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Patient-specific mathematical models of respiratory mechanics can offer substantial insight into patient state and pulmonary dynamics that are not directly measurable. Thus, they offer significant potential to evaluate and guide patient-specific lung protective ventilator strategies for acute respiratory distress syndrome (ARDS) patients. To assure bedside applicability, the model must be computationally efficient and identifiable from the limited available data, while also capturing dominant dynamics and trends observed in ARDS patients. In this study, an existing static recruitment model is enhanced by considering alveolar distension and implemented in a novel time-continuous dynamic respiratory mechanics model. The model was tested for structural identifiability and a hierarchical gradient descent approach was used to fit the model to low-flow test responses of 12 ARDS patients. Finally, a comprehensive practical identifiability analysis was performed to evaluate the impact of data quality on the model parameters. Identified parameter values were physiologically plausible and very accurately reproduced the measured pressure responses. Structural identifiability of the model was proven, but practical identifiability analysis of the results showed a lack of convexity on the error surface indicating that successful parameter identification is currently not assured in all test sets. Overall, the model presented is physiologically and clinically relevant, captures ARDS dynamics, and uses clinically descriptive parameters. The patient-specific models show the ability to capture pulmonary dynamics directly relevant to patient condition and clinical guidance. These characteristics currently cannot be directly measured or established without such a validated model.