### Macroscopic visual analysis

As a hemolytic agent flows through a tube, there is loss along the way with a constant flow cross section. Additionally, where the flow cross section changes, there is a local loss of flow at bifurcations in the tube. Therefore, the pressure and velocity fields at different positions along the tube differ. Regarding tube properties, the initial boundary conditions are controlled by the following equations:

$$ A=\frac{\pi }{4}{d}_i^2 $$

(11)

$$ \operatorname{Re}=\frac{\rho {ud}_h}{\mu } $$

(14)

Here, *d*_{i} is the tube outer wall diameter and *d*_{h} is the tube hydraulic diameter. It is assumed that these two values are equal, meaning the tube wall thickness is ignored. *A* is the tube cross-sectional area and *Z* is the tube wet circumference, meaning the circumference of the fluid flowing through the tube. According to the geometrical model parameters, the fluid property parameters and initial boundary value conditions given in first part of the main tube are *d*_{i} = *d*_{h} = 2 mm. The dynamic viscosity of the hemolytic agent is approximately *μ* = 1 × 10^{−3} Pa·s and the density is approximately ρ = 30 kg/m^{3}. This information is entered into the expressions above to obtain a Reynolds number expression, denoted as Re. From this expression, we can calculate the Darcy friction factor, denoted as *f*_{D}.

$$ {f}_D=8{\left[{\left(\frac{8}{\operatorname{Re}}\right)}^{12}+{\left({c}_A+{c}_B\right)}^{-1.5}\right]}^{\frac{1}{12}} $$

(15)

Here, *c*_{A} and *c*_{B} are the liquid concentrations at any two points A and B, respectively, which are expressed as follows:

$$ {c}_A={\left\{-2.57\ln \left[{\left(\frac{7}{\operatorname{Re}}\right)}^{0.9}+0.27\left(\frac{e}{d_h}\right)\right]\right\}}^{16} $$

$$ {c}_B={\left(\frac{37530}{\operatorname{Re}}\right)}^{16} $$

Here, *e* is the surface roughness (*e* = 0.0015 mm).

The fluid properties are defined by the following governing equations:

$$ \rho u\nabla \cdot \left({ue}_t\right)=-{\nabla}_tp\cdot {e}_t-\frac{1}{2}{f}_D\frac{\rho }{d_h}\left|u\right|u+F\cdot {e}_t $$

(16)

$$ {\nabla}_t\cdot \left( A\rho {ue}_t\right)=0 $$

(17)

After the flow trajectory of the fluid has been determined, the properties of the fluid are only related to time, where *F* is the volume force and *e*_{t} is the tangential vector of roughness over time.

The treatment of nondestructive tube joints is handled based on the following formula:

$$ {p}_{junction}={p}_i+\frac{1}{2}{\rho}_i{u}_i^2 $$

(18)

Here, *p*_{i} is the pressure of fluid *i*, *ρ*_{i} is the density of fluid *i*, and *u*_{i} is the velocity of fluid *i* flowing into a joint.

Based on the calculations above, we can obtain the three-dimensional pressure and velocity distribution results for a fluid flowing through a bifurcated tube with a height expression.

Figures 3 and 4 indicate that for a particular initial velocity and flow driving force, the initial pressure value is determined in the tube inlet stage and the fluid velocity change and pressure change through the entire tube network are virtually the same. Additionally, the hemolytic agent has a specific flow rate. With the extension of the tube and loss of energy consumption, the pressure gradually decreases and eventually trends toward zero. When the tube diameter is halved, accompanied by a local loss of energy consumption, the flow rate decreases. Because the fluid has an initial velocity, the flow velocity at the end of the nozzle will not drop to zero, but the fluid will slowly flow out of the drainage tube.

### Visual analysis of microstreamlines

For a given velocity and pressure in a bifurcated tube, the flow state is determined by the Reynolds number and Mach number of the liquid flowing through the tube. The parameters of the hemolytic agent and tube were fed into Eqs. (3) and (4) to calculate that the Reynolds numbers of the main tube and branch tubes are Re_{1} ≈ 0.3 < 2320 and Re_{2} ≈ 0.15 < 2320, respectively, while the Mach number of the fluid is *Ma* = 0.05 < 1. It can be determined that the flow of the hemolytic agent in both the main tube and branch tubes is laminar. The fluid flows at subsonic speeds, so its compressibility can be ignored. Therefore, the incompressible Navier-Stokes equation can be used to describe the flow of the hemolytic agent in the tube.

The COMSOL Multiphysics software was used to generate a coarser mesh controlled by the physics field acting on a bifurcated tube, as shown in the Fig. 5.

The initial boundary conditions ensure that the motion equation satisfies the non-slip tube wall condition of *u*|_{r} = 0 and that the wall resolution of the bifurcated drainage tube can be obtained, as shown in the Fig. 6.

Figure 6 reveals that the wall lift force everywhere in the bifurcated tube is much lower than 100 viscous units, meaning the flow can be considered to be well resolved on the wall.

The fluid properties governing the equation of laminar flow are defined as follows:

$$ \backslash \mathrm{rho}\backslash \mathrm{left}\left(\mathrm{u}\backslash \mathrm{cdot}\backslash \mathrm{nabla}\backslash \mathrm{right}\right)\mathrm{u}=\backslash \mathrm{nabla}\backslash \mathrm{cdot}\backslash \mathrm{left}\Big(-\mathrm{pI}+\mathrm{K}\backslash \mathrm{right}+\mathrm{F} $$

(19)

$$ \rho \nabla \cdot u=0 $$

(20)

Here, *K* is the viscous stress, which is expressed by the following formula:

$$ K=\mu \left[\nabla u+{\left(\nabla u\right)}^T\right] $$

∇ ⋅(−*pI* + *K*) is the diffusion term of the governing equation and *F* is the volume force, which is the source term for the governing equation. Equation (20) satisfies the incompressible condition of fluid flow.

The inlet and outlet stages of the tube are set to be fully developed flows, where the inlet stage is constrained by a velocity field with an average velocity of *u*_{av} = 0.05 m/s and the outlet is constrained by the average pressure. Therefore, the governing equation can be written as follows:

$$ {\displaystyle \begin{array}{l}u\cdot t=0\\ {}\left(- pI+K\right)n=-{p}_{grad}n\end{array}} $$

(21)

By combining this equation with the streamline visualization method [23], a three-dimensional streamline diagram of the liquid flow in the tube can be obtained (Fig. 7).

To facilitate observation, we enlarged the streamline diagram of the bifurcated drainage tube to obtain the following partially enlarged views.

Figure 8 presents a colorized streamline diagram of the local velocity amplitudes in the tube. The flow at the inlet stage is laminar, so the flow lines are distributed in parallel. In addition to the laminar flow at the root of each bifurcated tube, there are also turbulent flows, which are similar to the critical state of flow. As the fluid flows, the farther it travels from the inlet, the lower the fluid velocity.

### Hemolytic agent and hematoma diffusion simulation

The shape model of a hematoma was extracted from a computed tomography image of a patient with an intracranial hematoma and imported into the COMSOL Multiphysics software. The bifurcated drainage tube model was then imported and grid refinement of the drainage tube was performed as it acted on the hematoma, as shown in the figure below.

The refined grid structure shown in Fig. 9 can be controlled by users. The cell size is calibrated to match the target fluid dynamics. Under conventional predefined conditions, the cell grid parameter size is divided and the maximum cell size is 2.01 mm, the minimum cell size is 0.601 mm, the maximum cell growth rate is 1.15, the curvature factor is 0.6, and the narrow-area resolution is 0.7. The element size scaling factor for refining angles is 0.35 and smoothing is performed across all removed control entities. The number of iterations is four and the maximum element depth to be processed is four units. Analysis was performed on this refined mesh using calculations and a preliminary interaction diagram of the hemolytic agent and hematoma was obtained. The concentration difference between the hemolytic agent and hematoma is defined by the concentration of the initial boundary conditions so that the system produces a diffusion phenomenon driven by concentration differences.

For the fluid flow in the free-flow region, the steady-state Navier-Stokes equation can be applied, where the law of mutual diffusion is derived from Fick’s law.

$$ \frac{\partial {M}_1}{\partial t}=-{D}_{12}\frac{\partial {\rho}_1}{\partial x}\cdot A $$

(22)

In this section, the Maxwell-Stefan diffusion model is incorporated [24], convection and porous media mass transfer mechanisms are added, and the mass conservation equation is adopted to obtain a convection-diffusion continuity equation under the transfer of dilute species.

$$ {\displaystyle \begin{array}{l}\frac{\partial \left({\varepsilon}_p{c}_i\right)}{\partial t}+\frac{\partial \left(\rho {c}_{p,j}\right)}{\partial t}+\nabla \cdot {N}_i+u\cdot \nabla {c}_i={R}_i+{S}_i,\\ {}\kern5.5em {N}_i=-{D}_{e,i}\nabla {c}_i.\end{array}} $$

(23)

Here, *ε*_{p} is the porosity (*ε*_{p} = 0.21 was derived from the calculation process), *R*_{i} is the total diffusion rate of substance *i*, and *S*_{i} is the source term. The flux expression has the form of Fick’s law.

The fluid diffusion coefficient is an isotropic coefficient.

$$ {D}_{e,i}=\frac{\varepsilon_p}{\tau_{F,i}}{D}_{F,i} $$

(24)

Here, *τ*_{F, i} is the effective diffusion coefficient model of the Millington-Quirk model.

$$ {\tau}_{F,i}={\varepsilon}_p^{-\frac{1}{3}} $$

Figure 10 illustrates the concentration of the hemolytic agent on the wall surface of the tube when it drains into the hematoma over 9 s.

To obtain the diffusion concentration changes of the hemolytic agent and hematoma, the simulation target is placed in the interactive area between the hematoma and bifurcated drainage tube, and post-processing is performed on the COMSOL results to account for the concentration surface two-dimensional drawing group animation.

Figure 11 demonstrates that the overall concentration changes on the surface of the hematoma spread along the main tube of the bifurcated tube and gradually spreads outward. For a point on the surface of the hematoma, the smaller the radial distance to the main tube, the more thorough the diffusion.

The analysis above focused on concentration changes on the surface of the hematoma. To study the diffusion and visualization of the hemolytic agent and hematoma, it is necessary to perform two-dimensional cross-sectional analysis inside the hematoma with multiple branch tubes. The drainage tube and hematoma wall outside the hematoma were set to have no flux. We generated a two-dimensional cross section of the overall figure from the symmetry plane of the bifurcated drainage tube. Because there is a concentration gradient between the hemolytic agent and hematoma, after the hemolytic agent enters the hematoma through the bifurcated drainage tube, four times instances were selected randomly. The inter-diffusion dilution reactions observed at each node are illustrated in the figure below.

Figure 12 indicates that the initial hematoma concentration is *c*_{1} = 3 mol/m^{3} and the hemolytic agent concentration is *c*_{0} = 1 mol/m^{3}. The colors in the legends in this figure represent changes in the liquid concentration, which are driven by the diffusion process of the hemolytic agent into the hematoma. The lines with white arrows in the figure represent the flux. Changes in direction represent the diffusion process from the hematoma into the hemolytic agent, which is driven by the process of mutual diffusion of binary fluids.

In the case of *t* = 16.6 s, the mutual diffusion reaction is in the initial stages. When the hemolytic agent is drained into the hematoma, the ends of the bifurcated tube begin to change in concentration and flux. In the case of *t* = 66.6 s, with the continuous injection of the hemolytic agent, the agent and hematoma experience the inter-diffusion phenomenon. The hemolytic agent dilutes the concentration of the hematoma and the direction of the hematoma concentration flux points toward the orifices of the drainage tube. When *t* = 116.6 s, one can see that the central part of gradually completes the mutual diffusion reaction with the hemolytic agent. Overall, the hemolytic agent, the part of the hematoma far away from the drainage tube outlets, and the position of the dead corners with complex geometric shapes change more slowly. At *t* = 183.3 s, one can see that the third branch tube, fourth branch tube, and main tube outlet interact more clearly. The hemolytic agent flowing from these three nozzles plays a major role in diffusion. In contrast, the hemolytic agent flowing from the first two branch tubes has a slower diffusion effect in the hematoma.

To make the hemolytic agent spread evenly through the hematoma, it is necessary to improve the design of the bifurcated drainage tube based on this phenomenon. This is one future model application and prospect of this research.