Carcinogen Dose Response and Relative Risk
A carcinogen is a sustained bias that lowers the R19 switch barrier, so malignant-crossing rate is Kramers over that barrier and relative risk is rate(dose)/rate(0). The lung-carcinoma curve starts exactly at RR(0)=1, has no threshold (half a pack-year already gives RR>1), and is convex, reaching ~18× at 100 pack-years. Forced endpoints [consistency]; shape [code output]; absolute magnitude [O].
Barrier-lowering Beff = U(saddle)−U(basin) from the cubic s³−γs−h=0 gives RR = exp((B₀−Beff)/scale); RR(0)=1 [consistency], no threshold (slope √γ) [consistency], convex to ~18× at 100 pack-years [code output]; absolute magnitude [O].
A carcinogen lowers the R19 barrier
Cell fate is the same R19 bistable switch. A carcinogen is a sustained aberrant bias hc that pushes toward the malignant basin and lowers the barrier the normal basin must climb. The malignant-crossing rate is Kramers/Arrhenius over that barrier, rate ∝ exp(−Beff/scale), and relative risk is RR(dose) = rate(dose)/rate(0).
The barrier is derived, not postulated: the three critical points are the real roots of the cubic s³ − γs − h = 0 (three reals inside |h| < h* = 2(γ/3)^1.5), and Beff = U(saddle) − U(basin). Two exact identities follow and are checked at import.
| identity | value | grade |
|---|---|---|
| Beff(h=0) equals substrate barrier γ²/4 | 0.5691 | [consistency] |
| barrier-drop slope at the origin equals √γ | 1.228 | [consistency] |
Dose–response: monotone, convex, no threshold
Mapping cumulative tobacco exposure to bias gives a lung-carcinoma relative risk that starts exactly at RR(0) = 1, rises monotonically, and is convex (accelerating) — reaching 18.3× at 100 pack-years, the Doll & Peto order of magnitude. This value is a [code output] of two declared choices in repro/cardioresp/repro/_oncology/carcinogen_dose_response.py: scale = barrier/3, which caps RR near e3 ≈ 20, and the mapping of 100 pack-years to a fixed fraction of the spinodal; other choices give the same shape with other values. [observation] Lung-cancer relative risks of roughly 10–30 are observed in long-term heavy smokers (Peto et al. 2000, BMJ 321:323; Doll et al. 2004, BMJ 328:1519).
There is no threshold: even half a pack-year already gives RR = 1.021 > 1. The low-dose slope is √γ by the barrier algebra, so any positive exposure raises risk — the linear-no-threshold shape, a [consistency] result.
| pack-years | relative risk | ln RR |
|---|---|---|
| 0 | 1.00× | 0.000 |
| 5 | 1.23× | 0.205 |
| 10 | 1.50× | 0.405 |
| 20 | 2.20× | 0.788 |
| 30 | 3.15× | 1.148 |
| 40 | 4.41× | 1.484 |
| 50 | 6.03× | 1.796 |
| 60 | 8.02× | 2.082 |
| 80 | 13.02× | 2.566 |
| 100 | 18.32× | 2.908 |
Combination of carcinogens
Two carcinogens add their bias h. The model then predicts a combined risk that is 0.76× the product of the singles — sub-multiplicative at moderate dose, approaching multiplicative only in the low-dose limit. This is a falsifiable corollary; empirical agent-pair synergy varies.
What is open
The absolute relative-risk magnitude is grade [O]. Obstacle: the switch's effective noise temperature (scale) and the molecular dose-to-bias conversion are not fixed by substrate geometry and require an external population-hazard calibration — exactly as absolute organ size is open (relative size and order read from γ are also open [O], dna §RB). The forced endpoints (RR(0)=1, no threshold) and the convex shape stand independently of that calibration.