Wenxing Zhang

Jérôme Fehrenbach

Annaick Desmaison

Valérie Lobjois

Bernard Ducommun

Pierre Weiss
email: pierre.armand.weiss@gmail.com

Annaïck Desmaison

Structure tensor based analysis of cells and nuclei organization in tissues

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The advent of new imaging technologies allows observing biological samples with an unprecedented spatial, temporal and spectral resolution. It offers new opportunities to perform systematic studies of geometrical configurations of cells or nuclei in their micro-environment to better understand the fundamental processes involved in morphogenesis or tumor growth [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF][START_REF] Jorand | Deep and clear optical imaging of thick inhomogeneous samples[END_REF].

Due to the huge amount of data contained in large images, automatic procedures are however essential to assess cells properties such as location, size, orientation, aspect ratio, etc... The lack of robust, fast and universal procedures to provide such a geometric description is probably one of the main obstacles to exploit the full potential of images.

The mainstream approach to analyze image contents nowadays consists in segmenting each cell/nuclei independently [START_REF] Olivier | Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy[END_REF][START_REF] Pop | Extracting 3d cell parameters from dense tissue environments: application to the development of the mouse heart[END_REF]. A precise segmentation completely describes the geometrical contents of images and is often regarded as the best source of information one can hope for. However biological images often suffer from many degradations. For instance, in fluorescence microscopy, light scattering, absorption or poor signal to noise ratio strongly impair image quality, especially in 3D. In many situations it is therefore hopeless to perform a proper image segmentation. Moreover, in cases where large cell populations are investigated, a complete segmentation (i.e. a precise description of the objects boundaries) still brings more information than needed to understand the overall geometrical distribution.

In this paper we therefore pursue a somehow less ambitious goal. We adopt a macroscopic point of view and consider the biological sample as a continuous medium. This idea stems from mathematical models that describe tissues as continuous media such as incompressible fluids, elastic or viscoelastic materials [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF][START_REF] Ambrosi | On the mechanics of a growing tumor[END_REF][START_REF] Humphrey | A constrained mixture model for growth and remodeling of soft tissues[END_REF][START_REF] Humphrey | Review paper: Continuum biomechanics of soft biological tissues[END_REF][START_REF] Bonnet | Mechanical state, material properties and continuous description of an epithelial tissue[END_REF]. Our main contribution is to show through both theoretical and numerical results that the so-called structure tensor [START_REF] Förstner | A fast operator for detection and precise location of distinct points, corners and centres of circular features[END_REF][START_REF] Bigun | Multidimensional orientation estimation with applications to texture analysis and optical flow[END_REF][START_REF] Gösta | Signal processing for computer vision[END_REF], provides a fast, robust and precise enough tool to retrieve cells orientation and anisotropy in 2D and in 3D. We show the following original results:

• While the structure tensor is usually implemented to assess texture orientations, we show that it also allows quantifying precisely the anisotropy of cells or nuclei. This is done by analyzing the method behavior on fields of functions with ellipsoidal isosurfaces.

• The proposed mathematical analysis also allows quantifying the method bias. It shows that a very good estimation can be expected even when very few nuclei are locally similar (in the sense that they can be well approximated by the same ellipsoid).

• We show that the method is invariant under contrast changes.

We also perform various experiments to validate our theoretical findings. We assess the structure tensor efficiency on synthetic and real 2D and 3D images. Its output is compared to ground-truth obtained analytically in case of synthetic data or manually in case of real data. These comparisons show that the structure tensor allows to quickly assess cells organization at a large scale. We finish the paper by providing an example of application to the analysis of geometrical configurations of nuclei in multicellular tumor spheroids.

Related work

The structure tensor has already been used in various contexts of biological imaging. One of its main applications is coherence enhancing or diffusion [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF][START_REF] Pop | Extracting 3d cell parameters from dense tissue environments: application to the development of the mouse heart[END_REF] which usually allows improving images quality without degrading their geometrical content too much. It was also used to analyze geometrical features such as fibers orientations in 2D and 3D [START_REF] Palaniappan | Moving object segmentation using the flux tensor for biological video microscopy[END_REF][START_REF] Menzel | Towards an orientation-distribution-based multi-scale approach for remodelling biological tissues[END_REF][START_REF] Federico | Nonlinear elasticity of biological tissues with statistical fibre orientation[END_REF][START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF][START_REF] Fonck | Effect of aging on elastin functionality in human cerebral arteries[END_REF]. The works [START_REF] Rezakhaniha | Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy[END_REF][START_REF] Fonck | Effect of aging on elastin functionality in human cerebral arteries[END_REF] also come with an ImageJ plugin http://bigwww.epfl. ch/demo/orientation/. This plugin has a nice interface and can be used to reproduce some experiments of our paper. The authors of these two references mention that the structure tensor allows quantifying the orientation and the isotropy properties of a region of interest. However, the isotropy is defined in a way different from the present paper and the authors do not state precisely how this information relates to the image contents.

Paper organization

The rest of the paper is organized as follows. Notation is introduced in Section 2. Section 3 constitutes the theoretical part of the paper. We introduce the structure tensor and demonstrate its properties when applied to images that consist in fields of functions with ellipsoidal isosurfaces. In Section 4, we illustrate the method on synthetic and real 2D and 3D data.

Notation

For any x, y ∈ R d , the angle in degrees between x and y is denoted ∠(x, y), this angle lies in [0

• , 90 • ]. The ℓ p -norm of x ∈ R d is denoted x p and defined by x p := ( d i=1 |x i | p ) 1/p . The positive semidefiniteness of a matrix A is denoted A 0.
The spectral norm of a matrix A is denoted A 2 . We denote Ā := A/ A 2 the normalized version of A. The largest (resp. smallest) eigenvalue of A is denoted λ max (A) (resp. λ min (A)). The notation Id denotes the identity operator. Given a vector x ∈ R d , we let diag(x) denote a diagonal matrix whose diagonal elements are the entries of x. The Givens transform, denoted by R θ ij ∈ R d×d , represents a counter-clockwise rotation for an angle θ in the (i, j)-coordinates plane. In the particular case d = 2, it is abbreviated R θ .

Theoretical analysis

The structure tensor appeared in the field of image processing in the late 80's [START_REF] Förstner | A fast operator for detection and precise location of distinct points, corners and centres of circular features[END_REF] for the problem of interest point detection. It was then justified theoretically and popularized in different contexts such as interest point detection [START_REF] Förstner | A fast operator for detection and precise location of distinct points, corners and centres of circular features[END_REF][START_REF] Harris | A combined corner and edge detector[END_REF], texture analysis [START_REF] Bigun | Multidimensional orientation estimation with applications to texture analysis and optical flow[END_REF][START_REF] Gösta | Signal processing for computer vision[END_REF], representation of flow-like images [START_REF] Rao | Schinck. Computing oriented texture fields[END_REF], optical flow problems [START_REF] Bruce | An iterative image registration technique with an application to stereo vision[END_REF] and anisotropic or coherence enhancing diffusion [START_REF] Pop | Extracting 3d cell parameters from dense tissue environments: application to the development of the mouse heart[END_REF][START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF].

In this section, we first recall the definition of structure tensor, then show its capability of analyzing fields of locally coherent ellipses (in 2D) or ellipsoids (in 3D). The motivation for introducing fields of ellipses is that images such as Figure 3 are a rather good approximation of certain dense tissues such as microtumors. To the best of our knowledge, even though fields of locally coherent ellipsoids share some similarities with flow-like images, the proposed theoretical analysis and results are original and shed a novel light on the structure tensor.

Preliminary facts about the structure tensor

Let u : R d → R denote a grayscale image. In this paper, we restrict on the practical cases d = 2 and d = 3, although the theory is valid in any dimension. For ease of exposition, we assume that u ∈ C 1 (R d ) and has bounded partial derivatives. The function K ∈ L 1 (R d ) is a filter that satisfies the following conditions:

(i) K(x) ≥ 0, ∀x ∈ R d ; (ii) R d K(x) dx = 1. (1)
For any ρ > 0, we define K ρ the scaled version of K by

K ρ (x) := 1 ρ d K x ρ , ∀x ∈ R d . (2)
The conditions (1) still hold for any ρ > 0. In practice, K is usually a smoothing filter (e.g. a Gaussian) and K ρ is a scaled version at scale ρ.

The structure tensor of u, denoted by J ρ , is defined by

J ρ := K ρ ⋆ ∇u∇u T , (3)
where ∇ denotes the gradient operator and '⋆' is the convolution operation which acts independently on each component of the d × d tensor ∇u∇u T . Using the boundedness of the partial derivatives of u and the conditions (1), we have

| (J ρ (x)) i,j | ≤ R d |K ρ (x -y)| dy =1 max y∈R d |(∂ i u∂ j u)(y)| < +∞
for any x ∈ R d , which implies that J ρ is bounded in R d . Notice that the definition given in equation (3) differs slightly from that found in standard articles or textbooks such as [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF]. Therein, the image u is first convolved with a Gaussian filter and the filter K is assumed to be a Gaussian, see the discussion in Section 3.4.

A useful property of structure tensor is its positive semidefiniteness, which is a direct consequence of (1) and the convexity of the cone of symmetric positive semidefinite matrices.

Proposition 1 (Positive semidefiniteness) The structure tensor satisfies J ρ (x) 0 for any ρ > 0 and any x ∈ R d .

Structure tensor and a single ellipsoid

In this section, we analyze the structure tensor behavior on a simple image whose isosurfaces are concentric ellipsoids. We show that it allows recovering its principal orientations as well as the ratios between the length of the ellipsoid main axes.

Let The isosurfaces of ψ are ellipsoids in R d with semiaxes of length

ϕ : R + → R denote a C 1 function different from 0 satisfying ϕ ′ (0) = 0. Let A ∈ R d×d denote a symmetric positive-definite matrix with spectral decomposition A = U ΣU T , where Σ = diag(σ -2 1 , . . . , σ -2 d ) and U is orthogonal. Let x = (x 1 , . . . , x d ). Define ψ : R d → R by ψ(x) := ϕ(x T Ax), ∀x ∈ R d . (4)
σ i (i = 1, 2, • • • , d).
Example 1 Let ϕ : R → R denote the bump function

ϕ(t) = exp -1 1-t 2 , if |t| < 1; 0, otherwise, (5)
and let

U = cos θ sin θ -sin θ cos θ and Σ = σ -2 1 0 0 σ -2 2 . ( 6
)
With the above choices, the level lines of ψ in (4) are ellipses in R 2 (see Figure 1).

Proposition 2 Let u := ψ with ψ defined in (4). Assume that supp(ψ) ⊂ -1 2 , 1 2
d . Let K be the indicator of a unit disk:

K(x) = C d , if x 2 ≤ 1; 0, otherwise, (7)
where the normalizing constant C d is chosen so that the normalization condition (ii) of ( 1) is satisfied. Then, for all ρ ≥ d/2, we have Jρ (0) = Ā and if x is small enough then Jρ (x) = Jρ (0).

The above proposition leads to the following observations:

• First, for simple functions with ellipsoidal isosurfaces, a diagonalization of the structure tensor allows recovering the orientation matrix U as well as the matrix Σ up to a multiplicative constant. In 2D, it means that λ min (J ρ (0))/λ max (J ρ (0)) is the ratio between the principal axes.

• Second, since this result holds for any function ϕ, the method is contrast invariant, which is a highly desirable property. Contrast invariance basically indicates that the method should behave similarly on different imaging devices or when using different stainings.

• Third, the structure tensor is stable in the sense that this result holds not only at point 0, but also in a neighborhood of 0.

Structure tensor and fields of ellipsoids

In the previous section, we focussed on a very simple image u. We now turn to a slightly more realistic setting where u is the sum of functions with ellipsoidal isosurfaces and nonoverlapping support.

We use the same notation as in the previous section and assume that supp(ψ) ⊂ - 1 2 , 1 2 d . The Dirac comb, denoted by X, is defined by

X := x∈Z d δ x ,
where δ x is the Dirac delta function (see e.g. [START_REF] Ronald | The fourier transform and its applications[END_REF]). Consider the image

u u = X ⋆ ψ, (8)
where ψ is defined in [START_REF] Bonnet | Mechanical state, material properties and continuous description of an epithelial tissue[END_REF]. The image u consists of a function ψ replicated periodically over all R d . Note that the translated versions of ψ do not overlap since supp(ψ) ⊂ -1 2 , 1 2 d . Figure 2 illustrates such an image in the 2D case.

In Proposition 2, it is proven that Jρ (0) = Ā, which implies that J ρ (0) possesses adequate information to retrieve the orientation and anisotropy of u. However, that proposition holds under the following assumptions: (i) the image u has concentric ellipsoidal isosurfaces without neighbors; (ii) the ellipsoid's center should be known. Both requirements are hardly met in practical applications. We develop below a stronger theory stating that the structure tensor allows recovering the orientation and anisotropy at every point of the image domain. Namely, for the image u defined in [START_REF] Dabov | Image denoising by sparse 3-d transform-domain collaborative filtering[END_REF], we expect that Jρ (x) ≃ Ā holds at any x ∈ R d . The questions we tackle in this paragraph are the following: does the structure tensor provide an approximation of Ā at any point of the image domain? How many cells are necessary to reach a low approximation error?

The following proposition provides a preliminary answer.

Proposition 3 Let K be the function defined in [START_REF] Buades | A non-local algorithm for image denoising[END_REF] and u be the image defined in [START_REF] Dabov | Image denoising by sparse 3-d transform-domain collaborative filtering[END_REF].

Then Jρ (x) -Ā 2 = O(1/ρ) for all x ∈ R d .
Proposition 3 indicates that Jρ (x) ≃ Ā for any x ∈ R d and sufficiently large ρ. However, the asymptotic O(1/ρ) convergence rate is not compelling. The following theorem shows that much more attractive results can be obtained by using smoother kernels.

Theorem 1 Let K be the Gaussian function and ∀x ∈ R d :

K(x) = 1 (2π) d/2 exp - x 2 2 2 (9)
Jρ (x) -Ā 2 ≤ C exp - ρ 2 2 .
Surprisingly, the convergence of Jρ (x) to Ā is extremely fast if smooth filters are exploited. Theorem 1 implies that moderate values of ρ's should produce satisfactory orientation and anisotropy estimates, and it follows from numerical experiments that a value of ρ of the order of the radius of the objects of interest is sufficient. A closer inspection at the proof of Theorem 1 reveals that the smoothness of the filter function plays a key role to control the asymptotic convergence rate.

Theorem 1 is illustrated in the 2D case in Figure 2. It shows that the magnitude Jρ (x) -Ā 2 decreases to zero (up to numerical errors) extremely fast. An ellipse is around 8 pixels wide and a value of ρ around 5 provides results nearly as good as can be expected.

A note on pre-processing

As mentioned in Section 3.1, the structure tensor definition (3) is different from what is found in [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF]. The image u is usually pre-convolved with a Gaussian filter so as to improve the signal-to-noise-ratio. Before getting further in our investigation, let us illustrate the detrimental effect of this strategy.

Assume for simplicity that u = ψ and that ϕ(t) = exp(-t/2). The image u(x) = exp -x T Ax/2 is thus a Gaussian function with covariance matrix A -1 . If u is further convolved with a Gaussian filter as in [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Coherence-enhancing diffusion filtering[END_REF], we obtain

u σ := K σ ⋆ u,
where K is the Gaussian filter defined in [START_REF] Desmaison | Mechanical stress impairs mitosis progression in multicellular tumor spheroids[END_REF]. By exploiting the facts that the convolution of Gaussian functions is still a Gaussian and that the covariance matrices sum up, we obtain

u σ (x) ∝ exp - x T (A -1 + σ 2 Id) -1 x 2 ,
where Id is identity matrix. Consequently, pre-convolving u with a Gaussian filter shifts the eigenvalues of A by a quantity σ 2 . This, in turn, results in biased anisotropy estimates. For instance, in the 2D case, the structure tensor based anisotropy obtained using u σ is

λ -1 max (A) + σ 2 λ -1 min (A) + σ 2 1/2
, which is different from the ground truth anisotropy λmin(A) λmax(A)

1/2
. The simple analysis above shows that in practice, more advanced image denoising techniques that keep isosurfaces unchanged should be preferred over a simple convolution with a Gaussian filter. A wide choice is now available such as anisotropic diffusion, total variation denoising, frame based regularization or non-local methods [START_REF] Tschumperlé | Fast anisotropic smoothing of multi-valued images using curvature-preserving pde's[END_REF][START_REF] Leonid I Rudin | Nonlinear total variation based noise removal algorithms[END_REF][START_REF] Starck | The curvelet transform for image denoising[END_REF][START_REF] Buades | A non-local algorithm for image denoising[END_REF][START_REF] Dabov | Image denoising by sparse 3-d transform-domain collaborative filtering[END_REF].

Numerical validation

In this section, we conduct some numerical experiments to evaluate the structure tensors performance on synthetic and real data. The main biological problem addressed in our experiments is the analysis of geometrical configurations of nuclei in multicellular tumor spheroids. There are at least two reasons making this analysis relevant. First, tumor development is associated with a disorganization of the tissue. The role played by this disorganization is not well understood yet and it has been shown that it could have an impact on tumor cell behavior [START_REF] Weigelt | Unraveling the microenvironmental influences on the normal mammary gland and breast cancer[END_REF]. Second, among the key parameters involved in tumor growth, those related to mechanical forces seem to play a critical role [START_REF] Montel | Stress clamp experiments on multicellular tumor spheroids[END_REF][START_REF] Stylianopoulos | Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors[END_REF][START_REF] Desmaison | Mechanical stress impairs mitosis progression in multicellular tumor spheroids[END_REF]. The elongation of nuclei in a preferential direction is an indicator of local stresses [START_REF] Guilak | Compression-induced changes in the shape and volume of the chondrocyte nucleus[END_REF][START_REF] Versaevel | Spatial coordination between cell and nuclear shape within micropatterned endothelial cells[END_REF]. Assessing the anisotropy and orientation of nuclei in their micro-environment is therefore crucial to better understand tumor organization and mechanics.

Implementation details

Up to now, we only performed theoretical analyses in the continuous domain. In practice, the structure tensor should be adapted to the discrete setting.

The discrete gradient operator ∇ is defined by

∇ =    ∂ 1 . . . ∂ d   .
In all reported experiments, the partial differential operators ∂ i are defined as convolutions with discrete kernels. From an asymptotic point of view any kernel leading to a consistent discretization should provide good results. However, the kernel design turns out to be crucial to provide good orientation and anisotropy estimates. Key properties of discrete kernels are [START_REF] Weickert | A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance[END_REF]: i) rotation invariance, ensuring a reliable orientation estimation, ii) separability, ensuring faster computations and iii) no shift, implying the use of centered finite differences. Following these criteria, the authors of [START_REF] Weickert | A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance[END_REF] suggested to use the following filter in 2D: The gradient is computed in the space domain, while the convolution with K ρ is based on fast Fourier transforms. The overall computational complexity for an image with n pixels is therefore O(n log(n)). In practice, the structure tensor can be computed in near real-time for 2D images and takes a few seconds for 3D images. It can be very easily parallelized on multicore or GPU architectures. All codes were written in Matlab 7.9 and experiments were conducted on a Lenovo personal computer with Intel Core (TM) CPU 2.30GHZ and 8G memory.

h = 1 32

2D and 3D synthetic data

In order to validate the theory, we first concentrate on 2D and 3D synthetic tumor spheroids.

Synthesizing images

A tumor spheroid image u : R d → R + is synthesized by

u(x) = N i=1 ϕ (x -x c i ) T A i (x -x c i ) , ∀x ∈ R d , ( 10
)
where N is the number of nuclei in the spheroid; ϕ is the bump function defined in (5); and x c i ∈ R d is the i-th nucleus center. The centers are drawn at random in a d-dimensional sphere in such a way that the nuclei do not overlap.

• If d = 2, the matrix A i ≻ 0 is defined by

A i = R θi diag(σ -2 1,i , σ -2 2,i )(R θi ) T . ( 11
)
where R θi is the Givens transform (see Section 2); θ i is the phase angle of the radial line issued from the origin and crossing the nucleus center x c i , see Fig. 1.

• If d = 3, the matrix A i ≻ 0 is defined by

A i = R θi 1,2 R γi 2,3 diag(σ -2 1,i , σ -2 2,i , σ -2 3,i )(R θi 1,2 R γi 2,3 ) T , (12)
where θ i , γ i denote the compass and elevation angles of the radial line crossing the ellipsoid center.

The anisotropy of the i-th nucleus is defined by

α i := min k=1,2,••• ,d σ k,i max k=1,2,••• ,d σ k,i . (13)
In our 2D experiment, the anisotropy α i increases linearly from the image center to the outer layers of the sphere. For the 3D case, the anisotropy α i is set as constant. Figure 3(a) and Figure 6 (a)-(b) display the 2D and 3D synthetic spheroids respectively.

Measuring the performance

In order to assess the structure tensor efficiency, we evaluate the following quantities:

• Orientation. Let ṽi denote the eigenvector corresponding to the largest eigenvalue of J ρ (x c i ) and v i = (cos θ i , sin θ i ) denote the ground truth orientation. The following angles

∠(ṽ i , v i ), i = 1, 2, • • • , N, (14)
are used to evaluate the orientation accuracy in 2D. An angle close to 0 • indicates a good orientation estimation, while an angle close to 90 • is the worst possible estimate. For the 3D case, we choose for ṽi either the largest eigenvector (prolate spheroid) ot the smallest eigenvector (oblate spheroid).

• Anisotropy. The structure tensor based anisotropy is defined by αi =

λmin(Jρ(x c i )) λmax(Jρ(x c i )) 1/2
. The ratio

αi /α i , i = 1, 2, • • • , N, (15)
is used to quantify the estimated anisotropy accuracy, where α i is groundtruth anisotropy, see [START_REF] Gösta | Signal processing for computer vision[END_REF]. A ratio close to 1 indicates that the anisotropy is correctly evaluated.

• Spectral norm. In the experiments based on synthetic images, a matrix A(x) can be associated to every point of the image domain. We can therefore evaluate the spectral norm Jρ (x) -Ā(x) 2 everywhere and not only at the nuclei centers.

2D synthetic data

We report numerical results on the 2D synthetic spheroid displayed in Figure 3 We use the value ρ = 5 in (3), this value is to be compared with the small axis of the ellipses that is σ 1 = 10 and the large axis which ranges σ 2 from 10 to 32. An adequate value for ρ is of the order of the radius of the objects of interest. Figure 3 (e)-(f) show the structure tensor based orientations and anisotropies which should be compared to the ground truth ones in Figure 3 (b)-(c). The orientations and anisotropy are clearly very well estimated. This is somehow surprising since for this image, only a small number of cells share the same orientation and anisotropy locally. This favorable behavior is another illustration of the fast convergence speed obtained in Theorem 1.

To further quantify the accuracy of the results, the quantities ( 14)-( 15) are evaluated. The histograms of both quantities are displayed in Figure 4. They show that approximately 95% of estimated orientations have an angular error below 4 • . Similarly, approximately 65% of estimated anisotropies have a an error below 10%.

Finally, when using synthetic data, a ground-truth structure tensor A(x) can be defined at every point of the image domain. We can thus compare the four coefficients of the 2 × 2 matrix Ā(x) with the coefficients of Jρ . The distance Jρ -Ā 2 can also be evaluated at every point of the image domain, see Figure 5. We observe that the structure tensor approximates accurately the ground truth tensor.

3D synthetic data

We now test the structure tensor efficiency on 3D synthetic data generated by formula [START_REF] Förstner | A fast operator for detection and precise location of distinct points, corners and centres of circular features[END_REF], see Figures 6(a) for an oblate example and (b) for a prolate example. The structure tensor J ρ is computed with ρ = 5. The estimated orientations are displayed in Figure 6 (e) and (f) while the ground truth orien- tations are displayed in Figure 6 (c) and (d). The orientation is well retrieved. This is also confirmed by the histogram in Figure 7. Moreover, the histogram of anisotropy discrepancies show that approximately 70% of the anisotropies are evaluated with an error below 10%. Overall, these results confirm that the structure tensor is also very attractive to estimate anisotropies and orientations for 3D data.

Performance evaluation on real spheroid images

We now evaluate the structure tensor based orientation and anisotropy on a 2D real image. The image that we use in our experiment contains 1465 nuclei. The 'gold standard' reference was obtained manually. The scalar ρ is set to 10 in this section. This value was selected manually and corresponds roughly to the radius of the nuclei.

The gold standard and estimated orientations are displayed in Figure 8. The orientation and anisotropy errors are presented in Figure 9(a) and (b) respectively.

Let us emphasize that the gold standard is subject to many errors since i) a preferential orientation cannot be defined properly on isotropic cells and ii) fitting a thousand ellipses manually is subject to many errors. Moreover, real data strongly depart from the ideal models considered in Section 3 since (i) the SNR of real tumor spheroid image is typically low due to noise, blur, variations in illumination, etc; (ii) the nuclei geometry is only approximately ellipsoidal; (iii) the nuclei may overlap in 2D. Despite the rather poor quality of data and gold standard, the results fit remarkably well. Overall the orientation is evaluated with an error no larger 20 • while the anisotropy seldom exceeds 50% error. Finally, note that the preferential orientation is not well defined at the image center since cells are near isotropic.

In Figure 9(c) we plot the median, mean and 25% ∼ 75% percentiles of the angle error [START_REF] Guilak | Compression-induced changes in the shape and volume of the chondrocyte nucleus[END_REF] with respect to the distance to the spheroid boundary. The notation 'dist(x c i , ∂H)' stands for the Euclidean distance from the nucleus center x c i to the spheroid boundary. Figure 9(c) indicates that structure tensor provides accurate orientations near the spheroid boundary and less accurate in the center, since the median error increases from 7 • close to the boundary to 14 • near the center.

Application to drug effects analysis on spheroids

As a proof of concept, spheroids were treated with latrunculin A, an inhibitor of actin polymerization. The comparison of (a) and (e) in Figure 10 shows that this treatment induces a disorganization. We applied the structure tensor to extract the orientation and anisotropy maps in Figure 10 (c),(d),(g) and (h). The angle is measured with respect to the normal of the closest point of the boundary. In other words, an angle equal to 90 • means that the local orientation is parallel to the boundary, and an angle equal to 0 • means that the local orientation is normal to the boundary.

The results obtained show a decrease of both nuclei anisotropy and alignment with the spheroid boundary in the outer layers after treatment. This observation is confirmed by the graphs showed in Figure 11. In Figure 11(a) we compare the alignement of the nuclei with respect to the boundary with or without treatment. In Figures 11(b) and (c) we present the alignment with respect to the distance to the boundary. We observe that the mean angle at the origin is around 80 • for the control spheroid and 60 • for the treated spheroid, indicating that the drug tends to desalign the nuclei with respect to the spheroid boundary. Moreover, in Figure 11 (b), the gray zone is narrow at the origin, reflecting the fact that nearly all nuclei near are well aligned with the boundary in the outer layers. On the contrary, the gray zone at the origin of the graph in Figure 11 (c) is thick, indicating that the orientation is much more erratic.

In Figure 11(d) we present the distribution of anisotropy for the control and the treated spheroid. In Figures 11 (e) and (f) we present the anisotropy with respect to the distance to the boundary. We observe that the mean anisotropy near the spheroid boundary is approximately equal to 0.6 for the control spheroid and 0.8 for the treated spheroid, indicating that nuclei are more round for the treated case.

As a summary, our methodology allows quantifying both the change of alignment and the decrease of anisotropy and could therefore have interesting applications in high content throughput.

Outlook

We proposed an original theoretical analysis of structure tensors, justifying their use for evaluating orientations and anisotropies of cells or nuclei in 2D or 3D images. Our theoretical results were validated by numerical experiments, on synthetic and real data. Overall the present work shows that the structure tensor provides a fast, simple, robust and user friendly method for biomedical imaging. To finish, let us add a few remarks:

• First, cells or nuclei are not the only ellipsoid like objects and the method can actually be applied in any field where such objects appear (e.g. lobules in adipose tissues).

• Second, our results show that the structure tensor can be used as a preprocessing tool for segmentation methods. Knowing the rough orientation, anisotropy and volume of a cell can provide an initialization for more advanced segmentation techniques. For instance, it is fundamental to have a first rough segmentation in random based optimization methods [START_REF] Soubies | A 3d segmentation algorithm for ellipsoidal shapes. application to nuclei extraction[END_REF] .

• Finally, more and more mathematical models describe tissues as continuous media. The information contained in the structure tensor is itself continuous and therefore seems easy to use in data assimilation problems. This opens interesting perspectives to understand tissue biomechanics for instance. We plan to further investigate this problem in forthcoming works.

A Proof of Proposition 2

Simple calculation lead to ∇ψ(x) = 2ϕ ′ (x T Ax)Ax.

where the last equality follows from the fact that the functions are nonoverlapping. By exploiting the facts that J ρ = K ρ ⋆ J 0 and X = X, it comes that

Ĵρ = ( ∂ i ψ∂ j ψ) • X • K ρ 1≤i,j≤d =   k∈Z d ( ∂ i ψ∂ j ψ)(k) K ρ (k)δ k   1≤i,j≤d
.

This analysis provides a Fourier series decomposition of J ρ :

J ρ (x) =   k∈Z d ( ∂ i ψ∂ j ψ)(k) K ρ (k) exp(-2iπ x, k )   1≤i,j≤d
.

Moreover ( 1)-(ii) implies that K ρ (0) = 1, and for all i, j:

(J ρ (x)) i,j -( ∂ i ψ∂ j ψ)(0) = k∈Z d \{0} ( ∂ i ψ∂ j ψ)(k) K ρ (k) exp(-2iπ x, k ) ≤ k∈Z d \{0} ( ∂ i ψ∂ j ψ)(k) K ρ (k)
By denoting y = Σ 1 2 U T x, we obtain:

∂ i ψ∂ j ψ)(0) 1≤i,j≤d = R d (∂ i ψ∂ j ψ)(x) 1≤i,j≤d = 4A R d ϕ ′ (x T Ax) 2 xx T dx 1≤i,j≤d A T = 4|Σ -1 2 |AU Σ -1 2 R d ϕ ′ ( y 2 2 )yy T dx 1≤i,j≤d ∝Id Σ -1 2 U T A T ∝ A.
Since ψ ∈ C 1 with bounded support, c = ∂ i ψ∂ j ψ 1 < +∞. Therefore ∂ i ψ∂ j ψ ∞ ≤ c. Moreover, K ρ (k) = K(ρk) so that

k∈Z d \{0} ( ∂ i ψ∂ j ψ)(k) K ρ (k)| ≤ c k∈Z d \{0} | K ρ (k)| = c k∈Z d \{0} | K(ρk)| ∝ k∈Z d \{0} exp - ρ 2 k 2 2 2 .
We now remark that where

u(ρ) = j≥1 exp - ρ 2 j 2 2 ≤ j≥1 exp - ρ 2 j 2 = exp(- ρ 2
2 )

1 -exp(- ρ 2 2 )
.

Since u(ρ) is asymptotic to exp(-

ρ 2 2
) when ρ → ∞ the claim is proved.

Figure 1 :

 1 Figure 1: Representation of ψ in (4) with specific ϕ, U and Σ in (5)-(6). Left: the case σ 1 = 2, σ 2 = 1 and θ = 0 • . Right: the case σ 1 = 2, σ 2 = 1 and θ = 30 • .

Figure 2 :

 2 Figure 2: Left: image u defined in (8). The axes lengths of a single ellipsoid are respectively 8 and 4 pixels. Right: Jρ (x) -Ā 2 w.r.t. ρ.

  . to set ∂ 1 u = h ⋆ u and ∂ 2 u = h T ⋆ u. Using the same methodology in 3D, one can derive the following filter (using Matlab notation): :, :, 3) = h(:, :, 1).

Figure 3 :

 3 Figure 3: 2D synthetic data (a) 512 × 512 synthetic tumor spheroid (b) groundtruth orientations (c) ground-truth anisotropy (d) ground-truth orientations (e) Structure tensor based orientations (f) Structure tensor based anisotropy.

Figure 4 :

 4 Figure 4: Analysis of structure tensor based accuracy for the 2D test case. Histograms of orientations errors (left) and of anisotropy errors (right).

Figure 5 :

 5 Figure 5: Four coefficients of matrix Ā (left), four coefficients of matrix Jρ (center) and error map Jρ -Ā 2 (right).

Figure 6 :

 6 Figure 6: 3D synthetic data (size: 128 × 128 × 128). (a) spheroid with σ 1 : σ 2 : σ 3 = 5 : 5 : 2. (b) spheroid with σ 1 : σ 2 : σ 3 = 5 : 2 : 2. (c)-(d) ground truth orientations of spheroids (a) and (b) respectively. (e)-(f) structure tensor based orientations.

Figure 7 :

 7 Figure 7: Analysis of structure tensor based errors for 3D synthetic data from Figure 6. Histograms of orientations errors (left) and of anisotropy errors (right).

Figure 8 :

 8 Figure 8: Comparison of ground truth orientations (top) and estimated orientations (bottom) on spheroid sections. Nuclei are stained using DAPI. Images were acquired using epifluorescence microscope (LEICA DM5000) and 10X objective NA:0.3. Scale bar: 100µm.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Quantification of structure tensor based results for the 2D experimental test case. (a) histogram of orientation error, (b) histogram of anisotropy error, (c) structure tensor based orientations with respect to nuclei locations.

  (ρ) + O ρ→+∞ (u(ρ))

In Matlab, this result can be obtained by first thresholding the image and then using the quickhull algorithm[START_REF] Barber | The quickhull algorithm for convex hulls[END_REF] (convhull command).

Acknowledgement

The authors would like to thank the TRI-Genotoul and ITAV-USR3505 image facilities and all the members of ITAV for providing a nice research environment.

Funding This work was supported by the ANR SPH-IM-3D (ANR-12-BSV5-0008), the NNSFC Grant 11301055 and la Fondation pour la Recherche Médicale (équipe labelisée 2012). A. D. is recipient of a doctoral fellowship from the Association pour la Recherche contre le Cancer and is as student of the école de l'INSERM Liliane Bettencourt.

Therefore

But if ρ > d/2 then K ρ is constant on the domain where ψ does not vanish. Therefore by using the change of variable y = Σ 1 2 U T x, we obtain:

Since K ρ is constant on a neighborhood of the domain where ψ does not vanish, J ρ is invariant by small translations hence J ρ is locally constant around 0.

B Proof of Proposition 3

We denote B(x 0 , ρ) = {x | xx 0 2 ≤ ρ} the Euclidian ball centered at x 0 with radius ρ and S(x 0 , ρ) = {x | xx 0 2 ≤ ρ} the Euclidian sphere centered at x 0 with radius ρ. We have

The image u is a sum of (non-overlapping) replicates of ψ. From Proposition 2, we know that all the replicates of ψ with support included in B(x 0 , ρ) will have a contribution to J ρ (x) proportional to A while the ones with support intersecting S(x 0 , ρ) have a contribution that can be considered as a bias, whose components are bounded by

For sufficiently large ρ, |I| ∝ ρ d (the volume of a ball of radius ρ) while |J | ∝ ρ d-1 (the area of the sphere). Therefore

where B ρ is some bias of bounded amplitude. This implies that

C Proof of Theorem 1

The non-smoothed structure tensor is defined by


G Fabbri

S Federico

On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term

Keywords: Stochastic delay differential equations, Evolution equations in Hilbert space, Dynamic programming JEL Classification: C61, G11, O40 AMS 2010 Classification: 39A50, 34K30, 47D06, 60H99, 49L20

In the deterministic context a series of well established results allow to reformulate delay differential equations (DDEs) as evolution equations in infinite dimensional spaces. Several models in the theoretical economic literature have been studied using this reformulation. On the other hand, in the stochastic case only few results of this kind are available and only for specific problems.

The contribution of the present letter is to present a way to reformulate in infinite dimension a prototype controlled stochastic DDE, where the control variable appears delayed in the diffusion term. As application, we present a model for quadratic risk minimization hedging of European options with execution delay and a time-to-build model with shock.

Some comments concerning the possible employment of the dynamic programming after the reformulation in infinite dimension conclude the letter.

Introduction

The setting of optimal control problems with delays in the state and/or the control variable has been employed in the last decades to model a wide variety of economic and financial phenomena where the dynamics of the system has a natural memory of the past. We may quote, among others: the problem of growth models with time-to-build in production (see [START_REF] Asea | Time-to-build and cycles[END_REF][START_REF] Bambi | Endogenous growth and time to build: the AK case[END_REF][START_REF] Bambi | Optimal policy and consumption smoothing effects in the time-to-build AK model[END_REF]) or investment (see [START_REF] Kydland | Time-to-build and aggregate fluctuations[END_REF][START_REF] Tsoukalas | Time to Build Capital: Revisiting Investment-Cash Flow Sensitivities[END_REF]) or with delays in the learning-by-doing process (see [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF]); vintage capital models (see [START_REF] Boucekkine | Vintage capital and the dynamicsof the AK model[END_REF][START_REF] Fabbri | Solving optimal growth models with vintage capital: The dynamic programming approach[END_REF]); advertising models (see [START_REF] Feichtinger | Dynamic Optimal Control Models in Advertising: Recent Developments[END_REF][START_REF] Gozzi | On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects[END_REF][START_REF] Pauwels | Optimal Dynamic Advertising Policies in the Presence of Continuously Distributed Time Lags[END_REF][START_REF] Sethi | Sufficient Conditions for the Optimal Control of a Class of Systems with Continuous Lags[END_REF]).

Considering for simplicity the case of one-dimensional linear and autonomous state equation, the mathematical problem can be formulated in a quite wide generality as follows. Given d > 0 and two Borel measures µ, ν on the interval [-d, 0], consider the linear functionals, defined in the space of continuous functions from [-d, 0] to R,

Lf := 0 -d f dµ, Mf := 0 -d f dν.
Then we can consider the controlled differential equation

x ′ (t) = Lx t + Mu t , (1)
where x(•) is the state variable, u(•) is the control of the system belonging to a suitable class of functions, and where x t and u t denote, respectively, the trajectory of the state and of the control in the time interval [t -d, t], i.e. x t := x(t + ξ)| ξ∈[-d,0] and u t := u(t + ξ)| ξ∈ [-d,0] . Obviously, to give a precise sense to [START_REF] Asea | Time-to-build and cycles[END_REF], one needs, beyond completing it with initial conditions involving the past of the control and the state variable, also to impose some conditions to the operators L and M ensuring that (1) has a unique solution for each control u(•) in a suitable class of functions. We do not discuss that in detail and refer instead to Chapter 3, Part II of [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF]. We only notice that the nature of (1) -as soon as L and M are not trivial, i.e. µ and ν are not both (multiple of) Dirac measures concentrated at 0 -is basically infinite dimensional: to have an intuition of that, one can just think that (1) invokes, to state the dynamics, the past of the state and of the control variable, i.e. data which belong to (infinite dimensional) functional spaces. Now, given a dynamics in the form (1), one can consider the problem of optimizing a payoff functional of the form

T 0 g(t, x(t), u(t)) dt, T ∈ [0, +∞], ( 2
)
where g is a measurable function. We notice that one can consider also the case of a function g depending on x t and u t too, and the problem of optimizing (2) would be basically infinite dimensional even in the case of no delay in the state equation [START_REF] Asea | Time-to-build and cycles[END_REF].

Given the observation of the structural infinite dimensional nature of the problem we have done, one can understand the reason for which the mathematical literature has proposed infinite-dimensional formulations of the problem above to "absorb" (and remove) the delay, paying the price of a passage to the infinite dimensional setting. This is the first step to do if we want to try to employ the dynamic programming tools to tackle these problems. We observe that, when the delay is only in the state variable, i.e. ν is a multiple of δ {0} , the infinite dimensional representation is straightforward, just considering as infinite dimensional space the couple (x(t), x t ) (see [START_REF] Delfour | State theory of linear hereditary differential systems[END_REF][START_REF] Delfour | Hereditary differential systems with constant delays. II. A class of affine systems and the adjoint problem[END_REF][START_REF] Federico | HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, I: Regularity of Viscosity[END_REF][START_REF] Federico | HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, II: Verification and Optimal[END_REF]). On the other hand, when the control enters into the dynamics with delay, i.e. ν is not a multiple of δ {0} , then two approaches can be employed:

1. Consider as infinite dimensional state the triple (x(t), x t , u t )) (or the couple (x(t), u t ) if there is delay no delay in the state, i.e. µ is a multiple of δ {0} ). This leads to an infinite-dimensional control problem with boundary control, i.e. with unbounded control operator (see [START_REF] Ichikawa | Quadratic control of evolution equations with delays in the control[END_REF]).

2. Consider as infinite dimensional state a kind of "minimal summary" combining together through a linear transformation leading to the construction of the so called structural state, the current state x(t), the past state x t (if the delay in the state is present in the equation, i.e. µ is not a multiple of δ {0} ) and the past of the control u t (see [START_REF] Vinter | The infinite dimensional quadratic control problem for linear systems with state and control delay: an evolution equation approach[END_REF]). With respect to the latter one, this approach has the advantage that the infinite dimensional control problem has not a always the complication of becoming a boundary control problem, see e.g. [START_REF] Tacconi | Dynamic programming for optimal control problems with delays in the control variable[END_REF][START_REF] Vinter | The infinite dimensional quadratic control problem for linear systems with state and control delay: an evolution equation approach[END_REF].

A systematic treatment of the methods described above can be found in Chapter 3, Part II of [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF]. The aim of this letter is to try to understand how the above method can be extended to the stochastic case. We have to mention that attempts in this direction have already be done by some authors. For instance, [START_REF] Gozzi | On controlled linear diffusions with delay in a model of optimal advertising under uncertainty with memory effects[END_REF] consider the case of a stochastic system with delays in state and control, but where the noise is purely additive. This allows to generalize in a straightforward way the method described in point 2 above. On the other hand, [START_REF] Tankov | Finite-dimensional representation for controlled diffusions with delay[END_REF] consider the case when there is delay in the state both in the drift and the diffusion, but no delay in the control variable. Other cases seems not treated in the literature. A challenging case seems to be the case when the control variable appears delayed in the diffusion term. Several interesting aspects of the problem could be already discussed dealing with the prototype equation

dX(t) = u(t -d) dW (t), (3)
which may work as a guide to investigate the possibilities and the limits of an infinitedimensional representation along the direction described above for deterministic systems. In fact we will deal with a slightly more general equation that can be specified to obtain the state equations of the two motivating examples we will introduce below.

The main result is provided by Theorem 3.4 that allow to connect such controlled DDE with a suitable controlled stochastic evolution equation (without delay) in an infinite dimensional space. In Section 4 we will comment on the possibility of exploiting this result by means of a dynamic programming approach to the resulting infinite dimensional control problem. Now we provide two concrete examples to motivate our mathematical problem.

Hedging of European options with execution delay

Let us consider a complete filtered probability space (Ω, F , F := (F t ) t≥0 , P) on which a Brownian motion (W (t)) t≥0 is defined. We consider a standard Black-Scholes financial market, composed by a riskless asset with dynamics

dB(t) = rB(t) dt (4)
and a risky asset with dynamics

dP (t) = rP (t) dt + σP (t)(λ dt + dW (t)), P (0) = p > 0. (5)
Above r, λ, σ are positive constants representing the riskless spot rate, the risk premium and the volatility of the risky asset, respectively. Now, let d > 0 be a fixed execution delay time: at time t ≥ 0 the investor chooses, on the basis of the information F t , to allocate the amount of money u(t) ≥ 0 of its portfolio in the risky asset. This is the control process, which is an F-adapted process. However, due to the execution delay this order will be executed at time t + d when the price of the risky asset has changed (see [START_REF] Bruder | Impulse Control Problem on Finite Horizon with Execution Delay[END_REF] for the definition of this problem in a stochastic impulse control framework). Letting (X(t)) t≥0 be the Fadapted process representing the value of the portfolio and assuming that the portfolio is self-financing, we get the dynamics

dX(t) = (rX(t) + σλu(t -d)) dt + σu(t -d) dW (t), t ≥ 0. (6)
We notice that ( 6) is a controlled stochastic differential equation with delay in the control variable. As a delay equation, it requires, other than the specification of the initial state X 0 , also the specification of part of the past of the control variable as initial data. More precisely (6) requires the specification of u in the time interval [-d, 0), so that the complete state equation looks like

dX(t) = (rX(t) + σλu(t -d)) dt + σu(t -d) dW (t), t ≥ 0, X(0) = x I ; u(s) = u I (s), s ∈ [-d, 0). ( 7
)
Within the setting above, we can consider the problem of quadratic risk minimization in hedging a European option g(P (T )), where T > 0 is the exercise date:

inf u(•) E g(P (T )) -X(T ) 2 . ( 8
)
Remark 1.1 Since in [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF] the control variable appears only evaluated at time t -d and in the functional [START_REF] Delfour | State theory of linear hereditary differential systems[END_REF], the optimization problem ( 7)-( 8) can be actually seen as a stochastic optimal control probelm under partial information (in this case just a delayed information, simply setting ũ(t) = u(t -d)). This kind of problems can be approached also by means of the Malliavin Calculus as done, e.g., in [START_REF] Di Nunno | Optimal portfolio, partial information and Malliavin calculus[END_REF].

Time-to-build with shock

In the same stochastic framework defined in the previous example, let us consider a state process K(t) representing the capital stock of a certain enterprise at time t and a control process i(t) ≥ 0 representing the investment undertaken at time t to increase K. Assume that there is a time lag (time-to-build) d > 0 between the time when the investment is undertaken and the time when it becomes productive, and that there exists some randomness affecting the achievement of the investment, so that the dynamics of K can be described by the stochastic equation

dK(t) = i(t -d)( dt + σ dW (t)), (9)
where σ > 0 is a volatility constant measuring the uncertainty of achievement of the investment plans. The goal is to maximize the expected integral of the discounted future profit flow in the form

sup i(•) E ∞ 0 e -ρt (pF (K(t)) -C(i(t))) dt , σ > 0,
where F : R → R is a production function, p > 0 is the price of the produced good, and

C : R + → R is a cost function.
Remark 1.2 In this case the approach suggested in Remark 1.1 does not apply.

Structure of the letter

In this letter we will deal with the state equation ( 7) that, varying the parameters, gives as a particular cases (3) and ( 9). The letter is organized as follows. In Section 2 we present some preliminary facts and results, in Section 3 we see how we can rewrite (3) as an evolution stochastic equation in a suitable Hilbert space and in Section 4 we conclude with some remarks on the use of dynamic programming in the described setting.

Preliminary facts and results

In this section we reformulate the problem in infinite dimension. More precisely we reformulate the problem in the space

H := R × L 2 ([-d, 0]; R).
An element ψ in H is then a couple (ψ 0 , ψ 1 ), with ψ 0 ∈ R and

ψ 1 ∈ L 2 ([-d, 0]; R). H is a
Hilbert space when endowed with the inner product φ, ψ = (φ 0 , φ 1 ), (ψ 0 , ψ 1 ) :

= φ 0 ψ 0 + φ 1 , ψ 1 L 2 ([-d,0];R) .
The symbol • will denote the norm induced by the inner product •, • on H. In the following by M ′ we will denote the topological dual of a Banach space (M, • M ). When differently not specified, we shall consider the spaces endowed with their natural norms which make them Banach space, suppressing the norm in the notations.

On H, we consider the unbounded linear operator

A : D(A) ⊂ H -→ H, ψ = (ψ 0 , ψ 1 ) -→ (rψ 0 , Dψ 1 ), (10)
where

D(A) := {(ψ 0 , ψ 1 ) ∈ H | ψ 1 ∈ W 1,2 ([-d, 0]; R), ψ 0 = ψ 1 (0)} ⊂ H,
and D is the (closed) derivative operator on the Sobolev space W1,2 ([-d, 0]; R). It is worth recalling here that by well known Sobolev's embedding theorems, the space W 1,2 ([-d, 0]; R) is continuously embedded into the space of continuous function from [-d, 0] into R, so given an element of W 1,2 ([-d, 0]; R) there is a unique continuous representative of it. In the following we will always refer to this continuous representative when pointwise definitions are invoked. The space D(A) is a Banach space when endowed with the graph norm

ψ D(A) = ψ + Aψ , ψ ∈ D(A).
Identifying H with its topological dual we have the inclusions

D(A) ⊆ H = H ′ ⊆ D(A) ′ .
The operator A is the generator of a C 0 -semigroup on H -see e.g. Theorem 4.3, page 254 of [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF] -that we denote by e tA . The semigroup e tA acts as follows 1 on ψ = (ψ 0 , ψ 1 ) ∈ H:

e tA ψ = e rt ψ 0 , 1 [-d,(-t)∨(-d)] ψ 1 (t + •) + 1 [(-t)∨(-d),0] e r(t+•) ψ 0 . (11)
The adjoint operator A * is also a closed unbounded operator defined on a domain D(A * ) ⊂ H generating a C 0 -semigroup e tA * on H that we denote by e tA * . The operator A * and its domain can be characterized explicitly as follows (see e.g. Theorem 4.6, page 269 of [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF]):

A * : D(A * ) ⊂ H -→ H, ψ = (ψ 0 , ψ 1 ) -→ (rψ 0 + ψ 1 (0), -Dψ 1 ), (12)
where

D(A * ) = {(ψ 0 , ψ 1 ) ∈ H | ψ 1 ∈ W 1,2 ([-d, 0]; R), ψ 1 (-d) = 0} ⊂ H, Lemma 2.
1 For φ ∈ H, the adjoint semigroup e tA * acts as follows on a element φ = (φ 0 , φ 1 ) ∈ H:

e tA * φ = e rt φ 0 + e rt 0 -d e rs 1 [(-t)∨(-d),0] (s)φ 1 (s) ds, φ 1 (• -t)1 [-d+t∧0,0] (•) . (13)
Proof. Given φ and ψ in H, we have

e tA ψ, φ = e rt ψ 0 φ 0 + 0 -d 1 [-d,(-t)∨(-d)] (s)ψ 1 (t + s) + 1 [(-t)∨(-d),0] (s)e r(t+s) ψ 0 φ 1 (s) ds = ψ 0 e rt φ 0 + 0 -d ψ 1 (s)φ 1 (s -t)1 [(-d+t)∧0,0] (s) ds + ψ 0 e rt 0 -d e rs 1 [(-t)∨(-d),0] (s)φ 1 (s) ds = ψ 0 e rt φ 0 + e rt 0 -d e rs 1 [(-t)∨(-d),0] (s)φ 1 (s) ds + 0 -d ψ 1 (s)φ 1 (s -t)1 [(-d+t)∧0
,0] (s) ds ( 14) so we have the claim.

Lemma 2.2

The semigroup e tA * can be extended to a C 0 -semigroup to the space D(A) ′ and the semigroup e tA can be restricted to a C 0 -semigroup on D(A).

Proof. The claims follow by the general Semigroup Theory, see e.g. [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF] pages 202-204.

On the space D(A) we introduce the continuous linear functional

B : D(A) ⊆ H -→ R, ψ = (ψ 0 , ψ 1 ) -→ ψ 1 (-d). ( 15
)
The adjoint operator B * : R → D(A) ′ can be identified with (0, δ -d ) ∈ D(A) ′ , defined by

(0, δ -d )ψ = ψ 1 (-d), ψ = (ψ 0 , ψ 1 ) ∈ D(A),
where δ -d is the Dirac measure concentrated at -d: indeed

B * a = a(0, δ -d ) ∈ D(A) ′ , a ∈ R,
With an abuse of language we will confuse B * : R → D(A) ′ and (0, δ -d ) ∈ D(A) ′ . Given ψ = (ψ 0 , ψ 1 ) ∈ D(A), we have

Be tA ψ = ψ 1 (t -d), t ∈ [0, d], ψ 0 e r(t-d) , t > d. (16)
By Lemma 2.2, e tA * B * is again an element of D(A) ′ . In the following lemma we give an explicit expression for it.

Lemma 2.3

We have

e tA * B * = e (t-d)r 1 [d,+∞) (t), δ -d+t 1 [(-d+t)∧0,0] (•) , (17)
i.e.

e tA * B * = (0, δ -d+t ) if t ∈ [0, d), (e r-d , 0) if t ∈ [d, +∞). ( 18
)
Proof. We consider ψ ∈ D(A) -which implies in particular, by Lemma 2.

= 1 [-d,(-t)∨(-d)] (-d)ψ 1 (t -d) + 1 [(-t)∨(-d),0] (-d)e r(t-d) ψ 0 . ( 19
)
This yields the expression of e tA * B * when acting on a generic element ψ of D(A), yielding the expression of e tA * B * described in the claim.

Infinite dimensional representation of the delay differential equation

We want to find a way to reformulate [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF] as a stochastic evolution equation in H. First we state existence and uniqueness of solutions for [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF]. In the following given a Banach space M , the symbols L 2 loc ([0, +∞); M ) and L 2 loc ([0, +∞); M ) will denote the space of M -valued functions which are square integrable over compact intervals and, respectively, the space of M -valued functions which are essentially bounded. Proof. This is a straightforward application of classical results from the theory of SDEs with measurable coefficients. See, e.g., Chapter 1 of [START_REF] Zhou | Stochastic Controls: Hamiltonian Systems and HJB Equations[END_REF].

Given a ∈ R and v ∈ L 2 ([-d, 0]; R), we denote by Γ(a, v) the element of L 2 (Ω, F d , P; D(A) ′ ) defined as Γ(a, v) := a 0 + 0 -d σλv(τ )e -τ A * B * dτ + 0 -d σv(τ )e -τ A * B * dW (τ + d),
where a 0 ∈ H ⊂ D(A) ′ . The expression of Γ(a, v) provides the construction of the so called structural state (see [START_REF] Vinter | The infinite dimensional quadratic control problem for linear systems with state and control delay: an evolution equation approach[END_REF] and [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF]) in our stochastic case: at each time t, we expect to be able to describe the essential state of the system through Γ(X(t), u t ) ∈ L 2 (Ω, F t+d , P; D(A) ′ ). Note that, since our system does not depend on the past of the state, as in the deterministic case (see [START_REF] Bensoussan | Representation and Control of Infinite Dimensional System[END_REF], Chapter II.4) the structural state depends only on the present of the state and of the whole past of the control. Now denote by y I the random element of D(A) ′ defined as

y I := Γ(x I , u I ). (20)
Then y I corresponds to the structural state for the initial datum and it will be the initial datum of our evolution equation. Notice that it is a stochastic datum even though (x I , u I ) is deterministic.

In the space D(A) ′ we consider the following SDE

   dY (t) = (A * Y (t) + λσu(t)B * ) dt + σu(t)B * dW (t + d), Y (0) = y I . (21)
We call mild solution to (21) the D(A) ′ -valued process Y (t), t ≥ 0, by

Y (t) = e tA * y I + t 0 λσu(s)e (t-s)A * B * ds + t 0 σu(s)e (t-s)A * B * dW (s + d). (22)
We notice that Y (t) ∈ L 2 (Ω, F t+d , P; D(A) ′ ), ∀t ≥ 0, so Y is adapted with respect to (G t ) t≥0 := (F t+d ) t≥0 . Although a priori the process Y takes values in D(A) ′ , actually we have the following.

Proposition 3.2 Assume that u I ∈ L ∞ ([-d, 0]; R) and that u ∈ L ∞ loc ([0, +∞); L ∞ (Ω; R)).
Let Y be the mild solution to [START_REF] Sethi | Sufficient Conditions for the Optimal Control of a Class of Systems with Continuous Lags[END_REF]. Then Y (t) ∈ L 2 (Ω, F t+d , P; H) for all t ∈ [0, +∞). In particular Y (t) ∈ H a.s. for all t ∈ [0, +∞).

Proof. First we shall show that y I ∈ L 2 (Ω, F d ; H). By definition

y I ∈ L 2 (Ω, F d , P; D(A) ′ ) = (L 2 (Ω, F d , P; D(A))) ′ .
Due to Lemma 2.2, e tA * is a C 0 -semigroup on D(A) ′ and then there exists C, α > 0

e tA * L(D(A) ′ ) ≤ Ce αt . (23)
Taking into account ( 16) and ( 23), we can find c, c > 0 such that, for any ψ = (ψ 0 , ψ 1 ) ∈ L 2 (Ω, F d , P; D(A)), we have

y I , ψ L 2 (Ω;D(A)) ′ ×L 2 (Ω;D(A)) = E y I , ψ D(A) ′ ×D(A) ≤ c |x I | E|ψ 0 | 2 + E 0 -d Be -τ A ψu I (τ ) dτ + E 0 -d Be -τ A ψu I (τ ) dW (τ + d) ≤ c |x I | ψ 0 L 2 (Ω;R) + E 0 -d ψ 1 (-d -τ )u I (τ ) dτ + E 0 -d ψ 1 (-d -τ )u I (τ ) dW (τ + d) 2 1/2 ≤ c |x I | ψ 0 L 2 (Ω;R) + ψ L 2 (Ω;H) u I L ∞ + ψ L 2 (Ω;H) u I 1/2 L ∞ = c |x I | + u I L ∞ + u I 1/2 L ∞ ψ L 2 (Ω;H) , (24)
where we have suppressed the σ-algebra F d and the probability P| in the notations, we have set We prove now that the solution of the initial stochastic DDE ( 7) is indeed linked with the solution of the stochastic evolution equation [START_REF] Sethi | Sufficient Conditions for the Optimal Control of a Class of Systems with Continuous Lags[END_REF]. We need first a simple lemma that we will use to approximate the solution. Consider the functions

L ∞ := L ∞ ([-d, 0]; R),
(1, ϕ ε ) ∈ D(A), ϕ ε (s) := (1 + s/ε)1 [-ε,0] (s). ( 26
) and notice that (1, ϕ ε ) ε→0 -→ (1, 0), in H. ( 27
) Lemma 3.3 (i) For t ≥ 0 we have e tA (1, 0) = e rt , 1 [(-t)∨(-d),0] e r(t+•) . (28)
(ii) Let ϕ ε be the function defined in (26). We have

Be tA (1, ϕ ε ) =                0, t ∈ [0, d -ε], 1 + t-d ε , t ∈ (d -ε, d],
e r(t-d) , t > d.

(29)

Proof. Claim (i) follows from the explicit expression of e tA given in [START_REF] Federico | HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, I: Regularity of Viscosity[END_REF]. Claim (ii) follows from [START_REF] Fabbri | Solving optimal growth models with vintage capital: The dynamic programming approach[END_REF].

Theorem 3.4 Let the assumptions of Proposition 3.2 be verified. Let X(t) be the solution of [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF] and let Y (t) = (Y 0 (t), Y 1 (t)) be the mild solution of [START_REF] Sethi | Sufficient Conditions for the Optimal Control of a Class of Systems with Continuous Lags[END_REF]. Then Y 0 (t) = X(t) a.s. for all t ≥ 0.

Proof. Choose t > 0. Let ϕ ε be the function defined in Lemma 3.3. Using ( 22), ( 27) and (25), we can write (the limits below are intended holding a.s.)

Y 0 (t) = Y (t), (1, 0) = Y (t), lim ε→0 (1, ϕ ε ) = lim ε→0 Y (t), (1, ϕ ε ) = lim ε→0 e tA * x I 0 , (1, ϕ ε ) + 0 -d λσu I (τ )e (t-τ )A * B * dτ, (1, ϕ ε ) + 0 -d σu I (τ )e (t-τ )A * B * dW (τ + d), (1, ϕ ε ) + t 0 λσu(τ )e (t-τ )A * B * dτ, (1, ϕ ε ) + t 0 σu I (τ )e (t-τ )A * B * dW (τ + d), (1, ϕ ε ) . (30)
Passing to the adjoint,

Y 0 (t) = lim ε→0 x I 0 , e tA (1, ϕ ε ) + 0 -d λσu I (τ )Be (t-τ )A (1, ϕ ε ) dτ + 0 -d σu I (τ )Be (t-τ )A (1, ϕ ε ) dW (τ + d) + t 0 λσu(τ )Be (t-τ )A (1, ϕ ε ) dτ + t 0 σu(τ )Be (t-τ )A (1, ϕ ε ) dW (τ + d) . (31)
Now, setting u(τ ) = u I (τ ) for τ ∈ [-d, 0), using (28) and (29) and taking the limit we obtain

Y 0 (t) = x I e rt + t-d -d λσu(τ )e r(t-τ -d) dτ + t-d -d σu(τ )e r(t-τ -d) dW (τ + d) = t 0 λσu(τ -d)e r(t-τ ) dτ + t 0 σu(τ -d)e r(t-τ ) dW (τ ) = X(t), (32)
where the last equality follows from the fact that X solves the original state equation [START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF]. Thus we have the claim.

4 Some comments on the dynamic programming approach to the infinite dimensional formulation By Theorem 3.4, we can try to rewrite the optimal control problems we described in the introduction in a equivalent way as optimal control problems in H and then try to apply the tools of the dynamic programming to them. For example, let us focus on the optimization problem ( 7)-( 8) and define the value function at time t = 0 V (0, p, x I , u I ) := inf u(•)

E g(P (T )) -X(T ) 2 .

At least three nontrivial questions arise.

1. Let us consider the optimal control problem in H having as state equations ( 21) with deterministic initial datum y and (5). Then define the value function at time t = 0

V H (0, p, y) := inf

u(•) E g(P (T )) -Y 0 (T ) 2 . ( 33
)
When is it true the desirable equality V (0, p, x I , u I ) = E V H (0, p, y I ) ? (34) 2. Consider as initial time t ∈ [0, T ) and denote by P t,p the process with dynamics (5) starting from p at time t and by Y t,y,u(•) the process with dynamics (21) starting from the deterministic datum y at time t and under the control u(•). The dynamic programming principle formally writes as (with clear meaning of V H (t, •))

V H (t, p, y) = inf u(•)
E V H (τ, P t,p (τ ), Y t,y,u(•) (τ )) .

where τ ∈ [t, T ] is a stopping time. Is it always true?

3. What is the HJB equation associated to V H ? Clearly it should be derived formally by the dynamic programming principle. But this passes through the use of Itô's formula. However the variation dY (s) depends on dW (s+d), while the variation dP (s) depends on dW (s). So how does Itô's calculus apply to this situation? Clearly there are some specific cases where this difficulty disappears, for instance when g, hence V H , does not depend on p.

All these questions (and possibly more) are out of the scope of the present letter and left for future research. Probably positive answers can be obtained in some specific problems.

2 ,

 2 e tA ψ ∈ D(A) for all t ≥ 0 -and we write ψ, e tA * B * D(A)×D(A) ′ = e tA ψ, B * D(A)×D(A) ′ = e rt ψ 0 , 1 [-d,(-t)∨(-d)] (s)ψ 1 (t + s) + 1 [(-t)∨(-d),0] (s)e r(t+s) ψ 0 , B * D(A)×D(A) ′

Proposition 3 . 1

 31 Let (x I , u I ) ∈ H and let u ∈ L 2 loc ([0, +∞); L 2 (Ω; R)) be F-adapted. The SDE[START_REF] Albis | Business cycle fluctuations and learning-by-doing externalities in a one-sector model[END_REF] has a unique (adapted) strong solution X ∈ L 2 loc ([0, +∞); L 2 (Ω; R)) and such solution admits a continuous version.

  )e (t-s)A * B * ds + t 0 σu(s)e (t-s)A * B * dW (s + d)L 2 (Ω,F t+d ; H), ∀t ∈ [0, d].We can use the same arguments that we have used above for estimating the term 0 -d λσu I (τ )e -τ A * B * dτ + 0 -d σu I (τ )e -τ A * B * dW (τ + d) appearing in the expression of y I , getting the result for t ∈ [0, d].In particular one gets also Y (d) ∈ L 2 (Ω, F d , P; H). So, if now t ∈ [d, 2d], we observe first that Y (t) solvesY (t) = e (t-d)A * Y (d) + t-d 0 u(d + s)e (t-d-s)A * B * ds + t-d 0 u(d + s)e (t-d-s)A * B * dW (2d + s).and then we use once again the same kind of estimates to prove the claim for t ∈ [d, 2d]. Iterating the argument we conclude the proof. A fundamental consequence of Proposition 3.2 is that for the mild solution Y = (Y 0 , Y 1 ) of (21) we have Y 0 (t) = Y (t), (1, 0) , a.s. ∀t ≥ 0. (25)

  and we have used the Holder inequality to estimate the term in dτ and the Jensen inequality and the Ito isometry to estimate the term in dW (τ + d). Since L 2 (Ω, F d , P, P; D(A)) is a dense subset of L 2 (Ω, F d , P; H), by[START_REF] Zhou | Stochastic Controls: Hamiltonian Systems and HJB Equations[END_REF] we can extend y I to a bounded functional on L 2 (Ω, F d , P; H). Since we are identifying H with its dual, we conclude thaty I ∈ (L 2 (Ω, F d , P, P; H)) ′ = L 2 (Ω, F d , P, P; H ′ ) = L 2 (Ω,F d , P, P; H). now Y (t) whose expression is given in[START_REF] Tsoukalas | Time to Build Capital: Revisiting Investment-Cash Flow Sensitivities[END_REF] with t ∈ [0, d]. Since we have proved that y I ∈ L 2 (Ω, F d , P; H), also e tA * y I ∈ L 2 (Ω, F d , P; H). So, in order to prove the claim for t ∈ [0, d], we need to prove that

	Consider

Giorgio Fabbri, EPEE, Université d'Evry-Val-d'Essonne (TEPP, FR-CNRS 3126), Département d'Economie, E-mail: giorgio.fabbri@univ-evry.fr. The research of G. Fabbri has been developed in the framework of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01).

Salvatore Federico, Dipartimento di Economia, Management e Metodi Quantitativi, Univeristà di Milano, Italy. E-mail: salvatore.federico@unimi.it.

Hereafter, given f ∈ L

([-d, 0]; R), with a slight abuse of notation we consider f extended on [-d, +∞) asξ -→ f (ξ), if ξ ∈ [-d, 0], 0, if ξ > 0.


Alexandre Hinzpeter

Alix De Becdelièvre

Eric Bieth

Christine Gameiro

François Brémont

Natacha Martin

Bruno Costes

Catherine Costa

Abdel Aissat

Aurélie Lorot

MD Emmanuelle Girodon
email: emmanuelle.girodon@inserm.fr

Keywords: Cystic fibrosis, nasal polyposis, CFTR, alternative transcripts, transcription regulation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Identification of a novel 5' alternative CFTR mRNA isoform in a patient with nasal polyposis and CFTR

mutations.

1 Humu-2013-0257

Brief Report

Identification of a novel 5' alternative CFTR mRNA isoform in a patient with nasal polyposis and CFTR mutations Alexandre Hinzpeter 1,2 #, Alix de Becdelièvre 1,2,3 #, Eric Bieth 4 , Christine Gameiro 3 , François Brémont 5 , Natacha Martin 1 , Bruno Costes 1,2 , Catherine Costa 1,3 , Abdel Aissat 1,2 , Aurélie Lorot 1,2,6 , Virginie Prulière-Escabasse 1,2,6 , Michel Goossens 1,2,3 , Pascale Fanen 1,2,3 , Emmanuelle Girodon 1,3 * 1 U955, INSERM, Créteil, France; 2 Université Paris-Est, Créteil, France; 3 GH Henri Mondor, Nasal polyposis (NP) is a chronic inflammatory disease characterized by the presence of bilateral and diffuse nasal polyps. Although NP is common, its mechanisms of development still remain elusive. NP can be either primary, often associated with allergy, asthma, aspirin intolerance, or secondary, associated with another underlying disease, such as cystic fibrosis (CF; MIM# 219700). Prevalence of NP varies according to the studies, between 0.2-5.6% [START_REF] Alobid | SEAIC-SEORL. Consensus Document on Nasal Polyposis. POLINA Project[END_REF] and increases with age, with a peak incidence between the fourth and fifth decades of life.

In children, incidence of NP is lower, at about 0.1% in the general population [START_REF] Alobid | SEAIC-SEORL. Consensus Document on Nasal Polyposis. POLINA Project[END_REF], and often leads to suspect CF. Indeed, NP has been described in 6-57% of CF patients according to different studies [START_REF] Alobid | SEAIC-SEORL. Consensus Document on Nasal Polyposis. POLINA Project[END_REF][START_REF] Feuillet-Fieux | Nasal polyposis and cystic fibrosis(CF): review of the literature[END_REF]. CF is one of the [START_REF] Costa | A recurrent deepintronic splicing CF mutation emphasizes the importance of mRNA studies in clinical practice[END_REF][START_REF] Giordano | Molecular and Functional Analysis of the Large 5' Promoter Region of CFTR Gene Revealed Pathogenic Mutations in CF and CFTR-Related Disorders[END_REF]. Identifying such mutations in gDNA by sequencing is challenging, whereas seeking the defect first by the mRNA approach has shown some efficiency [START_REF] Faà | Characterization of a disease-associated mutation affecting a putative splicing regulatory element in intron 6b of the cystic fibrosis transmembrane conductance regulator (CFTR) gene[END_REF][START_REF] Costa | A recurrent deepintronic splicing CF mutation emphasizes the importance of mRNA studies in clinical practice[END_REF].

inserm-01044414, version 1 -23 Jul 2014

We describe here a 12y-old boy presenting with isolated severe nasal polyposis. He had no lung disease, as assessed by thoracic tomodensitometry and pulmonary function test, and no pancreatic insufficiency or other digestive symptoms. Sweat test was negative (11 to 18 mEq/L), and measurement of nasal potential difference was not feasible due to rhinosinusitis disease.

While no other clinical sign was related to a CFTR channel dysfunction, the early occurrence of polyposis led to the suspicion of atypical CF. Full sequencing of CFTR coding regions, intronexon boundaries and search for rearrangements only identified the heterozygous c.2551C>T, p.Arg851* (R851X) mutation located in exon 15 (14a according to the traditional nomenclature), of paternal origin (Figure 1A). As previously described, the c.2551C>T mutation induces two defects: the generation of a premature termination codon (p.Arg851*) in the full length mRNA and alternative splicing of exon 15 [START_REF] Hinzpeter | Alternative splicing of in-frame exon associated with premature termination codons: implications for readthrough therapies[END_REF]. This latter mRNA encodes CFTR-del831-871, a protein identified as unstable, mislocalized, non-glycosylated and non-functional [START_REF] Hinzpeter | Alternative splicing at a NAGNAG acceptor site as a novel phenotype modifier[END_REF].

CFTR mRNA studies were performed with a view to identifying a second, maternally inherited CFTR defect, from nasal epithelial cells of the index case and his parents. The qualitative study consisted of PCR amplification of cDNA in different fragments encompassing all CFTR exons [START_REF] Costa | A recurrent deepintronic splicing CF mutation emphasizes the importance of mRNA studies in clinical practice[END_REF]. This analysis only evidenced the abnormal exclusion of exon 15, caused by the c.2551C>T mutation, in the patient and his father (Figure 1A). Relative quantification of CFTR mRNA level was performed by real-time PCR with normalization to keratin 18 (KRT18), a marker of ciliated and secretory epithelial cells [START_REF] Linde | Nonsense-mediated mRNA decay affects nonsense transcript levels and governs response of cystic fibrosis patients to gentamicin[END_REF]), using Taqman probes (TaqMan FAM/NFQ-MGB probe format, CFTR: Hs00357011_m1, located at the boundary of CFTR exons 20/21; KRT18: Hs01920599_gH, Applied Biosystems), as previously described [START_REF] Hinzpeter | Alternative splicing at a NAGNAG acceptor site as a novel phenotype modifier[END_REF] (Supp. Materials and Methods). Results were controlled using a widely inserm-01044414, version 1 -23 Jul 2014 used housekeeping gene as another reference gene, Glyceraldehyde-3-Phosphate Dehydrogenase (GAPDH), [START_REF] Ramalho | Quantification of CFTR transcripts[END_REF] according to the instructions of the manufacturer (GAPDH:

AX-004253-00-0400, Thermo Scientific, Solaris qPCR assay). CFTR mRNA level was significantly reduced in the father to 38% ± 8% (p < 0.01) compared to eight different control individuals (100% ± 30%, n=8) tested for CFTR/KRT18 (Figure 1B). Such a diminution can be in part due to degradation of the mRNA form containing the c.2551C>T mutation, via the nonsense-mediated decay pathway (NMD) which was shown to vary between mutations and even between individuals carrying the same mutation [START_REF] Linde | Nonsense-mediated mRNA decay affects nonsense transcript levels and governs response of cystic fibrosis patients to gentamicin[END_REF]. Inter-individual variation in CFTR mRNA amount has also been documented, regardless of NMD [START_REF] Masvidal | Assessing the residual CFTR gene expression in human nasal epithelium cells bearing CFTR splicing mutations causing cystic fibrosis[END_REF]. Finally, no mutation that could impair CFTR mRNA expression was found in the patient's father after complete sequencing of the coding region and the promoter region (c.-3897 to c.-1). By contrast, surprisingly, the index case exhibited a significant (p<0.01) increase in CFTR mRNA to 160% ± 34% of the controls (n=8), similar to his mother (151% ± 32%, p<0.01) (Figure 1B). Similar results were obtained using GAPDH as a reference gene (57% in the father, 148% in the patient and 227% in the mother, as compared with two control individuals). Among the 1900 CFTR gene mutations reported, few have been described in the promoter region reaching down to 6kb, in CF and CFTR-RD patients [START_REF] Romey | A naturally occurring sequence variation that creates a YY1 element is associated with increased cystic fibrosis transmembrane conductance regulator gene expression[END_REF][START_REF] Giordano | Molecular and Functional Analysis of the Large 5' Promoter Region of CFTR Gene Revealed Pathogenic Mutations in CF and CFTR-Related Disorders[END_REF].

The first description of an increased amount of CFTR mRNA was suggested from in vitro experiments in the case of a complex allele, c.[-234T>A;1647T>G] (-102T>A in cis with p.Ser549Arg). In this case, the promoter mutation c.-234T>A was suspected to attenuate the effect of the c.1647T>G mutation by increasing the amount of the mutant channel that has a residual function [START_REF] Romey | A naturally occurring sequence variation that creates a YY1 element is associated with increased cystic fibrosis transmembrane conductance regulator gene expression[END_REF]. Very recently, Giordano et coll. described three novel promoter variants associated with an in vitro increased CFTR promoter activity, which were inserm-01044414, version 1 -23 Jul 2014

nonetheless supposed to attenuate as well the deleterious effect of other, undetected, in cis mutations [START_REF] Giordano | Molecular and Functional Analysis of the Large 5' Promoter Region of CFTR Gene Revealed Pathogenic Mutations in CF and CFTR-Related Disorders[END_REF]. Sequencing of the CFTR promoter gDNA, from c.-3897 to c.-1, was performed in our patient in order to identify a putative mutation affecting the regulation of transcription. A single heterozygous sequence variation was identified, c.-2954G>A (chr7:117,117,195G>A (GRCh7/hg19)) in the mother and the child, and not in the father (Figure 1C).

This variant has been recently reported, notably in the dbSNP 135 database as rs145885255:G>A, in the German and other European populations with an allelic frequency of 0.14% and 0.34%, respectively. Testing this mutation in a panel of 100 healthy individuals led to find a 0.5% allelic frequency in the French population (1/200 chromosomes). However, nasal epithelial cells from the only individual identified with this variant in this group were not available to study mRNA. The variant was not found in a series of 14 patients having nasal polyposis and only one identified CFTR mutation.

A functional assay was performed to assess if this variant was related to the increase in CFTR mRNA levels observed in the patient and his mother. We used two constructs of different lengths As an increased level of normal CFTR transcripts was not expected to be associated with CF or a CFTR-RD, we hypothesized the production of an abnormal transcript in its 5' segment, induced or favoured by the mutation and that could not be detected by our protocol set-up for CFTR mRNA studies, since the first 5' primer is located in exon 1. We thus looked for a 5' alternative transcript in nasal epithelial cells using the 5' Rapid Amplification of cDNA Ends (5'RACE)

technology (FirstChoice® RLM-RACE Kit, Life Technologies, Paris, France) (Supp. Materials and Methods). Analysis was performed in three unrelated controls without any CF symptom or CFTR mutation detected upon gDNA and cDNA analysis, from nasal cells for one control (Control 1) and primary cultures of nasal cells [START_REF] Prulière-Escabasse | Effect of neutrophil elastase and its inhibitor EPI-hNE4 on transepithelial sodium transport across normal and cystic fibrosis human nasal epithelial cells[END_REF] for the other two.

Results revealed the presence of two PCR products for Control 1 and only one PCR product in the other two after separation in a 2% agarose gel (Figure 2C). These products corresponded to known CFTR transcripts including exons 1 to 4 and differing by the length of the 5'UTR, as assessed by sequencing analysis: one initiation start site was at c.-71 [START_REF] Yoshimura | The cystic fibrosis gene has a "housekeeping"-type promoter and is expressed at low levels in cells of epithelial origin[END_REF], observed in the three controls, and the second one at c.-122 [START_REF] Riordan | Identification of the cystic fibrosis gene: cloning and characterization of complementary DNA[END_REF], observed only inserm-01044414, version 1 -23 Jul 2014

in Control 1 (Figure 2D and Figure 2E). In the patient, two isoforms were detected (Figure 2C), one containing exons 1 to 4 (initiation site at c.-71) and one containing exons 2 to 4, preceded by a sequence corresponding to the previously described exon 1a [START_REF] Koh | Characterization of the cystic fibrosis transmembrane conductance regulator promoter region. Chromatin context and tissue-specificity[END_REF][START_REF] Mouchel | Alternative 5' exons of the CFTR gene show developmental regulation[END_REF] plus 33 bp upstream (initiation site at c.-666, end at c.-401) (Supp. Figure S1). We called this novel exon 1a-L for 1a-Long (Figure 2D and Figure 2E). Alternative 5' CFTR transcripts have never been identified in CF or CFTR-RD. Two 5' alternative transcripts have been previously described, both containing exon -1a (c.-937 to c.-750) as the first exon: -1a, 1a, 2 and -1a, 2, firstly in the human T84 carcinoma cell line [START_REF] Koh | Characterization of the cystic fibrosis transmembrane conductance regulator promoter region. Chromatin context and tissue-specificity[END_REF] and further, notably in the human developing fetal lung [START_REF] Mouchel | Alternative 5' exons of the CFTR gene show developmental regulation[END_REF] (Figure 2E). Neither of these transcripts was predicted to encode a fully functional CFTR channel but they were hypothesized to have regulatory functions in the normal developing lung, with a substantial decreasing expression during gestation [START_REF] Mouchel | Alternative 5' exons of the CFTR gene show developmental regulation[END_REF][START_REF] Lewandowska | Multiple mechanisms influence regulation of the cystic fibrosis transmembrane conductance regulator gene promoter[END_REF]. We describe here for the first time in a patient, a novel, third 5' alternative isoform (1a-L, 2), which excludes exon -1a and resembles to these CFTR fetal isoforms. The 5' additional sequence of 33 bp contains one AUG codon, so that exon 1a-L contains two AUG codons and a CUG putative initiation codon. The use of the first AUG would lead to a stop UGA codon, 38 codons downstream, whereas the use of the second AUG and CUG codons would not lead to a functional CFTR protein, as shown for the previously described 5' alternative transcripts [START_REF] Lewandowska | Multiple mechanisms influence regulation of the cystic fibrosis transmembrane conductance regulator gene promoter[END_REF].

Although the link between the increased CFTR promoter activity and the presence of an abnormal transcript is not established, it is tempting to speculate that the existence of a 5' alternative CFTR mRNA isoform in our patient reflects a defect in the regulation of CFTR expression. It is however not clear if production of the abnormal CFTR transcript is induced by a pathological state such as proliferation of nasal cells or inflammation or, alternatively, if the 1a-inserm-01044414, version 1 -23 Jul 2014

L-containing transcript induces CFTR dysfunction. CFTR dysfunction could then be due to impaired translation and subsequent reduction of functional CFTR, the presence of the abnormal mRNA isoform per se, or a combination of both. Taken together, our observation highlights the importance of searching for such alternative transcripts in patients presenting either isolated NP or another CFTR-RD, as well as in patients carrying promoter variants recently shown to increase the CFTR promoter activity [START_REF] Giordano | Molecular and Functional Analysis of the Large 5' Promoter Region of CFTR Gene Revealed Pathogenic Mutations in CF and CFTR-Related Disorders[END_REF].

In conclusion, this case report could represent the first description of a CFTR disease associated with the presence of a novel 5' alternative transcript. This would thereby lead to search for a new mutation type, especially in patients with CFTR-RD. Such mutations could also act as modifiers in the framework of complex alleles. This observation finally illustrates the importance of mRNA studies in the new era of next-generation sequencing, which inevitably will lead to identify a great number of variants of unknown clinical significance.

  of the CFTR promoter for cross-validation(from c.-3068 to c.-47 and c.-3628 to c.-47), driving a luciferase reporter gene, as illustrated in Figure2A. Both WT and mutated plasmids were transfected in the bronchial BEAS-2B and CF IB3-1 epithelial cell lines and the Firefly luciferase activity was normalized to the Renilla luciferase activity (Supp. Materials and Methods). In BEAS-2B cells, the c.-2954G>A mutation significantly increased the promoter activity by 2-fold with the c.-3068 construct (2.3 ± 0.3, p < 0.03, n=3) and by 5-fold with the c.-3628 construct (5.6 ± 0.5, p < 0.004, n=5) (Figure2B). In IB3-1, the increase in transcriptional activity of the mutated promoter was confirmed by an 8-fold gain with the c.-3628 construct (8.0 inserm-01044414, version 1 -23 Jul 2014 ± 2.0, p < 0.02, n=6) (data not shown). Similar results were obtained in the intestinal Caco-2 cell line, where the c.-2954G>A mutation increased the promoter activity by 1.6-fold with the c.-3068 construct (1.6 ± 0.02, p < 0.003, n=3) and by 2-fold with the c.-3628 construct (2.6 ± 0.5, p < 0.004, n=3) (data not shown). This increased transcriptional activity could be the consequence of alteration of transcription factor binding sites, as suggested by in silico analysis of the concerned region. Nonetheless, the CFTR gene transcription regulation is somewhat complex, with multiple interplaying factors located outside the promoter region[START_REF] Mccarthy | The CFTR gene and regulation of its expression[END_REF]. The characterization of the underlying mechanism responsible for the increased transcriptional activity due to c.-2954G>A is under investigation.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. In vitro functional evaluation of the c.-2954G>A mutation and identification of a

patient with inclusion of exon 1a-L (middle), and in the human developing fetal lung, including exons -1a and 1a as previously described (lower). The arrows correspond to transcription start sites. Splicing events leading to either the wild-type transcript (solid line) or 5' alternative transcripts (dashed lines) are indicated. First and last nucleotides of the alternative exons are indicated.

Acknowledgments

The authors wish to thank B. Boissier, A. LeFloch and R. Medina for valuable technical assistance and J. Percodani and M.N. Calmels for patients' nasal cells sampling. Mario Ollero is acknowledged for re-reading the English of the manuscript. This work was supported by the French CF Foundation « Vaincre La Mucoviscidose ».

Disclosure Statement:

The authors declare no conflict of interest.

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