Radiation Oncology Physics

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 ⁶ electron volts (eV). 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 nucleus. Detailed atomic and nuclear models have been formalized that describe the character and interactions of atoms and nuclei.

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 1/2000 that of a proton or neutron and has a charge of −1. The electron is considered a fundamental particle that cannot be further divided into smaller constituents and, thus, has “point-like” behavior. 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 cosmological implications. More recently it has been determined that the neutrino mass, inferred through observed oscillations in the neutrino state from one type to another, is finite and 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 (5% of the material “stuff” we are familiar with), dark matter (27%), and dark energy (which comprises approximately 68% of the total mass-energy content of the universe). These “dark” components are unknown entities.

Particles are given two general names, based on their character. Fermions are the particles that make up matter as we know it. Bosons are the particles that mediate interaction forces. Fermions 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 1/2. Hadrons are heavy particles with two subclasses called mesons (middleweight, spin 0 or 1) and baryons (heavyweight, spin 1/2 or 3/2). 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 1/3, spin of 1/2, and other properties with whimsical, quark-deserving names. 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 (there are two), and gravitational forces, with respective relative strengths of 10 ¹ , 10 ⁻² , 10 ⁻¹³ , and 10 ⁻⁴² . Each force acts by exchange of its respective mediator particle: the gluon (strong), photon (electromagnetic), W and Z particles (weak), and graviton (which has not yet been discovered). Counting mediators, leptons, hadrons, and their antiparticles, there are about 170 fundamental and composite particles that make up matter. The model in which matter consists of fundamental particles as described previously is called the standard model . Table 6.1 summarizes the fundamental particles in the standard model, which are six quarks, six leptons, antiparticles for each quark and lepton, as well as the four mediator forces, except for gravity. Of great importance is the 2012 discovery of the “Higgs boson,” predicted in 1964 by Higgs, which is a heavy particle (~ 125 GeV in energy) responsible for the character of matter that we have named “mass.” The Higgs boson is placed by itself alongside the other entries to the standard model as the description of fundamental particles that make up matter (see Table 6.1 ). How fundamental particles fit together to make up other particles continues to be studied. A relatively new theory, called string theory , is a descriptor of how quarks and other fundamental particles are assembled with vibration (energy) states for string-like entities that give each fundamental particle its unique character. However, “flexible” as they may be, the strings still need additional dimensions called “branes” (after “membranes”) to enable their full character to be described. 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; 1/12 of the mass of the carbon nucleus) or in units of energy by conversion with Einstein's formula, E = mc ² . Table 6.2 gives the symbol, charge, mass, and stability for the electron, proton, neutron, and other particles of interest.

TABLE 6.2

Physical Characteristics of Selected Atomic and Nuclear Particles

Name	Symbol	amu	MeV	m e	Charge	Lifetime
Electron	e −	0.000549	0.511	1	−1	Stable
Positron	e +	0.000549	0.511	1	+1	10 −6 s
Proton	p	1.007276	938.256	1836.1	+1	Stable
Neutron	n	1.008665	939.550	1838.6	0	12 min
Neutrino	ν ,	< 2.4 × 10 -9	< 2.2 × 10 -6	< 4.3 × 10 -6	0	Considered Stable
Muon	µ	0.11320	105.659	206.4	−1	Unstable
Pion	±Π	0.14990	139.578	273.2	±1	Unstable

amu, Atomic mass unit; m _e , rest mass of the electron; MeV, million electron volts.

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. 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 isoto p es (p for proton), and they exhibit identical chemical characteristics (they are the same elements). Nuclei with the same N but different Z are called isoto n es (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 ^A X , such as ³ H and ¹⁹² Ir. 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. Fig. 6.2 shows the distribution of stable and naturally radioactive nuclides as a function of the number of neutrons and protons. In Fig. 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 . Note 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 Z s, 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.

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. 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 ¹ H, ¹² C, ³³ P, ³⁴ S, and ⁵⁹ Co. Radioactive nuclides for these elements, some naturally occurring and others man-made, include ³ H, ¹⁴ C, ³² P, ³⁵ S, and ⁶⁰ Co. Other radionuclides of interest to the medical field include ^99m Tc, ¹²⁵ I, ¹³¹ I, ¹³⁷ Cs, and ²²⁶ Ra.

Photons and Other Definitions

Interactions of ionizing radiation occur at the atomic and nuclear levels, where binding energies range from 10s of eV to 10 MeV, relatively 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 × 10 ⁸ m/s). Photons are also the mediators of charged particle bonds. A photon has wavelength λ, frequency ν, speed c = λν, and energy E = h ν; c is the speed of light, and h is Planck's constant ( h = 6.626 × 10 ⁻³⁴ m ² /kg/s). The electromagnetic spectrum consists of photons with wavelength, frequency, and energy ranges more than 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 ).

Radiation Production by Radioactive Decay

Teletherapy is the use of radioactive material, such as ⁶⁰ Co, 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, in which 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; this unique energy can be used to identify the daughter (and, therefore, the parent). ²²⁶ Ra, ¹³⁷ Cs, and most commonly ⁶⁰ Co have been used for teletherapy. While ²²⁶ Ra was chemically separated from naturally occurring ores, ¹³⁷ Cs and ⁶⁰ Co are manufactured and made available by neutron activation and as a by-product of fission after the invention of the nuclear reactor. Use of ⁶⁰ Co as a source of gamma rays for treatment was pioneered by H. E. Johns and represented a major step in obtaining high-energy photons above 1 MeV, termed “megavoltage” photons. 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. Specialized particle accelerators were required to produce potentials above 300 keV (e.g., betatrons and van de Graaff accelerators ).

Although of historical interest for most readers, ⁶⁰ Co teletherapy units have lower cost and a relatively simple design with lower requirements for their operating environment compared with LINACs. For these reasons, ⁶⁰ Co teletherapy devices continue to be used worldwide in regions with limited resources for funding and operating infrastructure. One manufacturer (Best Theratronics, Ltd., Ottawa, Ontario, Canada) states that, as of 2010, there were 45,000 daily radiation treatments delivered worldwide using this conventional ⁶⁰ Co teletherapy device.

In a conventional ⁶⁰ Co teletherapy unit ( Fig. 6.4 ), a cylindrical sealed-source capsule about 3 cm in diameter and 5 cm high contains pellets of ⁶⁰ Co. In each transformation, a ⁶⁰ Co nucleus decays to ⁶⁰ Ni, with the prompt emission of two gamma rays at 1.17 and 1.33 MeV each (1.25 MeV average). Typical activity is 6000 to 9000 Ci (2.22-3.33 × 10 ¹⁴ Bq) for dose rates of approximately 2 to 3 Gy/min at 80 to 100 cm from the source. A disadvantage is the constantly decreasing dose rate as a result of the decay of the ⁶⁰ Co source and the requirement for eventual replacement of the source. The source decays with a half-life of 5.27 years and is replaced every 5 to 7 years when the dose rate becomes “too low”; treatment times may be excessive and radiobiological effect for treatment efficacy may be compromised below 1 Gy/min.

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, additional collimator bars closer to the patient surface, can be used to reduce the beam penumbra, which is large because of the relatively large source diameter of approximately 3 cm. Maximum field size is 35 × 35 cm ² at 80 or 100 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 push bar (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 a nominal isocenter position of 80 cm from the source. Later models have a 100-cm isocenter and treatment distance. An additional degree of freedom is provided by a head swivel mechanism that allows the beam direction to be rotated away from the isocenter, if desired. A beam stopper may be used to intercept the beam for additional shielding of the exit beam, potentially reducing the amount of shielding needed in walls of the facility. 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, multivane collimation, and mechanical wedges for producing angled isodose distributions or tissue compensation. Other important current-day uses of gamma-ray beams from radioactive ⁶⁰ Co, discussed later, include gamma radiosurgery and gamma-magnetic resonance imaging (MRI) image-guided radiotherapy (IGRT).

Radiation Production by LINACs

In a LINAC, 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 LINAC enables convenient production of megavoltage x-rays in a relatively small device; its existence is directly related to the invention of the magnetron and the klystron during the development of microwave radar in World War II. LINACs are quite versatile, with x-ray and electron modes, multiple energies, and computer controls. These capabilities have led to the replacement of most ⁶⁰ Co teletherapy machines in the United States with LINACs.

The principle of operation for LINACs is to accelerate electrons through a waveguide by use of alternating microwave fields. 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 LINACs ( Fig. 6.5B ) 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 Fig. 6.5 for C-arm gantry-based LINACs, the most common design.

Major mechanical components are similar to those for teletherapy; two main designs are used for the mechanical support structures to implement C-arm geometries for the radiation beam. In the first design, a rigid, floor-mounted stand holds high-voltage microwave transport guides, cooling and control components, and supports the 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. 6.5B ). In a second design, the entire accelerator—including waveguide, high-voltage microwave transport guides, and other components—is contained within a large rigid cylinder. The cylindrical structure and all of its contents rotate on a horizontal axis about the isocenter (see Fig. 6.5C ). 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 or width of the rectangular x-ray radiation field. A maximum field size of 40 × 40 cm ² at 100 cm (isocenter) is common, with 180- to 360-degree rotation of the entire collimator assembly about the isocenter (see Fig. 6.3 ). Collimator settings are continuously variable, and jaw pairs can be operated in coupled or independent modes to produce symmetrical and asymmetrical fields about 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 LINACs, 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 because the table assembly is a separately added component to the rotating gantry structure.

Accessories and beam modifiers for conventional geometry LINACs 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 (made of steel) 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 common, using high Z alloy materials to provide primary x-ray beam attenuation of at least 97% (5 half-valve layers). 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 LINAC manufacturers ( Fig. 6.6 ). The MLC provides on-board, automated field-shaping capabilities, replacing standard manual 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 manufacturers’ MLC designs include singly and doubly focused leaf ends, in which 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, 4, 5, or 10 mm. MLCs also have capabilities for dynamic treatment, as required and discussed later for intensity-modulated radiotherapy (IMRT).

LINACs and their supporting technologies have become 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 LINACs 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 radiotherapy (IGRT). A wide range of IGRT designs is possible, as discussed later.

Although LINACs typically have had an isocentric rotating C-arm gantry, there are other commercialized megavoltage accelerator designs that use the highly stable computed tomography (CT)-like ring gantry along with integrated image guidance for treatment. One precedent-setting design, called tomotherapy, uses a ring gantry around which an X-band LINAC rotates, always pointed toward the rotational center (an isocenter). A binary (“on” or “off”) MLC 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. Other ring gantry designs use a cone beam 6-MV x-ray beam with full or dual-layer MLCs, along with kV cone-beam CT (CBCT) imaging for image guidance. In one design, the entire ring-gantry unit is gimbaled to enable noncoplanar beam geometries relative to the transverse plane of the patient support (table). In general, ring and C-arm gantry designs using cone-beam x-rays can have high efficiency for treatment compared with fan-beam or pencil-beam designs depending on field size and intensity modulation because of the increased amount of solid angle that can be irradiated—and, thus, treatment volume—per unit time.

Another unique LINAC geometry uses a nonisocentric format with an X-band LINAC fixed on the end of a high-precision robot that is computer controlled for a range of motions with multiple degrees of freedom. This device produces a relatively small-sized 6-MV x-ray beam that can be pointed at the target from various “nodes” in space—using a large collection of irradiation nodes from many directions generates the cumulative conformal dose distribution at the target. The x-ray beam can be shaped by circular collimators or a dynamic MLC for field sizes up to 6- to 10-cm widths.

Three differences in radiation sources for teletherapy and LINAC devices are fundamental. First, a ⁶⁰ Co teletherapy source is always “on,” emitting radiation. Radioactive decay always occurs and cannot be interrupted; to initiate irradiation, the source must be moved from the shielded condition to an unshielded condition (see Fig. 6.4 ). With a LINAC, 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 ⁶⁰ Co unit, two monoenergetic gamma rays are emitted with every decay to produce a discrete spectrum with peaks at 1.17 and 1.33 MeV. In contrast, a LINAC 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 LINAC 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 ⁶⁰ Co gamma rays occur because of an immediately preceding beta decay, the beta particles (electrons) are stopped by the radiation source enclosure, a metallic capsule that contains the radioactive material, and cannot be otherwise harnessed for treatment.

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 have included the betatron, van de Graaff accelerators, cyclotrons, the racetrack microtron, and synchrotrons and synchrocyclotrons similar to those developed for high-energy physics experiments.

Among high-energy charged particles, protons have become the most attractive for therapeutic uses, where proton energies of 220 to 250 MeV yield treatment depths of up to 30 to 40 cm. Although currently an expensive technology to build ($40-$200 million USD or higher) and maintain, the use of high-energy protons for treatment is increasing with approximately 80 operating proton facilities (28 in the United States, 52 sites in other nations, 11 with carbon ions), 45 sites under construction, and 25 in the planning stages worldwide. Typically, a large-scale proton facility has a single accelerator (e.g., cyclotron or synchrotron) with a beam line that is split to service two to four treatment rooms, each with either a fixed treatment beam or a movable gantry to aim the proton beam. An alternative design includes the use of a smaller (though still large) cyclotron combined with a short beam-line segment, all contained within a single treatment room (one cyclotron per treatment room). The in-room cyclotron is mounted on a large, double-legged gantry system that enables single-plane rotation about the patient, with a patient table that swivels to provide additional solid-angle coverage. Facility designs and technical aspects for proton therapy are available.

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 Fig. 6.8 . A number, N ₀ , of monoenergetic x-rays or gamma rays is incident to a slab of material, and a smaller number, N, is transmitted. The attenuated photons, numerically equal to N ₀ − 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.

Megavoltage photon interaction probabilities are less than 1, typically 2% to 4% per centimeter in tissue depending on the incident photon energy, atomic number of the interaction material, and interaction type. By physical law, the fractional number of unattenuated photons interacting per unit thickness of a material is constant, such that

$\frac{\Delta N/N}{\Delta x}=-\mu \text{or}\frac{\Delta N}{N}=-\mu \Delta x.$

In the equation, Δ N 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 ), Δ x is the thickness traversed by the photons for attenuation Δ N , and the minus sign indicates that the effect is negative, resulting in fewer photons—the attenuation process decreases the number of photons in a beam.

Constant fractional attenuation per unit thickness compounds over successive thicknesses, illustrated in Fig. 6.9 . This “fraction of a fraction” effect is nonlinear, and the integrated form of Eq. 1 yields an important relationship: attenuation in a continuous material is an exponential process:

$N\left(x\right)=N_0{e}^{-\mu x}\text{or}\frac{N\left(x\right)}{N_0}=e^{-\mu x}$

In Eq. 2 , N ₀ 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):

$I=I_0{e}^{-\mu x}\text{or}\frac{I}{I_0}=e^{-\mu x}\text{and X}=X_0{e}^{-\mu x}\text{or}\frac{X}{X_0}=e^{-\mu x}$

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.

Eq. 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 ( Fig. 6.10B ). A smaller attenuation coefficient results in less attenuation and a shallower attenuation curve. Different attenuation coefficients for different materials or different photon energies result in a different amount of attenuation for the same thickness traversed and attenuation curves that differ in their slopes ( Fig. 6.11 ).

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). The HVL is the thickness of a material that reduces the transmitted intensity to one-half of the original intensity. When , the thickness, x, is the HVL and the attenuation equation becomes

$\frac{I}{I_0}=\frac{1}{2}=e^{-\mu \text{HVL}}\text{}.$

By taking the natural log (ln) of each side, and with ln(1/2) ≈ 0.693, two important relationships come from this equation:

$\mu =0.693/\text{HVL}\text{and HVL}=0.693/\mu .$

Thus, 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 . Note that the units for the HVL are the reciprocal of the units for µ.

Because , and given a thickness of material in units of HVL ( x = n HVL , n equals the number of HVLs), the amount of attenuation can be computed by

$\frac{I}{I_0}={\left(\frac{1}{2}\right)}^n\text{}.$

In Eq. 5 , I / I ₀ is the fractional transmission, intensity, or exposure and n is the thickness of the material, expressed as the number of HVLs. This equation is valid for positive integer and noninteger values of n (e.g., n = 1, 2, 5.9, 100.1).

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

$\frac{I}{I_0}=\frac{1}{10}=e^{-\mu \text{TVL}}\text{and}\frac{I}{I_0}={\left(\frac{1}{10}\right)}^m\text{}.$

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

The 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 1/2. 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, additional 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, the HVL also changes, yielding different first and second HVLs, as defined in Fig. 6.12B . The monoenergetic case is shown for comparison. For a polyenergetic beam, HVL ₁ is always less than HVL ₂ because photons incident on HVL ₂ have a higher average energy, thus HVL ₁ is less than HVL ₂ . The greatest HVL is found at maximum depth, on the linear portion of the attenuation curve where beam hardening is the greatest. For polyenergetic photon beams, an effective attenuation coefficient (µ _eff ) can be calculated by determining an effective HVL over a region of interest and then calculating the µ _eff :

${\mu }_{\text{eff}}=0.693/{\text{HVL}}_{\text{eff}}\text{}.$

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 scale	1. Complex exponential curve on linear scale.
2. Linear curve on semi-log scale	2. Curved on semi-log scale.
3. Constant µ	3. Varying µ/µ eff can be determined.
4. Constant HVL	4. Varying HVL/HVL 1 , HVL 2 , 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 most common photon interactions: coherent scattering, photoelectric effect, Compton effect, pair production, and photodisintegration. Each interaction type has an independent interaction probability and contributes to the cumulative amount of attenuation. The total linear and mass attenuation coefficients are given by the sum of their components:

${\mu }_{\text{TOT}}={\mu }_{\text{COH}}+{\mu }_{\text{PE}}+{\mu }_{\text{CE}}+{\mu }_{\text{PP}}+{\mu }_{\text{PD}}$

and

$\mu /\rho |{}_{\text{TOT}}=\mu /{\rho }_{\text{COH}}+\mu /{\rho }_{\text{PE}}+\mu /{\rho }_{\text{CE}}+\mu /{\rho }_{\text{PP}}+\mu /{\rho }_{\text{PD}}\text{}.$

In these equations, TOT signifies the total coefficient and COH, PE, CE, PP, and PD refer to the respective five interactions. The photoelectric and Compton effects are the most important interactions for imaging with kilovoltage photons. 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.

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 mass attenuation coefficient varies as (1/Eγ) and Z , and the amount of coherent scattering is 3% to 8% or less in the conventional x-ray imaging (~ 100 kV) and therapy (MeV) energy ranges compared with other principal interactions. Diagnostic and security imaging techniques are being developed using low-energy x-ray beams (~ 30 kV) based on the changes in coherent scatter that occur for soft-tissue surfaces and interfaces.

Photoelectric effect.

The following actions occur in the photoelectric effect ( Fig. 6.14 ):

• 1.

  An incident photon with energy, Eγ = h ν , 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:

  $E_{\text{pe}}=E\gamma -E_B\text{}.$

  If E γ is less than E _B , the interaction cannot occur, but the interaction may occur with another orbital electron with a binding energy less than E γ. Depending on the incident photon energy and the orbital electron momentum at the time of interaction, the photoelectron can travel backwards toward the origin of the incident photon; however, the majority of ejected photoelectrons travel in the forward direction, especially as photon energy increases into the megavoltage range.

• 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 ):

  $E_{C_X}=E_{B1}-E_{B2}\text{}.$
  

• 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, , is transferred to one of the nearby orbital electrons without x-ray emission, and the electron, called an Auger electron , is ejected with energy:

  $E_{\text{AU}}=E_{\text{CX}}-E_{B2}=E_{B1}-2E_{B2}\text{}.$

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 the amount of characteristic x-ray emission, at a ratio 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/Eγ) and Z ³ , respectively. Mathematically, this is shown as follows, where C _PE is a proportionality constant:

$\mu /{\rho }_{\text{PE}}=C_{\text{PE}}{Z}^3/{\left(E\gamma \right)}^3\text{}.$

These dependencies for water are shown graphically in Fig. 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 , with the graphical representation called an absorption edge . In Fig. 6.16 , the K and L edges 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 graph.

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

$\mu /{\rho }_{\text{PE},2}=\mu /{\rho }_{\text{PE},1}{\left(E_{\gamma ,1}/E_{\gamma ,2}\right)}^3{\left(Z_2/Z_1\right)}^3\text{}.$

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

$\mu /{\rho }_{\text{PE},2}=\mu /{\rho }_{\text{PE},1}{\left(E_{\gamma 1}/E_{\gamma 2}\right)}^3\text{}.$

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

$\mu /{\rho }_{\text{PE},2}=\mu /{\rho }_{\text{PE},1}{\left(Z_2/Z_1\right)}^3\text{}.$

Compton effect.

In the Compton effect (see Fig. 6.15 ), several events occur:

• 1.

  An incident photon with energy, Eγ = 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:

  $E_{\text{ce}}=E_{\gamma }-E{\prime }_{\gamma }\text{}.$

• 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 are coupled owing to conservation of momentum and energy and can be described mathematically. The scattered photon has energy E ′γ = h ν′ given by

$h{\nu }^{\prime }=\frac{h\nu }{1+\alpha (1-\cos \theta )}=\frac{m_o{c}^2}{1+\left(\frac{1}{\alpha }-\cos \theta \right)}.$

In Eq. 12 , α = h ν/ m _o c ² 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 Fig. 6.15 .

The Compton electron has energy E _ce given by the equation

$E_{\text{ce}}=h\nu \frac{\alpha (1-\cos \theta )}{1+\alpha (1-\cos \theta )}$

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

$\cos \varphi =(1+\alpha )\tan \frac{\theta }{2}.$

Selected 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

  $h{\nu }^{\prime }=\frac{h\nu }{1+2\alpha }.$

  The 180-degree backscattered Compton photon energy will always be less than and will approach a maximum of 0.25 MeV (which equals 1/2 m _o c ² ) even as the incident photon energy becomes very large. The electron has maximum energy of E _ce = h ν − h ν′ = h ν (2α/[1 + 2α]) and travels in the forward direction.

• 3.

  A 90-degree Compton scattered photon has energy:

  $h{\nu }^{\prime }=\frac{h\nu }{1+\alpha }.$

The 90-degree scattered Compton photon energy will always be less than and will approach 0.511 MeV (which equals m _o c ² ) even as the incident photon energy becomes very large. The electron has energy of E _ce = h ν − h ν′ = h ν (α/[1 + α]), and it travels in a direction that depends on the incident photon energy.

For lower-energy incident photons, Compton electrons can scatter at most to 90 degrees from the forward direction—no backscatter greater than 90 degrees is possible—and they travel increasingly 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. The forward-peaked distribution, also contributed to by photoelectrons in the forward direction, 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 × 10 ²³ electrons per gram because the N/Z ratio is almost constant for most elements and materials. The exception is hydrogen, which, with one proton in the nucleus and no neutrons, has an electron density of around 6 × 10 ²³ 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 biological materials such as tissues. Mathematically, this relationship is

$\mu /{\rho }_{\text{CE}}=C_{\text{CE}}\text{}.$

In Eq. 17 , C _CE is almost a constant.

With its relatively constant mass attenuation coefficient for a range of materials, the attenuation by the Compton effect reduces to the following expression:

${\frac{I}{I_0}|}_{\text{CE}}=e^{-C_{\text{CE}}\text{px}}\text{}.$

For unit thickness of a material, the transmitted amount depends primarily 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 = h ν and E γ 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):

  $E_{\text{ep}}=E_{\gamma }-1.022\text{MeV}.$

  The electron and positron travel off in forward directions and do not have equal kinetic energies although their average (shared) energies are easily calculated.

• 2.

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

• 3.

  The positron, the antiparticle of an electron, slows down 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):

$\mu /{\rho }_{\text{pp}}=C_{\text{pp}}Z(E_{\gamma }-1.022\text{MeV}).$

Photodisintegration.

Several steps occur in photodisintegration ( Fig. 6.18 ):

• 1.

  An energetic photon of Eγ greater than 8 to 10 MeV interacts with the atomic nucleus.

• 2.

  The photon penetrates the nucleus and is absorbed. The energy deposition results in the emission of a nucleon—a neutron or proton. The nomenclature used is either (γ, p) or (γ, n) for a proton or neutron emitted from the nucleus, respectively.

• 3.

  Nucleon emission leaves a fragmented nucleus, prompting the name photodisintegration , which may also be unstable (i.e., radioactive).

A photon 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. It can be an important radiation safety consideration for both neutron contamination in treatment beams and 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 Fig. 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 effect 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 ).

Comparison of interaction dependencies shows why 30 to 100 kV energy photons give good contrast for diagnostic imaging: the photoelectric effect dominates and is quite sensitive to the atomic numbers of the materials being imaged. Materials with even slightly different Z s have good subject and image contrast because attenuation depends on the cube of the atomic numbers. Radiotherapy portal images with 6-MV photons, 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 biological materials (except for hydrogenous materials). Instead of imaging atomic number, the Compton effect images the density of a material, and subject contrast and image contrast are essentially representations of differences in material densities. A comparison of simulator and port images 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. 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 charged particle interacts at every opportunity. The quotient dE / dx (units of million electron volts per centimeter) is called the stopping power ; for charged particles, 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/g/cm ² ), 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:

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

In Eq. 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 (and energy) 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. 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. 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 previously, 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 cm ² /g), electrons are stopped sooner in low Z than in high Z materials. Fig. 6.21 shows these relationships for water and lead.

Radiative interactions of electrons 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 x-ray spectrum has a maximum energy equal to the initial electron energy and all other x-ray energies below this maximum 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 LINACs. 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 (x-ray production), resulting ultimately in heat deposition. At electron energies of 10 MeV, bremsstrahlung is a much more efficient process; approximately 50% of the interactions are collisional and approximately 50% radiative. At 100 keV, x-ray production is inefficient, whereas at 10 MeV, x-ray production is efficient. Above 10 MeV, bremsstrahlung x-ray 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/cm ³ (10 MeV ÷ [2 MeV/cm × 1.5]). These relationships use Eq. 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 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 Z ² and 1/ v ² discussed previously (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 . 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.

The proton finite range can be modulated through a physical attenuator or other means to decrease the range from the maximum. This process is essentially modulating the incident proton beam energy. If range modulation is done continuously while the beam is on, the Bragg peak will be swept over a range called the Spread Out Bragg Peak (SOBP) and will produce a high-dose plateau (see Fig. 6.22 ). The width of the SOBP 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 33.97 eV is required to produce one ion pair (ip) in air. This number is called the W value (W = 33.97 eV / ip for air), and it is almost independent of the radiation energy. W for other gases is similar to that of air, and its uniformity from material to material relates to atomic energy levels and capabilities for transfer of excitation energy. Note that several values for the W value have been determined over time; these variations are small and are responsible for the slight differences in the coefficients used for conversion of exposure to dose, as reviewed later.

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. For each interaction, it has origins as follows:

• 1.

  In coherent scattering, there is no ionization. However, the scattered photon is removed from the primary beam and might be able to undergo an ionizing interaction.

• 2.

  In the photoelectric effect, the inner shell electrons are ionized. The photoelectron is ejected and loses its energy by excitation and ionizations. Characteristic x-rays and Auger electrons are energetic enough to cause ionizations.

• 3.

  In the Compton effect, the outer shell electrons are ionized. The scattered photons and the Compton (recoil) electrons may be energetic enough to cause ionizations.

• 4.

  In pair production, there is no direct ionization of the atom. However, the electron and positron each have enough energy to cause ionizations, and annihilation radiation is energetic enough to cause ionizations.

• 5.

  In photodisintegration, there is no direct ionization of the atom. However, the remaining atom (with one fewer neutron or proton) may be unstable and decay by emission of ionizing particles. An ejected neutron may cause activation of other atoms, resulting in nuclear decay and ionizations, and an ejected proton will directly deliver dose.

If all of the created ions are collected and measured, the amount of energy deposited can be determined. This measurement of ionization is the basis for the determination of radiation dose.

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). Exposure has historical significance as the first quantitative definition of ionizing radiation quantity. The SI unit for exposure is a charge of 1 coulomb (C) per mass of 1 kilogram (kg): X = C/kg. The original unit of exposure, X, essentially as the first quantitative definition of radiation dose (circa 1933), is the roentgen (R) , named after the discoverer of x-rays, Wilhelm Conrad Roentgen:

$1R=2.58\times {10}^{-4}C/\text{kg}\text{of air}(=1\text{esu}/{\text{cm}}^3\text{in air}).$

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/cm ³ , was based on the e lectro s tatic u nit and is responsible for the odd units of the roentgen as now defined. Use of the roentgen as a radiation quantity remains quite common, particularly within the radiation protection field.

Several important concepts characterize exposure, X :

• 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-rays and gamma rays), not electrons or other particles.

• 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 within the volume of interest. The concept is illustrated in Fig. 6.23 . After interaction, ionization—made up of photoelectrons, Compton electrons, and pair production electrons—travels away from the interaction point in the direction of the incident photons and a portion actually leaves the volume of interest. To accurately measure the amount of ionization produced in the volume, 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 of energy flow is called electronic equilibrium or charged-particle equilibrium and is a required condition for measurement of exposure.

Dose

Dose, D , is the measurement of the amount of energy imparted per mass of material (energy per mass). 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 Gy = 100 rad. Although not an official unit, cGy is often used because of its numerical equivalency to the rad: 1 cGy = 1/100 Gy = 1 rad. Thus, 100 cGy = 1 Gy.

Dose does not have the restrictions of exposure. Therefore, the following two statements 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 that it describes energy absorbed, not attenuated. The process is illustrated in Fig. 6.24 for the following steps, using the original definition of exposure, the roentgen, and with the assumption of electronic equilibrium:

• 1.

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

  $X=1R.$

• 2.

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

  $\begin{array}{c}{D}_a=\text{WX}=33.97\text{eV}/\text{ip}\times 1R\\ \begin{array}{c}\left(\text{converting units}\right)=\left(33.97\text{eV}/\text{ip}\right)(1\text{ip}/1.6\times {10}^{-19}\text{coul})\\ \times (2.58\times {10}^{-4}C/\text{kg}/1R)\\ \times (1.6\times {10}^{-12}\text{erg}/\text{eV})\\ \times \left(1\text{kg}/1000g\right)\end{array}\\ \left(\text{converting to Gy}\right)=\left(87.6\text{erg}/g/R\right)\left(0.01\text{Gy}/100\text{erg}/g\right),\end{array}$

  thus,

  $D_a=0.00876\text{Gy}/R\text{for an exposure of}1R.$

  The coefficient of 0.00876 Gy/R is called the roentgen-to-gray conversion coefficient and is commonly expressed as the roentgen-to-centigray coefficient of 0.876 cGy/R. The important concept is that an exposure of 1 R, by definition, results in the energy release in air of 0.00876 Gy: 1 R → 0.00876 Gy (in air). This representation of exposure is called “air kerma”; it is the kinetic energies of charged particles released in air. Kerma and the Gray as a unit of dose are discussed later in greater detail.

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

  $D_a=0.00876\text{Gy}/RX\left(R\right).$

• 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 that

  $D_m={D_a\frac{{\mu }_{\text{en}}}{\rho }|}_a^m\text{},$

in which

${\frac{{\mu }_{\text{en}}}{\rho }|}_a^m$

is the mass energy absorption coefficient for the material, m,

${\frac{{\mu }_{\text{en}}}{\rho }|}_m$

divided by

${\frac{{\mu }_{\text{en}}}{\rho }|}_a$

the mass energy absorption coefficient for air, a.

Combining Eqs. 23 and 24 yields the following expressions:

$D_m={\frac{{\mu }_{\text{en}}}{\rho }|}_a^{m}0.876\text{cGy}/\text{RX}\left(R\right)$

and

$D_m=\text{fX}\left(R\right),$

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

$f={\frac{{\mu }_{\text{en}}}{\rho }|}_a^{m}0.876\text{cGy}/R.$

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 that is inserted into the radiation beam and the ionization collected in the chamber is converted to dose using exposure-to-dose calibration factors in a prescribed procedure that accounts for the properties of the ionization chamber and particular radiation beam energy and irradiation geometry.

The f factor is an important parameter, which varies as a function of the material and the photon energy, as shown for water, muscle, and bone in Fig. 6.25 . By its definition, the f factor for air is always 0.876 cGy/R. 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 4 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. Note the decrease in f factor for fat in the photoelectric region because of its lower Z as a result of its high hydrogen content.

For treatment with ionizing radiation, the amount of energy deposited is quite small per fraction of dose. It takes about 1 million cGy to raise the temperature of 1 cm ³ of water by 1° C. Conversely, for a daily treatment dose of 2 Gy (200 cGy), the increase in temperature per gram is only 0.0002° C. Thermal effects are nonexistent for fractionated and radiosurgical radiation treatment.

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. Fig. 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.

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 without interruption, independent of the absence or presence of ionizing radiation. The detector is not useful at this applied voltage.

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