This paper presents the methodology used in patient-specific calibration of a novel highly integrated model of the cardiovascular and pulmonary pathophysiology associated with Acute Respiratory Distress Syndrome (ARDS). We focus on data from previously published clinical trials on the static and dynamic cardio-pulmonary responses of three ARDS patients to changes in ventilator settings. From this data, the parameters of the integrated model were identified using an optimization-based methodology in multiple stages. Computational simulations confirm that the resulting model outputs accurately reproduce the available clinical data. Our results open up the possibility of creating in silico a biobank of virtual ARDS patients that could be used to evaluate current, and investigate novel, therapeutic strategies.