26.8. Observing the Bragg Peak

Table of Contents

Why a Bragg Peak Appears

In a homogeneous medium like water, a monoenergetic proton beam loses energy mainly through ionization and excitation of atoms along its path. At high energy, protons traverse matter with a relatively small energy loss per unit length. As they slow down, the stopping power increases and reaches a maximum near the end of their range. This produces a pronounced peak in the depth dose curve called the Bragg peak.

In your water phantom simulation, you have already set up a proton beam, a suitable physics list, and a way to record energy deposition in depth slices. The Bragg peak will appear directly in the depth dose result as a sharp maximum at a depth that depends on the initial proton energy.

At a conceptual level, the differential dose as a function of depth $z$ is proportional to the energy that the protons deposit in each thin slab of water. Far from the end of the range, the energy loss per unit length $dE/dx$ is moderate. As the protons lose kinetic energy, their stopping power rises, so the energy deposited per centimeter increases. Just before the protons come to rest, the stopping power reaches its largest value, so the dose in that final region is much higher than in the entrance region. After the protons stop, there is essentially no further energy deposition, which is why the dose falls off very sharply beyond the Bragg peak.

The exact shape and height of the peak depend on the incident proton energy, the physics processes included in the simulation, and the scoring geometry that you use. However, even a simple monoenergetic pencil beam in a uniform water phantom is enough to see a clean Bragg peak when you plot depth dose.

From Slice Energies to a Depth Dose Curve

To observe the Bragg peak numerically, you must transform the raw energy deposition recorded in each phantom slice into a depth dose curve. You already have a division of the water phantom into $N$ slices along the beam axis, and you accumulate the deposited energy $E_i$ in each slice $i$ over many events.

Each slice has a known volume. For example, if the water phantom is a box of cross section $A$ and thickness $L$ along $z$, and you divide it into $N$ equal slices, each slice has thickness $\Delta z = L/N$ and volume $V = A \Delta z$. The depth coordinate for slice $i$ can be taken as the slice center, for instance
$$
z_i = \left(i + \tfrac{1}{2}\right)\,\Delta z \quad \text{for} \quad i = 0, 1, \dots, N - 1.
$$

The average dose in slice $i$ per primary particle is then
$$
D_i = \frac{E_i / N_\text{events}}{\rho V},
$$
where $E_i$ is the total energy deposited in that slice over all $N_\text{events}$ primary protons, and $\rho$ is the density of water. In practice, for comparing the relative shape of the Bragg peak, it is often enough to consider the energy deposited per event per slice without converting to physical dose units.

In the Geant4 analysis system, you might fill a histogram or ntuple during the run. For a histogram, each bin corresponds to a depth slice. At the end of the run, you can write the histogram to an output file, for example a ROOT file, and then use ROOT to plot $E_i$ or $D_i$ vs $z_i$. The resulting curve will show the entrance region, a gradually rising dose, the Bragg peak near the end of the proton range, and a rapid falloff beyond that point.

If you choose a small number of slices, the Bragg peak will look broad and poorly resolved, because each slice averages energy over a large depth. To resolve the shape of the peak, you should use thinner slices along $z$, which increases the depth resolution at the cost of more scoring elements and potentially larger statistical fluctuations. You can improve the statistical precision of each bin by simulating more primary events.

The Bragg peak is identified as the depth where the dose per unit depth reaches its maximum and then drops sharply to nearly zero beyond that depth.

Visualizing the Bragg Peak with ROOT or Other Tools

Once your simulation has written the slice energy information to a file, the next step is to plot it. If you are using ROOT output, your depth dose can be stored in a 1D histogram, such as hDepthDose, with bin centers corresponding to the slice depths.

In a ROOT session, you can open the file, retrieve the histogram, and draw it as a function of depth. The key is to have the horizontal axis represent depth in water, usually in millimeters or centimeters, and the vertical axis represent the energy deposited per primary, or a normalized quantity such as normalized dose.

When you draw the histogram, the Bragg peak should appear as a narrow spike near the end of the proton range. You may wish to adjust the axis range and use a line or smooth drawing option in ROOT so that the structure of the peak is clearly visible. If the plot looks very noisy, you should increase the number of simulated events. If the peak appears too broad and smooth, consider reducing the thickness of each depth slice to increase the depth resolution.

A simple way to highlight the peak is to normalize the entire depth dose curve to the maximum bin content and then plot the normalized curve. This emphasizes the relative shape of the distribution instead of its absolute scale, and makes it easier to compare results from different beam energies or different simulation settings.

Always verify that the maximum of the plotted depth dose occurs inside the phantom and that there is a sharp drop in dose beyond the peak. A flat or slowly decreasing tail beyond the expected range often indicates a scoring or geometry mistake.

Checking the Bragg Peak Position and Shape

Once you can clearly see the Bragg peak in your depth dose curve, it is important to check whether it appears at a reasonable depth and whether the shape is physically plausible. For a given initial proton energy, tables of proton ranges in water are available in reference data compilations. The depth of the Bragg peak in your simulation should be close to this expected range.

For example, if you simulate a monoenergetic proton beam of a few tens or hundreds of MeV, the corresponding range in water is typically from a few millimeters up to several tens of centimeters. If your Bragg peak appears far deeper or shallower than expected, you should check the following aspects of your setup: the initial proton energy and direction in the primary generator, the material of the phantom, the total thickness of the phantom, and the physics list that you have selected.

The shape of the Bragg peak itself depends on several factors. Monoenergetic protons in a uniform material produce a relatively narrow peak. If the beam has an energy spread, or if you include multiple Coulomb scattering and nuclear interactions in your physics list, the peak will be broadened. Additionally, the finite slice thickness averages the dose over a finite depth and can smooth out the maximum. You can adjust your geometry, physics options, and beam parameters to see how these factors affect the peak.

From a practical perspective in proton therapy, clinicians often use a spread out Bragg peak. This is achieved by superposing several monoenergetic Bragg peaks at different energies to cover a target volume. In your basic water phantom example, you do not create a spread out Bragg peak, but the simple monoenergetic peak forms the foundation for understanding how clinical beams are shaped.

To interpret the Bragg peak correctly, relate its depth to the known proton range in water at the simulated energy, and interpret its width in terms of your beam energy spread, physics list, and scoring resolution.

Using the Bragg Peak to Refine Your Simulation

Observing the Bragg peak is not only a visual confirmation that your proton transport is working, it is also a useful tool for improving and validating your simulation setup.

First, the presence of a clear, sharp rise and fall in the depth dose distribution reassures you that your geometry and scoring are correctly aligned with the beam direction. If the peak is missing or the dose is nearly constant with depth, you may have an orientation problem or a mistake in assigning volumes for scoring.

Second, by comparing the simulated range of the Bragg peak to reference range tables, you can check whether the energy scale of your proton source and your material definitions are correct. A systematic shift in the peak depth could indicate a wrong material, an incorrect density, or an error in the beam energy.

Third, you can use the Bragg peak to investigate the impact of your choice of physics list on the energy loss behavior. Different electromagnetic and hadronic models can slightly alter the position and shape of the peak. For educational purposes, you can run the same simulation with different reference physics lists and overlay the resulting curves. Small differences in range and peak height will appear, which illustrates the importance of physics model selection for precision work.

Finally, the Bragg peak in a simple water phantom is an excellent starting point for more advanced studies. You can add inhomogeneities, such as layers of bone or lung tissue, and observe how the peak shifts and distorts. You can also change the beam energy to see how the range scales with energy. These explorations naturally lead toward realistic proton therapy simulations and toward the later topics of simulation validation and detector resolution that are covered elsewhere in this course.

Use the Bragg peak as a basic benchmark: if you cannot reproduce a realistic Bragg peak in water, you should not trust more complex proton simulations that build on the same setup.