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*
Friedrich Miescher Institute, Basel, Switzerland;
Aaron Diamond AIDS Research Center, Rockefeller University, New York, NY 10016; and
Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545
1 This work was supported by National Institutes of Health
Grants RR06555, AI28433 (to A. S. P.), and AI40387 (to
D. D. H.). S. B. gratefully acknowledges support from
the Novartis Research Foundation. Portions of this work were performed
under the auspices of the U.S. Department of Energy.
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Abstract |
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Introduction |
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TopAbstractIntroductionModels and ResultsDiscussionAppendixReferences |
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Models and Results |
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A). Let T be
the total number of cells in the compartment. Then a simple model for
the dynamics of cell turnover is
 | (1) |
has the
solution
 | (2) |
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B). However,
labeling may be inefficient so that a fraction, 
, of cells
will not incorporate label upon division. Then, the total
population sizes of labeled cells, L, and unlabeled cells,
U, change according to
 | (3) |
 | (4) |
results
from the assumption that an unlabeled cell, which picks up label and
divides, gives rise to two labeled progenies. If BrdU is nontoxic, we
can assume that the total input of cells from the source is not
affected by BrdU, i.e., we assume sU +
sL = s. Furthermore, the model assumes that the
incorporation of BrdU has no effect on the loss or proliferation rate
of a cell, since both are assumed to be the same for unlabeled and
labeled cells. If significant toxicity is associated with BrdU during
the experiment, then these assumptions may not be justified.
After a time, te, BrdU treatment is stopped. In
the absence of BrdU, division of an unlabeled cell results in two
unlabeled cells. However, the progeny of a labeled cell remains
labeled, albeit with on average half the amount of BrdU, since
chromosomes segregate independently into the daughter cells (Fig. 1C). After a sufficient number of divisions, some cells may
contain too little BrdU to be experimentally distinguishable from
unlabeled cells. However, as long as loss of label by dilution is
negligible, the dynamics of unlabeled and labeled cells after BrdU
treatment are given by
 | (5) |
 | (6) |
Equations 3–6 are a system of linear differential equations. In
general, one expects the source of labeled cells to vary in time, with
no labeled cells entering immediately after label is administered,
building up over time to a constant input rate. Here, we assume that
changes in the source rates are fast compared to the changes in the
fraction of labeled cells in the compartment we are measuring. We
discuss the effect of a nonconstant source further below.
For constant source rates, the fraction of labeled cells during and
after BrdU administration are given by
 | (7) |
=
p, otherwise = (1 - 2)p. The other
parameters are 
= sU/((d +
)T0), 
= (sU +
sL)/((d - p)T0), and 
= s'L/((d - p)T0). Equation 7
was derived for t 
te by solving for
U(t), noting that U0 =
T0 since at t = 0 all cells are
unlabeled, and using fL(t) = 1 -
U(t)/T(t), where T(t) = U(t) + L(t). When
BrdU is nontoxic to the source, T(t) is given by Equation 2.
For te t td,
L(t)/T0 was directly determined and divided by
T(t)/T0 to obtain fL(t).
Equation 7 is the general solution for the fraction of labeled cells.
If the cell population is in steady state prior to the administration
of BrdU, T0 = s/(d - p) (see Equation 2). In addition, if BrdU is nontoxic (i.e., s =
sU + sL = s'U
+ s'L), then = 1. If labeling is
perfect, = p. Under these three conditions, Equation 7 becomes
 | (8) |
= 1 -
(sU(d - p)/s(d + p)) and
A2 = s'L/s. The
parameters A1 and A2 have
an intuitive interpretation as the asymptotic levels that the fraction
of labeled cells approach during long labeling and delabeling periods,
respectively.
The proliferation and death rate appear in the argument of the
exponential functions in Equations 7, and 8, which makes possible their
estimation from experimental data. At first glance, one might have
expected that the fraction of labeled cells would increase only in
proportion to the proliferation rate p. However, this
fraction increases as the fraction of unlabeled cells decreases, and
unlabeled cells disappear by death and through the acquisition of label
during division. Thus, the exponent d + p in Equation 8
reflects the net loss rate of unlabeled cells during labeling.
Similarly, one might have expected that during delabeling the fraction
of labeled cells would decrease at a rate simply proportional to the
death rate d. However, because the progeny of labeled cells
is assumed to remain labeled, the loss of labeled cells occurs at rate
d - p and is reflected in the exponent in Equation 8.
The graphical interpretation of the exponent d + p in
Equation 8 is the ratio of the second to the first-order derivative of
fL(t) during labeling. Hence, d +
p influences the curvature of the function
fL(t) during labeling. However, if
A1 = 1, then d + p is also
the initial rate of increase of fL(t). The
graphical interpretation of the exponent d - p in
Equation 8 is the linear slope of the delabeling curve in a natural
logarithmic plot.
Inefficient labeling and label dilution
If all dividing cells incorporate BrdU during the labeling phase,
then = p and the estimation of p is
unencumbered by considerations of labeling efficiency. However, it is
unclear to what extent this assumption is justified. Rapidly dividing
cells such as the lymphocyte progenitors in the bone marrow typically
attain very high percentages of labeled cells in relatively short
periods of time (
90% in 3 wk (12)). This argues that
the efficiency of labeling in rapidly dividing cells may be high.
However, more slowly dividing cell populations often achieve much lower
levels of labeling. This could be either because a large fraction of
cells have never divided during the period of BrdU administration or
because a fraction of cells that did divide during BrdU administration
did not incorporate BrdU. The latter could be due to tissue
heterogeneity with respect to BrdU concentration or intermittent
troughs of BrdU bioavailability between dosings.
The assumption that after BrdU treatment is stopped all progeny of
dividing cells can be detected as BrdU-positive cells can be relaxed so
as to extend the validity of the theory beyond time
td. During the delabeling phase, the intensity
of the signal per cell on average halves with each cell division. To
investigate the effect of label dilution on the estimation of
p and d, assume that at all times a fraction,
, of the labeled cells have such a low intensity of BrdU labeling
that their daughter cells are below the threshold of detectability. The
modified dynamics of labeled and unlabeled cells after BrdU treatment
(t > te) then are:
 | (9) |
 | (10) |
Equations 3, and 4 and 9 and 10 can be solved in full generality,
but we give only the solution for the fraction of labeled cells under
the assumption that the total cell population size is constant during
and after BrdU administration:
 | (11) |
)p) and
Â2 = (s'L/s) (d -
p)/(d - (1 - 2)p).
Comparing Equations 8, and 11, we see that both functions have identical
structure. Call the exponent for t < te,
e1, and the exponent for te
t, e2. Using Equation 8, one can obtain p
from (e1 - e2)/2 and
d from (e1 + e2)/2.
Using the same procedure on Equation 11 shows that by neglecting the
loss of labeled cells due to label dilution and assuming perfect
labeling, we underestimate the proliferation rate by a factor 1/(1
- - ). The death rate will be underestimated if
> , maximally by a factor (1 - 2)/(1 - -
). If < , it will be overestimated maximally by a
factor (1 - 2)/(1 - - ). To give a
numerical example: Suppose that 20% of the dividing cells do not
incorporate BrdU during labeling and 10% of the labeled cells that
proliferate during delabeling produce daughter cells that cannot be
detected as labeled cells due to label dilution, then using Equation 8
instead of Equation 11 results in a 40% underestimate of p
and maximally a 15% underestimate of d.
These error estimates are based on the assumption that the fraction of labeled cells lost due to label dilution is constant. This assumption is reasonable as a first approximation, but may represent an oversimplification when the bulk of cells comes close to the detection threshold. A more detailed description of the delabeling of the labeled cell population can be formulated as a partial differential equation that describes the change in the frequency of cells as a function of BrdU-labeling intensity and time. Such elaborations are outside of the scope of this paper.
Nonconstant source terms
A limitation of Equations 7, and 8 is that the sources of unlabeled
and labeled cells are assumed to be constant during the experiment.
More realistically, these source terms may change over time. The source
of labeled cells, for example, may initially be small but increase
during labeling. Similarly, the source of labeled cells may decrease
during delabeling. In principle, one could take into account
nonconstant source terms; however, to do so requires detailed knowledge
of how these source terms vary with time. If the source is a population
of rapidly dividing cells, as one finds in the bone marrow or thymus,
then one might expect only a short initial transient before the source
rates became constant.
In general, the source need not be external, but could also be a
subpopulation of cells in the same compartment. For example, in
experiments, such as that of Mohri et al. (12), in which
labeling of peripheral blood mononuclear cells is studied, the source
may be a subpopulation of resting or slowly dividing cells. To see
this, consider "resting" cells, R, with slow turnover,
and activated cells, A, with fast turnover. A model for the
dynamics of these two subpopulations before, during, and after BrdU
administration is illustrated in Fig. 2.
The resting and activated cell populations proliferate at rates
p' and p and die at rates d' and
d, respectively (Fig. 2A). When resting cells
receive an activation signal they expand clonally and become activated
cells. We model this process, as suggested by De Boer and Noest
(13), and assume activation results in removal of resting
cells at rate a' and their replacement by activated cells at
rate a = a', where represents the size of the
generated clone. This model assumes that clonal expansion upon
activation is fast compared with the other processes being modeled, as
has been seen in some experimental systems (14, 15, 16, 17).
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B). Upon
activation, resting cells also incorporate BrdU, as activation results
in cell division. After treatment is stopped, labeled cells produce
labeled progeny (with half the intensity of BrdU) when they proliferate
(Fig. 2C), and unlabeled resting cells that receive an
activation signal produce unlabeled activated cells. Labeled resting
cells that become activated may produce either labeled activated cells
(as depicted in Fig. 2C) or if they undergo so many cell
divisions that the label intensity becomes too diluted to differentiate
them from unlabeled cells, they may produce unlabeled cells (data not
shown in Fig. 2).
In terms of ordinary differential equations, the dynamics before BrdU
treatment are
 | (12) |
 | (13) |
 | (14) |
 | (15) |
 | (16) |
would need to be
modified.
Comparing Equation 16 to Equation 8, we see that these equations are
very similar. If p' = 0, so that the resting population does
not divide unless stimulated into clonal expansion, then the equations
are identical with A1 =
AU(0)/(AU(0) + RU(0)).
However, if p' is not zero but simply small compared to
p + d, the initial dynamics of Equation 16 are
dominated by the exponent p + d as in Equation 8.
However, instead of converging toward a fixed asymptote, Equation 16
converges to 1 -
(RU(0)e-2p't)/(AU(0) +
RU(0)). In other words, first the activated cells
label at rate p + d, and later, once the activated
cells are labeled, the increase in the fraction of labeled cells will
be dominated by the labeling of resting cells, which label at a rate
2p'.
After treatment, the dynamics of labeled resting and activated cells
are given by
 | (17) |
 | (18) |
t < td, we find
RL(t) = RL(te) and
 |
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| (19) |
Comparing Equation 19 to the fraction of labeled cells after BrdU
treatment is stopped in Equation 8, we find that both equations have
identical structure and identical exponents, showing that a resting
population may indeed act as the "source" for the activated
population.
Turnover of T lymphocytes
To illustrate how this theory can be used to estimate proliferation and death rates of cell populations, we discuss experimental data that was collected to determine the turnover of lymphocyte subpopulations in uninfected and SIV-infected macaques (12). Infected and uninfected macaques received BrdU in their drinking water over a period of 3 wk. The percentage of BrdU-positive CD4+ and CD8+ lymphocytes in the blood was determined by flow cytometric analysis at weeks 0, 1, ... , 7, and 10.
The dynamics of labeling and delabeling of these lymphocyte populations
are given by Equations 3–6. The source of CD4+ and
CD8+ lymphocytes may represent the import of cells from the
thymus or extrathymic tissue, but could also be a resting or slowly
dividing subpopulation as described under Nonconstant source
terms. Since cells coming from these sources are likely to have
undergone a phase of rapid cell division shortly before their
appearance in the compartment being modeled, one would expect only a
short initial transient before the source rates became constant after
BrdU treatment is started or stopped. In these experiments, the
fraction of labeled cells is first measured 7 days after BrdU
administration, and measurements are continued for a number of weeks.
For this system, assuming that the source rapidly becomes constant is a
reasonable first step. Furthermore, no toxicity of BrdU was observed
clinically or by laboratory tests and the total population size of both
lymphocyte populations remained approximately constant during the
experiment. Furthermore, we assume that to a good approximation
labeling is 100% efficient, so that the conditions that allow one to
simplify Equation 7 into Equation 8 are met, and we use Equation 8 to
obtain estimates for the death and proliferation rates of
CD4+ and CD8+ T lymphocytes.
The data displayed in Fig. 3 show that
after BrdU treatment is stopped the fraction of labeled
CD4+ and CD8+ T cells declines in both the
infected and the uninfected animal. This decrease in the fraction of
labeled cells after BrdU treatment must be due to labeled cells being
lost either 1) because their death rate exceeds the sum of their
proliferation rate and rate of input from the source and/or 2) because
a fraction of cells have so little label that their progenies become
experimentally indistinguishable from unlabeled cells. Flow cytometric
analysis of the intensity of BrdU labeling suggests that the bulk of
cells have sufficiently high levels of BrdU that it would require five
to six cell divisions before the label is sufficiently diluted to reach
the threshold of detectability. If label dilution was the principal
cause for the loss of labeled cells after BrdU treatment is stopped, we
would expect to observe a lag before the fraction of labeled cells
begins to decrease noticeably. Because this is not observed (Fig. 3)
dilution of label seems unlikely as the principal reason for the loss
of labeled cells, and we conclude that the death rate exceeds the
proliferation rate. Since the cell populations are in steady state this
implies that the total source (of labeled and unlabeled cells) has to
make up for the difference between the death and proliferation rates.
This raises two questions: What percentage of cells are replenished
from the source per day and what acts as the source?
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was fit to the data from one SIV-infected and one uninfected
macaque using a nonlinear least squares regression method that employed
the subroutine DNLS1 from the Common Los Alamos Software Library, which
is based on a finite difference Levenberg-Marquardt algorithm. The
resulting best fitting theoretical curves and the data are shown in
Fig. 3. We fitted Equation 8 to the data assuming that the asymptotic
value during the delabeling phase, A2, is zero
(solid line), and assuming that it may have any value between 0 and 1
(dashed line). The estimates obtained from these two different fits
differ less for the death rate, d, than for the
proliferation rate, p. The estimates with
A2 = 0 using the data from the uninfected
macaque yielded p = 0.000. Lower and upper 68%
confidence bounds for each parameter estimate were calculated by a
bootstrap method (18). Using bootstrapping, we estimated
that with 68% confidence the values of p for
CD4+ and CD8+ cells are between 0.000 and 0.014
day-1 and 0.000 and 0.012 day-1,
respectively. This suggests that from this experiment we cannot obtain
precise estimates for the proliferation rate in slowly dividing
populations. The percentage of lymphocytes that come from the source per day is given by (d - p) x 100. This yields a daily replenishment rate of 1.9 and 2.9% for the CD4 and CD8 cells of the infected animal and 0.9 and 1.1% for the CD4 and CD8 cells of the uninfected animal (the values are taken from the fits represented by the solid line), respectively. CD4+ and CD8+ T lymphocytes mature in the thymus and possibly other tissues, such as the gut, which could thus represent the source of T lymphocytes. A daily replenishment rate of peripheral T cells from the thymus of a few percent may appear high, although it is not incompatible with other measurements (10) and the recent finding that the human thymus can generate new T cells throughout life (19, 20).
We have shown that a resting cell population could also act as the
source for the activated population. If this is the case, we need to
interpret p and d as the proliferation and death
rate of the activated subpopulation of CD4+ and
CD8+ T lymphocytes. The data fits represented by the solid
lines in Fig. 3 correspond to the case where after BrdU treatment
labeled resting cells upon activation produce only unlabeled activated
cells due to label dilution in a phase of rapid replication. The data
fits represented by the dashed lines correspond to the case where
labeled resting cells give rise to labeled activated cells after BrdU
treatment.
Alternative approaches for parameter estimation
There are also other approaches to analyze the flow cytometric data of BrdU uptake and washout to obtain parameter estimates. The flow cytometric data provide measurements for the intensity of labeling of individual cells. Therefore, instead of computing the fraction of labeled cells, one can also determine the mean and the total intensity of BrdU in labeled cells.
Consider the time after BrdU treatment has been stopped. The total
intensity of BrdU summed over all cells is affected by death but not by
proliferation of a cell. Death of a cell results, on average, in the
removal of one times the current mean intensity of the labeled cells.
Proliferation, however, conserves the total amount of BrdU, since half
of the total label of the mother cell is passed to each daughter cell.
Hence, the change of the total intensity of BrdU,
It, during delabeling is given by
 | (20) |
Death of a labeled cell has no effect on the mean intensity of label.
On average, a dying cell will remove the mean intensity of label from
the total intensity summed over the labeled cell population. At the
same time, death of a labeled cell will reduce the total number of
labeled cells by one. Hence, the mean intensity is not affected by cell
death. However, proliferation of a labeled cell results, on average, in
the removal of one cell with the current mean intensity and the
addition of two cells with half the mean intensity. Therefore,
replication results in reduction of the mean intensity. The decrease in
the mean intensity, Im, can be derived as
 | (21) |
To our knowledge, neither the mean nor the total intensity of BrdU labeling has been used for parameter estimation. However, these data can supplement the fraction of labeled cells and provide a better basis for estimation of cell lifetime parameters. However, care has to be taken in the design of the experiment to assure that the mean and total intensity is not affected by experimental variation such as antibody staining or cell numbers. Any such external variability may overshadow a significant signal in the change of BrdU intensity. In such cases it may be better to base the analysis on the changes of the fraction of labeled cells.
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Discussion |
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TopAbstractIntroductionModels and ResultsDiscussionAppendixReferences |
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We have developed a mathematical framework for the analysis of BrdU uptake and washout in proliferating cell populations. The theory we have developed is based on a simple one-compartment model, which is relevant, for example, to measurements of single cell populations in blood, such as CD4+ or CD8+ T cells. Clearly, more complex models can be developed that take into consideration multiple compartments and emigration between them. These compartments may be spatial, e.g., lymphoid tissue and blood, or represent subpopulations of cells defined by cell surface markers, e.g., CD4+ naive and memory T cells. Within the context of the one-compartment model, we have shown that the fraction of labeled cells, as well as the mean and total intensity of BrdU label in the cell population, can be used to obtain quantitative estimates for the rates of cell proliferation, cell death, and cell input from a source. We have given analytical solutions for populations that are growing, declining, or at equilibrium and we have provided an example of how kinetic parameters describing T cell kinetics can be estimated from BrdU data.
Mathematical modeling combined with quantitative experimentation provides a powerful set of tools for elucidating biological phenomena. More sophisticated cell kinetic models and improved labeling techniques, such as the use of the stable isotope-labeled precursors of DNA (21), will likely provide further insights into important problems such as the nature of T cell depletion in HIV-infected individuals.
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Appendix |
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t the total intensity is given by
 | (22) |
t describes the
total input of BrdU from the source during the time interval
t. During this time interval, a total number of
dtN(t) cells die, each of them removing on average once
the mean intensity Im(t) from the compartment.
Hence, -dtN(t)Im(t) reflects the total loss
of BrdU by cell death. Proliferation is accounted for by the terms
-ptN(t)Im(t) +
2ptN(t)Im(t)/2, which reflect the loss of cells
with mean intensity Im(t) and their replacement
by two progeny numbers that each contain on average half the mean
intensity. These two terms cancel, showing that proliferation does not
affect the total BrdU intensity. Using that
N(t)Im(t) = It(t), we derive
the differential equation for It(t):
 | (23) |
t is
given by
 | (24) |
 | (25) |
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Acknowledgments |
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Footnotes |
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2 Abbreviation used in this manuscript: BrdU, 5-bromo-2'-deoxyuridine; SIV, simian immunodeficiency virus.
Received for publication October 21, 1999. Accepted for publication March 1, 2000.
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