Multi-Scale Estimation of Cell Kinetics

Abstract

We introduce a methodology based on the Luria-Delbrück fluctuation model for estimating the cell kinetics of clonally expanding populations. In particular, this approach allows estimation of the net cell proliferation rate, the extinction coefficient, and the initial (viable) population size. We present a systematic approach based on spatial partitioning which captures the local fluctuations of the number and sizes of individual clones. However, partitioning introduces measurement error by inflating the number of clones, which is dependent on time and the degree of cell migration. We perform various in silico experiments to explore the properties of the estimators and we show that there exists a direct relationship between precision and observation time. We also explore the trade-off between measurement error and estimation accuracy. By exploring different scales of cellular fluctuations, from the entire population down to those of individual clones, we show that this methodology is useful for inferring important parameters in neoplastic progression.

1. Introduction

Mutations, clonal expansions, and (epi)genetic heterogeneity are the hallmarks of many types of neoplasms [1]. Although Nowell’s description of cancer as an evolutionary system [2] is widely accepted, key parameters of this description for measuring cell fitness remain generally undetermined. For example, what is the difference between cell replication and death rates (net cell proliferation rate)? What is the ratio between the cell death and replication rates (extinction coefficient)? What is the actual number of (viable) cells that contribute to future neoplastic generations (initial population size)? Knowing these parameters will help measure, manage, and further our understanding of neoplastic progression.

A number of experimental techniques have been devised to quantify the frequency of cell proliferation and cell death (apoptosis or necrosis), but have limitations [3, 4]. In particular, most labeling methods provide only relative measures (or proportions) of dividing or dying cells, but do not provide absolute rates, i.e. the number of events (cell divisions or cell deaths) per unit time. Moreover, many labels used, such as the mitotic index [5, 6], tritiated thymidine labeling index [7, 8], and 5-bromodeoxyuridine (BrdU) [9], have cytotoxic effects [10–12] and may lead to biased estimates. Furthermore, perhaps with the exception of continuous labeling methods, these assays only provide snapshots of cellular kinetics at the time the assay is performed.

The mathematical approach described here for quantifying cell kinetics has roots in the classical Luria-Delbrück fluctuation analysis [13], which was initially developed to study the natural selection of mutations. In an attempt to determine whether mutations arise spontaneously or from adaptation [14], Luria and Delbrück studied the origin of phage-resistant mutants when cultures of E. coli bacteria are seeded in the presence of bacteriophage T1. Surprisingly, the number of observed T1-resistant mutants varied greatly from culture to culture, which led to the conclusion that mutation to phage resistance is the result of rare spontaneous events. Several extensions of the original Luria-Delbrück model were developed over the years [14], including the Lea-Coulson [15], discretized [15, 16], Bartlett [16], Haldane [17, 18] extensions, and the generalized Luria-Delbrück formulation by Dewanji, Luebeck and Moolgavkar [19].

While the use of Luria-Delbrück-type fluctuation models has been limited to the estimation of mutation rates, the technique remains essentially untested for the estimation of cell kinetics parameters such as the net cell proliferation rate, extinction coefficient, and the initial (viable) population size in cell culture experiments. A mathematical framework for the quantitative analysis of the number of clones and their sizes has been developed previously by Dewanji et. al. [20] and was subsequently used for the analysis of enzyme-altered liver foci generated in animal experiments for the estimation of both mutation rates and cell kinetics [21–23]. However, information on the number and sizes of clones are not always available in cell culture experiments.

The limitations of assays with no or imprecise information on the number and sizes of individual clones signal a need for a generally applicable method which attempts to capture the net cell proliferation rate, extinction coefficient, and the initial population size. In this study we introduce such a methodology based on the generalized Luria-Delbrück model developed by Dewanji et. al. The strength of our approach is the capability of estimating cell kinetics in the absence of information on the number and sizes of individual clones.

We begin with an overview of the generalized Luria-Delbrück model followed by an introduction to the methodology, which discusses spatial partitioning of observation areas and the effects of measurement error on parameter estimation. We carry out a series of in silico experiments to explore the properties of the estimators, and show that they remain stable to a point where the measurement error introduced by the spatial partitions dominates the characteristic size fluctuations of the clones. For each model parameter, this leads to a range of spatial partitions in which the bias in absolute value is minimized. We also examine the relationship between estimation accuracy, precision and measurement error due to partitioning of in silico cell cultures. This is done by observing changes in the estimator properties both in time and under variable spatial dispersion of the cells in a clone. Finally, the methodology is demonstrated on data obtained from a Barrett’s Esophagus EPC-2 neoplastic cell line culture.

2. The Generalized Luria-Delbrück Model

The different variants of the Luria-Delbrück model are characterized according to the dynamics of the normal and mutant populations [14]. With the exception of the original formulation, the Luria-Delbrück models involve a stochastic component to either describe the dynamics of the normal cells, mutants, or both. Here we adopt a framework which allows for stochastic cell kinetics in the mutants. Specifically, let X(t) and Y(t) represent the population size of normal cells and mutated cells (mutants) at time t (in days), respectively. It is assumed that once a normal cell acquires a mutation with rate ν(s) per normal cell at time s, it undergoes a clonal expansion in the form of a birth-death process with birth rate α(t, s) (per mutant cell per day) and death rate β(t, s) (per mutant cell per day) to form a clone of size Y(t, s) at time t.

The total number of mutants at time t is given by the filtered Poisson process [24]

$$Y(t)=\{\begin{array}{ll}0,\hfill & M(t)=0\hfill \\ {\sum }_{i=1}^{M(t)}Y(t,s_i),\hfill & M(t)\ge 1\hfill \end{array},$$ (1)

where M(t) denotes the number of Poisson mutation events from normal cells with rate ν(t)X(t) at time t. According to Dewanji et. al. [19], the distribution of Y(t) takes the form

$$P_k(t):=\text{Prob}\{Y(t)=k\}=\sum _{i=0}^{k-1}\left(\frac{k-i}{k}\right)P_i(t)h_{k-i}(t),k=1,2,3,\dots$$ (2)

$$P_0(t):=\text{Prob}\{Y(t)=0\}=exp\left\{-{\int }_0^t\frac{\nu (s)X(s)}{G(t,s)+g(t,s)}ds\right\},$$ (3)

where

$$h_i(t):={\int }_0^t\nu (s)X(s)\frac{g(t,s)}{G{(t,s)}^2}{\left[\frac{G(t,s)}{G(t,s)+g(t,s)}\right]}^{i+1}ds,i=1,2,3,\dots$$ (4)

$$g(t,s):=exp\left\{-{\int }_s^t[\alpha (u,s)-\beta (u,s)]du\right\},$$ (5)

$$G(t,s):={\int }_s^t\alpha (u,s)g(u,s)du.$$ (6)

3. Methodology

3.1 Parameter Estimation

Here we describe a general in vitro experiment to determine the rates of both cell replication and cell death: x fluorescently labeled cells are seeded in each of y wells (petri dishes), cells in a specific medium are grown for t days, locations of labeled cells in a rectangular field are recorded and their spatial locations are digitized. In the case that individual clones can be distinguished, the final step is to obtain the total number of clones and their respective sizes; otherwise, count the total number of cells after t days. Some cells will not survive the initial seeding in such experiments due to damage during culturing or failure to adhere to medium. This set of procedures may be complemented with experimental methods of estimating net cell proliferation rate for validation of the estimates.

To keep the presentation simple, we assume constant birth and death rates, i.e. α(t, s) = α and β(t, s) = β. When a normal cell is initiated (or “mutated” in the Luria-Delbrück language), it is signaled to undergo clonal expansion. For the cell culture experiment described above, initiation can be considered as the initial seeding of cells that may undergo clonal expansion. This is expressed mathematically by defining ν(t)X(t) = ηδ(t), where η is the initial population size (initial number of viable cells) and δ(t) is the dirac delta function. This initial “pulse” seeding of cells has been previously applied to the modeling of gestational mutations in the context of a multistage theory of carcinogenesis [25]. Within this framework, the cell kinetics parameters we are interested in estimating are defined as ρ:=α − β (net cell proliferation rate), γ:=β/α (extinction coefficient), and η (initial population size). Under the constant parameter assumption, the probability generating function for the total population, Y(t), can be shown to be (see Appendix)

$$\mathrm{\Psi }(z,t):=\mathbb{E}[z^{Y(t)}]=\sum _{i=0}^{\infty }\text{Prob}\{Y(t)=i\}{z}^i=exp\left[\frac{(\gamma -1)\eta (z-1)}{(\gamma -z)e^{-\rho t}+(z-1)}\right].$$ (7)

Due to the computational inefficiency of the recursive form of the distribution P_k(t), k = 1, 2, … (equation (2)), we resort to the inverse Laplace transform approximation of the distribution introduced by Abate et. al. [26]. This method incorporates the Poisson summation formula [27], which uses a periodic function represented by its Fourier series to approximate distribution. The formula is also useful for controlling the discretization error that comes from the numerical approximation. It can be shown that the exact forms of the distribution P_k(t) for k = 0 and k = 1 are given by (see Appendix)

$$P_0(t)=e^{-\mathrm{\Lambda }(t)}$$ (8)

and

$$P_1(t)=e^{-\mathrm{\Lambda }(t)}·\frac{\eta {(1-\gamma )}^2{e}^{-\rho t}}{{(1-\gamma e^{-\rho t})}^2},$$ (9)

respectively, where $\mathrm{\Lambda }(t)=\frac{(1-\gamma )\eta }{1-\gamma e^{-\rho t}}$ is the rate of the Poisson process M(t), i.e. the number of clones at time t. According to Abate et. al. [26], equation (2) for k = 2, 3, … can be approximated by

$$P_k(t)\approx \frac{1}{2{kr}^k}\left\{\mathrm{\Psi }(r,t)+{(-1)}^k\mathrm{\Psi }(-r,t)+2\sum _{j=1}^{k-1}{(-1)}^j\text{Re}\left[\mathrm{\Psi }\left({re}^{\frac{i\pi j}{k}},t\right)\right]\right\},$$ (10)

where an error bound of 10^−h may be achieved by choosing r such that $r\approx {10}^{-\frac{h}{2k}}$. Furthermore, in the case where the total number of clones and their sizes are available, we use their respective distributions for the likelihood estimation of the model parameters ρ, γ, and η. They are respectively given by

$$P_m^{(N)}(t):=\text{Prob}\{M(t)=m\}=\frac{e^{-\mathrm{\Lambda }(t)}\mathrm{\Lambda }{(t)}^m}{m!},m=0,1,2,\dots$$ (11)

and

$$\begin{array}{l}{P}_w^{(S)}(t,s):=\text{Prob}\{Y(t,s)=w\}\\ =\frac{(1-\gamma )e^{-{\rho }^{(t-s)}}}{1-\gamma e^{-\rho (t-s)}}{\left[\frac{1-e^{-\rho (t-s)}}{1-\gamma e^{-\rho (t-s)}}\right]}^{w-1},w=1,2,\dots \end{array}$$ (12)

The superscripts N and S represent the number and sizes of individual clones, respectively. The derivation can be found in the Appendix. We point out that $P_w^{(S)}(t,s)$ is the size distribution for non-extinct clones.

In the absence of information on the number (M(t)) and sizes (Y(t, s)) of individual clones, we would like to explore the size fluctuations on ever shorter length scales down to those of individual clones. Furthermore, the need for precision in the parameter estimation requires a sufficient number of observations. To accomplish this we exploit the information hidden in local fluctuations of clone sizes and introduce the idea of spatial partitioning. Figure 2 shows a sample field spatially partitioned into N different ways, each time creating n × n subfields of identical dimension for n = 1, 2, …, N.

Figure 2.

Schematic representation of a single field spatially partitioned into N different ways where for each n = 1, 2, 3, …, N, n × n subfields with an identical dimension are created, each containing a fractional information of the entire field.

In the case where the number and sizes of individual clones are not available, an observation is referred to as the total number of cells within a particular subfield. Hence, a field with n × n subfields will have n × n independent observations. In order to accurately obtain the cell count within any given subfield, it is necessary to identify all the cells within that subfield, which requires knowledge of their spatial coordinates. To accomplish this task, we digitize the cells and their locations using the ImageJ^© software [28, 29].

Figure 3 shows four identical in silico fields with spatial partitions of n × n = 4 × 4, 0 × 10, 20 × 20, and 30 × 30, respectively. In this computer simulated experiment, the first progenitor cell of each clone is randomly seeded according to a spatial Poisson process, then the distance between any daughter cell and the progenitor cell from which it came is normally distributed with a variance proportional to clone size. Thus clones with a larger number of cells occupy a greater region. This is demonstrated in Figure 2 where blue, red, and green represent the three largest clones in decreasing cell counts. Although increasing the number of spatial partitions by fine-graining increases the number of observations, each subfield only contains a fraction of cell counts relative to the entire population. In addition, there is an increasing tendency for a clone to spread across multiple subfields for the finer partitions. We refer to these clones as partitioned clones. Thus, the question arises: how does excessive partitioning beyond the scale of local fluctuations of clone sizes affect parameter estimation? We address explore this question in more detail by simulations.

Figure 3.

Single field of an in silico cell culture with spatial partitions (a) 4 × 4, (b) 10 × 10, (c) 20 × 20 and (d) 30 × 30. Each cell in an initial viable population of η = 1000 undergoes a birth-death process with parameters α = 0.75, β = 0.5, respectively (hence ρ = 0.25 and γ = 0.67), and the field is obtained after t = 1 day. The blue, red and green regions represent the three largest clones in decreasing sizes. Increasing the number of spatial partitions increases the occurrence of partitioned clones.

3.2 Trade-Offs Associated with Partitioning a Single Field

In the absence of experimental replicates and without spatial partitioning (i.e., using a single 1 × 1 partition), no estimation is possible since a single data point is insufficient for estimating three parameters. Spatial partitioning, on the other hand, introduces subfields (or multiple observations) and allows the estimation of all three model parameters in a single experiment. However, the process of spatial partitioning increases the frequency of partitioned clones (see Figure 4), thereby artificially inflating clone number and skewing their sizes toward smaller clones. We refer to this phenomenon simply as measurement error.

Figure 4.

A closer look at partitioned clones and measurement error. A schematic field with (a) 5 clones is partitioned into (b) 2×2 and (c) 3×3 subfields. Partitioned clones (in blue) are created as clones spatially spread across multiple subfields, which artificially inflate the number of clones. The Luria-Delbrück theory regards the original 5 clones as 8 and 11 distinct clones under the 2×2 and 3×3 partitions cases, respectively, leading to biases in the parameters estimation.

The trade-off associated with spatial partitioning suggests that for each parameter, there exists a range of partitions with acceptable accuracy and precision for estimates. We refer to this set of partitions as the optimal partition range of the parameters. We will demonstrate the existence of such ranges via simulations, and show how they assist in identifying the set of optimal estimates in an in vitro cell culture.

4. in silico Experiments

Before applying the methodology to actual cell cultures, we explore the properties of the estimators for ρ (net cell proliferation rate), γ (extinction coefficient), and η (initial population size), defined as ρ̂_n, γ̂_n, and η̂_n, respectively, where n represents the n × n partition. In particular, we examine how the estimators behave in the following distinct in silico cell culture scenarios:

1. resolvable clones with no measurement error

2. non-resolvable clones with no measurement error

3. non-resolvable clones with measurement error

In this context, a cell culture with resolvable clones is one in which the number and sizes of individual clones are known, while measurement error refers to the artificial inflation of the number of clones due to partitioned clones (Figure 3). Hence, scenarios (I) and (II) are less relevant to the real world, but are nevertheless important for understanding the properties of the estimators. For each scenario, we obtain population dynamics by randomly seeding cells and simulating clonal expansion based on arbitrarily chosen but fixed parameters. Doing so allows us to acquire the desired cell culture information (number of clones, their respective sizes, or total cell counts) necessary to perform the maximum likelihood estimation. Finally, we study the effects of measurement error on parameters estimation.

4.1 Case (I): Resolvable Clones with No Measurement Error

In the first of three in silico cell cultures designed to explore the properties of the estimators, we restrict all daughter cells to be situated at the same location as the first progenitor cell from which they descended. Doing so uniquely assigns a clone to a single subfield, hence no measurement error is introduced. In addition, it is assumed that the number of clones and their sizes are known (resolvable) without measurement error, hence we can use the distributions given by equations (11) and (12) rather than the Luria-Delbrück distribution for the estimation of the cell kinetic parameters from such observations. Consider a field with n × n subfields, let Ñ_i and S̃_i = [S̃_1i, S̃_2i, …, S̃_Ñii] be the observed number and sizes of clones, respectively, of the i^th subfield for i = 1, 2, …, n × n at t = T̃, then the likelihood is

$$\mathcal{L}(\rho ,\gamma ,\eta \mid \{{\stackrel{\sim }{N}}_i,{\stackrel{\sim }{S}}_i,i=1,2,\dots ,n\times n\},\stackrel{\sim }{T})=\prod _{i=1}^{n\times n}\left\{P_{k={\stackrel{\sim }{N}}_i}^{(N)}(t=\stackrel{\sim }{T})\prod _{j=1}^{{\stackrel{\sim }{N}}_i}{P}_{k={\stackrel{\sim }{S}}_{ji}}^{(S)}(t=\stackrel{\sim }{T})\right\},$$ (13)

where $P_k^{(N)}(t)$ and $P_k^{(S)}(t)$ are given in equations (11) and (12). For the case that the individual clone number and sizes are known, it is immediately clear that different spatial partitions yield the same likelihood, and the estimators are independent of partition size. Indeed, simulations for different observation times reveal that the variances of the estimates decrease with increasing observation time (Figure 5). This observed effect may simply reflect the gain in efficiency of the estimators due to an increase in the number of stochastic events with time.

Figure 5.

Analysis of resolvable clones with no measurement error. For each time point t = 1, 2, 3, 4 and 5 days, 1000 independent fields of a cell culture with parameters ρ = 0.55 (net cell proliferation rate), γ = 0.75 (extinction coefficient) and η = 300 (initial population size) (dotted lines) are simulated and analyzed. For each parameter, the mean of the 1000 estimates is plotted along with the 95%CI. Since different spatial partitions produce the identical likelihood function, only the results for the 10 × 10 partition are shown.

4.2 Case (II): Non-Resolvable Clones and No Measurement Error

A layer of complexity is added to the simulation so that for each cell, only the spatial location is given, with no information on which clone it belongs to or how many clones are observed. In this case, the Luria-Delbrück distribution plays a central role in the parameter estimation and we show that it introduces a bias on the estimates. As in case (I), measurement error as a result of partitioned clones is still designed to be absent.

$$\mathcal{L}(\rho ,\gamma ,\eta \mid \{{\stackrel{\sim }{Y}}_i,i=1,2,\dots ,n\times n\},\stackrel{\sim }{T})\approx \prod _{i=1}^{n\times n}{P}_{k={\stackrel{\sim }{Y}}_i}(t=\stackrel{\sim }{T}),$$ (14)

where P_k(t) for k = 0, k = 1 and k ≥ 2 are given by equations (8), (9) and (10), respectively. Ψ(z, t) is given by equation (7) and $r_i\approx {10}^{-\frac{h}{2{\stackrel{\sim }{Y}}_i}}$ gives an accuracy of 10^−h for the i^th subfield, where i = 1, 2, …, n × n. In comparison to case (I), we expect the accuracy of the estimates to diminish for some partitions due to the lack of information on the number and sizes of individual clones. However, accuracy should increase as the partition is fine-grained since ultimately every subfield contains at most a single clone, thus representing case (I).

Figure 6 shows the accuracy of the estimates as a function of spatial partitions size. While the estimates for ρ and η show high accuracy independent of partition size, the estimate of the extinction parameter γ exhibits a downward bias for partitions n ≤ 5. An inspection of the distribution of the γ estimates reveals a bias in the mean for smaller sample sizes. For n > 5, the convergence can be attributed to the increasing number of informative subfields associated with the higher partitions, each capturing local fluctuations both in the number of clones and their sizes. Note, under this scenario {ρ̂_n|n = 2, 3, …}, {γ̂_n|n = 2, 3, …}, and {η̂_n|n = 2, 3, …} resemble statistically consistent sequences of estimators, as n → ∞. Also apparent is a bias in γ with smaller partitions, which is due to a lack of normality of the estimators for coarse-grained partitions (fewer subfields).

Figure 6.

Non-resolvable clones with no measurement error. For each observation time of t = 1, 2, 3, 4 and 5 days (rows), 1000 individual fields from a single cell culture with parameters ρ = 0.55 (net cell proliferation rate), γ = 0.75 (extinction coefficient) and η = 300 (initial population size) are simulated, each time obtaining an estimate. For each parameter, the mean of the 1000 estimates and its 95%CI are plotted versus the spatial partitions for n × n = 2 × 2, 3 × 3,…, 20 × 20.

In addition, the confidence intervals for ρ and γ decrease with spatial partitions but increase for the initial population size η. This can be explained as follows: consider a field with n × n subfields, let ξ̂_n be the estimated initial number of viable cells for each subfield, hence η̂_n = n²ξ̂_n is the estimated initial number of viable cells for the entire field. Then the variance (hence the confidence interval), var[η̂_n] = n⁴ · var[ξ̂_n], is amplified by a factor of n⁴ for increasing partition size n even though var[ξ̂_n] decreases with increasing spatial partitions (not shown). In spite of the lack of information on the number and sizes of individual clones, the information stored in the local fluctuations can be “re-captured” with fine-grained partitions in the absence of measurement error since in the limit the observation of the number of clones is either one or zero in each subfield.

4.3 Case (III): Non-Resolvable Clones with Measurement Error

Experimental evidence suggests that cells migrate inside the wells upon seeding. To account for the resulting spatial dispersion of cells, we assume that all daughter cells disperse randomly within a radial distance $\omega =\sqrt{A_{\text{clone}}/\pi }$ away from the first progenitor cell, where A_clone = S_cloneπσ² is the spatial area occupied by a clone having S_clone cells. 2σ is the average pairwise distance between the center of masses of any two cells, and we refer to σ as the dispersion factor. The spatial location of any daughter cell is relative to that of the progenitor. In particular, the x and y displacements of a daughter cell away from the progenitor are separately sampled from the normal distribution (0, ω²).

To better understand the effects of measurement error caused by partitioning on parameters estimation, we simulated a cell culture 1000 times for four different values of the cellular dispersion factor σ. Figure 7 shows this in silico 3-day culture partitioned into 10 × 10 subfields for the cellular dispersion factors σ = 5, 10, 15, and 20. We see that the frequency of occurrence of partitioned clones increases as the dispersion factor, hence the measurement error, increases.

Figure 7.

An in silico cell culture with parameters ρ = 0.55, γ = 0.75 and η = 300 observed after time t = 3 days post-initial seeding. The effect of measurement error is quantified by the cellular dispersion factor σ. The four plots are for (a) σ = 5, (b) σ = 10, (c) σ = 15, (d) σ = 20, and the blue, red and green regions in each case represent the three largest clones in decreasing sizes.

We expect the mean of the estimates in this in silico cell culture to deviate further from the true parameters as σ increases due to the increasing measurement error on the number of clones and their sizes. This is demonstrated in Figure 8, where the rows represent σ= 5, 10, 15, 20, and the horizontal and vertical axes represent the spatial partitions and the means of the 1000 estimates, respectively. For smaller σ, the estimates come closer to the true values in general. As in case (II), we see a reduction in accuracy for coarser spatial partitions. However, we now find that the bias for all three estimators increase for spatial partitions beyond n ≈ 7. This bias occurs because fine-graining increases the tendency for partitions to intersect the clones, thereby artificially inflating their number. Hence more clones are being analyzed via the Luria-Delbrück assumption than what are actually observed (measurement error).

Figure 8.

Effects of measurement error. For each dispersion factor of σ = 5, 10, 15 and 20 (rows), 1000 independent in silico experiments of a cell culture with parameters ρ = 0.55 (net cell proliferation rate), γ = 0.75 (extinction coefficient) and η = 300 (initial population size) are simulated, obtaining an estimate after t = 3 days. For each parameter, the mean of the 1000 estimates and its 95%CI are plotted versus the spatial partitions for n × n = 2 × 2, 3 × 3,…, 20 × 20.

For the extinction coefficient (γ), the measurement error increases the bias of the estimators as the dispersion factor (σ) increases. A similar behavior is observed (not shown) as the time of observation is increased for fixed σ. Furthermore, the confidence interval does not seem to be affected by the change in dispersion factors, suggesting that the variance of the estimators is independent of the measurement error. The bias due to measurement error seems to have a greater effect on γ and η than it does on ρ, indicating that the estimators for the net cell proliferation rate may be more robust under measurement error than those of the other two parameters.

For in silico cell cultures of this type, the optimal estimates for each parameter exist and may come from different partitions. We demonstrate next how these optimal estimates may be identified in an in vitro cell culture using data from a cell culture experiment.

5. Experimental Results

We demonstrate the methodology using data collected from Barrett’s Esophagus EPC-2 cell line. A field with 5, 772 cells was observed 21.167 hours following an initial seeding of ≈ 12, 000 cells. The field was spatially partitioned into sizes ranging from 2 × 2, 3 × 3,…, 20 × 20, where the coordinates of the cell positions were obtained via the ImageJ^© software [28, 29]. For each n × n partition, we obtained n² total subfields (observations), which were used in the likelihood analysis to produce one estimate (Figure 9). We used the Luria-Delbrück form of the likelihood (equation (14)) for analysis, since the number of clones and their sizes in the field were unknown. Figure 9 shows the estimates for the net cell proliferation rate (ρ), the extinction coefficient (γ), and the initial population size (η) in red, blue, and green, respectively.

Figure 9.

A single cell culture field from Barrett’s Esophagus EPC-2 cell line data. The cell culture images are reproduced in silico after cellular coordinates were obtained via ImageJ^©, and partitioned from 2 × 2, 3 × 3 up to 20 × 20. The number of clones and sizes are not resolvable, hence our methodology was used to estimate the cell kinetics parameters ρ, γ and η. For each parameter, one estimate was obtained for each partition.

As can be seen from Figure 9, the estimates for ρ and η show a similar behavior as the mean of the estimates for the in silico cell culture scenario (III) in section 4.3 and appear stationary over a narrow range of partitions. In contrast, the estimates of γ do not exhibit a clear plateau or shoulder as seen in Figure 8, but with increasing cellular dispersion factor, σ, the estimates of γ are increasingly biased downward as the partition size increases, overwhelming the small sample bias on the left. Thus, we conclude that the unbiased estimate of γ is larger than the observed maximum value of 0.96. On the other hand, the estimates using partitions less than 5 appear to suffer from small sample biases and tend to be unreliable. The lack of a stationarity in the estimates for γ further indicates that the observation time was not long enough. The qualitative agreement between experimental and in silico results suggests that our method is applicable to cell counts from single wells at a single time point. Moreover, we expect multiple experiments and multiple time points to improve the estimates.

6. Summary

We introduced a multi-scale image processing technique in conjunction with a Luria-Delbrück distribution for estimating cell kinetics parameters in the absence of information on the number and sizes of individual clones. We find that it is possible to estimate cell kinetics parameters via optimal spatial partitioning of the field of observation. More specifically, the estimated net cell proliferation rate, extinction coefficient, and initial population size are given by optimal spatial partitions which may differ from parameter to parameter. We show that there exists a trade-off between accuracy versus bias introduced by measurement errors due to the inability to assign clones uniquely to a particular subfield of a partition. For later observation times and for increasing measurement error due to cellular migration, the bias increases due to a higher occurrence of partitioned clones, thereby decreasing the accuracy of the estimates i.e., increasing the bias of the estimators.

Several aspects of the methodology presented can be improved. Properties of the estimators should also be explored when observations from multiple time points are available. This may lead to more accurate estimates since each time point provides a snapshot of a growing random population undergoing size fluctuations that are determined by the cell kinetics. Furthermore, spatial clustering algorithms might be helpful in improving the estimates. It would also be useful to validate estimates by independent measurements of cell kinetics using experimental techniques, such as BrdU for the measurement of the cell replication rates. The methodology described here for analysis of in vitro data may also be applied to cross sectional images of a neoplastic population in vivo, although stereological methods [30–32] may need to be invoked to transform the cross-sectional data into three dimensions.

The net cell proliferation rate, extinction coefficient, and initial population size are key parameters in understanding the evolutionary biology in neoplastic progression. We have shown that by exploring different scales of cellular fluctuations from the entire population down to those of individual clones, this approach can be useful for estimating these parameters in a clonally expanding population. Ultimately, measuring proliferation and death rates directly in competing cell lines simultaneously could further our understanding of their relative fitness and could potentially provide quantitative information that better characterizes the clonal evolution of neoplastic populations.

Figure 1.

The generalized Luria-Delbrück model. X(t): normal cells at time t (in days); ν(t): rate of Poisson mutation events per normal cell; Y(t, s): size of a single mutant clone at time t whose progenitor cell initiated at time s; α(t, s), β(t, s): birth and death rates per mutant cell of such a clone per day; Y(t): sum of the sizes of all such clones, and its distribution is a Luria-Delbrück distribution.

8. Appendix

8.1 Derivation of equation (7)

The probability generating function for a single clone with cell count Y(t, s) initiated at time s is given by Luebeck et. al. [23] to be

$$\varphi (z;t,s)=\mathbb{E}[z^{Y(t,s)}]=1-\frac{z-1}{(z-1)G(t,s)-g(t,s)},$$ (15)

where under constant birth and death rates, i.e., α(t, s) = α, β(t, s) = β,

$$g(t,s)=exp\left\{-{\int }_s^t(\alpha (u,s)-\beta (u,s))du\right\}=e^{-(\alpha -\beta )(t-s)},$$ (16)

$$G(t,s)={\int }_s^t\alpha (u,s)g(u,s)du={\int }_s^t\alpha e^{-(\alpha -\beta )(u-s)}du=\frac{\alpha }{\alpha -\beta }\left[1-e^{-(\alpha -\beta )(t-s)}\right].$$ (17)

By assuming ρ = α − β, γ = β/α, ν(t)X(t) = ηδ(t), the PGF for Y(t) [23] can be easily shown to be

$$\begin{array}{l}\mathrm{\Psi }(z,t)=\mathbb{E}[z^{Y(t)}]=exp\left\{{\int }_0^t\nu (s)X(s)[\varphi (z;t,s)-1]ds\right\}\\ =exp\left\{{\int }_0^t\eta \delta (s)[\varphi (z;t,s)-1]ds\right\}\\ =exp\{\eta [\varphi (z;t,0)-1]\}\\ =exp\left\{-\eta \frac{z-1}{(z-1)\frac{1}{1-\gamma }(1-e^{-\rho t})-e^{-\rho t}}\right\}\\ =exp\left\{\frac{(\gamma -1)\eta (z-1)}{(\gamma -z)e^{-\rho t}+(z-1)}\right\}.\end{array}$$ (18)

8.2 Derivation of Λ(t), equations (8) and (9)

According to Luebeck et. al. [23], M(t) is a Poisson process with rate

$$\mathrm{\Lambda }(t)={\int }_0^t\frac{\nu (s)X(s)}{G(t,s)+g(t,s)}ds=\frac{\eta }{\frac{1}{1-\gamma }(1-e^{-\rho t})+e^{-\rho t}}=\frac{(1-\gamma )\eta }{1-\gamma e^{-\rho t}}.$$ (19)

Equation (8) is easily derived as follows,

$$\begin{array}{l}{P}_0(t)=exp\left\{{\int }_0^t\nu (s)X(s)[\varphi (0;t,s)-1]ds\right\}\\ =exp\left\{-{\int }_0^t\frac{\nu (s)X(s)}{G(t,s)+g(t,s)}ds\right\}=e^{-\mathrm{\Lambda }(t)},\end{array}$$ (20)

and equation (9) follows immediately from the PGF Ψ(z, t),

$$\begin{array}{l}{P}_1(t)={\frac{\partial \mathrm{\Psi }(z,t)}{\partial z}|}_{z=0}\\ ={\mathrm{\Psi }(0,t)·\frac{\eta (1-\gamma )[(z-1)(1-e^{-\rho t})-((\gamma -z)e^{-\rho t}+(z-1))]}{{[(\gamma -z)e^{-\rho t}+(z-1)]}^2}|}_{z=0}\\ =e^{-\mathrm{\Lambda }(t)}·\frac{\eta {(1-\gamma )}^2{e}^{-\rho t}}{{(1-\gamma e^{-\rho t})}^2}.\end{array}$$ (21)

8.3 Derivation of equation (12)

According to Luebeck et. al. [23],

$$\begin{array}{l}{P}_k^{(S)}(t,s)=\text{Prob}\{Y(t,s)=k\}={\frac{1}{k!}\frac{d^k}{{dz}^k}\mathbb{E}[z^{Y(t,s)}]|}_{z=0}\\ =\frac{g(t,s)}{G(t,s)}{\left[\frac{G(t,s)}{G(t,s)+g(t,s)}\right]}^k\\ =\frac{e^{-\rho (t-s)}}{\frac{1}{1-\gamma }(1-e^{-\rho (t-s)})}{\left[\frac{\frac{1}{1-\gamma }(1-e^{-\rho (t-s)})}{\frac{1}{1-\gamma }(1-e^{-\rho (t-s)})+e^{-\rho (t-s)}}\right]}^k\\ =\frac{e^{-\rho (t-s)}}{\frac{1}{1-\gamma }(1-e^{-\rho (t-s)})+e^{-\rho (t-s)}}{\left[\frac{1-e^{-\rho (t-s)}}{1-e^{-\rho (t-s)}+(1-\gamma )e^{-\rho (t-s)}}\right]}^{k-1}\\ =\frac{(1-\gamma )e^{-\rho (t-s)}}{1-\gamma e^{-\rho (t-s)}}{\left[\frac{1-e^{-\rho (t-s)}}{1-\gamma e^{-\rho (t-s)}}\right]}^{k-1}.\end{array}$$ (22)