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Biophysically motivated efficient estimation of spatially isotropic component from a single, standard gradient recalled echo measurement
Sebastian Papazoglou1, Tobias Streubel1,2, Mohammad Ashtarayeh1, Kerrin Pine2, Evgeniya Kirilina2,3, Markus Morawski4, Carsten Jäger2, Stefan Geyer2, Martina F Callaghan5, Nikolaus Weiskopf2, and Siawoosh Mohammadi1

1Department of Systems Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg, Germany, 2Department of Neurophysics, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany, 3Department of Education and Psychology - Neurocomputation and Neuroimaging Unit, Freie Universität Berlin, Berlin, Germany, 4Paul Flechsig Institute of Brain Research, University of Leipzig, Leipzig, Germany, 5Wellcome Centre for Human Neuroimaging, UCL Institute of Neurology, London, United Kingdom

Synopsis

Gradient recalled echo-based $$$R_{2}^{*}$$$ measurements are sensitive to the degree of myelination of white matter fibres and their local orientation inside the magnetic field of the MR scanner. This orientation dependence has been observed experimentally and could be explained biophysically by anisotropic susceptibility of the myelin sheaths. In case of single, quantitative $$$R_{2}^{*}$$$ measurements the orientation dependence represents a potential confounder, since the observed $$$R_{2}^{*}$$$ would be biased by the sample’s orientation inside the scanner. Here, we propose an efficient method for separating $$$R_{2}^{*}$$$ into orientation dependent and independent components based on a biophysically motivated higher order $$$R_{2}^{*}$$$ decay model.

Introduction

$$$R_{2}^{*}$$$ measured by gradient recalled echo (GRE) sequences is a marker depending on the myelination of fibres 1, 2, which is also sensitive to local axonal fibre orientation $$$\theta$$$ relative to the main magnetic field of the MR scanner 3. While this could be used to map fibre direction using GRE measurements at multiple sample orientations 4, 5, a single GRE measurement is biased by the subject’s head position inside the scanner. For GRE based $$$R_{2}^{*}$$$ maps to be truly quantitative, this effect needs to be controlled for. Here, we propose an efficient method for decomposing $$$R_{2}^{*}$$$ into orientation independent (isotropic) and dependent (anisotropic) components, motivated by the biophysical hollow cylinder fibre model (HCFM), which explains orientation dependence by the anisotropic susceptibility of myelin sheaths 5, 6. In contrast to previous methods 4, 5, 6, 7, it requires only GRE data acquired at a single, unknown orientation of the sample.

Methods

Theory: Classically, the GRE signal is described by a mono-exponential decay, the logarithm of which may be written as a linear model

$$ ln(S(TE)) = ln(S(0)) -\beta_{1}^{(1)}TE ,\quad\quad\quad (1)$$

where $$$ln(S(0))$$$ is the signal at echo time $$$TE=0$$$ and the coefficient $$$\beta_{1}^{(1)}=R_{2}^{*}$$$ is the effective transverse relaxation time 8. Equation (1) can also be interpreted as the first order (in $$$TE$$$) approximation to a more complex signal expression as suggested in case of the HCFM 5, 6. Inspired by the second-order expansion of the predicted signal in the HCFM for TE<36 ms (at 7T) 5, we assume a quadratic signal model

$$ ln(S(TE)) = ln(S(0)) - \beta_{1}^{(2)}TE - \beta_{2}^{(2)}TE^{2} ,\quad\quad\quad (2)$$

where according to 5 $$$\beta_{1}^{(2)}$$$ is expected to be orientation independent (the isotropic component of $$$R_{2}^{*}$$$), while $$$\beta_{2}^{(2)}$$$ is expected to be orientation dependent following a $$$\sin^{4}\theta $$$-dependence (related to the anisotropic component of $$$R_{2}^{*}$$$). To validate the prediction following from the quadratic model, the orientation dependence of the components in equations (1) and (2) were investigated in an ex vivo human optic chiasm (OC), using separation of the parameters according to the well-established phenomenological model for $$$R_{2}^{*}$$$ 4, 5, 6, 7

$$\beta_{j}^{(\alpha)} = \beta_{j,iso}^{(\alpha)} + \beta_{j,aniso}^{(\alpha)}\sin^{4}\theta ,\quad\quad\quad (3)$$

with $$$\alpha =1, 2$$$ indicating model (1) or (2) and $$$j=1$$$ ($$$\alpha =1$$$) and $$$j=1, 2$$$ ($$$\alpha = 2$$$) denoting the order of the coefficient.

Sample: A human OC sample with adjoining optic nerves and tracts (OTs) was obtained at autopsy with prior informed consent (48 hrs postmortem, multiorgan failure) and approved by the responsible authorities. Following the standard Brain Bank procedures, blocks were immersion-fixed in (3% paraformaldehyde +1% glutaraldehyde) in phosphate-buffered saline.

MRI: GRE measurements were performed on a 7T Siemens Magnetom MRI scanner (Siemens Healthcare, Germany) using a custom RF coil with a diameter of 60 mm and the following protocol: 16 equally spaced echoes (3.4-53.5 ms, step-size 3.34 ms), repetition time TR = 100 ms, total acquisition time: 20:59 min. The measurement was repeated 16 times, using different orientations of the sample (figure 1). The two models (1) and (2) were inverted using customized tools 9, yielding $$$\beta_{j}^{(\alpha)}$$$- maps for each angle. All parameter maps were registered to the reference. Two regions-of-interest (ROIs) in the left and right OT were manually defined (figure 2), and equation (3) was fitted to the mean signals inside the ROIs for separating $$$\beta_{j,iso}^{(\alpha)}$$$ and $$$\beta_{j,aniso}^{(\alpha)}$$$.

Results

The results of the inversion according to (1) and (2) and the subsequent fits to (3) are shown in figure 3. For model (1), the well-known 4, 5, 6, 7 orientation dependence of $$$R_{2}^{*}$$$ (=$$$\beta_{1}^{(1)}$$$) was observed (figure 4). Model (2) showed orientation dependence mainly in the second-order coefficient $$$\beta_{2}^{(2)}$$$, while the first-order coefficient $$$\beta_{1}^{(2)}$$$ was orientation independent. This observation was complemented by the separation according to (3) showing that $$$\beta_{1}^{(2)}\approx\beta_{1,iso}^{(2)}$$$ and $$$\beta_{2}^{(2)}\approx\beta_{2,ansio}^{(2)}$$$ (figure 5). The anisotropic component of $$$R_{2}^{*}$$$, $$$\beta_{1,aniso}^{(1)}$$$, was in agreement with literature 4 whereas $$$\beta_{1,aniso}^{(1)}$$$ was smaller.

Discussion and Conclusion

The proposed method based on inversion of (2) allows for direct estimation of the isotropic $$$R_{2}^{*}$$$ component ($$$\beta_{1,iso}^{(1)}$$$) from a single, standard GRE measurement in terms of $$$\beta_{1,iso}^{(2)}$$$. Moreover, the orientation dependence of the second order in (2) suggests a direct correspondence of $$$\beta_{2}^{(2)}$$$ to the anisotropic part of $$$R_{2}^{*}$$$. Differences in the isotropic component of $$$R_{2}^{*}$$$ ($$$\beta_{1,iso}^{(1)}$$$ and $$$\beta_{1,iso}^{(2)}$$$) with respect to literature values 4 may be attributed to differences in the fixation protocol 10. Future work will have to demonstrate how the proposed method translates to measurements in vivo and at lower field strengths and validate its compatibility with biophysical models that account for the dependence of on other tissue components such as iron.

Acknowledgements

This work was supported by the German Research Foundation (DFG Priority Program 2041 "Computational Connectomics”, [AL 1156/2-1;GE 2967/1-1; MO 2397/5-1; MO 2249/3–1], by the Emmy Noether Stipend: MO 2397/4-1) and by the BMBF (01EW1711A and B) in the framework of ERA-NET NEURON. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 616905

References

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Figures

Optic chiasm sample embedded in agarose inside the acrylic sphere with markings for 16 orientations with respect to the main magnetic field (B0). The markings indicate directions equally distributed on a spherical triangle covering fibre direction $$$\theta$$$ between 0 and p/2. The left optic tract aligned with B0 defined $$$\theta =0$$$ and served as reference for coregistration of parameter maps. The arrow above the sphere points to the actual orientation of the sphere inside the magnet, while another arrow indicates the direction of the main field of the MR scanner.

ROIs are depicted in an axial and coronal view, covering fibres that are approximately aligned with a well-defined orientation within the left and right optic tract.

Mean values of the coefficients estimated according to equations (1) and (2) inside the ROIs shown in figure 2. Error bars represent standard deviations inside ROIs. A, B and C correspond to the left ROI, while D, E and F correspond to the right ROI. Results for equation (1) are shown in A and D. The results for equation (2) are shown in B, C, E and F. Dashed blue lines: fits according to equation (3). The numerical values of the corresponding isotropic and anisotropic components are summarised in table 2

Table summarising the correlation of the $$$\beta_{j}^{(\alpha)}$$$ and $$$\sin^{4}\theta$$$ quantified by Pearson’s $$$\rho$$$ ($$$p$$$- value).

Table summarising the numerical values of the fit parameters according to equation (3), shown as blue dashed lines in figure 3.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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