# Absolute quantification of somatic DNA alterations in human cancer

## Abstract

We describe a computational method that infers tumor purity and malignant cell ploidy directly from analysis of somatic DNA alterations. The method, named ABSOLUTE, can detect subclonal heterogeneity and somatic homozygosity, and it can calculate statistical sensitivity for detection of specific aberrations. We used ABSOLUTE to analyze exome sequencing data from 214 ovarian carcinoma tumor-normal pairs. This analysis identified both pervasive subclonal somatic point-mutations and a small subset of predominantly clonal and homozygous mutations, which were overrepresented in the tumor suppressor genes TP53 and NF1 and in a candidate tumor suppressor gene CDK12. We also used ABSOLUTE to infer absolute allelic copy-number profiles from 3,155 diverse cancer specimens, revealing that genome-doubling events are common in human cancer, likely occur in cells that are already aneuploid, and influence pathways of tumor progression (for example, with recessive inactivation of NF1 being less common after genome doubling). ABSOLUTE will facilitate the design of clinical sequencing studies and studies of cancer genome evolution and intra-tumor heterogeneity.

## Main

Defining chromosome copy number and allele ratios is fundamental to understanding the structure and history of the cancer genome. Current genomic characterization techniques measure somatic alterations in a cancer sample in units of genomes (DNA mass). The meaning of such measurements is dependent on the tumor's purity and its overall ploidy; they are hence complicated to interpret and compare across samples. Ideally, copy number should be measured in copies per cancer cell. Such measurements are straightforward to interpret and, for alterations that are fixed in the cancer cell population, are simple integer values. This is considerably more challenging than measuring relative copy number in units of diploid DNA mass in a tumor-derived sample.

Measuring somatic copy-number alterations (SCNAs) on a relative basis is straightforward using microarrays1,2,3,4,5 or massively parallel sequencing technology6,7; it has been the standard approach for copy-number analysis since the development of comparative genomic hybridization (CGH)8.

Inferring absolute copy number is more difficult for three reasons: (i) cancer cells are nearly always intermixed with an unknown fraction of normal cells (tumor purity); (ii) the actual DNA content of the cancer cells (ploidy), resulting from gross numerical and structural chromosomal abnormalities, is unknown9,10,11,12,13; and (iii) the cancer cell population may be heterogeneous, perhaps owing to ongoing subclonal evolution14,15. In principle, one could infer absolute copy numbers by rescaling relative data on the basis of cytological measurements of DNA mass per cancer cell16,17,18, or by single-cell sequencing approaches15. However, such approaches are not suited to support initial large-scale efforts to comprehensively characterize the cancer genome19.

We began focusing on this issue several years ago, initially developing ad hoc techniques20,21. We subsequently developed the fully quantitative ABSOLUTE method and applied it to several cancer genome analysis projects, including The Cancer Genome Atlas (TCGA) project. ABSOLUTE provides a foundation for integrative genomic analysis of cancer genome alterations on an absolute (cellular) basis. We used these methods to correlate purity and ploidy estimates with expression subtypes and to develop statistical power calculations and use them to select well-powered samples for whole-genome sequencing in several published22,23,24, and numerous ongoing projects, including breast, prostate and skin cancer genome characterization. Recently, we extended ABSOLUTE to infer the multiplicity of somatic point-mutations in integer allelic units per cancer cell.

Our purpose here is to (i) present the mathematical inference framework of the ABSOLUTE method, as well as experimental validation of its predictions; (ii) apply it to analyze a large pan-cancer data set, enabling characterization of the incidence and timing of whole-genome doublings during tumor evolution; and (iii) describe an integrated analysis of point-mutation and copy-number estimates and its application to ovarian carcinoma.

We describe three key mathematical features of ABSOLUTE. First, it jointly estimates tumor purity and ploidy directly from observed relative copy profiles (point mutations may also be used, if available). Second, because joint estimation may not be fully determined on a single sample, it uses a large and diverse sample collection to help resolve ambiguous cases. Third, it attempts to account for subclonal copy-number alterations and point mutations, which are expected in heterogeneous cancer samples.

We apply ABSOLUTE to conduct the first, to our knowledge, large-scale 'pan-cancer' analysis of copy-number alterations on an absolute basis, across 3,155 cancer samples, representing 25 diseases with at least 20 samples each. The analysis reveals that whole-genome doubling events occur frequently during tumorigenesis, ultimately resulting in mature cancers descended from doubled cells bearing complex karyotypes. Despite evidence that genome doublings can result in genetic instability and accelerate oncogenesis13,25,26, the incidence and timing of such events had not been broadly characterized in human cancer.

We then describe how estimates of tumor purity and absolute copy number allow us to analyze allelic-fraction values (the fraction of non-reference sequencing reads supporting a mutation) to distinguish clonal and subclonal point mutations, and to detect macroscopic subclonal structure in an ovarian cancer sample. Clonal events may be classified as homozygous or heterozygous in the cancer cells, guiding interpretation of their function. In addition, the ability to quantify integer multiplicity of point mutations aids in the relative timing of segmental DNA copy-number gains, as multiplicity values of greater than one imply that the point mutation preceded copy gain of the locus. Controlling for tumor purity and local copy-number allow such timings to be calculated more generally than in the special case of copy-neutral loss of heterozygosity27. Finally, our data allow characterization of somatic cancer evolution with respect to whole-genome doubling, which we demonstrate in ovarian carcinoma and associate with clinicopathological values.

## Results

### Inference of sample purity and ploidy in cancer-derived DNA

A conceptual overview of ABSOLUTE is shown in Figure 1. When DNA is extracted from a mixed population of cancer and normal cells, the information on absolute copy number per cancer cell is lost. The purpose of ABSOLUTE is to infer this information from the population of mixed DNA. This process begins with the generation of segmented copy-number data, which is input to the ABSOLUTE algorithm together with precomputed models of recurrent cancer karyotypes and, optionally, allelic fraction values for somatic point mutations. The output of ABSOLUTE then provides inferred information on the absolute cellular copy number of local DNA segments and, for point mutations, the number of mutated alleles (Fig. 1).

We begin by describing the inference framework used in ABSOLUTE. Suppose a cancer-tissue sample consists of a mixture of a proportion α of cancer cells (assumed to be monogenomic—that is, with homogenous SCNAs in the cancer cells) and a proportion (1– α) of contaminating normal (diploid) cells. For each locus x in the genome, let q(x) denote the integer copy number of the locus in the cancer cells. Let τ denote the mean ploidy of the cancer-cell fraction, defined as the average value of q(x) across the genome. In the mixed cancer sample, the average absolute copy number of locus x is α q(x) + 2(1 − α) and the average ploidy (D) is ατ + 2(1 − α), measured in units of haploid genomes.

The relative copy number (R) of locus x is therefore:

Because q(x) takes integer values, R(x) takes discrete values. The smallest possible value is (2(1 − α)/D), which occurs at homozygously deleted loci and corresponds to the fraction of DNA from normal cells. The spacing between values (α/D) corresponds to the concentration ratio of alleles present at one copy per cancer cell and zero copies per normal cell. Notably, if a cancer sample is not strictly clonal, copy-number alterations occurring in substantial subclonal fractions will appear as outliers from this pattern (Fig. 1b,e, arrows). Similar considerations have formed the basis for algorithms to infer purity and ploidy using allelic copy-ratios derived from single-nucleotide polymorphisms (SNP) microarrays28,29,30,31,32,33.

We extend absolute copy inference to encompass somatic point mutations as follows:

Here, sq represents the multiplicity of the point mutation, in integer values per cancer cell (which cannot exceed q(x)), and Ds = αq(x) + 2(1 − α). The values of F(x) correspond to the expected fraction of sequencing reads that support the mutation, which depend on the sample purity and absolute somatic copy number at the mutant locus, q(x).

The ABSOLUTE algorithm examines possible mappings from relative to integer copy numbers by jointly optimizing the two parameters α and τ (Fig. 1e–g; Supplementary Fig. 1; Online Methods equation (5)). In many cases, several such mappings are possible, corresponding to multiple optima.

To help resolve ambiguous cases, we used recurrent cancer-karyotype models based on large data sets (Supplementary Fig. 2; Online Methods equation (8) to identify the simplest (that is, most common) karyotype that can adequately explain the data. This method favors simpler solutions, while preserving the flexibility to identify unexpected karyotypes given sufficient evidence from the copy profile. Indeed, several unusual karyotypes, including near-haploid (<1.2n) and hyperaneuploid (>6n) genomes, were identified using ABSOLUTE (Supplementary Fig. 3).

Our implementation supports copy-number inference from either total or allelic copy-ratio data, such that array-CGH, SNP microarray or massively parallel sequencing data may be used. ABSOLUTE is available for download at http://www.broadinstitute.org/cancer/cga/ABSOLUTE.

### Validation

We validated the purity and ploidy predictions made by ABSOLUTE on Affymetrix SNP microarray data using several approaches: (i) direct ploidy measurement of 37 TCGA ovarian carcinoma samples by fluorescence-activated cell sorting34 (Fig. 2a); (ii) measurement of ploidy for 33 NCI60 cell lines based on spectral karyotyping35 (Fig. 2b,c); and (iii) DNA-mixing experiments, in which cancer cell lines were mixed with paired normal B lymphocyte–derived DNAs in varying mass proportions (Fig. 2d, Online Methods). We also evaluated a related computational method, ASCAT31, on these data (Fig. 2a–d and Supplementary Note). Although the results were broadly concordant with our estimates, ABSOLUTE achieved significantly more accurate results (Fig. 2a–d) on our validation data. Notably, we observed an apparent bias by ASCAT to underestimate the cancer cell fraction (Fig. 2 c,d), consistent with previous reports applying ASCAT in similar mixing experiments using Illumina SNP arrays31 (Fig. S4 therein).

Notably, the purity estimates produced by ABSOLUTE appeared to be more accurate for the bulk tumor than those derived from histological examination of frozen tumor sections (Online Methods, Fig. 2e). Estimates of the proportion of contaminating normal cells for 458 ovarian carcinoma samples34 produced by ABSOLUTE were strongly correlated with a molecular signature of genomic methylation (Online Methods) seen in leukocytes (r2 = 0.59, P < 2.2 × 10−16, Fig. 2e), but only weakly correlated with estimates of contamination from histological examination (r2 = 0.1, P = 2.4 × 10−12; Online Methods; Fig. 2e x-axis scale, Supplementary Fig. 4).

### Estimation of tumor purity and ploidy across cancer types

We used ABSOLUTE to analyze allelic copy-ratio profiles derived from SNP arrays from 3,155 cancer samples, comprising 2,791 tissue specimens and 364 cancer cell lines. This yielded predicted purity and ploidy values (Supplementary Table 1) and the segmented absolute allelic copy number of each tumor (Supplementary Table 2). The samples came from two TCGA pilot studies describing glioblastoma multiforme (GBM; 192 samples)21 and ovarian carcinoma (488 samples)34, as well as 2,445 profiles incorporated from a previous pan-cancer copy-number analysis36 (Online Methods; see Supplementary Table 1 for characteristics of each tumor sample). A minority of these samples (519 or 16.4%) could not be analyzed because they lacked clearly identifiable SCNAs, either because they were nearly euploid (nonaberrant), or were excessively contaminated with normal cells (insufficient purity) (Fig. 3a). Although sequencing data for somatic point mutations may have resolved these cases, such data were not available for the majority of samples in this cohort36.

For the 2,636 samples with detectable SCNAs, ABSOLUTE provided purity and ploidy calls for 92% of cases, and designated the remaining samples as 'polygenomic' (genomically heterogeneous) (Fig. 3a), (Online Methods and Supplementary Fig. 5). The fraction of called samples varied by disease type, from 34.6% (myeloproliferative disease; mostly nonaberrant genomes) to 96.7% (ovarian carcinoma; 100% aberrant genomes), with a median call-rate of 79.2% (Fig. 3a).

The distributions of estimated purity varied among cancer types, with the tested lung, esophageal and breast cancer samples being the least pure on average in our data set (Fig. 3b). The effect of contamination was readily visible in the copy ratios of impure tumor types (Supplementary Fig. 6). Distributions of estimated ploidy (Fig. 3c) were qualitatively consistent with those derived from previously obtained cytological data for each tumor type13.

### Power for detection of somatic point-mutations by sequencing

Both tumor purity and ploidy affect the local depth of sequencing necessary to detect point mutations. For example, suppose that a region is present at six copies with only one copy carrying a mutation in a sample that has 50% contamination with normal cells. In this case, only one of eight alleles at this locus (six from the cancer cells and two from the normal cells) carry the mutation (Supplementary Fig. 7a). We therefore expect that the mutation will be observed in only 12.5% of reads. Given this allelic fraction, local sequence coverage of 33-fold is required to detect the mutation with 80% sensitivity, assuming a sequencing error rate of 10−3 per base and a false-positive rate controlled at <5 × 10−7 (Online Methods, equation (9); Supplementary Fig. 7b).

Using ABSOLUTE's estimates of purity and genome-wide integer copy numbers, we can calculate the required coverage for powered detection of mutations present at specified allelic multiplicity per cancer cell. Similar considerations apply to detecting subclonal mutations, present in a fraction of cancer cells, by using fractional multiplicities (Supplementary Fig. 7c). We note that consideration of tumor purity in units of cells, rather than DNA fraction, is preferred for devising power calculations for sequencing experiments, because many somatic alterations of interest are expected to occur at a single copy per cancer cell.

We analyzed the distribution of purity and ploidy values in cancer samples analyzed for allelic copy number21,34,36 to determine an appropriate depth of sequencing coverage needed to detect clonal mutations with power 0.8 in each sample. For this purpose, we calculated the number of reads needed to detect a mutation present in one copy, at a locus present at the average copy number, given the sample's purity. (One could alternatively choose a particular percentile on the copy-number distribution.) For such a locus, we found that 30× local coverage would suffice for most samples (Supplementary Fig. 7d). By contrast, a locus of average copy number with a mutation carried in a subclone at 0.2 cancer-cell fraction would require coverage of 100-fold to allow detection in about half of the samples (Supplementary Fig. 7e). Using these calculations and the distribution of local coverage along the genome (which depends on the specific sequencing technology), one can determine the average coverage necessary to obtain sufficient power in a predefined fraction of the genome (e.g., >80% power in >80% of the genome).

We then examined whole-exome sequencing data (150 × average coverage) from 214 TCGA ovarian carcinoma samples34 to determine whether detection power was related to the number of mutations actually observed. For each sample, we calculated the proportion of loci for which the local coverage provided at least 80% power to detect mutations present at single copy in a subclone present at 0.05 cancer-cell fraction. Those samples with the lowest proportion of such well-powered loci tended to be those in which the fewest such mutations were detected (r2 = 0.24, P = 2.7 × 10−13; Supplementary Fig. 7f), suggesting that the failure to find such mutations was due to the lack of power. This result also demonstrates the importance of power calculations for characterization of the subclonal frequency spectrum.

### Multiplicity analysis of somatic point-mutations

We next used ABSOLUTE to convert the allelic fraction of mutations to cellular multiplicity estimates. For this purpose, we examined 29,268 somatic mutations identified in whole-exome hybrid capture Illumina sequencing37 data from 214 ovarian carcinoma tumor-normal pairs34 (Fig. 4a). Tumor purity, ploidy and absolute copy-number values were obtained from Affymetrix SNP6.0 hybridization data on the same DNA aliquot that was sequenced, allowing the rescaling of allelic fractions to units of multiplicity (Fig. 4a,b; Online Methods, equation (12)).

This procedure identified pervasive subclonal point-mutations in ovarian carcinoma samples. Although many of the mutations were clustered around integer multiplicities, a substantial fraction occurred at multiplicities substantially less than one copy per average cancer cell, consistent with subclonal multiplicity (Fig. 4b)).

Several lines of evidence support the validity of these subclonal mutations, including Illumina resequencing of an independent whole-genome amplification aliquot, which confirmed both their presence (Supplementary Fig. 8a,b), and that their allelic fractions corresponded to subclonal multiplicity values (Supplementary Fig. 8c,d). In addition, the mutation spectrum seen for clonal and subclonal mutations was similar (root mean squared error (RMSE) = 0.04, Fig. 4c), consistent with a common mechanism of origin. Power calculations showed that these samples were at least 80% powered for detection of subclonal mutations occurring in cancer-cell fractions ranging from 0.1 to 0.53, with a median of 0.19 (Supplementary Fig. 7e).

The distribution of subclonal multiplicity was similar in the majority of samples (Fig. 4b); it rapidly increased at the sample-specific detection limit and then decreased in a manner approximated by an exponential decay in the multiplicity range of 0.05 to 0.5 when pooling across all samples. In contrast, the high-grade serous ovarian carcinoma (HGS-OvCa) sample TCGA-24-1603 (Fig. 4d–f) showed evidence for discrete 'macroscopic subclones'. Rescaling of subclonal SCNAs (Fig. 4d) and point mutations (Fig. 4e) to units of cancer cell fraction (Fig. 4f) revealed discrete clusters near fractions 0.2, 0.3 and 0.6 (Fig. 4f), implying the alterations within each cluster likely co-occurred in the same cancer cells. We note that this combination of cell fractions sums to more than one, implying that at least one of the detected subclones was nested inside another.

We next used ABSOLUTE to analyze the multiplicity of both the reference and alternate alleles in order to classify point mutations as either heterozygous or homozygous in the affected cell fraction (Fig. 5a–c). We considered 15 genes with mutations recently identified in these data34, including five known tumor suppressor genes and five oncogenes (Fig. 5d). The frequency of homozygous mutations in known tumor suppressor genes and oncogenes was significantly different, with a significantly elevated fraction of homozygous mutations in the tumor suppressor genes (P = 0.006, Fig. 5d) and no homozygous mutations in the oncogenes (P = 0.012, Fig. 5d). This result provides evidence supporting CDK12 as a candidate tumor suppressor gene in ovarian carcinoma34, since 7 of 12 CDK12 mutations were homozygous (P = 6.5 × 10−5; Fig. 5d).

Overall, TP53 had among the greatest fraction of clonal, homozygous and 'multiplicity >1' mutations of any gene in the coding exome (Fig. 5e), demonstrating the clear identification of a key initiating event in HGS-OvCa carcinogenesis38 directly from multiplicity analysis.

### Whole-genome doubling occurs frequently in human cancer

For many cancer types, the distribution of total copy number (ploidy) was markedly bi-modal (Fig. 3c), consistent with chromosome-count profiles derived from SKY10,13. Although these results are consistent with whole-genome doubling during their somatic evolution, it has been difficult to rule out the alternative hypothesis that evolution of high-ploidy karyotypes results from a process of successive partial amplifications12.

To study genome doublings, we used homologous copy-number information—that is, the copy numbers, bi and ci, of the two homologous chromosome segments at each locus. By looking at the distributions of bi and ci across the genome, we could draw inferences regarding genome doubling. Immediately following genome doubling, both bi and ci would be even numbers. Following the loss of a single copy of a region, the larger of bi and ci will remain even, but the smaller would become odd. In fact, when we looked at high-ploidy samples, we discerned that the higher of bi and ci was usually even throughout the genome, consistent with their having arisen by doubling of the entire genome (Supplementary Fig. 9). Using simulations, we found that the observed profiles were unlikely to arise owing to SCNAs occurring in serial fashion at multiple independent chromosomes (P < 10−3).

Using such information, we classified samples into three groups, which we interpreted as corresponding to 0, 1 and >1 genome doubling events in the clonal evolution of the cancer. These three groups had modal ploidy values of 1.75, 2.75 and 4.0, respectively (Fig. 6a), and also segregated into three clusters by ploidy and mean homologous copy-number imbalance (Fig. 6b). We interpreted this as evidence of SNCAs occurring with net losses, interspersed with the genome doublings. This process resulted in intermediate ploidy values for the doubled clones (2.2–3.4n), with pervasive imbalance of homologous chromosomes (Fig. 6b).

The frequency of genome doubling varied across tumor types (Fig. 6c), reflecting differences in disease-specific biology and clinical progression status. Hematopoietic neoplasms (myeloproliferative disease, acute lymphoblastic leukemia) had nearly no doubling events, whereas glioblastoma multiforme, renal cell carcinoma, prostate cancer, various sarcomas, hepatocellular carcinoma and medulloblastoma all had 25% incidence of doubling. Genome doubling was more common in epithelial cancers, with colorectal, breast, lung, ovarian and esophageal cancers all having >50% incidence of doubling (Fig. 6c). Esophageal adenocarcinoma had the greatest doubling incidence, consistent with previous reports of frequent 4n populations at various stages of Barrett's esophagus progression39,40.

### Specific aneuploidies precede genome doubling

We then used ABSOLUTE to infer the temporal order of genome doubling in tumorigenesis, relative to SNCAs involving specific chromosome arms. In many cancer types, the fixation of arm-level SCNAs was inferred to occur before genome doubling, because both doubled and nondoubled samples had similar frequencies of specific arm-level SNCAs (Fig. 6d and Supplementary Fig. 10).

In glioblastoma multiforme samples, loss of heterozygosity involving chromosomes 9 and 10, and gains of chromosome 7 occurred at equivalent frequencies (Fig. 6d), demonstrating that the most common broad SCNAs in glioblastoma multiforme occur before genome doubling. Gain of chromosomes 19 and 20 was nearly exclusive to nondoubled samples, and several arms had greater frequency of loss of heterozygosity in doubled samples (Fig. 6d), suggesting that additional biological differences underlie these samples.

Because ABSOLUTE could not distinguish between ploidy 2N and 4N in cases with no observed SCNAs, we discarded such nonaberrant samples from our analysis (Fig. 3a). For many tumor types, such cases were rare, due to the tendency for chromosomal losses after doubling (Figs. 3c and 6a,b and Supplementary Fig. 9). The representation of specific cancer subtypes may be biased by differences in ascertainment, however.

In contrast to broad chromosomal alterations, focal SCNA events occurred at greater frequency in doubled genomes (Fig. 6e). Consistent with previous reports36,41,42, the observed frequency of focal SCNAs as a function of their length (L) followed power-law scaling: P(L) L−α, for L > 0.5 Mb (Fig. 6e). Genome doubling was associated with a larger overall number of SCNAs; however, we obtained estimates of α near 1 for each group (Fig. 6e), suggesting that the mechanism(s) by which they were generated did not greatly depend on ploidy.

### Genome doubling influences progression of ovarian carcinoma

We next sought to correlate whole-genome doubling occurrence in high-grade serous ovarian carcinoma with other genetic and clinical features. Genome-doubled samples showed a higher incidence of heterozygous mutations, but correcting for sample ploidy removed this effect (Fig. 7a), suggesting that the per-base mutation rates are equivalent. Clonal mutations at multiplicity >1 were approximately tenfold more prevalent in doubled samples; many of these events likely occurred before the doubling event. Genome-doubled samples had significantly lower frequencies of both of homozygous deletions (Fig. 7b) and of clonal homozygous mutations (Fig. 7c). We expect that many of the observed homozygous alterations in the doubled samples were fixed before genome doubling.

The lower incidence of homozygous mutations in genome-doubled samples may reflect the fact that more events are required to render a mutation homozygous in a genome-doubled sample (although the effect may be partially offset by a possible increase in genetic instability following doubling, for example, by centrosome duplication43). These considerations suggest that genome-doubled samples evolve by means of distinct trajectories, because inactivation of tumor suppressors may occur less frequently after doubling.

We note that 13 of the 15 detected point mutations in the tumor suppressor NF1 occurred in the 93 ovarian samples that had not undergone genome doubling (P = 0.002; Fisher's exact test), and these mutations were uniformly homozygous (data not shown). This is consistent with selection for recessive inactivation of NF1, a typical pattern for a tumor suppressor gene. It also suggests that nongenome-doubled ovarian carcinoma samples evolved through a distinct trajectory, rather than being precursors to doubled samples. If not, many NF1 mutations would be homozygous with multiplicity >1 in doubled samples, as is seen for TP53.

Finally, we noted that genome-doubled samples were associated with a significant increase in the age at pathological diagnosis (Fig. 7d) and with a significantly greater incidence of cancer recurrence (Fig. 7e).

## Discussion

Here we report the development of a reliable, high-throughput method to infer absolute homologous copy numbers from tumor-derived DNA samples, as well as multiplicity values of point mutations (ABSOLUTE). It may be possible to extend ABSOLUTE to other types of genomic alterations, such as structural rearrangements and small insertions and deletions, although this may require longer sequence reads to ensure accurate sequence alignment.

ABSOLUTE analysis of SCNAs demonstrated that many of the copy-number alterations analyzed were fixed in the cancer lineage represented in the sample (Fig. 3). This was recapitulated in ovarian cancer by somatic point-mutations, many of which were fixed at integer multiplicity (Fig. 4b). Classification of point mutations based on their multiplicities may help distinguish tumor suppressors and oncogenes (Fig. 5d). Knowledge of discrete tumor copy-states, subclonal structure and genome doubling status provides a foundation for further reconstruction of the phylogenetic relationships within a cancer and the temporal sequence by which a given cancer genome arose44,45,46.

ABSOLUTE provides a tool for the design of studies using genomic sequencing to detect variant alleles in cancer tissue samples, based on calculation of sensitivity to detect mutations as a function of sample purity, local copy number and sequencing depth (Supplementary Fig. 7). The high accuracy of tumor purity and ploidy estimates produced by ABSOLUTE, based on SNP microarray data (Fig. 2), makes it possible to determine the sequencing depth required for a given sample or to select suitable samples given a fixed sequencing depth. Such considerations are vital to the interpretation of subclonal point-mutations (Supplementary Figs. 7f and 10).

Analysis of the predicted absolute allelic copy-number profiles across human cancers produced by ABSOLUTE shed new light on cancer genome evolution. The observed SCNA profiles (Supplementary Fig. 9) were consistent with a common trajectory consisting of an early period of chromosomal instability followed by the emergence of a stable aneuploid clone, as previously described11. Our data further indicate that genome doublings occur in a subset of cancer cells already harboring arm-level SCNAs characteristic of the corresponding tumor type. The genomes of these cancers were therefore shaped by selection at chromosomal arm-level resolution before doubling and further clonal outgrowth (Fig. 6d and Supplementary Fig. 10).

These findings are broadly consistent with an earlier interpretation of primary breast cancer FACS/SKY profiles47, and has recently been recapitulated in studies of macro-dissected and ploidy-sorted cell populations14, and single-cell sequencing15 of primary breast tumors. We note that this model represents a departure from the idea that tetraploidization is an initiating event13,26,47,48,49,50. In addition, the association of genome doubling with epithelial lineage (Fig. 6c) and with age at diagnosis in ovarian carcinoma (Fig. 7d) is consistent with a recently described mechanism linking telomere crisis, DNA damage response, and genome doubling in cultured mouse embryonic fibroblasts49.

The analysis of clonality presented in this work offers a path forward for clinical sequencing of cancer, and provides the means to address recently reported concerns regarding intratumor heterogeneity14,15,45,46,51,52,53. Analysis using ABSOLUTE can identify alterations present in all cancer cells contributing to the DNA aliquot (Fig. 1), even if such clonal alterations correspond to the minority of observed mutations. Such alterations are candidate founding oncogenic drivers for a given cancer, which may be the preferred therapeutic targets. Further characterization of subclonal somatic alterations in cancer may become important for understanding variable response to targeted therapeutics, with the clonality of targeted mutations potentially affecting response level.

## Methods

### Inference of purity, ploidy, and absolute somatic copy-numbers.

Homologue-specific copy ratios (HSCRs; copy-ratio estimates of both homologous chromosomes) are preferred for analysis with ABSOLUTE, and were used for all analyses in this study. Although ABSOLUTE can be run on total copy-ratio data (e.g. from array CGH or low-pass sequencing data), we do not present such results here. The use of HSCRs reduces the ambiguity of copy profiles. For example, the total copy-ratio profile of a sample without SCNAs would be equivalent for ploidy values of 1,2,3, etc., however the HSCR profile would rule out odd ploidy values, since these would not be consistent with equal homologous copy-numbers. In addition, since subclonal SCNAs will generally affect only one of the two HSCR values in a given genomic segment, the ratio of clonal to subclonal SCNAs genome-wide is generally higher when considering HSCRs rather than only total copy-numbers.

HSCRs were derived from segmental estimates of phased multipoint allelic copy-ratios at heterozygous loci using the program HAPSEG54 on data from Affymetrix SNP arrays. As part of this procedure, haplotype panels from population linkage analysis (HAPMAP3)55 were used in conjunction with statistical phasing software (BEAGLE)56 in order to estimate the phased germline genotypes at SNP markers in each cancer sample. This increased our sensitivity for resolving genotypes, since it naturally exploited the local statistical dependencies between SNPs54. In addition, this allowed greater resolution of small differences between homologous copy-ratios, since phase information from allelic imbalance of heterozygous markers due to SCNA could be combined with the statistical phasing from the haplotype panels54.

### Identification and evaluation of candidate tumor purity and ploidy values.

We describe identification of candidate tumor purity and ploidy values and calculation of their SCNA-fit log-likelihood scores using a probabilistic model. This is accomplished by fitting the input HSCR estimates with a Gaussian mixture model, with components centered at the discrete concentration-ratios implied by equation (1). The model also supports a moderate fraction of subclonal events which are not restricted to the discrete levels. Candidate solutions are identified by searching for local optima of this likelihood over a large range of purity and ploidy values. This results in a discrete set of candidate solutions with corresponding SCNA-fit likelihoods (equation (1), Fig. 1e–g, Supplementary Fig. 1c–e).

The SCNA-fit scores quantify the evidence for each solution contributed by explanation of the observed HSCRs as integer SCNAs. These computations are independent for each sample. The input data consist of N HSCRs xi, i {1,..., N}. Each of these is observed with standard error σi, and corresponds to a genomic fraction denoted wi. Each of the xi is assumed to have arisen from either one of Q integer copy-number states: Q = {0,1,...,Q − 1}, or an additional state Z corresponding to subclonal copy-number. We refer to the collection of possible copy-states as S = Q Z. We define Q + 1 indicators s for the copy-state of each segment, with p(si) representing the probability of segment i having been generated from state sS. The integer copy-states of S are indexed by q Q; the non-integer state is denoted by z.

The expected copy-ratio corresponding to each integer copy-number q(x) in a tumor sample is given by equation (1). Note that when homologous copy-ratios are used, this equation becomes:

since HSCRs are measured relative to haploid concentrations, as opposed to the diploid values assumed by equation 1. D is related to tumor purity and ploidy (α and τ) (equation (1), Fig. 1). The observed xi are modeled with a mixture of Q Gaussian components located at μ = {μq Q} representing integer copy-states Q and an additional uniform component Z. The mixture Z allows segments to be assigned non-integer copy values so that subclonal alterations or artifacts do not dramatically impact the likelihood.

and denote the normal and uniform densities, respectively. The free parameter σH represents sample-level noise in excess of the HSCR standard-error σi, which might represent a moderate number of related clones in the malignant cell population, ongoing genomic instability, or excessive noise due to variable experimental conditions. The mixture weights θ = {θsS} specify the expected genomic fraction allocated to each copy-state. The parameter d represents the domain of the uniform density, corresponding to the range of plausible copy-ratio values (we used d = 7).

Some complication arises due to the fact that the data consist of copy-ratios calculated from a segmentation of the genome. For consistent interpretation, the mixture weights P(si|wi, θ) must be calculated for each segment separately, taking into account the variable genomic fraction wi. This is accomplished by constraining the canonical averages of genomic mass allocated to each copy-state to match those specified by θ:

where denotes the average over all configurations {si}, weighted by the function This density corresponds to the maximum entropy distribution over s subject to these constraints:

where s# indicates the order of state s in a sequence of copy-states, beginning with 0. Values of the Q Lagrange multipliers λ are determined via Nelder-Mead optimization of L2 loss:

This approximation allows for robustness of the SCNA-fit score to over-segmentation of the data. The likelihood of a given segment i is then calculated as:

and the full log-likelihood of the data is then:

We define the parameterization

which determines μ via equation (3). Candidate purity and ploidy solutions for a tumor sample are identified by optimization of equation (5) with respect to b and δτ. Calculation of equation (5) requires estimates of θ and σH, which are not known a priori. We make an approximation (scale-separation) assuming that locations of the modes of equation (5) are invariant to moderate fluctuations in these parameters. A provisional likelihood for each xi may then be calculated by

Candidate purity and ploidy solutions are then identified by optimization of

initiated from all points in a regular lattice spanning the domain of b and δτ. The parameter σP was set to 0.01 for this study. We verified that the above approximation identified modes equivalent to those obtained through a full Metropolis-Hastings Markov chain Monte Carlo (MCMC) simulation (data not shown). The approximation allows for much simpler computations to be used.

The SCNA-fit score for each solution is calculated after optimization of σH:

with the elements of θ calculated for each value of σH by:

Note that each i is a vector representing the posterior probability of each Q Q integer copy-states, corresponding to the copy-ratios (locations) μ.

Genome-wide absolute copy-profiles are over-determined with respect to DNA ploidy estimates. An alternate estimate of ploidy may be calculated as the expected absolute copy-number over the genome:

By definition, this quantity (τ̂g) is an alternate estimate of cancer ploidy (note an additional factor of 2 is added when HSCRs are used). Because τ̂g is a weighted average over discrete states in the modeled data, it is expected to be more robust to experimental fluctuations that shift or scale the copy-profile slightly. Note that, for this computation, the ij were calculated with so that the above expectation is over integer states only.

We verified that these estimates were generally close to the values of τ̂ obtained by optimization of the SCNA-fit likelihood (RMSE = 0.26, Supplementary Fig. 11a). However, we noted a relationship between the level of discordance between ploidy estimators and the mean of the calibrated data (Supplementary Fig. 11b). Noting that correctly calibrated copy-ratio data always has mean = 1, we examined whether the miscalibration was due to scaling bias in the data. We found that this model explained nearly two thirds of the discordance between the two estimates (corrected RMSE = 0.09, Supplementary Fig. 11c), by which we inferred that scaling biases dominated our miscalibration. This is significant, as such biases do not effect the estimation of tumor purity (Supplementary Fig. 11).

Two additional transformations of the copy-state locations μ are used when copy-ratios are measured using microarrays. The first of these accounts for the effect of attenuation with an isothermal adsorbtion model7:

where the value parameterizes the attenuation response in a given sample, and is estimated via HAPSEG. The second transformation is a variance stabilization for microarray data adapted from57:

where ση and σɛ represent multiplicative and additive noise scales for each microarray, estimated by HAPSEG. This transformation is applied to the marker-level data during estimation of the xi values, after which their distribution is approximately normal. The normal mixture component specified in (4) then becomes and the corresponding likelihood calculations are performed under these transformations.

### Karyotype models.

Additional information is generally required in order to reliably select the correct tumor purity and ploidy solution from the set of candidates identified by fitting the model in (4). In a given tumor sample, several combinations of theoretically possible purity, ploidy, and copy number values may map to equivalent copy ratios (Fig. 1e–g, Supplementary Fig. 1). Furthermore, the presence of subclonal SCNAs may result in a spuriously high ploidy solution with an implausible karyotype receiving a greater SCNA-fit likelihood by over-discretizing the copy profile, allowing their assignment to integer copy-levels (Supplementary Fig. 1c–e).

ABSOLUTE models common cancer karyotypes by grouping tumor sets according to similarities in their absolute homologous copy-number profiles (Supplementary Fig. 2). These models are constructed directly from the tumor data in a 'boot-strapping' fashion, whereby a subset of tumors with relatively unambiguous profiles (e.g., due to high purity values) is used initialize the models, iteratively allowing more tumors to be called, etc. Previous cytogenetic characterizations of human cancer were used to guide this process13. These models enable calculation of a karyotype likelihood, for each candidate purity/ploidy solution, reflecting the similarity of the corresponding karyotype to models associated with the specified disease of the input tumor sample (8). Integration of the SCNA-fit and karyotype likelihoods favors robust and unambiguous identification of the correct purity and ploidy values in many tumor samples (Fig. 1g, Supplementary Fig. 1e).The selection of a solution implying a less common karyotype requires greater evidence from the SCNA-fit of the copy profile.

Prior knowledge of karyotypes characteristic of a particular disease is summarized as a mixture of K multivariate multinomial distributions over the integer homologous copy-states Q = [0,7] of each chromosome arm. For a given candidate purity and ploidy solution, the corresponding segmental copy-state indicators for each segment i, ij, are summarized into estimates of the J arm-level homologous copy-numbers, denoted Ĉ. The karyotype log-likelihood score is calculated as:

where wi denotes the weight of each mixture component. The karyotype models Ki are J × Q SCNA probability matrices obtained by clustering arm-level homologous copy-states of modeled copy-profiles using the standard expectation-maximization (EM) algorithm58 for multinomial mixtures. This calculation identifies groups of disease subtypes with similar genomic copy profiles (Supplementary Fig. 2). Note that copy-states for both homologues of each arm are modeled (J = 78). Karyotype scores for samples with only total copy-ratio data are calculated using convolution of the multinomial probabilities for the two homologous chromosomes.

The number of clusters K for each disease was chosen by minimizing the Bayesian information criterion (BIC) complexity penalty: where indicates the sum of values over the N input samples, computed using K clusters. In order to avoid local minima, the EM algorithm was run 25 times for each value of K [2,8] with randomized starting points and the best model was retained.

The models were constructed in a semi-automated fashion by seeding with relatively unambiguous copy-profiles. As tumors were added, the use of recurrent karyotypes clearly identified the correct solutions of additional samples, etc. For example, LOH of chr17 occurs in nearly 100% of ovarian carcinoma samples34, allowing the model to learn that solutions implying LOH of chr17 are likely to be correct. In total, models for 14 disease types were created. Diseases with fewer than 40 samples called by ABSOLUTE were omitted from this procedure. In addition, a “master” model was created by combining called primary cancer profiles. This model was used for diseases with no specific karyotype model.

### Limitations of joint purity/ploidy inference from copy-profile data.

Accurate calibration of both the SCNA-fit and karyotype models to the true level of certainty implied by the data would allow for assignment of probabilities to each candidate solution; we do not believe that the models we have presented here sufficiently capture the complexity of cancer genomes to allow for such interpretations. Even with manual review, analysis with ABSOLUTE may occasionally result in incorrect interpretations, for example genome-doubling without subsequent detectable gains or losses would result in a solution implying half the true ploidy value, which in some cases may correspond to a plausible karyotype model. Alternately, when multiple subclonal SCNAs appear close to the midpoints between adjacent clonal peaks, a solution implying twice the true ploidy may be chosen. We note that samples with no reliably detected SCNAs could not be called in our framework (ploidy 2n or 4n; purity undetermined). Such samples were therefore excluded from downstream analysis (see below). Estimation of inference error-rate requires independent measurement of sample ploidy. Further validation experiments in diverse tumor types will help to clarify any disease specific caveats.

We note that the use of somatic mutation allelic-fractions, combined with the SCNA copy-ratios, generally allows for increased sensitivity for samples with few SCNAs. In addition, the mutation data helps distinguish genome-doubling ambiguity in purity/ploidy estimation, although it does not inform ambiguities of the type b′ = b + 2(1–α)/D (Fig. 1f, Supplementary Fig. 1d, equation (1)). Thus, combined analysis generally facilitates obtaining higher call-rates using ABSOLUTE (not shown).

Fortunately, many samples in our pan-cancer SNP array dataset were consistent with frequent SCNA both before and after genome doubling, enabling unambiguous inference for many samples without use of somatic point-mutation data. This aspect of cancer genome evolution was noted previously in breast cancer cytogenetic data47. We note that manual review of ABSOLUTE results was performed prior to generation of the FACS validation data or analysis of the NCI60 cell-line ploidy estimates (Fig. 2a,b).

### Identification of samples refractory to purity/ploidy inference.

In order to facilitate rapid analysis of many cancer samples used in this study, ABSOLUTE was programed to automatically identify copy profiles that cannot be reliably called and to classify them into informative failure categories (Fig. 3a), which were defined by the following criteria. Define as the sorted vector of posterior genome-wide copy-state allocations (), so that represents the greatest element of (the modal copy-state). This vector was constructed with θ0 replaced by 0 if θ0 <0.01 and b < 0.15, so that germline copy-number variants (CNVs) or regions of inherited homozygosity are not confused with small SCNAs implying very pure samples. The categories are then:

These criteria were applied to the top-ranked mode for each sample (combined SCNA-fit and karyotype scores). Several examples of each outcome are shown in Supplementary Figure 5. The above designations led to reasonably good concordance of automated calls with those obtained after manual review. We note that the use of somatic point-mutation data increases the calling sensitivity within these sample categories.

### Cancer cell-line DNA mixing experiment.

DNA extracted from two cancer cell lines (HCC38, HCC1143) was mixed with DNA from matched B-lymphocyte cell lines (HCC38BL, HCC1143BL) in various proportions, and hybridized to Affymetrix 250 K Sty SNP arrays. Stock DNA aliquots were created for each cell-line by normalization of DNA concentration to 50 ng/μl. Mixing of cancer and matched B-lymphocyte DNA to each required mixing fraction was done by volume.

### FACS analysis of primary tumor samples.

Formalin-fixed and paraffin-embedded blocks from ovarian serous carcinoma cases were available from tumor-sections corresponding to the frozen blocks from which DNA-aliquots were obtained for SNP-array hybridization. Multiple curls containing at least 70% tumor cell nuclei were cut to an aggregate thickness of 150 μm. Sections were disaggregated and labeled with propidium iodide (DNA stain). FACS was performed to determine ploidy.

### Determination of tumor purity via pathology review.

Frozen ovarian serous cystadenocarcinoma specimens were collected from multiple hospital tissue banks and maintained frozen in liquid nitrogen vapors. A tissue portion was created with two flanking H&E slides (arbitrarily named top and bottom) as follows: tissues were mounted in optimal cutting temperature media (OCT) and brought to −20 °C. A 4 μm frozen section (top slide) was cut with a cryostat (Leica Microsystems, Wetzlar, Germany). A specimen for molecular extraction was created by shaving 100 mg of tumor tissue from the tissue face with a scalpel, then a second 4 μm frozen section was cut (bottom slide). An H&E stain was conducted on both slide tissue sections using an Autostainer XL with integrated coverslipper (Leica). Digital images of slides were created at 20× resolution using a Scanscope XT (Aperio, Vista, CA, USA). Board-certified pathologists conducted the pathology review remotely via ImageScope software (Aperio). Pathologists initially reviewed each slide at low magnification to determine low power microscopic morphology, then increased magnification to 20× and reviewed 10 representative high power fields on each slide. Diagnosis of ovarian serous cystadenocarcinoma was verified, and tumor purity was determined as the proportion of tumor nuclei present compared to the total nuclei present on the slide. The tumor purity of the extracted specimen was calculated as the average purity score of the top and bottom slides. Quality control included a random review of 10% of slides by a second pathologist to verify consistency of reads.

### Leukocyte methylation signature.

DNA methylation data for 489 high stage, high grade serous ovarian tumors and eight normal fallopian tube samples was obtained from http://tcga.cancer.gov/dataportal/. In addition, buffy coat samples from two female individuals were obtained. All data were generated with Illumina Infinium HumanMethylation27 arrays, which interrogate 27,578 CpG sites located in proximity to the transcription start sites of 14,475 consensus coding sequencing in the NCBI Database (Genome Build 36). The level of DNA methylation at each probe was summarized with beta values ranging from 0 (unmethylated) to 1 (methylated)59.

The leukocyte methylation signature was derived as follows. Each probe was ranked by the difference in mean beta value in buffy coat and fallopian tube samples. We retained the 100 probes with the largest positive difference and the 100 with the largest negative difference between mean DNA methylation in normal fallopian tube tissues and peripheral blood leukocytes, designated BC and FT (buffy coat and fallopian tube enriched, respectively). Let Tik denote the beta value for probe k in tumor sample i. Let Bk denote the average beta value of buffy coat samples for each probe. Let Tk denote the minimum observed beta value across all tumor samples for the BC probes and the maximum for the FT probes. Denote by fB the fraction of buffy coat components in the sample, then we have the following equation for each probe: Solving this equation for fB The values of fB for each of the 200 probes in the signature were calculated and a kernel density estimate was obtained. The leukocyte signature was then calculated as the mode of this density estimate.

### Selection of data sets.

We analyzed 2,445 Affymetrix 250 K Sty SNP samples from a previous pan cancer survey36 containing 3,131 cancer samples. Because our processing of the data required use of the Birdseed algorithm60, external data sets lacking diploid PCR controls could not be used. In addition, cancer types with fewer than 20 samples were excluded. In addition, 680 Affymetrix SNP6.0 samples were taken from the TCGA GBM21 and HGS-OvCa34 studies, as well as 30 cell-line samples, bringing total sample count to 3,155. The complete table of cancer samples analyzed is available as Supplementary Table 1. The complete table of ABSOLUTE results is available as Supplementary Tables 1 and 2.

### Power calculation for somatic mutation detection in cancer tissue samples.

We develop a framework for calculation of statistical power for the detection of mutations. Power to detect a variant depends on the allelic fraction f and local depth of coverage n. To calculate power, we model the idealized scenario in which random sequencing errors occur uniformly with rate . We calculate a minimum number of supporting reads k such that the probability of observing k or more identical non-reference reads due to sequencing error is less than a defined false-positive rate (FPR):

where

Variants with ≥k supporting reads are then considered detected. We specified the sequencing error rate = 1 × 10−3 and FPR = 5 × 10−7 for all computations in this study. Power is then calculated as: where

We consider the case of detecting clonal somatic variants present at a single copy per cancer cell in cancer-tissue derived DNA samples. Given estimates of purity (α) and local absolute copy-number (qt), the allelic fraction of such variants is:

Power is calculated in such cases as Pow(n,δ).

In order to simplify the relationship between power and tumor purity/ploidy for presentation in Supplementary Figure 7, we considered the detection power of the expected locus, over the genome-wide copy average. Power as such is determined by the sample allelic index δτ = α/D, which is solely a function of tumor purity/ploidy (equation 1). Expected power is obtained by using allelic fraction f = δτ in equation (9). This calculation differs only in the substitution of expected genomic copy-number, i.e. ploidy (τ), for the local copy-number qt in equation (10).

Power for expected subclonal variants present in fraction sf of cancer cells is given by Pow(n,sfδτ). This calculation was used for Supplementary Figure 7c,e. Local copy calculations using Pow(n,sfδ) were used for Supplementary Figures 7f and 12.

### Detection of somatic point mutations in ovarian carcinoma.

We analyzed whole-exome hybrid capture Illumina sequencing (WES)37 data from 214 ovarian carcinoma tumor-normal pairs previously analyzed by the TCGA consortium34. We used the program muTect (K. Cibulskis et al., in preparation) We have used a newer version of the program muTect than used in previous analysis of this data34. The primary improvement in the new version is a reduction in the prior that somatic mutations be at an allelic fraction of 0.5, allowing greater sensitivity at low allelic-fraction mutations, such as clonal events in impure samples, or to subclonal mutations. This procedure resulted in 29,268 somatic mutations.

### Inference of point mutation multiplicity.

We develop a probabilistic model for inference of the integer multiplicities for both germline and somatic variants, based on knowledge of tumor purity and genome-wide absolute copy-numbers. Denote the absolute homologous copy-numbers at a mutant locus as q1 and q2, with q1q2. The possible multiplicities of germline variants are then:

where qt = q1 + q2. Under the assumption that all somatic point-mutations arise uniquely on a single haplotype, the possible multiplicities are:

Note that when only total copy-ratio data are available, q2 above is unknown, and qt is used instead.

Germline mutations are generally present in both the cancer and normal cell populations, with somatic copy-number alterations affecting the allelic fraction. A heterozygous variant in the germline, with multiplicity gq in the cancer genome, has allelic fraction:

whereas the allelic fraction of homozygous germline variants is 1 regardless of α. For somatic point mutations, the expected allelic fraction at multiplicity sq is fsq = sqδ, with δ as in equation (10).

Consider an observed somatic point-mutation of unknown copy sq sq, observed allelic fraction , and with n total reads covering the locus. The complete likelihood of may be represented as a mixture of Beta distributions corresponding to each element of sq, plus an additional component S corresponding to subclonal states:

where specify mixture weights for each state in sq and $w s c$ specifies the subclonal component weight. The subclonal component S is specified by composing a Beta distribution (modeling sampling noise) with an exponential distribution over subclonal cancer-cell fractions, having a single parameter λ:

Note the change of coordinates in the exponential component using δ; this allows modeling in consistent units of cancer-cell fractions, regardless of tumor purity and local copy-number (note this distribution is renormalized on the unit interval). The probability of a given integer copy-state sq may then be calculated as:

Similarly, the probability that a given mutation is subclonal is calculated as:

For the computations in this study, we fixed λ = 25, $w s q$ to 0.25, and $w s c$ to 0.75, which produced a fit to the combined-sample mutation-fraction distribution (Fig. 4b). The results presented in Figure 4 were robust to various settings.

Optimization of mixture components weights corresponding to integer somatic multiplicities may be accomplished in a manner similar to that described for the SNCA mixture model in equation (6). A Dirichlet prior may be specified as a vector of pseudo-counts equivalent to prior observations of each multiplicity value. Weights are then calculated as the mode of the posterior Dirichlet calculated from the observed counts. These computations are used to calculate a mutation-score likelihood for each purity ploidy mode when ABSOLUTE is run with paired SCNA and somatic point mutation data.

### Simulation of cancer-genome evolution to support genome-doubling inferences.

A simple simulation was performed to obtain P-values for the probability that an observed configuration of homologous copy-numbers could be produced from a serial process of independent gains and losses. Genome-wide homologous copy-numbers are summarized at chromosome-arm resolution as integer gains/losses (total of 78 states). We then fix the total number of gains/losses N for the sample, and calculate rates for each arm, which are normalized to probabilities. Simulation of the sample is performed by independently sampling N gains and losses from these probabilities. This was repeated 1,000 times for each sample, keeping track of the number of times M that the extent of even high homologous copy-number present in the observed sample was attained or exceeded. The P-value is then: if M > 0, otherwise P < 0.0001.

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## Acknowledgements

This work was supported by grants from the National Cancer Institute as part of The Cancer Genome Atlas project: U24CA126546 (M.M.), U24CA143867 (M.M.), and U24CA143845 (G.G.). S.L.C. and E.H. were supported by training grant T32 HG002295 from the National Human Genome Research Institute. In addition, D.P. was supported by the US National Institutes of Health (NIH) - 5R01 GM083299-14; D.A.L. by Department of Defense W81XWH-10-1-0222; T.Z. by NIH/National Institute of General Medical Sciences 5 T32 GM008313; R.B. by NIH K08CA122833 and U54CA143798. B.A.W. was supported by National Research Service Award grant F32CA113126.

## Author information

M.M. posed the concept of using allelic copy number analysis to assess tumor genomic purity. G.G. and S.L.C. conceived and designed the analysis. S.L.C. designed and implemented the ABSOLUTE algorithm, and performed data analysis. K.C. assisted with mutation calling and validation. E.H. performed analysis with ASCAT and assisted with multiplicity analysis. A.M. assisted with the determination of statistical power for the ovarian cancer data. T.Z. assisted with analysis of the SCNA length distribution versus genome doubling. R.C.O. and W.W. processed the SNP array experiments. W.W. designed and executed the DNA mixing experiment. H.S. and P.W.L. conceived and executed the leukocyte methylation signature analysis. B.A.W. contributed insight into allelic analysis of SNP array data. D.A.L. managed the FACS analysis of specimens for ploidy determination. All authors contributed to the final manuscript. D.P. and R.B. provided critical review of the manuscript. S.L.C., M.M., G.G. and E.S.L. wrote the manuscript. R.B., M.M. and G.G. provided leadership for the project.

Correspondence to Scott L Carter or Gad Getz.

## Ethics declarations

### Competing interests

M.M. is an equity holder in and consultant for Foundation Medicine, and is a consultant for, and receives research support from, Novartis.

## Supplementary information

### Supplementary Text and Figures

Supplementary Figures 1–12 and Supplementary Note (PDF 3420 kb)

### Supplementary Table 1

Pan-cancer analysis: sample characteristics and predicted purity / ploidy values (TXT 658 kb)

### Supplementary Table 2

Pan-cancer analysis: segmented absolute allelic copy numbers (TXT 27073 kb)

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Carter, S., Cibulskis, K., Helman, E. et al. Absolute quantification of somatic DNA alterations in human cancer. Nat Biotechnol 30, 413–421 (2012). https://doi.org/10.1038/nbt.2203

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