The 3D companion decides whether a printed cell survives with a static geometric check: distance from that cell's fixed position to the nearest outer surface or channel, compared once against a fixed diffusion-limit radius. This 2D cross-section instead solves an actual time-dependent diffusion–consumption PDE for dissolved oxygen through the growing slab, cell by cell, every frame — so a region can visibly start oxygenated right after deposition and starve gradually as the tissue above it thickens and consumption outpaces resupply, a dynamic the 3D snapshot-distance model cannot show at all.
τ = 8·η·v / d (wall shear stress in the nozzle, same as the 3D model)
viability ≈ 100%·e^(−τ/40) (shear-induced cell damage at extrusion)
∂O/∂t = D∇²O − k·O (oxygen diffusion + cellular consumption in printed tissue)
O = 1 in any medium/channel cell (open bath, Dirichlet boundary)
- Print speed — faster extrusion raises the shear stress cells feel passing through the nozzle (same formula as the 3D model).
- Nozzle diameter — a narrower nozzle prints thinner, higher-resolution layers (more, finer layer bands) but squeezes cells harder, raising shear stress; a wider nozzle deposits thick, fast, lower-resolution layers under less shear.
- Bioink viscosity — thicker bioink transmits more shear stress to cells at a given speed and nozzle size.
- Vascular channels — carved columns that stay open to the bath at every depth, acting as internal oxygen sources the diffusing field can reach instead of only the top and side surfaces — watch the field brighten around them once printing reaches that depth.
Real-world relevance: a printed construct's core keeps consuming oxygen after it's laid down, so viability is not decided at the nozzle alone — a slab that prints with low shear can still starve its interior minutes later if it has no vascular access, which is exactly what this diffusion field makes visible over time.