Silvia Capuani CNR-IPCF UOS Roma Physics Department Sapienza University of Rome Piazzale Aldo Moro 2, 00185 Rome, Italy

1. MR Measurements of Anomalous Diffusion indices α and μ from diffusion signalat varying of diffusion time and gradient strength may be a new probe to tissue features Silvia Capuani CNR-IPCF UOS Roma Physics Department Sapienza University of Rome Piazzale Aldo Moro 2, 00185 Rome, Italy silvia.capuani@roma1.infn.it

2. Summary Anomalous diffusion: What is that? Why should we study it? What indices of AD can we estimate experimentally? and how can α and μ be determined? Validation in phantoms Preliminary results obtained in biological samples and potential applications in vivo Conclusion and… critical review of previous literature

3. C T R W Normal. Gaussian ν=1 Subdiffusion, ν<1 t = waiting time r = displacement length Anomalous Diffusion <r2(t)> ≈ tν ν< 1 subdiffusion ν> 1 superdiffusion Superdiffusion, ν>1

4. Theory……very briefly The features of anomalous diffusion (AD) can be described by defining the motion propagator (MP) as the solution of fractional diffusion equations, which arise from CTRW model. These equations involve two fractional exponents αand µ which are the orders of the time and space fractional derivatives, respectively. 0<α<1 , 0<μ<2 As a consequenceα quantifies subdiffusion processes and µ quantifies superdiffusion processes. α=1 and µ=2 ordinary Brownian diffusion PFG Signal is proportional to FT of MP Remember! R. Metzler e J. Klafter, Physics Report, 339:1, 2000

5. How to determine α and  PGSE α μ Signal Signal Δ (s) q (m-1)

6. Validation in phantoms Materials: …to simulate biological tissues… ΔХ=| Х(H20) – Х(polystyrene)| =1.6 It is known that water diffusion in these kind of interconnected micro-pore systems is Brownian or subdiffusive. No superdiffusion!

7. Validation in phantoms μ ordered α ordered disordered α disordered The first experimental diffusion phase diagram in heterogeneous media as predicted by CTRW. M. Palombo, A. Gabrielli, S. De Santis, C. Cametti, G. Ruocco and S. Capuani, Phys Rev Lett, submitted

8. Validation in phantoms The role of magnetic susceptibility difference R=0.996 Gint = internal gradient quantifies ΔХ

9. (<z2(t)>)1/2 Pseudo superdiffusion Gi 2 (<z2(t)>)1/2 Δφ ≈|g| 0 Z Superdiffusion process ( μ <2 ) is spurious, not real but due to local ΔX (or Gi) at the interface between beads and water

10.  Parameter was already known… It is twice the stretching  parameter of the stretched exponential model introduced by Bennett et al. (Bennett KM, et al. MRM 2006;56:235-240) M. Palombo, MS thesis in Physics, Sapienza University of Rome

11. Imaging results in phantoms MD map (b=1000s/mm2) M map Local ΔX map H2 O 6 and10 µm TiO2 5µm 300nm 6 µm 6 µm 10 µm Mαmap Disorder degree map T2 weighted image M. Palombo, S. De Santis, A. Gabrielli, S. Capuani, manuscript in preparation

12. How to exploit pseudo-superdiffusion Mγ vs MD Mean values and SD obtained from 10 subjects Talamus (WM+GM) Internal capsule (WM) Splenium of CC (WM) S. De Santis, A. Gabrielli, M. Bozzali, B. Maraviglia, E. Macaluso and S. Capuani, MRM 2011 in press

13. Conclusion We measured AD indices α and μ in heterogeneous media using both Spectroscopic and Imaging modalities 0<α<1 quantifies subdiffusion and it measures the disorder degree of media It is possible to measure “α” by changing Δ at a fixed g strength in PFG experiment and fitting the signal decay with the FT of subdiffusion MP 0<μ<2 quantifies superdiffusion and it is equivalent to 0<<1 stretching parameter of the stretched-exponential model recently introduced to quantify non-Gaussian diffusion. µ=2 is measured by changing g strength at a fixed Δ and fitting the signal decay with FT of superdiffusion MP We demonstrate that, when investigating systems for which only subdiffusive or Brownian motion is expected, ΔХ within media emulates superdiffusion with measured μ values less than 2

14. In other words, due to the peculiar features of NMR technique, µ (or  ) does not quantify a real superdiffusion mechanism but quantifies local ΔХ! (*) We also demonstrate that  (or ) values are strongly correlated with local Gis which quantify ΔХ As a consequence, using the stretched-exponential model we can probe local ΔХ at the interface between different tissues, inferring on microstructural sizes of media. The real source of contrast in the stretched-exponential model is not AD but Gi! (*) Both α and γ maps show different contrasts compared to conventional MD maps, Being related to microstructural changes, α and γrepresent promising tools to detect pathological abnormalities in human tissues in vivo (*) critical revision of the findings related to AD

15. Marco Palombo PhD student in biophysics Physics Department Sapienza University of Rome, Italy E-mail: gani18@libero.it Silvia Capuani Silvia De Santis now in Cardiff (UK) Cardiff University Brain Research Imaging Centre (CUBRIC) E-mail: desantiss@cardiff.ac.uk Andrea Gabrielli Statistical Mechanics and Complexity Center CNR-ISC (Complex Systems Institute) Rome, Italy E-mail: andrea.gabrielli@roma1.infn.it Thank you silvia.capuani@roma1.infn.it

16. Erice, Sicily (Italy) May 25- June 1, 2011 One day meeting on NMR Anomalous Diffusion ADD Symposium which will take place in Erice-Sicily (Italy), within the context of the “IX workshop of the International School on Magnetic Resonance and Brain Function” (May 25- June 1, 2011). The primary objective of the NMR ADD (Anomalous Diffusion Discussion) Symposium is to improve the knowledge, the understanding and the potential application of NMR Anomalous Diffusion for the investigation of biological systems. silvia.capuani@roma1.infn.it