Model-based Multi-material Decomposition using Spatial-Spectral CT Filters

Abstract

Spectral CT with multiple contrast agents has been enabled by energy-discriminating detectors with multiple spectral channels. We propose a new approach that uses spatial-spectral filters to provide multiple beamlets with different incident spectra for spectral channels based on “source-side” control. Since these spatial-spectral filters yield spectral channels that are sparse, we adopt model-based material decomposition to directly reconstruct material densities from projection data. Simulation studies in three-and four-material decomposition experiments show the underlying feasibility of the spatial-spectral filtering technique. This methodology has the potential to facilitate imaging of multiple contrast agents simultaneously with relatively simple hardware, or to improve spectral CT performance via combination with other established spectral CT methods for additional control and flexibility.

I. INTRODUCTION

Spectral CT is an emerging technology that permits decomposition and density estimation for multiple material components within an image volume. In particular, spectral CT has enabled simultaneous imaging of multiple contrast agents for applications including simultaneous iodine-bismuth imaging for angiography/lung imaging [1] and three-agent iodine-gadolinium-bismuth imaging [2] for multiphasic renal [3] studies. A number of new contrast agents are also in development including gold nanoparticles for angiography [4], mammography [5], and targeted imaging of HeLa cells [6], lung adenocarcinoma [7], and colorectal liver metastasis [8]; xenon for lung ventilation [9]; bismuth sulphide nanoparticles for lymph nodes [10]; and tantalum oxide nanoparticles for cartilage [11]. However, most spectral CT has focused on single contrast agent imaging (e.g. iodine) – inevitably leading to systems that are optimized for that agent. With the emergence of new contrast agents and simultaneous imaging of multiple agents, there is need for sufficient flexibility in data acquisition to acquire high-quality spectral data for many contrast agents.

A number of different strategies have been investigated to enable spectral CT. Methods include the use of dual-sources [12], kV-switching [13], split-filters [14], dual-layer detectors [15], and energy-discriminating photon-counting detectors [16]. Many photon-counting detectors have the flexibility to provide several energy bins for spectral discrimination which enables multi-material decomposition. However, individually, most of the other methods typically only easily allow two different spectral channels limiting their use in multiple contrast agent studies. For example, “source-side” spectral variation is often limited to two x-ray sources, alternation between two energies in kV-switching, or two filters in a split-filter design where spectral filters cover exactly one-half of the fan-beam x-ray. With only two spectral channels, only two (or, three using a constraint) different materials may be estimated as part of a decomposition. Thus, a strategy for more control over the number and form of x-ray spectra available for data acquisition has the potential to enable and improve multi-material decomposition in CT.

In this work, we introduce a new concept wherein the x-ray beam is spectrally modulated across the face of the detector using a repeating pattern of filter materials. Such spatial-spectral filters allow for collection of many different spectral channels using “source-side” control. However, in contrast to other spectral techniques that provide mathematically complete projection data, spatial-spectral filtered data is sparse in each spectral channel – making traditional projection-domain [17] or image-domain [18] material decomposition difficult to apply. Thus, we adopt direct model-based material decomposition [19]–[21] which combines reconstruction and multi-material decomposition, and permits arbitrary spectral, spatial, and angular sampling patterns.

Spatial-spectral filters can be interpreted as an extension of split-filters methods [14] that divide the beam into two different spectra for each half-fan – instead, dividing the x-ray beam into several different beamlets. The proposed spatial-spectral approach also includes mechanical translation of the filter to vary spectral sampling patterns. Such an approach shares similarities with other recent “source-side” filtering innovations in CT acquisition including beam-shaping approaches using multiple aperture devices [22], interrupted-beam acquisitions for sparse data [23], and grating-oriented line-wise filtration for dual-energy CT [24].

In this paper we introduce the concept of spatial-spectral filtering for multi-material CT decomposition. Simulation experiments are conducted demonstrating the basic feasibility of the approach. Various spatial-spectral filter designs are explored and applied to multi-contrast imaging studies in simple digital phantom studies.

II. METHODS

A. Spatial-Spectral Filtering

An overview of the spatial-spectral filtering approach is shown in Figure 1. The main idea is to place a tiled filter with different materials in front of the x-ray source. For example, a repeating pattern of several material types can be used to shape the spectrum of a number of beamlets across the face of the detector. This kind of spectral encoding through spatial filtering is widely applied in optical imaging – e.g. Bayer filters [25] used in visible light detectors for color imaging. For x-ray imaging, practical filters may be constructed from materials with k-edges in the diagnostic range. This includes elements from Z = 50 (tin, k-edge at 29.2 keV) to Z = 85 (bismuth, k-edge at 90.5 keV). A sample repeating filter that uses a tiled pattern of bismuth, tungsten, and erbium filters is shown in Figure 1. The spectra produced by a 0.4 mm thick filter and a typical 120 kVp x-ray spectrum are shown in Figure 1B. Note the distinct edges in the spectra that coincide with the k-edges of those material filters. The exact number and combination of materials for filter construction can be optimized for particular contrast agents; however, it is straightforward to include several different filters for a diversity of spectral channels.

Fig. 1.

Illustration of the spatial-spectral filtering concept. A) The spatial-spectral filter is composed of a repeating pattern of attenuating materials used to shape the x-ray beam into beamlets with different spectra. Materials with k-edges in the diagnostic energy range are potentially of greatest utility for spectral shaping. B) In a sample spatial-spectral filter application, a 120 kVp spectrum emitted from the x-ray tube is modified with a repeating pattern of bismuth, tungsten, and erbium filters (0.4 mm thickness) to achieve three different spectral channels. The total fluence over all energies is normalized to the integral of the bismuth-filtered spectrum. C) Adding the ability to translate the spatial-spectral filter permits variation of the sampling pattern. A simple constant velocity linear translation as a function of rotation angle is shown.

While one could potentially implement spectral CT with a static spatial-spectral filter, this has the potential to have poor overall sampling. For example, for a static filter, the center of the object being scanning may only be probed with a single spectrum (based on the filter placed at the central ray). To provide increased flexibility in the spatial-spectral sampling pattern, we propose to translate the filter with rotation. Even a simple constant velocity linear translation will improve the sampling homogeneity across spectral channels. Such a linear translation and an illustration of the resulting projection data is shown in Figure 1C. Note that each spectral channel is sparse but all channels can be collected in a standard acquisition.

B. Model-based Material Decomposition

In order to reconstruct such sparse data, we will adopt a direct model-based material decomposition (MBMD) approach. The authors have previously developed a MBMD algorithm [26] based on related development in advanced high-resolution CT reconstruction [27]. This MBMD approach uses a forward model for projection data where mean measurements are

$$y_i={\sum }_e{s}_{e,i}\text{ exp}\left(-{\sum }_m{p}_{e,m}{[\mathbf{\text{A}}{x}_m]}_i\right)$$

where the i^th measurement is formed by projection (via A) of m material density maps, x_m, and scaling by the energy-dependent mass attenuation coefficients, p_e,m. Each measurement has an energy-dependent factor, s_e,i, which incorporates the local incident spectrum induced by the moving spectral-spatial filter as well as the detector sensitivity (e.g., energy-dependent scintillator stopping power, secondary quanta proportional to energy, etc.).

The objective function for the MBMD approach uses a Gaussian log-likelihood function and standard roughness penalty regularization for each material basis. The nonlinear objective function is solved using an optimization transfer approach based on separable surrogate functions. Details of the algorithm can be found in [26] and [27]. For all reconstructions/material decompositions in this work, 500 iterations of the algorithm are applied initializing the water material basis with a thresholded, binarized FBP image with a density of 1000 mg/ml, and all other material bases set to zero. All reconstructions used a 256 × 256 volume of 0.5 mm voxels.

C. Simulation Experiments

Two preliminary investigations of spatial-spectral filters are explored in this work. First, we consider a three-material decomposition example using a digital phantom composed of water, iodine, and gold – emulating a multiple contrast agent study with both iodinated contrast agent and gold nanoparticles (an emerging contrast agent). The digital phantom is illustrated in Figure 3A and includes a 10 cm cylinder of water and various concentrations of iodine (2.5–15 mg/ml) and gold (1–4 mg/ml) in 2 cm diameter cylinders. We simulated a CT system with 120 cm source-to-detector distance and 60 cm source-to-axis distance with an indirect energy integrating detector (with 600 μm thick CsI scintillator) with 512–0.5 mm flat-panel pixels, and rotated 360° in 360–1° increments. We used a 120 kVp source spectrum (prior to spatial-spectral filtration but with typical inherent filtration due to the housing, and shown in Figure 1B).

Fig. 3.

Three-material decompositions using spatial-spectral filters. A) The ground-truth digital phantom and a representative reconstruction/material decomposition for the Bi-Au-Lu-Er spatial-spectral filter. B) Error image volumes for each material basis for three different spatial-spectral filters: Bi-Au-Lu-Er (best 4-element filter); Bi-Au-Er (best 3-element filter for gold contrast); and Bi-Lu-Er (best 3-element filter for iodine contrast).

Various combinations of filtering materials were explored to determine their relative performance for the three-material water-iodine-gold decomposition. To keep the search space reasonable, we focused on five metals with k-edges in the diagnostic range, that are readily available, machinable, and with few physical hazards. These were bismuth (k-edge at 90.5 keV), gold (80.7 keV), tungsten (69.5 keV), lutetium (63.3 keV), and erbium (57.5 keV). For all investigations, a 0.25 mm thick filter was simulated passing 17.5%, 6.5%, 6.5%, 17.7%, and 8.3% of the incident 120 kVp spectra for each material, respectively. All data was normalized so that the bare-beam fluence for the bismuth-filtered beamlet was 5×10⁵ photons/pixel. Filters were designed so that each beamlet was 8 detector pixels wide and the spatial-spectral filter was translated 1 pixel per rotation angle. For these preliminary studies, perfect alignment of beamlets with pixel boundaries was presumed. All 3- and 4-element spatial-spectral filters possible with the above five materials were investigated. Root mean squared error (RMSE) was computed for material decomposition density estimates. Regularization parameters for material bases were chosen to be 3 × 10⁸, 5 × 10¹¹, and 1 × 10¹² for the water, iodine, and gold bases, respectively; and the Huber penalty delta was set to 10⁻³.

A second study considered a four-material decomposition for a digital phantom with water, iodine (2–8 mg/ml), gold (1–4 mg/ml), and gadolinium (1–4 mg/ml), which is illustrated in Figure 4B. The same CT system geometry, source, detector, and filter parameters were used. In this case all possible 4-element filters and the one 5-element filter from the five aforementioned filter materials were explored. RMSE was computed for each material decomposition. Regularization parameters for material bases were chosen to be 3 × 10⁸, 1 × 10¹¹, 1 × 10¹² and 1 × 10¹² for the water, iodine, gold, and gadolinium bases, respectively; and the Huber penalty delta was set to 10⁻³.

Fig. 4.

Four-material decompositions using spatial-spectral filters. A) Summary of the performance of all 4-element filters and the 5-element filter. B) Ground truth and estimated densities using the Bi-Ai-Lu-Er filter for the four-material decomposition experiment.

III. RESULTS/DISCUSSION

The results of the three-material decomposition with spatial-spectral filtering are shown in Figure 2. RMSE for both the iodine and gold contrast agents are reported and ranked. The best filter for estimating iodine density is the 3-element Bi-Lu-Er filter; whereas the best filter for estimating gold density is the Bi-Au-Lu filter. The 4-element Bi-Au-Lu-Er filter achieves the 2^nd best performance for both iodine and gold – suggesting that the 4-element filter provides some level of compromise, and gets the best of both 3-element filters. Note that the bismuth filter appears important for good performance. A reconstruction and error images associated with these three spatial-spectral designs is shown in Figure 3.

Fig. 2.

A summary of all spatial-spectral filter performance for the water-iodine-gold material decompositions. The RMSE for iodine and gold contrast agents are reported and ranked.

A summary of the four-material decomposition results are shown in Figure 4A. The best overall filter was the 4-element Bi-Au-Lu-Er filter (although the Bi-W-Lu-Er had very slightly better gadolinium estimates). Decomposition images for this filter are shown in Figure 4B including a colorized iodine-gold-gadolinium image. Again, the bismuth filter appears critical in delivering a good decomposition.

These preliminary studies suggest the feasibility of spatial-spectral filters to provide spectral CT using relatively simple hardware. The filters permit a larger number of spectral channels that traditional “source-side” modifications which facilitates material decompositions with multiple contrast agents. Moreover, spatial-spectral filters may be combined with other spectral CT approaches for additional advantages. For example, combinations may help to improve the relationship between density estimates and radiation dose, or to help improve the low concentration density estimation limits (especially important for highly specific and targeted contrast agents).

While these preliminary results are encouraging a number of ongoing studies are required to fully realize the spatial-spectral filter concept in a physical system. In particular, due to finite x-ray focal spot size, misalignment of beamlets with pixel boundaries, etc., more complete modeling of the transition regions between beamlets is required. Similarly, kV optimization and an exploration of tube efficiency and tube loading must be undertaken. Optimal sampling patterns driving filter actuation and the effects of regularization also need to be fully explored.