D Brizard
email: denis.brizard@ifsttar.fr

An impact test to determine the wave speed in SHPB: measurement and uncertainty

Post-processing the strain waves in Split Hopkinson Pressure Bars to get the stress-strain in the sample requires the knowledge of the characteristic wave speed c 0 in the measuring bars. In the context of metrology, the measurement uncertainty in the value of c 0 must be assessed. The aim is to minimize this uncertainty, which depends on the way c 0 is determined, as it has an impact on the uncertainty in the final stress and strain in the sample. The frequency domain method we introduce is based on an impact test on a single bar. The frequency spectrum of the impact response of the bar clearly exhibits the longitudinal resonant frequencies of the bar. The experimental dispersion curve is deduced from the spectrum and an optimization procedure was applied to determine the wave speed c 0 along with the Poisson's ratio and the uncertainty in the wave speed. This method gives a relative uncertainty in c 0 lower than 0.05 %: it is mainly related to the uncertainty in the length of the bar (which is hard to reduce when using standard meter tape), which prevails over the uncertainty in the resonance frequencies of the bar. A precise value of the wave speed c 0 and of the associated measurement uncertainty is an important step in the context of Split Hopkinson pressure bars if we want to precisely assess the final uncertainty in Split Hopkinson pressure bars test results, which is at present scarcely done.

Introduction

Split Hopkinson pressure bars are commonly used to measure the stress-strain relationship of materials in the intermediate range of strain rate (100 s -1 to 10 000 s -1 ) [START_REF] Kuhn | Mechanical Testing and Evaluation[END_REF]. The main components of a SHPB device are a striker Address(es) of author(s) should be given and two measuring bars [START_REF] Bazle | Hopkinson bar experimental technique: A critical review[END_REF]. The sample is placed between the two measuring bars and the striker hits the first measuring bar at a prescribed velocity. A strain wave propagates down the first measuring bar: part of the wave is reflected at the interface with the sample and part of the wave crosses the sample and propagates through the second measuring bar. Strain gauges are commonly used to measure the strain time series in each bar, they are often placed in the middle of the bars to avoid any overlay between the incident and reflected pulses. The stress and strain in the sample are computed from the incident, reflected and transmitted waves in the bars.

The value of the wave speed c 0 = E/ρ is important in the post-processing of the SHPB tests because it appears in the equations of the strain rate and strain in the sample as [START_REF] Li | Determination of the dynamic response of brittle composites by the use of the split Hopkinson pressure bar[END_REF] ˙ s (t) = -2c 0 l s r (t) (1)

s (t) = -2c 0 l s t 0 r (τ )dτ (2)
where l s is the initial length of the sample and r (t) is the reflected wave measured in the input bar. E is the modulus of elasticity and ρ the density of the measuring bars.

In the context of quality in measurement, it is also necessary to determine the uncertainty in c 0 as it is involved in the final uncertainty in the stress and strain in the sample. The general approach to derive the measurement uncertainty in the stress and strain in the sample is presented in [START_REF] Brizard | Estimating Measurement Uncertainty on Stress-Strain Curves from SHPB[END_REF], however this paper does not detail how to assess the uncertainty in the wave speed. The present paper aims to fill this gap.

Naturally it is possible to compute c 0 from the measured values of E -on a tensile testing machine-and Post-print : BRIZARD, Denis, 2019, An Impact Test to Determine the Wave Speed in SHPB: Measurement and Uncertainty, Journal of Dynamic Behavior of Materials, 1-8, DOI: 10.1007/s40870-019-00226-7 of ρ. However, the determination of the wave speed c 0 directly from a test made on SHPB is appealing, especially if other facilities are not available. The method described later in the present paper also allows the determination of Poisson's ratio ν.

In the literature of SHPB, the accurate measurement of velocity c 0 is said to be critical [START_REF] Bazle | Hopkinson bar experimental technique: A critical review[END_REF] [5], however few details are given on how accurate it should be. In the case of low attenuation bars, the measurement methods may be classified into three categories.

The intuitive method of measuring the transit time between a pulse and its reflection seems rather simple but it faces some overlooked issues. The propagation of longitudinal waves in bars of finite diameter is dispersive [START_REF] Dennison | The Velocity of Longitudinal Waves in Cylindrical Bars[END_REF], meaning that waves of different frequency ω have different phase velocity c(ω). This is described by the dispersion curve which gives the evolution of phase velocity with respect to either wave frequency or wave length (see Fig. 1 from [START_REF] Dennison | The Velocity of Longitudinal Waves in Cylindrical Bars[END_REF] for instance); by definition c 0 = c(0). The shape of the pulse travelling down the bar is therefore affected: it is hard to determine the beginning of the pulse and to find the precise time shift between two pulses with slightly different shapes. Possibly noisy measurement can make this even harder. Furthermore, Davies [START_REF] Davies | A Critical Study of the Hopkinson Pressure Bar[END_REF] pointed out that "the mean velocity of propagation of the disturbance is decreased by dispersion", which means that measuring the velocity c 0 from a transit time procedure may be erroneous.

To overcome the change in shape of the pulse travelling down the bar, Lifshitz and Leber [START_REF] Lifshitz | Data processing in the split Hopkinson pressure bar tests[END_REF] introduced an iterative procedure to determine the velocity c 0 from an incident and a reflected pulse. One of the pulses was shifted -taking into account of dispersive effects-and compared to the other one, the value of c 0 was then adjusted "until the best fit is obtained". They however made the assumption that the low frequency part of the dispersion curve is insensitive to the value of ν, and therefore did not discuss the determination of ν.

The third method for the measurement of c 0 is based on vibration natural frequency, it was employed by Davies [START_REF] Davies | A Critical Study of the Hopkinson Pressure Bar[END_REF], Kolsky [START_REF] Kolsky | An Investigation of the Mechanical Properties of Materials at very High Rates of Loading[END_REF] and Zemanek and Rudnick [START_REF] Zemanek | Attenuation and Dispersion of Elastic Waves in a Cylindrical Bar[END_REF]. This method requires a specific device for the excitation of longitudinal resonances. Othman and Gary [START_REF] Othman | Dispersion identification using the fourier analysis of resonances in elastic and viscoelastic rods[END_REF] proposed a method based on the analysis of the rod resonances from an impact test, however they only plot the experimental dispersion curve and do not deduce the values of c 0 and ν, nor do they treat measurement uncertainties.

The scope of this paper is limited to bars in which the attenuation is low, typically metallic bars. On the contrary, attenuation cannot be neglected in polymeric bars. Bacon [START_REF] Bacon | An experimental method for considering dispersion and attenuation in a viscoelastic Hopkinson bar[END_REF] proposed a non-parametric approach to obtain the experimental dispersion and attenuation curves of viscoelastic bars based on the transfer function between two observation points (or between an incident and a reflected pulse). Other bars identification procedures rather rely on parametric approaches: the aim is to determine a viscoelastic rheological model describing the behaviour of the material of the bars (see [START_REF] Butt | Parametric identification for material of viscoelastic SHPB from wave propagation data incorporating geometrical effects[END_REF] for example).

In this article we introduce a parametric method for precisely determining the velocity c 0 , along with Poisson's ratio ν, from a simple impact test that can be done on the SHPB setup. The method uses the impact spectrum rather than the impact time series. Some specific points of the dispersion curve are deduced from the resonance frequencies in the impact spectrum and they feed an optimization procedure for finding the values of c 0 and ν. Throughout all the process, measurement uncertainties are taken care of, in order to assess the final uncertainty in the wave speed.

This paper is organised as follows. The first part (Section 2) recalls the computation of the longitudinal resonant frequencies of a finite length rod. In Section 3, we derive the expression of the wave speed as function of resonance frequencies -used to plot the experimental dispersion curve-, compute the associated uncertainty and detail the optimization procedure to get the wave speed, the Poisson's ratio and the uncertainty in wave speed. Finally, in Section 4 the proposed method is applied on a practical example to illustrate its efficiency, it is compared to other methods and the influence of the uncertainty in c 0 on the SHPB stress-strain curve is illustrated.

Longitudinal modes of a finite length rod

Let L be the bar lenght. Let E, ρ and ν be respectively the Young modulus, the density and the Poisson's ratio of the bar.

Resonance frequencies of 1D rod

The wave equation of longitudinal waves in thin rods is [START_REF] Graff | Wave Motion in Elastic Solids[END_REF] (pp77, (2.1.7)))

∂ 2 u ∂t 2 + E ρ ∂ 2 u ∂x 2 = 0 (3)
where u(x, t) is the displacement at coordinate x and time t. We can define the wave speed of the bar as c 0 = E/ρ. Stationary solutions are of the form u(x, t) = U (t)Φ(x) and Eq. 3 becomes

Ü Φ + c 2 0 U Φ" = 0 (4)
where the dot denotes the derivative with respect to the time variable t and the prime denotes the derivative with respect to the space variable x. The previous equation can be reformulated as

Ü U = -c 2 0 Φ" Φ = -ω 2 , ∀x, ∀t (5)
where ω is the circular frequency. The longitudinal modes are therefore the solutions of

Φ" - ω 2 c 2 0 Φ = 0 (6) Let λ 2 = ω 2 /c 2 0 .
The solutions of Eq. 6 are of the form

Φ(x) = P sin(λx) + Q cos(λx) (7)
Φ (x) = λP cos(λx) -λQ sin(λx) (8)
where the constants P and Q are obtained from the boundary conditions, which are a null strain at x = 0 and

x = L Φ (0) = λP (9) Φ (L) = -λQ sin(λL) (10)
A null strain at the ends of the rod implies P = 0 and as we are interested in non trivial solution (ie. Q = 0) Eq. 10 becomes sin(λL) = 0. The longitudinal modes of the rod must therefore satisfy

λ n = nπ L , n ∈ N (11)
The eigenfrequencies of the rod are

f n = nc 0 2L , n ∈ N (12)

Antiresonances and observable resonances

Whatever the position of the strain gauge on the bar, some vibration modes are not observable. These are the modes which have a vibration node located at the same position as the gauge. The resonant frequency is then as antiresonant frequency. In our case the strain gauge is located at the middle of the bar in order to minimize wave overlay, the additional condition for antiresonant frequency is therefore

Φ (L/2) = 0 (13)
From Eq. 8 a non trivial solution still implies sin(λL/2) = 0. The antiresonances therefore verify

λ 2n = 2nπ L , n ∈ N (14)
The corresponding antiresonant frequencies are

f 2n = (2n)c 0 2L , n ∈ N (15)
The observable resonant frequencies are deduced from Eq. 12 and Eq. 15 and if the strain gauge is in the middle of the bar we have

f 2n+1 = (2n + 1)c 0 2L , n ∈ N ( 16
)
3 Measurement of wave speed, Poisson's ratio and uncertainty in wave speed from the impact test

Wave speed from impact test spectrum

Since in reality the propagation of longitudinal waves in the measurement bars is dispersive [START_REF] Dennison | The Velocity of Longitudinal Waves in Cylindrical Bars[END_REF] [7], we substitute c 0 by c exp (f ) in Eq. 16. The observable resonant frequencies of the impact spectrum therefore give the wave speed at some specific frequencies

c exp (f 2n+1 ) = 2Lf 2n+1 2n + 1 , n ∈ N (17)
In the following, this is called the experimental dispersion curve (see Fig. 3b).

Uncertainty in the value of the wave speed

We use here the notations defined in the GUM (guide to the expression of measurement uncertainty [START_REF]Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF]) and in the VIM (international vocabulary of metrology [START_REF]International vocabulary of metrology -Basic and general concepts and associated terms, 3rd edn[END_REF]). Let u(x 1 ) denote the standard -in the sense of standard deviation-measurement uncertainty in x 1 , it will be abbreviated as "uncertainty in x 1 ". The standard relative measurement uncertainty in x 1 is therefore u(x 1 )/x 1 , it will be abbreviated as "relative uncertainty in x 1 ". The relative uncertainty in the computed velocity is deduced from Eq. 17 and the set of rules for functional relationships defined in [START_REF] Farrance | Uncertainty of Measurement: A Review of the Rules for Calculating Uncertainty Components through Functional Relationships[END_REF]

u (c exp ) c exp 2 = u (L) L 2 + u (f 2n+1 ) f 2n+1 2 (18)

Optimization procedure on the experimental dispersion curve

The experimental dispersion curve (Eq. 17) -including measurement uncertainties (Eq. 18)-obtained from the impact test feeds an optimization procedure. The aim of the optimization is to find the values of the wave speed c 0 , width of corridor ∆c 0 and Poisson's ratio ν that

d sup + d mean + d inf ( 19
)
d mean = dist (c(c 0 , ν), c exp ) ( 20
)
d sup = dist (c(c 0 + ∆c 0 , ν), c exp + u(c exp )) ( 21
)
d inf = dist (c(c 0 -∆c 0 , ν), c exp -u(c exp )) (22)
where c(c 0 , ν) is the computed dispersion curve whose parameters are wave speed c 0 and Poisson's ratio ν. The computed dispersion curves are calculated by solving Pochhammer equation with a Newton algorithm (details are given in [START_REF] Brizard | Polynomial mode approximation for longitudinal wave dispersion in circular rods[END_REF] and [START_REF] Brizard | Uncertainty quantification and global sensitivity analysis of longitudinal wave propagation in circular bars. Application to SHPB device[END_REF]). The distance dist(c 1 , c 2 ) is the average absolute difference between the two curves c 1 and c 2 . The distance is evaluated at the experimental points and the average absolute difference is weighted with the inverse of the measurement uncertainty to avoid giving too much importance to the low frequency part of the experimental dispersion curve (see Fig. 3b).

The optimization algorithm is CMA-ES [START_REF] Hansen | The CMA Evolution Strategy: A Tutorial[END_REF], we used the Python version available online.

The computed corridor for the dispersion curve depends on three parameters: c 0 , ∆c 0 and ν (see Fig. 3a). The lower limit of the corridor is c(c 0 -∆c 0 , ν) and the upper limit is c(c 0 +∆c 0 , ν). The value of c 0 only affects the vertical position of the dispersion curve without affecting its shape. The value of ν affects the shape of the velocity drop (see [START_REF] Dennison | The Velocity of Longitudinal Waves in Cylindrical Bars[END_REF]).

No uncertainty in Poisson's ratio ∆ν was used in the optimization procedure. Indeed, varying ν for a given value of c 0 only moves the high frequency part of the dispersion curve, the very low frequency part of the dispersion curve -before the drop in velocity-being insensitive to the value of ν. So adding ∆ν to the optimization procedure would not increase the fit of the experimental corridor -as it tends to widen the corridor as the frequency increases-nor would it give a value of ∆ν other than zero. Indeed this was tested and the optimum ∆ν was around 10 -14 .

Application to 17-4PH steel bars

Impact test conditions

The present method is applied on the specific SHPB device designed and built at our laboratory, LBMC UMR_T9406 (Iffstar/UCLB). Armco 17-4PH precipitationhardening steel is used for the measuring bars.

The diameter of the bars is 31.75 mm, the bars were centerless ground to obtain a precise diameter, a good cylindricity and a low surface roughness. The end surfaces of the bars were machined on a lathe. The length of the bars is L = 3058 mm, it is measured with a class II measuring tape. Directive 2014/32/EU [21] (Annex X, Chapter 1) gives the maximum permissible error (MPE) between two non-consecutive scale marks as A + BL r + C, where A = 0.3 mm, B = 0.2 mm m -1 , C = 0.2 mm and L r is the value of the length rounded to the next whole metre (L r = 4 m here). The uncertainty in the length is u(L) = 1.3 mm, the relative uncertainty is 0.043 %.

The test for determining the wave speed c 0 is a single bar test: the strain wave is therefore reflected back and forth at the free ends of the bar. The high frequency content in the strain wave is ensured by the use of a short striker. The steel striker bar is 94.2 mm long and has a diameter of 30.0 mm. The speed of the striker bar just before impact is 23 m s -1 ; yet this speed is not important for the determination of c 0 as changing the striker velocity has no influence on the wave duration.

Strain gauges are used to measure the wave travelling in the bar ( [START_REF] Brizard | Estimating Measurement Uncertainty on Stress-Strain Curves from SHPB[END_REF] gives the full details). However, the method presented in this article does not depend on any specific method of measurement of the waves propagating in the bars. For example Photon Doppler velocimetry, recently introduced for SHPB [START_REF] Avinadav | Interferometrybased Kolsky bar apparatus[END_REF], could also be used.

The useful part of the signal is the one containing the multiple reflections of the wave in the bar (see Fig. 1). The duration of the time window is T =191 ms and during this period the strain wave travels approximately 160 times the length 2L. The uncertainty in the measured frequency u(f ) is taken as the resolution in the frequency domain because the peaks are very sharp (see Fig 2), it is therefore directly linked to the period T : u(f ) = 1/T .

Detection of the peaks in the frequency spectrum

The resonant frequencies are clearly visible in the frequency spectrum of the impact test (see Fig. 2). The peaks are automatically detected as the points with maximum amplitude between two antiresonances. The antiresonances are roughly determined using a first rough estimate of the wave speed c 0 and Eq. 15. Because the noise on the spectrum increases in high frequency, especially around antiresonances, the peak search zone is reduced to 60 % of the frequency interval between two successive antiresonances (the light red bands in Fig. 2 are discarded). : discarded peak search zone). Adapted from [START_REF] Brizard | Experimental measurement of value and associated uncertainty of wave speed in Hopkinson bars with an accurate and robust method[END_REF].

Wave speed and Poisson's ratio

Fig. 3b (blue curves) illustrates the experimental dispersion curve obtained from the post-processing of the impact spectrum with Eq.17. In fact, the impact test was repeated 5 times and all the corresponding experimental dispersion curves are displayed. These five tests give similar dispersion curves, except for the low frequency part where the discrepancy is noticeable.

The uncertainty in the velocity decreases as the frequency increases. This is due to the frequency term in Eq. 18: u(f 2n+1 ) is constant and taken equal to the frequency resolution. The term [u(f )/f ] 2 therefore decreases as the frequency increases. On the contrary, [u(L)/L] 2 is constant whatever the considered peak. From Eq. 18 it is clear that the uncertainty in c tends asymptotically to the value of the uncertainty in L as the frequency increases. The reduced uncertainty in the high frequency part of the curve is one of the interests of the present method.

The optimum dispersion curve and corridor is also plotted in Fig. 3b, the five tests were used in the optimization process. Practically, the five data sets were gathered as if it was only one dispersion curve. The agreement between the experimental curves and the optimum curve is very good. The optimization gives a wave speed of 5089.2 m s -1 , a Poisson's ratio of 0.287 and corridor width of ∆c 0 = 2.46 m s -1 . Not weighing the average for the distance criteria in the optimization gives the same values for c 0 and ν, only ∆c 0 is slightly higher and is equal to 2.7 m s -1 .

By examining Fig. 3b, we can see a close fit, in the high frequency part, between the experimental dispersion curve with measurement uncertainty and the optimum computed corridor: the uncertainty in the wave speed can be taken as the width of the corridor u(c 0 ) = ∆c 0 . Taking u(c 0 ) = 2.5 m s -1 gives a relative uncertainty in the wave speed equal to u(c 0 )/c 0 = 0.05 % In case a longer striker is used, the spectra are noisier and it is more difficult to automatically detect the resonant frequencies: dispersion curves are not as clean as with the shorter strikers. Indeed there are bumps on the experimental dispersion curve due to wrong peak picking in the impact spectrum. The optimization procedure using three impact tests 1 performed with a 610 mm long striker gave the following values: c 0 = 5089.8 m s -1 , ν = 0.290 and ∆c 0 = 2.44 m s -1 . The value for c 0 agrees with the one found with a short striker, concerning ν only the first two significant digit agree.

The proposed method does not allow to determine the uncertainty in the Poisson's ration u(ν). In SHPB tests, the Poisson's ratio is only required for the disper- sion correction procedure of the raw pulses measured in the bars. For the uncertainty quantification of the whole SHPB post-processing (see [START_REF] Brizard | Estimating Measurement Uncertainty on Stress-Strain Curves from SHPB[END_REF]), the uncertainty in the dispersion curve may then be taken as the uncertainty in c 0 only. The uncertainty in c 0 is also required to quantify the uncertainty in the strain rate and in the strain from Eq. 1 and Eq. 2. So the undetermined value of the uncertainty in ν does not impede the assessment of the uncertainty in the stress and strain from SHPB tests.

Using only the first resonant frequency to determine the wave speed would lead to a much greater uncertainty in c 0 (0.6 % in this case as opposed to 0.05 % with the optimization procedure) and would not enable the determination of ν.

Comparison with other methods

The material properties of 17-4PH steel in H1075 condition are given by the manufacturer (AKSteel) in two successive brochures. The 1994 brochure gives ρ = 7.81 g cm -3 , E = 201 000 MPa and ν = 0.291, from which c 0 = 5073 m s -1 . The 2018 brochure gives ρ = 7.81 g cm -3 , E = 197 × 10 3 MPa and ν = 0.27, from which c 0 = 5022 m s -1 . No measurement uncertainties are given in the brochures, so it is hard to assess u(c 0 ). The different values provided for E and ν highlight the necessity to characterize the bar material and not to rely on the tabulated data.

Because of the effect of dispersion, the transit timemethod is not suitable to determine the value of c 0 , be it by using the starting points of the wave or the peak of the wave. Both are affected by dispersion (see [START_REF] Graff | Wave Motion in Elastic Solids[END_REF], §2.5.4 and §8.4.1). Fox and Curtis [START_REF] Fox | Elastic Strain Produced by Sudden Application of Pressure to One End of a Cylindrical Bar. II. Experimental Observations[END_REF] however introduced a change in variable ((t -x/c 0 )/x 1/3 , where x is the propagation distance) which "should reduce the beginning portions of all strain-time records to a single curve". This was also used by Kaul and McCoy [25]. It is interesting to note that "the time of initial rise should vary inversely as the cube root of the distance of travel" [START_REF] Fox | Elastic Strain Produced by Sudden Application of Pressure to One End of a Cylindrical Bar. II. Experimental Observations[END_REF]: that is why the starting points of the wave are not suitable for the transit time-method.

Instead of the transit time-method, we propose a modified time-method to determine c 0 from the straintime records. It consists in finding the optimal value of time shift ∆t opt (within sampling accuracy of 0.2 µs) for which the waves best overlay in the modified time-plot, c 0 is then computed as 2L/∆t opt . The best overlay is illustrated by Fig. 4, in which the full strain-time record of Fig. 1a was used (the record duration of nearly 0.2 s corresponds to a propagation distance of nearly 1 km). The optimal value of c 0 is 5089.9 m s -1 , it is in accordance with the value from the experimental dispersion curve of Section 4.3. We observed that a small variation of ∆t opt of only a few samples clearly reduced the quality of the overlay of the pulses: that is why the value of c 0 can reasonably be given with 5 significant figures. However, with this transit time-method it is difficult to assess the uncertainty in c 0 and the value of ν cannot be determined.

Uncertainty corridor on the stress-strain curve with SHPB

In order to illustrate the influence of the uncertainty in the wave speed, the uncertainty corridor on the stressstrain curve of a typical SHPB test is computed for two values of u(c 0 )/c 0 , namely 0.6 % and 0.05 %.

We assume that the modulus of elasticity is computed from the wave speed and the density as E = ρc 2 0 . The following relationship holds for the uncertainty in 0.00004 0.00002 0.00000 0.00002 0.00004 0.00006 0.00008 modified time (t x/c0)/x Fig. 4: Normalized strain with respect to modified time (according to [START_REF] Fox | Elastic Strain Produced by Sudden Application of Pressure to One End of a Cylindrical Bar. II. Experimental Observations[END_REF]). For sake of clarity, only one wave (pulse propagating on a distance of 2L) out of 5 is plotted, and the abscissa is restrained to the compression part of the strain wave.

E (see [START_REF] Farrance | Uncertainty of Measurement: A Review of the Rules for Calculating Uncertainty Components through Functional Relationships[END_REF])

u (E) E 2 = u (ρ) ρ 2 + 4 u (c 0 ) c 0 2 (23)
The value of the density given by the manufacturer is ρ = 7.81 g cm -3 . Following [START_REF] Kirkup | An Introduction to Uncertainty in Measurement: Using the GUM (Guide to the Expression of Uncertainty in Measurement)[END_REF] (Rule 1, page 23), we can assume that the uncertainty in ρ is on its last significant digit: u(ρ) = 0.005. This gives u(ρ)/ρ = 0.06 %.

In case u(c 0 )/c 0 = 0.6 %, then u(E)/E = 1.2 %. In case u(c 0 )/c 0 = 0.05 %, then u(E)/E = 0.12 %. Both cases are used to compute the uncertainty corridor around the stress-strain curve of a copper sample tested with SHPB. All the details of the method to get the uncertainty corridor are given in [START_REF] Brizard | Estimating Measurement Uncertainty on Stress-Strain Curves from SHPB[END_REF]. The influence of u(c 0 ) on the width of the uncertainty corridor is noticeable (see Fig. 5).

The importance of the uncertainty corridor is quantified by the ratio of the area of the uncertainty corridor upon the area of the stress-strain curve. The stressstrain curve contour is closed by a point located on the x-axis at the vertical of the last point of the stress strain curve. This ratio is equal to 10.4 % when u(c 0 )/c 0 = 0.6 % and 3.5 % when u(c 0 )/c 0 = 0.05 %: the area of the uncertainty corridor is divided by 3 when the uncertainty in c 0 is divided by 10.

Conclusion

This article introduces a method for the determination of the wave speed c 0 , uncertainty in wave speed u(c 0 ) and Poisson's ratio ν of SHPB measurement bars. This method is based on a straightforward impact test on a single bar with a short striker, which is an easy experiment to perform on a SHPB device. From the impact test spectrum, the sharp peaks that correspond to the resonant frequencies of the bar are easily identified. These resonances are used to obtain some specific points of the experimental dispersion curve. Finally, the points of the experimental dispersion curve feed a simple optimization procedure which outputs the values of c 0 , u(c 0 ) and ν. With this method, the relative uncertainty in c 0 is of the order of magnitude of the relative uncertainty in the bar length, ie. around 0.05 %, because the uncertainty in the frequency is very small.

In the context of quality in measurement with SHPB, this is an important preliminary step required to assess the uncertainty in the strain and strain rate in the sample [START_REF] Brizard | Estimating Measurement Uncertainty on Stress-Strain Curves from SHPB[END_REF].

  best fit the experimental data including the uncertainty corridor, i.e. such that argmin c0,∆c0,ν

Fig. 1 :

 1 Fig. 1: Strain-time record for the impacted bar. Vertical lines | indicate a travel of the wave over a distance of: (a) 10 × 2L, (b) 2L.

Fig. 2 :

 2 Fig. 2: Frequency spectrum of the impact test (+: identified f 2n+1 peaks with the corresponding value of 2n + 1; |: rough antiresonant frequencies f 2n ;: discarded peak search zone). Adapted from[START_REF] Brizard | Experimental measurement of value and associated uncertainty of wave speed in Hopkinson bars with an accurate and robust method[END_REF].

  /s] optimum: c 0 = 5089.2, = 0.2868 & dc 0 = 2.46 optimization measurement (b) Experimental dispersion curve and optmization result

Fig. 3 :

 3 Fig. 3: Computed and experimental dispersion curves

Fig. 5 :

 5 Fig. 5: Influence of the relative uncertainty in c 0 on the stress-strain curve in the sample with SHPB (copper sample, signals are low-pass filtered at 200 kHz)

The number of resonance peaks is also18.

Acknowledgements

The author is grateful to Pr Eric Jacquelin and Pr Sylvie Ronel for their help and suggestions of improvement.


A Castellazzi

F Richardeau

A Borghese

F Boige

A Fayyaz

A Irace

G Guibaud

V Chazal

Gate-damage accumulation and off-line recovery in SiC power MOSFETs with soft short-circuit failure mode

This paper proposes the detailed analysis of the short-circuit failure mechanism of a particular class of silicon carbide (SiC) power MOSFETs, exhibiting a safe fail-to-open-circuit type signature. The results are based on extensive experimental testing, including both functional and structural characterisation of the transistors, specifically devised to bring along gradual degradation and progressive damage accumulation. It is shown that the soft failure feature is associated with degradation and eventual partial shorting of the gate-source structure. Moreover, partial recovery, induced here by ad-hoc off-line biasing, is observed on degraded components. The results indicate that it is a realistic new option for deployment in the application to yield enhanced system level robustness and system-level hopping-home operational mode capability, of great importance in a number of reliability critical domains, such as, for instance, transportation.

Introduction

Short-circuit robustness is a key application requirement for semiconductor power devices. The relative performance of silicon carbide (SiC) MOFETs and silicon (Si) IGBTs has been amply discussed and compared in a number of studies (see [START_REF] Castellazzi | SiC power MOSFETs performance, robustness and technology maturity[END_REF][START_REF] Boige | Ensure an original and safe "fail-to-open" mode in planar and trench power SiC MOSFET devices in extreme short-circuit operation[END_REF][START_REF] Chen | Study of short-circuit robustness of SiC MOSFETs, analysis of the failure modes and comparison with BJTs[END_REF], for example), which have pointed out a somewhat weaker withstand capability of SiC, as opposed to Si, and differing failure mechanisms. In particular, a twofold failure mechanism was highlighted in [START_REF] Romano | Influence of design parameters on the short-circuit ruggedness of SiC power MOSFETs[END_REF] for SiC, yielding either a fail-to-opencircuit (FTO, or soft) or fail-to-short-circuit (FTS, or hard) signature, depending on the energy dissipation rate and related temperature increase within the device.

A soft type failure is by far the preferred option in the application. Indeed, FTS behaviour typically implies failure propagation, loss of whole system functionality and potentially destructive events, with explosion-like characteristics; on the other hand, FTO behaviour results in the failed transistor shutting off: an intrinsic fault containment capability, which, in the case of parallel device or multi-chip modules based designs, still enables to operate the system, even if at reduced power ratings, without additional complex or costly circuitry [START_REF] Boige | Short-circuit robustness of parallel SiC MOSFETs and fail-safe mode strategy[END_REF]. Such an operational status is usually referred to as hopping-home mode and is a highly desirable feature of power conversion systems in some strategically relevant application domains, in particular transportation. For that reason, soft failing devices are of great interest and have already received dedicated attention in the recent past [START_REF] Fayyaz | Aging and failure mechanisms of SiC Power MOSFETs under repetitive short-circuit pulses of different duration[END_REF]: it was shown in particular that the failure is due to the progressive creation of cracks and aluminium (Al) diffusion leading to the partial shorting of the gate and source terminals, effectively reducing the gate-source bias voltage to values below the threshold voltage, thus turning off the device. Such effect was shown to interest mainly the inter-layer dielectric (ILD), a thicker oxide layer between gate and source terminals, which is directly in contact with the Al top metal layer, but not the thin field oxide region.

This paper extends the present understanding by more carefully controlling the progressive accumulation of damage and by discussing a very new observation, namely the possibility to recover device functionality after soft failure and re-impose a stable, even if slight degraded, characteristics by ad-hoc gate biasing conditions. Transient and static functional measurements are coupled with device structural analysis to yield in depth understanding of the phenomena and contribute to the development of SiC based power conversion systems with enhanced robustness and operability under critical conditions

Functional characterisation

Short-circuit ageing analysis

Fig. 1 shows the results of repetitive short-circuit tests on a commercial planar-gate 80 m -1200 V SiC discrete power MOSFET. Fig. 1 a) shows the drain current, ID; Fig. 1 b) the gate-source bias voltage, VGS, and Fig. 1 c) the corresponding gate current, IG. The pulse width is chosen to include a sufficient safety margin from the maximum device withstand capability (tPW = 10 µs compared to TSC MAX = 14 µs), so as to avoid failure. Still, the pulse duration is long enough to produce gradual damage accumulation: typically, for tPW < 6 µs, no permanent damage can be detected. As is evident from these results, the device characteristics are progressively degraded during the test: to a reduction of ID (Fig. 1a), corresponds a decreased VGS (Fig. 1b), as a result of increased gate leakage current (Fig. 1c). As highlighted by the authors for SiC power MOSFETs under short-circuit stress in [START_REF] Richardeau | SiC MOSFETs soft and hard failure modes: functional analysis and structural characterization[END_REF], creation of cracks and metal (aluminium, Al) diffusion lead to the partial shorting of the gate and source terminals. To support this hypothesis, thermal modelling and simulation of the chip temperature were carried out, as detailed in [START_REF] Boige | SiC power MOSFET in short-circuit operation: Electro-thermal macro-modelling combining physical and numerical approaches with circuit-type implementation[END_REF][START_REF] Richardeau | Circuit-type modelling of SiC power Mosfet in short-circuit operation including selective fail-toopen and fail-to-short modes competition[END_REF]. The model takes into account the temperature dependence of the conductivity and the mass heat of SiC and aluminium, as well as the solid-liquid phase transition of the Al top layer. Using structural computer aided design tools (specifically, Comsol™ was used here), the model received as input ID(t) waveform of Fig. 1a), with a constant VDS value of 600 V. Considering pulse number #1600, Fig. 2 shows the temperature rise of all the main layers of the chip during the short circuit. It is clearly visible that the Al layer is above its melting point with a high thermal energy injected into its liquid phase. Furthermore, according to [START_REF] Liu | Gate Failure Physics of SiC MOSFETs Under Short-Circuit Stress[END_REF], at the Al melting temperature, the thick oxide is already cracked because the mechanical stress at the upper corner greatly exceeds its mechanical strength.

Gate-source damage recovery

On the side of the gate ageing observation, a new observation can be made in the results of Fig. 1: a partial recovery effect can be observed in going from pulse #3000 to pulse #4600 (in these tests, a Keysight™ B1505A source-meterunit was used). Here, the recovery is not stable, and half-way through the pulse the device characteristics fall back to the previous degraded status. Because of its potential relevance for the application, this observation triggered further detailed analysis of the effect, specifically of the possibility to force and control the recovery and stabilise the device characteristics, even if at a slightly degraded level.

Based on the aforementioned findings of [START_REF] Fayyaz | Aging and failure mechanisms of SiC Power MOSFETs under repetitive short-circuit pulses of different duration[END_REF], a first attempt was made by statically sweep-biasing the degraded transistor (same device used previously for the tests of Fig. 1) in a VGS range  12 V, with different limit values of gate pulsed current IGSS. Fig. 3) summarises the results:

 Step 1 (blue curve) corresponds to the first traced curve, in which the gate current was limited at 10 mA: here, as VGS is increased beyond about 10.8 V, a sudden breakdown like characteristics is entered, with a steep increase of IGSS;  Step 3 (red curve) is a sweep with increased IGSS limit value to 30 mA: here, as VGS is increased beyond just 1 V a recovery effect is manifest, with IG dropping suddenly again.

 Step 2 (brown curve) is a second sweep carried out straight
These results point out that the device gate characteristics can be partially recovered by producing moderate heat generation in the gate region, which is ascribed to a fusing effect taking place, which effectively disconnects degraded portions of the chip from the remaining cells; in this case, the device reaches a new stable, though slightly degraded state. However, if the pulse width tPW is increased closer to the maximum duration Tsc max of the DUT, the level of damage accumulation is greater and for some devices analysed, the IG value after recovery was found to be too high for them to be still usable with typical gate-driver designs. In Fig. 3), at the end of the first series of low-voltage gate sweeping, the measured equivalent gate-source resistance was 2.4 k, corresponding to a gate-drive supply of 135 mW at 18 V. To try and get the entire gate recovery, a second series of transfer characteristics sweeping was carried out, the results of which are given in Fig. 4. Here, the gate bias voltage is increased up to the nominal voltage of 18 V at first and beyond such value subsequently. For reference, a Keysight™ B2902A SMU with 600 µs pulse duration was used for these tests; the effect of the pulse duration is discussed at the end of this section.

Fig. 4a) shows that, for VGS higher than 12 V and lower than 16 V, a chaotic transient behaviour appears when the limit gate current is moved from 5 to 12 mA (step 1 to step 6): a first metallic injection effect appears in the form of a soft breakdown of the oxide, following a fusing effect and a second metallic injection effect.

In Fig. 4b), by gradually increasing the limit gate current up to 30 mA and increasing the bias voltage from 16 V to 30 V, the chaotic behaviour disappears, giving way to a stable ohmic transfer characteristics. From an equivalent RGATE  1.8 k at 12 V in step 1, RGATE  1.3 k at 18 V in step 18, giving an acceptable gate-drive over-supply of 250 mW at 18 V to continue operation of the device in hopping-home mode. However, in Fig. 4c), in an attempt to further restore the gate oxide, the sweeping was repeated with a voltage bias higher than 18 V up to 30 V. A new stable ohmic state appears in this case with a lower RGATE  465 at 15V. Unfortunately, such value is too low, giving a gate-drive over-supply of 480mW at 15V. The preliminary qualitative conclusion can be drawn that the bias voltage and injected restoration energy values should be high enough to eliminate the main ohmic paths for the device to be usable at nominal gate voltage, but they should not be too high, so as not to re-metallize again the cracked oxide.

Finally, in Fig. 4c), a final attempt was made to induce gate recovery by increasing the pulse duration from 600 µs to 600 ms. Even in this case of high energy injection, the ohmic state of the gate remains stable and, although the equivalent gate resistance is lower in this case, it remains nevertheless at a value which makes the device further use in a real circuit still possible, as demonstrated in the next section.

Characteristics of partially recovered device

The device was then characterized with the same SMU for reusing. Fig. 5a) shows the transfer characteristics at VDS = 6 V, comparing a new device and the recovered device from Fig. 4c). For illustration, at VGS=12 V, the comparison shows a decrease of only 20 % in ID for the recovered device. The same behaviour is visible in Fig. 5b) on the output characteristics, which proves the potential re-usability of the device.

Structural characterisation and partial recovered device operation

The plastic case of the device was opened by laser ablation followed by terminal chemical etching on the encapsulation resin. Fig. 6 a) shows the degraded die, with a very clear indication of severe damage to the top metal having accumulated in the proximity of the external gate-ring source-metal structural boundary. Lock-in-thermography analysis was also carried out, Fig. 6 b), which confirmed the abnormal heating of the cracked gate fingers around the corner region, where mechanical stresses are likely to be highest during the short-circuit. After opening, the device was re-used within a circuit, at reduced drain-source bias, and it was still able of correct transient operation, as demonstrated by the results in Fig. 7, which refer to a single short-circuit pulse. It can be seen that the channel saturation current is reduced of 37 % after the recovery gate-source process described above, as a result of the lowering of the IDS(VGS) curve presented in Fig. 5 a).

Conclusion

The possibility to implement self-healing capability, remindful of film capacitor behaviour, in SiC power MOSFETs by ad hoc design and biasing is a new and very appealing feature, which can greatly change the present way to look at semiconductor device and power converter robustness, reliability and availability in application. This paper has shown that, following short circuit ageing stress and damage edge accumulation, it is still possible to partially restore the integrity of the gate structure by proper electrical biasing for a particular class of devices and contained stress and damage levels. Here, device recovery was implemented in an off-line setup by means of a simple pulsed mode source-meter-unit, by progressively increasing the gate current limit. This operation is only possible if the gate is not already damaged too severely. In addition, it was shown that the voltage and injected recovery energy should be high enough to eliminate the main ohmic paths to be used at nominal gate voltage, but not so high as to cause excessive metal diffusion within the cracked oxide, which would yield a permanent short-circuit between the control terminals. Further studies are being dedicated to the online implementation of the recovery capability in a real power conversion system.

Fig. 2 .

 2 Fig. 2. Fig. 2 1D chip temperature simulation in short-circuit operation for #1600: tPW = 10 s @VDS = 610 V -VGATE DRIVE = 18 V -RGATE EXT. = 47  -TCASE = 25 °C.

Fig. 1 .after

 1 Fig. 1. a), repetitive SC DUT progressive degradation, manifest as a decrease of ID after #1600 pulses with tPW = 10 s @VDS = 610 V -VGATE DRIVE = 18 V -RGATE EXT. = 47  -TCASE = 25 °C; b), corresponding VGS waveform; c) corresponding IGS waveform. In the results corresponding to pulse n. #4600, the device initially exhibits partial recovery, followed by degradation.

Fig. 3 .

 3 Fig. 3. Static characteristics of device degraded by repetitive shortcircuit: first sweeping biasing-gate test highlighting breakdown and recovery effects.

Fig. 4

 4 Fig. 4 IGSS(VGS) gate ohmic restoration analysis: a) 5 mA to 12 mA, b) 30 mA, c) high gate-source bias and final metallic crack. All graphs share the bottom x-axis and scaling, not repeated for better readability. VGS [V] 6 12 18 24 30 0

Fig. 5 Fig. 6 a

 56 Fig. 5 Transfer and direct behaviour comparison @ TCASE = 25 °C : a) IDS(VGS) @VDS = 6 V, b) IDS(VDS) curve.


Adrien Barton
email: adrien.barton@gmail.com

Ludger Jansen

Jean-François Ethier
email: ethierj@gmail.com

A Taxonomy of Disposition-Parthood

Keywords: . Disposition, Mereology, Causality 1.!Introduction Disposition, Mereology, Causality

We propose a taxonomy of mereological relations between dispositions, as well as a series of axioms relating the bearers, triggers or realizations of disposition-complexes and disposition-parts. We apply them on a use case to show how basic dispositions can be combined to express elaborate dispositions. We consider whether those relations of partial ordering satisfy axioms of composition and decomposition.

Introduction

Dispositions are realizable entities, that is, properties that can be realized by certain types of processes, but that are present even when they are not realized [START_REF] Jansen | Tendencies and other realizables in medical information sciences[END_REF][START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF]. The Basic Formal Ontology (BFO) defines a disposition as a realizable entity "that exists because of certain features of the physical makeup of the independent continuant that is its bearer" ( [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF], p. 178). To be realized, dispositions need to be triggered by some process: for example, a fragile vase has a disposition to break (its realization) when undergoing a strong shock (its trigger). Dispositions are useful to represent causal relations in BFO, as they provide their bearers with certain causal powers. In particular, when a disposition is triggered by t and realized by r, this implies that t causes r.

Röhl and Jansen [START_REF] Röhl | Representing dispositions[END_REF] propose a formalization of dispositions for single-track dispositions, that is, dispositions that have exactly one kind of realization given one specific bundle of triggering events and/or background conditions. Barton and Jansen [START_REF] Barton | A modelling pattern for multi-track dispositions for life-science ontologies[END_REF] suggest first steps towards a formalization of multi-track dispositions, taking into account that there could be several kinds of triggers as well as several kinds of realizations. For this, they introduce a relation of parthood between dispositions. The present paper will extend those results by distinguishing several mereological relations that can be defined for dispositions.

In the following, terms for classes will be written in italic (e.g. "Breaking"), and terms for instances in bold (e.g. "vase 0 "). A statement of the form "disposition d of b to R when T" T T should be interpreted as expressing (in the vocabulary of [START_REF] Röhl | Representing dispositions[END_REF]), when d is a

We begin in section 2 by introducing three case studies illustrating different kinds of disposition-parthood which we will axiomatize in section 3. Section 4 will demonstrate the expressive power of the combined use of these kinds of dispositionparthood. Section 5 discusses whether disposition-parthood relations are authentic mereological relations and conclude.

2.!Three case studies

The dispositional properties of an object and the way they are realized are often quite complex. In order to distinguish different varieties of such complexity, we introduce three case studies: the fragility of a vase, the solubility of a tablet, and the vulnerability of a human to a certain kind of poison.

2.1.!(Vase): Mod-part and fragility

The case study (Vase) serves as an illustration for the kind of disposition-parthood that has been discussed in [START_REF] Barton | A modelling pattern for multi-track dispositions for life-science ontologies[END_REF] under the more general label "d-part_of". We will call it "mod-part_of" here, in order to distinguish it from other varieties of disposition-parthood. This case study involves the fragile vase vase 0 , whose fragility can be formalized as a disposition. Three particular dispositions disp_break 0 , disp_crack 0 and fragility 0 inhere in vase 0 , defined as follows:

•! disp_break 0 is the disposition of vase 0 to Breaking when Strong_shock, •! disp_crack 0 is the disposition of vase 0 to Cracking when Light_shock, •! fragility 0 is the disposition of vase 0 to [Breaking when Strong_shock] and [Cracking when Light_shock].

We can introduce a relation of disposition-parthood mod-part_of such that:

•! disp_break 0 mod-part_of fragility 0 •! disp_crack 0 mod-part_of fragility 0

We use the modifier "mod", derived from "mode", because fragility 0 has two possible pathways, or modes, of being realized: via disp_break 0 or via disp_crack 0 . We will now introduce two alternative kinds of disposition-parthood relations.

2.2.!(Tablet): Add-part and solubility

The case study (Tablet) involves a medication tablet tablet 0 , composed of two halves half-tablet 1 and half-tablet 2 -that are both part_of tablet 0 . Let's introduce the following dispositions:

•! solubility 0 is the disposition of tablet 0 to Dissolving when Object_in_solvent,

•! solubility 1 is the disposition of half-tablet 1 to Dissolving when Object_in_solvent, •! solubility 2 is the disposition of half-tablet 2 to Dissolving when Object_in_solvent.

We can introduce another relation of disposition-parthoood add-part_of such that:

•! solubility 1 add-part_of solubility 0 •! solubility 2 add-part_of solubility 0

The modifier "add" is derived from "additive": the dissolving processes of both halftablet 1 and half-tablet 2 compose the realization of solubility 0 , and the triggers of the former two compose the trigger of the latter.

2.3.! (Poison): Chain-part and vulnerability

The We introduce another relation of disposition-parthoood chain-part_of such that:

•! vulnerability_heart 0 chain-part_of vulnerability_heart_&_life 0 •! heart_need 0 chain-part_of vulnerability_heart_&_life 0

We use the modifier "chain" because the realization of vulnerability_heart 0 triggers heart_need 0 , thus creating a causal chain.

3.!Axiomatization

As will become clear later, the relations add-part_of and chain-part_of share several commonalities. Thus, we introduce a relation aggreg-part_of, of which both are subrelations. We also introduce a general relation of dispositional-parthood noted disp-part_of, of which mod-part_of and aggreg-part_of are subrelations. All these relations are subrelations of the general part_of relation, which is axiomatized in the Relation Ontology [START_REF] Smith | Relations in biomedical ontologies[END_REF] (cf. Table 1).

part_of disp-part_of mod-part_of aggreg-part_of add-part_of chain-part_of Table 1: Taxonomy of parthood relations

The respective relations of proper parthood will be denoted by adding "proper_" in the name -e.g., proper_mod-part_of. We define a disp-complex (resp. mod-complex, aggreg-complex, etc.) as a disposition that has some proper disp-part (resp. mod-part, aggreg-part, etc.). I.e., for "[x]"="disp", "mod", "aggreg", "add" or "chain":

d 0 instance_of [x]-complex :⇔ (∃d, d proper_[x]-part_of d 0 )

3.1.!Bearer and disp-part_of

The bearer of a disp-part is a (proper or improper) part of the bearer of the disp-complex:

(DISP-BEARER)

d 1 disp-part_of d 0 ⇒ [(d 0 inheres_in b 0 ∧ d 1 inheres_in b 1 ) ⇒ b 1 part_of b 0 ]
In (Tablet), the bearer of solubility 1 (resp. solubility 2 ) is half-tablet 1 (resp. half-tablet 2 ), which is a part of tablet 0 . In (Vase), the bearers of fragility 0 , disp_crack 0 and disp_break 0 are the same, namely vase 0 . Similarly, in (Poison), the bearers of vulnerability_heart 0 , heart_need 0 , and vulnerability_heart_&_life 0 are the same, namely Jones. However, we will see later in section 4.2 with the domino case study that the bearer of a mod-part (resp. chain-part) is not always identical to the bearer of a modcomplex (resp. chain-complex).

3.2.!Realization and disp-part_of

We will now turn to the axioms concerning the realization of a disp-complex. In the general case, if a disp-complex is realized in a process, then at least one of its proper disp-parts is realized in a part of this process:

(DISP-REALIZATION) (d 0 instance_of Disp-complex ∧ d 0 realized_in r 0 ) ⇒ (∃d 1 , ∃r 1 , d 1 proper_disp-part_of d 0 ∧ r 1 part_of r 0 ∧ d 1 realized_in r 1 )
As we will now see, both mod-part_of and aggreg-part_of satisfy more specific axioms.

A mod-complex is realized in a process if and only if at least one of its mod-parts is realized in this very same process:

(MOD-REALIZATION) (d 0 instance_of Mod-complex ∧ d 0 realized_in r 0 ) ⇔ (∃d 1 , d 1 proper_mod-part_of d 0 ∧ d 1 realized_in r 0 )
In (Vase), fragility 0 is realized in a process (instance of Cracking or Breaking) if and only if either disp_crack 0 or disp_break 0 is realized in this process. Things are different for aggreg-parthood. If an aggreg-complex is realized in a process, then all its aggregparts are realized in a part of this process:

(AGGREG-REALIZATION) (d 0 instance_of Aggreg-complex ∧ d 0 realized_in r 0 ) ⇒ [∀d 1 , d 1 proper_aggreg-part_of d 0 ⇒ (∃r 1 , d 1 realized_in r 1 ∧ r 1 part_of r 0 )]
In (Tablet), if solubility 0 is realized in dissolving 0 , its aggreg-parts solubility 1 and solubility 2 are both realized, respectively in dissolving 1 and dissolving 2 (that are both parts of dissolving 0 ). In (Poison), if vulnerability_heart_life 0 is realized in cardiac_arrest 0 +death 0 , its aggreg-parts vulnerability_heart 0 and heart_need 0 are both realized, respectively in cardiac_arrest 0 and death 0 .

Note however that if an aggreg-part is realized, this does not imply anything for the realization of the aggreg-complex. For example, if half-tablet 1 dissolves, this does not imply the dissolving of tablet 0 -as half-tablet 2 may not dissolve (in case it is not put into a solvent). Similarly, in (Poison), Jones might undergo some cardiac arrest which is not due to the ingestion of poison, in which case heart_need 0 will be realized without vulnerability_heart_&_life 0 being realized (because it was not triggered by any ingestion of poison).

3.3.!Trigger and disp-part_of

We will now turn to the axioms concerning the triggers of a disp-complex. In general, if a disp-complex is triggered, then at least one of its proper disp-parts is triggered.

(DISP-TRIGGER) (d 0 instance_of Disp-complex ∧ d 0 has_trigger t 0 ) ⇒ (∃d 1 , ∃t 1 , d 1 proper_disp-part_of d 0 ∧ d 1 has_trigger t 1 )
Again, both mod-part_of and aggreg-part_of satisfy more specific axioms. A modcomplex is triggered by a process if and only if at least one of its mod-parts is triggered by this very process:

(MOD-TRIGGER) (d 0 instance_of Mod-complex ∧ d 0 has_trigger t 0 ) ⇔ (∃d 1 , d 1 proper_mod-part_of d 0 ∧ d 1 has_trigger t 0 )
In (Vase), fragility 0 is triggered by a process (an instance of Light shock or Strong shock) if and only if either disp_crack 0 or disp_break 0 is triggered by this process. However, if an aggreg-complex is triggered, then all its aggreg-parts are triggered:

(AGGREG-TRIGGER) (d 0 instance_of Aggreg-complex ∧ d 0 has_trigger t 0 ) ⇒ ∀d 1 , d 1 proper_aggreg-part_of d 0 ⇒ (∃t 1 , d 1 has_trigger t 1 )]
In (Poison), if vulnerability_heart_&_life 0 is triggered by ingestion_poison 0 , its aggreg-parts vulnerability_heart 0 and heart_need 0 are both triggered, respectively by ingestion_poison 0 and the cardiac_arrest 0 that followed. More specifically, if an addcomplex is triggered by a process, then any of its add-parts is triggered by a part of this process:

(ADD-TRIGGER) (d 0 instance_of Add-complex ∧ d 0 has_trigger t 0 ) ⇒ [∀d 1 , d 1 proper_add-part_of d 0 ⇒ (∃t 1 , d 1 has_trigger t 1 ∧ t 1 part_of t 0 )]
In (Tablet), if solubility 0 is triggered by tablet 0 _in_solvent, its add-parts solubility 1 and solubility 2 are both triggered, respectively by the processes half-tablet 1 _in_solvent and half-tablet 2 _in_solvent (that are both parts of the process tablet 0 _in_solvent).

Note that replacing add-part_of by chain-part_of in the axiom (ADD-TRIGGER) does not lead to a true statement: as a matter of fact, when vulnerability_heart_&_life 0 is triggered by poison_ingestion 0 , the disposition heart_need 0 is triggered by cardiac_arrest 0 , which is not a part of poison_ingestion 0 .

However, on top of (AGGREG-TRIGGER), we can state that the trigger of a chaincomplex is the trigger of one of its chain-parts:

(CHAIN-TRIGGER) (d 0 instance_of Chain-complex ∧ d 0 has_trigger t 0 ) ⇒ (∃d 1 , d 1 proper_chain-part_of d 0 ∧ d 1 has_trigger t 0 )
In (Poison), the trigger of vulnerability_heart_&_life 0 is poison_ingestion 0 , which is a trigger of vulnerability_heart 0 .

Note, however, that if an aggreg-part is triggered, this does not imply anything for the triggering of the aggreg-complex. For example, in (Tablet), if solubility 1 is triggered by half-tablet 1 _in_solvent, this does not imply the triggering of solubility 0 -as the solubility 2 of half-tablet 2 may not be triggered (in case half-tablet 2 is not put into a solvent). Similarly, in (Poison), Jones might undergo some cardiac arrest which is not due to the ingestion of poison, in which case heart_need 0 is triggered without vulnerability_heart_&_life 0 being triggered.

4.!Composition of various disposition-parthood relations

We will now show how we can use disposition-parthood relations to combine basic dispositions into mereological complexes, by developing case studies involving dominos.

4.1.!Domino arrangement

Let's consider five dominos domino i (for any integer i∈[0,4]) vertically placed on their short edge, such that they will fall if pushed. Each domino i is thus the bearer of a disposition unstability ←i that is realized by domino i falling to the left (a process that is an instance of the defined class Fall ←i ) and triggered when domino i is pushed to the left (Push ←i ); and the disposition unstability →i realized by domino i falling to the right (Fall →i ) when domino i is pushed to the right (Push →i ). It also has a disposition unstability →i /←i to fall to either side (Fall i equivalentTo (Fall ←i or Fall →i )) when pushed (Push i equivalentTo (Push ←i or Push →i )). We can characterize this by stating:

•! unstability →i /←i realized_in only Fall i •! unstability →i /←i has_trigger only Push i However, those two relations do not express the information that a Push ←i always leads to a Fall ←i , and that a Push →i always leads to a Fall →i . Therefore, it is more informative to say that unstability →i /←i has the two mod-parts unstability →i and unstability ←i and to ascribe the respective realizations and triggers to the mod-parts:

•! unstability →i mod-part_of unstability →i /←i •! unstability ←i mod-part_of unstability →i /←i

The five dominos 0 to 4 are arranged as illustrated on figure 1a below (dominos 5 to 8 from figure 1b will be considered later):

Figure 1a

Figure 1b domino 2 is placed to the right (when seen from the domino's side) of domino 1 , such that when domino 1 falls to the right, it pushes domino 2 to the right; therefore, in this configuration, the endphase of an instance of Fall →1 is also an instance of Push →2 . Let's define unstability →1,→2 as the disposition of domino 1 and domino 2 to fall to the right (a process named Fall →1∧→2 = Fall →1 + Fall →2 ) when domino 1 is pushed to the right (Push →1 ). We have:

•! unstability →1 ,→2 realized_in only Fall →1∧→2 •! unstability →1 ,→2 has_trigger only Push →1 However, those two relations do not express the information that there is a causal chain. Therefore, it is more informative to write that unstability →1,→2 is a chain-complex composed by two chain-parts unstability →1 and unstability →2 :

•! unstability →1 chain-part_of unstability →1,→2 •! unstability →2 chain-part_of unstability →1,→2
Suppose that domino 3 is placed next to domino 1 , such that when one of these falls, it does not push the other. Then, the system domino 1+3 = domino 1 + domino 3 has a disposition unstability →1:→3 of both dominos falling to the right (Fall →1∧→3 = Fall →1 + Fall →3 ) when both dominos are pushed to the right (Push →1∧→3 = Push →1 + Push →3 ):

•! unstability →1:→3 realized_in only Fall →1∧→3 •! unstability →1:→3 has_trigger only Push →1∧→3 However, those two relations do not express the information that when an instance push →1 + push →3 causes an instance of fall →1 + fall →3 , it is push →1 that causes fall →1 , and push →3 that causes fall →3 . Therefore, it is more informative to write that unstability →1:→3 is an add-complex with two proper add-parts unstability →1 and unstability →3 :

•! unstability →1 add-part_of unstability →1:→3 •! unstability →3 add-part_of unstability →1:→3 domino 4 is placed to the right of domino 3 (like domino 2 was placed to the right of domino 1 ), such that when domino 3 falls to the right, it pushes domino 4 to the right. The disposition to Fall →3∧→4 when Push →3 is noted unstability →3,→4 .

Finally, domino 0 is placed to the left of domino 1 , such that when domino 1 falls to the left, it pushes domino 0 to the left. The chain-complex disposition to Fall ←1∧←0 when Push ←1 is noted unstability ←1,←0 .

4.2.!Composition of mod-part_of and chain-part_of

We can now combine these dispositions into more complex dispositions. domino 1,2 = domino 1 + domino 2 falls to the right (Fall →1∧→2 ) when domino 1 is pushed to the right (Push →1 ), and domino 1,0 falls to the left (Fall ←1∧←0 ) when domino 1 is pushed to the left (Push ←1 ). We can introduce the corresponding disposition unstability (→1,→2) / (←1,←0) , which is the disposition of [Fall →1∧→2 when Push →1 ] and of [Fall ←1∧←0 when Push ←1 ] (see figure 2). That is, unstability (→1,→2) / (←1,←0) is a mod-complex composed by the mod-parts unstability →1,→2 and unstability ←1,←0 : We have pointed out earlier that the bearer of a mod-part (resp. chain-part) is a part of the bearer of a mod-complex (resp. chain-complex), but not necessarily the same bearer; as a matter of fact:

•! unstability →1,→2
•! The bearer of the chain-complex unstability ←1,←0 is domino 1,0 ; whereas the bearer of its chain-parts unstability ←1 and unstability ←0 are domino 1 and domino 0 , respectively. •! The bearer of the mod-complex unstability (←1,←0) / (→1,→2) is domino 0,1,2 ;

whereas the bearers of its mod-parts unstability (←1,←0) and unstability (→1,→2) are domino 1,0 and domino 1,2 , respectively.

4.3.!Further compositions

We could also express a large variety of dispositions thanks to the various parthood relations introduced here. For example, domino 1,3 has a disposition to [Fall →1∧→3 when Push →1∧→3 ] and to [Fall ←1∧←3 when Push ←1∧←3 ], which is the mod-complex unstability (→1:→3) / (←1:←3) , composed by the add-complexes unstability →1:→3 and unstability ←1:←3 .

To illustrate other configurations, consider the domino arrangement on Figure 1b. domino 5 is placed on the left of domino 6 and domino 7 , in-between, such that if domino 5 falls to the right, then both domino 6 and domino 7 also fall to the right. There is an underlying disposition to Fall →5∧→6∧→7 when Push →5 , which is the add-complex unstability (→5,→6):(→5,→7) composed by the chain-complexes unstability →5,→6 and unstability →5,→7 . Now, domino 8 is placed on the right of domino 6 and domino 7 , such that if either domino 6 or domino 7 falls to the right, then domino 8 also falls to the right. The underlying disposition to [Fall →6∧→8 when Push →6 ] and to [Fall →7∧→8 when Push →7 ] is the mod-complex unstability (→6,→8) / (→7,→8) composed by the chain-complexes unstability →6,→8 and unstability →7 ,→8 .

Suppose now that domino 8 is more massive, such that it needs both domino 6 and domino 7 to fall to the right for domino 8 to also fall to the right. The underlying disposition to Fall →6∧→7∧→8 when Push →6∧→7 is the chain-complex unstability (→6:→7) , →8 composed by the add-complex unstability →6:→7 as well as by unstability →8 .

Finally, we could compose dispositions in more intricate ways, mixing mod-part_of, add-part_of and chain-part_of -consider for example the disposition unstability [(→6:→7) , →8] / [(←6:←7) , ←5] to [Fall →6∧→7∧→8 when Push →6∧→7 ] and to [Fall ←6∧←7∧←5 when Push ←6∧←7 ].

5.!Discussion and conclusion

This paper has presented different ways to compose dispositions together: mod-parthood and aggreg-parthood, and the subrelations of the latter add-parthood and chain-parthood. All of them are subsumed by disp-parthood, which satisfy some axioms (DISP-BEARER), (DISP-TRIGGER) and (DISP-REALIZATION); and each of those subrelations also satisfy more specific axioms.

All these relations of disposition-parthood are partial orders: they are reflexive, transitive and anti-symmetric. To be authentic mereological relations, however, they also need to satisfy axioms of composition and decomposition (see [START_REF] Barton | A modelling pattern for multi-track dispositions for life-science ontologies[END_REF] for a few preliminary remarks concerning mod-part_of). It can easily be seen that composition axioms can be introduced at least for some disposition-parthood relations. For any two dispositions d 1 and d 2 (with bearers b 1 and b 2 ) there exists a mod-complex noted (d 1 /d 2 ) and an add-complex noted (d 1 :d 2 ), which inheres in b 1 +b 2 . On the other hand, there exist a chain complex (d 1 ,d 2 ) (also inhering in b 1 +b 2 ) only if the maximally specified realization class of d 1 has as part the class of triggers of d 2 . By introducing such axioms of composition, we commit ourselves to a lot of strange disposition complexes, such as complexes of dispositions of different bearers, that might even exist at different times. This, however, is no special feature of disp-parthood: the same phenomenon occurs in standard mereology. Moreover, the formal commitment to the existence of these mereological sums does not imply any commitment that all of them have causal or explanatory relevance.

Consider now axioms of decomposition. Although this would require further investigations, the following axiom of complementation [START_REF] Varzi | Mereology[END_REF] seems to be satisfied for P being any of the five relation of disposition-parthood mentioned above, and O the associated relation of overlapping: ¬Pyx → !z "w (Pwz ↔ (Pwy ∧ ¬Owx)) Note that our taxonomy of relations of disposition-parthood is not meant to be exhaustive. For example, one could consider the disposition realized by (only) domino 2 falling to the right when domino 1 is pushed to the right -not to be confused with the above disposition unstability →1,→2 realized by both domino 1 and domino 2 falling to the right when domino 1 is pushed to the right.

By way of conclusion we point out that we expect disposition-parthood to be a powerful tool to explore relations between medical entities that have been formalized as dispositions, such as diseases [START_REF] Scheuermann | Toward an ontological treatment of disease and diagnosis[END_REF][START_REF] Barton | The Cardiovascular Disease Ontology[END_REF] or medical risks [START_REF] Barton | What is a risk? A formal representation of risk of stroke for people with atrial fibrillation[END_REF].

  mod-part_of unstability (→1,→2) / (←1,←0) •! unstability ←1,←0 mod-part_of unstability (→1,→2) / (←1,←0) Moreover, by transitivity of the relation disp-part_of (of which mod-part_of and chain-part_of are subrelations), we can state that unstability (→1,→2) / (←1,←0) is a disp-complex composed by the disp-parts unstability →1 , unstability →2 , unstability ←1 and unstability ←0 .

Figure 2 .

 2 Figure 2. Conjoint use of mod-part_of and chain-part_of

heart_need 0 is the disposition of Jones to Death when Cardiac arrest, •! vulnerability_heart_&_life 0 is the disposition of Jones to

  Cardiac arrest + Death when participating in an instance of Poison ingestion.

	case study (Poison) involves a person Mr. Jones vulnerable to a given kind of poison,
	that causes cardiac arrest and consequently (cerebral) death. Let's introduce the
	following dispositions:
	•! vulnerability_heart 0 is the disposition of Jones to Cardiac arrest when Poison
	ingestion,
	•!

Acknowledgements

AB acknowledges financial support by the "Bourse de fellowship du département de médecine de l'université de Sherbrooke" and the CIHR-funded Quebec SPOR Support Unit.


Lieven Annemans

Jane K Stock

M John

Professor M John Chapman
email: john.chapman@upmc.fr

PCSK9 Inhibition, Atherosclerotic Cardiovascular Disease and Health Economics: Challenges at the Crossroads

HAL is

Preamble

Atherothrombosis, characterized by atherosclerotic lesion disruption with superimposed thrombus formation, is the primary cause of cardiovascular death, the leading cause of mortality world-wide. [START_REF]Cardiovascular diseases. Fact sheet[END_REF] The burden of atherosclerotic cardiovascular disease (ASCVD) is not, however, confined to mortality. With ageing populations, morbidity and disability associated with cardiovascular complications become increasingly relevant. Heart failure, the endpoint of a variety of cardiovascular disease trajectories, most notably ischaemic heart disease in developed regions [START_REF] Ziaeian | Epidemiology and aetiology of heart failure[END_REF] , atrial fibrillation, a complication of acute myocardial infarction (MI) in more than 10% of cases [START_REF] Kundu | Relation of atrial fibrillation in acute myocardial infarction to in-hospital complications and early hospital readmission[END_REF] , and stroke are key examples, as each confers a high risk for hospital (re)admission, multiple pharmacotherapeutic approaches, and interventional procedures. Moreover, improved management in the acute MI setting means that a greater proportion of individuals are surviving and therefore are at high risk of recurrent events. [START_REF]Cardiovascular diseases. Fact sheet[END_REF][START_REF] Jernberg | Cardiovascular risk in post-myocardial infarction patients: nationwide real world data demonstrate the importance of a long-term perspective[END_REF] All of these scenarios reinforce the premise that ASCVD is not just a health issue, but a major economic burden to society, with the global cost estimated to exceed one trillion US dollars by 2030. [START_REF]The global economic burden of non-communicable diseases[END_REF] Increasingly, this burden will be met by middle-income countries as their economies and populations grow and age.

Renewed efforts are needed to address the burden of ASCVD. Inherent to such efforts is a move to personalized management approaches that allow clinicians to target treatments to patients at highest risk. An urgent priority is to adapt public health approaches to risk estimation in order to reflect changing patient demographics. One solution may be incorporation of the SMART (Second manifestations of arterial disease) score, which enables clinicians to differentiate the risk of recurrent events in individuals with clinical ASCVD. [START_REF] Kaasenbrood | Estimated life expectancy without recurrent cardiovascular events in patients with vascular disease: The SMART-REACH Model[END_REF] This approach may facilitate precise targeting of innovative therapies to individuals at highest risk, in whom the benefits are likely to be commensurate with the cost.

The aim of this critical appraisal is to throw down the gauntlet to both health economists and to clinicians involved in the management of ASCVD patients in order to clearly identify the current challenges that are faced at the crossroads between the introduction of an innovative and highly efficacious lipid lowering therapy, i.e. proprotein convertase subtilisin/kexin type 9 (PCSK9) inhibition, and the health economic dimension (such as quality-adjusted life years

[QALYs] gained) intimately linked to such therapy.

Gauntlet No. 1: Clinical challenges

The challenges in ASCVD prevention are multi-tiered. From a public health perspective, promotion of cardiovascular health, specifically targeting smoking prevention/cessation, diet quality and physical inactivity, is a pre-requisite and cost-effective base component of any programme aimed at maintaining low cardiovascular risk. Such approaches are feasible to implement on a population-basis, require only a fraction (<5%) of total health spending even in low-income countries, and confer important societal benefits, not just in terms of cost savings for healthcare systems but also for loss of productivity associated with ASCVD disability and death. [START_REF]The global economic burden of non-communicable diseases[END_REF][START_REF] Barton | Effectiveness and cost effectiveness of cardiovascular disease prevention in whole populations: modelling study[END_REF] It is, however, evident that maintenance of low cardiovascular risk does not eliminate the development of preclinical ASCVD 8 , highlighting the need to target intervention with nonpharmacological approaches as early as possible, ideally to children or adolescents.

In both primary and secondary prevention settings, lowering low-density lipoproteincholesterol (LDL-C), a major modifiable cardiovascular risk factor, must be a central part of any programme for adults. This overarching principle is supported both by evidence for the cumulative arterial burden of LDL, and by genetic data for the beneficial impact of lifelong exposure to lower LDL-C levels on lifetime cardiovascular risk. [START_REF] Ference | Low-density lipoproteins cause atherosclerotic cardiovascular disease. 1. Evidence from genetic, epidemiologic, and clinical studies. A consensus statement from the European Atherosclerosis Society Consensus Panel[END_REF][START_REF] Ference | Variation in PCSK9 and HMGCR and risk of cardiovascular disease and diabetes[END_REF] One of the major challenges in the primary prevention setting is to identify and treat individuals at the highest risk, including those with familial hypercholesterolaemia (FH) before the onset of clinical complications, and non-FH individuals with pre-clinical atherosclerosis. Indeed, detection of subclinical carotid or coronary atherosclerosis with non-invasive imaging may be one cost-effective strategy to improve targeting of therapeutic intervention appropriately, with the potential for gains in healthy life years. [START_REF] Nordestgaard | Familial hypercholesterolaemia is underdiagnosed and undertreated in the general population: guidance for clinicians to prevent coronary heart disease: consensus statement of the European Atherosclerosis Society[END_REF] Yet is clear that clinical reality lags far behind what is required. [START_REF] Wiegman | Familial hypercholesterolaemia in children and adolescents: gaining decades of life by optimizing detection and treatment[END_REF] For individuals with clinical ASCVD, the major challenge is to reduce the risk of recurrent events and the associated burden of hospitalization, revascularization and intensive clinical management.

For both primary and secondary prevention, statins are established first-line therapy for preventing cardiovascular events and are cost-effective in individuals with clinical ASCVD; [START_REF] Collins | Interpretation of the evidence for the efficacy and safety of statin therapy[END_REF][START_REF] Ward | A systematic review and economic evaluation of statins for the prevention of coronary events[END_REF] the addition of ezetimibe provides further benefit. [START_REF] Cannon | Ezetimibe added to statin therapy after acute coronary syndromes[END_REF] Yet even on optimally tolerated statin therapy, with or without ezetimibe, a proportion of high-risk patients fail to attain guidelinerecommended LDL-C goal, in part due to lack of adherence, inter-individual variability in treatment response, and/or inability to tolerate statin therapy, in large part due to muscle symptoms. [START_REF] Reiner | Lipid lowering drug therapy in patients with coronary heart disease from 24 European countries--Findings from the EUROASPIRE IV survey[END_REF][START_REF] Ridker | Percent reduction in LDL cholesterol following high-intensity statin therapy: potential implications for guidelines and for the prescription of emerging lipid-lowering agents[END_REF][START_REF] Stroes | Statin-associated muscle symptoms: impact on statin therapy-European Atherosclerosis Society Consensus Panel Statement on Assessment, Aetiology and Management[END_REF] These individuals have unmet clinical needs, requiring additional potent therapeutic options to attain LDL-C goal and thus reduce the risk of recurrent cardiovascular events.

There is now evidence from randomized controlled trials (RCTs) that new treatments such as PCSK9 inhibitors -which lower LDL-C levels on top of intensive statin therapy in patients with clinical ASCVD-can regress atherosclerosis as exemplified by the GLAGOV trial, [START_REF] Nicholls | Effect of evolocumab on progression of coronary disease in statin-treated patients: The GLAGOV randomized clinical trial[END_REF] and equally improve clinical outcomes and survival significantly, with no evidence to suggest a LDL-C threshold for clinical benefit. [START_REF] Sabatine | Evolocumab and clinical outcomes in patients with cardiovascular disease[END_REF][START_REF] Schwartz | Alirocumab and cardiovascular outcomes after acute coronary syndrome[END_REF][START_REF] Giugliano | Clinical efficacy and safety of achieving very low LDL-cholesterol concentrations with the PCSK9 inhibitor evolocumab: a prespecified secondary analysis of the FOURIER trial[END_REF][START_REF] Szarek | Alirocumab reduces total nonfatal cardiovascular and fatal events in the ODYSSEY OUTCOMES Trial[END_REF] Moreover, the absolute benefit was greater in subjects at higher risk, i.e. with multivessel disease including peripheral arterial disease, [START_REF] Bonaca | Low-density lipoprotein cholesterol lowering with evolocumab and outcomes in patients with peripheral artery disease: insights from the FOURIER Trial (Further Cardiovascular Outcomes Research With PCSK9 Inhibition in Subjects With Elevated Risk)[END_REF] or with recent or recurrent MI. [START_REF] Sabatine | Clinical benefit of evolocumab by severity and extent of coronary artery disease[END_REF] These findings offer the prospect of attainment of guidelinerecommended LDL-C goal in all secondary prevention patients. Furthermore, significant improvement in all-cause mortality in acute coronary syndrome patients over a median follow-up of 2.8 years as recently reported in ODYSSEY OUTCOMES, [START_REF] Szarek | Alirocumab reduces total nonfatal cardiovascular and fatal events in the ODYSSEY OUTCOMES Trial[END_REF] is highly relevant to the longterm trajectory of chronic ASCVD disease in these very high-risk patients. As these highly effective injectable treatments are administered every 2 weeks or monthly, advantages for patient adherence and convenience appear evident.

It is well recognized, however, that there is often disparity between patients in a RCT and those seen in routine care, with the latter typified by a higher prevalence of polyvascular disease and comorbidities. Evidence-based recommendations support the use of innovative therapies such as PCSK9 inhibitors in these patients. [START_REF] Landmesser | Update of ESC/EAS Task Force on practical clinical guidance for proprotein convertase subtilisin/kexin type 9 inhibition in patients with atherosclerotic cardiovascular disease or in familial hypercholesterolaemia[END_REF] The key question then is how to translate these therapies to measures of societal benefit, taking account of the timing and long-term impact of these efficacious treatments on the trajectory of ASCVD. These uncertainties, of which cost effectiveness is a key component, have prompted the publication of a slew of analyses with varying conclusions. Some have indicated that the incremental cost of adding a PCSK9 inhibitor is prohibitive without significant discounting, [START_REF] Arrieta | Updated cost-effectiveness assessments of PCSK9 inhibitors from the perspectives of the health system and private payers: insights derived from the FOURIER Trial[END_REF][START_REF] Kazi | Updated cost-effectiveness analysis of PCSK9 inhibitors based on the results of the FOURIER trial[END_REF][START_REF] Fonarow | Cost-effectiveness of evolocumab therapy for reducing cardiovascular events in patients with atherosclerotic cardiovascular disease[END_REF] whereas others claimed that incorporation of PCSK9 inhibitor treatment in patients with clinical ASCVD or with heterozygous FH, would be below the benchmark or willingness to pay threshold for a QALY gained (typically $50,000/year). [START_REF] Shah | Pharmacoeconomics of PCSK9 inhibitors in 103 hypercholesterolemic patients referred for diagnosis and treatment to a cholesterol treatment center[END_REF][START_REF] Villa | Cost-effectiveness of evolocumab in patients with high cardiovascular risk in Spain[END_REF] These contrasting findings may be due to a number of factors, including the setting (US or a single country in Europe), drug costs, patient characteristics (clinical setting, baseline age, background therapy, and baseline LDL-C levels), the estimated LDL-C and event reductions (either modelled, or based on registry or trial data), the extent to which indirect costs (which are related to productivity losses due to cardiovascular events) were incorporated, and the time horizon and perspective of the health economic analysis.

Clinicians are clearly confused by the different approaches used in these analyses. Is it possible to simplify the health economics modelling in order to anticipate the projected impact of these innovative treatments on CVD trajectory over a decade or more?

Gauntlet No. 2: Health economic challenges

Health economic models have been used for decades to evaluate the potential economic benefit of innovative agents. Not surprisingly, these models have also been applied in assessment of the value of PCSK9 inhibitors. The prevailing perception among clinicians, however, is that the final cost per QALY seems to have 'fallen from the sky'. In large part this is due to the lack of 1) clear information on the parameters considered essential for integration into modelling scenarios, and 2) the necessary support to explain the results of these assessments. Furthermore, large differences in the results reported for different studies do not generate confidence. Many, many questions emerge when clinicians try to understand and interpret these models, as highlighted below.

• Is the considered target population a real-life population or clinical trial population?

Clinical trial populations tend to be more homogeneous than those seen in routine practice due to screening procedures that tend to exclude patients with multiple morbidities, who are likely to be more susceptible to adverse effects. Thus, the trial population may not reflect the clinical reality as shown by event rates that are usually lower than those seen in practice. Sometimes, however, the opposite is true if the clinical trial requires a minimal risk level for patient inclusion.

• Which clinical trial data have been used to express the benefits of a new treatment? Are treatment effects assumed beyond the duration of the trial, and if so, is there the same relative risk reduction, a waning effect or something else?

• How long is the prediction period and does this make clinical sense? Many models use a lifetime horizon which is often perceived by clinicians as beyond a realistic scope ("who are you to predict what's going to happen over the next 30-40 years?").

• How many events are predicted in the scenario without and with the innovation?

Clinicians prefer to see the predicted clinical benefit of the models, not just a cost per QALY. For instance, over a period of 5 years, how many myocardial infarctions are predicted in the scenario without the new medicine and how many in the scenario with the new medicine?

• What are the assumed consequences of the events and which management patterns have been applied? For example, if the patient suffers a stroke, what is the probability that this is disabling and how does this correspond to the clinical reality? How many patients are in permanent residential care following that stroke? What is the assumed follow up care?

• Which disease and treatment trajectories have been assumed? Will avoiding a MI lead to less heart failure in the future? And if so, will only heart failure of ischaemic origin be affected, or is there an impact on all types of heart failures (including valvular)?

• Are composite end points in clinical trials concealing benefit -or lack thereof -on specific events that are included in that endpoint? Should the composite endpoint strategy be revised in this context?

• Finally, and pertinent in the case of the PCSK9 inhibitors, what is the assumed impact on mortality when the duration of the clinical trial is too short or insufficiently powered to show a statistically significant effect? Will avoiding strokes lead to reduced cardiovascular mortality? What are the patterns of fatal events and mortality overall over time? In this context, even though the two completed trials with PCSK9 inhibitors were of relatively short duration (<3 years), an all-cause mortality benefit was observed in acute coronary syndrome patients. [START_REF] Szarek | Alirocumab reduces total nonfatal cardiovascular and fatal events in the ODYSSEY OUTCOMES Trial[END_REF] As long as health economists fail to clearly answer these questions, clinicians will be tempted to ignore the reported findings and, as a consequence, not adapt their practice.

A new health economics scenario for the 21 st century: for and with clinicians

In order to be relevant to clinicians, health economic models should comply with several requirements (see Box 1). Briefly, the model should be clinically validated, transparent, use real world data to provide event rates in the target population and RCT data to provide the relative reduction in risk of these events with any new therapy, and should also consider the clinical realities of the condition and its consequences. Moreover, if the real world population has different characteristics compared to the clinical trial population (older, more comorbidities, higher LDL-C,…) and the relative risk reduction has been shown to be influenced by some of these characteristics, then the relative risk needs to be adjusted. Also, if data on clinically relevant endpoints such as MIs and strokes are lacking, then the assumed relationship between the intermediate endpoint (in this case, LDL-C reduction) and the endpoints (cardiovascular events) should be documented and justified.

Health economic models should also avoid the pitfalls of overclaiming or underclaiming benefits. An example of the former is assumption that effects from LDL-C reduction impact directly the incidence of heart failure despite the lack of evidence for any such relationship. In contrast, underestimating the impact of stroke on quality of life has consequences, as for example, on the calculation of the incremental cost-effectiveness ratio (ICER) in US analyses when stroke was assumed to decrease quality of life by only 3%. [START_REF] Kazi | Updated cost-effectiveness analysis of PCSK9 inhibitors based on the results of the FOURIER trial[END_REF] A key question is how best to relate the impact of innovative treatments on the trajectory of ASCVD. In middle-aged individuals with a first MI, attainment of very low LDL-C levels with early initiation of a PCSK9 inhibitor on the background of statin or statin plus ezetimibe treatments has the potential for large gains in life expectancy and QALYs. Models should clarify what the health benefits are for those patients for whom more aggressive treatment really makes a difference. In those patients where MIs and strokes are prevented, what will be the impact on their lives? These benefits for the patient will undoubtedly translate to societal benefits arising from reduced morbidity and loss of productivity associated with cardiovascular complications.

If, however, the requirements for access to such new treatments are restrictive, then such a promising scenario is unlikely. [START_REF] Doshi | Prior authorization requirements for proprotein convertase subtilisin/kexin type 9 inhibitors across US private and public payers[END_REF] Indeed, some have suggested that despite their established clinical value, perceived costs and budgetary concerns relating to the use of PCSK9 inhibitors in the target patient groups have led to lower than expected uptake of these treatments. [START_REF] Sabatine | Proprotein convertase subtilisin/kexin type 9 (PCSK9) inhibitors: comparing and contrasting guidance across the Atlantic[END_REF] This is a simplistic approach, however, which does not integrate possible rebates and discounts that may be offered by the manufacturers; indeed, recent developments suggest that the budget impact of such treatments is lower than anticipated, [START_REF] Mallya | Budget impact analysis of PCSK9 inhibitors for the management of adult patients with heterozygous familial hypercholesterolemia or clinical atherosclerotic cardiovascular disease[END_REF] especially when weighed against the burden of cardiovascular complications.

Importantly, the outcomes of the model should be clearly reported and validated, using graphs that reflect the predicted number of clinical events over time in the model, both for the control arm (in this case, background statin or statin plus ezetimibe therapy) and the intervention arm (add-on PCSK9 inhibitor). The extent to which the model prediction for the control arm corresponds to the observed event rate in real world datasets should be assessed. If the model underestimates the event rates in the control arm (either single or composite events), then the absolute benefit of the better therapy will be smaller, which will in turn result in a worse costeffectiveness estimate, whereas if baseline event rates are overestimated, then this will result in more refined cost-effectiveness estimates. In either case, the model cannot be considered as validated.

A recommended approach is cross-validation. If the results of model X widely deviate from model Y, the input parameters in X can be modified sequentially to reflect those in Y. The extent to which the results of X evolve towards the results of Y for each modified parameter provides insights into the key drivers of the deviation in results. The onus is on health economists to clarify this, instead of making unconditional claims about the level of costeffectiveness of medicines.

Take home lessons

With increasingly finite resources, the use of innovative therapies needs to be balanced against their cost relative to the substantial burden of associated cardiovascular sequelae. Health economic modelling needs to adapt to the evolving characteristics of patients, in particular taking account of the impact of new therapies, as well as the timing of initiation of such treatments, on the trajectory of ASCVD (Summary Figure). By following these proposed recommendations, health economic models can become informative rather than create confusion. To achieve this, it is imperative that all stakeholders in the healthcare pathway are involved in the production and interpretation of health economic models to ensure appropriate patient access to innovative therapies for prevention of cardiovascular events.

Box 1 .

 1 Essential requirements of health economic models • Clinically validated structure which correctly reflects the (combination of) clinical outcomes in the target population • Transparency, with disclosure and justification of clinical and economic inputs and assumptions • Derived from real world and randomized controlled trial data • Use of age-adjusted event rates, with explanation of the adjustment • Reflect the clinical reality in terms of event and treatment sequences • Account for non-adherence

Acknowledgements:

The Authors are indebted to Dr Henry Ginsberg for insightful discussion.

Responsibility for all opinions, conclusions, and interpretation of data lies with the authors.

Funding: No author received any honoraria in writing this manuscript.

Conflict of interest statement: LA has received honoraria for lectures and advisory boards from Amgen and Sanofi. MJC has received research grants from Amgen, Kowa Europe and Pfizer, and honoraria for participation on speakers' bureau and advisory boards from Akcea, Alexion, Amarin, Amgen, Daiichi-Sankyo, Kowa Europe, Merck/MSD, Pfizer, Sanofi, Regeneron and Unilever. JKS reports no disclosures.

Summary Figure

Essential components of a health economic model for assessment of the cost-effectiveness of an innovative therapy for prevention of cardiovascular events in very high-risk patients.

Abbreviations: ASCVD atherosclerotic cardiovascular disease; MACE major adverse cardiovascular events; QALY quality adjusted life year; RCT randomized controlled trial