Michael W. Vogel^{1}, Ruben Pellicer-Guridi^{1}, Jiasheng Su^{1}, Viktor Vegh^{1}, and David C. Reutens^{1}

We explore the use of small permanent magnets moving along prescribed helical paths for spatial encoding in ultra-low field magnetic resonance systems based on Halbach arrays. A semi-analytical simulation method was developed to analyse different magnet path and orientations. For proof-of-concept, different helical magnet paths and lengths for one and two small magnets were considered to establish spatial encoding efficiency. We demonstrate that a single small encoding magnet moving around the sample in a single helical revolution can be used to generate 3D images via the method of back projection for image reconstruction.

**Introduction**

**Methods**

*The ULF-MRI instrument with the
encoding array*: Fig.1
illustrates the design of an SPMA based ULF-MRI system. It comprises four
concentrically arranged cylindrical magnet arrays: Array *A* with individually
rotatable magnets for switching the pre-polarization field **Bp** which generates sample magnetization; Arrays *B* and *C* for
generating the measurement field **Bm**;
and the Encoding Array *D* with two permanent magnets (a-b) that creates 3D
spatial encoding fields **Be**.

*Simulation environment*: COMSOL©
was used to estimate the temporal field evolution from pre-polarization
to measurement. Spin evolution, signal detection and image reconstruction simulations
were performed in MatLab©. A 3D cubic
cross-shaped digital phantom (8x8x8 voxels)(Fig.5) was employed using typical
soft tissue relaxation times at ULF (T1=100ms and T2=80ms)^{4}.

*
Acquisition strategy*:
The encoding field configurations are generated by Array* D*
with each encoding magnet moving along a prescribed path in discrete steps. For
a total of *Q* voxels in a sample and *N* time points, we used *Q/N*
different encoding field configurations. In our simulations we used 64 encoding
steps to solve the 512 voxels of the 3D phantom. We examined encoding field
configurations generated by magnet paths that were feasible for the ULF-MRI
instrument design. We considered the case of one and two identical magnets
moving along different path configurations (Fig.2&5).

*Evaluation of
encoding field configuration*: We aimed to maximize
the rank of the encoding matrix, which reflects the number of linearly
independent rows. We also aimed for a low condition number, which determines
the accuracy of the numerical matrix solvers.

*One encoding magnet*: Fig.3 shows the condition
number of the encoding matrix versus the encoding magnet orientation. Three
values for α_{2} were
selected: 180^{o} (Fig.3A), 100^{o} (Fig.3B) and 230^{o}
(Fig.3C). Condition number significantly increased as path length decreases but
varies by less than one order of magnitude for α_{3} between 240^{o}
and 360^{o}. Greater path length results in a lower standard deviation between reconstructed and phantom images:
standard deviation = 0.0231 for α_{3}=180^{o},
0.0221 for α_{3}=240^{o}
and 0.0200 for α_{3}=360^{o}
(Fig.4). Besides, a linearly descending trajectory improves performance (Fig.2).

*Two encoding magnets*: The optimal orientation angles for the two magnets are perpendicular to the magnet path (*φ*_{1}^{opt} and *φ*_{2}^{opt}~0^{o})
and parallel to the xy-plane (*θ*_{1}^{opt} and *θ*_{2}^{opt}~0^{o}).
The reconstructed images for each configuration are shown in Fig.5. The standard
deviations for configurations 1 and 2 were 0.0254 and 0.0287 respectively;
image quality was higher in the former.

Discussion

Shortening the path length reduced the step
size, leading to an increased linear
dependence between encoding field configurations. Hence, image quality is
degraded if the encoding magnet does not revolve completely around the sample.
For the configurations considered, the optimal magnet orientations were
perpendicular to both the motion path and the cylindrical surface of Array D.
This is a consequence of the torus-shaped magnetic field distribution of a
magnetic dipole^{5}.

The lowest condition numbers were achieved when z(α)
was a linear function of α. This is
attributed to the low helical path slopes for the nonlinear height variation
near the bottom (black curve, α_{2}=100^{o})
and the top (blue curve, α_{2}=240^{o};
see Fig.2) which lead to lower variation in the encoding field along the z-axis
and hence increased linear dependencies and higher condition numbers.

The generated encoding field strength in the field
of view ranged between 1-10µT, corresponding to a frequency spread of 43-430Hz,
well within the bandwidth of coil-based magnetometers^{6}.

Although only spiral paths with equidistant stopping points along a cylindrical surface were considered, the semi-analytical method can readily be extended to include any number of magnets moving along any prescribed paths.

1. Vogel, M. W., Giorni, A., Vegh, V., Pellicer-Guridi, R. & Reutens, D. C. Rotatable Small Permanent Magnet Array for Ultra-Low Field Nuclear Magnetic Resonance Instrumentation: A Concept Study. PloS one 11, e0157040 (2016).

2. Cooley, C. Z.
et al. Two‐dimensional imaging
in a lightweight portable MRI scanner without gradient coils. Magnetic resonance in medicine 73, 872-883 (2015).

3. Cooley, C. Z., Stockmann, J. P., Sarracanie, M., Rosen, M. S. & Wald, L. L. in Intl Soc Mag Res Med. 4192.

4. Zotev, V. S. et al. SQUID-based microtesla MRI for in vivo relaxometry of the human brain. IEEE Transactions on Applied Superconductivity 19, 823-826 (2009).

5. Cheng, D. K. Field and wave electromagnetics. (Addison-Wesley, 1989).

6. Pellicer-Guridi, R., Vogel, M. W., Reutens, D.
C. & Vegh, V. Towards ultimate low frequency air-core magnetometer
sensitivity. Scientific reports 7, 2269 (2017).

Concept
design of ULF-MRI instrument with permanent magnet arrays. Array *A* with 12
magnets switches the pre-polarisation field **Bp** by individual magnet rotation.
Shown here is the tangential magnetisation pattern (**Bp** = off). Array *B* (24
magnets) and array *C* (36 magnets) generate the measurement field to define the
Larmor frequency. Array *D* consists of two small permanent magnets (Ma1 and Ma2)
for 3D spatial encoding moving in helical paths along a cylindrical surface.

(A)
Three examples of 3D helical paths with linear (α_{2} =180^{o}, red) and
quadratic height variations z (α_{2} =100^{o}, black curve and α_{2} =240^{o}, blue curve)
are shown, each starting from initial angle α_{1} = 0^{o} to final angle α_{3} = 360^{o}.
The height varies from z(α_{1}) = -0.15m to z(α_{3}) = 0.15m. Each line segment
corresponds to one encoding step location and the magnet orientation, shown here
for θ = 0 and φ = 0. (B) 3D helical paths with linear height variation (α_{2} =
180^{o}) but different final angles α_{3} = 360^{o} (red), α_{3} = 240^{o} (blue curve)
or α_{3} = 180^{o} (black).

Condition
number of the encoding matrix vs magnet Ma1 orientation with helical path
parameters α_{1} = 0^{o} and α_{3} = 360^{o} and height variation from z(α_{1}) = -0.15 m and
z(α_{3}) = 0.15 m (A) Linear height variation (α_{2} = 180^{o}), (B) Nonlinear height
variation (α_{2} = 100^{o}). (C) Nonlinear height variation α_{2} = 230^{o}.

The
image quality dependence on path length for one encoding magnet, shown for
α_{3}=180^{o} (black), α_{3}=240^{o} (blue) and α_{3}=360^{o} (red), with constant encoding step
numbers. The height varies from z=-0.15m to z=0.15m. The image reconstruction
with the Kaczmarz method is shown after 10 iterations. The standard deviations
are 0.0231 (α_{3} = 180^{o}), 0.0221 (α_{3}=240^{o}) and 0.0200 (α_{3}=360^{o}).

Image
reconstruction for the encoding array with two magnets, Ma1 (black) and Ma2
(red). The magnet motions are indicated by the arrows for two configurations.
The height varies from z = -0.15m to z = 0.15m. The image reconstruction with
the Kaczmarz method is shown after 10 iterations. The standard deviations are
0.0254 (configuration 1) and 0.0287 (configuration 2).