Fast vascular reconstruction with MAP-EM method from few projections

Naoko Ohura', Koichi Ogawa' and Etsuo Kunieda2 Dept. of Electronic Informatics, College of Eng., Hosei Univ., Tokyo, Japan

Dept. of Radiology, School of Med., Keio Univ., Tokyo, Japan

Abstract It is necessary to know exactly the position and shape of

blood vessels when radiation therapy is done for intracranial arteriovenous malformation. Various three-dimensional reconstruction methods for brain blood vessels have been proposed on the basis of clinical importance. But it is very difficult to accurately reconstruct blood vessels of complex shape from a few two-dimensional angiograms. In this paper, we used a MAP-EM method to reconstruct three-dimensional brain blood vessels from a few angiograms. Moreover, we decreased the calculation time by introducing a new penalty function in this MAP-EM method. As a result, a high quality image was obtained with a few iterations than with conventional iterative methods.

I. INTRODUCTION Intracranial arteriovenous malformation is a fatal disease.

Nowadays it is often found without symptoms thanks to the development of new diagnostic techniques. Stereotaxy irradiation (Radiosurgery) is one of the treatment methods for this disease, and its effectiveness has been demonstrated. This method destroys diseased parts by irradiating the local vascular malformation region, and this radiation therapy may replace surgery. But the intracranial arteriovenous malformation has a very complex shape, and we should also clarify the blood Bow conditions (inflow and outflow) of these blood vessels. Therefore, a three-dimensional reconstruction method with both high spatial resolution and time resolution is required. The goal of our study is development of an accurate system for locating diseased parts in the treatment of intracranial arteriovenous malformations.

Three-dimensional imaging of brain blood vessels provides important information for the treatment of vascular lesions. Various methods have been proposed from the standpoint of clinical importance [1]-[5], but there is still no excellent reconstruction method. The biplane method [1]-[3], which is typical as a three-dimensional reconstruction method for blood vessels, uses two angiograms and connects the blood vessels referring to these two angiograms in a three-dimensional coordinate system. This method has a problem in that the exact position of a blood vessel cannot be determined if the blood vessel exists on a plane which is orthogonal to the angiogram. And there is the possibility of separating the blood vessels in succession as individual blood vessels or joining two individual blood vessels into one blood vessel. From the viewpoint of examination, it is not possible to acquire angiograms from a lot of directions because there is a limitation to the time in which the contrast and opaque media can be administered. On the other hand, if the image reconstruction is done from

a lot of projections by using CT (computed tomography), we can obtain a series of transaxial images of high spatial resolution [4]. But we need an X-ray CT scanner besides the angiographic system. In addition, the long data acquisition time degrades the temporal resolution of the reconstructed images. In this paper, we have proposed a method for iterative image reconstruction from a few projections (angiograms). It is generally known that the iterative approximation method can reconstruct a comparatively excellent image even though there is a limitation of the number of projection data. ML-EM (Maximum Likelihood-Expectation Maximization) [6]-[9] and MAP-EM (Maximum A Posteriori-Expectation Maximization) [IO]-[ 161 are typical iterative image reconstruction methods. MAP-EM can yield higher quality reconstruction images than ML-EM, suppressing the statistical noise by introducing a priori information such as the smoothness of an image. The basic advantages of these methods are the sum of the pixel values is equal to the sum of the measured projection data and all the reconstructed pixel values are positive as long as the pixel values of an initial image are positive. On the other hand, these iterative reconstruction methods have the disadvantage that they take a long time to converge. To solve this problem, we proposed a new MAP-EM method to shorten the calculation time in the reconstruction of three-dimensional blood vessels. This method separates the image reconstruction procedure into two stages, and a coarse image is reconstructed in the first stage by using a conventional MAP-EM method, and a fine image is reconstructed by using a MAP-EM method with a penalty function which is newly introduced. This penalty function maintains the continuity of blood vessels and is done by using a coarse image. To confirm the effectiveness of this method, numerical phantom was simulated. The results obtained showed that the mean absolute error in the reconstructed image in the second stage was smaller than that with the conventional MAP-EM method. Moreover, the images reconstructed from the actual angiograms of the wire phantoms had high :spatial resolution and this proved the effectiveness of the proposed method.

11. THREE-DIMENSIONAL IMAGE RECONSTRUCTION

A. MAP-EM method MAP-EM is a method for improving the image quality

by introducing inherent information in an image as a priori probability. The pixel value in an image reconstructed by

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MAP-EM is estimated as follows: function W(qj) defined in Eq. (7) speeds up the convergence.

Rr = cijXxJ (2) j E I i

where an image is composed of J pixels, and Xy is the expected value of pixel j at iteration n. Pi is the projection measured with the ith detector. cij is the conditional probability of photons emitted from pixel j and detected at detector a. Ii is a set of pixels which contribute to detector a, and Jj is a set of projections which affect pixel j, and RT is a projection estimated from the reconstructed image at iteration n. U(X7) is an energy function as shown in Eq. (3). This is the weighted summation of a potential function V ( r ; b ) where r is the distance between the pixel X j and the pixel Al.

U(Xjn) = wjlv(r ; 6 ) (3) 1ENj

where wjl is the weight between pixel j and 1, and this is the inverse of the distance between pixel j and 1. Nj is the neighborhood of pixel j and we considered the three-dimensional third neighborhood system (26 pixels around the pixel of interest). /3 is the parameter which controls the degree of regularization and 6 is the constant which penalizes the discrepancy between the two pixel values. Moreover, the potential function is the form whose first derivative shown in Eq. (4) takes the maximum value 1 at T = 6.

(4)

where 6 is the parameter which decides the degree of influence received from a pixel in a neighboring system.

d V ( r ; 6) 16(r/6) -- - dr (3 + ( r /6 )2 )2

B. ModiJied MAP-EM method It is very important to decrease the time required to

reconstruct a final image by MAP-EM. We introduced a penalty function to decrease the calculation time. This penalty function works according to pixel values, and is roughly reconstructed by a conventional MAP-EM method. The roughly reconstructed image has information on the continuity of blood vessels and this is very important in three-dimensional reconstruction. Therefore, image reconstruction by using this information can accelerate the convergence and preserve an exact three-dimensional shape. The proposed method reconstructs vascular vessels roughly until we can recognize the area of the vascular vessel from the images reconstructed by the conventional MAP-EM method. Now we define this image qj at mth iteration, i.e.,

qj = AT (5)

In the second stage the reconstruction is performed by using Eq. (6) from the iteration rn + 1. In this equation the penalty

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~ ( q j ) = exp(-a. q;) (7)

where y is a parameter which controls the weight of the penalty function in the estimation process.

111. SIMULATIONS

A. Simulation phantom To confirm the effectiveness of the proposed method

we compared the quality of the reconstructed images with that of the images reconstructed by ML-EM and MAP-EM. The simulation phantom (Fig. 1) consists of two spiral wires of different thickness bfit has the same value of 100. The area outside the wires has a value of 0. The image size was 64 x 64 x 64 voxels, and the voxel size was 0.5 x 0.5 x 0.5[cm3]. The diameters of these two wires were 0.5[cm] and 2.0[cm]. We compared the images reconstructed from a set of projection data for four directions (LAO-90deg., LAO-45deg., RAOd5deg. and Odeg.) or six directions (LAO-90deg., LAO-60deg., LAO-30deg., RAO-60deg., MO-30deg. and Odeg.). The number of projection bins in each view was 64 x 64. Moreover, parameters /3 and 6 were fixed at 100 and 10, respectively, and parameter m was set at 10.

0.

ll Numerical phantom

soirrce

Figure 1: Simulation phantom and data acquisition geometry.

B. Results The mean absolute error (MAE) between the original image

and a reconstruction image at each iteration was calculated as follows:

where the total number of pixels is J , and the value of pixel j of the original image and that of the reconstructed image are ATg and A;"", respectively. Fig. 2 shows the relation between the number of iterations and MAE. Fig. 2(a) shows the results reconstructed by four projections, and (b) shows those by six

I169

projections. The parameters used were y = 0.5 and a = 0.002 for the reconstruction by four projections, and were y = 1.2 and a = 0.003 for the reconstruction by six projections. From these graphs MAP-EM yields a much better image than ML- EM. Moreover, when the proposed technique was used from the tenth iteration, the MAE decreased more rapidly than with the conventional MAP-EM method. The reconstructed image at the 20th iteration obtained by the proposed technique are shown in Fig. 3. In this figure spiral wires are accurately reconstructed.

0.061 4

ML-EM

02- .. ....... -

0.06: Modified MAP-EM

10 15 26 16 15 20 0.041 0.04c

Iterations Iterations

(a> (b) Figure 2: Mean absolute error: (a) Number of projections : 4, and (b) Number of projections : 6.

Figure 3: Three-dimensional display of the wire phantom.

IV. EXPERIMENTS

A. Experiment phantoms Three-dimensional image reconstruction was performed

by using the experiment data obtained with a commercially available angiographic data acquisition system. The two phantoms consisting of aluminum wires of different diameters are shown in Fig. 4. The wire phantom (Blood vessel phantom A), which simulates a brain blood vessel, has a diameter of 2.5[mm]. The other phantom (Blood vessel phantom B), which has a diameter of 0.7[mm], simulates a complex intracranial arteriovenous malformation.

B. Data acquisition system The experiment was done with GE ADVANTX ACT, and

angiograms were acquired from LAO-90deg. to RAO-90deg. at intervals of 30deg. As a preprocessing, we subtracted a background image (vacant scan image) from an acquisition image and extracted the aluminum wire part. The data acquisition conditions were that the tube voltage of the X-ray was 70.0[kV], the tube current was 50.0[mA], the distance between the X-ray source and the image intensifier was

2.5mm 0.7mm

(a) (b) Figure 4: Experiment phantoms: (a) Blood vessel phantom A, (b) Blood vessel phantom B.

llO.O[cm], irradiation time was about 9.6[ms], and the field of view was 9.0[inch].

C. Results 1) Blood vessel phantom A

Image was reconstructed by using the angiogranis for five directions, i.e., LAO-90deg., LAO-60deg., LAO-30deg., RAO-30deg. and Odeg. The size of the angiograms was 318 x 318 pixels and the pixel size was 0.49 x 0.49[mm2/pixel], but we reduced the matrix size to 159 x 159 in order to reconstruct an image quickly. Fig. 5 shows the calculated projection data for the reconstructed three-dimensional image and the original angiograms measured with this angiographic equipment. The results obtained showed that the proposed method could reconstruct an accurate wire phantom. Fig. 6 is a three-dimensional display of the wire phantom, in which the reconstructed value was quantized into two levels (the threshold was 30). It is easily seen that the reconstructed image of the wire is almost the same as the photograph of the object taken from the same angle.

2) Blood vessel phantom B

Image was reconstructed by using the angiograms for five directions, i.e., LAO-90deg., LAO-60deg., LAO-30deg., RAO- 30deg. and Odeg. A simulated vascular malformation region was reconstructed only in a part (the image size was 135 x 135 pixels). The pixel size was 0.49 x 0.49[mm2/pixel]. Fig. 7 shows the calculated projection data for the reconstructed three- dimensional image and the original angiograms measured with this angiographic equipment. In addition, we made vertical density profiles of this reconstructed image (Fig. 8). The results obtained showed that our method could accurately reconstruct the complex shape of the aluminum wire of small diameter with high spatial resolution.

V. DISCUSSION In radiation therapy for intracranial arteriovenous

malformation, extracting an accurate three-dimensional image of the brain blood vessel and deciding the irradiation position are necessary to improve the effectiveness of the therapy. As for three-dimensional reconstruction of the brain blood vessels, various methods have been proposed from the standpoint of importance in medical practice. R. Ning et al. [4] had succeeded in the reconstruction of three-dimensional blood

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LA030' O0 RA030' LA030' 0" RA030"

Figure 5: Projection data calculated from a reconstructed image(a) and the original angiograms(b)

Figure 7: Projection data calculated from a reconstructed image(a) and the original angiograms(b).

0

Y

Figure 6: Photographs and three-dimensional display of a reconstructed image of the blood vessel phantom A.

vessel images with high spatial resolution from X-ray CT images. In Japan, X-ray CT is basically used for the purpose of diagnosis and is not commonly used with angiographic equipment. Bullitt et al. [ l ] proposed a biplane method, but accurate reconstruction of blood vessels from two orthogonal angiograms is sometimes very difficult, and they adopted the method of combining the MRA data with these angiograms. The advantage of our method is that this reconstruction method uses only a few projections. Our experiment results showed that projections for five directions were almost enough for reconstructing blood vessels. At the same time, a high spatial resolution image was achieved by this proposed method. Although the diameter of blood vessels varies, if we can reconstruct 1 - 3[mm] blood vessels, we can accurately locate lesions. We demonstrated in the experiments that the proposed method could accurately reconstruct the images of small diameter wire phantoms. In particular, overcrowded blood vessel phantom B was accurately reconstructed and this was also confirmed by its profiles. The time required for three-dimensional image reconstruction and display should be decreased in planning radiation therapy. We accomplished higher speed image reconstruction than with conventional

150 150

s 100 100

.% 50 50

- a a, -

0 0 0 40 80 120 0 40 80 120

position position

(a) (b) Figure 8: image: (a) LAO-45deg., (b) RAO-45deg.

iterative image reconstruction methods such as ML-EM and MAP-EM. In the simulation we set the number of iterations in the first stage at ten. The reason is that around the 10th iteration the rough vascular structure was established and the value was close to 100 which is the same value as in the true image. After this iteration number the vascular region and the background region are separating clearly as shown in Fig. 9. The scaling factor a of the penalty function can be designed to some degree by guessing the pixel value of blood vessels from the histogram of the pixel value for the reconstruction image. In this paper, the nonlinear distortion of these images caused by the data acquisition geometry was very little, because these phantoms were small. But higher quality image can be obtained by preprocessing the acquisition image, and we will introduce a correction technique, such as Schueler's approach [5], in our method in the future.

Density profiles in vertical direction of a reconstructed

,

VI. CONCLUSIONS We succeeded in accurate three-dimensional imaging

of blood vessels from a few projections with the proposed method. And we introduced the penalty function to speed up the convergence, and the results showed that the proposed method is suitable for clinical use.

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200 -

20 40 60 80 100 120 pixel value

ure 9: Pixel values V.S. iteration number.

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