Lethal-threshold crossing in a stochastic lesion–repair model: exact linear-quadratic coefficients, the Fokker–Planck limit and parameter identifiability

Objective. To derive the linear-quadratic (LQ) survival law from a stochastic model of lesion induction and repair with an effective lethal threshold, to determine what identifies its parameters, and to test it on published data. Approach. The number of unrepaired lethal-candidate lesions in a sub-nuclear domain, the unit whose lesions share one lethal threshold, follows a birth–death master equation with first-order repair rate μ and fluctuation intensities σ² (linear) and κ (multiplicative). Cell death is first passage to an effective threshold. Main results. The exact acute-irradiation survival probability gives explicit small-dose LQ coefficients; at leading order α/β=(σ²-μ+κ)/(μλ_d), where λ_d is the lesion yield per domain, independent of the threshold. The quadratic term needs no lesion interaction. The probability of not crossing the threshold is log-concave in lesion number, so a Poisson shock-model theorem makes -ln S of a homogeneous population of this model class convex in dose after acute exposure. Exact moment equations establish structural identifiability in principle from ideal lesion-kinetics data, but not from survival curves; computational experiments did not recover the number of domains in practice. On 20 independent survival series at 1–12 Gy no general advantage of the model over LQ was established: it was preferred in 5, LQ in 9, and extrapolated worse to lower doses in 16 of 19. Within laboratories, 6 and 10 MV curves, flattened or not, differed only by dose factors of 0.98–1.07 (0.96–1.08 in sensitivity analyses), consistent with beam quality acting through λ_d alone. In 91 low-dose datasets selected for hyper-radiosensitivity, the number showing a significant decrease of -ln S/D, inconsistent with convexity for a homogeneous population, depended on unreported error properties: 66 with plotted errors as standard errors and exact normalization, 18–81 across error models with normalization errors up to 5%. Significance. The main value of the model lies not in a more accurate description of survival but in its exact solvability and its experimental testability. The model links the LQ coefficients to measurable parameters of lesion kinetics. It predicts the time course of the variance of the lesion number and allows α/β to be calculated from kinetic data. Existing survival curves cannot test these predictions. Testing them requires joint measurements of lesion kinetics and survival in the same cells.