Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

Coatléven, Julien

doi:10.3233/asy-151294

Cited by 1 publication

(4 citation statements)

References 18 publications

(38 reference statements)

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“…On the mathematical side, we only provided a formal justification of the periodic model. A complete mathematical justification of the homogenization process is the object of a separate publication [11]. For the inverse problem, studying the uniqueness and robustness of determining the coefficients of the ODE model from noisy data is another future direction of research.…”

Section: Discussionmentioning

confidence: 99%

“…The nature of the homogenized model then would depend on the scaling for the proportionality coefficient with respect to the periodicity size parameter. A comprehensive presentation as well as a fully rigorous mathematical justification of all the possible limit models is the subject of a forthcoming work [11].…”

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confidence: 99%

“…More precisely, the dependence of κ ε on ε → 0 will influence the nature of the limit macroscopic model. A comprehensive presentation as well as a fully rigorous mathematical justification of all the possible limit models corresponding to a scaling κ ε = σ m ε m , for all m ∈ Z, is the subject of a forthcoming work [11]. We will concentrate in this article on the relevant case for dMRI, i.e., the scaling that provides the limit model that shows a "memory" effect, which corresponds with m = 1.…”

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confidence: 99%

“…The proof is given in [11] and uses the periodic unfolding method [9], extended to the time dependent cases.…”

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confidence: 99%

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A Macroscopic Model Including Membrane Exchange for Diffusion MRI

Coatléven¹,

Haddar²,

Li³

2014

SIAM J. Appl. Math.

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Diffusion Magnetic Resonance Imaging is a promising tool to obtain useful information on the microscopic structure and has been extensively applied to biological tissues. We establish a new macroscopic model from homogenization theory for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of intermediate water exchange across biological cellular membranes. Based on a particular scaling of the permeability condition modeling cellular membranes, this macroscopic model reproduces the memory effects often observed in experiments. Explicit formulae given by homogenization for the coefficients of this model emphasize their link to the relevant physiological quantities. In addition, we explicitly solve the macroscopic model to obtain an ODE model for the dMRI signal. This ODE model is numerically easy to invert, and the inverse problem of retrieving model coefficients from synthetic dMRI signal data is considered.

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Section: Discussionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

“…The proof is given in [11] and uses the periodic unfolding method [9], extended to the time dependent cases.…”

mentioning

confidence: 99%

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