doi: 10.1371/journal.pone.0121396.
eCollection 2015.
Tensor GSVD of patient- and platform-matched tumor and normal DNA copy-number profiles uncovers chromosome arm-wide patterns of tumor-exclusive platform-consistent alterations encoding for cell transformation and predicting ovarian cancer survival
Affiliations
- PMID: 25875127
- PMCID: PMC4398562
- DOI: 10.1371/journal.pone.0121396
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Tensor GSVD of patient- and platform-matched tumor and normal DNA copy-number profiles uncovers chromosome arm-wide patterns of tumor-exclusive platform-consistent alterations encoding for cell transformation and predicting ovarian cancer survival
PLoS One.
.
Abstract
The number of large-scale high-dimensional datasets recording different aspects of a single disease is growing, accompanied by a need for frameworks that can create one coherent model from multiple tensors of matched columns, e.g., patients and platforms, but independent rows, e.g., probes. We define and prove the mathematical properties of a novel tensor generalized singular value decomposition (GSVD), which can simultaneously find the similarities and dissimilarities, i.e., patterns of varying relative significance, between any two such tensors. We demonstrate the tensor GSVD in comparative modeling of patient- and platform-matched but probe-independent ovarian serous cystadenocarcinoma (OV) tumor, mostly high-grade, and normal DNA copy-number profiles, across each chromosome arm, and combination of two arms, separately. The modeling uncovers previously unrecognized patterns of tumor-exclusive platform-consistent co-occurring copy-number alterations (CNAs). We find, first, and validate that each of the patterns across only 7p and Xq, and the combination of 6p+12p, is correlated with a patient's prognosis, is independent of the tumor's stage, the best predictor of OV survival to date, and together with stage makes a better predictor than stage alone. Second, these patterns include most known OV-associated CNAs that map to these chromosome arms, as well as several previously unreported, yet frequent focal CNAs. Third, differential mRNA, microRNA, and protein expression consistently map to the DNA CNAs. A coherent picture emerges for each pattern, suggesting roles for the CNAs in OV pathogenesis and personalized therapy. In 6p+12p, deletion of the p21-encoding CDKN1A and p38-encoding MAPK14 and amplification of RAD51AP1 and KRAS encode for human cell transformation, and are correlated with a cell's immortality, and a patient's shorter survival time. In 7p, RPA3 deletion and POLD2 amplification are correlated with DNA stability, and a longer survival. In Xq, PABPC5 deletion and BCAP31 amplification are correlated with a cellular immune response, and a longer survival.
Conflict of interest statement
Figures
For each chromosome arm or combination of two chromosome arms, the structure of the tumor and normal discovery datasets (
1 and 
2) is that of two third-order tensors with one-to-one mappings between the column dimensions but different row dimensions. The patients, platforms, probes, and tissue types, each represent a degree of freedom. Unfolded into a single matrix, some of the degrees of freedom are lost and much of the information in the datasets might also be lost. We define a tensor GSVD that simultaneously separates the paired datasets into weighted sums of paired subtensors, i.e., combinations or outer products of three patterns each: Either one tumor-specific pattern of copy-number variation across the tumor probes, i.e., a tumor arraylet (a column basis vector of U
1), or the corresponding normal-specific arraylet (a column basis vector of U
2), combined with one pattern of variation across the patients, i.e., an x-probelet (a row basis vector of ), and one pattern across the platforms, i.e., a y-probelet (a row basis vector of ), which are identical for both the tumor and normal datasets (Equation 1). The tensor GSVD is depicted in a raster display, with relative copy-number gain (red), no change (black), and loss (green), explicitly showing the first through the 5th, and the 245th through the 249th 6p+12p x-probelets, both 6p+12p y-probelets, and the first through the 10th, and the 489th through the 498th 6p+12p tumor and normal arraylets. We prove that the significance of a subtensor in the tumor dataset relative to that of the corresponding subtensor in the normal dataset, i.e., the tensor GSVD angular distance, equals the row mode GSVD angular distance, i.e., the significance of the corresponding tumor arraylet in the tumor dataset relative to that of the normal arraylet in the normal dataset. The tensor GSVD angular distances for the 498 pairs of 6p+12p arraylets are depicted in a bar chart display, where the angular distance corresponding to the first pair of arraylets is ∼ π/4. For the 6p+12p combination of two chromosome arms, we find that the most significant subtensor in the tumor dataset (which corresponds to the coefficient of largest magnitude in ℛ1) is a combination of (i) the first y-probelet, which is approximately invariant across the platforms, (ii) the first x-probelet, which classifies the discovery set of patients into two groups of high and low coefficients, of significantly and robustly different prognoses, and (iii) the first, most tumor-exclusive tumor arraylet, which classifies the validation set of patients into two groups of high and low correlations of significantly different prognoses consistent with the x-probelet’s classification of the discovery set.
1 and
2) is that of two third-order tensors with one-to-one mappings between the column dimensions but different row dimensions. The patients, platforms, probes, and tissue types, each represent a degree of freedom. Unfolded into a single matrix, some of the degrees of freedom are lost and much of the information in the datasets might also be lost. We define a tensor GSVD that simultaneously separates the paired datasets into weighted sums of paired subtensors, i.e., combinations or outer products of three patterns each: Either one tumor-specific pattern of copy-number variation across the tumor probes, i.e., a tumor arraylet (a column basis vector of U
1), or the corresponding normal-specific arraylet (a column basis vector of U
2), combined with one pattern of variation across the patients, i.e., an x-probelet (a row basis vector of ), and one pattern across the platforms, i.e., a y-probelet (a row basis vector of ), which are identical for both the tumor and normal datasets (Equation 1). The tensor GSVD is depicted in a raster display, with relative copy-number gain (red), no change (black), and loss (green), explicitly showing the first through the 5th, and the 245th through the 249th 6p+12p x-probelets, both 6p+12p y-probelets, and the first through the 10th, and the 489th through the 498th 6p+12p tumor and normal arraylets. We prove that the significance of a subtensor in the tumor dataset relative to that of the corresponding subtensor in the normal dataset, i.e., the tensor GSVD angular distance, equals the row mode GSVD angular distance, i.e., the significance of the corresponding tumor arraylet in the tumor dataset relative to that of the normal arraylet in the normal dataset. The tensor GSVD angular distances for the 498 pairs of 6p+12p arraylets are depicted in a bar chart display, where the angular distance corresponding to the first pair of arraylets is ∼ π/4. For the 6p+12p combination of two chromosome arms, we find that the most significant subtensor in the tumor dataset (which corresponds to the coefficient of largest magnitude in ℛ1) is a combination of (i) the first y-probelet, which is approximately invariant across the platforms, (ii) the first x-probelet, which classifies the discovery set of patients into two groups of high and low coefficients, of significantly and robustly different prognoses, and (iii) the first, most tumor-exclusive tumor arraylet, which classifies the validation set of patients into two groups of high and low correlations of significantly different prognoses consistent with the x-probelet’s classification of the discovery set.
(a) Plot of the first 6p+12p tumor arraylet describes a pattern of tumor-exclusive and platform-consistent co-occurring CNAs across the combination of the two chromosome arms 6p+12p. The probes are ordered, and their copy numbers are colored according to each probe’s chromosomal band location. Segments (black lines) amplified and deleted include most known OV-associated CNAs that map to 6p+12p (black), including an amplification of KRAS and a deletion of PRIM2. CNAs previously unrecognized in OV (red) include a deletion of the p38-encoding MAPK14, and p21-encoding CDKN1A, and an amplification of RAD51AP1, a deletion of TNF, and focal amplifications of ASUN, ITPR2, and the 5’ ends of isoforms a and e, and exons 5 and 6 of SOX5. A high 6p+12p arraylet correlation is significantly correlated with a patient’s shorter survival time. (b) Plot of the first 6p+12p x-probelet describes the classification of the discovery set of patients into two groups of high (blue) and low (red) coefficients. A high 6p+12p x-probelet coefficient is significantly and robustly correlated with a patient’s shorter survival time. (c) Raster display of the 6p+12p tumor profiles, where medians of the profiles of the same patient measured by the two platforms were taken, with relative gain (red), no change (black), and loss (green) of DNA copy numbers. (d) Plot of the first 7p tumor arraylet describes a pattern of CNAs across the chromosome arm 7p. CNAs previously unrecognized in OV (red) include a focal deletion of RPA3 and an amplification of POLD2. A high 7p arraylet correlation is significantly correlated with a patient’s longer survival time. (e) Plot of the first 7p x-probelet describes the classification of the discovery set of patients into two groups of high (red) and low (blue) coefficients. A high 7p x-probelet coefficient is significantly and robustly correlated with a patient’s longer survival time. (f) Raster display of the 7p tumor profiles. (g) Plot of the first Xq tumor arraylet. CNAs previously unrecognized in OV (red) include a focal deletion of PABPC5 and an amplification of BCAP31. A high Xq arraylet correlation is significantly correlated with a patient’s longer survival time. (h) Plot of the first Xq x-probelet describes the classification of the discovery set of patients into two groups of high (red) and low (blue) coefficients. A high Xq x-probelet coefficient is significantly and robustly correlated with a patient’s longer survival time. (i) Raster display of the Xq tumor profiles.
(a) Kaplan-Meier (KM) curves of the discovery set of 249 patients classified by the 6p+12p x-probelet coefficient, show a median survival time difference of 11 months, with the corresponding log-rank test P-value < 10−2. The univariate Cox proportional hazard ratio is 1.7. (b) Survival analyses of the 249 patients classified by the 7p x-probelet coefficient. (c) The 249 patients classified by the Xq x-probelet coefficient. (d) The 249 patients classified by both the 6p+12p tensor GSVD and tumor stage at diagnosis, show the bivariate Cox hazard ratios of 1.5 and 4.0, which do not differ significantly from the corresponding univariate hazard ratios of 1.7 and 4.4, respectively. This means that the 6p+12p tensor GSVD is independent of stage, the best predictor of OV survival to date. The 61 months KM median survival time difference is about 85% and more than two years greater than the 33 month difference between the patients classified by stage alone. This means that the tensor GSVD and stage combined make a better predictor than stage alone. (e) The 249 patients classified by both the 7p tensor GSVD and stage. (f) The 249 patients classified by both the Xq tensor GSVD and stage. (g) KM curves of the validation set of 148 stage III-IV patients classified by the 6p+12p arraylet correlation, show a median survival time difference of 22 months, with the corresponding log-rank test P-value < 10−2, and the univariate Cox proportional hazard ratio 1.9. This validates the survival analyses of the discovery set of 249 patients. (h) Survival analyses of the 148 patients classified by the 7p arraylet correlation. (i) The 148 patients classified by the Xq arraylet correlation.
(a) KM curves of the discovery set of 249 patients classified by combination of the 6p+12p, 7p, and Xq x-probelet coefficients, show median survival times of 86, 52, and 36 months for the groups A, B, and C, respectively, with the corresponding log-rank test P-value < 10−3. (b) KM survival analysis of only the 218, i.e., ∼ 88% platinum-based chemotherapy patients in the discovery set, classified by combination of the three tensor GSVDs, gives qualitatively the same and quantitatively similar results to those of the analyses of 100% of the patients. This means that the combination of the three tensor GSVDs predicts survival in the platinum-based chemotherapy patient population. (c) KM curves of the validation set of 148 stage III-IV patients classified by combination of the 6p+12p, 7p, and Xq arraylet correlation coefficients, show median survival times of 72, 57, and 33 months for the groups A, B, and C, respectively, with the corresponding log-rank test P-value < 10−3. This validates the survival analyses of the discovery set of 249 patients. (d) KM survival analysis of only the 140, i.e., ∼ 95% platinum-based chemotherapy patients in the validation set, classified by combination of the three tensor GSVDs.
References
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- Ponnapalli SP, Golub GH, Alter O. A novel higher-order generalized singular value decomposition for comparative analysis of multiple genome-scale datasets. Stanford University and Yahoo! Research Workshop on Algorithms for Modern Massive Datasets (MMDS) (Stanford, CA: ). 2006; June 21–24.
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- Golub GH, Van Loan CF. Matrix Computations. 4th ed Baltimore, MD: Johns Hopkins University Press; 2012.
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