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Drug combination therapy against cancer has been a promising approach in precision oncology. In recent literature, most anticancer drug combinations (ADC) are estimated and derived from their independent drug response efficacy. Synergistic drug combination response is the most effective drug therapy for various cancer treatments. However, the ADC identification problem is challenging and practically infeasible to exhaustively screen out experimentally from the extensive ADCs. Therefore, computational approaches can be efficiently used to measure and identify drug combination responses in precision oncology. In recent decades, numerous computational approaches have been applied and proposed to predict the responses of clinical ADC using different pharmacological and multi-omics cancer data. Effective computational tools and approaches are needed to predict and measure ADC, address its challenges, and reduce complexity. We have reviewed state-of-the-art computational methods for ADC prediction in the recent decade. This review paper has provided an overview of synergistic ADC response and computational machine-learning approaches for ADC. A critical discussion of the advantages and limitations is also provided. Moreover, we have reviewed the recent existing drug combination resources for ADC response prediction and found the most influential computational method for anticancer drug combination response. Finally, we have compared different computational approaches using benchmark data for ADC responses and discussed the experimental results, limitations, and future direction of ADC responses in precision oncology.