Radiation Biology & Dosimetry

Ionizing Radiation in Biology

Ionizing radiation — X-rays, gamma rays, alpha particles, and beta particles — deposits energy in biological tissue, causing DNA damage, free radical formation, and cell death. Quantifying radiation exposure is essential in medical physics, radiotherapy, and radiation protection.

Absorbed Dose

The absorbed dose D is the energy deposited per unit mass of tissue:

$D = \frac{\text{Energy}}{\text{Mass}} \quad [\text{Gray (Gy) = J/kg}]$

Equivalent Dose

Different radiation types cause different degrees of biological damage. The equivalent dose H accounts for this via the radiation weighting factor W_R:

$H = D \times W_R \quad [\text{Sievert (Sv)}]$

Effective Dose

The effective dose E sums over all irradiated tissues, weighted by tissue sensitivity factor W_T:

$E = \sum_T W_T \cdot H_T \quad [\text{Sv}]$

Linear-Quadratic (LQ) Model

The LQ model describes cell survival fraction S after dose D:

$S = \exp\left(-(\alpha D + \beta D^2)\right)$

• α (Gy⁻¹): linear component — DNA double-strand breaks from single tracks
• β (Gy⁻²): quadratic component — breaks from two independent tracks
• α/β ratio: characteristic dose where linear and quadratic contributions are equal; typically 10 Gy for early-responding (tumor) tissues and 3 Gy for late-responding (normal) tissues

D₅₀: Dose for 50% Survival

Setting S = 0.5 and solving α·D + β·D² = ln 2:

$D_{50} = \frac{-\alpha + \sqrt{\alpha^2 + 4\beta \ln 2}}{2\beta}$

Your Task

Implement three functions:

1. equivalent_dose_Sv(D_Gy, W_R=1) — Equivalent dose in Sv
2. cell_survival_LQ(D_Gy, alpha=0.3, beta=0.03) — Cell survival fraction from the LQ model
3. d50_LQ_Gy(alpha=0.3, beta=0.03) — Dose (Gy) at which 50% of cells survive