StejskalTanner

The Stejskal–Tanner equation is a foundational relation in nuclear magnetic resonance (NMR) and diffusion magnetic resonance imaging (MRI) that describes how the observed signal decays due to molecular diffusion when pulsed magnetic field gradients are applied. It was introduced by Stejskal and Tanner in 1965 to quantify diffusion effects in NMR experiments and later became central to diffusion-weighted MRI.

The equation is S/S0 = exp(-b D), where S is the diffusion-weighted signal, S0 is the reference signal without diffusion weighting, D is the diffusion coefficient, and b is a parameter determined by the gradient pulses. For rectangular gradient pulses of duration delta and separation Delta, b = gamma^2 G^2 delta^2 (Delta - delta/3), with G the gradient strength and gamma the gyromagnetic ratio of the observed nucleus (for protons in MRI, gamma/2π ≈ 42.58 MHz/T). This relationship links the measured signal to the microscopic diffusion properties of the material or tissue.

In practice, the Stejskal–Tanner equation underpins diffusion-weighted imaging and diffusion tensor imaging, enabling the estimation of apparent diffusion coefficients (ADC) and providing contrast related to tissue microstructure. By acquiring data at multiple b-values, researchers obtain quantitative maps of diffusion behavior.

Limitations include its assumption of mono-exponential, Gaussian diffusion within a voxel and homogeneous D; real biological tissues often exhibit multiple compartments, restricted diffusion, perfusion effects, and exchange, which can cause deviations from the model. High b-values also reduce signal-to-noise ratio, complicating interpretation.

Despite these caveats, the Stejskal–Tanner framework remains a cornerstone of diffusion NMR and diffusion MRI, guiding both clinical imaging and material science investigations and forming the baseline for more advanced diffusion models.