The relaxation wall: experimental limits to improving MPI spatial resolution by increasing nanoparticle core size

MPI works by exploiting the nonlinear magnetization of the SPIO tracers used. At any given time during a scan, strong magnetic field gradients magnetically saturate the SPIO tracers in most of the imaging field-of-view (FOV), leaving only tracers near a field-free-region (FFR) unsaturated. Only these tracers are able to respond to a superposed AC driving magnetic field, typically at 20 kHz. As a result, the detected MPI signal from an inductive pick-up coil is dominated by the tracers at the FFR, thereby localizing the signal to the FFR and forming the basis of spatial encoding in MPI. Following the x-space approach, image reconstruction can be performed by gridding the MPI signal to the instantaneous location of the FFR (Goodwill and Conolly 2010). Because the SPIO magnetization curve is nonlinear, multiple harmonics of the drive frequency are generated with the application of the drive waveform. Reconstruction can also be performed in the frequency domain by obtaining a system function and then solving the inverse problem of the spectral image data (Rahmer et al 2009). Both methods have to discard signal at the fundamental frequency as a consequence of direct feedthrough from the excitation waveform and reconstruct from the higher harmonics. To scan the full FOV, strong electromagnets are used to slowly raster the FFR across the entire FOV. Hardware details of the MPI scanners such as the Philips-Bruker 3D Fast MPI demonstrator scanner (Rahmer et al 2015) and the Berkeley x-space scanners (Goodwill et al 2012b, 2012c) have been discussed in prior work.

Spatial resolution in MPI is closely linked to the shape of the M–H magnetization curve of the SPIO. A steeper M–H curve results in a narrower x-space PSF. This also implies stronger higher harmonics leading to a better modulation transfer function for frequency domain reconstruction. This results in a more well-posed inverse problem leading to better SNR and less artifacts in the reconstructed image and therefore improved effective resolution. For an ensemble of ideal superparamagnetic nanoparticles, each with magnetic moment m, the M–H curve can be described by the Langevin function as follows:

$$\begin{eqnarray}&&M(H)={Nm}\,{ \mathcal L }({k}_{\mathrm{sat}}H),\end{eqnarray} \tag{ 1 }$$

where ${ \mathcal L }(x)$ is the Langevin function which is analytically defined as ${\rm{\coth }}(x)-1/x$. The constant ${k}_{\mathrm{sat}}$ is determined by the magnetic properties of the nanoparticle and defined as:

$$\begin{eqnarray}&&{k}_{\mathrm{sat}}=\displaystyle \frac{{\mu }_{0}m}{{k}_{{\rm{B}}}T}.\end{eqnarray} \tag{ 2 }$$

The steepness of the M–H curve is closely linked to m which is defined as:

$$\begin{eqnarray}&&m=\displaystyle \frac{{M}_{\mathrm{sat}}\pi {d}^{3}}{6}.\end{eqnarray} \tag{ 3 }$$

The magnetic core diameter d has a profound impact on the M–H curve and, correspondingly, spatial resolution in MPI. Previous studies on SPIO sizes from 13 to 19 nm have shown improved MPI resolution with increasing core size to 19 nm (Ferguson et al 2009, 2010). Similarly, studies on the Feraspin series (size-fractionated Resovist) show improved MPI spectra with increased size-fraction (Ludwig et al 2012).

Rahmer et al (2009) showed that MPI spatial resolution should be inversely proportional to the cube of the magnetic core diameter using the system matrix MPI reconstruction method. Knopp et al (2011) also showed the width of the convolution kernel highly depends on the particle size. Precisely the same resolution limit was derived for the x-space MPI reconstruction method (Goodwill and Conolly 2010). For x-space reconstruction, the point spread function (PSF) that defines the spatial resolution of MPI is simply the derivative of the M–H curve. One can analytically solve for the full-width-half-maximum (FWHM), which is the standard Houston resolution criterion (Houston 1927). If the effects of magnetic relaxation are neglected, the 1D spatial resolution of MPI, for both system matrix (Rahmer et al 2009) and x-space MPI reconstruction (Goodwill and Conolly 2010), is

$$\begin{eqnarray}&&{\rm{\Delta }}x=\left(\displaystyle \frac{24\,{k}_{{\rm{B}}}\,T}{{\mu }_{0}\,\pi \,{M}_{\mathrm{sat}}}\right)\displaystyle \frac{1}{{{Gd}}^{3}},\end{eqnarray} \tag{ 4 }$$

where G is the gradient strength in T/(${\mu }_{0}$m). This equation shows that MPI spatial resolution should improve cubically with increasing magnetic core diameter. Because resolution only scales linearly with gradient strength G, we are limited by cost, power and cooling constraints when improving MPI spatial resolution with stronger gradients (above 5 T/(${\mu }_{0}$m)). The relationship of spatial resolution to core diameter and gradient strength is shown in figure 1. The gradient strength is also constrained by human safety limits, magnetostimulation and SAR limitations (Saritas et al 2013b). Hence, it is enormously important to investigate the limits of the resolution improvement by increasing magnetic core diameter, and this will be even more important for scaling up to a human MPI scanners.

There have been numerous works on modeling the MPI nanoparticle response (García-Palacios et al 1998, Weizenecker et al 2010, 2012, Graeser et al 2015). Weizenecker et al modeled the effect of anisotropy and frequency on the MPI performance for 20, 25, 30 nm particles. The study noted that high levels of anisotropy reduced the MPI performance to worse than the Langevin model prediction. Graeser et al extended the Weizenecker model for 30 nm particles and showed how proper superimposition of shape semi-axis and the crystal axes during particle synthesis can give optimal MPI performance. Other research groups (Croft et al 2012, 2016, Deissler et al 2014, Deissler and Martens 2015, Dieckhoff et al 2016, Dhavalikar et al 2016) have also modeled how MPI spatial resolution varies with NP size, and with drive field frequency and amplitude. In general, all these studies have noted that larger SPIOs have longer magnetic relaxation times. While there are a multitude of models concerning the exact physical causes of magnetic relaxation, from an imaging perspective we are most concerned about time-delays and spreading of the MPI signal that will directly impact the reconstructed image. As such, we will define magnetic relaxation using the most general definition: the non-instantaneous response of the ensemble magnetization to the applied field (Croft et al 2012). The general Néel and Brownian time constants involved in magnetic relaxation can be written as follows:

$$\begin{eqnarray}{\tau }_{{N}} & = & {N}(H)\cdot {\tau }_{0}\,\exp \left(\displaystyle \frac{{{KV}}_{{\rm{c}}}}{{k}_{{\rm{B}}}T}\right),\\ {\tau }_{B} & = & B(H)\cdot \displaystyle \frac{3\eta {V}_{{\rm{h}}}}{{k}_{{\rm{B}}}T},\end{eqnarray} \tag{ 5 }$$

where N(H) and B(H) are functions that account for the effect of applied field, H, on the time constants.

These equations show that both relaxation mechanisms have longer time constants when the nanoparticle grows larger. When these relaxation times approach the period of the excitation waveform, a noticeable blurring effect results that reduces achievable resolution. Hence, larger NPs should improve resolution, but relaxation-induced blurring of larger NPs may obviate the improvements in spatial resolution. Here, we expand upon prior modeling work (listed above) by performing the first experimental study (see figure 2 for MPI hardware used) on the MPI imaging performance of nanoparticles across a wide range of core sizes from 18 to 32 nm. By observing these two opposing effects in practice, we determine the practical limits to improving MPI spatial resolution by increasing core size.

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Imagion Biosystems PrecisionMRX^® SPIO nanoparticles (Imagion Biosystems, Inc. Albuquerque, NM, USA) with carboxylic acid coated outer shell and varying core diameters were used. The core is single crystalline magnetite (${{\rm{Fe}}}_{3}{{\rm{O}}}_{4}$) for all core sizes. This is experimentally confirmed by prior work (Vreeland et al 2015) with high resolution transmission electron microscopy and x-ray diffraction. A representative data sample is found in the supplementary information of this reference. The nanoparticle shape and hence the shape anisotropy remains approximately uniform and does not change significantly with core size. This is evidenced by the TEM (JEOL 1200EX) images in figure 3(a) showing uniform spherical shape across all core sizes. The high isoperimetric quotients tabulated in figure 3(b) show that the deviation from an ideal sphere with increasing core size is very low.

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To determine the particle core size distribution, synthesized nanoparticles were analyzed using small angle x-ray scattering (SAXS) with a Rigaku SmartLab diffractometer system and SmartLab Guidance system control software. Data analysis was performed using Rigaku NANO-Solver v3.5 software that uses a spherical particle shape with a Gaussian size distribution. The spherical shape assumption was experimentally confirmed by TEM analysis (see figure 3(b)). Statistically, SAXS is a more rigorous technique for measuring size distribution because it is possible to measure a large ensemble of particles with SAXS while TEM image analysis is practically limited to 100–1000 particles.

All SPIOs have same outer coating material and differ only in the magnetic core size. The core is covered by an inner monolayer of oleic acid coating and then encapsulated within a monolayer of a carboxylic acid functionalized amphiphilic polymer. The hydrodynamic diameter and zeta potential of all SPIOs was measured in deionized water pH 6.0 using a Malvern Zetasizer Nano ZS system. The zeta potentials of all SPIOs ranged between −40 and −50 mV. Hydrodynamic diameter scales with core size as described in figure 3(a). This scaling is necessary to maintain a large enough coating buffer to prevent inter-particle aggregation.

Magnetization measurements were collected using a Quantum Design MPMS-7 SQUID magnetometer. The precise mass of iron in the sample was determined using a colorimetric assay as described previously (Vreeland et al 2015). Magnetization curves were recorded from −4000 to +4000 kA m⁻¹ at 293 K. The field-dependent magnetization data was normalized to the mass of iron in the sample to determine the saturation magnetization (${\sigma }_{{\rm{sat}}}$). The measured saturation magnetization is ${M}_{{\rm{sat}}}=0.551\,{\rm{T}}/{\mu }_{0}$ which is $\sim 92 \%$ of the bulk magnetite value of $0.6\,{\rm{T}}/{\mu }_{0}$ (Cullity and Graham 2011). A representative M versus H magnetization plot (25.0 nm particles) is also plotted in figure 3(b). The exact synthesis and more characterization of the nanoparticles can be found in detail in the main article and supplementary information of (Vreeland et al 2015). The MPI performance of the SPIOs was also measured using a magnetic particle spectrometer/relaxometer described in (Tay et al 2016) as well as the x-space MPI scanner described in (Croft et al 2016). Unless otherwise stated, all experimental measurements employed $25\,\mu {\rm{l}}$ SPIOs at 5 mg Fe ml⁻¹.

The SPIOs of different core sizes were first tested on our arbitrary waveform relaxometer (AWR) which is described in detail in our prior work (Tay et al 2015, 2016). The AWR includes a non-resonant transmit coil design to enable frequency agility as well as arbitrary excitation waveforms This table-top system does not have gradient fields for signal localization; instead, a sinusoidal excitation and linear bias field are superimposed to test the aggregate response of a sample in the applied magnetic field space. This system can reconstruct a 1D point-spread function (PSF) characteristic of the entire sample supplied to the device. To interrogate the entire Langevin M–H curve and even up to the saturation regions of the SPIOs, we scan with the background (bias) field slowly decreasing from 60 to −60 mT across 0.2 s. We have demonstrated the AWR measured 1D PSFs agree with 1D PSFs obtained using our gradient-based imaging systems  (Tay et al 2016). To translate the AWR PSF to a true imaging PSF, we only need to divide the DC field by the gradient strength. Reconstruction uses the middle 60% of the time-domain signal within a half-period, centered about the sinusoidal zero-crossing in order to maximize SNR. The 60% value is chosen to be similar to that used in the Berkeley 3D Magnetic Particle Imager. For figure 4, the drive waveform is a 20.25 kHz, 20 mT sine wave to match the drive field of the 3D imaging scanner used. For figure 7, the drive frequency and amplitude are as specified in the figure.

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To verify the findings from the relaxometer experiments, we imaged a point source phantom by lining up 5 tubes (ID 0.7 mm, OD 1.3 mm) with $1\,\mu {\rm{l}}$ of 5 mg ml⁻¹ SPIO inside each as shown in figure 5. We used our 3D MPI scanner with a 7 T/(${\mu }_{0}$m) FFP selection field, comprised of permanent magnet pair. An MPI image is formed by rastering the sensitive field-free-point (FFP) across the entire FOV and gridding the received voltage in an inductive pick-up coil to the instantaneous FFP position. The FFP is shifted by electromagnetic shift coils in the x and y directions and by the drive coil in z. A robot arm moves the sample in discrete steps in z to compensate for limited amplitude of the transmit coil. The scanning setup is shown in figure 2(c), where the green trajectory represents the FFP location in a y-plane. A homogeneous transmit solenoid adds a 20.25 kHz, 20 mT sine wave excitation along the z-axis, thus making the FFP zigzag (in the z-axis) as it moves along the green trajectory. This is done to sample the regions between the green trajectory lines. The magnetic field gradient is 7 T/(${\mu }_{0}$m) in the x-axis, and 3.5 T/(${\mu }_{0}$m) in the y- and z-axes. The field of view is $14.56\times 4\times 3.75\,{\mathrm{cm}}^{3}$ and the total scan time is 9 min. Details on the 3D MPI scanner hardware and the image reconstruction algorithm have been published  (Goodwill et al 2012a, Lu et al 2013, Saritas et al 2013a).

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In this paper, we use the Houston criterion for spatial resolution (Houston 1927) that uses the FWHM of the PSF or imaging point source. The Langevin spatial resolution (red) curves plotted in figures 4 and 7 was calculated using the equation (4) (Langevin model of an ensemble of ideal magnetic nanoparticles) where the gradient strength G is 3.5 T/(${\mu }_{0}$m) to match the Berkeley Magnetic Particle Imager gradient. We use the experimentally measured nanoparticle ${\mu }_{0}{M}_{\mathrm{sat}}$ value of $0.551\,{\rm{T}}$ for these calculations. T is set as 293 K.

The AWR was used to obtain 1D PSFs of Imagion Biosystems PrecisionMRX^® nanoparticles with core diameters between 18 and 32 nm. The results in figure 3(a) show that while the PSF narrows when core diameter increases from 18 to 24 nm, this trend reverses after 25 nm. This is concomitant with an increasing displacement in the direction of the scan. This behavior is similar to results from prior work (Croft et al 2012, 2016) and is indicative of delay between the time-domain MPI signal and the applied field due to increasing magnetic relaxation. This results in blurring of the 1D PSF in the direction of the scan when the time-domain MPI signal is gridded to applied field values during reconstruction. As a result, the improved spatial resolution predicted by the steady-state Langevin PSF is not realized. The steady-state Langevin theory assumes the SPIOs instantaneously respond to the applied field. Figure 4(b) shows an increasing disparity between the Langevin model and the experimentally measured spatial resolution as core size increases. For example, the Langevin model predicts that 700 μm resolution is achievable with 32 nm SPIOs, while the experimental resolution achieved was around 7 mm, assuming a 3.5 T/(${\mu }_{0}$m) gradient.

We confirmed our relaxometer measurement results with an imaging scan using the Berkeley MPI scanner. The images are shown in figure 5. The same trend of worsening resolution beyond 24.4 nm is observed and measurement of the FWHM from a line plot (blue dashed line) through the image are in good agreement with the AWR results. The 32.1 nm particles are not visible because the large relaxation time constants leads to very low induced signal in the MPI receive coil (MPI signal $\propto \,\tfrac{{\rm{d}}{M}}{{\rm{d}}{t}}$). 32.1 nm particles are visible in the relaxometer because a much larger volume ($25\,\mu {\rm{l}}$) is used.

To verify that magnetic relaxation is the mechanism behind the limiting of spatial resolution as core size increases, the raw time-domain signal across a half-period of drive field was investigated. Any magnetic relaxation will cause a delay of the time-domain signal in the direction of the scan. In figure 6(a), while an increase in magnetic relaxation can be inferred from the slight delay of the peak going from 18.5 to 24.4 nm, the signal peak has an overall narrowing leading to overall better resolution. This suggests that the effect of Langevin physics dominates over magnetic relaxation in this range of core sizes.

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From 24.4 to 32.1 nm, however, there is a dramatic increase in peak delay (figure 6(b)). The signal is delayed by $\sim 6\,\mu {\rm{s}}$ and is spread out across almost twice the time as 24.4 nm. These results show that magnetic relaxation is dramatically more significant beyond 24.4 nm and clearly dominates over Langevin physics.

The raw MPI time-domain data shown here also provides an intuitive explanation of worsening resolution. Since the signal is spread out over more time for 27.4 and 32.1 nm, the signal is associated with more voxels in x-space reconstruction because signal(t) is directly gridded to the voxels that the FFP bypassed in that time. Similarly, for system matrix reconstruction, a spread-out signal peak corresponds to a steeper decay in the magnetic particle spectrometry spectra and thus a more poorly conditioned system matrix during image reconstruction.

Prior work has demonstrated that the MPI drive waveform significantly affects the MPI performance of SPIOs (Kuhlmann et al 2015, Croft et al 2016). Here, we investigate if low amplitude and low frequency approaches can help mitigate magnetic relaxation and achieve the Langevin model spatial resolution. For the low amplitude approach, the frequency was kept constant at the original 20.25 kHz while amplitudes of 20.0, 4.0 and 0.5 mT/${\mu }_{0}$ were used. For the low frequency approach, the amplitude was maintained at the original 20 mT/${\mu }_{0}$ while frequencies of 20.25, 2.0 and 0.4 kHz were used.

The results show that low amplitude approaches do not work well at larger core sizes. This can be attributed to the strong field strength dependence of Néel and Brownian time constants as described in equation (5) and  (Deissler et al 2014, Dieckhoff et al 2016), where the smaller AC drive amplitudes mean that the time constants remain relatively large throughout the entire AC period. In contrast, low frequency approaches work better at larger core sizes. The spatial resolution minima of the curve is shifted to 27.4 nm at 0.4 kHz and there is significantly improved spatial resolution by almost 2-fold from the optimal resolution at 20 kHz curve. This can be attributed to the fact that the temporal blurring has a lower impact on the time-domain raw MPI signal shape when the drive waveform period is longer, which in turn results in less blurring during image reconstruction as shown by Croft et al (2016). Overall, while both approaches help mitigate magnetic relaxation and improve spatial resolution, the general trend of worsening spatial resolution beyond 27.4 nm still remains. Notably, the fact that low frequency approaches work well at core sizes past the minima of the 20 kHz curve while low amplitude approaches do not (relaxation time constants are longer at low fields) lends further credence to the idea that increasing magnetic relaxation from larger core sizes is the main cause of worsening spatial resolution. In essence, these results suggest that magnetic relaxation is a significant spatial resolution barrier that is not easily resolved by current approaches.