Analysis of Relative Biological effectiveness in Proton therapy

Phys291 Project - June 2020


The relative biological effectiveness (RBE) of protons varies with multiple physical and biological factors. In proton therapy today, the clinically applied RBE is set to a constant value of RBE=1.1, although experimental results have shown the RBE to depend on multiple factors. To account for the potential effect of a variable RBE, numerous RBE models for proton therapy have been developed. In Eivind Rorvik's phd thesis "Analysis and Development of Phenomenological Models for the Relative Biological Effectiveness in Proton Therapy" (2019), such a model is developed.
In this project, the content of an experimental database will be analysed and categorized in terms of radiation quality, tissue types and number of data points and dose levels in each experiment. In addition, the database is divided into seven sub-sets, each containing datapoints with (α/β)x values in a specified interval. For each interval of (α/β)x values, a linear regression is done for RBEmax vs LETd, and the regression line is compared with the definition of RBEmax(LETd) form the Rorvik model.


1. Introduction
  1.1 RBE
  1.2 RBE Modelling from the Linear Quadratic Model
  1.3 Linear Energy Transfer
  1.4 The Rorvik Model
2. Methods
3. Results
4. Discussion and outlook

1. Introduction

1.1. RBE

Photon therapy have been in clinical use for a longer time than proton therapy, leading to a much higher level of clinical experience from the former modality. Since the radiation quality of protons differ from the one of photons, a conversion factor is needed in order to make the clinical knowledge from photon therapy applicable for proton therapy. To serve this purpose, the RBE is used. The RBE is a scaling factor showing the relative efficiency of the proton radiation compared to photons, defined as


where Dx is the absorbed physical dose deposited by the reference photon radiation, and Dp is the absorbed physical dose deposited by the proton radiation. Clinically, a constant RBE of 1.1 is recommended and applied for proton beams. However, experimental results have shown that the RBE of protons is dependent on multiple physical and biological factors, such as the deposited physical dose, irradiated tissue type, radiation quality, oxygen concentration and biological endpoint. These dependencies implies a variable RBE, not accounted for by clinically applying a constant RBE. This gives the possibility for underestimation of biological effect in proton therapy, potentially leading to complications for the patient.

1.2. RBE Modelling from the Linear Quadratic Model

There has been developed many mathematical models to describe the cell survival curve of tissue under the exposure of radiation. One of these are the linear quadratic model (LQ-model), which is a dose response model that can describe the effect of radiation on multiple endpoints. When a cell is exposed to ionizing radiation, the DNA in the cell nucleus is the main target. The damage on the DNA caused by the radiation can be divided into two types, single track events and two-track events. The single track events causes a non-repairable damage, while the damage from a two-track event is repairable. The mathematical description of the LQ model describing the survival fraction of cells after irradiation by a single dose D is written


where α and β are the LQ-model parameters. The first term in the exponential describes the initial slope of the cell survival curve, that is, the linear component caused by single track events. The second term describes the quadratic component caused by two-track events. The parameters are found by fitting a regression to experimental data. The ratio (α/β) is widely used to describe the fractionation sensitivity of different tissue types.

Phenomenological models are based on empirical data from in vitro proton irradiation of various cell lines and utilize the LQ-model with cell inactivation as the biological endpoint. The majority of the RBE models that have been developed for proton therapy belong to this group of models. They are all based on the same mathematical reasoning, starting with the fact that the survival fraction for proton radiation and the reference photon radiation are equal:


where SF(Dp) is the survival fraction of proton irradiation and SF(Dx) is the survival fraction of photon reference irradiation. To couple the RBE with the LQ-model, we can insert the mathematical description of the LQ-model, Equation (2) into Equation (3):


where Dp is the physical dose deposited by protons per fraction, α and β are the LQ-model parameters for protons, Dx is the physical reference photon radiation dose, and αx and βx are the LQ-model parameters of the reference photon radiation. Solving this equation for Dx and inserting the result into the definition of RBE given in Equation (1) yields


Now, the RBE of the proton beam is made a function of the LQ-model parameters and physical proton dose. In general, the RBE dependency on dose gives highest RBE for low doses and decreasing RBE with increasing doses. By evaluating Equation (5) at the upper and lower proton physical dose limits we get two equations for RBE values, one for very low doses, and one for very high doses:


Both the equations for the extreme RBE values are dependent on only one of the LQ-model parameters. Thus, they can be regarded as independent functions describing the extreme RBE at low and high doses. We can use equations (6) and (7) to reformulate equation (5) with respect to RBEmax and RBEmin:


where (α/β)x is equivalent to αxx and is the treatment fractionation sensitivity of the photon reference radiation. All LQ-based RBE models have Equation (8) in common, but the definition of the RBEmax and RBEmin functions differ between models. For example, the RBEmax and RBEmin functions can be reformulated as functions dependent on LETd, by making numerical assumptions for αx and βx.

1.3. Linear Energy Transfer

Linear energy transfer (LET) is a physical quantity describing the ionization density of the proton beam, given in units of keV/μm. LET is used to quantify the radiation quality of the beam, as it reflects the biological effectiveness of the radiation. It is not a directly measurable quantity. The experimental equivalent to LET is lineal energy (l). For beams consisting of ions, like proton radiation, the LET is strongly dependent on the beam energy. The LET for ion beams increases with decreasing energy. Clinical proton beams are not monoenergetic, as they consists of protons with different energies. The protons in the beam thus have a broad range of LET values, and the radiation quality of the beam can be described by a LET spectrum. Typically, the reported LET values are average LET values. One approach to obtain average values is the dose averaged LET (LETd). LETd are found from the dose weighted LET spectrum, in which each LET entry in the spectrum is weighted by its dose deposition. This is the most commonly reported and applied LET value in radiation therapy, and the LET values in the dataset used in this project are LETd values. Experiments have shown that the RBE of protons increase with increasing LETd.

1.4. The Rorvik model

In Rorvik's model, the RBEmax is defined as


The RBEmax dependence on LETd of a model is reflecting the general RBE dependence on LET. Thus, a comparison of an obtained regression-line for RBEmax vs LETd for experimental datapoints with a specific (α/β)x ratio, and the line obtained from the function for RBEmax(LETd) given in Equation (9) for the same datapoints, can give an indication of the accuracy of the model's estimation of the RBE.

2. Methods

The variables from the experimental database which is used for this project is (α/β)x, LETd and RBEmax. When the code is run, histograms showing the range and distribution in the database among all variables are made. The datapoints are then filtered based on the value of the (α/β)x ratio, and RBEmax is plotted against LETd for each interval of the (α/β)x values. A linear regression is done for the datapoints in each interval, and this is visualized in a plot containing the datapoints and the regression line for all the intervals. Next, Rorvik's function for RBEmax(LETd) (Equation (9)) is used to obtain a plot of RBEmax vs LETd for each interval of (α/β)x values, in accordance with his model. The (α/β)x values applied in the formula are the mean-values of the (α/β)x values for the datapoints in each interval. For comparison, the earlier obtained regression-lines, with their associated datapoints, and the Rorvik-lines are visualized for each interval of (α/β)x values.

The code, along with the original database and an edited version needed to run the code are availeble here: project.tar.xz

3. Results

The obtained histograms showing the range and distribution for the variables (α/β)x, LETd and RBEmax are shown in figure 1.

hist1 hist2 hist3
Figure 1: Range and distribution of (α/β)x, LETd and RBEmax.

The regression lines for RBEmax vs LETd, for seven different intervals of (α/β)x are shown in figure 2.

Figure 2: Regression lines for RBEmax vs LETd.

Figures 3-9 show the regression lines, plotted together with the Rorvik-lines. The Rørvik-lines were obtained from Equation 9, using the mean-values of the (α/β)x ratios of the datapoints in each interval.

rorvik1 rorvik2
Figure 3: Regression line and Rorvik-line for RBEmax vs LETd.       Figure 4: Regression line and Rorvik-line for RBEmax vs LETd.
rorvik3 rorvik4
Figure 5: Regression line and Rorvik-line for RBEmax vs LETd.       Figure 6: Regression line and Rorvik-line for RBEmax vs LETd.
rorvik5 rorvik6
Figure 7: Regression line and Rorvik-line for RBEmax vs LETd.       Figure 8: Regression line and Rorvik-line for RBEmax vs LETd.
Figure 9: Regression line and Rorvik-line for RBEmax vs LETd.

4. Discussion and outlook

The dependence of RBEmax on the (α/β)x ratio is clearly visible in figure 2. The regression line for the interval with (α/β)x ratios below 2.5 stand out from the other lines with its steep slope. It is clear that RBEmax as a function of LETd is influenced by dividing the datapoints into sub-sets based on the (α/β)x ratio.

We see that the Rorvik-line lies above the regression line for most values of LETd in figure 3 and 5, and for all values of LETd in fugure 4. In figure 6, we see that the regression line is lying above the Rorvik-line for all values of LETd. In figure 7, 8 and 9, the regression line lies above the Rorvik-line for most values of LETd. This indicates that the Rorvik model might over-estimate the RBEmax for lower values of the (α/β)x ratio, and under-estimate the RBEmax for the higher values of the (α/β)x ratio. In order to determine whether this could be the case or not, a more detailed investigation of the relations highlighted in this project must be carried out.

As an extention of this project, it would be interesting to use smaller intervals of (α/β)x when filtering the datapoints. This would make it possible to investigate the dependency of RBEmax on LETd in more detail. To obtain a sufficient number of datapoints in each interval, it would be prefered to use a larger database than the one used in this project. With the database used here, the number of datapoints in each interval would probably be too small to produce reliable regression lines. As pointed out earlier, the regression line for the interval with (α/β)x ratios below 2.5 stand out from the other lines. Thus, investigating the use of smaller intervals for the division of datapoints based on their (α/β)x value would be particularly interesting for datapoints with (α/β)x ratios inside and a bit above this interval.