This work targets investigators with an interest in quantitative susceptibility weighted imaging and diffusion weighted imaging with special emphasis on theoretical aspects of MR imaging and signal formation.

The susceptibility difference between air-filled alveoli and surrounding tissue causes microscopic magnetic field inhomogeneities that lead to a fast dephasing of the local magnetization. Yet, spin-bearing particles experience an averaged magnetic field, since they diffuse around the local field inhomogeneities. Thus, with increasing diffusion effects the dephasing process slows down. The free induction decay caused by diffusion and susceptibility effects is described by the Bloch-Torrey-equation. The first full analytic solution of this equation, solved on the surface of alveoli, is presented. It allows comparing theoretical predictions of the free induction decay with in vivo measurements in human lung tissue. Therefore, microscopic subvoxel information as e.g. the alveolar radius can be estimated. This will improve the quantitative diagnose of pulmonary emphysema or fibrosis.

A spherically shaped alveolus with radius $$$R$$$ in a static magnetic field $$$B_0$$$ is considered. Spin-bearing particles are localized on the surface of this alveolus [1]. The susceptibility difference $$$\Delta \chi$$$ between air-filled alveolus and surrounding tissue generates a local magnetic dipole field on the alveolar surface: $$$\omega(\theta) = \delta\omega [3\cos^2(\theta)-1]$$$, where $$$\theta$$$ is the angle to the static field $$$B_0$$$. The strength of the dipole field is given as $$$\delta\omega = \gamma B_0 \Delta\chi /3$$$, with the gyromagnetic ratio $$$\gamma$$$. Furthermore, the influence of diffusion on spin-bearing particles around the alveolus is characterized with diffusion coefficient $$$D$$$ and diffusion time $$$\tau = R^2/D$$$. The local transversal magnetization $$$m(\theta,t)= m_x(\theta,t)+i m_y(\theta,t)$$$ is described by the Bloch-Torrey-equation [2]: $$\frac{\partial}{\partial t} m(\theta,t) = [D\Delta – i\omega(\theta)] m(\theta,t).$$ The free induction decay $$$M(t)$$$ is obtained as a superposition of the local magnetization on the alveolar surface. Thus, solving the Bloch-Torrey-equation allows determining the free induction decay in dependence of microscopic tissue parameters.

The Bloch-Torrey-equation is solved by an eigenfunction expansion: $$m(\theta,t) = m_0 e^{-2i\delta\omega t}\sum\limits_{k=0}^{\infty} a_k PS_{2k,0}(\sqrt{3i\tau\delta\omega},\cos(\theta)) e^{- \lambda_{2k,0}(\sqrt{3i\tau\delta\omega})\frac{t}{\tau}},$$ where $$$PS_{2k,0}(\sqrt{3i\tau\delta\omega},\cos(\theta))$$$ and $$$\lambda_{2k,0}(\sqrt{3i\tau\delta\omega})$$$ denote spheroidal functions and corresponding eigenvalues, respectively, according to the notation of Meixner and Schäfke [3]. The eigenvalue spectrum is shown in Fig. 1 in dependence of the parameter $$$\tau\delta\omega$$$. The local magnetization depends on the angle $$$\theta$$$ and on time $$$t$$$ and is shown in Fig. 2. The free induction decay can be obtained as a superposition of the local magnetization over all positions on the spherical shell.

For experimental verification of the theoretical results the free induction decay in human lung tissue was measured with a PRESS-sequence in a cubic $$$(15mm)^3$$$ voxel. In Fig. 3 the experimental free induction decay is compared to the theoretical free induction decay assuming typical microscopic parameters. Both curves agree very well, especially for small times.

This work treats dephasing of the local magnetization under susceptibility and diffusion effects on a spherical shell. The local magnetization is governed by the Bloch-Torrey-equation, where the influence of susceptibility and diffusion effects is taken into account. As expected, the local magnetization exhibits the same symmetry as the local Larmor frequency. Due to this angular dependence of the local Larmor frequency, the eigenvalues are in general complex-valued. Therefore, the local magnetization as well as the free induction decay are dependent on both transversal components.

The free induction decay is monoexponential only for a large influence of diffusion (small values of $$$\tau\delta\omega$$$), since mainly the first summand with $$$k=0$$$ in the infinite sum contributes. With decreasing diffusion effects, more and more terms contribute and the free induction decay shows significant differences from a monoexponential decay.

The considered model is an appropiate description of human lung tissue in inspiration, since water molecues are mainly localized on the surface of the alveolus. Furthermore the averaged contribution of all neighboring alveoli vansihes in inspiration.

Since the free induction decay depends only on the parameter $$$\tau\delta\omega \propto \gamma B_0 \Delta\chi R^2/D$$$, the alveolar radius $$$R$$$ can be extracted from measured free induction decays [4]. Indeed, a calculated free induction decay and in vivo measurements in human lung tissue show an excellent agreement.

With the provided method it is possible to quantify the local mean alveolar size in an imaging voxel. This is highly relevant for the clinical application of lung MRI, since the acquisition of a free induction decay is a robust and fast imaging method.