ECMath CH12: Advanced Magnetic Resonance Imaging: Fingerprinting and Geometric Quantification


Project head Prof. Dr. Michael Hintermüller (1,2)
Staff Dr. Guozhi Dong (1)
Project associate Dr. Kostas Papafitsoros (2)
Project period 1 June 2017 - 31 December 2018
Affiliations (1) Humboldt-Universität zu Berlin
(2) Weierstrass Institute Berlin

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Background of the project

Magnetic resonance imaging

Magnetic resonance imaging (MRI) is a powerful imaging tool for a detailed visualization of soft tissues, leading to accurate diagnoses and thus to more effective therapies. It is based on the interaction of an externally controlled magnetic field B and the magnetization field $m$ of the tissue, both related by the Bloch equations [Wri97]
\begin{equation} \frac{\partial m}{\partial t}(t)=m(t)\times \gamma B(t)-\left (\frac{m_{x}(t)}{T_{2}}, \frac{m_{y}(t)}{T_{2}}, \frac{m_{z}-m_{eq}}{T_{1}} \right)^{\top}.\qquad\qquad \qquad (1) \end{equation} Here, $\gamma$ and $m_{eq}$ are known parameters while $T_{1}$, $T_{2}$ the longitudinal and transverse relaxations times respectively, depend on the tissue type and thus precise estimations of these quantities are necessary for its accurate description. Focusing on a specific tissue slice, the transverse magnetization component ($m_{x}$, $m_{y}$) is excited with radio frequency pulses, and on its way to equilibrium it precedes around the $z$-axis, emitting an electromagnetic signal which is then detected. The obtained data $D$, consist of an incomplete collection (by applying a projection $P$) of the coefficients of the Fourier transform $\mathcal{F}$ of the net magnetization $M=\rho m$ where $\rho$ is the matter density, i.e., $D=P\mathcal{F} M$, $M=\rho m$ where $m$ solves (1). The reconstruction of $M$ requires "inverting" the Fourier transform (in the spirit of an inverse problem). Due to undersampling, compressed sensing techniques have been recently employed for this final reconstruction step. In classical MRI, multiple excitation pulses are used to obtain a sufficiently large sample of coefficients for a reliable reconstruction. This typically results in long scan times as, after each excitation, sufficient time must be allowed for the magnetization to achieve equilibrium again.

Magnetic resonance fingerprinting

Magnetic resonance fingerprinting (MRF) has been recently introduced as a highly promising MRI acquisition scheme which allows for the simultaneous quantification of the tissue parameters (e.g. $T_{1}$, $T_{2}$ and others) using a single acquisition process [MGS+13]. In MRF, the tissue of interest is excited through a random sequence of pulses without needing to wait for the system to return to equilibrium between pulses. After each pulse, a subset of the signal's Fourier coefficients is collected, as in classical MRI, and a reconstruction of the net magnetization image is performed. These reconstructions suffer from the presence of artifacts since the Fourier coefficients are not fully sampled. The formed sequence of image elements is then compared to a family of predicted sequences (dictionary of fingerprints) each of which corresponds to a specific combination of values of the tissue parameters. This dictionary is computed beforehand by solving the Bloch equations. The idea is that, provided the dictionary is rich enough, every material element (voxel) can be then mapped to its parameter values. While first, very promising results have been obtained in biomedical engineering, many aspects of MRF remain widely open and require a proper mathematization for optimizing and robustifying the procedure.


Aim of the project

Quantitative mathematical model for MRF

MRF research is very recent in biomedical engineering. Like for many other imaging modalities, the further advance of the technology will strongly depend on its successful mathematical treatment. In the context of MRF this starts already with setting up a proper variational model as no such model is known, except for the one in [DPVW14] conceived in a specific discrete context only. Hence, our first aim is to establish a novel variational framework, modeling an image as a continuous object, taking into account desired regularity properties of the tissue parameters. Further, we will advance the MRF technology by incorporating regularization into the different stages of MRF acquisition and reconstruction.

We will investigate different regularisation procedures based on non-smooth derivative based type functionals, e.g. Total Variation (TV) and Total Generalised Variation (TGV), both in the reconstruction of the net magnetisation and in the fingerprint matching process. These regularization choices will be enhanced by very recently introduced statistics-based spatial weighted regularization, i.e., spatial dependence of the regularization parameters, leading to a better preservation of detailed features [HRWL17, HPR17]. Finally, since in MRF one is interested in a precise recovery of the parameter values, emphasis will be given minimizing any loss of contrast associated with the regularization.

The comparison of the reconstructed magnetization images and the fingerprints has been done so far by employing a quadratic distance function. This assumes a Gaussianity of data imperfections which however is inappropriate in the context of MRI where specific non-Gaussian degradation effects are taking place. Thus, one aspect of the project will be to develope proper data fidelity measures different to the maximum likelihood associated with a Gaussian distribution.

Development of fast solvers

Any type of regularization procedure must be performed efficiently due to the large amount of involved data. Currently, primal-dual first-order and variable splitting schemes are frequently employed for TV and TGV regularization; see e.g. [CP11] for TV and analogous methods for TGV. It is known that these solvers typically admit no function space analysis, hence, resulting in resolution (mesh) dependent convergence behavior; see [HRH14] for an investigation. Moreover, proper discretization must be employ to keep numerical artifacts (such as numerical diffusion near contours) minimal. In this vein, this project will develop quasi-second order methods by pursuing a Fenchel-dual approach along with an advancing semismooth Newton methods for the resulting dual variational problems; see [HK04] for TV regularization with scalar regularization parameter. Mathematically, this requires to overcome several challenges:
(i) For a proper dualization one has to investigate density results for convex sets in Banach spaces; compare [HR15].
(ii) The primal-dual framework needs to be established in to obtain a reconstructed image from dual quantities.
(iii) Convergence analysis of the semismooth Newton solver in a non-reflexive Banach space setting needs to be performed (ensuring image resolution independent convergence).
(iii) Proper primal-dual discretization needs to be devised to avoid numerical diffusion.


Geometric quantification in segmentation

Finally, we aim to investigate the uncertainty of MR-data based segmentation (not necessarily only related to MRF), given data along a known distribution of measurement errors. Concerning the latter, we assume that it can be represented as a finite sum of i.i.d. normally distributed random variables. We are in particular interested in a proper visualisation of the uncertainty in the segmentation in order to facilitate a decision support devise for medical practitioners. The underlying mathematical segmentation paradigm is the Mumford-Shah functional which contains the unknown boundaries of image segments as geometric quantities [HR04]. For the quantification of the geometric uncertainty, i.e., the uncertainty in segmentation, using the renowned Ambrosio-Tortorelli phase field approximation, the original sharp interface (contour) model will transformed into a purely functional context. The quantification will then be done with respect the posteriori related to the phase field variable using a variance reduction approach based on anthitetic sampling within a Monte Carlo framework.


Publications

No publications available yet.

References

[MGS+13] D. Ma, V. Gulani, N. Seiberlich, K. Liu, J.L. Sunshine, J.L. Duerk, M.A. Griswold, Magnetic resonance fingerprinting. Nature, 495(7440):187-192, 2013.
[Wri97] G.A. Wright, Magnetic resonance imaging. IEEE Signal Processing Magazine, 14(1):56-66, 1997.
[DPVW14] M. Davies, G. Puy, P. Vandergheynst, Y. Wiaux, A compressed sensing framework for magnetic resonance fingerprinting. SIAM Journal on Imaging Sciences, 7(4):2623-2656, 2014.
[HRWL17] M. Hintermüller, C.N. Rautenberg, T. Wu, and A. Langer, Optimal selection of the regularisation function in a weighted total variation model. Part II: Algorithm, its analysis and numerical tests. Journal of Mathematical Imaging and Vision, 2017.
[HPR17] M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Analytical aspects of spatially adapted total variation regularisation. Journal of Mathematical Analysis and Applications, 454(2):891-935, 2017.
[CP11] A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1):120-145, 2011.
[HRH14] M. Hintermüller, C.N. Rautenberg, J. Hahn, Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction. Inverse Problems, 30(5):055014, 2014.
[HK04] M. Hintermüller, K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM Journal of Applied Mathematics, 64(4):1311-1333, 2004.
[HK15] M. Hintermüller, C.N. Rautenberg, On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces. Journal of Mathematical Analysis and Applications, 426(1):585-593, 2015.
[HR04] M. Hintermüller, W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. Journal of Mathematical Imaging and Vision, 20:19-42, 2004.