Mammographie ultrasonore en champ proche
Near-Field Ultrasound Mammography
OPEN ACCESS
We introduce in this note a near-field formulation of the acoustic field scattered by a fluid object supposed to be weakly heterogeneous (Born approximation). This derivation is based on the Huygens-Fresnel principle that describes the scattered field as the result of the interferential scheme of all the secondary spherical waves. This derivation leads us to define a new Fourier transform that we name the Elliptical Fourier transform. The latter provides an elliptical spectrum whose harmonic components, the weighted elementary basis functions, have an elliptical spatial support. Based on these elliptical projections, we define the Elliptical Radon transform that allows us to establish a near-field extension of the Fourier Projection-Slice theorem. Thanks to these spectral and tomographic transforms, we show that it is possible to reconstruct either the impedance or the celerity maps of an acoustical model characterized in terms of impedance and celerity fluctuations. We observe that this formulation is very close to that one developed in the far field domain where the Radon transform pair is derived from an harmonic plane wave decomposition. This formulation allows us to introduce the Ductal Tomography, following the example of the Ductal Echography, that provides a systematic inspection of each mammary lobe, in order to reveal lesions at an early stage. In that aim, we develop 2D anatomic breast computer phantoms corresponding to an axial cross-section of the ductolubular structure in healthy and pathological situations. The goal is also, from the practitioner's point of view, to compare the recognized DE reference with the high potential tomographic approach.
Résumé
Nous introduisons par cette note une formulation originale en champ proche, du champ ultrasonore diffracté par un organe faiblement contrasté – le sein – (Approximation de Born). Cette formulation est fondée sur le principe d'Huygens-Fresnel qui construit ce champ sur la base d'un schéma interférentiel à partir des sources secondaires sphériques. Cette approche nous amène à définir une nouvelle transformée de Fourier dite «elliptique» dont le spectre (du même nom) s'établit sur une famille de fonctions de base – les composantes harmoniques – qui présentent un support spatial ellipsoïdal caractéristique des senseurs actifs multi-statiques. Ces projections elliptiques nous permettent de définir la transformée de Radon Elliptique qui fonde une extension en champ proche du théorème coupe-projection. Grâce à ces transformées spectrale et tomographique, nous montrons qu'il est possible de reconstruire les cartographies de fluctuation d'impédance en réflexion et de célérité en transmission. Nous observons que l'algorithme est très proche de celui universellement utilisé en champ lointain, basé sur une décomposition harmonique de type ondes planes. Cette formulation nous permet, à l'instar de l'échographie, de jeter les bases de la tomographie ductale offrant une inspection systématisée de chaque lobe en vue de la détection précoce du cancer du sein. Dans ce but, des fantômes numériques anatomiques 2D de sein correspondant à une coupe axiale de la structure ductolobulaire dans des situations saines et pathologiques sont développés, afin de valider les modèles précités d'analyse et d'inversion des données en champ proche. Ils permettent aussi une comparaison réaliste des procédés échographique (référence du radiologue) et tomographique, avec, pour cette dernière modalité, un avantage confirmé.
Ultrasound, diffraction tomography, near-field, anatomic breast phantom
Mots clés
Ultrasons, tomographie de diffraction, champ proche, fantôme anatomique de sein
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