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As the non-invasive nature of tomographic imaging had obvious benefits for the field of medicine, many significant advances in tomographic technology were driven by clinical research. One such form of tomographic imaging is x-ray computed tomography (CT) which uses projection images acquired from different locations around the object to compute the distribution of material densities inside the object. With the growth of tomographic imaging, additional technologies were developed to acquire images using a variety of waves from ultrasound to injecting radioactive tracers which emit these waves from inside a patient as in single-photon computed tomography (SPECT) and positron emission tomography (PET).
In a CT scanner system, an x-ray source and opposing detector typically rotate in a circle relative to the object being imaged as x-ray projection images are acquired at different angular positions. By modeling the attenuation of the incident x-rays by the object being imaged at different projection angles and finding an approximate inversion of this model, an estimate of the object’s interior could be produced. The development of modern CT imaging systems today was driven not only by innovation in the hardware design, but also by advances in the algorithms used to invert the poorly conditioned forward model encountered in CT imaging.Figure (ref:fig:intro_hist_1st_gen) shows a schematic of the first generation of CT scanner, which was built by EMI (Electric and Musical Industries Ltd.) in 1967 using the work of Allan M. Cormack and Godfrey N. Hounsfield cite:buzug_thorston_m._milestones_2008. In this design, a source and detector acquire a series of pencil-beam projections by translating together along a line in a plane orthogonal to the rotation axis. The source and detector then rotate before acquiring another series of projections along another line. This process is repeated for successive rotation angles until the system has rotated an entire $180{ˆ}$ and the source and detector have moved to each other’s starting position. For their work, Cormack and Housfield shared the 1979 Nobel Prize in Physiology or Medicine cite:hsieh_computed_2009.
The algorithms for reconstructing the tomographic image from these x-ray projections at different angles evolved concurrently with the different iterations of CT-scanner hardware. The projection data acquired from the first-generation scanner was necessarily digitized as the reconstructed image was computed by solving a linear system of equations, now known as algebraic reconstruction technique (ART) cite:hsieh_computed_2009. Though Radon’s theories that provide the basis for some of the analytic-based reconstruction algorithms were known at the time, the necessary computer hardware needed for a practical implementation had yet to be developed when Cormack and Hounsfield reconstructed their first CT image cite:buzug_thorston_m._milestones_2008.
Though the first-generation CT system was a groundbreaking achievement, the rasterized scanning of the pencil beam at multiple angles required approximately 4.5 minutes to acquire the projection information to reconstruct a single two-dimensional (2D) slice of the scanned object cite:hsieh_computed_2009. Long scan times are problematic as they increase the possibility of motion during scanning which creates motion contamination artifacts in the reconstruction. The second generation of CT scanners significantly reduced this acquisition time to about 30 seconds by replacing the pencil beam of x-rays with a fan beam and a detector array with approximately 30 detector elements. This new design, shown in Figure (ref:fig:intro_hist_2nd_gen), still required the source and detector to acquire projections using a linear translation before rotating to a new angular position and repeating the process cite:buzug_thorston_m._milestones_2008.
One of the most important factors driving the hardware development in CT-scanner technology was the issue of acquisition speed. As the reconstruction framework used a forward model that assumes the projections are acquired from a stationary object, the first two generations were only successful in imaging parts of the patient that were relatively stationary. As such, these early scanners were initially only used with the cranium as it is relatively motionless relative to the required acquisition time. Unlike the cranium, other anatomical sites such as the thorax and abdomen are greatly affected by cardiac and respiratory motion. The next generation of scanners were designed to reduce the acquisition time to under 20 seconds in order to image a patient’s abdomen in a single breath hold cite:buzug_thorston_m._milestones_2008.
The third generation of CT scanners, shown in Figure (ref:fig:intro_hist_3rd_gen), would soon become the dominant scanner design. This remains true today as most modern diagnostic CT imaging systems, such as the one shown in Figure (ref:fig:intro_ct_scan), are still based on the third generation design. Unlike the earlier generations of scanners, this design removed the need for the linear translation of the source and detector by increasing the size of the detector so that the entire patient could be illuminated at a given angle. By doing this, it was only necessary to acquire a single projection at each rotation angle which significantly reduced the acquisition time cite:hsieh_computed_2009.
The evolution from the first generation of scanners to the second and third generations of scanners was enabled by advances in both computer hardware and reconstruction algorithms for solving the inverse problem. As we will further discuss in the following chapter, analytic-based algorithms for solving the CT inverse problem gradually replaced the initial algebraic solution to the linearized forward model used by Cormack and Hounsfield. The most popular form of this implementation is known as filtered-backprojection (FBP) cite:buzug_thorston_m._two-dimensional_2008,hsieh_computed_2009.
The FBP approach to CT reconstruction was first implemented as the parallel-beam backprojection algorithm cite:buzug_thorston_m._two-dimensional_2008. This provided an analytic inverse to the acquisition method of the first generation of scanners where at a given angle, all of the projections are acquired as parallel incident x-ray beams. However, with the second and third generation of CT scanners, this imaging model was modified from the parallel-beam geometry to the fan-beam geometry to account for the divergent x-ray beam of a point-like x-ray source on an array of x-ray detectors. The new fan-beam FBP algorithm enabled the scanning geometry of the third generation of scanners which are still the backbone of clinical CT today cite:pan_why_2009.
The fourth generation of CT scanners was developed to eliminate ring artifacts that can appear in the third-generation CT scanners. These ring artifacts can occur when there is a mismatch in projection data of opposing rays along the same line in the patient which can result from misalignment of the moving detector. With a stationary ring of detectors, these ring artifacts are eliminated. However, with the advent of multi-slice detector technology which will be discussed in /Cone-beam CT and new scanning trajectories/, the engineering and cost requirements has led to fourth-generation scanners being phased out cite:hsieh_computed_2009.
The fifth generation of CT scanners, also known as electron-beam, computed-tomography (EBCT) scanners, was developed in the early 1980s for cardiac imaging. In order to acquire the projection data fast enough to ”freeze” cardiac motion (20-50 ms for a full rotation), it would be impossible to design a mechanical system that could rotate that quickly and withstand the centripetal force incurred at such high rotational velocity. Instead, this generation was designed to steer the electron beam onto the x-ray anode that was curved around the patient – effectively placing the patient inside the x-ray tube. The design is similar to the fourth generation in that the EBCT scanners have a fixed, partial-ring detector around the patient cite:buzug_thorston_m._milestones_2008,hsieh_computed_2009.
Though the fourth and fifth generation scanners are interesting manifestations of CT scanning technology, they are only included here for completeness and will not be discussed further. In the following section, we will look at a major development in reconstruction algorithm technology that cemented the third-generation CT scanner’s popularity. This algorithm development enabled the extension of the CT detector’s axial coverage allowing for volumetric image acquisition and reconstruction using the third-generation scanner design. This development also led to the development of a new cone-beam CT (CBCT) geometry which is the focus of this work.
For all the CT scanners discussed in the previous section /CT development history/, the only scanning trajectory utilized for CT was the circular rotation of the source and detector around the patient. This limitation was due to both the hardware geometry and the reconstruction algorithms that were initially focused on acquiring and reconstructing 2D-planar slices of the object being imaged. Unfortunately, this slice-by-slice acquisition and reconstruction framework was somewhat limiting in acquiring volumetric CT images.The use of new scanning trajectories to increase the volumetric imaging capabilities of CT began with the development of the spiral or helical CT reconstruction algorithm cite:kalender_spiral_1990,kudo_helical-scan_1991,katsevich_theoretically_2002,katsevich_exact_2004. By adding longitudinal translation of the patient couch through a third-generation scanner, it was possible to perform a helical trajectory of the source and detector around the patient. This made it possible to rapidly acquire multi-slice (or volumetric) CT of a patient using the existing diagnostic imaging hardware of the third-generation scanners.
Another approach to acquire volumetric tomographic images was to extend the CT detector array in the longitudinal direction by adding additional rows of detector arrays. These multi-array detectors helped to improve the interpolation procedure used for reconstructing the data acquired from a helical scan, and continue to be used in modern third generation CT scanners. As these multi-array detectors began to cover larger extents of the axial field of view (FOV), they eventually led to large flat-panel detector being used to acquire projection information. The flat-panel detector systems are now known as cone-beam CT (CBCT) systems to reflect the cone of x-ray illumination on these detectors as opposed to the fan-beam geometry of the earlier slice-by-slice scanners.
With the advent of CBCT scanners, efforts were made to extend the FBP algorithm to three dimensions (3D) cite:grangeat_mathematical_1991,kudo_derivation_1994,kudo_fast_1998,buzug_thorston_m._three-dimensional_2008. Though all of these methods attempted to find an analytic inverse to the forward-projection imaging model, they require exact Radon data, which is not provided by the circular trajectory routinely employed by third generation scanners. It was the development of a modified FBP algorithm by Feldkamp, Davis and Kress or FDK cite:feldkamp_practical_1984 (which we will discuss further in /Analytic-based reconstruction/) that made it possible to obtain a useful reconstruction from a circular scanning trajectory on a CBCT system.
X-ray technology is unique in how rapidly it was applied to the field of medicine following the discovery of x-rays by Wilhelm R\text{"o}ntgen in 1895. The next year in Chicago, Emil Grubbe built his own x-ray device which he began to use for therapeutic purposes cite:mukherjee_emperor_2010. Both diagnostic and therapeutic uses of radiation developed in concert throughout the 20^th century culminating in radiation treatment devices that combine low-energy CT imaging or magnetic resonance imaging (MRI) with high-energy treatment beams in image-guide radiation therapy (IGRT). A particularly popular method of delivering therapeutic radiation doses are linear accelerators (LINACs) that deliver powerful megavoltage (MV) treatment beams to diseased tissue.Though the type of particle and energy spectra of therapeutic radiation will vary depending on both the type and progression of the disease, the desired effect of the prescribed dose is to ablate the diseased tissue by inducing cell death in the cancer cells. Much as with traditional surgical techniques, there is also a simultaneous need to spare the healthy tissue while removing the diseased tissue. In the pursuit of achieving this balance between killing diseased tissue and sparing healthy tissue with the delivered dose, a variety of radiation therapy modalities have been developed.
The initial application of radiation for therapeutic purposes was initially limited to superficial lesions. As great care must be take to ensure the dose is delivered just to the target area, there must be a way to visualize the target of the delivered dose. In the early application of therapeutic radiation, physicians were limited to those pathologies that were externally visible. It was the Swedish neurosurgeon Lars Leksell who conceptualized using a high-energy photon beam to deliver a therapeutic dose deep inside the brain without the additional complications of invasive surgery cite:kushnirsky_history_2015.
The half a century between the discovery of x-rays and the use of them to treat internal structures was the need for imaging technology that made visualizing these internal targets feasible. The development of CT technology discussed in /CT development history/ provided the requisite volumetric information that allowed physicians to locate these internal lesions in the patient. This was required in order for the physicians to accurately target the lesions using the high-energy beams proposed by Leksell. He realized by distributing small-field beams around the target, we could create a high-dose deposition at a desired internal target location which led to him and his colleagues producing the GammaKnife in 1968 which consisted of 179 cobalt-60 sources distributed in a hemisphere around the patient’s cranium for stereotactic radiosurgery (SRS) cite:kushnirsky_history_2015.
The eventual acceptance of SRS in the United States in 1984 and the subsequent rise in popularity of the technique led to use of linear accelerators for delivering the therapeutic dose rather than the fixed cobalt-60 sources cite:kushnirsky_history_2015. LINACs were not only able to produce treatment beams that were comparable to those provided the cobalt-60 sources, but they had the additional benefits of being able to produces much higher energy spectra without the need to replace the radiation source as they decayed as cobalt-60 sources. However, despite the tomographic imaging used to plan the prescribed radiation dose, only radiographic projection imaging was used for setting up the patient at the treatment isocenter of these devices.
The addition of a LINAC-mounted, kV-imaging, cone-beam computed tomography (CBCT) system to the gantry-mounted clinical linear accelerator cite:jaffray_flat-panel_2002,letourneau_cone-beam-ct_2005,rahman_linac:_2015 helped this modality become the most popular form of image-guided radiation therapy (IGRT) cite:xing_overview_2006,bissonnette_quality_2012,dawson_advances_2007. The tomographic information provided in the kV energy range improves soft-tissue contrast resolution over that provided by the MV electronic portal imaging device (EPID) alone cite:jaffray_radiographic_1999. The LINAC-mounted, kV-imaging, CBCT system not only helps with patient setup and target verification, but it also allows the monitoring of the tumor response during treatment cite:oldham_cone-beam-ct_2005.
Figure (ref:fig:intro_linac) shows an annotated image of a Varian TrueBeam LINAC (Varian Medical Systems, Palo Alto, CA). On the patient couch is the CIRS Torso Phantom (Computerized Imaging Reference Systems, Norfolk, VA) aligned at the mechanical isocenter using the laser guidance system. Above the torso phantom to the left is the MV treatment head with the metallic accessory mount and beam exit window. Below the table to the left is the kV source which provides the kV x-rays for the kV-CBCT imaging system. Above the phantom to the right is the kV detector panel which acquires the projections through the phantom for the kV-CBCT imaging system. Both the kV source and kV detector are mounted on robotic position arms. Below the phantom to the right is the MV electronic portal imaging device, which is retracted in this image, for acquiring MV projections from the MV treatment beam. Finally all of these components are mounted on a rotating gantry which can rotate $360ˆ$ around the mechanical isocenter for a single rotation. A subsequent rotation must occur in the opposite direction as the gantry lacks the ability to make multiple rotations in the same direction like a diagnostic CT system due to the complexity of the MV LINAC design.
While there are many advantages to using LINAC-mounted CBCT imaging systems for IGRT, there are still technical limitations that negatively impact clinical utility, that could be alleviated by utilizing non-circular scanning trajectories with optimization-based reconstruction. One issue is the limited axial FOV coverage provided by the current detectors and circular scanning trajectory. Another issue is the increased potential of patient collisions with the rotating treatment gantry. In this work, we will focus exclusively on utilizing a generalized optimization-based reconstruction framework from arbitrary CBCT trajectories to address these clinical issues for IGRT. However, the framework itself is not necessarily restricted to IGRT and could be of potential use for a variety of other CBCT applications.
In this work, we will discuss an optimization-based, image-reconstruction framework that enables the use of new scanning trajectories. In particular, we will focus on how this approach was developed to address the two clinical shortcomings of limited axial FOV coverage and potential patient collisions with the LINAC gantry. By using these two examples, we will not only show the feasibility of using these non-circular trajectories, but also a potential solution to existing clinical needs.First, we will discuss the framework and considerations of using optimization-based reconstruction with different scanning trajectories in /Optimization-based algorithms/. Next, we will discuss the need for geometric calibration and discuss a method we developed to accomplish this for these trajectories in /Geometric calibration/. We will then review the use of new trajectories to address the limited axial FOV issue in /Axial field-of-view extension/ followed by using these trajectories to alleviate the issue of patient collisions in /Collision-avoiding trajectories/. Finally, we will summarize this work and discuss possible clinical considerations with this methodology in /Summary and conclusions/.
The increased flexibility in choosing different scanning trajectories allowed by optimization-based reconstruction methods provided two solutions to the issues of limited axial FOV coverage and potential patient collisions. For these two problems, we found that the existing limitations could be resolved by using a different scanning configuration. In each case, we proposed a trajectory that would solve the existing problem, and then we evaluated how well the optimization-based reconstructions compared to currently used clinical images.
Through the years of CT research, a fundamental question has always been how to move the source and detector of the imaging system relative to the object to obtain sufficient projection information to reconstruct a useful image. Part of this answer must take into account certain engineering limitations that go into building such a system. However, this is fundamentally a question that must address the requirements of the computational reconstruction algorithm used to assemble the image from the x-ray projections.There are two main classes of reconstruction algorithms. Analytic-based algorithms, such as FDK cite:feldkamp_practical_1984, represent an approximate solution to the inverse imaging problem, i.e., calculating the object function from its projections. Optimization-based algorithms represent the forward imaging problem as a linear system, and attempt to iteratively invert this system to find an object function that is consistent with the observed projections. Image reconstruction with optimization-based methods provides a robust framework for reconstructing from projections acquired with nonstandard trajectories designed to address specific CBCT limitations as they require no assumptions about the initial scanning trajectory.
The use of optimization-based methods for tomographic image reconstruction is a natural extension of linearizing the x-ray transform imaging model of a tomographic scan. Approaching the image reconstruction problem as a linearized imaging model has existed since the first CT system built by Cormack and Hounsfield. As discussed in the /Introduction/, they utilized the algebraic reconstruction technique (ART) to solve a system of equations created by the summation of the rays through the image pixel grid at each projection angle cite:herman_art:_1973.
Though the initial optimization-based image reconstruction with ART was successful in providing a solution to the inverse problem, the limited computational power available at the time proved to be an intractable limitation. Though the number of unknown variables in the system of equations associated with this 2D reconstruction problem is trivial by today’s standards, the lack of parallelization and other engineering limitations of transistors at the time were too onerous for the clinical workflow. This computational complexity was further increased when moving from two-dimension (2D), single-slice images to three-dimensional (3D), volumetric image reconstruction which introduces a greater number of unknowns. However, a recent renaissance of utilizing graphics processing units (GPUs) – technology once solely in the purview of video games – for scientific computation has made optimization-based methods temporally competitive with analytic-based methods cite:xu_accelerating_2005,sharp_gpu-based_2007.
Analytic-based reconstruction algorithms are formulated by explicitly finding an inverse to the X-ray transform \begin{equation} \label{eq:xray} g(\mathbf{r}_0,\hat{θ})=∫_0∞f(\mathbf{r}_0+t\hat{θ})dt, \end{equation} where the data function $g$ is acquired by integrating along the ray from the source at $\mathbf{r}_0$ in the direction $\hat{θ}$ through the object function $f$. In x-ray CT, this object function represents the distribution of the linear attenuation coefficients of the various materials within the object’s interior that provide exponential attenuation to the incident beam as it travels through the object as characterized by Beer’s law cite:buzug_thorston_m._fundamentals_2008.A fundamental problem with these reconstruction algorithms when
practically reconstructing
In the 1980s, work was done to directly solve the inverse problem for the cone-beam geometry cite:parker_optimal_1982-1,finch_cone_1985. By modeling the projection formation process as a Radon transform or an X-ray transform, reconstruction algorithms were formulated by finding an analytic-based inverse to the transform. However, for the inverse to be exact, it needed to meet strict requirements such as Tuy’s condition which states that every plane through the object must intersect the source trajectory cite:tuy_inversion_1983. Though there are non-circular trajectories such as the infinite-line trajectory which satisfy Tuy’s condition, the application of such trajectories for real scanning configurations are always approximations cite:sidky_volume_2005 which fail to satisfy the requisite geometry to provide an exact inverse.
The circular scanning trajectory that is ubiquitous in the clinic for
CBCT is one trajectory that fails to meet Tuy’s condition. The most
popular reconstruction algorithm for the circular CBCT trajectory is
the filtered-backprojection (FBP) algorithm proposed by Feldkamp,
Davis, and Kress (FDK) cite:feldkamp_practical_1984 which is still the
industry standard. FDK is only an exact inversion to the Radon
transform on the midplane containing the circular source trajectory.
For transaxial planes other than the midplane, a quasi-redundancy in
the scanning data is assumed. It is the violation of this assumption
which leads to cone-angle artifacts, an example of which is shown in
Figure (ref:fig:opt_defrise_fdk). These artifacts become more severe
at larger cone angles (the angle
The presence of cone-angle artifacts in FDK reconstructions from the incomplete data acquired with circular scanning trajectories led to research into inverse algorithms for cone-beam scans from theoretically complete trajectories such as a circle plus a line cite:zeng_cone-beam_1992. It became apparent in the reconstruction results that implementing these direct reconstruction algorithms did not produce the anticipated results cite:kudo_derivation_1994. Severe artifacts and numerical errors were found in the reconstructions due to factors such as truncation introducing high-frequency components that are amplified in the filtration process.
Analytic-based reconstruction algorithms are problematic in that they require a fixed scanning trajectory to formulate the inverse. When approximations are made for the inverse, as in FDK, deviations from these approximations lead to inconsistencies in the model and subsequently artifacts in the reconstruction such as the cone-angle artifacts shown in Figure (ref:fig:opt_defrise_fdk). In contrast, optimization-based reconstruction algorithms represent a more robust model of the image formation process cite:shepp_maximum_1982,han_optimization-based_2012,sidky_image_2008,sidky_accurate_2006,bian_evaluation_2010. As Figure (ref:fig:opt_defrise_mlem) shows, this can help reduce artifacts such as the cone-angle artifacts.Optimization-based reconstruction algorithms provide a more accurate
model of the DD imaging system that comprises both the digitized
projection images from the kV-imaging detector and the digitized
tomographic image calculated by the reconstruction program. The X-ray
transform of the object function can be represented as the linear
system
\begin{equation}
\label{eq:opt_ddsys}
\mathbf{g}=\mathcal{H}\mathbf{f},
\end{equation}
where
The optimization problem is formulated as an objective function based
on the actual data
In selecting the parameters for the reconstruction program, consideration must be given to the impact each parameter will have on the reconstructed image quality. As we investigated using optimization-based reconstruction for non-circular scanning trajectories, we selected parameters of our reconstruction program to mimic the relevant clinically-utilized parameters where applicable. For instance, our reconstruction resolution sizes are chosen to provide the same voxel sizes used by the clinical reconstruction software. However, these parameter choices are made only to provide comparisons to the current clinical image quality. This does not mean that these values are optimally selected, and for any clinically relevant evaluation, rigorous parameter optimization must be studied for the clinical imaging task cite:metz_basic_1978,xia_optimization-based_2016-2.
Previous work has shown that optimization-basaed algorithms can reconstruct clinically useful images under scanning conditions for which analytic-based FDK fails cite:han_optimization-based_2012,sidky_image_2007,sidky_accurate_2006. In applying optimization-based reconstruction to non-circular trajectories, we focus primarily on the well-understood maximum-likelihood expectation maximization (MLEM) cite:shepp_maximum_1982,dempster_maximum_1977. Though a variety of optimization-based reconstruction programs exist, we used the MLEM program to limit the number of parameters introduced by the reconstruction program, since new scanning trajectories already introduce additional parameters that impact the projection data and resulting tomographic reconstruction.
In IGRT, LINAC-mounted CBCT imaging systems such as Varian’s TrueBeam kV-imaging system now routinely provide patient image information. These images are used to check the patient alignment before delivering the radiation treatment. The circular rotation of the LINAC gantry defines the acquisition trajectory for the CBCT scan. While such a scanning trajectory provides sufficient information for an analytic-based reconstruction of the scan volume, there are a variety of limitations that arise from this work flow.Due to engineering and cost restrictions, the kV detector has a limited size. The TrueBeam system has a transaxial width of 40 cm and an axial height of 30 cm. This restricts the FOV that can be imaged in a traditional circular scan. While the offset detector technique cite:bian_optimization-based_2013,cho_cone-beam_1995 is commonly used to increase the transaxial FOV (for a 1.5X magnification, this is an increase in FOV diameter from 26.7 cm to 44.0 cm on the TrueBeam system with a 13 cm offset), the axial coverage is still very limited (20 cm for the same TrueBeam geometry) cite:pearson_non-circular_2010. The reason that the limited FOV has not been addressed by increasing the detector size is partially due to the industry’s reliance on the approximate FDK algorithm cite:pan_why_2009. As shown in Figure (ref:fig:opt_defrise), as the cone angle increases, artifacts near the end of the axial FOV become more severe.
Another problem with the current circular imaging trajectory is the potential for LINAC collisions with the patient cite:hua_practical_2004,nioutsikou_patient-specific_2003. Cases arise when the patient is positioned in the treatment position, a CBCT image cannot be acquired due to part of the patient being in the path of the LINAC’s trajectory (i.e., gantry clearance cannot be achieved). As the current FDK algorithm requires a trajectory with sufficient angular coverage, the patient must be moved to a position where the gantry can make an uninterrupted rotation around the patient. These workarounds can incur significant temporal costs in re-positioning the patient on the treatment couch for a new collision-avoiding setup. A robust scanning modality that could avoid these collision zones while providing sufficient tomographic information would alleviate these expensive re-positioning occurrences.
In both of these examples, the default circular trajectory prescribed by FDK is inadequate for obtaining the desired tomographic information. Furthermore, the disruption to the clinical workflow created by these limitations introduces bottlenecks into clinical efficiency which affects both the clinical staff as well as the patient’s comfort in the procedure. In the case of a potential patient collision, the inability to acquire the required trajectory can even result in forgoing the CBCT image. For these particular examples, we investigated ways in which new trajectories enabled by optimization-based reconstruction could alleviate the limitations imposed by the standard circular scan.
Though there has been previous work in developing analytic methods for addressing the reconstruction from some novel trajectories cite:katsevich_theoretically_2002,katsevich_image_2004,katsevich_image_2005,katsevich_formulation_2006, it could be clinically useful to enable reconstruction from an arbitrary, collision-avoiding trajectory. As the collision region (if one arises) is contingent on the patient’s size and treatment position, the imaging trajectory would vary on a per patient basis. As such, deriving the analytic inverse for each patient’s scanning trajectory would be impractical.Optimization-based reconstruction provides a generalized framework
enabling greater flexibility in reconstructing from projections
acquired with non-circular trajectories. Provided the geometry of each
view is correctly incorporated into the system matrix
For the problem of the limited axial coverage, the current clinical method of extending the FOV is to acquire two circular scans at different axial positions and reconstruct each circle independently using FDK before stacking the two volumes together cite:forthmann_adaptive_2009. Unfortunately, the increased distortion from cone-angle artifacts at large cone angles limits the axial separation between these two circles illustrated in blue and red in Figure (ref:fig:opt_fov_schematic). In addition limitations incurred by the failure in the FDK approximation at larger cone angles, there is an additional limitation that the support, or volume of the image space that can be reconstructed, allowed by analytic-based methods is restricted to the shaded regions of Figure (ref:fig:opt_fov_schematic).
The use of the two circles alone provides one interesting example of a trajectory where optimization-based reconstruction provides an advantage to the stacked-FDK method currently used. Unlike stacking two separate reconstructions together, it is possible to reconstruct the entire volume at once provided the system matrix is correctly calculated to reflect the acquisition of two circles in planes located at different axial positions relative to the patient. In addition to the reduced cone-angle artifacts already seen in optimization-based methods, reconstructing both volumes together provides additional information about the overlapping region between the circles that further helps to reduce the cone-angle artifacts.
In addition to improving the use of the two circles, the optimization-based framework allows for noncircular trajectories. Given that there needs to be a relative axial translation between the kV-imaging system and the patient, we investigated if there were any advantages to acquiring some projection views during the axial translation. Such trajectories that included an axial translational stage have been studied before and have the potential to further reduce the impact of cone-angle artifacts with both analtyic-based cite:zeng_cone-beam_1992,noo_stable_1996,johnson_feldkamp_1998,katsevich_image_2004 and optimization-based methods cite:davis_we-g-brf-07:_2014.
In the case of potential patient collisions with the LINAC gantry, a simple change in the scanning trajectory could be sufficient to prevent a collision. Much like the extended axial FOV case, optimization-based reconstruction is able to handle variations in the acquisition trajectory provided it is accurately reflected in the system matrix. As such, there are two different ways we studied where the scanning trajectory could be modified to avoid a collision.
If the patient collision were to occur with the kV detector (the closest component of the CBCT system to the patient), one possible way to avoid that collision would be to move the kV detector away from the patient at the collision region. Since the detector is mounted on a robotic arm, it should be possible to move the detector outward from the isocenter radially, increasing the diameter of both the collision-free region and of the scanning trajectory. This effectively changes the magnification for that region, but the reconstruction framework is able to reconstruct from all the views at both magnifications provided that everything is accurately modeled in the reconstruction problem.
The other trajectory modification that could solve this problem would
be to move the patient. As with the change in magnification, the
change in the patient position does not prevent reconstruction with
the optimization-based methods provided the patient motion is
correctly incorporated into the system matrix
We define a scanning trajectory as the sequence of source and detector
positions used to acquire each projection view. For all of the
trajectories we studied, the detector moves in diametric opposition to
the kV-imaging source though this is not a requirement of this
formulation. The coordinates of the trajectory are then defined
relative to a fixed origin in the patient. In a traditional scanning
configuration where the patient is stationary, the system matrix
$\mathcal{H}ij$ projects the object
When the patient is no longer fixed relative to the room-coordinate
system, (e.g. moving the treatment couch as the gantry rotates), the
image space $(\mathbf{f}\text{patient})$ is moving relative to the
room coordinate system for each projection view. As such, a change of
basis for the columns space of
weighting factor & schematic
| Couch | Detector | Gantry | kV Source | |
| Acquisition | CouchLat | ImagerLat | GantryAcceleration | Current |
| CouchLng | ImagerLng | StartAngle | FrameRate | |
| CouchRtn | ImagerOrigin | StopAngle | KVFilter | |
| CouchThickness | ImagerResX | PulseLength | ||
| CouchVrt | ImagerResY | SAD | ||
| CouchWidth | ImagerSizeX | SID | ||
| ImagerSizeY | Voltage | |||
| ScatterGrid | ||||
| Projection | CouchLat | ImagerDeltaLat | GantryRtn | SourceAngle |
| CouchLng | ImagerDeltaLng | SourceDeltaLat | ||
| CouchRtn | ImagerDeltaPitch | SourceDeltaLng | ||
| CouchVrt | ImagerDeltaRtn | SourceDeltaVrt | ||
| ImagerDeltaVrt |
To study these trajectories on a clinical, kV-imaging system, we implemented some of them on Varian’s TrueBeam system. The TrueBeam Developer Mode provides control of the kV imaging system to allow for motion control that is unavailable in clinical modes. Developer Mode provides a scriptable control interface that allows control of the gantry rotation, the kV-imaging arms, as well as the position of the treatment couch. By combining motions with all of these components, it is possible to acquire kV projection data from a variety of different interesting motions. From the acquisition, each projection is returned with self-reported nominal values that can be used to build the reconstruction system matrix. Table (ref:tab:opt_varian_header) shows a subset of these header variables pertaining to the kV imaging system.
The TrueBeam’s kV imaging system is illustrated in Figure
(ref:fig:intro_linac) in addition to the gantry, couch, and robotic
arms that can all be utilized to implement these trajectories. The
kV-imaging system itself consists of a Varian kV x-ray source
(GS-1542) and a 39.7 cm x 29.8 cm amorphous silicon flat-panel
detector (PaxScan 4030CB) with a
The first coordinate system shown in Figure (ref:fig:opt_coords_rad) is the radiation coordinate system; a basis in which the projection headers describe the projective geometry of the source onto the detector at each view. In this convention, the detector pixels can be converted into the physical units to describe their location relative to the source at each projection. As the source and detector rotate together with the gantry, this basis ignores the gantry rotation angle $\left(θg\right)$ at each view.
However, in order to determine the relationship of each projection to the other views, this radiation coordinate system must be transformed into a global coordinate system describing the ensemble of projections relative to the image space. The global coordinate system describing the TrueBeam room geometry is the International Electrotechnical Commission (IEC) 61217 coordinate system shown in Figure (ref:fig:opt_coords_iec). This coordinate system is designated by the IEC as the standard coordinate system for radiotherapy machines cite:international_electrotechnical_commission_radiotherapy_2011.
For a gantry angle of $θg=0ˆ$, the radiation-coordinate system
shown in Figure (ref:fig:opt_coords_rad) has the same basis as the
radiation coordinate system in Figure (ref:fig:opt_coords_rad). This
can be used to place the view-by-view header information into the IEC
basis ignoring the gantry rotation initially. Using the IEC basis
vectors $X_{\text{IEC}}$, $Y_{\text{IEC}}$, and $Z_{\text{IEC}}$ shown
in Figure (ref:fig:opt_coords_rad), the source and detector positions
for that view in homogeneous coordinates are then
\begin{equation}
\boldsymbol{r}\text{src,rad} = \begin{bmatrix}
\text{SourceVrt}
\text{SourceLng} \
\text{SourceLat} \
1
\end{bmatrix},
\label{eq:opt_src_rad}
\end{equation}
and
\begin{equation}
\boldsymbol{r}\text{det,rad} = \begin{bmatrix}
\text{ImagerVrt} \
\text{ImagerLng} \
\text{ImagerLat} \
1
\end{bmatrix},
\label{eq:opt_det_rad}
\end{equation}
respectively.
To then get these source and position vectors into the correct IEC
position, they are transformed via a rotation around the longitudinal
or $y\text{IEC}$ axis by the gantry angle $θg$ which is
\begin{equation}
\mathcal{R}\left(θg\right) = \begin{bmatrix}
\text{cos}\left(θg\right) & 0 & \text{sin}\left(θg\right) & 0
0 & 1 & 0 & 0 \
\text{-sin}\left(θg\right) & 0 & \text{cos}\left(θg\right) & 0 \
0 & 0 & 0 & 0
\end{bmatrix}.
\label{eq:opt_rot_rad_iec}
\end{equation}
By then applying this transform to each projection view geometry in the
radiation coordinate system, we then have
\begin{equation}
\boldsymbol{r}\text{src,IEC} = \mathcal{R}\left(θg\right)
\boldsymbol{r}\text{src,rad}
\label{eq:opt_det_iec}
\end{equation}
and
\begin{equation}
\boldsymbol{r}\text{det,IEC} = \mathcal{R}\left(θg\right)
\boldsymbol{r}\text{det,rad}
\label{eq:opt_det_iec}
\end{equation}
which are the view-by-view projection geometry in the IEC basis.
At this point, the system matrix
To scan a patient with a trajectory where either the imaging object or the isocenter itself are changing relative to each other during a scan, it is then necessary to make an additional transform the projection geometry into the basis of desired image space. Again, for IGRT, this is the image space of the patient’s treatment volume whose basis vectors are schematically depicted in Figure (ref:fig:opt_coords_img).
As an example of this, Figure (ref:fig:opt_coords_img) illustrates one
view where there is a translation vector ($\boldsymbol{r}\text{img}$) denoting a
transformation of the patient’s imaging isocenter away from the
mechanical isocenter of the imaging system. If this offset can be
determined for each projection view, the detector data can be
transformed into the basis of the patient’s imaging volume. Though
this transform need not be limited to translation, the form presented
here would then involve a final transform of the form
\begin{equation}
\mathcal{T}{\text{img,IEC}}\left(θg\right) = \begin{bmatrix}
1 & 0 & 0 & -rx,IEC
0 & 1 & 0 & -ry,IEC \
0 & 0 & 1 & -rz,IEC \
0 & 0 & 0 & 0
\end{bmatrix},
\label{eq:opt_rot_rad_iec}
\end{equation}
and the requisite transform of the system matrix would then be the
necessary transfrom $\mathcal{T}\text{IEC,patient}$ needed to
reconstruct from view-by-view shifts of the image object and the
imaging system into the fixed coordinate system of the patient’s image
space.
As we will show in the following chapters, this framework provides a very robust way to handle a variety of non-circular trajectories that have such shifts between the imaging system and the object. In our work the TrueBeam system, these shifts could arise from motion of the imaging arms relative to the patient, motion of the patient table, or even simultaneous motion of both.
Though we by no means wish to suggest that this framework is limited to a particular hardware implementation, or even simply to the purview of IGRT alone, the TrueBeam system and the practical issues that currently face the clinic with its use motivated this investigation. As we present this as a potential solution to some of the limitations in the clinic, we must demonstrate that in bringing the benefits of these new trajectories, we are not adversely impacting the subsequent image quality. We emphasize that image quality alone cannot ultimately determine the true value of any particular imaging modality or technique.It is imperative that for any translation of novel technology to a clinical setting to occur, rigorous clinical studies must be performed to determine the true impact any technology has on the real metric of performance which is the task-based utility. How that is defined is itself a challenging component in any field, but in medicine this must be given consideration to the patient outcome.
Image quality is itself one component of the myriad factors that must be considered when evaluating a clinical technique. As we discuss this framework of reconstructing from non-circular trajectories with optimization-based reconstruction, we will use image quality as a surrogate for clinical utility. Though the trajectories we will look at were formulated to address practical limitations of the current state of the art, the benefits to the clinical workflow must eventually be evaluated with the ultimate patient outcome. What we do posit is that if the framework can achieve reconstructions with image quality that is comparable to existing techniques that have already met the stringent clinical evaluation criteria, then the additional benefits allowed by the non-circular trajectories truly do have the potential to improve clinical utility.
As we look at some examples of different scanning trajectories that could be beneficial to the IGRT clinical workflow, we will use some of the following image quality metrics to compare reconstructions using this framework to the images of existing clinical techniques. We have attempted to select metrics based on existing clinical phantoms and image quality phantoms so as to reflect the potential image quality of these methods were they to be used clinically. Though some of the uses cases provide scanning configurations for which clinical image quality phantoms do not exist, we tried to use phantoms with direct clinical relevance.
The majority of the quantitative metrics we present come from scans of the Catphan 504 (The Phantom Laboratory, Salem, NY). This is a standard quality assessment (QA) phantom for clinical CT devices that provides a series of sections with different objects for calculating image quality metrics. We used the CTP 404 sensitometry module shown in Figure (ref:fig:opt_ctp404) and the CTP 528 spatial resolution module shown in Figure (ref:fig:opt_ctp528).
There are a number of ways to evaluate the spatial resolution from images of the Catphan phantom in a CT image. The CTP 528 module contains a circular array of bar patterns which we used to subjectively determine the highest frequency set which is resolvable. The same module also has two 0.28mm tungsten carbide beads simulating an impulse source from which a point spread and then modulation transfer function (MTF) can be determined. Furthermore, the MTF can be calculated from the bar patterns themselves cite:droege_practical_1982, as well as any suitably high contrast edge in the image cite:rossmann_point_1969.
While the use of MTF in CT has its challenges, notably the assumption of shift-invariance is not satisfied, it still can be useful when treated with some care. Each of the methods above has some advantages and disadvantages. The point source method can provide 3D directional estimates of the point-spread function (PSF), however it can also be sensitive to the location of the bead relative to the image grid with significant difference between a bead located totally within a single voxel or on the interface of many. The bar pattern based evaluation is a clear complement to the visual analysis, however the orientation of the bars relative to the grid will affect some frequencies differently than others which can result in atypical appearing MTF curves. Using an edge spread analysis on the circular phantom boundary provides many samples, at varying directions to the image grid which can be averaged out. It can be impacted by scatter or saturation in the air region near the phantom boundary, however this has not proven to be a significant factor in the images we have analyzed.
The image slice for analysis, the central slice here, is first
thresholded based on the image intensity, the connected component with
area of the appropriate size is isolated and any holes in the
thresholded region are filled. The center of this region is taken as
the phantom center and used as the origin of the coordinates for
analysis. The data are then resampled at high density, along radial
spokes at 8 angles chosen to avoid surface alignment marks, using a
linear interpolant from 5 mm inside to 5 mm outside the surface
boundary. The edge-spread function is then the mean
To characterize low-contrast resolution, we calculated the
contrast-to-noise ratio (CNR) using the polystyrene insert in the CTP
404 sensitometry module. These inserts have CT numbers which are the
closest to the water-like polymer that surrounds them. The metric is
defined as
\begin{equation}
\label{eq:lcv}
\text{CNR} = \frac{2\left|μ\text{roi}-μ\text{bkg}\right|}{σ_\text{roi}+σ_\text{bkg}}
\end{equation}
where
The last metric we evaluated was the reproducibility of the CT numbers in the images from the different scanning trajectories we investigated. For this we used the mean and standard deviations in ROIs for all the material inserts of the Catphan CTP 404 sensitometry module in addition to the polystyrene and background ROIs used for the low-contrast CNR calculations. We also evaluated three additional ROIs of the water-like background for a total of four background ROIs.
In addition to the CTP 504 Catphan modules, we also used the CIRS model 600 torso phantom shown in Figure (ref:fig:opt_cirs) to evaluate these trajectories with this non-circular scanning trajectory configuration. Though we primarily used this anthropomorphic phantom to produce clinically-relevant reconstructions in the use-case we envisioned for these proposed trajectories, we also extracted CT numbers from ROIs in some of the soft-tissue organs corresponding to aorta, liver and spleen in the phantom’s abdomen.
As our framework was developed in the context of addressing current limitations of using CBCT for IGRT, we wanted to evaluate the image quality obtained from examples of real-data scans of such non-circular trajectories to address two existing clinical issues. To do this, we use our framework to build the necessary reconstruction geometry transforms for the TrueBeam system and selected image quality metrics from standard clinical phantoms for comparing reconstructions from this framework against current clinical image quality. In the next two chapters, we will look at two specific examples of two types of non-circular trajectories that address two existing limitations and compare the subsequent image quality to the current clinical standard.
Given that part of the robust nature of optimization-based algorithms is the ability to handle the poorly-conditioned nature of the inverse problem…
Correctly modeling the geometric parameters of the image acquisition is a critical component of tomographic image reconstruction. This is true regardless of whether reconstruction is done with analytic-based or optimization-based methods. Any inconsistency between the real projection geometry and that used for image reconstruction creates artifacts in the reconstructed image cite:rougee_geometrical_1993,fahrig_three-dimensional_2000,noo_analytic_2000,smekal_geometric_2004,cho_accurate_2005,yang_geometric_2006,panetta_optimization-based_2008,daly_geometric_2008,li_generic_2010,wicklein_image_2012. An example of such an artifact is shown in Figure (ref:fig:geo_cal_catphan_example). This is no less true when using non-standard scanning trajectories. Thus we developed a calibration procedure that can accommodate the different scanning configurations including non-standard scanning trajectories and scenarios in which the object, source and detector are all moving during the scan.Previous work on geometric calibration for tomographic image reconstruction has approached the calibration problem via analytic cite:noo_analytic_2000,smekal_geometric_2004,cho_accurate_2005,yang_geometric_2006,daly_geometric_2008 and estimation cite:gullberg_estimation_1990,rougee_geometrical_1993,mitschke_optimal_2000,silver_determination_2000,panetta_optimization-based_2008 frameworks. Initial calibration efforts utilized optimization-based methods to determine the geometric offsets from projections of a known phantom geometry and nominal system setup. By framing the calibration as an optimization problem, the acquisition parameters were estimated in a way that minimized a cost function associated with improper modeling of the acquisition geometry.
These calibration methods (analytic-based methods included) usually rely on a known calibration phantom, which is typically a set of highly attenuating fiducials arranged in a specific pattern. After scanning the phantom with the system of interest, the detected fiducials are then compared to predicted positions based on the known geometry of the phantom and the nominal projection geometry. In the analytic-based approach, the view parameters are determined by solving for parameters that would transform the projection of the phantom to match the observed projection. In the optimization-based approach, geometric parameters are varied to improve the match between the projection of the modeled fiducials and the detected fiducials in the sinogram.
Both methods of performing geometric calibration have their own strengths and weaknesses. The biggest advantage of utilizing analytic-based calibration methods is that the sensitivity to initialization and the sensitivity to the order of parameter variation due to nonlinearity and coupling of parameters faced by estimation are avoided cite:smekal_geometric_2004. However, as with optimization-based reconstruction, optimization-based calibration methods are more flexible in providing calibration offsets for the novel trajectories that we studied.
Using previous work for optimization-based geometric calibration
cite:rougee_geometrical_1993,gullberg_estimation_1990,silver_determination_2000,
we developed a calibration method that utilizes a phantom with known
placement of highly attenuating fiducials. By scanning this phantom
and comparing the projections to the modeled forward-projection of a
mathematical model of the phantom, we can more accurately determine
the system matrix
Before attempting to determine any geometric errors in our scanning acquisition, we first modified the calculation of our system matrix to incorporate the geometry information provided by the TrueBeam system as discussed in /Varian coordinates/. In doing this, we took advantage of all the inherent geometry information that is provided with the current clinical system. This information then provided an initial estimate of the scanning geometry which we could then refine with the calibration information we extracted with our calibration protocol.
The first calibration phantom we fabricated for determining geometric offsets is shown in Figure (\ref{fig:geo_geocal}). The phantom is a 15.2 cm outer diameter acrylic tube with a spiral pattern of CT-spot fiducials placed 2.5 cm along the axial direction every $45ˆ$. When scanned, the CT spots are clearly visible in the projection images which is ideal for automating the fiducial detection in the data domain.However, we realized that using such a spiral calibration phantom creates a degree of ambiguity in the geometry of the projected fiducials. With both this phantom and additional calibration phantoms we created, too much symmetry in the phantom design leads to a rather challenging objective function. Given that only a small portion of the phantom is visible in any one projection view, excessive symmetry produces multiple minima in the objective function where a simple axial shift and rotation offset allows for multiple matches of the modeled fiducials and those in the real data. To avoid such complexity, a calibration phantom with intentional asymmetry is desirable so that the projected fiducials can be indentified and matched without ambiguity.
In addition to the necessary complexity created by this phantom, another concern for a calibration phantom is the uncertainty in the geometry of the phantom itself. Though the guide lines on the cylinder were inscribed with the lathe and its rotational stage, we placed the fiducials by hand. As we were trying to determine millimeter offsets with our calibration, this fiducial placement was suboptimal.
The phantom we then decided to use for calibration was the Isocal phantom created by Varian shown in Figure (ref:fig:geo_isocal). Additionally, the phantom is manufactured by Varian to help align the MV-treatment isocenter with the kV-imaging isocenter. The Isocal phantom directly addresses the two problems encountered with our first phantom. First, the phantom is designed with intentional asymmetry. The position of the beads on this phantom have a much tighter tolerance than that of our original phantom.
Figure (ref:fig:geo_cal_schematic) provides a schematic illustration of the Isocal phantom for a single view. Ideally, the nominal geometry used to calculate a single projection would produce the simulated projected fiducials in blue. However, as both our work and that of others has found, this is not usually the case cite:rougee_geometrical_1993,noo_analytic_2000,smekal_geometric_2004,cho_accurate_2005,yang_geometric_2006,li_generic_2010,wicklein_image_2012. Discrepancies between the reported geometry and the actual scanning geometry can arise from multiple sources in a given acquisition.
With a typical CBCT scan, deviations from the nominal geometry can occur in both the phantom’s setup (translation and rotation in all three dimensions) as well as that of the source and detector positions (due to translation and rotation deviations in the gantry, source, and detector). The collective impact of these various discrepancies will produce projection views for which the projected fiducials in the data domain do not match the simulated projections from the nominal geometry as shown by the red projected fiducials in Figure (ref:fig:geo_cal_schematic).
Starting with the nominal scanning geometry reported by the projection
metadata, we first build an initial projection matrix
Projection of the fiducial coordinates onto the detector needs to be
done in a coordinate system aligned with the detector’s frame vectors.
The source-to-detector distance needed for projection is the distance
along a direction normal to the detector plane, i.e. parallel to the
frame vector
With this new piercing point, it is possible to now construct a
transform that projects the fiducials as well as transforms them to
the detector basis. The transform to the detector basis is represented
by the homogeneous coordinate transform
\begin{equation}
\boldsymbol{G} = \begin{bmatrix}
u_i & u_j & u_k & -rs,x
v_i & v_j & v_k & -rs,y \
w_i & w_j & w_k & -rs,z \
0 & 0 & 0 & 1
\end{bmatrix}.
\label{eq:geo_gmat}
\end{equation}
where $\left[-r{s,x}, -r{s,y}, -r{s,z} \right]$ are the
room-coordinate components of the source position. Then using the
orthogonal ray component found in Equation (ref:eq:geo_along), the
homogeneous coordinate projection matrix is
\begin{equation}
\boldsymbol{P} = \begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & \frac{1}{L} \
0 & 0 & 0 & 0
\end{bmatrix}.
\label{eq:geo_pmat}
\end{equation}
Using these transforms so that they are pre-multiplied by the fiducial
position vectors, the combined transform is then
\begin{equation}
\boldsymbol{M} = \boldsymbol{G}\boldsymbol{P}.
\label{eq:geo_magicmat}
\end{equation}
which transforms a room coordinate point into the detector basis, and
then projects it onto the detector plane.
Finally, this information can be combined to create a single transform
of the fiducials in the global image coordinate system to the
projected spots on the detector in discretized detector bin
coordinates. First, the coordinates of the piercing point must be
calculated in the detector basis as
\begin{equation}
\vec{p}uv = \left(\vec{p}-\vec{r}_d\right)\boldsymbol{G},
\label{eq:geo_pierecuv}
\end{equation}
where $\vec{r}_d$ is the center of the detector in room coordinates.
With all this, the projection transform used to calculate the
projected fiducials in discretized detector bin coordinates is
\begin{equation}
\boldsymbol{X} = \boldsymbol{M}\boldsymbol{T} (\vec{p}uv)
\boldsymbol{S}\left( \left[\frac{1}{s\text{pix}},
\frac{1}{s\text{pix}}, 1 \right]\right) \boldsymbol{T} \left(
\left[\frac{u\text{len}}{2}+0.5, \frac{v\text{len}}{2}+0.5, 0
\right] \right),
\label{eq:geo_xproj}
\end{equation}
where
With the projection transform
As with other optimization-based calibration procedures, we iteratively vary the parameters corresponding to the geometric degrees of freedom (DOF) of the scanning trajectory. The phantom pose (position and orientation) is first allowed to vary in the room coordinate system to account for potential setup errors between the room coordinates and the modeled position of the phantom. Once the pose of the Isocal phantom is identified, then the source, detector, and patient couch translations and rotations are allowed to vary, and the cost of the simulated fiducial projections are calculated at each step. We use the Nelder-Mead simplex algorithm cite:lagarias_convergence_1998 to minimize the $L2\text{-norm}$ cost function.
Given that there are there are different combinations of couch, source
and detector motions that can cause the same change of the object
relative to the source and detector within the image coordinate
system, there are some degrees of freedom that can couple with others.
For instance, shifting the patient in the positive longitudinal
direction is effectively the same as allowing the source and detector
to move the same distance in the negative longitudinal direction. This
requires that only a few parameters are allowed to vary at once as
allowing too many parameters on this non-convex surface will often
produce nonphysical geometric corrections. Once the cost has been
minimized, the geometric offsets are used as the calibration
information for calculating the system matrix
For a new trajectory, this phantom is first scanned to identify any potential corrections to the parameters reported in the TrueBeam data headers. Though we find the self-reported position from the acquisition metadata to be very accurate, there are still some scanning configurations for which the additional refinement from our geometric calibration is critical for obtaining the best quality reconstruction. This is particularly true for scanning trajectories where the object and the kV imaging system move simultaneously.
Though the type of artifacts that are introduced by geometric offsets for circular scanning trajectories are relatively well known, this same sort of understanding is lacking for these new trajectories. To study how geometric offsets affect images reconstructed from these new trajectories, we will create a simulation catalog of artifacts produced by different geometric errors. By introducing intentional geometric inconsistencies in the reconstruction system matrix, we can characterize the artifacts that appear in the reconstruction compared to a numerically-exact inverse crime reconstruction.As one of our primary objectives in using these novel trajectories is to create an extended axial FOV image, we need to study how these geometric errors degrade the image quality along the axial direction. To ensure our simulation can adequately identify these artifacts, we will create a simulated phantom such as an axially extended version of the Catphan high resolution module. This will provide resolution metrics not only in the axial dimension, but also in the transverse planes as a function of axial position.
The simulation catalog of different artifacts that arise from geometric offsets will provide a guide to visually identify potential geometric errors based on the reconstructed image. This provide one way in which we can verify the effectiveness of our geometric calibration procedure. By incorporating the calibration information we obtain with the calibration, known geometric error artifacts should be reduced.
- Overview of the simulated work/analysis
Geometric errors introduce blurring at sharp boundaries in the image which increases the entropy. By reducing geometric errors with calibration, this blurring effect and subsequently entropy should reduced. For our non-circular trajectories, Wicklein’s conclusion can be verified readily with the images in our catalog of geometric errors. The image entropy of the correct-geometry reconstruction will be against the reconstructions with intentional geometric errors to determine if improved geometric modeling reduces the image entropy in Equation (\ref{eq:entropy}).
If the entropy calculations based on simulation agree with Wicklein’s findings, entropy would be reasonable metric to characterize the benefits and limitations of using the geometric offsets from the calibration phantom on different non-circular trajectory reconstructions. We would then use entropy as the metric to compare reconstructions with and without calibration. From this, we can not only verify the effectiveness of our calibration method with different non-circular trajectories, but also then characterize the impact additional geometric corrections have on image quality.
To evaluate the efficacy of our calibration procedure, we investigated its performance on calibrating both a standard, half-fan, circular trajectory where the couch is stationary as well as a virtual isocenter trajectory. For purposes of discussion, a virtual isocenter is where the couch moves simultaneously with the gantry rotation from a fixed point that we call the virtual isocenter. We will return to the example of the virtual isocenter trajectory in /Collision-avoiding trajectories/. For each of these trajectories, we used the same Developer Mode script to scan both the Catphan phantom and the Isocal phantom. We subsequently used the sinogram from the Isocal scan to extract calibration offsets for that particular trajectory using the calibration method described above.We reconstructed the Catphan scans from these two trajectories with and without the calibrations offsets. An isotropic image grid of 0.473 mm was used for each reconstruction with application of the half-fan weighting cite:bian_optimization-based_2013. For all reconstructions, 200 iterations of MLEM were used, as described in the /Generalized trajectory framework/ section.
After acquiring the circle and virtual isocenter trajectories of both the Catphan phantom and the Isocal phantom, we used our calibration procedure to find the geometric offsets for each view. Table (ref:tab:geo_cal_offs) shows a subset of the offsets given by our method for a gantry angles at approximately $90ˆ$ intervals. For these calibrations, we only allowed the optimization program to vary the detector’s lateral and longitudinal position as well as the source’s longitudinal position. With the virtual isocenter trajectory that also introduces the couch motion, there is a larger amount of variation in the offset magnitude than in the circle trajectory that has a stationary treatment couch.
| Trajectory | Gantry angle $\left[ˆ\right]$ | Det lat [mm] | Det long [mm] | Src long [mm] |
|---|---|---|---|---|
| Circle | -179.9 | -0.4 | -0.9 | 1.8 |
| -90.2 | 0.4 | -0.8 | 1.5 | |
| -0.2 | 0.5 | -0.8 | 2.0 | |
| 90.2 | -0.2 | -1.0 | 2.2 | |
| 180.2 | -0.4 | -1.0 | 1.8 | |
| Virtual isocenter | -180.1 | -0.3 | -0.6 | 2.5 |
| -90.0 | 1.6 | 0.1 | 0.7 | |
| 0.0 | 0.7 | -0.7 | 1.5 | |
| 90.0 | 0.1 | -1.5 | 3.0 | |
| 178.2 | -0.2 | -0.8 | 2.5 |
Figure (ref:fig:geo_cal_catphan_bar) shows the CTP 528 spatial resolution module slice from the reconstructions of both the circular scan (left column) and the virtual isocenter scan (right column). The top row shows the slice from the uncalibrated reconstruction using the nominal projection geometry from image metadata. The circle and virtual isocenter scans without calibration demonstrate that moving the treatment couch during the scan introduces additional geometric error over the standard circle scan which visually degrades spatial resolution.
The bottom row of Figure (ref:fig:geo_cal_catphan_bar) shows the same
slice from the corresponding trajectory with the geometric offsets
from the calibration procedure incorporated into the system matrix
Figure (ref:fig:geo_cal_cost) shows the $L2-\text{norm}$ of the distance between the simulated fiducial projections and the real fiducial projections acquired from the circle and virtual isocenter scans of the isocal phantom. We can see that the calibration did effectively reduce this cost from the nominal geometry (blue) to the calibrated geometry (green). This cost also reflects the same trend we see in the spatial resolution of the images shown in Figure (ref:fig:geo_cal_catphan_bar).
Comparing the the $L2-\text{norm}$ of the uncalibrated scans in Figure (ref:fig:geo_cal_cost), we see that there is far more disagreement between modeled and observed Isocal fiducial positions for the virtual isocenter scan than that of the circular scan, leading to more artifacts and loss of spatial resolution in the virtual isocenter reconstruction than in that of the circular scan. With the geometric calibrations applied, the cost for the virtual isocenter and circular trajectories is quite comparable, as is the spatial resolution.
- truebeam/170603_virtiso_circ_smth_catphan/dynmag/em/calibration_images.ipynb
- truebeam/170603_virtiso_circ_smth_catphan/dynmag/calibs/calib_analysis.ipynb
- 170603_virtiso_circ_smth_catphan/dynmag/calibs/calib_analysis.ipynb
It is important to understand that by introducing the motion of the couch in the virtual isocenter trajectory, there is the subsequent introduction of additional uncertainty associated with the geometry of the couch position. Both the visual appearance of the spatial resolution module in Figure (ref:fig:geo_cal_catphan_bar) as well as the cost function in Figure (ref:fig:geo_cal_sens_cost) prior to calibration reflect the additional uncertainty introduced by the couch motion. However, the ability to recover the bar phantom pattern and achieve a similar cost function after calibration as the circle trajectory demonstrate that this additional uncertainty can be accounted for with calibration.
It is also interesting to note the sinusoidal appearance of the cost function as plotted against gantry angle in Figure (ref:fig:geo_cal_sens_cost). The variation as a function of the gantry’s orientation is most likely due to the variation in the torque applied to the gantry by gravity. Though this gravitational impact has been studied before cite:sharpe_stability_2005, it is interesting to see the additional uncertainty this effect has when also combining the simultaneous couch motion into the trajectory.
One of the most challenging aspects of using an optimization-based calibration procedure such as ours is that different offset variables can couple with other parameters that geometrically impart the same effective offset relative to the image space coordinate system. An example of that would be a couch offset in one direction being equivalent to a detector offset in the opposite direction. While this ambiguity is exactly the same feature we utilize when implementing some of trajectories, i.e., an axial-FOV extension by either translating the couch or the imaging arms, it does complicate the calibration procedure.
In this work, we addressed by only allowing some parameters to vary while keeping other variables we know do have uncertainty fixed. For example in Table (ref:tab:geo_cal_offs), we only use offsets for the source and detector for both the circle and virtual isocenter trajectories. In this case, though we know the couch introduces additional uncertainty, the calibration procedure is able to incorporate these offsets into the relative offsets of the source and detector. As we are ultimately interested in obtaining a proper reconstruction of the object in the image space, reduction in cost in the objective function subsequently manifests as an improvement in the projection geometry relative to the image space.
In future calibration work, it would useful to address this ambiguity by constraining the optimization function in a way that would ensure much greater absolute accuracy of the calibration offsets. Though we did not do that here, this could be done to some degree by taking known tolerance of the LINAC components into account. For example, the tolerances of the couch position and the imaging arms could be used to provide the appropriate weighting for how much the different offset magnitudes are allowed to vary.
In developing our optimization-based geometry calibration procedure, we found that proper geometric calibration is critical to achieving optimal tomographic image quality. This is particularly true for more complicated trajectories where additional motion components such as that of the treatment couch introduce additional degrees of freedom in which geometric errors can arise. As shown in Figure (ref:fig:geo_cal_sens_cost), the additional motion of the couch with the simultaneous motion of the source and detector introduces a larger deviation from the nominal scanning geometry.The optimization-based calibration we used in this study provides a robust framework for calibrating arbitrary scanning trajectories. The ability to acquire view-by-view calibration information with this approach dovetails nicely with the optimization-based framework that enables the reconstruction from the different trajectories we studied in this research. Though many of the different analytic-based methods described in the literature could be adapted to some of these trajectories cite:noo_analytic_2000,smekal_geometric_2004, the benefit of the optimization-based framework for both reconstruction and geometric calibration comes from freedom to easily model and reconstruct from any desired trajectories as well as geometric offsets that deviate from the analytically prescribed model.
Though this does imply that calibration scans must be acquired for each scan of interest, there are optimization-based calibration methods similar to ours that attempt to extract calibration information with no a priori knowledge of the phantom cite:panetta_optimization-based_2008. Such calibration methods or built-in calibration markers in the table are potential ways in which it would be possible to avoid acquiring calibration information for every scan of interest. As we used the TrueBeam kV-CBCT system for our data acquisition, Varian’s Isocal phantom provided a convenient means of calibrating the imaging system as the LINAC use case already demands accurate calibration for treatment accuracy in addition to image quality alone.
In the following chapters, where we investigate particular
applications of these different trajectories, we will use our
calibration method with the Isocal phantom to more accurately model
the system matrix
- General approach seems to be to make the chapters presentations of different studies (papers/proceedings) and the subsequent results and conclusions that can be made.
- simulation study of axial FOV extension
- Trilogy scans of RANDO and Defrise phantom for axial FOV extension
- Similar results to cite:davis_verifying_2013 using RANDO and Defrise Trilogy scans
- AAPM talk using CLLC scan from TrueBeam
- AAPM poster for disk phantom metrics
- Varian meeting showing non-circular scans
A major limitation of LINAC-mounted CBCT kV-imaging systems is their axial coverage. This is primarily due to the detector size which is restricted by both cost and engineering concerns. Unlike modern diagnostic CT systems, the LINAC gantry is unable to perform more than a single rotation in a given direction. Without the ability to continuously rotate about the patient in the same direction, the helical scan solution to this limited-axial-FOV problem used by modern diagnostic CT systems is untenable.
When an extended axial FOV is needed in the IGRT clinic, the current practice is to acquire two circular scans centered at different axial positions. Once each independent volume has been reconstructed with an analytic-based FBP algorithm such as FDK, the two volumes are stacked to create the extended image. Though this stacked image does extend the axial coverage beyond that provided by a single circular scan, there are some limitations to this approach. We investigated if the non-circular trajectories with optimization-based methods described in /Generalized trajectory framework/ can provide a solution to these shortcomings.
The main problem with the current clinical approach is a limitation incurred by reconstructing each of the volumes with analytic-based reconstruction methods, such as FDK. When combining the volumes of two independently-constructed FDK volumes acquired at different axial positions, the overlap region between the two axial positions corresponds to the larger cone angles of the two independent volumes. As methods like FDK are known to suffer from cone-angle artifacts at the axial extremes of the reconstruction volume as shown in Figure (ref:fig:opt_defrise), this volume stacking approach abuts the regions of the two independent volumes most afflicted with cone angle artifacts against each other as shown in Figure (ref:fig:opt_fov_schematic).
Furthermore, the stacking of FDK volumes is also limited by the axial coverage allowed by the reconstruction algorithm. In Figure (ref:fig:opt_fov_schematic), everything bounded by the detectors for both circles at opposing views constitutes the image support of optimization-based methods like MLEM. For comparison, the support of the FDK reconstruction is limited to just the shaded regions of each circle.
As the axial spacing between the two circles is increased, the shadow zones cite:forthmann_adaptive_2009 corresponding to regions outside the support of the FDK reconstruction further contaminate the region of overlap. As the spacing between the circles is increased, these shadows regions encroach into the volume of the stacked FDK image. At the maximum axial spacing of 20 cm for the TrueBeam system, the shadow zones extend directly to the center of the image as shown in Figure (ref:fig:opt_fov_20cm) resulting in missing information in the stacked volume.
In addition to the shadow zone contamination, there is another
limitation that affects the stacked FDK volume approach. As the two
circular volumes are reconstructed independently, neither scan
benefits from mutually-shared information in the overlap region
between the two scans, which could potentially reduce the cone-angle
artifacts. For axial separations between the two circles
Taking advantage of the flexible reconstruction framework described in /General CBCT trajectory reconstruction framework with optimization-based algorithms/, data from more than a single circular scan can be reconstructed. Provided that the correct geometry of the acquisition trajectory is well understood and properly calibrated (e.g. using a calibration method such as that discussed in /Geometric calibration/), the system matrix of the image formation process can be calculated for arbitrary CBCT scanning configurations. Trajectories are not limited to the few cases of non-circular trajectories for which analytic inverse formulations exist such as the line cite:sidky_volume_2005, circle and line cite:zeng_cone-beam_1992,katsevich_image_2004, circle and arcs cite:zou_image_2005-1,katsevich_image_2005, and non-planar orbits cite:kudo_derivation_1994.
Using some of the trajectories enabled by optimization-based methods, we can address the problem of the limited axial coverage for LINAC-mounted CBCT kV-imaging systems. Rather than increasing the size of the detector, the source and detector motion can be extended in the axial direction, allowing projections to be obtained for axial positions beyond what is illuminated with current detector sizes and a circular scanning trajectory.
In this chapter, we study a few different trajectories that could address the limited axial coverage provided by LINAC-mounted kV-imaging systems. For the case of the TrueBeam system, the axial coverage of the treatment FOV is 40 cm while the CBCT coverage from a single circle is only 20 cm. This limitation can be problematic for patients with treatment volumes that extend axially beyond what is visible in a single circular acquisition cite:voong_dosimetric_2014. The trajectories we investigate in this chapter provide extended axial coverage. Our hypothesis is that data from scans with extended axial coverage can be reconstructed into extended FOV images with quality equivalent to current clinical scans using only a single circle and resultant limited axial FOV.
In order to obtain additional axial tomographic information from a patient beyond what is covered by the detector, there must be a relative shift along that axis between the patient and the imaging system. As described to in the /Generalized trajectory framework/ section, either the patient or the imaging system can shift along this direction to obtain the desired projection information as long as the motion is correctly reflected in the system matrix $\mathcal{H}$. With the additional projections along the axial direction, it is possible to extend the axial coverage of the tomographic image beyond that which is provided by a single circular scan.The current clinical method of obtaining an extended axial image involves stacking the FDK reconstructions of two circular scans at different axial positions. For this reason, the first class of trajectories we studied was a dual-circle trajectory shown on the left in Figure (ref:fig:fov_3d_trajectories). As the LINAC is only able to make one complete rotation of the gantry, this is implemented by acquiring a circular trajectory scan, applying the axial shift, and then acquiring a second circle by rotating the gantry in the opposite direction.
figures/fov/3d_trajectories.pdf
Another class of trajectories we studied was the circle-line-circle (CLC) trajectory shown in the middle of Figure (ref:fig:fov_3d_trajectories). Like the double circle trajectory, this trajectory consists of two circles at various axial positions, but with projections also acquired during the linear shift between the two axial positions of the circles for the CLC trajectory.
The last class of trajectories we studied will be referred to as the smooth trajectory which is shown at the right of Figure (ref:fig:fov_3d_trajectories). It is similar to the CLC trajectory in that it consists of two circles at two different axial positions with projections acquired along the axial shift component. The gantry motion is the same in that it performs two rotations in the opposite direction. The difference here is that the axial shift component is begun before the first gantry rotation is complete, and the second gantry rotation begins before the axial shift is complete. Unlike the double circle and CLC trajectories, this trajectory does not have a complete circle at either of the two axial positions.
**** notebooks :noexport: ***** truebeam/170816_circ_clc_smth_catphans/matlab/em/trajectories.ipynb
- plots of trajectories from real scan data
*** Simulation
As the current clinical procedure for obtaining an extended axial FOV CBCT is by stacking together the independently reconstructed images of two circular scans acquired at different axial locations, we first used simulations to determine the maximum axial spacing between the two circular trajectories allowed by this technique. To evaluate this, we compared the simulated results of stacking independently-reconstructed FDK images from two circular scans at various axial separations to the MLEM reconstruction of those same two circles reconstructed simultaneously as a single acquisition sinogram cite:davis_extended_2013.We simulated a Defrise-style phantom modeled with the 3D X-ray projection software TAKE cite:seger_matlab/c_2005. The phantom was composed of a 15.2 cm outer diameter acrylic cylinder with alternating density disks of Delrin and cork 0.5 cm thick. Due to the alternating density disks along the axial direction, this particular phantom has been acknowledged by the authors of FDK to be particularly susceptible to cone-angle artifacts cite:feldkamp_practical_1984. We used the TAKE software to forward project the phantom as well as generate a digitized “truth” phantom for calculating comparison metrics. The simulation program generates a forward projection from a specified trajectory given a mathematical definition of the phantom as well as its material properties and the spectrum generated by the x-ray source.
We created projection data for dual-circle trajectories that had variable axial separation between the two circles. With a 1.5x magnification factor and a 30 cm detector size along the axial direction (e.g., geometry equivalent to that of the TrueBeam kV-imaging detector), a single circular scan has a maximum axial coverage of 20 cm. Thus, the maximum spacing between the two circles is 20 cm as any separation larger than this creates a gap between the two imaging volumes. We therefore created trajectories with 10, 12, 14, 16, 18, and 20 cm separations between the planes of the source’s dual-circle trajectory.
In addition to the double circle trajectory, we also simulated projections from the CLC and smooth classes of trajectories shown in Figure (ref:fig:fov_3d_trajectories). We uniformly distributed 600 views over the entire trajectory. Of these 600 views, 20% were distributed along the axial translation stage.
For the extended-volume reconstruction using the stacked FDK, we
independently reconstructed each circular scan with FDK using a
standard Hann filter. The reconstruction image space consisted of a
For the MLEM reconstructions, the projection data were treated as a single sinogram to reconstruct the extended volume. After defining the extended image volume, we computed the system matrix for each of the different spacings and trajectories based on the trajectory of the source and detector. We used 100 iterations of the MLEM algorithm to find an estimate for the image.
Our initial evaluation of the images obtained from non-circular trajectories is simply a qualitative visual inspection which does provide an informative assessment of the variety of artifacts that occur for a given reconstruction. For a more rigorous evaluation of the images obtained from different trajectories, we will use mutual information (MI) cite:pluim_mutual-information-based_2003 and the universal quality index (UQI) cite:wang_universal_2002 to provide a quantitative assessment of the image similarity between the reference image and the images from different trajectories.
Simulating forward projections from this trajectory, we compared the images obtained from stacking together the independent FDK images to those obtained by reconstructing the two circles as a single trajectory with MLEM. We also compared stacked FDK images to reconstructions from the simulated CLC and smooth trajectories.
After identifying potential benefits of addressing the limited axial FOV using non-circular trajectories as described in the /Generalized trajectory framework/ section, we then evaluated how well this approach worked when implemented on our TrueBeam system using Developer mode as describe in the /Framework implementation with Varian TrueBeam kV-CBCT system/ section. By using the Developer Mode XML control schema, we created acquisition scripts that implemented gantry rotations with a component of axial translation between the patient and the kV-imaging system. The trajectory plots shown in Figure (ref:fig:fov_3d_trajectories) are from the actual trajectories of the three classes studied with a 17 cm gap between the two planes of the circular orbits.Though the imaging framework is agnostic to which component of the system effects the relative motion between the patient and the imaging system, there are some engineering limitations of the TrueBeam system that determine how the axial translation component of these trajectories is implemented. In the current TrueBeam implementation, the kV-imaging robotic arms cannot perform any translational movement while the gantry is rotating. As such, all of our axial translation were implemented by moving the treatment couch for all of the trajectories instead of translating the robotic arms. As the CLC trajectory requires no gantry rotation during the translation stage, we did acquire one CLC trajectory using translation of the robotic imaging arms for comparison.
The image quality that results from using these trajectories with optimization-based algorithms must be quantitatively evaluated for the different trajectories and spacings chosen. Given that contrast resolution is important to clinical utility cite:dawson_advances_2007, we wanted to characterize the low-contrast resolution as a function of axial position for the different trajectories and spacings. As the extended axial coverage we obtain with the kV imaging system using these methods is novel, there is not a standard phantom for characterizing low-contrast resolution as a function of axial position within a single scan. Figure (ref:fig:fov_catphan_sagittal) shows the limited coverage afforded by a single Catphan 504 phantom.
For this reason, we built a custom low-contrast disk phantoms that fit into an acrylic tube with extended axial coverage as shown in Figure (ref:fig:fov_tube_setup). The disks themselves, such as the one shown in Figure (ref:fig:fov_disk), are designed to provide similar metrics such as those obtained with the Catphan phantom’s low-contrast module CTP515 shown in Figure (ref:fig:fov_catphan_ctp515). Additionally, the largest holes are designed to hold the different electron density plugs from the Gammex (Middleton, WI) RMI tissue characterization phantom. By placing four of these disks in the tube, we can obtain these metrics as a function of axial position within a given reconstruction.
After some preliminary studies, we found the nominal 1% low-contrast inserts were too challenging to consistently identify on the LINAC’s kV-imaging system. Thus, we decided to use two Catphan 504 phantoms to provide features for obtaining image quality metrics. As we needed to acquire image quality metrics over the extended axial coverage, we placed the two Catphan phantoms end to end. This effectively provided a standard image-quality phantom that also spanned the extended axial-FOV volume enabled by these trajectories of interest.
For the all of the different trajectory scans, the double-Catphan configuration was positioned with one of the Catphan’s sensitometry module shown in Figure (ref:fig:opt_ctp404) aligned at imaging isocenter. As this module provides many of the features we used to calculate image metrics, we wanted to acquire metrics with this module placed so that it is in the plane of the source’s orbit for one of the two circles. By doing this, we could compare the image quality metrics from the different scanning trajectories against a clinical circular FDK scan where the image quality is evaluated on the plane where FDK satisfies Tuy’s condition. We acquired all three classes of trajectories (double circle, CLC, and smooth) with both full-fan and half-fan detector configurations.
For a LINAC system, it often is the case that the patient’s transverse slice volume exceeds the transverse FOV support allowed by the trans-axial coverage of the kV-imaging detector when it is centered with the piercing point of the source to detector ray in the middle of the detector. To overcome this limitation, a detector offset is given to the detector so that it is shifted in the lateral dimension in the radiation coordinate system shown in Figure (ref:fig:opt_coords_rad) cite:chang_asymmetric_1995,cho_cone-beam_1995. In doing this, the transaxial FOV is increased as this half-fan configuration acquires projections from half of the phantom or patient in the first $180ˆ$ of rotation and the second half of the phantom or patient projections in the second $180ˆ$ of rotation..
We also scanned the CIRS torso phantom shown in Figure (ref:fig:opt_cirs_setup). We aligned the phantom so that the center of phantom was placed at the midpoint between the axial position of the two circular components of the different trajectories. We again acquired the three classes of scanning trajectories, but only with a half-fan detector configuration as this larger phantom would incur truncation artifacts with a full-fan configuration.
In addition to acquiring the different trajectory scans of these phantoms, we also repeated the scans with the Isocal phantom to acquire geometric calibration information. As the Isocal phantom is designed to align the MV-treatment isocenter with the kV-imaging system’s isocenter, it has a fixed mounting point on the treatment couch. In order to extract the calibration information for the different trajectories, we positioned the Isocal phantom on the treatment table at the location where we placed the double Catphans and the CIRS torso phantom. As described in /Geometric calibration/, we used the Isocal scans to provide calibration corrections for all three trajectories with both the full-fan and half-fan detector configurations.
The clinical reconstruction software uses a larger voxel size for half-fan detector scans than for full-fan detector configurations. We therefore reconstructed the CIRS torso phantom onto a 0.836 mm isotropic voxel grid. However, as voxel size is a critical parameter of the reconstruction program, we reconstructed all of the scans of the double-Catphan configuration onto a 0.473 mm isotropic image grid for all detector configurations so that only the detector configuration would be the independent variable. As we will discuss in the next chapter, we selected 200 iterations of MLEM for all of our optimization-based reconstructions.
In addition to reconstructing the extended-axial-FOV sinograms using the optimization-based framework, we also reconstructed the superior and inferior circles of the double circle using Varian’s clinical FDK algorithm in iTools. In addition to the default reconstruction chain, we also reconstructed the two circles without Varian’s pre-processing chain that provides a scatter-correction to the projection data. We did this because no scatter correction modeling was implemented in our MLEM reconstructions. Though we had initially attempted to extract the pre-processed projection data from iTools to use in our reconstruction chain, we discovered that some of the non-standard motions we implemented with the LINAC could not be accommodated in iTools as it did not conform to the typical circular trajectory anticipated by Varian’s software.
For additional comparison, we also reconstructed the superior and inferior circle independently with our analytic-based reconstruction chain. We then stacked these two independent volumes together the same way as for the FDK reconstructions. Finally, as MLEM generally provides better spatial resolution that FDK, we also reconstructed the two circles without pre-processing using the iTools FDK with a sharp kernel as to increase the spatial resolution from the FDK reconstructions.
For the dual-circle-extended-line trajectory, it was possible to acquire this in Developer Mode by controlling just the c-arms. This trajectory shown in Figure (\ref{fig:cllc_traj}) acquires data from two circles 19 cm apart and during the translation of the source and detector from the superior to inferior circle. It also acquires an additional 4.5 cm linear scan above and below the two circles, which helps reduce the cone-angle artifacts at the edge of the FOV. Downsampling from this data set can provide scan data for the dual circle, the dual circle and line, and the dual circle extended line. We scanned the CIRS Torso Phantom (Computerized Imaging Reference Systems, Norfolk, VA) which contains low-contrast structures unlike the RANDO phantom.
One limitation of our version of Developer Mode is that it is not possible to move the source and detector arms during the gantry rotation. For this work, any desired arm translation during gantry rotation is mimicked with the couch translation whenever it will produce an equivalent relative translation. While this is not exactly equivalent, it allows a close approximation of such a trajectory. For this reason, the smooth trajectory discussed in the previous section was implemented with gantry rotation and couch translation.
We determined that though the two methods are comparable for the smaller axial gaps between the two circles that FDK can support, it would be clinically advantageous to evaluate larger axial gaps that are untenable for the stacked FDK method. Using the TrueBeam system, we then acquired scans from axial spacings of 17, 18,19, and 20 cm between the two circular planes of these trajectories. These larger spacings provided data for potential axial FOV’s of 37, 38, 39, and 40 cm respectively. However, after some preliminary results, we decided to use a 17 cm gap between the planes of the two circles for each of the three classes of trajectories shown in Figure (ref:fig:fov_3d_trajectories).
To acquire more scan information from the central region between the two circles, we also simulated two other trajectories. Both of these trajectories acquire additional view while translating the source and detector from the axial position of one circle to the other. One of these trajectories is a circle-line-circle trajectory where the translation phase of the source and detector between the two circles is a line as Figure (\ref{fig:traj_clc}) shows. The other trajectory shown in Figure (\ref{fig:traj_smth}) is a smooth trajectory that is similar to the circle-line-circle, but the gantry rotation and longitudinal translation are designed to avoid sharp transition points in the imaging trajectory.
| Trajectory | Axial gap [cm] | Detector configuration | Air scan | lognorm | Notes |
| Circle | Full | x | |||
| Circle | Half | x | |||
| CLLC | 20 | Full | 20 cm air | x | |
| CLLC | 19 | Full | ”” | x | |
| CLLC | 18 | Full | ”” | x | |
| CLLC | 17 | Full | ”” | x | |
| CLLC | 18 | Half | x | 2 of these for some reason | |
| SMTH | 20 | Full | x | ||
| SMTH | 19 | Full | x | ||
| SMTH | 18 | Full | x | ||
| SMTH | 17 | Full | x | ||
| SMTH | 18 | Half | x |
** Results
Our simulations suggested that the optimization-based reconstruction of the dual-circle trajectory is comparable to the clinical stacked FDK method when the axial spacing is small enough that the stacked FDK method is still able to yield a complete reconstruction. As shown in Figure (ref:fig:opt_fov_schematic), larger axial spacings $\left(d\right)$ between the two circles results in larger portions of the image volume falling outside of FDK’s support volume. For the geometry of our Defrise-style phantom and our TrueBeam kV-imaging system geometry, the stacked-FDK method and the optimization-based reconstruction were comparable up to a gap of approximately 14 cm. For larger axial gaps, the FDK shadow zone between the two volumes begins to infiltrate the reconstruction volume creating voids in the reconstruction volume.figures/fov/take/central_cbct_rmse.pdf
Figure (ref:fig:fov_take_rmse) shows a root-mean-square error (RMSE) comparison of the three classes of trajectories as well as the stacked FDK method with different axial spacings between the two planes of the circles. The volume for which the RMSE is calculated is the central volume between the two circles with a 20 cm axial length, which would be the region seen with a single circular scan at the midplane. The figure shows that for any extended volume spacing, the stacked-FDK reconstruction from two separate circles deviates the most from the truth, and it degrades with increasing separation.
Figure (ref:fig:fov_take_rmse) also shows that the optimization-based reconstruction of the same dual-circle trajectory is much closer to the truth, but also demonstrates degradation with increasing spacing between the two circles. Finally, the CLC trajectory and the smooth trajectories reconstructions remain relatively constant for increasing spacing, with the smooth trajectory being closer to the truth. The slices shown in Figure (ref:fig:fov_sim_take) visually agree with these results. Notice that in the center of the volume, both the CLC and smooth trajectories are able to recover most of the alternating disks by acquiring some projections in the region between the circles. Furthermore, by acquiring these projections in the overlap region while also rotating as with the smooth trajectory, there is an additional improvement in this central region. The double circle and line trajectory was left out of Figure (ref:fig:fov_sim_take) since the results were not visually distinguishable from the smooth trajectory reconstructions.
Figure (ref:fig:fov_dcatphans_itools) shows the central sagittal slice from the stacked volume of the iTools FDK reconstruction of the dual-Catphan phantom scanned with a full-fan detector configuration. This is representative of the current clinical method of acquiring an extended axial FOV image though the axial spacing exceeds the axial support of FDK. Figure (ref:fig:fov_dcatphans) shows the central sagittal slices of MLEM reconstructions of the same dual-Catphan configuration with each of the extended-axial-FOV trajectories in both the full-fan and half-fan detector configurations. The left column of the figure represents full-fan detector configuration scans, and the right column the half-fan detector configuration scans. From the top row to the bottom, these images correspond to the double circle, the CLC, and the smooth trajectory.
We first investigated the quantitative Hounsfield units (HU) from the different sensitometry modules. Figure (ref:fig:sensROIplot) shows the mean ROI value for each of the Catphan sensitometry modules for the stacked FDK and the stacked MLEM reconstructions as well as the extended-axial-FOV reconstructions. The height of each of the bars is the standard deviation of the ROI. The first plot in Figure (ref:fig:fov_sensROIfull) corresponds to the full-fan detector scans, and the bottom plot in Figure (ref:fig:fov_sensROIhalf) corresponds to the half-fan detector configuration.
In both Figures (ref:fig:fov_sensROIfull) and (ref:fig:fov_sensROIhalf), we can see there is a consistent reproducibility between the different scanning trajectories reconstructed with both the full-fan and half fan detector configurations. As the FDK reconstruction is from Varian’s iTools software, it serves as a reference for the performance of current clinical reconstructions available from this LINAC-mounted CBCT imaging system. We used the Hounsfield values from the clinical reconstruction in the sensitometry module to determine a map from electron density to CT number that we then used to convert the MLEM reconstructions to Hounsfield units.
To further investigate the relationship between the full-fan and half-fan configuration as well as the algorithm used, we plotted the ROI values for just the stacked circle trajectories. As the introduction of additional trajectory components will be applicable for the optimization-based framework, this comparison simply provides a baseline of how MLEM directly compares to the clinical standard. Figure (ref:fig:fov_sensROI_full_half) shows the ROI CT numbers of both the FDK and MLEM reconstructions for both the full-fan and half-fan detector configurations. Again, each bar is centered on the mean CT number, and the height corresponds to the standard deviation of the ROI. Again, these plots server for comparison purposes between scans rather than illustrating the true CT number as even the clinical iTools reconstructions show a bias given as the CT number for water is not mapped to zero HU.
figures/fov/rois/sensROIplot_full_half_fdk.pdf
We then studied the impact the different scanning configurations and algorithms had on the spatial resolution of the reconstruction. Though the MTF is a problematic metric of spatial resolution in CT due to its violation of linear shift invariance, it can provide a useful benchmark for evaluating the impact different scanning parameters could have on the reconstruction’s spatial resolution. By using different MTF metrics from different features in the Catphan, we could get an approximate characterization of how the different classes of trajectories reconstructed with our framework compare to the current clinical standard of stacked FDK.
The spatial resolution metrics we extracted from the dual-Catphan reconstructions from the different classes of trajectories are shown in Figure (ref:fig:fov_mtf_alg). This plot shows the different MTF values for the different algorithms with both the full and half-fan configuration. As the lack of spatial-shift invariance in CT makes using the MTF metric only a rough estimate of spatial resolution performance, there are a few reconstructions where the metric values are somewhat anomalous.
In addition to the spatial resolution metrics, we also used the CTP 404 sensitometry module to calculate the CNR of the acrylic, PMP, and polystyrene inserts. These are the inserts that provide the lowest contrast relative to the water-equivalent inserts as can be seen in the ROI measurements in Figure (ref:fig:sensROIplot). Figures (ref:fig:fov_acrylic_cnr), (ref:fig:fov_pmp_cnr), and (ref:fig:fov_poly_cnr) show the calculated CNR plots for the different algorithms and trajectories for both the full-fan and half-fan detector configurations for three different contrast plugs of acrylic, polymethylpentane (PMP), and polystyrene.
Finally, the reconstructed images of the anthropomorphic CIRS torso phantom provide a visual illustration of the three different trajectory classes. The rows in Figure (ref:fig:fov_ed) show the central sagittal, coronal, and transverse slices (from top to bottom respectively) of the CIRS torso phantom scanned with the three extended-axial-FOV trajectories using a half-fan detector configuration. Each column in the figure corresponds to one of the scanning trajectories.
- initial analysis that didn’t include the iTools FDK
rsync -avz remus:/data/amdavis/truebeam/170816_circ_clc_smth_catphans/figs/ .By increasing the separation between the two circular scans up to the maximum spacing of 20 cm, it is apparent that FDK has fundamental support limitations that restricts the acceptable distance between these two circles to less than the maximum 20 cm spacing due to the increasing size of the shadow zone with increased separation between the two planes of the circles. This is particularly problematic in that the infiltration of the shadow zones into the reconstruction volume occur in the region between the two circles. It is likely that if such an extended image volume were needed clinically, it would be axially centered on the region of interest. As such, using the stacked-FDK method for these larger axial volumes would place the region of interest directly in the overlap region where FDK is plagued by both cone-angle artifacts and the shadow zone.
Optimization-based reconstruction methods can use information from both circular scans simultaneously, leading to improved reconstruction of the image in the shadow zone as seen in Figure (ref:fig:fov_sim_take). However, as it can be seen in Figure (ref:fig:fov_catphan_sagittal), artifacts can still appear in the overlap region even with optimization based method. The streaking is particularly noticeable in these results for two reasons.
The first issue is the fact that there is a sharp density change in the axial direction much like the difficult Defrise-like phantom design we used in simulation. We made an effort to minimize this drastic density change from the Catphan to air and then directly back to the Catphan by inserting a urethane plug we machined to fit in the ends of the two phantoms. In addition to this, we also sandwiched a foam disk in the region to attempt to fill the flush concavities of the two Catphan tops.
Another problem with this particular data is related to contamination of the beam at the edge of the axial FOV from the collimator blade appearing in the FOV. Though this collimator blade would be slightly inconvenient in a typical circular scan, it would not be excessively problematic as it would appear in the region typically affected by cone-angle artifacts. Here however, as the edge of the axial FOV appears in the center of the image, this additional source of inconsistency exacerbates an already challenging feature to reconstruct with a CBCT system.
For future work on this particular use of non-circular trajectories, it would first be beneficial to ensure the collimator blades do encroach onto the detector. Especially along the axial extremes of the detector as that will contaminate what would probably be the region of most interest. It would also be nice to place an image quality module in the overlap region to evaluate the image quality that this method would be able to provide without the collimator blade contamination.
Despite the limitations of this study, the decision to place the image quality modules at the axial position of the circle’s source plane gives an upper limit on what FDK can reasonably achieve, even as a standalone circular scan comparison. By evaluating the two algorithms at a location where FDK does satisfy Tuy’s condition, we can see in the results that MLEM can do just as well if not better than FDK.
For example, in the comparison of the stacked, independently-reconstructed, dual-circle trajectories using both algorithms, it can be seen that FDk fails to be consistent with itself in the full-fan versus half-fan configuration. As Figure (ref:fig:fov_sensROI_full_half) shows, the CT numbers for the FDK with the different detector configuration do not agree with each other. The two detector configurations for MLEM however, not only are consistent with each other, but they also fall between the two extremes bracketed by the FDK results which ought to be the same number.
The MTF results shown in Figure (ref:fig:fov_mtf_alg) also reflect a consistency among the different classes of scanning trajectories reconstructed with MLEM. Though the difficulty of MTF metrics in CT introduce some nonsensical outiers, we can see that the MLEM reconstructions perform as well as the standard FDK, and that the sharper kernel does produce better spatial resolution. This is best illustrated by the top row of Figure (ref:fig:fov_mtf_alg) that shows the bar pattern MTF results which are more robust to the violation of linear-shift invariance.
Similarly, the CNR results shown in Figures (ref:fig:fov_acrylic_cnr), (ref:fig:fov_pmp_cnr), and (ref:fig:fov_poly_cnr) demonstrate the same consistency for the MLEM reconstructions of the different trajectory classes. As expected, the increased spatial resolution of the sharp kernel in FDK is accompanied by a loss of contrast. Other than the very low-contrast acrylic CNR, the FDK results using the two kernels bracket the MLEM reconstruction results. Again, we only wish to illustrate that these new trajectories with optimization-based reconstruction can perform as well as the current clinical standard.
Finally, it is important to point out that for the smooth trajectory, the half-fan configuration does not provide complete support when using the half-fan detector configuration. In a full-fan configuration, the typical π-plus-fan-angle angular coverage provides sufficient projection data. With the half-fan configuration however, this is not the case; so as the axial-translation component of the trajectory begins, there is insufficient support at the axial extremes of the entire extended image.
However, as both the simulation and experimental results show, distributing some of the angular coverage during the translation does help to improve the image quality in the overlap region. As we have mentioned, this would probably be the region of most interest if a technique like this were used in the clinic. Therefore, while the axial extremes of the extended volume are degraded, as can be seen in Figure (ref:fig:fov_ed) especially in the shoulders of the torso phantom, the streaking in the overlap region is significantly reduced as opposed to the double circle and CLC trajectories. Therefore, if using a half-fan configuration with such a trajectory, care must be take to determine which part of the tomographic image requires higher fidelity.
We found that the use of our optimization-based, non-circular scanning trajectories could successfully reconstruct extended-axial-FOV reconstructions that are comparable to the current clinical image quality achieved with a circular scan and FDK. In addition to being able to maintain comparable image quality in the region where FDK performs the best, our framework was able to extend the axial-FOV coverage beyond the axial separation distances allowed to FDK by the fundamental limitations of its support. Our hypothesis that our optimization-based reconstruction framework could reconstruct from extended axial coverage non-circular trajectories was correct. We also found that the image quality from these reconstructions was at least as good as the current clinical standard. Though additional studies need to be conducted to perform quantitative image quality analysis in the overlap region, we do know this framework provides a feasible means addressing the limited axial FOV coverage that currently affects the clinical use of CBCT in IGRT.- General approach seems to be to make the chapters presentations of different studies (papers/proceedings) and the subsequent results and conclusions that can be made.
- 2015 MIC virtual isocenter
- 2016 CT meeting dyanmic magnification
- 2016 MIC mixed magnification
- 2017 Varian dynamic magnification
Collision avoiding trajectories may be of particular concern in breast and lung cancer patients where the arm position increases the likelihood of potential collisions as shown in Figure (ref:fig:col_barbie_collision). Collisions also present a problem in treatment of posterior and lateral lesions in stereotactic body radiosurgery (SBRT). Similarly in prone breast treatments, where the target is near the couch top and a lateral couch translation is needed to bring the target to isocenter, collision with the contralateral side of the patient may occur. When collisions do occur, the angular range available for scanning is restricted and it is not possible to acquire a complete circular scan in the treatment position.
To avoid collisions with the LINAC head, it might be desirable to move the patient away from the gantry by translating the couch. To avoid collisions with the imaging panel, the patient might also be moved away from the panel. As many LINAC-mounted, kV-imaging panels have motion capabilities, another solution would be to move the imager away from the patient in the collision zone, which changes the imaging magnification for that portion of the scan. Given that the clearance distance of the kV-imaging panel is not much larger than that of the MV-treatment head, a collision avoidance solution should account for both of these components.
Here, we investigate trajectories that could allow the acquisition of sufficient projection information for a clinically useful image while avoiding a potential patient collision with the gantry such as those shown in Figure (ref:fig:col_barbie_collision). One trajectory that would avoid a patient collision with the MV-treatment head is a virtual isocenter trajectory, which increases the effective source-to-axis distance (SAD) for all gantry angles. By using this increased SAD for an imaging trajectory, the clearance between the patient and the MV-treatment head as the gantry rotates is increased and the collision is avoided.
The virtual isocenter trajectory utilizes synchronized gantry rotation and couch translation to maintain a fixed distance (“virtual SAD”) between the MV source and a chosen center of rotation (“virtual isocenter”) in the patient as shown by the red circle in Figure (ref:fig:col_virtual_iso). At the beginning of the scan, the patient is moved away from the LINAC head along the MV beam direction. As the gantry rotates, the couch moves continuously to maintain the specified separation as shown in Figure (ref:fig:col_virtual_iso). The virtual SAD can be chosen large enough such that collisions as shown in Figure (ref:fig:col_barbie_mv) are avoided; at this point in the trajectory, the couch would have moved far enough to the left to avoid the collision. Note that it is only the distance to the LINAC head that is increased in the virtual SAD technique; the distance from the kV source and detector to the patient and to each other are unchanged.
Another trajectory that could avoid a patient collision with the kV detector would be one during which either the patient or the detector is moved during the scan in the angular range of a collision. Either solution leads to changing kV-CBCT imaging magnification during the acquisition. Again, optimization-based reconstruction methods can readily handle such a change in magnification provided the projection information is correctly incorporated into the system matrix.
Finally, we study a trajectory that combines virtual isocenter and dynamic magnification trajectories to create a hybrid scanning acquisition that could alleviate collisions with both the MV-treatment head and the kV-CBCT detector. With such a trajectory, the potential collisions with both the LINAC head and the kV detector panel are resolved. We use such a trajectory as an example of a patient-specific scanning trajectory that could be implemented to resolve potential collisions with two components of the LINAC treatment system.
For the circular scans, the gantry made a full rotation about the patient with the treatment volume at a fixed mechanical SAD of 100 cm. For the virtual isocenter scans, the patient couch was translated continuously in the gantry rotation plane during gantry rotation to maintain a distance of 112 cm between the MV source and the chosen target point within the treatment volume (the “virtual isocenter”), rather than the mechanical SAD of 100 cm as shown in Figure (ref:fig:col_virtual_iso). We generated all of the scanning trajectories in this study using the Developer Mode 2.0 XML schema to define the positions of the gantry, the kV imaging arms, and the patient treatment table. The schema was also used to enable kV-projection imaging during the scan. We used a half-fan detector configuration with a 13 cm offset for the circular acquisition, and an equivalent offset for the virtual isocenter to obtain the same illumination.To increase the clearance between the kV detector and the patient, we
increased the radius of the kV detector with accompanying increase in
magnification of the kV imaging system. Currently, the detector
positioning arm cannot be moved while the gantry rotates. Thus, to
create projection datasets where the detector distance changes during
a scan, we acquired multiple scans using different detector positions
and subsequently spliced these together to create the sinograms of
interest with the corresponding system matrix
figures/col/collision_zones.pdf
To create the combined sinogram of a hypothetical collision-avoiding dynamic magnification scan, we replaced a $45ˆ$ region of the 1.5X circular scan with the corresponding angular range from scans at different magnifications. We chose this region to be centered on the angular position corresponding to the mannequin’s elbow in Figure (ref:fig:col_barbie_collision). Increasing the magnification in this region corresponds to increasing the clearance between the kV-detector and the patient. Increasing the magnification to 1.6X and 1.7X provides an additional 10 cm and 20 cm of clearance respectively. We also created an additional $35ˆ$ 1.7X bump magnification with $5ˆ$ transitions at a 1.6X magnification. The kV-detector collision zones of these dynamic magnification trajectories are shown in the middle plot shown in Figure (ref:fig:col_collision_zones).
The virtual isocenter imaging trajectory would alleviate the potential collision with the MV treatment head accessory mount shown in Figure (ref:fig:col_barbie_mv). The radius of the accessory mount from the mechanical isocenter is 41.7 cm. Using a virtual isocenter would increase the radius of the collision zone by 12cm to alleviate potential collisions (as shown in (ref:fig:col_collision_zones)). Utilizing a different virtual SAD would allow for additional clearance if necessary.
The last set of trajectories we studied combines the dynamic magnification with the virtual isocenter trajectory. As the collision radius with the MV treatment head and the kV detector are similar for the current clinical scan trajectories, collisions with either could arise. By combining the change in magnification with the virtual isocenter trajectory, both collision zones could be avoided. Table (ref:tab:col_trajectories) shows the different scans investigated in this study.
| Trajectory | Magnification |
|---|---|
| Circle | 1.5X |
| 1.5X & $45ˆ$ 1.6X bump | |
| 1.5X & $45ˆ$ 1.7X bump | |
| 1.5X & $35ˆ$ 1.7X bump, 1.6X transitions | |
| Virtual isocenter | 1.5X |
| 1.5X & $45ˆ$ 1.6X bump | |
| 1.5X & $45ˆ$ 1.7X bump | |
| 1.5X & $35ˆ$ 1.7X bump, 1.6X transitions |
As shown in Figure (ref:fig:col_visoDet), using a virtual isocenter
at a distance $Rv$ from the mechanical isocenter results in the
virtual isocenter projecting to a different location on the detector.
The red arrow in the image indicates the projection of the mechanical
isocenter onto the detector whereas the blue arrow indicates the
projective location of the virtual isocenter onto the detector. If
using a detector of width
If the lateral offset $mRv$ in the virtual-isocenter image space is the desired offset, then the lateral detector offset for the circular scan must be changed in the opposite lateral direction to achieve equivalent illumination of the object on the detector. Figure (ref:fig:col_cequiv) shows such an example of the lateral offset needed for achieving the equivalent illumination of the object on the detector as the virtual isocenter configuration shown in Figure (ref:fig:col_visoDet).
However, it is also possible to calculate the virtual-isocenter
lateral offset
Once the desired lateral offset
We reconstructed all of these fixed magnification and dynamic magnification scans from circular and virtual isocenter trajectories into the patient image space described by the imaging model in Equation (ref:eq:opt_linmodel_patient). The Catphan scans were reconstructed onto an isotropic voxel size of 0.473 mm. The CIRS torso scans were reconstructed onto an isotropic voxel size of 0.836 mm. As the circular acquisition with 1.5X magnification is the typical clinical acquisition trajectory, this provides a clinical reference volume for the reconstructions from the other scanning configurations.
figures/col/edge_spread_mtf_iteration.pdf
For the number of iterations in our reconstruction program, we selected 200 iterations as subsequent improvements in spatial resolution with further iterations were diminished. This is shown in Figure (ref:fig:col_mtf_vs_iteration) which plots as a function of iteration numnber the 25% and 50% crossing spatial frequencies of the modular transfer function (MTF) and the area under the curve (AUC) of the MTF. The MTF was found using the edge spread function from the MLEM reconstruction of the Catphan phantom acquired with a circular scan at 1.5X magnification.
- collisions figures
- ed analysis
figures/col/catphanDynmagQuarter.pdf
The visual similarity in the spatial resolution shown in Figure (ref:fig:col_recons_catphan) is reflected in the MTF metrics for all of the different scanning configurations. We compared MTF metrics derived from the PSF using the Catphan beads, the bar pattern shown in Figure (ref:fig:col_recons_catphan), and the edge-spread function (ESF) measure along an ensemble of radial lines. When comparing these MTF-based metrics between the circle and the virtual isocenter scans with different magnifications, we found no clear trend distinguishing the different trajectories.
Figure (ref:fig:col_catphan_cnr) shows the low-contrast CNR from the Catphan with these different imaging configurations which are also in good agreement with each other. In addition to using the material rods in the CTP 404 sensitometry module to calculate the CNR, we also compared mean values from each of the materials for the different scanning trajectories and magnifications. Figure (ref:fig:col_catphan_rois) shows the ROI means for these different scanning configurations. The height of the bar represents the standard deviation of the ROI. The first four bars for each material are from the circular scans, and the remaining four from the virtual isocenter scans.
Figure (ref:fig:col_recons_ed) shows an abdominal slice from the $200\text{th}$ iteration MLEM reconstruction of the CIRS torso phantom scanned with a 13 cm offset half-fan configuration. The layout of these images is the same as that in Figure (ref:fig:col_recons_catphan) with the top row showing magnification variations from the circular trajectory. The bottom row shows the corresponding magnifications from the virtual isocenter trajectory. The slice is the same as that in Figure (ref:fig:col_ed_rois) which we used to calculate values for three organ ROIs.
For the different organ ROIs, we recorded the means and standard deviations for the different scanning trajectories and magnifications. Figure (ref:fig:col_ed_rois) shows these mean values and the associated standard deviations. Though the circular scan variations fluctuate more than the virtual isocenter scans, the values are in agreement for the different organs of interest. As with Figure (ref:fig:col_catphan_rois), the first four points for each organ are from the circular scan, and the remaining four are from the virtual isocenter trajectory.
6.4.1.1 figs
import dicom
import glob
import numpy as np
import matplotlib.pyplot as plt
from IPython.core.debugger import Tracer
debug_here = Tracer()
def crop(center, width, res):
"""Crop image
Keyword Arguments:
center -- [x, y] of center
width -- deisred width (mm)
res -- res (mm/px)
"""
px_width = np.int(width*0.5/res)
xbox = [center[0]-px_width, center[0]+px_width]
ybox = [center[1]-px_width, center[1]+px_width]
return(xbox, ybox)
# window
window_hu = [-160, 240] # HU
window_em = [0.015, 0.026] # mm^1
# itools
catphan_dcm = dicom.read_file(('itools/161213_catphan_circ_1p5x_ringrem/IMG_0043.dcm'))
catphan_hu = catphan_dcm.pixel_array*catphan_dcm.RescaleSlope+catphan_dcm.RescaleIntercept
# crop
width = 230. # mm each way
res_hu = 0.473 # mm/px
center_hu = [np.int(catphan_hu.shape[0]*0.5), np.int(catphan_hu.shape[1]*0.5)]
winx, winy = crop(center_hu, width, res_hu)
catphan_hu = catphan_hu[winx[0]:winx[1], winy[0]:winy[1]]
plt.imsave('figures/catphan_itools_lcr.png', catphan_hu, vmin=window_hu[0], vmax=window_hu[1])
# recon
dir = "/home/amdavis/research/truebeam/150902_virtiso_cbct_catphan_ed/itools/img/"
files = sorted(glob.glob(dir+"*catphan*itools*{0:04d}*.raw".format(iter)))
catphan_circular = np.fromfile(files[0], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
catphan_virtiso = np.fromfile(files[1], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
catphan_circular = catphan_circular[...,
int(catphan_circular.shape[2]/2)]
catphan_virtiso = catphan_virtiso[...,
int(catphan_virtiso.shape[2]/2)]
# crop
res_hu = 0.473 # mm/px
center_circular = [np.int(catphan_circular.shape[0]*0.5), np.int(catphan_circular.shape[1]*0.5)]
winx, winy = crop(center_hu, width, res_hu)
catphan_circular = catphan_circular[winx[0]:winx[1], winy[0]:winy[1]]
catphan_virtiso = catphan_virtiso[winx[0]:winx[1], winy[0]:winy[1]]
plt.imsave('figures/catphan_circular_lcr_mlem_iter_{0:03d}.png'.format(iter), catphan_circular, vmin=window_em[0], vmax=window_em[1])
plt.imsave('figures/catphan_virtiso_lcr_mlem_iter_{0:03d}.png'.format(iter), catphan_virtiso, vmin=window_em[0], vmax=window_em[1])
# return('figures/catphan_itools_lcr.png')
# return('figures/catphan_circular_lcr_mlem.png')
# return('figures/catphan_virtiso_lcr_mlem.png')
fig = plt.figure()
ax1 = fig.add_subplot(131)
ax2 = fig.add_subplot(132)
ax3 = fig.add_subplot(133)
ax1.imshow(catphan_hu, vmin=window_hu[0], vmax=window_hu[1])
ax1.set_title('Circular iTools FDK')
ax1.axis('off')
ax2.imshow(catphan_circular, vmin=window_em[0], vmax=window_em[1])
ax2.set_title('Circular MLEM')
ax2.axis('off')
ax3.imshow(catphan_virtiso, vmin=window_em[0], vmax=window_em[1])
ax3.set_title('Virtual Isocenter MLEM')
ax3.axis('off')
fig.savefig('figures/catphan_comparison_iter_{0:03d}.png'.format(iter), bbox_inches="tight")
return('figures/catphan_comparison_iter_{0:03d}.png'.format(iter))import dicom
import numpy as np
import matplotlib.pyplot as plt
from phantoms import lcr
from IPython.core.debugger import Tracer
debug_here = Tracer()
catphan_dcm = dicom.read_file(('itools/161213_catphan_circ_1p5x_ringrem/IMG_0042.dcm'))
catphan_hu = catphan_dcm.pixel_array*catphan_dcm.RescaleSlope+catphan_dcm.RescaleIntercept
itools_lcr = lcr.CatphanLowContrast(catphan_hu, xres=0.473, x0=121.8/0.473,
y0=121.6/0.473, z=0, rot=-3.48, hu=True)
itools_lcr.save_contrast_calcs('data/', 'catphan_circular_itools')
# itools_lcr.show_rod_locs(vmin=-60, vmax=60)
# plt.savefig('figures/catphan_itools_lcr_locs.png')
# return('figures/catphan_itools_lcr_locs.png')
print(itools_lcr.contrast_1_0)
print(itools_lcr.contrast_0_5)
print(itools_lcr.contrast_0_3)Results from my reconstructions
import glob
import numpy as np
import matplotlib.pyplot as plt
from phantoms import lcr
from IPython.core.debugger import Tracer
debug_here = Tracer()
# get files we have
dir = "/home/amdavis/research/truebeam/150902_virtiso_cbct_catphan_ed/itools/img/"
files = sorted(glob.glob(dir+"*catphan*itools*{0:04d}*.raw".format(iter)))
catphan_circular = np.fromfile(files[0], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
catphan_virtiso = np.fromfile(files[1], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
catphan_circular = catphan_circular[...,
int(catphan_circular.shape[2]/2)]
catphan_virtiso = catphan_virtiso[...,
int(catphan_virtiso.shape[2]/2)]
circular_lcr = lcr.CatphanLowContrast(catphan_circular, xres=0.473,
x0=263.5, y0=261.5, z=0, rot=-3.,
hu=False)
virtiso_lcr = lcr.CatphanLowContrast(catphan_virtiso, xres=0.473,
x0=272, y0=259.5, z=0, rot=-3.,
hu=False)
circular_lcr.save_contrast_calcs('data/', 'catphan_circular_mlem_iter_{0:03d}'.format(iter))
virtiso_lcr.save_contrast_calcs('data/', 'catphan_virtiso_mlem_iter_{0:03d}'.format(iter))
# circ_fig = circular_lcr.show_rod_locs(vmin=0.02, vmax=0.021)
# virt_fig = virtiso_lcr.show_rod_locs(vmin=0.019, vmax=0.021)
# circ_fig.savefig('figures/catphan_circular_mlem_lcr_locs.png')
# virt_fig.savefig('figures/catphan_virtiso_mlem_lcr_locs.png')
# return('figures/catphan_circular_mlem_lcr_locs.png')
# return('figures/virtiso_circular_mlem_lcr_locs.png')
print("\n0.3% circular")
print(circular_lcr.contrast_0_3)
print("\n0.3% virtiso")
print(virtiso_lcr.contrast_0_3)
print("\n0.5% circular")
print(circular_lcr.contrast_0_5)
print("\n0.5% virtiso")
print(virtiso_lcr.contrast_0_5)
print("\n1.0% circular")
print(circular_lcr.contrast_1_0)
print("\n1.0% virtiso")
print(virtiso_lcr.contrast_1_0)
6.4.3.1.1.3.1 CNR
import glob
import numpy as np
import matplotlib.pyplot as plt
import pandas
import seaborn as sns
sns.set(font='serif', font_scale=0.8)
from itertools import cycle
lines = ["-","--","-."]
linecycler = cycle(lines)
from matplotlib.ticker import FormatStrFormatter
# setup plot params
sns.set_context("paper")
from phantoms import lcr
from IPython.core.debugger import Tracer
debug_here = Tracer()
# load previous calculations
circ_fdk = sorted(glob.glob("data/*itools*rods.pkl"))
circ_em = sorted(glob.glob("data/*circular_mlem*_iter_{0:03d}*rods.pkl".format(iter)))
virt_em = sorted(glob.glob("data/*virtiso*_iter_{0:03d}*rods.pkl".format(iter)))
# load 0.3%, 0.5%, 1.0% rod contrast
circ_fdk = [pandas.read_pickle(f) for f in circ_fdk]
circ_em = [pandas.read_pickle(f) for f in circ_em]
virt_em = [pandas.read_pickle(f) for f in virt_em]
# data sets
titles = ['0.3% contrast rods', '0.5% contrast rods', '1.0% contrast rods']
fig, axs = plt.subplots(3, 1, True)
for j, ax in enumerate(axs):
ax.errorbar(circ_fdk[j].index, 'cnr', yerr='cnr sigma', data=circ_fdk[j], label='FDK Circular', ls=next(linecycler))
ax.errorbar(circ_em[j].index, 'cnr', yerr='cnr sigma', data=circ_em[j], label='MLEM Circular', ls=next(linecycler))
ax.errorbar(virt_em[j].index, 'cnr', yerr='cnr sigma', data=virt_em[j], label='MLEM Virtual Isocenter', ls=next(linecycler))
ax.set_xlim([1, 16])
# ax.set_ylim([-3, 3])
ax.set_ylabel('CNR')
ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
ax.set_title(titles[j])
ax.legend(loc='best', fancybox=True, framealpha=0.5)
ax.set_xlabel('Rod diameter [mm]')
fig.savefig('figures/cnr_iter_{0:03d}.pdf'.format(iter))
fig.savefig('figures/cnr_iter_{0:03d}.png'.format(iter))
return('figures/cnr_iter_{0:03d}.png'.format(iter))
import glob
import numpy as np
import matplotlib.pyplot as plt
import pandas
import seaborn as sns
sns.set(font='serif', font_scale=0.8)
from itertools import cycle
lines = ["-","--","-."]
linecycler = cycle(lines)
from matplotlib.ticker import FormatStrFormatter
# setup plot params
sns.set_context("paper")
from phantoms import lcr
from IPython.core.debugger import Tracer
debug_here = Tracer()
# load previous calculations
circ_fdk = sorted(glob.glob("data/*itools*rods.pkl"))
circ_em = sorted(glob.glob("data/*circular_mlem*_iter_{0:03d}*rods.pkl".format(iter)))
virt_em = sorted(glob.glob("data/*virtiso*_iter_{0:03d}*rods.pkl".format(iter)))
# load 0.3%, 0.5%, 1.0% rod contrast
circ_fdk = [pandas.read_pickle(f) for f in circ_fdk]
circ_em = [pandas.read_pickle(f) for f in circ_em]
virt_em = [pandas.read_pickle(f) for f in virt_em]
# data sets
titles = ['0.3% contrast rods', '0.5% contrast rods', '1.0% contrast rods']
fig, axs = plt.subplots(3, 1, True)
for j, ax in enumerate(axs):
ax.errorbar(circ_fdk[j].index, '% contrast', yerr='% contrast sigma', data=circ_fdk[j], label='FDK Circular', ls=next(linecycler))
ax.errorbar(circ_em[j].index, '% contrast', yerr='% contrast sigma', data=circ_em[j], label='MLEM Circular', ls=next(linecycler))
ax.errorbar(virt_em[j].index, '% contrast', yerr='% contrast sigma', data=virt_em[j], label='MLEM Virtual Isocenter', ls=next(linecycler))
ax.set_xlim([1, 16])
ax.set_ylim([-3, 3])
ax.set_ylabel('% contrast')
ax.yaxis.set_major_formatter(FormatStrFormatter('%.1f'))
ax.set_title(titles[j])
ax.legend(loc='best', fancybox=True, framealpha=0.5)
ax.set_xlabel('Rod diameter [mm]')
fig.savefig('figures/percent_contrast_iter_{0:03d}.pdf'.format(iter))
fig.savefig('figures/percent_contrast_iter_{0:03d}.png'.format(iter))
return('figures/percent_contrast_iter_{0:03d}.png'.format(iter))
import dicom
import glob
import numpy as np
import matplotlib.pyplot as plt
from IPython.core.debugger import Tracer
debug_here = Tracer()
def crop(center, width, res):
"""Crop image
Keyword Arguments:
center -- [x, y] of center
width -- deisred width (mm)
res -- res (mm/px)
"""
px_width = np.int(width*0.5/res)
xbox = [center[0]-px_width, center[0]+px_width]
ybox = [center[1]-px_width, center[1]+px_width]
return(xbox, ybox)
# window
window_hu = [150, 1000] # HU
window_em = [0.022, 0.032] # mm^1
# itools
catphan_dcm = dicom.read_file(('itools/150902_catphan_full_low_contrast/hu/IMG_0075.dcm'))
catphan_hu = catphan_dcm.pixel_array*catphan_dcm.RescaleSlope+catphan_dcm.RescaleIntercept
# crop
width = 130. # mm each way
res_hu = 0.473 # mm/px
center_hu = [np.int(catphan_hu.shape[0]*0.5), np.int(catphan_hu.shape[1]*0.5)]
winx, winy = crop(center_hu, width, res_hu)
catphan_hu = catphan_hu[winx[0]:center_hu[0], winy[0]:winy[1]]
plt.imsave('figures/catphan_itools_sr.png', catphan_hu, vmin=window_hu[0], vmax=window_hu[1])
# recon
dir = "/home/amdavis/research/truebeam/150902_virtiso_cbct_catphan_ed/itools/img/"
files = sorted(glob.glob(dir+"*catphan*itools*{0:04d}*.raw".format(iter)))
catphan_circular = np.fromfile(files[0], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
catphan_virtiso = np.fromfile(files[1], 'f').reshape(125, 528, 528).swapaxes(1, 2).T
sr_slice = 93
catphan_circular = catphan_circular[..., sr_slice]
catphan_virtiso = catphan_virtiso[..., sr_slice]
# crop
res_em = 0.5 # mm/px
center_circular = [np.int(catphan_circular.shape[0]*0.5), np.int(catphan_circular.shape[1]*0.5)]
winx, winy = crop(center_circular, width, res_em)
catphan_circular = catphan_circular[winx[0]:center_circular[0], winy[0]:winy[1]]
catphan_virtiso = catphan_virtiso[winx[0]:center_circular[0], winy[0]:winy[1]]
plt.imsave('figures/catphan_circular_sr_mlem_iter_{0:03d}.png'.format(iter), catphan_circular, vmin=window_em[0], vmax=window_em[1])
plt.imsave('figures/catphan_virtiso_sr_mlem_iter_{0:03d}.png'.format(iter), catphan_virtiso, vmin=window_em[0], vmax=window_em[1])
# return('figures/catphan_itools_sr.png')
# return('figures/catphan_circular_sr_mlem.png')
# return('figures/catphan_virtiso_sr_mlem.png')
fig = plt.figure()
ax1 = fig.add_subplot(131)
ax2 = fig.add_subplot(132)
ax3 = fig.add_subplot(133)
ax1.imshow(catphan_hu, vmin=window_hu[0], vmax=window_hu[1])
ax1.set_title('Circular iTools FDK')
ax1.axis('off')
ax2.imshow(catphan_circular, vmin=window_em[0], vmax=window_em[1])
ax2.set_title('Circular MLEM')
ax2.axis('off')
ax3.imshow(catphan_virtiso, vmin=window_em[0], vmax=window_em[1])
ax3.set_title('Virtual Isocenter MLEM')
ax3.axis('off')
fig.savefig('figures/catphan_comparison_sr_iter_{0:03d}.png'.format(iter), bbox_inches="tight")
return('figures/catphan_comparison_sr_iter_{0:03d}.png'.format(iter))
As mentioned earlier, the virtual isocenter trajectory is designed to increase the distance between the gantry head and the patient by synchronized table translation and gantry motion. The geometry of the imaging arms remains unchanged, though the center of the image space is moved from the mechanical isocenter to the virtual isocenter. As shown in the previous section, this results in CBCT image quality that is comparable to a normal circular scan.
It is certainly possible to envision situations in which collisions with the kV imaging source or (more likely) the detector also occur. The system matrix in our optimization-based reconstruction method can incorporate changes in the position of the imaging arms as well as the patient position. Thus a trajectory with variable source-detector distance, caused by the detector moving to avoid a patient collision, can also be reconstructed. Such a variable magnification scan may be performed with the patient couch moving in a virtual isocenter trajectory, or with the patient couch fixed and rotation about the physical machine isocenter.
All virtual isocenter scanning in the present work was done in TrueBeam Developer Mode, which is a strictly nonclinical mode of operation. Simultaneous motion of couch, gantry and imaging arms is not fully supported by the TrueBeam; however, the motions required for the virtual isocenter scan, which involve only coupled gantry rotation and couch translation, are feasible in Developer Mode. Although the LINAC can clearly execute the required motions and acquire the images, virtual isocenter scanning is as yet not available as a clinical capability. In addition, these scans must be reconstructed using iterative optimization-based methods, rather than the current clinically available filtered back-projection method. A historic concern about optimization-based methods has been reconstruction speed. These are very computationally intensive programs, but with the recent availability of GPU-based processing, reconstruction times are more manageable.
Virtual isocenter trajectories and dynamic magnification are potentially useful as collision-avoiding alternatives to standard isocentric rotation, both in cases where arc treatments are being delivered and in cases where CBCT scanning is desirable, but a normal isocentric scan could cause gantry-patient collisions. Using optimization-based reconstruction methods, patient-specific, collision avoiding imaging trajectories that utilize virtual isocenter CBCT scans and different kV-detector magnifications can be reconstructed by incorporating the view-by-view imaging and patient geometry into the system matrix. Image quality, as characterized by spatial resolution and low contrast object detectability, is comparable for virtual isocenter scans and for standard isocentric scans using different magnification bumps to avoid kV-detector collisions. Thus, virtual isocenter CBCT scans and kV-detector magnification changes could be combined with optimization-based reconstruction as a useful clinical approach in collision-plagued situations. In this dissertation work, we developed a framework that leverages optimization-based reconstruction methods to enable generalized, non-circular, scanning trajectories in CBCT. Where analytic-based reconstruction methods require a prescribed trajectory in order to derive an analytic inverse, optimization-based methods require no such assumption regarding the trajectory by which the projection views were acquired. This agnosticism of optimization-methods to the order and geometry by which projections are acquired allows for dynamic, view-by-view modifications to the scanning trajectory that would be untenable if the scanning trajectory were fixed by necessity of the reconstruction program’s requirements.We describe our generalized framework for reconstructing from generalized scanning trajectories using optimization-based reconstruction methods in /Chapter 2/. Though we limited ourselves to MLEM to solve our reconstruction problem, we by no means imply that this framework is dependent on this algorithm. Not only can other optimization-based algorithms be plugged into this framework, more sophisticated optimization techniques may be better suited to this optimization problem. However, regardless of the optimization-based algorithm employed in this paradigm, it is always critical that the user ensure that the parameters of the algorithm are carefully selected for the desired task.
In addition to developing this framework, we also developed a general calibration procedure in /Chapter 3/ that would accommodate view-by-view corrections to the different non-circular trajectories we studied. We then proceeded to use this calibration protocol to make calibration corrections for the non-circular trajectory classes we studied in the subsequent chapters. Though optimization-based algorithms are very robust, the view-by-view agnosticism we utilize for our trajectory framework is only beneficial if the geometry describing each of the views in the system matrix is correct. As with all inverse problems, the answer provided by the inverse is only useful if you have solved the right problem.
As our investigation into different non-circular trajectories was motivated by existing clinical limitations currently encountered in using LINAC-mounted CBCT for IGRT, we then found two clinical examples where the ability to use non-circular trajectories could provide a solution. By first selecting two particular tasks to test our framework, we were able to constrain our investigation of the trajectory parameters that were relevant to solving these problems. Just as the parameters of the optimization program must be selected for the desired task, similar task-based considerations must be given to selecting what type of non-circular trajectory, if any, would be suited for intended application.
The first clinical limitation we studied in /Chapter 4/ is the problem of the limited axial FOV provided by LINAC-mounted CBCT systems. Since the axial coverage provided by the CBCT detector is limited by engineering constraints, we hypothesized that non-circular trajectories would be able to extend the axial FOV while maintaining image quality that is comparable to current clinical methods. In addition to studying the double circle trajectory that is comparable to the current clinical practice of stacking the independently reconstructed FDK volumes of two circles together, we also studied the circle-line-circle and smooth trajectory classes that include acquiring projections as the CBCT imaging system translates between the two planes of the circular components of the scan. We found that not only does our framework with MLEM perform as well as FDK in the portion of the reconstructed volume where FDK satisfies Tuy’s condition, but that it is also able to improve the image quality in the region between the planes of the source’s circular orbit. Additionally, our framework demonstrated that it is possible to extend the axial spacing between the planes of the two circles into distances that exceed the support of two stacked FDK volumes.
The second clinical problem in using LINAC-mounted CBCT for IGRT that we investigated in /Chapter 5/ is that of potential patient collisions with the LINAC when attempting to acquire a CBCT. For some of the treatment positions used for breast, lung, and head and neck cancer patients, the orientation of the patient on the treatment table puts them at risk of being hit by the LINAC as it rotates around the patient. In this study, we investigated different non-circular trajectories that could alleviate the different types of collisions potentially faced by these patients. One such trajectory would increase the radius of the kV-imaging detector in a potential collision zone to create a dynamic magnification trajectory. To avoid potential collisions with the LINAC treatment head, we studied a virtual isocenter trajectory that moved the patient couch while the gantry rotated to maintain a safe distance between the patient and treatment head to ensure no collision would take place. Finally, we created a trajectory that combined the virtual isocenter trajectory with the dynamic magnification in order to create a trajectory that could alleviate collisions with both the kV-detector and the treatment head. For each of these different configurations, we showed we could obtain reconstructions with image quality comparable to the standard clinical circular trajectory.
In conclusion, we found that our non-circular scanning trajectory framework using optimization-based reconstruction methods have the potential to address limitation of LINAC-mounted CBCT for IGRT. We selected specific non-circular trajectories that we knew had the potential to address real clinical limitations. In finding that the subsequent image quality was no worse than that achieved by the existing clinical methods, we demonstrated that these non-circular scanning trajectories have the potential to address limitations of CBCT in IGRT. Though we did not explicitly study any other application here, we do not think this framework is limited to these two applications or even solely to the purview of radiotherapy. In selecting the two examples shown here, we have only wished to demonstrate the potential utility enabled by non-circular trajectories with optimization-based reconstruction. As with all tomographic reconstruction problems, careful consideration must be given to the desired task and further task-based metric analysis must be performed to evaluate not only the performance of the chosen reconstruction chain, but also to that of the chosen scanning trajectory.
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