Model-Based 3-D X-Ray Induced Acoustic Computerized Tomography

Abstract

X-ray-induced acoustic (XA) computerized tomography (XACT) is an evolving imaging technique that aims to reconstruct the X-ray energy deposition from XA measurements. Main challenges in XACT are the poor signal-to-noise ratio and limited field-of-view, which cause artifacts in the images. We demonstrate the efficacy of model-based (MB) algorithms for three-dimensional XACT and compare with the traditional algorithms. The MB algorithm is based on iterative, matrix-free approach for regularized-least-squares minimization corresponding to XACT. The matrix-free-LSQR (MF-LSQR) and the non-iterative model-backprojection (MBP) reconstructions were evaluated and compared with universal backprojection (UBP), time-reversal (TR) and fast-Fourier transform (FFT)-based reconstructions for numerical and experimental XACT datasets. The results demonstrate the capability of MF-LSQR algorithm to reduce noisy artifacts thus yielding better reconstructions. MBP and MF-LSQR algorithms perform particularly well with the experimental XACT dataset, where noise in signals significantly affects the reconstruction of the target in UBP and FFT-based reconstructions. The TR reconstruction for experimental XACT are comparable to MF-LSQR, but takes thrice as much time and filters the frequency components greater than maximum frequency supported by the grid, resulting loss of resolution. The MB algorithms are able to overcome the challenges in XACT and hence are vital for the clinical translation of XACT.

I. Introduction

X-rays are one of the widely used tools in clinics these days, commonly employed for imaging the body to investigate the structures of the internal organs [1-3], as well as for radiotherapy for treating tumors by guiding intense X-ray beams towards the tumor to destroy the malignant cells [4-5]. X-rays, being ionizing radiation, are known to be carcinogenic [6], and hence, techniques for X-ray based medical procedures which minimize the radiation exposure are desirable. Computed tomographic scans (commonly known as CT scans) [1-3] are very commonly used in clinics to obtain three-dimensional (3D) images of the organ/region of interest (ROI) in the body. This is achieved by rotating the X-rays and detectors around the body and collecting multiple projection datasets. X-ray-induced acoustic computerized tomography (XACT) is a modernistic imaging technique that aims to reconstruct the X-ray energy deposition (XED) map from boundary X-ray-induced acoustic (XA) measurements.

Absorption of pulsed X-ray radiation in tissues leads to thermoelastic expansion which produces XA waves [7]. These waves can be sensed by ultrasonic transducers around the irradiated ROI and fed to an XACT algorithm to reconstruct the XED in the ROI. Propagation of acoustic waves is inherently a 3D phenomenon, and hence unlike CT scans, XACT can potentially facilitate 3D imaging from XA measurements obtained from X-ray excitation from one direction, thereby reducing the radiation exposure. Lower scattering of X-rays and XA waves in tissues enables XACT [7]-[27] to image deeper in the body. The potential clinical applications XACT is currently being researched for are dose-monitoring during radiotherapy [11,12] and tomographic imaging [13,14].

XACT was first demonstrated by Xiang et al. [7] in 2013, where a chicken tissue with targets embedded was irradiated with X-ray pulses and the XA signals were collected by rotating a transducer element around the tissue. The backprojection (BP) algorithm was employed to evaluate two-dimensional (2D) reconstructions showing the embedded targets. Over the past ten years, researchers have explored several potential tomographic imaging applications of XACT such as imaging microcalcifications for breast cancer diagnosis [14], imaging tumors using fiducial markers [15], and bone imaging [16]. XACT using a synchrotron based X-ray source was demonstrated in [17] which overcomes the limitations of the traditionally used X-ray sources such as portable tubes and the linear accelerators (Linac). Typically, XACT signals require multiple averaging and hence higher radiation exposure and scanning time. In a recent publication [18], we demonstrated the capability of 2D XACT reconstruction from ring array measurements with a single pulse excitation of the sample. One of the bottlenecks for in vivo XACT imaging in clinics is the poor SNR of the XA signals. In [19], Choi et al demonstrated in situ XACT using Gastrografin, a conventional CT contrast agent which improved the SNR of the XA signals.

Radiation dosimetry is another application that XACT researchers across the globe have delved into. The XED and hence the strength of the collected XA signals are directly proportional to the deposited dose in the tissue. Several articles [11,20-23] discuss computational studies (both in 2D and 3D) exploring the feasibility of radiation dosimetry using XACT. The early experimental studies demonstrated 2D maps of the XED by Linac X-ray pulse in water [24,25] and veal liver tissue [26] were reconstructed. In [12,27], Kim et al demonstrated the capability of dose monitoring in 3D by scanning a spherically focused transducer element along a line at multiple depths. All these numerical and experimental studies indicate the potential of XACT for medical imaging and dosimetry. Clinical X-rays have been demonstrated to produce reasonable XA signal-strength and XA measurements can easily be carried out in clinics by integrating a coupling medium and US detection grid.

The XACT setups in many of the aforementioned studies use a single element transducer for detecting the XA signals. Such transducers have larger aperture size and hence the collected XA signals have reasonable SNRs. However, data collection for imaging purposes needs the transducer to be scanned around the ROI thus increasing the acquisition time and the radiation exposure. To overcome this drawback, XACT can employ a detection array of small transducer elements facilitating multi-channel data acquisition (DAQ), but at the cost of decreased SNR.

The inverse problem corresponding to XACT is to reconstruct the XED map in an ROI from the XA signals collected at various spatial locations around the ROI. This is traditionally achieved using the BP algorithms [28]. BP algorithms, though computationally efficient, carry certain drawbacks such as limited-view, and noisy artifacts for low SNR XA data. Model-based (MB) algorithms on the other hand, not only ameliorate these artifacts but can also facilitate incorporating experimental attributes such as acoustic inhomogeneity, finite detector size, etc. In our recent article [29], we demonstrated the first MB, 2D XACT reconstructions from ring-array measurements. We also demonstrated how MB reconstructions can diminish the ring artifacts which occur because of the nonuniform response among the detectors as a result of the interference between the multichannel DAQ electronics [30]. In this work, we demonstrate MB 3D XACT reconstructions and compare them with the traditional reconstructions. The MB reconstruction algorithm employed in this work is first validated on the numerical measurements generated using the k-wave toolbox [31] in spherical and planar detection geometries and then used to reconstruct a lead target from experimental XACT measurements. This we believe is the first illustration of 3D XACT using the MB algorithm.

The physics of XACT is similar to that of photoacoustic or optoacoustic tomography (PAT/OAT) [28]. The main difference between these two modalities is that XACT employs X-ray radiation and aims to image the X-ray absorption-based contrast, while PAT relies on optical radiation (~ hundreds of nanometers wavelengths) and images the optical absorption contrast in the region of interest. The penetration depth of PAT is limited to a few centimeters due to high optical scattering in tissues. X-rays on the other hand penetrate much deeper and high energy X-rays can penetrate through the human body. Due to the shallow penetration depths, PAT is typically used for imaging soft tissues where the variation in sound speed is very small (<10%) [32]. XACT on the other hand can have bones within the region of interest where the sound speed is about twice of that in water and soft tissues [32]. Moreover, PAT typically uses lasers with nanosecond pulse duration as the source while the commonly used X-ray sources such as LINACs typically have a pulsewidth of several microseconds. The shorter pulse duration in PAT facilitates higher photoacoustic efficiency as well as higher frequency content in the acoustic signals and hence high resolution imaging. XACT signals on the other hand typically have lower SNRs and frequency content. The high penetration depth of XACT also contributes toward lowering the frequency content of the XA signals because the higher frequency components of the XA waves originating deep in the tissue are strongly attenuated. These drawbacks of XACT with respect to PAT motivate the need for superior image reconstruction algorithms than the backprojection based ones, which are commonly employed in XACT. One of the applications that XACT is being researched for is radiation dosimetry which seeks a quantitative XED map in the treatment organ during radiotherapy. This further emphasizes the need for advanced image reconstruction algorithms which can facilitate quantitative imaging.

MB reconstruction schemes are based upon a model that relates the XED with the XA signals measured at detectors and XACT can borrow the model used in PAT, a sister modality that shares similar physics. MB reconstruction algorithms have been reported to be very successful in PAT and a detailed review of the 3D PAT models and corresponding reconstruction algorithms can be found in [33]. Here, we limit our discussion to discrete-to-discrete (D-D) forward models with homogeneous sound speed, negligible acoustic attenuation, and point detector assumptions. PAT forward model is inherently a continuous-to-discrete (C-D) mapping because the photoacoustic measurements are discrete while the optical energy deposition (pressure source) is continuous. To be able to employ the model in optimization based tomographic inversion, an equivalent D-D mapping is desired. To achieve this, the heat/pressure source map $H(\overrightarrow{r})$ needs to be represented in the discrete domain: $H(\overrightarrow{r})\equiv \underset{¯}{h}$. This is achieved by using expansion functions that define the variation of the pressure source in each discrete element. Kaiser–Bessel (KB) window functions [34-36] and linear interpolation functions [37-40] are some of the commonly used expansion functions used in PAT. In [29,30], we employed linear interpolation functions for modeling and subsequent image reconstruction in 2D XACT. In this manuscript, we chose $H(\overrightarrow{r})$ to be constant within each voxel to avoid computing the expansion coefficients during the iterative image reconstruction algorithm. We observed that with a high angular discretization for computing the surface integral in the acoustic propagation operator (detailed later in section II A), our model matches reasonably well with the simulations done using the k-wave toolbox, a popular open source PAT toolbox [31]. All the measurements (forward data) used in the numerical studies in this paper were also computed using the k-wave toolbox to avoid inverse crime.

D-D forward model is typically expressed in the form: $\underset{¯}{p}=\mathbf{M}\underset{¯}{h}$, with $\mathbf{M}$ being the model matrix relating the discrete domain XA measurements $\underset{¯}{p}$ with $\underset{¯}{h}$, the discrete domain representation of the XED. For 3D problems as well as for problems involving high resolution and sampling frequency, the computation and assembly of the model matrix can be time-consuming and storing it can be computationally demanding. Moreover, evaluation of reconstruction commonly involves regularized inversion of $\mathbf{M}$ which further increases the computational demands [29]. In this paper, we overcome these challenges by using a matrix-free (MF) approach for solving the least-squares (LS) problem (MF-LSQR) corresponding to XACT. This method employs functions for computing $\mathbf{M}\underset{¯}{u}$ and ${\mathit{M}}^{\mathit{T}}\underset{¯}{v}$ (matrix-vector multiplications) in MATLAB’s LSQR (LS with QR factorization) function to iteratively compute the minimizer to the XACT LS problem. We also demonstrate a non-iterative MB reconstruction (known as model-backprojection (MBP) [29,41,42]) which is computationally efficient, but only provides the structural information in the ROI.

In this manuscript, the theory and numerical implementation of the XACT forward model and corresponding tomographic problem are discussed in section 2. Section 3 deals with the computational validation of the MB reconstruction schemes and the phantom XACT experiments are described in section 4. Concluding remarks are offered in section 5.

II. Mathematical formulation

A. Forward Problem

Propagation of the XA waves in acoustically homogeneous and negligibly attenuating media following XED under thermal and stress confinement is governed by the wave equation [28]:

$$\frac{{\partial }^{2}p(\overrightarrow{r},t)}{\partial t^2}-v^2{\nabla }^{2}p(\overrightarrow{r},t)=\Gamma H(\overrightarrow{r})\frac{\partial \delta (t)}{\partial t},$$ (1)

with $\Gamma$, $v$, and $H$ being the Gruneisen constant, the acoustic propagation speed, and the XED map. Solution to Eq. (1) after dropping the multiplicative constant can be written as: [29]

$$\begin{gathered} p(\overrightarrow{r},t)=\frac{\partial }{\partial t}\left(\frac{1}{vt}{\int }_{S(\overrightarrow{r},t)}H({\overrightarrow{r}}^{\prime })d{S}^{\prime }(t)\right); \\ \mid \overrightarrow{r}-{\overrightarrow{r}}^{\prime }\mid =vt, \end{gathered}$$ (2)

where $S(\overrightarrow{r},t)$ is a spherical surface centered at a detector (located at $\overrightarrow{r}$) such that its radius at time t is $\mathrm{R}(\mathrm{t})=\mid \overrightarrow{r}-{\overrightarrow{r}}^{\prime }\mid =vt$. Eq. (2) can further be simplified to:

$$p(\overrightarrow{r},t)==\frac{\partial }{\partial t}\left(vt{\int }_{S(\overrightarrow{r},t)}H({\overrightarrow{r}}^{\prime })d\Omega \right).$$ (3)

The ROI (commonly a cube/cuboid which contains( or is expected to contain) the $\mathrm{H}(\overrightarrow{r})$ map) is discretized into voxels and the XED is expressed in the discrete domain as: $H(\overrightarrow{r})\equiv \underset{¯}{h}$. The value of the XED $(\mathrm{h}(\overrightarrow{r}))$ is given by

$$\begin{gathered} h(\overrightarrow{r})=\sum _{k=1}^{N_v}{h}_k{N}_k(\overrightarrow{r}), \\ {N}_k(\overrightarrow{r})=\{\begin{array}{cc}1\hfill & if\overrightarrow{r}liesinsidethe{k}^{th}voxel\hfill \\ 0\hfill & otherwise\hfill \end{array}\phantom{\}}, \end{gathered}$$ (4)

with $N_v$ being the number of voxels in the domain.

To compute the integration over the spherical surface $S(\overrightarrow{r},t)$ in Eq. (3), a unit sphere is first discretized into M uniformly distributed triangular spherical elements, each characterized by their centroids {${\widehat{\beta }}_1,{\widehat{\beta }}_2,\dots ,{\widehat{\beta }}_M$}. The co-ordinates of the quadrature points for a detector located at $\overrightarrow{r}$ and at time t are ${\overrightarrow{\alpha }}_i=\overrightarrow{r}+\text{vt}{\widehat{\beta }}_i$. The elements contributing to the integral over the spherical arc in Eq. (3) are simply the ${\overrightarrow{\alpha }}_i\mathrm{s}$ lying inside the ROI (Fig. 1). To reduce the computational burden, $h(\overrightarrow{r})$ was considered to be constant over each contributing element (total $M_e$ elements (say)). Hence, the surface integral in Eq. (3) for the $m^{th}$ detector location (${\overrightarrow{r}}_m$) and $q^{th}$ time instant is:

$$\begin{gathered} I({\overrightarrow{r}}_m,t_q)={\int }_{S(\overrightarrow{r},t)}H({\overrightarrow{r}}^{\prime })d\Omega \approx \sum _{i=1}^{M_e}H({\overrightarrow{r}}_i)\Delta {\Omega }_i \\ =\sum _{i=1}^{M_e}\Delta {\Omega }_i\sum _{k=1}^{N_v}{h}_k{N}_k({\overrightarrow{r}}_i) \\ {I}^{(m,q)}={\sum }_{i=1}^{M_e}\Delta {\Omega }_i{h}_k^i, \end{gathered}$$ (5)

where $H({\overrightarrow{r}}_i)$ denotes the XED at the centroid of the $i^{th}$ contributing quadrature element, $\Delta {\Omega }_i$ is the solid angle this element subtends at ${\overrightarrow{r}}_m$, and $h_k^i$ denotes the XED value at the voxel that contains ${\overrightarrow{r}}_{ı}$. Eq. (5) can further be expressed as

$$I^{(m,q)}={\underset{¯}{v}}^{(m,q{)}^T}\underset{¯}{h};\underset{¯}{h}=[h_1,h_2,\dots ,h_N{]}^T.$$ (6)

and the integrals for all the detectors at $q^{th}$ time instant will be:

$$I^{(q)}={𝒱}^{(q)}\underset{¯}{h},{𝒱}^{(q)}={\left[{\underset{¯}{v}}^{(1,q)},{\underset{¯}{v}}^{(2,q)},\dots ,{\underset{¯}{v}}^{(N_d,q)}\right]}^T.$$ (7)

Fig. 1:

Demonstration of the method for computing the surface integral in Eq. (3).

In the same way, the ${𝒱}^{(q=1,\dots ,N_t)}$ can be stacked into matrix $\mathit{V}$ and the integration for all the detectors at all time instances will be

$$\underset{¯}{I}=\mathit{V}\underset{¯}{h},\mathit{V}={\left[{𝒱}^{(1{)}^T},{𝒱}^{(2{)}^T},\dots ,{𝒱}^{(N_t{)}^T}\right]}^T$$ (8)

The differentiation in Eq. (3) can be written as:

$$\begin{gathered} p(\overrightarrow{r},t)=\frac{\partial }{\partial t}[vtI(\overrightarrow{r},t)]=vt\frac{\partial }{\partial t}I(\overrightarrow{r},t)+vI(\overrightarrow{r},t), \\ \approx vI(\overrightarrow{r},t)+vt\frac{I(\overrightarrow{r},t+\Delta t)-I(\overrightarrow{r},t)}{\Delta t} \end{gathered}$$ (9)

The XA measurements at all the detectors at the $q^{th}$ time step, can thus be written as:

$${\underset{¯}{p}}^{(q)}={ℳ}^{(q)}\underset{¯}{h},$$ (10)

Where

$$\begin{gathered} {ℳ}^{(q)}=v{𝒱}^{(q)}+v{\mathrm{t}}^{(q)}\left[\frac{{𝒱}^{(q+1)}-{𝒱}^{(q)}}{\Delta t}\right] \\ =vq{𝒱}^{(q+1)}+v(1-q){𝒱}^{(q)}. \end{gathered}$$ (11)

The full XA measurement vector $\underset{¯}{p}$ (dimension: $(N_d.N_t)\times 1$) can thus be expressed as:

$$\underset{¯}{p}=\mathbf{M}\underset{¯}{h}.,,\mathbf{M}={\left[{ℳ}^{(1{)}^T},{ℳ}^{(2{)}^T},\dots ,{ℳ}^{(N_t{)}^T}\right]}^T$$ (12)

with $\mathbf{M}$ being the model-matrix (dimension: $(N_d.N_t)\times N$).

B. Inverse Problem

Corresponding tomographic problem is to evaluate the XED map ($\underset{¯}{\widehat{h}}$) in the ROI from the discrete time-domain XA data (${\underset{¯}{p}}_{{}^{meas}}$) collected at detectors placed around the ROI. This can be written in the form of a regularized LS minimization problem [42], [43]:

$$\underset{¯}{\widehat{h}}=\text{arg}\underset{\underset{¯}{h}}{\text{min}}\Vert {\underset{¯}{p}}_{meas}-\mathbf{M}\underset{¯}{h}{\Vert }_2^2+{\lambda }^2\Vert \mathbf{R}\underset{¯}{h}{\Vert }_2^2,$$ (13)

where $\Vert .{\Vert }_2$ is the $L^2\text{-}norm$. $\mathbf{R}$ and $\lambda$ denote the regularization matrix and the regularization parameter respectively. In this paper, we used the Laplacian regularization ($\mathbf{R}=\text{Incidence matrix}$). Solution to Eq. (13) can be obtained by evaluating:

$$\underset{¯}{\widehat{h}}=({\mathbf{M}}^T\mathbf{M}+{\lambda }^2{\mathbf{R}}^T\mathbf{R}{)}^{-1}{\mathbf{M}}^{\mathit{T}}{\underset{¯}{p}}_{{}^{meas}}.$$ (14)

For computationally demanding reconstruction problems (with large number of measurements or nodes in the ROI), evaluating and storing $\mathbf{M}$ needs a lot of time and memory. Therefore, evaluating the reconstructions via Eq. (14) gets unrealistic. The minimizer to Eq. (13) can also be evaluated using the iterative LSQR method which utilizes direct computations of $\mathbf{M}\underset{¯}{u}$ and ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$, instead of using $\mathbf{M}$ explicitly.

Algorithm 1: MF evaluation of $\mathbf{M}\underset{¯}{u}$

$$\begin{array}{cc}\hfill & \mathbf{procedure}\text{Computing}\underset{¯}{p}=\mathbf{M}\underset{¯}{u};\hfill \\ \hfill & \underset{¯}{p}=[](\text{initialize})\hfill \\ \hfill & \mathbf{for}m=1to{N}_d\mathbf{do}\hfill \\ \hfill & \mathbf{for}q=1to{N}_t\mathbf{do}\hfill \\ \hfill & \overrightarrow{\alpha }{}_i\leftarrow \overrightarrow{r}+\text{vq}\Delta t{\widehat{\beta }}_i\hfill \\ \hfill & \text{find}\overrightarrow{\alpha }{}_{ix}\mid <\frac{x_0}{2}\&\mid \overrightarrow{\alpha }{}_{iy}\mid <\frac{y_0}{2}\&\mid \overrightarrow{\alpha }{}_{iz}\mid <\frac{z_0}{2}\hfill \\ \hfill & \text{find the voxels in which each}\overrightarrow{\alpha }{}_i\text{lies in:}\hfill \\ \hfill & \text{say voxel}k(=1,2,\dots ,n)\text{carries}{l}_k\overrightarrow{\alpha }{}_i\mathrm{s}\hfill \\ \hfill & I_q={\sum }_{k=1}^n{\sum }_{i=1}^{l_k}\Delta {\Omega }_i{u}_k,\hfill \\ \hfill & \mathrm{end}\mathrm{for}\hfill \\ \hfill & \underset{¯}{\zeta }=[I_2,2I_3,3I_4,\dots ,(N_t-1)I_{N_t},0{]}^T\hfill \\ \hfill & \underset{¯}{\mu }=[0,-I_2,-2I_3,-3I_4,\dots ,-(N_t-1)I_{N_t}{]}^T\hfill \\ \hfill & \underset{¯}{p}\leftarrow {\left[{\underset{¯}{p}}^T,[v(\underset{¯}{\zeta }+\underset{¯}{\mu }){]}^T\right]}^T\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \mathbf{return}\underset{¯}{p}\hfill \\ \hfill & \mathrm{end}\mathrm{procedure}\hfill \end{array}$$

Algorithm 2: MF evaluation of ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{u}$

$$\begin{array}{cc}\hfill & \mathbf{procedure}\text{Computing}\underset{¯}{\omega }={\mathit{M}}^{\mathit{T}}\underset{¯}{u};\hfill \\ \hfill & \underset{¯}{\omega }=zeros(Nodes,1)](\text{initialize})\hfill \\ \hfill & \mathbf{U}=\text{reshape}(\underset{¯}{u},N_d\times N_t)\hfill \\ \hfill & \mathbf{for}m=1to{N}_d\mathbf{do}\hfill \\ \hfill & {\underset{¯}{u}}_d\leftarrow [\mathbf{U}(m,:){]}^T\hfill \\ \hfill & \mathbf{for}q=1to{N}_t\mathbf{do}\hfill \\ \hfill & \overrightarrow{\alpha }{}_i\leftarrow \overrightarrow{r}+\text{vq}\Delta t{\widehat{\beta }}_i\hfill \\ \hfill & \text{find}\overrightarrow{\alpha }{}_{ix}\mid <\frac{x_0}{2}\&\mid \overrightarrow{\alpha }{}_{iy}\mid <\frac{y_0}{2}\&\mid \overrightarrow{\alpha }{}_{iz}\mid <\frac{z_0}{2}\hfill \\ \hfill & \text{find the voxels in which each}\overrightarrow{\alpha }{}_i\mathrm{s}\text{lie in:}\hfill \\ \hfill & \text{say voxel}k(=1,2,\dots ,n)\text{carries}{l}_k\overrightarrow{\alpha }{}_i\mathrm{s}\hfill \\ \hfill & \gamma ={\underset{¯}{u}}_d(q)(1-q)+{\underset{¯}{u}}_d(q-1)(q-1)\hfill \\ \hfill & \underset{¯}{\in }=zeros(Nodes,1)\hfill \\ \hfill & \mathbf{for}k=1ton\mathbf{do}\hfill \\ \hfill & \in (k)=\gamma v{\sum }_{i=1}^{l_k}\Delta {\Omega }_i,\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \underset{¯}{\omega }\leftarrow \underset{¯}{\omega }+\underset{¯}{\in }\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \mathbf{end}\mathbf{for}\hfill \\ \hfill & \mathbf{return}\underset{¯}{\omega }\hfill \\ \hfill & \mathbf{end}\mathbf{procedure}\hfill \end{array}$$

The algorithms for matrix-vector multiplications: $\mathbf{M}\underset{¯}{u}$ and ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ are given in Algorithm-1 and Algorithm-2 respectively. We also demonstrate a non-iterative, model-based backprojection (MBP) reconstruction [42], which is computed using:

$$\underset{¯}{\widehat{h}}={\mathbf{M}}^{\mathit{T}}{\underset{¯}{p}}_{{}^{meas}}.$$ (15)

MBP is an approximation to Eq.(14), when ${\mathbf{M}}^T\mathbf{M}\ll {\lambda }^2{\mathbf{R}}^T\mathbf{R}$, with $\mathbf{R}=\mathbf{I}$ (Tikhonov regularization).

III. Numerical Studies

The detection geometries as well as the phantom considered in the numerical study are depicted in Fig. 2. To perform the numerical validation of the developed algorithms, the numerical XA measurements were generated using the k-wave toolbox [31], and the algorithms for performing reconstructions were coded and executed on MATLAB R2019a. All computations were performed on a workstation with AMD Ryzen Threadripper 3960X 24-Core Processor (3.79 GHz), 256 GB installed RAM on Windows 10, 64-bit operating system.

FIG. 2.

Schematic of the full and partial view setting in (a) spherical and (b) planar detection geometries. (c) 3D representation, (d) multiple slices (e) z=0 plane slice of the phantom used in the numerical study.

A. Validation of the MF-LSQR algorithms

The forward and adjoint operator evaluations $\mathbf{M}\underset{¯}{u}$ and ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ were validated for a simple 3D problem for which the matrix $\mathbf{M}$ could be computed and stored. In this problem, the region of interest (ROI) was chosen to be a cube of 1 cm edge length, discretized with 0.25 mm grid resolution (41 x 41 x 41 nodes). On the surface of a sphere of 4 cm, 80 detectors were placed and the 106 temporal measurements for each detector were evaluated at 2 MHz sampling frequency. For this problem, the model-matrix $\mathbf{M}$ of size (80*106 x 41³) was computed and stored. The pressure source was chosen to be a sphere of radius 2.5 mm, located at the center of the ROI. The discrete-domain representation of this pressure source was chosen as $\underset{¯}{u}$. The validation of the on-the-fly $\mathbf{M}\underset{¯}{u}$ function with the explicit matrix-vector product evaluation is demonstrated in Figs. 3(a) and (b).

FIG. 3.

(a) Pointwise matching of the model-matrix-based and on-the-fly $\mathbf{M}\underset{¯}{u}$ computation and (b) pointwise difference. Stem plots of (c) pointwise matching of the model-matrix-based and on-the-fly ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{u}$ computation and (d) pointwise difference. (e) Convergence of the relative residual with increasing iteration number, for an inverse crime scenario.

For the validation of the on-the-fly ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ function, $\underset{¯}{v}$ was chosen as a random vector of size (80*106 x 1) and the stem plots of the on-the-fly function and the explicit matrix-vector product ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ evaluation is demonstrated in Fig 3(c) and the difference between these quantities is shown in Fig. 3(d). The magnitudes of these differences are ~10⁻¹⁵ times smaller than the values which validate the on-the-fly ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ function.

For this problem setup, we also perform a reconstruction for the inverse crime scenario, where the measurements as well as the MF-LSQR reconstruction, both were evaluated on the same grid and using the same model without additive noise or employing regularization. Fig. 3(e) shows the convergence of the (log₁₀ of the) relative residuals for 177 iterations when the relative residual decreases <10⁻¹⁰.

B. Generating measurements using k-wave toolbox

As shown in Fig. 2, a cube with 2 cm edge length was chosen as the ROI, and consisted of a cross (amplitude 1) and eight spheres (amplitude 0.5) which were located in a larger cylinder (amplitude 0.2). We performed simulations for spherical as well as planar detection geometries, depicted in Fig. 2(a) and (b) respectively- (i) Spherical detection: Sphere with 4 cm radius, with 1280 elements uniformly spread over the sphere (S1), upper half of the sphere (z>0) with 630 detectors (S2), and a cup array with 312 detectors on the spherical surface with 1cm ≤ z ≤ 3cm (S3); (ii) Planar detection: 10 cm x 10 cm grid (P1) and 4 cm x 4 cm grid (P2), both with 2 mm separation between each detection elements. The 3D perspective view, multiple slices and the z=0 slice of the phantom used for the numerical validation is shown in Fig. 2(c),(d) and (e), respectively.

The computational domain was discretized into cubical voxels (size: h = 0.2mm). Such a discretization leads to the maximum supported frequency of f_max= v/2h = 3.75 MHz. The sampling frequency has therefore been chosen to be f_s = 10MHz, satisfying the Nyquist criterion (f_s>2f_max). The simulated signals were then contaminated with white Gaussian noise to have 5dB SNR in the measurements.

C. Evaluating the reconstructions

Universal backprojection (UBP) [28], time-reversal (TR), fast Fourier-transform (FFT)-based (both using the k-wave toolbox [31]) and the MB algorithms (MBP and Laplacian regularized MF-LSQR) were employed to evaluate the reconstructions at 0.25 mm grid resolution. The k-wave toolbox supports FFT-based reconstructions only for the planar detection geometry using the function ‘kspacePlaneRecon’ for 3D (corresponding 2D analog is ‘kspaceLineRecon’). For evaluating the MB reconstructions, the unit sphere (for numerically computing the integration over the spherical arc in Eq.(3)) described in section IIA, was discretized into M~5x10⁵ elements.

The stopping criterion and the strategy to tune the regularization parameter are essential for an iterative image reconstruction algorithm. The functional one aims to minimize (Eq. (13) in the manuscript) decreases with each LSQR iteration. The computation time for LSQR minimization is directly proportional to the number of iterations. Therefore, it is reasonable to enforce the stopping criterion when the change in the minimizers (reconstructions in our case) becomes negligible during the iterations. For the computational test cases considered in the manuscript, we observed the variation of the correlation coefficient [44] of the LSQR minimizer with respect to the solution from the previous iteration. The plot corresponding to test cases P2, S2 and S3 are provided in the supplementary material (Fig. SM1). Based on the plot we see >99.9% correlation between subsequent updates after ~10 LSQR iterations and it resulted reasonable reconstructions. Therefore, throughout the manuscript, we used 10 iterations for evaluating the MF-LSQR reconstructions. A similar number of iterations were used for model-based PACT evaluations in [40,42].

The regularization parameter ($\lambda$) is typically chosen based on the trade-off between the data fidelity $\Vert {\underset{¯}{p}}_{{}^{meas}}-\mathbf{M}\underset{¯}{h}{\Vert }_2$ and the regularization term $\Vert \mathbf{R}\underset{¯}{h}{\Vert }_2$. In medical imaging it is more common to choose regularization parameter that results in images that are maximally useful for specified diagnostic tasks. The L-curve method is one of the popular approaches for evaluating the regularization parameter for optimization problems [45]. L-curve plots the log of the two terms from the error functional (given in Eq.(13) in the manuscript): (1) log of the regularized solution norm $(\log (\Vert \mathbf{R}\underset{¯}{h}{\Vert }_2))$ versus the (2) log of the corresponding residual norm $\log \Vert {\underset{¯}{p}}_{{}^{meas}}-\mathbf{M}\underset{¯}{h}{\Vert }_2$ for multiple regularization parameters ($\lambda$) with $\underset{¯}{h}$ being solutions obtained for these $\lambda \mathrm{s}$. The optimal $\lambda$ is the one that corresponds to the corner of the L-curve. Therefore, the L-curve demonstrates the trade-off between minimizing these two terms, which is the basic idea behind regularization.

As evident from its standard definition, plotting an L-curve would need computing the solution $\underset{¯}{h}$ for multiple $\lambda \mathrm{s}$, which would require high computation time and is unrealistic. Alternatively, the aforementioned toolbox has a function ‘l_curve’ which takes the measurements (${\underset{¯}{p}}_{{}^{meas}}$) and the matrices U and s obtained from the compact form of the singular value decomposition (SVD) of the model-matrix $\mathbf{M}$, given as: M = U*diag(s)*V', where U is m x min(m,n), s is min(m,n)x 1, and V is n x min(m,n). However, the algorithm presented in this manuscript does not compute $\mathbf{M}$ explicitly, and hence this method cannot be used for evaluating the appropriate regularization parameter. Therefore, the regularization parameters were chosen by tuning based on our computational experience. Future research will focus on efficient evaluation of an appropriate regularization parameter for MF-LSQR based XACT reconstruction.

D. Reconstruction results

The TR, FFT, UBP, MBP and MF-LSQR reconstructions of the phantom obtained from the spherical and planar detection geometries (Fig. 2(a),(b)) are demonstrated in Fig. 4 and Fig. 5 respectively. The time taken for computing the reconstructions ($\tau$) and the correlation coefficients ($\rho$) [44] w.r.t. the original phantom are tabulated in Table 1. As expected, both the backprojection algorithms (UBP and MBP) and the FFT-based algorithm, being non-iterative in nature, are more efficient in evaluating the reconstructions than the MF-LSQR algorithm. The TR algorithm takes the longest for evaluating the reconstructions because it needs to evaluate wave-propagation for each time-step. The structures in the cross-section are reasonably reconstructed for all the algorithms. However, the UBP and FFT reconstructions carry strong noisy artifacts which are further amplified in the limited view settings and therefore have lower correlation w.r.t. the original phantom. The MBP does better than the UBP and FFT in terms of suppressing the noise and hence yields relatively better correlation coefficients in comparison to the UBP. As mentioned in section 1, MBP is not a quantitative reconstruction and this is indicated in the contrasts of the reconstructed entities. For example, the background region (0.5cm<∣x∣<1cm) where the true value is 0 unit, MBP reconstructs ~ 0.2 units. Similarly the contrast of the spheres and the cross w.r.t. the rectangular region (−0.5cm<x<0.5cm) in MBP reconstructions are ~1.6 and 3 respectively; corresponding true values are 2.5 and 5, which were found to be reasonably well reconstructed by UBP and MF-LSQR as these both are quantitative reconstruction algorithms. The MF-LSQR reconstructions were evaluated with Laplacian regularization, which assisted in ameliorating the noisy artifacts. TR reconstruction is based on the pseudospectral wave propagation [31] and restricts the frequency components higher than the maximum frequency supported by the spatial resolution (f_max = v/2h) which for h = 0.25 mm resolution, comes out to be 3 MHz. This inherent low-pass filtering leads to reduced noisy artifacts in the TR reconstructions. However, it also leads to the loss of information of the edges of the target which are encoded in the high frequency components (Fig. 4 column I and Fig. 5 column II). Moreover, the TR reconstructions were found to be highly susceptible to the limited view problem and the detector sparsity. To further support this point, the TR reconstruction for S1 geometry, with the pressure measurements interpolated all over the spherical detection surface is demonstrated in the supplementary material Fig. SM2. Across all numerical evaluations performed in the study, MF-LSQR yielded the best reconstructions and hence higher $\rho$ values w.r.t. the TR, FFT, UBP and MBP counterparts.

FIG. 4.

Spherical geometry (S1, S2, and S3) reconstruction results. Column-I (a,e,i,m,q,u):TR, column-II (b,f,j,n,r,v): UBP, column-III (c,g,k,o,s,w): MBP, and column-III: (d,h,l,p,t,x) MF-LSQR reconstructions.

FIG. 5.

Planar geometry (P1 and P2) reconstruction results. Column-I: (a,f,k,p) FFT-based, Column-II: (b,g,l,q) TR, Column-III: (c,h,m,r) UBP, column-IV: (d,i,n,s) MBP, and column-V: (e,j,o,t) MF-LSQR reconstructions.

TABLE. 1.

Computation time ($\tau$) and correlation coefficient ($\rho$) for numerical studies

View	Spherical Geometry								View	Planar Geometry
	TR		UBP		MBP		MF-LSQR			FFT		TR		UBP		MBP		MF-LSQR
	$\rho$	$\tau \mathrm{s}$	$\rho$	$\tau (\mathrm{s})$	$\rho$	$\tau (\mathrm{s})$	$\rho$	$\tau (\mathrm{s})$		$\rho$	$\tau \mathrm{s}$	$\rho$	$\tau (\mathrm{s})$	$\rho$	$\tau (\mathrm{s})$	$\rho$	$\tau (\mathrm{s})$	$\rho$	$\tau (\mathrm{s})$
S1	0.59	952	0.78	4.6	0.83	81	0.92	873	P1	0.58	13	0.20	2714	0.66	8.9	0.74	173	0.82	1971
S2	0.30	937	0.68	2.5	0.74	40	0.90	436	P2	0.37	2.3	0.41	763	0.44	1.7	0.46	39	0.52	570
S3	0.25	944	0.48	1.5	0.56	21	0.79	236

The ease of computation associated with the UBP makes it computationally most efficient. Although, MBP is also termed as ‘backprojection’, the operator corresponding to MBP evaluation given in Algorithm 2, is computationally more involving than the UBP, and hence takes longer to compute. Moreover, at this point, while our UBP implementation has been vectorized and made efficient, improving the model-based algorithms is a work in progress. We do expect to accelerate the MBP computation to achieve the performance similar to UBP. The MB evaluation of reconstructions on the other hand, involves rigorous computation and hence takes longer. However, MB algorithms offer the flexibility of incorporating regularization and experimental attributes such as finite detector size, acoustic heterogeneity, etc. in the model and hence are desirable. As discussed in Section II B, the MBP needs a single evaluation of ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$. The MF-LSQR is an iterative scheme and each iteration requires a {$\mathbf{M}\underset{¯}{u},{\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$} evaluation; ${\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$ being computationally more demanding. In our work, MF-LSQR algorithm was found to converge at ~ 10 iterates. Therefore, in all the testcases, MF-LSQR takes ~10-15 times longer than MBP.

IV. XACT experiment and reconstruction

The developed MB schemes were employed to evaluate reconstructions from experimental XACT data. The schematic of the setup is depicted in Fig. 6. A ‘+’ shaped lead target (thickness: 1/16 inch) was placed at the center of a phantom prepared using 3% agar (BactoTM, Becton, Dickinson and Company, NJ, the United States). This phantom was submersed in water and excited using a portable and battery driven pulsed X-ray source (XR200, Golden Engineering, IN, USA) with tube energy: 150 kVp and 40° beam divergence which emits X-ray pulses with 50 nanoseconds pulse-width at 10 Hz repetition rate. The averaged dose at the target located at ~ 30.5 cm from the source was estimated to be 2.6 mR/pulse for the diverging X-ray pulse. The x-ray source was shielded in a cage and was controlled through a remote from the outside. A 256-element planar array transducer (size: 4cmx4cm; Doppler Electronic Technologies Co. Ltd., Guangzhou, China) with 1 MHz central frequency was used to collect the XA signals. These signals were then directed to a customized 256-channel data acquisition (DAQ) device (Legion ADC, Photosound Technologies Inc., Houston, the United States) which had a 256-channel preamplifier integrated to enhance the SNR of the XA signals. A scintillator (Ce:Lu₂SiO₅ crystal, MTI Corporation, CA, USA) was placed behind the water tank to collect the X-rays exiting the water tank. The scintillator converts the collected X-rays to visible light. The photons were then detected by an integrated variable-vain, InGaAs avalanche photodetector (APD410C, Thorlabs, NJ, USA) which sends the trigger signal to a wave generator which further relays the trigger to the Legion ADC DAQ system as square pulsed with 4V strength for precise timing control.

FIG. 6.

Schematic of the XACT experiment.

The acquired experimental XA data were then used to evaluate the TR, FFT, UBP, MBP, and MF-LSQR-based XACT reconstructions. The results thus obtained are demonstrated in Fig. 7, which shows the 3D reconstructions (first row), maximum intensity projections (MIP- second row), and the reconstruction slice (at a plane 3.5 cm away from the detection grid- third row) from the TR, FFT, UBP, MBP, and MF-LSQR reconstructions, respectively. The evaluation times and contrast-to-noise ratios (CNRs) [46] are provided in Tab. 2. The CNRs (C1 and C2) were computed for two regions in the blue and yellow rectangles respectively, w.r.t. the background (green rectangle). The weak XACT signal is corrupted with strong noise. In addition to the low SNR, the detection geometry has very limited view for collection. Therefore, the resolution in the reconstructed images is poor as expected. The noisy artifacts overshadow the reconstructed target in the UBP reconstructions and hence UBP reconstruction has low CNRs in the slice as well as the MIP.

FIG. 7.

XACT experimental results: (a,b,c,d,e) 3D view, (f,g,h,i,j) maximum intensity projection (MIP), and (k,l,m,n,o) reconstruction slice (at a plane 3.5 cm away from the detection grid) from the TR, FFT-vased, UBP, MBP, and MF-LSQR reconstructions, respectively.

TABLE. 2.

Computation time ($\tau$) and CNRs (C1 and C2) for experimental XACT reconstructions

	FFT		TR		UBP		MBP		MF-LSQR
	Sliced	MIP	Sliced	MIP	Sliced	MIP	Sliced	MIP	Sliced	MIP
C1	0.80	0.23	1.69	2.47	0.50	0.20	0.84	1.15	1.19	1.90
C2	0.85	0.68	3.07	3.01	0.70	0.46	1.11	1.45	1.31	2.21
$\tau (\mathrm{s})$	18		2081		1.5		65		693

The FFT-based reconstructions perform marginally better than the UBP which also reflects in the CNRs given in Table 2. While the target is barely perceivable in the UBP reconstruction, it is clearly visible in both the MB and TR reconstructions, thus yielding higher CNRs as compared to the UBP. The Laplacian regularization has stronger noise ameliorating capability and hence the MF-LSQR reconstruction yields better CNRs than UBP, FFT, and MBP for the chosen regions. As mentioned earlier in section III D, TR reconstruction restricts the frequency components higher than the maximum frequency supported by the spatial resolution (f_max = v/2h) which for h = 0.25 mm resolution, comes out to be 3 MHz. This leads to reducing the noise in the reconstructions and hence TR yields the best CNR values among all the other reconstructions. However, it also leads to the loss of information of the edges of the target which are encoded in the high frequency components thus the resolution is adversely affected. It is evident in Fig. 8, where the reconstructed profiles (across four locations displayed by white lines in Fig. 7 (o)) are compared for MF-LSQR and TR algorithms. The FWHM of these profiles were also calculated by fitting them to Gaussians and displayed in the legends of each plot. For each of the plots, the FWHMs of TR reconstructions were ~15-35% larger than the MF-LSQR counterparts. Moreover, the TR takes about thrice as much time as the MF-LSQR does. In order to take higher frequency components into account, the spatial grid size (h) for TR needs to be decreased accordingly which will further increase the computation time. The Laplacian regularization has noise ameliorating capability and hence the MFLSQR reconstruction yields good CNRs for the chosen regions without compromising the resolution of the reconstructed structures.

FIG. 8.

Cross-sectional profile plots of the LSQR and TR reconstruction along (a) line 1 (b) line 2 (c) line 3 and, (d) line 4 shown in Fig. 7 (o). The FWHM of the profiles (given in each figure legend) were obtained by Gaussian fitting the reconstructed profiles.

The electromagnetic interference in the data acquisition is the main source of the noise in the XA signals. Additionally, the reflections of the head-wave and the XA waves between the phantom and detection array and inaccuracy in the co-ordinates of the elements in the planar detection grid also contribute to the artifacts in the XACT reconstructions.

V. Conclusion

Model-based algorithms were employed for evaluating 3D XACT reconstructions and their efficacy was demonstrated on numerical and experimental datasets. Unlike the traditional MB algorithms which require explicit computation and storage of the measurement-matrix ($\mathbf{M}$), the algorithm used (MF-LSQR) employs routines for on-the-fly evaluation of {$\mathbf{M}\underset{¯}{u},{\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$} to solve the Laplacian-regularized LS problems, thus reducing the computation times and memory costs corresponding to the evaluation and storage of $\mathbf{M}$. Algorithm for evaluating non-iterative MBP reconstruction is also provided. The MB schemes were used to evaluate the reconstructions for numerical and experimental XACT datasets which were compared with the traditional UBP reconstructions. Due to the regularization, MB algorithms were able to ameliorate the noisy artifacts, while the structures in UBP and FFT-based reconstructions were overshadowed by the noisy artifacts. The difference between the performance of MB and the traditional UBP and FFT-based algorithms is even more evident in the experimental XACT reconstructions, where the target is clearly visible in the MB reconstructions, while it is barely discernible in the UBP reconstruction and the FFT-based algorithm performing marginally better than the UBP.

The TR restricts frequency components higher than the maximum frequency supported by the spatial resolution which leads to reduced noise in the reconstructions. Therefore, TR yields the best CNRs for the experimental XACT. However, it also leads to the loss of information of the sharp structures and edges of the target which are encoded in the high frequency components thus the resolution is adversely affected. We found out that the FWHMs of TR reconstructions were ~15-35% larger than the MF-LSQR counterparts. The TR is based upon wave-propagation at each time step, and hence the evaluation of TR reconstruction takes much longer than the FFT, UBP, and MBP algorithms. For evaluating the experimental reconstructions, TR took about thrice as much time as the iterative MF-LSQR did. In order to take higher frequency components into account, the spatial grid size (h) for TR needs to be decreased accordingly which will further increase the computation time. Moreover, the simulation results also demonstrate that TR is more susceptible to the limited view problem and low detector density. The iterative MF-LSQR performed the best across all the testcases due to the superior noise suppression by the Laplacian regularizer without compromising the resolution of the reconstructed structures.

To the best of our knowledge, the results shown in this paper are the first demonstration of the model-based 3D XACT. The computational efficiency of the proposed MB algorithms is governed by the functions for {$\mathbf{M}\underset{¯}{u},{\mathbf{M}}^{\mathbf{T}}\underset{¯}{v}$} computations. Work is ongoing to accelerate these function using GPUs in order to facilitate high-resolution/real-time/large scale XACT imaging. Unlike the conventional reconstruction algorithms such as UBP, FFT-based, and TR, MB algorithms facilitate incorporating experimental attributes such as acoustic heterogeneity distribution, finite detector aperture, etc.. Integrating such attributes in the MB algorithms will be the focus of the future research, which we believe is the right step towards the clinical translation of XACT.