1.

INTRODUCTION

Cone-beam computed tomography (CBCT) is a common solution to provide in-room imaging in various forms of radiotherapy. Several commercial solutions are available, often as an imager mounted on the gantry of a linear accelerator. At the same time, custom solutions exist as well, in particular for hadrontherapy.¹ The National Center for Oncological Hadrontherapy (CNAO, Pavia, Italy) employs such a scanner: a custom CBCT scanner mounted on a Kawasaki robotic manipulator.² It provides high geometric reproducibility and is employed for both individual X-ray radiographs and CBCT acquisitions inside the treatment room before the delivery of the treatment fraction. The device was primarily intended for correcting the patient setup based on registration of the bony anatomy and cannot produce a sufficiently wide field-of-view (FOV) to avoid data truncation when targeting wider locations, e.g. the abdomen or the thorax. In this work, the FOV is defined as the region in the source trajectory plane for which all ray lines are acquired (at any position and direction). Adaptive radiotherapy using the acquired CBCT images is being investigated, but it is of clinical interest to accurately reconstruct complete axial slices in wide locations for qualitative and dosimetric assessments of interfractional variations. Like most CBCT scanners with a fixed flat panel detector (FPD), the conventional geometry of the system is such that for all source positions, the midline, defined by the point source and the center of the detector, is orthogonal to the rotation axis and intersects it at the center of the source trajectory.³ Axial FOV enlargement is typically achieved through a technique called half fan, suitable for filtered backprojection (FBP) reconstruction.⁴ Mechanically, half fan requires a detector capable of transverse displacements and an adjustable X-ray source collimation. Both features are not available in the aforementioned CNAO CBCT scanner. A feasible alternative is to perform a complete displaced center of rotation (DCoR) scan with a fixed tilt of the detector. This consists of a 360° scan where the midline is offset w.r.t. the axis of rotation. This offset can be realized by the CNAO CBCT scanner while the relative positions of the source and detector are unchanged and the same collimation can be used. Previous works dealt with DCoR and aimed at correcting it as a misalignment.⁵ Instead, we would like to use this geometry to enlarge the reconstructed FOV, as in Ref. 3. Such a trajectory is compatible with robotic manipulators as the one of the CNAO CBCT scanner and is easily programmable in the robot controller. However, bulky hardware in the treatment room limits the sampling arc of the C-arm to 220°. The objective of this work is to exploit a couple of short scans⁶ with complementary offset centers of rotation to obtain the same FOV as a single full 360° DCoR scan would produce.

Figure 1.

Short time-lapse depicting conventional motion of custom in-room CBCT imaging system at CNAO. On the bottom left, one can observe the base of the robotic couch, which prevents the full rotation of the C-arm.

2.

MATERIALS AND METHODS

2.1

Circular geometry with displaced center of rotation

We examine the scanner geometry in the central plane z = 0 containing the trajectory of the source S. The source and detector are attached to a C-arm which rotates around the isocenter I. We note SID and SDD the source-to-isocenter and source-to-detector distances which are fixed by construction. In the conventional operation, the source and detector rotate on a circle centered on I which is static. We note β the gantry angle between the y-axis and the midline (defined by the source S and the isocenter I).

The DCoR geometry makes use of the possibility given by the robotic manipulator to modify the position of I during the system rotation. With respect to the conventional operation, both the source and the FPD are offset from the center O in a direction orthogonal to the midline (and therefore parallel to the detector) as shown in Figure 2. We note τ the angle between the source-center line and the midline which is also the tilt angle between a conventional detector (orthogonal to the source-center line) and the DCoR detector. Like in Ref. 5 and unlike Ref. 3, τ is constant w.r.t. the gantry angle β. The angle τ is positive when I is on the positive side of the x-axis at gantry angle β = 0 as in Figure 2. The offset of the isocenter w.r.t. to the conventional geometry, i.e. the isocenter-to-center distance, is R_I = SID|tan τ|.

Figure 2.

Circular geometry defined as displaced center-of-rotation (DCoR) in the central plane. The FPD and source S are bound together while S travels the circle with radius R_S. The midline is offset from the center O of the source trajectory.

The source trajectory is still a circle, centered on O and defined by S(τ, β) = R_S(−sin(β − τ), cos(β − τ), 0), where is the radius of the circle. The isocenter trajectory is also a circle of center O such that I(β) = R_I(cos β, sin β, 0). The conventional geometry is when τ = 0, i.e. when R_I = 0 and R_S = SID. We note α_τ = α + |τ| the large fan angle captured by the detector in the central plane with α = arctan(N_uΔ_u/2 SDD) the fan angle captured by the detector w.r.t. the midline (Figure 2), N_u the number of pixels of the detector in the transaxial direction and Δ_u their spacing. If |τ| < α, the source-to-center line hits the detector and the DCoR strategy could achieve exact reconstruction in the central slice of the FOV with radius R_Fov = R_S sin α_τ. This is normally done with a full 360° scan but we propose to combine it with short scans.

2.2

Short scan

Let g_τ be the set of cone-beam projections acquired for a given DCoR geometry with an angle τ defined by the line integral

where (τ, β, u, v) is the line defined at gantry angle β and tilt angle τ by the source position S(τ, β) and the position of the detector pixel with coordinates (u, v).

If the projections were untruncated with a full fan α_u ∈ [−α_τ, α_τ] with α_u = τ + arctan(u/SDD), a short scan of π + 2α_τ would be sufficient to reconstruct from divergent rays in the central plane using Parker pre-reconstruction weights⁶

In Ref. 3, the authors provide an adaptation of the FDK algorithm⁷ to a full scan DCoR geometry and we note its application to a short scan:

This short scan DCoR reconstruction is only exact in the central slice and approximate elsewhere due to the cone-beam artifact stemming from incomplete data with a circular source trajectory.

2.3

Complementary short scans

We note that for two scans with opposite offsets τ₁ and τ₂ = −τ₁, the source rotates on the same circle since S(τ₁, β) = S(τ₂, β − 2τ₁). We therefore acquire the corresponding cone-beam projections g₁ and g₂ for the same arc of source trajectory using the two sets of gantry angle B₁ = [τ₁, π + 2α_τ1 + τ₁] and B₂ = [τ₂, π + 2α_τ2 + τ₂] which measure the same line integrals in the central plane

We refer to this composition of trajectories as complementary displaced centers-of-rotation (C-DCoR). The detectors are however in different positions with a small overlap (Figure 3). We follow the approach of Refs. 4 and 6 to manage redundancies with the weights

Figure 3.

Illustration of two projections of two complementary scans with τ₁ and τ₂ = −τ₁, with gantry angle β₂ = β₁ − 2τ₁ such that the source is at the same position S(τ₁, β₁, 0) = S(τ₂, β₂, 0) in the two arcs.

which are derived from the displaced detector weights of Ref. 8. We obtain the reconstruction formula

It can be shown that the reconstruction is exact in the central plane, as in (3). The result of (3) is however different in other planes but previous investigations have shown that the effect of the tilt is limited compared to the cone-beam artifacts in these planes.³

3.

EXPERIMENTS

The proposed C-DCoR trajectories have been tested on simulated projections of a Forbild thorax phantom using RTK.⁹ Geometrical parameters of the simulations mimick the CNAO scanner with SID = 1100 mm and SDD = 1600 mm. The detector has isometric spacing Δ_u = Δ_v = 0.388 mm and size N_u × N_v = 768 × 1024 pixels; we therefore have α = 5.32°. We set τ₁ = −τ₂ = 4.159° which results in R_I = 80 mm and R_S = 1102.91 mm. The same arc of the circle trajectory is covered with B₁ ≃ [−102°, 110°] and B₂ ≃ [−110°, 102°], which is larger than 180° + 2α_τ ≃ 199° and accounted for in the redundancy weights w_s (by replacing α_τ by half the scan range minus π in Ref. 2).

The reconstructed DCoR scans have a FOV diameter of 363 mm instead of 204 mm for the conventional geometry. The Forbild phantom is isometrically rescaled 0.8 times such that the thorax ellipse semi-minor and semi-major axes are 80 mm and 160 mm and fit in the DCoR FOV. The phantom is centered on O and 400 projections are generated along each complementary scan. The reconstructed scans have size 400 × 400 × 400 pixels with a resolution of 1 × 1 × 1 mm.

4.

RESULTS

The weight maps w_Pw₁ and w_Pw₂ and the partial reconstructions _τ1{w_Pw₁g₁} and _τ2{w_Pw₂g₂) are displayed in the central slice along with the corresponding weight maps in Figure 4. The resulting phantom reconstruction is qualitatively compared to the reference phantom in Figure 5. The images are masked to the FOV of the C-DCoR scan. In the latter, artifacts caused by strong attenuation in high density vertebrae along the z axis are visible in the sagittal plane. Intensity profiles are extracted from the middle of each slice along the colored lines in Figure 5 in the same figure. The difference between the reference image and the reconstructed image resulted in a mean absolute error (MAE) of 20.5(±24.5) HU inside the FOV. Slightly worse results are obtained when simulating and reconstructing a complete 400 projections half fan acquisition with a 130 mm detector offset with a MAE of 27.6(±25.5) HU (image not shown). Finally, we find equivalent results when simulating and reconstructing a conventional geometry acquisition with a large panel (1536 × 1024 pixels) on the same source trajectory with a MAE of 20.6(±24.0) HU (image not shown).

Figure 4.

Top row shows the weight maps w_pw₁ and w_pw₂ (grayscale range [0, 1]) for the two fan-beam sinograms of the central slice. Vertical dotted red lines indicate the limit for the offset detector weights w₁ and w₂ (Equation 5). Slanted blue dotted lines indicate the limits for Parker short scan weights (Equation 2). Bottom row shows partial reconstructions _τ1(w_pw₁g₁) and _τ2(w_pw₂g₂), the sum of which gives the axial slice shown at the left of the second row in Figure 5. The grayscale range is [−740, 920] HU.

Figure 5.

Top row shows central slices (axial, coronal, sagittal) extracted from the Forbild reference volume, with grayscale in Hounsfield Units. Central row shows the same slices from a C-DCoR image with an offset R_I = 80 mm. Grayscale level and window are set to 0 and 2000 HU, where -740 HU is the value in the lung and 920 HU is the vertebrae density. Note that axial slices are cropped in the antero-posterior direction for optimal display. Colored lines indicate where the intensity profiles (bottom row) have been evaluated.

5.

DISCUSSION AND CONCLUSIONS

We successfully developed a reconstruction algorithm to extend the FOV of the CNAO scanner. The C-DCoR acquisition can nearly double the FOV diameter. The method requires two short scans with a short movement of the isocenter and detector between the scans to ensure that the same source trajectory arc is imaged twice with different detector positions. With these arcs, we are able to produce CBCT images of similar quality as conventional short scan or displaced detector CBCT images. Most discrepancies in the reconstructed C-DCoR are caused by the phantom vertebrae and seem to be linked to the cone angle of incident X-rays. Physical effects such as scatter or geometrical fluctuations remain to be assessed on real data.