Compartment shape anisotropy (CSA) revealed by double pulsed field gradient MR

Abstract

The multiple scattering extensions of the pulsed field gradient (PFG) experiments can be used to characterize restriction-induced anisotropy at different length scales. In double-PFG acquisitions that involve two pairs of diffusion gradient pulses, the dependence of the MR signal attenuation on the angle between the two gradients is a signature of restriction that can be observed even at low gradient strengths. In this article, a comprehensive theoretical treatment of the double-PFG observation of restricted diffusion is presented. In the first part of the article, the problem is treated for arbitrarily shaped pores under idealized experimental conditions, comprising infinitesimally narrow gradient pulses with long separation times and long or vanishing mixing times. New insights are obtained when the treatment is applied to simple pore shapes of spheres, ellipsoids, and capped cylinders. The capped cylinder geometry is considered in the second part of the article where the solution for a double-PFG experiment with arbitrary experimental parameters is introduced. Although compartment shape anisotropy (CSA) is emphasized here, the findings of this article can be used in gleaning the volume, eccentricity, and orientation distribution function associated with ensembles of anisotropic compartments using double-PFG acquisitions with arbitrary experimental parameters.

Introduction

Heterogeneous specimens contain several different species of molecules, which may be in different phases. In one scenario, a population of enclaves within a solid matrix may be filled with a liquid whose molecules possess nuclear magnetism, hence are MR observable. When the time scale of the MR experiment is sufficiently long for a significant portion of the molecules to probe the pore-grain interface, the effect of molecular diffusion on the MR signal can be quantified to procure information about the porous structure.

The incorporation of pulsed gradients into standard MR pulse sequences [1] has made it possible to enhance the effects of diffusion on the MR signal intensity in a controllable manner. In its most widely employed form, these pulsed field gradient (PFG) experiments comprise one pair of gradients to encode displacements that occur between the application of the two pulses. Such experiments will be referred to hereafter as single-PFG acquisitions.

Among other information, the anisotropy of the pores can be obtained as a result of the restricting character of the solid host domain. However, in random media, the orientation of the pores is randomly distributed, making the single-PFG MR signal invariant to changes in the direction of the gradients. Consequently, the measured anisotropy in single-PFG acquisitions emerges from the interplay between the anisotropy of the compartments (hereafter referred to as compartment shape anisotropy, CSA) and the coherence in the population of these compartments (hereafter referred to as ensemble anisotropy, EA). Therefore, single-PFG MR provides a feasible means to characterize any coherence in the orientation of restricting domains.

This observation has proven to be particularly significant for biological applications of MR. In fact, the orientational dependence (anisotropy) of the diffusion-weighted MR signal intensity has been exploited to generate exquisite contrast between white- and gray-matter regions of the brain [2], [3]. Further, when there is significant anisotropy in the MR signal, the voxel-averaged orientation of the cells can be computed, and anatomical connections between different regions of the brain can be mapped [4], [5]. As established by experimental investigations, anisotropy observed in neural tissue specimens via single-PFG acquisitions is primarily a product of the interaction between cellular membranes and diffusing water molecules, suggesting the restricting character of the cellular membranes [6], [7]. Thus, such anisotropy is observed when the cells have an elongated shape. Moreover, any incoherence in the orientation of a collection of cells leads to a decrease in the observed anisotropy [3].

Although single-PFG has been successful in characterizing the coherence (EA) of pores with CSA, the deduced anisotropy information is compromised due to the influence of one type of anisotropy on the other. Therefore, decoupling various mechanisms of anisotropy could be very useful in obtaining information about the shape of the cells and, independently, an orientation distribution function for the population of cells within the voxel.

A natural extension of single-PFG acquisitions could decouple the effects of different mechanisms of anisotropy. This extension is achieved by applying two pairs of diffusion gradients [8] separated from each other by a mixing time, $t_{\mathrm{m}}$, and will be referred to as double-PFG acquisitions. A spin echo version of such a pulse sequence is illustrated in Fig. 1. Variants of this double-PFG sequence have been considered for and employed in a host of different applications [9], [10], [11], [12], [13], [14], [15], [16].

An early theoretical investigation of the effects of restricted diffusion on the double-PFG signal suggested that even for isotropically distributed pores, the MR signal intensity may be dependent on the angle between the two gradients of the double-PFG sequence [17]. This early work predicted that angular dependence at long mixing times would be observed only when the compartments were non-spherical. Moreover, at short mixing times, even spherical pores exhibited such an angular dependence. This result points to yet another mechanism of anisotropy, called microscopic anisotropy (μA). This kind of anisotropy was predicted to influence the signal even at very low diffusion weightings [17], and suggested a means of obtaining a signature for restricted diffusion conveniently, because the presence of μA is tantamount to the existence of restrictions.

Fig. 2 illustrates when these three different mechanisms of anisotropy may be encountered. When the cells are spherical, only μA can be observed. A randomly oriented population of anisotropic cells will, in addition, exhibit CSA. Finally, if the anisotropic cells have any orientational preference in their alignment, all three mechanisms of diffusion anisotropy—μA, CSA, and EA—coexist.

Diffusion-induced anisotropy of the signal is not the only signature for restricted diffusion that double-PFG experiments provide. Similar to the case in single-PFG acquisitions [18], the double-PFG signal, when plotted against the gradient strength, does not decay monotonically—a phenomenon called diffusion–diffraction [19]. In addition, double-PFG signal at short mixing times was predicted to become negative at a gradient strength smaller than that necessary to observe the non-monotonicity in single-PFG MR acquisitions. Moreover, unlike the diffraction wells in single-PFG acquisitions, the zero-crossing of the double-PFG signal decay curves was shown to be robust to the heterogeneity of the specimen [19]. These predictions were recently validated experimentally [20].

In a recent study [21], a theoretical treatment for the detection of μA via double-PFG experiments was presented. There, the emphasis was placed on the quadratic term in a Taylor series expansion of the signal, which exhibited μA. The analysis did not make any assumption regarding the experimental timing parameters, and significant changes in the estimates of compartment size were predicted if double-PFG data was analyzed using the formulations in Ref. [17], which assumed limiting values for the experimental timing parameters. More recently, Shemesh et al. confirmed experimentally that accurate size estimates are feasible only when the experimental parameters were accounted for, as long as the diffusion time is not extraordinarily short [22]. In Ref. [21], the need to decouple the effects of EA and μA was recognized and a scheme to do so was introduced.

Following Mitra’s recognition for the prospect of resolving CSA in a randomly distributed ensemble of pores [17], Cheng and Cory performed experiments on elongated yeast cells, and reported successful delineation of cell eccentricity using double-PFG MR at long mixing times [23]. In their analysis, the cells were assumed to be ellipsoids of revolution, and cell size and eccentricity were estimated by using data obtained by setting the angle between the two gradients to 0° and 90°. The brief analytical treatment in Ref. [23] made the same assumptions as in Ref. [17] regarding the experimental timing parameters. It is surprising that despite its far reaching implications, the double-PFG experiments have not been studied extensively to assess the CSA.

In this article, we provide a comprehensive theoretical treatment of the problem of double-PFG experiments by extending the theory of restricted diffusion to account for CSA as well. In the next section, we set the experimental timing parameters to their limiting values. The derivations are based on arbitrary pore shapes, although two symmetry conditions were imposed in obtaining a Taylor series representation of the signal intensity for simplicity. Explicit relations up to the fourth-order term in a Taylor series expansion of the signal are provided. Then more specific pore geometries consisting of spherical, ellipsoidal, and cylindrical pores are considered, for which solutions can be obtained for arbitrary gradient strengths. In the subsequent section, we derive a general solution of the double-PFG acquisition that takes all experimental conditions of the acquisition into account. These formulations are based on a recent generalization [24] of the multiple correlation function method [25], [26].