Atomic and Nuclear Structure, Particles, and Nomenclature

Matter is made up of atomic and nuclear particles that have interaction and binding energies from 1 to 10 million electron volts (MeV). The smallest structural subunit that retains the character of an element, the atom, has a nucleus consisting of one or more protons and zero or more neutrons and a surrounding cloud of orbiting electrons. In an atom with zero net electrical charge, the number of orbital electrons equals the number of protons. A simplistic representation shows the atom to be neatly arranged with neutrons and protons in the nucleus and electrons in uniform orbits (Fig. 6-1). In reality, the atom is a dynamic structure with defined energy states for the electron orbits and the nucleus. Detailed atomic and nuclear models have been formalized.^1,2

Figure 6-1 The atom has a nucleus composed of protons and neutrons. The nucleus is surrounded by a cloud of electrons in distinct energy levels called orbitals named s, p, d, and so on by chemists or called shells named k, l, m, and so on by physicists.

Neutrons and protons are the building blocks of the nucleus; hence, they are called nucleons. The two particles have similar rest masses (the mass of the particle at rest, when kinetic energy is zero) but different electrical charges. The neutron, symbolized by n, has no charge (neutron for neutral) and the proton, p, has a charge of +1. The electron, e, has an atomic mass about that of a proton or neutron and has a charge of −1. Besides the neutron, proton, and electron, other particles exist with unique masses and properties (e.g., spin). These include neutrinos, pions, muons, and others. The neutrino was first proposed as the neutral accompanying particle emitted in beta decay. There has been great interest in determining the mass of the neutrino because the cumulative neutrino mass may account for the “missing mass” in the Universe, with great cosmologic implications. More recently it has been determined that the neutrino mass, inferred through observed oscillations in the neutrino state,^3,4 is finite and also very small (close to zero). However, even with nonzero mass the total amount of neutrino mass is estimated to be insufficient to make up for the “missing” mass. The Universe is now stated to be made up of matter (4.6% the material “stuff” we are familiar with), dark matter (23%), and dark energy (which comprises over 70% of the total mass-energy content of the Universe).⁵ These “dark” components are unknown entities.

Particles are classified by their mass as leptons or hadrons. Leptons, which include the electron and neutrino, are “lightweight” particles with mass comparable to that of the electron and spin of . Hadrons are heavy particles with two subclasses called mesons (middleweight, spin 0 or 1) and baryons (heavyweight, spin or ). Fundamental, or elementary, particles are those that have no subparts and cannot be divided. All leptons are elementary particles; however, the neutron and proton are not and are instead made up of three fundamental particles each, called quarks, which have been observed through high-energy physics experiments. Quarks have positive or negative charge in integer increments of , spin of , and other properties with whimsical, quark-deserving names.¹ Combinations of quarks yield the neutron, proton, and all other hadrons. For instance, a proton is made of two up quarks and one down quark, whereas a neutron is made of one up and two down quarks, which explains their similar mass but difference in charge. Antimatter is real, and an antiparticle is defined as a particle with identical mass but opposite charge to the particle. An antiparticle for a neutral particle has opposite spin or internal charge (i.e., quark) compared with the particle.

Matter is held together by four fundamental forces that operate over certain ranges and with certain particles. They are the strong, electromagnetic (coulomb [C]), weak, and gravitational forces, with respective relative strengths of 10¹, 10⁻², 10⁻¹³, and 10⁻⁴².¹ Each force acts by exchange of its respective mediator particle: the gluon, the photon, the W and Z particles, and the graviton. Counting mediators, leptons, hadrons, and their antiparticles, there are about 170 fundamental and composite particles that make up matter.¹ The model in which matter consists of fundamental particles as described earlier is called the standard model.¹ Table 6-1 summarizes the fundamental particles in the standard model, which are six quarks, six leptons, six antiquarks, six antileptons, and also the four known forces. A relatively new theory called string theory is a descriptor of how quarks and other fundamental particles are assembled with vibration (energy) states for stringlike entities that give each fundamental particle its unique character.⁶ However “flexible” they may be, the strings still need additional dimensions called “branes” (after “membranes”) to enable their full character to be described.^7,8 Clearly, the fundamental properties of the Universe continue to be studied in basic physics research.

Particle mass can be expressed as the atomic mass unit (amu; of the mass of the carbon nucleus) or in units of energy by conversion with Einstein’s formula, E = mc². Table 6-2 gives the symbol, charge, mass, and stability for the electron, proton, neutron, and other particles of interest.

Combinations of nucleons form a variety of nuclei and determine the physical character of an atom. The number of protons in the nucleus, Z, is called the atomic number and determines the chemical properties of an atom and the atom’s identity as an element. The atomic number also equals the number of electrons in the neutral atom, with one electron per proton. The number of neutrons in the nucleus, N, is called the neutron number. Whole protons and neutrons constitute the nucleus, and Z and N have integer values. An atom’s mass number, A, is the sum of its neutrons and protons. The mass number (an integer) has a value close but not equal to the actual nuclear mass. Their values are similar but must not be confused. Nuclear mass is the noninteger sum of the masses of the individual particles minus their binding energies. Definitions for Z, N, and A are summarized in Table 6-3.

Nuclides and Radionuclides

Nuclides are atomic species made of different combinations of nucleons, and they may be classified by their number of protons, neutrons, or nucleons (Z, N, or A) and by their energy state. Nuclei with the same Z but different N are called isotopes (p for proton), and they exhibit identical chemical characteristics (they are the same elements). Nuclei with the same N but different Z are called isotones (n for neutron). Nuclei with the same A but different Z and N are called isobars. In a last category, nuclei with the same Z and N, and therefore A, but different nuclear energy states (i.e., excited vs. ground) are called isomers. Table 6-3 shows these classifications and example nuclides. A nuclide, or nuclear species, X, is denoted as

in which X is the chemical symbol for the element with atomic number Z, N is the neutron number, and A is the mass number. Because A = Z + N, the N value is often dropped to give the following form:

Because the atomic number determines the element’s name, represented by the chemical symbol X, the Z is also dropped to give the form ^AX. An alternative nomenclature uses the nuclide’s name followed by the mass number, such as hydrogen-3 and iridium-192.

Not all combinations of Z and N exist in nature or can be manufactured. Instead, certain combinations are possible, whereas other combinations cannot occur. Figure 6-2 shows the distribution of stable nuclides as a function of the number of neutrons and protons. Notice that at low Z, the ratio of neutrons to protons (N : Z) is about 1.0. Above Z = 20, stable nuclides have more neutrons than protons (N : Z > 1.0). At higher Zs, the stability of nuclei tends toward neutron-rich nuclides. It has been observed that nuclei with 2, 8, 20, 28, 50, 82, or 126 nucleons (protons and neutrons combined) are stable. These stability “magic numbers” relate to the filling of nuclear energy levels, similar to the complete filling of electron shells. Pairing of like nucleons also results in increased nuclear stability. There are 165 stable nuclei with an even number of both protons and neutrons, 57 stable nuclei with an even number of protons and odd number of neutrons, 53 stable nuclei with an odd number of protons and even number of neutrons, but only 6 stable nuclei with an odd number of both protons and neutrons.

Figure 6-2 Distribution of stable and naturally radioactive nuclides.

Data from Radiological Health Handbook. Bethesda, Md., U.S. Department of Health, Education, and Welfare, 1970.

Some nuclides are unstable and eventually transform to stable states by the emission of particles or energy. These nuclides are called radionuclides or radioactive species, because a particle or energy is given off during the nuclear transition. Unstable nuclides lie off the line of stability and will have a neutron or proton excess relative to the stable nuclide. In Figure 6-2, isotopes lie along a vertical axis, isotones lie along a horizontal axis, and isobars lie at 45 degrees to either axis, perpendicular to the line of N = Z. Modes of radioactive decay depend on the type of nucleon excess, whether neutron or proton, and are discussed later in more detail.

Stable nuclides include ¹H, ¹²C, ³³P, ³⁴S, and ⁵⁹Co. Radioactive nuclides for these elements include ³H, ¹⁴C, ³²P, ³⁵S, and ⁶⁰Co. Other radionuclides of interest to the medical field include ^99mTc, ¹²⁵I, ¹³¹I, ¹³⁷Cs, and ²²⁶Ra.

Photons and Other Definitions

Interactions of ionizing radiation occur at the atomic and nuclear level, where binding energies are small compared with macroscopic realms. Electromagnetic radiation, or photons, are particles that have wavelike qualities with zero mass, which transfer energy from one location to another by propagation of an electromagnetic wave at the speed of light, c (c = 3 × 10⁸ m/s). Photons are also the mediators of charged particle bonds. A photon has wavelength λ, frequency ν, speed c = λν, and energy E = hν; c is the speed of light, and h is Planck’s constant (h = 6.626 × 10⁻³⁴ m² kg/s). The electromagnetic spectrum consists of photons with wavelength, frequency, and energy ranges over 10 orders of magnitude (the range is really infinite). From low to high energy, there are radar waves, microwaves, infrared, light (visible photons), ultraviolet, x rays, and gamma rays. Photons are named by their wavelength (e.g., radar waves, microwaves), character (e.g., “purple”), and origins (e.g., x rays from the atom, gamma rays from the nucleus) (Table 6-4).

TABLE 6-4 Physical Definitions

Radiation Production by Radioactive Decay

Teletherapy is the use of radioactive material for production of an external beam of gamma rays for treatment at a distance from the radioactive source (tele, meaning “at a distance”). The term is historical and is in contrast to brachytherapy, where the radioactive source is placed in or on the treatment volume (brachy, meaning “close”). Gamma rays are emitted from a daughter nucleus formed after radioactive decay of an unstable parent nucleus. Each gamma ray has a unique energy that relates to the immediately preceding nuclear transformation, and this unique energy can be used to identify the daughter (and therefore the parent). ²²⁶Ra, ¹³⁷Cs, and most commonly ⁶⁰Co have been used for teletherapy. ¹³⁷Cs and ⁶⁰Co are manufactured and became available by neutron activation and as a by-product of fission after the invention of the nuclear reactor. Use of ⁶⁰Co as a source of gamma rays for treatment was pioneered by H.E. Johns⁹ and represented a major step in obtaining high-energy photons above 1 MeV. At that time, electronic means of photon production from high-energy x-ray tubes was limited to 300 keV maximum because of electrical arcing at higher accelerating potentials. Particle accelerators were required to produce potentials above 300 keV (e.g., betatrons¹⁰ and van de Graaff accelerators¹¹).

Although of historical interest for most readers, ⁶⁰Co teletherapy units have lower cost and a relatively simple design with lower requirements for their operating environment compared with linear accelerators. For these reasons ⁶⁰Co teletherapy devices are used worldwide in regions with limited resources for funding and operating infrastructure. A disadvantage is the constantly decreasing dose rate as a result of the decay of the ⁶⁰Co source and the requirement for eventual replacement of the source. In a conventional ⁶⁰Co teletherapy unit (Fig. 6-4), a cylindrical sealed-source capsule about 3 cm in diameter and 5 cm high contains pellets of ⁶⁰Co. In each transformation a ⁶⁰Co nucleus decays to ⁶⁰Ni, with the prompt emission of two gamma rays at 1.17 and 1.33 MeV each (1.25 MeV average). Typical activity is 9000 Ci (3.33 × 10¹⁴ Bq) for a dose rate of approximately 3 Gy/min at 80 cm from the source (2 Gy/min at 100 cm from the source). The source decays with a half-life of 5.263 years and is replaced every 5 to 7 years when the dose rate becomes “too low.”

Figure 6-4 In a typical ⁶⁰Co unit, the source moves from the shielded position (Off) into an unshielded position (On) to produce a treatment beam.

B, Courtesy Best Theratronics, Ltd., Ottawa, Canada.

In one of the most commonly available teletherapy unit configurations (Theratron Phoenix, Best Theratronics Ltd, Ottawa, Ontario, Canada) the source is stored in a shielded head of the machine, mounted on the end of a movable piston in a horizontal cylinder (see Fig. 6-4). On the initiation of treatment, the source is moved pneumatically to a position over an opening in the shield that allows a treatment beam to exit. A collimator consisting of interleaved bars of a high Z material is used to define the field size as the beam exits the shield port. Trimmer bars can be used to reduce the beam penumbra, which is large because of the relatively large source size of approximately 1.5 inches. Maximum field size is 35 × 35 cm² at 80 cm from the source. Irradiation time is measured and controlled by two independent timers. An end effect caused by the mechanical movement of the source, for which the effective irradiation time is less than the timer setting, is inherent and can be measured. Cross-hairs and a field light are used to delineate the central ray and field dimensions. There is a source-to-surface indicator. Source movement is designed so that in the event of treatment termination or device failure the source is automatically returned to the shielded condition. An emergency pushbar (T-bar) can be used to manually return the source to the shielded position, if necessary.

The machine has a rotatable gantry allowing 360-degree rotation of the source and an isocenter position of 80 cm from the source. Later models have a 100-cm isocenter. An additional degree of freedom is provided by a head swivel mechanism that allows the beam direction to be rotated away from isocenter, if desired. A beam stopper may be used to intercept the beam for additional shielding of the exit beam. The beam stop also acts as a counterweight for the head of the machine. There is a patient support assembly (treatment table) with vertical, longitudinal, lateral, and rotation motions. Beam modifiers include custom or standard field blocks and mechanical wedges for producing angled isodose distributions or tissue compensation.

The manufacturer (Best Theratronics, Ltd., Ottawa, Canada) states that as of the year 2010 there were 45,000 daily radiation treatments delivered worldwide using this conventional ⁶⁰Co teletherapy device.

Radiation Production by Linear Accelerators

In a linear accelerator, electrons are accelerated to high energy and are allowed to exit the machine as an electron beam or are directed into a high Z target to produce x rays by the bremsstrahlung interaction. The linear accelerator enables convenient production of megavoltage x rays in a relatively small device, and its existence is directly related to the invention of the magnetron and the klystron during the development of radar in World War II. Linear accelerators are quite versatile, with x-ray and electron modes, multiple energies, and computer controls. These capabilities have led to the replacement of most ⁶⁰Co teletherapy machines in the United States with linear accelerators.

The principle of operation for linear accelerators is to accelerate electrons down a waveguide by use of alternating microwave fields.¹² Two basic waveguide designs exist: standing wave and traveling wave. Waveguide length is a function of the maximum acceleration energy (longer is higher energy) and the frequency of the microwave field. The most common frequency for gantry-based linear accelerators (see later) is 2.998 GHz (S-band microwaves); however, specialized, shorter waveguides are possible at 9.3 GHz (X-band microwaves) that produce megavoltage energies above 6 MeV with acceptable dose rates for treatment. Major electronic components are described in Table 6-5 and are shown in Figure 6-5 for gantry-based linear accelerators, the most common design.

TABLE 6-5 Major Electronic Components of Linear Accelerators

* Components are shown in Figure 6-5.

† Components are not shown in Figure 6-5.

Figure 6-5 Electron linear accelerator schematic. Key components are shown that allow a beam of electrons to be accelerated, producing a treatment beam of electrons or x rays.

B, Courtesy Varian Medical Systems.

Major mechanical components are similar to those for teletherapy. A rigid, floor-mounted stand holds high voltage, microwave transport guides, cooling and control components, and supports a rotatable gantry holding the waveguide to allow 360-degree rotation of the source at an isocenter of 100 cm to enable multiple beam directions (see Fig. 6-3). In an alternative barrel-type design the entire accelerator—including waveguide, high voltage, microwave transport guides, and other components—is contained within a large rigid cylinder and the cylinder and all its contents rotate on a horizontal axis about the isocenter (this design not shown). A set of one or two pairs of high Z collimators (“jaws”), depending on the manufacturer’s design, provides at least 99.9% attenuation of the primary beam (0.1% transmission) and defines the length and/or width of the rectangular x-ray radiation field. A maximum field size of 40 × 40 cm² at 100 cm (isocenter) is common, with 180-degree rotation of the entire collimator assembly about the isocenter. Collimator settings are continuously variable, and jaw pairs can be operated in coupled mode to produce symmetric, rectangular fields or, in most modern machines, independently to produce asymmetric fields. Asymmetric operation may include travel of one jaw across the central axis. Cross-hairs and a field light are used to delineate the central ray and field dimensions, and there is a source-to-surface indicator.

Although rarely used for linear accelerators, a beam stopper may be used to intercept the beam for additional shielding of the exit beam when facility shielding is limited. Otherwise, internal counterweights provide balance to the gantry to offset the weight of the accelerator waveguide and high-density shielding required around the x-ray target. There is an isocentric patient support assembly (treatment table) with vertical, longitudinal, lateral, and rotational motions. Fine tolerances for the rotational axes and isocentricity of the gantry, collimator, and table are important for accuracy in patient treatments. The combined variation in rotation and coincidence of the three axes is typically maintained within a 1-mm diameter sphere. The patient support assembly is usually the most difficult rotational axis to set for isocentricity.

Although linear accelerators typically have had an isocentric rotating gantry, there are two other megavoltage accelerator designs available. One design, called tomotherapy, uses a highly stable CT-like ring gantry around which an X-band linear accelerator rotates, always pointed toward the rotational center (an isocenter).¹³ A binary (“on” or “off”) collimator intensity modulates a 6-MV x-ray fan beam as the accelerator waveguide is rotated. The treatment table is advanced in a stepwise or continuous motion for either slice-by-slice or helical fan beam treatment. Another approach uses a non-isocentric format with an X-band linear accelerator fixed on the end of a high-precision robot that is computer controlled for a range of motions with multiple degrees of freedom.¹⁴ This device produces a circular 6-MV x-ray beam that can be pointed at the target from various “nodes” in space—using a relatively large collection of irradiation nodes generates the cumulative dose distribution at the target.

Accessories and beam modifiers for conventional geometry linear accelerators are important for customizing the external beam field to an individual patient and include custom or standard shielding blocks to shape a patient’s treatment fields to minimize the amount of normal tissue treated or protect critical structures; electron applicators for defining electron fields at the patient surface; physical and “virtual” or “dynamic” wedges for producing angled isodose distributions; and compensators for shaping the dose distribution within a patient for desired dose uniformity. Custom block manufacturing for shaping of patient-specific fields was once very common, using high Z alloy materials to provide primary beam attenuation of at least 97% (five half-valve layers [HVLs]). The use of custom blocks has been almost totally replaced by a versatile collimator design called the multileaf collimator (MLC), which is now available from all linear accelerator manufacturers (Fig. 6-6). The MLC provides field-shaping capabilities, replacing standard hand or custom blocks by the use of a large number of adjustable high Z vanes, or leaves. The desired beam outline is shaped as a series of steps by positioning each leaf to approximate a continuous field edge (see Fig. 6-6B). Different manufacturer’s MLC designs include singly and doubly focused leaf ends, where the leaf ends match beam divergence either across the direction of leaf travel (single) or both across and with the direction of leaf travel (double), and projected widths at the isocenter of 3 mm, 4 mm, 5 mm, or 10 mm.¹⁵ MLCs also have capabilities for dynamic treatment, as discussed later for intensity-modulated radiation therapy (IMRT).

Figure 6-6 The multileaf collimator. A, Installed 80-leaf device for a Varian 2100C linear accelerator. B, Schematic of beam definition.

Linear accelerators and their supporting technologies have become very computer controlled, enabling better-optimized dose distributions, rapid treatment setup, and in-room verification of target location through remote sensing and image-guided approaches. In particular, most modern linear accelerators are hybrid imaging treatment devices fitted with a gantry-mounted kV or MV x-ray imaging device that is coincident with the treatment isocenter. This imaging device provides “in-room” imaging to confirm patient and target position immediately before treatment, in the process termed image-guided radiation therapy (IGRT). A wide range of IGRT designs is possible, as discussed later.

Three differences in radiation sources for teletherapy and linear accelerator devices are fundamental. First, a ⁶⁰Co source is always “on” in that radioactive decay always occurs; the source must be moved to an unshielded condition to initiate an irradiation (see Fig. 6-4). With a linear accelerator, no source is present until the unit is energized; the irradiation is on or off with the flip of an electronic switch initiated by the operator with a key or button at the treatment console. Second, their photon spectra are different (Fig. 6-7). In a ⁶⁰Co unit, two monoenergetic gamma rays are emitted with each decay to produce a discrete spectrum with peaks at 1.17 and 1.33 MeV. In contrast, a linear accelerator produces a continuous x-ray spectrum through the bremsstrahlung interaction. The spectrum has a maximum energy of E_max (i.e., accelerating potential) and all other photon energies down to zero. An average x-ray energy of approximately one-third E_max can be determined and is consistent with theoretical predictions for the continuous bremsstrahlung spectrum (see Fig. 6-7). Third, a linear accelerator can produce a treatment beam of electrons as well as photons. The accelerated electron beam is allowed to exit the machine under controlled conditions of scatter. Although ⁶⁰Co gamma rays occur because of a priori beta decay, the beta particles (electrons) are stopped by the source capsule and cannot be otherwise harnessed for treatment.

Figure 6-7 Discrete and continuous photon spectra. The discrete spectrum is characteristic of photons emitted by radioactive material. The continuous spectrum is characteristic of bremsstrahlung x-rays emitted by a linear accelerator. E_max, 10 MeV.

Radiation Production by Other Accelerators

Other accelerator techniques have been used to produce a variety of high-energy particles such as electrons, protons, neutrons, and higher Z ions. These techniques include the betatron,¹⁰ van de Graaff accelerators,¹¹ cyclotrons,¹⁶ the racetrack microtron,¹⁷ and accelerators¹⁸ used for high-energy physics experiments.

Among high-energy particles, protons have become the most attractive for therapeutic uses, where proton energies of 230 to 250 MeV yield treatment depths of up to 30 to 40 cm. Although currently a very expensive technology to build ($100 to $150 million) and maintain, the use of high-energy protons for treatment is increasing with 8 proton facilities in the United States and 33 sites worldwide (including a few for carbon ions).¹⁹ Typically, a fixed, single cyclotron has a beam line that is split to service one to four treatment rooms, each with a movable gantry to aim the proton beam. Alternative designs include the use of a smaller (though still large) cyclotron that is mounted on a very large double-legged gantry system that enables rotation about the patient, and a new novel design that uses the principle of dielectric wall acceleration in a relatively compact device (2.0 to 2.5 m length) that would potentially fit within a standard-size treatment room. Facility designs and technical aspects for proton therapy have been discussed.²⁰

Photon Interactions

Attenuation and Transmission

When incident on matter, ionizing photon radiation undergoes interactions with atomic electrons or nuclei. Interacting photons are removed from the primary beam, an effect called attenuation. Photons that do not interact and instead exit the material are called transmitted photons. Attenuation and transmission are illustrated in Figure 6-8. A number, N₀, of monoenergetic x rays or gamma rays is incident to a slab of material, and a smaller number, N, is transmitted. The attenuated photons, N₀ and N, are absorbed in the material or scattered in other directions. In narrow-beam geometry, N transmitted photons alone reach a detector as shown. In broad-beam geometry, scattered photons also reach the detector.

Figure 6-8 Attenuation and transmission. A number, N₀, of mono-energetic x or gamma rays is incident to a slab of material, and a smaller number, N, is transmitted. The attenuated photons, N₀ and N, are absorbed in the material or scattered in another direction.

Megavoltage photon interaction probabilities are less than 1 and vary with incident photon energy and the atomic number of the interaction material, depending on the interaction type. By physical law, the fractional number of unattenuated photons interacting per unit thickness of a material is constant, such that

Eq. 1

In the equation, ΔN is the change in the number of photons as the result of attenuation, N is the number of incident photons, µ is a constant that represents the constant fractional attenuation per thickness (called the linear attenuation coefficient, with units of length⁻¹ [cm⁻¹]), Δx is the thickness traversed by the photons for attenuation ΔN, and the minus sign indicates that the effect is negative, resulting in fewer photons.

Constant fractional attenuation per unit thickness compounds over successive thicknesses, illustrated in Figure 6-9. This “fraction of a fraction” effect is nonlinear, and the integrated form of Equation 1 yields an important relationship: attenuation in a continuous material is an exponential process:

Figure 6-9 Attenuation through successive slabs of material.

Eq. 2

In Equation 2, N₀ is the original number of incident photons, N(x) is the transmitted (unattenuated) number of photons, e is Euler’s constant (e ≈ 2.7), µ is the linear attenuation coefficient, and x is the thickness traversed by the photons.

Exponential attenuation is valid for monoenergetic photons and all homogeneous materials in narrow beam geometry and applies to other radiation quantities such as intensity (I) and exposure (X):

Eq. 3

The linear attenuation coefficient is unique for each photon energy and element or material but varies with absorber density. Attenuation coefficients are discussed more fully with radiation interactions.

Equation 2 can be graphed in linear or semi-log form (Fig. 6-10). In semi-log form, the attenuation curve is a straight line with a slope of −µ (see Fig. 6-10B). A smaller attenuation coefficient means less attenuation and a shallower attenuation curve. Different attenuation coefficients result in a different amount of attenuation for the same thickness traversed and attenuation curves that differ in their slopes (Fig. 6-11). A smaller attenuation coefficient results in less attenuation.

Figure 6-10 Graphic representation of attenuation. A, Linear plot. B, Semi-log plot. HVL, half-value layer.

Figure 6-11 Attenuation for two different materials. A, Linear plot. B, Semi-log plot. HVL, half-value layer.

Beam Quality

Beam quality is a term used to describe the amount of penetration by a photon radiation beam. One indicator of quality is beam energy, defined by accelerating potential, effective energy, or gamma ray energy. Another indicator is the half-value layer (HVL). HVL is the thickness of a material that reduces the transmitted intensity to one half of the original intensity. When I/I₀ = , the thickness, x, is the HVL, and the attenuation equation becomes

Eq. 4

Two important relationships come from this equation:

Given the linear attenuation coefficient, the HVL can be computed, and vice versa. For example, if µ = 0.10 cm⁻¹, the HVL = 0.693/0.10 cm⁻¹ = 6.93 cm, and if HVL = 10 cm, µ = 0.693/10 cm = 0.0693 cm⁻¹. Notice that the units for HVL are the reciprocal of the units for µ.

Because e^−µHVL = , and given a thickness of material in units of HVL (x = n HVL), the amount of attenuation can be computed by

Eq. 5

In Equation 5, I/I₀ is the fractional transmission, intensity, or exposure and n is the thickness of the material in number of HVLs.

Similar to HVL, the tenth-value layer (TVL) is the thickness required to reduce the number, intensity, or exposure by a factor of 10:

Eq. 6

In Equation 6, m is the thickness of the material given in number of TVLs.

HVL can be found graphically from the attenuation curve, whether on a linear or log plot (see Fig. 6-10) and for monoenergetic photons can be determined anywhere along the curve because µ is constant; the ratio of the two intensities must be . For polyenergetic photons, µ is not constant but instead decreases with increasing depth (Fig. 6-12A). The low-energy component of the beam spectrum is attenuated preferentially, compared with high-energy photons, because of increased attenuation at low energies. As depth increases, the ratio of high-energy to low-energy photons increases, resulting in increased beam penetration, or “beam hardening.” After a large number of low-energy photons are attenuated at depth, beam hardening is minimal and µ is essentially constant. The log attenuation curve begins steeply, has curvature, and then becomes linear at depth (see Fig. 6-12B). Because µ changes with depth, HVL also changes, yielding different first and second HVLs, as defined in Figure 6-12B. The monoenergetic case is shown for comparison. HVL₁ is always less than HVL₂ because photons incident on HVL₂ have a higher average energy:

Figure 6-12 Attenuation for polyenergetic photons. A, Variation in attenuation coefficient as a function of depth for polyenergetic photons. B, Attenuation curve for polyenergetic photons. HVL, half-value layer; TVL, tenth-value layer.

The greatest HVL is found at maximum depth, on the linear portion of the curve. For polyenergetic photon beams, an effective attenuation coefficient (µ_eff,) can be calculated. First, an effective HVL is found over a region of interest, and then µ_eff is calculated:

Characteristics of attenuation curves for monoenergetic and polyenergetic photons are summarized in Table 6-6.

TABLE 6-6 Characteristics of Attenuation Curves

Monoenergetic	Polyenergetic
1.Exponential curve on linear paper	1.Complex exponential curve on linear paper
2.Linear curve on semi-log paper	2.Curved on semi-log paper
3.Constant µ	3.Varying µ:µeff can be determined
4.Constant HVL	4.Varying HVL: HVL1, HVL2, TVL, and greatest HVL can be determined

HVL, half-value layer; TVL, tenth-value layer. See text for other terms.

Photon Interactions: X Rays and Gamma Rays

Attenuation of photon beams is the result of interactions in the intercepting material. There are five possible photon interactions: coherent scattering, photoelectric effect, Compton effect, pair production, and photodisintegration. Each interaction type has an independent interaction probability and contributes to the amount of attenuation. The total linear and mass attenuation coefficients are given by the sum of their components:

and

In the previous equations, TOT signifies the total coefficient and COH, PE, CE, PP, and PD refer to the respective five interactions. The photoelectric effect, Compton effect, and pair production are the most important interactions for megavoltage photons used for radiotherapy. Detailed presentations of these five interactions have been published.2,21,22

Coherent Scattering

In coherent scattering, a photon is scattered off an outer orbital electron with a change in direction and no change in energy (Fig. 6-13). At very low energies (<10 keV), the amount of coherent scattering can be large and attenuation can be high, even though there is no change in photon energy. The amount of coherent scattering is negligible in the diagnostic and therapy energy ranges compared with other principal interactions.

Figure 6-13 Coherent scattering.

Photoelectric Effect

The following actions occur in the photoelectric effect (Fig. 6-14):

1 An incident photon with energy, Eγ = hν, interacts with the inner orbital electron with binding energy E_B (most tightly bound). The interaction can occur with other orbital electrons, but the most probable interaction is with the innermost electron.

2 The photon is completely absorbed and no longer exists.

3 The orbital electron, now called the photoelectron, is ejected with kinetic energy, E_pe, equal to the photon energy minus the binding energy:

Figure 6-14 Photoelectric effect.

If Eγ is less than E_b, the interaction cannot occur, but the interaction may occur with another orbital electron with a binding energy less than Eγ.

4 Ejection of the orbital electron leaves a vacancy in the inner electron shell. This vacancy is filled by an electron from one of the outer orbitals, with the simultaneous emission of a characteristic x ray with an energy of E_cx equal to the difference of the two electron binding energies (Fig. 6-15):

Figure 6-15 Compton effect.

5 This process leaves a new vacancy in an outer orbital shell, which is filled by an electron from an orbital beyond, with emission of a second characteristic x ray of lower energy than the first. This cascade of vacancy creation, filling, and characteristic x-ray emission continues until the most outer orbital electron shell has a vacancy that is filled by a “free,” or unbound, electron.

6 A competing process to characteristic x-ray emission is the production of an Auger (pronounced “oh-jhay”) electron. In this process, the characteristic x-ray energy, E_cx, is transferred to one of the nearby orbital electrons with no x-ray emission, and the electron, called an Auger electron, is ejected with energy:

Characteristic x rays are so named because their energies are directly related to the unique energy levels of the electron orbits for an element. A material’s elemental composition can be determined by detecting its characteristic x rays. Characteristic x rays are named for the orbital electron transition that occurred. For instance, a K_α x ray results in the L shell electron filling the K shell vacancy, an M → K transition yields a K_β x ray, and an M → L transition yields an L_α x ray.

For an atom with five electron orbitals, the possible electron transitions and their characteristic x rays after a photoelectric interaction are L → K, M → L, N → M, O → N, M → K, N → K, O → K, N → L, O → L, and O → M. The most probable transitions are those between adjacent orbitals: L → K, M → L, N → M, and so forth. At the same time, Auger electron emission competes with characteristic x-ray emission, at a rate given by w, the fluorescence coefficient.

The photoelectric effect has a strong dependency on photon energy and atomic number of the material. The mass attenuation coefficient varies as (1/E_γ)³ and Z³, respectively. Mathematically, this is shown as follows, where C_PE is a proportionality constant:

Eq. 8

These dependencies for water are shown graphically in Figure 6-16. In the photoelectric effect, no interaction is possible until the photon energy is greater than the electron binding energy. After the binding energy is barely exceeded, the probability for interaction increases greatly, leading to a sharp increase in µ/ρ_PE, called an absorption edge. In Figure 6-16, the K and L edge for lead are seen, corresponding to photoelectric interactions for the K and L shell electrons. No absorption edges are shown for water; the binding energies are less than 1 keV and do not show up on the plot.

Figure 6-16 Total mass attenuation coefficients for lead and water. Individual interaction coefficients for water (dashed lines) also are shown.

Data from Radiological Health Handbook. Bethesda, Md., U.S. Department of Health, Education, and Welfare, 1970; and Evans RD: The Atomic Nucleus. Malabar, Fla., 1955, Robert E. Krieger Publishing.

The dependence on Z and E_γ can be used to approximate the photoelectric contribution in a different material, using the formula:

Eq. 9

If the material (Z) is constant but energy is changed, the new photoelectric mass attenuation coefficient is found by:

Eq. 10

If the photon energy (E_γ) is constant but the attenuating material is changed, the new photoelectric mass attenuation coefficient is found by:

Eq. 11

Compton Effect

In the Compton effect (see Fig. 6-15), several things occur:

1 An incident photon with energy, E_γ = hν interacts with a loosely bound, outer orbit electron.

2 The photon is scattered at some angle with reduced energy E′γ = hν′.

3 The orbital electron, now called the recoil or Compton electron, is ejected with kinetic energy, E_ce, equal to the difference between the incident and scattered photon energies:

4 Because the interaction is with the outer shell electron, which has negligible binding energy, there are no characteristic x rays or Auger electrons produced.

The distribution of energies and scattering angles for the Compton photon and electron can be described mathematically. The scattered photon has energy E′γ = hν′ given by

Eq. 12

In Equation 12, α = hν/m_oc² and represents the ratio of the incident photon energy to the rest mass of the electron, m_o, and θ is the angle of photon scattering as defined in Figure 6-15.

The Compton electron has energy E_ce given by the following equation:

Eq. 13

and a scattering angle, φ, that depends on the incident photon’s energy and its scattering angle, θ:

Eq. 14

A few scattering cases can be considered:

1 The minimum energy transfer occurs for a 0-degree photon scatter; there is no interaction, and the “scattered” photon has the same energy as the incident photon. The electron is scattered at 90 degrees (φ = 90 degrees) with zero energy.

2 The maximum energy transfer occurs for a direct hit with a backscattered photon (θ = 180 degrees) and yields a (minimum) scattered photon energy of

Eq. 15

The backscattered Compton photon energy will always be less than and will approach 0.25 MeV (which equals m_oc²) even as the incident photon energy becomes very large. The electron has maximum energy of E_ce = hν − hν′ = hν (2α/(1 + 2α)) and travels in the forward direction.

3 A 90-degree Compton scattered photon has energy:

Eq. 16

The 90-degree scattered Compton photon energy will always be less than and will approach 0.511 MeV (which equals m_oc²) even as the incident photon energy becomes very large. The electron has energy of E_ce = hν − hν′ = hν (α/(1 + α)), and it travels in a direction that depends on the incident photon energy.

Although isotopically distributed for lower-energy photons, Compton electrons scatter more and more in the forward direction as incident photon energy increases. Their average energy also increases from about 10 keV to 7 MeV as incident photon energy increases from about 100 keV to 10 MeV.¹¹ The forward-peaked distribution is responsible for the buildup region for megavoltage photon beams, as explained later.

The Compton effect has a slight dependency on incident photon energy, decreasing as energy increases through the megavoltage range, but the interaction probability is essentially constant over most of the megavoltage energy range (see Fig. 6-16). Compton scattering is independent of atomic number and is dependent on the number of electrons available, or electron density (electrons per gram). The electron density for almost all materials is constant at approximately 3 × 10²³ electrons per gram because N : Z is almost constant. The exception is hydrogen, which with one proton in the nucleus has an electron density of around 6 × 10²³ electrons per gram. Except for a small energy dependency and increased interaction probabilities for hydrogen-laden materials, the Compton mass attenuation coefficient is remarkably constant across energy and atomic number, especially for low Z biologic materials such as tissues. Mathematically, this relationship is

Eq. 17

In Equation 17, C_CE is almost a constant.

Attenuation by the Compton effect reduces to the following expression:

Eq. 18

For unit thickness, the transmitted amount depends only on material density, not atomic number and photon energy, as with the photoelectric effect.

Pair Production

In pair production (Fig. 6-17), the following steps occur:

1 An incident photon with energy E_γ = hν and Eγ greater than 1.022 MeV passes near a heavy nucleus and spontaneously disappears, creating an electron, e⁻, and a positron, e⁺, in its place. These two particles are called an electron-positron pair. The total kinetic energy of the electron-positron pair, E_ep, is equal to the photon energy minus the energy needed to create two electrons, or 1.022 MeV (the rest energy of an electron is 0.511 MeV):

Figure 6-17 Pair production.

The electron and positron travel off in no particular directions and do not have equal energies.

2 The electron gradually slows down and is stopped in the material.

3 The positron, the antiparticle of an electron, slows down very quickly and annihilates with a free electron, giving off two 0.511-MeV photons (called annihilation radiation) that travel in opposite directions (i.e., at 180 degrees).

The mass attenuation coefficient for pair production varies linearly with atomic number and incident photon energy (when the photon energy is above the threshold of 1.022 MeV):

Eq. 19

Photodisintegration

Several steps occur in photodisintegration (Fig. 6-18):

1 A very energetic photon of E_γ greater than 8 to 10 MeV interacts with the atomic nucleus.

2 The photon penetrates the nucleus and is absorbed, resulting in the emission of a neutron.

3 Neutron emission leaves a fragmented, possibly unstable nucleus (i.e., radioactive), prompting the name photodisintegration.

Figure 6-18 Photodisintegration.

An energy of greater than 8 to 10 MeV is required because the nuclear binding energies for nucleons are 8 to 10 MeV for most materials. Photodisintegration is the interaction responsible for neutron production for photon energies at 10 MeV and greater and can be an important consideration for facility shielding for photon beams of 15 MeV and greater.

Total Attenuation Coefficient

Each radiation interaction contributes its part to the total attenuation coefficient. Photons having the same energy can undergo any one of the five interactions when energetically possible. However, the probability for each interaction is different and in a particular energy range a particular interaction will dominate.

In Figure 6-16, the individual interaction and total mass attenuation coefficients are shown as a function of energy for lead and water. Notice the K edge in the curve for lead. It can be seen that different energy regions are dominated by particular interactions. For water, the photoelectric effect dominates for photon energies up to 60 keV, the Compton dominates from 60 keV to 10 MeV, and pair production dominates approximately above 10 MeV. For lead, the photoelectric effect dominates up to 700 keV, the Compton effect from 700 keV to 3 MeV, and pair production at 3 MeV and higher. An interaction’s region of dominance can be represented graphically to show relative importance as a function of energy (Fig. 6-19).

Figure 6-19 Relative importance of photon interactions as a function of photon energy.

Comparison of interaction dependencies shows why diagnostic energy photons give good contrast for imaging: the photoelectric effect dominates and is quite sensitive to the atomic numbers of the materials being imaged. Materials with even slightly different Zs have good subject contrast (and image contrast) because attenuation depends on the cube of the atomic numbers. Radiotherapy port films, however, have poor contrast because the dominant interaction is the Compton effect. There is no dependence on atomic number, and the effect is constant for most biologic materials (except for hydrogenous materials). Instead of imaging atomic number, the Compton effect images the density of a material, and subject contrast is essentially density differences. A comparison of simulator and port films is given later (see Fig. 6-49).

Charged Particle Interactions

Charged particles incident on matter undergo inelastic and elastic interactions with atomic electrons and nuclei, that is, other charged entities.2,12,13 Inelastic interactions include collisional and radiative processes and result in energy loss by the particle. In an elastic interaction the particle is scattered by an atomic electron or nucleus, resulting in a change of direction for the particle but no energy loss. The interaction probability for charged particles is effectively 1; an incident particle interacts at every opportunity. The quotient dE/dx (units of million electron volts per centimeter) is called the stopping power, and it describes the rate of energy loss, dE, that occurs over distance traveled, dx, for inelastic collisions. Stopping power is often expressed as the mass stopping power, dE/ρdx (units of million electron volts per gram per square centimeter), which is independent of the absorber density. Because energy loss is almost continuous and a particle has a particular kinetic energy, E_KE, the particle loses this amount of energy and then stops. The distance traveled is finite and is called the particle range; the particle can go no farther, and its kinetic energy is zero. For an absorber of density ρ, the particle range, r, can be calculated as follows:

Eq. 20

The energy lost through inelastic collisions depends on the particle mass, charge, and kinetic energy and on the mass and charge of the target atom, according to the formula:

Eq. 21

In Equation 21, z is the atomic number of the incident particle of mass M, V is its velocity, e is the electron charge, m_o is the electron mass, NZ is the number of electrons per cubic centimeter in the absorber, and F_Q is a complex function describing energy transfer per interaction.

Collisional energy losses increase by the square of the particle’s atomic number and as the incident particle velocity decreases. Increased atomic number results in a greater coulomb force, and decreased velocity increases the amount of interaction time, both leading to increased dE/dx. The energy transfer function, F_Q, is complex and varies with the type of interaction.² It accounts for the atomic mass of the absorber, ionization potential, and relativistic effects as V approaches the speed of light.

Light Charged Particle Interactions: Electrons

The electron mass is small compared with any atomic mass, and incident electrons undergo four types of particle interactions with a large amount of scattering.⁸ Collisional interactions result in energy loss of dE_COL/dx, causing ionizations or excited states (higher electron orbits) (Fig. 6-20). Collisional losses increase as the electron velocity decreases, as stated earlier, and decrease as the absorber atomic number increases. The decrease with absorber atomic number results from the decrease in the number of electrons per gram (NZ/ρ) as Z increases. For equivalent mass thicknesses (mass thickness is thickness divided by density, with units of square centimeters per gram), electrons are stopped sooner in low Z than in high Z materials. Figure 6-21 shows these relationships for water and lead.

Figure 6-20 Electron interactions. A, Excitation. B, Ionization of outer shell. C, Ionization of inner shell. D, Bremsstrahlung production.

Adapted from Johns HE, Cunningham JR: The Physics of Radiology, ed 3, Springfield, Ill., 1978, Charles C Thomas.

Figure 6-21 Collisional and radiative electron losses as a function of incident electron energy.

Adapted from Johns HE, Cunningham JR: The Physics of Radiology, ed 3, Springfield, Ill., 1978, Charles C Thomas.

Radiative interactions result in x-ray emissions. The incident electron penetrates the electron cloud and interacts with the nucleus’s positive electrical field, undergoing an abrupt deceleration with energy loss dE_RAD/dx and a change in direction (see Fig. 6-20D). The energy change, dE_RAD/dx, is released in the form of x rays, called bremsstrahlung (or braking) radiation. With an incident monoenergetic electron fluence, a continuous x ray spectrum is emitted because the probability of any energy loss, large or small, is equal per interaction. Successive bremsstrahlung interactions may occur as the electron loses its energy; a bremsstrahlung spectrum has a maximum energy equal to the initial electron energy and energies below to zero (see Fig. 6-7). Radiative interactions are important; they are the mechanism by which bremsstrahlung x rays are produced in diagnostic x-ray tubes and linear accelerators. Bremsstrahlung production increases with incident electron energy and the Z of the absorber (see Fig. 6-21).

The probability for collisional and radiative interactions depends on electron energy and the atomic number of the incident material (see Fig. 6-21). At electron energies of 100 keV and for high Z absorbers (e.g., x-ray targets), 99% of the interactions are collisional and 1% are radiative, resulting ultimately in heat deposition. At electron energies of 10 MeV, bremsstrahlung is a much more efficient process, and 50% of the interactions are collisional and 50% radiative. At 100 keV, x ray production is inefficient, whereas at 10 MeV, x ray production is efficient. Above 10 MeV, bremsstrahlung production exceeds collisional losses. It has been observed that dE_COL/dx increases as electron kinetic energy decreases below 1 MeV (see Fig. 6-21). As the electron slows down, it loses energy faster. Above 1 MeV, electrons lose about 2 MeV/cm traveling in water (dE_COL/dx ≈ 2 MeV/cm). A 10-MeV electron has a range of about 5 cm in water (10 MeV ÷ 2 MeV/cm). Density scaling can be applied for materials different from unit density. For example, a 10-MeV electron travels approximately 3.3 cm in bone of density of 1.5 g/cm³ (10 MeV ÷ [2 MeV/cm × 1.5]). These relationships use Equation 20 as their basis.

Heavy Charged Particle Interactions

Heavy charged particles, such as protons and alpha particles, experience mainly inelastic collisions. The rate of energy loss is high, resulting in short ranges. Trajectories in water or tissue are in the forward direction with little scattering; the particle mass is similar to that of the interacting material, and very few large-angle direction changes occur, in contrast to the large amount of scattering experienced by electrons. Heavy charged particles exhibit rapidly increasing and large energy losses near the end of their ranges, because of the dependency on Z² and 1/V² discussed earlier (see Eq. 21). This increase in energy loss results in a dramatic increase in ionization at the tail of the particle track length after a length of relatively constant loss, a phenomenon called the Bragg peak.^2,12,13 Bragg peaks are observable for protons (Fig. 6-22) and alpha particles. All charged particles exhibit a Bragg peak, including electrons; however, electrons are light enough such that multiple scatters occur and ionization paths are randomly oriented, blurring any observable effect.

Figure 6-22 Ionization curves for heavy particles. Protons are charged and exhibit a Bragg peak at the end of their range. Neutrons are uncharged and are exponentially attenuated.

Data for protons from Miller DW: Update on proton radiotherapy. In Mackie TR, Palta JR, editors: Teletherapy: Present and Future. Madison, Wisc., 1996, Advanced Medical Publishing.

The proton finite range can be modulated through a physical attenuator to decrease the range from the maximum. If range modulation is done continuously while the beam is on, the Bragg peak will be swept over a range and produce a high dose plateau (see Fig. 6-22). The width of the dose plateau can be designed to match a tumor size at a certain depth and demonstrates the attractiveness of protons for radiotherapy.

Ionization and Its Fate

Ionizing radiation is quantified by measuring the amount of ionization produced. The number of ions is directly proportional to the amount of energy imparted to a material. An average of 33.92 eV is required to produce one ion pair (ip) in air.²³ This number is called the W value (W = 33.92 eV/ip for air), and it is almost independent of the radiation energy. W for other gases is quite similar to that of air, and its uniformity from material to material relates to atomic energy levels and capabilities for transfer of excitation energy.²

In four of the five photon interactions, energy is transferred from incident photons to produce ionization or secondary processes. This energy transfer results in radiation dose, and for each interaction it has origins as follows:

If these ions are collected and measured, the amount of energy deposited can be determined.

Radiation Quantities and Units

Ionizing radiation is quantified using two important quantities: exposure and dose. A third quantity, kerma, is an important concept that relates to exposure and dose.

Exposure

Exposure, X, is the measurement of the amount of ionization produced by photon interactions per mass of air (charge per mass of air). The unit of exposure is the roentgen (R), named after the discoverer of x rays, Wilhelm Conrad Roentgen:

In the previous expression, C (coulomb) is a unit of charge (or ionization) such that 1 ip = 1.6 × 10⁻¹⁹ C. The original definition of charge per volume, 1 R = 1 esu/cm³, was based on the electrostatic unit and is responsible for the odd units of the roentgen as now defined.

Several important concepts characterize the roentgen:

1 It is defined for all ionizations, primary and secondary, when produced and measured in air.

2 It is defined only for ionizing photons (x and gamma rays), not electrons.

3 It is properly measured only under conditions of electronic equilibrium, and it is difficult to measure for photon energies higher than 3 MeV. Above this energy the electron range in air becomes too large for electronic equilibrium to be achieved practically.

Electronic Equilibrium

With electronic equilibrium, ionization lost downstream from a volume of interest is replaced by an equal amount from upstream, producing an equilibrium and an accurate ionization measurement. The concept is illustrated in Figure 6-23. After interaction, ionization (electrons) travels away from the interaction point in the direction of the incident photons and a portion actually leaves the volume. To accurately measure the amount of ionization produced, the lost kinetic energy of the escaping electrons must be replaced by an equal amount that enters the volume from upstream (see Fig. 6-23). This equilibrium is called electronic equilibrium or charged-particle equilibrium and is a required condition for measurement of exposure.

Figure 6-23 Ionization lost downstream from a volume of interest is replaced by an equal amount from upstream, producing electronic equilibrium and an accurate ionization measurement.

Dose

Dose, D, is the measurement of the amount of energy imparted per mass of material (energy per mass).¹¹ The unit of dose is the gray (Gy): 1 Gy = 1 J/kg. An earlier dose unit is the rad (rad): 1 rad = 100 erg/g = 0.01 Gy = 1 cGy; 1 Gy = 100 rad.

Dose does not have the restrictions of exposure, and the following two statements therefore apply:

1 Dose is valid for any ionizing radiation at any energy (e.g., x rays, gamma rays, e⁻, e⁺, n, p, α, π).

2 Dose is valid for any material and phase: solid, liquid, or gas.

In practice, dose to a medium is determined by measuring exposure and converting it to dose using the W value and the mass energy absorption coefficient, µ_en/ρ. The mass energy absorption coefficient is similar to the mass attenuation coefficient, except it describes energy absorbed, not attenuated. The process is illustrated in Figure 6-24 for the following steps:

1 Exposure, X, of 1 R is given to a small air-filled cavity contained in a material:

Figure 6-24 The dose measurement process. Exposure to a small mass of air is converted to the dose to medium.

2 Dose to air in the cavity is computed using the W value and the definition of the roentgen, converting energy units as needed:

so

Eq. 22

For X number of roentgens, the dose to air is calculated as follows:

Eq. 23

3 Dose to the material is calculated by converting dose to air, D_a, to D_m. Conceptually, this step replaces the air cavity with an identical volume of the material. It can be shown^21,24 that

Eq. 24

in which

is the mass energy absorption coefficient for the material,

divided by

the mass energy absorption coefficient for air.

Combining Equations 23 and 24 yields the following expressions:

Eq. 25

and

Eq. 26

in which the f factor, f, is calculated as follows:

Eq. 27

For any exposure, X(R), measured in air, the dose to a material substituted at that point can be found by multiplying the f factor by the exposure. In practice, the air-filled cavity is an ionization chamber. Ionization collected is converted to exposure and then dose, usually through a calibration procedure.

The f factor is an important parameter, and it varies as a function of the material and the photon energy, as shown for water, muscle, and bone in Figure 6-25. By its definition, the f factor for air is always 0.875. Water, muscle, and air have similar f factors because their effective atomic numbers are similar. Bone has a higher atomic number (by a factor of 2) and receives up to four times the dose that soft tissue would receive in the photoelectric region. This effect is responsible for high bone doses received during orthovoltage treatment as well as the contrast seen in diagnostic x-ray images. Notice the decrease in the f factor for fat in the photoelectric region because of its lower Z as a result of its high hydrogen content.

Figure 6-25 The f factor varies as a function of the material and the photon energy.

Data from Radiological Health Handbook. Bethesda, Md., 1970, U.S. Department of Health, Education, and Welfare.

For treatment with ionizing radiation, the amount of energy deposited is quite small. It takes about 1 million cGy to raise the temperature of 1 cm³ of water by 1° C. Conversely, for a daily treatment dose of 2 Gy (200 rad, or 200 cGy), the increase in temperature per gram is only 0.0002° C.

Radiation Detection and Measurement

There are different types of radiation detection and measurement instruments with particular characteristics that enable measurements of a certain kind to be performed. General classes of instruments are gas-filled detectors, scintillation detectors, other solid-state detectors, absolute dosimeters, and personnel dosimeters.

Gas-Filled Detectors

Gas-filled detectors operate on the basic principle of measuring exposure, that is, collecting ionization in air or a substitute gas. A gas-filled volume is defined as one that contains two oppositely charged plates or wires to collect the charge resulting from ionization in the gas. The reading is in units of charge (coulombs), exposure, or exposure rate. Figure 6-26A shows a simple parallel plate gas-filled detector and its components. As the voltage applied to the electrodes is increased from zero, the amount of charge collected increases (see Fig. 6-26B). The distinct voltage regions are expanded in Table 6-8. In general, gas-filled detectors require calibration before making radiation measurements.

Figure 6-26 Response of a gas-filled detector. A, A gas-filled detector collects ionizations from an irradiated volume. B, Variation of collected ionization as bias voltage is increased. Regions are explained in Table 6-8.

TABLE 6-8 Voltage Regions of Gas-Filled Detectors

Region	Description
I	The recombination region. The voltage is not high enough to separate the ion pairs and recombination occurs.
II	The saturation region. The voltage is high enough to collect almost 100% of the ionization (hence, “saturation”). Also called the ionization region.
III	The proportional, or gas amplification, region. The voltage is high enough to accelerate the ionized electrons to an energy that causes additional ionization, amplifying the actual amount of initial ionization by a factor M.
IV	The Geiger-Muller, or GM, region. The voltage is high enough that amplification by accelerated ions proceeds so that the entire gas in the detector is ionized each time a photon hits the detector. Whether low or high energy, the amount of ionization that occurs is the same. Thus, a GM counter emits a “click” for each photon seen and is really a photon counting device.
V	The continuous discharge region. The voltage is high enough to spontaneously ionize the detector gas. Once started, the ionization continues and continues. The detector is not useful at this applied voltage.

Ionization Chambers

Ionization chambers operate in region II (see Fig. 6-26B) and are an important type of radiation dosimeter as the principal device used for calibration of radiotherapy beams. The design most commonly used for photon beams is the thimble chamber, also called a Farmer chamber, which has a cylindrical geometry with central linear and outer cylindrical electrodes (Fig. 6-27A). An active volume of 0.6 cc is convention, although chambers with smaller or larger volumes are available for specialized purposes such as small field dosimetry or low-dose-rate measurements. Parallel-plate chambers are also used and are the recommended chamber geometry for electron beam dosimetry (see Fig. 6-27B). In general, ionization chambers require calibration before use. Ionization currents produced by radiation beams are very small, on the order of 10⁻⁹ amperes (A), and require a high-precision device called an electrometer for accurate measurement. An ionization chamber and electrometer require calibration before use and with a triaxial connecting cable are required tools for radiation beam calibration. Conditions for radiation beam calibrations are set by national protocols such as the Task Group (TG)-51 from the American Association of Physicists in Medicine¹⁵ and by national or state regulatory bodies. A requirement for operation of all ionization chambers is that the chamber be operated under conditions of electronic equilibrium. In disequilibrium conditions, the amount of ionization measured is incorrect, as will be the exposure or dose calculation.

Figure 6-27 Ionization chamber designs. A, Thimble chamber. B, Parallel-plate chamber. A buildup cap may be added to either chamber to ensure electronic equilibrium, depending on the measurement being performed.

Solid-State Detectors

Thermoluminescent Dosimeters

In a simple model, crystalline solids have two electron energy levels, called the valence and conduction bands. The valence band holds bound electrons, whereas the conduction band consists of free electrons. Energy states between these two bands are forbidden. When ionized, a bound electron may gain sufficient energy to jump to the conduction band, leaving a positive “hole” in the valence band. The conduction band electron and the hole can move through the lattice and eventually recombine, with the emission of light on recombination. Because of impurities in the lattice, some crystalline materials have the property of having long-lived “traps” that can hold an electron-hole pair in an excited (or unfilled) state. The electron-hole pair is created and excited into the trap by the absorption of ionizing radiation, which provides the energy to push the pair into the trap (Fig. 6-28). Heating the crystal enables the electron-hole pair to leave the trap and return to the de-excited state. With this transition, an amount of light is emitted that is proportional to the amount of radiation dose. Dose is measured by measuring the amount of light emitted. The two most common types of thermoluminescent dosimeter (TLD) are lithium fluoride (LiF, almost tissue equivalent) and calcium fluoride (CaF, not tissue equivalent). A calibration factor must be obtained before use as a dosimeter. TLD is used for in vivo patient dose monitoring during treatment and for personnel monitoring for radiation safety purposes.

Figure 6-28 Thermoluminescence. Ionizing radiation raises an electron-hole pair to an excited but trapped state. Addition of heat allows de-excitation, which occurs with the emission of light.

Film

Film is a solid-state detector and typically refers to silver-halide films that are sensitive to both light and ionizing radiation. The amount of optical density is proportional to the dose. The response is energy sensitive because film is relatively high Z (the silver content). Response curves are shown in Figure 6-29. Film is used to obtain relative dose distributions (i.e., isodose distributions) for treatment machine quality assurance tests for flatness, symmetry, radiation/light field congruence, and imaging.^16,25 A dose-response curve must be carefully measured, including constant processor control, to enable film to be used to measure actual (not relative) doses. Because of digital image receptors for photography and medical imaging with kV and MV photons (e.g., diagnosis and treatment verification), relatively high cost, and the development process required after exposure, medical uses of silver-halide radiographic film are almost nonexistent. For physics quality assurance and dosimetry purposes a new type of tissue-equivalent film, called radiochromic film, has become a powerful tool. Radiochromic film uses opacity changes from radiation-induced polymerization to indicate dose.²⁶ This film does not require developing but does have an optimal wavelength of light for reading. A digital flatbed three-channel (color) scanner can be used to read the radiochromic film opacity. Example uses in therapy beam quality assurance and dosimetry include small beam dosimetry, patient treatment plan verification (such as currently performed prior to a first IMRT procedure), and mechanical and radiologic quality assurance in radiosurgery. Radiochromic film is tissue equivalent and can be used over the full range of x-ray energies. Radiochromic film types are also available for different dose ranges, from 1 Gy up to 50 Gy. Currently, radiochromic film is not an acceptable medical imaging device because of its cost and lack of imaging contrast.

Figure 6-29 Film radiation response varies with film type, photon energy, and dose. A, Diagnostic photon energy response. B, Therapeutic photon energy response.

MOSFET Detectors

A semiconductor device called the metal oxide-silicon semiconductor field effect transistor, or MOSFET, can be used as a radiation detector for in vivo dosimetry and physics quality assurance. The p-type device generates electron-hole pairs during irradiation proportional to the dose delivered, and the number of holes, corresponding to dose, can later be read using an appropriate metering device. Advantages include very small size, which enables small field or “point” dose measurements, and the capability for being incorporated into both implantable dosimeters and surface-placed dosimeters for in vivo patient dosimetry.²⁷ MOSFET dosimeters are primarily used for photons and require calibration for the photon energy in use. Surface applications require an understanding of the build-up effect for proper assessment of surface dose.

Scintillation Detectors

Scintillation (light) detectors detect radiation by measuring the amount of light produced in special crystalline materials by the ionizing radiation. The most common material used is sodium iodine (NaI). The NaI crystal is coupled to a photomultiplier tube that amplifies the detection signal coming from the crystal (Fig. 6-30A). NaI detectors are extremely sensitive and detect background radiation easily. They are used to detect low-energy or low-activity (dose rate) sources. They are used in uncalibrated form; a pulse indicates a photon has been detected.

Figure 6-30 Scintillation detectors. Ionizing radiation creates light photons that are converted to electrons and amplified. A, Sodium iodine crystal and photomultiplier tube. B, Liquid scintillation detection.

Scintillation materials can also be made in liquid form. With this technique a radiation source (i.e., a sample of radioactive material) is put in direct contact with a liquid scintillator by immersion (see Fig. 6-30B). This intimate contact enables extremely small activities to be detected. Detection limits are in the microcurie and picocurie ranges. This method is used most for detection of very-low-energy beta particles for radioisotope tracer studies, environmental sampling, and isotope analysis of waste or other materials. The liquid scintillator is called a cocktail and includes a solvent and the scintillation material. Liquid scintillation detectors require calibration before use, including determination of background count rates.

Semiconductor Detectors: Germanium and Silicon

Germanium (GeLi) and silicon (SiLi) detectors are semiconductor materials that can be used to detect the energies of photons. These detectors are useful only at very low dose rates (i.e., small sample activities or low fluence rates) and are used to measure the photon energy spectrum, not dose.

Percent Depth Dose

The PDD is the ratio of the dose at depth, D_d, to the maximum dose, D_m, measured along the central axis of the beam and expressed as a percent. Points X and Y are the locations at depth, D_d and D_m, respectively, in Figure 6-32B. PDD is dependent on the depth, d, reference depth, d_m (the point of maximum dose), beam quality (or energy), E, source-to-surface distance, SSD, and field size at the surface, w. Mathematically,

Eq. 28

PDD is often expressed as a fraction rather than as a percent. PDD was the dose function first measured for diagnostic and therapy radiation beams because of the simplicity of measurement, and it remains a basic dose measurement for beam characterization. Inherent in a PDD measurement are the effects of attenuation by the material and the inverse square effect as the distance from the source is changed. PDD is used for photon and electron beams and is often used in tabular form for monitor unit calculations.

Tissue-Air Ratio

Given a fixed irradiation point in space, the ratio of the dose in phantom, D_x, to the dose in air, D_x′, at the same point is called the TAR (see Fig. 6-32A and B). TAR indicates how the dose in air is affected when the air is replaced by tissue. The field size at the point and source-to-point distance are constant; the only variable is the depth in phantom:

Eq. 29

TAR is essentially independent of SSD because scatter contributions are almost constant for fixed field sizes, regardless of SSD. This independence enables the use of the TAR over a wide range of clinically used SSDs without correction. TAR is used for photon beams only and is not valid for electron beams.

A special case of the TAR, called the backscatter factor (BSF), exists when the point of interest is the depth of maximum dose (when d = d_m). In this case, the TAR has a maximum value, and

Eq. 30

The BSF takes its name for historical reasons, from a time when low-energy photons were used and the point of maximum dose was on the surface. Any dose scattered to the surface from the phantom or patient was truly backscattered dose.

To determine the TAR and BSF, dose data are acquired at sampled points in phantom and air across and along the radiation beam central axis for the range of field sizes, depths, and SSDs or source-to-axis distances (SADs) clinically relevant.

Tissue-Phantom Ratio

TPR must be used for photon energies above about 4 MV, when in-air measurements required for the TAR are impractical. As with PDD and the TAR, the TPR is the ratio of two doses; however, both doses are made in phantom. The TPR is the ratio of dose D_x, measured at depth d, to dose D_x″, measured at a reference depth d″ (see Fig. 6-32B and C):

Eq. 31

Both measurement points are fixed in space such that the source-to-point distance and field size are the same at each point. The depth of point X is varied to generate TPRs over a range of depths for a particular field size. TPR is essentially independent of SSD because scatter contributions are almost constant for fixed field sizes, regardless of SSD. This independence enables the use of the TPR over a wide range of clinically used SSDs without correction. TPR is used for photon beams only and is not valid for electron beams.

When d″ is the depth of maximum dose, the TPR is called the tissue-maximum ratio (TMR). The TPR and the TMR are used in the same manner as the TAR in dose calculations. TMR is the most common data format used for monitor unit calculations in simple computer programs and manual (hand) calculations.

Primary and Scatter Dose: The Scatter-Air Ratio

Mayneord first proposed that PDD could be separated into primary and secondary (or scatter) components.¹² Later, Clarkson³⁰ proposed sector integration of scatter components in calculating the PDD at any point inside or outside an irregular field. This concept was later applied by Gupta and Cunningham to the TAR.³¹ The concept states that dose at a point is the sum of primary and scatter components. The primary dose results from photons that have interactions at the point of interest. The secondary dose results from photons and electrons that scatter to the point of interest from other interaction points. With this concept, TAR is given by the following expression:

Eq. 32

In the equation, TAR₀ is the TAR for zero field size, representing primary dose, and SAR is the scatter-air ratio, representing secondary dose. The TAR₀ is found by extrapolation of circular field TARs to zero field size. The SAR is calculated as the difference between the TAR and the TAR₀. For an irregular field, an effective TAR can be found that is the sum of the TAR₀ and an effective SAR for the irregular field:

Eq. 33

The irregular field SAR is found by sector integration:

Eq. 34

in which Δθ is the angular increment in radians for each sector. Separation of primary and secondary dose can be applied to any of the dose functions in a similar fashion. For instance, the TMR can be represented as TMR = TMR₀ + SMR.

The TAR/SAR and TMR/SMR functions are more commonly used than PDD in dose computation algorithms because fewer inverse square corrections are needed. The TAR and TMR are easily found for isocentric treatment geometries where the size of the treatment field at the isocenter is equal to the treatment field size at the center of the target volume, which simplifies hand (noncomputer) calculations.

Although PDD can be measured with an ionization chamber beyond d_m over the wide range of clinically used photon energies, the TAR cannot be measured when the photon energy exceeds about 3 MV because the range of secondary electrons exceeds the thickness of the buildup cap for the ionization chamber. At these energies, PDD is measured and converted to TPR and TMR by well-known relationships.^21,22 PDD remains the most important and fundamental dose function and performance specification to be measured for almost all physics quality assurance procedures for all external beam treatment devices.

Dose Characteristics

Photon beams used in radiotherapy have dose characteristics that vary primarily as a function of beam energy and treatment machine design. In the standard irradiation geometry (see Fig. 6-32B and C), PDD measurements are obtained in water along the central axis, beginning at the surface and proceeding to depth. A typical PDD curve shown in Figure 6-33 for megavoltage beams has the characteristics listed in Table 6-9.

Figure 6-33 A typical photon percent depth dose (PDD) curve is characterized by surface dose, a buildup region, a point of maximum dose, and an exponentially attenuated region. Standard PDD measurements are obtained along the central axis (dotted line), beginning at the surface and proceeding to depth.

TABLE 6-9 Characteristics of a Typical Percent Depth Dose Curve for Megavoltage Beams

Other important descriptors include an effective attenuation coefficient for depths beyond the maximum depth (d_max) and the HVL. The 10 cm PDD in water is most used as an indicator of beam quality, in place of the HVL, because the 10 cm PDD has clinical relevance and the HVL is a better indicator for shielding purposes.

Field size, flatness, symmetry, and sharpness of the beam edge can be measured with a dose profile (Fig. 6-34), which is a scanned dose measurement perpendicular to the central axis, across a beam (at a fixed depth, occurring along the horizontal line w_d in Fig. 6-32B and C). These and PDD measurements are most commonly obtained using a large water phantom with a three-axis scanning system to move a water-immersible ionization chamber or diode detector (Fig. 6-35). Dose profiles are normally made at several depths of interest (i.e., d_m, 5 cm, 10 cm, and 25 cm). Beam characteristics from a dose profile are listed in Table 6-10.

Figure 6-34 The dose profiles of a typical photon beam are characterized by shoulder and toe regions, with definitions for field width, flatness, and penumbra as indicated.

Figure 6-35 Irradiation geometries for isodose measurements. A, Beam orientation with phantom. B, Planes orthogonal to the central axis and isodose curves.

A, Courtesy IBA Dosimetry, Schwartzenbruck, Germany.

TABLE 6-10 Beam Characteristics from a Dose Profile

The flattening filter and beam steering determine flatness and symmetry in a linear accelerator. Flatness is defined at a reference depth, for instance at a 10-cm depth at the isocenter (SSD = 90 cm). A typical specification is for no more than 3% dose variation across the useful 80% of the field width (see Fig. 6-34). A radiation beam does not have a perfectly sharp edge but instead has a gradient from high to low dose with a rounded “shoulder” and “toe.” This penumbra region is named for geometric shadowing of part of the source, as is the case for ⁶⁰Co teletherapy beams, which have a large (2 cm) physical source size. However, the penumbra region for linear accelerator beams is mostly a radiation component of secondary electrons and photons scattered from within the field and from the collimators to outside the beam edges. This scatter degrades the beam edge; the dose from the shoulder region scatters outside the beam edge to form the toe. Penumbra is defined as the distance over which the dose falls from 80% to 20% of the central axis dose at the same depth, or at the width of the 90% to 10% dose gradient (see Fig. 6-34).

Various methods exist to measure a radiation beam along lines of equal dose, called isodose lines. In a three-axis water phantom scanner, after determining PDD along the central axis, a certain isodose level is identified in the standard geometry and then tracked to obtain a digital representation of the dosimetric shape of the beam. Figure 6-35 shows sample isodose curves for a plane containing the central ray of a beam and a plane perpendicular to the central ray. The modern water phantom scanning device is computer controlled and has robust software tools for two (2D)- and three-dimensional (3D) renderings and analysis of acquired dose distribution data.

Effect of Energy

Photon beam characteristics change with beam energy. Representative beam data are presented in Table 6-11, PDDs in Figure 6-36, and isodose curves in Figure 6-37. Several key observations should be considered:

Figure 6-36 Photon percent depth dose curves.

Data courtesy Wake Forest University Baptist Medical Center, Winston-Salem, N.C., and the Radiological Health Handbook. Bethesda, Md., 1970, U.S. Department of Health, Education, and Welfare.

Figure 6-37 Photon isodose curves, 10 × 10 cm² fields. A, 6 MV, cross plane. B, 18 MV, cross plane. C, 6 MV, orthogonal plane. D, 18 MV, orthogonal plane.

Data courtesy Wake Forest University Baptist Medical Center, Winston-Salem, N.C.

Effect of Field Size

Field size determines the amount of scatter present in the beam, with the following observations:

Figure 6-38 Relative dose as a function of photon field size.

Penumbra

Beam edge sharpness, or penumbra, is typically 7 to 12 mm in width and varies with the amount and energy of secondary electrons and scattered photons. As the amount of scatter increases, penumbra does likewise:

Beam Modifiers

Various devices can be put into a photon beam to modify the shape of the beam and its dose distribution. Collimators and custom blocks are used to shape the field, as previously discussed. Their dosimetric effect is to change the irradiated volume and therefore the amount of scatter.

Physical wedges produce skewed isodose lines at fixed angles across the central ray (Fig. 6-39) to compensate for beam angle of incidence, tissue shape, or the trajectory of other treatment beams. Historically, wedge angles of 15, 30, 45, and 60 degrees have been available using physical wedges made of steel or lead. Isodoses show the wedge angle is measured from a perpendicular to the central ray, typically at a depth of 10 cm. A physical wedge produces an attenuated beam that is decreased in absolute dose rate and slightly more penetrating due to beam hardening by the wedge. A commonly available alternative wedging system uses no physical wedge and produces angled isodose lines by scanning one of the independent collimator jaws across the field while the beam is on. This dynamic method does not result in hardening or attenuation of the beam and is called a dynamic, virtual, or soft wedge. Another method to produce an angled beam intensity is to use one physical wedge with the maximum wedge angle of 60 degrees that is left in the beam for various fractions of time to produce wedge angles from 0 (no time in beam) to 60 (full time in beam) degrees.

Figure 6-39 Isodose curves with physical wedges, 6-MV photons, and a 10 × 10 cm² field. A, A 15-degree wedge. B, A 30-degree wedge. C, A 45-degree wedge. D, A 60-degree wedge.

Data courtesy Wake Forest University Baptist Medical Center, Winston-Salem, N.C.

Compensators attenuate the beam in desired locations to provide dosimetric shaping. A variety of methods exist that include simple and complex approaches.¹³ Compensators have an important role in dose optimization schemes that include static applications³² as an alternative to dynamic techniques.