Original contributionStructural information revealed by the dispersion of ADC with frequency
Introduction
Diffusion MRI provides a non-invasive means to characterize tissue microstructure, and has been widely used to detect stroke and monitor tumor response to therapy [1], [2], [3], [4]. The apparent diffusion coefficients (ADCs) measured at different diffusion times are believed to reflect the structural hindrances and restrictions to free water movement at varying length scales [5]. Conventional pulsed gradient spin echo (PGSE) measurements of ADC in biological tissues usually involve relatively long diffusion times (20–40 ms), so the corresponding one-dimensional root mean square displacements (RMSD) of diffusion molecules are on the order of 10 μm. The ADC measured with PGSE is thus observed to correlate with cellularity in several types of tumors [2], [6], [7].
Oscillating gradient spin echo (OGSE) methods have been used to achieve much shorter effective diffusion times, and hence they are able to differentiate smaller structures with higher sensitivity [8], [9]. For example, the measured ADCs at high oscillating frequencies have been shown to convey microstructural variations at sub-cellular scales [10], which may help detect earlier tumor response to treatment before changes in tissue cellularity [11], [12], [13], [14]. Moreover, by varying the oscillating frequencies, an apparent diffusion spectrum can be obtained [9]. The manner in which ADC disperses with oscillating frequency provides information on tissue structure over a range of intrinsic length scales and in general may reflect several tissue properties, but some simple features of such spectra have proven empirically useful [5]. For example, the initial rate of change of ADC with frequency at low frequencies has been shown to correlate with axon sizes in white matter [15] and provide novel tissue contrast in images of mouse hippocampus and cerebellum [16], [17], [18]. However, despite increasing interest in applying frequency-dependent ADC to derive novel information on tissue [19], [20], [21], [22], the interpretations of ADC spectra are not always clear.
Restricted diffusion with generalized time-varying diffusion gradient waveforms has been studied previously [23], [24]. Specifically for water diffusion inside simple geometries using cosine-modulated gradient waveforms, analytical equations describing ADC as a function of frequency have been derived and validated [24], [25], [26]. In this study, the theory of water diffusion inside an impermeable cylinder or sphere is re-examined with emphasis on the rate of frequency-dependent changes in ADC at relatively low frequencies. From this, a simple relation between the rate and restricting size can be derived for limited ranges of parameters. Simulations and experiments illustrate this relation. This study may help better understand the information revealed by the behavior of ADC with frequency and suggests a novel type of parametric image that depicts structural dimensions.
Section snippets
Theory
For an OGSE sequence with a pair of cosine-modulated gradients on either side of a refocusing pulse, the diffusion weighting b-value is:
Here γ is the nuclear gyromagnetic ratio, G the maximum gradient amplitude, δ the gradient duration, and f the diffusion gradient oscillation frequency. Based on the equivalence of b-values between OGSE and PGSE sequences [8], the effective diffusion time (Δeff) for a cosine-modulated OGSE sequence is:
Using a Gaussian phase approximation [27]
Results
Fig. 1 shows the numerically calculated values of ADC, ΔfADC and RMSD(f0) based on Eqs. ((3), (4), (5)) for water diffusion within impermeable cylinders at two typical values of D. It is evident that ΔfADC is not a monotonic function of f. ΔfADC increases with increasing f at low frequencies but drops after reaching a maximum at f = f0. Fig. 1(b) and (f) suggest that f0 shifts to higher frequency as the free diffusion coefficient D increases. This is consistent with Eq. (3) which predicts, under
Discussion
This study re-examined restricted diffusion within an impermeable cylinder or sphere under cosine-modulated OGSE sequences. Our emphasis was to interpret the dispersion rate of ADC with oscillating frequency at relatively low frequencies. Whereas ADC spectra increase continuously with frequency, they also show inflections so that ΔfADC is not a monotonic function of frequency but reaches a maximum when the second derivative is zero. This occurs at a frequency that in practice is proportional to
Conclusion
In this study, the theory of water diffusion within an impermeable cylinder was re-examined to derive the relation between ΔfADC and restricting size. The results indicate that ΔfADC is not a monotonic function of the oscillating frequency, and the inflection frequency f0 is in theory an indicator of compartment size. Besides, ΔfADC passes through a maximum when the restricting radius R is close to the corresponding RMSD. The change of ΔfADC is a reliable indicator of the change of restricting
Acknowledgements
NIH Grants K25CA168936, R01CA109106, R01CA173593, and P50CA128323 funded this work.
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