User tutorial of the SynergyFinder Plus

Shuyu Zheng, Wenyu Wang, and Jing Tang Research Program in Systems Oncology, Faculty of Medicine, University of Helsinki

2024-05-01

Summary

Combinatorial therapies is one of the major strategies for improving treatment efficacy in treating cancer. High-throughput drug combination screening has the advantage of assaying a large collection of chemical compounds in search for promising drug pairs. Studies that utilizes drug combination screening have generated dynamic dose-response profiles that allow researchers to quantify the degree of drug-drug interactions at an unprecedented level. SynergyFinder R package is a software tool to analyze such pre-clinical drug combination data sets. It provides efficient implementations for

1.the popular synergy scoring models, including HSA, Loewe, Bliss, and ZIP to quantify the degree of drug combination synergy; 2. higher order drug combination data analysis and synergy landscape visualization for unlimited number of drugs in a combination; 3. statistical analysis of drug combination synergy and sensitivity with confidence intervals and p-values; 4. synergy barometer for harmonizing multiple synergy scoring methods to provide a consensus metric of synergy; 5. evaluation of synergy and sensitivity simultaneously to provide an unbiased interpretation of the clinical potential of the drug combinations.

To facilitate the use of the R package for the drug discovery community, we also provide a web server at https://synergyfinderplus.org/ as a user-friendly interface to enable a more flexible and versatile analysis of drug combination data. In the web application, in addition to the functions in R package, we provide the annotation of drugs and cell lines that are tested in an experiment, including their chemical information, targets and signaling network information.

1 Installation

Note: We recommend to install Bioconductor >= 3.13 and SynergyFinder >= 3.0.1 to access the new features described in this vignette.

To install the SynergyFinder package from Bioconductor, start R (>= 4.0) and then enter:

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("synergyfinder")

Load SynergyFinder to R environment

library(synergyfinder)

2 Input data

synergyfinder requires a data frame that describes the drug combination data set. The dose-response data is represented as a long table where each row represent one observation in the dose-response matrix.

The input table must contain the following columns (The column naming styles used by the old versions of synergyfinder or DrugComb, i.e. ‘Alternative Column names’ are accepted):

Required Columns Alternative Column names Description
block_id PairIndex, BlockId Identifier for the drug combination blocks.
drug1 Drug1, drug_row, DrugRow Name of the first tested drug.
drug2 Drug2, drug_col, DrugCol Name of the second tested drug.
conc1 Conc1, conc_row, ConcRow Concentration of first tested drug.
conc2 Conc2, conc_col, ConcCol Concentration of second tested drug.
response Response, inhibition, Inhibition Cell response to the drug treatment (%inhibition or %viability).
conc_unit ConcUnit Unit of concentration for drugs. This column could be replaced by multiple separated columns for each tested drugs (see table below), while different unit was used for measuring the concentrations.

There are several optional columns available for the higher-order drug combination data sets or the data sets using different concentration units for drugs.

Optional Columns Alternative Column names Description
conc_unit1 conc_r_unit Unit of concentration for the first drug. Used if the concentration units are not identical across the drugs tested in one block.
conc_unit2 conc_c_unit Unit of concentration for the second drug. Used if the concentration units are not identical across the drugs in test one block.
drug[n] Name of the n_th_ tested drug. For example, “drug3” for the third tested drug. Used for higher-order drug combination data point.
conc[n] Concentration of n_th_ tested drug. Used for higher-order drug combination data point.
conc_unit[n] Unit of concentration for n_th drug. Used if the concentration units are not identical across the drugs in test one block.

Note:

  1. The duplicated concentration combinations in one block (with the same “block_id”) will be treated as replicates.
  2. There is no restriction on the number of drug combinations (blocks) for the input file. The data should however, contain at least three concentrations for each drug in each block, so that sensible synergy scores can be calculated.
  3. synergyfinder allows for missing values in the dose-response matrix. The missing value will be automatically imputed by mice R package.

SynergyFinder provides an example input data (mathews_screening_data) in the package, which is extracted from a published drug combination screening study for the treatment of diffuse large B-cell lymphoma (DLBCL) [1]. The example input data contains two representative drug combinations (ibrutinib & ispinesib and ibrutinib & canertinib) for which the %viability of a cell line TMD8 was assayed using a 6 by 6 dose matrix design. To load the example data, please run:

data("mathews_screening_data")
head(mathews_screening_data)
#>   block_id  drug_row  drug_col conc_r  conc_c  response conc_r_unit conc_c_unit
#> 1        1 ispinesib ibrutinib   2500 50.0000  7.802637          nM          nM
#> 2        1 ispinesib ibrutinib   2500 12.5000  6.831317          nM          nM
#> 3        1 ispinesib ibrutinib   2500  3.1250 15.089589          nM          nM
#> 4        1 ispinesib ibrutinib   2500  0.7812 24.503885          nM          nM
#> 5        1 ispinesib ibrutinib   2500  0.1954 38.043076          nM          nM
#> 6        1 ispinesib ibrutinib   2500  0.0000 45.790634          nM          nM

3 Reshaping and pre-processing

We provide a function ReshapeData to reshape and pre-process the input data for further analysis:

Important parameters:

res <- ReshapeData(
  data = mathews_screening_data,
  data_type = "viability",
  impute = TRUE,
  impute_method = NULL,
  noise = TRUE,
  seed = 1)

The output data is a list containing following components:

str(res)
#> List of 2
#>  $ drug_pairs: tibble [2 × 7] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id  : int [1:2] 1 2
#>   ..$ drug1     : chr [1:2] "ispinesib" "canertinib"
#>   ..$ drug2     : chr [1:2] "ibrutinib" "ibrutinib"
#>   ..$ conc_unit1: chr [1:2] "nM" "nM"
#>   ..$ conc_unit2: chr [1:2] "nM" "nM"
#>   ..$ input_type: chr [1:2] "viability" "viability"
#>   ..$ replicate : logi [1:2] FALSE FALSE
#>  $ response  : tibble [72 × 5] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id       : int [1:72] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ conc1          : num [1:72] 2500 2500 2500 2500 2500 2500 625 625 625 625 ...
#>   ..$ conc2          : num [1:72] 50 12.5 3.125 0.781 0.195 ...
#>   ..$ response       : num [1:72] 92.2 93.2 84.9 75.5 62 ...
#>   ..$ response_origin: num [1:72] 92.2 93.2 84.9 75.5 62 ...

4 Synergy and sensitivity analysis

4.0 Baseline Correction

Typically, the models for estimating synergy or sensitivity of drug combinations expect that the drug’s effect to be expressed as continuous values ranged between 0% and 100%. However, in practice the readouts from the drug combination screening could be negative. In this cases, the technical error might be introduced in the dose-response data. The base line correction function is designed to adjust the “baseline” of the dose-response data closer to 0. Following is the main steps for the process of “baseline correction”:

  1. Extract all the single drug treatment (monotherapy) dose-response data from the combination.
  2. Fit the four-parameter log logistic model for each monotherapy with the data table from step 1.
  3. Pick the minimum fitted response value from all the models fitted in step 2 as “baseline”.
  4. Adjusted response value = observed response value - ((100 – observed response value) / 100 * baseline).

By setting the value of the parameter correct_baseline while calling function CalculateSynergy or CalculateSensitivity, user could chose:

4.1 Drug synergy scoring

The synergistic effect can be determined as the excess of observed effect over expected effect calculated by a reference models (synergy scoring models). All of the models make different assumptions regarding the expected effect. Currently, 4 reference models are available in SynergyFinder.

\[y_{HSA} = max(y_1, y_2)\] where \(y_1\) and \(y_2\) are the monotherapy effect of combined drug 1 and drug 2

\[ y_{Bliss} = y_1 + y_2 - y_1 \cdot y_2 \] where \(y_1\) and \(y_2\) are the monotherapy effect of combined drug 1 and drug 2

\[ \frac {x_1}{\chi_{Loewe}^1} + \frac{x_2}{\chi_{Loewe}^2} = 1 \]

where \(x_{1,2}\) are drug doses and \(\chi_{Loewe}^1,\ \chi_{Loewe}^2\) are the doses of drug 1 and 2 alone that produce \(y_{Loewe}\). Using 4-parameter log-logistic (4PL) curves to describe dose-response curves the following parametric form of previous equation is derived:

\[ \frac {x_1}{m_1(\frac{y_{Loewe}-E_{min}^1}{E_{max}^1 - y_{Loewe}})^{\frac{1}{\lambda_1}}} + \frac{x_2}{m_2(\frac{y_{Loewe}-E_{min}^2}{E_{max}^2 - y_{Loewe}})^{\frac{1}{\lambda_2}}} = 1 \] where \(E_{min}, E_{max}\in[0,1]\) are minimal and maximal effects of the drug, \(m_{1,2}\) are the doses of drugs that produce the midpoint effect of \(E_{min} + E_{max}\), also known as relative \(EC_{50}\) or \(IC_{50}\), and \(\lambda_{1,2}(\lambda>0)\) are the shape parameters indicating the sigmoidicity or slope of dose-response curves. A numerical nonlinear solver can be then used to determine \(y_{Loewe}\) for (\(x_1\), \(x_2\)).

\[ y_{ZIP} = \frac{(\frac{x_1}{m_1}) ^ {\lambda_1}}{(1 + \frac{x_1}{m_1})^{\lambda_1}}+\frac{(\frac{x_2}{m_2})^{\lambda_2}}{(1 + \frac{x_2}{m_2})^{\lambda_2}} - \frac{(\frac{x_1}{m_1})^{\lambda_1}}{(1 + \frac{x_1}{m_1})^{\lambda_1}} \cdot \frac{(\frac{x_2}{m_2})^{\lambda_2}}{(1 + \frac{x_2}{m_2})^{\lambda_2}} \] where \(x_{1, 2}, m_{1, 2}, and\ \lambda_{1, 2}\) are defined in the Loewe model.

For the input data without replicates, the statistical analysis for synergy scores of the whole dose-response matrix is estimated under the null hypothesis of non-interaction (zero synergy score). Suppose that \(S_1, S_2, ...S_n\) are the synergy scores for all the concentration combinations (excluding monotherapies) for the dose-response matrix with mean of \(\bar s\). The p-value is:

\[p = exp(-0.717z - 0.416z^2),\ where\ z = abs(\bar s')/\sqrt{\frac{1}{n - 1}\sum_{i = 1}^n(s_i' - \bar s')^2}\]

The function CalculateSynergy is designed to calculate the synergy scores. Important parameters are:

res <- CalculateSynergy(
  data = res,
  method = c("ZIP", "HSA", "Bliss", "Loewe"),
  Emin = NA,
  Emax = NA,
  correct_baseline = "non")

The output adds one data frame “synergy_scores” to the input R object. It contains:

The mean of synergy scores for the combination matrix (excluding the monotherapy observations) and the p-values for them are added to the “drug_pairs” table in the input R object.

res$drug_pairs
#> # A tibble: 2 × 15
#>   block_id drug1      drug2     conc_unit1 conc_unit2 input_type replicate
#>      <int> <chr>      <chr>     <chr>      <chr>      <chr>      <lgl>    
#> 1        1 ispinesib  ibrutinib nM         nM         viability  FALSE    
#> 2        2 canertinib ibrutinib nM         nM         viability  FALSE    
#> # ℹ 8 more variables: ZIP_synergy_p_value <chr>, HSA_synergy_p_value <chr>,
#> #   Bliss_synergy_p_value <chr>, Loewe_synergy_p_value <chr>,
#> #   ZIP_synergy <dbl>, HSA_synergy <dbl>, Bliss_synergy <dbl>,
#> #   Loewe_synergy <dbl>
str(res$synergy_scores)
#> tibble [72 × 13] (S3: tbl_df/tbl/data.frame)
#>  $ block_id     : int [1:72] 1 1 1 1 1 1 1 1 1 1 ...
#>  $ conc1        : num [1:72] 2500 2500 2500 2500 2500 2500 625 625 625 625 ...
#>  $ conc2        : num [1:72] 50 12.5 3.125 0.781 0.195 ...
#>  $ ZIP_fit      : num [1:72] 93.3 92.3 82.7 75.2 57.3 ...
#>  $ ZIP_ref      : num [1:72] 79.4 79.4 67.9 37.2 37.2 ...
#>  $ ZIP_synergy  : num [1:72] 13.9 12.9 14.8 38.1 20.1 ...
#>  $ HSA_ref      : num [1:72] 71.3 54.2 54.2 54.2 54.2 ...
#>  $ HSA_synergy  : num [1:72] 20.89 38.96 30.7 21.29 7.75 ...
#>  $ Bliss_ref    : num [1:72] 86.9 78.6 73.1 45.9 52.5 ...
#>  $ Bliss_synergy: num [1:72] 5.34 14.54 11.85 29.6 9.47 ...
#>  $ Loewe_ref    : num [1:72] 62.3 62.3 62.3 62.3 -14.9 ...
#>  $ Loewe_synergy: num [1:72] 29.9 30.8 22.6 13.2 76.9 ...
#>  $ Loewe_ci     : num [1:72] 0 0 0 0 0.0347 ...

4.2 Sensitivity scoring

SynergyFinder mainly calculates 3 sensitive scores: relative IC50 and relative inhibition (RI) for single drug treatment, and combination sensitivity score (CSS) for drug combinations.

\[y = y_{min} + \frac{y_{max} - y_{min}}{1 + 10^{\lambda(log_{10}IC_{50} - x')}}\] where \(y_{min}, y_{max}\) are minimal and maximal inhibition and \(x' = log_{10}x\)

knitr::include_graphics("./ri.png")
Figure 1. Concept of RI

Figure 1. Concept of RI

\[AUC = \int_{c_1}^{c_2}y_{min} + \frac{y_{max}-y_{min}}{1 + 10^{\lambda(log_{10}IC_{50}-x')}}dx'\] where \([c_1, c_2]\) is the concentration range of the foreground drug tested.[6]

The function CalculateSensitivity is designed to calculate the synergy scores. Important parameters are:

res <- CalculateSensitivity(
  data = res,
  correct_baseline = "non"
)

Following sensitivity scores are added to the “drug_pairs” table in the input R object:

sensitive_columns <- c(
  "block_id", "drug1", "drug2",
  "ic50_1", "ic50_2",
  "ri_1", "ri_2",
  "css1_ic502", "css2_ic501", "css")
res$drug_pairs[, sensitive_columns]
#> # A tibble: 2 × 10
#>   block_id drug1     drug2 ic50_1 ic50_2  ri_1  ri_2 css1_ic502 css2_ic501   css
#>      <int> <chr>     <chr>  <dbl>  <dbl> <dbl> <dbl>      <dbl>      <dbl> <dbl>
#> 1        1 ispinesib ibru…  2500    2.74  60.0  27.0       85.9      82.8   84.3
#> 2        2 canertin… ibru…   973.   1.44 -48.0  45.2      -35.0      -6.99 -21.0

5 Visualization

5.1 Dose-response curve

The PlotDoseResponseCurve function will plot the dose-response curve fitted by 4-parameter log logistic function for selected drug. Important parameters are:

This function will return an object of class recordedplot. User could use replayPlot to plot it. User could modify the parameter plot_setting to control the themes of some items in the plot.

for (i in c(1, 2)){
  PlotDoseResponseCurve(
    data = res,
    plot_block = 1,
    drug_index = i,
    plot_new = FALSE,
    record_plot = FALSE
  )
}

If parameter record_plot is set as TRUE, the function will return a plot object recorded by recordPlot.

5.2 Two-drug combination visualization

SynergyFinder provides 3 types (Heatmap, 2D contour plot, or 3D surface) of plots to visualize dose-response map or synergy map for two-drug combinations.

From the plots for dose-response landscape, user can assess the therapeutic significance of the combination, e.g. by identifying the concentrations at which the drug combination can lead to a maximal effect on cancer inhibition.

The landscape of drug interaction scoring is very informative when identifying the specific dose regions where a synergistic or antagonistic drug interaction occurs.

Important parameters shared by all the three functions are:

Following shows the examples for dose-response and ZIP synergy landscape.

5.2.1 Heatmap

If dynamic is set as FALSE, this function will return a ggplot object. User could use + theme() to adjust the plot theme.

Plot2DrugHeatmap(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "response",
    dynamic = FALSE,
    summary_statistic = c("mean",  "median")
  )

Plot2DrugHeatmap(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "ZIP_synergy",
    dynamic = FALSE,
    summary_statistic = c( "quantile_25", "quantile_75")
  )

5.2.2 2D contour plot

If dynamic is set as FALSE, this function will return a ggplot object. User could use + theme() to adjust the plot theme.

Plot2DrugContour(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "response",
    dynamic = FALSE,
    summary_statistic = c("mean", "median")
  )
Plot2DrugContour(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "ZIP_synergy",
    dynamic = FALSE,
    summary_statistic = c("quantile_25", "quantile_75")
  )

5.2.3 3D surface plot

If parameter dynamic is set as FALSE, this function will return an object of class recordedplot. User could use replayPlot to plot it.

Plot2DrugSurface(
  data = res,
  plot_block = 1,
  drugs = c(1, 2),
  plot_value = "response",
  dynamic = FALSE,
  summary_statistic = c("mean", "quantile_25", "median", "quantile_75")
)
Plot2DrugSurface(
  data = res,
  plot_block = 1,
  drugs = c(1, 2),
  plot_value = "ZIP_synergy",
  dynamic = FALSE,
  summary_statistic = c("mean", "quantile_25", "median", "quantile_75")
)

It is recommend to plot the 3D surface in dynamic (dynamic=TRUE) way, with which user could adjust the view point in an interactive way to show the surface at the best angle.

5.3 Plotting wrapper for two-drug combination

SynergyFinder provides two plotting wrapper for two-drug combination dose-response visualization (PlotDoseResponse) and synergy score visualization (PlotSynergy).

The function PlotDoseResponse combined the dose-response curve and dose-response heatmap in one figure. In addition to the parameters inherited from PlotDoseResponseCurve and Plot2DrugHeatmap, this function has the parameters for saving the plot as a file, including:

PlotDoseResponse(
  data = res,
  block_ids = c(1, 2),
  drugs = c(1,2),
  save_file = TRUE,
  file_type = "png"
)

The function PlotSynergy wrapped the Plot2DrugHeatmap, Plot2DrugContour and Plot2DrugSurface to draw the synergy score landscape and inherits the parameters from them. Parameter type is used to select the plot type (2D, 3D, or heatmap). The parameters for saving plot in files is same as those in PlotDoseResponse. This function will return a ggplot object. User could use + theme() to adjust the plot theme.

PlotSynergy(
  data = res,
  type = "2D",
  method = "ZIP",
  block_ids = c(1, 2),
  drugs = c(1,2),
  save_file = TRUE,
  file_type = "png"
)

5.4 Synergy barometer

SynergyFinder leveraged the implementation of the four models so that the expected response of non-interaction (i.e. \(y_{ZIP}\), \(y_{Loewe}\), \(y_{HSA}\), and \(y_{Bliss}\) described in section 4.1) could be derived. Since all the expected responses share the same unit as the observed response (i.e. % inhibition), SynergyFinder provides the function for plotting a synergy barometer to compare these values. The expected and observed responses for a given combination at a specific dose condition are positioned on the same scale. With such a tool as the synergy barometer, one can not only evaluate the degree of synergy of a specific model, but also easily understand the differences in the results among them. Ideally, a strong synergy should be concluded if the observed response goes beyond the expected responses of all four models.

The function PlotBarometer is designed to generate the barometer for given concentration combination in a matrix. The needle of the barometer points to the observed response value. The expected responses from different models are marked as the ticks on the color bar. The observed response and the concentration of the combined drugs are tested at the center of the barometer. This plot is available for high-order drug combination visualization.

Important parameters for this function are:

This function will return a ggplot object. User could add additional theme() layers to adjust the plot theme.

Following shows two examples for a synergy combination from block 1 at ispinesib 9.7656 nM + ibrutinib 50 nM and a antagonistic combination from block 2 at Canertinib 625 nM + Ibrutinib 12.5 nM

# Block1: ispinesib (drug1) 9.7656 nM + ibrutinib (drug2) 50 nM
PlotBarometer(
  data = res,
  plot_block = 1,
  plot_concs = c(9.7656, 50), 
  needle_text_offset = 2.5 # Move the texts below the needle
)
# Block2: Canertinib (drug1) 625 nM + Ibrutinib (drug2) 12.5 nM
PlotBarometer(
  data = res, 
  plot_block = 2, 
  plot_concs = c(625, 12.5),
  needle_text_offset = -2.5 # Move the texts above the needle
)

5.5 Barplot

SynergyFinder provides the function PlotMutiDrugBar to visualize the observed response, expected response, or synergy scores over the whole combination matrix. This plot is available for high-order drug combination visualization. It will generate a group of bar plots for concentrations of tested drugs and selected values for visualization. User could sort the whole bar group by the values in any panel or highlight certain concentration combination across all bar plots.

Important parameters for this function are:

This function will return a ggplot object. User could use + theme() to adjust the plot theme.

PlotMultiDrugBar(
  data = res,
  plot_block = 1,
  plot_value = c("response", "ZIP_synergy", "Loewe_synergy", "HSA_synergy", "Bliss_synergy"),
  sort_by = "response",
  highlight_row = c(9.7656, 50),
  highlight_label_size = 8
)

5.6 SS plot

The CSS indicates the efficacy of a drug combination, whereas the synergy score indicates the degree of interactions. To prioritize potential drug combinations, it is necessary to identify those with both higher CSS and higher synergy scores. The synergy scores have the same scale as the CSS that was developed earlier [6], thus allowing a direct comparison in an synergy-sensitivity (SS) plot .

The function PlotSensitivitySynergy is designed to draw the SS plot. It generates a scatter plot for all the combination blocks in the input data. The x-aix represents CSS, the y-axis represents selected synergy score and each point represents one combination block in the input data. The combinations locating in the upper-right corner have both higher CSS and higher synergy scores, which could be consider as a potential drug combination for further validation. Important parameters for this function are:

Following is an example for SS plot on the “mathews_screening_data” build-in data set.

PlotSensitivitySynergy(
  data = res,
  plot_synergy = "ZIP",
  show_labels = TRUE,
  dynamic = FALSE
)

6 Data with replicates

SynergyFinder has an build-in example data (ONEIL_screening_data) in the package. It is extracted from a pan-cancer drug screening study.[7] The example input data contains two representative drug combinations (MK-1775 & Niraparib and Paclitaxel & L-778123 free base) for which the %inhibition of a cell line OCUBM and NCIH2122 was assayed using a 5 by 5 dose matrix design with four replicates. To load the example data, please run:

data("ONEIL_screening_data")
head(ONEIL_screening_data)
#>   block_id   drug1     drug2 cell_line_name  conc1  conc2  response conc_unit
#> 1        1 MK-1775 Niraparib          OCUBM 0.0000  0.000 -0.000626        uM
#> 2        1 MK-1775 Niraparib          OCUBM 0.0000  0.223  2.127464        uM
#> 3        1 MK-1775 Niraparib          OCUBM 0.0000  0.775  2.268122        uM
#> 4        1 MK-1775 Niraparib          OCUBM 0.0000  2.750 24.505645        uM
#> 5        1 MK-1775 Niraparib          OCUBM 0.0000 10.000 44.475959        uM
#> 6        1 MK-1775 Niraparib          OCUBM 0.0325  0.000  2.371026        uM

6.1 Reshaping and pre-processing

The function ReshapeData automatically detect whether there are replicates in the input data. If there are replicates, this function calculates the statistics for the response value (both adjusted and original values) via bootstrapping.

Suppose that \(B\) (depending on parameter iteration) bootstrap dose-response matrix samples are drawn from the replicates and that \(r_1, r_2, ..., r_B\) are the estimates of the response (%inhibition) at certain concentration combination in the samples, with mean of \(\bar r\). The bootstrap standard error is determined as:

\[ SE = \sqrt{\frac{1}{B - 1}\sum_{i = 1}^B(r_i - \bar r)^2)} \] The 95% confidence interval for the observed response is approximately:

\[[\bar r - 1.96SE, \bar r + 1.96SE]\]

Meanwhile, at the whole dose-response matrix level, we provide an empirical p-value to asses the significance of the difference between the estimated average synergy score over the whole dose-response matrix and 0 %inhibition under the null hypothesis of non-response.

Letting \(r_1 ', r_2 ', ..., r_B '\) be the estimates of the average response over the whole dose matrix from the bootstrap samples, the p-value is:

\[p = exp(-0.717z - 0.416z^2),\ where\ z = abs(\bar r')/\sqrt{\frac{1}{n - 1}\sum_{i = 1}^n(r_i' - \bar r')^2}\]

res <- ReshapeData(
  data = ONEIL_screening_data,
  data_type = "inhibition",
  impute = TRUE,
  impute_method = NULL,
  noise = TRUE,
  iteration = 10, # Number of iterations for bootstrapping
  seed = 1
)

The output for this function is an R list with the similar structure as that from data without replicates. The differences are:

  1. The p-values for response value (adjusted and original input value) is added in data frame “drug_pair”.
  2. The response values in data frame “response” are the mean values over the replicates.
  3. A data frame “response_statistics” containing the statistics (mean value, standard error of mean and bounders for 95% confidence interval) for observed response values is added. The “n” column records the number of replicate for each observation.
str(res)
#> List of 3
#>  $ drug_pairs         : tibble [2 × 9] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id               : int [1:2] 1 2
#>   ..$ drug1                  : chr [1:2] "MK-1775" "Paclitaxel"
#>   ..$ drug2                  : chr [1:2] "Niraparib" "L-778123 free base"
#>   ..$ conc_unit1             : chr [1:2] "uM" "uM"
#>   ..$ conc_unit2             : chr [1:2] "uM" "uM"
#>   ..$ input_type             : chr [1:2] "inhibition" "inhibition"
#>   ..$ replicate              : logi [1:2] TRUE TRUE
#>   ..$ response_p_value       : chr [1:2] "1.50e-165" "3.45e-163"
#>   ..$ response_origin_p_value: chr [1:2] "1.60e-182" "< 2e-324"
#>  $ response           : tibble [200 × 5] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id       : int [1:200] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ conc1          : num [1:200] 0 0 0 0 0 0.0325 0.0325 0.0325 0.0325 0.0325 ...
#>   ..$ conc2          : num [1:200] 0 0.223 0.775 2.75 10 0 0.223 0.775 2.75 10 ...
#>   ..$ response       : num [1:200] -0.00125 2.12765 2.26729 24.50724 44.47629 ...
#>   ..$ response_origin: num [1:200] -0.000626 2.127464 2.268122 24.505645 44.475959 ...
#>  $ response_statistics: tibble [50 × 14] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id                : int [1:50] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ conc1                   : num [1:50] 0 0 0 0 0 0.0325 0.0325 0.0325 0.0325 0.0325 ...
#>   ..$ conc2                   : num [1:50] 0 0.223 0.775 2.75 10 0 0.223 0.775 2.75 10 ...
#>   ..$ response_sd             : num [1:50] 0.000462 0.000349 0.000819 0.001162 0.000827 ...
#>   ..$ response_mean           : num [1:50] -0.000624 2.127207 2.267631 24.505661 44.476357 ...
#>   ..$ response_origin_sd      : num [1:50] 0 0 0 0 0 ...
#>   ..$ response_origin_mean    : num [1:50] -0.000626 2.127464 2.268122 24.505645 44.475959 ...
#>   ..$ n                       : int [1:50] 4 4 4 4 4 4 4 4 4 4 ...
#>   ..$ response_sem            : num [1:50] 0.000231 0.000174 0.00041 0.000581 0.000413 ...
#>   ..$ response_ci_left        : num [1:50] -0.00136 2.12665 2.26633 24.50381 44.47504 ...
#>   ..$ response_ci_right       : num [1:50] 0.00011 2.12776 2.26893 24.50751 44.47767 ...
#>   ..$ response_origin_sem     : num [1:50] 0 0 0 0 0 ...
#>   ..$ response_origin_ci_left : num [1:50] -0.000626 2.127464 2.268122 24.505645 44.475959 ...
#>   ..$ response_origin_ci_right: num [1:50] -0.000626 2.127464 2.268122 24.505645 44.475959 ...

6.2 Drug synergy scoring

For input data with replicates, the function CalculateSynergy calculates the statistics for the synergy scores via bootstrapping. The methods is same as that was described in previous section which is for observed dose-response matrix. The differences are:

  1. The synergy score matrix calculated from bootstrap samples instead of the sampled dose response matrix is used to calculate the SE and 95% confidence interval.
  2. The empirical p-value is provided at the whole matrix level to assess the significance of the difference between the estimated average synergy score over the whole matrix and the expected synergy score of zero under the null hypothesis of non-interaction.
  3. While calculating average synergy score over the whole matrix, the scores for monotherapies are excluded.

The output for this function is an R list with the similar structure as that from data without replicates. The differences are:

  1. The p-values for synergy scores is added in data frame “drug_pair”.
  2. The values in data frame “synergy_scores” are the mean values over the bootstrap samples.
  3. A data frame “synergy_scores_statistics” containing the statistics (mean value, standard error of mean and bounders for 95% confidence interval) for values related with synergy scores is added.
str(res$drug_pairs)
#> tibble [2 × 17] (S3: tbl_df/tbl/data.frame)
#>  $ block_id               : int [1:2] 1 2
#>  $ drug1                  : chr [1:2] "MK-1775" "Paclitaxel"
#>  $ drug2                  : chr [1:2] "Niraparib" "L-778123 free base"
#>  $ conc_unit1             : chr [1:2] "uM" "uM"
#>  $ conc_unit2             : chr [1:2] "uM" "uM"
#>  $ input_type             : chr [1:2] "inhibition" "inhibition"
#>  $ replicate              : logi [1:2] TRUE TRUE
#>  $ response_p_value       : chr [1:2] "1.50e-165" "3.45e-163"
#>  $ response_origin_p_value: chr [1:2] "1.60e-182" "< 2e-324"
#>  $ ZIP_synergy_p_value    : chr [1:2] "4.34e-47" "3.96e-192"
#>  $ HSA_synergy_p_value    : chr [1:2] "1.58e-48" "< 2e-324"
#>  $ Bliss_synergy_p_value  : chr [1:2] "4.56e-37" "< 2e-324"
#>  $ Loewe_synergy_p_value  : chr [1:2] "1.91e-41" "< 2e-324"
#>  $ ZIP_synergy            : num [1:2] 35.6 -75.8
#>  $ HSA_synergy            : num [1:2] 41.6 -71.6
#>  $ Bliss_synergy          : num [1:2] 36 -77.4
#>  $ Loewe_synergy          : num [1:2] 38.1 -73.6
str(res$synergy_scores_statistics)
#> tibble [50 × 53] (S3: tbl_df/tbl/data.frame)
#>  $ block_id              : int [1:50] 1 1 1 1 1 1 1 1 1 1 ...
#>  $ conc1                 : num [1:50] 0 0 0 0 0 0.0325 0.0325 0.0325 0.0325 0.0325 ...
#>  $ conc2                 : num [1:50] 0 0.223 0.775 2.75 10 0 0.223 0.775 2.75 10 ...
#>  $ ZIP_fit_mean          : num [1:50] -0.00066 2.12711 2.26775 24.5057 44.47623 ...
#>  $ ZIP_fit_sd            : num [1:50] 0.00025 0.000191 0.000528 0.000466 0.000352 ...
#>  $ ZIP_fit_sem           : num [1:50] 0.0000789 0.0000604 0.000167 0.0001473 0.0001114 ...
#>  $ ZIP_fit_ci_left       : num [1:50] -0.00101 2.12683 2.26688 24.50496 44.47565 ...
#>  $ ZIP_fit_ci_right      : num [1:50] -0.000241 2.1274 2.268567 24.506312 44.476663 ...
#>  $ ZIP_ref_mean          : num [1:50] -0.00066 2.12711 2.26775 24.5057 44.47623 ...
#>  $ ZIP_ref_sd            : num [1:50] 0.00025 0.000191 0.000528 0.000466 0.000352 ...
#>  $ ZIP_ref_sem           : num [1:50] 0.0000789 0.0000604 0.000167 0.0001473 0.0001114 ...
#>  $ ZIP_ref_ci_left       : num [1:50] -0.00101 2.12683 2.26688 24.50496 44.47565 ...
#>  $ ZIP_ref_ci_right      : num [1:50] -0.000241 2.1274 2.268567 24.506312 44.476663 ...
#>  $ ZIP_synergy_mean      : num [1:50] 0 0 0 0 0 ...
#>  $ ZIP_synergy_sd        : num [1:50] 0 0 0 0 0 ...
#>  $ ZIP_synergy_sem       : num [1:50] 0 0 0 0 0 ...
#>  $ ZIP_synergy_ci_left   : num [1:50] 0 0 0 0 0 ...
#>  $ ZIP_synergy_ci_right  : num [1:50] 0 0 0 0 0 ...
#>  $ HSA_ref_mean          : num [1:50] -0.00066 2.12711 2.26775 24.5057 44.47623 ...
#>  $ HSA_ref_sd            : num [1:50] 0.00025 0.000191 0.000528 0.000466 0.000352 ...
#>  $ HSA_ref_sem           : num [1:50] 0.0000789 0.0000604 0.000167 0.0001473 0.0001114 ...
#>  $ HSA_ref_ci_left       : num [1:50] -0.00101 2.12683 2.26688 24.50496 44.47565 ...
#>  $ HSA_ref_ci_right      : num [1:50] -0.000241 2.1274 2.268567 24.506312 44.476663 ...
#>  $ HSA_synergy_mean      : num [1:50] 0 0 0 0 0 ...
#>  $ HSA_synergy_sd        : num [1:50] 0 0 0 0 0 ...
#>  $ HSA_synergy_sem       : num [1:50] 0 0 0 0 0 ...
#>  $ HSA_synergy_ci_left   : num [1:50] 0 0 0 0 0 ...
#>  $ HSA_synergy_ci_right  : num [1:50] 0 0 0 0 0 ...
#>  $ Bliss_ref_mean        : num [1:50] -0.00066 2.12711 2.26775 24.5057 44.47623 ...
#>  $ Bliss_ref_sd          : num [1:50] 0.00025 0.000191 0.000528 0.000466 0.000352 ...
#>  $ Bliss_ref_sem         : num [1:50] 0.0000789 0.0000604 0.000167 0.0001473 0.0001114 ...
#>  $ Bliss_ref_ci_left     : num [1:50] -0.00101 2.12683 2.26688 24.50496 44.47565 ...
#>  $ Bliss_ref_ci_right    : num [1:50] -0.000241 2.1274 2.268567 24.506312 44.476663 ...
#>  $ Bliss_synergy_mean    : num [1:50] 0 0 0 0 0 ...
#>  $ Bliss_synergy_sd      : num [1:50] 0 0 0 0 0 ...
#>  $ Bliss_synergy_sem     : num [1:50] 0 0 0 0 0 ...
#>  $ Bliss_synergy_ci_left : num [1:50] 0 0 0 0 0 ...
#>  $ Bliss_synergy_ci_right: num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_ref_mean        : num [1:50] -0.00066 2.12711 2.26775 24.5057 44.47623 ...
#>  $ Loewe_ref_sd          : num [1:50] 0.00025 0.000191 0.000528 0.000466 0.000352 ...
#>  $ Loewe_ref_sem         : num [1:50] 0.0000789 0.0000604 0.000167 0.0001473 0.0001114 ...
#>  $ Loewe_ref_ci_left     : num [1:50] -0.00101 2.12683 2.26688 24.50496 44.47565 ...
#>  $ Loewe_ref_ci_right    : num [1:50] -0.000241 2.1274 2.268567 24.506312 44.476663 ...
#>  $ Loewe_synergy_mean    : num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_synergy_sd      : num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_synergy_sem     : num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_synergy_ci_left : num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_synergy_ci_right: num [1:50] 0 0 0 0 0 ...
#>  $ Loewe_ci_mean         : num [1:50] NA NA NA NA NA ...
#>  $ Loewe_ci_sd           : num [1:50] NA NA NA NA NA ...
#>  $ Loewe_ci_sem          : num [1:50] NA NA NA NA NA ...
#>  $ Loewe_ci_ci_left      : num [1:50] NA NA NA NA NA ...
#>  $ Loewe_ci_ci_right     : num [1:50] NA NA NA NA NA ...

6.3 Sensitivity scoring

For input data with replicates, the function CalculateSensitivity calculates the statistics for the sensitivity scores (RI and CSS) via bootstrapping. The methods is same as that was described in section 6.1 which is for observed dose-response matrix. The differences are:

  1. The sensitivity scores are calculated from bootstrap samples instead of the sampled dose response matrix is used to calculate the SE and 95% confidence interval.
  2. As there is only one value for each sensitivity score (RI1, RI2, …, CSS1_IC502, …, CSS) for one dose-response matrix, the empirical p-value is calculated on the estimated scores instead of the mean values over whole matrix.
  3. As the unit for IC50 is not the %inhibition, the null hypothesis non-response is not adaptable to it. Currently, we don’t provide the p-value for IC50.
res <- CalculateSensitivity(
  data = res,
  correct_baseline = "non",
  iteration = 10 # Number of iterations for bootstrapping
)

The output for this function is an R list with the similar structure as that from data without replicates. The differences are:

  1. The sensitivity scores in data frame “drug_pair” are the mean values over the bootstrap samples.
  2. A data frame “sensitivity_scores_statistics” containing the statistics (mean value, standard error of mean and bounders for 95% confidence interval) for values related with synergy scores is added.
str(res$sensitivity_scores_statistics)
#> 'data.frame':    2 obs. of  39 variables:
#>  $ block_id           : int  1 2
#>  $ ic50_1_mean        : chr  "0.5" "0.00157428688872775"
#>  $ ic50_2_mean        : chr  "2.64214451636998" "0.375135166419377"
#>  $ ri_1_mean          : chr  "22.0254" "62.6112"
#>  $ ri_2_mean          : chr  "16.8071" "48.5743"
#>  $ css1_ic502_mean    : chr  "79.8737" "0.6425"
#>  $ css2_ic501_mean    : chr  "80.2739" "0.554"
#>  $ css_mean           : chr  "80.0738" "0.59825"
#>  $ ic50_1_sd          : chr  "0" "0.000000316559225685352"
#>  $ ic50_2_sd          : chr  "0.0000488815419007319" "0.00000624302934324253"
#>  $ ri_1_sd            : chr  "0.00544058820349433" "0.000421637021357797"
#>  $ ri_2_sd            : chr  "0.000316227766017224" "0.000483045891541954"
#>  $ css1_ic502_sd      : chr  "3.70723023443529" "1.14138874962818"
#>  $ css2_ic501_sd      : chr  "5.62189470730287" "1.83480189666351"
#>  $ css_sd             : chr  "3.47759128165203" "1.37756232466387"
#>  $ ic50_1_sem         : chr  "0" "0.000000100104816750499"
#>  $ ic50_2_sem         : chr  "0.0000154577007947269" "0.00000197421922239115"
#>  $ ri_1_sem           : chr  "0.00172046505340857" "0.00013333333333397"
#>  $ ri_2_sem           : chr  "0.000100000000000122" "0.000152752523165924"
#>  $ css1_ic502_sem     : chr  "1.17232913514555" "0.360938814451671"
#>  $ css2_ic501_sem     : chr  "1.77779920407227" "0.580215304865358"
#>  $ css_sem            : chr  "1.09971092211645" "0.435623456477419"
#>  $ ic50_1_ci_left     : chr  "0.5" "0.00157400274871024"
#>  $ ic50_2_ci_left     : chr  "2.64209193225961" "0.375124678423372"
#>  $ ri_1_ci_left       : chr  "22.018675" "62.611"
#>  $ ri_2_ci_left       : chr  "16.807" "48.574"
#>  $ css1_ic502_ci_left : chr  "73.6528" "-1.440475"
#>  $ css2_ic501_ci_left : chr  "70.2681" "-1.91945"
#>  $ css_ci_left        : chr  "74.8257" "-1.5825375"
#>  $ ic50_1_ci_right    : chr  "0.5" "0.00157495595803027"
#>  $ ic50_2_ci_right    : chr  "2.64222224300616" "0.375144201469938"
#>  $ ri_1_ci_right      : chr  "22.036075" "62.612"
#>  $ ri_2_ci_right      : chr  "16.807775" "48.575"
#>  $ css1_ic502_ci_right: chr  "85.108775" "2.219375"
#>  $ css2_ic501_ci_right: chr  "86.583725" "3.5085"
#>  $ css_ci_right       : chr  "84.1326375" "2.6700625"
#>  $ ri_1_p_value       : chr  "< 2e-324" "< 2e-324"
#>  $ ri_2_p_value       : chr  "< 2e-324" "< 2e-324"
#>  $ css_p_value        : chr  "1.11e-103" "6.77e-01"

6.4 Visualization

All the functions introduced in section 5 are viable on the data set with replicates. In addition, the statistics for response or synergy scores could be visualized in heatmap by calling function Plot2DrugHeatmap. User could used parameter statistic to select the statistics shown on the plot. Avaliable values are “sem” - sandard error of mean, and “ci” - 95% confidence interval.

# Standard error of mean
Plot2DrugHeatmap(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "response",
    dynamic = FALSE,
    statistic = "sem",
    summary_statistic = c("mean",  "median")
  )

# 95% confidence interval
Plot2DrugHeatmap(
    data = res,
    plot_block = 1,
    drugs = c(1, 2),
    plot_value = "ZIP_synergy",
    statistic = "ci",
    dynamic = FALSE,
    summary_statistic = c("quantile_25", "quantile_75")
  )

7 Higher-order drug combination screening

SynergyFinder has an build-in example data (NCATS_screening_data) in the package. It is extracted from a triple drug screening study on malaria.[8] The example input data contains two representative drug combinations (Piperaquine & Pyronaridine Tetraphosphate & Darunavir Ethanolate and Piperaquine & Pyronaridine Tetraphosphate & Lopinavir) for which the %inhibition of malaria was assayed using a 10 by 10 by 12 dose matrix design with four replicates. To load the example data, please run:

data("NCATS_screening_data")
head(NCATS_screening_data)
#>   block_id       drug1                       drug2                drug3   conc1
#> 1        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.60000
#> 2        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.30000
#> 3        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.15000
#> 4        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.07500
#> 5        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.03750
#> 6        1 Piperaquine Pyronaridine Tetraphosphate Darunavir Ethanolate 0.01875
#>   conc2 conc3 response conc_unit1 conc_unit2 conc_unit3
#> 1 0.075     0 3.079183         uM         uM         uM
#> 2 0.075     0 3.867034         uM         uM         uM
#> 3 0.075     0 1.937995         uM         uM         uM
#> 4 0.075     0 1.601282         uM         uM         uM
#> 5 0.075     0 4.680111         uM         uM         uM
#> 6 0.075     0 4.592774         uM         uM         uM

7.1 Reshaping and pre-processing

The utility of function ReshapeData on higher-order drug combination screening data is same as that on 2-drug combination screening data. Please check section 3 for more details about the parameters and output format.

res <- ReshapeData(
  data = NCATS_screening_data,
  data_type = "inhibition",
  impute = TRUE,
  impute_method = NULL,
  noise = TRUE,
  seed = 1
)

The output of this function:

str(res)
#> List of 2
#>  $ drug_pairs: tibble [2 × 9] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id  : int [1:2] 1 2
#>   ..$ drug1     : chr [1:2] "Piperaquine" "Piperaquine"
#>   ..$ drug2     : chr [1:2] "Pyronaridine Tetraphosphate" "Pyronaridine Tetraphosphate"
#>   ..$ drug3     : chr [1:2] "Darunavir Ethanolate" "Lopinavir"
#>   ..$ conc_unit1: chr [1:2] "uM" "uM"
#>   ..$ conc_unit2: chr [1:2] "uM" "uM"
#>   ..$ conc_unit3: chr [1:2] "uM" "uM"
#>   ..$ input_type: chr [1:2] "inhibition" "inhibition"
#>   ..$ replicate : logi [1:2] FALSE FALSE
#>  $ response  : tibble [2,400 × 6] (S3: tbl_df/tbl/data.frame)
#>   ..$ block_id       : int [1:2400] 1 1 1 1 1 1 1 1 1 1 ...
#>   ..$ conc1          : num [1:2400] 0.6 0.3 0.15 0.075 0.0375 ...
#>   ..$ conc2          : num [1:2400] 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 ...
#>   ..$ conc3          : num [1:2400] 0 0 0 0 0 0 0 0 0 0 ...
#>   ..$ response       : num [1:2400] 3.08 3.87 1.94 1.6 4.68 ...
#>   ..$ response_origin: num [1:2400] 3.08 3.87 1.94 1.6 4.68 ...

7.2 Drug synergy scoring

SynergyFinder extends the algorithms for calculate synergy scores to higher-order drug combination data sets. Following describes the expending of four major synergy reference models (“HSA”, “Bliss”, “Loewe”, and “ZIP”) to higher-order drug combinations.

Consider the response of a drug to be measured as a %inhibition \(y\) that ranges from 0 to 1, with a higher value indicating better efficacy. For a combination that involves \(n\) drugs, the observed combination response is denoted as \(y_c\), while the observed monotherapy response of its constituent drugs is \(y_{(i,i=1,...n)}\). The expected combination response is determined by the assumption of non-interaction.

\[y_{HSA}=max(y_1,…,y_i,…,y_n )\].

\[y_{Bliss}=1 - \prod_i(1-y_i)\].

Accordingly, the multi-drug synergy score for the observed combination response \(y_c\) can be determined as:

\[S_{HSA} = y_c - max⁡(y_1,…,y_i,…,y_n)\]

\[S_{Bliss} = y_c - (1-\prod_i(1-y_i )) = y_c - (\sum_i y_i - \sum_{i<j} y_i y_j + (-1)^{r+1} \times \sum_{i<j} \prod_j^r y_j +(-1)^{n+1} \times \prod_i y_i )\]

\[S_{LOEWE}=y_c - y_{LOEWE}, s.t.\sum_i (\frac{x_i}{f_i^{-1} (y_{LOEWE})}) = 1\]

For determining the ZIP-based synergy score, \(y_c\) needs to be replaced with the predicted average response \(\hat y_c\) given by the curve fitting models \(f_i'\) to make it comparable to \(y_{ZIP}\):

\[S_{ZIP} = \hat y_c - y_{ZIP} = \frac1n \sum_i f_i'(x_i) - (1- \prod_i(1-f_i(x_i)),\ where\ E_{min}(f_i')=f_{-i}'(x_{-i})\]

Namely, \(f_i'(x)\) stands for the log logistic model defined for the combination response at dose \(x\) of drug \(i\) when the other drugs are present. Furthermore, \(E_{min}\) of \(f_i'(x)\) is determined by \(f_{-i}'(x_{-i})\), which is the fitted curve of the combination response while drug i is absent. Note that the ZIP model captures the shift of potency for a drug combination in comparison to its monotherapy drugs, therefore, the ZIP model compares the difference of fitted models for the drug combination \(f_i'(x_i)\) and for the monotherapy drugs \(f_i(x_i)\).

The function CalculateSynergy is designed to calculate the synergy scores. The utility is same as that for 2-drug combination screening data. Please check section 4.1 for more details about the parameters and output format.

res <- CalculateSynergy(
  data = res,
  method = c("ZIP", "HSA", "Bliss", "Loewe"),
  Emin = NA,
  Emax = NA,
  correct_baseline = "non")
res$drug_pairs
#> # A tibble: 2 × 17
#>   block_id drug1       drug2   drug3 conc_unit1 conc_unit2 conc_unit3 input_type
#>      <int> <chr>       <chr>   <chr> <chr>      <chr>      <chr>      <chr>     
#> 1        1 Piperaquine Pyrona… Daru… uM         uM         uM         inhibition
#> 2        2 Piperaquine Pyrona… Lopi… uM         uM         uM         inhibition
#> # ℹ 9 more variables: replicate <lgl>, ZIP_synergy_p_value <chr>,
#> #   HSA_synergy_p_value <chr>, Bliss_synergy_p_value <chr>,
#> #   Loewe_synergy_p_value <chr>, ZIP_synergy <dbl>, HSA_synergy <dbl>,
#> #   Bliss_synergy <dbl>, Loewe_synergy <dbl>
str(res$synergy_scores)
#> tibble [2,400 × 14] (S3: tbl_df/tbl/data.frame)
#>  $ block_id     : int [1:2400] 1 1 1 1 1 1 1 1 1 1 ...
#>  $ conc1        : num [1:2400] 0.6 0.3 0.15 0.075 0.0375 ...
#>  $ conc2        : num [1:2400] 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.075 ...
#>  $ conc3        : num [1:2400] 0 0 0 0 0 0 0 0 0 0 ...
#>  $ ZIP_fit      : num [1:2400] 2.46 3.45 2.29 2.71 3.43 ...
#>  $ ZIP_ref      : num [1:2400] 101 101 101 101 101 ...
#>  $ ZIP_synergy  : num [1:2400] -98.2 -97.2 -98.3 -97.9 -97.2 ...
#>  $ HSA_ref      : num [1:2400] 99.3 99.3 99.3 99.3 99.3 ...
#>  $ HSA_synergy  : num [1:2400] -96.2 -95.4 -97.4 -97.7 -94.6 ...
#>  $ Bliss_ref    : num [1:2400] 99.3 99.3 99.3 99.3 99.3 ...
#>  $ Bliss_synergy: num [1:2400] -96.3 -95.5 -97.4 -97.7 -94.6 ...
#>  $ Loewe_ref    : num [1:2400] 105 105 105 105 105 ...
#>  $ Loewe_synergy: num [1:2400] -102 -102 -104 -104 -101 ...
#>  $ Loewe_ci     : num [1:2400] NaN NaN NaN NaN NaN NaN NaN NaN NaN NA ...

7.3 Sensitivity scoring

The calculation of RI and relative IC50 of monotherapies in higher-order drug combination is same as that is used in the 2-drug combination data. Please check section 4.2 for more details.

SynergyFinder extends the CSS score developed by Malyutina et al.[6] to higher-order drug combinations. For a combination that involves \(n\) drugs, the \(CSS_ij\) for each pair of “drug i + drug j” combination, where \(i,j \in {1, 2, 3, ..., n}\), is calculated while fixing the drug i (background drug) at its relative IC50 and let drug j be at varying concentrations (foreground drug). Drug j’s dose-response is modeled using 4-parameter log-logistic curve. The area under the log-scaled dose-response curve (AUC) is then determined according to:

\[AUC = \int_{c_1}^{c_2}y_{min} + \frac{y_{max}-y_{min}}{1 + 10^{\lambda(log_{10}IC_{50}-x')}}dx'\] where \([c_1, c_2]\) is the concentration range of the foreground drug tested.

The CSS for the whole dose-response matrix is defined as the average of all the pair-wise CSS:

\[CSS = \frac1n \sum_{i = 1}^n \sum_{j = 1}^n CSS_{ij}\] The function CalculatSensitivity is designed to calculate the sensitivity scores. The utility is same as that for 2-drug combination screening data. Please check section 4.2 for more details about the parameters and output format.

res <- CalculateSensitivity(
  data = res,
  correct_baseline = "non"
)
sensitive_columns <- c(
  "block_id", "drug1", "drug2", "drug3",
  "ic50_1", "ic50_2", "ic50_3", 
  "ri_1", "ri_2", "ri_3", 
  "css1_ic502", "css2_ic501","css1_ic503", "css3_ic502", "css2_ic503",
  "css3_ic502", "css")
res$drug_pairs[, sensitive_columns]
#> # A tibble: 2 × 17
#>   block_id drug1  drug2 drug3 ic50_1  ic50_2 ic50_3  ri_1  ri_2  ri_3 css1_ic502
#>      <int> <chr>  <chr> <chr>  <dbl>   <dbl>  <dbl> <dbl> <dbl> <dbl>      <dbl>
#> 1        1 Piper… Pyro… Daru… 0.0172 0.0168    7.40  37.0  71.7 102.        12.0
#> 2        2 Piper… Pyro… Lopi… 0.0204 0.00595   2.12  34.1  38.9  65.5       22.4
#> # ℹ 6 more variables: css2_ic501 <dbl>, css1_ic503 <dbl>, css3_ic502 <dbl>,
#> #   css2_ic503 <dbl>, css3_ic502 <dbl>, css <dbl>

7.4 Visualization

All the functions introduced in section 5 are viable on the higher-order drug screening data, though the functions for synergy or sensitivity landscapes, which are introduced in section 5.2 and 5.3, can only visualize two out of all combined drugs.

To visualize the dose-response and synergy landscapes of higher-order drug combination, SynergyFinder provides the function PlotMultiDrugSurface. It implements a dimension reduction technique that is based on multi-dimensional scaling, similar to the recent application in transforming numerical data into images.

For a drug combination in a high-dimensional dose space, with its coordinates of \(X = (x_1,...x_n)\), we utilize their rankings \(R=(r_1,...r_n)\) to determine the pairwise similarity between the instances of \(X\). We utilize multi-dimensional scaling to minimize the error of the pairwise distance in a two-dimensional space in which the synergy and sensitivity scores can be visualized as a landscape (Figure 2). Note that using the dose rankings as the input for multi-dimensional scaling can assure that the resulting two-dimensional coordinates are equally distanced among the neighboring dose conditions, thus making the visualization easier to interpret. Furthermore, for the case of two-drug combinations, the algorithm can converge to the actual dose rankings, thus preserving the consistency across all the orders of combinations.

knitr::include_graphics("./fig2.png")
Figure 2. Dimension reduction for the visualization of high-order drug combinations

Figure 2. Dimension reduction for the visualization of high-order drug combinations

The important parameters for function PlotMultiDrugSurface are:

The output of this function is a interactive plot generated by plotyly. It could be used on interactive platforms, such as R Shiny or Rmarkdown. The plane expanded by x-axis and y-axis is the two-dimensional space to which the data point is projected. The z-axis represents the value of selected plot_value. The landscape is visualized as a surface in the 3D space, the data points are visualized as dots. User could check the concentration combination and exact value for each data point by hovering mouse on certain dot.

PlotMultiDrugSurface(
  data = res,
  plot_block = 2,
  plot_value = "response",
  summary_statistic = NULL,
  distance_method = "mahalanobis",
  show_data_points = TRUE
)
PlotMultiDrugSurface(
  data = res,
  plot_block = 1,
  plot_value = "ZIP_synergy",
  summary_statistic = NULL,
  distance_method = "mahalanobis",
  show_data_points = FALSE
)

Citation

For use of SynergyFinder R package or web application:

Zheng,S., Wang,W., Aldahdooh,J., Malyutina,A., Shadbahr,T., Pessia,A. Tanoli.Z and Tang,J. (2022) SynergyFinder Plus: Toward Better Interpretation and Annotation of Drug Combination Screening Datasets. Genomics, Proteomics & Bioinformatics. https://doi.org/10.1016/j.gpb.2022.01.004.

For use of ZIP synergy scoring:

Yadav,B., Wennerberg,K., Aittokallio,T. and Tang,J. (2015) Searching for Drug Synergy in Complex Dose-Response Landscapes Using an Interaction Potency Model. Comput Struct Biotechnol J, 13, 504–513.

For how to harmonize the different synergy scoring methods:

Tang,J., Wennerberg,K. and Aittokallio,T. (2015) What is synergy? The Saariselkä agreement revisited. Front Pharmacol, 6, 181.

For general ideas of drug combination therapies:

Tang,J. (2017) Informatics Approaches for Predicting, Understanding, and Testing Cancer Drug Combinations. Methods Mol Biol, 1636, 485–506.

For retrieving the most comprehensive drug combination data resources and their sensitivity and synergy results by SynergyFinder, please go to DrugComb :

Zheng,S., Aldahdooh,J., Shadbahr,T., Wang,Y., Aldahdooh,D., Bao,J., Wang,W. and Tang,J. (2021) DrugComb update: a more comprehensive drug sensitivity data repository and analysis portal. Nucleic Acids Research, 10.1093/nar/gkab438.

Zagidullin,B., Aldahdooh,J., Zheng,S., Wang,W., Wang,Y., Saad,J., Malyutina,A., Jafari,M., Tanoli,Z., Pessia,A., et al. (2019) DrugComb: an integrative cancer drug combination data portal. Nucleic Acids Res, 10.1093/nar/gkz337.

For use of combination sensitivity score:

Malyutina,A., Majumder,M.M., Wang,W., Pessia,A., Heckman,C.A. and Tang,J. (2019) Drug combination sensitivity scoring facilitates the discovery of synergistic and efficacious drug combinations in cancer. PLOS Computational Biology, 15, e1006752.

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