Mathematical models can be used to generate high-fidelity simulations of the cardiopulmonary system. Such models, when applied to real patients, can provide valuable insights into underlying physiological processes that are hard for clinicians to observe directly. In this work, we propose a novel modelling strategy capable of generating scenario-specific cardiopulmonary simulations to replicate the vital physiological signals clinicians use to determine the state of a patient. This model is composed of a tree-like pulmonary system that features a novel, non-linear alveoli opening strategy, based on the dynamics of balloon inflation, that interacts with the cardiovascular system via the thorax. A baseline simulation of the model is performed to measure the response of the system during spontaneous breathing which is subsequently compared to the same system under mechanical ventilation. To test the new lung opening mechanics and systematic recruitment of alveolar units, a positive end-expiratory pressure (PEEP) test is performed and its results are then compared to simulations of a deep spontaneous breath. The system displays a marked decrease in tidal volume as PEEP increases, replicating a sigmoidal curve relationship between volume and pressure. At high PEEP, cardiovascular function is shown to be visibly impaired, in contrast to the deep breath test where normal function is maintained.