Biodegradable Implant Drug Elution — Radial Diffusion Field

Implant & Drug Parameters

Diffusion coefficient D₀ 0.030

Hydrolysis rate k_h (day⁻¹) 0.060

Erosion–diffusivity coupling β 3.0

Initial drug loading 900

Live State

Cumulative release

0.0%

Simulated time

0.0 d

D_eff (matrix)

0.030

Polymer M_w remaining

100.0%

Cumulative release fraction vs. time

How it works

This is a continuum, grid-based twin of the particle-based 3D version: instead of tracking individual drug particles doing a 3D random walk, this engine solves the diffusion PDE (Fick's second law) directly on a 2D finite-difference grid representing a cross-section through the implant and the surrounding release medium.

Fick's 2nd law (finite-volume form, per cell):
  ∂c/∂t = ∇·(D(x)∇c)
  c[i,j] += Δt/Δx² · Σ_neighbors (D̄ · (c_neighbor − c[i,j]))
  where D̄ is the average diffusivity between the two cells

Bulk hydrolysis (first-order, Siepmann-type):
  M_w(t) = M_w0 · exp(−k_h·t)

Erosion-coupled diffusivity (same law as the 3D model):
  D_eff(t) = D_0 · exp(β·(1 − M_w(t)/M_w0))

Drug is "released" once it diffuses past the implant's own
radius R into the surrounding medium region — the grid edge itself
is a reflective (closed-vessel) boundary, so total drug mass is
conserved and "release" means crossing R, exactly mirroring the
3D model's per-particle r > R test.

D₀ — intrinsic diffusivity through the intact polymer, shown here as the rate the teal concentration field spreads within the disk.
k_h — hydrolysis rate constant for M_w decay; faster hydrolysis raises D_eff sooner.
β — erosion–diffusivity coupling. At β = 0 the field spreads by pure Fickian diffusion (Higuchi-like √t release); larger β adds an erosion-triggered acceleration once the matrix has degraded.
Drug loading — sets the initial concentration amplitude only; because the diffusion equation is linear, it changes the field's visual contrast, not the release-fraction curve — the same statement the 3D model makes about particle count.

Both engines solve the same physics (Fick's second law coupled to first-order polymer erosion) — one via a stochastic particle random walk, this one via the deterministic continuum field the random walk converges to in the many-particle limit.

Drug concentration (in matrix) Diffused into release medium