A practical guide for model-based reconstruction in optoacoustic imaging

Optoacoustic (OA, photoacoustic) imaging capitalizes on the low scattering of ultrasound within biological tissues to provide optical absorption-based contrast with high resolution at depths not reachable with optical microscopy. For deep tissue imaging applications, OA image formation commonly relies on acoustic inversion of time-resolved tomographic data. The excitation of OA responses and subsequent propagation of ultrasound waves can be mathematically described as a forward model enabling image reconstruction via algebraic inversion. These model-based reconstruction methods have been shown to outperform alternative inversion approaches and can further render OA images from incomplete datasets, strongly distorted signals or other suboptimally recorded data. Herein, we provide a general perspective on model-based OA reconstruction methods, review recent progress, and discuss the performance of the different algorithms under practical imaging scenarios.

1 Introduction

Development, implementation and optimization of image formation algorithms is essential for advancing the performance of biomedical imaging modalities [1–3]. Optoacoustic (OA, photoacoustic) imaging has experienced an unprecedented growth in the last 10–15 years to become a powerful technology covering a large range of spatial and temporal imaging scales and providing otherwise-unavailable functional and molecular information deep from living tissues [4–8]. Rapid evolution of the OA technology has resulted in a myriad of embodiments based on different light delivery and ultrasound (US) detection methods [9–12]. Various types of sensors with detection bandwidth tailored to the desired resolution-depth range [13–16] and acquisition geometries (i.e. spatial distribution of US detectors relative to the sample) have further been developed [17–22]. Many of these systems have found use in biomedical research studies and clinical applications [23–28]. The great diversity of possible hardware designs represents an important advantage of the OA technology [29]. On the other hand, significant variability of image formation approaches, independently developed for each particular configuration [30–33], may compromise reproducibility and reliability of the reported experimental results thus hindering the development of standardized methodologies enabling accurate quantification of bio-markers [34].

OA is inherently a tomographic imaging modality. This implies that OA images can only be accurately reconstructed if signals are acquired at a set of locations enclosing the sample with sufficient angular coverage [35–37]. OA excitation is mainly performed with short (10⁻¹⁰–10⁻⁷ s) light pulses. In this case, the collected signals around the sample correspond to a solution of an initial value problem expressed as a Radon-type transform that depends on the arrangement and shape of US sensors [38–40]. Much like for x-ray CT, analytical inversion can be performed by means of a filtered back-projection (FBP) formula, which in practice is discretized to reconstruct an image from a finite number of signals [41–44]. However, accurate image reconstruction implies a more complex representation of the physical problem. The negative impact of insufficient spatial sampling and noise in tomographic imaging with FBP algorithms has long been recognized [45]. Note that the OA forward model corresponds to a different type of transform than that in x-ray CT, but the term FBP is also used in the OA literature as it is conceptually equivalent. The shape and detection bandwidth of the sensors as well as acoustic propagation effects cannot accurately be accounted for by FBP formulas, which have only been developed for specific acquisition or scanning geometries. Model-based (MB) inversion schemes, which determine the desired solution (reconstructed image) via minimization of a cost function, are known to overcome the major limitations of FBP algorithms [39, 46–48]. Apart from the so-called data fidelity term, the cost function can additionally include regularization terms penalizing unlikely solutions given the available a priori knowledge on the sample [49–56]. Also, the solution can be constrained e.g. not to have negative values having no physical meaning [57–59]. Another important aspect to consider is the fact that the admissible solutions of the tomographic inversion problem can be restricted by expressing the images as a combination of a set of basis functions. Proper selection of these functions along with regularization terms enable exploiting compressed-sensing-based concepts to reconstruct an image from a relatively low number of signals [60–62]. Generally, MB reconstruction provides a flexible framework for the reconstruction of OA images and is generally applicable for any physical configuration.

In this review article, we provide a practical guide for MB reconstruction in OA imaging. We focus on describing the enhanced performance with respect to alternative reconstruction algorithms rather than on the mathematical foundations, which are briefly discussed without loss of generality for short-pulsed excitation. We further show that MB reconstruction enables capabilities unattainable with conventional reconstruction approaches, such as super-resolution imaging beyond the acoustic diffraction barrier, image reconstruction using a single detector or high-resolution imaging through the skull bone.

2 The optoacoustic forward model

In practice, MB reconstruction is performed by considering a discrete-to-discrete linear model mapping a finite-dimensional representation of the optical absorption distribution to the collected signals at a set of locations and time points. The mathematical derivation of the OA forward model depends on the temporal profile of the light beam and can be done in the time or frequency domains [63–65]. Without loss of generality, we briefly describe below the time-domain model for standard short-pulsed excitation verifying the so-called stress and thermal confinement conditions [66] (Figure 1A). In this case, the temporal profile of the light beam can be approximated as a Dirac delta $\delta \left(t\right)$ and optical absorption results in an initial pressure rise $p_0\left(\mathrm{r}\right)=\Gamma \left(r\right)H\left(r\right)$, being $\Gamma \left(r\right)$ the dimensionless Grueneisen parameter and $H\left(r\right)$ the energy being thermalized per unit volume. Propagation of an US wave optoacoustically generated in soft tissues is mathematically described by [67]

$\frac{{\partial }^{2}p\left(r,t\right)}{{\partial t}^2}-{c\left(r\right)}^2\rho \left(r\right)\nabla \bullet \left(\frac{1}{\rho \left(r\right)}\nabla p\left(r,t\right)\right)=\mathrm{\Gamma }\left(r\right)H\left(r\right)\frac{\partial \delta \left(t\right)}{\partial t},(1)$

FIGURE 1

FIGURE 1. Time-domain optoacoustic (OA) forward model for short-pulsed excitation. (A) Schematic representation of the excitation of ultrasound (US) waves with light. An example of time-resolved OA signal is shown. For a uniform acoustic medium with a defined speed of sound, the time-of-flight linearly scales with the distance to the measuring location. (B) Discretization of the OA image by considering the initial pressure at a define grid of points. (C) Discrete-to-discrete OA forward model expressed in a matrix form mapping the initial pressure expressed as a column vector to the measured pressure at different time points also expressed as a column vector. The columns of the model-matrix correspond to the pressure signals for a unit initial pressure at the corresponding pixel (or voxel). Note that only one collected signal is shown and the model-matrix is generally built by considering multiple measuring locations.

being $p\left(r,t\right)$ the acoustic pressure field, $c\left(r\right)$ the speed of sound and $\rho \left(r\right)$ the mass density of the medium. Equation 1 corresponds to a wave equation with a well-defined source term that can be expressed as an initial value problem [67], hence a mathematical solution exists. Generally, the pixels (or voxels) of a Cartesian grid enclosing the sample are considered to build the basis functions used to approximate $H\left(r\right)$ e.g. via weighted interpolation (Figure 1B) [68, 69], i.e., Eq. 1 is approximated by

$\frac{{\partial }^{2}p\left(r,t\right)}{{\partial t}^2}-{c\left(r\right)}^2\rho \left(r\right)\nabla \bullet \left(\frac{1}{\rho \left(r\right)}\nabla p\left(r,t\right)\right)\approx \frac{\partial \delta \left(t\right)}{\partial t}{\displaystyle \sum }_{i=1}^N{p}_{0,i}{k}_i\left(r\right),(2)$

where $p_{0,i}$ is the initial pressure at the center of the ith voxel and $k_i\left(r\right)$ is the interpolation function. The solution of Eq. 2 can be expressed as

$p\left(r,t\right)={\displaystyle \sum }_{i=1}^N{p_{0,i}p}_i\left(r,t\right),(3)$

where $p_i\left(r,t\right)$ is the generated pressure field for a unit initial pressure rise at the ith voxel. An analytical expression of $p_i\left(r,t\right)$ can be derived in some cases, e.g. for a uniform non-attenuating acoustic medium [48]. Note that soft tissues are modelled as a heterogeneous fluid in Eq. 1. The wave equation can be significantly more complex for hard (solid) and/or absorbing tissues such as the skull bone [70]. However, an initial value problem can still be defined, thus the time-resolved $p_i\left(r,t\right)$ signals for all voxels can be obtained e.g. with numerical simulations [30]. Alternatively, these can be measured experimentally by scanning a sub-resolution absorber [71]. This approach further enables accounting for the response of the US transducer(s) used to collect the OA signals. In this case, $p_i\left(r,t\right)$ corresponds to the collected voltage signals rather than the pressure field, which are affected by the transducer response and can also be theoretically modelled [72–74]. The discrete-to-discrete OA forward model is built from the values of $p_i\left(r,t\right)$ for the positions of the transducer(s) and the sampling instants. Specifically, $p\left(r,t\right)$ at these positions and instants, expressed as a column vector $\mathbf{p}$, is given as a function of $p_{0,i}$ for the voxels of the grid, also expressed as a column vector ${\mathbf{p}}_{\mathbf{0}}$, as

$\mathbf{p}=\mathbf{A}{\mathbf{p}}_{\mathbf{0}}.(4)$

The model matrix $\mathbf{A}$ represents the discrete-to-discrete OA forward model. The columns of this matrix correspond to the $p_i\left(r,t\right)$ signals for each voxel of the grid (Figure 1C). If the reconstruction grid is selected to match the scanning geometry, translational and rotational scanning symmetries as well as axial symmetries of the transducer surface can be considered to build the model matrix [75–77]. In this manner, the $p_i\left(r,t\right)$ signals for all positions can be derived from that corresponding to a reference position. This facilitates storage of the model matrix in memory, which if often challenging, particularly for three-dimensional imaging.

3 The inverse problem

The OA forward model enables calculating the theoretical pressure (or transducer) signals as a linear function of the initial pressure distribution (sources). MB OA image reconstruction corresponds to the inverse problem aiming at estimating the initial pressure rise from the acquired signals. This is generally formulated as an optimization-based problem involving minimizing the energy functional defined as

${\mathbf{p}}_{\mathbf{0},\mathbf{s}\mathbf{o}\mathbf{l}}=\underset{{\mathbf{p}}_{\mathbf{0}}\left(\ge 0\right)}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left\{{\Vert {\mathbf{p}}_{\mathbf{m}}-\mathbf{A}{\mathbf{p}}_{\mathbf{0}}\Vert }_2+\lambda R\left({\mathbf{p}}_{\mathbf{0}}\right)\right\},(5)$

where ${\mathbf{p}}_{\mathbf{m}}$ are the measured signals expressed in a vector form. The first term of Eq. 5 corresponds to the data fidelity term driving the solution towards the observed data. $R\left({\mathbf{p}}_{\mathbf{0}}\right)$ corresponds to the regularization term, being $\lambda$ the regularization parameter. Optionally, ${\mathbf{p}}_{\mathbf{0}}$ can be constrained e.g. not to have negative values (${\mathbf{p}}_{\mathbf{0}}\ge 0$) in order to avoid solutions with no physical meaning [57]. Regularization is generally included in the inversion problem to reduce the effects of noise and data incompleteness and/or to incorporate prior knowledge on the image. The simplest and most standard regularization strategy to stabilize the inversion, known as Tikhonov regularization, is based on the L2 norm of the image, i.e., $R\left({\mathbf{p}}_{\mathbf{0}}\right)={\Vert {\mathbf{p}}_{\mathbf{0}}\Vert }_2$. In this case, the unconstrained solution of Eq. 5 is analytically expressed as

${\mathbf{p}}_{\mathbf{0},\mathbf{s}\mathbf{o}\mathbf{l}}={\left({\mathbf{A}}^{\mathrm{T}}\mathbf{A}+\lambda \mathbf{I}\right)}^{-1}{\mathbf{A}}^{\mathrm{T}}{\mathbf{p}}_{\mathbf{m}},(6)$

being ${\mathbf{A}}^{\mathrm{T}}$ the transpose of the model matrix. Note, however, that Eq. 6 is impractical in most cases due to the large size of the model matrix, particularly for high-resolution three-dimensional imaging. Thereby, the inversion problem in Eq. 5 is solved iteratively e.g. with algorithms based on steepest descent or conjugate gradient methods [58]. The matrix-vector multiplications involving the model matrix and its transpose are the most computationally demanding operations in iterative algorithms. In some cases, the model matrix cannot be stored in memory and the elements of the model matrix are calculated in each iteration to perform these operations. Graphics processing unit (GPU)-based parallelization of these operations is then essential to accelerate the inversion process. For this, efficient methods for on-the-fly calculation of the elements of the model matrix e.g. based on look-up tables are needed, since GPU-storage of the entire matrix is generally not possible [55, 74].

If a very large regularization term (strong Tikhonov regularization) is chosen to suppress artifacts and enhance signal-to-noise ratio (SNR), the term $\lambda \mathbf{I}$ becomes dominant over ${\mathbf{A}}^{\mathrm{T}}\mathbf{A}$, thus in Eq. 6 can be approximated as

${\mathbf{p}}_{\mathbf{0},\mathbf{s}\mathbf{o}\mathbf{l}}={\mathbf{A}}^{\mathrm{T}}{\mathbf{p}}_{\mathbf{m}}.(7)$

${\mathbf{A}}^{\mathrm{T}}$ is the algebraic adjoint (transpose) of the discrete-to-discrete OA forward model. The adjoint of the continuous operator can also be calculated prior to discretization, which enables exploiting fast wave propagation solvers [78]. Equation 7 can be interpreted as a model back-projection (MBP, non-iterative) reconstruction approach associated to the discrete-to-discrete model and enabling fast reconstructions. It can also be regarded as a cross-correlation with the theoretical signals generated by each voxel and can be interpreted in terms of a matched filter [79].

4 Outperforming filtered back-projection

MB (iterative) reconstruction methods were used to reconstruct the first x-ray CT images. However, they were quickly replaced by less complex FBP algorithms and were not implemented in commercial scanners until around 10 years ago [80]. The wide use of FBP algorithms in x-ray CT fostered the development of similar formulas for OA tomography [41, 42, 81, 82] whose discretized versions are still in common use. Particularly, GPU-based implementations of FBP algorithms enable real-time preview of three-dimensional data, which is essential for the clinical translation of the OA technology [83].

Sparse spatial sampling of signals is common in OA tomography since transducer arrays are routinely used for real-time imaging [84]. Indeed, ad-hoc design of US arrays with optimized OA performance generally focuses on increasing the angular coverage to avoid limited-view effects rather than on reducing the inter-element pitch [85]. Much like in x-ray CT, reduced spatial sampling is known to result in streak-type artefacts in the images. These artefacts, illustrated in Figure 2A, are a consequence of signals from strong absorbers such as blood vessels being back-projected into arcs, which average out for a sufficient number of measuring locations but can lead to strong noise in the images if the number of collected signals is relatively small. The capability of MB OA algorithms to mitigate this and other sources of noise has been demonstrated with the development of the first iterative algorithms [46]. Figure 2B shows an example of a cross-sectional MB algorithm significantly reducing the noise in the OA images of the mouse brain with respect to those obtained with FBP [86]. Iterative MB reconstruction could also clearly enhance the performance of full-body three-dimensional small-animal imaging systems [48]. In the example shown in Figure 2C, OA images reconstructed with FBP are clearly afflicted by noise even though a large number of signals (11,520) were used for reconstruction. The noise could be reduced by considering an optimization-based framework including a total-variation (TV) regularization term, i.e., $R\left({\mathbf{p}}_{\mathbf{0}}\right)={\Vert \sqrt{{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial x}\right)}^2+{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial y}\right)}^2+{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial z}\right)}^2}\Vert }_1$. The effects of streak-type artefacts in three-dimensional OA images are more prominent if these are acquired with hand-held scanners based on spherical arrays, currently being used in clinical trials [87, 88]. Indeed, even though three-dimensional imaging is possible with a single laser pulse, the number of signals that can be simultaneously collected with an array is generally significantly lower than that required according to the Nyquist spatial sampling criterion [89]. Figure 2D shows an example of an image of the finger vasculature of a healthy volunteer [55]. The MBP approach (Eq. 7) is shown to reduce the background noise of FBP images, while still begin able to provide real-time feed-back during the scans. Iterative MB inversion based on Tikhonov regularization clearly provides an enhanced imaging performance, but results in a longer computation time in the order of 5 s.

FIGURE 2

FIGURE 2. Performance comparison of model-based (MB) and filtered back-projection (FBP). (A) Example of streak-type artefacts produced in the reconstructed images as a function of the number of sensors considered. Adapted from [171] with permission from Nature Publishing Group. (B) Comparison of the cross-sectional images of the mouse brain rendered with FBP and MB. Adapted from [86] with permission from American Institute of Physics. (C) Comparison of the three-dimensional full-wave images reconstructed with FBP and MB based on total-variation regularization. Adapted from [48] with permission from Institute of Physics. (D) Comparison of the three-dimensional images of the finger vasculature of a healthy volunteer obtained with FBP, model back-projection (MBP) and MB based on Tikhonov regularization. Reprinted from [55] with permission from Institute of Electrical and Electronics Engineers.

MB reconstruction is more flexible than FBP as it is generally applicable regardless of the acquisition geometry. Moreover, it facilitates guiding the solution towards an image consistent with prior knowledge on the sample. For example, the images reconstructed with FBP are known to be affected by negative values with no negative meaning [90]. These can be avoided by including non-negative constraints in the optimization-based framework [57, 91, 92]. The regularization term in the energy functional to be minimized (Eq. 5) also plays an essential role in the contrast, resolution and overall quality of the image being rendered. For example, an L1-based regularization term in the image domain, i.e., $R\left({\mathbf{p}}_{\mathbf{0}}\right)={\Vert {\mathbf{p}}_{\mathbf{0}}\Vert }_1$, tends to set many voxels to zero (or close to zero) values. This was exploited to enhance the spatial resolution beyond the acoustic diffraction barrier [76, 92, 93]. TV-based regularization is also commonly used as it enables mitigating noise while preserving sharp edges [52, 53, 94–98]. Figure 3A displays a comparison of the images of a freely-swimming zebrafish larvae reconstructed with FBP and MB including a TV regularization term in the spatial and temporal domains, i.e., $R\left({\mathbf{p}}_{\mathbf{0}}\right)={\Vert \sqrt{{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial x}\right)}^2+{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial y}\right)}^2+{\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial z}\right)}^2+{k\left(\frac{\partial {\mathbf{p}}_{\mathbf{0}}}{\partial t}\right)}^2}\Vert }_1$. A sequence of images is simultaneously reconstructed with this approach following a sparse acquisition scheme [99]. MB reconstruction then enables significantly reducing the required number of signals for each laser pulse and thus increasing the achievable frame rate [100].

FIGURE 3

FIGURE 3. Model-based (MB) reconstruction from sparse data based on advanced regularization methods. (A) Performance of three-dimensional MB reconstruction based on total-variation (TV) regularization in the spatial and temporal domains. As a reference, the images reconstructed with filtered back-projection (FBP) are shown in the left column as a function of the number of time-resolved signals considered. The equivalent MB images are shown in the other columns. TVxN indicates that a sequence of N images is simultaneously reconstructed. Reprinted from [99] with permission from Optica Publishing Group. (B) Performance of the deep gradient descent (DGD) algorithm when considering signals sub-sampled by a factor of 4 in the spatial and temporal domains. The image obtained with DGD after 5 iterations (top right) is compared to the image obtained with the model back-projection (MBP) method (1 iteration, top left) and iterative TV-based MB algorithm (20 iterations, bottom left). As a reference, the image reconstructed with the same TV-based MB algorithm (20 iterations) from the full data is also shown (bottom right). Adapted from [101] with permission from Institute of Electrical and Electronics Engineers.

Defining the optimal regularization strategy is generally challenging as prior information may not be available and/or cannot be expressed by standard regularization terms. Also, regularizers known to provide a good performance in other imaging modalities may be suboptimal for OA. The MB performance can be enhanced by learning realistic information on the expected image content and how to best incorporate this in an iterative reconstruction approach. For this, convolutional neural networks have been suggested. These can be used e.g. to iteratively update the reconstructed OA image from the previous image and the gradient of the fidelity term [101]. The regularization effects are then learned from the data during training. The performance of this reconstruction approach, termed deep gradient descent (DGD), is shown in Figure 3B when using x-ray CT angiography images for training. The updated image after 5 iterations of the DGD method considering data sub-sampled by a factor of 4 is shown to clearly reduce the noise of the MBP image, corresponding to the first iteration (Figure 3B, top). The DGD can also provide more accurate results than standard iterative MB reconstruction based on TV regularization for the same data sparsity level (Figure 3B, bottom left), as validated with the equivalent TV-based image reconstructed from the full data (Figure 3B, bottom right).

5 Modelling the transducer response

FBP formulas are derived by considering the spatio-temporal pressure wavefield at a surface surrounding the sample. As mentioned above, OA signals are often acquired at a relatively sparse distribution of measuring locations. Also, standard piezoelectric US transducers have a finite size and a finite detection bandwidth, thus the collected signals generally differ from the acoustic pressure.

The effects of the transducer are generally characterized as a convolution with the so-called electrical impulse response (EIR) and the spatial impulse response (SIR). The EIR corresponds to the signal for an impulse-type wavefront incident normal to the surface of the transducer [102]. The SIR is associated to the finite aperture of the transducer and has long been known to degrade the tangential resolution in the reconstructed OA images [72, 103]. Both the EIR and the SIR can be incorporated into a MB reconstruction framework. As an example, Figure 4A displays the OA images reconstructed with the simulated signals for a flat transducer with 6 mm diameter circularly scanned around the sample with 50 mm radius [103]. Point absorbers at distances of up to 25 mm away from the scanning center could be accurately reconstructed with a MB algorithm accounting for the SIR, while the OA images reconstructed with FBP clearly show a progressive reduction in the azimuthal resolution at peripheral regions. For a finite-size transducer perfectly matched to the acoustic coupling medium, the SIR is approximated as the integral of the pressure on the active surface [104]. This enables estimating analytically the SIR of some types of sensors [72, 105]. Alternatively, the active surface of the sensor can be numerically approximated to estimate the SIR [47, 73, 106]. Figure 4B shows the OA microscopy images of the subcutaneous murine vasculature obtained with a broadband spherical polyvinylidene difluoride (PVDF) transducer [76]. A MB algorithm incorporating the SIR clearly outperformed a back-projection-based synthetic aperture focusing technique (SAFT). Specifically, the vascular network at different depths was more clearly resolved with the MB considering a L2-norm-based regularization term. Also, an increase in resolution beyond the barrier of acoustic diffraction was achieved with L1-norm-based regularization to the detriment of the number of vascular branches being observed. A similar enhanced performance was observed with MB reconstruction in other acquisition geometries [55, 107–110], which substantiates the general applicability of this approach. It is important to note that the estimated SIR is based on an approximation. Wave propagation and energy conversion within piezoelectric materials is generally more complex. A more accurate estimation of the spatially-dependent total impulse response (TIR) of the transducer can be performed by scanning a sub-resolution absorber [111, 112]. The TIR can then be incorporated into the MB reconstruction framework to further enhance the reconstruction accuracy [113]. The TIR may also be measured by considering the response of small particles flowing in blood [114–116], which may facilitate enhancing the imaging performance in vivo.

FIGURE 4

FIGURE 4. Accounting for the finite aperture of the ultrasound transducer. (A) Comparison of the optoacoustic (OA) images reconstructed with MB and filtered back-projection (FBP) algorithms considering the simulated signals for a flat transducer circularly scanned around the sample. Reprinted from [103] with permission from Optica Publishing Group. (B) Comparison of the OA microscopy images of the subcutaneous murine vasculature obtained with a FBP-based synthetic aperture focusing technique (SAFT, left), a MB algorithm considering a L2-norm regularization term (middle) and a MB algorithm considering a L1-norm regularization term (right). Reprinted from [76] with permission from Wiley Online Library.

6 Accounting for acoustic propagation effects

Direct inversion of a Radon-type transform into a FBP formula is only possible for a uniform non-attenuating acoustic medium. However, biological tissues are heterogeneous acoustic media further absorbing part of the acoustic energy. In soft tissues, the speed of sound can change within a range of approximately ±10% [117]. This mainly affects the time-of-flight of US waves, which results in a time-shift of the collected signals [118]. Other effects such as reflections, refractions or scattering can further result in strong artefacts in the images [119–121], but these are only prominent in other tissues featuring stronger acoustic mismatch such as bones or lungs. On the other hand, frequency-dependent acoustic attenuation causes both a decrease in intensity and a reduction in bandwidth of the collected signals, which in turn results in quantification errors and in a loss of spatial resolution [122, 123].

MB reconstruction frameworks provide a flexible means for compensating for all these effects. For uniform non-attenuating acoustic media, discrete-to-discrete time-domain OA forward models can be built by discretizing the Poisson-type integral corresponding to the solution of Eq. 1 [68]. The effects of speed of sound changes can then be corrected by modifying the integration surface (or curve) based on the induced time-of-flight changes [49]. Full-wave MB algorithms based on the exact OA wave equation have also been suggested to mitigate the effects of speed of sound heterogeneities [33, 78, 124, 125]. Also, so-called joint-reconstruction MB algorithms aiming at simultaneously recovering an OA image and the speed of sound distribution have been suggested [126, 127]. These algorithms were shown to be ill-conditioned and numerically unstable [128], but could be used to reconstruct cross-sectional images of mice by considering a low-dimensional parametrization of the sound speed distribution [129]. MB algorithms can also be used to reconstruct the speed of sound distribution from transmitted US waves [130]. Figure 5A shows a comparison of the OA images of the murine liver region obtained with MB algorithms assuming a uniform speed of sound and considering two different speed of sound values for the tissue and coupling medium, respectively [129]. TV regularization was used in both cases. The distortion induced in the images due to speed of sound changes is higher when using heavy water as coupling medium [131] or air-coupled transducers [132, 133]. Acoustic attenuation effects can also be accounted for with MB algorithms. These correspond to a convolution of the time-domain signals (columns of the model matrix) with an attenuation impulse response function [123]. Recently, cross-sectional OA tomography based on a full-ring array was combined with transmission US methods providing speed of sound and attenuation maps [134]. This information can then be exploited in MB reconstruction frameworks to advance the OA imaging performance. In principle, full-wave MB algorithms can account for any effect associated to acoustic propagation in strongly mismatched tissues. However, this approach is impractical unless accurate knowledge of the distribution of acoustic properties is available. The artefacts induced in the images can be mitigated by weighting the signals and the model matrix so that less distorted signals have a higher influence in the inversion algorithm [135]. This approach was initially suggested for FBP algorithms [136, 137]. Figure 5B shows a comparison of the coronal cross-sectional OA images of the mouse brain reconstructed with a standard MB algorithm and with a weighted MB (wMB) algorithm considering the probability that the signals are distorted by scattering events [138], indicating the enhanced performance achieved with the latter. Arguably, the greatest challenge in OA is to achieve high-resolution imaging through the skull bone, particularly in humans. Transcranial US propagation is known to be affected by reflections, refractions, mode conversion and other effects [139, 140]. OA forward models accounting for some of these effects have been suggested [141], but their applicability in vivo is challenged by the lack of accurate knowledge on cranial acoustic and dimensional properties. Recently, it has been demonstrated that the distortion of optoacoustically-generated waves is locally preserved after transcranial propagation [142]. A MB reconstruction algorithm can then be developed based on this memory effect (MBM). Figure 5C shows the enhanced performance achieved with the MBM algorithm with respect to a standard MB algorithm assuming a uniform speed of sound. The strong distortion induced by the skull is clear in the MB-reconstructed images, while high-resolution images of point absorbers could be obtained with the MBM algorithm.

FIGURE 5

FIGURE 5. Accounting for acoustic heterogeneities in the sample. (A) Comparison of the cross-sectional optoacoustic (OA) images of the mouse liver reconstructed with a model-based (MB) algorithm considering a uniform speed of sound (top) and a MB algorithm considering a different speed of sound within the mouse and the water coupling medium (MB2, bottom). Adapted from [129] with permission from Society for Industrial and Applied Mathematics. (B) Comparison of the cross-sectional OA images of the mouse brain reconstructed with a standard MB algorithm (top) and a MB algorithm weighted with the probability the collected signals are distorted by scattering or reflection events (wMB, bottom). Reprinted from [138] with permission from Wiley Online Library. (C) Transcranial OA images reconstructed with a standard MB algorithm (left) and with a MB algorithm built from a reference signal based on the OA memory effect (MBM, right) [142].

7 Compressed acquisition and reconstruction

MB frameworks are also particularly suitable for recovering signals or images from a few signal samples in the spatial and/or temporal dimensions. Mathematically, this corresponds to finding a solution of an underdetermined linear system and is the basis of compressed sensing methods [143]. Efficient recovery implies two conditions, namely that the signal (or image) exhibits sparsity in some domain and that the sampling matrix verifies the so-called restricted isometry property. The lack of speckle noise in OA images promotes sparsity, thus facilitates defining compressed acquisition and reconstruction schemes.

Compressed-sensing-based methods have been widely used in OA for reconstructing images from so-called partial data [50, 144–147]. Generally, this corresponds to signal acquisition at a spatially sparse location of sensors. Note that this is fundamentally different than the limited-view problem. Indeed, signals acquired under limited-view conditions lack sufficient information to achieve accurate OA reconstructions regardless the spatial and temporal sampling density [36]. An approach to recover information from partial data consists in formulating a MB problem to reconstruct the OA signals corresponding to dense spatial sampling and subsequently perform image reconstruction [83]. Alternatively, MB image reconstruction capitalizing on compressed-sensing principles can be performed, which generally implies L1-norm-based regularization terms [50, 144–146, 148, 149]. It is important to take into account that the advantage of sparse sampling is two-fold. On the one hand, compressed data acquisition significantly reduces the complexity of the transducer array and associated electronics, thus facilitates the development of low-cost OA imaging systems. On the other hand, the imaging rate can be significantly accelerated for a given data throughput capacity. Recently, OA imaging with a single time-resolved signal has been achieved by encoding the location of absorbers via complex propagation of the generated US waves. Three examples of systems based on this principle are shown in Figure 6. Figure 6A shows the OA images of the cortical murine vasculature reconstructed by considering the time-resolved signal acquired with a single-element transducer and single-shot excitation [150]. For this, US wave propagation through an ergodic relay causing multiple reflections was exploited. The model-matrix was experimentally calibrated by scanning a focused laser beam. Figure 6B shows the image seven sparsely-distributed synthetic hairs obtained with a different system based on a reverberant cavity also causing multiple reflections of US [151]. In this case, the signal from an individual synthetic hair was used for calibration. Figure 6C displays the image of a zebrafish larva obtained by exploiting acoustic scattering to physically encode the position of optical absorbers in the acquired signals [71]. For this, an imaging system based on ultrasound propagation through a randomized scattering medium was used. The model-matrix was experimentally calibrated by scanning an individual absorbing microsphere.

FIGURE 6

FIGURE 6. Model-based (MB) image reconstruction with time-resolved optoacoustic (OA) signals encoding the position of absorbers via complex propagation of ultrasound (US) waves. (A) Image of the cortical vasculature of the mouse brain obtained by exploiting US propagation through an ergodic relay. Adapted from [150] with permission from Nature Publishing Group. (B) Image of seven synthetic hairs sparsely distributed within an acoustically reverberant cavity causing multiple reflections of the US waves. Reprinted from [151] with permission from Optica Publishing Group. (C) Image of a zebrafish larva obtained by exploiting US transmission through a randomized scattering medium. Adapted from [71] with permission from American Physical Society.

8 Summary and outlook

In this article, we described the basis of both iterative and non-iterative MB reconstructions in OA imaging and reviewed recent work on this topic. The reported results, based on OA models independently developed by different groups, have systematically demonstrated an enhanced performance with respect to conventional reconstruction methods. In particular, the commonly-used FBP algorithms are often afflicted by noise and other artefacts, even when considering an arrangement of sensors for which an accurate FBP formula - derived from the inversion of a Radon-type transform - is available. Iterative methods provide a flexible means for mitigating these artefacts, further rendering a solution consistent with a priori knowledge on the sample. This is achieved by properly selecting regularization terms, constraints or number of iterations. Moreover, the spatial- and frequency-dependent response of the US transducer(s) as well as acoustic heterogeneities and attenuation can be accurately accounted for with MB methods. In this manner, it was possible to advance the OA imaging capabilities to a new level of performance e.g. by enabling breaking through the resolution limit dictated by acoustic diffraction. Considering the large variety of existing OA embodiments and the fact that the OA hardware is continuously being upgraded, new MB methods can be developed or adapted to new systems. Thereby, further research on MB reconstruction is expected.

The existence of a linear OA forward model stems from the thermoelastic generation mechanism of US within biological tissues, regardless the complexity of the acoustic media or the type and arrangement of US sensors [152]. We have shown that mathematical modelling of this effect results in a well-defined source term in the time-domain wave equation corresponding to short-pulsed excitation. Similarly, wave equations in the time or frequency domains can be derived for other types of time-varying light sources [153]. This is fundamentally different from pulse-echo US or other coherent imaging modalities based on back-scattered waves from different types of sources. For homogenous media, a solution of the wave equation exists and a discrete OA forward model, defined as a model matrix, can be theoretically derived from it. However, analytical derivation of the forward model is significantly more challenging when considering the propagation of US waves through complex media or through piezoelectric sensing materials. In this case, the model-matrix can be experimentally calibrated by collecting the OA signals generated by sub-resolution optical absorbers at a grid of points covering the region of interest. The number of signals required for experimentally building the model-matrix can be significantly reduced by considering the OA memory effect if the isoplanatic zone is relatively large due to symmetries in the scanning geometry [154]. Indeed, MB reconstruction based on an experimentally calibrated matrix enabled exploiting the complex propagation of US waves to encode the absorption distribution in a single OA signal or imaging through strongly mismatch acoustic media such as the skull bone.

MB methods are also commonly used for linear un-mixing of spectrally-distinctive chromophores such as oxygenated and deoxygenated haemoglobin from OA images acquired at multiple wavelengths [155, 156]. Multi-spectral (multi-wavelength) OA imaging is a powerful approach for molecular imaging applications provided accurate quantification of the bio-distribution of specific molecules is achieved [4, 5, 23, 157–159]. The combination of reconstruction and un-mixing steps into a single model-matrix has been shown to render more quantitative results if regularization parameters and non-negative constraints are properly selected [160]. Also, the model-matrix can incorporate the wavelength-dependent light fluence distribution within the sample, which can be estimated with different methods [161–165]. Thereby, the so-called spectral coloring effects, which hamper quantification of tissue oxygenation, can potentially be mitigated [156].

In recent years, deep learning has rapidly emerged as an appealing alternative for tomographic image reconstruction and processing. MB methods are sometimes categorized as knowledge-driven as they are based on minimizing the error between estimations of a physical model and actual data. On the contrary, deep learning methods are based on “reconstructing” the inversion model by minimizing the error between predicted and ground truth data. This data driven approach has similarities with experimentally-calibrated MB methods but the computationally efficiency can greatly be enhanced once the network has been trained. However, the ways deep neural networks process data are mathematically not fully understood with serious concerns raised regarding validity of the results. Nevertheless, in the OA field deep learning has been used for image quality improvement as well as for directly reconstructing images from the acquired signals [166–179]. Also, it was possible to train neural networks to learn the regularization term in an iterative MB inversion framework [101, 180, 181]. The latter hybrid method combines the advantages of both approaches further highlighting the importance of physical modeling for enhancing reliability and practical applicability of newly developed neural networks. The regularization performance in this case depends on the data quality as well as on the amount of data used for training. Combination of MB and deep learning methods is expected to emerge as a promising research direction.

In summary, MB reconstruction is a powerful tool for enhancing image quality and performance of OA imaging systems. Its efficacy derives from accurate physical modelling of US generation and propagation further accounting for experimental imperfections and other effects using either mathematical derivations or experimental calibrations. It has been demonstrated that MB approaches may facilitate super-resolution imaging, accurate image reconstruction from partial data as well as imaging through strongly aberrative media. The MB framework can further be exploited for processing multi-spectral or time-lapse data. Performance of MB approaches can be enhanced by incorporating neural networks, e.g. to act as regularizers.

Author contributions

XLD-B and DR wrote the manuscript.

Acknowledgments

XLD-B acknowledges funding support from the Helmut Horten Stiftung.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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