# The description of the MRI signal from the magnetization vector

I have been working on simulating the MRI signal and what I have done so far is simulate the actions of the RF pulse and the required localization gradients. So, in the simulation, I am looking at a gradient echo sequence and I have the RF pulse and the slice selection gradient which tips the spin magnetization in the XY plane. This is followed by a rephasing gradient (to take into account the dephasing due to the slice selection gradient).

After this, I have the PE gradient along the Y-direction and the frequency rewinder (to have a symmetrical echo) and the readout gradients along the X-direction.

In the simulator, I am tracking the magnetization vector during the whole process by solving the Bloch equations. Now, the part that I am confused about is how can I convert this to the MRI signal? At the end of the day for each spin, I have its magnetization at every time step. During the readout process, I would like to convert it to the MRI signal and I am not sure how to go about it.

You always measure the (vector) sum over all magnetization vectors in your object, $S(t) = \sum\limits_{(x,y)} M(x,y,t)$. This sum gives you an amplitude and a phase. That is your measured signal.
From the gradients that you simulate, you know the $k$-space position as a function of time, i.e. you have $k(t)$. Now you sample your signal at certain points and sort the signals $S(t_n)$ into the respective $k(t_n)$-space. A FFT should now convert your $k$-space into the image again.
• Thanks! it took me a while to figure this. So, basically the amplitude and the phase are the x and y components of the transverse magnetisation? – Luca May 31 '16 at 20:22
• The $x,y$-components can be rewritten to a complex form: $M_\perp = M_x + iM_y \Longrightarrow M_\perp = M\exp{i\phi}$ with $M = \sqrt{M_x^2 + M_y^2}$ and $\phi = \arctan{\frac{M_y}{M_x}}$ – M529 May 31 '16 at 20:42