Synopsis
In calibrated functional MRI (fMRI), the
parameter β is used to describe the non-linear dependence of the change in the
transverse relaxation rate on the susceptibility offset (Δχ) of blood relative
to tissue. Estimates of β at high field strengths have generally been assumed
or obtained in post-hoc analyses. Using simulations from vessel networks, we
present here a detailed description of β’s dependence on vessel radius. We show
that the estimate of β is highly dependent on the range of Δχ used in the fit.
This could have important implications for methods that propose to measure
β-values in vivo using a contrast agent to alter Δχ.Purpose
In calibrated functional MRI (fMRI),
the parameter β is used to describe the non-linear dependence of the change in
the transverse relaxation rate, ΔR
2*,
on the susceptibility offset (Δχ) of blood relative to tissue
1: ΔR
2* $$$\propto$$$ |Δχ|
β. This non-linear behaviour arises from the
differential dephasing experienced by spins diffusing through the field offsets
generated by vessels of varied radii. For small vessel radii, β is
approximately 2 and for large radii, β is approximately 1
2; however,
other than characterizing this dependence into vessels of small and large radii,
little has been done to examine the transition between the vessel sizes. Maps
of β across the brains of rats have recently been generated with the
administration of a contrast agent
3; therefore, given β’s vessel-size dependence, it should be possible to ascribe a
more quantitative physiological meaning to these maps. This study aims to
characterize the change in β as a function of vessel radius and for
distributions of vessel radii using simulations.
Methods
Following the methods originally used to
estimate β,4 we simulated the transverse MR signal decay from
networks of vessels and calculated ΔR2* at a single time point using ΔR2* =
-ln[S(TE)]/TE, where S(TE) is the signal magnitude at the echo time (TE). Δχ
was varied from 4π(0.01 – 0.1) ppm, which covers the range of Δχ encountered under
normal oxygen saturations and extends to values induced by contrast agents. B0
was set to 3 T and a gradient echo (GE) TE of 60 ms was used. The
simulations were repeated for a spin echo (SE) sequence with TE = 100 ms.
The simulations were conducted with the deterministic diffusion method in two-dimensions.5 Simulation networks were randomly populated to volume fractions of 2% with vessels of 1-μm radii. Vessel radii were systematically increased up to 100 μm by scaling the diffusion coefficient in the simulations by the appropriate factor and reusing the 1-μm networks.
For
every simulation, the function ΔR2(*) = a|Δχ|β
was fit to obtain estimates of β. This was performed using the full range of Δχ
that were simulated as well as just the range 4π(0.01 – 0.05) ppm, corresponding
to oxygen saturations of ~50% – 90%. To determine how β is affected by a
distribution of vessel radii, the simulations were repeated in networks that
were populated to 2% with radii drawn from one of two different distributions
of cortical vessel radii, referred to here as the Lognormal 6 and
Frechet 7 distributions (Fig. 1). These simulations were run at 3 T
and 1.5 T, in order to compare against currently used β-values.
Results
Fig.
2 shows plots of the average β vs. radius from the simulations. As expected for
the GE simulations, β is approximately 2 for the smallest radii and rapidly decreases
towards 1 with increasing radius. Like the GE β, the SE β is also approximately
2 for small radii and rapidly decreases with increasing radius; however, beyond
this, β no longer behaves monotonically. Fitting for β over the extended range of Δχ
decreased the fitted values for all radii and resulted in SE β-values less than
1 for intermediate vessel radii.
As
shown in Fig. 1, the two distributions of radii tested are quite different: the
Lognormal distribution is predominantly microvascular, with radii < 10 μm, and
the Frechet distribution peaks at ~10 μm and contains a substantial amount of
vessels with radii > 10 μm. This resulted in β-values that were systematically
greater in the Lognormal-distributed simulations. Table 1 summarizes the
β-values from the two distributions. The GE β-values from the
Frechet-distributed simulations tended to be closer to currently accepted
values at 1.5 T (β = 1.5) 1 and at 3 T (β = 1.3) 8.
Conclusion
Estimates
of β at high
field strengths (>1.5 T) have generally been assumed or obtained in
post-hoc analyses.
8 We have presented a detailed description of β’s
dependence on vessel radius for both GE and SE signals. We have shown that the
estimate of β is highly dependent on the range of Δχ used in the fit, in
agreement with similar analyses.
4,9 This could have important
implications
10 for methods that propose to perform calibrated fMRI
using β-values measured in vivo with contrast agent. We are currently
investigating β at ultra-high field strengths as well as the influence of intravascular
signal decay on β.
Acknowledgements
The authors would like to recognize financial
support from the Canadian Institutes of Health Research (FDN 143290) and the
Campus Alberta Innovates Program.References
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