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University of Southern California Dissertations and Theses
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Velocity-encoded magnetic resonance imaging: acquisition, reconstruction and applications
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Velocity-encoded magnetic resonance imaging: acquisition, reconstruction and applications
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Content
VELOCITY-ENCODED MAGNETIC RESONANCE IMAGING:
ACQUISITION, RECONSTRUCTION AND APPLICATIONS
by
Joao Luiz Azevedo de Carvalho
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
August 2008
Copyright 2008 Joao Luiz Azevedo de Carvalho
Dedication
To my little sister.
ii
Acknowledgements
I am deeply grateful to my advisor Professor Krishna Nayak for his support and guid-
ance. I also thank the members of my qualifying and dissertation committees { Professor
Richard Leahy, Professor Tzung Hsiai, Professor Gerald Pohost, Professor Urbashi Mitra,
and Professor Manbir Singh {, as well as Professor Antonio Ortega, and all EE and SIPI
sta members.
I also express appreciation to all current and former members of the USC Magnetic
Resonance Engineering Laboratory for their friendship and support. I specially acknowl-
edge Kyunghyun Sung, Jon-Fredrik Nielsen, Chia-Ying Liu, Taehoon Shin, Yoon-Chul
Kim, and Mahender Makhijani for very useful discussions and collaboration. I also thank
fellow EE591 colleague Antonio Ordonez.
I have greatly benetted from invaluable discussions and collaborations with several
researchers { including Thomas and Barbara Burke (Phantoms by Design Inc.), Ramdas
Pai, Padmini Varadarajan, Susana Perese, Gigi Youssef, Nilesh Ghugre, John Wood,
Adam Kerr, Julie DiCarlo, Juan Santos, Michael Lustig, Tolga Cukur, Michael Hansen,
Christopher Macgowan, Hervaldo Carvalho, Hee-Won Kim, Samuel Valencerina, Bob Hu,
John Oshinski, Ajit Yoganathan, Kartik Sundareswaran, and Lisong Ai { whom I thank
profusely.
iii
I profoundly thank the support of my loving family, specially my parents, my sister, my
grandmother, my cousin Diana, my uncle Vilson, my aunts Dilma and Svetlania, and the
Gargs. I also thank Prof. Adson da Rocha for invaluable support during the application
process, and throughout the course of my doctoral studies. I extend these thanks to the
Brazilian USC/UCLA community (\Perdidos") for their friendship { specially Alexandre,
Cristine, Denis, and Lisandra, for invaluable rst-year support {, to my good friend Bruno,
and to all my old friends from college (\Agadirre"), specially Fabio, Lea, Sara, Renato
(ah!), \Port", and Fabiano, for their constant companionship.
This work was supported by the National Institutes of Health (HL074332), the Amer-
ican Heart Association (0435249N), GE Healthcare, and the USC Graduate School.
iv
Table of Contents
Dedication ii
Acknowledgements iii
List Of Tables vii
List Of Figures viii
Abstract xv
Chapter 1: Introduction 1
1.1 Cardiovascular disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Doppler ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Magnetic resonance imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Phase contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Fourier velocity encoding . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline and contributions of this work . . . . . . . . . . . . . . . . . . . . 8
Chapter 2: MR
ow imaging 11
2.1 Basic principles of MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Mathematical formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Principles of MR
ow imaging . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Phase contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Fourier velocity encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1 FVE bipolar gradient design . . . . . . . . . . . . . . . . . . . . . 21
2.6 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Chapter 3: Slice-selective FVE with spiral readouts (spiral FVE) 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
Chapter 4: Accelerated spiral FVE 40
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 5: Reconstruction of variable-density FVE data 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2 In vitro experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.3 In vivo experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter 6: Measurement of carotid
uid shear rate using spiral FVE 72
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.1 The Frayne method . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.2 Simulating spiral FVE distributions from velocity maps . . . . . . 79
6.3 Spiral FVE vs. 2DFT phase contrast . . . . . . . . . . . . . . . . . . . . . 81
6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4 Feasibility of the spiral FVE/Frayne method . . . . . . . . . . . . . . . . 83
6.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 In vivo demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter 7: Concluding remarks 94
References 96
vi
List Of Tables
3.1 Scan parameters used in the dierent spiral FVE studies. . . . . . . . . . 27
4.1 Design parameters used to evaluate improvements from using variable-
density (VD) over uniform-density (UD) spiral trajectories in spiral FVE. 48
5.1 Variable-density FVE signal-to-error ratio (dB) comparison to a high-
resolution uniform-density reference. . . . . . . . . . . . . . . . . . . . . . 68
6.1 Scan parameters used in the in vivo studies. . . . . . . . . . . . . . . . . . 87
vii
List Of Figures
1.1 Cardiovascular diseases. (a) illustration of normal and diseased aortic
valves (copyright: Nucleus Communications); (b) illustration of normal
and stenotic aortic
ow (copyright: Mayoclinic); (c) illustration and ul-
trasound of a backward jet due to aortic regurgitation (copyright: Yale
University); (d) volume-rendered computerized tomography, showing a
stenosis in the left anterior descending coronary artery (copyright: Mas-
sachusetts General Hospital); (e) illustration of coronary stenosis (copy-
right: Sunao Watanabe); (f) illustration of carotid artery disease (copy-
right: SVS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Mitral valve
ow in a healthy volunteer, measured with Doppler ultra-
sound: (a) color Doppler; (b) spectral Doppler. . . . . . . . . . . . . . . . 4
1.3 Aortic
ow, measured with real-time phase contrast MRI [41]. . . . . . . . 5
1.4 Aortic
ow, measured with (a) real-time FVE, and (b) Doppler ultra-
sound [44]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Timing diagram (left) and correspondent k-space trajectories (right) for
(a) 2DFT [55], and (b) spiral acquisitions. . . . . . . . . . . . . . . . . . . 15
3.1 Spiral FVE pulse sequence. It consists of (a) slice selective excitation, (b)
velocity encoding bipolar gradient, (c) spiral readout, and (d) refocusing
and spoiler gradients. This timing corresponds to the healthy volunteer
cardiac studies in Figures 3.3 and 3.5. . . . . . . . . . . . . . . . . . . . . 26
3.2 Spiral FVE k-space sampling scheme. The dataset corresponding to each
temporal frame is a stack-of-spirals in k
x
;k
y
;k
v
space. Each spiral acqui-
sition corresponds to a dierent k
v
encode. . . . . . . . . . . . . . . . . . 26
viii
3.3 Comparison of artifacts in dierent view-orderings, in a 12-heartbeat spiral
FVE acquisition using single-shot spiral readouts. Each box represents the
acquisition of onek
v
level, during one imaging TR. A sliding window is used
to produce a new image every TR. Acquiring the k
v
levels in a sequential
fashion (a), ghosting artifacts appear shifted by 1/2 of the velocity eld-of-
view. By acquiring them in an interleaved fashion (b), the artifacts overlap
with the true
ow prole. The aortic valve
ow proles shown are from a
fully-sampled 24-heartbeat healthy volunteer acquisition, and 50% of the
data was appropriately discarded to simulate each of the view-ordering
schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Comparison of dierent view-orderings for multi-shot spiral FVE, in a
healthy volunteer carotid study. When two or more k
v
levels are acquired
during the same heartbeat (a), velocity distribution changes between con-
secutive TRs cause ghosting artifacts along the velocity axis (arrow). This
artifact is not seen if, in the same heartbeat, dierent spiral interleaves,
but only one k
v
encoding, are acquired (b). . . . . . . . . . . . . . . . . . 33
3.5 Comparison of the spiral FVE method with Doppler ultrasound, in healthy
volunteer studies: (a) aortic valve and (b) carotid artery. Peak velocity
and time-velocity waveforms are in good agreement. . . . . . . . . . . . . 33
3.6 Multiple
ow distributions obtained from a single spiral FVE dataset: (a)
heart; (b) neck. For each voxel in the images, a
ow distribution was
calculated, and the red and blue dots indicate voxels where ascending and
descending blood
ow was detected, respectively. The color intensity of
each dot indicates the highest velocity detected in that voxel in a particular
temporal frame (indicated by the white dashed lines). Multiple ROIs were
specied around dierent regions of the heart and the neck, and the
ow
distributions from voxels automatically selected from each ROI are shown. 35
3.7 Evaluation of spiral FVE in an aortic valve study of a patient with aortic
stenosis. Note the high-speed jet with a wide distribution of velocities. . . 35
3.8 Spiral FVE trade-os between temporal resolution, velocity resolution, and
breath-hold duration. Velocity resolution corresponds to a 600 cm/s eld-
of-view, temporal resolution corresponds to a 8.1 ms spiral readout, and
scan time corresponds to a single-shot spiral acquisition. The arrow in-
dicates the conguration used in the cardiac study in Figure 3.5 (2TR
temporal resolution, 24 velocity encodes). . . . . . . . . . . . . . . . . . . 36
ix
4.1 Proposed view-ordering scheme for accelerated spiral FVE (a) and corre-
spondent point-spread function (b). In (a), each color represents a dif-
ferent variable-density spiral interleaf, and darker tones indicate \views"
that were discarded in the partial Fourier experiments (i.e., need not be
acquired). Views aligned in k
v
are acquired sequentially throughout the
cardiac cycle. Views aligned in time (same cardiac phases) are acquired in
dierent heartbeats. In (b), each square shows the point-spread function in
x;y for a particular velocity-frequency (v-f) coordinate. The main aliasing
replicas are separated from the main-lobe by half the velocity eld-of-view,
and half the temporal frequency bandwidth, which reduces overlaps and
facilitates ltering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Temporal acceleration approach proposed for spiral FVE. The red, green
and blue ellipses illustrate the expected footprint of aortic valve
ow in
v-f space, more specically for plug
ow, stenotic
ow, and regurgitant
jets, respectively. Yellow dots represent the point-spread function for the
undersampling strategy (Figure 4.1). Grey ellipses represent aliasing com-
ponents. The aliasing at20 and40 Hz has a small footprint in v-f
space because it is composed mainly of static tissue or slow moving
ow,
while aliasing components at60 Hz are exact replicas of the main signal.
A 2D lter (dashed lines) is capable of removing the aliasing components
while preserving all signal content. . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Reconstruction
ow-chart for accelerated spiral FVE. Time-velocity his-
tograms (s(v;t)) from multiple regions-of-interest may be obtained from a
single spiral FVE dataset (S(k
x
;k
y
;k
v
;t)). Such ROIs are prescribed by
the operator, using a color-
ow video obtained from the same dataset. . . 47
4.4 Improvements from using variable-density spirals in the spatial images
(top), and correspondent time-velocity histograms (bottom). Each result
(a-e) correspond to its respective trajectory described in Table 4.1. O-
resonance eects in both spatial and time-velocity domains are observed
in the long readout results (a-b). Improved resolution and reduced o-
resonance provide better spatial localization. Consequently, less signal
from static material is observed in the time-velocity histograms in (d) and
(e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
x
4.5 Temporal acceleration results inv-f (left) andv-t (right) spaces: a) under-
sampled data; b) 2D lter; c) 2D lter and notch lter; and d) view-sharing.
The 2D lter (dashed lines) removes most of the aliasing components, and
the notch lter (dotted line) removes the remaining aliasing energy (solid
arrows). The proposed method (c) removes aliasing components with-
out noticeable loss of temporal resolution, as opposed to view-sharing (d),
which attenuates high temporal frequency components (dashed arrows)
and causes blurring along t (circled). Compare the v-f representation in
(a) with Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Accelerated spiral FVE results. Comparison between: (a) reference ac-
quired without using acceleration (36 heartbeats); (b) 2-fold acceleration,
using variable-density spirals (18 heartbeats); (c) 12-fold acceleration, us-
ing variable-density spirals combined with temporal acceleration (18 heart-
beats); and (d) 18-fold acceleration combining variable-density spirals,
temporal acceleration, and homodyne reconstruction (12 heartbeats). A
reduced velocity eld-of-view (200 cm/s) was used in (a) and (b) in or-
der to to restrict acquisition to a single breath-hold. Using a 1200 cm/s
velocity eld-of-view as in (c) and (d), the total acquisition time for (a)
and (b) would have been 216 and 108 heartbeats, respectively. All other
scan parameters are the same for all acquisitions. No signicant artifacts
were observed when comparing the reference dataset (a) with the 18-fold
accelerated result (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.7 Flow in multiple ROIs acquired in a single 12-heartbeat spiral FVE ac-
quisition using 18-fold acceleration. Very few artifacts are observed in the
dierent time-velocity histograms obtained throughout the heart (arrow).
Note the improvements in spatial resolution, velocity eld-of-view, and
temporal resolution, compared to Figure 3.6. . . . . . . . . . . . . . . . . 53
4.8 An alternative 2D-lter. Compared to the lter used in Figure 4.5a, this
lter would provide better temporal resolution for higher velocities, but
could also be less eective in removing aliasing artifacts. . . . . . . . . . . 55
4.9 An alternative view-ordering scheme, and comparison in v-f space. The
view-ordering scheme in (a) provides the aliasing pattern shown in (d),
while the scheme shown in Figure 4.1a provides the aliasing pattern shown
in (b). By applying a notch lter, it becomes clear that this alternative
scheme reduces the amount of overlap (circled) between aliasing and signal
components (c,e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
xi
5.1 Typical aortic valve velocity distributions (left) and correspondent k-space
representation (right). Data obtained in vivo using spiral FVE with uniform-
density sampling. (a) plug
ow (healthy volunteer); (b) turbulent/complex
ow (patient with aortic stenosis). There is clearly an inverse relationship
between velocity and k
v
footprints, which implies a trade-o between ve-
locity FOV and velocity resolution needs. . . . . . . . . . . . . . . . . . . 59
5.2 Variable-width sinc interpolation diagram, for variable-density FVE. High-
resolution components are more sparsely sampled then low-resolution ones,
resulting in a smaller unaliased velocity FOV. Using a sinc kernel of appro-
priate width for each k-space sample, the resulting apodization function
lters the corresponding aliasing, and the velocity resolution in the recon-
structed data varies across the velocity FOV. . . . . . . . . . . . . . . . . 62
5.3 Numerical simulations results for variable-density FVE, using = 4 (FOV
= 1200300 cm/s) and 14 k
v
samples. Velocity resolution is 86 cm/s for
uniform sampling and 33 cm/s for variable-density sampling. (a) simu-
lation of plug
ow and static material; (b) simulation of a
ow jet; (c)
simulation of a
ow jet with higher peak velocity. The proposed method
reduces undersampling artifacts considerably (arrows), specially for narrow
and moderately broad distributions. . . . . . . . . . . . . . . . . . . . . . 66
5.4 Evaluation of the proposed reconstruction method, using a pulsatile
ow
phantom. Data was acquired using variable-density real-time FVE with
= 2:75, and reconstructed using (a) conventional gridding and (b) the
proposed method. Undersampling artifacts (arrow) are signicantly re-
duced by the proposed method. . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 In vivo demonstration of the proposed reconstruction method, using spi-
ral FVE. Velocity distributions were measured through the aortic valve of
a healthy volunteer using: (a) uniform-density, large FOV; (b) uniform-
density, small FOV; (c) variable-density, reconstructed using conventional
gridding; (d) variable-density, reconstructed using the proposed method.
The proposed reconstruction reduces undersampling artifacts (arrows), and
shows velocity resolution equivalent to the small FOV uniform-density ref-
erence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
xii
6.1 Velocity proles measured in vivo with 2DFT phase contrast, at the carotid
bifurcation, and at peak
ow. It is not possible to achieve sucient SNR in
clinically practical scan time when estimating shear rates with phase con-
trast, as sub-millimeter spatial resolution is required. Averaging multiple
acquisitions (NEX) improves SNR, but this increases scan time, and also
may introduce motion-related issues, such as loss of eective spatial and
temporal resolutions. Scan parameters: 0.33x0.33x3 mm resolution, 30
ip angle, 80 cm/s venc, 37 ms temporal resolution, 2 minute acquisition
(120 heartbeats) per NEX. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2 Construction of the intra-voxel velocity prole v(r) from the FVE mea-
sured velocity distribution s(v), using the Frayne method. The volume
fraction at each velocity bin is converted into a position within the voxel,
using the algorithm described in Equation 6.1. Before the conversion, a
threshold is applied to the velocity distribution in order to address noise
rectication, and the signal amplitudes are compensated for saturation ef-
fects and blood/vessel wall signal dierences. A small velocity interval
(v
0
,v
1
) is used for estimating the shear rate (dv=dr). . . . . . . . . . . . . 79
6.3 In vitro comparison between time-velocity distributions derived from 2DFT
phase contrast (left) and those measured with spiral FVE (center-left).
Results from two representative voxels, selected near opposite walls of the
vessel's bifurcation, as indicated (upper-left corner). The dierence be-
tween measured and PC-derived histograms is also shown (center-right),
as well as the 4-fold magnied dierence (right). The signal-to-error ratio
(SER) between measured and PC-derived time-velocity distributions is in
the range of 10-12 dB within the lumen. The two datasets show good
visual agreement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Shear rates measured using the spiral FVE/Frayne method on simulated
velocity distributions obtained from a CFD simulation. The CFD-based
velocity map is shown on left. Within a region-of-interest, dened near
the wall-blood interface, the FVE-measured shear rates agree very well
with the true values (center). The spiral FVE/Frayne method was able to
estimate the shear rate with at least 30% accuracy for 95% of the voxels
within the region-of-interest (right). . . . . . . . . . . . . . . . . . . . . . 85
6.5 Spatial variation in carotid FSR along all three dimensions, near the carotid
bifurcation of subject #1. These results correspond to the cardiac phase
with the highest peak velocity, acquired 96 ms after the ECG trigger. The
common, external, and internal carotid arteries, and the jugular vein, are
indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
xiii
6.6 Oscillatory pattern of FSR along the cardiac cycle in the common carotid
artery of subject #2. These results correspond to the cardiac phases ac-
quired 48-168 ms after the ECG trigger, and to the most inferior slice
(labeled 0 mm in Figure 6.5, i.e., 10 mm below the carotid bifurcation). . 89
6.7 Repeatability evaluation of the spiral FVE/Frayne method. Subject #3
was imaged in two-dierent occasions, with a two-week interval. Regions
of low and high FSR match reasonably well in the two studies (arrows).
These results correspond to the cardiac phases acquired 48-120 ms after
the ECG trigger, and to the second-most inferior slice (labeled 5 mm in
Figure 6.5, i.e., 5 mm below the carotid bifurcation). . . . . . . . . . . . . 89
6.8 Proposed method for compensation of voxel shape eects. The measured
volume fractions (a) are obtained by normalizing the velocity distribution.
Then, the velocity prole is estimated using the Frayne method (b). This
prole is projected into a circle, representing the voxel shape (c). Volume
fractions are calculated from this circle (d). By dividing the measured vol-
ume fractions by the estimated volume fractions, a compensation function
is obtained (e). The adjusted volume fraction is obtained by multiplying
the measured volume fraction by this function (f). Finally, the adjusted
velocity prole is calculated using the Frayne method (g). Intra-voxel
weighting can also be compensated using this approach, simply by mul-
tiplying the projected prole (c) by the appropriate kernel (e.g., a jinc),
prior to calculating (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
xiv
Abstract
Cardiovascular disease is the leading cause of death and disability in the United States.
An important component of the assessment of cardiovascular disease is the visualization
and quantitation of cardiovascular
ow. The current gold standard for
ow measurement
is Doppler ultrasound. However, evaluation by ultrasound is inadequate when there is
fat, air, bone, or surgical scar in the acoustic path.
Magnetic resonance imaging (MRI) is a modality uniquely capable of imaging all
aspects of heart disease. The evaluation of valvular disease and intracardiac
ow will be
a necessary capability in a comprehensive cardiac MR examination. However, existing
MRI methods for
ow quantitation suer from partial volume eects or poor spatial
localization.
This work introduces spiral Fourier velocity encoding (spiral FVE), a new method for
rapid and non-invasive measurement of cardiovascular blood
ow using MRI. Spiral FVE
is uniquely capable of spatially separating static tissue from
owing blood in dierent
vessels, and of measuring multiple
ows in a single acquisition.
Spiral FVE can be improved using acceleration techniques such as variable-density
spirals, temporal acceleration, and partial Fourier reconstruction. Using these techniques,
xv
we demonstrate improvements such as higher spatial resolution, increased velocity eld-
of-view, reduced o-resonance eects, and higher temporal resolution. These are achieved
without increase in scan time, and without introducing signicant artifacts.
We also propose a new reconstruction scheme for variable-density FVE, which can
be used to achieve higher velocity resolution. The proposed reconstruction approach
signicantly reduces aliasing artifacts, and can, in principle, be applied to any FVE-
based method.
We also show that spiral FVE can be used for non-invasive estimation of carotid
uid
shear rate, an important predictor of atherosclerosis. Spiral FVE is uniquely suitable for
this application, as it is the only currently available FVE method capable of providing
fully-resolved spatial localization in clinically practical scan time.
No MR gold standard exists for assessing valvular disease and carotid
uid shear.
Spiral FVE oers full spatial localization and accurate
ow quantitation, which are crucial
in these applications. Using the proposed acceleration techniques, spiral FVE may be used
as a rapid and non-invasive alternative in addressing these important clinical problems.
xvi
Chapter 1
Introduction
1.1 Cardiovascular disease
Cardiovascular disease is the leading cause of death and disability in the United States
and most European countries. According to the American Heart Association, diseases of
the heart alone account for 40% of all deaths in the United States, with other diseases of
the cardiovascular system causing substantial further death and disability.
Most cardiovascular diseases are either caused by abnormal blood
ow or can be
diagnosed based on abnormal blood
ow. For example, valvular stenosis consists in a
narrowing or incomplete opening of any of the valves in the heart. This typically alters the
blood
ow, causing turbulent and/or complex
ow jets. Such jets display peak velocities
considerably higher than those of normal
ow, and a much broader range of
ow velocities.
Another form of valvular disease is valve insuciency, or regurgitation. This condition
occurs when a heart valve fails to close completely. This is also known as \leaky valve",
as
ow jets in the reverse direction are observed when no
ow should occur. Important
examples of valvular disease include aortic stenosis and aortic insuciency, as well mitral
1
stenosis and mitral insuciency. These correspond to stenosis and regurgitation in the
aortic valve (which connects the left ventricle to the aorta), and in the mitral valve (which
connects the left atrium to the left ventricle), respectively. The left ventricle is responsible
for pumping blood to the entire body. Thus, abnormal function in either of these valves
may lead to a series of problems. Flow jets are also a symptom of atherosclerosis, which
consists in the narrowing of an arterial vessel. Important examples include coronary
stenosis and carotid stenosis. The carotid arteries supply blood to the brain, so early
detection of carotid stenosis may prevent thrombotic stroke. Coronary artery disease
is the most common type of heart disease. The coronary arteries supply blood to the
muscular tissue of the heart (the myocardium), so early detection of coronary stenosis
may prevent myocardial infarction (heart attack). Myocardial infarction is the leading
cause of death for both men and women all over the world. A few cardiovascular diseases
are illustrate in Figure 1.1.
The visualization and quantitation of cardiovascular blood
ow is an important com-
ponent of the assessment of these diseases. For example, peak velocity measurements in
ow jets are used to estimate pressure drop, which is an indicator of the hemodynamic
load of a stenosis [74].
1.2 Doppler ultrasound
The current non-invasive gold standard for
ow quantitation is Doppler ultrasound. The
ultrasound equipment is relatively inexpensive, small, and portable. Measurements are
typically obtained in real-time, with excellent temporal resolution. The most popular
2
normal aortic valve
open closed
aortic valve dsease
open closed
a b
c
d e f
Figure 1.1: Cardiovascular diseases. (a) illustration of normal and diseased aortic valves
(copyright: Nucleus Communications); (b) illustration of normal and stenotic aortic
ow (copyright: Mayoclinic); (c) illustration and ultrasound of a backward jet due to
aortic regurgitation (copyright: Yale University); (d) volume-rendered computerized to-
mography, showing a stenosis in the left anterior descending coronary artery (copyright:
Massachusetts General Hospital); (e) illustration of coronary stenosis (copyright: Sunao
Watanabe); (f) illustration of carotid artery disease (copyright: SVS).
techniques for
ow assessment are color Doppler and spectral Doppler. In color Doppler
(Figure 1.2a), a gray-scale magnitude image of the vessel of interest is overlayed by a
color image which indicates the velocity measured at each location. In spectral Doppler,
which comprises pulsed wave and continuous wave Doppler, a time-velocity histogram is
used to represent the distribution of velocities within a region of interest as a function of
time (Figure 1.2b).
Evaluation by ultrasound is impossible when there is air, bone, or surgical scar in
the ultrasound path. Examination by ultrasound in obese patients is dicult, as the
overlying adipose tissue (fat) scatters the sound waves. Doppler
ow measurements may
be inaccurate when the ultrasound beam cannot be properly aligned with the vessel axis,
3
a b
Figure 1.2: Mitral valve
ow in a healthy volunteer, measured with Doppler ultrasound:
(a) color Doppler; (b) spectral Doppler.
requiring measured velocities to be \angle-corrected" by the operator. Peak-velocity
overestimation on the order of 18-40% have been reported in literature [31, 79], usually
due to spectral broadening at large insonation angles, and to Doppler gain settings.
1.3 Magnetic resonance imaging
Magnetic resonance imaging (MRI) is potentially the most appropriate technique for
addressing all aspects of cardiovascular disease examination, which includes assessing
myocardial function and perfusion, as well as visualizing and measuring blood
ow. MRI
overcome the acoustical window limitations of ultrasound, potentially allowing
ow mea-
surements to be obtained along any direction, and for any vessel in the cardiovascular
system. Magnetic resonance (MR) measurements are also less operator-dependent than
that of Doppler ultrasound, and the true direction of
ow can generally be precisely
measured. The main MR techniques for measuring
ow are phase contrast and Fourier
velocity encoding. These techniques are introduced below, and will be discussed in further
detail in Chapter 2.
4
1.3.1 Phase contrast
Phase contrast [58] is a technique in which a bipolar gradient (see Chapter 2) aligned
with the axis of
ow is used to obtain a velocity measurement associated with each pixel
(or \voxel") of the image. In practice, two acquisitions are used, and the rst moment of
the bipolar gradient is varied between measurements. The velocity estimate is obtained
from the phase dierence between the images obtained in each acquisition.
Phase contrast can be combined with cine MRI [25], in which short acquisitions and
some form of cardiac synchronization are used to produce images throughout the car-
diac cycle. The combined technique (cine phase contrast) can depict motion and
ow
throughout the cardiac cycle [54]. Alternatively, phase contrast can be combined with
real-time MRI [30,64], in which the encodings are applied sequentially, periodically, and
continuously. In real-time MRI, images are formed by sliding a window along the ac-
quired data and reconstructing an image for each position of the window. The display of
real-time phase contrast data is typically implemented as color overlay of
ow information
(phase dierence) over the anatomical (magnitude) image (Figure 1.3), which is called
real-time color
ow [53,65].
Figure 1.3: Aortic
ow, measured with real-time phase contrast MRI [41].
5
In phase contrast, data inconsistency, partial volume eects, and intravoxel phase
dispersion can lead to peak velocity underestimation [15,71]. Partial voluming is partic-
ularly problematic when
ow is highly localized and/or turbulent. When a large voxel
size is adopted to measure the
ow rate, not only may moving spins and stationary spins
coexist in a voxel, but also the velocity distribution of spins within a voxel may spread
over a wide range of velocities. This results in signal loss, distortion and erroneous veloc-
ity estimates. As a result, phase contrast imaging can not provide accurate peak velocity
measurements in turbulent and/or complex
ow jets. Such jets are commonly observed
in narrowed vessels, and in valves presenting stenotis and/or regurgitation.
1.3.2 Fourier velocity encoding
The limitations mentioned above can be overcome using Fourier velocity encoding
(FVE) [51]. FVE can be considered the MR equivalent to spectral Doppler. In this
technique, the full spectrum of velocities within each voxel is measured by phase-encoding
the velocity information in Fourier domain. Therefore, FVE is robust to partial voluming,
and
ow measurements from low spatial resolution images are still accurate [74]. FVE
shows satisfactory agreement with Doppler ultrasound [50]. However, it is typically not
used clinically, because the acquisition time required by this technique is in principle
considerably longer than that of phase contrast.
Dierent approaches to accelerating FVE have been proposed. One example is the
use of two-dimensional cylindrical excitation to restrict the eld-of-view to a beam that
can be imaged without phase encoding [18]. This approach makes it possible to perform
6
spatial encoding and velocity encoding simultaneously, and in a single pulse repetition
time (TR) [17, 33, 34, 44]. This allows FVE measurements to be obtained in real-time
(Figure 1.4). However, real-time FVE has problems related to the precise placement
of the imaging beam, especially when the region of interest (e.g., mitral valve) moves
during the cardiac cycle. Other problems include the large voxel size and low temporal
resolution.
Figure 1.4: Aortic
ow, measured with (a) real-time FVE, and (b) Doppler ultra-
sound [44].
FVE has also been accelerated by simply neglecting spatial encoding along one of the
spatial dimensions [19,29], or by acquiring velocity images with no spatial encoding other
than slice selection [22]. In these techniques, the velocity measurement is a projection of
all signal along a line or a plane in 3D space, respectively. As a consequence, both methods
have dynamic range issues, as the signal of
owing blood has to be distinguished from
all the background signal from static tissue observed along the projection. Furthermore,
these approaches are unable to resolve dierent sources of
ow that may co-exist in the
projected line or plane.
7
1.4 Outline and contributions of this work
Chapter 2 introduces the basic principles of magnetic resonance imaging, and present
a review of
ow imaging methods in MR, namely, phase contrast and Fourier velocity
encoding.
Chapter 3 introduces \spiral FVE", a novel method for MR
ow quantitation that
addresses the limitations discussed in the previous section. The proposed method pro-
vides fully-localized time-velocity distribution measurements, in a single acquisition, that
is one short breath-hold long (approximately 10 seconds). Spiral FVE uses conventional
slice-selective excitation [6, 55], which excites (selects) a thin slice of the body to be
imaged. The two-dimensional plane dened by this slice is imaged using spiral acquisi-
tions [2,49], which encode both spatial dimensions simultaneously. Therefore, no spatial
encoding is neglected, and measurements are fully localized in 3D space. 2D-resolved
spatial encoding allows for easy localization of the region of interest, and the ability to
automatically resolve multiple sources of through-plane
ow in the imaged eld-of-view,
without requiring static tissue suppression. This makes the proposed method uniquely
suitable for measuring
ow in small vessels (e.g., coronary arteries), and for estimating
uid shear rate [21], which could be used as a preventive examination for atherosclerosis.
Scan-plane prescription is performed using classic protocols, which is considerably less la-
borious than the beam-placing process used in real-time FVE. Semi-automatic algorithms
were designed for in-plane localization of the
ow.
8
Without acceleration, spiral FVE presents a few limitations: (1) insucient velocity
eld-of-view (the maximum range of velocities allowed without aliasing); (2) low in-
plane spatial resolution, which limits the ability of spatially localizing the
ow; (3) long
readouts, which causes spatial blurring at 3T, due to o-resonance eects [56]; and (4)
moderate temporal resolution, which may blur certain features of the
ow waveform. We
address these limitations in Chapter 4, by using the following acceleration techniques:
variable-density spirals [73], partial Fourier reconstruction [57], and temporal accelera-
tion [47, 75]. By combining these techniques, we achieved a total 18-fold acceleration in
spiral FVE.
In Chapter 5, we propose a new reconstruction scheme for FVE sequences that uses
variable-density sampling along the velocity dimension [17]. The variable-density sam-
pling scheme is incorporated into the spiral FVE pulse sequence, and the proposed re-
construction method is demonstrated both in vitro and in vivo. We also show that the
method can potentially be used with any FVE-based method. This is demonstrated by
applying the proposed method to datasets obtained using both spiral FVE and real-time
FVE.
In Chapter 6, we show that spiral FVE can be used for assessing carotid
uid shear.
Wall shear stress is widely believed to be a predictor of locations of formation and growth
of atherosclerotic plaque. However, there is no current gold standard for assessing shear
rates in vivo. Spiral FVE is uniquely suitable for this application, as it is the only
currently available FVE method capable of providing fully-resolved spatial localization
in clinically practical scan time. The proposed method is demonstrated in vivo, and in
simulations.
9
In Chapter 7, we present a summary of what was accomplished, and our concluding
remarks.
10
Chapter 2
MR
ow imaging
In this chapter we discuss the basic principles of magnetic resonance imaging, and present
a review of
ow imaging methods in MR. Phase contrast and Fourier velocity encoding
methods are discussed here with greater formalism.
2.1 Basic principles of MRI
Magnetic resonance imaging (MRI) is a modality uniquely capable of imaging all aspects
of heart disease, and is a potential \one-stop shop" for cardiovascular health assessment.
MRI can generate cross-sectional images in any plane (including oblique planes), and
can also measure blood
ow. The image acquisition is based on using strong magnetic
elds and non-ionizing radiation in the radio frequency range, which are harmless to the
patient.
The main component of a MRI scanner is a strong magnetic eld, called the B
0
eld.
This magnetic eld is always on, even when the scanner is not being used. Typically,
MR is used to image hydrogen nuclei, because of its abundance in the human body.
Spinning charged particles (or \spins"), such as hydrogen nuclei, act like a tiny bar
11
magnet, presenting a very small magnetic eld, emanating from the south pole to the
north pole. In normal conditions, each nucleus points to a random direction, resulting in
a null net magnetization. However, in the presence of an external magnetic eld (such as
the B
0
eld), they will line up with that eld. However, they will not all line up in the
same direction. Approximately half will point north, and half will point south. Slightly
more than half of these spins (about one in a million) will point north, creating a small
net magnetization M
0
, which is strong enough to be detected. The net magnetization
is proportional to the strength of the B
0
eld, so MRI scanners with stronger magnetic
elds (e.g., 3 Tesla) provide higher signal-to-noise ratio (SNR).
Another important component of the scanner are the gradient coils. There are typi-
cally three gradient coils (G
x
, G
y
, and G
z
), that produce an intentional perturbation in
theB
0
eld when turned on (\played"). This perturbation varies linearly along each spa-
tial direction (x,y andz), such that no perturbation is perceived at the iso-center of the
magnet when these gradients are used. In the presence of an external magnetic eld, the
spins rotate about the axis of that eld. B
0
is (approximately) spatially uniform, so all
spins initially rotate at the same frequency (the Larmor frequency), ! =
B
0
, where
is
the gyromagnetic ratio (
= 42.6 MHz/Tesla for hydrogen protons). However, when any
of the gradients is played, the magnetic eld becomes spatially varying, and so does the
rotation frequency of the spins. Therefore, G
x
,G
y
, andG
z
are used to frequency-encode
(or phase-encode) spatial position along the x, y and z directions, respectively.
The nal major component of the MR scanner is the radio-frequency (RF) coil. This is
used to transmit a RF \excitation" pulse to the body, and also to receive the frequency-
encoded signal from the \excited" portion of the body. In practice, independent coils
12
may be used for transmission and reception. The RF pulse is typically modulated to the
Larmor frequency. While B
0
is aligned with the z-axis (by denition), B
1
, which is a
very weak magnetic eld associated with the RF pulse, is aligned with the x-axis (also
by denition). When the RF pulse is played, some of the spins which are in resonance
with the RF pulse (i.e., rotating at the RF pulse's frequency) will now begin to rotate
around the x-axis (thus the name magnetic resonance). This tilts the net magnetization
towards the x-y plane, and the net magnetization will now have a component in the x-y
plane (M
xy
).
The RF pulse is typically designed to have a somewhat rectangular prole in Fourier
domain, centered at the modulation frequency (e.g., a modulated windowed sinc). This
implies that the RF pulse in fact contains a certain range of frequencies, thus all spins
rotating within that range become \excited", or tilted towards the x-axis. So, by playing
gradient(s) of an appropriate amplitude, and designing the RF pulse accordingly, one can
excite only a thin slice of the body, which correspond to the region containing all spins
that are in resonance with the RF pulse's range of frequencies. Excitation proles other
than \slices" may also be obtained (e.g., a pencil beam, or cylindrical excitation [33]), by
designing an appropriate gradient-pulse combination.
When the RF pulse is turned o, M
xy
starts to rotate (at the Larmor frequency)
around the z-axis, as the net magnetization starts to realign with B
0
. This rotating
magnetization generates an oscillating signal, which can be detected by the receive coil.
The frequency content of the received signal can be used to obtain spatial information
about the excited portion of the body. In order to frequency-encode spatial information,
gradients are also played during signal acquisition. These are called readout gradients.
13
For imaging a slice perpendicular to the z-axis (an axial image) , G
z
is played during
excitation (for slice-selection), and G
x
and G
y
are played during acquisition. These can
be switched, for acquiring sagittal or coronal images, or all three gradients may be used
during both excitation and acquisition to image oblique planes.
When the readout gradients are played, the acquired signal at a particular time instant
correspond to the sum of dierent sinusoidal signals generated by spins located at dierent
regions of the body, each rotating at dierent frequencies corresponding to their spatial
location. If an axial slice is being acquired, for example, the demodulated signal value
is equivalent to a sample of the Fourier transform M(k
x
;k
y
) of the cross-sectional image
m(x;y). In this case, by changing the amplitudes of G
x
and G
y
during acquisition, one
may acquire dierent samples of M(k
x
;k
y
). In fact, by playing G
x
and/or G
y
, one can
move along the k
x
-k
y
plane (which is known in MRI as k-space), collecting samples of
M(k
x
;k
y
). When enough samples of M(k
x
;k
y
) have been collected, ideally covering the
entire k-space, an inverse Fourier transform produces m(x;y).
The required coverage of k-space, and the number of samples, depend on the specied
spatial resolution and eld-of-view. For low spatial resolution imaging, only the central
portion of k
x
-k
y
needs to be sampled. For higher spatial resolution, the periphery of
k-space must also be covered. The eld of view is associated with the spacing between
samples. For a larger eld-of-view, k-space needs to be more densely sampled, requiring
an increased number of samples. If k-space is not suciently sampled, and the resultant
eld-of-view is not large enough to cover the entire object, overlap in spatial domain is
observed (aliasing).
14
Because signal amplitude is lost as the net magnetization realigns with B
0
(this is
called relaxation), multiple acquisitions (excitation + readout) may be needed in order
to cover the entire k-space. Dierent trajectories are more ecient in covering k-space
than others. For example, spiral imaging, which uses oscillating gradients to achieve spiral
k-space trajectories (Figure 2.1b), are generally faster than 2DFT imaging, i.e., require
fewer acquisitions. In 2DFT imaging, each acquisition readout acquires a single line of
k-space, sampling k
x
-k
y
in a Cartesian fashion (Figure 2.1a). This is generally slower,
but may be advantageous in some applications with respect to the nature of associated
image artifacts. The fashion in which RF pulses and gradients are played is called pulse
sequence. The time between acquisitions is called repetition time (TR).
a
b
G
z
G
x
G
y
RF
k
x
k
y
Figure 2.1: Timing diagram (left) and correspondent k-space trajectories (right) for
(a) 2DFT [55], and (b) spiral acquisitions.
15
2.2 Mathematical formalism
As discussed on section 2.1, the acquired MR signal s(t) at a particular time instant
corresponds to a sample of the Fourier transform M(k
x
;k
y
) of the cross-sectional image
m(x;y):
M(k
x
;k
y
) =
Z
x
Z
y
m(x;y)e
j2(kxx+kyy)
dxdy: (2.1)
The Fourier coordinates k
x
and k
y
vary with time, according to the zeroth moment
of the readout gradients G
x
and G
y
:
k
x
(t) =
2
Z
t
0
G
x
()d (2.2)
k
y
(t) =
2
Z
t
0
G
y
()d: (2.3)
These equations explain how the gradients can be used to \move" along k-space, as
discussed in the previous section. This formalism can be generalized for any combination
of the gradients G
x
, G
y
and G
z
as:
M(
~
k
r
) =
Z
~ r
m(~ r)e
j2
~
kr~ r
d~ r (2.4)
~
k
r
(t) =
2
Z
t
0
~
G
r
()d; (2.5)
where
~
G
r
is the oblique gradient resultant from the combination of the G
x
, G
y
and G
z
16
gradients, and ~ r is its correspondent axis, along which the linear variation in magnetic
eld intensity is perceived.
Given a spatial position function ~ r(t) and a magnetic eld gradient
~
G
r
(t), the mag-
netization phase is:
(~ r;t) =
Z
t
0
~
G
r
()~ r()d; (2.6)
For static spins,~ r(t) is constant (~ r), and this becomes:
=
~ r
Z
t
0
~
G
r
()d (2.7)
= 2
~
k
r
~ r; (2.8)
as in the exponential in equation 2.4.
2.3 Principles of MR
ow imaging
The basic principles of quantitative
ow measurement using magnetic resonance were rst
proposed by Singer [67] and Hahn [26] in the late 1950's. However, clinical applications of
MR
ow quantitation weren't reported until the early 1980's [52,54,68,77]. Current MR
ow imaging methods are based on the fact that spins moving at a constant velocity accrue
a phase proportional to the velocity times the rst moment of the gradient waveform along
the direction in which they are moving.
17
For spins moving along the ~ r-axis with a constant velocity~ v, and initial position ~ r
0
,
we can write~ r(t) = ~ r
0
+~ vt. Rewriting equation 2.6, for t =t
0
:
=
Z
t
0
0
~
G
r
(t) (~ r
0
+~ vt)dt (2.9)
=
~ r
0
Z
t
0
0
~
G
r
(t)dt +
~ v
Z
t
0
0
~
G
r
(t)tdt (2.10)
=
~ r
0
~
M
0
+
~ v
~
M
1
; (2.11)
where
~
M
0
and
~
M
1
are the zeroth and rst moments of the ~ r-gradient waveform at echo
time, respectively. Thus, if a gradient with null zeroth moment is used (e.g., a bipolar
gradient, aligned with~ v), the phase accrued for a constant velocity spin is =
~ v
~
M
1
.
Therefore, if a bipolar gradient waveform is played between the excitation and the
readout, the phase measured in a pixel of the acquired image is directly proportional to
the velocity of the spins contained within its correspondent voxel. However, factors other
than
ow (such as inhomogeneities of the magnetic eld) may cause additional phase
shifts that would cause erroneous interpretation of the local velocity [62].
2.4 Phase contrast
The phase contrast method addresses the problem mentioned above by using two gradient-
echo data acquisitions in which the rst moment of the bipolar gradient waveform is varied
between measurements [58]. The velocity in each voxel is measured as:
18
v(x;y) =
a
(x;y)
b
(x;y)
(M
a
1
M
b
1
)
; (2.12)
where
a
(x;y) and
b
(x;y) are the phase images acquired in each acquisition, and M
a
1
and M
b
1
are the rst moment of the bipolar gradients used in each acquisition.
2.5 Fourier velocity encoding
While phase contrast provides a single velocity measurement associated with each voxel,
Fourier velocity encoding (FVE) [51] provides a velocity histogram for each spatial loca-
tion, which is a measurement of the velocity distribution within each voxel.
FVE involves phase-encoding along a velocity dimension. Instead of only two acqui-
sitions, as in phase contrast, multiple acquisitions are performed, and a bipolar gradient
with a dierent amplitude (and rst moment) is used in each acquisition. Equation 2.10
can be rewritten as:
(~ r;~ v;t) = 2(
~
k
r
~ r +
~
k
v
~ v); (2.13)
where
~
k
v
is the velocity frequency variable associated with ~ v, and is proportional to the
rst moment of
~
G
r
(t):
~
k
v
=
2
~
M
1
: (2.14)
19
Each voxel of the two-dimensional image is associated with a distribution of veloc-
ities. This three-dimensional function m(x;y;v) is associated with a three-dimensional
Fourier space M(k
x
;k
y
;k
v
). Thus, an extra dimension is added to k-space, and multi-
ple acquisitions are required to cover the entire k
x
;k
y
;k
v
. In order to move along k
v
,
a bipolar gradient with the appropriate amplitude (and rst moment) is played before
the k
x
-k
y
readout gradients, in each acquisition. Placing the bipolar gradient along the
z-axis will encode through-plane velocities. Placing the bipolar gradient along x ory will
encode in-plane velocities. Oblique
ow can be encoded using a combination of bipolar
gradients along the x, y and z axes. The design of FVE bipolar gradients is discussed in
section 2.5.1.
Each acquisition alongk
v
is called a velocity encode. The number of required velocity
encodes depends on the desired velocity resolution and velocity eld-of-view (the maxi-
mum range of velocities measured without aliasing). For example, to obtain a 25 cm/s
resolution over a 600 cm/s eld-of-view, 24 velocity encodes are needed. The velocity
distributions along the cross-sectional image m(x;y;v) is obtained by inverse Fourier
transforming the acquired data M(k
x
;k
y
;k
v
). If cine imaging [25] is used, measurements
are also time resolved, resulting in a four-dimensional dataset: m(x;y;v;t).
The main drawback of FVE is scan time, as k
x
-k
y
should be fully sampled at eachk
v
plane. As discussed in section 1.3.2, dierent approaches to accelerating FVE have been
proposed. These techniques are typically inecient in spatially separating
owing blood
from nearby static tissue. Furthermore, they are not capable of resolving multiple
ows
in a single acquisition. The spiral FVE method, proposed in the next chapter, addresses
these limitations.
20
2.5.1 FVE bipolar gradient design
The size and shape of the velocity encoding bipolar gradients are calculated from the
specied velocity eld-of-view (v
FOV
) and velocity resolution (v
res
). The required number
of velocity encode samples isN =v
FOV
=v
res
. Let k
v
be the increment ink
v
for dierent
velocity encodes, then:
k
v
=
1
v
FOV
: (2.15)
Thus, if N encodes are acquired, the maximum value of k
v
being sampled is:
k
vmax
=
N
2
k
v
: (2.16)
From (2.15) and (2.16):
k
vmax
=
N=2
v
FOV
: (2.17)
The n-th sample of k
v
is k
v
[n] =k
vmax
+n k
v
, for n = 0; 1; 2; ; (N 1).
Plugging (2.16) into this expression, we get k
v
[n] =
N
2
k
v
+n k
v
= (nN=2)k
v
.
Then, from (2.15) we get:
k
v
[n] =
nN=2
v
FOV
: (2.18)
Now, letG
max
(t) be the bipolar gradient waveform correspondent tok
vmax
, then from
(2.14):
21
k
vmax
=
2
Z
TE
0
G
max
(t)tdt: (2.19)
Let us write k
v
[n] as a scaled k
vmax
:
k
v
[n] =
n
k
vmax
: (2.20)
The scaling factor
n
is thus
kv[n]
kvmax
. If we plug-in (2.17) and (2.18), we get:
n
=
nN=2
N=2
=
2
N
n 1: (2.21)
Finally, plugging (2.19) into (2.20), we get:
k
v
[n] =
2
Z
TE
0
[
n
G
max
(t)]tdt: (2.22)
Thus, the bipolar gradient G
n
(t) that corresponds to a particular k
v
[n] can be com-
puted as
n
G
max
(t), a scaled version of the largest bipolar. The bipolar gradientG
max
(t)
is formed by concatenating two identical unipolar subpulses, with area A, amplitude G,
and duration . For positive values of M
1
, we use a negative subpulse immediately fol-
lowed by a positive subpulse. For practical values of velocity resolution, the FVE bipolar
is an isosceles trapezoid. If maximum gradient amplitude and slew rate are used, G is 4
G/cm and rise time (t
r
) is 276 s. Solving (2.19), it can be shown that k
vmax
=
2
A.
Plugging in A =G( +t
r
), and solving for , we get:
=
t
r
2
+
1
2
s
t
r
2
+ 4
k
vmax
=2G
: (2.23)
22
Finally, is rounded up to the next multiple of 4 s (the sampling rate). By doing
that, the resultant rst moment (A) increases, soG is slightly reduced to adjustA, and
the resulting G
max
(t) matches the specied rst moment exactly.
2.6 Experimental setup
Most experiments were performed on a GE Signa 3T EXCITE HD system, with gradients
capable of 40 mT/m amplitude and 150 T/m/s slew rate, and a receiver with sampling
interval of 4s. Sequence designs were optimized for this scanner conguration. The body
coil was used for RF transmission in all studies. An 8-channel phase array cardiac coil was
used in cardiac studies, but data from only 1 or 2 elements were used in reconstruction.
Similarly, a 4-channel neck coil was used in carotid studies, but only 2 channels were used
in reconstruction. In phantom studies, a single channel 5-inch surface coil was used.
In the patient and phantom experiments presented in chapters 3 and 5, respectively,
a GE Signa 1.5T LX system with the same gradient and receiver conguration was used,
and acquisition was performed using a 5-inch surface coil.
The institutional review boards of the University of Southern California and Stan-
ford University approved the imaging protocols. Subjects were screened for magnetic
resonance imaging risk factors and provided informed consent in accordance with insti-
tutional policy.
23
Chapter 3
Slice-selective FVE with spiral readouts (spiral FVE)
This chapter introduces spiral Fourier velocity encoding (spiral FVE), a method for rapid
measurement of cardiovascular blood
ow using magnetic resonance imaging (MRI). We
present practical implementations for measuring blood
ow through the aortic valve
(heart) and carotid arteries (neck), and comparison with Doppler ultrasound, the current
non-invasive gold standard. We also show experiments for evaluating the most appropri-
ate view-ordering schemes for aortic and carotid
ow. The proposed method is demon-
strated in healthy volunteers and patients. We also propose algorithms for automatic
in-plane localization of the
ow.
3.1 Introduction
In order to address the limitations of existing
ow imaging methods (Chapter 1), we
propose the use of slice-selective FVE with spiral acquisitions. The proposed method is
capable of acquiring fully localized, time-resolved velocity distributions in a short breath-
hold. Scan-plane prescription is performed using classic protocols, and a semi-automatic
algorithm is used for in-plane localization of the
ow. The method is demonstrated in
24
vivo, and the results are qualitatively compared with Doppler ultrasound. Patient results
show that this technique can detect
ow distributions in stenotic jets.
Spiral FVE can acquire cardiac velocity distributions in 12 heartbeats, with 26 ms
temporal resolution and 25 cm/s velocity resolution over a 600 cm/s eld-of-view, using
single-shot spiral readouts. This time-velocity resolution is sucient for the diagnostic
assessment of valvular stenosis, where peak velocities are on the order of 200 to 600 cm/s.
The velocity eld-of-view, and/or the temporal, spatial and velocity resolutions can be
improved by increasing the acquisition time. For example, in carotid studies scan time
was increased to 48 heartbeats (without breath-hold) to achieve greater spatial resolution,
by using multiple spiral interleaves.
3.2 Methods
3.2.1 Imaging
The spiral FVE imaging pulse sequence (Figure 3.1) consists of a slice-selective excitation,
a velocity-encoding bipolar gradient, a spiral readout, and refocusing and spoiling gradi-
ents. The dataset corresponding to each temporal frame is a stack-of-spirals in k
x
;k
y
;k
v
space (Figure 3.2). The bipolar gradient eectively phase-encodes ink
v
, while each spiral
readout acquires one \platter" in k
x
;k
y
.
The excitation achieved a 5 mm slice thickness and 30
ip angle, with a 0.5 ms RF
pulse and 1 ms gradient. Through-plane velocity encoding was implemented using a large
bipolar pulse along the z direction that was scaled to achieve dierent k
v
encodes. The
bipolar gradients were designed as discussed in section 2.5.1. The velocity resolution is
25
RF
Gz
Gx
Gy
a b c d
1 ms 2.5 ms 8.1 ms 1.2 ms
Figure 3.1: Spiral FVE pulse sequence. It consists of (a) slice selective excitation, (b)
velocity encoding bipolar gradient, (c) spiral readout, and (d) refocusing and spoiler
gradients. This timing corresponds to the healthy volunteer cardiac studies in Figures
3.3 and 3.5.
k
v
k
x
k
y
1:
24:
Figure 3.2: Spiral FVE k-space sampling scheme. The dataset corresponding to each
temporal frame is a stack-of-spirals ink
x
;k
y
;k
v
space. Each spiral acquisition corresponds
to a dierent k
v
encode.
determined by the rst moment of the largest bipolar gradient, and the velocity eld-
of-view is determined by the increment in gradient rst moment for dierent velocity
encodes. A bipolar duration of 2.5 ms achieves a velocity resolution of 25 cm/s. Gradient
duration increases if velocity resolution is improved.
In cardiac scans, an 8.1 ms optimized uniform-density single-shot spiral acquisition [28]
acquires k
x
;k
y
at each velocity encode, and zeroth and rst moments are refocused in
0.5 ms. The readout and refocusing gradients were designed using free software developed
26
by Hargreaves (http://www-mrsrl.stanford.edu/brian/vdspiral/). A 0.65 ms spoiling
gradient [81] achieves a 6 phase-wrap over the slice thickness. The spoiling gradient was
not overlapped with the refocusing gradients, but this could be done to further shorten the
TR. In carotid scans, spatial resolution was increased by segmenting thek
x
;k
y
acquisition
in 4 spiral interleaves, across extended scan time. In this case, the spiral readout and
refocusing gradients durations are 7.6 ms and 0.9 ms, respectively. Total acquisition
time in this case is 4 times longer, and breath-holding was not used. The minimum TR
(approximately 13 ms) was used in all studies. Other scan dependent pulse sequence
parameters are listed in Table 3.1.
Table 3.1: Scan parameters used in the dierent spiral FVE studies.
Carotid Cardiac Cardiac
(Healthy) (Patient)
spiral interleaves 4 1 1
eld-of-view 20 cm 25 cm 20 cm
spatial resolution 2.5 mm 7 mm 6.5 mm
velocity eld-of-view 400 cm/s 600/800 cm/s 1200 cm/s
velocity encodes 24 24 32
velocity resolution 16.7 cm/s 25/33 cm/s 37.5 cms/s
TR 13.2 ms 12.8/12.5 ms 12.8 ms
k
v
encodes/heartbeat 1 2 4
interleaves/heartbeat 2 1 1
temporal resolution 26.4 ms 25.6/25 ms 51.2 ms
scan time 48 heartbeats 12 heartbeats 8 heartbeats
Prospective ECG gating was used to synchronize acquisitions with the cardiac cycle.
In cardiac studies, twok
v
levels were repeatedly acquired during each R-R interval in order
to resolve 25 to 35 cardiac phases and produce a cine dataset (Figure 3.3, discussed later).
In carotid studies, one k
v
level was acquired per heartbeat, and the spiral interleaves
27
were segmented across multiple heartbeats using a sequential interleaf order. The true
temporal resolution was 26 ms (2 TRs). Sliding window reconstruction [64] was used to
produce a new image every 13 ms.
3.2.2 Reconstruction
Reconstruction was performed in Matlab (Mathworks, Inc., South Natick, MA). Each
spiral interleaf is rst gridded [35] and inverse Fourier transformed to form an image x;y
for each temporal frame. This step converts the acquired data S
kx;ky
(k
v
;t) toS
x;y
(k
v
;t).
The operator manually denes a region of interest (ROI) in thex;y plane using the image
corresponding tok
v
= 0 andt = 0. Pixel intensities within the ROI are averaged at each
temporal frame, resulting in a 2D dataset: S
ROI
(k
v
;t) =
P
ROI
x;y
S
x;y
(k
v
;t). View-sharing
is then applied toS
ROI
(k
v
;t) to increase the number of temporal frames [64]. Saturation
eects [24] are compensated by normalizing the `
2
-norm of S
ROI
(k
v
) independently for
each temporal frame, which eectively normalizes each cardiac phase. In carotid studies,
signal from static tissue was discarded by subtracting the average value of S
ROI
(k
v
;t)
(along k
v
) from each temporal frame. S
ROI
(k
v
;t) is then zero-padded along the k
v
axis,
and an inverse Fourier transform produces s
ROI
(v;t). The time-velocity histogram for
the ROI isjs
ROI
(v;t)j, and for display purposes, smoother histograms are obtained by
interpolating along t [4]. Note that in carotid studies, which use multiple interleaves,
view-sharing is applied prior to gridding.
The reconstruction process can be repeated for each voxel, or for multiple regions of
interest, using the same data. The operator can be presented with a 2D color-overlay
image or video indicating voxels where high-speed
ow was detected, and use that as
28
reference to dene the regions of interest (see Figure 3.6, discussed later). In our imple-
mentation, the operator manually denes a large ROI over the heart or around the carotid
artery, and the best pixels within this ROI are selected to maximize the cost function
max
x;y
R
jvjv
1
R
RR=2
t=0
js
x;y
(v;t)j
2
dtdv
R
jvjv
2
R
RR=2
t=0
js
x;y
(v;t)j
2
dtdv
; (3.1)
wherev
1
andv
2
are thresholds that dene ranges of high and low velocities, respectively.
In words, the algorithm selects the pixels that maximize the ratio between the energy at
high velocities and the energy at low velocities, during the rst half of the cardiac cycle.
This cost function can be adjusted for dierent velocity ranges, or for dierent portions
of the cardiac cycle. For example, to detect regurgitant jets, a modied cost function
that locates high negative velocities during the second half of the R-R interval is:
max
x;y
R
vv
1
R
RR
t=RR=2
js
x;y
(v;t)j
2
dtdv
R
jvjv
2
R
RR
t=RR=2
js
x;y
(v;t)j
2
dtdv
: (3.2)
3.2.3 Experimental methods
The proposed method was evaluated in vivo, aiming at quantifying
ow through the
common carotid artery and the aortic valve. Scan-plane prescription was performed
using a real-time imaging sequence, and in-plane localization of the
ow was performed
using the proposed semi-automatic algorithm. The velocity eld-of-view was chosen based
on the targeted region of the body. For a severely stenosed heart valve, peak velocities
can reach up to 600 cm/s [22]. In the carotid arteries, peak velocities are slightly lower,
29
reaching up to 400 cm/s. As regurgitant jets don't overlap in time with forward
ow,
we used 600 cm/s and 400 cm/s as the velocity eld-of-view for imaging the aortic valve
and the carotids, respectively. These values could be increased by extending the scan
time, or by sacricing temporal, velocity and/or spatial resolutions. Scan parameters are
summarized in Table 3.1.
Two experiments were performed to determine the appropriate view-ordering and
interleaf-ordering schemes for both cardiac and carotid studies. In the rst experiment,
ow was measured through the aortic valve of a healthy volunteer, using one velocity
encode level per heartbeat, with a single-shot spiral readout, and over a 24-heartbeat
breath-hold. During reconstruction, 50% of the data was discarded in two dierent ways,
to simulate the sequential and interleaved view-ordering schemes and compare resulting
artifacts. In the second experiment,
ow was measured through the carotid artery of
another healthy volunteer, using a 4-interleaf spiral FVE acquisition. The measurement
was performed twice, and in each acquisition, a dierent view-ordering scheme was used.
In the rst acquisition, one spiral interleaf was acquired per heartbeat, and two dierent
k
v
levels were encoded throughout each R-R interval. In the second acquisition, two
spiral interleaves were acquired per heartbeat, encoding one k
v
level per cardiac cycle.
Reconstructed velocity histograms were compared with respect to data inconsistency
artifacts related to view-ordering.
In cardiac and carotid experiments, Doppler ultrasound was used as a gold standard
and was qualitatively compared with the proposed method. Aortic out
ow (at the valve
plane) was studied in 7 volunteers and 2 patients with aortic stenosis. Carotid
ow was
studied in 3 healthy subjects.
30
3.3 Results
Results from the cardiac view-ordering experiment are shown in Figure 3.3. When the
k
v
levels are acquired in a sequential fashion, ghosting artifacts due to data inconsistency
appear shifted by 1/2 of the velocity eld-of-view (Figure 3.3a). When they are acquired
in an interleaved fashion, the artifacts overlap with the true
ow prole (Figure 3.3b),
blurring the velocity histogram and making the detection of the peak velocity and other
ow related parameters less precise. Using the sequential ordering and an appropriate
velocity eld-of-view, the artifacts will not overlap with the
ow prole and may be easily
identied and masked out. Thus, we used the sequential view-ordering in all subsequent
cardiac acquisitions.
The results from the carotid view-ordering experiment are shown in Figure 3.4. It
can be noticed that the ghosting artifacts that arise in the velocity histograms when
acquiring two dierent velocity encode levels during the same heartbeat (Figure 3.4a)
do not appear when we used interleaf segmentation instead (Figure 3.4b). In contrast to
view-sharing alongk
v
, which causes ghosting in the velocity direction, view-sharing among
spiral interleaves introduces swirling artifacts in image domain, reducing the eective
unaliased spatial eld-of-view by a factor of 2. However, only moving spins (
owing
blood) experience these artifacts, and vessels on the same side of the neck are relatively
close to each other. Therefore, the unaliased eld-of-view is wide enough to enclose all
vessels on one side of the neck, so that quantitation of a vessel of interest will not be
disturbed by
ow from neighboring vessels (e.g. measurement of
ow in the left carotid
arteries will not be disturbed by
ow in the the left jugular vein). In order to suppress
31
•
•
•
• • • 1 2 1 2 1 2 1 2 HB #1
• • • 23 24 23 24 23 24 23 24 HB #12
26 ms
trigger
Frame #1 Frame #3
Frame #2 Frame #4
• • • 3 4 3 4 3 4 3 4 HB #2
• • • 5 6 5 6 5 6 5 6 HB #3
time (ms)
0 200 400 600 800
−300
−150
0
150
300
velocity (cm/s)
•
•
•
• • • 1 13 1 13 1 13 1 13
• • • 12 24 12 24 12 24 12 24
trigger
Frame #1 Frame #3
Frame #2 Frame #4
• • • 2 14 2 14 2 14 2 14
• • • 3 15 3 15 3 15 3 15
time (ms)
0 200 400 600 800
−300
−150
0
150
300
velocity (cm/s)
a
b
HB #1
HB #12
HB #2
HB #3
26 ms
Figure 3.3: Comparison of artifacts in dierent view-orderings, in a 12-heartbeat spiral
FVE acquisition using single-shot spiral readouts. Each box represents the acquisition of
one k
v
level, during one imaging TR. A sliding window is used to produce a new image
every TR. Acquiring the k
v
levels in a sequential fashion (a), ghosting artifacts appear
shifted by 1/2 of the velocity eld-of-view. By acquiring them in an interleaved fashion
(b), the artifacts overlap with the true
ow prole. The aortic valve
ow proles shown
are from a fully-sampled 24-heartbeat healthy volunteer acquisition, and 50% of the data
was appropriately discarded to simulate each of the view-ordering schemes.
signal from the opposite side of the neck, we separately reconstruct data from the left
and right neck-coil elements. This uses the receiver coil sensitivity prole to avoid spiral
view-sharing artifacts. In face of these observations, we used spiral interleaf view-sharing
in all subsequent carotid studies.
Representative in vivo results are compared with Doppler ultrasound in Figure 3.5.
Large ROI's were manually specied around the entire heart and the right side of the
neck, respectively, and the in-plane localization algorithm was able to successfully pin-
point voxels containing the
ows of interest. The MRI measured time-velocity histograms
32
time (ms)
0 200 400 600
time (ms)
0 200 400 600
−100
−50
0
50
100
velocity (cm/s)
a b
Figure 3.4: Comparison of dierent view-orderings for multi-shot spiral FVE, in a healthy
volunteer carotid study. When two or morek
v
levels are acquired during the same heart-
beat (a), velocity distribution changes between consecutive TRs cause ghosting artifacts
along the velocity axis (arrow). This artifact is not seen if, in the same heartbeat, dierent
spiral interleaves, but only one k
v
encoding, are acquired (b).
show good agreement with the ultrasound measurements, as the peak velocities and the
shape of
ow waveforms were comparable to those observed in the ultrasound studies.
c
0 200 400 600 800
−20
20
60
100
140
Ultrasound
Ultrasound
time (ms)
velocity (cm/s) velocity (cm/s)
a
b
−50
0
50
100
150
0 200 400 600 800
−50
50
150
−20
60
140
0 400 800 200 600
0 400 800 200 600
FVE
FVE
Figure 3.5: Comparison of the spiral FVE method with Doppler ultrasound, in healthy
volunteer studies: (a) aortic valve and (b) carotid artery. Peak velocity and time-velocity
waveforms are in good agreement.
Figure 3.6 illustrates spiral FVE's ability of detecting dierent regions of
ow from a
single dataset. A dierent
ow distribution was calculated for each voxel, and the distribu-
tions from single voxels from dierent ROIs are displayed. Each voxel was automatically
33
selected within the specied ROIs using the
ow localization algorithm described previ-
ously. Red and blue dots indicate voxels where ascending and descending blood
ow was
detected, respectively, and the color intensity of each dot indicates the highest velocity
detected in that voxel in a particular temporal frame. This sort of representation can
be used by the operator to facilitate the process of manually specifying the ROI. As the
data is temporally resolved, the operator can step through the cardiac cycle and visualize
the dierent regions of
ow during systole and diastole, for example. A modied in-plane
localization algorithm could be designed to correct for in-plane motion of the region of
interest during the cardiac cycle.
Figure 3.7 shows the time-velocity distribution measured through the aortic valve of
a patient with aortic stenosis. This result demonstrate that spiral FVE can accurately
detect complex
ow, as a high-speed jet with a wide distribution of velocities is clearly
visible.
3.4 Discussion
In spiral FVE, there is an important trade-o between velocity resolution, temporal
resolution, and scan time (Figure 3.8). This trade-o also involves other scan parameters,
such as velocity eld-of-view, number of spiral interleaves, spiral readout duration, spatial
resolution, and spatial eld-of-view. Velocity resolution can be improved in many ways,
such as increasing the breath-hold duration to acquire more k
v
levels, or by reducing
the velocity eld-of-view. Temporal resolution can be made as high as one TR duration
34
a
b
0 400 800
−200
0
200
0 400 800
−200
0
200
0 400 800
−200
0
200
−150
0
150
−80
0
80 0 400 800
150
−100
0
0 400 800
150
−100
0
0 400 800
150
−100
0
0 400 800
150
−100
0
Figure 3.6: Multiple
ow distributions obtained from a single spiral FVE dataset: (a)
heart; (b) neck. For each voxel in the images, a
ow distribution was calculated, and
the red and blue dots indicate voxels where ascending and descending blood
ow was
detected, respectively. The color intensity of each dot indicates the highest velocity
detected in that voxel in a particular temporal frame (indicated by the white dashed
lines). Multiple ROIs were specied around dierent regions of the heart and the neck,
and the
ow distributions from voxels automatically selected from each ROI are shown.
500
0
-200
velocity (cm/s)
0 250 500
time (ms)
Figure 3.7: Evaluation of spiral FVE in an aortic valve study of a patient with aortic
stenosis. Note the high-speed jet with a wide distribution of velocities.
35
(13 ms) by segmenting the k
v
encodes across additional R-R intervals (longer breath-
holds), or by compromising velocity resolution or eld-of-view. Spatial resolution can
be improved by reducing the spatial eld-of-view, or by increasing the number of spiral
interleaves, which would require compromising other scan parameters such as scan time,
temporal resolution and/or velocity resolution.
10 30 50 70
0
20
40
60
80
) s / m c ( n o i t u l o s e r y t i c o l e v
8 heartbeats
12 heartbeats
16 heartbeats
24 heartbeats
32 heartbeats
temporal resolution (ms)
Figure 3.8: Spiral FVE trade-os between temporal resolution, velocity resolution, and
breath-hold duration. Velocity resolution corresponds to a 600 cm/s eld-of-view, tem-
poral resolution corresponds to a 8.1 ms spiral readout, and scan time corresponds to a
single-shot spiral acquisition. The arrow indicates the conguration used in the cardiac
study in Figure 3.5 (2TR temporal resolution, 24 velocity encodes).
Artifacts and loss of temporal resolution due to view-sharing can be avoided or cor-
rected using dierent approaches. Acquiring multiple k
v
levels per heartbeat reduces
scan time, but causes blurring along the time axis and ghosting along the velocity axis.
Blurring is caused by the reduction in temporal resolution, and ghosting artifacts arise
when the velocity distribution changes between the acquisition of consecutive velocity
encodes (Figure 3.3). Although ghosting is not seen in the multiple interleave results if
appropriate view-ordering is used (Figure 3.4), the temporal resolution is still lower, and
this may cause blurring when the velocity distribution changes rapidly. Both ghosting
36
and blurring can be overcome by acquiring only one view per heartbeat, but this would
require increase in scan time or reduction in velocity resolution. As an alternative, these
artifacts may be corrected using techniques that exploit ecient use of k-t space, such as
UNFOLD [47,75] andk-t BLAST [76]. Applications of these methods have been demon-
strated to both cylindrical beam and slice-selective FVE [27,45], and similar approaches
can be directly applied to spiral FVE to reduce breath-hold duration even further, without
reduction in velocity or temporal resolution.
As the spiral readouts are considerably long, another potential issue in the proposed
method is blurring in image domain, due to o-resonance. Because SNR was not a
limiting issue for the applications we have presented, spiral FVE may perform better at
lower eld strengths where there is reduced o-resonance. At 3T, localized shimming
and o-resonance correction techniques can be used to reduce blurring. Furthermore,
readout duration can be reduced by decreasing the spatial resolution or eld-of-view, or
by using variable-density spirals [73]. Another alternative is to use multiple short spiral
interleaves, which would require longer scan times, but parallel imaging techniques [60,
66] can potentially accelerate acquisition if multi-channel receiver coils are used. This
approach also has the benet of allowing increase in the frame rate, as the number of
imaged cardiac phases is limited by the minimum TR. Another possible solution to the
o-resonance problem is the use of EPI trajectories, which produce dierent o-resonance
eects (geometric warping) [20], but are also more sensitive to artifacts from in-plane
ow
or motion.
Another noticeable artifact in spiral FVE is Gibbs ringing along the velocity dimen-
sion. These artifacts can be less noticeable if velocity resolution is increased, which would
37
also improve the ability to visualize features in the
ow waveform and the precision to re-
solve the peak velocity, but would require longer breath-holds. Alternatively, the velocity
resolution can be improved by using variable-density sampling along k
v
[17], or partial
k-space techniques [57]. Another approach to reducing ringing artifacts is to window the
k
v
samples before applying the inverse Fourier transform [5]. However, windows with
lower sidelobes generally have wider mainlobes, which would cause blurring along the
velocity axis and consequent reduction in velocity resolution.
One drawback of the proposed method is the requirement of cardiac gating and breath-
holding. Cardiac gating does not work well in patients with arrhythmias, and breath-
holding may cause hemodynamic changes and is not possible for some patients [44].
However, arrhythmia rejection [14] and respiratory gating schemes may overcome this
problems, at the cost of increased scan time.
3.5 Conclusions
We have demonstrated that spiral FVE can resolve fully-localized time-velocity his-
tograms in short breath-holds. Evaluation of cardiac aortic valve
ow was performed in
12-heartbeat breath-holds, and evaluation of carotid
ow was performed in 48-heartbeat
scans, with diagnostically useful temporal and velocity resolutions. The resulting peak
velocity and velocity waveforms were accurate compared to Doppler ultrasound. Patient
results show that this technique can measure
ow distributions in stenotic jets, detecting
multiple velocities within a voxel.
38
It was also demonstrated that spiral FVE can resolve multiple sources of through-
plane
ow in a single slice acquisition, which is particularly useful when evaluating heart
valves. A semi-automatic algorithm was developed for in-plane ROI selection, and can be
adapted for dierent
ows of interest. In carotid studies, spatial resolution was improved
using multiple spiral interleaves. Even higher spatial resolution can be achieved in order
to evaluate stenosis in smaller vessels such as the coronary arteries.
Preliminary volunteer and patient results indicate that the proposed method of spiral
FVE may have an important role in the rapid and accurate quantitation of abnormal
valvular
ow, abnormal vascular
ow, and congenital
ow defects using magnetic reso-
nance imaging.
39
Chapter 4
Accelerated spiral FVE
This chapters aims to improve the spiral FVE method, by (1) increasing the spatial reso-
lution, (2) reducing o-resonance eects, (3) increasing the velocity eld-of-view, and (4)
improving the temporal resolution, without increasing the scan time. This is made pos-
sible using three dierent techniques: variable-density sampling, temporal acceleration,
and partial Fourier reconstruction. A total 18-fold acceleration was achieved, without
introducing signicant artifacts. Improvements that would require a 216-heartbeat ac-
quisition were achieved in a single 12-heartbeat breath-hold. The accelerated method was
demonstrated in healthy volunteers.
4.1 Introduction
Several approaches to accelerating FVE imaging have been recently proposed. Hansen et
al. achieved 8-fold acceleration in 2DFT-based FVE usingk-t BLAST [27]. However, this
approach requires an additional breath-hold for acquisition of training data. MacGowan
et al. proposed using UNFOLD with a pre-processing step to accelerate real-time FVE
by 2-fold [45]. The proposed pre-processing step consists in an attempt to remove the
40
pulsatility of the
ow waveform, because pulsatile
ow contains high temporal frequency
components, which would otherwise be lost when the UNFOLD method { implemented as
a one-dimensional temporal low pass lter { is applied. This approach may not be appli-
cable in vivo (specially in patients), as obtaining an accurate estimate of
ow pulsatility
may be an issue when a distribution of velocities exist [10]. Gamper et al. achieved 6.8-
fold acceleration in 2DFT FVE using compressed sensing [23]. However, this approach
is very computationally intense, specially in the context of non-Cartesian imaging. Di-
Carlo et al. achieved 2-fold acceleration in real-time FVE using variable-density sampling
along the velocity dimension [17], and proposed using thresholding to mask aliasing arti-
facts due to undersampling. This may cause distortions, as
ow jets may present signal
amplitudes lower than the artifacts.
As described in Chapter 3, the spiral FVE method can measure aortic
ow in a
single 12-heartbeat breath-hold, without acceleration. Current limitations include: (1)
insucient velocity eld-of-view (400 cm/s); (2) low in-plane resolution (7 mm); (3) long
readouts (8 ms), which causes spatial blurring at 3T due to o-resonance eects, and also
increases the minimum TR (13 ms); and (4) moderate temporal resolution (26 ms), due
to the use of view-sharing, which causes blurring along the temporal dimension (t), and
ghosting along the velocity dimension (v).
This chapter aims to address all these limitations. The velocity eld-of-view was
increased to600 cm/s, the in-plane resolution was improved to 3.6 mm, the readout du-
ration was reduced to 4 ms, and the temporal resolution was improved to 9 ms. Without
acceleration, this would require a prohibitively long 216-heartbeat acquisition. However,
41
spiral FVE datasets are four-dimensional (k
x
,k
y
,k
v
,t), which makes this method partic-
ularly suitable for accelerated acquisition [27]. We achieved 18-fold acceleration, using a
combination of three dierent techniques: variable-density spirals [73], partial Fourier [57]
alongk
v
, and temporal acceleration through a specically designed implementation of the
UNFOLD method [47,75], which does not require pulsatility removal. Such acceleration
factor allows the improved acquisition to be performed in only 12 heartbeats, i.e., with-
out any increase in scan time compared to the original implementation. The proposed
acceleration method was demonstrated in healthy volunteers.
4.2 Methods
4.2.1 Acceleration
Variable-density sampling In order to improve the spatial resolution, we replaced
the single-shot uniform-density spiral readout with a 3-interleaf variable-density spiral ac-
quisition [73]. The readout duration was reduced from 8 to 4 ms in order to reduce spatial
blurring due to o-resonance, and to allow higher improvement in temporal resolution, by
reducing the minimum TR from 13 to 9 ms. Gradient waveforms were designed using free
software developed by Hargreaves (http://www-mrsrl.stanford.edu/brian/vdspiral/).
The eld-of-view (FOV) was varied linearly, from 25 cm at the center of k-space, to
6 cm at the periphery, achieving 3.6 mm spatial resolution. The FOV design parame-
ters may be optimally selected based on dierent criteria such as incoherence of spatial
aliasing, which can be specially useful if regularized iterative reconstruction is used [36].
However, such problem is out of the scope of this work.
42
Temporal acceleration In order to improve the temporal resolution and increase
the velocity eld-of-view (FOV
v
), we used a novel implementation of the UNFOLD
method [47, 75], specically designed for spiral FVE. UNFOLD reduces scan time by
making ecient use ofk-t space, and can be very successful in the context of spiral FVE
due to the high dimensionality of this imaging method (x-y-v-t, where x and y are in-
plane spatial dimensions). We designed a view-ordering scheme that reduces overlap in
v-f space (f denotes temporal frequency). It consists in alternating spiral interleaves
and k
v
encodes for each cardiac phase, according to Figure 4.1a. As a result, the point
spread function (in x;y;v-f space) will be such that aliasing replicas due to temporal
undersampling are separated from the main lobe both in velocity (by half the velocity
eld-of-view) and in temporal frequency (by half the sampling frequency) (Figure 4.1b).
Aliasing components also appear closer to the main lobe, but these are expected to be
localized inv-f space (i.e., narrow temporal frequency bandwidth, and with a small range
of velocities). This is because they are more likely to correspond to static or slow moving
spins, as their correspondent point spread functions in x,y spread around the periphery
of the spatial eld-of-view, but are null at the center, where the pulsatile
ow is located.
Valvular
ow is expected to have both a broad temporal frequency bandwidth and a
wide range of velocities, but its expected footprint in v-f space suggest that overlapping
with aliasing components will be minimal (Figure 4.2). While pusatile plug
ow contain
high temporal frequency components, typical peak velocities are low. On the other hand,
stenotic and regurgitant jets may present very high peak velocities, but are generally
temporally smooth, and therefore have a narrow temporal frequency bandwidth. Using
a two-dimensional lter that has a broad bandwidth (50 Hz) for low velocities (below
43
interleaf 1
interleaf 2
interleaf 3
kv encodes
time
a
b
400 200 0 −200 −400
−60
−600
−40 −20 0 20 40
−40 dB
−35 dB
−30 dB
−25 dB
−20 dB
−15 dB
−10 dB
−5 dB
0 dB
600
60
25 cm
25 cm
temporal frequency (Hz)
velocity (cm/s)
y
x
Figure 4.1: Proposed view-ordering scheme for accelerated spiral FVE (a) and correspon-
dent point-spread function (b). In (a), each color represents a dierent variable-density
spiral interleaf, and darker tones indicate \views" that were discarded in the partial
Fourier experiments (i.e., need not be acquired). Views aligned in k
v
are acquired se-
quentially throughout the cardiac cycle. Views aligned in time (same cardiac phases) are
acquired in dierent heartbeats. In (b), each square shows the point-spread function in
x;y for a particular velocity-frequency (v-f) coordinate. The main aliasing replicas are
separated from the main-lobe by half the velocity eld-of-view, and half the temporal
frequency bandwidth, which reduces overlaps and facilitates ltering.
150 cm/s), and a narrow bandwidth (15 Hz) for higher velocities, it is possible to
lter most of the aliasing components, while preserving signal content (dashed lines in
Figure 4.2). Some aliasing at20 and40 Hz may remain, but since its temporal
frequency bandwidth is likely to be narrow, as discussed above, it may be ltered using
a tight zero-phase one-dimensional notch lter along t.
44
) z H ( y c n e u q e r f
velocity (cm/s)
0 6
0
0 0 6 -
0 0 6
l a m r o n
w o l f
n o i t a t i g r u g e r
s i s o n e t s
0 6 - 0
Figure 4.2: Temporal acceleration approach proposed for spiral FVE. The red, green
and blue ellipses illustrate the expected footprint of aortic valve
ow in v-f space, more
specically for plug
ow, stenotic
ow, and regurgitant jets, respectively. Yellow dots
represent the point-spread function for the undersampling strategy (Figure 4.1). Grey
ellipses represent aliasing components. The aliasing at20 and40 Hz has a small
footprint inv-f space because it is composed mainly of static tissue or slow moving
ow,
while aliasing components at60 Hz are exact replicas of the main signal. A 2D lter
(dashed lines) is capable of removing the aliasing components while preserving all signal
content.
Using the lters described above, the eective temporal resolution of the reconstructed
signal is 9 ms. In theory, the temporal resolution is lower (30 ms) for velocity components
above150 cm/s. In practice, lower temporal resolution at high velocities may prove
to be unnoticeable, as high-velocity
ow is typically temporally smooth. However, this
hypothesis remains to be validated by patient studies. Nevertheless, if view-sharing [64]
was used instead, the temporal resolution would be reduced to 50 ms, for the entire range
of velocities.
A dierent view-ordering scheme and alternative k-t lters are proposed in the Dis-
cussion session.
45
Partial Fourier Acquisition time can be reduced by 30-40% using partial Fourier along
the velocity dimension. This consists in acquiring only slightly more than half of the k
v
encodings, and synthesize the missing data using homodyne reconstruction [57]. In our
experiments, we actually acquired all the k
v
encodings (in longer breath-holds), and
retrospectively discarded 33% of the data (dark-colored squares in Figure 4.1a) before
reconstruction. Thus, we were able to compare the partial Fourier results with the fully
sampled data.
4.2.2 Reconstruction
Reconstruction was performed in Matlab (Mathworks, Inc., South Natick, MA). The
reconstruction process is illustrated in Figure 4.3. The acquired data (S(k
x
;k
y
;k
v
;t))
is rst re-sampled onto a Cartesian grid using gridding [35] with a Kaiser-Bessel kernel
designed for the largest FOV (25 cm). Each spiral interleaf is gridded separately, and
inverse Fourier transformed to form a spatial image for its corresponding k
v
;t coordi-
nate (S(x;y;k
v
;t)). The data corresponding to the two central k
v
values (k
v
= 0 and
k
v
=
1
FOVv
s/cm) are separately ltered using a 6-tap moving average lter that eectively
implements view-sharing [64]. A color-
ow video [65] is obtained from the ltered data.
The operator draws one or multiple regions-of-interest (ROIs) over the video, and pixel
values of S(x;y;k
v
;t) within each ROI are averaged, resulting in multiple 2D datasets:
S
ROI
i
(k
v
;t) =
P
ROI
i
x;y
S
x;y
(k
v
;t). Each of these 2D datasets is ltered using the 2D lter
and the notch lter described above. Saturation eects [24] are compensated by normal-
izing the `
2
-norm of the data in each cardiac phase. The data is then zero-padded along
the k
v
axis, and homodyne reconstruction [57] is used to produce each s
ROI
i
(v;t). The
46
time-velocity histogram for each ROI isjs
ROI
i
(v;t)j, and for display purposes, smoother
histograms are obtained by interpolating along t [4].
S(k
x
,k
y
,k
v
,t) DFT
xy
S(x,y,k
v
,t)
6-tap MA FIR 6-tap MA FIR
S(x,y,0,t)
S(x,y,FOV
v
,t)
phase contrast | . |
color overlay
operator ROIs
∑
ROI
2D filter
notch
homodyne
spline
zero-pad
S
ROI1
(k
v
,t)
s
ROI1
(v,t)
color-flow video
2D filter
notch
homodyne
spline
zero-pad
S
ROI2
(k
v
,t)
s
ROI2
(v,t)
2D filter
notch
homodyne
spline
zero-pad
S
ROI3
(k
v
,t)
s
ROI3
(v,t)
gridding
-1
-1
normalize normalize normalize
| . | | . | | . |
Figure 4.3: Reconstruction
ow-chart for accelerated spiral FVE. Time-velocity his-
tograms (s(v;t)) from multiple regions-of-interest may be obtained from a single spiral
FVE dataset (S(k
x
;k
y
;k
v
;t)). Such ROIs are prescribed by the operator, using a color-
ow video obtained from the same dataset.
4.2.3 Experimental methods
In order to evaluate the use of variable-density spirals in spiral FVE for improving the
in-plane resolution and reducing blurring due to o-resonance eects, we experimented
with dierent spiral trajectories (Table 4.1). Multiple datasets were obtained, measuring
47
the
ow through the aortic valve of a healthy volunteer. In each dataset acquisition (one
breath-hold), a dierent trajectory from Table 4.1 was used. A small velocity eld-of-view
was used in order to to restrict each acquisition to a single breath-hold. The obtained
data was qualitatively compared in spatial and time-velocity domains.
Table 4.1: Design parameters used to evaluate improvements from using variable-density
(VD) over uniform-density (UD) spiral trajectories in spiral FVE.
a b c d e reference
interleaves 1 1 1 2 3 6
sampling UD VD VD VD VD UD
readout 8 ms 8 ms 4 ms 4 ms 4 ms 4 ms
FOV 25 cm 256 cm 256 cm 256 cm 256 cm 25 cm
resolution 7 mm 4.7 mm 7 mm 4.7 mm 3.6 mm 3.6 mm
heartbeats 8 8 8 16 24 N/A
For evaluating the proposed temporal acceleration scheme, we measured the
ow
through the aortic valve of another healthy volunteer using the trajectory described in
Table 4.1e, and the undersampled view-ordering scheme illustrated in Figure 4.1a. The
velocity resolution and eld-of-view were set to 33 and 1200 cm/s, respectively. The data
was acquired in an 18-heartbeat breath-hold, and was reconstructed using view-sharing,
the proposed 2D lter (Figure 4.2), and using the proposed notch lter on top of the
2D lter. Fully sampled reference datasets (without temporal undersampling) were also
obtained, using the trajectories (e) and (reference) in Table 4.1. Such datasets would
require 108 and 216 heartbeats, respectively, to be acquired. Thus, we used a small
velocity eld-of-view (200 cm/s), which reduced scan times to 18 and 36 heartbeats,
respectively.
Partial Fourier acceleration was evaluated by discarding 12 of the 36k
v
encodes before
reconstruction, as illustrated in Figure 4.1a.
48
4.3 Results
The results of the experiment using trajectories a-e in Table 4.1 are shown in Figure 4.4.
The result in (a) was obtained using the same low-resolution, long readout, uniform-
density spiral trajectory used in Chapter 3. Using a variable-density trajectory to improve
the spatial resolution without reducing the readout length proved inecient, because of
signicant spatial blurring due to o-resonance (b). Reducing the readout length, we
observed less o-resonance eects, but signal from static tissue was still perceived in the
time-velocity histogram (c). Better spatial resolution was needed to provide better spatial
localization of
ow (d,e). However, in order to achieve such improvement in resolution
while restricting the readout length to 4 ms, multiple interleaf acquisitions are required.
Without temporal acceleration, this would either cause increase in scan time or loss of
temporal resolution.
a b c d
time(ms)
velocity (cm/s)
e
0
180
0 500
-80
250
Figure 4.4: Improvements from using variable-density spirals in the spatial images (top),
and correspondent time-velocity histograms (bottom). Each result (a-e) correspond to
its respective trajectory described in Table 4.1. O-resonance eects in both spatial and
time-velocity domains are observed in the long readout results (a-b). Improved resolution
and reduced o-resonance provide better spatial localization. Consequently, less signal
from static material is observed in the time-velocity histograms in (d) and (e).
49
The results in Figure 4.5 demonstrate how the proposed temporal acceleration scheme
is capable of providing 6-fold acceleration in multi-interleaf spiral FVE. This is achieved
without noticeable loss of temporal resolution, and without introducing signicant arti-
facts. Figure 4.5a shows the undersampled data in both v-f and v-t domains (compare
this to Figure 4.2). Using the proposed 2D lter (dashed lines), the aliasing components
were almost completely removed, while all of the signal energy is preserved (b). The
notch lter (dotted line) removes most of the remaining aliasing at20 and40 Hz
(solid arrows), without introducing any signicant artifacts or loss of resolution (c). For
comparison, the result using view-sharing is shown in (d). This approach is equivalent
to a moving-average low-pass lter, which attenuates the high temporal frequency com-
ponents (dashed arrows), and causes loss of temporal resolution, perceived as blurring
along t (circled).
A comparison between the accelerated results and the fully sampled reference is shown
in Figure 4.6. Using three variable-density spiral interleaves (Table 4.1e), we achieved the
same in-plane resolution that would be achieved using 6 uniform-density readouts of same
length (reference trajectory in Table 4.1). Thus, 2-fold acceleration was achieved with this
approach (b), with no noticeable artifacts in the time-velocity histogram when compared
to the fully sampled reference (a). Additional 6-fold acceleration was achieved using the
proposed temporal acceleration scheme (c). While a fully sampled acquisition with the
same scan parameters would require 216 heartbeats, equivalent results were achieved in
only 18 heartbeats, which correspond to a 12-fold acceleration. As demonstrated in (d),
50
-60 -40 -20 0 20 40 60 0 800 400
frequency (Hz) time (ms)
velocity (cm/s)
a
b
c
d
600
400
200
0
-200
-400
-600
−40 dB −30 dB −20 dB −10 dB 0 dB
−40
−30
−20
−10
0
−40 −30 −20 −10 0
Figure 4.5: Temporal acceleration results in v-f (left) and v-t (right) spaces: a) under-
sampled data; b) 2D lter; c) 2D lter and notch lter; and d) view-sharing. The 2D lter
(dashed lines) removes most of the aliasing components, and the notch lter (dotted line)
removes the remaining aliasing energy (solid arrows). The proposed method (c) removes
aliasing components without noticeable loss of temporal resolution, as opposed to view-
sharing (d), which attenuates high temporal frequency components (dashed arrows) and
causes blurring along t (circled). Compare the v-f representation in (a) with Figure 4.2.
partial Fourier could be used to reduce the acquisition time to 12 heartbeats (i.e., by 1.5-
fold), which represents a total 18-fold acceleration. No signicant artifacts were observed
when comparing the reference dataset (a) with the 18-fold accelerated result (d).
Using the same 18-fold undersampled dataset used for Figure 4.6d, we reconstructed
multiple ROIs, prescribed around dierent
ows observed in a color-
ow video, as dis-
cussed in Figure 4.3. The result is presented in Figure 4.7, which shows that very few
artifacts are observed in the dierent time-velocity histograms obtained throughout the
51
0 400 800
time (ms)
600
0
-600
velocity (cm/s)
a b c d
Figure 4.6: Accelerated spiral FVE results. Comparison between: (a) reference ac-
quired without using acceleration (36 heartbeats); (b) 2-fold acceleration, using variable-
density spirals (18 heartbeats); (c) 12-fold acceleration, using variable-density spirals
combined with temporal acceleration (18 heartbeats); and (d) 18-fold acceleration com-
bining variable-density spirals, temporal acceleration, and homodyne reconstruction (12
heartbeats). A reduced velocity eld-of-view (200 cm/s) was used in (a) and (b) in order
to to restrict acquisition to a single breath-hold. Using a 1200 cm/s velocity eld-of-view
as in (c) and (d), the total acquisition time for (a) and (b) would have been 216 and
108 heartbeats, respectively. All other scan parameters are the same for all acquisitions.
No signicant artifacts were observed when comparing the reference dataset (a) with the
18-fold accelerated result (d).
heart. In order to further avoid artifacts, dierent 2D lters could be designed for each
ROI, based on typical characteristics of each targeted
ow. For example, the artifacts
observed in the descending aorta (arrow), could be reduced by more aggressively ltering
high positive-velocity components, as no ascending
ow is expected in that vessel.
When compared to Figure 3.6, the results in Figure 4.7 also illustrate all the dierent
improvements achieved in this work. The spatial resolution was improved from 7 mm to
3.6 mm, and o-resonance eects were reduced. The velocity eld-of-view was increased
from400 to600 cm/s, without loss of velocity resolution (33 cm/s). The temporal
resolution was improved from 26 ms to 9 ms, and ghosting artifacts due to view-sharing
were eliminated. Both acquisitions were performed in 12-heartbeat breath-holds.
52
velocity (cm/s)
0 400 800
−600
0
600
velocity (cm/s)
time (ms)
0 400 800
−600
0
600
velocity (cm/s)
0 400 800
−600
0
600
velocity (cm/s)
time (ms)
0 400 800
−600
0
600
Figure 4.7: Flow in multiple ROIs acquired in a single 12-heartbeat spiral FVE acquisition
using 18-fold acceleration. Very few artifacts are observed in the dierent time-velocity
histograms obtained throughout the heart (arrow). Note the improvements in spatial
resolution, velocity eld-of-view, and temporal resolution, compared to Figure 3.6.
4.4 Discussion
The results discussed above show that 18-fold acceleration can be achieved in spiral FVE,
without introducing signicant artifacts. Using variable-density sampling, partial Fourier
reconstruction, and a specically designed temporal acceleration scheme, we were able to
address all the main limitations of the method. We demonstrated improvements in spatial
resolution, velocity eld-of-view, and temporal resolution. We also reduced o-resonance
eects, by reducing the spiral readout length. This was achieved without increase in
acquisition time.
53
The acceleration factor achieved by the proposed temporal acceleration scheme (6-
fold) is specic to the scan parameters we chose. For example, if the spatial FOV or
the velocity FOV were smaller, or if the repetition time was longer, aliasing components
would potentially overlap more with the true signal, and ltering could be less eective.
Similarly, more overlapping could occur if coils with larger sensitivity regions were used.
The same would be true if variable-density sampling along k
v
(discussed in Chapter 5)
or a more aggressive variable-density spiral trajectory had been used. Furthermore, such
acceleration factor would not be achievable if only one or two spiral interleaves were used.
If more than three interleaves were used, a dierent view-ordering scheme would have to
be designed to avoid overlaps. Representation in v-f space such as used in Figure 4.1b
and 4.5a would certainly be a useful tool in this task.
The proposed temporal acceleration scheme makes assumptions about the footprint
of typical aortic valve
ow patterns in v-f space. Such hypothesis needs to be validated
by patient studies. Depending on the true footprint, dierent view-ordering schemes
and/or two-dimensional lters may be more eective in avoiding overlaps and ltering
aliasing components, respectively. For example, the lter proposed in Figure 4.8 would
provide better temporal resolution for higher velocities, but could also be less eective
in removing aliasing artifacts. Alternatively, the view ordering proposed in Figure 4.9
reduces the amount of overlap inv-f space, but could require a tighter lter then the one
proposed in Figure 4.8.
While partial Fourier acceleration may reduce the signal-to-noise ratio (SNR), the
other acceleration approaches we used do not necessarily aect the SNR of the time-
velocity histograms. As it can be observed in Figure 4.4e, aliasing artifacts due to
54
-60 -40 -20 0 20 40 60
frequency (Hz)
velocity (cm/s)
600
400
200
0
-200
-400
-600
Figure 4.8: An alternative 2D-lter. Compared to the lter used in Figure 4.5a, this
lter would provide better temporal resolution for higher velocities, but could also be less
eective in removing aliasing artifacts.
interleaf 1 interleaf 2 interleaf 3
kv encodes
time
-60 -40 -20 0 20 40 60
frequency (Hz)
velocity (cm/s)
600
400
200
0
-200
-400
-600
b c
d e
a
Figure 4.9: An alternative view-ordering scheme, and comparison in v-f space. The
view-ordering scheme in (a) provides the aliasing pattern shown in (d), while the scheme
shown in Figure 4.1a provides the aliasing pattern shown in (b). By applying a notch
lter, it becomes clear that this alternative scheme reduces the amount of overlap (circled)
between aliasing and signal components (c,e).
variable-density sampling do not overlap with the aortic valve. With respect to temporal
undersampling, while some aliasing energy may remain, the proposed lters retain most
of the signal energy, and lter some of the noise in the process (Figure 4.5). As SNR is not
a limiting issue in spiral FVE, even higher acceleration factors could be achieved if data
from multiple coil elements were used, by applying parallel imaging techniques [46,60,66].
55
4.5 Conclusions
In this chapter, we achieved 18-fold acceleration in spiral FVE imaging, using three dif-
ferent techniques: variable-density sampling, partial Fourier reconstruction, and a speci-
cally designed temporal acceleration scheme based on UNFOLD. The spiral FVE method
was improved with respect to spatial resolution, o-resonance robustness, velocity eld-
of-view, and temporal resolution, without increase in scan time. Without acceleration,
a 216-heartbeat spiral FVE acquisition would be required in order to achieve such im-
provements. This was achieved in only 12 heartbeats using the accelerated method. These
improvements were demonstrated in healthy volunteers, and no signicant artifacts were
observed.
56
Chapter 5
Reconstruction of variable-density FVE data
In Fourier velocity encoding (FVE) imaging, variable-density sampling along the velocity
dimension can be used to improve the velocity resolution and/or increase the veloc-
ity FOV to accommodate high-speed
ow jets. However, conventional reconstruction
methods such as gridding and direct Fourier transform (DrFT) do not adequately deal
with associated undersampling artifacts. This chapter introduces a novel reconstruction
scheme for variable-density FVE, which is eective in reducing such artifacts. The pro-
posed approach was evaluated using numerical simulations, and demonstrated both in
vitro and in vivo, using two dierent FVE techniques: spiral FVE (chapter 3) and real-
time FVE [17,33,34,44]. Velocity resolution was improved by 160% without introducing
signicant artifacts due to undersampling. In spiral FVE, this translates to 2.6-fold re-
duction in breath-hold duration. In real-time FVE, this translates to improved temporal
resolution and reduced o-resonance eects. The proposed reconstruction method can po-
tentially be applied to any FVE method that utilizes variable-density sampling, requiring
no modication to the imaging pulse-sequence.
57
5.1 Introduction
Magnetic resonance imaging can be accelerated using variable-density sampling of k-
space. Variable-density sampling is typically implemented by using a sampling pattern
that is dense at the center of k-space (low resolution) and sparse at the periphery (high
resolution), i.e., the eective eld-of-view (FOV) is varied from large at the center of
k-space to small at the periphery [73]. The basic assumption is that artifacts from un-
dersampling the periphery of k-space will not be signicant, as the energy of the high
resolution components is low compared to that of the low resolution components.
In FVE, variable-density sampling along the velocity frequency dimension (k
v
) can be
used to improve the velocity resolution and/or increase the velocity FOV to accommodate
ow jets [17], but the basic assumption of variable-density sampling is typically violated,
resulting in signicant aliasing artifacts. Conventional reconstruction methods such as
gridding [35] and direct Fourier transform (DrFT) [48] do not adequately deal with un-
dersampling. The resultant aliasing artifacts may be masked using an operator-dened
threshold, which needs to be adjusted for each dataset [17]. However, this approach may
lead to loss of clinically important information. This work introduces a new reconstruc-
tion method designed specically for variable-density FVE, which deals with associated
undersampling artifacts without signicant loss of diagnostic information.
Peak velocities for laminar and plug
ows are typically below 200 cm/s, which means
that a small velocity FOV is needed to image such distributions. However, high-resolution
components (in k
v
) are required to avoid blurring (Figure 5.1a). Flow jets may present
velocities up to 600 cm/s, requiring a wide velocity FOV. But jets typically contain a wide
58
distribution of velocities, which generally implies having little energy in the periphery of
k
v
(Figure 5.1b). When this is the case, such velocity distributions can be measured
using only low-resolution k
v
components, without noticeable blurring. Therefore, in typ-
ical velocity distributions, there is a clear trade-o between velocity FOV and velocity
resolution needs (Figure 5.1).
−0.03 0 0.03 −600 −300 0 300 600
velocity (cm/s) k
v
(s/cm)
a
b
Figure 5.1: Typical aortic valve velocity distributions (left) and correspondent k-space
representation (right). Data obtained in vivo using spiral FVE with uniform-density sam-
pling. (a) plug
ow (healthy volunteer); (b) turbulent/complex
ow (patient with aortic
stenosis). There is clearly an inverse relationship between velocity and k
v
footprints,
which implies a trade-o between velocity FOV and velocity resolution needs.
The proposed method reduces artifacts by providing maximum velocity resolution
only for a small portion of the velocity FOV. In order to achieve velocity-varying velocity
resolution, we use variable-width [16,61] sinc interpolation. A sinc kernel of appropriate
width provides
at apodization over the fully-sampled FOV, and automatic suppression of
59
aliasing components outside the correspondent FOV. With the proposed method, max-
imum velocity resolution is achieved at the center of the FOV, so the FOV is shifted
towards the center of the distribution, estimated from the two central k
v
samples. By
preserving high resolution at low velocities (where it is required), but still providing a
wide velocity eld-of-view (with lower but diagnostically sucient resolution), this recon-
struction scheme reduces the aliasing artifacts associated with variable-density sampling.
The method was evaluated using numerical simulations, and demonstrated both in
vitro and in vivo, using two dierent FVE techniques: spiral FVE (chapter 3) and real-
time FVE [17,33,34,44]. Real-time FVE (also known as MR Doppler or one-shot FVE)
utilizes a cylindrical excitation to restrict the eld-of-view to a one-dimensional beam.
An oscillating readout gradient simultaneously encodes spatial position and velocity along
the axis of the beam. We demonstrate 160% improvement in velocity resolution, with no
signicant artifacts. The proposed reconstruction method can potentially be used with
any FVE method that utilizes variable-density sampling, requiring no modication to the
imaging pulse-sequence.
5.2 Theory
The analytical solution to reconstruction of non-uniformly sampled k-space data is the
direct Fourier transform (DrFT) [48]. Jackson et al. [35] showed that gridding with
a Kaiser-Bessel convolution kernel is an almost exact approximation to the analytical
solution. However, DrFT and gridding reconstructions do not correct aliasing artifacts
that arise when the data is sampled below the Nyquist rate.
60
Such artifacts may be reduced by varying the resolution across the FOV. In variable-
density spiral imaging, this has been accomplished by Cukur et al. [16] by separately
reconstructing dierent portions of the k-space using dierent convolution kernel widths,
and deapodizing only the unaliased \disc" obtained from each portion. The radius of
each of these discs is dened by the spacing between the k-space samples in each of
these portions. By not deapodizing the aliased portion of the FOV outside each disc,
artifacts are eectively suppressed by the apodization function associated with each kernel
width [16]. A similar approach has been used by Rasche et al. in the context of variable-
density Cartesian sampling [61].
Instead of separately reconstructing and deapodizing dierent portions of the k-space,
we propose using variable-width sinc interpolation. Sinc interpolation is feasible along the
velocity axis, because the number of k
v
samples is typically quite small (usually 10-30)
and the interpolation is one-dimensional. Interpolation using a sinc kernel of appropriate
width provides
at apodization over the unaliased FOV, and automatic suppression of
aliasing components outside the correspondent FOV (Figure 5.2). As no deapodization
is required, we use a dierent sinc width for each k
v
sample. Using this approach, the
velocity resolution is maximum inside the smallest unaliased velocity FOV, and velocity
components inside this region do not induce any aliasing artifacts. Components far from
the center of the velocity FOV will increasingly lose velocity resolution, and may induce
artifacts. Thus, each sinc is multiplied by a linear phase that eectively shifts the velocity
FOV towards the center of the distribution [16]. This eectively eliminates most of the
aliasing artifacts, while preserving the eective velocity resolution.
61
k
v
n i a m o d
s e r - w o l s e r - i h s e r - i h
V O F g i b V O F l l a m s
s e r - i h s e r - i h
aliasing
artifacts
reconstructed
data
velocity domain
s e r - i h s e r - w o l s e r - w o l
convolution kernel
apodization
Figure 5.2: Variable-width sinc interpolation diagram, for variable-density FVE. High-
resolution components are more sparsely sampled then low-resolution ones, resulting in
a smaller unaliased velocity FOV. Using a sinc kernel of appropriate width for each k-
space sample, the resulting apodization function lters the corresponding aliasing, and
the velocity resolution in the reconstructed data varies across the velocity FOV.
Let S(k
v
) be the Fourier transform of the velocity distribution s(v), and S
n
be a
sample of S(k
v
) taken at k
v
=
n
, associated with a density compensation factor w
n
,
which can be calculated from the spacing between samples, i.e. w
n
= (
n+1
n1
)=2.
In conventional gridding, (k
v
), an estimate of S(k
v
), is obtained as:
(k
v
) =
X
n
w
n
S
n
(k
v
n
);
where (x) is a constant-width convolution kernel (e.g., Kaiser-Bessel, Gaussian).
62
Similarly, the proposed method can be summarized as:
(k
v
) =
X
n
w
n
S
n
n
(k
v
n
) hamm
k
v
n
maxjj
e
j2(kvn)vo
;
where:
n
(x) =w
1
n
sinc(x=w
n
) is a variable-width sinc kernel;
v
o
is the center of the distribution, estimated by taking the two central k
v
samples
and obtaining a phase-contrast velocity measurement, i.e. v
o
=
FOVm
2
]S
m
S
m+1
,
for m chosen s.t.
m
= 0;
hamm(x) = 0:540:46 cos(x) forjxj 1, 0 forjxj> 1, is a Hamming window,
which is applied to reduce ringing artifacts.
5.3 Methods
The proposed method was compared against conventional gridding and uniform-density
sampling. Conventional gridding reconstruction was used as reference for evaluation of
artifact reduction. Low-resolution uniformly-sampled datasets acquired using the same
number of k
v
samples were used to illustrate the improvement in velocity resolution
from using variable-density sampling. High-resolution uniformly-sampled datasets were
obtained for comparison, using a smaller velocity FOV. Signal-to-error ratio (SER) was
measured in all studies, and compared. The distributions were assumed to be null outside
the acquired range of velocities for the reference dataset obtained using a small velocity
FOV (e.g., Figure 5.5b).
63
5.3.1 Simulations
Numerical simulation experiments were used to evaluate the choice of the density reduc-
tion factor parameter (). This parameter is dened as the ratio between the largest
FOV (center of k-space) and the smallest FOV (periphery). Dierent values of were
evaluated. Two \numerical phantoms" were designed using windowed rect functions to
represent
ow jets with peak velocities of 300 and 500 cm/s, respectively. A third numeri-
cal phantom was designed using Gaussian distributions atv=0 and 150 cm/s to represent
static material and plug
ow, respectively.
5.3.2 In vitro experiments
Phantom experiments were performed with Tygon R-3603 tubing (1.75 cm inner diam-
eter) connected to a pulsatile blood pump for dogs/monkeys (model 1421; Harvard Ap-
paratus, Holliston, MA) [17]. The phantom was scanned using variable-density real-time
FVE [17], with 24 cm/s velocity resolution over a 480 cm/s velocity FOV ( = 2:75).
These data were provided by Dr. Julie C. DiCarlo.
5.3.3 In vivo experiments
In vivo data was acquired using spiral FVE (chapter 3). The scan plane was prescribed
perpendicular to the aortic valve of a healthy volunteer. Uniform and variable-density
FVE data was acquired with dierent velocity resolution and FOV values.
64
5.3.4 Reconstruction
The data was reconstructed in Matlab (Mathworks, Inc., South Natick, MA). Interpolated
k
v
data was obtained from the variable-density FVE data using conventional gridding
and the proposed reconstruction method, as described above. In conventional gridding, a
Hamming-windowed sinc kernel was used instead of a Kaiser-Bessel or a Gaussian kernel
in order to avoid noise amplication due to deapodization, and the kernel width was
designed for the largest FOV. For all datasets, post-processing was performed as follows.
Velocity distributions were obtained by zero-padding and inverse Fourier transforming
the interpolated k
v
data (for real-time FVE datasets, homodyne reconstruction [57] was
used). Each temporal frame was independently reconstructed. View-sharing [64] was used
to increase the number of temporal frames (spiral FVE only). Saturation eects [24] were
compensated by normalizing the `
2
-norm of the velocity distribution in each temporal
frame. Interpolation along t [4] was used for display purposes.
5.4 Results
Figure 5.3 shows representative results from the numerical simulations. For density re-
duction factors of up to = 4, few aliasing artifacts were observed in the results obtained
with the proposed method. Strong artifacts are seen in the results using conventional
gridding reconstruction. Velocity resolution improvement of 160% in comparison to uni-
form sampling is shown.
Figure 5.4 shows the results of the in vitro experiment, using real-time FVE. The
proposed method eectively cleared most of the aliasing artifacts, while maintaining the
65
a b c
true distribution
uniform sampling
gridding
proposed method
−600 −300 0 300 600 −600 −300 0 300 600 −600 −300 0 300 600
velocity (cm/s)
Figure 5.3: Numerical simulations results for variable-density FVE, using = 4 (FOV =
1200300 cm/s) and 14 k
v
samples. Velocity resolution is 86 cm/s for uniform sampling
and 33 cm/s for variable-density sampling. (a) simulation of plug
ow and static material;
(b) simulation of a
ow jet; (c) simulation of a
ow jet with higher peak velocity. The
proposed method reduces undersampling artifacts considerably (arrows), specially for
narrow and moderately broad distributions.
velocity resolution. Note that this
ow phantom displays a wide distribution of velocities,
similar to what would be observed in a
ow jet.
velocity (cm/s)
time (s)
0 3 6
−200
0
200 a
b
Figure 5.4: Evaluation of the proposed reconstruction method, using a pulsatile
ow
phantom. Data was acquired using variable-density real-time FVE with = 2:75, and
reconstructed using (a) conventional gridding and (b) the proposed method. Undersam-
pling artifacts (arrow) are signicantly reduced by the proposed method.
In vivo demonstration is shown in Figure 5.5. Flow through the aortic valve of a
healthy volunteer was measured three times, using spiral FVE. The scan time for each
acquisition was 14 heartbeats, with breath-hold. Uniform sampling was used in the
66
rst two acquisitions, and the velocity resolution was improved from 86 cm/s in (a) to
33 cm/s in (b), by reducing the the velocity FOV from 1200 to 467 cm/s. In the third
acquisition, variable-density sampling with = 4 was used to achieve 33 cm/s resolution
over a 1200 cm/s FOV. The data was reconstructed using conventional gridding (c) and
the proposed method (d). The proposed reconstruction scheme was able to clear the
aliasing artifacts, without any loss of velocity resolution compared to the small FOV
uniform-density reference.
) s / m c ( y t i c o l e v
time (ms)
0
−600
0
600
proposed
400
gridding
200
variable-density
small FOV large FOV
uniform-density
a b c d
−300
300
Figure 5.5: In vivo demonstration of the proposed reconstruction method, using spiral
FVE. Velocity distributions were measured through the aortic valve of a healthy volunteer
using: (a) uniform-density, large FOV; (b) uniform-density, small FOV; (c) variable-
density, reconstructed using conventional gridding; (d) variable-density, reconstructed
using the proposed method. The proposed reconstruction reduces undersampling artifacts
(arrows), and shows velocity resolution equivalent to the small FOV uniform-density
reference.
A quantitative evaluation of the simulated and in vivo experiments is presented in
Table 5.1. The proposed reconstruction method shows a consistently higher SER than
67
conventional gridding, when compared against a reference dataset acquired using uniform-
density sampling. The proposed scheme is specially SER ecient for distributions with
a narrow range of velocities. Nevertheless, using the proposed method, the SER for
distributions with a broad range of velocities is still higher than that of conventional
gridding.
Table 5.1: Variable-density FVE signal-to-error ratio (dB) comparison to a high-
resolution uniform-density reference.
Simulation In vivo
Plug Narrow Broad Healthy
ow distribution distribution Volunteer
Low-resolution 4 26 27 0
uniform-density
Conventional -1 7 10 0
gridding
Proposed 17 15 11 14
method
5.5 Discussion
Velocity resolution improvement of 160% was achieved in FVE using variable-density
sampling along k
v
. The reduction in aliasing artifacts using the proposed method was
drastic in comparison to results obtained using conventional gridding or DrFT. No loss
of velocity resolution was observed when compared to reference datasets using uniform-
density FVE. The proposed scheme has higher SER for distributions with a narrow range
of velocities, but it still provides better SER than gridding when the distribution contains
a broad range of velocities.
The velocity FOV and resolution were chosen as 1200 cm/s and 33 cm/s, respectively,
as we consider these to be diagnostically sucient for most applications. In the numerical
68
simulations, we found = 4 to be the appropriate density reduction factor for this
conguration, and for the expected range of peak velocities in the aortic valve. The
choice or needs to be validated by patient studies. Also, if the velocity FOV and/or
resolution are changed, or in a dierent application (e.g., coronary or carotid
ow), a
dierent value of might be more appropriate.
If the density reduction factor is set too high, artifacts and loss of velocity resolu-
tion may be observed, specially when the
ow distribution occupies a very wide range of
velocities. When the velocity distribution occupies only a small portion of the velocity
FOV (e.g., plug or laminar
ow, jets with low peak velocity), less artifact is observed. In
fact, if the velocity distribution is entirely accommodated inside the smallest re-centered
FOV, or if portions of the distribution outside the smallest re-centered FOV only contain
low-resolution features (i.e., not aected by loss of velocity resolution), then no artifacts
are observed. In typical distributions, plug and laminar
ows can be entirely accom-
modated inside the smallest FOV, and
ow jets have little energy in high-resolution k
v
(Figure 5.1). Thus, very few artifacts are observed using the proposed method. However,
aliasing artifacts and blurring at the edges of the distribution may be experienced if these
assumptions are violated.
In aortic valve
ow, stenotic and regurgitant jets occur in dierent cardiac phases.
This means that, typically, only one side of the velocity FOV is used at any given time.
Under this assumption, it would be possible to reduce scan time in uniform-density FVE
simply by using a smaller FOV, and then unwrapping the distribution in each cardiac
phase to correct for aliasing. For example, scan time could be reduced by a factor of 2
by reducing the FOV from 1200 cm/s (600 cm/s) to 600 cm/s (300 cm/s). As long as
69
the aliasing high-velocity components do not overlap with the low-velocity
ow, unwrap-
ping the distribution would be straightforward. The proposed reconstruction method for
variable-density FVE does not require unwrapping, and is capable of achieving acceler-
ation factors higher than 2 by exploiting an additional assumption:
ow jets have little
energy in high-resolution k
v
components (Figure 5.1b).
Variable-density sampling alongk
v
can be easily combined with variable-density sam-
pling along the spatial dimensions (k
x
-k
y
) [73], and with partial Fourier reconstruction [57]
along k
v
. In fact, this was demonstrated in this Chapter, as spiral FVE results were
obtained using variable-density spiral readouts (as demonstrated in Chapter 4), and real-
time FVE imaging is intrinsically a partial Fourier method and requires homodyne recon-
struction. However, temporal acceleration using two-dimensional k-t lters as proposed
in Chapter 4 is not as ecient when variable-density sampling alongk
v
is used. Variable-
density sampling causes the point-spread function inv-f space to spread along the velocity
axis, causing overlaps that may not be directly ltered.
5.6 Conclusions
We have proposed and evaluated a novel reconstruction scheme for variable-density FVE
which is capable of reducing artifacts due to under-sampling by using variable-width sinc
interpolation combined with dynamic FOV centering. This approach was evaluated in
vitro with real-time FVE, in vivo with spiral FVE, and in numerical simulations of typical
ow distributions. We demonstrated 160% improvement in velocity resolution, and no
signicant artifacts were observed. In spiral FVE, this translates to 2.6-fold reduction in
70
breath-hold duration. In real-time FVE, this translates to improved temporal resolution
and reduced o-resonance. The proposed method may potentially be used with any
FVE method, including 2DFT-based FVE, reduced spatial encoding FVE [19,22,29] and
cylindrical localized FVE (CLIVE) [18]. No modication to the imaging pulse sequence
other than variable-density encoding is required.
71
Chapter 6
Measurement of carotid
uid shear rate using spiral FVE
Arterial wall shear stress is widely believed to be a predictor of locations of formation
and growth of atherosclerotic plaque. However, there is no current gold standard for in
vivo assessment of
uid shear rates near arterial walls. In MRI, phase contrast is not
an adequate solution for measuring wall shear, due to partial volume and signal-to-noise
ratio (SNR) issues. In this chapter, we discuss the use of spiral FVE for assessment of
carotid
uid shear rate. This is an application in which spiral FVE can be uniquely
useful, as it provides fully-resolved spatial localization in clinically practical scan time.
The following experiments were conducted: (1) in vitro comparison of spiral FVE against
high-resolution/high-SNR 2DFT phase contrast, using a pulsatile carotid
ow phantom;
(2) evaluation of the FVE-based method for assessing
uid shear rate, using simulated
data from computational
uid dynamics; and (3) in vivo demonstration of the proposed
method.
72
6.1 Introduction
Carotid artery disease, or carotid atherosclerosis, is an important risk factor for stroke.
It consists in narrowing (stenosis) of the carotid arteries { which supply blood to the
brain { caused by fatty deposits (plaque) in the vessel walls. Plaque rupture may form
blood clots (thrombus). A thrombotic stroke occurs when clots break o,
ow down-
stream, and occlude a distal vessel, blocking blood
ow to a region of the brain. Plaque
formation occurs preferentially in areas such as the inner curvatures of arteries, or near
bifurcations [13].
Arterial wall shear stress (WSS) { the drag force acting on the endothelium as a result
of blood
ow { is widely believed to be a predictor of locations of formation and growth
of atherosclerotic plaque. WSS can be estimated as the product of wall shear rate (WSR)
and blood viscosity, where WSR is the radial gradient of blood
ow velocity (dv=dr)
at the wall. Low WSS [80] and highly oscillatory WSS [38] appear to be indicators of
high risk for development of atherosclerosis. High WSS has also been hypothesized as a
factor responsible for atherosclerotic lesions [72]. Therefore, WSS mapping may help in
verifying these hypotheses and answering questions about the causes of plaque growth
and rupture. Eventually, WSS mapping could become a screening test for atherosclerosis
prediction.
Available direct methods for measuring WSS are highly invasive and/or can only be
used in conjunction with in vitro models [1, 32, 40]. Indirectly, WSS can be estimated
based on extrapolation of the measured axial velocity prole near the vessel wall [42],
which can be measured by either ultrasound [8] or magnetic resonance imaging [59].
73
However, the accuracy of such methods is limited by the spatial resolution of the velocity
estimates. Particularly, phase contrast MRI is inadequate, due to partial volume [71] and
signal-to-noise ratio (SNR) issues and tradeos (Figure 6.1).
−80 −40 0 40 80
magnitude phase contrast
1 NEX 10 NEX
velocity (cm/s)
0 5 10 15 20 25 30
−80
−60
−40
−20
0
20
40
60
80
spatial dimension (mm)
velocity (cm/s)
0 5 10 15 20 25 30
−80
−60
−40
−20
0
20
40
60
80
spatial dimension (mm)
velocity (cm/s)
velocity profile
carotid
bifurcation
jugular
vein
Figure 6.1: Velocity proles measured in vivo with 2DFT phase contrast, at the carotid
bifurcation, and at peak
ow. It is not possible to achieve sucient SNR in clinically
practical scan time when estimating shear rates with phase contrast, as sub-millimeter
spatial resolution is required. Averaging multiple acquisitions (NEX) improves SNR, but
this increases scan time, and also may introduce motion-related issues, such as loss of
eective spatial and temporal resolutions. Scan parameters: 0.33x0.33x3 mm resolu-
tion, 30
ip angle, 80 cm/s venc, 37 ms temporal resolution, 2 minute acquisition (120
heartbeats) per NEX.
An alternative approach for estimating WSS is to reconstruct a complex
ow eld
via computational
uid dynamics (CFD) analysis of models derived from medical imag-
ing data [69, 70]. However, this method is very computationally intensive and time-
consuming, and its accuracy is limited by assumptions and simplications made about
the properties of blood and endothelium. Moreover, CFD simulations typically do not
take vessel wall compliance into account.
74
In 1995, Frayne et al. proposed an alternative method for non-invasively estimating
uid shear rate using MRI. The Frayne method [21] consists in measuring the distribution
of velocities contained within a voxel at the vessel wall, and then using the distribution
to reconstruct the velocity prole across the voxel. Compared with extrapolation-based
techniques, this approach has the advantage of providing eective spatial resolution much
smaller than the voxel size, as it is able to determine the position of the wall-blood
interface within the voxel with high accuracy. The velocity distribution is measured
using the Fourier velocity encoding (FVE) technique [51], which does not suer from
partial volume, and provides intrinsically high SNR. Due to acquisition time limitations,
the Frayne method was originally demonstrated only in vitro.
Although the scan time of 2DFT FVE is prohibitively long for clinical use, the re-
cently introduced spiral FVE method [12] shows promise as it is substantially faster.
In this work, we propose the use of spiral FVE for estimating carotid
uid shear rate
(FSR) in clinically practical scan times. We begin by discussing the blurring functions
associated with spiral FVE, and proposing a model for deriving FVE distributions from
high-resolution magnitude and velocity maps. We then use this model to compare velocity
distributions measured in a pulsatile carotid
ow phantom using the spiral FVE method
with those derived from a high-resolution/high-SNR 2DFT phase contrast measurement.
We also use this model to evaluate the feasibility of using spiral FVE, combined with
the Frayne method, for estimating carotid FSR. This is performed using simulated FVE
distributions derived from a CFD-based velocity map. Finally, we present the rst in
vivo results obtained using the Frayne method.
75
6.2 Theory
6.2.1 The Frayne method
The method for MRI-based FSR estimation proposed by Frayne et al. [21] consists in
obtaining the velocity distribution for a voxel at the vessel wall, and then using the
distribution to reconstruct the velocity prole across the voxel. The distributions are
converted into velocity proles with sub-voxel spatial resolution, and the reconstructed
proles are then used to obtain shear rate estimates.
Assuming spatially and velocity invariant signal intensities (which will later be ad-
dressed), the sum of signal intensity over all velocities at a single voxel is proportional
to the total volume of material within the spatial extent of that voxel (conservation of
volume) [21]. Furthermore, two assumptions can be made about the shape of the velocity
prole: (1) at the vessel wall, the
uid velocity is approximately zero, and (2) over the
spatial interval of a single voxel (x
o
,y
o
), the velocity prole is monotonically increasing
or decreasing. Using these assumptions, a step-wise discrete approximation v(r) to the
true velocity prole within a voxel can be obtained by converting the volume fraction at
each velocity { obtained by normalizing the `
1
-norm of the velocity distribution { into a
position within the voxel. This is performed according to the following algorithm, which
is demonstrated graphically in Figure 6.2.
v(r) =v
0
s.t. r
0
(v
0
) =
r
js(x
o
;y
o
;v
00
)j
1
v
0
X
v
00
=0
js(x
o
;y
o
;v
00
)j =r ; (6.1)
where r is the voxel diameter, i.e., the spatial resolution, and s(x;y;v) is the velocity
76
distribution (x and y are the in-plane spatial coordinates, and v denotes the velocity
axis).
However, before using this algorithm, it is important to address the spatial and ve-
locity variations in signal intensities discussed above, as well as noise rectication due to
the magnitude operation in Equation 6.1. [21]
Noise threshold To address noise eects, a threshold is applied to s(x
o
;y
o
;v) before
normalization. The appropriate threshold value must be determined by analyzing
the signal intensities in the velocity distribution for a range of velocities outside the
range of expected blood
ow [21]. In our implementation, this threshold is manually
specied, and only components that are below the specied threshold and outside
this expected range of velocities are set to zero.
Blood/vessel wall signal dierences Next, the velocity distribution is adjusted for
compensating signal dierences { due to dierences in
1
H density and relaxation
properties { between vessel wall and blood. This is performed by multiplying the
signal in the static bin by a correction factor. This factor is simply the ratio
between signals from a voxel containing only static or slowly moving blood and a
voxel containing only stationary wall material [21]. This adjustment assumes that
the signal in the static bin is only due to the wall tissue.
Saturation eects Finally, s(x
o
;y
o
;v) is compensated for signal saturation (
ow en-
hancement). Because TR 5T
1
, the magnetization does not fully recover between
RF pulses, resulting in signal saturation. As blood
ows through the slice, a portion
of the saturated spins are replaced with unsaturated spins. Within a certain range
77
of velocities (which is dened by the slice thickness and the TR), this results in spins
with lower velocities having lower signal intensity [24]. By assuming full-relaxation
by the end of each cardiac cycle, this eect can be easily simulated, and the velocity
distribution is adjusted by dividing each velocity bin by the correspondent saturated
intensity. This compensation curve is calculated for each cardiac phase. To reduce
saturation eects, RF excitation (and data acquisition) should be performed only
in the cardiac phases of interest, and the
ip angle should be made as small as SNR
permits. If a spin-echo sequence is used, out
ow signal loss must also be taken into
account [21], but that does not apply to this work as a gradient-echo sequence was
used.
After these three pre-processing steps are applied, the algorithm described in Equa-
tion 6.1 can be used to estimate the velocity prole (Figure 6.2). The FSR can be
estimated by prescribing a velocity interval (v
0
,v
1
) and then tting a rst-order polyno-
mial to the points of v(r) within this interval. Ideally, v
0
= 0 and v
1
= v (where v is
the velocity resolution), because we wish to estimate the velocity derivative at the blood-
wall interface. The shear rate estimate may improve SNR-wise as this velocity interval
becomes larger because of averaging across multiple velocity steps [21]. However, as the
interval becomes larger, the shear rate is averaged over a larger distance within the voxel
(lower spatial resolution) and may deviate from the true local shear at the wall. There-
fore, it is important to prescribe a reasonable (v
0
,v
1
) interval. In our implementation,
v
0
= v and v
1
30 cm/s are used for an initial assessment, and then the interval is
manually adjusted for selected voxels of interest. The same voxel-based approach is used
78
with respect to the noise threshold discussed above, with a xed threshold value being
used for the initial assessment.
0 10 20 30 40
−40
0
40
80
120
velocity (cm/s)
volume fraction (%)
0 0.35 0.7 1.05 1.4
−40
0
40
80
120
intra−voxel position (mm)
velocity (cm/s)
measured
corrected
v
0
v
1
thr eshold
velocity distribution velocity profile
Figure 6.2: Construction of the intra-voxel velocity prole v(r) from the FVE measured
velocity distributions(v), using the Frayne method. The volume fraction at each velocity
bin is converted into a position within the voxel, using the algorithm described in Equa-
tion 6.1. Before the conversion, a threshold is applied to the velocity distribution in order
to address noise rectication, and the signal amplitudes are compensated for saturation
eects and blood/vessel wall signal dierences. A small velocity interval (v
0
,v
1
) is used
for estimating the shear rate (dv=dr).
For accuracy, we will refer to the shear rate estimate over the (v
0
,v
1
) interval as FSR
instead of WSR, since dv=dr is being estimated near, but not at the wall, as discussed
above. Despite the potential underestimation of WSR due to this approximation, WSS
values estimated at a distance from the wall will not be too dierent from those at the
wall, because viscosity decreases towards the wall, and thus shear stress can be considered
as a continuum from the center of the vessel to the wall [63].
6.2.2 Simulating spiral FVE distributions from velocity maps
As no gold standard currently exists for measuring carotid FSR, simulations may be
useful in demonstrating the feasibility of the spiral FVE/Frayne method. Therefore, it
79
is important to be able to derive spiral FVE velocity distributions from high-resolution
magnitude and velocity maps, obtained with known gold standards.
Let m(x;y) and v
o
(x;y) be magnitude and velocity maps, respectively, measured
with ultra-high resolution and ultra-high SNR. These can be obtained, for example, by
using high resolution phase contrast MRI with multiple repetitions (NEX), or by CFD
simulation. Assuming innite spatial and velocity resolutions, the FVE dataset associated
with the object represented by such maps is:
s(x;y;v) =m(x;y)(vv
o
(x;y) ); (6.2)
where (v) is the Dirac delta function.
In spiral FVE, k-space truncation follows a cylindrical shape, i.e. circular along
k
x
;k
y
(with diameter 1=r), and rectangular along k
v
(with width 1=v). Therefore,
the associated object domain blurring can be modeled as a convolution of s(x;y;v) with
the correspondent blurring functions: jinc(
p
x
2
+y
2
=r) and sinc(v=v). Thus, the
nite resolution object is:
^ s(x;y;v) =
m(x;y) sinc
vv
o
(x;y)
v
jinc
p
x
2
+y
2
r
!
; (6.3)
where denotes convolution.
In 2DFT FVE, the spatial blurring function is sinc(x=x) sinc(y=y), and the
above equation would become:
80
^ s(x;y;v) =
m(x;y) sinc
vv
o
(x;y)
v
sinc
x
x
sinc
y
y
: (6.4)
6.3 Spiral FVE vs. 2DFT phase contrast
In this section, we present an in vitro comparison of velocity distributions measured with
spiral FVE to those derived from velocity and magnitude maps measured with 2DFT
phase contrast, using the model described in Equation 6.3. This comparison has two
objectives: (1) demonstrate that spiral FVE is capable of accurately measuring velocity
distributions, when compared with the current MR gold standard; and (2) demonstrate
that the proposed model for deriving spiral FVE velocity distributions from velocity maps
(Equation 6.3) is applicable in practice. In the following section, we will use this model for
evaluating the feasibility of using the spiral FVE/Frayne method for measuring carotid
FSR, thus the importance of this experiment.
6.3.1 Methods
Experiments were performed using a pulsatile carotid
ow phantom (Phantoms by Design,
Inc.). A slice perpendicular to the carotid bifurcation was prescribed, and through-plane
velocities were measured. A gradient-echo 2DFT phase contrast sequence with high
spatial resolution and high SNR (0.33 mm resolution, 10 NEX, 80 cm/s venc) was used
as a gold standard reference. Spiral FVE data with r = 3 mm and v = 10 cm/s
was obtained from the same scan plane. Acquisitions were prospectively gated, and the
same TR (9.2 ms),
ip angle (30
), slice prole (3 mm), temporal resolution (18 ms), and
81
pre-scan settings were used for both acquisitions. The total scan time was 40 minutes for
phase contrast, and 12 seconds for FVE.
A simulated spiral FVE dataset was derived from the phase contrast (PC) data using
the convolution model described in Equation 6.3. Registration between PC-derived and
measured FVE data was done by taking one magnitude imagem(x;y) from each dataset,
and then using the phase dierence in their Fourier transforms M(k
x
;k
y
) to estimate the
spatial shift between them. Amplitude scaling was performed by normalizing the`
2
-norm
of each dataset.
6.3.2 Results
Figure 6.3 shows measured and PC-derived time-velocity FVE distributions from two rep-
resentative voxels, selected near opposite walls of the vessel's bifurcation, as indicated.
The signal-to-error ratio (SER) between measured and PC-derived time-velocity distri-
butions was in the range of 10-12 dB within the lumen. We believe that imperfect regis-
tration between the two datasets, combined with spatial blurring (due to o-resonance)
in the measured spiral FVE data, may have contributed for this moderate SER. Never-
theless, measured and PC-derived datasets show good visual agreement. We consider the
observed agreement to be satisfactorily good for the purposes of this experiment.
In this phantom, only the lumen generates MR signal. Vessel wall and other static
materials in the phantom are not imageable, and appear dark (Figure 6.3, upper-left
corner). Therefore, it would not be possible to obtain shear rate estimates from the mea-
sured FVE data using the Frayne method. Thus, we compared the velocity distributions,
but not the actual shear rate estimates.
82
diff x4 PC FVE
velocity
time
diff
voxel 2
1 2
voxel 1
Figure 6.3: In vitro comparison between time-velocity distributions derived from 2DFT
phase contrast (left) and those measured with spiral FVE (center-left). Results from two
representative voxels, selected near opposite walls of the vessel's bifurcation, as indicated
(upper-left corner). The dierence between measured and PC-derived histograms is also
shown (center-right), as well as the 4-fold magnied dierence (right). The signal-to-error
ratio (SER) between measured and PC-derived time-velocity distributions is in the range
of 10-12 dB within the lumen. The two datasets show good visual agreement.
In the experiment presented in the next section, we will compare shear rates measured
using the Frayne method to the true shear rate values measured in a high resolution veloc-
ity map obtained from CFD simulations. Such experiment was not performed using the
phase contrast data from this experiment because the measured velocity maps provided
insucient SNR for accurately estimating a ground truth shear rate map, as discussed in
Figure 6.1.
6.4 Feasibility of the spiral FVE/Frayne method
In this section, we investigate the feasibility of using spiral FVE, combined with the Frayne
method, for estimating carotid FSR. As no gold standard currently exists for assessing
carotid FSR, a simulation will be used for this investigation. We will obtain simulated
83
spiral FVE distributions using the convolution model described in Equation 6.3 with a
velocity map obtained from a CFD simulation of carotid
ow. We will then compare
the shear rate values measured using the Frayne method to the true gradient values,
measured in the high-resolution velocity map.
6.4.1 Methods
The CFD data was provided by Lisong Ai and Prof. Tzung K. Hsiai, from the De-
partment of Biomedical Engineering of the University of Southern California, and was
obtained through simulations using Fluent (Ansys, Inc., Lebanon, NH). A carotid bifur-
cation geometry was designed, and the
ow simulation was done under steady sate with
maximum center-line velocity of 120 cm/s [3]. An axial map of through-plane velocities
was obtained and interpolated onto a Cartesian grid. The correspondent shear rate map
was calculated using rst-order 2D-polynomial weighted least-squares t, and this was
used as a gold standard reference.
The CFD velocity map was used as v
o
(x;y) in the model described by Equation 6.3
to obtain simulated spiral FVE data. The signal intensities were assumed to be uniform
throughout the object, i.e. m(x;y) = 1. The simulated spatial and velocity resolutions
were r = 1 mm and v = 12:5 cm/s, over a200 cm/s velocity eld-of-view (32
velocity encode levels).
The Frayne method was then used with the simulated velocity distributions to ob-
tain shear rate estimates. The noise threshold was set to 2.5% of the `
1
-norm of the
velocity distribution, and the shear rates were estimated within the (v
0
,v
1
) interval of
84
12.5-25 cm/s. No signal intensity correction was necessary, since the intensities are al-
ready spatially uniform in this simulation. The results were compared with the gold
standard shear rate map measured directly from the velocity map.
6.4.2 Results
The results of this experiment are shown on Figure 6.4. Within a region-of-interest,
dened near the wall-blood interface, the spiral FVE/Frayne method was able to estimate
the shear rate with at least 10% accuracy for 50% of the voxels, at least 20% accuracy
for 80% of the voxels, and at least 30% accuracy for 95% of the voxels (see histogram in
Figure 6.4).
0 50 100 150
velocity (cm/s)
velocity map
true shear rate FVE-measured
true shear rate
(windowed)
FVE-measured
(windowed)
difference (%)
0 10 20 30
difference (%)
difference
0
1000
2000
3000
4000
shear rate (s
-1
)
-60 0 30 60
0
5
10
15
difference (%)
histogram (%)
-30
Figure 6.4: Shear rates measured using the spiral FVE/Frayne method on simulated
velocity distributions obtained from a CFD simulation. The CFD-based velocity map
is shown on left. Within a region-of-interest, dened near the wall-blood interface, the
FVE-measured shear rates agree very well with the true values (center). The spiral
FVE/Frayne method was able to estimate the shear rate with at least 30% accuracy for
95% of the voxels within the region-of-interest (right).
85
This feasibility evaluation was not optimized with respect to spatial and velocity
resolutions, threshold value, and (v
0
,v
1
) interval. The parameters used in this simulation
were selected heuristically and ad hoc. Selecting the optimal set of parameters would be
extremely time-consuming, and also not benecial for practical purposes, as the optimal
set of parameters would not only be subject dependent, but also spatially and temporally
varying. For example, the optimal (v
0
,v
1
) interval in a region of low
uid shear would
probably be dierent to that in a region of high shear. Similarly, the optimal interval at
peak
ow would probably not be optimal in a cardiac phase with lower peak velocity.
In practice, the spiral FVE data should be acquired with spatial and velocity resolu-
tions as high as SNR and scan time permit. As discussed in the Theory section, a xed
threshold value and (v
0
,v
1
) interval may be used for an initial assessment, and then these
can be manually adjusted for selected voxels of interest, by analyzing their correspondent
velocity distributions and reconstructed velocity proles.
6.5 In vivo demonstration
In this section, we present a demonstration of in vivo estimation of carotid
uid shear
rate, using the spiral FVE/Frayne method. Three healthy subjects were imaged at 3T. To
demonstrate repeatability, one of the subjects was imaged twice, on dierent occasions,
with a two-week interval. The results show the variation in FSR along all three spatial
dimensions near the carotid bifurcation, and also the oscillatory pattern of carotid FSR
along the cardiac cycle.
86
6.5.1 Methods
Three healthy subjects were studied. Multiple cardiac phases were acquired, covering the
systolic portion of the R-R interval. One of the volunteers was imaged again two weeks
after the rst study, in order to evaluate repeatability. The heart rate of this volunteer
was 74 bmp during the the rst study, and 64 bmp during the second study.
Scan parameters are summarized in Table 6.1. Five slices perpendicular to the left
carotid bifurcation were prescribed, with no spacing between them. Each slice was im-
aged independently, using spiral FVE. Only through-plane velocities were measured. Lo-
cal gradient shimming was used, and acquisitions were prospectively ECG-gated. Re-
construction was performed in Matlab (Mathworks, Inc., South Natick, MA), and FSR
estimates for each slice/cardiac phase pair were obtained using the Frayne method. The
noise threshold was set to 2.5% of the`
1
-norm of the velocity distribution, and the shear
rates were estimated within the (v
0
,v
1
) interval of 5-25 cm/s. Only cardiac phases near
peak
ow were reconstructed. A smaller v
1
value should be used in cardiac phases with
lower peak velocities.
Table 6.1: Scan parameters used in the in vivo studies.
Excitation Spatial parameters Velocity parameters Temporal parameters
slices 5 interleaves 8 kv levels 32 TR 12 ms
thickness 5 mm readout 4 ms resolution 5 cm/s views/beat 2
coverage 2.5 cm density variable FOV -40:120 cm/s resolution 24 ms
ip angle 30
FOV 164 cm
ow-axis through-plane heartbeats 128/slice
TBW 2 resolution 1.4 mm density uniform scan time 2 min/slice
6.5.2 Results
The variation in FSR along all three spatial dimensions near the carotid bifurcation can
be appreciated in Figure 6.5 (results from subject #1). These results correspond to the
87
cardiac phase with the highest peak velocity. The FSR value is shown only for voxels in
which the measured FSR was less than 2000 s
1
.
0
200
400
600
800
1000
1200
1400
1600
1800
2000
wall shear rate (s
-1
)
z-axis
0 mm
5 mm
10 mm
15 mm
20 mm
CCA
JV
ICA
ECA
Figure 6.5: Spatial variation in carotid FSR along all three dimensions, near the carotid
bifurcation of subject #1. These results correspond to the cardiac phase with the highest
peak velocity, acquired 96 ms after the ECG trigger. The common, external, and internal
carotid arteries, and the jugular vein, are indicated.
The oscillatory pattern of FSR along the cardiac cycle in the common carotid artery
of subject #2 can be appreciated in Figure 6.6. It has been suggested that high oscillatory
patterns of wall shear stress may contribute to plaque growth [38], thus the importance
of obtaining temporally-resolved FSR estimates with high temporal resolution.
Subject #3 was imaged in two-dierent occasions, with a two-week interval. Al-
though dierences in hemodynamic conditions and slice prescription may contribute for
some dierences between the shear rate estimates, we did not observe visibly signicant
dierences (Figure 6.7). Specically, regions of low and high FSR match reasonably well
in these two studies (arrows).
88
2000
0
1000
500
1500
wall shear rate (s
-1
)
cardiac cycle
48 ms 72 ms 96 ms 120 ms 144 ms 168 ms
Figure 6.6: Oscillatory pattern of FSR along the cardiac cycle in the common carotid
artery of subject #2. These results correspond to the cardiac phases acquired 48-168 ms
after the ECG trigger, and to the most inferior slice (labeled 0 mm in Figure 6.5, i.e.,
10 mm below the carotid bifurcation).
72 ms 120 ms 144 ms
2000
0
1000
500
1500
wall shear rate (s
-1
)
cardiac cycle
48 ms
first acquisition two weeks later
Figure 6.7: Repeatability evaluation of the spiral FVE/Frayne method. Subject #3 was
imaged in two-dierent occasions, with a two-week interval. Regions of low and high
FSR match reasonably well in the two studies (arrows). These results correspond to the
cardiac phases acquired 48-120 ms after the ECG trigger, and to the second-most inferior
slice (labeled 5 mm in Figure 6.5, i.e., 5 mm below the carotid bifurcation).
6.6 Discussion
If the Frayne method is used with voxels containing only static tissue, or only plug-like
ow, the FSR estimate is incorrectly estimated as very high, instead of very low or zero
89
(see Figure 6.4, center). This happens because the algorithm is designed to measure the
FSR within the specied (v
0
,v
1
) interval. The shear rate is dened as dv=dr. If a xed
velocity interval is used for the entire image, thendv =v
1
v
0
for all voxels. However,dr
is directly proportional to the volume fraction within the (v
0
,v
1
) interval. In voxels where
low signal is detected within this interval { typically those containing only static tissue or
only plug-like
ow {,dr will be estimated as very small. Sincedv is xed, the FSR will be
incorrectly estimated as very high. However, this is absolutely acceptable, as we are only
interested in voxels at the wall-blood interface, which generally contain signal throughout
the (v
0
,v
1
) interval. Voxels in which the estimated FSR is outside the expected range of
FSR values can be simply ignored or not displayed, as they are typically not voxels of
interest, i.e., not located at the wall-blood interface.
A thin slice should be used if sucient SNR is available, specially when imaging at
the bifurcation, as signicant variations in FSR along thez-axis may occur within a thick
slice. Also, because only through-plane velocities are measured, the scan plane should be
made perpendicular to the vessel wall where FSR is to be estimated. If the scan plane
is slightly oblique to the wall, it should be possible to correct both dv and dr estimates,
as long as the carotid geometry relative to the imaged plane is known. However, the
oscillating spatial-encoding spiral readout gradients also intrinsically perform some in-
plane velocity phase-encoding. Therefore, artifacts could arise if in-plane velocities are
present in the voxels of interest, which would be the case for oblique planes.
Other general possible sources of error associated with the Frayne method, not directly
related to the use of spiral readouts, are discussed in [21]. These include: imprecisions
associated with measuring the shear rate near but not at the wall (choice of the (v
0
,v
1
)
90
interval); signal loss at the blood-wall interface due to susceptibility eects caused by
local magnetic eld inhomogeneity; and ringing and leakage along the velocity dimension
due to nite velocity resolution. Although not discussed in [21], nite spatial resolution
is also a potential source of error. Truncation in k-space results in blurring, which can
be modeled as a convolution of the object with a kernel (see Theory), which consists
in a main-lobe and also side-lobes. The side-lobes cause crosstalk between neighboring
voxels. The main-lobe causes intra-voxel weighting. The side-lobes may be reduced by
weighting the k-space data. However, this process reduces the eective resolution, and
also increases the intra-voxel weighting eect.
Voxel shape is also a potential source of error. Frayne et al. implicitly assume voxels
to be squares (which is approximately true in 2DFT imaging), and the radial velocity
gradient to be along one of the voxel axes (which is generally not true). In spiral FVE,
the voxel shape is circular, therefore no assumption needs to be made about the direction
of the velocity gradient. However, the volume fraction corresponding to each velocity
bin is no longer directly proportional to its associated radial distance (dr). In a circular
voxel, the relationship between volume fraction and radial distance varies with intra-
voxel position. Voxel shape eects could be compensated by adjusting dr values in an
initial estimate of the velocity prole { obtained as proposed in Theory { such that the
corresponding volume fractions in the true voxel shape would match those measured in
the velocity distribution (Figure 6.8). This approach could also be used to compensate
for intra-voxel weighting.
91
intravoxel position (mm)
intravoxel position (mm)
0 1 2 3
0
1
2
3
0
35
70
velocity (cm/s)
−10 20 50 80
0
6
12
18
velocity (cm/s)
volume fraction (%)
a
0 1 2 3
0
35
70
intravoxel position (mm)
velocity (cm/s)
true
estimated
b
c
−10 20 50 80
0
6
12
18
velocity (cm/s)
volume fraction (%)
measured
estimated
d
−10 20 50 80
0.7
1
1.3
1.6
velocity (cm/s)
compensation factor
e
−10 20 50 80
0
6
12
18
velocity (cm/s)
volume fraction (%)
measured
adjusted
f
0 1 2 3
0
35
70
intravoxel position (mm)
velocity (cm/s)
true
adjusted
g
Figure 6.8: Proposed method for compensation of voxel shape eects. The measured
volume fractions (a) are obtained by normalizing the velocity distribution. Then, the
velocity prole is estimated using the Frayne method (b). This prole is projected into
a circle, representing the voxel shape (c). Volume fractions are calculated from this
circle (d). By dividing the measured volume fractions by the estimated volume fractions,
a compensation function is obtained (e). The adjusted volume fraction is obtained by
multiplying the measured volume fraction by this function (f). Finally, the adjusted
velocity prole is calculated using the Frayne method (g). Intra-voxel weighting can also
be compensated using this approach, simply by multiplying the projected prole (c) by
the appropriate kernel (e.g., a jinc), prior to calculating (d).
6.7 Conclusions
We have demonstrated a method for non-invasive assessment of carotid
uid shear rate.
The spiral FVE technique was used with the method proposed by Frayne et al. for
estimating
uid shear rates. This method provides sub-voxel spatial resolution, but had
only been demonstrated in vitro. The rst in vivo results using the Frayne method
were presented. The results show the variation in carotid FSR along all three spatial
92
dimensions, and also along the cardiac cycle. The achieved temporal resolution was
sucient to capture the oscillatory pattern of carotid FSR, an important indicator of
atherosclerotic plaque growth and risk of rupture.
Spiral FVE is uniquely suitable for FSR estimation, as it is the only currently available
FVE method capable of providing fully-resolved spatial localization in clinically practical
scan time. We have shown that velocity resolutions measured with spiral FVE agree well
with those obtained with 2DFT phase contrast, the current MR gold standard. Also, we
proposed a model for deriving FVE data from high-resolution velocity maps, which can
be used for many simulation purposes. In this work, we used this model to demonstrate
that the spiral FVE/Frayne method accurately estimates carotid shear rates in simulated
data obtained using computational
uid dynamics.
The proposed spiral FVE/Frayne method can potentially help answering questions
about the causes of plaque growth and rupture, and could eventually be clinically useful
as part of a screening test for predicting carotid atherosclerosis.
93
Chapter 7
Concluding remarks
In this work, we have addressed the issue of non-invasive cardiovascular
ow quantitation
through magnetic resonance imaging (MRI). We addressed both imaging and reconstruc-
tion aspects, including accelerated acquisitions and reconstruction from undersampled
data. Possible clinical applications for the proposed methods include assessment of valvu-
lar stenosis and insuciency, and carotid artery disease, as demonstrated.
We proposed a new method for MRI
ow quantitation (spiral FVE), which is capable
of accurately capturing peak velocities in
ow jets due to stenosis or regurgitation. The
proposed method is capable of providing full spatial localization, and therefore may be
uniquely useful in imaging
ow in small vessels and estimating
uid shear rate. Spiral
FVE compared well against Doppler ultrasound, the current gold standard for cardio-
vascular
ow imaging. The method was demonstrated in both healthy volunteers and
patients. Dierent implementations were proposed and evaluated for imaging aortic
ow
and carotid
ow, respectively.
Using a combination of three dierent techniques (variable-density spirals, temporal
acceleration, and partial Fourier reconstruction), we are able to improve the spiral FVE
94
method by 18-fold. Improvements consisted of increased velocity eld-of-view, higher
spatial resolution, reduced o-resonance eects, and higher temporal resolution. The
improved acquisition was performed in only 12 heartbeats, whereas 216 heartbeats would
be necessary to achieve such improvements without acceleration. No signicant artifacts
were observed.
We also proposed a new reconstruction scheme for FVE imaging using variable-density
sampling along the velocity dimension. Numerical simulations showed that the proposed
method is very successful in reducing aliasing artifacts due to undersampling, even in
the presence of
ow jets. We incorporated the variable-density sampling scheme into the
spiral FVE pulse sequence, and demonstrated the proposed reconstruction method both
in vitro and in vivo. We also showed that the method can potentially be used with any
FVE-based method.
Finally, we combined the spiral FVE technique with the method proposed by Frayne
et al. for estimating
uid shear rates. This approach was used for for non-invasive
assessment of carotid shear rate. The rst in vivo results using the Frayne method were
presented. The method was shown capable of capturing the spatial variation in shear
rate along all three spatial dimensions, and also the oscillatory pattern along the cardiac
cycle. These are important indicators of risk for development of carotid artery disease.
Due to its high dimensionality (x;y;z;v;t), spiral FVE has great potential for high or-
ders of acceleration [11]. Several approaches have been proposed to drastically reduce the
acquisition time of FVE imaging without signicant loss of information [9,11,17,23,27,45].
Furthermore, if the subject can be successfully immobilized, and hemodynamic conditions
95
are relatively stable, higher accelerability can be achieved by using 3D imaging (a stack-
of-spirals in k
x
;k
y
;k
z
) [39], instead of single or multi-slice acquisitions. This would not
only increase SNR and allow for higher orders of parallel imaging acceleration [7, 78],
but also increase sparsity and allow even higher factors of temporal [37] and compressed-
sensing-based [43] acceleration.
As future work, we propose using accelerated spiral FVE for the assessment of coro-
nary
ow reserve, and for measuring pressure gradients in all four heart valves. We also
propose patient studies to:
verify the hypothesis that
ow jets are typically temporally smooth (Chapter 4);
determine the optimal density reduction factor () for variable-density FVE (Chap-
ter 5);
validate the in vivo assessment of carotid shear rate using the spiral FVE/Frayne
method, and nd the optimal set of resolution values and reconstruction parameters
(Chapter 6).
Magnetic resonance imaging is potentially the most appropriate technique for ad-
dressing all aspects of cardiovascular disease examination. The evaluation of valvular
disease and intracardiac
ow will be a necessary capability in a comprehensive cardiac
MR examination. The imaging and reconstruction techniques proposed in this work can
be an important contribution towards making such exam feasible. These methods may
also be useful in applications where no non-invasive gold standard is currently available,
such as studying and predicting carotid atherosclerosis.
96
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Abstract (if available)
Abstract
Cardiovascular disease is the leading cause of death and disability in the United States. An important component of the assessment of cardiovascular disease is the visualization and quantitation of cardiovascular flow. The current gold standard for flow measurement is Doppler ultrasound. However, evaluation by ultrasound is inadequate when there is fat, air, bone, or surgical scar in the acoustic path.
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Asset Metadata
Creator
de Carvalho, Joao Luiz Azevedo
(author)
Core Title
Velocity-encoded magnetic resonance imaging: acquisition, reconstruction and applications
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
05/28/2008
Defense Date
04/24/2008
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
aortic regurgitation,aortic stenosis,carotid artery disease,Fourier velocity encoding,FVE,MRI flow imaging,OAI-PMH Harvest,wall shear stress
Language
English
Advisor
Nayak, Krishna S. (
committee chair
), Hsiai, Tzung K. (
committee member
), Leahy, Richard M. (
committee member
), Pohost, Gerald M. (
committee member
)
Creator Email
jcarvalh@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m1249
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de Carvalho, Joao Luiz Azevedo
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(contributing entity),
University of Southern California Dissertations and Theses
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Repository Name
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Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
aortic regurgitation
aortic stenosis
carotid artery disease
Fourier velocity encoding
FVE
MRI flow imaging
wall shear stress