## Abstract

5-Bromo-2′-deoxyuridine (BrdU) is frequently used to measure the turnover of cell populations in vivo. However, due to a lack of detailed mathematical models that describe the uptake and loss of BrdU in dividing cell populations, assessments of cell turnover kinetics have been largely qualitative rather than quantitative. In this study, we develop a mathematical framework for the analysis of BrdU-labeling experiments. We derive analytical expressions for the fraction of labeled cells within cell populations that are growing, declining, or at equilibrium. Fitting the analytical functions to data allows us to quantify the rates of cell proliferation and cell loss, as well as the rate of cell input from a source. We illustrate this for the BrdU labeling of T lymphocytes of uninfected and SIV-infected rhesus macaques.

Studies of the cell cycle, DNA synthesis, and cell proliferation have benefited from the use of labeled DNA precursors. Use of radiolabeled precursors, such as tritiated thymidine, has been replaced by safer methods employing nonradioactive molecules, such as the base analog 5-bromo-2′-deoxyuridine (BrdU).^{2} BrdU is substituted stoichiometrically for thymidine in newly synthesized DNA (1) and cells harboring BrdU-substituted DNA can be easily detected by an immunofluorescent assay (2) or via flow cytometry (3, 4, 5). In the mouse, BrdU labeling has been used to study both B and T lymphocytes (5, 6, 7, 8, 9, 10, 11). Typically, BrdU is given and the fraction of cells that acquire BrdU label is measured as a function of time. After BrdU is withdrawn, one can measure the loss of BrdU-labeled cells with time. These measurements have been used to estimate the fraction of proliferating cells and cell life span. However, we shall show that BrdU measurements are not easy to interpret in the absence of a theory. For example, one might mistakenly interpret the rate at which a population of cells acquire BrdU as the rate of cell proliferation and the rate of loss of BrdU-labeled cells after BrdU is withdrawn as the rate of cell death. We shall show under ideal circumstances that for cell populations maintained at steady state, the rate at which the fraction of labeled cells increases is indicative of the sum of the per cell proliferation and death rates, whereas the rate of decay during the BrdU-free chase reflects the per cell death rate minus the per cell proliferation rate. To illustrate the use of the theory that we develop here, we shall interpret the experiments of Mohri et al. (12), in which BrdU labeling was used to measure the rate of T cell proliferation and death in uninfected and SIV-infected macaques.

## Models and Results

BrdU is administered to an animal for a period of time, which we call the labeling period. Cells that divide during the labeling period incorporate BrdU into their DNA and, in principle, can be detected. Here, we focus on the analysis of cells obtained from the peripheral blood and thus invoke the concept of a compartment. The number of cells of a particular type in the compartment may change because of input of cells into the compartment, cell proliferation within the compartment, and loss of cells due to death and/or emigration from the compartment (Fig. 1⇓*A*). Let T be the total number of cells in the compartment. Then a simple model for the dynamics of cell turnover is where s is the rate at which cells enter the compartment, and p and d are the per cell proliferation and death (and/or emigration) rates, respectively. Below, we assume that s, p, and d are constants, although under some circumstances they may be density dependent, i.e., functions of T. However, if the total population is changing slowly (or at steady state) then to a good approximation these rates can be assumed constant. Under these conditions, Equation 1 has the solution where T_{0} is the total size of the cell population at time t = 0. If the loss rate exceeds the proliferation rate, i.e., d > p, then in the long term the population approaches a steady state given by s/(d − p). If there is no source, then p must equal d for the population to have a non-zero steady state. Finally, if the proliferation rate exceeds the death rate, then the total population grows exponentially in the presence or absence of a source. Under such circumstances the potential density dependence of p and d may need to be reconsidered.

During the administration of BrdU, newly diving cells incorporate BrdU. When a cell divides, one strand of DNA in each chromosome is synthesized and, in the presence of BrdU, incorporates BrdU. Thus, if a sufficient concentration of BrdU is present, the progeny of all dividing cells will be scored as labeled (Fig. 1⇑*B*). However, labeling may be inefficient so that a fraction, ε, of cells will not incorporate label upon division. Then, the total population sizes of labeled cells, L, and unlabeled cells, U, change according to where the parameters s_{U} and s_{L} denote the rate of input of labeled and unlabeled cells from the source. The factor 2 in Equation 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 s_{U} + s_{L} = 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, t_{e}, 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. 1⇑*C*). 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 where s^{′}_{U} and s^{′}_{L} reflect the rate of input of unlabeled and labeled cells after BrdU administration is stopped. As above, if BrdU is nontoxic to the source then we have s^{′}_{U} + s^{′}_{L} = s.

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 where t = 0 is the start, t = t_{e} is the end of BrdU treatment, and t_{d} is the time at which dilution of label becomes a significant factor. For perfect labeling, p̂ = p, otherwise p̂ = (1 − 2ε)p. The other parameters are α = s_{U}/((d + p̂)T_{0}), β = (s_{U} + s_{L})/((d − p)T_{0}), and γ = s^{′}_{L}/((d − p)T_{0}). Equation 7 was derived for t ≤ t_{e} by solving for U(t), noting that U_{0} = T_{0} since at t = 0 all cells are unlabeled, and using f_{L}(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 t_{e} ≤ t ≤ t_{d}, L(t)/T_{0} was directly determined and divided by T(t)/T_{0} to obtain f_{L}(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, T_{0} = s/(d − p) (see Equation 2). In addition, if BrdU is nontoxic (i.e., s = s_{U} + s_{L} = s^{′}_{U} + s^{′}_{L}), then β = 1. If labeling is perfect, p̂ = p. Under these three conditions, Equation 7 becomes where A_{1} = 1 − α = 1 − (s_{U}(d − p)/s(d + p)) and A_{2} = s^{′}_{L}/s. The parameters A_{1} and A_{2} 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 f_{L}(t) during labeling. Hence, d + p influences the curvature of the function f_{L}(t) during labeling. However, if A_{1} = 1, then d + p is also the initial rate of increase of f_{L}(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 p̂ = 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 t_{d}. 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, ς, 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 > t_{e}) then are:

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: where Â_{1} = 1 − (s_{u}/s) (d − p)/(d + (1 − 2ε)p) and Â_{2} = (s^{′}_{L}/s) (d − p)/(d − (1 − 2ς)p).

Comparing Equations 8, and 11, we see that both functions have identical structure. Call the exponent for t < t_{e}, e_{1}, and the exponent for t_{e} ≤ t, e_{2}. Using Equation 8, one can obtain p from (e_{1} − e_{2})/2 and d from (e_{1} + e_{2})/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 − ε − ς). The death rate will be underestimated if ε > ς, maximally by a factor (1 − 2ς)/(1 − ε − ς). If ε < ς, it will be overestimated maximally by a factor (1 − 2ς)/(1 − ε − ς). 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. 2⇓*A*). 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 = γa′, where γ 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).

During BrdU administration, both resting and activated cells incorporate BrdU when they proliferate (Fig. 2⇑*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. 2⇑*C*), 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. 2⇑*C*) 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 We assume the resting cells form a constant pool of cells, i.e., that R is in steady state and thus p′ − d′ − a′ = 0. During BrdU treatment, the unlabeled populations obey with solution R_{U}(t) = R_{U}(0)e^{−2p′t} and A_{U}(t) = A_{U}(0)e^{−(p+d)t}. Because the fraction of labeled cells equals 1 minus the fraction of unlabeled cells, where we have assumed that the total population remains constant and hence equal to its initial value. If the population size is changing then the denominator in Equation 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 A_{1} = A_{U}(0)/(A_{U}(0) + R_{U}(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 − (R_{U}(0)e^{−2p′t})/(A_{U}(0) + R_{U}(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 and where we have assumed labeled resting cells upon clonal expansion produces labeled activated cells. Solving these equations for t_{e} ≤ t < t_{d}, we find R_{L}(t) = R_{L}(t_{e}) and where R_{L}(t_{e}) and A_{L}(t_{e}) are the population sizes of labeled resting and activated cells, respectively, when BrdU treatment is stopped. (If we assume that labeled resting cells produce unlabeled activated cells due to label dilution, the corresponding expression for the fraction of labeled cells is obtained by setting a equal to 0 in the above equation.)

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?

Equation 8 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, A_{2}, 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 A_{2} = 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) × 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, I_{t}, during delabeling is given by where s_{t} is the contribution to the total intensity of BrdU from the source during delabeling (see *Appendix*). If we assume that s_{t} = 0 during delabeling, then I_{t}(t) = I_{t}(t_{e})e^{−d(t−te}), where t = t_{e} is the time when BrdU treatment is stopped. Using this result, the death rate is easily estimated. However, the method does have the restriction that it assumes there is no loss of labeled cells due to label dilution.

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, I_{m}, can be derived as where s_{m} represents the rate of contribution from the source to the mean intensity during delabeling (see *Appendix*). If s_{m} = 0, the proliferation rate is given by the slope of a linear regression through the logarithm of the mean intensity of label during delabeling. Again, this derivation assumes that there is no loss due to label dilution.

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.

## Discussion

BrdU is commonly used to label dividing cells and to estimate cell kinetic parameters. We have shown that even under ideal conditions, where the labeling efficiency is 100%, loss of label by dilution can be ignored and the cell population size is constant, interpreting BrdU data so as to obtain cell proliferation and death rates is not straightforward. For example, the initial rate of increase in the fraction of labeled cells during BrdU administration does not simply reflect the cell proliferation rate, but, as we have shown, the sum of the death and proliferation rates. If the cell population is growing or declining, if labeling is imperfect, if dilution of intra- cellular label is significant, or if there is a source of unlabeled cells during BrdU administration, then the initial rate of increase in the fraction of labeled cells is more complicated to interpret.

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.

## Appendix

Assume that at time t the total and mean intensity of BrdU are given by I_{t}(t) and I_{m}(t). Then after a small time interval Δt the total intensity is given by where N(t) is the total number of labeled cells at time t and s_{t}Δt describes the total input of BrdU from the source during the time interval Δt. During this time interval, a total number of dΔtN(t) cells die, each of them removing on average once the mean intensity I_{m}(t) from the compartment. Hence, −dΔtN(t)I_{m}(t) reflects the total loss of BrdU by cell death. Proliferation is accounted for by the terms −pΔtN(t)I_{m}(t) + 2pΔtN(t)I_{m}(t)/2, which reflect the loss of cells with mean intensity I_{m}(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)I_{m}(t) = I_{t}(t), we derive the differential equation for I_{t}(t): The mean intensity of BrdU at the time t + Δt is given by Hence, the differential equation for I_{m}(t) is: where s_{m} = s_{t}/N(t) is the contribution to the mean intensity from the source.

## Acknowledgments

We thank J. Mittler for useful comments.

## Footnotes

- Received October 21, 1999.
- Accepted March 1, 2000.

- Copyright © 2000 by The American Association of Immunologists