The ability to accurately characterize particles in a tracer-based modality such as MPI is crucial, especially when experimenting with varying particle environments, but also to validate the quality of particles. To respond to this need we have developed our MPS system with a magnetometry mode. This mode is fairly similar to, and partially inspired by the "Superparamagnetic Quantifier" developed by Van De Loosdrech and colleagues (1) through our system has some notable hardware differences. Their paper has a thorough discussion of the working principle which we are describing here as "magnetometry".

Functional Principle

Our system operates by measuring the magnetic susceptibility of the particles within the Rx gradiometer. To do this we superimpose two magnetic fields. One high amplitude and low frequency, the other low amplitude and high frequency. Below are a series of figures illustrating this. In order the figures show:

1. Excitation waveform (External field)
2. A zoomed in section of the waveform showing the high frequency, low amplitude component
3. The acquired signal across time
4. The acquired signal when plotted versus the external field
5. An integration of signal vs field plot with the boundary condition that at zero eternal field, there is zero magnetization

The principle assumption this technique relies on is that the drive amplitude is sufficiently small such that the susceptibility (dM/dH) does not substantially change across this range-- it is a linearity assumption, which given the other assumptions we make, is fairly reasonable. The other big assumption is the lack of relaxation effects. While the linearity and relaxation-free assumptions are certainly worth keeping in mind, by making them, the hardware requirements and software complexity dramatically reduces while still providing the user with a DC magnetization response of the particles.

Effectively what happens in the analysis is the same thing as AM-radios. It is taking an amplitude modulated signal and taking out the carrier frequency (in this case the drive frequency) and looking at the signal underneath. In this case that signal is the susceptibility of the nanoparticles as a function of time. But by knowing the bias field, which is also a function of time, we can plot them against each other (Bias field as independent variable on X, and susceptibility as the dependent variable on the Y axis).

By integrating the susceptibility with respect to bias field, and imposing the boundary condition that there is zero remnant magnetization (no magnetization at H=0), we can get the roughly Langevin magnetization function describing the SPION's superparamagnetic behavior.

Note Regarding Application

For the chemist synthesizing nanoparticles, or those doing a first-pass at including SPION physics in a reconstruction, magnetometry is a great starting point. It describes the bulk of the information such as the FWHM magnetization (the "Full width at half maximum"), which roughly corresponds to the point spread function of the SPIONS in X-Space MPI. It also will tell the user what the total magnetic moment is, and that will roughly indicate how much signal one can expect from the particles (when compared to similarly saturated particles). Indeed, it is quite similar with regard to the resultant data to "vibrating sample magnetometry (VSM)", which is a widely accepted and used method in the field.

But, it is important to note there have been substantial assumptions made along the way, so SPIONs that appear to be good based on the DC magnetization function may either be better or worse than expected due to relaxation or other higher-order effects coming into play. By combining the information from magnetometry with relaxometry and spectroscopy, you can get a much fuller picture.

Citations

 (1) M. Van De Loosdrecht, et al. A novel characterization technique for superparamagnetic iron oxide nanoparticles: The superparamagnetic quantifier, compared with magnetic particle spectroscopy, 2019