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Pulses

Pulse sequence design is an area of research that touches many fundamental fields in science, including biology, chemistry, physics, and medicine. It is our perspective – one supported by many great discoveries – that methods from mathematical control theory and optimization are well suited for solving problems in pulse sequence design. These are challenging problems both from a scientific and engineering point of view. Although they are academically interesting problems they also have direct applications to our present daily lives – such as clinical MRI imaging – as well as cutting edge future technology – such as quantum computing.

The AML investigates a wide array of pulse sequence design problems. Like any scientific field, pulse sequence design can be simplified by making various assumptions about the system behavior, however, we find these assumptions are no longer valid if we want increasingly better results. Most of the system dynamics used in pulse sequence design can be modeled as spin systems, in which the system has a characteristic frequency. A common challenge in the design of pulses that manipulate these “spins” is that chemical interactions with the environment and surrounding molecules creates shifts in this characteristic frequency.  On a macro level, since each spin is shifted by a different amount, we observe the bulk frequency of the sample to take on a continuum of values, or band.  Designing pulses that take this band of frequencies into account are called broadband compensating pulses and are one of the many problems that the AML considers in its research.

Shown below are several of the RF pulses the AML has developed for enhancement of MRI and NMR imaging. The lefthand side of each animation shows the trajectories of the spin vectors on the unit sphere. And the righthand side shows the control pulses. Because the pulses are short-lived, we do not take into account any relaxation – or change in magnitude of the spin vectors. Therefore, each of the pulses executes various (and complicated) rotations of the spin vector on the unit sphere. Typically we specify an initial state, final state, and a quantity to be minimized. The systems here all illustrate a variation in the frequency of the imaged particle. The challenge is to steer all of these spins simultaneously from an initial state to a final state with only one set of controls, or pulses. For all of these scenarios, we start with systems oriented in the positive z-direction. Our aim is for all of the spins to come back together at the positive x axis.

Minimum Error Pulse:

Here we minimize the error between the desired final state and the resulting final spin after we apply the pulse.

Minimum Energy Pulse:

Here we minimize the energy used in the pulse. This is related to the magnitude of the amplitude of the pulse, where u and v are the x and y components of the pulse.

Minimum Time Pulse:

Here we minimize the time it takes to achieve the desired transition.

This material is based upon work supported by the National Science Foundation under Grant No. 0747877.