Medical Applications of Radiation Fields

Chapter: Physics of Radiation in Matter

General introduction

Energy and particles

In order to penetrate the patient tissues, the energy of the beam used, whatever type of particle it is composed of and whatever kind of mean of acceleration is used, assumes a particularly important meaning. Generally, such energy is very low in terms of joule, thus the units of measurement one uses is the electronvolt, eV, equal to 1.6·10^-16 J. However, a lot of energy is needed in order to accelerate such particle beams, hence multiples of the eV are used, from keV to GeV.

The energy implied in any medical treatment through radiation is the one needed to break molecular bonds, which is typically around 10 eV. In particular, we will focus, from now on, on a specified molecule: the DNA. Even if the issue is far more complex than that, we can ultimately identify such energy as the minimum requirement to destroy a tumoral cell DNA; however, more energy is needed to penetrate the patient tissues proximal to the tumor.

Another point we have to make regards the fact that only charged particles can be accelerated and, furthermore, that such particles interact vividly with the electrons of the medium through electromagnetic interactions. Moreover, a charged particle that reaches closer to the atomic nucleus can directly interact with it, via scattering or nuclear interactions, depending on the kind of particle and its energy. For example, light particles such as an electron will hit the nucleus, transferring very little energy, thus changing its direction and resulting in a more or less random path in matter.

Many different particles are typically used in medical treatment; hence it can be useful to give a brief depiction of them:

• Particle: e^-, Charge: -1, Mass: 511 MeV
• Particle: e⁺, Charge: +1, Mass: ~2000 m_e
• Particle: p, Charge: +1, Mass: ~2000 m_p
• Particle: n, Charge: 0, Mass: ~2000 m_n
• Particle: C⁶⁺, Charge: +6, Mass: ~22000 m_C
• Particle: π^-, Charge: -1, Mass: ~270 m_π
• Particle: γ, X

Radiation therapy

There are basically two kinds of radiation therapy: the one through the use of external sources, such as accelerators, called teletherapy, and the one which employs internal sources, placed inside the patient body, such as radiopharmaceuticals, which is called brachytherapy. The latter one is typically used when the tumor cannot be easily reached from the outside or it is particularly difficult to be irradiated, for example, because of the presence of cavities which change shape, such as the case of prostate and uterus.

The latter kind uses radiation sources that emit not so energetic particles, so that they lose all their energy inside the tumor itself, thus avoiding damage to the surrounding healthy tissues. This latter statement, in particular, is the basic principle of radiation therapy: one shall look to obtain the maximum damage to the tumoral cells while bringing the minimum one to healthy tissues. Many techniques have been developed in order to realize this principle, as we will be able to see in the rest of the course.

Radiation quantities

Absorbed dose

Let us consider a charged particle moving inside matter. Neglecting, for now, the possibility of nuclear interactions—usually there are high energy thresholds—, such particle will interact via electromagnetic force with the electrons of the atoms. Part of its kinematic energy will therefore be transferred in order to strip the electron and set it loose in the medium; such process is called ionization, and the energy needed for it is the ionization potential. If the energy is less, the electron is simply excited to a higher energy level while remaining bounded to the atom; in this case, the surplus energy is later lost via emission of electromagnetic rays, called characteristic X-ray. In any case, the charged particle is giving energy to the medium; notice that the same fate is followed by the electrons that have been set off, called secondary electrons.

Furthermore, if the particle in question is a heavy charged particle, the loss of energy is every time rather low; however, since the electromagnetic interaction spans long distances, this energy transfer can be considered a continuous process. All particles able to ionize matter transferring energy are called ionizing radiation. The energy deposit ε of such particles in a single interaction can be written as:

ε = T_in - T_out + Q_Δm

In which T_in and T_out indicate, respectively, the kinetic energy—rest energy excluded—of the incident ionizing particle and that of all the particles leaving the interaction; Q_Δm is, instead, the rest energy variation of the atom and of all particles involved in the interaction. This quantity, in a reaction, is given by:

2Q = Σ m_i,inc² - Σ m_j,outc²

And it is called the Q-value of the reaction. Such rest energy is converted, if positive, into the kinetic energy of the reaction products—we will see an example in the boron neutron capture technique.

Such energy deposit has to be defined for any point of the particle trajectory; thus, the whole energy deposited is simply equal to:

ε = Σ ε_ii

expressed in joule. Notice that ε is indeed a stochastic quantity, since the amount of energy transferred, hence the interaction that happens, is a probabilistic effect.

Another way of defining the energy imparted in matter is through the radiant energy R, in turn defined as the emitter, transferred, or received energy of a particle (rest energy excluded) in a given volume. Therefore, the energy deposit is defined as:

ε = R_in - R_out + Σ Q_ii

In which R_in and R_out are, respectively, the sum of the energies of all charged and uncharged particles entering and exiting the volume, while ΣQ is the sum of all the rest energy variations of nuclei and elementary particles in each interaction in the considered volume.

Both definitions of the energy deposit are equivalent. The first one is formulated through single interactions of ionizing particles and it is of better use in the so-called microdosimetry, the branch of dosimetry aiming at measuring the energy fluctuations in micrometric and submicrometric sites, where single events and stochastic behavior are fundamental. According to this definition, the imparted energy can be calculated through numerical simulations. The second definition is based on the sum of the energies of the radiation field and it is at the basis of classical dosimetry.

The following step is to define a quantity that deals with energy transfer inside a given volume. Such is the absorbed dose, D, defined as the ratio:

D = dε̄/dm

where dε̄ is the mean energy imparted by ionizing radiation to matter of mass dm. Its unit of measurement is the gray, Gy=1J/kg. It should be stressed that the absorbed dose is a deterministic, i.e., non-stochastic, quantity since it has been defined through a mean value. For this reason, the extent of the mass dm has to be taken such to ensure that the mean value of the energy deposited can be defined, hence such that the number of events in that volume should be sufficiently high; moreover, energy should be at best deposited homogeneously inside that volume. For these reasons, in very small volumes of the order of cellular dimensions (few μm) or of DNA dimensions (about 2 nm), this concept has no meaning.

Furthermore, the absorbed dose is a quantity that is not able to express the radiation damage, that is the biological effects of radiation. In fact, heavy charged particles lose their energy in a very short track, while lighter particles have longer tracks; furthermore, electromagnetic radiation is able to produce in matter secondary radiation—typically electrons—, spreading its energy deposits in a more or less wide area. One can easily imagine that a shorter track corresponds to a higher density of deposited energy, or ionization energy, hence to higher damage to the medium. Because of that, it is not a good quantity to characterize a medical treatment, since the biological effect depends on what is called the particle track, of which we will talk about in moments. Nevertheless, the absorbed dose is a physical quantity which is directly measurable at low cost, while radiation quantities are not. Moreover, it is simple to define and is easily understandable by people not so familiar with ionizing radiation.

Linear energy transfer

We will now consider that quantity that can distinguish between the different densities of energy deposition. Such quantity, defined as the infinitesimal energy loss per unit path along a charged particle track, is called linear energy transfer, or LET, or also stopping power. It can be important to point out the fact that such infinitesimal definition is needed, since, as we will derive, the LET does depend on the instantaneous energy of the particle.

The LET correlation with the ionization density is evident: high LET particles will have higher ionization density, while lower LET particles will ionize less. However, since the LET is still a deterministic quantity, it does not give us information about how such energy is deposited. For this reason, even if better than the absorbed dose, the LET is not the best quantity to be used in radiotherapy.

In tissue, the electron LET varies (depending on the energy) from 0.2 up to 30 keV·µm^-1, that of protons from fractions of keV·µm^-1 (e.g., 0.4 keV·µm^-1 at 250 MeV) up to about 100 keV·µm^-1 (for 90 keV protons), that of alpha particles from a few keV·µm^-1 (e.g., 1.6 keV·µm^-1 at 1 GeV) up to 250 keV·µm^-1 (for 650 keV alpha particles) and that of heavier nuclei (up to C) from a few keV·µm^-1 up to 4000 keV·µm^-1.

We will now try to determine an expression for the LET, considering a charged heavy particle—such as an hadron—which is interacting with the medium in which it is moving. Such particle can interact with matter in very different ways, e.g., ionization and excitations of atoms and molecules, Coulomb scattering with nuclei, inelastic reactions with nuclei and so forth and so on. In particular, we will need to distinguish between those collisions which transfer an amount of kinetic energy which is higher than the bonding energy of the electrons of the medium—called hard collision—and those that do not—soft collisions.

Of course, we are dealing with electromagnetic interaction between the hadron and one atom of the medium, in particular through Coulomb force between the hadron and the electrons. Such force is given by:

F = k 2zZe/d²

In which k is the Coulomb constant, ze the charge of the hadron colliding, Ze the one of the atom; d is the distance between the two particles.

The momentum transferred by the projectile to the target (i.e., an electron in this case) can be calculated by assuming that the projectile moves with its impinging direction at a distance b from the position of the target particle, assumed to be at rest. Therefore, this distance, called impact parameter, is the minimum one between the target electron and the projectile trajectory. Under this assumption:

F = k 2zZe/b²

Because of the great difference in mass, the projectile trajectory is not significantly influenced by the presence of the electron and moreover, the electron can be assumed to absorb the transferred momentum without changing its position during the collision, because the duration of the interaction is very short. Under these assumptions, the momentum acquired by the electron must be perpendicular to the projectile trajectory and, since the force transferred to the electron is:

F_⊥ = Ze∫ ⊥ dt = Ze ∫ ⊥ dx/v

where ⊥ is the electric field component at the electron position which is normal to the projectile trajectory and v is the projectile velocity, which can be considered constant during a single collision.

Now, according to Gauss theorem, the electric field flux through the cylindrical surface of radius b, length dx and axes on the projectile trajectory is given by:

∫_S ⊥ dS x , t) d x⋅n = ρ

In our case:

∫ ∫⊥ dx= ze ⊥ dx= πb

Therefore, we can write:

q = 2zZe/bv

This allows us to compute the transferred energy as:

E = q²/2m_e

Notice that the higher the atomic number of the projectile, the higher the transferred energy will be, thus the LET and the ionization/excitation density, thus the biological damage. For this reason, ions are used to treat patients.

Let's look towards the particle impinging direction. For a fixed impact parameter, the area of the circle with radius b and with the target at its center can be thought of as the equivalent area offered to the particle for energy transfers higher than a certain value. In other words, this area is the geometrical cross-section of the scattering reaction for energy transfers equal to or higher:

σ = π (q²/2m_e)

If the electron density is uniform (and equal to N×Z, where N is the number of atoms per unit volume) and the material is mono-atomic, the mean energy transferred along the particle trajectory between x and x+dx to the electrons (Z=-1) can be expressed as:

dE/dx = -NZ σ

where E and E are the minimum and the maximum energies that can be transferred to the electron during the collision. The maximum and the minimum impact parameters b and b correspond to E and E, respectively.

The integration limits b and b still have to be determined. Starting with b, which corresponds to the maximum transferred energy E, it could be demonstrated that in one elastic collision between two particles, the kinetic energy E transferred, deflecting the target particle of an angle θ, is:

E = 4mM 2 E T cosθ/c² (m+M)

where T is the kinetic energy of the impinging particle, M its mass, and m the one of the target. The maximum energy is transferred in a knock-on collision, hence at θ=0:

E = 4mM T/(m+M)²

Notice that if m=M, all the projectile energy is transferred to the target—such is the case of a proton-proton scattering interaction; if M>>m, instead, the energy transferred is:

E = 2mv²

As in our case of hadron-electron interaction. b can be assessed through the Heisenberg uncertainty principle, stating that the position and the momentum of a particle can be assessed with an uncertainty equal to h/2π. For the present case:

Δb Δq ~h

Accounting for relativistic corrections, not treated herein, the maximum energy transferred result is:

E_max = 2mv²/(1-β²)

Thus, from the uncertainty principle:

b = h/2mv(1-β²)

For estimating the maximum impact parameter b, the electrons cannot be considered as free, and the rigorous description of the interaction would require models based on quantum mechanics. At long distances from the target, we can assume a constant electric field over the atom. For simplicity, no distinction is made between ionization and excitation by introducing the mean excitation and ionization energy I, as the mean over all energies that may be absorbed by an atom of a certain type. I is also called ionization potential.

In the present case, the target atoms are assumed to respond as a classical system, absorbing electromagnetic waves, τ in a period, corresponding to the ionization potential I=h/τ. If the oscillation period of the atom is comparable with the collimation time the interaction occurs, hence the impulse transferred to the electron is:

J = F τ = k 2zZeb/v

Now, the maximum impact parameter is derived by the so-called adiabatic condition, stating that the projectile electric field perturbs the atomic electron without any energy absorption. Under this condition:

b ≈ v τ/2π

Furthermore, introducing the relativistic correction about the contraction of distances, we get:

b ≈ v τ/2π√(1-β²)

Finally, substituting b and b in -dE/dx, we obtain:

dE/dx = -NZ 4z²e⁴/2m_ev² ln(I/mv²)/(1-β²)

By considering further relativistic corrections (not treated herein), the Bethe-Bloch formula for charged hadrons is obtained:

dE/dx = -NZ 4z²e⁴/2m_ev² [ln(I/mv²)-ln(1-β²)-β²]

A further correction can be introduced when relativistic particles are considered; in fact, one can show that particle motion contracts the loci of the equipotential surfaces of the electric field along the direction of incidence and, correspondingly, it widens them in the transversal direction. Therefore, the more distant the collisions from the particle position, the higher importance they gain with increasing energy. The retarded potential theory demonstrates that the effective electric field is higher in the zone just traveled by the particle than in those that have to be traveled. Therefore, an imbalance of charges is produced, and this polarization field reduces the effective electric field of the projectile and shields the more distant atoms, which otherwise could be ionized or excited. In other words, the particle polarizes the medium by limiting the extension of its associated electric field and thus truncating part of the logarithmic increase.