GR Metrics Calculator and Browser

Commonly, relative cell count at the end of treatment is used to assess drug response. Across a range of concentrations, the measured cell count or a surrogate of it (e.g. measurement of ATP using CellTiter-Glo®) is normalized to the cell count of DMSO-treated controls grown on the same plate under the same conditions. For each condition (combination of cell line, drug, and drug concentration), we define the relative cell count as x(c)/x_ctrl, where x(c) is the count in the presence of drug and x_ctrl is the 50%-trimmed mean of the count for control cells. Based on a sigmoidal curve fitted to the relative cell count across different concentrations, one can define:

• IC₅₀ (50% relative cell count), which is the most commonly used metric but one that can only be defined if the relative cell count decreases during the assay to below 50% of the starting value;
• EC₅₀ (half-point of the sigmoidal curve);
• E_inf (the asymptotic efficacy), which differs from E_max (the maximal measured efficacy); and
• AUC (area under, or over, the curve), which captures both IC₅₀ and E_max to some extent and is more robust to experimental noise than other metrics.

More details on dose-response assays and traditional metrics can be found in Sebaugh (2010).

As an alternative to traditional metrics, we propose to use GR metrics, which are based on inhibition of the growth rate over the course of the assay and are independent of the division rate of the assayed cell lines.

Growth can be estimated at different concentrations based on the cell count at the time of treatment (x₀), the cell count in the untreated control (xctrl), and the cell count after treatment at concentration c (x(c)):

This equation assumes exponential growth and constant growth inhibition by the drug. Many phenomena such as drug efflux, homeostasis and adaptation can result in non-exponential growth. In such cases, time course assays and time-dependent GR values are better suited to evaluating drug response. Given measurements of cell count at different time points, time-dependent GR values are defined as:

GR(c) values (or time-dependent GR(c,t) values at a given time t) across a range of concentrations are fitted to a sigmoidal curve:

where the fitted parameters are:

• GR_inf: the effect of the drug at infinite concentration (GR_inf = GR(c→∞)). GR_inf lies between -1 and 1. Negative values correspond to cytotoxic responses (i.e., induction of cell death), and a value of 0 corresponds to a fully cytostatic response.
• h_GR: the Hill coefficient of the fitted curve, which reflects how steep the dose-response curve is. In practice, we typically constrain h_GR to a value between 0.1 and 5.
• GEC₅₀: the concentration at half-maximal effect. To avoid artefacts in curve fitting we constrain GEC₅₀ to be within two orders of magnitude higher and lower than the experimentally tested concentration range. (In practice, this is usually about 10^-7 to 10³ M.) If the fit of the curve is not significantly better than that of a flat curve (i.e., GR(c) ≡ GR_inf) based on an F-test with cutoff of p = 0.05, the response is considered flat, and the parameter GEC₅₀ is set to 0.

If time-dependent GR(c,t) values are used, GR metrics can be estimated at a different time points to study how sensitivity evolves over time.

Inferred drug response metrics

The GR₅₀ value is the concentration of drug at which GR(c = GR₅₀) = 0.5. If the value for GR_inf is above 0.5, the GR₅₀ value is not defined and is therefore set to +∞. By extension, other thresholds can be defined in a similar manner. For example, GR₁₀₀ corresponds to the concentration at which a drug is fully cytostatic: GR(c = GR₁₀₀) = 0.

GR_max is the maximum effect of the drug at the highest tested concentration. It lies between -1 and 1. A value of 0 corresponds to a fully cytostatic response, and negative values correspond to a cytotoxic response. GR_max can be estimated from the fitted curve or obtained directly from experimental data. (We often do the later.)

For time course data, all metrics are evaluated at each time point individually.

Area under the curve and over the curve (GR_AOC)

Another common metric for quantifying dose response is the area under the response curve (AUC), which is based on integrating the dose-response curve over the range of tested concentrations. In the case of GR curves, which can have negative values, it is more intuitive to use the area over the curve:

where GR(c_i) are measured GR values at discrete concentrations c_i. GR_AOC has the benefit that, in the case of no response, it has a value of 0. It is important to note that GR_AOC values (like conventional AUC) can only be used to compare responses evaluated across the same drug concentration range.

The GR_AOC value captures variation in potency and efficacy at the same time. The calculation of GR_AOC at discrete (experimentally determined) concentrations has the advantage that it does not require curve fitting and is therefore free of fitting artifacts. This is especially useful for assays where fewer than five concentrations are measured and curve fitting is unreliable. GR_AOC values are also more robust to experimental noise than metrics derived from curve fitting; e.g. GR_max is particularly sensitive to outlier values when directly obtained from data.

We used computer simulation (see below) to model drug response by three idealized cell lines having identical sensitivity to a partial, or complete cytostatic drug (i.e., a drug that arrests but does not kill cells), or a cytotoxic drug (i.e., a drug that kills cells) and different division times (T_d). For example, the lower quartile, median, and upper quartile of the division times for breast cancer cell lines has been reported as 1.8, 2.4, or 3.9 days, respectively (Heiser et al. (2012)). These values are similar to those of NCI-60 cells. In a slow-dividing cell line (T_d > 3.9 d), the total number of cells does not double in a typical three-day assay; thus E_max ≥ 0.5 and IC₅₀ are undefined. In the case of the two faster-growing cell lines, IC₅₀ and E_max values fall as division rate increases simply because cell number (or CTG value) is normalized to a drug-naïve control in which cell number increases as division time falls.

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To simulate the effect of division time on GR and conventional drug-response metrics under different assumptions about the degree of cytostasis or cell killing, we developed a theoretical model of drug response. To the first approximation, cell growth can be considered exponential, with drugs either decreasing the division rate or killing cells in a cell cycle-dependent manner:

where x is the cell count, k is the untreated growth rate (per day), c is the drug concentration, S_M is the maximal inhibitory effect, SC₅₀ is the concentration at half-maximal effect of drug, and h is the Hill coefficient. The growth rate k corresponds to the division rate k₀ as k = ln(2)⋅k₀ = ln(2)/T_d, where T_d is the division time. S_M can be larger than 1 to account for drugs inducing cell death at a specific phase of the cell cycle.

Integrating these equations for an assay of t days yields the cell count at concentration c:

where x₀ = x(t = 0) is the cell number at the time of treatment. Thus, the relative cell count is:

where x_ctrl ≡ x(0), and the normalized growth rate inhibition (GR value) is:

This equation for GR(c) is independent of the length of the assay t and the untreated growth rate k, and, thus, the metrics GR₅₀, GR_max, GR_AOC, and h_GR are also independent of t and k.

Through the accompanying webpages we provide online tools that allow users to upload their own endpoint data, calculate the corresponding GR values and GR metrics, and compare their results across the variables in the assay, such as cell line, agent (drug), time and replicate. Through the above Online GR Calculator webpage, we provide online tools that allow users to upload their own endpoint data, calculate the corresponding GR values and GR metrics, and compare their results across the variables in the assay, such as cell line, agent (drug), time and replicate. Python, R and MATLAB code for calculating GR value and GR metrics is also available in the gr_metrics repository on GitHub. Our R code is also available in the Bioconductor package GRmetrics.

Input data

The GRcalculator takes in either comma-separated (.csv) or tab-separated (.tsv) files with specified column names. See the step-by-step example. For more explicit instructions, go to the GRcalculator and click "Import Data File" and select a "GR Calculation Method" and "Input File Format" to see the exact columns needed for a particular input style.

Calculated metrics

The following GR metrics are calculated:

• GR₅₀: The concentration at which the effect reaches a GR value of 0.5 based on interpolation of the fitted curve.
• GR_max: The effect at the highest tested concentration.
• GR_inf = GR(c → ∞): The effect at infinite concentration based on extrapolation of the fitted curve, which reflects asymptotic drug efficacy. Note that GR_inf can differ from GR_max if the measured dose-response does not reach its plateau value.
• GEC₅₀, the drug concentration at half-maximal effect, which reflects the potency of the drug.
• h_GR: Hill coefficient of the sigmoidal curve, which reflects how steep the dose-response curve is.
• GR_AOC: The area over the dose-response curve, which is the integral of 1–GR(c) over the range of concentrations tested.

In addition, the online calculator reports the r-squared of the fit and evaluates the significance of the sigmoidal fit based on an F-test. Further information about these calculations is available through our GR tutorial. Traditional metrics based on relative cell count (e.g. IC₅₀, E_max, etc.) are computed as well for comparison.

The results of analysis of existing LINCS dose-response datasets using the GR model are available in the LINCS GR Data Browser webpage.

Analysis and visualization tools

The following tabs allow users to analyze and visualize their data and the calculated GR metrics:

• Data Tables displays the uploaded data and calculated GR values as well as the inferred GR metrics. Data can be downloaded in .csv or .tsv format.
• Dose-Response by Condition plots the data points and fitted dose-response curves for each condition. If no specific conditions are selected, all data will be overlaid on a single plot. To display a subset of the data, select specific conditions to display (e.g. a specific cell line or drug). Mousing over the plot displays the GR values and the corresponding concentration. Plots can be downloaded in a variety of image formats.
• Dose-Response Grid shows the fitted dose-response curves side by side and grouped according to the selected variables. ‘Selector variable’ and ‘grid variable’ define which plots are displayed. ‘Selector variable’ also defines the variable displayed in the selector box to the left of the plots.
• GR Metric Comparison shows a scatter plot of the a given GR metric across different conditions. Users can select which metric to plot relative to which experimental variable. Mousing over the plot displays the plotted GR metrics. Plots can be downloaded in a variety of image formats.

Feedback

We welcome your feedback on all of these tools and resources. Please visit our support page for additional resources and to provide feedback.

This website was designed and built by the following members of the LINCS-BD2K Data Coordination and Integration Center and the Harvard Medical School (HMS) LINCS Center based on analytical tools described in Hafner et al. (2016):

• Nicholas Clark¹
• Marc Hafner²
• Michal Kouril¹
• Mario Niepel²
• Elizabeth Williams²
• Jeremy Muhlich²
• Mario Medvedovic¹
• Peter Sorger²

¹LINCS-BD2K Data Coordination and Integration Center; ²HMS LINCS Center, Harvard Medical School

Further Links

• HMS LINCS Center
• LINCS-BD2K Data Coordination and Integration Center
• Sorger Lab on GitHub
• Bioconductor R package (GRmetrics)
• ICSB 2017 Workshop on Drug Response Measurement and Analysis