Silvia Capuani^{1}, Fabio Micalizzi^{2}, Simona Sennato^{3}, Giulio Costantini^{3}, Roberto Matassa^{4}, Dante Rotili^{5}, Lorenzo Correale^{2}, Francesca Giuffrida^{2}, Annalisa Caligiuri^{2}, Giovanni Familiari^{4}, and Andrea Gabrielli^{3}

^{1}Physics Dpt. Sapienza, CNR ISC, Rome, Italy, ^{2}Physics Dpt., Sapienza University of Rome, Rome, Italy, ^{3}Physics Dpt. Sapienza Roma, CNR ISC, Rome, Italy, ^{4}Anatomical, Histological, Forensic Medicine and Orthopedics Sciences, Sapienza University of Rome, Rome, Italy, ^{5}Drug chemistry and technology department, Sapienza University of Rome, Rome, Italy

The quantification of transient anomalous diffusion (tAD) parameters by NMR could provide higher sensitivity, resolution and complementary detailed information for detecting early changes due to pathological conditions than the conventional metric does. However, there are some questions and controversial issues which prevent the take-off of tAD NMR investigations in soft condensed matter and medical diagnostic field. To overcome these obstacles, we planned to investigate the potential and limits of NMR subdiffusion in calibration standards characterized by porous structured materials and diffusing probes matching or not characteristics length-scales of the porous matrix.

𝑆(Δ)=𝐴∙𝑒xp(−K

for a parameter quantification. PGSTE by varying g at different diffusion times t=D was used to obtain D(t) behavior. Structural and geometric parameters together with S/V and tortuosity τ were obtained and compared. Simulations with fast random walk (FRW) algorithm

Subdiffusion in poly-samples is due to multiple diffusion length-spaces.

Both water and liposomes in G50-Sephadex are characterized by a≈1, because their dimension does not match with the peculiar structural dimension of the Sephadex matrix. Conversely, diffusion of B12F12

However, attention must be paid to the limits of validity of the tAD theory.

1. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000).

2. M. Palombo, A. Gabrielli, S. De Santis, C. Cametti,G. Ruocco, S. Capuani, Spatio-temporal anomalous diffusion in heterogeneous media by nuclear magnetic resonance. 135, J. Chem. Phys. 034504 (2011).

3. M Palombo, A Gabrielli, VDP Servedio, G Ruocco, S Capuani, Structural disorder and anomalous diffusion in random packing of spheres. Sci Rep. 3, 2631 (2013)

4. Latour LL, Svoboda K, Mitra PP, Sotak CH. Time-dependent diffusion of water in a biological model system. Proc Natl Acad Sci USA. 91(4):1229 (1994)

5. V.G. Kiselev, Fundamentals of diffusion MRI physics. Nmr Biomed 30(3) (2017).

6. Novikov DS, Jensen JH, Helpern JA, Fiere-mans E. Revealing mesoscopic structural universality with diffusion. Proc Natl Acad Sci USA.111(14):5088 (2014)

7. Grebenkov DS A fast random walk algorithm for computing the pulsed-gradient spin-echo signal in multiscale porous media. J Magn reson 2, 208, 243--255 (2011)

α parameter as a function of the magnetic gradient strength g along the x-direction obtained in polydispersed (Poly) and monodispersed (mono) packed polystyrene beads.

In mono-sample the diameter of the beads is 10 μm, in poly-sample, the beads diameters are: 6, 10, 40, 80, 140 μm.

The graph shows that as the gradient strength increases it is necessary to decrease the time range considered to perform the data fit to the function shown in Eq.1.

S(Δ) data points for diffusing water, B12F12^{--} and liposomes in G50-Sephadex matrix. The water molecule has a linear dimension of about 0.2 nm, B12F12^{-- } is characterized by a diameter of about 2-3 nm and liposomes diameter is about 100nm. The hindered water enters and exits from all the beads pores (see Fig. 3) without constrictions, as well as the liposomes. Conversely, the B12F12^{--} has the same size as the dextran beads meshes. Therefore it will be trapped in the meshes for a greater time. For this reason, only the B12F12^{--} shows noticeable subdiffusion (α =0.76±0.03).

Details of G50-Sephadex hydrogel matrix. Sephadex beads are made of a network of chemically cross-linked dextrans. Each Sephadex bead of diameter around 40 to 160μm is characterized by surface pores from 1 to 10μm (Fig. 3A) and by a mesh of diameter around 3 nm ( Fig. 3B)