### INTRODUCTION

### Principles and Methodology

### Imaging Principles

_{L}of spins under a magnetic gradient can be described as:

##### (1)

${\scriptscriptstyle {\mathit{\omega}}_{\mathrm{L}}\left(\overrightarrow{\mathit{r}},\mathit{t}\right)=\mathit{\gamma}{\mathit{B}}_{\mathit{0}}+\mathit{\gamma}\Delta {\mathit{B}}_{\mathit{0}}+\mathit{\gamma}\overrightarrow{\mathit{r}}\left(\mathit{t}\right)\overrightarrow{\mathit{G}}\left(\mathit{t}\right)}$*B*

_{0}is the static magnetic field, Δ

*B*

_{0}is the local field inhomogeneity, and ${\scriptscriptstyle \overrightarrow{\mathit{G}}}$ is the magnetic field gradient. Assuming that the fluid velocity ${\scriptscriptstyle \overrightarrow{\mathit{v}}}$ is constant during the acquisition, the displacement ${\scriptscriptstyle \overrightarrow{\mathit{r}}}$ can be described as ${\scriptscriptstyle \overrightarrow{\mathit{r}}\left(\mathit{t}\right)=\overrightarrow{{\mathit{r}}_{\mathit{0}}}+\overrightarrow{\mathit{v}}\left(\mathit{t}-{\mathit{t}}_{\mathit{0}}\right)}$, where

*t*

_{0}is the excitation time and ${\scriptscriptstyle \overrightarrow{{\mathit{r}}_{\mathit{0}}}}$ is the displacement at

*t*

_{0}. Then, the phase shift of the fluid with the velocity ${\scriptscriptstyle \overrightarrow{\mathit{v}}}$ under the magnetic gradient can be obtained by integrating the ω

_{L}from

*t*

_{0}to the echo time (TE) as:

##### (2)

${\scriptscriptstyle \mathit{\varphi}\left(\overrightarrow{\mathit{r}},\mathit{T}\mathit{E}\right)={\mathit{\varphi}}_{\mathit{0}}+\gamma \overrightarrow{{\mathit{r}}_{\mathit{0}}}{\int}_{{\mathit{t}}_{\mathit{0}}}^{\mathit{T}\mathit{E}}\overrightarrow{\mathit{G}}\left(\mathit{t}\right)\mathrm{d}\mathrm{t}+\gamma \overrightarrow{\mathit{v}}{\int}_{{\mathit{t}}_{\mathit{0}}}^{\mathit{T}\mathit{E}}\overrightarrow{\mathit{G}}\left(\mathit{t}\right)\mathit{t}\mathit{d}\mathit{t}+...={\mathit{\varphi}}_{\mathit{0}}+\gamma \overrightarrow{{\mathit{r}}_{\mathit{0}}}{\mathit{M}}_{\mathit{0}}+\gamma \overrightarrow{\mathit{v}}{\mathit{M}}_{\mathit{1}}+...}$_{0}) is the background phase offset which is influenced by the field inhomogeneity. The second and third are the phase accumulations from the stationary and moving spins, respectively, under the magnetic gradient ${\scriptscriptstyle \overrightarrow{\mathit{G}}}$. The integral terms describing the influence of the magnetic gradient on the static and moving spins are named the zeroth and the first gradient moments,

*M*

_{0}and

*M*

_{1}, respectively.

_{1}values to remove the unknown phase offset. In 2D PC-MRI, two acquisition schemes are conventionally used. The first type of the acquisition scheme employs the flow-compensation gradient with the flow-encoding gradient to obtain the reference image. The second type of the acquisition employs the combination of the flowencoding gradient and the following gradient with the opposite polarity. Meanwhile, the no bipolar gradient for the reference image is rarely used. In 4D PC-MRI, the first acquisition scheme is mostly used, and the reference acquisition without the bipolar gradient is rarely used.

_{enc}or VENC, determines the maximum velocity by changing the difference in the first gradient momentum Δ

*M*as follows: Here, it can be noted that the flow velocity that is as high as the pre-determined VENC (cm/s) gives the phase shift of π.

_{1}### Image Acquisition and Analysis Procedures

#### Scan Parameters

#### Pre-Processing

#### Data Analysis

### Hemodynamic Quantification

*in vivo*. The following sections introduce the principles and physiological meanings of the hemodynamic parameters that can be obtained using 4D PC-MRI (Table 1). However, most commercial software does not cover all of these quantifications. Therefore, the development of in-house software is required for these quantifications.

#### Velocity and Flow Rate

#### Wall Shear Stress

##### (6)

${\scriptscriptstyle \tau ={\left.\mu \frac{\u018f\mathit{u}}{\u018f\mathit{n}}\right|}_{\text{wall}}}$^{2}when the units of u, n, and µ are m/s, m, and N·s/m

^{2}, respectively.

#### Turbulent Kinetic Energy

##### (7)

${\scriptscriptstyle {\sigma}_{\mathrm{i}}=\frac{1}{{\mathit{k}}_{\mathit{v}}}\sqrt{2\mathrm{l}\mathrm{n}\left(\frac{\left|{\mathit{S}}_{\mathit{0}}\right|}{\left|{\mathit{S}}_{\mathit{i}}\right|}\right)}}$_{i}indicates the standard deviation in the ith-direction (x, y, z), k

_{v}indicates the net motion sensitivity (k

_{v}= π / VENC), and |

*S*

_{0}| and |

*S*

_{i}| indicate the magnitude of the MRI signal obtained without and with flow sensitivity, respectively. The TKE per unit volume can be estimated from σ as follows:

##### (8)

${\scriptscriptstyle \mathit{\text{TKE}}=\frac{\mathrm{1}}{\mathrm{2}}\rho \sum _{\mathit{i}=\mathrm{1}}^{\mathrm{3}}{\sigma}_{\mathit{i}}^{\mathrm{2}}\left[\mathrm{J}/{\mathrm{m}}^{\mathrm{3}}\right]}$_{i}indicates the standard deviation in the ith-direction (x, y, z).

_{v}). The sensitivity of the TKE estimation is optimal when the signal magnitude ratio |S|/|S

_{0}| becomes approximately 0.6 (analytically: e

^{-1/2}) (16). However, the VENC value for obtaining |S|/|S0|–0.6 is usually less than the maximum velocity of the flow; consequently, the scan for TKE estimation usually results in phase aliasing. Therefore, at least two acquisitions with different VENC values are generally requested to obtain both the velocity field and TKE distribution.

#### Vortical Structure

### Vorticity

##### (9)

${\scriptscriptstyle \overrightarrow{\omega}=\nabla \times \overrightarrow{\mathit{v}}=\left(\frac{\u018f{\mathit{v}}_{\mathit{z}}}{{\u018f}_{\mathit{y}}}-\frac{\u018f{\mathit{v}}_{\mathit{y}}}{{\u018f}_{\mathit{z}}},\hspace{1em}\frac{\u018f{\mathit{v}}_{\mathit{x}}}{{\u018f}_{\mathit{z}}}-\frac{\u018f{\mathit{v}}_{\mathit{z}}}{{\u018f}_{\mathit{x}}},\hspace{1em}\frac{\u018f{\mathit{v}}_{\mathit{y}}}{{\u018f}_{\mathit{x}}}-\frac{\u018f{\mathit{v}}_{\mathit{x}}}{{\u018f}_{\mathit{y}}}\right)}$^{-1}.

### λ_{2}-Criterion

_{2}-criterion is the one of the most popular methods for identifying the vortical flow structure (58). This algorithm is based on the velocity gradient tensor J, where ${\scriptscriptstyle \mathrm{J}\equiv \nabla \overrightarrow{\mathit{v}}}$. Then, the velocity gradient tensor can be decomposed into the symmetric S and asymmetric Ω terms as follows:

##### (10)

${\scriptscriptstyle \mathit{S}=\frac{\mathit{J}+{\mathit{J}}^{\mathit{T}}}{\mathrm{2}},\hspace{1em}\omega =\frac{\mathit{J}-{\mathit{J}}^{\mathit{T}}}{\mathrm{2}}}$^{2}+ Ω

^{2}for each voxel results in three eigenvalues, λ

_{1}, λ

_{2}, and λ

_{3}, where λ

_{1}≥ λ

_{2}≥ λ

_{3}. Finally, the vortex flow region can be found where λ

_{2}is negative. Since this algorithm is Galilean invariant, the vortical flow structures can be identified even though the vortical flow is overlaid with the uniform translational velocity field.

### Critical Point Analysis

_{r}) and a pair of complex conjugate eigenvalues (λ

_{cr}± iλ

_{ci}) when the discriminant of its characteristic equation is positive. Then, λci

^{-1}represents the period required for a fluid particle to rotate along the λ

_{cr}axis (61). Therefore, the region with nonzero λ

_{ci}corresponds to the local vortical flow, and the magnitude of λ

_{ci}is related to the intensity of the vortical flow. This method is also Galilean invariant and contains information on the strength of the vortical flow, in contrast to the λ

_{2}-criterion (Fig. 6B).

#### Non-Invasive Estimation of Pressure Drop

*in vivo*. However, it is not preferred due to its invasiveness. Alternatively, the pressure gradient can be estimated by Doppler echocardiography, which is a non-invasive method for measuring the maximum velocity of the flow across the stenosis region and estimating the pressure gradient based on a simplified Bernoulli equation (64). However, this method does not provide the spatial and temporal variations in the pressure. In addition, the accuracy of the results can vary according to the flow conditions because the simplified Bernoulli equation assumes the flow is laminar without turbulence, and it also neglects the viscous energy dissipation (65).

##### (11)

${\scriptscriptstyle -\nabla \mathit{p}=\rho \left(\frac{\u018f\mathit{v}}{\u018f\mathit{t}}+\mathit{v}\xb7\nabla \mathit{v}\right)-\mu {\nabla}^{2}\mathit{v}}$*p*is the pressure, µ is the viscosity, ρ is the density, and

*v*is the velocity. Here, the gravitational term has been neglected. Because the temporal variations in the 3D velocity can be obtained from 4D PC-MRI, the spatial gradient of the pressure can be easily obtained using Eq. (11). Once the pressure gradients over the volumetric regions are obtained, the spatial and temporal variations in the relative pressure field can be obtained by solving the Poisson pressure equation. Notably, the pressure field obtained from Eq. (11) is a relative pressure field without the absolute pressure reference. Therefore, only a pressure gradient can be estimated from the present method without any reference pressure, and the absolute pressure value cannot be obtained.

### Clinical Applications

*in vivo*feasibility of the use of TKE quantification to detect cardiovascular flow abnormalities such as those caused by aortic valve flow and aortic coarctation (18). Recent clinical studies found that patients with dilated cardiomyopathy had higher TKE than healthy subjects (78). In addition, TKE quantification was used as a non-invasive tool for quantifying the severity of aortic stenosis because the TKE value in the ascending aorta was strongly correlated to index pressure loss (Fig. 13) (17).