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## NetLogo User Community Models

# A NetLogo Model For Fractionated Radiation Treatment

## WHAT IS IT?

This is a basic model of the effect of radiation treatment on healthy tissue and tumour tissue as predicted by the Linear Quadratic (LQ) model of cell survival. The healthy tissue is modelled in green and the cancer cells in red. As predicted by the LQ model, the two types of cells respond differently to radiation. The LQ model assumes the surviving fraction to be given by:

S = exp (-α*d -β*d<sup>2</sup>)

where d is the dose and α and β are free parameters used to fit the model to experimental results from the irradiation of different tissue types.

The "tissue sample" seen in the interface represents the interface between a tumour and the surrounding normal tissue.

This model can be used to develop an understanding of fractionated radiation treatment planning. During fractionated radiation treatment, a patient receives small doses of radiation over many treatments.

## HOW IT WORKS

Each cell has a probability of undergoing cell division; since cancerous cells divide more rapidly, their probability of dividing is higher. When the radiation is applied, each cell has a probability of dying, according to the LQ model. By adjusting the time between treatments and the dose per fraction, an ideal treatment can be developed such that a maximum about of healthy tissue remains after 99.9% of the cancer has been eliminated.

The model starts with 5000 of each type of cell, and the first fraction is applied on the first tick.

## HOW TO USE IT

The SETUP button creates the tissue sample. The GO button runs the treatment simulation.

The set-up options determine the type of treatment. The `Fractions` input sets the maximum number of fractions that are applied. The time between fractions is given by the `hours` input. The dose applied per fraction, in Gray, is given by the `dose` input.

The `alpha` and `beta` inputs can be used to adjust the parameters for the tumour.

## THINGS TO NOTICE

#### Interface layout

If you want to re-adjust the layout of the interface tab to better fit your computer screen, you can select each item and move it or adjust its size.

#### Plots

There are two plots included in the model. On the Survival plot, the number of each cell type is plotted for each tick, while on the logSurvival plot, the population is a semi-log plot, meaning the log of the population is plotted. Most of the plots of cell survival in the literature are on semi-log scales. Looking at the equation for cell survival in the LQ model, what is the benefit to a semi-log scale? Consider how the plots look when the populations are small.

#### Regrowth

Looking at the Survival plot, notice the increase in population between fractions. The tumour cells are more likely to divide and therefore have more regrowth between fractions.

## THINGS TO TRY

* Apply only a single fraction of radiation. Compare the ratio of surviving tumour and healthy cells directly after treatment using the Survival plot. What is the dose at which more tumour cells survive the radiation than healthy cells? Allow the simulation to continue running. What happens?

* Use the graphing software of your choice to plot the above equation for cell survival for alpha beta values of (0.1, 0.1) and (0.3, 0.04). Use a semi-log plot and a range of 0 to 8. Verify that the dose you found for the point at which equal amounts of tumour and healthy cells survive corresponds to the point where the two equations cross.

* Develop a fractionated treatment schedule by adjusting the time between doses and the dose amount. Set the number of fractions to 20 so that the program ends before the treatment ends. Try some of the following:
* A conventional treatment: 2 Gy every 24 hours
* Hyperfractionation: 1 to 1.5 Gy multiple times in a day
* Accelerated hyperfractionation: 1 to 1.5 Gy three times a day for one week
* Adjust the parameters until you find a treatment with the maximum number of surviving healthy tissue. Consider the plot you made in the last activity. Where does there seem to be the largest potential for keeping healthy tissue alive? Look at the gap between the two equations.

* Run a fractionated treatment a few times and determine the average number of treatments applied before the program stops. Run the model again with `Fractions` set to be one less than this number. What happens? How long after treatment does the tumour population overtake the number of healthy cells?

* The lifetime of a cell varies greatly between different types of tissues. This model uses a lifetime of 1 month for the cancer cells and 3 months for the healthy tissue. Try adjusting these values by changing the values of `lifeT` and `lifeH` within the `setup` command procedure in the code.

* Have the model also output the biologically effective dose (BED) in the Output box. BED is given by:

(numdoses * Dose * ( 1 + Dose / (alpha / beta) ))

The above is properly formatted syntax. Look at the code for printing the output at the beginning of the `go` command procedure and add this in. Note: `numdoses` is the actual number of treatments applied, NOT the maximum number you set with the `Fractions` variable.

* Select the dose slider and adjust the range of allowed values and experiment with doses even higher than 6 Gray. You might get an error. Why? Try and fix this.

* Adjust the `alpha` and `beta` values for the tumour cells to correspond to different types of cancer. Warning: you should first plot the survival curve for these new values and the values used for the healthy cells (0.1, 0.1) to check that they are reasonable.

## TROUBLE SHOOTING

#### The program takes too long to run

The program can take up to 10 minutes to run, especially when there is a long time between fractions. To help speed it up, deselect "view updates" at the top of the interface tab.

#### Runtime Error: "Can't take logarithm of 0"
This error occurs when high doses are used; the population decreases too rapidly for the program to end before reaching an error. Click "Dismiss" and run the simulation again. You may need to use a smaller dose, or adjust the cut-off point in the code.

#### Slider Stops Working
Select the dose slider by right clicking it, move it around the interface, and deselect it by clicking elsewhere on the screen. Repeat until it works.

## EXTENDING THE MODEL

* Develop treatment schedules other than evenly spaced fractionated treatments. Often patients will only receive treatment 5 days a week. Adjust the code to have radiation applied in fractions which are not equally spaced.

* Hypoxic cells are significantly less sensitive to radiation than well-oxygenated (oxic) cells (Steel, 2010). After irradiation, most of the oxic cells will be killed, leaving a tumour consisting of nearly entirely hypoxic cells. It has been found that after irradiation the tumour undergoes reoxygenation, which restores the ratio of hypoxic to oxic cells. It can take anywhere from minutes to days for a tumour to undergo reoxygenation. If the next fraction of radiation is applied while most of the tumour is still hypoxic, the radiation will be less effective. This sets a lower limit on the time between fractions and a treatment planner should be aware of balancing regrowth with reoxygenation. Expand the model to include reoxygenation by using the variable `hypoxic?`.

* Hypoxic cells are also more likely to undergo metastasis. Allow hypoxic cells to spread into the healthy tissue on the left side of the tissue sample.

## CREDITS AND REFERENCES

For a good overview of alpha-beta ratios and the LQ model, see:

Eye Physics, 2014. _Biological Models._ [online] Available at: <http://www.eyephysics.com/tdf/models.htm>

For a more in-depth explanation of the rationale behind the LQ model, and its validity compared to other models, see:

Brenner, D., 2009. Point: The linear-quadratic model is an appropriate methodology for determining iso-effective doses at large doses per fraction. _National Institutes of Health_, 18(4), pp.234–239.

Steel, G. ed., 2002. _Basic Clinical Radiobiology_. 3rd ed. London: Arnold.

This model was developed by Alexandra Kasper as part of ISCI 3A12 at McMaster University during the winter term of 2014. If you have questions about the model, e-mail alexandrakskasper@gmail.com.