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Mathematical Biosciences 167 (2000) 109±121
www.elsevier.com/locate/mbs

Estimation problems associated with stochastic modeling
of proliferation and di€erentiation of O-2A progenitor cells
in vitro
Alexander Zorin a, Margot Mayer-Proschel b, Mark Noble b,
Andrej Y. Yakovlev b,*
a

The Central Research Institute of Radiology, 70/4 Leningradskaya Str., Pesochny-2, St. Petersburg 189646,
Russian Federation
b
Department of Oncological Sciences, Huntsman Cancer Institute, University of Utah, 2000 Circle of Hope,
Salt Lake City, UT 84112-5550, USA
Received 11 January 2000; received in revised form 30 June 2000; accepted 17 July 2000

Abstract
Our previous research e€ort has resulted in a stochastic model that provides an excellent ®t to our experimental data on proliferation and di€erentiation of oligodendrocyte type-2 astrocyte progenitor cells at
the clonal level. However, methods for estimation of model parameters and their statistical properties still
remain far away from complete exploration. The main technical diculty is that no explicit analytic expression for the joint distribution of the number of progenitor cells and oligodendrocytes, and consequently

for the corresponding likelihood function, is available. In the present paper, we overcome this diculty by
using computer-intensive simulation techniques for estimation of the likelihood function. Since the output
of our simulation model is essentially random, stochastic optimization methods are necessary to maximize
the estimated likelihood function. We use the Kiefer±Wolfowitz procedure for this purpose. Given sucient
computing resources, the proposed estimation techniques signi®cantly extend the spectrum of problems
that become approachable. In particular, these techniques can be applied to more complex branching
models of multi-type cell systems with dependent evolutions of di€erent types of cells. Ó 2000 Elsevier
Science Inc. All rights reserved.
Keywords: Branching process; Oligodendrocytes; Progenitor cells; Maximum likelihood; Stochastic approximation

*

Corresponding author. Tel.: +1-801 585 9544; fax: +1-801 585 5357.
E-mail address: yak@hci.utah.edu (A.Y. Yakovlev).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 4 0 - 7

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

1. Introduction
Mathematical models of cell population dynamics are designed to understand how events within
the cell cycle of individual cells can produce the dynamics observed for populations of cells.
Oligodendrocyte type-2 astrocyte progenitor cells (O-2A progenitor cells) are known to be the
precursor cells that, under certain conditions, generate oligodendrocytes in cell culture. In our earlier
publications [1±4], we explored a possibility of modeling the process of oligodendrocyte generation
in vitro as a multi-type Bellman±Harris branching stochastic process. When developing that model,
it was assumed that the founding O-2A progenitor cell acquires the competence for di€erentiation
only after it undergoes a certain number of critical mitotic divisions. It was also assumed that
`competent' progenitor cell di€erentiates into an oligodendrocyte with probability 1 ÿ p in each of
the subsequent mitotic cycles. Since experimental observations strongly suggest that the population
of progenitor cells in each clone eventually becomes extinct, the probability p is expected to be less
than or equal to 0.5. More speci®cally, the model was based on the following assumptions:
(1) The process begins with a single progenitor cell cultured at time t ˆ 0. At the end of the
mitotic cycle, every progenitor cell either gives rise to two progenitor cells with probability p, or it
di€erentiates into one oligodendrocyte with probability 1 ÿ p. In other words it is assumed that
the time to division and the time to di€erentiation are identically distributed. This assumption is a
forced one, otherwise the model would have been non-identi®able.

(2) Oligodendrocytes and O-2A progenitor cells do not die during the time of the experiment.
(3) The founding progenitor cell acquires the competence for di€erentiation only after it undergoes a critical number, N, of mitotic divisions. Upon completion these divisions the behavior
of progenitor cells is modeled as a subcritical branching stochastic process. In other words the
division probability is equal to 1 for the ®rst N cycles and it is equal to some value p 6 0:5 for
the subsequent cycles, i.e., for those cycles occurring once the critical cycles have been completed.
The parameter N is treated as a non-negative random variable with a ®nite discrete support, and
its distribution is to be estimated from experimental data.
(4) The lengths of the mitotic cycle of the initiator cell and its descendants of the same type are
independent and identically distributed non-negative random variables with a common cumulative distribution function F …t† such that F …0† ˆ 0.
(5) Cultured cells do not migrate out of the ®eld of observation.
(6) The usual independence assumptions regarding the evolution of age-dependent branching
processes are adopted.
The mathematical formulation of the above model is as follows. Let Zi …t†; i ˆ 1; . . . ; n be the
number of cells of type i at time t. Consider the n-dimensional stochastic process
Z…t† ˆ …Z1 …t†; . . . ; Zn …t†† and introduce the joint probability generating function
U…t; s† ˆ …U1 …t; s†; . . . ; Un …t; s††;
where s ˆ …s1 ; . . . ; sn †. The components of U…t; s† are given by
X
PrfZ…t† ˆ k j Z…0† ˆ ei gsk ; i ˆ 1; . . . n; j s j 6 1;
Ui …t; s† ˆ

k

…1†
…2†

where ei ; 1 6 i 6 n; represents the vector whose ith component is 1 and whose other components are
0. The summation in (2) is over the set of all points in Rn with non-negative integer coordinates.

A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

111

Let hi …s† be the generating function of the cell progeny and Fi …t† the corresponding cell cycle
time distributions for each cell type. It is well known (see for example [5]) that the generating
functions Ui …t; s† ˆ 1; . . . ; n satisfy the following general system of integral equations
Z t
Ui …t; s† ˆ si ‰1 ÿ Fi …t†Š ‡
hi …U…t ÿ u; s†† dFi …u†; i ˆ 1; . . . ; n
…3†
0

allowing for arbitrary transformations between distinct cell types.
In our branching model of oligodendrocyte generation, the ®rst N successive mitotic cycles are
represented by N di€erent types of progenitor cells while type N ‡ 1 is introduced to represent
those progenitor cells that have completed the ®rst N cycles. We introduce also type N ‡ 2 to
represent oligodendrocytes. Based on the above assumptions the generating functions hi …s† are
speci®ed as follows:
hi …s† ˆ s2i‡1

hN ‡1 …s† ˆ

for i ˆ 1; . . . ; N ;

ps2N ‡1

hN ‡2 …s† ˆ sN‡2 :

‡ …1 ÿ p†sN ‡2 ;

…4†

Given Fi …t† ˆ F …t† and hi …s† is speci®ed by (4), the system (3) assumes the form
Z t
U22 …t ÿ u; s† dF …u†;
U1 …t; s† ˆ s1 ‰1 ÿ F …t†Š ‡
0
Z t
U23 …t ÿ u; s† dF …u†;
U2 …t; s† ˆ s2 ‰1 ÿ F …t†Š ‡
0

..
.
UN ‡1 …t; s† ˆ sN ‡1 ‰1 ÿ F …t†Š ‡ p

Z

0

t

U2N ‡1 …t ÿ u; s† dF …u† ‡ …1 ÿ p†

Z

0

…5†

t

U2N‡2 …t ÿ u; s† dF …u†;

UN ‡2 …t; s† ˆ sN ‡2 :
In the above formulas, the parameter N can be treated as a random variable. We allow for an
arbitrary distribution of N:
p0 ˆ PrfN ˆ 0g;

p1 ˆ PrfN ˆ 1g; . . . ; pc ˆ PrfN ˆ cg

…6†

with a ®nite discrete support f0; 1; . . . cg, where c is the maximum number of critical mitotic cycles
and the parameters p0 . . . ; pc and c are to be included in an estimation procedure.
The main problem one faces when applying the model to experimental data on clonal growth
and di€erentiation of progenitor cells is that no explicit analytic solution of the system of
equations (5) is available. Therefore, the form of the joint distribution of the number of progenitor cells and oligodendrocytes at any given time after the start of experiment remains unknown, thereby preventing the use of the maximum likelihood techniques for estimation of model
parameters. To circumvent this obstacle in our previous studies we took advantage of the fact that
analytic expressions for the ®rst two moments of the number of both types of cells can be derived,
given the distribution F …t† is in®nitely divisible. We used the least squares method to ®t the expected number of progenitors and oligodendrocytes to the corresponding sample mean values. In
a recent paper [4], a partial likelihood function, based on an embedded random walk model of

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

clonal growth and di€erentiation of O-2A progenitor cells, was introduced with the aim to estimate the distribution of residual critical mitotic cycles and the probability p, and a EM-algorithm
was developed for estimation purposes. Under certain conditions the partial likelihood function
yields consistent estimates of model parameters. The estimates of p; c and p0 ; . . . ; pc obtained by
this method were found to be in good agreement with those resulted from the parametric analysis

based on the least squares techniques.
Owing to ineluctable loss of information inherent in both approaches, a further search for
possible ways of implementing estimation techniques based on maximization of the complete
likelihood function for the observations under study is warranted. In the present paper, we resort
to computer-intensive simulation methods to attain these ends. The main idea of the proposed
method is described in Section 2. In Section 3, we test the method by computer simulation. We
analyze experimental data and discuss our ®ndings in Section 4.

2. The maximum likelihood infereence
Let Z…t† be the number of O-2A progenitor cells and Y …t† the number of oligodendrocytes at
time t. As noted in Section 1 explicit analytic expressions for the joint distribution fZ…t†; Y …t†g of
progenitors and oligodendrocytes at time t are not feasible and therefore the complete likelihood
of the data cannot be computed. However, we take advantage of the fact that a relatively simple
simulation counterpart of our mathematical model is available. To make our simulations faster
we decided on developing an original simulation program without resorting to the use of commercial simulation software; our computer programs were written in FORTRAN.
We used the gamma distribution with parameters a and b to model the distribution, F …t†, of the
mitotic cycle duration. Thus, the model depends on the vector of parameters h with the components p0 ; p1 ; . . . ; pc ; p; a; and b. For each parameter vector h; we simulated a ®xed number K, of
clones at each time point t, for which we have biological data, in accordance with an algorithm
almost identical to the GPSS=H procedure described at length in our earlier paper [3].
Each simulated clone starts with one cell at simulated time t ˆ 0: The stop time t is ®xed in

advance. The parameters, pl ; l ˆ 0; . . . ; c; of the distribution of the critical number of cycles, the
parameters a and b of the duration of the mitotic cycle, and the probability p of progenitor self
renewal are speci®ed at each step of the optimization procedure described below. Using a random
number generator, the critical number of mitotic cycles is chosen independently for each clone,
and the duration of the mitotic cycle is chosen independently for each cell in the clone, in accordance with speci®ed values of the model parameters. The growth of each clone is stopped at a
®xed time and the number of progenitors and oligodendrocytes present at that time is recorded.
The pairs fZ…t†; Y …t†g thus simulated for each parameter vector h are used to construct
^ i ; yj ; h; t† of the joint distribution w…zi ; yj ; h; t† ˆ PrfZ…t† ˆ zi ;
the empirical counterpart w…z
Y …t† ˆ yj g; zi ; yj 2 f0; 1; . . .g: In other words, the joint distribution w…zi ; yj ; h; t† is viewed as a
discrete distribution on the product set of non-negative integers.
^ i ; yj ; h; t† are estimated by the corresponding relative frequencies of each pair
The values of w…z
fZ…t†; Y …t†g: Since our experimental design implies that each clone is examined only once at time
tk ; k ˆ 1; . . . ; r; resulting in mutually independent observations, the following estimate of the
likelihood function is obtained

A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

^ ˆ

L…h†

YY
k

i;j

^ i ; yj ; h; tk †;
w…z

113

…7†

which is unbiased and consistent.
Despite the fact that the number of simulated clones in our procedure is very large (up to
250 000 per time point), there is still a chance that, while a given combination fZ…t†; Y …t†g is
observed in the biological experiment, it is not present in the corresponding simulated sample.
When maximizing the likelihood function, one should expect that the proportion of such mismatches tends to increase with increasing the deviation of the estimated value of h from its true
value h0 . To account for possible mismatches, the corresponding factors (for which no match can
be found in the simulated sample) in the likelihood function were replaced by the reciprocal of the
^ The sample size for
number of simulated clones, leading to our ®nal likelihood for simulations SL.
simulation was chosen so that very few mismatches occurred in a neighborhood of the ®nal parameter estimate.
To maximize the penalized likelihood SL…h†; we applied the Kiefer±Wolfowitz stochastic approximation procedure as outlined in [6,7]. This method has the advantage that it does not require
derivatives. Let d be the dimension of the parameter space and …ei †; i ˆ 1; . . . ; d be the unit basis
vectors in the Euclidean space of dimension d. A sequence hn of parameter vectors is generated
using the formula

d 
X
SL…hn ‡ cn ei † ÿ SL…hn ÿ cn ei †
hn‡1 ˆ hn ‡ an
ei ;
2cn
iˆ1
where an and cn are some sequences of positive constants chosen as to ensure convergence of the
sequence
P hn :
P
P 2 2
If n an < 1;
n an  cn < 1;
n an =cn < 1 and certain regularity conditions hold [7], the
sequence hn will converge almost surely to the maximum likelihod estimate ^h of the parameters.
Under more restrictive assumptions asymptotic normality will apply [8].

3. The likelihood pro®les
The proposed methodology was ®rst tested by computer simulations. We simulated 480 pairs
fZ…t†; Y …t†g to be used as biological data. In doing so, we employed the same simulation model as
the one incorporated into the estimation procedure. The simulations were conducted at the following values of model parameters: a ˆ 16; b ˆ 0:8; p ˆ 0:45; c ˆ 2 p0 ˆ 0:33; p1 ˆ 0:33: The
observation times chosen in the ®rst set of simulation experiments were 72, 96, 120, and 144 h. A
total of 200 000 cell clones (50 000 per time point) were simulated for estimating the likelihood
function SL…h† at each step of the procedure by Kiefer and Wolfowitz. This number of simulation
runs seems to be more than sucient in the light of the results presented in Fig. 1. This ®gure
shows that the marginal distributions of Z…t† and Y …t† at a ®xed time (t ˆ 96 h) di€er little when
estimated non-parametrically from simulated samples of sizes 1000 and 1 000 000, respectively.
However, the proportion of mismatches (see Section 2) in a sample of size 1000 is rather high, and
we performed all our analyses of actual biological data with 200 000 (50 000 per time point)
simulation runs per iteration in the Kiefer±Wolowitz optimization procedure. As an additional

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

Fig. 1. Simulated marginal distributions of the number of progenitor cells (a) and oligodendrocytes (b) at 72 h after
plating a single progenitor cell from a seven day old animal in the presence of thyroid hormone. The distributions are
estimated from 1000 (open triangles) to 1 000 000 (open squares) simulated clones, respectively.

assurance of suciency of the selected sample size we repeated some of our calculations with
larger numbers of simulated cell clones.
The estimation procedure produces good estimates of the parameters incorporated into the
model whenever the initial points are chosen to fall not too far from their true values. A sample
experiment of this type is given in Table 1. The maximum likelihood estimation of a; b and p
appears to be quite stable to the choice of the initial parameter vector. This is much less so for the
parameters p0 and p1 : To gain greater insight into why the estimation procedure displays this
dissimilar behavior with respect to di€erent parameters we constructed the likelihood pro®les for
all the parameters under study. Shown in Fig. 2 are the likelihood pro®les for the parameters
b a; p; and p0 evaluated at the observation times 72, 96, 120, 144 h in accordance with formula
(7). While the likelihood pro®les for b; a; p demonstrate a clear-cut parabolic maximum, the
pro®le for p0 is quite ¯at. The same is true for p1 (and consequently for p2 ˆ 1 ÿ p0 ÿ p1 ). This
means that the data fZ…t†; Y …t†g contain less information on the distribution of critical cycles than
that on the other model parameters. To see whether the information on pl can be enriched
through a more judicious choice of the observation points, we carried out the same simulation
experiment with t ˆ 24; 48; 72; and 96 h. The results of this experiment are presented in Fig. 3. It is
clear from Fig. 3 that, while the likelihood pro®le for the parameter p ¯attens the pro®le for the
Table 1
Parameters

a

b

p

p0

p1

Initial value
Estimated values
True values

12
16
16

0.400
0.796
0.800

0.300
0.439
0.450

0.300
0.332
0.330

0.400
0.355
0.330

A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

115

Fig. 2. The likelihood pro®les for the basic parameters: (a) b, (b) a, (c) p, and (d) p0 , given the observation times t ˆ 72 h,
96 h, 120 h, 144 h after plating (simulation study).

parameter p0 is almost insensitive to the shift in the time points at which the composition of each
cell clone is examined.

4. Results of data analysis
In this study, we analyzed the processes of proliferation and di€erentiation of O-2A progenitor
cells isolated from optic nerves of 7-day-old rats. This is the same set of experimental data as we
used in [3]. Puri®cation of O-2A progenitor cells was carried out as described previously in the
paper by Ibarrola et al. [9]. Brie¯y cells were immunopuri®ed using a speci®c antibody-capture
assay. Panning with the Ran-2 antibody was ®rst used to eliminate type-1 astrocytes, followed by
panning with an anti-galactocerebroside (GalC) antibody to eliminate oligodendrocytes. O-2A
progenitor cells in the remaining cell suspension were then positively selected by their binding to a
positive selection dish coated with the A2B5 monoclonal antibody which labels O-2A progenitor
cells. Once the unbound cells were washed o€, the bound cells were scraped free and plated at
clonal density on poly-L -lysine coated 80 cm2 tissue culture ¯asks in 10 ml of basal growth medium, which consists of chemically-de®ned medium supplemented with 10 ng/ml of platelet-derived growth factor [9]. Such cultures routinely consist of >99.5% pure O-2A progenitor cells.
Cells were plated at a density of 2500±3000 and thyroid hormone (T3) was added to about a half
of cell cultures. Only clones consisting of a single cell at day 1 after plating were scored for the

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

Fig. 3. The likelihood pro®les for the basic parameters: (a) b, (b) a, (c) p, and (d) p0 , given the observation times t ˆ 24,
48, 72, 96 h after plating (simulation study).

presence of progenitor cells and oligodendrocytes. The number of clones examined for the cell
type composition at each time after plating varied from 106 to 131. A total of 458 cell clones
grown in the absence of H3 and 465 clones with added H3 were examined in our experiments.
Composition of a clone was determined by application of well-established morphlogical criteria
to the identi®cation of progenitor cells and oligodendrocytes. O-2A progenitor cells are distinguished from oligodendrocytes by analysis of the number of processes per cell (progenitors are
almost all unipolar and bipolar, while oligodendrocytes are multipolar) and by size and shape of
cell bodies (progenitors have small ovoid cell bodies, while oligodendrocytes have large round cell
bodies). In cases where cell shape is ambiguous (i.e., in which a cell is ®rst visualized very early in
the process of di€erentiation into an oligodendrocyte), the clone was re-scored several hours later
(by which time di€erentiation was more advanced). In practice, such problems with interpretation
are very rare. Comparison of such morphological analysis with immunohistochemical con®rmation of cell identity has consistently revealed complete agreement between these two methods.
Reproducibility of the biological system under study was con®rmed in several independent series
of experiments.
When ®tting the model by the least squares technique [3], we required the estimates of
p0 ; . . . ; pc and c to be equal for experiments with and without H3, thereby reducing the number of
unknown parameters. In doing so we relied on some indirect evidence from experimental data that
thyroid hormone does not exert any signi®cant e€ect on the number of critical mitotic divisions.
Now that we have a much more ecient estimation procedure, this constraint may be removed.

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

Let us compare the two estimation procedures. The least squares ®t through the sample mean
numbers of progenitor cells and oligodendrocytes resulted in the following estimates:
c^ ˆ 2; p^0 ˆ 0; p^1 ˆ 0:39; p^2 ˆ 0:61: It was noted in [3] that estimate p^0 ˆ 0 is suspect, since the
proportion of clones represented by a single oligodendrocyte is 3.3% in experiments with hormone

Table 2
Parameters
Least squares estimates,
experiments without T3
Maximum likelihood estimates,
experiments without T3
Least squares estimates,
experiments with T3
Maximum lillihood estimates,
experiments with T3

s ˆ a=b

p

p0

p1

p2

25.3

p
r a=b
5.7

0.500

0

0.390

0.610

26.3

15.2

0.504

0.030

0.363

0.607

21.3

9.5

0.350

0

0.390

0.610

20.0

14.1

0.348

0.028

0.360

0.612

Fig. 4. Comparison of the observed distribution of progenitor cells (bars) with the simulated distribution (open
squares) at: (a) 72 h, (b) 96 h, (c) 120 h, and (d) 144 h after plating a single progenitor cell from a 7-day-old animal in the
presence of thyroid hormone. The simulation results are from 500 000 simulated clones.

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

Fig. 5. Comparison of the obseerved distribution of oligodendrocytes (bars) with the simulated distribution (open
squares) at: (a) 72 h, (b) 96 h, (c) 120 h, and (d) 144 h after plating a single progenitor cell from a 7-day-old animal in the
presence of thyroid hormone. The simulation results are from 500 000 simulated clones.

and 3.2% in experiments without hormone. When applying the maximum likelihood estimation
procedure we set c ˆ 2, this value having been strongly suggested by both the least squares and
the partial likelihood inference. We also used the earlier obtained least squares estimates of all
model parameters as the initial points in the search for a maximum of the likelihood function,
which is common practice in such settings. In this analysis, we simulated 50 000 clones for each
time point per iteration in the Kiefer±Wolfowitz optimization procedure.
The estimates of p0 ; p1 ; and p2 resulted from the maximum likelihood estimation procedure are
not too di€erent from the corresponding least squares estimates (Table 2), but the estimate of p0
looks much more plausible. What is also important to note is that the maximum likelihood estimates obtained from experiments with and without thyroid hormone are very close to each
other, thereby lending support to the viewpoint that a few initial mitotic cycles represent a special,
relatively insusceptible to external signals, stage of the development of O-2A cells in vitro.
in Table 2 are also the resultant estimates of the parameters p; s ˆa=b; and r ˆ
pGiven

a=b; where s is the mean mitotic cycle duration and r is the corresponding standard deviation.
These estimates coincide closely with the corresponding least squares estimates, with one exception: the maximum likelihood estimates of r tend to be much larger than their least squares
counterparts in both sets of experiments. Figs. 4±7 compare non-parametric estimates of the
marginal distributions for both types of cells with their parametric counterparts simulated at the

A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

119

Fig. 6. Comparison of the observed distribution of progenitor cells (bars) with the simulated distribution (open
squares) at: (a) 72 h, (b) 96 h, (c) 120 h, and (d) 144 h after plating a single progenitor cell from a 7-day-old animal in the
absence of thyroid hormone. The simulation results are from 500 000 simulated clones.

parameter values suggested by the maximum likelihood estimation. It is clear that the model
provides a good description of the observed data at all times after plating.
It was suggested in [3] that the e€ect of thyroid hormone is twofold: it reduces the mean duration of the mitotic cycle for progenitor cells, and it increases the probability of their transformation into oligodendrocytes. The results presented in Table 2 reinforce this implication. It
should be also emphasized that the expected mitotic cycle duration in the absence of H3 estimated
by both methods is in remarkable agreement with the results of time-lapse video recording in
independent experiments [10]. As to the standard deviation r; the least squares estimate appears
to be in better agreement with the time-lapse measurements reported in [10].
It follows from the above application that the proposed estimation procedure performs well,
yielding estimates that by and large are consistent with those obtained by other methods. Generally, the maximum likelihood estimators enjoy some appealing large sample properties (asymptotic unbiasedness, eciency, etc.) under quite mild regularity conditions [11]; the same
cannot be said about the least squares method in this setting. What makes the proposed version of
the maximum likelihood estimation procedure especially valuable is that it can be easily extended
to even more complex branching models of cell proliferatio kinetics. One of the most challenging
problems in this ®eld of research is modeling and analysis of branching processes with dependent
evolutions of cells. In particular, many interesting attempts have been made to evaluate stochastic

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A. Zorin et al. / Mathematical Biosciences 167 (2000) 109±121

Fig. 7. Comparison of the observed distribution of oligodendrocytes (bars) with the simulated distribution (open
squares) at: (a) 72 h, (b) 96 h, (c) 120 h, and (d) 144 h after platting a single progenitor cell from a 7-day-old animal in
the absence of throid hormone. The simulstion results are from 500 000 simulated clones.

models of cell proliferation with dependent cell cycle times [12±26]. A signi®cant portion of this
work has been focused on time-lapse microphotography data. However, substantial mathematical
diculties still present the most critical deterrent to the development of sound methods of statistical inference from clonal data generated by a discrete observation process. The proposed
simulation-based estimation procedure holds much promise in such studies because it can accommodate any type of interactions between individual cells within each clone.
Acknowledgements
The research of A.Y.Y. was supported by the John Simon Guggenheim Fellowship.
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