Maria Letizia Raffa

Vu-Hieu Nguyen

Philippe Hernigou

Charles-Henri Flouzat- Lachaniette

Guillaume Haiat
email: guillaume.haiat@univ-paris-est.fr

Stress shielding at the bone-implant interface: influence of surface roughness and of the bone-implant contact ratio

Keywords: bone-implant interface, surface roughness, stress-shielding, osseointegration, finite element modeling

Short and long-term stabilities of cementless implants are strongly determined by the interfacial load transfer between implants and bone tissue. Stress-shielding effects arise from shear stresses due to the difference of material properties between bone and the implant. It remains difficult to measure the stress field in periprosthetic bone tissue. This study proposes to investigate the dependence of the stress field in periprosthetic bone tissue on i) the implant surface roughness, ii) material properties of bone and of the implant, iii) the bone-implant contact ratio. To do so, a microscale 2-D finite element model of an osseointegrated boneimplant interface was developed where the surface roughness was modeled by a sinusoidal surface. The results show that the isostatic pressure is not affected by the presence of the bone-implant interface while shear stresses arise due to the combined effects of a geometrical singularity (for low surface roughness) and of shear stresses at the bone-implant interface (for high surface roughness). Stress-shielding effects are likely to be more important when the bone-implant contact ratio value is low, which corresponds to a case of relatively low implant stability. Shear stress reach a maximum value at a distance from the interface comprised between 0 and 0.1 time roughness wavelength λ and tend to 0 at a distance from the implant surface higher than λ, independently from bone-implant contact ratio and waviness ratio. A comparison with an analytical model allows validating the numerical results. Future work should use the present approach to model osseointegration phenomena.

Introduction

Cementless implants have been used clinically for more than forty years, leading to important progresses in orthopedic surgery. However, despite their routine clinical use, short and longterm implant failures still occur and remain difficult to anticipate. After surgery, the high contrast of mechanical properties between periprosthetic bone tissue and the implant biomaterial may causes bone resorption [START_REF] Paul | Strength requirements for internal and external prostheses[END_REF] , a phenomenon defined as "stress-shielding" that is visible radiographically [START_REF] Engh | Clinical consequences of stress shielding after porous-coated total hip arthroplasty[END_REF] . Stress-shielding may cause aseptic loosening and a loss of bone mineral density that increases the risk of periprosthetic fracture and makes revision surgery more difficult [START_REF] Kusano | Preoperative Canal Bone Ratio is Related to High-Degree Stress Shielding: A Minimum 5-Year Follow-Up Study of a Proximally Hydroxyapatite-Coated Straight Tapered Titanium Femoral Component[END_REF] . Although some studies on total hip arthroplasty found that stress-shielding may also have no clinical consequences [START_REF] Engh | Clinical consequences of stress shielding after porous-coated total hip arthroplasty[END_REF] , even at a very-long term with anatomic hydroxyapatite femoral components with high wear rates [START_REF] García-Rey | Outcome of a hemispherical porous-coated acetabular component with a proximally hydroxyapatite-coated anatomical femoral component: an update at 23 to 26 years' follow-up[END_REF] , monitoring and preventing stressshielding has been recognized as an important action to obtain long-term fixation of joint arthroplasties [START_REF] Kusano | Preoperative Canal Bone Ratio is Related to High-Degree Stress Shielding: A Minimum 5-Year Follow-Up Study of a Proximally Hydroxyapatite-Coated Straight Tapered Titanium Femoral Component[END_REF] .

Different authors have developed new biomaterials with lower mechanical properties compared to titanium alloy, leading to a reduction of stress-shielding related effects [START_REF] Arabnejad | Fully porous 3D printed titanium femoral stem to reduce stress-shielding following total hip arthroplasty[END_REF][START_REF] Korabi | On stress/strain shielding and the material stiffness paradigm for dental implants[END_REF] .

Long-term stability (also referred to as "secondary" stability) is obtained after the healing processes corresponding to osseointegration phenomena [START_REF] Albrektsson | Osteoinduction, osteoconduction and osseointegration[END_REF] . The implant primary stability is reinforced by the creation and maturation of newly formed bone tissue around the boneimplant interface (BII) [START_REF] Abrahamsson | Early bone formation adjacent to rough and turned endosseous implant surfaces. An experimental study in the dog[END_REF][START_REF] Berglundh | Bone healing at implants with a fluoride-modified surface: an experimental study in dogs[END_REF] . Osseointegration phenomena are stimulated by stresses applied to bone tissue through the BII and the bone microstructure has been shown to be aligned along the principal local loading direction, suggesting that mechanical stimuli act as local regulators [START_REF] Albrektsson | Osseointegrated oral implants: a Swedish multicenter study of 8139 consecutively inserted Nobelpharma implants[END_REF] . The quality of osseointegration phenomena is strongly influenced by the implant surface roughness [START_REF] Orr | Mechanisms of mechanotransduction[END_REF] , which can be obtained using different processes such as sand blasting [START_REF] Rønold | Effect of micro-roughness produced by TiO2 blasting-tensile testing of bone attachment by using coin-shaped implants[END_REF] , plasma spraying [START_REF] Wennerberg | Nanostructures and hydrophilicity influence osseointegration: a biomechanical study in the rabbit tibia[END_REF] or laser blasting [START_REF] Pető | Surface treatment of screw shaped titanium dental implants by high intensity laser pulses[END_REF] . Interfacial load transfer between implants and bone tissue determines osseointegration phenomena. Many biomechanical factors, including i) external loading, ii) implant material properties, geometry and surface properties and iii) periprosthetic bone tissue quality and quantity (defined by the bone-implant contact ratio, denoted as BIC) strongly affect stress and strain fields distribution around osseointegrated implants. All these intercorrelated features have a multiscale (from the nanometric scale up to the organ scale) and multitime (from seconds up to several months) nature and contribute to the evolution of the biomechanical properties of the BII. As a consequence, the mechanical and microstructural properties of bone tissue around the implant surface (at a distance of up to around 200µm from the implant) are determinant for the evolution of the implant stability as well as for the surgical success [START_REF] Götz | Effect of surface finish on the osseointegration of laser-treated titanium alloy implants[END_REF] .

Despite the development of experimental techniques based on acoustical methods [START_REF] Albrektsson | Osseointegrated oral implants: a Swedish multicenter study of 8139 consecutively inserted Nobelpharma implants[END_REF][START_REF] Deng | Influence of osseointegration degree and pattern on resonance frequency in the assessment of dental implant stability using finite element analysis[END_REF][START_REF] Michel | Assessing the acetabular cup implant primary stability by impact analysis: a cadaveric study[END_REF][START_REF] Vayron | Ultrasonic evaluation of dental implant osseointegration[END_REF] to characterize the BII properties, it remains difficult to measure the biomechanical properties of the BII in vivo and there is a lack of experimental data at the scale of 1 to 100 µm [START_REF] Hecht | Magnetic resonance imaging susceptibility artifacts due to metallic foreign bodies[END_REF] . In this context, finite element (FE) modeling represent an interesting modality because all parameters can be controlled and investigated independently, which is not possible when following an experimental approach. Numerical simulations have been extensively used to investigate the stress field in bone tissue at the scale of the implant (macroscale) [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][START_REF] Ryu | The influence of thread geometry on implant osseointegration under immediate loading: a literature review[END_REF][START_REF] Baggi | Stress-based performance evaluation of osseointegrated dental implants by finite-element simulation[END_REF][START_REF] Natali | Analysis of bone-implant interaction phenomena by using a numerical approach[END_REF][START_REF] Baggi | The influence of implant diameter and length on stress distribution of osseointegrated implants related to crestal bone geometry: a three-dimensional finite element analysis[END_REF][START_REF] Maceri | Optimal mechanical design of anatomical post-systems for endodontic restoration[END_REF] , allowing to investigate the influence of implant design and mechanical properties on the implant stability.

Nevertheless, the effects of changes of micromechanical properties (such as the BIC ratio, surrounding bone properties and the implant surface roughness) on the spatial variation of the stress field around the BII remain unclear.

The aim of this study is to investigate the dependence of the local stress field in bone tissue located around the BII on the implant mechanical properties and environment. To do so, a microscale 2-D FE model based on previous work by authors [START_REF] Raffa | Micromechanical modeling of the contact stiffness of an osseointegrated bone-implant interface[END_REF] is proposed. The model accounts for the effects of implant surface roughness, of the bone and implant stiffness and of the BIC ratio on the local stress field in bone tissue under tensile loading. The spatial variation of the stress field in the direction perpendicular to the implant interface is also investigated. The implant surface in contact with bone tissue was assumed to be rough and was modeled by a sinusoidal description, similarly as what was done in previous studies [START_REF] Johnson | The contact of elastic regular wavy surfaces[END_REF][START_REF] Ciavarella | Adhesive rough contacts near complete contact[END_REF][START_REF] Yastrebov | The contact of elastic regular wavy surfaces revisited[END_REF] . The onedimensional sinusoidal surface of the implant of amplitude 2∆ and wavelength λ was geometrically defined as:

Material and methods

Mechanical model

𝐹(𝑥) = ∆ [1 -cos ( 2π𝑥 λ )] , 𝑥 ∈ [0, λ 2 ] ( 1
)
where F is the coordinate of the implant surface as a function of the local abscissa x, as shown in Fig. 1c. Note that the standard parameters used in surface engineering to describe a surface roughness (i.e. the arithmetical mean roughness Ra and the mean spacing Sm) can be related to ∆ and λ by : R a ≅ 2∆ 𝜋 and S m ≅ λ. The waviness ratio ∆/λ was assumed as a model parameter varying between 0.01 and 0.5. We verified that computations realized with different values of ∆ and λ but identical values of waviness ratio ∆/λ gave the same results, which justifies the choice of the waviness ratio as a suitable parameter of the propsed model. The lowest limit (0.01) of the waviness ratio represents a "microscopic" roughness corresponding to an implant surface roughness obtained by sandblasting and/or acid etching, while the highest limit corresponds to a "macroscopic" roughness due to the implant threading, for example in the case of the pedicle screws [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] . The value 0.1 was taken as the reference value for ∆/λ in what follows.

Due to the symmetry of the proposed configuration, a region of interest (ROI) comprising a half period λ/2 of the BII (see Fig. 1b,c) was considered. The origin of the reference system was defined at the point O shown in Fig. 1c. The ROI shown in Fig. 1c comprised bone tissue and implant domains having a same thickness H = 1 mm along the y-direction.

The BIC ratio, which corresponds to the ratio of the bone in direct contact with the implant surface, was described by the parameter h defined in Fig. 1c. Due to the sinusoidal geometry of the implant roughness, the BIC ratio can be obtained geometrically as follows:

BIC = 𝐿 𝑃 𝐿 𝑇 = 1 - 𝐸( 2π𝑥 𝑇 λ |-4𝜋 2 ( Δ 𝜆 ) 2 ) 2𝐸(-4𝜋 2 ( Δ 𝜆 ) 2 ) , (2)
where LP and LT represent the arc length of the sinusoidal implant surface in contact with bone tissue and the total arc length of the implant boundary, respectively. d𝑡, which represent the complete and incomplete elliptic integral of the second kind, respectively. In Eq. ( 2), xT corresponds to the abscissa of the point T, as indicated in Fig. 1c.

The BIC ratio is a parameter of the proposed model and it assumed to vary between 5% and 80% [START_REF] Götz | Effect of surface finish on the osseointegration of laser-treated titanium alloy implants[END_REF] . For each BIC, ∆ and λ values the parameter h (see Fig. 1c) was calculated following this two-step procedure:

i. the value of abscissa x T was explicitly derived from Eq. (2);

ii. the obtained abscissa was injected in Eq. ( 1) to calculate the corresponding value of h.

Materials

All materials were assumed to be linear-elastic and to have homogeneous isotropic mechanical properties. Three different implant materials used in orthopedic applications were considered: i) a titanium alloy (Ti-6Al-4V) with a Young's modulus E of 113 GPa All materials had a Poisson ratio ν equal to 0.3.

Boundary conditions and assumptions

The boundaries conditions, represented in Fig. 1c, were symmetric (u 𝑥 = 0) on the boundaries parallel to the y-axis. The boundary condition was fixed (u 𝑦 = 0) on the lower boundary (y = -H) of the bone domain. A uniaxial normal tension 𝜎 0 = 25 MPa was applied on the upper implant boundary (y = 2Δ+H). Continuity in terms of displacement and normal stress fields holds at the contacting surfaces between bone and implant, which corresponds to a perfect contact condition. Since the out-of-plane dimension of the ROI was assumed to be larger than its height (2𝐻 + 2Δ) and the applied load was homogeneous, the assumption of plane strain was made.

Numerical simulations

A standard linear elastic problem was solved by the finite element (FE) method. All numerical analyses were carried out using Comsol Multiphysics ® (Stockholm, Sweden).

Mesh generation and convergence

The finite element mesh used to run the numerical simulation, which was slightly changed depending on the geometrical parameters, typically contained around 5700 second-order triangular Lagrange elements, leading to a global system with about 21700 degrees of freedom. The interpolation functions for the displacement field were quadratic.

A convergence study was performed to choose the suitable element size for all values of ∆/λ considered in the present study. Note that the mesh size had to be lower at the tip of the noncontact zone at the BII in order to better describe the stress concentration, which occurs especially for lower values of ∆/λ. The minimum element size was set equal to 3×10 -5 m.

With the chosen mesh size, the local error related to the displacement along y u 𝑦 , is

‖u 𝑦 𝑟 -u 𝑦 𝑎 ‖ 𝐿 2 ‖u 𝑦 𝑟 ‖ 𝐿 2
= 0.006%, where ‖‖ 𝐿 2 indicates the L 2 norm, u 𝑦 𝑟 and u 𝑦 𝑎 are respectively the reference displacement obtained with a finest mesh size and the approximated displacement calculated with the chosen mesh size (defined through the convergence study).

Parametric study and stress field indicators

Parametrical analyses were carried out in order to investigate the influence of the BIC ratio (between 5% and 80%) and of the waviness ratio ∆/λ (between 0.01 and 0. In order to characterize the spatial variation of the stress fields in bone tissue, two parameters were chosen: the maximal shear stress 𝜏 𝑚𝑎𝑥 and the isostatic pressure𝑝 𝑖𝑠𝑜 . Both parameters are associated to the principal stresses of the material as follows:

𝜏 max (𝑥, 𝑦 ̃) = 𝜎 𝑝1 -𝜎 𝑝3 2 , ( 3
)
𝑝 𝑖𝑠𝑜 (𝑥, 𝑦 ̃) = 𝜎 𝑝1 +𝜎 𝑝2 +𝜎 𝑝3 3 , (4)
where 𝑦 ̃= -𝑦 𝜆 is the non-dimensional coordinate indicating the distance from the implant surface (see Fig. 1c), x is the local abscissa and 𝜎 𝑝1 , 𝜎 𝑝2 , 𝜎 𝑝3 were the first, second and third principal stresses in bone tissue, respectively (with 𝜎 𝑝1 > 𝜎 𝑝2 > 𝜎 𝑝3 ).

The mean value of 𝜏 𝑚𝑎𝑥 and of 𝑝 𝑖𝑠𝑜 as a function of x was then determined for each values of 𝑦 ̃ following:

〈𝜏 𝑚𝑎𝑥 〉(𝑦 ̃) = 2 𝜆 ∫ 𝜏 𝑚𝑎𝑥 (𝑥, 𝑦 ̃) 𝜆 2 0 𝑑𝑥, (5)
〈𝑝 𝑖𝑠𝑜 〉(𝑦 ̃) = 2 𝜆 ∫ 𝑝 𝑖𝑠𝑜 (𝑥, 𝑦 ̃) 𝜆 2 0 𝑑𝑥. (6)
The average values 〈𝜏 𝑚𝑎𝑥 〉(𝑦 ̃) and 〈𝑝 𝑖𝑠𝑜 〉(𝑦 ̃) were used as indicators of the level of the stress field in periprosthetic bone tissue.

In order to compare the results obtained with different implant materials, the following parameters were defined:

∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 (𝑦 ̃) = 2 𝜆 ∫ |𝜏 𝑚𝑎𝑥 𝑇𝑖 (𝑥, 𝑦 ̃)-𝜏 𝑚𝑎𝑥 𝑍𝑟 (𝑥, 𝑦 ̃)| 𝜆 2 0 𝑑𝑥, (7)
∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 (𝑦 ̃) = 2 𝜆 ∫ |𝜏 𝑚𝑎𝑥 𝑇𝑖 (𝑥, 𝑦 ̃)-𝜏 𝑚𝑎𝑥 𝐵𝑃𝐴 (𝑥, 𝑦 ̃)| 𝜆 2 0 𝑑𝑥, (8)
where 𝜏 𝑚𝑎𝑥 𝑇𝑖 (𝑥, 𝑦 ̃) (respectively 𝜏 𝑚𝑎𝑥 𝑍𝑟 (𝑥, 𝑦 ̃) and 𝜏 𝑚𝑎𝑥 𝐵𝑃𝐴 (𝑥, 𝑦 ̃)) corresponds to the value of the maximum shear stress when considering titanium alloy (respectively Ti-Nb-Zr alloy and metal-polymer composite) as the biomaterial used for the implant (obtained by Eq. ( 3)). The indicator ∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 (𝑦 ̃) (respectively ∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 (𝑦 ̃)) corresponds to the difference between the maximum shear stress in bone when the implant is made of titanium alloy and the maximum shear stress in bone when implant is made of Ti-Nb-Zr alloy (respectively metal-polymer composite) averaged along x at each distance 𝑦 ̃ from the implant surface.

Note that ∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 (𝑦 ̃) and ∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 (𝑦 ̃) depend on the waviness ratio ∆/λ and are indicators of the reduction of the stress-shielding effect due to the modification of the implant material.

Results

Effect of the variation of the BIC ratio

〈𝑝 𝑖𝑠𝑜 〉/𝜎 0 was shown to weakly vary as a function of 𝑦 ̃. For all values of the BIC and of the waviness ratio ∆/λ, the maximum relative variation between 〈𝑝 𝑖𝑠𝑜 〉/𝜎 0 and 0.6191 was always lower than 0.07%, the value of 〈𝑝 𝑖𝑠𝑜 〉/𝜎 0 = 0.6191 corresponding to the solution obtained for the planar BII. Consequently, the variation of the isostatic pressure will not be studied in what follows and only the results in terms of the maximal shear stress will be considered. Figure 4 shows the variation of 𝜏 max (𝑥, 𝑦 ̃= 0.2) as a function of 𝑥 λ ⁄ for three values of the waviness ratios ∆/λ = 0.01, 0.1, and 0.5 and for a BIC ratio equal to 50%. Note that plotting 𝜏 𝑚𝑎𝑥 for 𝑦 ̃= 0.2 allows to be sufficiently far from the BII to avoid too strong effects of the crack tip, and sufficiently close from the BII to obtain a significant spatial variation of 𝜏 𝑚𝑎𝑥 .

For ∆/λ = 0.01, the maximum value of 𝜏 𝑚𝑎𝑥 is obtained for 𝑥/λ = 0.2, which can be explained by the influence of the stress concentration at the crack tip located at 𝑥/λ = 0.25.

However, for ∆/λ ≥ 0.1, the maximum value of 𝜏 𝑚𝑎𝑥 is obtained for x = 0 due to the stronger influence of the BII on the spatial distribution of shear stresses (see above and Fig. 3).

Figure 5 shows the variation of the normalized maximal shear stress 𝜏 𝑚𝑎𝑥 /𝜎 0 averaged over x as a function of 𝑦 ̃ for different values of ∆/λ and of the BIC. Figure 5 shows that the maximal value of 〈 𝜏 𝑚𝑎𝑥 𝜎 0 〉 decreases when the BIC increases. The results obtained for the different values of the waviness ratio when BIC = 5% are qualitatively similar, while the effect of the waviness ratio on the distribution of the maximum shear stresses is significant for higher values of the BIC ratio.

Effect of the bone Young's modulus Eb

As indicated in Section 2.1, two bone types have been considered in the present study, i.e. cortical bone tissue (with Eb = 2 GPa) and trabecular bone tissue (with Eb = 0.2 GPa).

Simulations considering both types of bone tissues were carried out for all values of BIC (between 5 and 80 %) and waviness ratio (between 0.01 and 0.5). The relative difference between the maximal shear stresses obtained with trabecular and cortical bone tissues is always lower than 1.9 %, which indicates that the bone material properties weakly affect the stress field distribution.

Effect of the implant stiffness

Figure 6 shows the variation of ∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 (𝑦 ̃) and ∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 (𝑦 ̃) corresponding to the average relative variation of the shear stress as a function of the waviness ratio ∆/λ for different values of 𝑦 ̃ and for a BIC ratio equal to 50%. The results show that the reduction of maximal shear stresses in bone tissue obtained with an implant made of Ti-Nb-Zr instead of Ti-6Al-4V is lower than that obtained with an implant made Ti-35BPA instead of Ti-6Al-4V. Moreover, ∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 and ∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 decrease as a function of 𝑦 ̃ and increase as a function of ∆/λ.

Discussion

A usual assumption made in implant biomechanics consists in considering that osseointegration phenomena mainly occur in a region of interest located at a distance lower than around 100-200µm [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] . Nevertheless, the distribution of the stress field around the BII at the scale of several hundred micrometers remains unclear due to a lack of experimental evidences. The originality of the present study is to investigate the spatial distribution of the local stress field in the periprosthetic bone tissue, which is an important determinant of the implant stability. To this aim, a microscale 2-D FE model of an osseointegrated BII taking into account the effects of the implant roughness, of bone and implant stiffnesses and of the BIC was developed. The influence of these parameters on the spatial variation of the stress field around the BII is investigated, which leads to an estimation of the region of interest where the presence of the implant influences the stress field and where osseointegration is thus likely to be affected by the implant surface roughness.

The stress fields in bone tissue has been characterized in terms of maximal shear stress 𝜏 𝑚𝑎𝑥 and isostatic pressure 𝑝 𝑖𝑠𝑜 associated to the principal stresses. The results obtained in the present study show that the isostatic pressure is not influenced by the implant surface roughness for all values of the waviness ratio and of BIC, which may be explained by the fact that 𝑝 𝑖𝑠𝑜 is related to the changes of the ROI volume that does not occur for the proposed model. However, the maximum shear stress is shown to be sensitive to the implant surface roughness. These results are consistent with the work by Anderson et al. 35 showing that a compressive loading condition acting on an elastic body with a wavy interface may induce a local state of shear stresses along the interface, even if a fully bounded interface is considered 35 .

Figures 2 and5 show the effect of the BIC on the maximal shear stress. The results show that stress-shielding effects are likely to be more important when the BIC value is low, which corresponds to a case of relatively low implant stability. These results emphasize the importance of maximizing the BIC ratio in order to maximize the chances of surgical success because of three reasons. First, increasing the BIC ratio is known to improve the implant stability, which is a strong determinant of implant success 45 . Second, increasing the BIC ratio is known to favor osseointegration phenomena, which are defined by multiscale and multitime phenomena leading to bone apposition around the implant surface. Based on the results obtained herein, it seems that bone ingrowth (i.e. bone apposition at the implant surface at the scale of the surface roughness) is more important compared to bone ongrowth (i.e. bone apposition on the implant surface at the scale of the implant, without considering the surface roughness). Third, as shown in the present study, stress-shielding effects, which are detrimental to the implant success, are more important when the BIC ratio decreases.

The maximum shear stress exhibits a maximum value comprised between 𝑦 ̃= 0 and 𝑦 ̃= 0.1 〉 as a function of 𝑦 ̃ may be explained by considering an analytical model corresponding to the Boussinesq theory 36 . As illustrated in Fig. 7a, the contact pressure induced by the implant surface is modeled by a linear load applied on a line of length B acting on a semi-infinite space corresponding to bone tissue, with an amplitude q that is taken equal to the normal stress averaged over the contacting BII derived from FE model. The value of B is equal to x T (see Fig. 1c). The maximum shear stress 𝜏 𝑚𝑎𝑥 𝑎 (𝑦 ̃) derived from this analytical model for x = 0 writes 36 :

(
𝜏 𝑚𝑎𝑥 𝑎 (𝑦 ̃) = 𝑞 sin 𝛼 𝜋 , (9)
where 𝛼 = 2 tan -1 (-𝐵 2 𝜆𝑦 ̃). The comparison between the analytical solution and the numerical solution for BIC = 5% and ∆/λ = 0.1 is shown in Fig. 7b. A good agreement is obtained for the amplitude and the position of the peak of the maximum shear stress. Note that the numerical solution does not tend to zero, as the analytical solution does, which is due to the confined boundary conditions assumed in the numerical model (see Fig. 1c). The slightly difference between analytical and the numerical results shown in Fig. 7b is due to the difference in terms of the distribution of the contact pressure, which is approximated to a linear distribution in the analytical case. A good agreement between analytical and numerical models is obtained also for other values of the BIC as can be appreciated by comparing Fig. 5 and Fig. 7c. The analytical model allows understanding the decrease of the peak value of

〈 𝜏 𝑚𝑎𝑥 𝜎 0
〉 when BIC increases (see Fig. 7c), which is due to an increasing contact area (contact length in 2-D case) between implant surface and bone tissue.

As shown in Figs. 3&4, the stress distribution around the BII strongly depends on the waviness ratio ∆/λ, which may be explained by the fact that the singularity at the tip of the non-contact zone is stronger for lower values of ∆/λ, as shown in Fig. 3. When ∆/λ = 0.5, the maximal shear stress 𝜏 𝑚𝑎𝑥 is almost uniformly distributed along the implant surface in contact with bone tissue, and the localization of the stresses at the tip is negligible. This last result is consistent with the "crack-like" behavior of the non-contact zone that is highlighted for waviness ratio lower than 0.03 [START_REF] Raffa | Micromechanical modeling of the contact stiffness of an osseointegrated bone-implant interface[END_REF] .

As shown in Fig. 5,

〈 𝜏 𝑚𝑎𝑥 𝜎 0 〉 -〈 𝜏 𝑚𝑎𝑥 𝐻 𝜎 0
〉 ≈ 0 when 𝑦 ̃≥ 0.8 for all BIC and waviness ratios, which indicates that the region in the surrounding bone tissue affected by the surface roughness is located at a distance lower than the amplitude of the implant roughness. This result is consistent with the belief that the evolution of the bone properties, at a distance of several hundred micrometers from the interface, are determinant for the implant success; since such distance corresponds to typical roughness wavelength used in many orthopedic implants [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] .

The effects of changes of the implant stiffness on stress-shielding phenomena was investigated in this study. Stress-shielding phenomena come from shear stresses at the BII generated by the strong gap of mechanical properties between bone tissue and the implant.

The results shown in Fig. 6 provide a quantitative indication of the decrease of stressshielding phenomena when using Ti-Nb-Zr alloy and a metal-polymer composite (Ti-35BPA) compared to titanium implant (Ti-6Al-4V). As expected, it is found a stronger decrease of stress-shielding effects for Ti-35BPA compared to Ti-28Nb-35.4Zr for all waviness ratio at all depth 𝑦 ̃ (see Fig. 6). ∆𝜏 ̅̅̅ 𝑇𝑖,𝑍𝑟 and ∆𝜏 ̅̅̅ 𝑇𝑖,𝐵𝑃𝐴 decrease as a function of 𝑦 ̃, which is due to fact that the maximal shear stress decrease as a function of 𝑦 ̃ for all implant roughness and BIC values (see Figs. 2&5).

This study has several limitations. First, a sinusoidal description of the implant surface roughness is used, similarly as papers [START_REF] Raffa | Micromechanical modeling of the contact stiffness of an osseointegrated bone-implant interface[END_REF]37 . This description constitutes a strong approximation and considering the real surface texture is likely to lead to different results.

However, comparable approaches have already been developed in contact mechanics [START_REF] Johnson | The contact of elastic regular wavy surfaces[END_REF][START_REF] Ciavarella | Adhesive rough contacts near complete contact[END_REF][START_REF] Yastrebov | The contact of elastic regular wavy surfaces revisited[END_REF][START_REF] Kozachok | Contact problem for wavy surfaces in the presence of an incompressible liquid and a gas in interface gaps[END_REF] and arbitrary irregular surface may be easily generated from a collection of sinusoidal surfaces, which should be done in the future. Second, bone material properties were assumed to be linearly elastic, homogeneous and isotropic and fluid-structure interactions were neglected, similarly as what was done in various previous FE-based numerical studies [START_REF] Baggi | Stress-based performance evaluation of osseointegrated dental implants by finite-element simulation[END_REF][START_REF] Natali | Analysis of bone-implant interaction phenomena by using a numerical approach[END_REF][START_REF] Baggi | The influence of implant diameter and length on stress distribution of osseointegrated implants related to crestal bone geometry: a three-dimensional finite element analysis[END_REF][START_REF] Maceri | Optimal mechanical design of anatomical post-systems for endodontic restoration[END_REF][START_REF] Rourke | Primary stability of a cementless acetabular cup in a cohort of patient-specific finite element models[END_REF] .

However, newly formed bone properties are heterogeneous [START_REF] Vayron | Evolution of bone biomechanical properties at the micrometer scale around titanium implant as a function of healing time[END_REF] and viscoelastic, which was neglected herein. The assumption of bone homogeneity implies a simplified modeling of the bone microstructure, which may include cavities at various scales [START_REF] Cowin | Bone mechanics handbook[END_REF][START_REF] Fritsch | Universal"microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity[END_REF] . Third, it was not possible to perform an experimental validation due to the difficulty of measuring stress distribution around the implant. Fourth, adhesion phenomena at the BII [START_REF] Rojek | Numerical simulation of bone-implant systems using a more realistic model of the contact interfaces with adhesion[END_REF][START_REF] Immel | A modified Coulomb's law for the tangential debonding of osseointegrated implants[END_REF] that may be important in particular at early stage of osseointegration were not taken into account, which should be considered in the future.

Conclusion

This study proposes a model of the influence of the implant surface roughness, of the stiffness of bone and implant and of the BIC on the local stress field in periprosthetic bone tissue. The proposed FE model shows that the influence of the surface roughness is limited to a distance corresponding approximately to the wavelength of the surface roughness, which is consistent with previous results. The results obtained in the present study emphasize the importance of maximizing the bone-implant contact ratio at the microscopic scale not only to improve osseointegration phenomena, but also to minimize stress-shielding effects that are shown to occur in bone tissue mostly in region of interests located at a distance of the order of the surface roughness from the implant surface.

Based on the computation of the stress field around the implant, future works should focus on introducing bone remodeling at the local scale in order to model osseointegration phenomena and the evolution of the BII.

2. 1

 1 .1. Geometry A 2-D micromechanical model of the contacting region between a cementless implant and bone tissue is proposed in what follows. The model comprises two half-spaces corresponding to an implant and to bone tissue respectively, separated each other by an irregular interphase (i.e. a zone comprising the implant surface, bone tissue in partial contact with the implant and a region filled with void in the simulation), as shown in Fig.1.

  LP and LT depend on the amplitude ∆ and the wavelength λ of the sinusoidal implant surface through the operators 𝐸(𝑧) = 𝐸 ( 𝜋 2 |𝑧) and 𝐸 = (𝑧|𝑚) = ∫ √1 -𝑚 sin 2 (𝑡) 𝑧 0

  [START_REF] Mishra | Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness[END_REF] ; ii) a Ti-Nb-Zr alloy (Ti-28Nb-35.4Zr) with E = 51 GPa 31 and iii) a metal-polymer composite (Ti-35BPA) with E = 4.4 GPa 32 . Two values of the bone Young's modulus Eb were tested in order to simulate i) cortical bone tissue: Eb = 2 GPa 33 and ii) trabecular bone: Eb = 0.2 GPa 34 .

  5) on the spatial variation of the stress field in periprosthetic bone tissue. For each configuration, two values of the bone Young's modulus (Eb = 0.2 and 2 GPa) and three different implant materials with Young's modulus E equal to 113 GPa, 51 GPa and 4.4 GPa were tested. The reference configuration corresponds to the following parameters: BIC = 50%, ∆/λ = 0.1, E = 113 GPa (Ti alloy) and Eb = 2 GPa (cortical bone).

Figure 2 3 . 2 Figure 3

 2323 Figure2shows the effect of the BIC on the variation of the normalized maximum shear stress

〉〉

  according to the BIC and to the waviness ratio ∆/λ) and then decreases as a function of 𝑦 ̃ and tends asymptotically to the value 𝜏 𝑚𝑎𝑥 𝐻 (see subsections 3.1 & 3.2). Note that for relatively high BIC values (> 50%), the maximum of 〈 𝜏 𝑚𝑎𝑥 𝜎 0 is always reached for 𝑦 ̃= 0 (i.e. at the implant surface) and 〈 𝜏 𝑚𝑎𝑥 𝜎 0 is then a strictly decreasing function of 𝑦 ̃. This last result provide an estimation of the region of interest where stress-shielding are likely to be significant. The aforementioned distribution of 〈 𝜏 𝑚𝑎𝑥 𝜎 0

Fig. 1

 1 Fig. 1 Geometrical configuration of the BII. (a): macroscopic description corresponding to a

Fig. 2

 2 Fig. 2 Variation of the normalized maximum shear stress 𝜏 𝑚𝑎𝑥 /𝜎 0 averaged over x as a

Fig. 3

 3 Fig. 3 Spatial variation of the maximal shear stress 𝜏 𝑚𝑎𝑥 (expressed in MPa) around the BII for BIC = 50% and three values of the waviness ratio ∆/λ = 0.01, 0.1, and 0.5. Implant and

Fig. 4

 4 Fig. 4 Variation of the maximal shear stress 𝜏 𝑚𝑎𝑥 as a function of 𝑥 λ ⁄ for 𝑦 ̃= 0.2. Three values of the waviness ratio ∆/λ = 0.01, 0.1, and 0.5 are considered. The BIC ratio is equal to

Fig. 5 29 Fig. 6

 5296 Fig. 5 Variation of the normalized maximal shear stress 𝜏 𝑚𝑎𝑥 /𝜎 0 averaged over x as a function of 𝑦 ̃. Three values of the waviness ratio ∆/λ = 0.01, 0.1, and 0.5 and four values of

Fig. 7

 7 Fig. 7 (a) Geometrical configuration corresponding to the Boussinesq theory; (b) comparison

  alloy for surgical implants. In: Medical Applications of Titanium and Its Alloys: The Material and Biological Issues. ASTM International. , dans ces solides, des efforts quelconques exercés sur une petite partie de leur surface ou de leur intérieur: mémoire suivi de notes étendues sur divers points de physique, mathematique et d'analyse. Vol. 4. Gauthier-Villars. 45. Mathieu, V., Vayron, R., Richard, G., et al. 2014. Biomechanical determinants of the stability of dental implants: influence of the bone-implant interface properties. J Biomech. 47(1):3-13.

	31. Xu W, Li M, Wen C, et al. 2018. The Mechanical Properties and In Vitro
	Biocompatibility of PM-Fabricated Ti-28Nb-35.4 Zr Alloy for Orthopedic Implant
	Applications. Materials. 11(4):E531.
	32. Okulov IV, Weißmüller J, Markmann J. 2017. Dealloying-based interpenetrating-
	phase nanocomposites matching the elastic behavior of human bone. Sci Rep. 7(1):20.
	33. Vayron R, Barthel E, Mathieu V et al. 2012. Nanoindentation measurements of
	biomechanical properties in mature and newly formed bone tissue surrounding an
	implant. J Biomech Eng. 134(2):021007.
	34. Wazen RM, Currey JA, Guo H, et al. 2013. Micromotion-induced strain fields
	influence early stages of repair at bone-implant interfaces. Acta Biomater. 9(5):6663-
	6674.
	35. Anderson E, Li R, Chew HB. 2014. Negative stiffness induced by shear along wavy
	interfaces. J Mech Phys Solids. 63:285-97.
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	mouvement des solides élastiques: principalement au calcul des déformations et des
	pressions que produisent

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).

Conflict of Interest: the authors declare that they have no conflict of interest.


Stephane Perrin

Yidenekachew J Donie

Paul Montgomery

Guillaume Gomard

Sylvain Lecler

Compensated Microsphere-Assisted Interference Microscopy

We propose an experimental method in microsphere-assisted interference microscopy that makes it possible to reconstruct surface topographies as much with high quantitative depth accuracy as when the transversal feature sizes are smaller than the classical diffraction limit. The full-field super-resolution interference microscope consists not only of one glass microsphere in the object arm but also of a second similar microsphere in the reference arm. By compensating the sphere aberrations in the two interference arms, the spatial resolution appears to be considerably increased. The increase in the three-dimensional spatial resolution allows for the topography reconstruction of 300-nm-width grating lines and 200-nm-diameter transparent nanopillars.

I. INTRODUCTION

Interference microscopy is a widely used noncontact optical technique enabling the reconstruction of the surface topography of microscopic objects [START_REF] Malacara | Optical Shop Testing[END_REF] for surface-quality inspection or surface-roughness measurement. This fullfield imaging technique, in which the lateral measurement field (ranging from a few hundred micrometers to several millimeters) depends on the combination of the microscope objective performance and the sensor chip size, is able to achieve a high axial sensitivity (approximately 5 nm). However, interference microscopy has a lateral resolving power that is not only limited by the diffraction of light [as in the case of two-dimensional (2D) optical microscopy] [START_REF] Lehmann | Coherence scanning and phase imaging optical interference microscopy at the lateral resolution limit[END_REF] but also by its ability to reconstruct the height map at the same time [START_REF] Giusca | Calibration of the scales of areal surface topography measuring instruments: Part 3. Resolution[END_REF]. Indeed, characterizing the performance of an instrument in order to retrieve the surface topography is more complex than simply estimating its 2D imaging performance [START_REF] Leach | Coupland[END_REF]. According to ISO 25178-600, the lateral-period limit of the topographic system is defined as the spatial period of a sinusoidal profile at which the height response of an instrument falls to 50% [START_REF]Geometrical product specifications (GPS)-Surface texture: Areal-Part 600: Metrological characteristics for areal topography measuring methods[END_REF]. Therefore, the depth accuracy depends on the lateral feature size. Typically, when the spatial frequency of the object appears to be close to λ (the wavelength of * stephane.perrin@unistra.fr light), the instrument response related to the depth measurement can no longer be considered as linear [START_REF] Leach | Coupland[END_REF][START_REF] De Groot | Fringe 2005[END_REF]. Combined with optical aberrations as well as the influence of the nature of light [START_REF] Goodman | Introduction to Fourier Optics[END_REF][START_REF] Horstmeyer | Standardizing the resolution claims for coherent microscopy[END_REF], the lateral resolution in three-dimensional (3D) surface profilometry is usually not higher than a few micrometers, hence considerably increasing the nonisotropic 3D spatial resolution. Recently, several techniques have been proposed to improve the lateral-resolution ability of 3D microscopy, for example, the development of structured-illumination 3D profilometry [START_REF] Lehmann | 3-D optical interference microscopy at the lateral resolution limit[END_REF] and of tomographic diffractive microscopy [START_REF] Haeberle | Tomographic diffractive microscopy: Basics, techniques and perspectives[END_REF][START_REF] Cotte | Marker-free phase nanoscopy[END_REF]. In 2009 and in 2011, 2D super-resolution imaging was experimentally demonstrated in air using nanoscale spherical lenses [START_REF] Lee | Near-field focusing and magnification through self-assembled nanoscale spherical lenses[END_REF] and glass microspheres [START_REF] Wang | Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope[END_REF], respectively.

Microsphere-assisted microscopy is a super-resolution imaging technique that has the advantage of being easy to implement, label free, full field, and fast [START_REF] Perrin | Label-Free Super-Resolution Microscopy[END_REF]. The technique consists in introducing a transparent microsphere, having a diameter ranging from 5 μm to 150 μm, in the imaging part of a classical white-light microscope. The microsphere acts as a magnifying optical component [START_REF] Perrin | Unconventional magnification behaviour in microsphere-assisted microscopy[END_REF], generating a super-resolved image that is collected by an objective lens of the microscope. Geometrical (e.g., the microsphere diameter) and optical (e.g., the illumination wavelength and refractive index of the microsphere) parameters thus have an influence on performance. For example, the nature of the image can be virtual or real depending on the contrast refractive index [START_REF] Sam Lai | Super-resolution real imaging in microsphere-assisted microscopy[END_REF][START_REF] Lecler | Photonic jet lens[END_REF]. A lateral resolution of 100 nm has been claimed in air [START_REF] Wang | Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope[END_REF][START_REF] Maslov | Resolution and Reciprocity in Microspherical Nanoscopy: Point-Spread Function versus Photonic Nanojets[END_REF], thus being equivalent to the resolution limit in structured-illumination microscopy and in tomographic diffractive microscopy. Nevertheless, the phenomenon behind the super-resolution remains not fully understood. It has been reported that the microsphere collects the nearfield information of the sample and then propagates it into the far field [START_REF] Hao | Microsphere based microscope with optical super-resolution capability[END_REF][START_REF] Ben-Aryeh | Increase of resolution by use of microspheres related to complex Snell's law[END_REF]. The immersion of high-refractiveindex microspheres in a surrounding medium has also been suggested to improve the imaging quality [START_REF] Darafsheh | Optical super-resolution by high-index liquid immersed microspheres[END_REF][START_REF] Li | Label-free super-resolution imaging of adenoviruses by submerged microsphere optical nanoscopy[END_REF]. More recently, microsphere-aided microscopy has been combined with dark-field illumination for the label-free visualization of translucent samples [START_REF] Perrin | Transmission microsphere-assisted darkfield microscopy[END_REF] and with interferometry for the high-resolution reconstruction of surface topography [START_REF] Wang | Three-dimensional super-resolution morphology by near-field assisted white-light interferometry[END_REF][START_REF] Perrin | Microsphere-assisted phase-shifting profilometry[END_REF] or of cell volume [START_REF] Aakhte | Microsphere-assisted super-resolved Mirau digital holographic microscopy for cell identification[END_REF]. These previously reported methods in microsphere-aided interference microscopy have been implemented by placing a glass microsphere in the object arm of a quantitative-phase interferometer, leading, first, to an unbalanced optical path difference between the arms and, second, to the generation of optical aberrations, as highlighted in Ref. [START_REF] Perrin | Microsphere-assisted phase-shifting profilometry[END_REF]. An imageprocessing method based on 2D data fitting has thus been required to remove the spherical component. Since then, it has been implemented assuming that the wave-front deformation provided by the microsphere is well known and that the microsphere is of high quality. However, at this scale, it appears to be complex to retrieve wave-front deformation by a microsphere. The image-processing algorithm, more complex to automate, is thus individually implemented, leading to a long processing time (e.g., about 20 min in our case, depending on the performance of the computer and the image sampling). Furthermore, despite an increase in the spatial resolution, the reconstruction of surface topography with a high height accuracy (e.g., reconstruction of the straight shapes of grating grooves) still remains a real challenge.

The method that we propose in this work allows for an accurate 3D profile reconstruction for straight shapes of grating grooves. Our microsphere-assisted interference microscope consists not only of one microsphere in the object arm, but also of a second similar microsphere in the reference arm. In this way, the wave-front distortion due to the optical aberrations is avoided and the optical path difference in the interferometer arms is compensated. The performance of compensated-dispersion microsphereaided interference microscopy is demonstrated in air using a coherent phase-shifting interference microscope in a Linnik configuration and in a reflection mode. Such a method has the advantage of not requiring the implementation of image processing, thus being of further promise in super-resolution surface metrology [START_REF] Leach | Applications of super-resolution imaging in the field of surface topography measurement[END_REF]. The areal topography of features having a spatial period smaller than λ is reconstructed with high axial and lateral accuracy [START_REF] Leach | Coupland[END_REF][START_REF]Geometrical product specifications (GPS)-Surface texture: Areal-Part 600: Metrological characteristics for areal topography measuring methods[END_REF]. We demonstrate that the imaging system can retrieve the details with a high-spatial resolution through nanometric axial sensitivity.

II. METHOD

Figure 1 shows the layout of the microsphere-assisted interference microscope in a Linnik configuration, thus allowing for the collection of the back-reflected wave fronts from both the sample and the reference mirror. The emitting beam from the light source (LED, λ 0 = 520 nm, λ = 31 nm) passes through a series of collimating and focusing lenses, as well as a pinhole for spatial filtering. The illumination beam is thus spatially coherent and temporally quasicoherent in order to reduce speckle generation and coherent noise [START_REF] Chavel | Optical noise and temporal coherence[END_REF]. With regard to the incident beam, 50% is reflected into the reference arm and 50% is transmitted into the object arm by a beam-splitter cube. Given the requirement of two similar microspheres for dispersion compensation, the object arm is therefore tilted at 90 • by a mirror in order to have a horizontal object plane for easier placement of the compensating microsphere. In this optical setup, the two microspheres are first randomly deposited on the surfaces of the object and the reference mirror. Despite being in contact with the sample, the latter is, however, not affected due to the light microsphere. In order to perform noninvasive measurements, an optical head can be introduced, allowing the microsphere to 014068-2 be held at a distance (less than the wavelength) from the surface [START_REF] Yan | Superlensing microscope objective lens[END_REF][START_REF] Huszka | Microsphere-based super-resolution scanning optical microscope[END_REF][START_REF] Li | Microsphere-based super-resolution scanning optical microscope[END_REF]. X -Y stages are used in the reference arm to match the transverse positions of the two similar microspheres. In each arm of the interferometer, the coupled microscope objective and microsphere allow the super-resolution phenomenon to be achieved with the same resolving power. The two microscope objectives (Mitutoyo Corporation) have a magnification factor of ×50 and a numerical aperture of 0.55. The condenser lens, placed in an afocal configuration with the microscope objectives, enables wide-field illumination of the microspheres. The soda-lime-glass microspheres (Cospheric LLC) have a diameter of 30 μm and a refractive index of 1.52. The diameter dispersion of the microspheres is guaranteed to be less than 10% and the sphericity to be more than 95%. The dispersion range is decreased by selecting only the same microspheres through the microscope objectives of the object and reference arms. This method leads to an error of the diameter dispersion of ±300 nm. Moreover, before performing a measurement, the surface of the microspheres is examined through the microscope objectives. In doing so, microspheres having imperfections or scratches (with a size larger than the resolution of the objectives) on the bottom are excluded. However, it appears to be complex to estimate the surface quality of the rear side of the microsphere. The selection of good microspheres is less time consuming than the implementation of image processing when only one microsphere is used in the object arm. Also, the selection needs to be applied only one time. The objective lenses then collect the virtual images of both the sample and the mirror, generated by the microspheres in the object and reference arms, respectively. The object and reference beams are superimposed by the beam-splitter cube. At the output of the interferometer, a tube lens, which forms an infinity-corrected optical system, finally directs the super-resolved images onto the detector (Basler AG). The lateral magnification factor from the microsphere reaches 4.5, as previously estimated [START_REF] Perrin | Unconventional magnification behaviour in microsphere-assisted microscopy[END_REF]. In the object plane, the size of the square pixels is equal to 31.5 nm, giving a high spatial sampling of the object features. To implement the π/2 phase shift, all the optical elements of the reference arm are axially displaced using a piezoelectric actuation system. In our case, the π/2 phase shift should lead to an increment of the piezoelectric device of 66.25 nm [START_REF] Malacara | Optical Shop Testing[END_REF]. However, in the presented results, the high numerical apertures of the microspheres must be considered in the axial increment to prevent wrong interpretations of the depth results [START_REF] Schulz | Errors in phase-measurement interferometry with high numerical apertures[END_REF]. This not being trivial with regard to microspheres (it is complex to estimate their numerical aperture), a specific calibration of the phase shifter has previously been made using a 1200-nm-period standard. Through a 30-μm-diameter microsphere in air, the π/2 phase shift now corresponds to an axial increment of 78 nm. The coherence function (or visibility) of the light source is constant at least within the axial stepping range. The series of interferograms make it possible to retrieve the phase-difference distribution between the object and the flat reference using a four-step phaseshifting algorithm [START_REF] Malacara | Optical Shop Testing[END_REF]. Fig. 1 shows the four interferograms used for the surface-topography reconstruction presented in Fig. 4(b). In addition, a 2D unwrapping algorithm based on least squares is implemented to remove the 2π jumps [START_REF] Takajo | Noniterative method of obtaining the exact solution for the normal equation in leastsquares phase estimation from the phase difference[END_REF], which result from the arctan function.

III. RESULTS

Figure 2 shows the transfer function of a ×50magnification microscope objective alone (without using microspheres) of the Linnik-based interference microscope as a function of the object transverse frequency. A positive Ronchi ruling with a period ranging from 5 μm to 200 nm is placed in the object arm of the interferometer. The shape of the contrast evolution corresponds to the shape of an aberration-free quasicoherent imaging system [START_REF] Goodman | Introduction to Fourier Optics[END_REF]. The objective lenses are thus assumed to be diffraction limited and the lateral-resolution limit equals 950 nm in air in the absence of microspheres. However, in interference microscopy, the lateral resolution depends not only on the transfer function of the transverse frequencies but also on its ability to transfer height information [START_REF] Giusca | Calibration of the scales of areal surface topography measuring instruments: Part 3. Resolution[END_REF][START_REF] Leach | Coupland[END_REF][START_REF] De Groot | Fringe 2005[END_REF]. In our case, the Linnik-based interference microscope alone is thus not able to reconstruct object features that have a period smaller than 1.8 μm, as shown in Fig. 3. The line features of the 50%-duty-cycle grating standard (SiMETRICS GmbH), which has a period of 2 μm and a depth of 200 nm, are indeed resolved, unlike the 600-nmperiod features. The height profiles in Fig. 3(c) highlight the resolution limit of the interference microscope without microspheres. The standard deviation σ of the phase distribution of a flat plane reaches 21 mrad, i.e., the axial sensitivity of the interferometer is 1.8 nm.

The addition of a 30-μm-diameter soda-lime microsphere in each interference arm considerably improves the lateral resolution of the imaging system. As such, the microsphere-assisted phase-shifting microscope is then able to reconstruct the 600-nm-period grating of the standard in air. Since no postprocessing of the data is required to remove the phase deformation from the microsphere, phase measurement can be used directly. Figure 4(a) demonstrates the ability of the imaging system to retrieve the square shape of the grating lines (solid blue curve) with high accuracy over the majority of the measured area [Fig. 4(b)]. The measured heights, i.e., 140 nm, are in agreement with the heights measured using atomic force microscopy (AFM), as shown by the dashed red curve in Fig. 4(a). However, while the results [Figs. 4(c.i) and 4(c.ii)] give the correct depth, AFM appears to be limited for the reconstruction of the straight shapes of the 300nm-width grooves. In fact, the convolution of the tip shape with the object results in a "V"-shape reconstruction of the square features and a wrong duty cycle of 18%. In fact, the width of the grooves of the standard equals the space between the grooves, as shown in Fig. 4(d) using scanning electron microscopy (SEM). In addition, the dotted black curve in Fig. 4(a) represents the depth profile measured with the microsphere-aided interference microscope having only one microsphere in the object arm, i.e., without aberration compensations. In this case, a 2D polynomial fit of the wave front is required to retrieve the topography. The axial sensitivity of the aberration-compensation interference nanoscope is slightly decreased when using two microspheres, going from σ = 1.8 nm (without any microspheres) to σ = 5.6 nm. By applying a median filter to the interferograms, it is possible to reach a sensitivity of 3.9 nm. The M-C curve, in Fig. 4(a), is plotted using raw data, leading to a noisier signal than the two other methods, M-NC and AFM, where the data are filtered using a 3 × 3 median filter and averaged by the round-trip scanning mode, respectively. The higher-frequency noise of the M-C curve can mainly be explained by nonzero optical path differences between the two microspheres at each transverse position, i.e., lateral nanometric disparities. Also, the obliquely incident beams on a straight grating can be subject to multiple reflections before being backreflected and collected by the microsphere, increasing the optical path. By compensating the optical path difference in both arms, the super-resolution phenomenon is therefore enhanced in comparison with the previous results reported in Refs. [START_REF] Wang | Three-dimensional super-resolution morphology by near-field assisted white-light interferometry[END_REF][START_REF] Perrin | Microsphere-assisted phase-shifting profilometry[END_REF][START_REF] Aakhte | Microsphere-assisted super-resolved Mirau digital holographic microscopy for cell identification[END_REF]. Nevertheless, such a high-quality measurement contributes to the clear visibility of the areas of artifact generation [green arrows in Fig. 4(b)]. In fact, the defocused patches are most likely due to imperfections on the microsphere surface and inside the microsphere (despite the microspheres being previously selected to have the same diameter, a high sphericity, and a high surface quality), leading to distortions, a potential reduction of the lateral field of view, empty areas, or wrong height values. These microsphere imperfections have a hundredsof-nanometers order of magnitude, unlike the nanometric imperfections behind the high-frequency noise.

The grating standards are then substituted for selfassembled and disordered nanopillars made of poly(methyl methacrylate), processed on a silicon substrate (Karlsruhe Institute of Technology, Germany). They are obtained by the polymer-blend-lithography method and developed as light-trapping substrates for photovoltaic applications (for more details, see Ref. [START_REF] Donie | Light trapping in thin film silicon solar cells via phase separated disordered nanopillars[END_REF]). The diameters of the obtuseangled nanopillars range from 200 nm to 450 nm and their average height is 80 nm (a non-negligible height dispersion, ranging from 70 nm to 100 nm, remains).

IV. DISCUSSION

Figure 6 shows the ability of compensated microsphereaided interference microscopy to retrieve the height distribution of a 50%-duty-cycle standard grating (SiMETRICS GmbH) using two microspheres having a diameter of 50 μm. Increasing the size of the microspheres allows the topography to be reconstructed over a larger lateral field of view and the influence of microsphere imperfections to be reduced (due to the lower sphericity). However, the super-resolving power of the imaging technique is lower [START_REF] Perrin | Label-Free Super-Resolution Microscopy[END_REF], thus providing the possibility of measuring a 800-nm-period grating in air, in this case.

V. CONCLUSION

An experimental method is proposed for the combination of microsphere-assisted microscopy with coherentphase-shifting interference microscopy in order to measure 3D topographies with a high quantitative accuracy. Similar 30-μm-diameter glass microspheres are introduced in each interference arm to compensate the optical aberration and to further remove the phase distortion from the microbead without the need for image processing. A full-field superresolution system in air, based on a Linnik configuration, makes it possible to perform subwavelength 3D imaging in the far field. Indeed, using a compensating microsphere, a lateral resolution of at least 400 nm is achieved in air using a 0.55-numerical-aperture microscope objective and quasicoherent illumination. An axial sensitivity of around 4 nm is reached. High-accuracy quantitative 3D measurements of a reference grating having a line width of 300 nm and transparent nanopillars are performed. The correct object shapes and the right height distributions are retrieved in air. Without microspheres, the interferometer would require an equivalent numerical aperture of 1.5, which is relatively complex to implement in interference microscopy. Due to the high numerical aperture of the imaging system, the phase-shift increments must be taken into account. The introduction of a microsphere holder in each interferometer arm in order to position the microspheres and to perform contactless measurements would be a significant research perspective. Moreover, microsphereaberration retrieval would make it possible to reduce the influence of sphere imperfections. Finally, the measurement errors and the repeatability depending on the dissimilarity of the diameter and the refractive index of spheres should be systematically investigated.

FIG. 1 .

 1 FIG. 1. The layout of the super-resolution interference microscope: LS, light source; SF, spatial filtering; C, condenser lens; MO, microscope objective; BF, back focal plane; S, sample; M, mirror; MS, microsphere; PA, piezoelectric actuator; TL, tube lens; D, detector.

FIG. 2 .FIG. 3 .

 23 FIG. 2. The contrast transfer function of the object arm of the interference microscope without any microspheres. The lateral resolution is limited by diffraction of light and the cut-off frequency equals 1.25 cycles/μm.

FIG. 4 .

 4 FIG. 4. The performance of the microsphere-aided interference microscope. (a) A comparison of the height profiles of the 600nm-period standard using aberration-compensated microsphereaided interference microscopy (M-C, solid blue curve) along the dashed white lines in (b), noncompensated microsphereaided interference microscopy (M-NC, dotted black curve), and AFM (dashed red curve). (b) High-resolution surface-topography reconstruction. (c.i) 2D and (c.ii) 3D views of the AFM topography reconstruction of the 600-nm-period standard. (d) A SEM direct image.

Figure 5 (FIG. 5 .

 55 FIG. 5. The characterization of a disordered array of nanopillars using microsphere-assisted interference microscopy. (a) The topography using AFM. (b) High-resolution surface-topography reconstruction using M-C. The two topographies shown in (a) and (b) are not performed at the same position. (c),(d) Height profiles along the black dashed lines in (a) and (b), respectively.

FIG. 6 .

 6 FIG.6. The characterization of a 800-nm-period standard grating using microsphere-assisted interference microscopy compensated by two 50-μm-diameter microspheres. (a) The height profile along the dashed white lines of (b) the high-resolution surface-topography reconstruction.

ACKNOWLEDGMENTS

This work received funding from SATT Conectus Alsace and was supported by the University of Strasbourg and by the Deutsche Forschungsgemeinschaft (DFG) through program DFG-SPP 1839, "Tailored Disorder." G.G. acknowledges funding by the Helmholtz Postdoctoral Program and the Karlsruhe School of Optics and Photonics (KSOP). We thank the French RENATECH network (FEMTO-ST institute, France) for fabricating the Ronchi calibration target and Amandine Elchinger for her constructive input in writing the manuscript. We also acknowledge help from the C 3 -Fab platform of the ICube laboratory.