Bloch-Torrey PDE in NMR and completely monotone functions.

In the first half of the talk I will review the theory of nuclear magnetic resonance (NMR), leading to the Bloch-Torrey PDE. I will then describe the pulsed-gradient spin-echo method for measuring the Fourier transform of the voxel-averaged propagator of the Bloch-Torrey equation.  This technique permits one to compute the diffusion coefficient in a voxel. For complex biological tissue, as in the brain, the standard model represents spin-echo as a multiexponential signal, whose exponents and coefficients describe the diffusion coefficients and volume fractions of isolated tissue compartments, respectively. The question of identifying these parameters from experimental measurements leads us to investigate the degree of well-posedness of this problem that I will discuss in the second half of the talk. We show that the parameter reconstruction problem exhibits power law transition to ill-posedness, and derive the explicit formula for the exponent by reformulating the problem in terms of the integral equation that can be solved explicitly. This is a joint work with my Ph.D. student Henry J. Brown.