Extraction of functional volumes from medical dynamic volumetric data sets

Compurerized Medical Imaging and Graphrcs, Vol. 17, Nos. 4/S, pp. 397-404, F’rinted in the U.S.A. All rights reserved.

1993

0895-611 l/93 $6.00 + .oO Copyright 0 1993 krgamon Pres Ltd.

EXTRACTION OF FUNCTIONAL VOLUMES FROM MEDICAL DYNAMIC VOLUMETRIC DATA SETS FrCdk-ique Frouin *l , Luc Cinotti2, Habib Benali’, Irsne Buvat’, Jean-Pierre Bazin’, Philippe Millet2, and Robert Di Paola’ ‘U66 INSERM

Institut Gustave-Roussy 39, rue C. Desmoulins 94805 Villejuif Cedex, France ‘CERMEP, 59 Boulevard Pinel, 69003 Lyon Cedex, France (Received 8 April 1993)

Abstract-A method based on factor analysis is presented to process dynamic volumetric (t + 3D) data sets acquired for flow, excretion, or metabolic studies. It estimates a reduced number of underlying physiological kinetics and their associated spatial distributions, corresponding to functional volumes, using dedicated algorithms. The global (t + 3D) approach is shown to be superior to the conventional one, which repeats estimations on each (t + 2D) data set, obtained for each slice or projection of the volume. Key Words: Factor analysis, Functional imaging, 3D Image processing, Multidimensional data, Physiological model, PET studies

INTRODUCIION

structures, such as cardiac wall motion, few were devoted to the so-called “functional imaging.” Factor Analysis of Dynamic Structures (FADS) was proposed to reduce without significant loss of information (t + 2D) data sets into a few kinetics and spatial distributions, corresponding to physiological structures (3-5). It combines the goals of parametric imaging by computing spatial distributions and region of interest (ROI) method by estimating underlying time signal curves. Moreover, its underlying model assumes that several structures can lie in a single pixel, and yield mixed kinetics. FADS algorithm consists in estimating first the underlying kinetics, then their associated spatial distributions. The extension of FADS, called Factor Analysis of Medical Image Sequences (FAMIS), to 4D data sets was first proposed by Frouin et al. (6). This paper presents new methods of estimation used by FAMIS, and assesses the validity of a (t + 3D) approach compared to a series of conventional (t + 2D) approaches, repeated for each slice or projection. The next section introduces the notation and the model to explain (t + 3D) data. Then, the estimation methodology is presented. A numerical simulation and two clinical applications are finally described and discussed.

In vivo temporal image sequences are acquired in the major part of medical image modalities to study noninvasively dynamic physiological mechanisms. A classification of such dynamic observations in three groups separates structure motion (as cardiac wall motion), flow and excretion, and metabolic studies (1). Except for some cases dealing with structure motion, all these observations require the injection of a substance (e.g., a radionuclide or a contrast medium). In this paper, only time image sequences obtained for flow or metabolism investigations are considered. Dynamic planar image sequences, referenced as threedimensional (t + 2D) data sets (one temporal dimension and two spatial dimensions) in Chameroy and Di Paola (2), included projections (planar scintigraphy, X-ray angiography) or single slice acquisitions (MRI, X-ray CT, PET). Dynamic volumetric acquisitions, referenced as four-dimensional (t + 3D or 4D) data sets (2), are the most appropriate way to investigate without ambiguity the volume of interest. They are not frequently performed due to technical difficulties related to the data acquisition process, and to the lack of subsequent data processing, which is more than ever necessary to extract relevant physiological information from this huge amount of data. If some 4D processing methods have been developed to study motion or deformation of physiological

NOTATION

AND MODEL

Notation Matrices M are represented by boldface upper case letters and column vectors v by boldface lower case letters. The elements of a matrix M are denoted M(i, j), i = 1, * - - Izi and j = 1, - - - nj. The components

* To whom correspondence should be addressed. 397

398

Computerized Medical Imaging and Graphics

of a vector v, v(k), k = 1, * - * nk. Row vectors are indicated by the transpose relation vT. A frame (i.e., a slice or a projection) of ni rows and nj columns is considered in its vector form p obtained by row scanning, with ni - nj components called pixels. A volumetric data set of n, slices or projections is represented in its vector form too, v of nk = n, components, by scanning the n, frames which compose the volume. Each component of v is called a voxel. A dynamic planar (t + 2D) data set consisting of n, frames is expressed as a matrix D of n, - nj rows and n, columns. A dynamic volumetric data set consisting of n, frames is expressed as a matrix U of nk rows and n, columns. Each row of both matrices D and U is a time signal intensity curve, denoted cT, having n, components. The generic name for a vector cT is a n-ixel. It is respectively defined as a 3-ixel (or trixel) for dynamic planar image sequences, and as a 4-ixel (or quadrixel) in case of (t + 3D) data sets. The external product vBcT of two vectors v and c of nk and nt components is defined by the matrix U of nk rows and n1columns, such that U(i,j) = v(i)*c(j). ' ml

' nj

Model statement The model is first expressed for (t + 2D) data sets. It is assumed that the physiological process under observation can be modelled by D exp = D + E,

(1)

where Dewprepresent noisy observations, D the theoretical noise-free process and E an error term. D(i, j ) and E(i, j ) are supposed to be uncorrelated for each i, j. Furthermore, the process is assumed to be reduced to a limited number n, of components, corresponding to n, rigid spatial structures ps in which the injected substance has a uniform kinetics c,: D=

5 P,@$.

(2)

s=,

This hypothesis is close to the assumption made in compartmental analysis, according to Atkins’ definition of a compartment as “a quantity of substance which has a uniform and distinguishable kinetics of transformation or transport” (7). However, a structure may correspond to several compartments, distinguishable at microscopical level, but not distinguishable at medical imaging macroscopical level. Modelling of (t + 3D) data set is an extension of the previous one with the additional assumption that kinetics are uniform not only in a projection or a slice, but also in the observed volume. It results: U exp = U + E,

(3)

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with the decomposition of U into n, kinetics c, and n, functional volumes v, :

u =

5 V,OCf,

(4)

s=l

which is equivalent to eq. (5) giving the definition of vs: PI

;

u=s

oc:.

S=l ( Pn, 1s

(5)

The problem addressed by FAMIS is the estimation of the n, kinetics cs, called physiological functions, and their associated spatial distributions v,. Positivity constraints are applied to both kinetics and spatial distributions (3). To avoid ambiguity between normalization coefficients of spatial distributions and kinetics, kinetics are arbitrarily scaled so that their areas are equal to 1. ESTIMATION

METHOD

FAMIS computes first n, factors, estimates of the underlying physiological functions c,. Once factors are available, the estimates of spatial distributions vs, called factor images or factor volumes, are then computed in the least square sense. FAMIS is based on an orthogonal stage which estimates a (n, - 1) dimensional subspace U containing the n, physiological functions, and an oblique stage to estimate the n, physiological functions under positivity constraints. Data preprocessing, consisting in clustering pixels into regular pattern, increases signal to noise ratio and reduces the number of elements to process (4, 8). Aurengo (9) introduced some adaptive clustering techniques to connect only spatially contiguous n-ixels, with statistically similar profiles. A more computing time efficient Clustering Algorithm of Medical Image Sequences (CAMIS), was proposed (10). It is based on mutual nearest neighbour criterion, which avoids drift effect (11). In case of scintigraphic data, the similarity is expressed as a statistical test (9, lo), based on Poisson noise distribution. A threshold and, if necessary, a manual segmentation are then applied to remove noisy n-ixels and “out of interest” regions. Resulting n-ixels are then submitted to an orthogonal analysis, based on the eigendecomposition of the variance covariance matrix U,‘,, - Uexp of the experimental preprocessed data. The orthogonal subspace U is generated by the eigenvectors corresponding to the (n, - 1) largest eigenvalues (4, 5). The oblique step is performed in the subspace U. As first expressed by Barber (4) positivity constraints on spatial distributions are geometrically expressed as

Functional volumes from volumetricdata sets??F. FROWN et al.

399

Table 2. Orthogonal step quality index after CAMIS preprocessing G

s,

SZ

DI

0.4

1.1

2.8

33.

4

1.0

10.

D3

2.1

udf

::d

180. 21.

s3

In the paper, the comparison of the (t + 3D) ap preach with a series of (t + 2D) analyses, the interest of CAMIS preprocessing, and the addition of knowledge introduced in the oblique step are studied. For dynamic studies, target-apex seeking method can be used when auxiliary measurements (e.g., blood samples) are available (20). EXPERIMENTAL

DESiGN

Numerical simulation

Fig. 1. Three simulated functional volumes v, , v2, and vg , and three corresponding physiological functions c, , c2, and c3. Each column of the figure represents the three slices of the numerical volume and its simulated kinetics. the inclusion of experimental trixels inside a polytope the apices of which are the estimates of the underlying physiological functions called factors. The algorithm was refined to take into account positivity constraints on factors (5). However, in most cases, positivity constraints are necessary but not sufficient. Many attempts to introduce additional constraints during the oblique step were developed. Some general methods propose to minimize entropy criterium (12) or volume of the factor polytope ( 13) under positivity constraints. Another class of methods introduces more physiological information. They are based on factor (14, 15) or factor image ( 16- 18) modelling. Buvat ( 19) proposed a “target-apex seeking” method to estimate factors which best fit a priori knowledge, provided that this knowledge is expressed by a criterion to minimize. This method is particularly well suited for energy studies. Table 1. Orthogonal step quality index after 4 X 4 pixel rectangular clustering preprocessing G

Sl

Dl

0.5

1.5

D2

1.7

9.3

D3

2.1

udP

* udf stands for undefined results.

s2

1.2 5.4 8.5

s3

0.4 320. 46.

Three volumetric data sets v, consisting of n, = 3 slices, each having h - nj = 4096 pixels, and three physiological kinetics c, of nt = 10 time samples (Fig. 1) were simulated. A (t + 3D) data set U was computed using eq. (5). Poisson noise was added to simulate scintigraphic process, producing a matrix U,,,. Three main questions were investigated: 1. interest of a global (t + 3D) analysis compared to three (t + 2D) analyses performed on each slice; 2. interest of CAMIS preprocessing compared to a conventional rectangular clustering; and 3. interest of introducing a priori knowledge concerning one physiological function. The first two points were assessed for both the orthogonal and the oblique steps, the last one only for the oblique analysis. To evaluate the orthogonal decomposition, the distance d(c,, U) of each physiological function c, to the orthogonal subspace U was computed. To assess the oblique rotation, the correlation (I,) of each factor with the corresponding physiological function was computed as well as the correlation (R,) of each factor volume or factor image with the underlying spatial distribution. Clinical examples

A (“F) fluoro-2-deoxyglucose ( 18FDG) consumption brain study was chosen as a first example. Data were acquired with a Time Of Flight (TOF) PET scanner during 1 h. Seven slices of 128 X 128 pixels were reconstructed from the list mode acquisition. Pixel size was equal to 2 mm and cutoff frequency was 0.18

Computerized Medical Imaging and Graphics

July-October/ 1993, Volume 17, Numbers 4/5

lb)

(4

Fig. 2. Factors and factor images estimated for the simulation study. (a) Case of (t + 3D) analysis. (b) Case of (t + 2D) analyses. Image sequence included 30 time samples: 12 10 s, 6 X 30 s, 5 X 60 s, 4 X 300 s, and 3 X 600 s. Blood samples were manually drawn from the radial artery, constituting an additional measurement. First the (t + 2D) data set associated with the fifth slice was conventionally processed. Then the global (t + 3D) acquisition was submitted twice to FAMIS: first, with positivity constraints alone, and second, with positivity constraints and using the auxiliary measurements as a priori knowledge. The second example is a brain blood flow study performed with [“O] water. Data were acquired with the same PET scanner during 4 min. Reconstruction parameters are similar to those of the first example except for the cut-off frequency (0.15 mm-‘) and the time sampling of the image sequence: 15 X 4 s, 6 X 10 s, 6 X 20 s. Blood samples were drawn from the

radial artery too, at the rate of 1 per 3 s. Four contiguous planes were submitted twice to FAMIS: first, with positivity constraints alone, and second, with positivity constraints and using the blood samples as a priori knowledge.

Table 3. Indices of FAME estimation quality after 4 X 4 pixel rectangular clustering preprocessing, and oblique rotation under positivity constraints

Table 4. Indices of FAME estimation quality after CAMIS preprocessing, and oblique rotation under positivity constraints

mm-‘. X

RESULTS

Numerical simulation To express the results, the global (t + 3D) analysis is denoted G; and the (t + 2D) analyses performed on the first slice S,, the second slice S2 and the third slice &. The factors and factor volumes which were computed by the (t + 3D) analysis are displayed in Fig. 2a, and the factors and factor images which result from the (t + 2D) analyses in Fig. 2b.

G

s,

r2 r3

0.99 1.00 0.91

RI R2 R3

0.90 0.99 0.66

G

S1

Sz

S3

1.oo

0.99

rl

r3

1.00 1.oo udf

1.oo

1.oo 0.86

1.00 0.92

1.oo 0.62

R, R2 R3

0.82 0.99 0.69

1.00 1.00 udf

0.9 1 0.98 0.68

0.55 0.99 0.60

rI r2

s2

&

1.00 1.a0 udf

1.00 1.00 0.96

1.00 0.97 -0.63

0.99 1.00 udf

0.94 0.98 0.69

0.99 0.89 0.51

Functionalvolumesfrom volumetricdata sets??F. FROWN Table 5. Indices of FAMIS estimation quality after CAMIS preprocessing, and oblique rotation under positivity constraints and a priori knowledge about the kinetics C3 G

s1

SZ

0.99

udf udf udf

I .oo 1.00 0.96

0.11

0.96 0.99 0.66

udf udf udf

0.95 0.98 0.61

0.97 0.96 0.64

rl

1.00

r2 r3

1.00

RI

g2 R3

s3

1.00 1.00

The two first tables express the quality of the orthogonal decomposition in case of a rectangular clustering (Table 1) and after CAMIS (Table 2). To make the interpretation of orthogonal subspace estimation easier, the ratios D, of the distances d(cs, U) to the ones obtained on noise-free data (i.e., without Poisson noise) are reported in Tables 1 and 2. As the orthogonal step aims at reducing noise in the experimental data, the ratios D, are assumed to be larger than 1, the best results in terms of separation of signal from noise are obtained when this ratio becomes close to 1. However, some ratios (Dt ) were found smaller than 1, due to computation noise which occurred for “noise-free” data, too. Tables 3, 4, and 5 present the correlations r, of the three factors with the three underlying physiological functions and the correlations R,of the factor volumes (or images) with the underlying spatial distributions (volumes or images) in the three configurations that we tested.

Clinical examples For the 18FDG brain study, two factors and factor volumes are estimated corresponding to a vascular component and a brain tissue component. The superimposition of vascular factors and soft tissue factors obtained in the three following configurations are presented respectively in Figs. 3a and 3b: 1. FAMIS applied to the 5th slice; 2. FAMIS applied to the (t + 3D) data set; and 3. FAMIS applied to the (t + 3D) data set using the blood sample constraint. From these curves, it appears that the estimation performed on the (t + 3D) data set is better compared to the one performed on the 5th slice. Indeed, the vascular factor is closer to the projection of the blood sample in the orthogonal subspace. Figure 4 shows the two factor volumes obtained in the (t + 3D) analysis, using the blood sample constraint. In this case, a better repartition of vascular and tissue activities is observed. Subcortical structures of the brain are clearly delineated and the statistical noise is diminished. For the (150) water brain study, two factors and factor volumes are estimated, too. They correspond to a vascular component and a brain tissue component (Fig. 5). When the positivity constraints are used alone (Fig. 5a), the vascular factor shows a peak corresponding to the inflow of the bolus, but it has an insufficient amplitude compared to the diffusion phase following its first pass. The associated factor volume primarily corresponds to the locations of major vessel sections. The brain tissue uptake factor is compatible with the 800

800

4 + +

600

401

et al.

1

1

SFl-SC SFl-CS SFl-AC

1

6(Do

M 4L”

I

”

0

loo0

2ooo

(4

3ooo

time

I

I

I

0

1000

2000

.

a-

sF2-SC

+ -)

sFzc5 SF;?-AC

I 3ooo

(W

Fig. 3. Kinetics estimated for the (1 SFDG) PET study. (a) Profiles of the vascular factors estimated on the 5th slice (SFl-CS), on the 7 slices (SFl-SC), and on the 7 slices using the blood sample constraint (SFl-AC). (b) Profiles of the brain tissue factors estimated on the 5th slice (SF2-C5), on the 7 slices (SF2-SC), and on the 7 slices using the blood sample constraint (SF2-AC).

ti Ae

Fig. 4. Factor volumes computed on the (t + 3D) data set using the blood sample constraint for the (18FI IG) PET study. Left columns: Vascular factor volume, right columns: Brain tissue factor volume.

(b) Fig. 5. Factors and factor volumes estimated for the I50 water brain study. Upper rows: Vascular factor volume, lower rows: Brain tissue factor volume. (a) Analysis with positivity constraints alone. (b) Analysis with both 1Jositivity constraints and arterial blood samples constraints.

Functional volumes from volumetric data sets ??F. FROWN

kinetics of a freely diffusible tracer. The factor volume represents the regions with larger uptake due to higher flow (i.e., the gray matter). However, these images remain patchy. When the arterial blood sample information is additionally used, results (Fig. 5b) show a substantial improvement. The vascular factor has an improved peak to diffusion plateau ratio. If the tissue uptake factor remains unchanged, the associated factor volume demonstrates a better contrast, and anatomical details become available in the central gray structures. DISCUSSION

etal.

403

netics is difficult to estimate without additional information because the kinetics is always mixed with the others when it is present in a voxel. For the PET examples, spill-over effect is high in the upper slices of the brain, where no vascular structures can be precisely delineated. The inclusion in the analysis of lower slices with more apparent vascular structures reduces this effect. Improved qualitative and quantitative results are expected by the addition of physiological knowledge. They should improve‘ the compartmental modelling step involved in PET studies.

AND CONCLUSION

The (t + 2D) acquisitions are generally processed by ROI or parametric imaging. The main limitations of ROI method lie in its intra- and inter-operator’s variability or the difficulty to set up algorithms that automatically locate the ROI at the correct position (2 1). The volumetric extension of ROI is defined as 3D regions or Volumes of Interest (VOI). In most cases, they correspond to the mathematical union of ROIs acquired on the different slices, because few techniques allow precisely delineated manual volumes, using a two-dimensional interface. The 3D extension of FAMIS can be easily derived because it is based on a pixel-by-pixel computation. The (t + 3D) FAMIS approach provides a better approximation of the orthogonal subspace than the (t + 2D) approaches (Tables 1 and 2). It enables better estimations of the n, physiological functions and spatial distributions, and by hypothesis provides identical factors for the three slices. The superiority of the (t + 3D) approach of FAMIS for estimating the orthogonal subspace, is due to statistical reasons, because more data are used for the estimation. These supplementary data are useful for the oblique analysis step, too, especially when they contain kinetics, which are close to the underlying physiological functions. CAMIS appears to be superior to 4 X 4 rectangular pattern in the global analysis, even if it seems equivalent in the (t + 2D) analyses (Tables l-4). This can be theoretically explained, because CAMIS clusters have less mixed kinetics, facilitating the estimation of the underlying pure kinetics (9, 10). In some cases, a (t + 2D) analysis (for the second slice of the example) can provide better factors and factor images than the (t + 3D) analysis (Tables 3 and 4), although this result is reversed for the orthogonal subspace estimation. This underlines the difficulties due to the oblique rotation algorithm, that remains the critical point of FAMIS. The introduction of a priori knowledge concerning one physiological function (~3) is in this case particularly relevant (Table 5). This ki-

Acknowledgments-We thank J.P. Serra and F. Lavenne of the CERMEP team for allowing us to acquire the PET data, and M. Di Paola of Institut Gustave-Roussy for computer programming and iconography. I. Buvat thanks the IFSBM (Institut de Formation Sup&ieure Biomkdicale, VilIejuif, France) and Sopha Medical (But, France) for supporting her Ph.D.

REFERENCES 1. Adam, W.E. A general comparison of functional imaging in nuclear medicine with other modalities, Semin. Nucl. Med. XVII: 3-17; 1987. 2. Chameroy, V.; Di Paola, R. Towards multidimensional medical image coding. In: Kim, Y., ed. Medical imaging VI: Image formation, capture and workstation. SPIE, 1653; 1992:261-272. 3. Bazin, J.P.; Di Paola, R.; Gibaud, B.; Rougier, P.; Tubiana, M. Factor analysis of dynamic scintigraphic data as a modelling method. An application to the detection of the metastases. In: Di Paola, R.; Kahn, E. eds. Information processing in medical imaging. INSERM. Paris: 88; 1980~345-366. 4. Barber, D.C. The use of principal components in the quantitative analysis of gamma camera dynamic studies. Phys. Med. Biol. 25:283-292; 1980. 5. Di Paola, R.; Bazin, J.P.; Aubry, F.; Aurengo, A.; Cavailloles, F.; Herry, J.Y.; Kahn, E. Handling of dynamic sequences in nuclear medicine. IEEE Trans. Nucl. Sci. 29: 1310-I 32 1; 1982. 6. Frouin, F.; Bazin, J.P.; Di Paola, M.; Jolivet, 0.; Di Paola, R. FAMIS: A software package for functional feature extraction from biomedical multidimensional images. Comput. Med. Imaging Graph. 16:81-91; 1992. I. Atkins, G.L. Multicompartment models for biological systems. London: Methuen; 1969. a. Di Paola, R.; Bazin, J.P.; Aubry, F.; Kahn, E.; de Talhouet, H.; Aurengo, A.; Cavailloles, F.; Herry, J.Y. Developments in data processing in nuclear medicine imaging. Izotoptechnika 25:94111; 1982. 9. Aurengo, A. Analyse factorielle des sequences d’images en Mbde&e Nucltaire. These d’Etat. Universite de Paris-Sud: 1989. 10. Buvat, I. Correction de la diffusion en imagerie scintigraphique. Ph. D. Univemitk de Paris XI; 1992. Il. Lebart, L. Programme d’agrkgation sous contraintes. Les cahiers de l’analyse des don&es. 3~215-278; 1988. 12. Nakamura, M.; Suzuki, Y.; Kobayashi, S. A method for recovering physiological components from dynamic radionuclide images using the maximum entropy principle: A numerical inves t&ion. IEEE Trans. Biom. Eng. 36:906-917; 1989. 13. Van Daele, M.; Joosten, J.; Devost, P.; Vandecruys, A.; Willems, J.L.; De Roo, M. A new vertex-finding algorithm for the oblique rotation step in factor analysis. Phys. Med. Biol. 36:77-85; 199 1. 14. Nijran, K.S.; Barber, D.C. Factor analysis of dynamic function studies, using a priori physiological information. Phys. Med. Biol. 31:1107-1117; 1986.

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15. Frouin, F.; Bazin, J.P.; Di Paola, R. Image sequence processing using factor analysis and compartimental modelling. In: Viergever, M.A., ed., Science and engineering of medical imaging. SPIE, 1137;1989:37-44. 16. Nijran, KS.; Barber, DC. The importance of constraints in factor analysis ofdynamic studies. In: de Graaf, C.N.; Viergever, M.A., eds. Information processing in medical imaging. New York: Plenum Press; 1988:521-529. 17. Samal, M.; Kamy, M.; Surova, H.; Marikova, E.; Dienstbier, Z. Rotation to simple structure in factor analysis of dynamic radionuclide studies. Phys. Med. Biol. 32:371-382; 1987. 18. Houston, A.S. The use of cluster analysis and constrained optimisation techniques in factor analysis of dynamic structures. In: Viergever, M.A.; Todd-Pokropek, A.E., eds. Mathematics and computer science in medical imaging. Berlin Heidelberg: Springer-Verlag; F39; 1988:49l-503. 19. Buvat, I.; Benali, H.; Frouin, F.; Bazin, J.P.; Di Paola, R. Target apex-seeking in factor analysis of medical image sequences. Phys. Med. Biol. 38:123-138; 1993. 20. Cinotti, L.; Frouin, F.; Le Bars, D.; Bazin, J.P.; Landais, P.; Millet. P.: Benali. H.: Mauguiere. F.; Di Paola. R. Auplication of constrained factor ‘analy& of dynamic structures (FADS) to model and quantify PET image sequences in phantom and brain studies. Proceedings of the 6th Mediterranean Conference on Medical and Biological Engineering. 1992. 21 Cosgriff, B. Region of Interest, ROI: Confidence in derived parameters? Nucl. Med. Comm. 6:305-309; 1985.

About the Author-FRgDgRIQUE FROWN was graduated from the Ecole Nationale Sun&ieure des T&communications (ENST), Paris, in 1986. She received the Ph. D. degree in Signal Processing from ENST in 1989. From 1986 to 1989 she attended the Institut de Formation Superieure Biomedicale, Villejuif, France to follow a theoretical training in medical and biological engineering. In 1990, she was a post-doctoral fellow at Sopha Medical Inc., But, France. She is currently Charge de Recherche of the Institut National de la Sante et de la Recherche MMicale (INSERM) in the Unite 66. Her main interests are biomedical image processing’and physiological modelling. About the Author-Luc CINO~~Ireceived a Chemical Engineer degree in 1973 and a Medical Doctor degree in 1980 in Paris. He worked in biophysics and nuclear medicine as an assistant in Paris-XII university hospital from 1980 until 1988. He is currently Professor of biophysics at the university of Lyon and technical director at the PET facility of Lyon. His research interests are the physiological modelling and its applications to clinical investigations in cardiology and neurology. Luc Cinotti is a member of the IEEE and of the Society of Nuclear Medicine.

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Volume 17, Numbers 4/5

About the Author-HABIB BENALIreceived his Ph.D. Degree in signal processing and dam analysis from the University of Rennes, France. From 1984 to 1987 he was with the University of Rennes where he worked as Lecturer in Computer Science andStatistics. From 1987 to 1989 he was an Ingenieur de Recherche at the Institut National de la Sante et de la Recherche Medicale (INSERM) in the Unite 287 of biostatistics. He joined the Unite 66 of the INSERM in 1989. His research interests include pattern recognition, data analysis and statistical modeling for morphological and functional biomedical imaging. About the Author-IRgNE BUVATjoined the Unite 66 of the Institut National de la Sante et de la Recherche Medicale (INSERM) in 1989 as a Ph.D. student in Physics. She was working on the development of factor analysis of spectral image sequences in nuclear medicine. Alter defending her Ph.D. in 1992, she joined the Department of Medical Physics of the University College London for a postdoctoral researche position under the direction of Andrew Todd Pokropek. Her interest includes the use of image processing and especially factor analysis of medical image sequences applied to quantitation in medical imaging. About the Author-JEAN-PIERRE BAZINworked at the Commissariat a 1’Energie Atomique (CEA) from 1961 to 1968, where he was involved in biological system modelling. He has been Ingenieur de Recherche of INSERM. in the Unite 66 since 1968. From this time. he has worked both on’compartmental modelling and on image se: quence processing. He has been particularly concerned with the development of Factor Analysis techniques applied to image sequence processing. About the Author-PHILIPPE MILLET received the MS degree in biomedical engineering. He is currently in a Ph. D. degree program in biomedical engineering in the Positron Emission tomography center of Lyon. His research interests include mathematical modeling, identification of the receptor model parameters and maps of receptor binding parameters in collaboration with the PET center in Orsay, France. About the Author-ROBERT DI PAOLAreceived the DES. in physics from the University of Toulouse, France in 1961. He joined the Nuclear Medicine Department at the Institut Gustave-Roussy, Villejuif, France in 1963, as attache de recherche of the Faculty of Medicine of Paris. Since 1966, he has been Attache, Charge and Directeur de Recherche of the Institut National de la Sante et de la Recherche M&&ale (INSERM) in the Unite of Clinical Radiology no66 of the INSERM. From 1986 he leads this Unite which deals with Morphological and Functional Biomedical Imaging. He has specialized in the processing of medical images, both two-dimensional and threedimensional with emphasis on functional imaging.