Frederik Laun^{1}

Following this lecture, the audience will understand the effect of magnetic susceptibility, flow, chemical shift, and motion on MRI. They will be able to use this knowledge to identify and minimize related artifacts or, alternatively, to measure the underlying effects.

The transversal magnetization of water protons in the main magnetic field $$$B_0=|(0,0,B_0)^T| $$$ rotates with the angular frequency $$$\omega_0=-\gamma B_0 $$$ around the z-axis. Here, $$$ \gamma $$$ is the gyromagnetic ratio. In the following, the reference frame rotating with $$$ \omega_0$$$ is used.

In this lecture, the following effects are considered:

**Chemical shift
**

Protons residing in a different
chemical environment than water have a slightly different rotation frequency,
i.e. $$$\omega \rightarrow \gamma B_0 (1+\sigma) $$$ with the chemical shift
$$$ \sigma$$$. E.g. for the dominant chemical
group in fat, the aliphatic CH_{2} group, a chemical shift of 3.5 ppm
arises with respect to water (1).

**Magnetic
susceptibility
**

Human tissue becomes slightly magnetized in a magnetic field according to the equation $$$\mathbf{M}=\chi \mathbf{H}$$$ with the magnetic susceptibility $$$\chi$$$ (2). $$$\chi$$$ is a rather small unitless quantity (roughly 1 ppm) and is larger or smaller than zero for paramagnetic or diamagnetic tissue, respectively. In general, the magnetic susceptibility and the magnetization dependent on position, i.e. they may be noted as $$$\mathbf{M}(\mathbf{r})$$$ and $$$\chi(\mathbf{r})$$$. A point source, i.e. an infinitesimal volume $$$dV$$$ at position $$$\mathbf{r}_1$$$ with magnetization $$$\mathbf{M}(\mathbf{r}_1)$$$, generates the additional magnetic field $$$\Delta B_{\textrm{point}}(\mathbf{r}-\mathbf{r}_1)=B_0(4\pi)^{-1}(3 \cos^2(\theta)-1)|\mathbf{r}-\mathbf{r}_1|^{-3}dV$$$. Here, $$$\theta$$$ is the angle between z-axis and $$$\mathbf{r}-\mathbf{r}_1$$$. These additional fields add up linearly if more than one point source is present. Consequently the complete field perturbation is $$$\Delta B(\mathbf{r})=\Delta B_{\textrm{point}}(\mathbf{r})*\chi(\mathbf{r})$$$, where “$$$*$$$” denotes a convolution. If all tissues had the same magnetic susceptibility, then $$$\Delta B(\mathbf{r})$$$ would have almost no spatial dependency. But if tissue interfaces – in particular tissue-air interfaces – are present, a stronger spatial dependency of $$$\Delta B(\mathbf{r})$$$ arises resulting in $$$\omega(\mathbf{r})=\gamma \Delta B(\mathbf{r}) $$$.

**
Flow
and motion**

Some spins in the human body move on a macroscopic scale and their position becomes dependent on time $$$t$$$, i.e. $$$\mathbf{r}\rightarrow \mathbf{r}(t)$$$ . For example, blood often flows approximately coherently with a velocity $$$\mathbf{v}$$$ so that $$$\mathbf{r}(t)=\mathbf{r}(0)+\mathbf{v}t$$$. If a magnetic field gradient $$$\mathbf{G}(t)$$$ is applied, then the angular frequency becomes $$$\omega(\mathbf{r})=\gamma \mathbf{G}(t) \cdot \mathbf{r}(t)=\gamma \mathbf{G}(t) \cdot (\mathbf{r}(0)+\mathbf{v}t)$$$.

**Chemical shift**

(a) Single line sequences. Applying a readout gradient $$$G_{\textrm{read}}$$$ along x-direction, the angular frequency of water protons is $$$\omega_{water}=\gamma G_{\textrm{read}}x$$$. The angular frequency of chemically shifted protons is $$$\omega_{CS}=\gamma G_{\textrm{read}}x+\gamma B_0 \sigma$$$. Detecting the signal, one does not know its origin, one just sees a signal component at angular frequency $$$\omega_{\textrm{measured}}$$$. In image reconstruction, the following positions are attributed to this angular frequency. For water: $$$x_{\textrm{water}}=\gamma^{-1}G_{\textrm{read}}^{-1}\omega_{\textrm{measured}}=x$$$ thus finding the true position. For the chemically shifted group: $$$ x_{\textrm{CS}}=\gamma^{-1}G_\textrm{{read}}^{-1}\omega_{\textrm{measured}}-G_{\textrm{read}}^{-1}B_0\sigma=x+\Delta x_{\textrm{CS}}$$$ with $$$\Delta x_{\textrm{CS}}=-G_{\textrm{read}}^{-1}B_0\sigma$$$ thus finding an incorrect position shifted by $$$\Delta x_{\textrm{CS}}$$$ in read direction.

(b) Echo planar imaging (EPI). In EPI, the shift $$$\Delta x_{\textrm{CS}}$$$ occurs dominantly along phase direction, since $$$G_{\textrm{phase}}$$$ is effectively much smaller than $$$G_{\textrm{read}}$$$. The shift is usually much larger than for single line sequences, which makes necessary a good fat suppression (see e.g. chapter 12 of (3)).

(c) Spectroscopy. For example, using free induction decay signal acquisition, the frequency composition of spins contained in the measured volume can be retrieved, which reveals the presence of molecules other than water (4).

**Magnetic
susceptibility**

(a) Phase images. Due to $$$\Delta B(\mathbf{r})$$$, an additional phase $$$\varphi(\mathbf{r},t)=\gamma \Delta B(\mathbf{r})t$$$ arises, which can be depicted in “phase images”.

(b) T2’ signal decay. If $$$\varphi(\mathbf{r},t)$$$ varies within a voxel, then this phase dispersion results in a reduced signal in this voxel. This effect becomes more pronounced at larger $$$t$$$. Spin echoes can be used to mitigate this effect if particle motion is negligible (static dephasing regime).

(c) Susceptibility-weighted imaging (SWI) (5,6). Images are generated by multiplying magnitude images with a function $$$f_\varphi(\varphi(\mathbf{r},t))$$$, which is, e.g., defined as: 1 for $$$\varphi>0$$$, and $$$\varphi\pi^{-1}$$$ for $$$\varphi<0$$$ using a high-pass filtered map of $$$\varphi(\mathbf{r},t)$$$. Consequently, the image is darkened for negative $$$\varphi(\mathbf{r},t)$$$, i.e. in veins, iron containing tissue, and hemorrhage.

(d) Quantitative susceptibility mapping (QSM). In QSM (7), phase images, i.e. $$$\varphi(\mathbf{r},t)$$$, are measured allowing to compute $$$\Delta B(\mathbf{r})=\gamma^{-1}t^{-1}\varphi(\mathbf{r},t)$$$. Then the equation $$$\Delta B(\mathbf{r})=\Delta B_{point}(\mathbf{r})*\chi(r)$$$ is used to compute $$$\chi(\mathbf{r})$$$. This involves the use of advanced post processing techniques taking care of issues such as phase unwrapping, background field removal, and the actual field-to-susceptibility inversion. QSM can be used, for example, to detect demyelination (8), calcifications (9) and to differentiate them from hemorrhage (10). Since $$$\chi(\mathbf{r})$$$ is a tensor quantity in myelinated tissue, it may be used to perform fiber susceptibility tensor imaging (STI) and fiber tracking (11).

(e) Image distortions. In the same manner as for chemical shifts, $$$\Delta B(\mathbf{r})$$$ can cause image distortions (replace $$$\Delta \chi_{CS}$$$ by $$$\Delta x_{\Delta B}(\mathbf{r})=-G_{read}^{-1}\Delta B(r)$$$).

**Flow
and motion**

(a) Flow compensation. Since $$$\varphi(\mathbf{r},t)=\gamma\int \mathbf{G}(t) \cdot (\mathbf{r}(0)+\mathbf{v}t)dt=\mathbf{k} \cdot \mathbf{r}(0) + \mathbf{k}_1 \cdot \mathbf{v}$$$ with $$$\mathbf{k}=\gamma\int \mathbf{G}(t) dt$$$ and $$$\mathbf{k}_1=\gamma \int \mathbf{G}(t) t dt$$$, adapting the temporal evolution of the applied gradients such that $$$\mathbf{k}_1=0$$$ suppresses the effect of flow on images (see e.g. chapter 5 of (12)). This also minimizes ghosts arising in phase direction originating from blood in major vessels.

(b) Measuring flow velocity. Performing two exams with different $$$\mathbf{k}_1$$$, e.g. by using additional bipolar gradient pulses, allows one to extract $$$\mathbf{v}$$$ from phase maps (see e.g. chapter 11 of (12)). This can be used to measure blood flow in major vessels.

(c) Triggering/gating. Repetitive body motions can be monitored externally (e.g. by ECG, pulse oximeter, breathing belts) or by means of MRI data (self-gating, see e.g. (13)). To suppress the effect of this motion, the signal is only acquired in the desired time intervals (triggering). Alternatively the signal is acquired without interruption, but only data acquired in the desired time intervals is used for image reconstruction (gating).

(d) Breathhold examinations. The signal is acquired in one or multiple breathholds thus minimizing the effect of breathing motion.

(e) Imaging Registration. Several MRI images that are acquired at different time points and that shall be combined can be registered. This is useful, for example, in high-resolution imaging (14) or in diffusion-weighted imaging.

(f) Prospective motion correction. Using navigator echoes, the read-phase-slice system can be adjusted during the examination to correct for rigid motion (see e.g.(15)).

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