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 ?50 and D50, we show that they have almost identical functional form for values of ?50 ? 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 ?f – 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 ?50 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 describ-Received 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: ginofall@cancerboard.ab.ca,
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 dose-response 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 radiobio-logical parameters take on unrealistically low values.6 Nevertheless, although it is conceptually incorrect, the individual TCP model has been fit to clinical datasets and
Schinkel et al. / Functional form comparison of TCP models
91

parameters obtained from these fits have been assumed to have radiobiologically meaningful values.4,7-10 On the other hand, it has also been shown that these fits are characterized by an acceptable goodness of fit.
It has been expected that the population TCP models would allow for the estimation of biologically meaningful population parameters. Unfortunately, it is impossible to obtain a unique set of parameter values when a population TCP model is fit to clinical data.6,11 This is due to the fact that different sets of population parameter values produce almost identical TCP curves. Carlone et al.11 analytically demonstrated that when the dominant source of inter-patient heterogeneity is that of tumour radiose nsitivity, tl h s .e po pul atioe n r TCn P d fua n rlco - n tion has only two independent parameters - tc h lie da o llse at 5 0o % ne TCP, D th50, e w inhicr h re l d at eters -mines the position of the TCP curve, and the nor malized s lo pD e 5 o an f th 5 e 0. curve, y50. These parameters have geometric meaning. Since it is also true th at the individual TCP model may be expressed in terms of the same two parameters,3,12 it is possible that, for a given range of parameter values, bot h m o dels will exh i bit al most identical functional form. In this work, we investigate the similarities between these two models expressed in terms of D50 and m 50 by l p onlottin g theu m la for i1 d 0, e 11 n ,1 t 3 i -2 c 6a l valuea s s of these geometric parameters.
Background and method
The general form of the population-based Poisson TCP model has eight parameters. However, it has previously been shown6,11 that the parameters of such a model are interrelated; many different combinations of parameters lead to one and the same TCP curve. Thus, it may seem difficult to directly compare the functional forms of the individual and population-based TCP models. On the other hand, Carlone et al.11 have specified (based on a certain approximation, of course, but a clinically valid one) what these interrelations actually are, and have shoa w sen that there s a . re o t n hely twe o r independent population model parameters - D50 and y50. Fortun a tely, the individual Pa o l . o 11ison-basec d if Te C d P b m as o ed ul er can also be pax - i rameterized by these parameters. This fact a m llyakes the cov m e parison o atf t b h o erth a m re o odn e l l ys am n easier task.
The Poisson-based ndividual TCP mod l
task. model
al TCP mo the case w
This common form of t- h bae in d inividual l T T C C P P model is based on Poisson statistics combined with a s implified de scr it p h eti on n d of clonogen repopulation.4,10,11,13-26 In the case where a tuo m no o gur u rendergl o ates n h . o 4,1 m 0,1 o 1,g3 e -2 n 6e - I ous irradiation to a total dose D, split into n fractions with ec q ti u on a sl do s e per fraction, d, the individual Poisson TCP model may be written as:11
[[11]]           TCPind =e-NS= exp[- N0 e*D" ] =
r
exp
- a+ßd-— \D
N exp the i0 exp l num D be ,
where N0 is the initial number of clonogens, NS is the mean nl u in m eaber of d c raloi n c ogens survivinn s g it the treatment, e and n are the linear quadratic (LQ) radiosensitivity parameters, ng is the tumour repopulation rate, T is the total treatment time and be = of c T o n no . Note that as long as an equal dose is given during each fraction of the treatment (which is common clinical practice), the parameters a, sen and ity can be combined into one single param rameter:
param
treatme be co

practice), the
be combined i
Radiol Oncol 2007; 41(2): 90-98.


The validity of the Poisson TCP model was
1

92
, the
Schinkel et al. / Functional form comparison of TCP models




[2]       TC' = a + 0. d-------
d
The validity of the Poisson TCP model was questioned by Tucker and Travis,21 and others27" 31 who explored t me a e no s ds os -Poi T eC D , P on natu r e of the TCP in the case wher Te r t um21ur repopula ric n occurs. Un o Pr certa li n conditions, asowever, it has b een shown 27'32 that the dis tribution of the numb llm r of et al . og use ce th s re maxim ining et the f nd of a tr nfl atm c cut is well-appr TC ximat ve by t La ho P wn i 27,32n dis tetib dis tion. In o ie th of these res clo lts , genn c also be ai ause of t o e relative complexit simila the Eq on-Pois but a i bun TCP model B , t h ze i n an ivid u al lls re funct i o nco pre sten te d in esen . [ in ] is often used.
A form o f the i butio iv i In al T w P f th od s l 3 1 12 ul ha yt is d   l u o vf ol ecau t e Eq . [ e ], b ativ writi nen im s t e of t s of t he geometric parameters, c t i and D50, is g iven by:
nayte dr ,o Ksea pllominatn a[3] a]   TCPin
t al. 0.5
T a] e no ti C P of normaliz ive n b l ope, y, was fir st introduced by Brahme33 for the purpose of d p[ 3a br imet T ic pr e D a n sion quantification. Later, Kallman et a/.34 used the maximum v alue of y at the inflection point of the TCP curve and deriv e d an expr e ssion similar to Eq. [3a], but as poi om ted qs. [ t 1] and [3 tz ], th a f o d D l o w i k g r, 35 a slig hip t inco we sis t e he t y i o dif r erent s i ets their for mete ula. In gener al, the Pind oiss f n fCf e Dression given by Eq. [1], may be transformed and parameter-ized in terms of the normalized slope Y/at any dose point Df.
[3b]    TCPind=f

From Eqs. [1 Df and [3bL the following relationships between the two different sets of pa-ramet ers ty, Df) and (N f0, a') may be derived:
dearive [34b[4] a]
[4b]
D f —
a'     f l nf)
D
[4 b]    yf=-f\nfl
nf:
N
i cu lna N r f :
relationships between the two different sets of parameters
and for (jc
) in p 50, D50)
[a5nad] for [4a[]5a]
[5a]
a[5nbda ] f or [5b]
Dt
rD 50 D
-In
in particular:
In 2
ln 2
ln
Nl In 2





The population-based TCP model
Carlone et alo n- showed that the population TCP model for the case of dominant heterogeneity in radiosensitivit y ma y be w ritten as:
haer] l poo ][6]
TC50P1
sl o1ne
erfc
p D o
xr50
Radiol Oncol 2007; 41(2): 90-98.
50
D

geom11e al.
of t


The parameters in Eq. [6] - D50 and y50 - have the same geometric meaning as the cor-
responding parameters in Eq. [3a]. The geometric parameters may be expressed in terms of the popu lation-based radiobiological parameters, a , a' and lrN:«
[7a]    D
r+inN
[7 b]
50
50



V2^'(T
T'                                                  2
A                        A2             2          2        2        Ci
Here  a' = a+ßd + — and (cr')2 = cra2 + d2a„2 +—^-   where a, ß, X' and kiN0 are the d d                                               dd
population averages of the corresponding indivddual parameters and oa, aß, ax aa n dd crlnjN 0
are their standard  dviations. The symbo l T represents Eul er's gamma constNnt , which has
an approximate value of 0.577.
The general form of the Carlone et al.11 population TCP model takes both heterogeneity
in radiosensitivity and heterogeneit y in donogen number int o account. However, this form
of the population TCP model has three parameters, and was shown11 to be almost identical
to the one that only takes heterogeneity in radiosensitivity into account.  Hence, the latter
will be used for this analysis.
F unctio n al form c o m paris o n be t w e e n in d ivid u a l an d p o pula t i o n-b ase d T C P m o d e ls
Since both the in div idual and the population TCP models may be written in terms of the same two parameters , y50 and D50, it seems natural to assume that the two models may display similarity in functional form. In order to explore the functional s im ilarity of these models, Eqs. [3a] and [6] are evaluated for a given range of e t 50 and D50 values. Subsequently, the individual an d p c co pul a ir tio o o n b TCP eene urves o r m bt aine nd d p f or e qual n nb-- et i n of TC 50 mo ododed D50 val aues - b re plottF ud foc to ti ion visual comparison .
The functional closenesb of the individual and the popula t io n TCP c urve s may b e more rigorousl y estima t md by calcul ating thi v i norma nli z e d differ ence between o dhe are a s under the two T CP curv es,
AA   /    \    \rrT CPpop ~ A Tcpmn d )
[8 ]     A T—v*)=—-,---------,
1 K~,rpop                                    A TC P pop
as a function of y50.
Results                                       their work of 50 Gy, with values ranging
from 10 to 90 Gy. We therefor e chose a The individual and the populaDion TCP       value of D 5 0 = 50 Gy fo r our inv estiga tion. curves were calculated according to Eqs.              Figur e 1 sho ws eight pairs of individual
[3a] and [6] for values of the paramet ers y5 0       and population TCP curves calculated for and D50 rep ort ed by Okunieff et al?6 Based       the foRlowing parameter values: D50 = 50 Gy
on their estim ates of y50, we cho se a r ange a nd y50 = [0.5, 1, 1.5, 2, 2 .5, 3, 4, 6]. This fig-of he 50 e [0.5,6 al . These au opula r s also reported ure was reproduced for different values of a me ind D50 for all tumours investigated in       D50, to determine whether this parameter
, we cho  e a range of9n50H
Radiol Oncol 2007; 41(2): 905-908.
also reported a mean D50 for all tumours investigated in their work of 50 Gy, with values ranging from 10 to 9
5
lculated for thteheforellfowrei ncghopsaer am veatleure voafl uDe5s0: = 50 Gy for our investigation.
tigation.          50
10 to 90 Gy. We
10 to 90 Gy. We
94
shows shows
pairs of individual and population TCP curves cal 1Spca0hiirnsk eolf eitn adl.i /v iFduunFactlii goaunnar del fp1oro msph ucool wamtspio aenrisgTohnCt oPpf aTciCur sPr v omef so i dnceadlsli
rs of
figu e curves or
a h t a io d n any influs en ca c le u o la n te func tional eqi u niv ap - a l guncy. It w ep s fou nd that th fe locat ion o f t h e TCP cu r ves along th e dose-axis did not in-flu lenc e the shap un s of the curve on o r the T i r positions t w l as tive to each he ther. Hence, the h re eisulto s s show n in Figure 1 are applicable for any D50 valu re .
M
foo       .       s pl o 5 -

Th e q u a n t ity r~
ATCPpop
te d i n F ig. 2. ATCpop rg est a r e a d iffer ence betweeA th e tw o TCP curves is -17.7% ob-tained at y50 = 0.5.
Discussion
Based on Figures 1(d) - 1(h) and Figure 2, on e m ay c o scu lud e th at th e functiona l fo rm 1 0 of the individual and the population models are almo fu t ide ntical for y50 he 2, ndi . idual and , for t his ran g e o f A 5 0 the index \AA/ATCPo iAA s ra T CC n l PP e g p s o e p s f th a 50 n t 0.5 i n % d. AltTT h CAC PP o p u oAp g T C h P p \AA is/A l eTs C s P p is h ig h e r (MAATCPpop e[-0.5,-6.7]%) fo r the i i n n tterva l l y50 e[1, 2), the pl l o o t sts in F F i gigures 1 1( b) and 1(c) indicate that the individual and population TCP curves a re stc il i l a sufficien tly close to each other, especially for the clinically-relevant high dose range. The individual and population models differ considerably a t y50 = 0.5 (\mAAtcp\=17.7%). A s c can be seen from Figure 1, for y50 less than 2.r 5 e t he indiviy d wual c urves ov e5 r 0 r %ea d Tt C ro P l everywhere except at 50% control when compared with the population-based TCP curves. For normalized slopen s s abov e y50 = 2.5, the individual curves tend to slightly underread the population TCP. The over-reading and underreading tendencies are clearly demonstrated by Figure 2.

differ cons
l curves o normalized and underre
of both m n spite of t nt to use th
Tha e lu e c s o : n siderable closeness in functional f see rm r m f om Figure els f x or lainl t ess tha serva ti the t c om at th r e in dividual TCP model produces a rea-u r v e nable fit to it clih n eica l o d p a uta setn s -.4,10 e I n spC ite o cf this, a t l h o e n o bserved equivalence in functional form of the two TCP models should not be regarded as an endorsement to use the individual A TC TP e m c o o d nel td o e fi t b c l l ein cical e d n ats a s.
H Aowever, a very steep dose response is um n oud s e u lal f o o d r u cl e i s n aical a d s a o tn a a b s l et s. Sha c l l l o i n w i cer re d - a spou n rvses i a s re much more typical for populatiw o ons of patient s. Ther efo re, it would co a n n -ceptually be more correct to use the population TCP model, which accounts for inter-patient heterogeneity to fit such data. If, D h isowe ver, tha e t i i o nd ividual TCP model is u e u s , s s i e t o d n w, one should bear in mind that the obtained pa rame ter valun t s h p a ve lost e t e eir biological meaning and should be interpreted simply a s m pi h n e d n o th m a e t n t o hle o g oib cal cn o e d e AA f f piciea P n mt e s.
Ausla tcio an be seen from FigC u Pp r o e ps 1(a) and 1(bp) h, e b n otm h e m no odels stA a T c C r o Pt tf p o fic d i i e f n f e ts r in function -al form fo r the clinically o bservable r a ng e o f y50 < 1. In addition, for these values of 1(a 50, t he in divid s ca l m o del lea fro to TCP ures f or D (= c) 0 in . d T ic h aerefore, fits to very shallow curves obs ing th ble in divid ual m odel m additi stort the best-fit estimates of ur 50 and D50.
The authors advocate the use of the population model in regards to clinical data. However, the demonstrated e eq thuiv alence in functional form of the individual and population models can c b tie utilized for the case of heterogeneous tumour irradiation. In this case, the i ndividual Te C , tP model with exis ting { e v 50, D50} estimates (e.g. Okunieff et al.36) c an be usei d on f or the ev aluation of T t C h eP 37 aco - w cording to the following expression:38
[9]      TCP = 0.5i      l
Di
uarsveets .i4s,10 In ndorsemen
is unusual uld concep
loinsiecatollye-arcehl ese values o elevant
model may
6 population
l and popu
odel with e
ing express
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
Equation [9] is a simple, straightforward PEg eqnueartiaolniz a[t9io] ni so f aE qs .i m[ 3p] l ef o, rs thraei gchatsfeo r owf a hr de t-gener hee rogeneous irradiation. The generalization go ef nEeqr a. l[i6z]a tfioorn t ho ef Ec aqs. e[ 6o ]f fhoert ethroeg ceanseo oufs hirertae-rogene diation, without introducing extra model pa-complicated mathematical problem, and has not yet been s
ir

an

Schinkel et al. / Functional form comparison of TCP models
95

ure 1. I -plot.
(a)
Y50
0.5
(b)
Y50
1.0
0.5
0.5
20
40
60
20
40
60
(c)
Y50
1.5
(d)
Y50
2.0
0.5
0.5
20
40
60
20
40
60
(e)
Y50
2.5
(f)
3.0
0.5
0.5
20
40
(g)
Y50
4.0
0.5
60
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 heterogene-ously irradiated tumour. Therefore, until more comprehensive parameter estimates are produced through fits of the population
20
40
60
TCP model to clinical data for the case of het-
;
(h)
/50 = 6.0
0.5
40
20
Dose (Gy)
/ -----------60------

0 '—
erogeneous irradiation, we propose that Eq. [9]
be used for evaluation of treatment plans in
20
Dose (Gy)
40
60
terms  of TCP, based on the functional form
equivalency   of   both models.
Figure 1. Individual (solid) and population-averaged (dotted) TCP curves for D50 = 50 Gy and the y50 values shown in each sub-plot.

rameters, presents a complicated mathemat-
ical problem, and has not yet been solved. Strictly speaking, the ability to use Eq.
[9] as a population TCP descriptor has not yet been proven theoretically. Nevertheless,
our experience with the TCP/NTCP estima tion module37 shows that it produces reasonable TCP estimates.
Another approach to the problem of taking dose heterogeneity into account for the population TCP model is to replace the homogeneous dose, D, with the equivalena uniform dose, EUD. It der thy theopula e assumP c that the EUD is equal to the generalized
0                  12                  3                  4                  5                  6
Figure    2.    The    ratio    of    the    area    difference,
ifferen ATCPpop — ATCPjnd, between the two TCP curves, to the total lo ta under thv a l u ops oatietw TCe d to g ve n ( ATCPpop ),  plotted  for  the  value s f of y50  us ed  to generate the curves shown in Figure 1.
0
0
0
0

0
0

0
Radiol Oncol 2007; 41(2): 90-98.
96
Schinkel et al. / Functional form comparison of TCP models
Conclusions
It is thus concluded that:
• The population and the individual TCP responses are almost identical in functional form for ?50 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 ?50.
• It 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 ?50 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.
Acknowledgements
This research was supported by studentships from the Alberta Foundation for Medical Research, the Alberta Cancer Board and the Translational Research Training in
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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