Sophie Loizillon
email: sophie.loizillon@gmail.com

Simona Bottani

Aurélien Maire

Sebastian Ströer

Didier Dormont

Olivier Colliot

Ninon Burgos

Apprimage Study Group

Transfer learning from synthetic to routine clinical data for motion artefact detection in brain T1-weighted MRI

Keywords: Quality Control, Clinical Data Warehouse, Deep Learning, Transfer Learning, Motion, MRI

Clinical data warehouses (CDWs) contain the medical data of millions of patients and represent a great opportunity to develop computational tools. MRIs are particularly sensitive to patient movements during image acquisition, which will result in artefacts (blurring, ghosting and ringing) in the reconstructed image. As a result, a significant number of MRIs in CDWs are unusable because corrupted by these artefacts. Since their manual detection is impossible due to the number of scans, it is necessary to develop a tool to automatically exclude images with motion in order to fully exploit CDWs. In this paper, we propose a CNN for the automatic detection of motion in 3D T1-weighted brain MRI. Our transfer learning approach, based on synthetic motion generation, consists of two steps: a pre-training on research data using synthetic motion, followed by a fine-tuning step to generalise our pre-trained model to clinical data, relying on the manual labelling of 5500 images. The objectives were both (1) to be able to exclude images with severe motion, (2) to detect mild motion artefacts. Our approach achieved excellent accuracy for the first objective with a balanced accuracy nearly similar to that of the annotators (balanced accuracy>80%). However, for the second objective, the performance was weaker and substantially lower than that of human raters. Overall, our framework will be useful to take advantage of CDWs in medical imaging and to highlight the importance of a clinical validation of models trained on research data.

INTRODUCTION

Recently, hospitals have created clinical data warehouses (CDWs) which gather medical images from thousands to millions of patients. [START_REF]Healthcare and Artificial Intelligence[END_REF] These resources represent an exceptional opportunity to develop computational tools. 2 In contrast to research datasets where acquisition protocols are well standardised, the quality of CDW images is highly heterogeneous. Images come from different hospitals over several decades and diverse machines were used with no homogenisation on the acquisition parameters. 3 Therefore, quality control (QC) is a fundamental first step before developing any machine learning project on a CDW. Lately, we have developed a framework for the automatic QC of T1-weighted (T1w) brain MRIs for a CDW based on noise, motion and contrast labels. While we obtained good results for the classification of images which are not proper 3D T1w brain MRI and for recognising low quality images, we encountered difficulties in detecting motion in the images. 4 MRIs are sensitive to motion induced by patient movement during the acquisition process. As they require a long acquisition time, subjects are more likely to move during the examination, which causes artefacts (blurring, ringing, ghosting or signal loss) in the reconstructed image. [START_REF] Wood | Truncation Artifacts in Magnetic Resonance Imaging[END_REF] Previously, we found that 25% of MRIs in the CDW were totally unusable, and almost a third had a very low quality especially due to motion. 4 A study conducted in a single hospital showed that the prevalence of repeating an MRI examination due to the presence of motion was up to 20% of all the acquisitions. 6 Beyond the cost that this represents for hospitals, these studies are highlighting the fact that many images present in the PACS are corrupted by motion. Therefore, it is important to be able to automatically exclude such images before conducting any study on a CDW. As motion quantification is a complex problem, particularly due to its sensitivity to many cofactors such as contrast, we suffer from a lack of dataset with reliable quantitative ground truths. Thus, some studies were based on synthetic motion to detect motion artefacts in a controlled way. [START_REF] Mohebbian | Classifying MRI motion severity using a stacked ensemble approach[END_REF][8][9][START_REF] Duffy | Retrospective correction of motion artifact affected structural MRI images using deep learning of simulated motion[END_REF] Despite the excellent results claimed in the literature, only few papers have attempted to validate their performance on data with real motion. Even when they did, their test sets were extremely limited and only composed of research data. 8,[START_REF] Duffy | Retrospective correction of motion artifact affected structural MRI images using deep learning of simulated motion[END_REF] It is thus unclear how they would perform on clinical routine data.

In this paper, we propose a transfer learning framework for the automatic detection of motion artefacts in 3D T1w brain MRI using a CDW. We generated synthetic motion in MR images of research databases to train a CNN classifier which was validated on synthetic and real motion artefacts. Our model was then generalised to clinical data with an effective transfer learning technique using 5000 manually labelled MRIs from a CDW.

MATERIALS AND METHODS

Dataset description

Three publicly available research datasets were used to train a CNN with synthetic motion artefacts and one routine clinical dataset was exploited for transfer learning and validation. The Alzheimer's Disease Neuroimaging Initiative (ADNI) study is a multisite study of elderly individuals with normal cognition, mild cognitive impairment, or Alzheimer's disease. [START_REF] Petersen | Alzheimer's Disease Neuroimaging Initiative (ADNI)[END_REF] The ADNI-1 phase included T1w MRIs acquired on 1.5 T scanners from different manufacturers (GE, Siemens, and Philips). Part of the metadata, the IPMOTION score indicates the absence of motion (0), or the presence of mild (1), moderate (2) or severe (3) motion artefacts in MRIs. This score was used to select motion free T1w MRI. We also created a small test set with MRI corrupted by severe, moderate or mild motion based on the IPMOTION score and the comments section. None of the ADNI images were acquired with gadolinium injection. We also used data from the MSSEG MICCAI challenge, which aim is to segment multiple sclerosis lesions and includes 53 patients across 4 different sites, 12 and from the Montreal Neurological Institute's Brain Images of Tumors (MNI BITE) database, which includes pre and postoperative MR images acquired from 13 brain tumour patients with different subtypes of gliomas. [START_REF] Mercier | Online database of clinical MR and ultrasound images of brain tumors[END_REF] We only considered the 3D T1w with gadolinium injection from these two datasets.

The clinical routine data comes from a large CDW containing all the T1w brain MRIs scanned in hospitals of the Greater Paris area (Assistance Publique-Hôpitaux de Paris). We used the same dataset as in our previous study, where we randomly selected 5500 images that were acquired on various scanners (Siemens, GE, Philips and Toshiba). 4 Motion artefacts were manually annotated as a three-grade level by two annotators. A score of (0) was given when no motion was seen, ( 1) when the structures of the brain were distinguishable despite the presence of motion and (2) when the structures were difficult to distinguish. Some of the 5500 images did not correspond to T1w brain MRI and were therefore not labelled with a motion score (straight reject). 4 If the users labelled differently a given MRI, the consensus grade was chosen as the maximum of the two grades. Dataset characteristics are shown in Table 1.

Image preprocessing

The T1w MR images were pre-processed using Clinica 14 and its t1-linear pipeline. First, a bias field correction was applied using the N4ITK method. 15 An affine registration to the MNI space was then performed. [START_REF] Avants | Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain[END_REF] Next, images were cropped to remove background resulting in images of size 169×208×179, with 1 mm isotropic voxels. 17 The Z-score method, which consists of subtracting the mean intensity of the entire image from each voxel value and dividing it by the corresponding standard deviation, was used to normalise the voxel intensities.

Proposed approach

We developed a transfer learning approach to detect motion artefacts in clinical images based on motion simulated on research images.

Head motion can be well approximated as rigid body motion which requires six degrees of freedom, comprising three translations and three rotations. [START_REF] Duffy | Retrospective correction of motion artifact affected structural MRI images using deep learning of simulated motion[END_REF] We used the Python library TorchIO and its RandomMotion function. [START_REF] Pérez-García | TorchIO: A Python library for efficient loading, preprocessing, augmentation and patch-based sampling of medical images in deep learning[END_REF] In this image-based approach, the assumption was made that the subject takes N t different positions during the acquisition. 8 The motion simulation algorithm follows these four steps:

1. N t rigid transformations of the motion free image were computed in the image domain. The three translation (t x , t y , t z ) and rotation (Θ x , Θ y , Θ z ) parameters were sampled for each transformation from a uniform distribution across the translation and rotation ranges given as input. 2. For the N t transformed images and the original one, the fast Fourier transform was estimated to transform the image domain into the k-space domain. 3. A new k-space was built by concatenating blocks for the N t different simulated positions. 4. To transform the k-space to the image domain, an inverse fast Fourier transform was calculated to produce the final motion corrupted synthetic image.

To classify motion artefacts, we implemented a CNN composed of five convolutional blocks and three fully connected layers (denoted as Conv5FC3) that proved successful in our previous work. 4 Each convolutional block is made of a convolutional layer, a batch normalisation layer, a ReLU activation function and a max pooling layer. We used the ADAM optimiser, the weighted binary cross-entropy loss and a batch size of 16. Our implementation was done using Pytorch and the ClinicaDL software. [START_REF] Thibeau-Sutre | Clini-caDL: An open-source deep learning software for reproducible neuroimaging processing[END_REF] Transfer learning was used to generalise our model from synthetic to routine clinical data. We fine-tuned our pre-trained model by allowing the training of the three fully connected layers of the Conv5FC3 model and freezing the weights of the other layers.

Experiments

At first, we aimed to test the ability of the proposed Conv5FC3 network to detect motion in research datasets using only synthetic motion while training. We performed a series of experiments on the research datasets, where we corrupted motion-free MRIs with different severities of motion to study the influence of the translation and rotation ranges. A first independent test set was created by randomly selecting 182 images over the three public datasets and corrupting 91 MRIs with different severities of motion (rotation: [2°, 8°]; translation: [2 mm, 8 mm])). The remaining images were split into training and validation using a 5-fold cross validation. Our model was also validated on a second small test set with 41 ADNI MRIs corrupted by real motion. Then, a second set of experiments was performed on the CDW dataset using transfer learning. We fine-tuned our pre-trained model using 5000 clinical images on two target tasks: the detection of severe (Motion01vs2) and mild (Motion0vs1) motion. We tested our method on an independent test set of 500 clinical MRIs.

RESULTS

Validation on research dataset

We evaluated the performance of our model trained on synthetic motion on our independent test set corrupted with different severities of motion artefacts. We studied the influence of the RandomMotion function parameters by performing several experiments with different motion severities. We first trained a model with synthetic severe motion by applying a large rotation and translation range ([6°,8°]; [6 mm,8 mm]). The balanced accuracy (BA) on our independent test set is excellent (98%). We also obtained very good results for smaller ranges of rotation and translation as reported in Table 2. Then, we evaluated the ability of these models to detect real motion. As mentioned in Section 2.1, we defined a test set according to the IPMOTION. Our models were perfectly able to detect motion on these images. In light of these experiments, we concluded that the most suitable model to continue our study was the one trained with rotation and translation ranges of [6°, 8°] and [6 mm, 8 mm], respectively.

Table 2. Results for the detection of synthetic and real motion in T1w brain MRI from research datasets. For the validation on synthetic motion, we report the mean and the empirical standard deviation across the five folds for the balanced accuracy (BA), specificity and sensitivity. For the detection of real motion, only the accuracy obtained by the best model of the 5-fold CV was reported as our independent test set contained only images with motion.

Cross-validation on synthetic motion Test on real motion

Application to the clinical data warehouse

The results obtained with the proposed transfer learning framework on our independent clinical test set are presented in Table 3. For the detection of severe motion, the classifier BA is almost as good as that of the annotators, which is defined as the average of the BA between each rater and the consensus. For the detection of mild motion the classifier BA is low (62.61%) and lower than that of the raters (73.21%). We compared the performance obtained with and without fine tuning. When applying the network trained on synthetic data directly on the clinical data, we observed a large drop in performance with a particularly low specificity for both tasks. A second comparison was performed between the proposed transfer learning framework and when training with the clinical data from scratch. Our transfer learning method achieved a gain of more than 10 percent points for the detection of severe motion. A much smaller improvement was observed for the detection of mild motion.

DISCUSSION AND CONCLUSION

In this work, we developed a transfer learning framework for the automatic detection of motion in 3D T1w brain MRI using a CDW. After having pre-trained a CNN to detect motion artefacts on images from research datasets corrupted with synthetic motion, we improved the generalisation ability of our network on a clinical dataset of 5000 images using fine-tuning. We validated the proposed approach by applying it on 500 manually labelled clinical MRIs. To the best of our knowledge, we are the first to propose a very large-scale validation using a CDW for motion artefact detection using synthetic motion. For the synthetic motion detection on the research datasets, our model achieved excellent results with a BA of 98.22%. Trained using synthetic motion only, the model had no difficulty generalising to real motion on a small research test set (BA: 100%) but it was not able to generalise to the CDW (BA: 60.26%). This poor result with a low specificity highlights both the critical importance of validating models trained on research datasets to clinical ones, but also the quality gap between research, where strict acquisition protocols are respected, and clinical data, which suffer from a lack of homogenisation of acquisition parameters. To overcome these issues, we proposed a transfer learning framework which achieved very good results for the detection of severe motion with a BA of 84.52%, which is nearly as good as that of the annotators (86.29%) and 10 percent points higher than when training the model from scratch.

Table 1 .

 1 Distribution of the sex and age over the research and clinical datasets. We grouped the three research datasets into a single row.

	Database	N patients N images Age in years [range] Sex (%F)
	Research databases	136	1222	72.52 [24, 90]	42.57%
	Clinical data warehouse	4177	5500	55.51 [18, 95]	55.39%

Table 3 .

 3 Detection of motion artefacts on the CDW. For both the detection of severe motion (Motion01vs2) and mild motion (Motion0vs1), we report: the agreement between human raters and the consensus (manual annotations), results of the proposed approach (pre-training on synthetic motion from research data and fine tuning on CDW), results when training on synthetic motion from research datasets without fine tuning and results when training from scratch on CDW.

			BA	Specificity Sensitivity
		Manual annotations	86.29%	-	-
		Fine-tuning on CDW (proposed) 84.52%	85.37%	83.67%
	Motion01vs2	Training on research dataset	60.26%	33.33%	87.19%
		Training from scratch on CDW	73.75%	49.58 %	97.93%
		Manual annotations	73.21%	-	-
		Fine-tuning on CDW (proposed) 62.61%	52.00%	73.23%
	Motion0vs1	Training on research dataset	53.18%	17.96%	88.57%
		Training from scratch on CDW	58.31%	33.39%	83.24%

ACKNOWLEDGMENTS

The research was done using the Clinical Data Warehouse of the Greater Paris University Hospitals. The authors are grateful to the members of the AP-HP WIND and URC teams, and in particular Stéphane Bréant, Florence Tubach, Jacques Ropers, Antoine Rozès, Camille Nevoret, Christel Daniel, Martin Hilka, Yannick Jacob, Julien Dubiel and Cyrina Saussol. They would also like to thank the "Collégiale de Radiologie of AP-HP" as well as, more generally, all the radiology departments from AP-HP hospitals. The research leading to these results has received funding from the Abeona Foundation (project Brain@Scale), from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute) and reference ANR-10-IAIHU-06 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6).

APPRIMAGE Study Group

Olivier Colliot, Ninon Burgos, Simona Bottani, Sophie Loizillon 1 Didier Dormont


Cécilia Tarpau

Javier Cebeiro

Geneviève Rollet

Mai K Id

Nguyen

Laurent Dumas

An analytical reconstruction formula with efficient implementation for a modality of Compton Scattering Tomography with translational geometry

Keywords: Analytic inversion, Compton Scattering Tomography, Image formation, Image reconstruction, Radon transform on double circle arcs

In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly implemented in the Fourier domain. Simulations are carried out to illustrate the efficiency of the proposed reconstruction algorithm.

Introduction

Compton Scattering Tomography (CST) is an imaging technique whose objective is to exploit wisely Compton scattered photons by the object to scan in order to reconstruct its electron density map. Since the early proposition of this type of imaging by Lale [START_REF] Lale | The examination of internal tissues, using gamma-ray scatter with a possible extension to megavoltage radiography[END_REF], Clarke [START_REF] Clarke | Compton-scattered gamma rays in diagnostic radiography[END_REF] and Farmer [START_REF] Farmer | A new approach to the determination of anatomical cross-sections of the body by compton scattering of gamma-rays[END_REF], studies about CST systems proved already promising results in medical imaging, for instance for the identification of lung tumours [START_REF] Redler | Compton scatter imaging: A promising modality for image guidance in lung stereotactic body radiation therapy[END_REF][START_REF] Jones | Characterization of Comptonscatter imaging with an analytical simulation method[END_REF], but also for earthquake engineering [START_REF] Gautam | Compton interaction tomography I. Feasibility studies for applications in earthquake engineering[END_REF], cultural heritage imaging [START_REF] Prado | Three-dimensional imaging of flat natural and cultural heritage objects by a Compton scattering modality[END_REF][START_REF] Harding | Compton scatter imaging: A tool for historical exploration[END_REF], landmine detection [START_REF] Hussein | On the use of radiation scattering for the detection of landmines[END_REF] and agricultural measurements [START_REF] Cruvinel | Compton scattering tomography for agricultural measurements[END_REF]. In fact, CST has made it possible to widen the fields of application for tomography to the imaging of one-sided large objects, because, in some configurations, sources and detectors can be placed at the same side of the object.

The proposition of CST systems is strongly related to the study of the associated integral transform, which models data acquisition [START_REF] Truong | Recent developments on Compton scatter tomography: theory and numerical simulations[END_REF][START_REF] Nguyen | Imagerie par rayonnement gamma diffusé[END_REF]. These integral transforms are generalizations of the classical Radon transform on lines studied by Radon [START_REF] Radon | Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten[END_REF] and Cormack [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications[END_REF]. While, in two-dimensional CST, manifolds are on circle arcs or double circle arcs, in three dimensions, data is acquired on toric surfaces. These circular geometries for the considered manifolds originate from the Compton effect. This physical phenomenon occurs when a photon emitted by the source with energy E 0 collides with an electron as it passes through matter. This photon is scattered, and deviated by an angle ω from its original direction. The photon looses also a part of its energy. The Compton formula gives us the one-to-one correspondence between the energy of scattered photons E(ω) and the related scattering angle ω

E(ω) = E 0 1 + E0 m c 2 (1 -cos(ω))
.

(1.1) m is the electron mass and c the speed of light. This relation ensures also that scattering sites of photons with identical energy E(ω) are located on a circle arc labelled by the scattering angle ω. Figure 1 illustrates the general functioning principle of a CST system. Several issues around the study of these generalized Radon transforms are then of interest, such as existence and uniqueness of the solution, its stability or the range conditions. The main important problem remains the reconstruction of the image, mostly coming from the proposition of analytical inverse formulas. In that way, the first studied two-dimensional CST system was made of a fixed line of detectors containing a fixed source [START_REF] Norton | Compton scattering tomography[END_REF]. See Figure 2a. The associated scanning circle arcs consist of a family of semicircles with a common fixed end point (the point source) and another extremity (the considered detector) on the line. Then, circular geometries for CST have been proposed, the first one considers a pair source -detector diametrically opposed in rotation around the object [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Rigaud | Novel numerical inversions of two circular-arc Radon transforms in Compton scattering tomography[END_REF][START_REF] Rigaud | Combined modalities of Compton scattering tomography[END_REF] and data acquisition is modelled by a Radon transform on circle arcs having a fixed source-detector chord length. See Figure 2b. The second consists also of a pair source -detector moving on a circle, but the distance between the two is no longer constant and depends on the scattering angle [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF] (Fig. 2c). With this modality, data measurement is performed on a family of circles orthogonal to the fixed circular path of the pair source -detector. In a third modality (see Fig. 2d), on the contrary, it was considered a set of detectors placed on a ring containing the source, thus obtaining a completely fixed CST modality [START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Tarpau | Compton scattering imaging system with two scanning configurations[END_REF][START_REF] Cebeiro | An interior Compton Scatter Tomography[END_REF][START_REF] Rigaud | Compton Scattering Tomography: Feature Reconstruction and Rotation-Free Modality[END_REF]. The corresponding Radon transform measures the contribution of the photons scattered at the points located on circles arcs having a common extremity (the point source) and another (the considered detector) on the detector ring. Some configurations, mentioned above, employ collimators to split up photons coming from different circle arcs. Geometries without collimation have also been studied. Consequently, for a given position of detector, the acquisition is performed on a family of double circle arcs and the amount of registered data is potentially doubled. This feature may also reduce the acquisition time and finally radiation exposition in comparison with a similar geometry with collimation. Among the proposed CST systems without collimation, one supposes a fixed source and a detector rotating around this source [START_REF] Tarpau | Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography[END_REF]. See Figure 2e. The other one, proposed in [START_REF] Webber | Compton scattering tomography in translational geometries[END_REF], is made of a source and a detector which translate along a line (Fig. 2f).

The direct reconstruction of volumes is also of interest with the proposition of three-dimensional CST systems. These modalities use uncollimated sources and detectors, and data acquisition is performed on toric surfaces. In many cases, these 3D modalities correspond to extensions of 2D systems. Circular geometries become thus spherical or cylindrical [START_REF] Webber | Three dimensional Compton scattering tomography[END_REF][START_REF] Rigaud | 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms[END_REF][START_REF] Cebeiro | On a three dimensional Compton scattering tomography system with fixed source[END_REF] and linear geometries become planar [START_REF] Webber | Compton scattering tomography in translational geometries[END_REF].

Here, we are interested in the two-dimensional modality proposed in [START_REF] Webber | Compton scattering tomography in translational geometries[END_REF]. The purpose of this article is to develop an alternative formulation suitable for the development of a faster and efficient reconstruction algorithm. The associated reconstruction algorithm will use only classical tools such as Fast Fourier Transform (FFT) algorithm.

The paper is outlined as follows. Section 2 recalls the general setup of the system and the model for data acquisition for the modality. Section 3 introduces the main result of the paper, that is, the alternative formulation for the inverse Radon transform on double circle arcs. Section 4 will give the discrete formulation of the forward operator, as well as the proposed strategy to reconstruct the object under study. Section 5 discusses the obtained simulation results with a study of the influence of some parameters on reconstruction quality.

2 Setup and measurement model of the CST system under study

Setup

The system under study is made of a source, assumed to be monochromatic, and a detector separated by a fixed distance from each other. The source and the detector move respectively on a horizontal line of equation z = 3 and z = 1. The horizontal position of the pair source-detector is labelled by x 0 (see Figure 3). Alternatively, this system may be sketched with fixed lines of sources and detectors that will be used in pair. The object, placed below the detector path, is scanned transversely. No collimation is used at the detector, hence the acquisition is performed on a family of double circle arcs (called toric sections in the original publication [START_REF] Webber | Compton scattering tomography in translational geometries[END_REF]) 1 . We parameterize these circle arcs and define the corresponding Radon transform in the next paragraph.

Modelling of data acquisition using the CST system

Given a scattering angle ω, data acquisition is performed on a family of double circle arcs of radius r where r = 1/ sin(π -ω) (or, equivalently, ω = π -arcsin (1/r)). For parameterization, these double circle arcs are obtained with the union of four half arcs denoted S j (x 0 , r), j ∈ {1, 2, 3, 4} of respective equation

x 1 = x 0 + r 2 -1 + r 2 -(z -2) 2 , x 2 = x 0 + r 2 -1 -r 2 -(z -2) 2 , x 3 = x 0 -r 2 -1 + r 2 -(z -2) 2 , x 4 = x 0 -r 2 -1 -r 2 -(z -2) 2 and z ∈]2 -r, 1[. See Figure 3.
The Radon transform which mathematically models data measurement with this CST system is then defined as follows : (2.1)

where ds refers to the elementary arc length measure on the considered double circle arc. Then, after computation of the arc length measure, we have the explicit reformulation for R D [25, Proposition 3.1] :

R D f (x 0 , r) = r 1 1 1 -z r 2   2 j=1 f 1 r 2 -1 + (-1) j r 1 - z r 2 + x 0 , z + f 1 -r 2 -1 + (-1) j r 1 - z r 2 + x 0 , z dz, (2.2)
where

f 1 (x, z) = f (x, 2 -z).
In the original study of this modality, the invertibility of the corresponding Radon transform as well as its analytical inversion formula has been established. The invertibility was proven using the theory of integral equations and resulted in a Volterra integral equation with a weakly singular kernel in the Fourier domain. This study also leads to a formulation for inversion formula as an integral transformation with a kernel computed iteratively. The numerical calculation of this kind of kernel may require high computational time and/or memory. Furthermore, as mentioned in the Remark 3.4 of the original paper, the proposed approach by Webber is severely ill-posed, particularly in terms of stability. Implementing such a method can lead to large instabilities, even when these are due to small changes in the data.

An alternative formulation for the inversion formula of the Radon transform on double circle arcs

In this section, we state the main result of the paper, a different formulation for the associated inversion formula, that will be easier to implement numerically. Let us introduce before some notations that will be used in the proofs.

Notations

It is useful to define the following transform pairs. Definition 2 (Fourier transform). Let f be a compactly supported function in R n . The n-dimensional Fourier transform of f , denoted f , is given by

f (ξ) = R n f (x)e -ix•ξ dx (3.1)
with ξ ∈ R n . The inverse Fourier transform is

f (x) = 1 (2π) n R n f (ξ)e ix•ξ dξ. (3.2)
Definition 3 (Fourier cosine transform [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]). Let f be a compactly supported function in R + . The Fourier cosine transform of f , denoted f c , is given by

f c (ξ) = 2 π ∞ 0 f (x) cos(xξ)dx (3.3)
with ξ ∈ R. The inverse Fourier cosine transform is

f (ξ) = 2 π ∞ 0 f c (ξ) cos(xξ)dξ. (3.4)
We define also the Hankel transform.

Definition 4 (Hankel transform [START_REF] Bracewell | Numerical transforms[END_REF]). Let f be a compactly supported function in R + . The zero-order Hankel transform of f is defined as

H 0 f (η) = ∞ 0 f (r)J 0 (ηr)rdr (3.5)
where J 0 stands for the Bessel function of the first kind of order 0.

Finally, we recall the integral representation of the Bessel function J 0 :

J 0 (x) = 1 2π π -π
e ix sin θ dθ.

(3.6)

Inversion formula

Proposition 1. Denoting Gf (x 0 , r) the operator whose Fourier transform according to the first variable is

Gf (ξ, r) = R D f (ξ, r) 2r cos (ξ √ r 2 -1) , (3.7)
if r > 1 and 0 when r ∈ [0, 1], the unknown function f is completely recovered from Gf as follows

f (x, z) = 1 4π ∞ -∞ e ixξ ∞ 0 H 0 Gf (ξ, ξ 2 + σ 2 ) cos (σ(2 -z))σdσdξ. (3.8)
Proof. With the change of variables s = z/r in (2.2) and taking the Fourier transform of R D respectively to variable x 0 , one gets

R D f (ξ, r) = ∞ -∞ dx 0 1 1/r ds re -ix0ξ √ 1 -s 2 •   2 j=1 f 1 r 2 -1 + (-1) j r 1 -s 2 + x 0 , rs + f 1 -r 2 -1 + (-1) j r 1 -s 2 + x 0 , rs   . (3.9)
With the second change of variables

x = x 0 ± √ r 2 -1 + (-1) j r √ 1 -s 2 , one gets R D f (ξ, r) = 4 1 1/r ds r √ 1 -s 2 f 1 (ξ, rs) cos (ξ r 2 -1) cos (ξr 1 -s 2 ) (3.10)
where f 1 stands for the one-dimensional Fourier transform relatively to the variable x.

Using relation (3.7), multiplying both sides of (3.10) by r • J 0 (ηr) with η ≥ 1 and integrating with respect to variable r, for r > 1, one recognizes the Hankel transform of Gf , denoted

H 0 Gf H 0 Gf (ξ, η) = 2 ∞ 1 dr 1 1/r ds r √ 1 -s 2 f 1 (ξ, rs) cos (ξr 1 -s 2 )J 0 (ηr). (3.11)
Then, with the double substitution (r = √

z 2 + b 2 , s = z/ √ z 2 + b 2 ), one gets H 0 Gf (ξ, η) = 2 ∞ 1 dz f 1 (ξ, z) ∞ 0 db cos (ξb)J 0 (η z 2 + b 2 ). (3.
12)

The result of the b-integral is given in the table [34, p. 55, eq. ( 35)]. Finally, one gets for 0 < ξ < η

H 0 Gf (ξ, η) = 2 ∞ 1 dz f 1 (ξ, z) 1 η 2 -ξ 2 cos (z η 2 -ξ 2 ) (3.13)
and

H 0 Gf (ξ, η) = 0 if η ≤ ξ. Let 0 < ξ < η. Then with the fact f 1 (x, z) = 0 for z ∈ [0, 1], ∞ 0 dz f 1 (ξ, z) cos (z η 2 -ξ 2 ) = η 2 -ξ 2 2 H 0 Gf (ξ, η). (3.14)
The left-hand side is the Fourier cosine transform of f 1 (ξ, z) according the variable z. We can then extract f 1 (ξ, z), applying the inverse cosine transform to (3.14)

f 1 (ξ, z) = 1 2 ∞ 0 dσH 0 Gf (ξ, ξ 2 + σ 2 ) cos (zσ)σ (3.15)
where σ = η 2 -ξ 2 . The final equation is obtained going back to variable f , and applying the inverse Fourier transform.

Remark 1. The projections Gf (3.7) contain zeros in the denominator, since the cosine function vanishes when ξ √ r 2 -1 = 2kπ ± π 2 , k ∈ Z. From (3.11) to the end of the demonstration, it was supposed that r is different from 1 + π 2ξ (2k + 1) . Furthermore, this may be a source of instability in the simulations. The addition of a regularization parameter for simulations is discussed in Section 4.2 to prevent this. Remark 2. Another reconstruction algorithm is also possible from the projections of Gf . This process is achieved performing geometric inversion. Geometric inversion is a mapping converting a point X into a point X such that XX T = q 2 , where q ∈ R * + is a constant value. The mapped point X has the same direction as the original point X but a distance of q 2 /||X|| to the origin of the considered coordinate system. As an example, geometric inversion converts circles passing through the origin into straight lines. In the present case, the Radon transform on double circle arcs is converted into a Radon transform on an apparent family of circle arcs of similar geometry as the one studied in [START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF][START_REF] Truong | Function reconstruction from reflection symmetric radon data[END_REF]. Although the inverse problem can be alternatively solved using geometric inversion, the approach we employ here is more straightforward.

Numerical formulations for the forward and inverse transform 4.1 Image formation

Let N SD be the number of positions for the pair source -detector and N r the number of double scanning circle arcs per sensor position. We denote x 0,k , k ∈ {1, ..., N SD } and r l , l ∈ {1, ..., N r } the discrete variables corresponding respectively to x 0 and r. The matrix of projection data R D f (x 0,k , r l ) is then computed, writing (2.2) under a discrete form, with the change of variables z = r cos θ

R D f (x 0,k , r l ) = r l ∆ θ • θ∈ arcsin 1 r l , π 2   2 j=1 f 1 (x 0,k + r 2 l -1 + (-1) j r l cos θ, r l sin θ) + f 1 (x 0,k -r 2 l -1 -(-1) j r l cos θ, r l sin θ)   , (4.1)
where ∆ θ is the sampling angular distance of θ. The above Cartesian parameterization allows having a constant distance between running points of the considered scanning circle arcs during simulations.

Image reconstruction

For image reconstruction, we need to compute the projections Gf in the Fourier domain according to (3.7). This expression contains zeros in the denominator. This may induce instabilities on reconstruction. For simulations, we add a small regularization parameter denoted

Gf (ξ, r) = R D f (ξ, r) 2r cos (ξ √ r 2 -1) 2 + cos (ξ √ r 2 -1) 2 . (4.2)
In terms of computational cost, the most demanding step in the implementation of (3.8) is the calculation of H 0 Gf . The idea is to establish a relation between the above operator with the Fourier transform of G ‡ f , defined as follows

G ‡ f (x, z) = ∞ -∞ Gf (x 0 , (x -x 0 ) 2 + z 2 ) dx 0 . (4.3)
We have now the following proposition.

Proposition 2. Let (σ, ξ) ∈ [0, ∞[×R. H 0 Gf (ξ, ξ 2 + σ 2 ) is related with the two-dimensional Fourier trans- form of the operator G ‡ as H 0 Gf (ξ, ξ 2 + σ 2 ) = 2π G ‡ f (ξ, σ). (4.4)
Consequently, from the inversion formula (3.8), it follows in the Fourier domain

f 1 (ξ, σ) = 2π 2 |σ| G ‡ f (ξ, σ). (4.5)
where f 1 is the two-dimensional Fourier transform of f 1 .

Proof. From the definitions of Fourier and Hankel transforms and with the integral representation of the Bessel function, one gets

H 0 Gf (ξ, ξ 2 + σ 2 ) = 1 2π ∞ -∞ dx 0 ∞ 0 dr Gf (x 0 , r) π -π e ir √ ξ 2 +σ 2 sin(θ) dθ e -iξx0 . (4.6)
There is an angle φ(σ, ξ) ∈ [0, 2π[ which corresponds to the angular coordinate of the point (σ, ξ), such that σ = ξ 2 + σ 2 cos (φ(σ, ξ)) and ξ = ξ 2 + σ 2 sin (φ(σ, ξ)). Using the property of periodicity of trigonometric functions, it follows that

H 0 Gf (ξ, ξ 2 + σ 2 ) = 1 2π ∞ -∞ dx 0 ∞ 0 dr Gf (x 0 , r) π -π
e irξ sin θ+σ cos θ dθ e -iξx0 . (4.7)

Changing variables x = r cos θ and z = r sin θ,

H 0 Gf (ξ, ξ 2 + σ 2 ) = 1 2π R 2 dxdz ∞ -∞ dx 0 Gf (x 0 , (x -x 0 ) 2 + z 2 ) e -i(xξ+zσ) (4.8)
The right-hand side of (4.8) is the two-dimensional Fourier transform of G ‡ , weighted by 2π (4.4). We are now able to reformulate the inversion formula (3.15) as

f 1 (ξ, z) = 2π ∞ -∞ dσ G ‡ f (ξ, σ)e izσ |σ|.
(4.9)

Taking the Fourier transform according to variable z to the above equation (4.9) leads to (4.5).

This leads to the reconstruction algorithm summed up in Algorithm 1.

Algorithm

Simulations results

The original object used for simulations is Derenzo phantom, an object made of multiple circles of different sizes.

The circles in the object also allow the study of the performance of the algorithm in front of different contrasts and spatial resolution, as well as its ability to reconstruct features locally tangent to lines of any slope. The unit length used here is the pixel. We suppose thus that the distance between the source and the detector paths of the modality is two pixels. The size of the object is N × N = 256 × 256 pixels in all simulations. Furthermore, given the linear geometry of this modality, there is no loss of generality to consider the object centred relatively to the z-axis.

Data acquisition

Data measurement is calculated according to (4.1). Figure 4 shows an example of data obtained for Derenzo phantom.

Influence of some parameters on reconstruction quality

In the following paragraphs, we study the influence of the other general parameters of the system such as the position of the object, the number of required positions for sensors or the number of scanning circle arcs. In addition to a visual comparison of the reconstruction quality, we propose here to measure quantitatively the error rate between the original object f 0 and the reconstruction f with the Normalized Mean Squared Error (NMSE) = ||f -f 0 || 2 2 /N 2 where ||.|| 2 refers to the 2-norm. The regularization parameter , which has to be small, was arbitrarily set to 0.01.

Position of the object relative to the detector path

We analysed here the influence of the position of the object on reconstruction quality. Firstly, the number of positions N SD for the pair source -detector and the number of scanning circles per position N r was chosen to largely satisfy the well-known condition [START_REF] Bracewell | Numerical transforms[END_REF] N SD × N r ≥ N 2 and are set arbitrarily to N SD = 1024 and N r = 1024. A convenient choice for these parameters will be discussed later. We performed various acquisition, modifying the gap δ between the detector path and the upper part of the object (see Figure 3). Consequently, the object is in the square of Cartesian coordinates x ∈ -N 2 + 1, N 2 ; z ∈ [-N -δ, -δ + 1 ). Figure 5 shows the reconstruction results for δ = 1, 26 and 51 pixels which correspond respectively a position for the object in the respective domains [-N -1, 0], [-N -26, -25] and [-N -51, -50] along the z-axis. The difference between the three reconstructions is on the top of the object. If the object is close to the line of movement of the detector, then this part is less well reconstructed. Indeed, this distance between the object and the detector path allows having arcs of circle tangent horizontally to this part of the object. An offset of δ = 51 seems to be a good trade-off between quality of reconstruction and the applicability of such a measure in practical use. For the rest of the simulations, δ is set to 51.

Number of necessary positions for the pair source-detector

We studied then the number of different positions required for a good quality of reconstruction. The influence of two running parameters is analysed, first, the farthest position x 0,max from the object for the source-detector pair (that is an array of length [-x 0,max , x 0,max ] for the source and detector paths) and the distance ∆ x0 between two adjacent positions of the pair.

Figures 6a, 6b and 6c show the reconstruction results when the farthest position from the object to the pair source detector is respectively 2N, 3N and 4N with a common ∆ x0 set to 1. Reconstruction from a domain [-x 0,max , x 0,max ] = [-2N, 2N ] appears to be blurred with strong artefacts in the upper parts of the image. For x 0,max = 3N and 4N , reconstruction quality seems to be visually equivalent, even if the NMSE for x 0,max = 4N is higher. This may be due to numerical approximations. For the rest of the simulation, x 0,max is set to 3N .

The influence of the distance ∆ x0 between two adjacent positions of the pair is now evaluated. Figures 6d, 6e and 6f show the result for ∆ x0 = 2, 1 and 0.5 pixels. This represents a respective amount of 0.5, 1 and 2 detectors per unit length. In Figure 6d (∆ x0 = 2), we can see streaks suggesting a lack of data for reconstructing the object. On the contrary, the doubling of the number of detectors between Fig. 6e and Fig. 6f does not bring a better quality of reconstruction. Consequently, the use of one detector per unit length seems to be a good trade-off, and the value will remain constant in the rest of the paper.

Number of scanning circles per position of the pair source-detector

The number of scanning circles necessary for reconstruction is now under study. In the same way, two parameters are of interest, that is, the maximum radius r max of the scanning double circle arcs to be taken and the discretization step ∆ r that have to be chosen. We first evaluate the consequences of the value of r max with three examples on Fig. 7a, 7b and 7c where r max is set respectively to 2N, 3N and 4N and ∆ r = 1. For r max = 2N , the reconstruction suggests a lack of data in front of the obtained results for r max = 3N and r max = 4N . Notice the higher NMSE for r max = 4N , probably due to numerical approximations.

We were finally looking for the appropriate discretization step for r, setting ∆r to 1, 2 and 4. Reconstruction results are shown respectively in Fig. 7d, 7e and 7f. Reconstructions with a large discretization step exhibit blur. Reconstruction results for 0.5 6d, 1 6e and 0.5 6f detector per unit length and x 0,max = 3N remains constant.

Discussions

The above simulations results show some interesting issues for the CST modality and the reconstruction quality that can be expected with such a system. First, this system is able to scan objects whose depth is largely oversized relative to the source-detector distance. However, this seems to be counterbalanced by the need for a consequent length for the source and detector linear paths, since sufficient reconstruction results appear for source and detector paths of length size six times greater than the object size. The necessary number of scanning circle arcs is also very important since, if we relate the values of r max with the scattering angles, the reconstruction quality is largely improved when r max is high whereas the angular distance between two values of r max is very small. Considering a larger distance between the source and the detector paths for the system may reduce partially the required amount of projection data.

Moreover, the geometry offers a sufficient reconstruction quality for every tangent with arbitrary slopes. However, one can notice that vertical slopes are slightly less well reconstructed than the other ones. This can be seen if we pay carefully attention to the left and right sides of the reconstruction. This may be problematic if the object to scan is essentially made of vertical features. One way to avoid this is to perform additional scans by rotating the object, if possible. Some artefacts that look like shadows around the circles which compose the object remains clearly visible. The issue of artefacts in CST has already been addressed in different manners, for instance in [START_REF] Webber | Microlocal analysis of a compton tomography problem[END_REF], where microlocal analysis was employed to alleviate artefacts in reconstructions from limited data. Moreover, in [START_REF] Webber | A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging[END_REF], a penalized iterative algorithm was developed for a mixed modality. Regarding our approach, it can be combined in a pipeline with post-processing stages based on machine learning, as we did in [START_REF] Ayad | Deep morphological network-based artifact suppression for limited-angle tomography[END_REF] for limited data issues in classical computed tomography. Some work is on the way.

Concluding remarks

In this article, we proposed a new reconstruction algorithm for a recently proposed CST modality with translational geometry. The algorithm can be numerically implemented efficiently and reconstructions exhibit good quality. A quantitative study of the required data has also been carried out. This study proved the ability for such a CST system to reconstruct larger objects than the system itself. This advantage is moderated by the fact that it is necessary to have an important amount of source-detector positions, which can make the acquisition time longer. An interesting issue of this work concerns a reconstruction algorithm for the three-dimensional extension of this system with planar paths for the source and the detector.

Figure 1 :

 1 Figure 1: General functioning principle of a CST system. Photons are emitted by source S, interact at sites M , and are recorded at site D. When a photon is detected carrying an energy E(ω 1 ) (resp. E(ω 2 )), the possible interaction sites lie on the upper (resp. lower) circle arc which subtends the angle (π -ω 1 ) (resp. (π -ω 2 )).

Figure 2 :

 2 Figure 2: Previous proposed CST modalities (a, b, c, d, e, f). (a): Fixed source and detectors placed on a line. (b): Rotating pair source-detector diametrically opposed. (c): Rotating pair source detector. (d): Fixed source and detectors placed on a ring. (e): Detector rotating around a fixed source. (f) Source-detector translating simultaneously along two parallel lines. In all figures: The source S is represented by a red point. The detector(s) D is (are) represented by blue point(s). The M , M i or M i in black, are running points and examples of scattering site. An example of trajectory for a photon whose scattering site is M is shown in purple. The corresponding scattering angle is denoted ω. The object to scan is represented in grey. The red continuous curves are the examples of scanning circles arcs. For (c), (e) and (f), the dashed circles (resp. lines) represents the circular (resp. linear) paths on which move the sensors.

Figure 3 : 4 j=1

 34 Figure 3: Setup and parameterization of the CST modality proposed in [25]. The source S and the detector are respectively represented by a red and a blue point. To make the difference between the four half-arcs, S 1 , S 3 and S 2 , S 4 are respectively depicted in red and green. Ω 1,2,3,4 denote the centres of the circles supporting the half-arcs S 1,2,3,4 . The point M is an example of a scattering site.

Figure 4 :

 4 Figure 4: (a) Original object: Derenzo phantom. (b) Corresponding acquired data for N SD = 2048 and N r = 1024. A distance of one pixel is left between the upper part of the image and the detector path (δ = 1, see 5.2.1).

  (a) Reconstruction for δ = 1 NMSE = 0.0112 (b) Reconstruction for δ = 26 NMSE = 0.0074 (c) Reconstruction for δ = 51 NMSE = 0.0061

Figure 5 :

 5 Figure 5: Reconstruction results of the Derenzo phantom 4a for δ = 1 (a), δ = 26 (b) and δ = 51 (c) pixel(s).

  (a) Reconstruction for x0,max = 2N . NMSE = 0.0084 (b) Reconstruction for x0,max = 3N . NMSE = 0.0049 (c) Reconstruction for x0,max = 4N . NMSE = 0.0061 (d) Reconstruction for 0.5 detectors per unit length. NMSE = 0.0046 (e) Reconstruction for 1 detectors per unit length. NMSE = 0.0049 (f) Reconstruction for 2 detectors per unit length. NMSE = 0.0049

Figure 6 :

 6 Figure 6: Evaluation of the number of source-detector positions on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for x 0,max = 2N 6a, 3N 6b and 4N 6c where ∆ x 0 = 1. Second row: Reconstruction results for 0.5 6d, 1 6e and 0.5 6f detector per unit length and x 0,max = 3N remains constant.

  (a) Reconstruction for rmax = 2N . NMSE = 0.0058 (b) Reconstruction for rmax = 3N . NMSE = 0.0040 (c) Reconstruction for rmax = 4N . NMSE = 0.0049 (d) Reconstruction for ∆r = 1. NMSE = 0.0040 (e) Reconstruction for ∆r = 2. NMSE = 0.0043 (f) Reconstruction for ∆r = 4. NMSE = 0.0043

Figure 7 :

 7 Figure 7: Evaluation of the number of scanning circles on reconstruction quality. First row: Reconstruction results of the Derenzo phantom 4a for r max = 2N 7a, 3N 7b and 4N 7c where ∆ r = 1. Second row: Reconstruction results for ∆ r = 1 7d, 2 7e and 4 7f detector per unit length and r max = 3N remains constant.

  1: Reconstruction of object f Data: RDf (x0, r), projections on double circular arcs of function f Result: f (x, y)1 Compute the one-dimensional Fourier transform of RDf (x0, r) using FFT relative to the first variable ; 2 Compute Gf (ξ, r) according to (4.2) and perform the inverse FFT to recover Gf (x0, r); 3 For each x0, interpolate the obtained data and sum on all values of x0 to have the back-projected data G ‡ f (x, z); 4 Perform the 2D FFT of G ‡ f (x, z) and weight by 2π 2 |σ|; 5 Compute the inverse FFT of the result to recover f ;

We position ourselves in the same frame of study as the original article, with the same working assumptions. Thus, first order Compton scattering is the only source of attenuation for radiation and data acquisition is performed with a pair source detector, assumed to be point-like. These conditions are common in the literature[16,

[START_REF] Webber | Three dimensional Compton scattering tomography[END_REF][START_REF] Webber | Microlocal analysis of a spindle transform[END_REF][START_REF] Cebeiro | New "improved" Compton scatter tomography modality for investigative imaging of one-sided large objects[END_REF][START_REF] Truong | Compton scatter tomography in annular domains[END_REF][START_REF] Tarpau | A new concept of Compton Scattering tomography and the development of the corresponding circular Radon transform[END_REF][START_REF] Tarpau | Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography[END_REF] and have been already discussed in[START_REF] Norton | Compton scattering tomography[END_REF][START_REF] Truong | Radon transforms on generalized Cormack's curves and a new Compton scatter tomography modality[END_REF][START_REF] Webber | Three dimensional Compton scattering tomography[END_REF][START_REF] Tarpau | Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography[END_REF].

Acknowledgements

We would like to thank Prof. T. T. Truong for stimulating discussions. C. Tarpau research work is supported by grants from Région Île-de-France (in Mathematics and Innovation) 2018-2021 and LabEx MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l'Incertitude et des Interactions) (No. ANR-11-LBX-0023-01).

J. Cebeiro research work is supported by a postdoctoral grant from the University of San Martín. He is also partially supported by SOARD-AFOSR (grant number FA9550-18-1-0523).


Christos Panagiotopoulos
email: panagiotopoulos@univmed.fr

Daniel J Repeta
email: drepeta@whoi.edu

Carl G Johnson

Characterization of methyl sugars, 3-deoxysugars and methyl deoxysugars in marine high molecular weight dissolved organic matter

Nuclear magnetic resonance (NMR) spectroscopy shows that > 50% of the carbon in marine high molecular weight dissolved organic matter (HMWDOM) from surface waters is a compositionally well-defined family of acylated polysaccharides that is conserved across ocean basins. However, acid hydrolysis of HMWDOM, followed by chromatographic analysis, recovers only 10-20% of the carbon as neutral, amino and acidic sugars. Most carbohydrate in HMWDOM therefore remains uncharacterized. Here, we use acid hydrolysis followed by Ag + and Pb 2+ cation exchange chromatography to separate HMWDOM hydrolysis products for characterization using 1-D and 2-D NMR spectroscopy. In addition to neutral sugars identified in past studies, we find 3-O-methylglucose, 3-O-methylrhamnose, 2-O-methylrhamnose and 2-O-methylfucose. We also find 3-deoxysugars to be present, although their complete structures could not be determined. Methyl sugars are widely distributed in plant and bacterial structural carbohydrates, such as cell wall polysaccharides, and their presence in HMWDOM suggests that structural carbohydrates may contribute to DOM in surface seawater. We find most HMWDOM carbohydrate is not depolymerized by acid hydrolysis and that the non-hydrolyzable component includes 6-deoxysugars.

Introduction

Dissolved organic matter (DOM) is a large reservoir of carbon that affects light penetration [START_REF] Nelson | Seasonal dynamics of colored dissolved material in the Sargasso Sea[END_REF][START_REF] Nelson | Chromophoric DOM in the Open Ocean[END_REF], microbial activity [START_REF] Williams | Microbial contribution to overall marine plankton metabolism: direct measurements of respiration[END_REF][START_REF] Azam | The ecological role of water column microbes in the sea[END_REF][START_REF] Ducklow | Oceanic bacterial production[END_REF], carbon-nitrogen-phosphorus storage [START_REF] Benner | Chemical composition and reactivity[END_REF][START_REF] Bronk | Dynamics of DON[END_REF][START_REF] Karl | Dynamics of DOP[END_REF][START_REF] Mopper | Photochemistry and the cycling of carbon, sulfur, nitrogen and phosphorus[END_REF], trace metal bioavailability [START_REF] Rue | The role of organic complexation on ambient iron chemistry in the Equatorial Pacific Ocean and the response of a mesoscale iron addition experiment[END_REF][START_REF] Wells | Marine colloids and trace metals[END_REF] and the retention of hydrophobic organic contaminants [START_REF] Guo | Composition and cycling of colloids in marine environments[END_REF] and references therein) in seawater. Although the influence of the dissolved organic matter reservoir on ocean biogeochemical cycles has been widely recognized, we have only a broad understanding of its chemical composition, sources or fate. Molecular level analysis of DOM hydrolysis products (simple sugars, amino acids and lipids) in the surface and deep ocean recovers only 4-11% and 1-2% of the total carbon respectively, and most DOM remains unidentified at this level of characterization [START_REF] Lee | Dissolved amino acids in the equatorial Pacific, the Sargasso Sea, and Biscayne Bay[END_REF][START_REF] Wicks | Carbohydrate signatures of aquatic macrophytes and their dissolved degradation products as determined by a sensitive high-performance ion chromatography method[END_REF][START_REF] Benner | Chemical composition and reactivity[END_REF][START_REF] Eglinton | Marine Organic Geochemistry[END_REF].

The introduction of large volume ultrafiltration techniques was a major advance in DOM studies and has made gram quantities of the high molecular weight fraction of DOM (HMWDOM) available for wet chemical and spectral analysis. Ultrafiltration through a 1 nm pore size membrane nominally retains organic matter of > 1 kDa MW, and routinely isolates 20-35% of total marine DOM. Marine HMWDOM is chemically distinct from humic substances and is enriched in carbohydrates [START_REF] Benner | Bulk chemical characteristics of dissolved organic matter in the ocean[END_REF]. 13 C NMR spectral analysis of HMWDOM in surface waters shows that > 50% of the carbon consists of compositionally welldefined acylpolysaccharides (APS) that are conserved across ocean basins [START_REF] Aluwihare | A major biopolymeric component to dissolved organic carbon in surface water[END_REF][START_REF] Eglinton | Marine Organic Geochemistry[END_REF]. However, after acid hydrolysis, only 10-20% of HMWDOM carbon (HMWDOC) can be recovered as simple neutral sugars (arabinose (I), fucose (II), galactose (III), glucose (IV), mannose (V), rhamnose (VI) and xylose (VII); see Appendix 1) using gas chromatography (GC) or high pressure liquid chromatography (HPLC; Panagiotopoulos and Sempe ´re ´, 2005 and references therein). Another 5-8% of HMW-DOC is recovered as acetic acid [START_REF] Aluwihare | A major biopolymeric component to dissolved organic carbon in surface water[END_REF], and a further 3-5% is recovered as amino acids [START_REF] Mccarthy | Major biochemical composition of dissolved high molecular weight organic matter in seawater[END_REF]. Lipids represent < 2% of the HMWDOC [START_REF] Mannino | Lipid composition in particulate and dissolved organic matter in the Delaware Estuary: sources and diagenetic patterns[END_REF][START_REF] Wang | Stable carbon isotopic compositions of high molecular weight dissolved organic matter from four US estuaries[END_REF]. It is notable that the distribution of major sugars released by acid hydrolysis of HMWDOM is nearly the same irrespective of where the sample is collected or the specific hydrolysis conditions used for depolymerization [START_REF] Sakugawa | Chemical studies on dissolved carbohydrates in water samples collected from North Pacific and Bering Sea[END_REF][START_REF] Mccarthy | Major biochemical composition of dissolved high molecular weight organic matter in seawater[END_REF][START_REF] Aluwihare | A major biopolymeric component to dissolved organic carbon in surface water[END_REF][START_REF] Borch | Concentration and composition of dissolved combined neutral sugars (polysaccharides) in seawater determined by HPLC-PAD[END_REF][START_REF] Panagiotopoulos | Analytical methods for the determination of sugars in marine samples: a historical perspective and future directions[END_REF]. Most of the carbohydrate in HMWDOM is therefore not characterized using conventional HPLC or GC techniques. Reconciling the discrepancy between the amount of carbohydrate measured using 13 C NMR spectroscopy and the amount of carbohydrate hydrolysis products measured with chromatographic techniques represents a major challenge for HMWDOM characterization.

Other approaches such as direct temperatureresolved mass spectrometry (DT-MS) and 15 N NMR indicate the presence of a much broader range of sugars than has been observed with GC and HPLC, including methyl sugars, methyl-deoxysugars, amino sugars and furfurals [START_REF] Boon | Molecular characterization of microgram amounts of oceanic colloid organic matter by direct temperature-resolved ammonia chemical ionization mass spectrometry[END_REF][START_REF] Minor | Estuarine organic matter composition as probed by direct temperature-resolved mass spectrometry and traditional geochemical techniques[END_REF][START_REF] Minor | Estuarine/marine UDOM as characterized by size-exclusion chromatography and organic mass spectrometry[END_REF][START_REF] Aluwihare | Two chemically distinct pools of organic nitrogen accumulate in the ocean[END_REF]. DT-MS studies characterize classes of sugars via diagnostic fragment ions but do not allow full identification of specific sugars or determination of their abundance. The 15 N NMR spectroscopy technique can be used to quantify nitrogen-containing sugars indirectly, but likewise does not give information on the presence or amounts of specific amino sugars. For example, [START_REF] Aluwihare | Two chemically distinct pools of organic nitrogen accumulate in the ocean[END_REF] inferred from 15 N and 1 H NMR analysis that 25-30% of HMWDOM occurs as unspecified N-acetyl-aminosugars. Chromatographic analysis has identified glucosamine and galactosamine in HMWDOM, but at concentrations of 6 2% HMWDOC [START_REF] Aluwihare | A major biopolymeric component to dissolved organic carbon in surface water[END_REF][START_REF] Aluwihare | Chemical composition and cycling of dissolved organic matter in the Mid-Atlantic Bight[END_REF]. Similar inconsistencies have also been reported for plankton and sinking particulate organic matter [START_REF] Hedges | Evidence for non-selective preservation of organic matter in sinking marine particles[END_REF][START_REF] Hedges | The biochemical and elemental compositions of marine plankton: a NMR perspective[END_REF]. Although a clear explanation for these discrepancies has not yet been found, two likely scenarios exist.

First, sugars other than neutral sugars, such as amino sugars, sugar alcohols, methylated sugars, uronic acids, sulfonated sugars, furfurals, etc., may constitute a considerable fraction of HMWDOM carbohydrate but lie outside the analytical window of the HPLC or GC techniques used for HMW-DOM characterization. A second likely explanation is that HMWDOM carbohydrate is simply not depolymerized with acid hydrolysis. In this study, we use Ag + and Pb 2+ ion chromatography to show that acid hydrolysis does not depolymerize most carbohydrate in HMWDOM. We also report the spectroscopic analysis of new methyl, deoxy and methyl deoxy sugars isolated from HMWDOM hydrolysis products. These sugars represent 2-5% of HMWDOM carbohydrate, but may also be present in the unhydrolyzed carbohydrate fraction.

Methods

Sampling

Seawater was drawn from the 15 m seawater intake at the Natural Energy Laboratory of Hawaii Authority in Kona, Hawaii in February 2002. The samples were filtered to remove bacteria and small particles using a cleaned (10% HCl) Suporflow dual stage (0.8 lm and 0.2 lm) Gelman polyether sulfone cartridge filter (Chisolm Corp., Lincoln, RI) fitted to an Advanta stainless steel housing. HMWDOM samples were collected using a large volume cross flow ultrafiltration system consisting of a stainless steel centripetal pump and membrane housings and a fluorinated high density polyethylene reservoir. The system was plumbed with teflon tubing and fitted with PVDF valves. We used four ultrafiltration membranes [90 ft 2 (8.4 m 2 ) each; Separation Engineering, Escondido, CA] with a 1 nm pore size that nominally retain organic matter of molecular weight greater than 1000 Da (> 98% rejection of vitamin B 12 ). Membranes were cleaned using isopropanol, detergent (0.01% micro), HCl (0.01 N) and NaOH (0.01 N), stored in sodium azide (0.55 mM) and rinsed with water and seawater immediately before use. The four filters were plumbed in parallel. Surface seawater (28,900 L) was concentrated over several days to approximately 40 L, frozen and returned to Woods Hole for further processing. The HMWDOM samples were concentrated to 2 L using a smaller system fitted with a 1 nm pore size, 10 ft 2 (0.93 m 2 ) membrane, desalted using repeated (•10) diafiltration with high purity, low carbon content, deionized water (Milli-Q water, 2 L) and reduced to 2 L. The final concentrate gave no visible precipitate (AgCl) when an aliquot of permeate was mixed with a 5 mg/50 ml solution of AgNO 3 . After diafiltration was complete, the sample (2 L) was lyophilized to a fluffy off-white powder (9.4 g) that was 35% by weight carbon with a C/N ratio of 13.

Sample preparation

HMWDOM samples (200-300 mg) were hydrolyzed in 2 M trifluoroacetic acid (TFA). Approximately 14.5 g (TFA) was added to 43 g of water in a 250 mL round bottomed flask. The solution was heated and HMWDOM added as a solution in approximately 2-3 mL water. The flask was flushed with He, sealed and placed in a sand bath at 120 °C. After 4 h, the dark brown solution was cooled, frozen at À20 °C and freeze dried. The dried sample was dissolved in 2-3 mL water and applied to a 6-8 mm i.d. column packed with 4 g anion exchange resin (OH À form, Bio-Rex Ò 5 resin, Bio-Rad Laboratories, Hercules, CA). The column was preconditioned by rinsing with 15 mL NaOH (0.5 M) followed by Milli-Q water until the pH returned to ca. 7. The column was drained of water and the sample was applied. Carbohydrates were eluted with 20-40 mL water, collected, concentrated to a small volume (few ml) using rotary evaporation and freeze dried. Sample recovery (by weight) after hydrolysis and ion exchange was 87%.

Chromatography

Chromatography was performed with a Waters pump (Model 510) and a Shodex RI-71 refractive index detector. Sample injection was performed with a manual Rheodyne injector fitted with a 100 lL sample loop. Carbohydrates were separated by degree of polymerization on sulfonated styrenedivinylbenzene gel analytical columns (30 cm • 7.8 mm i.d.; particle size 8 lm; cross linking 8%) in the silver or lead form (Supelcogel TM Ag or Pd, Supelco, Inc., Bellefonte PA). Column temperature was controlled using an oven (Tiberline TL-50) set at 80 °C. Sugars were eluted with water at a flow rate of 0.5 ml min À1 . Prior to use, the mobile phase was degassed with He for 30 min. The total weight of the sample injected in to the column did not exceed 10 mg/injection. Columns were regenerated at the end of each day by washing overnight with water at 0.2 ml min À1 . Retention times were measured using arabinose (I), fucose (II), galactose (III), glucose (IV), mannose (V), rhamnose (VI) and xylose (VII), 3-O-methylglucose (VIII) and DP (degree of polymerization, or number of monosaccahrides) 2-7 maltooligosaccharides (IX through XII; Sigma, St. Louis, MO, USA). Data acquisition and processing were performed with Agilent Chem-Station software (Agilent Technologies).

Mass spectrometry

Matrix assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF-MS) analysis was performed at the Center for Complex Carbohydrate Research in Athens, GA (USA), using a Voyager DE mass spectrometer (Applied Biosystems) operated in the positive ion mode. Aqueous solutions of samples were diluted 1:1 with 2,5-dihydroxybenzoic acid, applied to a MALDI plate and desorbed with a nitrogen laser having a guide wire reading of 0.005% and laser intensity of approximately 2630. Ions formed by the pulsed laser beam were accelerated through 20 kV with an acquisition range of 149-2000 Da. The mass spectrometer was calibrated with angiotensin I.

NMR spectroscopy

Spectra were recorded with a Bruker Avance DPX 400 MHz spectrometer using XWIN-NMR 3.0 software and an inverse broadband (BBI) 5 mm probe tuned for 13 C with a Z-gradient coil. Approximately 1-2 mg of the isolated compounds were dissolved in 1-2 ml D 2 O (99.9 atom% D, Aldrich) and dried under a N 2 stream. Dissolution/drying was repeated to ensure complete replacement of all exchangeable hydrogen with deuterium. Samples were finally dissolved in 0.75 ml D 2 O (freshly opened ampoules, > 99.96 atom% D) and transferred to 5 mm glass NMR tubes (Model 535-PP, Wilmad) using combusted (450 °C) NMR pipettes; 1 H chemical shift was set against the residual HOD signal (d 4.80 ppm) and 13 C chemical shifts were derived from the transmitter frequency.

The proton spectral width and file size were set to achieve ca. 0.3 Hz resolution, with subsequent equivalent zero filling bringing the final resolution to ca. 0.14 Hz. Experiments employed some suppression of the residual water signal, either a simple 30°pulse and presaturation during 1 s relaxation or WATERGATE (WATER suppression by GrAdient Tailored Excitation; Bruker sequence ''p3919gp'') sequence with gradient strengths reduced for minimal collateral suppression (particularly of the anomeric proton signals). Acquisition times ranged from 10 min to several days depending on the amount of material available. 1 H-1 H gs-COSY (gradient selected correlation spectroscopy; Bruker sequence ''cosygp'') spectra were acquired with a 45°mixing pulse to produce 512 or 1024 FID (free induction decay) of 32 or 64 summed scans, resulting in 2 • 8 Hz raw resolution and 2 • 2 Hz final resolution after zero filling and symmetrization. Interscan relaxation time was 3 s. Acquisition times were 15-30 h. The 1 H-13 C gs-HSQC (phase sensitive echo/antiecho-TPPI gradient-selected heteronuclear single quantum coherence; Bruker sequence ''invietgpsi'') spectra were acquired with delays optimized for 1 J 13C1H = 145 Hz; 1024 FIDs of 144 summed scans and no zero filling resulted in 2 • 24 Hz final resolution. Interscan relaxation time was 2 s and acquisition times were 4 days. All gradient experiments used 1500 lS 100 point sine shaped gradient pulses.

Results and discussion

Isolation of mono-and oligosaccharides from HMWDOM

To further investigate the composition of HMW-DOM carbohydrates, we used Ag and Pb cation exchange chromatography to separate oligosaccharides from monosaccharides released by acid hydrolysis. The mechanism of separation is based on the strength of the complex formed between the hydroxyl groups of the sugars and the metal (Ag + or Pb 2+ ) and the technique easily separates glycans via their degree of polymerization (DP) as shown in Fig. 1a [R t monosaccharides: (glucose, IV) > disaccharides (maltose, IX) > trisaccharides (maltotriose, X) > oligosaccharides (maltopentaose, XI; maltoheptaose, XII) > polysaccharides (laminarin, XIII)]. Monosaccharides are resolved via the sequence of axial and equatorial (ax, eq) vicinal hydroxyl groups that form the complexes with the metal cation [START_REF] Angyal | Complexes of metal cations with carbohydrates in solution[END_REF] can be manipulated by changing the complexing cation (K + , Ca 2+ , Ag + , Pb 2+ ) and resolution is generally improved when large cations are used (e.g. Pb 2+ or Ca 2+ ), because stronger complexes are formed. Fig. 1b depicts a typical chromatogram of hydrolyzed HMWDOM using two Ag columns in series. Unhydrolyzed or partially hydrolyzed HMWDOM fractions F 1 (59%, 11-18.5 min) and F 2 (15%, 18.5-23.6 min) were eluted before disaccharides and methyl sugars (F 3 ; 5%, 23.6-26.8 min), neutral sugars (F 4 ; 20%, 26.8-35 min) and other hydrolysis products (F 5 ; 2%, 35-40 min). The seven major neutral sugars characteristic of HMWDOM are eluted in fraction F 4 and collected in three sub-fractions (F 4a -F 4c ). Fraction F 4a includes glucose (27.8 min) and rhamnose (28.5 min), fraction F 4b xylose (29.5 min), galactose (30.5 min) and mannose (30.5), while fraction F 4c contains fucose and arabinose which co-elute at 33.1 min.

The 1 H NMR spectrum of fraction F 3 shows it to be a complex mixture of methylated, 6-deoxy-and 3-deoxysugars and the fraction was further separated into 10 additional fractions using four Pb cation exchange columns in series. The three largest peaks were collected as fractions F 3 A-C (Fig. 2a) and were purified one more time using four Ag columns in series. Fig. 2b depicts a typical separation of the F 3 B fraction purification. Only the largest peaks were collected for detailed compositional analysis (see below).

Characterization of methyl, methyldeoxy, and deoxysugars in HMWDOM

Fraction F 3 A

Separation of fraction F 3 A on four Ag columns yielded three peaks. The major peak was collected as fraction F 3 A2 and had an 1 H NMR spectrum with signals for the anomeric protons at d 5.27 ppm (H-1, a anomer: J 1,2 = 3.6 Hz) and 4.69 ppm (H-1, b anomer: J 1,2 = 7.9 Hz), H-C-OH protons at 3.34-3.93 ppm and one O-methyl (-OCH 3 ) group at 3.65 ppm (Table 1). Protons H-2 through H-6 were assigned from COSY experiments, while HSQC experiments indicated the presence of seven carbon atoms consistent with an O-methylated sugar. On the basis of its NMR spectrum, fraction F 3 A2 was identified as a mixture of a-and b-3-O-methylglucose (VIII). Additional confirmation was given by examination of the first order coupling constants and co-elution with authentic 3-O-methylglucose on HPLC using four Ag columns connected in series (retention time of 46.2 min).

Fraction F 3 A3 was an a/b mixture of at least three different sugars. The major anomers appeared as a broad singlet at 5.34 ppm (a anomer) and 4.82 ppm (b anomer). We were unable to measure the coupling constant for the b anomer due to interference with the solvent peak (water, 4.80 ppm). We observed a sharp O-methyl singlet at 3.68 ppm and the 1 H COSY NMR spectrum allowed us to partially assign two coupled spin systems: aH-1 (5.34 ppm) ! aH-2 (4.29 ppm) ! aH-3 (2.54, 1.88 ppm) ! aH-4 (4.17 ppm) and bH-1 (4.82 ppm) ! bH-2 (3.60 ppm) ! bH3 (2.32, 1.67 ppm) ! bH-4 (3.58 ppm; Table 1). Unlike other sugar fractions we isolated from HMWDOM, we observed for this sugar strong cross peaks in the COSY spectrum between the equatorial proton (H eq ) aH-3 eq at 2.59 ppm (14.9, 5.5 Hz) and the axial proton (H ax ) at aH-3 ax at 1.90 ppm (13.0, 5.5 Hz) with aH-2 at 4.29 ppm, and between bH-3 eq at 2.34 ppm (1.7, 14.4 Hz) and bH-3 ax at 1.70 ppm (1.2, 12.5 Hz) with bH-2 at 3.35 ppm that are characteristic of a 3-deoxysugar. Fig. 3 compares the 1 H COSY NMR spectrum for 3-deoxyglucose (top; XIV) and fraction F 3 A3 (bottom) and highlights the presence of the CH 2 protons as cross peaks in the 3.5-4.5 ppm • 1.5-2.6 ppm region (compare to Fig. 4). The large coupling con-stants of 12.5-14.4 Hz are characteristic of H eq /H ax coupling, while the smaller coupling constants are typical of H eq /H ax coupling to neighboring (H-2, H-4) protons. We were unable to identify the major compound in F 3 A3 beyond the level of an O-methyl-3-deoxysugar. Comparison with the NMR spectra for authentic 3-deoxyglucose shows that F 3 A3 is not O-methyl-3-deoxyglucose, based on differences in cross peaks at 3.5-4.5 ppm • 1.5-2.6 ppm (Fig. 3). We also identified a third deoxysugar which, from the 1 H COSY NMR spectrum, appears to have the system: bH-1 (4.69 ppm) ! bH-2 (3.34 ppm) ! bH-3 (3.54 ppm) ! bH-4 (2.62, 1.52 ppm). From the COSY spectrum, we assigned this as a 4-deoxysugar, but the assignment is not definitive.

Fraction F 3 B

Separation of F 3 B on four Ag columns resulted in two major peaks collected as fractions F 3 B1 and F 3 B2 (Fig. 2b). The 1 H NMR spectra showed the two fractions to be closely related. Fraction F 3 B1 was obtained as a mixture of a and b anomers, for which we were able to assign protons H 2-6 from the COSY spectrum and all carbons through HSQC experiments. The 1 H NMR spectrum of fraction F 3 B1 displayed signals for two anomeric protons at d 5.21 ppm (H-1, a-anomer; d, J 1,2 = 1.6 Hz) and 4.91 ppm (H-1, b-anomer; d, J 1,2 = 1.0 Hz), H-C-OH protons at 3.37-4.25 ppm, one O-methyl at 3.50 ppm and two methyls at 1.32 ppm (aH-6, J 5,6 = 5.4 Hz) and 1.34 ppm (bH-6, J 5,6 = 6.4 Hz) (Table 1). We noted the very small coupling constant for the b anomer, the a/b anomeric ratio > 1 and the smaller coupling constant of J 5,6 for the b relative to the a anomer, which are all characteristics of rhamnose (VI). First order coupling constants for a F 3 1.6 Hz;3.6 Hz;9.7 Hz;9.7 Hz;6.4 Hz) were within 1-2 Hz of authentic rhamnose (a-6-deoxymannose) reported by [START_REF] Bock | Nuclear magnetic resonance spectroscopy in the study of mono-and oligosaccharides[END_REF]; 1.6 Hz;3.5 Hz;9.5 Hz;9.5 Hz;6.2 Hz). We also noted the ca. 10 ppm downfield shift in the HSQC spectrum for C-3 (aC-3, 79.9; bC-3, 82.3) relative to rhamnose (aC-3, 71.1; bC-3, 73.8 ppm), indicating methylation at this site [START_REF] Deepak | Structure elucidation of oligosaccharides containing 2-deoxy sugars by NMR techniques[END_REF]. In the HSQC spectrum we observed a correlation between bH-3 (3.37 ppm) and a carbon at 82.4 ppm, which we assigned as bC-3, further establishing the site of methylation. There is a cross peak between H-4 (3.45 ppm) and a carbon at 71.4 ppm, which we assigned as bC-4, in good agreement with the values assigned by [START_REF] Popper | Primary cell wall composition of bryophytes and charophytes[END_REF] and [START_REF] Popper | 3-O-Methylrhamnose in lower land plant primary cell walls[END_REF] for 3-Omethylrhamnose released by hydrolysis of lower land plant primary cell walls. We therefore identified F 3 B1 as a mixture of a-and b-3-O-methylrhamnose (XV; Table 1). Mass spectrometry of F 3 B1 showed a molecular ion at m/z 176, in agreement with our identification. However, for aF 3 B1, protons aH-3 and aH-4 have the same chemical shift and for this anomer we are unable to establish the site of O-methylation (3-O-methyl vs 4-O-methyl). Our assignment of 3-O-methylation therefore rests on our assignments for the b anomer.

We used the same reasoning to identify F 3 B2 as a a/b mixture of 2-O-methylrhamnose (XVI; [START_REF] Zdorovenko | Location of the Omethyl groups in the O polysaccharide of Pseudomonas syringae pv. phaseolicola[END_REF][START_REF] Scha ¨ffer | The surface layer (S-layer) glycoprotein of Geobacillus stearothermophilus NRS 2004/3a[END_REF]. Fractions F 3 B1 and F 3 B2 have very similar NMR spectra, but fraction F 3 B2 was obtained as a mixture of at least four different sugars, leading to some ambiguity in our interpretation of the COSY and HSQC spectra. We were only able to make full assignments for the most abundant a anomer (a F 3 B2). However, due

α.β H-2 x H-3(a,e) H-3 x H-3 H-3,4 x H-3,4 α.β H-2,3 x H-3,4(a,e)
Fig. 3. 1 H NMR COSY spectra of 3-deoxyglucose (top panel) and fraction F 3 A2 (bottom panel). The 1-D 1 H NMR spectrum of 3-deoxyglucose (XIV) is projected on the X-and Y-axes of the top panel for reference. Cross peaks due to coupling of axial and equatorial protons H-3 between 1.5 and 2.6 ppm are characteristic of 2-, 3-, and 4-deoxysugars. Cross peaks in this region are not observed in the spectra of hexoses (glucose, galactose) and 6deoxysugars (see Fig. 4 for comparison).

to the complexity of the spectrum, our assignments of H-3 to H-5 should be viewed as tentative. We found two anomeric protons at 5.32 ppm (H-1, a-anomer; d, J 1,2 = 1.6 Hz) and 4.92 ppm (H-1, b-anomer; broad singlet), H-C-OH protons at 3.39-3.87 ppm, one O-methyl group at 3.51 ppm and two methyls at 1.31 ppm (J 5,6 = 5.3 Hz) and 1.33 ppm (J 5,6 = 6.7 Hz) for the a and b anomers, respectively. We found good agreement between our spectral data and 2-O-methylrhamnose as the terminal sugar in a glycan polymer [START_REF] Scha ¨ffer | The surface layer (S-layer) glycoprotein of Geobacillus stearothermophilus NRS 2004/3a[END_REF]. Because the 2-O-methylrhamnose reported by Scha ¨ffer et al. ( 2002) is glycosylated at C-1, we expected a difference in the chemical shift for H-1 and the C-1 chemical shift to appear downfield by 6-7 ppm of our value for F 3 B1 [START_REF] Deepak | Structure elucidation of oligosaccharides containing 2-deoxy sugars by NMR techniques[END_REF]. Other 1 H and 13 C values should remain near those for a-2-O-methylrhamnose (Table 1); 2-O-methlyrhamnose was previously identified in HMWDOM from the NE Atlantic Ocean using gas chromatography mass spectrometry of the alditol peracetate [START_REF] Aluwihare | Chemical composition and cycling of dissolved organic matter in the Mid-Atlantic Bight[END_REF].

(α/β)H-6 β H-1 α H-1 α O-methyl β O-methyl β H-2 α H-5 α H-
The mass spectrum of fraction F 3 B2 has a molecular ion at m/z 176, as expected for a methylated deoxysugar.

Fraction F 3 C

Separation of F 3 C on four Ag columns yielded one large peak collected as fraction F 3 C1. The 1 H NMR spectrum has major signals for the anomeric protons at d 5.49 ppm (H-1, a anomer, J 1,2 = 1.4 Hz) and 4.64 ppm (H-1, b anomer, J 1,2 = 8.0 Hz), H-C-OH protons at 3.23-4.24 ppm, two O-methyl groups at 3.54 and 3.66 ppm and two methyls at 1.26 and 1.29 ppm (J 5,6 = 6.5 Hz; Table 1). The strong coupling between the methyl at 1.26 ppm and the upfield quartet (J 5,6 = 6.5 Hz,) at 4.24 ppm is typical of an a fucose-like configuration. Protons H-2 through H-6 were assigned from COSY experiments, while HSQC experiments indicated the presence of fourteen carbon atoms, consistent with a mixture of a and b anomers of methylfucose (Fig. 4). In Table 1 we compare chemical shift data for F 3 C1 with 1 H NMR data we acquired for authentic fucose. We find excellent agreement between fucose and F 3 C1, with the greatest difference at H-2, suggesting 2-O-methylation in F 3 C1. We measured first order coupling constants for bF 3 C1 of H-1 (8.0 Hz), H-2 (9.6 Hz), H-3 (3.4 Hz), H-4 (1 Hz), H-5 (6.6 Hz) compared to H-1 (8.2 Hz), H-2 (10.0 Hz), H-3 (3.6 Hz), H-4 (0.8 Hz), H-5 (6.6 Hz) for authentic b-fucose [START_REF] Bock | Nuclear magnetic resonance spectroscopy in the study of mono-and oligosaccharides[END_REF]. Integration of the O-methyl and 6-methyl groups gave a 1:1 quantitative relationship and we note that C-2 signals (a C-2 = 77.7; b C-2 = 81.9) are shifted approximately 10 ppm downfield from C-2 in fucose (a C-2 = 68.6 ppm, b C-2 = 72.2 ppm), confirming O-methylation at C-2. On the basis of its NMR spectrum, we assigned F 3 C1 as a mixture of a and b 2-O-methylfucose (XVII). To the best of our knowledge, there are no reports of 2-O-methylfucose 1 H and 13 C NMR spectra with which to compare our data. However, NMR data for 2-O-methylfucosyl-containing heptasaccharide isolated from red wine rhamnogalacturonan II are in good agreement with our values [START_REF] Glushka | Primary structure of 2-O-methyl-a-L-fucose-containing side chain of the pectic polysaccharide, rhamnogalacturonan II[END_REF]; Table 1).

O-Methyl and deoxysugars in structural polysaccharides

The components 2-O-methylrhamnose and 3-Omethylrhamnose have been reported in structural polysaccharides from a large number of cyanobacteria [START_REF] Painter | Algal Polysaccharides[END_REF][START_REF] Shekharam | Carbohydrate composition and characterization of two unusual sugars from the blue green alga, Spirulina platensis[END_REF][START_REF] Ogawa | Isolation and identification of 2-O-methyl-L-rhamnose and 3-O-methyl-Lrhamnose as constituents of an acidic polysaccharide of Chlorella vulgaris[END_REF] and vascular plants [START_REF] Bacon | Apiose and mono-Omethyl sugars as minor constituents of the leaves of deciduous trees and various other species[END_REF][START_REF] Casagrande | Organic geochemical investigations in the Okefenokee swamp, Georgia: the fate of fatty acids, glucosamine, cellulose, and lignin[END_REF][START_REF] Popper | Primary cell wall composition of bryophytes and charophytes[END_REF][START_REF] Popper | 3-O-Methylrhamnose in lower land plant primary cell walls[END_REF][START_REF] Sassaki | Carbohydrate, glycolipid, and lipid components from the photobiont (Scytonema sp.) of the lichen, Dictyomema glabratum[END_REF]. These two sugars, together with 3-O-methylglucose and 2-O-methylfucose are common in lipopolysaccharides, part of the outer cell membrane of gram negative bacteria [START_REF] Kennedy | Bioreactive carbohydrates[END_REF][START_REF] Brade | Structure-activity relationships of bacterial lipopolysaccharides (endotoxins)[END_REF]. 2-O-methylfucose is also a constituent of rhamnogalacturonan II pectic polysaccharides, part of the primary walls of dicotyledenous and monocotyledenous plants and gymnosperms [START_REF] Glushka | Primary structure of 2-O-methyl-a-L-fucose-containing side chain of the pectic polysaccharide, rhamnogalacturonan II[END_REF] and 2-and 3-O-methylrhamnose, along with 2-and 3-O-methylfucose, have been putatively reported in cyanobacterial mats and in recent marine sediments, where they comprise up to 5-15 wt% of the total neutral monosaccharide yield [START_REF] Klok | Qualitative and quantitative characterization of the total organic matter in a recent marine sediment[END_REF][START_REF] Klok | Carbohydrates in recent marine sediments-II. Occurrence and fate of carbohydrates in a recent stromatolitic deposit: Solar lake[END_REF], as well as in humic and fulvic acids extracted from soil [START_REF] Ogner | Analysis of the carbohydrates of fulvic and humic acids as their partially methylated alditol acetates[END_REF]. Given the wide distribution of these methyl sugars in biological and geological materials, they provide little information about the specific source of HMWDOM carbohydrate. However, to the best of our knowledge they have only been reported to occur in structural polysaccharides, and not in storage polysaccharides. Their occurrence in HMWDOM suggests that specific degradation-resistant structural polysaccharides contribute to DOM.

Only one methyl sugar, 2-O-methylrhamnose, has been previously reported as a trace constituent of HMWDOM [START_REF] Aluwihare | Chemical composition and cycling of dissolved organic matter in the Mid-Atlantic Bight[END_REF] and we are not aware of reports of 3-deoxysugars in HMW-DOM. Methyl and 6-deoxysugars may nevertheless be important in the uncharacterized fraction of HMWDOM and might serve to give the 1 H NMR spectrum of HMWDOM its characteristic appearance [START_REF] Aluwihare | A major biopolymeric component to dissolved organic carbon in surface water[END_REF]. Fractions F 1 and F 2 from our Ag column separation of acid-hydrolyzed HMWDOM were eluted with PDP 3 glucans, which have a nominal molecular weight > 0.5 kD (Fig. 1b). 1 H NMR spectra of F 1 (Fig. 5) and F 2 display none of the spectral fine structure characteristic of DP 1-4 mono-and oligosaccharides that would be expected if these fractions contained significant amounts of degraded simple sugars. The two fractions represent 74% of the total HMW-DOM, demonstrating that commonly used hydrolytic protocols [2N TFA; 1-6 N HCl (Repeta, unpublished results)] do not depolymerize most HMWDOM. The recalcitrance of HMWDOM carbohydrate towards acid hydrolysis explains most of the discrepancy between NMR measurements of HMWDOM carbohydrate and GC or HPLC measurements of simple sugars. Previous GC and HPLC analyses of HMWDOM hydrolysis products have also missed methyl sugars identified in our current study, further lowering the amount of carbohydrate reported using these methods.

Hydrolysis resistant carbohydrate in HMWDOM

Qualitative information on the composition of HMWDOM carbohydrate that is resistant to acid hydrolysis can be obtained from the NMR spectra of fraction F 1 . The COSY spectrum of hydrolyzed HMWDOM fraction F 1 (Fig. 1b) shows strong cross peaks for 6-deoxysugars (1.2-1.4 ppm • 3.3-4.4 ppm; Fig. 5). The most intense cross peaks appear at 1.34 • 4.32 ppm and 1.23 • 3.79 ppm, similar to a and b fucose (1.26 • 4.24 ppm and 1.29 • 3.83 ppm, respectively). The COSY spectrum further shows at least eight distinct a anomeric and four distinct b anomeric protons. These protons could represent a mixture of different a-and b-linked sugars or positional isomers, but nevertheless suggest a structurally complex polysaccharide.

We did not observe cross peaks in the COSY spectrum that are characteristic of lipids. No cross peaks were observed between 1.3 ppm (-CH 2 -) and signals < 1 ppm (CH 3 ) or 1.5-3 ppm (-CH 2 -OH; -CH 2 -COOH) that we expect to see if ether or ester bound lipids were present. The COSY spectrum of HMWDOM does not suggest a major contribution from lipids to the peak at 1.3 ppm in the 1 H NMR spectrum. In support of this, [START_REF] Quan | Chemical characterization of dissolved organic matter (DOM) in seawater: structure, cycling, and the role of biology[END_REF] noted the production of acetic acid as a major oxidation product of HMWDOM previously treated with sodium periodate and ascribed this to the oxidation of 6-deoxysugars at C-5. [START_REF] Quan | Chemical characterization of dissolved organic matter (DOM) in seawater: structure, cycling, and the role of biology[END_REF] did not recover lipids from HMWDOM periodate oxidation products, even though lipids should have survived the reaction conditions. [START_REF] Quan | Chemical characterization of dissolved organic matter (DOM) in seawater: structure, cycling, and the role of biology[END_REF] data support the COSY NMR assessment that lipids are only a minor component of HMWDOM. If we assume that all the carbon represented by the 1.3 ppm peak is from 6-deoxysugars, then 6-deoxysugars represent 25-30% of the total uncharacterized HMWDOM carbohydrate.

Conclusions

Previous characterization of HMWDOM using DT-MS has shown that HMWDOM carbohydrate includes methyl-and methyldeoxysugars. We confirm these results with the identification of 2-and 3-O-methylrhamnose, 2-O-methylfucose and 3-Omethylgucose in HMWDOM. We also isolated fractions that have NMR spectral characteristics of 3-and 4-deoxysugars, although the full structure of these components could not be determined. We are not aware of reports of O-methylrhamnose and O-methylfucose in algal or bacterial storage polysaccharides, but these sugars are commonly found in structural polysaccharides from higher plant cell walls and our analyses point to degradation-resistant cell wall polysaccharides as an impor- tant component of HMWDOM. Separation of HMWDOM after acid hydrolysis using Ag + and Pb 2+ cation exchange chromatography shows that most carbohydrate is not depolymerized using commonly used hydrolysis protocols. New strategies need to be developed if HMWDOM is to be characterized at the level of constituent simple sugars. The 2-D NMR techniques provide some insight into the composition of the non-hydrolyzed component and show it to be a heterogeneous mixture containing up to 25-30% 6-deoxysugars, most likely rhamnose and fucose.

VII

Fig. 2 .

 2 Fig. 2. (a) Separation of fraction 3 (F3; Fig. 1b) with cation exchange HPLC-RI. Conditions: four Pb columns in series; column temperature, elution mode, flow rate and detection as for Fig. 1. (b) Separation of F 3 B fraction on four Ag columns in series; NB expansion of 35-55 min section. No peaks detected before 35 min. Conditions same as for Fig. 1.

  Fig. 4. 1 H NMR COSY (top) and HSQC spectra (bottom) of fraction F 3 C1 (2-O-methylfucose; XVII). The 1-D 1 H NMR spectrum of F 3 C1 is projected on the X-and Y-axes of the top panel for reference. Cross peaks in COSY spectrum are due to coupling between adjacent protons and are used to assign signals in the 1-D spectrum. 1 H NMR spectra assignments are then used to determine carbon chemical shifts in the HSQC spectrum (lower panel).

Fig

  Fig.5. 1 H NMR COSY spectrum of fractionF1 (11.8-18.5 min). The spectrum displays numerous cross peaks for a and b anomeric protons(aH-1, 5.23-5.52; bH-1, 4.54-4.64) and a/bH-2 protons (3.22-3.80 ppm) indicate heterogeneity in either the monosaccharide composition or glycosidic linkages of HMW-DOM carbohydrate. Strong cross peaks are also apparent between 1.23-1.42 ppm and 3.79-4.38 ppm from 6-deoxysugars that are part of the non-hydrolyzable fraction of HMWDOM carbohydrate.

  -30 laminarin XIII C. Panagiotopoulos et al. / Organic Geochemistry 38 (2007) 884-896 Associate Editor-P. Schaeffer

  . The column selectivity

						Glucose
		Laminarin	Maltoheptaose	Maltopentaose	Maltotriose	Maltose
	5	10	15	20	25	30	min
							F 4
							F 4b
						F 4a		F 4c
			F 1		F 2	F 3		F 5
	5	10	15	20	25	30	35	40	min
	Fig. 1. (a) Separation of glucose (IV; 28.2 min), maltose (IX;
	22.6 min), maltotriose (X; 19.3 min), maltopentaose (XI;

15.8 min), maltohepatose (XII; 14.2 min) and laminarin (XIII; 12.8 min) with Ag + cation exchange chromatography (80 °C, two columns in series). Sugars were eluted with high purity water at 0.5 ml min À1 and detected with a refractive index (RI) detector. (b) Typical separation of hydrolyzed HMWDOM using same columns and conditions.

Table 1 1

 1 H and 13 C NMR chemical shifts for HMWDOM sugars a,b

	Fraction/compound	Nucleus	1	2	3	4	5	6; 6 0	O-Methyl
	a-F3-A2	H	5.27	3.63	3.53	3.53	3.88	3.88; 3.81	3.65
		C	92.5	71.4	83.1	69.4	71.9	60.9	60.3
	3-O-Methyl-a-glucose c	H	5.25	3.64	3.53	3.53	3.89	3.88; 3.81	3.68
		C	92.4	71.4	83.1	69.4	71.9	60.9	60.3
	b-F3-A2	H	4.69	3.34	3.34	3.53	3.53	3.93; 378	3.65
		C	96.3	73.9	85.8	69.4	76.3	61.1	60.3
	3-O-Methyl-b-glucose c	H	4.70	3.35	3.34	3.53	3.53	3.94; 3.77	3.68
		C	96.3	73.9	85.7	69.4	76.2	61.1	60.3
	a-F3-A3	H	5.34	4.29	2.54; 1.88 d	4.17	ND i	ND i	3.68
		C	92.9						60.5
	b-F3-A3	H	4.82	3.60	2.32; 1.67 d	3.58	ND i	ND i	3.68
	a-F3-B1	H	5.21	4.23	3.53	3.53	3.94	1.32	3.50
		C	94.3	66.8	79.9	71.6	68.8	17.2	56.5
	a-3-O-Methylrhamnose e	H	5.16	4.18	3.48	3.46	3.89	1.27	3.45
		C	94.3	66.9	79.9	71.6	68.7	17.2	56.5
	b-F3-B1	H	4.91	4.25	3.37	3.45	3.47	1.34	3.50
		C	94.0	67.3	82.4	72.5	ND	17.2	56.5
	b-3-O-Methylrhamnose e	H	4.86	4.18	3.31	3.39	3.41	1.29	3.45
		C	93.9	67.4	82.3	71.3	72.5	17.2	56.5
	a-F3-B2	H	5.32	3.60	3.87	3.39	3.87	1.30	3.51
		C	91.1	81.2	70.1	73.0	68.6	17.2	59.1
	a-2-O-Methylrhamnose f	H	5.17	3.70	3.89	3.37	3.81	1.26	3.45
		C	99.8	81.0	70.6	73.1	70.0	17.4	59.4
	a-F3-C1	H	5.49	3.52	3.93	3.86	4.24	1.26	3.54
		C	89.8	77.7	68.9	72.4	66.6	15.8	57.8
	a-2-O-Methylfucose g	H	5.65	3.45	4.00	3.81	4.37	1.17	3.47
		C	97.6	79.9	70.7	74.4	68.8	17.9	59.8
	a-Fucose h	H	5.25	3.83	3.92	3.85	4.25	1.27
		C	92.7	68.5	69.7	71.3	66.8	16.0
	b-F3-C1	H	4.64	3.23	3.73	3.79	3.83	1.29	3.66
		C	96.4	81.9	73.0	71.9	71.2	15.8	60.6
	b-Fucose h	H	4.60	3.50	3.70	3.79	3.86	1.29
		C	96.7	72.2	73.4	71.9	72.3	16.0

a Numbering of C 1-6 and H1-6 given in Appendix 1. b Chemical shifts for fractions isolated from HMWDOM in bold face type. c From authentic 3-O-methylglucose. d H(equatorial); H(axial). e Popper et al. (2004) [Data for pure a and b-3-O-methylrhamnose; trivial name acofriose]. f Scha ¨ffer et al. (2002) [Data for a-2-O-methylrhamnose as terminal sugar in a glycan polymer]. g Glushka et al. (2003) [Data for a-2-O-methylfucoheptasaccharide]. h From authentic fucose.

i Not determined.

C.Panagiotopoulos et al. / Organic Geochemistry 38 (2007) 884-896

Acknowledgements

We thank Jan War and the staff at the Natural Energy Laboratory in Kona, Hawaii for assistance in providing facilities for sample collection. We also thank Dr. Parastoo Azadi and the staff of the analytical services program at the Complex Carbohydrate Research Center in Athens, Georgia for mass spectrometric analysis. We are also grateful to P. Schaeffer, J. S. Damste ´and G. Wolf for valuable comments, advice and suggestions. Funding was provided by the Ocean Carbon Sequestration Research Program, Biological and Environmental Research (BER), US Department of Energy Grant DEFG0200ER62999 and the National Sciences Foundation Chemical Oceanography Program Grant OCE 9818654. C.P. received support through the Postdoctoral Fellowship Program of the Woods Hole Oceanographic Institution and DJR support through the Stanley Watson Chair in Oceanography.

Appendix 1

Structures of methyl-deoxy-and methyldeoxysugars. Positions 1-6 are noted in bold numerals on structures II and XVII; H ax and H eq in b-3-deoxyglucose (XIV) denote configuration (axial and equatorial) of H-3 protons.