Compute apparent diffusion coefficient

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^{- b D} 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 for a given material (in the case of brain scanning, this means a particular position in the brain) for a given protocol. ^[1] The b-value, more formally known as the "diffusion-weighting factor", is a fixed value for a given pulse sequence. ^[2]

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. ^[3]

D = - \frac{\ln (S (b_{1}) / S (b_{2}))}{b_{1} - b_{2}}

Testing a Protocol for Accuracy

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 $b = 0 s / m m^{2}$ (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.

Another technique that seeks to test protocol accuracy but offers less detail than the above is to fit tensors to the DWIs and then check the trace of the tensors in the ventricles. The trace should be approximately equal to the natural diffusion coefficient of CSF, 3.1x10^-3 mm²/s. Ideally, both the more detailed check and the simple tensor check should be part of an acquisition and processing pipeline.

Testing Two Protocols for Consistency

Briefly: make sure all your scans from two different protocols report the same D in the CSF.

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:

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

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 $d_{μ}$ falls in the tails of the distribution of $d_{μ}^{*}$ . Otherwise, we accept it.

If the two protocols are consistent with each other, we expect to see $d_{μ}$ close to the center of the $d_{μ}^{*}$ 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

Briefly: make sure your scans report the same D in the CSF for every gradient direction.

In addition to having $D = 3.1 \times 1 0^{- 3} m m / s^{2}$ , the CSF in the ventricles also exhibits isotropic diffusion, since the motion of water molecules 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 $b = 0 s / m m^{2}$ 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.

Another technique that seeks to test for isotropic response but offers less detail than the above is to fit tensors to the DWIs and then check the FA of the tensors in the ventricles. The FA should be very close to zero. Ideally, both the more detailed check and the simple tensor check should be part of an acquisition and processing pipeline.

Notes

↑ A protocol specifies fixed values for the echo time (TE) and repetition time (TR) of the radio-frequency MRI pulses. Proton density (PD) and the $T_{1}$ and $T_{2}$ MR relaxation times are physical properties of tissue that remain constant over the time span of an MRI scan. For the most basic diffusion MRI scan technique, single PGSE,
$S_{0} \equiv P D (1 - e^{- T R / T_{1}}) e^{- T E / T_{2}} .$
Thus we see that for single PGSE, $S_{0}$ is a constant factor in the equation for the diffusion MRI signal in a given material. Other techniques have other $S_{0}$ equations, but they are similarly fixed for a given material.
↑ b may be computed analytically for simple pulse sequences but it may also be determined experimentally by solving the diffusion MRI signal equation with measurements of a physical sample with known D.
↑ We say "apparent" coefficient because, when our voxel of interest contains brain tissue, we are measuring the behavior of water under the influence of the tissue microstructure. Theoretically, the diffusion coefficient of free water is constant at a given temperature. The $D$ that we compute is therefore just the diffusion coefficient that the water appears to have when its molecules are hindered and restricted by microscopic structures.

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.

[1] A protocol specifies fixed values for the echo time (TE) and repetition time (TR) of the radio-frequency MRI pulses. Proton density (PD) and the $T_{1}$ and $T_{2}$ MR relaxation times are physical properties of tissue that remain constant over the time span of an MRI scan. For the most basic diffusion MRI scan technique, single PGSE,
$S_{0} \equiv P D (1 - e^{- T R / T_{1}}) e^{- T E / T_{2}} .$
Thus we see that for single PGSE, $S_{0}$ is a constant factor in the equation for the diffusion MRI signal in a given material. Other techniques have other $S_{0}$ equations, but they are similarly fixed for a given material.

[2] y be computed analytically for simple pulse sequences but it may also be determined experimentally by solving the diffusion MRI signal equation with measurements of a physical sample with known D.

[3] We say "apparent" coefficient because, when our voxel of interest contains brain tissue, we are measuring the behavior of water under the influence of the tissue microstructure. Theoretically, the diffusion coefficient of free water is constant at a given temperature. The $D$ that we compute is therefore just the diffusion coefficient that the water appears to have when its molecules are hindered and restricted by microscopic structures.

[1]

[2]

[3]

Compute apparent diffusion coefficient

Contents

Testing a Protocol for Accuracy

Testing Two Protocols for Consistency

Testing a Protocol for Isotropic Response

Notes

References

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Compute apparent diffusion coefficient

Testing a Protocol for Accuracy

Testing Two Protocols for Consistency

Testing a Protocol for Isotropic Response

Notes

References

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