Physical basis of ion beam therapy

The most prominent feature of charged particle beams for their use in radia-tion therapy is the inversed dose profile, i.e. the increase of energy deposiradia-tion with penetration depth, which represents the most evident difference respect to electromagnetic radiation. As already mentioned, the increase of ionization den-sity with range was first described for α particles in a publication by Bragg in 1903 [23], and was later confirmed for protons and heavier ions by Wilson[14].

In the interaction of photons with the matter the energy loss is mainly due to three processes: photoelectric effect, Compton scattering and electron-positron

18

pair production. The relative probability of one of these interactions is a function of the incident photon energy and the atomic number Z of the absorbing mate-rial. On the other hand, the energy of the ions is mostly transferred to the target electrons that are emitted as δ-electrons (see section A.1). More than three quar-ter of the dissipated energy is used for the ionization process and only 10 to 20

% for the target excitation [24]. The interaction strength is directly correlated with the interaction time. At high energies the energy transfer to the target is small but grows when the particles are slowed down. The different physical processes which characterize respectively electromagnetic radiation and charged particles directly affect the different depth dose profiles. In Figure 1.4 (on the left) the depth dose profiles of proton and ion beams is compared to those of pro-tons, neutrons and electromagnetic radiation.

Figure 1.4. On the left: comparison of depth dose profiles of photon, neutron, proton and carbon ion beams; the “inverse” dose profile of protons and ions is clearly visible in con-trast to the less favourable profiles of electromagnetic radiation and neutrons. On the rigth: Construction of an extended Bragg-peak by superimposition of single Bragg-peaks of different energy.

For low-energy X-rays the stochastic absorption by photoelectric and Comp-ton processes yields an exponential decay of absorbed dose with penetration. For greater photon energies the produced Compton electrons are strongly forwardly scattered and transport some of the transferred energy from the surface to greater depth, yielding an increase in dose in the first few centimetres. For high energy

1.2 Rationale of ion beam therapy

19 electron Bremsstrahlung, which is mostly used in conventional therapy, this maximum is shifted a few centimetres from the surface of the patient’s body sparing the very radio-sensitive skin. In contrast, the energy deposition of charged particles like protons or heavier ions shows a completely different trend.

When ions enter an absorbing medium, they are slowed down by losing their ki-netic energy. The specific ionization increases with decreasing particle velocity, giving rise to a sharp maximum in ionization near the end of the range. Thus the depth-dose distribution is characterized by a relatively low dose in the entrance region (plateau) near the skin and a sharply elevated dose at the end of the range, which in this case is finite and energy dependent (peak). The ballistic pre-cision of charged particles respect to electromagnetic radiation is evident. In-deed, in the surrounding healthy tissues before or just behind the Bragg peak the dose released is minimized as more as possible respect to the target volume and a better compromise is achieved.

However, a monoenergetic beam with a narrow Bragg peak makes possible to irradiate a very small, localized region within the body with an entrance dose lower than that in the peak region [26]. In practical use in therapy, the tumour volume to be treated is normally much larger than the width of the Bragg peak and the lateral spot of the particle beam. In order to fill the target volume with the necessary amount of stopping power particles the peak has to be “spread out”

in the longitudinal direction. This is achieved by superimposing several Bragg peaks of different depths obtained by appropriate selection of distribution of ion energies. The resulting dose depth distribution, known as spread-out Bragg peak (SOBP) shows an “extended” Bragg peak area which has to accurately overlap the target volume (Figure 1.4, right). The use of ion beams allows tumor con-form treatments of enhanced quality respect to those obtained by conventional radiotherapy [25]. In fact, even if the peak-plateau ratio decreases for SOBP re-spect to a pristine Bragg peak, the final result is still satisfying if we look at depth dose profile for photon beams.

The interesting behaviour of charged particles is due to their electromagnetic interactions with the matter. Within the range of therapeutically relevant ener-gies of some hundred AMeV down to rest, the process of energy loss is

domi-20

nated by inelastic collisions with atomic electron. The average energy loss per unit path length, the so called electronic stopping power, is well described by the Bethe-Block formula [27-29]:

with Z and v respectively the charge and velocity of the projectile. Details on the dependencies of the formula and relative corrections are given in section A.1.1.

In the energy range of interest in hadrontherpy, dE/dx is mainly dominated by the term 1/v² ≈ 1/E. Moreover, at very low energies, when the projectile velocity is comparable to the velocity of the orbital electron in the materials, there is a high probability that the projectile picks up these electrons. Thus the charge changes and Z has been replaced by Z_eff (the effective projectile charge) which is well described by the Barkas formula [30]:

)

with β = v/c. Considering these dependences, at not relativistic energies, the en-ergy loss rate grows up as the kinetic enen-ergy of the projectile decreases along the penetration depth, in particular in the last few millimetres of the particle path where it shows a much steeper rise. That is the reason why the distribution of the ionizing density produced by the charged particle along the track is characterized by a rather constant plateau, followed by a sharp maximum towards the end, where gives rise at the Bragg peak. Anyway, at the end of the path the stopping power drops because of the rapid reduction of the effective charge Z_eff to very low energy values. The relation between electronic stopping power and particles energy is shown in Figure 1.5 for energy range and ions which are of interest in hadrontherapy. As expected, each particle exhibits a dE/dx curve which is dis-tinct from the other particle types.

Once the stopping power is known, it is possible to calculate the range R of a charged particle. It is the distance it travels before coming to rest. The reciprocal of the stopping power is the distance travelled per unit energy loss.

1.2 Rationale of ion beam therapy

21 Figure 1.5. Energy loss of different particles as function of the energy; as the atomic number increases the stopping power grows up.

Therefore the range R(E) of a charged particle having kinetic energy E is the integral of the reciprocal of the negative stopping power [34]:

∫

This equation provides a well defined value but in reality the energy loss processes are affected by statistical fluctuations which are responsible of a dis-persion of the path length (range straggling).

Moreover, fluctuations in energy loss and multiple scattering processes yield an almost Gaussian energy loss distribution f(∆E) given by [31][32]:

( )

where σ is the straggling parameter which expresses the half-width at the (1/e)-th height [33]. Hence, statistical fluctuations of energy loss cause a smearing of the range of the stopping particle beam and, consequently, a larger width of the Bragg peak experimentally measured. Range straggling effects for ion beams vary approximately inversely to the square-root of the atomic mass and increase

22

as the penetration depth grows up. Indeed, at the same penetration depth heavier ions show narrower Bragg peaks and also a steeper distal fall-off, which has a positive effect on the final level of conformation of the radiation to the tumour.

A typical fragmentation tail is also well distinguished in case of heavy ions, but this aspect with its implication will be discussed ahead (Figure 1.6).

Figure 1.6. Depth dose distribution of photons, protons and carbon ions of 254 AMeV and 300 AMeV; comparison with 135 MeV protons (having the same range value) shows the difference in the Bragg peak width. Moreover, higher incident energies (i.e.

higher ranges) correspond to higher peak widths.

For greater particle energies and longer penetration the half width of the Bragg maximum becomes larger and the height smaller. Typical values for Car-bon ions are given in Table 1.1 [35].

Energy (MeV/u) 90 180 270 330

Range (mm) 21.3 82.8 144.3 200.5

FWHM (mm) 0.7 2.3 5.0 7.0

Table 1.1. Typical values for carbon ions Bragg curve.

Anyway, more significant for clinical applications are the consequences of

1.2 Rationale of ion beam therapy

23 multiple scattering processes in the lateral scattering of the beam, especially when the beam has to pass near critical structures which sometimes can be placed adjacent to the tumour volume. Multiple scattering of an incident ion stems from the small angle deflection due to collisions with nuclei of the trav-ersed material. Numerous small angle deflections in an ion beam lead to lateral spreading of the incident ions away from the central trajectory resulting in larger divergence of the beam. Elastic Coulomb scattering dominates this process with a small strong-interaction scattering correction. The angular distribution of the scattered particles is roughly Gaussian for small deflection angles, and the mean beam deflection is approximately proportional to the penetration depth. The Coulomb scattering of the projectiles is described very precisely in the theory of Molière [36] [37]. Measurements of proton scattering confirmed this theory [38]

and a parameterization for small angle scattering having an angular distribution f(α) [39]:

where σα is the standard deviation, p the momentum, Lrad the radiation length and d the thickness of the material. At low energy, multiple coulomb scattering in-creases and, in general, the effects are more evident for lighter ions. In Figure 1.7 (left) the lateral scattering of a therapy beam is compared for 21 MeV pho-tons, for protons and carbon ions of a range of 14.5 cm in water. The comparison clearly that for protons the lateral scattering exceeds the photon value for pene-tration depth larger than 7 cm. The lateral deflection of carbon beams is better than 1 mm up to a penetration depth of 20 cm. As expected, carbon ions exhibit a less pronounced lateral deflection, and they are well confined respect to proton beams. This fact represents another advantage of the clinical use of carbon ion

24

beams and it also contributes to an enhanced ballistic precision.

Figure 1.7. On the left: comparison of the lateral scattering of photon, proton and carbon beams as function of the penetration depth [35]. On the right: comparison between lat-eral distribution of dose deposited by proton and carbon ion beams having approximately the same range; the comparison clearly shows the improved selectivity of carbon ion beams respect to protons [40].

Effects on lateral broadening are much more evident looking at the so called apparent penumbra, which is the sharpness of the lateral dose fall-off [40].

Heavier ion beams exhibit sharper lateral dose fall-offs at the field boundary than lighter ions: in Figure 1.7 the penumbras of proton and carbon beams are compared. The penumbra width increases essentially linearly with the penetra-tion depth of the beam. For low-Z ions, such as protons, sharpest dose fall-offs are obtained when the final collimator is at the surface of the patient. For higher-Z ion beams, such as carbon ion beams, active scanning techniques without col-limations will produce narrow penumbras.

So far, physical effects related to electromagnetic interaction has been dis-cussed. When a heavy ion beam passes through beam line elements and finally traverses the tissues, nuclear reactions take place as well, which cause a small but not negligible amount of nuclear fragments. The result is a reduced number of primary particles reaching the tumour which are replaced by a certain number of secondary particles. An interesting potential for quality control arises from nuclear fragmentation, which by far compensate for the disadvantages, discussed

1.2 Rationale of ion beam therapy

25 ahead. Concerning the advantages, the stripping of one or two neutrons from the projectile ¹²C yields the positron emitting isotopes ¹¹C and ¹⁰C with half-lifes of 20 min and 19 sec, respectively. The stopping point of these isotopes can be monitored by measuring the coincident emission of the two annihilation gamma quanta following the β⁺ decay. In general, most of the lighter fragments have the same velocity as the primary ions at the collision [42]. The range of these frag-ments is given by the formula:

2 fragments and the projectiles, respectively [25]. Hence, the range of carbon iso-topes is only slightly shorter than that of the primary particle:

)

Thus, from the measured distribution of the annihilation quanta, the range of the stopping particles can be controlled and compared to the calculated range in the treatment planning, providing an in-situ beam monitoring using Positron Emission Tomography (PET). In fact, it is further expected that the spatial distri-bution of β⁺-activity induced by heavy-ion beams is strongly correlated with the corresponding dose distribution. Even if carbon beams mostly produce ¹¹C and

10C nuclei via projectile fragmentation, also ¹⁵O nuclei are produced from the target [41].

In Figure 1.8 depth-dose distributions of β⁺-emitters and primary particles in case of proton and carbon ion beams are shown. In contrast to the proton irradia-tion, the maximum of β⁺-activity produced by carbon ions is clearly visible. In this second case, the β⁺-activity shows a prominent maximum, shortly before the peak, formed by ¹¹C and ¹⁰C fragments of the ¹²C projectiles. Hence, all the ¹²C isotopes are stopped before the Bragg peak and the sharp fall-off in the β⁺ -activity distribution clearly indicates the position of the Bragg peak [43].

26

Figure 1.8. Calculated depth distributions of deposited energy (dash-dotted curves) and β⁺activity (histograms) for (a) 110 MeV protons and (b) 212.12 AMeV ¹²C nuclei in the PMMA phantom. The distributions of ¹¹C, ¹⁰C and ¹⁵O nuclei are shown by the long-dashed, dashed and dotted histograms, respectively, and their sum is shown by the solid-line histogram. The distribution of actual e⁺-annihilation points is shown by the solid-line histogram [43]. All the distributions have been calculated with the Monte Carlo code Geant4.

PET monitoring is of great interest in the hadrontherapy field because pro-vides a direct measurement of the beam distribution inside the patient and it represents as well another great advantage coming from the exploitation of car-bon ions in radiotherapy. The calculation of projectile range is, in fact, a critical point in treatment planning because human body is composed of a large variety of materials with different densities (bones, muscles, fat, air-filled cavities, etc).

Unfortunately, from another point of view beam fragmentation represents also the main disadvantage of using carbon ion beams for tumour treatments. Ion beams suffer nuclear reactions by interacting with the elements present along the

1.2 Rationale of ion beam therapy

27 beam line as well as inside the tissue itself, as already discussed. The first con-tribution can be opportunely reduced, and it is strictly dependent on the beam delivery system used; active systems are in this sense advisable (see 1.4.1). The second one is an intrinsic contribution and therefore not eliminable, but it is im-portant to know in details the effects on the delivered dose. According to the equation (1.7) the lighter fragments have a longer range than the primary parti-cles and, thus, are responsible for the undesired dose behind the Bragg peak, usually called tail. In Figure 1.9, the normalized depth-dose distributions in case of SOBP are showed for proton, carbon and neon ion beams having the same range and the tails are clearly visible for ions.

Figure 1.9. Comparison of spread out Bragg peak (SOBP) for proton, carbon and neon beams with the same range in water. Tails due to fragmentation are evident for ion beams and more dramatic for neon ion beams [19].

The increasing of the dose just boyind the paek strongly depends on the mass of the ion: in this specific case, it is of the order 15% of the dose in the SOBP for ions like carbon and oxygen, while it can reach 30% in the case of neon ions.

This is one of the reasons why, at least from the physical point of view, it is not justified to use ions heavier than oxygen for a really conformal therapy.

More-28

over, also biological reasons can be address for the exclusion of very heavy ions, as discussed in the next section. Considering also the surface dose in percentage in the plateau region, carbon ions represent a good compromise.

However, the effects of fragmentation have to be carefully taken into ac-count in treatment planning also because of the different biological effects char-acterizing the secondary particles produced, which give rise to a mixed radiation field. The study of fragmentation of carbon ion beam and the calculation of the fragments contribution in terms of dose and ionization density is a key point in hadrontherapy and it represents one of the aims of this work (see chapters 4 and 5). Indeed, for a volume irradiated by a parallel beam of ions, the absorbed dose D can be expressed as function of the ion fluence Φ and the stopping power (-dE/dx) by:

where ρ is the density of the stopping material. Because of the fragmentation processes, the particle fluence decreases with the penetration distance according to the relation:

where Φ(0) is the entrance fluence and µ is the linear attenuation coefficient, proportional to the total microscopic reaction cross-section σ for the ion-tissue interaction [34]. In principle, the dose distribution from each beam could be summed to obtain the total dose distribution. In practical radiotherapy it is not possible, because absorbed dose must be modified by a radiation weighting fac-tor that is energy dependent and changes according to the ion species considered.

The composition of this very complex particle field has to be known for dose op-timization in heavy ion therapy, in order to take correctly into account the global biological effect in the tissue, due to secondary as well as primary particles.

Trying to summarize what as been discussed, these are the main advantages showed by carbon ion beams from the physical point of view [45]:

- Carbon ions deposit their maximum energy density in the Bragg peak at the

1.2 Rationale of ion beam therapy

29 end of their range, where they can produce severe damage to the cells while sparing both the transversely adjacent and deeper located healthy tissues.

- They penetrate the patient with minor lateral scattering and longitudinal straggling respect to photon irradiation as well as proton beams. Moreover, being charged, they can easily be formed as narrow focused and scanning single beams of variable penetration depth, so that any part of a tumour can be accurately irradiated with optimal precision.

- The location where the dose is deposited by carbon ions can be determined by means of on-line positron emission tomography (PET), which permits exploitation of the millimetre precision of a focused carbon beam, crucial in case of target close to or inside critical structures.

Figure 1.10. Comparison of the planned dose distributions of a carcinoma in the front

Figure 1.10. Comparison of the planned dose distributions of a carcinoma in the front