In silico microdissection of microarray data from heterogeneous cell populations
© Lähdesmäki et al; licensee BioMed Central Ltd. 2005
Received: 28 October 2004
Accepted: 14 March 2005
Published: 14 March 2005
Very few analytical approaches have been reported to resolve the variability in microarray measurements stemming from sample heterogeneity. For example, tissue samples used in cancer studies are usually contaminated with the surrounding or infiltrating cell types. This heterogeneity in the sample preparation hinders further statistical analysis, significantly so if different samples contain different proportions of these cell types. Thus, sample heterogeneity can result in the identification of differentially expressed genes that may be unrelated to the biological question being studied. Similarly, irrelevant gene combinations can be discovered in the case of gene expression based classification.
We propose a computational framework for removing the effects of sample heterogeneity by "microdissecting" microarray data in silico. The computational method provides estimates of the expression values of the pure (non-heterogeneous) cell samples. The inversion of the sample heterogeneity can be facilitated by providing accurate estimates of the mixing percentages of different cell types in each measurement. For those cases where no such information is available, we develop an optimization-based method for joint estimation of the mixing percentages and the expression values of the pure cell samples. We also consider the problem of selecting the correct number of cell types.
The efficiency of the proposed methods is illustrated by applying them to a carefully controlled cDNA microarray data obtained from heterogeneous samples. The results demonstrate that the methods are capable of reconstructing both the sample and cell type specific expression values from heterogeneous mixtures and that the mixing percentages of different cell types can also be estimated. Furthermore, a general purpose model selection method can be used to select the correct number of cell types.
Recent developments in high-throughput genomic technologies have revolutionized the approaches aimed at understanding biological systems and emphasized the need for computational and systems biology research. Microarray analysis, for instance, can provide massive amounts of information about a biological sample by simultaneously measuring thousands of transcript levels. Application of such methodologies has already yielded important molecular insight into cellular phenotypes under various experimental conditions  and provided new knowledge about the development and treatment of human diseases, such as cancers [2–4]. During the last several years, microarray technology has undergone continued improvement with better quality control in the overall measurement process, ranging from hybridization conditions to image processing techniques . Nevertheless, to fully harness the power of the microarray technology to study biological materials such as cancer tissues, one has to deal with a source of measurement variability that comes from the biological materials themselves, which rarely consist of homogeneous cell populations. For example, except for a few types of immune-privileged tissues such as the brain, most solid tumor tissues contain infiltrating lymphocytes as a result of the immune response. Most tumor tissues also contain endothelial cells as part of the necessary vasculature systems that provide nutrients for the tumor cells. The complexity of this problem is that different tumor tissues contain different proportions of these non-tumor cells. Therefore, if tumor tissues are used without consideration of such a mixing phenomenon, measurement of differential gene expression will certainly be confounded by the heterogeneous cell populations. In some studies , pathologists carefully evaluated the tissues and only selected tissues with more than a certain percentage of tumor cells. This prescreening step, however, results in the exclusion of many tumor tissues for the study and contributes to the small sample size problem in some of the studies. Alternatively, laser capture microdissection (LCM) technology can be used to purify the tumor cells from mixed populations . This approach has been very successful in DNA-based studies because of the relatively high stability of DNA. However, for microarray studies, which require less stable RNA, LCM has seen limited success because it is much more challenging to maintain RNA stability during the microdissection process. Other drawbacks of LCM are that such procedures are time-consuming and yield insufficient quantities of RNA, thus requiring multiple amplification steps that may confound quantitative inferences from gene expression data.
A recent paper by Ghosh  introduced a mixture model based framework for determining differential expression in the presence of mixed cell populations. In this study, we aim at reconstructing the actual expression values of the pure cell types from the heterogeneous mixtures. That is, we develop a computational method for removing the effect of mixing from heterogeneous samples and to microdissect microarray data in silico. Similar analytical approaches have been previously proposed by Lu et al. , Stuart et al.  and Venet et al. . Lu et al. focused on estimating the fraction of cells in different phases of the cell cycle whereas Stuart et al. considered the problem of estimating the cell type specific expression patterns over all samples. Here we focus on estimating both the sample and cell type specific expression values using carefully controlled microarray experiments. The inversion of the 'cell mixing effect' can be made appreciably easier by providing estimates of the mixing percentages of different cell types in each measurement, which can be measured by an experienced pathologist. The entire process does not hinge upon such measurements, however, as the mixing percentages can be estimated within the modeling framework. Venet et al.  introduced some preliminary methods and results for tackling the same problem as we consider here. In particular, they used a similar regression based framework as in  and as we do. We also consider the problem of selecting the correct number of cell types using the cross-validation model selection framework.
The measured mixing percentages. The measured mixing percentages (RKO/normal) in the five heterogeneous samples.
Inversion of sample heterogeneity
The first goal is to invert the mixing effect caused by sample heterogeneity. We apply the linear model developed in Materials and methods Section to the heterogeneous microarray data. The obtained results are presented below.
The results are more easily appreciated when only the most significant PCA component is shown. As discussed above, the variation in the most significant PCA component is due to the mixing effect. The results in Figure 2 (a) are as in Figure 1, but now shown in 1-dimension in order to facilitate the interpretation. Results in Figure 2 (b), in turn, are as in Figure 2 (a) except that the inversion was done using only the samples #2, #3, and #4. This represents a more difficult and realistic case, since fewer mixtures are available.
When comparing Figure 2 (a) with Figure 2 (b), one can conclude that the method performs slightly better when more samples are used to estimate the true expression profiles – a result that was expected. Overall performance, however, is good in both cases. The estimated expression values for the pure colon cancer (RKO) are close to the mixture #1, as it should be since the mixture #1 corresponds to a measurement of the pure colon cancer. Similarly, the estimated expression values of the pure lymphocytes are close to the mixture #5 as well as to all of the reference samples (note that samples used in the reference channel (Cy5) are the same lymphocytes as the ones used in the mixtures). In Figure 1 and 2 (a), the most significant PCA component and the two most significant PCA components explain about 70.0% and 81.9% of the total variation in the data, respectively. For the reduced data, for which the results are shown in Figure 2 (b), a slightly smaller fraction of the variance is explained, namely about 67.3% and 81.2%. The results obviously depend on the optimality criterion for which we used the standard least squares. Less outlier sensitive results can be obtained with robust regression methods, such as the Huber estimator with the iteratively reweighted least squares implementation [12, 13] or median based regression methods [12, 14]. The robust methods provided similar global results, but improved results for some individual genes that contained one or more outliers.
Optimization of mixing percentages
The estimated mixing percentages. The found optimal values of the mixing percentages.
Again, the estimated expression values for the pure colon cancer cells (RKO) are close to those from mixture #1, as it should be, since mixture # 1 corresponds to a measurement of the pure colon cancer cells. Similarly, the estimated values from the lymph node sample are close to those from mixture #5 as well as to all of the reference samples. In Figures 4 and 5 (a), the first PCA component and the first two PCA components explain about 69.2% and 81.7% of the total variation in the data. For the reduced data, for which the results are shown in Figure 5 (b), the fractions of variance explained are about 65.1% and 80.4%. Although the fraction of variance explained is slightly smaller than without the optimization of the mixing parameters, the optimized mixing parameters provide a better fit to the data.
Above we were only interested in estimating the expression values of the pure cell types. Often it is also useful to assess the confidence intervals of the obtained expression estimates. For that purpose, we consider two methods: one based on Gaussian approximation and the other using bootstrap. (For more details, see Materials and methods Section.)
For illustration purposes, Figure 6 shows the 90% estimated confidence intervals for a set of genes by pooling each of them with the 50 closest genes. The horizontal and vertical axes correspond to the fraction of lymph node cells and the normalized expression value, respectively. In other words, the different heterogeneous mixtures are placed on the x-axis according to the corresponding mixing fractions. The vertical lines at x = 0 and x = 1 expand over the maximum of the two confidence intervals. In most of the cases the two confidence intervals are in good agreement. The confidence intervals can be tightened by measuring more heterogeneous mixtures.
The proposed inversion methods for the sample heterogeneity were also tested on standard non-replicated microarray data by treating the replicated measurements for each gene as individual "genes." The obtained results were qualitatively similar with the ones shown above and only slightly more variable. In a similar fashion, we examined the effect of low quality replicates on the heterogeneity inversion. Slightly less variable results were obtained with a method  that detects and removes unreliable replicates prior to the averaging. A drawback of such unreliable spot detection is that, without any missing value estimation method, some of the genes will be excluded from further analysis.
Selection of the number of cell types
It is known that the heterogeneous mixtures used in our experiments consist of only two cell types. However, in general case, heterogeneous mixtures may contain an unknown number of cell types. In those cases, it is useful to assess the validity of the model (i.e., the number of cell types) as well. As introduced in Materials and methods Section, the linear mixing model can be extended to incorporate more than just two cell types. We use a general purpose cross-validation for model selection. In particular, we apply the so-called leave-one-out cross-validation (LOOCV) and test the one, two, and three cell type models. (For more computational details, see Materials and methods Section.)
The estimated mixing percentages for the three cell type model. The found optimal values of the mixing percentages for the three cell type model.
the 3rd cell type
This paper presents an inversion method for the effects of sample heterogeneity. The proposed method is successfully applied to a carefully controlled microarray data consisting of five different heterogeneous mixtures of lymph node and colon cancer samples. The results demonstrate that both the sample and cell type specific expression values can be reconstructed from heterogeneous mixtures. In some situations, such as cancer metastases in the lymph node, lymphocytes constitute a major cell type beside tumors. Hence, with careful sample preparation, the two cell type model can directly be applied to such cases. For unknown heterogeneous mixtures obtained from more complex cancer samples, the analysis may be a bit more difficult. For example, contaminating cells may include several cell types, such as fibroblasts, endothelial cells, macrophages and lymphocytes. As the proposed method can be applied to any cell types and to any number of cell types, the method works in principle in more complex cases as well. Requirement for the number of measurements necessary for reliable inversion, however, increases together with the number of cell types present in the sample.
We have emphasized that proper inversion of the mixing effect results in more accurate expression values of the pure cell types. While this is true, it must be noted that clinically relevant information may also be incorporated into other populations than the pure (cancer) cells. For example, the degree of lymphocyte infiltration may be clinically important and could be used to complement microarray analysis. However, for comparative microarray analysis, it is important to make comparisons between homogeneous samples so as to minimize the confounding influence of different proportions of contaminating cell types.
Application of 'in silico microdissection' to detection of differentially expressed genes
In order to illustrate the above 'in silico microdissection' in practice, consider the following (hypothetical) experimental setting. Given the three middle mixture measurements (#2, #3, and #4), a goal is to identify a set of genes which are differentially expressed between the colon cancer and the lymph node samples. In a simple approach, often used in practice, the most heterogeneous sample would be discarded since it is measured to contain about 56% (resp., 44%) of colon cancer cells (resp., lymphocytes), thus giving no direct discriminative information about the underlying two samples. For illustration purposes, let us measure the expression difference of a given gene between these two samples using the fold-change, i.e., the expression value of the i th gene in the colon cancer sample, , is regarded as being differentially over-expressed (resp., under-expressed) if the ratio of to the expression value of the same gene in the lymph node sample, , is at least 2 (resp., smaller than 1/2). Of course, in practice, more sophisticated methods for detecting differential expression, including correction for multiple testing, should be used. However, for illustrative purposes, this example will suffice. Since only the heterogeneous samples are available, without any inversion of the mixing effect, one must compare the mixture measurement and . Figure 7 shows some example genes whose expression difference (i.e., the fold-change) between the two heterogeneous samples is within the given threshold (above 1/2 and below 2), but after the 'in silico microdissection,' the expression difference exceeds even a more stringent criterion (approximately 4-fold-change). The measured mixing percentages are used in the estimation (see Table 3). It is clear from this example that the proposed method is able to correctly detect differential expression even from heterogeneous samples, especially when the direct use of such samples may fail to find differential expression. Indeed, the conclusions we can draw based on the red stars are consistent with those that are based on the true homogeneous samples represented by blue squares in Figure 7.
As is evident from the example above, heterogeneity in the biological sample preparation can hinder further statistical analysis steps. Not only can the heterogeneity blur the identification of differentially expressed genes, it can also cause contrary effects. Presence of a considerable percentage of additional cell types can result in the identification of differentially expressed genes that may be unrelated to the biological question being studied. Similarly, irrelevant gene combinations can be discovered in the case of gene expression based classification. For an illustration, see  where the authors analyzed a colon cancer data set contaminated with muscle cells.
Although the microarray technology has been improved during the recent years, the measurements are still moderately noisy. The easiest and the most widely used approach for improving the measurement quality is to capture replicated measurements. This may become costly because each additional measurement requires an extra spot on the array, or an extra array. An alternative approach based on so-called composite microarrays was introduced in , where several different oligos representing different genes are printed on each spot. The multiplexing results in a mixing effect similar to the one introduced in this manuscript, and the phenomenon can be inverted to get the reconstructed expression values for single genes. The benefit is to obtain more replicated measurements without proportionately increasing the number of printed spots. Closely related ideas have also been introduced from an error-correcting microarray design point of view in . The standard non-repeated microarray method does not tolerate "drop-outs": if a spot is badly corrupted and its intensity cannot be read, the expression value of the corresponding gene will be missed. Khan et al. showed that a certain amount of "drop-outs" can be recovered from the multiplexed samples, thus providing more error-resilient measurements. Following the methods developed in [17, 18], instead of multiplexing individual genes on spots, one may wish to multiplex different samples on arrays, thus allowing a fault-tolerant recovery of expression values in the case of corrupted array(s). As a future extension, one can also consider multiplexing both the genes on spots and the samples on arrays. Similar methods for inverting the sample heterogeneity have also been studied in the context of time-series gene expression measurements in [19, 20], where the fundamental mixing effect is not due to the different tissue types present in the sample, but due to the loss of synchrony of the cell population. It would be worthwhile to simultaneously study the sample heterogeneity and the loss of synchrony in the future.
In this paper, we proposed a computational framework for removing the effects of sample heterogeneity. In addition to providing estimates of the expression values of the pure (non-heterogeneous) cell samples, the proposed computational methods can also be used to estimate the mixing percentages of different cell types. Furthermore, we also proposed a way of applying general-purpose model selection method for the selection of the correct number of cell types. Application of the proposed methods to a carefully controlled cDNA microarray data obtained from heterogeneous samples shows that the computational methods can invert the effect of sample heterogeneity and, at the same time, estimate the mixing percentages of the different cell types. Furthermore, a general purpose model selection method can be used to select the correct number of cell types.
Materials and methods
RNA isolation, microarray production, and microarray hybridization were carried out as described previously in . RNA from normal human lymph node was purchased from a commercial source (Stratagene, La Jolla, CA). Five μ g aliquots of total RNA from normal lymph node and RKO colon cancer cell line were reverse transcribed using Superscript II RT (Invitrogen, Carlsbad, CA) in conjunction with oligodT-T7 primers according to the manufacturer's suggested protocol. The second strand was synthesized using 10U E. coli DNA ligase (vendor), 40U E. coli DNA polymerase I (vendor), and 2U E. coli Rnase H (vendor). This reaction was stopped with EDTA and then cleaned with Qiagen's PCR Purification kit (Qiagen, Valencia, CA). The double stranded cDNA was then amplified by an in vitro transcription reaction (Ambion, Austin, TX) and cleaned with Qiagen's Rneasy kit (Qiagen, Valencia, CA). Each amplified cRNA sample was then quantitated using a Beckman DU640 spectrophotometer (Beckman, Fullerton, CA). Five μ g amplified cRNA from Stratagene's normal lymph node was labeled with Cy5 for each microarray hybridization. Mixtures of appropriate volumes of cRNA from normal lymph node and RKO were labeled with Cy3 in a reverse transcription reaction using Superscript II RT (see Table 3). Labeled samples were co-hybridized overnight at 60°C in a humidified incubator on a cDNA microarray containing 4704 human genes in duplicate produced in-house. The 4704 genes represent most of the known genes in the cDNA library we used to generate the microarrays. For the purpose of this study, the identity of the genes is not very important since we only study the general effect of sample heterogeneity. As the mixing effect is the same for all the genes, we expect to have similar results when the whole genome arrays are used. Slides were scanned with an LS-IV laser scanner (Genomic Solutions, Ann Arbor, MI). In total, five different heterogeneous mixtures were measured. The measured mixing percentages are shown in Table 3.
The microarray data consists of five different heterogeneous mixtures of lymph node and colon cancer samples which are hereafter abbreviated as normal and RKO, respectively. (For more details, see Microarray production Section above and Table 3.) The gene expression data set was preprocessed as follows. The replicated background-subtracted signal intensities were averaged and log2-transformed, and the dye-bias effect was corrected in the log2-domain using the standard lowess smoothing-based normalization (see e.g. ) with smoothing parameter f = 0.7. Because the averaging effect (source of heterogeneity) takes place on the molecular level, the phenomenon must be modeled using the absolute expression values. Therefore, after the correction of the dye-bias, the data were transformed back to the original domain using the inverse of the log2-transformation. Correspondingly, single-channel data were used for further analysis. In order to mitigate the between array variability, the data were further standardized for each array and the two channels separately.
Modeling sample heterogeneity
Inversion of sample heterogeneity
The first objective is to invert the mixing effect shown in Equation (1), that is, to obtain estimates for the expression values of the pure colon cancer cells and the pure lymphocytes. In practice, however, the measured expression values, y i , include one or more sources of noise. By making some distributional assumptions, one could use standard model-based estimation methods. However, in order to avoid making additional modeling assumptions, we prefer to use a general purpose least squares method to estimate the gene expression levels corresponding to the pure samples.
Let the number of genes be n and assume that one has measured the expression values for K different heterogeneous mixtures. Thus, one has measurements , 1 ≤ i ≤ n, 1 ≤ k ≤ K. Let us also assume for now that the mixing percentages are known or have been measured. For the i th gene the sample heterogeneity can be expressed as (excluding all noise terms)
When including all n genes, the above model can be rewritten as
where 0 denotes the K-by-2 zero matrix. Let the block matrix in Equation (2) above be denoted as Ã Assuming the column rank of A is full, the well-known least squares solution is given by
The Gauss-Markov theorem says that the standard least squares solution is indeed the best linear unbiased estimate if the noise in the measurements is additive and i.i.d. with constant variance. However, a common observation is that the homoscedasticity does not always hold for microarray data, but instead, the noise variance depends on the underlying signal intensity [23, 24]. Such heteroscedasticity may decrease the power of the inversion method shown in Equation (3). Fortunately, the structure of the matrix Ã ensures that the inversion can also be performed for each gene separately. Consequently, it is not necessary for the homoscedasticity to hold globally. Indeed, all we need to assume is that the noise variance is approximately constant for each gene separately.
Also note that, in this two cell type model, no prior knowledge about the expression values of either of the two cell types is needed since the method estimates the expression values for both of the two cell types. The same is also true for more general models including more cell types, assuming the model is sufficiently over-determined (see also Selection of the number of cell types Section below).
Optimization of mixing percentages
In practice, the mixing percentages must be measured by some means. Therefore, they are also likely to contain some error. So, overall, one would like to estimate not only the expression values for the pure cell types but also the most likely value of the mixing percentages.
Assuming the model in Equation (2) is sufficiently over-determined, the mixing parameters can be adjusted computationally, too. Let us again consider the case where only two different cell types are mixed. Note that K denotes the number of different heterogeneous mixtures measured. Therefore, the regression matrix Ã in Equation (2) has only K free parameters. Since the number of expression values to be estimated is 2n, the total number of free parameters in Equation (2) is 2n + K. The number of equations in Equation (2) is Kn. Hence, the model is over-determined if Kn > 2n + K, which, for a fixed n ≥ 3, holds if K > 2. (Note that in our case we have measured five different heterogeneous mixtures, i.e., K = 5.) As above, no assumptions on the noise distributions are being made and we use the least squares method. This results in the following optimization problem
A similar optimization problem was also introduced in .
It is useful to assess the confidence intervals of the obtained expression estimates. As explained above, the Gauss-Markov theorem is applied gene-wise that greatly alleviates the issue of heteroscedasticity. Should the noise variance σ2 be constant, then the variance of the estimated expression values would be . Due to the special structure of the matrix Ã (i.e., the gene-wise inversion of the mixing effect), the variance of the estimated expression values for the i th gene can be expressed as
Selection of the number of cell types
This study was partially supported by Tampere Graduate School in Information Science and Engineering (TISE), Academy of Finland, the Tobacco Settlement Fund to M. D. Anderson Cancer Center as appropriated by the Texas Legislature, a generous donation from Kaddorie Foundation, a grant from the Goodwin Fund, and the Cancer Center Support Grant from NCI/NIH.
- Spellman PT, Sherlock G, Zhang MQ, Iyer VR, Anders K, Eisen MB, Brown PO, Botstein D, Futcher B: Comprehensive identification of cell cycle-regulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization. Mol Biol Cell 1998, 9: 3273–3297.PubMed CentralView ArticlePubMedGoogle Scholar
- Golub T, Slonim D, Tamayo P, Huard C, Gaasenbeek M, Mesirov J, Coller H, Loh M, Downing J, Caligiuri M, Bloomfield C, Lander E: Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 1999, 286: 531–537. 10.1126/science.286.5439.531View ArticlePubMedGoogle Scholar
- van't Veer LJ, Dai H, van de Vijver MJ, He YD, Hart AA, Mao M, Peterse HL, van der Kooy K, Marton MJ, Witteveen AT, Schreiber GJ, Kerkhoven RM, Roberts C, Linsley PS, Bernards R, Friend SH: Gene expression profiling predicts clinical outcome of breast cancer. Nature 2002, 415: 530–536. 10.1038/415530aView ArticleGoogle Scholar
- Zhang W, Ramdas L, Shen WP, Song WS, Hu L, Hamilton SR: Apoptotic response to 5-fluorouracil treatment is mediated by reduced polyamines, non-autocrine fas ligand and induced tumor necrosis factor receptor 2. Cancer Biol Ther 2003, 2: 572–578.View ArticlePubMedGoogle Scholar
- Zhang W, Shmulevich I, Astola J: Microarray Quality Control. John Wiley and Sons; 2004.View ArticleGoogle Scholar
- Fuller GN, Rhee CH, Hess K, Caskey L, Wang R, Bruner JM, Yung WKA, Zhang W: Reactivation of insulin-like growth factor binding protein 2 expression in glioblastoma multiforme: a revelation by parallel gene expression profiling. Cancer Res 1999, 59: 4228–4332.PubMedGoogle Scholar
- Emmert-Buck MR, Bonner RF, Smith PD, Chuaqui RF, Zhuang Z, Goldstein SR, Weiss RA, Liotta LA: Laser capture microdissection. Science 1996, 274: 998–1001. 10.1126/science.274.5289.998View ArticlePubMedGoogle Scholar
- Ghosh D: Mixture models for assessing differential expression in complex tissues using microarray data. Bioinformatics 2004, 20: 1663–1669. 10.1093/bioinformatics/bth139View ArticlePubMedGoogle Scholar
- Lu P, Nakorchevskiy A, Marcotte EM: Expression deconvolution: a reinterpretation of DNA microarray data reveals dynamic changes in cell populations. Proc Natl Acad Sci USA 2003, 100: 10370–10375. 10.1073/pnas.1832361100PubMed CentralView ArticlePubMedGoogle Scholar
- Stuart RO, Wachsman W, Berry CC, Wang-Rodriguez J, Wasserman L, Klacansky I, Masys D, Arden K, Goodison S, McClelland M, Wang Y, Sawyers A, Kalcheva I, Tarin D, Mercola D: In silico dissection of cell-type-associated patterns of gene expression in prostate cancer. Proc Natl Acad Sci U S A 2004, 101: 615–620. 10.1073/pnas.2536479100PubMed CentralView ArticlePubMedGoogle Scholar
- Venet D, Pecasse F, Maenhaut C, Bersini H: Separation of samples into their constituents using gene expression data. Bioinformatics 2001, 17: S279–287.View ArticlePubMedGoogle Scholar
- Rousseeuw PJ, Leroy AM: Robust Regression and Outlier Detection. John Wiley; 1987.View ArticleGoogle Scholar
- Holland PW, Welsch RE: Robust regression using iteratively reweighted least-squares. Commun Stat Theory Methods 1977, A6: 813–827.View ArticleGoogle Scholar
- Rousseeuw PJ: Least median of squares regression. J A Stat Assoc 1984, 79: 871–881.View ArticleGoogle Scholar
- Hao X, Sun B, Hu L, Lähdesmäki H, Dunmire V, Feng Y, Zhang S-W, Wang H, Wu C, Wang H, Fuller GN, Symmans WF, Shmulevich I, Zhang W: Differential gene and protein expression in primary breast malignancies and their lymph node metastases as revealed by combined cDNA microarray and tissue microarray analysis. Cancer 2004, 100: 1110–1122. 10.1002/cncr.20095View ArticlePubMedGoogle Scholar
- Ben-Dor A, Bruhn L, Friedman N, Nachman I, Schummer M, Yakhini Z: Tissue classification with gene expression profiles. J Comput Biol 2000, 7: 559–584. 10.1089/106652700750050943View ArticlePubMedGoogle Scholar
- Shmulevich I, Astola J, Cogdell D, Hamilton SR, Zhang W: Data extraction from composite oligonucleotide microarrays. Nucleic Acids Res 2003, 31: e36. 10.1093/nar/gng036PubMed CentralView ArticlePubMedGoogle Scholar
- Khan AH, Ossadtchi A, Leahy RM, Smith DJ: Error-correcting microarray design. Genomics 2003, 81: 157–165. 10.1016/S0888-7543(02)00032-0View ArticlePubMedGoogle Scholar
- Lähdesmäki H, Huttunen H, Aho T, Linne M-L, Niemi J, Kesseli J, Pearson R, Yli-Harja O: Estimation and inversion of the effects of cell population asynchrony in gene expression time-series. Signal Process 2003, 83: 835–858. 10.1016/S0165-1684(02)00471-1View ArticleGoogle Scholar
- Bar-Joseph Z, Farkash S, Gifford DK, Simon I, Rosenfeld R: Deconvolving cell cycle expression data with complementary information. Bioinformatics 2004, 20(Suppl 1):I23-I30. 10.1093/bioinformatics/bth915View ArticlePubMedGoogle Scholar
- Shmulevich I, Hunt K, El-Naggar A, Taylor E, Ramdas L, Laborde P, Hess KR, Pollock R, Zhang W: Tumor specific gene expression profiles in human leiomyosarcoma: an evaluation of intratumor heterogeneity. Cancer 2002, 94: 2069–2075. 10.1002/cncr.10425View ArticlePubMedGoogle Scholar
- Quackenbush J: Microarray data normalization and transformation. Nat Genet 2002, (Suppl 32):496–501. 10.1038/ng1032Google Scholar
- Huber W, von Heydebreck A, Sultmann H, Poustka A, Vingron M: Variance stabilization applied to microarray data calibration and to the quantification of differential expression. Bioinformatics 2002, 18(Suppl 1):S96–104.View ArticlePubMedGoogle Scholar
- Durbin BP, Rocke DM: Variance-stabilizing transformations for two-color microarrays. Bioinformatics 2004, 20: 660–677. 10.1093/bioinformatics/btg464View ArticlePubMedGoogle Scholar
- Efron B, Tibshirani RJ: An introduction to the bootstrap. New York: Chapman & Hall; 1993.View ArticleGoogle Scholar
- Hastie T, Tibshirani R, Friedman J: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer-Verlag; 2001.View ArticleGoogle Scholar
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