First-pass perfusion cardiac MR (FP-CMR) is becoming essential for evaluating myocardial ischemia. However, FP-CMR requires ECG-gating and breath-holding, leading to a trade-off between spatial resolution and coverage. Moreover, perfusion abnormalities are often identified visually by highly trained operators. Recently, quantitative FP-CMR and compressed sensing (CS) have been proposed to reduce operator-dependency and moderately accelerate acquisitions, respectively. Here, a model-based reconstruction is proposed to directly estimate quantitative myocardial perfusion maps from highly undersampled acquisitions. Thus, allowing for higher spatial resolution and coverage than indirect methods, where dynamic images are reconstructed using CS and quantitative maps are obtained subsequently using tracer-kinetic modeling.