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Attix

CHAPTER 10. Cavity Theory

Jenghwa Chang, Ph.D. D.A.B.R.

Associate Professor Radiation Medicine, Hofstra Northwell School of Medicine at Hofstra University

Associate Adjunct Professor Physics and Astronomy, Hofstra University

J Chang, PhD, DABR

OVERVIEW

• This is the initial theoretical discussion on considerations for the use of a dosimeter to measure quantities from which dose may be computed in tissues

• Various types of dosimeters are available – each of which have specific advantages and disadvantages

• Ion chambers and calorimeters are the standard dosimeters for absolute dosimetry

• Does placing the dosimeter affect the traversing beam properties? What assumptions must be made? What corrections must be made?

J Chang, PhD, DABR

Use a radiation detector as a dosimeter

• Signal 𝑀det ∝ 𝐷det (mean dose to the sensitive material of the detector)

• Relation between 𝑀det and 𝐷det is determined from calibration, and will be discussed later .

• Assuming 𝐷det is known, want to find 𝐷med, or “mean dose to the undisturbed medium at the detector location”

• Undisturbed medium: when the detector is absent.

J Chang, PhD, DABR

Figure 9.1 The general situation of a detector

introduced into a medium, yielding Ode! for a given

irrad iation of quality Q and then being converted

into the dose 0 med at the reference point P in the

absence of the detector by multiplying by the cavity- theory factor fo'

Cavity theory

• The detector can be considered as a cavity inserted into the (uniform) medium of interest.

• The word cavity derives from the era when gas-filled ionization chambers dominated the development of the subject and ion chamber behavior was often referred to as 'cavity ionization’.

• Consequently, the theory that relates 𝐷det to 𝐷med, became known as cavity theory. The aim is to determine:

𝑓med,det,Q = 𝐷med

𝐷det 𝑄

for any type of detector for radiation quality Q.

J Chang, PhD, DABR

Bragg-Gray Theory

J Chang, PhD, DABR

Bragg-Gray Conditions

1. Thickness of g-layer is thin compared with the range of charged particles that its presence doesn’t perturb the charged-particle field.

– 𝑤 and 𝑔 must have very similar scattering properties

– Reasonably achieved for heavy charged particles which don’t scatter much; more challenging for electrons

2. All dose deposited in the g-cavity must be entirely from charged particles crossing it.

– All charged particles must originate outside the cavity

– Challenging for neutrons especially for hydrogenous gases due to high interaction cross -section

• CPE not required !

J Chang, PhD, DABR

Review: Calculation of Absorbed Dose for thin foil

• Let a parallel beam of charged particles of fluence Φ particles/cm2 and energy T0 strike a thin foil of mass thickness rt g/cm

2 perpendicularly; let foil thickness be a small percentage of the particle range

• Assumptions:

a. Collision stopping power Τ𝑑𝑇 𝜌𝑑𝑥 𝑐 is constant and depends on T0 b. Every particle passes straight through the foil, i.e., scattering is negligible

c. Effect of d rays is negligible: foil is thick enough or d ray equilibrium exists

• 𝐷 = Φ 𝑑𝑇

𝜌𝑑𝑥 𝑐 [MeV/g]

= 1.602 × 10−10Φ ∙ 𝑑𝑇

𝜌𝑑𝑥 𝑐 Gy

J. Chang, PhD, DABR

Φ, T0

BRAGG GRAY THEORY

• Measure the dose to medium 𝐷𝑤 with a gas filled ion chamber.

• 𝐷𝑔 : dose to the gas within ion chamber. Can we connect 𝐷𝑤 with 𝐷𝑔 ?

• If the fluence of identical charged particles of kinetic energy 𝑇 is continuous across the interface, i.e., Φ𝑤 = Φ𝑔 = Φ, then

𝐷𝑤 =Φ𝑤 ∙ 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 (MeV/g) , 𝐷𝑔 =Φ𝑔 ∙

𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 (MeV/g)

⟹ 𝐷𝑤

𝐷𝑔 =

Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑤

Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔 ⟹

𝐷𝑤

𝐷𝑔 =

Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑤

Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

• The ratio of dose to medium versus gas is the ratio of the respective mass collision stopping powers!

J Chang, PhD, DABR

What dose “𝒘” (in 𝑫𝒘) mean?

• The notation “𝑤” here means:

– The wall of the dosimeter, which can be

– Thick: produces all electrons entering the cavity, or

– Thin: produces a fraction of these electrons.

• The medium “𝑥” is

– Usually not the same as “𝑤” but is assumed initially

– To “match 𝑤” to simplify the discussion, that is,

– All electrons entering the cavity are produced in “𝑤”.

• For thick/matched wall, 𝑥 has no direct effect on 𝑔 other than modulate the Ψ that reaches 𝑤.

J Chang, PhD, DABR

r RS V

General form: for differential energy distribution 𝜱𝑻, average mass collision stopping power

• For a spectrum Φ𝑇 we calculate the average mass collision stopping power Τҧ𝑆 𝜌 𝑐,𝑤 by averaging over the differential fluence at each energy

• 𝑚 ҧ𝑆𝑤=

0׬ 𝑇max Φ𝑇

𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇

0׬ 𝑇max Φ𝑇𝑑𝑇

= 1

Φ ׬

𝑇max Φ𝑇 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇 =

𝐷𝑤

• 𝑚 ҧ𝑆𝑔=

0׬ 𝑇max Φ𝑇

𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇

0׬ 𝑇max Φ𝑇𝑑𝑇

= 1

Φ ׬

𝑇max Φ𝑇

𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇 =

𝐷𝑔

• 𝐷𝑤

𝐷𝑔 = 𝑚

ҧ𝑆𝑤

𝑚 ҧ𝑆𝑔

= 0׬

𝑇max Φ𝑇 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇

0׬ 𝑇max Φ𝑇

𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑤

𝑑𝑇 ≡ 𝑚 ҧ𝑆𝑔

𝑤 ∴ 𝐷𝑤 = 𝐷𝑔 ∙ 𝑚 ҧ𝑆𝑔 𝑤 (general form of

𝐷𝑤

𝐷𝑔 =

𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔

)

J Chang, PhD, DABR

Calculate Dose from BG cavity measurements

• In the ion chamber, let the mass of the gas be 𝑚 and charge collected 𝑄 due to the charged particles crossing it. Then

𝐷𝑔 = 𝑄

𝑚

𝑊

𝑒 𝑔 ⟹ 𝐷𝑤 = 𝐷𝑔 ∙ 𝑚 ҧ𝑆𝑔

𝑤 = 𝑄

𝑚

𝑊

𝑒 𝑔 ∙ 𝑚 ҧ𝑆𝑔

𝑤

• Dose to the medium immediately surrounding of the cavity

– Can be calculated from charges produced inside the cavity, however

– ‘𝑚’ may be effectively < actual mass of gas: some of the volume may not be actively involved in measuring the charge

– Measured 𝑄 < actual 𝑄 due to charges recombination

– Medium ‘𝑤’ may be ‘wall’ of the ion chamber, which complicates the calculation

– Φ𝑇 is generally unknown

J Chang, PhD, DABR

The role of CPE, 𝜱 or 𝚿

• Presence of CPE or homogenous Ψ (indirect photon energy fluence)

– not required once Φ𝑤 = Φ𝑔 = Φ𝑇 or Φ

– Φ𝑇 or Φ is determined by interactions of Ψ with 𝑤, not with 𝑔

• If CPE does exist in the neighborhood of a point of interest in 𝑤:

– Can assume the insertion of a dosimeter (cavity+wall) at the point may not perturb the “equilibrium spectrum” of Φ𝑇 , and energy fluence of Ψ

– Provides additional simplifications for dosimetry, e.g., 𝐾𝑐 = Ψ Τ𝜇𝑒𝑛 𝜌,

– “Equilibrium spectrum” of Φ𝑇 can be derived from Ψ under CPE.

• The application of cavity theory will be limited if

– CPE does not exist at the point of measurement and

– Φ𝑇 or Ψ are significantly perturbed by the presence of inserted dosimter.

J Chang, PhD, DABR

BG Corollary 1: two different gases, one wall

• Volume ‘𝑉’, wall medium ‘𝑤’, filled with two types gases:

gas ‘𝑔1’ density ‘𝜌1’: 𝐷1 = 𝑄

𝜌1𝑉 Τ𝑊 𝑒

1 = 𝐷𝑤 ∙ 𝑚 ҧ𝑆𝑤

𝑔 1 ⇒ 𝑄1 =

𝜌1𝑉

Τ𝑊 𝑒 𝑔1

𝐷𝑤 ∙ 𝑚 ҧ𝑆𝑤 𝑔

gas ‘𝑔2’, density ‘𝜌2’: 𝐷2 = 𝑄

𝜌2𝑉 Τ𝑊 𝑒

2 = 𝐷𝑤 ∙ 𝑚 ҧ𝑆𝑤

𝑔 2 ⇒ 𝑄2 =

𝜌2𝑉

Τ𝑊 𝑒 𝑔2

𝐷𝑤 ∙ 𝑚 ҧ𝑆𝑤 𝑔

Identical Irradiations ⇒ 𝑄2

𝑄1 =

𝜌2𝑉𝐷𝑤

𝜌1𝑉𝐷𝑤

Τ𝑊 𝑒 1

Τ𝑊 𝑒 2

𝑚 ҧ𝑆𝑤 𝑔

same Φ𝑇 𝜌2

𝜌1

𝑊

𝑒 𝑔2

𝑔1

𝑚 ҧ𝑆𝑔

𝑔 2

• Τ𝑄2 𝑄1 is the same for different chamber wall materials:

– As long as Φ𝑇 crossing the cavity is not significantly different, which

– Could happen if, e.g., only Compton Interactions occur leading to similar spectra

– If Φ𝑇 altered significantly by different walls ⇒ different 𝑚 ҧ𝑆𝑔 1

𝑔 2 ⇒ different Τ𝑄2 𝑄1

J Chang, PhD, DABR

BG Corollary 2 : one gas, two different walls, CPE

• Two ion chambers: ‘𝑉1’, ‘𝑤1’ and 𝑉2’, ‘𝑤2’ , filled with the same gas.

• Identical g or X-ray irradiations Ψ producing CPE at cavity:

𝐷𝑤1 = CPE

𝐾𝑐 𝑤1 ⇒ Ψ Τ𝜇𝑒𝑛 𝜌 𝑤1

= 𝐷1 ∙ 𝑚 ҧ𝑆𝑔 𝑤1 =

𝑄 1

𝜌 𝑉 1

Τ𝑊 𝑒 𝑔

∙ 𝑚 ҧ𝑆𝑔 𝑤1

𝐷𝑤2 = CPE

𝐾𝑐 𝑤2 ⇒ Ψ Τ𝜇𝑒𝑛 𝜌 𝑤2

=𝐷2 ∙ 𝑚 ҧ𝑆𝑔 𝑤2 =

𝑄 2

𝜌 𝑉 2

Τ𝑊 𝑒 𝑔

∙ 𝑚 ҧ𝑆𝑔 𝑤2

⇒ 𝑄2

𝑄1 =

𝑉2

𝑉1

Τ𝜇𝑒𝑛 𝜌 𝑤2

Τ𝜇𝑒𝑛 𝜌 𝑤1

𝑚 ҧ𝑆𝑔 𝑤1

𝑚 ҧ𝑆𝑔 𝑤2

sameΨ, Φ𝑇 𝑄2

𝑄1 =

𝑉2

𝑉1 ∙

𝜇𝑒𝑛

𝜌 𝑤1

𝑤2 ∙ 𝑚 ҧ𝑆𝑤2

𝑤 1

• 𝑄2

𝑄1 is the same irrespective of the gas if the equilibrium spectrum Φ𝑇

crossing the cavity is the same (e.g., Compton interactions)

J Chang, PhD, DABR

Implication of BG Corollary 2

• Let 𝑉1 = 𝑉2 = 𝑉, and, 𝑤2 = 𝑥 the medium.

• The thickness of the wall:

– Is larger than the range of secondary electrons so

– CPE exists at the center of the cavity

• Ψ and CPE are not perturbed significantly at cavity:

𝐷𝑤1 = CPE

𝐾𝑐 𝑤1 = Ψ Τ𝜇𝑒𝑛 𝜌 𝑤1

= 𝑄

𝜌 𝑉 1

Τ𝑊 𝑒 𝑔

∙ 𝑚 ҧ𝑆𝑔 𝑤1

𝐷𝑤2 = CPE

𝐾𝑐 𝑤2 = Ψ Τ𝜇𝑒𝑛 𝜌 𝑤2

𝐷𝑥 = 𝐷𝑤2 = CPE

𝐷𝑤1 𝜇𝑒𝑛

𝜌 𝑤1

𝑤2 =

𝑄 1

𝜌 𝑉 1

Τ𝑊 𝑒 𝑔

∙ 𝑚 ҧ𝑆𝑔 𝑤1 𝜇𝑒𝑛

𝜌 𝑤1

𝑤2

– Basic dosimetry equation using an inserted ion chamber

J Chang, PhD, DABR

r RS V

𝐷𝑔 𝐷𝑤

𝐷𝑥

Spencer derivation of B-G

• Consider a small cavity filled with 𝑔, surrounded by a homogeneous 𝑤 that

– Contains a homogeneous source emitting 𝑁𝑇0 identical charged particles per gram,

– Each with the same kinetic energy 𝑇0 (MeV)

• When traveling through the medium, these primary charged particles will – Lose energy: a spectrum of primary charged particles with KE from 0 to 𝑇0, and

– Eject electrons: d rays, energetic enough to produce their own tracks

• The cavity is assumed to be far enough from the outer limits of 𝑤 that – CPE exists at the center of 𝑤 when the cavity is absent. However,

– When the cavity is present, CPE might be significantly perturbed or nonexistent (why?)

• Both B-G conditions are assumed to be satisfied by the cavity, and bremsstrahlung generation is assumed to be absent

J Chang, PhD, DABR

Equilibrium fluence spectrum under CPT

• Assuming the bremsstrahlung generation is absent, the absorbed dose at any point in the undisturbed medium 𝒘 where CPE exists:

𝐷𝑤 = CPE

𝐾𝑤 = 𝑁𝑇0 𝑇0 (MeV/g)

• At such point, an equilibrium fluence spectrum Φ𝑇 𝑒 for the primary

charged particles exists so that the absorbed dose can be calculated as

𝐷𝑤 = 0׬ 𝑇0

Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑤 𝑑𝑇 = 𝑁𝑇0 𝑇0 = 0׬

𝑇0 𝑁𝑇0 𝑑𝑇 ⟹ Φ𝑇

𝑒 = 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

– The above derivation is based on CSDA of primary charged particles so

– Φ𝑇 𝑒 is the spectrum for primary charged particles only and

– d rays are not included

J Chang, PhD, DABR

Example of an equilibrium fluence spectrum

• Φ𝑇 𝑒 =

𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

#𝑒

𝑔

Τ𝐽∙cm2 𝑔 =

#𝑒

𝐽∙cm2

• d rays are not included

• Φ𝑒 = ׬ 0

𝑇0 Φ𝑇

𝑒 𝑑𝑇 #𝑒

cm2

= ׬ 0

𝑇0 𝑁𝑇0 𝑑𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

= 𝑁 ׬ 0

𝑇0 𝑑𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 = 𝑁𝑇0 ℜ𝐶𝑆𝐷𝐴

= 𝜌𝑁𝑇0 ℜ𝐶𝑆𝐷𝐴

𝜌

#𝑒

cm3 cm ,

which is independent of 𝜌.

J Chang, PhD, DABR

𝒎 ത𝑺𝒘

𝒈 under CPE

• Assuming the same spectrum on both sides:

𝐷𝑔 = 0׬ 𝑇0 Φ𝑇

𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑔 𝑑𝑇 = ׬

𝑇0 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇

𝜌𝑑𝑥 𝑔 𝑑𝑇 = 𝑁𝑇0 0׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

• Ratio of the dose in the cavity 𝑔 to that of the medium 𝑤 𝐷𝑔

𝐷𝑤 =

𝑁𝑇0 𝑁𝑇0 𝑇0

׬ 0

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇 =

𝑇0 ׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇 = 𝑚 ҧ𝑆𝑤

𝑔

• Consistent with the general (non CPE) form of the original B-G theory:

𝑚 ҧ𝑆𝑤 𝑔

Φ𝑒 0׬

𝑇max Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇

Φ𝑒 0׬

𝑇max Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇

CPE 𝑁𝑜 𝑏𝑟𝑒𝑚.

0׬ 𝑇0

𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇

𝜌𝑑𝑥 𝑔 𝑑𝑇

0׬ 𝑇0

𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇

𝜌𝑑𝑥 𝑤 𝑑𝑇

= 0׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

𝑇0

J Chang, PhD, DABR

Generalized to accommodate bremsstrahlung generation

• 𝐷𝑤 = 0׬ 𝑇0

Φ𝑇 𝑒 𝑑𝑇/𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇 =

CPE 𝐾𝑐 𝑤 = 𝑁𝑇0 𝑇0 1 − 𝑌𝑤 𝑇0

= 𝑁𝑇0 𝑇0 − 𝑁𝑇0 𝑇0𝑌𝑤 𝑇0 = 0׬ 𝑇0 𝑁𝑇0 𝑑𝑇 − 𝑁𝑇0 𝑇0

𝑇0 ׬

𝑇0 𝑑𝑇/𝜌𝑑𝑥 𝑟,𝑤

𝑑𝑇/𝜌𝑑𝑥 𝑤 𝑑𝑇

= ׬ 0

𝑇0 𝑁𝑇0 1 − 𝑑𝑇/𝜌𝑑𝑥 𝑟,𝑤

𝑑𝑇/𝜌𝑑𝑥 𝑤 𝑑𝑇 = ׬

𝑇0 𝑁𝑇0 𝑑𝑇/𝜌𝑑𝑥 𝑐,𝑤

𝑑𝑇/𝜌𝑑𝑥 𝑤 𝑑𝑇 ⟹

• Φ𝑇 𝑒 =

𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

remains unchanged

• 𝐷𝑔 = 0׬ 𝑇0 Φ𝑇

𝑒 𝑑𝑇/𝜌𝑑𝑥 𝑐,𝑓 𝑑𝑇 = 𝑁𝑇0 0׬ 𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

• 𝐷𝑔

𝐷𝑤 =

𝑁𝑇0 0׬ 𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

𝑁𝑇0 𝑇0 1−𝑌𝑤 𝑇0 =

𝑇0 1−𝑌𝑤 𝑇0 ׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇 = 𝑚 ҧ𝑆𝑤

𝑔

J Chang, PhD, DABR

Averaging of stopping powers

• In a real interaction (e.g., Compton), the produced electrons are not monoenergetic but with an energy spectrum 𝑇0

• Extend a single starting energy 𝑇0 to a distribution 𝑇 (spectrum): stopping power must be integrated over the distribution 𝑇

𝐷𝑤 = 𝑇0=0׬ 𝑇max

𝑁𝑇0 𝑇0 1 − 𝑌𝑤 𝑇0 𝑑𝑇0

𝐷𝑔 = 𝑇0=0׬ 𝑇max

𝑑𝑇0 𝑇=0׬ 𝑇0

Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔

𝑑𝑇 = ׬ 𝑇0=0

𝑇max 𝑁𝑇0 𝑑𝑇0 𝑇=0׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

𝐷𝑔

𝐷𝑤 =

0׬ 𝑇max 𝑁𝑇0 𝑑𝑇0 0׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

𝑇0=0׬ 𝑇max 𝑁𝑇0 𝑇0 1−𝑌𝑤 𝑌0 𝑑𝑇0

≡ 𝑚 Ӗ𝑆𝑤 𝑔

averaging over each 𝑇0 distribution, and over 𝑇 for each 𝑇0-value.

J Chang, PhD, DABR

Alternative derivation: Averaging of stopping powers

• 𝑇0 ∈ 0, 𝑇max , contribute 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

for 𝑇0 ⟹ Φ𝑇 𝑒 = ׬

𝑇0=𝑇

𝑇max 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇0

• 𝑌 𝑇0 = 1

𝑇0 ׬

𝑇0 𝑑𝑇/𝜌𝑑𝑥 𝑟

𝑑𝑇/𝜌𝑑𝑥 𝑑𝑇 =

𝑇0 ׬

𝑇0 1 −

𝑑𝑇/𝜌𝑑𝑥 𝑐

𝑑𝑇/𝜌𝑑𝑥 𝑑𝑇 = 1 −

𝑇0 ׬

𝑇0 𝑑𝑇/𝜌𝑑𝑥 𝑐

𝑑𝑇/𝜌𝑑𝑥 𝑑𝑇

𝐷𝑤 = 𝑇=0׬ 𝑇max

Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇 = ׬

𝑇=0

𝑇max ׬

𝑇0=𝑇

𝑇max 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇0 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑤 𝑑𝑇

= ׬ 𝑇0=0

𝑇max 𝑑𝑇0 𝑇=0׬

𝑇0 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑤

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇 = ׬

𝑇0=0

𝑇max 𝑁𝑇0 𝑑𝑇0 𝑇=0׬

𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤,𝑐

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

= ׬ 𝑇0=0

𝑇max 𝑁𝑇0 𝑇0 1 − 𝑌𝑤 𝑇0 𝑑𝑇0

𝐷𝑔 = 𝑇=0׬ 𝑇max

Φ𝑇 𝑒 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇 = ׬

𝑇=0

𝑇max ׬

𝑇0=𝑇

𝑇max 𝑁𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑑𝑇0 𝑑𝑇

𝜌𝑑𝑥 𝑐,𝑔 𝑑𝑇

= ׬ 𝑇0=0

𝑇max 𝑑𝑇0 𝑇=0׬ 𝑇0 𝑁𝑇0

Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇 = ׬

𝑇0=0

𝑇max 𝑁𝑇0 𝑑𝑇0 𝑇=0׬ 𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

J Chang, PhD, DABR

Average energy approximation

• The general expression 𝐷𝑤

𝐷𝑔 = 𝑚

ҧ𝑆𝑤

𝑚 ҧ𝑆𝑔

= 0׬

𝑇max Φ𝑇 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑔𝑑𝑇

0׬ 𝑇max Φ𝑇 Τ𝑑𝑇 𝜌𝑑𝑥 𝑐,𝑤𝑑𝑇

≡ 𝑚 ҧ𝑆𝑔 𝑤 is a

slowly varying function so average energy ത𝑇 may be used instead.

• When CPE does not exist, calculate

– ത𝑇 = 0׬

𝑇max Φ𝑇𝑇𝑑𝑇

0׬ 𝑇max Φ𝑇𝑑𝑇

= 0׬

𝑇max Φ𝑇𝑇𝑑𝑇

Φ , and then

– Look up the tabulated Τ𝑑𝑇 𝜌𝑑𝑥 for ത𝑇 and calculated the ratio for 𝑤 and 𝑔

• If CPE exists from charged particles of mean starting energy ത𝑇0:

– Look up the tabulated Τ𝑑𝑇 𝜌𝑑𝑥 for Τത𝑇0 2 and calculate the ratio to get 𝑚 ҧ𝑆𝑔 𝑤

– For Compton, use ത𝑇0 = ℎ𝜈 𝑒𝜎𝑡𝑟

𝑒𝜎

J Chang, PhD, DABR

• Energy not transferred to electrons is carried away by scattered photons.

𝑒𝜎 = 𝑒𝜎𝑡𝑟 +𝑒𝜎𝑠 • Average fraction of incident photon

energy given to and average energy of the Compton recoil electron:

𝑇

ℎ𝜈 = 𝑒

𝜎𝑡𝑟

𝑒𝜎 ⇒ 𝑇 = ℎ𝜈 𝑒

𝜎𝑡𝑟

𝑒𝜎

Review: ENERGY TRANSFER COMPTON

24 J. Chang, PhD, DABR

• For ℎ𝜈0= 1.6 MeV – half of the photon energy is transferred to the electron ( = 0.8 MeV)

FIGURE 7.7. Mean fraction ( Τ𝑇 ℎ𝜈) of the incident photon's energy given to the recoiling electron in Compton interactions, averaged over all angles (right ordinate). Also,

mean fraction ( Τℎ𝜈′ ℎ𝜈) of energy retained by the scattered photon (left ordinate)

Average energy for MV beams (Compton dominant)

1. Equivalent monoenergetic photon energy, k for y MV: find ratio ത𝑘/𝑦 MV in Figure 9.11.

2. The mean initial secondary

electron energy ത𝑇0 = ത𝑘 𝑒𝜎𝑡𝑟

𝑒𝜎 .

3. The mean energy in the eqilibrium slowing-down spectrum of the secondary electrons ത𝑇 is Τത𝑇0 2.

4. Calculate 𝑚 ҧ𝑆𝑔 𝑤 using stopping the

power for Τത𝑇0 2.

J Chang, PhD, DABR

Figure 9.11 Monte Carlo calculated values of the ratio ത𝑘/𝑦 , where k is the equivalent

monoenergetic photon energy that gives the same value of 𝑠w,air BG as the megavoltage

spectrum of y MV, and of the ratio 𝐸𝑒/𝑦, where 𝐸𝑒/𝑦 is the value of the single electron

energy such that ΤΤ𝑆el 𝐸e 𝜌 w Τ𝑆el 𝐸e 𝜌 air = 𝑠w,air BG for the megavoltage beam (for

comparison, the 𝐸𝑒/𝑦 value for a 60Co beam is also included, where y, in this case, is 1.25

MeV). These ratios may vary slightly depending on the spectral distribution corresponding to a given nominal MV. (Data from Andreo and Nahum (1985).)

Cavity Ionization as a Function of Wall Material

• Relative ionization measured per unit volume of

– Air in flat, guard-ringed ion chambers with

– Equilibrium-thickness walls of C, Al, Cu, Sn, and Pb,

– Exposed to the 412-keV g rays from 198Au.

• Ionizations normalized to graphite wall:

– Pb/C, SnIC, CuIC, and Al/C chambers,

– For various air-gap widths.

• B-G 2nd corollary:

– 𝑄𝑤2

𝑄𝐶 =

𝑉𝑤2

𝑉𝐶 ∙

Τ𝜇𝑒𝑛 𝜌 𝑤2

Τ𝜇𝑒𝑛 𝜌 𝐶

∙ 𝑚 ҧ𝑆𝑤2 𝐶 a constant

– 𝑄𝑤2/𝑉𝑤2

𝑄𝐶/𝑉𝐶 =

Τ𝜇𝑒𝑛 𝜌 𝑤2

Τ𝜇𝑒𝑛 𝜌 𝐶

∙ 𝑚 ҧ𝑆𝑤2 𝐶 ∝ 𝑚 ҧ𝑆𝑤2

𝐶

J Chang, PhD, DABR

Experiments had shown deviations from B-G theory

• Ionization ↓ as the air-gap thickness ↑

• More pronounced as Z increases.

• Partly due to a gradual loss of lateral CPE:

– Electrons scattering out of the chamber edges are not fully replaced by

– Electrons generated by g-ray interactions in the 2.5-cm-wide guard-ring area, made of the same wall material

J Chang, PhD, DABR

FIGURE 10.3. Comparison of measured ionization densities (solid curves) in flat air-filled ion chambers having various wall materials and adjustable gap widths, with Bragg-Gray theory (tick marks at left) and Spencer theory (dashed curves), for 198Au g ays. (After Attix, De La Vergne, and Ritz, 1958.)

Experiments had shown deviations from B-G theory

• 𝑄𝑤2/𝑉𝑤2

𝑄𝐶/𝑉𝐶 =∝ 𝑚 ҧ𝑆𝑤2

𝐶 , But

– Measured curves tend higher values as the cavity size is reduced,

– Do not approach the B-G values

– More pronounced for high Z

• For walls of high atomic number d-ray production becomes an issue

– B-G theory 𝑚 ҧ𝑆𝑤2 𝐶 is evaluated assuming CSDA for primary particles only. In fact,

– d-rays cross the cavity with the rest of electrons, and

– Change spectrum, enhancing low energy part, particularly for high Z material

• Spencer cavity theory comes significantly closer to experimental results

J Chang, PhD, DABR

Refresh: Mass Collision Stopping Power for heavy particles

J. Chang, PhD, DABR

• 𝛽 = 𝑣

𝑐

• 𝑇 = 𝑀0𝑐 2 1

1− Τ𝑣2 𝑐 2 − 1

= 𝑀0𝑐 2 1

1−𝛽2 − 1

• 𝛽 = 𝑣

𝑐 = 1 −

1+ 𝑇

𝑀0𝑐 2

Refresh: Mass Stopping Powers vs. Energy and Z

J. Chang, PhD, DABR

SPENCER-ATTIX CAVITY THEORY

• BG theory did not accurately predict ionization in air cavities especially when walls with high Z were used.

• d rays not addressed – these affect the equilibrium spectra by adding more low energy terms (which havehigher stopping power)

• Spencer-Attix approach was to account for this by relating the size of the cavity to be crossed with the mean energy D associated with those electrons that had a projected range matching the cavity dimension.

• The S-A approach gave better agreement with experimental observations and applies the same conditions required for the BG theory.

J Chang, PhD, DABR

Spencer Cavity Theory

• Goals: to account for d-rays and cavity size effect

• Starts with two B-G conditions (narrow g region and dose produced by crossing particles) and two additional assumptions (existence of CPE and absence of bremsstrahlung generation)

• Introduces mean energy D, needed to cross the cavity

J Chang, PhD, DABR

= Τ𝑑𝐸𝑖𝑛 𝑑𝑥 𝑐 × 𝑑 = 𝐸𝑖𝑛

Fast and slow groups

• Based on their energy T, electrons in a spectrum are divided into “fast” (T> D) and “slow” (T< D) groups

– Fast electrons have enough energy to cross the cavity if they strike it.

– Slow electrons are assumed to have zero range, i.e., to drop their energy "on the spot" where their kinetic energy falls below D. Hence they are assumed

– NOT to be able to enter the cavity, nor to transport energy.

• The stopping power was thus modified to restrict the energy losses to less than this value D relative to the cavity to be crossed, i.e., the mass restricted stopping power

J Chang, PhD, DABR

Restricted Stopping Power

• To calculate dose to small objects or thin foils traversed by charged

particles, unless CPE exists for d-rays, the formulae Φ 𝑑𝑇

𝜌𝑑𝑥 𝑐 where Φ

is the fluence of primary charged particles overestimates the dose.

• If there is no d-ray CPE, then then 𝑑𝑇

𝜌𝑑𝑥 𝑐 should be calculated for

– Soft collisions plus hard collisions with energy below a cut off energy D that permit local energy deposition in the small object / thin foil, and

– Consider both primary and d-ray electrons in the dose calculation.

• This leads to the notion of restricted stopping power 𝑑𝑇

𝜌𝑑𝑥 ∆

34 J. Chang, PhD, DABR

Refresh: calculation of Restricted Stopping Power

• To calculate restricted mass collision stopping powers, the term 𝑇max is replaced by D in the various expressions used till now.

𝑑𝑇

𝜌𝑑𝑥 ∆ = ׬

𝑇𝑚𝑖𝑛 ′

𝐻 𝑇′𝑄𝑐

𝑠𝑑𝑇′ + ׬ 𝐻

∆ 𝑇′𝑄𝑐

ℎ 𝑑𝑇′ = 𝑚𝑆 𝑇, Δ

• For heavy charged particles: 𝑑𝑇

𝜌𝑑𝑥 ∆ = 𝑘 ln

2𝑚0𝑐 2𝛽2∆

𝐼2 1−𝛽2 − 2𝛽2 −

2𝐶

𝑍

• For electrons and positrons: 𝑑𝑇

𝜌𝑑𝑥 ∆ = 𝑘 ln

𝜏2 𝜏+2

2 Τ𝐼 𝑚0𝑐 2 2

+ 𝐺 ± 𝜏, 𝜂 − 𝛿 − 2𝐶

𝑍

35 J. Chang, PhD, DABR

Equilibrium spectrum including d-rays

• For monoenergetic electron beam with 𝑇0 emitting 𝑁 particles per gram through a homogeneous medium w the absorbed dose is expressed in terms of restricted stopping power

𝐷𝑤 = CPE

𝑁𝑇0 = Δ׬ 𝑇0

Φ𝑇 𝑒,𝛿

𝑚𝑆𝑤 𝑇, Δ 𝑑𝑇

• The equilibrium spectrum including d-rays

– Φ𝑇 𝑒,𝛿

= 𝑁𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 in comparison to Φ𝑇

𝑒 = 𝑁

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 for primary only, where

– 𝑅 𝑇0, 𝑇 – ratio of differential electron fluence, including d-rays to that of primary electrons alone

J Chang, PhD, DABR

Ratio of the Differential Electron Fluences

• Spectrum enhanced many-fold at low electron energies

J Chang, PhD, DABR

Equilibrium Spectrum

J Chang, PhD, DABR

FIGURE 10.4. Equilibrium spectrum of 64Cu  rays in copper. The “primary” curve is the equilibrium spectrum of primary 𝛽−- and 𝛽+-particles emitted by the distributed source. The "secondary" curve, extending into the solid curve labeled "theory", is the d-ray contribution calculated by means of the factor 𝑅 𝑇0, 𝑇 . The solid curve combines the primaries and calculated secondaries. The points were measured with an electrostatic spectrometer. (After McConnell et al., 1964. Reproduced with permission from H. H. Hubbell, Jr. and the Oak Ridge National Laboratory.)

𝑫𝒈 and 𝑫𝒘 with Spencer Cavity Theory

• Φ𝑇 𝑒,𝛿

= 𝑁𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

• Taking into account adjustment for electron spectrum, dose to the wall

𝐷𝑤 = CPE

𝑁𝑇0 = Δ׬ 𝑇0

Φ𝑇 𝑒,𝛿

∙ 𝑚𝑆𝑤 𝑇, Δ 𝑑𝑇 = 𝑁 Δ׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚𝑆𝑤 𝑇, Δ 𝑑𝑇

• D regulates the cavity size; for D =0

𝐷𝑤 = 0׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚𝑆𝑤 𝑇, Δ 𝑑𝑇 = 𝑁𝑇0 ⇒ 𝑅 𝑇0, 𝑇 ≥

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

𝑚 𝑆𝑤 𝑇,Δ

• Dose to the cavity

𝐷𝑔 = 𝑁 Δ׬ 𝑇0

Φ𝑇 𝑒,𝛿

∙ 𝑚𝑆𝑔 𝑇, Δ 𝑑𝑇 = 𝑁 Δ׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚𝑆𝑔 𝑇, Δ 𝑑𝑇

J Chang, PhD, DABR

Ratio of doses in cavity and wall

𝐷𝑔

𝐷𝑤 =

𝑁 Δ׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚

𝑆𝑔 𝑇,Δ 𝑑𝑇

𝑁 Δ׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚

𝑆𝑤 𝑇,Δ 𝑑𝑇 =

Δ׬ 𝑇0 𝑅 𝑇0,𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 ∙ 𝑚

𝑆𝑔 𝑇,Δ 𝑑𝑇

𝑇0

• Works well for small cavities (electron range is much larger than the cavity size)

• If actual spectrum of crossing charged particles is known, can replace the ratio term

• Compared with B-G general form

𝐷𝑔

𝐷𝑤 =

0׬ 𝑇0 Τ𝑑𝑇 𝜌𝑑𝑥 𝑔

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤 𝑑𝑇

𝑇0 = 𝑚 ҧ𝑆𝑤

𝑔

J Chang, PhD, DABR

Comparison of Τ𝑫𝒈 𝑫𝒘 bt. B-G and S-A Theories

• Better agreement between Spencer and B-G for large cavity sizes

• Effective atomic number is ~7.0 for air

• Among all the wall medium, carbon is closest to air and has the best

match for 𝐷𝑔

𝐷𝑤 .

J Chang, PhD, DABR

Summary Spencer Cavity Theory

• The Spencer cavity theory gives somewhat better agreement with experimental observations for

– Small cavities than does simple B-G theory, by taking account of

– d-ray production and relating the dose integral to the cavity size

• However, it still relies on the B-G conditions, and therefore fails to the extent that they are violated

• In particular, in the case of cavities that are large (i.e., comparable to the range of the secondary charged particles generated by indirectly ionizing radiation), neither B-G condition is satisfied

J Chang, PhD, DABR

Burlin Cavity Theory

• g-ray cavity theory, for intermediate cavity size

J Chang, PhD, DABR

e1 – crossers e2 – starters e3 – stoppers e4 – insiders

Small Intermediate Large

When is ratio of dose equal to ratio of 𝝁𝒆𝒏

𝝆 ?

• If the same photon energy fluence Ψ is present in media A and B having two different average energy absorption coefficients, the ratio of absorbed doses under CPE conditions in the two media will be given by:

𝐷𝐴

𝐷𝐵 =

CPE 𝐾𝑐 𝐴

𝐾𝑐 𝐵 =

𝜇𝑒𝑛 𝜌 𝐴

𝜇𝑒𝑛 𝜌 𝐵

= 𝜇𝑒𝑛

𝜌 𝐵

𝐴

• Doses 𝐷𝐴 and 𝐷𝐵 can differ due to either different atomic compositions A and B, or radiation spectra

J. Chang, PhD, DABR

CROSSERS, STARTERS, STOPPERS AND INSIDERS Do uncharged particles outside w interact in g?

J Chang, PhD, DABR

Only “Crossers” NO 𝑚

ҧ𝑆𝑤 𝑔SMALL CAVITY

BRAGG –GRAY

“Crossers” – e1 “Starters” – e2 “stoppers” – e3 “Insiders” - e4

YES 𝑚 ҧ𝑆𝑤 𝑔

and 𝜇𝑒𝑛

𝜌 𝑤

𝑔 INTERMEDIATE CAVITY

Non-Uniform Dose in Cavity

“stoppers” – e3 only affect a thin layer Mostly “Insiders” - e4

YES 𝜇𝑒𝑛

𝜌 𝑤

𝑔 LARGE CAVITY

Non-Uniform Dose in Cavity

Assumptions Burlin Cavity Theory

1. The media 𝑤 and 𝑔 are homogeneous.

2. A homogeneous g-ray field exists everywhere throughout 𝑤 and 𝑔. (This means that no g-ray attenuation correction)

3. Charged-particle equilibrium exists at all points that are farther than the max electron range from the cavity boundary

4. The equilibrium spectra of secondary electrons generated in 𝑤 and 𝑔 are the same

J Chang, PhD, DABR

Assumptions Burlin Cavity Theory

5. The fluence of electrons entering from the wall is attenuated exponentially as it passes through the medium g, without changing its spectral distribution.

6. The fluence of electrons that originate in the cavity builds up to its equilibrium value exponentially as a function of distance into the cavity, according to the same attenuation coefficient  that applies to the incoming electrons

J Chang, PhD, DABR

Exponential-decay and -build up in Burlin Cavity Theory

• Φ𝑤 𝑒 , Φ𝑔

𝑒 : equilibrium wall, gas fluence of

electrons

• Φ𝑤 , Φ𝑔 : buildup wall, gas fluence of electrons

• Φ𝑤 + Φ𝑔 = Φ𝑤 𝑒 = Φ𝑔

𝑒 : Fano theorem

• 𝐿 = 4𝑉

𝑆 : mean chord of length,

– 𝑉: cavity volume, 𝑆: surface area S

– For sphere: 𝐿 = 4×

3 𝜋𝑟3

4𝜋𝑟2 =

3 𝑟

– For disk with thickness 𝑡 : 𝐿 = 4𝑡𝜋𝑟2

2𝜋𝑟2+2𝜋𝑟𝑡 =

2𝑡𝑟

𝑟+𝑡

• 𝑙: distance (cm) of any point in the cavity from the wall, along 𝐿

J Chang, PhD, DABR

FIGURE 10.6. Illustration of the exponential-decay and -buildup assumption in the Burlin cavity theory. The equilibrium wall fluence of electrons, Φ𝑤

𝑒 , is shown decaying exponentially as they progress into a homogeneous cavity for which the wall w and cavity g media are assumed to be identical. The electrons under consideration are only those flowing from left to right. The buildup of the cavity-generated electron fluence Φ𝑔 follows a

complementary exponential, asymptotically approaching its equilibrium value Φ𝑤

𝑒 = Φ𝑔 𝑒 .

Burlin Cavity Theory

• Cavity relation accounts for 2 sources of electrons depositing dose ഥ𝐷𝑔

𝐷𝑤 = 𝑑 ∙ 𝑚 ҧ𝑆𝑤

𝑔 + 1 − 𝑑

𝜇𝑒𝑛

𝜌 𝑤

𝑔

• ഥ𝐷𝑔 : the average absorbed dose in the cavity medium 𝑔;

• 𝐷𝑤 = 𝐾𝑐 , the absorbed dose in - medium 𝑤 under CPE conditions (i.e., not within electron range of the cavity);

• 𝑚 ҧ𝑆𝑤 𝑔

: the mean ratio of mass collision stopping powers, obtained either on the basis of the B-G or Spencer theory; and

• 𝜇𝑒𝑛

𝜌 𝑤

𝑔

: the mean ratio of the mass energy-absorption coefficients

J Chang, PhD, DABR

Parameters 𝒅 and 𝟏 − 𝒅

• Parameter 𝑑 and 1 − 𝑑 represent the average value of ΤഥΦ𝑤 ഥΦ𝑤 𝑒 and

ൗഥΦ𝑔 ഥΦ𝑔 𝑒 throughout the cavity:

𝑑 ≡ ഥΦ𝑤

Φ𝑤 𝑒 =

0׬ 𝐿

Φ𝑤 𝑒 𝑒 −𝛽𝑙𝑑𝑙

0׬ 𝐿

Φ𝑤 𝑒 𝑑𝑙

= 1−𝑒 −𝛽𝐿

𝛽𝐿

1 − 𝑑 ≡ ഥΦ𝑔

Φ𝑔 𝑒 =

0׬ 𝐿

Φ𝑔 𝑒 1−𝑒 −𝛽𝑙 𝑑𝑙

0׬ 𝐿

Φ𝑔 𝑒 𝑑𝑙

= 𝛽𝐿+𝑒 −𝛽𝐿−1

𝛽𝐿

where 𝑙 is the distance (cm) of any point in the cavity from the wall, along a mean chord of length 𝐿

• Parameter 𝑑 ≈ 1 for small 𝐿 (or small cavity).

• 𝑑 ≈ 0 for large 𝐿 (or large cavities).

J Chang, PhD, DABR

Nonhomogeneous case where g  w, (𝜱𝒘 𝒆 ≠ 𝜱𝒈

𝒆 )

• If the -value of the cavity medium for the wall electrons is not the same as for the cavity-generated electrons, due to a difference in spectral distributions, then in general

ഥΦ𝑔

Φ𝑔 𝑒 ≡ 𝑑′ ≠ 1 − 𝑑

and hence 𝑑′ + 𝑑 ≠ 1

• The Burlin theory ignores this possible source of error in adopting assumptions 5 and 6

J Chang, PhD, DABR

Parameter 

• Theory works well for wide range of cavity sizes and materials

• Parameter  estimated for air-filled cavity:

𝛽 = 16𝜌

𝑇max−0.036 1.4

𝑒−𝛽𝑡max = 0.01 or 0.04 ⟹ 𝛽 = − ln 0.01

𝑡max or

− ln 0.04

𝑡max

• 𝜌: air density,

• 𝑇max: max starting energy,

• 𝑡max: max electron penetration depth

J Chang, PhD, DABR

Burlin Cavity Theory Verification

• LiF thermoluminescence dosimeters (TLDs) each 0.1 g/cm2 thick: – Stacked four per layer in 1, 2,3,5, and 7

layers: small to large cavity

– Sandwiched between equilibrium- thickness walls of various media: either LiF, polystyrene, Al, Cu, or Pb

– Irradiated by 60Co g-rays perpendicularly

– Data normalized to the homogeneous case where the wall medium also consisted of solid LiF.

J Chang, PhD, DABR

FIGURE 10.7. 60Co gr-ay experiment to test the Burlin theory as applied to LiF TLD chips, each 0.38 x 3.18 x 3.18 mm3, r = 2.64 g/cm3, stacked four per layer in 1, 2,3,5, and 7 layers. The CPE buildup layer and backscattering medium were both made of the same wall material, either LiF, polystyrene, Al, Cu, or Pb. The spacer was adjusted to equal the TLD stack thickness, and for the results presented here was made of LiF to produce a semi-infinite one-dimensional cavity. (After Ogunleye, et al., 1980. Reproduced with permission of The Institute of Physics, U.K.)

Relative dose measurements

• Dose distribution across the cavity of increasing size

– LiF ℜ𝐶𝑆𝐷𝐴 =0.71 g/cm 2

which is ~7 TLDs.

– Normalized to LiF measurements

J Chang, PhD, DABR

Small -----------→ Intermediate ---------------→ Larger

Burlin Cavity Theory Verification

• Good agreement for polystyrene and aluminum wall media

• In higher-Z materials electron backscattering is important

J Chang, PhD, DABR

FIGURE 10.9. Comparison of the Burlin theory (solid curves). with the experiment referred to in Figs. 10.7 and 10.8. The application of the theory in this case, as described in the text, differs from that of Ogunleye et al. (1980).

Small → Intermediate → Larger

Summary

J Chang, PhD, DABR

Accuracy comparison

• 1 MeV photons incident on a small cylindrical cavity in a homogeneous water phantom

• Gold standard: direct Monte Carlo (MC)

• Three different cavity integrals: B-G, S-A and Burling

• Various cutoff energy D

– Ranging from 1 to 200 keV.

– More accurate for smaller D.

J Chang, PhD, DABR

FANO THEOREM

• In practice the requirement for small cavity is ignored by matching atomic numbers of wall and cavity materials.

• Theorem statement: In an infinite medium of given atomic composition exposed to a uniform field of indirectly ionizing radiation, the field of secondary radiation is also uniform and independent of the density of the medium, as well as of density variations from point to point.

• Proof employs radiation transport equations.

J Chang, PhD, DABR

Example of an equilibrium fluence spectrum

Φ𝑇 𝑒 =

𝑁

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

Φ𝑒 = ׬ 0

𝑇0 Φ𝑇

𝑒 𝑑𝑇

= ׬ 0

𝑇0 𝑁𝑑𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

= 𝑁 ׬ 0

𝑇0 𝑑𝑇

Τ𝑑𝑇 𝜌𝑑𝑥 𝑤

= 𝑁ℜ𝐶𝑆𝐷𝐴

= 𝜌𝑁 ℜ𝐶𝑆𝐷𝐴

𝜌 (particles/cm2)

Which is independent of 𝜌.

J Chang, PhD, DABR

FANO THEOREM

• Fano ‘worked around’ BG conditions for small cavities instead by matching g and w layers in terms of atomic composition.

• But his work ignored the polarization effect which is substantial for megavoltage x-rays

• So generally applicable under 1 MeV his theorem states:

• A loose interpretation: If we were to consider halving the density of a medium with homogeneous sources in a section of the medium, then the number of particles emitted would also halve, and their range would double, making the fluence the same as if we didn’t halve it.

J Chang, PhD, DABR

Dose near interfaces between dissimilar media

FIGURE 10.11 Variation of electron fluence with distance from a copper-carbon interface irradiated perpendicularly by 27

60Co g rays (Dutreix and Bernard, 1966). Solid curves: Ionization measured in a thin, flat air layer as it is gradually traversed through the interface by addition and removal of copper and carbon foils at the air-gap walls. Dashed curves: Electrons arising in copper. Dash-dotted curves: Electrons arising in carbon.

The arrows indicate the photon direction in each case: left to right in A, right to left in B and C. Fcu is the fraction of the equilibrium fluence of electrons that flow with a component in the g ray direction in the copper; Bcu is the backscattered component. In the carbon the backscattered component is small, and is assumed to be negligible here. (Reproduced with permission from J. Dutreix and The British Journal of Radiology.)

J Chang, PhD, DABR

FC BCu

FCu

Depends on relative atomic numbers

BCU

FCU

BC : negligible.

FCFCU

BCU

Dose near interfaces between dissimilar media

J Chang, PhD, DABR

Al Al

Dose near interfaces between dissimilar media

• A minimum is observed just beyond the interface when the photons go from a higher-Z to a lower-Z medium

• A maximum is observed just beyond the interface when the photons go from a lower-Z to a higher-Z medium

• Tissue-bone interface is an example

J Chang, PhD, DABR