## 5 Medical Imaging-Based Kinematics
### Magnetic Resonance Imaging (MRI)
#### 5.2.2 Phase-Contrast  
Phase-contrast (PC) MRI utilizes the intrinsic phase of the magnetic signal to retrieve kinematic information. When a magnetic field gradient is applied to a body, the spins of the protons develop a phase shift that is proportional to its relative velocity. When two consecutive and opposing gradients are applied, stationary protons will show no phase shift. However, moving protons will show different degrees of phase shifting as they change their position with respect to the gradient [183]. This information can then be used to encode the velocity and displacement of protons. Since the kinematic information is encoded in the phase information, and thus is independent of image markers, this technique allows measurement at scales below pixel-size resolution [12, 149]. However, the technique is sensitive to Eddy currents, concomitant gradients (Maxwell terms), and nonlinearities in the gradient field. These effects increase the signal-to-noise ratio (SNR) and produce offset errors that are both spatial- and time-dependent. SNR has been shown to increase with the magnetization energy and has been estimated to range between 20 for 1.5 T scanners to around 60 for 7 T scanners. Offset error correction requires the implementation of rectification algorithms in the postprocessing stage [184, 185]. 2D CINE PC-MRI and 4D Flow MRI  
The application of consecutive opposite magnetic field gradients is known as a bipolar gradient. After a bipolar gradient is applied, the net phase shift of static protons is zero, so only the mobile protons will show a phase shift. From the latter, faster protons will experience a greater difference in applied gradients as they physically move longer distances than slower protons, which in turn produces greater phase shifts. The end result is that the phase shift is proportional to the proton velocity. However, because phase angles are limited (from 0 to 2 \(\pi\)), only a certain range of velocities can be directly quantified [183]. That is, for a given gradient, there is a maximal velocity that can be measured before aliasing occurs, called the encoding velocity (VENC). Encoding velocity is inversely proportional to the magnitude of the gradient; thus, by manipulating the strength of the gradient, it is possible to manipulate the range of velocities that can be encoded. Setting the encoding velocity is a tradeoff between the risk of aliasing and the minimum measurable velocity by the discrete scale [186]. Standard 2D Cine PC MRI, typically applied to estimate through-plane velocity, has become part of clinical practice in the treatment of cardiovascular disease, specifically, for the calculation of flow in large arteries and their main branches, cardiac output, and quantitative assessment of regurgitation and retrograde flows. This technique is also used for the qualitative assessment of flow patterns in large arteries and heart chambers. PC MRI data is usually recorded in DICOM format images with 8-bit or 16-bit pixels, that is 256 or 65,536 possible discrete levels, respectively. Phase data is typically encoded within 4095 values for the whole 0 to 2 \(\pi\) range, with pixel sizes around 1.5 mm [187]. However, standard 2D PC MRI can only provide the dimensional component of velocity perpendicular (through-plane) or parallel (in-plane) to the imaging plane, and thus is inadequate to estimate relevant hemodynamic metrics requiring three-dimensional flow information, such as vorticity and wall shear stress [188; 189]. The logical evolution of this technique led to 4D flow MRI, which allows the volumetric and temporal resolution of three orthogonal components of velocity. This is achieved by applying consecutive bipolar gradients to three orthogonal directions on stacked planes. This requires the collection and processing of a significantly greater amount of data (three spatial dimensions and three velocity directions over several timesteps through the cardiac cycle), thus requiring special approaches to keep reasonable scanning times.