1. Diffusion Tensor MRI From Deterministic to Probabilistic Modelling
Marta Morgado Correia
MRC Cognition and Brain Sciences Unit
4th December 2009

2. Concept of Molecular Diffusion (1)
Water molecules in their liquid and gaseous states undergo thermally driven random motion.
Molecules collide with each other and their surroundings, which results in a random walk behaviour known as Diffusion.
In a free medium, molecular displacements obey a 3D Gaussian distribution.
Figure 1 - Three random walks originating from a common starting point.

3. Concept of Molecular Diffusion (2)
During diffusion water molecules move in brain tissues bouncing off, crossing or interacting with many tissue components, such as cell membranes, fibres and macromolecules.
The movement of water is impeded by these obstacles, and the diffusion displacement distribution is no longer Gaussian.
The non-invasive observation of the water diffusion-driven displacement distributions in vivo provides unique clues to the fine structural features and geometric organization of neural tissues and also to changes in these features with physiological and pathological states.
Figure 2 – In biological tissues, obstacles modulate the free diffusion process. Diffusion of molecules can be restricted in closed spaces (A), such as cells. Diffusion might also be hindered by obstacles that result in tortuous pathways (B). Exchange between compartments also slows down molecular displacements (C).

4. Isotropic vs Anisotropic Diffusion
Free diffusion (no obstacles) occurs equally in all directions. This is called isotropic diffusion.
If the water diffuses in a medium having barriers, the diffusion will be uneven. Barriers can be many things (cell membranes, molecules, axons, etc), but in white matter the principal barrier is the myelin sheath of axons.
Bundles of axons provide a barrier to perpendicular diffusion and a path for parallel diffusion along the orientation of the fibres. This is termed anisotropic diffusion.
Diffusion trajectory
Isotropic
Diffusion (free water)
Anisotropic Diffusion
(coherent axonal bundle)

5. Modelling Diffusion in the Brain
Traditional single-fibre approach: Diffusion Tensor Model (DTI)
Multiple fibre approaches
The multi-tensor model
Diffusion Spectrum Imaging (DSI)
QBall Imaging
Spherical Deconvolution
Persistent Angular Structure (PAS)

6. Diffusion Tensor Model (1)
Diffusion is a 3D process and water molecules mobility in tissues is not necessarily the same in all directions.
Organized fibrous tissues, such as muscle and cerebral white matter, demonstrate anisotropic diffusion.
The direction of greatest diffusion corresponds to the fibre axis direction.
The Diffusion Tensor (DT) describes:
the magnitude of the water diffusion;
the degree of diffusion anisotropy;
the orientation of the anisotropy.

7. Diffusion Tensor Model (2)
In a DT MRI experiment, the diffusion signal due to only one fibre can be described by the Single Tensor Model:
and it is a function of the diffusion weighting
parameter, b, and the direction of the encoding
gradient, .
The DT is a 3x3 symmetric matrix:
Figure 3 – Profiles obtained for the Single Tensor Model.
b = 1000
b = 2000
b = 3000

8. The Diffusion Ellipsoidz y x
Rotation l1 l3 l2
(a)
(b)
(c)

9. Apparent Diffusion Coefficient (ADC)
Measurements of the DT and its components have been found to have several applications in the human brain.
Apparent Diffusion Coefficient (ADC) map: ADC is the average value of the rate of diffusion on each voxel. ADC maps contain very useful information for detecting and evaluating brain ischemia and stroke.
Figure 4 – ADC maps of a healthy brain (left) and a stroke patient (right).

10. Fractional Anisotropy (FA)
Fractional Anisotropy (FA) map: FA is a measure of the degree of diffusion anisotropy, and it is calculated from the diffusivity constants l1, l2, l3.
FA produces high values for white matter (highly anisotropic) and low values for grey matter(isotropic).
It has been used to study white matter in terms of morphology, disease and trauma, brain development and neurosurgical planning.
Figure 5 – FA maps of a healthy brain (left) and a brain tumour patient (right).

11. Colour coded FA maps
Let v1 designate the longest axis of the diffusion ellipsoid.
v1 can be identified with the main direction of diffusion.
This directional information can be added to our FA map using a colour code:
Red indicates directions in the x axis: right to left or left to right.
Green indicates directions in the y axis: front to back or back to front.
Blue indicates directions in the z axis: foot-to-head direction or vice versa.
Figure 6 – Colour coded FA map.

12. Tractography
Once direction v1 has been calculated for all voxels, the trajectories of water molecules can be reconstructed using a method similar to the children’s activity “connect the dots”: we connect each voxel to the adjacent one toward which the fibre direction, v1, is pointing.

13. Tractography in the Brain
Figure 7 – Fibre tracks obtained for a dataset of a healthy volunteer using simple streamlining (FACT).

14. The Need for Probabilistic Modelling
The information provided by DTI can be very useful for the characterisation of brain white matter.
However, the estimated tensor can be highly dependent on noise.
In fact, if the noise is known to be on a level that can significantly affect the outcome of the model fit, then the uncertainty in the model parameters needs to be characterised if any useful information is to be extracted from the data.
Probabilistic modelling can be used to estimate a probability distribution function (PDF) for the DTI model parameters.
The standard deviation (s.d.) of this PDF is a good marker for confidence in the results.

15. DTI Model Parameters Single tensor model: Model parameters: and
Deterministic approach – we look for the set of parameters which best fit the data by using a least-squares fitting routine.
Alternatively, Markov Chain Monte Carlo (MCMC) techniques have been successfully used to estimate a probability distribution function (PDF) for the DTI model parameters.

16. MCMC Methods (1) MCMC methods are based on Baye’s Theorem:
where w represents the vector of model parameters.
The prior term P(w) offers an opportunity for scientists to include knowledge they have about the expected values of the parameters.
The term P( data | w ) gives the probability of observing the data given a sampled set of parameters, and it is dependent on the model used. For the diffusion tensor model:

17. MCMC Methods (2)
Instead of producing a single set of parameters MCMC methods produce a PDF for each parameter. For example:
The standard deviation of these PDFs is a good marker for confidence in the results.
FA maps, ADC maps, etc., can be obtained by taking the average of each PDF as the most likely value of the model parameters.

18. PDF for fibre orientation
For each voxel, we can obtain a PDF for the fibre orientation, by combining samples from the PDFs for q and f:
Regions of one-fibre populations have very narrow distributions, while regions of crossing fibres show greater variability.

19. Probabilistic Tractography1
For each sample of the directional PDF we can produce a track (or streamline).
If we repeat this for a large number of samples, the probability of voxels A and B being connected can be calculated by dividing the number of streamlines that reach B, by the total number of streamlines generated from A. A

20. Probabilistic Tractography in the Brain
Figure 8 – Probabilistic tractography dataset obtained for a healthy volunteer.

21. Deterministic Modelling Probabilistic Modelling
Summary
Deterministic Modelling
Very sensitive to noise.
The uncertainty due to noise is not characterised.
Hard to ensure positivity of eigenvalues.
Fast.
Separate runs of the fitting method generate the same results.
Probabilistic Modelling
More robust in the presence of noise.
Uncertainty of model parameters well characterised by the s.d. of the PDF.
Positivity of eigenvalues and other prior knowledge can be included.
Very slow.
Results vary significantly between different runs if stability of the Markov chains is not guaranteed.

22. Thanks!