L. Tugan Muftuler^{1,2}, Nikolai J. Mickevicius^{3}, Andrew S. Nencka^{2,4}, and Eric S. Paulson^{3}

Our group recently reported a new fast radial imaging method called Highly Accelerated Projection Imaging (HAPI) with coil sensitivity encoding. We demonstrated that radial projections acquired at specific angles and at high resolution resulted in a well-conditioned matrix equation. In the previous work, the performance of HAPI was demonstrated with simulations and a simple phantom scan using an 8-channel receive array. In the study presented here, the HAPI method was tested in vivo with volunteers.

Rapid MRI techniques require sampling the k-space as fast as possible. For further acceleration, only a portion of the k-space is sampled and missing data is synthesized using parallel imaging techniques. We
recently reported a new fast radial imaging method called *Highly
Accelerated Projection Imaging* (HAPI)^{1}
that is capable of reconstructing an image from a small number of radial projections. Radial
projections acquired at specific angles and at high resolution resulted in a
well-conditioned matrix equation. In the previous work, the performance of HAPI was demonstrated with simulations and a
phantom scan. Here we present in vivo abdominal imaging results and compare with other conventional reconstruction methods.

Given high resolution radial k-space data, the 1D inverse Fourier transform of each spoke yields a projection of the imaged object. The isofrequency paths formed by the frequency encoding gradients are referred to as rays. The measured projections for each RF coil can be modeled as the integral along each ray of the object (x) times the coil sensitivity profiles (C) and times the fraction of the ray (F) covering each reconstructed voxel. The object can therefore be inversely solved for by solving the following equation.

*x = argmin*_{x }||*FCx - p*||^{2}_{2} + λ || *Tx* ||_{1}

_{ }

The
quality of the solution is dependent on the condition of the encoding matrix * A = FC*.
This condition depends on the acquired projection angles and the ratio of the
acquired resolution to the reconstructed resolution, as we published earlier

The study was approved by the IRB and written consent was obtained. Stack-of-stars
spoiled gradient echo data were acquired on a 1.5T MR-Linac (Unity, Elekta
Instruments) with an 8-channel receive coil. The data were acquired at a base
resolution of 512x512 with an in-plane FOV of 340 mm. For a central slice, five
reconstructions from 16 spokes were performed at a reconstructed resolution of
128x128: non-uniform FFT (NUFFT)^{2},
CG-SENSE^{3}, CG-SENSE with TV regularization^{4}, HAPI, and
HAPI with TV regularization. In the non-HAPI cases, the images were
reconstructed at 512x512 and resampled to 128x128. For HAPI reconstruction, the
optimal projection angles of {7 14 18 22 28 32 36 56 60 64 68 72 80 110
116 129} degrees were chosen based on the trends seen in Fig.1.

[1] Ersoz A, Arpinar VE, Muftuler LT. Highly accelerated projection imaging with coil sensitivity encoding for rapid MRI. Med Phys 2013;40:022305. doi:10.1118/1.4789488.

[2] Fessler JA, Sutton BP. Nonuniform fast fourier transforms using min-max interpolation. IEEE Trans Signal Process 2003;51:560–74. doi:10.1109/TSP.2002.807005.

[3] Pruessmann KP, Weiger M, Börnert P, Boesiger P. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med 2001;46:638–51. doi:10.1002/mrm.1241.

[4] Block KT, Uecker M, Frahm J. Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. Magn Reson Med 2007;57:1086–98. doi:10.1002/mrm.21236.

Fig.1. Effect of projection angle and the
number of samples per projection on the condition of the encoding matrix **A**. (a) Normalized eigenvalue
distribution for various number of samples per projection (Ns) values
(single projection at 20^{o}, 64×64 image reconstruction); (b) Condition
number of *A* calculated for different
projection angles and three different Ns
values (64×64 image reconstruction).

Fig.2.
In Vivo HAPI Results. Shown here are the reference, non-uniform fast Fourier
transform (NUFFT), conjugate gradient sensitivity encoding (CG-SENSE), CG-SENSE
with total variation (TV) regularization, HAPI, and HAPI with TV regularization
reconstructions.