Decay Constant

The decay of a large collection of nuclei can be described as a statistical phenomenon. At any moment in time, the number of nuclei disintegrating per unit of time is proportional to the number of nuclei available. Mathematically,

where N is the number of radioactive nuclei and λ is the decay constant. The minus sign indicates that the number of nuclei is decreasing with time. Integration of this differential equation leads to the following equation:

where N₀ is the number of radioactive nuclei initially present, N(t) represents the number remaining at time t, and e is the base of the natural logarithm (2.71828).

The rate of decay, previously referred to as dN/dt, is also called the activity of a radioactive substance, referred to by the symbol A. Then A = − λN and

where A₀ is the initial activity and A(t) is the activity at time t. The unit of activity is the Curie (Ci) or the Becquerel (Bq), which is 1 disintegration per second:

The Ci was originally defined to be the activity associated with 1 g of radium, although the number of disintegrations per second from 1 g of radium is now known to be somewhat less than 3.7 × 10¹⁰.

Half-Life and Mean Life

The half-life (T_½) of a radioactive substance is the time required for a population of atoms to decay to half the original number. That is,

Solving this equation,

the mean or average life (T_λ) is the time for the number of atoms to decay to 1/e of the initial number, so

The mean life can also be thought of as the time required for all the atoms to decay if the decay rate could be maintained at its initial value.

Modes of Radioactive Decay

Just as electrons have energy levels related to the orbit or shell they occupy, nucleons (protons and neutrons) are thought to occupy shells that represent discrete energy levels within the nucleus. When nucleons are excited into higher levels, they can subsequently decay back to their initial energy level, emitting a photon in the process. This process is called γ decay. Because of the very strong nature of nuclear forces, these excited states are very short-lived, with half-lives on the order of 10⁻¹⁵ seconds. If, in the process of leaving the nucleus, the γ-decay photon transfers enough energy to an inner shell electron to eject it from the atom, the process is called internal conversion. The subsequent cascade of an outer shell electron into the vacated hole in the inner shell releases a characteristic x-ray photon with energy equal to the difference between the energy levels of the inner and outer shells. In some situations, the characteristic x-ray is absorbed and re-emitted by the atom in the form of an ejected orbital electron. Such electrons, emitted in lieu of x-rays, are called Auger electrons.

The decay of a nucleus accompanied by the ejection of an electron (e⁻) or positron (e⁺) is called β decay. In β (−) decay, a nucleus that has an excess of neutrons can reduce its energy by converting a neutron into a proton, an electron, and an antineutrino (which has energy but no mass). In β (+) decay, a neutron-deficient nucleus can become more stable by converting a proton into a neutron, a positron, and a neutrino. The typical disintegration energy of 1 to 3 MeV is shared between the emitted particles, the neutrino (antineutrino), and positron (electron). For heavy nuclei, the average binding energy of the nucleons inside the nucleus may be less than 10 MeV. Half-lives for β decay can range from a few seconds to many years.

A second way in which neutron-deficient nuclei can gain stability is by the process of electron capture. In this process, an electron in one of the inner shells is captured by the nucleus, thus, transforming a proton into a neutron. That is, a proton plus the captured electron is converted to a neutron plus a neutrino. In general, the inner electron orbital vacancy is quickly filled with an electron from an outer shell, resulting in the emission of characteristic x-rays or Auger electrons. Very heavy nuclei with Z > 82 frequently decay with the emission of an α particle, which is identical to the helium nucleus and is composed of two protons and two neutrons tightly bound together. The binding energy of this configuration is particularly high, making this decay mode a very efficient way to decrease the energy and increase the stability of a heavy nucleus. From a specific nuclide, the α decay is monoenergetic, that is, the α particle is emitted with a single, well-defined energy. Typically, α particles are emitted with kinetic energies of between 5 and 10 MeV.

Radioactive Decay Series and Equilibrium

A total of 106 elements are known. Of these, the first 92 occur naturally; that is, they are either stable or radioactive with half-lives long enough that they can still be found naturally in trace amounts on the earth. Elements 93 to 106 are very unstable and can only be produced artificially in accelerators with heavy ion bombardment of heavy nuclei. Lead, with Z = 82 is the heaviest known element that has completely stable isotopes. All naturally occurring isotopes of elements with Z between 83 and 92 are members of one of three radioactive series referred to as the uranium (Z = 92) series, the actinium (Z = 89) series, and the thorium (Z = 90) series. The name of each series refers to the parent, a long-lived nuclide that decays into a daughter nuclide, that then decays into another daughter, followed by successive transformations that continue until a stable isotope of lead is reached.

In each series, the half-life of the parent is longer than any of the half-lives of any of the daughter nuclides. This phenomenon leads to a state of transient or secular equilibrium. Since each daughter is less stable than the original parent, the activity of the daughter will increase with time, until it approaches the activity of the long-lived parent. If the half-lives are very different, the activities will become essentially identical after a number of daughter half-lives, and this is called secular equilibrium. An example is shown in Fig. 7-3. If the half-lives of the parent and daughter are very similar, transient equilibrium is reached in a few half-lives, after which the daughter has an activity in excess of the parent activity but decays with the same rate as the limiting parent decay rate. An example of this phenomenon is shown in Fig. 7-4. For either case, after a time that is long compared to the half-life of the shorter-lived nuclide,

FIGURE 7-3 • Example of secular equilibrium in the decay of ²²⁶R_a to ²²²R_n.

(From Khan FM: The physics of radiation therapy, ed 3, Philadelphia, 2003, Lippincott Williams & Wilkins, p 19.)

FIGURE 7-4 • Example of transient equilibrium in the decay of ⁹⁹Mo to ^99mTc. This figure is drawn as if 100% of the ⁹⁹Mo decays to ^99mTc. In fact, only about 88% of the ⁹⁹Mo atoms decay in this fashion, and in actuality the ^99mTc activity curve lies slightly below the ⁹⁹Mo curve.

(Redrawn from Khan FM: The physics of radiation therapy, ed 3, Philadelphia, Lippincott Williams & Wilkins, p 18.)

where A_i and T_i refer to the activity and half-life of the daughter (d) or parent (p).

Characteristic X-Rays

When electrons are incident on target atoms, they can ionize those atoms by depositing sufficient energy to eject an inner shell electron. The inner shell vacancy is subsequently filled by an outer shell electron, causing the emission of a characteristic x-ray with energy equal to the difference between the binding energies of the inner and outer shells. This process is diagrammed in Fig. 7-5. Although this process can take place in both low-Z and high-Z atoms, only for high-Z atoms are the binding energies sufficient to produce radiation in the x-ray portion of the electromagnetic spectrum. For example, the binding energy of K-shell electrons in tungsten, a common target for x-ray tubes and accelerators, is about 70 keV, whereas for aluminum, it is only about 1.5 keV. As mentioned earlier, the characteristic x-ray energy is occasionally transferred directly to an orbital electron, leading to the production of an Auger electron.

FIGURE 7-5 • Illustration of the production of characteristic radiation.

(From Khan FM: The physics of radiation therapy, ed 3, Philadelphia, Lippincott Williams & Wilkins, p 35.)

Bremsstrahlung X-Rays

When high-energy electrons interact directly with the electromagnetic field of a target nucleus, they are deflected and lose a portion of their energy due to deceleration. The energy lost is then emitted in the form of radiation called bremsstrahlung (literally, braking radiation).

Since the incident electron can lose any portion of its energy in this process, the energy of these bremsstrahlung x-rays can vary continuously from nearly zero to the full energy of the incident electron. The angle of the x-rays, relative to the beam direction, depends on the energy of the electron. At low energies typical of those used for diagnostic radiology, the x-ray has equal probability of being emitted in any direction, whereas at high energies typically used for radiotherapy, the x-rays are emitted preferentially in the forward direction. These facts explain the use of thick targets and 90-degree angles between electrons and x-rays for diagnostic applications and thin transmission targets for radiotherapy.

The probability of bremsstrahlung production varies with Z² of the target material, whereas the efficiency of x-ray production depends on the product of Z and E, the energy of the electrons. At 100 keV in a tungsten target, only about 1% of the incident energy of the beam is converted into x-rays, the remainder ending up in heat.

In general, both bremsstrahlung and characteristic x-rays are present when electrons strike a target. Fig. 7-6 shows x-ray spectra for a thick tungsten target for electron energies of from 65 keV to 200 keV. Note the appearance of the characteristic x-rays on top of the bremsstrahlung spectrum once the electron energy is in excess of the K-shell binding energy of 70 keV. Any material in the x-ray beam, including the target itself, will absorb some of the energy of the beam. The effect of such absorption is to preferentially filter out the low energy portion of the bremsstrahlung spectrum. The process of preferentially attenuating the low-energy component of the beam is referred to as “beam hardening.” When the beam is hardened, the average energy of the x-rays increases at the expense of reduced intensity.

FIGURE 7-6 • Spectral distribution of x-rays calculated for a thick tungsten target. Dotted curves are for no filtration and solid curves are for a filtration of 1 mm aluminum. Note the characteristic radiation peaks on top of the continuous bremsstrahlung spectra.

(From Khan FM: The physics of radiation therapy, ed 3, Philadelphia, Lippincott Williams & Wilkins, p 36.)

Coherent Scattering

Coherent scattering, also called classical scattering, is a low-energy phenomenon in which the electromagnetic field of the incident photon interacts with the field of a bound electron in the atom without any transfer of energy. The bound electron is set into oscillation by the passing photon, but then reirradiates the energy of oscillation back to the outgoing photon. The net effect is only a change in direction for the photon. Since there is no transfer of energy, this process is of little importance to radiotherapy.

Photoelectric Effect

In the photoelectric effect, the energy of the photon is totally absorbed by the atom and subsequently transferred to an orbital electron, which is then ejected from the atom with an energy equal to the original energy of the photon minus the binding energy of the electron. The vacancy created by the ejection of the photoelectron from a shell gets filled by an electron from a shell with lower binding energy, followed by the emission of a characteristic x-ray with energy equal to the difference in binding energies of the two shells. Again, an Auger electron can be emitted in lieu of the x-ray in some cases.

The most likely angle of ejection of the photoelectron relative to the incident photon direction is 90 degrees for low-energy photons (50 keV or less), becoming smaller (more forward) as the photon energy increases. The mass attenuation coefficient for the photoelectric effect is proportional to Z³/E³, where Z is the atomic number of the target atom and E is the energy of the photon.

The photoelectric effect is important at diagnostic x-ray energies of 100 keV and is the basis for the radiographic contrast of bone versus soft tissue. However, because of the 1/E³ energy dependence, the photoelectric effect is relatively unimportant at typical radiotherapy energies of several MeV.

Compton Scattering

Compton scattering is a process in which the incident photon interacts with an orbital electron as if it were a free particle, since the binding energy is small compared to the photon energy. The dynamics of the interaction can be described as a typical particle–particle scattering interaction whereby the photon transfers some of its energy to the electron and is scattered at an angle φ relative to the incident direction. The electron is ejected at an angle θ relative to the forward direction. The energy of the outgoing Compton-scattered photon is equal to the difference between the incident photon energy and the energy transferred to the electron. A diagram of the Compton process is shown in Fig. 7-8.

FIGURE 7-8 • Illustration of the Compton effect.

(From Khan FM: The physics of radiation therapy, ed 3, Philadelphia, Lippincott Williams & Wilkins, p 67.)

As with any particle–particle scattering process, energy and momentum conservation can be applied to obtain relationships between energy and angle.

From conservation of energy:

From conservation of momentum:

and

where v_e is the velocity of the electron, m_e is the rest mass of the electron, c is the speed of light, and

Solving these three simultaneous equations, we obtain the following relationships:

where α = hν_o/ m_ec² and f = (1 − cos φ).

From these equations, we can examine several limiting cases of the Compton effect. A grazing hit corresponds to θ = 90 degrees, in which case the photon is undeflected, and continues in the forward direction. A direct hit corresponds to θ = 0 degrees, meaning that φ = 180 degrees, or in other words, the photon scatters backwards. For intermediate situations, for example, when the photon is deflected by 90 degrees, the angle and energy of the electron depend on the energy of the photon. When α is much greater than 1 (high energy), θ is small and E is essentially equal to hν_o. When α is much less than 1 (low energy), the photon energy remains nearly unchanged and the Compton process approaches that of coherent scattering.

The mass attenuation coefficient for Compton scattering is independent of Z, decreases slowly with photon energy, and is directly proportional to the number of electrons per gram, which varies by only 20% from the lightest to the heaviest elements (with the exception of hydrogen). Therefore, in the energy region where Compton processes dominate, the attenuation of the beam will vary according to the integrated density of the material traversed. This fact is responsible for the relatively poor contrast observed in portal verification films exposed with megavoltage x-rays exiting from the irradiated patient.

Pair Production

Pair production is an interaction of a photon with the electromagnetic field of a nucleus in which the energy of the photon is converted into an electron (e⁻) and a positron (e⁺). Since the rest mass of each of these particles is 0.511 MeV, the threshold energy for pair production is 1.02 MeV. The total shared kinetic energy of this pair of particles is just the photon energy minus 1.02 MeV. The mass attenuation coefficient for pair production increases logarithmically with energy above the threshold, and is proportional to Z². Subsequent to the production of the electron-positron pair, the positron has a high probability of combining with a free electron during the process of energy loss in the medium and converting the combined e⁺-e⁻ mass into a pair of annihilation photons, each of energy 0.511 MeV, that will leave the annihilation region in opposite directions.

Photodisintegration

At very high energies, the photon can deposit so much energy into the nucleus that partial or complete disintegration of the nucleus takes place. Since the binding energies of nucleons within the nucleus tend to be typically 7 MeV or higher, this process is of little importance at therapy energies. However, photodisintegration is a source of low-level neutron production that must be considered when designing radiation shielding around high-energy linear accelerators.

Relative Importance of Interaction Types

The total mass absorption coefficient is the sum of the coefficients for coherent, photoelectric, Compton, and pair production. For low-Z targets such as tissue (average Z = 7), Compton processes are overwhelmingly dominant throughout the range of energies used in therapy. Fig. 7-9 shows the relative importance of photoelectric, Compton, and pair production as a function of photon energy and the atomic number of the target material.⁴ For tissue, Compton dominates between approximately 30 keV and 30 MeV, whereas for lead, photoelectric dominates up to approximately 800 keV and pair production above about 5 MeV. Because of its calcium content, bone has a higher absorption by photoelectric and pair production. This leads to high bone doses for orthovoltage and kilovoltage and for higher energy accelerator beams.

FIGURE 7-9 • Relative importance of the three main modes of energy loss for photons as a function of energy and atomic number of the medium.

(From Hendee WR: Medical radiation physics: roentgenology, nuclear medicine and ultrasound, Chicago, 1979, Year Book Medical Publishers, p 115.)

Electrons

Electrons lose energy to the medium by two processes, ionization and radiation. Ionization processes are interactions with the atomic electrons, whereas radiation losses are interactions with the field of the nucleus, leading to bremsstrahlung x-ray production. Whereas the microscopic interactions are well understood, the macroscopic description of the energy and range of the particles at any point in the medium is not simple. This is because the electrons are very much lighter than the atomic nuclei. Therefore, the electron can lose a very large fraction of its energy in a single process and can be deflected by very large angles. This means that even if the electron beam is monoenergetic when entering a medium, there will be a large amount of range-straggling, or a large variation from electron to electron as to where, in the phantom, the electron will stop. Fig. 7-10 shows a plot of absorbed dose as a function of depth for monoenergetic electrons incident on water. There is a depth beyond which the dose is almost zero, where all the incident electrons have been stopped and where the remaining dose is due to bremsstrahlung x-rays produced by the electrons in the medium. This is in sharp contrast to the exponential falloff of dose for x-rays. This ranging out of electrons in matter is responsible for the popularity of electrons for treatment of superficial disease.

FIGURE 7-10 • Depth-dose curve for monoenergetic electrons incident on water. R₅₀ is the mean range and R_p is the extrapolated or practical range.

(From Johns HE, Cunningham JR: The physics of radiology, ed 4, Springfield, IL, 1983, Charles C. Thomas.)

Protons and Heavier Ions

The use of protons and heavy ions for radiotherapy was pioneered in the 1970s and 1980s at the Massachusetts General Hospital in collaboration with the now decommissioned Harvard Cyclotron Laboratory and at the Lawrence Berkeley National Laboratory in Berkeley, California.^5–15 Since then, several proton and charged-particle facilities have been built around the world, a list of which can be found on the website for the Particle Therapy Co-Operative Group (PTCOG; http://ptcog.web.psi.ch/). Similar to electrons, protons lose energy primarily by electromagnetic interactions with the atomic electrons. A major difference between protons and electrons is that protons are much heavier than electrons and therefore lose only a very small fraction of their energy in an individual interaction, scattering minimally in the process. Protons lose energy at an increasing rate as they slow down, yielding an enhanced region of energy deposition called the Bragg Peak just before they stop. One can predict very accurately the depth at which the protons will come to rest if the initial energy of the protons and the electron density of the material traversed are known. The absence of exit dose beyond the intended target makes protons nearly ideally suited for optimal physical dose delivery. A typical depth dose distribution for a clinical, energy-modulated proton beam, compared to that of a 10-MV bremsstrahlung x-ray beam, is shown in Fig. 7-11. A small percentage of proton interactions is via nuclear interactions with the nucleus. These rare interactions are qualitatively different than electromagnetic interactions and are thought to be responsible for enhancing the cell kill per unit dose by about 10% to 20% over that of x-rays.¹⁶

FIGURE 7-11 • Depth-dose curve for a modulated proton beam of 160 MeV compared to that for a 10-MV bremsstrahlung x-ray beam. The region of the flat dose is referred to as the spread-out bragg peak (SOBP).

Heavy ions such as helium, carbon, neon, and argon nuclei have also been used in radiotherapy.^{9–12,14,15,17,18} Similar to protons, they lose energy by interacting with the atomic electrons. Being even heavier than protons, they scatter even less and have even more rapid dose falloff outside the central beam and even faster falloff of dose beyond the end of range. In addition to these electromagnetic interactions, heavy ions have a probability of interacting with the nucleus via nuclear interactions that increases with the mass of the incident ion. In particular, as the heavy ions begin to slow down, they tend to suffer nuclear interactions in which they lose a very large amount of their energy in a single event. This high density of deposited energy kills cells in a far more efficient way per unit dose than x-rays. These particles are said to have a high radiobiologic efficiency (RBE). For more information on these particles, the reader is referred to the literature.¹³

Contact Therapy Machines

X-ray machines that operate at potentials of 40 to 50 kV are referred to as contact units. They typically operate at tube currents of 2 mA and the beams are usually filtered with 0.5 to 1.0 mm aluminum in order to remove the very low energy x-rays in the beam. Treating at a typical source-to-skin (SSD) distance of 2 cm, the dose in this beam drops off to 50% of its surface value in less than 5 mm of water or soft tissue. Such a beam would be useful only for the most superficial of targets.

Superficial Therapy Machines

Units with x-ray beams produced by electrons with energies between 50 and 150 kV are usually referred to as superficial therapy units. These units normally are filtered with 1 to 4 mm aluminum and treated at distances of 20 cm SSD. The 50% depth in water or soft tissue in this energy range would be typically 1 to 2 cm. By using thicker filters, it is possible to further harden the beam and move the 50% dose somewhat deeper with a reduction in dose rate. Superficial units can be very useful for treatment of skin lesions, using either regular fields defined by interchangeable cones or irregular fields defined with custom lead cutouts.

Orthovoltage Therapy Machines

Orthovoltage x-ray units are defined as those that operate in the 150 to 300 kV range. Typical SSDs are 50 cm, field sizes up to 20 × 20 cm and filters of 1 to 4 mm of copper. Depths of 50% are usually between 5 and 7 cm depending on filter thickness and field size. Regular fields are defined with detachable cones or adjustable collimators and irregular fields with lead cutouts or special hand blocking. Before 1950, these units were the workhorses of radiation therapy. Although useful for treating disease in thin sections of the body such as the neck, in thicker areas of the body, where tumors may lie 10 to 15 cm below the surface, the dose to normal tissue can be quite high when using beams in this energy range. Very few of these machines remain in current use in radiation therapy centers.

Teletherapy Machines

The megavoltage era began with the introduction of the ⁶⁰Co teletherapy machine into radiotherapy clinics beginning in 1951.^34,35 The development of nuclear reactors in the late 1940s made possible the production of small ⁶⁰Co sources with specific activities (in Curies per gram) high enough to produce clinically acceptable dose rates of more than 1 Gray (Gy) per minute at a typical treatment distance of 80 cm from the source. These machines quickly became the standard of radiotherapy because of their simplicity of design and operation, low cost, and availability. The two photons produced by the decay of ⁶⁰Co have energies of 1.17 and 1.33 MeV, yielding a depth-dose curve that falls to 50% of maximum in about 10 cm of water or tissue. In addition, the relatively high energy of the photons leads to “skin sparing” due to a reduced entrance dose that builds up for a distance of 5 mm (the maximum range of the secondary electrons from ⁶⁰Co photons) to a maximum value. The half-life of ⁶⁰Co is 5.27 years, making the source change a relatively infrequent requirement. With the highest specific activities currently achievable, dose rates of at least 1.5 Gy per minute for a field size of 40 cm × 40 cm can be achieved at 80 cm source-to-treatment distance.

Aside from the requirement to replace the sources every few years, additional disadvantages to ⁶⁰Co isotope machines include the need to shield against continuous leakage radiation and the lack of field flatness for the largest field sizes. In addition, the physical size of the source capsule, typically 1.5 cm to 2.0 cm in diameter, is much larger than the effective source size available on any machine that uses an electron beam (typically 4 mm or less for linear accelerators). This large source size contributes to a geometric penumbra (the penumbra is the area at the edge of the beam where the dose falls from high dose to low dose) that can be significantly wider than with accelerators.

Before the development of ⁶⁰Co isotope machines, both radium and cesium sources were used for teletherapy treatments. However, the limitations of low specific activity for radium and low energy and limited specific activity for cesium limited their usefulness for general radiotherapy.

Betatrons

The next chronological development in treatment machines is the betatron, first developed by D. W. Kerst at the University of Illinois³⁶ primarily for physics experimentation. Electrons with energies of up to 45 MeV have been produced in betatrons for radiotherapy and used either directly or to produce bremsstrahlung x-ray beams. The massive size of these machines, their high cost, and relatively low dose rate combined to limit their usefulness in radiation therapy.

Linear Accelerators

The use of microwaves to accelerate electrons to high energies for radiotherapy was first demonstrated in Great Britain in 1953.³⁷ This became possible primarily because of the development of high-power microwave generators for military radar use during World War II. The major components of a typical medical linear accelerator (linac) are shown in a block diagram in Fig. 7-13. The acceleration of the electron beam takes place in the accelerator wave guide, consisting of a stack of cylindrical cavities with a hole through the center, into which resonant electromagnetic waves of frequency in the microwave range (∼3000 MHz) have been coupled. The electron beam is created and preaccelerated to approximately 50 keV in a conventional electrostatic electron gun, injected into the resonating waveguide, spatially bunched and accelerated through interactions with the electromagnetic field in the individual cavities, and then emerges from the accelerating structure as a narrow pencil beam that can be magnetically bent or focused onto a scattering foil (for electron treatments) or a bremsstrahlung target (for x-ray treatments) in the treatment head of the linac. For low energies of 6 MeV or less, the beam can be magnetically focused in the forward (vertical) direction onto the axis of the treatment head, whereas at higher energies, the beam is usually bent through a 90-degree or 270-degree angle before entering the treatment head since the accelerating structure is much longer for high energies and is frequently oriented horizontally to save space. For more details about the acceleration process, the reader is referred to several excellent review articles in the literature.^38–40

FIGURE 7-13 • A block diagram of a typical medical linear accelerator.

(From Khan FM: The physics of radiation therapy, ed 3, Philadelphia, 2003, Lippincott Williams & Wilkins, p 43.)

Although the details of the acceleration process can be quite technical, the shaping of the beam in the treatment head is both important and conceptually easy to understand. Fig. 7-14