## 2 Materials and Methods
### Geometrical Model and Mesh Generation  
Digital Imaging and Communications in Medicine data obtained by CT angiography in two patients with right oral cancer with the linguofacial trunk (patients A and B) were imported into the 3D Slicer software (4.10.2, The Slicer Community, [[http://www.slicer.org](http://www.slicer.org)]([http://www.slicer.org](http://www.slicer.org))). In 3D Slicer, the vascular tissue was extracted by applying thresholding functions. The analysis region for the CFD was extracted manually. The models for patient A included the CCA, ICA, superior thyroid artery (STha), LA, FA, occipital artery (OA), maxillary artery (MA), and superficial temporal artery (STA) (Figure 2a). In addition to these eight branches, the models for patient B included the ascending pharyngeal artery (APA) and postauricular artery (PAA) (Figure 2b). The diameters of the inlet of the CCA in patient A model and patient B model were 7.0 and 6.9 mm, respectively. The centerlines of the ECA and linguofacial trunk were presented using the VMTK module in 3D Slicer. Then, a catheter with a diameter of 1.3 mm was created, assuming a catheter used clinically (Figure 2c), and placed according to the centerline using Markups to Model function. Models A0 and B0 and models A1-A5 and B1-B3 were generated to mimic the CIAC and SSIAC, respectively. The catheter position was defined as follows (Figure 3a,b). Figure 2: Geometrical model. (**a**) Vessel model for patient A. (**b**) Vessel model for patient B. (**c**) Modeled catheter. APA, ascending pharyngeal artery; CCA, common carotid artery; ECA, external carotid artery; FA, facial artery; ICA, internal carotid artery; LA, lingual artery; MA, maxillary artery; OA, occipital artery; PAA, postauricular artery; STA, superficial temporal artery; STha, superior thyroid artery. 1. The catheter tip was placed in the main trunk of the ECA (models A0 and B0). 2. The catheter tip was placed at the bifurcation point of the LA and FA (models A5 and B3). 3. The catheter tip was pulled toward the central site of the linguofacial trunk and placed every 2 mm along the centerline of the linguofacial trunk (models A1-A4 and B1-B2). In all models, the observation point was set in the LA and FA and named as P\({}_{\text{LA}}\) and P\({}_{\text{FA}}\), respectively (Figure 3a,b). The zones within a radius of 2 mm from the center of the catheter tip, P\({}_{\text{LA}}\), and P\({}_{\text{FA}}\) were defined as the catheter zone, LA zone, and FA zone, respectively. These zones were used to calculate the concentration gradients of fluids (section \"Numerical condition\"). A smaller number of models were created for patient B than for patient A, because the length of the linguofacial trunk in patient B was shorter than that in patient A. These generated models for the vessel and catheter were imported into Mimics 18 (Materialise, Leuven, Belgium) in Standard Triangle Language (STL) format, and the vessel and catheter were combined using Boolean operators. STL data were imported into 3-matic 10.0 (Materialise, Leuven, Belgium), and the CCA, ICA, and ECA branches were segmented. A total of six models for patient A and of four models for patient B were imported into the ICEM CFD (2019 R1, ANSYS Inc., Canonsburg, PA, USA) for the mesh generation. The mesh size was determined based on our previous study [14]. Figure 4 shows the mesh for model A1. The mesh in the present study was composed of tetrahedral cells in the artery core and from 4 to 7 prismatic layers in the region near the artery wall and the catheter wall. The inside of the catheter was treated as a hollow tube without mesh. The average lengths of the cell edge in patient A models and patient B models were 0.11 and 0.13 mm, respectively. The average total numbers of cells for patients A and B were 1,756,023 and 2,623,782, respectively. Figure 3: Catheter tip location. (**a**) The catheter tip position (A1–A5) and observation points (P\({}_{\text{FA}}\) and P\({}_{\text{LA}}\)) of the LA and FA in model A.