1

Cavity theory

Eirik Malinen

Cavity

• Cavity theory, narrow sense: convert “dose to detector” to “dose to medium”

• Cavity theory, broad sense: dose distribution in inhomogeneous media

med

cav

γ, n, e-, …

2

Absorbed dose in γ irradiated thin foil, CPE

Energy transferred:

xΔ

xhNRRRRRR

tr

outin

CPE

eoutouteinintr

Δ=

−=−−+=

μν

ε γγγγ

)( ,,,,,,

γ e− e−

( )νγ

hNRin

=, einR , eoutR , γ,outR R: Radiant energy

= N(hν) for monoenergetic photons

Absorbed dose in γ irradiated thin foil, CPEAbsorbed dose (no brehmsstrahlung)

If brehmsstrahlung:

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ=

ΔΔΨ

=Δ

===ρμ

ρμμνε trtrtrtr

xAxA

mxhN

mKD )(

⎟⎟⎠

⎞⎜⎜⎝

⎛==

ρμΨ en

cKD

νμμ

hT

tr =

3

Energy loss from electrons• Stopping power:

• Collision stopping power: Scol

• Restricted stopping power: LΔ

∫ ⎟⎠⎞

⎜⎝⎛=+==

max

min

E

E

totradcol dE

dEdEnSS

dxdTS σρ

∫ ⎟⎠⎞

⎜⎝⎛=

max

min

E

E

colcol dE

dEdEnS σρ

∫Δ

Δ ⎟⎠⎞

⎜⎝⎛=

minE

col dEdE

dEnL σρ

n: number of atomicelectrons per gram

Absorbed dose in thin foil, electrons

Energy loss <ΔT> → energy imparted ε ?

→ Brehmsstrahlung, δ rays, path lengthening

Brehmsstrahlung: Srad

ℜ<<xΔ

xΔRange of charged particle

4

Path lengthening due to multiple scattering

Scattering power:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Δ=

dxdxcos)cos(

2θρθ

xΔ

θ

θcosx'x Δ

=Δ

dxd 2θ

δ rays• Energetic, secondary electrons• Significant range compared to foil thickness• Results from high energy transfers (included in Scol)

Maximum energy transfer:

22

12

ββ−

= cmE emax

Heavy ions

2/TEmax =

Electrons

5

δ raysEnergy imparted for charged particles:

δ particle equilibrium

δPE requirements: homogeneous medium andδPE always present under CPE

δδε ,outp,out,inp,in RRRR −−+=

p,outp,in,out,in RRRR −=⇒= εδδ

xΔ

pℜ<<ℜδ

primary

δ rays• Since β is low for heavy charged particles in the MeV-

region, Emax is low

• β=0.1 (e.g. 38 MeV α-particles) gives Emax=10 keV

• Range of 10 keV electrons in water: 2.5 μm

• →δ-electrons deposit their energy locally, and δ-equilibrium may often be present

• Range of 1 MeV electrons: 0.5 cm

• →δ-equilibrium may not obtained for high energy electron beam

6

Absorbed dose

• Under δPE (foil sandwiched, short ), no path lengthening, no brehmstrahlung:

mD ε=

AN,S

AN

xAxNS

VxNSD

xNSRRR pp,outp,in

====⇒

==−=

ΦρΔρ

ΔρΔ

ΔΔε

δℜ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρΦ S D Fluence of primary electrons

Absorbed dose, thick foil, heavy particles

• The average dose may be found by:– Calculating the residual range: ℜres = ℜin-L– Find the energy Tout corresponding to ℜres

– Imparted energy is: ΔT = Tin-Tout

– Dose:

L

L

Tin Tout

N T TDm LΔ Δ

= = Φρ

7

REMAINDER: ELECTRONS

Foil placed in vacuumδ rays with T > Δ lost from foil (δPE absent):

Better formulation (?) :δ rays with rpr > Δx lost from foil

⎥⎥⎦

⎤

⎢⎢⎣

⎡=−−= ∫ dE

dEdEnNRRR

minE,outp,outp,in

Δ

δσρε

⎟⎟⎠

⎞⎜⎝

⎛=

ρΦ ΔL D

Projected range, expectation value of farthest depth of penetration

8

LΔ

airwater

Δ=10 keV

Range and projected range, electrons

9

Spectrum of charged particles, δPE presentΦTdT : number of primary electrons cm-2 in [T, T+dT ]Minimum energy: 0Maximum energy: Tmax

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛=⇒

maxT

0TT

SdTDSdTdDρ

Φρ

Φ

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

maxT

0T dTSD

ρΦ

Partial δPEElectron beams: constant fluence of secondary, low energy electrons with T < ΔEnergetic secondary electrons added to total fluence:

Particles either assigned to radiation field or to energy imparted

∫ ⎟⎟⎠

⎞⎜⎝

⎛= +

maxTp

T dTLDΔ

Δδ

ρΦ

? pT

δΦ +

10

Bragg-Gray cavity theory

B-G conditions:1. Charged particle fluence is not perturbed by cavity2. Absorbed dose entirely due to charged particles

wall

cav

e-,p, ....

wallwall

cavcav

S D

S D

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρΦ

ρΦ

cav

wall

col

wall

cav S DD ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⇒

ρ

Bragg-Gray cavity theory

11

Bragg-Gray-LaurenceLaurence: incorporated slowing down spectrum of charged particles generated in the wall

wall

0T

T

00

wallT

T

00

T

0 wallT

00

CPET

0 wallT

Sn 0dTnS

dTndTS

TndTS

0

00

0

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒=⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∫

∫∫

∫

ρ

Φρ

Φ

ρΦ

ρΦ n0 monoenergetic electrons per gram

with kinetic energy T0

Bragg-Gray-LaurenceFurthermore:

ΦT: track length distribution

dxn

dxdTdTn

SdTndT 00

wall

0T ρρ

ρ

Φ ==

⎟⎟⎠

⎞⎜⎜⎝

⎛∝

12

Bragg-Gray-LaurenceSlowing down spectrum of primary electrons in water

Bragg-Gray-LaurenceThe total fluence:

Independent of density!

Dose to cavity:

CSDACSDA

TT

T VNn

)/S(dTndT ℜ=ℜ=== ∫∫ ρρ

ΦΦ 00

00

0

00

∫∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

000 T

0

cav

wall0

T

0

wall

cav0

T

0 cavTcav dTSn dT

S

S

ndTSDρ

ρ

ρρ

Φ

13

Bragg-Gray-LaurenceUnder CPE or homogeneous source distribution:

Thus:

00, TnKDwall

enwallc

CPE

wall =⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ==

ρμ

∫∫

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛

=0

0

T

0

cav

wall000

T

0

cav

wall0

cavwall dTS

T1

Tn

dTSnD

ρρ

Spencer-Attix cavity theoryKey issue: develop general cavity dose equationS-A for monoenergetic electron source:1) The electron spectrum, originating in the wall only, must

include secondary electrons2) Stopping power in the cavity → restricted stopping power3) The ”dose integral” must be integrated from Δ to T0:

B-G conditions must still be fulfilled

∫Δ

ΔΦ=0

)/(T

cavTcav dTLD ρ

14

Spencer-Attix cavity theory

Two-group theory:

– T ≥ Δ have range ~ cavity dimension

– T ≤ Δ have zero range

Spencer-Fano equilibrium spectrum

δ ray energy range 1

Møller cross section

15

Spencer-Fano equilibrium spectrumAssume spectrum generated in the wall on the form:

nδ ?

May be shown that total differential fluence is:

wall

0T S

nn

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

ρ

Φ δ

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∫

0

2

0 1T

T'TT 'dT)T,'T(K

)/S(n Φρ

Φ

Spencer-Fano equilibrium spectrumIntroduced quantity RT; ratio of total to primary fluence

RT dependent on medium and energy of electron source

( )∫+=⇒

=

0

2

1

1

T

T wall'TT

Twall

T

'dT/S

)T,'T(KRR

R)/S(

ρ

ρΦ

16

Spencer-Fano equilibrium spectrum

Spencer-Attix: Absorbed doses at CPE

∫

∫

=

==

0

0

T

cavitywall

Tcavity

T

wallwall

T00wall

dTLS

RD

dTLS

RTnD

ΔΔ

ΔΔ

ρ

ρ

,

,

)/(

)/(

17

Experiment vs. theory

Air cavity

Burlin cavity theory

Abs

orbe

d do

se

wall

cavity

cav

en⎟⎟⎠

⎞⎜⎜⎝

⎛ρμΨ

wall

en⎟⎟⎠

⎞⎜⎜⎝

⎛ρμΨ

18

Burlin cavity theory

cavwall wall cav cav

wall

Burlin cavity theoryCavity with dimensions << electron range: S-A theory:

Cavity with dimensions >> electron range: CPE-theory:

( )

( )

cav

wallT

wallwall,T

T

cavwall,T

wall

cav L

dT/L

dT/L

DD

⎟⎟⎠

⎞⎜⎝

⎛==

∫

∫ρ

ρΦ

ρΦΔ

ΔΔ

ΔΔ

0

0

( )( )wallen

caven

wall

cav

DD

ρμρμ

//

=

19

Burlin cavity theoryGeneral theory for intermediate sized cavities:

d: average attenuation of electrons generated in the wall crossing the cavity

cav

wall

en

cav

wallwall

cav d1LdDD

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ρμ

ρΔ )(

LeLd

Le

dx

dxed

LL

L

Lx

ββ

β

βββ

111

0

0 −+=−⇒

−==

−−−

∫

∫

Burlin cavity theoryβ: effective electron attenuation coefficientEmpirical expression:

tmax: depth at which 1 % of electrons can traveltmax/ℜCSDA ≈ 0.9 low Ztmax/ ℜCSDA ≈ 0.8 intermediate Ztmax/ ℜCSDA ≈ 0.7 high Z

040.e maxt ≈−β

20

Burlin cavity theory - assumptions• Wall and cavity homogenous• No significant γ attenuation• CPE exists• Spectrum of δ rays equal in wall and cavity• Electrons generated in wall are exponentially attenuated within cavity• Electrons generated in cavity increase exponentially

Burlin cavity theory – experiment vs theory

60Co γ raysPerspex dosimeter in tin wallCorrected for attenuationNormalised to 8 cm

21

X ray dose to bone marrow

Monoenergetic X raysExcess dose compared to soft tissue only:

tissue

bone

entissue

bonebone

tissue d1SdDD

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛≈

ρμ

ρ)(

onlytissue

tissue

DD

bone

tissue

enCPE

onlytissue

bone

DD

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρμ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

tissue

bone

entissue

bone

bone

tissue

en

only tissue

tissue d1SdD

Dρμ

ρρμ )(

bone

Trabecular spacew/ bone marrow

X ray dose to bone marrow

tissue

bone

S⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

tissue

bone

en⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

22

X ray dose to bone marrowMean electron energy :

Range in soft tissue:

μμν trhT =

X ray dose to bone marrowDirectional dependence of electron flux?For convex cavity and isotropic field, the mean chord length is:

Spherical cavities:

SVL /4=

rL34

=

23

X ray dose to bone marrow

X ray dose to bone marrow, Monte Carlo

24

Interface dosimetry

1 MeV γ

MC simΦAl

ΦC

Interface dosimetry

1 MeV γ

MC sim

25

Interface dosimetryTotal equilibrium fluence, secondary electrons, CPE:

Remember that

CSDAn ℜ=Φ 0

ρμ

νμμ

ρμν

ρμ

Nn

hnNhTn enenenCPE

=⇒

≈=Ψ=

0

00

Interface dosimetryTherefore, fluence ratio, medium 1 and 2 becomes:

1 MeV γ rays:= 0.45 MeV , = 0.064 cm-1 , = 0.061 cm-1

= 0.186 g/cm2 , = 0.211 MeV cm2/g

ΦAl/ΦC ≈ 1.10, against 1.14 for MC

( )121

22

1CSDAℜ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ΦΦ

ρμ

TC⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

Al⎟⎟⎠

⎞⎜⎜⎝

⎛ρμ

Cℜ Alℜ

26

Interface dosimetryAt the interface, transition from Φ1 to Φ2

Simplistic vector representation:

Forward/backward ratio depend on medium

forwardbackward

ΦFΦB

Backscatter ratio

27

Interface dosimetrySimplistic considerations – total fluence ΦB+ΦF

At interface, Φtot ≈ ΦB,2 + ΦF,1

ΦF,1ΦB, 1ΦF,2ΦB, 2

Φ1 Φ2

ℜ1 ℜ2

Depth (cm)