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.

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.

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.

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.