Lionel Desgranges
email: lionel.desgranges@cea.fr

Gianguido Baldinozzi
email: gianguido.baldinozzi@centralesupelec.frfischer@ill.fr

Henry E Fischer

Gerard H Lander
email: ¶lander@ill.fr

Temperature dependent anisotropy in the bond lengths of UO 2 as a result of phonon-induced atomic correlations

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Temperature dependent anisotropy in the bond lengths of UO2 as a result of phonon-induced atomic correlations

Introduction

Thermal expansion is a phenomenon known to all. Although thermal expansion cannot be treated formally within the well-known harmonic approximation, and requires anharmonic potentials, the implicit assumption when a solid expands is that all the interatomic bond lengths expand in concert, or more precisely, that all average interatomic distances track the overall lattice expansion monotonically. Dramatic exceptions may occur for crystalline framework structures containing rigid structural units that are flexibly bonded to each other [START_REF] Goodwin | Colossal Positive and Negative Thermal Expansion in the Framework Material Ag3[END_REF][START_REF] Tucker | Local structure in ZrW2O8 from neutron total scattering[END_REF], or else close to the melting temperature where diffusive motion and changes in coordination number may play a strong role.

Recent experiments by Skinner [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF], using total-scattering x-ray diffraction, and by Prieur et al. [START_REF] Prieur | Peculiar Thermal Behavior of UO2 Local Stucture[END_REF], using x-ray absorption spectroscopy, have suggested that above room temperature the well-known compound UO 2 (fcc fluorite structure) exhibits an unusual anisotropy in the distance distribution of O atoms around a given U atom, such that the radius r(U-O) of the first coordination shell decreases with increasing T, whereas the U-U distance distribution remains symmetric and tracks the lattice thermal expansion. No fundamental explanation that can lead to such observations has been advanced, although molecular dynamics (MD) simulations are consistent with a U-O potential that is mostly ionic with some evidence of covalency [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF][START_REF] Benmore | Topological ordering in liquid UO2[END_REF], a qualification that is supported by Density Functional Theory studies by Wen et al. [START_REF] Wen | Density Functional Theory Studies of the Electronic Structure of Solid State Actinide Oxides[END_REF].

In the present paper we report results from total-scattering neutron diffraction experiments [START_REF] Desgranges | Characterisation of cuboctahedra in UO2+x as function of temperature[END_REF][START_REF] Desgranges | Local atomic and magnetic order in uranium oxide at low temperature[END_REF] using the D4 diffractometer [START_REF] Fischer | D4c: A very high precision diffractometer for disordered materials[END_REF] at the Institut Laue-Langevin (ILL), Grenoble, France, on a polycrystalline sample of UO 2 over a range of temperature from 50 to 1023 K. Our results strongly support the earlier studies, and show additionally that the anisotropy effect in U-O distances is present at all temperatures, albeit stronger at higher T . We then give a description of the possible phonon-induced atomic correlations that could produce this effect. We suggest that such effects as reported here may exist in a large range of compounds where light and heavy atoms are combined in a simple atomic structure. The possible consequences to the materials properties, especially at higher temperatures, may prove to be significant.

Experiments

The details of the sample and neutron diffraction techniques are given in the Supplementary Information (SI). We have employed two types of diffraction data analyses. The first, and the most well-known, is that of Rietveld refinement of the diffraction patterns, i.e. considering only the integrated intensity of Bragg peaks, which consist almost entirely of elastic neutron scattering. As such, the crystallographic structure deduced from Rietveld refinement represents a time-averaged structure, following the Van Hove correlation function formalism (see also section 3). In addition, Rietveld refinement discards all diffuse scattering between Bragg peaks, such that local deviations from the long-range average structure are ignored. All this results in an atomic structure deduced from Rietveld refinement that has been averaged over time and space, which is well adapted to crystallographic considerations. Our first Rietveld results appeared to show that the oxygen atoms located at ( ) or equivalent positions are displaced from these highly symmetric coordinates towards the octahedral voids of the fluorite structure by a small distance δ along equivalent 111 directions [START_REF] Desgranges | What Is the Actual Local Crystalline Structure of Uranium Dioxide, UO 2 ? A New Perspective for the Most Used Nuclear Fuel[END_REF], and that δ is temperature-dependent, reaching a value of δ ≈ 0.017 at 1200K. This corresponds to a shift of ≈ 0.16 Å from the symmetrical position (δ = 0, x = 1 4 ). Crucially, a displacement of this magnitude in the space-time averaged structure probed by Rietveld amounts to a significant long-range static structural change that would certainly give rise to a series of new reflections at positions forbidden by the high symmetry of the fluorite lattice's fcc structure (space group F m3m). Previous diffraction work on single crystals of UO 2 [START_REF] Faber | Neutron diffraction study of UO2. Antiferromagnetic state[END_REF][START_REF] Wilkins | Direct observation of electric-quadrupolar order in UO2[END_REF] have observed such "satellite" reflections below T N (= 30 K), but not above; so, this interpretation of the Rietveld results must be rejected.

Instead, we should move beyond the space-time averaged structure of Rietveld, and that brings us to our second diffraction data analysis method, known as pairdistribution function analysis (PDF-analysis), which considers all of the measured diffraction intensity, whether diffuse or not, elastic or not, and thus is qualified as a total-scattering method. Originally used for diffraction on liquids/glasses for which no long-range periodic order exists [START_REF] Fischer | Neutron and x-ray diffraction studies of liquids and glasses[END_REF], PDF-analysis is now regularly applied to disordered crystalline or nano-crystalline materials [START_REF] Egami | Underneath the Bragg peaks: structural analysis of complex materials[END_REF][START_REF] Terban | Structural Analysis of Molecular Materials Using the Pair Distribution Function[END_REF]. The diffuse scattering in diffraction results from local atomic correlations, either from static disorder or from dynamic disorder such as phonons or thermal-diffuse scattering (TDS). The PDF(r) is then obtained via Fourier transform of the total-scattering diffraction pattern (xray or neutron or electron) and thus retains all information about atomic positional correlations, whether static or dynamic, principally for short interatomic distances r as limited by the neutron coherence volume (about 60 Å FWHM diameter for D4). Since a diffraction technique integrates over all energy transfers between beam and sample, its characteristic time scale of measurement is inversely proportional to the incident beam energy, and for the D4 neutron diffractometer this "snapshot time" is about 100 times smaller than the characteristic time of atomic motion. The PDF(r) thus represents an ensemble-average of quasi-instantaneous local atomic correlations that exist in the sample, and more precisely, the amplitude of PDF(r) is proportional to the probability of finding an atom at a distance of r from an average atom at the origin. The integrated area of a peak in PDF(r) is proportional to the coordination number at that peak position. The r-space resolution of the PDF(r) is inversely proportional to the maximum wave-vector transfer Q max which for the D4 instrument is about 23.5 Å-1 , giving ∆r = 0.16 Å FWHM and reducing Fourier truncation effects to a minimum. Like all diffraction techniques, PDF-analysis cannot distinguish between static and dynamic correlations. At the high T for our experiments on UO 2 however, it is most likely that observed changes in local atomic configurations (via PDF-analysis) are dynamic in nature, perhaps resulting from identifiable optic phonons, since long range diffusion of O should not occur in the T -range of our experiments.

We show in Fig. 1 our results for the pair-distribution function PDF(r) derived from total-scattering neutron diffraction over a broad range of temperature for a UO 2 polycrystalline sample.

Our neutron diffraction PDF(r) results are in good agreement with the x-ray diffraction PDF(r) results of Skinner et al. [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF] on UO 2 acquired over a temperature range from ≈ 1100 K to the melting temperature (T m ) of 3270 K, with Q max = 21 Å-1 . The higher incident energies of x-rays compared to neutrons allows for even shorter snapshot times, but the low-Z oxygen atoms contribute much less than the U atoms to the x-ray diffraction intensity, as compared to the case of neutron diffraction. Skinner et al. [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF] calculate running coordination numbers that show convincingly that the observed contraction of the first coordination shell of O atoms around an average U atom, i.e. a downward shift in the position of the r(U-O) peak in PDF(r), is accompanied by a loss in members of that shell due to O atoms making excursions to higher r. Prieur et al. [START_REF] Prieur | Peculiar Thermal Behavior of UO2 Local Stucture[END_REF] have performed extended x-ray absorption fine structure (EXAFS) measurements on UO 2 at a number of temperatures, and this technique also gives a picture of the quasiinstantaneous local structure. Their results for r(U-O)(T ) match those of Skinner et al. [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF]. An anisotropic peak broadening of the r(U-O) distribution, similar to the one observed in our experiments, was reported in a neutron total scattering experiment on UO 2 at 898 K and modelled using a continuous static distribution of distances in a reverse Monte-Carlo large-box approach by Palomares et al. [START_REF] Palomares | Oxygen point defect accumulation in single-phase UO2+x[END_REF].

The different temperature dependence of the ensemble-averaged U-O and U-U distances (defined as the peak positions r(U-O) and r(U-U) in the PDF) is immediately evident from Figure 2. Further data analyses are described in the supplementary information. Note that the determination of r(U-O) and r(U-U) becomes increasingly difficult at higher T , as the peaks become very broad and overlapping -a reason we have truncated the analysis at 1023 K.

Model for anisotropic bond lengths

Within the Van Hove formalism, the neutron scattering amplitude of a crystal whose atoms have positional fluctuations u R (t) (phonon amplitudes) around time-averaged positions R is described by the space-time correlation function G(r, t) that can be space-Fourier transformed to the correlation function S(q, t), which in turn can be expressed in terms of the eigenvectors of the dynamical matrix of a crystalline sample. The time-Fourier transform of S(q, t) yields S(q, ω), which can be measured directly by neutron or x-ray scattering. When the analysis is performed in the experimental space (q, ω), as for a single-crystal experiment, large spatial separations dominate the contributions to S(q, ω), whose oscillations therefore average out to an asymptotic value, the Debye-Waller factor. On the other hand, analysis of G(r, t) in real-space is dominated by short-range spatial correlations of atomic positions within the coherence volume of the scattered quanta (as determined by beam optics), and the Debye-Waller factor then does not provide a good description of the positional fluctuations u R (t).

When phonon amplitudes are small, as in a renormalization process, the phonon self-energy modifications can, in principle, be described by perturbation theory of the anharmonic Hamiltonian expressed by a set of phonons with amplitudes u and with frequencies ω of the unperturbed harmonic problem. The important feature of the dynamics, when we are not in a soft mode picture, is that the relevant phonon frequency is sufficiently high and it can be described by a truncated MacLaurin's series (Taylor's at q 0 = 0) as ω 2 = ω 2 0 + α 2 q 2 + . . . . In the classic high-temperature regime (k B T hω) the phonon amplitude u can be derived from the fluctuation-dissipation theorem:

u 2 ∝ k B T ω 2 0 + α 2 q 2 (1)
The correlation function of the amplitudes of the phonon u at two atom sites i and j is then given by the Ornstein-Zernike function:

u i u j ∝ 1 R i -R j exp -ω 0 R i -R j α (2)
and here the correlation length ξ = α ω 0 decreases with increasing phonon frequency. For this reason, even if all phonons become active at high temperature and contribute to the total phonon density of states, the weakly dispersing low-energy phonons contribute the most to correlated displacements, whereas the atomic correlations from higher-energy phonons can be safely described as uncorrelated. An estimate of the correlation length ξ can be obtained for the zone-boundary: for an optic band having a dispersion γ and lattice parameter a, we have ξ -→ a π 2γ ω 0 , which amounts to a few Å. This means that in a total-scattering experiment (as we have performed) the atomic correlations resulting from these dynamic positional fluctuations contribute to the pair-distribution function PDF(r) at distances rarely exceeding the third neighbours.

The phonon dispersion curves for UO 2 were first measured in pioneering experiments in 1965 by Dolling et al. [START_REF] Dolling | The crystal dynamics of uranium dioxide[END_REF] More recently, the phonons (and their linewidths) have been measured as a function of temperature up to 1200 K. We reproduce in Figure 3a the phonon modes along the [001] direction as measured by Pang et al. [START_REF] Pang | Phonon Lifetime Investigation of Anharmonicity and Thermal Conductivity of UO2 by Neutron Scattering and Theory[END_REF].

The only vibrational modes in the figure with energy < 40 meV are the acoustic bands (∆ 1 and ∆ 5 ), a fraction of the highly dispersive ∆ 3 branch, and the weakly dispersing ∆ 5 branch related to the IR-active transverse mode. The phonon density of states (PDOS) measured at 295 K and 1200 K (Figures 1 and2 in Pang et al. [START_REF] Pang | Phonon density of states and anharmonicity of UO2[END_REF]) actually show a series of peaks in the vibrational PDOS, one of which is close to 35 meV. The highly dispersive components with a longitudinal character do not contribute effectively to the correlated-amplitude features (branches ∆ 1 and ∆ 3 ), whereas the two doubly degenerate transverse branches (∆ 5 ) (TO1 and TO2) can provide some correlated-amplitude contributions. In particular, this is the case of the weakly dispersing ∆ 5 branch going from Γ - 4 to X + 5 (TO1) at about 30 meV that provides the strongest contribution to the correlated-amplitude effects in the total scattering pairdistribution function. Most of this contribution seems to originate from phonons close to the X + 5 point. The atomic displacements corresponding to the phonon branches X + 5 at the Xpoint see the uranium atoms at rest, whereas the O atoms vibrate along 111 directions. The displacements are schematically displayed in Figs. 3b and3c, where we see that the octahedral voids of the fluorite structure permit such vibrational excursions along 111 .

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The crucial importance of the X + 5 phonon is that its action results in a separation of the bonds into two different lengths. Assuming a lattice parameter of 5.47 Å, the eight U-O distances in the radial distribution function are initially at 2.3685 Å (a 3 4 ) and the O-O distances at 2.735 Å (a/2). Assuming an amplitude of 0.0053 • [111] (equivalent to a displacement of 0.05 Å) for the correlated displacements of the O atoms, there are six U-O distances at 2.35 Å and two at 2.415 Å, thus shortening and skewing the distance distribution, whereas the peak of the O-O distances is only slightly changed. Furthermore, as the U atoms are at rest, the U-U distances are unaffected by this phonon correlation.

Discussion and Conclusion

Our neutron PDF(r) results for UO 2 strongly support the earlier x-ray PDF(r) results by Skinner et al. [START_REF] Skinner | Molten uranium dioxide structure and dynamics[END_REF] as well as the EXAFS results by Prieur et al. [START_REF] Prieur | Peculiar Thermal Behavior of UO2 Local Stucture[END_REF]. All three techniques measure an ensemble-average of quasi-instantaneous local atomic structures, and all three show that r(U-O), corresponding to the peak in the nearest-neighbor U-O distance distribution, decreases with increasing T as the distribution becomes asymmetric (see Figures 1 and2). In contrast, the U-U distance distribution remains symmetric, and r(U-U) increases as expected with T . We have presented a plausible model for these observations, based on the eigenvectors of the phonon modes, showing that the ∆ 5 transverse optic phonon is capable of splitting the eight nominally equivalent U-O distances (see Fig. 3) into a bimodal distribution that has six O atoms at a slightly shorter-than-nominal distance, and two O atoms at a slightly longer-than-nominal distance, as they make vibrational excursions along 111 towards the octahedral voids of the fluorite structure. Although a dynamic process, this distance splitting is clearly visible in the x-ray and neutron PDF(r) as an increasingly asymmetric U-O distance distribution acquiring a high-r tail as T increases. We find that the contraction of the first U-O coordination shell represented by r(U-O) accelerates above ≈ 400 K, which corresponds well to an increase in the population of the ∆ 5 transverse optic phonon having energy ≈ 35 meV (≈ 400 K).

Asymmetric/anisotropic nearest-neighbour distance distributions due to atomic vibrations are indicative of anharmonic potentials, and it is well known that anions in the fluorite structure are subject to such potentials. These effects manifest themselves as modifications of the Debye-Waller factors observed in Bragg peak refinement methods, as was first detected in CaF 2 itself by Strock & Batterman [START_REF] Strock | X-ray study of anharmonic vibrations in calcium fluoride[END_REF]. Willis and collaborators found similar effects in UO 2 [START_REF] Willis | Re-analysis of Single-Crystal Neutron-Diffraction Data on UO2 Using Third Cumulants[END_REF]. Interestingly, Willis initially interpreted the singlecrystal neutron data with a model having a displacement of the O atoms along 111 in the space-time average structure, i.e. measurable Bragg peak intensities [START_REF] Willis | Neutron diffraction studies of the actinide oxides II Thermal motions of the atoms in uranium dioxide and thorium dioxide between room temperature and 1100 C[END_REF]. However, as we have discussed in Section 2, this interpretation predicts new Bragg reflections, which have never been observed. Anharmonicity also has other consequences that have been observed, noticeably a strong broadening of the phonon linewidths as a function of temperature [START_REF] Pang | Phonon Lifetime Investigation of Anharmonicity and Thermal Conductivity of UO2 by Neutron Scattering and Theory[END_REF][START_REF] Pang | Phonon density of states and anharmonicity of UO2[END_REF], as well as effects on the vibrational entropy and specific heat [START_REF] Bryan | Impact of Anharmonicity on The Vibrational Entropy and Specific Heat of UO2[END_REF].

Anion amplitudes of low frequency optical phonons were singled out as an important ingredient to assist superionic conduction in binary and ternary compounds [START_REF] Wakamura | Roles of phonon amplitude and low-energy optical phonons on superionic conduction[END_REF]. Whereas Wakamura's model addresses specifically diffusion and the behaviour of zonecentre phonons, it can be straightforwardly extended to phonons at the zone boundary that are relevant to the present study. The parameter with specific importance is the damping constant of the vibrational modes involving only displacements of the anions. This is indeed the specific configuration of X + 5 phonon, and of part of the related ∆ 5 branch. Following Wakamura's arguments, in systems with large anions (as O ions) the damping constant for cations is significantly smaller than the one for anions. The direct consequence of the model is a specific and significant broadening of the phonons involving anion displacements compared to modes involving the heavier cations. In real space, this translates into an increased capacity of O ions to ramble around their equilibrium positions. The predicted behaviour of zone-centre modes correlates with this model in many superionic conductors, and, not surprisingly, the extension to zoneboundary correlates impressively with the present case of UO 2 as well. Indeed, Pang et al. [START_REF] Pang | Phonon density of states and anharmonicity of UO2[END_REF] provide irrefutable experimental evidence that the ∆ 5 branch and in particular X + 5 is overdamped, whereas the other phonon branches behave normally. These zone boundary effects agree with the extension of Wakamura's model.

We do not imply that the current effect is related to a superionic transition, because the model is applied to phonons at different points of the Brillouin zone, but the two effects share some common ingredients. Therefore, we can reasonably assume that the correlation effects discussed in this paper are not specific to UO 2 , but rather relevant to a broader class of systems that can have an intersection with ionic conductors, in particular with those that involve large anions and cations.

Figure 1 :

 1 Figure 1: Neutron PDF(r) results from 50 to 1023 K. Note the increased broadening of the U-O peak with increasing temperature as it becomes more asymmetric and acquires a high-r tail, all which suggests that the potential well softens and that anharmonic contributions are present. The r-space resolution is about 0.16 Å (FWHM) for a Q max = 23.5 Å-1 .

Figure 2 :

 2 Figure 2: Experimental results for U-O and U-U distances in UO 2 as determined from various techniques, as a function of temperature. Red symbols refer to U-U distance, left-hand scale. Filled diamonds: data from neutron PDF analysis. Empty diamonds; values from lattice parameters derived by Rietveld analysis of neutron data. Dashed line: values derived from x-ray lattice parameters recommended by Martin [17]. Black symbols refer to U-O distances, right-hand scale. Filled squares; values derived from neutron PDF analysis (this work). Open circles; values derived by x-ray PDF analysis and given by Skinner et al. [3]. Open squares: values derived from EXAFS experiments by Prieur et al. [4].

Figure 3 :

 3 Figure 3: a) Schematic of phonon dispersion curves, as measured by Pang et al. [19] in the [001] direction. b) Schematic representation of the correlated displacements of the oxygen atoms generated by the X + 5 phonon at the X-point. In this figure, the octahedral coordination around each uranium atom (larger gray spheres) is emphasized. c) Schematic representation of the correlated displacements of the oxygen atoms generated by the X + 5 phonon at the X-point. In this figure, the tetrahedral coordination around each oxygen atom (smaller red spheres) is emphasized.


USTHB Saif Eddine Khelifa

Algeria Lyes Khelladi

Miloud Bagaa

Yassine Hadjadj-Aoul

Semi-invertible Convolutional Neural Network for Overall Survival Prediction in Head and Neck Cancer

The paper addresses the issue of overall survival prediction in head and neck cancer as an effective mean of improving clinical diagnosis and treatment planning. A new solution is proposed using semi-invertible convolutional networks. Our model exploits the 3D features of computed tomography (CT) scans to enrich the dataset used in the learning phase and thereby improve the prediction accuracy. This is achieved by designing a first architecture featuring a combination of a CNN classifier with a fully convolutional network pre-processor. The latter has been replaced in the second solution by an invertible network to deal with the memory constraints noticed in the first architecture. Obtained results showed that both architectures have led to considerable improvements in terms of prediction accuracy (0.75) compared to state-of-the-art solutions.

I. INTRODUCTION

Head and Neck Cancers (HNC) are among the most common cancers. The HNC patients are characterized by a poor prognosis [START_REF] Hoesseini | Physicians, "clinical prediction of survival in head and neck cancer patients in the palliative phase[END_REF]. Of note, the survival rates of incurable HNC are concise. The overall survival rates of HNC cancer vary significantly according to the location and stage of the tumor. These make this type of cancer a source of utmost concern to the patients and physicians. Therefore, early diagnosis and prognosis are essential for the proper management of HNC patients [START_REF] Alabi | Comparison of nomogram with machine learning techniques for prediction of overall survival in patients with tongue cancer[END_REF]. Several attempts have been made for patients' prognosis estimation. For instance, the experience of the oncologists is well-positioned to estimate the prognosis of the patients [START_REF] Alabi | Comparison of nomogram with machine learning techniques for prediction of overall survival in patients with tongue cancer[END_REF]. However, oncologists are concerned about the possibility of making inaccurate prognoses. Additionally, this approach can be subjective and increases the risk of bias [START_REF] Alabi | Comparison of nomogram with machine learning techniques for prediction of overall survival in patients with tongue cancer[END_REF], [START_REF] Vlaev | Game relativity: how context influences strategic decision making[END_REF]. The traditional statistical analysis method was outperformed by disruptive technology such as the sub-field of artificial intelligence, that is, deep machine learning in the prognosis of cancer outcomes [START_REF] Tseng | The application of data mining techniques to oral cancer prognosis[END_REF], [START_REF] Zhu | Comparison between artificial neural network and cox regression model in predicting the survival rate of gastric cancer patients[END_REF].

The deep learning technique leverages the improved computational power to modify the widely known artificial neural network architecture. It is a specialized machine learning technique that uses a multi-layered neural network [START_REF] Chu | Deep learning for clinical image analyses in oral squamous cell carcinoma:a review[END_REF]. This multilayered architecture is able to learn the potentially complex and hidden relationships between various variables contained in the input data. Consequently, the learned relationships between these variables are used to define the output of interest, such as the overall survival of HNC patients. The deep learning technique has been reported to have significant importance in the proper management of head and neck cancer [START_REF] Howard | Machine learning-guided adjuvant treatment of head and neck cancer[END_REF]. The study by Howard et al. [START_REF] Howard | Machine learning-guided adjuvant treatment of head and neck cancer[END_REF] has reported the potential of deep learning techniques to identify the patients that could benefit from chemo-radiation.

The study, accomplished by Diamant et al. [START_REF] Diamant | Deep learning in head neck cancer outcome prediction[END_REF], has explored the potential of deep learning techniques to enhance the performance of traditional radiomics [START_REF] Diamant | Deep learning in head neck cancer outcome prediction[END_REF]. The latter has used computed tomography (CT) scans as the primary modality rather than other modalities, such as magnetic resonance and positron tomography. Formally, these scans are fundamental, and each modality provides different information on the tumor. CT scans are obtained from an X-ray study that produces 3D cross-sectional images of the body. Therefore, this modality can be used to reveal the tumor volume and the blood vessels that are feeding the tumor, and hence it provides more details on the tumor.

In this work, we examine the use of deep learning models applied to pre-treatment CT scans to predict overall survival in head and neck cancer. The proposed approach aims at improving the predictive accuracy for the proper management of this cancer through effective treatment and informed clinical decisions. Furthermore, we explore the potential of having 3D inputs (CT scans volume) instead of having 2D inputs (CT scans slice) to enrich the features extraction using a deep learning network by adding the spatial information between slices in the Z space. We devised two solutions for head and neck cancer survival prediction. While the first solution relies on a fully convolutional network, the second one uses the semi-invertible fully convolutional network to reduce memory utilization. The obtained results demonstrate the efficiency of each solution for achieving its design goals.

The remainder of this paper is organized as follows. First, section II briefly describes the most recent related works in the scope of predicting overall survival for HNC patients. Section III describes the employed dataset and the methods used for data filtering and pre-processing. Section IV presents the limitations of architectures based on datasets of 3D CT scans and the solutions to tackle those limitations. In section V, we present the proposed deep learning model for overall survival prediction. Section VI provides a comparative discussion considering results obtained in previous studies. Additionally, it emphasizes the clinical implications of the overall survival prediction of head and neck cancer. Finally, section VII concludes this article.

II. RELATED WORKS In [START_REF] Vallires | Data from head-neck-pet-ct[END_REF], a Deep-learning convolutional neural network is used to predict radiotherapy outcomes (overall survival) on HNC cancer using 2D CT scans as inputs. In [START_REF] Diamant | Deep learning in head neck cancer outcome prediction[END_REF], the solution relies on selecting scan slices containing the most meaningful information on the tumor. This was possible by performing Gross Tumor Volume (GTV) contours that are manually delineated by experts. Using these contours, CT volumes were converted to 2D images based on the largest primary GTV lesion area slice selection. Most recently, a combination of CT and Positron Emission Tomography (PET)was most recently were employed to improve the overall survival prediction in radiotherapy treatment for HNC patients [START_REF] Le | A normalized fully convolutional approach to head and neck cancer outcome prediction[END_REF]. Both volumes of PET and CT scans were converted into 2D images using the largest primary GTV lesion area slice selection, which resulted in a 2D network model with two blocks. The first block is a fully convolutional network that acts as a pre-processor for image normalization using the techniques proposed [START_REF] Drozdzal | Learning normalized inputs for iterative estimation in medical image segmentation[END_REF]. The second block is the Resnext classifier inspired by the work presented in [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF]. The combination of these two blocks led to notable improvements in overall survival prediction for radiotherapy treatments.

All solutions presented above rely on 2D images as input and adopt classifiers of the largest primary GTV lesion area in the slices. However, the constraints of being sealed into labeled data (GTVp GTVn) have led to a restriction in the amount of data used by the learning architecture and reduced the prediction accuracy. In fact, manual delineation of GTV contours and data labeling induce considerable costs in terms of financial budgets and time spent, which limits its application on big data size.

In our work, we explore the possibility of using the whole 3D images as data inputs in order to enrich the dataset used with additional spatial information between slices along the Zaxis. We propose a fully convolutional network-based solution that uses 3D images to predict the overall survival in head and neck cancer. The proposed approach does not assume any data labeling process via GTV contours. This approach enables us to omit the need for manual processing by experts and its related cost and potential mistakes. Nevertheless, deep learning models with 3D images may suffer from two main issues namely, depth uniformity and memory constraints. In order to mitigate these limitations, we devise a semi-invertible fully convolutional network-based solution that dramatically reduces memory utilization.

In the next section, we emphasize these two problems and discuss the possible approaches to solve them. Lessons learned from the next section will be of great help in the design of our second solution.

III. DATASET PREPROCESSING A. Dataset

The dataset used in our study consists of imaging, radiotherapy, and clinical data of 627 head and neck squamous cell carcinoma (HNSCC) patients, treated at MD Anderson Cancer Center (MDACC). This dataset is publicly available through the TCGA website. Out of the 627 patients, 215 patients had computed tomography (CT) scans with non-enhanced contrast. Thus, these patients were used in our analysis. These CT scans in our dataset were provided in Digital Imaging and Communications in Medicine (DICOM) format [13]. This latter represents the standard format for medical images stored in healthcare systems.

B. Pre-processing methods 1) Filtering CT scans: As a standard, the DICOM format provides certain attributes regarding the stored scan. Accession and instance numbers are examples of these important attributes used in the filtering of the obtained CT scans. Traditionally, a single CT scanning event is indicated by one accession number. However, if the first scan is of insufficient quality, a second scan may be obtained. In this case, these two volumes from a single scanning event would have the same accession number. Controversially to the accession number that can be the same for the different scan events for the same patient, a "serial number" is unique to every scan event, even if it was done on the same patient. Furthermore, the final scans called the "original scans" can be filtered using the ImageType field in the DICOM that may equal to "Original" if the scan is an original, otherwise the high-quality scan for the same patient is selected. Therefore, to enable effective filtering of the CT scans, a volume of slices with the same serial number and ImageType ("original") was arranged to construct a proper volume. The aforementioned process was important for the proper collation of the appropriate CT scans. This process is followed by the actual pre-processing of the CT scans.

2) Pre-processing CT scans: Collected slices with the same serial number and original image type were stacked together using the Instance Number attribute of the DICOM. The Instance Number indicates the order of the slice in Z-space. Of note, the CT scanner's measurements are represented with Hounsfield Units (HU). The raw pixel values in the DICOM standard had undergone a linear transformation to the Hounsfield unit to enable efficient disk storage. This transformation should be reversed to obtain pixel values in Hounsfield Units once again. The DICOM standard attributes, such as RescaleSlope and RescaleIntercept are needed to linearly transform the raw pixel values stored in the DICOM pixel array into Hounsfield units. Subsequently, the pixel values were clipped to [-1000 HU, +1000 HU], as depicted in Figure 1, which represents practical lower and upper limits of the HU scale, corresponding to the radio densities of air and bone, respectively.

Further pre-processing of the scans (slices) were done by eliminating noises like patient clothes and bed through a denoising technique as depicted in figure 2.

Since the data include the volume of slices with the same serial numbers, it was important to ensure that each pixel represents a consistent volume throughout the whole data set. To achieve this, important attributes of the DICOM header The Z-spacing (craniocaudal spacing) can be inferred by subtracting the z-position values of adjacent slices reported in ImagePosition patient. Taken together (x, y, z) spacing indicates the physical volume in cubic millimeters, whereby a single voxel represents a pixel in 3D space. Unfortunately, these physical distances vary according to the scan parameters, such that the physical size of each voxel is different for different patients. Furthermore, the (x, y) spacing is often different from the z-spacing for a single scan (Figure 3) To achieve consistent physical distance meaning of all pixels in the dataset, all the CT scans in our dataset were resampled to 0.8 × 0.8 × 0.8mm. Finally, all volumes were normalized.

IV. LIMITATIONS AND SOLUTIONS OF 3D ARCHITECTURES

3D architectures are heavy in terms of memory. They always suffer from memory problems during the training phase, especially with large 3D inputs like CT scans volumes. The latter also have their own problems which are depth uniformity and memory usage.

A. Depth uniformity

This is a common issue when dealing with 3D CT scans. Each patient had 3D images (CT scans) formed by a number of 2D images (slices), which is known as the depth of the 3D image. The depth can vary from one patient to another according to the CT scans. This variation imposes an additional problem in the design of the CNN architecture. That is, CNN can learn other non-important features along with the important features. Consequently, it becomes necessary to uniform the depth of the 3D images. An example of the adopted approaches for handling the depth uniformity is to randomly pick up 20 slices from the middle of each 3D image. Unfortunately, this method results in a waste in the number of slices which induces a considerable loss of invaluable information. One of the interesting approaches to handle depth uniformity while preventing the wasting of slices is the use of 3D techniques mentioned in [START_REF] Shivdeo | Comparative evaluation of 3d and 2d deep learning techniques for semantic segmentation in ct scans[END_REF]. To prepare the input data for training V-Net, the volumes of the CT scans should be normalized to the same depth. To satisfy this condition and without losing any information, CT volumes are split into smaller volumes that match the input dimensions, as depicted in figure 4. For a CT scan having 601 slices with a stack size of 32 and number of overlapping slices as 20 (overlap factor = 0.625), we get a list of 49 stacks having dimensions (512, 512, 32). The first stack will have indices from 0 to 32, which means that the first input data point for V-Net will be a CT sub volume from the 1st slice to the 32 nd slice. The second input data point will be a sub volume of the CT from index 12 to 44, both inclusive. As they have 20 overlapping slices, the second stack starts from the 12 th index and not the 33 rd , and so on for the rest of the stacks. For the last stack, however, the indices are (576, 608), the 7 extra slices are the added paddings to keep the volume of the stack compatible with the 3D model's input dimension. During the inference using this technique, the problem of depth uniformity can be solved.

B. Memory constraint

The mini-batch technique described previously solved the depth uniformity and allowed also partially alleviate the memory problem. Indeed, By taking only small stacks, we can save the input memory consumption. However, for further improvement in terms of memory usage, invertible networks are known to be a judicious choice. In fact, invertible networks dramatically minimize memory consumption. For this reason, we have adopted invertible UNet iUNets in our solution.

C. Basic highlight on iUNets and invertible layers

A fully invertible UNet described in [START_REF] Etmann | iunets: Fully invertible u-nets with learnable up-and downsampling[END_REF] is similar to conventional UNets with two paths (downsampling/upsampling). The difference resides in the fact of having invertible layers instead of classical layers. Those invertible layers are then followed by a split operator to split channels by a factor of 2. The first half of the channels goes to the upsampling, while the second half is copied to the upsampling path and concatenated with upsampled layers at each level. Also, this architecture has an invertible upsampling and downsampling instead of classical upsampling and downsampling, (Figure 4 [START_REF] Etmann | iunets: Fully invertible u-nets with learnable up-and downsampling[END_REF]).

Having invertible mapping functions will save memory usage. Instead of storing all the activation functions, only the last function will be stored. Then an inverse of this function will be calculated to obtain previous activation functions during backpropagation phase (Table I) 1) Invertible Layer(Additive coupling): Additive coupling is simple invertible layers that represent the main components of iUNets. They are conceptually similar to the residual layer and their inverse mapping can be computed just as fast as their forward mapping. These layers are employed to compute features within each specific resolution of the iUNet, as depicted in figure 5. The main idea behind additive coupling is splitting the activation's x (input) and y (output) into two groups of channels. i.e (x 1 , x 2 ) for x and (y 1 , y 2 ) for y. This is in order to define the invertible mapping xy via y 1 y 2 = x 2 = x 1 +F (x 2 ), where the F can be a simple arbitrary mapping that maps from the space of x 2 to the space of x 1 . In practice, F can be as simple as a sequence of convolutional layers with batch normalization. This function will be defined in next section.

V. METHODOLOGIES

In this section, two methods were used to predict overall survival using 3D input data (CT scan volumes) while elim-inating the constraint of having limited labeled data (GTV contours).

A. Fully Convolutional Network

As mentioned earlier, the employed 3D architecture is inspired by the work of Le et al. [START_REF] Le | A normalized fully convolutional approach to head and neck cancer outcome prediction[END_REF] using V-Nets (fully convolutional network) [START_REF] Milletari | V-net: Fully convolutional neural networks for volumetric medical image segmentation[END_REF] as a pre-processor. Drozdzal et al. [START_REF] Drozdzal | Learning normalized inputs for iterative estimation in medical image segmentation[END_REF] have previously shown that the V-net technique acts as an image normalization on medical images for liver tumor segmentation, as depicted in Figure 6. Then, it is followed by a fully convolution classifier inspired by the ResNeXt architecture [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF], as depicted in Figure 7. This architecture was implemented in 3D format to make use of 3D inputs and to have more information in 3D space as illustrated in Figure 8. The inputs of this architecture were a 3D CT volumes with 3D technique as mentioned in [START_REF] Shivdeo | Comparative evaluation of 3d and 2d deep learning techniques for semantic segmentation in ct scans[END_REF] to deal with depth uniformity problem. However, this architecture suffers from the memory problem since 3D UNet (V-NET) [START_REF] Milletari | V-net: Fully convolutional neural networks for volumetric medical image segmentation[END_REF] is a heavy network. In order to solve this problem, a part of the 3D UNet was replaced by an Invertible UNet to reduce the memory consumption, which led us to a Semi-Invertible Fully convolutional network.

B. Semi-Invertible Fully Convolutional Network

In this study, a new architecture called the Semi-Invertible Fully Convolutional Network is suggested. The latter is more efficient in terms of memory utilization. It is built through replacing the pre-processor block with an inverted pre-processor (3D Invertible UNet) with 4 invertible downsampling blocks followed by 4 invertible upsampling blocks, as shown in Figure 9. Each block consists of a number of invertible layers at each level. Each invertible layer includes a coupling function F , which is a sequence layer that consists of 3x3x3 convolutional layer followed by SELU function for regularization purposes (Figure 10). Then, it is followed by a classifier that consists of 3D Resnext (Figure 7) blocks with two 3x3x3 bottleneck layers, with a filter growth factor of 2, and cardinality of 32 and residual connections. Each connection is finalized by global average pooling [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF], and fully connected layers are used to output the binary survival class, as depicted in figure 11. The prediction of the overall survival of HNSCC patients was considered as a classification problem in this study. For the classification task (0:survival , 1:death), different parameters were taken for each proposed method as shown in Table II. The proposed architecture is implemented using Python and Pytorch library [17].

A. Data Augmentation and Preparation

Remarkably, our target output variable, i.e. 0:survival , 1:death was imbalanced. In our case, the number alive was significantly higher than the number of death cases. While the number alive is 458, the number of death cases is 169.

To handle this incidence of imbalanced dataset, oversampling technique is used. In an attempt to handle the data imbalance through an oversampling technique, a rotation of 20 degrees and random flip with a probability of 50% were applied. This method allowed to fill the gap between the 2 targets, such that each class has 458 cases.

After augmenting the input dataset, an important phase was to apply the mini-batch 3D technique discussed earlier, as illustrated in Figure 4. This last was performed in order to converge towards the optimal value of the mini-batch that has undergone several changes during the training figure 12.

At this stage, each patient has a CT scan volume ranging between 300 and 750 2D slices (after re-sampling). By applying the 3D technique formula on the minimum volume, 24 sub-volumes were extracted with depth equals to 32, and this number of sub-volumes was taken as "mini-batch = 24". Consequently, the model can accept 24 sub-volumes of depth 32 and resolution 512×512 at once (feed-forward pass). Based on this value (i.e., Mini-batch = 24), having a patient's size volume higher then 300 can give a higher number of subvolumes (i.e., higher then 24 sub-volume).

For example, the volume size of 700 slices, with 512 × 512 resolution produces 57 sub-volumes with a depth of 32 and a resolution of 512 × 512. In order to keep the mini-batch at 24 sub-volumes, the 48 is divided by 2, which results 2 × 24 subvolumes. Then, the network has a 24 sub-volumes two times (two forward passes). On the other hand, having a volume size that is not divided by 24 can be processed by taking the quotient of the division and multiplying it by 24. i.e 24 x quotient this will result in how many sub-volumes should be taken from the middle of the sub-volumes.

B. Training Architectures and effect of invertible preprocessor

In both Fully Convolutional Architecture and Semiinvertible Architecture, the performance of the network is III. Despite of the accuracy of the Fully Convolutional architecture that equals to 0.70 (Figure 13(a)), the model suffered from performance and memory constraints. Using the Semiinvertible UNet , as depicted in Table III, led to an increase of the mini-batch size to 24. It also led to a significant improvement in the accuracy to 0.75 (i.e., 75%), as depicted in Figure 13(b).

Comparing the loss curve of both networks, we noticed that the Semi-invertible network loss was slightly enhanced, compared to the first architecture. However, the graph is not quite perfect because of the unrepresentative data points in the data validation set. In addition, the data augmentation phase (oversampling) was not sufficient to cover the huge gap between the labels (alive, dead).

VII. CONCLUSION

The present study examined the use of a deep learning model for the estimation of overall survival in head and neck cancer patients. In this study, we explored the application of semi-invertible UNets for survival prediction in head and neck cancer. The overall accuracy obtained by our solution demonstrated the potential of some emergent deep learning techniques in building models to assist clinicians in the proper treatment of HNC patients. Nevertheless, the use of deep learning architectures for overall survival estimation in head and neck cancer are still facing multiple challenges that are worth considering. For example, it is important to empower the developed models with additional techniques that help in explaining and interpreting the predicted results, which will foster their adoption in actual clinical settings.

Figure 1 :Figure 2 :

 12 Figure 1: Hounsfield unit for each structure

Figure 3 :

 3 Figure 3: Voxel(3d pixel) Presentation from two different patients

Figure 4 :

 4 Figure 4: The 3D mini-batch technique In this 3D technique, there are 3 variables that affect the stack or sub-volumes creation, which are: i) The CT volume that presents the number of slices in the CT scan; ii) Stack size that presents the desired number of slices in the sub-volume; iii) The overlap factor can be calculated as follows: Overlap factor = Number of overlapping slices Stack size

Figure 5 :

 5 Figure 5: Additive coupling layers as implemented in this library. The input and output are split into two groups of channels. Here, the computational block F can be any sequence of e.g. convolutional layers with normalization. Note that the roles of x1 and x2 are reversed afterward.

Figure 6 :

 6 Figure 6: Preprocessor 3D unet (V-net)

Figure 8 :

 8 Figure 8: Fully convlutional architecture

Figure 9 :Figure 10 :

 910 Figure 9: The invertible pre-processor (3D invertible UNet) with custom additive coupling (invertible layer)

Figure 11 :

 11 Figure 11: The Semi-Invertible architecture

Figure 12 :

 12 Figure 12: Choosing the right mini-batch

  Figure 13: A loss comparison between the fully convolutional and semi-invertible solutions evaluated using the accuracy metric defined as follow: Accuracy = Number of correct predictions Total no of predictions with cross-entropy loss function: Cross-entropy = -n i=1 m j=1 y i,j log(p i,j ) During the training phase, both architectures are used in order to demonstrate the effect of adding invertible UNet on the overall survival accuracy, as shown in TableIII.

Table I :

 I Comparison between U-net and Invertible U-net

	Depth Conventional Invertible UNet Ratio
	5	3.17GB	0.85GB 26.8%
	10	5.90 GB	1.09 GB	18.4%
	20	11.4 GB	1.57 GB	13.8%
	30	16.8 GB	2.06 GB	12.2%

Table II :

 II Models parameters

	Network	Mini-batch Batch Learning rate
	Model (Invertible UNet)	24	1 0.0002
	Model (Classic UNet)	1	1 0.0006

Table III :

 III Observing the effect of invertible UNet on the accuracy of our model to predict overall survival

	Network	Memory usage(GB) Mini-batch Accuracy
	Model (Invertible UNet)	9.902	24	0.75
	Model(Classic UNet)	14,756	1	0.70