A proton deposits little dose on entry, a sharp maximum where it stops, and essentially zero exit dose. Photons attenuate exponentially and exit through the patient. Range follows the Bortfeld (1997) power law:
Protons don't care what a tissue is — only how much it slows them. Stopping-power ratio maps geometry to water-equivalent depth:
Adipose (ρ≈0.92, SPR≈0.94) advances the peak ≈0.06 mm/mm → the scheduler shifts that layer's energy down; muscle (ρ≈1.06) retards it → energy up. At 12% BF this patient has ≈9 mm subcutaneous adipose and a ≈13 mm rectus wall, but a ≈26 mm paraspinal band on posterior fields — every ray gets its own energy-layer schedule, recomputed each optimizer pass.
Fat and muscle respond identically; density differences are purely kinematic (range modulation), never biological sensitivity. All doses here are Gy(RBE).
Stage 1 — projected AdaGrad on per-spot fluence weights; PV constraint via an augmented-Lagrangian penalty that escalates while violated. Stage 2 — a coordinate search edits the computed scanning plan (energy layers ±1.5 MeV, gantry ±4°), keeping only edits that lower the objective; every edit is logged.
The anterior path has least WEPL, but its distal fall-off (80→20% over ~3–5 mm) then points straight at the portal vein. Under ±2.5% range uncertainty a 1σ overshoot converts that fall-off into near-full dose inside the vessel. The optimizer prefers geometrically longer posterior/right-oblique paths with stable density whose distal edges terminate inside T₁/T₂.
Multi-field dilution: each of 3 fields carries ~⅓ of target dose, so a ~25% entrance plateau deposits ≲8% of Drx along its own channel; the 100% cloud exists only where all SOBPs intersect — on the targets.
| Structure | Model value | Basis |
|---|---|---|
| Torso cross-section | 30.0 × 21.2 cm ellipse | lean male, waist ≈84 cm |
| Subcutaneous adipose | ≈9 mm, SPR 0.94 | 12% BF |
| Abdominal wall / paraspinal | ≈13 mm / ≈26 mm, SPR 1.06 | athletic musculature |
| Liver | right-lobe-dominant, SPR 1.05 | ≈1500 mL organ |
| Portal vein (OAR) | Ø 14 mm @ porta, Dmax ≤ 12 Gy(RBE) | main trunk |
| IVC / aorta | Ø 18 / 16 mm | standard |
| L1 + ribs | SPR 1.62 | cortical/trabecular |
| Stomach gas | SPR ≈ 0.001 | large range effect |
| T₁ (seg VIII) / T₂ (seg VII) | Ø 9 mm / Ø 8 mm | right-lobe lesions |
Default fields 170° / 250° / 305° (0° = anterior AP, increasing toward patient-left): posterior + right obliques that avoid stomach gas, cord and the PV distal-edge alignment. Not a clinical TPS.
| Structure | Dmean | Dmax | Metric | Value | Status |
|---|---|---|---|---|---|
| T₁ | – | – | D95 | – | – |
| T₂ | – | – | D95 | – | – |
| Portal vein | – | – | Dmax≤lim | – | – |
| Uninvolved liver | – | – | V₃₀ | – | – |