A composite likelihood approach to the analysis of longitudinal clonal data on multitype cellular systems under an age-dependent branching process.
ABSTRACT A recurrent statistical problem in cell biology is to draw inference about cell kinetics from observations collected at discrete time points. We investigate this problem when multiple cell clones are observed longitudinally over time. The theory of age-dependent branching processes provides an appealing framework for the quantitative analysis of such data. Likelihood inference being difficult in this context, we propose an alternative composite likelihood approach, where the estimation function is defined from the marginal or conditional distributions of the number of cells of each observable cell type. These distributions have generally no closed-form expressions but they can be approximated using simulations. We construct a bias-corrected version of the estimating function, which also offers computational advantages. Two algorithms are discussed to compute parameter estimates. Large sample properties of the estimator are presented. The performance of the proposed method in finite samples is investigated in simulation studies. An application to the analysis of the generation of oligodendrocytes from oligodendrocyte type-2 astrocyte progenitor cells cultured in vitro reveals the effect of neurothrophin-3 on these cells. Our work demonstrates also that the proposed approach outperforms the existing ones.
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ABSTRACT: Stem and precursor cells play a critical role in tissue development, maintenance, and repair throughout the life. Often, experimental limitations prevent direct observation of the stem cell compartment, thereby posing substantial challenges to the analysis of such cellular systems. Two-type age-dependent branching processes with immigration are proposed to model populations of progenitor cells and their differentiated progenies. Immigration of cells into the pool of progenitor cells is formulated as a non-homogeneous Poisson process. The asymptotic behavior of the process is governed by the largest of two Malthusian parameters associated with embedded Bellman-Harris processes. Asymptotic approximations to the expectations of the total cell counts are improved by Markov compensators.Mathematical Population Studies 10/2012; 19(4):164-176. · 0.38 Impact Factor
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ABSTRACT: Two-type reducible age-dependent branching processes with inhomogeneous immigration are considered to describe the kinetics of renewing cell populations. This class of processes can be used to model the generation of oligodendrocytes in the central nervous system in vivo or the kinetics of leukemia cells. The asymptotic behavior of the first and second moments, including the correlation, of the process is investigated.Pliska. Pliiska. 01/2011; 20:81-108.
Article: Quantifying T lymphocyte turnover.[Show abstract] [Hide abstract]
ABSTRACT: Peripheral T cell populations are maintained by production of naive T cells in the thymus, clonal expansion of activated cells, cellular self-renewal (or homeostatic proliferation), and density dependent cell life spans. A variety of experimental techniques have been employed to quantify the relative contributions of these processes. In modern studies lymphocytes are typically labeled with 5-bromo-2'-deoxyuridine (BrdU), deuterium, or the fluorescent dye carboxy-fluorescein diacetate succinimidyl ester (CFSE), their division history has been studied by monitoring telomere shortening and the dilution of T cell receptor excision circles (TRECs) or the dye CFSE, and clonal expansion has been documented by recording changes in the population densities of antigen specific cells. Proper interpretation of such data in terms of the underlying rates of T cell production, division, and death has proven to be notoriously difficult and involves mathematical modeling. We review the various models that have been developed for each of these techniques, discuss which models seem most appropriate for what type of data, reveal open problems that require better models, and pinpoint how the assumptions underlying a mathematical model may influence the interpretation of data. Elaborating various successful cases where modeling has delivered new insights in T cell population dynamics, this review provides quantitative estimates of several processes involved in the maintenance of naive and memory, CD4(+) and CD8(+) T cell pools in mice and men.Journal of Theoretical Biology 01/2013; · 2.35 Impact Factor
Biostatistics (2011), 12, 1, pp. 173–191
Advance Access publication on August 23, 2010
A composite likelihood approach to the analysis of
longitudinal clonal data on multitype cellular systems
under an age-dependent branching process
RUI CHEN, OLLIVIER HYRIEN∗
Department of Biostatistics and Computational Biology,
University of Rochester Medical Center, 601 Elmwood Avenue, Rochester, NY 14642, USA
MARK NOBLE, MARGOT MAYER-PR¨OSCHEL
Department of Biomedical Genetics, University of Rochester Medical Center,
601 Elmwood Avenue, Rochester, NY 14642, USA
A recurrent statistical problem in cell biology is to draw inference about cell kinetics from observations
collected at discrete time points. We investigate this problem when multiple cell clones are observed
longitudinally over time. The theory of age-dependent branching processes provides an appealing frame-
work for the quantitative analysis of such data. Likelihood inference being difficult in this context, we
propose an alternative composite likelihood approach, where the estimation function is defined from the
marginal or conditional distributions of the number of cells of each observable cell type. These distri-
butions have generally no closed-form expressions but they can be approximated using simulations. We
construct a bias-corrected version of the estimating function, which also offers computational advantages.
Two algorithms are discussed to compute parameter estimates. Large sample properties of the estimator
are presented. The performance of the proposed method in finite samples is investigated in simulation
studies. An application to the analysis of the generation of oligodendrocytes from oligodendrocyte type-2
astrocyte progenitor cells cultured in vitro reveals the effect of neurothrophin-3 on these cells. Our work
demonstrates also that the proposed approach outperforms the existing ones.
Keywords: Bias correction; Cell differentiation; Composite likelihood; Discrete data; Monte Carlo; Neurotrophin-3;
Oligodendrocytes; Precursor cell; Stochastic model.
In stem cell biology, there exists considerable interest in studying signals that may modulate or alter the
processes that regulate the formation of tissues during development or repair. Understanding such pro-
cesses has important clinical implications as it may offer potential means to restore or maintain tissue
∗To whom correspondence should be addressed.
c ? The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: email@example.com
174R. CHEN AND OTHERS
and organ functions. One of the challenges in conducting detailed analysis of the effects of these sig-
nals, however, is the general lack of quantitative approaches that allow analysis of different aspects of
progenitor or stem cell function.
of the best studied precursor cell populations (e.g. Noble and others, 2004). These progenitor cells may
either divide into 2 new progenitor cells or differentiate into terminally differentiated oligodendrocytes,
which produce the myelin sheaths that enwrap axons in the central nervous system. The differentiation
of progenitor cells into oligodendrocytes needs to be properly regulated to maintain normal function of
the central nervous system. In addition to the function of O-2A progenitor cells in vivo, their ability to
grow and differentiate in vitro has established these cells as an important tool in modern cell biology
that allows the study of the most basic cellular functions (i.e. self-renewal by division, differentiation,
and death) at the clonal level in tissue culture (e.g. Ibarrola and others (1996); Smith and others, 2000).
The development of a typical clone of these cells cultured in vitro started from a single O-2A progenitor
cell is displayed in Figure 1. It may be altered in multiple ways following exposure to agents, such as
cytokines, drugs, or toxicants, and it is of great interest to biologists to determine and quantify these
changes (Dietrich and others, 2006).
Over the past decade, a great deal of attention has been paid to the development of a statistical frame-
work for gaining a quantitative understanding of the biological processes that govern the division of O-2A
progenitor cells and their differentiation into oligodendrocytes (Yakovlev, Boucher, and others, 1998;
Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others, 2000; von Collani and others, 1999;
Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and others, 2005a,b, 2006; Hyrien,
2007). These publications used multitype age-dependent branching processes as a means to model the
temporal development of cell clones. The resulting methods provided quantitative insights into the gen-
eration of oligodendrocytes from cultured O-2A progenitor cells. For instance, Yakovlev, Boucher, and
others (1998) observed that the generation of oligodendrocytes was regulated by a combination of cell-
intrinsic factors and environmental signals that may modulate the probability of differentiation of O-2A
progenitor cells. Other studies based on similar quantitative approaches (Yakovlev, Boucher, and others,
1998; Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others, 2000; von Collani and others,
Fig. 1. Oligodendrocyte development under in vitro conditions. When cultured under appropriate growing conditions,
any O-2A progenitor cell will either divide into 2 new O-2A progenitor cells or it will differentiate into a single
terminally differentiated oligodendrocyte. O-2A progenitor cells may also die. Oligodendrocytes are terminally dif-
ferentiated cells that arise from O-2A progenitor cells. These cells are not able to divide and they ultimately die (i.e.
disintegrate and disappear from the population). Cell death was not observed in our NT-3 experiment.
Composite likelihood for branching processes
1999; Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and others, 2006) confirmed that
thyroid hormone induces the differentiation of O-2A progenitor cells into oligodendrocytes (Barres and
others, 1994; Ibarrola and others (1996); Ahlgren and others, 1997; Rodr´ ıguez-Pe˜ na, 1999), and that the
decision of O-2A progenitor cells to either divide or differentiate is made early in the cell cycle, if not
before, rather than at the end (Hyrien and others, 2005a, 2006, 2010).
Most of these previous studies were concerned either with clonal experiments in which every cellular
clone was observed only once in the course of the experiment thereby yielding independent observations
(Yakovlev, Boucher, and others, 1998; Yakovlev, Mayer-Pr¨ oschel, and others, 1998; Yakovlev and others,
2000; von Collani and others, 1999; Boucher and others, 1999, 2001; Zorin and others, 2000; Hyrien and
others, 2005a,b, 2006, 2010; Hyrien, 2007) or with time-lapse experiments (Hyrien and others, 2006),
where the complete history of each clone was fully recorded. While time-lapse experiments provide com-
plete information about the temporal development of each clone, clonal experiments are far less time-
consuming. This is the reason why clonal analyses remain the experiment of choice to investigate the
ability of agents to impact the regulation of precursor cell functions. There exists a third experimental
setting in which each cell clone is observed longitudinally over time and which offers a compromise be-
tween the above 2 experiments. In this experimental setting, the family trees are not completely observed,
but several snapshots of the temporal development of each individual clone are obtained at multiple time
points thereby providing more information than independent clonal experiments. We shall refer to these
alternate experiments as longitudinal clonal experiments (as opposed to “independent” clonal experiments
where each cell clone is examined only once).
Longitudinal clonal data are not independent across time points, and their analysis presents new chal-
lenges. Some of the estimation methods that have been proposed for independent clonal data could also
apply to the longitudinal setting. Since maximum likelihood estimation remains generally unattainable
with age-dependent branching processes, simulation-based approaches have been adopted by some
authors as a means to construct viable estimators. For instance, Hyrien and others (2005a,b) and Hyrien
(2007) considered the utility of a simulated pseudolikelihood approach. This method relies solely on the
expectation and variance–covariance functions of the process. It is attractive because it provides consistent
and asymptotically Gaussian estimators, while remaining easy to implement, even for longitudinal clonal
data. The disadvantage of this estimator is that it does not enjoy optimality properties, and, for instance,
it is not as efficient as the method of maximum likelihood (Hyrien, 2007). The simulated maximum like-
lihood estimator proposed by Zorin and others, 2000 could be more efficient than the simulated pseudo
maximum likelihood estimator but its implementation faces serious limitations that are amplified when
cell clones are observed repeatedly over time. Specifically, it is very time-consuming, and it suffers from
mismatches, as described by Zorin and others (2000). We shall discuss these limitations in greater details
in Section 4 when motivating the construction of the proposed estimator.
This article proposes an alternative approach that provides a compromise between the statistical effi-
ciency of the maximum likelihood estimator and the computational advantage of the simulated maximum
pseudolikelihood estimator. The proposed method is based on a composite likelihood approach (some-
times also referred to as method of pseudolikelihood). The use of composite likelihood traces back to
Besag (1974). It has been investigated in other settings by several authors, including Azzalini (1983),
Lindsay (1988), Heagerty and Lele (1998), Cox and Reid (1987), Chandler and Bate (2007), Varin and
Vidoni (2005), Hyrien and Zand (2008), to name a few. Composite likelihoods can be defined as estimat-
ing functions formed by combining together likelihood objects that remain tractable for the problem at
hands. Our work shows that composite likelihood estimators provide a viable solution to the analysis of
longitudinal clonal data.
The proposed method relies on a complex stochastic process. Simpler statistical approaches could be
invoked to describe the time course of cell counts and assess whether specific treatment conditions alter
the kinetics of the cell population. The level of sophistication of the proposed method is required for
176R. CHEN AND OTHERS
gaining a quantitative insight into the processes of division and differentiation of precursor cells because
the events of interest (division and differentiation) are not observable during clonal experiments.
The remainder of this article is organized as follows. Section 2 introduces the data structure. Section 3
presents a branching process model of the generation of oligodendrocytes from cultured O-2A progenitor
cells. Section 4 describes simulated composite likelihood estimation as applied to longitudinal clonal
data. The finite-sample properties of the proposed method are investigated via simulations in Section 5.
Section 6 presents an application to the analysis of the proliferation of O-2A progenitor cells and their
differentiation into oligodendrocytes exposed to neurotrophin-3 (NT-3) in tissue culture.
2. LONGITUDINAL CLONAL DATA
Clonal data on the generation of oligodendrocytes in vitro
A typical longitudinal clonal experiment conducted on oligodendrocytes begins by plating initiator O-2A
progenitor cells in culture dishes at a density that is low enough so these cells (and the pools of subsequent
progenies) will not interact with each other. Over time, these progenitor cells divide into O-2A progenitor
cells and/or differentiate into oligodendrocytes to give rise to individual cell clones. These clones may
contain O-2A progenitor cells or oligodendrocytes, or, as is more typically the case, a mixture of both cell
types. O-2A progenitor cells and oligodendrocytes are morphologically distinguishable, which enables
experimentalists to count separately the number of O-2A progenitor cells and the number of oligodendro-
cytes contained in any given clone through visual examination using a microscope. The location of each
clone in the dish is identified for subsequent follow up, and the composition of each clone is examined
repeatedly over time so one can observe how the numbers of O-2A progenitor cells and the number of
oligodendrocytes change over time.
An example of longitudinal clonal data
We performed a longitudinal clonal experiment, as described above, to investigate the impact of NT-3 on
the processes of division and differentiation of O-2A progenitor cells. We purified O-2A progenitor cells
isolated from the corpus callosum tissue of 7-day old rats as described previously (Ibarrola and others,
1996) and plated cells at a clonal density in defined medium supplemented with NT-3 at 20 ng/ml. No
platelet-derived growth factor (PDGF) was added to the culture medium at any time. In this experiment,
n = 40 clones were followed for up to 6 days. Every clone was generated by a single O-2A progenitor
cell. The composition of each clone was examined daily, so ti = (1,2,3,4,5,6) in this particular ex-
periment (with time being expressed in days). The number of O-2A progenitor cells and the number of
oligodendrocytes were reported at each time point for every clone. Half of these clones were cultured in
the presence of NT-3, and the other half was cultured without NT-3.
Figure 2 shows the histograms for the number of progenitor cells (left panels) and for the number of
oligodendrocytes (right panels) from day 0 to day 6 (from top to bottom). At the start of the experiment
(day 0), all clones contained exactly one O-2A progenitor cell and zero oligodendrocyte. The composi-
tion of each clone evolved over time according to whether O-2A progenitor cells and their subsequent
progenies divided or differentiated. In this particular experiment, all clones growing without NT-3 con-
tained between 0 and 6 O-2A progenitor cells and between 0 and 4 oligodendrocytes. In the presence of
NT-3, the number of O-2A progenitor cells ranged between 0 and 9, and the number of oligodendrocytes
between 0 and 6.
The primary objectives of our experiment were to assess and investigate the effects of NT-3 on the
proliferation of O2-A progenitor cells and their differentiation into oligodendrocytes. A visual comparison
of the histograms of the number of O2-A progenitor cells and of the number of oligodendrocytes clearly
Composite likelihood for branching processes
Fig. 2. Histograms for the number of O-2A progenitor cells (PCs) and for the number of oligodendrocytes along with
the fitted model in the absence and in the presence of NT-3.
suggests that the numbers of O-2A progenitor cells were substantially smaller in clones treated with NT-3
starting from day 2. In contrast, the numbers of oligodendrocytes appeared similar among treated and
untreated clones at all time points. The marginal distributions of the number of O-2A progenitor cells
(respectively, oligodendrocytes) at any time point in clones treated with and in clones treated without
NT-3 could be compared using (e.g.) Wilcoxon rank sum statistics. In order to assess the overall effect of
NT-3, irrespective of time, and avoid issues associated with multiple testing, the resulting p-values could
be combined using Fisher’s combination method. Since the individual p-values are dependent across time
points, an overall p-value could be computed using a permutation testing approach, where clones (not
just individual observations) are randomly assigned to one group or the other so the dependencies among
observations are properly accounted for when performing the test.
We assessed the overall effect of NT-3 on the numbers of O-2A progenitor cells and on the numbers of
oligodendrocytes separately using this approach. Our test suggested that the numbers of O-2A progenitor
cells were significantly different depending upon whether they had been cultured with or without NT-
3 (p = 0.02), but it did not detect any significant difference among the numbers of oligodendrocytes
(p = 0.98). Thus, this analysis would suggest that NT-3 had a significant impact on the number of O-2A
progenitor cells but not on the number of oligodendrocytes, at least for the first 6 days of culture.
The conclusion of these analyses is of interest in itself because it reveals the existence of a potential
effect of NT-3 on the regulation of the processes of division and of differentiation of O-2A progenitor
cells. From a biological standpoint, however, it remains limited in scope because it neither offers any
mechanistic interpretation on how these processes might have actually been altered by NT-3 nor quanti-
fies the effects of NT-3 on cellular functions. A number of biological reasons could be invoked to explain
the observed effects of NT-3. For instance, O-2A progenitor cells exposed to NT-3 might have differen-
tiated more frequently in the presence of NT-3, causing the number of O-2A progenitor cells to decrease