# Vector Modulator Based Active Compensation
of Direct Feedthrough in Magnetic Particle Imaging

Bilal Tasdelen1,2, Mustafa Utkur1,2, Asli Alpman1,2 Can Baris Top3, Ergin
Atalar1,2,4, ,
and Emine Ulku Saritas1,2,4 Manuscript received XX May 2024; revised XX
September 2024; accepted XX October 2024. This work was supported by the
Scientific and Technological Research Council of Turkey under Grant TUBITAK
120E208. The authors would like to thank Ecrin Yagiz for her help during the
MPS experiments. Preliminary versions of this work were presented at the 10th
International Workshop on Magnetic Particle Imaging (IWMPI 2020) and the
Annual Meeting of International Society for Magnetic Resonance in Medicine
(ISMRM 2021). (Corresponding authors: Bilal Tasdelen, Emine U. Saritas) Bilal
Tasdelen was with Department of Electrical and Electronics Engineering,
Bilkent University, Ankara, Turkey. He is now with Department of Electrical
and Computer Engineering, University of Southern California, Los Angeles, CA,
USA. (e-mail<EMAIL_ADDRESS>Mustafa Utkur was with Department of
Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey. He
is now with Department of Radiology, Boston Children’s Hospital and Harvard
Medical School, Boston, MA, USA. Asli Alpman was with Department of Electrical
and Electronics Engineering, Bilkent University, Ankara, Turkey. She is now
with Department of Electrical Engineering and Computer Sciences, University of
California, Berkeley, Berkeley, CA, USA. Can Baris Top is with ASELSAN
Research Center, Ankara, Turkey. (e-mail<EMAIL_ADDRESS>Ergin Atalar
is with Department of Electrical and Electronics Engineering, Bilkent
University, Ankara, Turkey. (e-mail<EMAIL_ADDRESS>Emine U.
Saritas Department of Electrical and Electronics Engineering, Bilkent
University, Ankara, Turkey. (e-mail<EMAIL_ADDRESS>1Department of
Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey
2National Magnetic Resonance Research Center (UMRAM), Bilkent University,
Ankara, Turkey  3ASELSAN Research Center, Ankara, Turkey  4Neuroscience
Program, Aysel Sabuncu Brain Research Center, Bilkent University, Ankara,
Turkey

###### Abstract

In magnetic particle imaging (MPI), simultaneous excitation and signal
acquisition leads to direct feedthrough interference. While this interference
can be mitigated up to some extent with passive compensation, its time-varying
nature necessitates active compensation methods to achieve the sensitivity
levels needed for applications such as stem cell tracking. We have recently
proposed an active compensation technique for MRI, which uses a vector
modulator and a lookup-table-based algorithm for reducing the direct
feedthrough in the analog domain. Here, we adapt this technique to MPI,
demonstrating a successful recovery of the fundamental frequency and a
significant increase in detection sensitivity.

###### Index Terms:

Magnetic particle imaging, direct feedthrough, coupling, interference, active
compensation, vector modulator.

##  I Introduction

Magnetic Particle Imaging (MPI) is a rapidly developing modality that images
the spatial distribution of magnetic nanoparticles (MNPs) by exploiting their
nonlinear magnetization curves [1, 2]. MPI has excellent potential for
numerous clinical applications such as angiography [3], stem cell imaging[4],
viscosity mapping[5, 6], functional imaging[7], and inflammation imaging [8,
9].

In MPI, a sinusoidal drive field (DF) is applied to excite the MNPs. Since
MNPs align with the applied DF almost instantaneously, excitation and signal
acquisition are inherently simultaneous in MPI. Furthermore, the pulse
transmitted from the drive coil couples to the receive coil and induces a
direct feedthrough signal [10]. Because this direct feedthrough can be orders
of magnitude larger than the MNP signal, it can easily saturate the
preamplifier and reduce the dynamic range, jeopardizing MPI’s promise of high
sensitivity imaging [11]. As a first measure of reducing the direct
feedthrough, gradiometer receive coils are widely utilized in MPI to minimize
the mutual inductance between the drive and the receive coils [12, 13, 14,
15]. While gradiometer receive coils passively reduce the direct feedthrough
signal, their tuning is prone to degradation and drift due to factors such as
vibration and heating. As a result, even after passive cancellation, the
residual feedthrough can easily exceed the MNP signal. A typical secondary
measure is to acquire background measurements to be subtracted from the
imaging signal. Although this simple approach can further reduce the direct
feedthrough, system drift during long scan durations can cause significant
temporal variations in its efficacy [16, 17, 18]. Due to these difficulties,
the fundamental frequency is almost always filtered out of the received signal
[10].

Another approach for reducing the direct feedthrough is via active
compensation (AC), where the time-dependent direct feedthrough is actively
monitored and subtracted from the received signal at the analog stage. Time-
dependent direct feedthrough (also known as self-interference in
telecommunications) is a challenging but ubiquitous issue. Therefore, AC is an
active and vital area of research not just in MPI [19, 16], but in other
fields such as telecommunications (as full-duplex radio problem)[20, 21, 22,
23, 24] and MRI (as concurrent excitation and acquisition problem) [25, 26,
27, 28, 29, 30, 31]. Indeed, the ability to actively reduce the direct
feedthrough would be extremely useful for almost all applications of MPI. In
imaging, it can enable the fundamental frequency to be preserved, which in
turn can improve the image signal-to-noise ratio (SNR) and thereby the
detection sensitivity [16]. In addition, preserving the fundamental frequency
can circumvent the image artifacts associated with the recovery of the
fundamental frequency in x-space reconstruction [17]. Furthermore, the
increase in SNR can enable imaging at lower DF strengths. As a result, cheaper
systems can be built with less stringent power and cooling requirements and
with less emphasis on building a carefully tuned gradiometer coil, increasing
the availability and robustness of MPI.

Previously in MPI, AC was performed by utilizing a second independent signal
source [19, 16]. Although this approach provides simplicity and flexibility
for creating the cancellation signal, it also requires a precise knowledge of
the DF waveform. Furthermore, in the case that transmit noise is dominant,
addition of an independent transmit source can significantly increase the
received noise level [28]. Furthermore, if the DF gets distorted due to
heating, vibrations or other non-linear distortions, this method will fail to
compensate.

In this work, we propose a vector modulator (VM) based AC method to address
the direct feedthrough problem in MPI. The proposed VM is integrated to the
MPI system and is digitally controlled, empowering the MPI system with a fully
automated AC process that does not require any manual interventions. In
addition, we propose a fast look-up table-based algorithm that automatically
and dynamically compensates for the time-varying direct feedthrough. The
proposed AC method and the algorithm are demonstrated via extensive
simulations and experiments in a magnetic particle spectrometer (MPS) setup.
The simulation results show that the additional feedthrough cancellation
reaches up to 70 dB at high direct feedthrough levels. In addition, the
experiments show that the proposed method provides 39 dB of additional
cancellation in 500 milliseconds, successfully recovering the MNP signal at
the fundamental frequency and significantly improving the detection
sensitivity.

##  II Methods

###  II-A Direct Feedthrough Signal

The direct feedthrough signal in MPI stems from the non-zero mutual inductance
between the closely wound DF coil and receive coil. Usually, the receive coil
is configured as a gradiometer coil to reduce the mutual inductance.
Theoretically, for the case of a perfect construction, it is possible to zero
out the mutual inductance. In practice, however, it is extremely challenging
to achieve this goal due to possible constructional flaws.

Regardless of the configuration, the effect of the mutual inductance is
considered as the coupling of the DF waveform to the receive side, attenuated
and phase shifted. If we consider a sinusoidal DF with a center frequency
$f_{0}$ and envelope $A(t)$, the received direct feedthrough at $f_{0}$ can be
written as:

$s(t)=k(t)\cdot A(t)\cdot\sin(2\pi f_{0}t+\phi(t)),$ (1)

where $k(t)$ and $\phi(t)$ are the attenuation coefficient and the phase shift
stemming from mutual coupling, respectively, which can both vary with time.
The main reasons for this time-variance of the mutual inductance are the
heating of the DF coil due to applied power, temperature fluctuations in the
environment, vibrations, or other small movements in the system. It should be
noted that the system becomes more susceptible to these effects as the mutual
inductance becomes smaller.

One approach to reduce this direct feedthrough is actively removing it from
the received signal using the knowledge of the DF signal. This can be done
either in the digital or in the analog domain. Digital cancellation of the
feedthrough is advantageous in terms of both system complexity and cost, as it
is not encumbered by the challenges of hardware design. However, it fails to
accommodate for several outstanding issues such as the reduction of the
receive chain dynamic range, unforeseen deviations of the DF due to hardware
imperfections caused by amplifier nonlinearity, and interference from external
sources into the DF signal.

###  II-B Active Compensation Circuitry

In this work, an AC scheme is proposed that estimates the parameters $k$ and
$\phi$ and creates a replica of the direct feedthrough signal to remove it in
the analog domain. Here, both $k$ and $\phi$ are assumed to be stable during
short intervals of signal acquisition, but may drift over time. The
aforementioned replica of the direct feedthrough will be referred to as the
“compensation signal”. To adjust the compensation signal according to the
parameters $k$ and $\phi$, we propose a VM circuit that can modify both the
amplitude and phase of its input, as shown in Fig. 1. This circuit contains
two branches, where each branch employs a phase shifter and an attenuator: The
phase shifter provides up to $\pi$ relative phase between the input and the
output, whereas the attenuator adjusts the relative amplitude of the
compensation. As a result, the compensation provided by each branch
approximately covers a complex half-plane in phasor domain. These two branches
are then combined with weighting, enabling approximately full coverage of the
entire complex plane. Finally, the resulting compensation signal is subtracted
from the received signal.

Figure 1: (a) System diagram of the setup and the vector modulator. Red arrows
show the places where digital rheostats are deployed. (b) Photograph of the
gradiometer coil and the pick-up coil. (c) Photograph of the vector modulator
and the controller unit.

The phase shifters in Fig. 1 were realized as first-order all-pass filters
[32]. The relative phase of an all-pass filter in a low-pass configuration at
the inverting input can be calculated as:

$\angle H(i\omega)=-2\>\arctan(\omega RC),$ (2)

where $\omega$ is the radial frequency. In a high-pass configuration, on the
other hand, there is an additional $\pi$ phase in Eq. 2. Hence, by fixing C
and sweeping R via a rheostat, we can cover phases from 0 to $-\pi$ with a
low-pass configuration and from 0 to $\pi$ with a high-pass configuration.
Here, for a given set of available R values of a rheostat, C needs to be
chosen to ensure that a sufficient range of phases are covered while being
sampled densely enough. Moreover, since the phase is frequency dependent, the
operating regime needs to be taken into account.

In this work, linearly varying 8-bit digital rheostats (MCP4352, Microchip)
with 5 k$\Omega$ maximum resistances were used for implementing the phase
shifters and the attenuators, and C was chosen as 22 nF to provide the
necessary phase coverage at $f_{0}$ = 10 kHz. The attenuators controlled the
gain/attenuation for each branch of the weighted adder, which was implemented
as an inverting Op-Amp (TL074) summing amplifier. An external PC calculated
the digital codes for the rheostats, and an Arduino microcontroller translated
these codes into serial commands to configure the rheostat states.

As shown in Fig. 1a, at the input of the VM circuit, the feedthrough signal
was sensed from the DF coil via a pick-up coil with an inductance of 313
$\mathrm{\SIUnitSymbolMicro H}$. This relatively large inductance provided
high sensitivity to the DF signal while ensuring minimal interaction with the
receive coil. An inverting Op-Amp (TL074) amplifier was used to scale the gain
of the VM and isolate the input impedance due to the pick-up coil from the
phase shifters. Finally, the output of the VM was subtracted from the received
signal using the differential input of a low noise amplifier (LNA; Stanford
Research Systems SR560).

###  II-C Active Compensation Algorithm

To boost the performance of the VM, we propose an AC algorithm that can
control the VM via its digital inputs (i.e., the wiper states of the 8-bit
digital rheostats) and minimize the residual feedthrough. Such an algorithm
needs to be fast enough to accommodate a time-varying feedthrough (i.e.,
system drift), while being precise enough to minimize the residual feedthrough
to a sufficiently low level. A non-linear integer optimization algorithm can
be run to determine the digital inputs that minimize the residual feedthrough,
by measuring only the signal at the output of the differential LNA in Fig. 1.
Importantly, this measurement does not require the explicit knowledge of
either the feedthrough or the VM output. However, such optimization algorithms
typically require many consecutive measurements to converge, leading to a very
time-consuming operation, which is especially problematic if the feedthrough
drifts quickly over time. To avoid this convergence issue, we adopt a fast,
system characterization based approach in this work.

1) Look-up Table for System Characterization: In theory, the output phase and
amplitude of the VM for a given input signal can be estimated, since the
transfer function of this circuit can be derived using the component values
for fixed digital inputs. However, a better characterization of the system is
necessary for various reasons such as the tolerances in the circuit
components, which can limit the performance of the VM. Here, the most
prominent nonidealities are due to the values of the resistors and wiper
resistances in the digital rheostats, which may have up to 1% error as stated
in the datasheet. Another prominent reason necessitating a better
characterization is the system drift. To mitigate these issues, we propose a
measurement based characterization procedure.

Figure 2: (a) Amplitude and phase measurements, in the case where the state of
each rheostat is swept one by one, while the other rheostats are disconnected.
With each rheostat having $N$ = 256 states, a total of $4N=2^{10}$
measurements were conducted by digitally controlling the rheostat states.
These measurements were then used to create the look-up table constellations
that map the phasor of the output contribution by each branch. Graphical
representation of the (b) coarse adjustment and (c) fine adjustment steps of
the LUT algorithm in phasor domain. For each step, the black arrows represent
the measured initial feedthrough phasor, the green dashed arrow shows the
selected phasor for subtraction, the solid green arrow shows the subtraction
operation. The purple arrow shows the residual feedthrough after each
adjustment step. Note the scale difference along the radial direction between
the coarse and fine adjustment steps in (b) and (c). These plots show highly
undersampled versions of the estimated constellation, which contained $2^{17}$
points for each step (i.e., $N^{2}=2^{16}$ points each for top and bottom
branch constellations).

Here, we refer to the entire possible input-output mapping of the VM in phasor
domain as the “constellation”. Assuming that each rheostat has
$N\in\mathbb{N}$ different wiper states (i.e. $N$ possible resistance values),
$N^{4}$ measurements would be needed to fully characterize the constellation
given that the circuit contains 4 rheostats. However, if we assume that the
two branches of the VM are decoupled from each other (a reasonable assumption
considering the Op-Amps at the input/output of each branch), the overall
transfer function acting on the input is equal to the vector summation of the
transfer functions of each branch. Hence, to characterize the modulator, one
can disconnect one of the branches and measure the amplitude and phase shift
with respect to the input while sweeping the rheostats in the connected
branch, covering the constellation in $2N^{2}$ measurements instead.

In this work, we propose going one step further and sweeping each rheostat one
by one. First, the bottom branch is disconnected, and the attenuator rheostat
on the top branch is kept at a fixed wiper state while the phase shifter
rheostat wiper is swept. Next, for the same branch, the phase shifter rheostat
is kept fixed and the attenuator rheostat is swept. Finally, this procedure is
repeated for the bottom branch, with the top branch disconnected. In sharp
contrast to $N^{4}$ or $2N^{2}$ measurements that would otherwise be needed,
this approach requires only $4N$ measurements to cover the entire
constellation. For the circuit described in Section II-B that uses 8-bit
rheostats with $N=2^{8}$, the results of the characterization measurements are
given in Fig. 2(a). The experimental details of these measurements are
provided in Section II-E below. Here, by digitally controlling the rheostat
states, a total of $4N=2^{10}$ measurements were rapidly conducted (as opposed
to $N^{4}=2^{32}$ or $2N^{2}=2^{17}$ measurements). Next, the unmeasured part
of the constellation is estimated using the following equation:

$G_{vm}(x_{1},x_{2},x_{3},x_{4})=G_{1}(x_{1},x_{2})+G_{2}(x_{3},x_{4})$ (3)

where

$\displaystyle G_{1}(x_{1},x_{2})$ $\displaystyle=G_{\phi_{1}}(x_{1})\cdot
G_{a_{1}}(x_{2})$ (4) $\displaystyle G_{2}(x_{3},x_{4})$
$\displaystyle=G_{\phi_{2}}(x_{3})\cdot G_{a_{2}}(x_{4})$ (5)

Here, $x_{1},x_{2},x_{3},x_{4}\in{1...N}$ are the wiper states determining the
resistance value of the 4 rheostats, $G_{vm}$ is the estimated transfer
function of the entire VM, and $G_{1}$ and $G_{2}$ are the estimated transfer
functions of the top and bottom branches, respectively. In addition,
$G_{\phi_{1}},G_{a_{1}}$ and $G_{\phi_{2}},G_{a_{2}}$ are the measured
transfer functions of the phase shifter and attenuator in the top and bottom
branches, respectively. A visual representation of the estimated constellation
can be seen in Fig. 2(b). Finally, the top branch and bottom branch parts of
the estimated constellation are stored in a look-up table (LUT). Note that
this procedure only needs to be performed once for a given VM circuit.

2) Look-up Table Based AC Algorithm: To compensate for the feedthrough that
occurs during an actual experiment, we propose a LUT-based AC algorithm that
can be divided into two parts: coarse adjustment and fine adjustment. In the
coarse adjustment process, first, the initial direct feedthrough is measured
with the VM disabled. Then, the VM is initialized with fixed digital inputs
and its output is measured. Using these two measurements, a complex-valued
scaling factor that describes the amplitude and phase differences between the
constellation in the LUT and the VM output is estimated, and the constellation
is scaled accordingly. After that, the branch suitable for the coarse
adjustment is chosen as the one with the smaller distance between the initial
feedthrough and the mean phase of the constellation for that branch. In the
chosen branch’s constellation, a compensation phasor is sought that will
minimize the distance between the residual and the densely covered part of the
other branch’s constellation, under the constraint that the residual will lie
in the support of the other branch’s constellation. The rheostat wipers for
the coarse adjustment branch are updated accordingly.

Next, a fine adjustment process is performed. First, the residual feedthrough
after the coarse adjustment (i.e., the coarse residual) is measured. Note that
it could also be estimated from the previous step using the LUT and the known
initial feedthrough. Another search is performed in the constellation in which
the coarse residual lies, to seek a compensation phasor that minimizes the
final residual. Then, the rheostat wipers for the fine adjustment branch are
updated accordingly. The graphical demonstrations of the coarse and fine
adjustments can be seen in Fig. 2(b) and (c), respectively.

After applying the LUT algorithm, if the resulting residual feedthrough is not
satisfactory, an iterative local search can be performed in the vicinity of
the resulting residual. In this work, this local search is done by
incrementing or decrementing each rheostat wiper one-by-one and measuring the
new residual until either another minimum for the residual is reached or the
residual starts to increase. This version of the algorithm is referred to as
LUTit, where the appended ”it” indicates the additional iterative local
search.

###  II-D Simulations

Simulation studies were performed in phasor domain to demonstrate the
performance of the VM circuit and the proposed AC algorithm. First, the
frequency response of the VM circuit was simulated in SPICE. The AC algorithm
simulations, performed in MATLAB, utilized the experimentally measured and
estimated LUT (see Fig. 2), which was assumed to be ideal and noise-free.
First, Monte Carlo simulations were performed with 5000 repetitions. For each
repetition, LUT algorithm ran with random feedthrough inputs, with independent
and uniformly distributed real and imaginary parts. The cancellation
performance was first evaluated without any noise, and then with white
Gaussian noise added to both the measurements and the LUT at an interference-
to-noise ratio (INR) of 40 dB to mimic the INR observed in the experiments.
Next, the cancellation performance was evaluated as a function of INR at 20
different INR levels ranging between 14-94 dB, with white Gaussian noise of
matching standard deviations added to both the measurements and the LUT at
each INR level. At each INR level, Monte Carlo simulations were performed with
1000 repetitions.

###  II-E Experiments

The proposed technique was demonstrated on an in-house arbitrary waveform
magnetic particle spectrometer (MPS) device [33]. Shown in Fig. 1b, this MPS
device featured a DF coil with a small inductance of 3.8
$\mathrm{\SIUnitSymbolMicro H}$with 0.8 mT/A sensitivity [34]. The gradiometer
receiver coil had 8.8 $\mathrm{\SIUnitSymbolMicro H}$inductance and had a
manual adjustment knob for fine-tuned decoupling of the direct feedthrough.

The signal at the LNA output (i.e., the difference between the received signal
and the VM output) was digitized at 2 MS/s by a data acquisition card
(National Instruments PCIe-6374), which also generated and sent the DF pulses
to a power amplifier (AETechron 7224). Thanks to the low inductance of the DF
coil, the power amplifier was directly connected to DF coil without the need
for impedance matching. The DF was at $f_{0}$ = 10 kHz with amplitudes ranging
between 5-22.5 mT, depending on the experiment performed. Finally, the VM (see
Fig. 1c) was powered by batteries to ensure that no external interference
coupled from the powerline to the received signal through the circuit. The
entire setup (both the MPS and the VM) were controlled via MATLAB.

With this experimental setup, 4 different types of experiments were performed:
system characterization experiments, AC algorithm performance experiments,
dynamic AC experiments, and MNP detection sensitivity experiments. All
experiments, except for the sensitivity experiments, were performed with an
empty MPS chamber.

1) System Characterization Experiments: To create the LUT for AC,
characterization experiments were performed using a DF with $f_{0}$ = 10 kHz
frequency and 13.7 mT amplitude. This DF amplitude was chosen to maximize the
VM input without heating the drive coil, to avoid rapid system drift. As
described in Section II-B, this DF was sensed by the pick-up coil and fed to
the VM circuit. The VM output was measured using the single-ended mode of the
LNA with an acquisition duration of 100 ms. Then, the attenuator and phase
shifter rheostat wipers were swept one-by-one on the selected branch, while
the other branch was disconnected using the disable function of the rheostats.
Next, the output amplitude and phase of each measurement were estimated from
the Fourier transforms of the acquired signals. This procedure was repeated
for each branch and each rheostat one-by-one, resulting in $4N=2^{10}$
measurements for the case of four 8-bit rheostats. Finally, the unmeasured
phasors were estimated from these acquisitions using Eq. 3 to create a LUT
with $2N^{2}=2^{17}$ entries. The entire characterization process took less
than 5 minutes.

2) AC Algorithm Performance Experiments: The performances of the proposed LUT-
based algorithms were compared with the following calibration-free nonlinear
integer optimization algorithms:

  * •

Genetic Algorithm (GAL): This algorithm tries to minimize the direct
feedthrough by applying a natural selection process [31]. Briefly explained,
GAL randomly generates candidate inputs for the VM, evaluates their
performance at each iteration, and selects a subset of high performance inputs
from which a new generation of candidate inputs are created. This process
continues until convergence is achieved. For the experiments in this work, the
population size for GAL was set as 10, which means that the algorithm
generated 10 different candidates at each iteration, performed measurements of
the residual to determine the cancellation performance of each candidate, and
then selected the best ones to create the new generation of 10 candidates.

  * •

Surrogate Optimization (SOP): This algorithm approximates the complex transfer
function of the circuit in terms of simpler surrogate functions using several
input/output measurements [35]. For the experiments in this work, SOP
performed 8 measurements of the residual at the two-end wiper states of each
rheostat to initialize the algorithm, followed by an iteration of measurements
using the estimated optimal inputs. The approximated transfer function is
updated at each iteration to determine the optimal inputs.

Experiments were performed to compare the cancellation performances and the
execution times for the proposed LUT and LUTit algorithms, and the comparison
methods GAL and SOP. Each algorithm was implemented in MATLAB to digitally and
automatically control the rheostat states, without any manual intervention.
During the experiments, 20 independent runs were conducted for each algorithm,
with the VM states randomly reset at the beginning of each run. For all of the
algorithms, 50 $\mathrm{ms}$ acquisition time was used for direct feedthrough
measurements. For these experiments, the DF was at $f_{0}$ = 10 kHz with 6.8
mT amplitude.

3) Dynamic AC Experiments: The ability of the proposed method to dynamically
reduce the direct feedthrough was demonstrated by inducing rapid system drift
via applying high power to heat up the DF coil. Accordingly, a relatively high
22.5 mT amplitude DF was applied at $f_{0}$ = 10 kHz, with 100 ms pulses and
50 ms intervals throughout 20 seconds. During the first 5 seconds, the AC was
kept off. Then, for the following 15 seconds, the direct feedthrough level was
monitored during the 100 ms DF heating pulses. Whenever the feedthrough
exceeded a threshold of -70 $\mathrm{dB}\text{\,}\mathrm{V}$, the algorithm
kicked off automatically to reduce it down. The acquisition duration for
monitoring and heating was set as 100 $\mathrm{ms}$, whereas the duration of
each acquisition needed for the LUT or LUTit AC algorithms (see Section II-C)
was set as 10 $\mathrm{ms}$.

To demonstrate the performance of the proposed AC method under different
initial feedthrough levels, the above-mentioned experiment was repeated with 3
different initial cancellation levels by adjusting the knob of the gradiometer
receive coil to different positions.

4) MNP Detection Sensitivity Experiments: To demonstrate the improvements
achieved by the proposed AC method in an MPS setup, MNP samples were prepared
using 3 different commercially available nanoparticles: Vivotrax (Magnetic
Insight, USA), Resovist (Bayer, Germany), and Perimag (Micromod GmbH, Germany)
with undiluted iron concentrations of 5.5 $\mathrm{mg}\text{/}\mathrm{mL}$, 28
$\mathrm{mg}\text{/}\mathrm{mL}$, and 25 $\mathrm{mg}\text{/}\mathrm{mL}$,
respectively. For each MNP, a dilution series at 9 different concentrations
were prepared at [2, 10, 20, 50, 100, 200, 400, 1000] fold dilutions. With a
total volume of 20 $\mathrm{\SIUnitSymbolMicro L}$ each, the resulting samples
had total iron masses ranging between 0.05-250 $\mu$g. The DF was applied at
$f_{0}=10~{}$\mathrm{kHz}$$ at 2 different amplitudes of 5 mT and 10 mT, where
the system drift was relatively slow. For each vial, first, the LUTit version
of the proposed AC algorithm was applied using 30 ms acquisitions. Then, the
rheostat states were fixed to the optimal values determined by the algorithm.
Next, 10 repeated empty-chamber baseline measurements were acquired, followed
by 10 repeated measurements of the MNP signal. The acquisition times for the
baseline signal and MNP signal were 20 ms each. This procedure was then
repeated with the AC turned off.

The signal strength at the harmonics, as well the root-mean-squared (RMS)
value of the acquired signal were investigated with and without AC. Since the
MNP signal at all harmonics should be proportional with the MNP concentration,
$y=ax$ line was fit to the signal as a function of concentration for each
nanoparticle type at each harmonic. The corresponding $R^{2}$ values were
calculated to observe whether the measurements followed the expected linear
trend. The baseline measurements were used to calculate the baseline signal
level at each harmonic, which had contributions from both the direct
feedthrough and noise. The signal levels that were at least one standard
deviation (std) above the mean of the baseline were considered as being
successfully detected.

##  III Results

Figure 3: (a) Frequency response plots of cancellation and feedthrough,
simulated by SPICE. Important values are annotated on the plots. (b) Histogram
of the Monte-Carlo simulations with and without measurement noise added. (c)
Median, 25th and 75th percentile of additional cancellation with respect to
interference-to-noise (INR) ratio.

The results of the VM circuit frequency response simulations are given in Fig.
3(a). Comparing the feedthrough amplitude and the residual amplitude at the
targeted $f_{0}=10~{}$\mathrm{kHz}$$, the proposed VM circuit can successfully
achieve at least 40 dB cancellation (exceeding 50 dB at exactly
$10~{}$\mathrm{kHz}$$) with 8-bit digital rheostats used in the design.
Another observation from Fig. 3(a) is, the bandwidth of the cancellation at
$f_{0}=10~{}$\mathrm{kHz}$$ was quite narrow, with a 6 dB bandwidth of 1.7
kHz. This narrow bandwidth was expected due to the disparity between the phase
responses of the VM and the direct feedthrough.

In Fig 3(b), the results of the Monte-Carlo simulation with 5000 repetitions
are plotted, showing the histogram of the additional cancellation provided by
the LUT-based AC algorithm. In the absence of noise (i.e., at infinitely large
INR), the algorithm provides 68$\pm$13 dB (mean$\pm$std) additional
cancellation at $f_{0}$. Even in the presence of a realistic level of noise at
INR = 40 dB, the algorithm provides 35$\pm$6 dB an additional cancellation at
$f_{0}$. Next, Fig. 3(c) shows the additional cancellation with respect to
INR, where the error bars indicate the median and the $25^{th}-75^{th}$
percentiles of the Monte Carlo simulation results with 1000 repetitions at
each INR level. As expected, the additional cancellation performance increases
with INR, reaching a mean value of 67 dB at INR levels above 80 dB, where the
interference strongly dominates over the noise. On the other hand, the
performance deteriorates at low INR, slowly converging to an additional
cancellation of 0 dB as INR approaches 0. Note that at INR = 0, there is no
interference but only noise, which cannot be cancelled with or without AC. The
additional cancellation level stays above 0 dB for INR$>$25 dB (corresponding
to an interference that is at least 18 times larger than the noise),
indicating that the proposed VM circuit and the LUT-based AC algorithm can
ensure successful cancellation of the direct feedthrough in realistic MPI
settings.

Figure 4: Comparison of different algorithms in terms of (a) additional
cancellation provided and (b) execution time per run. The red lines show the
median values, the boxes show 25th-75th percentile values, and the whiskers
show the extents of the measurements.

The results for the VM system characterization experiments were already
presented in Fig. 2. Accordingly, Fig. 2a shows that rheostats 1 and 3 (i.e.,
the phase shifters in the top and bottom branches, respectively) both have
flat amplitude responses and provide 0 to $-\pi$ and 0 to $\pi$ phase
coverages, respectively. On the other hand, rheostats 2 and 4 (i.e., the
attenuators in the top and bottom branches, respectively) both have flat phase
responses with over 50-fold variation in their amplitude responses. From these
$4N=2^{10}$ measurements, the estimated LUT constellation with $2N^{2}=2^{17}$
entries is demonstrated in Fig. 2b, showing that the VM ensures sufficient and
dense coverage of a range of phases.

The experiment results comparing the performances of the proposed LUT and
LUTit algorithms with SOP and GAL are shown in Fig. 4, for 20 independent runs
of each algorithm. As seen in Fig. 4(a), LUTit had the highest additional
cancellation performance among the tested algorithms. GAL was slightly better
than LUT, whereas SOP outperformed both LUT and GAL. The additional
cancellation levels were 39$\pm$6 dB for GAL, 46$\pm$6 dB for SOP, 36$\pm$3 dB
for LUT, and 47$\pm$5 dB for LUTit (mean$\pm$std). By comparing the
interference level at $f_{0}$ with the noise at non-harmonic frequencies, the
INR level of these experiments was estimated as approximately 45 dB.
Accordingly, we can conclude that the additional cancellation provided by
LUTit in these experiments agrees well with the simulation results in Fig
3(c).

Next, comparing the execution times of the AC algorithms in Fig. 4(b), both of
the proposed algorithms were significantly faster than SOP and GAL. LUTit and
LUT were at least 8-fold and 26-fold for faster than SOP, respectively. The
execution times were 28$\pm$8 s for GAL, 18$\pm$4 s for SOP, 0.5$\pm$0.03 s
for LUT, and 1.4$\pm$0.2 s for LUTit (mean$\pm$std). This major improvement in
execution time for the proposed AC algorithms is due to the system
characterization based approach adopted in this work. Overall, the results in
Fig. 4 clearly demonstrate the superior performance of the LUT-based
algorithms, as they provide competitive cancellation levels at significantly
reduced execution times. The results also indicate that, LUTit should be
preferred over LUT in the cases where the system drift is slow and a longer
execution time can be allocated, as it provides improved cancellation
performance.

Figure 5: Results of the heating experiments. Green dashed boxes marks the
time intervals where the AC is running. Red dashed line shows the threshold
-70 dBV. AC is activated at the 5 second mark.

Figure 5 shows the results of three different dynamic AC experiments for LUT
and LUTit. Here, the gradiometer knob of the MPS receive coil was adjusted to
a different position at the beginning of each experiment, as reflected with
the differences in the initial feedthrough levels across experiments. Here,
due to the heating of the DF coil under a relatively large 22.5 mT DF
amplitude, the direct feedthrough drifts over time relatively quickly. Despite
the rapid system drift, both of the proposed AC algorithms successfully keep
the feedthrough under the desired threshold of -70
$\mathrm{dB}\text{\,}\mathrm{V}$. The algorithm runtimes for each cancellation
cycle were measured as 100 ms for LUT and 340 ms for LUTit, respectively, with
the additional time stemming from the local search iterations for LUTit. Note
that LUTit did not provide an apparent increase in the additional
cancellation, because the rapid drift of the direct feedthrough throughout the
increased runtime of LUTit caused the chosen solution to diverge from the
optimal solution. In fact, if the drift occurs extremely rapidly, LUTit can
potentially perform worse than LUT for the same reason. In contrast, the drift
was relatively slow for the results in Fig. 4, where the system temperature
remained constant and LUTit provided 11 dB additional cancellation over LUT on
average.

Figure 6: Magnitude of the measured signal’s fundamental frequency, 3rd and
5th harmonics for each dilution and particle when (a) 5 mT and (b) 10 mT DF is
applied. Figure 7: RMS magnitude of the measured signal and RMS when $f_{0}$
is filtered for (a) 5 mT and (b) 10 mT DF is applied.

Finally, the results of the MNP detection sensitivity experiments are given in
Figs. 6-7. In Fig. 6, the acquired signal strength at $f_{0}$, $3f_{0}$ and
$5f_{0}$ are plotted as a function of the iron mass in the MNP sample at 5 mT
and 10 mT DF amplitudes for 3 different nanoparticle types, without and with
AC. The mean and the std of the baseline measurements are also marked.
Likewise, Fig. 7 shows the RMS value of the acquired signal up to the 5th
harmonic with and without including the signal contribution at $f_{0}$.
$R^{2}$ values of the line fits were higher than 0.99 for the fundamental
frequencies where AC was employed, verifying that the linear trend is
preserved. Finally, in Table I, Fe masses where the line fits cross with
baseline mean is shown to determine the minimum detectable Fe mass for each
case. By looking at the RMS signal when AC is on, compared to AC off filtered
RMS (RMS-$f_{0}$) case, up to 2.8 and 1.6-fold increase is observed for 5 mT
and 10 mT DF, respectively.

TABLE I: Detectable Fe masses ($\mathrm{\SIUnitSymbolMicro g}$). |  | Vivotrax | Resovist | Perimag
---|---|---|---|---
|  | AC Off | AC On | Improvement | AC Off | AC On | Improvement | AC Off | AC On | Improvement
5 $\mathrm{mT}$ | RMS | 30 | 0.67 |  $\times$ 44.8 | 17.49 | 0.56 |  $\times$ 31.2 | 17.19 | 0.49 |  $\times$ 35.1
RMS-$f_{0}$ | 1.92 | 2.02 |  $\times$ 0.95 | 1.04 | 1.35 |  $\times$ 0.83 | 0.90 | 0.92 |  $\times$ 0.98
$|f_{0}|$ | 38.5 | 0.36 |  $\times$ 107 | 23.02 | 0.32 |  $\times$ 71.9 | 24.8 | 0.31 |  $\times$ 80
$|3f_{0}|$ | 1.04 | 0.99 |  $\times$ 1.05 | 0.57 | 0.79 |  $\times$ 0.72 | 0.51 | 0.56 |  $\times$ 0.91
$|5f_{0}|$ | 2.72 | 2.59 |  $\times$ 1.05 | 1.43 | 1.65 |  $\times$ 0.87 | 1.17 | 1.08 |  $\times$ 1.08
10 $\mathrm{mT}$ | RMS | 20.35 | 0.38 |  $\times$ 53.5 | 14.55 | 0.36 |  $\times$ 40.4 | 13.76 | 0.28 |  $\times$ 49.1
RMS-$f_{0}$ | 0.64 | 0.62 |  $\times$ 1.03 | 0.47 | 0.37 |  $\times$ 1.27 | 0.34 | 0.34 |  $\times$ 1
$|f_{0}|$ | 29.66 | 0.28 |  $\times$ 105.9 | 23.64 | 0.29 |  $\times$ 81.5 | 25.51 | 0.28 |  $\times$ 91.1
$|3f_{0}|$ | 0.46 | 0.45 |  $\times$ 1.02 | 0.29 | 0.29 |  $\times$ 1 | 0.28 | 0.28 |  $\times$ 1
$|5f_{0}|$ | 0.72 | 0.67 |  $\times$ 1.07 | 0.38 | 0.36 |  $\times$ 1.05 | 0.32 | 0.31 |  $\times$ 1.03

##  IV Discussion

In this work, we proposed an active compensation method that can reduce direct
feedthrough to the point where the fundamental frequency of highly diluted
particles can be recovered. This allowed us to detect such particles, or
increased the total signal acquired from these particles, which in turn
increased the SNR.

Simulations in Fig. 3 suggests that, the proposed vector modulator together
with the proposed algorithm is able to reduce the direct feedthrough
consistently and in a variety of INR conditions. By looking at the INR versus
additional cancellation plot in Fig. 3(c), we can also say that the proposed
method is suitable where the feedthrough is at least 25 dB above the noise
floor.

Algorithm comparison study shows that, even though GAL and SOP were able to
provide at least 30 and 40 dB cancellation, respectively, in regard to the
execution times, they were significantly slower than the proposed algorithms
LUT and LUTit. In cases where near real-time operation is desired, or scan
time can be an issue (e.g. imaging), both LUT and LUTit can be used without
sacrificing much additional isolation.

In the case that a calibration-free algorithm is needed, SOP is more suitable
for the task rather than GAL, as SOP provides both higher additional
cancellation and lower execution time than GAL. It should be noted that the
parameters of SOP and GAL such as the population size and initial population
range for GAL were set so that they would work well for different scenarios.
More careful choice of these parameters can increase the performance of these
algorithms.

By heating experiments in Fig. 5, we have shown that how direct feedthrough
drifts over time, and how our proposed method can keep the feedthrough under
the threshold. Moreover, in this case, we did not observe any improvement over
LUT when we used LUTit. This is expected, since with heating, we expect the
feedthrough to change quickly over time. Hence, in the case that feedthrough
is expected to vary quickly, LUT can be preferred over LUTit.

The algorithm runtime was dependent on the acquisition time. On average, it
was measured as 100 ms and 330 ms for LUT and LUTit, respectively when the
acquisition time for AC was 10 ms and, 480 ms and 1.4 s for LUT and LUTit
respectively when the acquisition time for AC was 50 ms. The execution times
are bottle-necked by the four acquisitions that are used to measure the
feedthrough, system gain, coarse and fine compensation results. The execution
time can be reduced by reducing the overhead associated with these
acquisitions.

In this paper, we only focused on the direct feedthrough cancellation at a
single frequency, i.e., the fundamental harmonic. However, any source of
nonlinearity in the transmit chain can also lead to feedthrough at the higher
harmonics, especially if sufficient passive cancellation can not be achieved
by the gradiometer receive coil. Although the proposed AC system can slightly
decrease the feedthrough at the harmonics ($\sim 6\text{ dB}$), it may not be
sufficient if the feedthrough at the higher harmonics is substantial. For such
cases, increasing the bandwidth can increase the practicality of the proposed
device significantly. The main reason behind the narrow bandwidth is the
disparity between the phase responses of the vector modulator and the
feedthrough signal. One possible solution to this issue is the addition of a
phase shifter after the receive coil in order to control the phase response of
the received signal. However, this circuit would need to be ultra-low noise to
not degrade the SNR of the received signal. Furthermore, the warped phase
response of the particle signal due to the phase shifter would need to be
corrected during postprocessing.

Although not investigated in this work, another issue that needs to be
considered for the future work is the added noise coming from the vector
modulator itself. For this experiment configuration, slight ($\sim 0.5\text{
dB}$) increase of the noise floor was observed, which was deemed negligible.
However, if a system with higher sensitivity is used (e.g., an LNA with lower
noise and higher gain, or less noise coming from the receive coil), the vector
modulator may become the dominant source of noise. Fortunately, noise from the
vector modulator can be easily reduced by employing lower noise Op-Amps, or by
designing the circuit in a noise-optimized fashion. Also, addition of an LNA
before the vector modulator can readily solve the noise problem.

Our system provides robustness to time-varying changes on the direct
feedthrough, especially if they are coming from the transmit side, as we sense
the compensation signal from the transmit coil. However, our system is still
sensitive to the changes occurring at the receive side, more prominently at
the gradiometer. We have observed that, mechanical stability of the connecting
cables and coils are essential to sustain the low feedthrough level. The
necessity of removing the particle to measure the feedthrough level presents a
challenge, as there is no straightforward way of observing whether the
feedthrough increased meanwhile. Moreover, if the operator is not careful,
inserting the sample into the coil is observed to increase the feedthrough
significantly. Thus, to solve this issue, the coil and the connector cables
need to be stabilized properly. Furthermore, a method to measure the
feedthrough even if the particle is present in the coil could increase the
practicality of the system significantly, especially for the longer scans.

In the future, phase shifters, or weighted summers can be replaced with their
passive counterparts to reduce circuit complexity, and achieve better
linearity. However, in this case, impedance of these passive filters needs to
be taken care of. Usage of Op-Amp based filters allowed us to design the
filter without considering the impedance matching and power losses.

To increase the additional cancellation, the number of bits in the rheostats
can be increased, or more branches can be added. Adding more branches would
increase the degree-of-freedom when covering the constellation, however, it
would also increase the complexity. Furthermore, the proposed LUT-based
algorithms would need to be modified, as they currently assume that the
constellation is covered in two opposing parts.

A better way of increasing the cancellation performance would be optimizing
the circuit for a specific coil. Especially, if the system will be applied to
a fixed setup where the coils are not expected to move, such as an MPI
scanner, we expect the feedthrough to wander in a small section of the
constellation. Hence, the rheostat values and the capacitance values of the
phase shifters can be chosen so that, the relevant section of the
constellation is more densely sampled. This can also allow us to decrease the
number of bits to achieve the same additional cancellation performance.

The search that performed after LUT-based algorithm completes is currently a
simple search that does not utilize the phase and amplitude information of the
residual feedthrough. If a better search is implemented, that only iterates
towards to minimize the residual feedthrough could dramatically decrease the
required number of iterations, hence close the gap between LUT and LUTit.

The proposed algorithm can be further improved, especially to achieve a better
cancellation performance for the lower INR cases. For example, currently,
which branch’s constellation will be used is determined by the angular
distance of the feedthrough to the mean angle of the constellation. However,
especially for the low INR cases, this method fails and decides in favor of
the wrong constellation. A support vector machine can be used to decide in
which constellation the feedthrough falls into. Furthermore, currently coarse
adjustment is done by minimizing the distance of the residual to a point where
we expect the opposite constellation to be dense. Instead, we can choose the
point where we want the coarse residual to be by also considering a density
map of the opposite constellation.

##  V Conclusion

An active compensation method utilizing a digitally controlled vector
modulator and a lookup-table-based cancellation algorithm is proposed to solve
the time-varying direct feedthrough problem in MPI. This method can reduce the
feedthrough by more than 40 dB in less than half a second, recover the
fundamental frequency of the MPI signal, and enable a significant improvement
in sensitivity.

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