Frederik Laun^{1}

Links between measured signal, the temporal profile of the diffusion-encoding gradients and spatial features of the measured probes will be explained. Among others, the temporal gradient profiles "short-short", "long-long", "long-short", and gradient profiles with multiple gradient pulses will be discussed.

**Two
gradient pulses 1: short-short**

Diffusion encoding is achieved by application of two gradient pulses of opposite polarity that are played out so quickly that particle motion is effectively “frozen” during their application. Consequently, only the particle motion in the time interval $$$\Delta$$$ between the gradient pulses is of relevance (1,2).

**Two
gradient pulses 2: long-long**

In reality, the application of short gradient pulses is challenging due to limited available gradient amplitude. Thus, diffusion encoding is achieved by “long” gradients of a non-zero duration $$$\delta$$$ and separation $$$\Delta-\delta$$$ . Particle motion during the gradient pulse must be taken into account (3).

**Two
gradient pulses 3: long-short
**

The first gradient pulse is long and the second gradient pulse is short (4). Their durations and amplitudes must meet the following condition: $$$ \delta_{long}G_{long}=\delta_{short}G_{short}$$$.

**Three or more short gradient pulses
**

Diffusion encoding is achieved by three or more short gradient pulses (5-7).

**Further temporal gradient profiles**

In general, basically any temporal profile $$$G(t)$$$ can be used for diffusion encoding (8). Several temporal profiles that have been described in the literature will be shortly presented.

**Two
gradient pulses 1: short-short
**

For two gradient pulses in the short pulse approximation, the measured signal equals the Fourier transform of the volume-averaged diffusion propagator. This makes possible the measurement of many propagator-derived metrics such as the time-dependent diffusion coefficient $$$D(t)$$$ and the diffusional kurtosis $$$K(t)$$$ (9). The behavior of diffusion propagator, $$$D(t)$$$, and $$$K(t)$$$ will be discussed for open and closed domains. Moreover it will be discussed that the magnitude of the form factor of the domain can be retrieved (10,11).

**Two
gradient pulses 2: long-long
**

For two long gradient pulses, the measured signal equals the Fourier transform of the volume-averaged center of mass (COM) propagator (3). Here, “COM” refers to the COMs $$$x_{com,1}$$$ and $$$x_{com,2}$$$ of the particle trajectories during first and second gradient, i.e. $$$x_{com,1}=\frac{1}{\delta}\int_0^\delta x(t)dt$$$ and $$$x_{com,2}=\frac{1}{\delta}\int_\Delta^{\Delta+\delta} x(t)dt$$$ . Several effects that occur with long gradient pulses will be discussed including edge enhancement (12). Differences between closed and open domains will be explained and it will be discussed to which extent long-long can be regarded as an approximation for short-short (13).

**Two
gradient pulses 3: long-short
**

The same mathematical framework as for long-long can be used. It will be discussed how the average shape of an ensemble of closed pores or cells can be detected by means of long-short gradient pulses making possible to perform diffusion pore imaging (14).

**Three or more short gradient pulses
**

Dividing the signal obtained with short-short-short and short-short gradient profiles can be used to estimate the shape of point-symmetric pores (15). By means of a recursive reconstruction, the shape of arbitrary domains (16) and of periodic lattices can be retrieved (17). Differences between the pore imaging techniques relying on multiple short gradient pulses and on long-narrow gradient pulses will be discussed.

**Further temporal gradient profiles
**

Some properties of further temporal gradient profiles will be summarized and discussed.

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2. Stejskal EO, Tanner JE. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J Chem Phys 1965;42(1):288-292.

3. Mitra PP, Halperin BI. Effects of Finite Gradient-Pulse Widths in Pulsed-Field-Gradient Diffusion Measurements. J Magn Reson Ser A 1995;113(1):94-101.

4. Laun FB, Kuder TA, Semmler W, Stieltjes B. Determination of the Defining Boundary in Nuclear Magnetic Resonance Diffusion Experiments. Physical review letters 2011;107(4).

5. Mitra PP. Multiple Wave-Vector Extensions of the NMR Pulsed-Field-Gradient Spin-Echo Diffusion Measurement. Physical Review B 1995;51(21):15074-15078.

6. Özarslan E, Basser PJ. MR diffusion - "diffraction" phenomenon in multi-pulse-field-gradient experiments. J Magn Reson 2007;188(2):285-294.

7. Shemesh N, Jespersen SN, Alexander DC, Cohen Y, Drobnjak I, Dyrby TB, Finsterbusch J, Koch MA, Kuder T, Laun F, Lawrenz M, Lundell H, Mitra PP, Nilsson M, Ozarslan E, Topgaard D, Westin CF. Conventions and nomenclature for double diffusion encoding NMR and MRI. Magnetic resonance in medicine 2016;75(1):82-87.

8. Grebenkov DS. NMR survey of reflected Brownian motion. Reviews of Modern Physics 2007;79(3):1077-1137.

9. Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med 2005;53(6):1432-1440.

10. Callaghan PT, Coy A, Macgowan D, Packer KJ, Zelaya FO. Diffraction-Like Effects in NMR Diffusion Studies of Fluids in Porous Solids. Nature 1991;351(6326):467-469.

11. Cory DG, Garroway AN. Measurement of translational displacement probabilities by NMR: an indicator of compartmentation. Magnetic resonance in medicine 1990;14(3):435-444.

12. De Swiet TM. Diffusive Edge Enhancement in Imaging. Journal of Magnetic Resonance Series B 1995;109(1):12-18.

13. Zielinski LJ, Sen PN. Effects of finite-width pulses in the pulsed-field gradient measurement of the diffusion coefficient in connected porous media. Journal of magnetic resonance 2003;165(1):153-161.

14. Laun FB, Kuder TA, Wetscherek A, Stieltjes B, Semmler W. NMR-based diffusion pore imaging. Physical review E, Statistical, nonlinear, and soft matter physics 2012;86(2 Pt 1):021906.

15. Shemesh N, Westin CF, Cohen Y. Magnetic resonance imaging by synergistic diffusion-diffraction patterns. Phys Rev Lett 2012;108(5):058103.

16. Kuder TA, Laun FB. NMR-based diffusion pore imaging by double wave vector measurements. Magnetic resonance in medicine 2013;70(3):836-841.

17. Laun FB, Muller L, Kuder TA. NMR-based diffusion lattice imaging. Physical Review E 2016;93(3).