TPMS Gyroid Lattice & Porous Bone Ingrowth Prepress Guide
Comprehensive guide to implicit level-set Gyroid surfaces, Gibson-Ashby modulus scaling, stress shielding mitigation in titanium orthopedic implants, and 2D CAM vector slice contour generation for LPBF / DMLS additive manufacturing.
1. Triply Periodic Minimal Surfaces (TPMS) Mathematics
Triply Periodic Minimal Surfaces are mathematically defined continuous implicit surfaces possessing zero mean curvature ($H = \frac{\kappa_1 + \kappa_2}{2} = 0$) and periodic symmetry in all three spatial Cartesian directions ($X, Y, Z$). The Schoen Gyroid surface is expressed via trigonometric approximations:
$$F(x, y, z) = \sin\left(\frac{2\pi x}{L}\right)\cos\left(\frac{2\pi y}{L}\right) + \sin\left(\frac{2\pi y}{L}\right)\cos\left(\frac{2\pi z}{L}\right) + \sin\left(\frac{2\pi z}{L}\right)\cos\left(\frac{2\pi x}{L}\right) = c$$
Sheet-based Gyroids are formed by bounding the domain between two parallel level sets: $-t/2 \le F(x, y, z) \le +t/2$.
2. Modulus Matching & Stress Shielding Elimination
Solid titanium implants ($E \approx 110\text{ GPa}$) induce severe stress shielding when bonded to human trabecular bone ($E = 0.5 - 3.5\text{ GPa}$) or cortical bone ($E = 12 - 18\text{ GPa}$), causing bone resorption and implant loosening. By engineering the relative density of the Gyroid lattice, the apparent elastic modulus ($E^*$) is tailored exactly to match host bone:
$$\frac{E^*}{E_s} = C_E \left(\frac{\rho^*}{\rho_s}\right)^{n_E} \quad \text{and} \quad \frac{\sigma^*}{\sigma_y} = C_\sigma \left(\frac{\rho^*}{\rho_s}\right)^{n_\sigma}$$
For sheet Gyroids in Ti-6Al-4V ELI: $C_E \approx 0.88$, $n_E \approx 1.82$, $C_\sigma \approx 0.72$, $n_\sigma \approx 1.50$.
3. Clinical Bone Ingrowth (Osteointegration) Windows
Biological cell migration, vascular angiogenesis, and rapid bone tissue infiltration depend strictly on pore interconnectivity and pore throat diameter ($D_{\text{pore}}$):
| Pore Size ($D_{\text{pore}}$) | Biological Response | Relative Density ($\rho^*/\rho_s$) | Recommended Unit Cell ($L$) |
|---|---|---|---|
| < 200 µm | Fibrous tissue encapsulation (vascular starvation) | > 65% | < 1.5 mm |
| 300 – 600 µm | Optimal osteogenesis & capillary ingrowth (Gold Standard) | 25% – 45% | 2.5 – 4.5 mm |
| 600 – 900 µm | High bone infiltration, trabecular interlocking | 15% – 30% | 4.5 – 7.0 mm |
| > 1000 µm | Reduced osteoblast bridging rate, lower strength | < 15% | > 8.0 mm |
4. 2D Vector Slicing & CAM Hatch Generation
In Laser Powder Bed Fusion (LPBF / DMLS), 3D implicit Gyroid solids are sliced into planar 2D layer boundaries ($\Delta z = 30 - 60\text{ µm}$). The resulting intersection profiles consist of smooth, closed spline contours that are converted directly to DXF vector entities for galvo-scanner laser hatching:
// DXF Layer Slice Output for LPBF Galvo Scanner
0
SECTION
2
ENTITIES
0
POLYLINE
8
GYROID_SHEET_CONTOUR_Z0450
66
1
70
1 // Closed loop spline polygon
0
VERTEX
8
GYROID_SHEET_CONTOUR_Z0450
10
4.2518
20
12.8391
0
VERTEX
8
GYROID_SHEET_CONTOUR_Z0450
10
4.4102
20
13.1205
0
SEQEND
0
ENDSEC
0
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