The additive damage model: a mathematical model for cellular responses to drug combinations

Abstract

Mathematical models to describe dose-dependent cellular responses to drug combinations are an essential component of computational simulations for predicting therapeutic responses. Here, a new model, the additive damage model, is introduced and tested in cases where varying concentrations of two drugs are applied with a fixed exposure schedule. In the model, cell survival is determined by whether cellular damage, which depends on the concentrations of the drugs, exceeds a lethal threshold, which varies randomly in the cell population with a prescribed statistical distribution. Cellular damage is assumed to be additive, and is expressed as a sum of separate terms for each drug. Each term has a saturable dependence on drug concentration. The model has appropriate behavior over the entire range of drug concentrations, and is predictive, given single-agent dose-response data for each drug. The proposed model is compared with several other models, by testing their ability to fit 24 data sets for platinum-taxane combinations and 21 data sets for various other combinations. The Akaike Information Criterion is used to assess goodness of fit, taking into account the number of unknown parameters in each model. Overall, the additive damage model provides a better fit to the data sets than any previous model. The proposed model provides a basis for computational simulations of therapeutic responses. It predicts responses to drug combinations based on data for each drug acting as a single agent, and can be used as an improved null reference model for assessing synergy in the action of drug combinations.

Keywords: Cellular pharmacodynamics; Chemotherapy; Cytotoxicity; Dose–response; Drug synergy.

Figure 1

Schematic illustration of additive damage model. A. The threshold of total damage required to kill a cell is distributed in the cell population according to a probability density function. Total damage is the sum of the damages resulting from each treatment. The area under the curve corresponding to the total damage represents the fractional cell kill. B. Survival fraction (relative to controls) as a function of total damage is given by one minus the cumulative frequency distribution of cell damage.

Figure 2

Comparison of AIC_c values for candidate models, when applied to 45 data sets from the literature. The cell lines, drugs and data source are given in Table 2. Data sets are ordered from highest to lowest number of data points. A. Results for 24 in-vitro data sets for platinum-taxane drug combinations acting on cancer cell lines. Right axis is for data set number 1 only. B. Results for 21 in vitro data sets for various drug combinations (other than platinum-taxane) acting on cancer cell lines.

Figure 3

Comparison of model predictions with experimental data for the additive damage model (A,B) and the Chou-Talalay multiple inhibitor model (C,D), for a paclitaxel-cisplatin combination data set Engblom et al. (22) (A,C), and the ZD1839-trastuzumab combination data of Nakamura et al. (30) (B,D). In each plot, dashed lines indicate experimental data and solid lines indicate best-fit model predictions.

Figure 4

Survival as a function of cellular damage. A. Data of Liu et al 2009 (32); H322 human lung cancer cells; gemcitabine and OBP-31 (data set #45 of Table 2). Predicted values are based on model fits to the data for each drug acting as a single agent, without using the combination data. B. Data of Mohi et al 2004 (29); Ba/F-BCR/ABL WT cells; imatinib and rapamycin (data set #44 of Table 2). C. Data of Teicher et al (26); MCF-7 cells; thiotepa and cyclophosphamide (data set #31 of Table 2). D. Data of Hadaschik et al(21); J82 bladder cancer cells; cisplatin and paclitaxel (data set # 3 of Table 2). In B, C and D, model fits are based on all available data for single agents and combinations.

Figure 5

Estimates of combination index based on data for combination of OBP-301 virotherapy with gemcitabine (32). “MOI” = multiplicity of infection. A. Combination index from present additive damage model. B. Combination index from Calcusyn program (38).