  Compressed sensing takes advantage of low-dimensional signal structure to
reduce sampling requirements far below the Nyquist rate. In magnetic resonance
imaging (MRI), this often takes the form of sparsity through wavelet transform,
finite differences, and low rank extensions. Though powerful, these image
priors are phenomenological in nature and do not account for the mechanism
behind the image formation. On the other hand, MRI signal dynamics are governed
by physical laws, which can be explicitly modeled and used as priors for
reconstruction. {1}These explicit and implicit signal priors can be
synergistically combined in an inverse problem framework to recover sharp,
multi-contrast images from highly accelerated scans. Furthermore, the
physics-based constraints provide a recipe for recovering quantitative,
bio-physical parameters from the data. This article introduces physics-based
modeling constraints in MRI and shows how they can be used in conjunction with
compressed sensing for image reconstruction and quantitative imaging. We
describe model-based quantitative MRI, as well as its linear subspace
approximation. We also discuss approaches to selecting user-controllable scan
parameters given knowledge of the physical model. We present several MRI
applications that take advantage of this framework for the purpose of
multi-contrast imaging and quantitative mapping.
