The dynamics of genome regions are associated to the functional or dysfunctional behaviour of the human cell. In order to study these dynamics it is necessary to remove all perturbations coming from movement and deformation of the nucleus, i.e. the container holding the genome. In literature models have been proposed to cope with the transformations corresponding to nuclear dynamics of healthy cells. However for pathological cells, the nucleus deforms in an apparently random way, making the use of such models a non trivial task. In this paper we propose a mapping of the cell nucleus which is based on the matching of the nuclear contours. The proposed method does not put constraints on the possible shapes nor on the possible deformations, making this method suited for the analysis of pathological nuclei

De Vos, Winnok

De Vylder, Jonas

Philips, Wilfried

English

Ghent University Academic Bibliography

2D MAPPING OF CELL NUCLEI BASED ON CONTOURWARPINGJonas De Vylder1,∗ , Winnok De Vos2, Wilfried Philips11Department of Telecommunications and Information Processing, IBBT - Image Processing and Interpretation Group,Ghent University, St-Pietersnieuwstraat 41, B-9000 Ghent, Belgium2Department of Molecular Biotechnology, Bio-imaging and Cytometry Unit,Ghent University, Coupure Links 653, B-9000 Ghent, BelgiumABSTRACTThe dynamics of genome regions are associated to the func-tional or dysfunctional behaviour of the human cell. In orderto study these dynamics it is necessary to remove all per-turbations coming from movement and deformation of thenucleus, i.e. the container holding the genome. In litera-ture models have been proposed to cope with the transfor-mations corresponding to nuclear dynamics of healthy cells.However for pathological cells, the nucleus deforms in anapparently random way, making the use of such models anon trivial task. In this paper we propose a mapping of thecell nucleus which is based on the matching of the nuclearcontours. The proposed method does not put constraints onthe possible shapes nor on the possible deformations, makingthis method suited for the analysis of pathological nuclei.1. INTRODUCTIONThe internal organisation of the human cell nucleus in spaceand time is essential for its function. Within the limited spaceof the nucleus the entire genome as well as many proteins areaccommodated in a non-random manner. Of particular inter-est are telomeres, the ends of chromosomes, which show aspatiotemporal behaviour that is functionally relevant to thecell and organism. Telomeres are arranged in distinct pat-terns and display mobility regimes at different time scales.Diverse biological processes such as telomere maintenance,cell ageing and apoptosis, i.e. cell death, are associated withspecific telomere movements [11]. Likewise, altered telom-ere dynamics and by expansion nuclear protein dynamicsare associated with specific diseases such as laminopathiesand cancer [8, 4]. Therefore, quantitative studies of nucleardynamics may help revealing novel mechanisms of dysfunc-tion and disease.However, analyses of dynamics in time-lapse micro-scopic image data sets are hampered by global cell motionand deformation. This superimposes a motion on the submi-cron dynamics that needs to be removed. Gladilin et al. haveproposed a mapping based on a spherical model [5]. Thismodel can cover for most of the cellular and nuclear displace-ments (translation, rotation and small affine transformations)that occur in normal cells during interphase. However, dur-ing cell division and in certain pathological conditions, suchas laminopathies, the nuclear shape alters dramatically [4]. InFig. 1.a an example of such a pathological nucleus is shown.As can be seen is the spherical model no longer valid. Hencea method is required that allows reliable motion measure-ments of subnuclear features in a deformable volume without∗ Corresponding author: phone: +32 9 264 3416, email:jonas.devylder@telin.ugent.beimposing any shape constraints.In this paper we propose a new mapping technique whichmaps two nuclei between consecutive time points withoutimposing any constraint such as a spherical shape. Insteadwe will extract the contours out of segmented micrographs.These contours will be matched in such a way that we re-trieve a point to point correspondence. Based on this match-ing a mapping of the full nucleus is calculated using polyhar-monic splines. This paper is arranged as follows: in the nextsection we describe a simple segmentation technique, whichcan be used to extract the nucleus out of micrograph. Sec-tion 3 provides a detailed description of the contour match-ing algorithm. The interpolation using polyharmonic splinesis described in section 4. In section 5 the validation resultsare explained and discussed. Section 6 recapitulates and con-cludes.2. SEGMENTATIONSince the proposed method estimates the topological changesbased on the deformation of the nucleus, we first need to seg-ment the nucleus. In this paper we start from fluorescent 3Dmicrographs captured by a confocal microscope. We projectthe 3D micrograph on a 2D image. This is done by takingthe average in the z direction, i.e.I(x,y) =1nn∑z=1M(x,y,z) (1)where I is a 2D image, M is the 3D micrograph and n is thez-dimension of M. An example of such a z-stack projection isshown in Fig. 1.a. Depending on the microscope system it isof course possible to start immediately from 2D micrographssuch as e.g. widefield images.The 2D image is now thresholded using Otsu’s thresh-olding [10]. This method calculates the threshold in order tominimize the variance of the foreground, σ f , and the vari-ance of the background, σb, i.e.t = argmint(p f (t)σ f + pb(t)σb)(2)where p f (t) is the probability that a pixel in the image ishigher than t, while pb(t) calculates the probability the pixelis lower than t. The result of Otsu thresholding can be seenin Fig. 1.b. Due to noise, not all desired pixels are consid-ered to be foreground, e.g. part of the nucleus, whereas somebackground pixels are considered to be foreground. Missingforeground pixels are generally in the vicinity of detectedforeground pixels. By using a morphological closing these(a) (b) (c) (d) (e)Figure 1: An example of the different steps in the segmentation algorithm. (a) shows the projection of the 3D micrographon a 2D image. (b) demonstrates the result of Otsu thresholding. (c) and (d) are respectively the morphological opening andclosing. (d) shows the filtered contour plotted on top of the image.missing pixels are added to the foreground. The falsely de-tected background pixels are discarded using a morphologi-cal opening, as can be seen in Fig 1.c-d. Finally the borderdelineating the segment is filtered with a Gaussian kernel,i.e.we assume the nucleus has a smooth surface. The final resultof the segmentation can be seen in Fig. 1.e.This segmentation algorithm is developed for a specificdataset which consists of isolated cell nuclei which havegood contrast compared to the background. This is off coursedependent on the application and the dyes used for the fluo-rescent micrographs. The method for nuclei mapping pro-posed in this paper can also be used with more complex im-ages, but would then acquire a more suitable segmentationtechnique such as described in [7, 9, 12, 2].3. CONTOURWARPINGIn this section contours delineating the nucleus in subsequentframes will be matched, i.e. points belonging to the contourin a frame will be linked to their counterpoints on the contourin the next frame. In order to do so, we calculate a signatureout of the contour, then based on this signature the matchingis done.3.1 Contour SignaturesConsider the contour c(t) = (x(t),y(t)) with t ∈ N andc(N+ 1) = c(1), where N is the number of samples . Fromthis contour, different signatures can be calculated:• a complex signature:s(t) = x(t)+ iy(t)where i2 =−1• the centroid distance signature:s(t) =√(x(t)− xc)2+(y(t)− yc)2where (xc,yc) is the centroid of c• a direction signature, i.e. direction of the tangent line ata point t:s(t) = cos−1(y(t+1)−y(t−1)‖c(t+1)−c(t−1)‖)0 50 100 150 200 250 300 350 4000100200300coordinates (pixels)  real partimaginary part0 50 100 150 200 250 300 350 4006080100Distance (pixels)0 50 100 150 200 250 300 350 40001234Angle (radians)Figure 2: An example of different signatures. These are thesignatures for the contour shown in Fig. 1.e. From top tobottom the complex, the centroid distance and the directionsignature are shownFor a more detailed overview of signature functions we referto [6, 13]. In Fig. 2 an example is shown of the definedsignatures. This are the signatures for the nucleus shown inFig. 1.e. These signatures are from top to bottom: complex,centroid distance and the direction signature.3.2 Signature MatchingNote that each point on a signature exactly corresponds toone point on the curve. So matching contour points is equiv-alent to matching points on a signature. For the matchingof two signatures, we propose a matching based on DynamicTime Warping or dog-man distance. The DTW distance be-Samples SamplesFigure 3: An example of matching two curves. On the lefta naive matching where the matching is done using the pa-rameterization. On the right the matching with the minimalcumulative difference between the samples. It is this match-ing which will correspond with the warping used for DTW.tween two signals is analogous of a man and a dog each wak-ing with a different path, i.e. the signals. The dog can walkon a different speed by giving it a longer leash. Fig. 3 showson the left two paths, where both paths are scanned at thesame speed, so man and dog are walking on the same speed.On the right of the figure, different speeds result in a a bettermatch, i.e. the cumulative difference between these samplesis less. The dog-man distance is then the shortest possiblecumulative difference between both paths, by changing thescanning speed, i.e.dDTW (s1(.),s2(.)) = minγ ∑t‖s1(t)− s2(γ(t))‖ (3)where γ(.) is a warping function, i.e. any monotonic func-tion mapping [1,N] on [1,N]. This can be calculated usingdynamic programming:D(i, j) = d(i, j)+min(αD(i, j−1),αD(i−1, j),D(i−1, j−1)) (4)with d(i, j) = ‖s1(i)− s2( j)‖ and α a real number greateror equal then one. If α equal to one, all warping functionsare considered equally good. In order to penalize warpingfunctions where to many points of one signal are matched toa single point in the other signal, one can set α > 1.The distance itself is of little importance to us, but thewarping function resulting in the minimal distance is, i.e.γ(.). Using linear interpolation the signals can be consid-ered continuous, based on the warping function the signalswill be resampled in such a way that there is a unique pointto point correspondence. Consider the left case of Fig. 4,where one point, s(t) of the signal is matched with multiplepoints in the other signal. First define s(t − 1) and s(t + 1)the last predecessor and first successor of s(t) with a uniquematch, e.g. s′(t−1) and s′(t+1) respectively. The point s(t)will be replaced by the number of points between s(t−1) ands(t+1): for each point s′(i) between s′(t−1) and s′(t+1) anew sample is added on s(.). These new samples are calcu-lated in such a way thatal(s′(.), t−1, i)al(s′(.), t−1, t+1) =al(s(.), t−1, i)al(s(.), t−1, t+1) (5)where al(k(.), l,m) is the arc length from point l to m overthe curve k(.).S(t)S(t+1)S(t-1)S’(a)S’(t+1)S’(t-1)S’(b)S(t+1)S(t-1)S’(a)S’(t+1)S’(t-1)S’(b)S(b)S(a)Figure 4: An example of resampling the curves based on thewarping function of DTW4. CELL MAPPINGBased on the previous contour matching we get for eachpoint on the contour at frame t a unique point on the con-tour at frame (t + 1). So for each point on the contour weassume to know the exact position in the next frame. For themapping of points not on the contour, P= (x,y), we proposeinterpolation with biharmonic splines:Pt+1 = a+axx+ayy+N∑j=1w j‖Pt −c( j)‖ (6)where a,ax,ay and w j are a set of weighting coefficients.These coefficients are calculated in such a way that:• points on the contour are mapped to the correspondingpoints according to the contour matching• the weighting vector w is orthogonal to x and y, i.e.N∑i=1wixi =N∑i=1wiyi = 0 (7)These weighting coefficients can be found by solving a lin-ear system of equations, which can be done in a fast way asproposed by Beatson et al. [1].5. RESULTSTwo datasets where used for the validation of the proposedmethod. Both datasets contain an isolated fibroblast cell nu-cleus of a patient with a homozygous nonsense mutation inlamin A/C gene. These are cells where the nucleus can havebig deformations [4]. The datasets where captured using con-focal controlled light-exposure microscopy [3], resulting inimages of 288×288 pixels. These cells were transfected witha construct expressing a fluorescent fusion protein targetingthe telomeres (TRF2-mCitrine). Non-bound protein diffusesfreely throughout the nucleus and allows for demarcating theboundaries for contour identification. For both validationsets the telomeres where manually tracked by the author us-ing MTrackJ1. The centroids of these tracked telomeres willserve as ground truth for the validation. The telomeres themself might move locally, but this small local motion, e.g. 1to 2 pixels, is neglectable compared to the big motion intro-duced by deforming nuclei.In Fig. 5 three examples of the mapping are shown. Thered and blue curves are the contours of a nucleus in subse-quent time frames. The red and blue dots are the centroids1http://www.imagescience.org/meijering/software/mtrackj/Figure 5: Three examples of mapping telomeres from a frame to a subsequent time frame. The red and blue curves representthe nucleus contour in the concerning frames. The he red and blue dots are the telomere centroids in the frame correspondingto respectively the red and blue curve. The green dots are the mapping of the red dots based on the warping of the red andblue contour.of the telomeres corresponding to respectively the red andblue contours. The green dots are the location of the reddots mapped with the proposed technique using the centroiddistance signatures. So the green dots are the prediction ofthe location of the blue dots. As can be seen does the map-ping closely approximate the real telomere location in thenext time frame.In Table 1 the results are shown for the full datasets. Thefirst column shows the dataset. In the first dataset 12 telom-eres where tracked during 12 frames. The second datasetconsists of 11 telomeres which were tracked for 11 framesIn the second column the used signature is shown. Distancemeans the non processed data, i.e. the centroid coordinatesare used as they are extracted out of the images, without anymapping. The third column shows the average Euclidean dis-tance between the telomeres in the one frame and where thetelomeres are ought to be according to the mapping of thelocation of the telomere in the previous frame. This distanceor error is expressed in pixels. In the next column the me-dian error is shown, also in pixels. The fifth column showsthe variance of the error. For the sixth column the telomeresof a frame where matched with the telomeres in the previousframe using the Iterative Closest Point (ICP) method. Theamount of false matches for the full sequence are mentionedin this column. The last columns give measures of the im-provement of the proposed techniques compares to using theunprocessed data, i.e. the ”Distance” row. Both the averageas the median improvements are shown. And for both mea-surements this is expressed both absolute, i.e. the distancein pixels that the proposed method works better, as relative,i.e. the ratio of the error using the proposed method over theerror using the non processed data.The proposed method results in a significant improve-ment compared to not mapping the nuclei as can be seen inthe table, although the amount of improvement differs forboth datasets. The centroid distance and direction signatureperform best for both datasets. Since the absolute error issimilar for both signatures, the direction signature is slightlyin favor based on the ICP results.6. CONCLUSIONIn this paper a new mapping algorithm is proposed for themapping of cell nuclei. This mapping does not imposeany shape constraints, which allows it to map cell nucleiof pathological cells. The proposed technique works in twosteps. First the nucleus contours are matched. Three differ-ent matching techniques are defined and tested. Based onthis contour matching the full nucleus is mapped using bi-harmonic interpolation. The proposed method was tested ontwo real datasets, which both show significant improvement.7. ACKNOWLEDGMENTJonas De Vylder is funded by the Institute for the Promotionof Innovation by Science and Technology in Flanders (IWT).REFERENCES[1] R. K. Beatson, M. J. D. Powell, and A. M. Tan. Fastevaluation of polyharmonic splines in three dimensions.IMA Journal of Numerical Analysis, 27:427–450, 2007.[2] X. W. Chen, X. B. Zhou, and S. T. C. Wong. Automatedsegmentation, classification, and tracking of cancer cellnuclei in time-lapse microscopy. IEEE Transactions onBiomedical Engineering, 53(4):762–766, 2006.[3] W. H. De Vos, R. A. Hoebe, G. H. Joss, W. Haffinans,S. Baatout, P. Van Oostveldt, and E. M. M. Manders.Controlled light exposure microscopy reveals dynamictelomere microterritories throughout the cell cycle. Cy-tometry Part A, 75A(5):428–439, 2009.[4] W. H. De Vos, F. Houben, R. A. Hoebe, R. Hennekam,B. van Engelen, E. M. Manders, F. C. Ramaekers, J. L.Broers, and V. O. P. Increased plasticity of the nuclearenvelope and hypermobility of telomeres due to the lossof a-type lamins. Biochimica et Biophysica Acta (BBA)- General Subjects, In Press, Corrected Proof:–, 2010.[5] E. Gladilin, S. Goetze, J. Mateos-Langerak,R. Van Driel, R. Eils, and K. Rohr. Shape normaliza-tion of 3d cell nuclei using elastic spherical mapping.Journal of Microscopy-Oxford, 231(1):105–114, 2008.data set method av. err. med. err. σ2 ICP err. av. improvement med. improvementabsolute relative absolute relativenucleus 1Distance 7.7 7.2 12 15Complex 3.8 3.1 7.4 4 3.8 53% 3.3 49%Centroid dist. 2.5 2.0 2.6 3 5.2 48% 4.9 28%Direction 3.0 2.7 4.0 0 4.7 50% 4.0 35%nucleus 2Distance 8.2 7.7 11.5 30Complex 6.8 6.9 10.4 23 1.5 89% 1.5 86%Centroid dist. 6.3 5 17.3 16 2 87% 2.3 69%Direction 5.2 4.7 7.5 8 3.1 67% 2.8 65%Table 1: Error measurements of the proposed methods[6] V. V. Kindratenko. On using functions to describe theshape. Journal of Mathematical Imaging and Vision,18(3):225–245, 2003.[7] G. Li, T. Liu, J. Nie, L. Guo, J. Chen, J. Zhu, W. Xia,A. Mara, S. Holley, and S. T. C. Wong. Segmentation oftouching cell nuclei using gradient flow tracking. Jour-nal of Microscopy-Oxford, 231(1):47–58, 2008.[8] S. F. Louis, B. J. Vermolen, Y. Garini, I. T. Young,A. Guffei, Z. Lichtensztejn, F. Kuttler, T. C. Y. Chuang,S. Moshir, V. Mougey, A. Y. C. Chuang, P. D. Kerr,T. Fest, P. Boukamp, and S. Mai. c-myc induces chro-mosomal rearrangements through telomere and chro-mosome remodeling in the interphase nucleus. Pro-ceedings of the National Academy of Sciences of theUnited States of America, 102(27):9613–9618, 2005.[9] S. G. Megason and S. E. Fraser. Digitizing life atthe level of the cell: high-performance laser-scanningmicroscopy and image analysis for in toto imag-ing of development. Mechanisms of Development,120(11):1407–1420, 2003.[10] N. Otsu. A threshold selection method from gray-levelhistograms. IEEE transactions on Systems, Man andCybernetics, 9:62–66, 1976.[11] V. Raz, B. J. Vermolen, Y. Garini, J. J. M. Onderwater,M. A. Mommaas-Kienhuis, A. J. Koster, I. T. Young,H. Tanke, and R. W. Dirks. The nuclear lamina pro-motes telomere aggregation and centromere peripherallocalization during senescence of human mesenchymalstem cells. Journal of Cell Science, 121(24):4018–4028, 2008.[12] P. K. Yan, X. B. Zhou, M. Shah, and S. T. C. Wong.Automatic segmentation of high-throughput rnai fluo-rescent cellular images. IEEE Transactions on Informa-tion Technology in Biomedicine, 12(1):109–117, 2008.[13] D. S. Zhang and G. J. Lu. Review of shape represen-tation and description techniques. Pattern Recognition,37(1):1–19, 2004.

Mapping of cell nuclei based on contour warping

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Mapping of cell nuclei based on contour warping

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