2.2.1

BB Extraction

The BB locations were identified in each projection by performing a 2D correlation between the line integral images and a disc-shaped kernel of roughly the same size as the BBs. The highest correlations represent the approximate location of the centroids of the BBs. This location is further refined by computing the centroid of pixel values in a square region-of-interest (ROI) about the initial location estimate. Predicted BB locations based on linear interpolation across adjacent frames were used to identify individual BBs and maintain continuity. For scans where the BBs had very low contrast against background features such as metal and thick bones, the extraction was initialized by manually selecting the approximate centroid locations on the first image.

2.2.2

Geometry Optimization

With accurately calibrated geometry, backprojected rays of the centroid of a BB should intersect at an infinitesimal point in space. With an inaccurate geometry, these rays may not intersect. Thus, we can potentially estimate the true geometry by minimizing the mean Euclidean distance from the BB centers to corresponding backprojected rays, or, the reprojection error (RPE):

where n is the projection index; k is the BB index; Ω_n are parameters of the view-dependent system geometry; p contains all 3D BB center locations; L_k (Ω_n) is the line equation of the ray backprojected from the k-th BB centroid on the n-th projection view using geometry Ω_n; d [p, L] is the Euclidean distance from point p to line L.

We presume that the BB center locations are unknown (e.g. no prior scan or calibration) and also need to be estimated. We start with an approximate geometry (e.g. based on rough encoder positions or commanded location) as an initial guess. In this case, backprojected lines will likely not intersect at a point but instead a larger region that is generally close to the true BB location in 3D. For each pair of backprojected rays, we calculate the nearest point between these rays - the midpoint of the line segment orthogonal to and connecting both rays. We form a point cloud by computing the nearest points between all ray pairs, and the mean location of the point cloud is used as the estimated BB center: where is the mean of all nearest points formed from N rays backprojected using geometry Ω. We iteratively update and Ω by minimizing the RPE to jointly estimate BB point cloud centers and the system geometry:

where i is the iteration number. Note that the solutions of this objective will have fixed points where the point clouds have shrunk to infinitesimal size. It is possible that such a scenario is not the true geometry; however, such a geometry should provide accurate representations at those points. In this work, the optimization is performed using the MATLAB (MathWorks Inc.) function fmincon.

2.4.1

Phantom Design

The test phantom (Figure 4A) consisted of a 3D-printed cervical spine placed in a plastic cylinder filled with plastic spheres of variable sizes for background clutter. Eight steel BBs of 3.17 mm diameter were affixed to the side of the container. BB placement followed a roughly spiral fashion to reduce the chance of overlapping BBs in projections; however, precise locations were unknown a priori.

2.4.2

Bench Experiment Implementation

The test bench includes an X-ray tube (Varex Rad-94), a flat-panel detector (Varex PaxScan 4343CB), a 6 degree-of-freedom (DoF) hexapod (PI H-900K Series) and rotation stage (PI PRS-200) (Figure 2), which enables precise emulation of arbitrary source-detector trajectories. In previous work,⁷ inaccuracies in positioning of a Siemens Artis Zeego were identified. We found systematic inaccuracies in the system encoded positions in LAO/RAO angle, CRAN/CAUD angle, isocenter, and a slight in-plane panel rotation around its normal axis, but also found that the C-arm was sufficiently rigid that we could assume a fixed source-detector distance and piercing point.^7,9 In our bench studies, we emulated a C-arm system whose source remained fixed relative to the detector but had similar isocenter shifts and angular inaccuracies. These inaccuracies were added to an ideal sawtooth orbit (Figure 1A) and were realized by the hexapod using 3 DoF linear motions and 3 DoF rotations in roll, pitch, and yaw. Figure 3 illustrates the error in each DoF added to each frame. The error amplitudes in yaw (rotation) and pitch (tilt) were equal to the angular step size in each axis. A sinusoidal error pattern that was previously observed during data acquisitions (likely caused by gravity-induced sagging) was also added. The spatial shifts were generated with a sine wave of amplitude 8 mm plus random noise of ±2 mm for an exaggerated level of uncertainty as compared with the clinical system. A normal circular scan was also acquired and each scan had 500 frames.

Figure 2.

Photo of bench setup and diagram of the 6 DoF motion space of the hexapod stage.

Figure 3.

Motion errors added to the sawtooth orbit in Fig. 1 for the test bench experiment. Top plot: angular errors added in roll, pitch, and yaw. Bottom plot: spatial shift of iso-center added in x, y, and z axes.

2.5.1

Phantom Design

To test and quantify the performance of the new calibration method on the Zeego C-arm, we used a cylindrical calibration phantom as proposed in,¹³ which contained 16 steel BBs, with the addition of a thin tungsten wire of 0.13 mm diameter suspended in the middle to probe in-plane image fidelity (Figure 4B).

Figure 4.

Phantoms used in the three experiments. (A) C-spine phantom, showing the clear container, the plastic spheres inside, and steel BBs taped on the outside; the 3D-printed cervical spine is not visible due to the spheres. (B) Diagram of the cylindrical phantom adding a tungsten wire to a Cho calibration phantom with BBs. (C) Torso phantom with Sawbones spine, pedicle screws (green ovals), and steel BBs affixed with black tape.

2.5.2

Siemens Artis Zeego implementation

The DCArc orbit (Figure 1B) was performed manually with the bedside joystick controller, and projections were acquired in fluoroscopy mode. The orbit was acquired in three parts: two full tilted circles in LAO/RAO and one arc in CRAN/CAUD. For comparison, a standard circular scan was also acquired. The acquisition process required no modification to the system. The raw images were extracted with a dedicated software tool.

2.6.1

Phantom Design

To further test the proposed workflow, we imaged a torso phantom with Sawbones spine (Figure 4C). Four pedicle screws (Evolution Surgical, Sydney, Australia) were placed into three vertebrae (L2-4). Ten BBs of diameter 2 mm were taped to the surface of the torso phantom.

2.6.2

Siemens Artis Pheno implementation

We combined non-circular orbits with an upright weight-bearing setup — positioning that could provide more diagnostic information for spine diseases.^{1, 3, 14} Since the phantom was sat upright on the bed, the C-arm’s CRAN/CAUD tilt axis now functioned as the primary LAO/RAO gantry rotation. Due to motion range limitation, in order to achieve the 210^o arc in the multi-arc orbit (Figure 1C), we manually rotated the phantom approximately 90^o between two identical 120^o arc scans at a 10^o tilt. Another 120^o arc was acquired at a – 20^otilt, and a tilting arc between – 25^o and 30^o was acquired.