3.3 Analyse quantitative des fibres estim´ees
3.3.2 Et en cardiologie ?
A notre connaissance, il n’y a pas de travail sur la quantification des fibres cardiaques en IRMTD.
Plusieurs articles ont valid´e l’IRMTDde par son ad´equation avec l’histologie [Scollan 98,
Hsu 98]. Hsu et al. proposent une ´etude comparative entre des mesures sur l’orientation des fibres myocardiques estim´ees par IRMTD et des mesures histologiques, les donn´ees
provenant d’un bloc de myocarde canin ex vivo. Une excellente corr´elation est observ´ee entre les mesures d’angle des fibres en histologie et les mesures angulaires circonf´erentielles
3.3. ANALYSE QUANTITATIVE DES FIBRES ESTIM ´EES 59
et transmurales de ces mˆemes fibres en IRMTD. L’´etude de Scollan et al., men´ee sur sept
coeurs de lapin perfus´es-arrˆet´es, fait ´etat de mesures de diff´erences angulaires minimes entre les deux techniques (de l’ordre de 5◦ avec un s´equence IRMTD de type spin-echo et
de l’ordre de 0.5◦ avec un s´equence IRMTD de type fast spin-echo).
Une premi`ere ´etude quantitative sur des cartes de diffusion scalaires a ´et´e men´ee sur un coeur porcin ex vivo dans [Wu 07] ; le but ´etant de caract´eriser l’infarctus ant´erieur du ventricule gauche en terme d’alt´erations structurales des fibres cardiaques. Les zones infarcies pr´esentent une chute de la FA de pr`es de 40%, une augmentation du coefficient de diffusion apparent (ADC en anglais) et des modifications de l’angle des fibres.
Enfin, la d´esorganisation architecturale des fibres myocardiques associ´ee `a la cardiomy- opathie hypertrophique (CMH) est ´etudi´ee dans [Tseng 06]. Cinq patients porteurs d’une CMH ont b´en´efici´e d’une IRMTD avec double synchronisation (cardiaque et respiratoire).
Leur m´ethodologie IRMTDin vivo est tir´ee de [Dou 02]. Ils ont observ´e une d´esorganisation
r´egionale de l’orientation des fibres et une chute de la FA, qui caract´erisent tr`es bien la physiopathologie de la CMH : les fibres myocardiques ne sont plus dispos´ees en un r´eseau orient´e mais de fa¸con anarchique, avec des cellules musculaires tr`es hypertrophi´ees.
Dans ce contexte, le but de la classification et de la quantification des fibres car- diaques est de mieux comprendre l’architecture du coeur humain ainsi que sa fonction sous-jacente. A noter que les faisceaux de fibres cardiaques n’ont pas la mˆeme struc- ture que les faisceaux de la substance blanche du cerveau : une exploration manuelle des faisceaux cardiaques est une tˆache difficile car le myocarde n’est pas segment´e en des r´egions anatomiques ind´ependantes comme l’est le cerveau. Cependant, nous cherchons `a d´emontrer dans cette th`ese que des r´egions sp´ecialis´ees existent dans l’architecture my- ocardique et que les r´esultats d’une classification sur l’ensemble des fibres cardiaques pour- raient fournir une approximation de telles r´egions.
Deuxi`eme partie
Contributions m´ethodologiques
Chapitre
4
Comparison of regularization methods for
human cardiac diffusion tensor MRI
1
Carole Frindel, Marc Robini, Pierre Croisille and Yue-Min
Zhu
Sommaire
4.1 Introduction . . . 65 4.2 Regularization methods . . . 66 4.2.1 Regularization operating on DW images . . . 66 4.2.2 Tensor field regularization . . . 69 4.3 Experimental setup . . . 71 4.3.1 Synthetic data . . . 71 4.3.2 Real data acquisition . . . 73 4.3.3 Evaluation methods . . . 74 4.4 Results . . . 75 4.4.1 Synthetic data . . . 75 4.4.2 Real data . . . 77 4.4.3 Tractography . . . 79 4.5 Conclusions . . . 83
1Medical Image Analysis, May 3 ; 13 :405-418, 2009
R´esum´e – L’Imagerie par R´esonance Magn´etique de tenseur de diffusion (DT-MRI en anglais) est une technique d’imagerie qui gagne en importance dans les applications clin- iques. Cependant, les travaux concernant le coeur sont encore peu nombreux. L’application de cette modalit´e d’imagerie `a des coeurs humains in vivo requiert, en effet, une acquisi- tion rapide des donn´ees afin de minimiser les art´efacts dus aux mouvements cardiaques et respiratoires et d’am´eliorer le confort du patient, souvent au d´etriment de la qualit´e des images. Il en r´esulte des images pond´er´ees en diffusion (DWI en anglais) corrompues par le bruit, qui peut avoir un impact significatif sur la forme et l’orientation des tenseurs et conduit `a des donn´ees de diffusion qui ne sont pas toujours adapt´ees `a la tractographie. Ce travail compare des approches de r´egularisation qui op`erent soit sur les images pond´er´ees en diffusion soit sur les champs de tenseurs de diffusion (DT en anglais). Les exp´eriences men´ees sur des donn´ees synth´etiques montrent que, pour un rapport signal sur bruit (SNR en anglais) ´elev´e, les m´ethodes op´erant sur les images pond´er´ees en diffusion produisent les meilleurs r´esultats, elles r´eduisent sensiblement la propagation de l’erreur due au bruit `
a travers l’ensemble des calculs propres `a la chaˆıne de traitements de l’IRMTD. Toutefois,
lorsque le rapport signal sur bruit est faible, les m´ethodes de r´egularisation de Cholesky et Log-Euclidienne avec un terme d’attache aux donn´ees de type Ricien g`erent le biais introduit par le bruit Ricien et assurent la sym´etrie et la positivit´e des tenseurs de diffu- sion. Les r´esultats bas´es sur une s´erie de seize coeurs humains ex vivo montrent que les diff´erentes m´ethodes de r´egularisation ont tendance `a donner des r´esultats ´equivalents.
Abstract – Diffusion tensor MRI (DT-MRI) is an imaging technique that is gain- ing importance in clinical applications. However, there is very little work concerning the human heart. When applying DT-MRI to in vivo human hearts, the data have to be ac- quired rapidly to minimize artefacts due to cardiac and respiratory motion and to improve patient comfort, often at the expense of image quality. This results in diffusion weighted (DW) images corrupted by noise, which can have a significant impact on the shape and orientation of tensors and leads to diffusion tensor (DT) datasets that are not suitable for fibre tracking. This paper compares regularization approaches that operate either on diffusion weighted images or on diffusion tensors. Experiments on synthetic data show that, for high signal-to-noise ratio (SNR), the methods operating on DW images produce the best results ; they substantially reduce noise error propagation throughout the diffu- sion calculations. However, when the SNR is low, Rician Cholesky and Log-Euclidean DT regularization methods handle the bias introduced by Rician noise and ensure symmetry and positive definiteness of the tensors. Results based on a set of sixteen ex vivo human hearts show that the different regularization methods tend to provide equivalent results.
4.1. INTRODUCTION 65
4.1
Introduction
Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) – which measures the dif- fusion of water molecules along various directions within tissues – provides unique and biologically relevant information without invasion. This information includes parameters that help to characterize physical properties of tissue constituents, tissue microstructure and its architectural organization. The construction of the diffusion tensor distribution re- quires the acquisition of a set of diffusion-weighted (DW) images associated with diffusion sensitization along Ng non-collinear gradient directions (Ng ≥ 6). More specifically, it is
possible to estimate a 3×3 symmetric positive definite matrix T (x, y, z) (the diffusion ten- sor) at each location (x, y, z) that characterizes the diffusion process. The diffusion tensor is related to the DW measurements S : Si (i = 1, . . . , Ng) according to the Stejskal-Tanner
diffusion equations : Si = S0exp(−bgiT giT), where gi is the diffusion encoding gradient
direction associated to Si, S0 is the MR measurement without diffusion sensitization, and
the constant b is the diffusion weighting factor.
DT-MRI is particularly subject to noise for two reasons. First, since multiple DW im- ages are needed, each individual image has to be acquired relatively quickly, thus reducing the signal-to-noise ratio (SNR). Second, DT-MRI measures physical properties (water dif- fusion) that require careful treatment of the noise. Therefore, image processing techniques to remove noise in the DW images or in the estimated tensor field are important, especially to perform streamlining tractography. The methods investigated here apply either to raw data (DW images) or to diffusion tensor fields. We do not consider the methods that apply to principal direction fields since we are interested in working as closely as possible to DW images to prevent noise propagation.
In DW image regularization, the denoising process is either applied to each DW image independently (e.g., [Parker 00] and [Basu 06]) or takes coupling between the different DW images into account to synchronize their evolution (e.g., [Vemuri 01] and [McGraw 04]). The common idea is to introduce prior information about the solution in order to smooth the DW images while preserving relevant details. One generally uses a non-quadratic reg- ularization term on the intensity gradient modulus. During this process, large gradients are preserved while small gradients are smoothed, which allows preservation of both edges and local coherence of DW-images. A common idea to restore multivalued images is to use classical scalar anisotropic diffusion on each Si of the set of DW images S ( [Parker 00]).
However, this scheme is criticizable since each DW image Si evolves independently with
different smoothing geometries. To take into account coupling between the different DW images, [Vemuri 01] proposed a weighted TV-norm regularization method to smooth the multivalued image S : the diffusion corresponds to scalar Total Variation-(TV)-norm reg- ularization (applied independently on each Si) weighted by a coupling term, which is the
same for all Si in order to ensure their synchronized evolution.
Another class of regularization methods operates directly on tensor fields. However, tensor computing is difficult due to some limitations of standard Euclidean calculus. Dif- fusion tensors do not form a vector space since they are symmetric positive definite ma- trices whose space is restricted to a convex half cone ( [Pennec 06]). Therefore, special care must be taken in order not to reach the boundaries of the non-linear tensor space, which leads to null or negative eigenvalues. To overcome these limitations, [Wang 04] pro- posed to parameterize the diffusion tensor T (x) by its Cholesky factor F (x) ; smoothing the tensor’s Cholesky factor in the Euclidean framework insures that T (x) = F FT(x) is positive, but definiteness is not insured since null eigenvalues are still possible. [Pennec 06] recently developed another approach to solve the definiteness problem : the tensor space
is replaced by a Riemannian manifold where 3× 3 matrices with null or negative eigenval- ues are at infinite distance from any tensor. It involves a new metric family, the so-called Log-Euclidean metrics, which amounts to classical Euclidean computations in the domain of matrix logarithms. This method takes into account prior knowledge about the diffusion tensor itself – like symmetry and positive definiteness – and prevents the tensor swelling effect that is observed when using the Euclidean framework. Note that there exists other matrix-valued image smoothing techniques that are not covered in this work. We refer the interested reader to a survey by [Weickert 02].
The aim of this work is to focus on the application of regularization methods to human cardiac DT-MRI datasets. This kind of study has never been conducted before, although it is of great interest to better comprehend the impact of DT-MRI in cardiological diagnosis. In fact, to apply DT-MRI to the in vivo heart, the data have to be acquired rapidly to minimize artefacts due to cardiac and respiratory motions, often at the expense of image quality. This results in DT-MRI datasets with very low SNR (much lower than in neurology), that are not suitable for fibre tracking or for building a statistical heart model ( [Peyrat 07]). Therefore, there is a need of reliable algorithms to improve image and tensor fields. We investigate the characteristics and limitations of six regularization methods : three operating on DW images ( [Basu 06] ; [Parker 00] ; [Vemuri 01]) and three operating on diffusion tensor fields ( [Tschumperl´e 01b] ; [Fillard 07] ; [Wang 04]). For each of them, we make two different noise assumptions : Gaussian and Rician. The real nature of the noise in MR magnitude images is known to be Rician ( [Henkelman 85]). However, the Rician distribution can be approximated by a Gaussian distribution when the SNR is sufficiently high ( [Gudbjartsson 95] ; [Sijbers 98b]).
In order to evaluate the solutions produced by these methods, we focused on the following three representations of the DT-MRI processing pipeline : tensors, parametric maps and fibres. Evaluations were performed using different metrics : the Frobenius norm for tensors, the ℓ1-norm for parametric maps and for the helix angle and the sheet angle
for fibres, which are standards in cardiology to characterize the heart architecture. The paper is organized as follows. Section 2 formulates the essential aspects of regu- larization methods and their underlying assumptions. Section 3 presents our framework for comparing the methods. The results on synthetic and real data are exposed in Section 4. Finally, conclusions are given in Section 5.