Kiran Thapaliya^{1}, Viktor Vegh^{1}, Steffen Bollmann^{1}, and Markus Barth^{1}

Quantitative assessment of model parameters (water fraction and frequency shift) estimated using a multi-compartment model can be useful to study tissue properties in white matter. In this work, we utilise multi-compartment models for multi-echo gradient echo data acquired at 7T. We investigate the variation of model parameters that could potentially be affected by differences in tissue microstructure in the corpus callosum. We further study the effect of different models (number of compartments and parameters) on the estimation of tissue parameters. We show that the tissue parameters vary across the sub-regions of the corpus callosum and are effected by different modelling choices.

The study was approved by the university human ethics
committee and written informed consent was obtained from ten healthy
participants (aged 30-41). The data were
acquired using a 3D GRE-MRI sequence on a 7T whole-body MRI research scanner
(Siemens Healthcare, Erlangen, Germany) with a 32 channel head coil (Nova Medical,
Wilmington, USA) using the following parameters: TE1=2.04ms with
echo spacing of 1.53ms and 30 echoes, TR=51ms, flip-angle=20^{o},
voxel-size=1mm$$$\times$$$1mm$$$\times$$$1mm and matrix
size=210$$$\times$$$168$$$\times$$$144. Individual channel data were processed
before images were combined.^{$$$6$$$} A brain mask for
each participant was created using FSL BET.^{$$$7$$$} iHARPERELLA (http://people.duke.edu/~cl160, STI
Suite^{$$$8$$$}) was used to compute
tissue phase for each echo. The CC was manually segmented into three and seven sub-regions,
respectively, using a standardised template^{$$$9$$$} (Fig.1) to assess
variation of tissue parameters across the CC and for a comparison with
literature values. The three regions and whole CC were used to investigate the distribution
of frequency shifts (Fig.2), whereas seven regions segmentation was used to
estimate water fractions and frequency shifts (Fig.3). We also performed a voxel
based analysis of CC (Fig.3). Signal fitting was performed using a three
compartment (7 parameter) and two compartment (5 parameter) model::

$$ S(t)=A_{1}e^{-\left(\frac{1}{T_{2,1}^*}+i2\pi\Delta f_{1}\right)t}+A_{2+3}e^{-\left(\frac{1}{T_{2,2}^*}+i2\pi\Delta f_{2}\right)t} (1)$$

$$S(t)=A_{1}e^{-\left(\frac{1}{T_{2,1}^*}+i2\pi\Delta f_{1}\right)t}+A_{2}e^{-\left(\frac{1}{T_{2,2}^*}+i2\pi\Delta f_{2}\right)t}+A_{3}e^{-\left(\frac{1}{T_{2,2}^*}+i2\pi\Delta f_{3}\right)t} (2)$$

where $$$A_{1}$$$, $$$A_{2}$$$ and $$$A_{3}$$$ are water fraction for the myelin, axonal, and extracellular compartments, or or combined axonal and extracellular compartments $$$(A_{2+3})$$$ respectively, and corresponding $$$T_{2,1}^*$$$,
$$$T_{2,2}^*$$$ and
$$$\Delta f_1$$$, $$$\Delta f_{2}$$$ and $$$\Delta f_{3}$$$ are the compartment relaxation times and frequency shifts. In Eqs. 1 and 2, the relaxation time of the myelin compartment was set to
7 ms4, and the relaxation time of the axonal and
extracellular compartments were made the same^{$$$10$$$}. Parameter fitting was performed in MATLAB (MathWorks, Natick, MA) using nonlinear
curve fitting method (lsqnonlin). The
model performance was assessed by computing the standard error rate.

Fig.2 shows the cumulative frequency distribution across the three CC sub-regions and the whole CC in all participants. The frequency distribution for myelin, axonal and extracellular spaces derived from the three compartment model is similar across the three regions of CC, whereas the two compartment model showed variation of myelin frequency distribution between the three sub-regions of the CC (blue arrows in Fig.2). Both, two and three compartment models showed similar results for the whole CC.

The voxel based analysis and 7-ROI results are shown in Fig.3. We found that myelin water fraction and frequency shift estimates were very similar at the anteiror and posterior end of the CC for both models. The variation of the axonal frequency shift had a similar trend across the sub-regions with a stronger variation for the three compartment model. These results are also reflected in the CC maps using the voxel based analysis (Fig.3, right hand side). The error rate of the fit was less than 10% and 15% for both models. Significant differences between two models (p<0.0005) across the CC were found for myelin water fractions and axonal frequency shifts.

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Fig. 1 Illustration of the location of the definition of seven
(shown on the left side) and three (shown on the right side) regions of
interest across the corpus callosum which were used to study tissue
characteristics via signal compartmentalisation.

Fig.2 shows the frequency distribution across three
regions and the whole CC accumulated over all 10 participants. The first row
shows the myelin, axonal, and extracellular frequency distribution across genu,
mid-body, splenium and whole CC estimated from the three compartment model and
simulation result from Sati et al.^{$$$$2$$$} $$$\Delta f_1$$$, $$$\Delta f_{2}$$$ and $$$\Delta f_{3}$$$ are the myelin (blue
line), axonal (red line), and extracellular (green line) frequency shifts in
Hz. The
second row shows the frequency distributions for the two compartment model (axonal
and extracellular now as red line).

Fig.3: ROI based
fitting results obtained for the seven regions of the corpus callosum using the
proposed two (blue lines) and three (green lines) compartment models across ten
subjects (error bars reflect SD over subjects). The maps of corresponding tissue
parameters in a single subject is shown on the right side. MWF=myelin water
fraction, $$$\Delta f_{ms}$$$, =myelin frequency shift, $$$\Delta
f_{afs}$$$, =axonal frequency shift, $$$\Delta f_2compart$$$= frequency shift from two compartment model, $$$\Delta f_3compart$$$= frequency shift from three compartment model in Hz.