Shani G. Radiation Dosimetry: Instrumentation and Methods

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150 Radiation Dosimetry: Instrumentation and Methods

and 0.6 cm

, respectively. The N

gas

for each of the chambers

is 4.90  10

(PTW 430), 4.75  10

(PTW 1663), 4.04

 10

(NEL 484), and 4.01  10

Gy/C (NEL 1626).

The polarity error is deﬁned as the fractional error

between the reading taken at positive bias and the correct

reading

(3.129)

In order to compare the polarity error for the different

ionization chambers, the measured data was ﬁt using a

non-linear regression of the model

(3.130)

where

P.E.  polarity error, E

 mean energy at depth,

and A, B, C  ﬁtting parameters.

This model resulted in a better correlation coefﬁcient

than either a second- or third-order polynomial ﬁt. The

FIGURE 3.64 The polarity error as a function of mean energy at depth (MeV) is shown for the (a) Farmer, (b) Holt, (c) PS-033,

and (d) scanning chambers. The characteristic curve resulting from a third-order polynomial ﬁt to the data is also shown for each

chamber. Data for the ﬁrst three chambers were taken using a 15

 15-cm

square ﬁeld in a polystyrene phantom with 100-cm

source-to-surface distance. Data for the scanning chamber was taken using an identical setup in a water phantom. (From Reference

[66]. With permission.)

polarity error







------------------------------









------------------------------

-------------------------------------------











-----------------------------



P.E.

AB E

()



------------------------------------



Ch-03.fm(part 2) Page 150 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 151

model uses the Levenberg-Marquardt method to combine

the steepest descent method and a Taylor series–based

method for non-linear optimization. Although this model

will approach zero polarity error at large mean energies,

the model adequately ﬁts in the region of lower mean

energies where the polarity error is most pronounced.

VII. DETECTOR PERTURBATION

Two effects perturb the electron ﬂuence compared to that

at the same position in the undisturbed medium. One of

them is due to an imbalance between the number of

electrons that come into the gas cavity through the side

walls and the number of electrons already in the cavity

that leave it. This is due to the fact that the average

angular divergence of the former is greater than that of

the latter. This has become known as in-scattering and

will cause the ﬂuence in the cavity to increase. The other

effect is that the ﬂuence of electrons entering the front

wall of a plane-parallel cavity is hardly affected by the

scattering in the air, whereas in the medium the ﬂuence

continues to build up, owing to scattering. This effect

results in a lower ﬂuence in the gas cavity than in the

medium at the same depth. Thus, the two effects act in

opposite directions. [69]

The second, or ﬂuence build-up, effect is generally

taken care of by referring the reading of the chamber to

an effective point of measurement which differs from that

of the chamber center.

The ﬁrst effect exceeds 3% for Farmer-type chambers

for below 8 MeV and is one of the principal reasons

why plane-parallel chambers are recommended in low-

energy electron beams.

In coin-shaped gas cavities, the in-scattering perturbation

in the electron ﬂuence is conﬁned to a region close to the

edges of the cavity. Therefore, by constructing the chamber

with a guard ring of sufﬁcient width, it is possible to exclude

the contribution to the ionization signal from this edge region.

Backscatter plays a role in the response of certain

plane-parallel chambers. It should be part of any theoret-

ical treatment of the variation of the overall perturbation

factor

with electron energy and with chamber material.

The overall perturbation factor p

will be written as the

product of two factors:

(3.131)

where

cav

refers exclusively to the perturbation due to the

air cavity, i.e., predominantly in-scattering effects, and p

wall

refers to any effects due to the composition of the chamber

wall, which could include possible backscatter effects.

A series of chambers with different guard ring widths,

each with the same cavity height of 2 mm, were compared

to each other in a 6-MeV beam (Figure 3.65). It can be

seen that for widths greater than 3 mm, the chamber per-

turbation remained constant. p

cav

is unity for all guard ring

widths greater than 3 mm.

Values for the ﬂuence perturbation correction factor

were determined by Wittkamper et al. [71], in a number

of electron beams for the commonly applied PTW/Markus

plane-parallel chamber and a cylindrical NE 2571 farmer

chamber. These data were determined relative to the

cav

wall



FIGURE 3.65 Experimental study of the effect of guard ring width on the perturbation factor for various plane-parallel chamber

designs. The points labelled (1) and (3) are for the PTW/Markus chamber and the NACP chamber, respectively. Points (2), (4), and

(5), represent specially constructed chambers of the PTW/Roos design with guard ring widths of 2, 4, and 8 mm, respectively. The

measurements were made at the depth of dose maximum in a 6-MeV beam against a chamber with an ultrabroad (8-mm) guard ring,

with the normalization of the response quotients obtained in a 20 MeV beam. [70] (From Reference [8]. With permission.)

Ch-03.fm(part 2) Page 151 Friday, November 10, 2000 11:59 AM

152 Radiation Dosimetry: Instrumentation and Methods

NACP plane-parallel chamber as a function of the mean

energy at depth E

In recent codes of practice the formalism to determine

the absorbed dose in high-energy electron beams is based

on an air kerma calibration of the ionization chamber and

application of the Bragg-Gray equation. This leads to the

following expression:

(3.132)

where D

is the absorbed dose to water in the electron

beam at the depth of the effective point of measurement,

M is the instrument reading corrected for temperature,

pressure, recombination, polarity and humidity, N

is the

air kerma calibration factor, g is the fraction of the energy

of secondary-charged particles converted to bremsstrahlung

in air at the calibration quality, and  k

is the product of

correction factors to be applied for the determination of

the absorbed dose to the air in the ionization chamber

cavity from the air kerma in the photon calibration beam

in the absence of the chamber:

(3.133)

w,air

is the restricted stopping-power ratio of water to air and

 p

is the product of correction factors to be applied to the

measurements in the water phantom at the user’s quality:

(3.134)

The wall and central electrode corrections p

wall

and p

cel

are

negligible for cylindrical ionization chambers commonly

applied in electron beam dosimetry. If measurements

are performed using the effective point of measurement,

the displacement correction p

can also be omitted, and

the only correction factor in the formalism to be applied

during phantom measurements in electron beams is the

ﬂuence perturbation correction factor p

For cylindrical chambers p

was determined experi-

mentally by Johansson et al. [72] for a number of beam

energies. p

is usually given as a function of the mean

electron energy at depth z, .

The plane-parallel chambers are not calibrated in

terms of air kerma in a

Co gamma-ray beam but are

calibrated in a high-energy electron beam against a cylin-

drical chamber. In this way, the product

(3.135)

is determined. The ﬂuence perturbation correction factor

of plane-parallel chambers in the calibration electron

beam is thus incorporated in the calibration factor.

By applying Equations (3.132), (3.133), and (3.134)

twice, it follows that

(3.136)

TABLE 3.16

Characteristics of the Reference and Plane-Parallel Ionization Chambers and

Build-Up Caps

NE 2571 NACP PTW/Markus

Chamber Type

Cylindrical

0.6 cm

Farmer

Plane-

Parallel

Plane-Parallel

Cavity Length 24 mm Plate

distance

2 mm 2 mm

Dimensions Diameter 6.3 mm Diameter 16.4 mm 6 mm

Chamber wall Material

Thickness

graphite

65 mg cm

–2

Front wall

material

Thickness

Graphite

90 mg cm

–2

Graphited

polyethylene

2.3 mg cm

–2

Side wall

material

Polystyrene Graphited

PMMA

Electrode Material aluminum Material Graphited Graphited

Diameter 1 mm Diameter polystyrene

10 mm

polystyrene

5.4 mm

Build-up cap Material PMMA Material Graphite Graphite

Thickness 558 mg cm

–2

Thickness 540 mg cm

–2

540 mg cm

–2

Guard ring Width 3.2 mm 0.7 mm

NE, Nuclear Enterprises Ltd.; NACP, Scanditronix, Dosetek (Calcam); PTW, Physikalisch Technische

Werkstatte.

Source: From Reference [71]. With permission.

1 g() k

wair,

 p



k

att

cel

stem



 p

wall

cel



k

 p

f ,x

att

cel

()

ref

att

cel

()

-----------------------------

()

ref

()

----------------

wall

cel

()

wall

cel

()

-----------------------------------



Ch-03.fm(part 2) Page 152 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 153

where the subscripts x and ref refer to the chamber with

unknown ﬂuence perturbation correction values and to the

reference chamber, respectively.

In Figure 3.66 the resulting p

values are given for the

PTW/Markus chambers as a function of the mean energy

at the depth of the effective point of measurement, which

is the depth of dose maximum.

In Figure 3.67 p

values for the 0.6-cm

NE2571 cylin-

drical chambers are given as a function of the mean energy

at depth. In the same ﬁgure, the data derived from Johans-

son et al. [72] are depicted as a function of .

The p

values reported in this work are obtained by

averaging the results of at least ﬁve individual measurement

series. The error bars indicated in Figures 3.66 and 3.67 are

one standard deviation, derived by quadratic summation of

the statistical uncertainties of the ionization chamber read-

ings in the electron beams M

and M

ref

and at the air-kerma

calibrations N

k,x

and N

N,ref

. A contribution to the overall

FIGURE 3.66 Fluence perturbation correction values for the PTW/Markus plane-parallel chamber as a function of the mean energy

at depth. The results are normalized to the NACP chambers. The error bars represent one standard deviation. The solid curve represents

a ﬁt through the data points of this work and is adopted in the NCS Code of Practice (1989). The broken curve gives a ﬁt through

all data points in the ﬁgure. (From Reference [71]. With permission.)

FIGURE 3.67 Fluence perturbation correction values for the NE2571 cylindrical Farmer-type chamber as a function of the mean

energy at depth. The results are normalized to the NACP chambers. The error bars represent one standard deviation. The solid curve

represents a ﬁt through the data points of this work and is adopted in the NCS Code of Practice (1989). (From Reference [71]. With

permission.)

Ch-03.fm(part 2) Page 153 Friday, November 10, 2000 11:59 AM

154 Radiation Dosimetry: Instrumentation and Methods

uncertainty of random and systematic errors in the k

and

correction factors of the reference and second chamber

in Equation (3.136) was not taken into account.

The solid curve in Figure 3.66, which is given by

(3.137)

is an exponential ﬁt to the data determined in this work,

weighted with the one standard uncertainty.

A second exponential term was added to make a best

ﬁt to all data given in Figure 3.66. This curve is the broken

curve and is given by

(3.138)

Figure 3.67 shows that for the cylindrical 0.6-cm

ionization chamber, good agreement exists between

Wittkämper et al. p

values and those determined by

Johansson and colleagues, while Wittkämper et al. [71]

values are computed using the Andreo and Brahme

tables. A partial linear and partial exponential ﬁt was

applied to the data determined in our work, which is

depicted in Figure 3.67 by the solid curve. The analytical

expression of the ﬁt is given by

(3.139)

(3.140)

In electron beam dosimetry, the perturbation effect in

the medium by the ionization chamber cavity is accounted

for by introducing a replacement correction factor, p

repl

Another perturbation correction factor, denoted as p

wall

, is

due to the materials of the walls of the parallel-plate cham-

ber differing from the phantom material. Because of the

difﬁculties in separating these two components, Reft and

Kuchnir [73] measured the overall perturbation factor, p



repl

wall

. A distinct advantage of parallel-plate ionization

chambers over cylindrical chambers is that, p

has been

shown to be close to unity at the standard calibration

depth, d

max

At the same depth in the phantom, they equate the

percentage depth dose (PDD) measured with the NACP

chamber to that measured with the other chambers, to

derive an equation for the overall perturbation factor:

(3.141)

Therefore,

(3.142)

where is the ratio of the corrected measured

charges at depths d and d

max

, respectively, and

represents the restricted mass collision stop-

ping power ratio of water to air evaluated at depth d

divided by its value at depth d

max

. In this derivation they

have assumed that p

is unity for the NACP chamber and

the restricted stopping power ratios cancel because the

measurements are performed at the same depths in the

phantom. For the parallel-plate chambers, p

was taken as

1.00 at d

max

with one exception, the Markus chamber. For

this chamber there are experimental data showing that, at

energies below 12 MeV, p

measured at d

max

decreases

with decreasing energy.

Using the NACP chamber as the reference detector,

was obtained for the other three ionization chambers.

The data as summarized in Table 3.18 are obtained by

using Equation (3.142) and show that for the depths

slightly greater than d

ref

, p

is unity for the parallel-plate

chambers within the experimental uncertainties.

For the Markus chambers, results show that p

increases signiﬁcantly with an increasing dose gradient.

This behavior is illustrated in Figure 3.68. There are a

number of studies showing that this chamber over-responds

 1.00) at d for low-energy electrons. The results

reported here show that it under-responds (p

 1.00) in

regions beyond d

max

. The response of the Markus chamber

is reduced in the decreasing portion of the depth ionization

TABLE 3.17

Detector Characteristics

Entrance wall

Type Model Material

Thickness

(g/cm

)

Gap

(cm)

Guard

(cm)

Diameter

(cm)

NACP ... Graphite 0.09 0.2 0.3 1.0

ATTIX 449 Kapton 0.0048 0.1 1.35 1.27

MARKUS 30–329 Polyethylene 0.0023 0.2 0.02 0.53

FARMER 30–351 Acrylic 0.059 0.25 ... ...

Source: From Reference [73]. With permission.

1 0.041e

0.40E





1 0.041e

0.40E



0.036e

0.72E





0.952 0.0028 E

for E

12 MeV

1 0.18e

0.21E



for E

12 MeV

PDD d, E

()Q

NACP

L 



()

air

[]

max



L 



()

air

()

[]

max



()

d,ch

()

max

,ch

[]Q

NACP

[]

max

[]

max



Q[]

max

L 



()

air

[]

max

Ch-03.fm(part 2) Page 154 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 155

curve. This behavior is summarized in Figure 3.69; the dose

gradient is shown in parentheses.

VIII. DISPLACEMENT CORRECTION

For measurements free in air as commonly employed in

calibration situations, the variation in the radiation ﬁeld

from a point source is governed by the inverse square law.

Inside a phantom, the effective point of measurement can

be displaced relative to the free-in-air condition due to the

replacement of phantom material by the cavity of the

ionization chamber. A correction has to be made for

changes in attenuation and scattering due to this replace-

ment. This can be achieved in terms of a radial displace-

ment using an effective point of measurement or by taking

the geometrical center of the chamber as the measuring

point and applying a displacement correction factor. Such

a factor is essentially the same as the perturbation factor.

Displacement corrections for in-phantom measure-

ments with ionization chambers for the purpose of mam-

TABLE 3.18

Summary of Data Used to obtain p

for the Other Ionization Chambers Using Equation (3.142) and the

NACP Chamber as the Reference Detector

(MeV)

(cm)

(MeV)

PDD/d

(%/mm)

NACP Attix Markus Farmer Attix Markus Farmer

22 5.3 10.9 0.4 0.8913 0.8929 0.8993 0.9033 0.998 0.991 0.981 (0.983)

22 6.4 8.8 0.8 0.8150 0.8177 0.8203 0.8326 0.997 0.994 0.973 (0.977)

18 5.3 7.0 1.0 0.8850 0.8834 0.8881 0.8993 1.002 0.996 0.968 (0.972)

18 5.8 6.1 1.3 0.8264 0.8244 0.8251 0.8382 1.002 1.002 0.970 (0.969)

22 9.0 3.6 2.3 0.4184 0.4232 0.4239 0.4350 0.989 0.987 0.956 (0.961)

18 7.6 2.7 3.1 0.4018 0.4075 0.3931 0.4175 0.986 1.022 0.947 (0.958)

15 6.1 2.5 3.8 0.4948 0.4876 0.4660 0.5079 1.015 1.062 0.951 (0.957)

12 4.9 1.9 4.5 0.4796 0.4679 0.4398 … 1.025 1.085

…

This value was obtained using 0.985 for (P

)

d max

from TG 39.

The values in parentheses for the Farmer chamber are taken from Table VIII in the TG-21 protocol.

Source: From Reference [73]. With permission.

FIGURE 3.68 A plot of p

vs. the dose gradient for the Markus chamber. The values in parentheses are the energies at depth. (From

Reference [73]. With permission.)

Q[]

max

Ch-03.fm(part 2) Page 155 Friday, November 10, 2000 11:59 AM

156 Radiation Dosimetry: Instrumentation and Methods

mography are large and represent a major correction to

consider for dose determinations. Experimental data on

displacement corrections depend to a large degree on the

model used to extrapolate to zero cavity radius. Calcula-

tions of displacement correction factors using a Monte

Carlo code for different cavity shapes were carried out by

Zoetelief et al. [74]

For photons employed in mammography, experimental

displacement correction factors obtained by linear extra-

polation to zero cavity radius differ considerably from

unity; e.g., k

is about 0.7 for a 0.6-cm

Baldwin-Farmer

(Nuclear Enterprises Ltd., UK, type BF2571) ionization

chamber. [75] Non-linear extrapolation to zero cavity

radius for mammography radiation qualities would lead to

different displacement correction factors and their associ-

ated errors. Some of the experimental data at different

cavity radii suggest a nonlinear dependence of the reading

on cavity size. There is, however, a lack of extrapolation

models. In addition, for photons as employed in mammog-

raphy, information is not available for phantom materials

FIGURE 3.69 A plot of p

vs. energy at d

max

and at the depth for the Markus chamber. The values in parentheses are the dose

gradients in % per mm. (From Reference [73]. With permission.)

FIGURE 3.70 Lateral dose relative to dose in the center of the cavity for calculations inside spherical cavities of different radii (,

4 mm;

, 8 mm; , 12 mm; and , 16 mm) placed with their geometric centers at 30-mm depth in PMMA phantoms. Fits are

made according to Equation 3.143. (From Reference [74]. With permission.)

Ch-03.fm(part 2) Page 156 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 157

other than polymethylmethacrylate (PMMA) and is

restricted to cylindrical and spherical ionization chambers.

There is only limited knowledge of the dependence of

displacement correction factors on depth in phantom. [74]

Calculated dose values at points along the central lat-

eral axis (a line through the center of the cavity parallel

to the x-axis) are given in Figure 3.70 for the spherical

cavities located at 30-mm depth in the PMMA phantom.

A decrease in calculated dose values with increasing lat-

eral distance, l, from the center of the cavity is found.

This is due to an increase in the amount of phantom

material traversed by the photons with increasing lateral

distance from the center of the cavity. Due to partial lack

of side scatter, the calculated dose values at the maximum

lateral distance are still somewhat higher (up to about

10%) than the dose at the same depth in the homogeneous

phantom. Also shown in the ﬁgure are ﬁts using STAR-

PAC, a least-squares program for nonlinear functions

[76], according to

(3.143)

The results of calculations of dose in the cavities rel-

ative to dose at 30-mm depth in the homogeneous phan-

toms, (D(r)/D(0)), for spherical cavities of different radii

placed with their centers at 30-mm depth inside the

PMMA, A-150 plastic, and EVA/28 phantoms are shown

in Figure 3.71 as a function of cavity radius.

It is evident from Figure 3.71 that the exponential ﬁts

are superior.

When parallel-plate chambers are used for dosimetry

in electron ﬁelds, the AAPM dosimetry protocol recom-

mends a value of 1.0 for the replacement correction factor,

repl,pp,E

, until further data become available. P

repl,pp,E

for

ﬁve commercially available parallel-plate chambers was

measured by Reft and Kuchnir [77] as a function of elec-

tron energy from a nominal value of 5.5 to 22 MeV by

comparison with a cylindrical chamber whose P

repl, cyl,E

was obtained from data in the protocol. It was found that

for three of the chambers, P

repl,pp,E

is independent of

energy, consistent with unity within one or two standard

deviations (s.d.). For the fourth chamber, P

repl,pp,E

is sim-

ilarly consistent with one above 10 MeV but decreases at

lower energies, while for the ﬁfth one it shows a system-

atic drop with decreasing energy.

For equal dose to the homogeneous medium at the

effective point of measurement, the following relation

holds:

(3.144)

where

M is the measured charge corrected for temperature,

pressure, polarity effect, and ion collection efﬁciency. P

repl

and P

wall

are the replacement and wall correction factors,

and pp and cyl refer to the parallel-plate and cylindrical

chambers, respectively. The effect of chamber polarity is

evaluated from the ratio of charges measured with positive

and negative polarizing voltages. For measurements in a

photon ﬁeld, the AAPM protocol considers P

repl,pp,x

equal 1, and when a cylindrical chamber is used at d

max

repl,cyl,x

 1, also. With the exception of the NACP cham-

ber, which has a graphite entrance wall, all measurements

were taken in a phantom matched to the chamber wall

material. Therefore, P

wall

 1.0. Equation (3.144) then

reduces to:

(3.145)

where the notation is used to represent the ratio

)

cyl

(M

)

. For the NACP chamber, was

used; it varied from 0.995 to 0.997 over the energy range

investigated. In the case of the 22-MeV electron beam

irradiation, values of

repl,cyl,E

are available only for the

Farmer chamber. The Exradin cylindrical chamber, there-

fore, was not used with the electron beams. Following

recommendations of the AAPM protocol, P

wall

 1 for all

chambers and P

repl,pp,E

 1 at high electron energies.

Referring to Equation (3.144),

(3.146)

Using Equation (3.144) and (3.145), Reft and Kuchnir

found that for the matched pair of chambers in homoge-

neous media,

(3.147)

where

E and X refer to electron and photon ﬁelds, respec-

tively. All of the measurements were made in phantom

with a source-to-surface distance of 100 cm and the effec-

tive point of measurement of each chamber placed at

max

A 10  10-cm

ﬁeld deﬁned at the phantom surface was

used at nominal electron energies of 5.5, 6, 9, 12, 15, 18,

and 22 MeV from a Cl 2500 linear accelerator.

Results for P

repl

as a function of electron energy for

the ﬁve chambers are summarized in Table 3.19.

As seen in Figure 3.72, the experimentally determined

values of P

repl,pp,E

for the Holt, NACP, and Exradin cham-

bers are constant over the energy range investigated. The

respective average values are 0.985, 0.980, and 0.989 with

a standard deviation of 0.5% and an overall uncertainty

of 1% (63% conﬁdence level).

The TG-21 protocol analyzes the replacement correc-

tion factor in two components: (a) gradient correction and

(b) electron ﬂuence correction. The gradient correction,

according to the protocol, arises because, on the descend-

ing portion of a depth-dose curve, the proximal surface of

a cylindrical cavity intercepts an electron ﬂuence that is

l() 1 dl bl



gas, pp

repl, pp

wall, pp

gas, cyl

cyl

repl, cyl

wall, cyl



gas, pp

gas, cyl

Mx[]

cyl



[]

cyl

wall

1

gas, pp

gas, cyl

repl, E

[]

cyl

[]

cyl



repl, E

[]

cyl

[]

cyl

 Mx[]

cyl

()P

wall,pp, X



Ch-03.fm(part 2) Page 157 Friday, November 10, 2000 11:59 AM

158 Radiation Dosimetry: Instrumentation and Methods

more intense than that at the measurement depth when the

chamber is removed.

The protocol assumes that P

repl

for photons, under the

calibration conditions, is mainly due to the dose gradient

and that electron ﬂuence corrections are not required since

there is a transient electronic equilibrium at the center of

the chamber. In the case of electron beams, the gradient

correction is ignored as the chamber is positioned at the

depth of maximum dose, but an electron ﬂuence correction

is used as a function of beam energy and cavity diameter.

For the plane-parallel chamber,

repl

is assumed to be unity

for both photon- and electron-beam calibration. The pro-

tocol does not address the problem of measuring dose

distributions with ionization chambers that require

replacement corrections. [78]

In an idealized case where a parallel beam of electrons

(or secondary electrons generated by photons) enters the

chamber from a direction perpendicular to the chamber

axis, it has been shown theoretically that the displacement

in the effective point of measurement is given by 0.85r

( 8r/3



). Experimental data show that the actual values

lie between 0 and 0.75r, depending upon beam energy,

in the energy range of medium energy x-rays to 42-MV

x-rays. In the megavoltage range of photon beams, the mea-

surements of Johansson et al. [72] show good agreement

between depth-dose curves measured with plane-parallel

and cylindrical chambers if a constant shift correction of

0.75r is used for the cylindrical chambers.

The TG-21 protocol equates P

repl

or ‘‘gradient correc-

tion” to photon attenuation corrections, calculated by Cun-

ningham and Sontag [79] for cylindrical phantoms of the

same size as the chamber that displaces the medium. These

factors are used to correct for change in energy ﬂuence of

photons at the center of the chamber compared to the

energy ﬂuence that would exist at that point in the absence

of the chamber. Under conditions of transient electronic

equilibrium, the change in photon energy ﬂuence is

equated to the change in electron ﬂuence, which is the basis

of P

repl

corrections.

repl

can then be thought of as the ratio of dose to the

medium at the depth of central axis of chamber to that at

the depth of the effective point of measurement.

Mathematically: [78]

(3.148)

FIGURE 3.71 Absorbed dose calculated for spherical cavities with radii of 4, 8, 12, and 16 mm, placed with their geometric centers

at 30-mm depth inside the PMMA phantom, relative to dose in the homogeneous phantom,

D(r)/D(0), as a function of cavity radius.

(a) Exponential ﬁts; (b) linear ﬁts. (From Reference [74]. With permission.)

repl

d() Dd()Dd 0.75r()

Ch-03.fm(part 2) Page 158 Friday, November 10, 2000 11:59 AM

Ionization Chamber Dosimetry 159

TABLE 3.19

Summary of Results for P

repl

Nominal

Energy

(MeV)

repl,cyl

(TG #21)

Chamber

Type P

repl,pp,E

E-(MeV) E

5.5 4.7 2.5 0.957 N 3.404 1.022 0.978

M 10.34 0.998 0.955

H 0.641 1.032 0.988

C 1.175 1.044 0.999

E 1.027 1.027 0.983

6 5.2 3.1 0.959 N 3.411 1.024 0.982

M 10.42 1.006 0.965

H 0.641 1.032 0.990

C 1.167 1.036 0.994

E 1.024 1.024 0.982

9 8.0 4.7 0.964 N 3.385 1.016 0.979

M 10.56 1.019 0.982

H 0.637 1.026 0.989

C 1.186 1.053 1.015

E 1.027 1.027 0.990

12 11.0 6.4 0.970 N 3.349 1.005 0.975

M 10.49 1.013 0.983

H 0.628 1.011 0.981

C 1.168 1.037 1.006

E 1.021 1.021 0.990

15 14.0 7.6 0.974 N 3.356 1.007 0.981

M 10.49 1.013 0.987

H 0.627 1.010 0.983

C 1.174 1.043 1.016

E 1.018 1.018 0.992

18 17.0 12.3 0.986 N 3.306 0.992 0.978

M 10.30 0.994 0.980

H 0.617 0.994 0.980

C 1.168 1.037 1.022

E 1.001 1.001 0.987

22 20.6 18.4 0.994 N 3.303 0.991 0.985

M 10.31 0.995 0.989

H 0.616 0.992 0.986

C 1.168 1.037 1.031

E 1.005 1.005 0.999

X-MV X

4 … … 1.000 N 3.332 … …

M 10.36 … …

H 0.621 … …

C 1.126 … …

E 1.000 … …

Note: The values for in column 7 were obtained by dividing the electron charge ratio, , by

the photon ratio, , according to Equation (3.147), for

X  4 MV. The last column is the product of values

in columns 4 and 7. N

 NACP; M  Markus; H  Holt; C  Capintec; EExradin.

Source: From Reference [77]. With permission.

M[]

cyl

repl,E

[]

cyl

repl

[]

cyl

[]

cyl

Mx[]

cyl

Ch-03.fm(part 2) Page 159 Friday, November 10, 2000 11:59 AM