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In this chapter the basic principles of MR imaging is presented. We will start at the demodulated complex signal of the precessing spins as it was at the end of the Signal detection section:
::{EQUATION()}
\label{NMR_signal_again}
S \propto \omega_0 \int \mathrm{d^3r} \hspace{3pt} \mathrm{e}^{ - \tfrac{t}{T_2 (\mathbf{r})} } M_{\bot} (\mathbf{r} , 0) \mathcal{B}^{rec}_{\bot} (\mathbf{r} ) \mathrm{e}^{ - \mathrm{i} \big ( \delta \omega t + \theta_B (\mathbf{r} ) - \phi (\mathbf{r} , t ) \big ) }
{EQUATION}(1)::
For the simplicity of the description, from now on we will assume the followings:
1) Appropriate RF coil, i. e. spatially independent {EQUATION()}$\mathcal{B}^{rec}${EQUATION} and {EQUATION()}$\theta_B${EQUATION}
2) The proportionality factors in (1) are summed up in one factor {EQUATION()}$\Lambda${EQUATION}
3) Relaxation effects are negligible since the relevant imaging process is fast related to their characteristic times
4) the signal demodulation is perfect, i. e. {EQUATION()}$\delta \omega = 0${EQUATION}
Provided these the signal takes the following form:
::{EQUATION()}
\label{NMR_signal_simplier}
S = \Lambda \omega_0 \mathcal{B}^{rec}_{\bot} \int \mathrm{d^3r} \hspace{3pt} M_{\bot} (\mathbf{r} , 0) \mathrm{e}^{ \mathrm{i} \phi (\mathbf{r} , t) }
{EQUATION}(2)::
Without a detailed explanation (from statistical physics, expected value of the grand canonical ensemble) the equilibrium value of the magnetization can be estimated as:
::{EQUATION()}
\label{Curie_suscept}
M_0 = \rho_0 \frac{S(S+1) \gamma^2 \hbar^2}{3 k_B T} B_0
{EQUATION}(3)::
The factor in (3) is the well-known Curie susceptibility with {EQUATION()}$S=\frac{1}{2}${EQUATION} for proton and {EQUATION()}$\rho_0${EQUATION} is the spatial spin density. As {EQUATION()}$M_{\bot}${EQUATION} in equation (2) means the amplitude of the transverse component right after the 90° pulse, it can be well approximated with the equilibrium magnetization {EQUATION()}$M_0${EQUATION} expressed in (3). Therefore, our signal can be written as the following.
::{EQUATION()}
\label{NMR_signal_simplier2}
S = \Lambda \omega_0 \mathcal{B}^{rec}_{\bot} \frac{1}{4} \frac{\gamma^2 \hbar^2}{k_B T} B_0 \int \mathrm{d^3r} \hspace{3pt} \rho_0 ( \mathbf{r} ) \mathrm{e}^{ \mathrm{i} \phi (\mathbf{r} , t) }
{EQUATION}(4)::
This can be simplified by introducing the so-called effective spin density {EQUATION()}$\rho ( \mathbf{r})${EQUATION}:
::{EQUATION()}
\label{effective_spin_density}
\rho ( \mathbf{r}) = \Lambda \omega_0 \mathcal{B}^{rec}_{\bot} \frac{1}{4} \frac{\gamma^2 \hbar^2}{k_B T} \rho_0 ( \mathbf{r}) B_0
{EQUATION}(5)::
::{EQUATION()}
\label{signal_phased_sum}
S = \int \mathrm{d^3r} \hspace{3pt} \rho ( \mathbf{r} ) \mathrm{e}^{ \mathrm{i} \phi (\mathbf{r} , t) }
{EQUATION}(6)::
As can be seen, the measureable signal is finally made up of the phase-correct sum of the effective spin density.
This is the point where move to the concept that fundamentally distinguishes imaging tecniques from traditional NMR experiments - the idea of gradient fields, and their effect to the magnetization phase.
If the main field {EQUATION()}$B_0${EQUATION} is perfectly homogeneous and the signal demodulation is complete then the phase term in (6) becomes time-independent as all the spins will precess with the same Larmor frequency. Moreover, if we define the starting time to be at the end of the 90° pulse, or as also referred, the excitation, then all the spins will point to the same direction, meaning all of them will have the same phase at {EQUATION()}$t=0${EQUATION}. In this case the phase term will also be spatially independent.
::{EQUATION()}
\label{phi_homogen}
\phi (\mathbf{r} , t) = \phi (\mathbf{r} , 0) = \phi_0
{EQUATION}(7)::
However if our main field is not the same everywhere then the phase of a certain spin will change in time with the local difference of the larmor frequency:
::{EQUATION()}
\label{phi_definition}
\phi (\mathbf{r} , t) = \int\limits_0^t \Delta \omega (\mathrm{r} , t') \mathrm{d} t' = - \gamma \int\limits_0^t \big [ B(\mathbf{r},t') - B_0 \big ] \mathrm{d} t'
{EQUATION}(8)::
The idea is the following. Let's add an additional field to the main field {EQUATION()}$B_0${EQUATION} which is not homogeneous but varies linearly in space. This new field is referred to as the gradient field and is characterised by its spatial gradient {EQUATION()}$\mathbf{G}${EQUATION}.
::{EQUATION()}
\label{B_definition}
\mathbf{B} = \big ( 0, 0, B(\mathbf{r} , t) \big )
{EQUATION}(9)::
::{EQUATION()}
\label{gradient_definition}
\mathbf{G} (t) = \nabla B (\mathbf{r}, t)
{EQUATION}(10)::
::{EQUATION()}
\label{B_definition_with_gradient}
B(\mathbf{r}, t) = B_0 + \mathbf{G}(t) \mathbf{r}
{EQUATION}(11)::
With this gradient field existing the spin phases will develop in a manner defined by the gradient field.
::{EQUATION()}
\label{phi_gradient}
\phi (\mathbf{r} , t) = \int\limits_0^t \Delta \omega (\mathrm{r} , t') \mathrm{d} t' = - \gamma \int\limits_0^t \big [ B(\mathbf{r},t') - B_0 \big ] \mathrm{d} t' = - \gamma \mathbf{r} \int\limits_0^t \mathbf{G} (t') \mathrm{d} t'
{EQUATION}(12)::
Here we define a new vector {EQUATION()}$\mathbf{k}${EQUATION} which will eventually play an extremely important role in the imaging process.
::{EQUATION()}
\label{k_vector}
\mathbf{k} \equiv \frac{\gamma}{2 \pi}\int \limits_0^t \mathbf{G} (t') \mathrm{d} t'
{EQUATION}(13)::
As shown, this {EQUATION()}$\mathbf{k}${EQUATION} is defined by the time integral of the gradient and therefore can be easily manipulated by switching different gradients on and off.
With this new vector {EQUATION()}$\mathbf{k}${EQUATION} the spin phases can be expressed quite simple:
::{EQUATION()}
\label{phi_gradient2}
\phi (\mathbf{r} , t) = - 2 \pi \mathbf{k} \mathbf{r}
{EQUATION}(14)::
By substituting this into (6) we get an astonishing outcome:
::{EQUATION()}
\label{signal_fourier}
S = \int \mathrm{d^3r} \hspace{3pt} \rho ( \mathbf{r} ) \mathrm{e}^{- \mathrm{i} 2 \pi \mathbf{k} \mathbf{r} } = \mathcal{F}\left\{ \rho ( \mathbf{r} ) \right\}
{EQUATION}(15)::
The result in (15) is of enormous importance. It tells us that by the use of appropriate gradient fields the connection between the acquired signal and the spatial effective spin density, or in other words, the image of the sample or the patient, becomes the well-known and easy-to-compute operation, the Fourier transformation. In this interpretation the previously defined vector {EQUATION()}$\mathbf{k}${EQUATION} gains a new meaning: this is the spatial frequency, the conjugate of the spatial coordinate {EQUATION()}$\mathbf{r}${EQUATION}.
Therefore to get an image we just have to apply the gradient fields in a way to achieve a sufficient set of {EQUATION()}$\mathbf{k}${EQUATION} vectors and then perform an inverse Fourier transform on the acquisited signal. In this way our signal is no longer a function of time, but a function of the spatial frequency {EQUATION()}$\mathbf{k}${EQUATION}.
::{EQUATION()}
\label{signal_inverse_fourier}
\rho (\mathbf{r}) = \mathcal{F}^{-1} \left\{ S(\mathbf{k}) \right\}
{EQUATION}(16)::
To put this concept into a form easier to imagine, the alignment of the spins before and after the application of a gradient for a certain time is illustrated in Figure 1. On the left, before the gradient effect, all the spins point to the same direction, i. e. they all have the same phase. On the right we can see the spins after some time in the upwards-pointing gradient field and therefore folded into a helix. This is because the magnetic field and thus the Larmor frequency vary linearly along the direction of the gradient, causing the spins to accumulate a phase depeding linearly on their position. If the gradient field continiues to exist the helix will have more and more twists on the same length as time elapses, accordingly to the growth of the spatial frequency.
{img fileId="3175" thumb="y" rel="box[g]" width="500" imalign="center" align="center" desc=Figure 1. Effect of the gradient field. Left: Aligned spins before the gradient. Right: Spins folded into a helix after the application of a gradient field pointing upwards.}