Julian Emmerich^{1,2} and Sina Straub^{1}

In
this work, we examine the effect of two different species of susceptibility
producing-structures within one voxel to deduce whether the relaxation rate is
proportional to the sum of the absolute product between volume fraction and
susceptibility value. Furthermore,
the effect of diffusion on the *R _{2}’*
relaxation rate beyond the static dephasing regime is analyzed.

Recently,
*R _{2}’* was proposed as an additional source of information in the
processing pipeline of quantitative susceptibility mapping (QSM) to achieve a
separation of positive and negative susceptibility sources within one voxel

In a gradient echo experiment, signal loss can be described through $$$S(TE)\propto e^{-R_2^*\cdot TE}$$$
, where *TE* represents the echo time and *R _{2}*

We considered a simulation volume (1000×1000×1000 pixel) with
embedded spherical susceptibility sources possessing susceptibilities *χ+*
and *χ-. *Numbers and radii of these perturbers were
labeled with *N _{+}, N_{-},
R_{+}* and

For the simulation neglecting
diffusion, the signal intensity at echo time *TE* was calculated as the sum of
complex signals over the entire simulation volume *V*.
The signal was calculated for the echo times *TE*=0:5:140 ms for
different settings of *N, R* or *χ*. *R _{2}*’ was
determined by a mono exponential fit to the simulation data.

For the simulations including diffusion, the signal was
calculated for *TE*=0:0.25:100 ms for* N*=150000 particles as the complex sum
along their diffusion paths. In a first simulation, the diffusion constant *D*
was varied in the range of *D*=0,...,2.0 μm²/ms for particle radii ranging from
0.1 μm to 30 μm (fixed volume fraction *η*=0.01 and *Δχ _{±}* =
±4 ppm). In the second simulation,

Figure 1 exemplarily shows the susceptibility distribution of positive and negative spheres and the corresponding field distribution. Results for the simulation with varying susceptibility are shown in Figure 2 (solid lines). In the figure, the proportion of positive/negative susceptibility spheres increase/decrease linearly from left to right. Simulated relaxation rates incease linearly with the absolute susceptibility value of the spheres and they are independent from the proportion of positive and negative susceptibility within the simulated volume. In Figure 3 the radii of the spheres incease linearly, causing a cubic increase in volume fraction while background susceptibility and susceptibility of the spheres remained constant. The relaxation rate shows a cubic dependence on the radius of the spheres and therefore a linear dependence on the volume fraction.

Figure 4 shows the results for the
simulation including diffusion effects. In the simulations, large
deviations from the SDR limit for small particle sizes can be observed. The
relaxation rate in the presence of diffusion converges towards the SDR limit
for particle radii larger than *r*=0.5 μm
for *D*=0.1 μm²/ms and *r*=3 μm for *D*=1.6 μm²/ms.
In
Figure 5, results for a constant absolute volume fraction (sum of positive and
negative susceptibility sphere volume) are shown. There is no change in
relaxation rate behavior when changing the ratio of positive and negative
spheres within the simulation volume. In the limit of large particle radii (*r*>3 μm), the simulated relaxation rate again converges towards the SDR
limit.

It was shown that in the SDR as well as
under the presence of diffusion effects the relaxation rate *R _{2}’* of a
volume containing spherical susceptibility-producing structures of opposite
sign depends solely on the overall absolute susceptibility content of the
measurement volume.
The diffusional motion can be seen as
an averaging process when particles move through the inhomogenous field
prodiced by the spheres. This effect will be promoted by either small spheres
or large diffusion constants.

Pending the quantitative experimental
validation of these findings, *R _{2}’* can become a valuable information
source in the challenge of separating positive and negative susceptibility
sources in QSM reconstruction.

1. Lee, J., et al., Separating positive and negative susceptibility sources in QSM. Proc. Intl. Soc. Mag. Reson. Med. , 2017. 25: p. 0751.

2. Yablonskiy, D.A. and E.M. Haacke, Theory of NMR signal behavior in magnetically inhomogeneous tissues: the static dephasing regime. Magn Reson Med, 1994. 32(6): p. 749-63.

3. Brown, R.W., et al., Magnetic resonance imaging : physical principles and sequence design. Second edition ed. 2014, Hoboken, New Jersey: Wiley Blackwell.