The quantity kerma (K) (kinetic energy released in the medium) is defined as “the quotient of dE_tr by dm, where dE_tr is the sum of the initial kinetic energies of all the charged ionizing particles (electrons and positrons) liberated by uncharged particles (photons) in a material of mass dm” (1).

The unit for kerma is the same as for dose, that is, J/kg. The name of its SI unit is gray (Gy).

For a photon beam traversing a medium, kerma at a point is directly proportional to the photon energy fluence Ψ and is given by

where is the mass energy transfer coefficient for the medium averaged over the energy fluence spectrum of photons. As discussed in Section 5.4,

where /ρ is the averaged mass energy absorption coefficient and is the average fraction of an electron energy lost to radiative processes. Therefore,

A major part of the initial kinetic energy of electrons in low-atomic-number materials (e.g., air, water, soft tissue) is expended by inelastic collisions (ionization and excitation) with atomic electrons. Only a small part is expended in the radiative collisions with atomic nuclei (bremsstrahlung). Kerma can thus be divided into two parts:

where K^col and K^rad are the collision and the radiative parts of kerma, respectively. From Equations 8.8 and 8.9,

and

In Chapter 6, the quantity exposure was defined as dQ/dm where dQ is the total charge of the ions of one sign produced in air when all the electrons (negatrons and positrons) liberated by photons in (dry) air of mass dm are completely stopped in air.

Exposure is the ionization equivalent of the collision kerma in air. It can be calculated from K^col by knowing the ionization charge produced per unit of energy deposited by photons. The mean energy required to produce an ion pair in dry air is almost constant for all electron energies and has a value of = 33.97 eV/ion pair (2). If e is the electronic charge (= 1.602 × 10-¹⁹ C), then is the average energy required per unit charge of ionization produced. Since 1 eV = 1.602 × 10-¹⁹ J, = 33.97 J/C. Exposure (X) is given by

From Equations 8.10 and 8.12,

The relationship between absorbed dose (D) and the collision part of kerma (K^col) is illustrated in Figure 8.1 when a broad beam of photons enters a medium. Whereas kerma is maximum at the surface and decreases with depth, the dose initially builds up to a maximum value and then decreases at the same rate as kerma. Before the two curves meet, the electron buildup is less than complete, and

where β is the quotient of absorbed dose at a given point and the collision part of kerma at the same point.

Because of the increasing range of the electrons, complete electronic equilibrium does not exist within megavoltage photon beams. However, conceptually electronic equilibrium would exist if it were assumed that photon attenuation is negligible throughout the region of interest. Then

At depths greater than the maximum range of electrons, there is a region of quasie-quilibrium called the transient electron equilibrium in which

In the transient equilibrium region, β is greater than unity because of the combined effect of attenuation of the photon beam and the predominantly forward motion of the electrons. Because the dose is being deposited by electrons originating upstream, one can think of a point somewhere upstream at a distance less than the maximum electron range from where the energy is effectively transported by secondary electrons. This point has been called the “center of electron production” (3). Since the center of electron production is located upstream relative to the point of interest, the dose is greater than kerma in the region of transient electronic equilibrium.

The relationship between absorbed dose and photon energy fluence Ψ at a point is given by

Suppose D₁ is the dose at a point in some material in a photon beam and another material is substituted of a thickness of at least one maximum electron range in all directions from the point; then D₂, the dose in the second material, is related to D₁ by

The factor β has been calculated for ⁶⁰Co and other photon energies for air, water, polystyrene, carbon, and aluminum (4,5). The results show that the value of β varies with energy, not medium. A fixed value of β = 1.005 has been used for ⁶⁰Co in conjunction with ion chamber dosimetry (6).

For further details of the relationship between absorbed dose and kerma and its significance in dosimetry, the reader is referred to a paper by Loevinger (4).

Determination of absorbed dose from exposure is readily accomplished under conditions of electron equilibrium. However, for energies in the megavoltage range, the electron fluence producing absorbed dose at a point is characteristic of photon energy fluence some distance upstream. Consequently, there may be appreciable photon attenuation in this distance. The calculation of absorbed dose from exposure when rigorous electronic equilibrium does not exist is much more difficult, requiring energy-dependent corrections. Therefore, the determination of exposure and its conversion to absorbed dose is practically limited to photon energies up to ⁶⁰Co. In the presence of charged particle equilibrium (CPE), the dose at a point in any medium is equal to the collision part of kerma; that is, β = 1. Dose to air (D_air) under these conditions is given by (see Equation 8.12)

Inserting units:

Since 1 cGy = 10⁻² J/kg,

From Equation 8.20, it is seen that the conversion factor from roentgen to cGy for air, under the conditions of electronic equilibrium, is 0.876.

In the presence of full CPE (i.e., β = 1 in Equation 8.17), the absorbed dose (D) to a medium can be calculated from the energy fluence Ψ and the weighted mean mass energy absorption coefficient, :

Suppose Ψ_air is the energy fluence at a point in air and Ψ_med is the energy fluence at the same point when a material other than air (medium) is interposed in the beam. Then, under conditions of electronic equilibrium in either case, the dose to air is related to the dose to the medium by the following relationship:

where A is a transmission factor that equals the ratio Ψ_med/Ψ_air at the point of interest.

From Equations 8.19 and 8.22, we obtain the relationship between exposure to air and absorbed dose to a medium:

Again, if X is in roentgens and D_med is in cGy, we have

The quantity in brackets has frequently been represented by the symbol f_med so that

where

The quantity f_med or simply the f factor is sometimes called the roentgen-to-rad conversion factor. As the above equation suggests, this factor depends on the mass energy absorption coefficient of the medium relative to the air. Thus, the f factor is a function of the medium composition as well as the photon energy.

A list of f factors for water, bone, and muscle as a function of photon energy is given in Table 8.1. Since for materials with an atomic number close to that of air, for example, water and soft tissue, the ratio varies slowly with photon energy (˜10% variation from 10 keV and 10 MeV), the f factor for these materials does not vary much over practically the whole therapeutic range of energies. However, bone with a high effective atomic number not only has a much larger f factor between 10 and 100 keV, but also the f factor drops sharply from its maximum value of 4.24 at 30 keV to about 1.0 at 175 keV. This high peak value and rapid drop of the f factor are the result of the photoelectric process for which the mass energy absorption coefficient varies approximately as Z³ and 1/E³ (see Chapter 5). At higher photon energies where the Compton process is the predominant mode of interaction, the f factors are approximately the same for all materials.

Strictly speaking, in the Compton range of energies, the f factor varies as a function of the number of electrons per gram. Since the number of electrons per gram for bone is slightly less than for air, water, or fat, the f factor for bone is also slightly lower than for the latter materials in the Compton region of the megavoltage energies. Of course, the f factor is not defined beyond 3 MeV since the roentgen is not defined beyond this energy.

As discussed in Chapter 6, a cavity ion chamber is exposure calibrated against a free-air ion chamber or a standard cavity chamber, under conditions of electronic equilibrium. For lower-energy radiations such as x-ray beams in the superficial or orthovoltage range, the chamber walls are usually thick enough to provide the desired equilibrium, and therefore, the chamber calibration is provided without a buildup cap. However, in the case of higher-energy radiations such as from cobalt-60, a buildup cap is used over the sensitive volume of the chamber so that the combined thickness of the chamber wall and the buildup cap is sufficient to provide the required equilibrium. This buildup cap is usually made up of acrylic (same as Plexiglas, Lucite, or Perspex) and must be in place when measuring exposure.

Suppose the chamber is exposed to the beam (Fig. 8.2A) and the reading M is obtained (corrected for air temperature and pressure, stem leakage, collection efficiency, etc.). The exposure X is then given by

where N_x is the exposure calibration factor for the given chamber and the given beam quality. The exposure thus obtained is the exposure at point P (center of the chamber-sensitive volume)
in free air in the absence of the chamber (Fig. 8.2B). In other words, the perturbing influence of the chamber is removed once the chamber calibration factor is applied.

Consider a small amount of soft tissue at point P that is just large enough to provide electronic equilibrium at its center (Fig. 8.2C). The dose at the center of this equilibrium mass of tissue is referred to as the dose in free space. The term “dose in free space” was introduced by Johns and Cunningham (7), who related this quantity to the dose in an extended tissue medium by means of tissue-air ratios (to be discussed in Chapter 9).

Equation 8.25 can be used to convert exposure into dose in free space, D_f.s.:

where A_eq is the transmission factor representing the ratio of the energy fluence at the center of the equilibrium mass of tissue to that in free air at the same point. Thus, A_eq represents the ratio of the energy fluence at point P in Figure 8.2C to that at the same point in Figure 8.2B. For cobalt-60 beam, A_eq is close to 0.99 (7) and its value approaches 1.00 as the beam energy decreases to the orthovoltage range.

Equations 8.27 and 8.28 provide the basis for absorbed dose calculation in any medium from exposure measurement in air. A similar procedure is valid when the exposure measurement is made with the chamber imbedded in a medium. Figure 8.3A shows an arrangement in which the chamber with its buildup cap is surrounded by the medium and exposed to a photon energy fluence Ψ_b at the center of the chamber (point P). If the energy of the beam incident on the chamber is such that a state of electronic equilibrium exists within the air cavity, then the exposure at point P, with the chamber and the buildup cap removed, is given by

X = M · N_x

The exposure thus measured is defined in free air at point P due to energy fluence Ψ_c that would exist at P in the air-filled cavity of the size equal to the external dimensions of the buildup cap (Fig. 8.3B). To convert this exposure to absorbed dose at P in the medium, the air in the cavity must be replaced by the medium (Fig. 8.3C) and the following equation is applied:

D_med = X · f_med · A_m

or

where A_m is the transmission factor for the photon energy fluence at point P when the cavity in Figure 8.3B is replaced by the medium. If Ψ_m is the energy fluence at P in the medium, the factor A_m is given by Ψ_m/Ψ_c. A_m has been called the displacement factor.

The above equation is similar to Equation 8.28 except that A_m is used instead of A_eq. However, the difference between A_m and A_eq is small for a tissue equivalent medium since the equilibrium mass of tissue to which A_eq applies is only slightly smaller than the mass of the medium displaced by a typical small ion chamber with its buildup cap.

An interesting question arises in regard to the necessity of the buildup cap being left on the chamber when making measurements in a medium such as water. If the chamber has been calibrated for exposure in air with its buildup cap on (to achieve electronic equilibrium) and if a significant part of the cavity ionization is the result of electrons produced in the buildup cap, then replacing the buildup cap with the medium could alter the chamber reading. This substitution of a layer of medium for the buildup cap could change the electronic and photon fluence incident on the chamber wall by virtue of differences in the composition of the medium and the material of the buildup cap. However, in practical calibration measurements, no significant differences have been observed when exposing the chamber in water with and without the acrylic buildup cap. Day et al. (8) added Perspex sheaths up to 5 mm in thickness to a Baldwin-Farmer ionization chamber irradiated at a depth of 5 cm in a water phantom using radiations from ¹³⁷Cs to 6 MV. The readings differed only by less than 0.5%.

The term stopping power refers to the energy loss by electrons per unit path length of a material (for greater details, see Section 14.1). An extensive set of calculated values of mass stopping powers has been published (15,16). As mentioned earlier, to use stopping power ratios in the Bragg-Gray formula, it is necessary to determine a weighted mean of the stopping power ratios for the electron spectrum set in motion by the photon spectrum in the materials concerned. Methods for calculating average stopping powers (S) for photon beams have been published (17). Several authors have worked out the theory of the stopping power ratio for an air-filled cavity in a medium such as water under electron irradiation. A good approximation is provided by the Spencer-Attix formulation (11,18), which uses a restricted mass stopping power in Equation 8.30, defined as

where Φ(E) is the distribution of electron fluence in energy and L/ρ is the restricted mass collision stopping power with Δ as the cutoff energy.

The “primary electrons” (original electrons or electrons generated by photons) give rise to ionization as well as “secondary electrons” or σ rays. The effects of the latter are accounted for in the Spencer-Attix formulation by using an arbitrary energy limit, Δ, below which energy transfers are considered dissipative; that is, the secondary electron of energy less than Δ is assumed to dissipate its energy near the site of its release. Thus, when the integration is performed (Equation 8.31) to obtain the energy deposited in the cavity by the electron fluence, the lower energy limit should be Δ, greater than zero. For ion chambers it must have a value of the order of the energy of an electron that will just cross the cavity. The value of Δ for most cavity applications in ion chambers will lie between 10 and 20 keV.

The Spencer-Attix formulation of the Bragg-Gray cavity theory uses the following relationship:

where is the average restricted mass collisional stopping power of electrons. Tables A.1, A.2, A.3, A.4 and A.5 in the Appendix give for various media and various photon and electron energies.

The quantity J_g in Equation 8.32 can be determined for a chamber of known volume or known mass of air in the cavity if the chamber is connected to a charge-measuring device. However, the chamber volume is usually not known to an acceptable accuracy. An indirect method of measuring J_air is to make use of the exposure calibration of the chamber for ⁶⁰Co γ-ray beam. This in effect determines the chamber volume.

Consider an ion chamber that has been calibrated with a buildup cap for ⁶⁰Co exposure. Suppose the chamber with this buildup cap is exposed in free air to a ⁶⁰Co beam and that a transient electronic equilibrium exists at the center of the chamber. Also assume initially that the chamber wall and the buildup cap are composed of the same material (wall). Now, if the chamber (plus the buildup cap) is replaced by a homogeneous mass of wall material with outer dimensions equal to that of the cap, the dose D_wall at the center of this mass can be calculated as follows:

where is the ratio of electron fluence at the reference point P (center of the cavity) with chamber cavity filled with wall material to that with the cavity filled with air. This correction is applied to the Bragg-Gray relation (Equation 8.29) to account for change in electron fluence.

As discussed by Loevinger (4),¹ Φ in the above equation can be replaced by Ψ, provided a transient electron equilibrium exists throughout the region of the wall from which secondary electrons can reach the cavity. Therefore,

If D_air is the absorbed dose to air that would exist at the reference point with the chamber removed and under conditions of transient electronic equilibrium in air, we get from Equation 8.18:

where is the ratio that corrects for the change in photon energy fluence when air replaces the chamber (wall plus cap).

From Equations 8.34 and 8.35, we get

Also, D_air (under conditions of transient electronic equilibrium in air) can be calculated from exposure measurement in a ⁶⁰Co beam with a chamber plus buildup cap, which bears an exposure calibration factor N_x for ⁶⁰Co γ rays:

where k is the charge per unit mass produced in air per unit exposure (2.58 × 10^-4 C/kg/R), M is the chamber reading (C or scale division) normalized to standard atmospheric conditions, A_ion is the correction for ionization recombination under calibration conditions, and P_ion is the ionization recombination correction for the present measurement.

Standard conditions for N_x are defined by the standards laboratories. The National Institute of Standards and Technology (NIST) specifies standard conditions as temperature at 22°C and pressure at 760 mmHg. Since exposure is defined for dry air, humidity correction of 0.997 (for change in with humidity) is used by the NIST, which can be assumed constant in the relative humidity range of 10% to 90% for the measurement conditions with minimal error (19). Thus, the user does not need to apply additional humidity correction as long as it is used for dry air.

From Equations 8.36 and 8.37:

The product , which represents a correction for the change in J_air due to attenuation and scattering of photons in the chamber wall and buildup cap. This factor has been designated as A_wall in the American Association of Physicists in Medicine (AAPM) protocol (6). Thus, Equation 8.38 becomes

Now consider a more realistic situation in which the chamber wall and buildup cap are of different materials. According to the two-component model of Almond and Svensson (20), let α be
the fraction of cavity air ionization owing to electrons generated in the wall and the remaining (1 – α) from the buildup cap. Equation 8.39 can now be written as

or

J_air = k · M · N_x · A_wall · β_wall · A_ion · A_α · P_ion

where Aα is the quantity in the brackets of Equation 8.40.

The fraction α has been determined experimentally by dose buildup measurements for various wall thicknesses (Fig. 8.4). In addition, it has been shown (21) that α is independent of wall composition or buildup cap, as long as it is composed of low-atomic-number material.

Since J_air is the charge produced per unit mass of the cavity air, we have

where V_c is the chamber volume and ρ_air is the density of air normalized to standard conditions. Comparing Equations 8.40 and 8.41, we have