In patients with dense breasts or at high risk of breast cancer, dynamic contrast enhanced MRI (DCE-MRI) is a highly sensitive diagnostic tool. However, its specificity is highly variable and sometimes low; quantitative measurements of contrast uptake parameters may improve specificity and mitigate this issue. To improve diagnostic accuracy, data need to be captured at high spatial and temporal resolution. While many methods exist to accelerate MRI temporal resolution, not all are optimized to capture breast DCE-MRI dynamics. We propose a novel, flexible, and powerful framework for the reconstruction of highly-undersampled DCE-MRI data: enhancement-constrained acceleration (ECA). Enhancement-constrained acceleration uses an assumption of smooth enhancement at small time-scale to estimate points of smooth enhancement curves in small time intervals at each voxel. This method is tested in silico with physiologically realistic virtual phantoms, simulating state-of-the-art ultrafast acquisitions at 3.5s temporal resolution reconstructed at 0.25s temporal resolution (demo code available here). Virtual phantoms were developed from real patient data and parametrized in continuous time with arterial input function (AIF) models and lesion enhancement functions. Enhancement-constrained acceleration was compared to standard ultrafast reconstruction in estimating the bolus arrival time and initial slope of enhancement from reconstructed images. We found that the ECA method reconstructed images at 0.25s temporal resolution with no significant loss in image fidelity, a 4x reduction in the error of bolus arrival time estimation in lesions (p < 0.01) and 11x error reduction in blood vessels (p < 0.01). Our results suggest that ECA is a powerful and versatile tool for breast DCE-MRI.