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

Stochastic modeling in toxicokinetics.

Application to the in vivo micronucleus assay

Ollivier Hyrien

a,*

, Virginie Kl

es

a

, Didier Concordet

b

, Michel Bonneu

c

,

Michel Laurentie

a

, Pascal Sanders

a

a_{Agence Fran}

c

ßaise de SecuriteSanitaire des Aliments, Laboratoire d'Etudes et de Recherches sur les Medicaments Veterinaires et les Desinfectants, BP 90203, 35302 Fougeres cedex, France

b_{Ecole Nationale V}

eterinaire de Toulouse, UniteAssociee INRA de Physiopathologie et Toxicologie Experimentales, 23 chemin des Capelles, F31076 Toulouse cedex, France

c

Equipe GRIMM, Universitede Toulouse II, IUT-B, BP73, 31073 Blagnac, France

Received 24 January 2000; received in revised form 13 September 2000; accepted 2 October 2000

Abstract

A stochastic model for the in vivo micronucleus assay is presented. This model describes the kinetic of the rate of micronucleated polychromatic erythrocytes induced by the administration of a mutagenic compound. For this, biological assumptions are made both on the erythropoietic system and on the mechanisms of action of the compound. Its pharmacokinetic pro®le is also taken into account and it is linked to the induced toxicological e€ect. This model has been evaluated by analyzing the induction of micronuclei is mice bone marrow by a mutagenic compound, 6-mercaptopurine (6-mp). This analysis en-abled to make interesting remarks about the induction of micronuclei by 6-mp and to put to light an unsuspected wavy kinetic by optimizing the experimental design of the in vivo micronucleus assay. Ó ₂₀₀₁ Elsevier Science Inc. All rights reserved.

Keywords: Markovian process; Branching process; Non-homogeneous birth and death process; Toxicokinetic; Micronucleus assay; Erythropoietic system

1. Introduction

The in vivo micronucleus assay is a test widely used to evaluate the mutagenicity of chemical compounds. It is performed by scoring damaged cells, i.e., micronucleated cells, in bone marrow

www.elsevier.com/locate/mbs

*_{Corresponding author. Tel.: +33-2 99 94 78 78; fax: +33-2 99 94 78 80.} E-mail address:o.hyrien@fougeres.afssa.fr (O. Hyrien).

0025-5564/01/$ - see front matter Ó _{2001 Elsevier Science Inc. All rights reserved.}

post-injection (PI). Micronuclei are extranuclear chromosomal fragments and their induction in cells of the bone marrow by chemical agents or radiation is well documented [1±5]. It is necessary to perform this test under optimal conditions to bring potential mutagenicity to light, but no suitable approach to ®nd these has been yet established. Even contradictory results are found in the literature for some chemicals tested with the in vivo micronucleus assay, as for instance the isoproturon [6,7]. This problem may be due to di€erences between experimental designs. It is therefore of interest to develop a mathematical model for achieving a better understanding of the erythropoietic system and of the induction of micronuclei. Then, by that mean, one can perhaps expect to improve the sampling design. The objective of this paper is to develop a mathematical model that describes the kinetic of the rate of micronucleated polychromatic erythrocytes which is the observed parameter when performing the in vivo micronucleus assay. To achieve this, basic biological assumptions are taken into account by the use of three kinds of stochastic processes.

The model describes ®rst the following cellular system. After the administration and arrival in the bone marrow, one molecule can induce damages in precursor cells, namely proerythroblasts and erythroblasts. This population of precursor cells is denoted by P_{I in Section 2 to simplify.}

Each cell of P_{I can either give birth to two new o€springs, of the same type, after a generation}

period, or transforms into one di€erentiated cell after a maturation period, by expelling its nu-cleus. Hence, this population of cells has been described by a branching process. Cell cycles are regulated by a cascade of reactions and although their duration is not the resultant of a me-moryless process, the generation and maturation periods have been assumed to be exponentially distributed in the model. This approximation enabled nevertheless to develop a tractable model that was informative enough to reveal an unexpected kinetic of the rate of micronucleated cells induced by 6-mercaptopurine (6-mp) (see Section 3). The transformation of cells of P_{I is}

irre-versible. In that case, the cell is eliminated of P_{I and gets into the population of polychromatic}

erythrocytes (PCE) which is denoted byP_{II to simplify. Damages induced previously in}P_{I keep}

present and become easily detectable inP_{II. The molecule is supposed to have no activity on the}

cells ofP_{II. In addition, these cells have not the possibility to divide themselves and after a}

dif-ferentiation delay they leaveP_{IIand turn into normochromatic erythrocytes (NCE). For the same}

reasons as above, the di€erentiation delay of cells of P_{II has been assumed to be exponentially}

distributed. This led to describe the number of cells of P_{II by a birth and death process. This}

cellular system is depicted on Fig. 1.

In addition to these ®rst two stochastic processes used to describe the dynamic of the di€erent types of cells, a third stochastic process is de®ned to model the induction of micronuclei in cells of

P_{I. It is a bidimensional markovian process. Its ®rst component describes the time course of the}

number of damages in a cell lineage. Its second component enables to take into account a possible cellular death due to the presence of micronuclei. Since the intensity of the induction of mi-cronuclei depends on the pharmacokinetic pro®le of the compound, the transitions of this third stochastic process are related to the time course of the concentrations of the compound.

Numerical experiments have been conducted to get theoretical predictions of the rate of micro-nucleted PCE and they have been compared to experimental data. These simulations predicted waves in the kinetics of micronucleated PCE that were not suspected before. A second experi-mental in vivo micronucleus assay was then performed which con®rmed the theoretical predic-tions.

The model is connected with stochastic models used in carcinogenesis studies, such as for in-stance the model of Yakovlev et al. [8] who described spontaneous regression of tumor within a similar cell kinetic. Tsodikov and Muller [9] developed a model of carcinogenesis for fractionated and continuous exposure. Yakovlev and Polig [10] presented a model allowing for cell killing. The erythropoietic system has also been presented and described in many papers [11±15]. Many mathematical models have been elaborated to achieve a better understanding of the erythropoiesis or of the formation of micronuclei in bone marrow. Mary et al. [12] have investigated a mathe-matical analysis of bone marrow erythropoiesis and derived estimates of cell kinetic parameters. Ludwikow and Ludwikow [13] have developed a bicompartmental model describing the forma-tion of micronucleus in PCE. Their model is based on the assumpforma-tion that the chemical acforma-tion is constant during a periodT. The model developed here is more ¯exible and more realistic since it enables to take into account a time-changing exposure due to the variation of the compound concentrations with time, and it also adapts to its route of administration. Furthermore, it is able to take into consideration a variety of mechanisms of action that can ¯uctuate during the cell cycle for instance. Thus, it addresses to a large number of compounds.

This paper is organized as follows. The theoretical model is developed in Section 2. The branching process modeling the dynamic of the populationP_{I is detailed in Section 2.1. Next, in}

Section 2.2, the bidimensional markovian process used to describe both the number of mi-cronuclei contained in a cell lineage of P_{I and its state of live or death is given, and the}

proba-bilities for a precursor cell leavingP_{Iat any timet>0 to be both micronucleated and alive (resp.}

to be both non-micronucleated and alive) are deduced. In Section 2.3, two birth and death processes are presented. They describe the numbers of micronucleated and healthy PCE and their rates of birth are de®ned with the probabilities determined in Section 2.2. A model of the rate of micronucleated PCE at any time t>0 is then established. In Section 3, this theoretical model is

evaluated by analyzing the induction of micronuclei by 6-mp in mice bone marrow. To achieve this, a few steps are needed to specialize the model to this compound. In Section 3.1, the time course of the plasma concentrations of 6-mp is modeled. In Section 3.2, estimates of the pa-rameters of the cell cycle kinetic are presented. In Section 3.3, the mechanism of action of the 6-mp is presented and included into the model. In Section 3.4, numerical experiments are carried out and compared to observed data. The results are discussed in Section 3.5. Section 4 concludes the paper. The proofs of the results are given in Appendix A.

2. The stochastic model

The model presented in this paper, which describes the kinetics of the rate of micronucleated cells ofP_{II, is developed in this section. Recall that this model is based on three kinds of stochastic}

processes: a branching process modelingP_{I(see Section 2.1), a markovian process modeling both}

the number of micronuclei contained in a given cell of a cell lineage and its state of live or death (see Section 2.2), and two birth and death processes modeling the population of healthy and damaged cells ofP_{II (see Section 2.3).}

2.1. A branching process for the growth of P_I

In this section a branching process modeling the stochastic growth ofP_{I is de®ned and some}

terminology used through out the paper is introduced.

First,P_{I…t†denotes the collection of cells of}P_{Iat timet. Every cell of}P_{Ican either split up into}

o€springs after a generation time or leave the population after a maturation time without di-viding. Thus, a cell born at time 0 has a random generation time G with the probability distri-bution g…x† ˆPr…G6x†. At the end of this generation time it is replaced by exactly two similar cells of age 0. Similarly, a cell born at time 0 has also a random maturation time M with the probability distributionm…x† ˆPr…M6x†. At the end of this maturation time, the cell irreversibly leaves the population without dividing. For all time t, the generation and maturation times are supposed to be independent of the state of the population or of its past history. The so de®ned process is an age-dependent branching process. In general it is not markovian except whenGand

M are exponential, which is supposed in this paper. This simpli®ed version of the model was developed to obtain tractable mathematical results. Although obviously wrong from a biological point of view, the assumption that the generation and maturation times are exponentially dis-tributed allowed to reach our goal: it enabled concluding remarks to be made about the kinetics of micronucleated PCE with 6-mp and to improve the sampling design of this in vivo micronucleus assay (see Section 3).

In the sequel, for any cellC;C1 denotes the mother cell ofC, i.e., the cell which gives birth toC

after one division, C2 denotes the mother cell ofC1 and so on.

De®nition 1.Pick a cell of P_{I, denoted byC. For all k2}N_{, a family} F_{k…C† is the collection of}

cells

whereC0 ˆCfor convenience. For allk2N, a family historyHk…C†is the collection of random

The set F_{k…C† is the cell lineage initiated by the cell Ck and it follows the family tree of C}

backwards up to the kth generation. The index k represents the number of successive divisions realized byCk and its descendants to give birth to C.

For any timet>0 and for any cellCt_2P

®rst cell of the lineage exposed to the chemical compound. The integer K0 represents then the

number of divisions that have occurred in the cell-line F

K0…C

t_{† in the interval of time [0,t]. It is}

non-deterministic as soon as the generation and maturation phases are random. Its distribution function is time-dependent. Let Tt

k be the time of division of Ctk, k ˆ1;. . .;K0;T0t the time of

di€erentiation ofCt _andTt

K0‡1 the time of birth ofC

t

K0. Note that the identity below is satis®ed for

all timet>0

Many authors are used to deriving the equations of similar systems without using these times of division. We do not proceed that way because further applications require the decomposition of the intervals‰Tt

k;Tkt‡1‰ into subintervals (cf. Section 3). An example of cell lineage is presented in

Fig. 1: the family of Ct _{is composed of three cells, F} K0…C

t_{† ˆ fC}t

0;Ct1;Ct2g. The compound is

administrated at time 0 and the cellCt

2is then exposed to its toxicity up to the end of its generation

phase which occurs at timeTt

2. This cell divides then into two new o€springs and the cell-lineage

F_K 0…C

t_{† is exposed to the compound through the cell C}t

1. Since two divisions are necessary to

obtain Ct _{from C}t

2,K0 ˆ2.

2.2. Modeling the induction of damages inP

I

In this section, an N _f0_;1_g-valued stochastic process is developed. It is denoted by

Z…u† ˆ …Z1…u†;Z2…u††, where u runs in the time domain [0;t]. Note that t is ®xed here. The ®rst

component of this random vector models the number of damages in a cell-lineageF_K 0…C

t_{†at time}

u, whereCt _{is a cell leavingP}

I at a positive timet. The second component describes the possible

death of this cell-lineage: by convention the cell-lineage is alive at timeu ifZ2…u† ˆ0 and dead if

Z2…u† ˆ1. This death process refers to death due to the presence of damages in the cell and not to

the disparition of cells from the populationP_{Iafter division or di€erentiation. In order to simplify}

2.1 holds also for cells that have died. Their division could be stopped, but this choice, even if surprising, retains a meaning because Z2…u† marks the dead lineages.

It is also supposed that Z…0† ˆ …0;0†, that is any cell is both alive and healthy when the compound is administrated. The objective in this section is to derive the probability that the cell

Ct _{is both alive and healthy at time t, that is}

Pr…Z…u† ˆ …0;0† jZ…0† ˆ …0;0††

and the probability thatCt _{is both alive and damaged at time t, that is}

Pr…Z…u† ˆ …n;0†;n>0jZ…0† ˆ …0;0††:

To this end, the transitions for this process are ®rst de®ned on any semi-open intervals‰Tt k‡1;T

t k‰,

for allk ˆ0;. . .;K0. Next, the transitions at the timesTkt that involve changes in the dynamic of

Z1…u† are considered. For all timeu and for all h>0, set

Pu;h…k1;l1;k2;l2† ˆPr…Z…u‡h† ˆ …k2;l2† jZ…u† ˆ …k1;l1††:

For allk ˆ0;. . .;K0 and for all u2 ‰Tkt‡1;Tkt‰, the probabilities for the transitions of this process

from time u to timeu‡h, withh>0, are de®ned according to the following three cases. Case1:Z…u† ˆ …0;0†. At timeu, the cell is healthy and alive. It remains in that state up to time

u‡h with probability

Pu;h…0;0;0;0† ˆ1ÿF…u†h‡o…h†;

whereF is a positive valued function describing the formation of the damages. The probability that a damage appears during that interval of time and that the cell stay alive is

Pu;h…0;0;1;0† ˆF…u†h‡o…h†:

On the contrary if the cell keeps healthy at time u‡h, there is no cellular death, that is

Pu;h…0;0;0;1† ˆ0:

The other transitions are assumed to occur with a probability of ordero…h†.

Case 2: Z…u† ˆ …n;0†, with n>0. In this case, the cell is alive and damaged at time u with exactly ndamages. First, the probability that the cell remains in that state at time u‡h is

Pu;h…n;0;n;0† ˆ1ÿ …F…u† ‡m†h‡o…h†;

wheremis a positive constant describing the death process of the damaged cells. The probability that exactly one damage appears and that the cell is still alive at timeu‡h is

Pu;h…n;0;n‡1;0† ˆF…u†h‡o…h†:

The probability that the cell dies but that no damage appears is

Pu;h…n;0;n;1† ˆmh‡o…h†:

The other transitions are assumed to occur with a probability of order o…h†.

Case3:Z…u† ˆ …n;1†, withnP0. Here, the cell is dead at timeuand possibly damaged ifn>0. The states…n;1† are supposed to be absorbing, that is

One can easily remark that Z1…u† is non-decreasing on each interval ‰Tkt‡1;Tkt‰, but at the times

Tt

k;kˆ1;. . .;K0, the dynamic of the processZ1…u† changes. Indeed, at division the damages are

randomly allocated in the two o€springs. So, the transitions of the process are described according to the following two cases:

(i) in case of an alive lineage, for allk 2 f1;. . .;K0g, for all n>0 and for alliˆ0;. . .;n

Pr…Z…T_kt† ˆ …i;0† jZ…T_ktÿ† ˆ …n;0†† ˆc…n;i†;

wherec…;†is a given application satisfying the condition Pn

iˆ0c…n;i† ˆ1;

(ii) in case of a dead lineage

Pr…Z…Tt

k† ˆ …n;1† jZ…T tÿ

k † ˆ …n;1†† ˆ1:

Fig. 2 gives an example of trajectory of the processZ1…u†which increases by jumps on the intervals

of time ‰0;Tt

2‰;‰T2t;T1t‰ and ‰T1t;T0t‰, and which possibly decreases when a division occurs. When

leavingP_{I at timet, C}t _{is damaged and contains one damage.}

The function F links the toxicological e€ect and the amount of chemical, denoted byQ…t†. Its choice depends on the characteristics and on the mechanism of action of the chemical. Suppose for instance that the probability of appearance of a damage between times t and t‡h is pro-portional to the amount of chemical at timet, then

F…t† ˆKQ…t†;

whereK is a parameter for the toxicity of the chemical.

The distribution function c…n;i† describes the way the damages are allocated in the two o€-springs at division. For instance, if each of theZ1…T_ktÿ†damages are allocated in each of the two

cells with probability 1/2,Z1…Tkt† is a binomial random variable with parametersZ1…Tktÿ†and 1/2.

In that case, for alln>0;c…n;i† ˆCi n2ÿn.

De®ne now for all timet>0 the following events: for allx2 ‰0;tŠ,E_{x† ˆ fZ2…x† ˆ0g, meaning}

that the cell-lineageF_K 0…C

t_{† is alive at time x, and A…x† ˆ fZ}

1…x†>0jE…x†g, meaning that the

Fig. 2. This ®gure gives an example of trajectory ofZ1…u†. This process describes the number of microunclei contained in a cell lineage di€erentiating at timet…ˆTt

0†. After administration of the compound at time 0, two divisions occur at times Tt

2 and T1t. At time 0, Z1…u† ˆ0 that is the cell lineage is healthy. On each time domain

‰0;Tt

2 ‰and‰T t i‡1;T

t

cell-lineage is damaged at time x given that it is also alive. For all k2 f0;. . .;K0g and for all

x0 _{2 ‰T}t

k;tŠ, de®ne also the timezmax…x0† ˆmaxfz2 ‰0;x0Š=Z1…z† ˆZ1…zÿ† ‡1g, which is the time of

appearance of the last damage in the cell lineage and

B_k…x0_{† ˆ fzmax…x}0_{† 2ŠT}t

meaning both that the last damage of the cell-lineage at timex0_{has appeared between times T}t k‡1

and Tt

k and that the cell lineage keeps micronucleated up to time x0, given the cell lineage is still

alive at that time. The events

C

mean that at least one damage has occurred between the timesTt

k‡1andTkt. The complementary of

The following notation has been adopted

Pr0…† ˆPr… jZ…0† ˆ …0;0††: On each time domain ‰Tt

k‡1;Tkt‰, the process Z1…x† ÿZ1…T t

k‡1† is a non-homogeneous markovian

point process, with rate F…x† [16]. Therefore, Z1…x† ÿZ1…Tkt‡1† has the Poisson distribution with

and the following lemma can be stated.

Lemma 2. Let F be a bounded measurable function for all time t. Then,8k 2 f0;. . .;K0g,

By use of Lemmas 1 and 2, one derived Theorem 1. It gives the probability for a cell that ceases cycling at timeT_kt to be damaged conditional on the historyH_K

0…C

t_{†and given that the cell is still}

Theorem 1. For all timet>0; consider a cell leaving P_{I at time t, denoted by C}t_{; and its family}

Theorem 2 provides the probability for a cell to die during its cycle.

Theorem 2.For all time t>0, consider a cell leaving P_{I at time t, denoted by C}t_{, and its family}

The probability for a cell leaving P_{I at timet to be damaged is deduced from Theorem 2:}

Corollary 1. For all timet>0; consider a cell leavingP_{I at time t, denoted by C}t_{; and its family}

By use of Theorem 1 and Corollary 1 one can derive the probability for a cell Ct _{to be both}

healthy and alive when leaving P_{I, conditional on}H

K0…C

and the probability for the cellCt_{to be both damaged and alive when leavingP}

I, conditional on

H

K0…C

Pr0 Z1…t†

ÿ

>0;Z2…t† ˆ0jHK0…C

t_†

ˆPr0 Z1…t†

ÿ

>0jH

K0…C

t_†;Z

2…t† ˆ0

Pr0 Z2…t†

ÿ

ˆ0jH

K0…C

t_†

: …2†

The marginal probabilities Pr0…Z…t† ˆ …0;0††and Pr0…Z1…t†>0;Z2…t† ˆ0† are obtained by taking

the mathematical expectation of (1) and (2) with respect to the Tt

k;kˆ0;. . .;K0. No explicit

formulas for these quantities are available and their computations require numerical technics, such as Monte Carlo simulation method.

2.3. Some birth and death processes for the growth of P_II

This section deals with the population of polychromatic erythrocytes denoted by P_{II: These}

cells cannot divide since they are anucleated and after a time of di€erentiation they leaveP_II:This

population is made up of two types of cells, say type healthy if they are not micronucleated and type damaged if they are micronucleated. Both of them come from the populationP_{I after they}

di€erentiate by expulsion of their main nucleus. The population of damaged (resp healthy) cells is denoted by P1

II (resp. P 2

II†: Here, two birth and death processes are used to describe the time

course of the size of the two subpopulations of cells and a mathematical model predicting the kinetic of the rate of damaged cells of P_{II is derived [17,18].For all time t, consider a birth and}

death process denoted by N…t† with conditional probabilities for its transitions Pr…N…t‡h† ˆn‡1jN…t† ˆn† ˆk…n;t†h‡o…h†;

Pr…N…t‡h† ˆnÿ1jN…t† ˆn† ˆl…n;t†h‡o…h†:

The other transitions are supposed to occur with a probability of order o…h†: Therefore Pr…N…t‡h† ˆnjN…t† ˆn† ˆ1ÿ …k…n;t† ‡l…n;t††h‡o…h†:

The functionsk…n;t†andl…n;t†are non-negative and de®ne respectively the birth and death rates of the process.

Whenk…n;t† ˆk…t† and l…n;t† ˆl…t†n for all time t, with k…t† and l…t† two non-negative real valued functions, the processN…t†is a queueing process with in®nitely many servers, and the state of the system is interpreted as the length of a queue for which the inter-arrival times have an exponential distribution with parameterk…t† and the service times have an exponential distribu-tion with parameter l…t† [19±22]. In the special case k…t† ˆk and l…t† ˆl withk and lpositive constants, N…t† is the homogeneous M=M=1 queue. In the sequel, birth and death rates of the form

8n2N_;8t2R k…n;t† ˆk…t†

l…n;t† ˆln

are considered, where k…t† is a real valued function of time and la positive constant. Any birth and death process N…t† of that kind, with time dependent birth rate, is a non-homogeneous queueing processMt=M=1[20,21]. The class of birth and death processes de®ned above is denoted

by C_.

In the context of queueing systems, Keilson and Servi [21] proved that ifN…0† is Poisson, the distribution of any birth and death processN…t†ofC_{is Poisson for all timet>0: In the present}

compound is administrated. Without mutagenic compound, the induction of micronuclei is possible by a process of natural mutation but it remains a rare event. Therefore, N…0† can be assumed to follow a Poisson distribution. Hence one can focus attention on the mathematical expectation ofN…t†; denoted by f…t†; which is the parameter of such variate. For all time t>0;

f…t†is governed by the linear di€erential equation d

dtf…t† ˆk…t† ÿlf…t†; …3†

whose solutions are

f…t† ˆeÿlt _f…0†

‡ Z t

0

elx_k…x†dx

:

Consequently, in case of a time homogeneous process, i.e., ifk…t† ˆk, thenf…t† ˆk=lfor all time

t. The sum of independent processes is stable in C_{, that is if N1…t† and N2…t† are independent}

processes ofC_{, their sum N1…t† +N2…t† is still a birth and death process of}C_{. Consider now two}

sequences of independent birth and death processes ofC_{, denoted by fN}i

j…t†;i2Ng with

math-ematical expectations ffi

j…t†;i2Ng forjˆ1,2. Suppose that for all i2N and for alljˆ1,2 the

random variablesNi

j…0†are Poisson and thatSj…t† ˆlimm!1 Pimˆ1fji…t†=mexists for allt. Then, by

applying the law of large numbers to a sequence of independent Poisson random variables, the following convergences are obtained:

1

m

Xm

iˆ1

N₁i…t†!a:s:S_1…t†

and

1

m

Xm

iˆ1

…N₁i…t† ‡N₂i…t††!a:s:S_{1…t† ‡}S_2…t†:

Therefore, the rate of damaged cells inP_{II converges as follows:}

Pm iˆ1N1i…t†

Pm

iˆ1…N1i…t† ‡N2i…t††

!Pr_S S1…t†

1…t† ‡S2…t†

…4†

asmgoes to in®nity.

In order to model the dynamic of the two subpopulations P1

II and P 2

II; the bone marrow is

decomposed intomdistinct and homogeneous areas, denoted byfAi;iˆ1;. . .;mg:In each areaAi

the number of damaged (resp. healthy) cells is described by a birth and death process denoted by

Ni

1…t† (resp. N2i…t†† with identical birth and death rates k1…n;t† and l1…n;t†; (resp. k2…n;t† and

l₂…n;t†) satisfying the equations

8n2N_;8t>0 k1…n;t† ˆk1…t† …resp:k2…n;t† ˆk2…t††;

l₁…n;t† ˆln …resp:l₂…n;t† ˆln†;

wherek1…t†andk2…t† are non-negative valued functions andlis a positive constant. In each area

Ai and for all timet; the expected number of damaged and healthy cells, denoted respectively by

fi

d

dtf1…t† ˆk1…t† ÿlf1…t† and

d

dtf2…t† ˆk2…t† ÿlf2…t†:

LetNT

1…t† (resp. N2T…t†† denotes the number of cells ofP 1

II (resp. P 2

II† at time t. These stochastic

processes are de®ned by

N₁T…t† ˆX m

iˆ1

N₁i…t† and N₂T…t† ˆX m

iˆ1

N₂i…t†:

Since the sum of independent processes is stable in C_;NT

1…t† and N2T…t† are birth and death

pro-cesses ofC_{;with respective birth and death ratesmf1…t† andl;mf2…t†andl. Their mathematical}

expectations, denoted byfT

1 …t† and f2T…t†, respectively, are therefore de®ned by

f₁T…t† ˆmf1…t† and f2T…t† ˆmf2…t†:

The integermis assumed to be large enough to justify the approximation of the stochastic process

N₁T…t†=…N₁T…t† ‡N₂T…t††by the functionf₁T…t†=…f₁T…t† ‡f₂T…t††:This approximation is ensured by (4) and seems to be realistic since the number of PCE in bone marrow is large. Therefore, the time course of the rate of damaged cell ofPII is described by the function

R…t† ˆ f1…t†

f1…t† ‡f2…t†

: …5†

Moreover the rates of birth of the processes Ni

1…t† and N2i…t† are taken of the form

k1…t† ˆkP1…t† and k2…t† ˆkP2…t†;

wherekis an arbitrary positive constant representing the rate of birth of cells ofP_{II in each area}

Ai;iˆ1;. . .;m:The function P1…t† is the rate of damaged cells di€erentiating at time twhile the

function P2…t† represents the rate of healthy cells di€erentiating at time t. In fact, P1…t† is the

probability for a cell ofP_{I di€erentiating at timet to be both alive and damaged}

P1…t† ˆPr…Z1…t†>0;Z2…t† ˆ0†:

In a similar way,P2…t†is the probability for a cell ofPIdi€erentiating at timetto be both alive and

healthy

P2…t† ˆPr…Z1…t†>0;Z2…t† ˆ0†:

The expressions of these probabilities are given in Section 2.2. This completes the presentation of the toxicokinetic model whose performance are evaluated in the next section by analyzing the induction of micronuclei by 6-mp.

3. Analysis of the induction of micronuclei by 6-mercaptopurine

6-mp plasma concentrations (see Section 3.1). It is also based on hypotheses made on mechanisms of action of the compound; they are given in Section 3.3. A part of the parameters are estimated by literature values (see Section 3.2). A preliminary assay has been conducted (see Section 3.4) which gives the observed rates of micronucleated PCE at several times PI. The remaining pa-rameters are then chosen so that the theoretical kinetic is close to these experimental data. Nu-merical experiments have been performed with the model. They are presented in Section 3.4 and they predict a succession of waves that do not appear on the observed kinetic. A second exper-imental assay was then planned to check these predictions. The results are discussed in Section 3.5.

3.1. Pharmacokinetic analysis

Forty eight mice were allocated into nine groups. Each group of 5 mice received a single dose of 6-mp at the dose rate of 50 mg/kg body weight (bw) by intraperitoneal (IP) route. All animals of one group were killed at the same sampling time and blood was sampled. Data were analyzed using a compartmental approach. Based on Akaike's information criterion [23] a bicompart-mental model with an absorption phase was selected to ®t the data

C…t† ˆAexp…ÿat† ‡Bexp…ÿbt† ÿ …A‡B†exp…ÿct†;

whereC…t† is the plasma chemical concentration at timet; a, b and c are, respectively, the dis-tribution, the elimination and the absorption rates. The ®ve parameters of this equation have been estimated by the use of a least square criterion. Their estimates were respectively A^ˆ2796:9;a^ˆ

0:205;B^ˆ27:9;b^ˆ0:029 and ^cˆ0:211.

Fig. 3 shows the plasma concentrations and the ®tted data of 6-mp after intraperitoneal ad-ministration to mice. The maximal concentrations were rapidly reached, about 5 min PI and were about 45lg=ml. The plasma concentrations decreased rapidly and the terminal half live was estimated at about 3 h.

3.2. The in vivo micronucleus assay

The in vivo micronucleus assay is usually performed on PCE. It consists in scoring the number of micronucleated PCE in mice bone marrow. These PCE are precursors of the red blood cells. They cannot divide themselves since they are anucleated. The dynamic of their population is similar to that of the populationP_{II presented in Section 2.3 and the rate of micronucleated PCE}

is modeled by Eq. (5). On the contrary, precursors of these PCE are dividing cells that can either divide into two new precursors or di€erentiate and turn into PCE. Their growth is closed to that of populationP_{I described in Section 2.1 by a branching process. The generation phases of these}

precursors are made up of four distinct and successive periods calledG1, DNA synthesis,G2 and

mitosis. Cole et al. [2] estimate the mean time duration of these periods and of the maturation period. Their estimates are, respectively,d^G1 ˆ1 h,d^syˆ7:5 h, d^G2 ˆ1:5 h, d^miˆ1 h (giving an

estimate of the mean time duration of the generation periodd^Gˆ11 h) andd^M ˆ10 h. This last

The PCE are anucleated except in case of chromosomal mutation which leads to the formation of micronuclei in their precursors. These micronuclei are DNA strands. They may be induced spontaneously by natural mutation: between 0 and 0.5% of the PCE among mice are usually micronucleated by that way. This phenomenon is supposed to be stationary. The administration of a mutagenic compound can also increase their induction. The action of the chemical is sup-posed to be independent of the stage of the precursor, that is of the number of remaining cycles before the nucleus expulsion: in other words, a cell which has to divide itself two times before transforming into PCE has the same probability to be micronucleated at the end of its cycle than a cell which has only one division to realized before transforming into PCE, provided these cells are in the same cycle position. On the contrary, the action of the chemical agent on PCE can not only di€er during the four periodsG1, DNA synthesis,G2 and mitosis but it can also be limited only to

a part of them.

In Section 2.2, the probability for a PCE to be micronucleated only by the chemical compound was derived. To take into account the induction of micronuclei by natural mutation, a little modi®cation of the model is required. Denote byP…t† the probability for a cell just transformed into PCE at timetto be micronucleated. Letp0 andp…t†denote the probabilities for this cell to be

micronucleated respectively by natural mutation and by the chemical agent. Suppose ®rst that micronuclei induce no cellular death, that ismˆ0, and that the natural and the chemical processes of micronuclei induction are independent. Then, it follows from the identity

P…t† ˆPr…fmicronucleated by natural mutationg [ fmicronucleated by the chemicalg†;

that a new PCE is micronucleated with probability

P…t† ˆp0‡ …1ÿp0†p…t†:

In case of cellular death, it is still possible to take into account the induction of micronuclei by natural mutation. This can be done by adding a constant in the probabilities of transition Pu;h

given in Section 2.2 and it is also required biological assumptions on the induction of natural micronuclei that are not done in the present paper.

3.3. The 6-mercaptopurine

The 6-mp is a mutagenic compound inducing micronuclei in nucleated cells of the red blood lineage. The chemical is supposed to induce damages mainly on precursor cells that are in phase of DNA synthesis [24]. For simplicity the probability of formation of micronuclei is assumed to be proportional to the amount of 6-mp in plasma and, or equivalently, to the plasma concentrations denoted by C…t†. The model includes also a delay needed by the chemical to hit its target. This delay is denoted bys. Therefore, the functionF of Section 2.2 is given by

F…t† ˆ K₀SC…tÿs† if the cell is in DNA synthesis at time_otherwise t;

;

whereKS denotes the parameter describing the toxicity of the 6-mp on cells that are in phase of

DNA synthesis.

3.4. Numerical experiments and observed data

A ®rst sampling design has been planned to get a preliminary data set about the in vivo mi-cronucleated PCE induced in mice by 6-mp. Eleven groups of ®ve animals were administrated with 50 mg/kg bw of 6-mp by IP administration. Each animal gave a single observation and each group gave ®ve observations for a given sampling time. The data are plotted in Fig. 4 which represents the experimental time course of the rate of micronulceated PCE. After a lag time of about 20 h, the rate of micronucleated PCE reaches a maximal value of about 3% 42 h PI. This peak is single and the rate then returns to its original level of about 0.3%.

Theoretical predictions of the kinetics of the rate of micronucleated cells have been computed with the mathematical model under the previous assumptions. Some parameters were estimated by using the preliminary study: the empirical mean of controls provided p^0 ˆ0:3% and the lag

time in Fig. 4 gave d^G2‡d^mi‡d^M‡s^ˆ24 h. Indeed, this lag time is the sum of mean time

du-ration of aG2phase, a mitosis phase, a maturation phase and the delay…s†needed by the chemical

to hit its target. To simplify, the cellular death was not taken into account for these simulations

micronuclei induced within a cell cycle can be approximated with a Poisson distribution and it has a small probability to be larger than one. Therefore, c…1;0† ˆc…0;0† ˆ1=2 has been chosen to describe the allocation of the micronuclei at time of division.

The rate of micronucleated PCE, that is the functionR…t†given in eq. (5), after administration of a dose of 6-mp at the rate of 50 mg/kg bw has been simulated several times, with di€erent parameter values. These values were taken in a neighborhood of those coming from the literature or from the preliminary assay. Fig. 5 shows one of them. After a lag time, the rate increases quickly and then returns to its natural level. The simulated curve presents also a wave following the time of maximum rate. This wave is not very marked because of a too large variability in transit time distributions and, in fact, a succession of waves may be suspected. They did not appear in the kinetic of the ®rst experimental design perhaps because the sampling times were to distant from each others.

Therefore from this remark, a second assay has been performed for checking the accuracy of these predictions. The experimental design was the same as in the preliminary study except for the sampling times: the rate of micronucleated PCE has been observed every four hours, from 20 to 92 h PI. The data are plotted on Fig. 6 which gives the observed rates of micronucleated PCE in function of time and their empirical means per sampling times, joined by a solid line. These results are discussed in the following section.

3.5. Discussion

In both studies the observed lag time is about 20 h and agrees with previous results obtained for 6-mp [4]. This lag time represents ®rst the delay needed by the 6-mp to hit its action site and to be metabolized to its active form, and secondly the time for precursor cells that are in last DNA synthesis to cease cycling and next to expel their main nucleus for producing PCE. No data were available in the literature for the ®rst delay. Therefore it is dicult to estimate a priori this pa-rameter. Estimates of the mean time duration of maturation phases are generally situated between 6 and 10 h [3,4,25,26]. Jenssen and Ramel [1] found that the time from DNA synthesis step to nucleus expulsion lasts about 12 h, meaning time for G2 and mitosis was 2±6 h. Here G2 and

mitosis gathered was assumed to last in average 2.5 h, considering data from Cole et al. [2]. Therefore, from these references, intracellular distribution and metabolism of the 6-mp in the bone marrow lasted at least 8 h in average.

In both experimental results, the maximum rate of micronucleated PCE was near 3%. This frequency is less than values given in literature: 6% in Hayashi et al [4]. Several arguments may explain this di€erence. First, the examination of PCE has not been fully standardized from a laboratory to another and there exists a reader e€ect. Each reader applies his own criteria to conclude whether or not a PCE is micronucleated. Anyway, the rules generally applied are

sucient to ensure a good reproducibility in terms of `positive' or `negative' response owing to the use of internal controls. Second, the rates of micronucleated PCE displayed large interindi-vidual variations. Therefore, one can expect that such variations may also exist among mice strains. These variations can be explained by the variability of the animal sensitivity. Many factors are involved in these di€erences as the various speed of metabolism or of absorption for example.

The existence of successive decreasing peaks in the kinetics of micronucleated PCE was sus-pected with the simulations. Several factors are involved in these theoretical results. First, these simulations have been performed under the assumption that only precursor cells in phase of DNA synthesis are exposed to the chemical. This hypothesis leads to an alternance of exposure and non-exposure periods during the generation and maturation phases. Second, as mentioned in Section 3.1, the 6-mp is quickly eliminated from blood. So, the window of exposure is relatively short compared with length of cell cycles. Therefore, some precursor cells are hardly exposed to the 6-mp explaining that new micronucleated PCE arrive with waves in the population of PCE. Moreover, it has also been supposed that exposed cells could be at one division or more from the nucleus expulsion. Then, at each division, the probability for a cell to be micronucleated is divided by two. This last point explains that the peaks decrease progressively. The desynchronisation of precursors, modeled with exponentially distributed generation and maturation phases, lead to a ¯attened kinetic in Fig. 5.

Experimental kinetics of micronucleated PCE rates of the second study present also a suc-cession of waves as predicted by the theoretical model. To our knowledge, nobody remarked successive peaks as these. As they have not been detected in the ®rst study, one can expect that they could not be detected unless the protocol design is speci®cally built to detect them. Note that these waves can explain divergent results. Indeed, if the sampling times are placed in the trough between two waves, a statistically signi®cant toxic e€ect is more dicult to underscore or need more observations.

The observed delay between two successive peaks is di€erent from the predicted one. This gap represents the time that separates two successive DNA synthesis steps, i.e., one generation time. In the present experiment this time was about 24 h. Data taken from the literature and used to get the theoretical simulations give a cellular cycle duration of about 10 or 12 h [2,3,11,12]. Two main hypotheses can explain this di€erence. First, it is known that compounds acting on the DNA synthesis phase can perturbate and stretch out cell cycle delay [3]. Second, it can be supposed that cell cycles can be separated with cell rest periods, called `G0 phases'.

This complex phenomenon has been described for early precursors of PCE, but to our knowledge it has not been documented for the 3 or 4 last stages of these cells [27]. The last predicted micronucleated PCE rate is higher than the observed one, but cell death process was not taken into consideration for the simulations. Otherwise, this di€erence would have probably decreased. Be that as it may, the experimental data seem to con®rm the assumption on which the model has been built, postulating that the chemical agent can induce damages in every nucleated precursors, independently of their stage. Some authors have omitted this hypothesis and they have estimated parameters of the formation of micronucleated PCE considering that only last stage precursors were exposed to the chemical. This position led probably to biased estimates.

The model was built on the assumption that generation and maturation times were expo-nentially distributed. This assumption is usually made for memoryless systems: in case of cell cycles it is clearly an approximation leading to a more or less precise description of the reality. However, this hypothesis has proved to be sucient in the particular case for improving the description of the kinetic of the rate of micronucleated PCE and for optimizing the experimental design.

4. Conclusion

A mathematical model has been developed to predict the kinetics of the rate of micronucleated PCE in mice after administration of a chemical compound. This model is made up of stochastic processes and it is based on basic biological assumptions. Each of its parameters has a biological meaning. This enables to interpret easily the results by drawing a parallel with the biological phenomenons. This model takes both into account the kinetic behaviour of the erythropoietic system, the pharmacokinetic pro®le of the chemical and its mechanisms of action on precursor cells. Hence, it establishes a link between the toxicological e€ect induced by the compound and the time course of its concentrations.

the use of simulations, the model has guided the experimenter for improving a preliminary pro-tocol and for choosing appropriate sampling times to put to light a wavy kinetic. The observed kinetic was similar to the theoretical one and it has enabled to make interesting remarks about the formation of micronuclei in precursor cells as well as about the erythropoiesis. By the use of experimental data, the model can also provide a quantitative analysis of both the mechanisms of action of the compound and of the erythropoietic system.

The simulated kinetics plotted on Fig. 7 has been performed with sˆ11:5 h and with a phase G0 of length 13 h. The empirical means of the observed data per sampling times are

symbolized with …† and the bars indicate three standard errors of the empirical means. Pres-ently, the waves appear better on the simulated kinetics. A graphical examination of the curve enables to see that the model overestimates precursor cells desynchronization since the real generation and maturation phases are not exponentially distributed. An improvement of the model can be expected nevertheless by changing this assumption and by considering non-markovian models that are much more complex than the model developed here. It has also been assumed that accumulation of damage in a cell does not have any e€ect on cell survival. This hypothesis can be modi®ed in such a way that the residual survival shortens as damages in-creases. These changes would probably improve the ®t and provide better estimates of the parameters.

Appendix A. Proofs

Proof of Lemma 1. Proof of (i)

If Z1…0† ˆ0 and ifZ1…T_kt†>0, then there exists at least one time in Š0;T_ktŠ for whichZ1…u†

in-creases. Let zmax…Tkt† denotes the last of these times satisfying the following identities:

Z1…zmax…T_kt†† ˆZ1…zmax…T_kt†ÿ† ‡1;

and the result stated in (i) of Lemma 1 follows becauseB_i…Tt

k† is equivalent to the event

fzmax…T_kt† 2ŠT_it_‡1;T

The proof is straightforward since the Tt

Proof of Theorem 1. In this proof, the probabilities are implicitly conditional on E_Tt

Moreover, by the use of the probabilities of transitions at time Tt

Expression of P1: Write ®rst

in damaged cells only. Therefore the ®rst probability in the right-hand term equals one and

P1ˆPr Z1…T_kt†

which can be also written as follows

P1ˆPr Z1…T_kt_ÿÿ1† So, the following expression for P1 is obtained

P1ˆ …1ÿh…T_kt††exp ÿ

Expression of P2: Proceed here by considering the decomposition

P2ˆP21P22;

First, by using the time of induction of a micronucleus between timesTt

Second, by de®nition of h…Tt

The result stated in Theorem 2 follows from these four expressions.

Proof of Corollary 1.The proof of this Corollary is straightforward by the use of the markovian

property of the process Z2…t†:

The authors are very grateful to Professor Jean Deshayes (University of Rennes 1) and to Dr Jean Michel Poul (AFFSSA Fougeres) for fruitful discussions. They are also indebted to the editor and the reviewers whose comments have led to the improvement of the manuscript.

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