Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
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ABSTRACT
It has been recently proposed that computer-simulated phantom images can be used to evaluate methods for fMRI preprocessing. It is widely recognized that Gradient-Echo Echo Planar Imaging (EPI), the most often used technique for fMRI, is strongly affected by field inhomogeneities. Accurate and realistic phantom images for use by the fMRI community for software evaluation and training must incorporate these distortions and account for the effects of head motion and respiration on the distortions. A method to generate realistic distortions caused by field inhomogeneity for the generation of an fMRI phantom is presented in this paper. Changes in field inhomogeneity due to motion are studied by means of adding motions to the brain model and calculating the induced field map numerically rather than measuring it experimentally. A fast analytic version of an MR simulation is used to generate distorted EPI images based on the calculated field maps. The new generated fMRI phantoms can be used to evaluate processing algorithms for fMRI study more accurately. We can appreciate the importance of distortions for fMRI phantom generation by simulating a distortion-free image and adding distortions afterwards. Validations are performed by comparing the calculated field maps with measured ones. In addition, we show the similarities between a simulated fMRI phantom and real EPI image from our MR scanner.
Keywords : Magnetic Resonance, Neurological, Other (simulaton)
1. INTRODUCTION
Gradient-echo echo planar imaging (GE-EPI) is widely employed in fMRI due to its high sensitivity to susceptibility variations on the microscopic scale such as those due to blood oxygen level dependent (BOLD) signal and its fast imaging speed. Unfortunately, it suffers from patient-induced field inhomogeneity caused by macroscopic susceptiblity variation at air/tissue and bone/tissue interfaces, which, because of its long TE and signal readout time, results in significant geometric distortion and signal loss. As a result, both accurate registration between anatomical images and
EPI images and the detection of BOLD signals are compromised.
Computer-generated phantoms provide a way of evaluating new processing algorithms for fMRI data analysis such as motion correction 1 . A realistic fMRI phantom should characterize noise, motion, and functional activation. It is important to model image distortions and signal loss due to magnetic field inhomogeneities as well. It is known that the variations in field inhomogeneity depend on the orientation of the head relative to the applied static field 2 . Thus, it is not correct simply to rotate and translate the measured field map of one image and apply it to differently rotated images in

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006) the fMRI time series. The purpose of this work is to simulate realistic susceptibility induced distortions for fMRI phantom generation and examine the impact of motion on the changes in image distortion.
2. METHOD
Given a known distribution of tissue susceptibility, a realistic field inhomogeneity can be calculated 3-7 . Susceptiblity artifacts are caused primarily by the susceptiblity difference across air and tissue interfaces. Hence, to obtain an approximate susceptibility distribution, an air-tissue model needs to be generated. The typical series of EPI images exhibits varying orientations of one image relative to another. To generate a realistic phantom series, the orientation parameters of a real series can be estimated or modeled. The same orientations can then be assigned to the phantom series, with the caveat that the induced field maps should be recaculated if new images corresponding to new positions are to be generated from an initial image.
We start with the numerically calculated field inhomogeneity distribution
D
( )
and an initial distortion-free image with the goal to generate an image with induced image distortion and signal loss. A time series can be created after we apply the orientation parameters to the first distortion-free image, recalculate the field maps, and repeat the same process to simulate distortion and signal loss.
We first examine the image formation process, to develop an approximate method to distort the image analytically. A single shot 2D GE-EPI signal at position z
1
can be expressed as
 x
, k y
, z
1


A

P z
 x y z e n t y m t x
TE ) / T
2
( , ,
1
) e

( x
 yk y
) e i ) e

( , ,
1
)( n t y m t TE ) dxdydz (1) where
 is the gyromagnetic ratio,
 x y z
1 is the proton density at x y z
1
, P z
1
is the slice selection profile, A is a constant. k x
 m
  x
, y
 t y
are the k-space components, m and n are the corresponding sampling indices, , t y
are the sampling time intervals between adjacent k-space points in the x and y direction respectively, TE is the echo time, and  k o
is a constant k-space shift caused by system delay. T2 , the transverse relaxation time constant, is used here instead of T2* since the additional “*” parts of T2* are explicitly included in the last exponential term in Eq. (1). The
 sign means that the corresponding exponential term changes sign for opposite k-space trajectory for odd echos and even echos. To simplify our analysis, we assume there is negligible distortion in the slice selection direction and the slice selection profile is a prefect rectangular function. Thus, P z
1
can be replaced with 1.
The ideal signal would have the form,
 x
, k y
, z
1

= A
  
/
2
( , ,
1
) e

( x
 yk y
) dxdydz , and, as a result, the inverse Fourier transform would get back exactly the proton density weighted by T
2 decay. However, there are other factors in Eq.(1) that perturb the result. Fundamentally, it is the introduced extra phase modulation along with the alternating k-space sampling trajectory (expressed by the

sign) that cause ghosting, distortions, and signal loss artifacts in EPI images.
The alternating sign changes in the exponential term from odd echos to even echos are reflected in e
  x
/ T
2
( , ,
1
) , e i ) , e
 
( , ,
1
)( x
)
, which cause amplitude and phase discontinuities in the phase encoding direction 8,9 . This gives rise to the well known “N-over-2” ghosting artifacts in image space. Normally these artifacts can be reduced or minimized using reference scans, and thus will not be simulated in our EPI simulation. Therefore our initial distortion-free image can be simulated using the following equation.

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
 x
, k y
, z
1


A
  x y z e n t y
TE ) / T
2
( , ,
1
) e

( x
 yk y
) dxdydz (2)
If we have as input a distribution of proton-density and T2 for each tissue, we can generate signals using Eq. (2). After taking the Fourier transform, we can produce a pure T2 -weighted distortion-free image. We can simulate distorted EPI images by running a complete simulation using Eq. (1). This approach is, however, relatively time consuming for fMRI phantom generation, where large numbers of volumes need to be generated for each time series. We thus run this simulation only once, while generating the rest of images based on this initial undistorted image using the following derived analytical approximation.
Ignoring T
2 decay, we will have the signal in a form as follows,
 x
, k y
, z
1


A
  x y z e

( x
 yk y
) e i

( , ,
1
)
 y e
In Eq. (3),
  y
is a phase shift at position x y z
1 i

( , ,
1
) dxdydz (3)
that is proportional to the inhomogeneity 
( , , )
1
at that point, which, after the inverse Fourier transform of the signal, results in a shift in the phase encoding direction, y , in image space. The term,
 
B x y z TE
1
, which is absent in SE-EPI image, causes an echo shift for each voxel 10 and is the source of intravoxel dephasing. It is not a problem when the image formation is in continual space, where each reconstructed spin will be modulated with an extra position dependent phase after the Fourier transform is taken. The amplitude of the transform image will get the true image back exactly. In the discrete case, however, the image intensity for each voxel is actually the result of a weighted average of the complex image over the voxel, or, more precisely, a convolution between the complex image and a non-delta point-spread function. Since 
B varies with position, spins in different positions within a voxel experience different fields, and, as a result they accumulate different phases during kspace sampling. The convolution will cause an in-plane intravoxel dephasing, which is reflected in the image as signal loss (meaning signal diminution). Because we are dealing with 2D multislice GE-EPI image, the measured image intensity still remains in the k-space with respect to the third dimension. However, because the image intensity is a result of an integral over slice selection direction,
 
B x y z TE
1
also causes phase dispersion in the z direction. The signal loss results from the intravoxel dephasing in all three directions, but the z direction dephasing plays a major role because of the larger voxel dimension compared with the other two dimensions 11,12 .
Taking the Fourier transform of the signal, the reconstructed image intensity at x y z
1 1 1 can be represented as


1
,
1
,
1
 
C

 
1
 x
, y
, z
1


 x
, y
, z
1

, x y
1
,
1
 e

 x
 y k y
 dk dk x y
(4)

C
 z z
1
1

 v z v z
2
2
 
(
1
, , ) ' where
J y
1
 y
1
 
( ,
1
, )FOV y
  t y
B x y z y
( , , )FOV
1 1 1 y
  t y
/ 2

/ 2

represent the shifted voxel position in phase encoding direction,
is the Jacobian that expresses the change intensity that accompanies the stretching or shrinking of the image,

B y
is the gradient of

B along the phase encoding direction, FOV y
is the field of view,
, y
, z
are voxel size, and C is a constant.

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
If we assume that 
B is varies linearly within a voxel, then 
B at voxel x y z
1 1 1
can be represented as
  
B x y z o
( ,
1
, )
1 1
 
B x x
 x
1
)
 
B y y
 y
1
)
 
B z z
( '
 z
1
)
(5) where 
B x y z o
( , , )
1 1 1
is the constant value at x y z
1 1 1
, 
B x
,

B y
, and 
B z
are the gradients along the three respective directions.
If a rectangular truncation function is assumed in k-space, the point spread function in image space will have the form of a sinc function. If we neglect contributions to a voxel’s intensity arising from neighboring voxels, we find that i b

1
,
1
,
1
 
D v x

1 v y

1 v z

1 z
1 z
1

 v v z
2 z x x
1
1

 v v x
2 y y
1
1

 v y v y
2
2
  i

(

B o
(
1
, ,
1
)

B x
( '
 x
1
)

B y
( '
 y
1
)

B z
( '
 z
1
)) TE 
J

1
 x
 x
1
) / v x
 y
 y
1
) / v dx dy dz ' (6) where D is a constant. If the further assumptions are made that
 have i b

1
,
1
,
1
 
D v x

1 v y

1 v z

1 
( ,
1
, )
and J are each constant within a voxel, we

B o
(
1
, , ) 
 x
1 x
1

 v v x
2
2 x e
 y y
1
1

 v y v y
2
2 e

B x
( '

1
)

B y
( '

) 
 x
 x
1
) / v dx '
 y
 y
1
) / v dy '

(7)
 z
1 z
1

 v v z
2
2 z e

B z
( '

1
) dz '
We define f
  x
1 x
1

 v x v x
2
2 e g
  y
1 y
1

 v v
2
2 y y e

B x
( '

)

B y
( '

) 
 x
 x
1
) / v dx x
) ' y
 y
1
) / v dy y
) ' (8) h
  z
1 z
1

 v z v z
2
2 e

B z
( '

) dz ' in Eq. (8) as “dephasing factors”, because they are intensity factors that account for the effect of image intravoxel dephasing. The factors , are in-plane dephasing factors, and h is the through-plane dephasing factor. All three of

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006) them are real numbers in the range from 0 to 1. Given the field inhomogeneity at the center of each voxel, the pixel shift, the Jacobian, and all three dephasing factors can be calculated. The factor e

B o
(
1
,
1
,
1
)
can be removed by taking the magnitude. Thus, we have for the magnitude image, i b

1
,
1
,
1
 
D

1

1 v x v y v z

1

( ,
1
, )

1 x y z J fgh (9)
We can use Eq. (9) to generate EPI images with simulated distortions and signal loss included. Rather than simulating distorted EP images using a complete MR simulation from the former description, which is very time consuming, we use the approximate analytic expressions described in Eq. (9) to calculate the geometric distortion ( intensity change due to stretching or shrinking ( J y
1
relative to y
1
), the

1
), and signal loss caused by intravoxel dephasing for each voxel given its field inhomogeneity ( fgh ).
For a healthy volunteer subject, MR scans were acquired on a Philips Intera Achieva 3.0 T scanner. A 3D T
1
weighted anatomical image with matrix size of 256

256

170 and voxel size 1x1x1mm 3 was obtained. A second volume was acquired using slice selection. We call this volume the “2D anatomical image”. The 2D anatomical image, which is also
T
1
weighted, has 256

256

28 reconstruction resolution, voxel size 0.9375mm

0.9375mm

4.5mm, and a 0.4mm slice gap. A measured field map was obtained based on the well known techniques for mapping the B0-field described in 13,14 .
A single-shot GE-EPI sequence was then performed 127 times for an fMRI time series with 28 slices and 80

78 scan resolution (interpolated to 128

128), 1.875mm

1.875mm

4.5mm
voxel size with a 0.4mm gap. Presumably, the 2D anatomical image has the same orientation and location as the first EPI images. We then align the 3D anatomical image with the 2D anatomical image.
Two different brain models are then constructed based on the aligned 3D anatomical image. Firstly, it is segmented into
CSF, white matter, gray matter, etc. to provide a voxel-based brain model. We then take the proton density and T
2
values for these tissues as inputs and run a complete simulation to generate a T
2
weighted distortion free image. Secondly, the aligned the 3D anatomical image is segmented into two components air and tissue. This second segmentation when provided with the susceptibility of the components represents an approximate susceptibility distribution of the human head. We treat the susceptibility of air as zero and the susceptibility of all tissue, including bone, as equal to that of water, which is 9.05e-6. From that distribution we calculate the field inhomogeneity map

B using a previously published numerical method 3 . The calculated field map is used in Eq. (9).
In order to show the necessity of field map recalculation when the head is in different orientations, we calculated two different field maps with the head of the subject in differing orientations. For comparison, we also simply rotated the field map from orientation 1 to orientation 2.
3. RESULTS
Figure 1 shows a calculated field map comparision between a rotated field map and the field map of an equivalently rotated human head. The maps are not the same, as shown by the absolute difference shown in Figure 1(c). This result demonstrates that it is not correct simply to rotate and translate the field map of one image and use it to correct other rotated and translated images. The model distribution of air and tissue used for the field map calculation is shown in
Figure 2. The distribution can be constructed by segmenting the MR images using a simple threshold. To assess the capability and feasibility of our field map calculation, we show comparisons between a numerically calculated field map and an experimentally measured one for the same volunteer in Figure 3. It shows that the field inhomogeneities at the air-tissue interfaces in the calculated map field agree well with those in the measured field map. A simulated distortionfree EP image, a simulated distorted EP image, and a real EPI volume are shown in Figure 4. We can observe similar image distortion and signal loss around sinuses in the simulated images compared with the real EP images. All calculations and measurements are based on the same subject.

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
Figure 1. Calculated field map comparision between rotated field map (a) and the field map of rotated human head (b). The absolute difference image, |(a) − (b)|, is shown in
(c). The difference between (a) and (b) shows that it is not correct simply to rotate and translate the field map of one image and apply it to a rotated and translated image.
(a) (b) (c)
Figure 2. A binary image that represents the air-tissue model is used to calculate the field map. Tissue, including bone, is given intensity 1; air is given intenensity 0. Approximate air-versus-tissue distributions can be constructed by segmenting the available MRI data using a simple threshold.
x 10
-6
4
2
0
-2
-4

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
Figure 3. Comparison of calculated map to measured map. First row is calculated. Second row is measured.
Intensity is proportional to magnetic field. Scale bar is in Hz( g times tesla). Noise in empty regions has been masked to a uniform gray value. Similar field inhomogeneities can be observed at the air-tissue interfaces of both calculated field map and measured one.
Figure 4. (a) Simulated distortion-free images using MR simulator, (b) Analytically simulated EP images with distortion and signal loss, (c) Corresponding real EP images from scanner. We can find similar image distortions and signal loss in both the simulated images and real EP images.

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
3. DISCUSSION
Preprocessing techniques are widely used in various fMRI packages to correct for effects unrelated to brain activation such as motion and image distortion. It is necessary to evaluate the accuracy of each technique, and appropriate test cases
(“phantom” studies) are essential for rigorous testing. The computer generated phantom presented here can be used as a gold standard and thereby provides a feasible solution to the problem of an evaluation of these techniques. The phantom can be constructed from real MR images, but simulation needs to be involved considering the complexity of object motion and motion-correlated induced field inhomogeneity artifacts. Thus, it is desired to have a method to simulate real motion and artifacts. In this paper, we create a distortion-free EPI image using a realistic MR simulator, and then we derive an approximate analytic method to simulate realistic susceptibility artifacts. We divide these artifacts into two kinds. One is the geometric distortion in the phase-encoding direction and its associated intensity distortion represented by the Jacobian. The other is the signal loss arising from both in-plane and through-plane intravoxel dephasing. Results from phantom generation experiments indicate the simulation captures the main artifacts observed in real fMRI data including both distortions and signal loss.
The calculated field map and simulated EPI image, however, exhibit some disagreement with the real measured field map and real EPI image in some details. Some of the disagreement is probably due to the fact that we segment the brain tissues only into air and soft tissue. While the difference between these two components is known as the main source of susceptibility-induced field inhomogenity, bone and soft tissue interfaces should also be considered. In this study, we did not model because of the difficulty of separating bone from air in the available MR images. Also we ignore B
0
field inhomogneity from sources other than susceptiblity in our simulation. The real measured field map is expected to include field inhomogeneity from other sources, such as hardware and shimming errors. EPI images also have image distortion caused by time dependent field inhomogeneities like eddy currents, which cannot be measured from GE image-based field measurement or by our field map calculation.
It is observed that signal loss comes mainly from through-plane intravoxel dephasing. It is reported that in-plane local field gradients also contribute to the image artifacts 15 . Hence, it is reasonable to model the intravoxel dephasing in all three directions. In much of the literature, the slice-selection profile is assumed to be a rectangular function, but it is not difficult to model other functions such as the Gaussian function 15 . While a first-order approximation in the slice-selection direction is assumed in this study, it maybe better to use higher order approximation, because the EPI image has a slice thickness as large as 7mm in some cases. We can accommodate the higher order approximations by calculating or measuring field maps at a higher resolution than that of the relatively low resolution EPI volumes. It would be more accurate to employ a linear least squares curve-fitting to calculate a local field gradient for each voxel as in 16 .
In-plane intravoxel dephasing is caused by a convolution between a complex image and a point-spread function. In Eq.
(6), we assume that this convolution covers only one voxel. It is unknown how much accuracy is lost by this assumption.
We are in the process of removing this assumption, and its effect will be evaluated in our future work.
4. CONCLUSION
This work employs an MR simulator to generate an fMRI phantom with the inclusion of both motion and susceptibility artifacts. Two different brain models are constructed to assist field map calculation and image simulation. A numerical method is employed to calculate field inhomogeneities due to susceptibility differences. Simulated distorted EP images are generated with the assistance of MR simulation and a fast analytic method. The distortions induced by susceptibility are modeled in this way to generate a more realistic fMRI phantom than those previously published.
Our preliminary results demonstrate that the calculated field matches the experimentally measured one quite well. The simulated EP image also shows similarity to the real EP image in terms of susceptibility induced distortion and signal loss. These computer generated phantoms will provide a test bed for pre-processing algorithms that are designed to compensate for motion and susceptibility artifacts.

Preprint of paper to appear in Proc. SPIE Medical Imaging 2006 San Diego, CA, (Feb 2006)
ACKNOWLEDGEMENTS
The authors wish to acknowledge support from NIH grant R01-NS46077, the Vanderbilt Intramural Discovery Grants
Program and the Vanderbilt University Institute of Imaging Science.
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