Parsimonious modeling of skeletal muscle perfusion: Connecting the stretched exponential and fractional Fickian diffusion

David A. Reiter, Fatemeh Adelnia, Donnie Cameron, Richard G. Spencer, Luigi Ferrucci

Research output: Contribution to journalArticlepeer-review

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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 D , 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 D , 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.

Original languageEnglish
Pages (from-to)1045-1057
Number of pages13
JournalMagnetic Resonance in Medicine
Issue number2
Early online date16 Mar 2021
Publication statusPublished - Aug 2021


  • anomalous diffusion
  • fractional calculus
  • hyperemia
  • intravoxel incoherent motion
  • microvascular volume
  • superdiffusion

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