A Challansonnex

J Casalinho

P Perré

Non-Fickian diffusion in biosourced materials: prediction of the delay between relative humidity and moisture content

Keywords: building, coupled transfer, energy, experiment, fading memory, macroscopic formulation, renewable materials

This work focuses on non-Fickian diffusion in biosourced building materials. To study these memory effects in this kind of materials, an original and automatic device was conceived and built. It allows a series of samples to be submitted, simultaneously, to a perturbation of RH at its front face and to record the evolution of sample mass and of RH at its back face. When analyzed using a classical heat and mass transfer model, the experimental data depicts a delay between RH and moisture content, the RH evolution being too fast. To address this problem, a new formulation able to model those types of behaviors using memory functions was used. To test the predictive potential of the new formulation, these memory functions were not identified but taken from a previous work, where they were determined experimentally using minute samples. A full set of experiments performed with Low Density Fiberboard (LDF) and Medium Density Fiberboard (MDF) was simulated using this new formulation feed by the experimental memory functions. The good agreement found between simulated and experimental results proves the predictive potential of the new macroscopic formulation.

Notations

The main notation are summarized in tables 1 to 3.

Introduction

In recent decades, there has been a strong interest in bio-based materials in the construction industry. They can be found in many forms and used in many applications, including solid wood for the structure, fibrous materials (wood fibres, straw, etc.) for insulation or hemp concrete for mixed use. This interest is partly due to the low carbon footprint of these materials of plant origin (hence with carbon capture) and their renewable nature [START_REF] Florentin | A life-cycle energy and carbon analysis of hemp-lime bio-composite building materials[END_REF][START_REF] Pawelzik | Critical aspects in the life cycle assessment (LCA) of bio-based materials -Reviewing methodologies and deriving recommendations[END_REF][START_REF] Peñaloza | Exploring the climate impact effects of increased use of bio-based materials in buildings[END_REF][START_REF] La Rosa | Environmental impacts and thermal insulation performance of innovative composite solutions for building applications[END_REF]. But it is also due to their hygrothermal properties. Indeed, it has been shown that, thanks to their high hygroscopicity, these materials can help to regulate ambient humidity and thus improve indoor air quality or even reduce energy consumption [START_REF] Osanyintola | Moisture buffering capacity of hygroscopic building materials: Experimental facilities and energy impact[END_REF][START_REF] Woloszyn | The effect of combining a relative-humidity-sensitive ventilation system with the moisture-buffering capacity of materials on indoor climate and energy efficiency of buildings[END_REF][START_REF] Orosa | Improving PAQ and comfort conditions in Spanish office buildings with passive climate control[END_REF]. However, when it comes to simulating their dynamic behaviour at the building or even laboratory level, significant differences are observed with the experiments [START_REF] Jacques | Dynamic coupling between vapour and heat transfer in wall assemblies : Analysis of measurements achieved under real climate[END_REF][START_REF] Kalamees | Hygrothermal calculations and laboratory tests on timber-framed wall structures[END_REF][START_REF] Künzel | Simulation of indoor temperature and humidity conditions including hygrothermal interactions with the building envelope[END_REF]. These differences can have different origins, one of them being the parameters that feed the building energy simulations [START_REF] Wang | Hygrothermal performance of cross-laminated timber wall assemblies: A stochastic approach[END_REF][START_REF] Spitz | Practical application of uncertainty analysis and sensitivity analysis on an experimental house[END_REF]. For instance methodological errors in the vapor diffusivity measurement [START_REF] Duforestel | Des transferts couplés de masse et de chaleur à la conception bioclimatique: recherches sur l'efficacité énergétique des bâtiments[END_REF] or the non consideration of the hysteresis of the sorption isotherm [START_REF] Kwiatkowski | Modelling of hysteresis influence on mass transfer in building materials[END_REF][START_REF] Rémond | The gripped-box model: A simple and robust formulation of sorption hysteresis for lignocellulosic materials[END_REF] are part of the issues currently explored. For fibrous insulation materials, significant errors in the kinetics of water vapour transfer within the materials can be explained by the absence of local thermodynamic equilibrium [START_REF] Krabbenhoft | Double porosity models for the description of water infiltration in wood[END_REF][START_REF] Nyman | A numerical method for the evaluation of non-linear transient moisture flow in cellulosic materials[END_REF][START_REF] Perré | Multiscale aspects of heat and mass transfer during drying[END_REF][START_REF] Perre | Multiscale Modeling of Drying as a Powerful Extension of the Macroscopic Approach: Application to Solid Wood and Biomass Processing[END_REF]. Indeed, these materials consist of two phases with contrasting characteristics. The gaseous phase which is highly diffusive and well connected and the solid phase (the fibers) which is poorly diffusive. The microscopic diffusive time inside the fibers can therefore be of the same order of magnitude than the macroscopic diffusive time, which explains the possibility of non-local equilibrium. The consequences of this phenomenon are manifold. One can cite the dependency with thickness of the macroscopic vapor diffusion coefficient in transient state [START_REF] Rémond | Mass diffusivity of low-density fibreboard determined under steady-and unsteady-state conditions : Evidence of dual-scale mechanisms in the diffusion[END_REF], the different values obtained when measured in transient or steady state [START_REF] Challansonnex | Mass diffusivity determination of various building materials based on inverse analysis of relative humidity evolution at the back face of a sample[END_REF] or non-physical values of the identified water vapour diffusivity [START_REF] Perré | Determination of the Mass Diffusion Coefficient Based on the Relative Humidity Measured at the Back Face of the Sample during Unsteady Regimes[END_REF]. The presence of a phase shift between relative humidity and moisture content, as reported in [START_REF] Lelievre | Hygrothermal behavior of bio-based building materials including hysteresis effects: Experimental and numerical analyses[END_REF] is also likely to be explained by this phenomenon. Indeed, because of the delayed diffusion inside the fiber, RH is ahead of moisture content. Consequently, in building energy simulations, this phenomenon generates errors on the quantity of vapour transferred and on the moisture quantity stored inside the walls. In turn, due to the latent heat of evaporation, this induces important errors in the energy demand.

The first objective of this work was to establish a reliable experimental dataset to unquestionably confirm the existence of memory effects at the scale of centimetric samples. Our strategy is to prove that a simple macroscopic model is not able to fit the experiment, which requires at least two independent physical parameters to be measured simultaneously. To achieve this goal, an original and automatic device was conceived and built. It allows a series of samples to be tested simultaneously. Each sample is submitted to a perturbation of RH at its front face and the evolution of sample mass and of RH at its back face are continuously recorded.

The data collected by this new device confirms the memory effects : a classical heat and mass transfer model is not able to predict correctly, at the same time, the evolution of averaged moisture content and the RH at back-face.

To overcome this problem, a new macroscopic formulation able to consider the memory effects using kernel functions was used [START_REF] Perré | Coupled heat and mass transfer in biosourced porous media without local equilibrium : a macroscopic formulation tailored to computational simulation[END_REF]. This formulation is embedded in a comprehensive model considering the coupling between heat and mass transfer (H&M). The comprehensive formulation is solved by the in-house computational code TransPore [START_REF] Perré | A 3-D version of TransPore: a comprehensive heat and mass transfer computational model for simulating the drying of 34 porous media[END_REF][START_REF] Perré | The Proper Use of Mass Diffusion Equations in Drying Modeling : Introducing the Drying Intensity Number[END_REF]. In a previous study [START_REF] Challansonnex | Non-Fickian diffusion in biobased materials : experimental determination of the memory function using minute samples[END_REF], very thin samples of biosourced materials were submitted to variations in ambient RH while their mass were recorded. The memory functions that represent the microscopic diffusive behavior of the materials were defined from these experiments by inverse analysis.

In the present work, these memory functions were used as input data in the new macroscopic model. Finally, just one parameter was identified using the

Identified diffusivity

Modeling Experiment

Materials and methods

Materials

The materials that were studied in this work are low density fiberboard (LDF) and medium density fiberboard (MDF). LDF is a fibrous material used for insulation. It is made of wood fibres agglomerated by their bonding properties during the so-called wet process [START_REF] Suchsland | Fiberboard manufacturing Practices In the United States[END_REF]. Its hygrothermal properties have been widely studied [START_REF] Kawasaki | Manufacture and properties of ultralow-density fiberboard[END_REF][START_REF] Czajkowski | Thermal properties of wood-based panels : specific heat determination[END_REF][START_REF] Lux | Macroscopic thermal properties of real fibrous materials : Volume averaging method and 3D image analysis[END_REF][START_REF] Vololonirina | Characterization of hygrothermal properties of wood-based products -Impact of moisture content and temperature[END_REF][START_REF] Almeida | Hygroscopic behaviour of lignocellulosic materials: Dataset at oscillating relative humidity variations[END_REF]. In this work, panels of LDF with different thickness and densities are studied, all produced by Steico (Munich, Germany).

Testing different thickness is important in the context of this research as the effect of non local equilibrium should be greater with thin panels. Indeed, the macroscopic characteristic time increases as the thickness squared and eventually becomes higher than the microscopic characteristic time. Therefore, the material becomes progressively homogenisable.

The three different panels characteristics are as follow, all densities being measured at 40% RH:

• a 20mm thick panel with a density of 145kg.m -3 ,

• a 10mm thick panel with a density of 255kg.m -3 ,

• a 20mm thick panel with a density of 222kg.m -3 .

Note that although the two last panels have different densities, they are sold under the same commercial name.

Contrary to LDF, MDF is made of wood fibres tied by an adhesive : fibres, urea-formaldehyde resin and paraffin wax are processed at high temperature and pressure to manufacture panels [START_REF] Ganev | Effect of pannel moisture content and density on moisture movement in MDF[END_REF]. This produces a structural material mostly used for internal decoration and furniture. Its hygrothermal properties were also widely studied [START_REF] Sonderegger | Thermal conductivity and water vapour transmission properties of wood-based materials[END_REF][START_REF] Ye | Properties of medium density fiberboards made from renewable biomass[END_REF]. In this work, a 10mm-thick panel with a density of 655kg.m -3 was studied.

Disk-shaped samples with a diameter of 72mm were cut from those panels.

As in previous studies [START_REF] Agoua | A double climatic chamber used to measure the diffusion coefficient of water in wood in unsteady-state conditions: Determination of the best fitting method by numerical simulation[END_REF], the samples were coated with an aluminum foil glued to the lateral faces by epoxy resin to avoid lateral water vapor leakage.

A photograph of the different types of samples is depicted in Fig. 2.

Experimental setup

The principle of the measurement is to impose a sudden perturbation of RH at the front face of the sample which will create a transient diffusive flux through

Climatic Chamber

This in-house climatic chamber, built from an insulated container (Froid Loisir Equipement, Saillans, France), is divided in two volumes by a vertical plate: an operational chamber that contains the setup and a narrow rear chamber for temperature and relative humidity control. The temperature control is ensured by a regulated heating resistance and the relative humidity is ensured by an air flow forced against a water film set at the dew point temperature.

The water is pumped from a thermocryostat (Ministat230, HUBER, Offenburg, Germany) to the top of the rear chamber where it flows along a plate. Three ventilators ensures the air flow. Two RH/temperature sensors (Sensirion SHT75, HDI Electronics, Pérois, France) are placed at opposite corners of the climatic chamber in order to record its average conditions. In One can observe that, once the conditions are constant, the regulation allows very stable conditions to be obtained, both in RH and temperature.

Sample supports

The sample support was designed and manufactured to fulfill several functions : i) to allow the sample to be weighted without perturbation, ii) to insure a temporary contact for RH measurement and iii) to insure a good exposure of the sample front-face to the chamber conditions (Fig 6). The holder and imperviousdisk designs are based on previous studies ( [START_REF] Perré | Determination of the Mass Diffusion Coefficient Based on the Relative Humidity Measured at the Back Face of the Sample during Unsteady Regimes[END_REF], [START_REF] Agoua | A double climatic chamber used to measure the diffusion coefficient of water in wood in unsteady-state conditions: Determination of the best fitting method by numerical simulation[END_REF]). The impervious-disk which has the same diameter as the sample has a small cavity which contains a SHT75 temperature/humidity sensor. This disk is inserted against the sample inside the holder whose internal diameter is larger. To ensure air tightness, a rubber tube whose internal diameter is smaller than the sample has been previously fixed inside the holder. A small hole in the holder lateral face allows a vacuum pump to be plugged between the rubber tube and the holder internal face. The under-pressure presses the rubber against the holder face, so that the sample/disk assembling can be inserted. When the vacuum pump is unplugged, the rubber tube squeezes the lateral face of the sample/disk ensuring airtightness is spite of dimensional changes of the sample. The sensor is linked to the target at the top of the support which transmits the information during the temporary contact. It is important to mention that two precautions are required to ensure perfect air-tightness : two coats of resin epoxy are applied on the lateral face of the sample before the test and some void grease is put between the rubber tube and the epoxy surface when placing the sample into the support. This airtightness was checked using an impervious sample (a PVC sample) as tested material . In this case, the change of RH value at back face following a sudden change of RH in the climatic chamber is hardly detectable (typically 0.1 % of RH change over 5 hours).

The support is perfectly centered inside the crown sample holder thanks to a conical shape. This ensures two functions :

• to avoid any friction when the crown gets downwards for weighing (Fig 4b),

• to ensure an accurate contact between the target and the contactors when the crown moves upwards for RH measurement (Fig 4a).

General assembly

The general assembly can be observed in To preserve the electronic balance from the chamber conditions, the balance (EP6102C, Ohaus, Greifensee, Switzerland, precision of 0.01g) is placed beneath the climatic chamber. A stable and adjustable support secured on the balance plate holds a glass rod crossing the chamber wall without friction. The internal weighing plate is then fixed on this rod. A glass rod presents a good stiffness together with a low thermal conductivity, required to avoid condensation nearby the chamber wall.

Experimental Protocol

The whole setup is remotely controlled using a software developed in Microsoft Visual Basic. For each sample, a first acquisition of RH at the back The RH value used as raw data is the average of the measurements before and after weighing and is allocated to the weighing time. Once all samples have been treated, the rotation stage comes back to its reference position. The device is ready for another data collection.

In this work two samples of each kind were tested (referred as sample A and B). Thus, a total of eight samples were tested during two different experiments (four samples per experiment). In the first experiment, materials with a lower macroscopic diffusive time were tested: LDF 20mm thick with a density of 145kg.m -3 and LDF 10mm thick with a density of 255kg.m -3 . In the second experiment 20mm thick LDF with a density of 222kg.m -3 and MDF were tested.

In order to avoid hysteresis effects and yet ensure the reproducibility of the

Determination of the external heat transfer coefficient

The heat and mass transfer coefficients between the sample surface and the climatic chamber play a role in the global resistance to transfer [START_REF] Perré | The Proper Use of Mass Diffusion Equations in Drying Modeling : Introducing the Drying Intensity Number[END_REF]. Consequently, its effect may interfere with the internal resistances in the identification procedure. In order to be consistent with our predictive strategy, we decided to experimentally determine the resistance to transfer at the interface rather than an indirect determination by inverse analysis. Assuming the analogy between heat and mass transfer in the boundary layer [START_REF] Bird | Transport phenomena[END_REF], the heat transfer coefficient can be deduced from the drying rate during the constant drying rate (CDR) period provided this period is long enough to assume a steady temperature field.

Obviously, as the transfer coefficients depend on the configuration (velocity and geometry), the experimental configuration must be exactly the same as in the actual experiments. To ensure this constraint while obtaining a long CDR period, a PVC disk was covered on its front face with a sponge saturated with water and placed in a sample support. During a well-established CDR period, the product is at constant temperature and the heat flux is just used for water evaporation. Assuming the heat transfer coefficient h to be the same on both sides of the sample, the energy balance during a steady CDR period reads as:

L v Q v = S(h + (h -1 + /λ) -1 )(T d -T w ) (1)
Where L v is the latent heat of vaporization of water, Q v the drying rate determined by weighing, S the surface area of the sample, λ the thermal conductivity of PVC, the thickness of the PVC disk, T w the wet bulb temperature and T d the dry bulb temperature. Using equation ( 1) during a well established CDR period allowed the value of h to be determined (h = 8.9W.K -1 ). This value will be used in what follows as input parameter. Consistently, the same value, together with the thermal resistance of the PVC disk, is used on the back-face of the sample. This is important to consider correctly the heat and mass transfer coupling inside the sample [START_REF] Perré | On the importance of heat and mass transfer coupling for the characterization of hygroscopic insulation materials[END_REF].

Theoretical model

Memory functions

Memory effects occurring in biosourced materials are explained by two additional effects : possible dual-cale phenomena arising when the conductive phase is not the storage phase and molecular relaxation arising at the macromolecular level. In a theoretical paper [START_REF] Perré | Coupled heat and mass transfer in biosourced porous media without local equilibrium : a macroscopic formulation tailored to computational simulation[END_REF], we proved that both effects could be transferred as additive effects at the macroscopic scale. Based on this result, the cumulative memory function was characterized from experiments performed on minute samples [START_REF] Challansonnex | Non-Fickian diffusion in biobased materials : experimental determination of the memory function using minute samples[END_REF]. The global memory function ϕ(t) is defined as the evolution of the moisture content (water in the storage phase) after a sudden change from RH ini to RH f in . Considering that, in practical situations, RH varies continuously rather than stepwise, the variation of moisture content should be expressed through a convolution product:

X(t) -X ini = t 0 ϕ(t -τ ) ∂X eq ∂τ dτ ( 2
)
where X is the moisture content and X eq is the equilibrium moisture content, defined by water activity. In equation 2,

∂X eq ∂τ dτ = dX eq (3)
represents the variation of equilibrium moisture content during the infinitesimal time interval dτ .

In order to ease the computational solution, the function ϕ(t) was defined with the following expression:

ϕ(t) = 1 -α 1 exp(- t τ 1 ) -α 2 exp(- t τ 2 ) ( 4
)
where τ 1 and τ 2 are the time constants and α 1 and α 2 the respective proportions. The use of exponential function allows the convolution product to be treated by internal variables defined by simple ODEs [START_REF] Perré | Coupled heat and mass transfer in biosourced porous media without local equilibrium : a macroscopic formulation tailored to computational simulation[END_REF]. the parameter values of the memory function, as determined for a RH jump from 40 to 60 % in [START_REF] Challansonnex | Non-Fickian diffusion in biobased materials : experimental determination of the memory function using minute samples[END_REF], are summarized in Table 4. These values were used as 270 input data in all simulations of the present work.

Macroscopic formulation

The macroscopic formulation that was used in this work is based on a classical set of equations taking into account coupled heat and mass in porous media.

More details about it can be found in [START_REF] Perré | A 3-D version of TransPore: a comprehensive heat and mass transfer computational model for simulating the drying of 34 porous media[END_REF][START_REF] Perré | The Proper Use of Mass Diffusion Equations in Drying Modeling : Introducing the Drying Intensity Number[END_REF]. Those equations were modified to consider the absence of thermodynamic equilibrium [START_REF] Perré | Coupled heat and mass transfer in biosourced porous media without local equilibrium : a macroscopic formulation tailored to computational simulation[END_REF]. In the new formulation, equation ( 2) is used to express the derivative of moisture content with respect to time. Thus, the set of macroscopic equations becomes :

Moisture conservation ρ s ∂ ∂t ( t 0 ϕ(t -τ ) ∂X eq ∂τ dτ ) + ∇ • (ρ v vg ) = ∇ • (ρ g f D v • ∇ω v ) (5)
Energy conservation

∂ ∂t ε g (ρ v h v + ρ a h a ) + ρ b h b + ε s ρ s h s + ∇ • ((ρ v h v + ρ a h a )v g ) = ∇ • (λ ef f ∇T + ρ g f D v (h v ∇ω v + h a ∇ω a )) (6)
Air conservation

∂ (ε g ρ a ) ∂t + ∇ • (ρ a vg ) = ∇ • (ρ g f D v ∇ω a ) ( 7
)
Transport equations

vg = - Kk g µ g (∇p g -ρ g ∇ψ g ) (8)
Boundary conditions

J w | x=0 + • n = k c M v ln 1 -x ∞ 1 -x v | x=0 J q | x=0 + • n = h ( T | x=0 -T ∞ ) P g | x=0 + = P atm (9)
One should note that the value of the partial pressure of water vapour is related to the equilibrium moisture content X eq through the sorption isotherm (or water activity) p v = p vs (T ) × a w (T, X eq ).

The comprehensive set of equation proposed in [START_REF] Perré | Coupled heat and mass transfer in biosourced porous media without local equilibrium : a macroscopic formulation tailored to computational simulation[END_REF] was adapted to the configuration of interest, hence simplified, thanks to the following assumptions:

• since the samples remain in the hygroscopic domain, all liquid water contributions have been discarded,

• due to the particular morphology of the studied materials, a very large majority of the transfer occurs in the gaseous phase. Consequently, the transport terms of bound water have also been discarded,

• the effective diffusivity is expressed as a proportion of the binary diffusivity of vapour in air :

D ef f = f D v ,
where f is a dimensionless diffusivity tensor. Note that along one given direction, f = 1/µ where µ is the vapour resistance ratio used for building materials.

Using a suitable computational strategy, this set of equations is able to simulate the coupling between heat and mass occurring during transfer in building materials while taking into account the absence of thermodynamic equilibrium.

It allows three independent variables to be computed in space and in time (for example, temperature or enthalpy, moisture content, air density or gaseous pressure).

Inverse analysis

The principle of the inverse procedure is to feed the computational code with known parameters and to determine the unknown parameters by minimising an objective function. As our intend is to prove the prediction potential of the new formulation, only two parameters were identified : the initial moisture content X ini identified only to reduce the residue and the dimensionless diffusivity f , whose identification results in a characterisation of the material. The sample characteristics (thickness, density, porosity) were simply measured and supplied to the computation code. The sorption isotherm and the effective permeability come from recent papers [START_REF] Almeida | Hygroscopic behaviour of lignocellulosic materials: Dataset at oscillating relative humidity variations[END_REF][START_REF] Ai | A novel device to measure gaseous permeability over a wide range of pressures: Characterisation of slip flow for Norway spruce, European beech, and wood-based materials[END_REF]. The thermal conductivity is determined by a mixture law using the gaseous and solid volume fractions and their conductivities [START_REF] Louërat | Heat and moisture diffusion in spruce and wood panels computed from 3-D morphologies using the Lattice Boltzmann method[END_REF]. The relevance of the value given by this law was confirmed in [START_REF] Perré | On the importance of heat and mass transfer coupling for the characterization of hygroscopic insulation materials[END_REF]. Using the actual chamber conditions (temperature and RH) recorded during the test and the external heat transfer coefficients as determined in section 3.7, the physical model computes all variable fields versus time. It is therefore able to give the mean water content of the sample X and the RH at its back face.

This allows two objective functions S X and S RH to be derived to quantify the difference between experimental and predicted values using the mean square distance as norm:

S X = N i=1 [X * exp (t i ) -X * pred (t i )] 2 N ( 10
)
S RH = N i=1 [RH exp (t i ) -RH pred (t i )] 2 N ( 11
)
where X * exp is the dimensionless moisture content from the experiment, X * pred is the dimensionless moisture content predicted with estimated parameters, RH exp is the experimental value of RH, RH pred is predicted value of RH, t i is the time of the i th acquisition and N the number of instants used in the minimization. The dimensionless value of moisture content was used to correct the small differences that may appear between the sorption isotherm supplied to the code and the equilibrium moisture content obtained for each individual sample.

Accurately defined here thanks to the long duration of our experiments, it varies from 0 for the equilibrium moisture content at RH ini to 1 for the equilibrium moisture content at RH f in .

The global objective function S is a linear combination of those two functions:

S = β × S X + (1 -β) × S RH (12)
A value of β = 0.2 was used as best compromise between the two variables.

It corrects for the difference in variation range of the two variables (0.2 for RH and 1 for the dimensionless MC). A Nedler-Mead algorithm was used to minimise this objective function and find the best set of parameters (X ini and f ). Here, the identification starts five hours before the RH jump. At this moment the moisture content is already stabilized at its equilibrium value. The identification ends 50 hours after the perturbation for the two more diffusive samples, 130 hours after for the less diffusive samples. In both cases, the samples were almost at equilibrium.

Results and discussion

Without memory effects, the simulation uses a classical, yet comprehensive, set of coupled heat and mass transfer. In the case of LDF at 145kg.m -3 , the most diffusive material, the RH information travels very fast from the climatic chamber to the sample core, faster than diffusion inside fibres. Dual-scale effects are obvious and it is not possible to fit the two pieces of information (RH back and X) simultaneously. The identification procedure simply tries to decrease the global objective function by a compromise between the two variables. As a result, the RH evolution is too slow and the moisture content too fast (green curves in Fig. 8). With an average value of 0.332, the dimensionless factor identified without memory effects is lower than the published values for this material, either measured in steady-state experiments [START_REF] Rémond | Mass diffusivity of low-density fibreboard determined under steady-and unsteady-state conditions : Evidence of dual-scale mechanisms in the diffusion[END_REF][START_REF] Vololonirina | Characterization of hygrothermal properties of wood-based products -Impact of moisture content and temperature[END_REF] or predicted by upscaling from the real pore morphology [START_REF] Louërat | Heat and moisture diffusion in spruce and wood panels computed from 3-D morphologies using the Lattice Boltzmann method[END_REF].

On the contrary, when using the new formulation with memory effect, the model is in very good agreement with the experimental data for both RH and moisture content (blue curves in Fig. 8). Thanks to the memory effects, the identified diffusivity (average value of 0.573) is now consistent with literature data, even though the experiment is performed in transient state. Keeping in mind that the memory functions were not identified but measured on minute samples, these good results (simultaneous prediction of two variables and consistent value of f ) confirm that the new formulation perfectly accounts for all important physical phenomena. In particular, this includes the coupling between heat and mass transfer and the fading memory resulting from dual-scale effects and molecular relaxation.

For each sample, the identification of the effective diffusivity was performed without and with memory effects. The identified dimensionless diffusivities f as well as the residues between the model and the experiment for RH (S RH ) and dimensionless moisture content (S X ) are summarised in Table 5. In order to focus on the transient period during which the formulation matters, the residues were calculated over the first hours after the RH step (20 hours for the two most diffusive materials and 35 hours for the others). In general, the residues are dramatically reduced when using the memory function with an average decrease by a factor 7 for S RH and 1.5 for S X . The lower factor obtained for S X is partly explained by the noise of the experimental mass. Anyway, as the additional information embedded in the new formulation is not an additional degree of freedom, these good results confirm the previous comment regarding the 20-mm thick LDF sample. It demonstrates that the observed phase shift originates from the microscopic diffusive behaviour and validates the new formulation. For each material, one might notice, sometimes, a significant scattering of the identified values of f between samples A and B. This is not surprising, as important scattering of vapour diffusivity has already been reported in literature for those types of biosourced materials [START_REF] Vololonirina | Characterization of hygrothermal properties of wood-based products -Impact of moisture content and temperature[END_REF]. Thus, the determination of an average macroscopic parameter suitable for building energy simulation would require more repetitions, which the experimental device makes it possible to do in a simple way. One has however to mention that this scattering is due to the variability of the material morphology at the spatial scale of laboratory characterisation : neglecting the heat and mass coupling or the dual scale effects would add an important systematic error to this variability. Such a biais would 385 never be corrected, whatever the number of tested samples.

The results obtained for the 10-mm thick LDF samples with a density of 255kg.m -3 depict the same trend (Fig. 9). Once again, the classic formulation is not able to fit both quantities whereas the fit is excellent when using the memory function. In spite of the change in density and morphology, the same memory 390 function works perfectly. This is consistent as, whatever its density, the LDF has the same microscopic composition. This also proves that the memory functions are representative of microscopic effects, that just need to be characterised once.

Besides, the morphology influences the macroscopic behaviour: with an average value of 0.304, the identified diffusivities are lower for this material than the previous one. Again, it is in good agreement with what was measured for similar materials [START_REF] Rémond | Mass diffusivity of low-density fibreboard determined under steady-and unsteady-state conditions : Evidence of dual-scale mechanisms in the diffusion[END_REF]. The residues reduction when accounting for memory effects is similar to the previous material (a factor 5 for S RH and 1.4 for S X ).

The 20-mm thick LDF sample with a density of 222kg.m -3 suggests similar comments (Fig 10). Again, the agreement is excellent with memory effects. The average value of the identified diffusivity (0.404) lies between the two previous ones which is consistent with the material densities. However, one can note that even though the residues are still lower when using memory function, the difference between the two simulations is less important than for 10-mm thick samples (reducing factors of 3.6 for S RH and 1.4 for S X ). The memory effect is then less important in this case. Since this macroscopic diffusive time increases as the thickness squared and with the smaller f value compared with 145kg.m -3 LDF samples, more time is allowed to the fibre to approach equilibrium. The dual-scale mechanisms progressively vanished at increasing thickness. A further increase of the thickness would have also reduced the effect of molecular relaxation (the longest time constant of Table 4). In spite of this fact, the use of memory functions still considerably ameliorates the results at short times. This is of great interest for building materials submitted to day and night alternation.

In the case of MDF, the simulation results with or without memory effects are similar (Fig 11). When using the memory functions, the average residues only decrease by 9% for S RH and 20% for S X . This is not surprising as the high density of MDF dramatically reduces the diffusion at the macroscopic level: the average f value (0.064) is one order of magnitude lower than those obtained for LDF. Thus, even with a thickness of 10 mm, the macroscopic characteristic time is long enough to ensure local equilibrium. However, one must note that the model does not fit the experimental RH values. Even with memory effects, the residues remain important compared to the previous materials. Based on our predictive approach, one can affirm that this discrepancy can not be explained by microscopic effects, as those effects are embedded in the memory function determined on minutes samples. Therefore, the origin of this difference has yet to be explained, formulated and predicted. As MDF is a structural material, we believe that the stress profile in the thickness could be the explanation: during adsorption, swelling occurs at surface while the sample core is still close to the initial MC with little dimensional change. Depending on the mechanical behaviour, the external part of the sample is therefore under compression and the sample core under tension. Besides, compressive stress or contrained deformations are known to decrease the equilibrium moisture content of wood or lignocellulosic products [START_REF] Goulet | Phénomènes de second ordre de la sorption d'humidité dans le bois au terme d'un conditionnement de trois mois à température normale. Seconde partie : Essais du bois d'érable à sucre en compression radiale[END_REF][START_REF] Simpson | Moisture changes induced in red oak by transverse stress[END_REF][START_REF] Mårtensson | Mechano-sorptive effects in wooden material[END_REF]. This coupling between local scale (stress field on sorption isotherm) and global scale (stress profile along the sample thickness) is likely to give rise to another effect able to produce a fading or permanent memory. Its time constant would be tied to the macroscopic parameters, including thickness, while its amplitude would be tied to both macroscopic and microscopic effects. This could explains the very long time constant observed in wood for example [START_REF] Wadsö | Describing non-Fickian water-vapour sorption in wood[END_REF].

Conclusion and perspectives

This work is based on experimental results obtained with an original in-house device. This device was conceived to control the environment (temperature and relative humidity, RH) and to follow the evolution of several samples, by measuring simultaneously the RH on their back-face and their mass. Results are presented for LDF samples of different densities and thicknesses and for MDF samples. The experimental results were analysed by an inverse method using a comprehensive H&M model with non-Fickian effects. The following conclusions can be drawn :

• A classic H&M model is not able to predict correctly the evolution of both RH and moisture content for those materials, which denotes the presence of non-local equilibrium,

• For LDF, the model using the new formulation remarkably predicts the evolution of both quantities when it is fed with memory functions determined on micro-samples,

• This novel and validated formulation allows the materials to be accurately characterised in transient state, This work proves that the new formulation with non-Fickian effects is able to predict the diffusive behaviour of biosourced materials. This formulation should be used either at the lab scale for charaterisation or at the building scale to predict the energy demand.

Figure 1 :

 1 Figure 1: Principle of the identification procedure

Figure 2 :

 2 Figure 2: Photograph of the samples: From left to right and top to bottom: LDF-255kgm -3 -10mm, LDF-145kgm -3 -20mm, LDF-222kgm -3 -20mm, MDF-655kgm -3 -10mm

  Fig 5, one can see the evolution of the average RH and temperature inside the climatic chamber while performing a stepwise change in RH from 40 to 60 %. It takes ca. 10 minutes for the thermocryostat to reach the targeted temperature. At this time, the change in RH already reached 90 % of its total change. The targeted value is obtained after ca. one hour, when the temperature has decreased to its equilibrium value.

  Fig 3a. The crown holding all samples can move vertically to ensure weighing and RH measurement and can rotate to deal successively with each sample. This is ensured respectively by an electromechanical actuator (EMC32-12, Rexroth, Lohr am Main, Germany) driven by a DC motor and a precision rotation stage (M-URM80, Newport Corporation, Irvine, United States). Three vertical positions are secured: i) the upper position for temporary contact with the RH sensor, (Fig 4a), ii) the lower position to put the sample on the weighing plate (Fig 4b) and the intermediate position allowing a free rotation of the crown.

Figure 3 :

 3 Figure 3: Left:a) CAD representation of the setup with: 1)Electromechanical cylinder, 2)Horizontal motor, 3)Crown, 4)Weighing system, 5)Structure for RH acquisition, Right:b) Photograph of the whole setup in the climatic chamber

Figure 4 :

 4 Figure 4: Left a): Sample support in upper position, the contact between the target (in red) and the contactors allows the acquisition of RH at the back face of the sample, Right b): Sample support in lower position, the sample support is placed on the weighing plate (in red).

Figure 5 :

 5 Figure 5: Evolution of RH and temperature inside the climatic chamber during a RH step from 40 to 60 %

Figure 6 :

 6 Figure 6: Left a) Exploded view of the 3D model with: 1) Target, 2) Cone, 3) Rod, 4) Upper part, 5) Holder (with rubber tube inside), 6) Spacer, 7) Impervious disk, 8) Sample, Right b) Photograph of the sample support.

Figure 7 :

 7 Figure 7: Example of raw data obtained for MDF: actual RH value in the climatic chamber, RH at the back face and sample mass.

Figure 8 :

 8 Figure 8: Comparison between simulation and experiment for sample A of LDF-20mm-145kg.m -3 : (a) RH evolution at the back face (top), (b) dimensionless average moisture content (bottom).

Figure 9 :

 9 Figure 9: Comparison between simulation and experiment for sample A of LDF-10mm-255kg.m -3 : (a) RH evolution at the back face (top), (b) dimensionless average moisture content (bottom).

Figure 10 :

 10 Figure 10: Comparison between simulation and experiment for sample A of LDF-20mm-222kg.m -3 : (a) RH evolution at the back face (top), (b) dimensionless average moisture content (bottom).

Table 1 :

 1 List symbols (Latin letters)

	Symbol	Name

X

solid moisture content (dry basis) -

Table 2 :

 2 List of symbols (Greek letters)

	Symbol	Name	Unit
		proportion terms of memory
	α		-
		function
	ε	volume fraction	-
	ϕ	memory function	-
	λ	thermal conductivity	W.m -1 .K -1
	µ	vapour resistance ratio	-
	ρ	density	kg.m -3
	∂	partial derivative	-
	∇	gradient	-
	∇ •	divergence	-
	ω	mass fraction	-
	ψ	gravitational potential	s -2
		time constants of memory func-
	τ		s
		tion

Table 3 :

 3 Subscripts and superscripts

	Subscripts	Meaning
	a	air
	b	bound water
	ef f	effective property
	eq	at equilibrium
	exp	experimentally obtained
	g	relative to the gaseous phase
	pred	predicted by model
	s	solid
	v	water vapour
	vs	saturated water vapour
	w	liquide water
	∞	at large distance from interface
	Superscript Meaning
	ψ	averaged of variable ψ over the REV
	ψ	intrinsic average of ψ over phase

Macroscopic model with new formulation previously measured memory functions Microscopic parameters : New experimental device measuring RH at back face and mass evolution Inverse method Simulations Experimental results

Table 4 :

 4 Values of the coefficients defining the memory function for both materials (identified

	Since LDF and MDF

  With memory 0.341 8.41 × 10 -3 3.32 × -2

	Material	Sample	Formulation	f	S RH	S X
			Classic	0.366 1.62 × 10 -2 2.90 × -2
	LDF	A	With memory 0.667 2.11 × 10 -3 2.09 × -2
	20 mm 145 kg • m -3	B	Classic With memory 0.480 3.17 × 10 -3 2.32 × -2 0.298 1.92 × 10 -2 3.65 × -2
			Classic	0.171 1.94 × 10 -2 3.55 × -2
	LDF	A	With memory 0.306 2.94 × 10 -3 2.68 × -2
	10 mm 255 kg • m -3	B	Classic With memory 0.302 6.00 × 10 -3 2.70 × -2 0.169 2.25 × 10 -2 4.08 × -2
			Classic	0.347 1.34 × 10 -2 2.55 × -2
	LDF	A	With memory 0.478 2.95 × 10 -3 2.15 × -2
	19 mm 234 kg • m -3	B	Classic	0.267 1.93 × 10 -2 4.29 × -2
			Classic	0.0482 1.81 × 10 -2 3.55 × -2
	MDF	A	With memory 0.0497 1.79 × 10 -2 2.50 × -2
	10 mm 655 kg • m -3	B	Classic With memory 0.0785 1.82 × 10 -2 2.80 × -2 0.0752 2.23 × 10 -2 3.12 × -2

Table 5 :

 5 Identified diffusivity and residues with or without memory functions


MD Ali Amad
email: ali.amad@kcl.ac.uk

MD Joaquim Radua

MD Guillaume Vaiva

Williams Phd

MD Thomas Fovet

Similarities between Borderline Personality Disorder and Post traumatic Stress Disorder: evidence from Resting-State Meta-Analysis

Keywords: borderline personality disorder, post-traumatic stress disorder, resting state, age-dependant neuroplasticity, trauma

Borderline personality disorder (BPD) and post-traumatic stress disorder (PTSD) are common psychiatric disorders. The nature of the relationship between BPD and PTSD remains controversial, but it has been suggested that these disorders should brought closer because of their many similarities.

We thus performed a quantitative meta-analysis of resting-state functional imaging to assess similarities in the brain activation across BPD and PTSD diagnostic groups.

Overlap analyses revealed decreased activation in the left and right precuneus of both BPD and PTSD groups when compared to control subjects. BPD showed significant increased, but PTSD showed decreased activation, relative to control subjects, in the anterior cingulate/paracingulate gyri and in the left superior frontal gyrus. Complementary overlap analyses on a subgroup of studies with similar sex and age distribution partially confirmed the main results as the same pattern of functional activation in the anterior cingulate and in the left superior frontal gyrus were found.

Our findings are in agreement with the hypothesis that BPD and PTSD share common neuropathological pathways.

INTRODUCTION

Borderline personality disorder (BPD) is a common mental disorder characterized by a pervasive pattern of emotional lability, impulsivity, interpersonal difficulties, identity disturbances, and disturbed cognition (e.g., depersonalization, derealization, and hallucinations) [START_REF] Lieb | Borderline personality disorder[END_REF].

BPD is estimated to occur in 0.5-5.9% of the general population [START_REF] Grant | Prevalence, correlates, disability, and comorbidity of DSM-IV borderline personality disorder: results from the Wave 2 National Epidemiologic Survey on Alcohol and Related Conditions[END_REF][START_REF] Lenzenweger | DSM-IV Personality Disorders in the National Comorbidity Survey Replication[END_REF] and is likely to be diagnosed in early adulthood [START_REF] Lenzenweger | DSM-IV Personality Disorders in the National Comorbidity Survey Replication[END_REF]. Follow-up studies show a decrease in impulsivity with age [START_REF] Stevenson | Diminished Impulsivity in Older Patients With Borderline Personality Disorder[END_REF][START_REF] Zanarini | The Subsyndromal Phenomenology of Borderline Personality Disorder: A 10-Year Follow-Up Study[END_REF], but associated mood disorders and interpersonal difficulties appear to be persistent and chronic [START_REF] Zanarini | The Subsyndromal Phenomenology of Borderline Personality Disorder: A 10-Year Follow-Up Study[END_REF]. BPD is commonly comorbid with other psychiatric disorders, notably mood disorders, anxiety disorders, substance abuse, and other personality disorders [START_REF] Skodol | The borderline diagnosis I: psychopathology, comorbidity, and personality structure[END_REF][START_REF] Tomko | Characteristics of Borderline Personality Disorder in a Community Sample: Comorbidity, Treatment Utilization, and General Functioning[END_REF].

Post-traumatic stress disorder (PTSD) can develop following exposure to extremely traumatic events (e.g. physical and sexual assaults, accidents, natural disasters). PTSD is characterized by the persistence of intense, distressing, and fearfully avoided reactions to reminders of the triggering event, alteration of mood and cognition, distressing and intrusive memories and nightmares of the trauma, irritability and hypervigilance. PTSD occurs in 5-10% of the population and is also associated with suicidal behaviour [START_REF] Shalev | Post-Traumatic Stress Disorder[END_REF][START_REF] Yehuda | Post-traumatic stress disorder[END_REF]. Both BPD and PTSD are associated with high mortality and morbidity, due to suicide, frequent hospitalization, substance use, psychiatric comorbidity, considerable economic burden and poor quality of interpersonal relationships [START_REF] Bisson | Post-traumatic stress disorder[END_REF][START_REF] Skodol | The borderline diagnosis I: psychopathology, comorbidity, and personality structure[END_REF].

The nature of the relationship between BPD and PTSD remains controversial but several arguments suggest that these disorders should brought closer because of their many similarities [START_REF] Cloitre | Distinguishing PTSD, Complex PTSD, and Borderline Personality Disorder: A latent class analysis[END_REF][START_REF] Frías | Comorbidity between Post-Traumatic Stress Disorder and Borderline Personality Disorder: A Review[END_REF][START_REF] Gunderson | The phenomenological and conceptual interface between borderline personality disorder and PTSD[END_REF][START_REF] Lewis | Borderline personality or complex posttraumatic stress disorder? An update on the controversy[END_REF].

First, BPD and PTSD share numerous clinical features, particularly major disturbances in emotional and affect regulation, impulse control, reality testing, interpersonal relationships, self-evaluation and sense of identity [START_REF] Gunderson | The phenomenological and conceptual interface between borderline personality disorder and PTSD[END_REF][START_REF] Lewis | Borderline personality or complex posttraumatic stress disorder? An update on the controversy[END_REF][START_REF] Resick | A critical evaluation of the complex PTSD literature: Implications for DSM-5[END_REF]. Second, the role of trauma is obviously essential for PTSD but it also plays a major role in BPD. Indeed, the prevalence of childhood trauma is so frequently associated with BPD (either neglect (92%), sexual abuse (40%-70%) or physical abuse (25%-73%) [START_REF] Zanarini | Severity of reported childhood sexual abuse and its relationship to severity of borderline psychopathology and psychosocial impairment among borderline inpatients[END_REF]) that a causal relationship can be suggested [START_REF] Ball | Borderline personality disorder and childhood trauma: Evidence for a causal relationship[END_REF]. Individuals with BPD are also probably vulnerable to develop PTSD [START_REF] Golier | The Relationship of Borderline Personality Disorder to Posttraumatic Stress Disorder and Traumatic Events[END_REF]. Furthermore, recent studies seem to indicate that BPD shares a genetic vulnerability with PTSD as FKBP5 variants -an important functional regulator of the glucocorticoid receptor complex-are associated with PTSD [START_REF] Zannas | Gene-environment interactions at the FKBP5 locus: sensitive periods, mechanisms and pleiotropism[END_REF]-and BPD [START_REF] Amad | FKBP5 gene variants and borderline personality disorder[END_REF][START_REF] Martín-Blanco | The role of hypothalamus-pituitary-adrenal genes and childhood trauma in borderline personality disorder[END_REF], which appears to be consistent with the hypothalamus-pituitary-adrenal (HPA) axis dysregulation highlighted in both disorders [START_REF] De Kloet | Assessment of HPA-axis function in posttraumatic stress disorder: pharmacological and non-pharmacological challenge tests, a review[END_REF][START_REF] Drews | Hypothalamic-pituitary-adrenal axis functioning in borderline personality disorder: A meta-analysis[END_REF][START_REF] Schumacher | HPA axis regulation in posttraumatic stress disorder: A meta-analysis focusing on potential moderators[END_REF][START_REF] Wingenfeld | Borderline personality disorder: Hypothalamus pituitary adrenal axis and findings from neuroimaging studies[END_REF]. Finally, from a neuro-functional perspective, BPD and PTSD are supposed to share the same abnormalities. Indeed, abnormalities in fronto-limbic networks, particularly a limbic hyperactivity and a diminished recruitment of frontal brain regions, have been suggested for both BPD and PTSD (Krause-Utz et al., 2014;[START_REF] Patel | Neurocircuitry models of posttraumatic stress disorder and beyond: a meta-analysis of functional neuroimaging studies[END_REF]. Interestingly, BPD and PTSD recently showed, in several resting-state functional meta-analysis some common features. Both disorders were associated with an increased activity in inferior and superior frontal gyrus, respectively involved in response inhibition, working memory and attention [START_REF] Amad | Resting-state meta-analysis in Borderline Personality Disorder: Is the fronto-limbic hypothesis still valid?[END_REF][START_REF] Wang | Altered resting-state functional activity in posttraumatic stress disorder: A quantitative meta-analysis[END_REF]. In the same studies, resting-state activation in other regions were found to be differently associated to both disorders such as the anterior cingulate cortex that showed an increased activation in BPD versus healthy controls, but a decreased activation in PTSD versus healthy controls.

To investigate the relationship between BPD and PTSD from a neuro-functional perspective, we performed a functional quantitative meta-analysis to compare results across diagnostic groups and to assess functional similarities that may reflect common neuropathological pathways in these disorders.

MATERIAL AND METHODS

All brain imaging studies of the resting-state studies which have compared brain activation in patients with BPD or PTSD versus healthy controls were included in two separate meta-analyses (one for BPD and one for PTSD). To identify eligible studies, a systematic search, in respect to the PRISMA statement [START_REF] Moher | Preferred Reporting Items for Systematic Reviews and Meta-Analyses: The PRISMA Statement[END_REF], was conducted on the Medline and ISI Web of Knowledge up to April 2018 using the following search term combinations: (1) ''neuroimaging'', ''fMRI'', ''PET'', "ALFF", " ReHo", " rCBF", (2) ''resting-state'', ''default network'' and (3) the terms ''borderline personality disorder'' or "post-traumatic stress disorder". Reviews and meta-analyses were cross-referenced to identify studies that were missed in the literature searches.

Studies were only selected according to the following inclusion criteria: 1) studies were published in an English-language peer-reviewed journal; 2) used at least one of the functional imaging techniques (fMRI, PET, or SPECT), 3) enrolled patients with BPD or PTSD compared to controls, 4) performed a whole brain analysis and reported activation coordinates in standard space (MNI or Talairach), 5) the diagnosis was made according to the current diagnostic criteria for both disorders. Also, we ensured that the same threshold throughout the whole brain was used within each included study, in order to avoid bias toward liberally thresholded brain regions. Authors were contacted for unpublished data including t-maps from the original studies. Article titles and the abstracts of studies identified from the searches were screened and excluded for the following reasons: not written in English, used seed-voxel-based analysis procedures, reported only region of interest results, review article, opinion, or hypothesis article; and samples with a disorder other than BPD or PTSD. The literature search strategy is summarised in the flow chart presented in Supplementary Figures 1 and2.

The functional neuroimaging meta-analyses were performed using the anisotropic effect size version of Seed-based D Mapping (formerly Signed Differential Mapping, AES-SDM, www.sdmproject.com) [START_REF] Radua | Anisotropic Kernels for Coordinate-Based Meta-Analyses of Neuroimaging Studies[END_REF][START_REF] Radua | A new meta-analytic method for neuroimaging studies that combines reported peak coordinates and statistical parametric maps[END_REF]. This method allows the inclusion of both peak information (coordinates and t-values) and statistical parametric maps to create whole-brain effect size and variance maps, which are then used to perform voxel-wise random effects meta-analyses.

Peak coordinates and effect-sizes (e.g. t-values or z-scores) of statistically significant differences between patients and controls at the whole-brain level were extracted from each dataset. A standard MNI map of the resting-state differences was then separately recreated for each dataset using an anisotropic Gaussian kernel. The mean map was finally generated by voxel-wise calculation of the random-effects mean of the dataset maps, weighted by the sample size, intradataset variability, and between-dataset heterogeneity. All analyses were conducted using the grey matter template included in AES-SDM (template sampling size of 2x2x2 mm3 voxels). The default SDM kernel size and thresholds were used (full width at half maximum [FWHM] = 20 mm, p < 0.005, peak Z value of >1, cluster size of >10). To assess heterogeneity between studies, I2 values were extracted from the meta-analytic peaks and interpreted according to [START_REF] Higgins | Cochrane Handbook for Systematic Reviews of Interventions[END_REF]. Jackknife sensitivity analyses, consisting of iteratively repeating the meta-analysis excluding one study at a time, were also conducted, for both BPD and PTSD, to examine the robustness of the main meta-analytic output. Meta-regression analyses were performed to study the effects of age and gender. Theses analyses were thresholded more conservatively (p < 0.0005) to minimize spurious findings. Publication bias was also examined by using the Egger test.

The multimodal analysis function of the AES-SDM statistical package also allows conjunction analyses to be performed, which enabled us to identify regions where both patient groups show common differences with respect to controls, while taking into account error in the estimation of the magnitude of these differences [START_REF] Radua | A general approach for combining voxel-based meta-analyses conducted in different neuroimaging modalities[END_REF]. The default parameters of SDM were used (peak p < 0.00025, cluster size of >10).

The main meta-analyses were performed for BPD and all the PTSD studies. Importantly, neural activity abnormalities in patients with PTSD compared to controls can appear dramatically different depending on whether individuals without trauma exposure (non-trauma exposed controls; NTC) or individuals with trauma exposure who did not develop PTSD (trauma-exposed controls; TEC) are employed as experimental controls [START_REF] Disner | Spontaneous neural activity differences in posttraumatic stress disorder: A quantitative restingstate meta-analysis and fMRI validation[END_REF]. To account for such heterogeneity, the analyses were also performed using, both by merging all the PTSD studies and by performing separate analysis according to the control group. Finally, as BPD and PTSD patients represent different socio-demographic characteristics, especially for age and gender, we conducted a subgroup analysis of studies with similar sex and age distribution (more than half females and mean age less than 40 years).

RESULTS

Studies characteristics

Borderline personality disorder

Nine studies comparing patients and healthy controls (Supplementary Table 1) were included. These studies included a total of 235 patients and 214 healthy controls. Patients' mean age was 28,5 years (SD=5.3) and 86.5 % were female. The mean age of healthy control participants was 30.4 years (SD=6.3), and 83.6 % were female. No information about the history of trauma and its characteristics (age and type) was available.

Post-traumatic stress disorder

Eighteen studies comparing patients and healthy controls (Supplementary Table 2) were included. These studies included a total of 433 patients and 619 healthy controls. Patients' mean age was 39.4 years (SD=8.78) and 41.3 % were female. The mean age of healthy control participants was 40.5 years (SD=7.8), and 39.4 % were female.

Meta-analysis

BPD vs healthy controls

Functional differences in BPD relative to healthy controls are shown in Table 1 andFigure 1. We found an increased activation in patients with BPD relative to healthy controls in the anterior cingulate, paracingulate gyri and in the left inferior frontal gyrus. Reduced activation was found in the right precuneus. Heterogeneity between studies was substantial in the anterior cingulate cortex (I2 = 63 %), moderate in the left inferior gyrus (I2 = 47 %) and was not seen in the right precuneus.

PTSD vs healthy controls

Functional differences in PTSD relative to healthy controls are shown in Table 2 andFigure 2. We found an increased activation in patients with PTSD relative to healthy controls in the left supramarginal, the right anterior insula, the left cerebellum, the right parahippocampal gyrus, the right dorso-lateral prefrontal cortex (dlPFC) and the median cingulate and paracingulate gyri.

Reduced activations in patients with PTSD relative to healthy controls were found in the right Heschl gyrus, the right lingual gyrus and inferior and middle frontal gyri. Heterogeneity between studies was low for all the brain regions found significant (I2 < 25%). The results of the meta-analysis performed by separating the control group in TEC and NTC are detailed in Table 3 and4.

Reliability analysis

Jackknife sensitivity analyses showed that the main findings were highly replicable across combinations of datasets for both disorders. Indeed, the increased activity in BPD patients in anterior cingulate cortex and left inferior frontal gyrus were robust as they were significant in all but one combination. Results in the right precuneus were preserved throughout all combinations of the data sets. The activation in patients with PTSD were also highly replicable, being preserved in 16 of 24 combinations of the data sets. See Supplementary Table 3 and 4 for details.

Age and gender effects

For BPD, meta-regression analysis suggested that a higher percentage of females had significantly increased activation in the left inferior frontal gyrus (MNI coordinates = -44, 42, 10; SDM = 2.697; p < 0.0001, 78 voxels) and in the anterior cingulate (MNI coordinates = 6,44,12 ; SDM = 2.444 ; p < 0.0001; 51 voxels) and a decreased activation the middle occipital gyrus (MNI coordinates = 34,-72,30; SDM = -2.739, p < 0.0001; 726 voxels). Age was associated with an increased activation in the right middle occipital gyrus (MNI coordinates = 34, -72, 32; SDM =2.802, p < 0.0001, 758 voxels) and a decreased activation in the middle frontal gyrus (MNI coordinates = -44,44,10; SDM = -2.470; p < 0.0001; 172 voxels).

For PTSD, meta-regression analysis suggested that a higher percentage of females had significantly increased activation in the right fusiform gyrus (MNI coordinates = 20,-38,-14; SDM =

Publication bias analysis

For BPD, there was no publication bias for these clusters (p-values for Egger tests > 0.15). For PTSD, only the cluster encompassing the dlPFC showed a publication bias (p = 0.003).

Overlap analyses

Overlap analyses (Figure 3) revealed a significant shared decreased activation in both BPD and PTSD groups relative to control subjects in left and right precuneus (MNI coordinates = 0, -46, 38; 606 voxels). BPD showed significant increased, but PTSD decreased activation relative to control subjects in the anterior cingulate/paracingulate gyri and in the left superior frontal gyrus (MNI coordinates = 2, 44, 20; 1205 voxels). The overlap analyses by using specifically TEC or NTC for PTSD studies are presented in Tables 5 and6.

Complementary overlap analyses were also performed on a subgroup which only included studies with more than half females and a mean age of less than 40 years. Several regions were identified where BPD showed significant increased but PTSD decreased activation relative to control subjects in the anterior cingulate/paracingulate gyri and in the left superior frontal gyrus (MNI coordinates = 2, 38, 0; 506 voxels), but also in the left insula and inferior frontal gyrus (MNI coordinates = -36, 12, 6; 2169 voxels) and in the left hippocampus (MNI coordinates = -26, -8, -24;

DISCUSSION

Here, we report findings from two distinct resting-state functional meta-analyses conducted in BPD and PTSD. For BPD, we found an increased activity in patients relative to healthy controls in the anterior cingulate cortex (ACC) and in the left inferior frontal gyrus, and a reduced activity in the right precuneus, which complements previously published functional meta-analyses using the same method [START_REF] Amad | Resting-state meta-analysis in Borderline Personality Disorder: Is the fronto-limbic hypothesis still valid?[END_REF][START_REF] Visintin | Mapping the brain correlates of borderline personality disorder: A functional neuroimaging meta-analysis of resting state studies[END_REF]. These results contrast with the established fronto-limbic hypothesis of BPD which postulates a decreased activity of frontal brain regions and a limbic hyperactivity. In fact, the brain regions found here may correspond to specific networks involved in pain processing [START_REF] Kluetsch | Alterations in default mode network connectivity during pain processing in borderline personality disorder[END_REF] and dissociative states (i.e. disintegration of perception, consciousness, identity and memory). Dissociation, characterized by subjective detachment from overwhelming emotional experience during and after a trauma, is known to be frequently found (up to two thirds) of BPD patients [START_REF] Vermetten | Trauma and dissociation: implications for borderline personality disorder[END_REF]. Interestingly, the regions identified here have been associated in participants with PTSD while in dissociative states [START_REF] Lanius | Brain activation during script-driven imagery induced dissociative responses in PTSD: a functional magnetic resonance imaging investigation[END_REF]. We also found an increased activity in the left inferior frontal gyrus a critical brain region for response inhibition and for successful implementation of inhibitory control over motor responses [START_REF] Swick | Left inferior frontal gyrus is critical for response inhibition[END_REF] which has also been associated with attentiondeficit/hyperactivity disorder (ADHD) [START_REF] Van Rooij | Altered neural connectivity during response inhibition in adolescents with attentiondeficit/hyperactivity disorder and their unaffected siblings[END_REF] another disorder characterised by impulsivity and which is often associated with BPD. Finally, one can also suggest that, as several studies about functional neuroimaging findings of emotion processing tasks have shown some evidence of the fronto-limbic hypothesis [START_REF] Schulze | Neural Correlates of Disturbed Emotion Processing in Borderline Personality Disorder: A Multimodal Meta-Analysis[END_REF], the resting-state imaging may not be the most accurate modality to demonstrate or refute it.

Regarding PTSD, our results are also highly consistent with a recent resting-state functional meta-analysis using the same method [START_REF] Wang | Altered resting-state functional activity in posttraumatic stress disorder: A quantitative meta-analysis[END_REF], even after adding 3 new neuroimaging studies. We found an increased activation in the anterior insula, the cerebellum, the cingulate and paracingulate gyri and the parahippocampal gyrus and a decreased activation in the Heschl and lingual gyri. When the analyses were performed with specific control groups (i.e. NTC or TEC), our results also confirmed previous studies: PTSD patients show (i) increased activation in the anterior insula and decreased activation in the dorso-medial prefrontal cortex when compared with NTC, (ii) increased activation in the ventro-medial prefrontal cortex when compared with TEC [START_REF] Wang | Altered resting-state functional activity in posttraumatic stress disorder: A quantitative meta-analysis[END_REF].

In this study, we have also directly compared BPD and PTSD to identify common and distinct patterns of functional modulations. We showed that the two conditions share similar patterns of decreased activation in left and right precuneus. We have also identified that the anterior cingulate/paracingulate gyri and left superior frontal gyrus are affected in both conditions (increased activation for BPD patients and decreased activation for PTSD subjects, relative to control subjects).

These results were partially confirmed by complementary analysis on a more homogeneous subgroup regarding age and gender and when the analyses were performed using specific control groups (TEC or NTC).

Reduced activation in precuneus was found in both BPD and PTSD. The precuneus, which is part of the medial parietal cortex, plays a central role to link the limbic structures with parietal regions. This region is also involved in autobiographical memory and in self-referential processing, namely first-person perspective and experience of agency [START_REF] Cavanna | The precuneus: a review of its functional anatomy and behavioural correlates[END_REF]. Precuneus activity has been demonstrated as related to trauma memory generalization [START_REF] Hayes | Reduced hippocampal and amygdala activity predicts memory distortions for trauma reminders in combat-related PTSD[END_REF], and flashbacks [START_REF] Whalley | An fMRI investigation of posttraumatic flashbacks[END_REF]. Thus, alterations in the precuneus found in BPD and PTSD patients may be related to altered self-referential and memory processes, such as memory deficits, intrusions or flashbacks.

Anterior cingulate/paracingulate gyri and left superior frontal gyrus were also identified to be affected in both conditions. The anterior cingulate gyrus is part of the medial prefrontal cortex (PFC) which exerts inhibitory control over stress responses and emotional reactivity partly through connections with the amygdala. Interestingly, patients with PTSD also exhibit decreased ACC volumes [START_REF] Bromis | Meta-Analysis of 89 Structural MRI Studies in Posttraumatic Stress Disorder and Comparison With Major Depressive Disorder[END_REF], that seem to be consequences of the course of PTSD rather than a pre-existing risk factor [START_REF] Kasai | Evidence for acquired pregenual anterior cingulate gray matter loss from a twin study of combat-related posttraumatic stress disorder[END_REF]. Functional imaging studies involving patients with PTSD also found decreased activation of the medial PFC in response to trauma stimuli, such as trauma scripts, combat pictures and sounds, trauma-unrelated negative narratives, fearful faces, emotional stroop (for review see [START_REF] Sherin | Post-traumatic stress disorder: the neurobiological impact of psychological trauma[END_REF]). Even if there are discordant findings, reduced activation of the medial PFC was associated with PTSD symptom severity in several studies. Successful SSRI treatment has also been shown to restore medial PFC activation patterns [START_REF] Shin | Amygdala, medial prefrontal cortex, and hippocampal function in PTSD[END_REF].

Regarding BPD, higher levels of glutamate in the ACC were found in patients when compared with healthy controls, and there are positive correlations between glutamate levels and impulsivity [START_REF] Hoerst | Correlation of Glutamate Levels in the Anterior Cingulate Cortex With Selfreported Impulsivity in Patients With Borderline Personality Disorder and Healthy Controls[END_REF]. It has also been shown that the hyperactivation of ACC was correlated with hippocampal activity, abnormal functioning of the memory system, poorer performance in retrieval tasks related to traumatic or emotionally significant memories [START_REF] Schmahl | A positron emission tomography study of memories of childhood abuse in borderline personality disorder[END_REF].

We believe that our results are in line with the hypothesis that the conceptual interface between BPD and PTSD could be highlighted by the 'age-dependent neuroplasticity' framework [START_REF] Amad | The age-dependent plasticity highlights the conceptual interface between borderline personality disorder and PTSD[END_REF] which postulates that the same experience can differentially affect the brain plasticity depending on one's age [START_REF] Kolb | Searching for the principles of brain plasticity and behavior[END_REF]. In this perspective, it has then been proposed that trauma during childhood could favour the development of BPD [START_REF] Amad | Genetics of borderline personality disorder: systematic review and proposal of an integrative model[END_REF] while trauma in adulthood could increase the risk of PTSD [START_REF] Amad | The age-dependent plasticity highlights the conceptual interface between borderline personality disorder and PTSD[END_REF]. However, this hypothesis could not be tested since no BPD studies included data about the trauma history of these patients. Interestingly, there is a current debate about the distinction between PTSD, BPD and complex PTSD (PTSD plus several symptoms such as emotional dysregulation, negative selfcognitions and interpersonal relationship problems). Complex PTSD has even been recently proposed as a distinct clinical entity in the WHO International Classification of Diseases, 11th version. It has also recently been suggested that the relationship between these three disorders could follow a continuum of trauma and biological severity on a spectrum of common clinical features. These disorders could then be classified together under trauma-related disorders instead of distinct clinical diagnoses [START_REF] Giourou | Complex posttraumatic stress disorder: The need to consolidate a distinct clinical syndrome or to reevaluate features of psychiatric disorders following interpersonal trauma?[END_REF]. However, one of the most frequent, and important, criticism when considering BPD as a trauma spectrum disorder is that the existence of trauma is not necessary or sufficient to explain the development of the disorder [START_REF] Lewis | Borderline personality or complex posttraumatic stress disorder? An update on the controversy[END_REF].

One of the main limitations of this study corresponds to the very different sociodemographic characteristics of BPD and PTSD patients included, especially for age and gender as highlighted by the meta-regression analysis. Accordingly, we conducted a complementary analysis focusing on studies with more than half females and a mean age of less than 40 years. This complementary analysis partially confirmed our main results as we found the same pattern of functional activation in the ACC and in the left superior frontal gyrus between BPD and PTSD.

Another possible limitation is the use of different statistical approaches such as independent component analysis, regional homogeneity, etc. to assess resting-state in the different studies included [START_REF] Kühn | Resting-State Brain Activity in Schizophrenia and Major Depression: A Quantitative Meta-Analysis[END_REF]. As soon as more resting-state studies on BPD and PTSD patients are published, new analysis should be performed including studies with the same statistical approach and future research will also need to confirm our results. Cautious interpretation is also needed because BPD and PTSD have a potentially high clinical heterogeneity and have been associated with many comorbidities, such as depression which can have a significant effect on the resting brain activity (e.g. [START_REF] Mulders | Resting-state functional connectivity in major depressive disorder: A review[END_REF]). Unfortunately, data presented in the different studies included in these meta-analyses did not allow to include these potential confounding factors in our analyses. Our results therefore can hardly be attributed solely to BPD or PTSD. In the same way, no sufficient information about medication was available to perform complementary analysis. Future functional neuroimaging research in this field should directly compare BPD and PTSD patients in studies specifically designed to compare these populations using appropriate paradigms (e.g. task or restingstate). The participants included should be carefully assessed to explore specific dimensions and refined phenotypes (e.g., type and number of trauma, age of exposure, psychotic features or impulsivity). From an epidemiological point of view, the clinical profile of BPD patients with and without a history of childhood abuse could also be directly compared as it seems that the clinical profiles of BPD patients appear to differ according to the severity of childhood trauma [START_REF] Zanarini | Severity of reported childhood sexual abuse and its relationship to severity of borderline psychopathology and psychosocial impairment among borderline inpatients[END_REF]. Ideally, these analyses would be performed in large well-characterized epidemiological cohorts powerful enough to detect BPD patients without a history of trauma. Lastly, genetic studies could also help to highlight the similarities/differences between BPD and PTSD, by, for example, comparing shared polygenic risk between these disorders.

In conclusion, this study sheds new light on the neurobiology of BPD, which has been little investigated despite its high prevalence (0.5-5.9% of the general population) and severity [START_REF] Amad | Genetics of borderline personality disorder: systematic review and proposal of an integrative model[END_REF] whilst also highlighting the functional similarities between BPD and PTSD. Our findings help reinforce the hypothesis that BPD shares conceptual and phenomenological similarities with PTSD.

One difference between these disorders could be the age at which trauma occurs, differentially affecting the resting brain function and thus the psychiatric symptoms [START_REF] Amad | The age-dependent plasticity highlights the conceptual interface between borderline personality disorder and PTSD[END_REF].

FIGURES AND TABLES CAPTIONS

  6.166; p ~ 0; 143 voxels) and in the superior temporal gyrus (MNI coordinates = 48,12,-16; SDM = 5.629; p ~ 0; 109 voxels) and a decrease in the left angular gyrus (MNI coordinates = -42,-62,34; SDM = -5.493; p ~ 0; 45 voxels). Age was associated with an increased activation in the right supplementary area (MNI coordinates = 10,-2,46; SDM = 4.518; p ~ 0; 1401 voxels) and in the right frontal superior gyrus (MNI coordinates = 22,-2,48; SDM = 2.170; p < 0.0001; 25 voxels) and a decreased activation in the left olfactory cortex (MNI coordinates = -18,4,-14; SDM = -2.018, p < 0.0001; 381voxels) and in the left lingual gyrus (MNI coordinates = -16,-98,-12; SDM = -1.546; p < 0.0001; 127 voxels).

Figure 1 .

 1 Figure 1. Regions of increased (red) and decreased (blue) activation at rest in patients with borderline personality disorder compared with healthy controls. Statistical maps are thresholded at p < 0.005 and k > 10.

Figure 2 .

 2 Figure 2. Regions of increased (red) and decreased (blue) activation at rest in patients with posttraumatic stress disorder compared with healthy controls. Statistical maps are thresholded at p < 0.005 and k > 10.

Figure 3 .

 3 Figure 3. Clusters showing significant shared decreased activation in both BPD and PTSD (purple) groups relative to control subjects in left and right precuneus (MNI coordinates: 0, -46, 38; 606 voxels). Other regions (in green) were also identified where BPD showed significant increased but PTSD decreased activation relative to control subjects in the anterior cingulate/paracingulate gyri and in the left superior frontal gyrus (MNI coordinates: 2, 44, 20; 1205 voxels).

Table 1 .

 1 Clusters showing differences between BPD and healthy controls.

Table 2 .

 2 Clusters showing differences between PTSD and all healthy controls.

Table 3 .

 3 Clusters showing differences between PTSD and trauma exposed controls (TEC).

Table 4 :

 4 Clusters showing differences between PTSD and non-trauma exposed controls (NTC).

Table 5 .

 5 Clusters showing similar activation in BPD and PTSD (trauma exposed controls, TEC).

Table 6 .

 6 Clusters showing similar activation in BPD and PTSD (non-trauma exposed controls, NTC).

				SDM-Z	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	BPD > Controls
	-10 56	22	1.647	0.000304401	805	32	L+R	Anterior cingulate / paracingulate gyri
	-54 10	16	1.371	0.002392530	113	44, 48	L	Left inferior frontal gyrus
	BPD < Controls
	6	-62 44	-2.268	0.000009835	2043	7	R	Precuneus

Table 1 . Clusters showing differences between BPD and healthy controls.

 1

	MNI coordinates (x, y, z)	SDM-Z	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	PTSD > Controls
	-56 -38 32	2.346	0.000039041	584	40, 48	L	Supramarginal gyrus
	48	20 -10	1.869	0.000604749	437	38	R	Anterior Insula
	-26 -28 -30	1.706	0.001354516	166	30	L	Cerebellum (lobule IV / V)
	-4	-12 42	1.750	0.001094818	154	23	L+R	Median cingulate / paracingulate gyri
	30 -24 -30	1.689	0.001469672	80	20	R	Parahippocampal gyrus
	38	42	38	1.999	0.000295222	65	9	R	dlPFC
	PTSD < Controls
	40 -20	4	-2.131	0.000078440	1625	48	R	Heschl Gyrus
	20 -74	0	-2.202	0.000050783	164		R	Lingual Gyrus
	52	22	38	-1.664	0.001235783	35	44	R	Inferior and Middle Frontal Gyri

Table 2 . Clusters showing differences between PTSD and all healthy controls.

 2

	MNI coordinates (x, y, z)	SDM-Z	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	PTSD > TEC
	-52 -32 32	2.561	0.000011563	1020		L	Supramarginal Gyrus
	0	50 -14	2.180	0.000144601	731	11	L	vmPFC
							46
	30	48	32	2.609	0.000008643	386		R	dlPFC
	58 -54 24	1.665	0.001578987	296	22	R	Angular gyrus
	PTSD < TEC
	46 -18	2	-1.759	0.000753582	1111		R	Heschl Gyrus
	20 -74	0	-2.897	0.000001013	322		R	Lingual Gyrus
	28 -68 -18	-1.629	0.001604378	80	19	R	Cerebellum (lobule VI)
	58	2	-26	-1.656	0.001384020	74	21	R	Temporo-parietal Junction
	50	22	42	-1.676	0.001234770	46	44	R	Middle Frontal Gyrus
	-54 18	30	-1.536	0.002774477	17	44	L	Inferior Frontal Gyrus

Table 3 .

 3 Clusters showing differences between PTSD and trauma exposed controls (TEC).

	MNI coordinates (x, y, z)	SDM-Z	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	PTSD > NTC
	28 -32 -24	2.246	0.000024676	1128	BA 20	Right	Parahippocampal Gyrus
	46	20	-6	2.111	0.000076294	703		Right	Anterior Insula
	-4	-14 44	1.627	0.001889586	144	BA 23	Left	Median Cingulate / Paracingulate Gyri
	-20 -26 -30	1.587	0.002381325	27	30	Left	Cerebellum (lobule IV / V)
	0	14 -24	1.518	0.003511727	12	11	Left	Gyrus Rectus
	PTSD < NTC
	4	54	-6	-1.664	0.001134872	236	10	Right	vmPFC
	46 -76 28	-1.732	0.000762343	144	39	R	Precuneus

Table 4 :

 4 Clusters showing differences between PTSD and non-trauma exposed controls (NTC).

	MNI coordinates (x, y, z)	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	PTSD pos -bpd pos (increased activity in both ptsd and bpd)
	-46 -6	4	< 0.00025	649	48	L	Left rolandic operculum, Left insula, Left superior temporal gyrus
	56 -52	30	< 0.00025	364	39,40	R	Right angular gyrus
	PTSD neg -bpd neg (decreased activity in both ptsd and bpd)
						None
	PTSD pos -bpd neg (increased activity in ptsd and decreased in bpd)
	-6	-24	30	< 0.00025	311	23	L+R	median cingulate / paracingulate gyri
	PTSD neg -bpd pos (decreased activity in ptsd and increased in bpd)
	-4	58	22	< 0.00025	396	10, 32	L	Left superior frontal gyrus, medial
	-24 50	26	< 0.00025	290	46	L	Left middle frontal gyrus, Left superior frontal gyrus, dorsolateral

Table 5 .

 5 Clusters showing similar activation in BPD and PTSD (trauma exposed controls, TEC).

	MNI coordinates (x, y, z)	P	Cluster size / voxels	Brodman areas	Side (L/R)	Brain areas
	PTSD pos -bpd pos (increased activity in both ptsd and bpd)
						None
	PTSD neg -bpd neg (decreased activity in both ptsd and bpd)
	4	-72	46	< 0.00025	570	7	L+R	Precuneus
	PTSD pos -bpd neg (increased activity in ptsd and decreased in bpd)
	24 -28 -14	< 0.00025	834			Right median network, cingulum
	-2	-18	30	< 0.00025	440	23	L+R	median cingulate / paracingulate gyri
	PTSD neg -bpd pos (decreased activity in ptsd and increased in bpd)
								Right superior frontal gyrus,
	4	48	4	< 0.00025	827	10, 32	L+R	medial, Left anterior cingulate
								/ paracingulate gyri
	-48 -2	2	< 0.00025	608	48	L	Left rolandic operculum

Table 6 .

 6 Clusters showing similar activation in BPD and PTSD (non-trauma exposed controls, NTC).

Acknowledgements

No Acknowledgement

Conflict of interests

The authors declare no potential conflict of interest.

Contributors

AA, JR, TF participated in the conception and design of the study; AA and TF participated in the acquisition of data; AA, JR, TF performed the analyses; AA and TF wrote the first draft of the manuscript. All authors participated in the writing and revision of the successive drafts of the manuscript and approved the final version.

Role of the Funding source

No funding source to declare.