After bioprinting, cells inside a construct depend entirely on diffusion for oxygen until new blood vessels grow in — there is no perfusion yet. Oxygen enters from the culture medium at the surface and is consumed by cells throughout the tissue. At steady state this is a spherically-symmetric reaction–diffusion balance:
D · (1/r²) d/dr(r² dc/dr) = Q (where c > 0)
Boundary: c(R) = C_s (medium concentration at the surface)
Symmetry: dc/dr(0) = 0 (no flux at the centre)
Necrosis: c = 0 and consumption stops once local O₂ is exhausted
This simulator solves that equation numerically (Gauss–Seidel relaxation on a radial grid, re-solved live as you move the sliders) rather than using a fixed formula. D is fixed at a typical soft-tissue oxygen diffusivity (~1500 µm²/s). Whenever the diffusion-limited penetration depth is shorter than the construct radius R, oxygen is fully consumed before reaching the centre and a necrotic core forms — cells there are shown dark/red. Cells above the Ccrit threshold (green) stay viable; cells between are hypoxic but alive (amber).
- Construct radius R — thicker prints have farther to diffuse; this is the single biggest lever on viability.
- Consumption rate Q — higher cell density or more metabolically active cell types consume O₂ faster.
- Medium O₂ (Cs) — hypoxic incubators or poor media exchange lower the surface boundary value.
- Necrosis threshold (Ccrit) — the O₂ concentration below which a given cell type becomes hypoxic/dies; varies by cell line.
Real-world relevance: this ~150–200 µm diffusion limit is exactly why bioprinted tissue constructs beyond a few hundred microns thick need sacrificial vascular channels, perfusion bioreactors, or pre-vascularization — without them the core of any sufficiently thick print starves before it can be rescued by ingrowing vessels.