Abstract
Invitro chemosensitivity experiments are an essential step in the early stages of cancer therapy development, but existing data analysis methods suffer from problems with fitting, do not permit assessment of uncertainty, and can give misleading estimates of cell growth inhibition. We present an approach (bdChemo) based on a mechanistic model of cell division and death that permits rigorous statistical analyses of chemosensitivity experiment data by simultaneous estimation of cell division and apoptosis rates as functions of dose, without making strong assumptions about the shape of the doseresponse curve. We demonstrate the utility of this method using a largescale NCIDREAM challenge dataset. We developed an R package “bdChemo” implementing this method, available at https://github.com/YiyiLiu1/bdChemo.
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Introduction
In the early stages of cancer therapy development, potencies of candidate compounds are usually tested in vitro through chemosensitivity studies^{1,2}. Researchers treat cultured tumor cells with different concentrations of compounds, and the numbers of cells remaining after a followup time T are recorded via fluorescent signal intensities that measure general metabolism levels or enzymatic activities^{3}. Compounds that achieve desired tumor inhibition effects within dose ranges that are not considered clinically toxic are identified for further optimization and then tested with animal models and clinical trials^{1}. Given the significant investment required to bring drug candidates to preclinical and clinical stages^{4}, screening and selecting the most promising candidates from chemosensitivity studies is essential for drug development.
Conventionally, the growth inhibition response of a cell line to a compound is modeled with a sigmoid curve. The most commonly used are the Gompertz^{5,6} and logistic curves^{7,8}. Figure 1a (left and middle panels) illustrates a typical dataset (cell line AU565 treated with compound 4HC^{6}), with fitted Gompertz and logistic curves. Concentrations needed to achieve certain levels of inhibition effects, such as GI_{50} (growth of the cell population is inhibited by half), TGI (growth of the cell population is eliminated) and LC_{50} (half of the initial cell population is eliminated) are then estimated as assessments of the compound’s potency^{9}.
However, this approach suffers from problems that may hinder its utility in chemosensitivity evaluation. First, it relies on a parametric form of the growth curve. Different compounds may affect cell growth in physiologically distinct ways makes it unreasonable to believe that all inhibition patters, which result from complex interactions between compounds and cells, could be modeled with the same parametric form (Fig. 1b, the first two panels). A newer nonparametric method called grofit^{10} provides a framework for fitting more flexible doseresponse curves using spline smoothing. However, all these methods, by fitting a single growth curve, only deliver information about the combined effects of the compound on cell birth and death processes, while understanding these responses individually is critical to designing experiments that more elaborately investigate the compound’s mechanism of action^{11}. Moreover, cancer therapy development often begins by designing compounds that target pathways either inhibiting cell division or inducing cell death separately^{12}, so it would be helpful to separately discern these effects from earlystage chemosensitivity experiments. Finally, existing approaches consider only “point estimation” of GI_{50}/TGI/LC_{50}, as a summary of the compound inhibition effects without providing measures of uncertainty for these estimated quantities.
To overcome these limitations, we describe a new analysis approach (bdChemo) for chemosensitivity studies. This method specifies a mechanistic model of stochastic cell growth, fits semiparametric doseresponse curves without strong assumptions on their functional forms, and separately estimates dosespecific “birth” and “death” rates for the compound. For a given compound, we assume that a cell line’s percell birth and death rates, λ and μ, are timehomogenous and depend on the log_{10} concentration z, of the compound; a cell community with size k has aggregate populationlevel birth and death rates kλ(z) and kμ(z), respectively. Such a system is known as Kendall, or birthdeath, process^{13} (BDP). When the BDP accurately characterizes of the dynamics of cellular response to the compound, the estimated rates of birth and death may have a mechanistic interpretation as “cell division” and “apoptosis” rates, respectively. To avoid unnecessary assumptions on the dosedependent shapes of λ(z) and μ(z) and allow a flexible relationship between dose and response, we employ a semiparametric Bayesian approach by assigning Gaussian process^{14} priors, which treats a regression function on a continuous domain as an infinitedimensional random variable. The method assumes that any finite marginal distribution follows a multivariate Gaussian distribution without restrictions on the parametric form of the regression function. We estimate the doseresponse relationships of the percell birth and death rates λ(z) and μ(z) as well as other model parameters using a Markov Chain Monte Carlo (MCMC) algorithm^{15}. Uncertainty in estimates of birth and death rates is appropriately propagated into uncertainty in summary statistics like GI_{50}, TGI and LC_{50}.
We demonstrate the utility of this method on a largescale chemosensitivity dataset from NCIDREAM challenge containing cell population size measurements on 53 breast cancer cell lines treated by 28 compounds. In the original work, the authors fit a Gompertz curve^{5} for each experiment and calculated GI_{50} to quantify the sensitivity of the cell line to the compound. We apply bdChemo to the dataset and estimate the posterior mean as well as the 95% equal quantile credible intervals (CI) of chemosensitivity summary statistics GI_{50},TGI and LC_{50}, and cell birth and death rates, λ(z) and μ(z), for each compound and cell line combination. We compare the results of the proposed method with those obtained by conventional Gompertz and logistic curve fitting approaches as well as grofit.
Results
We analyze the data from NCIDREAM drug sensitivity prediction challenge^{6} to demonstrate the utility of estimates produced by the proposed method. This dataset contains doseresponse measurements of 28 compounds on 53 breast cancer cell lines. Cells were treated with 9 doses of each compound in triplicate and cell counts at 72 h post treatment were measured using the Cell Titer Glo assay. In the original work, the authors fit a Gompertz curve^{5} for each experiment (a cell line treated by a compound) and calculated GI_{50} (a point estimate without uncertainty evaluation) to quantify the sensitivity of the cell line to the compound. We summarize the posterior mean and 95% equal quantile credible intervals (CI) of λ(z), μ(z), GI_{50},TGI and LC_{50} returned by bdChemo in Supplementary Table S1 and Figure S1.
bdChemo fits doseresponse curve flexibly
Conventional Gompertz and logistic curve fitting approaches rely on specified parametric forms of the doseresponse curve. These parametric forms may not be flexible enough to describe the observed doseresponse data due to the complex interactions between compounds and cell lines. Compared to the restricted parametric curve fitting approaches, bdChemo does not put strong assumptions on the functional forms of the doseresponse curve, and hence provides a flexible and datadriven approach to study the effect of a compound on a cell line.
Figure 1a and b depict two examples of different growth patterns modeled by the four approaches. For compound 4HC working on cell line AU565 (Fig. 1a), the growth curve follows a sigmoid shape. In such case, all four approaches produced good fits. However, for compound Everolimus working on cell line SUM52PE (Fig. 1b), where there is a plateau around the waist of the growth curve, the two sigmoid curve fitting approaches (the first two panels) could not capture this trend, while grofit (the third panel) and bdChemo (the fourth panel) generated curves more concordant with the data.
bdChemo provides separate estimates of cell birth and death rates
Together, cell division and apoptosis, as functions of compound dose, determine the dynamics of cell line response. For these distinct cellular processes, modelbased estimation of compounds’ effects on cell birth and death rates can serve as a screening tool for candidate compounds and provide guidance for hypothesis generation and experiment design to study compounds’ cellular mechanisms of action. We depict the percentage changes between the posterior means of birth/death rates at the largest and the smallest tested concentrations in Figure 2a (all compounds) and Figure S2 (individual compound). The overall growth, birth and death curves of four examples are included in Figure 2b–e as further illustrations. When treated by a compound, most cell lines in this study show decreased birth and increased death rates under higher concentrations (top left part of Figure 2a; an example in Figure 2b), while a few exhibit only decreased birth rates or increased death rates alone (top right and bottom left parts in Figure 2a; examples in Figure 2c,d). Not all compounds in this study appear to inhibit cell growth, resulting in a few other points with large birth rate increment or death rate decrement effect (bottom right part in Figure 2a; an example in Figure 2e).
These scenarios may suggest different underlying mechanisms of action. For example, in several other cancers, Imatinib has been reported to kill tumor cells by decreasing the activity of tyrosine kinase enzymes^{16}, whereas Cetuximab was known to hinder uncontrolled tumor cell division as an EGFR inhibitor^{17}; here we observe similar effects on breast cancer cell lines: for the compound Imatinib and cell line BT474 (Figure 3a), birth rate λ(z) is relatively stable with respect to dose z in the tested range, while death rate μ(z) first stays steady but then increases rapidly when dose z becomes large; in contrast, for Cetuximab working on cell line HCC1806 (Figure 3b), birth rate λ(z) decreases while death rate μ(z) stays stable. Therefore, we may hypothesize that compound Imatinib mainly works by inducing cell apoptosis on BT474 while compound Cetuximab is more likely to target on blocking cell division on HCC1806, and design experiments to investigate the effects of these compounds on cell apoptosis and divisionrelated pathways, respectively, to better understand their mechanisms on these cell lines. Similar cell death induction and birth inhibition effects of Imatinib and Cetuximab are also observed on most other cell lines in this dataset (Figure S2). In addition, some other compounds have consistent patterns across cell lines. For example, Mebendazole, TCS PIM11, QNZ and MG132 demonstrate both birth inhibition and death induction effects on most cell lines. Other compounds like 4HC, Doxorubicin, Olomoucine II, Valproate, Baicalein, Methylglyoxal and IKK 16 increase cell death rates on most cell lines, but their effects on cell birth rates differ by cell line.
bdChemo considers uncertainty in chemosensitivity evaluations
Point estimates of compound potency summary statistics, such as GI_{50}, on different experiments can be similar even if the growth curves have distinct patterns. In Figure 4a, we plot the lengths of the 95% credible intervals (CI) against the means for log_{10}GI_{50} estimated on all experiments from the NCIDREAM data. There are many cases where means are comparable but lengths of credible intervals differ greatly. For example, on cell line 21NT, compounds Nelfinavir and 4HC have GI_{50} values 1.33 × 10^{−5}M and 1.29 × 10^{−5}M respectively, which are close (especially in log_{10} scale the difference is negligible), yet Nelfinavir has a 95% CI [1.17 × 10^{−5},1.53 × 10^{−5}]M, much narrower than that of 4HC, [8.87 × 10^{−6},1.74 × 10^{−5}]M. These differences result from the distinct patterns of their growth curves: compared to the sharp drop of the growth curve under treatment of Nelfinavir (Figure 4b left panel), the growth curve under treatment of 4HC declines much more slowly as compound concentration increases (Figure 4b right panel), resulting in greater uncertainty about the location of GI_{50}.
Comparing point estimates of a summary statistic for different compounds can lead to conclusions about their relative potency that lack statistical validity. For instance, on cell line MDAMB415, compounds Disulfiram and Imatinib have GI_{50} values 6.55 × 10^{−6}M and 2.28 × 10^{−5}M, respectively (Figure 4c), based on which, a conclusion may be drawn that MDAMB415 cells are more sensitive to Disulfiram. However, as the GI_{50} CI of Disulfiram is [1.36 × 10^{−10},2.67 × 10^{−5}]M, covering that of Imatinib, [2.00 × 10^{−5},2.62 × 10^{−5}]M, there is no evidence supporting the statistical significance of such difference. In practice, summary concentrations are often computed to quickly compare a large number of compounds. However, rigorous statistical hypothesis testing for differences in cell line responses requires taking into account uncertainties in summary statistics used for comparison.
Discussion
Statistically rigorous analysis of chemosensitivity experiment data is of great importance in cancer therapy development. The major assumptions of the proposed model, bdChemo, based on the birthdeath process^{13} (BDP), are that the percell birth and death rates of a cell line under compound treatment are timehomogenous functions of the compound concentration, and the populationlevel birth and death rates are products of the cell community size and the percell birth and death rates, respectively. Unlike standard analysis methods that rely heavily on specific functional forms of the doseresponse curve, bdChemo employs a semiparametric Bayesian approach in function estimations to avoid restrictive assumptions. Although other nonparametric methods have been proposed for doseresponse curve fitting, bdChemo provides biologically motivated estimates of dosedependence in cell birth and death rates separately, in addition to estimating the combined effects of the compound on cell birth and death processes, delivering richer information that may guide subsequent experimental work. The method takes uncertainty into account when providing chemosensitivity summary statistics, such as GI_{50}, TGI and LC_{50}, to facilitate sound comparisons of compounds’ effects in tumor cell growth inhibition.
We applied bdChemo, as well as two conventional sigmoidcurve fitting approaches (logistic and Gompertz) and R package grofit^{10}, to NCIDREAM drug sensitivity prediction challenge^{6} data, where doseresponse measurements of 28 compounds on 53 breast cancer cell lines were provided. The results show that when the doseresponse curve does follow a sigmoid pattern, bdChemo produces estimates similar to conventional methods; but when the doseresponse curve deviates from a sigmoidal shape, bdChemo and nonparametric spline smoothing can better capture growth inhibition dynamics. We observe different patterns of cell birth inhibition and death induction effects for difference cell line/compound combinations; while some compoundcell line combinations have doseresponse curves that look similar, the underlying mechanism of action may be different. Separate estimates of cell birth and death rate estimations provided by bdChemo, hence, might be utilized in hypothesis generation and experimental design. In addition, even when the point estimates of GI_{50} are similar, credible intervals sometimes differ substantially.
While bdChemo can fit curves flexibly and provides mechanistic inferences about the dosedependent respond of cell birth and death rates, the approach is subject to limitations. First, the Kendall process framework assumes that cells undergo birth and death independently, with the same percell rates; when there are k cells, the populationlevel birth and death rates are kλ(z) and kμ(z) respectively. The model does not accommodate populationlevel effects, which could result in more complicated nonlinear rates λ(k,z) and μ(k,z). More sophisticated models for population effects of cell response may be warranted when biologically motivated. Second, we analyzed each experiment (compoundcell line combination) independently. Jointly modeling inhibition responses by compound and cell line could exploit information sharing across experiments, resulting in greater statistical precision in estimates, especially when the number of doses is small.
Finally, we point out potential issues that may arise when applying bdChemo in empirical data analysis. We observed some large credible intervals of λ and μ in our analysis, which is mainly caused by the small number of experimental replicates (three at each dose). Since the mean value of the cell numbers only contains information regarding the combined effects of the birth and death processes, the separate identifiability of the birth and death rates mainly comes from the variance of cell numbers at different doses. Therefore, a larger number of replicates is desirable for increased accuracy in estimates of λ and μ. Additionally, we observed a few outliers in which λ and μ are estimated to be much larger than elsewhere (Tables S2 and S3). These estimates are driven by large cell count variations in these experiments. For a comprehensive evaluation of our method’s performance on datasets of varying qualities, we analyzed all NCIDREAM experiments here. However, in practice, some quality control procedures in data preprocessing, as typically conducted in conventional chemosensitivity analysis^{18}, might also be necessary.
Methods
Conventional approaches
Conventionally a cell line’s response to a compound is modeled with a sigmoid curve. The most commonly used include Gompertz curve^{5,6}
and logistic curve^{7,8}
where g(z) is the final cell count at log_{10} concentration, z (usually with unit log_{10} M).
bdChemo
Model
We use birthdeath process (BDP) to model cell growth in the experiment. In a general BDP^{19}, where N(t) stands for the number of particles at time t, given N(t) = k(k ≥ 1), the birth rate
and the death rate
are timehomogeneous but dependent on the number of particles k. A simple linear BDP known as Kendall Process assumes λ_{ k } = λk and μ_{ k } = μk (we refer λ and μ as per cell birth and death rates, respectively)^{13}. For Kendall Process, the transition probability is^{20}
where
and
For a fixed t, N(t) is a random variable whose distribution is fully specified by P_{ ab }(t). The mean and variance of N(t) given N(0) = n_{0} are^{19,21}
and
respectively.
As indicated above, to calculate P_{ ab }(t) involves computing a large number of combinatorial numbers, making it computationally infeasible in real applications^{22}. Therefore, we use normal distribution with matched mean (m_{ t }) and variance (v_{ t }) as an approximation to reduce the computational cost. This approximation is accurate when initial cell counts are large, as they generally are in chemosensitivity experiments (see the Supplementary Note for a technical justification). Since in one chemosensitivity study, treatment duration t is usually fixed and same for all experiments (compounds), we omit the notation t for simplicity and interpret the new λ and μ as per cell birth and death rates for the entire experiment duration.
For a given compound working on a given cell line, we assume λ and μ are functions of log_{10} concentration, z, of the compound, so the mean and variance of cell counts are
and
respectively.
Note, although the mean as a function of λ and μ is only affected by their difference, the variance has a term λ + μ. Therefore, mean and variance of cell counts together provides information to identify λ and μ separately. An interesting example is illustrated by the difference between quiescence (λ ≈ 0 and μ ≈ 0) and matched birth and death rates (λ ≈ μ). In both cases, we would expect the mean cell count to be unchanged, m(n_{0}, z) ≈ n_{0}. However, if both λ and μ are close to 0, the variance v(n_{0},z) would also be very small; if the rates are large and nearly equal, λ ≈ μ ≫ 0, the variance is expected to be much larger.
Our model also takes experimental errors into account by assuming that the measured cell counts X_{ z } = N_{ z } + ∈_{ z } and ∈_{ z } ∼ N(θ,σ^{2}), where θ and σ^{2} are the mean and variance of background noises, respectively.
Therefore, the likelihood of observing data D = {(n_{01}, z_{1}, x_{1}), (n_{02}, z_{2}, x_{2}), …, (n_{0n}, z_{ n }, x_{ n }), e_{1}, …, e_{ q }} (n_{0i}’s are the initial cell population sizes, (z_{ i }, x_{ i })’s are compound concentrations and corresponding cell count measurements at followup, and e_{ i }’ s are independent background noise measurements) is
where f_{ N } is the density function of Gaussian distribution.
Note that we do not require the initial cell population sizes n_{0i} to be the same across different compound concentrations, but we treat them as known quantities here. In reality, initial cell population sizes may be uncertain in some experiments. However, with only one number provided in most experiment data, it is difficult to assign a distribution to the initial cell count. Moreover, uncertainty in the initial count mainly comes from variation in cell density and the amount of cell solution injected into each well, both of which, under appropriate experimental operation, could be controlled at a relatively low level. Although we do not model its randomness directly, the term σ^{2} models the variance from background noise can be interpreted to reflect variation in initial cell counts to some extent. For example, if two datasets have all other conditions similar, whereas one has a much larger estimate of σ^{2}, we might suspect that the large σ^{2} is induced by a poorly controlled initial cell seeding. If future experiments provide more data to quantify its fluctuations and suggest the necessity of treating it as a random variable, we may add another layer of randomness to n_{0} under this Bayesian hierarchical model framework.
To avoid unnecessary assumptions on λ(z) and μ(z) and allow more flexible estimations, we employ a semiparametric Bayesian approach by assigning Gaussian process^{14} priors to ϕ_{ λ }(z) = log(λ(z)) and ϕ_{ μ }(z) = log(μ(z)), i.e.
and
Note, unlike conventional approaches, under this Gaussian process framework, we impose no shape constraints including monotonicity on the curves.
Hyperparameters \({\alpha }_{\lambda },{\alpha }_{\mu },{\tau }_{\lambda }^{2},{\tau }_{\mu }^{2},{l}_{\lambda }^{2},{l}_{\mu }^{2},\theta \) and σ^{2}are assigned priors
and
Fitting algorithm
We utilize Gibbs sampler embedded with Metropolis updating^{15} to draw posterior samples for parameters \({\varphi }_{\lambda },{\varphi }_{\mu },{\alpha }_{\lambda },{\alpha }_{\mu },{\tau }_{\lambda }^{2},{\tau }_{\mu }^{2},{l}_{\lambda }^{2},{l}_{\mu }^{2},\theta \) and σ^{2} (see Supplementary Note for details).
Data availability
The dataset analyzed during the current study is available at http://www.nature.com/nbt/journal/v32/n12/full/nbt.2877.html. The code used to perform the analysis is available as an R package “bdChemo” at https://github.com/YiyiLiu1/bdChemo.
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Acknowledgements
We are grateful to Dr. Hongyu Zhao, Dr. David Stern, Dr. Christos Hatzis, Li Zeng and Michael Klein for helpful discussions and suggestions for the manuscript. F.W.C. was supported by NIH grant NICHD 1DP2HD09179901 .
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Y.L. and F.W.C. conceived the research. Y.L. implemented the method and performed the data analysis. Y.L. and F.W.C. wrote the manuscript.
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Liu, Y., Crawford, F.W. Estimating dosespecific cell division and apoptosis rates from chemosensitivity experiments. Sci Rep 8, 2705 (2018). https://doi.org/10.1038/s41598018210175
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DOI: https://doi.org/10.1038/s41598018210175
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