Gaussian approximation in the theory of MR signal formation in the presence of structure-specific magnetic field inhomogeneities. Effects of impermeable susceptibility inclusions

Abstract

A detailed theoretical description of the signal formation in the presence of mesoscopic structure-specific magnetic field inhomogeneities is presented in the framework of the Gaussian phase distribution approximation for two geometrical models of the field inhomogeneity sources—impermeable spheres and infinitely long cylinders. Analytical expressions for free induction decay (FID) and spin echo (SE) signal attenuation functions Γ( t)∼−ln S( t) are obtained and comparison with the case of unrestricted diffusion (susceptibility inclusions with freely permeable surfaces) is provided. For short times, the leading term in the FID signal attenuation function is proportional to t 2 similar to the case of unrestricted diffusion; the next term behaves as t 3 as compared to t 5/2 for the “permeable” case. For the SE signal, the leading term is proportional to t 3 as compared to t 5/2 for unrestricted diffusion. It is shown that the t 3 approximation can be used for an adequate description of the SE signal only for extremely short times compared to a characteristic diffusion time. In the long-time limit, the attenuation function in the impermeable and permeable sphere model contains not only terms linear in time, but also important terms proportional to t 1/2. In the cylindrical geometry, the leading term in the long-time expansion of the attenuation function is proportional to tln t for both the permeable and impermeable models. Application to description of MR in biological tissues signal in the presence of blood vessel networks and contrast agents is discussed. The validity criterion of the Gaussian approximation is also proposed.