We have been active in investigating the link between measured diffusion-encoded signal and diffusion restrictions. These works include:

• The development of the diffusion pore imaging (DPI) technique [1-5]. DPI solves the inverse problem in diffusion MR for diffusion in closed pores or cells, i.e. it allows measuring the exact average cell shape in a volume that contains an ensemble of pores or cells.
• The development of the diffusion lattice imaging (DLI) technique [6]. It solves the inverse problem for MR diffusion with periodic boundaries. The measurement process is based on double diffusion encodings. It appears to be the first solution to the inverse problem of MR diffusion for non-closed diffusion restrictions.
• An analysis of the diffusion short-time limit. It was shown that the second order correction term of the series expansion of the diffusion coefficient vanishes if the diffusion-encoding is flow-compensated, i.e. if its first gradient-moment is zero [7]. It was shown that this finding also translates to oscillating diffusion-encoding gradients [8]. Consequently, the important first order correction term – which is tightly linked to the surface-to-volume ratio of the diffusion restrictions – can be measured with higher accuracy if flow-compensated diffusion-encoding gradients are used.
• An analysis of the matrices that describe the perturbation caused by the diffusion-encoding fields. These matrices arise if the time-evolution of the Bloch-Torrey equation is described in a basis of the Laplace operator. It was shown that the matrices of general fields can be easily derived from the matrices of linear fields [9].
• Validation of theoretical predictions using phantoms made of acrylic glass with hyperpolarized xenon as diffusing medium [10-12]    

[1] Laun FB, Kuder TA, Semmler W, Stieltjes B.
Determination of the defining boundary in nuclear magnetic resonance diffusion experiments.
Phys Rev Lett. 2011 Jul 22;107(4):048102.

[2] Laun FB, Kuder TA, Wetscherek A, Stieltjes B, Semmler W.
NMR-based diffusion pore imaging.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Aug;86(2 Pt 1):021906.

[3] Kuder TA, Laun FB.
NMR-based diffusion pore imaging by double wave vector measurements.
Magn Reson Med. 2013 Sep;70(3):836-41.

[4] Laun FB, Kuder TA.
Diffusion pore imaging with generalized temporal gradient profiles.
Magn Reson Imaging. 2013 Sep;31(7):1236-44.

[5] Kuder TA, Laun FB.
Effects of pore-size and shape distributions on diffusion pore imaging by nuclear magnetic resonance.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Aug;92(2):022706.

[6] Laun FB, Müller L, Kuder TA.
NMR-based diffusion lattice imaging.
Physical Review E, 2016. 93(3).

[7] Laun FB, Kuder TA, Zong F, Hertel S, Galvosas P.
Symmetry of the gradient profile as second experimental dimension in the short-time expansion of the apparent diffusion coefficient as measured with NMR diffusometry.
J Magn Reson. 2015 Oct;259:10-9.

[8] Laun FB, Demberg K, Nagel AM, Uder M, Kuder TA.
On the Vanishing of the t-term in the Short-Time Expansion of the Diffusion Coefficient for Oscillating Gradients in Diffusion NMR.
Frontiers in Physics, 2017. 5.

[9] Laun FB.
Restricted diffusion in NMR in arbitrary inhomogeneous magnetic fields and an application to circular layers.
J Chem Phys. 2012 Jul 28;137(4):044704.

[10] Kuder TA, Bachert P, Windschuh J, Laun FB.
Diffusion pore imaging by hyperpolarized xenon-129 nuclear magnetic resonance.
Phys Rev Lett. 2013 Jul 12;111(2):028101.

[11] Demberg K, Laun FB, Windschuh J, Umathum R, Bachert P, Kuder TA.
Nuclear magnetic resonance diffusion pore imaging: Experimental phase detection by double diffusion encoding.
Phys Rev E. 2017 Feb;95(2-1):022404.

[12] Demberg K, Laun FB, Bertleff M, Bachert P, Kuder TA.
Experimental determination of pore shapes using phase retrieval from q-space NMR diffraction.
Phys Rev E. 2018 May;97(5-1):052412.