Parsimonious Modeling of Skeletal Muscle Perfusion: Connecting the Stretched Exponential and Fractional Fickian Diffusion

Purpose: To develop an anomalous (non-Gaussian) diffusion model for characterizing skeletal muscle perfusion using multi-b-value DWI.

Theory And Methods: Fick's first law was extended for describing tissue perfusion as anomalous superdiffusion, which is non-Gaussian diffusion exhibiting greater particle spread than that of the Gaussian case. This was accomplished using a space-fractional derivative that gives rise to a power-law relationship between mean squared displacement and time, and produces a stretched exponential signal decay as a function of b-value. Numerical simulations were used to estimate parameter errors under in vivo conditions, and examine the effect of limited SNR and residual fat signal. Stretched exponential DWI parameters, α and , were measured in thigh muscles of 4 healthy volunteers at rest and following in-magnet exercise. These parameters were related to a stable distribution of jump-length probabilities and used to estimate microvascular volume fractions.

Results: Numerical simulations showed low dispersion in parameter estimates within 1.5% and 1%, and bias errors within 3% and 10%, for α and , respectively. Superdiffusion was observed in resting muscle, and to a greater degree following exercise. Resting microvascular volume fraction was between 0.0067 and 0.0139 and increased between 2.2-fold and 4.7-fold following exercise.

Conclusions: This model captures superdiffusive molecular motions consistent with perfusion, using a parsimonious representation of the DWI signal, providing approximations of microvascular volume fraction comparable with histological estimates. This signal model demonstrates low parameter-estimation errors, and therefore holds potential for a wide range of applications in skeletal muscle and elsewhere in the body.