1. An Optimal Design Method for MRI Teardrop Gradient Waveforms
Tingting Ren
Joint work with Christopher Anand
and Tamás Terlaky

2. Overview Introduction Original Optimization Model
Improved Optimization Model
Conclusion
Future work
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3. Introduction Radio Frequency (RF) pulses excite the spins
Field gradients spatial encode the spins
Large uniform static RF coil receives a signal
Large uniform static
external magnetic field
Signal: Amplified
Digitized
Fourier-Transformed
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4. The Conventional Spatial Frequency Patterns and 2D Imaging Methods
Echo Planar Imaging
Projection
Reconstruction
2DFT Imaging
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5. Fast Imaging : acquire a greater portion of k-space per signal
Interleaved
spiral
Square
spiral
Spiral
Resample
Inverse FT
Fast Imaging : acquire a greater portion of k-space per signal
readout.
The gradient system must be able to generate the trajectory.
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6. The Theory of Teardrop
Steady State Free Precession imaging (i.e. SSFP) requires
balanced gradient waveforms.
Conventionally, this is done by adding read, slice and phase
rewinders.
waste time
generate high dB/dt
By using a non-raster trajectory beginning and ending in the
center of k-space, a teardrop readout requires neither read nor
phase dephase/rephase lobes, increasing scan-time efficiency.
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7. The Advantages of Teardrop
objective = radius = resolution
designed to follow an interleaved spiral like trajectory leaving the center of k-space, and returning on the mirror image trajectory to the center of k-space .
Family of waveforms, increasing scan time efficiency.
100% readout ( no rewinders).
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8. We acquired ungated, free breathing cardiac images using non optimal teardrop readout sequence. The image displays good blood/septum contrast to noise. We can expect better result with our optimal teardrop readout trajectory.
X-section of Dr. Anand
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9. Original Optimization Model
The design goal of the teardrop is to maximize resolution when obey the Nyquist sampling condition for the given FOV, and remain within the gradient strength and slew limits of the given system.
Parameter Sets:
A discrete series of point-by-point gradient amplitudes:
A set of points in k-space:
From Fourier transform, we know
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10. Objective Constraints Gradient System Hardware Limitations
For a fixed FOV, the basic technique for fast imaging is to acquire larger portion on k-space per signal readout. So our objective function is
if is an even number
if is an odd number
Constraints
Gradient System Hardware Limitations
Objective or
Amplitude Limits
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11. Slew or Rise Time Limits
Gradient Start and End Amplitudes
The beginning and the end of k-space trajectory
Constrains to keep a teardrop k-space trajectory
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12. The Spiral Constraints
So the constraint is
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13. By using atan function, we can simplify this constraint significantly.
Making the readout gradient motion-insensitive requires that
we zero the first moment, which is a global constraint
First Moment Nulling
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14. Model
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15. Results We used five nonlinear softwares: IPOPT, LOQO, KNITRO,
SNOPT and FILTER to solve the model with various parameters.
We got almost the same optimal results in each case. Here is the
optimal trajectory where = 0.04 T/m, = 150 T/m/s,
TR = 4ms, and N=50 ( 101 sampling points ).
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16. Rotating the associated gradient profiles is equivalent to rotating
the trajectory around the center of k-space. Together the
combined views cover k-space, as evidenced by a partial set of
views ( three interleaves ).
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17. Improved Optimization Model
Add a new set of points in k-space as design variables:
Replace the spiral constraints by two new constraints:
(1)
(2)
Recalculate the quadratic constraints, solve the optimization problem again.
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18. How to calculate  at each step
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19. Model
Repeat {
Recalculate } until obj < ( previous_obj + tolerance)
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20. Results
Objective Function = 52490 25 31 30
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21. Objective Function = 93481 31 32 25
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22. Objective Function = 32 33 25
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23. Objective Function = 25 33 32
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24. Objective Function = 33 32 25
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25. Objective Function = 25 32 33
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26. Conclusions
We have modeled 2D teardrop design using nonlinear and second order cone methods including
Global constraints ( moment nulling )
Commercial solves can solve both models quickly
Good foundation for 3D model.
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27. Future Work Generalize 2D model to 3D model.
We may add other constraints to the model, like second
moment nulling, which would reduce image artifacts
caused by pulsatile flow, just as first moment nulling
reduces the effect of constant flow.
Test on MRI machine.
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28. Thank you!
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