Mathieu Hatt
email: hatt@univ-brest.fr

Mohamed Majdoub

Martin Vallières

Florent Tixier

Catherine Cheze

Le Rest

David Groheux

Elif Hindé

Antoine Martineau

Olivier Pradier

Roland Hustinx

Catherine Cheze Le Rest

Elif Hindié

Remy Perdrisot

Remy Guillevin

Issam El

Dimitris Visvikis

FDG PET uptake characterization through texture analysis: investigating the complementary nature of heterogeneity and functional tumor volume in a

Keywords: 18 FDG-PET/CT, heterogeneity, textural features, metabolically active tumor volume, prognosis

ABSTRACT

Intra-tumor uptake heterogeneity in 18 F-FDGPET has been associated with patient treatment outcomes in several cancer types. Textural features (TF) analysis is apromising method for its quantification. An open issue associated with the use of TF for the quantification of intratumoral heterogeneity concerns its added contribution and dependence on the metabolically active tumor volume (MATV), which has already been shown as a significant predictive and prognostic parameter.Our objective was to address this question using a largercohort of patients covering different cancer types.

Methods: Asingle database of 555 pre-treatment 18 F-FDG PET images (breast, cervix, esophageal, head&neck andlung cancer tumors) was assembled. Fourrobust and reproducibleTF-derived parameters wereconsidered.The issues associated with the calculation of TF using co-occurrence matrices(such as the quantization and spatial directionality relationships) werealso investigated. The relationship betweenthese featuresand MATV, as well as among the features themselves was investigated using Spearman rank coefficients, for different volume ranges.The complementary prognostic value of MATV and TF was assessed through multivariate Cox analysis in the esophageal and NSCLC cohorts.

Results: A large range of MATVs was included in the population considered (3-415 cm 3 , mean=35, median=19, SD=50). The correlation between MATV and TF varied greatly depending on the MATVs, with reduced correlation for increasing volumes. These findings were reproducible across the different cancer types. The quantization and the calculation method both had an impact on the correlation.Volume and heterogeneity were independent prognostic factors (P=0.0053 and 0.0093respectively) along with stage (P=0.002) in NSCLC, but in the esophageal 18 F-FDG Positron Emission Tomography/Computed Tomography (PET/CT) is a powerfultoolfor diagnosis and staging inoncology [START_REF] Krause | FDG PET and PET/CT[END_REF]. Its use intherapy assessment [START_REF] Wahl | From RECIST to PERCIST: evolving considerations for PET response criteria in solid tumors[END_REF][START_REF] Herrmann | F-FDG-PET/CT in evaluating response to therapy in solid tumors: where we are and where we can go[END_REF]isincreasing.Within this context more and more emphasis is being given to image-derived indices [START_REF] Visvikis | The age of reason for FDG PET image-derived indices[END_REF]. On the one hand, features extracted from PET images, including metabolically active tumor volume (MATV), mean standardized uptake value (SUV mean ) and total lesion glycolysis (TLG) have providedpotentially higher prognostic value than standard maximum SUV (SUV max ) in various cancer types [START_REF] Van De Wiele | Predictive and prognostic value of metabolic tumour volume and total lesion glycolysis in solid tumours[END_REF].On the other hand, more recently the heterogeneity of 18 F-FDG uptakes within tumorshas been associated with treatment failure [START_REF] Visvikis | The age of reason for FDG PET image-derived indices[END_REF][START_REF] Lambin | Radiomics: extracting more information from medical images using advanced feature analysis[END_REF][START_REF] Chicklore | Quantifying tumour heterogeneity in 18F-FDG PET/CT imaging by texture analysis[END_REF][START_REF] Davnall | Assessment of tumor heterogeneity: an emerging imaging tool for clinical practice?[END_REF].Proposed approaches for the assessment of intra-tumor activity distribution heterogeneity includevisual evaluation [START_REF] Miller | Improved prognostic value of 18F-FDG PET using a simple visual analysis of tumor characteristics in patients with cervical cancer[END_REF], SUV coefficient of variation (SUV COV ) [START_REF] El Naqa | Exploring feature-based approaches in PET images for predicting cancer treatment outcomes[END_REF], area under the curve of the cumulative histogram (CH AUC ) [START_REF] Van Velden | Evaluation of a cumulative SUV-volume histogram method for parameterizing heterogeneous intratumoural FDG uptake in non-small cell lung cancer PET studies[END_REF],fractals [START_REF] Miwa | FDG uptake heterogeneity evaluated by fractal analysis improves the differential diagnosis of pulmonary nodules[END_REF]or textural features(TF)analysis [START_REF] El Naqa | Exploring feature-based approaches in PET images for predicting cancer treatment outcomes[END_REF][START_REF] Tixier | Intratumor heterogeneity characterized by textural features on baseline 18F-FDG PET images predicts response to concomitant radiochemotherapy in esophageal cancer[END_REF].The lattercan provide a number of parametersquantifyingtumor heterogeneity at the scales of voxels or groups of voxels. A recent study, based on the use of one of these parameters (local entropy calculated from co-occurrence matrices) has suggested that a minimum MATV of 45cm 3 is required to provide an estimate of heterogeneity independent of MATV confounding effects [START_REF] Brooks | The effect of small tumor volumes on studies of intratumoral heterogeneity of tracer uptake[END_REF]. However, this studyinvestigateda single heterogeneity parameter, ina single cancer type, and used an image quantization schemewith>150 grey levels. Another recent study investigating the relationship between MATV and TF [START_REF] Orlhac | Tumor texture analysis in 18F-FDG PET: relationships between texture parameters, histogram indices, standardized uptake values, metabolic volumes, and total lesion glycolysis[END_REF]has included a relatively small number of tumors (including patients who already had distant metastases) and cancer types,without explicitly reporting on the lesion sizes considered or making conclusions regarding the minimum tumor volume that should be considered. In addition, this study used Pearson correlation to testlinear relationship between MATV and TF, which would missnon-linear trendsthat may exist between these parameters.

To our knowledge, the potential interactionbetween MATV and TF has not been previously considered within the context of patient outcome prognosis.

This study was therefore designed to investigate in detail, and across a large number of primary tumors and cancer types, the relationship between tumor MATV and derived heterogeneity measurements using TF, in order (a) to determine whether a minimum MATV should be consideredin such analyses and (b)if tumor heterogeneity quantified through TF could provide complementary prognostic value relative to MATV.

MATERIALS AND METHODS

FDG PET images

Severalpatient cohorts wereretrospectively collectedin a dataset of 555 The institutional review boards of each involved group in this work approved this retrospective study and the requirement to obtain informed consent was waived.

Image Analysis

MATVswere firstdelineatedusing the Fuzzy Locally Adaptive Bayesian (FLAB) algorithm [START_REF] Hatt | A fuzzy locally adaptive Bayesian segmentation approach for volume determination in PET[END_REF], which has been previously validated for accuracy and robustness using simulated and clinical datasets,including homogeneous and heterogeneous MATVs [START_REF] Hatt | Accurate automatic delineation of heterogeneous functional volumes in positron emission tomography for oncology applications[END_REF][START_REF] Hatt | Impact of tumor size and tracer uptake heterogeneity in (18)F-FDG PET and CT nonsmall cell lung cancer tumor delineation[END_REF]. FLAB was used withtwo or three classes in order to adequately cover the entire MATV, including low uptake areas.The H&N tumors from McGill were manually delineated by an experienced radiation oncologist on fused PET/CT images as part of their radiotherapy treatment planning.

The differences in scanner, reconstruction and delineation between the H&N McGill dataset and the other cohorts were taken into account by only considering features that were shown to be the most robust with respect to PVEand segmentation [START_REF] Hatt | Robustness of intratumour 18 F-FDG PET uptake heterogeneity quantification for therapy response prediction in oesophageal carcinoma[END_REF] or reconstruction settings [START_REF] Galavis | Variability of textural features in FDG PET images due to different acquisition modes and reconstruction parameters[END_REF], as well as test-retest reproducibility [START_REF] Tixier | Reproducibility of tumor uptake heterogeneity characterization through textural feature analysis in 18F-FDG PET[END_REF].

Consequently,the present analysis included fourTFs, which have been previously

shown to have a predictive and prognostic value in different cancer types.The two local TFscalculated using co-occurrence matrices(13)were entropy (E)and dissimilarity (D). D and E were calculated according to two different methods: 1) using 13 matrices, one for each spatial direction, followed by averagingthe values calculated separately in each matrix, and2) using only one matrix taking into account all 13 directions simultaneously without an averaging step.From a conceptual point of view, the second method is more accurate, as it fully describes the 3D co-occurrence properties within the volume. The first method results in taking an average from fewer co-occurrence measurements, thus information could be lost and the complexity of the distribution of grey levels may not be optimally captured. On the other hand, averaging could artificially reduce the effect of the residual noise from the quantization process. The parameters related to the first method (averaging 13 matrices) will be denoted as E 13 and D 13 , whereas the parameters from the second method (1 matrix) will be denoted as E 1 and D 1 .

RegionalTFs calculated using size-zone matrices (13)were high intensity large area emphasis (HILAE) and zone percentage (ZP).

Other TFs were not included in the present analysis because of high correlation with D and/or E (correlation>0.8), or previously shown poor reproducibility/robustness (19-21) (supplemental Table 2).

Although an optimal quantization of 64 grey levels was identified in our previous work concerning the reproducibility [START_REF] Hatt | Robustness of intratumour 18 F-FDG PET uptake heterogeneity quantification for therapy response prediction in oesophageal carcinoma[END_REF], we also investigated the impact of the quantization pre-processing step by considering values from 4 to 256.

Finally, SUV max , SUV mean , and SUV COV were includedfor comparison purposes.

Statistical Analysis

Statistical analyses were performed using Medcalc TM (MedCalc Software, Belgium).

Spearman rank correlation (rs) was used to study relationships between parameters, since such relationships are non-linear and all these parameters are frequently not normally distributed. In order to assess the potential complementarity and/or additional clinical value of MATV and derived heterogeneity parameters, a survival analysis was performed in theesophageal and NSCLC cohorts for which overall survival (OS), follow-up and other clinical data were available (Supplemental Tables3 and 4), which was not the case for the other cohorts. Association with OS was assessed using univariate Cox proportional hazard regression including features as continuous variables (i.e. no dichotomization). Correction for multiple testing was performed using the false discovery rate Benjamini-Hochberg step-up procedure. It consists in declaring positive discoveries at level α (here α=0.05), among the k=1…K tested variables ordered according to their p-values p in increasing order, those ranked above the one satisfying the condition p(k)≤ 𝑘 𝐾 × α [START_REF] Benjamini | Controlling the false discovery rate: apractical and powerful approach to multiple testing[END_REF].After univariate analysis, a stepwise multivariate Cox analysis(significant variables are entered sequentially, then removed if they become non-significant) was performed to identify independent prognostic factors. Kaplan-Meier survival curves (with optimal cut-off values determined through ROC curves analysis) were subsequently used to assess the prognostic value of previously identified independent factors. Median survival, percentage of deaths in each group and hazard ratios(HR) were reported for each risk factor separately and for their combination, in order to quantify anyimprovedpatient stratification.

RESULTS

Correlation betweenparameters

To avoid confusion, absolute rs values are reported, although correlation direction results can be found in figure 2. Threeimportant observations can be emphasized.

First, significant detailsregarding the grey-levels distribution are lost when using a quantization <32 (Fig. 3), and the quantization had an important impact on the correlation between volume andE 1 , decreasing from almost 1 for a quantization of 256, to <0.6 for quantization<64. Correlation with MATV was much lower for E 1 compared to E 13 , except for quantization <16. In contrast, dissimilarity (either D 1 or D 13 ) was insensitive to the quantization value (rs~0.8) and there was no difference between D 1 and D 13 .The correlation ofHILAE and ZP withMATV was very sensitive to the quantization, although contrary to entropy, rs increased from <0.5 to >0.8 with decreasing quantization (Fig. 3).For quantization=32, the correlation with volume was >0.75 and <0.85 for all TFs except E 1 (0.3), whereas withquantization=64 there was a wider range (rs<0.2 for HILAE to rs≈1 for E 13 ), and several TFs had a correlation <0.7, suggesting a higher potential of complementary information with respect to MATV for 64 than 32. Quantization=64 was also previously shown to provide the highest TFs' reproducibility [START_REF] Tixier | Reproducibility of tumor uptake heterogeneity characterization through textural feature analysis in 18F-FDG PET[END_REF] and robustness [START_REF] Hatt | Robustness of intratumour 18 F-FDG PET uptake heterogeneity quantification for therapy response prediction in oesophageal carcinoma[END_REF].A quantization into 64 grey-levels wasthus considered for the rest of the analysis, as it represents the best compromise between sufficient sampling of voxel SUVs, preservation of original intensities information, andpotential complementary information with respect to MATV.

Secondly, significant correlations were found amongst almost all features considered.

In addition, MATV correlated with those features, as well as with SUV max and SUV mean (Fig. 2). The correlation between the co-occurrence matrix-derived features themselves and their respective correlation with MATV were sensitiveto theuse of asingle matrix compared to averaging 13 directional matrices. This was particularly true for entropy. The correlation between D 13 and E 13 was 0.76, whereas the correlation between D 1 and E 1 was 0.18. Correlation of D 13 and E 13 with MATV was 0.80 and 0.96, whereas it was 0.82 and 0.56 for D 1 and E 1 respectively (Fig. 2).

Thirdly, correlation between D 1 , E 1 , ZP, and HILAE with MATV ranged from 0.17 to 0.96, suggesting that a substantial amount of complementary information with respect to MATV may be found in some of these heterogeneity quantification features, similarly to SUV measurements which exhibited correlations of 0.31 to 0.42.

Figures 4 and 5 provide visual representations of the distributions of TFs with respect

to MATV.Supplemental figures 1-4 provide similar graphsfor each cancer type, for other TFs and quantization values. The difference between E 13 and E 1 was important (Fig. 4A-B), the calculation using only one matrix leading to a much tighter distribution with a smaller range (5.6-8 for E 1 vs. 3.9-7.3 E 13 ) of higher values (7.1±0.4 vs.

5.7±0.8), and with substantially lower correlation with volume (0.56vs. 0.96). The difference between D 13 and D 1 (Fig. 4C-D) was less important but nonetheless led also to a tighter distributionfor D 1 with a smaller range (3.5-24.1 vs. 0.9-30.5) of higher values (12.6±3.8 vs. 10.7±3.7) but with a similar correlation with MATV (0.82vs.

0.80).

By restricting the analysis to larger tumor volumes (from ≥10cm 3 to ≥60cm 3 using 5cm 3 steps), it was found thatthe correlation between TFs and MATV tended to decrease substantially with ranges of increasing volumes (Fig. 6). This was observed for all TFs but was especially true for entropy.The only exception to this observation was HILAE since its correlation with MATV was low even when considering the entire dataset.E 13 correlation with MATVdropped from 0.96 when considering the entire range of volumes, to <0.3 when considering only tumors >60cm 3 . E 1 dropped to <0.25 for volumes >10cm 3 . Similar observations were made for D 13 with rs>0.8 when considering all tumors, dropping <0.6 in tumors >15cm 3 . Although the overall correlation of D 1 with MATV was slightly higher than for D 13 , it was also more rapidly reduced with increasing MATV (Fig. 6). The same analysis for ZP led to similar observations, witha reduced correlation with increasing MATV when considering larger tumors, from 0.68for all tumors to 0.5 for those >15cm 3 .

The relationships between MATV and TFs were similar across the different tumor types, although measured correlations varied: cervix tumors included mostly large tumors (MATV>20cm 3 ), whereas the other tumor types included a larger number of tumors with MATV<10cm 3 (Fig. 1). The resulting correlation between E 1 and MATV was not significant for cervix cancer tumors (rs<0.003, P>0.9), whereas it wasfor esophageal, NSCLC, H&N, and breast tumors (rs=0.80, 0.39, 0.35, and 0.74 respectively). For dissimilarity (D 1 ) and ZP, smaller differences were observed, with rs from0.73 and 0.63 (cervix) to 0.93 and 0.83 (NSCLC) for D1 and ZP respectively.

HILAE correlations with MATV were <0.2 for all tumor types.

Survival prognosis in esophageal and NSCLC cohorts

In the esophageal cohort, median OS was17.0 months (range 1.0-71.0, mean 22.0), with 44 patients still alive at last follow-up. The trend for MATV did not reach statistical significance (p=0.0315) and none of the variableswere prognostic factorsexcept D 1 (p=0.0016) (Table 1), therefore no multivariate analysis was performed. However,if dichotomized with optimal cut-off values in the K-M analysis, both D 1 (HR 1.92, p=0.0052)and MATV (HR 1.66, p=0.0375) could differentiate survival curves(Fig. 7A). Adding volume and heterogeneity increased HR to 2.02 (p=0.0024, 95% CI 1.22-3.34) with 23months (N=64) vs.10months (N=48) median OS (supplemental table 5, Fig. 7A).

In the NSLC cohort, medianOS was 18.4 months (range 1.1-57.4, mean 27.4), with 39 patients still alive at last follow-up. In the univariate analysis, all variables were significantly associated with OS, expect age, histology, smoking history and SUV max (Table 1).The multivariate analysis retained clinical stage (P=0.0018), MATV (P=0.0053) and heterogeneity (E 1 , P=0.0093) as independent prognostic factors.MATV allowed for the highest differentiation (HR=2.8), whereas stage and heterogeneity led to lower HRs (2.3 and 2.1 respectively). Adding stage to MATV did not increase stratifying power (HR=2.84), whereas adding heterogeneity increased HR to 3.55, with 49monthsvs. 9.1months median OS. The addition of the three led to the highest HR of 3.81(supplemental table 6, Fig. 7B). Survival curves were also evaluated according to a 3-valued score (MATV<35cm

DISCUSSION

There is an increasing interest in the use of PETimagetextural featuresforquantification of intra-tumorheterogeneity [START_REF] Visvikis | The age of reason for FDG PET image-derived indices[END_REF][START_REF] Lambin | Radiomics: extracting more information from medical images using advanced feature analysis[END_REF].Few studies investigated the relationships between tumor volume and TFs [START_REF] Orlhac | Tumor texture analysis in 18F-FDG PET: relationships between texture parameters, histogram indices, standardized uptake values, metabolic volumes, and total lesion glycolysis[END_REF][START_REF] Hatt | Robustness of intratumour 18 F-FDG PET uptake heterogeneity quantification for therapy response prediction in oesophageal carcinoma[END_REF][START_REF] Willaime | Quantification of intratumour cell proliferation heterogeneity using imaging descriptors of 18F fluorothymidine-positron emission tomography[END_REF].

Most studies using textural features considered volumes>3-5cm 3 , assuming that PET could not characterize heterogeneity on smaller volumes due to its limited spatial resolution. A recent theoretical analysis suggestedthat volumes>45cm 3 shouldbe consideredto avoid volume related confounding effects [START_REF] Brooks | The effect of small tumor volumes on studies of intratumoral heterogeneity of tracer uptake[END_REF]. However, thisanalysisconsidered a single parameter (entropy), calculated on 2D co-occurrence matricesover 2 spatial directions followed by averaging, and using a quantizationvalue >150 [START_REF] Brooks | The effect of small tumor volumes on studies of intratumoral heterogeneity of tracer uptake[END_REF].

In this work, we addressed the question of the minimum functional volume that could be considered and investigated the potential complementary prognostic value between volume and heterogeneity.We investigated the influence of the quantization pre-processing and of thetextural featurescalculation methodology, investigating more thoroughly the relationships between heterogeneity and functional volume in a substantially larger tumors' database, covering large tumor volume ranges and different cancer types.Considering a patient cohort with variable cancer types and complementary volume ranges allowed providing a better picture of the relationship between the measured feature and its corresponding volume.

Our results partly confirm those of recent studies. Indeed, severaltextural featureswere found to be highly correlated with the volumefrom which they were calculated.In addition, high correlations were found between most of these parameters.We observed differences in the heterogeneity-volume distributions as a function oftumor type, although these differences can beexplained by differences in volumeranges for each cancer patient cohort, rather than the histology or heterogeneity specific to each tumor type. For instance, the distribution of textural features with respect to volume in the cervix tumors was significantly different than the others because it included only large tumors (>20cm 3 )relative to the other patient cohorts considered that also included smaller tumor volumes.

Our results emphasize that(i) the relationship and the level of correlation is not the same for all features, (ii) the level of correlation tends to decrease substantially when consideringlargertumor volumes, (iii) the calculation method and the quantization step both have an impact on the relationshipand level of correlation between volume and the feature, and (iv) volume and heterogeneity can have complementary prognostic value: in the NSCLC cohort, heterogeneity and volumewere identified as independent prognostic factors and hazard ratios were shown to increase from <2.9 to >3.8 when adding these risk factors for patient stratification. Although multivariate analysis could not be performed in the esophageal cohort, combining heterogeneity with volume led to an increased although non-statistically significant different stratification. NSCLC tumor volumes were much larger than esophageal ones (58±77cm 3 , median 34, range 3-415 vs. 25±27cm 3 , median 15, range 3-140), which is likely why heterogeneity and volume had higher complementary prognostic values in NSCLC.

When considered together, these results point to the potential added prognostic value of tumor heterogeneity quantified with textural feature, although regarding Kaplan-Meier curves, cut-off values found with ROC analysis are probably over-fitted and specific to the data, and thus validation in independent cohorts will be required.

The quasi-linear relationship between entropy (either E 1 or E 13 ) and volume in the range3-10cm 3 (rs>0.9, Fig. 4) suggests that entropyfor volumes<10cm 3 cannotprovide complementary information. However, when choosing appropriate quantization and co-occurrence matrix calculation, the correlation between volume and entropy for volumes in the range 10-50cm 3 was much lower. The minimal volume to consider may therefore be closer to 10cm 3 than 45cm 3 , a value previously suggested based on calculations carried out after a quantization into >150 grey-levels and using several co-occurrence matrices followed by averaging [START_REF] Brooks | The effect of small tumor volumes on studies of intratumoral heterogeneity of tracer uptake[END_REF]. In addition, for other TFs not considered in that last study, the correlation was weaker, even for volumes between 3 and 10cm 3 . It was only 0.4 and 0.6 for D 13 and D 1 respectively, and 0.3 and 0.2 for ZP and HILAE. Therefore, instead of excluding patients with volumes below the proposed 10cm 3 threshold from such analysis, we rather recommend to report the correlation between volume and heterogeneity and highlight their complementary value as tumor volumes increase. Indeed, the correlation with volume decreased substantially for all textural features when considering larger volumes.Larger tumors are known to exhibit higher hypoxia, necrosis or anatomical and physiological complexity at the microscopic and macroscopic scales, which logically translates to higher FDG uptake spatial distribution complexity and consequentlyassociated heterogeneity quantification. A significant correlation between quantified heterogeneity (viaany method) and its corresponding volume is therefore to be expected in a standard cohort of patients covering a large range of tumor sizes. Prospective clinical studies with optimized pre-defined imageacquisition settings and heterogeneity analysis protocols, as well as experimental studies demonstrating that tumor properties on a scale comparable with the PET resolution can actually lead to identifiable image textural features could benefit from the results

of the present study and should now be conducted.

One important aspect of this study is that the 555 PET images in our databasecame from six cohorts acquired in differentcenters, albeit with certain homogeneity in acquisition parameters (same scanner, reconstruction algorithm, voxel size), except for one. This variability was handledby restricting the analysis to robust features only,as to minimize the associated impact.Indeed, excluding the H&N cohort with a different imaging protocol did not change the results. On the other hand, the inclusion of a small number of heterogeneity quantification metrics may be considered as a limitation. However, the four textural featuresincludedhave been previously shown to be the most reproducible and robust amongst those shown to have a predictive and prognostic value in different cancer types. We also restricted the prognosis analysis to the esophageal and NSCLC cohorts because clinical and survival data for the other cohorts were not available forenough patients to allow formultivariate analysis.

CONCLUSIONS

Most of textural features considered to quantify intra-tumor heterogeneity were found to be significantly correlated with tumor volume. However, our detailed analysis also suggests that heterogeneity quantification through textural features may potentially provide valuable clinical complementary information in addition to functional volume, especially for tumors above 10cm 3 , with increasing complementary prognostic value for larger volumes. In 112 esophageal patients, heterogeneity was found to have some prognostic value that was not significantly improved when combined with volume, whereas in a cohort of 101 NSCLC patients, heterogeneity, volume and stage were independent prognostic factorsthat allowedincreased stratification of patients when combined.

Fig. 5B ,

 5B Fig.5B, Fig.7C), highlighting the higher complementary value of heterogeneity and

Figure 1

 1 Figure 1

Figure 7 Figure captions Figure 1 .

 7captions1 Figure 2

Figure 2 .

 2 Figure 2.Illustration of quantization (A) and impact on the correlation between TF and

Figure 3 .

 3 Figure 3. Spearman rank correlationsbetweenparameters(555 tumors). Red: [0.8,1.0]

Figure 4 .

 4 Figure 4. Distributions of E 1 (A), E 13 (B), D 1 (C) and D 13 (D) (quantization=64) with

Figure 5 .

 5 Figure 5. Distributions of HILAE (A) and ZP (B) (quantization=64) with respect to

Figure 6 .

 6 Figure 6. Absolute Spearman rank correlation with MATV for each TF

Figure 7 .

 7 Figure 7. Kaplan-Meier curves using volume and heterogeneity for(A) 112

  18

Table 1 .

 1 Prognostic value for OS in the NSCLC and esophagealcohorts assessed using Cox regression for univariate and multivariate analysis

			Esophageal	NSCLC
	Parameters	p-value Univariate Univariate Multivariate p-value
		Surgery	0.3654	0.0084	NRM
		Age	0.1861	0.0441 ¥	-
	Clinical	Sex Smoker	0.1795 0.4511	0.0227 0.8997	NRM -
		Histology	0.2154	0.3041	-
		Stage	0.0391 ¥	0.0003	0.0018
	Volume and SUV	MATV SUV max SUV mean	0.0315 ¥ 0.2781 0.6008	0.0008 0.0599 0.0256	0.0053 -NRM
		D 13	0.0405 ¥	0.0046	NRM
		D 1	0.0016	0.0027	NRM
	Heterogeneity (TF)	E 13 E 1	0.1087 0.3922	0.0002 0.0287	NRM 0.0093
		HILAE	0.2596	0.0132	NRM
		ZP	0.4391	0.0005	NRM
		NRM: not retained in the model

¥

: Not significant after correction for multiple testing

This work has received a French government support granted to the CominLabs excellence laboratory and managed by the National Research Agency in the "Investing for the Future" program under reference ANR-10-LABX-07-01.


Patrick Maillé
email: patrick.maille@telecom-bretagne.eu

Bruno Tuffin
email: bruno.tuffin@inria.fr

How Do Content Delivery Networks Affect the Economy of the Internet and the Network Neutrality Debate?

This paper investigates the economic impact and strategies of Content Delivery Networks (CDNs), Internet actors that reduce the capacity needs in the backbone network and improve the quality perceived by users. We consider so-called push and pull models where the traffic is paid by the sender or the receiver, respectively, as well as the situation where the CDN is (vertically) integrated to, i.e., owned by, an Internet Service Provider (ISP). We then discuss the implication of CDNs into the network neutrality debate, another issue forgotten by researchers and regulators.

Introduction

Content delivery networks (CDNs) are large distributed systems of servers deployed within the Internet, aimed at putting data closer to end users to offer better quality of service (QoS) and availability [START_REF]Content Delivery Networks[END_REF]. The role of CDNs is to produce the highest performance for content accessed by users, some typically demanding applications being video delivery or gaming. They are therefore of interest for content/service providers, but also for network providers since reducing the load (congestion) on the network: the most popular pieces of content, being stored at the edge of the network, do not need to be downloaded from their source upon each request.

But the telecommunications economics literature barely addresses the role of CDNs, although they have become major actors of the Internet. While there has been a huge amount of work on CDNs from a technical point of view, the literature on the economics of CDNs is more limited. Existing works focus on the optimal pricing strategies of CDNs [START_REF] Hosanagar | Service adoption and pricing of content delivery network (CDN) services[END_REF], or discuss if and how different CDNs within a network should cooperate when setting their caching strategies [START_REF] Atzmon | Replacement policies for a distributed object caching service[END_REF][START_REF] Christin | Economics-informed design of CDNs[END_REF][START_REF] Chun | Selfish caching in distributed systems: A game-theoretic analysis[END_REF][START_REF] Hosanagar | Content Delivery Networks[END_REF][START_REF] Hosanagar | Optimal pricing of content delivery network (CDN) services[END_REF]. But the interactions between CDNs, ISPs, and CPs, have received little attention, while they have a major impact all the different actors.

This paper is a first step in the direction of modeling and analyzing the relations between CDNs and other Internet actors:

• We extend a previous model of ISP competition by including CDNs into the picture, to study how they interact and what their best strategies are.

• We consider both the so-called pull model -where the ISP requesting content has to pay the ISP hosting it -and the push model -where the transfer fee is paid by the ISP hosting content to the one hosting end users; in addition the CDN can be an independent entity or be owned and managed by an ISP. For each case, we illustrate the impact of CDNs on ISPs' revenues and on fairness between CPs. • We then discuss the impact of CDNs on the network neutrality debate [START_REF]Net Neutrality or Net Neutering: Should Broadband Internet Services be Regulated[END_REF][START_REF] Maillé | Internet governance and economics of network neutrality[END_REF][START_REF] Wu | Network neutrality, broadband discrimination[END_REF], in which CDNs have also barely been addressed although they reduce traffic transit, thereby addressing in part the concerns that ISPs express in the debate.

Model

Extending the model in [START_REF] Coucheney | Network Neutrality Debate and ISP Inter-Relations: Traffic Exchange, Revenue Sharing, and Disconnection Threat[END_REF], we consider two ISPs (A and B), in competition for end users. We assume a continuum of content, whose total mass normalized to 1 represents the total volume downloaded per time unit (i.e., the mass of each piece of content is proportional to its popularity). Let x (resp., 1 -x) be the proportion of content that is directly connected to the Internet through A (resp., B).

At each ISP, we can have a CDN, indexed by the name of its host ISP, caching some of the content rooted at the other ISP to improve QoS and avoid potential transfer fees between ISPs. Each CDN can be an independent actor (ex: Akamai), or a service implemented by the hosting ISP (the CDN is then said vertically integrated ).

Figure 1 represents the actors and their economic relations. Users are treated too as a continuum (individuals having a negligible impact) of total mass 1. The price p A (resp. p B ) is the flat-rate subscription fee of ISP A (resp. B), and t is a unit price for the traffic transferred between ISPs: t = 0 corresponds to the classical peering agreement with no fee, while t = 0 means a paid transit. Finally, only a portion of the downloaded content is cached by the CDNs, we denote those quantities by y A ≤ 1 -x at CDN A and y B ≤ x at CDN B.

Users' choice of ISP (if any) is through a standard attraction model [START_REF] Ben-Akiva | Discrete choice analysis[END_REF]: the proportion of users choosing Option i ∈ {A, B, 0} -0 meaning no subscriptionis

σ i (p A , p B ) def = (x i /p i ) α (x A /p A ) α + (x B /p B ) α + 1/p α 0 , (1)
where α > 0 is a sensitivity parameter, p 0 represents the cost of not benefitting from any content, and x i is the average perceived quality with ISP i ∈ {A, B}.

Content is reachable with high quality γ > 1 (from the local ISP or its host CDN) or low quality 1 (from the distant ISP), yielding

x A = 1 + (x + y A )(γ -1); x B = 1 + (1 -x + y B )(γ -1).
Each ISP i ∈ {A, B} focuses on net benefits, stemming from: 1) end-users subscriptions, the corresponding incomes being proportional to the market share σ i and the price p i ;

2) the potential gains/costs from hosting a CDN:

if the CDN is an independent actor (Akamai for instance), we assume the CDN pays a per-unit-of-volume fee r to the ISP for delivering content from the CDN cache servers to the ISP users. -If the CDN role is managed by the ISP itself (something increasingly happening), r is the per-unit-of-volume revenue the ISP gets from remote CPs for serving as a CDN. We also add a storing cost c i depending on y i , that can be considered convex since the most popular contents will be cached in priority.

3) And the potential transit fees between ISPs: content associated to ISP A (resp. B) and transferred to B (resp. A) being proportional to x -y B (resp. 1 -x -y A ) and to ISP B (resp. A) market share, the net (possibly negative) amount ∆ A,B of traffic from A to B is

∆ A,B = (x -y B )σ B (p A , p B ) -(1 -x -y A )σ A (p A , p B ).
-In the Pull model, the transfer fee is paid by the ISP hosting end users to the one hosting content, i.e., the service is asked by end users. Let t be the per-unit-of-volume price. This leads to a revenue t∆ A,B for A, from B (and an equal cost for B). -Push model : the transfer fee is paid by the ISP hosting content to the one hosting end users, i.e., the service is asked by content providers. The same type of expression can be used with t < 0.

Regrouping, the ISP utilities per time unit are:

U A (p A , p B ) = σ A (p A , p B )p A + t∆ A,B + ry A -c A (y A ), (2)
U B (p A , p B ) = σ B (p A , p B )p B -t∆ A,B + ry B -c B (y B ).
We neglect the transfer cost of files from (remote) content providers to CDNs, since that transfer is done only once.

3 How much content to cache?

We focus here on the amount of cached content that benefits the most to the ISP hosting the CDN.

Pull model and independent CDN

From (2), if ISP i hosts a CDN increasing its cached content: i) ISP i attracts more users because providing a better QoS (i.e., σ i increases), ii) ISP i gains more from the CDN (ry i increases with y i ), iii) ISP i pays less for transit because "pulling" less traffic. U i thus increases with y i , ISPs should then let CDNs cache as much content as possible.

Push model and independent CDN

Now the ISP still gains from CDN payments and increased attractiveness to users, but loses from transfer fees since its competitor will have to "push" less traffic. Figure 2(a) shows the values of U A when y A (∈ [0, 1 -x]) varies, for three values of r and with α = 1.5, x = 0.6, t = -1, y B = 0.3, γ = 1.5, p 0 = 1, p A = 1.3, p B = 1.7. Note that c A = c B = 0, since the storage cost is borne by the CDN. Depending on the parameters, it can be optimal to cache nothing, the whole content, or an intermediate amount, but we expect the optimal cached amount to increase with r, because the ISP's revenue from the CDN increases.

Pull model and integrated CDN

We take here the same values as for Figure 2(a), except that now t = 1 (pull model) and that the ISP bears storage costs, of the form c A (y A ) = (y A ) 3/2 . Because of that cost, we end up too with a trade-off. Figure 2

(b) displays U A in terms of y A (∈ [0, 1 -x])
. Note here that because c ′ A (0) = 0, the derivative of the utility is positive at 0 (the other effects improve revenue), hence there are always incentives to cache some content. Again, when r is large enough, it becomes beneficial to cache all content.

Push model and integrated CDN

With the same set of values as for Figure 2(a), but with the storage costs c A (y A ) = (y A ) 3/2 , we studied the revenue U A when y A varies. The curves have a shape similar to those of Figure 2(a), we omit them due to space constraints.

Even in the push model, an ISP can have an interest in using an integrated CDN if r is large enough.

Caching and its impact on revenues

Figure 3 (top) plots the optimal cached amount y A and the corresponding utility U A in terms of x, for each of the four situations, adding the curve of U A when no CDN exists. Note it is not relevant to compare the numerical values between integrated and independent: r is taken the same (a cost c A is added to the integrated case), but it represents a price paid by the CDN in the independent case and a price paid by CPs in the integrated case. Remark in Figure 3 (top) that for the push/integrated case, y A is 0 or close to 0. In the push/independent case, caching a non-negligible proportion is better for small x but the optimal cached part quickly goes to 0 as x increases. As seen before, all B content (i.e., y A = 1-x) is cached in the pull/independent case. For the pull/integrated model the optimal y A increases with x when y a < 1 -x. Note that U A increases with x in the pull cases, but the trend is to decrease in the push ones, suggesting there is no interest for ISPs to attract CPs for the latter. Moreover the no-CDN curve is very close to the integrated one in the push case, because y opt A is close to 0. Caching does not yield a large revenue improvement in the pull/integrated case either (while it does for the independent case), but the gain can be significant with different parameters. Figure 3 (bottom) displays the same metrics, but when the price r varies. For small values of r, in both push cases there is no incentive to cache content, because gains are not sufficient, but above a threshold caching becomes beneficial. The revenue logically increases in r too (in cases with CDN).

Fairness concerns

We now focus on the impact of CDNs on CPs: Is there a difference of treatment among CPs according to the ISP they are attached to? To answer, we use the (aggregated) perceived quality Q i of content associated to ISP i, that is γ for the customers of ISP i and for those of j = i if the content is cached in j's CDN, and 1 otherwise. This gives:

Q A = γσ A + γσ B y B /x + σ B (1 -y B /x) Q B = γσ B + γσ A y A /(1 -x) + σ A (1 -y A /(1 -x)).
Note that for any set of parameters, quality does not depend on the type of model (push/pull, integrated or not). Taking the values (when fixed) of Section 3.2 (adding y A = 0.2), we display We observe a difference of treatment depending on the proportion of content hosted by ISPs (note that we have considered heterogeneous parameters). When x is small here, content hosted in B perceived better (because almost all content of B is cached in A, the y i being fixed, while after some point the quality in A becomes better. This illustrates the impact of CDNs on perceived quality, and that CPs may be disadvantaged if associated with a given ISP. Figure 4 (right) shows that as the cached amount in A increases, the quality Q B of content in B increases, but Q A also increases (even if to a lesser extent), because σ A increases.

Q A and Q B in

Discussion: CDN and neutrality

The question of how CDNs impact the network neutrality debate hasn't been addressed much, but introducing them into the picture reshapes the economic relationships between actors. Several remarks on that matter can be made from our model:

1) For the pull model, the ISP has interest in accepting the operation of an external CDN, even without charging it. In all other cases, there are conditions on the set of parameters for this to occur, but the ISP often has an interest in hosting a CDN (integrated or not).

2) Hosting a CDN induces less traffic within the network, thus reduces the pressure to increase capacities.

3) The CDN operation fees are paid by CPs (to improve their QoS), which somehow is a shift in payment with respect to the side payments requested by ISPs in the neutrality debate. But this fee can hurt innovation exactly as side payments would: can regulators accept that and still prevent ISPs from imposing side payments? 4) Only the most frequently downloaded content is cached by CDNs. If the service is proposed to all (distant) CPs, this makes a differentiation between CPs, which is in a sense against the neutrality principle, because packets are not treated the same. One can also argue that it makes a difference of treatment between local and distant CPs. In any case, the CDN activity requires attention, and its conformity depends on the definition of neutrality.

5) The case r = 0 (no payment) would not create any neutrality problem. But as we have seen, this is acceptable for the ISP only for the pull model with an external CDN.

6) But we assumed here that CPs always prefer to use the CDN services, something to be further studied.

We believe that this work highlights the need for CDNs economic modeling in the Internet and for a deeper investigation of their impact on actors, fairness, and neutrality.

Conclusions and perspectives

This paper is a first step in the modeling and analysis of CDNs' economics, in interaction with end users, content providers and ISPs, in order to start discussing their influence into the network neutrality debate. We have presented several situations, depending on whether the CDN is vertically integrated with the ISP or not, combined with the so-called push and pull models, where the traffic is paid by the sender or the receiver respectively. We have determined how the CDN can compute the optimal amount data to be cached in each situation (which of course depends on the selected parameters) and the impact on revenue. The impact on the fairness in terms of treatment between the different content providers is then described.

We observed that revenue-oriented CDNs may treat content providers differently, which could go against principles of neutrality and hurt innovation. Again, conclusions are case dependent, since different parameter choices can lead to different conclusions.

As extensions of the work, we would like to investigate the viability of CDNs' business in neutral and non-neutral contexts, and to study the case of more complex topologies involving more ISPs, CDNs in competition, as well as more content providers. The case when ISPs pay for caching will also deserve attention.

Figure 1 .

 1 Figure 1. Relations between users, ISPs, and CPs

Figure 2 .

 2 Figure 2. ISP A revenue versus cached amount yA

Figure 3 .

 3 Figure 3. ISP A optimal yA and utility UA

Figure 4 .

 4 Figure 4. Content quality depending on host ISP