We further demonstrate how using several NMF components enables accurate cancer detection on data from various early stage cancers (AUC = 0.96). Finally, we show that NMF, when applied across genomic regions, can be used to discover fragment length signatures associated with open chromatin.Sequencing of cell-free DNA (cfDNA) is currently being used to detect cancer by searching both for mutational and non-mutational alterations. Recent work has shown that the length distribution of cfDNA fragments from a cancer patient can inform tumor load and type. Here, we propose non-negative matrix factorization (NMF) of fragment length distributions as a novel and completely unsupervised method for studying fragment length patterns in cfDNA. Using shallow whole-genome sequencing (sWGS) of cfDNA from a cohort of patients with metastatic castration-resistant prostate cancer (mCRPC), we demonstrate how NMF accurately infers the true tumor fragment length distribution as an NMF component - and that the sample weights of this component correlate with ctDNA levels ( Circulating cell-free DNA (cfDNA) is rapidly emerging as an important biomarker – most notably in cancer and pregnancy. In the cancer setting, the detection of tumor cell derived DNA fragments containing somatically acquired mutations can reveal the presence of cancer. Most approaches rely on the detection of mutations using deep, targeted sequencing of a few genomic regions known to harbor driver mutations for the cancer type of interest . While dIt is, however, possible to get more information out of cfDNA data than just genetic variants. A key difference between cfDNA data and ordinary sequence data is that cfDNA is fragmented in vivo by a combination of enzymatic and non-enzymatic processes. Most importantly, during apoptosis, DNA is enzymatically cut between nucleosomes, and hence the lengths and positions of cfDNA fragments reflect the epigenetic state of the cell-types of origin . FurtherThe fragment length distribution has been a major focus of studies searching for non-mutational signals in cfDNA. Early studies used quantitative PCR with competitive primer sets targeting amplicons of different lengths to study the cfDNA fragment length distribution in cancer using either human-mouse xenografts or blood cfDNA representing different clinical states . Yet, whA more detailed picture of how cancer manifests in the fragment length distribution of cfDNA was later obtained using short-read sequencing based approaches. First, Jiang et al. used sequencing to determine that hepatocellular carcinoma is associated with a left shift in the cfDNA fragment length distribution . Later oDELFI approach, Cristiano et al. added a genomic dimension to cfDNA fragment length analyses by computing the ratio of short (100–150 bp) to long (151–220 bp) fragments in 5 MB windows along the genome and showed how these capture nucleosomal distances, which in turn reflect chromatin state (In their in state . Using tswh:1:rev:cf9ed4240b74c866f62b3da2cdb4f0bbceb7f551).In this manuscript, we introduce a new computational method based on non-negative matrix factorization (NMF), an unsupervised learning method, for simultaneously determining the contributions of different cfDNA sources (e.g. background and tumor) to a sample along with the fragment length signatures of each source . The metsignature matrix, has as many columns as the original matrix and represents the preference of observing each fragment length for each cfDNA source. The other matrix, the weight matrix, has as many rows as the input matrix and represents the contributions of each cfDNA source to each sample. The number of cfDNA sources is a hyperparameter that needs to be set in advance.Our approach begins by computing cfDNA fragment length histograms in a series of samples based on paired-end read alignments and then uses these to construct a matrix with cfDNA fragment counts such that each row corresponds to a sample and each column to a specific fragment length (see n=93) or at disease progression after first line treatment (n=34) or later treatments (n=15) with some individuals (n=36) sampled at multiple timepoints. The observed fragment length distributions of high and low ctDNA samples differed  in 142 plasma samples from 94 patients with metastatic castration resistant prostate cancer (mCRPC). Samples were collected either before the initiation of first line treatment (differed  and resediffered . Assumindiffered .To confirm the ability of NMF to separate tumor and background (i.e. non-tumor) sources, we performed targeted, deep sequencing  for a subset of 86 mCRPC patients using a panel of genes related to prostate cancer  . We thenr=0.75, r=0.83, We next sought to investigate whether the estimated tumor signature weights are related to the blood ctDNA fraction. We, therefore, compared the sample weights of the tumor fragment length signature against sample ctDNA fractions estimated based on driver variant allele fractions (VAFs) obtained from the targeted sequencing data  as well as estimates obtained using ichorCNA , which uses CNVs . The estWe also explored using more cfDNA sources in the NMF. More specifically, we ran NMF for up to four components and empirically tested adding the weights of different combinations of NMF components (i.e. assuming different tumor sources) by comparing them with the driver VAF-based estimates . Only mar=0.68, r=0.79, The performance of our method was superior to using the ratio of short-to-long fragments (100–150 bp vs. 151–220 bp) proposed by Cristiano et al. and analogous to the PCR-based methods as judged against driver VAF-based ctDNA fraction estimates r=0.68, . ichorCNTo see if fragment length signatures can detect the presence of cancer fragments from different cancer types, we reanalyzed the data from the DELFI study . We obtaWe then tested whether using more signatures would improve this classification. We used a linear Support Vector Machine (SVM) to see if we could separate cancer and control samples based on their signature weights for a given number of signatures. The results showed that adding more signatures significantly improved the AUC see . The clabin-wise NMF signatures shown in The DELFI method also uses fragment length information to detect cancer samples, but rather than looking at the total length distribution, it looks at fragment lengths in bins along the genome . The DELWe then wanted to test whether using the NMF weight in each bin as input features for a classification method would be superior to using the DELFI ratios. We estimated the first bin-wise signature’s weight in each bin in each sample and used these weights as input to an SVM and assessed classification performance across different bin sizes. The results in In this article, we propose non-negative matrix factorization (NMF) of fragment length distributions as a new tool in the cfDNA-seq analysis toolbox. A key objective in cfDNA-seq analyses in the cancer setting is to determine the amount of circulating tumor DNA – most importantly whether any tumor DNA is present at all. This carries immense importance both for early detection of cancer in the screening setting – and for detection of relapse after treatment, due to residual disease after surgery or resistance to chemotherapy.Most analyses have focused on using mutational signals for determining ctDNA load through estimation of variant allele fractions for SNVs in deep, targeted cfDNA-seq data  – or CNVsignatures and their weights in each sample. Using sWGS cfDNA-seq data from a cohort of patients with metastatic prostate cancer, we show that NMF estimated with two components discovers a signature, which accurately matches the true tumor fragment length distribution and exhibits many of the characteristics previously associated with ctDNA. The weights of this signature correlated strongly with ctDNA levels – nearly as good as ichorCNA - without using any information about variants or ctDNA levels. Importantly, similar results were obtained when using deep, targeted cfDNA-seq. Furthermore, subsampling experiments revealed that NMF was markedly more robust when less data is available than ichorCNA. This may indicate that NMF is more robust than ichorCNA at lower tumor fractions. However, the lack of low ctDNA samples is a limitation of the current study and further experiments either based on samples with lower tumor burdens, spike-ins or in silico dilution are needed to confirm this assertion as it could not be directly tested with the data available in this study. A lack of healthy controls for the mCRPC cohort data meant that we were not able to perform spike-in or in silico dilution experiments in this study. Finally, as the fragment length signal is likely orthogonal to any mutational signal, it may be possible to combine these lines of information to obtain a better, joint estimate of tumor load.We propose NMF as a general analytical framework for working with cfDNA-seq fragment length distributions. NMF enables us to simultaneously estimate fragment length The unsupervised nature of NMF implies little risk of overfitting and the ability to inspect the fragment length profiles of each signature provides full transparency of the method in contrast to many other approaches such as the supervised learning by Random Forest strategy applied by Mouliere et al. for determining ctDNA levels. Transparency is important because using non-mutational information carries the risk of using information that is not directly linked to the presence of ctDNA in blood, but instead reflects, e.g. an ongoing immune response. This in turn may impact a models’ ability to generalize to unseen data and clinicians’ trust in the model. Using NMF, we were able to verify that the fragment length profile of the signature correlating with ctDNA levels does indeed match the length distribution of fragments containing mutations and hence that the model is directly measuring ctDNA load. Hence, the combination of unsupervised learning and transparency suggest that NMF constitutes a robust modeling framework for cfDNA-seq length spectra.The data from the prostate cancer cohort generally contained patients with a high tumor burden and we wished to also test the models applicability in a screening context characterized by low ctDNA load and also look at different cancer types. We obtained access to the data from the DELFI study, which contains cfDNA-seq data from a range of cancer types and primarily from patients with early stage disease . The sigWe note that a number of technical and biological factors in the DELFI study design may have inflated our classification performance. For instance, cases were generally older than controls and hence the observed differences in the fragment length distribution between cases and controls may partially reflect underlying comorbidities associated with higher age.On the DELFI dataset, we furthermore demonstrated that using several NMF components enabled accurate cancer detection of early stage cancers – on par with the original DELFI results. We obtained the best classification results by using 30 or more signatures, and an even larger number of signatures could likely be relevant for larger or more heterogeneous datasets. Indeed, we speculate that the difference in gain from using more signatures between the mCRPC cohort, where two signatures worked well, to the DELFI study reflects a larger degree of heterogeneity in the multi-cancer DELFI study. Even when more components are required, using a supervised model with tens of parameters rather than hundreds as in the genomic window model makes overfitting less likely.bin-wise training). The learned chromatin state signatures turned out to correlate better with open chromatin as measured using ATACseq than the DELFI ratio, but did not yield a clear improvement in classification performance over the DELFI method. We speculated that the lack of classification improvement could be due changes between cases and controls not related to chromatin status. We therefore investigated whether the bin-based approach could be improved by instead inferring the signature weights of the sample-wise trained NMF model in each genomic bin. To our surprise, this model outperformed both the bin-wise NMF and the DELFI ratio across all bins sizes, which may indicate that the DELFI classification signal is not purely a chromatin state signal but in part caused by CNVs in the tumors or other cancer-specific distorsions.Finally, we investigated whether the NMF approach could improve upon the genomic bin-based approach proposed by Cristiano et al. We first showed how NMF can discover fragment length signatures of different chromatin states when trained across genomic bins, where the fragment length histogram in each bin has been aggregated across multiple samples  or at disease progression after first line treatment (n=34) or later treatments  with 36 patients having multiple samples taken .A total of 142 EDTA-blood samples were collected from 94 patients with metastatic castration-resistant prostate cancer at Aarhus University Hospital and Regional Hospital of West Jutland between April 2016 and August 2019. Samples were collected either before the initiation of first line treatment (2 EDTA tubes (Beckton Dickinson) and processed within 2 hr (stored at 4°C until processing). To separate plasma from cellular components, EDTA blood samples were centrifuged at 2000–3000 g for 10 min (20°C) and plasma stored in cryo tubes (TPP) at –80°C until cfDNA extraction. Plasma samples (2.0–4.5 ml) were thawed at room temperature and centrifuged at 3000 g for 10 min (20°C). cfDNA was extracted on a QIAsymphony robot (Qiagen) using the QIAamp Circulating Nucleic Acids kit (Qiagen) as described by the manufacturer. Extracted cfDNA was stored in LoBind tubes (Eppendorf AG) at –80°C until further analysis (<1 month). cfDNA concentration was determined by droplet digital PCR (ddPCR) using a QX200 AutoDG Droplet Digital PCR System (Bio-Rad) according to the manufacturer’s instructions as previously described (Blood samples were collected in BD Vacutainer Kescribed . Germlinhttp://www.bioinformatics.babraham.ac.uk/projects).Libraries for next generation sequencing were prepared using the Kapa Hyper Library Preparation Kit (KAPA Biosystems) with 7.0–50.0 ng cfDNA (median 30.5 ng) or 50 ng for germline DNA as input. xGen CS-adapters – Tech Access (IDT-DNA) with Unique Molecular Identifiers (UMIs) on both strands and primers containing unique indexes on both strands were used to generate indexed libraries. Plasma libraries were pooled equimolarly and paired-end sequenced (2 × 151 bp) on an Illumina Novaseq instrument (S-prime flowcell), generating 12.66–34.30 million read pairs/ sample (median: 19.99) corresponding to coverages from 0.36 to 0.93X (median: 0.60X). Fastq files were demultiplexed using bcl2fastq (v2.20.0.422) and quality checked using fastQC and fastqScreen  with random initialization, multiplicative updates and the Kullback–Leibler loss function. For the bin-wise NMF experiments, the NMF was repeated across 20 different random initializations and the fit achieving the lowest loss selected. The estimated fragment length signature and weight matrices were scaled to sum to one for each signature and sample, respectively. Software for calculating fragment lengths and applying NMF is available under the MIT license .https://github.com/grenaud/libbam/blob/master/subsamplebamFixedNumberSingleFirstMate.cpp; swh:1:rev:c210c1ce1d4f0e1fe07fcbb4a438f9a7a2e3cf9b). Briefly, this program subsamples a specific number of fragments while taking into account paired-end information. The subsampled bam files were then used as input for NMF and ichorCNA and for each subsampling level, we calculated the average Pearson correlation between the NMF/ichorCNA tumor fraction estimates and the VAF based tumor fraction estimate.To determine the sequence coverage required to estimate tumor fraction using NMF  we subsaTo determine the number of samples required to reliably estimate fragment length signatures , we repeTo determine accuracy on hold-out samples , we repeWe first divided the genome into bins of 250 kb and estimated the fragment length distributions for each sample in each bin. We then calculated the mean number fragments in each bin across the controls. To avoid bins with possible mapping problems, we excluded bins where the mean number of fragments in the controls were less than the median-2*IQR or greater than the median + 2*IQR. Comparison with ATACseq data was performed using ENCODE track ‘ENCFF603BJO’  lifted from hg38 to hg19.The classification results based on multiple fragment length signatures and the classification results on chromatin state signatures and DELFI ratios were calculated using linear SVMs implemented in R using tidymodels and kernlab. Before training, the values for each sample was standardized by subtracting the mean and dividing with the standard deviation. Accuracy and AUC were calculated using repeated 10-fold cross-validation (50 repeats). Scripts used for the analysis of the DELFI data are available in A total of 86 indexed libraries were subjected to deep targeted sequencing based on ichorCNA ctDNA% estimates from plasma cfDNA libraries. A previously designed gene panel that captures regions in the human genome commonly altered in PC was used for capture . LibrariSomatic variants were called using VarDict (v. 1.6) , StrelkaAdapter sequences were trimmed using Cutadapt (v1.16)  and pairCtDNA fractions were also estimated from the targeted sequencing data. First, the tumor cell purity (i.e. tumor cell fraction) was calculated from somatic mutations with moderate or high impact. In brief, for each sample the somatic mutation with the highest VAF was used (i.e. clear driver mutation). If multiple mutations had similar VAFs (±2%), the median VAF of these mutations were used instead. A total of 1–2 mutations per sample were used to estimate purity. Accounting for LOH status of the mutation(s), the tumor cell purity was estimated as follows: purity = 2*VAF (no LOH) or purity = 2/(1/VAF + 1) (LOH). Next, to obtain ctDNA fractions, the tumor cell purity was adjusted for tumor ploidy: ctDNA fraction = tumor cell purity * tumor ploidy/. Here, normal purity was set to 1-tumor cell purity and normal ploidy was set to 2. Tumor ploidy estimates were obtained from PureCN . The authors introduce non-negative matrix factorization to analyze shallow WGS sequencing data to detect cell-free DNA fragments diagnostic of cancer. This is an area of active research and the authors add a potentially very useful unsupervised approach to analyze such data. public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.Our editorial process produces two outputs: (i) Decision letter after peer review:eLife. Your article has been reviewed by 3 peer reviewers, and the evaluation has been overseen by Y M Dennis Lo as the Senior/Reviewing Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Alain Thierry (Reviewer #1).Thank you for submitting your article "Discovering fragment length signatures of circulating tumor DNA using Non-negative Matrix Factorization" for consideration by The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.Essential revisions:1. More information on the source of cancer patient plasma, characteristics of the patients, especially in respect to treatments at time of blood sampling, should be included. A flow chart summarising the overall results might be useful.2. The authors can consider citing some of the older work analyzing cfDNA fragment size distribution using PCR-based approaches.3. The authors showed exactly the same illustrative fragment size profile in Figure 1  as those presented earlier by Sanchez et al. . Consequently, they should at least mention in a few sentences the observations made by Jiang et al. and by Sanchez et al.  in order to validate their assumption.4. As one could argue that the signatures of cfDNA fragment length had been discovered through various previously published work, in light of the previous concerns, term 'discovery' is perhaps inappropriate. Hence, we would suggest replacing the word 'discovery' by another term such as 'detection' or 'assessment'.5. In the introduction, it would be better to describe a bit more details about tumor DNA length characteristics that were reported previously, so that readers would have a more comprehensive understanding of what is already known in the prior art and why the analysis of tumor-specific length profile in this study is important.6. The authors should compare whether an NMF component would give a better correlation with tumor DNA estimated by ichorCNA, compared with the short DNA proportion (i.e.% of plasma DNA < 150 bp) in the overall size profile.7. It seems somewhat intriguing that the classification power between cases and controls improved as the number of components used in the NMF process. However, the authors did not see an improvement in the correlation between tumor DNA fraction and the weight of one component reflecting the tumor contribution. These observations should be discussed and clarified in a revised manuscript.8. The AUC was calculated using repeated 10-fold cross-validation. It would be better to show the boxplot of AUC values for a given number of NMF components .9. It will be informative to estimate the minimum number of samples is required for performing an accurate NMF analysis.10. The authors claim their model is directly measuring ctDNA load. Since an untargeted (unsupervised) model to assess the tumor fraction for monitoring purposes would be of great value, it would be interesting to see if changing levels of ctDNA can be captured by changing weights of the putative cancer Signature 1. Since their mCRPC cohort included patients with follow-up samples (at therapy start and progression) it is not clear why this has not been tested.11. It should be mentioned that ichorCNA can only provide informative results at ctDNA fraction of 3% and higher. At higher tumor fractions the fragment signature signal might be relatively strong, explaining the good correlation, but what about samples with lower fractions? Please indicate the tumor fractions  assessed with ichorCNA and based on the VAFs of the identified mutations for all patients.12. The authors claim that using NMF they were able to verify that the fragment length profile of the signature correlating with ctDNA levels does indeed match the length distribution of fragments containing mutations. However, although they included PBMCs in their mutation analysis the authors did not correct for clonal hematopoiesis (CH). Recent studies showed that up to 20% of identified mutation are related to CH and that such mutations are also found in driver genes for solid tumors. Since there might be a size difference of mutated fragment originating from the tumor or PBMCs, respectively, this statement is overrated. In case their sequencing approach was not sensitive enough to detected CH-related variants this should at least be mentioned.13. In Figure 2c, the shift towards smaller fragment is only seen for the dinucelosomes peak but not for the mononucleosomal fragments. How do the authors explain this (if not by contamination of CH-related variants)?14. The authors demonstrated that NMF was more robust when less data was available than the ichorCNA algorithm, but they lacked to provide convincing data that NMF as a quantitative measure is more sensitive with respect to tumor fraction. To this end, in silico dilution experiments where they computationally mixed healthy fragments and high ctDNA fraction sample(s) to a ctDNA fraction of their choice (with repetitions to achieve a more robust es-timation) should be performed.15. If there are no CNA or tumor fraction >3%, ichorCNA can literally not be used to determine ctDNA fraction, so the feasibility of the dilution approach to benchmark ichorCNA is not clear to the reviewer . Moreover, low depth can be overcome by additional sequencing whereas the tumor fraction cannot be enriched by more reads.16. Using NMF the authors demonstrate that there are 2 major contributors to fragment length distribution and the addition of further signatures does not add anything. However, this statement is conflicting with results of the linear SVM classification case/control approach . A plateau of classification accuracy is reached when using around 30 signatures, which suggests this would be a good trade-off for number of signatures to reach high accuracy (AUC).17. Figure 3d: Interestingly, open chromatin is associated with a higher fraction of dinucleosomal fragments. How do the authors explain this? Figure 3: the reviewer is not sure what the authors want to show here with Pearson's correlation coefficient, this does not look like a linear relationship.18. The paragraph on "epigenetic fragment length" could be better presented. I would re-phrase "epigenetic state". Both the DELFI ratio and the fragment length signature are surrogates for the chromatin state (open or closed). First the author talks about bin-wise extraction of two signatures. Why is it, that reducing bin size gives better classification results? ~50 kb bins seem to be best to predict ATACseq results. Is there literature that describes open/closed chromatin in this way? Then they correlate the weights from that with ATACseq results. And finally, they use their bin-wise signature extraction results/weights as input for cancer/control classification. Could be split in two parts for clarity: one that talks about correlation with ATACseq results and one that discusses classification results.19. cfDNA fragment length profiles are particularly prone to pre-analytical  and analytical (clustering efficiency of various sequencings machines) factors. How do preanalyitcal/analytical factors affect the classification? The authors mention in the discussion that changes in the size profiles from their mCRPC cohort and the DELFI samples vary, which might be attributed due to such factors. Yet such confounding event will impact the robustness of such approaches and may prevent a widespread implementation of fragmentomics. This should be further discussed.20. How can the authors exclude that the above-mentioned differences are attributed to higher/lower ctDNA fractions in the metastatic CRPC and the early stage DELFI samples? Even in Figure 2a it seems that samples with higher tumor fraction have more dinucleosomal fragments. Separate fragmentation profile plots for each stage should be shown; it would be interesting whether such profiles show any differences for stage I and stage II patients.21. The authors could have checked in the DELFI cohort whether there is a (physiologic fragment length variation) between the seven cancers types? Would there always be the same two signatures coming up if the only consider one tumor type? Based on Supplementary Figure 4 this seems not to be case. In addition, there is a huge overlap of healthy individuals and cancer patients for signature 1? Please also present Supplementary Figure 4 with Signature 2. Can we really assume that this signature is the one related to cancer?22. Abbreviations should be defined at the beginning for better readability.23. Resolution of figures need to be improved. Comprehensive legends should also be included for Supplementary figures.24. Matching colors for better readability are recommended.25. Specify average read pairs for cfDNA samples and corresponding PBMCs.26. Excel Supplementary file: Supplementary table 2 should be mentioned in the manuscript.27. ctDNA (%) on the x-axis in Figures 2 is calculated from targeted mutation analysis, right? How was est.ctDNA % in the Supplementary figures 2&3 calculated?28. What is the original h matrix in NMF in Figure 2f? There is a normal h matrix described as weight matrix, but I haven't seen the definition of what with "original" h matrix is meant.29. Assumptions for linear correlation using Pearson are not mentioned, I don't know whether they are met at all. Looking at the figure it does not seem like they are met. Normality of the variables should at least be checked.30. How is NMF performing on previously unseen data? This part is missing. The authors could have checked for overfitting in this way for example calculating the reconstruction error on a test dataset.31. It is not clear whether cross validation was used only for the classification task or also to evaluate the NMF decomposition. Are those signatures representative?32. It would be nice to see the position of the excluded bins based on +- 2* IQR. They already excluded the ENCODE blacklist regions. Other outliers may be real biological signal especially in some type of cancer where copy number alteration play an important role.Reviewer #1 (Recommendations for the authors):Renaud et al. aims at developing a new computational method based on NMF relying on the difference in the fragment size profiles of cfDNA extracted from plasma of healthy individuals and metastatic castration-resistant prostate cancer patients. The manuscript is clear and the drawings very helpful. While data are convincing, the reviewer has a few major concerns that should be necessarily addressed:Major concerns:1. The authors generally poorly refer to the source of cancer patient plasma. Limitations of the work, especially method performance, is slightly described in particular about cancer stages, or other cancers. In addition, characteristics of the patients are missing, especially in respect to treatments at time of blood sampling. Lastly, a flow chart is missing, enabling to evaluate successful rates.2. Citation of critical reports or issues are missing when introducing this work and placing it into the development of methods for pan cancer screening. Below are concerns that need to addressed:– The authors are citing reports on cancer vs healthy cfDNA fragment size distribution published after 2016 years. Most of the pre-omics era studies on distinguishing cancer to healthy individuals by fragment size distribution were based on the q-PCR analysis, while this is not mentioned. The first observation was made by Mouliere et al.  who studied by fractional size distribution with nested Q-PCR systems "the size distribution profile of ctDNA fragments in plasma" and indicated that "the size distribution profile of ctDNA fragments can be used to discriminate between healthy and cancer patients". This team later showed that mutant fragments are shorter than wild type counterparts , Lo's group later confirmed this observation by showing strong but somewhat different differences in regards to size ranges Jiang et al.  or more recently in Sun Kun et al. .– Renaud et al. work as well than those cited in the manuscript such as Cristiano et al., relies on Mouliere et al.  first observation. Note, Cristiano et al. cited this report as well as Sanchez et al. , Jiang et al., and Underhill (Plos Gen. 2015). Finally, Sanchez et al.  refined Mouliere et al. (2011) observation by using sWGS from double and single stranded DNA library preparations. The authors should necessarily add these citations.– The authors showed exactly the same illustrative fragment size profile in Figure 1  as those presented earlier by Sanchez et al. . Consequently, they should at least mention in a few sentences the observations made by Jiang et al. and by Sanchez et al.  in order to validate their assumption.3. In light of the previous concerns, the title should be corrected, as the term discovery is improper in regards to the study. The signatures were previously made; the method used here, solely allowed to combine fragment length markers in an optimal setting while developing a technology. This is already important since this method appears very performant, and the reviewer is impressed. Discovery should be replaced by another term such as detection or assessment.Reviewer #2 (Recommendations for the authors):Specific comments for authors to address:1) In the introduction, it would be better to describe a bit more details about tumor DNA length characteristics that were reported previously such as https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1006162, https://www.pnas.org/content/112/11/E1317, https://www.pnas.org/content/115/46/E10925, https://stm.sciencemag.org/content/10/466/eaat4921.abstract, therefore the readers would have a more comprehensive understanding of what is already known in the prior art and why the analysis of tumor-specific length profile in this study is important.2) The authors should compare whether an NMF component would give a better correlation with tumor DNA estimated by ichorCNA, compared with the short DNA proportion (i.e.% of plasma DNA < 150 bp) in the overall size profile.3) It seems a bit confused to the reviewer that the classification power between cases and controls improved as the number of components used in the NMF process. But the authors did not see the improvement in the correlation between tumor DNA fraction and the weight of one component reflecting the tumor contribution. Such observation should be discussed and clarified.4) The AUC was calculated using repeated 10-fold cross-validation. It would be better to show the boxplot of AUC values for a given number of NMF components .5) It will be informative to estimate the minimum number of samples is required for performing an accurate NMF analysis.Reviewer #3 (Recommendations for the authors):1) The authors claim their model is directly measuring ctDNA load. Since an untargeted (unsu-pervised) model to assess the tumor fraction for monitoring purposes would be of great value, it would be interesting to see if changing levels of ctDNA can be captured by changing weights of the putative cancer Signature 1. Since their mCRPC cohort included patients with follow-up samples (at therapy start and progression) it is not clear why this has not been tested.2) It should be mentioned that ichorCNA can only provide informative results at ctDNA frac-tion of 3% and higher. At higher tumor fractions the fragment signature signal might be rel-atively strong, explaining the good correlation, but what about samples with lower frac-tions? Please indicate the tumor fractions  assessed with ichorCNA and based on the VAFs of the identified mutations for all patients.3) The authors claim that using NMF they were able to verify that the fragment length profile of the signature correlating with ctDNA levels does indeed match the length distribution of fragments containing mutations. However, although they included PBMCs in their mutation analysis the authors did not correct for clonal hematopoiesis. Recent studies showed that up to 20% of identified mutation are related to CH and that such mutations are also found in driver genes for solid tumors. Since there might be a size difference of mutated fragment originating from the tumor or PBMCs, respectively, this statement is overrated. In case their sequencing approach was not sensitive enough to detected CH-related variants this should at least be mentioned.4) In Figure 2c, the shift towards smaller fragment is only seen for the dinucelosomes peak but not for the mononucleosomal fragments. How do the authors explain this (if not by contamination of CH-related variants)?5) The authors demonstrate that NMF was more robust when less data is available than the ichorCNA algorithm, but they lacked to provide convincing data that NMF as a quantitative measure is more sensitive with respect to tumor fraction. To this end, in silico dilution ex-periments where they computationally mix healthy fragments and high ctDNA fraction sample(s) to a ctDNA fraction of their choice (with repetitions to achieve a more robust es-timation) should be performed.6) Also, if there are no CNA or tumor fraction >3%, ichorCNA can literally not be used to de-termine ctDNA fraction, so the feasibility of the dilution approach to benchmark ichorCNA is not clear to me . Moreover, low depth can be overcome by additional sequencing whereas the tumor fraction cannot be enriched by more reads.7) Using NMF the authors demonstrate that there are 2 major contributors to fragment length distribution and the addition of further signatures does not add anything. However, this statement is conflicting with results of the linear SVM classification case/control ap-proach . A plateau of classification accuracy is reached when using around 30 signatures, which suggests this would be a good trade-off for number of signatures to reach high accuracy (AUC).8) Figure 3d: Interestingly, open chromatin is associated with a higher fraction of dinucleosomal fragments. How do the authors explain this? Figure 3: I am not sure what the authors want to show here with Pearson's correlation coefficient, this does not look like a linear relationship.9) The paragraph on "epigenetic fragment length" could be better presented. I would rephrase "epigenetic state". Both the DELFI ratio and the fragment length signature are surrogates for the chromatin state (open or closed). First the author talks about bin-wise ex-traction of two signatures. Why is it, that reducing bin size gives better classification results? ~50 kb bins seem to be best to predict ATACseq results. Is there literature that de-scribes open/closed chromatin in this way? Then they correlate the weights from that with ATACseq results. And finally, they use their bin-wise signature extraction results/weights as input for cancer/control classification. Could be split in two parts for clarity: one that talks about correlation with ATACseq results and one that discusses classification results10) cfDNA fragment length profiles are particularly prone to pre-analytical  and analytical (clustering efficiency of various sequencings machines) factors. How do preanalyitcal/analytical factors affect the classification? The authors mention in the discussion that changes in the size profiles from their mCRPC cohort and the DELFI samples vary, which might be attributed due to such factors. Yet such confounding event will impact the robustness of such approaches and may prevent a widespread implementation of frag-mentomics. This should be further discussed.11) How can the authors exclude that the above-mentioned differences are attributed to high-er/lower ctDNA fractions in the metastatic CRPC and the early stage DELFI samples? Even in Figure 2a it seems that samples with higher tumor fraction have more dinucelosomal frag-ments. Separate fragmentation profile plots for each stage should be shown; it would be interesting whether such profiles show any differences for stage I and stage II patients.12) The authors could have checked in the DELFI cohort whether there is a (physiologic fragment length variation) between the seven cancer types? Would there always be the same two signatures coming up if the only consider one tumor type? Based on Supplementary Figure 4 this seems not to be case. In addition, there is a huge overlap of healthy individuals and cancer patients for signature 1? Please also present Supplementary Figure 4 with Signature 2. Can we really assume that this signature is the one related to cancer?eLife. Your revised article has been evaluated by Detlef Weigel (Senior Editor) and a Reviewing Editor.Thank you for resubmitting your work entitled "Unsupervised discovery of fragment length signatures of circulating tumor DNA using Non-negative Matrix Factorization" for further consideration by The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:Reviewer #1 (Recommendations for the authors):Essential revisions:1. The authors partially answered to the concerns but did not add a flow chart to ease reading. In addition, they did not precise the pre-analytics such as delay between blood draw and plasma isolation, storage, centrifugation steps and speeds,… This would validate quantitative data accuracy and may influence fragment size.2. Please replace "discovery" with "detection" in the title.3. The authors should describe in more detail how they concluded that the NMF model can be reliably estimated (as well on which criteria) using as few as 20 samples.4. The authors should indicate whether they have used a spiking strategy to evaluate whether their model is directly measuring ctDNA load, or whether they use another ctDNA techniques to compare. Otherwise, their evaluation is not complete, and this should be indicated in the limitations of the study in the discussion.5. The authors answered to this concern by showing adequate data but omitting to discuss them. This is of importance since this cutoff appears as a critical limitation of the model.6. Explanations from authors are reasonable. However, this discrepancy as well as these explanations should be inserted in the text.7. The authors weakly addressed this concern. In particular, one of the major criticism of the Cristiano work is the selection of the healthy individuals: Are they age matched? Are they fully healthy? Are their plasma absolutely examined as those of cancer individuals? Any difference about pre-analytics or analytics between cancer types? Please discuss. Finally, please be cautious about fragment size cancer signature specificity. Essential revisions:1. More information on the source of cancer patient plasma, characteristics of the patients, especially in respect to treatments at time of blood sampling, should be included. A flow chart summarising the overall results might be useful.The timing of blood samples has been clarified in the results and methods sections of the manuscript. In brief, most samples were collected either before starting first line treatment or following relapse after first line treatment with a small minority of samples collected following relapse from later treatments.2. The authors can consider citing some of the older work analyzing cfDNA fragment size distribution using PCR-based approaches.We thank the referee for pointing out the lack of representation of the PCR-based studies of the cfDNA fragment length distribution in our introduction. We have now updated the introduction section to also cover four PCR-based studies.3. The authors showed exactly the same illustrative fragment size profile in Figure 1  as those presented earlier by Sanchez et al. . Consequently, they should at least mention in a few sentences the observations made by Jiang et al. and by Sanchez et al.  in order to validate their assumption.et al., 2016 and Sanchez et al., 2021.Again, we thank the referee for pointing out the missing references to two sequencing-based studies of cfDNA fragment length signals in cancer. We have updated the introduction and Results sections with references to both Jiang 4. As one could argue that the signatures of cfDNA fragment length had been discovered through various previously published work, in light of the previous concerns, term 'discovery' is perhaps inappropriate. Hence, we would suggest replacing the word 'discovery' by another term such as 'detection' or 'assessment'.It was not our intention to indicate that we were the first to “discover” fragment length signatures of circulating tumor DNA. Instead, we wished to indicate that our method was able to discover tumor signature(s) without using e.g. mutations or xenografts for identifying tumor-derived cfDNA fragments. We have updated the title, introduction and Discussion sections to better reflect this.5. In the introduction, it would be better to describe a bit more details about tumor DNA length characteristics that were reported previously, so that readers would have a more comprehensive understanding of what is already known in the prior art and why the analysis of tumor-specific length profile in this study is important.We agree with the referee that prior art was insufficiently represented in the previous submission. We have included several additional references to both PCR and sequencing based studies of the cfDNA fragment length distribution in cancer. Finally, we have now included a more thorough motivation for our NMF model as an important contribution to the field. In brief, previous studies have relied either on xenografts or detailed knowledge about mutations e.g. from targeted sequencing to accurately determine the tumor fragment length distribution. NMF enables us to do this in a completely unsupervised manner and hence independently of mutational or clinical information.6. The authors should compare whether an NMF component would give a better correlation with tumor DNA estimated by ichorCNA, compared with the short DNA proportion (i.e.% of plasma DNA < 150 bp) in the overall size profile.We tested using the ratio of short-to-long fragments (100-150bp vs. 151-220bp) proposed by Cristiano et al. This yielded a correlation with the VAF based estimates of ctDNA fraction of 0.68 on the shallow WGS dataset and hence significantly lower than the NMF based estimate (r=0.75). We have added a comment to the main text and a supplementary figure .7. It seems somewhat intriguing that the classification power between cases and controls improved as the number of components used in the NMF process. However, the authors did not see an improvement in the correlation between tumor DNA fraction and the weight of one component reflecting the tumor contribution. These observations should be discussed and clarified in a revised manuscript.On the mCRPC data, using more than two components does in fact improve the correlation – albeit only slightly . Using more than two components involves a supervised element as we need to find out which components are tumor associated and sum up their contributions. In order to avoid this, and also to have a more interpretable model, we settled on the model with two components. On the DELFI data, two components were clearly insufficient. We attribute this to greater diversity in the DELFI data due to the many different cancer types included as well as potential technical differences. We have clarified this in both the results and Discussion sections of the manuscript.8. The AUC was calculated using repeated 10-fold cross-validation. It would be better to show the boxplot of AUC values for a given number of NMF components .We now show boxplots to reveal the variation in AUC values between repeats.9. It will be informative to estimate the minimum number of samples is required for performing an accurate NMF analysis.We have now included tests of the robustness of our method to the number of samples. Across a range of dataset sizes, we repeatedly sampled a dataset with the appropriate number of samples and trained an NMF model. For each trained model, we then compared the estimated signatures with those estimated using the full datasets. Using this approach, we found that the NMF model can be reliably estimated using as little as 20 samples. These results have been incorporated in the Results section and a new supplementary figure .10. The authors claim their model is directly measuring ctDNA load. Since an untargeted (unsupervised) model to assess the tumor fraction for monitoring purposes would be of great value, it would be interesting to see if changing levels of ctDNA can be captured by changing weights of the putative cancer Signature 1. Since their mCRPC cohort included patients with follow-up samples (at therapy start and progression) it is not clear why this has not been tested.We thank the referee for suggesting to use serially sampled data to assess our method. Unfortunately, all the samples from the mCRPC cohort were collected either at baseline (i.e. before treatment) or upon progression with no samples collected at response. We thus do not have a clear a priori expectation of the direction of change of the ctDNA fraction across time and hence cannot use serial information to evaluate our method. Serially collected samples are also available in the DELFI cohort. A small subset of the subjects in the DELFI cohort have serial samples that were also examined using targeted sequencing in a previous publication. We have now added a supplementary figure that compares the prediction probabilities obtained by our classification method using 30 fragment length signatures to the maximum driver mutation allele fraction observed in the targeted sequencing .11. It should be mentioned that ichorCNA can only provide informative results at ctDNA fraction of 3% and higher. At higher tumor fractions the fragment signature signal might be relatively strong, explaining the good correlation, but what about samples with lower fractions? Please indicate the tumor fractions  assessed with ichorCNA and based on the VAFs of the identified mutations for all patients.We have added the requested information about ichorCNA and VAF-based ctDNA% estimates in the mCRPC dataset both as summary statistics  in the Results section and as violin plots in the supplement .12. The authors claim that using NMF they were able to verify that the fragment length profile of the signature correlating with ctDNA levels does indeed match the length distribution of fragments containing mutations. However, although they included PBMCs in their mutation analysis the authors did not correct for clonal hematopoiesis (CH). Recent studies showed that up to 20% of identified mutation are related to CH and that such mutations are also found in driver genes for solid tumors. Since there might be a size difference of mutated fragment originating from the tumor or PBMCs, respectively, this statement is overrated. In case their sequencing approach was not sensitive enough to detected CH-related variants this should at least be mentioned.We do make sure that the ctDNA estimates from targeted sequencing are not biased by clonal hematopoiesis. During variant calling, we compare the cfDNA sequence data with sequence data from PBMCs. And after variant calling, we manually inspect all the driver mutations that are used to estimate ctDNA fraction and make sure that no alternative reads are seen in the PBMC data. The PBMC sequence data has a mean coverage of 306X (range: 253-728X). We have now added the mean PBMC sequence depth to the methods section and have specified that we check that no alternative reads are seen in the germline during the manual inspection of the driver variants.13. In Figure 2c, the shift towards smaller fragment is only seen for the dinucelosomes peak but not for the mononucleosomal fragments. How do the authors explain this (if not by contamination of CH-related variants)?The reviewers are correct that the shift towards lower fragment sizes is more pronounced for the nucleosomal peak but we do also see a small shift in the mononucleosome peak. This plot is only based on reads that overlap known driver mutations and is therefore based on a small number of fragments from a few positions in the genome, which explains the differences between the global fragment length distributions and the distributions seen in this plot. As explained in the response above, we are quite certain that the mutations we look at are not CH-related variants. A possible explanation for the differences in the fragment length distributions is that the global distribution could reflect that the ctDNA fragments tend to come from other parts of the genome than the non-cancer cfDNA fragments but in Figure 2c, we restrict the fragments to come from the same positions.14. The authors demonstrated that NMF was more robust when less data was available than the ichorCNA algorithm, but they lacked to provide convincing data that NMF as a quantitative measure is more sensitive with respect to tumor fraction. To this end, in silico dilution experiments where they computationally mixed healthy fragments and high ctDNA fraction sample(s) to a ctDNA fraction of their choice (with repetitions to achieve a more robust es-timation) should be performed.in silico dilution experiments would be informative regarding the robustness of NMF versus other methods at different ctDNA levels. Unfortunately, we do not have any healthy controls available for the mCRPC cohort and hence cannot perform the requested experiments. We have added a comment about this limitation to the discussion and have also made it clear that more experiments will be needed in order to determine which method, NMF or ichorCNA, is most sensitive.We agree with the referee that 15. If there are no CNA or tumor fraction >3%, ichorCNA can literally not be used to determine ctDNA fraction, so the feasibility of the dilution approach to benchmark ichorCNA is not clear to the reviewer . Moreover, low depth can be overcome by additional sequencing whereas the tumor fraction cannot be enriched by more reads.in silico dilution as also suggested by a referee in comment #14. Unfortunately, no healthy controls were available in the mCRPC cohort and hence we had instead to rely on subsampling experiments to prove the robustness of methods.As the vast majority of samples have ctDNA levels above 3% as judged using driver VAFs and as our subsampling experiments do not change this, we believe our conclusions still stand. We agree with the referee that additional sequencing can overcome low depth but not low tumor fraction. A more ideal experiment would be to instead lower the ctDNA fractions by 16. Using NMF the authors demonstrate that there are 2 major contributors to fragment length distribution and the addition of further signatures does not add anything. However, this statement is conflicting with results of the linear SVM classification case/control approach . A plateau of classification accuracy is reached when using around 30 signatures, which suggests this would be a good trade-off for number of signatures to reach high accuracy (AUC).This questions was also raised by another referee (comment #7) and we agree that our presentation of results regarding the optimal number of NMF components was not sufficiently clear in the previous version of the manuscript. This has now been clarified in the manuscript. In brief, using more than two components on the mCRPC data did not yield significant improvement in ctDNA correlation, whereas the gain was much bigger on the DELFI data. We speculate that this is due to higher heterogeneity  in the DELFI data.17. Figure 3d: Interestingly, open chromatin is associated with a higher fraction of dinucleosomal fragments. How do the authors explain this? Figure 3: the reviewer is not sure what the authors want to show here with Pearson's correlation coefficient, this does not look like a linear relationship.We have surveyed the literature to see if previous articles have mentioned that open chromatin is associated with a higher fraction of dinucleosomal fragments but have not found any previous reports of this. We agree that it is an interesting observation but unfortunately we do not have a good explanation for why it should be the case.We agree that Figure 3e did not show a linear relationship and have now changed the figure and the text so that we use Spearman's rank correlation coefficient instead of Pearson’s correlation coefficient.18. The paragraph on "epigenetic fragment length" could be better presented. I would re-phrase "epigenetic state". Both the DELFI ratio and the fragment length signature are surrogates for the chromatin state (open or closed). First the author talks about bin-wise extraction of two signatures. Why is it, that reducing bin size gives better classification results? ~50 kb bins seem to be best to predict ATACseq results. Is there literature that describes open/closed chromatin in this way? Then they correlate the weights from that with ATACseq results. And finally, they use their bin-wise signature extraction results/weights as input for cancer/control classification. Could be split in two parts for clarity: one that talks about correlation with ATACseq results and one that discusses classification results.We appreciate the suggestions for improving the section and have tried to do so. Instead of writing about “epigenetic state” we now use “chromatin state” and we have split the section in two as suggested. With regards to why a smaller bin size leads to better classification results we can see that the ATACseq values do fluctuate to some degree from bin to bin even with the smaller bins sizes. So if the classification results are achieved through prediction of chromatin state then it makes sense that the smaller bins that better capture the small range changes in chromatin state would be best. We are not aware of any existing literature besides the 2019 DELFI article that deals with this issue.19. cfDNA fragment length profiles are particularly prone to pre-analytical  and analytical (clustering efficiency of various sequencings machines) factors. How do preanalyitcal/analytical factors affect the classification? The authors mention in the discussion that changes in the size profiles from their mCRPC cohort and the DELFI samples vary, which might be attributed due to such factors. Yet such confounding event will impact the robustness of such approaches and may prevent a widespread implementation of fragmentomics. This should be further discussed.We agree with the referee that technical factors may significantly impact the fragment length distribution and hence make it hard for fragment length based models to generalize to new data with different technical characteristics. We have now added an additional statement regarding this limitation to the discussion.20. How can the authors exclude that the above-mentioned differences are attributed to higher/lower ctDNA fractions in the metastatic CRPC and the early stage DELFI samples? Even in Figure 2a it seems that samples with higher tumor fraction have more dinucleosomal fragments. Separate fragmentation profile plots for each stage should be shown; it would be interesting whether such profiles show any differences for stage I and stage II patients.Our statements regarding potential technical differences between mCRPC and DELFI were based on comparing the estimated NMF signatures , which – hopefully – eliminates the effect of different ctDNA levels. That said, it cannot be ruled out that the large differences in ctDNA levels between mCRPC and DELFI affects the signature estimates or that other, biological differences (e.g. stage effects) contribute. We have now clarified this in the manuscript and included a “spaghetti plot” showing fragment length distribution for the different stages in the DELFI data . The stage plot confirms differences across stages, albeit subtle, which likely reflect increased ctDNA% at higher stages.21. The authors could have checked in the DELFI cohort whether there is a (physiologic fragment length variation) between the seven cancers types? Would there always be the same two signatures coming up if the only consider one tumor type? Based on Supplementary Figure 4 this seems not to be case. In addition, there is a huge overlap of healthy individuals and cancer patients for signature 1? Please also present Supplementary Figure 4 with Signature 2. Can we really assume that this signature is the one related to cancer?To address this, we reran NMF separately for each cancer type. The AUC was highest for colorectal cancer patients (0.93) and lowest for gastric cancer (0.56). Full results are shown in Figure 3 —figure supplement 3a. Furthermore, we sought to determine whether the inferred signatures differ from one cancer type to the next. The different inferred signatures were plotted . Apart from gastric cancer, which happens to be the one with the lowest AUC, most fragment length signatures do not seem to change depending on the cancer type.22. Abbreviations should be defined at the beginning for better readability.Done.23. Resolution of figures need to be improved. Comprehensive legends should also be included for Supplementary figures.We have now uploaded all figures as good quality pdf figures instead of just the low-quality inlined figures. Furthermore, we have improved the appearance of the supplementary figures and provided them with comprehensive legends.24. Matching colors for better readability are recommended.We have now tried to use the same color scheme in all figures.25. Specify average read pairs for cfDNA samples and corresponding PBMCs.We have now added a summary of the number of reads pairs for all samples sequencing in the mCRPC cohort.26. Excel Supplementary file: Supplementary table 2 should be mentioned in the manuscript.We thank the referee for pointing out the omission of these references. Supplementary tables 1 and 2 contain cfDNA fragment length counts for all samples for sWGS and targeted sequencing, respectively. Furthermore, VAF- and CNV-based estimates of the ctDNA% are included in both tables. References to the two tables have now been included in the manuscript.27. ctDNA (%) on the x-axis in Figures 2 is calculated from targeted mutation analysis, right? How was est.ctDNA % in the Supplementary figures 2&3 calculated?We apologize for not making this sufficiently clear. All ctDNA% estimates used for validation were based on VAFs from targeted mutation analyses unless it is explicitly specified that it is based on CNVs / ichorCNA. We have now clarified this in all figures.28. What is the original h matrix in NMF in Figure 2f? There is a normal h matrix described as weight matrix, but I haven't seen the definition of what with "original" h matrix is meant.“Original h matrix” referred to using signatures estimated on the full dataset to estimate the weights on the subsampled data. We agree with the referee that this was not clear in the previous version of the manuscript. We have now removed that line from Figure 2f in order to improve readability of the subsampling analysis.29. Assumptions for linear correlation using Pearson are not mentioned, I don't know whether they are met at all. Looking at the figure it does not seem like they are met. Normality of the variables should at least be checked.We thank the reviewers for bringing this omission to our attention. Since the results in Figure 3e do not show a linear relationship we have now changed the figure and the text so that we use Spearman's rank correlation coefficient instead of Pearson’s correlation coefficient.30. How is NMF performing on previously unseen data? This part is missing. The authors could have checked for overfitting in this way for example calculating the reconstruction error on a test dataset.To address this, we modified our subsampling analysis to repeatedly divide the data into two, a training and testing set. The average correlation of the weights estimated on the training set with ctDNA% was 0.75, whereas the average correlation on the testing set was 0.74 thus indicating that the NMF is robust to novel data .31. It is not clear whether cross validation was used only for the classification task or also to evaluate the NMF decomposition. Are those signatures representative?We do not retrain the NMF components in each fold in the cross validation. The results mentioned in the response to question 9 and shown in Figure 2 —figure supplement 5 show that signatures inferred from 90% of the data are indistinguishable from signatures inferred from the whole data set.32. It would be nice to see the position of the excluded bins based on +- 2* IQR. They already excluded the ENCODE blacklist regions. Other outliers may be real biological signal especially in some type of cancer where copy number alteration play an important role.We exclude bins that are outliers with regards to depth in controls. It is the same bins that are removed in all samples and we do not look at the case samples when we determine which bins to remove. So biological signals from copy number alterations in the cancer samples do not affect which bins are removed.The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:Reviewer #1 (Recommendations for the authors):Essential revisions:1. The authors partially answered to the concerns but did not add a flow chart to ease reading. In addition, they did not precise the pre-analytics such as delay between blood draw and plasma isolation, storage, centrifugation steps and speeds,… This would validate quantitative data accuracy and may influence fragment size.We agree with the referee that the technical description of cfDNA processing was not completely clear in the previous version of the manuscript and we have now added extra details regarding this to the methods section. The description in the “Sample processing and DNA extraction” subsection now includes all the requested details about the pre-analytics. Besides the main overview in Figure 1 we have now added a flow chart giving an overview of the different analyses and experiments presented in Figure 2 and Figure 3 as supplementary figures to Figure 2 and Figure 3 respectively.2. Please replace "discovery" with "detection" in the title.The title has been changed as requested.3. The authors should describe in more detail how they concluded that the NMF model can be reliably estimated (as well on which criteria) using as few as 20 samples.We agree with the referee that the subsampling procedure and criteria used for assessing reliability was not clear enough in the previous version of the manuscript. We have now added more thorough descriptions to both the results and methods sections. The methods section now includes a subsection called “Subsampling experiments” containing the details about the robustness experiments presented in Figure 2f, Figure 2 supp. 5 and Figure 2 supp.6.4. The authors should indicate whether they have used a spiking strategy to evaluate whether their model is directly measuring ctDNA load, or whether they use another ctDNA techniques to compare. Otherwise, their evaluation is not complete, and this should be indicated in the limitations of the study in the discussion.We agree with the referee that using spike-ins could have been a valuable add on to the present study, however this was not done due to lack of healthy controls for the mCRPC cohort. We have clarified this in the Discussion section of the manuscript. Please see the response to questions #11 for more details.5. The authors answered to this concern by showing adequate data but omitting to discuss them. This is of importance since this cutoff appears as a critical limitation of the model.We have added the following text to the discussion acknowledging the limitations of the current study:“However, the lack of low ctDNA samples is a limitation of the current study and further experiments either based on samples with lower tumor burdens, spike-ins or in silico dilution are needed to confirm this assertion as it could not be directly tested with the data available in this study. A lack of healthy controls for the mCRPC cohort data means that we were not able to perform spike-in or in silico dilution experiments in this article”6. Explanations from authors are reasonable. However, this discrepancy as well as these explanations should be inserted in the text.We believe that the text added in response to the previous question covers these concerns as well.7. The authors weakly addressed this concern. In particular, one of the major criticism of the Cristiano work is the selection of the healthy individuals: Are they age matched? Are they fully healthy? Are their plasma absolutely examined as those of cancer individuals? Any difference about pre-analytics or analytics between cancer types? Please discuss. Finally, please be cautious about fragment size cancer signature specificity.We agree with the referee that technical or biological factors may have affected our results. We have now added the statement below to the discussion:“A number of technical and biological factors in the DELFI study design may have inflated our classification performance. For instance, cases were generally older than controls and hence the observed differences in the fragment length distribution between cases and controls may partially reflect underlying comorbidities associated with higher age.”We are unaware of any specific differences in how samples from cases and controls were processed in the DELFI study and hence feel unable to comment further on this in the manuscript.


Recent discoveries of distinct molecular subtypes have led to remarkable advances in treatment for a variety of diseases. While subtyping via unsupervised clustering has received a great deal of interest, most methods rely on basic statistical or machine learning methods. At the same time, techniques based on graph clustering, particularly clique-based strategies, have been successfully used to identify disease biomarkers and gene networks. A graph theoretical approach based on the paraclique algorithm is described that can easily be employed to identify putative disease subtypes and serve as an aid in outlier detection as well. The feasibility and potential effectiveness of this method is demonstrated on publicly available gene co-expression data derived from patient samples covering twelve different disease families. This is especially true in cancer. Lung cancers, for example, fall into two overall types based on histological characteristics: small cell lung cancer (SCLC) and non-small cell lung cancer (NSCLC). Although histological classification remains crucial, significant advances in the treatment of NSCLC over the last decade have centered around the development of therapies targeting subtypes at the molecular level, such as those defined by genetic mutations . In partk-means and hierarchical clustering, have long been used to identify sets of genes or samples that exhibit similar expression patterns [S. cerevisiae (baker’s yeast) gene co-expression data can be found in [Given the potential impact of disease subtype identification, it is not surprising therefore, that the search for effective clustering methods has become an intense area of interest. Traditional approaches, such as patterns –11. Machpatterns –14. Latepatterns –17. Meanfound in , where ifound in  was founfound in , for exafound in  are hobbfound in , proteomfound in , epigenefound in  and the found in , as wellfound in , diabetefound in , allergifound in  and commfound in , and evefound in . NeverthTo address this gap, we describe an initial study of putative subtypes based on molecular signatures using the paraclique method. Our technique is general and applies easily to other types of data such as protein interaction, metabolite abundance, or DNA methylation profiles, but we focus our experimentation on gene co-expression data thanks to its relative quality and ubiquity. In addition to subtype discovery, we will show how our techniques can be used to help pinpoint potential outliers, providing an automatic means for the identification of suspected data collection errors such as mislabeled samples or misdiagnosed patients. We hasten to observe, however, that biological variation can be inscrutable, inconsistent, and unpredictable. No method is therefore likely to be extraordinarily accurate. We will address this and related issues in the sequel.This paper is organized as follows. In the next section, we provide a brief review of the paraclique algorithm and discuss details of our workflow for subtyping in gene co-expression data. In 2.g. An illustration of paraclique construction with g = 2 is provided in Clique-centric methods have long been used in a wide variety of applications . On realp-values for the differential expression of genes between case data and control data were calculated using the Benjamini-Hochberg method [p-values less than or equal to 0.1 were retained. The motivation for such a filter was to restrict attention to genes of potential interest in the differential diagnosis of disease. After all, we wanted to concentrate on potential disease subtypes and not be distracted by irrelevant subgroups such as age, ethnicity, or hair color. Once filtering was complete, we focused our attention only on case data, and reversed the roles of variables and correlations. We therefore calculated pairwise Pearson correlation coefficients between samples (not genes), and across their corresponding lists of expression levels (not patients). We then thresholded the resultant correlation matrix using spectral methods as in [In this effort, we were mainly concerned with case-control transcriptomic data, for which we applied an initial filtering step to limit the effects of confounding factors. False discovery rate adjusted g method  accessedg method . Only thds as in  and consg = 1 throughout this effort. Depending on the data under study, however, crisper results may naturally be anticipated with fine tuning. In [g = 5 was found to produce superior ontological enrichments when studying non-obese diabetic mice as a model of type 1 diabetes mellitus.For consistency, and because this work mainly represents a proof of concept, we set the glom term to ning. In , for exa3.3.1.We applied this novel analytical approach to a dozen sets of publicly available gene co-expression data obtained from the Gene Expression Omnibus (GEO). These data were selected because they provide a wide cross-section of human disease, and because each has both a case and a control group for the aforementioned filtering task. Our investigation into the effectiveness of this proposed new methodology was focused on two guiding questions: (1) Are these tools capable of reliably and robustly identifying putative subtypes? (2) Are these subtypes appropriate to the associated disease as supported by biological evidence from clinical, published, analytical or other orthogonal information source(s)?The answer to the first question seems to be an unequivocal yes. As summarized in p-values for the top 100 differentially expressed genes in each paraclique. These results and their corresponding GO categories are summarized in p-values ranging from 1.1 × 10−4 for asthma to 4.92 × 10−46 for prostate cancer. Next, we performed a literature search to check the top scoring genes for involvement in known subtypes. As such, this is at best a hit-or-miss proposition, and one depending for each disease on whether the research community has studied subtyping issues, found results, and published them in venues that we were able to search. Despite these obstacles, however, we found strong evidence in print to support our putative subtype decompositions for four of the diseases we studied. These are asthma, breast cancer, chronic lymphocytic leukemia, and colorectal cancer.The second question is considerably more difficult to answer because it depends on the availability of alternate, non-transcriptomic data sources. We therefore followed a two-pronged approach in putative subtype comparisons. First, we calculated GO enrichments and their associated 3.2.3.2.1.The incidence of asthma in the U.S. has been on the rise for two decades. It is currently estimated that nearly one in ten children under 18 are asthmatic. The risk for some groups is based largely on ethnicity (particularly African American and Puerto Rican), with incidence among those with lower socioeconomic status rising as high as one in six .GEO series GSE4302 data were derived from the Affymetrix Human Genome U133 Plus 2.0 Array, and are designed to identify genes associated with response to corticosteroid treatment in asthmatics . They coFiltering reduced the number of probes from 54,676 to 2322. Our method produced three paracliques with respective sizes 31, 8 and 3 that were stable until they began to merge as the threshold was lowered below 0.93. The 100 most differentially expressed genes across the two larger putative subtypes included CLCA1, periostin, and ovalbumin, which are all known to serve as markers of a Th2-high endotype of asthma .3.2.2.Genetic factors have long been known to play a significant role in breast cancer. Studies have shown that in families with at least four breast cancer cases, most can be linked to mutations in either BRCA1 or BRCA2 genes ,44. MoreGEO series GSE10810 data were also derived from the Affymetrix Human Genome U133 Plus 2.0 Array, although values for only 18,382 probes were provided. This study was designed to investigate links between gene co-expression and phenotypic breast cancer differences , and conFiltering reduced the number of probes to 11,531. Our tools produced two paracliques of size 22 and 5 that persisted to a threshold of 0.8 and left four tumor samples unclassified. The 100 most differentially expressed genes between these putative subtypes include SLC39A6, S100a4, AGR3, Cd24, and epcam, all of which have been reported in the literature as biomarkers for distinct breast cancer phenotypes –51.3.2.3.Chronic lymphocytic leukemia is one of the most common types of leukemia, with pathogenesis characterized by an overproduction of neoplastic B cells in the blood-stream. The current median age at diagnosis is 65, with males affected more often than females . ChronicGEO series GSE8835 data were instead derived from the Affymetrix Human Genome U133A Array with 22,283 probes, and were designed to study the effects of chronic lymphocytic leukemia on T cells in peripheral blood . The stuFiltering reduced the number of probes to 1338. At a threshold of 0.8, our tools produced two paracliques of size 4 and 18, leaving two samples unclassified. The most differentially expressed genes across these two putative subtypes included ZAP-70, previously identified as the best discriminator of Ig-mutated and Ig-unmutated chronic lymphocytic leukemia .3.2.4.The incidence of colorectal cancer has been in decline since the mid 1980s . DespiteGEO series GSE9348 data were again derived from the Affymetrix U133 Plus 2 array, and were intended to search for transcriptomic signatures of early stage colorectal cancer that is prone to metastasis . The stuFiltering reduced the number of probes from 54,675 to 22,968. At a threshold of 0.87, our tools produced two paracliques of size 63 and 5, covering all but two of the case samples. The list of 100 genes most differentially expressed between these two putative subtypes include Cd24, identified as a prognostic marker for colorectal cancer  as well 3.3.The experimental effort just described suggests that our methods have the potential to identify both known and novel subtypes, as based on biologically relevant genetic signatures. The lack of any widespread established ground truth, however, places a limitation on any in-depth interpretation of these results. In an effort to address this shortcoming, we identified two sets of publicly available data on GEO that include metadata labeling in the form of known subtyping information. These are based on gastric cancer and non-small cell lung cancer. As our intent is to identify and contrast novel subtypes in disease, our metric of interest is patient stratification.3.3.1.GEO series GSE35809 data, from the Affymetrix Human Genome U133 Plus 2.0 Array, were derived from 70 primary gastric tumors intended for use as a validation set for subtype classifier testing –64. The Filtering was irrelevant, because no healthy tissues were studied that could be used as controls. At a threshold of 0.955, paraclique produced subsets of size 29 and 16 and performed admirably. All but one of the invasive samples it classified were placed in the first paraclique, while all but one of the proliferative samples it classified were placed in the second. Metabolic samples proved only slightly more challenging, with 75% of those classified placed in the first paraclique. See 3.3.2.GEO series GSE10245 data, also from the Affymetrix Human Genome U133 Plus 2.0 Array with 54,675 probes, were derived from 40 adenocarcinoma tumors and 18 squamous cell carcinoma tumors, NSCLC’s two most prevalent subtypes. These data were intended to provide a basis for studying co-expression differences between these two cancers .Filtering was again irrelevant. Paraclique performed quite well on these data too. At a threshold of 0.94, it produced three subsets of size 26, 12 and 8. Roughly 74% of the adenocarcinoma samples were placed in the first paraclique while the second contained none, and 80% of the squamous cell carcinoma samples were placed in the second paraclique while the third had none. Again, see 3.3.3.kmeans  and hclust .We sought to compare this basic and untuned version of the paraclique algorithm with well-known strategies such as k-means and hierarchical clustering, as implemented in core-R through the functions Results for the k-means method were mixed. It proved extremely successful on the gastric cancer data. There it divided samples into two subsets of size 26 and 44. All but one of the invasive samples were placed in the first cluster, while all of the proliferative and all but one of the metabolic samples were placed in the second. However, k-means failed completely on the non-small cell lung cancer data. Samples were divided into subsets of size 28 and 30, with both the adenocarcinoma and the squamous cell carcinoma samples spread almost evenly across these two clusters. The hierarchical approach was also a rather uneven performer. On the gastric cancer data, it divided samples into two subsets of size 33 and 37. While all of the proliferative samples found their way to the second cluster, the first contained 84% of the invasive and about 73% of the metabolic. On the non-small cell lung cancer data, it divided samples into subsets of size 9 and 49. All the adenocarcinoma samples were admirably grouped in the second cluster, but the squamous cell carcinoma samples were not convincingly stratified at all, with exactly half placed in each cluster. These results are also summarized in As demonstrated by these experiments, the paraclique methodology can provide excellent patient stratification, further motivating the use of graph theoretical methods to differentiate samples based on their underlying genetic signatures. Such stratification is not perfect, of course, nor should we expect it to be given data limitations and biological variability. Moreover, unlike techniques such as k-means and hierarchical clustering, patients are not forced into a cluster under paraclique, as is evidenced by the 25 samples it left unclassified in the gastric cancer data. We suggest therefore that the tools we have described here may be best suited to fast screening tasks, for example, when transcriptomic data are relatively easy to obtain. Once clinical and/or additional forms of data have been collected, histological and other more laborious techniques will likely help provide more comprehensive subtyping of entire patient populations.4.The methodology just described is readily extensible to automating the task of outlier detection. This follows from the observation that an outlier would be expected to appear as its own distinct subtype, and not reside in a paraclique of even modest size. Although detection may be accomplished with our algorithms in several ways, we endorse the use of thresholding, as follows. A normalized threshold of 0.0 will of course produce a single large clique, and a threshold of 1.0 will generally yield an edgeless graph, under the assumption that no two samples are perfectly correlated. As the threshold value is lowered from 1.0, the effect on cliques and paracliques is slightly nuanced. As more and more edges are added, cliques and paracliques will get larger but also begin to merge. If a vertex consistently fails to join any of these dense subgraphs, then the sample it represents is flagged as a potential outlier. The process is illustrated in While this approach has intuitive appeal, we conducted a series of six experiments using known misclassifications to test its limitations. We formed test instances by introducing data from one randomly chosen healthy sample into data from the case samples for breast cancer (GSE10810), chronic lymphocytic leukemia (GSE8835), colorectal cancer (GSE9348), lung cancer (GSE7670), pancreatic cancer (GDS4102), and prostate cancer (GSE6919). For the breast, colorectal, lung, and pancreatic cancer sets, we observed that the normal sample was either the last or the next to last vertex to be drawn into the final paraclique. For the chronic lymphocytic leukemia and prostate cancer sets, we found instead that the healthy sample fell into a large paraclique early on and stayed there. From our previous experience with outlier detection , these o5.We have developed and described a disease classification strategy based on the paraclique algorithm that can identify putative subtypes, segregating samples based on signatures in their molecular profiles. Although our tools are easily applicable to many types of biological data, we have focused on gene co-expression data largely thanks to their overall quality and availability. We have analyzed high throughput data taken from a dozen different disease samples obtained from the Gene Expression Omnibus and sought to validate the significance of our findings by reviewing the literature and examining ontological enrichment for the biological relevance of genes differentially expressed across putative subtypes. We also performed testing over data augmented with phenotypic information for known subtypes. Overall, the results of this study indicate a strong utility for this approach in the confirmation of known, and the discovery of novel, disease subtypes. Additionally, we described the extension of our methodology to the task of outlier identification. By iteratively lowering the threshold and re-running the paraclique algorithm, we can detect samples resistant to subtype coalescence. Such a finding can point to critical clinical errors such as tissue misclassifications and/or patient misdiagnoses. Throughout, our aim has been to employ scalable, cutting edge graph theoretical methods that can help automate the disease subtyping process, which can in turn accelerate the pace of discovery and lead to improvements in targeted therapies.IL-6 and IL-10. See [We emphasize that this exploratory effort has focused exclusively on unsupervised techniques and tools that require no prior knowledge. To keep things simple, we even refrained from fine tuning the glom term for each dataset. This, therefore, bolsters the argument that the methods we have espoused are really quite effective. In large-scale clinical applications, however, techniques such as feature selection and paraclique anchoring will almost surely prove helpful to narrow the focus on genes or other variables of interest and their disease-associated relationships. In the context of community-acquired pneumonia, for example, we have previously found it advantageous to anchor paraclique analytics at the interleukin genes -10. See .To place these results in proper context, we note that any subtyping method based on tissue morphology or molecular signatures is almost certain to be highly imperfect. In addition, this holds true whether it be implemented in silico or conducted manually by a human pathologist. A 2015 study  underscoFinally, we wish to emphasize that this work represents but a first step in determining the utility of paraclique in the molecular subtyping of disease. Although clique-based methods have been used as a basis for tasks such as biomarker detection and gene network elucidation, disease subtyping has received surprisingly little attention. In future work, it would thus be interesting to see systematic comparisons of this and other emergent sub-typing technologies. Numerous other research directions beckon. For example, we would like to gain a better understanding of the impacts of improved feature selection, and see extensions of the basic method to multiple heterogeneous data types, an area that has attracted a flurry of recent attention –69. Collalgorithms-14-00063-s001