We introduce a novel, generalized tracer kinetic model selection framework to quantify microvascular characteristics of liver and tumor tissue in gadoxetate-enhanced dynamic contrast-enhanced MRI (DCE-MRI). Our framework includes a hierarchy of nested models, from which physiological parameters are derived in 2 regimes, corresponding to the active transport and free diffusion of gadoxetate. We use simulations to show the sensitivity of model selection and parameter estimation to temporal resolution, time-series duration, and noise. We apply the framework in 8 healthy volunteers (time-series duration up to 24 minutes) and 10 patients with hepatocellular carcinoma (6 minutes). The active transport regime is preferred in 98.6% of voxels in volunteers, 82.1% of patients' non-tumorous liver, and 32.2% of tumor voxels. Interpatient variations correspond to known co-morbidities. Simulations suggest both datasets have sufficient temporal resolution and signal-to-noise ratio, while patient data would be improved by using a time-series duration of at least 12 minutes. In patient data, gadoxetate exhibits different kinetics: (a) between liver and tumor regions and (b) within regions due to liver disease and/or tumor heterogeneity. Our generalized framework selects a physiological interpretation at each voxel, without preselecting a model for each region or duplicating time-consuming optimizations for models with identical functional forms.