Exploring how limited nutrients and diffusion gradients shape the morphology of developing tumors.
Oxygen and glucose diffuse in from healthy tissue, but the outer cell layers consume them first. The resulting gradient decides where cells live and where they die.
This is diffusion-limited growth. The result is a viable rim of proliferating cells on the outside and a necrotic core where nutrients cannot reach.
A small protrusion on the tumor surface reaches further into nutrient-rich tissue, grows faster, and amplifies into a branch — the Mullins-Sekerka instability. The same mechanism shapes snowflakes and mineral deposits.
Blue bars mark living cells, charcoal bars mark necrotic cells, and the green line tracks average nutrient from the center outward.
Figure 1. A simulated tumor growing in a nutrient field. The background color represents nutrient concentration. Watch how branches form as the tumor surface destabilizes. Click anywhere to seed a new colony; switch regimes to compare compact fronts with diffusion-starved branches.
Branching maximizes the surface-to-volume ratio, exposing more of the tumor to the nutrient field.
The nutrient concentration \(c\) follows a diffusion equation with a consumption term (shown in the figure above): \(\partial_t c = D\nabla^2 c - kc\), where \(D\) is the diffusion coefficient and \(k\) is the consumption rate by the cells. In our simulation, the growth probability of a surface cell is proportional to the local concentration \(c\).