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A new formula for normal tissue complication probability (NTCP) as a function of equivalent
uniform dose (EUD)
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2008 Phys. Med. Biol. 53 23
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IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 53 (2008) 23–36
doi:10.1088/0031-9155/53/1/002
A new formula for normal tissue complication
probability (NTCP) as a function of equivalent
uniform dose (EUD)
Gary Luxton, Paul J Keall and Christopher R King
Department of Radiation Oncology, Stanford University School of Medicine,
875 Blake Wilbur Drive, Stanford, CA 94305, USA
E-mail: [email protected]
Received 9 May 2007, in final form 18 September 2007
Published 12 December 2007
Online at stacks.iop.org/PMB/53/23
Abstract
To facilitate the use of biological outcome modeling for treatment planning,
an exponential function is introduced as a simpler equivalent to the Lyman
formula for calculating normal tissue complication probability (NTCP). The
single parameter of the exponential function is chosen to reproduce the Lyman
calculation to within ∼0.3%, and thus enable easy conversion of data contained
in empirical fits of Lyman parameters for organs at risk (OARs). Organ
parameters for the new formula are given in terms of Lyman model m and TD50,
and conversely m and TD50 are expressed in terms of the parameters of the new
equation. The role of the Lyman volume-effect parameter n is unchanged from
its role in the Lyman model. For a non-homogeneously irradiated OAR, an
equation relates dref, n, veff and the Niemierko equivalent uniform dose (EUD),
where dref and veff are the reference dose and effective fractional volume of the
Kutcher–Burman reduction algorithm (i.e. the LKB model). It follows in the
LKB model that uniform EUD irradiation of an OAR results in the same NTCP
as the original non-homogeneous distribution. The NTCP equation is therefore
represented as a function of EUD. The inverse equation expresses EUD as a
function of NTCP and is used to generate a table of EUD versus normal tissue
complication probability for the Emami–Burman parameter fits as well as for
OAR parameter sets from more recent data.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
In three-dimensional conformal (3D) and particularly in intensity-modulated radiotherapy
(IMRT) treatment planning, volumetric control of the dose distribution is determined through
0031-9155/08/010001+21$30.00 © 2008 Insititute of Physics and Engineering in Medicine Printed in the UK
23
24
G Luxton et al
a multi-factorial, quantitative decision-making process. Dose distributions routinely involve
partial irradiation of organs at risk (OARs), and OARs in the vicinity of a treatment target
are often subjected to high-dose partial irradiation, with considerable potential for treatmentrelated complications. To avoid these risks, a planning decision might be made that results
in an unnecessary reduction in dose to part of the target, possibly causing loss of effective
palliation or probability of cure. Existing models for tumor control probability (TCP) might
then be used to provide numerical estimates of those effects. In general, a planner chooses
a balance between minimizing partial-volume irradiation of certain OARs to intermediate
and high doses against partially irradiating other OARs, losing dose uniformity in target
volumes or trimming geometric margins of the high-dose region around a defined target. To
be objective in selecting a treatment plan, one would ideally engage a quantitative model to
calculate complication probabilities for the various OARs. Such models exist, but are often
not brought to bear in a planning process in part because parameters for particular organs are
not considered well established or because models may be awkward to use. A tool to simplify
calculating normal tissue complication probability (NTCP) can aid in the overall treatment
planning process by facilitating estimates of the likelihood of adverse outcomes and together
with tools for calculating TCP could promote an increase in both extent and sophistication of
use of current treatment delivery capabilities.
2. Methods
2.1. Purpose of study, background and outline of method
The present work is aimed at facilitating the use of quantitative modeling of biological effects
in treatment planning. This we attempt to do by constructing a new, simplified formula
for one of the most widely employed phenomenological models for NTCP, then by deriving
tissue parameters for the new equation. Our method is developed for the Lyman probit model
(Lyman 1985) with the Kutcher–Burman (K–B) reduction algorithm (Kutcher and Burman
1989, Kutcher et al 1991) for handling the general case of inhomogeneous organ irradiation, a
model collectively known as the LKB model. The K–B reduction is a method for calculating
the single effective fractional volume corresponding to irradiation to a particular reference
dose, and thereby determining the NTCP from a formula of Lyman for NTCP from a partialvolume irradiation (Lyman 1985).
We first provide a new derivation for the fact that in the LKB model, for a nonhomogeneously irradiated OAR, the uniform dose to the entire structure that would result
in the same NTCP can be represented by the quantity referred to by Niemierko (1997, 1999)
as the equivalent uniform dose (EUD). The same quantity had been introduced by Mohan et al
(1992) as ‘effective dose’, and a method was described for deriving the above result, but the
details of the derivation were omitted. The EUD as introduced by Niemierko (1997) was
defined as the uniform dose that resulted in the survival of the same number of clonogens
as the non-homogeneously irradiated tumor. The term was generalized by Niemierko (1999)
to be associated with a concept of tolerance dose for non-homogeneously irradiated normal
structures, and he proposed that the EUD for a general dose–volume histogram (dvh) be given
by the generalized mean dose. The formula for the generalized mean dose is of the same form
as the equation that appeared in Mohan et al (1992).
We next observe that the Lyman formula for a uniformly irradiated OAR can be represented
by an analytic approximation, specifically an exponential of a second-degree polynomial in
dose. Using the new formalism we illustrate how model tissue parameters can readily be
calculated for tissues previously fit to the LKB model. From the new formula and model
A new formula for NTCP as a function of equivalent uniform dose
25
tissue parameters, one can, by quite simple means, calculate the EUD for any pre-selected
level of OAR complication probability. The method is used to generate a table of EUD across
the complete range of NTCP, using the Lyman model parameters of Emami–Burman (Burman
et al 1991), as well as more recently fitted Lyman model parameters. A brief discussion and
some speculation are offered regarding possible interpretation of the analytic formula.
2.2. Lyman model
In the Lyman model (Lyman 1985), NTCP for uniform irradiation of an organ to dose D is
given by
" u
!
2
e−t /2 dt
(1)
NTCP = c(u) = 1/ (2π ) ·
−∞
where
u = (D − TD50 )/(m · TD50 ),
(2)
TD50 (ν) = TD50 (1) · ν −n .
(3)
m is a dimensionless parameter and TD50 is the whole organ dose for which NTCP is 50%.
For the case of uniform irradiation of a fractional volume ν to dose D, Lyman (1985) gives
the NTCP by the same formula with TD50 replaced by a partial-volume-dependent parameter
TD50 (ν), given by
The exponent, with n > 0, is the parameter that determines volume dependence and
TD50 (1) ≡ TD50 , the value for uniform organ irradiation. For the sake of brevity in the
following, when the meaning is clear from the context, we shall simply abbreviate TD50 (1) as
TD50 . The fractional volume ν is written as ν = V /Vref where Vref is a reference volume for
the OAR, usually taken to refer to the entire volume of the OAR.
2.2.1. Lyman–Kutcher–Burman (LKB) model. Kutcher and Burman (1989) developed a
volume-reduction algorithm for the Lyman model for an inhomogeneously irradiated OAR,
the resulting model conventionally referred
# to as the LKB
$ model.% In the LKB model, for
each irradiated fractional sub-volume νj , j = 1, . . . , k, j νj = 1 irradiated to dose dj and
(j )
reference dose dref , there corresponds a partial effective volume νeff . The partial effective
volume is defined as that volume which, if it were the only volume irradiated and it were
irradiated to dose dref , would result in the same NTCP in the Lyman model as if the volume
νj were the only volume irradiated and it had been irradiated to dose dj (Kutcher and Burman
1989). Then
'1
&
dj n
(j )
.
(4)
veff = νj ·
dref
In the LKB model, the total effective fractional volume irradiated to the dose dref that would
give the same NTCP as the inhomogeneously irradiated OAR is given as the sum of all the
effective sub-volumes in the dvh:
νeff =
k
(
(j )
νeff .
(5)
j =1
Explicitly, the LKB model gives the NTCP by (1) with the variable u, given by
u = (dref − TD50 (νeff ))/(m · TD50 (νeff )).
(6)
26
G Luxton et al
In K–B (Kutcher and Burman 1989, Kutcher et al 1991), dref was taken to be the maximum
dose in the dvh, which ensures that νeff < 1. That choice is arbitrary, however, and as shown
in appendix A.1, NTCP is the same for any choice of dref in the LKB model. A different
%
%
would result in a different effective volume νeff
given by (A.3), resulting in the
choice dref
same NTCP when substituted in equations (3), (6) and (1).
In the LKB model, NTCP is uniquely determined from the dvh. Consider now the
case of a uniformly irradiated OAR. Since in this case, the Lyman model gives NTCP as a
monotonically increasing function of dose, it follows that given an NTCP calculated by LKB,
there is a unique uniform dose E that corresponds to this value of NTCP. In appendix A.2, we
show that E is the quantity called the generalized EUD introduced by Niemierko (1999).
3. Results
3.1. Relationship between the LKB variables and the EUD
It is shown in appendix A.2 that the EUD for an OAR calculated by the generalized Niemierko
formula yields a dose which, if applied uniformly to the entire volume of the OAR, would
result in the same NTCP as the effective volume Kutcher–Burman dvh reduction algorithm,
calculated for any reference dose. Equation (A.8) of appendix A.2 is quoted here as (7):
n
.
EUD ≡ E = dref · νeff
(7)
This general property of the EUD suggests that a simplified formula for NTCP in terms of
EUD dose E might prove useful as a mathematical tool for comparing treatment plans. The
Lyman representation of NTCP in terms of the error function is not a simple formula inasmuch
as it is expressed in the form of an integral, and is somewhat cumbersome to use. We shall see
below that there is a simpler formula whose dose dependence is a very close approximation
to that of the Lyman model.
3.2. An approximation for the Lyman formula
The most generally accepted feature of the phenomenological Lyman NTCP formula is that
its shape is sigmoidal as a function of dose. Another property is that it is symmetric about
the dose value for 50% complications, a feature that has not been tested directly, but that
is in no apparent contradiction to clinical experience. The sigmoidal shape has provided a
useful basis for fitting clinical data on treatment complications, and the fitted parameters m, n
and TD50 obtained from data from treatments performed using conventional fractionation of
1.8–2 Gy per fraction (Burman et al 1991) represent a distillation of considerable empirical
clinical experience. Any alternative phenomenological formulation of the NTCP should
preserve the sigmoidal dose–response model and, as a practical matter, should preserve in
some form the data contained in the published fitting parameters. The formula to be introduced
below will be seen to satisfy this criterion.
Consider now the function ϕ(u) defined in (8) as a candidate to be used as an approximation
for c(u) of (1) for u ! 0, (i.e. for dose D ! TD50 ):
ϕ(u) =
1
2
eκu−κ
2 2
u /2
(8)
where u is defined in (2). The expression for ϕ(u) assumes the value 0.5 for u = 0, the same
as c(u). As in LKB, for non-homogenous irradiation the quantity D of (2) would be replaced
by E. For u > 0, i.e. (D > TD50 ) we extend the definition of ϕ(u) by the equation
ϕ(u) = 1 − ϕ(−u).
(9)
A new formula for NTCP as a function of equivalent uniform dose
27
NTCP (%)
100
1
2
3
4
5
6
80
Graph
1
2
3
4
5
6
κ
0.4
0.5
0.75
1
Lyman Eq.
2
Exponential:
60
u < 0:
2 2
ϕ (u ) = 1 exp(κ u − κ u )
u > 0:
ϕ (u ) = 1 − ϕ (−u )
2
2
40
Lyman Equation:
NTCP = c ( u ) = 1/ (2π ) ⋅
20
u
∫
e
()
2
− t
2
dt
−∞
u = ( E − TD50 ) /( m ⋅ TD50 )
0
-4
-2
0
u
2
4
Figure 1. The single-parameter quadratic exponential form of (8) and (9) results in sigmoid-shaped
curves with slopes varying according to the value for parameter κ. Plotted are curves for five κ
values (0.4, 0.5, 0.75, 1.0, 2.0). The Lyman function is also plotted for comparison.
This imposes the same symmetry about u = 0, i.e. D = TD50 as in the Lyman model. In
particular, the first and second derivatives are respectively symmetric and antisymmetric about
u = 0, i.e. ϕ % (u) = ϕ % (−u) and ϕ %% (u) = −ϕ %% (−u), properties also of the Lyman function
c(u). The form of (8) with its extension to u > 0 by (9) ensures continuity for the second
derivative of ϕ(u) at u = 0, where ϕ %% (0) = 0. Higher even-order derivatives, however, are
discontinuous at u = 0. The function ϕ(u) is plotted in figure 1 for several values for κ > 0.
It can be seen that ϕ(u) displays sigmoidal behavior as a function of the variable u starting at
0 for u → −∞ increasing monotonically, reaching the value 0.5 as u passes through 0 and
tending toward unity as u → ∞. Just as in the Lyman model, values of u < − m1 correspond
to negative values of dose D or E, and have no physical meaning.
3.2.1. Selecting the parameter κ for agreement with the Lyman equation. To enable the new
formula to be easily used with the modeling information established by previous fits of organ
complication data to the parameters of the Lyman model, we select the parameter κ in (8) so
that NTCP = ϕ(u) closely reproduces the values from the Lyman equation (1). For the sake
of simplicity, we define κ by equating the NTCP of (8) to that of the Lyman model at a single
point. For further simplicity, we choose the point u = −1. This value corresponds to a dose
of NTCP of 15.9%, which is in a region of high clinical interest for the NTCP. From (B.3) of
appendix B we obtain κ ≈ 0.8154, and as will be seen, turns out to offer a good fit.
The fit with this value is shown in figure 2. Deviations between the Lyman equation and
the linear-quadratic exponential form are virtually indiscernible on the linear plot, although
small deviations can be seen with a logarithmic presentation, as in the inset to figure 2. In this
fit, the maximum difference between ϕ(u) and c(u) is 0.0033, i.e. 0.33%.
28
G Luxton et al
100
30
NTCP (%)
NTCP (%)
10 0
80
10
Exponential:
3
Lyman formula
Exponential
1
0.3
60
0.1
-3
-2
-1
0
1
u
2
u < 0:
ϕ ( u ) = 1 exp(κ u −
u > 0:
ϕ (u ) = 1 − ϕ (−u )
2
κ 2u 2
2
)
κ ≈ 0.8154
3
Lyman Equation:
40
NTCP = c ( u ) = 1/ (2π ) ⋅
u
∫e
−∞
−
( )dt
t2
2
u = ( D − TD50 ) /( m ⋅ TD50 )
20
Lyman formula
Exponential
0
-3
-2
-1
0
1
2
u
3
Figure 2. Exponential single-parameter second-degree polynomial fit to Lyman formula. Inset:
exponential fit depicted on a semi-log plot. Curves for parameter κ = 0.8154.
3.2.2. NTCP as a function of EUD. For an OAR with Lyman parameters m and TD50, u and
E are linearly related by (2), with E serving as the uniform dose D. Expression (8) for the
NTCP ϕ(u) ≡ %(E) can therefore be rewritten as a function of E as follows:
where
%(E) = e(AE−BE
2
−C)
(10)
'
&
1
κ2
A= κ+
m mTD50
(11a)
B=
(11b)
κ2
2m2 TD250
and
κ
κ2
1 A2
+
+
.
(11c)
=
ln
2
−
m 2m2
2 4B
To summarize, (8) and (9) with parameter κ ≈ 0.8154 represent an approximation to the Lyman
formula (1) for c(u) that is accurate to within 0.33% over the entire range −∞ < u < ∞.
Equations (10) and (11a)–(11c) represent the NTCP in terms of E and the Lyman parameters
mand TD50 . Since A, B and C are not all independent, C can be expressed in terms of A
and B [equation (11c)]. Inverse formulae for mand TD50 in terms of A and B are given in
appendix C.
For E > TD50 , u > 0, and we use (9)
C = ln 2 +
%(E) ≡ ϕ(u) = 1 − ϕ(−u) = 1 − 12 e−κu−κ
2 2
u /2
%
%
= 1 − e(A E−B E
2
−C % )
%
(12)
.
%
From (2) with E in place of D and from the last equality in (12), one can calculate A , B and
C% in terms of m and TD50 or, equivalently, in terms of A and B. The calculations are given in
appendix C.
A new formula for NTCP as a function of equivalent uniform dose
29
3.3. Tissue parameters and EUD dose levels for complication rates
The formulae of (10) for E < TD50 and (12) for E > TD50 allow straightforward computation
of E corresponding to pre-selected levels of complication. Thus for a selected value
p = %(E) < 0.5 of the NTCP, (10) applies, and taking the natural logarithm of both
sides results in a quadratic equation for E with the solution
A − [A2 − 4B(C + ln p)]1/2
.
2B
For NTCP ≡ p = %(E) > 0.5, applying the same method to (12) results in
E=
(13)
A% + [A%2 − 4B % (C % + ln(1 − p))]1/2
.
(14)
2B
The conditions E < TD50 and E > TD50 for solutions of (10) and (12), respectively, restrict
solutions for the two quadratic equations so that there is a unique solution for each.
E=
3.4. EUDs for pre-determined complication rates using Emami parameters
The Lyman model tissue parameters were fitted by Burman et al (1991) to data on severe
complications from treatment, such as pneumonitis for lungs, stricture or perforation for
esophagus, liver failure for liver and necrosis, proctitis, stenosis or fistula for rectum. These
data and the selection of endpoints were compiled from treatments at conventional fractionation
of 1.8–2 Gy per fraction by Emami et al (1991), and the fitted parameters are known as the
Emami or Emami–Burman parameters. From these one can obtain the corresponding OAR
quantities A, B, C and A% , B % , C % , and in table 1, we give the values of A, B, C, A% and
C % for the Emami OARs. The quantity B % = B. For ease of reference we include the
Emami parameters in table 1. The new formalism enables calculation of the value of EUD
corresponding to any complication level for the various OARs of Emami by using (13) and
(14). The EUDs corresponding to selected levels of NTCP are given in table 2 for the
Emami–Burman parameterized OARs.
3.5. Application to organs at risk using modified Emami parameters
A number of reports have appeared which analyze treatment complication data for various
organs in terms of the LKB model, for example, Seppenwoolde et al (2003), Belderbos et al
(2005), Eisbruch et al (1999), Rancati et al (2004), Dawson et al (2002), Chapet et al (2005),
Peeters et al (2006), Cheung et al (2004) and Kwa et al (1998b). These papers which appeared
after the early report of Emami et al (1991) present analyses to determine new best-fit values for
LKB model parameters. We have selected a sampling of several such recent LKB parameter
fits from the literature for several organs, namely, esophagus (Belderbos et al 2005), parotid
(Eisbruch et al 1999), lungs combined as a single organ (Seppenwoolde et al 2003) and
rectum (Rancati et al 2004), and we have included these in tables 1 and 2 for comparison.
Each of the studies that were selected for inclusion in the tables was based on outcomes from
more than 100 patients and found fitted LKB parameters that differed substantially from the
Emami–Burman values.
4. Discussion
The phenomenological Lyman model gives the complication probability (NTCP) for an organ
as a sigmoid-shaped function of the dose to which it is uniformly irradiated. We have seen
30
G Luxton et al
A%
C%
Table 1. Lyman model n, m and TD50 , and A, B, C,
and
parameters [equations (10)
and (12)], for OARs fitted by Burman et al (1991) for tissue complications from treatments with
conventional fractionation. Last four rows: parameters from more recent data from the references
cited.
OAR parameters for
equation (10)
OARs and Emami–Burman parameters
Parameters
equation (12)
OAR
n
m
TD50
A
B
C
A%
C%
Bladder
Brachial plexus
Brain
Brain stem
Cauda equina
Colon
Ear-1, acute serous otitis
Ear-2, chronic otitis
Esophagus
Femoral head and neck
Heart
Kidney
Larynx-cartilage necrosis
Larynx-laryngeal edema
Lens
Liver
Lungs (both combined)
Optic nerve
Optic chiasm
Parotid
Rectum
Retina
Rib cage
Skin
Small intestine
Spinal cord
Stomach
Thyroid
TM joint and mandible
0.5
0.03
0.25
0.16
0.03
0.17
0.01
0.01
0.06
0.25
0.35
0.7
0.08
0.11
0.3
0.32
0.87
0.25
0.25
0.7
0.12
0.2
0.1
0.1
0.15
0.05
0.15
0.22
0.07
0.11
0.12
0.15
0.14
0.12
0.11
0.15
0.095
0.11
0.12
0.1
0.1
0.17
0.075
0.27
0.15
0.18
0.14
0.14
0.18
0.15
0.19
0.21
0.12
0.16
0.175
0.14
0.26
0.1
80
75
60
65
75
55
40
65
68
65
48
28
70
80
18
40
24.5
65
65
46
80
65
68
70
55
66.5
65
80
72
0.7796
0.7063
0.5831
0.6115
0.7063
1.1339
0.8747
1.2655
0.9171
0.8149
1.5551
2.6659
0.3972
1.6135
0.6745
0.8747
1.0225
0.6115
0.6115
0.5446
0.4373
0.3494
0.2788
0.7567
0.5649
0.3966
0.6115
0.1622
1.0367
4.293 × 10−3
4.104 × 10−3
4.104 × 10−3
4.015 × 10−3
4.104 × 10−3
9.083 × 10−3
9.235 × 10−3
8.719 × 10−3
5.942 × 10−3
5.464 × 10−3
1.443 × 10−2
4.240 × 10−2
2.348 × 10−3
9.235 × 10−3
1.408 × 10−2
9.235 × 10−3
1.709 × 10−2
4.015 × 10−3
4.015 × 10−3
4.849 × 10−3
2.309 × 10−3
2.180 × 10−3
1.630 × 10−3
4.712 × 10−3
4.293 × 10−3
2.455 × 10−3
4.015 × 10−3
7.684 × 10−4
6.413 × 10−3
35.58
30.58
20.91
23.48
30.58
35.58
20.91
46.11
35.58
30.58
42.09
42.09
16.99
70.67
8.27
20.91
15.48
23.48
23.48
15.48
20.91
14.19
12.11
30.58
18.78
16.21
23.48
8.75
42.09
0.5942
0.5251
0.4019
0.4323
0.5251
0.8643
0.6029
1.0014
0.6991
0.6058
1.2153
2.0835
0.2602
1.3417
0.3389
0.6029
0.6527
0.4323
0.4323
0.3476
0.3014
0.2173
0.1646
0.5626
0.3796
0.2564
0.4323
0.0837
0.8102
20.76
16.99
10.03
11.83
16.99
20.76
10.03
28.95
20.76
16.99
25.78
25.78
7.40
48.92
2.23
10.03
6.42
11.83
11.83
6.42
10.03
5.61
4.35
16.99
8.58
6.89
11.83
2.47
25.78
Lyman model and A, B, C, A% , C % parameters from data of references cited
Esophagusa
Parotidb
Lungs (both combined)c
Rectumd
0.69
1
0.99
0.23
0.36
0.18
0.37
0.19
47
28.4
30.8
81.9
0.1574
0.8821
0.2292
0.2773
1.161 × 10−3
1.272 × 10−2
2.560 × 10−3
1.373 × 10−3
5.52
15.48
5.33
14.19
0.0610
0.5631
0.0861
0.1725
0.99
6.42
0.92
5.61
a
Belderbos et al (2005).
Eisbruch et al (1999).
c Seppenwoolde et al (2003).
d Rancati et al (2004).
b
that the LKB algorithm gives the same NTCP as uniform organ irradiation to dose E, where
E is the equivalent uniform dose (EUD) of Niemierko (1999). Expressing NTCP in terms of
EUD represents a step toward simplifying the conceptual framework for modeling probability
of expected complications using dvhs from a proposed treatment plan. A further step in the
A new formula for NTCP as a function of equivalent uniform dose
31
Table 2. EUD (Gy) corresponding to indicated NTCP of OARs with Lyman parameters fitted
by Burman et al (1991) for tissue complications from treatments with conventional fractionation.
Last four rows: EUD for NTCP from parameters of more recent data as cited.
EUD (Gy) for indicated NTCP
OAR
0.01 0.02 0.03 0.05 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Bladder
Brachial plexus
Brain
Brain stem
Cauda equina
Colon
Ear-1 acute
Ear-2 chronic
Esophagus
Femur
Heart
Kidney
Larynx
Larynx
Lens
Liver
Lungs (both, as single organ)
Optic nerve
Optic chiasm
Parotid
Rectum
Retina
Rib cage
Skin
Small intestine
Spinal cord
Stomach
Thyroid
TMJ and mandible
58.7
53.3
38.3
43.0
53.3
40.4
25.5
50.1
49.9
46.2
36.4
21.2
41.2
65.5
6.3
25.5
13.8
43.0
43.0
26.0
51.0
35.2
33.5
49.7
33.7
38.4
43.0
29.7
54.6
72.6
67.5
52.5
57.4
67.5
49.9
35.0
59.8
61.7
58.5
44.0
25.7
60.0
75.0
13.9
35.0
20.8
57.4
57.4
39.1
69.9
54.7
56.0
63.0
47.6
56.8
57.4
62.6
66.0
75.4
70.3
55.3
60.3
70.3
51.9
36.9
61.8
64.1
61.0
45.5
26.6
63.8
76.9
15.5
36.9
22.2
60.3
60.3
41.7
73.8
58.6
60.6
65.7
50.4
60.5
60.3
69.2
68.3
77.8
72.8
57.8
62.7
72.8
53.5
38.5
63.5
66.1
63.1
46.8
27.3
67.0
78.5
16.8
38.5
23.4
62.7
62.7
43.9
77.0
61.9
64.5
67.9
52.8
63.6
62.7
74.8
70.2
80.0
75.0
60.0
65.0
75.0
55.0
40.0
65.0
68.0
65.0
48.0
28.0
70.0
80.0
18.0
40.0
24.5
65.0
65.0
46.0
80.0
65.0
68.0
70.0
55.0
66.5
65.0
80.0
72.0
82.2
77.2
62.2
67.3
77.2
56.5
41.5
66.5
69.9
66.9
49.2
28.7
73.0
81.5
19.2
41.5
25.6
67.3
67.3
48.1
83.0
68.1
71.5
72.1
57.2
69.4
67.3
85.2
73.8
84.6
79.7
64.7
69.7
79.7
58.1
43.1
68.2
71.9
69.0
50.5
29.4
76.2
83.1
20.5
43.1
26.8
69.7
69.7
50.3
86.2
71.4
75.4
74.3
59.6
72.5
69.7
90.8
75.7
87.4 91.4
82.5 86.6
67.5 71.6
72.6 76.8
82.5 86.6
60.1 62.8
45.0 47.8
70.2 73.0
74.3 77.7
71.5 75.1
52.0 54.2
30.3 31.6
80.0 85.4
85.0 87.8
22.1 24.3
45.0 47.8
28.2 30.2
72.6 76.8
72.6 76.8
52.9 56.7
90.1 95.5
75.3 81.0
80.0 86.5
77.0 80.9
62.4 66.4
76.2 81.5
72.6 76.8
97.4 106.9
78.0 81.3
55.8
31.0
36.7
90.0
61.2 68.9
32.7 35.0
40.3 45.5
94.9 102.0
61.4
55.9
40.9
45.7
55.9
42.2
27.3
51.9
52.2
48.5
37.8
22.1
44.8
67.3
7.7
27.3
15.2
45.7
45.7
28.5
54.6
38.8
37.8
52.2
36.4
41.8
45.7
35.9
56.7
63.0
57.6
42.6
47.4
57.6
43.3
28.4
53.1
53.6
49.9
38.7
22.6
47.0
68.4
8.6
28.4
16.0
47.4
47.4
30.0
56.8
41.2
40.4
53.8
38.0
44.0
47.4
39.8
58.1
65.2
59.9
44.9
49.7
59.9
44.9
29.9
54.6
55.5
51.9
40.0
23.3
50.0
69.9
9.8
29.9
17.1
49.7
49.7
32.1
59.9
44.3
44.1
55.9
40.2
47.0
49.7
45.1
59.9
68.6
63.4
48.4
53.2
63.4
47.2
32.2
57.0
58.3
54.9
41.8
24.4
54.6
72.2
11.7
32.2
18.8
53.2
53.2
35.3
64.5
49.0
49.5
59.1
43.6
51.5
53.2
53.1
62.7
0.9
EUD (Gy) from data of references cited
Esophagusa
6.1 11.2 14.3 18.6 25.1 32.8
Parotidb
16.0 17.6 18.5 19.8 21.8 24.1
Lungs (both, as single organ)c 3.3 6.7 8.8 11.7 16.1 21.3
Rectumd
44.3 48.9 51.9 55.8 61.8 68.9
38.2
25.8
24.9
73.8
42.8
27.1
28.0
78.0
47.0
28.4
30.8
81.9
51.2
29.7
33.6
85.8
a
Belderbos et al (2005).
Eisbruch et al (1999).
c Seppenwoolde et al (2003).
d Rancati et al (2004).
b
process of modeling NTCP in the LKB model has been found in a formula for NTCP as
a second-degree polynomial exponential function of E that may be simpler to use than the
Lyman equation.
LKB model parameter fits for an organ have also been reported for restricted sets of
patients, grouped according to different endpoints, or according to the presence or absence
of previous treatment, such as whether prostate patients had previously undergone abdominal
surgery (Peeters et al 2006). Patients have been grouped by other medical factors, such as for
32
G Luxton et al
example, whether a partial liver irradiation resulted from treatment of a primary or a metastatic
liver tumor (Dawson et al 2002). This approach suggests an area of application for the new
formalism. Studies of different levels of complication or groupings of patients could expand
the definition of what constitutes a population or type of treatment complication data that
could be fitted directly to (10). From organ parameters fitted to the new formula, one could
then obtain Lyman parameters if desired, by means of (C.1) and (C.2) of appendix C.
5. Summary
We have found a formula to represent NTCP as a function of EUD, and this formula may
well be useful. The equation is an exponential of a second-degree polynomial of the EUD.
In the general case of inhomogeneously irradiated OARs, normal tissue effects have been
seen to be equally well represented by this new exponential formula as by the LKB model.
Transformation formulae have been derived to connect organ parameters for the exponential
with the Lyman parameters m and TD50 . Tables of OAR parameters have been given, derived
from published LKB model fits to the Emami OAR complication data and from LKB model
fits to organ complication data from several recent studies. Simple equations have been
given for the EUD that corresponds to any pre-selected level of NTCP. These equations have
been applied to create a table of EUDs for different levels of complication probability for
conventionally fractionated treatment of tissues for which LKB model parameters have been
fitted.
Appendix A
A.1. NTCP in the LKB Model is independent of the choice of dref
In the LKB calculation, for a given reference dose dref , the effective volume is given by (4)
and (5) from the text as
'1
&
k
k
(
(
dj n
(j )
νeff =
νeff =
νj ·
.
(A.1)
dref
j =1
j =1
%
. This would correspond to a different
Consider now a different choice of reference dose dref
%
effective volume, νeff . From (A.1) and equation (4) of the text, one can write
'1
&
k
k
(
(
dj n
%(j )
%
νeff =
νj ·
.
(A.2)
νeff =
%
dref
j =1
j =1
This can be expanded as
'1 &
'1
'1 k
'1
'1
&
&
&
&
k
(
dj n
dj n
dref n
dref n (
dref n
%
=
νj ·
·
=
·
ν
·
=
· νeff
νeff
j
%
%
%
dref
dref
dref
dref
dref
j =1
j =1
or
%n
%
n
νeff
· dref
= νeff
· dref .
From the LKB reduction, the NTCP for the choice of reference dose
equations (1) to (3) of the text, with the variable u:
or
%
%
%
− TD50 (veff
))/(m · TD50 (veff
))
u = (dref
%)#
%
# %
%−n
%−n
m · TD50 (1) · veff
.
− TD50 (1) · veff
u = dref
(A.3)
%
dref
is given by
(A.4a)
(A.4b)
A new formula for NTCP as a function of equivalent uniform dose
33
n
νeff
%n
νeff
%
Substituting dref
= dref ·
from (A.3) into (A.4b), the variable u may be written as
&
'*
n
%
#
veff
%−n
%−n
or
u = dref · %n − TD50 (1) · veff
m · TD50 (1) · veff
veff
#
%)#
%
−n
−n
u = dref − TD50 (1) · veff
m · TD50 (1) · veff
.
(A.5)
which is the form for u in the LKB reduction with the choice of reference dose dref . This
proves the result that the NTCP in the LKB reduction is independent of the choice of reference
dose.
A.2 In the LKB model, the NTCP of an inhomogeneously irradiated OAR is equal to NTCP
for uniform irradiation of the OAR to a dose equal to the Niemierko EUD
The EUD is obtained by computing the dose contributions from N sub-volumes of equal
fractional size 1/N unequally irradiated to dose dj (j = 1, 2, . . . , N) according to the
generalized mean (Niemierko 1999, Abramowitz and Stegun 1964, p 10), with the parameter
a:
 a1

N
(
1
da .
(A.6)
EUD =  ·
N j =1 j
The sub-volumes may be considered voxels, and by summing over voxels irradiated to the
same dose, the equation can be written for k unequal fractional sub-volumes (Mohan et al
1992, Kwa et al 1998a, Niemierko 1999):

 a1
k
k
(
(
νj = 1,
as
EUD = 
νj · dja  .
(A.7)
νj , j = 1 . . . k,
j =1
j =1
Consider now the general case of an OAR with Lyman volume-dependence parameter n, and
make the identification a = n1 . Then, from (A.7) and abbreviating EUD by the symbol E
n
n 

1
k
k


(
(
(dj ) n
1
1
n
νj · (dj ) n  =
νj ·
EUD ≡ E = 
1 · (dref )


(dref ) n
j =1
j =1



n
' n1 n
&
k
k
(
(
d
j
(j
)
n
 = dref · 
= dref · 
νj ·
νeff  , i.e. E = dref · νeff
.
d
ref
j =1
j =1
(A.8)
Now consider the Lyman model in which the entire volume is irradiated to the dose E, derived
from the dvh using (A.8). Then the parameter u in equation (2) of the text would be given by
u=
−n
· TD50
ν n · dref − TD50
dref − νeff
E − TD50
= eff
=
.
−n
m · TD50
m · TD50
m · νeff · TD50
or, using equation (3) of the text:
u=
dref − TD50 (νeff )
.
m · TD50 (νeff )
(A.9)
Now, (A.9) is identical to equation (6) in the text for the parameter u in the Kutcher–Burman
reduction algorithm of the Lyman model. Therefore, for an inhomogeneously irradiated OAR,
34
G Luxton et al
the equivalent uniform dose defined according to the generalized EUD formula (A.7) with the
parameter a = n1 gives the same NTCP as the LKB dvh reduction procedure. The NTCP is the
same as when the OAR is subjected to uniform irradiation to dose E in the Lyman model. The
result has been derived by Mohan et al (1992), but to our knowledge the present derivation
has not previously been published.
Appendix B
B.1. Calculation of parameter κ
As explained in the text, we have elected to approximately fit the function ϕ(u), defined in (8),
to the Lyman NTCP function c(u), defined in (1), by selecting the parameter κ > 0 to force
the value of ϕ(u) to be equal to that of c(u) at the point u = −1. Thus,
&
'
" −1
κ2
1
1
t2
exp −κ −
=√
e− 2 dt.
(B.1)
2
2
2π −∞
Taking the natural logarithm of both sides, and using symmetry and the change of variable
y = √t 2 , we obtain
&" −1
'6
5
5
" ∞ 2 6
κ2
1
1
−t 2
−t
2
−ln 2 − κ −
e dt
= ln √
e 2 dt
= ln √
2
2π
2π 1
−∞
9:
7 8
'<
;
&
" √1
2
1
1
2
2
e−y dy
= −ln 2 + ln 1 − erf √
1− √
= ln
2
π 0
2
where erf(x) =
Therefore,
√2
π
=x
0
2
e−t dt is the error function (Abramowitz and Stegun 1964, p 297).
>
# %?
κ 2 + 2κ = −2 ln 1 − erf √12 .
There is only one solution to equation (B.2) that satisfies κ > 0:
>
#
# %%?1/2
κ = 1 − 2 ln 1 − erf √12
− 1.
(B.2)
(B.3)
Thus, κ ≈ 0.8154.
Appendix C
C.1. Formulae for m and TD50 in terms of parameters A and B
Straightforward algebraic manipulation of equations (11a)–(11c) results in the following
solutions for Lyman organ parameters m and TD50 in terms of parameters A and B:
m=
2κB
=
√
A 2B − 2B
TD50
√
A − 2B
=
.
2B
and
κ
√A
2B
−1
,
(C.1)
(C.2)
A new formula for NTCP as a function of equivalent uniform dose
35
C.2. Formulae for parameters A% , B% and C% in terms of m and TD50
From (2), with EUD E substituted for dose D,
and from (12),
(C.3)
u = (E − TD50 )/(m · TD50 ),
1
2
exp(−κu − κu2 /2) = exp(A% E − B % E 2 − C % ).
(C.4)
Inserting (C.3) into (C.4) and taking logarithms results in
κ2
√
κ
2κ
=A−
= A − 8B
mTD50
mTD50
A% =
m2 TD
B% =
κ2
=B
2m2 TD250
50
−
(C.6)
@
3 A2
2
κ2
κ
= ln 2 + +
−A
−
C = ln 2 +
2m2
m
2 4B
B
%
where (C.1) has been used to substitute for
(C.5)
κ
m
(C.7)
in (C.7).
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