A quantitative understanding of the advantages of nanoparticle-based drug delivery vis-à-vis conventional free drug chemotherapy has yet to be established for cancer or other diseases despite numerous investigations. Here, we employ first-principles cell biophysics, drug pharmaco-kinetics, and drug pharmaco-dynamics to model the delivery of doxorubicin (DOX) to hepatocellular carcinoma (HCC) tumor cells and predict the resultant experimental cytotoxicity data. The fundamental, mechanistic hypothesis of our mathematical model is that the integrated history of drug uptake by the cells over time of exposure, which sets the cell death rate parameter, and the uptake rate are the sole determinants of the dose response relationship. A universal solution of the model equations is capable of predicting the entire, nonlinear dose response of the cells to any drug concentration based on just two separate measurements of these cellular parameters. This analysis reveals that nanocarrier-mediated delivery overcomes resistance to the free drug because of improved cellular uptake rates, and that dose response curves to nanocarrier mediated drug delivery are equivalent to those for free-drug, but "shifted to the left;" that is, lower amounts of drug achieve the same cell kill. We then demonstrate the model's general applicability to different tumor and drug types, and cell-exposure time courses by investigating HCC cells exposed to cisplatin and 5-fluorouracil, breast cancer MCF-7 cells exposed to DOX, and pancreatic adenocarcinoma PANC-1 cells exposed to gemcitabine. The model will help in the optimal design of nanocarriers for clinical applications and improve the current, largely empirical understanding of in vivo drug transport and tumor response.