Diffusion weighting b-value ($b=\int q(t{)}^{2}dt$)

The diffusion weighting b-value quantifies the encoding applied by a magnetic field gradient waveform in a diffusion measurement. For a refocused experiment in one spatial direction, defineHere $\gamma$ is the angular gyromagnetic ratio, in radians per second per tesla, and $G_{\mathrm{eff}}$ includes the sign reversal caused by the refocusing pulse. The echo time is $T$, with $q(T)=0$ for refocusing of stationary spins. Thus $q$ has units ${\mathrm{m}}^{-1}$ and $b$ has units $\mathrm{s}{\mathrm{m}}^{-2}$. The squared encoding history matters even though its final value is zero.^[ref]

For unrestricted, isotropic Gaussian diffusion with constant diffusion coefficient $D$, the otherwise matched signal ratio is $S/S_0=e^{-bD}$. Here $S_0$ uses the same pulse timing and detection conditions with the applied diffusion-encoding gradients removed. The diffusion weighting b-value describes the experiment's weighting; $D$ describes the assumed sample model.

For an original unit check, $1000\mathrm{s}\mathrm{m}{\mathrm{m}}^{-2}=10^9\mathrm{s}{\mathrm{m}}^{-2}$. With $D=10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the ideal attenuation is $e^{-1}\approx 0.368$. Quoting a numerical $b$ without its units can therefore cause a millionfold scaling error.

The Stejskal-Tanner equation gives $b=(\gamma g\delta {)}^2(\Delta -\delta /3)$ for its equal rectangular pulses. More general waveforms require their own integral. For direction-dependent diffusion, a scalar $b$ alone cannot specify the complete directional encoding and response; a tensor description is needed.