In today's lab we will examine the basic principles of Magnetic Resonance Imaging (MRI). Review the MRI lectures MRI - introduction, principles, excitation, relaxation and MRI - spin-echo, Fourier imaging . A very rich information source on the topic of MRI with a lot of illustrations and animations offers MRI Questions. The relationship between the Bloch equations and the relaxation times is shown here.

• [3 pts] Repeated excitation Examine the time courses of magnetization for different materials while varying the sequence parameters.
• [2 pts] Resonance frequency of spins Compute the gradient pulse strength so that the excitation affects only a limited space.

MATLAB scripts necessary for the homework are here.

Visualize magnetization time course during repeated excitation by $90^\circ$ pulse. Examine the magnetization time courses for materials with different values of $T_1, T_2$ for different values of repetition time TR and echo time TE. Consider combinations of low and high values of TE or TR.

Use the weighting.m script to visualize the time course of the transversal magnetization $M_{xy}$ (red curve) and longitudinal magnetization $M_z$ (blue curve) for a sequence of $90^\circ$ RF pulses. The resulting visualization contains two parts, both of them show the time courses of magnetization for two materials (solid and dashed lines). In the top part both the simulated materials have the same $T_1$ times but different $T_2$ times, in the bottom part the materials have the same $T_2$ times but different $T_1$ times. The vertical lines indicate the repetition time TR (solid line) and echo time TE (dashed line).

The amplitude of the echo measured at the echo time TE with repeated excitation with repetition time TR can be simplified by the equation

$$U = \rho \cdot (1 - e^{-\frac{TR}{T_1}}) \cdot e^{-\frac{TE}{T_2}}$$

Do the observation agree with this equation? Try to vary the parameters TE and TR (try combinations of low and high values) and find out which of the three factors in the equation for a given combination affects the signal amplitude. Do the results correspond to the magnetization time courses with repeated excitation? What are the implications for MRI (T1-weighted and T2-weighted).

Using gradients in the magnetic field we can locally change the resonance frequency of spins. We can excite only atoms in a selected area by using an excitation pulse that contains frequencies in certain range. This is called the frequency encoding.

Run the selection.m script. The visualization produced by the script shows the excitation pulse. What frequency range does this excitation pulse contain? Consider only the frequencies with the magnitude greater than the half of the maximum magnitude (an approximation by a rectangular pulse).

The second visualization shows the magnetization on several points located on the x axis (the y and z coordinates are zero). With the default values the result of the selection.m script will look different.

In our case the excitation pulse is given and we will try to adjust gradient in the direction of the x-axis.

The goal is to excite only the area in the range $[-2~mm, 2~mm]$ on the x axis. To excite only the selected points the resonance frequency at the $2~mm$ coordinate has to be equal to the maximal frequency of the RF pulse. Use the following equation to compute the slope of the magnitude of magnetic field.

$f = 42.58\cdot 10^6 \cdot B$

where f is the Larmor (resonance) frequency, $42.58~MHz \cdot T^{-1}$ is the gyromagnetic ratio of hydrogen and B is the magnitude of the magnetic field. This formula serves only to estimate the gradient field.

The actual magnetic field in an actual scanner is the superposition of the static $B_0$ and several (linear) gradient fields; however in our example there is no stationary magnetic field, otherwise the relationship between the Larmor frequency and magnitude of the magnetic field would not work.

Be careful with the units of measurement – the frequency is in kHz, the coordinate of the point where you are looking for the magnetization B is $2~mm$. The input to the Grad variable in the selection.m script is in [T/mm].

So the procedure could be as follows:

1. Estimate the maximum frequency from the spectrum in the right part of Figure 1.
2. Compute the gradient of the magnetic field $b(x)$ using the formula above.
3. Adjust the gradient according to the size of the area you want to excite.
4. Input the adjusted slope to the selection.m script and produce the results.

Adjust the gradient and produce a time course for the range $[-3~mm, 3~mm]$. In your report, show how you computed the gradient magnitude and include also the resulting time courses of magnetization (the second visualization from the selection.m script).

More information and illustrations to the frequency encoding can be found at Questions and answers in MRI and Video lectures.

The simulator relies on mex files in Matlab, which link Matlab function to fast programs written in C. The bundle of functions for this lab contains mex files for many operating systems (32 and 64bit Windows, 32 and 64 bit Linux, Mac), but sometimes the old mex files don't work. In that case you can create your own mex files in Matlab by navigating to the simulator folder (blochsim) and executing:

 mex bloch.c -compatibleArrayDims