Solutions of the Bloch equations for MRT

In summary, the conversation is about the relationship between the upper equations and the Bloch equations. It is confirmed that the upper equations are special solutions of the Bloch equations and they apply when there is no RF being transmitted. It is also clarified that the upper equations are valid after the end of a 90° excitation pulse as long as no further RF pulse is irradiated. The lower three solutions are the magnetization component from the perspective of the detector coil and have slight differences compared to the upper equations. These differences include assuming that the spins are on-resonance and the phase of the transverse component is constant, as well as assuming a longitudinal magnetization of 0 at t=0.
  • #1
Derbyshire
4
0
Hello all,

I have a question about the relationship between
1691101354726.png

1691101337116.png

resp.
1691101317270.png


and the Bloch equations
1691101375370.png
.

Are these upper equations special solutions of the Bloch equations? If yes, under what condition(s) do the solutions hold?

Thanks in advance for helpful support!
 
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  • #2
Yes, those are the free precession equations. They apply when there is no RF being transmitted.
 
  • #3
Thank you. So the upper 5 equations are valid e.g. after the end of a 90° excitation pulse as long as no further RF pulse is irradiated?
 
  • #4
Derbyshire said:
Thank you. So the upper 5 equations are valid e.g. after the end of a 90° excitation pulse as long as no further RF pulse is irradiated?
Yes, that is right
 
  • #5
Thanks Dale,

one more question: i assume that the lower three solutions are the magnetization component from "view" of the detector coil, right?
 
  • #7
Thanks!
 

FAQ: Solutions of the Bloch equations for MRT

What are the Bloch equations in the context of Magnetic Resonance Imaging (MRI)?

The Bloch equations are a set of differential equations that describe the dynamics of nuclear magnetization in the presence of magnetic fields. In MRI, they are used to model how the net magnetization vector of spins in a sample evolves over time under the influence of external magnetic fields, including the main magnetic field and any applied radiofrequency pulses.

How do you solve the Bloch equations for a given magnetic resonance experiment?

Solving the Bloch equations typically involves numerical methods due to their complexity, especially when time-dependent fields are involved. Analytical solutions are possible for simple cases, such as constant or piecewise constant fields. The solutions provide the magnetization vector as a function of time, which can be used to predict the signal detected in MRI.

What initial conditions are commonly used when solving the Bloch equations?

Initial conditions for the Bloch equations usually assume that the magnetization vector is aligned with the main magnetic field (typically the z-axis). This corresponds to the thermal equilibrium state of the spins before any radiofrequency pulses are applied. Mathematically, this is represented as M(0) = (0, 0, M0), where M0 is the equilibrium magnetization.

How do relaxation times T1 and T2 affect the solutions of the Bloch equations?

Relaxation times T1 and T2 are critical parameters in the Bloch equations. T1 (longitudinal relaxation time) describes the rate at which the magnetization vector returns to its equilibrium value along the z-axis. T2 (transverse relaxation time) describes the dephasing of the magnetization in the xy-plane. These times determine how quickly the magnetization recovers and loses coherence, respectively, and significantly influence the MRI signal and image contrast.

Can the Bloch equations be extended to include more complex effects such as diffusion or inhomogeneous fields?

Yes, the Bloch equations can be extended to include additional effects such as diffusion and inhomogeneous magnetic fields. For example, the Bloch-Torrey equations incorporate diffusion by adding a diffusion term to the standard Bloch equations. Inhomogeneous fields can be modeled by allowing the magnetic field terms to vary spatially. These extensions are important for accurately modeling the behavior of spins in more complex and realistic scenarios.

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