Variable Rate Selective Variable Rate Selective Excitation Radio Frequency Excitation Radio Frequency Pulse in Magnetic Resonance Pulse in Magnetic Resonance

ImagingImaging

Stephen Stoyan Stephen Stoyan

OverviewOverview

o MRI BackgroundMRI Backgroundo ModelModelo ResultsResultso Future WorkFuture Work

MRI Background: MRI Background: Basics of MRIBasics of MRI

Radio Frequency (rf) pulses excite the Radio Frequency (rf) pulses excite the samplesample

Field gradients spatial encode the sampleField gradients spatial encode the sample Large uniform static rf coil receives a signalLarge uniform static rf coil receives a signal external magnetic fieldexternal magnetic field

Signal: AmplifiedSignal: Amplified DigitizedDigitized

Fourier-TransformedFourier-Transformed

MRI Background: MRI Background: MagnetizationMagnetizationo Nuclei with odd atomic weight and/or odd atomic number posses Nuclei with odd atomic weight and/or odd atomic number posses

an angular momentum.an angular momentum.o Any electrically charged particle which moves creates a Any electrically charged particle which moves creates a

magnetic field called a magnetic moment.magnetic field called a magnetic moment.o An ensemble of nuclei produce a ‘spin system.’ An ensemble of nuclei produce a ‘spin system.’

o When an external magnetic field is applied the magnetic When an external magnetic field is applied the magnetic moments align in the direction of the field.moments align in the direction of the field.

MRI Background: MRI Background: MagnetizationMagnetizationo Magnetization is the net vector quantity of the magnetic Magnetization is the net vector quantity of the magnetic

moments of each nuclei in a given unit volume or voxel.moments of each nuclei in a given unit volume or voxel.

o Given an external magnetic Given an external magnetic field, magnetic moment field, magnetic moment vectorsvectors rotate around the rotate around the axis of the field. axis of the field.

o This secondary spin is This secondary spin is termed, Precession.termed, Precession.

MRI Background: MRI Background: PrecessionPrecession

o The speed of proton precession The speed of proton precession is referred as, Precessional is referred as, Precessional FrequencyFrequency..

o Stronger magnetic fields Stronger magnetic fields constitute higher precessional constitute higher precessional frequencies.frequencies.

o The frequency at which the nucleus will absorb energy is The frequency at which the nucleus will absorb energy is described in the Larmor equation.described in the Larmor equation.

MRI Background: MRI Background: InteractionsInteractionso The equation for torque on a magnetic moment due to an external The equation for torque on a magnetic moment due to an external

magnetic field,magnetic field,

o Making the substitution into ,Making the substitution into ,

Proton interactionsProton interactionso Spin-lattice interactions: A magnetic moments minimum Spin-lattice interactions: A magnetic moments minimum

energy state is in the energy state is in the direction of the direction of the external external magnetic field.magnetic field.

o Spin-spin interactions: Magnetic moments experience local Spin-spin interactions: Magnetic moments experience local fields of their neighbours fields of their neighbours

and the and the applied field. applied field.

MRI Background:MRI Background: Bloch Equation Bloch Equationo Combining proton interactions into equation produces Combining proton interactions into equation produces

the Bloch equation,the Bloch equation,

where,where,

Model: Model: General rf PulseGeneral rf Pulseo In processing an image a precise radio frequency (rf) pulse is In processing an image a precise radio frequency (rf) pulse is

applied in combination with a synchronized gradient.applied in combination with a synchronized gradient.

o An rf pulse at the Larmor frequency excites a voxel of protons An rf pulse at the Larmor frequency excites a voxel of protons into the transverse plane.into the transverse plane.

o Gradients produce time-altering magnetic fields of linear-Gradients produce time-altering magnetic fields of linear-varying magnitude.varying magnitude.

Model:Model: VERSE Pulse VERSE Pulseo The Variable Rate Selective Excitation (VERSE) rf pulse is a transverse The Variable Rate Selective Excitation (VERSE) rf pulse is a transverse

excitation with a fraction of the field strength. excitation with a fraction of the field strength. o By decreasing the duration of each sample and uniformly distributing By decreasing the duration of each sample and uniformly distributing

signal amplitude, the VERSE pulse reduces SAR (Specific Absorption signal amplitude, the VERSE pulse reduces SAR (Specific Absorption Rate).Rate).

o Subsequently our objective becomes,Subsequently our objective becomes,

Model: Model: GradientGradiento The gradient is set to have linear-varying magnitude.The gradient is set to have linear-varying magnitude.

o represents the transverse plane at a particular position represents the transverse plane at a particular position depending on its specific coordinate valuedepending on its specific coordinate value..

Model: Model: Coordinate PositionsCoordinate Positionso Set and restrict to be a finite subset of , then Set and restrict to be a finite subset of , then

partition the constraint into coordinate position values partition the constraint into coordinate position values and .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:Model: S Sinin and S and Soutout

o Magnetization vectors in will be tipped by an angle of .Magnetization vectors in will be tipped by an angle of .

o Magnetization vectors in will not be tipped and remain at Magnetization vectors in will not be tipped and remain at

the initial magnetization valuethe initial magnetization value..

Model: Model: Rotating Frame of ReferenceRotating Frame of Reference

o The main super-conducting magnet, , induces a The main super-conducting magnet, , induces a

rotating frame of reference.rotating frame of reference.

Model: Model: Coordinate PositionsCoordinate Positionso Now external magnetization is a function of Now external magnetization is a function of

coordinate positions .coordinate positions .

o and are independent of .and are independent of .o The same notation must be incorporated into net The same notation must be incorporated into net

magnetization.magnetization.

Model: Model: Bloch EquationBloch Equationo Since VERSE pulses have short sampling times there is no proton Since VERSE pulses have short sampling times there is no proton

interactions, hence, from the Bloch equation:interactions, hence, from the Bloch equation:

Model: Model: Gradient and Slew RateGradient and Slew Rate

o Slew rate or gradient-echo rise time, identifies how fast a magnetic Slew rate or gradient-echo rise time, identifies how fast a magnetic gradient field can be ramped to different field strengths.gradient field can be ramped to different field strengths.

o For our problem we bound gradient and slew rate, .For our problem we bound gradient and slew rate, .

Model:Model: Optimization Problem Optimization Problemo The semi-infinite nonlinear optimization problem,The semi-infinite nonlinear optimization problem,

Results: Results: InitializationsInitializations

o 5 Slice Problem:5 Slice Problem:

Results: Results:

Results:Results:

Results:Results:

Results:Results:

Results:Results:

Results:Results:

Results:Results:

Results:Results:

Future WorkFuture Worko Use 5 slices to interpolate 15 slices.Use 5 slices to interpolate 15 slices.o Add spin-lattice and spin-spin proton interactions.Add spin-lattice and spin-spin proton interactions.o Add rotation into the equations.Add rotation into the equations.o Investigate other variations of VERSE pulses. Investigate other variations of VERSE pulses. o Test on MRI machine.Test on MRI machine.