The goal of relaxometry is to understand the dynamics of SPIONs being excited similarly to how they are when imaging. Unlike magnetometry, in relaxometry, the drive field amplitude will be on the order of 10mT, so most of the assumptions that make were made in magnetometry mode fall apart. Notably, there can be no linear approximation in this mode of operation, and hysteresis plays a prominent role. Because there is not this linear approximation in the analysis, relaxometry can provide data such as relaxation time constants which can play a role in reducing the blurring in image reconstruction. On the other hand, Relaxometry has higher hardware demands-- it requires a higher sampling rate to get satisfactory results, requires both coils operating at high power, and the excitation field (both drive and bias) need to be measured with high fidelity.

For some good places to start reading about this technique see the following sources (in no particular order):

• Relaxation in X-Space Magnetic Particle Imaging, L. Croft et al. 2012
• Effective Viscosity of Magnetic Suspensions, M. Shliomis, 1972
• Benchtop magnetic particle relaxometer for detection, characterization and analysis of magnetic nanoparticles, N.Garrault et al. 2018
• Ferrohydrodynamic relaxometry for magnetic particle imaging, P. Goodwill et al. 2011

For the most part, it is identical to the magnetometry mode where there are two fields superimposed: the excitation field which yields signal from the particles, and a second field which biases the magnetization of the SPIONs. In this mode, the excitation field should be roughly that which you plan to image with, and the biasing field should fully saturate the particles. For most SPIONs which work for MPI, drive amplitudes are ~6-18mT, and the particles generally saturate by ~50mT. Additionally, while the drive field should be sinusoidal, there is considerably more flexibility in the bias field. If you do choose to use a TDA7293, or another similar audio amplifier that is AC coupled, it is probably easiest to use a sine wave at the lowest allowable frequency. The lower frequencies will induce less voltage in the Rx (ideally everything is decoupled, but practically it wont be), and make the analysis cleaner.

When operating the system a few streams of data come in. From the primary Rx coil you collect data in three states: * Data (from the pre-amp) with the sample inside the Rx coil and full-strength drive and full-strength biasing fields * Data (from the pre-amp) with the sample inside the Rx coil and no drive field and full-strength biasing field. In a well-decoupled system and filled samples, this should be very small. But in a sample with a bubble or if it is horizontal (not wall mounted) motion of the particles/bubbles can cause substantial signal. * Data (from the pre-amp) with the sample outside the Rx coil and full-strength drive field and no biasing field. This is just the "drive feed-through".

By subtracting the latter two data vectors from the first you get the signal due to the nanoparticles with respect to the drive coil. I will call this the "Signal" from here on out.

The script also acquires the signal from the current-sense modules for the drive and bias coils. These currents are multiplied by a scaling factor (pre-defined, ideally from a measurement) to translate the currents into the magnetic field where the sample is. This is the "Total Field"

The total field is used to find the field "velocity" through differentiation with respect to time. The signal is divided by the velocity to get the "Velocity corrected signal", BUT ONLY AT TIMES WHERE THE VELOCITY IS ABOVE 1/2 MAX. This is important, because at times when the (noisy) data is divided by the velocity terms near-zero, you get extremely high amplitude noise. The reason this must be done is the voltage at any given instant is proportional to dB/dt through the coil's enclosed area. But that dB/dt term is a composition of Mu(the susceptibility of the substance) as well as dH/dt, and the goal is to elucidate the external field dependence of Mu. Therefore if we write signal voltage as being proportional to (Mu x dH/dt), and divide by dH/dt you get the signal voltage / (dH/dt) is proportional to Mu, and that division is the velocity correction.

By then plotting the velocity corrected signal against the external field and looking at the envelope of the plot, the positive and negative envelopes each give data about the different drive field directions, and the integral of each of these envelopes with respect to external field is the magnetization.