A sculptor looks at a block of marble and sees the figure within, waiting to be freed from excess material. Instead of adding clay, he removes everything that is not part of the form. The authors of a new study on magnetic resonance took a similar approach: rather than complicating the tissue model by adding ever more details, they stripped away the superfluous to reveal the essence.
Instead of describing each magnet individually, the authors propose working with the averaged signal from the entire cube. They introduce a general nonlinear field that governs the return of magnetization to thermal equilibrium and require four things from it: consistency with rotations around the longitudinal axis, uniqueness and stability of the equilibrium point, and strict dissipativity, meaning monotonic decrease of a measure of deviation from equilibrium. These requirements are physically natural, yet they suffice to fully describe all admissible relaxation fields.
The authors also provide a constructive method: by choosing three functions, two positive and one that sets the rotation, one can build an infinite family of admissible nonlinear models. The classical Bloch model emerges as a special case when the additional functions are zero and the coefficients are constant, equal to the inverse relaxation times. New examples demonstrate non-exponential recovery of longitudinal magnetization, which is observed in real tissues, and coupled relaxation, where the rate of transverse decay depends on the longitudinal deviation.
Previously, it was unclear how to generalize the Bloch equation to heterogeneous tissues without making the model unmanageably complex. Now a complete class of admissible nonlinear fields that preserve the key physical properties of the classical model has been obtained. The secret is that the authors formulated four physical postulates and mathematically derived all vector fields satisfying them. The practical benefit is still theoretical: it provides a foundation for building new models of MRI signals that could improve quantitative measurements in tissues. Like a sculptor who removes excess to reveal the figure, restricting the class of admissible dynamics by physical principles narrows the search space and guides future experiments.
The classical Bloch equation was proposed by Felix Bloch in 1946 as a phenomenological description, without derivation from microscopic physics. The new theory follows the same spirit: it postulates physical principles and derives admissible forms of dynamics, rather than trying to model every spin.