The dynamics of complex diseases are governed by intricate interactions of myriad factors. Drug combinations, formed by mixing several single-drug treatments at various doses, can enhance the effectiveness of the therapy by targeting multiple contributing factors. The main challenge in designing drug combinations is the highly nonlinear interaction of the constituent drugs. Prior work focused on guided space-exploratory heuristics that require discretization of drug doses. While being more efficient than random sampling, these methods are impractical if the drug space is high dimensional or if the drug sensitivity is unknown. Furthermore, the effectiveness of the obtained combinations may decrease if the resolution of the discretization grid is not sufficiently fine. In this paper, we model the biological system response to a continuous combination of drug doses by a Gaussian process (GP). We perform closed-loop experiments that rely on the expected improvement criterion to efficiently guide the exploration process toward drug combinations with the optimal response. When computing the criterion, we marginalize out the GP hyperparameters in a fully Bayesian manner using a particle filter. Finally, we employ a hybrid Monte Carlo algorithm to rapidly explore the high-dimensional continuous search space. We demonstrate the effectiveness of our approach on a fully factorial Drosophila dataset, an antiviral drug dataset for Herpes simplex virus type 1, and simulated human Apoptosis networks. The results show that our approach significantly reduces the number of required trials compared to existing methods.