The Physics of Diffusion MRI

Water Molecules Are Always Moving

Every water molecule in your body is in constant random motion, bouncing off other molecules billions of times per second. This is Brownian motion, named after Robert Brown who first observed it in 1827 (though he was watching pollen grains, not water molecules).

In a glass of pure water, this motion is completely random — a molecule is equally likely to move in any direction. Over time, the average displacement of many molecules follows a Gaussian distribution described by the diffusion coefficient $D$:

$$\langle r^2 \rangle = 6Dt$$

where $\langle r^2 \rangle$ is the mean squared displacement and $t$ is the observation time. For free water at body temperature, $D \approx 3 \times 10^{-3}$ mm$^2$/s, which gives an RMS displacement of about 30 micrometers in 50 ms. In brain tissue, where membranes and myelin restrict movement, the effective diffusion coefficient is much lower ($D \approx 0.7\text{–}1.0 \times 10^{-3}$ mm$^2$/s) and water molecules travel roughly 10–20 micrometers in the same time — about the width of a cell.

Biological Tissue Restricts Diffusion

In the brain, water is not free. It bumps into cell membranes, myelin sheaths, organelles, and macromolecules. These barriers restrict and hinder water movement, and the pattern of restriction tells us about the underlying tissue architecture.

Three Types of Diffusion in the Brain

Type	Where	What Happens
Free (isotropic)	CSF, ventricles	Water moves equally in all directions — no barriers. $D \approx 3.0 \times 10^{-3}$ mm$^2$/s
Hindered	Extracellular space	Water can move in all directions but is slowed by obstacles — diffusivity reduced by tortuosity
Restricted	Inside axons, cells	Water is trapped within membranes and can only move along certain directions. $D \approx 0.7\text{–}1.0 \times 10^{-3}$ mm$^2$/s

White Matter Is Anisotropic

White matter axons are long, thin cylinders wrapped in myelin — a fatty insulating sheath. Water inside an axon can move freely along its length but is blocked from moving perpendicular to it by the myelin membrane. Water between axons is also preferentially channeled along the fiber direction.

This creates anisotropic diffusion — water moves farther along the fiber axis than perpendicular to it. This anisotropy is the signal that diffusion tensor imaging exploits. In terms of eigenvalues (which we define below), this means $\lambda_1 \gg \lambda_2 \approx \lambda_3$: fast diffusion along the fiber, slow diffusion across it.

Measuring Diffusion

A standard MRI scanner measures total signal from hydrogen atoms (protons in water). To make the signal sensitive to diffusion, we add diffusion-encoding gradients — brief magnetic field gradients that dephase and rephase the proton spins.

The Stejskal-Tanner Pulsed Gradient Spin Echo

The classic diffusion-encoding sequence applies two gradient pulses separated by a diffusion time. This is the Stejskal-Tanner experiment (Stejskal & Tanner, 1965):

1. 90° RF pulse: Excites the protons.
2. First gradient pulse (duration $\delta$, strength $G$): Imparts a position-dependent phase shift to every proton — protons at different locations get different amounts of phase twist.
3. Diffusion time ($\Delta \approx 50$ ms): Protons are allowed to diffuse freely.
4. Second gradient pulse (identical to the first): Reverses the phase shift — but only perfectly for protons that have not moved.
5. Signal acquisition: Measure the remaining signal.

Stationary protons receive equal and opposite phase shifts from the two gradient pulses, so their net phase is zero and they contribute full signal. Moving protons are at a different position during the second gradient pulse, so the reversal is imperfect — they retain residual phase, causing destructive interference and signal loss. More movement means more signal loss.

The Stejskal-Tanner Equation

This signal attenuation is described mathematically:

$$S = S_0 \, e^{-b \cdot D}$$

where:

• $S_0$ is the signal without diffusion weighting (b=0 image)
• $b$ is the b-value, which controls how sensitive the measurement is to diffusion
• $D$ is the apparent diffusion coefficient along the gradient direction

The b-value itself is determined by the gradient parameters:

$$b = \gamma^2 G^2 \delta^2 \left(\Delta - \frac{\delta}{3}\right)$$

where $\gamma$ is the gyromagnetic ratio (2.675 $\times$ 10$^8$ rad/s/T for hydrogen), $G$ is the gradient strength, $\delta$ is the gradient pulse duration, and $\Delta$ is the time between the two gradient pulses (Stejskal & Tanner, 1965).

The b-value

The b-value determines how strongly the image is weighted by diffusion:

b-value (s/mm$^2$)	Sensitivity	Signal Level	Typical Use
0	None — anatomical image	High	Reference image for normalization
~1000	Moderate	Medium	Standard DTI — good balance of diffusion contrast and SNR
~2000–3000	High	Low	Advanced models (CSD, NODDI) — stronger angular contrast
5000+	Very high	Very low	Specialized microstructural models

Higher b-values give stronger diffusion weighting (better contrast between different tissue types) but lower signal-to-noise ratio. Most DTI studies use b=1000 s/mm$^2$ as a good compromise.

Gradient Directions

Each diffusion-weighted volume measures diffusion along one specific direction in 3D space. To fully characterize the 3D diffusion profile at each voxel, you need measurements along multiple directions:

• Minimum: 6 non-collinear directions (enough to fit the tensor)
• Typical: 30–64 directions (better angular resolution, more robust tensor fit)
• High angular resolution (HARDI): 60–128+ directions (needed for crossing fiber models)

The gradient directions are stored in a .bvec file — three rows of numbers specifying the x, y, z components of each gradient direction.

The Diffusion Tensor

At each voxel, the diffusion tensor is a 3$\times$3 symmetric positive-definite matrix that describes the 3D diffusion profile:

$$\mathbf{D} = \begin{pmatrix} D_{xx} & D_{xy} & D_{xz} \\ D_{xy} & D_{yy} & D_{yz} \\ D_{xz} & D_{yz} & D_{zz} \end{pmatrix}$$

This matrix has 6 unique elements (the diagonal plus the three off-diagonal elements), which is why you need at least 6 gradient directions plus a b=0 image to solve for them.

Eigendecomposition

The tensor is decomposed into three eigenvectors (directions) and three eigenvalues (magnitudes):

• $\lambda_1$ (largest eigenvalue): Diffusion rate along the principal direction — corresponds to the fiber axis in white matter
• $\lambda_2$: Diffusion rate along the second axis
• $\lambda_3$ (smallest eigenvalue): Diffusion rate along the third axis — perpendicular to the fiber

Tensor Ellipsoid Shapes

The eigenvectors define the orientation of an ellipsoid that visualizes the diffusion profile. The shape of this ellipsoid tells you about the tissue:

Ellipsoid Shape	Eigenvalue Pattern	Where You See It	Typical FA
Prolate (cigar)	$\lambda_1 \gg \lambda_2 \approx \lambda_3$	Coherent white matter (corpus callosum, corticospinal tract)	0.7 – 0.9
Oblate (pancake)	$\lambda_1 \approx \lambda_2 \gg \lambda_3$	Crossing fiber regions (centrum semiovale)	0.3 – 0.5
Sphere	$\lambda_1 \approx \lambda_2 \approx \lambda_3$	Isotropic diffusion (CSF, ventricles, gray matter)	~0

A prolate (cigar-shaped) ellipsoid indicates a single dominant fiber direction — this is what you see in large white matter tracts. An oblate (pancake-shaped) ellipsoid often indicates fiber crossings where two or more tracts intersect in a voxel. A spherical ellipsoid means diffusion is equal in all directions — characteristic of CSF or gray matter.

Scalar Metrics

From the eigenvalues, we compute the metrics used in most DTI studies:

Fractional Anisotropy (FA)

$$FA = \sqrt{\frac{3}{2}} \cdot \frac{\sqrt{(\lambda_1 - \bar{\lambda})^2 + (\lambda_2 - \bar{\lambda})^2 + (\lambda_3 - \bar{\lambda})^2}}{\sqrt{\lambda_1^2 + \lambda_2^2 + \lambda_3^2}}$$

where $\bar{\lambda} = (\lambda_1 + \lambda_2 + \lambda_3) / 3$ is the mean eigenvalue. FA ranges from 0 (isotropic — equal diffusion in all directions) to 1 (strongly directional diffusion along a single axis).

Mean Diffusivity (MD)

$$MD = \frac{\lambda_1 + \lambda_2 + \lambda_3}{3}$$

The average rate of diffusion across all three axes, regardless of direction.

Axial Diffusivity (AD)

$$AD = \lambda_1$$

Diffusion rate along the primary axis (the largest eigenvalue).

Radial Diffusivity (RD)

$$RD = \frac{\lambda_2 + \lambda_3}{2}$$

Average diffusion rate perpendicular to the primary axis.

The FA formula measures the variance of the eigenvalues relative to their magnitude. If all three eigenvalues are equal ($\lambda_1 = \lambda_2 = \lambda_3$), there is zero variance and FA = 0. If one eigenvalue is much larger than the others, the variance is high and FA approaches 1. FA was first defined by Basser & Pierpaoli (1996).

For more on what these metrics mean biologically, see What is DTI?.

Limitations of the Single-Tensor Model

The diffusion tensor assumes that diffusion at each voxel can be described by a single ellipsoid — one principal fiber direction. This works well in regions with coherently organized fibers (e.g., the corpus callosum).

However, approximately 60–90% of white matter voxels contain multiple fiber populations — crossing, kissing, or fanning fibers (Jeurissen et al., 2013). In these voxels, the single tensor averages the different fiber orientations into a misleading intermediate direction. This is why:

• FA is artificially reduced at fiber crossings (the averaging makes the ellipsoid more spherical)
• The principal eigenvector points in a direction that does not correspond to any actual fiber
• Tractography based on the principal eigenvector fails at crossings

Advanced models address this limitation:

Model	What It Does	Required Data
CSD (Constrained Spherical Deconvolution)	Estimates a fiber orientation distribution at each voxel	1+ shells, many directions
BedpostX	Estimates multiple discrete fiber orientations with uncertainty	Multi-shell preferred
NODDI	Models neurite density and orientation dispersion separately	2+ shells
DKI (Diffusion Kurtosis Imaging)	Captures non-Gaussian diffusion behavior	2+ shells

See Multi-Shell Analysis for more on these methods.

References

• Stejskal EO, Tanner JE (1965). Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. The Journal of Chemical Physics, 42(1), 288-292.
• Basser PJ, Mattiello J, LeBihan D (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66(1), 259-267.
• Basser PJ, Pierpaoli C (1996). Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance, Series B, 111(3), 209-219.
• Le Bihan D, Breton E (1985). Imagerie de diffusion in vivo par résonance magnétique nucléaire. Comptes Rendus de l'Académie des Sciences, Série II, 301(15), 1109-1112.
• Jones DK (2010). Diffusion MRI: Theory, Methods, and Applications. Oxford University Press.
• Tournier JD, Mori S, Leemans A (2011). Diffusion tensor imaging and beyond. Magnetic Resonance in Medicine, 65(6), 1532-1556.
• Jeurissen B, Leemans A, Tournier JD, Jones DK, Sijbers J (2013). Investigating the prevalence of complex fiber configurations in white matter tissue with diffusion magnetic resonance imaging. Human Brain Mapping, 34(11), 2747-2766.