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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5518_Библиотеки_им_академика_М_И_Перельмана.pdf
Quantitative Radiobiology for Proton Therapy
Figure 2.5. Plot of isoeffective doses and total treatment time to match the same effect (or BED) as a single
dose of 12.5 Gy given by megavoltage protons or cobalt gamma rays given in 0.5 hr. Note that this applies
only in the case of the specified parameters used for a variable RBE.
delivered instantaneously has a BED of around 75.76 Gy
, but when the time is
[2.47]
extended to 0.5 hr this value becomes 61.66 Gy. The isoeffective dose is 10.71 Gy for
a proton beam with an assumed RBE
= 1.75 and RBE
max
= 1.09 (note that this
min
would be 11.36 Gy if a ‘standard’ 1.1 RBE correction for the 12.5 Gy dose). For
further extensions of time, isoeffective doses may be individually solved for this BED
value, as shown in figure 2.5. It should be noted that the total doses required to
maintain an isoeffect for α/β = 2.47 Gy (as might be the case in very slow-growing
tumours) increases with treatment time to match the required BED. If this is not
done then tumour control could be reduced, although the fall in BED should be
protective on normal tissues. The doses should be recalculated using more specific
tumour α/β ratio values. Where normal tissue toxicity is of great concern, an elective
decision to deliver a treatment fraction or individual fractions over a slightly longer
time may be appropriate.
Readers may wish to consider using BEDs from their own dose protocols, which
could be a BED given, say, over a designated time, such as a half-hour or 1 hr using
photons, and then finding the solution for the equivalent dose of protons at relevant
LET and RBE values using the equations given above and also in chapters 8 and 9.
In principle, these refinements could be added to treatment planning systems. The
textbook by Wigg (2008) contains comprehensive summaries of dose-rate radiobiology and practical applications, as does the shorter review by Dale & Jones
(1998); Fowler (2012) provides a more recent review of DNA repair half-times.
Suffice it to say that tumour repair half-time rates may be shorter than in some
normal tissues, but with some reduced repair capacity if there is an absence of a
repair pathway, especially the recombination repair (RR) mechanism. Further work
needs to be done on this important topic.
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Quantitative Radiobiology for Proton Therapy
2.2.8 Closely spaced fractions
Unlike the dose-rate effect where f(μT) terms cause reduced effects with increasing
time, closely spaced fractions incur higher effects by imposing a multiplicative factor
(1 + h) to replace the in place of the f(μT) term: readers should note that sometimes a
term h is used where h > 1, which can cause confusion.
The standard BED is then
+
dh
1
⎛
⎜⎟
=+
BED 1
nd
⎝
()
ab
⎞
.2.28
/
⎠
()
To find h values based on the mean inter-fraction interval ( f ), formulations and
tables are available in Dale (1985) and Jones et al (2001). As with the dose-rate
effect, the overall influence of a higher LET causing larger increments in α will
diminish the magnitude of closer fraction spacing to some extent, but this aspect
needs careful assessment since breakdown of normal tissue tolerances have occurred
with closely spaced fractions in the past, in late-reacting normal tissues such as the
spinal cord.
The high-LET BED equation for incomplete repair becomes
⎛
=+
BED RBE
nd
⎜⎟
max
⎝
()
+
dh
1.RBE
ab
/
2
min
⎞
.2.29
()
⎠
2.2.9 Hypoxia
For hypoxia studies, the separate oxygen enhancement ratios for the α
terms can be incorporated as OERα(defined as the ratio
b
b
hypoxic
oxic
), as multipliers of the dose, so that
⎛
⎜⎟
=+
BED . OER
nd
⎝
d
.OER
a
ab
/
b
ratio
a
oxic
) and OERβ(the
a
hypoxic
⎞
.2.30
⎠
and β
L
()
The same replacements can in principle be used within the high-LET BED equation,
as
d
.OER .RBE
⎛
=+
BED . RBE . OER
nd
⎜⎟
max
a
⎝
b
ab
/
2
min
⎞
.2.31
()
⎠
Care must be taken to use the appropriate OER, which will itself diminish with LET.
This aspect has been explored by several authors (Furusawa et al 2000, Wenzl &
Wilkens 2011, Bopp et al 2016).
The effect of hypoxia in high-LET radiation can be seen in graphical presentations of the work of Furusawa et al (2000). Not only are the OER
by hypoxia, but also the LET
turnover points for RBE occur at much higher values
U
values lowered
α
2-18
L
Quantitative Radiobiology for Proton Therapy
of LET. One example from Jones (2022) is shown here as figures 2.6(a) and (b) for
carbon and neon ions on human HSG cells.
The phenomenon of extreme radioresistance occurring with ultra-high dose rates
(FLASH exposures) in moderately hypoxic cells is thought to be caused by high
Figure 2.6. (a) and (b): Relationships between LET and both OER(α), which is the ratio of the α
radiosensitivities in oxic and hypoxic conditions, and the ï¡radiosensitivity parameter in HSG cells irradiated
with carbon and neon ions in oxic (black data points, fitted line and curve) and hypoxic (grey data points and
fitted curve) conditions. The blue data points and curve (and in (a) the brown line) show the change in OER(α)
with LET. (Data from Furusawa et al (
2000), no standard errors provided.)
2-19
v
v
Quantitative Radiobiology for Proton Therapy
oxygen depletion rates as a consequence of rapid ionisation and free-radical
interactions which cause severe hypoxia. Intriguingly, the intensification of dose
rate may also increase RBE
whilst more severely reducing RBE
max
(as in Jones
min
2022) due to increased α and sharply reduced β values. These effects will be
considered in further detail in chapters 9 and 14.
2.2.10 Very low doses
Deviations from the LQ model in fast-growing cells with a high G2 cell cycle phase
content exist, and have been described as a phenomenon of ‘low-dose hypersensitivity’, thought to be related to induced repair, in other words induction of
repair enzyme systems with increasing dose. Such cells will have high α/β ratios.
These have been modelled empirically by Joiner et al (2001), as in Marples et al
(2004), but more mechanistically using the Michaelis–Menten enzyme kinetics
approach and induction of additional enzyme capacity in Jones & Morgan (2007).
This is relevant to high-LET radiations as the extreme initial slope represents a state
of enhanced radiosensitivity, represented by an increase in the α coefficient, and so
represents the equivalent of a high-LET situation where there is an increased yield of
non-repairable damage, and which does not exhibit a slightly higher SF at doses up
to 1.5 Gy before resuming the normal LQ survival curve shape. For intermediate
LET values one can assume intermediate patterns of this phenomenon. In most
practical applications the doses will exceed the dose range (0–1.5 Gy at the most) in
which this phenomenon operates.
These formulations can be transformed to higher-LET situations by replacing α
with either α
or αL.R
H
, and β by either βHor βL.R
max
2
. The low-LET
min
formulations are
d
a
d
rbr
[] ()
=+−
res2res
fe f eSF overall . 1 . ,
ab−− −−
dd
2
2.32
()
where
rr=+ −xx1
res
()
2.33
()
and
ab
v
V
dd
++
Kdd
Mmax
2
+
ab
2
=
r=
,
2.34
()
where f is the fraction of cells in the G2 phase, and x is the fraction of DNA repair
that is active at zero dose. The saturation of repair rate (ρ) is described by
where
Values of f = 20%, K
is the rate at dose d.
The change in effective K
=+ −
KK eK K.
MM
= 10, K
init
from a minimum to maximum value is described by
M
jd
−
()
MMeff max min max
= 0.01 and j = 6 were used to fit data in the above
min
/V
,
max
2.35
()
source reference.
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Quantitative Radiobiology for Proton Therapy
Figure 2.7. Theoretical plot of the effect of increasing the radiosensitivity (assumed here due to LET) causing
progressive loss of low-dose hyper-radiosensitivity. The black curve has α = 0.1 Gy
α = 0.7 Gy
the cell line (here α
−1
, and red α = 1.0 Gy−1. The orange line represents αU, the maximum possible radiosensitivity for
= 1.3 Gy−1).
U
−1
, blue α = 0.4 Gy−1, grey
The more pragmatic equation developed by Joiner et al (2001), shows the
effective change in α with dose as
⎡
⎢
⎣
⎛
a
⎜⎟
r
⎝
a
s
⎛
⎜⎟⎜⎟
a
rC
⎝
⎞
⎞
⎠
⎠
d
⎛
⎞
d
⎝
⎠
⎤
2
b=− +− −−d
dSF exp 1 1 . exp . 2.36
⎥
()
⎦
This is of considerable interest since as αr(the ‘normal’ LQ parameter) becomes
larger and approaches α
(the more sensitive parameter, which describes the
s
initial steep slope of the phenomenon), then the plotted function will shift towards
the steep initial slope (as shown in figure 2.7). This may have relevance to highLET radiations since the α
probably reflects the high-LET condition (or the αU,
s
the ultimate value of α at high LET, discussed in later chapters 8 and 9, since
there is effectively no repair at such low dose, just as there is reduced or absent
repair at high dose with high-LET radiations depending on the LET value).
Consequently, α
can be replaced by αH(or by αr.RBEmax) up to a limit of αU,at
r
any high LET. So, at the LET–RBE turnover positions representing maximum
efficiency:
2-21
Quantitative Radiobiology for Proton Therapy
⎡
⎢
⎣
⎛
a
⎜⎟
H
⎝
a
U
⎛
⎜⎟⎜⎟
a
HC
⎝
⎞
⎞
⎠
⎠
d
⎛
⎞
d
⎝
⎠
⎤
2
b=− +− −−d
dSF exp 1 1 . exp . 2.37
H
⎥
⎦
()
Failure to account for low-dose hypersensitivity (LDH) can influence the
interpretation of RBE. For example, the study by Mara et al (2020) in Austria
compared several models for RBE estimation in cell lines with different α/β. Cells
with high α/β values (which correlates with faster growth rates) did show LDH, and
the experimental dose range included low-dose values in the LDH range, resulting in
altered α values which contribute to RBE. The rank order of models changed
completely for the cells with low α/β values, where the LDH phenomenon did not
occur (and where the models used in chapters 8 and 9 provided the best estimates of
the measured RBE). This shows that experiments should be more carefully designed
to avoid inclusion of doses within the LDH range.
2.2.11 Higher doses per fraction
There continues to be discussion about deviations from the LQ model at higher
doses, even if these may be artefacts due to experimental conditions. In different cell
systems and growth conditions, deviations have been observed at doses above the 6–
10 Gy range. This effect can occur at least partly because of the statistical issues that
arise for lower SFs in colony-forming assays where the number of colonies growing
are smaller in number. Nevertheless, there are also some theoretical reasons why
some apparent straightening of the cell survival curve might occur with increasing
dose.
Three approaches are presented here.
1. The first assumes that second or third lethal chromosomal breaks on the
same chromosome result in ‘wasted dose’, being incapable of causing
additional lethality, with resulting high-dose inefficiency of the survival
curve.
2. The second is a pragmatic model which assumes a final slope beyond a fixed
dose. Alternative approaches include that of Park et al (2008), who suggested
methods for a gradual transition of the dose–response from the LQ to
increasing linearity at higher doses. It is claimed that such approaches may
be better suited for assessing high-dose radiotherapy, but others (e.g. Fowler
2008, 2010) suggest that caution needs to be exercised in their application.
3. The third is based on the effect of time protraction to deliver a higher dose,
allowing more sub-lethal damage repair during the radiation exposure.
Careful assessment of the time taken to deliver higher doses should be
made in the interpretation of data sets and whether the change in time is
sufficient to allow significant changes in cell survival.
To return to the two models presented here: they are shown graphically with a
comparison to the standard LQ model in figure 2.8.
The yield of lethal injury is conventionally represented by the number of lethal
chromosome breaks per cell, broadly represented by αd + βd
2-22
2
, as initially
Quantitative Radiobiology for Proton Therapy
Figure 2.8. Three theoretical survival curves taken to high doses capable of causing tumour sterilisation. The
curves appear to deviate around 10 Gy. The standard LQ model shows progressive curvature, the LQ-M
(chromosomal) model 1 gradually straightens and the LQ-L (or LQ-linear) model 2 follows a constant slope
after a dose of 10 Gy has been achieved.
Table 2.1. Derivation of chromosomal number considerations in apparent straightening of the cell survival
curve with increasing dose.
• The nuclear chromosome number is M.
• Total number of lethal chromosome breaks per cell, E = αd + βd2.
• Expected average number of lethal breaks per chromosome at dose d = (αd+βd2)/M.
• Probability of no breaks on an individual chromosome = exp[−(αd+βd2)/M].
• Probability of any number of lethal breaks on an individual chromosome = 1−exp[−(αd+βd2)/
M].
• The average number of lethally damaged chromosomes per cell becomes M(1−exp[−(αd+βd2)/
M]) = E
• So the surviving fraction becomes SF = exp[−EM].
.
M
determined by Douglas Lea in 1942 (Lea and Catcheside 1942). Since some of these
breaks will be second or third breaks on the same chromosomes (this being likely
since some chromosomes are larger than others), such an effect would require
modification of the yield of ‘effective’ lethal chromosomal events, defined as lethal
breaks restricted to only one chromosome (second, third, etc., breaks on the same
chromosome representing no further lethality risk). This makes a difference when
Poisson statistics are considered, and such an interpretation is given in table 2.1.
The model provides a relatively simple formulation but requires radiosenstivity
and chromosomal number input. Straightening of the LQ curve occurs at reasonable
doses in the 6–10 Gy range depending on the radiosensitivities and scaling used. This
2-23
Quantitative Radiobiology for Proton Therapy
model may be appropriate for normal tissue with a normal chromosome content of
M = 46. Rapidly dividing cells have the added complexity of chromatids, and
tumour cells may possess greater than 46 chromosomes per cell. Another complexity
is that chromosomal volumes are variable, so some chromosomes will be more
susceptible than others to random radiation damage. The chances of lethal breaks
and multiple such breaks on the smallest chromosomes can probably be ignored, so
one might compromise on there being an effective smaller input number of M to the
model. A number of around 36–40 is suggested. Some unpublished work showed
that a greater number of cell survival curves were better fitted by this model than the
standard LQ model, but the difference was not statistically significant. However, the
model is attractive for normal tissue modelling purposes and may be biologically
more realistic than the assumption of an arbitrary dose beyond which a straight line
is assumed, to be considered next.
Since the chromosomes are of variable volume, random lethal break distributions
will favour the larger ones, so there is likely to be an ‘effective value’ which is less
than the 46 chromosomes of a cell. This is not known at the present time but is
potentially obtainable by experiment. A disadvantage of this model is that it is not
possible to use the BED concept with accuracy: this is because series expansion of
the exponential term does not converge well on the pure function, and there is
increasing dependence on knowledge of α and β terms rather than α/β.
Consequently, graphical methods are necessary to determine the reduction of effect
or the equivalent dose to produce the same effect, and then apply BED equations.
The method in its present form may be a useful research tool since by fitting
appropriate data sets the effective value of M could be obtained.
Jones & Dale (2015) presented an alternative simple approach, inclusive of a BED
approach, which assumes that the transition from pure-LQ cell kill to purely
logarithmic (i.e. straight-line) cell kill is abrupt rather than gradual and occurs at
a specific fractional dose (c) after which the same slope is maintained. In terms of the
number (L) of lethal lesions created.
At low fraction doses (less than c) equation (2.1) holds, i.e. E = αd + βd
to a limit of E = αc + βc
2
, but at a dose of c the gradient given by the first
2
and up
differential coefficient of the linear quadratic equation (α + 2βc) at dose c, so that
for doses which exceed c (i.e. d−c):
ab ab=+++ −Ec c cdc2,
2
()()
2.38
()
which provides a constant slope for d > c.
Following exactly the same procedures as were applied in the basic BED
derivation (see equations (2.1)–(2.5)), the BED for N fraction treatments involving
fraction doses greater than c is given by
2
⎛
=+ ++ −
Nc
N
⎜⎟
⎝
cc
ab ab
//
() ()
⎛
⎜⎟
⎝
2-24
2
⎞
() ()
⎠
⎞
dcBED 1
,2.39
⎠
which simplifies to
Quantitative Radiobiology for Proton Therapy
⎛
⎜⎟
=+ −
BED 1 2 . 2.40
Nd
N
⎝
cc
⎛
ab
/
()
⎝
⎞
⎞
d
⎠
⎠
()
For purposes of illustration, we assume that the crossover from LQ cell kill to linear
(logarithmic) cell kill occurs at 10 Gy, i.e. c = 10. Then, again following the
steps used earlier, for a single-fraction treatment to be isoeffective to a conventional
30 × 2 Gy treatment (BED = 72 Gy
Then, 1 1
×× + − =d
):
10
10
⎛
⎜⎟
⎝
10
⎛
2
⎝
10
⎞
⎞
d
72,
⎠
⎠
i.e. the single-fraction isoeffective tumour dose (d ) is 27.3 Gy (see the 22.3 Gy
derived above for the ‘pure-LQ’ case).
In the case of a late-reacting normal tissue with α/β = 3 Gy (for 60 Gy in 30
fractions, BED = 100 Gy
), then solve
3
⎛
×× + − =dd1
⎜⎟
⎝
10
10
⎛
2
3
⎝
⎞
⎞
100,
⎠
⎠
which provides a single-fraction dose of 17.4 Gy (see the 15.9 Gy derived for the
‘pure-LQ’ case).
The effect of changing the dose at which the transition occurs is shown in
figure 2.9.
High-LET formulations for each method.
1. Chromosomal M model: use the BED formulation containing RBE
RBE
instead of the standard BEDs:
max
max
and
Figure 2.9. Changing the transition points on the LQ survival curve at 8 Gy (red), 10 Gy (blue) and 12 Gy
(green).
2-25
Quantitative Radiobiology for Proton Therapy
2
=−−+
EM ad dMor
MHH
1Exp
(
=−− +
EM Exp RD dM
MLL
[
()
1..RBE,
(
b
/
]
)
ab
[
()
max min
22
/
]
)
where M is the ‘effective number of chromosomes’.
2. For the straightening of the LQ curve after a dose c, it can be shown by
similar rearrangements as in the standard model case that
2
⎛
=+−
BED RBE
Nd
N max
⎜
⎝
RBE
c
min
⎛
ab
()
⎜
/
⎝
c
⎞⎠⎞
22.41
d
⎠
()
It is important to understand that standard BED equations (those assuming that the
LQ model holds for indefinitely large fraction doses) will always underestimate the
isoeffective doses required for treatments involving high fraction doses. However,
caution must be taken in applying this alternative and untested BED approach, but
it might be useful for those who wish to analyse their data and attempt to isolate
what is the best dose for c in particular situations such as normal tissue type, or
tumour class.
In practice, because the dose–response curves ‘straighten out’ (compared with the
increasing curvature predicted by the LQ model), the isoeffect doses calculated using
the above methodology provide lower tolerance or isoeffective dose estimates
compared with the standard LQ approach. To that extent the standard BED
calculations tend to ‘fail-safe’; both the required tumour and normal tissue doses will
be underestimated, but the degree of underestimation is probably greater in the latter
case. This is because the late-reacting normal tissue dose–response curve has greater
curvature (and lower α/β ratio) than has the tumour. In the case of slow-growing
tumours with α/β ratios that are close to that of the late-reacting normal tissue, there
should be less difference. However, the LQ model will continue to underestimate the
effect and will fail-safe with respect to normal tissue late-reacting effects. This must
be understood by the physicians and physicists involved in the treatment planning
process. These methods may of course be useful in obtaining correlations with
clinical data sets.
The inclusion of normal tissue and tumour repopulation effects, and further more
detailed considerations on fractionation, as well as the increasingly important topic
of re-treatment of recurrent tumours are discussed further in chapter 12.
The third potential mechanism can be illustrated by plotting cell survival curves
for different exposure times, as shown in figures 2.10(a) and (b), where treatment
exposure time extension leads to reduced survival and apparent ‘straightening’ of the
survival curve. The degree of change in survival is much greater in the case of low α/
β ratios.
2.3 The α/β ratio and its choice for modelling particle therapies
Models can only be sufficiently informative for clinical decision making if their input
parameters are reasonably accurate and if the model is used in a fail-safe way,
2-26