NMR Advanced methodologies to investigate water diffusion in materials and biological systems

NMR Advanced methodologies to investigate water diffusion in materials and biological systems PhD Candidate _Silvia De Santis PhD Supervisors _dott. Silvia Capuani _prof. Bruno Maraviglia

Outlook Introduction: _Water Diffusion _Experimental technique: NMR Diffusion NMR: applications to the study of complexity in materials and biological systems _Diffusion in biological environments _Diffusion in colloidal glasses Diffusion NMR in the framework of multi-quantum coherences Project outline

Introduction_Water Diffusion Water diffusion is the principal phenomenon regulating the system dynamics at the mesoscopic lengthscales Interacting water: free water _Simple Colloidal Suspensions _Complex Biological Systems

Introduction_NMR NMR allows the characterization of the diffusive dynamics S(t) = f (N,T 1,T 2,J,CS,D ) S(b)=S 0 exp(-bd) b=γ 2 G 2 δ 2 (Δ-δ/3) δ δ Diffusion-sensitized sequence

Introduction_NMR Diffusion Coefficient <r 2 > = 6Dt _homogeneous and isotropic mean Apparent Diffusion Coefficient _restricted diffusion: D depends on the diffusion time Diffusion Tensor _anisotropic diffusion: D depends on the observation direction

Diffusion Tensor Imaging Introduction_DTI ( )! ln # S TE " S( 0) $ & = ' % 3 3 (( i=1 j =1 b ij D ij eff! D D D " xx xy xz # $ = # yx yy yz $ D D D D # D D D $ % zx zy zz & D! D 0 0 " 1 # $ = 0 D2 0 # 0 0 D $ % 3 & _mean diffusivity _fractional anisotropy

Introduction_DTI Courtesy of Alberto Bizzi Isotropy VS Anisotropy

Outlook Due to its sensibility to the environment where the water protons diffuse, Diffusion NMR is peculiarly suitable for the study of complex systems _biological matter_porous systems_colloids_etc. It is perhaps necessary to overcome the classical description in order to take into account the influence of the material/tissue microstructure properly

Application_Biological environments The hypothesis under which the description of the signal obtained from a diffusion-sensitized sequence can be made by a single exponential decay, is the Gaussian shape of the motion propagator S(b) = S 0 exp (!bd) Water diffusion in biological tissues is often non-monoexponential as seen experimentally, especially at high b-values The inability of fitting the signal as a single exponential function may be linked to the failure of the gaussian approximation

Application_Biological environments How to solve this problem? Empirically introduced stretched exponential form of the signal decay: It is possible to map the space depending on the anisotropy of the anomalous exponent: Hall, MRM 59 (2008) 447

Application_Biological environments Cortical and deep GM can be differentiated from WM and CSF Hall, MRM 59 (2008) 447

Application_Biological environments Our proposal: 1. The signal decay expression must be obtained theoretically using the Continuous Time Random Walk theory and choosing the suitable distribution of the waiting times between 2 adjacent steps to take into account trapping phenomena 2. The parametric maps must be based on tensorial quantities to obtain information independent on the direction in which the measurements are performed

1. Exact signal decay expression Application_Biological environments Hall and Barrick Formally the two description are different, but in an usual diffusion sensitized sequence the diffusion time is kept fixed

2. Tensor reconstruction Application_Biological environments In Gaussian diffusion approximation D is a tensor: In the case of subdiffusion instead: It is not correct to consider γ a tensor!

2. Tensor reconstruction Application_Biological environments Genu of the CC Human brain Conventional γ depends on the direction of the applied gradient!

Results_Bone marrow Lipid Weakly bound water Bulk water Liquid Phase Solid Matrix 2/γ S. De Santis and S. Capuani. Proc. Intl. Soc. Mag. Reson. Med. 17 (2009) H. H. Ong et al. Proc. Intl. Soc. Mag. Reson. Med. 17 (2009)

Results_Human Brain MD FA _Conventional Diffusion Tensor MNG NGA _NonGaussian Diffusion Tensor S. De Santis and S. Capuani. Proc. Intl. Soc. Mag. Reson. Med. 17 (2009)

Results_Human Brain MD*10-3 mm 2 /s FA MNG ANG ROI1 0.86±0.34 0.83±0.10 175±71 0.87±0.13 ROI1L 0.82±0.23 0.78±0.11 159±70 0.85±0.19 ROI2 0.83±0.09 0.37±0.10 217±50 0.46±0.23 ROI3 0.96±0.20 0.19±0.07 409±296 0.40±0.32

Perspectives_Human Brain To implement the theoretical description of the phenomenon of the non-gaussian diffusion To investigate anomalous water diffusion on phantoms simulating the bone marrow (PolyStyrene spheres in water at different concentration and polidispersity conditions) To apply a clinical protocol (already approved by the S.Lucia Foundation ethical committee) to investigate human brain morphology by means of the non-gaussian tensor, both in diseased and healthy subjects

Application_Colloidal glasses Laponite is a synthetic disc-shaped crystalline colloid AGING Anisotropic diffusion Restricted diffusion

Preliminary results show that the arrested state is characterized by water anisotropic dynamics _Laponite 3% w/w D x =(2.152±0.005)*10-9 m 2 /s D y =(2.151±0.006)*10-9 m 2 /s D z =(2.125±0.008)*10-9 m 2 /s _Laponite 1.4% w/w Application_Colloidal glasses _Laponite 2.4% w/w D x =(2.159±0.009)*10-9 m 2 /s D y =(2.170±0.003)*10-9 m 2 /s D z =(2.131±0.005)*10-9 m 2 /s G // z D=(2.08±0.01)*10-9 m 2 /s G // x D=(2.11 ±0.01)*10-9 m 2 /s

Application_Multiquantum coherences Diffusion may also be important in understanding the contrast due to the Multiquantum Coherences (MQCs) Conventional loss of coherence effects in liquids are mainly due to the dipolar interactions, which are long-ranged Dipolar interaction are averaged out thanks to: _molecular diffusion at short distances _simmetry at long distances SHORT RANGE LONG RANGE Capuani, MRM 46 (2001) 683

Application_Multiquantum coherences Diffusion may also be important in understanding the contrast due to the Multiquantum Coherences (MQCs) If the simmetry is broken, for example with a linear gradient, it is possible to re-introduce long range dipolar interactions The peculiarity of this mechanism is the possibility of tuning the characteristic distance at which the spins interact (porous system) Capuani, MRM 46 (2001) 683

Application_Multiquantum coherences Quantum-mechanical description: High temperature approximation CRAZED Sequence Lee J. Chem. Phys. 105 (1996) 3

Application_Multiquantum coherences The origin of the obtained contrast is still unclear since molecular diffusion during the experimental time has been taken into account only in the classical formalism and only a posteriori and/or numerically BUT Diffusion is critical when dealing with porous system! Trabecular bone network in calf spongy bone samples Our aim is to investigate porous systems with known distances to elucidate the role of diffusion in MQCs refocused signal In the meanwhile our aim is also the development of a theoretical description of MQCs signal decay including diffusive dynamics S. De Santis et al. Proc. Intl. Soc. Mag. Reson. Med. 17 (2009) De Santis et al. Phys Med Biol submitted

In spite of their widespread popularity, Diffusion-based NMR techniques still offer several possibilities of innovative applications For my PhD thesis my aim will be: Project Outline to apply conventional diffusion sequences to obtain further information about the arrested state of Laponite to clarify the role of diffusion in the multiquantum contrast, in order to exploit its peculiar suitability for porous systems description to develop the formalism of diffusion out of the ideal condition and apply the non-gaussian DTI protocols to biological systems (like the brain and the bone marrow) where the environment experimented by the water protons cannot be modelled as entrapment-free

Thanks for your attention!

Additional Material_Link between NMR signal and gaussian propagator Spin phase:! ( t) = " B 0 t + " g! # rt!!" = #$! g % E!! g! r '&! r ( ) The NMR signal is proportional to the modulation given by: ( ) = " ( r) ( ) ( ) # # P r! r! ',! exp '( i$% g! r! & r! ' ) * d! rd! r ' exp( i!" ) If the probability is independent of the spin position and defining:! q = 2! ( ) "1 #$! g E!! q ( ) ( ) = P! R,! " exp % i2# q! $ R! & ' ( d R! E! q FT ( ) ( ) P! R

Additional Material_Link between NMR signal and gaussian propagator P ( R,!! ) = 4"D! $ & % ' ) 4D! ( ( ) # 3 2 exp # R2 Gaussian E!! q ( ) ( ) = P! R,! " exp % i2# q! $ R! & ' ( d R! S( g) = S 0 ( )exp % &!" 2 # 2 g 2 $D' ( Or in terms of b-value, where the finite duration δ of the gradient pulse is taken into account: S(b) = S 0 exp (!bd) b =! 2 g 2 " 2 (# $ " 3 )

Additional Material_CTRW Theory applied to subdiffusion Discrete random walk: W j ( t +!t) = 1 2 W ( t) + 1 j "1 2 W ( t) j +1!W!t W j ( t +!t) = W j ( t) +!t "W j "t W j ±1 = K 1! 2 ( t) = W x,t!x 2 W x,t + O ([!t] 2 ) ( )2 ( ) ±!x "W "x +!x 2 ([ ] 3 ) " 2 W + O!x "x 2 ( ) K 1 = lim!x"0,!t "0 (!x) 2 2!t For long times, i.e. large number of steps: W ( x,t) = 1 4!K 1 t # x2 exp % " $ 4K 1 t & ( Gaussian propagator ' (central limit theorem)

Additional Material_CTRW Theory applied to subdiffusion Validity of the Gaussian Approximation Existence of : The first two moments of the pdf describing the normalised distance covered in a jump event and the variance <x>, <x 2 > The mean time span between any two individual jump events Δt or in terms of the spin system: Particles trajectories must be a pure random walk with no memory effects with respect to probability and direction The displacements must be unrestricted on the time scale of the experiments There is no mutual obstruction of the diffusing particles

Additional Material_CTRW Theory applied to subdiffusion Jump to Continuous Time Random Walk: Δx and Δt should be drawn from a proper pdf! ( x,t) ( ) = dt" ( x,t)! x # $ 0 T =! ( ) " dtw t t 0 ( ) = dx" ( x,t) w t +# %# $ +! # 2 = " dx$ x x 2 %! ( ) T diverges and Σ 2 is finite: subdiffusion Σ 2 diverges and T is finite: superdiffusion Metzler and Klafter,Phys Rep 339 1 (2000)

Additional Material_CTRW Theory applied to subdiffusion Diverging pdf for t: # " ( )! A! 1+! & w t % ( 0 <! < 1 $ t ' Working in the Fourier-Lapalace space it s possible to find the fractional diffusion equation!w!t = " 0 D t 1"# K #! 2 ( )!x 2 W x,t Riemann-Liouville operator x 2 ( t) = 2K! "( 1+!) t!

Additional Material_CTRW Theory applied to subdiffusion Propagator: P( x,t) = 0 1 4!K " t " H 1,2 1 $ x ' 4!K " t " & % K " t " ) ( *,,, 4K " t ", + 2,0 x 2 # 1#" 2#" $ & 1 # k % 2, k ' - ) / ( / $ 1 ( 0,1), & % 2,1 '/ )/ (. *, exp # 2 # " $ " ', & ), 2 % 2 ( + " 2#" $ x & % K " t " ' ) ( # 1 1# " 2 - / / /. x! K! t!

Additional Material_CTRW Theory applied to subdiffusion P( x,t)! x " #1 exp #A x 2" FT ( ) S q,t ( )! exp "Bq 2# t ( ) Substituting! = 1 2 " # P( x,t) = *, exp ) 2 ) " # " &, % (, 2 $ 2 ' + 1 # x & 4!K " t " % $ K " t " ( ' " 2)" # x % $ K " t " ) 1)" 2)" & ( ' ) 1 1) " 2 - / / /. FT S( q,t)! exp ("Bq # t) 0 <! = 2" < 1 Chen, Soliton, Fractal, & Chaos 28 923 (2006)

Additional Material_CTRW Theory applied to subdiffusion S ( q,t)! exp ("Bq # t) b = qt S ( b) = S ( 0)exp (!Ab " ) (the A factor contains also t 1-γ respect to b, but in our experiments t is kept fixed)