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Mathematical Biosciences 169 (2001) 27±51
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Stochastic modeling in toxicokinetics.
Application to the in vivo micronucleus assay
Ollivier Hyrien a,*, Virginie Kles a, Didier Concordet b, Michel Bonneu c,
Michel Laurentie a, Pascal Sanders a
Agence Francßaise de S
ecurit
e Sanitaire des Aliments, Laboratoire d'Etudes et de Recherches sur les M
edicaments
V
et
erinaires et les D
esinfectants, BP 90203, 35302 Foug
eres cedex, France
Ecole Nationale V
et
erinaire de Toulouse, Unit
e Associ
ee INRA de Physiopathologie et Toxicologie Exp

erimentales,
23 chemin des Capelles, F31076 Toulouse cedex, France
c
Equipe GRIMM, Universit
e de Toulouse II, IUT-B, BP73, 31073 Blagnac, France

a

b

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 enabled 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

*

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.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 5 2 - 3

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

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 PI in Section 2 to simplify.
Each cell of PI 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 nucleus. 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 memoryless 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 PI is irreversible. In that case, the cell is eliminated of PI and gets into the population of polychromatic
erythrocytes (PCE) which is denoted by PII to simplify. Damages induced previously in PI keep
present and become easily detectable in PII . The molecule is supposed to have no activity on the
cells of PII . In addition, these cells have not the possibility to divide themselves and after a differentiation delay they leave PII and turn into normochromatic erythrocytes (NCE). For the same
reasons as above, the di€erentiation delay of cells of PII has been assumed to be exponentially
distributed. This led to describe the number of cells of PII 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
PI . 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 micronuclei 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.
The usefulness of the model, that is its ability to analyse the induction of micronuclei and
possibly to improve the experimental design of the in vivo micronucleus assay, has been evaluated
through the study of a mutagenic compound, 6-mp. The theoretical model applies in principle to
any chemical and for this it was required to adapt it to this particular compound by specializing
the mechanistic assumptions and also by modeling the plasma concentrations. The parameters of
the model were estimated by values coming either from the literature or from a preliminary assay.

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

29

Fig. 1. This ®gure represents the processes of generation and maturation of the precursor cells.

Numerical experiments have been conducted to get theoretical predictions of the rate of micronucleted 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 experimental in vivo micronucleus assay was then performed which con®rmed the theoretical predictions.
The model is connected with stochastic models used in carcinogenesis studies, such as for instance 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 mathematical analysis of bone marrow erythropoiesis and derived estimates of cell kinetic parameters.
Ludwikow and Ludwikow [13] have developed a bicompartmental model describing the formation of micronucleus in PCE. Their model is based on the assumption that the chemical action is
constant during a period T . 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 population PI is detailed in Section 2.1. Next, in
Section 2.2, the bidimensional markovian process used to describe both the number of micronuclei contained in a cell lineage of PI and its state of live or death is given, and the probabilities for a precursor cell leaving PI at any time t > 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

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

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 parameters 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 of PII , is developed in this section. Recall that this model is based on three kinds of stochastic
processes: a branching process modeling PI (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 of PII (see Section 2.3).
2.1. A branching process for the growth of PI
In this section a branching process modeling the stochastic growth of PI is de®ned and some
terminology used through out the paper is introduced.
First, PI …t† denotes the collection of cells of PI at time t. Every cell of PI can either split up into
o€springs after a generation time or leave the population after a maturation time without dividing. Thus, a cell born at time 0 has a random generation time G with the probability distribution g…x† ˆ Pr…G 6 x†. 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 distribution m…x† ˆ Pr…M 6 x†. 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 when G and
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 distributed 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 cell C; C1 denotes the mother cell of C, i.e., the cell which gives birth to C
after one division, C2 denotes the mother cell of C1 and so on.
De®nition 1. Pick a cell of PI , denoted by C. For all k 2 N, a family Fk …C† is the collection of
cells
Fk …C† ˆ fC0 ; C1 ; . . . ; Ck g;

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

31

where C0 ˆ C for convenience. For all k 2 N, a family history Hk …C† is the collection of random
non-negative values
Hk …C† ˆ fdC ; G1 ; . . . ; Gk g;
where Gi denotes the generation time of the cell Ci 8i ˆ 1; . . . ; k, and

GC if C divides itself at age GC ;
dC ˆ
MC if C differentiates at age MC :
The set Fk …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 by Ck and its descendants to give birth to C.
For any time t > 0 and for any cell C t 2 PI di€erentiating at t, FK0 …C t † denotes the family of C t
initiated by CKt 0 . By convention, this cell CKt 0 is the cell of FK0 …C t † belonging to PI …0†, that is the
®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 FK0 …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 Tkt be the time of division of Ckt , k ˆ 1; . . . ; K0 ; T0t the time of
di€erentiation of C t and TKt 0 ‡1 the time of birth of CKt 0 . Note that the identity below is satis®ed for
all time t > 0
t ˆ MCt ‡

K0
X

Gtk ÿ TKt 0 :

kˆ1

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
t
‰ into subintervals (cf. Section 3). An example of cell lineage is presented in
the intervals ‰Tkt ; Tk‡1
Fig. 1: the family of C t is composed of three cells, FK0 …C t † ˆ fC0t ; C1t ; C2t g. The compound is
administrated at time 0 and the cell C2t is then exposed to its toxicity up to the end of its generation
phase which occurs at time T2t . This cell divides then into two new o€springs and the cell-lineage
FK0 …C t † is exposed to the compound through the cell C1t . Since two divisions are necessary to
obtain C t from C2t , K0 ˆ 2.
2.2. Modeling the induction of damages in PI
In this section, an N  f0; 1g-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-lineage FK0 …C t † at time
u, where C t is a cell leaving PI at a positive time t. The second component describes the possible
death of this cell-lineage: by convention the cell-lineage is alive at time u if Z2 …u† ˆ 0 and dead if
Z2 …u† ˆ 1. This death process refers to death due to the presence of damages in the cell and not to
the disparition of cells from the population PI after division or di€erentiation. In order to simplify
the model, dead cells have been let divide; the process of cells proliferation presented in Section

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

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
C t is both alive and healthy at time t, that is
Pr…Z…u† ˆ …0; 0† j Z…0† ˆ …0; 0††
and the probability that C t is both alive and damaged at time t, that is
Pr…Z…u† ˆ …n; 0†; n > 0 j Z…0† ˆ …0; 0††:
t
; Tkt ‰,
To this end, the transitions for this process are ®rst de®ned on any semi-open intervals ‰Tk‡1
for all k ˆ 0; . . . ; K0 . Next, the transitions at the times Tkt that involve changes in the dynamic of
Z1 …u† are considered. For all time u and for all h > 0, set

Pu;h …k1 ; l1 ; k2 ; l2 † ˆ Pr…Z…u ‡ h† ˆ …k2 ; l2 † j Z…u† ˆ …k1 ; l1 ††:
t
; Tkt ‰, the probabilities for the transitions of this process
For all k ˆ 0; . . . ; K0 and for all u 2 ‰Tk‡1
from time u to time u ‡ h, with h > 0, are de®ned according to the following three cases.
Case 1: Z…u† ˆ …0; 0†. At time u, the cell is healthy and alive. It remains in that state up to time
u ‡ h with probability

Pu;h …0; 0; 0; 0† ˆ 1 ÿ F …u†h ‡ o…h†;
where F 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† ˆ F …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† ˆ 0:
The other transitions are assumed to occur with a probability of order o…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 n damages. First, the probability that the cell remains in that state at time u ‡ h is
Pu;h …n; 0; n; 0† ˆ 1 ÿ …F …u† ‡ m†h ‡ o…h†;
where m is 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 time u ‡ h is
Pu;h …n; 0; n ‡ 1; 0† ˆ F …u†h ‡ o…h†:
The probability that the cell dies but that no damage appears is
Pu;h …n; 0; n; 1† ˆ mh ‡ o…h†:
The other transitions are assumed to occur with a probability of order o…h†.
Case 3: Z…u† ˆ …n; 1†, with n P 0. Here, the cell is dead at time u and possibly damaged if n > 0.
The states …n; 1† are supposed to be absorbing, that is
Pu;h …n; 1; n; 1† ˆ 1:

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

33

t
One can easily remark that Z1 …u† is non-decreasing on each interval ‰Tk‡1
; Tkt ‰, but at the times
Tkt ; k ˆ 1; . . . ; K0 , the dynamic of the process Z1 …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 all k 2 f1; . . . ; K0 g, for all n > 0 and for all i ˆ 0; . . . ; n

Pr…Z…Tkt † ˆ …i; 0† j Z…Tktÿ † ˆ …n; 0†† ˆ c…n; i†;
where c…
;
† is a given application satisfying the condition
(ii) in case of a dead lineage
Pr…Z…Tkt † ˆ …n; 1† j Z…Tktÿ † ˆ …n; 1†† ˆ 1:

Pn

iˆ0

c…n; i† ˆ 1;

Fig. 2 gives an example of trajectory of the process Z1 …u† which increases by jumps on the intervals
of time ‰0; T2t ‰; ‰T2t ; T1t ‰ and ‰T1t ; T0t ‰, and which possibly decreases when a division occurs. When
leaving PI at time t, C t is damaged and contains one damage.
The function F links the toxicological e€ect and the amount of chemical, denoted by Q…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 proportional to the amount of chemical at time t, then
F …t† ˆ KQ…t†;
where K 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 the Z1 …Tktÿ † damages are allocated in each of the two
cells with probability 1/2, Z1 …Tkt † is a binomial random variable with parameters Z1 …Tktÿ † and 1/2.
In that case, for all n > 0; c…n; i† ˆ Cni 2ÿn .
De®ne now for all time t > 0 the following events: for all x 2 ‰0; tŠ, E…x† ˆ fZ2 …x† ˆ 0g, meaning
that the cell-lineage FK0 …C t † is alive at time x, and A…x† ˆ fZ1 …x† > 0 j E…x†g, meaning that the

Fig. 2. This ®gure gives an example of trajectory of Z1 …u†. This process describes the number of microunclei contained
in a cell lineage di€erentiating at time t…ˆ T0t †. After administration of the compound at time 0, two divisions occur at
times T2t and T1t . At time 0, Z1 …u† ˆ 0 that is the cell lineage is healthy. On each time domain
t
‰0; T2t ‰ and ‰Ti‡1
; Tit ‰; i ˆ 0; 1; Z1 …u† increases by jumps. On the contrary, the process can decrease at times Tit ; i ˆ 0; 1; 2;
as shown here.

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

cell-lineage is damaged at time x given that it is also alive. For all k 2 f0; . . . ; K0 g and for all
x0 2 ‰Tkt ; tŠ, de®ne also the time zmax …x0 † ˆ maxfz 2 ‰0; x0 Š=Z1 …z† ˆ Z1 …zÿ † ‡ 1g, which is the time of
appearance of the last damage in the cell lineage and
t
; Tkt ‰ and 8y 2 ‰Tkt ; x0 Š; Z1 …y† > 0 j E…x0 †g
Bk …x0 † ˆ fzmax …x0 † 2ŠTk‡1
t
meaning both that the last damage of the cell-lineage at time x0 has appeared between times Tk‡1
t
0
and Tk and that the cell lineage keeps micronucleated up to time x , given the cell lineage is still
alive at that time. The events
t
Ck ˆ f9z 2ŠTk‡1
; Tkt Š=Z1 …z† ÿ Z1 …zÿ † ˆ 1 j E…Tkt †g
t
mean that at least one damage has occurred between the times Tk‡1
and Tkt . The complementary of
c
Ck is denoted by Ck .

Lemma 1. For all k ˆ 0; . . . ; K0 and for all i ˆ k; . . . ; K0 , the events A…Tkt †; Bk …Tkt †, and Ck satisfy
the following relations:
…i†

A…Tkt † ˆ

K0
[
Bi …Tkt †;
iˆk

…ii†

8i; j 2 fk; . . . ; K0 g; with i 6ˆ j; Bi …Tkt † \ Bj …Tkt † ˆ ;;

…iii† 8i > k; Bi …Tkt † ˆ Ci

iÿ1
\

Ccn \ f8j ˆ k; . . . ; i; Z1 …Tjt † > 0g:

nˆk

The following notation has been adopted
Pr0 …
† ˆ Pr…
j Z…0† ˆ …0; 0††:
t
t
; Tkt ‰, the process Z1 …x† ÿ Z1 …Tk‡1
† is a non-homogeneous markovian
On each time domain ‰Tk‡1
t
point process, with rate F …x† [16]. Therefore, Z1 …x† ÿ Z1 …Tk‡1 † has the Poisson distribution with
parameter
Z x
t
F …s† ds
; x† ˆ
K…Tk‡1
t
Tk‡1

and the following lemma can be stated.
Lemma 2. Let F be a bounded measurable function for all time t. Then, 8k 2 f0; . . . ; K0 g,
t
; Tkt ††:
Pr0 …Ck j HK0 ; Z2 …Tkt † ˆ 0† ˆ 1 ÿ exp…ÿK…Tk‡1

By use of Lemmas 1 and 2, one derived Theorem 1. It gives the probability for a cell that ceases
cycling at time Tkt to be damaged conditional on the history HK0 …C t † and given that the cell is still
alive at that time, that is fZ2 …Tkt † ˆ 0g.

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

Theorem 1. For all time t > 0; consider a cell leaving PI at time t, denoted by C t ; and its family
history HK0 …C t †: Then, for all k ˆ 0; . . . ; K0 ;
Pr0 …Z1 …Tkt † > 0 j HK0 …C t †; Z2 …Tkt † ˆ 0†
Z
K0
X
ÿ

ˆ
E Ckn 1 ÿ exp ÿ

!!

Tnt

F …x† dx

t
Tn‡1

nˆk

ÿ

exp

Z

Tkt

!

F …x† dx ;

Tnt

Q
where Ckn ˆ niˆk …1 ÿ c…Z…Titÿ †; 0†† and where the expectation is taken with respect to the
…Titÿ ; i ˆ k; . . . ; K0 †:
Theorem 2 provides the probability for a cell to die during its cycle.
Theorem 2. For all time t > 0, consider a cell leaving PI at time t, denoted by C t , and its family
history HK0 …C t †. Then, for all k ˆ 1; . . . ; K0 ;
t
Pr0 …Z2 …Tkÿ1
† ˆ 0 j HK0 …C t †; Z2 …Tkt † ˆ 0†

ˆ

t
h…Tkt † exp…ÿm…Tkÿ1

ÿ

Tkt ††

t
ÿ u† ÿ
ÿ m…Tkÿ1

 exp

‡ …1 ÿ
Z

Z

h…Tkt ††

u

F …x† dx

Tkt

!

t
Tkÿ1

F …u†

Tkt

du ‡ exp

ÿ

Z

t
Tkÿ1

Tkt

!!

F …x† dx

;

where h…Tkt † ˆ Pr0 …Z1 …Tkt † > 0 j HK0 …C t †; Z2 …Tkt † ˆ 0†:
The probability for a cell leaving PI at time t to be damaged is deduced from Theorem 2:
Corollary 1. For all time t > 0; consider a cell leaving PI at time t, denoted by C t ; and its family
history HK0 …C t †: Then, for all k ˆ 1; . . . ; K0 ;
0

ÿ

t




0

Pr Z2 …t† ˆ 0 j HK0 …C † ˆ Pr



Z2 …TKt 0 †



t
†ˆ0 :
ˆ 0 j HK0 …C t †; Z2 …Tk‡1

t



ˆ 0 j HK0 …C † 

KY
0 ÿ1
kˆ0

ÿ
Pr0 Z2 …Tkt †

By use of Theorem 1 and Corollary 1 one can derive the probability for a cell C t to be both
healthy and alive when leaving PI , conditional on HK0 …C t †; and on fZ…0† ˆ …0; 0†g


ÿ


ÿ


0
0
t
t
Pr Z…t† ˆ …0; 0† j HK0 …C † ˆ 1 ÿ Pr Z1 …t† > 0 j HK0 …C †; Z2 …t† ˆ 0
…1†
ÿ


Pr0 Z2 …t† ˆ 0 j HK0 …C t † ;
and the probability for the cell C t to be both damaged and alive when leaving PI , conditional on
HK0 …C t †; and on fZ…0† ˆ …0; 0†g

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O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

ÿ


ÿ


Pr0 Z1 …t† > 0; Z2 …t† ˆ 0 j HK0 …C t † ˆ Pr0 Z1 …t† > 0 j HK0 …C t †; Z2 …t† ˆ 0
ÿ


Pr0 Z2 …t† ˆ 0 j HK0 …C t † :

…2†

The marginal probabilities Pr0 …Z…t† ˆ …0; 0†† and Pr0 …Z1 …t† > 0; Z2 …t† ˆ 0† are obtained by taking
the mathematical expectation of (1) and (2) with respect to the Tkt ; k ˆ 0; . . . ; 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 PII
This section deals with the population of polychromatic erythrocytes denoted by PII : These
cells cannot divide since they are anucleated and after a time of di€erentiation they leave PII : 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 population PI after they
di€erentiate by expulsion of their main nucleus. The population of damaged (resp healthy) cells is
denoted by P1II (resp. P2II †: 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 PII 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† ˆ n ‡ 1 j N…t† ˆ n† ˆ k…n; t†h ‡ o…h†;
Pr…N …t ‡ h† ˆ n ÿ 1 j N…t† ˆ n† ˆ l…n; t†h ‡ o…h†:
The other transitions are supposed to occur with a probability of order o…h†: Therefore
Pr…N…t ‡ h† ˆ n j N …t† ˆ n† ˆ 1 ÿ …k…n; t† ‡ l…n; t††h ‡ o…h†:
The functions k…n; t† and l…n; t† are non-negative and de®ne respectively the birth and death rates
of the process.
When k…n; t† ˆ k…t† and l…n; t† ˆ l…t†n for all time t, with k…t† and l…t† two non-negative real
valued functions, the process N …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 parameter k…t† and the service times have an exponential distribution with parameter l…t† [19±22]. In the special case k…t† ˆ k and l…t† ˆ l with k and l positive
constants, N …t† is the homogeneous M=M=1 queue. In the sequel, birth and death rates of the
form

k…n; t† ˆ k…t†
8n 2 N; 8t 2 R
l…n; t† ˆ ln
are considered, where k…t† is a real valued function of time and l a positive constant. Any birth
and death process N…t† of that kind, with time dependent birth rate, is a non-homogeneous
queueing process Mt =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 if N …0† is Poisson, the
distribution of any birth and death process N …t† of C is Poisson for all time t > 0: In the present
case, the random variable N…0† denotes the number of damages contained in the cell when the

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

37

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 of N …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
f …t† ˆ k…t† ÿ lf …t†;
dt

…3†

whose solutions are


Z t
f …t† ˆ eÿlt f …0† ‡
elx k…x† dx :
0

Consequently, in case of a time homogeneous process, i.e., if k…t† ˆ k, then f …t† ˆ k=l for all time
t. The sum of independent processes is stable in C, that is if N1 …t† and N2 …t† are independent
processes of C, 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 of C, denoted by fNji …t†; i 2 Ng with mathematical expectations ffji …t†; i 2 Ng for j ˆ 1,2. Suppose thatPfor all i 2 N and for all j ˆ 1,2 the
m
random variables Nji …0† are Poisson and that Sj …t† ˆ limm!1 iˆ1 fji …t†=m exists for all t. Then, by
applying the law of large numbers to a sequence of independent Poisson random variables, the
following convergences are obtained:

and

m
1 X
a:s:
N1i …t†!S1 …t†
m iˆ1
m
1 X
a:s:
…N1i …t† ‡ N2i …t††!S1 …t† ‡ S2 …t†:
m iˆ1

Therefore, the rate of damaged cells in PII converges as follows:
Pm
N1i …t†
S1 …t†
Pr
Pm iˆ1
!
i
i
S1 …t† ‡ S2 …t†
iˆ1 …N1 …t† ‡ N2 …t††

…4†

as m goes to in®nity.
In order to model the dynamic of the two subpopulations P1II and P2II ; the bone marrow is
decomposed into m distinct and homogeneous areas, denoted by fAi ; i ˆ 1; . . . ; mg: In each area Ai
the number of damaged (resp. healthy) cells is described by a birth and death process denoted by
N1i …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

k1 …n; t† ˆ k1 …t†
…resp:k2 …n; t† ˆ k2 …t††;
8n 2 N; 8t > 0
l1 …n; t† ˆ ln
…resp:l2 …n; t† ˆ ln†;
where k1 …t† and k2 …t† are non-negative valued functions and l is a positive constant. In each area
Ai and for all time t; the expected number of damaged and healthy cells, denoted respectively by
f1i …t† and f1i …t†; satisfy the linear di€erential equation given in Eq. (3), that is

38

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

d
d
f1 …t† ˆ k1 …t† ÿ lf1 …t† and
f2 …t† ˆ k2 …t† ÿ lf2 …t†:
dt
dt
Let N1T …t† (resp. N2T …t†† denotes the number of cells of P1II (resp. P2II † at time t. These stochastic
processes are de®ned by
m
m
X
X
N1T …t† ˆ
N1i …t† and N2T …t† ˆ
N2i …t†:
iˆ1

iˆ1

Since the sum of independent processes is stable in C; N1T …t† and N2T …t† are birth and death processes of C; with respective birth and death rates mf1 …t† and l; mf2 …t† and l. Their mathematical
expectations, denoted by f1T …t† and f2T …t†, respectively, are therefore de®ned by
f1T …t† ˆ mf1 …t†

and f2T …t† ˆ mf2 …t†:

The integer m is assumed to be large enough to justify the approximation of the stochastic process
N1T …t†=…N1T …t† ‡ N2T …t†† by the function f1T …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 of PII is described by the function
R…t† ˆ

f1 …t†
:
f1 …t† ‡ f2 …t†

…5†

Moreover the rates of birth of the processes N1i …t† and N2i …t† are taken of the form
k1 …t† ˆ kP1 …t† and k2 …t† ˆ kP2 …t†;
where k is an arbitrary positive constant representing the rate of birth of cells of PII in each area
Ai ; i ˆ 1; . . . ; m: The function P1 …t† is the rate of damaged cells di€erentiating at time t while 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 of PI di€erentiating at time t to be both alive and damaged
P1 …t† ˆ Pr…Z1 …t† > 0; Z2 …t† ˆ 0†:
In a similar way, P2 …t† is the probability for a cell of PI di€erentiating at time t to be both alive and
healthy
P2 …t† ˆ Pr…Z1 …t† > 0; Z2 …t† ˆ 0†:
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
In this section, the performance of the model developed in Section 2 is evaluated by analyzing
the induction of microunclei in mice after a dose of 6-mp is administered. For this, it is required to
assign values to the parameters of the model and to de®ne the application F used in Section 2.2 for
linking the amount of chemical and the probability of damage induction at any time t P 0: Here,
the function F is ®rst de®ned from a pharmacokinetic model describing the time course of the

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

39

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 parameters are then chosen so that the theoretical kinetic is close to these experimental data. Numerical 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 experimental 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 bicompartmental model with an absorption phase was selected to ®t the data
C…t† ˆ A exp…ÿat† ‡ B exp…ÿbt† ÿ …A ‡ B† exp…ÿct†;
where C…t† is the plasma chemical concentration at time t; a, b and c are, respectively, the distribution, 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^ ˆ 2796:9; a^ ˆ
0:205; B^ ˆ 27:9; b^ ˆ 0:029 and c^ ˆ 0:211.
Fig. 3 shows the plasma concentrations and the ®tted data of 6-mp after intraperitoneal administration to mice. The maximal concentrations were rapidly reached, about 5 min PI and were
about 45 lg=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 population PII 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 population PI described in Section 2.1 by a branching process. The generation phases of these
precursors are made up of four distinct and successive periods called G1 , 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 ˆ 1 h, d^sy ˆ 7:5 h, d^G2 ˆ 1:5 h, d^mi ˆ 1 h (giving an
estimate of the mean time duration of the generation period d^G ˆ 11 h) and d^M ˆ 10 h. This last
value is refuted by Hart and Hartley-Asp [3] who obtained an estimate of about 5 h for the same
parameter. These values give an approximate idea of the values of the model parameters that are
obviously subject to variability.

40

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

Fig. 3. This ®gure shows the plasma concentration pro®le of 6-mp after a single intra peritoneal injection of 50 mg/kg
bw of 6-mp to mice (observed data (*) and predicted values (solid line)): 6-mp is quickly eliminated from plasma.

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 supposed 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 periods G1 , 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 by P …t† the probability for a cell just transformed
into PCE at time t to be micronucleated. Let p0 and p…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 is m ˆ 0, and that the natural and the chemical processes
of micronuclei induction are independent. Then, it follows from the identity

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

41

P …t† ˆ Pr…fmicronucleated by natural mutationg [ fmicronucleated by the chemicalg†;
that a new PCE is micronucleated with probability
P …t† ˆ p0 ‡ …1 ÿ p0 †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 by s. Therefore, the function F of Section 2.2 is given by

KS C…t ÿ s† if the cell is in DNA synthesis at time t;
F …t† ˆ
0
otherwise;
where KS 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 micronucleated 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 ˆ 0:3% and the lag
time in Fig. 4 gave d^G2 ‡ d^mi ‡ d^M ‡ s^ ˆ 24 h. Indeed, this lag time is the sum of mean time duration of a G2 phase, 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
…m ˆ 0†. In order to get a maximal frequency of about 3%, KS ˆ 0:007 was chosen. It has been set
k ˆ 100 hÿ1 , arbitrarily, and l ˆ 0:1 hÿ1 according to data from Tarburtt and Blackett, which
give an estimate of about 10 h for the life span of a PCE. The observed frequencies of micronucleated PCE stay under a low level (about 3.5%). Hence, the distribution of the number of

42

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

Fig. 4. This ®gure gives the kinetics of the rate of micronucleated PCE after a single intra peritoneal injection of 50 mg/
kg bw of 6-mp to mice (®rst experimental design; observed data (*) and their empirical means …†). After a lag time the
rate of micronucleated PCE quickly increases and then return to its natural level.

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† ˆ c…0; 0† ˆ 1=2 has been chosen to
describe the allocation of the micronuclei at time of division.
The rate of micronucleated PCE, that is the function R…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.

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

43

Fig. 5. This ®gure shows the theoretical kinetics of the rate of micronucleated PCE after a single intra peritoneal injection of 6-mp to mice simulated under the assumption that the generation and maturation periods are exponentially
distributed. The curve suggests the existence of successive decreasing peaks in the kinetics.

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 parameter. 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

44

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

Fig. 6. The ®gure displays the observed kinetics of the rate of micronucleated PCE after a single intra peritoneal injection of 50 mg/kg bw of 6-mp to mice (second experimental design; observed data (*) and their empirical means …†).
This second assay detects the succession of decreasing peaks suggested by the simulated rate plotted on Fig. 5.

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 interindividual 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 suspected 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 nonexposure 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.

O. Hyrien et al. / Mathematical Biosciences 169 (2001) 27±51

45

Experimental kinetics of micronucleated PCE rates of the second study present also a succession 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 exponentially distributed. This assumption is usually made for memoryless systems: in case of cell
cycles it is clearly an approxim