Exercise 1: SLR Pulse Design¶

Welcome! This first exercise will cover the design of SLR pulses for MRI using Python.

Documentation for all pulse design tools can be found here.

Prior to beginning this exercise (and at the beginning of all additional exercise notebooks) we will need to import the packages we will use:

Problem 1a: design an inversion pulse¶

Large-tip inversion pulses cannot be effectively designed by the small-tip-angle approximation alone, and should instead be designed by iterative methods or the SLR method, which we will use here.

In SigPy, basic SLR pulses are designed using the sigpy.mri.rf.slr.dzrf function, modeled after John Pauly's SLR design tools.

For this example, we want to design an inversion pulse with a sharp edge to invert magnetization(flip the sign of Mz).

• Using dzrf(), design an inversion pulse with a duration of 128 samples.
• To start, give your pulse 8 zero crossings.
• Use a filter type that concentrates the pulse energy at the end of the pulse.
• Design the pulse to have at maximum 1% ripple in both the passband and the stopband.
• Plot the pulse using pl.LinePlot()

Problem 1b: simulate the inversion pulse's profile¶

• Simulate and plot the Mz profile created by this pulse.

This is done in 2 steps:

• First, find get the a and b Cayley-Klein coefficients that correspond to the RF pulse using rf.sim.abrm(). You'll want to use spatial locations ranging from -2TB to 2TB.
• Second, you need to make a conversion from the Cayley-Klein parameters to the magnetization response. For an inversion pulse (assuming M₀ is initially oriented entirely along M_Z), the relationship for M_Z's final state is:

For this example you can just take M₀ = [0,0,1].

If done correctly, you should see a profile of mainly 1's (uninverted magnetization) with a segment in the middle with a value of -1 (successfully inverted magnetization).

Problem 1c: alter the time-bandwidth product¶

• Now, go back and rerun the code above, but change the time-bandwidth product to 4. Note the change in the sharpness of the inversion profile!
• While keeping the time-bandwidth product at four, change the design filter to a least-squares filter 'ls' and re-plot the magnetization profile.
• Increase the allowable stopband ripple level to 25% while keeping the previous settings, and re-plot the magnetization profile.

Problem 1d: root-flip the inversion pulse¶

For the final problem of exercise 1, we will explore the rf.mri.slr.dzrf() source code, and design a root-flipped SLR inversion pulse using what we find there. We can compare this to the standard SLR inversion pulse designed in 1a. The slides from the educational session discussing the inside workings of dzrf() will be a helpful guide.

Root flipping can be performed using the rf.slr.root_flip() function. This function is used in the following fashion:

[pulse, bRootFlipped] = rf.slr.root_flip(b, d1, flip, tb)

To design the root flipped pulse, we will need b (the SLR b polynomial), d1 (the passband ripple level), flip (the flip angle of our pulse in radians) and tb (our time bandwidth product). All of these are knowns, aside from b.

Your assignment is to write code to design b by extracting 4 lines of code from the rf.slr.dzrf() source code. Once this is done, root flip the pulse using root_flip().

Step #1: You will want 1 line of code from dzrf() that calculates new filter ripples from the desired magnetization d1 and d2

Step #2: You will want 2 lines of code from dzrf() that design a minphase b.

Step #3: You will want to end with a 4th line of code from dzrf() that multiplies bsf by b.

Step #4: Once you have written these 4 lines of code to design b, use the root_flip() function to root flip the pulse. Plot its magnetization profile using the same code that you used in problem 1b!