Objective: The blood flow mechanical characteristics is very important for analyzing the behavior of blood vessels. They can be used as an indicator that characterizes  many cardiovascular diseases. For narrow blood vessels, the pressure difference between the stenosis is a significant feature, which also provides a method for diagnosis. Clinically measurement of artery stenosis is more varying creative catheter measurements, but the disadvantage of creative measurement is obvious. This paper, based on this preliminary exploration of a method of non-invasive measurement of non-invasive measurement of 4D Flow MRI technology, is intended to avoid wounds and potential risks that have creative measurements.
Methods: 4D-flowPCMRI technology was used to collect magnetic resonance images of the target vessels, and the images containing phase information, namely three-dimensional spatial velocity information, were obtained. The velocities in the three directions X, Y, and Z, as well as 12-20 anechoic time phases, are then taken as input parameters into the Navier-Stokes (NS) equations to calculate the pressure gradient. Imaging parameters were: field of view(FOV)=( 181×199mm)²; flip angle (FA)=12°; echo time(TE)= 2.71-3ms; repetition time (TR)= 19.04-43.83ms; in-plane spatial resolution=(0.76-1.04×0.76-1.04mm)²;slice thickness=0.75-2mm; Venc=120cm/s. 4D-flow phase contrast technology was used to obtain mathematical relationship between phase and velocity: Δ^ϕ=π v/VENC. After setting the appropriate VENC value, accurate blood flow velocity can be obtained according to the phase value[1].Bringing the blood flow velocity into the NS equation as an input value:-∂P/∂x_i=ρ(∂v_i/∂t)+ρ[v₁(∂v_i/∂x₁)+v₂(∂v_i/∂x₂)+v₃(∂v_i/∂x₃)]-[(∂^2v_i)/(∂x₁^2)+(∂^2v_i)/(∂x₂^2)+(∂^2v_i)/(∂x₃^2)]-F_i),for conclulating pressure gradient, the numerical method of forward difference is used to calculate the partial derivative. Before calculating the pressure gradient, simplify the NS equation. Since the blood vessel is collected horizontally and has stenosis, the external force term and the bivalent derivative term are temporarily ignored, that is, the following simplified NS equation is used for calculation:-∂P/∂X_i=ρ(∂v_i/∂t)+ρ[v₁(∂v_i/∂x₁)+v₂(∂v_i/∂x₂)+v₃(∂v_i/∂x₃)],using numerical methods to accumulate path integrals: P=∑_i [(∂P/∂x)_i+(∂P/∂y)_i+(∂P/∂z)_i],for the pressure[2]. This article analyzed 3 patients with vertebrobasilar artery stenosis and 3 healthy control volunteers. To evaluate the feasibility of non-invasive measurement of vascular pressure gradient based on 4D flow MRI images in this study.
Results and Discussion: Maximum velocity, minimum velocity and average velocity of the three patients with Vertebrobasilar artery stenosis are follows: maximum velocity=2.04m/s, 1.84m/s, 1.86m/s; minimum velocity=0.16m/s, 0.12m/s, 0.09m/s; average velocity=0.91m/s, 0.96m/s, 0.43m/s. The corresponding average pressure difference were 51.7mmHg, 27.3mmHg, 4.5mmHg. And for the healty volunteers, these above were:maximum velocity=0.78m/s, 0.78m/s, 1.05m/s;minimum velocity=0.24m/s, 0.29m/s, 0.30m/s; average velocity=0.41m/s, 0.46m/s, 0.44m/s. The corresponding average pressure difference were 2.7mmHg, 0.05mmHg, 1.5mmHg.
   According to the calculated pressure difference, the blood flow pressure difference of patients with vascular stenosis is significantly increased, which is in line with the law of fluid mechanics.
   From this preliminary result, we can see that it is feasible to calculate the relative pressure difference of stenosis vessels using the image of velocity information obtained by 4Dflow magnetic resonance technology combined with NS equation. However, our study is too simplified. At first, we only consider the cases of vertical blood vessel sampling (which can reduce the interference of noise on the path and obtain relatively accurate pressure difference), but in fact, we ignore the viscous term, and blood definitely belongs to the viscous liquid. Therefore, based on these two points, our research is still immature, and we need to take more general and in-depth consideration of the related properties and conditions of computational fluid dynamics. In our subsequent study of relevant literature and research, we found that the results of pressure field obtained from velocity field would be more accurate under the strict coordination with the method of computational fluid dynamics. However, relevant experiments and algorithms are still in the further exploration and research, I believe that they will be implemented soon, so that our results and conclusions are more accurate and reliable, and more usable.
Conclusion: According to our current preliminary results, images containing velocity information obtained based on 4Dflow magnetic resonance technology combined with fluid dynamics NS equation can be used to detect the relative differential pressure of the narrowed basilar artery to a certain extent. Moreover, we believe that more accurate and reliable results can be obtained after further application of the method combined with computational fluid dynamics, and the pressure field data of any blood vessel can be obtained without being limited to vertical blood vessels.
Reference：
[1]. _BOOK Handbook of MRI Pulse Sequences.
[2]. Deng, Z., et al., Noninvasive measurement of pressure gradient across a coronary stenosis using phase contrast (PC)‐MRI: A feasibility study. Magnetic Resonance in Medicine, 2017. 77(2): p. 529-537.

4D flow MRI; Non-invasive measurement; pressure gradient.