Apparent diffusion coefficient and the tensor model

Denis Le Bihan applied diffusion pulse sequence to clinical MR imaging studies in 1980's and presented the first diffusion images in August 1985 at the Society of Magnetic Resonance in Medicine meeting. He was the first to map diffusivity in human brain [1] and showed that diffusion MRI (previously diffusion NMR) is feasible. To simplify the exponential relation between diffusivity and diffusion weighted signals. He proposed using a "b factor" to simplify the diffusion signal pattern:

whereas S is the DWI signal. S0 is the DWI signals without diffusion weighting (also known as the b0 signal)

The apparent diffusion coefficient, as explained by its naming, is the diffusion coefficient "appearing" from the observation of heterogeneous environment (e.g. biological tissues). 


Apparent diffusion coefficient has numerous applications in imaging diagnosis.


1. Biological tissue has a microscopic structure that limits diffusion. The exponential relation does not hold if there is restricted diffusion.
2. ADC is an ensemble measurement of how fast water diffuses in a complex diffusion environment. Consequently, in an environment mixed with free diffusion and restricted diffusion, ADC only reveals an overall measurement and is thus b-value dependent. 
3. The ADC measurement is directionally dependent. The measurement parallel to white matter bundle is higher than that perpendicular to the bundle.


1. b-value is a product of three parameters related to diffusion sensitization, including diffusion time, diffusion gradient strength and duration.
2. Apparent diffusion coefficient is the "measured diffusion coefficient" of water in biological tissues.

Diffusion tensor imaging

ADC may not be uniform at all orientations, and there is a need to model the difference of ADC at different orientationsPeter Basser introduced diffusion tensor imaging in 1990's [2][3]. He used a tensor to model the orientation distribution of the apparent diffusion coefficient. The tensor model is by far the most popular diffusion model used. To understand the tensor model, one may compare it to a 3D Gaussian distribution.

Started from one-dimensional ADC estimation:

Here the diffusion gradient g come with an additional unit vector of it orientation, "g hat", and the diffusivity D can be replaced by a 3 x 3 matrix, known as the tensor.

This equation has an identical format as a 3D Gaussian distribution:

The principle direction can be calculated by eigenanalysis.

Figure: Jellison, Brian J., et al. "Diffusion tensor imaging of cerebral white matter: a pictorial review of physics, fiber tract anatomy, and tumor imaging patterns."American Journal of Neuroradiology 25.3 (2004): 356-369.

Eigen analysis on tensor yields:

1. Principle fiber direction
2. Axial diffusivity
3. Radial diffusivity
4. Mean diffusivity
5. Fractional anisotropy


DTI allows for measuring diffusivity at any 3D orientation and provide principle direction that tells axonal direction. It also provides diffusivity-based metrics for characterizing diffusion biophysics.


1. Diffusion tensor assumes that the diffusion is Gaussian distribution, but, in reality, the diffusion in biological tissue is often restricted by microscopic structures. 
2. Diffusion tensor cannot model multiple fiber populations.
3. Diffusion tensor also inherits the limitation of ADC: b-value dependency.


1. A diffusion tensor is a 3D presentation of the diffusivity.
2. A tensor can be analyzed to get the principle direction.
3. A diffusion tensor assumes Gaussian distribution. It cannot model restricted diffusion or multiple axonal directions.

1. How to estimate ADC using a diffusion MRI data set?
2. Is the ADC estimated affected by T1W, T2W, or spin density?
2. What are the pros/cons of using a low b-value acquisition for ADC estimation? How about a high b-value acquisition?
3. In human brain, white matter ADC obtained from b1 = 1000 mm2/sec and one b0 is ~0.8 x 10^-3 mm2/sec. The ADC value acquired from = 3000 sec/mm2 is ~ 0.6 x 10^-3 mm2/sec. Why is there a difference in ADC measurement?
4. What is the minimum requirement of diffusion images for calculating a tensor?
5. Is the diffusion tensor affected by T1w, T2w, or spin density? Does it scales with spin density? 
6. Does T1w, T2w, spin density affect FA and diffusivity measurement?
7. How do eddy current artifact, susceptibility artifact, motion artifact, and fat sat artifact affect DTI reconstruction result? 


1. Reconstruct DTI from DWI (follow instructions in Reconstruction(DTI, QBI, DSI, GQI, QSDR)) and generate FIB files. 
2. Load the FIB file in STEP3 fiber tracking to visualize FA map, ADC map, fiber direction map. Is the FA map noisy? Where are the noises coming from?
3. Choose a brain region where you know there are crossing fibers. What does the fiber direction map show in that region?
4. Delete one or two DWI in the data set and reconstruct DTI again. Does this affect the results? Why or why not? 


[1] 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. Radiology161(2), 401-407. [link]
[2] Basser, P. J., Mattiello, J., & LeBihan, D. (1994). Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance, Series B103(3), 247-254. [link]
[3] Basser, P. J., Mattiello, J., & LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical journal66(1), 259. [link]