ChargedParticle Interactions in Matter III

1
Charged-Particle Interactions in Matter III

• Calculation of Absorbed Dose

2
Dose in Thin Foils

• Consider a parallel beam of charged particles of
  kinetic energy T0 perpendicularly incident on a
  foil of atomic number Z
• We assume that the foil is thin enough so that
• the collision stopping power remains practically
  constant and characteristic of T0, and
• every particle passes straight through the foil,
  that is, scattering is negligible

3
Dose in Thin Foils (cont.)

• At the same time we will assume that
• the net kinetic energy carried out of the foil by
  ? rays is negligible, either because the foil is
  thick compared to the average ?-ray range, or
  because the foil is sandwiched between two foils
  of the same Z to provide CPE for the ? rays
• Backscattering may be ignored, as it
  insignificant for heavy particles, and the
  average energy deposited by electrons in a thin
  foil is practically the same whether they are
  backscattered or transmitted

4
Dose in Thin Foils (cont.)

• For heavy charged particles it is usually
  feasible to satisfy all of these requirements
  reasonably well if the foil thickness is only a
  few percent or less of the range
• For electrons assumption b is the weakest, but
  may still give an adequate approximation in low-Z
  foils
• Corrections for failure of each of these
  assumptions will be addressed later

5
Dose in Thin Foils (cont.)

• The energy lost in collision interactions by a
  fluence of ? (charged particles/cm2) of energy T0
  passing perpendicularly through a foil of mass
  thickness ?t (g/cm2) is
• where (dT/?dx)c (MeV cm2/g particle) is the
  mass collision stopping power of the foil medium,
  evaluated at T0, and ?t is the particle
  pathlength through the foil

6
Dose in Thin Foils (cont.)

• Under assumption c the energy thus lost by the
  particles remains in the foil as energy imparted
• Hence the absorbed dose in the foil can be gotten
  by dividing this energy by the mass per unit area
  of the foil
• in which the foil thickness ?t cancels,
  leaving the dose as simply the product of fluence
  and mass collision stopping power

7
Dose in Thin Foils (cont.)

• This cancellation is very important, meaning that
  the dose in the foil is independent of its
  thickness as long as the particles travel
  straight through and do not lose enough energy to
  cause the stopping power to change significantly
• Within these limitations, even tilting the foil
  away from the perpendicular does not alter the
  dose

8
Estimating ?-ray Energy Losses

• In the case where the foil is comparable in
  thickness to the range of the ? rays produced in
  it, assumption c may not be satisfied unless the
  target foil is sandwiched between buffer foils
  of the same material
• Otherwise ? rays leaving will carry out energy,
  and other ? rays from adjacent but dissimilar
  materials may carry a different amount of energy
  in, producing a non-CPE situation for the ? rays
  in which the dose may differ from that given by
  the equation above

9
Estimating ?-ray Energy Losses (cont.)

• If such a foil is isolated so that only the
  primary charged particles (no ? rays) are
  incident on it, one can estimate the dose in it
  by modifying the equation
• The mass collision stopping power is replaced by
  the corresponding restricted stopping power,
  (dT/?dx)?
• Here (dT/?dx)? is that portion of the collision
  stopping power that includes only the
  interactions transferring less than the energy ?

10
Estimating ?-ray Energy Losses (cont.)

• Thus if one chooses ? to be the energy of those ?
  rays having, say, lttgt ?t, then one discards all
  the energy given to ? rays having projected
  ranges greater than the foil thickness
• This will roughly account for the energy carried
  out of a thin isolated foil by ? rays, and
  provide an improved estimate of the average dose
  remaining in the foil

11
Estimating Path Lengthening Due to Scattering in
the Foil

• The average pathlength of heavy charged particles
  penetrating a foil in which only a few percent of
  the incident kinetic energy is lost is not
  significantly longer than a straight path through
  the foil in the direction of the entering
  particles
• This is evident from the fact that the entire
  range of protons is usually not more than 3
  greater than the projected range
• Therefore a correction to the equation for path
  lengthening is not necessary for heavy particles

12
Path Lengthening Due to Scattering (cont.)

• For electrons, however, significant path
  lengthening results from multiple scattering, and
  a correction may be indicated
• That is, the factor t in the numerator of the
  equation, which represents the mean electron
  pathlength traversed, becomes greater than the
  foil thickness t in the denominator, and should
  be given a modified symbol t
• Thus t/t becomes greater than unity, and
  constitutes a correction factor to take account
  of path lengthening

13
Path Lengthening Due to Scattering (cont.)

• The following figure gives values of 100(t -
  t)/t, the mean percentage path increase of
  electrons traversing a foil of mass thickness ?t
  (g/cm2)
• In order to make the figure common to all foil
  media, the foil thickness is normalized by
  dividing it by the radiation length of the
  medium, which is the mass thickness in which
  electron kinetic energy would be diminished to
  1/e of its original value due to radiative
  interactions only

14
Percentage increase in mean electron pathlength
relative to normalized foil thickness ? foil
mass thickness ?t divided by the radiation length
of the medium
15
Radiation Lengths for Selected Elements
16
Mean Dose in Thicker Foils

• In foils that are thick enough to change the
  stopping power significantly (i.e., cause failure
  of assumption a, but not to stop the incident
  particles) one makes use of charged-particle CSDA
  range tables instead of stopping-power tables to
  calculate the average absorbed dose, which of
  course will no longer be uniform in depth through
  the foil
• ?-ray effects may be neglected (that is,
  assumption c is satisfied), since the foil
  thickness is now large compared to most ?-ray
  ranges

17
Mean Dose in Thicker Foils (cont.)

• Assumption b, however, requiring straight tracks
  through the foil, will not be satisfied for this
  case, especially for electrons
• As pointed out before, the resulting
  path-lengthening error is small (1) for heavy
  particles, and that correction will not be
  discussed here

18
Dose from Heavy Particles

• Using appropriate heavy-particle range tables,
  one first enters the table to find the CSDA range
  (g/cm2) of the incident beam of particles having
  kinetic energy T0, in the appropriate foil
  material
• The foil mass thickness in the beam direction is
  then subtracted, to find the residual CSDA range
  of the exiting particles
• The range table is again entered to find the
  corresponding residual kinetic energy, Tex,
  interpolating as necessary

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Dose from Heavy Particles (cont.)

• Thus the energy spent in the foil by each
  particle is
• and the energy imparted per unit
  cross-sectional area of particle beam is
• where ? is the fluence

20
Dose from Heavy Particles (cont.)

• The average absorbed dose is then obtained by
  dividing this energy by the mass thickness ?t if
  the beam passes through perpendicularly, or
  ?t/cos? if the beam makes an angle ? with the
  perpendicular to the foil plane
• Thus

21
Dose from Electrons

• In this case we combine the technique of using
  range tables with that in which the path
  lengthening is corrected for
• A further complication arises from the effect of
  bremsstrahlung production on the range
• To avoid needless complication let us assume that
  the beam is perpendicularly incident it will be
  obvious from the preceding section how to modify
  the calculation for a tilted foil

22
Dose from Electrons (cont.)

• The first step is to estimate the true mean
  pathlength for the electrons
• If the foil is too thick to be covered by the
  diagram above, this method is probably inadequate
  and computer radiation transport calculations
  should be employed
• However, the percentage path lengthening in
  thicker foils may be roughly estimated by
  noticing that it is proportional to foil
  thickness in this approximation

23
Dose from Electrons (cont.)

• Using electron range tables, one enters at the
  incident kinetic energy T0 and obtains the
  corresponding CSDA range
• From this the true mean pathlength of the
  electrons is subtracted to obtain the residual
  range of the exiting electron
• The table is then reentered to obtain the
  residual kinetic energy Tex
• The energy lost by the particle is just T0 - Tex

24
Dose from Electrons (cont.)

• Some of this energy is carried away by
  bremsstrahlung x-rays, which can usually be
  assumed to make a negligible contribution to the
  energy imparted (or the dose) in the foil
• To estimate the production of x-rays, the
  radiation yield column in the Berger-Seltzer
  tables in Appendix E is employed
• As explained before, the radiation yield Y(T) of
  an electron of kinetic energy T is the fraction
  of T that is spent in radiative collisions as the
  electron slows down and stops

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Dose from Electrons (cont.)

• Consequently the energy fraction spent in
  collision interactions is 1 Y(T)
• The energy spent in collision interactions in the
  foils is
• where Y(T0) and Y(Tex) are obtained from
  column 6 in Appendix E
• For a fluence ? the average dose in the foil of
  mass thickness ?t is given in grays by

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Mean Dose in Foils Thicker than the Maximum
Projected Range of the Particles

• If the charged particles cannot penetrate through
  the foil of mass thickness ?t, then there will be
  a layer of unirradiated material beyond their
  stopping depth
• If ? particles/cm2 of energy T0 are
  perpendicularly incident and backscattering is
  negligible, then the energy imparted in the foil
  per cm2 equals the energy fluence (except for the
  correction for radiative losses)
• where the radiation yield Y(T0) is zero for
  heavy particles

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Foils Thicker than the Maximum Projected Range
(cont.)

• The average absorbed dose in the foil is given by
• The dose of course changes radically with depth
  in the foil, as will be discussed shortly

28
Foils Thicker than the Maximum Projected Range
(cont.)

• If the radiative losses are considerable and the
  foil thickness is great enough, the dose
  throughout the foil may be significantly enhanced
  by the resulting x-ray field
• A very crude estimate of the reabsorbed fraction
  of the energy invested in these x-rays can be
  gotten by calculating
• where ?en/? for the foil material is to be
  evaluated at some mean x-ray energy, say 0.4T0
  for thick target bremsstrahlung

29
Foils Thicker than the Maximum Projected Range
(cont.)

• Multiplying the Y(T0) by the above exponential
  term roughly corrects for x-ray absorption,
  assuming the rays must pass through half the foil
  thickness to escape
• An accurate treatment of this problem requires
  computer calculations, taking account of x-ray
  distributions vs. angle and energy

30
Electron Backscattering

• As noted before, the effect of particle
  backscattering on dose calculation has been
  neglected so far
• For heavy particles this is justified by the fact
  that they are seldom scattered through large
  angles
• For electrons, backscattering due to nuclear
  elastic interactions can be an important cause of
  dose reduction, especially for high Z, low T0,
  and thick target layers

31
Electron Backscattering (cont.)

• In this connection, an infinitely thick foil with
  respect to the backscattering of an incident beam
  of charged particles will be provided by a
  thickness of tmax/2
• Particles penetrating beyond that depth in a
  thicker layer obviously cannot return to the
  surface

32
Electron Backscattering (cont.)

• Electrons incident on a thin foil, in which a
  backscattering event is equally likely to occur
  in the first or the last infinitesimal layer of
  the foil, require no backscattering correction to
  the absorbed dose
• On average, backscattering can be assumed to
  occur in the midplane of the foil
• The energy spent in the foil by an electron
  reflected from the midplane is the same as if it
  passed straight through without backscattering
• The energy distribution vs. depth in the foil is
  thus shifted toward the entry surface, but the
  average absorbed dose through the foil remains
  the same to a first approximation

33
Electron Backscattering (cont.)

• For thicker foils a backscattering correction
  requires a knowledge of what fraction of the
  incident energy fluence is redirected into the
  reverse hemisphere
• For electrons perpendicularly incident on
  infinitely thick layers, this fraction may be
  called the electron energy backscattering
  coefficient, ?e(T0, Z, ?)

34
Electron Backscattering (cont.)

• The measurement of ?e is best accomplished by
  calorimetry, comparing the known incident energy
  flux (i.e., the number of primary electrons
  multiplied by their individual energy) with the
  heating of the target
• An example of such data for the energy range T0
  1 to 3.5 MeV is shown in the following figure

35
Fraction ?e of incident energy flux carried away
by backscattered electrons
36
Electron Backscattering (cont.)

• For lack of additional data on electron
  backscattering, one can make use of information
  on backscattered-electron numbers as an upper
  limit on the backscattered energy
• For electrons with incident energies T0 ? 1 MeV,
  the backscattering coefficient ?(T0, Z, ?) has
  been given as
• which applies for T0 at least up to 22 MeV,
  although it tends to underestimate small values
  of ? ? 2

37
Electron Backscattering (cont.)

• This formula predicts that ? increases with Z and
  decreases with increasing electron energy
• For electrons below T0 1 MeV, this equation
  probably underestimates ?
• ?e(T0, Z, ?) should be less than ?(T0, Z, ?),
  because each backscattered electron has less
  energy than it had when it was incident on the
  scattering material

38
Dose vs. Depth for Charged-Particle Beams

• Panes a, b, and c of the following figure show
  how the number of charged particles penetrating
  through a layer of some absorbing medium varies
  with the layer thickness
• The variation of absorbed dose vs. depth in a
  medium shows quite different characteristics
• The shape of this function depends on the
  particle type and energy, the medium being
  penetrated, and the geometry of the beam

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The Bragg Curve

• Heavy charged particles (protons and heavier)
  penetrating a material in which nuclear
  interactions are negligible show a dose-vs.-depth
  distribution in the shape of the classical Bragg
  curve, as illustrated in the following figure
• This is a consequence of the ? T02 dependence of
  the range at low energies, which in turn results
  from the ? ?-2 dependence of the stopping power

41
Dose vs. depth for 187-MeV protons in water,
showing Bragg peak
42
The Bragg Curve (cont.)

• This means that if a particle spends the first
  half of its initial kinetic energy along a
  pathlength x, the remaining half of the energy
  will be spent in distance ? x/3, thus crowding
  the spatial rate of energy expenditure toward the
  end of the track
• The dose decreases from its maximum as the
  particles run out of energy and stop
• This descending limb of the Bragg curve roughly
  coincides with the corresponding curve of
  particles vs. depth

43
The Bragg Curve (cont.)

• The highly localized dose maximum shown in the
  figure suggests the possible usefulness of such a
  beam for delivery of therapeutic doses of
  ionizing radiation to tumors at some depth in the
  body while minimizing dose to overlying normal
  tissues
• The Bragg peak needs to be smeared out in depth
  if tumors even 1 cm in diameter are to be
  uniformly dosed
• Such devices as oscillating wedges can be used to
  produce a distribution of incident energies,
  resulting in a roughly square-topped Bragg peak,
  but at the expense of increasing the plateau
  dose level relative to the Bragg peak dose

44
The Bragg Curve (cont.)

• Negative pions are captured by atoms of tissue
  when they stop, causing the atomic nuclei to emit
  neutrons, ?-rays, and heavy charged particles
• The latter particles, being of relatively short
  range, enhance the dose in the vicinity of the
  Bragg peak
• The following figure shows the resulting enhanced
  Bragg curve, in comparison with the corresponding
  curve for positive pions that are not captured

45
Dose vs. depth in water for 65-MeV positive and
negative pion beams
46
Dose vs. Depth for Electron Beams

• As noted before, the small mass of electrons
  makes them scatter easily
• As a result, they do not give rise to a Bragg
  peak near the end of their projected range as
  heavy particles do
• Instead, a diffuse maximum is reached at roughly
  half of the maximum penetration depth, as shown
  in the following figure for broad beams of
  electrons of several incident energies

47
Dose vs. depth in water for broad electron beams
of the indicated incident energies
48
Dose vs. Depth for Electron Beams (cont.)

• An electron beam is defined as broad if its
  radius upon entry is at least equal to its CSDA
  range
• The following figure shows the effect of
  decreasing the radius below that value (indicated
  as ? in the figure), for a 10-MeV beam
• The curve shape is evidently affected very
  strongly

49
Dose vs. depth in water for circular electron
beams of radius r at incidence
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Calculation of Absorbed Dose at Depth

• At any point P at depth x in a medium w where the
  charged-particle fluence spectrum is known, the
  absorbed dose can be calculated as
• where ?x(T) is the differential
  charged-particle fluence spectrum, excluding ?
  rays, in particles/cm2 MeV (dT/?dx)c,w is the
  mass collision stopping power for medium w, in
  units of MeV cm2/g particle, given as a function
  of kinetic energy T T is in MeV

51
Calculation of Absorbed Dose at Depth (cont.)

• The exclusion of ? rays from ?x(T) is based on
  the assumption that CPE exists at P for the ?
  rays
• Thus any energy carried out of a small volume
  around P by ? rays will be replaced by other ?
  rays from elsewhere
• The use of the mass collision stopping power,
  rather than a restricted stopping power, is
  consistent with this assumption

52
Calculation of Absorbed Dose at Depth (cont.)

• ?-ray CPE requires that the medium and the
  particle fluence be homogeneous within the
  maximum ?-ray range from P
• In practice this assumption is usually adequately
  satisfied because most ? rays tend to have short
  ranges (? 1 mm) in condensed media

53
Calculation of Absorbed Dose at Depth (cont.)

• The problem of determining ?x(T) at the point of
  interest is, of course, nontrivial, generally
  requiring radiation-transport calculations for a
  good solution
• However, an estimate can be obtained rather
  easily from range tables for a monoenergetic
  plane-parallel beam of heavy charged particles
  incident on a homogeneous medium, since
  scattering and energy straggling are small
  effects
• One enters the range tables at initial energy T0,
  determining the range ?

54
Calculation of Absorbed Dose at Depth (cont.)

• From this the depth x is subtracted to determine
  the remaining range ?r of the particle when it
  reaches depth x
• Then the table is reentered at range ?r to
  determine the remaining kinetic energy Tr
• The particle fluence ?x at depth x in this simple
  case is taken to be the same as the ?0 incident
  on the surface (neglecting nuclear interactions),
  and all particles are assumed to have energy Tr
  (MeV)

55
Calculation of Absorbed Dose at Depth (cont.)

• Thus the integral can be dispensed with, and the
  dose (Gy) is given by
• where ?0 is in particles/cm2 and (dT/?dx)c,w
  is the mass collision stopping power for the
  medium w, evaluated at Tr
• This method begins to fail when x approaches the
  particle range, making ?x lt ?0

56
Calculation of Absorbed Dose at Depth (cont.)

• For the case of a broad beam of monoenergetic
  electrons of energy T0 gt 1 MeV perpendicularly
  incident on a semi-infinite homogeneous low-Z
  medium, one can roughly estimate the most
  probable energy of the electrons at depth
• Since the range is proportional to the kinetic
  energy for megavolt electrons, the modal energy
  decreases from T0 to 0 approximately linearly
  with depth as x goes from 0 to the range ?
• However, the electron fluence at depth is not
  easily estimated, mainly because of multiple
  scattering

57
Calculation of Absorbed Dose at Depth (cont.)

• The problem of measuring dose in a medium by
  inserting a small sensor or probe (e.g., a cavity
  ion chamber) at the point of interest involves
  cavity theory
• The usefulness of the above method for estimating
  the electron modal energy vs. depth will become
  useful with in-phantom dosimetry, as it provides
  an effective energy at which stopping-power
  ratios (used in cavity theory) can be evaluated