This is a direct 2D finite-difference solution of the same physics as the 3D probe view, but computed a different way: instead of one lumped ODE for a single front radius, this engine timesteps the full 2D heat-conduction field around a cross-section of the probe and lets the freezing front emerge on its own from the numbers.
Latent heat is handled with the apparent (equivalent) heat-capacity method: near the freezing point the tissue's effective heat capacity is boosted by a physically-derived factor so that energy is consumed absorbing/releasing latent heat instead of just changing temperature, exactly reproducing a moving Stefan front without tracking its position explicitly:
ρc·∂T/∂t = k·∇²T − w·ρc·(T − T_body) (Pennes bioheat form)
c_app(T) = c · [ 1 + (L_f / (c·ΔT_mush)) · band(T) ]
L_f = latent heat of fusion of tissue water (334 kJ/kg)
ΔT_mush = width of the freezing "mushy zone" around 0°C
w = perfusion rate (blood flow heat-sink)
The probe boundary is held at the set temperature by a fixed (Dirichlet) inner circle; the domain edge is held at core body temperature (37°C), and the whole field is timestepped explicitly with enough substeps per frame to stay numerically stable at every probe-temperature setting.
- Probe temperature — colder argon flow pushes the front outward faster and deepens the lethal isotherm (verified: −186°C reaches a larger ice front than −140°C at the same elapsed time).
- Tissue perfusion — blood flow resupplies heat as a volumetric sink term and shrinks both the ice front and the lethal margin — the reason large vessels near a tumor are a treatment risk.
- Freeze-thaw cycles — the field never resets between cycles, so residual cold from the first freeze carries into the second; combined with a lowered lethal threshold per completed cycle, this reproduces the clinical practice of standard 2-cycle protocols killing more tissue at the same radius.
Drag the field view to pan, scroll/pinch to zoom. The lower panel plots the live radial temperature profile straight out of the grid — the same curve the 0°C and −40°C isotherm rings on the field are computed from.