Off-resonance artifact correction for magnetic resonance imaging: a review

Melissa W. Haskell1,2, Jon-Fredrik Nielsen3, Douglas C. Noll3

1Electrical Engineering and Computer Science, University of Michigan, Ann
Arbor, MI, United States
2Hyperfine Research, Guilford, CT, United States
3Biomedical Engineering, University of Michigan, Ann Arbor, MI, United States

## Abstract

In magnetic resonance imaging (MRI), inhomogeneity in the main magnetic field
used for imaging, referred to as off-resonance, can lead to image artifacts
ranging from mild to severe depending on the application. Off-resonance
artifacts, such as signal loss, geometric distortions, and blurring, can
compromise the clinical and scientific utility of MR images. In this review,
we describe sources of off-resonance in MRI, how off-resonance affects images,
and strategies to prevent and correct for off-resonance. Given recent advances
and the great potential of low field and/or portable MRI, we also highlight
the advantages and challenges of imaging at low field with respect to off-
resonance.

##  1 Introduction

Magnetic resonance imaging (MRI) is an important clinical tool for diagnosis
and intervention, as well as the leading modality for noninvasively imaging
the _in vivo_ human brain. At the core of the MR imaging experiment is a
polarizing magnetic field, referred to as the $B_{0}$ field or the main
magnetic field, that aligns the magnetic spins within an object. These spins
will precess at a resonant frequency linearly proportional to the external
$B_{0}$ field, and in the majority of MRI experiments it is ideal to have as
homogeneous a $B_{0}$ field as possible so that all spins precess at the same
frequency. In practice, many factors can lead to $B_{0}$ inhomogeneity, thus
leading to a non-uniform precession frequency of the spins, and this is called
_off-resonance_. Off-resonance can lead to artifacts in MR images, including
signal loss, geometric distortions, and blurring.

Off-resonance artifacts have been reported since the early days of clinical
MRI, along with the problems they present in interpreting MR images [1, 2, 3].
Geometric distortions that warp the shape of anatomy create challenges in many
clinical applications, including stereotactic localization [4], MRI-guided
biopsy [5], and MRI-guided radiation therapy [6, 7]. Metal implants can lead
to severe geometric distortions, and also can create a total loss of signal
around the metal object [8]. In MRI based neuroscience research, uncorrected
off-resonance artifacts can lead to a loss in quality of data, most notably in
the field of functional MRI (fMRI) [9, 10].

Because of these problems, many approaches exist in MRI workflows to mitigate
off-resonance artifacts. Magnetic field shimming, where dedicated shim coils
apply additional fields to offset and cancel out main field inhomogeneity, is
automated and standard on most vendors. From a sequence perspective, one of
the simplest solutions to avoiding signal loss from off-resonance is to use
spin echo-based sequences [11, 12], which refocus signal at the echo time
(TE). The expanded use of parallel imaging [13, 14] has also allowed for
smaller voxels in standard imaging protocols. Smaller voxels reduce signal
loss because the magnetic fields within the voxels have less variation in
frequency, hence less signal cancellation. There is also less $B_{0}$
distortion with parallel imaging, because data can be acquired in shorter
readouts with higher effective bandwidths in k-space.

However, not all $B_{0}$ artifacts can be avoided with these approaches. The
magnetic field off-resonance map, called a $B_{0}$ _fieldmap_ or _fieldmap_
for short, inside the human body often has spatial variations that are much
higher order than the 2${}^{\text{nd}}$ order spherical harmonic shim
corrections available on some systems, leaving the potential for residual off-
resonance in hard to shim anatomy. Additionally, certain sequences require
gradient echo imaging with longer readouts to generate the desired image
contrast or acquisition speed, and are therefore more prone to off-resonance
artifacts (e.g., echo planar imaging (EPI) [15]). Longer TEs are important for
sequences such as blood oxygen level dependent (BOLD) contrast fMRI [16], to
allow for dephasing of the spins to present as $T^{*}_{2}$ contrast.
Susceptibility weighted imaging (SWI) and quantitative susceptibility mapping
(QSM) also require longer TEs to generate magnetic susceptibility contrast,
and these sequences are useful tools to investigate pathologies such as
intracranial hemorrhage, traumatic brain injury, stroke, neoplasm, and
multiple sclerosis [17, 18]. Further, even when the desired contrast does
allow for the use of spin-echo imaging and most $B_{0}$ induced signal loss
can be avoided, geometric distortions can still be present due to unwanted
gradients that add to the imaging gradients, resulting in k-space encoding
errors.

It is also important to consider off-resonance artifacts and potential
correction strategies when working with MRI scanners that have more
inhomogeneous $B_{0}$ fields than high-field superconducting magnets.
Recently, there have been many advances in portable MRI at lower fields to
perform _in vivo_ human imaging [19, 20, 21, 22, 23]. These new technologies
are important for increasing the accessibility of MRI [24], but they generally
have less uniform $B_{0}$ fields than superconducting magnets and off-
resonance correction approaches are needed. Low field MRI scanners built with
permanent magnets can also experience large field drifts due to temperature
induced field changes, so dynamic tracking of the $B_{0}$ field strength is
sometimes needed during imaging.

This review is structured by first describing the sources of off-resonance in
MR imaging in Section 2 and their effects on imaging in Section 3. Section 4
details ways to estimate a $B_{0}$ fieldmap using pulse sequence, hardware,
and optimization based approaches. Section 5 discusses strategies to prevent
and correct for $B_{0}$ artifacts, and Section 6 concludes the paper. All code
for the simulations shown here is available online at
https://github.com/fmrilab/B0-review-2022, including example code to perform
model-based image reconstruction [25] that incorporates a $B_{0}$ fieldmap
[26] using the Michigan Image Reconstruction Toolbox (MIRT) [27].

##  2 Sources of magnetic field off-resonance

###  2.1 Main field inhomogeneity

####  2.1.1 Modern superconducting magnet homogeneity

The $B_{0}$ polarizing fields of modern high-field superconducting magnets,
most often at field strengths of 1.5T or 3T, are generally very uniform, and
after shimming with superconducting shim coils and passive shim elements are
on the order of 1ppm uniformity over a large imaging volume. Because of this,
artifacts from susceptibility, chemical shift, and metal implants (discussed
in later sections) are the dominant sources of $B_{0}$ artifacts in high-field
superconducting MRI scanners, and off-resonance artifacts from the magnet
itself are minimal.

####  2.1.2 New magnet designs for lower cost and portability

As more light-weight, low-cost, and portable MRI designs are being developed,
permanent magnets and resistive electromagnets are being chosen to create the
polarizing fields. Both permanent and resistive magnet designs are much less
expensive to produce than superconducting magnets, and, importantly, are much
less expensive and easier to operate and maintain. However, these magnets are
generally less spatially homogeneous and may require $B_{0}$ field correction
even for routine imaging. Permanent magnets have the advantage that they
require no power or cooling mechanism to operate, but they can be very
sensitive to temperature induced field drifts, which often need to be
monitored during scanning. Resistive magnets are not as temperature sensitive,
but power consumption and cooling will need to be taken into consideration and
will potentially limit the field strength and homogeneity.

There are examples both in industry and academia of newer permanent magnetic
designs. Currently available commercially scanners include a 64 mT portable
brain scanner (https://www.hyperfine.io), a 1T scanner for the neonatal
intensive care unit (https://www.aspectimaging.com), and a low-field system
for dedicated prostate imaging and biopsy guidance (https://promaxo.com).
Academic groups have created permanent magnet designs using cylindrical
Halbach geometries [28, 29, 30], dipole magnet geometries [20, 19], and single
sided designs [31]. Unlike superconducting magnets, these scanners have field
homogeneities in the range of tens of ppm (for smaller FOV extremity imaging)
to hundreds or thousands of ppms for head imaging, but have been shown to
successfully image human subjects [21, 22, 20, 19].

Resistive magnets at low field have also been explored. Obungoloch et al.
designed and constructed a prepolarized MRI scanner that acquired phantom
images at 2.66mT with a polarizing field of 27mT, and a field homogeneity of
5000 ppm, similar to the level of inhomogeneity in the permanent magnet
Halbach designs [32].

###  2.2 Magnetic susceptibility

One of the largest sources of $B_{0}$ inhomogeneity during routine _in vivo_
MRI is magnetic susceptibility. Magnetic susceptibility (sometimes called
volume magnetic susceptibility) [33], denoted as $\chi$, is the property of a
material to either strengthen ($\chi>0$) or weaken ($\chi<0$) an applied
magnetic field, based on its internal magnetic polarization, $M$, as follows:

$\chi=M/B_{0}$ (1)

Magnetic susceptibility is unitless since it is the ratio of two fields, but
it is often expressed in parts per million (or ppm) for convenience in the
context of MRI. For example, molecular oxygen, O2, has a magnetic
susceptibility of roughly +1ppm, which means that when in the presence of a
1.5T, or 64 MHz, external magnetic field, its internal polarization will be
roughly 1.5$\mu$T, or 64 Hz.

If an object has non-uniform magnetic susceptibility, it will lead to a non-
uniform magnetic field, even within the most uniform of applied fields
generated by today’s modern superconducting magnets. In human imaging, most
biological tissues are weakly diamagnetic ($\chi<0$) and oppose the applied
field, but oxygen, present in the lungs, sinuses, and other air cavities, is
weakly paramagnetic ($\chi>0$) and strengthens the applied field. This leads
to spatially varying $B_{0}$ fields within the body, such as the one shown in
Fig. 2.

Looking at Eq. (1), one can see that as the applied magnetic field $B_{0}$
increases, the internal polarization, $M$, will also increase. This means that
at higher field strengths the amount of off-resonance will increase, and
therefore the susceptibility artifacts will be more severe [34]. For a similar
_in vivo_ fieldmap at 3T to the 7T fieldmap shown in Fig. 2, as well as the 3T
vs. 7T _in vivo_ images with corresponding amounts of susceptibility
artifacts, we point the reader to Fig. 5 of Stockmann & Wald, 2018 [35]. Due
to the scaling with field strength, it is sometimes convenient to describe
field susceptibility in terms of ppm, where the typical fieldmap often less
than 1 ppm for most of the brain, but can grow to about 2 ppm at air/brain
interfaces.

Magnetic Property |  Direction of $M$ Relative to External Field |  Relative Magnetic Susceptibility ($\chi$, in ppm) |  Example Materials
---|---|---|---
Diamagnetism |  Opposite |  -10 |  Water, most biological tissues
Paramagnetism |  Same |  +1 |  Molecular oxygen, O2
Superparamagnetism |  Same |  +5000 |  SPIO contrast agents
Ferromagnetism |  Same |  $>$10,000 |  Iron, steel
Figure 1: Magnetic Susceptibility. Adapted from https://mriquestions.com/what-
is-susceptibility.html [36] Figure 2: $\boldsymbol{B_{0}}$ magnetic field in
the brain at 7T with 2${}^{\text{nd}}$ order shimming. This fieldmap shows the
spatially varying patterns of the $B_{0}$ field in the human brain
(superimposed over an anatomical image). Even after 2${}^{\text{nd}}$ order
shimming has been applied to flatten the field, higher order field
inhomogeneities persist. Arrows show that near the air-filled, and therefore
paramagnetic, sinuses, the $B_{0}$ field is higher than areas in the center of
the mostly diamagnetic brain tissue. _(Images reused from Stockmann & Wald,
2018 [35] with permission.)_ Figure 3: Metal artifact from 50mT to 7T. Here we
show Figure 7 from Van Speybroek, et al., 2021 [37] where the “worst-case”
scenario images for four types of metal implants are demonstrated. _(Images
reused via open access license.)_

###  2.3 Chemical shift

Another source of off-resonance is the chemical shift in resonant frequency of
different tissues in the body. The most relevant source of chemical shift is
from fat, which has a shift in frequency of -3.5ppm relative to water [38]. At
1.5T (64 MHz), this corresponds to a shift in frequency of 64MHz $\times$
-3.5ppm $=$ -224 Hz, and at 3T a shift of -448 Hz. The earliest accounts of
chemical shift image artifacts describe enhanced or diminished borders of
organs within the body that have tissue boundaries containing fat [39, 40].
This is due to the fat signal shifting within the image because it is off-
resonant compared to water (Eq. (7)). The directions and angles of the slice
selection and frequency encoding axes can also affect chemical shift
artifacts, leading to them sometimes being present and sometimes not, which
can confound diagnosis and affect treatment [41]. Hood et al. [38] provide a
good review of how chemical shift artifacts impact clinical interpretation of
images. The authors also describe what are referred to as chemical shift
artifacts of the 2${}^{\text{nd}}$ kind, which are not shifts in image space
of the voxel, but rather a cancellation of the out-of-phase water and fat
components in the voxel that leads to a loss of total signal (see Hood et al.
Fig. 10). These out-of-phase images at a given TE can be compared to images
with different TEs where fat is visible, allowing the clear identification of
fat.

###  2.4 Metal implants

While magnetic susceptibility differences drive many sources of $B_{0}$
inhomogeneity as described in Section 2.2, the $B_{0}$ effects of metal
implants are different, mainly due to the sheer magnitude of the $B_{0}$
shifts, leading to effects that often require advanced measures to compensate.
Most metal implants in common use today are labeled as MRI-conditional, which
indicates that MRI scanning is acceptable provided the specified conditions
for safe use are met [42, 43]. Allowable or safe use, however, does not mean
that the quality of the images is not impacted by susceptibility effects
around the implant.

Hagreaves et al. [8] is a review on sources and artifacts due to metal
implants as well as a compendium on the various approaches to mitigate the
artifacts. Jungmann et al. [44] also reviews mitigation methods and provides
numerous imaging examples. As with all susceptibility-induced $B_{0}$ changes,
these effects scale with field strength. Notably, lower magnetic fields result
in smaller distortions and lower field strength clinical scanners are often
preferred when scanning structures proximal to metal implants, as seen in Fig.
3. Further, the lower resonant frequency at lower field reduces the safety and
imaging impact of RF ($B_{1}$) inhomogeneity around metal implants.

##  3 Impact on Imaging

###  3.1 MRI Signal Equation Incorporating Off-Resonance

In an MRI experiment, the magnetic spins being imaged precess at resonant
frequency $\omega$:

$\omega=\gamma B$ (2)

where $\gamma$ is a spin’s gyromagnetic ratio and $B$ is the magnetic field
strength at the spin location. For an 1H proton in water for which
$\gamma/2\pi=42.577$ MHz/T, at a magnetic field strength $B=$ 3T, the spins
precess at roughly 128 MHz. Note that throughout this paper we may use “field”
or “frequency” depending on context, but the reader can convert from a
precession frequency in Hz to a magnetic field in T or vice versa by
multiplying or dividing by $\gamma/2\pi$.

To create an image in an ideal MRI experiment, first, the object is placed in
a spatially uniform polarizing field, $B_{0}$. The 1H spins throughout the
entire object will now be precessing at $\omega_{0}$. Next, spatially varying
magnetic fields, $G_{x}$, $G_{y}$, and $G_{z}$, referred to as “gradient
fields” or simply “gradients”, are applied to linearly change the magnetic
field as a function of spatial position, which therefore changes the
precession frequency of spins at different locations, i.e.,
$\omega(x,y,z)=\gamma(B_{0}+G_{x}x+G_{y}y+G_{z}z)$. By applying different
gradient fields over time, spins at different locations can be encoded, the
most common method being Fourier space, or $k$-space, encoding. The MRI signal
as a function of these time-varying gradients can be described using the
signal equation:

$s(t)=\int m(x,y)e^{-i2\pi[k_{x}(t)x+k_{y}(t)y]}dxdy\hskip
10.0pt\text{,}\hskip
10.0ptk_{\alpha}(t)=\frac{\gamma}{2\pi}\int_{0}^{\tau}G_{\alpha}(\tau)d\tau$
(3)

where $s(t)$ is the time-varying MRI signal, $m(x,y)$ is the magnetization at
location $(x,y)$, and $k_{x}(t)$ and $k_{y}(t)$ are the $k$-space sampling
locations. For brevity, the equation above and the rest of the image encoding
mathematics will be in 2D, but it can be extended to 3D as well by adding a
third $z$ term to the exponent in Eq. (3).

By having a spatially uniform polarizing $B_{0}$ field, the time-varying
signal $s(t)$ is only a function of the applied gradients. However when
$B_{0}$ is not spatially uniform, i.e., $B_{0}(x,y)=B_{0,ideal}+\Delta
B_{0}(x,y)$, we must add a term to the signal equation for the _additional_
(and generally unwanted) spatially dependent phase, $\phi(x,y)$, accrued over
time $t$ due to $\Delta B_{0}(x,y)$:

$\phi(x,y,t)=\gamma{\Delta B_{0}(x,y)t=\Delta\omega_{0}(x,y)t=2\pi\Delta
f_{0}(x,y)t}$ (4)

The phase can now be added to the signal equation using a complex exponential
of the product of the fieldmap and the time, $t$:

$s(t)=\int
m(x,y)e^{-i\Delta\omega_{0}(x,y)t}e^{-i2\pi[k_{x}(t)x+k_{y}(t)y]}dxdy$ (5)

This leads to a breakdown of the encoding properties of the MRI signal
equation, since now there will be additional phase in the object that is not a
result of the $k$-space encoding. Depending on the length of the readout, the
type of $k$-space acquisition, and other scan parameters, the excess phase
accrued due to off-resonance can lead to various types of artifacts such as
signal loss, distortions, and blurring.

Figure 4: Signal Loss from Spin Dephasing. A. Voxel view of spin dephasing.
Spins with different off-resonance will gain different amounts of phase during
signal readout leading them to “fan out”, and eventually completely cancel
each other. B. A gradient echo image is shown for two different slices at
three different echo times (TE). At the later echo times, signal loss due to
dephasing is present as indicated by the arrows.

Signal Loss

When spins in a voxel have different resonant frequencies due to rapid spatial
variations in the $B_{0}$ field, spins will dephase as seen in Fig. 4 (images
in Fig. 4 were acquired from a healthy subject after obtaining IRB approved
written informed consent). Immediately after RF excitation the spins are all
aligned, but during the course of the readout the spins dephase and there is
destructive interference in the voxel, referred to as $T^{*}_{2}$ decay. The
most common way to deal with this signal loss is through the use of spin
echoes [11]. In a spin echo sequence, the spins are flipped into the
transverse plane, and then after time TE$/2$, a 180∘ RF pulse flips the spins
about one of the transverse axes. Then, after precessing for time TE/2, the
spins will refocus at the echo time, TE. Spin echoes are very robust overall,
and because of this spin echo based sequences have been the pulse sequence of
choice for many of the academic low field scanners [21, 22, 20, 32], as well
as the commercially available 64 mT portable brain scanner [23].

Image Encoding Artifacts: Distortions & Blurring

Spin echoes can mitigate much of the off-resonance based signal loss, but even
using spin echo sequences, significant off-resonance artifacts can be present
based on image encoding errors, such as distortion [45], blurring, and pile-
up. Looking at Eq. (5), the reduction of intravoxel dephasing by using a spin
echo leads to an apparent increase in the $m(x,y)$ term, but the spatial
encoding in the exponential of that equation will still be affected by $B_{0}$
effects. When there is a non uniform fieldmap like the one in Fig. 2, there is
a breakdown in the encoding since it is not only scanner gradient $k$-space
encoding that is contributing to a voxel’s phase, but also phase accrued due
to off-resonance. We can factor out 2$\pi$ and combine the exponentials into
one term as follows:

$s(t)=\int m(x,y)e^{-i2\pi[\Delta f_{0}(x,y)t+[k_{x}(t)x+k_{y}(t)y]]}dxdy$ (6)

If $[k_{x}(t)x+k_{y}(t)y]>>\Delta f_{0}(x,y)t$, $s(t)$ will accurately encode
the object at $m(x,y)$ for k-space location $(k_{x}(t),k_{y}(t))$. However, if
there is large off-resonance (or long time $t$), the phase pattern associated
with the encoding will be warped, as seen in Fig. 5.

Figure 5 shows the ideal vs. the actual encoding pattern, i.e., the phase of
the complex exponential in Eq. (6), to acquire the sample at
$(k_{x},k_{y})=(0,0.426\text{ cm}^{-1})$. Note that Fig. 5 shows the phase
pattern superimposed on the object _for a single point in k-space_ , and the
encoding error is only a function of the time since RF excitation, $t$, and
the off-resonance fieldmap, $\Delta f_{0}(x,y)$, not the sampling trajectory
used. The specific geometric distortions in image space will be a function of
what these encoding errors were _across all k-space points_ , and will depend
on the sample ordering, with later samples having more excess phase accrued
due to off-resonance. As described in more detail below, for Cartesian
sequences this encoding error normally manifests in geometric distortions, and
for non- Cartesian sequences, such as spiral, this will result in both
distortions and blurring.

###  3.2 Utility of long readouts

In an examination of Eq. (4), we can see that off-resonance phase accrual is
linearly dependent on the time since excitation. This can be split into two
parts: the phase accumulation to the echo time (TE) and phase accumulation
during k-space scanning. The first term contributes to signal loss in the
$m(x,y)$ term in gradient echo imaging. The second term contributes to k-space
distortions and is effects, as shown in Fig. 5. Thus, long readouts for
imaging will show increased distortions. That said, there are advantages to
having longer readouts in terms of SNR, $k$-space scanning efficiency, and
overall acquisition speed.

####  3.2.1 SNR

For a given repetition time (TR), flip angle, and spatial resolution, the
signal to noise ratio (SNR) is proportional to the square root of the total
acquisition time, $T_{A/D}$ [46], at least to a first approximation. The total
acquisition time here is defined the sum of the length of all acquisitions
contributing to a given image. One can also define SNR efficiency as the SNR
divided by the square root of overall time to acquire an image. Thus, as one
increases the time spent acquiring data in each TR, say by filling in dead
times, the SNR efficiency also increases. This is, however, often a minor
effect.

####  3.2.2 k-space scanning

Perhaps the most important advantage of a long readout is for speeding up the
image acquisition process, by acquiring more k-space data with each RF
excitation. In general, reduced study duration is beneficial for patient
comfort and for economic reasons. Beyond that, accelerated imaging is also
enabling for many applications. While there are numerous methods to accelerate
imaging, notably parallel imaging, k-space scanning with long readouts also
plays an important role in reducing scan time. For example, in dynamic imaging
applications like cardiac imaging, there may be advantages to acquiring a
larger part of k-space with each excitation so that higher resolution images
can be acquired during a breath hold. In neuroimaging, the ability to “freeze”
motion has led to single shot k-space scanning, most commonly using echo
planar imaging (EPI), to be the dominant methods for acquiring diffusion
weighted and BOLD functional brain images. These are but a few examples where
long readout and k-space scanning are useful in MRI.

Figure 5: Ideal vs. actual encoding phase at a single k-space location. A.
Simulated off-resonance fieldmap in Hz. B. 1D plots of the encoding fields as
a function of position, where the red dotted line correspond to the dotted red
line in A. C. The ideal k-space encoding phase for
$(k_{x},k_{y})=(0,0.426\text{ cm}^{-1})$. D. The actual k-space encoding phase
for $(k_{x},k_{y})=(0,0.426\text{ cm}^{-1})$. Combined with k-space encoding
errors at other k-space sampling locations, this phase error will lead to off-
resonance image artifacts.

###  3.3 Cartesian imaging: spin-warp and EPI

In Cartesian imaging, data is sampled along straight line segments along
$k_{x}$. The term “Cartesian” implies that the samples are regularly spaced,
though in practice the $k_{x}$ dimension is sometimes sampled along the
trapezoidal gradient ramps (in addition to the plateau) for increased scan
efficiency. Since the invention of MRI, Cartesian imaging has remained the
“bread and butter” MRI technique despite its relatively low scan-time
efficiency compared to non-Cartesian imaging, since (i) it allows fast and
simple reconstruction using inverse Fast Fourier Transforms (iFFT), (ii) image
artifacts due to off-resonance and chemical shift are often easy to recognize
and “read through”, and (iii) geometric distortions can be corrected using
fast and widely available methods.

####  3.3.1 Spin-warp imaging

To understand the impact of off-resonance and chemical shift on Cartesian
imaging, consider first the case of “spin-warp” imaging, where a single line
along $k_{x}$ is sampled after each RF excitation, with constant gradient
amplitude $G_{x}$. In this case the $k_{x}$-encoding term in Eq. (6) can be
written as $k_{x}(t)=\frac{\gamma}{2\pi}G_{x}t$ and (since the acquisition
bandwidth along $k_{y}$ is effectively infinite) the exponent in Eq. (6)
becomes $-i2\pi k_{x}[x+\delta x]-i2\pi k_{y}(t)y$ where

$\delta x=\frac{2\pi\Delta f_{0}(x,y)}{\gamma G_{x}}.$ (7)

In other words, a voxel at position $x$ with off-resonance frequency $\Delta
f_{0}$ will appear at $x+\delta x$ in the reconstructed image. For typical
readout bandwidths, $\delta x$ is negligible for most tissue types at clinical
field strengths, but can be on the order of a voxel width, e.g., fat. As a
consequence, the acquisition bandwidth for most clinical sequences is kept
sufficiently high to avoid significant fat shift. For example, a common
acquisition bandwidth is +/- 32 kHz (64 kHz total). With the fat chemical
shift at 3T (448 Hz), the corresponding image shift it 1.75 voxels for a 256
readout sample. Note that Eq. (7) ignores any $B_{0}$ inhomogeneity within the
voxel centered at location $(x,y)$, and hence does not describe any signal
loss that may occur at a given echo time (TE).

####  3.3.2 Echo-planar imaging

In echo-planar imaging (EPI), multiple lines at different $k_{y}$-encoding
levels are acquired after RF excitation. This is done by playing a small
y-gradient trapezoid “blip” after the end of each $k_{x}$ line, and then
traversing the subsequent line in reverse direction (non-flyback EPI).
Assuming regular sampling along $k_{y}$ with subsequent “echoes” spaced
$\tau_{s}$ apart in time, and ignoring the (typically small) shift $\delta x$
along the frequency-encoding direction, the exponent in Eq. (6) becomes
$-i2\pi k_{x}x-i2\pi k_{y}[y+\delta y]$ where

$\delta y=\frac{2\pi\Delta f_{0}(x,y)}{\gamma\overline{G_{y}}}$ (8)

is the spatial shift (along $y$) in the reconstructed image. Here,
$\overline{G_{y}}$ is the average gradient amplitude of the $y$ blip over the
echo time spacing $\tau_{s}$, which is much smaller than the readout gradient
amplitude $G_{x}$ in Eq. (7). As a result, EPI can produce substantial spatial
shifts along the phase encoding (PE) direction, here $k_{y}$. However if the
user were to change the PE direction, the spatial shifts would also change
direction, as these off-resonance induced spatial shifts are primarily in the
PE direction.

A useful way to think about these EPI distortions is to express the average
gradient in the PE direction in units of Hz/cm, as follows:

$\overline{G}_{y,Hz/cm}=\frac{\gamma\overline{G_{y}}}{2\pi}=1/(\tau_{s}*FOV/R)$
(9)

where $\tau_{s}$ = echo spacing and R = in-plane acceleration. Typical values
are about 66 Hz/cm for high-resolution fMRI (2mm isotropic resolution,
$\tau_{s}$ = 0.75 ms, FOV = 20 cm, R = 1), 100 Hz/cm for low-resolution fMRI
($\tau_{s}$ = 0.50 ms, FOV = 20 cm, R = 1), and about 133 Hz/cm for DTI with
2mm resolution and in-plane acceleration ($\tau_{s}$ = 0.75 ms, FOV = 20 cm, R
= 2). The spatial shift along the PE direction (Eq. (8)) is then found by
dividing the local off-resonance $\Delta f_{0}$ by the average gradient
expressed in Hz/cm.

####  3.3.3 Interleaved and readout mosaic segmented EPI

EPI comes in several multi-shot variants that make various tradeoffs between
image artifacts and scan efficiency. In addition to geometric distortions as
just described, another consideration is $T_{2}^{*}$ decay during the echo
train, which acts as a k-space filter resulting in blurring in image space.

In interleaved EPI, the area of each $k_{y}$-encoding blip, and hence
$\overline{G_{y}}$, is increased, making each EPI train undersampled by a
factor equal to the number of segments (shots) $N_{s}$. This reduces the
geometric shifts, and the time allowing for $T_{2}^{*}$ decay, by that same
factor $N_{s}$. Interleaved EPI is typically implemented by shifting the start
of subsequent EPI segments by a time $n/\tau_{s}$ where $n=0,...,N_{s}-1$ is
the shot number, to avoid abrupt phase jumps in $k_{y}$-space due to off-
resonance. A potential drawback of interleaved EPI is shot-to-shot variability
in the acquired data, resulting from, e.g., physiological fluctuations or
subject motion.

An alternative scheme useful for diffusion MRI is readout mosaic segmented
EPI, where only a portion (“blind”) of $k_{x}$-space is acquired during each
shot, while sampling all desired $k_{y}$ locations per shot [47, 48]. The
advantage of this approach is that it preserves the regular sampling along
$k_{y}$ in the presence of shot-to-shot k-space shifts due to rotational
motion in the presence of diffusion-encoding gradients [49]. By introducing a
small overlap between blinds along $k_{x}$, such shifts need not introduce any
gaps along $k_{x}$ as well. A drawback of this approach is that the echo time
spacing is not reduced by the full segmentation factor $N_{s}$ due to finite
gradient rise time, and hence the reduction in geometric shift is somewhat
smaller than for interleaved EPI for a given $N_{s}$.

###  3.4 Non-Cartesian acquisitions

While much of MRI carried out on clinical scanners uses Cartesian acquisitions
like spin-warp or EPI, there are numerous useful applications that use non-
Cartesian acquisitions. This is a broad class of image acquisitions that
typically do not acquire data along parallel lines in k-space. Some aspects
are similar to Cartesian imaging, for example, the need to acquire data over a
prescribed area in k-space for the desired spatial resolution and to acquire
k-space data with sufficient sample density to prevent aliasing. The nature of
the off-resonance artifacts, however, are quite different. Below, we discuss
these effects for radial line and spiral imaging, but note that there are yet
other non-Cartesian trajectories, e.g., rosettes [50], that have quite
complicated off-resonance behavior that is beyond the scope of this review.

####  3.4.1 Radial line and spiral imaging

Radial line and spiral imaging are useful in a number of applications. For
example, radial line acquisitions in 2D or 3D have substantial utility in
zero- and ultra-short TE (ZTE and UTE) imaging [51, 52], angiography [53], and
contrast enhanced abdominal imaging [54]. Spiral imaging has been shown to be
useful in functional MRI [55], cardiac imaging [56], and brain imaging [57].

Spiral and radial line imaging have off-resonance features that are similar to
each other, but are quite different from spin-warp and EPI. For each line of a
radial line acquisition, there is a shift in the radial direction in the form:

$\delta r\equiv\frac{2\pi\Delta f_{0}(x,y)}{\gamma G_{r}}$ (10)

where $G_{r}$ is the strength of the radial gradient. Since the direction of
the radial gradient is constantly changing and ultimately points in all
directions, the net effect is not a single shift, but a shift in all
directions. More specifically, any off-resonant point in the object maps to a
ring-like response of radius $\delta r$. This is equivalent to saying that the
point spread function (PSF) or impulse response is this ring-like function,
the radius of which, $\delta r$ is proportional to the off-resonance
frequency, $\Delta f_{0}(x,y)$ . For a complex image, this manifests as a blur
of radius $\delta r$.

For spirals, the off-resonance response is again a radial shift in all
directions, with amplitude:

$\delta r_{s}\approx\frac{2\pi\Delta f_{0}(x,y)}{\gamma\overline{G_{r}}}$ (11)

where $\overline{G_{r}}$ is the average outward component of the spiral
gradients. One can also infer the average outward component with
$\overline{G_{r}}=2\pi k_{max}/T$, where $T$ is the length of the spiral
readout. Now, any off-resonant point in the object maps to a ring-like
response of radius $\delta r_{s}$ and as with radial imaging, this is
proportional to the off-resonance frequency, $\Delta f_{0}(x,y)$ . As can be
seen from the alternate $\overline{G_{r}}$ expression, the blur is also
proportional to length of the spiral readout, $T$. Again, for a complex image,
this manifests as a blur of radius $\delta r_{s}$. For spiral imaging, the
relationship that the blur is a ring of radius $\delta r_{s}$ is only
approximate because the average outward component typically varies during the
readout making the actual blur function a bit more complicated. Still, the
main principle is that longer readouts make the off-resonance image blurring
worse.

Figure 6: Image simulations: Cartesian vs. non-Cartesian. Here we show the
different off-resonance effects from EPI vs. spiral acquisition. In the EPI
image, geometric distortion of the brain is visible with a distortion
sensitivity of 75 Hz/cm (see Eq. (9)), a typical value, resulting in the brain
being stretched compared to the original Cartesian spin-warp image (blue
arrows). In the spiral image, there is a large amount of blurring, noted by
the red arrow, but there is less geometric distortion than in EPI.

The off-resonance behavior also extends to three dimensions. For examples, ZTE
imaging and angiography applications often use a 3D radial line acquisition.
While spirals don’t extend directly to 3D, there are 3D trajectories like 3D
cones, trajectories that spiral outward while rotating along two axes
(sometimes called yarn trajectories), and spiral-radial hybrids [58] that have
off-resonance properties similar to spirals. In these cases, we consider
induced blur to be occurring in all three dimensions, that is, the PSF or
impulse response is an annular sphere with radius $\delta r$ (or $\delta
r_{s}$). Again, greater off-resonance and longer readouts increase the radius
of the blurring function.

The ways that off-resonance affects EPI and spiral images at 3T is shown in
Fig. 6 (images in Fig. 6 were acquired from a healthy subject after obtaining
IRB approved written informed consent). As expected the artifacts in the
spiral image are primarily blurring, and the artifacts in the simulated EPI
image are primarily geometric distortions along the phase encode (vertical)
direction. Additional details on the simulation can be found at
https://github.com/fmrilab/B0-review-2022.

###  3.5 Imaging near metal implants

Though not ferromagnetic, the susceptibility differences between metals and
tissues are substantially larger than the differences between tissues and the
surrounding air. These large susceptibility differences, in combination with
the often large mass of these implants, lead to similar, but magnified,
$B_{0}$ effects. For example, much of GRE imaging is infeasible in close
proximity to implants, except for the very shortest TEs, meaning that most
studies use SE or Fast/Turbo Spin Echo (FSE/TSE) acquisitions. Similarly,
rapid imaging methods like EPI or spiral imaging with implants will typically
produce artifacts that are too severe to correct. There are often severe
geometric distortions in the readout (frequency encoding) direction associated
with implants, including signal voids and piling up artifacts. Another common
issue is the distortion of the slice profiles, where there can be spatial
displacement several slice thicknesses in magnitude, distorted slice profiles
like potato chips, and variable slice thicknesses. In both frequency encode
and slice effects, the spatial distortions are proportional to the frequency
offset divided by the bandwidth per pixel (or slice thickness).

##  4 Estimating _B_ 0 Field Inhomogeneity

###  4.1 MRI sequences for acquiring fieldmaps

Before correction for $B_{0}$ inhomogeneity artifacts, it is often necessary
to acquire and estimate the fieldmap. The simplest way to estimate a $B_{0}$
fieldmap is to create a pulse sequence that acquires two sequential gradient
echo images at different TEs, and then subtract their relative phases to
obtain the amount of precession at that voxel over the echo time difference
$\Delta$TE. The phase difference is then scaled by $\Delta$TE to get the
fieldmap:

$\Delta\phi(x,y)=\angle m(x,y,\mathrm{TE}_{1})-\angle m(x,y,\mathrm{TE}_{2})$
(12) $\Delta\omega_{0}(x,y)=\frac{\Delta\phi(x,y)}{\Delta\mathrm{TE}}$ (13)

This is the basic idea behind $B_{0}$ fieldmapping in MRI, but there are many
ways to estimate a fieldmap within this general approach. The choice of
$\Delta$TE will affect the quality of the fieldmaps, and has been
quantitatively evaluated in prior work [59]. Normally gradient echo images at
short TEs are used to avoid distortions in the underlying fieldmap, but for
applications using EPI, there has also been work in using EPI-based fieldmaps
[60]. While EPI fieldmaps will be distorted given the longer TEs required,
they have the advantage that they will be distorted in the same way and also
experience the same eddy currents as the EPI data used in the imaging
experiment, potentially allowing for easier look up of the off-resonance at
any voxel.

Another consideration when calculating fieldmaps is phase wrap. A longer
$\Delta$TE is desirable because it will increase the SNR of the phase
difference measurement. However, when $\Delta$TE is increased, more phase wrap
can occur. Fortunately, many tools such as FSL [61] include phase unwrapping
algorithms that are useful when calculating off-resonance field maps, such as
the automated method introduced by Jenkinson in 2003 [62].

There have also been recent approaches to rapidly acquire $B_{0}$ fieldmaps
along with transmit coil patterns ($B_{1}^{+}$), receive coil sensitivity
($B_{1}^{-}$), and eddy current maps in as little as 11s at
1$\times$2$\times$2mm3 resolution at 3T [63]. This kind of approach is
especially useful for model-based image reconstructions that incorporate
$B_{0}$ (see Section 5.4), since $B_{1}^{-}$ maps are often needed as well.

There are requirements that the fieldmaps be reasonable accurate and not too
noisy. As a general rule, the error/noise level should be below the level of
observable degradation of the point spread function. One rule of thumb is for
the error and noise levels to be below 0.25 cycle of phase over the readout.
For example, if you want to correct the field effects for a 25 ms readout, 40
Hz offset will produce one cycle of phase over the readout and thus, we would
want worst case errors to be below 10 Hz. Often times, the fieldmaps are
smoothly varying exec pt for small air/tissue boundaries. As such, it is often
expedient to collect fieldmaps at lower spatial resolution (2-3 mm) than the
images to be corrected (1-2 mm).

###  4.2 Hardware methods

In addition to measuring a fieldmap using a sequence, it is possible to employ
external hardware and/or built in scanner hardware to monitor changes in off-
resonance. Generally these types of hardware tracking methods will be used for
dynamic $B_{0}$ correction (see Section 5.5) after an initial $B_{0}$ map has
been measured using a fieldmapping sequence. One option is to use NMR field
probes [64, 65, 66], which measure the changes in phase of the MRI signal
within small external samples. These probes are now commercially available
(https://skope.swiss/) and have been used in applications such as diffusion
MRI, fMRI, and ultra high field imaging. Another approach is to use FID
measurements from a multicoil receive array, which can encode changes in
signal based on the different spatial locations of the coils. By combining
with a reference image from each coil, Wallace et al. showed that FIDs can be
used to dynamically track up to 2${}^{\text{nd}}$ order field changes [67].

###  4.3 Optimization and smoothing methods

After multi-TE data has been acquired, Eq. (13) is the simplest way to
calculate a fieldmap, but the maps are often improved by using some form of
smoothing or regularization due to the difficulty of getting $B_{0}$ values
near the edges of anatomy. In the original TOPUP paper [68], a smoothed
version of the fieldmap that is based on polynomial fitting was used. The work
by Hutton et al. [59], in addition to examining $\Delta$TE, also
quantitatively evaluates how the amount of smoothing used affects fieldmap
estimates.

There are also optimization-based approaches that use a multi-term objective
function, where the phase differences between multiple TEs is a data fit term
and another regularization term penalizes rapidly varying and sudden field
changes [69]. This method has been expanded upon to incorporate a more
efficient optimization algorithm and the ability to input multicoil data [70].
Additionally, it is possible to jointly optimize for a fieldmap and an image,
and these approaches are described in Section 5.4 on model based image
reconstruction.

##  5 Correction Strategies

In this section we discuss strategies to correct for off-resonance artifacts
in MRI. Many of the methods described could fit into multiple categories, but
we have broadly categorized them to help in describing the different options
available to MRI users, as outlined in Figure 7.

Artifact Prevention (Section 5.1) |  $\boldsymbol{B_{0}}$ Correction Methods (Sections 5.2-5.6)
---|---
• Hardware based – passive & active shimming – lower-field MRI • Sequence based – spin echoes – shorter readouts: parallel imaging, multishot imaging – fat suppression / nulling (STIR) – smaller voxels: parallel imaging, simultaneous multislice |  • EPI/Cartesian focused – Image Space: TOPUP – PSF • Non-Cartesian focused – Direct/Analytical: conjugate phase, SPHERE – Autofocusing • Model based image reconstruction including $B_{0}$ • Dynamic corrections • Corrections for larger $B_{0}$ shifts around metal implants – View angle tilting – MAVRIC – SEMAC
Figure 7: Common $B_{0}$ artifact mitigation strategies.

###  5.1 Artifact Prevention & Reduction

####  5.1.1 Hardware based artifact mitigation

The most straightforward way to prevent $B_{0}$ artifacts is to have as
uniform a field as possible before starting a scan, and the process of
refining the uniformity of the $B_{0}$ field is referred to as _shimming_
[71]. Shimming can be done either by improving the static $B_{0}$ field
hardware (i.e., with superconducting shim magnets or ferrous passive elements)
or by actively changing the currents in the scanner’s electro-shim coils,
which include the gradient coils. An example of passive shimming at low field
is the work of Liu, et al., where on their 50mT scanner they originally had a
2000 ppm peak-to-peak uniformity over a 24cm diameter spherical volume (DSV)
within their SmCo dipole magnet, but after adding smaller passive shim magnets
their uniformity was increased to 250 ppm [20]. Automated shimming is a
standard part of a clinical MRI pre-scan calibration (at least for the
constant linear shims terms) and for most users this will be automatic, but
for certain types of acquisitions, such as MR spectroscopy that requires a
very uniform field [72], the user may need to do manual shimming as well.

In addition to shimming, using a lower magnetic field is a direct way to
reduce $B_{0}$ artifacts. As was discussed in Section 2, $B_{0}$ field changes
from susceptibility and chemical shift scale with field strength, so these
artifacts are generally worse at higher field. There are some $B_{0}$ artifact
mitigation strategies that are harder at low field due to decreased SNR, but
in general going to lower field will reduce signal loss and encoding artifacts
from an inhomogeneous field.

####  5.1.2 Sequence based artifact mitigation

When designing an MRI sequence, parameters can be adjusted to reduced the
effects of off-resonance. Using a short readout time (high acquisition
bandwidth) will limit the amount of extra phase accrued due to off-resonance
and therefore reduce artifacts. The use of parallel imaging techniques [13,
73] is an effective way to reduce the readout length of a single-shot
acquisition (such as EPI or single-shot spiral), by allowing the user to
collect less k-space data each shot. If using longer readouts, spin echoes and
spin echo trains will refocus spins and avoid much of the $T_{2}^{*}$ signal
loss [11, 12]. Chemical shift artifacts can be avoided by using a short time
inversion recovery (STIR) sequence that nulls recovering fat signal, inserting
fat saturation modules before water excitation (an RF excitation at the fat
frequency and then a large crusher gradient), or by using a spectrally
selective RF pulse centered around the water resonant frequency [74]. Further,
while this generally has the downside of longer scan time due to additional
encoding, using smaller voxels will decrease the possible signal loss due to a
more local neighborhood of spins being averaged in each voxel. Parallel
imaging and simultaneous multislice imaging [14] are effective strategies to
reduce voxel size without increasing scan time.

###  5.2 EPI

Figure 8:  Illustration of EPI distortion correction using Eq. (14). (A)
Ground-truth (undistorted) object and (B) fieldmap. (C,D)
$\boldsymbol{K}_{+,i}$ and $\boldsymbol{K}_{-,i}$ for the dashed lines in
(E,F). (E,F) Distorted images $\boldsymbol{f}_{+}$ and $\boldsymbol{f}_{-}$.
(G) Distortion correction attempt using only $\boldsymbol{f}_{+}$. (H)
Successful distortion correction using Eq. (14). _(Images reused from
Andersson et al., 2003[75] with permission.)_

Several methods for correcting geometric and intensity distortions in EPI have
been proposed (see, e.g., [76, 77] for recent overviews), and here we focus on
three of the more popular ones: image-based correction using an acquired
$B_{0}$ map, image-based correction using two reference scans with opposite
phase-encode blip directions, and the point-spread function (PSF) mapping
approach. These have all proven to be useful for EPI distortion correction in
fMRI and diffusion imaging.

####  5.2.1 Fieldmap based correction

One popular approach is based on the separate acquisition of an undistorted
$B_{0}$ map (e.g., using spin-warp 3D GRE), that is used to calculate the
shift $\delta y$ (Eq. (8)) for each spatial location [68]. The pixel values in
the reconstructed (distorted) image are then shifted along the phase-encode
direction by $-\delta y$. This is followed by an intensity correction step
based on the spatial gradient of $\delta y$ [68]. Potential drawbacks of this
approach include inaccurate correction near object edges due to inaccurate
$B_{0}$ estimates (partial-volume effects), and inability to resolve
overlapping pixels.

####  5.2.2 Opposite phase-encoding scans

Another widely used approach that also operates on magnitude images is TOPUP
[75], that uses as input two EPI images that differ only in the sign of the
phase-encoding blips (i.e., traversing $k_{y}$ in the positive and negative,
or bottom-up and top-down, directions). The TOPUP model is

$\begin{bmatrix}\boldsymbol{f}_{+,i}\\\
\boldsymbol{f}_{-,i}\end{bmatrix}=\begin{bmatrix}\boldsymbol{K}_{+,i}\\\
\boldsymbol{K}_{-,i}\end{bmatrix}\boldsymbol{x}_{i}$ (14)

where $\boldsymbol{x}_{i}$ is the $i^{\textrm{th}}$ column (along y) of the
undistorted true image $\boldsymbol{x}$, $\boldsymbol{f}_{+}$ and
$\boldsymbol{f}_{-}$ are the distorted bottom-up and top-down images,
respectively, and $\boldsymbol{K}_{+,i}$ and $\boldsymbol{K}_{-,i}$ are real-
valued square matrices that map the pixel locations in the true image column
$\boldsymbol{x}_{i}$ to the observed image columns $\boldsymbol{f}_{+/-,i}$.
In the absence of $B_{0}$ inhomogeneity, $\boldsymbol{K}_{+,i}$ and
$\boldsymbol{K}_{-,i}$ would be the identity matrix $\boldsymbol{I}$.

With knowledge of $B_{0}$ (which determines $\boldsymbol{K}_{+,i}$ and
$\boldsymbol{K}_{-,i}$), Eq. (14) can be inverted to obtain a least-squares
estimate $\boldsymbol{\hat{x}}$ of the true image, with the benefit that
pixels that overlap in one of the images (causing either
$\boldsymbol{K}_{+,i}$ or $\boldsymbol{K}_{-,i}$ to be singular) are separated
in the other, which enables the underlying true image to be resolved.

In the absence of $B_{0}$ inhomogeneity, $\boldsymbol{\hat{x}}_{i}$ is simply
the average of $\boldsymbol{f}_{+,i}$ and $\boldsymbol{f}_{-,i}$. Furthermore,
the TOPUP model can be used to estimate $B_{0}$ from the two EPI scans
(obviating the need for a separately acquired $B_{0}$ map), by enforcing
consistency between the observed images $\boldsymbol{f}_{+,-}$ and the
predicted images $\boldsymbol{\hat{f}}_{+,-}$ created by distorting
$\boldsymbol{\hat{x}}$ using Eq. (14):

$\boldsymbol{\hat{{b}_{i}}}=\mathop{\mathrm{arg\,min}}_{\boldsymbol{b}_{i}}\,\biggl{|}\begin{bmatrix}\mathbf{f}_{+,i}\\\
\mathbf{f}_{-,i}\end{bmatrix}-\begin{bmatrix}\mathbf{K}_{+,i}(\boldsymbol{b}_{i})\\\
\mathbf{K}_{-,i}(\boldsymbol{b}_{i})\end{bmatrix}\boldsymbol{\hat{x}}_{i}(\boldsymbol{b}_{i})\biggr{|}_{2}^{2}$
(15)

where $\boldsymbol{b}_{i}$ parametrizes the displacements (using the notation
from [75]), and we have made the dependence of $\boldsymbol{\hat{x}}_{i}$ on
$\boldsymbol{b}_{i}$ explicit. Knowledge of $\boldsymbol{b}_{i}$ yields an
estimate of the fieldmap $B_{0}$, which can then be used to create an
undistorted image using the approach described above. Advantages of this
approach include estimation of $\delta y$ (Eq. (8)) from fast EPI scans, and
avoiding potential inaccuracy issues in the directly acquired $B_{0}$ maps.
However, like fieldmap based correction, TOPUP cannot easily correct for
signal pile-up in regions with severe $B_{0}$ inhomogeneity. While there are
some limits to how much correction is possible, this approach has the
important practical advantage that it can operate on magnitude images after
simultaneous multislice reconstruction and coil combination.

####  5.2.3 Point-spread function mapping

Another way to obtain $\delta y(\boldsymbol{r})$ is based on measuring the
point-spread function (PSF) along $y$ [78, 79, 80]. This can be done by
performing a calibration prescan wherein one adds an additional $k_{y}$
encoding blip prior to the EPI readout, and repeats the measurement with
different blip areas. For a 2D EPI acquistion, this additional “constant-time”
encoding yields a 3D dataset where the spatial profile in the constant-time
encoding dimension is the PSF [78]. The pixel shift can then be obtained as
the center (peak) of the PSF [80]. An advantage of this approach is that the
shift is estimated with the same EPI readout as the imaging sequence to be
corrected, and hence experiences and accounts for the same eddy currents and
concomitant gradients.

###  5.3 Non-Cartesian

Off-resonance effects lead to burring and other distortions for non-Cartesian
acquisitions, and this is often more objectionable than the geometric
distortions seen in Cartesian imaging. Thus, it is often necessary to do some
sort of off-resonance correction in order to produce clinically useful images.
This section describes common approaches for this correction that are
applicable to radial line, spiral, and related k-space trajectories.

####  5.3.1 Direct/Analytical Methods

##### Conjugate phase (CP) reconstruction

In the absence of off-resonance, the signal equation in Eq. (6) can be seen as
a simple Fourier transform and a good image reconstruction will be simply an
inverse FFT. For a non-Cartesian acquisition, one can implement an approximate
iFFT as:

${\hat{m}(x,y)}=\frac{1}{M}\sum_{i}w_{i}\,y_{i}\,e^{i2\pi(k_{x,i}x+k_{y,i}y)}\approx
F^{-1}\\{y(t)\\}$ (16)

where $i$ is the index over $M$ k-space samples $y_{i}=y(t_{i})$ at k-space
locations $(k_{x,i},k_{y,i})$, and $w_{i}$ is the sample density compensation
term that accounts for uneven distribution of k-space samples common in non-
Cartesian methods. In practice, this expression is commonly implemented by
some sort of interpolation in the k-domain to a Cartesian grid (known as
gridding) followed by iFFT operators or by using integrated functions like the
NUFFT [81].

Figure 9:  CP and MB reconstruction at 50mT. Compared to a standard iFFT
reconstruction, CP and MB reconstructions both correct for geometric
distortions due to off-resonance, especially in the right side of the brain
where the magnetic field off-resonance is stronger. MB also shows more uniform
signal intensity. _(Images reused from Koolstra, et al., 2021[82] via open
access license.)_

When considering off-resonance effects, a good image reconstruction may be to
compensate for the off-resonance phase accumulation in the image
reconstruction at every time point and for each location $(x,y)$ [83, 84] as
shown here:

${\hat{m}(x,y)}=\frac{1}{N}\sum_{i}w_{i}\,y_{i}\,e^{i2\pi(\Delta
f_{0}(x,y)t_{i}+k_{x,i}x+k_{y,i}y)}$ (17)

where the $\Delta f_{0}(x,y)t_{i}$ term is known as the conjugate phase (CP)
correction for the off-resonance (see Fig. 9 for an image example).

There are limitations to the CP reconstruction, notably in that it is not a
true inverse, but rather, an approximation.

In particular, this approach fails to make high quality image reconstructions
when the magnetic field ($\Delta f_{0}(x,y)$) varies rapidly in space and
therefore the ring-like PSF will also vary rapidly in space. If two PSFs
overlap in space and these PSF functions are different, then this approach
cannot fully correct for both blurring functions. An alternate way to think
about this is that rapidly varying $\Delta f_{0}$ corresponds to a local
gradient, and therefore, the k-space is distorted for locations with large
local gradients [85]. The distorted local k-space limits the effectiveness of
the CP reconstruction. Model-based image reconstruction, described below, is
more robust to address rapid variations in the field map.

##### Efficient approximations

While Eq. (16) has fast implementations with gridding or the NUFFT, Eq. (17)
is no longer an inverse Fourier transform due to the $\Delta f_{0}(x,y)t$
term. Still, there are computationally efficient implementations that result
from decomposing this term into a sum of terms that vary only in space and
terms that vary only in time. The exponential can be modeled as a summation of
basis functions [86, 87]:

$e^{i2\pi\Delta f_{0}(x,y)t}\approx\sum_{l=1}^{L}b_{l}(t)c_{l}(x,y)$ (18)

where $b_{l}$ and $c_{l}$ represent the basis functions for the time and image
space, respectively. Note that frequency, $\Delta f_{0}(x,y)$ , varies only
with space.

The first such implementation was the so-called time-segmented approximation
[88] to the CP correction term, where $b_{l}=b_{l}(t-\bar{t}_{l})$ is a family
of time-varying weighting functions centered at $\bar{t}_{l}$ and
$c_{l}=e^{i2\pi\Delta f_{0}(x,y)\,\bar{t}_{l}}$ is a spatially varying phase
function corresponding to $\bar{t}_{l}$. This approximation to Eq. (17) is:

${\hat{m}(x,y)\approx\sum_{l=1}^{L}F^{-1}\\{b_{l}(t-\bar{t}_{l})\,y(t)\\}e^{i2\pi\Delta
f_{0}(x,y)\,\bar{t}_{l}}}$ (19)

where $F^{-1}$ is the gridding or NUFFT inverse transform of windowed segments
of $y(t)$. A related form is known as the multifrequency interpolation (MFI)
method [89, 90], which modulates the acquired data at frequencies $f_{l}$
using $b_{l}=e^{i2\pi f_{l}\,t}$ and interpolates between modulated images
using $c_{l}(x,y)$. This approximation to the CP reconstruction is:

${\hat{m}(x,y)\approx\sum_{l=1}^{L}F^{-1}\\{e^{-i2\pi
f_{l}\,t}\,y(t)\\}\,c_{l}(\Delta f_{0}(x,y)-f_{l})}.$ (20)

Observe that $L$ Fourier transforms are implemented in both of these
approximations, so while they are about $L$ times slower than Eq. (16), they
are still substantially faster than the discrete Fourier transform
implementation of Eq. (17).

Both the time-segmented approximation and the MFI methods perform similarly
and can be viewed as low-rank approximations to the $e^{i2\pi\Delta
f_{0}(x,y)t}$ term. The interpolation functions can be selected optimally
according to different optimality criteria [90, 86].

##### Deblurring with deconvolution

While the above approaches implement off-resonance correction as part of the
image reconstruction, this can also be implemented in the image domain using
deconvolution [91]. Provided the spatial extent of the PSF or blurring
function is small, one can deblur the images using a space variant
deconvolution, where the deconvolution kernel is guided by the acquired
fieldmap. In Ahunbay et al. [91], it was shown that in some circumstances the
deconvolution kernel is separable in $x$ and $y$, which can greatly reduce the
computational burden.

##### Simulated phase evolution rewinding (SPHERE)

An alternate approach to field correction that is similar to the CP approach
is to apply the CP correction to the forward Fourier transform rather than the
inverse transform. This general approach is known as simulated phase evolution
rewinding (SPHERE) [92]. The approach starts with a reconstruction (iFFT) that
does not apply an off-resonance phase correction, e.g., Eq. (16), and follows
it with a forward transform back to the k-domain, however, during the forward
Fourier transform, the conjugate phase correction is applied:

${y_{corr}(t)=\sum_{x,y}{\hat{m}(x,y)}\,e^{-i2\pi(\Delta
f_{0}(x,y)t+[k_{x}(t)x+k_{y}(t)y])}}$ (21)

where the sign of the complex exponent $-\Delta f_{0}(x,y)$ is opposite of
that in the signal equation Eq. (6), thus applying the conjugate phase to
compensate for the off-resonance phase accumulation. This is followed by a
simple iFFT to produce the final image. As with the CP reconstruction, the
$\Delta f_{0}(x,y)t$ term means the forward transform is no longer a Fourier
transform, but the time segmented approximations and the MFI can be used to
implement this in a computationally efficient manner. Schomberg’s review [93]
provides a detailed mathematical analysis of CP and SPHERE methods, along with
fast implementation strategies and timing comparisons of the different
approaches.

####  5.3.2 Autofocusing

One unique aspect of non-Cartesian imaging is that the PSF may allow for
autodetection of off-resonance based on features of the image. Conceptually,
it is like an autofocus camera where different parts of the image have
different off-resonance corrections to produce a fully deblurred image. This
is done by examining image features rather than through the use of a fieldmap.
In [94], a multi-frequency reconstruction was used, in which non-Cartesian
images were reconstructed to produce a family of images where different parts
of each image are unblurred. An image metric based on a measure of how closely
the phase of low spatial frequency images aligned with the phase of high
spatial frequencies was used to determine, for each voxel $(x,y)$, which
modulated image was sharpest. This measure works well for spiral imaging
because the low spatial frequencies are acquired earlier than high spatial
frequencies, and off-resonance will lead to these components being out of
phase. In recent work [95], real time speech MRI with spiral readouts were
automatically deblurred using a sharpness metric of features in the vocal
tract. Interestingly, these autofocus methods can produce a fieldmap based on
the selected image.

##### Machine Learning

One challenge to autofocusing is the selection of an appropriate measure of
image sharpness. Because of the predictable and reproducible nature of the
off-resonance effects in non-Cartesian imaging, it is possible to train a
neural network to learn and remove image blurs. In [96], body imaging with 3D
cones trajectories and long readouts were deblurred by a convolution neural
network (CNN) that directly maps blurred images to unblurred images acquired
with shorter readouts (but an overall longer scan time). This type of approach
has also been applied to speech MRI, where a longer readout image was
deblurred using a CNN, allowing for longer k-space readouts and therefore
higher temporal visualization of speech in the hard to correct areas around
the mouth and sinuses [97]. In another approach, a CNN is trained to estimate
the fieldmap directly, and then uses the fieldmap within a model based image
reconstruction to produce unblurred images [98].

###  5.4 Model based image reconstruction incorporating _B_ 0

As the use of techniques such as parallel imaging and compressed sensing has
grown, performing MRI reconstruction using model based image reconstruction
(MBIR) [25] has grown in popularity. MBIR has the advantage that it can
incorporate additional operations into a model of the MR imaging system beyond
standard Fourier encoding, and many groups have found that incorporating a
fieldmap into their imaging model can greatly aid in the reduction of $B_{0}$
artifacts. MBIR can also serve as the foundation for hybrid physics+machine
learning approaches that combine an imaging physics model with learned
parameters, often referred to as “unrolled” algorithms [99].

In MBIR, a so called “forward model” is created that mathematically describes
the transform from an image in vector form, denoted as $\boldsymbol{x}$, to
acquired data, $\boldsymbol{y}$, using a matrix that here we will denote as
$\boldsymbol{A}$. This can be thought of as a linear algebra representation of
Eq. (3) where data from all timepoints has been combined into a single vector
$\boldsymbol{y}$, and can be written as:

$\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}$ (22)

Note that we have changed the letters representing the image and data from Eq.
(3) to align with common linear algebra nomenclature. In the simplest
optimization approach, once we have a forward model, we can then optimize for
the image using a linear least squares fit (since the dominant noise in MRI is
Gaussian) as follows:

$\boldsymbol{\hat{x}}=\mathop{\mathrm{arg\,min}}_{\boldsymbol{x}}\frac{1}{2}\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{y}\|_{2}^{2}+\lambda
R(\boldsymbol{x})$ (23)

where $R(\boldsymbol{x})$ is a regularizer chosen by the user, and $\lambda$
is a tuning parameter to balance between the data fit term and the
regularization. A common choice of regularizer in MRI is a total variation
(TV) term, but there are many different options for regularizers and
alternative optimization algorithms to linear least squares, as discussed in a
recent overview by Fessler [25].

In MRI, the simplest possible forward model would be to have $\boldsymbol{A}$
equal a discrete Fourier transform matrix. Additional common terms that are
added include multicoil sensitivity maps and undersampling operators. The
addition of an off-resonance field map is more complex, as the way the off-
resonance affects the encoding forward model changes with time, as is shown in
the first exponential term in Eq. (5), and it requires approximating the full
forward model and applying corresponding optimization algorithms to leverage
the efficiencies of these models. Solving a minimization problem of the form
shown in this section, as well as the direct methods discussed in Section
5.3.1, are often referred to as “inverse problems”, and there are extensive
prior work and tools available for these types of problems that can be applied
to MRI.

####  5.4.1 Efficient approximation of an expanded _B_ 0 forward model

To fully model the off-resonance exponential term in Eq. (5) in matrix form
would be computationally prohibitive, but lower order approximations of the
exponential can be made to efficiently model off-resonance [26]. As described
in prior work [86, 87] and Eq. (18), the exponential can be modeled as a
summation of basis functions. Using discretized versions of the $L$ basis
functions ($b_{il}$ and $c_{lj}$), the matrix vector multiplication of
$\boldsymbol{Ax}$ at row $i$ (i.e., k-space data point $i$) now becomes:

$[\boldsymbol{A(\boldsymbol{\omega})x}]_{i}\approx\sum_{l=1}^{L}b_{il}\left[\sum_{j=1}^{N}(x_{j}c_{lj})e^{-i2\pi(\vec{k}_{i}\cdot\vec{r}_{j})}\right]$
(24)

where $x_{j}$ is the magnetization (image value) at voxel index $j$, $N$ is
the number of voxels, $\vec{k}_{i}\cdot\vec{r}_{j}$ is the inner product of
the $i^{\text{th}}$ k-space sampling location with the image space location of
the $j^{\text{th}}$ voxel index, and $\boldsymbol{A}(\boldsymbol{\omega})$
denotes that $\boldsymbol{A}$ now contains the fieldmap. Here,
$\boldsymbol{A}(\boldsymbol{\omega})$ is implemented efficiently with $L$
NUFFTs. Additionally one can expand this to jointly optimize for the image and
the fieldmap, assuming there is enough temporal variation in the k-space
samples:

$\boldsymbol{\hat{x},\hat{\omega}}=\mathop{\mathrm{arg\,min}}_{\boldsymbol{x,\omega}}\frac{1}{2}\|\boldsymbol{A}(\boldsymbol{\omega})\boldsymbol{x}-\boldsymbol{y}\|_{2}^{2}+\lambda_{1}R(\boldsymbol{x})+\lambda_{2}R(\boldsymbol{\omega})$
(25)

Sutton et al. performed this type of joint optimization using a spiral-
in/spiral-out k-space trajectory [100] to calculate the fieldmap at every
timepoint in an fMRI timeseries, allowing for dynamic $B_{0}$ correction of
respiration induced phase oscillations. Recent work has also incorporated
multicoil information to aid in the joint reconstruction of the fieldmap and
image using undersampled spiral-out and EPI trajectories [101].

Other works have expanded the forward model
$\boldsymbol{A}(\boldsymbol{\omega})$ to allow for a more accurate imaging
model. Joint optimization of the fieldmap and $R_{2}^{*}$ values has been
performed to acquire quantitative BOLD information [102]. The imaging model
can also be expanded to describe the differences in signal between water and
fat, and to jointly optimize for water images, fat images, and the $B_{0}$ map
[103]. More recently, a method to model through-voxel dephasing has been
presented, that uses the gradient of the fieldmap within each voxel to
estimate dephasing and recover lost signal [104].

In addition to joint optimization, it is also possible to incorporate
information from external sensors such as NMR probes to dynamically model the
$B_{0}$ changes that occur during an MRI scan [105]. This dynamic $B_{0}$
tracking approach has recently allowed the acquisition of 0.8mm $\times$ 0.8mm
resolution brain imaging with a 23cm FOV at 7T by using field probes during
the 57ms long single-shot spiral readouts [106], which would produce serve
artifacts without $B_{0}$ modeling.

Even with fast approximations, it should be noted that MBIR generally takes
longer to run than direct approaches such as CP due to its iterative nature,
and can also be more complex to implement. That said, the potential
improvement can be substantial, and the utility of MBIR’s improved accuracy
will depend on the application and requirements of the imaging protocol (i.e.,
reconstruction time).

####  5.4.2 Model based reconstruction for low field

Model based image reconstructions have also proven valuable at low field,
including recently published results showing improved _in vivo_ human brain
images from a portable low field scanners using MBIR [21, 82]. In Koolstra et
al., the forward model includes a $B_{0}$ fieldmap in a conjugate phase
reconstruction (CP) as well as in a model based (MB) reconstruction (see Fig.
9). Additionally, they model gradient nonlinearities. While not the focus of
this paper, gradient nonlinearities can be important to incorporate at low
field, and those are actually the terms that are modeled in the MBIR of the
low field work by Cooley et al., 2021 [21] not the $B_{0}$ field itself.
Inclusion of gradient nonlinearities into the model will affect the k-space
encoding term in equation 24, and shows the flexibility of the MBIR approach
and the benefits of using it at low field.

###  5.5 Dynamic correction and motion considerations

Dynamic off-resonance correction models the time-varying changes in the
$B_{0}$ field during a scan, modeling the temporal updates either using a low
rank spatial representation of the field (i.e., spherical harmonics) or an
entire fieldmap. Dynamic $B_{0}$ correction can be done either prospectively,
where the $B_{0}$ field is updated during the scan (this is often referred to
as dynamic shimming) or retrospectively, where different frames (e.g.,
individual frames in an fMRI timeseries or diffusion images at different
$b$-values) are reconstructed assuming a different fieldmap.

For prospective correction, dynamic shimming can be done either in a
volumetric approach, where the shim is updated during the scan to optimize
homogeneity over the whole region of interest (this will require real time
measurement of the $B_{0}$ field changes), or in a slice-by-slice manner where
the optimal shims for each slice are determined, and then the shims are
updated during the scan depending on which slice is being acquired (one can
measure and dynamically update slice shims in real time, or optimal slice
shims are determined prior to the experiment). Prospective correction without
spatial information can also be useful to dynamically measure and update the
center $B_{0}$ frequency in real time, for example in applications such as
interventional MRI [107] and Oscillating Steady State Imaging (OSSI) fMRI
[108], which has been shown to be very $B_{0}$-sensitive [109].

In terms of hardware to update the shims, one option is to use built in
scanner shim coils (separate from the gradient coils), which generally shim up
to 1${}^{\text{st}}$ or 2${}^{\text{nd}}$ order but can go up to
3${}^{\text{rd}}$ order on some ultra high fields systems. When using built in
shim coils, eddy currents should be taken into consideration as those can
cause artifacts, and there are also potentially limits on the speed at which
the shims can be updated using those coils. There are also more recent methods
that dynamically shim using multicoil receive arrays that can also generate a
DC current [110, 111], one of which by Stockmann et al. is aptly named an
“AC/DC” coil. These methods take advantage of the higher order shim fields
generated from a multicoil array, due to it being located much closer to the
object than scanner shim coils and having more elements to fine tune a desired
field shape.

Retrospective approaches have also proven useful, and are helpful when dealing
with motion during a scan. When a subject moves within the scanner, it will
often result in a change of the $B_{0}$ fieldmap, requiring dynamic distortion
correction. In diffusion MRI, techniques have been developed to correct for
both motion changes and distortion changes during data acquisition [112, 113].
Dynamic distortion correction is also important in fMRI experiments, and many
groups have shown the benefits of estimating $B_{0}$ dynamically, either using
raw k-space data [100], phase measurements from the images themselves [114,
115] or from multicoil FID measurements [116], and then using the information
in the reconstruction. Additionally, the potential clinical benefit of dynamic
off-resonance correction has been show in the application of presurgical fMRI
at 7T [117].

Dynamic $B_{0}$ correction is also relevant in low field imaging, especially
in permanent magnet designs where there can be field drift during a scan [28,
19, 22]. In these approaches, a field probe or other methods are used to
monitor drift during the acquisition, and then these measurements were used
during the image reconstruction. On the other end of the field spectrum,
dynamic $B_{0}$ correction has proven useful at ultra-high field, where there
are large susceptibility artifacts [118, 119, 35].

Figure 10:  Comparison of standard spin echo imaging (top row), VAT (middle),
and SEMAC (bottom row) methods for a patient with a spinal implant. The right
column is reformatted so slice direction is along the horizontal axis. The SE
image is badly distorted in both frequency and slice directions, the VAT
images are partially corrected in the frequency direction with substantial
slice distortion, while the SEMAC image is corrected in both directions.
_(Images reused from[120] with permission.)_

In MR spectroscopy, which is very sensitive to off-resonance artifacts [72],
dynamic shimming is also very helpful. Spinal cord fMRI, including combined
brain+spinal cord fMRI, can also benefit from dynamic shimming, because the
effects of respiratory induced magnetic susceptibility changes are much more
severe when imaging closer to the lungs [121, 122]. Finally, in applications
such as speech MRI, where the data is inherently time varying, dynamic $B_{0}$
correction is important, since the $B_{0}$ field pattern will change as a
subject speaks during the exam [123, 95, 97].

###  5.6 Correction for metal artifacts

As described above, the source of $B_{0}$ distortions around metal implants is
largely the same as other sources of susceptibility-induced $B_{0}$ shifts,
except larger in magnitude. Most imaging around implants uses SE (and
variants) and uses traditional spin-warp type acquisitions. This has the
benefit of there being no geometric distortion in the phase encoding
direction, but this still leaves artifacts in the readout and slice selection
directions. The first step to minimize artifacts is to maximize the bandwidth
per resolution element in each case, by increasing the receiver bandwidth for
frequency encoding readouts and increasing the RF pulse bandwidth. While both
have costs, these are useful strategies for reducing distortions. Similarly,
one can use geometric distortion correction methods, like fieldmap based
methods and others described above in Section 5.2.1, but these are ineffective
for many of the larger distortions seen with metal implants.

####  5.6.1 View angle tilting

One clever approach for reducing in-plane distortions in the frequency
direction is called view-angle tilting (VAT) [124]. Note that slices are
shifted during the slice excitation, and shifted in the slice direction by an
amount proportional to $B_{0}$-induced frequency shift. Similarly, during
frequency encoding the voxels are shifted in the readout direction
proportional to that frequency shift. The view angle tilting works by applying
the slice select gradient again during frequency encoding. If the slice
gradient is the same amplitude as that used during excitation, then spins will
be centered at the slice resonant frequency and not the original frequency
shift. As a result, the frequency encoding sees the correct frequency
associated with each spatial location along the read direction and distortions
are removed. While effective, this approach is somewhat limited in that the
frequency encoding should be about the same length as the RF pulse, which
unfortunately makes the readouts very short at the expense of SNR. One could
use a longer (narrow bandwidth) RF pulse, but this leads to the slices being
shifted by even larger amounts, so generally, high bandwidth RF pulses and
frequency encoding are used. The main limitation of VAT is the residual and
uncorrected distortion in the slice direction.

####  5.6.2 Robust methods for distortion correction

The extreme challenge of large $B_{0}$ shifts around metal implants has lead
to several new methods for encoding images and the frequency offsets at the
same time. One popular method is called multiacquisition variable-resonance
image combination (MAVRIC) [125], in which slices are not excited at all, but
rather narrow frequency bands are excited. Since there is no slice selection,
3D encoding is required to prevent aliasing in slice and phase encoding
directions. This is usually combined with fast spin echo/turbo spin echo
methods to improve acquisition speed. Because a narrow band of frequency is
excited, there is limited distortion of the images, governed by the readout
bandwidth per pixel. To fill in the image holes, many (often overlapping)
frequency bands must be excited and for each the exact shift correction for
the frequency offset is applied. This creates many 3D volumes, which are then
combined to create a volume that is undistorted in all three dimensions. The
disadvantage of this approach is that it has poor SNR efficiency because
shifted frequency bands do not improve the SNR of other frequency bands.

As mentioned above, the main issue with VAT is that the slice distortions are
not corrected. Another comprehensive approach for distortion correction that
addresses this issue is called slice-encoding for metal artifact correction
(SEMAC) [120], in which 2D slices are excited, but for each a 3D acquisitions
is acquired so the distorted 3D profile of each excited slice can be
visualized. VAT is used to correct for in-plane distortion and the 3D shifts
are spliced together into a fully corrected 3D volume. As with MAVRIC, the SNR
efficiency of this method is also poor. Figure 10 demonstrates how regular
spin-echo images are badly distorted in both in-plane and slice directions,
how VAT mostly corrects for the in-plane distortions but not the slice
direction effects, and that SEMAC corrects for both in-plane and slice
distortions. There are MAVRIC-SEMAC hybrid approaches [126] as well that can
improve image quality and efficiency.

##  6 Conclusions

Off-resonance artifacts are widespread in MRI, but fortunately there are many
strategies to reduce and correct for them. Here we have described how off-
resonance can come from a variety of sources, such as inherent $B_{0}$ magnet
inhomogeneity, magnetic susceptibility of tissue, chemical shift, and metallic
implants. The effects of these inhomogeneities depend on the type of sequence
used (spin echo vs. gradient echo, Cartesian vs. non-Cartesian) and therefore
the correction strategies must be different as well. We outlined the most
common strategies for $B_{0}$ correction in Cartesian sequences (e.g., EPI),
including fieldmap-based, TOPUP, and PSF approaches, as well as common
strategies for non-Cartesian sequences (e.g., spiral), including direct (e.g.,
conjugate phase) and autofocusing methods. The effects and correction
strategies for metallic implants also require a special approach, because of
the very large magnetic susceptibility difference between metal and
surrounding tissue. We also discussed how model based image reconstruction
(MBIR) and dynamic correction can be used to correct for off-resonance in
multiple types of sequences and applications. Throughout we highlighted the
ways that low field MRI can be easier to work with in terms of off-resonance
effects (less magnetic susceptibility and chemical shift effects), how it can
be more challenging (nonuniform $B_{0}$ field), and the substantial progress
that has been made in correcting for $B_{0}$ effects at low field. Overall,
the type of off-resonance artifact reduction and/or correction strategy will
depend on the application, and future developments in the field of off-
resonance correction will continue to enable new applications and use cases
for MRI.

## Acknowledgements

The authors would like to thank Dr. Jeffrey Fessler for helpful discussions on
model based image reconstructions. Images from Figure 2 are reprinted from
Neuroimage, Vol 168, Stockmann & Wald, “In vivo B0 field shimming methods for
MRI at 7 T”, Pages 71-87, Copyright 2018, with permission from Elsevier.
Images from Figure 8 are reprinted from Neuroimage, Vol 20, Andersson et al.,
“How to correct susceptibility distortions in spin-echo echo-planar images:
application to diffusion tensor imaging”, Pages 870-888, Copyright 2003, with
permission from Elsevier. Images from Figure 10 are reprinted from Magnetic
Resonance in Medicine, Vol 62, Issue 1, Lu et al., “SEMAC: Slice encoding for
metal artifact correction in MRI”, Pages 66-76, Copyright 2009, with
permission from John Wiley and Sons. This work was supported by the National
Institutes of Health Grants F32EB029289, R01EB023618, U01EB026977, and
U24NS120056.

## Competing Interests

Melissa W. Haskell is employed by Hyperfine, Inc.

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