Breaking β: Understanding the β-Value in Calibrated Functional MRI
Avery J.L. Berman1,2 and Bruce Pike2

1Montreal Neurological Institute, McGill University, Montreal, QC, Canada, 2Department of Radiology and Hotchkiss Brain Institute, University of Calgary, Calgary, AB, Canada

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, ΔR2*, on the susceptibility offset (Δχ) of blood relative to tissue 1: ΔR2* $$$\propto$$$ |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

1. Davis T.L., Kwong K.K., Weisskoff R.M., et al. Calibrated functional MRI: mapping the dynamics of oxidative metabolism. Proc Natl Acad Sci USA, 1998;95(4):1834-9.

2. Ogawa S., Menon R.S., Tank D.W., et al., Functional brain mapping by blood oxygenation level-dependent contrast magnetic resonance imaging. A comparison of signal characteristics with a biophysical model. Biophys J, 1993;64(3):803-12.

3. Shu C.Y., Rothman D., Sanganahalli B.G., et al. Quantitative β mapping for high-field calibrated fMRI in rat brain. ISMRM23 Toronto 2015;3937.

4. Boxerman J.L., Hamberg L. M., Rosen B.R., et al. MR Contrast Due to Intravascular Magnetic-Susceptibility Perturbations. Magn Reson Med, 1995;34(4):555-566.

5. Pannetier N.A., Debacker C.S., Mauconduit F., et al. A simulation tool for dynamic contrast enhanced MRI, PLoS One, 2013;8(3):e57636.

6. Lauwers F., Cassot F., Lauwers-Cances V., et al. Morphometry of the human cerebral cortex microcirculation: general characteristics and space-related profiles, Neuroimage, 2008;39(3):936-48.

7. Germuska M.A., Meakin J.A., and Bulte D.P. The influence of noise on BOLD-mediated vessel size imaging analysis methods, J Cereb Blood Flow Metab, 2013;33(12):1857-63.

8. Uludag K., Dubowitz D.J., Yoder E.J., et al. Coupling of cerebral blood flow and oxygen consumption during physiological activation and deactivation measured with fMRI, Neuroimage, 2004;23(1):148-55.

9. Martindale J., Kennerley A.J., Johnston D., et al. Theory and generalization of Monte Carlo models of the BOLD signal source, Magn Reson Med, 2008;59(3):607-18.

10. Griffeth V.E. and Buxton R.B., "A theoretical framework for estimating cerebral oxygen metabolism changes using the calibrated-BOLD method: modeling the effects of blood volume distribution, hematocrit, oxygen extraction fraction, and tissue signal properties on the BOLD signal," Neuroimage, 2011;58(1):198-212.

Figures

Figure 1: The two sampled distributions of vessel radii used in the simulations. The mean and median radii in the Lognormal distribution were 4.64 μm and 4.12 μm, respectively, and in the Frechet distribution, they were 14.7 μm and 11.6 μm.

Figure 2: β’s dependence on vessel radius in GE (left) and SE (right) simulations. In both cases, β’s dependence on the range of Δχ used in the fitting was also examined. Using a maximum Δχ of 4π·0.05 ppm corresponds to the more naturally occurring range of Δχ’s produced by partially deoxygenated blood.

Table 1: Mean fitted β-values from simulations using two different realistic distributions of cortical vessel radii. Fits were either calculated over the range Δχ = 4π(0.01 – 0.05) ppm or 4π(0.01 – 0.10) ppm.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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