Elena Bergamini

Pélagie Guillon

Valentina Camomilla
email: valentina.camomilla@uniroma4.it

Hélène Pillet

Wafa Skalli

Aurelio Cappozzo

Trunk Inclination Estimate During the Sprint Start Using an Inertial Measurement Unit: A Validation Study

Keywords: biomechanics, sport, motion analysis, Kalman filter, Micro-Electrical-Mechanical Systems

The proper execution of the sprint start is crucial in determining the performance during a sprint race. In this respect, when moving from the crouch to the upright position, trunk kinematics is a key element. The purpose of this study was to validate the use of a trunkmounted Inertial Measurement Unit (IMU) in estimating the trunk inclination and angular velocity in the sagittal plane during the sprint start. In-lab sprint starts were performed by five sprinters. The local acceleration and angular velocity components provided by the IMU were processed using an adaptive Kalman filter. The accuracy of the IMU inclination estimate and its consistency with trunk inclination were assessed using reference stereophotogrammetric measurements. A Bland-Altman analysis, carried out using parameters (minimum, maximum, and mean values) extracted from the time histories of the estimated variables, and curve similarity analysis (correlation coefficient > 0.99, Root Mean Square Difference < 7 deg) indicated the agreement between reference and IMU estimates, opening a promising scenario for an accurate in-field use of IMUs for sprint start performance assessment.

Introduction

An effective start is crucial for successful sprint running performance and directly affects 60 to 400 m sprint times. [START_REF] Čoh | The biomechanical model of the sprint start and block acceleration[END_REF][START_REF] Harland | Biomechanics of the sprint start[END_REF] To date, research on sprint start performance has mainly focused on the lower part of the body. In particular, a medium antero-posterior spacing between the feet on the blocks, 1,3 short reaction times, [START_REF] Čoh | The biomechanical model of the sprint start and block acceleration[END_REF][START_REF] Fortier | Starting block performance in sprinters: a statistical method for identifying discriminative parameters of the performance and an analysis of the effect of providing feedback over a 6-week period[END_REF][START_REF] Mero | Biomechanics of sprint running -a review[END_REF] and large forces, powers and impulses exerted on the blocks [START_REF] Kugler | Body position determines propulsive forces in accelerated running[END_REF][START_REF] Slawinski | Segment-interaction in sprint start: analysis of 3D angular velocity and kinetic energy in elite sprinters[END_REF][START_REF] Mero | Reaction-time and electromyographic activity during a sprint start[END_REF] were shown to benefit 10 and 20 m sprint times.

While the importance of the trunk movement during the sprint start has been acknowledged by expert coaches, [START_REF] Jones | Coaching sprinting: expert coaches' perception of race phases and technical constructs[END_REF] only a few studies have focused on the upper part of the body. [START_REF] Slawinski | Segment-interaction in sprint start: analysis of 3D angular velocity and kinetic energy in elite sprinters[END_REF][START_REF] Slawinski | Kinematic and kinetic comparisons of elite and well-trained sprinters during sprint start[END_REF][START_REF] Čoh | Kinematic and kinetic parameters of the sprint start and start acceleration model of top sprinters[END_REF][START_REF] Mero | A biomechanical study of the sprint start[END_REF][START_REF] Natta | Caractérisation des comportements posturo-cinétiques en sprint[END_REF] In particular, trunk inclination during the "set" position was found to correlate with times at 10 and 20 m, [START_REF] Čoh | Kinematic and kinetic parameters of the sprint start and start acceleration model of top sprinters[END_REF] while during the block start phase, from the "on your marks" to block clearing, [START_REF] Jones | Coaching sprinting: expert coaches' perception of race phases and technical constructs[END_REF] trunk angular velocity was reported to discriminate between high and medium level sprinters. [START_REF] Natta | Caractérisation des comportements posturo-cinétiques en sprint[END_REF] None of these studies, however, monitored the trunk during the pick-up phase, i.e. the phase ranging from block clearing to the upright position. [START_REF] Jones | Coaching sprinting: expert coaches' perception of race phases and technical constructs[END_REF] Such monitoring is allowed in-the-field by wearable Inertial Measurement Units (IMUs), embedding a three axes accelerometer and gyroscope. So far, IMUs have been used

for sprint running analysis to estimate foot-ground contact times [START_REF] Purcell | Use of accelerometers for detecting footground contact time during running[END_REF][START_REF] Bergamini | Estimation of temporal parameters during sprint running using a trunk-mounted inertial measurement unit[END_REF] and lower leg rotational kinematics, [START_REF] Channels | Determination of rotational kinematics of the lower leg during sprint running using accelerometers[END_REF] while trunk motion was assessed with IMUs only during daily-life activities or under slow flexion-extension, lateral bending and torsion. [START_REF] Faber | Optimal inertial sensor location for ambulatory measurement of trunk inclination[END_REF][START_REF] Wong | Trunk posture monitoring with inertial sensors[END_REF] The methodology used in these studies, however, cannot be extended to the sprint start for two reasons. First, the movements of the soft tissues separating the IMU from the skeleton, [START_REF] Forner-Cordero | Study of the motion artefacts of skin-mounted inertial sensors under different attachment conditions[END_REF] to be considered as an artifact, are larger during sprint running compared to daily-life activities, to the extent that they may jeopardize the outcome. [START_REF] Liu | A mechanical model to determine the influence of masses and mass distribution on the impact force during running[END_REF] Second, the accuracy of the IMU orientation estimate, based on sensor-fusion algorithms, depends on the relative importance given to the acceleration, which is more reliable during quasi-static phases, and to the angular velocity, prevailing during jerked phases. [START_REF] Sabatini | Quaternion-based extended Kalman filter for determining orientation by inertial and magnetic sensing[END_REF] This relative importance reverses when dealing with the sprint start as opposed to daily motor acts, requiring sensor-fusion algorithms to be specifically adapted and tested.

Within this framework and to supplement track and field coaches, the purpose of this study was to validate the use of a trunk-mounted IMU in estimating the trunk inclination and angular velocity during the sprint start. To this aim, an adapted sensor-fusion algorithm was tested and the reliability of the IMU estimates was assessed by comparison with stereophotogrammetric data.

Methods

Experimental set up and data acquisition

Five male sprinters (age: 23.8±0.8 y; mass: 72.4±3.8 kg; stature: 1.79±0.07 m) gave their informed consent to participate in the study, which received ethical approval. The athletes were currently competing over 100 or 200 m and their best times for 100 m ranged from 11.21 to 11.50 s. Each sprinter, wearing his running shoes without spikes, was asked to perform four sprint starts from the starting-blocks, individually set and embedded in the floor of an indoor laboratory (12 m length). The block start phase and the first three steps of each start were analyzed.

The athletes were equipped with an IMU (FreeSense, Sensorize Ltd, Italy) containing a 3D accelerometer and a 3D gyroscope (± 6 g and±500 deg•s -1 of full range, respectively; 100 samples•s -1 ) providing linear acceleration and angular velocity components, respectively along and about the axes of a unit-embedded frame (IMU local reference frame: L IMU ). The IMU was mounted so that its axes were parallel to the trunk anatomical axes (Figure 1). The unit data were transmitted via Bluetooth® to a laptop computer. Careful attention was paid to the fixation of the IMU on the lower back trunk (L2 level) to limit its oscillations relative to the skeleton. [START_REF] Forner-Cordero | Study of the motion artefacts of skin-mounted inertial sensors under different attachment conditions[END_REF] To this aim, an ad-hoc elastic belt was used along with a memory foam material placed between the paravertebral muscles and the IMU.

To validate the IMU estimates, a nine-camera stereophotogrammetric system (Vicon MX3, Oxford, UK, 200 samples•s -1 ) was used. Four retro-reflective markers were attached to the IMU and five on the subjects' trunk (Figure 1) to determine the orientation of the unit and of the trunk, which was considered to be rigid and described as a line joining shoulder and hip joint centers. [START_REF] Jones | Coaching sprinting: expert coaches' perception of race phases and technical constructs[END_REF] A synchronization task (sudden trunk flexion-extension from standing) was performed at the beginning of each trial.

Data analysis

To remove random noise, marker and IMU data were low-pass filtered using a 40points moving average filter (smooth function, loess method, Matlab ® , MathWorks, MA). A global inertial reference frame (G IMU ) was defined in a static phase aligning the x-axis of L IMU with the gravity vector (Figure 2). The orientation matrix of L IMU with respect to G IMU was computed through an adaptive Kalman filter, [START_REF] Jurman | Calibration and data fusion solution for the miniature attitude and heading reference system[END_REF] ad hoc designed to combine the information provided by the accelerometer and the gyroscope during the static and non-static phases of the analyzed motor task. The ratio between two noise sources associated to the two sensors (r gyro/acc ) was adaptively modified during the sprint start to suit the characteristics of the movement. In particular, a threshold (t) was defined for the difference between predicted and measured values of the system state, [START_REF] Jurman | Calibration and data fusion solution for the miniature attitude and heading reference system[END_REF] i.e. the IMU inclination in the sagittal plane.

Below t, r gyro/acc was set to use the accelerometer and the IMU inclination was computed through a quaternion based approach; 23 above t, r gyro/acc increased in favour of the gyroscope, and the unit inclination was estimated by integrating the angular velocity around the y-axis, with the initial conditions obtained by the accelerometer-based estimates. Ad hoc trials were performed to select the following initial values of the Kalman filter parameters: 24 t = 0.5 deg, and r gyro/acc equal to 1 and 100 deg•s•m -1 , below and above the threshold t, respectively.

To obtain reference values, a stereophotogrammetric local reference frame (L S ) was defined (Figure 2) and its orientation in the stereophotogrammetric global reference frame (G S ) was obtained in each instant of time. In order to compare the orientation of L S and L IMU in the same global reference frame, the latter had to be expressed with respect to G S . To this aim, the time-invariant rotation matrix relating the stereophotogrammetric and the IMU global reference frames was calculated, assuming that L S was coincident with L IMU at time zero (Figure 2). To compare the IMU orientation with the whole trunk orientation, a trunk anatomical frame (A TR ) was defined (Figure 1) and its orientation with respect to G S obtained in each instant of time. Finally, Tait-Bryant angles (axis mobile rotation sequence: yxz) were calculated with respect to G S , from the orientation of L IMU , L S and of the whole trunk A TR .

The rotation about the y-axis, referred to as angular displacement β, was further considered.

Four phases were identified using β: on your marks (OYM), transition (TNS), set (SET), and pick-up (PCK) (Figure 3). The average β values during OYM and SET (β OYM and β SET ), the variation from β OYM to β SET (Δβ), and the peak angular velocities during TNS and PCK (ω TNS and ω PCK ) were computed from each curve. The absolute difference between each parameter set, as obtained from L IMU and L S (IMU accuracy), and from L IMU and A TR (IMUtrunk consistency), was then evaluated (Δ IMU and Δ TR ).

Statistical analysis

To assess both IMU accuracy and IMU-trunk consistency, the β curve as obtained from L IMU was compared to those obtained from L S and A TR , respectively. To this aim, the difference between β OYM obtained from the L IMU and A TR was computed (β OFF = 18±4 deg) and proved to be consistent within subjects (P = .675). Thus, β OFF was removed from the trunk curves, and the Pearson's correlation coefficient (r) and the Root Mean Square Difference (RMSD) were computed to assess the curve similarity. RMSD was also expressed in percentage of Δβ (RMSD % ). The same coefficients were assessed for the PCK phase alone (r PCK , RMSD PCK , RMSD %PCK ), which is the most affected by inertial factors. After a normal distribution test (Shapiro-Wilk test), the effect of the factor athlete was verified on all parameters and absolute differences by a repeated-measures one-way ANOVA, and descriptive statistics (mean±Standard Deviation (SD)) was performed (SPSS Inc. 17.0, Chicago, IL, USA; alpha = 0.05). Moreover, to assess the agreement between IMU and stereophotogrammetry, Bland-Altman analysis of Δ IMU and Δ TR was performed, [START_REF] Bland | Agreement between methods of measurement with multiple observations per individual[END_REF][START_REF] Atkinson | Statistical methods for assessing measurement error (reliability) in variables relevant to sports medicine[END_REF] and the absence of heteroscedasticity [START_REF] Nevill | Assessing agreement between measurements recorded on a ratio scale in sports medicine and sports science[END_REF] was verified.

Results

For the IMU accuracy, β curves from L IMU and L S presented a RMSD % lower than 4±3 % and a high correlation (r) during the whole trial and the PCK phase alone (Table 1).

After removing β OFF , which was found to be different among athletes (P = .001), similar results were obtained for the IMU-trunk consistency (Table 1).

All parameters and absolute differences proved to be normally distributed. While all estimated parameters differed among athletes (P < .001), no statistical differences resulted for Δ IMU and Δ TR (P > .15). Δ IMU and Δ TR for the angular displacement parameters (β OYM , β SET , Δβ) were lower than 1 deg and 4 deg, respectively, while Estimated values of Δ IMU and Δ TR for ω PCK were found to be lower than 9 and 6 deg•s -1 , and decreased to 8 and 4 deg•s -1 for ω TNS (Table 2). Bland-Altman plots indicated a good agreement for both IMU accuracy and IMU-trunk consistency. Symmetry of the confidence interval of Δ IMU and Δ TR was observed for all estimated parameters, indicating that no systematic error was present. Moreover, no heteroscedasticity was found for all parameters included in Δ IMU and Δ TR (overall average correlation = 0.09±0.41).

Discussion

The reliability of a lower-trunk mounted IMU in estimating the trunk inclination and angular velocity during the sprint start was assessed. Although in the present study, athletes were forced to complete the task in a relatively short distance, the average peak angular velocity of the trunk (A TR : ω PICK-UP = 198±31 deg•s -1 ) was comparable with previous literature results [START_REF] Slawinski | Segment-interaction in sprint start: analysis of 3D angular velocity and kinetic energy in elite sprinters[END_REF][START_REF] Natta | Caractérisation des comportements posturo-cinétiques en sprint[END_REF] (170±60 deg•s -1 and 186±48 deg•s -1 ).

The accuracy of the estimate of the IMU rotation about one of its local axes was supported by the agreement between the unit and the reference estimates. Such agreement proves that the main limitations concerning the use of IMUs [START_REF] Forner-Cordero | Study of the motion artefacts of skin-mounted inertial sensors under different attachment conditions[END_REF][START_REF] Liu | A mechanical model to determine the influence of masses and mass distribution on the impact force during running[END_REF] do not have a detrimental effect on the unit inclination estimate, even during the most jerked phase of the sprint start. In particular, the efficacy of the implemented adaptive Kalman filter is attested by the strong curve similarity, [START_REF] Cohen | Statistical power for the behavioral sciences[END_REF] which also suggests that the site and method of the unit attachment were effective in limiting the effects of the soft tissue movements.

The IMU and trunk inclinations were in agreement during the whole trial and during the PCK phase alone. However, they presented an initial offset, β OFF , which had a negligible variability over repeated measurements, but was subject-dependent. This offset (18±4 deg) depends on the trunk model commonly used by coaches and adopted in the study, i.e. rigid trunk. After β OFF removal, Δ TR increased moving from the OYM (0±1 deg) to the SET phase (4±4 deg), indicating a change in the athletes' kyphosis and neck extension and further supporting the inadequateness of a rigid trunk model. On the one hand, these results suggest the importance of trunk movement in determining the sprint start performance given the key role of the spine muscles during explosive motor tasks, [START_REF] Bobbert | Effects of muscle strengthening on vertical jump height: a simulation study[END_REF] in terms of strength as well as of control. On the other hand, a comparison between the IMU orientation and the lower trunk, where the unit was located, would reasonably reduce, or even reset β OFF, and would be independent from changes in the upper spine. Indeed, strong correlations (r) obtained for all athletes and trials suggest that tracking only the lower trunk should not prevent describing the overall start strategy. Moreover, from a field point of view, the variation and rate of variation of the inclination are considered far more interesting than the relevant absolute values.

In conclusion, the study shows that a single IMU positioned on the lower back trunk provides reliable angular displacements and angular velocity in the sagittal plane during both the block start and the beginning of the pick-up phase of the sprint. The results support the possibility for coaches to be supplemented with a reliable and wearable instrument to collect information about the sprint start performance of their athletes directly in the field.

Figure 1 -

 1 Figure1-IMU, belt, memory foam material, and marker location. The markers placed on the IMU (LI, LS, RI, RS), trunk (C7) and pelvis, (RPSIS, LPSIS, RASIS, LASIS) are indicated, as well as the trunk anatomical reference frame, A TR : O: midpoint between LPSIS and RPSIS; x-axis: joining C7 and O, positive downward; z-axis: directed as the vector product between the x-vector and the vector LASIS-RASIS; y-axis: orthogonal to the x-z plane.

Figure 2 -

 2 Figure 2 -Local (L) and global (G) frames for the inertial unit (IMU) and the stereophotogrammetric system (S).S

Figure 3 -

 3 Figure 3 -Raw acceleration (A) and angular velocity (B) data, for a randomly chosen subject and trial, measured along and about the IMU local axes (L IMU , depicted in the bottom left corner of panel A: x axis (solid line), y axis (dashed line) and z axis (dotted line). Angular displacement β (C), obtained for the same subject and trial from the IMU local frame, L IMU , (solid line), from the stereophotogrammetric local frame, L S , (dashed line) and from the trunk anatomical reference frame, A TR , (dotted line). Static phases (OYM, SET, shaded intervals) and non-static phases (TNS, PCK, white intervals) are shown. The angular displacement parameters (β OYM , β SET and Δβ) are also indicated. β was considered to be zero when the unit was in a horizontal position; clockwise rotations correspond to positive angles.

Table 1 -

 1 Curve similarity analysis: mean±SD of the correlation coefficient (r), the Root Mean Square Difference (RMSD) and the RMSD expressed in percentage of Δβ (RMSD % ), computed to assess IMU accuracy and IMU-trunk consistency, relative to the whole task and to the PCK phase alone.

		IMU accuracy	IMU-trunk consistency
	r	0.994 ± 0.013	0.998 ± 0.002
	r PCK	0.995 ± 0.015	0.998 ± 0.001
	RMSD [deg]	3 ± 2	3 ± 2
	RMSD % [%]	4 ± 3	5 ± 3
	RMSD PCK [deg]	3 ± 3	3 ± 2
	RMSD %PCK [%]	5 ± 3	6 ± 4

Table 2 -

 2 Parameter analysis: descriptive statistics (mean ± SD) of each estimated parameter and of the absolute differences between the IMU and the stereophotogrammetric sets of parameters, for both IMU accuracy (Δ IMU ) and IMU-trunk consistency (Δ TR ).

		L S	L IMU	A TR	IMU accuracy	IMU-trunk consistency
					Δ IMU	Δ TR
	β OYM [deg]	-29 ± 6	-29 ± 6	-29 ± 6	1 ± 1	0 ± 1
	β SET [deg]	12 ± 7	12 ± 6	9 ± 6	1 ± 1	4 ± 4
	Δβ [deg]	43 ± 8	43 ± 7	39 ± 5	1 ± 1	4 ± 4
	ω TNS [deg/s]	98 ± 18 106 ± 19	93 ± 18	8 ± 4	4 ± 6
	ω PCK [deg/s]	-192 ± 29 -201 ± 25 -198 ± 31	9 ± 6	6 ± 13

Acknowledgements

The authors wish to acknowledge Dr. Marco Donati for his technical support in developing the Kalman filter, and John McCamley for his contribution to the refinement of the manuscript.

This research was supported by grants of the Università Italo-Francese (Call Galileo 2011, Call Vinci 2008, awarded to E. Bergamini) and of the LLP program Higher Education (Erasmus, awarded to P. Guillon). The financial support of the Department of Human Movement and Sport Sciences, University of Rome "Foro Italico" and of the Laboratoire de Biomécanique -Arts et Métiers ParisTech is also gratefully acknowledged.


Pierre-Olivier Goffard
email: pierreolivier.goffard@axa.fr.

Stéphane Loisel
email: stephane.loisel@univ-lyon1.fr.

Denys Pommeret
email: denys.pommeret@univ-amu.fr.

A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

Keywords: compound Poisson model, ultimate ruin probability, natural exponential families with quadratic variance functions, orthogonal polynomials, gamma series expansion, Laplace transform inversion

A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretisation step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.

Introduction

A non-life insurance company is assumed to be able to follow the financial reserves' evolution associated with one of its portfolios in continuous time. The number of claims until time t is assumed to be an homogeneous Poisson process {N t } t≥0 , with intensity β. The successive claim amounts (U i ) i∈N * , form a sequence of positive i.i.d. continuous random variables and independent of {N t } t≥0 , with density function f U and mean µ. The initial reserves are of amounts u ≥ 0, and the premium rate is constant and equal to p ≥ 0. The risk reserve process is therefore defined as

R t = u + pt - Nt i=1 U i ,
the associated claims surplus process is defined as S t = u -R t . In this work, we focus on the evaluation of ultimate ruin probabilities (or infinite-time ruin probabilities) defined as

ψ(u) = P inf t≥0 R t < 0|R 0 = u = P sup t≥0 S t > u|S 0 = 0 . (1.1)
This model is called a compound Poisson model (also known as Cramer-Lundberg ruin model) and has been widely studied in the risk theory literature. For general a background about ruin theory, we refer to [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF] and [START_REF] Asmussen | Ruin Probabilities[END_REF].

A useful technique in applied mathematics consists of determining a probability density function from the knowledge of its Laplace transform. We give here a brief review of the literature involving numerical inversion of Laplace transform and ruin probability approximations. In a few particular cases, the inversion of the Laplace transform associated with ruin probabilities is manageable analytically and leads to closed formula. But in most cases numerical methods are needed. The Laguerre method is an old established method based on the 1935 Tricomi-Widder Theorem. The recovered function takes the form of a sum of Laguerre functions derived through orthogonal projections. The numerical inversion of Laplace transform using Laguerre series has been originally described in [START_REF] Weeks | Numerical inversion of laplace transforms using laguerre functions[END_REF] and improved in [START_REF] Abate | On the laguerre method for numerically inverting laplace transforms[END_REF]. In the wake of Laguerre series method, we found attempts in the actuarial science literature to write probability density functions as sum of gamma densities. For instance the early work of Bowers [START_REF] Bowers | Expansion of probability density functions as a sum of gamma densities with applications in risk theory[END_REF] that gave rise to the so-called Beekman-Bowers approximation for the ultimate ruin probability, derived in [START_REF] Beekman | Ruin function approximation[END_REF]. The idea is to approximate the ultimate ruin probability by the survival function of a gamma distribution using moments fitting. Gamma series expansion has been employed in [START_REF] Taylor | Representation and explicit calculation of finite-time ruin probabilities[END_REF] and later in [START_REF] Albrecher | On a gamma series expansion for the time dependent probability of collective ruin[END_REF]. The authors highlight that it is useful to carry out both analytical calculations and numerical approximations. They show that the direct injection of the gamma series expression into integro-differential equations leads to reccurence relations between the expansion's coefficients and therefore characterize them. They focus on the finite-time ruin probability but the results are valid in the infinite-time case by letting the time t tend to infinity. We explain later that our method, within the frame of ruin probabilities approximation, is closely related to the Laguerre method and represents in fact an improvement. The numerical inversion via Fourier-series techniques (Fast Fourier Transform) received a great deal of interest. These techniques have been presented for instance in [START_REF] Abate | The fourier-series method for inverting transforms of probability distributions[END_REF] within a queueing theory setting. For an application within an actuarial framework, we refer to [START_REF] Embrechts | Some applications of the fast fourrier transform algorithm in insurance mathematics[END_REF] and [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF] Chapter 5 Section 5.5.

There is also a great body of literature dealing with Laplace transform inversion linked the Hausdorff moment problem. Probability density function are recovered from different kind of moments. The use of exponential moments and scaled Laplace transform is presented in [START_REF] Mnatsakanov | A note on recovering the distribution from exponential moments[END_REF] and has been performed for ruin probabilities computations in [START_REF] Mnatsakanov | Approximation of the ruin probability using the scaled laplace transform inversion[END_REF]. In the work of Gzyl et al [START_REF] Gzyl | Determination of the probability of ultimate ruin probability by maximum entropy applied to fractional moments[END_REF], the maximum entropy applied to fractional exponential moments is employed to determine the probability of ultimate ruin. Recently, Albrecher et al. [START_REF] Albrecher | On the efficient evaluation of ruin probabilities for completely monotone claim distributions[END_REF] and Avram et al. [START_REF] Avram | On moments based padé approximations of ruin probabilities[END_REF] discuss different methods for computing the inverse Laplace transform. The first one consider a numerical inversion procedure based on a quadrature rule that uses as stepping stone a rational approximation of the exponential function in the complex plane. The second one implements and reviews much of the work done using Padé approximants to invert Laplace transform. There are several usual techniques for calculation of ultimate ruin probabilities. We want to mention a classical iterative method that we will use for comparison purposes. The so called Panjer's algorithm introduced in [START_REF] Panjer | Recursive evaluation of a family of compound distributions[END_REF], has been widely used in the actuarial field.

One can find an application to the computation of the probability of ultimate ruin in [START_REF] Dickson | A review of panjer's recursion formula and it's applications[END_REF].

The method that we propose here consists in an orthogonal projection of the defective probability density function, associated with the probability of ultimate ruin, with respect to a reference probability measure that belongs to the Natural Exponential Families with Quadratic Variance Function. The desired defective PDF takes the form of an infinite serie of orthonormal polynomials (orthogonal with respect to the aforementioned reference probability measure). The coefficients of the expansion are defined by a scalar product and are computed from the Laplace transform or equivalently from the moments of the distribution. Ruin probability approximations are obtained through truncation of the infinite serie followed by integration. This method permits the recovery of functions from the knowledge of their Laplace transform. Once the set of coefficients of the expansion has been evaluated, ultimate ruin probabilities can be approximated for any initial reserve. It is easy to implement, does no necessitate large computation time and is competitive in terms of accuracy. The approximation of the ruin probability allows manipulations such as integration or reinjection in formulas to derive approximations of distributions that governed other quantities of interest in ruin theory. For instance, the probability density function of the surplus prior to ruin involves the ultimate ruin probability, we refer to [START_REF] Dickson | On the distribution of the surplus prior to ruin[END_REF][START_REF] Gerber | On the time value of ruin[END_REF] for more details. This work is also a theoretical background in view of a future statistical application. Many papers deals with statistical estimation of ruin probabilities when observations of the claim sizes are available. The use of a Laplace inversion formula as basis for a nonparametric estimator has already been employed for ruin probabilities estimation in [START_REF] Mnatsakanov | Nonparametric estimation of ruin probabilities given a random sample of claims[END_REF][START_REF] Zhang | On a nonparametric estimator for ruin probability in the classical risk model[END_REF], scaled laplace transform and the maximum of entropy may offer the same possibility and would be based on empirical estimation of the moments. The definition of the coefficients in our method are based on quantities that are well adapted to empirical estimations. By pluging in the estimators of the coefficients, we will obtain a nonparametric estimator taking the form of an orthogonal serie. We aim to investigate ruin probability statistical estimation in future work, for now we only consider ruin probability approximations. In Section 2, we introduce a density expansion formula based on orthogonal projection within the frame of NEF-QVF. Our main results are developed in Section 3: the expansion for ultimate ruin probabilities is derived, a sufficient condition of applicability is given and the goodness of the approximation is discussed. Section 4 is devoted to numerical illustrations. Just like what is done in [START_REF] Gzyl | Determination of the probability of ultimate ruin probability by maximum entropy applied to fractional moments[END_REF], we compare our method to other existing method, namely Panjer's algorithm, Fast Fourrier Transform and scaled Laplace transform inversion.

Polynomial expansions of a probability density function

Let F = {P θ , θ ∈ Θ} with Θ ⊂ R be a Natural Exponential Family (NEF), see [START_REF] Barndorff-Nielsen | Information and exponential Families in Statistical Theory[END_REF], generated by a probability measure ν on R such that

P θ (X ∈ A) = A exp{xθ -κ(θ)}dν(x) = A f (x, θ)dν(x),
where

• A ⊂ R, • κ(θ) = log R e θx dν(x) is the Cumulant Generating Function (CGF),
• f (x, θ) is the density of P θ with respect to ν.

Let X be a random variable P θ distributed. We have

µ = E θ (X) = xdF θ (x) = κ ′ (θ), V(µ) = Var θ (X) = (x -µ) 2 dF θ (x) = κ ′′ (θ). The application θ → κ ′ (θ) is one to one. Its inverse function µ → h(µ) is defined on M = κ ′ (Θ).
With a slight change of notation, we can rewrite F = {P µ , µ ∈ M}, where P µ has mean µ and density f (x, µ) = exp{h(µ)x -κ(h(µ))} with respect to ν. A NEF has a Quadratic Variance Function (QVF) if there exists reals v 0 , v 1 , v 2 such that

V (µ) = v 0 + v 1 µ + v 2 µ 2 . (2.1)
The Natural Exponential Families with Quadratic Variance Function (NEF-QVF) include the normal, gamma, hyperbolic, Poisson, binomial and negative binomial distributions. Define

P n (x, µ) = V n (µ) ∂ n ∂µ n f (x, µ) /f (x, µ), (2.2)
for n ∈ N. Each P n (x, µ) is a polynomial of degree n in both µ and x. Moreover, if F is a NEF-QVF, {P n } n∈N is a family of orthogonal polynomials with respect to P µ in the sense that

< P n , P m >= P n (x, µ)P m (x, µ)dP µ (x) = δ nm ||P n || 2 , m, n ∈ N,
where δ mn is the Kronecker symbol equal to 1 if n = m and 0 otherwise. For the sake of simplicity, we choose ν = P µ 0 . Then, f (x, µ 0 ) = 1 and we write

P n (x) = P n (x, µ 0 ) = V n (µ 0 ) ∂ n ∂µ n f (x, µ) µ=µ 0 . (2.3)
We also consider in the rest of the paper a normalized version of the polynomials defined in (2.3) with Q n (x) = P n (x)/||P n ||. For an exhaustive review regarding NEF-QVF and their properties, we refer to [START_REF] Morris | Natural exponential families with quadratic variance functions[END_REF]. We will denote by L 2 (ν) the space of functions square integrable with respect to ν.

Proposition 1. Let ν be a probability measure that generates a NEF-QVF, with associated orthonormal polynomials {Q n } n∈N . Let X be a random variable with density function dP X dν with respect to ν. If dP X dν ∈ L 2 (ν) then we have the following expansion

dP X dν (x) = +∞ n=0 E(Q n (X))Q n (x). (2.4)
Proof. By construction {Q n } n∈N forms an orthonormal basis of L 2 (ν), and by orthogonal projection we get

dP X dν (x) = +∞ n=0 < Q n , dP X dν > Q n (x). It follows that < Q n , dP X dν > Q n (x) = Q n (y) dP X dν (y)dν(y) × Q n (x) = Q n (y)dP X (y) × Q n (x) = E(Q n (X))Q n (x).
Denote by f ν and f X the probability density functions of ν and X respectively. Proposition 1 gives an expansion of f X that takes the following simple form

f X (x) = +∞ n=0 a n Q n (x)f ν (x), (2.5)
where {a n } n∈N is a sequence of real number called coefficients of the expansion in the rest of the paper, {Q n } n∈N is an orthonormal sequence of polynomials with respect to ν, and f ν is the PDF of ν. The polynomials {Q n } n∈N are of degree n in x and can therefore be written as Q n (x) = n i=0 q i,n x i . Using this last remark, we rewrite the coefficients of the expansion as

a n = E(Q n (X)) = n i=1 q i,n E X i (2.6) = n i=1 q i,n (-1) i d i f X (s) ds i s=0 , (2.7)
where f X (s) = e -sx dP X (x) is the Laplace transform of the random variable X. The approximation of the PDF of X is simply obtained by truncation of the infinite serie in (2.5). We need to choose a NEF-QVF then a member of the choosen family characterized by its parameters. These choices are to be made wisely so as to ensure the validity of the expansion and to reach an acceptable level of accuracy that goes along with an order of truncation as small as possible. In the light of the expression of the coefficients (2.6), it seems natural to consider a statistical extension that would lead to a nonparametric estimator of the probability density function. However, it is of interest to start with the probabilistic problem as it is the theoretical basis for a statistical application. The next section shows how to use the orthogonal polynomials and NEF-QVF framework to approximate ruin probabilities.

3 Application to the ruin problem

General formula

The ultimate ruin probability in the Cramer-Lundberg ruin model is the survival function of a geometric compound distributed random variable

M = N i=1 U I i , (3.1)
where N is an integer valued random variable having a geometric distribution with parameter ρ = βµ p , and (U I i ) i∈N * is a sequence of independent and identically distributed nonnegative random variables having CDF F U I (x) = 1 µ x 0 F U (y)dy. The distribution of M has an atom at 0 with probability mass P (N = 0) = 1 -ρ. The probability measure that governs M is

dP M (x) = (1 -ρ) δ 0 (x) + dG M (x), (3.2)
where dG M is the continuous part of the probability measure associated to M which admits a defective probability density function with respect to the Lebesgue measure. We denote by g M the defective probability density function. The ultimate ruin probability is then obtained by integrating the continuous part as the discrete part vanishes

ψ(u) = P (M > u) = +∞ u dG M (x).
Theorem 1. Let ν be an univariate distribution having a probability density function with respect to the Lebesgue measure, and that generates a NEF-QVF. Let {Q n } n∈N be the sequence of orthonormal polynomials with respect to ν.

If dG M dν ∈ L 2 (ν) then ψ(u) = +∞ n=0 a n +∞ u Q n (x)dν(x), (3.3)
where {a n } n∈N is defined as in (2.7). Recall that a n = n i=1 q i,n (-1) i d i g M (s)

ds i s=0
and

Q n (x) = n i=0 q i,n x i . Proof.
We simply apply Proposition 1 to get the result.

Approximation with Laguerre polynomials

We derive an approximation for the ultimate ruin probability, using Theorem 1, combined with truncations of the infinite series (3.3). For K ∈ N, we will denote by

ψ K (u) = K n=0 a n +∞ u Q n (x)dν(x), (3.4)
the approximated ruin probability with truncation order K. In practice, as the distribution of M is supported on R + , we will choose the gamma distribution with mean parameter m and scale parameter r, that is:

dν(x) = f ν (x)1 R + (x)dλ(x) = x r-1 e -x/m Γ(r)m r 1 R + (x)dλ(x).
The associated orthogonal polynomials are the generalized Laguerre polynomials. By definition, they satisfy the following orthogonality condition

+∞ 0 L r-1 n (x)L r-1 m (x)x r-1 e -x dx = n + r -1 n δ nm .
The polynomials involved in the ruin probability approximation in (3.4) are the generalized Laguerre polynomials with a slight change in comparison to the definition given in [START_REF] Szegö | Orthogonal Polynomials[END_REF], namely

Q n (x) = (-1) n n + r -1 n -1/2 L r-1 n (x/µ). (3.5)
Remark 1. The Laguerre functions are defined in [START_REF] Abate | On the laguerre method for numerically inverting laplace transforms[END_REF] as

l n (x) = e -x/2 L n (x), x ≥ 0. (3.6)
The application of the Laguerre method consists in representing g M as a Laguerre serie

g M (x) = +∞ n=0 a n l n (t). (3.7)
One can note that the representation (3.7) is close to the expansion proposed in this paper, indeed it is exactly the same if we choose for our expansion r = 1 and m = 2. The difference lies in the possibility to change the parameter in our expansion. We will see later that the parametrization is of prime importance.

The defective probability density function associated to G M has the following expression

g M (x) = +∞ n=1 (1 -ρ)ρ n f * n U I (x). (3.8)
By Taking the Laplace transform of (3.8), we get

g M (s) = (1 -ρ)ρ f U I (s) 1 -ρ f U I (s) , (3.9)
with f U I (s) = e -sx f U I (x)dx is the Laplace transform of f U I . The Laplace transform of the claim size distribution appears in the formula. This fact limits the application to claim sizes distributions that admit a well defined Laplace transform, namely light-tailed distributions. We want to mention that this problem might be reconsidered once the study of the statistical extension will be done. In [START_REF] Mnatsakanov | A note on recovering the distribution from exponential moments[END_REF], approximations of the ruin probability in case of heavy tail claim amounts are derived from the associated nonparametric estimator computed with simulated data.

Integrability condition

We illustrate here how the applicability of the method is subject to the parametrization i.e. the choice of m and r. The parametrization permits in this problem to ensure the integrability condition. First we define the adjustment coefficient as the unique positive solution of the so-called Cramer-Lundberg equation

m U I (s) = 1 ρ , (3.10)
where

m U I (s) = +∞ 0 e sx f U i (x)dx is the moment generating function of U I . The integra- bility condition dG M dν ∈ L 2 (dν) is equivalent to +∞ 0 g M (x) 2 e -x/m x 1-r dx < ∞. (3.11)
In order to ensure this condition, we need the following results.

Theorem 2. Assume that U I admits a bounded density function and that the equation (3.10) admits a positive solution, then for all x ≥ 0 g M (x) ≤ C(s 0 )e -s 0 x , (3.12)

with s 0 ∈ [0, γ) and C(s 0 ) ≥ 0, where γ is the adjustment coefficient.

Proof. In order to prove the theorem we need the following lemma regarding the survival function F U of the claim sizes distribution.

Lemma 1. Let U be a non-negative random variable with bounded density function f U . Assume there exists s 0 > 0 such that m U (s 0 ) < +∞. Then there exists A(s 0 ) > 0 such that for all x ≥ 0 F U (x) ≤ A(s 0 )e -s 0 x . (3.13)

Proof. As m U (s 0 ) < +∞, we have

m U (s 0 ) -1 = +∞ 0 (e s 0 x -1)f U (x)dx = s 0 +∞ 0 x 0 e s 0 y f U (x)dydx = s 0 +∞ 0 e s 0 y F U (y)dy ≥ s 0 x 0 e s 0 y F U (y)dy ≥ F U (x)(e s 0 x -1).
thus, we deduce that ∀x ≥ 0

F U (x) ≤ ( m U (s 0 ) -1 + F U (x))e -s 0 x . (3.14)
The equation (3.10) is equivalent to

ρ m U (s) = 1 + sµ. (3.15)
The fact that γ is a solution of the equation (3.10) implies that m U (s) < +∞, ∀s 0 ∈ [0, γ) and by application of Lemma 1, we get the following inequality upon the density function of

U I f U I (x) = F U (x) µ ≤ B(s 0 )e -s 0 x . (3.16)
In view of (3.8), it is easily checked that g M satisfies the following defective renewal equation,

g M (x) = ρ(1 -ρ)f U I (x) + ρ x 0 f U I (x -y)g M (y)dy. (3.17)
We can therefore bound g M as in (3.12),

g M (x) ≤ ρ(1 -ρ)f U I (x) + +∞ 0 f U I (x -y)g M (y)dy ≤ ρ(1 -ρ)B(s 0 )e -s 0 x + B(s 0 )e -s 0 x +∞ 0 e s 0 y g M (y)dy = (ρ(1 -ρ) + g M (-s 0 ))B(s 0 )e -s 0 x = C(s 0 )e -s 0 x .
The application of Theorem 2 yields a sufficient condition in order to use the polynomial expansion.

Corollary 1. For 1 m < 2γ and r = 1, the integrability condition (3.11) is satisfied. We note the importance of the choice of the parameter m. The Laguerre method, briefly described in Remark 1, does not offer the possibility of changing the parameter. In the next subsection, we shed light on another key aspect of parametrization.

On the goodness of the approximation

The approximation is obtained through the truncation of an infinite serie. Obviously, the higher the order of truncation gets, the better the approximation is. Our goal is to work out an efficient numerical method that combines high accuracy and small computation time. We want to minimize the number of coefficients to compute so the coefficients' sequence must decrease as fast as possible. First, we assume that dG M dν ∈ L 2 (ν) in order to apply the method. Consequently

< dG M dν , dG M dν >= +∞ n=0 a 2 n < +∞, (3.18)
and the sequence of coefficients is approaching 0. The question is how fast is the decay?

To answer this question we use generating function theory, just like what is done in [START_REF] Abate | On the laguerre method for numerically inverting laplace transforms[END_REF].

We take the Laplace transform of g M defined as a polynomial expansion.

g M (s) = +∞ 0 e -sx +∞ n=0 a n Q n (x)f µ (x)dx = +∞ n=0 a n +∞ 0 e -sx Q n (x)f µ (x)dx.
The orthonormal polynomials {Q n } n∈N are Laguerre's one

Q n (x) = (-1) n n+r-1 n L r-1 n x µ , (3.19)
where

L r-1 n (x) = n i=0 n+r-1 n-i (-x) i
i! , we refer to [START_REF] Szegö | Orthogonal Polynomials[END_REF] for this last definition. We then have

g M (s) = +∞ n=0 a n +∞ 0 e -sx (-1) n n+r-1 n L r-1 n x m x r-1 e -x/m Γ(r)m r dx = +∞ n=0 a n (-1) n n+r-1 n +∞ 0 e -sx n i=0 n + r -1 n -i (-x) i i!m r x r-1 e -x/m Γ(r)m r dx = +∞ n=0 a n (-1) n n+r-1 n n i=0 (n + r -1)! (n -i)!(r + i -1)!(r -1)!i! (-1) i +∞ 0 e -sx x r+i-1 e -x/m m r+i dx = +∞ n=0 a n (-1) n n + r -1 n n i=0 n i (-1) i +∞ 0 e -sx x r+i-1 e -x/m m r+i Γ(r + i) dx = +∞ n=0 a n (-1) n n + r -1 n n i=0 n i (-1) i 1 sm + 1 r+i = +∞ n=0 a n (-1) n n + r -1 n 1 sm + 1 r sm sm + 1 n = +∞ n=0 a n I ν (n)G ν (s)H ν (s) n (3.20) = G ν (s)B(H ν (s)), (3.21)
where B(z) = +∞ n=0 b n z n is the generating function of the sequence {b n } n∈N with b n = I ν (n)a n , ∀n ∈ N. The generating function B is expressed with the Laplace transform of g M via a change of variable, we get

B(z) = g M (H -1 ν (z)) G ν (H -1 ν (z)) = (1 -z) -r g M z m(1 -z) . (3.22)
In generating function theory, it is possible to study the decay of a sequence by considering its radius of convergence. If the radius of convergence of B(z) is greater than one then the sequence of coefficients admits a decay that is geometrically fast. Non-geometric convergence occurs when B admits a singularity on the unit circle {z : |z| = 1}. The parameters of the expansion permits to alter the form of the generating function in order to make it simpler. Sometimes it gets so simple that an exact formula is obtained. The two following examples may help convince the reader.

Example 1. In [START_REF] Abate | On the laguerre method for numerically inverting laplace transforms[END_REF], several attempts are made to expand PDF when the convergence is not geometrically fast. One of those cases is the expansion of a simple Gamma PDF, recall that the PDF is given by

f X (x) = x α-1 e -x/β β α Γ(α) , (3.23)
with associated Laplace transform

f X (s) = 1 1 + βs α .
(3.24)

The generating function of the coefficient defined in (3.22) is

B(z) = m α (1 -z) α-r (m -z(β -m)) α . (3.25)
It is easily check that taking r = α and m = β lead to B(z) = 1. Then a 0 = 1 and a n = 0, ∀n ≥ 1.

Example 2. The exact ultimate ruin probability for the classical ruin model is available in a closed form when the claim sizes are governed by an exponential distribution. In this particular case, the integrated tail distribution is also an exponential distribution with the same parameter as the claim amounts. Assume that the claim sizes are exponentially distributed with parameter δ, the Laplace transform of the defective PDF g M is

g M (s) = ρ 1 + δ (1-ρ) s . ( 3

.26)

The generating function of the coefficients is then

B(z) = ρm(1 -z) 1-r m -z δ 1-ρ -m . (3.27)
It is easily check that taking r = 1 and m = δ 1-ρ lead to B(z) = ρ. Then a 0 = ρ and a n = 0, ∀n ≥ 1. Thus g M (x) = ρ 1-ρ δ e -1-ρ δ x and ψ(u) = ρe -1-ρ δ u which is indeed the exact ultimate probability in the studied case.

This two examples shows how to play on the parameters to make the hole thing looks better. We see that this method is a mathematical tool to perform analytical Laplace transform inversion. The choice of the parameter is not automatic as one has to look at the generating function and choose wisely the parameters to have good results. It happens that the generating function B takes a tedious form which leave us with a difficult calll regarding the parameters.

Computation of the coefficients of the expansion

The coefficients are obtained from the derivative of the generating function defined in (3.22)

a n = I ν (n) n! d n dz n B(z) z=0 . (3.28)
Direct evaluation is doable using a computational software program. However if the expression of B is tedious then one might use an approximation procedure. The derivative can be expressed via Cauchy contour integral

a n = I ν (n) 2πi Cr B(z) z n+1 dz, (3.29)
where C r is a circle about the origin of radius 0 < r < 1. We make the change of variable z = re iw to get

a n = I ν (n) 2πr n 2π 0 B(re iw )e -iw dw. (3.30)
The integrals in (3.30) are approximated through a trapezoidal rule

a n ≈ a n = I ν (n) 2πr n 2n j=1 (-1) j ℜ(B(re πji/n )) = I ν (n) 2πr n    B(r) + (-1) n Q(-r) + 2 n j=1 (-1) j ℜ(B(re πji/n ))    ,
where ℜ(z) denotes the real part of some complex number z. The goodness of this approximation procedure is widely studied in [START_REF] Abate | On the laguerre method for numerically inverting laplace transforms[END_REF].

Numerical illustrations

We analyse the convergence of the sum in our method toward known exact values of ruin probabilities with gamma distributed claim sizes. For those claim sizes distribution we have explicit formulas that allow us to assess the accuracy of our approximated ruin probabilities. The goodness of the approximation depends on the order of truncation K.

Our method also enables us to approximate ruin probabilities in cases that are relevant for applications but where no formulas are currently available. A comparison is done with the results obtained with the Fast Fourrier Transform, the scaled Laplace transform inversion and Panjer's algorithm. We plot the difference between the exact ruin probability value and its approximation ∆ψ(u) = ψ(u) -ψ Approx (u).

We considerered Γ(α, β) and U [0, 1] distributed claim sizes. Regarding the ruin model settings, we fix a safety loading at 20% which is a standard value. The results are displayed on Figure 1 and 2 where the claim amount are Γ(1/2, 1/2)-distributed and Γ(3, 1)-distributed.

The generating function of the coefficients does not take a simple form and it is really hard to choose an optimal parametrization. We then choose r = 1 and m = 1/γ in order to ensure the convergence. We also set K = 40, which sounds like a good compromise between accuracy, computation time and numerical difficulties. The Panjer algorithm is applied in its basic form as we are trying to recover probabilities of a geometric compound distribution. The distribution of the random variable M , defined in (3.1), is approximated as follows

P (M = nh) ≈ g n = ρ 1 -ρf 0 n j=1 f j g n-j ∀n ≥ 1, (4.2)
where h is the bandwidth and

f j = F U I jh + h 2 -F U I jh -h 2 , ∀j ∈ N.
This arithmetization design has been recommended in a recent paper [START_REF] Embrechts | Panjer recursion versus fft for compound distribution[END_REF]. The algorithm is initialized with g 0 = G N (0), where G N is the probability generating function of N also defined in (3.1). The probability of ultimate ruin is then approximated by

ψ(u) = 1 - ⌊u/h⌋ i=0 g i .
The bandwidth is set to h = 0.01. The scaled Laplace transform technique has been presented in [START_REF] Mnatsakanov | A note on recovering the distribution from exponential moments[END_REF]. The ultimate ruin probability is approximated by is the Laplace transform of the ruin probability. We set α = 27 and b = 1.25 as it is recommended in [START_REF] Mnatsakanov | Approximation of the ruin probability using the scaled laplace transform inversion[END_REF]. Regarding the Fourier series method, we apply exactly the procedure described in [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF] Chapter 5 Section 5.5. The fast Fourrier transform offers the best accuracy for the two examples, our method is competitive in comparison to the two other methods. In the case of uniformly distributed claim amounts, we do not have an exact formula for the ultimate ruin probability. As the fast Fourier transform did a tremendous job on the two other cases we decide to take the ruin probabilty approximated via the FFT as a benchmark value to assess the accuracy of the three other methods. We also plot the ultimate ruin probability obtained with each method, see Figure 3. One can note that the polynomial expansion give again satisfying results.

Conclusion

Our method provides a very good approximation of the ruin probability when the claim sizes distribution is light-tailed. We obtained a theoretical result that ensures the validity of our expansions. As expected, the numerical results show the superiority of the Fourier series based method in term of accuracy. Nevertheless, our method provides an approximation of a simple form for the hole ruin function and allows reinjection to derive approximations of other quantities of interest in ruin theory. Another advantage is the possibility of a statistical extension that will lead to a nonparametric estimation of ruin probabilities just like the scaled Laplace transform inversion and maximum entropy methods. The great results in terms of accuracy are promising and it will be interesting to consider statistical application. It is also worth noting that this method can be easily adapted to a multivariate problem. The inversion of a bivariate Laplace transform will be at the center of a forthcoming paper.

  f U I (s)

Figure 1 :

 1 Figure 1: Difference between exact and approximated ruin probabilities for Γ(1/2, 1/2)-distributed claim sizes

Figure 2 :

 2 Figure 2: Difference between exact and approximated ruin probabilities for Γ(3, 1)-distributed claim sizes

Figure 3 :

 3 Figure 3: Difference between FFT approximation and approximated ruin probabilities with other method for U [0, 1]-distributed claim sizes

Acknowledgement

The authors also would like to thank H. Albrecher and X. Guerrault for helpful discussion and an anonymous referee for his comments which improve greatly the quality of this paper. This work is partially funded by the Research Chair Actuariat Durable sponsored by Miliman, and the insurance company AXA France.


Anna Mezzapesa

Cyrille Orset

Laurent Plawinski

Loic Doeuvre

Sara Martinez De Lizarrondo

Guglielmina Chimienti

Denis Vivien

Alexandre Mansour

Sabrina Matà

Gabriella Pepe

Eduardo Anglés-Cano
email: eduardo.angles-cano@inserm.fr

Plasminogen in cerebrospinal fluid originates from circulating blood

Keywords: Blood-cerebrospinal fluid barrier, Inflammation, LPS, Plasminogen

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

Vessel wall fibrinolytic activity and pericellular proteolysis in tissues requires plasminogen binding and transformation into plasmin at the surface of fibrin, cells or the extracellular matrix by either the tissue-type plasminogen activator (tPA) or the urokinase-type plasminogen activator (uPA) [START_REF] Das | Plasminogen and its receptors as regulators of cardiovascular inflammatory responses[END_REF]. The binding of plasminogen to cells or fibrin is a lysine-dependent mechanism that can be inhibited by lysine analogues such as tranexamic acid (TXA). In the vasculature, plasmin formation results in blood clot dissolution [START_REF] Rijken | New insights into the molecular mechanisms of the fibrinolytic system[END_REF], whereas in tissues, plasmin is responsible for pericellular proteolysis accompanying cell migration, angiogenesis, wound healing, tissue remodelling and inflammation [START_REF] Plow | The functions of plasminogen in cardiovascular disease[END_REF][START_REF] Li | Plasminogen activator/plasmin system: a major player in wound healing?[END_REF][START_REF] Garcia-Touchard | Extracellular proteases in atherosclerosis and restenosis[END_REF]. Pericellular proteolysis in the central nervous system (CNS) is involved in development, regeneration of nervous tissues, neuronal and synaptic plasticity and the inflammatory response [START_REF] Sappino | Extracellular proteolysis in the adult murine brain[END_REF][START_REF] Zhang | Extracellular proteases: biological and behavioral roles in the mammalian central nervous system[END_REF][START_REF] Monard | Cell-derived proteases and protease inhibitors as regulators of neurite outgrowth[END_REF][START_REF] Gur-Wahnon | The plasminogen activator system: involvement in central nervous system inflammation and a potential site for therapeutic intervention[END_REF]. In humans, plasminogen deficiency is associated with neural disorders including congenital hydrocephalus, periventricular nodular heterotopias and Dandy-Walker malformation [START_REF] Klammt | Identification of three novel plasminogen (PLG) gene mutations in a series of 23 patients with low PLG activity[END_REF]. Trace amounts of plasminogen have been reported in normal human cerebrospinal fluid (CSF), whereas raised concentrations have been detected in patients with meningitis [START_REF] Newman | A method for detecting and estimating plasminogen in cerebrospinal fluid[END_REF], subarachnoid haemorrhage [START_REF] Vermeulen | Source of fibrin/fibrinogen degradation products in the CSF after subarachnoid hemorrhage[END_REF] and multiple sclerosis [START_REF] Hammack | Proteomic analysis of multiple sclerosis cerebrospinal fluid[END_REF]. However, the question has not been settled as to whether plasminogen in CSF originates from circulating blood or is expressed in the CNS, as suggested by the presence of mRNA in mouse brain [START_REF] Basham | Plasminogen expression in the neonatal and adult mouse brain[END_REF][START_REF] Zhang | Plasminogen has a broad extrahepatic distribution[END_REF] or by its synthesis by rat microglial cells in culture [START_REF] Nakajima | Production and secretion of plasminogen in cultured rat brain microglia[END_REF].

The CNS is protected from the bloodstream by the blood-brain barrier (BBB) and the blood-CSF barrier [START_REF] Wolburg | Choroid plexus: biology and pathology[END_REF][START_REF] Saunders | Barriers in the brain: a renaissance?[END_REF]. The BBB is the barrier separating the brain interstitial fluid compartment from the general circulation (endothelial tight junctions, basal lamina of endothelial cells and astrocyte pedicles). It selectively limits penetration of a variety of noxious substances and supplies the brain with nutrients. The blood-CSF barrier separates the CSF compartment from blood through choroid plexus epithelial cells, tight junctions, a basal membrane and the endothelium. It restricts the passage of harmful substances from the blood into the CSF secreted across the choroid plexus epithelial cells into the brain's ventricular system [START_REF] Saunders | Barriers in the brain: a renaissance?[END_REF]. Because the blood-CSF barrier is more permeable than the BBB, many plasma proteins enter the cerebrospinal liquid (through pinocytosis or active transport). An impairment of the blood-CSF barrier thus leads to an increase in the concentration of proteins in the CSF.

Because diffusion of plasma proteins into the CNS is a selective process, we sought to demonstrate that circulating plasminogen may cross the blood-CSF barrier and can be detected in the CSF. For that purpose, we studied control rats and rats with lipopolysaccharide (LPS)-induced systemic inflammation. Our results show that circulating plasminogen enters the CSF space during blood-CSF barrier dysfunction induced by systemic inflammation, thus suggesting that plasminogen found in the CSF of patients with inflammatory neurological disorders originates from circulating blood.

Materials and methods

Reagents and proteins

The enhanced chemiluminescence reagent kit was obtained from Bio-Rad Laboratories (Hercules, CA, USA). The chromogenic substrate selective for plasmin (methylmalonyl)-hydroxyprolylarginine-para-nitroaniline (CBS0065) was purchased from Stago (Asnières-sur-Seine, France). Escherichia coli LPS serotype 0111:B4, trans-4-(aminomethyl)cyclohexane-1-carboxylic acid (or tranexamic acid, TXA), amiloride and 4′,6-diamidino-2-phenylindole (DAPI) and Evans blue were obtained from Sigma Chemical Co (St Louis, MO, USA). Alexa Fluor 488 succinimidyl ester was purchased from Life Technologies (Carlsbad, CA, USA). Goat anti-collagen type IV (Col IV) used for immunohistochemical analyses was obtained from Southern Biotech (Birmingham, AL, USA), and donkey anti-goat antibody F(ab′)2 fragments linked to tetramethylrhodamine isothiocyanate were purchased from Jackson ImmunoResearch (West Grove, PA, USA). Glu-plasminogen, plasmin and peroxidaseconjugated monoclonal antibody against plasminogen kringle 1 (CPL15-PO) were prepared and characterized as described previously [START_REF] Fleury | Characterization of the binding of plasminogen to fibrin surfaces: the role of carboxy-terminal lysines[END_REF][START_REF] Wiman | Activation of human plasminogen by an insoluble derivative of urokinase: structural changes of plasminogen in the course of activation to plasmin and demonstration of a possible intermediate compound[END_REF][START_REF] Montes | Development and clinical application of a new ELISA assay to determine plasmin-α2-antiplasmin complexes in plasma[END_REF]. A rabbit anti-mouse plasminogen polyclonal antibody was kindly provided by HR Lijnen (University of Leuven, Belgium).

Patient sampling

A total of 42 patients with neurological diseases attending the Neurological Clinic of the Careggi University Hospital, Florence, Italy, were admitted in this study. Informed consent was obtained according to the Declaration of Helsinki. The Careggi University Hospital Review Board approved the protocol. Diagnoses were based on clinical, laboratory and magnetic resonance imaging data according to the International Classification of Diseases, Tenth Revision, and the Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition-Text Revision [START_REF]American Psychiatric Association: Diagnostic and Statistical Manual of Mental Disorders, Volume text revision[END_REF]. Patients with Parkinson's disease, brain tumour, epilepsy and alcohol or other substance dependence were excluded. All CSF samples were obtained by lumbar puncture and were immediately used for routine laboratory analyses, which included erythrocyte and leucocyte differential cell counts, total protein concentration, albumin and immunoglobulin G (IgG) levels and agarose isoelectric focusing for IgG oligoclonal bands. CSF samples with increased lymphocyte counts (>5 mm 3 ) or with blood contamination (erythrocyte concentration >50 cells/mm 3 ) were excluded. The CSF to serum albumin quotient (Qalb) was calculated and used to evaluate blood-CSF barrier integrity. Qalb > 0.007 was considered a marker of blood-CSF barrier dysfunction [START_REF] Tibbling | Principles of albumin and IgG analyses in neurological disorders. I. Establishment of reference values[END_REF][START_REF] Brettschneider | Isolated blood-cerebrospinal fluid barrier dysfunction: prevalence and associated diseases[END_REF]. The remainder of each sample was stored in aliquots at -80°C for further analysis.

Animal experimental model

Experiments were performed using male Wistar rats (280 to 350 g) in accordance with the directives of the Council of the European Communities (86/609/EEC) and the French Agriculture and Forestry Ministry for handling animals (decree 87-848).

Purification and labelling of rat plasminogen

To monitor plasminogen uptake by cells and tissues in vivo in the rat model of inflammation, we used fluorescence-labelled native plasminogen purified from rat plasma. Blood was obtained by cardiac puncture of anaesthetized animals and collected in 0.129 M sodium citrate. The plasma was separated from blood by centrifugation, and plasminogen was isolated by lysine affinity and sieving chromatography as described previously [START_REF] Fleury | Characterization of the binding of plasminogen to fibrin surfaces: the role of carboxy-terminal lysines[END_REF]. The purified plasminogen had a relative molecular weight (M r = 92,000 Da) similar to that of human plasminogen and was labelled with Alexa Fluor 488 dye (A488-Pg) according to the instructions of the manufacturer.

Rat model of systemic inflammation

Rats were assigned to either LPS treatment (L) or the saline control group (C) (n = 3 rats per group (Figure 1). Groups L1, L2 and L3 received an intraperitoneal injection of LPS (1 mg/kg) diluted in saline solution, and control groups C1, C2 and C3 received only the saline solution. This dosage of LPS has already been demonstrated to be able to induce brain inflammation in the rat [START_REF] Jeong | Systemic LPS administration induces brain inflammation but not dopaminergic neuronal death in the substantia nigra[END_REF]. Groups L1 and C1 received no other treatment. Groups L2 and C2 received a tail vein injection of A488-Pg (1 mg) 18 hours after LPS or saline administration. Group L3 received a tail vein injection of 10 mg/kg TXA 30 minutes after LPS administration, followed by a subcutaneous dose of 100 mg/kg TXA. After 18 hours, each rat received a tail vein injection of A488-Pg (1 mg) preincubated for 15 minutes with 0.1 M TXA, followed by a subcutaneous injection of 100 mg/kg TXA. Group C3 received identical doses of TXA, but not of A488-Pg. This dosage schedule of TXA was necessary to ensure neutralization of lysine-binding sites in both the endogenous and injected plasminogen. At the end of the scheduled treatments (24 hours), samples were collected and animals killed for further studies.

Sample collection

The anaesthetized rats were placed in a stereotaxic frame and secured with ear bars, and a midline incision in the skin was made up to the head area to permit easy access to the cisterna magna. The needle, which was connected to a draw syringe, was inserted horizontally and centrally into the cisterna magna for CSF collection. The CSF sample was slowly drawn into the syringe, and the colour of the CSF was closely observed to avoid any possible blood contamination. Approximately 100 μl of CSF was collected from each animal. After a short centrifugation step (3 minutes at 5,000 g, 4°C) all CSF samples were immediately flash-frozen in liquid nitrogen and stored at -80°C until use.

Blood was drawn by cardiac puncture using a 1.2 × 40-mm gauge needle syringe containing a 0.1 volume of 0.129 M sodium citrate. Urine was collected by puncturing the bladder. Plasma was separated from blood by centrifugation at 1,500 g for 15 minutes. All the samples were kept on ice during collection and kept frozen at -80°C until use.

Evans blue permeability assay

Rats (n = 3 per group) treated with saline (group C1) or LPS (group L1) alone or supplemented with TXA (groups C3 and L3) received a tail vein injection of Evans blue dye (4% in saline) 4 hours before being killed. After transcardiac perfusion with heparinized saline, the brains and kidneys were harvested and plunged into cold saline until fluorescence measurements were performed. Evans blue dye permeability was monitored by ex vivo near-infrared fluorescence (NIRF) imaging using the IVIS 200 imaging system (Caliper Life Sciences, Hopkinton, MA, USA).

Fluorescence microscopy

After the rats were killed, parts of the brains were postfixed with 4% paraformaldehyde in 0.1 M phosphatebuffered saline, pH 7.4, at 4°C for 18 hours, followed by 24 hours in 20% sucrose and then frozen in isopentane.

For immunocytochemistry, cryostat brain sections (8 μm) were collected on polylysine-coated slides and incubated overnight with a goat anti-Col IV primary antibody (1:1,500), which was revealed with donkey anti-goat antibody F(ab′)2 fragments linked to tetramethylrhodamine isothiocyanate (1:500). The cells were then counterstained with DAPI, mounted with Fluoprep (Dako, Glostrup, Denmark) and observed under an epifluorescence microscope. Images were digitally captured with a Leica DM6000 microscope-coupled CoolSnap camera (Leica Microsystems, Wetzlar, Germany) and visualized with METAVUE 5.0 software (Molecular Devices, Sunnyvale, CA, USA). The detection of exogenously administered A488-Pg in those samples was also studied.

Detection of functional plasminogen in cerebrospinal fluid

Transformation of CSF plasminogen into plasmin was measured using uPA (5 IU/ml) and a plasmin-selective chromogenic substrate (0.75 mM CBS0065), as previously described [START_REF] Bangert | Assay of functional plasminogen in rat plasma applicable to experimental studies of thrombolysis[END_REF]. In this system, the initial velocity of p-nitroaniline released from CBS0065 is proportional to the amount of plasmin(ogen). The ability of TXA to impair plasminogen binding to the cell membrane was investigated as described previously [START_REF] Ho-Tin-Noé | Role of plasminogen activation in neuronal organization and survival[END_REF].

Western blot analysis

Human CSF samples (7 μg of total protein) and rat CSF, urine and plasma (diluted 1:50) samples (10 μl) were electrophoresed in 8% SDS-PAGE gel under nonreducing conditions. Proteins were transferred onto a polyvinylidene fluoride membrane, and plasminogen was revealed using a monoclonal antibody directed against human plasminogen (CPL15-PO, human CSF samples) or a rabbit anti-mouse plasminogen polyclonal antibody (rat samples). The amount of plasminogen was expressed as the number of pixels directly detected using Image-Quant TL 7.0 image analysis software with the Image-Quant LAS 4000 imaging system (GE Healthcare Life Sciences, Pittsburgh, PA, USA) and expressed as nanograms per microlitre by reference to a known amount of plasminogen (15 ng, 10 μl) electrophoresed under similar conditions. In rat samples, the amount of plasminogen was expressed as the number of pixels directly detected using the image analysis software.

Statistical analysis

Data are representative of at least three independent experiments and are expressed as median (25th to 75th

Results

Lipopolysaccharide-induced increase in plasminogen in rat cerebrospinal fluid

Endogenous plasminogen was detected by Western immunoblotting in plasma, CSF and urine of both LPS-and saline-treated control rats (groups L1 and C1, respectively) (Figure 2A). The LPS injection produced no significant variation in the amount of circulating plasminogen as compared to controls (0.1-fold increase in injected versus control rats) (Figure 2B). In contrast, we detected higher plasminogen levels in CSF and urine from LPS-treated rats (fourfold increase in CSF, P = 0.0002; twofold increase in urine, P = 0.0071 (both by Mann-Whitney U test)) (Figure 2B). Rat CSF plasminogen was efficiently transformed into plasmin; its active concentration appeared to be directly related to the amount of plasminogen detected in CSF samples by Western immunoblotting (P = 0.0045, Mann-Whitney U test) (Figure 2C).

Lipopolysaccharide-induced increase in barrier permeability

The aforementioned data suggest that vascular permeability was modified following the LPS injection. This hypothesis was investigated by testing permeability to Evans blue dye. Ex vivo fluorescent and NIRF images showed higher brain fluorescence in the LPS-treated rats (group L) as compared to controls (group C), indicating an increase in barrier permeability to the dye due to blood-CSF barrier impairment (Figure 3A). Modifications in the glomerular filtration barrier were also detected with the Evans blue test (Figure 3B). Higher kidney fluorescence was observed in LPS-treated rats (group L) as compared to controls (group C). The administration of TXA (groups C3 and L3) was without effect on barrier permeability.

In vivo evidence of the circulating origin of plasminogen in cerebrospinal fluid

To obtain in vivo evidence of the extravasation of plasminogen from blood, animals were challenged with exogenous A488-Pg 18 hours after the LPS (group L2) or saline injection (group C2). A488-Pg appeared in the CSF and urine of all LPS-treated rats, indicating that blood plasminogen crosses the barrier during its presence in the circulation (with a half-life of about 2 days) (Figure 4A) and integrates the CSF exchanged at least three times daily [START_REF] Brown | Molecular mechanisms of cerebrospinal fluid production[END_REF]. Actually, the presence of circulating A488-Pg was clearly visualized by fluorescence microscopy only in the choroid plexus of LPS-treated rats (Figure 4B). Figure 4C is a magnified view of the choroid plexus showing the presence of A488-Pg and the vessel wall detected by immunocytochemistry of Col IV. Of note, A488-Pg is absent in the saline-treated control.

Typically, binding of plasminogen by cells is mediated by a lysine-dependent mechanism via lysine residues encompassed in cellular receptors and lysine-binding sites located in plasminogen kringle domains [START_REF] Das | Plasminogen and its receptors as regulators of cardiovascular inflammatory responses[END_REF]. To explore whether the passage of plasminogen through the blood-CSF barrier implicates a lysine-dependent mechanism, we measured cellular uptake of A488-Pg in the presence of the lysine analogue TXA. Figure 5 shows that plasminogen remained increased in the CSF of rats treated with TXA, thus excluding uptake via known plasminogen receptors harbouring carboxy-terminal lysine residues [START_REF] Das | Plasminogen and its receptors as regulators of cardiovascular inflammatory responses[END_REF].

To substantiate the relevance of these in vivo rat studies to human pathology, we investigated the presence of plasminogen in the CSF of patients with inflammatory and noninflammatory disorders. Plasminogen antigen was detected in all CSF human samples (Figure 6). Higher concentrations of plasminogen were found in Guillain-Barré (See figure on previous page.) Figure 2 Determination of plasminogen in rat plasma, cerebrospinal fluid and urine. Samples were obtained from rats challenged with saline (group C1) versus rats challenged with lipopolysaccharide (LPS) (group L1) (see flowchart in Figure 1). (A) Representative immunoblot obtained for plasminogen detection using a rabbit antibody to mouse plasminogen. Electrophoresis of reference rat plasminogen (Pg) in an equal volume (10 μl) of plasma diluted 1:50, cerebrospinal fluid (CSF) and urine. (B) Box-and-whisker plot of plasminogen in plasma, CSF and urine after immunoblot and densitometric analyses. Results represent median (25th to 75th interquartile range (IQR)). *P = 0.0002 and §P = 0.0071 in CSF and urine (Mann-Whitney U test), respectively. (C) Box-and-whisker plot of measurement of the activation of plasminogen in CSF samples (chromogenic substrate assay). Results represent median (IQR) of the velocity of plasmin formation. #P = 0.0045 (Mann-Whitney U test). mOD: milli optical density. syndrome (GBS) (median = 1.28 ng/μl (25th to 75th interquartile range (IQR) = 0.66 to 1.59), n = 15) as compared to multiple sclerosis (MS) (median = 0.3 ng/μl (IQR = 0.16 to 0.61), n = 19) and noninflammatory neurological diseases (NINDs) (median = 0.27 ng/μl (IQR = 0.18 to 0.35), n = 8) (P < 0.0001, one-way ANOVA). The pairwise comparison yielded significant differences in GBS versus MS (P < 0.05) and GBS versus NINDs (P < 0.05), both analysed with Bonferroni's posttest. GBS patients had significantly higher values of CSF total protein, albumin and IgG, as well as higher Qalb, whereas MS patients had higher index values of IgG (QIgG/Qalb) (see Additional file 1: Table S1).

Discussion

Plasminogen is synthesized mainly by the liver and then is distributed to via the systemic circulation [START_REF] Bohmfalk | Plasminogen is synthesized by primary cultures of rat hepatocytes[END_REF]. The concentration of plasminogen in circulating blood is relatively high (1.5 to 2 μM) and constant, as plasminogen does not behave as an acute phase reactant such as fibrinogen. Therefore, inflammatory conditions or the injection of an inflammatory agent such as LPS do not increase the circulating concentration of plasminogen. However, a mechanistic link between inflammation and the blood-CSF barrier dysfunction has been established [START_REF] Begley | International Brain Barriers Society: Inflammation and the brain barriers[END_REF], which may explain an increased transfer of plasminogen from circulating blood to CSF. Indeed, trace amounts of plasminogen have been found in the CSF of patients with noninflammatory diseases [START_REF] Wu | Plasminogen in normal and abnormal human cerebrospinal fluid[END_REF][START_REF] Newman | The use of fibrinolytic activators in meningitis and similar conditions[END_REF]. By using an LPS-induced model of systemic inflammation in rats, we found that plasminogen is present at significantly higher concentrations in the CSF of LPS-treated rats as compared to controls. Accordingly, LPS-treated rats had increased blood-CSF barrier permeability, as demonstrated with Evans blue dye. CSF plasminogen is efficiently transformed into plasmin, thus suggesting that transfer of plasminogen from circulating blood to the CSF could be a source for plasmin formation in the nervous system. Furthermore, using A488-Pg, we found that this labelled exogenous rat plasminogen crosses the blood-CSF barrier. The half-life of plasminogen in the circulation is about 2 days, whereas the CSF is produced and exchanged by cells of the choroid plexus at least three times daily [START_REF] Brown | Molecular mechanisms of cerebrospinal fluid production[END_REF]. Thus, circulating A488-Pg could be detected in the CSF and the choroid plexus of LPS-treated rats. These in vivo data suggest that circulating plasminogen enters the CSF at the choroid plexus. This enormous epithelial/endothelial surface area is potentially available for untoward leakage of plasma proteins into CSF [START_REF] Damkier | Cerebrospinal fluid secretion by the choroid plexus[END_REF][START_REF] Johanson | The distributional nexus of choroid plexus to cerebrospinal fluid, ependyma and brain: toxicologic/pathologic phenomena, periventricular destabilization, and lesion spread[END_REF]. Because LPS is also known to induce interference with the integrity of the glomerular filtration barrier [START_REF] Xu | TNF-mediated damage to glomerular endothelium is an important determinant of acute kidney injury in sepsis[END_REF], we also investigated the passage of plasminogen through the renal glomerulus. We detected modifications in the glomerular filtration barrier with the Evans blue test and high concentrations of both endogenous plasminogen and A488-Pg in the urine of LPS-treated rats as compared to trace amounts found in control rats.

Binding of plasminogen to carboxy-terminal lysine residues of membrane receptors is a well-known mechanism governing its capture and concentration onto the cell surface, where it can be either transformed into plasmin or internalized [START_REF] Das | Plasminogen and its receptors as regulators of cardiovascular inflammatory responses[END_REF]. Because these receptors could potentially be involved in the transfer of plasminogen from blood to CSF, we tested the possibility that TXA, a lysine analogue, might prevent plasminogen from entering the CSF. Occupation of the lysine-binding site of plasminogen by TXA prevents binding to lysine residues of membrane glycoprotein receptors. However, TXA did not inhibit A488-Pg transfer through the blood-CSF barrier, thus suggesting that other binding sites might be involved in plasminogen uptake. For instance, a peptide derived from the plasminogen-binding site domain 1 of M6PIGF2R, peptide fragment 18 to 36, induces plasminogen internalization [START_REF] Leksa | Dissecting mannose 6-phosphate-insulin-like growth factor 2 receptor complexes that control activation and uptake of plasminogen in cells[END_REF]. Collectively, these findings demonstrate that, during inflammation, plasminogen is transported from circulating blood into the CSF of rats via the choroid plexus. Interestingly, the high plasminogen CSF content detected in LPS-treated versus control rats parallels the significantly higher concentration of plasminogen found in the CSF of patients with blood-CSF barrier dysfunction determined on the basis of a high Qalb. In agreement with previously published data [START_REF] Wu | Plasminogen in normal and abnormal human cerebrospinal fluid[END_REF][START_REF] Newman | The use of fibrinolytic activators in meningitis and similar conditions[END_REF], we found that plasminogen is present in trace amounts in noninflammatory CSF samples. However, we found significantly higher concentrations in the CSF of patients with blood-CSF barrier dysfunction. Our present data are in accord with our previous demonstration of the presence of apolipoprotein(a), a glycoprotein homologous to plasminogen in the CSF of patients with blood-CSF barrier dysfunction [START_REF] Pepe | Lipoprotein(a) in the cerebrospinal fluid of neurological patients with blood-cerebrospinal fluid barrier dysfunction[END_REF].

Although there are limitations to translating results obtained in experimental animals to human diseases, our data are suggestive of a similar mechanism for plasminogen transfer from blood to CSF and may be of relevance to patients with inflammation-derived CSF barrier impairment.

Conclusions

We have demonstrated that plasminogen is increased in the CSF of patients with a dysfunctional blood-CSF barrier. We have reproduced, in an in vivo rat model of systemic inflammation, similar modifications of barrier function, and we also have demonstrated that plasminogen from circulating blood enters the CSF space. Altogether our data strongly suggest that the increase in CSF plasminogen concentrations detected in patients obeys a mechanism by which circulating plasminogen crosses the altered blood-CSF barrier.

Figure 1

 1 Figure1Diagram of rat inflammatory model and study design. Rats were separated in six groups (three rats per group). Groups L1, L2 and L3 were challenged with 1 mg/kg lipopolysaccharide (LPS), and the control groups (C1, C2 and C3) were challenged with saline. Both LPS and saline were injected intraperitoneally. *Preincubated with tranexamic acid (TXA); i.v., Intravenous injection of TXA (10 mg/kg) or plasminogen labelled with Alexa Fluor 488 dye (A488-Pg) (1 mg); s.c., Subcutaneous injection of TXA (100 mg/kg).

Figure 2 (

 2 Figure 2 (See legend on next page.)

Figure 3

 3 Figure 3 Blood-cerebrospinal fluid barrier and glomerular barrier permeability to Evans blue dye. Ex vivo fluorescent (F) and near-infrared fluorescent (NIRF) images of brains and kidneys from rats injected with Evans blue dye 24 hours after lipopolysaccharide (LPS) treatment according to the outline in Figure 1. The measurement was performed in perfused brains 24 hours after Evans blue dye injection. C1, Saline-challenged rats; L1, LPS-challenged rats; C3, Saline/tranexamic acid (TXA); L3, LPS/TXA; Neg control, Rat without Evans blue dye; BF, Bright field. (A) Brain NIRF images. (B) Kidney NIRF images. The high fluorescence intensity observed in LPS-treated rats (L1 and L3) compared to saline-treated rats (C1 and C3) indicates leakage of the blood-cerebrospinal fluid barrier (A) or the glomerular filtration barrier (B). No differences were observed between saline-treated (C1) and TXA-treated (C3) animals, indicating no effect of TXA on extravasation of Evans blue dye.

Figure 4

 4 Figure 4 In vivo evidence of the circulating origin of plasminogen in cerebrospinal fluid. Samples were obtained from rats injected with plasminogen labelled with Alexa Fluor 488 dye (A488-Pg). Group L2 rats were challenged with lipopolysaccharide (LPS), and group C2 was challenged with saline (see flowchart in Figure1). (A). An equal volume (10 μl) of either plasma diluted 1:50 or of cerebrospinal fluid (CSF) or urine was electrophoresed, and fluorescence (F) in the gel was directly revealed using ImageQuant TL 7.0 image analysis software (upper panel). The gel was then transferred onto a polyvinylidene fluoride membrane and detected by Western blotting with a rabbit antibody to mouse plasminogen (WB, lower panel). Representative samples are shown. (B) Micrograph showing the presence of circulating A488-Pg (indicated by arrows) in the choroid plexus of an LPS-treated rats detected by direct fluorescence microscopy. 4′,6-diamidino-2-phenylindole (DAPI) staining (blue) indicates cell nuclei. (C) Magnified images of choroid plexus of saline-and LPS-treated rats showing the presence of circulating A488-Pg (indicated by yellow arrows) only in the LPS condition, as detected by direct fluorescence microscopy. DAPI staining (blue) indicates cell nuclei, and collagen type IV (Col IV, red) is used as a vessel marker.

Figure 5

 5 Figure 5 Effect of tranexamic acid on plasminogen transfer through the blood-cerebrospinal fluid barrier and the renal glomerulus. Rats with lipopolysaccharide (LPS)-induced systemic inflammation received an intravenous injection of plasminogen labelled with Alexa Fluor 488 dye (A488-Pg) alone (group L2) or in the presence of sustained concentrations of tranexamic acid (TXA) (group L3) (see flowchart in Figure 1). Plasminogen was detected by immunoblotting. (A) Representative immunoblot of rat plasminogen (Pg), plasma diluted 1:50 cerebrospinal fluid (CSF) and urine samples using a rabbit antibody to mouse plasminogen. Experiments were performed as indicated in Figure 2 with an equal volume (10 μl) of each sample. (B) Representative column bars of plasminogen in plasma, CSF and urine after immunoblot and densitometric analysis.

Figure 6

 6 Figure 6 Quantification of plasminogen in human cerebrospinal fluid. Cerebrospinal fluid (CSF) samples were collected from patients with Guillain-Barré syndrome (GBS, n = 15), multiple sclerosis (MS, n = 19) and noninflammatory neurological disease (NIND, n = 8). (A) Representative human CSF immunoblot showing peroxidase-conjugated monoclonal antibody directed against plasminogen kringle 1 (CPL15-PO). Electrophoresis of human plasminogen (Pg) (M r = 92,000 Da), 10 μl of human plasma diluted 1:50 and 7 μg of total protein in CSF. (B) Box-and-whisker plot of human CSF plasminogen after immunoblot analysis and densitometric analysis. Results represent median (interquartile range) (P < 0.0001, one-way analysis of variance). *P < 0.05 for GBS versus MS and §P < 0.05 for GBS versus NIND (Bonferroni's multiple-comparisons posttest).

  interquartile range (IQR)). The Mann-Whitney U test was used to compare values obtained in treated versus control rats. For human experiments, one-way analysis of variance (ANOVA) was performed to compare the pa-

	tient groups. Bonferroni's multiple-comparisons posttest
	was run for the pairwise comparison of groups. Statis-
	tical significance was set at P < 0.05. A specific statistical
	package for exact nonparametric inference (Stata Statis-
	tical Software release 9 (2005); StataCorp, College Sta-
	tion, TX, USA) was used.

http://www.jneuroinflammation.com/content/11/1/154

Acknowledgments

We acknowledge financial support from ADISU Puglia, Consorzio Interuniversità per le Biotecnologie, Italy, and Società Italiana di Biochimica e Biologia Molecolare for the mobility to AM. This study was funded by Inserm (National Institutes for Health and Biomedical Research, France) and the Lower Normandy Regional Council.

Additional file

Additional file 1: Table S1. Study groups and CSF biochemical parameters.

Abbreviations

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Ana Mezzapesa performed the research; collected, analysed and interpreted data; and participated in manuscript drafting. GP and GC analysed and interpreted data and participated in manuscript drafting. SM recruited patients, collected cerebrospinal fluid, analysed and interpreted data and participated in manuscript drafting. CO and SM performed animal experiments and analysed data. DV revised and commented on the manuscript and managed funding. LP, LD and Alexandre Mansour performed chromogenic and immunoblot assays and participated in the interpretation and discussion of the results. EAC conceived and designed the study, analysed and interpreted the data, and wrote the manuscript. All authors read and approved the final manuscript.