A sponge slowly watered from a watering can gets soaked all the way through. But if you dump the same bucket of water on it all at once, the middle gets wet while the edges stay almost dry. That's roughly how a material with nonlinear absorption behaves: the more intense the flux, the more it gets stuck at the point of impact.

In this new study, that image is turned into a minimal model. A two-dimensional medium of 250×250 cells, like a field divided into ten-centimeter squares, receives energy from a source at the center. The energy spreads and is absorbed according to one of four laws: constant, linear, saturating, or threshold. Two delivery modes of the same total dose are compared: continuous, like slow dripping, and pulsed, like a sudden splash.

For all nonlinear laws, the pulsed mode reduces the total absorbed energy in the ring-shaped region around the center: in the very core, absorption increases by about one and a half percent, but in the surroundings it drops. For the linear and threshold laws, the effect is stronger than for the saturating one. This is the qualitative explanation for the tissue-sparing action of FLASH therapy: the tumor at the center gets its due, while the healthy tissue around it gets less.

The dependence on pulse intensity turned out to be non-monotonic. If the absorption threshold is low, an overly intense pulse can even increase absorption in the surrounding region. At a high threshold, the benefit reaches a plateau. This agrees with experiments: the FLASH effect weakens or disappears when the critical dose is exceeded.

The sponge that absorbs differently depending on whether you pour slowly or splash all at once turned out to be more than just a metaphor. It suggests how to tailor pulse intensity to a specific tissue: if the absorption threshold is known, you can enhance the sparing effect. And it reminds us that in the physics of energy transport, sometimes a single nonlinearity is enough to change the whole picture.