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CitThCavity Theory, Stopping -Power Ratios, Correction Factors. Alan E. Nahum PhD

Physics Department Clatterbridge Centre for Oncology Bebington, Wirral CH63 4JY UK ((@[email protected] )

AAPM Summer School, CLINICAL FOR RADIOTHERAPY, 21-25 June 2009, Colorado College, Colorado Springs, USA 1 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

313.1 INTRODUCTION

3.2 “LARGE” DETECTORS

3.3 BRAGG-GRAY CAVITY THEORY

3.4 STOPPING-POWER RATIOS

3.5 THICK-WALLED ION CHAMBERS

363.6 CORRECTION OR PERTURBATION FACTORS FOR ION CHAMBERS

373.7 GENERAL CAVITY THEORY

3.8 PRACTICAL DETECTORS

3.9 SUMMARY 2 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. Accurate knowledgg(p)e of the (patient) dose in is crucial to clinical outcome

For a given fraction size

100 TCP

80 (%)

60 Therapeutic Ratio NTCP 40 & NTCP 20 TCP

0 20 40 60Dpr 80 100

Dose (Gy) 3 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

Detectors almost never measure dose to medium directly. Therefore, the interpretation of detector reading requires dosimetry theory - “cavity theory”

4 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

 D  especially when converting med from calibration at Q to f Q    1 D measurement at Q2 5  det Q A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

Also the “physics” of depth -dose curves:

6 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. Dose computation in a TPS

Terma cfKf.

7 First we will remind ourselves of two key results which relate the particle fluence, , to deposition in the medium.

Under charged-particle equilibrium (CPE) conditions, the in the medium, Dmed, is related to the photon (energy) fluence in the medium, h med ,by CPE  en  Dmed  Kc med  h Φmed     med for monoenergetic of energy h, and by

h CPE max dΦ   (h )  D  h med  en  dh med    0 dh   med for a spectrum of photon , where (en/)med is the mass-energy absorption coefficient for the medium in question. 8 For charged particles the corresponding expressions are: δeqm.  Scol  Dmed  Φmed     med

where (Scol/)med is the (unrestricted) electron mass collision stopping pow er for the mediu m. and for a spectrum of electron energies: δeqm. Emax dΦ  S (E)  D  med  col  dE med    0 dE   med Note that in the charged-particle case the requirement is: –ray equilibrium 9 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. “Photon” Detectors

MEDIUM

DETECTOR

DETECTOR

CPE  en  Ddet  Kc   h Φmed   det   10   det Thus for the “photon” detector: CPE  en  Ddet  det     det and for the medium: CPE  en  Dmed  med     med and therefore we can write: D CPE Ψ  /  f Q med,z  med,z en med Ddet Ψ det en / det 11 The key assumption is now made that the photon energy fluence in the detector is negligibly different from that present i n th e undi st urb ed medi um at th e positi on of th e detector i.e. det = med,z and thus: D  /  f Q med,z  en med Ddet en /  det which is the well-known en/ -ratio usually written as med    f Q  en       det and finally for aspectrum of photon energies: Emax  dΦmed,z  en (E)  med E   dE  h     dE     en   0 med    Emax dΦ   det med,z  en (E)  Eh   dE    12 0 dE   det A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

The dependence of the en/ -ratio on photon energy for water/medium

13 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

“Electron” detectors (Bragg-Gray)14 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

15 BRAGG-GRAY COCAVITY THEORY

We have seen that in the case of detectors with sensitive volumes large enough for the esta blis hmen t o f CPE, the ra tio Dmed/Ddet iiis given b(by (en/)med,det fthfor the case o fiditlf indirectly . In the case of charged particles one requires an analogous relation in terms of stopping powers. If we assume that the electron fluences in the detector and at the same depth in the medium are given byby det and med respectively, then according to EquEqu. 4343 wewe must be able to write

Dmed med (Scol / )med  (45) Ddet det(/)Scol  det

17 For the more practical case of a spectrum of electron energies, the stopping-power ratio must be evaluated from Emax   E (()/)SEcol  med d E Dmed o (48)  E Ddet max   E (()/)SEcol  det d E o where the energy dependence of the stopping powers have been made explicit and it is understood that E refers to the undisturbed medium in both the numerator and the denominator. It must be stressed that this is the fluence of primary electrons only; no delta rays are involved (see next section). For reasons that will become apparent in the next paragraph, it is convenient to denote the stopping-power ratio evaluated according to Equ. BG 48 by s meddd,de t [12]. The problem with -rays

medium

e-

e-

 gas  e- 

e- 

19 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. The Spencer-Attix “solution”

medium

gas 

Emax   E (()/)LE medd E E ()(()/) Scol med D med   Emax Ddet   E (()/)LE detd E Ecol ()(()/) S det 20  In the previous section we have neglected the question of delta-ray equilibrium, which is a pre-requisite for the strict validity of the stopping-power ratio as evaluated in Equ. 43. The original Bragg -Gray theory effectively assumed that all collision losses resulted in energy deposition within the cavity. Spencer and Attix proposed an extension of the Bragg-Gray idea that took account, in an approximate manner, of the effect of the finite ranges of the delta rays [11]. All the electrons above a cutoff energy , whether primary or delta rays, were now considered to b e part of th e fl uence spectrum inc ident on th e cavi ty. All energy losses below  in energy were assumed to be local to the cavity and all losses above were assumed to escape entirely. The local energy loss was calculated by using the collision stopping power restricted to losses less than ,L, L (see lecture 3). This 2-component model leads to a stopping-power ratio given by [12,13]:

Emax tot  tot med  (E) L (E) /  dE   ()S () /    L   E med  med E col med         Emax   gas tot (E) L (E) /  dE  tot ()S () /    E med  gas E col gas  BRAGG-GRAY CAVITY

The Stopp ing-PRiPower Ratio smed, det :

Emax  (S (E) / ) dE D  E col med med  o D Emax det  (S (E) / ) dE  E col det o The Spencer -Attix formulation: Emax   (L (E) / ) dE  ()(S () / )  D  E  med E col med med   D Emax det  (L (E) / ) dE  ()(S () / )   E  det E col det  A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. When is a cavityygg “Bragg-Gray? In order for a detector to be treated as a Bragg–Grayy( (B–G))y cavity there is reall yyy only one condition which must be fulfilled: – The cavity must not disturb the charged particle fluence (including its distribution in energy) existing in the medium in the absence of the cavity. In practice this means that the cavity must be small comppg,ared to the electron ranges, and in the case of photon beams, only gas- filled cavities,,, i.e. ionisation chambers, fulfil

this. 23 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

A condition is generally added: The absorbed dose in the cavity is deposited entirely l by b th e chdharged particl es crossi ng it . This implies that any contribution to the dose due to photon interactions in the cavity must be negligible. Essentially it is a corollary to the first condition. If the cavity is small enough to fulfil the first condition then the build-up of dose due to interactions in the cavity material itself must be negligible; if this is not the case then the charged particle fluence will differ from that in the undisturbed medium for this very reason.

24 A third condition is sometimes erroneously added:

Charged Particle Equilibrium must exist in the absence of the cavity.

Greeningg( (1981 ) wrote that Gra y’s ori ginal theor y re quired this. In fact, this “condition” is incorrect but there are historical reasons for finding it in old publications.

CPE is not required but what is required, however, is that the stopping-power ratio be evaluated over the charged-particle (i.e. electron) spectrum in the medium at the position of the detector.

Gray and other early workers invoked this CPE condition because they did not have the theoretical tools to evaluate the electron fluence spectrum (E in the above expressions) unless there was CPE. (but today we can do this using MC methods). 25 Do air-filled ionisation chambers function as Bragg-Gray cavities at KILOVOLTAGE X-ray qualities?

26 Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon beams Phys. Med. Biol. 36 413-428 The commonly used air-filled irradiated by a megavoltage photon beam is the clearest case of a Bragg-Gray cavity. However for typical ion chamber dimensions for kilovoltage x-ray beams, the percentage of the dose to the air in the cavity due to photon interactions in the air is far from negligible as the Table, taken from [9], demonstrates: Ma C-M and Nahum AE 1991 Bragg-Gray theory and ion chamber dosimetry for photon beams Phys. Med. Biol. 36 413-428 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

313.1 INTRODUCTION

3.2 “LARGE” PHOTON DETECTORS

3.3 BRAGG-GRAY CAVITY THEORY

3.4 STOPPING-POWER RATIOS

3.5 THICK-WALLED ION CHAMBERS

363.6 CORRECTION OR PERTURBATION FACTORS FOR ION CHAMBERS

373.7 GENERAL CAVITY THEORY

3.8 PRACTICAL DETECTORS

3.9 SUMMARY 30 STOPPING-POWER RATIOS

31 air 1.6 adipose tissue bone (compact) 1.5 Graphite LiF edium 1.4 photo-emulsion mm PMMA 1.3 Silicon

o, water to 121.2 )-rati 

/ 1.1 col S ( 1.0

0.9 0010.01 0100.10 1001.00 10. 00 100. 00 Electron energy (MeV) 32 Mass collision stopping-power ratio, water/air 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 . 0100 10 1 0.1 Electron kinetic energy (MeV) delta=10 keV unrestricted 33 Depth variation of the Spencer-Attix water/air stopping-power ratio, sw,air, for =10 keV, derived from Monte Carlo generated eltlectron spec tra for monoenerge tic, p lane-paralle l, broa d electron beams (Andreo (1990); IAEA (1997b)).

1.15 1 3 5 7 10 electron energy (MeV) 14 18 20

r 25 i 1101.10 30

w,a 40 50

1.05 ratio, s rr

1.00 ng-powe ii

0.95 stopp

0.90 0 4 8 12162024 depth in water (cm) S 35 36 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. “THICK-WALLED” ION CHAMBERS

med wall CPE  en   L  D (C)  D (C)  Δ med wall   & Dwall (C)  Dair (C)       wall  37 air A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

wall med D (C)  L     f Q med   Δ   en  D (C)       air  air  wall

Thick-walled cavity chamber free-in-air where the air volume is known precisely (Primary Standards Laboratories):

wall air D  L     K (C)  air     en  K' air 1 g       air  air  wall 38 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

CORRECTION FACTORS FOR ION CHAMBERS ((p)measurements in phantom)

39 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors. Are real , practical Ion Chambers reallyygg Bragg-Gray cavities?

40 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

med Dmed ,C Q  Dair LΔ / air  Pi i

med Dmed,C Q  Dair LΔ / air Prepl Pwall Pcel Pstem

FARMER chamber (distances in millimetres) 41 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

THE EFFECT OF THE CHAMBER WALL

42 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

43 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

The Almond-Svensson (1977) expression:

 med wall  med   en   LΔ   LΔ        1         wall  air   air Pwall  med  L   Δ      air 44 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

45 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

46 THE EFFECT OF THE FINITE VOLUME OF THE GAS CAVITY

A. E. Nahum: Cavity Theory, 47 Stopping-Power Ratios, Correction Factors A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

48 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

49 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

FLUENCE PERTURBATION?

Electrons

50 E z A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

Φmed Peff  Φcav Pfl

Johansson et al used chambers of 3, 5 and 7-mm radius and found an approximately linear relation

between (1 - Pfl) and cavity radius (at a given energy).

Summarising the experimental work by the Wittkämper, Johansson and colleagues, for the

NE2571 cyli n dr ica l c ham ber Pfl increases steadily from ≈ 0.955 at E z = 2 MeV, to ≈ 0.980 at E z = 10 MeV, and to ≈ 0. 997 at E z = 20 MeV.

51 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

52 GENERAL CAVITY THEORY We have so far looked at two extreme cases: i) detectors whichwhich areare largelarge comparedcompared toto thethe electronelectron ranges in which CPE is established (photon radiation only) ii) dtdetec tors w hihhich are sma llll compare dtd to ththe e llec ttron ranges and which do not disturb the electron fluence (Bragg- Gray cavities)

Many situations involve measuring the dose from photon (o r ) radiation usingusing detectorsdetectors whichwhich fallfall intointo neitherneither ofof thethe above categories. In such cases there is no exact theory. However, so-called General Cavity Theory has been developed as an approximationapproximation.

In essence these theories yield a factor which is a weighted mean of the stopping-power ratio and the mass-energy absorption coefficient ratio: det det D  L     det  d  Δ   1 d  en  D       med  med  med

where d is the fraction of the dose in the cavity due to electrons from the medium (Bragg-Gray part), and (1 - d) is the fraction of the dose from photon interactions in the cavity (“large cavity”/photon detector part) Paul Mobit EGS4

CaSO4 TLD discs, 0.9 mm thick Photon beams A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

SUMMARY OF KEY POINTS

56 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

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A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

59 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

60 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

61 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

62 A. E. Nahum: Cavity Theory, Stopping-Power Ratios, Correction Factors.

Thank you for your attention

63