# Real-Time Prediction of Transarterial Drug Delivery Based on a Deep Convolutional Neural Network
## 2 Methods
### Data Preparation  
#### 2.1.1 Physical Model and Numerical Simulation  
Our work considers two idealized geometries of straight and bifurcated blood vessels and a third case representing a human hepatic artery system. (See Figure 1a-c). All the cases are with circular cross-section with centers in the symmetry plane. The drug is injected at the intersection of inlet section and the symmetry plane with uniform injection velocity and direction. Therefore, we only simulate the domain in the symmetry plane, reducing the 3D problem to 2D for economy of computational cost. The information of the circumferential component is thereby ignored, which has much less influence than that of axial and radial components. The geometry of the human hepatic artery system used in the current work is created from one of the most common anatomical configurations [10] and has been investigated in several studies [25; 26; 27]. The geometry parameters are derived from the published articles of [PERSON] et al. [28] and [PERSON] et al. [29].  
We consider blood to be an incompressible Newtonian fluid, and solve the following continuity and linear momentum equations:  
\[\
abla\cdot\mathbf{u}_{f}=0 \tag{1}\]  
\[\rho\Big{(}\frac{\partial\mathbf{u}_{f}}{\partial t}+\Big{(}\mathbf{u}_{f}\cdot\stackrel{{ \rightarrow}}{{\
abla}}\Big{)}\mathbf{u}_{f}\Big{)}=-\
abla p+\
abla\cdot\mathbf{ \tau}+\mathbf{f} \tag{2}\]  
Figure 1: (**a**) Straight vessel geometry; (**b**) bifurcated blood vessel geometry; (**c**) a representative hepatic artery system and the velocity waveform in CHA, t1(=0.1 (t/T)) is the injection time.  
where \(\mathbf{u}_{f}\) (m/s) is the velocity of blood flow, \(t\) (s) is the time, \(p\) (Pa) is the pressure, \(\rho\) (kg/m\({}^{3}\)) is the density of blood, \(\mathbf{f}\) (N) is the body force per unit volume and \(\mathbf{\tau}\) (Pa) is the shear stress tensor. The main diameter of the vessel is 6 mm. The mean velocity was 0.26 m/s, corresponding to a [PERSON]'s number of 479.3. Newtonian fluid assumption is used in this study referring to the literature [30; 31; 32], as the influence of non-Newtonian behavior flow is not significant in large arteries.  
We assume that the drug transfer in the blood does not involve any physiological chemical processes and the mass of the liquid drug is mainly transferred by convection, as the influence of diffusion is relatively weak [9; 33; 34]. The scalar transport model is used to represent the drug concentration field:  
\[\frac{\partial s}{\partial t}+\
abla\cdot\left(\mathbf{u}_{f}s\right)=0 \tag{3}\]  
where \(s\) is the volumetric concentration of the liquid drug.  
The velocity waveform [10] is presented in subfigure of Figure 1c, where T represents the cardiac cycle and t = 0.1 T is the start of injection time, with the injection lasting 2 T. This model was proposed by [PERSON] et al. [35] and applied to the similar study of targeted injection [11; 36; 37].  
The PISO algorithm, one of the pressure-velocity coupling methods, is used and the IcoFoam solver in OpenFOAM is modified by embedding the above drug concentration control equation for numerical simulation.  
ICEM CFD is used to generate the mesh of the computational domains for the three types of blood vessels. The length of the grid in the hepatic artery is set to 0.005 times the inner diameter and we use a refined mesh in both the high curvature areas (bifurcated part) and the near-wall areas. The maximum and minimum distances between the first layer grid and the wall are 0.0133 mm and 0.0089 mm, respectively. The total number of grids is 143,730. Figure 2 illustrates the computational mesh near the RHA and LHA bifurcation.