Functional form comparison between the population and the individual ...

Radiol Oncol 2007; 41(2): 90-98.

doi:10.2478/v10019-007-0016-7

Functional form comparison between the population and the individual Poisson based TCP models Colleen Schinkel1,2, Nadia Stavreva2, Pavel Stavrev2, Marco Carlone2,3 and B. Gino Fallone1-3 1 Department of Physics, University of Alberta, 2 Department of Medical Physics, Cross Cancer Institute; 3 Department of Oncology, University of Alberta, Edmonton, Alberta, Canada

In this work, the functional form similarity between the individual and fundamental population TCP models is investigated. Using the fact that both models can be expressed in terms of the geometric parameters γ50 and D50, we show that they have almost identical functional form for values of γ50 ≥ 1. The conceptual inadequacy of applying an individual model to clinical data is also discussed. A general individual response TCP expression is given, parameterized by Df and γf – the dose corresponding to a control level of f, and the normalized slope at that point. It is shown that the dose-response may be interpreted as an individual response only if γ50 is sufficiently high. Based on the functional form equivalency between the individual and the population TCP models, we discuss the possibility of applying the individual TCP model for the case of heterogeneous irradiations. Due to the fact that the fundamental population TCP model is derived for homogeneous irradiations only, we propose the use of the EUD, given by the generalized mean dose, when the fundamental population TCP model is used to fit clinical data. Key words: radiotherapy dosage; Poisson distribution; dose-response relationship, models, statistical, TCP

Introduction In the decades following the introduction of the first individual TCP model by Munro and Gilbert,1 the distinction between the individual and population response has often been disregarded and individual TCP models have been fit to clinical datasets. The necessity of describReceived 01 June 2007 Accepted 20 June 2007 Correspondence to: B. Gino Fallone, Ph.D., Department of Medical Physics, Cross Cancer Institute, 11560 University Ave., Edmonton, Alberta, T6G 1Z2, Canada. Tel: (780) 432-8750, Fax: (780) 432-8615; Email: [email protected],

ing the impact of population heterogeneity on dose-response has lead to the development, by a number of authors, of population-based tumour control probability (TCP) models.2-5 It has been shown that the presence of population heterogeneity leads to a doseresponse curve that is flattened relative to the individual dose-response curve. If an individual TCP model is fit to a population dataset, the biological meaning of the parameter estimates is lost – the radiobiological parameters take on unrealistically low values.6 Nevertheless, although it is conceptually incorrect, the individual TCP model has been fit to clinical datasets and

91

Schinkel et al. / Functional form comparison of TCP models

parameters obtained from these fits have Background and method been assumed to have radiobiologically meaningful values.4,7-10 On the other hand, The general form of the population-based it has also been shown that these fits are Poisson TCP model has eight parameters. characterized by an acceptable goodness However, it has previously been shown6,11 of fit. that the parameters of such a model are It has been expected that the population interrelated; many different combinations TCP models would allow for the estimaof parameters lead to one and the same tion of biologically meaningful population TCP curve. Thus, it may seem difficult to parameters. Unfortunately, it is impossible directly compare the functional forms of to obtain a unique set of parameter valthe individual and population-based TCP ues when a population TCP model is fit to models. On the other hand, Carlone et al.11 clinical data.6,11 This is due to the fact that have specified (based on a certain approxidifferent sets of population parameter valmation, of course, but a clinically valid ues produce almost identical TCP curves. one) what these interrelations actually are, based that TCP there models.are Ononly the other Carlone et al.11 analytically demonstrated and have shown two hand Carlon that when the dominant source of interindependent population model parameters clinically valid one) these interrelations patient heterogeneity is that of tumour – D50 and γ50 . Fortunately, thewhat individual radiosensitivity, the On population TCP funcTCP(based module also approximation, be pabased TCP models. the other hand Carlone et Pooison-based al.11 have specified on acan certain of c model parameters – D and J50. Fortunately, t tion has only two independent parameters rameterized by these parameters.50This fact one)TCP, whatDthese interrelations are, and shown that only two – clinically the dose valid at 50% deter- actually makes the have comparison of there both are models an independen 50, which based TCP model could be parameterised too. mines the position of the TCP curve, and easier task. parametersslope – D50 of and the J50. Fortunately, themodel normalized curve, γ50. these two parameters are also the parameters in which the individ 5 These parameters have geometric meanThe Poisson-based individual model based TCP parameterised too. Hence making the comparison of both TCP models an easier task. ing. Since it model is alsocould true be that the individual TCP model may be expressed in terms of This common form of the individual The Poisson-based individualTCP TCP model 5 the same two parameters,3,12 it is possible model is based on Poisson statistics comthat, for a given range of parameter values, bined with This a simplified description of common form of the individual TCP mo 4,10,11,13-26 both models will exhibit almost identical clonogen repopulation. In the The Poisson-based individual TCP model 4,10,11,13-26 functional form. In this work, we invescase where a clonogen tumour undergoes homogenerepopulation. In the case w tigate the similarities between these two ous irradiation to a total dose D, split into This common form of the individual TCP model is based on Poisson statistics combined a simplified de n fractions dosewith per fraction, d, the models expressed in terms of D50 and γ50 n fractions with equal with doseequal per fraction, d, byclonogen plottingrepopulation. them for 4,10,11,13-26 identicalInvalues individual Poisson TCP model may be the caseofwhere the a tumour undergoes homogeneous irradiation to a total dose these geometric parameters. written as:11 NS n fractions with equal dose per fraction, d, the individual Poisson may bee written as:11N 0 e D Ed D OT 10 TCP TCP exp [1] model ind

>

10

@

Oc · § ª ¨ D Ed ¸ D º d ¹ exp « N 0 e © N initial D cD @ , of clonogens, N 0 expnumber where» Nexp is >the «¬ »¼ 0 linear quadratic (LQ) radiosensitivity parame where N0 is the initial number of clonogens, NS is the mean number of clonogens surviving

[1] [1]

TCPind

>

exp N 0 e D Ed D OT

e NS

@

thewhere treatment, and βnumber are the quadratic (LQ) radiosensitivity parameters, λ isthethe N0 is theαinitial of linear clonogens, NS is the mean number of clonogens surviving treatment, D a as an equal do tumour repopulation rate, T is the total treatment time and O c O T n . Note Notethat thatasaslong long aslinear an equal dose (LQ) is given during eachparameters, fraction ofOthe treatment is common quadratic radiosensitivity is the tumour (which repopulation rate, T clinical is the total treatme practice), the parameters α, β and λ′ can be combined into one singlethe parameter: practice), parameters D, E and Oc can be co O c O T n . Note that as long as an equal dose is given during each fraction of the treatment (which is comm Oc D c Oncol D 2007; Ed 41(2): . 90-98. [2] parameter: Radiol one single practice), the parameters D, E and Oc can be combined into15 d

15

[2]

D c D Ed

Oc d

.

The validity of the Poisson TCP model was

O c O T n . Note that as long as an equal dose is given during each fraction of the treatment (which is comm Schinkel et al. / Functional form comparison of TCP models 92 practice), the parameters D, E and Oc can be combined into one single parameter:

15

5 5 20 5 5 5

5 10 10 10 10 10

10

15 15 15 15 15

15

[2][2] [3a] [3a]

Oª 2cJ 50 § D · º D c D Ed exp ª«¬« 2lnJ 502. §¨¨© 1 DD50 ·¸¸¹º»¼»

TCPind TCPind

0.5exp «dln 2 ¨¨ 1 D ¸¸ » . 0.5 ª¬« 2J §© D50 ·¹º¼» .

The validity of the Poisson TCP model was questioned by Tucker and Travis,21 and others27exp « 50 ¨¨ 1 ¸¸ » 31 who explored the non-Poisson 21 ¬« ln 2 © D50 ¹ ¼» 33 where nature of questioned the TCP inby theTucker case tumour repopulation [3a] TCP 0.5Poisson The validity ofnormalized the was andtheTravis, and others27-31 precision who explored ª 2J slope, · ºҏ,.model § TCP D J for purpose of dosimetric qua The notion ofind ¸ » was first introduced by Brahme exp « 50 ¨¨ 1 27,32 ¸ 33 lnslope, 2 © D50 J occurs. Under certain ¬«conditions, however, it has been shownfor the that the distribution of ¹ ¼»ҏ, was first introduced by Brahme purpose of dosimetric precision quan The notion of normalized [3a] TCPind 0.5 . Poisson nature inthe theremaining case where occurs. certain however, the number of of clonogen cells attumour theofend is Under well-approximated by derived Later, Kallman et the al.34TCP used value Jrepopulation atofthea treatment inflection point of the TCP conditions, curve and an i 33 Jmaximum ҏ, was first introduced by Brahme for the purpose of dosimetric precision quan ThePoisson notion ofdistribution. normalized slope, the view of these results, and also because of the relative complexLater, 27,32 Kallman et al.34 used maximum value of J at the inflection point of the TCP curve and derived an ª 2J 50In º · § the D ¸¸ » ¨¨ 1 exp « 33 35 remaining shown that the distribution the number ofindividual clonogen atpresented the end of treatment well-approx Dof 2slope, ҏ,out introduced bycells Brahme forinconsistency the purpose of dosimetric precision qua Theofnotion of [3a], normalized ity the non-Poissonian the TCP Eq. [1] isis 50 ¹J¼»models, © TCP ¬« lnpointed similar toTCP Eq. but byfirst Bentzen and Tucker, a function slight isain present in their formula. 34 as [3a] 0 . 5 . was ind et al. Later, Kallman used the maximum value of J at the inflection point of the TCP curve and derived an e 35 similarused. to Eq. [3a], but as pointed out by Bentzen and Tucker, a slight inconsistency is present in their formula. often 34 In view of these 3,12 the Poisson distribution. results, and also because of the relative complexity of the non-Poisso Later, Kallman al. used given the maximum value at the35inflection point of the TCP curve an APoisson form ofTCP theetexpression individual TCP model that isJ equivalent to Eq. [1], but written in terms the by Eq. [1], mayofbe transformed and parameterized in terms ofand the derived normalized similar to Eq.TCP [3a],expression but as pointed by Bentzen and a slight inconsistency isinpresent in the their formula. I thethe Poisson given Eq. [1], beTucker, transformed and terms of normalized Jout ҏ, by was first by Brahme33 forparameterized the purpose of dosimetric precision quan The notion of normalized slope, of geometric parameters, γ50 and D50 ,introduced ismay given by: models, theEq. individual TCPpointed function presented in Eq. [1] is often 35 used. similar [3a], a slight inconsistency is present in their formula. any dosetopoint Df: but as ª 2J § D · º out by Bentzen and Tucker, ¸¸ » by Eq. [1], may be transformed and parameterized in terms of the normalized exp « 50 ¨¨ 1given the Poisson TCP expression 34 D ln 2 any dose point D : « » 50 Later, value of Jat the inflection point of the TCP curve and derived an ¼ [3a] Kallman [3a] TCPind etf 0al. .5 ¬used© the ¹maximum . 3,12 the Poisson TCP expression givenº byTCP Eq. model [1], may be transformed and parameterized terms ofinthe normalized A form of the individual that is equivalent to Eq. [1], butinwritten terms of the ª · § J any dose point Df: exp « f ¨ 1 D ¸ » 35 33 similar to Eq. [3a], but as pointed out by Bentzen and Tucker, a slight inconsistency is present in their formula. I ¨ ¸ The notion of normalized ª« f Jlnf f §© DDf slope, ·¹º¼» .γ, was first introduced by Brahme for the purpose of ¨ 1 ¸» exp «¬ 33 [3b] TCP fD any dose point D : 34 parameters, J and , is given by: ind ¨ ¸ f 50 50 f ln f D J , ҏ was first introduced by Brahme for the purpose of dosimetric precision quan The notion of normalized slope, « » f ¹¼ dosimetric precision ¬quantification. Later, Kallman et al. used the maximum value of γ at [3b]Poisson TCP f ª J f §© given ind expression D ·¸ º by Eq. [1], may be transformed and parameterized in terms of the normalized the TCP ¨ 1TCP the inflection pointexpof « the » curve and derived an expression similar to Eq. [3a], but as ¨ ¸ 34«¬ f ln f © D f ¹ »¼ 35 a value Later, Kallman etBentzen Jat thethe inflection point sets of curve (and derived an ª used º ·maximum § the J f and [3b] TCP fal. pointed out slightofinconsistency present inthe their formula. DTucker, indand JIn From Eqs. [1]by [3b], relationships between twoisdifferent of TCP parameters ¸» f, Df) and (N0, D exp « the¨ following 1 any dose point D : f « f ln f ¨© D f ¸¹ ¼» ¬ From [1]Poisson and [3b], the following relationships between the two sets ofand parameters (Jf, Df) and (N0, D general, the expression given by Eq. [1], may be different transformed parameter[3b] Eqs. TCP f TCP 35 similar Eq. ind [3a], butnormalized as pointed out by Bentzen Tucker, a slight inconsistency is present in their formula. I ized in to terms of the slope γf at anyand dose point D derived: f: From Eqs. [1] and [3b],ª the following relationships between the two different sets of parameters (Jf, Df) and (N0, D ·º Jf § derived: ¨ 1 D ¸ » exp « ¨ ¸ the Poisson TCP expression given by Eq. [1], may be transformed parameterized in terms of(Jthe normalized f ln f D f ¹ ¼» From [1] and f[3b], relationships between the twoand different sets of parameters ¬« the© following f, Df) and (N0, [3b]Eqs. TCP [3b] ind 1 derived: § N0 · [4a]dose point D f D1 : ln§¨¨ N ·¸¸ any derived: Dfc ln f0 ¸¹ the following relationships between the two different sets of paFrom Eqs. and ¨¨© ln[3b], [4a] D f [1] ¸ c D ln f From Eqs.(γ[1] and [3b], ¹ following relationships between the two different sets of parameters (Jf, Df) and (N0, D rameters §© (N Nthe f, Df1) and 00,·α′) may be derived: [4a] ln¨¨ ª J f ¸¸§ D · º Df D1c ©exp § ln «« fNfln§0f¹·¨¨ 1N D f·¸¸ »» derived: ¬ [4a] [4a] TCP [3b] f ¨¨f ln [4b] JDf f indDfc ln ln ln§¨¨f ¸¸© N 0 ·¸¸¹.¼ ¹ f0 ¸¹ . [4b] J f f ln ©f ln¨¨© ln ¸ ln §© · Nf0 ·¹ § 1 N [4b] f ln [4b] ¨¨0 ¸ following ¸¸ . . relationships between the two different sets of parameters (Jf, Df) and (N0, D From Eqs.JDf[1] and ¨¨f lnthe [4a] ln[3b], ©f§ ln ¸¹ Nf 0 ¹· and for (J50,f D50D) cin particular: ln © f ln¨¨ [4b]for (JJ f, D ) finlnparticular: ¸¸ . and and for50(γ5050, D50) in particular: derived: © ln f ¹ and for (J50, D50)1in particular: §N · ln ¨ N 0§ ¸ N 0 · [5a] 50 1 §©f ln f in [4b] JD ln§particular: ln0¨2 ·¹· ¸¸ . c f D J , D ) and for ( 5050 501 ln ¨ N¨ D [5a] [5a] D 0 ¸ln f ¨ [4a] ln c 2© ¸ ¹ D1 ¨© ln f N 0f ·¹¸¹ c ©§ ln D ln ¨ D50 [5a] ¸ D1c2 ©§§lnNN200¹·· ln ) inln particular: and for (JJ50 D50,50D50ln [5a] [5b] [5b] 2 ln¨§¨ N ¸·¸ . . J 50 D2c ln©¨©lnln§202¹¸¹N. · [5b] ln¨2 ¹ 0 ¸ . [4b] J f ln2f 2ln f§© ln N¨ · f ¸ J 50 [5b] 1 ln§¨ N 0©0 ·ln ¸. ¹ D50 ln2 2ln ¨©§ lnN2¸¹· [5a] 0 2 ¹¸ . model The J 50 D c ln©¨ln TCP [5b]population-based 2 © ln 2 ¹ D1150)showed in particular: and for (Jet50,al. Carlone that the population TCP model for the case of dominant heterogeneThe population-based TCP model Nmay ln 2 §TCP ity inpopulation-based radiosensitivity be written as: The 0 ·model ln ¨ J 50 [5b] ¸. 2 §© N ln0 2·¹ 11 1 ª 1 D50 Carlone et al. showed that the §population ·º TCP model for the case of dominant heterogeneity in radiosensitivi The population-based TCP ln D [5a] ¨ ¸ Smodel 50 11 [6] TCP 1¸» . TCP model for the case of dominant heterogeneity in radiosensitivi [6] Jthe c ©erfc 50 ¨population ln 2«that ¹ Carlone et al.popDshowed 2 TCP ¹¼ ¬ model© D The population-based written as: 11 Carlone et al. showed that the population TCP model for the case of dominant heterogeneity in radiosensitivi written as: Radiol Oncol 2007; 41(2): 90-98. N 2 11ln § · 0that the population TCP model for the case of dominant heterogeneity in radiosensitiv Carlone The et showed ln ¨ in J al.parameters . [6] – D50 and J50 – have the same geometric meaning as the corresponding parame [5b] ¸Eq. written as: 50 The population-based 2 © TCP ln 2 ¹model written as: [3a]. The geometric parameters may be expressed in terms of the population-based radiobiological parameters, D Carlone et al.11 showed that the population TCP model for the case of dominant heterogeneity in radiosensitivi

2 © D ¹D ¬ in ¼5050 and have the same geometric meaning as the corresponding par The parameters in Eq. [6] D and –––––have The [6] and J50 have the same geometric meaning as the corresponding pa The parameters in Eq. [6] D andJJJ50 J50 havethe thesame samegeometric geometricmeaning meaning asthe corresponding pa Theparameters parameters inEq. Eq. [6]–––––D D and have the same geometric meaning as the corresponding The parameters in Eq. [6] 50 50 50 50 50 ªthecorresponding ºpar 1 as § D50 ·para 1¸ » . [6]be Sthe J 50population-ba TCPpop inin erfc «of ¨ The may [3a]. The geometric parameters may be expressed terms of the population-ba [3a]. The geometric parameters may beexpressed expressed interms terms of the population-b andgeometric J50 – haveparameters the same geometric meaning as the corresponding parame The parameters in Eq. [6][3a]. – D50 2 © D ¹¼ ¬ [3a]. The geometric parameters may be expressed in terms of the population-based radiobiological parameter [3a]. The geometric may be expressed in terms of the population-based radiobiological parameters [3a]. The geometric parameters may be expressed in terms of the population-based radiobiological parameter [3a].The Theparameters geometricparameters parameters may be expressed in terms of the population-based radiobiological paramete [3a]. The geometric parameters may be expressed in terms of the population-based radiobiological parameter paramete and J – have the same geometric meaning as the corresponding parame in Eq. [6] – D 50 form comparison of TCP models Schinkel et al. / 50 Functional 93 11 . lnln NN000:expressed ::1111 lnNbe [3a]. The geometric parameters may in terms of the population-based radiobiological parameters, D The parameters in Eq. [6] – D50 and 11 11 11 ln :1111 ln NN lnNN ln N 0000:geometric 0::: [3a].ln The parameters may be expressed in terms of the population-based radiobiological parameters, D The parameters in Eq. [6] – D and γ – have the same geometric meaning as the cor11 50 50 ln N 0 : ***lnln lnNNN000 in Eq. [3a]. The geometric parameters may be expre d J50 – have the same11geometric meaning 5as the corresponding parameters DD 55 [7a] [7a] [7a] D505050 ln N 0 : responding parameters in Eq. [3a]. The geometric may be expressed in terms of cc ln N DDDcparameters *****ln ln NN lnN ln N 00000 [7a] D 55555 [7a] D [7a] D [7a] D [7a] D 50 50 50 50 50 11 c c c D * ln NDD D the radiobiological parameters, and ln N 0 ::11 ressed in ofpopulation-based the population-based parameters, D c , V c and 0Dc c radiobiological 5 terms [7a] D50 * Dlnc N 0 DDDc cc [7a] D50 5 [7a] [7b] JJJ505050 [7b] [7b] cD DD ccccc DDD 22S 2SSVVVc cc * ln N 0 [7b] [7b] [7b] [7b] JJJJ50J50505050 [7b] 5 [7a] D50 D c 222S VVVVVccccc 22SSSS Dc [7b] [7b] J 50 2 2DScV c VVOOcOcc22 OOOc cc 222 222 222 222 V [7b] J 50 c c c c c c Here and where d D D E d V V V Here and where an d D D E d V V V Here and whereDDD, ,, EEE, ,, OOOc cc an an d D D E d V V V DDD EEE 22222 2S V c OOOOcOcccc ddcd222 VdVdVVV D 22222 22222 22222 22222 d OOOcOcOcc c c c c c ccc DD Here and where and ln are the population aaa Here and where and are the Here VVVc cc c VVVVVDDDDD dddddVVVVVEEEEE 2 22222 where DDDDD,,,,,EEEEE,,,,,JOOOO50cOcccand Here and where and ln NN are the population HereDD andVV where[7b] andln lnNN arethe thepopulation populationav Here and where and ln N are the population DDD DDDEEEEdEdddd and 00000 are V O c ddddd O c ddddd 2 2 2 c the population avera 2 2S Vare c c c Here and where , E , O and ln N d D D E d V V V D D E individual and corresponding parameters and are their sts corresponding individual parameters and population averagesd of the corresponding individual parameters and andVVVlnlnlnNNN0000 are aretheir theirsta corresponding individual parametersand andVVVDDD, ,,VVVEEE,,,0VVVOOcOccand 2 VdO c 2 Oc 2 2 2 c The c are their deviations. Γ represents Euler’s gamma constant, which has parameters Herecorresponding and parameters where , E , O and ln N are the population avera d 2VVVV D c standard D E d individual Vparameters Vsymbol individual D , V , V and V are their standard deviations. The symbo corresponding and , V , V and V are their standard deviations. The symbo and V , V , V and V are their standard deviations. The symb corresponding individual parameters and , V , V and V are their standard deviations. The symb corresponding individual parameters and , V , V and V are their standard deviations. The symbo corresponding individual and D EV 0 NNN DDDDD lnln N ln ln ln 0000 0N d of 0.577. dEEEE2E OOOcOcOcc c an approximate value O c of Euler’s constant, which an approximate 0.577. Euler’s gamma constant, which has an approximate value 0.577. Euler’s which has antheir approximate value of 0.577. Vgamma , V O c and V ln Nhas are standard deviations. corresponding individual parameters and gamma D11, V E constant, c 2 symbol VThe and Dboth V D 2 d*2 E dvalue of 0 Here D c The general form of the Carlone et al. population TCP model takes heterogeneity d Euler’s gamma constant, which has an approximate value of 0.577. Euler’s constant, which an approximate Euler’s gamma constant, which has an approximate value of 0.577. Euler’sgamma gamma constant, whichhas has anV approximate value of0.577. 0.577. Euler’s gamma constant, which has an approximate value of 0.577. , V , V value and of V are their standard deviations. The symbol * corresponding individual parameters and

0 radiosensitivity and heterogeneity in clonogen number into account. However, this form V O c 2 in The form of the etet The general form of the Carlone al. 10 population TCP model take take Thegeneral general form ofthe theCarlone Carlone etal. al.111111population 10 populationTCP TCPmodel modeltakes take Euler’s an approximate value of 0.577. 11 to where gamma O c and which ln10 N 0 has are the population averages of D , E ,constant, has three parameters, and was shown be almost identical 2 of the population TCP model corresponding individual parameters and V D 11 11 11 d10 11 11 The general form of the Carlone et al. 10 population TCP model takes both heterogeneity in radios The general form of the Carlone et al. population TCP model takes both heterogeneity in radios The general form of the Carlone et al. population TCP model takes both heterogeneity in radio The general form of the Carlone et al. 10to population TCP model takes both heterogeneity in radio The general form of the Carlone et al. 10 population TCP model takes both heterogeneity in radio 10 Euler’s gamma constant, which has an approximate value of 0.577. the one that only takes heterogeneity in in radiosensitivity into account. Hence, the latter heterogeneity into account. this form heterogeneity in clonogen number into account. However, this form of the popu heterogeneity inclonogen clonogennumber number into account.However, However, this formof ofthe thepopu pop The for general form of the Carlone et al.11 population TCP model takes both heterogeneity in radiosens 10 will be used this analysis. V are their standard deviations. The symbol * represents Euler’s gamma constant, which has an approx heterogeneity in clonogen number into account. However, this form of the population TCP model has three pppp D , V E , V O c andheterogeneity ln N in clonogen number into account. However, this form of the population TCP model has three heterogeneity in clonogen number into account. However, this form of the population TCP model has three heterogeneity inclonogen clonogennumber numberinto intoaccount. account.However, However,this thisform formof ofthe thepopulation populationTCP TCPmodel modelhas hasthree threepa heterogeneity in 0

D

2

2 d 2V E

E

Oc

ln N

10

11 The general form of thewas Carlone et111111 al. population TCP model takes both heterogeneity in radiosens toto almost toto only takes inin was shown be almost identical the one that only takes heterogeneity wasshown shown tobe be almostidentical identical tothe theone onethat that only takesheterogeneity heterogeneity inrara r heterogeneity in clonogen number into account. However, this form of the population TCP model has three param 11 11 11 11 11 was shown to be almost identical to the one that only takes heterogeneity in radiosensitivity into account. H to to heterogeneity in H was shown to be almost identical to the one that only takes heterogeneity in radiosensitivity into account. H wasshown shown tobe bealmost almostidentical identical tothe theone onethat thatonly onlytakes takes heterogeneity inradiosensitivity radiosensitivity intoaccount. account.et H was shown to be almost identical to the one that only takes heterogeneity in radiosensitivity into account. H form comparison between individual and population-based TCP models oximate valueFunctional of was 0.577. The general ofinto the Carlone 10 heterogeneity in clonogen numberelected into account. However, this form of the population TCPform model has three param toto our analysis. elected use the latter for our analysis. elected touse usethe thelatter latterfor for our analysis. was shown11 to be almost identical to the one that only takes heterogeneity in radiosensitivity into account. Hence Since bothto the individual and the population TCP models may be written in terms of the elected to use the latter for our analysis. elected the our analysis. elected to use the latter for our analysis. elected touse use thelatter latterfor for our analysis. elected use the latter for our analysis. 11to in clonogeninto number into accoun was shown to be takes almostboth identical to the oneinthat only takes heterogeneity in radiosensitivity account. Hence et al.11 population TCP model heterogeneity radiosensitivity and heterogeneity same two parameters, γ and D , 50 50 elected to use the latter for our analysis. it seems natural to assume that the two models may display similarity in functional form. In order to explore the was functional ofidentical these to the one shown11similarity to be almost elected to use for ourTCP analysis. unt. However, this form of the the latter population model has three parameters, and

models, Eqs. [3a] and [6] are evaluated for a given range of γ50 and D50 values. Subsequently, Functional comparison between and TCP 15 Functional form comparison between individual and population-based TCP mo 15 Functional form comparison between individual andpopulation-based population-based TCPmod mo 15 TCP the individual and population curvesform obtained for equal setsindividual of toγ50 useand the D latter for ourare analysis. 50 values e that only takesFunctional heterogeneity in comparison radiosensitivity into account. Hence, we have electedTCP Functional form comparison between individual and population-based TCP models 15 form between individual and population-based models 15 Functional form comparison between individual and population-based TCP models Functional form comparison between individual and population-based TCP models 15 Functional form comparison between individual and population-based TCP models 15 15 plotted for visual comparison. form comparison individual population-based TCP models 15 Functional both individual TCP may be Since both the individual and the population TCP models may be written ter ter Since boththe theand individual andthe thepopulation population TCPmodels models may bewritten writteninin interm te The functional closenessbetween of Since the individual and the and population TCP curves may be more estimated by calculating the normalized difference between the areas under the Since both the individual and the population TCP models may be written in terms of the same two parameter Since the individual and TCP models may written in of two Since both the individual and the population TCP models may be written in terms of the same two paramete Sinceboth both the individual andthe thepopulation population TCP models maybe be written interms terms ofthe thesame same twoparameter paramete Since both the individual and the population TCP models may be written in terms of the same two paramete Functional form comparison between individual and population-based TCP models 15 rigorously itititseems seems natural assume that the two models may display similarity functi seemsnatural naturaltoto toassume assumethat thatthe thetwo twomodels modelsmay maydisplay displaysimilarity similarityinin infunctio funct two SinceTCP bothcurves, the individual and the population TCP models may be written in terms of the same two parameters, J5 Functional form comparison between individu 15 ATCPpopthat ATCP seems natural to assume that the two models may display similarity in functional form. In order to explore itititititseems the two seems natural to assume that the two models may display similarity in functional form. In order to explore seems naturalto toassume assume that the twomodels modelsmay maydisplay displaysimilarity similarityin infunctional functionalform. form.In Inorder orderto toexplore explor seems natural to assume that the 'natural A ind two models may display similarity in functional form. In order to explore Since the similarity population TCP models mayEqs. be written in of the same two J5 J 50 and [8][8] both the individual , of models, [3a] [6] evaluated for aaagiven similarity of these models, Eqs. [3a] and [6] are evaluated for given rang similarity ofthese these models, Eqs. [3a]and andterms [6]are are evaluated forparameters, givenrang ran , it seems natural to assume that the two models may display similarity in functional form. In order to explore the ATCPpop ATCPpop similarity of these models, Eqs. [3a] and [6] are evaluated for a given range of J and D values. Sub similarity ofof models, aaaagiven ofof JJ similarity of these models, Eqs. [3a] and [6] are evaluated for given range of JJ and D values. Sub similarityTCP ofthese these models,Eqs. Eqs.[3a] [3a]and and[6] [6]are areevaluated evaluatedfor for given range of and D values.Subs Su similarity these models, Eqs. [3a] and [6] are evaluated for given range and values. Sub dual and population-based models 50 50 50 50 50 50 50and 50values. 50 50 Since both range the individual andDD the population it seems natural to assume that theindividual two models may displayTCP similarity inobtained functional form. order to explore the and population equal sets of individual and population TCP curves obtained for equal sets of and val individual and population TCP curves obtained forJ equalIn sets ofJJJ andDD D505050valu va 50 50 50and as a function of models, γ50. similarity of these Eqs. [3a] and [6] are evaluated forcurves a given range for of 50 and D50 values. Subsequ individual and population TCP curves obtained for equal sets of J and D values are plotted for visual comp individual and population TCP curves obtained for equal sets of J and D values are plotted for visual comp individual and population TCP curves obtained for equal sets of J and D values are plotted for visual com individual and population TCP curves obtained for equal sets of J and D values are plotted for visual com individual and population TCP curves obtained for equal sets of J and D values are plotted for visual comp 50 50 50 50 50 50 50 50 50 50 seems natural that the two mode asmay a function of Jin 50. models, n TCP modelssimilarity be written terms of Eqs. the same J50 and for D50,a itgiven of these [3a] two andparameters, [6] are evaluated range ofJto50assume and D50 values. Subsequ individual and population TCP for equal sets of J50 and D values are and plotted visual comparis The closeness of population TCP 20 The functional closeness of the individual and the population TCP cc 20 Thefunctional functional closeness ofthe the individual andthe thefor population TCPcu 20 curves obtained 50individual Results their work of 50 Gy, with values ranging similarity of these models, Eqs. [3a] and [6 The functional closeness of the individual and the population TCP curves may be more rigorously 20 dels may display similarity in functional form. In order to explore the functional The functional closeness of the individual and the population TCP curves may be more rigorously 20 The functional closeness of the individual and the population TCP curves may be more rigorousl 20 The functional closeness of the individual and the population TCP curves may be more rigorous 20individual functional closeness of the individual and of theJpopulation TCP curves may for be visual more rigorously andThe population TCP curves obtained for equal sets are plotted comparis 50 and D50 values calculating difference between the calculating the normalized difference between the areas under the TCP curv from 10 to 90 Gy. We the therefore chose atwo calculatingthe thenormalized normalized difference between theareas areasunder under thetwo twoTCP TCPcurv cur The functional closeness of the individual and the population TCP curves may be more rigorously est 20 individual and population TCP curves obtaine The individual and the population TCP value of D = 50 Gy for our investigation. calculating the normalized difference between the areas under the two TCP curves, calculating the normalized between the calculating the normalized difference between the areas under the two TCP curves, 50the calculating therange normalized difference between theareas areasunder under thetwo twoTCP TCPcurves, curves, calculating the normalized difference between the areas under the two TCP curves, [6] are evaluated for a given ofJ50difference and D50 values. Subsequently, the The functional closeness of the andFigure the population TCP curves be more rigorously est 20 curves were according to individual Eqs.the areas 1 two shows pairsmay of individual calculating the calculated normalized difference between under the TCPeight curves, [3a] and [6] for values of the parameters γ and population TCP curves calculated for of the indi ned for equal sets of J50 and D50 values are plotted for visual50comparison. 20 The functional closeness calculating the normalized difference between the areas under the two TCP curves, 36 Based and D reported by Okunieff et al. the following parameter values: D = 50 Gy s. [3a] and of the parameters Results 5 [6] for values 50 50 on their estimates of γ50, we chose a range and γ50 = [0.5, calculating 1, 1.5, 2, 2.5, 3, 4, 6]. This figthe normalized difference between dividual and the population TCP curves may be more rigorously estimated by ose a range of J 50 >0.5, 6@. These These authors authors also reported ure was reproduced for different values of The individual and the population TCP curves were calculated according to Eqs. [3a] and [6] for values of the a mean tumours investigated in D50, to determine whether this parameter 50 for en the areas under the D two TCPall curves, 36 with values ranging 10 to 90by Gy. We J50 andfrom D50 reported Okunieff et al. Based on their estimates of J50, we chose a range of J >0.5, 6@ . The

50 Radiol Oncol 2007; 41(2): 90-98.

also reported a mean D50 for all tumours investigated in their work 5 of 50 Gy, with values ranging from 10 to 9

lculated for the following parameter therefore chose a valuevalues: of D50 = 50 Gy for our investigation.

for all tumours investigated ininvestigation. their workfrom of 5010Gy, withGy. values lso reported a mean D therefore chose value of Dof50 50 = 50 Gywith for our umours investigated in50atheir work Gy, values ranging to 90 We ranging from 10 to 90 Gy. We chose a value of D50 = 50 Gy for our investigation. therefore chose a value of D50 = 50 Gytherefore for our investigation. stigated in their work of 50 Gy, with values ranging from 10 to 90 Gy. We herefore a value of D50 = 50 Gy for our investigation. 0 Gy for chose our investigation. Figure 1 shows eight Schinkel pairs of individual and population TCP curves calculated for the following parameter va 10 al. / Functional comparison of TCP models 94 Figure Figure 1 shows eight pairs of individual TCP curves cal 1 shows eight 10 pairs ofetindividual and form population TCP curves calculatedand for population the following parameter va 10 nvestigation. high dose range. The individual and population models d Figure 1and shows eight ofcurves individual and 3, population TCP curves for for the following parameter rs of individual for the This following parameter values: D50 = 50 Gypopulation and J50 pairs = TCP [0.5, 1, 1.5,calculated 2, 2.5, 4, 6]. figure wascalculated reproduced different values of Dvalues: 50, to deter 50 4,Gy6].and J50figure = [0.5,was 1, 1.5, 2, 2.5, 3, 6]. Thisvalues figure of was D50 = 50 Gy and J50 = [0.5, 1, 1.5, 2,D2.5, This reproduced for4,different D50reproduce , to deter 50 = 3, al and population calculated for the following parameter The values: had TCP any curves influence on functional equivaconsiderable closeness in functional D5, Gy J50This =parameter [0.5, 1, 1.5, 2.5, 3, 4, 6]. figure was reproduced forfound different of D502.5 , to determine 2, 50 2.5, 3, and 4, 6]. figure was2, reproduced for different values of from D50, to determine 50 = whether this had any influence onThis functional equivalency. It was that the location of the TCP curves J50values than the seen 1, on forthat lency. It was found that the location of the form of both models explains the observation whether this parameter had any influence functional equivalency. Itindividual was founa whether this parameter had any influence on functional equivalency. ItFigure was found theless location of the TCP curves 4, 6]. This figure was reproduced fordose-axis different values ofinD50, to that determine TCP curves along the did not the individual TCP model produces a reawhether parameter hadnot anyinfluence influence onfound functional equivalency. It found that the location of4,10the Hence, TCP curves along show the functional dose-axis did the shapes of the the curves oroftheir positions relative to each other. the results fluencethis on equivalency. It was that location thewas TCP curves along the population-based TCPpositions curves. For n the dose-axis did not influence the ofdatasets. curves or their relative the dose-axis influence the shapes oforthetheir curves orcompared their positions relative tothe each other. the results show fluence did the not shapes of the curves sonable fitwith to shapes clinical InHence, spite of nctional equivalency. Itrelative was found that the location of thethe TCP curves along positions to each other. Hence, this, the observed equivalence in functional dose-axis did1curves not influence ofvalue. the curves theirHence, positions each in other. Hence, the results shown in figure are applicable forshapes any Drelative eheshapes of the or their the positions to eachorother. therelative results to shown 50 figure 1 are for applicable for of any D50two value. figure 1 are applicable any D150are value. slightly underread the population TCP. not The be overreading an results shown infor Figure applicable form the TCP models should e curves or their positions relative to each other. Hence, the results shown in gure 1 are applicable for any D value. any D value. regarded as an endorsement to use the indivalue. 50 50 'A vidual model to fitdifference clinical data. 'ATCP 15 [8]) is plotted in Fig. 2. The largest area between the two TCP curv Thequantity quantity 'A J 50 (Eq. The considerable closeness in 2. functional form 5 in The (Eq. [8]) [8]) is isThe plot[8])difference is plotted in Fig. The largest areao quantity 15 J 50 (Eq. plotted Fig.However, 2. TheJ largest area between the two TCP curv The quantity ATCP 15 50 a(Eq. very steep dose response is pop A A 'A TCPpop TCPpop 4,10 (Eq. [8]) area is plotted in Fig. 2. The largest areaclinical difference the two TCP curves is Theisquantity (Eq. [8]) ted in Fig. 2. 2.JThe largest area difference unusual for data sets.fitShallower replotted in Fig. The largest difference between the two TCP curves is between 50 model produces a reasonable to clinical datasets. In ATCPpop between the two TCP curves is -17.7% obsponses are much more typical for popula-17.7% obtained at J = 0.5. 50area difference between the two TCP curves is plotted in Fig. 2. The largest -17.7% obtained at J50 = 0.5. -17.7% obtained at J50 = 0.5.

tained at γ50 = 0.5.

tionsTCP of patients. Therefore, it would as contwo models should not be regarded an endorsemen ceptually be more correct to use the population TCP model, which accounts for interHowever, a very steep dose response is unusual Discussion patient heterogeneity to fit such data. If, Discussion however, the individual TCP model is used, Discussion Discussion for populations of patients. Therefore, it would concep Based on Figures 1(d) – 1(h) and Figure 2, one should bear in mind that the obtained Discussion Discussion Based onmay Figs.conclude 1(d) – 1(h) and Figure 2, one forms may10 conclude that the functional forms oftheir the individual andsuch the popul parameter values have lostheterogeneity biological one that the functional for inter-patient fit data. Figs. 1(d) – accounts 1(h) Figure 2, one may thattothe functional Based on Figs. 1(d) – 1(h) and FigureBased 2, oneonmay conclude that and the functional forms of conclude the individual and the popul Discussion meaning and should be interpreted simply the individual and the2,population modelsthat the functional forms of the individual and the population Based on one Figs.of 1(d) – 1(h) and one may conclude gure may conclude thatFigure the functional of the individual and of theJ population asmind phenomenological coefficients. models are almost identical for J 50 >2forms 20 2, , 6@ . Indeed, for this range the index 'A parameter ATCPpop is less thanhave 0.5%.lost Alth that obtained values 50the > models are almost identical for 20 J50 2index , 6@from . Indeed, for this range of J0.5%. inde > @ models are almost identical for 20 J 2 , 6 Indeed, for this range of J 'A Figures ATCP is1(a) less than are almost identical for . Indeed, 50 the Alth 50the 50 As can be seen and pop may conclude that the functional forms of the individual and the population models are almost identical forofJ 50of J>502the ,the 6@ .index Indeed,'for range of Jthan the0.5%. index 'A start ATCP is less in than 0.5%. Although 1(b), models to differ functionfor this range index A Athis Although for this range γ50 50 both coefficients. 50 >2, 6@ . Indeed, TCPpop is less phenomenological pop > @ > ' A A is higher ( ' A A 0 . 5 , 6 . 7 % ) for the interval J 1 , 2 , the plots in Figures 1(b) 1(c) ind al form for the clinically observable range of and TCP 50 TCP deed, for ' this the index A popATCP less pop ofthan less 0.5%. @% ) in >1, 2ind , higher ( 'A ATCPJpop >1, 02. 5,, the 6.7plots forFigures the interval J 1(c) @% is) 0.5%. A is Arange isJ50 higher ( 'AAlthough A'TCP >pop ' 0.Ais 5, A 6.7popthan for theAlthough interval 1(b) and TCP 50 TCPpop pop γ50 < 1. In addition, for these values of γ50, 50 As can be seen from Figures 1(a) and 1(b), forthe the >1, 21(c) , theindicate ' A>ATCP 'A interval ATCP J>500 .5>,1, 26 .7, @the % ) plots for interval J 50individual and plots in Figures 1(b)>and 1(c) indicate b 0.5pop , 6is.higher 7@higher % ) for( the in Figuresthe 1(b) leads to TCP 0forfor that the individual and poppopulation TCP curves are still sufficiently close tomodel each other, especially theDclinically-rel the individual and population curves areespecially still sufficiently to each individual curves are still closeTCP to fits each for the close clinically-rel = 0. Therefore, toother, very shallow curves interval , the plotsTCP Figures 1(b) >1, 2 population 7@% ) forthat thethe interval J 50 and the plots ininthat Figures 1(b) and sufficiently 1(c) indicate observable range of J50 < 1. In addition, for these values o using the individual model may distort the hat thecurves individual and population are still especially sufficiently to each other, especially for the clinically-relevant TCP are still closecurves to each other, the clinically-relevant and 1(c)sufficiently indicate TCP that the individual and forclose best-fit estimates of γ50 and D50. are still sufficiently close TCP to each other,are especially for the 15 clinically-relevant fits to very shallow curves using the individual model may population curves still sufficiently The authors advocate the use of the population model in regards to clinical data. close to each other, especially for the clini6 6 However, the demonstrated equivalence in The authors advocate the use of the population cally-relevant high dose range. The individ6 functional form of the individual and popuual and population models differ considerequivalence in functional form for of the lation models can be utilized theindividual case of and popu heterogeneous tumour irradiation. In this differ considerably ably atat J50 = 0.5 ( 'A ATCPpop 17.7% ).. As Ascan be irradiation. In this case, themodel individual TCP model with e case, the individual TCP with existcan be seen from Figure 1, for γ50 less than ing {γ50, D50} estimates (e.g. Okunieff et al.36) l curves overread TCPindividual everywherecurves except overread at 50% control 37 2.5 the TCP whenthe of the TCPevaluation accordingoftoTCP the 37 following express canevaluation be used for ac38 cording to the following expression: everywhere except at 50% control when normalized slopes above J = 2.5, the individual curves tend to

17.7% obtained at J50 = 0.5.

50

compared with the population-based TCP and underreading tendencies are clearly demonstrated 20 curves. For normalized slopes abovebyγ Figure = 2. [9][9] 50

ª 2J

TCP

0.5

§

D ·º

¦ vi exp «« ln502 ¨¨© 1 D50i ¸¸¹ »» i

¬

¼

Equation [9] is a simple, straightforward

2.5, the individual curves tend to slightly

of both models explains the observation that the individual TCPgeneralization of Eq. [3] for the case of hetEquation [9] is a simple, straightforward generalizatio underread the population TCP. The over-

erogeneous irradiation. The generalization generalization of Eq. the case of heterogeneous ir of Eq. [6] for the case[6] of for heterogeneous irradiation, without introducing extra model pacomplicated mathematical problem, and has not yet been s

n spite of this, the observed equivalence in tendencies functional form reading and underreading are of the

demonstrated nt to use the clearly individual TCP model toby fitFigure clinical2.data. Radiol Oncol 2007; 41(2): 90-98.

l for clinical data sets. Shallower responses are much more typical

eptually be more correct to use the population TCP model, which

95

Schinkel et al. / Functional form comparison of TCP models Figures

1

1

(a)

J50 = 0.5

J50 = 1.0

0.5

0.5

0 1

20

40

0

60

1

(c)

20

40

60

40

60

40

60

40

60

(d)

0

60

1

(e)

J50 = 2.5

20

(f)

J50 = 3.0

0.5

0.5

2 40

60

(g)

40

(% )

0.5

Dose (Gy)

1

-2

J50 = 4.0

20

0

-6 -8

20

0 (h)

0.5

-4 60

2

-2J50 = 6.0

(% )

20

0

T C P po p

0

0

40

0.5

0

1

20

J50 = 2.0

0.5

T C P po p

TCP

J50 = 1.5

1

mean dose (GMD), as is usually done.39,40 Unfortunately, this approach introduces a third model parameter, and knowledge of its value for each tumour type would then be needed in order to use this model to calculate TCP for a heterogeneously irradiated tumour. Therefore, until more comprehensive parameter estimates are produced through fits of the population TCP model to clinical data for the case of heterogeneous irradiation, we propose that Eq. [9] be used for evaluation of treatment plans in terms of TCP, based on the functional form equivalency of both models.

(b)

-4 0

-6 -8

20

Dose (Gy)

Figure 1. Individual (solid) and population-averaged (dotted) TCP curves for D = 50 Gy and the g50 values shown in each sub-plot.

-10 -12

$ A/ A

-10

2 0

-12

-2

T C P po p

(% )

rameters, presents a complicated mathemat-14 -4 -14 ical problem, and has not yet been solved. -6 12 -16 Strictly speaking, the ability to-16 use Eq. -8 [9] as a population TCP descriptor has not -10 -18 Nevertheless, yet been proven theoretically. -12 -18 1 0 our experience with the TCP/NTCP estima0 1 2 2 3 3 4 4 5 5 6 -14 G tion module37 shows that it produces reaG 50 -16 50 sonable TCP estimates. -18 0 1 2 3 4 5 6 Another approach to the problem of G 50 taking dose heterogeneity into account for Figure 2. The ratio of the area difference, the population TCP model is to replace the Figure 2. The ratio of the area difference, 'A ATCPpop ATCPind ,, between betweenthe thetwo two TCP curves, to the to TCP curves, Figure 2. The of the Figure area2.difference, 'A A A A AATCPind , between the two TCP curve homogeneous dose, D, with theratio equivalent The ratio of the area difference, 'TCPpop , between the two TCP curves, to the total area to the total area under the population TCP curve plotted for (the values totogenerate theshown curves underEUD. the population TCP curve ( ATCPpop A ), plotted forof the J values of J used generate the curves in Figureshow under the), population TCP curve 50 used uniform dose, It may then be assumed used to generate the cu under the population TCP 1.curve ( ATCPpop ),, plotted plotted for forthe thevalues valuesofofJ50γ50 used to 1. that the EUD is equal to the generalized generate the curves shown in Figure 1. 1. $ A/ A

-plot.

$ A/ A

ure 1. Individual (solid) and population-averaged (dotted) TCP curves for D50 = 50 Gy and the 50 J50 values shown in each

TCPpop

TCPpop

TCPind

50

Radiol Oncol 2007; 41(2): 90-98.

96


Conclusions It is thus concluded that: •T he population and the individual TCP responses are almost identical in functional form for γ50 belonging to the interval [1, 6]. If each of these models were fit to the same clinical dataset, they would produce statistically indistinguishable values of the parameters D50 and γ50. • I t is conceptually incorrect to use the individual TCP model to fit clinical data. • Until reliable estimates of the population TCP parameters for the case of heterogeneous tumour irradiation are obtained, the individual TCP model (Eq. [9]) with existing D50 and γ50 estimates could be used for TCP evaluations in this situation. • The case of a shallow dose-response relationship, which is usually observed clinically, can be explained by the presence of significant inter-patient heterogeneity. The population TCP model should be used to fit such data, as it accounts for this heterogeneity. If, however, the individual TCP model is used, the estimated parameter values should be interpreted simply as phenomenological coefficients. • A steep dose-response relationship indicates the presence of a relatively small interpatient heterogeneity. Though it is highly improbable to observe such dose-responses clinically, the individual TCP model may be applied to such data for the purpose of estimating biological parameters, as the individual parameters would retain some biological meaning in this case.

Cancer program (the Canadian Institutes for Health Research), as well as the Alberta Cancer Board Research Initiative Program Grant RI-218.

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This research was supported by studentships from the Alberta Foundation for Medical Research, the Alberta Cancer Board and the Translational Research Training in

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