The contraction of the heart is coordinated electrically, thanks to conducting links between the cells. Structural damage, due to aging or disease, can break these links and cause dangerous arrhythmia. To better understand these arrhythmia we need models in which the individual cells and their connections are resolved. For a whole heart, with billions of cells, this means trillions of model elements with many degrees of freedom each. The MICROCARD project aims to run such models on exascale computers. It covers every aspect of this challenge from code transformations to numerical schemes, mesh creation, and real-life use cases.