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n
Quantitative Radiobiology for Proton Therapy
of just above 1 in the case of beams used in medical applications. However, for these
applications it is rare to experience LET values greater than the turnover point since
treatment volumes are covered by a mixture of LET values from parts of individual
particle beams, which have lower LET values proximal to the Bragg peak and higher
values within and especially towards the end of the peaks.
The method follows some simple steps, each with numerical parameters given in
chapters 8 and 9. In brief:
1. LET
is obtained from LETU= 30.5 + S/k (1 − exp[−k (z − 1]),
U
[S = 114.4, k = 0.51].
2. α
is found from αU= A/j (1 − exp[−j αL]). [ A = 10.6, j = 3.9].
U
3. β
is found from βU= B/i (1 − exp[−i βL]). [ B = 3.2, i = 40].
U
4. Values of α
and βHat any particular value of LET (LETx) are found from
H
the following:
(a) A relatively simple equation representing efficiency changes between
the control (megavoltage photon) radiation LET value and the LET
value, as a change in radiosensitivity from αLto be a larger value of α
(α
) at any higher value of LET (LETx) up to a maximum possible
H
value of α given by α
. The same LETUis also assumed for β (in
U
order to preserve symmetry with dose). So, we obtain
U
−
LET LET
aaa=+
HC
bb bb=+
HC
xC
−
LET LET
UC
−
LET LET
xC
−
LET LET
UC
⋅−
()
UC
⋅−
()
UC
,
.
(b) A simple equation representing inefficiency for all LET values above
LET
, but converted into an efficiency equation, as
U
−
LET LET
⎛
⎜⎟
aaa=+−
HC
⎝
⎛
bb bb=+−
HC
⎜⎟
⎝
5. The RBE
and RBE
max
min
6. For the RBE, corrected for dose per fraction, it is necessary to solve an
isoeffect equation where α
LdL
which is then divided by the d
xU
LET
−
LET LET
xC
LET
⎞
()
⋅−1
U
U
UC
⎠
⎞
()
⋅−1
UC
⎠
,
.
values can then be obtained.
+ βLd
to give the RBE at any value of dH. In terms
H
2
= αHdH+ βHd
L
2
is solved for dL,
H
of BED, this means solving the isoeffect equation BED of control radiation,
which is equal to the BED of particle radiation, i.e.
2
⋅
min
k
d
L
⎞
.
⎠
d
⎛
d
L
⎝
L
⎞
+= +
1RBE
k
⎠
⎛
nd
⎜⎟
L
⎝
RBE
max
13-5
Quantitative Radiobiology for Proton Therapy
The RBE is then the solution of this equation for dL, then divided by dH.
Examples of this method to obtain RBE, fitted to various data sets, which include
changes in dose (or surviving fraction), have been given in chapters 8 and 9.
13.2.2 Assessment of BED changes after an error
From estimates of the changes in α and β described above, the operative RBE
and RBE
in equation (13.2) above can be determined.
min
max
The intended, delivered and corrected BEDs can then be calculated providing the
LET values and dose for these states are known. Any excess or deficit in BED is then
obtained as the difference between the intended and delivered BEDs in both tumour
and normal tissue. Any unintended overdosage or underdosage can then be
compensated for by a new, modified dose to provide the originally intended tumour
BED.
The above system applies to monoenergetic particles, or a small range of particle
energies, which is reasonable for modelling purposes at this stage and could in
principle be extended to a spectrum of LET values in accordance with energy ranges,
by an averaging process.
With the above relatively simple to operate method for predicting changes in
RBE due to changes in both LET and dose, it becomes possible to tentatively assess
the clinical impact of changes in Bragg peak positioning during particle therapy, and
for any ion species. For example, the intended BED at a LET of TL
nTd
(RBE
I
values of α
+ RBE
max
and βHfor the operative LET can then be obtained from the steps 1–5
H
2
TdI/(α/β)L) = nTdI(αH/αL+ (αH/βL)TdI/(α/β)L). The
min
will be given by
I
in the list given in section 2.1 of this report.
For a different dose Td
at a different LET value TLE, the BED for n fractions is
E
then
nTd
nTd
E(αH/αL
(RBE
E
+ (αH/βL)TdE/(α/β)L), which is the same as
RBE
2
TdE/(α/β)L).
min
max
+
For example, consider protons, helium ions and carbon ions for an underdose
of 50% of t he intended dose. The plots shown in figures 13.1(a)– (f) show
estimated changes in BED per fraction for a range of LET values up to the
turnover po int of the LET–RBE relationship. It shoul d be noted that the lower
range of these values pertain to clinical treatments, due to overlapping of Braggpeak and non-Bragg-peak regions of the beams. It sh ould b e note d that the
highest BEDs are found for the lowest α
because of the greater relative change in α when α
RBE
max
values.
radiosensitivity parameter values,
L
is small, which causes higher
L
An International Commission on Radiation Units and Measurements (ICRU)
matrix for particle beams has already been presented in chapter 4. This system
should be borne in mind, since shifts in dose (and LET) from one ICRU sub-volume
to another will be considered below.
13-6
Quantitative Radiobiology for Proton Therapy
Figure 13.1. Plots (a)–(f) of biological effective doses (BED) for different values of linear energy transfer (LET)
for different ionic species at two values of relatively low doses, with variations in the α
parameter, which is labelled as α in the graphs; β is kept constant in this study.
radiosensitivity
L
13.2.2.1 Simulations of treatment-delivery errors by modelling
Two simple modelling exercises, each for a population of 100 individuals, are shown
below in figures 13.2(a) and (b). These show dose-response curves for tumour control
probability (TCP) with increasing number of underdosed fractions ranging from
three to 15 for intended 2 Gy photon fractions (solid lines) and the same for carbon
ions (dashed lines). Figure 13.2(a) shows the situation where the same part of the
tumour (10% of its volume) is underdosed to 0.6 Gy rather than 2 Gy (these doses
being the physical doses used for both photons and carbon ions, for simplicity). It
can be seen that the fall in TCP with increasing number of erroneous fractions is
steeper for carbon ions than for photons. In figure 13.2(b), random underdosage
is allowed, where 40% of the tumour volume is underdosed to the same amount.
13-7
Quantitative Radiobiology for Proton Therapy
Figure 13.2. (a) and (b): Plot of TCP against number of 2 Gy fractions for variable numbers of erroneous
treatments (solid curves photons, dashed curves carbon ions), where the same 10% of the tumour volume is
randomly underdosed. Assumptions made: RBE
radiosensitivity parameter (mean 0.3 Gy
and a constant value of β = 0.03 Gy
−1
, with standard deviation 0.05 Gy−1) for a tumour with 5 × 108 cells
−2
(repopulation not included).
max
= 3, RBE
= 1.25, using random sampling for the α
min
The falloff in TCP is not so significant in this case for both types of radiation,
although the slope is again steeper for carbon. The steeper slope is due to a greater
proportion of the curative dose being given in fewer fractions with carbon ions, so in
such cases it is more important to avoid underdosage.
13-8
Quantitative Radiobiology for Proton Therapy
These two examples have not considered changes in LET. Changes in dose, LET
and RBE are considered further below.
13.2.3 Worked examples of errors and their correction
Next some quantitative, though simplified, worked examples will be considered:
A. Half of the volume of a tumour is underdosed by 50% for three fractions,
during which the average LET falls from 45 to 15 KeV μm
−1
. The
prescribed tumour dose was 70 Gy-Eq in 15 fractions using carbon ions.
1. A squamous cell carcinoma where for megavoltage photons α = 0.35
−1
, β = 0.04 Gy−2and the prescription RBE was 2.5.
Gy
2. A prostate cancer where for megavoltage photons α = 0.18 Gy
β = 0.035 Gy
−2
and the prescription RBE was 2.
−1
Calculate compensatory doses to the underdosed half of the tumour for the remainder of
treatment in order to maintain isoeffective tumour control.
B. What is the potential impact on a critical volume of normal tissue planned
to receive a dose of 40 Gy in 15 fractions, with the assumption of an
operative RBE = 2, which in the past has represented tolerance? Then, for
three fractions the dose is erroneously increased by 50% with an increment
in average LET from, say, 15 to 45 Kev μm
serious complications per unit rise in BED above tolerance, and where for
megavoltage photons α = 0.1 Gy
−1
, β = 0.03 Gy−2?
−1
, assuming a 1% increase in
,
13.2.3.1 Answers
The RBE
and RBE
max
can be found from the changes in α and β parameters
min
with LET, which can then be used within BED equations together with the
low-LET α/β value. An ultimate LET
value of around 230 keV μm−1is used
U
for carbon ions, compatible with the data of Weyrather et al (1999) and the
linear relationship between LET, α and β for values less than LET
A.1 and A2
The results of calculations, using the equations provided above, which estimate
the changes in the radiosensitivity parameters to provide RBE
RBE
for the different LET values, are shown in table 13.1. These values
min
are placed within the BED equations below where appropriate.
Note that the intended physical dose per fraction was 70/15 = 4.667 Gy-Eq,
which when divided by the assumed RBE provides a physical dose of 4.667/
2.5 = 1.866 Gy in example (1), but 4.667/2 = 2.33 Gy for example 2.
For worked example A1.
1. Proceed as follows, with α/β = 8.75 Gy.
2. The intended tumour BED, with a dose per fraction of 70/2.5/15 = 1.867
Gy, was
3. 70/2.5(1.892 + 1.035
2
.1.867/8.75) = 59.375 Gy
[8.75]
.
4. The BED after the three-fraction underdosage to the affected part of the
tumour was
U
max
.
and
13-9
Quantitative Radiobiology for Proton Therapy
Table 13.1. Parameters for worked examples A1 and 2.
A1: SCC (α = 0.35, β = 0.040) A2: Prostate (α = 0.18, β = 0.035)
Ar LET = 45 α
Ar LET = 15 α
Ar LET = 53 α
U
α
H
RBE
α
U
β
H
RBE
H
RBE
β
H
RBE
H
RBE
β
H
RBE
= 2.035
= 0.662
= 1.892
max
= 0.0555
= 0.0429
= 1.892
min
= 0.454
= 1.297
max
= 0.041
= 1.012
max
= 0.780
= 2.051
max
= 0.0434
= 1.04
max
5. 3 × 0.778 (1.297 + 1.0122. 0.778/8.75) = 3.855 Gy
α
U
α
H
RBE
α
U
β
H
RBE
α
H
R
BEmax
β
H
RBE
α
H
RBE
β
H
RBE
= 1.402
= 0.406
= 2.258
max
= 0.045
= 0.037
= 1.027
min
= 0.255
= 1.419
= 0.0359
= 1.0128
max
= 0.447
= 2.482
max
= 0.0372
= 1.031
max
[8.75]
.
6. This means that 59.375 – 3.855 = 55.488 Gy is required from the remaining
12 treatments. The dose per fraction is then obtained from the solution of
7. 12 d(2.051 + 1.041
2
.d/8.75).
8. This gives d = 2.01 Gy.
For worked example A.2.
1. The intended tumour BED was 70/2(2.26 + 1.02722.33/5.14) = 95.76
Gy
2. The BED given after the three-fraction underdosage was
3. 3 × 2.33 × 0.5 (1.42 + 1.01
4. There is a deficit of 95.76 – 5.78 = 89.98 Gy
[5.14]
.
2
. 2.33 × 0.5/5.14) = 5.78 Gy
.
[5.14]
[5.14]
.
5. This can be compensated for in the next 12 fractions, now with RBE
parameters at a higher value of LET, by solving
6. 12 d(2.48 + 1.03 d
2
/5.14) = 89.98, where d = 2.5 Gy.
For worked example B.
1. For the critical volume of normal tissue, the megavoltage photon α of 0.1
2. The intended BED is 38.97 Gy
−1
, the relevant parameter changes are given in table 13.2.
Gy
, using a dose per fraction of 1.33 Gy, and
[3]
this represents tolerance using the fi xed value of RBE given. The 150%
dose error will cause the dose per fraction to rise to around 2 Gy.
3. The excess BED over the three incorrect fractions, if uncorrected, would be
13-10
Quantitative Radiobiology for Proton Therapy
Table 13.2. Calculated parameters for worked example B.
Planned treatment
Baseline parameters
α
= 0.1 (Gy−1)
L
β
= 0.033 (Gy−1)
L
(LET = 15 keV μm−1)
= 0.906
α
U
α
= 0.15
H
RBE
β
β
RBE
max
= 0.04
U
= 0.0334
H
min
= 1.50
= 1.007
4. 3 × 2 (2.49 + 1.022.2/3) − 3 × 1.33 (1.50 + 1.012× 1.33/3) = 11.33 Gy
By error
(LET = 45 keV μm−1)
α
= 1.906
U
α
= 0.249
H
RBE
β
β
RBE
max
= 0.04
U
= 0.0343
H
min
= 2.49
= 1.02
[3]
5. If toxicity increases by 1% per unit BED, then an 11.3% increase in toxicity
might occur if no correction is applied.
The above steps are informative, and useful for teaching and research purposes, but
there are many pitfalls within the process, more so than in the much simpler
corrections sometimes necessary for errors arising in photon-based therapy.
13.2.4 The potential impact of erroneous fractions on tumour control
A further study is based on problem A1 above, but assumes that the same half of the
volume of a tumour has been underdosed on one or more sequential fractions. This
has been simulated in greater detail with estimation of tumour control probabilities
(TCPs). Plots of the numbers of erroneous treatment against the expected TCP are
shown in figure 13.3. Here several assumptions of different tumour RBE values have
been made in addition to the RBEs predicted by the model described in this report
(purple data points). Detailed inspection will show that the dose of 70 Gy in 15
fractions is very forgiving for treatment errors in the case of photons (the green data
points). This is because the tumour can be cured with doses of around 40–45 Gy in
15 fractions, but the excess dose guarantees a high cure rate even when dose delivery
is imprecise for up to 10 fractions. When the tumour RBE of 2.5 (as stated in the
problem) is itself incorrect, as shown by the red data points, the physical dose has
been excessively reduced, and even if all treatments are accurately delivered the TCP
is reduced to around 0.4 (or a control rate of 40%), with a sharp falloff in TCP if
only one or two erroneous treatments are made. For more than two erroneous
treatments the TCP is effectively zero. For lower assumed values of tumour RBE,
the prescribed dose is not so reduced, allowing increasing TCP and also improving
the numbers of erroneous treatments that can be made before causing a significant
reduction in TCP.
In reality, higher doses do oppose radioresistance regardless of cause, whether
intrinsic (e.g. genetic) or acquired (e.g. hypoxia). This example is sufficient to
demonstrate that higher doses are protective against at least some forms of error in
dose misplacement during particle therapy. Such higher doses can only be permitted
.
13-11
Quantitative Radiobiology for Proton Therapy
Figure 13.3. Plots of tumour cure probability against number of erroneous fractions where the same half of a
tumour is underdosed by 50%. TCP is calculated using standard Poisson statistics with the same radiobiological assumptions as in question A1, with altered radiobiological parameters as in table
clonogens per tumour. No repopulation factor has been used since the overall time is constant and less than the
approximately 28 day onset time of significant repopulation in a squamous cell carcinoma.
13.1, with 10
9
if normal tissue tolerances of smaller volumes of critical organs at risk are respected,
for example if the planning target volume (PTV) includes normal tissue whose
functional contribution can be lost, but without appreciable detriment in quality of
life.
There are other situations in oncology, for example the treatment of radiosensitive childhood tumours, where dose escalation to very high levels may not be
acceptable since normal tissue tolerances of nearby structures have to be respected,
for example the eye or spinal cord. In the context of more radioresistant tumours, for
example a craniopharyngioma where optic nerve and optic chiasmal structures may
extend into the clinical target volume/PTV, dose escalation, if used at all, may have
to be limited to a part of the tumour. In such situations, there is a risk that tumour
underdosage may reduce tumour control, which justifies the use of detailed image
guidance, especially if non-ionising radiation impacts on patient throughput.
Just as in airports, so in radiotherapy departments, delays occur due to a variety
of reasons which are often connected with patient safety. Sometimes, interruptions
during a treatment fraction occur due to accelerator malfunction or the beam
delivery process during therapy, although this is infrequent. In practice, a failed
quality assurance (QA) check before treatment has commenced leads to a delay in
beginning or resuming treatments. Delays are sometimes simulated as having a
Gaussian distribution, but they are naturally bounded at the lower limit (because of
more serious causes of failure, which need much longer times to be corrected), so will
tend to have a log-normal distribution. Interruption of a treatment fraction requires
a much more complicated approach, as shown above. Only if the treatment
13-12
Quantitative Radiobiology for Proton Therapy
interruption is less than 0.5 hr should the remainder of the dose be applied without
specific radiobiological correction. This is because DNA is repaired rapidly (in the
central nervous system the fast component repair has a half-time of 12 min and the
slow component just over 2 hr; see chapter 2). The correction, if adopted
comprehensively, will require input from physicists, clinical radiobiologists with
an interest in modelling, and the responsible clinician.
The best that could be aimed for at the present time would be a 24 hr break in
treatment, although longer intervals may be anticipated in some instances. There are
convenient ways to overcome longer delays after erroneous treatments have been
delivered, if these occur early in a treatment course of many fractions, since the usual
fractionation can be continued a day later, leaving the correction of the erroneous
fraction (or fractions) to a later time, towards the end of a treatment course. This
would allow the ‘correcting team’ a longer time to consider the problem; this may
avoid the situation where a rapid but incorrect decision is taken after an erroneous
fraction has been partly or totally given. In situations where voxel-by-voxel
compensations are attempted this would require automated computing, with input
of the relevant radiobiological parameters and many reiterations to obtain a
reasonable compromised solution.
Specific training in the clinical radiobiology area is required for several disciplines. Clinicians should be made more aware of the LET-dose-RBE issues and be
encouraged to practice simple calculations with their physics and radiobiology
colleagues.
This problem should be discussed in depth by the relevant bodies that govern the
use of radiotherapy in each country. At the present time, full correction may take a
physicist 4–6 hr and the clinician 1–2 hr to discuss; if the problem requires several
iterations these times will be far longer, taking perhaps a whole day or more.
Should a treatment error not be corrected until after the intended overall
treatment time has elapsed, then repopulation-related calculations may also be
necessary, as described in chapter 12. This adds a further level of complexity where
additional compromises will be necessary because the correct solution to preserve
the same tumour control will not be the same as for normal tissue complications.
13.2.4.1 General discussion
The causes of failure in achieving local tumour control with radiotherapy are
numerous, including intrinsic and acquired radioresistance, patient-related factors
and dose delivery errors. At the present time, much attention is being given to QA
measures regarding dose and the accuracy of its delivery for each treatment. Further
measures which minimise potential errors, such as adaptive radiotherapy (ADR),
are by now in routine use in some treatment centres (Schwartz et al 2012). In ADR,
the radiation fields are checked (using one or more of a variety of imaging
procedures) at either frequent exposures, or even at each exposure, and the treatment
field directions/sizes modified to take into account changes in patient and tumour/
normal tissue position and dimensions.
The Bragg peak position is dependent on the initial energy of the particle and its
rate of energy loss with increasing tissue depth in the body; these parameters
13-13
Quantitative Radiobiology for Proton Therapy
normally include some degree of variation, resulting in variations in the dose range.
So, dose deposition consequently can be uncertain, especially if there are extremely
non-homogenous tissue electron density variations in regions containing marked
changes in chemical composition. CPT is consequently more sensitive in terms of
changes in dose placement during therapy. Even loss or gain in patient weight during
treatment, or daily setup errors, could influence particle range, causing simultaneous
tumour underdosage and an increase in normal tissue dose compared to what was
intended. Use of ultrasonic measurements of distances between skin and bone on a
daily basis prior to entering the treatment room would help to reduce, for example,
errors due to loss/gain in subcutaneous fat.
It is important to consider the potential effects of serial underdosage, even if this
only affects part of a tumour volume and to study what the influence of this will be
on tumour control following conventional therapy and CPT.
The results show that increasing numbers of repeated underdosage of the same
section of a tumour can have deleterious consequences for tumour control
probabilities.
Practical calculations should include the possibility of lower RBEs due to lower
LET values in underdosed regions using the passive scattering technique. This would
need to be done on a voxel-by-voxel basis, and is beyond the scope of the present
study. A further important issue is the finding that for scanned beams, which allow
intensity-modulated proton therapy, there can be higher LET values (and so RBE
values) just outside the PTV, where dose is reduced (Grassberger et al 2011). Both
low dose and higher LET will lead to a higher RBE.
Adequate QA and vigilance is required during radiotherapy to ensure that
systematic underdosage does not occur. This could occur due to a variety of reasons
ranging from dose/particle numbers and energy/ranges as well as errors due to
scattering or scanning of the beam. Patient setup errors, tissue depth and density
variations, tumour shrinkage and other effects such as tumour motion with
respiration may all contribute.
These issues impinge on medical ethics: how much of the uncertainties in the
physical and biological aspects of particle therapy should be discussed with
individual patients as part of informed consent? Should full informed consent
include these in comparison with photon-based techniques? Should more phase III
randomised studies be done because of the influence of these uncertainties? Will
patients who might develop disappointing results after particle therapy take legal
action on the basis that they were not fully informed as a natural human right,
encoded in European law?
The results presented are dependent on the validity of the linear quadratic model
and the various assumptions made. It will be important to collect more data on
errors or interruptions during particle therapy, the net effect on LET distortions and
estimated RBE changes, their correction details, and to correlate this information
with tumour control outcomes and also toxicity since overshooting of Bragg peaks
may result in excess normal tissue dosage. A pan-European database could be
developed for this purpose.
13-14