9.2 RBE uncertainties

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Quantitative Radiobiology for Proton Therapy

must also be understood by the inevitable fact that displacement of Bragg peak position away from the International Commission on Radiation Units and Measurements (ICRU) target volumes might increase normal tissue toxicity outside the deﬁned planning target volume (or PTV), but would be expected to be accompanied by reduced tumour control: there is no evidence that this has occurred, so it is reasonable to invoke RBE as being the main culprit.

RBE is of paramount importance since it determines the dose given to the patient in equivalent-Gy. Essentially, if the RBE is incorrect the proton dose may be too low or too high, with consequent effects in each patient.

RBE depends on multiple factors, as discussed in the previous chapter. This multi-factorial basis of RBE implies that it cannot be a constant value, but must be a continuous variable (as is height and weight in humans). Particularly relevant are the radiobiological properties of the control low LET of conventional megavoltage radiotherapy, which govern the numerator of the RBE deﬁnition. Report 16 of the ICRU (1970) states that the mean LET for megavoltage radiations (electrons and photons) is around 0.2–0.3 keV μm

−1

. From the shapes of the cell-survival or bioeffect curves at low and high LET, it is inevitable that the numerator dose of the RBE ratio will increase to a greater extent than the denominator dose for a given isoeffect when dose per fraction is changed. An idealised example of cell survival isoeffects can be seen in ﬁgure 1.3 in chapter 1.

Proton therapy has two potential Achilles heels (unlike only one susceptible heel in the original Homeric poem). They are (1) physical dose placement uncertainty and (2) biological effect (RBE) uncertainties. First it is necessary to describe some of the physical uncertainties before considering the RBE issues in greater detail. The physical issues have already been discussed in chapter 1.

9.2 RBE uncertainties

The RBE concept is so often either misunderstood or underestimated in terms of complexity. It is vital to understand its multi-factorial dependency, already listed above. All clinicians should understand that if the RBE is incorrect, so also will be the dose given to the patient, sometimes by a percentage change that could exceed normally acceptable treatment plan dose variations, or the legally permitted range of dose due to errors in beam delivery (Dale et al 2009). RBE also has an impact on proton range uncertainties (Carabe et al 2012). A reported reduction in RBE at long range (Calugaru et al 2011) is possibly connected with the increasingly separating tracks, as discussed in chapter 1, and has important implications which need further experimentation (see chapters 11 and 14).

Proton RBEs have similarities with those for heavier ions than protons, but with some differences in scale. High proton RBEs have been found in various experiments, in high-LET parts of the beam (Belli et al 2000, Britten et al 2013, Marshall et al 2016). Also, the rise in RBE per unit increase in LET is larger at lower LET values for protons compared to all heavier ions (probably because of the smaller Katz radius of ionisation associated with each track; see chapter 1), which is directly proportional to the nuclear charge (Z), and protons have the lowest Z number of 1. So, protons are more efﬁcient at increasing RBE at lower values of LET. This

9-3

Quantitative Radiobiology for Proton Therapy

causes the turnover of RBE with LET at around 30.5 in Belli’s experiments, rather than at between 100–120 and 180–220 keV μm

−1

for protons, helium and carbon ions, respectively, as shown in the previous chapter. The potential for high RBE values occurring at deceptively low LET values consequently arises.

The essential clinical features of RBE are its inevitable dependency on LET, dose and tumour/tissue type. Changes in proton RBE with dose per fraction and tissue type have not been as extensively researched when compared with the range of experimental systems used for fast neutrons. In the latter case there was a clear dependency on tissue type: fast-proliferating, acute-reacting tissues (with high α/β ratios) had much smaller changes in RBE with dose than were the case for far more slowly proliferating, late-reacting tissues (with low α/β ratios) (Carabe-Fernandez

et al 2007, 2010). The collected proton experiments, and the expedient decision to

allocate an overall RBE of 1.1, representing a median value, to all tissues and tumours (regardless of their α/β ratio) is discussed in chapter 7; in order for proton therapy to improve, this must be reconsidered. This should be coupled with more detailed experimental studies and analysis of human clinical outcomes, including radiological changes in normal tissues for LET-RBE effects.

9.3 Description of the quantitative model

To determine the RBE, the same equations as in the preceding chapter 8 are used, but with a proton Z number of 1. Further details can be found in two publications (Jones 2015a, 2017) concerning the link between the low-LET reference values for α and β and their high-LET counterparts, and in the supplementary data in the most recent reference.

The efﬁciency of cell killing, in terms of the α and β parameters, are both assumed to rise linearly with LET to a maximum at around 30.5 keV μm and to a maximum level which is related to the low-LET (reference, or control radiation) α and β parameters (respectively termed α

and βL, but where α/β is used it

will refer to the low-LET state). The model considers increments in α and β that are proportional to LET, but using a saturation effect which limits the maximum possible increase in α and β, as given in the previous chapter. Some comments regarding the value of 30.5 keV μm

−1

are given in Jones (2019) especially the comments of Dr P O’Neill regarding proton ranges, since at this and high LET values the particle range can become insufﬁcient to reach the cellular nucleus in experimental conditions when using monoenergetic protons. Contrasting data using fast neutrons (see chapter 5 ) suggest that the proton LET

should be round 62 keV μm−1, but this would be in a

beam where protons are being released stochastically along neutron tracks and so at any point within a cell such that range restrictions will not be as applicable.

Since the proton LET–RBE turnover point (deﬁned as LET (Belli et al 2000), and the increments in RBE (and consequently the radiosensitivity parameters) with LET appear to be linear (Belli et al 2000, Sørensen et al 2011), then with increasing dose the overall RBE will fall in proportion to the shift between the high-LET α and β parameters as shown in ﬁgure 9.2 (a more complete version of this graphic is given as ﬁgure 8.1 in chapter 8).

−1

, as seen in ﬁgure 9.1,

) is at 30.5 keV μm

−1

9-4

Quantitative Radiobiology for Proton Therapy

Figure 9.1. Data of Belli et al showing α rising linearly with linear energy transfer (LET). The least-squares

ﬁtted line is α strain of V-79 cells used, where 0.12 Gy

= 0.12 + 0.02 LET for data points below 31 keV μm−1, which will only be the case for the

−1

is the control low-LET αLvalue.

Figure 9.2. Schematic diagram of the linear increase change in α and β with LET. Increasing dose will increase the proportion of β-mediated cell kill and so cause a lower overall RBE (itself determined by the increments in α (changing RBE

) and in β (changing RBE

max

), and so the relative proportions of RBE

min

max

and RBE

min

which contribute to the overall RBE at any value of LET within this range. The increase in β with LET is always smaller than that in α. So, with increasing dose the overall RBE follows the direction of the arrow. The numbers used in this ﬁgure are not intended to represent any particular cell system.

9-5

Quantitative Radiobiology for Proton Therapy

The model outputs, u sing a baseline (control) LET of 0.22 keV μm−1for megavoltage photons, are now displayed in the form of ﬁgures 9.3(a)–(e), where the ﬁxed parameter was β = 0.03 Gy tables 9.1–9.5, where stochastic estimates are obtain ed by assuming mean α values of 0.07 Gy

−1

for a α/β of 2 and 3 Gy for late-reacting normal tissues,

and reference radiation α values of 0.275, 0.375 and 0.55 Gy

−2

(at the reference low LET), and

−1

for tumours with α/β ratios of, respectively, 4.5, 10 and 25 Gy. In all cases a standard deviation of 10% of the mean was used in Mathematica for random-sampling procedures using 1000 samples, from which the mean RBE and the 1% and 99% quantile ranges were found.

Figure 9.3. (a)–(e) Plots of proton RBE with dose per fraction at speciﬁed LET values.

9-6

Quantitative Radiobiology for Proton Therapy

Table 9.1. α/β = 2 Gy: central nervous system late effects with mean α = 0.07 Gy−1(SD = 0.02).

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 1.25 1.10

(1.08, 1.11)

1.12 (1.08, 1.14)

1.15 (1.13, 1.18)

1.18 (1.16, 1.21)

1.21 (1.18, 1.24)

1.42 (1.37, 1.48)

1.80 (1.7, 1.9)

d = 1.5

d = 1.8

d = 2

d = 2.5

d =

d = 5

d = 10

d = 12.5

1.09

(1.07, 1.10)

1.08

(1.07, 1.09)

1.07

(1.06, 1.09)

1.06

(1.05, 1.08)

1.06

(1.05, 1.07)

1.04

(1.03, 1.05)

1.02

(1.01, 1.03)

1.02

(1.01, 1.03)

1.11

(1.10, 1.13)

1.10

(1.09, 1.12)

1.10

(1.08, 1.11)

1.08

(1.07, 1.10)

1.07

(1.06, 1.09)

1.05

(1.04, 1.07)

1.03

(1.02, 10.5)

1.03

(1.02, 1.04)

1.14

(1.12, 1.16)

1.13

(1.11, 1.15)

1.12

(1.10, 1.14)

1.10

(1.09, 1.12)

1.09

(1.07, 1.11)

1.06

(1.05, 10.8)

1.04

(1.03, 1.06)

1.03

(1.02, 1.05)

1.17

(1.14, 1.19)

1.15

(1.13, 1.17)

1.14

(1.12,1.16)

1.12

(1.10, 1.15)

1.11

(1.09, 1.13)

1.08

(1.06, 1.10)

1.05

(1.03, 1.07)

1.04

(1.02, 1.06)

1.19

(1.16,1.22)

1.17

(1.15, 1.20)

1.16

(1.14, 1.19)

1.14

(1.12, 1.17)

1.13

(1.10, 1.15)

1.09

(1.07, 1.11)

1.05

(1.04, 1.08)

1.05

(1.03, 1.07)

1.38

(1.33,1.44)

1.35

(1.30, 1.40)

1.33

(1.28, 1.38)

1.29

(1.24, 1.34)

1.25

(1.21, 1.31)

1.18

(1.14, 1.23)

1.11

(1.08, 1.12)

1.10

(1.06, 1.15)

SD = standard deviation, d = dose per fraction (Gy), LET = linear energy transfer (keV.μm−1).

1.72

(1.63, 1.82)

1.66

(1.57, 1.75)

1.62

(1.53, 1.71)

1.54

(1.46, 1.64)

1.48

(1.41, 1.58)

1.35

(1.28, 1.44)

1.22

(1.15, 1.31)

1.19

(1.12, 1.28)

Table 9.2. α/β = 3 Gy: general normal tissue late effects with mean α = 0.07 Gy−1(SD = 0.02).

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 1.25

d = 1.5

d = 1.8

d = 2

1.12

(1.10, 1.13)

1.11

(1.09, 1.12)

1.10

(1.09, 1.11)

1.09

(1.08, 1.11)

1.15

(1.13, 1.17)

1.14

(1.12, 1.16)

1.13

(1.11, 1.15)

1.12

(1.11, 1.14)

1.19

(1.17, 1.21)

1.18

(1.15, 1.20)

1.16

(1.14, 1.18)

1.15

(1.13, 1.17)

1.23

(1.20, 1.25)

1.21

(1.18, 1.23)

1.19

(1.17, 1.22)

1.18

(1.16, 1.20)

1.26

(1.23, 1.29)

1.24

(1.21, 1.27)

1.22

(1.19, 1.25)

1.21

(1.18, 1.24)

1.52

(1.46, 1.58)

1.48

(1.42, 1.54)

1.44

(1.39, 1.50)

1.42,

(1.37, 1.47)

1.98

(1.88, 2.09)

1.91

(1.80, 2.01)

1.83

(1.73, 1.93)

1.79

(1.69, 1.88)

(Continued)

9-7

Quantitative Radiobiology for Proton Therapy

Table 9.2. (Continued )

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 2.5

1.08

(1.07, 1.10)

1.11

(1.09, 1.13)

1.14

(1.12, 1.16)

1.16

(1.14, 1.18)

1.19

(1.16, 1.21)

1.37

(1.32, 1.43)

1.70

(1.61, 1.79)

d = 3

d = 5

d = 10

d = 12.5

1.08

(1.06, 1.09)

1.06

(1.05, 1.07)

1.04

(1.03, 1.05)

1.03

(1.02, 1.04)

1.10

(1.08, 1.11)

1.07

(1.06, 1.09)

1.05

(1.04, 1.06)

1.04

(1.03, 1.05)

1.12

(1.10, 1.14)

1.09

(1.07, 1.10)

1.06

(1.04, 1.07)

1.05

(1.04, 1.07)

1.15

(1.12, 1.17)

1.11

(1.09, 1.13)

1.07

(1.05, 1.09)

1.06

(1.04, 1.08)

1.17

(1.14, 1.19)

1.13

(1.10, 1.50)

1.08

(1.06, 1.10)

1.07

(1.05, 1.09)

1.34

(1.29, 1.39)

1.25

(1.21, 1.30)

1.17

(1.13, 1.21)

1.15

(1.11, 1.19)

1.64

(1.55, 1.73)

1.48

(1.40, 1.56)

1.32

(1.25, 1.40)

1.28

(1.21, 1.36)

Table 9.3. α/β = 4.5 Gy: tumours with mean α = 0.275 Gy−1(SD = 0.0275).

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 1.25

d = 1.5

d = 1.8

1.09

(1.08, 1.10)

1.08

(1.07, 1.09)

1.08

(1.07, 1.09)

1.12

(1.10, 1.13)

1.11

(1.09, 1.12)

1.10

(1.09, 1.11)

1.14

(1.12, 1.16)

1.13

(1.11, 1.15)

1.12

(1.11, 1.14)

1.17

(1.15, 1.20)

1.16

(1.14, 1.18)

1.15

(1.13, 1.17)

1.20

(1.17, 1.23)

1.19

(1.16, 1.21)

1.17

(1.15, 1.19)

1.41

(1.35, 1.46)

1.38

(1.33, 1.43)

1.35

(1.30, 1.40)

1.79

(1.69, 1.89)

1.73

(1.64, 1.82)

1.68

(1.59, 1.76)

d = 2

d = 2.5

= 3

d = 5

d = 10

d = 12.5

1.07

(1.06, 1.08)

1.07

(1.06, 1.08)

1.06

(1.05, 1.07)

1.04

(1.03, 1.05)

1.03

(1.02, 1.03)

1.02

(1.02, 1.03)

1.10

(1.08, 1.11)

1.09

(1.07, 1.11)

1.08

(1.07, 1.09)

1.06

(1.05, 1.07)

1.03

(1.03, 1.04)

1.03

(1.02, 1.03)

1.12

(1.10, 1.14)

1.11

(1.09, 1.12)

1.10

(1.08, 1.11)

1.07

(1.06, 1.08)

1.04

(1.03, 1.05)

1.03

(1.03, 1.04)

1.14

(1.12. 1.16)

1.13

(1.11, 1.14)

1.11

(1.10, 1.13)

1.08

(1.07, 1.10)

1.05

(1.04, 1.06)

1.04

(1.03, 1.05)

9-8

1.16

(1.14, 1.19)

1.15

(1.12, 1.17)

1.13

(1.11, 1.15)

1.10

(1.08, 1.11)

1.06

(1.04, 1.07)

1.05

(1.04, 1.06)

1.33

(1.29, 1.38)

1.30

(1.26, 1.34)

1.27

(1,23, 1.31)

1.20

(1.17, 1.23)

1.12

(1.09, 1.14)

1.10

(1.08, 1.12)

1.64

(1.56, 1.73)

1.58

(1.50, 1.65)

1.52

(1.49, 1.59)

1.38

(1.33, 1.44)

1.23

(1.19, 1.28)

1.19

(1.15, 1.23)

Quantitative Radiobiology for Proton Therapy

Table 9.4. α/β = 10 Gy: moderately radiosensitive tumours with mean α = 0.375 Gy−1(SD = 0.0375).

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 1.25

1.09

(1.8, 1.11)

1.12

(1.10, 1.14)

1.15

(1.13, 1.18)

1.18

(1.15, 1.21)

1.21

(1.18, 1.25)

1.44

(1.37, 1.51)

1.87

(1.73, 2.01)

d = 1.5

d = 1.8

d = 2

d = 2.5

= 3

d = 5

d = 10

d = 12.5

1.09

(1.07, 1.11)

1.09

(1.07, 1.10)

1.08

(1.07, 1.10)

1.08

(1.07, 1.09)

1.07

(1.06, 1.09)

1.06

(1.05, 1.07)

1.04

(1.03, 1.05)

1.04

(1.03, 1.04)

1.12

(1.11, 1.14)

1.11

(1.09, 1.13)

1.11

(1.09, 1.13)

1.09

(1.10, 1.12)

1.10

(1.08, 1.11)

1.08

(1.06, 1.09)

1.05

(1.04, 1.07)

1.05

(1.04, 1.06)

1.15

(1.12, 1.17)

1.14

(1.12, 1.16)

1.14

(1.11, 1.16)

1.13

(1.11, 1.15)

1.12

(1.10, 1.14)

1.10

(1.08, 1.11)

1.07

(1.05, 1.08)

1.06

(1.05, 1.07)

1.18

(1.15, 1.21)

1.17

(1.14, 1.20)

1.16

(1.14, 1.19)

1.15

(1.13, 1.18)

1.14

(1.12, 1.17)

1.12

(1.10, 1.14)

1.08

(1.06, 1.10)

1.07

(1.06, 1.08)

1.20

(1.17, 1.24)

1.19

(1.16, 1.23)

1.19

(1.16, 1.22)

1.18

(1.15, 1.20)

1.17

(1.14, 1.20)

1.13

(1.11, 1.16)

1.09

(1.08, 1.11)

1.08

(1.06, 1.10)

1.42

(1.35, 1.49)

1.40

(1.34, 1.47)

1.39

(1.33, 1.45)

1.36

(1.30, 1.42)

1.34

(1.28, 1.40)

1.28

(1.23, 1.32)

1.19

(1.16, 1.23)

1.17

(1.14, 1.20)

1.83

(1.70, 1.96)

1.79

(1.66, 1.92)

1.76

(1.64, 1.89)

1.71

(1.60, 1.82)

1.66

(1.55, 1.77)

1.54

(1.45, 1.62)

1.37

(1.30, 1.44)

1.33

(1.27, 1.39)

Table 9.5. α/β = 25 Gy: very radiosensitive tumours with mean α = 0.55 Gy−1(SD = 0.055).

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 1.25

d = 1.5

d = 1.8

d = 2

1.08

(1.06, 1.10)

1.08

(1.06, 1.10)

1.08

(1.06, 1.09)

1.08

(1.06, 1.09)

1.10

(1.08, 1.13)

1.10

(1.08, 1.13)

1.10

(1.08, 1.12)

1.10

(1.08, 1.12)

1.13

(1.10, 1.16)

1.13

(1.10, 1.16)

1.13

(1.10, 1.15)

1.12

(1.10, 1.15)

1.15

(1.12, 1.19)

1.15

(1.12, 1.19)

1.15

(1.12, 1.18)

1.15

(1.11, 1.18)

1.18

(1.14, 1.22)

1.18

(1.14, 1.22)

1.17

(1.14, 1.21)

1.17

(1.13, 1.21)

1.38

(1.30, 1.47)

1.37

(1.29, 1.46)

1.37

(1.29, 1.45)

1.36

(1.28, 1.44)

1.77

(1.60–1.94)

1.75

(1.59, 1.92)

1.73

(1.58, 1.90)

1.72

(1.57, 1.88)

(Continued)

9-9

Quantitative Radiobiology for Proton Therapy

Table 9.5. (Continued )

Dose (Gy) LET = 1 LET = 1.25 LET = 1.5 LET = 1.75 LET = 2.0 LET = 4.0 LET = 8.0

d = 2.5

1.07

(1.06, 1.09)

1.10

(1.08, 1.12)

1.12

(1.09, 1.15)

1.14

(1.11, 1.18)

1.17

(1.13, 1.20)

1.35

(1.27, 1.43)

1.69

(1.55, 1.85)

d = 3

d = 5

d = 10

d = 12.5

1.07

(1.06, 1.09)

1.06

(1.05, 1.08)

1.05

(1.04, 1.06)

1.05

(1.04, 1.06)

1.09

(1.07, 1.12)

1.08

(1.07, 1.10)

1.07

(1.05, 1.08)

1.06

(1.05, 1.08)

1.12

(1.09, 1.14)

1.10

(1.08, 1.13)

1.08

(1.07, 1.10)

1.08

(1.06, 1.10)

1.14

(1.11, 1.17)

1.12

(1.10, 1.15)

1.10

(1.08, 1.12)

1.09

(1.07, 1.11)

1.16

(1.13, 1.20)

1.14

(1.11, 1.18)

1.12

(1.09, 1.14)

1.11

(1.08, 1.13)

1.34

(1.26, 1.41)

1.30

(1.24, 1.37)

1.24

(1.19, 1.30)

1.22

(1.17, 1.27)

1.67

(1.53, 1.82)

1.59

(1.47, 1.72)

1.47

(1.38, 1.58)

1.44

(1.35, 1.53)

The steps taken to estim ate the RBE are as follows. The RBE calc ulat ion is based on the following scaling approach, an extension of the approach taken by Wilkens & Oelfke (2004)asinJones(2015b, 2017), which includes the relevant parameters:

−

bb bb=+

LET LET

aaa=+

−

With LETU, the slope of α and β with LET is found from the maximum radiosensitivity values:

(

a=−−/Qj j L1exp. ,

[]

)

and

bb=−−/Ru u L.1 exp .

()(

[]

)

If LETUis not known, then use the highest available LET (let this be LETS, and use this to replace LET then replace α

RBE is obtained by solving for d

+ βLd

LdL

, provided the radiosensitivity values at LETSare known, which

and βU).

in the general isoeffect equation:

= αHdH+ βHd

aaab bb

−+ + +dd

, and then dividing by the dHto give

()

LH H L

9-10

Quantitative Radiobiology for Proton Therapy

As an alternative to these tables, and with greater ﬂexibility in terms of input parameter choices, as well as allowing LET and dose per fraction values on a continuous scale rather than discrete values, the graphical user interface used in the previous chapter 8 can be modiﬁed for estimating proton RBEs and for isoeffective dose calculations (ﬁgure 9.3 and tables 9.1–9.4).

These tabulated estimates and those in ﬁgures 9.3(a)–(e) all show increasing RBE with LET and an inverse relationship between RBE and dose per fraction. Greater complexity is introduced by varying the α/β ratios. It again can be seen that increases in LET above, say, 1.5 keV μm

−1

can be accompanied by signiﬁcant increments in RBE. The lowest α/β ratio systems give the largest RBE at near-zero dose, but then the values fall sharply, the rate of change depending on α/β. High-α/β systems have much ﬂatter curves, showing less variation in RBE with dose per fraction. The curves cross over such that larger RBE values can exist for systems with high α/β ratios compared with low α/β ratios when dose per fraction is sufﬁciently large; it is important to note that the lowest RBE values then occur in the most fraction-sensitive tissues, suggesting that high dose per fraction can be relatively protective. This is because of enhanced repair capacity and fraction sensitivity in the case of low α/β ratios for the photon (numerator) part of the RBE deﬁnition and the converse for the high-LET denominator dose. Also, cells with low α/β, with the highest RBE

, will asymptotically approach RBE

max

with dose at a

min

faster rate owing to the higher proportion of β-related cell killing than in cells where α/β is high.

Many of the RBE values shown exceed the conventional value of 1.1. Of particular concern is that for low-α/β systems such as brain spinal cord and late reactions, the RBE values are highest when dose per fraction is small. This ﬁnding represents a warning to clinicians, who should consider either using a suitably modiﬁed RBE or changing the tolerance of key organs at risk in such situations. Recognition of changes of RBE with tissue and tumour α/β ratios, and appropriate use of these ratios to the clinical setting will be important in PBT.

The method of estimation has assumed ﬁrm knowledge of the LET

value for

protons. The sensitivity of this value is considered in chapter 14. Should the slope of the radiosensitivities, or RBE, with increasing LET be known by experiment or clinical data, it would be possible to estimate the RBE more directly, as mentioned in chapter 8 (section 3.5). Without such data, the more generic method has to be pursued, or alternative approaches based on the low-LET α/β ratio, although such a method may incur some inaccuracies for high dose per fraction if β is assumed to be invariant with LET.

9.4 RBE graphical examples

Interactive graphics for respectively estimating proton RBE with dose per fraction and for estimating the degree of normal tissue sparing required for a deﬁned BED isoeffect for different LET values are given in ﬁgures 9.4 and 9.5.

By solving for clinically relevant isoeffects (50 Gy in 25 fractions for spinal cord and optic chiasm, etc., and 60 Gy in 30 fractions for cortical brain) in the

9-11

Quantitative Radiobiology for Proton Therapy

Figure 9.4. RBE as a function of proton physical dose, dhi. The dashed line highlights a constant value of RBE = 1.1. The orange point on the RBE curve corresponds to the respective RBE value for a given dose, shown with a default d

https://josh-will-moore.shinyapps.io/InteractivePlots_Jones_IOP/, where input variable may be changed on the

relevant cursors. Credit: Joshua Moore.

= 2 Gy here. For a fully interactive version of this ﬁgure can be found at the link

Figure 9.5. The relationship of the central nervous system biological effective dose (for BED values appropriate for spinal cord, optic chiasm and brain stem tolerances) and the degree of proton-beam normal tissue sparing with variable operative LET values. The three horizontal lines show BED levels of 100, 90 and 80 Gy[2], in order to represent different levels of CNS tissue radiotolerance according to medical histories. Higher tolerance levels are sometimes permitted for small irradiated volumes according to local protocols. For a fully interactive version of this ﬁgure, see Credit: Joshua Moore.

https://josh-will-moore.shinyapps.io/InteractivePlots_Jones_IOP/.

biological effective dose (BED) equations with LET-RBE effects included (see chapter 6), the total doses required to maintain each isoeffect are given in ﬁgures 9.6(a) and (b), respectively.

9-12

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