Parallel MRI reconstruction

Parallel MRI (pMRI) is a way to increase the speed of the MRI acquisition by skipping a number of phase-encoding lines in the k-space during the MRI acquisition. Data received simultaneously by several receiver coils with distinct spatial sensitivities are used to reconstruct the values in the missing k-space lines. Our task is to propose and implement a robust pMRI algorithm that will reconstruct the original image using a set of images with incomplete information. We focus on the minimizing of the presence of noise in the reconstructed image and also on removing of the aliasing artifacts from the reconstructed image (artifacts caused by skipping some phase-encoding lines in the k-space during the acquisition).

Introduction

Project team - Jan Petr¹, Jan Kybic¹, Václav Hlaváč¹, Michael Bock², Sven Müller²

1 - Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Karlovo náměstí 13, Praha 2, 121 35, Czech Republic. petrj5@cmp.felk.cvut.cz
2 - Division of Medical Physics in Radiology, German Cancer Research Center, Im Neuenheimer Feld 280, D-69120 Heidelberg, Germany

Magnetic Resonance Imaging (MRI)

Noninvasive imaging method used in medicine.

Imaged object is placed in a strong magnetic field.
All protons in the tissue align with the direction of the magnetic field.
The protons are excited to a higher energy state using a radio-frequency electromagnetic pulse.
Excited protons return back to the energy equilibrium.
The accepted energy is retransmitted back and can be measured.


MRI scanner	MRI image of human brain

The electromagnetic pulse have to have an exact frequency (called the resonance frequency - in order of MHz) that depends on the chemical properties of the tissue, strength of the main magnetic field and temperature.
The spatial position of the signal cannot be resolved -> The signal need to be spatially encoded in order to retrieve images.
Magnetic gradient fields are used to locally change the resonance frequency -> The frequency of the MRI signal become dependent on the spatial position of the signal source.
A k-space image is formed by measuring the retransmitted signal. The k-space image corresponds to the image in the Fourier space.
The real image of the object is obtained by Fourier transform of the k-space image (it resolves the correspondence of the frequency and spatial position of the signal).

	2D FFT ====>
k-space image		final image

Parallel MRI (pMRI)

In MRI, signal is usually received by a single receiver coil with an approximately homogeneous sensitivity over the whole imaged object.
In pMRI, MRI signal is received simultaneously by several receiver coils with varying spatial sensitivity -> This brings more information about the spatial position of the MRI signal.


homogeneous coil	coil with varying sensitivity	coil with varying sensitivity

The task of pMRI is to speed up the acquisition in order to:

be able to image dynamic processes without major movement artifacts (i.e. reduce the speed of the acquisition so the movement during the acquisition time does not cause significant artifacts),
shorten the MRI acquisition time that could be very long (for example - acquisition of a high resolution 3D scan may take up time in order of minutes).

The bottleneck of the MRI acquisition is the number of retrieved lines in k-space and the time needed to acquire one line in k-space.
In pMRI, only a fraction 1/M of k-space lines is acquired while preserving spatial resolution.

The acquisition is M times faster.
It causes an aliasing in the images - M points from the original image overlaps over themselves in the image with aliasing.


Half of the k-space lines retrieved	image with aliasing

Linear combination of at least M images with aliasing retrieved by the coils with varying sensitivity is used to reconstruct the original image (the coil configuration is supposed to be suitable for pMRI reconstruction - the coil sensitivities should be distinct, all parts of the imaged slice should be covered by at least one coil with reasonable SNR in this part of the slice).
The parameters of the reconstruction are estimated using the exact knowledge of the coil sensitivities.

	Aliasing =====>

		+ Reconstruction ==========> +
	Aliasing =====>

Parallel MRI reconstruction using Implicit B-spline regularization (PROBER)

A novel method for pMRI reconstruction.
The parameters of the reconstruction transformation (linear combination of the images with aliasing) is estimated in the image space:

(1)

where

are the parameters of the linear combination for the l-th coil. Ŝ is the reconstructed image. S_l^A is the image with aliasing acquired by the l-th coil. N_y is the resolution in the y direction. M is the acceleration factor, L is the number of the receiver coils used and

= N_y/M.

The parameters are estimated directly from the low resolution images without aliasing S_l. A correlation between the ideal image Ŝ and the images from the small coils is used as the criterion to estimate the reconstruction coefficients

(2)

The parameters estimated using this equation should satisfy the condition, that the values in the original image are reconstructed using only the values in coil images with the same coordinates because the image values S(x,y+mk) and S(x,y+m'k) for different m and m' should be independent.

A B-spline approximation of the reconstruction coefficients is used in order to reduce the number of unknown parameters:

Reduces the speed of the estimation.
Makes the method more robust to noise.

(3)

where g_ijl are the B-spline coefficients and

and

are the B-spline basis function in the x and y direction. N_i and N_j splines are used in the x and y direction.

The aim is to estimate the reconstruction parameters to minimize the square reconstruction error (square difference between the ideal image and the reconstructed image). The error equation is formed using the equation (2).
Minimization of the error leads to a set of linear equations that are solved in the least squares sense.

Conclusion

The method was implemented in Matlab for offline reconstruction and in C++ for the online reconstruction in the MRI scanner.
The results of the current implementation were compared with the results of the methods mSENSE and GRAPPA.

The artifact level and noise level in all reconstructed images is comparable in most cases.
The method still need to be speeded up and the noise level in the reconstructed images need to be reduced.


PROBER SNR=25dB	GRAPPA SNR=26dB	mSENSE SNR=27dB

Publications

J. Petr, J. Kybic and V. Hlaváč. Parallel MRI Reconstruction using B-splines. Czech Technical University in Prague. Technical Report CTU-CMP-2004-09. July 2004. [ PDF ]
J. Petr. Parallel Magnetic Resonance Imaging Reconstruction - PhD Thesis Proposal. Technical Report CTU-CMP-2005-03. February 2005. [ PDF ]
J. Petr, J. Kybic, S. Müller, M. Bock and V. Hlaváč. Parallel MRI reconstruction using B-spline approximation. ISMRM 2005.