germinal center analysis with the tools of mathematical morphology on graphs

@ 1993 Wiley-Liss, Inc. Cytometry 14:848-861 (1993)

Germinal Center Mathematical

Analysis With the Tools of Morphology on Graphs’

Eric Raymond, Martine Raphael,2 Michel Grimaud, Luc Vincent, Jacques Louis Binet, and Fernand Meyer

Centre d’Ecologie Cellulaire, HBpital Pitie-salpetriere, 75651 Paris Cedex 13 (E.R., M.R., J.L.B.); Centre de Morphologie Mathematique, Ecole nationale superieure des mines de Paris, 77300 Fontainebleau,

(M.A., L.V., F.M.) France Received for publication August 25, 1992; accepted May 28, 1993

Only few studies devoted to quantitative analysis of tissue architecture have been performed. The analysis of neighborhood relationships between cells, using graphs and mathematical morphology (MM), constitutes one approach. We propose to analyse quantitatively the architecture of a tissue with the tools of MM on graphs. The use of graphs seems best suited to take into account the neighborhood relationships between cells, inde- pendently from their mutual distances: two cells are considered as neighbors if no interfering cell is placed in a given sense between them. Such neighboring cells are linked by an arc in a graph. On such a graph, all tools of MM may be applied. We investigate two of them in the present work (1) the distance transforms permits to analyse the repartition of a cellular population A relative to population B, (2) the size distribution permits to analyse the tendancy of a cell population to form clusters. We have applied this method using Gabriel’s graph, derived from the Voronoi diagram, to determine the “zone of influence” of a cell.

We have analysed the neighborhood relationships between cells in germinal centers (GC) from lymph nodes. Twelve hyperplastic GCs from follicular hyperplasia (FH) and 5 neoplastic GCs from small cleaved cell follicular lymphoma SCCFL) have been studied; 2 pm hema-

toxylin-eosin plastic embedded sections have been analysed. Cell nuclei have been identified manually by the observer by giving a numerical and image color code. Gabriel’s graphs have been constructed with all centrofollicular cells. Subgraphs with only lymphoid cells, large cells or small cleaved cells have also been studied.

The application of MM transformations on the graphs using software package Morphograph has allowed a quantitative description of cell distribution in the tissue. The distances of centrofollicdar lymphoid cells from the periphery of the GCs (mantle zone), from nonlymphoid cells as macrophages located within the GCs and between large lymphoid cells and small lymphoid cells have been de- termined in FH and SCCFL. Using iterative closings and openings, we have detected aggregates of small and large cells and characterized their size and distribution in the GCs. The application of this method on lymph node biopsy has allowed a quantitative description and comparison of GCs in different patholog- ical conditions. o 1993 Wiley-Liss, Inc.

Key terms: Mathematical morphology on graph, image analysis, morphometry, follicular hyperplasia, follicular lymphoma

In histopathology, many cases need a quantitative description to obtain reproducible diagnoses, classifica- tions and histoprognos~c grading (l).kor thkse reasons

been developed. a quantitative spatial anal-

‘This work was presented at the XV Congress of the International Society for analytical cytology, August 25-30,1991, Bergen, Norway.

dex 13, France.

Objective quantitative Of m o r p h o l o ~ have ”Address reprint requests to M. Raphael Centre &6cologie cellulaire, HBpital, Pitie-Salpetriere, 47 Bd de l’HBpita1, 75651 Paris Ce-

ysis of tissue architecture and relationships between

GERMINAL CENTER ANALYSIS WITH GRAPHS 849

cells has not been described yet, especially in lymph nodes.

We have studied the organization of germinal center (GC) cells from reactive follicular hyperplasia (FH) and from predominantly small cleaved cell follicular lymphoma (SCCFL). In some cases the distinction between FH and SCCFL remains difficult to draw, complemen- tary investigations such as immune markers and mo- lecular studies being needed. In such cases morphologic quantitative analysis could be helpful (18,19,23). The cytology and architectural organisation of cells in typical FH and SCCFL have been described by Rappaport and Lennert (11,19). In FH, GCs are characterized by a zonal structure composed of different populations of follicular center cells. Lympho’id cells are large non- cleaved cells (centroblasts), small and large cleaved cells (centrocytes), and immunoblasts. Nonlympho‘id cells are histiocytes, macrophages containing tingible bodies and antigens presenting cells such as dendritic reticulum cells. The architectural structure of the GCs in FH has been described with a dark zone mainly composed of large cells or centroblasts and a light zone composed of small cleaved cells or centrocytes.

Most morphometric analyses have been devoted to cell or nuclei characterization of follicular center cells in reactive FH and SCCFL (13,14,20,21). In contrast to a precise quantitative morphologic description of GCs that individualized a zonal structure, only a few studies have been carried out about quantitative architectural analysis of lymph nodes (4,5).

In this study, we analyse quantitatively the architecture of a tissue with the tools of mathematical morphology (MM) on graphs. The use of graphs seems best suited to take into account the neighborhood relationships between cells, independent of their mutual distances. Two cells x and y are considered as neighbors if no interfering cell z is placed between them in the following sense: the disk for which cells x and y are the extremities of a diameter should be empty from interfering cells. If we link by an arc all cells sharing this property, we get Gabriel’s graph. Vincent (25,261 has shown how all tools of MM may be applied on graphs. We use distance transforms and size distributions for the analysis of the architecture of GC. Distance transforms allow us to analyse the repartition of a cellular population A relative to another population B. Size distribution allows us to analyse the tendency of a cell population to form clusters. The application of MM to graph was used to quantify the topographic organisation of cells in normal and neoplastic GCs.

INTRODUCTION TO MATHEMATICAL MORPHOLOGY AND GRAPHS

Neighborhood Graph of a Population of Cells We see in the following section how the image is

acquired. At this point, we suppose that we must char- acterize a two-dimensional section of a tissue architecture. A number n of cell types coexist in the tissue; after identification, each cell nucleus is flaged by a

pixel in its center, the grey-tone attributed to this pixel being characteristic of the cell type.

As soon as all pixels are identified, we construct a graph, expressing their neighborhood relationships. For the construction of the graph, we make no difference between cell types but pool them all together. Let V be the set of cell markers.

By graph (G) we mean a pair (V,E) where V = { u l , u2, . . . , un} is a finite set of objects called vertices and E = { e l , e2, . . . , em} is a family of pairs (ui, u j ) of vertices designated by arcs. G (V,E) is assumed to be a nonoriented graph without loops. We designate as N,(u) the set of neighbors of v E V, N,(v) = {v’ E V, (v,v’> E E}.

There are several ways to define a neighborhood graph. Vincent (25) has listed them. A classical way to construct a graph is to define the graph of the k-nearest neighbors (16,17). This procedure is not satisfactory for our purpose because two cells may be linked even if other cells are placed between them. This is contradic- tory with our goal to express the direct relationships between cells.

We will come closer to this goal if we use the graphs deriving from Vorono’i diagrams.

Let us briefly recall the definition of the Voronoi diagram of V. For any p E V, the Vorono’i polygon Z(p) associated with p is the locus of points that are closer to p than to any other point of V.

Z(p) = {m E R2, V q E v\cp}, distance (m,p) < distance

Z(p) is called the “zone of influence’’ of point p . These regions split up the plane into a net that is called the Vorono’i diagram of V, and denoted V or (V). The Vorono’i tesselation associates each object of u E V with its zone of influence. Furthermore, it indicates the neighboring relationships between objects of V.

(m,q)).

The graphs we consider now are: Delaunay triangulation. The edges of this graph are

defined by all pairs of points (p,q) in V2, whose Vorono’i polygons Z(p) and Z(q) are adjacent, i.e., share an edge. Thus the Delaunay triangulation is often called the dual graph of the Voronoi diagram.

Gabriel’s graph. Such as we have used it, Gabriel’s graph is derived from the Voronoi diagram. For every pair of points (A,B) in the plane, we first define D(A,B) as the closed disc admitting segment [A,B] as one of its diameters. The edges of Gabriel’s graph are defined by all pairs of points (p,q) in V2, such that there is no other point of V in D(p,q), i.e.,

V m E V\@,q}, dist(m,p)’ + dist(m,q)’ > dist(p,qI2, dist being here the Euclidean distance in the plane (Fig. 1).

Relative neighborhood graph. Similary, let us define for every pair (A,B) of points in the plane Lun (A,B) as the intersection of open discs having [A,Bl as radius and centered in A and B, respectively. The edges of relative neighborhood graph are then defined by all pairs of points (p,q) in V2, such that there is no other point of V in Lun (p,q), i.e.,

850 RAYMOND ET AL.

ec

FIG. 1. Construction of Gabriel's graph. The positions of cells were identified. The Voronoi diagram was constructed. Two points a and b were connected by an edge of Gabriel's graph if their zones of influence were adjacent and if the segment [abl crossed the common boundary of each zones of influence. Points b and c are not neighbors in Gabriel's graph.

V m E v\b,q}, dist(m,p) 2 dist(p,q)

or dist(m,q) 3 dist(p,q).

Figure 2 compares for the same set of points, the Delaunay triangulation, Gabriel's graph, and the relative neighborhood graph. We have rejected the De- launay triangulation, because points that hardly seem to be neighbors may be linked; such is the case for the two outmost pixels in the graph. On the contrary, the relative neighborhood graph is obviously too poor. Ga- briel's graph seems to express best the type of neighborhood relationships that may have a biological relev- ence. For the construction of the graph, see Vincent (26).

Morphological Transformations on Graphs Definitions and notations. Let G = (V,E) be a

graph having V as its set of vertices and E as its set of edges. G is here supposed to be a nonoriented 1-graph without loops. In fact, none of these conditions are re- quired for processing graphs morphologically. How- ever, the whole set of problems we are concerned with does not involve more general graphs than these. In- deed, the graphs we most often use reflect symmetric

neighborhood relationships between objects or areas in an image. Let u E V be a vertex of G. We denote N,(u) the set of neighbors of u:

The discrete distance induced by E on the set V is denoted d,: two vertices v1 and v2 are at a distance n from one another if the shortest path between v1 and v2 in E is of length n. A binary graph on G is defined as a mapping from V into (0;l) and a decimal graph on G is a mapping from V into R. Figure 3 presents an example of a binary and of a decimal graph in our sense (the black vertices of the binary graph have value 1, whereas the white ones have value 0). Similarly, the grey tones associated with the vertices of the decimal graph stand for numerical values. Roughly speaking, V plays the role of the image's pixels in classical morphology, whereas E defines the neighborhood relationships, i.e., the underlying grid.

In our case, all cell nuclei will be nodes of the graph. If we represent the subset of all cells with the same type as white nodes and all other cells as black nodes, we get a binary graph. If we assign to each node its label, we get a grey-tone graph. We also get a grey-tone graph if we assign to each node the result of a mea-

N,(u) = {u' E V, (u,u') E E}.

GERMINAL CENTER ANALYSIS WITH GRAPHS 85 1

FIG. 2. (a) Delaunay triangulation; (b) Gabriel’s graph; (c) relative neighborhood graph.

FIG. 3. (a) Binary graph and (b) decimal grey-tone graph defined on the same graph structure.

surement made on the corresponding cell, for instance, the cell area. The result of some image transformations on binary graphs will also yield grey-tone graphs.

Basic transformations: erosions, dilations, closings, and openings. Now let, g be a binary or decimal graph on G. The dilation of size 1 of g, denote D’ (9) is the graph defined on G by:

D(g)V-.R v + rnaxk(u’1, u’ c N,(u) U {u } }

Figure 4 shows an example of a dilation of size 1 of a binary graph. The erosion of size 1 of g, denote El (g) has a dual definition:

V u E V, E(g)(u) = rnin k ( u ‘ ) , u’ E N,(u) U {u}}.

To perform a dilation or an erosion of size n E N of g, we simply iterate n dilations or erosions of size one. This yields the graph D” (9) (resp. En (g)).

On the left side of Figure 4, we see two cluster of

cells. The dilation of this binary graph has produced bigger clusters on the right. Our aim with the definition of opening is precisely to give this notion of size a meaning.

Let us consider an individual node. This node is the smallest nonempty object. We consider it as a circle with a radius 0. If we dilate this node by a size 1 dilation, we add all its first neighbors. This produces a circle with a radius 1. Similarly, a size n circle will be the result of an n times repeated size 1 dilation of a given node. Obviously, since we are on a graph, which is not invariant by translation, the shape of a circle will vary with its position.

We are now able to define the size of a binary object with the help of these circles. Obviously an object X will be bigger than an object Y if X contains a size r circle, and Y does not.

A size r opening applied to a binary object X will

852 RAYMOND ET AL.

FIG. 4. Dilation of size 1 of a binary graph.

\d a v b

FIG. 5. (a) Binary graph; (b) erosion size 1 of the binary graph, pixels in light grey are suppressed; (c) dilation of size 1 of graph b (black points), opening of size 1 yields the set made of black or dark grey nodes.

yield an object X, made of all size r circles included in X. Hence X, is a subset of X, which may be constructed in two steps. In a first step, one detects the centers of all

size r circles included in X. This is nothing else than the size r erosion of X: E'(X). Each of these centers is then dilated by a size r dilation, which means replaced


FIG. 6. (a) Let X be the binary set of black nodes; (b) dilation size 1 of the set X adds the pixels a t distance 1 (black nodes), dilation size n adds the pixels at distance n (grey-tone values).

by a size r circle. Hence the size r opening is obtained by a size r erosion followed by a size r dilation: OYX) = D'(E'(X)).

In Figure 5a, a binary set is shown in black nodes. A size 1 erosion suppresses all pixels indicated in light grey (Fig. 5b). The size 1 dilation of the remaining black points reintroduces a certain number of nodes; these nodes are in dark grey in Figure 5c. The size 1 opening yields the set made of black or dark grey nodes indicated in Figure 5c. We see that some narrow parts of the initial set have vanished.

Openings of increasing sizes have the same behaviour as sieves of increasing meshes. The result of a size r opening applied to a set X may be seen as the part of X, which does not pass through the r-meshed sieve. The properties of the opening are:

(1) increasing: if X is included in Y, then O(X) is included in O(Y); (2) idempotent: sieving twice does not change the re- sul t O(O(Xj) = O(X); (3) antiextensive: the result of an opening is a subset of the initial set; (4) granulometric behaviour. If we apply two sieves one after the other, the result is the same as using only one sieve, the one with the biggest mesh:

r > t implies O'(Ot(X)) = Ot(O'(X)) = OYX). The closing is the dual operation of the opening.

Closing a binary graph is obtained in three steps: (1) the binary graph is complemented (white nodes become black and black nodes white), (2) an opening is applied to the new graph, and (3) the resulting graph is again complemented.

Distance transform. The distance transform applied to binary graph will produce a grey-tone graph with the same nodes. Each node belonging to the binary set will have a distance 0. Each other node will have a grey-tone value equal to the distance between this node and the set X (by distance we mean the min-

imal number of arcs one has to travel along before reaching the set XI. The distance transform is easily expressed in terms of dilations. Each node of X is a t a distance 0 from itself. The size 1 dilation of the set X adds to the set X the pixels at distance 1. The size 1 dilation of the preceding set adds the nodes at distance 2, and so on. In Figure 6 , we present on the left a binary set and on the right the distance transform to this set. The successive distances are indicated with brighter and brighter grey-tone values.

The distance transform is actually constructed with an algorithm, which is much faster than successive size 1 dilations.

The distance transform is used in order to study the repartition of a population of cells B with respect to a population of cells A. We start by the construction of the distance transform to the population A on the graph of all cells. It is then easy to identify all nodes that have a given grey-tone for the distance transform and that simultaneously contain a B-type cell. Count- ing all such nodes yields in a very simple way the number of B-type cells that are at a given distance from a cell A. Repeating this operation for all distances allows one to obtain the complete distance distribution of cells B with respect to cells A.

MATERIALS AND METHODS Lymph Node Biopsies

Lymph node biopsies from 6 patients with chronic lymphadenopathy diagnosed as typical reactive FH and 3 patients with SCCFL were obtained from cervi- cal and axillary areas. The etiology of the hyperplasia was unknown, but human immunodeficiency virus (HIV) serology was negative in all patients. Tissue specimens were fixed in neutral formalin, embedded in epoxy resin; 2 km sections were stained with hematox- ylin-eosin. Twelve GCs of reactive FH and 5 neoplastic follicles of SCCFL were analyzed. Sections were ob-

854 RAYMOND ET AL.

FIG. I .

100

80

60

40

20

0

GERMINAL CENTER ANALYSIS WITH GRAPHS

Comparison between FH and SCCFL

Mean number of cells

75,3

21

5

19

Small Cells Large Cells Histiocytes Ly m p ho cy tes

FH: Foilicular Hyperplas ia . SCCFL: Small Cleaved Cel l Follicular Lymphoma

FIG. 8. Distribution of centro-follicular center cells

served on a microscope Orthoplan-Leitz with a x400 magnification.

Acquisition of Images The microscope was connected to a high definition

CCD camera and to a SM-90-Bull computer. The im-

FIG. 7. Construction of Gabriel's graph from the lymph node tissue specimens. (a) Digitized image of a germinal center (GC) in follicular hyperplasia (FH). The complete reconstruction of the images of the GC was obtained using several square images (256 x 256 pixels). The identification of cells was achieved on the video-screen by the observer with a color code (large cells: blue; small cells: green; lymphocytes: yellow, histiocytes: brown; macrophages: red). (b) Voronoi diagram of a GC. The Voronoi' diagram introduces periphery artefacts (*I. (c) Neighborhood Gabriel's graph of a GC in FH. In a graph, the cells were symbolized by a color spot and the neighborhood relationships between them by the edges. (d) Aggregates size T1 of large cells in FH (blue and pink clusters). (e) Gabriel's graph of SCCFL. Neoplastic cells (pink) were infiltrated into the mantle zone lymphocytes (yellow).

855

ages were analysed on the video screen on 256 grey scales (8 bits). The size of GCs displayed on the microscope was larger than the field of the CCD camera (256 X 256 pixels). Therefore, the complete reconstruction of the image of the GCs was obtained by juxtaposing several square 256 x 256-pixel images on the video screen (Fig. 7a). We acquired the image (256 x 256 pixels) of the center of the GCs on the video screen. The other images were placed around the first one on the screen manually by the observer using the microscope stage. The acquisition and the recording of GC images were performed when perfect juxtaposition of images from different parts was obtained. Different sizes were obtained: 512 x 512 pixels (4 square images) in 3 cases, 768 x 768 pixels (9 square images) in 13 cases, and 1,024 x 1,024 pixels (16 square images) in 1 case.

Identification of Cells The identification of cells of the same population was

made on the video screen by the observer with a nu-

856 RAYMOND ET AL.

merical and image color code. This code was given according to the origin of the cells. The cells were marked at the nuclei centers by color spots. It was possible to enlarge the image with a zoom, allowing a better recognition of cells.

Construction of Gabriel’s Graph

Gabriel’s graph of GCs was constructed on the digitized image labelled by the spot of color code.

The image of GCs labelled by the spots was thresh- olded. A binary image was obtained showing only where nuclei centers, manually marked with the numerical code, were located. Nonlabelled spots were zero. The Voronoi diagram was constructed with the software Morphograph from the zones of influence of the spots (Fig. 7b) according to the algorithms pub- lished elsewhere (26). Gabriel’s graph was obtained from the Voronoi’ diagram in linear time Oh) . Two points P and Q in V were connected by an edge of Ga- briel’s graph if their zones of influence were adjacent and if the segment [PQ] crossed the common boundary of each zone of influence. In our study’s Gabriel’s graphs of GCs, cells were represented by vertices and the neighborhood relationships between them were represented by the edges of the graph (Fig. 7c).

Mathematical Morphology Applied to the Graphs: Distance Transforms and Granulometry

The distance transform (DT) between cell A and cell B (vertices A and B) in a graph was defined as the minimal number of edges between A and B. We calculated and compared the histograms of DTs of small and large cells from the subgraph of the mantle zone lymphocytes and from the subgraph of macrophages in FH and in SCCFL.

Morphological transformations (22) as Erosion, Dila- tion, Closing, Opening, as previously discribed (25) were applied to the graphs (26).

Iterative openings and closings were used to detect and analyse clusters of cells. However, the opening is not very robust. If there is only one foreign cell in a cluster of identical cells, the cluster varies with a much smaller opening. For this reason, it makes sense to perform first a small closing in order to close the small holes (containing other types of cells) within the clusters before studying their granulometry . We defined the size TO aggregates as the set of vertices remaining after one closing. The granulometry was obtained by measuring the number of cells remaining after openings of increasing sizes. In this way we defined the size n aggregates in a graph, as connected components of the set of vertices remaining after an opening by a size n object. The size of an aggregate reflected of the den- sity of cells gathered in a localized area. In our study, we compared the aggregates of small and large cells in FH and in SCCFL (Fig. 7d).

(a) Mean number of cells

200 -

150 ‘ 1 1

- 50 0 6 10 16 20

Distance from the GMZL

(b) Mean number of cells

‘ I 800 -

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1 600 1

.-d -200 - 0 5 10 16 20

Distsnce from the GMZL -

FIG. 9. Distance transform of centro-follicular cells from the subgraph of the mantle zone lymphocytes (GMZL). (a) Distribution of cells in FH. (b) Distribution of cells in SCCFL. The number of neoplastic cells infiltrated in the mantle zone was calculated for a DT = 1.

RESULTS Distribution of Follicular Center Cells

The number of cells in a GC was obtained by count- ing the vertices in the GC graphs. In FH results of the average values of each cell subpopulation were 55% (* 8.75) for the small cells, 21% (2 6.57) for the large cells, 19% (5 9.1) for the centro-follicular lymphocytes and 5% (i 2.7) for the histiocytes. In SCCFL distribution was 75.3% (i 3.7) for the small cells, 12.3% (* 3.75) for the large cells, 11% (5 0.29) for centrofollicular lymphocytes and 0.4% (2 0.49) for histiocytes (Fig. 8).

Distance Transform of Centro-Follicular Cells The topographic relationships between cells were

calculated with the DT as defined previously. Centro- follicular cells were located in the graph by the DT from the subgraph of the mantle zone lymphocytes (GMZL) and from the subgraph of macrophages (GHM).

GERMINAL CENTER ANALYSIS WITH GRAPHS

(a) Small cells

857

(b) Large cells


loor

L - -2 ~- 1 - i -20 L 0 5 10 15 20

Distance from the GMZL

(C) Cent ro - follicular Ly m p hocy tes


4 0 1 I

L

0 5 10 15 Distance from the GMZL

(d) Histiocytes - Macrophages

Mean number of cells Mean number of cells

35 I

J

20

-- -

L - 1 I I 1 -~ . I - -5 ' 5 10 15 20 0 5 10 15 20

-5 O I ~

0 Distance lrom the QMZL Distance lrom the GMZL

FIG. 10. Distance transform of each subpopulation of cells from the graph of the mantle zone lymphocytes (GMZL) in follicular hyperplasia.

DT of Centro-Follicular Cells from the GM- ZL. The GMZL was composed of lymphoid cells surrounding the graph of centro-follicular cells. In the graph of all centro-follicular cells, the cells near the lymphocytes of the GMZL were characterized by DT = 1 and cells further from GMZL had DT > 1.

From the mantle zone (DT = 0) to the middle of GCs (DT > lo), the DT of GC cells were on lines (y = ax + b) with -9.5 < a < -6.8 (Fig. 9a). When we compared the disposition of all the centro-follicular cells from GMZL in FH and in SCCFL (Fig. 9b), we observed an average number of cells DT= 1 lower in FH (n = 153 f 40.7) than in SCCFL (n = 544 ? 178.6). The disposition of cells for DT > 1 was approximately the same in FH and SCCFL.

The DT of every subpopulation of cells from GMZL was calculated in FH and SCCFL. The DTs from GMZL of small and large cells were identical to those of populations of centro-follicular cells (Fig. 10a,b). These results suggest a homogeneous repartition from the periphery of GCs to the center. Lymphocytes were

preferentially observed (mean number of cells 80%) a t 2 < DT < 5 (Fig. 10c). Tingible bodies macrophages were preferentially observed at DT = 1 near the GMZL (Fig. 10d).

In SCCFL the curves of small and large cells were identical with all the centrofollicular center cell populations, showing most of the cells near the GMZL with DT= 1 (Fig. lla,b). Lymphocytes were observed a t 5 < DT < 8 (Fig. l l c ) and macrophages at DT = 1 (Fig. l l d ) .

DT of Centro-Follicular Cells from GHM. In FH the DTs of all the centro-follicular cells from the GHM showed a curve with a peak at DT = 2 (Fig. 12a). Each subpopulation of cells, i.e., small, large cells, lymphocytes located in GCs, showed the same repartition. The average DT between histiocytes was 4. There were on average 7 (f 3.2) cells surrounding histiocytes (at DT= 1) composed by 55% (2 8.6) of small cells, 21% (* 7.6) of large cells and 24% (f 13) of lymphocytes.

The results obtained in SCCFL showed differences from FH. The curve did not display any peak (Fig. 12b). The different subpopulations of cells were distributed

858 RAYMOND ET AL.


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(C) Cen tro-follicular lymphocytes


r

(d) Histiocytes - Macrophages


‘r

I ’ - -1 J

0 6 10 15 20 0 5 10 15 20 Distance from the O W L Distance lrom the GMZL

FIG. 11. Distance transform of each subpopulation of cells from the subgraph of the mantle zone lymphocytes (GMZL) in small cleaved cell follicular lymphoma.

along the same curves. It was not possible to calculate the average DT between histiocytes (10 < DT < 301, which suggests a heterogeneous repartition of histiocytes and macrophages. Histiocytes were surrounded by 7 (k 3.36) cells (at DT= 1) composed of 73% (* 12.5) of small cells, 16% (2 5.77) of large cells and 11% (2 10.6) of lymphocytes.

The Cell Aggregates Granulometry was used to detect aggregates of small

and large cells in FH and SCCFL. In FH we studied the number and the size of aggregates of small and large cells (Fig. 13a). For small cells the average numbers of aggregates were 43 (* 24) for size TO, 11 (* 4) for size T1,2 (* 1) for size T2 and 1 (+ 1) for size T3. For large cells the average numbers of aggregates were 64 (It: 11) for size TO, 2 (+ 1) for size T1 and there were no aggregates for sizes T2 and T3. Results were the same for the 3 sizes of GCs.

In SCCFL the numbers and the sizes of aggregates were analysed in the same way (Fig. 13b). For the

small cells the average numbers of aggregates were 69 (+ 8) for size TO, 30 (+ 12) for size T1, 6 ( 2 2) for size T2, and 1 (k 1) for size T3. For the large cells the average numbers of aggregates were 139 (* 15) for size TO and there were no aggregates for sizes T1, T2, and T3.

These results have shown a difference between FH and SCCFL for size T1 aggregates suggesting that clusters of large cells could be detected in FH and not in SCCFL.

DISCUSSION In this study we applied a new method using Gabri-

el’s graph to describe the architectural organisation of cells in hyperplastic and neoplastic follicles of lymph nodes.

In the literature, architectural, and cytologic features of FH and SCCFL have been described and compared by conventional morphology. Differentiation between FH and SCCFL can sometimes be particularly difficult, strong reproducible quantitative criteria be-


(a) Distances from GHM in FH

Mean number of cells 6 0 0 .

4001 ~

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300"\ 100 t

_ _ ~ o r

-100- -2

0 10 20 30 40 Distance from the GHM

(b) Distances from GHM in SCCFL

Mean number of cells 160-

100-

I

40 k

- 2 0 ' 0 10 20 30 40

Distance from the GHM

FIG. 12. Distance transform of centro-follicular cells from the suh- graph of macrophages (GHM). Results in follicular hyperplasia (a) and small cleaved cells follicular lymphoma (b) have shown differences in the neighborhood organization of macrophages. Those results suggest a homogeneous repartition of macrophages in FH hut not in SCCFL.

ing needed (19). The aspect of GCs located in the outer cortex of the lymph node changes in size and shape after antigenic stimulation (1,4-6,8). In secondary follicles, the zonal structure of GCs is characterized by a dark zone where densely packed large non cleaved cells are found with abundant mitotic activity and a light zone with many small cleaved cells facing the marginal sinus. Few immunoblasts, plasma cells, and active macrophages are present.

Comparative architectural analysis of SCCFL re- veals a nodular architecture with predominantely small cleaved cells. Neoplasic follicles are homogeneous varying moderatly in size and shape; mitotic activity and phagocytosis are rare (11,19,23).

Various computer-assisted or controlled image anal- ysers have been used to carry out morphometric studies of lymph nodes tissue in the last 20 years (2,12,24). One approach consisted in describing lympho'id cells. Mor-

phologic and morphometric nuclear analysis of reactive GCs and center cell lymphoma have been performed allowing good criteria as size, shape, perimeter, nuclear irregularity to differentiated reactive and neo- plastics cells (5-7,14,20,21). Immunohistochemistry methods have been used in the quantification of cellular composition in reactive lymph nodes and lymphoma, giving further information on the proliferation and differentiation stages of cells (3,9,28). Some studies have analysed the architectural organisation of cells in lymph nodes using the measurement of zones such as T- and B-dependent areas. Nathwani e t al. (18) have developed morphologic criteria for the differentiation of FH and SCCFL with architectural features using a low magnification. The best criteria were a higher number of follicles per unit of surface in lymphoma, a faded architecture with involvement of the mantle zone of lymphocytes and no polarity of follicles compared with reactive hyperplasia, but the quantification of the architectural organisation of cells in GCs has not been analysed in these studies.

The graph method we have applied was initially used in fractography to simulate the crack in a slice in po- rous materials (27). There are only few studies using graphs and MM in histopathology (15). Recently Kay- ser et al. (10) described an application of the minimal spanning tree to the description of human lung carcinoma.

We studied plastic embedded materials from typical FH and SCCFL. Cells were marked manually by the observer to avoid any incorrect recognition of cell populations. The graphs stemming from the Voronoi diagram are connected and planar. The construction of the Delaunay triangulation (dual graph) derives immedi- ately from the Voronoi diagram but introduces a great number of peripheral artefacts. Gabriel's graph seems to be more interesting to minimise those artefacts cre- ated at the periphery of the graph. To study the topographic repartition of cells in FH and SCCFL, we applied MM to the graphs; distance transform between cells and closing to isolated aggregates in different areas. The linear distribution of cells from the mantle zone to the middle of the GCs confirms that GCs in FH and SCCFL could first be taken as circular in an ap- proximation for the 3 sizes of GCs studied. The differences observed in the repartition of cells for DT = 1 and DT = 2 in SCCFL can be explained by a great number of neoplastic cells that infiltrated the mantle zone (Fig. 7e). The DTs of cells from the graph of histiocytes yield information about the regularity of the repartition of histiocytes and their closest neighborhoods. The use of graphs and distance transforms applied to the graph allowed us to obtain quantitative information that had not been obtained with classical morphometry.

Opening was used to isolate small and large aggregates of cells. The results showed that there were size T1 aggregates of large cells in FH and not in SCCFL. These aggregates of large cells were located in the dark area of GC in reactive FH. In contrast, the number of

860 RAYMOND ET AL.

a: Aggregates of cells in follicular hyperplasia

Nbs of aggregates 100 3

I: 1

TO T I Sizes 0 1 the aggregates

0

sc

0

b: Aggregates of cells in follicular lymphomas Nbs of aggregates

sc

w

€4

~ ~ _ _

LC sc

w

__

LC sc

LC

T2 T3

w

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Sizes of the aggregates

FIG. 13. Granulometry was applied to the graphs to regroup small and large cells into clusters called aggregates. (a) The different sizes of aggregates of small (SC) and large cells (LC) in follicular hyperplasia (FH). (b) The different sizes of aggregates of small (SC) and large cells (LC) in small cleaved cell follicular lymphoma (SCCFL). Aggregates of LC size T1 were detected in FH (arrow) but not in SCCFL.

aggregates of small cells was higher in SCCFL than in FH. This result can be explained by a complete infil- tration of neoplastic cells in SCCFL. Our results were

compatible with the known morphology of GC organisation in FH and in SCCFL. We have shown that dark and light areas of GCs in FH could be recognised on a

GERMINAL CENTER ANALYSIS WITH GRAPHS 86 1

graph with the quantification of the number of aggregates and the number of cells.

The comparison of GCs of FH and neoplastic nodules of SCCFL with graphs and MM shows differences even if only a small number of slides is studied in this work. The study of a greater number of cases would be nec- essary to confirm that this new method could be helpful in some cases where discrimination between hyperplasia and lymphoma is difficult. Thus the neighborhood relationships between cells in different tissue could be investigated with this new method in a different area of pathology.

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germinal center analysis with the tools of mathematical morphology on graphs

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