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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
In this way, the efficiency of cell kill per unit dose will increase linearly with LET,
leading up to maximum efficiency (defined as 100%) at LET
For values of LET beyond the turnover point (where LET > LET
.
U
), the additional
U
energy transferred does not contribute to extra lethality, but is wasted. That is, the
excess energy (LET
− LETU) beyond the optimal released energy is wasted.
x
Consequently, inefficiency, expressed in energy terms by energy wasted/energy
released, or 1 − (LET
− LETU)/(LETx− LETC), increases with LET. To express
x
this in terms of efficiency, the engineering relationship of % efficiency = 100 − %
inefficiency is used, and the α
Accordingly, the equation for α
aaa=+−
HL
valueisthenscaledbetweenαUand αL.
H
for LET > LETUthen changes to be
H
−
x
LET LET
⎛
⎜⎟
1
⎝
−
LET LET
xC
U
⎞
. , 8.9
(()
UL
⎠
⎞
−a
⎟
⎠
which effectively expresses a progressive reduction in α with further increasing LET.
In this way, if LET
is 180, LETC= 1 and LETUis 120, the value of αUat the
x
turnover point of 100% efficiency will fall to 1 − (180 − 120)/180, which provides
around 67% efficiency. For a LET
of 240, we obtain 1 − (240 − 120)/240, which is
x
50% efficient. These efficiencies are, of course, relative to a normalised value of 100%
at the turnover point.
Similar equations are used to provide β
, by proportionate scaling between β
H
and βU. These are obtained by simple replacement of αL, αHand αUby βL, βHand
, respectively, in equations (8.7) and (8.9) and are not reproduced here.
β
U
The estimated RBE is then obtained using the isoeffect equations given below in
section 8.3.5.
Whereas the above method can be used to tentatively estimate the RBE for any
cellular system based on generic assumptions using only the input value of the lowLET radiosensitivity parameters. Alternatively, if there is available data sufficient to
estimate the slope of the radiosensitivity parameters with LET then it is easy to
estimate the RBE for any LET value within the range of the experiments, as
described in the next section.
L
8.3.4 An alternative method which does not use LET
but the slope of the
U
radiosensitivity or measured RBE increments with increasing LET (up to the
turnover point)
For RBE measurements at specified doses or bioeffects (such as surviving fraction),
simple linear fitting should be sufficient, although in some instances a non-linear fit
may be required if the relationship deviates from linearity. One cannot easily then
predict the RBE at different dose or effect levels in the absence of radiosensitivity
information.
Where detailed radiosensitivity measurements have been made, it is easy to
estimate the slopes of α and β increasing with LET (noting that some cell lines show
little or no increase in β). If the slopes are, respectively, S
and Sβfor each
α
radiosensitivity parameter, then
8-9
E
Quantitative Radiobiology for Proton Therapy
aa bb=+ − =+ −
HL xCHL xC
SS. LET LET , and . LET LET ,
ab
8.10
()
then the RBEs can be estimated at any dose by applying the equations in the
following section.
8.3.5 The RBE at any specified dose per fraction
The reduction in RBE with reduced surviving fraction and increasing dose is
obtained by the solution of the following isoeffect equation for high- and lowLET radiations at a dose d
ab a b+= +dd dd.
LL L
and dH, for low and high LET, respectively:
L
22
HH H
L
H
()
8.11
The solution for dLis then divided by dHto provide the RBE, as shown elsewhere
(Jones 2015a)
For clinical isoeffect calculations, the solution of the following biological effective
dose (BED) equations are used for the low and high LET:
nd
L
⎛
+= +
1RBE
⎜
⎜
⎝
⎞
d
L
a
() ()
b
L
⎟
⎟
⎠
md
⎛
⎜
H
max
⎜
⎝
2
RBE .
min
a
b
L
⎞
d
H
,8.12
⎟
()
⎟
⎠
where n and m are the respective number of fractions for the low and high LET.
TheRBEparametersarereplacedbyLET (and the new parameters given in
the sequ ence of equations described above) and then solved for d
. Total doses
H
to provide the same BED can then be calculated for different numbers of
fractions.
8.4 The graphical results
8.4.1 Radiosensitivity data
In the previous edition, a different method for estimation of LET
ion was used (as described in Jones & Hill 2019), which used a linear function for the
increase in RBE with LET and a reciprocal function for its later decrease, as is
explained next.
For experiments where RBE results are available, a similar approach is used, with
the reference RBE being 1 and by using simpler symbolism, with a linear relationship for LET values up to LET
(LET
− LETC) and
U
ΔRB
, and where x replaces LETx− LETC, U replaces
U
is the increment in RBE. So that can be replaced as
=+ Δ
RBE 1 RBE.
x
U
=+mxOr more simply as RBE 1 , 8.14()
where m is the gradient.
8-10
values for each
U
8.13()
Quantitative Radiobiology for Proton Therapy
The LET –RBE relationship beyond LETUrequires further replacements. These
assume that since LET
replaced by x and LET
is then large compared to LETC, that LETx− LETCcan be
x
by u (since LETCis small compared to LETU). Then
U
−Δxu
()
=+ −
RBE 1 1 RBE, 8.15
⎛
⎝
⎞
x
⎠
which can be simplified as
=+Δ
RBE 1 RBE.
u
x
And more simply as
()
8.16()
=+
RBE 1 ,
k
8.17()
x
where the coefficient k determines the displacement of the reciprocal function.
Equations (8.14) and (8.17) can be used to fit LET-RBE data sets and estimate m
and k.
The intersection point where LET
= LETUis then found from equating (8.13)
x
and (8.17), so that
+=+m
1 LET 1
U
k
LET
.
U
()
8.18
Then,
U
m
k
=
LET .
()
8.19
Both k and m can be estimated by least-squares fitting of each of the LET-RBE data
sets given in Jones & Hill (2019). Also, k/m should be related to Z in order for there
to be a specific LET
for each Z. Then
U
= fZLET , 8.20
U
()
()
where f(√Z) is some more complex function that contains √Z or which might
closely resemble such a function. The experimental relationship between Z and
LET
is shown in figure 8.3, using pooled data for helium and heavier ions, and this
U
can be fitted by the equations given on each graphic below.
A range of potential fitted equations did not accommodate the practical LET
value of the protons at 30.5 keV μm−1with the results obtained with the heavier
ions, but a good fit with the square-root function is obtained for helium and heavier
ions (see figure 8.3) by excluding the protons due to the range issues described
further below. Even if alternative equations such as 62.5 + 34 √(Z − 1) are fitted,
although the R
with a proton LET
2
= 0.995 and parameter p values <0.01, this would be compatible
value of 62.5 keV μm−1, as estimated from fast neutron studies
U
(Jones 2021), but the overall fitting is not as good as with 88.75 + 25.84√Z (as used
U
8-11
Quantitative Radiobiology for Proton Therapy
Figure 8.3. The experimental data relationships between Z and the LETUvalue for different ions; the data are
shown as black points. The black point at 30.5 keV μm
thought to include cellular range limitations. The fitted equation is provided above the graphic with a statistical
summary of R
in figure 8.3). If, alternatively, the experimental data includes the alternative value of
62.5 keV μm
and parameter p values <0.0002, the suggested LET
around 80 keV μm
2
= 0.999 and p values of <0.0002 for the numerical parameters.
−1
for protons, the equation 79.7 + 29.4√(Z − 1) provides R2= 0.995
−1
, a value found in some experimental studies (but where the use
of deuterons complicated the interpretation; Folkard et al 1996). It seems best to
accept the practical value of 30.5 keV μm
−1
is the proton value found by Belli et al, which is
value for protons is then
U
−1
for a pure proton beam due to the
intracellular range limitations.
The relationships between α
and αUfor various ions are shown in figure 8.4,
L
fitted to data from the literature (with data where negative β values were obtained
excluded).
The fitted equation is shown in the figure, but also with a least-squares fit for a
linear no-intercept relationship of α
= 6.47⋅αL(p < 0.001, R2= 0.899) for α
U
values less than 0.35 Gy−1, the more radioresistant part of the radiosensitivity
spectrum. These relationships, between α
and αUand βLand βU, can only be
L
sought approximately due to data limitations (since there has been no formal study
of these relationships), but from the existing data it is possible to make estimations
from studies which include these parameters, especially if α
and βHare close to α
H
and βU. Also, if LETUis known with reasonable accuracy, then
L
U
LET
aa=−
UxC
LET
U
,
x
where αxis the α parameter at LETx, and αCis the reference radiation α value (or
α
).
low
In the case of a non-monoenergetic fast neutron beam the LET is not known with
precision since there is a LET spectrum to consider. However, the Clatterbridge fast
8-12
Quantitative Radiobiology for Proton Therapy
Figure 8.4. Ion-beam relationships between radiosensitivity parameters at low and high LET at the turnover
point (α
bars are not available for all data used.
is here αU) and fitted by the parameters shown, with α
H
being the reference radiation α value. Error
low
neutron data (Warenius et al 1994), which show the relationship between αL(for
values up to 0.8 Gy
−1
) and αHand between βLand βH, are shown in figures 8.5(a)
and (b), respectively. In each case, the linear and non-linear fits are not significantly
different (p > 0.05), although the residuals are smallest for the non-linear equations,
which also have the advantage of not extrapolating to infinitely high radiosensitivity
values. Since these neutron RBE values are high relative to mid spread-out Bragg
peak (SOBP) protons, and are mostly caused by recoiled protons, it is reasonable to
assume that they will reflect the conditions near to the proton Bragg peak. For
further discussion on this topic, see chapter 5.
8.4.2 Fits to experimental RBE data sets
The model is superimposed to the experimental data sets, using the different cell lines
of Barendsen (figure 8.6), Weyrather et al (figures 8.7(a) and (b)) and Todd
(figures 8.8(a) and (b)).
The data of Barendsen (1968) used monoenergetic deuterium or helium (alpha)
particles in one human cell type, with highly symmetrical curves which turnover at
around 110–120 keV μm
−1
. In this case (see figure 8.6), the model fits the data
reasonably well at all levels of surviving fraction. However, since this data set exists
as plotted graphical surviving fraction results without access to the original data,
there is inevitable uncertainty in assessing the low- and high-LET α and β values,
which makes the RBE determination even more difficult. The plot was obtained by
assessment using α
= 0.16, αU= 1.31, βL= 0.046 and βU= 0.15 obtained by crude
L
measurements of survival curves and RBE plots, each on logarithmic and linear
scales, drawn by artists, and which contain displacements of many data points for
convenience of display, but the data set is better fitted by α
= 0.03 and βU= 0.08, as shown in figure 8.6. The Barendsen data set suffers from
β
L
= 0.15, αU= 1.35,
L
retrospective inaccuracies in estimating parameters from diagrams in publications
8-13
Quantitative Radiobiology for Proton Therapy
Figure 8.5. (a) and (b): 64 MeV fast neutron relationships between low- and high-LET radiosensitivity
parameters. Linear no-intercept and non-linear least-squares fits are, respectively, (a) α
α
= 5.37/3.68(1 − e
H
−3.68. αlow
), (b) αH= 1.57⋅αLand 2.29/23.57(1 − e
−23.57. αlow
).
= 2.72α
H
low
and
8-14
Quantitative Radiobiology for Proton Therapy
Figure 8.6. Monoenergetic alpha particle data of Barendsen (with large points indicating 50% survival,
medium-sized points 10% survival and the smallest points 5% survival, using the proposed model to provide fit
lines, with black indicating use of parameters α
β
= 0.15 Gy−2, and grey using parameters αL= 0.15 Gy−1, αU= 1.35 Gy−1, βL= 0.03 Gy−2and βU= 0.08
U
−2
Gy
. The thickest lines show the 5% survival.
= 0.16 Gy−1, αU= 1.31 Gy−1, βL= 0.046 Gy−2and
L
rather than use of the raw data, but the graphic shows the sensitivity of the model to
the input parameters.
The critical dependency of each RBE limit on the ratio of α and β at low and high
LET, respectively, demonstrates the importance of obtaining the most accurate
possible data, rather than depending on published material which does not contain
precise surviving fraction outcomes. The Barendsen (1968) data set also suggests a
higher value of LET
based on Z in pooled data, at around 127 instead of 103 keV μm
predicted to be 1.18 Gy
for alpha particles than obtained above using the formula
U
−1
by equation (8.3). This illustrates the uniqueness of each
−1
; also, the αUis
data set and the distorting effect of pooling of data from different laboratories using
different cell systems, etc.
The important carbon-ion data of Weyrather et al (1999), from GSI, which covers
a broader range of LET values, shows an apparently constant turnover point for
different cell types and surviving fractions (figure 8.7). The data are published with
the LQ radiosensitivities, although the ions have a small variation in their LET
spectrum (with a maximum spread of less than 5% for the highest LET values, which
reduces further with decreasing LET). So, it is unlikely that energy and LET spread
contribute to the deviations from the modelled curves seen at lower LET values. The
RBE values found at low LET values seem higher than expected, possibly due to
biological sample variation, especially since irradiations were performed using two
different accelerator systems (for LET values above and below 100 keV μm
−1
)in
different laboratories and presumably at different times. These data, although very
informative, inevitably contain greater heterogeneity than the data of Barendsen
8-15
Quantitative Radiobiology for Proton Therapy
Figure 8.7. (a) and (b): Model-fitted data of Weyrether et al for carbon ions for three different cell lines and
doses, coded in the same way as for figure
cells and and (b) for V-79 cells.
8.4 with respect to line thickness and surviving fraction, (a) for CHO
(1968), and the data are less well fitted. Another more stochastic approach is to use a
Poisson function, which will be presented in a further publication.
In the case of Todd’s(1967) multi-ion data, a range of different monoenergetic
ions were used (protons, deuterium, helium, lithium, boron, carbon, nitrogen,
oxygen, neon and argon), which implies that there will be at least nine different
curves, one for each Z value, and each with unique turnover points. Such
heterogeneous data were fitted surprisingly well by allocating a unique turnover
point for each ion species before estimation of the RBE, as shown in figure 8.8(a),
followed by the RBE estimations for each ionic species, as shown in figure 8.8(b).
8-16
Quantitative Radiobiology for Proton Therapy
Figure 8.8. (a) and (b): In (a) the approximate LET–RBE relationships are plotted for the ions used by Todd
and is useful to interpret figure (b), where the data points are displayed with the modelled estimations of the
present Z-based model to determine LET
The largest-sized points are for surviving fraction SF = 50%, the intermediate-sized points are for SF = 10%
and the smallest-sized points are for SF = 1%. The ions used and their LET (keV μm
and αUand βU, respectively, with LQ model correction for dose.
U
−1
) values are given,
respectively, in parentheses: deuterium (6.5), helium (25), lithium (55), boron (165), carbon (220), nitrogen
(300), oxygen (385), neon (580) and argon (1940). Observed RBE data are printed as black points, with
estimated RBE values as grey points. Starting on the left-hand side the first two black points are for 250 and
50 kVp x-rays, respectively, followed by deuterons, etc.
8-17
v
Quantitative Radiobiology for Proton Therapy
8.4.3 Applications of the model to clinical radiobiology
Not only is it possible to estimate RBE for any LET and dose, but also tentative
assessments of changes in total dose required for different fractionation schedules
using protons, helium and carbon ions become feasible, as shown in figures 8.9(a)–(c).
The variations in LET are representative of the wide expected clinical ranges for nonBragg-peak regions, SOBPs of different sizes, and for scanned beams. It should be
noted that the changes in total dose required with number of fractions (and
consequently dose per fraction) are remarkably similar for the respective LET ranges
used. This indicates the importance of LET mapping as well as dose mapping in the
clinic, since RBEs and consequently changes in total dose with fractionation can be
the same for a wide range of ions, as determined by their Z value and LET.
Such graphics can be achieved by the interactive process as in figure 8.10.
8.5 Further investigations: properties of LET
U
The estimated kinematic results for the LETUpositions found in Jones & Hill (2019)
are reproduced in table 8.1. These were acquired using the Stopping and Range of
Ions in Matter (or SRIM) software (Ziegler et al 2008). They show unique features
for each ion in terms of all parameters for each LET
value obtained from
U
experimental data sets, in terms of kinetic energies, relativistic velocity (β), velocity
(
) and the various combinations of these studied where Z and atomic mass (A) were
included. The index vA/Z
suggests that each of its sub-parameters contributes to LET
2
(nm fs−1), appears to show the least variation, which
, and the presence of A
U
may imply that mass is important (perhaps due to disruptive sonic effects on DNA
in an aqueous medium).
Another interesting finding is that LET
than those for the LET values occurring at the Bragg peak (termed LET
values have higher relativistic velocities
U
M
), as
shown in figure 8.11, which indicates that the turnover of RBE occurs physically
before the depth of a Bragg peak. This finding seems to be consistent with increasing
Z, and remarkably the distance between LET
and LETMincreases with Z number,
U
as shown in table 8.2.
These results show that the physical separation differences between LET
LET
are probably trivial for protons, and even for carbon ions the difference is not
M
and
U
thought to be significant in treatment planning of therapy. Much larger differences
occur with heavier ions and with iron in particular.
It has also been shown by plotting the data of Furusawa et al (2000) that LET
values increase in hypoxic conditions (see chapter 2) and it has been suggested that
values may reduce with intensification of dose rate (Jones 2022), as tentatively
LET
U
modelled for protons in chapter 9. Further work is required using carefully
controlled dose rate and hypoxic experiments (see chapter 14).
The ranges of particles at specific energies is also of interest. Particle ranges for a
range of Z values in comparison with cell thickness is shown in figure 8.12, where
there is a significant limitation for protons. Range limitations probably contribute to
the experimental LET
value being as low as 30.5 keV μm−1for monoenergetic
U
protons (as discussed in chapter 9).
U
8-18