dataesr/scientific-paragraphs-categorization huggingface dataset:
The objective function of the coupled analysis between the two image datasets is as follow In order to solve the above optimization problem, we can calculate its gradient and solve it by using any first order optimization algorithm of  [40]. Next, this paper will discuss the gradient of the objective function, the first item of the function (  11 ) is written as the second item of the function (  11 ) is recorded as Let S = A (1) , A (2) ,. , A (N ) ,and the specific forms of the partial derivative of f 1 with respect to A (i) are as below where i = 1, 2,. , N. The matrices A and V ∈ R M×R are factor matrices obtained by matrix decomposition of the matrix Y. The specific forms of the partial derivative of f 2 with respect to A (i) and V can be com- puted as Combined with the above calculation results, we can calculate the objective function f.
Acknowledgments This work was granted access to the  HPC resources of Aix-Marseille Université  financed by the project  Equip@Meso  ( ANR-10-EQPX-29-01 ) of the program " Investissements d'Avenir " supervised by the  Agence Nationale de la Recherche. The authors thank  Jean Iwaz  for the revision of the final draft of this manuscript.
In section 5, the magnetic gap-labelling conjecture for principal solenoidal tori is proved in full generality as an application of the previous result, by showing that the Chern character is indeed integral in this principal case. The MGL conjecture remains open for general solenoidal tori, see comments after the statement of Theorem 2. Acknowledgements. The authors thank Dennis Sullivan for useful discussions and for providing an interesting topological proof of the integrality of the Chern character for tori, which is not included in the paper. MB thanks the French National Research Agency for support via the ANR-14-CE25-0012-01 (SINGSTAR), and VM thanks the Australian Research Council for support via the ARC Laureate Fellowship FL170100020.
Quadrivalent influenza vaccine (QIV), which includes two influenza A subtypes (H1N1 and H3N2) and two influenza Funding: This study was supported by Sanofi Pasteur. IQVIA was funded by Sanofi Pasteur for data collection and preparation of the manuscript. MS is an IQVIA employee. The funder provided support in the form of salaries for authors JLLB, FPA and HB. They played a role in the study design, data collection, and manuscript preparation. Competing interests: I have read the journal's policy and the authors of this manuscript have the following competing interests: PC has received scientific consultancy from Sanofi Pasteur. ER has received funding for scientific consultancy and speaker fees, as well as congress attendance grants from Sanofi Pasteur, GlaxoSmithKline, Merck Sharp and Dohme and Pfizer.
8 days  [34]. We assumed that the vaccination campaign was completed ahead of the start of the epidemic with a progressively increasing coverage starting on week 44 and ending on week 50 each year. Asymptomatic infections are also accounted for and assumed to represent a third of all infections  [34]. The economic model is an age-structured decision tree model developed in Excel (Fig  1 ). The age-stratified symptomatic influenza cases (output of the dynamic model described above) were used as inputs to the economic model. Based on available economic data, the age groups were re-categorized (0-1 years, 2-4 years, 5-14 years, 15-19 years, 20-49 years, 50-64 years 65-69 years, 70-74 years and 75+ years), using age-distribution data of the Spanish population  [35]. Influenza cases were stratified between non-high-risk (NHR) and high-risk (HR) patients, based on the presence of underlying medical conditions  [36] , and were subsequently divided into four categories based on medical outcomes (no medical attention, outpatient visit, hospital admission, and death), with different complication probabilities for each category  [37].
Costs As it is recommended in Spain, a discount rate of 3% was used for costs and health outcomes  [51]. Medical costs. The cost of outpatient visits in adults was obtained from the eSalud database  [44] , as well as the cost of outpatient visits in children (<18 years), which also include the cost of physician visits for otitis media and the cost of physician visits for pneumonia or other complications  [44]. Hospitalization cost and cost per death were obtained by age group from the National Health System hospital admission's registry (ICD-9 487 and 488)  [41]. Patients who died during an episode of hospitalization have an extra cost, based on hospital admission's registry (ICD-9 487 and 488) with an exitus discharge; the average cost of all those patients was calculated. Medication cost was calculated as the average cost of influenza antiviral products, obtained from the Spanish official webpage BotPlus 2.
0  [45]. Vaccination costs. The total yearly number of administered vaccinations was calculated by multiplying the age-specific coverage rates with the corresponding population sizes, as already present in the underlying dynamic model. Vaccine prices of TIV and QIV for the public and private markets were obtained from the Spanish official database BotPlus 2. 0  [45]. No administration costs were included in this analysis because these were assumed to be equal in both alternatives. Indirect costs. Days of productivity loss due to outpatient visit or hospitalization were only assigned to adults (�18 years) and were obtained from Galante et al. [46]. We used the friction costs method approach to count productivity losses  [53]  and labor elasticity adjustment factor  [54] , as detailed in a previous publication  [32].
Vaccine efficacy Vaccine efficacy against influenza A and B was calculated by strain (A/H1N1, A/H3N2, B/Victoria and B/Yamagata). In all cases, it was estimated on the basis of Dı ´az-Granados et al. [38]  and by age (CDC unpublished data). The same relative efficacy by age was assumed between different strains and between NHR and HR individuals. The model considered cross-protection between B lineages, estimated from mismatched efficacy data of Dı ´az-Granados et al. [38]. Potential heterosubtypic immunity between A/H1N1 and A/H3N2 has not been documented.
This study has limitations given that different assumptions were necessary. Probability of flu complications and QALY loss for influenza B were not available, so we used data for influenza in general, although no significant differences between influenza A and B have been shown in clinical burden  [14]. Because inter-individual contact data for the Spanish population were not available, we used an Italian contact matrix  [33]  assuming that population contact characteristics between Spain and Italy are similar. We also used data related to the natural history of influenza from other countries, however it is widely accepted that influenza natural history is similar among countries. Finally, despite the use of a dynamic model, which represents the recommended method for infectious disease vaccination programs  [65] , our approach did not capture other vaccination benefits like e.
Acknowledgements I thank  Giovanni Felder  for truly helpful discussions,  Pavel Etingof  and  Travis Schedler  for truly helpful e-mail exchanges, and  Philip Boalch  for both. This research began while the author was a member of the  Laboratoire de Mathématiques d'Orsay , and was supported by a temporary research and teaching assistant contract; it was concluded while the author was a member of the  Department of Mathematics at the ETH of Zürich , and was supported by the  National Centre of Competence in Research "SwissMAP-The Mathematics of Physics" of the Swiss National Science Foundation.
A fixed gamma distribution for the generation time interval with mean of 20 days and standard deviation of 7. 4 days was assumed to be known for all methods  (29). We initiated the analysis 6 weeks into each epidemic, fitting to the first 6, 9, 12, 15, etc. , weeks (approximating to 2, 3, 4, 5, etc. , generation times), up to the peak of the epidemic, simply defined as the week with the maximum number of cases from the entire epidemic curve. The performance of each method was assessed at each of these stages, with methods being fit to an increasing number of data points. We used a number of metrics to assess method performance: bias, calculated as the absolute difference between the estimated and true R 0 values; coverage, calculated as the proportion of times the estimated 95% confidence intervals included the true R 0 value; Pearson correlation coefficient (PCC); width of 95% confidence intervals; and the root mean square error (RMSE).
Funding This work is jointly funded by the  UK Medical Research Council (MRC)  and the  UK Foreign, Commonwealth & Development Office (FCDO) under the  MRC/DFID  Concordat agreement and is also part of the EDCTP2 programme supported by the  European Union  ( MO, CAD, AC, ID ); the  Fondation Mathématiques Jacques Hadamard (CH) ; the  Imperial College Undergraduate Research Opportunity Programme (CH) ; the  Imperial College Junior Research Fellowship ,  Wellcome Trust  and the  Royal Society  [grant  213494/Z/18/Z ] (ID). Potential Conflicts of Interest A. C. reports grants from  Medical Research Council, NIHR , and  USAID , outside the submitted work. I. D. reports grants from  Wellcome Trust, Medical Research Council ,  Janssen R&D , and  Natural Environment Research Council , outside the submitted work.
METHODS Incidence data were simulated using a deterministic compartmental SEIR (susceptible, exposed, infectious, recovered) model of transmission, assuming closed populations of variable sizes and homogeneous mixing; random, time-constant basic reproduction numbers (R 0 ); a variable number of infections seeding the epidemics; underreporting; and different levels of stochasticity in the reporting and infection process (details are given in Supplementary Methods S1. Estimating an average, fixed R 0 , we assessed the performance of each statistical method by fitting it to the simulated epidemic data at sequential time points in the epidemic growth phase, in line with a real-time outbreak analysis scenario. A detailed description of each of the 7 statistical methods is given in Supplementary Methods S2.
A fixed gamma distribution for the generation time interval, with a mean of 20 days and standard deviation of 7. 4 days, was assumed to be known for all methods  [19]. We initiated the analysis 6 weeks into each epidemic, fitting to the first 6, 9, 12, and 15 weeks, and so on (approximating to 2, 3, 4, 5, etc. generation times), up to the peak of the epidemic, simply defined as the week with the maximum number of cases from the entire epidemic curve. The performance of each method was assessed at each of these stages, with methods being fit to an increasing number of data points. We used a number of metrics to assess method performance: bias, calculated as the absolute difference between the estimated and true R 0 values; coverage, calculated as the proportion of times the estimated 95% confidence intervals (CIs) included the true R 0 value; Pearson correlation coefficient; width of 95% CIs; and the root mean square error (RMSE).
However, no works have uncovered scalings laws from large cancer patient datasets. Here we addressed this question under the initial hypotheses that malignant tumours would scale between the metabolic requirements of coordinated tissues governed by minimal energy principles (leading to an exponent  3/4) and that of independent uncoordinated units (exponent  1). Tumour cells exhibit high metabolic requirements to sustain an upregulated proliferation. Nutrients such as glu-cose and, to a lesser extent, glutamine are mostly used to fuel biomass formation and macromolecule synthesis  [10]. Deregulated glucose uptake by tumour cells, known as the Warburg effect, constitutes the basis of positron-emissiontomography/computed-tomography (PET/CT)-based imaging by means of the radioactive tracer 18 F-fluorodeoxyglucose ( 18 F-FDG), widely used in clinical oncology  [11].
To study the relationship between tumour metabolic rates and volume we collected data of different cancer types imaged at diagnosis with 18 F-FDG PET/CT. Tumours were segmented and their total lesion activity (TLA) and metabolic tumour volume (MTV) calculated. TLA and MTV were computed as the product of each voxel volume within the tumour by its measured standardized uptake value (SUV) and as the summed volume of the segmented tumour voxels, respectively. Our first goal was to determine whether a dependence of the form TLA ∼ MTV  could be identified. Figure  1  shows log-log plots of MTV versus TLA for patients with: locally advanced breast cancer (LABC), head and neck cancer (H&NC, stages II-IV), non-small-cell lung cancer (NSCLC, stages I-III) and rectal cancer (RC, stages III-IV) (see 'Methods' for more patient data).
The first one would be an increase of the Warburg phenotype leading to a less efficient use of glucose. Also, the presence of immune cells and inflammation within the tumour region could be a contributing factor. However, since glucose is mostly used to satisfy the proliferation demands  [10, 11] , we suspected that an increase of the proliferation rate with size was probably the main underlying cause. To clarify this, we gathered data from glioma patients (grades II-IV) imaged at diagnosis with 18 F-Fluorocholine PET ( 18 F-FCHOL), and from breast cancer patients (stages II-IV) imaged at diagnosis with 3'-deoxy-3'-18 Ffluorothymidine PET ( 18 F-FLT). These two radiotracers reflect choline and thymidine metabolism and are related to cell proliferation  [12, 13]. The obtained scaling exponents were  = 1.
The data supporting these models comes from patient-derived cell lines cultured in vitro or, else, from either allotransplantation of murine cells into syngeneic immunocompetent inbred mice or from xenotransplantation of patient-derived cells into immunocompromised mice. These models have a number of shortcomings when compared with their human counterparts. They display loss of genetic heterogeneity and irreversible changes in gene expression due to long-term in vitro propagation  [21]  and exhibit a rapid non-autochthonous growth that results in a perturbed tissue architecture with alterations in the vascular, lymphatic and immune compartments. To investigate whether explosive tumour growth could be observed in cancer patients, we looked for longitudinal imaging datasets of untreated tumours.
Data of this type is scarce since growing tumours are typically either treated or -as in the case of palliative care patients-not followed up by imaging. Most available datasets had either incomplete information, no volumetric imaging and/or very few time points. Mandonnet and colleagues  [22]  studied the growth dynamics of untreated WHO grade II gliomas, Van Havenbergh  [23]  analysed petroclival meningiomas, and Heesterman et al  [24]  head and neck paragangliomas. Growth dynamics consistent with sublinear scalings were observed for those slowly growing tumours. To further confirm this idea, we collected longitudinal volumetric growth data from a set of lung hamartomas, the most frequent benign lung tumour type, and found a best fit of Eq. (  1 ) with  = 0. 5 ± 0. 2 (Fig. 2(d) ). Hence, not all human tumours manifest an explosive growth.
We also collected imaging datasets of patients bearing tumours that were either malignant initially or became malignant over the course of the disease (see 'Methods' for a description of the patient datasets). The first one was a set of brain metastases in which one of the lesions was either below target definition or left without therapy due to medical reasons. A second set comprised initially WHO grade II gliomas that underwent surgery and then received no other treatment for long periods. The third was a set of patients enrolled in a lung cancer screening program. After detec- tion of lung nodules with no signs of malignancy they where followed up by low-dose CT scans. Many of these tumours accelerated their growth until a point at which further therapeutical actions were taken. Finally, we included a subset of petroclival meningiomas that showed signs of atypical behaviour (cases 5, 6, 9, 11 of Fig.
3. The total tumour load in the animals showed similar behaviour, with  = 1. 25 (Extended Data Fig. S1 ). In a second set of experiments, we injected primary glioma cells closely resembling the dynamics observed in patients  [26]  and expressing the luciferase reporter gene into the brains of nude mice. One month after the injection, weekly monitoring of the animals was started, measuring the total flow to assess tumour growth. The optimal exponents obtained were also  = 1. 25. Thus, a sustained increase in proliferation is supported both by the allometric scaling laws and the morphological longitudinal growth data during the tumour's natural history. We suspected that evolutionary dynamics could be the underlying process. Via genomic instability, driver gene mutations can confer, to subpopulations of clonal cells, somatic fitness advantages over other cells within the same tumour, and contribute to higher proliferation rates.
Scaling laws are very intriguing properties of physical and biological systems that shed light on their dynamics. They have a fundamental value but are often of limited applicability. We hypothesized that, once a scaling law of the form  =   is set as a reference for a specific cancer type, those with radiotracer uptake higher than the reference level, as defined by the scaling law, could be more aggressive than those with lower activity. Thus, we computed the distance with respect to a reference scaling law (DSL) for each tumour  and dataset for which survival information was available, via DSL  = TLA  -MTV   , and compared two sets with different DSL for the whole range of values of the prefactor , as described in Methods. Figure  4  summarizes our results for a fixed exponent  = 5/4 in four patient cohorts with distinct cancer types.
Methods Patients and image acquisition. Several patient datasets were included in our study. Patient subgroups 1-6 were used for the construction of the scaling laws (1,2,4,5 also for the survival studies). Patient subgroups 7-10 were used for the study of the longitudinal tumour volumetric dynamics. Overall survival (OS) was determined as the time from pretreatment imaging to death or last follow-up. Breast Cancer Patients (subgroup 1). Patients were participants of a multicenter prospective study approved by the Institutional Review Board (IRB). Written informed consent was obtained from all patients. The inclusion criteria were: (1) newly diagnosed locally advanced breast cancer with clinical indication of neoadjuvant chemotherapy, (2) lesion uptake higher than background, (3) absence of distant metastases confirmed by other methods previous request of the PET/CT for staging, and (4) breast lesion size of at least 2 cm.
54 patients (18% lobular carcinoma, 82% ductal carcinomas, 100% women, age rank 25-80, median 50 years) were included in this dataset. The TNM data were: 54% T2, 18% T3, 28% T4; 28% N0, 55% N1, 6% N2, 11% N3; 100% M0. PET/CT examinations were performed on the same dedicated whole-body PET/CT scanner (Discovery DSTE-16s, GE Medical Systems) in three-dimensional (3D) mode. The acquisition began 60 minutes after intravenous administration of approximately 370 MBq (10 mCi) of 18 F-FDG. The image voxel size was 5. 47 mm × 5. 47 mm × 3. 27 mm, with a slice thickness of 3. 27 mm and no gap between slices. Head & Neck cancer patients (subgroup 2). These patients were obtained from The Cancer Imaging Archive (TCIA)  [34]  Head-Neck-PET-CT collection (H&N1 data set)  [35]. This cohort was composed of patients with primary squamous cell carcinoma of the head-and-neck (stages I-IV).
76 consecutive patients from this subset satisfying the inclusion criteria (availability of pretreatment PET studies, well-defined primary tumour and lesion size larger than 2. 0 cm) were included in our study. Data for the cohort were: age rank 18-84, median 62 years; 63 male, 13 female;  13   Eligible patients had FDG-PET scans obtained on a hybrid PET/CT scanner (Discovery ST, GE Healthcare) within 37 days before treatment (median: 14 days). A median of 584 MBq (range: 368-715) was injected intravenously. Imaging acquisition of the head and neck was performed using multiple bed positions with a median of 300 s (range: 180-420) per bed position. The slice thickness resolution was 3. 27 mm for all patients and the median in-plane resolution was 3. 52 × 3. 52 mm 2 (range: 3. 52-4. 69).
Rectal cancer (subgroup 3). A retrospective observational study (SCALAWS: Scaling laws, shape factors and fractal measures in human cancers) was designed and approved by the IRB of the participating institutions. Inclusion criteria were: histological confirmation of advanced rectal cancer diagnosis, availability of pretreatment PET/CT and lesion size larger than 2 cm. A total of 23 rectal cancer patients (16 male, 7 female, age rank 54-80, median age 72 years) from the period October 2007 to October 2009 were included in the study. PET protocol and machine were as in subgroup 1. Lung cancer (subgroup 4). 175 patients (153 men, 22 women, age rank 41-84, median 65 years) were included in the SCALAWS study from a dataset of lung cancer patients that received surgery in the period June 2007 to December 2016.
PET machine was as in subgroup 1. PET adquisition was initiated 40 min after the intravenous administration of 185 MBq of 18 F-Fluorocholine. A brain scan was performed starting with a low-dose CT transmission study (modulated 120 kV and 80 mA) without intravenous contrast followed by 3D emission study with an acquisition time of 20 min (one single bed), voxel size of 2. 3 × 2. 3 × 3. 3 mm in a matrix of 128 × 128 and reconstructed using the CT images for attenuation correction and applying an iterative reconstruction algorithm. Breast cancer patients (subgroup 6). Pretreatment 18 F-FLT PET/CT scans of patients of the ACRIN 6688 observational study available in the TCIA (ACRIN-FLT-Breast) were included in the study  [36]. This dataset included histologically confirmed breast cancer patients (100% women), of them 46.
6 mm, 0. 438-0. 575 mm  resolutions, and 0. 8 -1. 3 mm spacing between slices. Lung cancer patients (subgroup 8). Patients included were participants in the study SCALAMATH. Five patients (3 men, 2 women) were included. Three of them were diagnosed as adenocarcinoma and two as squamous cell carcinomas. Age rank was 60-72 years (median 68). All of them were initially stage I tumours and progressed without treatment. We drew from the database of follow-up screenings in I-ELCAP between 2008 and 2019, which were performed according to a common protocol  [37]  using low-dose CT (LDCT). Enrollment was limited to those aged 50 years or older, with a smoking history of at least 10 pack-years, no previous cancer and general good health. Participants harboring parenchymal solid or part-solid non calcified nodule with at least three or more follow-up CTs were identified according to specified criteria in the A total of 22 imaging studies were used (range of studies per patient  [3] [4] [5] [6].
4. ). In case of discordance a third evaluator revised the images. In the visual evaluation, a PET scan was considered as positive if any uptake, higher than normal tissue background, was detected. Only positive PET scans were considered for tumour segmentation, i. e. those having a SUVmax larger than twice the background activity readings. PET images in DICOM files were imported into the scientific software package Matlab (R2018b, The MathWorks, Inc. , Natick, MA, USA). The tumour PET images were manually placed in a 3D box and then semi-automatically delineated using a grey-level threshold chosen to identify the metabolic tumour volume. Segmentations were corrected manually slice by slice as in  [41]. All segmentations were performed by at least a nuclear medicine physician and an imaging engineer, both with more than 5 years of experience in tumour segmentation.