Nteract Ons Of Charged Part Cles W Th Matter Biology Essay

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Charged particles are fundamental to the medical use of radiation. Even if the primary radiation is a photon beam, it is the charged particles, known as secondary radiation in such cases, that cause the biological effect, whether it is cell killing or other changes that may eventually induce cancer. In fact, charged particles are often termed ionising radiation, and photons (and neutrons) termed non-ionising or indirectly ionising.

The generation of x-rays, i.e. bremsstrahlung, is a charged-particle interaction. Radiotherapy is sometimes delivered by primary charged particle beams, usually megavoltage electrons, where electron interactions with matter are obviously crucial (Mayles et al. 2007).

Collision Losses

Coulomb interactions with the bound atomic electrons are the principal way that charged particles (electrons, protons, etc.) lose energy in the materials and energies of interest in radiotherapy. The particle creates a trail of ionisations and excitations along its path. Occasionally, the energy transfer to the atomic electron is sufficient to create a so-called delta ray (or δ-ray) which is a (secondary) electron with an appreciable range of its own. This is schematically illustrated in Figure 2.1 (Evans 1955, ICRU 1970).

The physical model of the Coulomb interaction between the fast charged particle and a bound electron in the medium is shown in Figure 2.2. The electron is assumed to be free, and its binding energy assumed to be negligible compared to the energy it receives. The incoming electron is moving at a speed V in a direction opposite to axis x. The primary particle imparts a net impulse to the bound electron in a direction perpendicular to its path (Nahum 1985).

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Figure 2.1. Diagrammatic representation of the track of a charged particle in matter

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Figure 2.2. Interaction between a fast primary charged particle and a bound electron

Using classical, non-relativistic collision theory, from Newton's second law, (i.e. the change in momentum is equal to the impulse [the time integral of the force]) and from the Coulomb law for the force between charged particles, it can be shown that the energy transfer Q is given by:

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where b, the distance of closest approach, is known as the impact parameter; m is the mass of the electron; z is the charge on the primary particle; V is the velocity of the primary particle; and the constant, k is constant appearing in the Coulomb-force expression (=8.9875Ã-109 Nm2C-2). It should be noted that the mass of the primary particle does not enter into Equation 3.1, which equally applies to electrons, protons (that both have z=1), and other heavier charged particles. Equation 3.1 leads to the following classical expression for the cross-section per electron, differential in the energy transfer Q:

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The full relativistic, quantum-mechanical cross-section for Coulomb interactions between free electrons, due to Møller (1932), is:

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where T is the electron kinetic energy; ε=Q/T is the energy transfer in units of the electron kinetic energy; τ=T/mec2 is the kinetic energy in units of the electron rest mass; V is the electron velocity.

The Møller expression (Equation 2.3) is valid provided that the electron energy is much greater than the binding energies of the atoms in the medium. The binding energies also set a lower limit to the energy transfer possible.

Collision Stopping Power

The occurrence of very frequent, small energy losses along the path of any charged particle in matter leads naturally to the concept of stopping power, defined as the average energy loss, dE, per unit distance, ds, along the track of the particle. This is usually expressed as the mass collision stopping power, written (1/ρ)(dE/ds)col or Scol/ρ, which is calculated from

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where NA is Avogadro's number, and Z and A are atomic and mass number, respectively. Qmax for an electron with kinetic energy E0 is equal to E0/2. The evaluation of the minimum energy transfer Qmin represents a major difficulty.

The full quantum-mechanical expression for the electron mass collision stopping power (Berger and Seltzer 1964; ICRU 1984a, 1984b) is given by

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where

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and the extra quantities not defined so far are

mec2, rest mass energy of the electron

β=V/c

re, electron radius (=e2/mec2=2.818Ã-10-15 m)

I, mean excitation energy

δ, density-effect correction

The mean excitation energy or potential, I, is an average of the transition energies Ei weighted by their oscillator strengths fi according to the following:

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It is effectively the geometric mean of all the ionisation and excitation potentials of the atoms in the absorbing medium; it is, of course, the more exact counterpart of Bohr's mean characteristic frequency that was discussed above. In general, I cannot be derived theoretically except in the simplest cases such as monoatomic gases. Instead, it must be derived from measurements of stopping power or range. The most recent values of I, based largely on experimental data, are given in ICRU (1984b). For example the best current estimate of the I-value for water is 75.0 eV. Generally, the I-value increases as Z increases (Table 2.1).

Table 2.1. Mean excitation energies, I, and other quantities relevant to the evaluation of the collision stopping power of selected human tissues and other materials of

dosimetric interest

Material

I (eV)

Density (g cm-3)

Adipose tissue (ICRP)

63.2

0.558468

0.920

Air (dry)

85.7

0.499190

1.205Ã-10-3

Bone, compact (ICRU)

91.9

0.530103

1.850

Bone, cortical (ICRP)

106.4

0.521299

1.850

Ferrous-sulphate dosimeter solution

76.3

0.553282

1.024

Lithium fluoride

94.0

0.462617

2.635

Muscle, skeletal (ICRP)

75.3

0.549378

1.040

Muscle, striated (ICRU)

74.7

0.550051

1.040

Photographic emulsion

331.0

0.454532

3.815

PMMA (lucite, perspex)

74.0

0.539369

1.190

Polystyrene

68.7

0.537680

1.060

Water (liquid)

75.0

0.555087

1.000

The essential features of the mass collision stopping power are retained in the following simplified expression:

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Comparing this expression with Equation 2.5, the increase at decreasing subrelativistic energies due to the (1/V2) factor can be identified. This is simply explained by the fact that slow electrons spend more time going past an atom than do fast ones, and consequently the impulse is greater and thus more energy is lost. At relativistic energies, there is a more gradual increase in the stopping power which is known as the relativistic rise (Figure 2.3 Nahum (1985)).

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Figure 2.3. Collision energy loss as a function of electron kinetic energy

Density Effect

The density or polarization effect (Fermi 1940; Sternheimer 1961; ICRU 1984a) reduces the value of Scol at relativistic energies in condensed media via the term δ in Equation 2.5 and Equation 2.8. It is connected to the relativistic rise in the stopping power. If the stopping medium has a high density, (i.e. condensed media as opposed to gases) then the electric field seen by the atoms distant from the fast particle track is reduced due to the polarization of the intervening atoms (as illustrated in Figure 2.4 (Nahum 1985)). Consequently, the contribution of these distant collisions to the stopping power will be reduced. This reduction in collision stopping power is known as the polarization or density effect.

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Figure 2.4. Schematic explanation of the mechanism of the density effect

Figure 2.5 shows the variation of Scol/ρ with energy for air and water, two substances with similar atomic compositions and similar I-values. The relativistic rise in the collision stopping power in the condensed medium is much less pronounced compared to that in the gas because of the density effect. Consequently, the ratio of mass stopping powers, water to air, is strongly energy dependent above around 0.5 MeV (Nahum 1983).

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Figure 2.5. The variation of mass collision stopping power with electron kinetic energy for air and for water

Electron Stopping-Power Data for Substances of Medical Interest

Table 2.1 lists the relevant parameters for various human tissues and some other substances of dosimetric interest taken from ICRU (1984a) as well as for water as a comparison. It can be seen that the I-values all fall between 73 eV and 75 eV with the exception of adipose tissue (high hydrogen content) and bone with its high calcium content. In fact, the I-value is approximately proportional to the mean atomic number. Given the similarity of the values of I, , and the (mass) density (the latter being involved in the density-effect correction δ) in the table, the values of (Scol/ρ) must also be very similar. This is very convenient, as it means that the electron energy loss over a given distance in the body can be derived from that in water by simply multiplying by the density, assuming that radiation losses are also very similar.

The values of the (mass) stopping-power ratio, smed,air, for various substances of interest in medical dosimetry, as a function of electron kinetic energy in the megavoltage region are shown in Figure 2.6. The ratio is virtually independent of energy except for that of air; this is very convenient for dosimeter response evaluation and treatment planning purposes. These medium-to-water stopping-power ratios are likely to find direct application in the conversion of Monte-Carlo-derived dose distributions in patients to water-equivalent doses (Siebers et al. 2000).

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Figure 2.6. Ratios of mass collision stopping powers, medium to water, for various substances of medical and dosimetric interest

PS: Polystyrene, PMMA: Polymethyl methacrylate (perspex).

Restricted Stopping Power

In restricted stopping power, only energy transfers below a certain value Δ are included and it is calculated by setting Qmax equal to Δ in Equation 2.4. The full expression is again given by Equation 2.5 and Equation 2.6, but with the F(τ) term modified to:

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The expression for the restricted stopping power still includes the density-effect correction factor δ.

Radiative Losses (Bremsstrahlung)

The acceleration of the electrons in the strong electric field of a nucleus leads to the production of bremsstrahlung. The acceleration is proportional to the nuclear charge, Z, divided by the mass, m, of the moving particle. The intensity of radiation produced is then proportional to (Z/m)2. This is a relatively unimportant energy loss mechanism below about 10 MeV in low-Z materials, and it is completely negligible for heavy charged particles. The cross-section, σrad, for this totally non-classical process is extremely complicated. One significant feature is that, very approximately:

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Therefore, on average, the losses will be appreciably larger than for collisions. This means that considerable energy-loss straggling due to radiation losses can be expected.

Radiation Stopping Power

In an exactly analogous fashion to that for collision losses in the previous section,

a radiative stopping power, (dE/ds)rad or Srad, and also a mass radiation stopping power

(Srad/ρ) can be defined. The general form of the mass radiative stopping power for high energies (complete screening: τ >> 1/α Z1/3) is given by:

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where a is the fine structure constant (α ≈ 1/137). From an inspection of Equation 2.11, it can be seen that the radiative stopping power increases almost linearly with kinetic energy in the MeV region, in contrast to the weak logarithmic energy dependence of the collision stopping power. Approximately, it can be written:

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where B is a very slowly varying function of E and Z. The factor, Z2/A, causes an increase in Srad/ρ for higher Z (Figure 2.7).

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Figure 2.7. A comparison of the mass radiative and mass collision stopping powers, Srad/ρ and Scol/ρ respectively, for carbon, copper, and lead

Radiatiion Yield

A useful quantity is the fraction of the initial electron energy, E0, that is lost to bremsstrahlung in slowing down to rest. This fraction is known as the Radiation Yield, Ï’(E0), and is given by:

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The dependence of ϒ(E0) on E0 and on Z is approximately linear, which closely corresponds to the relation between Srad/ρ, E, and Z.

The radiation yield is involved in calculating the dosimetric quantity, g, which is the fraction of energy transferred (by photons) to a medium in the form of electron kinetic energy that is subsequently re-radiated as bremsstrahlung.

Angular Distribution of Bremsstrahlung Photons

The angular distribution of the emitted photons is very strongly forward-peaked at relativistic electron energies with a mean value θ ≈ mc2/E where E is the total energy of the electrons. This forward-peaking is the reason for the flattening filter in a linear accelerator treatment head.

Total Energy Losses

Total Stopping Power

The collision and radiative stopping powers are frequently summed to give the total stopping power, written (dE/ds)tot or Stot:

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Figure 2.8 shows the total mass stopping power (labelled "Total Loss"), mass collision stopping power, and several restricted mass collision stopping powers (Δ = 10 keV, 1 keV and 100 eV) for water against electron kinetic energy E for values between 10-5 MeV and 104 MeV. It can be seen that Stot/ρ varies slowly with E over the energy range of primary interest in radiotherapy (from 1.937 MeV cm2 g-1 at 4 MeV to only 2.459 MeV cm2 g-1 at 25 MeV).

Several features should be noted:

Radiation losses only become important above around 10 MeV in water

The relativistic rise in the collision losses is small because of the density effect

Collision losses restricted to Δ<10 keV only result in a modest reduction in stopping power compared to the unrestricted Scol which emphasizes the predominance of very small losses

The approximate value for the electronic stopping power in water in the MeV region is around 2 MeV cm-1; the value in tissue is very similar.

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Figure 2.8. Mass stopping power in water for electrons

Energy-Loss Straggling

It is important to realize that stopping power is an average value for the energy loss per unit distance. Fluctuations will occur about this mean value in any real situation. This gives rise to what is known as energy-loss straggling (Figure 2.9) (Berger and Wang 1988).

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Figure 2.9. Energy broadening because of energy-loss straggling after the passage of a monoenergetic electron beam (energy E0) through a thin absorber

Continuous-Slowing-Down-Approximation (CSDA) Range

Charged particles lose energy in a quasi-continuous fashion along their tracks in matter, eventually coming to rest. This means that, unlike photons with their exponential attenuation, charged particles do have a finite, reasonably well-defined range. Mathematically it has been found convenient to define the so-called continuous-slowing-down-approximation (CSDA) range, r0 in the following fashion:

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This represents the average pathlength travelled in coming to rest by a charged particle having kinetic energy, E0. Note that for electrons, as opposed to heavy charged particles, this will always be considerably greater than the average penetration depth because of the marked angular deflections that electrons suffer in slowing down. The CSDA range r0 is approximately proportional to E0 in the therapeutic energy range because of the relatively slow variation of Stot in this energy range.

Elastic Nuclear Scattering

When a charged particle passes close to the atomic nucleus, at a distance much smaller than the atomic radius, the Coulomb interaction will now be between the fast particle and the nuclear charge rather than with one of the bound electrons. In the case of electrons, this causes appreciable changes of direction, but almost never (with the exception of the bremsstrahlung process) any change in energy. The scattering is basically elastic, the energy lost being the negligible amount required to satisfy momentum conservation between the very light electron and the positively charged nucleus.

This interaction process is essentially Rutherford scattering with differential cross-section:

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Application to an Electron Depth-Dose Curve

Figure 2.10 illustrates the physics of electron interactions as they apply to electron beams used in radiotherapy; it shows three different depth-dose curves obtained through Monte-Carlo simulation, corresponding to different approximations about electron transport physics for a 30 MeV broad, monoenergetic, and parallel electron beam in water (Nahum and Brahme 1985, Seltzer 1978).

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Figure 2.10. The effect of various approximations on the electron depth-dose curve for a broad, 30 MeV electron beam in water (CSDA range r0 = 13.1cm)

The curve labelled "CSDA straight ahead" corresponds to straight tracks and shows the Bragg Peak, normally associated with heavy charged particles; this extremely simple approximation illustrates very clearly the behaviour of the total stopping power Stot as the electron energy gradually decreases with depth. The gradual decrease in dose with depth mirrors the decrease in total stopping power with falling electron energy. At an energy close to that of the electron rest mass (0.511 MeV), however, the collision stopping power goes through a minimum and then rises rapidly (principally because of the 1/β 2 term in Equation 2.5.

The "CSDA multiple scattering" curve involves directional changes through multiple scattering (Equation 2.16), but does not involve any secondary particle transport or any simulation of energy-loss straggling. The increase in dose away from the surface is entirely due to the increasing average obliquity of the electron tracks with depth and the fact that the beam is broad (i.e. there is lateral scattering equilibrium); this is sometimes known as scatter buildup). At around z/r0=0.7, the planar fluence starts to decrease as electron tracks begin to reach their end. The maximum occurs as a result of a competition between the scatter build-up and the decrease in the planar fluence because of electrons reaching the end of their range.

The "No knock-on transport" curve does not include the generation and transport of knock-on electrons or delta rays, but it does include radiative losses (i.e. bremsstrahlung) and one sees the so-called bremsstrahlung tail beyond the practical range. Also, now the slope of the dose falloff is much reduced; this is primarily due to the incorporation of energy-loss straggling.

Finally, the unlabelled full curve corresponds to a simulation including the full electron transport physics (of most relevance in this energy region). The effect of simulating δ-ray transport is clearly seen in the build-up close to the surface; this is analogous to the much more pronounced one in megavoltage photon beams where the ranges of the mainly Compton electrons are significantly greater than those of the predominantly low-energy δ-rays.

INTERACTIONS OF PHOTONS WITH MATTER

Photon interactions are stochastic (i.e. random) by nature. Unlike electrons, they may undergo a few, one, or no interactions as they pass through matter. In each interaction, secondary ionising particles are created. These may be charged particles (usually electrons) or uncharged particles (usually photons). The charged particles deposit their energies close to the interaction site and contribute to the local energy deposition, whereas, secondary photons may be transported some distance before interacting.

Secondary photons are important because they contribute to the photon fluence inside and around an irradiated body and to dose when they interact and produce secondary electrons. The relative importance of secondary photons depends on the energies of the primary photons. In external beam therapy using megavoltage beams, the dominant contribution to the absorbed doses within the patient is due to primary photons.

Photon Interaction Cross-Sections

Photons interact with various target entities such as atomic electrons, nuclei, atoms or molecules. The probability of interaction with a target entity is usually expressed in terms of the cross-section σ. The type of target for the interaction is marked, when necessary, by adding an index to σ. Therefore, eσ and aσ designate the cross-section per electron and per atom, respectively. The relation between them is given by as=ZÃ-es where Z is the atomic number of the atom.

Photon interactions can be characterized as absorption or scattering processes. In a full absorption process, the incoming photon loses all its energy and the energy is transferred to the target entity. Secondary particles are emitted during or subsequently to the interaction. In a full scattering process, an incoming photon interacts with a target entity and its direction of motion, energy and momentum may be changed because of this interaction. The photon, however, is not absorbed, and changes of energy and momentum are governed by the laws of relativistic kinematics. The main absorption processes are photoelectric (pe) absorption, pair (pair), and triplet (trip) production. The main scattering processes are coherent (coh) and incoherent (incoh) scattering. Nuclear photo-effect (phn) is an absorption process that is mostly neglected but needs to be considered in some cases. The total interaction crosssection, independent of which process occurs, is the sum of the cross-sections for the individual processes:

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The unit of cross-section is m2. Although it does not belong to the International System of Units, the barn (1 barn = 10-24 cm2 = 10-28 m2) is still frequently used.

In a scattering process, the distribution of scattered photons may not be isotropic, but may instead be anisotropic in some fashion related to the direction of the incoming photon and its polarisation. In order to quantify such effects, the cross-section is regarded as a function of the solid angle Ω in the direction of the scattered photon and the concept of the differential crosssection dσ/dΩ is introduced. The differential cross-section is defined in a way analogous to the total cross-section with (dσ/dΩ)dΩ related to the probability that the photon scatters into solid angle dΩ. It follows that:

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where θ is the scattering (polar) angle, and ϕ is an azimuthal angle. In many situations, the scattering will, on average, have no azimuthal dependence, and the equation can then be written

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Equation 2.19 may also be written

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where

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The quantity dσ/dθ is also referred to as a differential cross-section.

Photoelectric Absorption

Photoelectric absorption is illustrated in Figure 2.11. In this process, an incoming photon interacts with an atom and is absorbed. An atomic electron is ejected with kinetic energy T from one of the atomic shells. Its kinetic energy is given by:

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Here, hν is the energy of the interacting photon, and EB is the binding energy of the atomic electron. The process cannot occur with a free electron. The atom is needed in order to conserve momentum. Because of the heavy mass of the nucleus, the energy transferred to the atom is negligible.

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Figure 2.11. Photoelectric absorption

In general, the cross-section σpe for photoelectric absorption increases strongly with decreasing photon energy. Figure 2.12 shows this cross-section for lead. The cross-section displays a series of discontinuities at energies corresponding to the binding energies of the electrons in the atomic shells. These discontinuities are known as absorption edges. Below the absorption edge, the photon does not have sufficient energy to liberate an electron from the shell. At energies just above the edge, the photon has sufficient energy to liberate the electron. Therefore, the cross-section abruptly increases because the number of electrons that can take part in the absorption process increases. The absorption edge is most pronounced at the K shell in a high atomic number material. The L-shell has three sub-shells and, correspondingly, three absorption edges are seen in Figure 2.12 at the energies 13.04 keV, 15.20 keV, and 15.86 keV of the L sub-shells in lead. At energies above the K absorption edge, about 80% of the interactions take place in the K shell.

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Figure 2.12. The total photoelectric absorption cross-section for lead as a function of photon energy

The cross-section for photoelectric absorption depends strongly on atomic number. Above the K absorption edge, the cross-section per atom as a function of photon energy and atomic number is approximately given by

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The cross-section increases as the fourth power of the atomic number and is inversely proportional to the third power of photon energy. This points to the strong impact of this process at low photon energies, particularly, at high atomic numbers.

The angular distribution of the photoelectrons is peaked at angles of π/2 to the forward direction at low photon energies, but it becomes increasingly forward directed as the photon energy increases.

After photoelectric absorption, a vacancy is left in the atomic shell. This vacancy is subsequently filled with an electron from an outer shell. The energy released is equal to the difference in the binding energies of an electron in the two shells (e.g. Ek-EL in a transition from the L to the K shell). The energy released is carried away either by the emission of a photon or an electron. The photon is known as a characteristic x-ray because of its fixed energy determined by the atomic number of the atom and the shells involved. Characteristic x-rays are isotropically emitted. At energies immediately above an absorption edge, they may carry a substantial fraction of the incident photon energy. Electrons emitted after electronic rearrangement are known as Auger electrons. They are also isotropically emitted. The kinetic energy of an Auger electron is equal to the energy released in the transition minus its binding energy.

Compton Interaction and Scattering Processes

In a scattering process, the photon changes its direction of motion. If its energy is reduced, the scattering is called incoherent. The scattering may also occur without energy loss and is then referred to as coherent scattering. The terms elastic and Rayleigh scattering have also been used for this process.

For photon energies that considerably exceed the binding energies of the atomic electrons, the kinematics of the scattering process is usually described by considering the target electron to be free and at rest at the moment of collision. In this case, the scattering is incoherent because the photon will lose energy upon being scattered. At lower photon energies, the binding energies of the atomic electrons cannot be neglected. The photon can then scatter from individual bound electrons (incoherent scattering) or from all the bound electrons together, scattering in phase (coherent scattering). In the latter case, the whole atom takes part in the scattering process to conserve momentum.

Incoherent Scattering

In incoherent scattering, the photon transfers part of its energy to an atomic electron that is ejected from the atomic shell. The process was first described by Compton who assumed the electron to be free and at rest at the moment of collision. In this approximation, the process is also known as Compton scattering. The kinematics of Compton scattering is illustrated in Figure 2.13.

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where κ = hν/(m0c2) and m0 is the rest mass of the electron.

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Figure 2.13. Scattering angles and energies for Compton scatter

The cross-section for Compton scattering is named after Klein and Nishina who first derived an expression for its value. The differential Klein-Nishina cross-section per electron is given by

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At low energies (hν→0), this reduces to

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This cross-section is known as the classical Thomson differential cross-section. The total Klein-Nishina cross-section per electron may be obtained by integrating Equation 2.25, substituting for hν' using Equation 2.24. The result is:

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The differential Klein-Nishina cross-section is shown in Figure 2.14.

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Figure 2.14. Cross-sections for Compton scattering from free electrons

The influence of electron binding on the incoherent scattering cross-section is usually quantified by the incoherent scattering function S(x,Z). The differential scattering cross-section for incoherent scattering per atom is then given by

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The incoherent scattering function is generally assumed to be a function of the momentum transfer and the atomic number, Z. It is tabulated in terms of the momentum transfer related quantity x given by

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where λ is the wavelength of the primary photon.

Coherent Scattering

In coherent scattering, the photon is collectively scattered by the atomic electrons. Essentially, no energy is lost by the photon as it transfers momentum x to the atom while being scattered through the angle θ. The scattering from the different electrons is in phase, and the resultant angular deflection is determined by an interference pattern characteristic of the atomic number of the atom. The differential cross-section for coherent scattering is obtained as the product of the differential Thomson scattering cross-section and the atomic form factor F squared

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The atomic form factor is, like the incoherent scattering function, a universal function of x. It takes its maximum value in the forward direction (θ=0) where F(0,Z)=Z. It decreases to zero as x increases; with increasing momentum transfer x, it gets increasingly difficult for all electrons to scatter in phase without absorbing energy.

Pair and Triplet Production

Pair production is illustrated in Figure 2.15. In pair production, the photon is absorbed in the electric field of the nucleus. An electron (negatron)-positron pair is created and emitted with the sum of their kinetic energies, T-+T+, being determined by the requirement for conservation of energy

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Figure 2.15. Pair production

From the above Equation 2.31, it is clear that the process has a threshold value of 2m0c2 (1.02 MeV), the minimum energy required to create two electrons. On average, the electron-positron pair about equally shares the kinetic energy available.

The process of pair production may also occur in the electric field of an atomic electron. The atomic electron will recoil with sufficient energy to be ejected from the atomic shell. Three electrons appear as a result of the interaction and, accordingly, the process is called triplet production. Triplet production has an energy threshold at 4m0c2 (2.04 MeV).

The cross-section for pair production in the nuclear field is zero below threshold. It then rapidly increases with increasing energy and, well above threshold, varies approximately as the square of the nuclear charge Z. The cross-section for triplet production, at energies above threshold, approximately varies as Z.

Nuclear Photoeffect

When the photon energy exceeds that of the binding energy of a nucleon, it can be absorbed in a nuclear reaction. As a result of the reaction, one or more nucleons (neutrons and/or protons) are ejected. The cross-section for the nuclear photoeffect depends on both the atomic number, Z, and the atomic mass, A, and thus on the isotopic abundance in a sample of a given element. The cross-section has an energy threshold, and it is shaped as a giant resonance peak. The peak occurs between 5 and 40 MeV, depending on the element, and it can contribute between 2% (high-Z element) and 6% (low-Z element) to the total cross-section.

The Total Atomic Cross-Section

The total atomic cross-section and its partial cross-sections are given in Figure 2.16 for the elements carbon (Z=6) and lead (Z=82).

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Figure 2.16. The total and partial cross-sections for carbon (a) and lead (b) for photon energies from 10 keV to 100 MeV

Macroscopic Behaviour

Photons incident on an absorber will either interact in it (producing secondary electrons and/or scattered photons) or else pass through it without interacting. The number of photons transmitted undisturbed through an absorber of thickness t of a given element and density can, for mono-energetic photons, be derived in the following way (see Figure 2.17). The number of primary photons, dΦ, interacting in a thin layer dx at depth x is proportional to the thickness of the layer and the number of photons incident on the layer so that

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The linear attenuation coefficient m is a property of the material and depends on photon energy. The minus sign indicates that photons are removed from the beam. Integrating the equation from x=0 to x=t gives the number, Φ(t), of primary photons that are transmitted through the absorber. This number decreases exponentially with increasing thickness t according to

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with Φ0=Φ(0) the number of incident photons.

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Figure 2.17. Calculation of photon transmission through a slab of matter

The linear attenuation coefficient is the probability per unit length for interaction and is related to the total atomic cross-section, σtot, through the relation

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where N is the number of target entities per unit volume. It is given by N=(NA/A)ρ.

The mass attenuation coefficient, μ/ρ, obtained by dividing μ with ρ, is independent of the actual density of the absorber and makes this quantity attractive for use in compilations.

The penetration power of a photon beam is commonly expressed by means of the mean free path. This is defined as the average distance, , travelled by the photon before it interacts. For mono-energetic photons it is given by

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Principles and Basic Concepts in Radiation Dosimetry

The accurate determination of absorbed dose is crucial to the success of radiotherapy because of the relatively steep sigmoidal dose-response curves for both tumour control and normal-tissue damage. There are many different steps involved in the determination of the absorbed dose distribution in the patient. One of the most important of these involves measurements with a detector (often termed a dosimeter) in a phantom (often water, sometimes water-like plastic) placed in the radiation field.

Definitions of Dosimetric Quantities

Absorbed Dose

ICRU (1980, 1998) defines absorbed dose as the quotient of by , where is the mean energy imparted by ionising radiation to matter of mass :

(2.1)

The unit of absorbed dose is the "gray" which is 1 Joule per kilogram (J kg-1); the old unit is the "rad" which is 10-2 gray (sometimes referred as a centigray).

The energy imparted, ε, by ionising radiation to the matter in a volume is defined by ICRU (1980, 1998) as:

(2.2)

where

is the sum of the energies (excluding rest mass energies) of all those charged and uncharged ionising particles that enter the volume (known as the radiant energy)

is the sum of the energies (excluding rest mass energies) of all those charged and uncharged ionising particles that leave the volume, and

is the sum of all changes (decreases: positive sign, increases: negative sign) of the rest mass energy of nuclei and elementary particles in any nuclear transformations that occur in the volume.

Figure 6.2 illustrates the concept of energy imparted. In the left part of the figure which represents a Compton interaction within the volume V, the energy imparted is given by

(2.3)

where is the kinetic energy of the charged particle-of initial kinetic energy T-upon leaving the volume V. Note that the photon does not appear as this is not emitted within the volume V. The term is not involved here.

C:\Users\Turkay\Desktop\Figures\Fig2.1.png

Figure 2.1. An illustration of the concept of the energy imparted to an elementary volume by radiation

The volume on the right in the figure involves γ-ray emission () from a radioactive atom, pair production (kinetic energies T1 and T2), and annihilation radiation as the positron comes to rest. The energy imparted in this case is given by

(2.4)

where

(2.5)

In Equation 2.4, the zero is the Rin term; '1.022 MeV' is the Rout term comprising the two annihilation γ-rays. The γ-ray energy arises from a decrease in the rest mass of the nucleus; the term is due to the creation of an electron-positron pair, i.e. an increase in rest mass energy. Finally the term is due to the annihilation of an electron and a positron.

Kerma and Exposure

The quantity kerma, can be thought of as a step towards absorbed dose. It is conceptually very close to exposure, the first radiation quantity to be formally defined (Greening 1981). All practising hospital physicists will come across kerma (in air in a cobalt-60 γ-ray beam) in the context of calibrating ionisation chambers at a National Standards Laboratory. The formal definition (ICRU 1980, 1998) follows:

The kerma, K, is the quotient dEtr by dm, where dEtr is the sum of the initial kinetic energies of all the charged ionising particles liberated by uncharged ionising particles in a material of mass dm:

(2.6)

The units of kerma are the same as for absorbed dose, i.e. J kg-1 or gray (Gy). Kerma applies only to indirectly ionising particles which, for our purposes, almost always mean photons, although neutrons also fall into this category.

Exposure is conceptually closely related to air kerma. Exposure, usually denoted by X, is the quotient of dQ by dm where dQ is the absolute value of the total charge of the ions of one sign produced in air when all the electrons (negatrons and positrons) liberated by photons in air of mass dm are completely stopped in air. Until the late 1970s, all ionisation chambers were calibrated in terms of exposure; subsequently this was replaced by air kerma.

Figure 2.2 illustrates the concept of kerma (and exposure). It is the initial kinetic energies that are involved; the eventual fate of the charged particles (i.e. if they do or do not leave the elementary volume), does not affect kerma. In the volume in the figure, the initial kinetic energies of the two electrons labelled e1 contribute to the kerma, as both were generated in the volume. The fact that one of these electrons leaves the volume with a residual kinetic energy T is irrelevant. None of the kinetic energy of the electron entering the volume with kinetic energy T contributes to kerma as this electron was generated outside the volume.

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Figure 2.2. Illustration of the concepts of kerma and exposure

It is important to realise that kerma includes the energy that the charged particles will eventually re-radiate in the form of bremsstrahlung photons. Kerma can be partitioned as follows (Attix 1979):

(2.7)

where c refers to collision losses and r to radiation losses. The collision kerma Kc is related to the (total) kerma by

(2.8)

The quantity g is the fraction of the initial kinetic energy of the electrons that is re-radiated as bremsstrahlung (in the particular medium of interest).

Exposure and air kerma can be related to each other as follows. Multiplying the charge dQ by the mean energy required to produce one ion pair, divided by the electron charge, i.e. W/e, yields the collision part of the energy transferred, i.e. and therefore

(2.9)

or

(2.10)

Particle Fluence

To calculate absorbed dose we require quantities that describe the radiation field; these are known as 'field quantities' (Greening 1981). Particle fluence is a very important basic quantity, involving the number of particles per unit area. The concept is illustrated in Figure 2.3.

C:\Users\Turkay\Desktop\Figures\Fig2.3.png

Figure 2.3. Characterization of the radiation field at a point P in terms of the radiation traversing a sphere centred at P

Let N be the expectation value of the number of particles striking a finite sphere surrounding point P (during a finite time interval). If the sphere is reduced to an infinitesimal one at P with a cross sectional area of dA, then the fluence Φ is given by

(2.11)

which is usually expressed in units of m-2 or cm-2 (ICRU 1980, 1998). Fluence is a scalar quantity-the direction of the radiation is not taken into account.

We will be meeting fluence, differential in energy, often written as ΦE:

(2.12)

in which case the (total) fluence is given by:

(2.13)

It should be noted that fluence can also be expressed as the quotient of the sum of the track lengths Δs of the particles crossing the elementary sphere and the volume of the sphere (Chilton 1978):

(2.14)

This form is extremely useful when considering so-called cavity integrals which involve evaluating the fluence averaged over a volume.

Energy Fluence

Energy fluence is simply the product of fluence and particle energy. Let R be the expectation value of the total energy (excluding rest mass energy) carried by all the N particles in Figure 6.4. Then the energy fluence Ψ is given by (ICRU 1980, 1998)

(2.15)

If only a single energy E of particles is present, then and .

In a similar fashion, one can define the above quantities per unit of time, i.e. fluence rate and energy fluence rate.

Planar Fluence

Planar fluence is the number of particles crossing a fixed plane in either direction (i.e., summed by scalar addition) per unit area of the plane. One can also define a vector quantity corresponding to net flow but this is of little use in dosimetry because it requires scalar, not vector, addition of the effects of individual particles.

Planar fluence is a particularly useful concept when dealing with beams of charged particles. In certain situations, e.g. at small depths in a parallel electron beam, one can say that the planar fluence remains constant as the depth increases; whereas, fluence generally increases due to the change in direction of the electron tracks. The difference between the two quantities is illustrated in Figure 2.4.

C:\Users\Turkay\Desktop\Figures\Fig2.4.png

Figure 2.4. Schematic 2D illustration of the concept of planar fluence. The number of particles crossing the two horizontal lines (of equal length) is the same. This shows that the planar fluence in the original beam direction remains constant, whereas the fluence, illustrated by the total track length in the two circles, is much greater on the downstream side of the scattering medium.

Relations between Fluence and Dosimetric Quantities for Photons

Relation between Fluence and Kerma

Consider the schematic Figure 2.5 showing N photons, each of energy E, crossing perpendicularly a thin layer (of material med) of thickness dl and area dA. To extract energy from particle tracks and transfer it to the medium, we require an interaction coefficient. We can use the ICRU (1980, 1998) definition of the mass energy-transfer coefficient μtr/ρ:

(2.16)

C:\Users\Turkay\Desktop\Figures\Fig2.5.png

Figure 2.5. Illustration of N radiation tracks (photons) of energy E crossing a thin layer of material med, thickness dl, area dA, mass dm, density ρ

Identifying the fraction of incident radiant energy dRtr/R as dEtr/(NE), and making a simple rearrangement, we have

(2.17)

Then, dividing both sides by the mass of the layer dm and rearranging again:

(2.18)

Replacing dm by ρdV we can write:

(2.19)

where we recognise the left hand side as kerma (in medium med) and the expression in the square brackets as the sum of the track lengths divided by the volume, resulting in

(2.20)

or in terms of energy fluence Ψ:

(2.21)

Note that the perpendicular incidence in Figure 2.5 was assumed for simplicity; Equation 2.20 and Equation 2.21 are valid for arbitrary angles of incidence.

Up to this point, we have confined ourselves to particles crossing the thin layer with a single energy. In the more practical case of a spectrum of energies, described by the fluence differential in energy, ΦE, we evaluate the Kmed from:

(2.22)

where the energy dependence of (μtr/ρ)med has been shown explicitly. Therefore, we have now arrived at a relationship connecting kerma and fluence for photons.

It will also be necessary to calculate the collision kerma, Kc from photon fluence. To do this, one replaces the mass energy transfer coefficient μtr/ρ by the mass energy absorption coefficient μen/ρ, where energy absorbed is defined to exclude that part of the initial kinetic energy of charged particles converted to bremsstrahlung photons. The two coefficients are related by:

(2.23)

which is naturally the same factor that relates Kc and K. It therefore follows that:

(2.24)

and, similarly, for the integral over ΦE in the case of a spectrum of incident photons:

(2.25)

Relation between Kerma and Absorbed Dose

Having established a relationship between kerma and fluence in the previous section, if absorbed dose can be related to kerma then a relationship will finally be established between absorbed dose and fluence for photons. However, the absorbed dose Dmed in medium med concerns the (mean) value of energy imparted to an elementary volume, whereas kerma concerns energy transferred as the charged particles can leave the elementary volume (or thin layer), taking a fraction of the initial kinetic energy with them. This is illustrated in Figure 2.6. Note that the quantity denoted in the figure by is the net energy transferred and excludes that part of the initial kinetic energy converted into bremsstrahlung photons. It is equal to as we have seen above.

In Figure 6.7, let the energy imparted to the layer be denoted by 3, the (net) kinetic energy leaving the layer be denoted by and the (net) kinetic energy entering the layer on charged particles be denoted by . Then, from Equation 6.2, we have:

(2.26)

If now the electron track that leaves the layer is replaced by an identical track that enters the layer we can write:

(2.27)

C:\Users\Turkay\Desktop\Figures\Fig2.6.png

Figure 2.6. Schematic illustration of how secondary electrons created by photons can transfer (kinetic) energy to a thin layer, leave that layer, and also enter the layer from outside. If the energy leaving the layer () is exactly replaced by energy entering (), then charged particle equilibrium is said to exist and absorbed dose can be equated to collision kerma. This is schematically represented on the figure, where the full length of the track of a secondary electron is exactly equal to the partial tracks of electrons (in bold) within the layer limits.

It then follows that:

(2.28)

The equality between energy leaving and energy entering on charged particles is known as charged particle equilibrium (CPE). This can be realised under certain circumstances.

Dividing both sides of Equation 6.27 by the mass of the layer or volume element, and

changing from stochastic to average quantities, we can write:

(2.29)

which is a very important result: under the special condition of charged particle equilibrium, the absorbed dose is equal to the collision kerma.

Consequently, replacing kerma by absorbed dose in Equation 2.24 and Equation 2.25, for monoenergetic photons it follows that

(2.30)

and, similarly, for the integral over ΦE in the case of a spectrum of incident photons:

(2.31)

Equation 6.30 and Equation 6.31 are very important relationships in radiation dosimetry.

Charged Particle Equilibrium

Charged particle equilibrium (CPE), also known as electronic equilibrium, is said to exist in a volume V in an irradiated medium if each charged particle of a given type and energy leaving V is replaced by an identical particle of the same energy entering V.

There are, however, many situations where there is a lack of CPE. Strictly speaking, it is impossible to divide the lack of CPE into separate components (i.e. longitudinal or lateral disequilibrium). However, this distinction may be useful for a better understanding, especially for high energy photons where the secondary electrons are primarily peaked forward.

Figure 2.7 illustrates schematically how charged particle equilibrium can actually be achieved in a photon beam. Naturally, the figure is a huge over-simplication as in practice there will be a whole spectrum of secondary electron energies and directions. However, the arguments are not essentially altered by showing only one electron starting in each voxel, (labelled A to G in the figure) and travelling in a straight line. In each voxel, one electron is generated and therefore, the kerma will be constant as it is assumed here that there is no or negligible photon attenuation. Only a fraction of the electron track deposits energy in voxel A; therefore, the dose is low and there is clearly no replacement for the part of the track that leaves the volume. In voxel B a new electron starts but here there is also part of the electron track which started upstream in voxel A. Hence the dose is higher than in voxel A; in voxel C the dose is higher still. However, in voxel D, where the electron which started in voxel A comes to rest, all the sections of an electron track are present. This means that the sum of the kinetic energies leaving this volume must be exactly balanced by the sum of the kinetic energies entering and remaining in the volume, i.e. CPE is first attained in D. Subsequent voxels (E, F, G, etc.) will contain patterns of electron tracks identical to those in D and, therefore, CPE must also apply in these volumes. At the depth of voxel D, the absorbed dose is now equal to kerma (strictly this ought to be collision kerma) and this will also be the case in the subsequent voxels E, F, G, etc., in the absence of photon attenuation.

C:\Users\Turkay\Desktop\Figures\Fig2.7.png

Figure 2.7. Diagram showing the build up to charged particle equilibrium for the idealised case of no attenuation of the photon beam and one straight electron track generated in each slab labelled A to G

In the situation of a photon beam true CPE is strictly impossible to achieve in practice. Attenuation means that the photon fluence does not remain constant and, therefore, the number of secondary particles (electrons) starting at different depths also cannot be constant. Table 2.1 illustrates the degree of photon attenuation in water thicknesses ensuring transient electronic equilibrium for photon beams of different energies. The degree of CPE failure increases as the photon energy increases. Consequently, experimental determination of exposure (now air kerma) is not attempted at maximum photon energies above approximately 3 MeV. Even below this energy, small corrections have to be made for the effect of photon attenuation.

Table 2.1. Approximate thickness of water required to establish transient charged particle equilibrium. For bremsstrahlung beams of different maximum energies; the final column gives approximately the attenuation of photons in that thickness of water.

Maximum Energy of Photons (MeV)

Approximate Thickness of Water for Equilibrium (mm)

Approximate Photon Attenuation (%)

0.3

0.1

0.03

0.6

0.4

0.1

1

0.8

0.3

2

2.5

0.8

3

8

2

6

15

4

8

25

6

10

30

7

15

50

9

20

60

11

30

80

13

Even though strict CPE may not exist, in many situations it is very well approximated, such as at depths beyond the dose maximum in media irradiated by photons below around 1 MeV in energy. At higher energies, the equals sign in Equation 2.29 can be replaced by a proportionality sign. This is then termed transient charged particle equilibrium (TCPE):

(2.32)

The quantities K, Kc and dose D are plotted as a function of depth in Figure 6.8. The build-up region and the region of TCPE, where the D and Kc curves become parallel to each other are illustrated.

If radiative interactions and scattered photons are ignored it can be shown (Greening

1981) that:

(2.33)

where μ is the common slope of the D, K and Kc curves and is the mean distance the secondary charged particles carry their energy in the direction of the primary rays while depositing it as dose (Attix 1986). This constant of proportionality between dose and collision kerma is usually denoted by β, i.e.

(2.34)

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Figure 2.8. Variation of kerma, K, collision kerma, Kc, and dose, D, with depth in a beam of indirectly ionising radiation such as a photon beam

Another situation where the dose is different from the collision kerma because CPE is not achieved is for beams with very small cross sections. Figure 2.9 and Figure 2.10 show the results from a Monte-Carlo simulation (using the EGSnrc code) of depth-dose variation for a cylindrical cobalt-60 γ-ray beam (energies 1.17 and 1.33 MeV, equally weighted) incident perpendicularly on the end of a cylinder of water with radius 20 cm and length 50 cm. The dose has been scored along the central axis in cylindrical columns of increasing radii, r, with 1 mm depth intervals for the first centimetre then 0.5 cm down to

10 cm depth (the results at greater depths are not shown).

C:\Users\Turkay\Desktop\Figures\Fig2.9.png

Figure 2.9. Cobalt-60 γ-ray beam in water incident on a water cylinder (radius 20 cm, length 50 cm): depth-dose curves obtained from Monte-Carlo simulation (108 photon histories for each case) when the equivalent beam radius, r, is 0.05 cm. The upper curve, with no transport of secondary electrons, corresponds to the kerma. The lower curve includes full electron transport and shows clearly the lack of CPE.

In Figure 2.9 the radius of the scoring volume is very small, 0.05 cm. The upper curve corresponds to no electron transport (which is achieved by setting the electron transport cutoff to 0.5 MeV); thus the quantity scored corresponds exactly to (water) kerma. Upon switching electron transport back on (by simply reducing the electron transport cutoff to a suitably low value: 50 keV was chosen) the buildup of dose in the first 3 mm, due to electron transport, can now be observed. At the same time, however, the absolute value of the dose has fallen by more than a factor 2. The radius of only half a millimetre is in fact equal to only a fraction of the ranges of the highest energy secondary electrons created (virtually 100% of these will be due to Compton interaction) which is insufficient for the establishment of CPE or, put another way, the electrons will predominantly leave the scoring regions without being balanced by an equal number entering them, as the beam is so narrow. The absorbed dose is consequently much lower than the kerma in this very special geometry.

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Figure 2.10. Depth-dose curves obtained for cobalt-60 γ-ray beams from Monte-Carlo simulation (108 photon histories) with equivalent radii, r, of 0.5 and 10.0 cm incident on a water cylinder (radius 20 cm, length 50 cm) with full electron transport (ECUT-KE=50 keV). The increased dose and reduced effective attenuation for the larger radius is due to photon (Compton) scatter.

In Figure 2.10, the equivalent beam radii, r, are 0.5 cm and 10.0 cm, respectively. The depth-dose curve for the larger radius lies considerably above that for the smaller one. The latter was chosen to be large enough for the establishment of (partial) CPE. The reduced effective attenuation at the larger beam radius is explained by an increasing build-up of Compton-scattered photons with depth. The dependence of depth-dose curves on field size is a well-known and important phenomenon in external-beam radiotherapy.

It must be stressed that for large collimated beams where CPE is achieved on the beam axis, it is still partially lost at the edge, which contributes to enlarging the penumbra region.

Relation between Fluence and Dose for Electrons

Stopping Power and Cema

Referring back to Figure 2.5, consider that we now have N electron tracks incident perpendicularly on the thin layer of medium med and thickness dl, etc. Instead of the mass energy-transfer coefficient for photons, for charged particles the quantity of relevance is stopping power, the energy lost per unit track length and we denote this energy by dEl to distinguish it from dEtr, which is used for indirectly ionising radiation. We are interested in energy locally deposited in the thin layer so it is clearly appropriate to employ the collision stopping power, Scol, rather than the total stopping power as the latter would include the energy lost in the form of bremsstrahlung that would escape the thin layer (this is analogous to the difference between kerma and collision kerma). Thus we can write

(2.35)

Note that, unlike the analogous Equation 2.17 for photons, we do not need the energy of the particles. Dividing both sides by the mass of the layer dm and expressing this as ρdV on the right hand side, we obtain

(2.36)

which can be rearranged to give:

(2.37)

where, as in the case of indirectly ionising radiation, the quantity in the square brackets is the fluence Φ and therefore:

(2.38)

Until recently, there was no equivalent of kerma for the case of charged particles. However, the quantity cema, converted energy per unit mass, was proposed by Kellerer et al. (1992). ICRU (1998) defines cema as the energy lost by charged particles, excluding secondary electrons, in electronic collisions in a mass dm of a material. 'Secondary electrons' refer to the delta rays generated by the incident primary electrons, and their kinetic energies have already been included in dEl. Therefore, cema is equal to dEl/dm and, consequently, to the product of electron fluence and mass collision stopping power.

Cema is not necessarily equal to absorbed dose, as some of the delta rays can leave the thin layer, just as secondary electrons can do in the case of the primary radiation being photons (Figure 2.6). To involve absorbed dose, it must follow that any charged particle kinetic energy leaving the thin layer or elementary volume is replaced by an exactly equal amount entering the layer and being deposited in it or imparted to it. Consequently, it must be assumed that there is delta-ray equilibrium in order to be able to equate cema with absorbed dose and therefore we can write, for a medium m:

(2.39)

or, in the case of polyenergetic electron radiation:

(2.40)

where ΦE is the fluence, differential in energy.

Delta-Ray Equilibrium

Naturally, delta-ray equilibrium must always exist if charged particle equilibrium exists. However, where the primary radiation consists of charged particles CPE can never be achieved except in the rather special case of uniformly distributed b sources in a large medium. For the beams of high-energy electrons used in radiotherapy, the energy of the primary electrons decreases continuously with depth and hence there cannot be equilibrium. However, the ranges of the delta rays are predominantly extremely short and almost all the energy transferred through collision losses, i.e. the cema, is deposited locally. One need only look at how close the ratio LΔ/Scol is to unity for very small values of Δ to be convinced of the above (Figure 3.8), i.e. most of the collisions result in very small energy losses and electrons with these low energies have extremely short ranges. Thus, deltaray equilibrium is generally fulfilled to a high degree in media irradiated by electron beams.

One situation where delta-ray equilibrium is definitely not a good approximation, however, is very close to the phantom surface in an electron beam. The appreciable range in the forward direction of the most energetic delta rays results in a small but discernible delta-ray build up (see Figure 3.16).

Cavity Theory

When a measurement is made with a detector, the detector material will, in general, differ from that of the medium into which it is introduced. The signal from a radiation detector will generally be proportional to the energy absorbed in its sensitive material and thus to the absorbed dose in this material, Ddet.

The detector can be thought of as a cavity introduced into the uniform medium of interest; this name stems from the fact that gas-filled ionisation chambers dominated the development of the subject (Greening 1981) and the associated theory, which relates Ddet to Dmed, is known as cavity theory. In its most general form, the aim of cavity theory is to determine the factor fQ given by

(2.41)

for an arbitrary detector 'det', in an arbitrary medium 'med', and in an arbitrary radiation quality Q (photons or electrons). Figure 2.11 illustrates the situation schematically.

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Figure 2.11. The general situation

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