Protons that enter human tissue, or a water tank as substitute, slow down and eventually stop. The “Bethe-Bloch” theory of this slowing and stopping process was developed in the 1930s and refined subsequently.² It predicts the rate of slowing down, or energy loss, as a function of the incident particle type and energy and the composition of the stopping material. By a simple extension, it therefore predicts the mean range of a given beam in a given material to an accuracy of 1% to 2%. For example, 160-MeV protons (whose speed is approximately half the speed of light) have a mean range of approximately 17 cm in water. These protons do not, however, all stop at exactly 17 cm, because stopping is the statistical outcome of thousands of collisions with atomic electrons. The approximately 1% fluctuation in the range of any given proton is called range straggling.

Protons lose energy faster as they slow down. The dose is greatest just before they stop. This leads to the Bragg peak in the depth-dose distribution, arguably the protons most important characteristic for radiation therapy. The peak becomes sharper, and the peak/entrance dose ratio larger, as incident proton energy decreases.

If protons enter the water tank in exactly the same direction we will find them, after a few centimeters, going in slightly different directions by a degree or so. This gradual
spread in their angular distribution is known as multiple Coulomb scattering because it arises from thousands of small electrostatic deflections by atomic nuclei. The accepted theory was published by Molière in 1947.³ It predicts the characteristic multiple scattering angle as a function of the incident particle type and energy and the composition of the scattering material, and holds even if the particle loses nearly all its energy. It has been tested experimentally to a few percent.⁴

The distribution of angles after multiple scattering is nearly a Gaussian, the bell-shaped curve familiar from statistics. What Molière’s theory allows us to calculate for any given case is the width of that Gaussian. The theory also predicts a large departure from the Gaussian shape for large angles, but that represents so few protons that we can ignore it in designing passive systems

Like stopping, scattering in a given material increases as the proton slows down, but according to a different mathematic formula. The net scattering when a proton passes through a thin sheet of material is roughly proportional to the square-root of the thickness of the sheet, whereas the net energy loss is directly proportional to the thickness.

Because stopping arises from interactions with atomic electrons, whereas scattering arises from nuclei, they depend differently on the target material. In a relative sense, materials with low Z (atomic number) are more effective in slowing protons down, whereas materials with high Z, scatter protons much more strongly. If we want to scatter a beam with minimum energy loss, we should use a high-Z material such as lead. To decrease proton energy with minimum scatter, we should use a low-Z material such as plastic or beryllium. By combining high-Z and low-Z materials in a “binary degrader” we can control both energy loss and scattering.

Besides stopping and scattering, protons can collide headon with an atomic nucleus. The resulting collision yields an excited residual nucleus, secondary protons and neutrons and (less frequently) heavy charged fragments such as α particles. Unlike stopping and scattering, the detailed description of all this is exceedingly complicated. Nuclear reactions are rare; approximately 20% of 160-MeV protons suffer one, compared to the thousands of other interactions, before they stop in water.

Fortunately for us, the decay products have such short ranges (they need to share the kinetic energy of the original proton) and make such large angles with the beam direction that they rarely reach the target region. Therefore, we can ignore nuclear reactions except for the diminution of the primary beam. They do not affect the target dose distribution, only its absolute value.

If the target has greater extent in depth, we need a range modulator to spread out the Bragg peak in depth in addition to a scatterer. The range modulator is typically a propeller-shaped object which rotates so as to put successively thicker layers of plastic into the beam. Each layer pulls back the Bragg peak in the target by an amount equal to the water equivalent thickness of the plastic. The thickness of each layer and the relative time it spends in the beam are carefully calculated so that the resulting spread-out Bragg peak (SOBP) will be flat. The arrangement is shown schematically in Figure 5A.2, with one extra feature. Suppose the distal edge of the PTV is shallower than the range of the beam after scattering. We can move the entire dose distribution forward by means of an additional piece of plastic, a “range shifter”.²

We now have three quantities characterizing the dose distribution: the useful radius r, depth d, and modulation m. These must be such as to cover the PTV.

If the scattering in the combined range shifter and modulator is insufficient to achieve lateral conformation to the PTV, we can add some lead or brass and compensate by removing some plastic. Usually this computation is different for each modulator step. It yields an energy compensated range modulator.

Figure 5A.2 shows an upstream modulator, so called because it also serves as part of the scatterer. The modulator’s scattering, if it cannot be compensated, must be taken into account when designing the dwell-time of each step. Historically, the range modulators were placed downstream much nearer the patient.⁵ Scattering in the modulator can then be ignored because of the small distance between it and the patient. Not only is the modulator easier to design, but the same modulator can be reused with various scattering systems.

An obvious disadvantage is that, being near the patient, the range modulator must be large enough to cover the entire treatment field, not merely the incident beam. That makes it difficult to change remotely and unsuitable for a gantry. Less obviously, a downstream modulator diffuses the effective proton source and makes the edges of the dose distribution less sharp as discussed in the following text. Figure 5A.3 shows an upstream and a downstream range modulator.

Range modulation is accomplished by putting extra material into the beam in a carefully programmed way. Instead of prefabricating the “propellers” discussed in the preceding text, one could simply insert individual constant thickness degraders and scatterers into the beam under the control of a computer. A popular arrangement employs plastic degraders and lead scatterers in binary sequences of thickness, so that any desired combination can be made up. Finally, we can use linear or helical wedges, or an adjustable water column.

All of these methods have been used. The SOBP is created step-wise in a process called lamination. A single set of hardware can produce any desired SOBP, eliminating the need for a large assortment of modulators. The mechanics and software are, however, more complicated. Because of the changeover time between degraders, lamination delivers the SOBP sequentially as a function of range and only once or twice per treatment session. This increases sensitivity to organ motion but only along the depth dimension as the lateral dimension is produced by scattering over the entire treatment field. By contrast, a spinning upstream modulator can easily paint the entire SOBP ten times per second.

If we use rotating modulators, how many do we need to cover all clinical requirements? A given modulator only works well over a limited incident energy span, and we need one for each SOBP modulation width. At first glance we might need hundreds of modulators!

The number can be greatly reduced by noting that a modulator properly designed for full modulation (m = d) can yield any lesser m if the beam is turned off during part of each revolution so as to eliminate the appropriate number of proximal dose layers that contribute dose below m. This method has two costs. First, the average dose rate is less than it would be with a purpose-built modulator because the beam is turned off part of the time. Second, the system is more complex and requires more checks. The delivery of the SOBP depends critically on synchronization of the beam current with the modulator.

A further reduction in the number of modulators can be obtained by varying the beam current during the modulator cycle. The built-in dwell times, defined by the angular width of the modulator steps, are only the first approximation to the desired exposure for each step, the rest being achieved by beam current modulation. The cost, as ever, is yet more engineering and more checks. In particular, the cyclotron output current detector must be very linear and stable.

Beam current modulation with a dynamic range from zero to full beam current provides full dynamic control of the creation of an SOBP depth-dose profile almost independent of a particular range modulator. This feature exists at the Francis H. Burr Proton Therapy Center (FHBPTC), formerly the Northeast Proton Therapy Center, where just four modulators cover all clinical requirements in the second gantry room.

The patient aperture (see Fig. 5A.5) is a brass beamstop with a hole shaped to the outer projection of the target in the beam’s eye view. This, as in photon radiotherapy, eliminates protons heading outside the PTV. It is impractical to block all protons in this way. That would require a wall of brass which could not be brought very near the patient (see the section “Lateral and Distal Dose Falloff” in the following text). Therefore the snout (see Fig. 5A.6) not only supports the aperture and range compensator near the patient, but is carefully designed to block unwanted protons, as are other nozzle components further upstream.

The range compensator (Fig. 5A.5) is a plastic block with material cut away in a complex shape. It is carefully aligned with the aperture and the patient’s PTV and tailors the dose in depth by shifting more or less proton range depending on what part of the PTV, and upstream tissues, a particular proton ray is aimed at. In this way, it can compensate for three effects simultaneously: patient external variation, the distal PTV surfaces (with respect to the beam axis), and tissue heterogeneities in the patient. Compensators may be designed for diverse clinical goals: Guaranteeing target coverage in the face of alignment errors, patient and internal organ motion, and assuring that neighboring critical structures are spared.¹¹

Figure 5A.6 shows the combined effect of a very simple aperture and range compensator. Note the “ears” of unwanted high dose proximal to the target. These are a consequence of constant range modulation over the field. Magnetic scanning can eliminate this and similar problems.