Team:GA State SW Jiaotong/HCAM
Hydrogel compression activation model
In terms of modeling design，ShanghaiTech_China solicited relevant design ideas from the modeling team of GA_State_SW_Jiaotong. And we helped them to construct the hydrogel compression activation model during the project.
Model introduction
In order to verify that the hydrogel, after being filled into the bone joints, is subjected to external compression due to the compression of the bone joints and thus reaches a state of its own activation, we have designed the following model.
We assume that the hydrogel is in the state shown in Figure 1 after being filled into the bone joints. The hydrogel is compressed on both sides and the hydrogel shows a cross-linked state of positive tetragonality. As the bone grows, the ends of the hydrogel are subjected to spatial compression, which is generated at both ends because the hydrogel is highly elastic. According to a review of relevant data [1], the compression force of ordinary hydrogels varies with the curve shown in Figure 2, i.e. the compression force gradually increases exponentially as the compression ratio increases.
Fig. 1 Model introduction diagram
Figure 2 Hydrogel compression force versus compression deformation curve
Based on the above information, if we can calculate that the compressive force acting on the hydrogel can reach the strength of the force required to activate itself at a small compressive deformation, it can be directly demonstrated that the hydrogel can achieve the conditions to activate itself after it has been filled into the bone joints.
Model building and solution
Modelling of the hydrogel compression curve
Although the variation curve between the hydrogel compression force and the compression deformation has been obtained, there is no more unified functional relationship, therefore, we take the points of Figure 2, after which we use Spline interpolation method to construct the hydrogel compression curve, taking 12 points evenly, as shown in Table 1 (see Appendix Figure 1 for the point taking method).
| No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Strain (%) | 0 | 12 | 20 | 32 | 40 | 52 | 60 | 72 | 80 | 84 | 88 | 92 |
| Com_stress | 0 | 2 | 5 | 10 | 13 | 19 | 23 | 33 | 48 | 62 | 80 | 108 |
Table 1 Compression curve points taken
For the above taken points, the Spline interpolation method is used for interpolation (this method is relatively basic, no need to explain the principle, to find, "Calculation methods" interpolation chapter that is, a brief description: on the basis of ordinary segmental interpolation, plus the second-order guide smooth constraint), MATLAB programming calculations to obtain Figure 3, where the interpolation effect is very good, the code see the Appendix.
Figure 3 Hydrogel compression curve Spline interpolation results
Establishment of the hydrogel compression model
After obtaining the compressive deformation curve of the hydrogel, the internal force analysis of the hydrogel was carried out. The micro-element of the hydrogel is assumed to be a square (square in two dimensions), as shown in Figure 4.
Fig. 4 Hydrogel micro-element
When subjected to squeezing, the forces are shown in Figure 5.
Fig. 5 Force diagram of the micro element when subjected to extrusion
After that, as the degree of extrusion increases, the process of change is shown in Figure 6, i.e. it gradually becomes a straight line.
Fig. 6 Dynamic process of extrusion deformation of the micro-element
A complete force decomposition of Fig. 5 is shown in Fig. 7. It is not difficult to find that the hydrogel, when squeezed, should really be subjected to a force F1(or F2, both are equal, here is an example of F1). The relationship between this and the compressive force F is shown in equation 1:
Where θ is the angle between F and F1.
Figure 7 Micro-element force analysis
It can also be seen from Figure 6, in the compression process, the angle θis also changing, after consulting the data, different materials in the process of axial compression, the process of change will be different. In this regard, we assume that the compressive deformation rate of the hydrogel is consistent with the variation of the pinch angle during compression. That is, when the hydrogel compression deformation is S%, the angle θ of clamping at this time is:
In summary, the hydrogel in the process of compression, the relationship between its real force and compressive deformation is:
Integration of the model
In summary, the hydrogel compression curve obtained by interpolation in 2.1 is brought into the hydrogel compression model in 2.2 to obtain the true hydrogel force versus compression deformation curve as shown in Figure 8, and the code is shown in the Appendix.
Figure 8 Realistic hydrogel force compression curve
Model results
According to the given hydrogel activation curve, as shown in Figure 9. It is not difficult to find that the activation rate basically reaches 100% when the pressure reaches 80 mmHG, at which point the corresponding pressure P is:
Fig. 9 Hydrogel activation curve
According to the change curve in Figure 8, it is found that when the hydrogel compresses Strain=35%, the compression force it withstands is 10.7783 KPa, i.e. the threshold value of 100% hydrogel activation is reached.
In summary, the hydrogel can activate itself when it compresses itself by 35%, proving that the hydrogel can reach the condition of activating itself after being filled into the bone joints.
Appendices
References
[1] Kim, J., et al., Ultra-Tough and Super-Swelling Poly(vinyl alcohol)/Poly(AAm-co-AA Sodium Salts) Double Network Hydrogels. Macromolecules, 2021. 54(5): p. 2439-2448.
Point taking method
Code: (where the generated image, with personal modifications, has different information from the article in the figure notes, but the data information is identical)
