Antibiotic transport through the extracellular polymeric substance (EPS) that binds a biofilm together is a classic reaction–diffusion problem. At steady state, Fick's second law with a first-order sink (degradation by matrix enzymes, e.g. β-lactamase, plus non-specific binding to EPS polymers) reduces to:
D · d²C/dx² = k · C
Steady-state solution (semi-infinite slab):
C(x) = C₀ · exp(−x / λ)
Penetration length:
λ = √(D / k)
This is the same functional form as electromagnetic skin depth or neutron shielding — an exponential attenuation length set by the ratio of transport speed to removal rate. A bacterium at depth x only dies if the antibiotic concentration reaching it, C(x), stays ≥ its Minimum Inhibitory Concentration (MIC) for long enough. Because λ often shrinks to a few micrometres for β-lactams against a mature biofilm, the innermost cells can be fully antibiotic-susceptible in a test tube yet survive completely intact inside the biofilm — not through mutation, but through a physical transport bottleneck.
- C₀ slider — the antibiotic concentration bathing the outer surface, in multiples of the planktonic MIC.
- D slider — effective diffusivity through the EPS matrix (lower than in free water; dense polysaccharide/eDNA slows diffusion).
- k slider — combined degradation + binding rate of the matrix; higher k (e.g. more β-lactamase-producing cells) shrinks λ sharply since λ ∝ 1/√k.
- L slider — total biofilm thickness; once L ≫ λ, the core is essentially unreachable regardless of dose.
- The exponential decay is animated as it advances into the biofilm after each dose, and every bacterium instance is recolored live from the concentration field at its own depth compared against its MIC.
Real-world relevance: this diffusion-reaction limitation is one of the three recognised physical/physiological mechanisms of biofilm-associated antibiotic tolerance (alongside metabolically dormant persister cells and altered gene expression in the biofilm phenotype), and it is why chronic infections on catheters, prosthetic joints and in cystic-fibrosis lungs routinely survive doses that easily clear the same species grown planktonically.