NMR Magnetic Moment: Understanding Bloch Equations and Relaxation Terms

[the_kid]

Let M be the magnetic moment of a system. Below are the Bloch equations, including the relaxation terms.

dM_x/dt=({\bf M} \times \gamma {\bf H_0})_x-M_x/T_2
dM_y/dt=({\bf M} \times \gamma {\bf H_0})_y-M_y/T_2
dM_z/dt=({\bf M} \times \gamma {\bf H_0})_z+(M_{\infty}-M_z)/T1

At t=0, {\bf M}=(0,0,M_{\infty}).

{\bf H_0}=H_0 {\bf k'} where primed coordinates are in the lab frame.

Now suppose an on resonance pulse is applied along the i direction of the rotating frame for T_{pi/2}=0.005 milliseconds, then it is turned off to watch the free induction decay. T_2=5 milliseconds, T_1=5000 milliseconds.

So, naturally we will have nutation due to the pulse, T_2 decay of the transverse magnetization, and T_1 recovery of the longitudinal magnetization. Due to the timescales, they will proceed sequentially.

I'm trying to sketch the time evolution of the above three components of the magnetic moment in both the rotating frame and lab frame. I'm supposed to zoom in on the interesting regions; i.e. where the aforementioned behavior occurs. I'm having some trouble understanding how these processes are affected by the parameters. Could anyone assist me with this?

 

[Naty1]

This is a stab in the dark, but maybe the illustration here helps?

http://en.wikipedia.org/wiki/MRI#Applications

 

[Dale]

Could anyone assist me with this?

Do you have Mathematica? I have a notebook for calculating the Bloch equations.

 

[the_kid]

Do you have Mathematica? I have a notebook for calculating the Bloch equations.

I do have Mathematica!

 

[Dale]

I do have Mathematica!

Here you go. The notebook has two methods of calculating the Bloch equations for a single isochromat. Both methods are in the rotating frame, not the lab frame. You will either need to modify them or treat the lab frame as a really big off resonance.

The first assumes that the RF pulses are very short compared to the relaxation times and off-resonance frequency. I.e. they can be represented by simple tips. It is based on matrix multiplication. So in your case you would make one matrix representing the RF pulse followed by one representing the free precession then you would calculate the final magnetization by: M_F = FP \cdot RF \cdot M_I. This can give you closed-form analytical solutions. Note that it has a strange approach. The magnetization vectors have 4 dimensions: {Mx, My, Mz, M0} where M0 is the fully-relaxed longitudinal magnetization. So a fully relaxed isochromat of unit magnetization would be {0,0,1,1}.

The second uses the numerical differential equation solver and doesn't make any assumptions. With that one you express things like RF and gradients in terms of functions that are passed as options to the solver. I have several examples. My approach would be to do the rotating frame using the numerical solver, show that there is no appreciable relaxation during the RF pulse, and then use the matrix solver to get an analytical solution for the lab frame. The numerical solver will have trouble with all of the cycles for the lab frame.

Enjoy.

 

[the_kid]

I'm still having some trouble visualizing these processes...

 

[Dale]

To visualize the best thing is to consider only the rotating frame and to make some simplifying assumptions.

First, for an RF pulse consider T1 and T2 to be infinite and consider H0 to be purely along the x axis. That just causes M to rotate about x.

Then, for T2 decay consider T1 to be infinite and H0 to be zero. That just causes any transverse magnetization to decay.

Then, for T1 regrowth consider T2 to be really short and H0 to be zero. That just causes longitudinal magnetization to relax.