Summary of Linear Quadratic Model in Radiotherapy

## Introduction Radiotherapy fractionation and tumour response models describe how total radiation dose, dose per fraction, and biological sensitivity interact to determine tumour control and normal tissue effects. This guide explains key concepts such as the Linear-Quadratic (LQ) model, BED/TE/EQD2 conversions, limits of models, and sources of variability that matter when planning or interpreting radiotherapy schedules. ## Basic concepts and definitions > **Definition:** The Linear-Quadratic (LQ) model relates cell kill to dose using linear ($\alpha$) and quadratic ($\beta$) components of damage. > **Definition:** Biologically Effective Dose (BED) quantifies the biological effect of a fractionated regimen relative to single-dose effects. > **Definition:** Equivalent Dose in 2 Gy fractions (EQD2) is the total dose in 2 Gy fractions that gives the same biological effect as a regimen with fraction size $d$ and total dose $D$. ## The LQ model and TE/BED/EQD2 ### Mathematical expression The total effect (TE) in LQ terms can be written proportional to $E/\beta$, often expressed as: $$TE = \frac{E}{\beta} = \left(\frac{\alpha}{\beta} + d\right) D$$ If we want the same effect as a 2 Gy fractionation, set $$\left(\frac{\alpha}{\beta} + d\right) D = \left(\frac{\alpha}{\beta} + 2\right) D_{2\mathrm{Gy}}$$ and solve for the equivalent dose $D_{2\mathrm{Gy}}$ (EQD2). ### EQD2 formula Rearranging gives the commonly used EQD2 expression: $$EQD2 = D \frac{\left(\frac{\alpha}{\beta} + d\right)}{\left(\frac{\alpha}{\beta} + 2\right)}$$ > **Definition:** $\alpha/\beta$ is the dose at which the linear and quadratic contributions to cell kill are equal; it is a tissue- or tumour-specific sensitivity parameter. ### Practical example A regimen: total dose $D=66\ \mathrm{Gy}$ in fractions $d=1.5\ \mathrm{Gy}$. Compute EQD2 for two $\alpha/\beta$ values. For $\alpha/\beta = 10$: $$EQD2 = 66 \frac{(1.5 + 10)}{(2 + 10)} = 66 \frac{11.5}{12} = 63\ \mathrm{Gy}$$ For $\alpha/\beta = 1$: $$EQD2 = 66 \frac{(1.5 + 1)}{(2 + 1)} = 66 \frac{2.5}{3} = 55\ \mathrm{Gy}$$ These show how tissue/tumour sensitivity ($\alpha/\beta$) strongly changes the EQD2 value. ## Limits and cautions of the LQ model - At low dose per fraction (below $1\ \mathrm{Gy}$) there may be **low-dose hypersensitivity** that increases biological effect more than predicted by LQ, so LQ may underestimate effect. - At very high dose per fraction the LQ assumption of continuously bending survival curves may fail; experimental curves often become asymptotically linear at high doses, so LQ may be inaccurate. - Always apply BED/TE/EQD2 with caution and awareness of dose range and tumour/tissue type. Did you know that low-dose hypersensitivity has been reported in some systems and can make ultra-fractionated regimens more biologically effective than predicted by simple LQ extrapolation? ## Patient-to-patient and intra-tumour variability ### Observations - Survival fraction at 2 Gy (SF2) varies widely between cell lines and between patients. Example SF2 values: $0.3$, $0.4$, $0.5$, $0.6$, $0.7$. - Variability in SF2 and in tumour cell number alters the steepness of the tumour control probability (TCP) curve for a patient group. - Stratifying patients by risk factors or biomarkers can yield steeper, more predictable TCP curves for each subgroup. > **Definition:** Tumour Control Probability (TCP) is the probability of eradicating all clonogenic tumour cells at a specified dose. ### Consequences - Heterogeneity increases uncertainty in the TCD50 (dose for 50% control), producing large error bars in experimental and clinical estimates. - Despite variability, clinical altered-fractionation outcomes often fit model predictions within typical clinical dose ranges. Fun fact: Experimental studies show that tumours transplanted into pre-irradiated tissue can have altered radiation response, demonstrating a ‘‘tumour bed effect’’ that changes the TCD50. ## Tumour bed effect - Pre-irradiat