## 2 Materials and Methods
### Extrusion Process Theory and Simulation Model of Printing Process  
#### 2.3.1 Extrusion Process Theory Model  
For the fabrication of biological scaffolds, it is of significant importance to strictly select the parameters of needle movement speed, air pressure, fiber spacing, and temperature and so on. These changes affect the mechanical properties of scaffolds and the viability of implanted cells, such as line width and spacing, which influence the pore size and porosity of scaffolds, and thus affect the mechanical properties of scaffolds and the proliferation and attachment of binding cells in scaffolds. The cryogenic printing head can be simplified, as shown in Figure 3a. The sodium SA-GEL solution was pre-stored in the syringe. When pressure \(P\) was applied to the top of the needle, the composite materials were extruded from the needle. The Biopltter was able to adjust the air pressure in the range of 0-5 Bar, and the diameter of the conical needle outlet was in the range of 0.2-1 mm. In Figure 3a(ii), \(P\) is the pressure applied to the needle of the biographical plotter, \(P_{e}\) is the outlet pressure of the needle end, \(r_{1}\) and \(r_{0}\) are the radius of the entry and exit of the needle respectively, \(\theta\) is the half-cone angle. Because the diameter of the barrel was much larger than that of the conical needle, and the drop speed of the liquid level in the barrel was much lower than that of the fluid in the conical needle, the change in the liquid level height in the barrel was neglected. To further simplify the model, we made the following assumptions in a pneumatic distribution system:  
1. The composite solution was in-compressible. 2. The flow of compound solution in needle was stable and fully developed. 3. The slip between the composite solution and the needle wall was significant. 4. The diameter of the barrel was much larger than that of the conical needle, so the pressure loss in the barrel could be neglected. 5. The slight loss caused by the entrance effects and accessories could be neglected. The pressure difference between the top and the outlet of the needle can be expressed as (1):  
\[\Delta P\ =\ P\ -\ P_{e} \tag{1}\]  
The tangent value of the \(\theta\) can be expressed as (2):  
\[\tan\theta=\frac{r_{1}-r_{0}}{l} \tag{2}\]  
Figure 2: Scaffold printing process. (**a**) CAD model construction and sliced layers generation; (**b**) Different pattern assigned before printing; (**c**) 3D-Biopltter. The total flow equation of needle mixed solution can be written as (3):  
\[Q\ =\ Q_{1}\ +\ Q_{2} \tag{3}\]  
where \(Q\) is total flow, \(Q_{1}\) is shear flow; \(Q_{2}\) is slip flow. The force balance equation of the pressure and shear stress of a conical needle can be written as (4):  
\[\pi r^{2}dp\ =\ 2\pi r\tau\sec\theta\cos\theta dl \tag{4}\]  
where: \(\tau=\frac{4Q_{1}}{\pi r^{3}}\), \(dl\ =\ \frac{dr}{\tan\theta}\). Given that the effects of accelerated shear flow are ignored, the pressure difference can be calculated as (5):  
\[\Delta P\ =\ \int_{r_{0}}^{r_{1}}\frac{2\pi r}{\pi r^{2}}\tau\sec\theta\cos \theta dl\ =\ \frac{2K}{(3n\tan\theta)\Big{(}\frac{4Q_{1}}{\pi r_{0} }\Big{)}^{n}\Big{[}1\ -\ \Big{(}\frac{r_{0}}{r_{1}}\Big{)}^{3n}\Big{]}} \tag{5}\]  
\(Q_{1}\) and \(Q_{2}\) can be deduced from (6):  
\[Q_{1}\ =\ \frac{\pi r_{1}{}^{3}r_{0}{}^{3}}{4\Big{[}\frac{3n\tan\theta \Delta P}{2K(r_{1}{}^{3n}-r_{0}{}^{3n})}\Big{]}^{\frac{1}{n}}} \tag{6}\]  
\[Q_{2}\ =\ V_{slip}\Big{(}\pi r^{2}\Big{)} \tag{7}\]  
Figure 3: The printing process model and the prediction model of scaffold porosity. (**a**) simplified model of printhead; (**b**) morphology change of extruded gel; (**c**) unit structure of scaffold. where \(V_{slip}\) is slip velocity, which can be expressed as (8):  
\[V_{slip}\ =\ \frac{\beta\tau_{w}{}^{\frac{1}{n}}}{K^{\frac{1}{n}}} \tag{8}\]  
where \(\beta\) is the slip layer thickness and \(\tau_{w}\) is the shear stress on the needle wall. \(K\) is the coefficient of mixed solution consistency.