Fick's first law of diffusion: the substance transfer J through an area is proportional to the gradient of concentration c (i.e. δc/δx):
where D is diffusivity, also termed diffusion coefficient, has a unit of mm2/s. The higher the diffusivity, the faster they diffuse.
The self-diffusion of water molecules follows the Brownian motion.
Einstein  derived the root mean squared displacement of a self-diffusion as , where D is diffusivity, and d is also termed the diffusion distance. Once should note that diffusivity is more like a velocity measurement (distance over time) not a quantity measurement (quantity over time). High diffusivity implies faster diffusion, not necessarily a large quantity of diffusion.
The self-diffusion estimated here assumes (1) no structural restriction (2) homogeneity in the diffusion environment.
1. Diffusion coefficient, a.k.a diffusivity, is a velocity measurement quantifying how fast substance diffuses.
2. Diffusion distance: the root mean squared displacement of free self-diffusion.
The effect of diffusion on the MR signal was mentioned by Hahn in 1950 . Carr and Purcell in 1954  furthered derived the mathematical framework and measured the diffusion coefficient using a Car-Purcell-Meiboom-Gill (CPMG) echo train sequence with background diffusion gradient. This can be applied to study how fast molecules (e.g. water, glycerol) diffuses by themselves, and the measurement is thus called self-diffusion coefficient. A more popular approach, the Stejskal-Tanner sequence, also known as the pulse-gradient spin echo (PGSE) sequence, was developed in 1960's to measure the diffusion coefficient (Fig. 1A). The modified version of this sequence is still used today in most of the clinical scanners. The diffusion signals (S) and diffusivity (D) have the following relation.
δ: diffusion-encoding gradient duration
Δ: the diffusion time
g: diffusion-encoding gradientγ: gyromagnetic ratio
S0: diffusion signal without diffusion weighting (the b0 signal)
The diffusion coefficient can thus be measured by acquiring diffusion signals with difference gradient strength |g|.
Fig.1 (A) the Stejskal-Tanner sequence (B) Diffusion-encoded stimulated echo sequence. The diffusion time (Δ) equals TE/2 plus the mixing time (TM) and is thus much longer than TE. (C) Stimulated echo sequence with bipolar diffusion-encoding gradients. Bipolar gradients lengthen the TE significantly and become limiting. (D) Stejskal-Tanner sequence with bipolar diffusion-encoding gradients.
source: Sosnovik, D. E., Wang, R., Dai, G., Reese, T. G., & Wedeen, V. J. (2009). Diffusion MR tractography of the heart. Journal of Cardiovascular Magnetic Resonance, 11(1), 1-15. [link]
The self-diffusion coefficient can be measured using an MR experiment.
Assumption of free diffusion.
1. Diffusion sequence uses gradients to encode/sensitize diffusion.
2. Diffusion is presented by an exponential signal attenuation.
3. The diffusion signal attenuation is determined by diffusion coefficient, diffusion time, and the strength/duration of the diffusion sensitization gradient.
The diffusion pulse sequence developed in 1960s allowed for measuring the diffusivity of free water diffusion using an NMR experiment, and it was yet to be combined with MRI (Lauterbur, 1973) to spatially map the distribution of diffusion coefficient. This idea gave birth to diffusion-weighted imaging at the mid-1980's , and Michael Moseley introduced its an important role in the diagnosis of ischemic stroke .
Diffusion MRI allows for spatial mapping of the diffusion signals.
The combination of a diffusion sequence and k-space imaging makes diffusion MRI possible. However, due to the additional diffusion gradients, diffusion MRI often has prominent eddy current artifact, susceptibility artifact, and fat saturation artifact (note that these artifacts also exist in other types of MRI sequence).
Eddy current induced by gradient switch affects the signal readout. In diffusion-weighted images, it results in shearing of the image volume. The solutions are (1) correcting the partial distortion by a linear transformation, (2) using a bipolar pulse to for diffusion encoding (Fig. 1D).
Diffusion gradient encoding increases echo time and results in greater susceptibility artifact. A longer diffusion time also increases echo time. The solutions are Solution: (1) Using a stronger gradient coil to reduce encoding duration. (2) Use a stimulated echo sequence that allows for long diffusion time and short TE. (But SNR is reduced by half in stimulated echo)
Fat saturation artifact becomes prominent as b-value increases. Fat signals can be greatly reduced using fat suppression. However, diffusion MRI relies on signal attenuation (high b-value, low signal) to quantify diffusion coefficient, but fat does not diffuse. As a result, in high b-value images, signals from water are gone while the fat signals remain. Consequently, it causes errors in diffusion analysis. The solutions are (1) adjust FOV to avoid wrapped around of the fat saturation (not practical for high-resolution imaging), (2) use a better fat saturation sequence.
The figure shows the fat saturation artifact in b0 (b) and FA map (c).
source: Yeh, P. H., Oakes, T. R., & Riedy, G. (2012). Diffusion tensor imaging and its application to traumatic brain injury: basic principles and recent advances.
1. Diffusion MRI usually suffers from the susceptibility artifact, fat saturation artifact, and eddy current artifact.
2. The eddy current artifact can be corrected using a software tool (provided that the SNR is good enough) or a set of specially designed diffusion gradients.
1.What is the relation between diffusion time/diffusion encoding gradient and the diffusion signals?
2.A study uses capillary tubes to create a diffusion phantom, and the diameter of the tube is larger than axon. How would you setup the diffusion time and diffusion gradient strength?
3.Why does a longer diffusion time require a longer TE in the Stejskal-Tanner PGSE sequence?
4.Are diffusion signal mostly T1- or T2-weighted? Why?
5.Scan A has TE=100 ms whereas scan 2 has TE=130 ms. What is the SNR difference in diffusion MRI at white matter? (white matter T2 = 69 ms, T1 = 1080 ms)
6.Why there are more eddy current artifact, susceptibility artifact, and fat saturation artifact in diffusion MRI?
1.Load an example diffusion MRI data set (see instructions in Read DICOM, NIFTI, Bruker 2dseq, or Varian FDF Files). Open the SRC file created in STEP and switch to source tab to inspect the raw DWI.
2.Compare the b0 with one DWI. Which regions brighten up? Why?
3.Point out the eddy current artifact, susceptibility artifact, motion artifact, fat sat artifact.
 Einstein, Albert. Investigations on the Theory of the Brownian Movement. Courier Corporation, 1956. (pdf)
 E L Hahn (1950). Spin echoes. Physical Review, 80:580–594.
 Carr, H. Y., & Purcell, E. M. (1954). Effects of diffusion on free precession in nuclear magnetic resonance experiments. Physical review, 94(3), 630. [link]
 Stejskal, E. O., & Tanner, J. E. (1965). Spin diffusion measurements: spin echoes in the presence of a time‐dependent field gradient. The journal of chemical physics, 42(1), 288-292. [link]
 Le Bihan, D., Breton, E., Lallemand, D., Grenier, P., Cabanis, E., & Laval-Jeantet, M. (1986). MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology, 161(2), 401-407. [link]
 Moseley, M. E., J. Kucharczyk, J. Mintorovitch, Y. Cohen, J. Kurhanewicz, N. Derugin, H. Asgari, and D. Norman. "Diffusion-weighted MR imaging of acute stroke: correlation with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats." American Journal of Neuroradiology 11, no. 3 (1990): 423-429. (link)