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kbansal
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- Homework Statement
- So my professor in my "physics of medical imaging for engineers" class has given us the option for extra credit if we do our own research and describe the underlying quantum mechanics and math behind the various stages of MRI imaging and contrast them with the less detailed explanations we have been given in class. I've attached what I've written so far below.
I am an engineer, not a physicist, but I figured I'd give it a shot after reading some textbooks and other physics resources online.
After looking around for a while this is the explanation I've come up with so far. I have a feeling it's not really correct (or phrased correctly), but I would really like to know if I'm on the right track. Because I have no other way of getting this checked. I'm probably going to just not do the extra credit if my explanation is really wrong. I feel like the information I'm giving isn't really what he's asking for and will just make me look really dumb. Any help would be greatly appreciated.
The last paragraph is just over the signal processing so you don't need to read that if you don't want. In theory I'm supposed to understand that well already lol.
- Relevant Equations
- {\displaystyle {\hat {H}}=-\mathbf {m} \mathbf {B_{0}} =-{\tfrac {\hbar }{2}}\gamma \sigma _{z}\mathbf {B_{0}} =-{\tfrac {\hbar }{2}}\omega _{0}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}
{\displaystyle {\hat {H}}=-\mathbf {m} \mathbf {B_{0}} =-{\tfrac {\hbar }{2}}\gamma \sigma _{z}\mathbf {B_{0}} =-{\tfrac {\hbar }{2}}\omega _{0}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}
{\displaystyle P_{12}={\frac {|\omega _{1}^{2}|}{|\Delta \omega ^{2}+\omega _{1}^{2}|}}\sin ^{2}[{\sqrt {\omega ^{2}+\Delta \omega ^{2}}}t/2]}
"B0 is a static magnetic field (produced by a superconducting magnet) that initially causes the protons in the body to align with the field and precess at the larmor frequency along the z axis .
From a mathematical perspective this precession around the B0 axis occurs due to the time evolution operator (which in this case is found by solving the time dependent Schrodinger equation using the Hamiltonian based on the magnetic moment of a ½ spin particle) being applied to the spin state of the proton.
From a general perspective, RF coils are used to perturb these precessing protons periodically with magnetic field B1 and cause resonance when the energy of the pulse is equal to the of the larmor frequency at that slice. The RF pulse causes the protons to go from aligning along the Z axis to the transverse plane. In order for the proton to return back to equilibrium along the z axis a radio wave (at the same energy of the larmor frequency) must be emitted.
From a mathematical perspective The B1 magnetic field is pulsed from RF coils at a frequency which gives the maximum probability that the eigenstate of the original unperturbed Hamiltonian (which is based on the larmor frequency and B0) will transition to an eigenstate of a new two state Hamiltonian after perturbation. The probability that a transition will occur between the states of the 2 state system at a given frequency is described by the Rabi oscillation equation for particles of ½ spin. Resonance (the emission of a radio wave from protons going back to equilibrium with B0) only occurs when B1 oscillates at the same rate/energy as the dipoles precessing due to B0.
The spatial information of the MRI image is gotten by using gradient coils along x,y, and z. These gradients select the image slice by varying the larmor frequency at each position and making it depend on space rather than having a uniform larmor frequency across the body. So only protons that are part of the slice are what resonate. The initial spatial information is formalized in K space as Kx and Ky of the signal. Kx and Ky, along with the coordinates x and y are used in the inverse Fast Fourier Transform with respect to signal S(Kx, Ky). This inverse FFT is proportional to the proton density in space, which is what gives the spatial information. The FFT uses a discrete Fourier transform because all of the data for the MRI signal is a fully known discrete set of values. The resonance signal of each slice is detected by the RF coils, and the frequency that the RF coils send to the precessing protons changes with each slice depending on how the gradient modifies the larmor frequency."
From a mathematical perspective this precession around the B0 axis occurs due to the time evolution operator (which in this case is found by solving the time dependent Schrodinger equation using the Hamiltonian based on the magnetic moment of a ½ spin particle) being applied to the spin state of the proton.
From a general perspective, RF coils are used to perturb these precessing protons periodically with magnetic field B1 and cause resonance when the energy of the pulse is equal to the of the larmor frequency at that slice. The RF pulse causes the protons to go from aligning along the Z axis to the transverse plane. In order for the proton to return back to equilibrium along the z axis a radio wave (at the same energy of the larmor frequency) must be emitted.
From a mathematical perspective The B1 magnetic field is pulsed from RF coils at a frequency which gives the maximum probability that the eigenstate of the original unperturbed Hamiltonian (which is based on the larmor frequency and B0) will transition to an eigenstate of a new two state Hamiltonian after perturbation. The probability that a transition will occur between the states of the 2 state system at a given frequency is described by the Rabi oscillation equation for particles of ½ spin. Resonance (the emission of a radio wave from protons going back to equilibrium with B0) only occurs when B1 oscillates at the same rate/energy as the dipoles precessing due to B0.
The spatial information of the MRI image is gotten by using gradient coils along x,y, and z. These gradients select the image slice by varying the larmor frequency at each position and making it depend on space rather than having a uniform larmor frequency across the body. So only protons that are part of the slice are what resonate. The initial spatial information is formalized in K space as Kx and Ky of the signal. Kx and Ky, along with the coordinates x and y are used in the inverse Fast Fourier Transform with respect to signal S(Kx, Ky). This inverse FFT is proportional to the proton density in space, which is what gives the spatial information. The FFT uses a discrete Fourier transform because all of the data for the MRI signal is a fully known discrete set of values. The resonance signal of each slice is detected by the RF coils, and the frequency that the RF coils send to the precessing protons changes with each slice depending on how the gradient modifies the larmor frequency."
