Patient-specific medical simulation holds the promise of determining tailored medical treatment based on the characteristics of an individual patient (for example, using a genotypic assay of a sequence of DNA). Decision-support systems based on patient-specific simulation can potentially revolutionize the way that clinicians plan courses of treatment for various conditions, ranging from viral infections to arterial abnormalities. Basing medical decisions on the results of simulations that use models derived from data specific to the patient in question means that the effectiveness of a range of potential treatments can be assessed before they are actually administered, preventing the patient from experiencing unnecessary or ineffective treatments. We illustrate the potential for patient-specific simulation by first discussing the scale of the evolving international grid infrastructure that is now available to underpin such applications. We then consider two case studies, one concerned with the treatment of patients with HIV/AIDS and the other addressing neuropathologies associated with the intracranial vasculature. Such patient-specific medical simulations require access to both appropriate patient data and the computational resources on which to perform potentially very large simulations. Computational infrastructure providers need to furnish access to a wide range of different types of resource, typically made available through heterogeneous computational grids, and to institute policies that facilitate the performance of patient-specific simulations on those resources. To support these kinds of simulations, where life and death decisions are being made, computational resource providers must give urgent priority to such jobs, for example by allowing them to pre-empt the queue on a machine and run straight away. We describe systems that enable such priority computing.