Optimization of Drive Parameters for Resolution, Sensitivity and Safety in Magnetic Particle Imaging

Abstract

Magnetic Particle Imaging is an emerging tracer imaging modality with zero background signal and zero ionizing radiation, high contrast and high sensitivity with quantitative images. While there is recent work showing that the low amplitude or low frequency drive parameters can improve MPI’s spatial resolution by mitigating relaxation losses, the concomitant decrease of the MPI’s tracer sensitivity due to the lower drive slew rates was not fully addressed. There has yet to be a wide parameter space, multi-objective optimization of MPI drive parameters for high resolution, high sensitivity and safety. In a large-scale study, we experimentally test 5 different nanoparticles ranging from multi to single-core across 18.5 nm to 32.1 nm core sizes and across an expansive drive parameter range of 0.4 – 416 kHz and 0.5 – 40 mT/μ₀ to assess spatial resolution, SNR, and safety. In addition, we analyze how drive-parameter-dependent shifts in harmonic signal energy away and towards the discarded first harmonic affect effective SNR in this optimization study. The results show that when optimizing for all four factors of resolution, SNR, discarded-harmonic-energy and safety, the overall trends are no longer monotonic and clear optimal points emerge. We present drive parameters different from conventional preclinical MPI showing ~ 2-fold improvement in spatial resolution while remaining within safety limits and addressing sensitivity by minimizing the typical SNR loss involved. Finally, validation of the optimization results with 2D images of phantoms was performed.

I. Introduction

MAGNETIC Particle Imaging is an emerging tracer imaging modality [1], [2] characterized by nanomolar tracer sensitivity [3], [4], absolute linear quantitation with respect to the tracer [5], depth-independent resolution and SNR, and the use of safe, non-ionizing superparamagnetic iron oxide (SPIO) tracers [6], [7]. MPI has utility for cardiovascular imaging [8] and real-time surgical instrument steering [9]. Other important applications include stem cell tracking [5], cancer imaging [10], acute gut bleed detection [11],lung imaging [12], [13], stroke and brain imaging [14], [15], and image-guided hyperthermia-therapy [16], [17].

Current preclinical MPI scanners uses drive parameters of 25 kHz at 14 mT/μ₀ amplitude (Philips-Bruker preclinical demonstrator system) [18], 20 kHz at 20 mT/μ₀ amplitude (University of California, Berkeley scanner) [10] and recently 45 kHz at 20 mT/μ₀ (Magnetic Insight preclinical system) [19]. While recent work has shown that low amplitude [20], [21] or low frequency drive parameters [22], [23] can improve MPI’s spatial resolution by mitigating relaxation effects, there is a significant concomitant drop in sensitivity due to the slower magnetic slew rates that was not fully addressed. Other work has looked at optimizing the MPI drive parameters but focusing mainly on safety [24], [25]. Studies involving a wide range of nanoparticles [26] have used only a single drive parameter (fixed f₀ and amplitude).

In this work, we present a wide parameter space, multi-objective optimization of MPI drive parameters for all three metrics of high resolution, high sensitivity and safety. Towards this goal, we expand upon prior work by experimentally optimizing MPI drive parameters across a parameter range of 0.4 – 416 kHz and 0.5 – 40 mT/μ₀. We also utilize a wide range of nanoparticle core sizes (18 nm to 32 nm) and types. Finally, 2D FFL-projection imaging data confirms that our novel optimum waveform shows improved performance. This optimization study will be relevant to MPI systems designers since a fixed operating drive frequency and amplitude needs to be selected before designing the MPI transmit and receive hardware.

II. Theory

The theory of MPI has been previously detailed [1], [27], [28], and briefly, MPI works by constructing a sensitive low-field region surrounding a null point in the middle of a strong (> 1 T/m) magnetic field gradient. The null point is rastered around the field-of-view, producing nonlinear magnetization response in superparamagnetic iron oxide (SPIO) when it passes through.

A. Effect of drive parameters on Spatial Resolution

Spatial resolution in MPI depends on the nanoparticle size [28], [23] as well as relaxation-induced blurring. SPIOs may not have an instantaneous magnetization response to the MPI drive waveform and the resultant temporal delays (ensemble magnetization relaxation) spreads the temporal response of the SPIO causing blurring. This also leads to poorer condition numbers during deconvolution, increasing noise when producing a good resolution image. Lower amplitudes have been shown to reduce blurring by limiting the reconstruction spatial mapping blur [21] as follows:

$$\Delta x\approx \alpha \Delta x_{base}+\beta \Delta x_{blur}$$ (1)

where the blurring term, Δx_blur has been shown [21] to be a strong function of both the phase lag of the signal, ϕ as well as the drive field amplitude H_amp, leading to the trend that lower drive field amplitudes improve spatial resolution (see equation below). G is the gradient field strength in T/m.

$$\Delta x_{blur}\approx \frac{ln(2)\varphi (H_{amp},f_0)H_{amp}}{G}$$ (2)

These equations show that the blurring term can be reduced by reducing the phase lag term (indirectly via drive frequency or amplitude) or directly by drive amplitude. Lower drive amplitudes decrease the H_amp term, directly limiting the worse-case extent of x-space spatial mapping blur [21]. Lower drive frequencies (f₀) decrease phase lag ϕ because the time delays caused by magnetic relaxation are now smaller relative to the longer drive period. In general, lowering the magnetic drive slew rate by amplitude or frequency improves spatial resolution (Fig. 1b). Notably, ϕ is not a linear function of H_amp or f₀ [29], which adds complexity considering that increasing H_amp and f₀ improve MPI sensitivity.

Fig. 1. Experimental data demonstrating the general effect of the MPI drive parameter slew rate on sensitivity and spatial resolution.

(a) Because MPI uses inductive pick-up coils, the MPI signal is proportional to drive parameter slew rate. However, we do not observe a perfectly linear gain in sensitivity as the ensemble magnetization of the SPIOs lags behind the drive parameter slew rate due to (ensemble) magnetic relaxation. (b) Magnetic relaxation at high drive parameter slew rates leads to blurring of the MPI signal as shown by the broadening of the point-spread-function (PSF). Thus, low slew rates are better for resolution but high slew rates are better for SNR and sensitivity, resulting in a SNR/resolution trade-off that can be optimized.

B. Effect of drive parameters on the Signal-to-Noise Ratio

MPI uses inductive pick-up thus the signal is directly proportional to the rate of change of magnetic flux [1], [28] set by the relaxation-modulated magnetization rate of the SPIO tracer:

$$\begin{gathered} s(t)=B_1\frac{d{\Phi }_B}{dt} \\ s(t)=B_{1}m\rho (x)\ast d\mathcal{L}[kGx(t)]∕dt\ast r(t,H_{amp},f_0) \\ s(t)=B_{1}m\rho (x)\ast d\mathcal{L}[kH]∕dH\cdot dH∕dt\ast r(t,H_{amp},f_0) \\ {s}_{max}\propto d\mathcal{L}[kH]∕dH{\mid }_{H=0}\cdot 2\pi H_{amp}{f}_0\cdot R_{particle}(H_{amp},f_0) \end{gathered}$$ (3)

where Φ_B is magnetic flux, B₁ is the receiver coil sensitivity, ρ(x) is the spatial distribution of the magnetic nanoparticles, $\mathcal{L}$ is the langevin function, k is a nanoparticle specific constant (k = μ₀m/k_BT), G is the magnetic gradient in T/m. r(t) is a relaxation convolution kernel of the specific nanoparticle used, for instance, the Debye model where r(t) = (1/τ)exp(−t/τ)u(t), and where u(t) is the Heaviside step function. R_particle is a particle-specific function taking values between 0 and 1, that attenuates the maximum MPI peak signal s_max as a function of H_amp, f₀. In the ideal case when relaxation effects are neglected, R_particle = 1 and the peak signal strength is directly proportional to both drive frequency f₀ and amplitude H_amp. In practice due to relaxation effects, R_particle is less than unity and could be worsened by changes in f₀ and H_amp. Thus, MPI signal is linear with drive frequency initially, but starts to deviate at high drive frequencies due to the increasing impact of relaxation effects (Fig. 1a).

Regarding noise in MPI, while initial MPI hardware has similar coil and preamp noise contributions, recent work has focused on noise matching to achieve coil noise dominance [30], [28], [31]. Our analysis thus proceeds assuming (1) coil noise dominance, (2) AC resistance and thermal noise is not frequency-dependent due to the use of appropriate litz wire, and (3) a fixed number (N_harm) of signal harmonics are acquired. Thus, we have:

$$\begin{gathered} <n>\approx \sqrt{(e_n^2+4k_B{T}_{coil}{R}_{coil})BW} \\ \approx \sqrt{(e_n^2+4k_B{T}_{coil}{R}_{coil})N_{harm}{\Delta }_f} \end{gathered}$$ (4)

where e_n is the pre-amplifier total noise contribution in $V∕\sqrt{Hz}$ and BW is the signal bandwidth. Δf is the small, fixed and equal bandwidth that is kept around each harmonic. Typically, the leftover regions between harmonics are removed in post-processing as they do not have signal and contribute only noise [21]. After this digital comb filtering, the total bandwidth is a constant N_harmΔf and is thus f₀ invariant.

With the above assumptions, from Eqn. 4, we observe that noise is approximately invariant with drive frequency and amplitude, and thus the SNR of MPI is proportional to the peak signal:

$$\begin{gathered} {\text{SNR}}_{\max }\approx \frac{s_{max}}{<n_{pp}>} \\ \propto d\mathcal{L}[kH]∕dH{\mid }_{H=0}\cdot 2\pi H_{amp}{f}_0\cdot R_{particle}(H_{amp},f_0) \end{gathered}$$ (5)

This implies that for the purposes of this paper, sensitivity S, defined as the peak signal in V per g of nanoparticle (with pre-amplifier gain of 1), is a good metric to compare SNR between different nanoparticles and drive parameters.

C. Effects of drive parameter on Safety

The MPI drive waveform can cause magnetostimulation and tissue heating (SAR) problems and thus f₀, H_amp must be kept within human-safe limits [25], which are thus set as optimization constraints. Fig. 3 illustrates this process.

Fig. 3. Surface plots illustrating the process for multi-variate optimization of the MPI drive parameter based on experimental data.

The data is obtained from our previously validated device [32] – the arbitrary waveform relaxometer. (a) Resolution surface plot. (b) Sensitivity surface plot, showing opposing trend to resolution. (c) The objective function plot is a weighted sum of both resolution and SNR surface plots. The relative weighting of resolution versus SNR is detailed in the Methods III.E. (d) Drive parameters that violate safety limits for humans are set to zero. The result is clear maxima (optimal point) for MPI performance accounting for all 3 metrics of resolution, sensitivity and safety. The red point indicates the optimal point for Vivotrax™.

III. Methods

A. Berkeley Arbitrary Waveform Relaxometer

The Berkeley Arbitrary Waveform Relaxometer measures the point-spread-function characteristic of MPI tracers and it was validated in prior work [32], [23]. The untuned nature of the device enables the use of drive parameters between DC – 400 kHz with amplitudes up to 40 mT/μ₀ and even square (pulsed) drive waveforms [33]. We exploit the frequency-flexibility of this device to optimize the MPI drive parameter across a wide parameter space that nearly covers the entire space within SAR and magnetostimulation safety limits (see Fig. 4c). Like other MPI spectrometers and relaxometers, this device does not utilize magnetic gradients and thus only measures the dynamic magnetization characteristics of the MPI tracer. However, the point-spread-function (PSF) characteristic can predict for imaging performance such as sensitivity and resolution. Prior work has confirmed this by showing that the resolution predicted by the PSF for this device correlates well with the actual resolution achieved on the 3D MPI scanner [32]. Although addition of a 3.5 T/m gradient is not technically necessary to obtain the resolution measurements in this optimization study, addition of the gradient (Fig. 4b) enables acquisition of representative show-shot images using the new drive parameters.

Fig. 4. MPI hardware used in this study.

(a) Photo of the Tabletop MPI scanner. (b) 3D rendering of the Tabletop MPI scanner. This scanner is derived from the Arbitrary Waveform Relaxometer described in previous work [32] with modifications such as permanent magnets to generate a 3.5 T/m FFL and selection field coils to enable small FOV imaging. (c) Because the same drive coil and drive transmit chain was maintained, the modified device retains the frequency flexibility enabling coverage of most of the drive parameters within human safety limits. This allows our study to cover an expansive parameter range of 0.4 – 416 kHz and 0.5 – 40 mT/μ₀. The accuracy of resolution measurements from our tabletop device was validated in prior work [32].

B. Magnetic Nanoparticles

Imagion Biosystems PrecisionMRX SPIO nanoparticles (Imagion Biosystems, Inc. Albuquerque, NM,USA) with carboxylic acid coated outer shell and varying core diameters were used for this study. The core diameters used are 18.5 nm, 24.4 nm, 27.4 nm and 32.1 nm where the former two are classified as small core tracers and the latter two are classified as large core tracers. Details regarding the performance and characterization of these nanoparticles are detailed in prior work [23]. In addition, dextran-coated Ferucarbotran multicore clustered SPIOs (Vivotrax™, Magnetic Insight Inc., Alameda, CA, USA) were used due to its similar structure and performance to Resovist™, a gold-standard for conventional MPI. The physical individual core size is 4 nm [1] but the multiple cores act together as if it is a 15 – 16 nm single-core particle [34]. For all AWR 1D measurements, 30μl of SPIOs at 5 mg/ml was used (0.150 mg Fe total sample).

C. Calculation of Resolution

We use the Houston criterion for spatial resolution in this study. The metric used is the full-width-half-maximum (FWHM) of a point source. Because relaxometers measure the MPI response to a varying applied field, the FWHM (in mT/μ₀) can be converted to spatial resolution (in mm) for x-space reconstruction by simply dividing by the MPI gradient to-be-used (in T/m). Here, we use 7 T/m to match the Berkeley 3D MPI scanner.

D. Calculation of Sensitivity

Sensitivity is measured by the height of the MPI signal peak from relaxometer data divided by the iron mass of the MPI tracer used to give a V/g Fe metric. Justification of the choice of this metric for sensitivity is detailed in the Theory Section II.B. The pre-amplifier gain used was accounted for so the final V/g Fe would be the actual MPI signal with a pre-amplifier gain of 1.

E. Optimization Process

The optimization process is graphically detailed in Fig. 3. The objective function plot is a weighted sum of both resolution and SNR surface plots. The relative weighting of resolution versus SNR is based on the estimated SNR cost needed to improve resolution two-fold by deconvolution approaches. Regarding this topic, prior data from Knopp et al. [27] suggested that an approximate 10-fold increase in SNR is needed to enable reconstruction with 2-fold better spatial resolution. With this estimated 2:10 equivalence value ratio, we set the objective function as z(r, s) = log2(r₀/r) + log10(s/s₀) where r₀ and s₀ are defined as the worst resolution and sensitivity respectively achieved for that nanoparticle within the entire drive parameter range used in the optimization.

Subsequently, we also assess shifts in MPI signal spectra energy distributed across the different harmonics when drive parameter is changed. As the fundamental (first harmonic) is analog-filtered and discarded, a higher amount of spectra energy fraction shifting into the discarded first harmonic will directly reduce the remaining spectra energy in higher harmonics that is usable, and thus affect the effective received signal. To calculate the dimensionless ratio of signal energy at the fundamental f₀ versus all higher harmonics, the following equation was used:

$$\frac{E_{f0}}{E_{higher}}={\int }_{f_0-\Delta f}^{f_0+\Delta f}\widehat{x}(f{)}^{2}df÷\left(\sum _{i=1}^N{\int }_{f_i-\Delta f}^{f_i+\Delta f}\widehat{x}(f{)}^{2}df\right)$$

where $\widehat{x}$(f) is the Fourier transform of the MPI time-domain signal x(t), Δf = 0.5 kHz and N = 21. Details of the harmonic energy shift accounting process are shown in Fig. 5.

Fig. 5. Changes in drive parameter affect the shape of the harmonic spectra, impacting relative SNR by changing the fraction of signal lost in the discarded first harmonic.

(a) Harmonic spectra changes for 24.4 nm SPIO as a result of drive parameter changes. (b) Computation of the relative SNR as a result of harmonic shifts. Low amplitude waveforms have a lower amount of useable signal energy after discarding the first harmonic signal, which is typically necessary during filtering of the first harmonic feedthrough. In contrast, low frequency waveforms reduces the relaxation impact, leading to less blurring and a richer harmonic spectra with more signal energy in the higher harmonics. Due to differences in use-able harmonic signal left, there is a relative SNR impact tabulated in the right column. (c,d) Objective function before and after accounting for this effect. Lower drive amplitudes are valued worse, showing a significant dip in contour shape. However, the overall optimal point does not shift, in part due to the unchanging safety boundary.

Finally, to assess the sensitivity of the objective function to variation in the resolution:sensitivity relative weighting, a 2:10 ratio, a 2:2 ratio and 2:1000 ratio was also tested and the results tabulated in Fig. 6f.

Fig. 6. Optimization results for 5 different nanoparticles.

The objective function is a weighted sum of both resolution and SNR surface plots as illustrated in Figure 3. Prior data from Knopp et al. [27] suggests that about a 10-fold increase in SNR is traded-off to reconstruct with 2-fold better spatial resolution, which we use as relative weighting of SNR vs resolution in our objective function: z(r, s) = log2(r₀/r) + log10(s/s₀) where r₀ and s₀ are defined as the worst resolution and sensitivity achieved for that nanoparticle within the entire drive parameter range used in the optimization. The relative SNR between different harmonic spectra (see previous figure) was also accounted for. (a – c) Optimization results for smaller magnetic core sizes show similar optima at ~ 5 mT and ~ 70 kHz. The 5 mT point reflects the diminishing returns of decreasing amplitude on improving spatial resolution (see Fig. 2b). The 70 kHz limit is set by SAR safety limits [25] since the optima is at the boundary. (d–e) Larger core sizes show a markedly different optima zone at ~ 14 mT and ~ 1 kHz, reflecting the high sensitivity to drive frequency due to long relaxation time constants impacting blurring and SNR due to incomplete magnetization. (f) Summary of optima for different nanoparticles. The optima are relatively robust to changes in objective function formula as significant variations only cause minor shifts in the optimal point. The base case (central column) uses z(r, s) = log2(r₀/r) + log10(s/s₀) while the 2:2 column has SNR valued equal to resolution (z(r, s) = log2(r₀/r) + log2(s/s₀), and 2:1000 is z(r, s) = log2(r₀/r) + log1000(s/s₀). These results suggest MPI systems can be designed with just two tuned operating frequencies corresponding to the [70 kHz, 5 mT] and [1 kHz, 14 mT] optima and achieve good performance even with a wide range of nanoparticle sizes.

IV. Results

A. Low Amplitude Only and Low Frequency Only Drive Parameters

The results in Fig. 2 show that in general, lower frequencies and lower amplitudes improve resolution as predicted by prior work. However, this comes at a significant cost of decreased sensitivity. This is an unfavorable trade-off with diminishing returns. For instance, for Vivotrax™, an 8-fold drop in amplitude is needed to improve resolution from approximately 1.6 mm to 0.9 mm. This shows that the strategy of decreasing solely amplitude or frequency to improve MPI performance is not desirable and underscores the importance of finding the optimum parameters so as to minimize sensitivity loss while obtaining as much resolution improvement as possible.

Fig. 2. Experimental data demonstrating that existing state-of-the-art methods [21], [22] – low drive amplitude in top panel and low drive frequency in bottom panel – improves spatial resolution at a large cost in sensitivity.

Top Panel: Experimental results from varying the drive amplitude (0.5 – 40 mT/μ₀) while keeping drive frequency constant at 20 kHz. (a) Point Spread Function (PSF) measurements. The inset is for visual comparison of peak-normalized PSF shape and width. (b) Effect on spatial resolution for 5 nanoparticle core sizes. (c) Effect on sensitivity. Bottom Panel: Experimental results from varying the drive frequency (0.5 kHz – 100 kHz) while keeping drive amplitude constant at 20 mT/μ₀. Error bars (n=3).

B. Optimal Drive Parameters for Small Core Size SPIOs

Since amplitude has a dominant effect of resolution, a possible strategy is to use low amplitudes but high frequencies so as to use the high frequencies to compensate for the sensitivity loss from the low amplitudes. Here we experimentally investigate this strategy. The results in Fig. 2b show that for Vivotrax™, a small core sPIO, spatial resolution reaches its best at around 5 mT/μ₀ with only marginal improvements as drive amplitude is dropped further. This data thus supports the resolution model shown in Eqn. 1 and Eqn. 2 [21], where drive amplitude mitigates the relaxation-induced blurring by limiting the blur to a smaller spatial area i.e. a 5 mT/μ₀ drive amplitude can be converted to a fixed length in space by dividing by the linear 7 T/m gradient. Since only the blur is mitigated while the steady-state point spread function is unchanged, the spatial resolution asymptotes to the steady-state spatial resolution as drive amplitude is decreased. Due to diminishing returns, there is no need to reduce drive amplitude to extremely low values and 5 – 8 mT/μ₀ is a good trade-off point.

While spatial resolution changes significantly with amplitude, there is a minimal change with drive frequency in the region of 20 – 100 kHz as shown in Fig. 2e, lending support to the strategy of increasing drive frequency to compensate for sensitivity loss from reducing drive amplitude. This also indicates that small magnetic core sizes are relatively frequency-insensitive within the 20 – 100 kHz region because the smaller magnetic cores (for the same magnetite material), have generally shorter Néel and Brownian relaxation constants [35] and can keep up with the driving waveform.

This low amplitude high frequency strategy was experimentally tested with a wide variety of drive parameters (Fig. 6a-c). The results show good overall MPI performance (weighted sum of both resolution and SNR) at low amplitude and high frequency parameters. After considering drive parameter safety constraints, an optimal region of 5.0 – 8.0 mT/μ₀ and 45 – 100 kHz emerges with peak value around 70 kHz and 5.0 mT/μ₀.

It must be noted that the 18.5 nm nanoparticle has a wider spread of optimal region along the safety limits border. One possible reason is that the steady-state point spread function for this smallest core particle is broader and thus the blurring mitigation asymptotes earlier at higher drive amplitudes beyond 10 mT/μ₀ rather than needing 5 mT/μ₀. As such, the optimal region extends farther due to the low variability in terms of fold-change of the spatial resolution objective relative to the SNR objective which is improved significantly at higher drive amplitudes due to the shifting of harmonic energy.

C. SNR Impact of Shifts in Harmonic Energy Into and Away from the First Harmonic

When the drive parameter is changed, the signal energy content may shift from the higher harmonics towards the lower harmonics as shown by experimental data in Fig. 5a. Because the time-domain magnetization response changes with drive parameter, the frequency-domain harmonic spectrum must change too. In the case of lowering drive amplitude, this spectral change is due to the magnetization curve going from a non-linear response towards linearity as the drive amplitude approaches the limit of zero. Current MPI scanners use an analog filter to remove the direct feedthrough at the drive frequency f₀ and therefore the SPIO signal at the first harmonic or fundamental (f₀) is also lost. Because more of the SPIO signal energy exists at f₀ at low drive amplitudes, this means that there is effectively less raw signal available after filtering (on top of the effects of Faraday’s law). Although this lost f₀ information can be recovered robustly [36], effective SNR is dependent on raw signal received and is still decreased by the analog filtering.

We account for this by considering the relative total harmonic energy not in the filtered-out first harmonic and comparing this value between drive parameters as illustrated in Fig. 5b. We adjust the original first-harmonic-included sensitivity surface plot by multiplying by the relative SNR values in Fig. 5b before computing the objective function surface plot. A before-and-after example of this process is shown in Fig. 5c and d respectively. These example plots for the experimental measurements of the 24.4 nm SPIO show shape modification of the optimization surface plot, with objective function (z-value) decreases at lower drive amplitudes and increases at higher drive amplitudes. While at first glance the optimal region still remains approximately the same, it can be noted that low frequency high amplitude drive parameters have a notable increase in objective function value. While still lower than the optimal point, high amplitude waveforms are now more viable due to the SNR impact of shifting harmonic energy towards the remaining higher harmonics. Finally, we account for this effect during the optimization process for all nanoparticles in Fig. 6.

D. Optimal Drive Parameters for Large Core Size SPIOs

In contrast to the frequency-insensitivity of small-core SPIOs, the large-core SPIOs show increased sensitivity to drive frequency, shifting the optimal region towards the lower frequencies as shown in Fig. 6d and e. This is likely due to longer relaxation time constants present in these larger cores [35] which are unable to keep up with the high drive frequencies. As such, a low amplitude high frequency strategy of using high frequencies to regain lost sensitivity quickly runs into diminishing returns on sensitivity as these larger SPIOs are unable to rotate fast enough. The achieved sensitivity quickly deviates from the expected trend as shown in Fig. 2f and thus the strategy of increasing frequency to compensate for lower drive amplitudes is shown to backfire for larger core sizes. Due to the harmonic energy shifting effect, higher drive amplitudes have an advantage in effective SNR. Thus, in contrast to smaller cores, a low frequency high amplitude strategy is optimal for larger cores (Fig. 6d-e).

E. Accounting for Incomplete Magnetization in the Spatial Resolution Model

In equation 5, the MPI signal increases with higher drive frequency or drive amplitude according to Faraday’s law for an inductive receive coil. However, depending on the relaxation time constant, slight temporal blurring may be exacerbated into incomplete magnetization when overly high drive frequencies are used.

Incomplete magnetization due to overly-high drive frequency negatively impacts MPI sensitivity through two processes: (1) the steady-state magnetization change normally elicited from the drive amplitude H_amp is not achieved resulting in less magnetization change; (2) this reduced magnetization change M is spread out temporally over the drive period as opposed to having most change at the center or zero-crossing of the drive period, resulting in an attenuated MPI inductive signal due to smaller peak dM/dt. In these cases, the sensitivity equation from Eqn. 3 can be approximated, after removing SPIO and hardware-specific constants, and expanding the function R_particle(f₀, H_amp) as follows:

$$S\propto H_{amp}{f}_0\left(1-\text{exp}\left(-\frac{1∕f_0}{\tau (H_{amp})}\right)\right)$$ (6)

where τ(H_amp) is the averaged, effective time constant over the half-period that depends on H_amp due to field-dependence of the time constant. As the period of the drive waveform becomes small enough to approach the relaxation time constant at H_amp, the MPI sensitivity drops rapidly due to incomplete magnetization of the SPIO. This shows that increasing frequency only improves MPI sensitivity provided that the drive period is still significantly larger than the relaxation τ. It is ineffective once drive frequency approaches τ (H_amp).

In addition, incomplete magnetization also increases blurring to worsen the spatial resolution. While the weighted sum approximation of Croft et al. (Eqn.1) holds true for lower drive frequencies and for SPIOs with relatively low amounts of magnetic relaxation, we have found large discrepancies for larger core SPIOs (Fig. 7b). The discrepancy cannot be fully accounted for by the phase lag term in the origina model. These resolution discrepancies result from reshaping of the point-spread-function as shown in Fig. 7a. We term this effect ”peak depression” as the blurring results from a diminished signal peak value while the shoulder regions remain unaffected. This effectively worsens resolution as the magnitude of the blurring which is mostly at the shoulders increases relative to the point signal at the peak position.

Fig. 7. Incomplete magnetization is the main cause for large shift in drive optima for larger core SPIOs. This is due to the double impact of the newly described peak depression blurring effect (details in IV.E) in addition to loss in magnetization-based signal for SNR.

(a) Experimental point spread functions (PSF) showing the different factors affecting resolution in the model (Eqn. 7). The PSFs are normalized to a peak value set by Faraday’s law. (b) The original model for spatial resolution does not match well with data for larger core size SPIOs (27.4 and 32.1 nm). (c) When we additionally account for the peak depression effect by adding γΔx_peakdepr. (calculated in Eqn. 7), the model shows much better agreement with the experimental data, especially for 27.4 and 32.1 nm. Overall, for larger core SPIOs, the double impact of loss of magnetization signal for SNR as well as ‘peak depression’ blurring for resolution results in the large sensitivity of the optimization surface plot to frequency.

To address the cause of this effect, it must be noted that there is a large variation of the relaxation time constant when the SPIO experiences near-zero versus high values of absolute applied field, H [37], with low-field time constants being several times longer than at-high-field. As a result of this field-dependent variation in time constant τ (even within a single drive waveform half-period), the received MPI point spread function is ‘reshaped’ to have less signal close to the point source location (x = 0 mm). Essentially, the MPI signal received when the field-free-point passes through the point source location at x = 0 mm is lower due to long relaxation time constants, while MPI signal received when the field-free-point moves nearby the point source location (x = 2 – 3 mm) is relatively unaffected due to short relaxation time. In other words, incomplete magnetization does not cause a uniform, scalar attenuation of the magnetization curve but rather attenuates the zero-crossing region much more than other parts of the magnetization curve. Rather than mere scaling of the original shape, the resultant particle magnetization curve is now reshaped to be gentler resulting in poorer resolution. We name this peak depression’ as the point-spread-function of the magnetization curve has only the peak height depressed as illustrated by experimental data in Fig. 7a. This peak depression effect that reduces steepness of the particle magnetization curve close to H = 0 is expected to also impact system matrix reconstruction. This is because the properties of realistic particles are introduced into the system function by a convolution-type operation leading to a blurring of the high-resolution components. This introduces a resolution limit which is determined by the steepness of the particle magnetization curve [2]. Notably, the phase lag measurement alone is unable to fully capture this effect because it does not consider magnitude changes. Thus a lagged signal that fully magnetizes would give a similar phase lag value to a lagged signal that does not fully magnetize.

To account for this, we propose to modify the Croft et al. model by adding a new ”peak depression” term to the original spatial resolution equation (Eqn.1):

$$\begin{gathered} \Delta x\approx \alpha \Delta x_{base}+\beta \Delta x_{blur}+\gamma \Delta x_{peakdepr} \\ \Delta x\approx \alpha \Delta x_{base}+\beta ln(2)\frac{{\varphi }_N{H}_{amp}}{G}+\gamma \Delta x_{base}\left(\frac{S_{ss}}{S}-1\right) \end{gathered}$$ (7)

where ϕ_N is the normalized phase lag (ϕ_N = ϕ/(0.5π) where 0.5π is set as maximal phase delay) of the magnetization as defined in Eq. 2, where γ is a new weighting factor for the incomplete magnetization term, S is the measured peak sensitivity, and S_ss is the expected steady-state peak sensitivity. α = 1.0, β = 1.67, γ = 0.05 gave the best-fit results using nonlinear least-squares regression.

The modified resolution model shows much better agreement with the experimental data as shown in Fig. 7c, suggesting that the incomplete magnetization factor is important. The combination of the following 4 factors thus well-describes MPI resolution over a large parameter space of 0.5 – 400 kHz and 0.5 – 40 mT/μ₀: (1) ideal (base) resolution defined by Langevin theory that is dependent on particle (effective) magnetic core size (2) Relaxation delays of the SPIO magnetization causing blurring (3) Blurring amplification by mapping the MPI signal to a wider region in space due to larger amplitudes / wider MPI trajectories (4) Blurring amplification caused by incomplete magnetization of the SPIO by the “peak depression” effect.

F. 2D Phantom Images of Optimal Drive Parameters

Fig. 8 shows 2D images comparing different drive parameters. The same absolute levels of Gaussian noise was added in post-processing to all images to illustrate the SNR differences between the drive parameters. The images verify that the optima at 70 kHz and 5 mT/μ₀ maintains the improved resolution of low amplitude approaches while minimizing sensitivity losses due to the compensating effect of increasing drive frequency.

Fig. 8. 2D Verification of improved MPI performance.

(a) Diagram of phantom used for experimental 2D images. (b) Measured point spread functions (PSF) of optimal drive parameters vs. standard MPI and low-amplitude-only parameters. (c) Experimental 2D images obtained on the benchtop scanner (relative arbitrary units A.U.). Low and high levels of Gaussian noise (A.U.) was added equally to all images in post-processing (σ = 0.025 and 0.25 respectively). Optimized drive parameter (70 kHz, 5 mT/μ₀) performs best overall with ~ 1.7× improved spatial resolution matching low-amplitude-only approaches, yet has minimal loss in sensitivity (≤ 20% drop) rather than the expected 4-fold drop. Deviation of standard MPI 2D image from PSF is due to blurring of the two unresolved point sources into an inflated pixel intensity. (d) Comparison of 5 nanoparticle types (error bars n=3, x-space MPI) shows optimized parameter sets consistently have the overall best performance considering both SNR and resolution.

V. Discussion

A. Optimization for all Three Key Metrics of Resolution, Sensitivity and Safety produces Clear Optima

While prior work has focused on individual metrics of MPI such as resolution [27], [21], [22], sensitivity [3], and safety [25], this study considers all three factors simultaneously in the optimization. As a result, novel optima was discovered at 5.0 – 8.0 mT/μ₀ and 50 – 100 kHz for the commonly used small-core-size SPIOs, which is significantly different from the typical 20 kHz and 20 mT/μ₀ used in most of MPI literature. The improved performance relative to conventional MPI was verified by 2D images in Fig. 8.

B. Optima are Similar across Nanoparticles and Differ Only Between the Small and Large-Core Classification

Fig. 6 shows that similar optimization results were obtained across the wide variety of nanoparticle types and sizes tested. For instance, the results for Vivotrax™, 18.5 nm, and 24.4 nm are largely similar. However, the optimization results changes dramatically when going to the large-core group of IB 27.4 nm and IB 32.1 nm, but these two have similar optima. Practically, this implies that just two optimal drive parameters – one at 70 kHz, 5 mT/μ₀ and one at 1 kHz, 14 mT/μ₀ – can enable optimal performance for a wide range of nanoparticle sizes and nanoparticles types. Because MPI imaging hardware is typically tuned to operate with a narrow range of drive frequencies, isolating just two optimal frequencies for versatile use will be valuable to inform future MPI systems design.

C. Relevance across Reconstruction Methods

While the optimization results presented are based on point spread function measurements using x-space reconstruction [28], similar results should be expected using system matrix reconstruction. First, prior work has shown that the point spread function can be derived from and is equivalent to the spectral magnetization response for system matrix reconstruction [38]. Second, the dominant trends of low amplitude and low frequency for resolution improvement observed here are shown to be appear for both reconstruction methods ( [20], [21] for amplitude, [22], [23] for frequency). Furthermore, our optimization process references the deconvolution analyses from prior system matrix reconstruction work [27], [2], [39] during objective function calculation. The 10-fold SNR = 2-fold resolution equivalence that we used was approximated referencing Knopp et al. and Werner et al. [27], [39] which states improving spatial resolution in system matrix reconstruction comes at the cost of a significant loss in SNR. In addition, the harmonic energy shift analysis used (Fig. 5) which discriminates for waveforms having strong higher harmonics is similar to system matrix reconstruction where the stronger the higher order harmonics, the better it is for spatial resolution and SNR [22], [38]. Finally, variation in the weights of the objective function formula (weak sensitivity analysis) only gives minimal shifts in the optima (Fig. 6f) which remain along the safety boundary, suggesting that these optima positions may be dominated by the safety boundary which is invariant across reconstruction method.

D. Implementation on Preclinical Scanners and Path to Clinical Application

As the optimization process sets the human MPI safety limits as a constraint (Fig. 3, the optima results are safe for clinical use. One possible safety challenge at higher drive frequencies, not covered by SAR and magnetostimulation, is the heating of the nanoparticle itself. To address this, we refer to prior work showing that nanoparticle heating is almost completely suppressed when magnetically saturated under MPI gradient, and only nanoparticles at the field-free-point heat [17], [16], [40], [41]. In a typical field-of-view (4 cm x 4 cm x 4 cm) with 1 mm³ voxels, the trajectory of the field-free-point or line is expected to spend less than 1% of total scan time at any individual nanoparticle position, and therefore heating of nanoparticles in the subject is suppressed for 99% of the scan duration, leading to a very low levels of overall heating.

The main implementation challenge will be shifting the transmit and receive bandwidth to higher drive frequencies of 50 – 100 kHz. Other MPI studies on multifrequency MPI hardware [42] will prove relevant to this effort, especially to cover both large-core and small-core optima (1 kHz and 70 kHz f₀ respectively). For hardware, increasing the frequency of the drive magnetic field will increase equipment, power and cooling costs for the drive coil which are also important considerations when optimizing the drive parameters for MPI. While hardware was not explicitly part of the optimization calculations in this study, we note that the safety limits that was part of the optimization calculations indirectly sets a power limit for MPI hardware and thus our evaluated parameter set cannot require more power than the existing MPI hardware running at 20 kHz 20 mT/μ₀. For example, consider the following points at the very boundary of the safety limits constraint: [0.1 kHz 100 mT/μ₀], [1 kHz 30 mT/μ₀], [70 kHz 5 mT/μ₀], [300 kHz 1 mT/μ₀]. It can be seen that all of these parameter sets, if implemented on the same drive coil, consume equal or less power than existing MPI at 20 kHz 20 mT/μ₀. In essence, safety constraints require a reduction in drive amplitude when drive frequency is increased, therefore this offsets any hardware issues like power or cooling arising from drive frequency increase. Receive bandwidth does not necessarily need to be increased from conventional as the use of lower drive amplitudes shifts more signal energy to lower harmonics (Fig. 5a), enabling the use of a smaller number of recorded harmonics than normal.

VI. Conclusion

Expanding on prior work focusing on individual MPI metrics, we present an optimization study for all three metrics of resolution, SNR and safety, across 5 nanoparticle types and a wide parameter space of 0.4 – 400 kHz, 0.5 – 40 mT/μ₀. Novel optima emerge at 5.0 – 8.0 mT/μ₀ and 45 – 100 kHz for the commonly-used small core SPIOs. These new drive parameters of [1 kHz, 14 mT/μ₀] for large-core and [70 kHz, 5 mT/μ₀] for small-core differ significantly from the 20 - 25 kHz range currently used in preclinical MPI. Recent work has shown that, from a MPI safety perspective, clinical MPI should shift to higher frequencies at 150 kHz [43], [44], [45]. The conclusions from our study thus agree with prior work but additionally considers spatial resolution and SNR via experimentally testing 5 different nanoparticle sizes. By considering safety, resolution and SNR simultaneously, we hope that this study will be informative to future MPI scanner design for improved imaging performance.