Nucleus modelling and segmentation in cell
clusters
Jesús Angulo
CMM-Centre de Morphologie Mathématique, Mathématiques et Systèmes, MINES
Paristech; 35, rue Saint Honoré - 77305 Fontainebleau cedex, FRANCE
[email protected]
Summary. This paper deals with individual nucleus modelling and segmentation,
from fluorescence labelled images, of cell populations growing in complex clusters.
The proposed approach is based on models and operators from mathematical morphology. Cells are individually marked by the ultimate opening and then are segmented by the watershed transformation. A cell counting algorithm based on classical results of Boolean model theory is heuristically used to detect errors in segmenting clustered nuclei.
1 Introduction
High content screening (HCS) refers to technological platforms for parallel
cells growing in multi-well plates (or in other supports as cell on chip) and fluorescent labelling of proteins of interest (immuno-fluorescence with antibodies,
GFP-tagged proteins), together with image capture by automated microscopy
and subsequent cell image analysis [5]. HCS is of interest for the discovery of
new cellular biology mechanisms (i.e., using siRNA), new pharmaceuticals
(i.e., mass screening of potential active molecules) or for the development of
new tests for diagnostic/prognostic, for toxicology tests (i.e., evaluation of
different compounds at different concentrations). Cell image segmentation [3]
to define individual cells is the most critical step to achieve a robust high
throughput system which will be able to process thousands of cell images
without needing a manual interaction. Errors in segmentation process may
propagate to the feature extraction and classification.
Many image processing algorithms have been proposed for cell segmentation, however segmentation of cell populations which grow in complex clusters
is still a challenging issue [6]. This paper deals with individual nucleus modelling and segmentation of cell clusters from fluorescence labelled images. The
proposed approach is based on models and operators from mathematical morphology [7], a non-linear image processing methodology which is proven to be
a very powerful tool in biomedical microscopy image analysis. Cell images
2
Jesús Angulo
used in this study represent only the nuclear content (a DNA marker is used
for fluorescence labelling). This is the most interesting case study since once
the nuclei are detected and segmented the other cell markers can be easily
quantified.
2 Morphological model of cell population
A cell population can be modelled as a realization I of Poisson points i ∈ I of
intensity θ in R2 , i.e., points implanted in the space independently one of the
others according to a constant density θ. Let us consider C cell as a compact
random set, centred at the origin, which represents the “individual cell”. For
each point i ∈ I a realization of C cell is generated and implanted at associated
point i, denoted Cicell . The union Cpopul of the Cicell :
Cpopul = ∪i∈I Cicell ,
is by definition a realization of a Boolean set where the different Cicell are
mutually independent. They can touch each other and overlap, and consequently can constitute cell clusters. In random set theory, the complement
c
Cpopul
is named “pores” set of the “grains” set Cpopul . The cell population
Cpopul is observed by an image associated to the microscopic field Z under
study. However, this model only involves binary images, and in practice, the
fluorescence images are scalar functions with values in the set T of grey levels. Consequently the realization of a individual cell is a random function ficell
whose support is the random set Cicell and the observed fluorescence image is
fpopul = ∨i∈I ficell .
As it is shown below, the binary model Cpopul ∈ P(E) is used in a heuristic
way for counting the cells in segmented clusters, which are obtained by the
deterministic segmentation of the scalar model fpopul (x) (x ∈ E, where E ⊂
R2 is the pixel space of the bounded field Z).
3 Individual nucleus segmentation
Fig. 1(a) gives a typical example of cell nuclei population growing in overlapped clusters. Our purpose is to use automated watershed segmentation [1]
to build the contours of individual cells. For a precise segmentation, watershed transformation wshed(g, mrk) needs a scalar function of contour energy
g and a marker for each cell mrk. The function g is calculated using the
morphological gradient, defined as the difference between the dilation and the
erosion [7], i.e., g = δB1 (fpopul ) − εB1 (fpopul ), where the structuring element
B1 is an unitary disk. The other required ingredient is the function providing
the inner markers.
Nucleus modelling and segmentation in cell clusters
(a)
(b)
(c)
(d)
3
Fig. 1. Example of individual nucleus segmentation from a field Z of a fluorescence labelled cell population: (a) Original image fpopul , (b) Ultimate opening using
hexagons Ult-γB (fpopul ), (c) Image of regional maxima Max (Ult-γB (fpopul )), (d)
Watershed segmentation using the maxima as inner markers (the outer marker is
the SKIZ of the inner markers) on the gradient of fpopul .
As a first approximation, we consider that the cell are modelled by balls
and consequently the support of the realization ficell is a circular random
set of radius ri . According to their cell cycle phase, the nuclei have different
radius but their distribution can be bounded in an appropriate interval. The
size distribution of the image structures can be studied using the notion of
granulometry [4]. A granulometry is a one-parameter family of openings Γ =
(γBn )n≥0 according to the structuring element (i.e., shape probe) B of size
n such that γBn follows the absorption law; i.e., ∀n ≥ 0, ∀m ≥ 0, γBn γBm =
γBm γBn = γBmax(n,m) . The opening γBn (f ) = δBn εBm (f ) is an increasing, antiextensive and idempotent operator [7]. Based on the notion of granulometry,
4
Jesús Angulo
the ultimate opening Ult-γB operator has been recently introduced [2]. Let
us consider the numerical residual operator associated to a discrete family of
openings defined as follows
¡
¢
Ult-γB (f )(x) =
sup
γBk (f )(x) − γBk+1 (f )(x) .
nmin ≤k≤nmax
It replaces the initial image f (x) by a union of the most significant cylinders
included in the sub-graph of the initial function. A significant cylinder is the
biggest and highest cylinder covering every point of the image. The application
of the ultimate opening to the image fpopul allows to adjust a maximal cylinder for each cell of the clusters, Fig. 1(b). The computation of the regional
maxima of Ult-γB (fpopul ) provides an appropriate inner marker for individual
cells, mrki (x) = Max (Ult-γB (fpopul )), Fig. 1(c). The outer markers mrko of
the nuclei, which constrain the segmentation, are defined as the skeleton by
influence zones [7, 1] (i.e., the voronoı̈ diagram) computed as the watershed of
the distance function of the complement of inner markers image. Using both
markers, mrk(x) = mrki (x) ∨ mrko (x), the application of the watershed lead
to the final cell contours, Fig. 1(c). Note that from a practical viewpoint, the
balls used to approach the cells are a family of hexagons.
4 Stochastic nucleus counting
As we can observe from the example of Fig. 1, the algorithm described above
segments properly well separated cells and most of cells in the clusters. However, to avoid obtaining clusters of various cells instead of individual cells,
our approach loses some nuclei. We propose now a second alternative, which
involves to relax the watershed segmentation, replacing the SKIZ by the image border as outer marker. The associated segmentation produces clusters of
nuclei and we consider by hypothesis that the cell population is determined
by the union of the detected clusters: Cpopul = ∪j∈J Ccluster,j . A full quantification involves an algorithm for counting cells in each segmented cluster
Ccluster,j .
Let us start by the following classical theorem [4] from the theory of
Boolean Random Closed Sets:
Pr{B ⊂ X c } = e−θA(X
0
⊕B)
which characterises the probability that the compact set B is contained in
the pores X c , where A(X 0 ⊕ B) is the average surface area of the primary
grain X 0 dilated by the set B. In particular, if B is reduced to a single point,
Pr{B ⊂ X c } becomes the porosity q (proportion of pores) and the relation is:
0
q = e−θA(X ) ⇔ θ = −
log q
A(X 0 )
Nucleus modelling and segmentation in cell clusters
5
Fig. 2. Two examples of quantified population of nuclei (in yellow). The number
close to each cluster indicated the counted nuclei by the Boolean formula: NZj .
Even when the structure is not Boolean, the Central Limit Theorem suggests
to use this result a priori.
Considering our problem, a probabilistic algorithm for counting partly
covering nuclei in the binary set of cluster j, Ccluster,j , is given by the following
formula
| Zj |
NZj = {number of nuclei in Zj } = −
log(q)
C cell
c
where q is the porosity of set Ccluster,j (i.e., q = A(Ccluster,j
)), | Zj | is the
area of the field Zj under study and C cell mean area of individual nucleus.
The equation is valid specifically if θ is constant in each cluster j. To better
match the Boolean model, the image field Zj of each cluster is the bounding
box containing the set Ccluster,j . The mean area of a individual nucleus is estimated from some isolated nuclei in the population. In fact, this value can be
learned and fixed from representative segmented cells of several populations.
Fig. 2 provide two examples of populations of segmented clusters counted by
the Boolean formula.
5 Conclusions and perspectives
A full automated segmentation algorithm for clustered nuclei in fluorescence
labelled images has been presented. Any parameter is required since the application of a granulometry is able to adaptively identify each region candidate
to be a nucleus, which is then segmented by watershed algorithm. In fact, the
only a priori datum is the shape used for the size distribution, a circle in our
case. For other cells presenting a more elongated nuclear shape, the ultimate
6
Jesús Angulo
opening can be implemented using families of ellipses of variable orientation
and eccentricity, which will lead to a better nucleus adjustment.
A probabilistic algorithm for counting the number of nuclei in a cluster
has been also presented. From our results, we state that the number of nuclei
obtained by the Boolean model is more robust than a simple ratio of surfaces.
It can be used to verify the appropriateness of the segmentation for each
cluster and eventually, to detect the wrong segmented cluster.
The result of the ultimate opening, see Fig. 1(b), produces a random function which describes each cell by a cylinder such as ficell (x) = ti if x ∈ Cicell ,
otherwise ficell (x) = 0, where ti is the fluorescence intensity of nucleus i.
Indeed, we expect to introduce in forthcoming research a direct cell modelling and counting, without passing by a binary image, using the theory of
Boolean functions [8]. However, this application need a more deep modelling
of scalar nuclei images, including the study of variation of the florescence intensity which seems be dependent on the DNA nucleus status but also on the
effect of cell aggregation. The present algorithms are suitable for static cell
culture images. Spatial modelling which includes the time dimension should
be necessary for analysis of cell culture kinetics using time-lapse images. The
challenging issue is to model how are formed the clusters and how do they
evolved in time.
References
1. Beucher, S., Meyer, F.: The Morphological Approach to Segmentation:
The Watershed Transformation. In: Dougherty, E. (ed) Mathematical
Morphology in Image Processing, 433–481. Marcel Dekker (1992)
2. Beucher, S.: Numerical residues. Image and Vision Computing, 25, 405–
415 (2007)
3. Carpenter, A.E., Jones, T.R., Lamprecht M.R., et al.: CellProfiler: image
analysis software for identifying and quantifying cell phenotypes. Genome
Biology, 7, R100 (2006)
4. Matheron, G.: Random sets and integral geometry. Wiley, New York
(1975)
5. Neumann, B., Held, M., Liebel, U., et al.: High-througthput RNAi screening by time-lapse imaging of live human cells. Nature Methods, 3, 385–390
(2006)
6. Nilsson, B., Heyden, A.: Segmentation of complex cell clusters in microscopic images: application to bone marrow samples. Cytometry A., 66(1),
24–31 (2005)
7. Serra, J.: Image Analysis and Mathematical Morphology. Vol. 1.. Academic Press, London (1992)
8. Serra, J.: Boolean random functions. Journal of Microscopy, 156(1), 41–
63 (1989)