Depth-dose simulator: cyclotron acceleration to spread-out Bragg peak tumor coverage
Proton therapy begins with stripping electrons from hydrogen gas to produce bare protons, then accelerating them to therapeutic energies of 70–230 MeV using a circular particle accelerator. The physics of how a fast charged particle loses energy in matter — the Bethe–Bloch stopping power — is the foundation for everything that follows in this simulator.
Two accelerator designs dominate clinical proton therapy:
Cyclotron (most common): protons spiral outward between two D-shaped electrodes ("dees") inside a strong, fixed magnetic field. An oscillating RF voltage across the gap kicks the particle to higher speed on every half-turn; as speed increases, the orbit radius grows, tracing an outward spiral. A fixed-energy (isochronous) cyclotron always extracts protons at one energy — typically the maximum, ~230–250 MeV — and a range-shifting energy selection system downstream degrades and filters the beam to the energy needed for each treatment layer.
Synchrotron: protons circulate in a fixed-radius ring while both the bending magnetic field and RF frequency are ramped up together in synchrony as the particle accelerates. This allows the machine to extract a variable energy directly, pulse by pulse, without a mechanical degrader — at the cost of a slower, pulsed beam-spill cycle rather than the cyclotron's continuous beam.
A cyclotron extracts one fixed energy and shaves it down with a graphite degrader; a synchrotron changes its own magnetic field to extract the exact energy needed, cycle by cycle. Both routinely reach the 70–230 MeV window used clinically.
The rate at which a charged particle loses energy per unit path length — the stopping power, −dE/dx — is described by the Bethe–Bloch equation:
−dE/dx = (4πe⁴z²Nₑ / mₑv²) × [ln(2mₑv²/I) − ln(1−β²) − β²]
where v is the proton's instantaneous velocity, z its charge, Nₑ the electron density of the medium, mₑ the electron mass, and I the mean ionization potential of the tissue.
The key term is the 1/v² prefactor: as the proton slows down, stopping power rises approximately as the inverse square of its velocity. A fast proton entering tissue interacts only briefly with each electron it passes and deposits energy slowly; a slow proton near the end of its path lingers near each atom far longer, transferring much more energy per unit distance. This is the entire physical origin of the Bragg peak explored in Stage 3.
Proton range in tissue scales approximately as R(cm) ≈ 0.0022 × E(MeV)^1.77. At the low end, 70 MeV protons penetrate only about 4 cm — useful for shallow targets such as the eye. At the high end, 230 MeV protons reach roughly 32–33 cm, enough to treat deep-seated tumors such as prostate or skull-base lesions.
Because range depends so steeply on energy, treatment planning selects — layer by layer — the precise energies needed to place many pristine Bragg peaks exactly where they are needed, and no further. This is the starting point for the spread-out Bragg peak built in Stage 4.
Once extracted, protons travel through beamline magnets, a nozzle, and finally into the patient. Inside tissue, each proton undergoes tens of thousands of tiny Coulomb interactions with the electron clouds of atoms along its path — each collision stripping away a small amount of kinetic energy while barely deflecting the proton's direction.
Unlike photons, which are absorbed or scattered in discrete, probabilistic events, protons lose energy in a quasi-continuous fashion: they are heavy enough (1836× the electron mass) that a single electron collision barely changes their direction, but the sheer number of collisions along the path — tens of thousands per centimeter of tissue — steadily bleeds away kinetic energy.
This process is called electronic (or "collisional") stopping, and it is the mechanism captured by the Bethe–Bloch formula from Stage 1. Because energy loss accumulates gradually and predictably, the proton's residual range at any point along the track can be computed in advance — which is exactly what makes proton therapy dose calculable and reproducible.
While energy loss is dominated by interactions with electrons, protons also undergo many small-angle elastic deflections off atomic nuclei — multiple Coulomb scattering (MCS). Each individual deflection is tiny, but their cumulative, random-walk effect broadens the beam laterally as depth increases.
For a clinical 230 MeV beam, the lateral penumbra grows to only a few millimeters by the end of range — far less lateral spread than the beam broadening seen with electron beams, and one reason proton fields can be shaped with sharp lateral margins alongside the sharp distal (depth) fall-off explored in Stage 3.
Immediately after entering tissue, a proton is moving fast, so 1/v² is small and stopping power is low. As it penetrates deeper it slows gradually, and stopping power creeps upward — producing the modest, slowly rising "entrance plateau" seen in the depth-dose curve, typically depositing only 30–45% of the eventual peak dose per centimeter along most of the track.
This low-dose entrance region is clinically valuable: healthy tissue proximal to the tumor (skin, muscle, other organs the beam passes through en route) receives substantially less dose than it would from a comparable photon beam, whose dose is highest right near the entrance.
The Bragg peak is the defining physical signature of charged-particle therapy: a sharp, localized spike of dose deposited right where the proton runs out of energy, followed by a near-total absence of dose beyond it. No conventional photon beam can reproduce this shape, and the difference is the single biggest reason proton therapy exists as a clinical discipline.
As a proton's velocity approaches zero near the end of its track, the Bethe–Bloch 1/v² term grows very large — stopping power rises sharply over the final few millimeters of travel. Because most protons in a beam have nearly identical initial energy, they also run out of range within a narrow band of depth (range straggling of only a percent or two of the total range), so this surge of energy loss from many particles piles up at almost the same depth.
The result is the Bragg peak: dose per unit depth rises 2–3× above the entrance plateau value in the last centimeter of the track, then collapses to near zero within a few millimeters beyond it, since the protons are now fully stopped and simply are not there anymore to deposit further dose.
A pristine Bragg peak's distal 80%→20% dose falloff is typically only 2–6 mm — steep enough that a proton beam can spare tissue millimeters beyond a tumor almost entirely, something no photon field can achieve.
Megavoltage X-ray (photon) beams lose intensity by a completely different mechanism: each photon either interacts (via Compton scattering, mostly) in a single probabilistic event or it does not. Averaged over billions of photons, this produces a smooth exponential attenuation, I(d) = I₀e^(−μd), with no natural stopping point — a photon beam that has traversed the tumor keeps depositing a shrinking but nonzero dose all the way to the patient's exit surface.
Photon depth-dose curves do show a brief buildup near the skin surface (as secondary electrons accumulate before reaching equilibrium) followed by that continuous exponential falloff — but there is no analog to the sudden stop a finite-range charged particle produces.
The clinical consequence of the Bragg peak's shape is called the dosimetric advantage of proton therapy: tissue beyond the tumor — the "distal" or "exit" side of the beam — receives almost no dose, whereas the same tissue in a photon plan continues to receive a meaningful fraction of the prescribed dose.
This translates directly into a lower total ("integral") dose delivered to the patient's body per unit of tumor dose delivered — often cited as a 2–3× reduction in integral dose for proton plans versus photon plans covering the same target. Stage 5 of this simulator visualizes that difference directly by overlaying the two depth-dose curves and shading the extra dose a photon beam delivers past the tumor.
A single pristine Bragg peak is only a few millimeters wide — far too narrow to cover a real tumor, which is typically several centimeters thick. Treatment planning solves this by summing many pristine peaks of different energies and weights to produce a spread-out Bragg peak (SOBP): a flat, uniform dose plateau spanning the tumor's full extent.
Passive scattering (older technology): a spinning range-modulator wheel, machined with steps of varying thickness, is placed in the beam path. As it rotates, the beam repeatedly passes through thicker or thinner sections, rapidly cycling the proton energy (and hence range) hundreds of times per second. Combined with a fixed weighting built into the wheel's step widths, this sums many pristine peaks into one static SOBP shape, which is then further shaped by a physical aperture and range compensator for each patient.
Pencil-beam scanning (modern, active technique): a narrow proton "pencil beam" is magnetically steered point by point across the tumor cross-section, one thin energy layer (one depth "slice") at a time, depositing a precisely weighted spot of dose at each position before the accelerator changes energy and paints the next depth layer. No physical wheel or aperture is required — the whole 3-D dose distribution is built up spot by spot and layer by layer, directly under computer control.
Pencil-beam scanning can shape dose in three dimensions without any patient-specific hardware, enabling intensity-modulated proton therapy (IMPT) — the proton analog of IMRT — for markedly better conformity to irregular tumor shapes.
Simply adding equal-intensity pristine peaks at evenly spaced depths does not produce a flat plateau — because every peak also contributes its own low-dose entrance plateau to all the shallower depths behind it, the shallowest layers need much less weight than the deepest one, which contributes to everything in front of it as well as forming the distal edge.
Treatment planning systems solve an optimization problem (often built on iterative or analytical solutions dating back to the classic Bortfeld ridge-filter design) to find per-layer weights that combine into a plateau flat to within about ±2.5% across the full target width — steep dose rises at the proximal and distal edges bracketing a nearly constant dose region matched to the tumor's exact depth extent.
The "Energy Layers (SOBP)" control in this simulator approximates that treatment-planning process directly: each additional layer adds one more mono-energetic pristine peak, spread between the maximum range set by the beam-energy slider and a shallower range. More layers both widen the flat plateau (covering a thicker tumor) and smooth its top, mirroring how real proton centers use more energy layers for thicker targets and fewer layers for a shallow, disc-like tumor near a single depth.
The side-by-side comparison of proton and photon depth-dose curves is where the physics translates directly into patient benefit. A flat SOBP plateau across the tumor with near-zero exit dose, set against a photon beam's continued exponential falloff, defines exactly which patients gain the most from proton therapy — and explains why proton centers remain expensive and comparatively rare.
"Integral dose" is the total energy deposited in the patient's body — proportional to the area under the depth-dose curve integrated over the whole irradiated volume, not just the tumor. Because a photon beam keeps depositing dose past the target on its way out of the body (and often needs multiple crossing beams to spare surrounding tissue, each adding its own entrance and exit dose elsewhere), photon plans typically deliver 2–3× more integral dose to the patient than a proton plan prescribing the same tumor dose.
Lower integral dose matters most where healthy tissue tolerance is the limiting factor: pediatric patients with decades of life ahead, tumors adjacent to critical structures, or patients who have already received prior radiation and have little remaining normal-tissue tolerance.
Proton dose is not simply "physical dose" — biological damage per unit of absorbed physical dose (Gray) differs slightly from photons. Clinically, protons are assigned a single generic Relative Biological Effectiveness (RBE) of 1.1, meaning 1 Gy of proton dose is treated as biologically equivalent to 1.1 Gy of photon dose ("Gy(RBE)" or "cobalt Gray equivalent").
This generic factor is a simplification: RBE actually varies along the beam path, rising toward the distal edge of the Bragg peak where protons are moving slowest and linear energy transfer (LET) is highest — exactly the region where the tumor's distal margin often sits, an active area of ongoing research and biological-dose modeling.
Proton therapy's sharp distal falloff and low integral dose are most valuable in:
• Pediatric brain and CNS tumors — sparing growth plates, cochlea, and cognitive structures from unnecessary dose in a still-developing brain and skeleton • Re-irradiation — treating a recurrence in a field that has already received a prior course of radiation, where photon exit dose could exceed cumulative normal-tissue tolerance • Ocular melanoma — the eye's small size and proximity to the optic nerve and lens make a beam with millimeter-scale range control and near-zero exit dose especially well suited • Skull-base and paraspinal tumors near the spinal cord or brainstem
Against these benefits, proton centers remain far more expensive to build and operate than conventional linear accelerators (roughly $100–200 million for a multi-room facility, versus a few million for a linac), require a cyclotron or synchrotron plus multi-ton rotating gantries, and number only around 120 worldwide — so access is limited relative to the far more widespread photon radiotherapy infrastructure.
The clinical decision to use protons instead of photons ultimately weighs a real, physics-grounded reduction in dose to healthy tissue against significantly higher cost and limited facility access — proton therapy is targeted at cases where that dose-sparing has the greatest expected clinical benefit.
| Product | Indication | Trial Design | Key Result |
|---|---|---|---|
| Entrance dose | ~30–45% of peak | Slow-rising plateau, low dE/dx at high velocity | Lower skin/entry dose than photon |
| Target (plateau) dose | ~100%, flat ±2.5% | Summed weighted pristine Bragg peaks (SOBP) | Uniform tumor coverage |
| Distal falloff | 2–6 mm, 80%→20% | 1/v² stopping power surge at end of range | Sharp sparing beyond tumor |
| Exit / distal healthy tissue | <1–2% (proton) vs 30–60% (photon) | Finite range vs. continuous exponential attenuation | No exit dose — key proton benefit |