REVIEW OF SCIENTIFIC INSTRUMENTS 83, 033708 (2012)
An x-space magnetic particle imaging scanner
Patrick W. Goodwill, Kuan Lu, Bo Zheng, and Steven M. Conolly
Department of Bioengineering, University of California, Berkeley, California 94720-1762, USA
(Received 30 July 2011; accepted 28 February 2012; published online 22 March 2012)
Magnetic particle imaging (MPI) is an imaging modality with great promise for high-contrast, high-
sensitivity imaging of iron oxide tracers in animals and humans. In this paper, we present the first
x-space MPI hardware and reconstruction software; show experimentally measured signals; detail
our reconstruction technique; and present images of resolution and “angiography” phantoms. © 2012
American Institute of Physics. [http://dx.doi.org/10.1063/1.3694534]
Magnetic particle imaging (MPI) is an imaging modal-
ity that directly detects magnetic nanoparticles1and shows
extraordinary promise as a safe substitute for iodinated or
gadolinium contrast angiograms, especially for patients with
poor kidney function who have difficulty in excreting these
standard contrast agents. The technique uses the clever obser-
vation that the nonlinear magnetic characteristics of a mag-
netic nanoparticle can be used to generate an image whose
resolution is defined by the magnetic properties of an iron ox-
ide nanoparticle tracer and by the strength of an applied mag-
netic field gradient. This means that the native imaging reso-
lution is not defined by the wavelength of the radiation used to
interrogate the magnetic nanoparticles. For perspective, MPI
excitation frequencies are typically below 25 kHz, which cor-
responds to a wavelength in the body of about 1 km. MPI
resolution, on the other hand, can be more than 6 orders of
magnitude finer than the wavelength and is measured in mil-
limeters. Imaging is performed using time-varying magnetic
fields, and so the technique does not use ionizing radiation.
The human body is diamagnetic and so tissue induces no MPI
signal. Moreover, tissue is completely transparent to the low-
frequency magnetic fields used in MPI. Hence, MPI is ideal
for detecting nanoparticle contrast agents with no background
and zero depth attenuation.
We believe MPI is ideally suited for imaging blood ves-
sels. An ideal angiography method would see only tracer
and not tissue. Tissue is visible in x-ray and magnetic res-
onance imaging (MRI), and results in a background signal
that can obscure clinically important vessel details, especially
in a projection imaging technique like x-ray angiography.
Hence, physicians typically rely on catheterized arterial in-
jections, with approximately 30-fold higher contrast concen-
tration than venous injections, to achieve adequate contrast.
Since MPI only detects tracer, MPI angiography would have
exceptional contrast with no background. Further, the iodine
used in x-ray fluoroscopy and Gd used in MRI can pose a
risk for patients with chronic kidney disease (CKD). Iodine,
used in x-ray fluoroscopy and CT Angiography, is the dom-
inant tracer for clinical angiography and places millions of
patients at risk for contrast-induced nephropathy (CIN).2Al-
though rare (<1%), severe CIN can require dialysis and is as-
sociated with high in-hospital mortality (36%) and high two
year mortality (81%).3,4Importantly, the iron oxide tracers
used in MPI are processed in the liver and do not affect the
Three MPI reconstruction techniques have been shown
experimentally. The most published technique is harmonic-
space MPI, which uses a system matrix that is comprised
of the Fourier components of the temporal signal for every
possible location of a point source.1,7–12Reconstruction is
achieved through regularization and matrix inversion tech-
niques such as singular value decomposition or algebraic re-
construction. This inversion can be complex since the size of
the system matrix is large and contains millions of elements.
The second technique is a narrowband technique, which re-
constructs harmonic images into a composite image using
a modified Wiener deconvolution.13,14The third technique,
which we use in this paper, is x-space reconstruction.15,16
X-space offers several advantages over harmonic-space MPI
matrix reconstructions.11Specifically, x-space MPI is experi-
mentally proven to generate Linear Shift-Invariant (LSI) im-
ages, as well as real-time image reconstruction speed as it in-
volves only division by a scalar to reconstruct each point in
the image. Importantly, x-space MPI makes no attempt to de-
convolve the MPI signal to improve resolution over the reso-
lution determined by the physics of the nanoparticles and field
gradient, and thus avoids the significant noise gain of decon-
volution (see the noise gain in Ref. 17).
space technique with significant differences in the pulse se-
quences, signal processing, and image reconstruction. This
paper details the construction of a small scale MPI scan-
ner, pulse sequence, signal processing, and image reconstruc-
tion necessary to form a native x-space MPI image. We also
demonstrate the LSI properties of x-space MPI by imaging a
resolution phantom and an angiography phantom.
II. X-SPACE THEORY OF MPI
We recently introduced a comprehensive systems theory
for MPI, x-space MPI.15,16X-space MPI finds that the imag-
ing equations that govern MPI are similar to those seen in the
k-space theory of MRI, with the dominant difference that the
x-space signal is in the spatial domain and the k-space signal
is in the spatial frequency domain.18,19
X-space MPI theory requires three assumptions:14–16
First, we can generate a strong magnetic field gradient
0034-6748/2012/83(3)/033708/9/$30.00© 2012 American Institute of Physics
033708-2 Goodwill et al. Rev. Sci. Instrum. 83, 033708 (2012)
FIG. 1. Two strong opposing magnets produce a field gradient with a
millimeter-scale field free point (FFP) at the isocenter where SPIO nanopar-
ticles are not magnetically saturated. Magnetic particles are fully saturated
elsewhere. The FFP can be moved across the sample using additional mag-
netic fields or mechanically in a scanning trajectory to produce an image.
(>2.5 T/m) with a unique null point (see Fig. 1). Second,
super-paramagnetic iron oxide nanoparticles (SPIOs) can be
adiabatically aligned and saturated with an applied magnetic
field greater than about 5 mT. Last, low-frequency nanoparti-
cle signals filtered out during signal detection are recoverable.
The problem of lost low-frequency signals is not unique to x-
space MPI; system matrix reconstruction must contend with
the lost information as well.
It is possible to construct a magnetic gradient that forms
a null point, of field free point (FFP – see Fig. 1). The FFP
is a millimeter-scale spheroid where the field is below a criti-
cal threshold field required to magnetically saturate the SPIO
tracer. At all positions in the magnet except at the FFP, SPIOs
aremagnetically saturated,andproduce notime-varying mag-
netic field. At the FFP, SPIOs are unsaturated and can produce
a time-varying magnetic field.
The position of the FFP in the scanner can be moved
in what is known as a “pulse sequence.” The combination
of mechanical translation, time-varying uniform fields, and a
gradient effectively shifts the FFP. Imaging occurs when the
FFP is rapidly shifted across the sample using a uniform field,
causing the magnetization of SPIOs passing through the FFP
to unsaturate as well as flip, inducing a voltage in a receive
coil. The receive chain detects the magnetic response of the
nanoparticles (while rejecting the excitation signal). Further
processing is necessary to recover the particle signal at the
excitation frequency in order to result in LSI images.16
Mathematically, x-space theory15,16describes the MPI
process in 1D as
s(t) = B1m ρ(x) ∗ h(x)
where ρ(x) [particles/m3] is the nanoparticle distribution and
xs(t) is the instantaneous position of the FFP. A number of
constants govern the magnitude of the received signal, includ-
ing the sensitivity of the receive coil B1[T/A], the magnetic
moment of the nanoparticles m [Am2], the gradient strength
G [A/m/m], and the field sufficient for saturation of the mag-
netic nanoparticle tracer, Hsat[A/m]. The point spread func-
tion (PSF) in 1D is the derivative of the Langevin function
and has the simple form
The 1D PSF is similar to a Lorentzian function,15and has a
full-width at half-maximum (FWHM) that becomes finer with
increasing gradient G and decreasing Hsat. Neglecting relax-
ation effects, Hsat∝ d−3, where d is the magnetic nanoparticle
ing nanoparticle diameter.
Once we have the MPI signal, we form the x-space MPI
image in a simple two-step process of velocity compensation
followed by gridding of the received signal to the instanta-
neous position of the FFP. Following reconstruction, the re-
sulting image equation is
ˆ ρ(x(t)) =
˙ xs(t)= ρ(x) ∗ h(x).
Thus, x-space theory describes how to obtain a native x-space
MPI image, ˆ ρ(x) that is a convolution of the nanoparticle den-
sity, ρ(x) with a PSF, h(x). The MPI signal and image equa-
tions tell us that the MPI signal is simply a sampling of the na-
tive MPI image at the instantaneous position of the FFP. Then,
by knowing the position of the FFP, we can reconstruct a na-
tive MPI image by velocity compensating and gridding the
received signal. Because the resulting image can be described
using a convolution with a well-behaved PSF, the resulting x-
space MPI technique is LSI. The x-space MPI process in 1D
can be seen visually in Fig. 2.
We can extend the x-space formulation to three-
dimensions, which we have described in our second x-space
paper,16where we find that the 3D PSF shape is anisotropic,
and rotates with the FFP velocity vector (see Fig. 3). The goal
of the scanner described in this paper is to validate the 1D and
3D theoretical results.
III. IMAGING HARDWARE
A. Main field gradient
MPI requires a main field gradient, also known as a se-
lection field,1,11to form the FFP. We can linearly improve
resolution (and shrink the size of the FFP) by increasing the
magnetic field gradient strength. Due to the properties of the
nanoparticles, resolution improves cubically with increasing
nanoparticle core diameter.11,15,16,20
033708-3 Goodwill et al. Rev. Sci. Instrum. 83, 033708 (2012)
(c) Signal / X-Space Reconstruction
(a) MPI Scanning in One Dimension
(b) Time Varying Magnetization
FIG. 2. Overview of the x-space MPI imaging process. (a) Let us consider a
one-dimensional phantom and rapid movement of the FFP across the sample.
(b) As the FFP passes over iron oxide nanoparticles in the sample, the total
magnetic moment, M(t), changes in a non-linear manner. (c) An inductive
receive coil detects the time-varying magnetic moment as a voltage, s(t). The
received signal is then converted into a native x-space MPI image using a
two-step process of velocity compensation and gridding to the instantaneous
position of the FFP.
We have designed our imager to work with a com-
mercially available nanoparticle tracer, Resovist (Bayer-
Schering), which produces a signal similar to a nanoparticle
with 17 ± 3.4 nm core diameter.16,20We have found that a
6.0 T/m (3.0 T/m transverse to the bore) gradient is reason-
able to build with permanent magnets using the Maxwell-like
permanent magnet configuration shown in Fig. 4. Our gradi-
ent is built using two opposed, high-grade (N45), NdFeB ring
magnets (ID = 8.89 cm, OD = 14.6 cm, THK = 3.2 cm)
mounted on G10 fiberglass composite structural plates. The
magnets generate a 6.0 T/m gradient down the bore, and 3.0
T/m transverse to the bore. The reduced gradient strength in
the transverse axis is fundamental to Maxwell’s equations.16
With Resovist, this system has a measured resolution of
≈ 1.6 mm down the bore. Because of the reduced transverse
field gradient strength and since we do not excite in a trans-
verse axis, x-space theory predicts that the transverse resolu-
tion is 7.4 mm, or 460% worse than the collinear PSF.16The
addition of transverse transmit and receive coils (in the x and
y axes) would improve the transverse resolution to ≈3.2 mm.
Native MPI Image Formation in 2D
FIG. 3. The MPI process can be generalized into two- and three-dimensions,
but with a point spread function, h(x), that changes orientation depending on
the FFP velocity vector, ˙ xs(t).
An important practical note is the tolerances required for
MPI imaging. The gradient linearity need not be better than
5%, as gradient non-linearity results in image distortion simi-
lar to rubber sheet distortion from MRI gradients. Because of
the small field of view (FOV) of this magnet, we currently ne-
glect this distortion. Further, several Kelvin temperature vari-
ation in the gradient magnets are well tolerated. This is in
contrast to the milliKelvin temperature stability required for
NMR/MRI applications with NdFeB magnets since the field
coefficient of rare Earth magnets is typically 1 ppt/K.
B. Analog signal chain
The analog signal chain is broken down into two parts
The excitation field, also known as a drive field,1,11rapidly
moves the FFP across the sample to produce a nanoparticle
tects and isolates the nanoparticle signal while rejecting the
large excitation field. Both subsystems are shown in Fig. 5.
The goal of the transmit chain is to excite the sample with
a pure sinusoid with no energy content above the excitation
frequency. To this end, we have designed the transmit-receive
filters shown in Fig. 2. The transmit filter is composed of a
third order low-pass filter that is power matched to a resonant
transmit coil. The transmit filter achieves 60 dB isolation at
×2 the fundamental frequency and 65 dB isolation at ×3 the
The resonant transmit coil (f0= 19 kHz) is wound with
10 gauge square magnet wire and is driven by a high power
linear amplifier (LVC5050, AE Techron) filtered by a third-
order low-pass filter. The transmit coil generates 30 mTpp
with a peak power output of 5 kW, most of which is dissipated
as heat in the water-cooled eddy current shield. We employ a
2.5 mm thick copper cylinder to shield the RF transmit and re-
ceiver coils from the permanent magnet (PM) gradient. This
crucial passive shield greatly diminishes harmonics produced
any produced harmonics from being picked up in the receiver
coil. The RF-gradient shield is also strong enough to serve
as a structural component. The shield requires active water
cooling because most of the transmit energy is deposited in
the shield due to eddy current heating. The peak current and
voltage in the transmit coil are approximately 100 A and
180 V. The transmit coil can heat up from resistive heating
during extended scanning sessions, and so we have filled the
space between the water cooled shield and the transmit coil
with a thermally conductive epoxy.
The goal of the receive chain is to receive a wide
bandwidth particle signal while suppressing the fundamen-
tal frequency. The fundamental frequency must be suppressed
from the received spectrum because it is over 140 dB, or
10 × 106times, larger than the nanoparticle signal. The sys-
tem described in this paper achieves better than 110 dB of
suppression, leaving considerable room for improvement.
The receive coil is wound in a gradiometer-like con-
figuration inside the transmit coil to minimize total shared
flux. The signal from the receive coil is notch filtered by a
033708-4 Goodwill et al.Rev. Sci. Instrum. 83, 033708 (2012)
Water Cooled Magnetic Shield
Aluminum BoltCooling Tube
FIG. 4. Hardware overview. (a) 3D MPI scanner with 2 cm × 2 cm × 4 cm FOV. The excitation coil generates a 30 mT peak-to-peak oscillating magnetic field
at 19 kHz. The NdFeB magnets generate a gradient of Gz= 6.0 T/m down the imaging bore. (b) Photograph of the small-scale x-space MPI scanner. The free
bore before addition of the transmit and receive coils is 8.4 cm. The scanner is potted in epoxy to eliminate vibration.
fourth-order resonant notch, and amplified by a battery-
powered low noise pre-amplifier (Stanford Research Sys-
tems SRS560). The signal is further conditioned by a noise
matched 8th order analog Butterworth high-pass filter (HPF)
(F3dB= 25 kHz, Stanford Research Systems SIM965), fol-
lowed by a second stage of amplification (Stanford Research
System SIM911). The signal is digitized at 1.25 MSPS (Na-
tional Instrument, NI-6259), digitally phase corrected, low
pass filtered at 200 kHz, and gridded to the instantaneous po-
sition of the FFP.
The tolerance of the transmit and receive systems to mag-
netic field inhomogeneity is quite reasonable. We calculate
that the system achieves no better than 5% transmit and re-
ceive magnetic field homogeneity within the imaging region,
and yet the resulting image artifacts are benign.
IV. PULSE SEQUENCE
The path of the FFP through x-space is considered the
MPI pulse sequence. In this scanner, we have implemented a
raster pulse sequence (see Fig. 6) for ease of implementation.
Any pulse sequence that covers the entire FOV is adequate
but optimal sequences are possible to cover the design space
of superior SNR, speed, avoidance of dB/dt stimulation, and
prevention of undue stress to the transmit hardware.
In the raster pulse sequence implemented here, we apply
a 30 mTpp excitation waveform at 19 kHz that rapidly moves
the FFP ±2.5 mm in the z axis while simultaneously translat-
ing the sample mechanically in the x axis by up to 2 cm. This
covers a region of 5 mm × 2 cm, which we term a partial FOV
The pFOV is crucial to meet safety limits as it allows
us to scan sub-regions of the image within instrumentation
limits, SAR limits, and magneto-stimulation limits, and then
stitch the resulting sub-regions into a full image. For exam-
ple, translation of ±2.5 mm does not cover the entire sample.
To image a larger FOV, we acquire multiple overlapping par-
tial FOVs by stepping in the z axis. We later stitch the par-
tial FOVs together to form a full image across the desired
FOV (see Sec. V C), which is up to 10 cm in the z axis.
Magneto-stimulation limits place an upper bound on the size
of a partial FOV, and the 30 mTpp field used in this scan-
ner already exceeds the limits for magneto-stimulation for a
theoretical human-sized MPI chest scanner.15While we use
mechanical translation of the sample in this small system,
it is straightforward to slowly move the FFP electronically
to cover the full FOV using high power, current-controlled
Receive Coil and Pre-amplifier
High Power LPF
FIG. 5. Analog signal chain.
033708-5 Goodwill et al.Rev. Sci. Instrum. 83, 033708 (2012)
(a) Electronic FFP Movement in z
(b) Mechanical FFP Movement in z
(c) Mechanical FFP Movement in x
(d) Overall Pulse Sequence
FIG. 6. Two-dimensional pulse sequence used in the Berkeley x-space scan-
ner. (a) Rapid movement in z of ±2.5 mm occurs at 19 kHz through the
use of a resonant transmit coil, moving the FFP at approximately 200 m/s in
the z axis. The FFP displacement is proportional to the current in the trans-
mit coil. This is a schematic representation as the actual movement as over
12 × 103cycles occur during the scanning period. (b) The sample is me-
chanically translated down the bore in the z axis in steps of 2.5 mm per scan
for a 50% partial FOV overlap. (c) The sample is mechanically rastered dur-
ing the scan across the FOV in the xaxis. (d) The full 2D pulse sequence in
real space. For a 3D scan, we mechanically step the sample in the y axis in a
similar manner to the stepping in the z axis.
electro-magnets, which we have implemented in the past in a
narrowband mouse imaging system.13,14
A. Phase recovery filter
The received signal has corrupted phase because it is
conditioned in the analog domain with a HPF before digiti-
zation (Fig. 2). This filter removes what can be significant
interfering signals from the fundamental excitation sinusoid,
mains noise, and 1/f noise. Removing these prevents satura-
tion of later stages of amplification. Unfortunately, this com-
plicates reconstruction because x-space MPI requires linear
phase throughout the receive system as non-linear receive
phase causes an image artifact in the spatial domain.
In the analog domain, our filter is an eighth-order But-
terworth HPF, whose transfer function we can write as Hhpf
1010 20 2030 30404050 5060 6070
10 1020 2030 3040 4050 5060 6070
analog filtering in the receive chain. (Bottom) Recovered phase following
inverse filtering. The stop-band also benefits with improved rejection.
(ejω). We can then write the filtered analog signal as
˜S(ejω) = Hhpf(ejω) S(ejω),
where S(ejω) is the signal from the receive coil, and˜S(ejω) is
the filtered analog signal. Butterworth filters are simple to de-
sign in the digital domain,21and we can readily calculate the
analog filter’s digital counterpart, which has the same transfer
function as the analog filter, Hhpf(ejω). We reverse the non-
linear phase accrued from the analog HPF by re-filtering with
its digital counterpart, but in reverse-time. This is equivalent
to filtering with the non-causal filter Hhpf(e−jω). Then, the
cascaded analog and digital filters have the straightforward
ˆS(ejω) = Hhpf(e−jω)˜S(ejω)
This is shown graphically in Fig. 7, which shows that the total
system has zero-phase accrual and a deeper stop-band. This
concept can be considered a modification of zero-phase fil-
tering (see problem 5.68 in Ref. 21) in which a digital filter
is applied twice, once in forward direction, and once in the
backwards direction. However, in the case presented here, one
direction is in the analog domain and the reverse direction oc-
B. Gridding and velocity compensation
Gridding is the process of sampling the time domain sig-
nal into the spatial image domain, introduced to MPI in our
first x-space paper.15Because the pulse sequence controls the
location of the instantaneous position of the FFP, we can as-
sign the received signal to a specific location in space. This
straightforward and simple technique does not require regu-
larization, optimization techniques, or prior knowledge of the
magnetic response of the tracer such as what is required in
The received signal must also be velocity compensated
because the induced signal is proportional to the instanta-
033708-6 Goodwill et al. Rev. Sci. Instrum. 83, 033708 (2012)
neous FFP velocity. Then, the gridded, velocity compensated
ˆ ρ(x(t)) =
In practice, we interpolate and average ˆ ρ(x) using a nearest
neighbor or linear interpolation.
During reconstruction in this scanner, we have assumed
that the FFP velocity unit vector is constant. This assumption
is reasonable because of the speed at which the resonant trans-
mit coil moves the FFP when compared to the speed we me-
chanically move the sample. In this system, we calculate that
the FFP moves at approximately 200 m/s. Mechanical move-
ment of the FFP occurs three orders of magnitude slower, and
so it is reasonable to assume the FFP velocity unit vectorˆ˙ xsis
The x-space process of gridding and velocity compensa-
tion can be compared directly to k-space sampling in MRI.
In MRI, the received signal corresponds to the instanta-
neous position in k-space;18in MPI the received signal cor-
responds to the instantaneous position in x-space. To form
a k-space image in MRI, the gridded signal must be density
compensated.22Similarly, in MPI the gridded signal must be
velocity compensated. Interestingly, because MRI and MPI
signal acquisition occur in different domains, MRI k-space
artifacts lead to coherent artifacts over the entire image while
artifacts in x-space MPI translate to local defects.
C. Image assembly
The analog and digital high-pass filters remove all low-
frequency information in the received signal and break the
LSI properties of the system. We approach the problem of the
loss of low frequencies by considering the HPF of the time
domain signal as a loss of low spatial frequency information.
For temporal frequencies near the fundamental frequency, our
experimental data showthatthelostsignalcorresponds toadc
offset across the FOV. Surprisingly, this means that if we ac-
quire overlapping partial FOVs, we can calculate the dc offset
between partial FOV images that maximizes continuity and
ensures zero signal outside of the FOV. To form a composite
image, we simply add together the overlapped partial FOVs
weighted by the SNR of each point. This technique is shown
in Fig. 9.
D. Image deconvolution
Image deconvolution is not necessary in x-space MPI to
form a native MPI image. However, the native x-space MPI
PSF is similar to a Lorentzian distribution, and so a point
source will have long tails that add a grey background to the
image. For the low resolution imager described in this pa-
per, Wiener deconvolution can provide a way to remove the
background and visually improve the image. In practice, we
perform Wiener deconvolution in k-space where each k-space
point is calculated23
˜ ρ(k) =
|H(k)|2+ 1/SNRˆ ρ(k)
using an estimate of the image SNR. We used moderate
Wiener deconvolution with a SNR cutoff of 50 in k-space
so as to minimize image artifacts that become significant
when over-deconvolving. The point spread function input
into Wiener deconvolution can be measured experimentally
or calculated theoretically (see Appendix). In the deconvo-
lution shown here, we have used the theoretically calcu-
lated PSF, which we can consider a form of model based
deconvolution.9,10In the future as we build higher resolu-
tion MPI imagers, we hope that deconvolution will not be
VI. EXPERIMENTAL RESULTS
In Fig. 8, we show the phase corrected signal for a point
source sample translated through the origin. We see the am-
plitude of the signal envelope changes slowly as we scan the
position of the FFP along the y axis. This envelope corre-
sponds to the Normal signal envelope in the multidimensional
x-space MPI theory.16
In Fig. 9, we show how it is possible to assemble mul-
tiple partial FOVs into a full FOV of arbitrary size. Fig. 9
(top) first shows that the phase corrected, velocity corrected,
gridded signal is missing the dc component. Using a simple
continuity algorithm, we obtain the dc corrected signal in Fig.
9 (middle). Assembling the dc corrected partial FOVs results
in a high quality, native MPI image seen in Fig. 9 (bottom).
In Fig. 10, we see the correspondence between the the-
oretical and measured collinear component of the PSF. The
measured FWHM is 1.6 mm down the bore of the imager,
and 7.4 mm transverse to the bore.
In Fig. 11, we show the line scan of a resolution phan-
tom containing point sources separated by 1 mm, 2 mm, and
3 mm. As can be seen, the 1 mm spaced samples are not re-
solvable as the spacing between them is less than the native
resolution of the system (FWHM ≈ 1.6 mm).
In Fig. 12, we demonstrate the potential of x-space MPI
for imaging complex phantoms by imaging an “angiography”
phantom. The phantom is composed of 300 μm ID tubes
filled with undiluted Resovist tracer, and covers an area of
FIG. 8. Measured signal showing phase corrected signal from a single scan
across a point source in z and y. (Top) The amplitude changes slowly as we
scan 1.5 cm in y. (Bottom) Time-slice near y = 0 showing the raw signal as
we rapidly scan ±2.5 mm in z. Total scan time of 650 ms.
033708-7 Goodwill et al. Rev. Sci. Instrum. 83, 033708 (2012)
z axis [cm]
a 400 μm wide Resovist point source phantom without baseline correction.
(Middle) Experimental data with baseline correction. (Bottom) The assem-
bled image recovers the linearity across the full FOV.
approximately 1.5 cm × 2 cm. The phantom aims to mimic
the branching of coronary arteries in the heart, and is a test
of the ability of the imaging system to image a continuous
nanoparticle density as opposed to point sources that can be
easier to deconvolve.
A. Linear and shift invariance
The recovery of the dc shifts in individual partial FOVs is
crucial to producing a native x-space MPI image that is LSI.
The importance of LSI properties in an imaging system can-
not be overstated. For the native images produced by x-space
MPI, linearity ensures image brightness is linear with concen-
tration, and shift invariance ensures image brightness does not
change with position of the object. With LSI, we then know
that the images produced accurately reflect the concentration
z position [cm]
x position [cm]
z position [cm]
x position [cm]
FIG. 10. Comparison between theoretical and measured collinear compo-
nent of the PSF. The measured FWHM is 1.6 mm along the imager bore
and 7.4 mm transverse to the imager bore. Total imaging time of 28 s not
including robot movement. The resolution of the PSF can be improved by
increasing the main gradient field strength.
Δx=1 mm Δx=2 mmΔx=3 mm
FIG. 11. Line scan of a linear resolution phantom with point sources sepa-
rated by 1 mm, 2 mm, and 3 mm. As can be seen, the 1 mm spaced samples
are not resolvable as the spacing between them is less than the native resolu-
tion of the system (FWHM ≈ 1.6 mm).
and distribution of the nanoparticles in the sample being im-
The signal received by the x-space scanner has high SNR
and repeatability. Our early (non-optimized) prototype scan-
ner presented here can detect 200 ng of tracer in a single
voxel, but we believe the electronics could be improved in
z position [cm]
x position [cm]
z position [cm]
x position [cm]
FIG. 12. X-space MPI generates a native MPI image. Further processing of
the native image can be done with standard image processing techniques.23
(a) Acrylic phantom with 300 μm ID tubing containing undiluted Resovist
tracer. (b) Native x-space MPI image. (c) Native MPI image deconvolved
using Wiener deconvolution. Total imaging time of 28 s, not including robot
033708-8 Goodwill et al.Rev. Sci. Instrum. 83, 033708 (2012)
future hardware revisions to reach physical sensitivity lim-
its. With our current voxel size of 1.6 mm × 7.4 mm × 7.4
mm, this corresponds to a sensitivity limit of approximately
2 μmol/L Fe. In theory, the sensitivity of the MPI technique
pushed to its theoretical limit would approach 20 nmol/L Fe
sensitivity in a single pixel.1,15Certainly, significant work on
both the tracer as well as the hardware must occur before any
group approaches this limit.
The primary technical difficulty in the system has been
to build a high power transmit system with low total har-
monic distortion (THD), high rejection of noise, and high re-
jection of interference above the excitation frequency. For the
phantoms shown here, the detected second and third harmon-
ics of the excitation frequency are approximately half of the
magnitude of the MPI signal from our phantoms. To counter-
act this distortion, we first take a baseline with no sample in
the bore and subtract this from the received signal. However,
the feedthrough drifts and so still remains our dominant noise
The source of the interfering harmonics of the excitation
signal has a complex origin. The transmit chain is driven by a
linear amplifier with better than 80 dB THD. The transmit fil-
ter achieves better than 60 dB isolation at frequencies greater
than double the transmitted frequency. However, the distor-
tion detected in the receive coil is significantly higher than
what would be predicted by these two measures. We believe
that the interfering signal instead arises from distortion in the
capacitors in the resonant transmit coil and in the transmit
filter, as well as mechanical vibrations. We are actively work-
ing to reduce this distortion in future scanners. We note that
the mechanical vibrations can be significant. During develop-
ment of the system, at times we tuned the transmit scanner to
frequencies in the audible range, which can be painfully loud
because of the transmit coil Lorentz-force interaction with the
main field gradient. However, at the current 19 kHz excitation
frequency, the system is inaudible because it is beyond the
range of human hearing. As MPI is scaled to humans, how-
ever, the optimal excitation frequency for imaging in the chest
nears the audible range and may require an audible system to
reach the optimal SNR while remaining within SAR limits.15
The native resolution of the system is of vital importance
to building a clinically relevant imaging system. We estimate
the resolution of this small-scale MPI scanner to be 1.6 mm
down the bore (see Fig. 10) using the Houston criteria for
resolution, which states that the resolution is approximately
the FWHM of a point source.24This is exactly what we
would expect from the formula from Rahmer et al.11and from
Goodwill et al.15,16,20if we assume the particles in Resovist
are approximately 17 nm in diameter. In Fig. 11, we see a res-
olution phantom that shows an image of point sources spaced
below the resolution limit, slightly above the resolution limit,
and firmly above the resolution limit. The scan of the resolu-
tion phantom shows that the Houston criteria for resolution is
an appropriate measure for resolution.
MPI resolution is a fundamental property of the gradi-
ent strength and nanoparticle properties, and is independent
of the bore size. The low resolution of the system transverse
to the bore of 7.4 mm is a limitation of this specific scanner
and not the MPI technique. The low resolution arises from the
weaker transverse gradient (3 T/m) and the single excitation
coil and receive coil pair oriented down the bore. The clear-
est method to improving the resolution without a new magnet
design would be to build excitation and receive coils in the
transverse axes, which would improve the transverse resolu-
tion to 3.2 mm.
Current MPI systems re-purpose a magnetic nanoparti-
cle tracer used in MRI for the detection of liver cancer.25
There are differing views on the effective diameter of Reso-
vist when fit to a Langevin model for magnetic nanoparticles.
In Refs. 1 and 17, the authors indicate Resovist acts like a
30+ nm magnetic nanoparticle. In a second work by the same
group,26experimental results indicate that the effective di-
ameter is approximately 17 nm. Our recent results15,16,20as
well as the imaging results in this paper lend support for the
smaller 17 mm diameter. Since Resovist is not optimized for
MPI, there remains significant room for improvement27as the
resolution looks to improve with the cube of the nanoparticle
As we scale MPI imagers to human sizes, our sensitivity
will be limited by dB/dt and SAR limits.15X-space MPI the-
ory works well as we scale to larger sizes since we can choose
the partial FOV to be the maximum size allowed by dB/dt or
SAR safety limits and then assemble the full FOV from the
partial FOV scans. While similar in goal to the “focus field”
approach of Philips,28we use a very different methodology,
including our pulse sequence, image reconstruction through
gridding, as well as our image assembly algorithm.
We also do not see any hardware limits that would pre-
vent scaling MPI to human sizes beyond our ability to gen-
erate magnetic field gradients sufficient for high resolution
imaging across a large volume. Reasonable human-sized MPI
magnet gradients constructed using electromagnets with iron
returns or permanent magnets are limited to gradient field
strength of approximately <2.5 T/m. Unfortunately, such a
scanner would suffer from low native resolution with exist-
ing nanoparticle tracers.15,16,20For the gradient field strength
of the scanner shown in this paper (6 T/m), a human magnet
would require a superconducting magnet gradient, and would
put system costs and complexity similar to that of a modern
superconducting MRI system.
We believe that MPI will find its place in rapid angiogra-
phy, cell tracking, and cancer detection through dynamic con-
trast enhancement. To do these tasks, the overall technique
must give a signal that linearly changes with tracer quantity
and gives the same signal no matter where in the animal the
tracer resides. In short, the technique must be LSI.
033708-9 Goodwill et al. Rev. Sci. Instrum. 83, 033708 (2012)
In this paper, we have reviewed the generalized x-space
theory, described the construction of a small scale MPI
scanner, and demonstrated the necessary signal processing
to result in a native x-space MPI image. We believe that
the x-space theory enables LSI imaging, which we have
demonstrated experimentally but are still working to prove
theoretically. As such, we believe that the x-space theory
is an important step in the development of MPI theory and
hardware. Here we focused on critical hardware components
which have never been published.
Significant work remains before x-space MPI can be-
come pre-clinically and clinically relevant. Critically, we be-
lieve that at present x-space MPI resolution is not competitive
with small animal MRI and CT imaging, where resolution
is routinely finer than 250 μm. However, this deficit can be
improved through the development of new tracers designed
for MPI27,29and the construction of stronger magnetic field
This work was supported in part by CIRM Tools and
Technology Grant RT1-01055-1, and a UC Discovery grant.
the authors and do not necessarily represent the official views
of CIRM or any other agency of the State of California. This
work was supported in part by Grant Number 1R01EB013689
from the National Institute of Biomedical Imaging and Bio-
engineering. The content is solely the responsibility of the
authors and does not necessarily represent the official views
of the National Institute of Biomedical Imaging and Bioengi-
neering or the National Institutes of Health.
APPENDIX: CALCULATION AND MEASUREMENT
OF THE PSF WITH LOGNORMAL
Here, we describe how x-space predicts the shape of the
point spread function (PSF). To calculate the theoretical PSF,
we must consider the nanoparticle distribution. Specifically, if
we assume the receive coil is collinear with the transmit coil,
we can calculate the collinear component of the point spread
where we integrate across the distribution of diameters d, fXis
a lognormal distribution function with mean μ and standard
deviation σ,26m(d) =π
ment of a single nanoparticle, and H−1
[m/A] is the inverse of the saturation field of the magnetic
nanoparticle. We weight the signal by m(d) because the sig-
nal induced in the receive coil from a single nanoparticle in-
creases with the nanoparticle’s magnetic moment. We also
weight the signal by H−1
creases as the Langevin curve becomes sharper.15Thus, ne-
glecting relaxation, doubling the diameter of a nanoparticle
increases the induced signal by a factor of 26.
By imaging a phantom smaller than the native resolu-
tion of the system, we can experimentally measure the point
sat(ud) fX(ud;μ,σ)ˆ˙ xs· h(ud,x)ˆ˙ xsdud,
6Msatd3[Am2] is the magnetic mo-
sat(d) = μ0m(d)/kBT
sat(d) because the induced signal in-
spread function of the system. As shown in Fig. 10, the the-
oretical PSF visually matches the experimentally measured
PSF. The shape of the PSF is not surprising given the govern-
ing principle behind MPI. From the adiabatic assumption, we
can assume that the magnetic nanoparticle remains aligned
with the locally experienced magnetic field vector. Introduc-
ing a single nanoparticle, we see that the magnetic moment
de-saturates, flips, and re-saturates to follow the FFP as it
passes over the nanoparticle. Since we inductively detect a
signal, the change in magnetic moment as well as flipping of
the moment induces a signal “blip” in the receiver coil. If we
move off axis so that the FFP does not pass directly over the
nanoparticle, we see that the nanoparticle flips slower, and so
the signal “blip” is smaller and more spread out. Thus, the
PSF has the best signal and resolution when the FFP passes
directly over the nanoparticle, and is wider when we no longer
pass directly over the nanoparticle.
1B. Gleich and J. Weizenecker, Nature(London) 435, 1214 (2005).
2R. W. Katzberg and C. Haller, Kidney Int., Suppl. S3 (2006).
Am. J. Med. 103, 368 (1997).
4P. McCullough, J. Am. Coll. Cardiol. 51, 1419 (2008).
5E. A. Neuwelt, B. E. Hamilton, C. G. Varallyay, W. R. Rooney, R. D.
Edelman, P. M. Jacobs, and S. G. Watnick, Kidney Int. 75, 465 (2009).
6M. Lu, M. H. Cohen, D. Rieves, and R. Pazdur, Am. J. Hematol. 85, 315
7B. Gleich, J. Weizenecker, and J. Borgert, Phys. Med. Biol. 53, N81 (2008).
8T. Knopp, S. Biederer, T. Sattel, J. Weizenecker, B. Gleich, J. Borgert, and
T. M. Buzug, Phys. Med. Biol. 54, 385 (2008).
9T. Knopp, T. F. Sattel, S. Biederer, J. Rahmer, J. Weizenecker, B. Gleich,
J. Borgert, and T. M. Buzug, IEEE Trans. Med. Imaging. 29, 12 (2010).
10T. Knopp, S. Biederer, T. Sattel, J. Rahmer, J. Weizenecker, B. Gleich,
J. Borgert, and T. Buzug, Med. Phys. 37, 485 (2010).
11J. Rahmer, J. Weizenecker, B. Gleich, and J. Borgert, BMC Med. Imaging,
9, 4 (2009).
12J. Weizenecker, J. Borgert, and B. Gleich, Phys. Med. Biol. 52, 6363
13P. Goodwill and S. Conolly, in Proceedings of the First International Work-
shop on Magnetic Particle Imaging, Lubeck, Germany, 2010, edited by
T. M. Buzug (World Scientific Publishers, 2010).
14P. Goodwill, G. Scott, P. Stang, and S. Conolly, IEEE Trans. Med. Imaging
28, 1231 (2009).
15P. W. Goodwill and S. M. Conolly, IEEE Trans. Med. Imaging 29, 1851
16P. Goodwill and S. Conolly, IEEE Trans. Med. Imaging 30, 1581 (2011).
17T. Knopp, S. Biederer, T. F. Sattel, M. Erbe, and T. M. Buzug, IEEE Trans.
Med. Imaging 30, 1284 (2011).
18D. B. Twieg, Med. Phys. 10, 610 (1983).
19S. Ljunggren, J. Magn. Reson. 54, 338 (1983).
20P. Goodwill, A. Tamrazian, L. Croft, C. Lu, E. Johnson, R. Pidaparthi,
R. Ferguson, A. Khandhar, K. Krishnan, and S. Conolly, Appl. Phys. Lett.
21A. Oppenheim and R. Schafer, Discrete-Time Signal Processing, 2nd ed.
(Prentice Hall, 1999).
22J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, IEEE Trans.
Med. Imaging 10, 473 (1991).
23R. C. González and R. E. Woods, Digital Image Processing (Prentice Hall,
24W. Houston, Phys. Rev. 29, 478 (1927).
25J. Ferrucci and D. Stark, Am. J. Roentgenol. 155, 943 (1990).
26S. Biederer, T. Knopp, T. Sattel, K. Lüdtke-Buzug, B. Gleich, J.
Weizenecker, J. Borgert, and T. Buzug, J. Phys. D: Appl. Phys. 42, 205007
27R. M. Ferguson, K. R. Minard, and K. M. Krishnan, J. Magn. Magn. Mater.
321, 1548 (2009).
28J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke, and J. Borgert, Phys.
Med. Biol. 54, L1 (2009).
29R. M. Ferguson, K. R. Minard, A. P. Kandhar, and K. M. Krishnan, Med.
Phys. 38, 1619 (2011).