Somaie Salajeghe1, Paul Babyn2, Logi Vidarsson3, and Gordon E. Sarty1
1Biomedical Engineering, University of Saskatchewan, Saskatoon, SK, Canada, 2Medical Imaging, University of Saskatchewan, Saskatoon, SK, Canada, 3LT Imaging, Toronto, ON, Canada
Synopsis
Portable MRI can be possible by eliminating gradient coils and B0
homogeneity requirements. Relaxing the B0 homogeneity requirements
leads to non-uniform B0 field. In-homogeneous B0 fields
have the potential to encode spatial information in one direction for use in
novel image encoding schemes. We investigated the possibility of image
reconstruction of the signal from a non-uniform rotating magnetic field and two
rotating RF receivers. Our results indicate that this is a feasible
approach. Purpose:
Conventional Magnetic Resonance Imagers (MRI)
use a uniform main magnetic field ($$$B_{0}$$$) to polarize the sample being
imaged and separately superimposed gradient fields for image encoding.
Inhomogeneous $$$B_{0}$$$ fields have the potential to encode spatial
information in one direction for use in novel image encoding schemes [1,2]. In
this study we explore the feasibility of image reconstruction of the signal
from a non-uniform rotating magnetic field and two rotating RF receivers.
Theory/Methods:
A Halbach magnet was used to
generate a non-uniform radially-varying $$$B_{0}$$$ field, such that intended
image plane was perpendicular to the axis of the magnet. The $$$B_{0}$$$ field
was measured experimentally and a polynomial was fit to the data. Two separate saddle
receiver coils were located $$$180^0$$$ from each other. The receiver coils
were fixed relative to the magnet and both were rotated around the imaged
object using a stepper motor (see Fig 1). The rotation therefore achieved
spatial encoding in the angular direction primarily through the variation of
the receive $$$B_{1}$$$ field relative to the object. In the
experiment reported here, 32 angular positions were used and 23 transmission
frequencies were used for radial encoding in the inhomogeneous $$$B_{0}$$$
field. To model the sensitivity of the receiver coils, we used the Biot Savart law
to calculate the $$$B_{1}$$$ field of each receive coil for different positions
in the field of view (FOV). The detected signal from object $$$x$$$, indexed by
transmission frequency $$$\omega$$$ and magnet position $$$\alpha$$$ may be
modeled as:
$$y(\omega,\alpha)=\int\int
x(r,\theta) A(\omega,\alpha;r,\theta)\: dr \: d\theta$$
where A is the encoding matrix and $$$(r,\theta) $$$ are polar
coordinates. A Riemann sum approximation of Eq. (1) was used to compute the
simulated signal and after rearrangement, it can be written as: $$$[y]=[A][x]$$$.
The encoding matrix for each excitation, at frequency $$$\omega$$$ (for radial
encoding) and rotation $$$\alpha$$$, can be calculated as
$$A(\omega,\alpha)=B_{\rm{weight}}(\omega,\alpha)B_{1}(\alpha)e^{\triangle
B_{0}(\omega,\alpha)\gamma\triangle T}$$
where we now discuss each term. $$$ B_{1}(\alpha)$$$
is the component of the RF receiver field perpendicular to the $$$B_{0}$$$ direction
which changes relative to the imaged object at each magnet angular position and
can be calculated as
$$B_{1}=\sqrt{B_{1x}^{2}+B_{1y}^{2}}
\sin(\phi_{B_1}-\phi_{B_0})$$
Where $$$\phi_{B_1}$$$ is the phase
of $$$B_{1}$$$ field in each angular position and $$$\phi_{B_0}$$$ is the phase
of $$$B_{0}$$$ field. For each excitation the slice thickness (radially) will
be a sinc function which means the nuclei with the same resonant frequency will
be excited completely and then others according to how far they are from the Larmor
frequency will be excited. Therefore we multiply the coil sensitivity with a
weighing matrix. The weighting matrix for each excitation and rotation is
$$B_{\rm{weight}}(\omega,\alpha)=\mid
\frac{\sin(\triangle B_{0}(\omega,\alpha))}{\triangle B_{0}(\omega,\alpha)}\mid$$
where $$$\triangle B_{0}$$$ is the
difference of $$$B_{0}$$$ field from the Larmor frequency field:
$$\triangle
B_{0}(\omega,\alpha)=B_0(\alpha)-B_{\omega}(\omega)$$
where
$$$ B_0(\alpha)$$$ is the measured $$$B_{0}$$$ field at each rotation angle and
$$$ B_{\omega}(\omega)$$$ is the Larmor frequency magnetic field as determined
by the transmitted frequency. The image $$$[x]$$$, from $$$[y]=[A][x]$$$, was
reconstructed using a constrained least squares method with Tikhonov
regularization. Let $$$M$$$ be the number of frequencies that we want to transmit at
each rotation, let $$$R$$$ be the number of angular positions we want to rotate the
magnet to (main magnet and the receive coils rotate together), let $$$C$$$ be the
number of the receive coils and let $$$N \times N$$$ be the number of the
pixels of the image. Then matrix $$$ [A]$$$ is of size of $$$MRC \times N^2$$$,
$$$[y]$$$ is a vector of size of $$$MRC \times 1$$$ and, $$$[x]$$$ is a vector of
size of $$$N^{2} \times 1$$$.
Results/Discussion:
Our results indicate the feasibility of
reconstructing images from non-uniform rotating magnet. The reconstructed images
from experimental data, however, include artifacts which are likely due to the
high sensitivity of the reconstruction method to the assumed $$$B_{0}$$$ and $$$B_{1}$$$ field values. There were, as yet unquantified,
errors associated with measuring the $$$B_{0}$$$ field. As well, the calculated $$$B_{1}$$$ field may be not be a good enough
approximation to the actual field.
Conclusion:
We have made progress in the realization of a
novel new approach to MRI that does not rely on active magnetic gradient
fields. Improvements are expected when we have better knowledge of the $$$B_{0}$$$
and $$$B_{1}$$$ fields produced by the constructed hardware.
Acknowledgements
This work was support by an NSERC Discovery Grant to GES.References
[1] Cooley CZ
et al. Magn Reson Med. 73:872–883 (2015).
[2] G.Sarty Magn Reson Imag. 33:304–311 (2015) .