# Scaffold Modelling:

## Finite Element Analysis (FEA) of Scaffold Macro-architecture

### Introduction:

Optimising the design of a biomaterial spinal scaffold is crucial and requires the consideration of a range of factors that include, albeit are not limited to: macro-architecture (the overall shape of the scaffold), microarchitecture (such as pore size), and degradation (the amount of time the scaffold remains within the body to successfully facilitate regrowth).

##### Why is macro-architecture important?

Different macro-architectures have been shown to have an influence on the success of scaffolds, with even acellular scaffolds having a strong positive outcome when having specific architecture; ‘open-path’ designs allow better guidance with less material (as opposed to cylindrical designs) and permit the extension of nerve fibres across the entire defect length (Wong, et al., 2008). Conversely, as opposed to exploring regrowth and guidance, this report focusses solely upon the mechanical properties of each scaffold design presented by Wong et al.: cylinder, tube, channel, open-path with core, and open-path without core.

One of the principal design concerns for a tissue engineering scaffold is ensuring that the mechanical properties match the native tissues and surroundings as closely as possible. The elastic modulus and yield strength are deemed the most important parameter in relation to scaffolds due to their relationship with load-bearing (Ali & Sen, 2017). Generally, an implant should have shape retention (i.e. a similar elastic modulus to the spinal cord such that it doesn’t become permanently deformed under stress) (Senatov, et al., 2016). Specifically, the match of the Young’s Modulus to that of the spinal cord is further required so that there is sufficient contact between the regenerative scaffold and the host grey matter (ref note). Experimental methods of defining scaffold mechanics are expensive, and instead, computational approaches are appealing for research (Ali & Sen, 2017). Subsequent sections of this report delve into the von Mises stress, strain and displacement. However, it has been shown that cells respond differently to substrate stiffness (Breuls, Jiya & Smit, 2008), and so this could be a future avenue of FEA analysis and research.

FEA (Finite Element Analysis) is a numerical type of analysis that provides a computational method of approximately solving partial differential equations (PDEs), of which model a real-life problem – via means including Euler’s method for example (Zienkiewicz, 1991). The problem, or in this case scaffold geometry, is broken up into a finite number of smaller regions (known as elements) – with the vertices of such being referred to as nodes. This combination of elements and nodes make up the finite element mesh – within this report, a triangular mesh was utilised. After generating the mesh, the loads acting on the body are modelled as forces applied to nodes (Boccaccio, et al., 2011). Following this, the solver finds the deformation of the model while being exerted to this load and then calculates the strains throughout the mesh (which is the relative deformation for each element). Finally, using known material property values, the stress can be computed for each element. Typically, these results are then displayed in a gradient plot.

The von Mises stress is essentially used to determine if a material will yield or fracture and is a widely used theoretical stress comparator. The von Mises yield criterion states that: if the von Mises stress of a material under a load is greater than the yield limit of the same material under simple tension, the material will yield. It is best applied to ductile materials, of which PCL falls under. Simply, the von Mises stress is a combination of principle normal stresses from the Cauchy Stress Tensor.

This report serves as an exploration into the mechanical properties of different scaffold macro-architectures, utilising Finite Element Analysis, to evaluate von Mises properties.

### Method

Computer-aided design (CAD) models of each scaffold macro-architecture were designed using Autodesk Inventor software, all of which having uniform dimensions of 11mm × ⌀ = 7mm (of the outermost circular element of each scaffold) to mimic the size of a human cervical cyst region (Koffler, et al., 2019). The microarchitecture of the implants was not considered within these tests to ensure that the true mechanical implications of each different macrostructure were reflected – and so all were modelled as smooth, dense polycaprolactone (PCL). However, within the final scaffold design, there will be a porous structure and hence the material properties will change. There is limited literature concerning all of the specific properties of PCL, and so the values used were as follows:

Table 1: Physical properties of polycaprolactone (PCL)Measurement | Value/Reference |
---|---|

Yield Strength (Limit) | 17.82 MPa (Ragaert, De Baere, Degrieck, & Cardon, 2014) |

Tensile Strength (Limit) | 34 MPa (Ragaert, De Baere, Degrieck, & Cardon, 2014) |

Compressive Strength (Limit) | 10 MPa (Ragaert, De Baere, Degrieck, & Cardon, 2014) |

Elastic Modulus | 4.3 MPa (MATWEB) |

Poisson’s Ratio | 0.442 (Lu, et al., 2014) |

To determine the von Mises properties, each scaffold had constraints on the flat, circular faces – under the assumption that the scaffold should be attached to the spinal cord on each side such that there is no displacement in the vertical direction. A simple gravitational load (g = 9806.65 mms&sup-2) was applied vertically, with the vertical faces of the scaffold constrained based on our assumption that there would be minimal movement in this direction. The analysis of the behaviour of the scaffold was carried out on Autodesk Inventor NASTRAN. In this case, no other loads were added, due to the lack of literature regarding such; the spinal cord’s mechanical response to confined compression has not yet been investigated (Yu, 2019).

When a material is subjected to a body force - in this case gravity - stress occurs. Within a linear model, the FEA solver assumes linearity between stress and strain; when the load is removed the object will return to its original state. This poses an issue when the stress applied is beyond the linear region of a material’s properties. In this case, with a linear model, the linear relationship between stress and strain is extrapolated – when in reality the situation may be much different, especially within a plastic material. Consequently, the stress assumed by the model may become much greater than the tensile limit (the point of mechanical failure). This is depicted within Fig. 1.

Subsequently, the method selected for this study was bi-linear material analysis coupled with non-linear static analysis. This bi-linearity accounts for the non-linearity of PCL; beyond the yield strength the rigidity of the material is different from the elastic range, and deformations of the scaffold may occur. Conversely, NASTRAN does not consider change in material properties over time (e.g. temperature), and so it must be noted that the properties evaluated in this report are subject to change in vivo – especially within the case of biodegradation. The application of true plastic material analysis was not possible, due to the lack of experimental values of stress/strain within literature – plastic material analysis requires three data points: the origin, the yield stress and another value between these. Despite this, with the given values within Table 1 a more realistic simulation was computed as opposed to linear analysis alone.

### Results

Table 2: Overall results for the simulations that were carried outCylinder | Tube | Open Path with Core | Open Path without Core | Channel | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Property | Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |

SVM Stress (MPa) | 1.11E-05 | 7.25E-04 | 1.23E-05 | 6.71E-04 | 6.61E-06 | 6.37E-04 | 1.09E-05 | 6.55E-04 | 1.01E-05 | 6.76E-05 |

SVM Strain | 2.47E-06 | 1.62E-04 | 2.75E-06 | 1.50E-04 | 1.48E-06 | 1.42E-04 | 2.43E-06 | 1.46E-04 | 2.26E-06 | 1.51E-04 |

Displacement (mm) | 0 | 2.30E-04 | 0 | 2.63E-04 | 0 | 3.04E-04 | 0 | 5.37E-04 | 0 | 2.60E-04 |

Applied Force (N) | 6.13E-08 | 2.78E-06 | 4.60E-08 | 2.71E-06 | 1.24E-08 | 1.97E-06 | 2.96E-08 | 2.07E-06 | 3.56E-08 | 2.56E-06 |

#### Applied Force

#### Stress

#### Strain

#### Displacement

### Discussion

The blue areas of the simulations represent areas of lower stress / strain / applied force / displacement, and the red areas are ares of higher values. Overall, in relation to von Mises Stress, none of the scaffolds exceeded the yield strength of 17.82MPa – indicating that each scaffold is a viable candidate. Due to the large plasticity region of PCL, it is advised that the comparison is made with respect to the yield strength rather than the tensile strength (34.1MPa) (Ragaert, De Baere, Degrieck, & Cardon, 2014). Therefore, under gravitational load none of the scaffolds would be deemed unsuitable due to permanent deformation. As each of the scaffolds fit the basic criterion (without reaching material thresholds), in order to determine the optimal scaffold type mechanically, each simulation result (von Mises stress, von Mises strain, displacement and applied force) was compared as shown in Table 3.

Table 3: Comparison of properties for each of the scaffold typesSVM Stress | SVM Strain | Displacement | Applied Force | Total | |
---|---|---|---|---|---|

Tube | 3 | 3 | 3 | 4 | 13 |

Channel | 4 | 4 | 2 | 3 | 13 |

Cylinder | 5 | 5 | 1 | 5 | 16 |

Open path with core | 1 | 1 | 4 | 1 | 7 |

Open path without core | 2 | 2 | 5 | 2 | 11 |

We analysed our results and determined that the open path with core scaffold was the optimal design to choose from the five scaffolds when considering the crucial mechanical properties. Each scaffold has 4 parameters which are scored in a range between 1 and 5 with 1 being the highest rank. The lowest total score is best. The open path with core achieved a score of 7. This result agrees with Wong et al., which also found the open path with core to be the most suitable as it supported white matter tracts (central core), allowed extension of myelinated fibres, maintained the defect size in a period of 3 months and axonal regeneration was observed.

## PCL Spinal Scaffold: Chosen Macro-Architecture Undergoing a Cough Simulation

Coughing and sneezing can have a substantial effect on the spinal subarachnoid space (SAS) (with a stenosis present) on the cyst. From (Martin & Loth, 2009), we obtained values for transmural pressure. Transmural pressure is the pressure difference across a hollow structure; it is the pressure gradient across the vessel wall. The compression of the syringomyelia is caused by the transmural pressure force. This force will compress the cyst and our scaffold. It is vital to model these forces on our open path with core scaffold to observe and evaluate its performance.

The results are displayed below. They show that when considering gravitational load and the simulation of a cough, the scaffold is capable of withstanding the force and will not break. The majority of the scaffold is in the blue region, resulting in low stresses. The areas with higher stresses are in red. However, this is not a cause for concern as this is in the order of magnitude of 5.87E-04. We took the largest value of TP from Fig 2. which was equivalent to 3.06641E-03 kPa. We applied this pressure load in the radial direction with the axial ends constrained.

# References:

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