Modeling Based Image Reconstruction in Time-Resolved Contrast-Enhanced Magnetic Resonance Angiography (CE-MRA) F.T.A.W. Wajer½ , J. van den Brink¾ , M. Fuderer¾ , D. van Ormondt½ , J.A.C. van Osch½ and R. de Beer½ ½
Delft University of Technology, Department of Applied Physics P.O. Box 5046, 2600 GA Delft, The Netherlands Phone: +31 (0)15 2786394 Fax: +31 (0)15 2783251 E-mail:
[email protected] ¾
Philips Medical Systems, Best, The Netherlands
Abstract— We have worked on the measurement protocol and related image reconstruction of 4D data sets in the field of time-resolved Contrast-Enhanced Magnetic Resonance Angiography (CE-MRA). The method aims at improving the interpolation of sparsely sampled -space data. It is based on exploiting prior knowledge on the time courses (TCs) of the image pixels, when monitoring the uptake of contrast agents in the blood vessels. The result is compared with an existing approach called 3D-TRICKS. Keywords— Magnetic resonance angiography, contrast agent, image reconstruction, scan-time reduction, modeling of pixel time courses.
I. I NTRODUCTION
from the blood is obtained. Since the passage of the contrast agent is captured as a function of time in a series of 3D images, it is a time critical measurement. Therefore it is of paramount importance to keep the MRI scan time as short as possible. Our method for reducing the scan time is based on “not collecting all data” needed for reliable 3D image reconstructions. As a consequence of this sampling strategy no standard FFT-based techniques are allowed to arrive at images of sufficient quality. In contrast, other techniques must be used, which in our approach is “employing prior knowledge on the pixel time courses” of the images.
Recently we have published on a method for reducing the scan time in the field of three-dimensional (3D) timeresolved Contrast-Enhanced Magnetic Resonance Angiography (CE-MRA) [1]. CE-MRA is a 3D Magnetic Resonance Imaging (3D-MRI) technique for visualizing the vascular system. It is based on using the passage of a contrast agent through the arteries and the veins.
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Fig. 2. Division of 3D -space into four blocks labeled A, B, C and D, as proposed in 3D-TRICKS [2].
II. M ETHODS
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Fig. 1. Three-dimensional -space. Sampling is performed along the arrows in the -direction.
During this passage the local ½ relaxation time is shortened. With a proper chosen measurement only MRI signal
A. Introduction In 3D-MRI the data values in -space are acquired along trajectories, which in case of cartesian sampling distributions are equidistant straight lines. In Figure 1 these cartesian trajectories are depicted as arrows in the -direction. Several of such trajectories are required to fill -space. The total measurement time for one 3D -space is equal to the
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number of trajectories times the repetition time (TR). Typical values are Ý Þ and TR = 10 ms, resulting in a measurement time of about 20 s. It is clear that this 20 s is too long to measure a series of 3D -spaces during passage of a contrast agent. A straightforward way to accomplish scan-time reduction is to ommit the measurement of a number of trajectories. In the next subsection we will describe how this is done in a sampling strategy called 3D-TRICKS.
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Fig. 3. 3D-TRICKS measurement protocol [2]. At each sampling point in time only one block A,B,C or D of -space is measured. The sequence DACABA is repeated. The empty boxes should be interpolated from the measured ones prior to image reconstruction.
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B. 3D-TRICKS In three-Dimensional Time-Resolved Imaging of Contrast KineticS (3D-TRICKS) reduction of MRI scan time is achieved by measuring only parts of the 3D -space. To avoid loosing spatial resolution, previously measured -space parts in the 4D data set are borrowed. The following steps can be distinguished in the original 3D-TRICKS approach [2]: ¯ “Subdivision” of the 3D -space into four blocks, labeled A, B, C and D (see Figure 2). ¯ Measurement of only “one single block” at each time point, following the sequence shown in Figure 3. ¯ Estimation of the data values in the empty blocks. In 3D-TRICKS this is accomplished by “linear interpolation” between the acquired blocks. ¯ 3D FFT based image reconstruction at each time point. The 3D-TRICKS measurement protocol, presented in Figure 3, makes it clear that the inner blocks A (the center of -space) are measured most. Furthermore it shows that the sequence DACABA is repeated, causing a five times larger empty gap between successive B, C and D blocks than A blocks. We have anticipated that estimating the data values of the empty outer blocks by means of linear interpolation might be problematic. In the next subsection we propose an alternative way to deal with the missing data. It is based on exploiting the behaviour in the time direction of the individual pixels (the pixel time courses (TCs)).
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Fig. 5. Fit of a sum of two gamma-variate functions (noiseless curves) to a TC of a pixel in an artery (left peak) and a vein (right peak).
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Since in CE-MRA we are dealing with “time series” of 3D -spaces (and related 3D images) it seems a logical step to investigate whether the behaviour as a function of time somehow can be included in the sampling- and image reconstruction strategy. In this context a first decision to be made is, in which domain are we exploiting the time
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behaviour? That is to say, are we looking in -space or in image-space as a function of time? Subdivision of 3D -space into 3D-TRICKS blocks Measurement of a single block at each time point
Zero filling in the truncation dimension towards full number of points 3D-FFT
TCs could even be modeled by a mathematical function (in the example by a sum of two gamma-variates [4]). Because of the observations, just described, we have decided to exploit the time behaviour solely via the image domain. In doing so we have defined a sequence of samplingand image reconstruction steps as visualized in the block scheme of Figure 6. Concerning the block scheme the following can be said. Since the A blocks are measured most (see Figure 3) it seems a logical step to interpolate in some way the TCs of and subsequently impose that time behaviour on , and . The latter amounts to solving the equation (only shown for ):
Interpolation of TCs of images
Interpolation of TCs of , and images by imposing the TCs of Adding , , and images
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Once the linear parameters ½ and ¾ have been obtained, they can be used to calculate the images at the other times (indicated by ):
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Hence an important step in the method is that in some way the missing time points of must be estimated. We have found that the number of A points is sufficient to allow “linear interpolation”, as was proposed in the original 3D-TRICKS method [2].
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Fig. 6. Block scheme of sampling- and image reconstruction steps for time resolved 4D CE-MRA.
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In principle, due to the linearity property of the Fourier transform [3], there should be no difference when observing the TCs via -space coordinates or image-space pixels. However, in practice of real-world CE-MRA data sets there is a complication since the contribution of the image domain “background”, or in other words of the pixels outside the blood vessels, are influencing the time behaviour in -space. The latter is illustrated in Figure 4, where we show some TCs for a 4D CE-MRA data set as depicted via the center of -space. Details about the data set are given in the section Results and Discussion. When explaining Figure 4 we assume for a moment that the -space is the result of reverse Fourier transforming the image domain. In doing so in the top part of Figure 4 some TCs are shown “with the contributions from all pixels”. In the bottom part of the figure, on the other hand, the TCs of the same -coordinates are shown, but now “without the contributions from the background pixels”. It can be seen that a somewhat clearer contrast agent uptake is visible. In Figure 5 we show two TCs as obtained via the image domain. One pixel is located in an artery and the other in a vein. The time behaviour is now so pronounced that the
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time (seconds) Fig. 7. The TC of an pixel after (solid) using all data, (dashed) using 3D-TRICKS and (dotted-solid) imposing the time behaviour of the A part on that of the B part.
In order to illustrate that imposing the time behaviour of pixels can yield an improvement we compare in Figure 7 the TC of an pixel as realized in three ways. The solid line is the true TC, obtained by using all data of the real-world 4D data set mentioned earlier. The dashed line results from simulating the 3D-TRICKS approach, that is to say after reducing the data set into the 3D-TRICKS blocks and performing linear interpolation. Finally, the dotted-solid line is the result of imposing the TC from the A part on that of the B part. It is clear that the latter is
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much closer to the true TC than the linearly interpolated one. III. R ESULTS AND D ISCUSSION A. Simulation of a 4D CE-MRA data set In order to establish whether the approach, described in the previous section, really works we have applied the method to a “simulated” 4D CE-MRA data set. That is to say, we simulated the 3D vascular system by sets of arteries and veins as shown in the top of Figure 8 and, in addition, we simulated the uptake of a contrast agent by the TCs shown in the bottom of that figure. Subsequently, we applied the protocol described by the block scheme of Figure 6.
measured according to the 3D-TRICKS protocol, really can benefit from our interpolation approach for the outer space blocks. In the figure most of the norm points for 3DTRICKS are considerable higher (i.e. larger differences with the true image) than the points for the new approach. Summing the norms over all time points and taking the 3D-TRICKS result as the 100% reference, it is found that the sum for our method is 42% of that of 3D-TRICKS. 5 4 3
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Fig. 9. Performance of 3D-TRICKS and the new method for the simulated 4D CE-MRA data set. Along the vertical axis is displayed the “norm” (see text) of the differences between the absolute pixel values of the reconstructed image and the true image. The horizontal axis concerns the first 40 seconds of the time domain. (dashed) Result of 3D-TRICKS. (solid) Result of the new method.
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Fig. 8. Simulation of a 4D CE-MRA data set. (top) Crosssection through the 3D image domain. (bottom) The simulated TC of an artery (solid) and a vein (dashed).
In Figure 9 we present the result for 3D-TRICKS and our approach, as obtained for the simulated 4D CE-MRA data set. Along the vertical axis the so-called “norm” is displayed, which is the square root of the sum of the squared differences between the absolute pixel values of the reconstructed image and the true image. The latter is known, of course, because we are dealing with a simulated image. The horizontal axis concerns the first 40 seconds of the time domain of the contrast agent uptake (the total time is about 62 seconds). Figure 9 demonstrates, that a 4D CE-MRA data set,
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time (seconds) Fig. 10. Performance of 3D-TRICKS and the new method for the real-world 4D CE-MRA data set. (dashed) Result of 3DTRICKS. (solid) Result of the new method.
B. A real-world 4D CE-MRA data set The next step to do is to apply the new approach to a real-world 4D CE-MRA data set. To that end we used a measurement that concerned the vascular system of the Ý human neck. The 4D acquisition matrix Ü Þ . This amounts for the
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complex-valued -space to a data set of about 87 MB, taking into account that the data values are stored in double precision. To increase the temporal resolution of the measurement, the SENSitivity Encoding (SENSE) method was employed. This method realizes a reduced-sampling strategy by using an array of receiver coils [5]. In this way for the data set at hand an inter-frame time of 1.6 s could be accomplished which means a total measurement time of about 62 s. This time is long enough to visualize the enlightenment of both the arteries and the veins. In order to handle the kind of data set just described we have worked with a PC containing a 1.1 GHz AMD Athlon CPU and having a 1.5 GB SDRAM memory. Furthermore, the hard-disk capacity is 150 GB and the system can operate under either RedHat Linux or Windows NT. When composing this system we have assumed that it must be capable of accessing something like three times the size of the resulting image domain matrix during image reconstruction. A typical case would be, for instance, an reconstruction matrix (when assuming short integers for the pixel values).
C. An improvement of the new interpolation approach The performance results, shown in Figure 10, were realized by using the original 3D-TRICKS -space blocks (Figure 2) and measurement protocol (Figure 3) [2]. In order to find out, whether we could improve the results for the real-world data set, we have varied some parameters of the approach: ¯ The A, B, C and D -space regions were changed into true 3D rectangular blocks (instead of blocks only dividing the dimension; see again Figure 2). ¯ The order of sampling the A, B, C and D blocks was varied, leading to an optimum sequence for the current realworld data set. In fact, more B blocks were introduced at the expense of the C and D blocks. 0.5 0.4 0.3 0.2
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In Figure 10 the performance of 3D-TRICKS and the new method is compared for the real-world 4D CE-MRA data set. It can be seen that now the improvement of the new interpolation method is only present for a few time points around 13 seconds. This is the moment that the contrast agent starts to pass the arteries (see again Figure 5). For most of the other time points the performance of 3DTRICKS is even slightly better. Summing the norms over all time points and taking the 3D-TRICKS result as the 100% reference, it is found that the sum for our method is 102% of that of 3D-TRICKS. In the next subsection we will describe how we have tried to improve this somewhat disappointing result for the real-world data set.
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Dividing the -space into true 3D rectangular blocks introduced some minor improvement for both 3D-TRICKS and our approach. Measuring more B blocks realized a larger improvement, however more for 3D-TRICKS than for our interpolation method (see Figure 11). This has led us to the idea that somehow our assumption about the time behaviour of the A, B, C and D blocks is not true for all pixels. Indeed, after visualization of many pixel TCs we could demonstrate that for certain pixels the contribu-
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tions from the various -domain blocks may have different shapes. The latter is illustrated in Figure 12. In order to be able to account for different time behaviours we finally have applied a mixed interpolation approach: ¯ Both the A blocks and C blocks are linearly interpolated (i.e. according to 3D-TRICKS). For A this was already the case. For C this was decided because they only represent a small signal strength (see Figure 12). ¯ The B and C blocks are interpolated either according to 3D-TRICKS or to our approach, depending on the kind of time behaviour (see Figure 13). 0.15
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time (seconds) Fig. 13. TCs of pixel (55,49,7) (in an artery; see also top of Figure 12). The curves are from the full, original, dataset (dottedsolid), from the 3D-TRICKS result (dashed) and from the modelbased result (solid). (top) The B part. (bottom) The C part. It is clear that in this example the B part should be interpolated according to 3D-TRICKS and the C part to our approach.
IV. C ONCLUSIONS We have worked on improving the measurement protocol and related image reconstruction method, when sampling 4D data sets in the field of time-resolved ContrastEnhanced Magnetic Resonance Angiography (CE-MRA). The method takes as a starting point the 3D-TRICKS approach and aimes at improving the interpolation of sparsely sampled outer -space blocks. This is realized by employing prior knowledge delivered by the pixel time courses (TCs). Compared to the original 3D-TRICKS protocol [2] we can conclude that: ¯ For a simulated CE-MRA data set the new interpolation approach works rather well. A gain in performance of image reconstruction of about 58% is obtained. ¯ Dividing the -space into true 3D blocks, instead of blocks that are only dividing the dimension, delivers a minor gain in performance. ¯ For the real-world CE-MRA data set a larger gain in performance is obtained by collecting more inner -space blocks than outer blocks as a function of time. The 3DTRICKS method benefits more from that than our method. Compared to the original 3D-TRICKS measurement protocol the new 3D-TRICKS gains about 14% and our new interpolation approach 12%. ¯ The assumption that outer -space blocks deliver the same kind of time contribution to the TCs of pixels than inner blocks is not always met. This can be the case, for example, for pixels near blood-vessel walls and pixels in between neighbouring arteries and veins. ¯ An improvement of our interpolation method is obtained by carrying out a mixed approach concerning the interpolation of the sparsely sampled -space blocks. That is to say, some blocks are linearly interpolated (i.e. according to 3D-TRICKS) and others are interpolated according to our approach (i.e. invoking the TCs of inner blocks on that of the outer ones). This yielded a gain in performance, when compared to the original 3D-TRICKS, of about 15%. ACKNOWLEDGEMENTS This work is supported by the Dutch Technology Foundation (STW, project DTN 4967). R EFERENCES [1]
The result for this mixed approach is visualized in Figure 11 in the form of the plus points. For a number of time points they represent the lowest norm, or in other words they have the best performance. However, when summing the norms over all time points the differences are rather small. Taking the original 3D-TRICKS approach of Figure 10 as the 100% reference, the sums for our original approach, the new 3D-TRICKS and the mixed approach are 88%, 86% and 85%, respectively.
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