A study on the section sensitivity profile in multi-row-detector spiral CT

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1 Journal of X-Ray Science and Technology (3) IOS Press A study on the section sensitivity profile in multi-row-detector spiral CT Ge Wang a, Mark Madsen a, Katherine Redford a, Shiying Zhao b and Michael W. Vannier a a CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Hawkins Drive, Iowa City, Iowa 54, USA b Department of Mathematics & Computer Science, University of Missouri St. Louis, 8 Natural Bridge Road, St. Louis, MO , USA Tel.: ; Fax: ; ge-wang@uiowa.edu Abstract. The section sensitivity profile (SSP) was well understood in the case of single-row-detector spiral CT. With the introduction of multi-row-detector spiral CT and the transition into cone-beam spiral CT, a revisit to the SSP issue becomes necessary. In this paper, the SSP of multi-row-detector spiral CT is formulated for the half-scan interpolation method at any transverse position. Based on the SSP formula, numerical simulation is performed to quantify the characteristics of the SSP with the number of detector rows up to 4. It is shown that the SSP varies as a function of the pitch and the number of detector rows. Given an appropriate selection of the pitch and the number of detector rows, the SSP does not change very much over the field of view in terms of the mean, the slice thickness, and the skewness of the SSP. Although in general applications the SSP at the gantry iso-center can be used as the representative of the SSP family, for more accurate analyses the spatial variation of the SSP must be taken into account. Keywords: Multi-row-detector spiral/helical CT, half-scan interpolation, section sensitivity profile (SSP), image quality, pitch effect, moment analysis. Introduction The newest technology in medical X-ray CT is multi-row-detector spiral/helical CT [ 3]. A multirow-detector scanner features several adjacent parallel rows of detectors, allowing simultaneous acquisition of data from multiple transverse slices. The tendency toward increasing the number of detector rows continues, and eventually cone-beam spiral CT, which uses a truly D detector array, will become a reality [4 7]. Mutli-row-detector/cone-beam spiral CT is advantageous for larger scanning range in shorter time with higher longitudinal image resolution, and has important medical and other applications. The image quality of multi-row-detector spiral CT has yet to be fully explored, especially when the number of detector rows is large [ 3]. A key aspect of image quality is the slice sensitivity profile (SSP) [8 ]. The SSP can be defined as the longitudinal profile of the point spread function (PSF). The slice thickness is derived from the SSP as the measure of longitudinal image resolution. The asymmetric geometry of spiral CT raw data interpolation, both in single- and multi-row-detector cases, makes the SSP dependent upon the position in the field of view. Although the SSP and its spatial variation was investigated in the single-row-detector case [8 ], the SSP issue has not been systematically addressed in the multi-row-detector case /3/$8. 3 IOS Press. All rights reserved

2 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT The purpose of this paper is to model the SSP of multi-row-detector spiral CT, compare with experimental results, and derive optimization guidelines. In the next section, the SSP formula is derived for multi-row-detector CT using the same approach as used in the single-row-detector case. Also, the slice thickness measure is discussed in an axiomatic framework for image resolution measurement, along with a brief review on moment analysis. In the third section, numerical simulation is described. Then, moment features of the SSP are presented. In the last section, relevant issues are discussed.. Methods.. SSP formulation In the case of single-row-detector spiral CT, the SSP was derived as a function of the transverse location in the field of view []. Using the same approach, the SSP of multi-row-detector spiral CT is similarly formulated in the following. Let us assume that N rows of detectors are contiguous along any row. Furthermore, the response of a detector to a point source is modeled as F (d) = D rect( d D ), () where d is the longitudinal relative distance between the detector center and the point source, D is the longitudinal detector collimation, and rect(.) is a rectangular function {,x [ rect(x) =, );, otherwise. In reconstruction of a transverse slice in incremental CT, if the longitudinal relative distance between a point source and the reconstruction plane is not greater than D, the image reconstructed from projection data of the point source should be the image of the same point source independent of its transverse position. Therefore, the SSP in incremental CT should be F (z), z is the longitudinal coordinate. Because of the spiral scanning pattern, an interpolation process is needed to synthesize planar projection data sets for reconstruction of transverse slices [ 3]. Due to this interpolation process, the SSP in multi-row-detector spiral CT is degraded compared to F (z). The extended half-scan interpolation method, which is also referred to as the extended 8 linear interpolation (LI) method [,], gives the thinnest slice thickness of all the commonly used raw data interpolation methods, therefore this half-scan method is chosen for this study. In the extended half-scan interpolation method [,], a set of planar projection data is obtained via linearly interpolating raw projection data associated with two nearest neighboring X-rays. This interpolation method requires that the two neighboring rays must be from different sides of a reconstruction plane. Unlike the case of single-row-detector spiral CT, in which two neighboring rays are always in opposite directions for the half-scan interpolation [,4], with multi-row-detector spiral CT the halfscan interpolation may be done based on two neighboring rays of the same orientation as long as they are from both sides of the reconstruction plane and the closest to it. As shown in Fig., the x-y plane denotes the reconstruction plane (the central plane of a slice to be reconstructed), θ the normal direction of a projection ray l in the x-y plane, z and z the longitudinal central coordinates of the X-ray fan-beams from N (N =4in Fig. ) rows of detectors when they are positioned to radiate direct and opposite rays respectively. Because of the small cone-beam angle in

3 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT 3 z t Direct rays y Virtual X-ray sources z z In-plane ray l Reconstruction plane θ x Real X-ray source Opposite rays Helical scanning locus Fig.. Reconstruction coordinate system and variables used for the extended half-scan interpolation in multi-row-detector spiral CT. multi-row-detector spiral CT [], fan-beams defined by detector rows are assumed to be in parallel. As a result, direct and opposite X-rays can be regarded as being sent from multiple virtual X-ray sources at corresponding longitudinal levels, instead of from a single real X-ray source on the scanning locus, also illustrated in Fig.. As shown in Fig., there is a point source at (r, φ, z), β and β are the gantry rotation angles associated with the direct and opposite rays corresponding to the point source with the normal direction θ. Furthermore, let T be the table increment per gantry rotation, P the pitch defined as P = T ND. () It is acknowledged that another definition of pitch P = T D is also used in the literature. Although which pitch definition to use is a matter of convention, we prefer using Eq. () because with that definition our results can be more easily summarized. Let us reconstruct the point source (r, φ, z) at the z =plane. As shown in Fig., the contribution to the reconstruction at (r, φ, ) due to the synthesized parallel-beam projection with a normal direction θ is proportional to the following: P θ (z)=w(θ)f (z z )+( w(θ))f (z z ), (3) where z and z are longitudinal coordinates of the direct and opposite rays, which are the closest to the in-plane ray l, through (r, φ, ) with the normal direction θ, and w(θ) = z (θ) z (θ) z (θ). (4) The relationship between z, z and z, z can be expressed as follows: { z =min{z (z,i,j) z (z,i,j) z,z }, z =max{z (z,i,j) z (z,i,j) z <,z < }, (5)

4 4 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT t y Reconstruction plane Direct rays ( r, ϕ, z ) X-ray source In-plane ray l θ r ϕ β Opposite rays x β Fig.. In the half-scan interpolation, a projection value associated with an in-plane ray is synthesized by linearly interpolating raw data from two nearest neighboring rays that are from different sides of the reconstruction plane. { z = z + (N )D +(i )D + jpnd, z = z + (N )D (6) +(i )D + jpnd, i =,,,N, j = M, M +,,,,,M, PND ND + M = PND = P +. (7) 4 Note that M is introduced to make sure that multiple scanning turns are taken into account when nearest neighboring rays are selected for the interpolation. The formulas for computing z and z in terms of θ can be written as { z = D π β (θ), z = D π β (8) (θ), where { β (θ)=θ ( π α(θ)) = θ + α(θ) π, β (θ)=(θ + α(θ) π ) π α(θ)=θ α(θ) 3π, (9) α(θ) =sin ( r cos(θ φ) R ), The total response due to the point source at (r, φ, z) is immediately obtained as follows: P (z)= π θ +π θ [ w(θ)f (z D π β (θ)) + ( w(θ))f (z D π β (θ)) ] dθ, ()

5 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT 5 y In-plane ray l t Reconstruction plane ( r, ϕ) r θ ϕ x X-ray source Fig. 3. Normal direction of the initial upper ray that passes through a point source can be expressed in terms of the transverse position of the point source. where θ, as shown in Fig. 3, is the normal direction of an initial reference ray, corresponding to the moment when the longitudinal coordinate of the center of the N detector rows becomes zero, ( ) θ = π r sin φ tan R r cos φ. () Equation () is the basis for analysis of the SSP over the field of view. To compute the SSP, a point source is placed on the z =plane, then multiple slices at different z levels are reconstructed to form the longitudinal response profile through the point source. Geometrically, reconstruction of a point source at (r, φ, ) on the z = z plane is equivalent to reconstruction of a point source at (r, φ, z ) on the z =plane in an appropriately rotated and longitudinally translated system x -y -z, φ = φ π z T. ().. Slice thickness and moment-based features Image spatial resolution characterizes the ability of an imaging system to separate two impulse-like signals. Image resolution measurement is a fundamental issue of imaging science and a practical requirement for assessment of a multi-row-detector spiral CT scanner. There are various resolution measures in the literature, such as the Rayleigh criterion, the cut-off-frequency of modulation transfer

6 6 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT SSP.5 SSP Z Z (a) (b) SSP.5 SSP Z Z (c) (d) Fig. 4. Representative section sensitivity profiles. (a) SSP at (, ), (b) (5, ), (c) (, 5), and (d) ( 5, ) in unit of mm. function, the full-width-at-half-maximum, the full-width-at-tenth-maximum, and the standard deviation of the PSF. An axiomatic approach was recently proposed for quantification of image resolution [5]. In the D case, the standard deviation of the PSF was uniquely identified as the measure of the choice. Therefore, in this work we use the standard deviation of the SSP to describe the slice thickness with multi-row-detector spiral CT. For convenience of readers, we summarize the axiomatic foundation of the standard deviation measure as follows [5]. It is assumed that a D imaging system can be modeled as o(z) =p(z) i(z), where p(z) denotes a PSF, i(z) and o(z) the input and output, respectively. Let R[p(z)] denote a resolution measure. The convention is that the smaller the value of R[p(z)], the finer detail the system (3)

7 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT SSP Mean.3.. St. Dev Y 5 3 X Y 5 3 X 3 (a) (b) Relative Variation.5 SSP Skewness Y 5 3 X Y 5 3 X 3 (c) (d) Fig. 5. Representative distributions of SSP features over a half of the field of view. (a) Mean, (b) standard deviation as measure of slice thickness, (c) relative variation in slice thickness, and (d) skewness. can resolve. The following axioms are formulated:. R[p(z)] is a continuous function of p in the distribution sense (continuity),. R[p(t c)] = R[p(z)] (translation), 3. R[p(ct)] = R[p(z)] c and R[cp(z)] = R[p(z)] (scaling), 4. R[p (z) p (z)] = F (R[p (z)],r[p (z)]) (combination), where c is any positive constant, and F a functional applying to all PSFs. According to the continuity axiom, image resolution is not sensitive to measurement noise. With the translation axiom, an image resolution measure solely depends on structures themselves in the field of view, instead of when or where these structures are imaged. By the scaling axiom, given a shape of a PSF the resolution measure is inversely proportional to the extent of the PSF, but invariant through any linear contrast transformation. The combination axiom requires that a resolution measure should extract all the information from a PSF for the purpose of image resolution measurement. Consequently, there must be a unique relationship between the resolution measure of a composite system consisting of sub-systems

8 8 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT. Average SSP Mean.5..5 Slice Thickness at Isocenter N.5 P N.5 P (a) (b) Average SSP Skewness.3.. Average SSP Skewness N.5 P N.5 P (c) (d) Fig. 6. Representative relationships with the number of detector rows and the pitch. (a) Average SSP mean, (b) slice thickness at the isocenter, (c) average of relative variation in slice thickness, and (d) average of skewness. in serial and resolution measures of these sub-systems. Based on the four axioms, it was proved that an image resolution measure must be proportional to the standard deviation of the PSF. Use of the standard deviation of the SSP is a special case of geometric moment analysis []. The SSP moment of order n can be expressed as: m n (r, φ) = z n P (z)dz, (4) where n =,,,. In our application, moments of order -3 carry most information on SSP features of interest. Specifically, m (r, φ) is the mean of an SSP, which is the local longitudinal position of a transverse slice. The standard deviation σ of an SSP can be directly obtained from m (r, φ) and m (r, φ), σ(r, φ)= m (r, φ) m (r, φ). (5) Finally, m (r, φ), m (r, φ) and m 3 (r, φ) can be combined to compute the skewness, which characterizes the degree of asymmetry of an SSP. While the mean and the standard deviation are dimensional

9 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT 9 quantities, the skewness τ is nondimensional: τ(r, φ)= ) 3 P (z)dz. (6) ( m (r,φ) z σ(r,φ) 3. Results Based on Eq. (), the SSPs were numerically computed and analyzed. In the simulation, the sourceto-origin distance was set to 6 mm, and the diameter of the reconstruction region 5 mm. That is, a 45 fan-angle was assumed. Figure 4 shows SSPs at four special positions in the reconstruction plane, which are (, ), (5 mm, ), (, 5 mm) and ( 5 mm, ) respectively, for D =mm, N =4and P =.5. This pitch value corresponds to 4.5 as defined in earlier studies [ 3]. Figure 5 plots the mean, the standard deviation, the relative variation of the slice thickness and the skewness, with the same imaging parameters as for Fig. 4. The reference for computation of the relative variation was the slice thickness at the gantry isocenter. Figure 6 summarizes the averages of the mean, the standard deviation, the relative variation, and the the skewness as a function of the scanning pitch and the number of detector rows. A multi-row-detector spiral CT scanner Aquilon (Toshiba Corp., Tokyo, Japan) was available in our Department. The CT system can be configured to provide 4 slices with.5,,, 3, 4, 5 and 8 mm collimation, respectively. Although it would be the optimal to validate our results by performing phantom experiments directly on this scanner, the interpolation details of the reconstruction software are not transparent to users. However, an extensive performance study on this type of the scanner was already reported [], which includes slice thickness measures using the half-scan interpolation method. A comparison between the reported measures and our simulated data indicates a satisfactory agreement. Specifically, it was reported in Fig. 7 of [] that the through-isocenter profiles of the extended multi-rowdetector spiral CT methods are almost the same as that of the full-scan interpolation method in the case of single-row-detector CT. This observation was precisely reproduced in all our simulation tests. 4. Discussion and conclusion In our numerical simulation, it has been observed that the SSP varies as a function of the scanning pitch and the number of detector rows. For example, the slice thickness depends on the number of detector rows and the pitch in an oscillation fashion, as shown in Fig. 3(b). Given an appropriate selection of the pitch and the number of detector rows, the SSP does not change very much over the field of view in terms of the mean, the slice thickness, and the skewness of the SSP. Our data have shown that the relative variation can be made less than % for a 49 fan-angle. Note that the presentations of the results are restricted on the positive half of the x-y plane. This restriction causes no loss of generality, because the SSP is anti-symmetric with respect to the x axis. This property is consistent with our geometrical intuition, and can be shown in a way similar to that used in an earlier study []. In addition to the half-scan interpolation method, there are various other interpolation methods possible [,,,6]. Although in this study we have focused on the half-scan interpolation method to provide insight into the characteristics of the SSP in multi-row-detector spiral CT, our methodology can be applied/adapted to investigate the features of the SSP with other interpolation methods, such as the extended full-scan interpolation method []. Different interpolation methods not only have effects on the slice thickness but also influence the image noise [,,4]. Spatial variation of the noise is an important

10 G. Wang et al. / A study on the section sensitivity profile in multi-row-detector spiral CT aspect of image quality assessment in multi-row-detector spiral CT, but it is beyond the scope of this work, and will be pursued in future. When wide-angle cone-beam spiral CT scanners emerge in the future, the fan-beams defined by individual detector rows will no longer be in parallel, but our analytic approach can still be applied to study the SSP. One way for such an analysis is to work in the Feldkamp-type reconstruction framework [4, 7,8]. The essence of the Feldkamp-type reconstruction can be formulated in two steps: () conebeam to fan-beam data conversion via a cosine correction, and () fan-beam reconstruction via filtered backprojection. Consequently, after wide-angle cone-beam data associated with direct and opposite X-rays are weighted by appropriate cosine factors, these rays can still be regarded as from fan-beams that are in parallel to the gantry plane. Hence, the SSP can be similarly computed and analyzed. The analysis on the condition under which the cone-beam angle can be neglected is an interesting question. A heuristic consideration is that whether the cone-beam angle is significant or not also depends on the detector collimation. The smaller the collimation, the less significant the cone-beam angle effect, the other conditions being equal. Generally speaking, our simulation results should be more accurate when the number of detector rows is smaller. It is emphasized that if the number of detector rows is large, SSP analysis results obtained under the parallel-beam assumption would be only approximate. In conclusion, the SSP of multi-row-detector spiral CT has been formulated for the extended half-scan interpolation method at any transverse position. Based on the SSP formula, numerical simulation has been performed to quantify the characteristics of the SSP with the number of detector rows up to 4. Although in general applications the SSP at the gantry iso-center can be used as a representative of the SSP family, for more accurate analyses the spatial variation of the SSP must be taken into account. Acknowledgement This work was supported in part by grants from the National Institutes of Health (DC359). References [] K. Taguchi and H. Aradate. Algorithm for image reconstruction in multi-slice helical CT, Med. Phys. 5 (998), [] H. Hu, Multi-slice helical CT: scan and reconstruction, Med. Phys. 6 (999), 5 8. [3] G. Wang and M.W. Vannier, The effect of pitch in multi-slice spiral/helical CT, Med. Phys. 6 (999), [4] G. Wang, T.H. Lin, P.C. Cheng and D.M. Shinozaki, A general cone-beam reconstruction algorithm, IEEE Trans. Medical Imaging (993), [5] S. Schaller, T. Flohr and P. Steffen, An efficient Fourier method for 3-D Radon inversion in exact cone-beam CT reconstruction, IEEE Trans. Med. Imaging 7 (998), [6] H. Kudo, F. Noo and M. Defrise, Cone-beam filtered-backprojection algorithm for truncated helical data, Phys. Med. Biol. 43 (998), [7] F. Noo, M. Defrise and R. Clackdoyle, Single-slice rebinning method for helical cone-beam CT, Phys. Med. Biol. 44 (999), [8] A. Polacin, W.A. Kalender and G. Marchal, Evaluation of section sensitivity profiles and image noise in spiral CT, Radiology 85 (99), [9] W.J. Davros, B.R. Herts, J.J. Walmsley and N.A. Obuchowski, Determination of spiral CT slice sensitivity profiles using a point response phantom, J. of Computer Assisted Tomography 9 (995), [] G. Wang and M.W. Vannier, Spatial variation of section sensitivity profile in helical CT, Med. Phys. (994), [] W.A. Kalender, W. Seissler, E. Klotz and P. Vock, Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation, Radiology 76 (99), [] C.R. Crawford and K.F. King, Computed tomography scanning with simultaneous patient translation. Med. Phys. 7 (99),

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