Mixture Model Analysis of DNA Microarray Images

Transcription

1 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? Mixture Mode Anaysis of DNA Microarray Images K. Bekas, N. P. aatsanos, A. Likas and I. E. Lagaris Abstract In this paper we propose a new methodoogy for anaysis of microarray images. First a new gridding agorithm is proposed for determining the individua spots and their borders. Then, a aussian Mixture Mode (MM) approach is presented for the anaysis of the individua spot images. The main advantages of the proposed methodoogy are modeing fexibiity and adaptabiity to the data, which are we known strengths of MM. The maximum ikeihood () and maximum a posteriori () approaches are used to estimate the MM parameters via the Expectation Maximization (EM) agorithm. The proposed approach has the abiity to detect and compensate for artifacts that might occur in microarray images. This is accompished by a mode-based criterion that seects the number of the mixture components. We present numerica experiments with artificia and rea data where we compare the proposed approach with previous ones and existing software toos for microarray image anaysis and demonstrate its advantages. Keywords: DNA microarray image anaysis, microarray gridding, aussian mixture modes, maximum ikeihood, maximum a posteriori, Markov random fieds, Expectation- Maximization agorithm, cross-vaidated ikeihood I. INTODUCTION DNA microarrays [1] are used to measure the expression eves of thousands of genes simutaneousy over different time points and different experiments. In microarray experiments, the two mna sampes to be compared are reverse transcripted into cdna and then hybridized simutaneousy to a gass side. The end product of a comparative hybridization experiment is a scanned array image, where the measured intensities from the two fuorescent reporters have been coored red () and green () and overaid. This array image is structured with intensity spots ocated on a grid and must be scanned to determine how much each probe is bound to the spots when stimuated by a aser. Yeow spots have roughy equa amounts of bound cdna from each sampe and so have equa intensity in the and channes (red + green = yeow). ene expression data derived from arrays measure spots quantitativey and can be used further for severa anayses [2], [3]. It has been shown [1] that background correction is an important task in the anaysis of microarray images. This is necessary in order to remove the contribution in intensity which is not due to the hybridization of the cdna sampes to the spotted DNA. The and intensities of a perfect microarray image depend ony on the dye of interest. However, due Manuscript received?? ; revised??? The authors are with the Department of Computer Science, University of Ioannina, Ioannina, reece (e-mai: {kbekas,gaatsanos,ary,agaris}@cs.uoi.gr) to system imperfections and the microarray image generation process, the resuting images, in addition to background fuorescence, contain aso other types of undesired signas which are termed in the rest of this paper as artifacts. The correction of such artifacts is crucia to making accurate expression measurements, because unike background fuorescence their spatia ocation is unknown and can ead to errors propagated to a subsequent stages of the anaysis [4]. Processing microarrays images requires two tasks. First, the individua spots and their borders are determined. This process is aso known as gridding. Second, each spot is anayzed to determine the corresponding gene expression eve. A number of software toos have been introduced that are avaiabe either commerciay or for research ony purposes for the anaysis of the microarray images [1], [5], [6], [7]. These toos use simpe gridding methods, which are based either on a grid with uniform ces, or on manua specifications of the spot borders. For spot anaysis some existing toos assume circuar spots for exampe, the ScanAyze [6] and the enepix [7]. Others use simpistic oca threshoding based techniques, for exampe the Spotfinder [5]. Histogram-based custering methods have been aso proposed for spot segmentation [8], [9], [10]. However, these methods use the we known K-means and the K-medoids agorithms that do not adapt we to irreguary based custers and do not utiize a the avaiabe prior knowedge about the data. Furthermore, a previous proposed methods correct ony for background fuorescence and ignore the presence of artifacts. The main contributions of this work are two; first, a new automatic gridding scheme and second, the appication of aussian mixture modes (MM) for anayzing microarray spot images [4]. This aows to bring on bear to this probem a the known advantages and powerfu features of the MM methodoogy, such as adaptabiity to the data, modeing fexibiity and robustness, that make it attractive for a wide range of appications [11]. The proposed methodoogy consists of three main steps. First, the new scheme for determing the individua spot borders in a microarray image is presented. This method does not require any human intervention and is very simpe and fast. It is hierarchica in nature since it first uses the goba and then the oca properties of the microarray image, thus it is aso very robust. Second, after determing the spot boundaries, the probabiity density of each spot pixes is modeed using a MM with K components. Two scenarios are possibe. First, K = 2 in which case two components are used corresponding to pixes abeed as background and foreground. Second, K = 3

2 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? when in addition to background and foreground we have pixes which are abeed as artifacts. The identification of the appropriate vaue of K is accompished using the crossvaidated ikeihood criterion [12]. This can be considered as artifact detection and correction mechanism, since when K = 3 an artifact is identified which is ignored in the subsequent anaysis of this spot. Two approaches are proposed for estimating the MM parameters. The first one is based on the Expectation-Maximization (EM) agorithm [11] for maximum ikeihood () estimation of the parameters, whie the second on a maximum a posteriori () formuation. The atter takes aso into account prior knowedge about the spatia assignment of the pixe abes using a Markov andom Fied (MF) mode [13]. Finay, based on the custering resuts, the means of the background and foreground aussian components are used to cacuate the normaized og-ratio for the fuorescence intensities (og 2 /). This task constitutes the reduction step of our approach and characterizes quaitativey each spot by finding its corresponding gene expression vaue. The rest of this paper is organized as foows: In section 2 we present the proposed technique for automatic gridding. Section 3 describes the two MM approaches for spot image segmentation and the mode-based criterion for estimating the number of mixture components. In section 4 we present numerica experiments that test the proposed gridding and custering methodoogies and compare them to existing software packages for microarray image anaysis, as we as to recenty pubished methods. For this purpose we used both artificia data, where the ground truth is known, together with rea data. Finay, we present our concusions in section 5. II. AUTOMATIC MICOAAY IDDIN The process of determining the spot boundaries is frequenty refered to as gridding. A variety of microarray gridding methods have been previousy suggested in the iterature. They determine individua spot boundaries either with user-defined anchor points [6] and semi-automated geometric techniques [10], or with compex methods that are computationay expensive [14]. Since typica microarray images contain hundreds or thousands of spots, a practica gridding method must be fuy automatic, fast and simpe. The proposed gridding method uses a scheme that combines goba and oca segmentation mechanisms for defining the boundaries of each microarray spot. It initiay creates goba boundaries, which are horizonta and vertica straight ines spanning the entire image. To define the goba boundaries we add the sums of the and intensities aong the rows and coumns of the microarray image. The resuting signas have mutipe peaks each corresponding to the coordinates of a spot center. We use the mid point of two successive peaks of the row and coumn sums to define the goba horizonta and vertica boundaries, respectivey. Fig. 1 (a) iustrates this process for a 5 5 grid. In the next step, the goba boundaries are refined. The horizonta boundary between spots S(i, ) and S(i + 1, ) ow Sums Vertica Sums (a) Fig. 1. (a). These signas are obtained by summing up the rows and coumns of both and channes for a 5 5 grid structure. Mid points of successive peaks define the horizonta vertica goba borders, respectivey. (b). The goba borders (dotted ines) are refined (soid ines) based on the oca sums. The signas on the eft and above the microarray image are the oca row and coumn sums, respectivey. is refined by ocating the minimum of the sum of the rows (within the goba boundary) of the and intensities of these spots. In the same spirit, the vertica boundary between spots S(i, ) and S(i, + 1) is refined by ocating the minimum of the coumns (within the goba boundary) sums of the and intensities of these spots. This procedure is repeated in a row-by-row or coumn-by-coumn fashion, scanning the entire microarray image. Fig. 1 (b) iustrates an exampe of the goba border refinement process. It must be aso noted that in many cases the coor channes are not aigned with each other. In such cases one can use image aignment agorithms prior to the gridding task, see for exampe [15], [16], [17]. III. MIXTUE MODELS FO SPOT ANALYSIS Spot anaysis refers to the task of abeing each pixe of a spot as background (B), foreground (F), and artifact (A). This can be viewed as a custering probem which is tacked using MM. Let x i = [x i, xi ]T (i = 1,..., N) denote the ith pixe vaue in a spot area, where the and correspond to the red and green intensities, respectivey. In other words, the segmentation is appied to the coor image and not to each coor seperatey. MMs [11] represent density functions as a convex combination of K aussian component densities φ(x θ ) = N(x µ, Σ ), where µ is the mean and Σ the covariance matrix of the th aussian, according to the formua f(x i Ψ K ) = π φ(x i θ ). (1) =1 The parameters 0 π 1 represent the mixing weights satisfying that K =1 π = 1, whie Ψ K is the vector of a unknown parameters of the mode, i.e. Ψ K = [π 1,..., π K, θ 1,..., θ K ], with θ = [µ, Σ ]. Having found the parameters of the MM, the posterior probabiities that the ith pixe is assigned to the component is given by P( i) = π φ(x i µ, Σ ). (2) π φ(x i µ, Σ ) =1 (b)

3 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? Therefore, the ith pixe is assigned to the abe with the argest posterior probabiity (P( i) > P( i) ). A. Maximum Likeihood () Estimation of MM Parameters A common approach for estimating the mode parameters of the MM (Eq. 1) is based on maximization of the ikeihood () L(X Ψ K ) = og f(x i Ψ K ) = og{ π φ(x i θ )}. =1 The EM agorithm is a popuar method for estimation since it is simpe to impement and guarantees convergence to a oca maximum of the ikeihood function [11]. Starting from an initia guess of the mode parameters Ψ K, at each iteration (t) the EM agorithm proceeds in two steps. The E-step, where the posterior probabiities are computed = π(t) φ(x i µ (t), Σ(t) ) φ(x i µ (t), Σ (t) =1 π (t) ) (3), (4) and the M-step, where the mode parameters are updated π (t+1) Σ (t+1) = = 1 N, µ (t+1) = (x i µ (t+1) z i(t) x i )(x i µ (t+1) ) T, (5). (6) In image segmentation the spatia adacency of pixes with the same abe is an important prior information that coud be aso taken into account [18], [19]. Since the approach does not provide this capabiity, an aternative method for maximum a posteriori () estimation of MM parameters wi be described next. However, before we address this probem, we wi eaborate on the probem of seecting the number of the mixture components K, and see how it fits in the proposed microarray image anaysis methodoogy. B. Cross-vaidated Likeihood for Artifact Identification The appication of the EM agorithm to MM requires knowedge of the number of the mixture components K used in the mode. Since previous approaches for microarray spot anaysis assume 2 abes, background (B) and foreground (F), it is reasonabe to consider MMs with K = 2. However, this assumption cannot hande the existence of artifacts which must aso be taken into account, see spots in Fig. 7. In this case an additiona custer appears in the data, therefore they are better modeed by a MM with K = 3. This effect can be visuaized by comparing the scatter pots in the Fig. 6 with those in Fig. 8. Thus, the artifact detection probem corresponds to a mode order seection probem between a 2-component or a 3-component MM. Cross-vaidated ikeihood [12] provides an efficient mode order seection framework for MMs. Foowing this scheme, a K-component mode is evauated by spitting the data in u disoint partitions (fods) X s, s = 1,..., u (of approximatey equa size). For each fod we estimate the Ψ s K parameters of a MM with K components using the dataset X {X s }. Then, we cacuate the ikeihood of this mode L(X s Ψ s K ) using X s as a test set. Next L(X s Ψ s K ) is averaged over the u fods in order to obtain the cross-vaidated evauation for the K-component mode CV K = 1 u L(X s Ψ s u K). (7) s=1 The CV K vaue is computed for the two candidate vaues K = {2, 3} and we seect the mode order with the argest CV K. It must be noted that in our experiments we have seected u = 10 for the number of fods. When K = 3 (existence of artifacts) the criterion used to determine which one of the three is the artifact custer is the aggregate variance in a dimensions. In other words, the custer with the argest Tr(Σ ) is considered as artifact. C. Maximum A Posteriori () Estimation of MM Parameters According to this approach [13], the probabiities π i = P( position i) of the pixe ocated at the ith position is assigned to the th abe are considered as additiona mode parameters that satisfy the constraints: 0 π i 1 and K =1 πi = 1. By denoting as Π = {π1,..., π N } the set of probabiity vectors and Θ = {θ 1,..., θ K } the set of aussian component parameters, the density function is given by f(x i Π, Θ) = πφ(x i i θ ). (8) =1 Spatia adacency of pixe abes is taken into account by using a suitabe prior density function for the parameter set Π. This is given by the Markov andom Fied (MF) mode [18], [13], [19] p(π) = 1 N Z exp( U(Π)), and U(Π) = β V Ni (Π), (9) where Z is a normaizing constant, and β a reguarization parameter. The function V Ni (Π) is the cique potentia function of the pixe abe vectors {π m } within the neighborhood N i (horizontay, verticay, and diagonay adacent pixes) to the ith pixe and is computed as foows V Ni (Π) = g(u i,m ), u i,m = π i π m 2 = (π i π m ) 2. m N i =1 (10) The function g(u) must be nonnegative and monotonicay increasing [18] and we used g(u) = (1 + u 1 ) 1.

4 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? iven the above prior density (Eq. 9), a posteriori ogdensity function can be formed as foows p(π, Θ X) = og f(x i Π, Θ) + og p(π), (11) and maximized for the estimation of the mode parameters Π, Θ. The EM agorithm can aso be used for this case [13]. The E-step is given by (t) = πi =1 π i(t) φ(x i µ (t), Σ(t) ) φ(x i µ (t), Σ (t) ), (12) whie the M-step requires the maximization of the foowing og-ikeihood [13] Q (Π, Θ Π (t) Θ (t) ) = og(φ(x i θ ))} β =1 z{og(π i ) i + m N i g(u i,m ). (13) This gives update equations for the parameters of the component densities, µ and Σ simiar to those of Eq. (6) of the -approach of the MM. However, the maximization of the function Q with respect to the abe parameters {π i } does not ead to cosed form update equations, since we must take into account the constraints: 0 π i 1 and K =1 πi = 1. Due to this difficuty, a eneraized EM scheme was adopted in [13] based on an iterative radient Proection method. For this approach, the gradient of the function is first proected onto the hyperpane of the constraints, and then a ine search is performed aong the direction of the proected gradient to find the parameters {π i} that maximizes the Q function. Here we use an improved M-step in order to maximize Q with respect to π i by formuating the probem as a constrained convex quadratic programming (QP) probem. We found that this is advantageous, since it provides a better and faster update rue for estimating abe parameters {π i} that meets a the avaiabe constraints [20]. A more detaied description of the M-step for this method is given in Appendix A. IV. EXPEIMENTAL ESULTS A variety of experiments have been performed to evauate the proposed methodoogy for the anaysis of DNA microarray images. The test images 1 used were artificiay created or obtained from pubicy avaiabe microarray databases described in [2] and [3]. 1 Coored version of the images can be downoaded from TABLE I PEFOMANCE OF THEE IDDIN METHODS USIN TEN (10) SPOT AAYS. Proposed Spotfinder ScanAyze Perfect (%) Margina (%) Incorrect (%) A. ridding experiments At first, we tested the proposed gridding technique for partitioning grid structures into distinct spot areas. In order to obectivey evauate and compare our method the foowing experimenta study was contacted: We appied our gridding method, and two other widey used microarray image anaysis toos, the Spotfinder [5] and the ScanAyze [6], to ten (10) spot arrays, (arbitrariy) seected from ten (10) different rea microarray images. Thus, in tota, neary 3500 spots were used in this experiment. Each method was evauated by visuay inspecting the gridding resuts and assigning each spot to one of three categories: perfecty, marginay and incorrecty gridded. A spot was perfecty, marginay, or incorrecty gridded if the entire, at east 80%, or ess than 80% of the spot area was contained in the assigned grid. The resuts of this study are shown in Tabe I. These resuts ceary indicate that our method determines the spot areas more accuratey than the two other methods. It must be aso noted that the Spotfinder and ScanAyze methods are based on manua gridding. More specificay, the size of the spot array is first defined. Then a rectange is paced manuay on the image. Based on the provided dimensions the rectange is divided into equa rectanguar or circuar ces each corresponding to the region of a spot. Thus the outcome of the gridding process for these methods is user dependent, whie our method is fuy automated. In these experiments, we tried to the best of our abiity to optimize the resuts obtained by the Spotfinder and ScanAyze toos. In Fig. 2 we provide the gridding resuts with one of the ten spot arrays using our approach as we as the two other image anaysis toos, the ScanAyze and Spotfinder. We aso provide more detaied gridding resuts for individua spots in the first coumn of Figures 5 and 7. B. Spot anaysis experiments After identifying the spot regions, we used the proposed MM-based approach to anayze each spot region. More specificay, the procedure we foowed consists of the foowing four stages: 1) Seect the number of components K of the MM mode using the cross-vaidated ikeihood method. In other words, test for the presence (K = 3) or absence (K = 2) of artifacts in a spot. 2) Estimate the parameters of the K-component MM mode using the or technique and abe each spot pixe with one of the K abes. 3) If K = 3, the artifact component (A) of the MM is identified by using the maximum variance criterion.

5 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? (a) (b) (c) Fig. 2. Comparative gridding resuts of our method (a) with two widey used microarray image anaysis toos: (b) the Spotfinder and (c) the ScanAyze. Then, the remaining two custers are abeed as F and B using the criterion µ F > µ B. 4) Cacuate the expression vaue of the corresponding gene according to the normaizing ogarithmic ratio: r = og 2 ( µf µb µ F ). µb For comparison purposes we have aso impemented two other methods proposed in [8], [9] for spot custering, namey the K-means agorithm and the partitioning around medoids () method. These two methods do not provide mode seection capabiities, and thus ony two custers (K = 2) were considered, B and F. At this point it shoud be aso noted that fitering, such as ow-pass or median, coud be used for noise remova in a separate step prior to segmentation [9]. In our methodoogy, the proposed approach provides a coherent framework for segmentation in which noise fitering is impicity integrated. Furthermore, it uses a MM to mode the data and thus, unike fitering, it aso adapts to their statistics. 1) With artificia spot images: In order to obectivey compare the proposed MM based methodoogy with previous ones we conducted Monte-Caro simuations using artificiay created spots for which the ground truth is known. The artificia spots were constructed with known mean intensities for the red () and green () channes both for the background (M B ) and the foreground (M F ). Then, the images were corrupted with additive white aussian noise at ten different eves. For statistica significance, the experiment at each noise eve was repeated ten times with different noise reaizations. Two criteria were used to evauate the methods tested: a) the cassification (segmentation) error defined as the percentage of mis-cassified pixes after custering, and b) the mean squared error (MSE) of the ratio ˆr, as estimated by each method over the ten repetitions of each experiment, with respect to the true ratio r true = (M F MB )/(MF MB ), i.e. MSE = t=1 (ˆr t r true ) 2. Cassification error (%) SN (DB) (a) Mean squared error of ratio Fig. 3. (a) cassification error and (b) mean squared error of ratio versus SN using artificia spot images. SN (DB) (b) SN=8DB SN=6DB SN=4DB r = r = r = r = r= r= r= r= r = r = Fig. 4. Segmentation maps and fuorescent ratios at different SNs using three artificia spot images The MSE from the true ratio was used as a comparison metric since, as mentioned previousy, this ratio is the feature used for further anaysis of microarray data. In Fig. 3 (a), (b) we show the resuting cassification error and MSE curves as functions of the noise eve to iustrate the performance of the four methods. In both curves, the x-axis corresponds to the signa-to-noise ratio (SN) cacuated in decibe units, whie the y-axis in Fig. 3 (b) is in ogarithmic scae. These resuts, demonstrate that the MM-based r = r = 0.394

6 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? method outperforms a other methods. Furthermore, at a SN eves, both the and the MM-based approaches provide both better segmentation accuracy and MSE vaues compared to the other methods, with these differences being quite significant at ow SN eves. In Fig. 4 three exampes are dispayed corresponding to three different SN eves showing the segmentation and the ratio vaue for each one of the compared methods. It must be noted that in the above experiments a custering methods were identicay initiaized. Furthermore, parameter β = 1 was used for a cases. 2) With rea spot images: We aso tested the proposed spot anaysis methodoogy with rea data. Figures 5 and 7 iustrate the resuts obtained for severa rea spot exampes. In each case we present the image segmentation resuts after abeing the pixes using each of the compared approaches. The spot segmentation map is constructed by setting the intensity vaue of each pixe equa to the mean vaue of the custer that is assigned to. In the case of the proposed approach, three different segmentation maps are presented that correspond to three vaues (0.01, 0.1, 1.0) for the reguarization parameter β of the ibbs prior (Eq. 9). In tota, for each spot we provide six segmentation maps aong with the corresponding fuorescent ratios. More specificay, Fig. 5 represents comparative resuts from five spot exampes where no artifacts were detected according to the cross-vaidated ikeihood criterion, i.e. K = 2. In cases where the shape of spots is not reguar and their contour is not round (mosty due to retrieva of the microarrayer s spotting pin), both MM-based methods generate more reguar foreground regions in comparison with the K-means and custering approaches. To better comprehend the behaviour of the different custering methods, we present in Fig. 6 four scatter pots of the and pixe intensities for the spot S 2 after abeing using MM with the (-MM), the (-MM), the K-means and the methods, respectivey. The main disadvantage of the K-means and methods is that they are restricted to use as error metric the L 2 distance from the mean or median of the custer. Thus, they generate custers which are separabe by simpe borders as shown in Figures 6, (c) and (d). In contrast, MM-based methods generate eipsoida custers with compex boundaries as shown in Figures 6, (a) and (b). As a resut, the K-means and methods in this exampe tend to overestimate the background custers and provide spots with background whoes, whie the MM-based methods provide more uniform spots. Fig. 7 iustrates comparative resuts with another four spot exampes that correspond to cases where an artifact was detected, i.e. K = 3. After abeing, the artifact pixes are excuded from the cacuation of the fuorescent ratios. In the absence of an artifact correction methodoogy, the K- means and the methods erroneousy cassify these pixes as foreground since the contribution of the artifact pixes is significant. The differences in the fuorescent ratios r, among these methods is noticeabe. For exampe, in the case of spots S 3 and S 5 of Fig. 7, the K-means and methods produce a ratio cose to zero (r = 0), since they consider as foreground -MM (a) K-means (c) -MM (b) Fig. 6. Pot of a pixe vaues of spot S 2 of Fig. 5 after abeing them with -MM (a), -MM (b), K-means (c) and methods (d), respectivey. The eipsoida custers resuting from the MM approaches and the inear boundary between the two custers in the K-means case are aso shown. -MM (a) K-means (c) (d) -MM (b) Fig. 8. Pot of pixe vaues in spot S 3 of Fig. 7 after abeing with - MM (a), -MM (b), K-means (c) and methods (d), respectivey. The eipsoida custers resuting from the MM approaches and the inear boundary between the two custers in the K-means case are aso shown. the (yeow) artifact pixes. On the other hand, the proposed -MM and -MM approaches, detect the presence of the artifact and generate more reaistic foreground regions. Thus, the produced fuorescent ratios of about r = 0.7 and r = 0.45 seem to be more reaistic for the spots S 2 and S 3, respectivey. We aso present in Fig. 8 four pots of the and pixe intensity vaues for these two spot areas after abeing pixes with the four approaches being compared. Again, the enhanced data fitting capabiities of the MMbased approaches are obvious. Another point to make in our experimenta study concerns the comparison between the -MM and -MM estimators. The resuts in Figures 5, 7 show that both approaches yied simiar resuts in terms of the fuorescent ratios. However, they do not produce the same segmentation maps. For ow vaues of the reguarization parameter β (β 0.01) both (d)

7 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? Origina -MM -MM K-means Existing image β = 0.01 β = 0.1 β = 1.0 toos enpix: S 1 r = r = r = r = r = r = Spotfinder: enpix: S 2 r = r = r = r = r = r = Spotfinder: enpix: S 3 r = r = r = r = r = r = Spotfinder: enpix: S 4 r = r = r = r = r = r = Spotfinder: Fig. 5. ratio. enpix: S 5 r = r = r = r = r = r = Spotfinder: Comparative resuts for 5 rea microarray spots without artifacts. For each method we give the segmentation map and the estimated fuorescence Origina -MM -MM K-means Existing image β = 0.01 β = 0.1 β = 1.0 toos enpix: S 1 r = r = r = r = r = r = Spotfinder: enpix: S 2 r = r = r = r = r = r = Spotfinder: enpix: S 3 r = r = r = r = r = r = Spotfinder: Fig. 7. enpix: S 4 r = r = r = r = r = r = Spotfinder: Comparative resuts for 4 rea microarray spots with artifacts. For each method we give the segmentation map and the estimated fuorescence ratio.

8 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? Origina enepix Spotfinder Origina enepix Spotfinder image image Origina - - K-means image MM MM S 1 S 3 (Fig. 5) r = r = (Fig. 5) r = r = r = r = r = r = S 5 S 1 (Fig. 5) r = r = (Fig. 7) r = r = r = r = r = r = S 2 S 4 (Fig. 7) r = r = (Fig. 7) r = r = Fig. 9. Cacuated fuorescent ratios for 6 spot exampes using the enepix and the Spotfinder microarray image toos. r = r = r = r = r = r = r = r = methods generate identica segmentation maps. As the vaue of β grows in -MM, the contribution of the prior term increases and generates smoother foreground and background regions. Thus, it eiminates isoated foreground pixes ocated in background regions. Whie the vaue of the parameter β must be tuned, in our experiments we observed that a β vaue in the range [0.1, 1.0] gives satisfactory resuts. From this point of view, the -MM approach can be viewed as a method for noise reduction in the sense that it eiminates the effects of the microarray manufacturing imperfections. In Fig. 9 we show some comparisons for spot quantification between the proposed method and two existing image anaysis toos, more specificay the enepix [7] and the Spotfinder [5]. Comparisons with the ScanAyze [6] were not incuded since enepix uses the same principe for spot segmentation. From Fig. 9 it is cear that the circe used in enepix is not representative on many occasions, when the spot is irreguary shaped or when artifact isets are present, of the spot area. In other words, the anaysis provided by enepix is based ony on the spatia properties of the spot and does not take into consideration the intensity of the pixes. For exampe, in spot S 5 shown in Figures 5 and 9 the circe used by enepix misses competey the cresent shaped spot which the proposed method captures quite accuratey. This is aso refected in the arge difference of the fuorescent ratios provided by these methods. Aso in spot S 4 in Figures 7 and 9 it is cear that the region seected by enepix segmentation as foreground incudes pixes that our agorithm abes as artifact and this is aso refected in the computed fuorescent ratios. Simiary, the threshoding based agorithm used in Spotfinder in certain instances of irreguar spots and spots with artifacts produces fauty segmentations, see for exampe spots S 1 in Figures 5 and 7, respectivey. In these spots aso the fuorescent ratios provided by Spotfinder and our method are significanty different. Finay, the ast series of experiments uses an interesting famiy of microarray images provided by Agient Technoogies that have a specific imperfections: the spots in these images athough perfecty circuar, contain sometimes artifacts in their perimeter. Agient provides anaysis software that ignores the r = r = r = r = Fig. 10. Five exampes of Agient Technoogies images. The segmentation resut together with the cacuated ratio vaue are provided for each custering method. perimeter of the spot based on what is caed as the Cookie Cutter agorithm [21]. We tested the proposed methodoogy with such images 2 and found that it is abe to detect the presence of artifacts in these spots using the cross-vaidation criterion. Furthermore, it cassifies as artifact a don t ike region which is not taken into account during the ratio cacuation. For comparison purposes, we aso provide the segmentation and the ratio r resuts using the K-means and the agorithms. Since the cross-vaidation method is specific to the MM, ony two custers were used in these methods. In Fig. 10 we show five spot exampes of this type of images. It is interesting to notice the considerabe difference in the r ratios obtained by the proposed methodoogy with respect to the other methods for certain spot cases (e.g. case 5). V. CONCLUSIONS In this paper we have proposed a new fuy automated approach for the anaysis of microarray images. First we describe a new hierarchica gridding procedure based on the vertica and horizonta proections of the coor images. This approach is simpe, automatic, and provides better resuts compared with popuar existing toos. However, the main novety of this work is the proposed MM-based methodoogy for spot image segmentation. Two methods for estimating the MM parameters are presented: the and a. Both approaches are based on the EM agorithm. A cross-vaidated ikeihood criterion is aso used to seect the number of components of the MM. This provides the capabiity to detect and correct artifacts in the spot area. As our experiments demonstrated, the proposed methodoogy produces better and more accurate 2 Test images were downoaded from

9 IEEE TANSACTIONS ON MEDICAL IMAIN, VOL.??, NO.??,?? resuts in terms of segmentation maps and fuorescence ratios as compared with existing software toos and other custering methods proposed in previous works. APPENDIX A: AN M-STEP FO ESTIMATIN THE PAAMETES π i To maximize Q (Eq. 13) with respect π i we set its derivative equa to zero and obtain the foowing quadratic expression [ ] [ ] 4β ġ(u i,m ) (π) i 2 4β ġ(u i,m )π m (π) i z i = 0, m N i m N i (14) where ġ(u) indicates the derivative. Let us denote with a the positive root of the above equation. The probem can be formuated as foows: iven a vector a K with eements a 0 and the hyperpane K =1 y = 1, find the point y on the hyperpane with y 0 that is cosest to a. This defines the foowing constrained convex quadratic programming (QP) probem: 1 min y 2 subect to (y a ) 2 =1 y = 1 and y 0, = 1,..., K. =1 (15) In order to sove this QP probem severa approaches can be empoyed such as active-set methods and penaty-barrier methods. For this purpose, we have impemented an active-set type of method [20] where we expoit the fact that the Hessian is the identity matrix which in turn eads to cosed form expressions for the Lagrange mutipiers. The detaied steps for soving this QP probem are given in the next Agorithm 1. EFEENCES [1] Y. H. Yang, M. J. Buckey, S. Duboit, and T. P. Speed, Comparison of Methods for Image Anaysis on cdna Microarray Data, Journa of Computationa and raphica Statistics, vo. 11, pp , [2] A. A. Aizadeh, M. B. Eisen, and et. a, Distinct types of diffuse arge B-ce ymphoma identified by gene expression profiing, Nature, vo. 403, pp , [3] J. Mata,. Lyne,. Burns, and J. Baher, The trancriptiona program of meiosis and sporuation in fission yeast, Nature enetics, vo. 32, pp , [4] K. Bekas, N. P. aatsanos, and I. eorgiou, An Unsupervised Artifact Correction Approach for the Anaysis of DNA Microarray Images, in Proc. IEEE Internationa Conf. on Image Processing (ICIP), vo. 2, (Barceona), pp , Sep [5] P. Hegde,. Qi, K. Abernathy, and et. a, A Concise uide to cdna Microarray Anaysis, Biotechniques, vo. 29, pp , [6] M. B. Eisen, ScanAyze [7] I. Axon Instruments, enepix Pro Documentation [8] D. Bozinov and J. ahnenfuhrer, Unsupervised Technique for obust Target Seperation and Anaysis of DNA Microarray Spots through Adaptive Pixe Custering, Bioinformatics, vo. 18, no. 5, pp , [9]. Nagaraan, Intensity-Based Segmentation of Microarray Images, IEEE Trans. on Medica Imaging, vo. 22, no. 7, pp , Agorithm 1 : A sequentia convex QP agorithm Input: a K Output: y K 1 : min K y 2 =1 (y a ) 2 s.t. 1 and y 0 Set D = K and v = 1, = 1,..., K 1. Cacuate y = 1,..., K as : if v = 1 then 1 v a =1 y = a + D ese {v = 0} y = 0 end if 2. Check for termination if y 0 = 1,..., K then STOP end if 3. Update v = 1,..., K and D as: if y < 0 then v = 0 and D = D 1 end if 4. o to step 1. K =1 y = [10] A. W.-C. Liew, H. Yang, and M. Yang, obust Adaptive Spot Segmentation of DNA Microarray Images, Pattern ecognition, vo. 36, pp , [11]. M. McLachan and D. Pee, Finite Mixture Modes. New York: John Wiey & Sons, Inc., [12] P. Smyth, Mode Seection for Probabiistic Custering using Cross- Vaidated Likeihood, Statistics and Computing, vo. 10, pp , [13] S. Sanay-opa and T. J. Hebert, Bayesian Pixe Cassification Using Spatiay Variant Finite Mixtures and the eneraized EM Agorithm, IEEE Trans. on Image Processing, vo. 7, no. 7, pp , [14] M. Katzer, F. Kummert, and. Sageter, A Markov andom Fied Mode of Microarray ridding, in Proc. ACM Symposium on Appied Computing (SAC), (Mebourne, Forida), pp , [15] H. S. Baird, The Skew Ange of Printed Documents, in Proc. Conf. of the Society of Photographic Scientists and Engineers, pp , [16] C. Bowman,. Baumgartner, and S. Booth, Automated Anaysis of ene-microarray Images, in Proc. IEEE Canadian Conference on Eectrica and Computer Engineering (CCECE), pp , [17] P. Bacsy, ridine: Automatic rid Aignment in DNA Microarray Scans, IEEE Trans. on Image Processing, vo. 13, no. 1, pp , [18] P. J. reen, Bayesian econstructions from Emission Tomography Data Using a Modified EM Agorithm, IEEE Trans. on Medica Imaging, vo. 9, no. 1, pp , [19] Y. Zhang, M. Brady, and S. Smith, Segmentation of Brain M Images Through a Hidden Markov andom Fied Mode and the Expectation- Maximization Agorithm, IEEE Trans. on Medica Imaging, vo. 20, no. 1, pp , [20] K. Bekas, A. Likas, N. P. aatsanos, and I. E. Lagaris, A Spatiay- Constrained Mixture Mode for Image Segmentation, IEEE Trans. on Neura Networks (to appear), [21] Agient Technoogies, Agient Feature Extraction Software