Introduction

The CFAcoop package equips you with functions to analyze data from the clonogenic assay (also called ‘Colony Formation Assay’) in presence or absence of cellular cooperation. Thus, this package allows you to robustly extract clonogenic survival information for your cell lines under a given treatment.

This vignette is meant to enable you to process your data following the method first presented in Brix et al., Radiation Oncology, 2020: “The clonogenic assay: robustness of plating efficiency-based analysis is strongly compromised by cellular cooperation”. Therefore, the data presented in Figure 2 in Brix et al., which is provided within the package, is used to illustrate how to use the CFAcoop package. Further, it is shown, how the survival fractions of cooperative cell lines would look like, if cellular cooperation was ignored.

Cellular Cooperation

In order to avoid confusion, cellular cooperation should be defined. Understanding this concept comes easy, when starting from the opposite. Clonogenic growth of non-cooperative cells is independent of cell density. Thus, there is a constant relationship between cells seeded \(S\) and colonies counted \(C\): \[C = a \cdot S,\] where \(a\) corresponds just to the conventional plating efficiency (\(PE = \frac{C}{S}\)).

Now, cellular cooperation refers to the benefit cells can have from their surrounding neighbors which results in a non-constant relation of cells seeded and colonies counted (For details, see Brix et al., Radiation Oncology, 2020). The probability of clonogenic growth for a single cell increases with the number of surrounding cells to cooperate with. It has turned out that generalizing the equation above by a parameter \(b\) adequately models the colonies counted of cooperative and non-cooperative cell lines: \[C = a \cdot S^b.\] In this model, \(b = 1\) gives the non-cooperative case and \(b > 1\) corresponds to cooperative growth.

In short, a cell line is called cooperative, if \(b > 1\).

Clonogenic Survival

Conventionally, clonogenic survival at a given treatment \(x\) was determined as the ratio of colonies counted \(C_x\) and the cells seeded \(S_x\) scaled to the plating efficiency of a reference \(PE_0 = \frac{C_0}{S_0}\) \[SF'_x = \frac{\frac{C_x}{S_x}}{PE_0}.\]

The new method now shifts the focus of the survival fraction directly to the number of cells needed to be seeded under the two conditions (treated and untreated) in order to achieve an identical expectation of the number of colonies formed \(C\). Essentially, the new method does not focus on the number of colonies formed after growth in different cell densities, but on the number of seeded single cells with clonogenic potential before growing to identical colony numbers.

\[SF_x(C) = \frac{S_0(C)}{S_x(C)} = exp\left( \frac{log\left(\frac{C}{a_0}\right)}{b_0} - \frac{log\left(\frac{C}{a_x}\right)}{b_x}\right)\]

Obviously, for \(b_x = b_0 = 1\) the equivalence \(SF_x(C) \equiv SF'\) holds for all \(C\), and thus, the non-cooperative case is well covered by the new method. Importantly, the conventional determination of clonogenic survival is heavily compromised by cellular cooperation, if present.

Getting Started

The data as presented in Figure 2 in Brix et al.  is included in the package in form of a `data.frame` CFAdata. It can be loaded and summarized by:

library(CFAcoop)
data("CFAdata")
summary(CFAdata)
#>      cell.line   Cells seeded           0 Gy             1 Gy       
#>  T47D     :54   Min.   :   14.68   Min.   :  0.00   Min.   :  0.00  
#>  MDA-MB231:84   1st Qu.:  146.78   1st Qu.:  8.00   1st Qu.: 10.00  
#>  A549     :72   Median : 1000.00   Median : 35.50   Median : 35.00  
#>  HCC1806  :48   Mean   : 3980.14   Mean   : 79.55   Mean   : 83.97  
#>  SKBR3    :80   3rd Qu.: 4641.59   3rd Qu.:114.00   3rd Qu.:125.50  
#>  SKLU1    :56   Max.   :31622.78   Max.   :535.00   Max.   :510.00  
#>  BT20     :60                      NA's   :214      NA's   :214     
#>       2 Gy             4 Gy             6 Gy             8 Gy       
#>  Min.   :  0.00   Min.   :  0.00   Min.   :  0.00   Min.   :  0.00  
#>  1st Qu.: 10.00   1st Qu.:  7.50   1st Qu.:  4.00   1st Qu.:  1.00  
#>  Median : 35.00   Median : 29.00   Median : 19.00   Median :  7.00  
#>  Mean   : 85.31   Mean   : 70.01   Mean   : 56.17   Mean   : 30.21  
#>  3rd Qu.:119.00   3rd Qu.:105.00   3rd Qu.: 73.25   3rd Qu.: 30.00  
#>  Max.   :480.00   Max.   :400.00   Max.   :450.00   Max.   :250.00  
#>  NA's   :219      NA's   :215      NA's   :214      NA's   :213

Assess Cellular Cooperation

Key to the robust analysis of clonogenic analysis data is the modeling of the cellular cooperation. We assume that the underlying functional dependency of seeded cells and counted colonies is of the form \[C = a \cdot S^{b},\] where \(b\) indicates the degree of cellular cooperation (\(b = 1\) is implicitly assumed for the PE-based approach).

The coefficient \(b\) is estimated in a linear regression model \[log(C) = log(a) + b \cdot log(S) + \varepsilon, \varepsilon \sim \mathcal{N}(0,\sigma^2). \]

The function `pwr_reg(seede, counted)` provides this regression and returns a `summary.lm` object.

Note that the analysis of cellular cooperation is restricted to the range of seeded cells, where at least one colony was observed. Outside this range, the attempt of studying clonogenic survival based on no observed colony counts is not reasonable and thus, `pwr_reg` will remove those data points from analysis. Thus, it is strongly recommended to use the averaged data for regression. By doing so, the range of the independent variable of the regression is widened. (Removing only those replicates with no colonies at one or few specific cell densities would bias the model fitting.)

data <- subset.data.frame(x = CFAdata, subset = CFAdata$cell.line == "BT20")
data <- aggregate(x = data[, -1], by = list(data$`Cells seeded`), FUN = "mean", na.rm = TRUE)
par_0 <- pwr_reg(seeded = data$`Cells seeded`, counted = data$`0 Gy`)
par_0$coefficients
#>              Estimate Std. Error   t value     Pr(>|t|)
#> (Intercept) -9.031773  0.7538225 -11.98130 2.169497e-06
#> lnS          1.758593  0.1172839  14.99433 3.864744e-07
plot(x = log10(data$`Cells seeded`), y = log10(data$`0 Gy`),xlim = c(2,3.5))
abline(a = log10(exp(1)) * par_0$coefficients[1, 1], b = par_0$coefficients[2, 1])

With the results of this function, we can also test for cellular cooperation. Note, that the p-value in the coefficients table corresponds to the null hypothesis \(b = 0\), but we are interested in the null hypothesis of \(b = 1\). Thus, we find our p-value of interest by computing

p_value <-
  (1 - pt(
    q = (par_0$coefficients[2, 1] - 1) / par_0$coefficients[2, 2],
    df = par_0$df[2]
  ))

Thus, BT20 is higly cooperative (\(b = 1.76\), \(\hat{\sigma}_b = 0.12\), \(p < 0.001\)).

Determine Clonogenic Survival Fractions

In this package, the survival fraction \(SF(C)\) for clonogenic survival is calculated as the number of cells that need to be seeded without treatment divided by the number of cells needed to be seeded with treatment for obtaining the same expectation of colonies counted \(C\). \[ SF(C) = \frac{S_0(C)}{S_x(C)} = exp\left( \frac{log\left(\frac{C}{a_0}\right)}{b_0} - \frac{log\left(\frac{C}{a_x}\right)}{b_x} \right)\]

Given two parameter sets of clonogenic assay data, the clonogenic survival can be calculated as:

data <- subset.data.frame(x = CFAdata, subset = CFAdata$cell.line == "BT20")
data <- aggregate(x = data[, -1], by = list(data$`Cells seeded`), FUN = "mean", na.rm = TRUE)
par_0 <- pwr_reg(seeded = data$`Cells seeded`, counted = data$`0 Gy`)
par_4 <- pwr_reg(seeded = data$`Cells seeded`, counted = data$`4 Gy`)
calculate_sf(par_ref = par_0, par_treat = par_4, C = 20)
#>        20 
#> 0.4308404

Determine Uncertainty of Survival Fractions

The survival fraction at \(C\) for treatment \(x\) is calculated by the function \[SF_x(C) = \frac{S_0(C)}{S_x(C)} = exp\left( \frac{log\left(\frac{C}{a_0}\right)}{b_0} - \frac{log\left(\frac{C}{a_x}\right)}{b_x}\right).\]

Since the SF-values are solely dependent on the estimated parameters in the power regression (and the chosen \(C\)), the inherent uncertainty can be assessed via parametric bootstrapping (e.g. using the package mvtnorm to generate parameter sets according to the variance-covariance matrix of the fit), or by following the laws of error propagation (First-Order Taylor-Series Approximation).

We choose the analytic approximation. In order to build meaningful uncertainty intervals (i.e. respect that survival fractions will never be below zero), we work on the log-scale and transform the boundaries to the linear scale at the end.

For the sake of a shorter notation, we write: \[g = log(SF_x(C)) = \frac{d-\alpha_0}{b_0} - \frac{d-\alpha_x}{b_x} \]

According to \(\Sigma_g \approx J \Sigma_pJ^T\), where \(J\) denotes the Jacobian of \(g\) and \(\Sigma_p\) the variance-covariance matrix of the estimated parameters \(\alpha = log(a)\) and \(b\) at \(0\) and \(x\) Gy, we find \[\sigma_g^2 \approx \frac{1}{b_0^2} z_0 \Sigma_0 z_0^T + \frac{1}{b_x^2} z_x \Sigma_x z_x^T \] with \[ z_x = \left(\begin{matrix}1 & \frac{d-\alpha_x}{b_x}\end{matrix}\right) \] and \[\Sigma_x = \left(\begin{matrix} \sigma_{\alpha_x}^2 & \sigma_{\alpha_x b_x}\\ \sigma_{\alpha_x b_x} & \sigma_{b_x}^2 \end{matrix} \right).\]

(1) The non-cooperative case

For calculating the survival fraction, plating efficiencies are required. Plating efficiencies (\(C/S\)) are calculated easily as:

data(CFAdata)
data <- subset.data.frame(x = CFAdata, subset = CFAdata$cell.line == "T47D")
data <- aggregate(x = data[, -1], by = list(data$`Cells seeded`), FUN = "mean", na.rm = TRUE)
PE_x <- data$`4 Gy` / data$`Cells seeded`
PE_0 <- data$`0 Gy` / data$`Cells seeded`
plot(x = rep(c(0, 1), each = 18), y = c(PE_0, PE_x), lty = 0, ylim = c(0,0.5),xlim = c(-0.1,1.1),
     xlab = "treatment", ylab = "C / S", axes = FALSE, main = "T47D")
axis(side = 1,at = c(0,1),labels = c("0 Gy","4 Gy"))
axis(side = 2,at = seq(0,0.5,0.1))

Now, which values are to be compared?

When there is no effect of the cell density, as assumed by the conventional approach (not taking cellular cooperation into account), each combination (of a \(C_{0 Gy}/S_{0 Gy}\)- and \(C_{4 Gy}/S_{4 Gy}\)-value) is equally reliable.

Thus, the full set of all combinations is:

SF_resample <- rep(PE_x, each = length(PE_0)) / rep(PE_0, times = length(PE_x))
hist(SF_resample, breaks = 25,xlim = c(0.12,0.25),xlab = "(C_4/S_4) / (C_0/S_0)",
     main = "valid PE-based SF'-values")

Without the assessment of cellular cooperation, conventional calculation of an \(SF'\)-value, corresponds to picking randomly a sample from the distribution shown in the histogram above.

A comparison of the range of this distribution and the calculated uncertainties of the new method shows, that for non-cooperative cell lines such as T47D, there is no big difference in this variability/uncertainty.

range(SF_resample,na.rm = TRUE)
#> [1] 0.1322039 0.2383178
as_nc_0 <- analyze_survival(RD = data[,-1],C = 20)
as_nc_0$uncertainty[4,c(3,5,6)]
#>          SF     lb.SF     ub.SF
#> 4 0.1914189 0.1577691 0.2322457

(2) The cooperative case

Now, making the same comparison as in (1) for the cooperative cell line BT20 shows the disastrous effect of ignoring the coefficient \(b\), when it is in fact different from \(1\).

data(CFAdata)
data <- subset.data.frame(x = CFAdata, subset = CFAdata$cell.line == "BT20")
data <- aggregate(x = data[, -1], by = list(data$`Cells seeded`), FUN = "mean", na.rm = TRUE)
PE_x <- data$`4 Gy` / data$`Cells seeded`
PE_0 <- data$`0 Gy` / data$`Cells seeded`
plot(x = rep(c(0, 1), each = length(PE_x)), 
     y = c(PE_0, PE_x), lty = 0, ylim = c(0,0.08),xlim = c(-0.1,1.1),
     xlab = "treatment", ylab = "C / S", axes = FALSE, main = "BT20")
axis(side = 1,at = c(0,1),labels = c("0 Gy","4 Gy"))
axis(side = 2,at = seq(0,0.08,0.02),las = 1)

SF_resample <- rep(PE_x, each = length(PE_0)) / rep(PE_0, times = length(PE_x))
hist(SF_resample, breaks = 100,xlim = c(0,10),xlab = "(C_4/S_4) / (C_0/S_0)",
     main = "valid PE-based SF'-values")

The range of the PE-based \(SF'\)-values does not correspond to the uncertainty of the \(SF(20)\)-values:

range(SF_resample,na.rm = TRUE)
#> [1] 0.02511682 8.53525274
as_c_4 <- analyze_survival(RD = data[,-1],C = 20)
as_c_4$uncertainty[4,c(3,5,6)]
#>          SF     lb.SF     ub.SF
#> 4 0.4308404 0.3518497 0.5275645

Even though the survival fraction can be accurately estimated under consideration of cellular cooperation the PE-based approach fails in returning a trustworthy estimate of the fraction of cells losing their potential due to the treatment (see histogram).

In particular, the average of PE-based \(SF'\) calculations does not asymptotically tend to a meaningful value. In case of strong cellular cooperation, the PE-based calculated \(SF'\)-value is heavily affected by this cellular cooperation and the treatment effect of interest is degraded to a side effect.

Conclusion from (1) and (2)

Before calculating PE-based survival fractions, one must check whether there is cellular cooperation or not.

Essentially, to decide, whether you have a cooperativity issue or not, you need to conduct the same analysis and to generate the same data that is necessary to solve this issue anyways.