The lecture provides researchers and clinicians who use or are planning to use dMRI to quantify the diffusion process and/or tissue microstructure with the basic tools to extract relevant features from the diffusion-weighted signal. The language of the dMRI community regarding signal modelling and representation is introduced. Examples of both signal representations going beyond the Gaussian diffusion regime, and model parameter estimation, are used to give intuition of how these concepts are relevant in the context of tissue microstructure. By solving exercises, the audience will gain intuition into the concepts discussed in the lecture.
Following this lecture, the audience will:
This lecture will cover the following topics:
Feature extraction from the dMRI signal: Representations vs models
From the set of diffusion-weighted images, useful features need to be extracted that ideally reflect characteristics of the diffusion process or tissue microstructure. A wide range of approaches have been proposed, that can broadly be classified as mathematical representations or biophysical models. Representations aim to accurately describe the signal with as few parameters and assumptions as possible; these parameters do not carry any physical meaning. Models are simplified pictures of physical reality, where assumptions are necessary to reduce the amount of parameters down to the most relevant ones [1].
Some interesting features: EAP, dODF, fODF
Features-of-interest that can be extracted from representations and/or models include the apparent diffusion- or spin displacement probability density function (PDF, or ensemble average propagator EAP), diffusion orientation distribution function (dODF), and fiber orientation distribution function (fODF) [5]. The EAP provides information on the average probability of particles to have a particular displacement during a given diffusion time [2]. The dODF contains information on the probability of diffusion in a particular direction regardless of the displacement. The fODF more directly represents the distribution of fiber directions and hence generally relies on a model of the diffusion within a single fiber population.
Mathematical representations
We will discuss mathematical representations, and will focus on the cumulant expansion:
Cumulant expansion
Related to the Taylor expansion in mathematics, which allows the approximation of smooth functions as polynomials in a sufficiently small neighbourhood of a given point, the Cumulant expansion allows to approximate the logarithm of the dMRI signal as powers of the diffusion gradient in the vicinity of the zero-gradient point [3]. It can then be derived that these expansion coefficients are reflective of diffusion properties of the medium.
Diffusion Tensor Imaging (DTI) and Diffusion Kurtosis Imaging (DKI)
The diffusion coefficient and its 3D extension (diffusion tensor) are already introduced in the first lecture of this educational. We will see that the diffusion tensor is related to the cumulant expansion up to the second order. The kurtosis tensor, which is a fourth-order tensor with 15 independent components, contributes to the fourth-order cumulant [3,4]. We will discuss features that can be derived from this tensor and how these have been related to tissue microstructure.
Convergence radius
The Taylor expansion converges for small gradients but diverges for large gradients, which means that the Cumulant expansion is only useful up to a certain diffusion weighting, referred to as the radius of convergence. This radius is dependent on the underlying diffusion process, and we will derive values in the case of two non-exchanging compartments [3].
How to get from signal to features
Representations and models express the signal as a function of the features-of-interest. If the features are known, the signal can be computed by filling the values in the expression; this is called the forward problem. The inverse problem then considers the task of finding values for the features-of-interest given the signal, and is a much harder problem to solve.
Noise
Noise is a result of imperfections in the measurement, which complicates solving the inverse problem. Noise and the noise floor in MRI have already been discussed in the third lecture of this educational, and these concepts are briefly recapped here.
Parameter estimation
Parameter estimation tries to solve the inverse problem in the presence of noise. Often this is done by minimising the residuals, which is the difference between the measurement and the signal after filling in predictions for the parameters. Parameter estimation is an active area of research in dMRI, and we will discuss examples of parameter estimation in the context of DTI and DKI.
Constraints
Noise and artefacts can result in parameter estimates that are physically implausible; for example, the eigenvalues of the diffusion tensor should be positive. Obtaining physically plausible estimates for DKI is even more challenging. Therefore, imposing constraints is sometimes necessary; we will discuss constraints in the context of DTI and DKI.
Signal modelling
DTI, DKI, and other representations describe features of the signal and/or the diffusion process. Such features have shown to be very useful to detect changes in tissue microstructure, and we will show some examples of that. However, interpreting a change in such features in terms of a change in microstructure is challenging. We will provide here a brief introduction of microstructural models as a sum of Gaussian compartments (the biexponential that was used to derive values for the convergence radius is an example of this). A more extensive discussion on microstructural modelling will be the focus of the afternoon sessions of this educational.
[1] Novikov, Dmitry S., Valerij G. Kiselev, and Sune N. Jespersen. "On modeling." Magnetic resonance in medicine 79, no. 6 (2018): 3172-3193.
[2] Callaghan, Paul T. Translational dynamics and magnetic resonance: principles of pulsed gradient spin echo NMR. Oxford University Press, 2011.
[3] Kiselev, Valerij G. "The cumulant expansion: an overarching mathematical framework for understanding diffusion NMR." Diffusion MRI (2010): 152-168.
[4] Jensen, Jens H., Joseph A. Helpern, Anita Ramani, Hanzhang Lu, and Kyle Kaczynski. "Diffusional kurtosis imaging: the quantification of non‐gaussian water diffusion by means of magnetic resonance imaging." Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 53, no. 6 (2005): 1432-1440.
[5] Tournier, J. , Mori, S. and Leemans, A. (2011), Diffusion tensor imaging and beyond. Magn. Reson. Med., 65: 1532-1556.