Jia-Shun Liu

Yi-Kun Zhang

Hui Tang

Li-Bo Zhang

Ben-Qiang Yang

Ying Yan

Li-Min Luo

Yang Chen

Rollback Reconstruction for TDC Enhanced Perfusion Imaging *

Keywords: Rollback reconstruction, CBCT, Time resolution, Time density curve 50

Tomographic perfusion imaging is a significant function imaging modality for stroke diagnosis. However, the low rotational speed of C-arm is a challenge for the application of perfusion imaging in C-arm CBCT(from 6 to 8 seconds per circle). The traditional reconstruction methods can not remove the artifacts caused by the slow rotational speed or can not acquire enough sample points to restore the time density curve(TDC). This paper presents a dynamic rollback reconstruction method for CBCT. The proposed method could improve the temporal resolution by increasing the sample points used for calculating the TDC. Combined with of the existing research results, the algorithm allows the possibility of using slow rotating scanners for perfusion imaging purposes. In experiments, the proposed method is compared with other dynamic reconstruction algorithms which is based on standard reconstruction, temporal interpolation approach. The presented algorithm could improve the temporal resolution without increasing x-ray exposure time or contrast agent.

I. INTRODUCTION

As one of leading causes of death, stroke may bring serious long-term disability. According to the statistics, there are about 2.5 million new stroke cases in China every year, and about 1.7 million patients die from stroke. In acute stroke treatment, time is life. Perfusion CT is a well-accepted method for stroke diagnosis[1-3] whose aim is to assess the blood supply of the tissue which has already found its way into clinical routine. Together with perfusion MRI(magnetic resonance imaging), they represent the primary imaging techniques for patients with symptoms of stroke. After injecting the contrast agent, the tissue was scanned multiple times in succession to obtain the TDC[4-6] of each slice. The curve reflects the changes of contrast agent concentration in tissues[7-9], i.e. the changes of blood supply in tissues.

Hence, perfusion CT can be used to identify tissue regions that can be salvageable and may contribute to stroke therapy.

However, there are still some challenges for perfusion CT.

One limitation is that perfusion CT could only produce functional maps of a limited number of slices. CBCT(Cone Bean CT) [START_REF] Fieselmann | Interventional 4-D C-arm CT perfusion imaging using interleaved scanning and partial reconstruction interpolation[END_REF][START_REF] Ganguly | Cerebral CT perfusion using an interventional C-arm imaging system: cerebral blood flow measurements[END_REF] can be used to tackle the problem. The application of large-area detectors allows users to conduct perfusion CBCT studies on the three-dimensional regions of interest.

However, there are two problems to be addressed.

The first is known as the temporal resolution problem [START_REF] Montes | Dynamic cone-beam reconstruction for perfusion computed tomography[END_REF].

In order to reconstruct an image at any time t, we need for each projection at projection angle α. However, the source

46

When using the same interpolation method to fit the time den-47 sity curve, fewer sample points will lead to inaccurate results.

48

The degraded time density curve will mislead the assessment 49 of the patient's condition. construction result at a certain time point, Grangeat and coworkers performed linear regression on the closest k values of the time series. This method was called k-mode linear regression. K-mode linear regression effectively reduced the impact of the temporal resolution problem, but did not solve the time sampling problem: it limited the attenuation of the contrast agent during the scanning process from the original to the current, but the number of sample points could not been improved. Although a sample point could be obtained every t π by the rearrangement method, it was still not enough. Under the CBCT scanning protocol, only a dozen sample points could be obtained. At present, there is no good way to solve the time sampling problem. From the perspective of data collection, Fieselmann [START_REF] Fieselmann | A dy-551 namic reconstruction approach for cerebral blood flow quan-552 tification with an interventional C-arm CT[END_REF] proposed a method: when collecting data, the patient was injected with the contrast agent twice.

After the second injection of the contrast agent, a time offset was added and data acquisition was not performed immediately. This time offset should be half the time of a single scan. In this way, the time sample points obtained are doubled. However, this method has great drawbacks. First, it did not solve the temporal resolution problem. More importantly, the contrast agent is harmful to human body, especially the liver and kidneys, and some patients may also have allergic reactions. Therefore, multiple injections of the contrast agent is obviously not recommended. In addition, the author did not prove whether the first injection of contrast agent will be left in the patient's body and whether it could influence the results of the second injection of the contrast agent. Jie Tang [START_REF] Tang | A novel temporal recovery tech-556 nique to enable cone beam CT perfusion imaging using an in-557 terventional C-arm system[C]//Medical Imaging[END_REF] proposed a temporal recovery method to recover time density curves to enable C-arm CBCT perfusion studies. This method did not increase the number of sample points, but proposed a more excellent fitting method. In their method, two optimization methods (CG and Bregman) were used to solve the proposed temporal recovery problem.

Yinsheng Li [START_REF] Li | An Enhanced SMART-RECON 561 Algorithm for Time-Resolved C-arm Cone-Beam CT Imag-562 ing[END_REF] introduced a new technique referred to as enhanced SMART-RECON(eSMART-RECON) to enhance the temporal performance in a multi-sweep CBCT data acquisition protocol. However, this method is an iterative reconstruction method, which consumes a significant amount of computing resources. In addition to the traditional methods, with the rise of deep learning and neural network, it has also been combined with perfusion. For example, Zhu [START_REF] Zhu | Temporally downsampled cere-565 bral CT perfusion image restoration using deep residual learn-566 ing[END_REF] proposed an optional way to reduce CTP imaging radiation dose, that is downsampling 30-pass images to 15 passes in temporal domain and then restoring them to 30 passes with the deep residual CNN model. It was also a method of recovering sample points. However, deep learning and neural network need a lot of training data to ensure the accuracy of the results, and the anti-risk ability is poor. If a little interference is added, the result of the network may be wrong.

In this article, we address the problem of dynamic reconstruction within the context of perfusion CT and presented a dynamic rollback reconstruction approach for perfusion CBCT to improve the time density curve. And combining it with the existing reconstruction method, it retains the advantages of the original method. We took the contrast agent attenuation caused by its propagation into account and assume that there was no movement or deformation during the scan-ning process. Then the temporal interpolation performed on lowing formula to obtain Ppara (θ, g, s) :

Ppara (θ, g, s)dθ = [ R 2 -g 2 R 2 -g 2 + s 2 • P para (θ, g, s)] * h(g) (1)
cone angle κ and h(g) is the filter function. It is necessary 181 to multiply the projection data by the cosine weight since 182 Ppara (θ, g, s) are the parallel-beam projections.

183

(3)Back projection reconstruction 184 Finally, the voxel (x, y, z) is reconstructed using the cor-185 responding filtered parallel-beam projections Ppara (θ, g, s).

186

The tomographic images can be obtained after all the voxels 187 are reconstructed. This process can be formulated as follows: due to the linear nature. In the same way, the interpolation 211 process also could be performed after the back-projection.

212

The computational complexity could be reduced since the 223 P BB j (X, t) = H 3 (P BB j (X, t π N (2j+1) )) j = 0, 1, ..., N -1

H 3 (x) = [(1 + 2 x -x 0 x 1 -x 0 )y 0 + (x -x 0 )y 0 ]( x -x 1 x 0 -x 1 ) 2 + [(1 + 2 x -x 1 x 0 -x 1 )y 1 + (x -x 1 )y 1 ]( x -x 0 x 1 -x 0 ) 2 (
(5)

The required results could be obtained by accumulating the partial block back-projections at the given time point. The FDK reconstructed results of a full scan could be got through adding all the PBBs:

F (x, y, z) = N -1 j=0 P BB j (X, t) (6)
The partial block back-projection approach reduced the data inconsistency in the projection domain. Even though the projections in a block were acquired at different times, the prob- (2)Rollback 2π 3 , in other words, j = j + 1

296

(3)Repeat steps 1 and 2;

297 It can be expressed as:

298 F π 2 +j• π 3 = π+j• π 3 0+j• π 3 Ppara (θ, g, s)dθ (7) 299
The number of reconstructed images will increase as the 300

319

(1)Rebinning cone-beam projections to parallel-beam pro-320 jections:

P cone (β, m, n) → P para (θ, g, s) (8)
(2)Take N=6, and the rollback angle is π 6 , reconstruction of 323 PBBs with T-FDK:

324 P BB j = π 3 +j• π 6 0+j• π 6 Ppara (θ, g, s)dθ j = 0, 1, ... (9)

325

(3)Combination of the samples of the jth and the (j+6)th 326 PBBs; 327 P BB j (X) = {P BB j , P BB j+6 , P BB j+12 , ...} j = 0, 1, ..., 5

328

(4)Interpolation of PBBs values; 329 P BB j (X, t) = H 3 (P BB j (X)) j = 0, 1, ..., 5

F 1 (x, y, z) = j=0,2,4 P BB j (X, t) and F 2 (x, y, z) = j=1,3,5 P BB j (X, t) (11)
µ i (t) = (50 + 50sin(2πv i t))HU (13)
In this case, µ is the HU value, and v is the frequency.

  turns with a restricted rotational speed ω. When the speed 28 ω is slow, it means the contrast agent attenuation during the 29 scanning process cannot be ignored. The scanning time of 30 traditional perfusion CT is limited to two seconds per circle.31In such a short time, the attenuation of the contrast agent is 32 negligible. So direct reconstruction will not influence the final 33 result, i.e., the recovery of the time density curve. However, 34 the scanning time of CBCT is from 6 to 8 seconds per circle, 35 which is much longer than the traditional perfusion CT. The 36 concentration of the contrast agent in blood has a great in-37 fluence on reconstruction quality. For example, it may result 38 in the artifacts or inaccurate values in the reconstructed im-39 ages. This will directly influence the subsequent production 40 of perfusion parameter maps. The second problem is called 41 the time sampling problem. After injecting the contrast agent, 42 the acquisition time of traditional perfusion CT and CBCT is 43 almost equal. However, due to the different rotational speed 44 ω, the total number of CBCT perfusion is much less than tra-45 ditional perfusion CT. That leads to a lack of sampling points.

124

  the partial block back-projections was employed to increase 125 the number of sampling points. Experiments demonstrated 126 the promising performance of the proposed rollback recon-127 struction algorithm. 128 II. METHODS AND MATERIALS 129 A. The T-FDK method 130 3D reconstruction from 2D projections obtained along a 131 single circular source trajectory is most commonly calculated 132 using the FDK algorithm. However, the FDK algorithm can 133 not well preserve the reconstruction accuracy when the cone 134 angle is large. In 2000, Grass et al. proposed an alternative 135 approach based on a cone-beam to parallel-beam rebinning 136 step, a corresponding rebining step into a rectangular virtual 137 detector plane and a filtered backprojection. The computa-138 tional complexity was lower than in Feldkamp's original ap-139 proach. As shown in Fig.1, the fan-beam data acquired along 140 a circular source-detector trajectory can be rearranged into 141 sets of fans of rays. Similar to the algorithm of rearranging 142 fan-beam into parallel-beam, we can rearrange cone-beams 143 X-ray emitted from different positions of the ray source's cir-144 cular trajectory into parallel-beams with the same cone angle 145 and parallel to each other in space. Although cone-beams X-146 ray and parallel-beams X-ray are not completely correspond-147 ing, they can be corresponded one by one by interpolation. 148 The X-ray source moves along a circular trajectory of ra-149 dius R, and the cone-beam projection datas P cone (β, m, n) at 150 various angles β are collected on the flat panel detector. Then 151 these projection data are rearranged into the parallel-beam 152 projections, which are used to reconstruct the tomographic 153 images of the object. As shown in Fig.2, the ray received by 154 virtual detector 1 can be converted to a corresponding ray on 155 virtual detector 2. The projection of the rearranged parallel-156 beam on the X-Y plane is perpendicular to the virtual detector 157 2. The projection value of the rearranged parallel-beam on the 158 virtual detector 2 can be expressed by P para (θ, g, s), where θ 159 is the projection angle of the parallel beam, g and s are the 160 horizontal and vertical coordinates of the rearranged paral-161 lel beam on the virtual detector 2, respectively. The specific 162 process can be divided into three steps: data rearrangement, 163 weighted filtering, and back-projection reconstruction. 164 (1)Data rearrangement 165 First, the cone-beam projections P cone (β, m, n) are rear-166 ranged into parallel-beam projection data P para (θ, g, s). Dur-167 ing transforming the (β, m, n) space to (θ, g, s) space, not 168 every parallel-beam projection has the corresponding cone-169 beam projection as the detector collected data are discrete. 170 Usually, the nearest interpolation or linear interpolation are 171 employed to tackle the above problem. Different interpola-172 tion methods will influence the spatial resolution of the re-173 constructed volume.

  174(2)Weighted filtering 175 Then, the rearranged parallel-beam projection data 176 P para (θ, g, s) is weighted and filtered according to the fol-

Fig. 1 .

 1 Fig. 1. The fan beam can be rearranged into a parallel beam.

188FFig. 3 .207

 3 Fig. 3. Partial block back-projection.

213

  calculated values only need to be accumulated after back-214 projection. And if the interpolation was not performed for 215 each projection but partial block back-projection, the compu-216 tational complexity could be further reduced. Therefore, a 217 full scan was equally divided into N parts. In this way, the 218 values of the jth and the (j + N/2)th partial block back-219 projections were combined to a unique time sequence and a 220 continuous signal estimated by interpolation. In the experi-221 ment, hermite interpolation was used, and the interpolation 222 process was as follows:

4) 224

 224 Where x 0 and x 1 was the positions of two adjacent points 225 of the point to be interpolated, y 0 and y 1 correspond to the 226 dependent variables of x 0 and x 1 , and y 0 and y 1 was the cor-227 responding derivatives. The steps of interpolation of PBBs was summarized as follows:

Start: j = 0 293 ( 1 )

 2931 lem of time resolution was mitigated since the time between angular intervals are shorter than the time of a short-scan reconstruction. The linear regression estimation had compensated for the temporal evolution. In the experiments, Pau et al. also proved that the data inconsistency would be significantly reduced as N increased. But the calculation cost would also increase synchronously. The TIA-TFDK algorithm is equivalent to the FDK algorithm at N=1. Furthermore, Pau et al. concluded that the reconstruction quality and time could be well balanced when N=6.C. Rollback reconstructionThe purpose of perfusion CBCT is to obtain an image sequence based on projection data acquired in several full rotations. However, the time sampling problem may lead to severe degradation of the time density curve. We adopt the rollback reconstruction strategy to address this problem. The cone-beam projection P t is indexed by the time t. The source rotates at a constant angular speed ω from time t = 0. The acquisition could be performed either in continuous mode (the x-ray source is always on) or in discontinuous mode (the xray source is switched off regularly). We assume that the region of interest is within the FOV (field of view) of the cylindrical detector. The projections at all angles are needed to reconstruct the image frame at any time t. However, the sampling points are discrete in practice. Thus, at the time t α , we only could acquire the projection at the angular position α. The same projection can be acquired again at time t α + T 2π after the source has finished a full rotation. Hence, at the angular position α, we could obtain a discrete sequence P tα = P tα+T2π . This dynamic acquisition process can be interpreted as sampling the time-dependent projection P t with a period of T s = T 2π .Since the cone-beam projections have been rearranged into parallel-beam projections, the 180°reconstruction condition is extanded from the 2D to the 3D case. Hence, we can get two reconstruction results after a full scan process. The one is reconstructed using the projection data of 0 -π and the other is reconstructed with the projection data of π -2π. Here we take the middle time as the reference time point, that is, t π/2 represents the time point of the scanning of 0 -π and t 3π/2 278 represents the time point of the scanning of π -2π. There-279 fore, after a single full scan, we can obtain the reconstructed 280 images at two time points: t π/2 and t 3π/2 . This dynamic 281 acquisition process can be interpreted as sampling the time-282 dependent projection P t with a period of T s = T π . These 283 sampling can be used to plot the density curve. Ideally, the 284 time density curve will be closer to the reference dynamic 285 curve as the number of sampling points increases. There-286 fore, we propose the rollback reconstruction method to in-287 crease the sampling points while not increasing the radiation 288 dose. Specifically, we could achieve a fixed angle rollback by 289 reusing the projection data have been used for reconstruction. 290 The following steps have explained the rollback reconstruc-291 tion method in detail. 292 Starting from 0°, reconstruct the angle range from 0 + 294 j • π 3 to π + j • π 3 , and record the result as t π 2 +j• π 3 295

Fig. 4 .

 4 Fig. 4. Rollback reconstruction.

Fig. 5 .

 5 Fig. 5. Rollback reconstruction with TIA-TFDK.

332( 6 )Fig. 6 .

 66 Fig. 6. Phantom A.

Fig. 7 .

 7 Fig. 7. The time density curves of simulated results. (a):vi = 0.133, (b):vi = 0.267, (c):vi = 0.4, (d):vi = 0.534, (e):vi = 0.667, (f):vi = 0.8.

Fig. 8 .

 8 Fig. 8. (a) is the result of FDK, (b) is the error between the FDK result and the original image, (c) is the result of RBTFDK,and the result of TIA-TFDK is the same, (d) is the error between the RBTFDK result and the original image, (e) is the original image.

385Fig. 7 .Fig. 9 .

 79 Fig.7. They demonstrate that with the increase of frequency 389

Fig. 10 .Fig. 8

 108 Fig. 10. Perfusion maps. From top to bottom are the results from FDK, TIA-TFDK, RBTFDK and ground truth, respectively. (a): CBF, (b): CBV, (c): MTT, (d): TTP

  435

2

  and Table 3 listed the root mean square error (RMSE) and 436 mean absolute percentage error (MAPE) of different meth-437 ods. Both the RMSE and MAPE show the promising perfor-438 mance of the proposed RBTFDK algorithm in reconstruction 439 accuracy preservation.

444

  As mentioned above, The original 512 × 512 × 30 data vol-445 ume for each slice location was downsampled to the dataset 446 of size 512 × 512 × 7. Compared with the traditional FDK 447 algorithm, TIA-TFDK adds the following calculation steps:

  This work was supported in part by the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant KYCX17 0104; in part by the China Scholarship Council under Grant 201706090248; in part by the State's Key Project of Research and Development Plan under Grants 2017YFC0109202, 2017YFA0104302 and 2018YFA0704102; in part by the National Natural Science Foundation under Grants 61871117, 61828101, and 31571001; and in part by the Science and Technology Program of Guangdong under Grant 2018B030333001.

* † Yang Chen, chenyang.list@seu.edu.cn

Table 1 .

 1 The perfusion parameters of different methods(mean).

		Ground truth	FDK	TIA-FDK	RBTFDK
	CBF	29.19	32.64	32.26	33.01
	CBV	1.67	3.83	2.10	2.09
	MTT	3.50	7.21	3.70	3.46
	TTP	4.81	9.58	5.97	5.60

Table 2 .

 2 The RMSE of perfusion parameters from different methods.

		FDK	TIA-FDK	RBTFDK
	CBF	36.60	29.18	26.35
	CBV	34.54	25.94	20.04
	MTT	42.53	46.98	42.38
	TTP	47.00	41.80	34.50

Table 3 .

 3 The MAPE of perfusion parameters from different methods.

		FDK(%)	TIA-FDK(%)	RBTFDK(%)
	CBF	39.23	32.48	27.65
	CBV	46.50	29.76	21.99
	MTT	40.22	32.95	28.64
	TTP	146.60	76.79	66.67
	data rearrangement, partial block back-projection and the in-

(5)Accumulation:

where R 2 -g 2 R 2 -g 2 +s 2 represents the cosine value of the beam terpolation step. After combining rollback reconstruction, the 450 computational complexity increases. Table 4 shows the re-451 construction times of the three algorithms with a PC with a 452 Intel Core i5 processor at 3.0 GHz. Indeed, the computational