COMPARISON OF SINGLE SHOT METHODS FOR R2* ESTIMATION
HRISHIKESH DESHPANDE
BIOMEDICAL ENGINEERING
ABSTRACT
In functional MRI (fMRI), typically a series of images are acquired in rapid succession in control and stimulation cycles to detect functional changes in regions of the
brain. These changes are signified by changes in tissue R2*, caused by the Blood Oxygenation Level Dependent (BOLD) mechanism. In most fMRI experiments R2* changes
are not directly measured, but are inferred from signal changes from a series of R2*weighted EPI images.
A potentially more accurate method of measuring changes in R2* is the multiplegradient echo EPI method (MEPI), which acquires separate EPI images at several echo
times following a single excitation (1). A recently introduced technique for R2* measurement, single-shot parameter assessment from signal encoding (SS-PARSE) maps R2*,
frequency and local magnetization M0 by modeling the local signal and solving an inverse problem.
Because it explicitly models local phase evolution during the signal, it is inherently free of off-resonance geometric distortion. The sensitivity and accuracy of the activation maps generated from fMRI block design experiment depend critically on the ability
to measure the small changes in R2* representing differences between control and stimulation states. The changes in R2* between the control and stimulation states need to be
consistent to get meaningful activation maps. To compare the temporal variability of R2*
0
estimates for MEPI and SS-PARSE, studies of repeatability were performed using phantoms with a range of R2* values typically found in human brain tissues.
After analysis, the performance of SS-PARSE was found to be consistently superior to MEPI in all the experiments. Within the SS-PARSE modality itself, the sequence
with lower variation in R2* estimation was deemed to be the most advantageous in fMRI
experiments.
1
INTRODUCTION
Magnetic resonance imaging (MRI) was first employed for non invasive imaging
of the human body in 1977 (1). Initially the focus of MRI was on scanning the anatomy.
Later, with the discovery of the blood oxygenation level dependent (BOLD) effect, MRI
became a tool for observing the areas of the brain that are active while performing a specific task (2). This branch of MRI studying the functional changes brain came to be
known as functional MRI (fMRI).
The BOLD effect is a function of tissue relaxation rate R2* (3). In order to correctly assess the brain activity in functional maps, it is sufficient to detect relative changes in R2*. For more quantitative functional mapping, an accurate and reliable method for
measurement of R2* would be useful. A recently developed technique called the multiple gradient echo-echo planar imaging (MEPI) measures R2* from signal changes over
several echoes following a single excitation pulse (4). However, since this technique
computes each individual image neglecting phase changes during the signal, it is sensitive to off-resonance frequencies within the object, possibly leading to severe geometric
distortion. (5).
A new technique, single-shot parameter assessment by retrieval from signal encoding (SS-PARSE), solves an inverse problem to estimate R2* parameter values directly
from the signal instead of detecting serial signal changes measuring it by fitting to the
decay over the course of serial frames (6). It uses a reconstruction algorithm that employs
2
a parametric signal model and estimates the encoded parameters using an iterative estimation process. The primary parameter of interest in our case is the tissue relaxation rate
R2*.
In fMRI studies using a block design method, single shot images are acquired during control and stimulus periods. The activation maps are computed from a series of control and stimulus images (7). If these images are consistent we get accurate activation
maps. If not, the accuracy of these maps can be quite questionable.
In our experiments we compare the temporal performance of R2* estimation for
the MEPI and SS-PARSE techniques using gradient echo multiple shot (GEMS) technique as the gold standard. We also tested the behavior of the MEPI and SS-PARSE under off-resonance conditions to examine the robustness of these two single shot methods,
and investigate the behavior of SS-PARSE technique at different gradient strengths in
order to determine what would be the optimal range to use in fMRI experiments.
This thesis is divided into five chapters. After this introduction, chapter 2 gives a
brief theoretical background about the techniques we use in our comparison study. In
chapter 3 we discuss the experimental design setup and the data acquisition techniques.
Results and statistical analysis from our experiments are presented and discussed in chapter 4. The results and ways to improve our results are discussed in this chapter. Chapter 5
includes the summary and conclusions drawn from this work.
3
1. BACKGROUND
1.1. Image acquisition using MRI
MRI is an imaging modality which uses signal obtained through the nuclear magnetic resonance (NMR) phenomenon to construct images. In the field of medicine, we are
mostly interested in looking at the NMR signal from water and fat, the major hydrogen
containing components of the human body.
When placed in a magnetic field of strength B0, a proton with a net spin can absorb a photon, of frequency υ. This frequency is a function of the gyromagnetic ratio (γ)
and the external field and is known as the Larmor frequency (8).
υ = γ B0
(1)
If an external field such as a radiofrequency (RF) pulse at Larmor frequency is introduced into the system, it excites the spin systems causing them to precess around their
axis and eventually return to their state of equilibrium. While returning to their state of
equilibrium, these proton spins emit energy, which is detected as signal during experiments (8).
The net magnetization observed during the process can be considered to consist of
two components, the longitudinal component Mz and the transverse component Mxy. The
rate at which Mxy decays to zero is the spin-spin relaxation time and is described by the
time constant T2 as:
M xy (t )  M xy (0)e t / T2
4
(2)
The rate at which Mz to returns to a steady state magnetization M0 is described by
time constant T1 and can be described as:
M z  M 0 (1  e t / T1 )
(3)
The relaxation time constants for magnetization along the transverse and longitudinal component can be considered to be an exponential decay. The transverse magnetization is a combination of two factors one of which is the molecular interactions leading
to the pure T2. Other factor that contributes to this decay is the variation in the field B0,
which is called impure T2 or T2’. The combined effect of pure and impure T2 is what results in the decay of transverse component of magnetization, and is the time constant T2*.
The relationship between the time constants responsible for transverse magnetization is
given as:
1 / T2  1 / T2  1 / T2'
*
(4)
During an experiment the signal obtained by the MRI system is in the spatialfrequency domain. The digitized data in the spatial-frequency domain is considered to be
in the k-space. This data is encoded in the frequency domain and is best represented by a
Fourier transformed data matrix. Through the application of an inverse Fourier transform
(IFT) the signal in frequency domain can be reconstructed into an image Im in spatial
domain.
Im ( x , y )   S ( k (t ))e 2i ( k ( t ) x  k ( t ) y ) dk 2
k (t )    G (t ' )dt '
5
(5)
(6)
A set of linear gradients in x, y, and z planes are used to modify the local signal to
indicate its location once the RF pulse has been applied. These linear gradients G determine the path of k(t) by what is termed as a k-trajectory.
These gradients can be used in a number of ways. They can be used to define a
plane of acquisition. Rapid imaging in which all data are acquired following a single excitation pulse is called the single shot method (9). The other method is the multiple shot
imaging method where several shots (rf pulses) are required in order to gather sufficient
data for an image. (9). Single shot techniques are much faster than multiple shot techniques.
1.2. BOLD effect
Figure 1.
Mechanism of BOLD effect.
For several years, researchers have attempted to devise better methods to understand the workings of the human brain. The last few decades saw techniques such as electroencephalography (EEG), magnetoencephalography (MEG) and position emission tomography (PET) gain momentum in their use in the study of brain function. However,
6
most of these methods lacked either the spatial or temporal resolution that was required to
understand the brain functions more accurately. With the discovery of BOLD effect, the
use of fMRI in studying the functions of brain gained widespread popularity.
The BOLD mechanism gives an explanation of how a change in stimulus gives
rise to change in acquired signal (2). The series of events from the onset of stimulus until
the final change in the NMR signal is quite intriguing. A typical stimulus consists of
some task that needs to be performed by the subject while in the scanner. Applying
stimulus leads to increase in brain activity in order to accomplish that task. This increased
brain activity is sustained by increased oxygen consumption and increase in blood flow to
the region that is performing the given task. Increased blood flow results in the reduction
of deoxyhemoglobin in that region, reducing the magnetic susceptibility of blood vessels
and the surrounding tissues. Reduction in magnetic susceptibility leads to reduction in the
relaxation rate R2* of the brain tissues (8).
Thus the changes in relaxation rate R2* are associated with brain activity. In order
to study the brain functions by means of BOLD fMRI, it is essential to obtain images that
have information pertaining to the R2* changes, i.e. the images need to R2* weighted.
1.2.1.
Significance of R2* in fMRI
fMRI based neuroimaging techniques allow us to detect the brain areas which are
involved in a task, a process or an emotion. A single shot technique is used to sequentially acquire brain images every few seconds over several minutes. fMRI cannot detect absolute activity of brain regions. It can only detect difference of brain activity between
7
several conditions. During the fMRI image acquisitions, the patient or subjects are asked
to alternately perform several tasks or a stimulus is provided to trigger several processes
or emotions (7). Each of these conditions is repeated several times and can be separated
by rest periods.
Since we are measuring the brain activity by computing signal changes, any signal change whose source is other than a brain task would lead to inaccuracy in the final
activation maps. Since the signal changes are sensitive to R2* changes, it follows that any
errors in R2* weighting would be lead to errors in the activation maps. It is thus critical to
have reliable means of R2* weighting and measurement.
1.2.2. R2* measurements
Accuracy of quantitative fMRI studies depends on the accuracy of R2* measurements. R2* can be measured using both the single shot and multiple shot techniques (11).
Even though the conventional multiple shot methods are used less frequently in fMRI
studies, they can serve as a valuable tool by serving as a gold standard in measurement of
some of the parameters of interest in fMRI.
1.3. Methods of R2* measurement
1.3.1.
Gradient echo multiple shot (GEMS)
The GEMS sequence, as the name suggests, is a multiple shot imaging method. In
this method each line in the k-space is scanned sequentially after excitation by a rf pulse.
8
This method relies on the formation of an echo by switching the linear gradients (8). The
GEMS technique allows us to take multiple acquisitions with a set of closely spaced echo
times. The signal at different echo times is weighted with R2* relaxation parameter making it possible to estimate it as an unknown from the equation:
S  S 0 e  R2 t
*
,
(7)
where S is the signal at individual echo time t. Gradient echo multiple shot images have a
significantly better signal to noise ratio (SNR) than the single shot methods.
1.3.2. MEPI
The encoding scheme employed in MEPI is quite similar to the one used in
GEMS sequence. The main difference between the two is that GEMS is a multiple shot
method, while MEPI is a single shot method.
In MEPI each line in the k-space is scanned rapidly following excitation by a single rf pulse. The sampling takes place over four echoes. These four echoes have similar
characteristics as the echoes obtained with GEMS and R2* can be calculated by monoexponentially fitting the signal using equation 7.
Measurement of M0, R2* and frequency f in a single shot poses a challenge for the
conventional single shot MRI techniques since it requires traversing the entire k-space
within a single echo time TE to obtain a complete image (8). Measuring f requires very
closely spaced TE values to accurately measure frequencies within reasonably wide
bandwidth. On the other hand accurate measurement of net relaxation rate requires widely spaced TE values (12).
9
Figure 2.
k,t-space plot of MEPI pulse sequence over four echoes (Donald Twieg,
Southeastern Magnetic Resonance Conference 2008)
Since the MEPI sequence fails to keep track of changes in phase within each individual EPI scan, it is impossible to obtain reliable parameter maps in the presence of
strong background gradients and off-resonance due to geometric distortions and intensity
distortions.
1.3.3. SS-PARSE
The signal model used in GEMS and MEPI is a static model described as:
s (t )   M ( x)e 2ik (t ) x dx
10
(8)
It assumes that no signal decay or phase evolution occurs during the acquisition
time of the imaging data. This assumption gives rise to numerous image errors, like geometric distortion due to field inhomogeneities or off-resonance frequencies (8), N/2
ghosting and filtering effects due to differential R2* weighting of spatial frequency components (13).
SS-PARSE addresses these issues by using a continuous signal model which
tracks the distribution M0 as it evolves across the k-space (6). This is accomplished by
spatially encoding the signal as a function of linear gradient waveforms and parallel receiver coils, which gives relative amplitude for each location along with phase changes
over time. The continuous signal model can be expressed as:
s (t )   M 0 ( x)e [ R2 ( x ) i 2f ( x )]t e 2ik ( t ) x dx
Sd (n)   x  y M 0 ( x, y)e( R2 ( x, y )i 2f ( x, y ))nt e2ik n x dx
*
(9)
(10)
The discretized parametric model can be assumed to lie along the x, y grid (6).
Using this continuous model, it is possible to encode magnetization and relaxation
rate, as well as frequency. By doing so, the model overcomes the limitations posed by
background gradients and off-resonance. This makes the signal acquired and parameters
estimated using SS-PARSE immune to geometric distortion or ghosting artifacts.
The discretized signal model forms an inverse problem in which we have to estimate the parameters from spatio-temporally encoded signal. Each data sample is the sum
of the deterministic signal and measurement noise, usually assumed zero-mean and independently distributed.
11
The solution is a set of unique parameter values found iteratively to match the actual observed signal. In our case, we design a progressive length conjugate gradient
(PLCG) search which iteratively modifies the parameter values that need to be estimated
(6). The iteration is performed over incremental lengths of signal data with a goal of minimizing the least square residual below a desired tolerance level (14). The algorithm iteratively modifies the unknown parameter values M0, R2* and f, in order to minimize the
least-square residual below a desired tolerance level.
1.4. Gradient tables and k-trajectory
The k-trajectory required for acquiring SS-PARSE data is very different from the
conventional Cartesian trajectories used in MEPI and GEMS. SS-PARSE makes use of a
rosette trajectory which samples the center of k-space multiple times as it traverses the
sampling space (6). Sampling the center of k-space multiple times gives us multiple samples along the time axis from which to estimate the frequency of the local signal.
The trajectories result from continually varying gradients that are orthogonal to
each other (15). We have to know the system parameters and make some assumptions
before we can proceed with the development of gradient waveforms. The k-trajectory for
SS-PARSE was computed within a k-space radius of kf=2.82 cm-1. A 2.82 cm-1 k-disc
with this radius has the same area as a square 5 cm-1 x 5 cm-1 Cartesian grid, with spatial
resolution comparable to 64x64 single-shot MEPI for a 12.8 cm field of view (FOV).
These gradient waveforms were generated using:
𝑘𝑓 =
12
𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
2∗𝐹𝑂𝑉
(5)
𝑘𝑟 = 1.1284 ∗ 𝑘𝑓
𝜔1 = 𝛾.
(6)
𝐺𝑚𝑎𝑥
(7)
𝑘𝑟
41
𝜔2 = (𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛) . 𝜔1
(8)
1
∆= 𝛾.𝐺
(9)
𝑚𝑎𝑥 .𝐹𝑂𝑉
𝑁𝑠𝑎𝑚𝑝𝑙𝑒𝑠 =
𝑛 = 1:
𝑡𝑎𝑐𝑞
(10)
∆
𝑁𝑠𝑎𝑚𝑝𝑙𝑒𝑠
(11)
∆
Using these, we get the complex gradient waveform values,
k r (((1   2 )(cos(1   2 ).n 
G s (1 : n) 
((1   2 )(cos(1   2 ).n 

2

2
)  (i. sin(1   2 ).n 
)  (i. sin(1   2 ).n 
2

2

2
))  ...
)))
(12)
These gradient tables require further resampling in order to conform to the specific dynamic range of the digital to analog converter of the system.
Since actual k-sampling locations depart from nominal values in system-specific
fashion, the effective k, t locations of the signal samples were determined experimentally
by the method of Zhang et al. (16). We call this the calibration process. Along with the
values of kx and ky at each time, calibration also determines the phase, Φ(t), a timevarying spatially constant field offset attributable largely to eddy currents (16).
1.5. Selection of maximum gradient amplitudes
One of the parameters that is used in the development of gradient waveforms is
the maximum gradient amplitude. The choice of gradient amplitude affects a number of
13
parameters, especially the sampling density. Using lower gradient strength yields fewer
sample data points. It also results in the k-trajectory crossing the origin less frequently.
The effect of number of data points is also visible in solving the inverse problem
used to generate the parameter maps. A higher number of data points makes it feasible for
the PLCG algorithm to search for solutions on longer data lengths. This enables the algorithm to converge more accurately on the x, y spatial grid. One would come to a logical
conclusion that using larger gradient strength waveforms for data acquisition would result
in parameter maps that are more accurate than the ones obtained by using lower gradient
strength waveforms. However, using higher gradient strengths can only improve the results to a certain point after which the performance begins to deteriorate.
The bandwidth (BW) of a system is a measure of the range between the highest
and lowest frequency of the signal that could be acquired. For analog signals, BW is the
frequency range in which the signal's Fourier transform is nonzero (17). The receiver BW
is the range of frequencies accepted by the receiver to sample the acquisition data. It depends upon the strength of gradients that are being used and the sampling rate of the system.
BW  SamplingRate
(13)
Since Nsamples is directly proportional to gradient strength, we can see that higher
gradient strengths result in wider receiver BW. Signal is proportional to the number of
samples acquired during the fixed time of acquisition, i.e., signal is proportional to bandwidth, while the standard deviation of the noise is proportional to the square root of the
bandwidth. Thus the SNR of the system is proportional to the square root of the bandwidth:
14
SNR  BW
(14)
From equations 9, 13 and 14 we can see that if the gradient amplitude is at its
maximum throughout the signal readout, the SNR of the system is proportional to the
square root of the gradient strength being used.
SNR  G max
(15)
We are now faced with a choice between better parameter estimation by selecting

larger gradient strengths which will give us longer data lengths, and having to deal with
noise as a result of higher gradient amplitudes. The analytical selection of an optimal
range is difficult to obtain y since it would require finding solution to a large non-linear
problem. This problem can be addressed empirically by evaluating the performance over
a range of gradient amplitudes in order to determine the range over which we can have
good spatial resolution and parameter estimation as well as have a good ability to handle
noise.
1.6. Comparison between the single shot methods
Activation maps in fMRI studies rely upon accuracy and stability of R2*. In order
to be proven useful, a single shot method must be capable of producing consistent and
accurate R2* maps.
In SS-PARSE we need to investigate the accuracy and consistency over a wide
range of gradient strength in order to determine which one would give the best trade-off
between SNR and data length.
15
The performance of the single shot methods, MEPI and SS-PARSE, is done by
using GEMS as the gold standard. GEMS is a multiple shot method. This allows us to use
a small acquisition BW thus giving us high SNR. The ability to obtain images with high
SNR with closely stacked TE values in GEMS, justifies its use as the gold standard.
Taking GEMS as a reference, we investigate the temporal standard deviation
TSD) of R2* in MEPI and SS-PARSE over a set of acquisitions. We also look at the accuracy of these R2* maps by comparing them with GEMS. In order to be useful of fMRI
studies, these methods must demonstrate their usability over the range of typical R2* values found in human brain tissues. Thus we need to examine the performance of these two
methods over a range of R2* values that can be typically found in human brain.
Achieving best possible shims is not possible in every human fMRI experiment
because of time constraints, and even then, the best possible shimming job typically
leaves significant field inhomogeneities within the imaged region, simply because of the
complex patterns of susceptibility gradients within the tissues. This means that there is
bound to be some level of frequency off-resonance in every fMRI study. Sometimes the
off-resonance can be large enough to cause serious deterioration in acquired data. This
makes it necessary to determine how the chosen single shot methods fare under severe
off-resonance conditions.
16
2. METHODS
The comparison study was performed on phantoms using MEPI and SS-PARSE,
using GEMS as the gold standard. Gradient waveforms were generated to get rosette trajectories which were then used for SS-PARSE trajectory calibration and data acquisition.
Experimental phantoms were constructed to give R2* ranges that would be comparable to
those found in brain tissues. Data was collected using GEMS, MEPI and SS-PARSE sequences on a Varian 4.7 T vertical bore scanner. Data analysis was performed remotely
on a cluster that utilized 16 AMD64-K8 processors with 64 GB of RAM (@Xi Computer
Corp.). SS-PARSE reconstruction made use of the PLCG algorithm to estimate the parameter maps. Temporal standard deviation of the relaxation rates, R2*, of SS-PARSE
and MEPI were compared on basis of converged pixels. The accuracy of R2* estimation
on a pixel by pixel basis and over a region of interest was also performed. Behavior of the
single shot methods was observed under severe off-resonance condition to test their robustness. SS-PARSE was analyzed over a wide range of gradient amplitudes to find an
optimal operating range that gave the best trade-off between SNR and parameter estimation.
17
2.1. Gradient waveform generation for SS-PARSE
The k-trajectories for SS-PARSE (see equation 12) were computed within a kspace radius of kf=2.82 cm. A k-disc with a 2.82 cm radius has the same area as a square
5 cm-1 x 5 cm-1 Cartesian grid, with spatial resolution closely comparable to 64x64 single-shot EPI for the same 12.8 cm FOV. Using kf=2.82 cm, FOV=12.8 cm, and resolution of 64x64, gradient waveforms were generated using equation 12.
Figure 3.
10 ms section of a gradient waveform for 2.9 G/cm
The gradient strength used in the case of MEPI is 2.29 G/cm. We investigated the
k-trajectories for SS-PARSE with gradient amplitudes that were lower as well as higher
than the MEPI gradient amplitude. To do so, we generated gradient tables for seven different amplitudes; 1.9 G/cm, 2.29 G/cm, 2.5 G/cm, 2.9 G/cm, 3.2 G/cm and 3.8 G/cm.
These gradient tables were scaled to ±32767 to correspond to the dynamic range of the
digital to analog converter amplifier of the Varian 4.7 T system. The gradient waveforms
18
must start and end at zero amplitude, making it necessary to add a ramp-up section at the
beginning of the gradient waveform which starts at zero and smoothly transitions to starting values of the two orthogonal readout gradients. Similarly we need to add a rampdown section at the end of the gradient waveform which smoothly transitions from the
end of the waveform to zero.
The gradient waveforms are designed for an acquisition period of 69 ms, which
includes 65 ms of the actual gradient waveform with 2 ms of ramp-up and ramp-down
intervals.
2.2. Phantoms
2.2.1. Calibration phantom
The phantom used for calibration experiments is a plastic sphere with an internal
volume of 25 cc. Selecting a phantom of appropriate dimensions is crucial for calibration.
The phantom should be large enough to give a good, noise free signal, while at the same
time it should be small enough to avoid excessive through-slice dephasing of the signal.
The calibration phantom is filled with water. Every precaution is taken to ensure
that there are no air bubbles trapped within the sphere. The presence of air bubbles leads
to severe signal decay due to susceptibility induced intra-voxel dephasing (18). Calibration over a phantom which has susceptibility related artifacts results in poor SNR in the
calibration data. It also leads to incorrect computation of k-trajectory and local phase
changes over time.
19
2.2.2. Fourtube phantom
Figure 4.
Fourtube phantom
Glass vials (5 cm long, 2.1 cm internal diameter) were filled with 1.5% agar gel
and doped with copper sulphate (10 mg/100 ml to 60 mg/100 ml) and Sephadex G-25
beads to give R2* relaxation rates in between 15 sec-1 and 45 sec-1 (19). The glass vials
were suspended in a glass jar filled with water so as to minimize the susceptibility artifacts. The jar has 4 vials suspended in it. The range of R2* relaxation rates is chosen
based upon the R2* values in gray and white matter observed in the human brain at 4.7T
(19).
The lid on top of the jar has holes drilled in four quadrants. Each hole has the
same diameter as the vials. This makes it possible to switch the vials with difference relaxation rates for different studies.
20
2.3. Calibration
The behavior of gradient waveforms during acquisition is dependent upon the system parameters during the time of acquisition. If hardware settings remain the same, the
gradient waveform continues to behave the same way on a particular system. Since actual
k-sampling locations depart from nominal values in system-specific fashion, the effective
k, t locations of the signal samples were determined experimentally by the method of
Zhang et al. (16). Along with the values of kx and ky at each time, this procedure also
determines the phase, Φ(t), a time-varying spatially constant field offset attributable
largely to eddy currents.
Figure 5.
Calibrated k-trajectories for four different gradient amplitudes.
21
All the gradient waveforms are calibrated to get the rosette trajectories and their
corresponding time varying phase information. Figure 6 shows the difference in sampling
density for at different gradient amplitudes when sampling over the same area.
Figure 6.
Coverage of 3.8 G/cm gradient amplitude k-trajectory over time.
Once we have the calibrated k-trajectories and the phase information, we can use
these trajectories to acquire experimental data. These trajectories and phases can be assumed to be stable as long as no changes are made to the system hardware.
2.4. Phantom data acquisition
Experimental data was collected on a 4.7 T 60 cm-vertical-bore Varian primate
MRI system (Varian Inc., Palo Alto, CA) using a stripline resonator quadrature head coil
(Insight Neuroimaging, Worcester, MA).
22
Several sets of phantom data are collected using three different pulse sequences.
The GEMS acquisitions consisted of a series of images 128x128 voxels, 12.8 cm FOV,
3mm thickness, repetition time (TR) 200 ms, and 16 echo times (TE): 5, 8, 10, 12, 15, 20,
25, 30, 35, 40, 45, 50, 55, 60, 65, 70 ms. MEPI acquisitions were acquired at 4 different
TE; 22.3 ms, 66.8 ms, 96.4 ms and 124.2 ms, following a single RF pulse. They had 50
repetitions of 64x64 voxel images, 12.8 cm FOV, 3mm slice thickness, and TR of 5 sec.
SS-PARSE acquisitions consisted of 50 repetitions, 12.8 cm FOV, 3 mm slice thickness,
TR of 5 sec, and pulse sequences with maximum gradient amplitude of 1.9 G/cm,
2.29G/cm, 2.5 G/cm, 2.9 G/cm, 3.2 G/cm, 3.5 G/cm and 3.8 G/cm.
2.5. Data Analysis
Data analysis was done in Matlab (Version 7.5, The Mathworks Inc., Natick,
MA). SS-PARSE parameter maps were estimated using the PLCG algorithm. Some of
the parameters that had to be set were:
1. Start (startx) and end (endx) points of the trajectory: These points are chosen by
observing the k-trajectory and noting the positions of the first and last echoes.
2. Swoop length (N1): The number of samples between two consecutive echoes is
the swoop length of that particular trajectory.
3. Data lengths (NLIST): Data lengths are integral multiples of swoops that are progressively incremented while running the PLCG. The data lengths have to set empirically for data sets at different gradient amplitudes.
23
4. Tolerances (FLIST): Each data length has to be associated with a tolerance level.
The tolerance levels set a limit on the minimum accuracy of estimation required
for a particular length of data before the length is incremented. The choice of
these parameters is subject to the gradient amplitude being currently used. Tolerance values have to be set empirically.
5. Initial frequency estimate (offr): PLCG is capable of estimating frequencies over
a wide range. However in practical settings there can be a huge frequency drifts
across the slice arising due to number of reasons. For this reason it is necessary to
specify an initial frequency which might help in convergence at more points in the
x, y grid. Even this initial frequency estimate has to be determined empirically.
6. Scaling (ffac): In order to obtain correctly determine the values of M0, R2* and f,
it is sometimes necessary to scale the FID signal by a factor. For a given object
size in a FOV, the scaling factor usually remains the same.
24
2.6. Parameter evaluation
2.6.1. R2* calculation in GEMS
Figure 7.
GEMS images at 16 different TEs.
25
GEMS images are obtained at 16 different TEs. Figure 9 shows the GEMS images
obtained from a fourtube phantom at echo times between 5ms and 70ms. The R2* values
that need to be computed lie in the region within the glass vials. In order to compute the
R2* values, we select the ROI as shown in by the green circles in Figure 9. The signal
within this ROI is monoexponentially fitted using Equation 2, with TEs used as t. By applying a non-linear fit to the signal intensities corresponding to their TEs, we obtain the
R2* maps over those FOVs.
This gives us a pixel-by-pixel map of R2*. By averaging over all the pixels within
the ROI, we get average R2* over the entire region. These R2* values are used as the
gold standard in comparison of single shot methods.
2.6.2. R2* calculation in MEPI
Figure 8.
MEPI images at four TEs.
Using the single shot MEPI sequence to obtain R2* weighted images provides
much less flexibility than in GEMS. In MEPI, we don’t have fine control over TEs like
we had for GEMS. Images can only be acquired at discrete times, which are odd multiples of the first TE (see Figure 11). In order to obtain four echoes in MEPI, we have to
26
acquire the signal over a period of 150 ms. If the system is properly shimmed and if the
R2* values aren’t too high, we can get reasonably good data until the fourth echo. However if the system is poorly shimmed or if the phantom has large R2* values, the signal
acquired at last TE decays to amplitudes that are comparable to the amplitude of system
noise. Typical SNR values observed at the four echoes for R2* of 42 sec-1 were 39 dB,
32 dB, 28 dB and 25 dB under good shimming conditions.
Figure 9.
MEPI signal sampling at specific echo times.
The method for computing R2* in MEPI is the same as the one that has been described for GEMS. The difference here is that the R2* is computed for each of the repetitions over the series of 50 repetitions. We proceed to find the average R2* over the ROI
over the entire series as well as the pixel-by-pixel map of R2* averaged over the time series, but not over the ROI. Finally we find the TSD of R2* on a pixel-by-pixel basis over
the ROI.
27
2.6.3. Analysis of SS-PARSE parameter maps
Figure 10.
R2* estimated from SS-PARSE (Gmax=2.9 G/cm) over 50 repetitions
Parameter estimation in SS-PARSE is accomplished using the PLCG algorithm.
The parameters estimated from the SS-PARSE experimental data are M0, R2* and f. Just
like MEPI, the estimation process has to be carried out over all the 50 repetitions. Once
we have the parameter maps, we need to compute separate out the R2* values over the
ROI over 50 repetitions. This process was carried out for data sets from all seven gradient
amplitudes.
28
2.7. Comparison
2.7.1. Comparison of R2* estimates
Once we have the pixel-by-pixel and averaged R2* maps over the ROI, we compare the accuracy of R2* obtained from MEPI and SS-PARSE using the R2* values from
GEMS as the reference. By arranging the R2* estimates in ascending order of GEMS
R2* estimates, we are able to compare the relative differences between the two single
shot methods as well as their accuracy in absolute terms.
Images are reconstructed from the experimental data sets. All the images are
scaled to 64x64 pixel resolution in order to facilitate an one on one comparison. We selected a ROI with a 5 pixel radius that lay completely within the area covered by the
tubes. This gave us 80 pixels for each ROI. Pixel-by-pixel R2* calculations are performed for SS-PARSE, GEMS and MEPI. From the 18 studies we had a total of 72 different values for R2*. Among these we selected 20 ROIs which had R2* values in the
range of 15 sec-1 to 45 sec-1.
The performance of SS-PARSE at different gradient strengths was compared with
MEPI using F-test for each pixel contained within the ROIs. This statistical method can
help us in determining if the distributions in SS-PARSE and MEPI have equal variances
(20). The null hypothesis for the F-test was that there was no difference in the variances
of R2* distributions obtained using the two methods at a 95% confidence interval.
The rejection of null hypothesis at a given pixel indicates that SS-PARSE estimations at that pixel location are better compared to that obtained using MEPI. This test was
repeated for all the gradient strengths used in the study.
29
Next, in order to compare the accuracy of the two single shot methods, we compared R2* values for each pixel from MEPI and SS-PARSE with that in GEMS. To test
for accuracy we used the ratio:
| R2* MEPI  R2*GEMS |
| R2* SSPARSE  R2*GEMS |
(16)
This ratio was computed for each pixel in the selected ROIs. A ratio of more than
1 implies that estimation of R2* by SS-PARSE at that pixel was more accurate compared
to the estimation of R2* done using MEPI. The percentage of pixels for which this ratio
was more than one was calculated for the each of the gradient amplitudes used in SSPARSE acquisitions.
2.7.2. Comparison between the TSDs of single shot methods
The TSDs computed from the R2* maps in SS-PARSE and MEPI are used for
comparing the temporal behavior of these two methods. We compare the temporal performance of SS-PARSE and MEPI on a pixel-by-pixel basis, using the pixel values of
GEMS R2* as a reference. The TSDs from the two single shot methods are compared on
an ascending R2* scale from GEMS.
To compare the temporal performance, we computed the temporal standard deviation of estimated R2* values from MEPI and SS-PARSE at the chosen gradient amplitude. This was done by computing the standard deviation of each pixel within the ROIs
along the time axis, i.e. standard deviation of pixels within the ROI was computed over
50 acquisitions in the time series. Standard deviation is a measure of variability of the
data set [ref]. The standard deviation of R2* indicates how much the relaxation rates vary
30
over time. Since it’s a phantom experiment, we would ideally expect to see R2* remain
constant across all acquisitions. However this isn’t the case and we do observe some variation in R2* over the course of acquisitions. A higher value of TSD would indicate more
variations in R2* and a lower TSD would indicate higher stability.
To compare the two methods, we subtract the TSD of SS-PARSE from TSD of
MEPI and plot it against the GEMS R2* value at that pixel (see Figure 21).
2.7.3. Comparing performance at off-resonance
One set of data was acquired under severe off-resonance by intentionally
deshimming the shim coils. This caused the field B0 to be inhomogeneous over the entire
imaging volume. The parameter estimation of data sets acquired with SS-PARSE was
performed for all the gradient amplitudes. The purpose of this exercise was to see if there
is any geometric distortion caused due to field inhomogeneities.
The data acquired under severe off-resonance conditions is observed for its ability
to reconstruct M0 maps. Parameter analysis for these maps is not possible owing to the
severe geometric distortion that’s prevalent in the Cartesian methods under offresonance.
3. RESULTS AND DISCUSSION
3.1. Comparing the accuracy of R2* estimation
From a total of 1600 pixels, the number of pixels at which the null hypothesis was
rejected was 241, 307, 468, 547, 485, 338 and 214 for gradient strengths of 1.9 G/cm,
2.29 G/cm, 2.5 G/cm, 2.9 G/cm, 3.2 G/cm, 3.5 G/cm and 3.8 G/cm respectively.
31
The rejection of null hypothesis was observed to be the highest in the SS-PARSE
acquisition when the acquisitions were done at 2.9 G/cm.
In the comparison of error ratios, the percentage of pixels with a ratio greater than
1 were 61.3%, 64.2%, 66.4%, 67.6%, 65.6% and 61.2% for gradient amplitude strengths
of 1.9 G/cm, 2.29 G/cm, 2.5 G/cm, 2.9 G/cm, 3.2 G/cm, 3.5 G/cm and 3.8 G/cm respectively. Based on these percentages we determined that R2* estimation using SS-PARSE
was most accurate in comparison to MEPI when the gradient strength used during acquisition was 2.9 G/cm.
From the test for variances and test for accuracy compared to the gold standard,
we determined that the R2* estimation by SS-PARSE acquired with a gradient amplitude
of 2.9 G/cm had a much better performance as compared to MEPI than the rest of the
gradient strengths used.
Figure 11.
SS-PARSE parameter maps at different gradient amplitudes.
32
Figure 12.
R2* distribution of MEPI and SS-PARSE at Gmax=1.9 G/cm and the distribution of accuracy ratios.
Figure 13.
R2* distribution of MEPI and SS-PARSE at Gmax=2.29 G/cm and the
distribution of accuracy ratios.
33
Figure 14.
R2* distribution of MEPI and SS-PARSE at Gmax=2.5 G/cm and the distribution of accuracy ratios.
Figure 15.
R2* distribution of MEPI and SS-PARSE at Gmax=2.9 G/cm and the distribution of accuracy ratios.
34
Figure 16.
R2* distribution of MEPI and SS-PARSE at Gmax=3.2 G/cm and the distribution of accuracy ratios.
Figure 17.
R2* distribution of MEPI and SS-PARSE at Gmax=3.5 G/cm and the distribution of accuracy ratios.
35
Figure 18.
R2* distribution of MEPI and SS-PARSE at Gmax=3.8 G/cm and the distribution of accuracy ratios.
From Figures 14-20, it can be seen that estimation accuracy increases as we go
from 1.9 G/cm to 2.9 G/cm. It may be due to denser sampling which takes place at higher
gradient amplitudes, leading to more accurate parameter estimates. With increasing gradient amplitudes, we also increase the system noise; however the higher number of sampled points at higher gradient amplitudes may offset the effect due to noise.
As gradient amplitude increases beyond 2.9 G/cm, noise begins to overshadow
the improvements brought about by increased number of sampled data points. From that
point onwards, the performance of SS-PARSE begins to deteriorate. At 3.8 G/cm the accuracy of R2* estimates from SS-PARSE are only marginally better than those calculated
from MEPI acquisitions.
36
The parameter estimates from data sets that are acquired using gradient waveforms at 2.9 G/cm are more accurate than those acquired with the other gradient amplitudes.
3.2. Comparison of TSD
For the pixels where the difference is positive, SS-PARSE performed better than
MEPI, whereas at pixels with negative value of MEPI performed better. From Figure
21(d) we notice that the difference between TSDs is greater than zero for majority of the
pixels at low R2* values. The percentage of pixels with a difference greater than zero was
79.3%, 82.9%, 83.4%, 85.2%, 86.1%, 87.4%, 88.3%, 89.4%, 90.8%, 91.5%, 92.7%,
93.1%, 93.5%, 94.2%, 94.8%, 95.6%, 96.0%, 96.3%, 96.9% and 97.3% for average
GEMS R2* values ranging between 15 sec-1 to 45 sec-1. We notice that the performance
of SS-PARSE compared to MEPI improves with increasing values of R2* over the range
that we observed.
37
Figure 19.
TSD plots of SS-PARSE (at Gmax = 2.9G/cm) and MEPI (a) R2* GEMS
vs. TSDSS-PARSE. (b) R2* GEMS vs. TSDMEPI (c) R2* GEMS vs. TSDSS-PARSE
shown by blue dots and R2* GEMS vs. TSDMEPI shown by red dots. (d) TSDMEPI –
TSDSS-PARSE vs. R2* GEMS.
3.3. Performance under off-resonance
The data set acquired under severe off-resonance by intentionally deshimming the
shim coils showed a rapid decay in the signal as compared to the sets that were obtained
using well shimmed coils. The images from GEMS acquisition didn’t show geometric
distortion but instead it showed significant amount of ghosting even at lower echo times.
38
Figure 20.
SS-PARSE parameter maps in presence of field inhomogeneities.
Parameter maps were computed with some adjustments in variables used in the
PLCG algorithm. We noticed that there is no geometric distortion in the parameter maps
for any of the gradient amplitudes used for acquisition (see Figure 22).
On the other hand, we notice a significant amount of distortion in the MEPI echo
series (see Figure 23). This is because standard Fourier reconstruction for the MEPI sequence doesn’t keep track of frequency changes over the k-space. Off-resonance and the
resulting field inhomogeneities lead to geometric distortion and significant reduction in
signal at higher echo times. It becomes impossible to compute the relaxation rate in presence of geometric distortion and loss of signal strength at the third and fourth echo times.
Figure 21.
MEPI images at 4 echo times in presence of field inhomogeneities.
39
Figure 22.
Reconstructed image from GEMS, SS-PARSE (Gmax=2.9 G/cm) and
MEPI in presence of field inhomogeneities.
In order to demonstrate the performance under off-resonance conditions, we compared the images from GEMS, SS-PARSE and the first echo of MEPI. By drawing a ROI
over one of the tubes on images reconstructed from all three modalities, we notice that
there is an one to one correspondence between the GEMS and SS-PARSE images. The
ROI in the image from the first echo time in MEPI fails to match the ROI in GEMS. This
shows that it is almost impossible to use MEPI in the presence of field inhomogeneities,
but SS-PARSE can still be used to obtain the parameter maps.
3.4. Discussion
R2* is an important parameter to consider when we are conducting experiments
that make use of the BOLD effect, thus making it an important parameter in fMRI experiments. Accuracy and temporal reliability of R2* play a big role in fMRI experiments.
Through carefully designed experiments, we conducted comparison studies of two singleshot pulse sequences to evaluate their feasibility in fMRI studies. One of these methods,
40
MEPI computes the R2* values using signal changes at different echo times. The other
method, SS-PARSE estimates the parameters by solving an inverse problem.
Parameter estimation in SS-PARSE is done using the PLCG algorithm. PLCG requires us to select an optimum set of parameters in order to minimize the least squared
residuals and generate M0, R2* and frequency maps. These parameters have to be determined empirically for each set of experiments that we conduct. However once we arrive
at these parameters, PLCG can estimate the parameter maps for the entire time series.
Comparing the experimental results of MEPI and SS-PARSE provides us a lot of
information. We notice that accuracy and TSD of MEPI is quite good at lower values of
R2* and its results are comparable to those obtained with SS-PARSE. As R2* increases
we begin to see a gradual decline in the performance of MEPI. This can be attributed to
declining signal strength from the third and fourth echoes in regions with high R2* values. Such conditions make the calculation of R2* somewhat unreliable when R2* is
higher. But high R2* values are unavoidable in fMRI studies. Consequences of using
MEPI to generate activation maps in brain regions with high R2* can result in erroneous
maps.
Data acquisition for SS-PARSE involves acquiring data over a range of gradient
strengths. Lower gradient amplitudes have fewer samples and lower noise levels. This
results in parameter maps with a lower spatial resolution with some smoothing observed
along the boundaries with different R2* values. This would make it somewhat difficult to
distinguish between some of the brain regions. The accuracy of parameter estimation is
also lower at lower gradient amplitudes because of fewer samples.
41
With an increase in gradient amplitudes there is an improvement in the spatial
resolution and parameter accuracies. However this improvement in resolution and accuracy with increasing gradient amplitudes continues only up to a certain point. At higher
gradient amplitudes the acquisition BW also becomes higher, causing more noise to be
added into the system. Due to higher number of acquired samples, SS-PARSE is able to
handle this increased noise and give an improvement in R2* maps until the gradient amplitude reaches 2.9 G/cm. Beyond 2.9 G/cm noise begins to put a limitation on the level
of accuracy and reliability we can expect from SS-PARSE.
From the parameter maps generated using the experimental data we noticed that
as we progressed from gradient amplitude of 2.5G/cm to 3.2G/cm, there was a performance peak at 2.9 G/cm. While the performance of SS-PARSE in comparison to MEPI
was best at 2.9 G/cm it is not the best achievable performance that one would observe on
a 4.7 T MRI system. In practice, we expect the optimal performance at a gradient amplitude that is somewhere between 2.5 G/cm and 3.2 G/cm.
Among the parameters that are encoded into the signal, one is local frequency. By
keeping track of local frequencies, SS-PARSE can create reliable parameter maps even
when the scanner is poorly shimmed during a particular study. This ability is absent in the
conventional fMRI sequences which make use of the Fourier transform alone for signal
encoding and reconstruction. Thus, when we have a poorly shimmed system, the data acquired using Cartesian methods becomes quite unusable. However under the same conditions, SS-PARSE would yield results that are completely usable in the generation of activation maps. However there is a theoretical limit to which SS-PARSE can track changes
in frequency, and it is the sampling frequency observed at k=0 in a rosette trajectory
42
which is typically a few kilohertz. In practical setup, this limit can be much lower because of poor conditioning of the inverse problem.
PLCG is an iterative algorithm which can takes somewhere between 3 to 4
minutes to converge for moderate lengths of data (10000 to 15000 sample points). However convergence times are subject to the choice of stopping tolerances used in the algorithm. Using very tight tolerances would require tens of minutes for the algorithm to converge with a very little improvement in the accuracy of the maps. However convergence
times for tight stopping tolerances can be lowered by using faster processors and by parallelizing the PLCG algorithm.
SS-PARSE is quite dependent on the stability of the scanning hardware. It is necessary to determine the behavior of the k-trajectory and the local phase information over
time in the actual system. This is done by the process of calibration. Once we obtain this
information, we continue to get repeatable performance using the calibrated parameters in
the PLCG algorithm as long as the hardware settings remain unchanged. If there is a
change in gradient hardware settings, such as change in the RF amplifiers, the gradient
amplifiers or analog-to-digital converter cards or digital-to-analog converter cards, the
old calibration information can no longer yield reliable results. Any change in gradient
hardware settings necessitates recalibration to determine the path of k-trajectory and the
phase information.
43
4. CONCLUSIONS
Two single shot methods, MEPI and SS-PARSE were tested for R2* estimation
accuracy and temporal reliability, using a multiple shot method, GEMS, as the gold
standard. Gradient waveforms at seven gradient amplitudes were developed to test the
performance of SS-PARSE. Data acquisition was done using a fourtube phantom containing aqueous mixtures with R2* ranges typically observed in brain tissues at 4.7T.
M0, R2* and frequency maps were estimated using the PLCG algorithm. The
comparison of accuracy of R2* estimation in SS-PARSE and MEPI was conducted using
R2* from GEMS as the gold standard. The R2* accuracy in MEPI was comparable with
SS-PARSE at lower values of R2*, however at higher values of R2*, SS-PARSE was
significantly more accurate than MEPI. Accuracy of R2* estimation in SS-PARSE was
tested using seven different gradient amplitudes ranging between 1.9 G/cm to 3.8 G/cm.
As gradient amplitudes went higher the performance of parameter estimates improved
until it reached 2.9 G/cm. As the gradient strength increases beyond 2.9 G/cm the performance gradually deteriorates.
SS-PARSE is capable of estimating R2* more reliably than MEPI over a number
of repetitions. When used in fMRI studies, SS-PARSE would estimate the R2* values
more accurately and reliably during the control and stimulus cycles, thus giving activation maps that are more accurate than those obtained using MEPI.
fMRI studies performed with poor shimming can introduce geometric distortions
in MEPI acquisitions. This can render the study completely useless. However, since SS44
PARSE keeps track of change in local frequencies over time, it is capable of estimating
parameter maps even in the presence of severe off-resonance frequencies.
SS-PARSE makes use of PLCG algorithm for parameter estimation. Convergence
of PLCG algorithm requires the adjustment of several variables that are determined heuristically. Over time, as more knowledge is gained about the estimation process, it may be
possible to set these variables in a deterministic manner. Doing so would enable its integration into the system software enabling its use in clinical systems.
45
5. REFERENCES
1.
Lauterbur, P.C., Image Formation by Induced Local Interactions: Examples of
Employing Nuclear Magnetic Resonance, Nature, 1977. 242: 190–191.
2.
Ogawa, S., et al., Oxygenation-sensitive contrast in magnetic resonance image of
rodent brain at high magnetic fields. Magn Reson Med 14: 68-78.
3.
Kwong, K.K., et al., Dynamic Magnetic Resonance Imaging of Human Brain Activity During Primary Sensory Stimulation. PNAS 89: 5675-79.
4.
Mansfield P. Multi-planar image formation using NMR spin echoes. J Phys C,
1977;10:L55–8.
5.
Noll, D.C., et al., A homogeneity correction method for magnetic resonance imaging with time-varying gradients., IEEE Trans Med Imaging. 1991;10(4):629-37.
6.
Twieg DB, Parsing local signal evolution directly from a single-shot MRI signal:
a new approach for fMRI. Magn Reson Med. 2003 Nov;50(5):1043-52.
7.
Lange, N., Statistical approaches to human brain mapping by functional magnetic
resonance imaging. Statistics in Medicine. 1996:15, 389-428.
8.
Haacke, E.M.,et al., Magnetic Resonance Imaging:Physical Principles and Sequence Design. New York: Wiley, 1999.
9.
Speck O., et al., High resolution single-shot EPI at 7T. MAGMA. 2008 ar;21(12):73-86.
10.
Arfken G., Mathematical Methods for Physicists. Orlando, FL: Academic Press.
1985;794-810.
11.
Péran, P., et al., Voxel-based analysis of R2* maps in the healthy human brain. J
Magn Reson Imaging. 2007 Dec;26(6):1413-20.
12.
Winkelmann, S., et al., Simultaneous imaging and R2* mapping using a radial
multi-gradient-echo (rMGE) sequence. J Magn Reson Imaging. 2006
Oct;24(4):939-44.
13.
Farzaneh, F., et al., Analysis of T2 limitations and off-resonance effects on spatial
resolution and artifacts in echo-planar imaging. Magn Reson Med 1990: 14:123139.
14.
Kearfott, R.B., et al., On stopping criteria in verified nonlinear system or optimization algorithm. ACM TOMS. 2000; 26(3): 373-389.
15.
Bucholz, E.K., et al., Multispectral imaging with three-dimensional rosette trajectories. Magn Reson Med. 2008 Mar;59(3): 581-9.
46
16.
Zhang, Y.T., et al., A Novel k-space Trajectory Measurement Technique. Magn
Reson Med 1998; 39:999-1004.
17.
Crawford, C.R., et al., Detection and elimination of wide bandwidth noise in MRI
signals. Magnetic Resonance Imaging. 1997:15(3): pp. XIV-XIV(1)
18.
De Guio, F., et al., Signal decay due to susceptibility-induced intravoxel dephasing on multiple air-filled cylinders: MRI simulations and experiments. MAGMA.
2008 Jul;21(4):261-71.
19.
Peters, A.M., et al., T2* measurements in human brain at 1.5, 3 and 7T. Magnetic
Resonance Imaging. 2007: 25: 748–753.
20.
Mario F. Triola, M.F., Essentials of Statistics, Pearson, 2006.
47