D Michel

A Zidna

A new deterministic heuristic knots placement for B-Spline approximation

Keywords: Dense and noisy data, B-Spline approximation, Knot placement, Discrete derivatives, Discrete angular variation, Curvature, Handwriting compression

In this paper, we propose an adaptive knot placement algorithm for B-Spline curve approximation to dense and noisy 2D data points. The proposed algorithm is based on a heuristic rule for knot placement. It consists in constructing a distribution knot function by blending geometric criteria such as discrete derivatives, discrete angular variations and curvature. It has been successfully compared to three well known methods for approximating various noisy functions and sets of data in handwriting context.

Introduction

Curve fitting is a fundamental problem in many fields, such as computer graphics, image processing, shape modeling and data mining. It has become a fundamental tool in Reverse Engineering, where very dense data point sets are acquired from physical objects in order to refine a digital model of such objects. Usually, the underlying function of data is difficult to approximate through a simple function. The B-Spline approximating functions are widely used as they are very convenient and can represent a large variety of shapes. This is due to their powerful mathematical properties such as local modification, projective invariance and convex hull property. The knots placement of the approximation B-spline curve plays an effective role in order to recover the underlying curve of the data points. When the given data points present considerable curvature variation of the underlying curve, the reconstructed approximation curve may be significantly different from the original curve if the knots are not distributed according to the varying of the curvature. The approximation of functions by B-Splines is greatly improved if the knots are treated as free variables. In many applications, curve approximation methods are iterative process. In the conventional approach to this problem, there are two ways for knots placement. One starts with the minimum (or the maximum) of knots and increase (or reduce) the number of knots to satisfy the error bound specified between the original curve and the approximation curve [START_REF] Burchard | Splines (with optimal knots) are better[END_REF][START_REF] Jupp | Approximation to data by splines with free knots[END_REF][START_REF] Rice | The approximation of functions[END_REF]. Unfortunately, this process fails to automatically generate a good knot vector and is time consuming if the initial knots are not well chosen.

For some years now, the trend in this topic has been to develop stochastic methods (genetic algorithms, swarm particle algorithms, etc.). They have been proved efficient in regard of the accuracy of the solution. Goldenthal et al. [START_REF] Goldenthal | Spline curve approximation and design by optimal control over the knots[END_REF] consider to solve the problem as a minimization of a cost function that depends on the knots. They propose to use a genetic algorithm. In the same way, the authors propose a real coded genetic algorithm for data fitting with a B-Spline [START_REF] Yoshimoto | Data fitting with a spline using a real coded algorithm[END_REF]. In [START_REF] Arslan | Automatic knot adjustment using an artificial immune system for B-spline curve approximation[END_REF], the authors propose to use artificial immune system (AIS) strategy to determine the locations and counts of interior knots of B-spline curves that fit a given data. Their benefits of using AIS is to accelerate the convergence to the objective function. In [START_REF] Gálvez | Efficient particle swarm optimization approach for data fitting with free knot B-splines[END_REF], in order to automatically compute an appropriate location of knots, the authors propose to use the particle swarm optimization (PSO) paradigm. This scheme yields very accurate results, even for curves with singularities and/or cusps. In [START_REF] Valenzuela | Evolutionary computation for optimal knots allocation in smoothing splines[END_REF][START_REF] Idais | Optimal knots allocation in the cubic and bicubic spline interpolation problems[END_REF], a Multi-Objective-Genetic Algorithm has been developed, to optimize both the number of knots and its optimal placement for cubic or bicubic interpolating B-Splines. This technique is able to avoid the large number of local minima existing in the problem of random knots placement. But all of these methods suffer a huge drawback : the CPU-time needed to achieve good results.

More recently, In [START_REF] Laube | Learnt knot placement in B-spline curve approximation using support vector machines[END_REF], the Support Vector Machines (SVMs) method is used to determine suitable knot vectors for B-spline curve approximation. The input for SVMs is a sequential points cloud, and the output from SVMs is the points set with signed scores that measures the quality of each location. Here the score is based on geometric and differential geometric features of points. But there are many other related works. For instance in [START_REF] Kang | Knot calculation for spline fitting via sparse optimization[END_REF], the authors propose an optimal method to fix the location and the number of knots in curve fitting with B-Splines by solving a sparse optimization problem. The objective is to minimize the total jump in the third derivatives on the knots. In [START_REF] Luo | Knot calculation for spline fitting based on the unimodality property[END_REF], the authors propose a framework for computing the number of knots and the appropriate positions for spline fitting based on the active knot paradigm. Starting with an initial dense knot vector, they remove the inactive knots and the remaining ones become the candidate knots. In practice, They also adopt strategies to automatically adjust the number and location of candidate knots. The proposed method significantly improves the one described in [START_REF] Kang | Knot calculation for spline fitting via sparse optimization[END_REF]. However, it is very sensitive to noise due to the sensitivity of the third derivative. In [START_REF] Zhang | B-spline surface fitting with knot position optimization[END_REF], a heuristic algorithm is presented to approximate scattered data with a B-spline surface. An heuristic criterion for optimal knot positions in the fitting problem is formulated as an optimization problem according to the geometric feature distribution of the input data. Then, the coordinate descent algorithm is used for the optimal knot computation. In [START_REF] Park | B-spline curve fitting based on adaptive curve refinement using dominant points[END_REF], the authors introduce the concept of dominant points which leads to an efficient algorithm. This approach is more detailed in the following as we used it as a method to compare our algorithm with. In this paper, we propose a deterministic heuristic method to set the knots. It is based on the blending of different geometric features of the input data such as discontinuities of successive derivatives, curvature and angular variations of derivatives. It has been successfully compared to a classical gradient method, to a PSO algorithm and to Park method [START_REF] Park | B-spline curve fitting based on adaptive curve refinement using dominant points[END_REF]. This paper is organized as follows. In section 2, we introduce the B-spline curve approximation. Then, in section 3, the new method is fully described. Section 4 deals with experiments, two benchmarks of data clouds are set. The first one consists in a series of six noisy non parametric functions. The second one mainly consists in naturally noisy handwritten words. Computational time and error measuring are discussed. Finally, in section 5, the conclusion is given and further work is discussed.

B-spline approximation

Let us denote the cloud of data P as follows : P = {(t 0 , P 0 ), . . . , (t N , P N )} where the t i are the time coordinates of the points and the P i are the space coordinates of the points. That is, {t i } i=0...N is an increasing sequence of real numbers and the P i belong to R d (2 ≤ d). By an affine transform, the finite sequence {t 0 , . . . , t N } can always be replaced by a sequence which entirely lies in [0, 1] with t 0 = 0 and t N = 1. In the following, we assume it is the case. Furthermore, we assume that, by linear interpolation, it is always possible to calculate a "missing point" in between two consecutive points of P . The general problem discussed here is to design a B-Spline curve C [START_REF] Boehm | Inserting new knots into B-spline curves[END_REF][START_REF] Boor | A Practical Guide to Splines[END_REF] which closely approximates a finite cloud of points P . These data points are supposed to lie on some unknown curve t → L(t) and are sorted along the time axis. They were obtained from some physical measures, hence noisy in the general case. The obvious purpose of this endeavour is data compression since the amount of necessary points dramatically drops when passing from a discrete sampled curve to a B-Spline curve. A B-Spline curve C of order k is a piecewise polynomial of degree k -1 defined by a set of m + 1 control points {c i } i=0...m and by a set of knots {u i } i=0...m+k . The greater the value of k, the smoother the curve C. The c i are defined over the space and the u i are real numbers distributed along the time axis. The problem addressed by this paper can be expressed as follows : Given m, k and a noisy discrete sampled curve P : calculate the knots then the control points of a B-Spline curve that minimizes the error (L2-norm) between the given data curve and the resulting B-Spline curve. Once the knots are set, the control points can straightforwardly be found using a well conditioned linear system based on L2-Inner product. The challenging part is to calculate the knots for this is a highly non linear problem.

B-Spline model

Usually, C is given by the following definition : Let m and k be two integer numbers such that 1 ≤ k ≤ m + 1. Let U = {u 0 , u 1 , u 2 , . . . , u m+k } be a finite sequence of real numbers such that : u i ≤ u i+1 for all i = 0 . . . m + k -1 and u i < u i+k for all i = 0 . . . m. Then C : R → R d is defined as follows :

t → c(t) = m i=0 c i N k i (t) (2.1)
where the coefficients c i are given in the d-dimensional euclidian space (in the following, we assume that d = 2) and the t → N k i (t) are the basis of all the B-Spline functions over U . They are defined as follows:

N 1 i (t) = 1, for u i ≤ t < u i+1 , 0, otherwise, (2.2)
and for 2 ≤ r ≤ k, we have

N r i (t) = t -u i u i+r-1 -u i N r-1 i (t) + u i+r -t u i+r -u i+1 N r-1 i+1 (t). (2.3)
Under these hypotheses, the c i are called control points, the u i knots and U a knot vector. The outer knots u 0 , . . . , u k-1 and u m+1 , . . . , u m+k are usually set such that u 0 = . . . = u k-1 = 0 and u m+1 = . . . = u m+k = 1. This ensures that c(0) = c 0 and that c(1) = c m , that is, endpoints of the curve C coincide with endpoints of the polygonal chain c 0 . . . c m . Under these assumptions, the knot vector U is said clamped. In the following, all knot vectors comply with this property.

Fitness function

The input of the problem is P and the output is m, k, U and {c i } i=0...m which ensure that C fits as best as possible the cloud P . In the following, we always assume that the cloud P is dense in regard to the set of control points of C, that is, N m. Once m, k and U are set, the coefficients c i are determined through solving linear systems of equations (least square method, quasi-interpolation techniques, etc.). The hardest part is how to define U . Since it is a solution of a highly non linear system of equations, U , that is, the inner knots u k , . . . , u m must be set otherwise. As we are dealing with an optimization problem, a convenient way to describe the whole process is to define a fitness function F . Let m, k and P be fixed, then F : [0, 1] m-k+1 → R + is defined as follows :

(u k , . . . , u m ) → F (u k , . . . , u m ) = |L -C| 2 |L| 2 (2.4)
where

|L-C| 2 |L| 2
is the squared relative error between the two curves t → C(t) and t → L(t). Let us detail how F is computed.

Computing the control points

c 0 , . . . , c m
First, by linearly interpolating P = {(t 0 , P 0 ), . . . , (t N , P N )}, we deduce the continuous curve t → L(t). This is made possible for P is supposed to be dense. Then, we consider the inner product in the vector space of continuous functions over [0, 1]. That is, given two continuous functions f : [0, 1] → R and g : [0, 1] → R, the inner product of f and g is < f, g >= 1 0 f (t)g(t)dt. Then the control points c 0 , . . . , c m are calculated by projecting t → L(t) on the vector space of B-Spline functions. Let c(t) = m i=0 c i N k i (t) be that projection. Then ∀i = 0 . . . m, we have

< N k i , L >=< N k i , c >=< N k i , m j=0 c j N k j >
Finally, we get :

< N k i , L >= m j=0 < N k i , N k j > c j ∀i = 0 . . . m (2.5)
which defines a very well conditioned (m + 1) × (m + 1) linear system of which the solutions are the c i . We solve it using Gauss-Jordan global pivot reduction. Strictly speaking, in the three equations above, <, > refers to an integral but not always to the inner product since the left member < N k i , L > is actually not a scalar. We write it this way for the sake of conciseness. The system (2.5) was preferred over least squares systems and over quasi-interpolating systems because of the better accuracy of its solutions. Once the c i are calculated, the curve t → c(t) is entirely defined. Then the error (2.4) is simply calculated using the norm deduced from the inner product, that is |f | 2 =< f, f >.

The heuristic knots placement method

The purpose is to set the m -k + 1 inner knots u k , . . . , u m over ]0, 1[ such that they form a nondecreasing sequence of real numbers where at most k -1 consecutive values can be identical. Common good sense suggests that the density of the knots should be proportional to the local level of irregularity of the curve L. In high curvature parts of the curve or in the vicinity of cusps the density of knots should increase. To achieve this, we build a continuous and increasing function t → V (t), [0, 1] → [0, 1] of which instantaneous slope aims to be proportional to instantaneous irregularity of the curve L.

Building the distribution knot function t → V (t)

Clearly, V defines a bijection from [0, 1] to [0, 1]. Then, to compute the knots, we uniformly spread m -k + 1 values v k , . . . , v m along ]0, 1[ and we get the knots by use of the inverse of V . That is,

u i = V -1 (v i ) ∀i = k, . . . , m.
V -1 is used to apply a distortion of [0, 1]. It "squeezes" the knots on highly irregular parts of the underlying curve L. If L is a straight line, the distribution is uniform. Let v be a continuous and non identically null function t → v(t), [0, 1] → [0, +∞[ of which instantaneous value is proportional to the magnitude of instantaneous irregularity of L. Let I = 1 0 v(t)dt then we define

V (x) = x 0 v(t)dt I
Clearly, x → V (x) verifies all the desired properties. v is built by combination of criteria extracted from successive derivatives of L with respect to time. So we first need to extrapolate derivatives of any order from L.

Calculating successive derivatives of L

Let t → q(t) be a such extrapolated derivative of some order of L. We assume that q is defined by interpolating a discrete and dense cloud of points Q = {(t 0 , Q 0 ), . . . , (t n , Q n )} with n ≤ N where the t i denote the time positions and the Q i the corresponding space positions, this hypothesis holds for L. Then, analogously, we define the derivative t → q(t) by interpolating a cloud of points R = {(s 0 , R 0 ), . . . , (s n-1 , R n-1 )} which is defined as follows :

time position : s i = ti+1+ti 2 ∀i = 0, . . . , n -1 space position : R i = Q i+1 -Q i ti+1-ti ∀i = 0, . . . , n -1 (3.1)
Starting from L and recursively applied, this discrete differentiation provides the successive derivatives of L.

The accuracy of the results is quite good thanks to the initial hypothesis of high density cloud. Throughout the iterations, a margin grows on both endpoints of the curve but the density remains constant.

Angular variations of the successive derivatives

Again, let t → q(t) be a discrete derivative of some order of L defined by the cloud Q = {(t 0 , Q 0 ), . . . , (t n , Q n )}. Then we can define the discrete angular variation t → α(t) of q by interpolating a cloud of points A = {(s 0 , a 0 ), . . . , (s n-1 , a n-1 )} which is defined as follows :

time position : s i = t i+1 +t i 2 ∀i = 0, . . . , n -1 space position : a i = |angle(Qi,Qi+1)| ti+1-ti ∀i = 0, . . . , n -1 (3.2)
In the following, when q is the i th derivative of L, t → α(t) is denoted t → α i (t) . We can give an intuitive meaning of t → α 1 (t) : if L is smooth and does not contain any inflexion point then

1 0 α 1 (t)dt 2π
is the number of loops of the curve L. If L is a straight line, then t → α 1 (t) is identically null.

Criteria

Any of the following functions constitutes a good candidate to represent v(t) :

• norm of the discrete i th derivative : t → d i dt i L(t) ∀i ≥ 1
• discrete angular variations of the i th derivative : t → α i (t) ∀i ≥ 1

• unsigned curvature :t → |ρ(t)|
For any hairpin turn, cusp, gap and more generally any irregularity on L can be detected by a change of magnitude of one or several of those functions. In the following, those functions are referred to as criteria.

Let us notice that in [START_REF] Razdan | Knot placement for B-spline curve approximation[END_REF][START_REF] Li | Adaptive knot placement in b-spline curve approximation[END_REF], the authors use a similar approach but only by considering the third criterion.

Blending the criteria

To take more benefit from the above criteria it is useful to blend together two or more of them since they can show different kinds of irregularities in different parts of the curve. Without loss of generality, let us detail how we blend two of them : Let t → a(t) and t → b(t) be two criteria respectively defined by the two time-space clouds {(t 0 , a 0 ), . . . , (t n , a n )} and {(t 0 , b 0 ), . . . , (t n , b n )}. Since they can be resampled, we can always suppose that both clouds have identical time sequences. We blend a and b by making local convex linear combinations. The operation produces the time-space cloud {(t 0 , r 0 ), . . . , (t n , r n )} defined as follows :

r i = α i a i a + β i b i b ∀i = 0 . . . n (3.3)
where :

α i = A i A i + B i β i = B i A i + B i A i = |a i -a| σ a B i = b i -b σ b
with a, b respectively denoting the population means of {a 0 , . . . , a n } and {b 0 , . . . , b n } and σ a , σ b respectively denoting the standard deviations of {a 0 , . . . , a n } and {b 0 , . . . , b n }. Equation (3.3) was found empirically, it is motivated by two facts : The farther a coefficient a i , (resp.b i ) deviates from the mean a, (resp. b), the more it means an irregularity. Therefore, its influence on the result should be proportional to this deviation. This influence should be local. Obviously, the purpose of α i and β i is only to perform a convex linear combination and the normalization of the coefficients, i.e. the divisions by the means is meant to give all the criteria an equal influence on the resulting blended criterion, no matter their magnitude. Eventually, the function t → v(t) is built as a continuous function by interpolation of the time-space cloud {(t 0 , r 0 ), . . . , (t n , r n )}. Experimentally, as criteria, we used euclidean norms of derivatives from order one up to order 5, angular variations from order one up to 5, plus the curvature function which was calculated using the classical formula : We blent from one to three criteria together. The maximum number of functions t → v(t) built this way is :

ρ(t) = det( L(t), L(t)) | L(t)| 3 . Table 1: Test functions used f 1 (t) = 90 1 + e -100(t-0.4) t ∈ [0, 1] f 4 (t) = sin(t) + 2e -30t 2 t ∈ [-2, 2] f 2 (t) =        1 0.01 + (t -0.3) 2 0 ≤ t < 0.6 1 0.015 + (t -0.65) 2 0.6 ≤ t ≤ 1 f5(t) = sin(2t) + 2e -16t 2 + 2 t ∈ [-2, 2] f 3 (t) = 100 e |10t-5| + (10t -5) 5 500 t ∈ [0, 1] f 6 (t) =          4t 2 (3 -4t) 0 ≤ t < 0.5 4 3 t(4t 2 -10t + 7) - 3 2 0.5 ≤ t ≤ 0.
11 1 + 11 2 + 11 3
= 231. Each one of the functions was used to provide a vector of inner knots which is an input for the fitness function F . Among the population of inner knot vectors produced this way, the best inner knot vector (u k , . . . , u m ) was then found by selecting the one yielding the smallest value of F . This knot vector was finally refined by a gradient descent.

Experimental results

All development was done by the authors in C++11. The integrals involved in the different calculus were made using trapezoidal rule. Thanks to C++ features, the matrix calculations were easily performed by massive use of operators overloading, template entities and functors (for details, see [START_REF] Allain | Jumping into C++, Programming[END_REF]). All computations were made on a laptop equipped with a processor Intel core i7-8550U CPU. Our method has been compared to three other methods, two deterministic and a stochastic one. The first one consists of launching a gradient descent starting from a uniform distribution of the knots. The second one is the Particle Swarm Optimization algorithm described in [START_REF] Gálvez | Efficient particle swarm optimization approach for data fitting with free knot B-splines[END_REF] and the third is the heuristic algorithm described in [START_REF] Park | B-spline curve fitting based on adaptive curve refinement using dominant points[END_REF].

Implementation details of the comparison methods

The gradient descent method was applied to the fitness function F (2.4). The partial derivatives of F are not analytically known (let alone F has not been proved to be differentiable), therefore they were computed using the classical approximating formula :

∂F ∂u i F (u k , . . . , u i-1 , u i + h, u i+1 , . . . , u m ) -F (u k , . . . , u m ) h ∀i = k, . . . , m (4.1)
Table 3: Error results of our algorithm, gradient descent, PSO algorithm and Park method for the six noisy functions, w and the handwritten words "@", "blue", "love", "key" and treble clef. with h being a small positive real number. The starting point used to initiate the gradient descent method was set to the uniform distribution.

The PSO algorithm used was the one described in [START_REF] Gálvez | Efficient particle swarm optimization approach for data fitting with free knot B-splines[END_REF], we also used the same parameters. Throughout the iterations, the sequences (u k , . . . , u m ) have to remain non decreasing and confined to the interval [0, 1].

Both conditions are hardly met by the algorithm though this issue was not discussed in the original paper. So we made our own adjustments to fix it. After each iteration : every knot u is replaced by u -u where . is the floor function and all the knots are sorted to ascending order. The same arrangement was done on the gradient descent method. The Park method is the heuristic algorithm described in [START_REF] Park | B-spline curve fitting based on adaptive curve refinement using dominant points[END_REF]. The method is based on the idea of dominant points which are a subset of the initial given data points P i . Then the knots are computed as mean values of k -1 consecutive dominant points. The algorithm consists in selecting the m + 1 best dominant points. The algorithm is composed of two steps. The first step determines the dominant seed points set. It consists of a few pre-selected data points which maximize the curvature. These points are used to build a first coarse B-spline approximation of the curve L. The second step consists in repeated locally refining the approximation by selection of new dominant points. For both steps of the algorithm (seed points initiation and adaptive refinement) the instantaneous curvature was calculated the same way we did in our method (i.e. by use of the two first discrete derivatives of t → L(t)) since the results are very accurate. For our algorithm, the knot vector (u k , . . . , u m ) was found following the method described in subsection 3.5. This knot vector was then refined by a gradient descent. Quite good results were obtained with 10 to 20 iterations. The accuracy of the four methods (Gradient, PSO, Park and our algorithm) was simply measured by the value of the fitness function F which is the squared relative error between the raw sampled curve and the resulting B-Spline curve. It is therefore based on the norm induced by the inner product on functions. That is, the accuracy of a stochastic method cannot easily be defined since its results vary with every execution.

Benchmark

Half of the benchmark of sampled curves used to make the comparisons is the one provided in [START_REF] Gálvez | Efficient particle swarm optimization approach for data fitting with free knot B-splines[END_REF]. It consists in a series of six non parametric functions denoted f 1 to f 6 in the following. First the tests were performed on the smooth functions then on noisy versions of them. A cubic B-Spline curve made of 19 control points (i.e. m = 18) was also added to the benchmark. It is shaped as a handwritten "w" with an additional angular point. It has been sampled then been made noisy in the same way as were f 1 , . . . , f 6 . In the following, this curve is simply denoted as w. To complete this benchmark, five handwritten words "@", "blue", "love", "key" and treble clef-therefore naturally noisy -were added. They were obtained from a software developed by the authors and running on an Android device. The clouds respectively contain 185, 418, 355, 175 and 250 points. So, altogether, we ran 72 tests (18 curves × 4 methods). The six functions f 1 , . . . , f 6 are summarized in table 1, their corresponding curves are drawn on table 2. They were uniformly sampled one hundred times over their domains. Then they were made noisy by adding a gaussian noise N (0, σ 2 ) produced by use of Box-Muller method [START_REF] Box | A note on the generation of random normal deviates[END_REF]. The different values of σ for f 1 , . . . , f 6 and w were respectively 0.005, 0.005, 0.005, 0.01, 0.02, 0.005 and 0.005.

Results and discussion

Table 3 shows the results of the four methods applied to the six noisy functions f 1 , . . . , f 6 , w and the handwritten curves "@", "blue", "love", "key" and treble clef. In table 3, d i ,α i , ρ respectively denote the norm of the i th derivative of L, the angular variation of the i th derivative of L, the unsigned curvature of L. For those 12 curves, the applicant pool of criteria to blend from were euclidean norms of derivatives from order one up to order five, angular variations from order one to five, plus the curvature function. Widening the range of derivatives would yield more accuracy but would be too time-consuming. Eventually, order five was an acceptable trade-off between time and accuracy. Since the raw curves are noisy, the criteria have been extracted from smoothed versions of the initial curves. This was made with a band diagonal matrix. In the following, we compare the four methods in terms of relative errors then in terms of CPU time, Table 3 summarizes the comparison accuracy results of our algorithm, the descent method, the particle swarm algorithm and Park algorithm. The third column in these tables shows which combination of criteria produced the best inner-knot vector for our algorithm. Table 3 points out that for the considered examples, the proposed algorithm gives the best results.

The comparison against PSO algorithm shows that the results are slightly better with handwritten words and twice as better with the synthetic curves. The PSO algorithm yields goods results either the curves are synthetic or handwritten. Unfortunately, it sometimes crashes because likely the Schoenberg-Whitney condition is not satisfied. Let us note that it is hard to compare deterministic methods with stochastic methods. How many attempts should we do with the latter ? If we allow more attempts with stochastic methods then deterministic methods should be allowed more evaluations too.

The comparison against Gradient descent shows that the results are similar with handwritten words and twice as better with the synthetic curves. The experimental results show that Gradient descent sometimes misses the global minimum and converges to a local minimum. For example see the curve f 1 (fig. 1).

The comparison against Park algorithm shows that the improvement factor accuracy is ranging from 2 to 10. Following the authors advice, the value of r which measures the compromise between curvature and length was set to 0.8 for all examples. Though the authors also suggested to set the number of dominant seed points to 10, we set it to 4 or 6 since small values usually yielded more accurate results. Let us notice that Park method is very sensitive to the number of dominant seed points. The error magnitude can grow by a factor of two when choosing a wrong number of seed points. For example, the experimental results show that with the curve @, when m = 9 and r = 0.8, the error grows from 0.0137 to 0.268 when the seed number varies from 4 to 8. As a general rule, with Park method, one must take more internal knots to achieve the same error level as our method does.

To make a fair comparison of the four methods, we allow them to make the same amount of computations of the value of the fitness function F which is, by far, the most CPU time-costly step of each algorithm. For the above tests, the number of computations of F was always set between 200 and 700. That is, this is meaningful for Gradient descent, PSO and our algorithm which are time-consuming methods but not for Park method which does not require an evaluation of F and therefore is instantaneous compared to the three others (see table 4).

Table 4 and figure 10 show that Park method is the fastest method. As it was predictable, PSO algorithm is the slowest method. Our algorithm and Gradient descent show similar results. Park method is instantaneous but if one needs several attempts to manually tune the values of parameters and number of seed points, one looses all the benefit of the immediacy of the method. Obviously, our algorithm is slower than Park method since it exhaustively and automatically examines combinations of many geometric criteria.

In both algorithms, the computed knots depend on local geometric properties of the considered data curve. Park method empirically selects two criteria : length and curvature. On the contrary, our algorithm tries all combinations of many geometrical criteria which include the norm of the first derivative (which is correlated to length) and curvature. That means that Park algorithm can be seen as a particular case of our algorithm. One can note that Park method is not well suited to discontinuous curves as shown on curves f 2 (see fig. 2) and f 6 (see fig. 6), neither to non smooth functions as shown in f 3 (see fig. 3). This could be explained by the fact that Park method is based on a discrete model, therefore it does not take into account the information in between two consecutive data points. It also means that the algorithm depends on the sample rate since the increase of it would reduce this problem. For the same reason, the accuracy of Park method is very sensitive to the number of seed points. The sensitivity to the sample rate is less obvious in our method since our model is continuous. Since our algorithm distorts an initially uniform distribution, it is very unlikely that it would result in an ill-conditioned inner-knot vector. This robust behaviour shows as well with gradient-descent method and Park method but not with PSO algorithm. Our algorithm does not need to manually set any parameters. It automatically selects the best parameters when constructing the function v. The main feature of our algorithm is that it never degenerates : It is always reliable whereas the three others are not.

Conclusion

In this paper, we propose an heuristic method for knot placement in B-spline approximation. It is based on blending geometric characteristics of the data. It has been successfully compared to three other methods 11 : Gradient descent, Particle swarm optimization and Park method. The proposed algorithm gives good accuracy results and it is more reliable than the other ones. Unlike the other ones, it is self contained since it does not need to set any parameters. The experimental results show that our algorithm is faster that PSO algorithm, but much slower than Park algorithm which is instantaneous. However, the latter needs more internal knots to achieve the same accuracy as our algorithm does. The main idea has been validated by the tests but some steps of the algorithm might be improved. Our algorithm is sensitive to noise and how to select the relevant criteria is the main issue. We are a bit blind about which criterion to use and which one to rule out. In further work, we shall try to use principal component analysis to see if any general rule appears about the criteria. We also wish to generalize this method to surface approximation.

Figure 1 :

 1 Figure 1: Approximation curves (red) of the original noisy curve f1 (blue) produced by the different methods. From left to right : Knot placement of 4 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 2 :

 2 Figure 2: Approximation curves (red) of the original noisy curve f 2 (blue) produced by the different methods. From left to right : Knot placement of 10 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 3 :

 3 Figure 3: Approximation curves (red) of the original noisy curve f3 (blue) produced by the different methods. From left to right : Knot placement of 5 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 4 :

 4 Figure 4: Approximation curves (red) of the original noisy curve f 4 (blue) produced by the different methods. From left to right : Knot placement of 8 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 5 :

 5 Figure 5: Approximation curves (red) of the original noisy curve f5 (blue) produced by the different methods. From left to right : Knot placement of 6 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 6 :

 6 Figure 6: Approximation curves (red) of the original noisy curve f6 (blue) produced by the different methods. From left to right : Knot placement of 6 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 7 :

 7 Figure 7: Approximation curves (red) of the original curve blue (blue) produced by the different methods. From left to right : Knot placement of 22 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 8 :

 8 Figure 8: Approximation curves (red) of the original curve love (blue) produced by the different methods. From left to right : Knot placement of 22 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 9 :

 9 Figure 9: Approximation curves (red) of the original curve trebleclef (blue) produced by the different methods. From left to right : Knot placement of 10 inner knots using Distribution function V (x), Our algorithm, Gradient descent, PSO algorithm, Park algorithm

Figure 10 :

 10 Figure 10: (left) CPU time and (right) Relative errors results of the four methods for the twelve curves.

Table 2 :

 2 Curves corresponding to the six functions f1, . . . , f6

Table 4 :

 4 CPU time results of our algorithm, gradient descent, PSO algorithm and Park method for the six noisy functions, w and the handwritten words "@", "blue", "love", "key" and treble clef.

	curve	m		CPU time (in s.)
			our algorithm	gradient descent	PSO algorithm[1]	Park algorithm[2]
	f 1	7	2.65	2.45	5.2	0.017
	f 2	13	2.9	2.8	6.2	0.02
	f 3	8	2.7	2.2	4.9	0.028
	f4	11	3.3	1.8	6.21	0.031
	f5	9	2.88	2.67	5.84	0.015
	f 6	9	2.6	2.52	5.87	0.017
	w	17	3.97	3.77	7.51	0.037
	@	12	7.45	6.84	10.8	0.076
	blue	25	38.9	36.1	48	0.93
	love	25	30	28	39	0.5
	key	23	10	9.3	15.7	0.093
	treble clef	13	12	11	16	0.27


Jerome Filippi

Amandine Rubio

Virgine Lasserre

Jean Maccario

Stephanie Walrand

Nathalie Neveux

Servane Le Plénier

Xavier Hébuterne

Luc Cynober

Christophe Moinard
email: christophe.moinard@univ-grenoble-alpes.fr

Dose-dependent beneficial effects of citrulline supplementation in short bowel syndrome in rats

Keywords:

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

Introduction

Short bowel syndrome (SBS) occurs when there is insufficient length of small intestine to maintain adequate protein-energy, fluid, electrolyte, or micronutrient balance without parenteral support [START_REF] Pironi | Definitions of intestinal failure and the short bowel syndrome[END_REF]. SBS most frequently occurs following extensive surgical resection of the intestine, congenital defect, or disease-associated loss of absorption. Current lines of research involve the development of pharmacological treatments and/or the use of specific nutrients with pharmacological properties to potentiate post-resectional gut adaptation and reduce parenteral nutrition requirements [START_REF] Jeppesen | New approaches to the treatments of short bowel syndrome-associated intestinal failure[END_REF].

Citrulline is a non-protein amino acid synthesized for systemic use almost exclusively by the enterocytes of the proximal small bowel. It escapes splanchnic extraction.

Citrulline is mainly metabolized by the kidney and converted into arginine [START_REF] Bahri | Citrulline: from metabolism to therapeutic use[END_REF][START_REF] Moinard | Citrulline: a new player in the control of nitrogen homeostasis[END_REF][START_REF] Papadia | Citrulline in health and disease. Review on human studies[END_REF][START_REF] Breuillard | Citrulline and nitrogen homeostasis: an overview[END_REF][START_REF] Cynober | The 2009 ESPEN Sir David Cuthbertson. Citrulline: a new major signaling molecule or just another player in the pharmaconutrition game?[END_REF]. As there are profound impairments of arginine metabolism in SBS [START_REF] Osowska | Impairment of arginine metabolism in rats after massive intestinal resection: effect of parenteral nutrition supplemented with citrulline compared with arginine[END_REF][START_REF] Wakabayashi | Arginine becomes an essential amino acid after massive resection of rat small intestine[END_REF], citrulline makes a good candidate to generate arginine and improve nutritional status in SBS. However, to date, there is only one clinical study of citrulline supplementation in SBS patients [START_REF] Jirka | Effect of oral citrulline supplementation on whole body protein metabolism in adult patients with short bowel syndrome: A pilot, randomized, double-blind, cross-over study[END_REF]. In 9 SBS patients with good nutritional status in the late phase of intestinal adaptation-which is likely too late-and with near-normal baseline citrulline homeostasis, a 7-day oral supplementation with citrulline (0.18 g . .kg .-1 day -1 ) enhanced citrulline and arginine bioavailability but had no anabolic effect on whole-body protein metabolism. However, the experimental data and this clinical study are not necessarily contradictory. Indeed, work carried out by Jourdan et al. [START_REF] Jourdan | Citrulline stimulates muscle protein synthesis in the post-absorptive state in healthy people fed a low-protein diet -A pilot study[END_REF] in healthy subjects found that citrulline intake significantly increases muscle protein synthesis but does not affect protein synthesis at whole-body level.

Likewise, Kuçi et al [START_REF] Kuci | Citrulline and muscle protein homeostasis in three different models of hypercatabolism[END_REF] showed that in critically-ill rats, citrulline effect varies according to type of injury and to the skeletal muscle under study, with a limited effect at whole-body level. This result connects with the findings of Osowska et al. [START_REF] Osowska | Citrulline modulates muscle protein metabolism in old malnourished rats[END_REF], who showed that old malnourished rats re-fed with a citrulline-enriched diet had higher muscle protein synthesis and lower hepatic protein synthesis than control animals. These data may explain the lack of citrulline effect at whole-body level. Other hypotheses proposed by Jirka et al [START_REF] Jirka | Effect of oral citrulline supplementation on whole body protein metabolism in adult patients with short bowel syndrome: A pilot, randomized, double-blind, cross-over study[END_REF] to explain their disappointing results included the patients' good nutritional status, the late phase of intestinal adaptation, and the normal baseline citrulline level.

To date, there is no data available on the most appropriate therapeutic range and the dose-response relationship of CIT in the acute-phase SBS population. Here, to address this gap, we used a model of intestinal insufficiency mimicking SBS disease, i.e. gut-resected rats, to characterize the action of citrulline in a dose-ranging study design, focusing on muscle nitrogen homeostasis.

Materials and Methods

Animals:

Male Wistar rats (n=46) (230-250g, 2-month-old) were used. Before surgery, the rats were acclimatized for five days in metabolic cages with free access to standard laboratory chow (Safe, Epinay-sur-Orge, France) and water.

Surgical procedures:

As previously described [START_REF] Osowska | Citrulline increases arginine pools and restores nitrogen balance after massive intestinal resection[END_REF], the rats were fasted for 12 hours before surgery, then anesthetized by isoflurane inhalation (3% isoflurane) using a regulated airflow apparatus (Minerve, Esternay, France). Enterectomy was performed on rats in the citrulline and control groups (see below for definition) by removing 80% of the small intestine, leaving 10% of the proximal jejunum and 10% of the distal ileum. Eight rats were sham-operated (transection).

For both resected and transected animals, continuity was restored with an end-to-end anastomosis using a single running silk 6-0 suture. The gastrostomy tube was placed immediately following intestinal resection or transection. A silicone tube (Tygon, size 0.51 mm; Fisher Bioblock Scientific, Illkirch, France) was introduced into the lumen of the stomach and the remaining end of the catheter was tunneled subcutaneously to the back of the neck and attached to a spring coil-swivel mechanism (Harvard Apparatus, Les Ulis, France) allowing the rat to move freely in the cage. Before the surgical procedure, all the rats received a painkiller (Temgesic ® ; Schering-Plough) subcutaneously at a dose of 0.05 mg/kg of body weight.

Postoperative care:

Recovery period (D-5 to D0): Rats were housed individually in metabolic cages and allowed a five-day recovery period. On the first two days they had free access to a 5% glucose solution. On day 3, glucose was withdrawn, and the rats had access to standard laboratory chow (UAR A04, Protein 16%, Arginine content: 9.8g.kg -1 Dietex, France) and water. On day 4, enteral nutrition was introduced at a flowrate of 0.5 mL/h using a Harvard infusion pump (pump 11; Harvard Apparatus), and the rats also had free access to chow and water.

Flowrate was increased gradually to a maximum on day 5 corresponding to an intake of 2 g N kg .-1 day -1 [START_REF] Bahri | Citrulline: from metabolism to therapeutic use[END_REF]. From day 5 until the end of the experiment, rats received only enteral nutrition and had free access to water.

Experimental groups:

The rats were randomly assigned to one of the following groups (from D0 to D10):

• The citrulline groups (CITRULLINE) consisted of rats receiving standard enteral nutrition (Sondalis; Nestlé Health Science, 1 kcal.mL -1 and 4.5 mg Arg-bound protein .mL -1 ) supplemented with citrulline at different doses: o 0.5 g . .kg .-1 day -1 (n=9) o 1.0 g . .kg .-1 day -1 (n=7) o 2.5 g . .kg .-1 day -1 (n=8) o 5.0 g . .kg .-1 day -1 (n=8)

• The control group (n = 6) and the sham group (n = 6) consisted of rats receiving standard enteral nutrition.

To keep all groups isonitrogenous (2 g N.kg .-1 day -1 ) and isocaloric (200 kcal kg .-1 day -1 ) to the CITRULLINE (5 g kg .-1 day -1 ) group, standard enteral nutrition was supplemented with a mixture of amino acids consisting of alanine, asparagine, glycine, serine, histidine, and proline in equimolar amounts (see Table 1, supplementary data).

Rats were weighed daily. Total enteral nutrition was administered for 10 days. Enteral nutrition was stopped two hours before decapitation.

Tissue removal:

Blood Blood was sampled in heparinized tubes and then swiftly centrifuged.

Liver

The abdominal cavity was opened. The liver was immediately removed and weighed, and a sample was cut off, frozen in liquid nitrogen, and stored at -80°C until analysis.

Jejunum and ileum

The intestinal mucosa was washed with cold NaCl (0.9%), reverted and scraped, rapidly frozen in liquid nitrogen, and stored at -80°C until analysis.

Muscles

Five muscles of the hindlimbs (extensor digitorum longus (EDL), gastrocnemius, soleus, plantaris, and tibialis anterior) were rapidly removed, weighed, frozen in liquid nitrogen, and stored at -80°C until analysis. These five muscles were selected because they differ widely in their function and fiber type and show different metabolic responses to stress [START_REF] Hasselgren | Protein metabolism in different types of skeletal muscle during early and late sepsis in rats[END_REF].

Parameters studied and analytical methods:

2.6.1. Amino acid concentrations in plasma and EDL muscle As previously described [START_REF] Osowska | Citrulline increases arginine pools and restores nitrogen balance after massive intestinal resection[END_REF], plasma was deproteinized with a 30% (w/v) sulfosalicylic acid solution. Supernatants were stored at -80°C until analysis. Tissues were ground and deproteinized with a 10% trichloroacetic acid solution containing 0.5 mM EDTA.

Supernatants were stored at -80°C until analysis. Amino acids were measured by ion exchange chromatography using an amino acid autoanalyzer. The results of our participation in the European Quality Control Scheme (ERNDIM) confirm the accuracy of our amino acid determinations.

Fractional synthesis rate of proteins

Just before euthanasia, each rat received a subcutaneous injection of a large dose of L-[ 13 C]valine (99 atom%, 300 µmol/100 g, Cambridge Isotope Laboratories, Andover, MA) to flood the protein synthesis precursor pool. The tracer incorporation time within the groups was different for each rat. In each group, a rat was killed at 20, 25, 30, 35, 40, 45 or 50 min after the tracer injection. Thus, the same kinetics of incorporation from 20 to 50 min was performed for each group of animals. We used the kinetics of tracer incorporation in the mitochondrial proteins, myosin and actin to calculate their synthesis rates (see the equation below). A 200 mg piece of tibialis anterior was used for isolation of mitochondrial proteins, myosin and actin, as previously described [START_REF] Ouhabi | Mitochondrial ATP synthesis in permeabilized cells: assessment of the ATP/O values in situ[END_REF]. L-[ 13 C]-valine enrichment in hydrolyzed proteins was measured using a GC-C-IRMS system (µGas System, Fisons Instruments, VG Isotech, Middlewich, UK). Amino acids in the tissue fluid were derivatized, and valine enrichments were used as precursor pool to calculate fractional synthesis rates (FSR) as previously described [START_REF] Osowska | Citrulline modulates muscle protein metabolism in old malnourished rats[END_REF] (Supplementary data, table 2). Another set of four rats per regimen was used to determine basal isotopic abundance in mitochondrial proteins. The FSR of mitochondrial proteins, actin and myosin was calculated using the following equation: FSR = (Ei x 100) /(Eprec x t), where Ei is the enrichment as atom percent excess of 13C derived from combustion of valine from proteins at time t (minus basal enrichment), Eprec is the mean enrichment in the precursor pool (tissue fluid [13C] valine), and t is the incorporation time in hours [START_REF] Osowska | Citrulline modulates muscle protein metabolism in old malnourished rats[END_REF]. Data are expressed as %/h.

Preparation of muscle lysates

Frozen muscles (tibialis) were ground in liquid nitrogen. Powdered muscles were weighed and homogenized in 10 volumes of buffer containing 150mM NaCl, 10mM Tris-HCl pH7.4, 5mM EDTA, 1% NP40, phosphatase inhibitors (20mM sodium fluoride, 1mM sodium pyrophosphate, 25mM sodium glycerophosphate, 1mM sodium vanadate), and protease inhibitors (complete protease inhibitor cocktail, Roche, Meylan, France). The samples were continuously homogenized (1 hour, 4°C) and then centrifuged at 1000 g for 30 min at 4°C. Supernatants were aliquoted and stored at -80°C until analysis. Protein concentrations were determined using the bicinchoninic acid method (BC Assay UP 40840A kit Uptima, Interchim, France) 2.6.4 Immunoblotting: determination of the phosphorylation state of the principal actors in mTORC1 Total protein (50 µg) in SDS sample buffer (Laemmli Sample Buffer, Biorad) was heated to 95°C, and the proteins were separated by SDS-PAGE using a denaturing gel. After electrophoresis, proteins were transferred to nitrocellulose (GE-Healthcare) and then blocked for 1 h with a blocking buffer (10mM Tris-HCl pH8.0, 150mM NaCl, 0.05% Tween20, 5% nonfat dry milk). Nitrocellulose blots were incubated overnight at 4°C with primary antibodies that recognize phosphorylated forms. After washing, membranes were incubated with horseradish peroxidase-conjugated secondary antibodies for 1 h in TBS/T under gentle shaking. The blots were developed using the enhanced chemiluminescence detection system (GE Healthcare) according to the manufacturer's protocol. Films were scanned and quantified using the Bio Imaging System (Syngene). After quantification of the relative intensity of the phosphorylated forms, membranes were stripped using a stripping buffer (0.5% acetic acid) and then probed with primary antibodies that recognize total forms.

Statistical analysis

Data are expressed as means ± SEM. Comparisons between sets of data were performed using one-way analysis of variance (ANOVA) followed by the Duncan test (all analysis were performed on parameters measured at euthanasia and are statistically independent). When statistical analysis did not show a dose-dependent effect of citrulline, we analyzed the global effect of citrulline supplementation versus no citrulline, results of all supplemented rats were pooled and compared to results of unsupplemented resected rats (controls).

Statistical analyses (ANOVA, Duncan test, correlation, quadratic analysis, linear regression etc.) were performed using Statview software (Cary, NC). Threshold of significance was set at p<0.05.

Results

Body weight

There was no linear relationship between citrulline dose and body weight gain at day 10 (Wd10). However, a polynomial regression model evidenced a significant relationship between citrulline supplementation dose and body weight (Figure 1): W(d10) =78.2+11.9(dose)-2(dose^2) +0.6W(d0); p=0.02. Furthermore, a significant linear relationship was found when the citrulline dose of 5 g/kg/d was excluded: W(d10) =60.8+7.5(dose)+0.7W(d0), p=0.004.

Liver weight

Liver weight was lower in the resected animals compared to the sham group (sham vs all resected rats: 8.95 g vs 7.65 g, p=0.038). This difference was not corrected by citrulline supplementation at any dose (Table 1).

Muscle weight

Intestinal resection led to a significant lowering in soleus muscle mass and protein content (Table 1) which was partly restored after citrulline supplementation: soleus weight in unsupplemented resected rats (controls) vs all citrulline groups: p=0.029. However, we were unable to find a clear relationship between citrulline dose and muscle mass for any muscle (EDL, tibialis, soleus, plantaris anterior, gastrocnemius) or between citrulline supplementation and protein content of EDL and soleus muscles (in absolute weight or in percent of muscle mass). This was mainly due to major inter-individual variability in the group of rats receiving the highest dose of citrulline. A linear regression model using the plasma citrulline logarithm showed a 20% variation in tibialis anterior weight according to plasma citrulline: W= 0.05log(plasma citrulline)+0.033; p=0.02 (Figure 2).

Intestinal adaptation

The variation of jejunal and ileum weight according to citrulline load is presented in Table 1.

The variation of jejunal weight was significantly related to plasma citrulline: W=0.05log(plasma citrulline)+97.12; p=0.035. However, there was no significant association between ileum weight and plasma citrulline, and no significant association between mucosal protein content of the intestines and citrulline dose or plasma concentrations.

Amino acid concentrations in plasma and muscles

Plasma amino acid concentrations:

Intestinal resection resulted in a lowering of plasma citrulline compared with the sham group (control vs sham: 49 vs 75 µmol/L, p=0.002) and did not have any significant influence on any of the other AA concentrations. Plasma concentrations of citrulline, arginine and ornithine rose dose-dependently with citrulline supplementation (Table 2).

Muscles amino acid concentrations:

EDL levels of citrulline, arginine and ornithine tend to be lowering on average by 21-27% following intestinal resection (control vs sham, p=0.07-0.08). Citrulline supplementation significantly raised levels of citrulline, arginine and ornithine in the EDL muscle in a dosedependent manner (p<0.001, Table 2). Likewise, we found highly significant positive associations between plasma citrulline and EDL levels of citrulline (r=0.96, p<0.001), arginine (r=0.73, p<0.001) and ornithine (r=0.79, p<0.001) (Figure 3).

Evaluation of muscle protein synthesis

Protein FSR tended to rise after intestinal resection, but the difference was only significant for sarcoplasmic FSR (sham versus unsupplemented resected rats: 0.08%/h vs 0.12%/h, p=0.014) and myofibrillar FSR (0.1%/h vs 0.15%/h, p=0.028). FSR levels tended to return to baseline levels with citrulline supplementation (myofibrillar FSR: control versus all citrulline-supplemented groups: p=0.018; p>0.05 for total FSR, mitochondrial FSR and sarcoplasmic FSR) (Figure 4). No dose-dependent effect of citrulline were observed.

Evaluation of mTORC1 pathway activation

Intestinal resection did not significantly modify the amounts of phosphorylated forms of AKT, 4E-BP1 and eIF-4E, S6K1 or S6 ribosomal protein (S6RP) in the tibialis. However, citrulline supplementation led to a significant lowering in the activated forms of S6K1 (p=0.003) and S6RP (p=0.003), regardless of citrulline dose (Figure 5). Furthermore, we found a significant positive association between myofibrillar FSR and S6K1 (r=0.614, p=0.02) and S6RP (r=0.601, p=0.023). No dose-dependent effect of citrulline were observed.

Discussion

Citrulline is renowned for its homeostatic effect on nitrogen balance, and recent literature has consistently shown that this effect was directed towards skeletal muscles [START_REF] Jourdan | Citrulline stimulates muscle protein synthesis in the post-absorptive state in healthy people fed a low-protein diet -A pilot study[END_REF][START_REF] Osowska | Citrulline modulates muscle protein metabolism in old malnourished rats[END_REF][START_REF] Bouillanne | Impact of 3-week citrulline supplementation on postprandial protein metabolism in malnourished older patients: The Ciproage randomized controlled trial[END_REF][START_REF] Goron | Synergistic effects of citrulline supplementation and exercise on performance in male rats: evidence for implication of protein and energy metabolisms[END_REF][START_REF] Ham | L-citrulline protects skeletal muscle cells from cachectic stimuli through an iNOS-dependent mechanism[END_REF][START_REF] Buckinx | Effect of high-intensity interval training combined with L-citrulline supplementation on functional capacities and muscle function in dynapenic-obese older adults[END_REF], by activation of the mTORC1/PI3K/MAPK pathway [START_REF] Plenier | Citrulline directly modulates muscle protein synthesis via the PI3K/MAPK/4E-BP1 pathway in a malnourished state: evidence from in vivo, ex vivo, and in vitro studies[END_REF] and by modulation of muscle energy metabolism [START_REF] Goron | Citrulline stimulates muscle protein synthesis, by reallocating ATP consumption to muscle protein synthesis[END_REF]. We thus focused on muscle nitrogen homeostasis in our animal model of SBS, and aimed to find a dose-ranging effect of citrulline supplementation.

In our study, citrulline supplementation generated dose-dependent rises in arginine and ornithine contents in both plasma and muscles, modulated weight, and enhanced intestinal adaptation. Furthermore, we demonstrated that this beneficial effect was dose-dependent, mainly from 0.5 to 2.5g kg .-1 day -1 of citrulline supplementation.

Citrulline had a specific effect on muscles, i.e. on muscle weight, AA content, thus corroborating the results of previous studies [START_REF] Jourdan | Citrulline stimulates muscle protein synthesis in the post-absorptive state in healthy people fed a low-protein diet -A pilot study[END_REF][START_REF] Osowska | Citrulline modulates muscle protein metabolism in old malnourished rats[END_REF][START_REF] Bouillanne | Impact of 3-week citrulline supplementation on postprandial protein metabolism in malnourished older patients: The Ciproage randomized controlled trial[END_REF][START_REF] Goron | Synergistic effects of citrulline supplementation and exercise on performance in male rats: evidence for implication of protein and energy metabolisms[END_REF][START_REF] Ham | L-citrulline protects skeletal muscle cells from cachectic stimuli through an iNOS-dependent mechanism[END_REF]. However, the action of citrulline on muscle FSR in our resected animals was the opposite of what was expected. First, intestinal resection resulted in a rise in skeletal-muscle protein FSR despite a decrease in muscle mass. Among the possible explanations for this FSR variation, we posit that it could reflect the onset of an intestinal adaptation process. Second, this FSR variation tended to be normalized by citrulline supplementation, in particular for myofibrillar and sarcoplasmic FSR. These results were not artifacts as they correlated with the mTOR pathway response to citrulline, with a major reduction in the activated phosphorylated forms of S6k1 and S6R concomitant with the lower values of FSR in citrulline-supplemented rats. We advance two explanations to account for these unexpected results:

1) Our experimental model used a continuous enteral feeding mode. It has been shown that continuous enteral feeding had a much less stimulatory effect on muscle and visceral protein synthesis than intermittent bolus feeding [START_REF] Gazzaneo | Intermittent bolus feeding has a greater stimulatory effect on protein synthesis in skeletal muscle than continuous feeding in neonatal pigs[END_REF][START_REF] El-Kadi | Viscera and muscle protein synthesis in neonatal pigs is increased more by intermittent bolus than by continuous feeding[END_REF]. Furthermore, [START_REF] Davis | Bolus vs. continuous feeding to optimize anabolism in neonates[END_REF] demonstrated that only intermittent bolus feeding could elicit a pulsatile pattern of amino acid-induced translation initiation, thereby promoting protein anabolism to a much greater extent than continuous enteral feeding [START_REF] Davis | Bolus vs. continuous feeding to optimize anabolism in neonates[END_REF]. Likewise, mTORC1 is mainly activated in the postprandial (bolus-style) state. 2) In patients with early-phase SBS, feeding fails to decrease proteolysis, in contrast to what is physiologically observed in healthy subjects. Note that the improvement in intestinal function and nutritional status achieved by rhGH treatment in SBS is mediated by the decrease in proteolysis in response to feeding [START_REF] Seguy | Growth hormone enhances fat-free mass and glutamine availability in patients with short-bowel syndrome: an ancillary double-blind, randomized crossover study[END_REF]. We hypothesize that the same could occur in work. The rise in muscle FSR after resection could reflect an increased protein turnover due to excess proteolysis, and the stimulatory effect of citrulline on muscle protein synthesis could, at least in part, be mediated by a decrease in proteolysis.

This potential effect of CIT on proteolysis was also hinted at in a proteomics study [START_REF] Faure | Citrulline enhances myofibrillar constituents expression of skeletal muscle and induces a switch in muscle energy metabolism in malnourished aged rats[END_REF].

Note that the systemic relation between protein synthesis and muscle mass is inconstant, as it was previously shown that CIT could increase protein synthesis without increasing muscle mass [START_REF] Goron | Synergistic effects of citrulline supplementation and exercise on performance in male rats: evidence for implication of protein and energy metabolisms[END_REF].

In conclusion, our results confirm that citrulline has potential benefit in SBS as it promotes body weight gain, preserves muscle trophicity, and enhances intestinal adaptation in our model of resected rats. Here we demonstrated that this beneficial effect was dose-dependent, with a maximal beneficial effect for a citrulline dose of 2.5 g/kg/day. Citrulline reversed the rise in protein synthesis in skeletal muscle induced by intestinal resection through the mTOR pathway, prompting the hypothesis that citrulline could reverse the post-resection modification in protein turnover. This new approach to the pharmacological properties of citrulline in SBS warrants further research, but we anticipate it could help clinicians develop valuable clinical studies in the early-adaptation phase of SBS.

Figure and table legends

  One of the authors (CM) is authorized by the French government (No. 75522) to use this experimental model of surgery. Animal care complied with French regulations for the protection of animals used for experimental and other scientific purposes (D 2001-486) and with European Community regulations (Official Journal of the European Community, L538 12:18:1986)..

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: A polynomial regression model on rat weight at D10 (W10) and citrulline dose administered in sham and resected rats. X-Axis: Citrulline dose (g.kg -1 .d -1 ), Y-Axis: Weight

  Figure 3

Table 1 :

 1 Organ weight and organ protein content in the sham group and in groups of resected and CIT-supplemented rats

Table 2 :

 2 Plasma and EDL amino acid contents in the sham group and in groups of resected and CIT-supplemented rats

Table 1 :

 1 Organ weight and organ protein content in the sham group and in groups of resected citrulline-supplemented rats (from 0 to 5g.kg -1 .d .-1 ) by enteral route during 10 days Values are means ± SEM. ANOVA + Tukey-Kramer test. Values with different superscripts are statistically different at p<0.05.

			Sham	Control	0.5 CIT	1 CIT	2.5 CIT	5 CIT
			(n=6)	(n=6)	(n=9)	(n=7)	(n=8)	(n=8)
	Soleus	Weight (mg)	106±5 a	90±3 b	101±3 a,c	96±7 c	103±13 a,c	104±4 a,c
		Protein (mg)	11.7±0.8 a 9.0±1.0 b 11.7±1.2 a 11.8±1.5 b 11.7±1.0 b	10.6±0.8 b
	EDL	Weight (mg)	110±4	106±4	104±3	104±6	103±3	109±6
		Protein (mg)	13.2±1.1	11.2±0.5	12.0±0.9	12.1±0.7	12.9±0.9	13.0±1.3
	Gastrocnemius	Weight (mg)	1175±54	1087±67	1113±36	1090±84	1119±37	1133±78
	Tibialis	Weight (mg)	450±20	417±19	426±13	422±28	437±15	473±39
	Plantaris	Weight (mg)	194±9	181±9	175±9	194±13	180±14	192±13
	Liver	Weight (g)	9.0±0.7 a	8.0±0.7 b	7.6±0.5 b	7.3±0.5 b	7.5±0.3 b	8.0±0.3 b
		Protein (mg)	NA	914±75	987±112	1095±212	955±86	1142±188
	Jejunum	Weight (mg/10 cm)	118±16	108±21	94±14	93±11	107±17	161±44
		Protein (mg/10 cm))	4.3±0.3	4.6±1.1	4.0±0.4	5.8±0.5	4.5±0.9	6.0±1.1
	Ileum	Weight (mg)	105±9	123±21	88±14	122±20	128±29	123±30
		Protein (mg)	63±10	53±10	54±9	74±9	58±8	54±14

Table 2 :

 2 Plasma and EDL amino acid contents (respectively in µmol/L and µmol/g) in the sham group and in groups of resected citrulline-supplemented rats (from 0 to 5g.kg -1 .d .-1 ) by enteral route during 10 days 2±0.02 a 0.16±0.04 a 0.4±0.08 b 0.66±0.16 c 2.67±0.31 d 9.14±0.73 e Ornithine 0.15±0.03 a 0.11±0.02 a 0.21±0.04 b 0.39±0.11 c 0.83±0.17 d 1.89±0.13 e arginine 0.06±0 a 0.05±0.01 a 0.07±0.02 a 0.09±0.02 a 0.18±0.03 b 0.58±0.14 c Values are means ± SEM. ANOVA + Tukey-Kramer test. Values with different superscripts are statistically different at p<0.05.

	Plasma	Sham	Control	0.5 CIT	1 CIT	2.5 CIT	5 CIT
	µmol/L	(n=6)	(n=6)	(n=9)	(n=7)	(n=8)	(n=8)
	Citrulline	75±3 a	49±2 b	91±15 a	110±22 c	372±40 d	1265±192 e
	Ornithine	55±4 a	47±4 a	54±4 a	71±10 b	157±15 c	330±46 d
	Arginine	94±8 a	79±8 a	114±10 b	147±21 c	299±24 d	528±35 e
	EDL
	µmol/g
	Citrulline	0.

Acknowledgments

We thank Metaform-langues for the English proofreading.

Funding sources

This study received funding support from the French Ministry of Research and Technology (EA 4466).

CRediT author statement

Jérome

Filippi: Resources, Methodology, Investigation, Validation. Amandine Rubio: Data curation, Writing-Original draft preparation. Virginie Lasserre: Methodology, Data curation, Validation. Jean Maccario: Data curation, Validation. Stéphane Walrand: Methodology, Investigation, Validation. Nathalie Neveux: Methodology, Investigation, Validation. Servane Le Plenier: Methodology, Investigation, Validation. Xavier Hébuterne: Methodology, Conceptualization, Validation. Luc Cynober: Conceptualization, Methodology, Supervision, Project Administration, Funding Acquisition, Validation. Christophe Moinard: Conceptualization, Methodology, Supervision, Project Administration, Funding Acquisition, Writing-Reviewing and Editing, Validation.

Conflict of interest statement

JF, CM, SLP and LC are Citrage shareholders. Other authors have no conflict of interest.