A new formula for normal tissue complication probability (NTCP) as a function of equivalent

uniform dose (EUD)

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Phys. Med. Biol. 53 (2008) 23–36 doi:10.1088/0031-9155/53/1/002

A new formula for normal tissue complicationprobability (NTCP) as a function of equivalentuniform 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: gluxton@stanford.edu

Received 9 May 2007, in final form 18 September 2007Published 12 December 2007Online at stacks.iop.org/PMB/53/23

AbstractTo facilitate the use of biological outcome modeling for treatment planning,an exponential function is introduced as a simpler equivalent to the Lymanformula for calculating normal tissue complication probability (NTCP). Thesingle parameter of the exponential function is chosen to reproduce the Lymancalculation to within !0.3%, and thus enable easy conversion of data containedin empirical fits of Lyman parameters for organs at risk (OARs). Organparameters 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 newequation. The role of the Lyman volume-effect parameter n is unchanged fromits role in the Lyman model. For a non-homogeneously irradiated OAR, anequation relates dref, n, veff and the Niemierko equivalent uniform dose (EUD),where dref and veff are the reference dose and effective fractional volume of theKutcher–Burman reduction algorithm (i.e. the LKB model). It follows in theLKB model that uniform EUD irradiation of an OAR results in the same NTCPas the original non-homogeneous distribution. The NTCP equation is thereforerepresented as a function of EUD. The inverse equation expresses EUD as afunction of NTCP and is used to generate a table of EUD versus normal tissuecomplication probability for the Emami–Burman parameter fits as well as forOAR 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 involvepartial irradiation of organs at risk (OARs), and OARs in the vicinity of a treatment targetare often subjected to high-dose partial irradiation, with considerable potential for treatment-related complications. To avoid these risks, a planning decision might be made that resultsin an unnecessary reduction in dose to part of the target, possibly causing loss of effectivepalliation or probability of cure. Existing models for tumor control probability (TCP) mightthen be used to provide numerical estimates of those effects. In general, a planner choosesa balance between minimizing partial-volume irradiation of certain OARs to intermediateand high doses against partially irradiating other OARs, losing dose uniformity in targetvolumes or trimming geometric margins of the high-dose region around a defined target. Tobe objective in selecting a treatment plan, one would ideally engage a quantitative model tocalculate complication probabilities for the various OARs. Such models exist, but are oftennot brought to bear in a planning process in part because parameters for particular organs arenot considered well established or because models may be awkward to use. A tool to simplifycalculating normal tissue complication probability (NTCP) can aid in the overall treatmentplanning process by facilitating estimates of the likelihood of adverse outcomes and togetherwith tools for calculating TCP could promote an increase in both extent and sophistication ofuse 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 effectsin treatment planning. This we attempt to do by constructing a new, simplified formulafor one of the most widely employed phenomenological models for NTCP, then by derivingtissue 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 Burman1989, Kutcher et al 1991) for handling the general case of inhomogeneous organ irradiation, amodel collectively known as the LKB model. The K–B reduction is a method for calculatingthe single effective fractional volume corresponding to irradiation to a particular referencedose, and thereby determining the NTCP from a formula of Lyman for NTCP from a partial-volume irradiation (Lyman 1985).

We first provide a new derivation for the fact that in the LKB model, for a non-homogeneously irradiated OAR, the uniform dose to the entire structure that would resultin 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 thedetails of the derivation were omitted. The EUD as introduced by Niemierko (1997) wasdefined as the uniform dose that resulted in the survival of the same number of clonogensas 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 normalstructures, and he proposed that the EUD for a general dose–volume histogram (dvh) be givenby the generalized mean dose. The formula for the generalized mean dose is of the same formas the equation that appeared in Mohan et al (1992).

We next observe that the Lyman formula for a uniformly irradiated OAR can be representedby an analytic approximation, specifically an exponential of a second-degree polynomial indose. Using the new formalism we illustrate how model tissue parameters can readily becalculated 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-selectedlevel of OAR complication probability. The method is used to generate a table of EUD acrossthe complete range of NTCP, using the Lyman model parameters of Emami–Burman (Burmanet al 1991), as well as more recently fitted Lyman model parameters. A brief discussion andsome 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 isgiven by

NTCP = c(u) = 1/!

(2!) ·" u

"#e"t2/2 dt (1)

where

u = (D " TD50)/(m · TD50), (2)

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 " to dose D, Lyman (1985) gives

the NTCP by the same formula with TD50 replaced by a partial-volume-dependent parameterTD50("), given by

TD50(") = TD50(1) · ""n. (3)

The exponent, with n > 0, is the parameter that determines volume dependence andTD50(1) $ TD50, the value for uniform organ irradiation. For the sake of brevity in thefollowing, when the meaning is clear from the context, we shall simply abbreviate TD50(1) asTD50. The fractional volume " is written as " = V/Vref where Vref is a reference volume forthe 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 avolume-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, foreach irradiated fractional sub-volume "j ,

#j = 1, . . . , k,

$j "j = 1

%irradiated to dose dj and

reference dose dref , there corresponds a partial effective volume "(j)eff . The partial effective

volume is defined as that volume which, if it were the only volume irradiated and it wereirradiated to dose dref , would result in the same NTCP in the Lyman model as if the volume"j were the only volume irradiated and it had been irradiated to dose dj (Kutcher and Burman1989). Then

v(j)eff = "j ·

&dj

dref

' 1n

. (4)

In the LKB model, the total effective fractional volume irradiated to the dose dref that wouldgive the same NTCP as the inhomogeneously irradiated OAR is given as the sum of all theeffective sub-volumes in the dvh:

"eff =k(

j=1

"(j)eff . (5)

Explicitly, the LKB model gives the NTCP by (1) with the variable u, given by

u = (dref " TD50("eff))/(m · TD50("eff)). (6)

26 G Luxton et al

In K–B (Kutcher and Burman 1989, Kutcher et al 1991), dref was taken to be the maximumdose in the dvh, which ensures that "eff < 1. That choice is arbitrary, however, and as shownin appendix A.1, NTCP is the same for any choice of dref in the LKB model. A differentchoice d %

ref would result in a different effective volume " %eff given by (A.3), resulting in the

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 amonotonically 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, weshow 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 Niemierkoformula yields a dose which, if applied uniformly to the entire volume of the OAR, wouldresult 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):

EUD $ E = dref · "neff . (7)

This general property of the EUD suggests that a simplified formula for NTCP in terms ofEUD dose E might prove useful as a mathematical tool for comparing treatment plans. TheLyman representation of NTCP in terms of the error function is not a simple formula inasmuchas it is expressed in the form of an integral, and is somewhat cumbersome to use. We shall seebelow that there is a simpler formula whose dose dependence is a very close approximationto 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 thatits shape is sigmoidal as a function of dose. Another property is that it is symmetric aboutthe dose value for 50% complications, a feature that has not been tested directly, but thatis in no apparent contradiction to clinical experience. The sigmoidal shape has provided auseful basis for fitting clinical data on treatment complications, and the fitted parameters m, nand TD50 obtained from data from treatments performed using conventional fractionation of1.8–2 Gy per fraction (Burman et al 1991) represent a distillation of considerable empiricalclinical experience. Any alternative phenomenological formulation of the NTCP shouldpreserve the sigmoidal dose–response model and, as a practical matter, should preserve insome form the data contained in the published fitting parameters. The formula to be introducedbelow will be seen to satisfy this criterion.

Consider now the function #(u) defined in (8) as a candidate to be used as an approximationfor c(u) of (1) for u ! 0, (i.e. for dose D ! TD50):

#(u) = 12 e$u"$2u2/2 (8)

where u is defined in (2). The expression for #(u) assumes the value 0.5 for u = 0, the sameas c(u). As in LKB, for non-homogenous irradiation the quantity D of (2) would be replacedby E. For u > 0, i.e. (D > TD50) we extend the definition of #(u) by the equation

#(u) = 1 " #("u). (9)

A new formula for NTCP as a function of equivalent uniform dose 27

Graph ! 1 0.4 2 0.5 3 0.75 4 1 5 Lyman Eq. 6 2

u-4 -2 0 2 4

NT

CP

(%)

0

20

40

60

80

100

123456

( )2

21/ (2 )

tu

NTCP c u e dt"#

#$= = % &

( )

50 50( ) /( )= # %u E TD m TD

( )2 21 exp( )

2 2uu u !' != #

Lyman Equation:

( ) 1 ( )u u' '= # #

0 :<u

Exponential:

0 :>u

Figure 1. The single-parameter quadratic exponential form of (8) and (9) results in sigmoid-shapedcurves with slopes varying according to the value for parameter $ . Plotted are curves for five $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. Inparticular, the first and second derivatives are respectively symmetric and antisymmetric aboutu = 0, i.e. #%(u) = #%("u) and #%%(u) = "#%%("u), properties also of the Lyman functionc(u). The form of (8) with its extension to u > 0 by (9) ensures continuity for the secondderivative of #(u) at u = 0, where #%%(0) = 0. Higher even-order derivatives, however, arediscontinuous at u = 0. The function #(u) is plotted in figure 1 for several values for $ > 0.It can be seen that #(u) displays sigmoidal behavior as a function of the variable u starting at0 for u & "# increasing monotonically, reaching the value 0.5 as u passes through 0 andtending toward unity as u & #. Just as in the Lyman model, values of u < " 1

mcorrespond

to negative values of dose D or E, and have no physical meaning.

3.2.1. Selecting the parameter $ for agreement with the Lyman equation. To enable the newformula to be easily used with the modeling information established by previous fits of organcomplication data to the parameters of the Lyman model, we select the parameter $ in (8) sothat NTCP = #(u) closely reproduces the values from the Lyman equation (1). For the sakeof simplicity, we define $ by equating the NTCP of (8) to that of the Lyman model at a singlepoint. For further simplicity, we choose the point u = "1. This value corresponds to a doseof NTCP of 15.9%, which is in a region of high clinical interest for the NTCP. From (B.3) ofappendix B we obtain $ ' 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 andthe linear-quadratic exponential form are virtually indiscernible on the linear plot, althoughsmall deviations can be seen with a logarithmic presentation, as in the inset to figure 2. In thisfit, the maximum difference between #(u) and c(u) is 0.0033, i.e. 0.33%.

28 G Luxton et al

u-3 -2 -1 0 1 2 3

NT

CP

(%)

0

20

40

60

80

100

Lyman formulaExponential

( )2

21/ (2 )tu

NTCP c u e dt"#

#$= = % &

( )

50 50( ) /( )= # %u D TD m TD

Lyman Equation:

( ) 1 ( )' '= # #u u

0 :u <

Exponential:

0 :>u

u-3 -2 -1 0 1 2 3N

TC

P (%

)

0.1

0.3

1

3

10

30

100

Lyman formulaExponential ( )

2 21 exp( )2 2

uu u !' != #

0.8154! (

Figure 2. Exponential single-parameter second-degree polynomial fit to Lyman formula. Inset:exponential fit depicted on a semi-log plot. Curves for parameter $ = 0.8154.

3.2.2. NTCP as a function of EUD. For an OAR with Lyman parameters m and TD50, u andE are linearly related by (2), with E serving as the uniform dose D. Expression (8) for theNTCP #(u) $ %(E) can therefore be rewritten as a function of E as follows:

%(E) = e(AE"BE2"C) (10)

where

A =&

$ +$2

m

'1

mTD50(11a)

B = $2

2m2TD250

(11b)

and

C = ln 2 +$

m+

$2

2m2= ln 2 " 1

2+

A2

4B. (11c)

To summarize, (8) and (9) with parameter $ ' 0.8154 represent an approximation to the Lymanformula (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 parametersmand TD50. Since A, B and C are not all independent, C can be expressed in terms of Aand B [equation (11c)]. Inverse formulae for mand TD50 in terms of A and B are given inappendix C.

For E > TD50, u > 0, and we use (9)

%(E) $ #(u) = 1 " #("u) = 1 " 12 e"$u"$2u2/2 = 1 " e(A%E"B %E2"C %). (12)

From (2) with E in place of D and from the last equality in (12), one can calculate A%, B% andC% in terms of m and TD50 or, equivalently, in terms of A and B. The calculations are given inappendix 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 computationof E corresponding to pre-selected levels of complication. Thus for a selected valuep = %(E) < 0.5 of the NTCP, (10) applies, and taking the natural logarithm of bothsides results in a quadratic equation for E with the solution

E = A " [A2 " 4B(C + ln p)]1/2

2B. (13)

For NTCP $ p = %(E) > 0.5, applying the same method to (12) results in

E = A% + [A%2 " 4B %(C % + ln(1 " p))]1/2

2B. (14)

The conditions E < TD50 and E > TD50 for solutions of (10) and (12), respectively, restrictsolutions for the two quadratic equations so that there is a unique solution for each.

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 severecomplications from treatment, such as pneumonitis for lungs, stricture or perforation foresophagus, liver failure for liver and necrosis, proctitis, stenosis or fistula for rectum. Thesedata and the selection of endpoints were compiled from treatments at conventional fractionationof 1.8–2 Gy per fraction by Emami et al (1991), and the fitted parameters are known as theEmami or Emami–Burman parameters. From these one can obtain the corresponding OARquantities A,B,C and A%, B %, C %, and in table 1, we give the values of A,B,C,A% andC % for the Emami OARs. The quantity B % = B. For ease of reference we include theEmami parameters in table 1. The new formalism enables calculation of the value of EUDcorresponding 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 theEmami–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 variousorgans 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 appearedafter the early report of Emami et al (1991) present analyses to determine new best-fit values forLKB model parameters. We have selected a sampling of several such recent LKB parameterfits 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) andrectum (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 frommore than 100 patients and found fitted LKB parameters that differed substantially from theEmami–Burman values.

4. Discussion

The phenomenological Lyman model gives the complication probability (NTCP) for an organas a sigmoid-shaped function of the dose to which it is uniformly irradiated. We have seen

30 G Luxton et al

Table 1. Lyman model n, m and TD50, and A, B, C, A% and C% parameters [equations (10)and (12)], for OARs fitted by Burman et al (1991) for tissue complications from treatments withconventional fractionation. Last four rows: parameters from more recent data from the referencescited.

OAR parameters for ParametersOARs and Emami–Burman parameters equation (10) equation (12)

OAR n m TD50 A B C A% C%

Bladder 0.5 0.11 80 0.7796 4.293 ( 10"3 35.58 0.5942 20.76Brachial plexus 0.03 0.12 75 0.7063 4.104 ( 10"3 30.58 0.5251 16.99Brain 0.25 0.15 60 0.5831 4.104 ( 10"3 20.91 0.4019 10.03Brain stem 0.16 0.14 65 0.6115 4.015 ( 10"3 23.48 0.4323 11.83Cauda equina 0.03 0.12 75 0.7063 4.104 ( 10"3 30.58 0.5251 16.99Colon 0.17 0.11 55 1.1339 9.083 ( 10"3 35.58 0.8643 20.76Ear-1, acute serous otitis 0.01 0.15 40 0.8747 9.235 ( 10"3 20.91 0.6029 10.03Ear-2, chronic otitis 0.01 0.095 65 1.2655 8.719 ( 10"3 46.11 1.0014 28.95Esophagus 0.06 0.11 68 0.9171 5.942 ( 10"3 35.58 0.6991 20.76Femoral head and neck 0.25 0.12 65 0.8149 5.464 ( 10"3 30.58 0.6058 16.99Heart 0.35 0.1 48 1.5551 1.443 ( 10"2 42.09 1.2153 25.78Kidney 0.7 0.1 28 2.6659 4.240 ( 10"2 42.09 2.0835 25.78Larynx-cartilage necrosis 0.08 0.17 70 0.3972 2.348 ( 10"3 16.99 0.2602 7.40Larynx-laryngeal edema 0.11 0.075 80 1.6135 9.235 ( 10"3 70.67 1.3417 48.92Lens 0.3 0.27 18 0.6745 1.408 ( 10"2 8.27 0.3389 2.23Liver 0.32 0.15 40 0.8747 9.235 ( 10"3 20.91 0.6029 10.03Lungs (both combined) 0.87 0.18 24.5 1.0225 1.709 ( 10"2 15.48 0.6527 6.42Optic nerve 0.25 0.14 65 0.6115 4.015 ( 10"3 23.48 0.4323 11.83Optic chiasm 0.25 0.14 65 0.6115 4.015 ( 10"3 23.48 0.4323 11.83Parotid 0.7 0.18 46 0.5446 4.849 ( 10"3 15.48 0.3476 6.42Rectum 0.12 0.15 80 0.4373 2.309 ( 10"3 20.91 0.3014 10.03Retina 0.2 0.19 65 0.3494 2.180 ( 10"3 14.19 0.2173 5.61Rib cage 0.1 0.21 68 0.2788 1.630 ( 10"3 12.11 0.1646 4.35Skin 0.1 0.12 70 0.7567 4.712 ( 10"3 30.58 0.5626 16.99Small intestine 0.15 0.16 55 0.5649 4.293 ( 10"3 18.78 0.3796 8.58Spinal cord 0.05 0.175 66.5 0.3966 2.455 ( 10"3 16.21 0.2564 6.89Stomach 0.15 0.14 65 0.6115 4.015 ( 10"3 23.48 0.4323 11.83Thyroid 0.22 0.26 80 0.1622 7.684 ( 10"4 8.75 0.0837 2.47TM joint and mandible 0.07 0.1 72 1.0367 6.413 ( 10"3 42.09 0.8102 25.78

Lyman model and A, B, C, A%, C% parameters from data of references cited

Esophagusa 0.69 0.36 47 0.1574 1.161 ( 10"3 5.52 0.0610 0.99Parotidb 1 0.18 28.4 0.8821 1.272 ( 10"2 15.48 0.5631 6.42Lungs (both combined)c 0.99 0.37 30.8 0.2292 2.560 ( 10"3 5.33 0.0861 0.92Rectumd 0.23 0.19 81.9 0.2773 1.373 ( 10"3 14.19 0.1725 5.61

a Belderbos et al (2005).b Eisbruch et al (1999).c Seppenwoolde et al (2003).d Rancati et al (2004).

that the LKB algorithm gives the same NTCP as uniform organ irradiation to dose E, whereE is the equivalent uniform dose (EUD) of Niemierko (1999). Expressing NTCP in terms ofEUD represents a step toward simplifying the conceptual framework for modeling probabilityof 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 fittedby 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 0.9

Bladder 58.7 61.4 63.0 65.2 68.6 72.6 75.4 77.8 80.0 82.2 84.6 87.4 91.4Brachial plexus 53.3 55.9 57.6 59.9 63.4 67.5 70.3 72.8 75.0 77.2 79.7 82.5 86.6Brain 38.3 40.9 42.6 44.9 48.4 52.5 55.3 57.8 60.0 62.2 64.7 67.5 71.6Brain stem 43.0 45.7 47.4 49.7 53.2 57.4 60.3 62.7 65.0 67.3 69.7 72.6 76.8Cauda equina 53.3 55.9 57.6 59.9 63.4 67.5 70.3 72.8 75.0 77.2 79.7 82.5 86.6Colon 40.4 42.2 43.3 44.9 47.2 49.9 51.9 53.5 55.0 56.5 58.1 60.1 62.8Ear-1 acute 25.5 27.3 28.4 29.9 32.2 35.0 36.9 38.5 40.0 41.5 43.1 45.0 47.8Ear-2 chronic 50.1 51.9 53.1 54.6 57.0 59.8 61.8 63.5 65.0 66.5 68.2 70.2 73.0Esophagus 49.9 52.2 53.6 55.5 58.3 61.7 64.1 66.1 68.0 69.9 71.9 74.3 77.7Femur 46.2 48.5 49.9 51.9 54.9 58.5 61.0 63.1 65.0 66.9 69.0 71.5 75.1Heart 36.4 37.8 38.7 40.0 41.8 44.0 45.5 46.8 48.0 49.2 50.5 52.0 54.2Kidney 21.2 22.1 22.6 23.3 24.4 25.7 26.6 27.3 28.0 28.7 29.4 30.3 31.6Larynx 41.2 44.8 47.0 50.0 54.6 60.0 63.8 67.0 70.0 73.0 76.2 80.0 85.4Larynx 65.5 67.3 68.4 69.9 72.2 75.0 76.9 78.5 80.0 81.5 83.1 85.0 87.8Lens 6.3 7.7 8.6 9.8 11.7 13.9 15.5 16.8 18.0 19.2 20.5 22.1 24.3Liver 25.5 27.3 28.4 29.9 32.2 35.0 36.9 38.5 40.0 41.5 43.1 45.0 47.8Lungs (both, as single organ) 13.8 15.2 16.0 17.1 18.8 20.8 22.2 23.4 24.5 25.6 26.8 28.2 30.2Optic nerve 43.0 45.7 47.4 49.7 53.2 57.4 60.3 62.7 65.0 67.3 69.7 72.6 76.8Optic chiasm 43.0 45.7 47.4 49.7 53.2 57.4 60.3 62.7 65.0 67.3 69.7 72.6 76.8Parotid 26.0 28.5 30.0 32.1 35.3 39.1 41.7 43.9 46.0 48.1 50.3 52.9 56.7Rectum 51.0 54.6 56.8 59.9 64.5 69.9 73.8 77.0 80.0 83.0 86.2 90.1 95.5Retina 35.2 38.8 41.2 44.3 49.0 54.7 58.6 61.9 65.0 68.1 71.4 75.3 81.0Rib cage 33.5 37.8 40.4 44.1 49.5 56.0 60.6 64.5 68.0 71.5 75.4 80.0 86.5Skin 49.7 52.2 53.8 55.9 59.1 63.0 65.7 67.9 70.0 72.1 74.3 77.0 80.9Small intestine 33.7 36.4 38.0 40.2 43.6 47.6 50.4 52.8 55.0 57.2 59.6 62.4 66.4Spinal cord 38.4 41.8 44.0 47.0 51.5 56.8 60.5 63.6 66.5 69.4 72.5 76.2 81.5Stomach 43.0 45.7 47.4 49.7 53.2 57.4 60.3 62.7 65.0 67.3 69.7 72.6 76.8Thyroid 29.7 35.9 39.8 45.1 53.1 62.6 69.2 74.8 80.0 85.2 90.8 97.4 106.9TMJ and mandible 54.6 56.7 58.1 59.9 62.7 66.0 68.3 70.2 72.0 73.8 75.7 78.0 81.3

EUD (Gy) from data of references cited

Esophagusa 6.1 11.2 14.3 18.6 25.1 32.8 38.2 42.8 47.0 51.2 55.8 61.2 68.9Parotidb 16.0 17.6 18.5 19.8 21.8 24.1 25.8 27.1 28.4 29.7 31.0 32.7 35.0Lungs (both, as single organ)c 3.3 6.7 8.8 11.7 16.1 21.3 24.9 28.0 30.8 33.6 36.7 40.3 45.5Rectumd 44.3 48.9 51.9 55.8 61.8 68.9 73.8 78.0 81.9 85.8 90.0 94.9 102.0

a Belderbos et al (2005).b Eisbruch et al (1999).c Seppenwoolde et al (2003).d Rancati et al (2004).

process of modeling NTCP in the LKB model has been found in a formula for NTCP asa second-degree polynomial exponential function of E that may be simpler to use than theLyman equation.

LKB model parameter fits for an organ have also been reported for restricted sets ofpatients, grouped according to different endpoints, or according to the presence or absenceof previous treatment, such as whether prostate patients had previously undergone abdominalsurgery (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 metastaticliver tumor (Dawson et al 2002). This approach suggests an area of application for the newformalism. Studies of different levels of complication or groupings of patients could expandthe definition of what constitutes a population or type of treatment complication data thatcould be fitted directly to (10). From organ parameters fitted to the new formula, one couldthen 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 maywell 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 beenseen 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 exponentialwith the Lyman parameters m and TD50. Tables of OAR parameters have been given, derivedfrom published LKB model fits to the Emami OAR complication data and from LKB modelfits to organ complication data from several recent studies. Simple equations have beengiven for the EUD that corresponds to any pre-selected level of NTCP. These equations havebeen applied to create a table of EUDs for different levels of complication probability forconventionally fractionated treatment of tissues for which LKB model parameters have beenfitted.

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

"eff =k(

j=1

"(j)eff =

k(

j=1

"j ·&

dj

dref

' 1n

. (A.1)

Consider now a different choice of reference dose d %ref . This would correspond to a different

effective volume, " %eff . From (A.1) and equation (4) of the text, one can write

" %eff =

k(

j=1

"%(j)eff =

k(

j=1

"j ·&

dj

d %ref

' 1n

. (A.2)

This can be expanded as

" %eff =

k(

j=1

"j ·&

dj

dref

' 1n

·&

dref

d %ref

' 1n

=&

dref

d %ref

' 1n

·k(

j=1

"j ·&

dj

dref

' 1n

=&

dref

d %ref

' 1n

· "eff

or

" %neff · d %

ref = "neff · dref . (A.3)

From the LKB reduction, the NTCP for the choice of reference dose d %ref is given by

equations (1) to (3) of the text, with the variable u:

u = (d %ref " TD50(v

%eff))/(m · TD50(v

%eff)) (A.4a)

or

u =#d %

ref " TD50(1) · v%"neff

%)#m · TD50(1) · v%"n

eff

%. (A.4b)

A new formula for NTCP as a function of equivalent uniform dose 33

Substituting d %ref = dref · "n

eff" %n

efffrom (A.3) into (A.4b), the variable u may be written as

u =&

dref · vneff

v%neff

" TD50(1) · v%"neff

' *#m · TD50(1) · v%"n

eff

%or

u =#dref " TD50(1) · v"n

eff

%)#m · TD50(1) · v"n

eff

%. (A.5)

which is the form for u in the LKB reduction with the choice of reference dose dref . Thisproves the result that the NTCP in the LKB reduction is independent of the choice of referencedose.

A.2 In the LKB model, the NTCP of an inhomogeneously irradiated OAR is equal to NTCPfor 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 equalfractional size 1/N unequally irradiated to dose dj (j = 1, 2, . . . , N) according to thegeneralized mean (Niemierko 1999, Abramowitz and Stegun 1964, p 10), with the parametera:

EUD =

+

, 1N

·N(

j=1

daj

-

.

1a

. (A.6)

The sub-volumes may be considered voxels, and by summing over voxels irradiated to thesame dose, the equation can be written for k unequal fractional sub-volumes (Mohan et al1992, Kwa et al 1998a, Niemierko 1999):

"j , j = 1 . . . k,

k(

j=1

"j = 1, as EUD =

+

,k(

j=1

"j · daj

-

.

1a

. (A.7)

Consider now the general case of an OAR with Lyman volume-dependence parameter n, andmake the identification a = 1

n. Then, from (A.7) and abbreviating EUD by the symbol E

EUD $ E =

+

,k(

j=1

"j · (dj )1n

-

.n

=

/0

1

k(

j=1

"j · (dj )1n

(dref)1n

· (dref)1n

23

4

n

= dref ·

+

,k(

j=1

"j ·&

dj

dref

' 1n

-

.n

= dref ·

+

,k(

j=1

"(j)eff

-

.n

, i.e. E = dref · "neff .

(A.8)

Now consider the Lyman model in which the entire volume is irradiated to the dose E, derivedfrom the dvh using (A.8). Then the parameter u in equation (2) of the text would be given by

u = E " TD50

m · TD50= "n

eff · dref " TD50

m · TD50= dref " ""n

eff · TD50

m · ""neff · TD50

.

or, using equation (3) of the text:

u = dref " TD50("eff)

m · TD50("eff). (A.9)

Now, (A.9) is identical to equation (6) in the text for the parameter u in the Kutcher–Burmanreduction 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 theparameter a = 1

ngives 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. Theresult has been derived by Mohan et al (1992), but to our knowledge the present derivationhas not previously been published.

Appendix B

B.1. Calculation of parameter $

As explained in the text, we have elected to approximately fit the function #(u), defined in (8),to the Lyman NTCP function c(u), defined in (1), by selecting the parameter $ > 0 to forcethe value of #(u) to be equal to that of c(u) at the point u = "1. Thus,

12

exp&

"$ " $2

2

'= 1)

2!

" "1

"#e" t2

2 dt. (B.1)

Taking the natural logarithm of both sides, and using symmetry and the change of variabley = t)

2, we obtain

"ln 2 " $ " $2

2= ln

51)2!

&" "1

"#e

"t22 dt

'6= ln

51)2!

" #

1e

"t22 dt

6

= ln

712

8

1 " 2)!

" 1)2

0e"y2

dy

9:

= "ln 2 + ln;

1 " erf&

1)2

'<

where erf(x) = 2)!

= x

0 e"t2dt is the error function (Abramowitz and Stegun 1964, p 297).

Therefore,

$2 + 2$ = "2 ln>1 " erf

# 1)2

%?. (B.2)

There is only one solution to equation (B.2) that satisfies $ > 0:

$ =>1 " 2 ln

#1 " erf

# 1)2

%%?1/2 " 1. (B.3)

Thus, $ ' 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 followingsolutions for Lyman organ parameters m and TD50 in terms of parameters A and B:

m = 2$B

A)

2B " 2B= $

A)2B

" 1, (C.1)

and

TD50 = A ")

2B

2B. (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,

u = (E " TD50)/(m · TD50), (C.3)

and from (12),12 exp("$u " $u2/2) = exp(A%E " B %E2 " C %). (C.4)

Inserting (C.3) into (C.4) and taking logarithms results in

A% = $2

m2TD50" $

mTD50= A " 2$

mTD50= A "

)8B (C.5)

B % = $2

2m2TD250

= B (C.6)

C % = ln 2 +$2

2m2" $

m= ln 2 +

32

+A2

4B" A

@2B

(C.7)

where (C.1) has been used to substitute for $m

in (C.7).

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