# diffusion processes

### Internal gradient distributions: A susceptibility-derived tensor delivering morphologies by magnetic resonance | Scientific Reports

Gonzalo A. Álvarez, Noam Shemesh & Lucio Frydman

Scientific Reports **7**, 3311 (2017)

doi:10.1038/s41598-017-03277-9

Nuclear magnetic resonance is a powerful tool for probing the structures of chemical and biological systems. Combined with field gradients it leads to NMR imaging (MRI), a widespread tool in non-invasive examinations. Sensitivity usually limits MRI’s spatial resolution to tens of micrometers, but other sources of information like those delivered by constrained diffusion processes, enable one extract morphological information down to micron and sub-micron scales. We report here on a new method that also exploits diffusion – isotropic or anisotropic– to sense morphological parameters in the nm-mm range, based on distributions of susceptibility-induced magnetic field gradients. A theoretical framework is developed to define this source of information, leading to the proposition of internal gradient-distribution tensors. Gradient-based spin-echo sequences are designed to measure these new observables. These methods can be used to map orientations even when dealing with unconstrained diffusion, as is here demonstrated with studies of structured systems, including tissues.

### Diffusion-assisted selective dynamical recoupling: A new approach to measure background gradients in magnetic resonance | J. Chem. Phys. (2014)

**Diffusion-assisted selective dynamical recoupling: A new approach to measure background gradients in magnetic resonance**

Gonzalo A. Álvarez, Noam Shemesh and Lucio Frydman

J. Chem. Phys. **140**, 084205 (2014); http://dx.doi.org/10.1063/1.4865335

Dynamical decoupling, a generalization of the original NMR spin-echo sequence, is becoming increasingly relevant as a tool for reducing decoherence in quantum systems. Such sequences apply non-equidistant refocusing pulses for optimizing the coupling between systems, and environmental fluctuations characterized by a given noise spectrum. One such sequence, dubbed Selective Dynamical Recoupling SDR [P. E. S. Smith, G. Bensky, G. A. Álvarez, G. Kurizki, and L. Frydman, Proc. Natl. Acad. Sci. **109**, 5958 (2012)], allows one to coherently reintroduce diffusion decoherence effects driven by fluctuations arising from restricted molecular diffusion [G. A. Álvarez, N. Shemesh, and L. Frydman, Phys. Rev. Lett. **111**, 080404 (2013)]. The fully-refocused, constant-time, and constant-number-of-pulses nature of SDR also allows one to filter out “intrinsic” T1 and T2 weightings, as well as pulse errors acting as additional sources of decoherence. This article explores such features when the fluctuations are now driven by unrestricted molecular diffusion. In particular, we show that diffusion-driven SDR can be exploited to investigate the decoherence arising from the frequency fluctuations imposed by internal gradients. As a result, SDR presents a unique way of probing and characterizing these internal magnetic fields, given an a priori known free diffusion coefficient. This has important implications in studies of structured systems, including porous media and live tissues, where the internal gradients may serve as fingerprints for the systems composition or structure. The principles of this method, along with full analytical solutions for the unrestricted diffusion-driven modulation of the SDR signal, are presented. The potential of this approach is demonstrated with the generation of a novel source of MRI contrast, based on the background gradients active in an ex vivo mouse brain. Additional features and limitations of this new method are discussed.

© 2014 AIP Publishing LLC

### Measuring small compartment dimensions by probing diffusion dynamics via Non-uniform Oscillating-Gradient Spin-Echo NOGSE NMR | J. Magn. Reson. (2013)

**Measuring small compartment dimensions by probing diffusion dynamics via Non-uniform Oscillating-Gradient Spin-Echo NOGSE NMR**

Noam Shemesh, Gonzalo A. Álvarez, Lucio Frydman.

Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel.

J. Magn. Reson. **237**, 49–62 (2013).

Highlights:

•NOGSE, a novel diffusion MR approach for measuring pore sizes, is presented and analyzed.

•NOGSE is based on a constant time and a constant number of oscillating gradients.

•Experiments on microstructural phantoms, spinal cords and brains, validate NOGSE.

Abstract:

Noninvasive measurements of microstructure in materials, cells, and in biological tissues, constitute a unique capability of gradient-assisted NMR. Diffusion–diffraction MR approaches pioneered by Callaghan demonstrated this ability; Oscillating-Gradient Spin-Echo OGSE methodologies tackle the demanding gradient amplitudes required for observing diffraction patterns by utilizing constant-frequency oscillating gradient pairs that probe the diffusion spectrum, Dω. Here we present a new class of diffusion MR experiments, termed Non-uniform Oscillating-Gradient Spin-Echo NOGSE, which dynamically probe multiple frequencies of the diffusion spectral density at once, thus affording direct microstructural information on the compartment’s dimension. The NOGSE methodology applies N constant-amplitude gradient oscillations; N − 1 of these oscillations are spaced by a characteristic time x, followed by a single gradient oscillation characterized by a time y, such that the diffusion dynamics is probed while keeping N − 1x + y ≡ TNOGSE constant. These constant-time, fixed-gradient-amplitude, multi-frequency attributes render NOGSE particularly useful for probing small compartment dimensions with relatively weak gradients – alleviating difficulties associated with probing Dω frequency-by-frequency or with varying relaxation weightings, as in other diffusion-monitoring experiments. Analytical descriptions of the NOGSE signal are given, and the sequence’s ability to extract small compartment sizes with a sensitivity towards length to the sixth power, is demonstrated using a microstructural phantom. Excellent agreement between theory and experiments was evidenced even upon applying weak gradient amplitudes. An MR imaging version of NOGSE was also implemented in ex vivo pig spinal cords and mouse brains, affording maps based on compartment sizes. The effects of size distributions on NOGSE are also briefly analyzed.

Keywords:

Restricted diffusion; Oscillating gradients; OGSE; Microstructure; Magnetic resonance imaging; CNS; Gradient echoes; Selective dynamical recoupling

Graphical abstract:

### Random – but not quite: exploiting quantum decoherence as a tool for characterizing unknown systems | Seminar

**SEMINAR at IV Quantum Information Workshop – Paraty 2013**

Wednesday, August 14th 2013

See comments of the talk at the Paraty 2013’s Blog: Decoupling system and environment.

**SEMINAR at CBPF, Rio de Janeiro – Brazil, August 20th, 2013**

**SEMINAR at FaMAF, Córdoba – Argentina, August 27th, 2013**

**Abstract**

The ability to understand and manipulate the dynamics of quantum systems that interact with external degrees of freedom is a major challenge for fundamental quantum physics and its diverse applications, e.g., quantum information processing (QIP) or precision measurements. Progress in dynamical decoupling has lead to new ways to “protect” quantum bits from external degrees of freedom. Sometimes, however, a little bit of “recoupling” –i.e., exposure to the unknowns of the surrounding medium– can help. In this seminar, I will present a series of experimental methods implemented in NMR where by varying the “protection” given to the quantum states of ½-spins (qubits) can lead to new tools for characterizing unknown systems. In particular, I will show how Dynamical Decoupling noise spectroscopy can probe the spectrum of the environmental noise in order to find optimal methods for protecting the qubits [1]. In a new twist, I will present a method termed Selective Dynamical Recoupling (SDR), where suitable designed pulse sequence applied to the spins can selectively address specific information from the probed systems. SDR can be used to measure coupling strengths to the environment via oscillatory modulations that can serve for example to probe chemical identities derived from chemical shifts [2]. Alternatively, SDR can be designed to selectively measure the correlation time of the environmental noise where its value can be useful to probe diffusion processes in restricted spaces to extract the sizes of pores or cells in a non-invasive manner [3]. Applications of this new and simple approach can be found in materials sciences and biology and in particular it can be useful for investigating the nature of tissue compartmentalization *in vivo*, in manners which eventually could be useful in human and clinical settings.

[1] G.A. Alvarez, and D. Suter. Phys. Rev. Lett. **107**, 230501 (2011).

[2] P.E.S. Smith, G. Bensky, G.A. Alvarez, G. Kurizki, and L. Frydman. Proc. Natl. Acad. Sci. U. S. A. **109**, 5958 (2012).

[3] G.A. Alvarez, N. Shemesh, and L. Frydman. Phys. Rev. Lett. **111**, 080404 (2013).

### Random – but not quite: exploiting quantum decoherence as a tool for characterizing unknown systems | Paraty 2013

**SEMINAR at IV Quantum Information Workshop – Paraty 2013**

Wednesday, August 14th 2013

See comments of the talk at the Paraty 2013’s Blog: Decoupling system and environment.

**Abstract**

The ability to understand and manipulate the dynamics of quantum systems that interact with external degrees of freedom is a major challenge for fundamental quantum physics and its diverse applications, e.g., quantum information processing (QIP) or precision measurements. Progress in dynamical decoupling has lead to new ways to “protect” quantum bits from external degrees of freedom. Sometimes, however, a little bit of “recoupling” –i.e., exposure to the unknowns of the surrounding medium– can help. In this seminar, I will present a series of experimental methods implemented in NMR where by varying the “protection” given to the quantum states of ½-spins (qubits) can lead to new tools for characterizing unknown systems. In particular, I will show how Dynamical Decoupling noise spectroscopy can probe the spectrum of the environmental noise in order to find optimal methods for protecting the qubits [1]. In a new twist, I will present a method termed Selective Dynamical Recoupling (SDR), where suitable designed pulse sequence applied to the spins can selectively address specific information from the probed systems. SDR can be used to measure coupling strengths to the environment via oscillatory modulations that can serve for example to probe chemical identities derived from chemical shifts [2]. Alternatively, SDR can be designed to selectively measure the correlation time of the environmental noise where its value can be useful to probe diffusion processes in restricted spaces to extract the sizes of pores or cells in a non-invasive manner [3]. Applications of this new and simple approach can be found in materials sciences and biology and in particular it can be useful for investigating the nature of tissue compartmentalization *in vivo*, in manners which eventually could be useful in human and clinical settings.

[1] G.A. Alvarez, and D. Suter. Phys. Rev. Lett. **107**, 230501 (2011).

[2] P.E.S. Smith, G. Bensky, G.A. Alvarez, G. Kurizki, and L. Frydman. Proc. Natl. Acad. Sci. U. S. A. **109**, 5958 (2012).

[3] G.A. Alvarez, N. Shemesh, and L. Frydman. Phys. Rev. Lett. (2013) – in press. arXiv:1305.2794.