Modeling an equivalent b‐value in diffusion‐weighted steady‐state free precession

Purpose Diffusion‐weighted steady‐state free precession (DW‐SSFP) is shown to provide a means to probe non‐Gaussian diffusion through manipulation of the flip angle. A framework is presented to define an effective b‐value in DW‐SSFP. Theory The DW‐SSFP signal is a summation of coherence pathways with different b‐values. The relative contribution of each pathway is dictated by the flip angle. This leads to an apparent diffusion coefficient (ADC) estimate that depends on the flip angle in non‐Gaussian diffusion regimes. By acquiring DW‐SSFP data at multiple flip angles and modeling the variation in ADC for a given form of non‐Gaussianity, the ADC can be estimated at a well‐defined effective b‐value. Methods A gamma distribution is used to model non‐Gaussian diffusion, embedded in the Buxton signal model for DW‐SSFP. Monte‐Carlo simulations of non‐Gaussian diffusion in DW‐SSFP and diffusion‐weighted spin‐echo sequences are used to verify the proposed framework. Dependence of ADC on flip angle in DW‐SSFP is verified with experimental measurements in a whole, human postmortem brain. Results Monte‐Carlo simulations reveal excellent agreement between ADCs estimated with diffusion‐weighted spin‐echo and the proposed framework. Experimental ADC estimates vary as a function of flip angle over the corpus callosum of the postmortem brain, estimating the mean and standard deviation of the gamma distribution as 1.50·10-4 mm2/s and 2.10·10-4 mm2/s. Conclusion DW‐SSFP can be used to investigate non‐Gaussian diffusion by varying the flip angle. By fitting a model of non‐Gaussian diffusion, the ADC in DW‐SSFP can be estimated at an effective b‐value, comparable to more conventional diffusion sequences.

: In DW-SSFP, repeated application of RF pulses decomposes the magnetization into a series of coherence pathways, which are sensitized to the diffusion gradient during transverse periods. Here we show 5 example coherence pathways. The spin-echo pathway (A), stimulated-echo pathway (B) and long stimulated-echo pathway (C) only survive for 2 TRs in the transverse plane, the condition for the two-transverse-period approximation (1). These pathways all experience the same q-value, but have different diffusion times, defined as ∆ = 1 · TR (A), 2 · TR (B), and 4 · TR (C).
For the full Buxton model (1), this condition is no longer required, and pathways can experience cumulative sensitization to the diffusion gradients over multiple TRs, such as the spin-echo pathway in (D), in addition to pathways that generate multiple echoes over their lifetime (E). This leads to pathways with different q-values, in addition to weighting of the signal by T 2 S1 (A)    Table S1: Acquisition protocols for the T 1 , T 2 and B 1 maps. Before processing, a Gibbs ringing correction was applied to the TIR and TSE data (2). T 1 and T 2 maps were derived assuming mono-exponential signal evolution. The B 1 map was obtained using the methodology described in (3)

Supporting Derivations
The two-transverse-period approximation with a gamma distribution of diffusivities From Equation 1 in the main text: where S 0 is the equilibrium magnetization, E 1 = e − TR T 1 , E 2 = e − TR T 2 , α is the flip angle, n is the number of TRs between the two transverse-periods for a given stimulated-echo, A 1 = e −q 2 ·TR·D , D is the diffusion coefficient and q = γGτ , where γ is the gyromagnetic ratio, G is the diffusion gradient amplitude and τ is the diffusion gradient duration. Separating this expression into spin-echo (SE) and stimulated-echo (STE) pathways: and:

SE term
Integrating over the SE term with a gamma distribution of diffusivities: [S5] Therefore: [S6]

STE term
Integrating over the STE term with a gamma distribution: Evaluating the summation term, considering Equation 3 in the main text: Pulling D m from the numerator and q 2 · TR · D 2 s from the denominator : Rearranging and defining m = n − 1: [S10] The summation term is in an equivalent format to the the Lerch Transcendent (4), defined as: where z = E 1 cos α, s =

2E1
. [S18] As E 1 and S SSFP are positive, the numerator is less than 0 when we consider the negative solution.