This is the physics behind in vitro sedimentation, diffusion and dosimetry (ISDD) modelling: the administered nanoparticle dose is not the dose the cells actually receive. Particles suspended in the media column above a cell monolayer are transported by two competing mechanisms:
Stokes settling: v_s = (2/9) · (ρ_p − ρ_f) · g · r² / μ
Stokes–Einstein diff.: D = k_B·T / (6π·μ·r)
Transport (1D): ∂C/∂t = D·∂²C/∂y² − v_s·∂C/∂y
with a reflecting boundary at the media surface (y=0) and an absorbing boundary at the cell monolayer (y=H), integrated from a uniform initial dose. Each particle drawn here is a real Langevin walker obeying exactly that drift-plus-diffusion equation — the same one-dimensional transport equation the 3D version renders as an orbitable scene, shown here instead as a column plus its own live concentration-depth histogram and dose-delivery time series, since the underlying physics was already one-dimensional (depth only).
- Diameter & material — set particle radius r and density ρ_p, which enter v_s through r² and Δρ, and D through 1/r.
- Agglomeration state — real suspensions rarely stay as single primary particles; agglomerates grow the effective hydrodynamic radius (fractal scaling, r_eff ≈ r₀·N^(1/D_f)) while entraining fluid, so their effective density sits between the particle and the media. Bigger agglomerates diffuse more slowly but usually settle much faster — the classic ISDD finding that aggregation state, not primary size alone, controls the cell dose.
- Media viscosity — serum proteins and polymers raise μ, slowing both v_s and D.
- Péclet number Pe = v_s·H/D compares advective settling to diffusive spreading — Pe ≫ 1 means gravity dominates (dense, large particles), Pe ≪ 1 means diffusion dominates (small, light particles) and transport is much slower.
Watch the concentration-depth panel flatten toward the monolayer and the dose-vs-time curve bend from linear (diffusion-limited) toward saturating exponential (settling-dominated) as Pe rises — the same reason a heavy gold nanoparticle and a light polystyrene bead at the same mass concentration expose cells to very different actual particle numbers over the same assay window.