T 2 Relaxation

3.5.2.1

Intuitive Description of T ₂ Relaxation

The complete MR signal from spinal cord tissue contains contributions from protons in water and from protons in nonaqueous tissue. It has been demonstrated that the signal from water in central nervous system tissue has T ₂ relaxation times greater than 10 ms and that the signal from nonaqueous tissue decays to zero in one ms or less. MRI scanners can readily assess signals with T ₂ times longer than 10 ms; therefore, it is possible to accurately characterize T ₂ relaxation of the water in the spinal cord by measuring the T ₂ decay curve.

T ₂ relaxation in the spinal cord is complicated by the fact that within a typical imaging voxel volume of a few cubic millimeter, structure at a cellular level is inhomogenous. For example, glial cells have single plasma membranes while myelinated neurons contain many membranes in close proximity. This microscopic heterogeneity results in different relaxation behaviors for water protons in different environments within a single voxel. From perusal of high resolution cross-sectional electron micrographs of CNS tissue ( Figure 3.5.1 ), one can visualize three distinct water reservoirs: intracellular water, extracellular water, and water trapped between the bilayers of the myelin sheaths. Because water interactions with nonaqueous tissue are most pronounced in myelin due to the tightly confined space between myelin bilayers, myelin water in spinal cord has a relatively short T ₂ time of 10–20 ms. Intra- and extracellular water, which is not in as close association with nonaqueous tissue as myelin water, have a longer T ₂ . Although intra- and extracellular spaces are physically different, they have similar T ₂ s of around 70–90 ms at 1.5 T and cannot be measured separately. Therefore, T ₂ relaxation measurements in spinal cord yield two T ₂ components, one at 10–15 ms and one at approximately 80 ms. The fraction of MR signal in the myelin water component is known as the myelin water fraction (MWF). This fraction, as expected, is substantially larger in white matter than gray matter. In previous studies on fixed brain samples, MWF was found to correlate strongly with histological stains for myelin lipids. The MWF in white matter of the spinal cord is approximately twice as large as MWF values in brain, which may reflect physiological variations. The term myelin water imaging has been used to describe quantitative T ₂ relaxation studies focused on the myelin water component.

In addition to the presence of multiple water environments, an even further complication in the spinal cord is that boundaries between water environments are blurred due to translational diffusion. Water molecules in pure water or dilute solutions undergo random motion, moving an average distance of R = (6 Dτ ) ^1/2 in time τ , where D is the water diffusion coefficient. For a typical MR image obtained with an echo time (TE) = 60 ms, free water in a solution moves over 30 microns. The barriers present in the spinal cord reduce water diffusion coefficients on average to about 1/4 that in free water, and water molecules move on average about 20 microns in 60 ms. Due to diffusion, parameters derived by MR techniques are averaged over their values during the timescale of the measurement. See Chapter 3.1 for more detailed information of diffusion and diffusion anisotropy in the cord.

3.5.2.2

Mechanistic Description of T ₂ Relaxation

For a simple spin system consisting of a molecule with a single proton site, T ₂ is understood quantitatively in terms of fluctuating magnetic fields produced by adjacent protons undergoing molecular motions. When protons are located on molecules that reorient rapidly due to molecular tumbling and translational diffusion, they produce oscillating magnetic fields. Molecular motions, which are driven by the inherent kinetic energy of the molecules, are characterized by a correlation time, τ _c , which is the time required for the molecule to undergo a substantial reorientation. The fluctuating magnetic fields caused by these molecular motions cause T ₂ relaxation. The dependence of T ₂ on the correlation time of molecular motions and the Larmor frequency is quantitatively expressed in Eqn (3.5.1) . The constant, K , is related to the strength of the interactions between adjacent protons.

Equation (3.5.1) shows that T ₂ is sensitive to rapid molecular motions at the Larmor frequency, ω _o , and twice the Larmor frequency. T ₂ is also influenced by low frequency motions that cause the term 3/2 τ _c to become large, thereby explaining why T ₂ is particularly sensitive to slow motions. It turns out that for very fast motions, like free water tumbling, the 3/2 τ _c term is negligible and T ₂ = T ₁ , which holds for relaxation in cerebrospinal fluid (CSF) (where T ₁ is defined as $1/T_1=K\left({\tau }_{\text{c}}/\left(1+{{\omega }_{\text{o}}}^2{{\tau }_{\text{c}}}^2\right)+4{\tau }_{\text{c}}/\left(1+4{{\omega }_{\text{o}}}^2{{\tau }_{\text{c}}}^2\right)\right)$ ). In the presence of slow fluctuations below the Larmor frequency (64 MHz at 1.5 T), T ₂ is shorter than T ₁ .

The model for relaxation described herein holds for simple spin systems such as water solutions but is unable to quantitatively characterize relaxation in the spinal cord, which contains multiple proton environments. Relaxation times for water in CNS tissue are strongly influenced by interactions between water and nonaqueous protons attached to lipids, proteins, and nucleic acids. In the spinal cord, water protons interact with other protons attached to molecules moving at a wide range of different frequencies (e.g., low frequencies arising from very slow macromolecular reorientations to high frequencies arising from fast small molecule tumbling). As a consequence, T ₂ times are shorter than T ₁ times.

3.5.2.3

Measurement of T ₂ Relaxation

The first step in a quantitative T ₂ study is the acquisition of high-fidelity T ₂ decay curves. The T ₂ decay curve is a plot of MR signal versus TE (time to echo). The Carr–Purcell–Meiboom–Gill sequence, which consists of a 90° excitation pulse followed by a series of equally spaced 180° refocusing pulses, is the most common technique for measuring T ₂ times in vivo, typically with 32 echoes from TE of 10 ms to TE of 320 ms. Estimated T ₂ times may depend upon the echo spacing of the multi-echo sequence. The echo spacing dependence of T ₂ arises due to events that occur during the interval between the 180° refocusing pulses, for example, diffusion of water between regions of different magnetic susceptibility. Producing accurate T ₂ decay curves in vivo is challenging because it requires accurate refocusing pulses in the presence of magnetic field and radio-frequency field inhomogeneities. An effective solution for single slice approaches (discussed following) is to apply composite rectangular radio-frequency refocusing pulses flanked by large gradient pulses that alternate in sign and decrease in height with echo time. The composite radio-frequency pulses provide robust 180° flip angles and the gradient pulses are required to eliminate contribution to the signal from spins located outside the selected slice. An example of a T ₂ decay curve from spinal cord is shown in Figure 3.5.2 .

Initial in vivo implementations of T ₂ measurements in both brain and spinal cord made use of the previously described rectangular refocusing pulses combined with large-gradient crusher lobes. This approach was sufficient for proof of concept and for addressing focused research questions; however, because it required over 20 min to produce a single slice, it had limited clinical applicability. This 2D multi-echo single slice pulse sequence cannot simply be extended to a multislice acquisition because the various microscopic environments in CNS tissue are affected differently by magnetization transfer resulting from the off-resonance slice selective refocusing pulses from neighboring slices. Several different approaches have since been employed to make myelin water imaging available in a clinically relevant imaging time. These include: a 3D-multiple spin echo sequence, a 3D gradient echo sequence, and a TE-prepared spiral sequence.

An additional technique that has been proposed to assess myelin water uses steady state sequences. The steady state approach for extracting T ₂ information from spinal cord is conceptually very different from the multiple spin echo approach. Multicomponent-driven equilibrium single pulse observation of T ₁ and T ₂ (mcDESPOT) makes use of three sequences: spoiled gradient echo, inversion-prepared spoiled gradient echo, and balanced steady state free precession. The strength of the mcDESPOT approach is the relatively faster and higher resolution data acquisition; each sequence is run at several different flip angles, and the entire protocol takes about 26 min for cervical cord coverage with 1 × 1 × 1.5 mm ³ isotropic voxels. However, a disadvantage of the mcDESPOT approach is the complex and somewhat constrained analysis, discussed in the following section.

Most myelin water imaging to date has been carried out in brain, and application to the spinal cord presents significant challenges. For example, the well-documented magnetic field inhomogeneities in spinal cord (see Chapter 2.1 ) are expected to make it very difficult, if not impossible, to use T ₂ ^∗ -based approaches. To date, only multiple spin echo and steady state sequences have been reported for measuring myelin water in spinal cord.

3.5.2.4

Analysis of T ₂ Relaxation

For single-component T ₂ relaxation, the T ₂ decay curve follows:

However, as mentioned before, the water environment in the spinal cord is inhomogenous. The T ₂ decay curve contains contributions from myelin water and also intra- and extracellular water; therefore, the T ₂ analysis must deal with multiple T ₂ components. Equation (3.5.2) then becomes:

Inversion of Eqn (3.5.3) for estimation of component amplitudes and relaxation times, S _i and T _2i , is an ill-posed problem, which means that there is not a unique solution. Quite different combinations of S _i s and T _2i s can fit the data equally well; this is a consequence of the fact that exponential functions, unlike sine waves, do not form an orthogonal basis set. The most conservative approach (and the most common approach in the literature) for solving Eqn (3.5.3) is to minimize the number of a priori assumptions about the solution. The non-negative least squares (NNLS) approach assumes only that (1) the S _i s are positive and (2) Eqn (3.5.3) fits the data. It makes no assumptions about the number of exponential components; this is a fundamental requirement for robust decay curve fitting, since the number of contributing components in the spinal cord is unknown a priori, especially in the presence of pathology. The NNLS algorithm uses χ ² minimization to fit the relaxation decay curve to a large number of T ₂ components. NNLS produces a T ₂ distribution consisting of a few discrete spikes; however, most investigators prefer a smooth distribution. A continuous distribution can be achieved by minimizing χ ² as well as a regularizer. A common regularizer is the sum of the squares of the solution amplitudes—the so-called “small model”. An example of a T ₂ distribution from spinal cord tissue can be seen in Figure 3.5.2 .

The myelin water fraction is defined as the area under the myelin water peak of the T ₂ distribution divided by the total area under the T ₂ distribution. At 1.5 T, MR signal, S ( T ₂ ), with T ₂ between 10 and 40 ms, is assigned to myelin water. At 3 T, this region has been contracted to 10–35 ms to minimize contributions from intra- and extracellular water. The geometric mean T ₂ (GMT ₂ ) time for each peak is estimated by evaluating the following equation:

where S ( T ₂ ) is the T ₂ distribution and the integral limits are chosen to cover the T ₂ peak of interest. For the intra/extracellular water peak, the integral is usually acquired from 40 to 200 ms. The GMT ₂ is the mean T ₂ on a logarithmic scale. An interesting property of the GMT ₂ is that it is the reciprocal of the GM(1/ T ₂ ).

An alternative approach to dealing with multicomponent relaxation curves is to use nonlinear curve-fitting techniques. In many applications, researchers have simultaneously fit T ₁ and T ₂ relaxation times. The nonlinear fitting approach requires more a priori information; however, if the a priori information is accurate, this approach may result in more robust solutions.

At magnetic fields larger than 1.5 T, it is well known that, due to dielectric effects, the B ₁ field in tissue is not homogenous. As a result, the refocusing pulses in the multi-echo train are not necessarily 180°, and the decay curve no longer follows Eqn (3.5.3) . In particular, the early echoes have amplitudes that exhibit an odd/even oscillation. Fortunately, it is possible to understand these oscillations quantitatively in terms of stimulated echoes, which occur after magnetization is temporarily placed along the z axis by suboptimal 180° pulses. In fact, it is possible to extract not only the T ₂ distribution but also the B ₁ distribution by making use of the extended phase graph algorithm.

The analysis of mcDESPOT data, instead, involves fitting the data with a constrained model with, typically, two water components. The fitting process is complex, involving at least six parameters, i.e., T ₁ , T ₂ , relative amplitudes, and residence times for each component. This fitting process, which may employ a genetic algorithm or stochastic regional contraction, does converge, yielding myelin water maps as well as relaxation time maps and B ₀ and B ₁ maps. For reasons that are not yet understood, in brain mcDESPOT results differ substantially from multiple spin echo values, but in the spinal cord the two approaches yield similar findings. However, the mcDESPOT analysis method and results derived from it must be interpreted with caution, as discussed in a very recent paper by Lankford and Does. The authors produced theoretical calculations of the Cramer-Rao lower bounds of the variance of fitted model using a variety of model system parameters, meant to mimic those expected in human white matter. The results indicated that mcDESPOT signals acquired at feasibly attainable signal-to-noise ratios cannot provide parameter estimates with useful levels of precision. While precision can be greatly improved by constraining solutions with a priori information, this could lead to biased parameter estimates. The authors concluded that mcDESPOT-derived estimates of two-pool model parameters cannot yet be unambiguously related to specific tissue characteristics.

The use of T ₂ relaxation to learn about the spinal cord is an emerging field; to our knowledge only 12 papers have reported quantitative measurements of T ₂ in the spinal cord. The following sections summarize preclinical and human T ₂ relaxation studies to date.

3.5.3.1

Initial Observation of Myelin Water Signal in the Spinal Cord

The first spinal cord T ₂ relaxation study that observed a myelin water signal was conducted in the early 1990s by Stewart et al. ( Table 3.5.1 ). At the time of this work, the usefulness of T ₂ relaxation to characterize pathology was under debate, but the authors postulated that with not only careful and systematic measurement, but also appropriate analysis, MR relaxation times could indeed yield important information about tissue and tissue pathology. In particular, the goal of their work was to demonstrate that quantitative MR can be used to distinguish the different types of pathology seen in multiple sclerosis (MS). They used a 4320 echo CPMG pulse sequence to measure proton T ₂ values in the spinal cord from Hartley guinea pigs inoculated to produce experimental allergic encephalomyelitis (EAE), the animal model of MS. In order to collect a very large number of points for the relaxation decay curve and negate the possible effects of localization techniques, experiments were conducted ex vivo on a modified Bruker SXP 4-100 NMR spectrometer operating at 2.1 T (90 MHz for protons). Up until that point, the majority of previous studies had restricted the tissue proton relaxation decay curves to be composed of one, two, or three discrete exponential components. However, as recent studies analyzed relaxation decay curves in terms of continuous distributions of relaxation times, the authors suggested that since this approach makes fewer a priori assumptions, it may be more appropriate when trying to characterize the complex nature of tissues.

Stewart et al. analyzed their data using both a nonlinear functional minimization program (Minuit) where decay curves were fitted by the sum of a specified number of exponential components, the optimum number of which was chosen to be the minimum number that adequately described the data, and using a non-negative least squares algorithm (NNLS) whereby no a priori assumptions were introduced about the number of contributing components. Using Minuit, three exponentials provided the best fit to spinal cord data (10 ms (13%), 76 ms (57%), 215 ms (30%)). However, not only did discrete NNLS analyses give lower χ ² (goodness of fit) values than the nonlinear Minuit discrete treatment, but two spikes were often observed on either side of the T ₂ obtained using Minuit, suggesting a distribution of T ₂ amplitudes rather than a discrete value. Smooth NNLS solutions consisted of two broad peaks, a small peak with a T ₂ near 10 ms and a larger peak near 100 ms. The smooth solutions were examined for features that could possibly distinguish between the normal and abnormal tissue states. The average relaxation time was calculated in a variety of ways including the median, mode, and weighted arithmetic, geometric, and harmonic means. Of these, the weighted geometric mean seemed to be the best indicator of pathology. Stewart et al. determined the marker with the best diagnostic potential was the “integral ratio” of the larger secondary peak size to the smaller fast-relaxing component.

Stewart et al. concluded that the smooth NNLS results support a model with two distinct reservoirs of water in CNS tissue: one with a T ₂ near 10–20 ms with approximately 17% of the total water, and the other a broad peak with T ₂ from about 50 to 300 ms, which included all the other water. The assignment of the T ₂ peak near 10 ms to myelin water was the first of its kind for spinal cord tissue. At the time, findings were in agreement with two other ex vivo studies that examined human and cat brain. Furthermore, the amplitude of the myelin water signal decreased with the extent of demyelination, as evidenced by histology, and the absence of change in the short T ₂ component after tissue homogenization was consistent with evidence from the literature that homogenization leaves the myelin layers intact. The usefulness of quantitative T ₂ assessment was however not deemed to be limited to the study of the myelin water signal, as Stewart et al. stated that the width of the longer T ₂ peak associated with the bulk of the water in CNS tissue is a qualitative measure of inhomogeneity in cell size, cytoplasmic content, and extracellular space in the tissue.

3.5.3.2