A disk, not a sphere — a genuinely different equation. The 3D version of this simulator treats the tumor as a sphere and evaluates Baxter & Jain's closed-form radial solution directly. This 2D twin instead models a flat cross-section (a disk) through the tumor and numerically solves the same governing PDE on a 2D grid — it does not just flatten the 3D camera view.
Both start from the identical steady-state Darcy/Starling balance, but circular (2D) symmetry gives a different closed form than spherical (3D) symmetry:
Governing PDE (both): ∇²P − α²P = −α²P0, α = √(Lp·S/V / K)
3D sphere (closed form): P(r) = P0·[1 − sinh(αr)/(αr) ÷ sinh(αR)/(αR)]
2D disk (closed form): P(ρ) = P0·[1 − I0(αρ) ÷ I0(αR)] (I0 = modified Bessel fn.)
Rather than evaluating that Bessel-function formula directly, this engine solves ∇²P − α²P = −α²P0 with over-relaxed Gauss–Seidel (SOR) on a 96×96 finite-difference grid, masked to the disk (P = 0 fixed outside the tumor rim) — a genuine numerical field solve, re-run whenever conductivity changes. Because the equation is linear in P0, the solver only needs to re-relax the unit-forcing shape once per K change; P0 then scales the converged field instantly.
The resulting pressure field's true 2D gradient (∂P/∂x, ∂P/∂y) drives Darcy advection v = −K·∇P on every particle, sampled by bilinear interpolation at the particle's own (x, y) — particles drift in the actual plane, not along a single radial line. Brownian motion is added independently on both axes from the Stokes–Einstein diffusivity D = kBT / 6πηrp, exactly as in the 3D model.
- P0 slider — scales the converged pressure field linearly; higher P0 flattens the interior gradient further and traps more particles perivascularly.
- Conductivity K — changes α, so it retriggers a fresh grid relaxation; low K shrinks the convective rim and adds drag.
- Particle diameter — larger particles diffuse slower (D ∝ 1/d) via the same Stokes–Einstein law, so they depend more on the convective rim.
The radial profile chart samples the converged 2D grid field along a ray from the vessel to the rim — it should trace the same shape as the disk's I0-Bessel closed form, just produced by relaxation instead of evaluating the special function.