In precision medicine, a common problem is drug sensitivity prediction from cancer tissue cell lines. These types of problems entail modelling multivariate drug responses on high-dimensional molecular feature sets in typically >1000 cell lines. The dimensions of the problem require specialised models and estimation methods. In addition, external information on both the drugs and the features is often available. We propose to model the drug responses through a linear regression with shrinkage enforced through a normal inverse Gaussian prior. We let the prior depend on the external information, and estimate the model and external information dependence in an empirical-variational Bayes framework. We demonstrate the usefulness of this model in both a simulated setting and in the publicly available Genomics of Drug Sensitivity in Cancer data.