulations. The results show that the Bayesian approximation error method can be used to reduce artifacts in reconstructed images due to unknown domain shape.

Quantitative photoacoustic tomography: modeling and inverse problems.

Significance: Quantitative photoacoustic tomography (QPAT) exploits the photoacoustic effect with the aim of estimating images of clinically relevant quantities related to the tissue's optical absorption. The technique has two aspects: an acoustic part, where the initial acoustic pressure distribution is estimated from measured photoacoustic time-series, and an optical part, where the distributions of the optical parameters are estimated from the initial pressure. Aim: Our study is focused on the optical part. In particular, computational modeling of light propagation (forward problem) and numerical solution methodologies of the image reconstruction (inverse problem) are discussed. Approach: The commonly used mathematical models of how light and sound propagate in biological tissue are reviewed. A short overview of how the acoustic inverse problem is usually treated is given. The optical inverse problem and methods for its solution are reviewed. In addition, some limitations of real-life measurements and their effect on the inverse problems are discussed. Results: An overview of QPAT with a focus on the optical part was given. Computational modeling and inverse problems of QPAT were addressed, and some key challenges were discussed. Furthermore, the developments for tackling these problems were reviewed. Although modeling of light transport is well-understood and there is a well-developed framework of inverse mathematics for approaching the inverse problem of QPAT, there are still challenges in taking these methodologies to practice. Conclusions: Modeling and inverse problems of QPAT together were discussed. The scope was limited to the optical part, and the acoustic aspects were discussed only to the extent that they relate to the optical aspect.

Truncated Fourier-series approximation of the time-domain radiative transfer equation using finite elements.

The radiative transfer equation (RTE) is widely accepted to accurately describe light transport in a medium with scattering particles, and it has been successfully applied as a light-transport model, for example, in diffuse optical tomography. Due to the computationally expensive nature of the RTE, most of these applications have been in the frequency domain. In this paper, an efficient solution method for the time-domain RTE is proposed. The method is based on solving the frequency-domain RTE at multiple modulation frequencies and using the Fourier-series representation of the radiance to obtain approximation of the time-domain solution. The approach is tested with simulations. The results show that the method can be used to obtain the solution of the time-domain RTE with good accuracy and with significantly fewer computational resources than are needed in the direct time-domain solution.

Assembly of fluorophore J-aggregates with nanospacer onto mesoporous nanoparticles for enhanced photoacoustic imaging.

Many fluorophores, such as indocyanine green (ICG), have poor photostability and low photothermal efficiency hindering their wide application in photoacoustic (PA) tomography. In the present study, a supramolecular assembly approach was used to develop the hybrid nanoparticles (Hy NPs) of ICG and porous silicon (PSi) as a novel contrast agent for PA tomography. ICG was assembled on the PSi NPs to form J-aggregates within 30 min. The Hy NPs presented a red-shifted absorption, improved photothermal stability, and enhanced PA performance. Furthermore, 1-dodecene (DOC) was assembled into the NPs as a 'nanospacer', which enhanced non-radiative decay for increased thermal release. Compared to the Hy NPs, adding DOC into the Hy NPs (DOC-Hy) increased the PA signal by 83%. Finally, the DOC-Hy was detectable in PA tomography at 1.5 cm depth in tissue phantom even though its concentration was as low as 6.25 µg/mL, indicating the potential for deep tissue PA imaging.

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.

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography.

Diffuse optical tomography (DOT) utilises near-infrared light for imaging spatially distributed optical parameters, typically the absorption and scattering coefficients. The image reconstruction problem of DOT is an ill-posed inverse problem, due to the non-linear light propagation in tissues and limited boundary measurements. The ill-posedness means that the image reconstruction is sensitive to measurement and modelling errors. The Bayesian approach for the inverse problem of DOT offers the possibility of incorporating prior information about the unknowns, rendering the problem less ill-posed. It also allows marginalisation of modelling errors utilising the so-called Bayesian approximation error method. A more recent trend in image reconstruction techniques is the use of deep learning, which has shown promising results in various applications from image processing to tomographic reconstructions. In this work, we study the non-linear DOT inverse problem of estimating the (absolute) absorption and scattering coefficients utilising a 'model-based' learning approach, essentially intertwining learned components with the model equations of DOT. The proposed approach was validated with 2D simulations and 3D experimental data. We demonstrated improved absorption and scattering estimates for targets with a mix of smooth and sharp image features, implying that the proposed approach could learn image features that are difficult to model using standard Gaussian priors. Furthermore, it was shown that the approach can be utilised in compensating for modelling errors due to coarse discretisation enabling computationally efficient solutions. Overall, the approach provided improved computation times compared to a standard Gauss-Newton iteration.

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.

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.

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.

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.

Approximation errors and model reduction in three-dimensional diffuse optical tomography.

Model reduction is often required in diffuse optical tomography (DOT), typically because of limited available computation time or computer memory. In practice, this means that one is bound to use coarse mesh and truncated computation domain in the model for the forward problem. We apply the (Bayesian) approximation error model for the compensation of modeling errors caused by domain truncation and a coarse computation mesh in DOT. The approach is tested with a three-dimensional example using experimental data. The results show that when the approximation error model is employed, it is possible to use mesh densities and computation domains that would be unacceptable with a conventional measurement error model.