Carcinogen Dose-Response: a Barrier-Lowering Kernel

A carcinogen is a sustained aberrant drive that lowers the R19 escape barrier, so the malignant-crossing rate is Kramers-like and the dose-response RR(dose) is monotone and convex (accelerating). With the natural barrier scale it collapses to a universal law RR = e^dose/dose* (RR≈2.46 at 90% of the yield drive) — shape verified [V], radiation anchor cited [L], absolute incidence open [O].

Modelling the carcinogen as a barrier-lowering drive gives RR(dose)=rate(dose)/rate(0) that rises convexly with dose (RR≈1.57 at 0.45, 2.46 at 0.90 of the spinodal drive). The lineage γ cancels in the ratio, so the NORMALISED dose-response is the universal convex law RR=e^dose/dose*. Radiation RR for bone sarcoma is cited [L]; absolute incidence and the genetic-dominant aetiology are open [O].

One kernel for cell fate and cancer

The cell-fate switch that emerges each organ is reused for oncology. A carcinogen acts as a sustained aberrant drive h_c that lowers the escape barrier out of the healthy basin; the malignant-crossing rate is Kramers-like, rate(h_c) = rate₀·e^{−ΔV_eff/scale}, with ΔV_eff→0 as the drive approaches the spinodal. The relative risk RR(dose) = rate(dose)/rate(0) is therefore monotone and convex — the signature of barrier-limited escape.

With the natural Kramers scale (the barrier height ΔV = γ²/4) the lineage γ cancels in the ratio, and the normalised dose-response collapses to a single parameter-free law RR = e^dose/dose*, where dose* is the spinodal drive. The curve gives RR≈1.57 at 45% and 2.46 at 90% of dose*. The SHAPE (monotone, convex, RR>1) is verified on the substrate [V].

Lineage mapping (measured γ)

Osteosarcoma is modelled on the osteoblast-lineage switch RUNX2 (bone, γ=1.2414); soft-tissue sarcoma on the myogenic-lineage switch MYOD1 (muscle, γ=1.4933) — MYOD1 is the canonical rhabdomyosarcoma marker. Because γ cancels in RR, both sites share the same convex normalised curve; γ sets only the absolute drive scale.

Two further sites, with an honest dose axis

The same kernel extends to chondrosarcoma (cartilage lineage, SOX9 γ=1.4598) and Ewing sarcoma (a bone-associated small-round-cell tumour). For both, the driver is NOT an environmental exposure: chondrosarcoma is driven by IDH1/2 neomorphic metabolism (the 2-HG oncometabolite) and Ewing by the constitutive EWSR1–FLI1 fusion. The kernel still reproduces the barrier-lowering SHAPE, but the dose axis is then the NORMALISED aberrant drive, not a cited exposure dose, and that distinction is recorded as an open item [O]. Ewing's cell of origin is debated (mesenchymal / neural-crest), so RUNX2 is only a bone-mesenchyme context stand-in — the lineage assignment itself is graded [O].

What is cited, and what is NOT claimed

Radiation relative risk for bone sarcoma rising with cumulative dose is the cited anchor [L] (radium dial painters; Tucker et al. 1987; UNSCEAR). The open items [O], not overclaimed: (1) absolute incidence in cases per person-year — the kernel gives RELATIVE risk only, the baseline hazard is not fixed by the substrate; (2) osteosarcoma is predominantly GENETIC (RB1, TP53/Li-Fraumeni, Paget), so the dose-response applies only to the radiation-attributable fraction; (3) chemical soft-tissue-sarcoma dose-response has no clean cited anchor (confounded, low incidence); (4) for chondrosarcoma and Ewing the aetiology is metabolic/fusion-genetic, so there is no environmental exposure dose-response and the dose axis is the normalised drive only.