L.    G u y     R A G U I N ,   P h . D.

A quantitative analysis of q-space MRI data (QUAQ) is presented to extract the physical parameters of diffusion in fiber networks and resolve fiber crossings, e.g. for brain white matter fiber tracking. This method is based on the analytical expression for the normalized echo attenuation E(q) in cylindrical pores measured by pulsed-gradient MRI in the short pulse approximation, which reflects the restricted diffusion process inside the fibers. An alternative method (SQUAQ) consists in approximating E(q) by a sum of Gaussian distributions, which corresponds to the assumption that the diffusion process in the fibers reduces to unrestricted anisotropic diffusion. Additionally we require that diffusion in the plane perpendicular to the fiber be isotropic. These two methods are validated and compared both numerically and experimentally, and the influence of the distribution of the input measurements in q-space (number of gradient strengths and orientations) on the results is investigated. This is a collaborative effort that includes Dimitrios Karampinos (Ph.D. candidate, Department of Mechanical Engineering, advisor: John Georgiadis), Diego Hernando (Ph.D. candidate, Department of Electrical and Computer Engineering, advisor: Zhi-Pei Liang) and Luisa Ciobanu (Ph.D., Biomedical Imaging Center, Beckman Institute for Advanced Science and Technology). An conference paper was accepted at the 3rd European Medical & Biological Engineering Conference, IFMBE European Conference on Biomedical Engineering (EMBEC'05) that took place in Prague in November 2005, and an abstract has been accepted for a poster at ISMRM 2006, 14th Scientific Meeting in Seattle, Washington, USA, May 6–12, 2006.

Introduction

In the last decade, diffusion-weighted magnetic resonance imaging (DW-MRI) has been used to track fibers in soft biological tissues such as brain white matter non-invasively [1]. DW-MRI produces maps of the echo attenuation that depend on the local diffusion process and are indexed by a value in q-space corresponding to the experimental parameters [2]. Different modalities exist to extract valuable information from DW-MRI data: diffusion tensor imaging (DTI) reconstructs the local diffusion tensor [1]; diffusion spectrum imaging (DSI), a.k.a. q-space imaging (QSI), extracts the local distribution of the spin displacements, a.k.a. the 3D average propagator, based on the (extensive) sampling of the echo attenuation in q-space [3]. The fiber orientation is obtained from the preferred diffusion and thus larger displacements along fibers. Lin et al. [3] showed that DTI cannot resolve crossing fibers, while DSI is a promising albeit time-consuming method. More complex methods have also been developed recently to elucidate fiber crossings: composite hindered and restricted model of diffusion (CHARMED [4]), q-ball imaging (QBI [5]), and high angular resolution diffusion imaging (HARDI [6]). In the latter, the signal is acquired with high angular resolution in q-space and then decomposed as a discrete mixture of Gaussian diffusion processes in slow exchange. This amounts to a multi-modal version of DTI. In this work, we present a quantitative analysis of q-space MRI data (QUAQ) that merges q-space MRI with the physics of the diffusion process inside a network of cylindrical fibers, and quantitatively estimate the characteristics of the system (diffusion constants and fiber orientations). In addition, we compare the physically correct QUAQ algorithm to a simplified version akin to DTI that we label SQUAQ, both numerically and experimentally using a synthetic fiber-crossing diffusion phantom.

Methodology

A simple but qualitative model would be to assume that the average propagator for a single fiber is described by a Gaussian distribution. This means that diffusion in a single channel can be interpreted as an unrestricted anisotropic diffusion process. Then, the signal expression for a single fiber population is also a Gaussian distribution as a function of q. This is the assumption on which DTI is based, and the experimental data are used to obtain the "apparent" diffusion tensor. By assuming that diffusion in the transverse plane is isotropic, the diffusion tenosr is characterized by only four parameters: two angles defining the fiber orientation, and two diffusion coefficients, longitudinal and trasnverse to the fiber. When multiple fiber populations are assumed to be present in the voxel of interest under the previous assumptions, the normalized echo attenuation becomes a multi-Gaussian decomposition of the MR signal. SQUAQ then consists in fitting the normalized echo attenuation measured experimentally to this multi-Gaussian decomposition. A Levenberg-Marquardt algorithm is used to solve the nonlinear least-squares minimization problem of fitting the echo attenuation data to the analytical formula.

It is known however that the propagator for a single fiber cannot in general be approximated by a Gaussian distribution, a more sophisticated description of the diffusion process is necessary. Based on the analytical results for cylindrical fibers by Callaghan [7] which assumes short pulsed gradients (duration of gradients, delta, is much less than the diffusion time, Delta), an analytical formula is derived for the normalized echo attenuation for a fluid inside M cylindrical fiber populations of radius a_m (m in [1,M]). To account for the layered structure of the brain white matter fibers, the diffusion process within the fiber is assumed to be anisotropic with parallel diffusivity D_|| and transverse diffusivity D_perp. E(q,Delta) given by Eq. (8) from [8] depends on the experimental parameters (delta, Delta, diffusion gradient g) and the physical parameters of the problem (a_m, D_||, D_perp, theta_m, psi_m, f_m) where theta_m and psi_m are the two angles that define the orientation of the mth fiber in 3-D, and f_m its signal-weighted volume fraction. As for SQUAQ, a Levenberg-Marquardt algorithm is used to solve the nonlinear least-squares minimization problem. The fitting parameters are (D_||, D_perp, f_m, theta_m, psi_m), while the fiber sizes a_m are assumed to be known (e.g. via histology). The infinite series in Eq. (8) are truncated, which is allowed since the exponential terms in the series vanish as n and k increase. Note that these terms vanish faster if the diffusion time Delta is large w.r.t. a_m^2/D_perp. The level of truncation can be estimated from the assumed a_m value, an estimated D_perp value and the known value of Delta. This fitting procedure constitutes the QUAQ algorithm to reconstruct the fibers pixel-per-pixel based on q-space MRI data.

Experimental Results A 2 cm x 2 cm x 1 cm phantom was built using two sets of perpendicular crossing PTFE fiber populations (Cole-Parmer Instrument Company, Vernon Hills, IL, USA) with a_1 = a_2 = 50 microns, filled with water (D_|| = D_perp = 1.6 x 10^{-9} m^2/s at 11.5ºC and placed in an Varian 14.1 Tesla NMR imager. Copper sulphate was added to the water to decrease the relaxation time T_1, and the solution was degassed to prevent air bubbles inside the fibers. A high-resolution SE image of the phantom is shown in Figure 1(a). The fiber bundles consist of two fibers wrapped around a Plexiglas substrate shaped appropriately, hence the thin fibers on the left side of the cross. The volume fractions of the two crossing fiber bundles at their intersection are v_1 = v_2 = 0.5. A 2-D pulsed-gradient stimulated-echo sequence was used to obtain diffusion-weighted images of the phantom with the following experimental parameters: FOV = 2.5 cm x 2.5 cm, slice thickness = 1 cm, 16 x 16 Fourier points, TR/TE = 1500/14 ms, delta = 5 ms, Delta = 250 ms, eight diffusion gradient strengths g = 0, 1, 2, 3, 3.5, 4, 4.5, and 5 G/cm (i.e. Ng = 7 non-zero g values), Nth = 30 orientations according to the minimum energy spatial distribution obtained in [9]. A coarse resolution is used to emulate the temporal and spatial constraints for clinical imaging, c.f. Figure 1(b). Thus, each pixel contains a bundle of fibers and the crossing of the fibers is not resolved. The QUAQ and SQUAQ reconstructions require 4 fitting parameters for single fibers (D_||, D_perp, theta_1, psi_1), and 7 for two crossing fibers (f_1, D_||, D_perp, theta_1, psi_1, theta_2, psi_2). For QUAQ, the series in Eq. (8) are truncated at n = 3 and k = 6. Figure 1. MR images of the experimental phantom (FOV = 2.5 cm x 2.5 cm): (a) high-resolution SE image (256 x 256 resolution, TR/TE = 1000/10 ms); (b) low-resolution stimulated-echo image (16 x 16 resolution, TR/TE = 1500/14 ms).

(a) Spin-echo image. (b) Stimulated-echo image.

(a) QUAQ with 210 q values.	(c) QUAQ with 30 q values.
(b) SQUAQ with 210 q values.	(d) SQUAQ with 30 q values.

References

Download EMBEC'05 paper

Application to a section of human pons

(d) Section of human pons: QUAQ fiber reconstruction using 90 q values (30 directions and 3 q values).

(e) Section of human pons: high resolution spin-echo image (in-plane pixel size: 100 x 100 microns).