Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol169.Issue1.2001:

Iterated birth and death process as a model of radiation cell

survival

Leonid G. Hanin

*

Department of Mathematics, Idaho State University, Pocatello, ID 83209-8085, USA

Received 6 April 2000; received in revised form 8 September 2000; accepted 18 September 2000

Abstract

The iterated birth and death process is de®ned as ann-fold iteration of a stochastic process consisting of the combination of instantaneous random killing of individuals in a certain population with a given sur-vival probabilityswith a Markov birth and death process describing subsequent population dynamics. A long standing problem of computing the distribution of the number of clonogenic tumor cells surviving a fractionated radiation schedule consisting of n equal doses separated by equal time intervals sis solved within the framework of iterated birth and death processes. For any initial tumor sizei, an explicit formula for the distribution of the numberM of surviving clonogens at moment safter the end of treatment is found. It is shown that ifi ! 1ands ! 0 so thatisn _{tends to a ®nite positive limit, the distribution of} random variableMconverges to a probability distribution, and a formula for the latter is obtained. This result generalizes the classical theorem about the Poisson limit of a sequence of binomial distributions. The exact and limiting distributions are also found for the number of surviving clonogens immediately after the nth exposure. In this case, the limiting distribution turns out to be a Poisson distribution. Ó _{2001 Elsevier}

Science Inc. All rights reserved.

Keywords: Birth and death process; Branching process; Clonogenic tumor cell; Fractionated cancer radiotherapy; Limiting distribution; Probability distribution; Probability generating function; Tumor recurrence

1. Introduction

The purpose of this work is to solve, under natural biological assumptions, the following problem of major theoretical and practical importance in radiation oncology that was posed in [1] as early as in 1990: What is the distribution of the number of clonogenic tumor cells surviving

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E-mail address:hanin@isu.edu (L.G. Hanin).

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fractionated radiation?Lack of theoretical results leading to a closed-form analytic expression for this distribution, even under very simplistic models of cell kinetics between radiation exposures, was the most critical deterrent to evaluating and increasing the ecacy of existing clinical prac-tices of cancer treatment and to developing relevant methods of statistical data analysis.

We proceed from the following model of tumor population kinetics, see e.g. [2]. A tumor initially comprising a non-random number i of clonogenic cells is exposed to a fractionated ra-diation schedule consisting ofninstantaneously delivered equal dosesDseparated by equal time intervalss. It is assumed that every clonogen survives each exposure to the doseDwith the same probability sˆs…D†, given that it survived the previous exposures, and independently of other clonogens. Observe that it is not supposed that all cells forming a tumor are necessarily clono-genic. We also assume that irradiated tumor cells are killed instantaneously. According to a common radiobiological convention, a tumor cell is killed if it is incapable of producing a viable clone. Between the exposures, clonogens proliferate or die spontaneously independently of each other with constant ratesk>0 andmP0 (de®ned in the sense of Markov processes), respectively. Finally, it is assumed that cell death between radiation exposures is instantaneous, and that de-scendants of a clonogenic cell are clonogenic.

Similar models arise when a tumor cell population is exposed to chemical or immunological agents, as well as in problems of population dynamics in the presence of emigration and external or internal killing factors.

Suppose that radiation exposures occur at time moments 0;s;_{. . .};…nÿ1†s. Let M be the random number of clonogens (including the original clonogenic cells and their descendants) that are alive at time ns, that is, at time s after the nth exposure. Random variable (r.v.) M can be viewed as the ®nal state of a stochastic process de®ned as then-fold iteration of the combination of exposure to the doseDwith the homogeneous birth and death Markov process with the birth rate kand death rate m. Thisn-stage stochastic process will be referred to in what follows as the iterated birth and death process. Also, denote byLthe number of surviving clonogensimmediately afterdelivery of thenth fraction of radiation. Although r.v.Lis biologically more natural thanM, the latter is more tractable mathematically.

Alternatively, r.v.sM and L can be characterized in terms of the Galton±Watson branching process (see e.g. [3, Chapter 1], and [4, Chapter 2]). Denote by Zs the state at time s of the

ho-mogeneous birth and death Markov process with the birth rate k, death ratem, and initial pop-ulation size iˆ1 (that can be also viewed as a continuous time branching process with the probabilities of no and two o€spring equal to m=…k‡m† and k=…k‡m†, respectively). Then the distribution of r.v. M coincides with the distribution of the nth generation size in the Galton± Watson process with the initial population size i and the o€spring distribution de®ned as the distribution of a r.v. that is equal to Zs with probability s and to 0 with probability 1ÿs.

Sim-ilarly, r.v. L is equidistributed with the size of the …nÿ1†st generation in the Galton±Watson process where the initial population size follows the binomial distributionB…i;s†and the o€spring distribution isB…Zs;s†. In what follows, however, we will use the iterated birth and death process

approach because it is more straightforward and technically simple.

irradiation schedule is equal tosn_{. Hence,M ˆL, and these r.v.s follow the binomial distribution} B…i;sn_{†. The initial numberiof clonogenic cells in a tumor is typically very large (1 cm}3 _{of a solid} tumor contains about 109 _{cells [5]; also, a clinically detectable tumor is estimated to contain at} least 105 _{clonogenic cells probably ranging up to 10}9 _{cells or even more [6]). Furthermore, the} survival probabilitysn_{is small because the doseD, usually 1}

±2 Gy, and the number of fractionsn, normally 20±40, are chosen to achieve just that. Therefore, the distributionB…i;sn_{†is close to the}

Poisson distribution with parameter hˆisn_{. It is this Poisson approximation that is used for}

planning fractionated cancer radiotherapy in current clinical practice, see e.g. [2,7,8].

The validity of this Poisson approximation was ®rst questioned in [1]. The idea of this work was that due to cell proliferation the distribution of the number of surviving clonogens must deviate from the binomial distributionB…i;sn_{† which makes the approximating Poisson distributionP…h†}

not well justi®ed. Extensive numerical simulations conducted in [1] con®rmed that in the absence of cell death between radiation exposures (mˆ0) the mean and variance of r.v.Lare distinct from those predicted by the binomial distributionB…i;sn_{†. On this basis, the authors made a conclusion}

that the distribution of r.v.Lcannot be approximated by a Poisson distribution. In response to this work, it was suggested in [9] to use branching stochastic processes as an adequate tool for studying the distribution of r.v.L(for an extensive discussion of this idea, the reader is referred to [10, pp. 31±40]) but the model proposed in this work was too simplistic. Numerical algorithms for computing the extinction probabilityp_~0in the casemˆ0 are described in [8,11]. The work [11] also contains a more general algorithm for numerical evaluation of the probability generating function (p.g.f.) of r.v.L. A numerical procedure for computing the extinction probabilityp_~0 within a more elaborate model accounting not only for cell division but also for cell loss, cell di€erentiation, and cell cycle e€ects is presented in [6].

The paper [2] represents the ®rst attempt at theoretical approach to describing the distribution of the number of surviving clonogens. It contains explicit formulas forp0:ˆPr…Mˆ0† and the p.g.f. of r.v. M obtained within the framework of homogeneous birth process. The following problems are therefore of theoretical and practical interest.

Problem 1.Find explicit computationally feasible formulas for the distribution of the ®nal state

M of the n-fold iterated birth and death process and for its counterpart L.

Problem 2.Compute the limiting distributions of r.v.sM and Lwheni ! 1; s ! 0 andisn !

h; 0<h<1:

In the present communication, we solve Problems 1 and 2. For an at length discussion of biomedical and statistical implications of the results of this work, the reader is referred to [12]. A model based on our ®ndings was successfully applied to statistical analysis of data on prostate cancer post-treatment survival [13].

2. Probability generating function of the iterated birth and death process

In this section, we will compute p.g.f. of the ®nal stateMof then-fold iterated birth and death process with i initial cells. As a consequence, we will ®nd p.g.f. of the number L of surviving clonogens immediately after thenth exposure. Since p.g.f. is an indispensable tool for analysis of Markov processes with the state space Z‡ (see e.g. [14, pp. 184±192]), we ®rst recall some well-known facts about them.

For a non-negative integer valued r.v.X, denote byu_X its p.g.f. de®ned by

u_X…z†_:ˆE…zX† ˆX

1

mˆ0

Pr…X ˆm†zm; jzj61;

whereEstands for the expectation. The probabilities Pr…X ˆm†are related to the p.g.f. ofXby

Pr…X ˆm† ˆD

m_u X…0†

m! ; mP0: …1†

In particular, u_X…0† ˆPr…X ˆ0†. Also observe that u_X…1† ˆ1;

EX ˆu0_X…1†; …2†

and

VarX ˆu00_X…1† ‡u0_X…1† ÿ ‰u0_X…1†Š2; …3†

where Var denotes the variance. The expectation and variance ofXcan also be expressed through the logarithmic derivatives of u_X as follows:

EX ˆ …lnu_X†0…1† and VarX ˆ …lnu_X†00…1† ‡ …lnu_X†0…1†: …4†

For a functionuwhose range is contained in the domain, we denote byu…n†_{itsn-fold composition} with itself.

We start our calculation of p.g.f. u_M with the case iˆ1; nˆ1: Since each clonogenic cell survives irradiation in doseDwith the probabilitys, the number of survivors for a single clonogen exposed to the dose Dis a Bernoulli r.v. with p.g.f.

b…z† ˆ1ÿs‡sz: …5†

According to our assumptions, expansion of a surviving clonogenic cell is governed by a ho-mogeneous birth and death process with the birth ratekand death ratem. It was shown in [15] (see also [16, pp. 165±166]) that the state of such process at times, that is, the size Sat time sof the clone emerging from a single cell, has a generalized geometric distribution of the form

Pr…S ˆ0† ˆr; Pr…S ˆm† ˆ …1ÿr†…1ÿq†qmÿ1; mP1:

Parameters r and q, 06r; q<1; of this distribution are related to the birth and death rates through the formulas

rˆm…1ÿa†

kÿam and qˆ

k…1ÿa†

where it is denoted hereaftera_:ˆe…mÿk†s_{: P.g.f. of r.v. Sis given by}

u_S…z† ˆr‡X

1

mˆ1

…1ÿr†…1ÿq†qmÿ1zmˆrÿ …r‡qÿ1†z

1ÿqz : …7†

Observe thatkÿamˆ0 only in the critical case kˆm, that is, when aˆ1. In this case,

rˆqˆ ks

1‡ks and uS…z† ˆ

ks‡ …1ÿks†z

1‡ksÿksz ;

which coincides with the corresponding limits in (6) and (7). It follows from (2) and (7) that

ES ˆu0_S…1† ˆ1ÿr

1ÿqˆa

ÿ1_{: …8†}

The numberXof o€spring of a single irradiated clonogenic cell at timesafter exposure has p.g.f.

uˆ1ÿs‡su_S ˆbu_S, that is, in view of (5) and (7),

u…z† ˆ1ÿs‡srÿ …r‡qÿ1†z

1ÿqz ˆ

1ÿs…1ÿr† ÿ ‰qÿs…1ÿr†Šz

1ÿqz : …9†

Therefore, r.v.X follows a generalized geometric distribution. By (2) and (8) we have

l_:ˆEX ˆu0…1† ˆsu0_S…1† ˆs

aˆse …kÿm†s

: …10†

Observe that p.g.f. u of r.v. X given in (9) is a fractional linear function. Conversely, every fractional linear p.g.f. of the form …aÿbz†=…cÿdz† with c; d 6ˆ0; c6ˆd; emerges from a gen-eralized geometric distribution.

A standard argument about p.g.f. of a branching process (see e.g. [14, pp. 190±192]) implies that

u_M ˆu…n†i

: …11†

To compute the functionu…n† _{explicitly, we need a more convenient formula for the p.g.f. u} ex-pressed in (9) through parameterss…1ÿr† andq. In the generic casea6ˆ1 we set

x_:ˆ1ÿs…1ÿr†

q ˆ

kÿmaÿs…kÿm†

k…1ÿa† ; …12†

while in the critical caseaˆ1 we de®ne by continuityx_:ˆ …1ÿs‡ks†=…ks†:It is easy to see that

xis a ®xed point of the functionu: Since

1ÿxˆ…sÿa†…kÿm†

k…1ÿa† ; …13†

we have 0<x<1 for l>1; xˆ1 for lˆ1; and x>1 for 06l<1:Observe that

qˆ lÿ1

lÿx and s…1ÿr† ˆ

l…1ÿx†

lÿx :

This allows us to rewrite (9) in terms of parametersl and xas follows:

u…z† ˆx…lÿ1† ÿ …xlÿ1†z

An explicit expression for the function u…n† _{is given in formula (15) below. Its remarkable} feature is that the action of the semigroup _Z‡ on the function u written in form (14) by com-positions is simplyn7!ln_{; n2Z‡:An equivalent but more cumbersome formula can be found in}

[9, p. 9], and (with a proof) in [17, pp. 296±298], [18, p. 7], and [19, p. 57]. In the case mˆ0;

formula (15) can be found in a slightly di€erent form in [2].

Proposition 1. Suppose that l6ˆ1. Then, for every natural number n,

u…n†…z† ˆx…l

n_{ÿ1† ÿx…l}n_ÿ1†z

ln_{ÿxÿ …l}n_ÿ1†z : …15†

Proof. We will use induction inn. For nˆ1, Eq. (15) coincides with (14). Suppose it holds for some nP1: Then we have by (14) and (15)

u…n‡1†…z† ˆ …u…n†u†…z† ˆx…l

n_{ÿ1† ÿ …xl}n_ÿ1†u…z†

ln_{ÿxÿ …l}n_ÿ1†u…z† ˆ

AnÿBnz CnÿDnz

;

where after an elementary calculation we ®nd that

An ˆx…lÿx†…lnÿ1† ÿx…lÿ1†…xlnÿ1† ˆx…1ÿx†…ln‡1ÿ1†;

Bn ˆx…lÿ1†…lnÿ1† ÿ …xlÿ1†…xlnÿ1† ˆ …1ÿx†…xln‡1ÿ1†;

Cnˆ …lÿx†…lnÿx† ÿx…lÿ1†…lnÿ1† ˆ …1ÿx†…ln‡1ÿx†;

Dnˆ …lÿ1†…lnÿx† ÿ …xlÿ1†…lnÿ1† ˆ …1ÿx†…ln‡1ÿ1†:

Sincel6ˆ1;we also havex6ˆ1. Therefore, the above formulas forAn; Bn; Cn and Dnlead to the

expression foru…n‡1† _{required in (15). This completes the proof of Proposition 1.}

Remark 1. By passage to limit in (15) asl ! 1, we ®nd that forlˆ1

u…n†_{z† ˆ}nk…1ÿa† ÿ ‰nk…1ÿa† ÿa…kÿm†Šz

nk…1ÿa† ‡a…kÿm† ÿnk…1ÿa†z :

According to (11) and (15), p.g.f. of r.v.Mis given by

u_M…z† ˆ x…l

n_{ÿ1† ÿ …xl}n_ÿ1†z

ln_{ÿxÿ …l}n_ÿ1†z

i

: …16†

Thinking of the number of iterationsn as being ®xed, we will also writeu_M in the form

u_M…z† ˆ aÿbz

cÿdz

i

; …17†

where

a:ˆx…ln_{ÿ1†; b}

:ˆxln_{ÿ1; c}

:ˆln_{ÿx; d}

Using (17) and (18) we evaluate the extinction probability following result obtained in [2]:

p0 ˆ

From (16) and (19) we conclude that in the case iˆ1 r.v.s M and L follow generalized geo-metric distribution. In the case of continuous irradiation with an arbitrary dose rate, p.g.f. for the number of surviving clonogens at the end of treatment and the corresponding tumor control probability were computed within the framework of continuous time birth and death process in [20].

LetUbe anynon-negative integer valued r.v. with a p.g.f. of the form

DM :ˆadÿbcˆln…1ÿx† passage to appropriate limits in (26) and (29).

3. Distribution of the ®nal state of the iterated birth and death process

In this section, we are going to compute the distributionpm; mP0;of any non-negative integer

valued r.v.V with a p.g.f. of form (23). In particular, our argument applies to r.v.sMand L. We write the fractional linear function involved in (23) as follows:

aÿbz

Di€erentiating this equality mtimes and setting zˆ0 we obtain on the basis of (1) that

pmˆ

Proposition 4) that will allow us to ®nd in Section 4 limiting distributions of r.v.sMandLwhen

Then it follows from (31) that

pmˆ

From (32) we see readily that

P0…x† ˆ …1‡x†i: …34†

Indeed, functionsPmandQmdepend oni:However, since in this section iis assumed to be ®xed,

this dependence will not be shown notationally.

In the next statement we ®nd a simple recurrence relation for polynomialsPm:

Proposition 2.

Proof.According to (34) and (35), relation (38) holds for mˆ0. In view of (32)

Pm‡1…x† ˆ

Di€erentiating this equality we obtain

xmÿ1Pm‡1…x† ˆ 1

m‡1‰mx

mÿ1_P

m…x† ‡xmPm0…x†Š;

From Proposition 2 we derive the following result about the functionsQm; mP0:

Proposition 3. fQmg1_mˆ0 is a sequence of polynomials satisfying the following recurrence relation:

Qm‡1…x† ˆ 1

m‡1‰…m‡ix†Qm…x† ‡x…1‡x†Q 0

m…x†Š; mP0: …39†

Proof.Using (36) we express the recurrence relation (38) in terms of the functionsQmas follows:

…1‡x†iÿmÿ1Qm‡1…x† ˆ this implies that every functionQm; mP0; is a polynomial of degreem. Proposition 3 is proved.

Our next statement provides the required formula for the polynomialsQm. Fori>m;it o€ers a

signi®cant computational advantage over formula (32) for the polynomials Pm:

Proposition 4.

ukˆ

Therefore, we continue (43) to ®nd through a simple calculation that

vkˆ

This completes the proof of Proposition 4.

The following result solves Problem 1 formulated in Section 1.

Theorem 1. Let V be a non-negative integer valued r.v. the p.g.f. of which has form(23)with some natural number i (in particular, this is the case forV ˆM andV ˆL). Then for the distribution of V we have

Now invoking formulas (33), (36) and (45) we ®nd that

4. Limit theorem for the iterated birth and death process

The following theorem establishes the form of the limiting distribution of r.v.Masi ! 1and

s ! 0. ParametersnP1,s>0; k>0 andmP0 are assumed to be ®xed. Formˆ0; k ! 0, this theorem gives the classical Poisson limit of a sequence of binomial distributions.

Theorem 2. Suppose thati ! 1 ands ! 0 so that there exists a limit

h_:ˆlim isn; 0<h<1: …46†

Then the distribution of r.v. M converges to the probability distribution pˆ fpmg

1

mˆ0 given by

pmˆeÿAqÿmrm; mP0; …47†

where

qˆ kÿam

k…1ÿa†; …48†

Aˆh…qÿ1†

an_q ˆ

h…kÿm†

anÿ1_kÿam†; …49†

r0 ˆ1,

rmˆ

Xm

kˆ1

mÿ1

mÿk

_dk

k!; mP1; …50†

and

dˆh…qÿ1†

2

an_q ˆ

h…kÿm†2

kanÿ2_{1ÿa†…kÿam†}: …51†

Proof.It follows from (12) and (48) that under conditions of the theorem x ! q. Note that

qÿ1ˆa…kÿm†

k…1ÿa†>0;

whenceq>1. Recall formulas (10) and (18) to ®nd that

a ! ÿq; b ! ÿ1; c ! ÿq: …52†

Also, by (24), (46), (52) and (51)

iDM

bc !

h…qÿ1†2

an_q ˆd: …53†

Observe that according to (44)

pmˆp0

b a

m

Qm

DM bc

First, we compute the limit forp0 ˆ …a=c†i. Note that by (18)

compare with (49). Next, invoking Proposition 4, we write

Qm

Passage to limit in this equation with (53) taken into account yields

Qm

It remains to show thatp is a proper probability distribution, that is,

X1

wherej:ˆmÿk. Observe that the last sum represents the binomial series

X1

Therefore, using (51) we continue (58):

u_p…z† ˆeÿAX

Remark 3. For the iterated pure birth process (mˆ0) , the limiting distribution of r.v. M under conditions of Theorem 2 is given by (56) with

qˆ 1

Remark 4.In view of formulas (4) Eq. (59) leads to the following expressions for the expectation and variance of the limiting distribution pof r.v. M:

Epˆ h

The same results arise when we pass to limit in formulas (25) and (26) for the expectation and variance of r.v.M when i ! 1 and s ! 0 under condition (46).

By contrast to r.v.M, the limiting distribution of r.v.Lunder the same conditions is a Poisson distribution.

Proof.Proceeding from (21) we easily ®nd that

~

Next, it follows from (20), (27) and (46) that

sÿ1b~

with (62) taken into account leads together with (61) to the desired conclusion (60).

Remark 5. It follows from (28) and (29) that under conditions of Theorem 2, EL and VarL

5. Numerical example

Theorems 2 and 3 lead to the following natural question: What are the rates of convergence of the distributions of r.v.s M and L to the limiting distributions(47) and(60), respectively? Sample calculations described below were designed to shed some light on this problem for r.v.L. A more thorough theoretical and numerical investigation will be a matter of another publication.

It follows from Theorem 3 that the distributionPLof r.v.Lcan be approximated by the Poisson

distribution P…h~† with

~

h_:ˆisn=anÿ1_{: …63†}

These two distributions were compared graphically and by means of computing the probability metric

d…PL;P…h~††:ˆsup mP0

Pr…L

ˆm† ÿeÿh~h~

m

m!

…64†

in the case when no cell death occurs between exposures (mˆ0) and for the number of fractions

nˆ30. The common expected valueh~of the two distributions was taken to be equal to 2; in other words, it was assumed that on the average two clonogens are alive after delivery of the last fraction of radiation.

The distributions PL and P…h~† were graphed for aˆ0:9 and iˆ105;107. The corresponding

valuesof the survival probability per fraction was computed from (63). The exact distributions of r.v. L for iˆ105_;107 _{and the Poisson approximationP…h}~_{† are represented on Fig. 1 by lines 1} (solid), 2 (dotted) and 3 (dashed), respectively. For easier visualization, all three discrete distri-butions were smoothed. Fig. 1 shows that, for large i, the Poisson distribution P…h~† provides a good ®t to the distribution PL.

The distance (64) was computed for aˆ0:7;0:8;0:9;0:95 andiˆ10N _{with N ˆ5;6;7;8.}

Re-sults of this calculation given in Table 1 con®rm that, for largei, the Poisson distributionP…h~†is close to PL. Furthermore, we conclude that the distance (64) monotonically tends to 0 when i ! 1for ®xedaand monotonically decreases asa ! 1 for ®xed i. Observe that in the limiting case aˆ1 …kˆ0†, that is, in the absence of cell proliferation and spontaneous cell death, (64) represents the distance between the binomial distributionsB…i;sn_{†and its Poisson approximation} P…h†; hˆisn_{. According to the Law of Rare Events (see e.g. [14, pp. 279±281]), in this case} d…PL;P…h††6is2n ˆh~

2

=i. Therefore, the distance (64) between the generalized negative binomial distribution PL and the Poisson distribution P…h~† for a<1 is larger than between the

corre-sponding binomial distributionB…i;sn_{†and P…h† in the limiting caseaˆ1.}

Table 1

The distance between the exact distributionPLand the approximating Poisson distributionP…h~†…nˆ30; mˆ0; h~ˆ2†

i aˆ0:7 aˆ0:8 aˆ0:9 aˆ0:95

105 _{0.177 0.122 0.062 0.031}

106 _{0.143 0.098 0.049 0.025}

107 _{0.119 0.080 0.040 0.020}

6. Discussion

Our results solve the main problem raised in [1] and followed up in [6,11]. According to Theorem 1, the exact distribution of the numberL of surviving clonogens belongs to the family (44) of generalized negative binomial distributions. This family does not contain binomial dis-tributions. However, as discovered in Theorem 3, the distribution of r.v. Lstill converges under the natural condition (46) to the Poisson distribution (60). Therefore, the distribution of r.v.Lcan be approximated by the Poisson distributionP…h~† withh~ˆisn_=anÿ1_{. Note that h}~_{is not equal to} hˆisn_{unlessaˆ1, that is,kˆm. This explains why the Poisson distributionP…h†was rejected in}

[1,6,11] on the basis of numerical computations which made the authors think that anyPoisson approximation is impossible. We observe also that the Poisson approximationP…h† is valid not only in the absence of cell proliferation and death (kˆmˆ0) but also in the more general case when their rates are equal (kˆm).

It is interesting to mention that, although the distributions of r.v.s M and L are quite close when the interdose intervalsis small, the limiting distribution (47) of r.v.Munder the condition (46) is not Poisson and never reduces to it.

The model of fractionated radiation cell survival discussed in this work depends on six pa-rametersn; s; i; s; k; m, the last four of which are unobservable. More importantly, the number of surviving clonogens is unobservable as well. To make use of the computed distribution of the size of the surviving fraction of clonogens, one has to relate it to an observable endpoint, like the time to tumor recurrence or death caused by a speci®c recurrent cancer. In what follows, we will

Fig. 1. Exact distributionPL foriˆ105 (solid line 1),iˆ107 (dotted line 2), and the approximating Poisson

focus on the ®rst possibility. Speci®cally, letTbe the time to tumor recurrence counted from the moment of delivery of the last fraction. It is assumed that r.v.T is absolutely continuous.

There are two competing models of post-treatment tumor development. According to the

clonogenic modelintroduced in [21], a recurrent tumor arises from a single clonogenic cell. Every surviving clonogen can be characterized by a latent time (called theprogression time) during which it could potentially propagate into a detectable tumor. It is assumed additionally that progression times of surviving clonogens are i.i.d. r.v.s. Suppose that the numberLof surviving clonogens is equal tom. Then for the observed timeTto tumor recurrence we haveT ˆmin16j6mTj;whereTj

is the progression time of thejth clonogen. Letu_L…z† ˆP1

mˆ0p~mzmbe the p.g.f. of r.v.L:It follows

from the assumptions of the clonogenic model that the conditional survival function

FTjLˆm…t†:ˆPr…T >tjLˆm†; tP0; of r.v. T is given by FTjLˆmˆF1m; where F1 is the common survival function of the progression times of surviving clonogens. Then

FT…t† ˆ

X1

mˆ0

FTjLˆm…t†Pr…Lˆm† ˆ

X1

mˆ0 ~

p_mF₁m…t† ˆu_L…F1…t††; tP0: …65†

Alternatively, the cumulative model proceeds from the assumption that all surviving clonogenic cells contribute to the resulting recurrent tumor, see for example [5]. Assume additionally that their contributions are identically distributed (but not necessarily independent). For mP0; de-note byrm the conditional hazard function of r.v. Tgiven that Lˆm: Then

FTjLˆm…t† ˆexp

ÿ

Z t

0

rm…u†du

: tP0:

By our assumptions, rmˆmr; where r is the common hazard function associated with the

con-tributions of the surviving clonogens to the resulting detectable tumor. ThenFTjLˆmˆF2m; mP0; whereF2 is the survival function corresponding to the hazard rater, and hence

FT…t† ˆ

X1

mˆ0 ~

pmF2m…t† ˆuL…F2…t††; tP0: …66†

Comparing (65) and (66) we conclude that, surprisingly, the distribution of the time Tto tumor recurrence within both models of post-treatment tumor development has the same form

FT…t† ˆuL…F…t††; tP0; …67†

whereF is the survival function of an absolutely continuous non-negative r.v. Suppose, in par-ticular, that r.v.L follows the Poisson distribution P…A†. Then we derive easily from (67) that

FT…t† ˆeÿAF…t†; tP0; …68†

whereF :ˆ1ÿF is the corresponding cumulative distribution function. Although this form of the recurrence time distribution was originally suggested in [21] in conjunction with the clonogenic model of post-treatment tumor development, it proved to be instrumental for statistical estima-tion of the probabilitypof tumor cure (also termed thecure rate) in many other settings [7,22±26]. Observe that according to (68),pˆeÿA_{;and the positivity ofpis due to the fact thatp~}

Signi®cance of formula (67) is two-fold. First, it suggests that knowledge of the entire distri-bution of the number of surviving clonogens (not only of the tumor control probability p_~0) is critical for developing biologically motivated post-treatment survival models. Second, assuming some parametric form for the function Fand using for the distribution of r.v. Leither its exact form (44) or the Poisson approximation (60) one can in principle estimate unknown quantitiesi;s

and especially the all-important kinetic parameters k and m from the observed times to tumor recurrence. Making use of the exact distribution of r.v.Lseems to be especially promising because it incorporates the known doseDper fraction through survival probability s, and this allows to develop a regression model accounting for the variation of the dose D and other clinically im-portant covariates. Pursuing this line of reasoning presupposes, indeed, investigation into iden-ti®ability properties of the resulting survival model.

Acknowledgements

The author is thankful to Drs A.Yu. Yakovlev (University of Utah) and M. Zaider (Memorial Sloan-Kettering Cancer Center) for their insightful comments on various issues raised in this paper, and to Dr A. Zorin for his help in producing Fig. 1. The author is also grateful to the referees for their valuable suggestions.

References

[1] S.L. Tucker, H.D. Thames, J.M.G. Taylor, How well is the probability of tumor cure after fractionated irradiation described by Poisson statistics? Radiat. Res. 124 (1990) 273.

[2] W.S. Kendal, A closed-form description of tumour control with fractionated radiotherapy and repopulation, Int. J. Radiat. Biol. 73 (1998) 207.

[3] T.E. Harris, The Theory of Branching Processes, Springer, Berlin, 1963.

[4] P. Jagers, Branching Processes with Biological Applications, Wiley, London, 1975.

[5] M. Klein, R. Bartoszynski, Estimation of growth and metastatic rates of primary breast cancer, in: O. Arino, D.E. Axelrod, M. Kimmel (Eds.), Mathematical Population Dynamics, Marcel Dekker, New York, 1991, p. 397. [6] S.L. Tucker, Modelling the probability of tumor cure after fractionated radiotherapy, in: M.A. Horn, G. Simonett,

G. Webb (Eds.), Mathematical Models in Medical and Health Sciences, Vanderbilt University, Nashville, 1999, p. 1.

[7] A.Y. Yakovlev, A.D. Tsodikov, Stochastic Models of Tumor Latency and their Biostatistical Applications, World Scienti®c, Singapore, 1996.

[8] J. Deasy, Poisson formulas for tumor control probability with clonogen proliferation, Radiat. Res. 145 (1996) 382. [9] A.Yu. Yakovlev, Comments on the distribution of clonogens in irradiated tumors (Letter to the Editor), Radiat.

Res. 134 (1993) 117.

[10] L.G. Hanin, A.Y. Yakovlev, L.V. Pavlova, Biomathematical Problems in Optimization of Cancer Radiotherapy, CRC, Boca Raton, 1994.

[11] S.L. Tucker, J.M.G. Taylor, Improved models of tumour cure, Int. J. Radiat. Biol. 70 (1996) 539.

[12] L.G. Hanin, M. Zaider, A.Yu. Yakovlev, Distribution of the number of clonogens surviving fractionated radiotherapy: a long-standing problem revisited, Int. J. Radiat. Biol., to appear.

[13] M. Zaider, L.G. Hanin, A.D. Tsodikov, M.J. Zelefsky, S.A. Leibel, A.Yu. Yakovlev, A survival model for fractionated radiotherapy with an application to prostate cancer, preprint.

[15] D.G. Kendall, On the generalized birth and death process, Ann. Math. Stat. 19 (1948) 1.

[16] D.R. Cox, H.D. Miller, The Theory of Stochastic Processes, Chapman and Hall, New York, 1984. [17] S. Karlin, A First Course In Stochastic Processes, Academic Press, New York, 1966.

[18] K.B. Athreya, P.E. Ney, Branching Processes, Springer, New York, 1972. [19] S. Asmussen, H. Hering, Branching Processes, Birkhauser, Boston, 1983.

[20] M. Zaider, G.N. Minerbo, Tumor control probability: a formulation applicable to any protocol of dose delivery, Phys. Med. Biol. 45 (2000) 279.

[21] A.Yu. Yakovlev, B. Asselain, V.-J. Bardou, A. Fourquet, T. Hoang, A. Rochefordiere, A.D. Tsodikov, A simple stochastic model of tumor recurrence and its application to data on premenopausal breast cancer. in: B. Asselain, M. Boniface, C. Duby, C. Lopez, J.-P. Masson, J. Tranchefort (Eds.), Biometrie et Analyse de Donnees Spatio-Temporelles, vol. 12, Rennes, France: Societe Francßaise de Biometrie, ENSA, 1993, p. 66.

[22] A.D. Tsodikov, B. Asselain, A. Fourquet, T. Hoang, A..Y. Yakovlev, Discrete strategies of cancer post-treatment surveillance. Estimation and optimization problems, Biometrics 51 (1995) 437.