This paper presents a data-driven approach to generating virtual patients using mathematical models of physiological processes. Such models often contain a large number of tunable parameters that must be calibrated to capture the observed characteristics of each real patient in a dataset. By sampling from this parameter space, potentially new virtual patients can be generated. However, it is often the case that the resulting set of virtual patients contains members that exhibit physiologically unrealistic behavior. In the present work, we employ a practically important case study on the modeling of cardiovascular responses to hemorrhage and fluid resuscitation in order to demonstrate that subject-specific characteristics observed in a dataset can be alternatively represented within a highly compressed latent parameter space without significant losses in calibration error for each real patient. Then, we show that by sampling from this latent parameter space, it is possible to generate new virtual patients that also exhibit physiologically realistic behavior.