Compute apparent diffusion coefficient

An example plot of exponential decay predicted by the Stejskal-Tanner model of the diffusion-weighted MR signal in one direction over a range of b values. Two observations are marked by red circles; the apparent diffusion coefficient is computed by fitting the model to these observations.

The equation for the diffusion MRI signal at b-value $b$ (measured in units of s/mm²) in a free fluid with diffusion coefficient $D$ (measured in mm²/s) is

$S(b)=e^{-bD}{S}_0$

where $S_0$ is a value that depends on fixed tissue properties and imaging parameters, and is equal to the signal value at $b=0$. $S_0$ is a constant at any particular position for a given protocol. The b-value, more formally known as the "diffusion-weighting factor", is a fixed value for a given pulse sequence.

If we take measurements in the same voxel with the same magnetic gradient at two different b-values, say $b_1$ and $b_2$, we can compute $D$, the apparent diffusion coefficient (ADC) in that voxel.

$D=-\frac{\ln (S(b_1)/S(b_2))}{b_1-b_2}$

Testing a Protocol for Accuracy

A histogram of ADC values computed from a protocol with one image at b=0 s/mm² and 64 images at b=1000 s/mm² for 54 voxels inside the ventricles. The mean of this distribution is about 2.7x10^-3 mm²/s, which seems a bit too low.

Briefly: make sure your scans report the correct D in the CSF.

One use of the ADC computation is making sure that the parameters we believe apply to a given protocol give us a physically realistic interpretation of the resulting diffusion-weighted image. It is well-established that the diffusion coefficient of cerebrospinal fluid (CSF) at normal body temperature is approximately 3.1x10^-3 mm²/s. Our diffusion MRI measurements should confirm this expected value.

Let's say we have $N$ volumes with $b=0\mathrm{s}/\mathrm{m}{\mathrm{m}}^{\mathrm{2}}$ (for many of our protocols, $N=1$) and $M$ volumes at other b-values, that all the volumes have been co-registered, and that we can segment out $P$ voxels that are clearly within the ventricles and should therefore contain signal only from CSF. For each CSF voxel, we can compute $(N\cdot M)$ measured ADC values for the CSF, and therefore we can compute $(N\cdot M\cdot P)$ ADC estimates overall. These will form a distribution, hopefully around the expected diffusion coefficient value of 3.1x10^-3 mm²/s. If the mean value is too far off (relative to the standard deviation), either there's a problem with the protocol or the b-values have been reported incorrectly.

Testing Two Protocols for Consistency

Often we would like to be able to compare results from data gathered using different imaging protocols. As a first check, we should make sure that the images from both are scaled properly (that is, that they measure a known physical phenomenon, like diffusion of free CSF, in the same way); otherwise the images simply can't be compared to each other.

Say we have two protocols, A and B. We have a sample acquisition from each one, and, following the directions in the previous section, we've computed all $Q_A\equiv (N_A\cdot M_A\cdot P_A)$ CSF ADC estimates for protocol A, and $Q_B\equiv (N_B\cdot M_B\cdot P_B)$ for B. These are two "populations" of measurements, and we wish to determine if there is a statistically significant difference between the populations. To do so, we use a statistical permutation test as follows:

1. Compute the true difference of the means of these populations: $d_{\mu }\equiv {\mu }_A-{\mu }_B$.
2. Now treat all of the $Q_A+Q_B$ measurements as though they were the same.
3. Randomly reassign $Q_A$ of the measurements into a "fake" population A^*, and assign the remaining $Q_B$ of them to B^*.
4. Compute the difference of the means of the "fake" populations: $d_{\mu }^{*}\equiv {\mu }_{A^{*}}-{\mu }_{B^{*}}$.
5. Repeat steps 2 and 3 many times and build up a distribution of fake population means $d_{\mu }^{*}$.

For this statistical test, the null hypothesis is that A and B are the same population. We reject that hypothesis with some degree of certainty if $d_{\mu }$ falls in the tails of the distribution of $d_{\mu }^{*}$. Otherwise, we accept it.

If the two protocols are consistent with each other, we expect to see $d_{\mu }$ close to the center of the $d_{\mu }^{*}$ distribution. If it's in the tails, the protocols disagree about the diffusion coefficient of free CSF, which is essentially a physical constant. Either one or both of the protocols has a problem in this case.

Testing a Protocol for Isotropic Response

In addition to having $D=3.1\times 10^{-3}\mathrm{m}\mathrm{m}/{\mathrm{s}}^{\mathrm{2}}$, the CSF in the ventricles also exhibits isotropic diffusion, since its Brownian motion is unrestricted and unhindered in all directions. We can check to make sure that a protocol accurately measures the diffusion in the CSF as isotropic.

As above, let our acquisition have $N$ volumes with $b=0\mathrm{s}/\mathrm{m}{\mathrm{m}}^{\mathrm{2}}$ and $M$ volumes at other b-values, with $P$ CSF voxels segmented out. We can compute $(N\cdot P)$ ADC estimates, and therefore a mean and standard deviation, per direction. All of these means should be approximately equal. If this is not the case, the protocol does not have isotropic response and any computations based on data collected with it will be invalid.

References

• S. Mori and J. Zhang. Principles of Diffusion Tensor Imaging and Its Applications to Basic Neuroscience Research. Neuron 51, pp. 527--539, September 2006.
  • Accessible overview that forms the basis of this article.
• P.J. Basser and D.K. Jones. Diffusion-Tensor MRI: Theory, Experimental Design, and Data Analysis --- A Technical Review. NMR in Biomedicine 15, pp. 456--467, 2002.
  • A more technical overview of DTI.
• J. Mattiello, P.J. Basser, and D LeBihan. Analytical Expressions for the b Matrix in NMR Diffusion Imaging and Spectroscopy. J. Magn. Resonance Series A 108, pp. 131--141, 1994.
  • A very early (and therefore dense and technical) analytic derivation of the b matrix in diffusion imaging.
• D. Gullmar, J. Haueisen, and J.R. Reichenbach. Analysis of b-value Calculations in Diffusion Weighted and Diffusion Tensor Imaging. Concepts in Magn. Resonance Part A 25, pp. 53--66, 2005.
  • A more recent analysis and refinement of the b matrix calculation.