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Towards Factor Analysis Exploration Applied to Positron Emission Tomography Functional Imaging for Breast Cancer Characterization

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Towards Factor Analysis Exploration Applied to Positron Emission Tomography Functional Imaging for Breast Cancer Characterization 2011 8th International Multi-Conference on Systems, Signals & Devices TOWARDS FACTOR ANALYSIS EXPLORATION APPLIED TO POSITRON EMISSION TOMOGRAPHY FUNCTIONAL IMAGING FOR BREAST CANCER CHARACTERIZATION Wafa REKIK, Ines KETATA, Lamia SELLAMI Khalil CHTOUROU, Mohamed BEN SLIMA, Su RUAN & Ahmed BEN HAMIDA Advanced Technologies for Medical and Signals, ATMS–LETI Laboratory, Sfax University, Tunisia Nuclear Medicine service, CHU Sfax hospital, Sfax University, Tunisia CReSTIC Laboratory (EA 3804), Reims University, France LECLERC Center (Centre de Lutte Contre le Cancer), Dijon, France ABSTRACT distinguish a necrosed tumor and a residual active disease. That’s why, nuclear medicine is essential to This paper aims to explore the factor analysis when provide complementary information in oncology research. applied to a dynamic sequence of medical images Using radiopharmaceuticals or radiotracers being obtained using nuclear imaging modality, Positron accumulated in the tumoral cells, it allows the early Emission Tomography (PET). This latter modality allows detection of cancer before reaching adverse pathological obtaining information on physiological phenomena, stages. An example of tracer, glucose settles immediately through the examination of radiotracer evolution during on tumoral cells considered as glucose consumer. The use time. Factor analysis of dynamic medical images of tracer in medical imaging favors the determination of sequence (FADMIS) estimates the underlying organ morphology and follow-up of radio-isotope along fundamental spatial distributions by factor images and the blood vessels or lymphatic ways. associated so-called fundamental functions (describing In nuclear medicine, PET is considered as the most the signal variations) by factors. This method is based on sophisticated, the most expensive and the most effective an orthogonal analysis followed by an oblique analysis. in oncology. Sequences of PET images are very important The results of the FADMIS are physiological curves for therapy and patient monitoring during and after showing the evolution during time of radiotracer within treatment. homogeneous tissues distributions. This functional Region of interest (ROI), parametric imaging, analysis of dynamic nuclear medical images is considered compartmental analysis and factor analysis are analysis to be very efficient for cancer diagnostics. In fact, it could tools that process functional images for extracting be applied for cancer characterization, vascularization as efficiently relevant information from theses sequences. well as possible evaluation of response to therapy. In this paper, we focus on factor analysis for PET imaging. It aims to diagnose breast cancer being the most Key Words— factor analysis, Positron Emission damaging, the most common and holding the interest of Tomography (PET), factor images, factor, orthogonal the majority of research in medicine. analysis, oblique analysis, cancer diagnostic. 2. FACTOR ANALYSIS PRINCIPLE Factor analysis (FA) of dynamic images sequences is a 1. INTRODUCTION method of analysis that processes simultaneously curves representing the temporal variation of the signal in each Today, medical imaging domain knows considerable pixel. It summarizes the information underlying theses evolutions in medicine. It has an essential role in curves into a small number of images visualizing the diagnosis and therapeutic follow-up of multiple different compartments of the study area and the pathologies. Using anatomic imaging modalities, such as associated kinetics. MRI, ultrasound, allow visualizing anatomy of the human FA uses the assumption that each curve is a linear body. It can also provide functional, metabolic and combination of a few kinetics called factors. Moreover, quantitative information by using dynamic methods, such some a priori information are used such as the as fMRI, Scintigraphy or PET. approximate number of physiological compartments and Anatomic imaging modalities cannot detect tumors the positivity of the signal on each compartment. [1] having size overtaking the cm3. Frequently, it cannot 978-1-4577-0411-6/11/$26.00 ©2011 IEEE 3. FACTOR ANALYSIS MODEL The SVD of data expresses each noise-free kinetic as a linear combination of P vectors associated with P The model solved by FA is a linear superposition model singular values. S is the space spanned by the Q described by following Equation: eigenvectors associated with the Q largest eigenvalues. (1) 4.1.1. Principal Component Analysis (PCA) xi (t) ak i)( fk (t) i)( k The PCA considers kinetics as N individuals and their With: values over time as P variables. It reduces the number of i (i=1,…,N): index of a pixel variables by projecting individuals onto Q orthogonal vectors UQ (t). PCA can be described through the xi: curve recorded for a pixel k (k=1,...,K) : index of a factor following steps: [4] A P dimensional mean vector is calculated from the N fk: fundamental kinetic th kinetics according to following relation: ak: coefficient of the k physiological compartment (i): noise recorded for the ith pixel N 1 x (t) x (t .) . (3) Curves recorded for each pixel are linear moy i N combination of a small number of fundamental kinetics i1 having a physiological interpretation. For each pixel, the A centered matrix X (N,P) is computed where the coefficient associated to each fundamental kinetic is the S rows are vectors difference associated with kinetics: proportion of the signal in that pixel following the fundamental kinetic. The coefficients give the kth (4) physiological compartment associated to the kth i (t) xi (t) xmoy (t). fundamental kinetic. [2] Fundamental kinetics are called "factors", while the The covariance matrix R is constructed as follows: associated images are called "factor images". R Xst .Xs. (5) 4. FACTOR ANALYSIS MODEL SOLUTION The eigenvalues ( ) associated with In factor analysis, only kinetics are considered to be λi (i 1..P) known. The physiological compartments and associated eigenvectors (Ui, i=1...P) of the matrix R are computed. kinetics must be estimated. Moreover, noise recorded for The new coordinates of individuals are their projections each pixel had to be considered. To estimate all these onto the new orthogonal basis according to following unknown functions, the resolution method may use three relation: types of prior knowledge: noise properties for estimating (6) , information related to the shape of the fundamental VP Xs.U p . kinetics and spatial distribution of physiological compartments. The centred matrix Xs can be then expressed as: The resolution of the FA model proceeds in two steps. The first one is an orthogonal analysis to estimate (7) noise-free data. The second one is an oblique analysis to Xs VP.P.U P determine physiological functions and associated compartments. [2] where ΛP is the diagonal matrix (λ1, λ2... λP). Hence, we can express the matrix X as: 4.1. Orthogonal analysis (8) X 1N.xmoy VP. P.U P We consider a sequence of P functional images of N pixels taken at instants (t , t ... t ). We denote X (of size 1 2 P where 1 is a (N,1) matrix of ones. By retaining only the NxP) as the original data matrix. The original data x (t) N i Q first principal components, we obtain: belongs to a P dimensional space, and noise-free data x (t)-ε (t) belongs to a Q (Q<P) dimensional space ‘S’. i i (9) Orthogonal analysis aims to estimate the space ‘S’. It X 1N.xmoy VQ.Q.UQ is based on singular value decomposition (SVD) of data that diagonalizes the covariance matrix of X. This matrix which yields to: involves two metrics ‘W and M’. [3] X V..U (10) Co variance matrix X tWXM. (2) With We compute the Q eigenvectors UQ associated with the Q largest eigenvalues λq of the matrix C. [5] 1 xmoy1 ... xmoyP We calculate VQ according to the following equation: V ... V Q U ... UQ (11) ' 1 (20) 1 VQ (Y 1N Ymoy ).M.UQ .Q With 1... 0 (12) 0 -1 (21) Q M Q diag ( 1 ,..., q ) Each noise-free kinetic may be expressed then as [4] Hence, each free-noise kinetic may be expressed as: Q Q x (t) x (t) .v (i).u (t) (13) i moy q q q y (t) y (t) .v (i).u (t) (22) q1 i moy q q q q1 4.1.2. Correspondance Analysis (CA) 4.1.3. Oblique analysis The CA is based on an optimal normalization using The oblique analysis aims to identify physiological statistical properties of signal and noise. It is described by functions and associated compartments in noise-free data the following procedure [5]: space. It is based on constraints and uses an iterative We calculate the following variables: approach to obtain the basis having physiological P N N P meaning. [1] The goal of this iterative algorithm is to perform a xi. xij x. j xij x.. xij (14) j1 i1 j 1 j1 linear transformation of the noise-free data space basis to express new components having physiological sense. The We define the matrix Y (N,P) issued from the constraint used is the positivity of physiological functions optimal normalization of the data matrix X [5] : and associated compartments. [6] It aims to transform the orthogonal basis (Eq. (23)) to perform a decomposition of the sequence into factors and xi factorials images. yi (15) xi. y (t) a (i).f (t) with a (i) 0 f (t) 0 We calculate the following variables: i k k k k (23) k x x x. j x Orthogonal analysis gives a noise-free data matrix i. 1. ..P (16) i diag ( ,..., ,..., ) described by the following equation [6]: x.. x.. x.. x.. The mean vector is defined as follow: Y 1N .Ymoy VQ .Q .UQ (24) N y Oblique analysis aims to represent it as a i i x combination of factors (matrix F) and factorial images i1 x 1. j x.P (17) ymoy N ( ,..., ,..., ) (matrix A) with respect to the constraints (A>0 and F>0). x.. x.
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