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I am trying to understand the imaging principles behind MRI and I was looking at some lecture slides found here

Specifically, I am looking at slide 41 where we look at some of the equations regarding the MRI signal.

I understand that when a proton is subjected to an external magnetic field, it precesses with an angular frequency which is proportional to the field strength. This is given by the Larmor equation.

$$ \omega = \gamma B_0 $$

Ignoring the other gradient components (to altar the main magnetic field at some spatial points), we see that the instantaneous phase would just be the integral of this frequency:

$$ \phi(t) = \int \gamma B_0 dt $$

Now the signal given by one of the spin particles (elemental signal in the lecture) is given by:

$$ s(t) = \rho \exp (i \phi(t)) $$

where $\rho$ is the magnetization from the particle. I do not understand why this exponential term is there. Basically, why does the MRI signal contribution from a particle have this form? It is not intuitive to me at all. I think it might have something to do with the quadrature detection but I am not convinced at all at the moment.

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MRI signal is always complex and it is related with signal demodulation. The detected signal is multiplied by a sinusoid or cosinusoid with frequency equals to $\omega_0 +\delta \omega$, respectively leading to the real and imaginary channels. You can find the complete algebra at $Haacke,\ Magnetic\ resonance\ imaging$ chapter 7.3.3

Phase is really useful in MRI as it leads to informations about magnetic susceptibility distribution in the sample which found application in the diagnosis of many diseases, such as ALS and MS.

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