Yunsang Kwak

Vu-Hieu Nguyen

Yoann Hériveaux

Pierre Belanger

Junhong Park

Guillaume Haïat
email: guillaume.haiat@cnrs.fr

Ultrasonic assessment of osseointegration phenomena at the bone-implant interface using convolutional neural network

Keywords: Bone-implant interface, osseointegration, quantitative ultrasound, Finite element method, Convolutional neural network. Running title: Implant ultrasound assessment with neural network PACS numbers: 43.80.Ev, 43.80.Vj, 43.20.Gp, 43.20.El

Although endosseous implants are widely used in the clinic, failures still occur and their clinical performance depends on the quality of osseointegration phenomena at the bone-implant interface (BII), which are given by bone ingrowth around the BII. The difficulties to ensure clinical reliability come from the complex nature of this interphase related to the implant surface roughness and the presence of a soft tissue layer (non-mineralized bone tissue) at the BII. The aim of the present study is to develop a method to assess the soft tissue thickness at the BII based on the analysis of its ultrasonic response using simulation based-convolution neural network (CNN). A large-annotated dataset was constructed using a 2-D finite element model in the frequency domain considering a sinusoidal description of the BII. The proposed network was trained by the synthesized ultrasound responses and was validated by a separate dataset from the training process. The linear correlation between actual and estimated soft tissue thickness shows excellent R 2 values equal to 99.52% and 99.65%, and narrow limit of agreement corresponded to [-2.56, 4.32 µm] and [-15.75, 30.35 µm] for the microscopic and macroscopic roughness respectively, supporting the reliability of the proposed assessment for the osseointegration phenomena.

I. Introduction

Although implants are now widely used in dental and orthopedic surgery, there remain risks of failure which may have dramatic consequences and are difficult to anticipate. The surgical success depends on the quality of osseointegration phenomena, which corresponds to the formation of bone tissue in contact with the implant [START_REF] Haiat | Effects of biomechanical properties of the boneimplant interface on dental implant stability: from in silico approaches to the patient's mouth[END_REF]. During osseointegration phenomena, collagen-rich tissue forms around the implant and is then mineralized, which corresponds to a transformation from soft tissue into a solid material [START_REF] Moerman | Structural and mechanical characterisation of the peri-prosthetic tissue surrounding loosened hip prostheses. An explorative study[END_REF]. Aseptic loosening, which depends on osseointegration phenomena, is the most common cause of surgical failure of endosseous implant surgery [START_REF] Pilliar | Observations on the effect of movement on bone ingrowth into porous-surfaced implants[END_REF]. Therefore, the non-invasive assessment of osseointegration phenomena is important in order to maximize the clinical success of such intervention [START_REF] Khan | Osseointegration and more-A review of literature[END_REF]. However, it remains difficult to assess the evolution of the implant stability, which is given by the biomechanical properties of the bone-implant interface (BII) [START_REF] Mathieu | Biomechanical determinants of the stability of dental implants: Influence of the bone-implant interface properties[END_REF].

Different biomechanical techniques have been proposed to investigate the BII properties because techniques using X-rays and Magnetic Resonance Imaging have strong limitations due to the presence of metal [START_REF] Shalabi | Evaluation of bone response to titanium-coated polymethyl methacrylate resin (PMMA) implants by X-ray tomography[END_REF][START_REF] Gill | Assessment of MRI issues at 3-Tesla for metallic surgical implants: findings applied to 61 additional skin closure staples and vessel ligation clips[END_REF]. To investigate the BII properties, impact methods ( [START_REF] Van | The Periotest method for determining implant success[END_REF][START_REF] Schulte | The Periotest method[END_REF][START_REF] Michel | Assessing the acetabular cup implant primary stability by impact analyses: a cadaveric study[END_REF][START_REF] Goossens | Development of an acoustic measurement protocol to monitor acetabular implant fixation in cementless total hip Arthroplasty: A preliminary study[END_REF][START_REF] Dubory | A cadaveric validation of a method based on impact analysis to monitor the femoral stem insertion[END_REF] and resonance frequency analysis [START_REF] Meredith | Quantitative determination of the stability of the implant-tissue interface using resonance frequency analysis[END_REF][START_REF] Georgiou | Accurate diagnosis of hip prosthesis loosening using a vibrational technique[END_REF][START_REF] Pastrav | In vivo evaluation of a vibration analysis technique for the per-operative monitoring of the fixation of hip prostheses[END_REF] have been employed but remain limited in terms of resolution and of sensitivity to investigate the biomechanical properties of the BII [START_REF] Aparicio | Validity and clinical significance of biomechanical testing of implant/bone interface[END_REF][START_REF] Seong | Correlations between physical properties of jawbone and dental implant initial stability[END_REF]. Approaches using quantitative ultrasound (QUS) seem promising to retrieve information on the BII because it is non-invasive, easy to use and nonionizing. Preliminary studies using coin-shaped implant models have evidenced a significant decrease of the reflection coefficient of the BII as a function of healing time [START_REF] Mathieu | Influence of healing time on the ultrasonic response of the bone-implant interface[END_REF][START_REF] Fraulob | Quantitative ultrasound assessment of the influence of roughness and healing time on osseointegration phenomena[END_REF], which can be explained by increases of the bone-implant contact (BIC) ratio, of the bone Young's modulus [START_REF] Vayron | Nanoindentation measurements of biomechanical properties in mature and newly formed bone tissue surrounding an implant[END_REF] and of bone mass density [START_REF] Mathieu | Micro-Brillouin scattering measurements in mature and newly formed bone tissue surrounding an implant[END_REF][START_REF] Vayron | Evolution of bone biomechanical properties at the micrometer scale around titanium implant as a function of healing time[END_REF] as a function of healing time. The sensitivity of the reflection coefficient of the BII to compressive stresses was also investigated [START_REF] Hériveaux | elastography of the boneimplant interface[END_REF]. Moreover, in vitro (Vayron et al., 2018b) and in vivo (Vayron et al., 2014c;Vayron et al., 2018a) studies have proven the potentiality of QUS techniques to evaluate dental implant stability and have evidenced a better resolution compared to the resonance frequency analysis technique, which is currently employed in the clinics. Despite the interesting performances of QUS techniques to retrieve information on the BII properties, experimental approaches are limited when it comes to understand the determinant of the QUS response of the BII because of the various parameters such as surface roughness, bone quality and bone quantity that are impossible to control and that evolve in parallel.

Acoustical modeling is the only way to gain a further understanding of the interaction between ultrasonic waves and the BII. Therefore, numerical studies considering 2-D and 3-D finite element modeling [START_REF] Vayron | Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment[END_REF][START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF][START_REF] Dorogoy | Modeling ultrasonic wave propagation in a dental implant-Bone system[END_REF][START_REF] Kwak | Rattle noise source localization through the time reversal of dispersive vibration signals on a road vehicle[END_REF] have been used to understand wave propagation in a dental implant but considered the BII as fully bonded, neglecting the effect of partial contact and of surface roughness. More recently, a finite element model was developed in the time domain to investigate the sensitivity of the ultrasonic response to surface roughness properties of the BII and to osseointegration processes [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. The implant roughness was first modeled with an idealized sinusoidal profile [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF] and actual implant roughness profiles (Hériveaux et al., 2019a), and the 3D case was eventually considered [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. However, it still remains difficult to perform the inversion of the ultrasound signal in order to determine the biomechanical properties of the BII. Moreover, the computation cost of the time-domain finite element simulation is still high, thus preventing constructing a sufficiently large database to run data-driven approaches.

Data-driven methods are powerful approaches to solve the inverse problem and have significant advantages because they can be adapted to various conditions by using annotated largescale dataset, which requires detailed understanding of the parametric sensitivity. Artificial neural networks with supervised learning have been used successfully in many engineering domains [START_REF] Khan | Classification and diagnostic prediction of cancers using gene expression profiling and artificial neural networks[END_REF][START_REF] Shalabi | Evaluation of bone response to titanium-coated polymethyl methacrylate resin (PMMA) implants by X-ray tomography[END_REF]. In particular, the convolutional neural network (CNN) method showed an excellent performance in terms of image recognition and classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]. Moreover, various applications of the CNN method have been presented including structural health monitoring and source localization [START_REF] Kwak | Acoustic Source Localization via Predicting Propagated Waves on Complex Systems using Time Reversal and Deep Neural Networks[END_REF][START_REF] Toh | Review of Vibration-Based Structural Health Monitoring Using Deep Learning[END_REF]. However, collecting large-scale dataset in the implant research field is not straightforward unlike measuring visual data such as images. CNNs with artificial imaging processes such as coloring, distorting and noising were presented to augmented training dataset numerically [START_REF] Antoniou | Data augmentation generative adversarial networks[END_REF][START_REF] Perez | The effectiveness of data augmentation in image classification using deep learning[END_REF][START_REF] Salamon | Deep convolutional neural networks and data augmentation for environmental sound classification[END_REF][START_REF] Kwak | Convolutional neural network trained with synthetic pseudo-images for detecting an acoustic source[END_REF]. Inspired by researches for the image recognitions, the spectrogram and spacetime matrices were employed to convert input acoustic data into relevant two dimensional color figures [START_REF] Costa | An evaluation of convolutional neural networks for music classification using spectrograms[END_REF][START_REF] Vesperini | Localizing speakers in multiple rooms by using Deep Neural Networks[END_REF]. The aim of this numerical study is to assess osseointegration phenomena by inverting ultrasonic signals using a 2-D convolutional neural network (CNN). Osseointegration phenomena are determined based on the estimation of the soft tissue thickness present at the BII that is obtained by analyzing the interaction of the ultrasonic waves with the BII. To do so, the forward model was modified compared to [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF] since it was derived in the frequency domain and then translated into the time domain, which has the advantage of significantly reducing the computation cost, allowing to construct an annotated large dataset. The CNN based-assessment method was then applied to the aforementioned dataset to assess the osseointegration at the microscopic and macroscopic roughness scales. Our algorithm was verified by separating the validation dataset from the training dataset.

II. Material and methods

A. Description of the problem

We consider a two-dimensional model which consists of three domains, corresponding to the implant (Titanium alloy, Ti-6Al-4V), cortical bone tissue and soft tissue (see Fig. 1a). The incident wave is assumed to be as a pulsed plane wave propagating from up-to-down vertical direction. Following the in vitro experiment setup described in [START_REF] Mathieu | Influence of healing time on the ultrasonic response of the bone-implant interface[END_REF], the objective is to determine the radiofrequency signal corresponding to the wave reflected by the BII.

After the implant insertion, collagen-rich tissues form around the implant and are then mineralized, which corresponds to a transformation from soft tissue into a solid material during osseointegration phenomena [START_REF] Moerman | Structural and mechanical characterisation of the peri-prosthetic tissue surrounding loosened hip prostheses. An explorative study[END_REF]. Based on this transformation, we assumed that the soft tissue thickness decreases when osseointegration phenomena progress, which is a similar model than that used in (Raffa et al., 2019;Raffa et al., 2020) to model osseointegration phenomena in the static regime. Moreover, we have shown experimentally that the ultrasound reflection coefficient of the BII is very sensitive to the properties of bone tissue located around the BII [START_REF] Mathieu | Influence of healing time on the ultrasonic response of the bone-implant interface[END_REF][START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Effect of the roughness parameters[END_REF][START_REF] Fraulob | Quantitative ultrasound assessment of the influence of roughness and healing time on osseointegration phenomena[END_REF].

Due to the symmetry of the problem, the streamlined model described in Fig. 1 was introduced similarly as in our previous studies [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. Only a single half-sine period of the interface was considered (Figure 1b), which is sufficient to simulate the propagation of the acoustic wave using symmetrical boundary conditions in the interfaces perpendicular to the direction x. The implant surface roughness was modeled by a sinusoidal function of amplitude h and half-period L.

The interface roughness was analyzed by considering two different scales: denoted microscopic and macroscopic roughness in what follows. The microscopic roughness corresponds to the situation obtained by sandblasting and acid etching [START_REF] Strnad | Corrosion rate of sand blasted and acid etched Ti6Al4V for dental implants[END_REF] of the implant surface, while the implant geometries such as threading of dental implant and pedicle screws correspond to the macroscopic roughness. For the microscopic case, the reference values of the surface roughness amplitude h and half-period L are 5 µm and 50 µm, respectively [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]. For the macroscopic case, the reference values of the surface roughness amplitude h and half-period L are 360 µm and 900 µm, respectively [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF].

Soft tissue was considered between the bone and the implant (see Figure 1a) in order to model non-mineralized fibrous tissue that may be present at the BII in the case of nonosseointegrated implants or just after surgery. The soft tissues thickness W was defined as the distance between the highest point of the implant profile and the bone layer (see Figure 1b). The progression of the osseointegration phenomena leads to a decrease of the soft tissue thickness W. The size of the implant and bone domains (Hb and HTi, respectively) were much larger than the characteristic lengths of the interface roughness so that they may be considered as semi-infinite domains. Lengths of the implant HTi and cortical bone Hb domains in vertical direction are 1 mm in the macroscopic case and 50 µm in the microscopic case, which corresponds to two times the wavelength in the y direction. In this study, we assumed that all materials have linear isotropic elastic behavior. The material properties of all media considered in the numerical simulation were taken from [START_REF] Njeh | An in vitro investigation of the dependence on sample thickness of the speed of sound along the specimen[END_REF][START_REF] Pattijn | The resonance frequencies and mode shapes of dental implants: Rigid body behaviour versus bending behaviour. A numerical approach[END_REF][START_REF] Pattijn | Resonance frequency analysis of implants in the guinea pig model: influence of boundary conditions and orientation of the transducer[END_REF][START_REF] Haïat | Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: Application to axial transmission[END_REF] (Table I).

Table I. Material properties used in the numerical simulations. Values taken from [START_REF] Njeh | An in vitro investigation of the dependence on sample thickness of the speed of sound along the specimen[END_REF][START_REF] Pattijn | The resonance frequencies and mode shapes of dental implants: Rigid body behaviour versus bending behaviour. A numerical approach[END_REF][START_REF] Pattijn | Resonance frequency analysis of implants in the guinea pig model: influence of boundary conditions and orientation of the transducer[END_REF][START_REF] Haïat | Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: Application to axial transmission[END_REF].

Cp (m•s -1 ) Cs (m•s -1 ) ρ(kg•m -3 )
Soft tissue (liquid) 1500 10 1000 Titanium 5810 3115 4420

Cortical bone tissue 4000 1800 1850

B. Time-domain computation of the reflection coefficient of the bone-implant interface

In this study, we are interested in the determination of the reflection coefficient of the BII that is evaluated based on the amplitudes of the incident and reflected waves in the time-domain. As it has been done experimentally, the average amplitude of the reflected waves was evaluated over a horizontal surface located at a given distance from the BII.

Implant

To solve the aforementioned problem, the approach developed in our previous studies [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF] consisted of i) considering a finite element discretization used to derive the system of dynamic equations and ii) using direct time integration schemes. The finite element simulation was performed by using the Comsol Multiphysics software (COMSOL AB, Stockholm, Sweden) using the 𝛼-generalized scheme.

However, such an approach involving a direct time-integration is computationally expensive, hampering their implementation into deep learning-based analyses because of the significant number of realizations required. To reduce computation time, another computational procedure is

proposed in what follows. The dynamic problem is firstly solved in the frequency domain, and the solution of reflected wave is then reconstructed in the time domain using the inverse Fourier transform.

C. Resolution in the frequency domain

For each subdomain #i (i = 1,2,3 correspond to titanium implant, soft tissues and cortical bone respectively) defined in the previous subsection, the dynamic equations in the frequency domain read:

-𝜔 2 𝜌 (i) 𝒖 (𝑖) -𝑑𝑖𝑣 𝝈 (i) = 𝟎 ( 1 )
where 𝜌 is the mass density, 𝒖 is the displacement vector and 𝝈 is the stress tensor in the subdomain #i. The constitutive relation using Hooke's laws is given by:

𝝈 (i) = λ (i) tr(𝛆 (i) ) 𝑰 + 2 μ (i) 𝛆 (i) , ( 2 )
where 𝜺 = 𝟏 𝟐 (grad𝒖 + grad𝒖 𝑇 ) is the strain tensor in the subdomain #i.

The continuity of displacement and stress fields lead to the following boundary conditions at the interface between subdomain #i and #k:

{ 𝒖 ( i ) = 𝒖 ( k ) 𝝈 ( i ) 𝒏 = 𝝈 ( k ) 𝒏 at interface (𝑖 -𝑘) ( 3)
where {i, k} = {1,2}, {1,3} or {2,3} and 𝒏 is the unitary vector normal to the interface.

Here, we considered an incident wave propagating in the y-direction, perpendicularly to the BII (see Figure 1). The displacement field 𝒖 (1) in the subdomain #1 can be decomposed into two parts representing the incident and reflected wave fields as follows:

𝒖 (1) = 𝒖 𝐼 + 𝒖 𝑅 , ( 4
)
where 𝒖 𝐼 = {0, 𝑈 𝐼 𝑒 𝑗𝑘 𝑝 (1) 𝑦 } in which 𝑘 𝑝 (1) and 𝑈 𝐼 are the wavenumber and the amplitude of the incident P-wave, respectively; 𝑘 𝑝 (1) = 𝜔/𝑐 𝑝 (1) . As 𝒖 𝐼 is supposed to be known, we substituted Eq.

(4) into Eqs. (1-2) to formulate a system of equations with respect to 𝒖 𝑅 as:

-𝜔 2 𝜌 (1) 𝒖 𝑅 -𝑑𝑖𝑣 𝝈 R = 𝟎, ( 5
) 𝝈 R = λ (1) tr(𝛆 R ) 𝑰 + 2 μ (1) 𝛆 R ( 6 )
In Eqs.

(5-6), all the terms related to 𝒖 𝐼 did not appear because 𝒖 𝐼 is also a solution of the dynamic equation ( 1). However, 𝒖 𝐼 cannot be neglected in the continuity conditions which are now expressed by:

𝒖 𝑅 + 𝒖 I = 𝒖 ( k ) (𝝈 R + 𝝈 I )𝒏 = 𝝈 ( k ) 𝒏 at intefaces (1 -𝑘) 𝑤𝑖𝑡ℎ 𝑘 = {2,3}
where:

𝝈 I = [ 2𝑗𝑘 𝑝
(1) 𝜇 (1) 𝑈 𝐼 𝑒 𝑗𝑘 𝑝

(1) 𝑦 0 0 𝑗𝑘 𝑝 (1) (𝜆 (1) + 𝜇 (1) )𝑈 𝐼 𝑒 𝑗𝑘 𝑝

(1) 𝑦 ] ( 7 )

The radiation conditions at the infinity read 𝒖 R → 𝟎 when 𝑦 → +∞ , ( 8)

𝒖 ( 3 ) → 𝟎 when 𝑦 → -∞ , ( 9 )
The system of equations ((Eqs.

(5-9)) was implemented using the finite element software COMSOL Multiphysics (Stockholm, Sweden). As the domain's size should be finite, to prevent the non-physical reflected waves from the upper and lower boundaries (Eqs. 8&9), two absorbing layers are introduced at 𝑦 = 𝐻Ti and at 𝑦 = -𝐻b. By introducing absorbing layers, the radiation conditions can be represented artificially, resulting in smoothly attenuated radiative waves. With the presence of absorbing layers, at the upper boundary of the implant domain (Ω (1) ) and at the lower boundary of bone domain (Ω (3) ), fixed boundaries are imposed:

𝒖 R = 𝟎 at 𝑦 = 𝐻Ti , ( 10)

𝒖 ( 3 ) = 𝟎 at 𝑦 = -𝐻b , ( 11 )
In addition, the symmetry conditions also impose that 𝑢 𝑥 = 0 at the lateral boundaries 𝑥 = 0 and 𝑥 = 𝐿.

In each subdomain, the element size was chosen to be equal to 𝜆 𝑚𝑖𝑛 20 ⁄ , where 𝜆 𝑚𝑖𝑛 denotes the shortest wavelength in the considered domain (i.e. the wavelength of shear wave). The implant and bone subdomains were meshed by structured quadrangular quadratic elements, and the soft tissue subdomain was meshed with triangular quadratic elements. Solving the system (Eqs.

5-9) leads to the solutions of 𝒖 R , 𝒖 ( 2 ) , 𝒖 ( 3 ) as a function of 𝒖 I and 𝜔 . The evaluation of the reflection coefficient requires calculating the average displacement over a horizontal section of the implant domain which is located at an ordinate 𝑦 𝑚 from the bone-implant interface. In the frequency domain, this average displacement is given by:

Φ 𝑅 (𝜔) = 1 𝐿 ∫ 𝑈 2 𝑅 (𝑥, 𝑦 𝑚 , 𝜔)𝑑𝑥 = 𝐻(𝜔)𝑈 𝐼 𝐿 0 (𝜔) ( 12 )
where 𝐻(𝜔) denotes the transfer function and 𝑈 𝐼 is the amplitude of the incident wave.

D. Determination of the solution in the time-domain

The incident plane wave was given by a uniform vertical stress 𝜎 22 𝐼 which has a broadband ultrasonic pulse form defined by:

𝜎 22 𝐼 (𝑡) = 𝐴𝑒 -4(𝑓 𝑐 𝑡-1) 2 sin(2𝜋𝑓 𝑐 𝑡) , ( 13 )
where A indicates an arbitrary constant and fc = 10 MHz indicates its central frequency chosen based on the QUS device developed by our group to assess dental implant stability [START_REF] Mathieu | Micro-Brillouin scattering measurements in mature and newly formed bone tissue surrounding an implant[END_REF][START_REF] Vayron | Variation of the ultrasonic response of a dental implant embedded in tricalcium silicate-based cement under cyclic loading[END_REF]Vayron et al., 2014a;Vayron et al., 2014c;Vayron et al., 2018a;Vayron et al., 2018b). The frequency response corresponding to the average displacement Φ 𝑅 (𝜔) of the reflected wave at the surface located at 𝑦 𝑚 can be expressed as:

Φ 𝑅 (𝜔) = 𝐻(𝜔)𝑆(𝜔)/(𝑗𝜔 𝜌 (1) 𝐶 𝑝 (1) ) ( 14
)
where 𝐶 𝑝 (1) denotes the compressional velocity in the subdomain #1 and 𝑆(𝜔) denotes the Fourier transform of 𝜎 22 𝐼 (𝑡) given by 𝑆(𝜔) = ∫ 𝜎 22 𝐼 (𝑡) . 𝑒 -𝑗𝜔𝑡 𝑑𝑡.

Zero-padding was performed in the frequency domain to obtain a high-resolution time response as follow:

Φ * (𝑛∆𝜔) = { Φ R (𝑛∆𝜔) 0 Φ 𝑅 ((𝑁 * -𝑛)∆𝜔) 0 < 𝑛 < 𝑁 -1 𝑁 < 𝑛 < 𝑁 * -𝑁 + 1 𝑁 * -𝑁 + 2 < 𝑛 < 𝑁 * -1 , ( 15
)
where ∆𝜔 is the frequency step of the simulation, N is the number of discrete data obtained by the simulation, N * is the extended number of data by the zero-padding. The time-domain response with higher resolution 𝑠(𝑡) can be calculated by the inverse Fourier transform of Φ * . Notably, the frequency bandwidth analyzed herein was limited to 30 MHz and the frequency step of the numerical model was set to 0.5 MHz (N = 60), resulting in 2 µs for the time duration and 33.3 ns for the time step. Considering that a tight time step is necessary for a satisfactory numerical analysis for the BII [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF], the extended number N * was set to 840 to increase time resolution of the response by 14 times, resulting in 28 µs for the time duration and 2.4 ns of the adjusted time step. Accordingly, we obtained the high-resolution time response efficiently from only 60 simulated data in the frequency domain, while the simulation based on the direct timeintegration would be computationally more expensive.

The reflection coefficient 𝑅 was determined based on the ratio of the amplitudes of the reflected and incident waves in the time domain. The first signal corresponds to the uniform incident vertical displacement, noted 𝑠 𝑖 (𝑡). The second signal corresponds to 𝑠 𝑟 (𝑡). The maximum amplitude of the modulus of the Hilbert's transform (corresponding to the envelope) of 𝑠 𝑖 (𝑡) and 𝑠 𝑟 (𝑡) were noted 𝐴 𝑖 and 𝐴 𝑟 , respectively. The reflection coefficient R in amplitude then writes:

𝑅 = 𝐴 𝑟 𝐴 𝑖 ( 16 )

E. Composition of the training dataset

To synthesize the training dataset, the ultrasound simulations were performed with the range of variation of the BII roughness parameters, considering the results obtained from our previous studies [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. In the case of the microscopic roughness, the training data set was constructed with various values of h comprised between 0 and 10 µm with a step of 1 µm and of W comprised between 0 and 100 µm with a step of 0.5 µm. In the case of the macroscopic roughness, the training data set was constructed with various values of h comprised between 310 and 410 µm with a step of 10 µm and of W comprised between 0 and 1.5h with a step of 0.01h. To employ separate training and validation dataset, all reference cases for both roughness scales (h = 5 µm for the microscopic scale

F. Layer configuration of the 2-D convolution neural network

The layer structure of the 2-D CNN employed herein is illustrated in Figure 2. The structure consists of two convolution layers. We used the Adam optimization algorithm for training our network because it requires little hyperparameter tuning effort for achieving adequate results [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF]. A mini-batch size was 120, and 100 epochs were conducted. 32 convolution filters were used in the first and the second convolution layer. The size of the convolution kernel was 4 by 4 with strides of two. To prevent change of the size of images, zeros were added at edges [START_REF] Chetlur | cudnn: Efficient primitives for deep learning[END_REF]. The batch normalization was performed with each convolution layer [START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF]. The rectified linear unit (ReLU) was used as the activation function [START_REF] Nair | Rectified linear units improve restricted boltzmann machines[END_REF]. The max pooling layer was performed with a 4 by 4 matrix and strides of two. After the second max pooling layer, 10 % of neurons were dropped out to prevent over-fitting [START_REF] Hinton | Improving neural networks by preventing co-adaptation of feature detectors[END_REF]. The flatten-layer was used before the first fully connected layer. The layer structure ends with two fully connected layers, which have 32 neurons et al., 2015). In other words, we can examine the reliability of networks from the visualized information. To examine the training process with the dataset built by the BII forward model, the feature map of each convolution layer was analyzed for the reference roughness cases of both scales and compared between different soft tissue thickness cases.

The bias of the estimation and the 95% limits of agreement (LOA), which has been widely used for evaluating regression results [START_REF] Park | mHealth spectroscopy of blood hemoglobin with spectral superresolution[END_REF], were computed to examine the estimated soft tissue thickness by the proposed assessment network. The bias 𝑚 ̅ corresponds to the averaged value of the differences between the estimated and actual soft tissue thicknesses. The 95% LOA is defined as:

LOA = 𝑚 ̅ ± 1.96√∑ (𝑑 𝑖 -𝑚 ̅ ) 2 𝑛 𝑖=1 (𝑛 -1) ⁄
where 𝑑 is the difference between the estimated and actual soft tissue thicknesses, and n is the total number of validation dataset. The linear correlation R between two thicknesses are evaluated such that:

𝑅 = 𝐸[( 𝑊 𝑎 -𝐸(𝑊 𝑎 ))(𝑊 𝑒 -𝐸(𝑊 𝑒 )) ] √ 𝐸 [(𝑊 𝑎 -𝐸(𝑊 𝑎 )) 2 ] • 𝐸 [(𝑊 𝑒 -𝐸(𝑊 𝑒 )) 2 ] ⁄
where E(…) indicates an averaged value, 𝑊 𝑎 and 𝑊 𝑒 are the actual and estimated values of the soft tissue thickness, respectively.

III. Results

A. Validation of the forward model

Figure 3 shows the comparison between the radiofrequency signals corresponding to the ultrasound responses of the bone-implant interface for h = 360 µm and L = 900 µm (macroscopic case) and obtained using the proposed model (in the frequency domain) and using the method described in [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF] (in the time domain). A good agreement is obtained between the two models, which constitutes a validation of the forward model. [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF].

B. Spectrogram of the rf signals

Figure 5 shows the spectrograms of the signal corresponding to the reflection of the ultrasonic wave on the BII when the soft tissue thickness varies in the microscopic case (Figures 5a-d) and in the macroscopic case (Figures 5e-h

C. Feature map of the proposed network

We looked over the feature map of the convolutional layer for the reference cases of the microscopic and macroscopic roughness cases, which was excluded from the training dataset. For the microscopic roughness, the two representative cases corresponding to W = 50, 100 µm were

IV. Discussion

A large-scale annotated dataset with diverse conditions should be considered to employ deep learning methods. However, accumulating a large-scale dataset in biomechanical engineering may be difficult because i) of the difficulty to obtain an important number of samples due to ethical requirements and ii) it is impossible to control all parameters affecting the ultrasound response of the BII since they vary simultaneously [START_REF] Fraulob | Quantitative ultrasound assessment of the influence of roughness and healing time on osseointegration phenomena[END_REF] as a function of healing time. Here, we present a method based on the deep convolutional neural network to assess osseointegration phenomena. The method is trained with a synthesized dataset corresponding to numerical results.

The originality of the proposed approach lies on hybridizing CNN and numerical simulation in order to assess osseointegration phenomena. Our method relies on the estimation of soft tissue thickness in the microscopic and macroscopic roughness cases based on the analysis of the signal corresponding to the reflection of an ultrasonic wave on the BII. To do so, a numerical model of the interaction between an ultrasonic wave and the BII has been validated by comparison with a model developed previously in the time domain [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF]Hériveaux et al., 2019a;[START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. The advantage of the present numerical model is that i) only 60 simulated (respectively 52 min) were needed with our previous model [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF] in the time domain.

Although the training dataset employed for the proposed data-driven approach is relatively small (3600 computations), the reliability of the method was examined by employing an independent validation dataset. The estimations (Figure 7) were implemented with the dataset of the reference surface roughness cases, which was excluded from the training process. Moreover, the results obtained for both scales exhibit excellent R 2 values (99.53 and 99.65%) and narrow

LOAs ([-2.56, 4.32 µm] and [-15.75, 30.35 µm]) supports the reliability of the proposed CNN method for assessing the effects of osseointegration phenomena. More importantly, the numerical approach under consideration can be used as a complementary tool to compose the data-driven approach for implant clinical studies, which remains difficult to access using experimental approaches only.

Experimental approaches have already shown that the reflection coefficient of the BII is sensitive to healing time [START_REF] Mathieu | Influence of healing time on the ultrasonic response of the bone-implant interface[END_REF], to the implant surface roughness [START_REF] Fraulob | Quantitative ultrasound assessment of the influence of roughness and healing time on osseointegration phenomena[END_REF] as well as to the mechanical loading applied to the BII [START_REF] Hériveaux | elastography of the boneimplant interface[END_REF]. Based on the aforementioned experimental results, an experimental set-up was developed to assess dental implant primary and secondary stability (Vayron et al., 2014a;Vayron et al., 2014c;Vayron et al., 2018a;Vayron et al., 2018b). However, although ultrasound techniques show great potentiality to evaluate dental implant stability, it remains difficult to control the parameters varying simultaneously during osseointegration phenomena, hampering its practical applications.

Furthermore, no inversion approach had been carried out and the present study shows the feasibility of applying a model-based inversion technique by using CNN to retrieve quantitative information on the BII. Such an approach should be applied to experimental data in the future.

Bone (and newly formed bone) tissues are complex materials which may be modeled as heterogeneous, anisotropic viscoelastic and poroelastic media. In the present work, which is the first tentative use of CNNs method in the context of BII characterization using ultrasound, we chose to use the simplest model to describe the bone mechanical behavior because we aim at focusing on the BIC ratio, which is the main parameter of interest. A more detailed description of bone behavior (and in particular taking into account bone anisotropy) would require a much larger set of parameters in the model, and consequently, will increase significantly computational cost.

Moreover, it is expected that bone anisotropy has a relatively weak effect of the ultrasound response of the BII because ultrasound at 10 MHz is highly attenuated in bone [START_REF] Sasso | Frequency dependence of ultrasonic attenuation in bovine cortical bone: an in vitro study[END_REF][START_REF] Sasso | Dependence of ultrasonic attenuation on bone mass and microstructure in bovine cortical bone[END_REF] and should not propagate over a large distance.

This study has several limitations. First, a 2-D sinusoidal function is used to describe the implant surface roughness, similarly to what was done in [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF], which constitutes an important approximation allowing to reduce the number of parameters affecting the ultrasonic response of the BII. However, this sinusoidal description may be considered as realistic in the macroscopic case because it models implant threading present in dental implants specifically.

For the microscopic case, it has been shown that an actual implant surface profile may be replaced by a sinusoidal profile without modifying significantly the dependence of its ultrasonic response as a function of the soft tissue thickness (Hériveaux et al., 2019a). Although considering 3-D surface function would be more relevant to describe the ultrasound response of the BII, the computational cost required to perform the direct simulations using the 3-D model is significantly higher than for 2-D simulation. Moreover, we have shown that the 2-D approximation may also be considered as realistic compared to the 3-D case [START_REF] Hériveaux | Reflection of an ultrasonic wave on the bone-implant interface: Comparison of two-dimensional and three-dimensional numerical models[END_REF]. The two aforementioned studies justify the use of a 2-D sinusoidal description of the implant surface roughness. Second, only normal incidence of the ultrasonic wave is reported, because it corresponds to the experimental situation of interest [START_REF] Mathieu | Influence of healing time on the ultrasonic response of the bone-implant interface[END_REF][START_REF] Fraulob | Quantitative ultrasound assessment of the influence of roughness and healing time on osseointegration phenomena[END_REF] and future studies should also consider the normal incidence. Third, this study only considered a center frequency of 10 MHz for the incident wave because it corresponds to the frequency used in the QUS device developed by our group (Vayron et al., 2018a). Fourth, we only considered the first reflection of the ultrasound waves on the BII. Analyzing the entire reflection signals (i.e. the coda) would require heavy computation cost and this aspect is left to future studies.

V. Conclusion

FIG 1 .

 1 FIG 1. (a) Schematic illustration of the geometrical configuration of the BII considered in the numerical simulation. (b) Description of the streamlined model.

  and h = 360 µm for the macroscopic scale) were excluded from the training dataset. The excluded dataset was then employed only to validate our method, resulting in a ratio of the validation data to the training data equal to 0.1.The ultrasound radiofrequency signals obtained using the numerical model for each set of parameters (h, W) corresponding to a given roughness and state of osseointegration were transformed into time-frequency spectrograms using short-time Fourier transform (STFT). In this study, the spectrograms were used as the input data of the 2-D CNN. We employed the rectangular window function for calculation of the spectrograms, which are not post-processed to reflect the magnitude of the radio frequency signals as they are. Considering the resolution of the spectrogram in time and frequency domains, we chose the window size and overlap ratio to be 2 7 and 7/8, respectively, resulting in 0.04 µs time step and 3.125 MHz frequency step. Moreover, since the emitted signal is a pulse wave with a center frequency of 10 MHz (see Figure3), the spectrogram above 1 µs and 37.5 MHz in time and frequency domain respectively were excluded from the analysis.

  FIG 2. Layer structure of the proposed CNN to assess the soft tissue thickness W. W e corresponds to the estimated soft tissue thickness.

FIG. 3 .

 3 FIG. 3. Radiofrequency signals corresponding to the ultrasound responses of the bone-implant interface for h = 360 µm, L = 900 µm (macroscopic case) and W/h = (a) 0, (b) 0.5, (c) 1. The solid

Figure 4

 4 Figure 4 shows the variation of the reflection coefficient R for different values of the roughness amplitude h as function of the soft tissue thickness W in the microscopic case (L = 50 µm) and as a function of W/h in the macroscopic case (L = 900 µm). The dashed lines correspond to the results obtained in (Hériveaux et al., 2018) (in the time domain) and the symbols indicate the results obtained using the proposed model in the frequency domain. As shown in Figure 4, a good agreement is obtained between the two models. The average relative error is equal to 1.8 % in the microscopic case and to 1.4 % in the macroscopic case, which may be explained by numerical errors. Again, Figures 3&4 clearly support the validity of the numerical model developed herein.

  FIG. 5. Spectrograms of the ultrasonic signals as a function of the soft tissue thickness. (a)-(d) For the microscopic case, h = 5 µm, L = 50 µm and (a): W = 0 µm, (b): 25 µm, (c): 50 µm, (d): 100 µm. (e)-(h) For the macroscopic case, h = 360 µm, L = 900 µm and (e): W/h = 0, (f): 0.5, (g): 1.0, (h): 1.5.

FIG. 6 .

 6 FIG. 6. Representative feature maps of the convolution layers for the reference BII roughness cases, separated from the training dataset. (a)-(d) For h = 5 µm, L = 50 µm for the microscopic roughness, the results from the first and the second convolution layers for the case of W = 50 µm ((a)&(c), respectively) and 100 µm ((b)&(d), respectively). (e)-(h) For h = 360 µm, L = 900 µm for the macroscopic roughness, the results from the first and the second convolution layers for the case of W/h = 0.5 ((e)&(g), respectively) and 1.0 ((f)&(h), respectively).

  frequency domain are needed to obtain the signal in the time domain and ii) a single transfer function of the BII can be used to obtain simulated reflections for different acoustical sources by a simple convolution calculation instead of simulations for all cases. The computation time necessary to simulate the ultrasonic response of the BII in the microscopic (respectively macroscopic) case using the model developed herein is equal to 40 sec (respectively 3 min), which allows us to form the large-annotated training dataset with affordable time cost, whereas 12 min

A

  simulation-based convolutional neural network method was derived in order to assess osseointegration phenomena at the BII. The large-annotated dataset was formed by taking advantage of an efficient ultrasound simulation of the BII. The proposed numerical model in the frequency domain allowed employing numerous simulations efficiently and was validated by a comparison with another model developed in the time domain. The synthesized ultrasound responses were converted into spectrograms and trained to the proposed network structure. The feature maps clearly demonstrated evident differences according to the soft tissue thickness, supporting the reliability of the training process on the proposed network. The reported assessment network for the osseointegration phenomena supports the feasibility of ultrasound analysis to be transformed to noninvasive applications for the bone-implant domain. Moreover, the present approach opens new paths in the development of model-based inversion methods to retrieve the properties of the BII. the overall research. Y.K., V.H.N and G.H. mainly wrote the manuscript. All of the authors discussed the results and the manuscript.

FIG 1 .FIG 2 .FIG. 3 .FIG. 4 .

 1234 FIG 1. (a) Schematic illustration of the geometrical configuration of the BII considered in the numerical simulation. (b) Description of the streamlined model.

FIG. 5 .FIG. 6 .FIG. 7 .

 567 FIG. 5. Spectrograms of the ultrasonic signals as a function of the soft tissue thickness. (a)-(d) For the microscopic case, h = 5 µm, L = 50 µm and (a): W = 0 µm, (b): 25 µm, (c): 50 µm, (d): 100 µm. (e)-(h) For the macroscopic case, h = 360 µm, L = 900 µm and (e): W/h = 0, (f): 0.5, (g): 1.0, (h): 1.5.

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 682001, project ERC Consolidator Grant 2015 BoneImplant), the National Research Foundation of Korea (NRF) (No. 2019K1A3A1A21031280) and the Partenariat Hubert Curien (PHC) STAR 2019 program (No 43572QM).

Competing interests

All of the authors declare no competing interests.

Author contributions

Y.K. developed the analytical model and conducted the analyses. Y.H. and P.B. worked on the finite element analysis. J.P. helped to analyze the neural network. V.H. N. and G.H. supervised


Vasil Penchev
email: vasildinev@gmail.com

Modal History versus Counterfactual History: History as Intention

Keywords: counterfactual history, historiography, hermeneutical circle, historical objectivity, historical phenomenon, historical reduction, historical subjectivity, history as intention, modal history

The distinction of whether real or counterfactual history makes sense only post factum. However, modal history is to be defined only as ones' intention and thus, ex-ante. Modal history is probable history, and its probability is subjective. One needs phenomenological "epoché" in relation to its reality (respectively, counterfactuality). Thus, modal history describes historical "phenomena" in Husserl's sense and would need a specific application of phenomenological reduction, which can be called historical reduction. Modal history doubles history just as the recorded history of historiography does it. That doubling is a necessary condition of historical objectivity including one's subjectivity: whether actors', ex-ante or historians', post factum. The objectivity doubled by ones' subjectivity constitutes a "hermeneutical circle".

I MODAL HISTORY AND HISTORY AS INTENTION

Here are a few main obstacles for history to be an objective science: History is usually classified as a humanitarian science. It can be compared with other humanitarian science such as philosophical metaphysics being unfalsifiable definitively. History is traditionally also unfalsifiable, but in a different sense. Its subject is postulated not to admit "conditional mood" as far as all historical facts are granted to be unique without any alternatives. However, Popper's falsifiability as a condition of any objective science needs for any historical fact to be at least thinkable also as a refutable conjecture.

However, some historians state that conditional mood is "forbidden" in history: If a fact granted to be historical and thus real is admissible not to have happened, one can suggest an alternative course of history, which also should not happen as far as the alternative version of the historical fact at issue would have many causal consequences further: they would not happen in general, though. Following the same idea, one would be to substitute any real fact by a relevant bifurcation containing at least one alternative different from that which has taken place really. Then, the branch after any bifurcation of those is to be written down right in the "forbidden" conditional mood.

Counterfactual history can be considered as an example of history in a conditional mood. Then one can define "counterfactual history" as an alternative branch of history, unreal and written in the discourse of conditional mood. However, counterfactual history seems to be rather literature than science, even not to speak about an objective science: Indeed, the only tool relevant to writing counterfactual history is the author's imagination and even, "wild fantasy" without any rational procedure of verification. So, the falsifiability, for which counterfactual history was introduced, turns out to be unattainable again in counterfactual history not less than in the standard historiography.

Rather unexpectedly, but quantum mechanics can offer a paradigm for history to be an objective science:

Anyway, one can admit that the failure is due to the paradigm of classical objectivity being irrelevant to historical objectivity. Indeed, classical objectivity suggests a "transparent observer" not influencing definitively on the fact at issue. However, any historian "observes" the corresponding fact only by discourse describing the fact not transparently: any historical fact exists only as recorded being past definitively.

So, the generalized objectivity of quantum mechanics including the "apparatus" (respectively the "observer") can be more relevant to history than the "transparent" paradigm of classical science. For example, those histories being many rather than a single one can occur in different Feynman's pathways:

Meaning the paradigm of quantum mechanics to history, one can suggest many histories not interacting with each other, but ongoing as if simultaneously. One may involve the visualization of the interpretation "by many worlds and histories" in quantum mechanics, in which any world and its history is equally real and an objective probability can be assigned to each of them according to the probability of the corresponding alternative in its beginning. At last, the assigned probability can be interpreted as quantum (i.e. both objective and subjective), and the history of the world, as a Feynman pathway.

One can distinguish real, alternative, and counterfactual history from each other. Meaning the idea of many histories in Feynman's pathways, one can distinguish real, alternative and counterfactual histories as follows:

"Real history" is a single one definitively, and the observer is within it: the historical description is internal, including both observer and discourse within the observed and described.

"Alternative history" is a real history, however, described by an external observer out of it: e.g. the history of China observed from Europe.

"Counterfactual history" is similar to an alternative history, but unlike it not being real: e.g. the history meant by Philip Dick's "The Man in the High Castle".

The distinction of historical actors versus scientists, historians is obvious. Two fundamentally inverse viewpoints to history exist: that of historical actors involved in history and creating it versus that of scientists, historians only describing it "outside" and observing it by the distance of time. The former viewpoint is ex-ante, and the latter is post factum. Nonetheless, both are partly similar to each other, sharing ones' human subjectivity being inseparable from the historical process itself and by itself. As psychology, history shares troubles about its relation to the modern cognitive episteme directed to the gapped distinction of the philosophical "subject and object" (or the Cartesian "mind and body").

One can infer further the distinction of "subjective history" versus "objective history". Indeed, following the dualistic paradigm of science and cognition, one can distinguish disjunctively "subjective history" from "objective history", namely:

There always exists some historical process by itself and independent of any human subjectivity.

There exist humans such as historical actors or scientists related directly to the historical process at issue: the former ones can influence it, but the latter ones cannot. So, history as science refers only to those who cannot influence on historical facts just according to the dualistic paradigm.

Once one has involved "subjective history", "modal history" can be situated within it. "Modal history" can be introduced as a kind of subjective history also and not worse even within the framework of Cartesian dualism. Historians study objectively the way for the historical actors' views, opinions, outlooks, conceptions, ideas, judgements, etc. to have been influencing the historical process "by itself" therefore changing it more or less essentially. So, those "mental facts" can be considered in accordance with the modern paradigm of scientific objectivity. An objective modal history of mental historical facts seems to be possible, not less that of the happened historical facts.

Furthermore, one can distinguish the objectivity of history by the historians versus that by the actors' modal history. Thus, the standard paradigm of scientifity applied as to history suggests for scientists, historians to be transparent observers studying only objective facts whether material or mental and not influencing them in any way.

However, the historians choose which facts to be studied as well as the discourse by which to be represented: so, the same history written by different historians especially in different epochs can be essentially and even absolutely different. Thus, the subjectivity of historians can be not less crucial than actors' as to the written history: one can introduce the historians' modalities as another subject of "modal history".

Even more, historians' modality can be equated to actors' modality both being studied by modal history even in a way not to be distinguishable from each other. Nonetheless, those modalities conserve a polar opposition: the former is ex-ante, but the latter is post factum. The one can be transformed into the other only by reversing the direction of time being the "normal" one only for the actors. For example, the conception of "hermeneutical circle" can link them to each other as a shared "hermeneutical modality", to which each of them is only the one "half".

Following the same trace, the hermeneutics of intention would be relevant. One can complement the hermeneutics of fact by the hermeneutics of intention and even consider them as complementary to each other. However, the "hermeneutics of intention" is an unambiguous concept being able to mean as an interpretation of intention by historians, i.e. post factum, as the intention itself as an interpretation of fact, i.e. ex-ante. The latter hermeneutics is related to historians' interpretation of the fact as dual to it and able to represent it ex-ante. The former hermeneutics is also related to the same historians' interpretations as dual but being co-temporal to it, i.e. again ex-ante.

However, the opposite viewpoint of intention as hermeneutics is not less relevant. One can consider intention as hermeneutics in detail. The historical actors are deprived of the opportunity to observe the historical process as a whole, as accomplished, or as historians are able. They are involved immanently in the historical process, but they can influence it therefor. A "metaphor of equation" can elucidate the interrelation. "Historical discourse can be equated to historical intention": on virtue of doubling reality whether past (the former) or future (the latter).

Intention is doubling. One can state that historical intention does the same as historical discourse, namely doubling reality though ex-ante or post factum correspondingly. So, one can concentrate on the necessity of any historical fact to be doubled as a definitive condition of it, i.e. being just a historical fact. That doubling is necessary fundamentally: to constitute the fact in relation to the totality or as a "particular totality". Indeed, doubling the fact by its counterpart whether external as its discursive image or internal as intentions to be influenced on the future is its transcendental condition, i.e. to be relatable to the totality.

Transcendentality can be demonstrated by intention as well. Still one metaphor might be coined: the historians are "Brecht historical actors" (and the historical actors themselves ar "Stanislavski ones"), or: "Stanislavski historians going beyond portraying by discourse in order to become actors literally". Then, one can understand intention as a key to transcendentality for all those involved in the historical process as its actors or as if "actors recreating it by Stanislavski's method". The transcendental viewpoint to history or its philosophy identify historical actors and historians as two modifications of the same transcendental essence.

The concept of "transcendental intention" can be a relevant generalization. Intention being a general key to transcendentality making it accessible "inside of the totality" can be conceptualized as "transcendental intention" or the "intention of transcendentality". As if the transcendentality itself is gifted by the human capability to intend. The necessary condition of intention is one's temporality, i.e. being within time. So, the idea of transcendental intention gives the transcendentality though being not less atemporal, i.e. out of time, by the property to be within it in virtue of the totality of the totality.

II FOLLOWING HUSSERL'S PHENOMENOLOGY AS A METHODOLOGY OF HISTORY

Discourse and intention can be as opposed as unified. The scientists historians can be as opposed as unified with the historic actors as to any historic process or facts. The aspect of their unification can be thought of as an underlying structure, the inseparable "discourse & intention". The temporality of the same structure turns out to be reversible necessarily since it has unified the description "post factum" (for discourse and directed "backwards" in time) with representation "ex-ante" (for intention and directed forwards in time). The state of "discourse & intention" seems to be a coherent state in the sense of quantum mechanics.

Husserl will be followed explicitly in a few papers (1927,1911) of his.

One can think of a new "epoché" opposed to Husserl's in a sense. Husserl's phenomenology uses the concept of "epoché" unifying reality & conscious, or the Cartesians "body" & "mind", or German classical philosophy's "subject" and "object" meaning the concept of phenomenon as what "remains" regardless of which member be chosen (the former or the latter) of any opposition of those. That kind of "epoché" can be called objective and opposed to a conjectural "subjective epoche", which is to be introduced. It should mean an epoché in relation to "which subject": for example, whether the historical subjects are historical actors and their intentions or scientists, historians with their corresponding discourses.

Then, the objectivity of history can be defined by a relevant "epoche". If one grants that kind of "subjective epoché" as to history, historical action and historical study would turn out to be indistinguishable from each other. Any historical action can be considered as a corresponding historical discourse or transformable into it, and vice versa as well. The objectivity of history is to be defined as being regardless of "which subject", first of all, which kind: whether historians or actors, but furthermore as regardless of which among any representatives of both groups. In other words, the objective historical cognition is to be independent of its subject whether involved into and within the historical process at issue or not.

Meaning the same viewpoint, the standard historiography turns out to be subjective. Meaning that objectivity of history in relation to "subjective epoché" as in the previous paragraph: the alleged "objectivity of the standard historiography" is to be re-estimated as seeming, ostensible. It privileges the viewpoint of the historians, post factum, but hides or even imposes a "taboo" on how the "historical facts" and "relevant discourse" had been chosen. Both "historical facts" and "relevant discourse" suggest an implicit subject and thus, subjectivity is even "tabooed". Consequently, the objectivity of the standard historiography is alleged, ostensible, and grounded on an arbitrarily subjectivity withal forbidden for criticism, discussion, or even articulation.

Thus, the concept of "epoché" for history seems to be necessary. Meaning that real subjectivity of the standard historiography, the necessity of "subjective epoché" for history can be justified as follows. Objectivity as being independent of any subject means for history to be invariant to the choice of subject, especially whether actors or scientists, but postulating the availability of subject as necessary for historical cognition for its specific engagement with the investigated process. Thus, the classical ideal of scientific objectivity of the "transparent observer" is inapplicable as to historical cognition: its observer influences on the studied necessarily. So, its objectivity is forced to consist in being independent of that influence just by means of that "subjective epoché". As Husserl's epoché implies his concept of phenomenon, the modified "subjective epoché" would be to imply a kind of "subjective phenomenon", which can be called "historical phenomenon" in the present context. What remains after Husserl's "epoché to reality" is his "phenomenon" being a new interpretation of the namesake concept of transcendentalism in the final analysis though it seems to be quite different at first glance.

So, involving his doctrine of phenomenology called to study right "phenomena", Husserl rejected Cartesian dualism and the cognitive episteme of Modernity at all therefore. Indeed, all phenomenology would be to be located in the gap between "body" and "mind", "subject" and "object", and thus, inaccessible in the framework of dualistic episteme.

Once the concept of "historical epoche" is involved as a corollary and application of Husserl's epoché as to the particular reality of historical facts, this admits an analogical continuation to "historical phenomenon". It can be defined explicitly as what remains from a historical fact if its necessary subjectivity identifies the actors' intentions ex-ante and the historians' discourses post factum. "Historical phenomenon" just as Husserl's "phenomenon" rejects the modern dualistic episteme for it would be "zero", i.e. would not make sense within its framework.

Husserl suggested the concepts of eidetic, phenomenological and transcendental reductions. If one considers "epoché" as an counterpart of "phenomenological reduction", the relevant idea can be extended: (1) to mathematics and its foundations ("eidetic reduction"); (2) to philosophy and its foundations ("transcendental reduction"); and (3) even to pure, phenomenological psychology being a possible foundation of the science of psychology (again "phenomenological reduction" but understood already as "psychological reduction" in the present context).

One can introduce "historical reduction" following Husserl's reductions. Thus, after transforming Husserl's "epoché" into "historical epoché" and meaning Husserl's lesson of extending "phenomenological reduction" into "psychological, transcendental and eidetic reduction", one can attempt still one application, now towards history: "historical reduction":

Its result is to be "historical phenomenon" already defined. It can be also explicitly directed to that description of historical fact independent of its author, and especially of author's belonging whether to historical actors or to historians.

One can link the concept of hermeneutic circle to both "historical phenomenon" and "historical reduction". Then, the ontological "hermeneutic circle" involved by Heidegger (1927) and applied to philosophy and methodology of history by Gadamer (1960) can be understand in the present context so:

"Hermeneutic circle" means the unification of "intention" (ex-ante) and "discourse" (post factum), and even, their identification. Thus, "hermeneutic circle" is both a pathway and working method for historical reduction progressing to historical phenomenon.

Writing means doubling. Derrida (1967;1967a) was the first philosopher who paid attention to the ontological necessity of writing. Writing means what is written to be doubled regardless of being real or not. Thus, writing is a practical and common procedure of Husserl's "epoché" realizing any text (implicitly written) to be understandable philosophically as a phenomenon and the phenomenology itself to be the ontological foundation of human culture being written or documented definitively. Consequently, history as a part of human culture and science need be written therefore doubling the described historical facts in discourse.

Writing implies information and communication, then, time. Any writing including historical represents a series of choices which are conserved and reproducible. Thus, a relevant quality or quantity of information can be assigned to that writing as a text though it refers not only to the explicit text (being "logocentric" by Derrida's term) but rather to all non-happened alternatives of each choice, and therefore constituting its context. Writing implies still one, withal fundamental and transcendental dimension of information: it is always a choice for something whether a fact or an idea to be written, i.e. doubled and thus transformed into a phenomenon.

The doubling of historical process by its record brings it out of temporality therefore giving it a meaning independent of time and permanently conserving it (furthermore, as a text being still one and different historical fact coinciding with itself). Thus, writing, regardless of what exactly has been written, is a properly philosophical procedure and method able to generate ontology of anything just out of bringing out of time and describing it "dioptrically" or "relatively": once, within time, and twice, out of time.

Physics, especially quantum mechanics and the theory of quantum information, implicitly means the converse ontology of creation: anything as information and out of time to be investigated as a physical process within time.

The necessity of written history follows doubling. After Gadamer, from Heidegger's hermeneutics as fundamental ontology to the hermeneutic methodology of history, one can think of the necessity of written history as to any historical research and its extremely restricted subject. It needs discourse (whatever it be) as a relevant mirror image of its actors' subjectivity embedded in their intentions. The choice of a certain discourse corresponds to a statement about the influence distribution among all historical actors relevant to what has happened really ("factually"). Then, the practical necessity of written history consists in the importance of actors' intentions as an inseparable part of historical fact, only by which it can be understood in one way or another.

History is to be situated between modal intention and written fact. Meaning that necessity of written history being both ontological and methodological, one can see history as crucified on the opposition of modal intention & written fact. The actors' modal intention and the historians' written fact can be considered as complementary (in the generalized sense of Niels Bohr) therefore following opposite time directions: ex-ante versus post factum correspondingly. History as an objective science is able to exist only by the tension of modal intention and written fact, but doomed to Sisyphus's work for reconciling them.

III INSTEAD OF CONCLUSION: KINDS OF MODAL HISTORY

Modal history can be considered from the viewpoint of intentions, but not less, from that of facts. Modal history seems to be opposed to historiography studying established and unconditional facts just only after they have happened as such. Thus, modal history is to correspond to actors' intentions describing them as a kind of special mental facts as well as their influence to shape the real historical process. Modal history should consider history ex-ante and by means of actors' intentions realized in the real history only partly, and therefore, the measure of that "partly" can be accepted to correspond to the quality of modality as to modal history. Modal history can be considered as the history of those intentions influenced the real course of history more or less.

Modal history can study the facts of intentions as they have been written in advance as programs, plans, ideas of change, manifestos or as memoirs of actors, serving to concentrate a fan of individual intentions into a single direction therefore able to influence history by their unification. Modal history can study the relation of those facts of intentions to real historical facts. Furthermore, modal history can investigate the doubling of facts as both modal and real to be a necessary condition for history to happen.

At last, modal history can be seen as shaped by the tension between a fact and its intention. Modal history can be understood as the history itself if the latter is situated in the tension of intention and fact: "crucified" between the normal course of time, ex-ante, and the reverse time of interpretation, post factum. Modal history would reconcile them and their complementarity in hermeneutical circles, whether in philosophical and historical hermeneutics or by the hermeneutical historical methodology suggested by Gadamer. Furthermore, modal history can research the correspondence or transition of historical intentions into historical discourses sharing similar modality.

Modal history can be compared with historiography. Modal history offers still one viewpoint to history and different from that of the standard historiography: a binocular viewpoint to it. Thus, modal history is crucially necessary for establishing the objectivity of history because the objectivity of history cannot be that of classical science and its "transparent observer". Its creators are involved in it, and even its historians write it in addition though post factum, therefore creating it again, but invisibly and restricting themselves to the "facts".

A few ideas of modal history can be concluded. Modal history can be understood and accordingly defined in different ways even partly incommensurable to each other: (1) modal history of intentions relevant to history; (2) modal history of historical discourses;

(3) modal history of both doubling and objectivity as to historical facts; (4) modal history as the hermeneutics of historical facts; (5) modal history of historical modalities.