Novel C-arm based cone-beam CT using a source trajectory of two concentric arcs

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1 Novel C-arm based cone-beam CT using a source trajectory of two concentric arcs Joseph Zambelli a, Brian E. Nett a,shuaileng a, Cyril Riddell c, Barry Belanger d,guang-hong Chen a,b a Department of Medical Physics, University of Wisconsin-Madison, WI 53704; b Department of Radiology, University of Wisconsin-Madison, WI 53792; c GE Healthcare, 283 rue de la Minière, Buc, France; d GE Healthcare, 3000 North Grandview Boulevard, Waukesha, WI ABSTRACT The first results from an interventional C-arm based computed tomography system where a complete source trajectory was used are presented. A scan with two arcs which are joined approximately at the center of their paths (CC trajectory) is utilized here. This trajectory satisfies Tuy s sufficiency condition for a large volume, but is not well populated with PI-lines. Therefore, a non-pi-line based reconstruction method is required. The desire for high dose efficiency led to the selection of an equal weighting based method. An FBP type reconstruction algorithm which was derived for two orthogonal concentric circles was utilized for reconstruction. The concept of a virtual image object was used to relate the projections from the two acquired non-orthogonal arcs to projections of a virtual object from two orthogonal arcs. Geometrical calibration is vital when performing tomography from an interventional system, and was incorporated here with the use of a homogeneous virtual projection matrix. The results demonstrate a significant reduction in cone-beam artifacts when the complete source trajectory is utilized. Keywords: CT, SYS 1. INTRODUCTION Interventional C-arm systems provide image guidance for minimally invasive interventional procedures where high contrast objects such as bony structure and iodinated vasculature are often the primary imaging targets. Recently, C-arm based cone-beam computed tomography (CT) has been released as commercial product by major manufacturers. Several investigators in academic environments have also implemented C-arm based conebeam CT systems 1 6 using either a large area flat-panel detector or an x-ray image intensifier (XRII). The potential gains offered by improved low contrast resolution in an interventional setting have significant clinical applications. One known limitation of these acquisitions is the presence of cone beam artifacts due to the utilization of a single circular arc trajectory, which is an incomplete source trajectory. 7 Interventional C-arm systems offer the capability for reconstruction from arc-based source trajectories via multiple rotations of the source and detector (i.e. a complete trajectory). However, given the presence of the patient bed and the large C-arm, each system will have limitations on the achievable source trajectories. Thus, any arbitrary source trajectory consisting of multiple arcs which lie on the surface of a sphere is not necessarily achievable in practice. A scan with two arcs which are joined approximately at the center of their paths (CC trajectory) is utilized here (Figure 1(c)). Several exact shift-invariant image reconstruction algorithms have been derived for circle and arc based trajectories Each of these reconstruction algorithms is based on the existence of generalized PI-lines. However, as noted by Pack and Noo, 12 there are many trajectories which satisfy Tuy s sufficiency condition but which may not be fully populated with PI-lines; the CC trajectory is of this type. Therefore, the PI-line based reconstruction algorithms that have been derived for circle and arc based trajectories may not be used here, and a new algorithm is required. Additionally, in clinical practice the delivered x-ray dose must be made as low as reasonably possible. In this work an equal weighting algorithm is utilized given its high dose utilization. 13, 14 In the first implementation of this two arc trajectory we will use an approximate algorithm. The All scientific correspondence should be addressed to G.-H. Chen via gchen7@wisc.edu

2 algorithm used here was derived for the case of two complete concentric circles, 14 but in this case data is present for only 210 of each circle. A more complex exact algorithm for the case of the two arc source trajectory has recently been developed, 15 and will be used in future research on the clinical system. A unique feature of this work is that the derived algorithms actually reconstruct a virtual object which is related to the real object by an affine transform. 14, 16 A critical component of any C-arm based CT implementation is the geometrical calibration. In this work the projection matrix formalism is utilized, where the trajectory is assumed to be sufficiently close to the ideal trajectory so that the calibration is taken into account solely in the backprojection step. To accomplish the geometrically corrected backprojection, we introduce the concept of the virtual projection matrix, which relates points in the virtual object to their conic projection onto the virtual detector. The initial results demonstrate the utility of the new calibration method, as well as a significant reduction in cone-beam artifacts in reconstructed images, as demonstrated through a scan of a Defrise type phantom (e.g. a stack of acrylic plates) on the GE Healthcare (Waukesha, WI) Innova 4100 system. (a) y (b) z x (c) Figure 1. The positioning arm parked at 15 (a), where the gantry spins along the path denoted by the solid arc. The positioning arm parked at 15 (b), where the gantry spins along the path denoted by the dashed arc. The complete source trajectory generated by these two motions of the C-arm gantry (c). 2. METHODS 2.1. Reconstruction The image reconstruction is carried out here using a shift-invariant filtered backprojection cone-beam algorithm for the source trajectory of two concentric circles using an equal weighting scheme. 14 The algorithm was derived

3 using Katsevich s general framework. 17, 18 The source trajectory consisting of two concentric and non-orthogonal circles may be parameterized as: y µ h (t) =R(cos t, sin t, 0), t [0, 2π) (2.1) y v µ (t) =R( sin µ cos t, sin t, cos µ cos t), t [2π, 4π), (2.2) where R is the radius of the trajectory, t is the view angle parameter, µ is the tip angle from vertical of the tilted source plane, h indicates the horizontal arc of the source trajectory lying in the xy plane, and v indicates the tilted arc of the source trajectory. The algorithm was derived for the case in which the source trajectory planes are orthogonal (i.e. µ =0). For reconstruction from non-orthogonal source planes an affine transform is used to relate the acquired trajectory to the trajectory of two orthogonal arcs. y s µ (t) =U y s (t), s = h, v (2.3) U = 1 0 sin µ cosµ,u 1 = 1 0 tanµ cos µ. (2.4) A virtual object may be reconstructed from the orthogonal trajectory using data collected from the nonorthogonal trajectory. The projection data must first be rebinned onto a corresponding virtual detector. 14 In order to perform the geometrical calibration the homogeneous projection matrices which map a point in the virtual object to a point on the virtual detector are utilized, as discussed below. Horizontal Arc Acquire Projection Data Rebin to the virtual detector Differentiation Four Hilbert Filtered Sets Q 1 h Q 2 h Q 3 h Q 4 h XY plane backprojection from Filtered Virtual Detector w/ Virtual P Matrix Reconstructed Virtual Object Affine Transform Reconstructed Real Object (Horz Contribution) Tilted Arc Acquire Projection Data Rebin to the virtual detector Differentiation Four Hilbert Filtered Sets Q 1 v Q 2 v Q 3 v Q 4 v YZ plane backprojection from Filtered Virtual Detector w/ Virtual P Matrix Reconstructed Virtual Object Affine Transform Reconstructed Real Object (Tilt Contribution) Reconstructed Real Object Figure 2. A schematic chart demonstrating the data flow for the reconstruction algorithm used in this work. The steps used to reconstruct an image voxel are given in Figure 2. An assumption is made that the trajectory the C-arm traces out is close enough to the ideal geometry that geometric corrections in the reconstruction are only made during the backprojection step. The rebinning step to the virtual detector is performed using the relations below 14 u u v sin t sin µ = D D + v cos t sin µ, v cos µ v = D D + v cos t sin µ, (2.5) where (u,v ) are the detector coordinates in the original detector and (u, v) are the coordinates in the virtual detector. Additionally a weighting of 1/λ(u, v, t) is applied in transforming to the virtual detector where λ(u, v, t) = 1 2v sin µ(u sin t D cos t) D 2 + u 2 + v 2. (2.6)

4 Although the horizontal arc of the source trajectory is unaltered when transforming from the real to the virtual trajectory, a rebinning operation is still required for the data acquired from the horizontal trajectory. This rebinning ensures that the projections correspond to those of the virtual object rather than the real object. 14 A differentiation operation is first performed on the data in the virtual detector. Next, three different Hilbert filtering operations are performed, linear combinations of which yield four sets of filtered data Q s 1,Q s 2,Q s 3,andQ s 4. After the data has been filtered the backprojection operation is carried out using the virtual projection matrices. If the backprojection operation is carried out over a volume larger than the object support, the virtual object may be reconstructed, and an affine transform will yield the real image object. This path is given for illustration purposes in Figure 2. However, in our implementation the virtual object is never actually reconstructed. Rather, in order to reconstruct a point in the real object, the coordinates of the desired point in the real object are transformed to the corresponding point in the virtual object. The backprojection operation is performed with this given virtual point from the virtual detector, and the result is assigned directly to the desired point in the real image object. In this manner no additional rebinning operation is introduced in image space, leaving only a single rebinning in the detector plane. The final reconstruction is achieved by simply summing the contributions from the horizontal and vertical circles System Calibration Since the C-arm undergoes slight mechanical deformations during rotation, to achieve maximal image quality the system geometry must be measured over a sample acquisition and incorporated into the reconstruction algorithm. To accomplish this, conic projection matrices (P matrices) are determined by first projecting a set of markers of known geometric configuration onto the detector (Figure 4(a)). The method used here incorporates a helical calibration phantom and a nonlinear optimization procedure to calculate the P matrix elements based on knowledge of the marker positions within the calibration phantom and their projected location on the detector. 19 For an FDK-type reconstruction, 20 knowledge of the projected detector locations of the real image points is sufficient information for reconstruction. 21 However, the algorithm implemented here requires backprojection for a source trajectory of two orthogonal circles from a virtual detector into a virtual object. Thus, the real P matrices (P R ), defined for a source trajectory of two non-orthogonal arcs and the real object and detector, can no longer be used for backprojection. Instead, a virtual projection matrix (P V ), defining the projection of a point in the virtual object onto the the virtual detector is used. Estimation of P V proceeds as follows. The coordinates of the known marker positions in the real image space from the calibration phantom are transformed to coordinates in the virtual object by multiplication with U 1 from Eq The projected detector locations of the markers in the real object are extracted from the original projections, and then transformed to the appropriate locations on the virtual detector using the relations (D + v cos t sin µ)+(dv sin t sin µ) u = u, v = D D D cos µ v cos t sin µ. (2.7) The projection matrix P V is then determined via a completely analogous optimization procedure to that used for calculation of the P R matrix relating the real image volume and real detector in the single arc case, s u s v = P V s x ỹ z 1 v, (2.8) where u, v are the virtual detector coordinates, and x, ỹ, z are the virtual object coordinates. Note that only P V is used in the backprojection operation from the two arc trajectory, and only P R is used for the single arc trajectory (FDK). The source coordinates for the real acquisition trajectory may be calculated directly from P R, 21 while the virtual source trajectory is calculated from P V. Examples of measured source trajectories are given in the results section. Additionally, the source trajectory parameters which are assumed to be global (i.e. R, D, and µ) are estimated from the P matrices. The mean value of the parameters R and D was determined with a P matrix-based estimation routine for the nine parameters describing thec-armgeometry. 19 The tip angle µ is

5 estimated by fitting a plane to the source coordinates from each gantry spin and then determining π/2 minus the measured dihedral angle between the two planes. The angulation of the positioning arm yields projections of the helix in which two markers may superimpose over each other for some projections. This degeneracy in the projections complicates the calibration procedure over the widely-used case of a single arc with an axis of rotation collinear to that of the longitudinal axis of the calibration phantom; where one-to-one correspondence between the markers and their projections is ensured for all views. The calibration procedure in the case of the two-arc trajectory proceeds by first placing the calibration phantom at an orientation in which the markers positions satisfy the one-to-one correspondence criterion, and an initial P matrix is calculated. The C-arm is then slewed to the starting point of desired trajectory to be calibrated. Based on the initial P matrices, the P matrix of each subsequent projection is extrapolated from those prior, and used for assignment of the projected marker location to calibration phantom location. Assignment of markers from projections taken over the trajectory of interest proceeds in an identical manner. All markers projecting to locations too close in proximity to be accurately assigned are discarded from use in the optimization procedure Acquisition In order to advance the technology more rapidly, it is desirable that new trajectories implemented on clinical C- arm systems not require dramatic changes in system hardware. In this work the GE Healthcare Innova 4100 flat panel based interventional C-arm was utilized for data acquisition. This interventional C-arm system acquires cone-beam projection data from a detector constantly opposed to the x-ray source. For the single arc acquisition, the trajectory was generated by performing a gantry spin about the axis defined by the S-I axis of the patient table. The two arc trajectory is achieved via a gantry spin at two different orientations of the positioning arm (i.e. the arm which is mounted to either the ceiling or the floor), see Figure 1(a)-(b). This source trajectory is effectively two circular segments that are joined approximately in the center of each respective arc (Figure 1(c)). Parameter Value Source to isocenter distance (SID) 726 mm Source to detector distance (SDD) 1200 mm Cone angle 19 Detector readout pixel pitch 400 µm Binned pixel pitch 800 µm Readout array format Binned array format Frame Rate 30 fps Number of projections (each arc segment) 420 Gantry angular increment 0.5 Gantry angular velocity 15 /s Gantry angular range 209 Vertical plane tip angle (µ) 61 Table 1. Nominal values for system parameters used in this work. In the results presented here projection data over the central 37 cm 37 cm of the detector was used for reconstruction. During each spin acquisition 420 views of projection data were acquired. The tip angle of the tilted plane from vertical was µ =61. All acquisitions used x-ray source parameters of 60 kv, 100 ma, 5 ms pulse width, 0.6 mm focal spot size, and 2.5 mm Al filtration. Additional system parameters are listed in Table 1.

6 3. RESULTS 3.1. System Calibration A representative acquired projection of the calibration phantom is shown in Figure 3(a), while the same view after rebinning to the virtual detector is shown in Figure 3(b). The horizontal arc is shown as a solid line and the tilted arc is shown as a dashed line. v v u u (a) (b) Figure 3. Measured projection from the real source trajectory (a), and the corresponding rebinned virtual projection (b). Two arc source trajectory, X component Two arc source trajectory, Y component Two arc source trajectory, Z component X s (mm) 0 Y s (mm) Z s (mm) Projection Number Projection Number Projection Number (a) (b) (c) Figure 4. Measured components of the two arc acquisition trajectory as calculated from P R, X component (a), Y component (b), and Z component (c). The measured acquisition trajectory for the reconstruction algorithm proposed here, as calculated from P R, is shown in Figure 4(a)-(c). A comparison of the acquisition source trajectory and the virtual source trajectory used for backprojection is shown in Figure 5. As desired, the virtual source trajectory shown in Figure 5(b) is seen to be that of two orthogonal arcs. The calibration method presented has proved to be robust for the two arc trajectory, with discrepancies between projected marker location and reprojected marker location comparable to that achieved in the single arc case (e.g. mean reprojection error 0.2 native detector pixels in each case) Reconstruction A Defrise type phantom was scanned, as this phantom accentuates the presence of cone-beam artifacts in the final reconstructed images. This phantom was constructed from five acrylic disks separated by polyurethane foam spacers. The disks are 14 cm in diameter, 4.7 mm thick, and spaced at 2.9 cm intervals. The overall

7 z z µ=61 x x y y (a) (b) Figure 5. Comparison between the acquisition trajectory and virtual trajectory used for backprojection. Threedimensional representation of the acquisition trajectory as calculated from P R, (a). The trajectory for the virtual source as calculated from P V,(b). length of the phantom is 12.2 cm. For the present system geometry, this phantom subtends a cone angle of approximately 10. Images were reconstructed of the xz plane of the phantom. The detector matrices were downsampled onto a image matrix with an effective pixel linear dimension of 0.8 mm. The reconstruction was performed using an image matrix with voxels of a linear dimension of 0.43 mm. The standard of reference for comparison is the single arc source trajectory. For this trajectory, reconstruction was performed with the standard FDK 20 22, 23 algorithm using a Parker-type short scan weighting; this trajectory is known to yield severe cone-beam artifacts (Figure 6). In both the ramp and Hilbert filtering operations no additional frequency roll-off window was utilized. Due to the rebinning procedure and differences in filtering kernels the spatial resolution of these two algorithms is not yet matched. However, in our preliminary study the aim is to demonstrate cone-beam artifact reduction. In order to highlight the reduction in cone-beam artifacts between the single arc scan and the new trajectory it is not necessary that the spatial resolution of the two algorithms be matched. Figure 6. Comparison between single arc trajectory (FDK) implementations. Left: Ideal geometry assumed without calibration, Right: Calibrated using P R. To illustrate the necessity of geometric calibration, reconstructed images from a single arc trajectory without and with calibration are shown in Figure 6, and reconstructions without and with calibration for the two arc trajectory are shown in Figure 7. Note the increased importance of the calibration procedure for the two arc

8 Figure 7. Comparison between two arc trajectory implementations. Left: Ideal geometry assumed without calibration, Right: Calibrated using P V. trajectory, as contributions from both arc segments must sum together coherently. We next compare the FDKtype reconstruction with the reconstruction algorithm proposed here using a two arc source trajectory. In Figure 8 the reconstruction images are given in an identical window where the entire range of images values is included. Figure 9 uses an extremely compressed grayscale, but maintains identical window parameters between the two images. Significant reduction in the cone-beam artifacts may be appreciated in both Figure 8 and Figure 9. Figure 8. Full window comparison of the xz plane of a Defrise phantom. Left: single arc trajectory (FDK), Right: two arc trajectory. Figure 9. Narrow window comparison of the xz plane of a Defrise phantom. Left: single arc trajectory (FDK), Right: two arc trajectory.

9 4. CONCLUSION Interventional C-arm systems are now frequently equipped with flat-panel imagers and several manufacturers have released clinical computed tomography products on these systems using a single arc source trajectory. Cone-beam artifacts remain an inherent problem when performing volumetric reconstruction from a single arc trajectory on such systems. In order to mitigate these artifacts a source trajectory of two concentric arcs has been proposed. An analytic filtered backprojection approach was used to reconstruct images from this trajectory. Although the algorithm used here is approximate and will not provide exact quantitative reconstruction, the results demonstrate a significant reduction in cone-beam artifacts when compared with the state-of-the-art techniques. Calibration techniques based on a virtual object concept were employed. These techniques will be directly applicable to system calibration in conjunction with other reconstruction algorithms that utilize the virtual object concept to relate the acquisition trajectory to a trajectory for which a reconstruction algorithm exists. Initial reconstruction results demonstrate a significant reduction in cone-beam artifacts when reconstruction is performed based on the two arc source trajectory. REFERENCES 1. D. A. Jaffray and J. H. Siewerdsen, Cone-beam computed tomography with a flat-panel imager: initial performance characterization, Med. Phys. 27, pp , B. Chen and R. Ning, Cone-beam volume CT breast imaging: Feasibility study, Med. Phys. 29, pp , J. M. Boone, T. R. Nelson, K. K. Lindfors, and J. A. Seibert, Dedicated breast CT: radiation dose and image quality evaluation, Radiology 221, pp , J. H. Siewerdsen, D. J. Moseley, S. Burch, S. K. Bisland, A. Bogaards, B. C. Wilson, and D. A. Jaffray, Volume CT with a flat-panel detector on a mobile, isocentric C-arm: pre-clinical investigation in guidance of minimally invasive surgery, Med Phys 32, pp , Jan R. Fahrig, A. J. Fox, S. Lownie, and D. W. Holdsworth, Use of a C-arm system to generate true threedimensional computed rotational angiograms: preliminary in vitro and in vivo results, AJNR Am J Neuroradiol 18, pp , Sep G.-H. Chen, J. Zambelli, B. E. Nett, M. Supanich, C. Riddel, B. Belanger, and C. A. Mistetta, Design and development of C-arm based cone-beam CT for image-guided interventions: Initial Results, SPIE Proc. Med. Imag. 6142, H. K. Tuy, An inverse formula for cone-beam reconstruction, SIAM J. Appl. Math 43, pp , A. Katsevich, Image reconstruction for the circle-and-arc trajectory, Phys. Med. Biol. 50, pp , G. H. Chen, T. Zhuang, S. Leng, and B. E. Nett, A showcase of exact cone-beam image reconstruction algorithms for circle-based trajectories, Proc. 8th Int. Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp , Y. Zou, X. Pan, and E. Sidky, Theory and algorithms for image reconstruction on chords and within regions of interest, J. Opt. Soc. Am. A 22, pp , J. D. Pack, F. Noo, and R. Clackdoyle, Cone-beam reconstruction using backprojection of locally-filtered projections, IEEE Trans. Med. Imag. 24, pp , J. Pack and F. Noo, Cone-beam reconstruction outside R-lines using the backprojection of 1-D filtered data, Proc. 8th Int. Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, pp , D. L. Parker, V. Smith, and J. Stanley, Dose minimization in computed tomography overscanning, Med. Phys. 8, pp , T. Zhuang, B. E. Nett, S. Leng, and G. H. Chen, A cone-beam reconstruction algorithm for two concentric arcs using an equal weighting scheme, Phys.Med.Biol.51, pp , T. Zhuang, S. Leng, and G. H. Chen, Cone-beam reconstruction algorithm for two concentric arcs using an equal weighting scheme, Proc. 9th Int. Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (submitted), 2007.

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