CHAMBER VOLUME

Measurement of Absorbed Dose

B. CHAMBER VOLUME

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 measur- ing J_air is to make use of the exposure calibration of the chamber for ⁶⁰Co g-ray beam. This in effect determines the chamber volume.

104 Part I Basic Physics

Consider an ion chamber that has been calibrated with a buildup cap for ⁶⁰Co exposure. Sup- pose 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:

D_wallJ_air

#

aW e b

#

aL

rb

air

wall

#

(_cav)_air^wall (8.33) where (_cav)_air^wall 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 in the above equation can be replaced by , provided a transient electron equilibrium exists throughout the region of the wall from which secondary electrons can reach the cavity. Therefore,

D_wallJ_air~aW e b~aL

rb

air wall

~(cav)_air^wall (8.34)

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:

D_airD_wall~abm_en r b

wall air

~(_chamb)_wall^{air (8.35)}

where (_chamb)wallair 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 D_airJ_air~aW

e b^~aL rb

air wall

~abm_en r b

wall air

~(_cav)_air^wall~(_chamb)_wall^air (8.36) 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 expo- sure calibration factor N_x for ⁶⁰Co g rays:

D_airk~M ~N_x~aW

e b^~b_air~A_ion~P_ion (8.37) 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 ioniza- tion 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 W 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:

J_airk~M~N_x~(_cav)_wall^air ~(_chamb)_air^wallb_wall~A_ion~aL rb

wall air

~am_en r b

air wall

~P_ion (8.38) The product (_cav)_wall^air ~(_chamb)_air^wall equals (_wall)_air^wall, 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

J_airk~M~N_x~A_wall~A_ion~aL rb

wall air

~am_en r b

air wall

~P_ion (8.39)

Now consider a more realistic situation in which the chamber wall and buildup cap are of dif- ferent materials. According to the two-component model of Almond and Svensson (20), let a be

1 Electron fluence at P with the cavity filled with wall material is proportional to _wall at P. With the air cavity in place, the electron fluence at P is proportional to the mean photon energy fluence at the surface of the cav- ity, which can be taken as equal to _air at the center of the cavity.

82453_ch08_p097-132.indd 104 1/7/14 6:55 PM

ChaPter 8 Measurement of absorbed Dose 105 the fraction of cavity air ionization owing to electrons generated in the wall and the remaining (1 – a) from the buildup cap. Equation 8.39 can now be written as

J_airk~M~N_x~A_wall~b_wall~A_ioncaaL rb

wall air

~ am_en r b

air wall

(1a)aL rb

cap air

~am_en r b

air

capd^~P_ion (8.40) or

J_airk~M~N_x~A_wall~b_wall~A_ion~A_a~P_ion where Aa is the quantity in the brackets of Equation 8.40.

The fraction a has been determined experimentally by dose buildup measurements for vari- ous wall thicknesses (Fig. 8.4). In addition, it has been shown (21) that a 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 J_airM~P_ion

r_air~V_c (8.41)

where V_c is the chamber volume and r_air is the density of air normalized to standard conditions.

Comparing Equations 8.40 and 8.41, we have

V_c 1

k~r_air~N_x~A_wall~b_wall~A_ion~A_a (8.42) C. EFFECTIVE POINT OF MEASUREMENT

C.1. Plane-Parallel Chambers

Since the front plane (toward the source) of the air cavity is flat and is exposed to a uniform fluence of electrons, the point of measurement is at the front surface of the cavity. This would be strictly true if the electrons were monodirectional and forward directed, perpendicular to the cavity face. If a significant part of the cavity ionization is caused by back-scattered electrons, the point of measurement will shift toward the center. If the plane-parallel chamber has a small plate separation and the electron fluence is mostly forward directed, it is reasonable to assume that the point of measurement is the front surface of the cavity.

C.2. Cylindrical Chambers

Electrons (from an electron beam or generated by photons) transversing a cylindrical chamber of internal radius r will enter the sensitive volume of the chamber (air cavity) at different distances from the center of the chamber. Dutreix and Dutreix (22) showed that theoretically the point of measurement for a cylindrical chamber in a unidirectional beam is displaced by 0.85r from the center and toward the source. Derivation of this value is instructive in understanding the concept and is, therefore, presented here.

Figure 8.5 shows a cross section of a cylindrical chamber exposed to a parallel, uniform, and forwardly directed fluence Φ of electrons. For an electron entering the chamber at point A, the

Chamber wall thickness (g/cm²) 0 .10

.20 .40 .60

Fraction of electrons originating in chamber wall (α) .80 1.00

.20 .30 .40 .50 .60 25 MV 10 MV 4 MV

2 MV ₆₀

Co

Figure 8.4. the fraction, a, of cavity ionization due to electrons generated in the chamber wall, plotted as a function of wall thickness. (From Lempert GD, Nath r, Schulz rJ. Fraction of ionization from electrons arising in the wall of an ionization chamber. Med Phys. 1983;10:1, with permission.)

106 Part I Basic Physics

point of measurement is at a distance X above the center. Considering electrons entering the cham- ber at point A, the effective point of measurement is influenced by the number of electrons entering through a surface area ds at A of the chamber and the track length of these electrons in the cavity.

Thus, the effective point of measurement, X_eff, can be determined by weighting the displacement X by the number of electrons (Φ · ds cos u) entering the chamber and the track length (2X):

X_eff1^u0p/2x~(2x)~~ cos u~ds 1u0

p/22x~~ cos u~ds (8.43)

Substituting X = r cos u and ds = rdu,

X_effrC10

p/2cos³u du 10

p/2cos ²u duS8r/3p0.85r (8.44) The above theoretical result is modified under actual irradiation conditions as some of the elec- trons enter the chamber at oblique angles.

The shift in the point of measurement takes place because of the cylindricality of the chamber cavity. If there is a gradient of electron fluence across the cavity (as in the exponential falloff of the depth–dose curve), a shift in the point of measurement will result in a “gradient correction”

to the dose measured at a point corresponding to the center of the chamber (to be discussed).

8.5. CALIBRATION OF MEGAVOLTAGE BEAMS: