Variable Rate Selective Excitation Radio Frequency Pulse in Magnetic Resonance Imaging Stephen Stoyan

Overview oMRI Background oModel oResults oFuture Work

MRI Background: Basics of MRI Radio Frequency (rf) pulses excite the sample Radio Frequency (rf) pulses excite the sample Field gradients spatial encode the sample Field gradients spatial encode the sample Large uniform static rf coil receives a signal Large uniform static rf coil receives a signal external magnetic field external magnetic field Signal: Amplified Signal: Amplified Digitized Digitized Fourier-Transformed Fourier-Transformed

MRI Background: Magnetization MRI Background: Magnetization oNuclei with odd atomic weight and/or odd atomic number posses an angular momentum. oAny electrically charged particle which moves creates a magnetic field called a magnetic moment. oAn ensemble of nuclei produce a ‘spin system.’ oWhen an external magnetic field is applied the magnetic moments align in the direction of the field.

MRI Background: Magnetization MRI Background: Magnetization oMagnetization is the net vector quantity of the magnetic moments of each nuclei in a given unit volume or voxel. oGiven an external magnetic field, magnetic moment field, magnetic moment vectors rotate around the vectors rotate around the axis of the field. axis of the field. oThis secondary spin is termed, Precession. termed, Precession.

MRI Background: Precession MRI Background: Precession oThe speed of proton precession is referred as, Precessional is referred as, Precessional Frequency. Frequency. oStronger magnetic fields constitute higher precessional constitute higher precessional frequencies. frequencies. oThe frequency at which the nucleus will absorb energy is described in the Larmor equation.

MRI Background: Interactions MRI Background: Interactions oThe equation for torque on a magnetic moment due to an external magnetic field, oMaking the substitution into, Proton interactions oSpin-lattice interactions: A magnetic moments minimum energy state is in the direction of the external magnetic field. oSpin-spin interactions: Magnetic moments experience local fields of their neighbours and the applied field.

MRI Background: Bloch Equation MRI Background: Bloch Equation oCombining proton interactions into equation produces the Bloch equation, where, where,

Model: General rf Pulse Model: General rf Pulse oIn processing an image a precise radio frequency (rf) pulse is applied in combination with a synchronized gradient. oAn rf pulse at the Larmor frequency excites a voxel of protons into the transverse plane. oGradients produce time-altering magnetic fields of linear- varying magnitude.

Model: VERSE Pulse oThe Variable Rate Selective Excitation (VERSE) rf pulse is a transverse excitation with a fraction of the field strength. oBy decreasing the duration of each sample and uniformly distributing signal amplitude, the VERSE pulse reduces SAR (Specific Absorption Rate). oSubsequently our objective becomes,

Model: Gradient oThe gradient is set to have linear-varying magnitude. o represents the transverse plane at a particular position depending on its specific coordinate value.

Model: Coordinate Positions Model: Coordinate Positions oSet and restrict to be a finite subset of, then partition the constraint into coordinate position values and. : Coordinate positions that are “in” the slice. : Coordinate positions that are “in” the slice. : Coordinate positions that are “outside” of the slice. : Coordinate positions that are “outside” of the slice.

Model: S in and S out oMagnetization vectors in will be tipped by an angle of. oMagnetization vectors in will not be tipped and remain at the initial magnetization value.

Model: Rotating Frame of Reference Model: Rotating Frame of Reference oThe main super-conducting magnet,, induces a rotating frame of reference. rotating frame of reference.

Model: Coordinate Positions Model: Coordinate Positions oNow external magnetization is a function of coordinate positions. o and are independent of. oThe same notation must be incorporated into net magnetization.

Model: Bloch Equation Model: Bloch Equation oSince VERSE pulses have short sampling times there is no proton interactions, hence, from the Bloch equation:

Model: Gradient and Slew Rate oSlew rate or gradient-echo rise time, identifies how fast a magnetic gradient field can be ramped to different field strengths. oFor our problem we bound gradient and slew rate,.

Model: Optimization Problem oThe semi-infinite nonlinear optimization problem,

Results: Initializations Results: Initializations o 5 Slice Problem:

Results: Results:

Future Work oUse 5 slices to interpolate 15 slices. oAdd spin-lattice and spin-spin proton interactions. oAdd rotation into the equations. oInvestigate other variations of VERSE pulses. oTest on MRI machine.