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

Overview Introduction Original Optimization Model Improved Optimization Model Conclusion Future work 11/14/2018

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 11/14/2018

The Conventional Spatial Frequency Patterns and 2D Imaging Methods Echo Planar Imaging Projection Reconstruction 2DFT Imaging 11/14/2018

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. 11/14/2018

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. 11/14/2018

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). 11/14/2018

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 11/14/2018

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 11/14/2018

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 11/14/2018

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 11/14/2018

The Spiral Constraints So the constraint is 11/14/2018

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 11/14/2018

Model 11/14/2018

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 ). 11/14/2018

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 ). 11/14/2018

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. ´ ´ ´ ´ ´ 11/14/2018

How to calculate  at each step 11/14/2018

Model Repeat { Recalculate } until obj < ( previous_obj + tolerance) 11/14/2018

Results Objective Function = 52490 25 31 30 11/14/2018

Objective Function = 93481 31 32 25 11/14/2018

Objective Function = 141852 32 33 25 11/14/2018

Objective Function = 179544 25 33 32 11/14/2018

Objective Function = 189799 33 32 25 11/14/2018

Objective Function = 191367 25 32 33 11/14/2018

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. 11/14/2018

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. 11/14/2018

Thank you! 11/14/2018