Elsevier

Magnetic Resonance Imaging

Volume 33, Issue 9, November 2015, Pages 1083-1090
Magnetic Resonance Imaging

Original contribution
Structural information revealed by the dispersion of ADC with frequency

https://doi.org/10.1016/j.mri.2015.06.009Get rights and content

Abstract

Diffusion MRI provides a non-invasive means to characterize tissue microstructure at varying length scales. Temporal diffusion spectra reveal how the apparent diffusion coefficient (ADC) varies with frequency. When measured using oscillating gradient spin echo sequences, the manner in which ADC disperses with gradient frequency (which is related to the reciprocal of diffusion time) provides information on the characteristic dimensions of restricting structures within the medium. For example, the dispersion of ADC with oscillating gradient frequency (ΔfADC) has been shown to correlate with axon sizes in white matter and provide novel tissue contrast in images of mouse hippocampus and cerebellum. However, despite increasing interest in applying frequency-dependent ADC to derive novel information on tissue, the interpretations of ADC spectra are not always clear. In this study, the relation between ADC spectra and restricting dimensions are further elucidated and used to derive novel image contrast related to the sizes of intrinsic microstructures.

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:b=γ2G2δ4π2f2

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:Δeff=14f

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.

References (48)

  • J. Xu et al.

    Mapping mean axon diameter and axonal volume fraction by MRI using temporal diffusion spectroscopy

    Neuroimage

    (2014)
  • M.E. Moseley et al.

    Diffusion-weighted MR imaging of acute stroke: correlation with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats

    AJNR Am J Neuroradiol

    (1990)
  • D.M. Patterson et al.

    Technology insight: water diffusion MRI--a potential new biomarker of response to cancer therapy

    Nat Clin Pract Oncol

    (2008)
  • M. Zhao et al.

    Early detection of treatment response by diffusion-weighted 1H-NMR spectroscopy in a murine tumour in vivo

    Br J Cancer

    (1996)
  • J.C. Gore et al.

    Characterization of tissue structure at varying length scales using temporal diffusion spectroscopy

    NMR Biomed

    (2010)
  • E. Squillaci et al.

    Correlation of diffusion-weighted MR imaging with cellularity of renal tumours

    Anticancer Res

    (2004)
  • T. Sugahara et al.

    Usefulness of diffusion-weighted MRI with echo-planar technique in the evaluation of cellularity in gliomas

    J Magn Reson Imaging

    (1999)
  • M.D. Does et al.

    Oscillating gradient measurements of water diffusion in normal and globally ischemic rat brain

    Magn Reson Med

    (2003)
  • E.C. Parsons et al.

    Temporal diffusion spectroscopy: theory and implementation in restricted systems using oscillating gradients

    Magn Reson Med

    (2006)
  • D.C. Colvin et al.

    Effects of intracellular organelles on the apparent diffusion coefficient of water molecules in cultured human embryonic kidney cells

    Magn Reson Med

    (2011)
  • D.C. Colvin et al.

    New insights into tumor microstructure using temporal diffusion spectroscopy

    Cancer Res

    (2008)
  • J. Xu et al.

    Characterizing tumor response to chemotherapy at various length scales using temporal diffusion spectroscopy

    PLoS One

    (2012)
  • J. Xu et al.

    Influence of cell cycle phase on apparent diffusion coefficient in synchronized cells detected using temporal diffusion spectroscopy

    Magn Reson Med

    (2011)
  • J. Xu et al.

    Dependence of temporal diffusion spectroscopy on axon size in white matter tracts of rat spinal cord

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