Non-linear phase evolution and non-exponential magnitude decay are observed when examining the variation of signal from white matter regions with echo time in high-field GE imaging. This behaviour results from the interference of signals from microstructural compartments with different frequency offsets [1,2]. It can be characterised by using a three-pool model [1-3] which incorporates the contributions from the external, myelin and axonal compartments :

$$F(t)=A_ae^{i2{\pi}f_at-\frac{t}{T_{2a}^{*}}}+A_me^{i2\pi f_mt-\frac{t}{T_{2m}^{*}}}+A_ee^{i2{\pi}f_et-\frac{t}{T_{2e}^{*}}} \qquad\qquad (1)$$

where $$$a$$$, $$$m$$$ & $$$e$$$ denote the axonal, myelin and external pools, $$$A_{a,m,e} $$$ are the relative signal amplitudes, $$$T_{2a,m,e}^{*}$$$ are the $$$T_{2}^{*}$$$ relaxation times and $$$f_{a,m,e}$$$ are the frequency offsets. Frequency difference mapping (FDM) [3-5] takes advantage of this non-linear temporal evolution of the phase in GE sequences to produce images that carry information about white matter microstructure, with a particular sensitivity to the rapidly decaying signal from the myelin pool. Here, we applied GE imaging at 7T to 10 subjects who were each scanned six times. The resulting data demonstrate the capability of FDM and show that fits of the FD and magnitude data to the three-pool model provided insight into the variation of microstructure along the corpus callosum.

The signal measured in a GE sequence takes the form :

$$S(t)=S_{0}e^{i\Omega t}e^{i\phi_{0}}F(t) \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\quad(2)$$

where $$$\Omega$$$ & $$$\phi_{0}$$$ represent the effects of non-local field sources and time-independent phase offsets respectively, with $$$F(t)$$$ as defined in Equation (1). To observe the non-linear signal evolution, it is necessary to remove the dependence on $$$\Omega$$$ & $$$\phi_{0}$$$. $$$\phi_{0}$$$ can be eliminated by dividing each echo by $$$S(\text{TE}_1)$$$:

$$S'(\text{TE}_n )=\frac{S({\text{TE}}{_n})}{S(\text{TE}_1)}=e^{-iΩ(n-1)∆{\text{TE}}}×\frac{F(\text{TE}_n)}{F(\text{TE}_1)} \qquad\qquad\qquad(3) $$

Non-local field effects, $$$\Omega$$$, can be removed by dividing $$$S'(\text{TE}_n )$$$ by $$$(S'(\text{TE}_2))^{n-1}$$$:

$$S''(\text{TE}_n)=\frac{S'(\text{TE}_n )}{(S'(\text{TE}_2))^{n-1}}=\frac{F(\text{TE}_n )}{F(\text{TE}_1 )}×\left(\frac{F(\text{TE}_1)}{F(\text{TE}_2)}\right)^{n-1} \qquad (4)$$

yielding a signal that is sensitive to the non-linear phase evolution, which is predominantly driven by the rapid decay of the signal from the myelin pool.

Using a Philips Achieva 7T MR scanner, 10 healthy subjects underwent a series of single-slice, sagittal multi-echo GE scans (slice thickness=5mm, resolution=1mm² ,FOV=224x224mm², TE₁=2.4ms, ΔTE=2.4ms, # of echoes=20, TR=140ms, flip angle=25^o, # of averages=10, acquisition time=314s). The slice was positioned on the mid-line, spanning a portion of the corpus callosum where the fibres are oriented perpendicular to B₀. Each subject was scanned six times. Frequency difference maps were generated using the method outlined above, with an additional step involving fitting and subtraction of a term describing linear phase variation in the read-direction (foot-head), resulting from small differences of echo position in the acquisition window. Magnitude and frequency difference curves for 5 ROIs over the corpus callosum [6] (Figure 1) were fitted to Equations (1&4) for each subject.Figure 2 shows the evolution of the frequency difference with TE for a single representative subject, while Figure 3 shows FD maps for the 10 different subjects at TE=12 ms. Figure 4 shows average magnitude and frequency difference curves from all subjects for the 5 ROIs spanning the corpus callosum. Figure 5 shows the three-pool model parameters that provide the best fit to the experimental data from the 5 regions of the corpus callosum in the 10 subjects.Figures 2 and 3 show that corpus callosum and other structures such as the superior cerebellar peduncles, in which nerve fibers are oriented perpendicular to the field appear consistently hypointense in the FD maps. The negative frequency difference results from the decay of the signal from the myelin compartment in which the average frequency offset is positive (Figure 5), with further variation at late echo times resulting from interference of signals from the axonal and external compartments. It is evident from Figure 4 that the genu and splenium display a more rapidly decaying magnitude signal than the central regions of the corpus callosum, and also show a greater change of frequency with TE. The three-pool model parameters shown in Figure 5 are generally in good agreement with those previously obtained by fitting a triple-exponential model to 7T data from the splenium of the corpus callosum [1] and the pool amplitudes are also in correspondence with values measured in the optic radiations at 3T [7] using a similar approach. Analysis of the variation of parameter values across the different regions of the corpus callosum shows that differences in the frequency of the myelin and axonal compartments are the strongest contributory factors to the measured difference in signal evolution in the genu and splenium compared with the central regions of the corpus callosum. These frequency offsets are sensitive to the fiber g-ratio and anisotropic susceptibility of myelin [3].