☢️Radiobiology Unit 6 – Cell Survival Curves & Linear-Quadratic Model

Key Concepts

• Cell survival curves depict the relationship between radiation dose and the fraction of cells that survive exposure
• The linear-quadratic (LQ) model is a mathematical model used to describe cell survival curves
• The LQ model assumes that cell killing occurs through two mechanisms: single-hit events (linear component) and double-hit events (quadratic component)
• The $\alpha/\beta$ ratio is a key parameter in the LQ model that represents the dose at which linear and quadratic components of cell killing are equal
• The $\alpha/\beta$ ratio varies among different cell types and tissues (e.g., 10 Gy for early-responding tissues, 3 Gy for late-responding tissues)
• Fractionation in radiotherapy exploits differences in $\alpha/\beta$ ratios between tumor and normal tissues to maximize therapeutic ratio
• The LQ model has limitations, particularly at high doses (>10 Gy) where it may overestimate cell killing

Cell Survival Basics

• Cell survival is the ability of cells to maintain viability and proliferative capacity after exposure to radiation
• Radiation-induced cell death occurs primarily through DNA damage, which can lead to cell cycle arrest, apoptosis, or mitotic catastrophe
• The probability of cell survival decreases exponentially with increasing radiation dose
• Clonogenic assays are used to measure cell survival by assessing the ability of single cells to form colonies after irradiation
  • Cells are seeded at low densities, irradiated, and incubated for several days to allow colony formation
  • Colonies consisting of 50 or more cells are counted as survivors
• Cell survival is influenced by various factors, including cell type, cell cycle phase, oxygen concentration, and radiation quality (LET)
• Radiosensitivity refers to the susceptibility of cells to radiation-induced damage and varies among different cell types and tissues
• Radioresistance is the ability of cells to withstand higher doses of radiation without significant loss of viability

Understanding Survival Curves

• Cell survival curves are semi-logarithmic plots with radiation dose on the linear scale and surviving fraction on the logarithmic scale
• The shape of the survival curve reflects the underlying mechanisms of cell killing and repair
• The initial slope of the survival curve represents the intrinsic radiosensitivity of cells, determined by the $\alpha$ component in the LQ model
• The shoulder region of the survival curve indicates the ability of cells to accumulate and repair sublethal damage
  • Wider shoulders suggest greater repair capacity and radioresistance
  • Narrower or absent shoulders indicate higher radiosensitivity
• The $D_0$ (or $D_{37}$) is the dose required to reduce cell survival to 37% and is a measure of the overall radiosensitivity
• The extrapolation number ($n$) is the intercept of the survival curve with the y-axis and reflects the width of the shoulder
• Survival curves can be used to compare radiosensitivity between different cell lines, tissues, or treatment conditions

Linear-Quadratic Model Explained

• The linear-quadratic (LQ) model is a mathematical description of cell survival curves based on the assumption of two components of cell killing
• The linear component ($\alpha D$) represents single-hit events, where a single radiation track causes lethal damage
• The quadratic component ($\beta D^2$) represents double-hit events, where two separate radiation tracks interact to cause lethal damage
• The LQ model is expressed as: $S = e^{-(\alpha D + \beta D^2)}$, where $S$ is the surviving fraction and $D$ is the radiation dose
• The $\alpha/\beta$ ratio is the dose at which the linear and quadratic components contribute equally to cell killing
  • Higher $\alpha/\beta$ ratios (>10 Gy) indicate a more linear survival curve and greater sensitivity to fractionation
  • Lower $\alpha/\beta$ ratios (<5 Gy) indicate a more curvy survival curve and less sensitivity to fractionation
• The LQ model predicts that fractionating a given dose into smaller doses per fraction will result in higher cell survival due to repair of sublethal damage between fractions
• The biological effective dose (BED) is a concept derived from the LQ model that allows comparison of different fractionation schemes based on their expected biological effect

Practical Applications

• The LQ model is widely used in radiotherapy treatment planning to optimize dose fractionation and predict tumor control probability (TCP) and normal tissue complication probability (NTCP)
• Hypofractionation (larger doses per fraction) is often used for tumors with high $\alpha/\beta$ ratios to achieve greater tumor cell killing while sparing late-responding normal tissues
• Hyperfractionation (smaller doses per fraction) is sometimes used for tumors with low $\alpha/\beta$ ratios to reduce late normal tissue toxicity
• The LQ model can guide the selection of appropriate dose-fractionation schedules based on the $\alpha/\beta$ ratios of target tissues and organs at risk
• Isoeffective dose calculations using the LQ model allow comparison of different fractionation schemes in terms of their expected biological effects
• The LQ model has been used to develop alternative fractionation strategies, such as accelerated fractionation and hypofractionated stereotactic body radiotherapy (SBRT)
• The LQ model has also been applied to other areas of radiobiology, such as predicting the effects of low-dose radiation exposure and optimizing radioprotection strategies

Mathematical Foundations

• The LQ model is based on the target theory of radiation action, which assumes that critical targets (e.g., DNA) must be inactivated to cause cell death
• The linear component of the LQ model arises from the assumption that a single radiation track can cause lethal damage with a probability proportional to dose
• The quadratic component of the LQ model arises from the assumption that two separate radiation tracks can interact to cause lethal damage with a probability proportional to the square of the dose
• The $\alpha$ and $\beta$ parameters in the LQ model are cell- and tissue-specific and can be estimated from experimental survival data using regression analysis
• The LQ model can be extended to incorporate additional factors, such as the effects of dose rate, radiation quality (LET), and hypoxia
• The generalized LQ model includes a time factor ($G$) to account for the effects of cell proliferation and repopulation during fractionated radiotherapy
• The BED formula is derived from the LQ model and is given by: $BED = D \times (1 + \frac{d}{\alpha/\beta})$, where $D$ is the total dose and $d$ is the dose per fraction

Experimental Methods

• Clonogenic assays are the gold standard for measuring cell survival and generating survival curves
• Cells are seeded at low densities in culture dishes, irradiated with varying doses, and incubated for several days to allow colony formation
• Colonies consisting of 50 or more cells are counted as survivors, and the surviving fraction is calculated as the ratio of the number of colonies formed to the number of cells seeded, corrected for plating efficiency
• Flow cytometry can be used to assess radiation-induced apoptosis, cell cycle arrest, and DNA damage (e.g., γH2AX foci)
• In vivo tumor growth delay assays can be used to evaluate the response of tumors to radiation in animal models
• Immunohistochemistry can be used to assess radiation-induced changes in tumor and normal tissue morphology, proliferation, and apoptosis
• Molecular techniques, such as Western blotting and qPCR, can be used to investigate the underlying mechanisms of radiation response and identify potential biomarkers of radiosensitivity

Limitations and Criticisms

• The LQ model assumes a homogeneous cell population and does not account for tumor heterogeneity or the presence of radioresistant subpopulations
• The LQ model may overestimate cell killing at high doses (>10 Gy) due to the phenomenon of hypersensitivity and induced radioresistance (HRS/IRR)
• The LQ model does not explicitly incorporate the effects of cell cycle redistribution, reoxygenation, and repopulation during fractionated radiotherapy
• The $\alpha/\beta$ ratios used in clinical practice are based on limited experimental data and may not accurately reflect the radiosensitivity of individual tumors or patients
• The LQ model does not account for the potential effects of concurrent chemotherapy or targeted agents on radiation response
• Some studies have suggested that alternative models, such as the linear-quadratic-linear (LQL) model or the universal survival curve (USC), may provide better fits to experimental data, particularly at high doses
• The applicability of the LQ model to hypofractionated and stereotactic radiotherapy has been debated, as these treatments often involve high doses per fraction that may violate the assumptions of the model