Aymen Laadhari
email: aymen.laadhari@ku.ac.ae

Ahmad Deeb

A Finite Element Approach For Modeling Biomembranes In Incompressible Power-Law Flow

We present a numerical method to model the dynamics of inextensible biomembranes in a quasi-Newtonian incompressible flow, which better describes hemorheology in the small vasculature. We consider a level set model for the fluid-membrane coupling, while the local inextensibility condition is relaxed by introducing a penalty term. The penalty method is straightforward to implement from any Navier-Stokes/level set solver and allows substantial computational savings over a mixed formulation. A standard Galerkin finite element framework is used with an arbitrarily high order polynomial approximation for better accuracy in computing the bending force. The PDE system is solved using a partitioned strongly coupled scheme based on Crank-Nicolson time integration. Numerical experiments are provided to validate and assess the main features of the method.

INTRODUCTION

This paper is concerned with the numerical study of the time-dependent dynamics of biomembranes in a surrounding Newtonian and non-Newtonian flow. The coupled fluid-membrane problem is highly nonlinear and time consuming.

Blood is a very complex fluid. Its rheology at the macroscopic scale depends both on the individual dynamics of its embedded entities and their fluid-structure interactions at the microscopic level. Red blood cells, referred to as RBCs, represent its main cellular component; They are responsible for the supply of oxygen and the capture of carbon dioxide. In the laboratory, giant unilamellar vesicles (diameter ≈ 10µm) are biomimetic artificial liquid drops, used in vitro and in silico to study the RBCs. Understanding the dynamics of RBCs in flow remain a difficult problem in the field of computational physics and at the theoretical level as well, consequently leading to a growing interest in the past two decades. In the published literature, several works have covered the areas of experimental biology [START_REF] Song | Characterization of stress-strain behaviour of red blood cells (RBCs), part II: response of malaria-infected RBCs[END_REF], theoretical biology [START_REF] Safran | Statistical Thermodynamics of Surfaces, Interfaces and Membranes[END_REF], physics [START_REF] Kaoui | Lateral migration of a two-dimensional vesicle in unbounded poiseuille flow[END_REF][START_REF] Keller | Motion of a tank-treading ellipsoidal particle in a shear flow[END_REF][START_REF] Choi | Fluctuations of red blood cell membranes: The role of the cytoskeleton[END_REF] and applied mathematics [START_REF] Dziuk | Computational parametric Willmore flow[END_REF][START_REF] Barrett | Numerical computations of the dynamics of fluidic membranes and vesicles[END_REF].

From a mechanical continuum perspective, Canham [START_REF] Canham | The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell[END_REF], Helfrich [START_REF] Helfrich | Elastic properties of lipid bilayers: theory and possible experiments[END_REF] and Evans [START_REF] Evans | Bending resistance and chemically induced moments in membrane bilayers[END_REF] independently introduced in the early 1970s a model to describe the mechanics of lipid bilayer membranes, where cellular deformations are driven by the principal curvatures. This results in a highly nonlinear membrane force with respect to shape, see a mathematical derivation for a generalized energy functional based on shape optimization in [START_REF] Laadhari | On the equilibrium equation for a generalized biological membrane energy by using a shape optimization approach[END_REF].

Different methods have been developed to study the dynamics of biomembranes in a Newtonian flow. We can distinguish the level set method [START_REF] Cottet | Eulerian formulation and Level-Set models for incompressible fluid-structure interaction[END_REF][START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF][START_REF] Doyeux | Simulation of two-fluid flows using a finite element/level set method. Application to bubbles and vesicle dynamics[END_REF], the phase field method [START_REF] Du | A phase field approach in the numerical study of the elastic bending energy for vesicle membranes[END_REF], the immersed boundary method [START_REF] Hu | An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier-Stokes flows[END_REF], the boundary integral method [START_REF] Rahimian | Dynamic simulation of locally inextensible vesicles suspended in an arbitrary two-dimensional domain, a boundary integral method[END_REF], the parametric finite elements [START_REF] Barrett | Numerical computations of the dynamics of fluidic membranes and vesicles[END_REF], and the lattice Boltzmann method [START_REF] Kaoui | Two-dimensional vesicle dynamics under shear flow: Effect of confinement[END_REF]. From a numerical point of view, iterative and fully explicit decoupling strategies for the membrane-fluid problem are the most used techniques [START_REF] Doyeux | Simulation of two-fluid flows using a finite element/level set method. Application to bubbles and vesicle dynamics[END_REF][START_REF] Salac | Reynolds number effects on lipid vesicles[END_REF]. An explicit treatment of the bending force usually leads to numerical instability problems and severe time step limitations, depending on the local mesh size and bending stiffness. However, only few works devised semi-implicit [START_REF] Barrett | Numerical computations of the dynamics of fluidic membranes and vesicles[END_REF] or fully implicit time integration schemes [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF][START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF]. Although stability is improved, a high computational burden is generally obtained with implicit strategies. Other interesting decoupling strategies can be found in [START_REF] Valizadeh | Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach[END_REF][START_REF] Laadhari | Implicit finite element methodology for the numerical modeling of incompressible two-fluid flows with moving hyperelastic interface[END_REF][START_REF] Laadhari | An operator splitting strategy for fluid-structure interaction problems with thin elastic structures in an incompressible newtonian flow[END_REF][START_REF] Torres-Sánchez | Modelling fluid deformable surfaces with an emphasis on biological interfaces[END_REF].

While blood flow behaves like Newtonian fluid in larger diameter arteries at high shear rates, it exhibits non-Newtonian behavior in small diameter arteries with low shear rates at the microscopic scale [START_REF] Cokelet | The rheology of human blood-measurement near and at zero shear rate[END_REF]. Non-Newtonian rheology is mainly due to polymerization and the underlying mechanisms leading to the activation and deactivation of platelets and the interactions between different microscopic entities. Blood viscosity tends to increase at low shear rates as RBCs aggregate into a roller shape. The Casson, Power-Law, and Quemada models are the most widely used generalised Newtonian rheologies for blood [START_REF] Neofytou | Non-newtonian flow instability in a channel with a sudden expansion[END_REF][START_REF] Copley | Apparent viscosity and wall adherence of blood systems[END_REF]. To our knowledge, such models have not yet been studied for the current problem. In this work, we consider a quasi-Newtonian power law model to describe the hemorheology.

The aim of this paper is to study the dynamics of biomembranes in a complex non-Newtonian incompressible viscous flow. In order to keep a reasonable computational cost compared to a fully mixed formulation, we design a penalty method to account for the local inextensibility of the membrane. Various higher-order finite element approximations are used to better approximate the bending force. We present a set of numerical examples to validate and show the main features of the method.

MATHEMATICAL SETTING

Membrane model

The deformations of the membrane allow minimizing the Canham-Helfrich-Evans [START_REF] Canham | The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell[END_REF][START_REF] Helfrich | Elastic properties of lipid bilayers: theory and possible experiments[END_REF] bending energy while preserving the local inextensibility of the membrane. Let H be the mean curvature, corresponding to the sum of the principal curvatures on the membrane. In the two-dimensional case, the membrane minimizes the bending energy given by:

J(Ω) = k b 2 ∂ Ω (H(Ω)) 2 ds, (1)
where k b ≈ 10 -20 /10 -19 kg m 2 s -2 is the bending rigidity modulus. The energy is a variant of the Willmore energy [START_REF] Willmore | Riemannian geometry[END_REF]. Let T be the final time of the experiment. For any time t ∈ [0, T ], Ω(t) ⊂ R d , d = 2, 3, is the interior domain of the membrane Γ(t) = ∂ Ω(t), assumed Lipschitz continuous. The membrane is embedded in the domain Λ which is large enough so that Γ(t) ∩ ∂ Λ = / 0, see Fig. 1. Hereafter, the dependence of Ω and Γ upon t is dropped to alleviate notations.

For a membrane with fixed topology, the Gauss-Bonnet theorem [START_REF] Feng | Finite element modeling of lipid bilayer membranes[END_REF] states that the energy term weighted by k g is constant and can be ignored. The spontaneous curvature helps describe the asymmetry of phospholipid bilayers at rest, e.g. when different chemical environments exist on either side of the membrane. We assume H 0 = 0. Let n and ν be the outward unit normal vector Γ(t) and on ∂ Λ, respectively. We introduce the surface gradient

∇ s • = (Id -n ⊗ n) ∇•, surface divergence div s • = tr(∇ s •) and surface Laplacian ∆ s • = div s (∇ s •),
where Id is the identity tensor. The expression and derivation of the bending force using shape optimization tools can be found in [START_REF] Laadhari | On the equilibrium equation for a generalized biological membrane energy by using a shape optimization approach[END_REF].

Membrane deformations are subject to specific constraints. Fluid incompressibility is assumed, this is div u = 0 in Λ. In addition, RBCs are phospholipid bilayers with local membrane inextensibility. This corresponds to a zero surface divergence, i.e. div s u = 0 over Γ, that helps preserve the local perimeter. Global perimeter conservation follows from Reynolds' lemma [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF]. As a consequence, a saddle point formulation results in a membrane surface force that balances the jump in hydrodynamic stress tensor and appears in the right side of (3g).

Level set description

The motion of the membrane is followed implicitly in a level set framework as the zero level set of a function ϕ. For t ∈ ]0, T [, ϕ is initialized by a signed distance ϕ 0 to Γ(0) and satisfies the transport equation (3a), with u the advection vector and ϕ = ϕ b on the upstream boundary Σ -= {x ∈ ∂ Λ : u • ν(x) < 0}. Geometric quantities such as n = ∇ϕ/|∇ϕ|, H = div s n and bending force are coded in terms of ϕ and are then extended to the entire computational domain Λ. Over time, a redistancing problem is resolved to maintain the signed distance property lost by advection [START_REF] Laadhari | Fully implicit finite element method for the modeling of free surface flows with surface tension effect[END_REF]. Indeed, a too large or too small gradient of ϕ close to Γ deteriorates the precise compting of the surface terms. Let ε be a regularization parameter. We introduce the regularized Heaviside H ε and Dirac δ ε functions:

H ε (ϕ) =        0, when ϕ < -ε 1 2 1 + ϕ ε + 1 π sin πϕ ε , when |ϕ| ε, 1, otherwise and δ ε (ϕ) = dH ε dϕ (ϕ).
Given a function ζ defined on Γ and its extension ζ to Λ, surface integrals are approximated as follows:

Γ ζ (x) ds ≈ Λ |∇ϕ| δ ε (ϕ) ζ (x) dx.

Governing equations

We assume constant densities ρ i and ρ o inside and outside of the membrane, respectively. Let us introduce the fluid velocity u and the pressure p which represent a Lagrange multiplier corresponding to the incompressibility constraint on Λ. Analogously, a position-dependent surface tension λ helps imposing the local inextensibility constraint on Γ.

Let D(u) = (∇u + ∇u T )/2 be the shear strain rate tensor, so the fluid Cauchy stress tensor is σ = T -pI where T is the stress deviator. The normal stress jump [σn] + -= σ + nσ -n on Γ describes the interactions of the membrane with the surrounding fluid [START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF], while the stress discontinuity is calibrated by (3f). For a simple shear flow, u b is the shear rate on Σ D ⊂ ∂ Λ, while natural boundary conditions are prescribed on Σ N ⊂ ∂ Λ.

We assume a quasi-Newtonian power-law model [START_REF] Neofytou | Non-newtonian flow instability in a channel with a sudden expansion[END_REF] where the nonlinear constitutive equation expresses the stress deviator with a power-law viscosity function as

T = 2η |D(u)| 2 D(u), with η (γ) = Kγ (υ -1)/2 , for all γ ∈ R, (2)
where υ > 0 and K are the power index and consistency index, respectively. According to [START_REF] Walburn | A constitutive equation for whole human blood[END_REF], υ = 0.7755 < 1 (i.e. a shear thinning fluid) and K = 14.67 × 10 -3 Pa s for normal blood samples obtained using a multiple regression technique. The Newtonian case υ = 1 corresponds to a linear stress-strain relationship that reduces the viscosity function η(γ) = K to a constant. By analogy with the Newtonian case, K = µ i and K = µ o stand for the values of the consistency index in the intra-and extra-membrane domains, respectively. We perform a dimensionless analysis. Let U be the maximum velocity on Σ D and D the diameter of a circle having the same membrane perimeter. We consider the dimensionless Reynolds number Re = ρ o UDµ -1 o which expresses the ratio between the inertial and viscous forces, and the capillary number Ca = µ o D 2 Uk -1 b which compares the flow force to the bending resistance of the membrane. Furthermore, the parameter β = µ i /µ o represents the ratio of consistency indices and corresponds to the viscosity ratio with respect to extracellular viscosity in the Newtonian case. The regularized dimensionless viscosity function is:

µ ε (ϕ)|D(u)| υ-1 = (H ε (ϕ) + β (1 -H ε (ϕ))) |D(u)| υ-1 .
Following [START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF], we choose ρ i = ρ o . Let σ ε stand for the regularized Cauchy stress tensor. The dimensionless reduced area Ξ 2d = 4π|Ω|/|Γ| 2 ∈]0, 1] compares the area of the interior domain to that of a circle with the same perimeter. The dimensionless coupled problem writes: find ϕ, u, p and λ such that

∂ t ϕ + u.∇ϕ = 0 in ]0, T [×Λ (3a) Re (∂ t u + u.∇u) -div σ ε (D(u), p, ϕ) = 0 in ]0, T [×(Λ\∂ Ω) (3b) div u = 0 in ]0, T [×Λ (3c) div s u = 0 on ]0, T [×∂ Ω (3d) [u] + -= 0 on ]0, T [×∂ Ω (3e) [σ ε n] + -= ∇ s λ -λ Hn + (2Ca) -1 2∆ s H + H 3 n on ]0, T [×∂ Ω (3f) ϕ = ϕ b on ]0, T [×Σ - (3g) u = u b on ]0, T [×Σ D (3h) σ.ν = 0 on ]0, T [×Σ N (3i) ϕ(0) = ϕ 0 in Λ (3j) u(0) = u 0 in Λ. (3k)
Let ε λ = 10 -8 be the penaly parameter. To make the method straightforward to implement from any Level Set / Navier-Stokes solver and considerably reduce the size of the linear system to be solved, the inextensibility constraint is relaxed by introducing a penalty term. Indeed, the corresponding minimization problem should be approximated by another minimization problem by penalizing the local inextensibility constraint for the velocity (3d). See analogous penalty method for other applications in [START_REF] Janela | A penalty method for the simulation of fluid -rigid body interaction[END_REF].

To overcome instability problems when solving the level set equation using the standard Galerkin method, there are a variety of stabilization methods such as the streamline diffusion method, the subgrid viscosity method and the Streamline Upwind Petrov-Galerkin (SUPG) method used in this work. The latter introduces a stabilization term by adding a diffusion in the streamline direction.

We introduce the functional spaces of admissible velocity u, pressure p and level set ϕ:

V(u b ) = v ∈ H 1 (Λ) d : v = u b , on Σ D , Q = q ∈ L 2 (Λ) : Ω q = 0 , X(ϕ b ) = ψ ∈ W 1,∞ (Λ) ∩ H 1 (Λ) : ψ = ϕ b , on Σ -.
To reduce a derivation order of ϕ when evaluating the bending strength, we use the Green formula on a closed surface. See e.g. [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF]. Testing with appropriate test functions and integrating (3b) over Ω and Λ\Ω separately, the variational problem writes:

Find u ∈ C 0 ]0, T [, L 2 (Λ) d ∩L 2 ]0, T [, V(u b ) , p ∈ L 2 ]0, T [, Q , and ϕ ∈ C 0 ]0, T [, L 2 (Λ) d ∩L 2 ]0, T [, X (ϕ b ) such that Re Λ ∂ u dt + u • ∇u • v + Λ 2µ ε (ϕ)|D(u)| υ-1 D(u) : D(v) + 1 ε λ Λ div s (u) div s (v)|∇ϕ|δ ε (ϕ) - Λ p div v + 1 2Ca Λ δ ε (ϕ)|∇ϕ| 2∇ s H • ∇ s (n • v) -H 3 n • v = Σ N σν • v, ∀v ∈ V(0), (4a) Λ q div u = 0, ∀q ∈ Q, (4b) Λ ∂ ϕ ∂t ψ + Λ (u • ∇ϕ) ψ + Λ ξ (τ; ϕ, ψ) = 0, ∀ψ ∈ X (0) . (4c)
Here, ξ (τ; ϕ, ψ) stands for the SUPG stabilisation term and τ is a stabilization parameter defined element wise to control the amount of diffusion.

NUMERICAL APPROACH

The interval [0, T ] is divided into N sub-intervals [t n ,t n+1 ) with 0 n N -1 of constant step ∆t . For n > 0, u n , p n and ϕ n are computed by induction to approximate u, ϕ and p at t n . We use the Crank-Nicolson scheme for the time discretization of (3a) and (3b) without the need to bootstrap the initial conditions. The choice of this scheme was for its simplicity to implement and being a second order one-step integrator. The discretized (3a) writes

ϕ n+1 = ϕ n + ∆t 2 u • ∇ϕ n+1 + u • ∇ϕ n in Λ.
For the spatial discretization, we consider a partition T h of Λ consisting of geometrically conformal open simplicial elements K. We define the mesh size as the diameter of the largest mesh element h = max h K with K ∈ T h . We consider a Taylor-Hood finite element approximation for u and p. After using a surface Green's transformation, the evaluation of the Canham-Helfrich-Evans force requires a third-order derivative in ϕ which induces numerical oscillations when using lower-order polynomial approximations. To avoid introducing additional mixed variables and additional equations as in [START_REF] Laadhari | Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods[END_REF], higher degree polynomials are considered for the discretization of ϕ because the bending force requires its fourth order derivatives. For the SUPG method, the streamline diffusion parameter is chosen numerically proportional to the local mesh size, this is τ K = Ch K / max {|u| 0,∞,K , tol/h K }, where C is a scaling constant and tol/h K helps to avoid division by zero. To overcome the instability problems induced by an explicit decoupling, we consider a partitioned implicit strategy based on a fixed point algorithm, as detailed in Alg. 1.

Algorithm 1 Fluid-membrane coupling 1: n = 0: let ϕ 0 and u 0 being given 2: for n = 0, . . . , N -1 do

3: Initialize u n+1,0 = u n , ϕ n+1,0 = ϕ n 4:
while e k < 10 -6 do 5:

Compute ϕ n+1,k+1 using u n+1,k 6:

Compute u n+1,k+1 , p n+1,k+1 using ϕ n+1,k+1 7:

Compute the error

e k = |u n+1,k+1 -u n+1,k | 1,2,Λ /|u n+1,k | 0,2,Λ + |ϕ n+1,k+1 -ϕ n+1,k | 0,2,Λ /|ϕ n+1,k | 0,2,Λ 8:
end while Example 1: Reversible Vortex -Grid convergence.

H ε (ϕ h ) -H ε (π h ϕ) 0,2,Ω 10 -1 10 -2
Simulations were performed using FEniCSx [START_REF] Alnaes | The FEniCS project version 1.5[END_REF]. To evaluate the capability of the level set solver for high-order finite elements, necessary afterwards for an accurate assessment of highly nonlinear bending force, we consider a reversible vortex test case featuring large deformations of the interface. The computational domain is Λ = [0, 1] 2 . A circular interface of radius R = 0.15 initially centered at (0.7, 0.7) is stretched into thin filaments which are coiled like a starfish by a vortex flow field. The deformations are periodic and the stretching of the membrane unravels before the interface regains its circular shape after a period at t = T . The maximal deformation ψ occur at t = T /2, with ψ = 3 and T = 1 in numerical computations. Similar 2D and 3D test cases are widely used to test interface tracking methods. We follow LeVeque's test [START_REF] Leveque | High-resolution conservative algorithms for advection in incompressible flow[END_REF] (Example 9.5) and consider a velocity field at x = (x, y) T ∈ Λ given by u(t, x) = -2 sin(ψπx) 2 sin(ψπy) cos(ψπy) cos(πt/T ), 2 sin(ψπy) 2 sin(ψπx) cos(ψπx) cos(πt/T ) T .

The spatial accuracy of the finite element numerical approximations is studied by computing the errors in L 2 (Λ) norm on successively refined meshes with respect to an exact reference solution π h ϕ at t = T , where π h represents the Lagrange interpolation operator. Errors are calculated after one stretching period. For k the degree of the polynomial approximation, the time step ∆t = h k is chosen small enough not to significantly influence the overall accuracy. Fig. 2 reports the convergence of calculated errors with respect to the mesh size for several polynomial finite element approximations. Convergence rates are also displayed, showing for instance an almost second-order accuracy for k = 1 and fifth-order accuracy for k = 4. from [START_REF] Salac | Reynolds number effects on lipid vesicles[END_REF] and [START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF] (Re = 10 -3 , Ca = 100), [START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF] (Ca = 9) and [START_REF] Kraus | Fluid vesicles in shear flow[END_REF] (Ca = 10). results in [START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF][START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF] and measurements in [START_REF] Kantsler | Orientation and dynamics of a vesicle in tank-treading motion in shear[END_REF].

Example 2: Dynamics of the biomembrane in Newtonian and quasi-Newtonian flows.

We first proceed to a quantitative validation with some experimental and numerical results available in the literature in the case of a purely Newtonian flow. We set υ = 1, a viscosity contrast β = 1, Ca = 10 2 and Re = 9 × 10 -3 . More details on the physiological values of the Reynolds number at the level of RBCs are available in [START_REF] Salac | Reynolds number effects on lipid vesicles[END_REF]. The membrane follows a tank-treading type movement, called TT, where it reaches a steady state characterized by a fixed angle of inclination; The surrounding fluid continues its rotation tangentially to the membrane. We consider different values of the reduced areas Ξ 2d ∈ [0.6, 1], and calculate the angle of inclination at equilibrium θ . Fig. 3 and Fig. 4 plot the change in θ /π against Ξ 2d in both Newtonian and quasi-Newtonian cases for different values of the viscosity ratio β . The results are compared with those of Kraus et al. [START_REF] Kraus | Fluid vesicles in shear flow[END_REF], Zhao et al. [START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF], Salac et al. [START_REF] Salac | Reynolds number effects on lipid vesicles[END_REF] and Laadhari et al. [START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF], showing good overall consistency. However, note that the values obtained with υ = 0.7755 fit slightly better compared to those of the Newtonian model which are a little higher than the other curves. This cannot be confirmed at all, given the setting of different dimensionless values such as Re and Ca in the different experiments. An in-depth study is in progress and will be the subject of a forthcoming work. Simulations are now performed using a different ratio β = 2.7 in the non-Newtonian case. We calculate the angle of inclination θ /π and compare with some numerical [START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF] and experimental [START_REF] Kantsler | Orientation and dynamics of a vesicle in tank-treading motion in shear[END_REF] results available only for larger reduced areas. Fig. 3(right) shows close but slightly higher equilibrium angles when the shape of the membrane becomes close to a circle. The deviations can be mainly due to the non-Newtonian model, but also to the different values of the confinement levels and the boundary conditions used in the different works.

According to an experimental systematic study on individual individual red cells in a simple shear flow, a change in dynamics occurs when the viscosity ratio exceeds a critical value depending on the reduced area [START_REF] Fischer | The red cell as a fluid droplet: Tank tread-like motion of the human erythrocyte membrane in shear flow[END_REF]. This is the tumbling regime, noted by TB, which is characterized by the periodic rotation of the membrane around its axis. A well-known empirical model was developed by Keller and Skalak [4]. This dynamics was obtained in the simulations with the non-Newtonian model, see Fig. 5 and Fig. 6 for the snapshots of the TT and TB dynamics obtained with the same set of parameters but with β = 1 and β = 10, respectively.

CONCLUSION

We have presented in this paper a relatively simple method for simulating the dynamics of an individual red blood cell, or inextensible biological membrane in general, in a surrounding incompressible non-Newtonian flow that better describes the hemorheology in small capillaries. We validated our framework using high-order finite element approximations in the case of a membrane in a simple shear flow. Simulations have shown that the method is capable of capturing the basic cellular dynamics, namely the well-known tank treading and tumbling motions. This is part of a larger ongoing work to explore the dynamics of red blood cells in small capillaries, while accounting for cell elasticity [START_REF] Gizzi | A three-dimensional continuum model of active contraction in single cardiomyocytes[END_REF] in non-Newtonian surrounding flow.

FIGURE 1 :

 1 FIGURE 1: Sketch of the membrane Γ embedded into a computational domain Λ, while Ω is the inner region.

9 :

 9 Update u n+1 = u n+1,k+1 , ϕ n+1 = ϕ n+1,k+1 10:

FIGURE 2 :

 2 FIGURE 2: Reversible vortex. (Left) Snapshots showing the interface deformations at t ∈ {0, 0.25, 0.57, 0.75, 0.875, 1} with = 0.01. (Right) Spatial convergence in L 2 norm for high-order finite element approximations.

FIGURE 3 :

 3 FIGURE 3: TT regime: Change in θ * /π with respect to Ξ 2d for a viscosity ratio β = 1. Comparisons with resultsfrom[START_REF] Salac | Reynolds number effects on lipid vesicles[END_REF] and[START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF] (Re = 10 -3 , Ca = 100),[START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF] (Ca = 9) and[START_REF] Kraus | Fluid vesicles in shear flow[END_REF] (Ca = 10).

FIGURE 4 :

 4 FIGURE 4: TT regime: Change in θ * /π with respect to Ξ 2d for β = 2.7. Comparison of non-Newtonian model withresults in[START_REF] Laadhari | Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods[END_REF][START_REF] Zhao | The dynamics of a vesicle in simple shear flow[END_REF] and measurements in[START_REF] Kantsler | Orientation and dynamics of a vesicle in tank-treading motion in shear[END_REF].

FIGURE 5 :

 5 FIGURE 5: TT regime for Ξ 2d = 0.68, β = 1, Ca = 4 × 10 4 and Re = 9 × 10 -3 at t ∈ 0, 0.125, 0.25, 0.5, 1, 2 .

FIGURE 6 :

 6 FIGURE 6: Snapshots showing a membrane in TB regime for Ξ 2d = 0.68, β = 10, Ca = 4 × 10 4 and Re = 9 × 10 -3 , at times t ∈ 0, 0.13, 0.25, 0.5, 1.25, 2, 2.25, 2.38, 3, 4, 4.5, 4.7 , respectively.

ACKNOWLEDGMENTS

The authors acknowledge financial support from KUST through the grant FSU-2021-027.


Geneviève Rouleau
email: genevieve.rouleau.chum@ssss.gouv.qc.ca

Quan Nha Hong

Navdeep Kaur

Marie-Pierre Gagnon

José Côté

Julien Bouix-Picasso

Pierre Pluye

Systematic Reviews of Systematic Quantitative, Qualitative, and Mixed Studies Reviews in Healthcare Research: How to Assess the Methodological Quality of Included Reviews?

Keywords: methodological quality assessment, systematic reviews of systematic reviews, systematic mixed studies reviews, systematic qualitative reviews, systematic quantitative reviews

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Background

More and more healthcare researchers are interested in combining qualitative, quantitative and mixed methods studies and their evidence into systematic reviews. The healthcare field of research has a long tradition in producing systematic reviews [START_REF] Hong | Systematic reviews: A brief historical overview[END_REF].With the number of published systematic reviews increasing each and every day, researchers are now interested in combining many systematic reviews into one single document, i.e. systematic reviews of systematic reviews (SRSRs) [START_REF] Hartling | A Descriptive Analysis of Overviews of Reviews Published between 2000 and 2011[END_REF][START_REF] Page | Epidemiology and Reporting Characteristics of Systematic Reviews of Biomedical Research: A Cross-Sectional Study[END_REF][START_REF] Pieper | Overviews of reviews often have limited rigor: A systematic review[END_REF][START_REF] Pollock | What guidance is available for researchers conducting overviews of reviews of healthcare interventions? A scoping review and qualitative metasummary[END_REF]. SRSRs provide a wide picture of the topic under consideration; they are "one-stop shopping for decision makers" (Hartling et al., 2014 p. 488) because all systematic reviews are synthesized together. SRSRs can be fine-tuned to focus on particular populations and/or interventions; they can combine various outcomes, and they are efficient [START_REF] Aromataris | Summarizing systematic reviews: Methodological development, conduct and reporting of an umbrella review approach[END_REF][START_REF] Becker | Chapter 22: Overviews of reviews[END_REF][START_REF] Hartling | Systematic reviews, overviews of reviews and comparative effectiveness reviews: A discussion of approaches to knowledge synthesis. Evidence-Based Child Health[END_REF].

There are challenges in conducting SRSRs including : the overlapping between systematic reviews (primary studies appearing in more than one review, leading to potential duplication of some findings in SRSRs); the misalignment between the scope of the systematic reviews and SRSRs question(s); the variable quality of reporting and methodological quality within systematic reviews; the lack of granularity in reported information; and the assessment of methodological quality of included systematic reviews [START_REF] Ballard | Risk of bias in overviews of reviews: A scoping review of methodological guidance and four-item checklist[END_REF]Lunny et al., 2016;[START_REF] Pieper | Overviews of reviews often have limited rigor: A systematic review[END_REF][START_REF] Pollock | What guidance is available for researchers conducting overviews of reviews of healthcare interventions? A scoping review and qualitative metasummary[END_REF]. One of the key steps when conducting SRSRs is to assess the methodological quality/risk of bias in the included systematic reviews (Lunny et al., 2018). We will focus only on the methodological quality at the systematic reviews level because we are interested in the tools required to assess the methodological quality when various types of systematic reviews are included in an SRSR. This paper builds on the literature on SRSRs and methodological quality (e.g., Jimenez et al., 2018a;Lunny et al., 2017Lunny et al., , 2018;;[START_REF] Shea | AMSTAR 2: A critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF]Whiting et al., 2016a) and is based on our experiences of conducting systematic reviews of systematic qualitative, quantitative, and mixed studies reviews [START_REF] Rouleau | Impacts of information and communication technologies on nursing care: An overview of systematic reviews (protocol)[END_REF][START_REF] Rouleau | Effects of E-Learning in a Continuing Education Context on Nursing Care: Systematic Review of Systematic Qualitative, Quantitative, and Mixed-Studies Reviews[END_REF][START_REF] Rouleau | Effects of e-learning in a continuing education context on nursing care: A review of systematic qualitative, quantitative and mixed studies reviews (protocol)[END_REF]Rouleau, Gagnon, Côté, Payne-Gagnon, Hudson, & Dubois, 2017) and of undertaking methodological, empirical, and conceptual works in the field of systematic reviews and mixed methods [START_REF] Hong | Convergent and sequential synthesis designs: Implications for conducting and reporting systematic reviews of qualitative and quantitative evidence[END_REF]Hong & Pluye, 2018a;[START_REF] Pluye | Opening-up the definition of systematic literature review: The plurality of worldviews, methodologies and methods for reviews and syntheses[END_REF].

From these sources of knowledge, we observed the following issues: (a) there is no attempt to provide simple and clear typology in the field of tertiary level of research to describe SRSRs that includes systematic quantitative, qualitative, and/or mixed studies reviews; and (b) there is little explicit guidance in assessing the methodological quality in systematic qualitative and mixed studies reviews compared to quantitative systematic reviews which are included in SRSRs.

Practical guidance is explicit for authors of SRSRs who assess methodological quality in systematic quantitative reviews that include randomized and non-randomized designs [START_REF] Shea | AMSTAR is a reliable and valid measurement tool to assess the methodological quality of systematic reviews[END_REF][START_REF] Shea | AMSTAR 2: A critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF]Whiting et al., 2016a). In fact, three critical appraisal tools are commonly used: A Measurement Tool to Assess Systematic Reviews (AMSTAR, [START_REF] Shea | AMSTAR is a reliable and valid measurement tool to assess the methodological quality of systematic reviews[END_REF] and AMSTAR 2 [START_REF] Shea | AMSTAR 2: A critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF]; as well as Risk of Bias in Systematic Reviews (ROBIS, Whiting, 2016b;Whiting et al., 2016a). Recently, one tool was published for assessing methodological quality of mixed studies reviews: the Mixed Methods Systematic Reviews appraisal tool (MMSR, Jimenez et al., 2018b, 2018a). These developments in systematic mixed studies reviews are still in their infancy. To our knowledge, no tool exists for assessing the methodological quality in systematic qualitative reviews: that is why we claim that guidance is needed for different types of systematic reviews included in SRSRs.

Aim and scope

This methodological discussion paper will address the aforementioned knowledge gaps.

The aim of this paper is twofold. The first is to describe a typology of SRSRs including the combination of various systematic reviews (quantitative, qualitative, and mixed studies reviews) and their corresponding evidence (quantitative and qualitative). Here the focus is not to describe in detail all types of SRSR methodologies and synthesis approaches but rather to define what is meant by SRSR and their different types in a simple manner. The second aim is to explore all criteria pertaining to three critical appraisal tools (AMSTAR 2, ROBIS, and MMSR) and their potential applicability and limitations for assessing methodological quality of systematic qualitative and mixed studies reviews. We will also propose recommendations for all criteria used in these tools that could be adapted to match with qualitative and mixed studies reviews. This paper does not focus on the quality of how well an SRSR is conducted (the reporting quality), appraisals which would involve other levels of assessment and guidance (M. [START_REF] Pollock | Preferred Reporting Items for Overviews of Reviews (PRIOR): A protocol for development of a reporting guideline for overviews of reviews of healthcare interventions[END_REF].

Levels of research related to three types of systematic review of systematic reviews

SRSR, considered as the tertiary level of research [START_REF] Biondi-Zoccai | Introduction[END_REF], refers to a review of reviews guided by a clearly formulated question that uses systematic and explicit methods to identify, select, and critically appraise relevant systematic reviews and to collect and synthesize findings pertaining to these reviews (adapted from [START_REF] Booth | Systematic Approaches to a Successful Literature Review[END_REF]. It differs from the secondary level of research [START_REF] Biondi-Zoccai | Introduction[END_REF] referring to a review guided by a clearly formulated question that uses systematic and explicit methods to identify, select, and critically appraise relevant research and to collect and analyse findings deriving from primary studies. One of the differences between SRSRs and systematic reviews is the level of synthesis: the former synthesizes findings from systematic reviewswhile the latter synthesizes the findings from primary research studies (Gough, Oliver, et al., 2012;Smith et al., 2011). The primary level of research (Biondi-Zoccai, 2016) refers to research based on observation, experiment, or simulation rather than on reasoning or theory alone [START_REF] Abbott | The Causal Devolution[END_REF][START_REF] Porta | A dictionary of epidemiology[END_REF]. Each level of research contains and produces their own set of data. In a SRSRs, data are the findings of included systematic reviews [START_REF] Hong | Convergent and sequential synthesis designs: Implications for conducting and reporting systematic reviews of qualitative and quantitative evidence[END_REF]. In a systematic review, data are the findings of included primary studies. For Gough, Thomas and Olivier (2019), a review, whether narrative or numerical, is systematic as long as it meets the tenets of research, i.e., rigor and transparency. Other systematicrelated adjectives are used, including explicit, reproducible, comprehensive, detailed, and transparent [START_REF] Krnic Martinic | Definition of a systematic review used in overviews of systematic reviews, meta-epidemiological studies and textbooks[END_REF]. These adjectives are applicable to primary studies, reviews of studies, and review of reviews [START_REF] Gough | Clarifying differences between reviews within evidence ecosystems[END_REF].

SRSRs can differ depending on the systematic reviews included. We are proposing the following three types of SRSRs (Table 1 and Figure 1) using descriptive typology, partially informed by [START_REF] Hong | Convergent and sequential synthesis designs: Implications for conducting and reporting systematic reviews of qualitative and quantitative evidence[END_REF]: 1) Systematic review of systematic qualitative reviews; 2) Systematic review of systematic quantitative reviews; 3) Systematic review of systematic quantitative, qualitative, and mixed studies reviews. Many reasons justify the use of a typology: a) to define and illustrate the main concept of this paper, i.e. SRSRs, and to highlight a novel type of SRSRs: Systematic review of quantitative, systematic qualitative and mixed studies reviews; b) to understand at a glance (with the figure 1) the included type of systematic review pertaining to each type of SRSRs to lay the groundwork of the assessment of methodological quality; c) to help readers distinguish the levels of research;

and d) to understand where the challenges are encountered. Currently, the existing critical appraisal tools to assess the quality of systematic reviews were mainly developed for systematic quantitative reviews. However, the typology clearly shows that there exist other types of systematic reviews (systematic qualitative reviews and systematic mixed studies reviews and) for which there is a need to adapt existing tools or develop new ones.

The assessment of methodological quality in systematic reviews as a key step for authors of SRSRs

For authors of SRSRs, their judgment about methodological quality refers to a concern about how well (the included) systematic reviews are conducted [START_REF] Pieper | Overviews of reviews often have limited rigor: A systematic review[END_REF]. The terms risk of bias, threats to validity, critical appraisal, and quality assessment are also commonly used to refer to methodological quality. Risk of bias can be defined as the systematic flaws in the design or conduct of a systematic review that can bias the results of both the systematic review and the SRSRs and that can occur at all steps of the systematic review process (Whiting et al., 2016a). The assessment of methodological quality of included systematic reviews is a critical step when undertaking SRSRs. The results of such assessment, i.e. low, moderate and high quality, are an important element allowing to interpret findings and drawing conclusions from an SRSR [START_REF] Pollock | What guidance is available for researchers conducting overviews of reviews of healthcare interventions? A scoping review and qualitative metasummary[END_REF].

Methodological quality is considered as one of the three dimensions of quality, together with conceptual and reporting quality (Hong & Pluye, 2018a). Conceptual quality is associated with a clarity of concept/construct and a clear understanding of a concept and a phenomenon given the depth of description provided. Reporting quality concerns the extent, if any, to which a paper offers sufficient details and information about the design, conduct, and analysis of the primary study, systematic reviews, or SRSRs (adapted from [START_REF] Huwiler-Müntener | Quality of Reporting of Randomized Trials as a Measure of Methodologic Quality[END_REF]. The Enhancing the QUAlity and Transparency Of Health

Research Network (EQUATOR Network, 2019) website proposes more than 400 reporting guidelines and checklists to support authors in improving transparency in the reporting of various types of health research (studies, systematic reviews and SRSRs). All reporting guidelines that have been published [START_REF] Bougioukas | Preferred reporting items for overviews of systematic reviews including harms checklist: A pilot tool to be used for balanced reporting of benefits and harms[END_REF][START_REF] Bougioukas | Reporting guidelines on how to write a complete and transparent abstract for overviews of systematic reviews of health care interventions[END_REF] or are being developed at the tertiary level of research (M. [START_REF] Pollock | Preferred Reporting Items for Overviews of Reviews (PRIOR): A protocol for development of a reporting guideline for overviews of reviews of healthcare interventions[END_REF] are for systematic reviews of quantitative systematic reviews. There are reporting guidelines available for authors conducting qualitative syntheses methodologies, such as the eMERGe [START_REF] France | Improving reporting of meta-ethnography: The eMERGe reporting guidance[END_REF] and the ENTREQ (Tong et al., 2012). However, these reporting guidelines for systematic reviews of quantitative systematic reviews and for qualitative syntheses methodologies do not capture the dimension of critical appraisal that is of interest in this paper, namely, the methodological quality of systematic reviews included in a SRSRs.

Assessing such methodological quality requires to determine how to perform this endeavour, i.e. selecting the appropriate tool to do so (Lunny et al., 2018). We faced challenges when we assessed and compared the quality of systematic quantitative, qualitative and mixed studies reviews included in SRSRs (Rouleau, Gagnon, Côté, Payne-Gagnon, Hudson, & Dubois, 2017). These challenges were due to the lack of unequivocal practical guidance, and insufficient tools, to assess the quality of systematic qualitative and mixed studies reviews.

Two examples supporting challenges in assessing the methodological quality in systematic quantitative, qualitative, and mixed studies reviews

We present here two examples in which we conducted SRSRs. These examples supported the experienced challenges in assessing systematic quantitative, qualitative and mixed studies reviews included in an SRSRs. These challenges were the "cue to action" to initiate a reflection about how we can leverage on existing tools to apprehend the assessment of methodological quality of various types of systematic reviews included in an SRSRs.

The first example targets an SRSRs aimed to summarize the effects of information and communication technologies on nursing care. We included 22 systematic reviews that were of different types: mixed studies reviews (n=12); quantitative reviews (n=9); and qualitative reviews (n=1) (Rouleau, Gagnon, Côté, Payne-Gagnon, Hudson, & Dubois, 2017). We used the AMSTAR [START_REF] Shea | Development of AMSTAR: a measurement tool to assess the methodological quality of systematic reviews[END_REF][START_REF] Shea | AMSTAR is a reliable and valid measurement tool to assess the methodological quality of systematic reviews[END_REF] to assess the methodological quality of systematic qualitative, quantitative, and mixed studies reviews, even if the AMSTAR was only designed for systematic reviews including randomized controlled trials. At that time, no other tool was designed to assess the methodological quality of systematic qualitative and mixed studies reviews. Consequently, some criteria of the AMSTAR did not fit into the specificities of systematic qualitative and mixed studies reviews. Thus, we slightly adapted some of these criteria. However, this process was still disadvantageous for systematic qualitative and mixed studies reviews as they started with a lower score (i.e., low methodological quality). The systematic reviews with the highest methodological quality, represented by an AMSTAR score of 9 or higher on 11, were mostly the systematic quantitative reviews (except one mixed studies review).

In another SRSR, we used two tools to assess the methodological quality and risk of bias in systematic quantitative and mixed studies reviews: the ROBIS (Whiting et al., 2016a) and the AMSTAR 2 [START_REF] Shea | AMSTAR 2: A critical appraisal tool for systematic reviews that include randomised or non-randomised studies of healthcare interventions, or both[END_REF]. The AMSTAR 2 and the ROBIS were applied to 22

SRs, including 11 systematic quantitative reviews and 11 systematic mixed studies reviews [START_REF] Rouleau | Effects of E-Learning in a Continuing Education Context on Nursing Care: Systematic Review of Systematic Qualitative, Quantitative, and Mixed-Studies Reviews[END_REF]. Out of 22 SRs, a total of 9 (41%) were at low risk of bias; 8 (36%)

were at high risk of bias; and 5 (23%) had an unclear risk of bias. Again, all the systematic mixed studies reviews assessed with ROBIS had an unclear or a high risk of bias and 10/11 scored a critically low or low level of confidence with AMSTAR 2. By the time we published this paper, the MMSR tool had been published (Jimenez et al., 2018a). This tool is designed to critically appraise systematic reviews that integrate quantitative and

qualitative evidence to answer various questions regarding the effectiveness of development interventions (Jimenez et al., 2018a). The description of ROBIS, AMSTAR, AMSTAR 2 and MMSR are presented in Textbox 1 and their criteria are described in Table 2.

Here is the summary of what we found in our SRSRs: 1) In both of them, half of the systematic reviews included were mixed studies reviews (and one was a qualitative review); 2) This led to suggest a new type of SRSR, which is systematic review of systematic quantitative, qualitative and mixed studies reviews; 3) Given this novel type of SRSRs, there is a need of guidance of how to assess the methodological of qualitative and mixed studies reviews, such guidance being still embryonic (especially for qualitative reviews) compared to quantitative reviews; 4) Some criteria pertaining to the AMSTAR, AMSTAR 2 and ROBIS were inadequate for appraising the quality of systematic qualitative and mixed studies reviews and demonstrated flaws in application.

Indeed, the criteria underpinning these tools come with their own set of epistemological and methodological assumptions and reasoning that are meaningful for systematic reviews aligned with those positions. For example, ROBIS and AMSTAR 2 tools include a criterion in which review authors make a judgment on whether appropriate methods were used for statistical combination of results if meta-analysis was performed. This criterion is well suited for quantitative reviews but is not adapted to fit with the specificities of qualitative and mixed studies reviews.

Textbox 1. Description of critical appraisal tools

The ROBIS contains 21 signaling questions divided into the following three phases: i) the assessment of relevance (optional); ii) the identification of concerns with the review process, in which bias can be introduced in four domains: study eligibility criteria; identification and selection of studies; data collection and study appraisal; and synthesis and findings; and iii) overall judgment about risk.

The AMSTAR is an 11-item tool used to assess the methodological quality of quantitative reviews using mainly randomized controlled trials (RCT) designs.

The AMSTAR 2 is a 16-item instrument that provides detailed and comprehensive assessment of SRs that include randomised or non-randomised studies of healthcare interventions.

The MMSR tool has been developed to "categorise and critically appraise systematic reviews that incorporate quantitative and qualitative evidence to answer different questions about the effectiveness of development interventions (Jimenez et al., 2018a p. 411)".

Comparison of criteria pertaining to three critical appraisal tools and their applicability for appraising qualitative and mixed studies reviews

We are now entering the main argument of our paper (i.e. the second aim of the paper). The contribution of this methodological discussion paper is to compare three critical appraisal tools and their criteria to show the gaps when applying them to qualitative and mixed studies reviews; and to make suggestions that would allow adapting those criteria to make them usable for qualitative and mixed studies reviews.

This approach was undertaken in three steps: 1) listing and regrouping all the criteria pertaining to AMSTAR 2, ROBIS, and MMSR, and classifying them according to the eight main steps of conducting a systematic review [START_REF] Becker | Chapter 22: Overviews of reviews[END_REF]Liberati, 2009); 2)

examining each criterion and making a judgment about their applicability and limitations for appraising the quality of systematic qualitative and mixed studies reviews; 3) making preliminary recommendations and adding commentaries to adapt some criteria so as to make them potentially applicable for systematic qualitative and mixed studies reviews (see Table 2). At the first step, the classification of criteria in main steps of a systematic review reflects our interest in providing guidance for authors of SRSRs who have to perform the assessment of methodological quality at the secondary level of research, and not at the tertiary one.

We discussed issues around the approach of comparing the tools and their criteria until consensus had been reached within our group, composed of qualitative, quantitative, and mixed methods researchers (mentors and trainees) who have a background in nursing, occupational therapy, public health, community health, and family medicine. Our assumptions were as follows: some criteria and their corresponding bias (e.g., unambiguousness of eligibility criteria) described in existing tools, such as ROBIS, pertain

to "generic" steps of conducting SRs (e.g., defining the review question and eligibility criteria), regardless of whether they are qualitative, quantitative, or mixed studies reviews.

In this case, the unambiguousness of eligibility criteria is applicable for systematic qualitative and mixed studies reviews.

Insert Table 2 about here

Summary of preliminary observations and recommendations

Most criteria pertaining to steps 0 through 4 (see Table 2) are applicable to qualitative and mixed studies reviews: assessment of relevance, definition of the review question and identification of eligibility criteria; application of extensive and comprehensive search strategies; identification and selection of SRs; and data extraction. However, one criterion retrieved from MMSR suggests a multicomponent review which is the mix of methods [START_REF] Gough | Clarifying differences between reviews within evidence ecosystems[END_REF] that is operationalized in the MMSR tool for answering different subquestions: "specification of a separate systematic review question for each review component such as the intervention design, the implementation processes, the participants, and the intervention/program effects (Jimenez et al., 2018a)." This criterion is then partially applicable to all types of systematic reviews depending on the review questions, and subsequently, methods and corresponding data that would or would not help explain the effectiveness of interventions/programs. Some criteria from AMSTAR 2 and the ROBIS contain vocabulary that is not adapted for qualitative and mixed studies reviews, such as pre-defined analysis, risk of bias, heterogeneity, meta-analysis, and publication bias (see the criteria pertaining to step 5, step 6, and step 8 in Table 2). We recommend renaming/adapting existing criteria to fit with the epistemological and methodological tradition of systematic qualitative and mixed studies reviews so as to include all types of systematic reviews. For example, the criterion "Justification of meta-analysis" could be renamed "Justification of the synthesis approach for the combination of results." The MMSR tool also includes criteria for integrating qualitative and quantitative evidence (see Step 7 and corresponding criteria in Table 2), which are applicable for authors interested in answering different systematic review questions about the effectiveness of programs using a logic model.

Contribution to the Field of Mixed Methods Research Methodology

Over the past few years, researchers have shown interest in conducting SRSRs. However, existing methodological guidance is predominant for conducting, interpreting, and reporting systematic reviews of systematic quantitative reviews [START_REF] Hunt | An introduction to overviews of reviews: Planning a relevant research question and objective for an overview[END_REF]Lunny et al., 2017Lunny et al., , 2018;;[START_REF] Pollock | Selecting and implementing overview methods: Implications from five exemplar overviews[END_REF][START_REF] Pollock | What guidance is available for researchers conducting overviews of reviews of healthcare interventions? A scoping review and qualitative metasummary[END_REF][START_REF] Pollock | Chapter V: Overviews of reviews[END_REF]. Tools to assess methodological quality have been developed for quantitative reviews included in a SRSRs.

This guidance is still lacking and is imperative for systematic reviews of systematic qualitative reviews (Typology 1, Table 1) and for systematic reviews including systematic quantitative, qualitative, and/or mixed studies reviews (Typology 3, Table 1), thereby highlighting the contribution of our paper to the field of mixed methods research methodology in identifying these gaps and needs for further work. Such type-3 SRSRs would be challenging to conduct, interpret, and report considering the combination of qualitative and quantitative findings involved. The step of synthesizing, presenting, and summarizing the qualitative and quantitative findings of systematic reviews would need special consideration. We strongly recommend building this reflection based on previous works [START_REF] Frantzen | Meta-integration for synthesizing data in a systematic mixed studies review: Insights from research on autism spectrum disorder[END_REF][START_REF] Hong | Convergent and sequential synthesis designs: Implications for conducting and reporting systematic reviews of qualitative and quantitative evidence[END_REF]Hong & Pluye, 2018a;[START_REF] Sandelowski | Mapping the Mixed Methods-Mixed Research Synthesis Terrain[END_REF] related to systematic mixed studies reviews in order, for example, to adapt the synthesis methods at the tertiary level of research. In other words, we have something to learn from works that have been done at the secondary level of research in combining various types of primary studies (quantitative, qualitative and mixed methods); to expand the field at the tertiary level of research, i.e. systematic reviews combining the findings of quantitative, qualitative and mixed studies reviews.

Our methodological discussion paper is original and contributes to this field for the following two reasons: First, it provides a typology centered on SRSRs including the combination of various types of systematic reviews and their corresponding mixed types of evidence. Second, it enables an analysis of common and distinctive criteria of three critical appraisal tools by making a judgment about their potential applicability to systematic qualitative and mixed studies reviews.

Our paper contributes to one of the three dimensions of quality: the methodological quality of systematic reviews included in a SRSRs, specifically of systematic quantitative, qualitative, and mixed studies reviews. The other dimensions of quality related to SRSRs (Hong & Pluye, 2018a), i.e., the reporting and conceptual dimensions, could be the subject of further research. For example, how should we report systematic reviews of systematic qualitative, quantitative, and mixed studies reviews? Based on our past experience, it is very challenging to evaluate and compare the quality of systematic reviews from different epistemological and methodological traditions. To do so clear guidance is needed in two ways. There is a need to develop or adapt a critical appraisal tool to assess the methodological quality of systematic qualitative reviews. There is also a need to explore whether the criteria of the MMSR tool (Jimenez et al., 2018a, b) could be expanded to answer research questions other than those pertaining to impacts and program evaluation.

This paper could be useful to feed methodological support organizations and any other research teams interested in this novel type of tertiary level research, i.e. systematic reviews of systematic qualitative, quantitative, and mixed studies reviews.

Other authors are invited to continue the reflection so as to eventually adapt or develop a unified/integrated tool that could enable the assessment of systematic qualitative, quantitative, and mixed studies reviews on a common ground. This tool could include a set of criteria named in a generic way (e.g., pre-published protocol, justification of synthesis method) that could be applicable regardless of the types of systematic reviews in question.

A guidance document could be made available for authors of SRSRs to explain how to apply and to interpret the methodological quality based on the specificities of each type of systematic review, as provided by Whiting et al. to facilitate the use of the ROBIS tool (2016b). Considering that there are existing and available tools, we should use them and adapt the criteria and their rating guidance as needed.

Limitations and opportunities

In this paper, we focused on typology of SRSRs that include various types of systematic reviews, themselves including various empirical studies with quantitative and qualitative data. We used a "classical/traditional" way of categorizing the various levels of research (Figure 1), i.e. by types of studies: empirical studies, systematic reviews and SRSRs.

However, we considered that systematic reviews of systematic quantitative and qualitative reviews must include systematic mixed studies reviews (Table 1 and Figure 1) in order to extract and synthesize, respectively, quantitative and qualitative findings from the second level of research (systematic reviews). We are aware that recent developments suggest that we should consider not only the types of reviews/SRSRs, but also the broader context of the evidence ecosystem in which the reviews are produced and used, including a consideration of key dimensions as mentioned earlier (e.g., aims, methodologies, and structure of systematic reviews /SRSRs) [START_REF] Gough | Clarifying differences between reviews within evidence ecosystems[END_REF]Gough, Thomas, et al., 2012).

These authors suggest that "for all of the many types of review question and types of systematic reviews, there can be many levels of evidence standards for making different evidence claims based on those reviews (Gough et al., 2019, p.7)." Indeed, the quality and relevance of evidence claims can be assessed within a "dimensions of difference framework": 1) the SRSR method used in the selection and synthesis of the research evidence; 2) systematic reviews included in an SRSR; and 3) evidence produced in the SRSRs [START_REF] Gough | Evidence Standards: A dimensions of difference framework for appraising justifiable evidence claims[END_REF]. We acknowledge that this methodological discussion paper illustrates only a small portion of the complexity of SRSRs that are embedded in a dynamic and changing evidence ecosystem. Otherwise, it has covered only empirical evidence (i.e.

synthesizing research evidence from systematic reviews in SRSRs) as part of the typology and the methodological quality while many other ways of knowing exist (such as theoretical knowing and administrative and individual participant data) [START_REF] Gough | Clarifying differences between reviews within evidence ecosystems[END_REF][START_REF] Nutley | What counts as good evidence? Alliance for Useful Evidence[END_REF].

Putting together the criteria deriving from three existing critical appraisal tools was a first attempt to provide an initial structure in order to reflect on their applicability to appraise the quality in systematic qualitative and mixed studies reviews. What could be done in further work is to use the template suggested in Tables 2 and3 and to systematically analyze the systematic review variations included in an SRSR based on these key dimensions: aims or question, approaches (e.g., ontological and theoretical assumptions), methodologies, structure, and components of the systematic reviews [START_REF] Gough | Clarifying differences between reviews within evidence ecosystems[END_REF]Gough, Thomas, et a., 2012). This exploration should be done at the systematic review level while assessing methodological quality of included systematic reviews for authors of SRSRs.

There are also opportunities for further works in developing reporting guidelines for systematic review of systematic qualitative, quantitative and mixed studies reviews by considering existing published checklists, as mentioned previously (e.g. ENTREQ, eMERGE). We also recommend to expand the works surrounding methodological quality for other fields outside health care (e.g. social science).

Conclusion

This paper has emphasized the importance of using a descriptive and simple typology for eliciting the different types of systematic reviews included in SRSRs. Having a clear and consistent typology is the foundation to extend a wider application of SRSRs, to address broad research synthesis questions, and to assess methodological quality. We have argued that researchers need to consider the combination of systematic qualitative, quantitative, and mixed studies reviews and the integration of qualitative and quantitative evidence for planning, conducting, and reporting SRSRs. For example, we considered that the effects of information and communication technologies on nursing care can be studied through quantitative and qualitative lens (Rouleau, Gagnon, Côté, Payne-Gagnon, Hudson, & Dubois, 2017). These effects may be perceived by nurses as support for their practice and provision of nursing care. Depending on the scope of SRSRs and the question(s) of interest, we have also argued that integrating many types of systematic reviews and a combination of evidence can help increase the understanding of a given phenomenon and provide a comprehensive answer to a complex review question. Finally, our work can help advance the field of assessing the methodological quality of systematic qualitative, quantitative, and mixed studies reviews on a common ground. We have no doubt that the development and validation of a consolidated critical appraisal tool for authors of SRSRs will be of great value.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. The use of systematic and explicit methods to identify, select, and critically appraise relevant systematic qualitative and mixed studies reviews and to collect and synthesize qualitative findings pertaining to these systematic reviews. Some authors also used the terms "Meta-review of systematic reviews of qualitative studies" (May et al., 2016) and "Meta-Review of Qualitative Systematic Reviews" [START_REF] Pearce | Experiences of Self-Management Support Following a Stroke: A Meta-Review of Qualitative Systematic Reviews[END_REF]. Toye et al. ( 2017) referring to "mega-ethnography", representing a method of synthesis rather than a type of SRSR.

2) Systematic review of systematic quantitative reviews

The use of systematic and explicit methods to identify, select, and critically appraise relevant systematic quantitative and mixed studies reviews and to collect and synthesize quantitative findings pertaining to these systematic reviews.

3) Systematic review of systematic quantitative, qualitative, and mixed studies reviews

The use of systematic and explicit methods to identify, select, and critically appraise relevant systematic quantitative, qualitative, and/or mixed studies reviews and to collect and analyse quantitative and qualitative findings pertaining to these systematic reviews.

Examples of this type of SRSR have been published [START_REF] Rouleau | Effects of E-Learning in a Continuing Education Context on Nursing Care: Systematic Review of Systematic Qualitative, Quantitative, and Mixed-Studies Reviews[END_REF][START_REF] Rouleau | Effects of e-learning in a continuing education context on nursing care: A review of systematic qualitative, quantitative and mixed studies reviews (protocol)[END_REF].

Note:

a The typology is partially informed by [START_REF] Hong | Convergent and sequential synthesis designs: Implications for conducting and reporting systematic reviews of qualitative and quantitative evidence[END_REF] b These definitions have in part been adapted from [START_REF] Booth | Systematic Approaches to a Successful Literature Review[END_REF]: The use of systematic and explicit methods to identify, select, and critically appraise relevant systematic reviews and to collect and synthesize findings pertaining to these systematic reviews. o Bias is commonly understood to be a concept drawn from the quantitative research paradigm and to be incompatible with the philosophical underpinnings of qualitative enquiry (Jimenez et al., 2018a).

Assessment of risk of bias/methodological quality by two independent reviewers -We recommend a clear and consistent use of terms.

o For example, in the MMSR tool, the term "risk of bias" is used interchangeably both for systematic QUANT (criterion B.2) and QUAL reviews (criterion C.4), along with the term "quality or critical appraisal".

Synthesizing and presenting findings

Pre-defined analyses are reported (e.g. in a protocol) or departures are explained -We recommend renaming this criterion to make it more neutral and inclusive for all types of SRs: "Planned analyses are reported (e.g., in a protocol) and differences between planned and performed analyses explained." o The planning of pre-defined analyses or departures relies on a quantitative tradition. The word "departure" underlines that deviations from a protocol are sources of bias or errors, introduced by reviewers during the selection of analyses and analysis methods. o As currently named, this criterion is appropriate for SRs of RCTs with metaanalysis but not for systematic QUAL reviews such as meta-ethnography.

Heterogeneity addressed in the synthesis -We recommend adapting the term "heterogeneity" when referring to systematic QUAL reviews in order to fit with this particular epistemological and methodological tradition. o For example, [START_REF] Pearce | Experiences of Self-Management Support Following a Stroke: A Meta-Review of Qualitative Systematic Reviews[END_REF] suggests this question: Do authors discuss convergence and divergence within the QUAL primary study findings?

Justification of meta-analysis (if applicable) for the combination of results

-We recommend renaming this criterion retrieved from the three tools to make it more neutral and inclusive for all types of SRs: "Justification of the synthesis approach for the combination of results." o Meta-analysis is a synthesis approach used in QUANT reviews. In QUAL reviews, other approaches are used, such as meta-synthesis.

In MSR, convergent and sequential synthesis designs are used (Hong & Pluye, 2018a).

Robustness of findings with funnel plot or sensitivity analyses -We recommend adapting the criterion to match with the vocabulary and tradition of systematic QUAL reviews and appropriate methods to ensure the "robustness" of findings. o In systematic QUAL reviews, some other terms can be used to refer to the robustness of findings, such as credibility, congruity, relevance, adequacy of data, credibility, dependence, and trustworthiness (e.g. see (Hong & Pluye, 2018a;[START_REF] Lewin | Using Qualitative Evidence in Decision Making for Health and Social Interventions: An Approach to Assess Confidence in Findings from Qualitative Evidence Syntheses (GRADE-CERQual)[END_REF][START_REF] Munn | Establishing confidence in the output of qualitative research synthesis: The ConQual approach[END_REF].

Integrating qualitative and quantitative evidence

Use of a program theory and/or logic model -We recommend adapting these criteria to take into account different synthesis designs (i.e. sequential exploratory, sequential explanatory, convergent) [START_REF] Pluye | Opening-up the definition of systematic literature review: The plurality of worldviews, methodologies and methods for reviews and syntheses[END_REF] and research questions outside of the program evaluation field.

The way these criteria are formulated are applicable for authors interested in answering different SR questions about the effectiveness of Incorporation of qualitative evidence in SR design

Analysis of intermediate and endpoint outcomes along causal chain

Use of qualitative evidence in causal chain analysis programs/interventions using a program theory/logic model and when a sequential explanatory design is used (i.e. results of the quantitative synthesis inform the qualitative synthesis).

Incorporation of qualitative evidence in other aspects of the analysis -These criteria are applicable for MSRs. However, it could be reformulated to also consider the incorporation of quantitative evidence because a mixed studies review can also start with collecting qualitative evidence, and then, incorporating quantitative evidence (sequential exploratory).

Integration of quantitative and qualitative evidence to form conclusions and suggest implications

Interpreting findings and drawing conclusions

Impact of heterogeneity on the results of SRs -As suggested earlier, the term "heterogeneity" needs to be adapted when referring to systematic QUAL reviews.

Investigation of publication bias and discussion of its likely impact on results

-We recommend developing and/or adapting specific guidance to make this criterion applicable for systematic QUAL reviews and MSRs.

o The guidance provided in ROBIS and in AMSTAR is formulated under a QUANT lens. o The examination of publication bias and its impact is a practice mainly seen in QUANT tradition. In health sciences, QUAL research and mixed methods are often rejected for publication given their perceived "low priority" (e.g., see [START_REF] Greenhalgh | An open letter to The BMJ editors on qualitative research[END_REF]. o The impact of publication bias for systematic QUAL reviews and MSRs can be even higher, compared to systematic QUANT reviews. Considering the hypothesis that few QUAL studies are included in a SR, missing one or more of these QUAL studies may seriously undermine the trustworthiness of the systematic QUAL reviews or MSRs due to unpublished papers that would contain meaningful results. Note: QUAL = qualitative; QUANT= quantitative; MSRs= mixed studies reviews; RCTs = randomized controlled trials; SRs= systematic reviews.

Figure 1 .

 1 Figure 1. Different Levels of Research, and Typology related to Systematic Reviews of Systematic Reviews

Table 1 .

 1 Liberati, A. (2009). The PRISMA Statement for reporting systematic reviews and meta-analyses of studies that evaluate health care interventions: Explanation and elaboration. PLOS MEDICINE, 6(7), e1000100. https://doi.org/10.1371/journal.pmed.1000100 Three types of Systematic Reviews of Systematic Reviews

	Lunny, C., Brennan, S. E., McDonald, S., & McKenzie, J. E. (2016). Evidence map of studies
	evaluating methods for conducting, interpreting and reporting overviews of systematic
	reviews of interventions: Rationale and design. Systematic Reviews, 5.
	https://doi.org/10.1186/s13643-015-0178-0
	Lunny, C., Brennan, S. E., McDonald, S., & McKenzie, J. E. (2017). Toward a comprehensive
	evidence map of overview of systematic review methods: Paper 1-purpose, eligibility,
	search and data extraction. Systematic Reviews, 6. https://doi.org/10.1186/s13643-017-
	0617-1

Lunny, C., Brennan, S. E.,

McDonald, S., & McKenzie, J. E. (2018)

. Toward a comprehensive evidence map of overview of systematic review methods: Paper 2-risk of bias assessment; synthesis, presentation and summary of the findings; and assessment of the certainty of the evidence. Systematic Reviews, 7(1), 159. https://doi.org/10.1186/s13643-018-0784-8 May, C. R., Cummings, A., Myall, M., Harvey, J., Pope, C., Griffiths, P., Roderick, P., Arber, M.,

Boehmer, K., Mair, F. S., & Richardson, A. (2016)

. Experiences of long-term life-limiting

Table 2 .

 2 Perceived Applicability of Critical Appraisal Tools to Assess Methodological Quality of Systematic Qualitative and Mixed Studies Reviews

	Criteria included	ROBIS and criteria	AMSTAR 2	MMSR criteria	Is the criterion perceived as	Is the criterion
	in 8 steps of a SR a	(Whiting et al., 2016a, b)	criteria (Shea et	(Jimenez et al., 2018a,	applicable for systematic	perceived as applicable for
			al., 2017)	b)	QUAL reviews?	systematic
						MSRs?
	0. Assessing the relevance
	Does the question addressed by the review
	SR match the SRSRs question (e.g. for the	Phase 1	No	No	Yes	Yes
	PICO or the SPICE)?
	1. Defining

the SR question and eligibility criteria for including SRs

	Pre-published protocol	1.1	2	A1	Yes	Yes
	Definition of the SR questions and eligibility criteria	1.1	1	A2, A3, A4.1	Yes	Yes
	Appropriateness of eligibility criteria for SR question	1.2	No	A3, A4.1	Yes	Yes
	Specification of a separate SR question for
	each review component, including the
	intervention design, the implementation	No	No	A2	Partially yes	Partially yes
	processes, the participants, and the
	intervention/program effects
	Unambiguousness of eligibility criteria	1.3	No	No	Yes	Yes
	Appropriateness of all restrictions in
	eligibility criteria based on SR characteristics	1.4	3	A3, A4.2	Yes	Yes
	Appropriateness of any restrictions in
	eligibility criteria based on sources of	1.5	No	A4.2, A5	Yes	Yes
	information
	2. Applying

extensive and comprehensive search strategies using a variety of information sources

	Comprehensiveness of the search strategies	2.1, 2.3	4	A4.2	Yes	Yes
	Appropriateness of restrictions based on date, publication format, or language	2.4, 2.5	4	A4.2 et A.5	Yes	Yes
	3. Identifying

potentially relevant SRs and selecting the relevant ones

		4. Extracting data
	Data extraction performed by two independent reviewers	3.1	6	B1	Yes	Yes
	List of all included SRs	No	No	A6	Yes	Yes
	List of all excluded SRs as well as their justifications	No	7	A6	Yes	Yes
	Use of methods to address dependency in
	findings at the between study level and within individual studies (e.g. multiple	3.2	No	A7	Yes	Yes
	outcomes reported)
	Use of all relevant SR results in the synthesis	3.3	No	D4,D5,D6	Yes	Yes
	5. Collecting
	Screening of title/abstracts and/or full texts by two independent reviewers	3.1	5	A6	Yes	Yes

and presenting data on risk of bias/methodological quality of pertinent individual studies within included SRs

	Use of an appropriate technique and/or
	appropriate criteria by authors of SRs for
	assessing the risk of bias/methodological	3.4	9	B.2, C.4	Yes	Yes
	quality of pertinent individual studies
	within included SRs
	Assessment of risk of bias/methodological quality by two reviewers	3.5	No	B.2, C.4	Yes	Yes
	6. Synthesizing and presenting findings
	Description and presentation of detailed characteristics of SRs	3.2	8	B.1, C.2	Yes	Yes
	Inclusion of all studies in the synthesis	4.1	No	No	Yes	Yes
	Pre-defined analyses are reported (e.g. in a protocol) or departures are explained	4.2	No	No	Partially yes	Partially yes
	Appropriateness and clarity of synthesis approach	4.3	No	B.3, C.5	Yes	Yes
	Heterogeneity addressed in the synthesis	4.4	11, 14	B.4	No	Partially yes
	Justification of meta-analysis (if applicable) for the combination of results	4.5	11, 12	B.4, B.5	No	Partially yes
	Robustness of findings with funnel plot or	4.5	15	B.7	No	Partially yes
	sensitivity analyses
	7. Integrating qualitative and quantitative evidence
	Use of a program theory and/or logic model	No	No	D.1	Partially Yes Partially yes

Table 3 .

 3 Recommendations to Adapt Criteria of Critical Appraisal Tools in Assessing Methodological Quality of Systematic Qualitative and Mixed Studies Reviews

Criteria included in steps of a SR a Recommendations 1. Defining the SR question and eligibility criteria for including SRs

	Specification of a separate SR question for each	-If it is a multicomponent review (Gough et al.,
	review component, including the intervention	2019) that includes broad questions and a
	design, the implementation processes, the	combination of methods and data making it
	participants, and the intervention/program effects	possible to explain the effectiveness of
		interventions/programs, then this criterion can be
		applicable for all types of SRs depending on the
		review questions. If the SR question(s) fall
		outside the program evaluation field, this
		criterion would need to be formulated in a more
		inclusive way so that other SR questions can fit
		in.
	5. Collecting

and presenting data on risk of bias/methodological quality of pertinent individual studies within included SRs Use

  of an appropriate technique and/or appropriate criteria by authors of SRs for assessing the risk of bias/methodological quality of pertinent individual studies within included SRs -We suggest avoiding the exclusive use of the term "risk of bias" and refer to "methodological quality" in order to avoid disciplinary or worldview jargon.

Smith, V., Devane, D., Begley, C. M., & Clarke, M. (2011). Methodology in conducting a systematic review of systematic reviews of healthcare interventions. BMC Medical Research Methodology, 11, 15. https://doi.org/10.1186/1471-2288-11-15 The EQUATOR Network. (2019) a Authors of systematic reviews of systematic reviews assess the methodological quality and risks of bias in included SRs. We could also assess risks of bias/methodological quality and other dimensions of quality, such as reporting, at a systematic review of systematic reviews level, but this is beyond the scope of this methodological discussion paper.


Isabelle Marty
email: isabelle.marty@ujf-grenoble.fr

Julien Fauré

Anne Fourest-Lieuvin

Stéphane Vassilopoulos

Sarah Oddoux

Julie Brocard

Triadin: What possible function 20 years later?

During the last 20 years, the identification of triadin function in cardiac and skeletal muscle has been the focus of numerous studies. First thought as the missing link between the ryanodine receptor and the dihydropyridines receptor and responsible of skeletal type excitation-contraction coupling, the current hypothesis on triadin function has slowly evolved, and triadin is envisaged now as a regulator of calcium release, both in cardiac and skeletal muscle. Nevertheless, none of the experiments performed up to now gave a clear cut view of what triadin really does in muscle. The problem became more complex with the identification of multiple triadin isoforms, having possibly multiple functions. Using a different approach from what has been done previously, we obtained new clues about the function of triadin. Our data point to a possible involvement of triadin in reticulum structure, in relation with the microtubule network.

Introduction

Excitation-contraction (EC) coupling, the process which results in muscle contraction after electrical stimulation, is performed by a macromolecular protein complex. The core of this complex is composed of two calcium channels, the dihydropyridines receptor (DHPR) and the ryanodine receptor (RyR). Around these two channels are organized a number of other proteins (calsequestrin, triadin, junctin, …), and an efficient muscle contraction relies on the synchronous work of all these proteins. One peculiar feature of this complex is that it is anchored in two different membranes: the T-tubule membrane, an invagination of the plasma membrane into the cytoplasm, that contains the DHPR, and the sarcoplasmic reticulum terminal cisternae that contains the RyR. In the skeletal muscle, it has been demonstrated that both channels, even anchored in two different membranes, are physically associated [START_REF] Marty | Biochemical Evidence for a complex involving Dihydropyridine Receptor and Ryanodine Receptor in Triad Junctions of Skeletal Muscle[END_REF]. The function of the calcium release complex is therefore based upon the precise organization of the triad (close apposition of two sarcoplasmic reticulum terminal cisternae with one T-tubule) and the implantation of the complex in this membrane structure. This extremely precise organization has to be maintained in a cell, the skeletal muscle fibre, which undergoes large deformations during each contraction. Triadin has first been identified as a 95kDa protein, specifically localized in the triad of the skeletal muscle, thought to connect DHPR to RyR [START_REF] Kim | Isolation of a terminal cisterna protein which may link the dihydropyridine receptor to the junctional foot protein in skeletal muscle[END_REF]. Since its identification in 1990, the specific function of triadin has been the matter of numerous studies, in cardiac as well as in skeletal muscle, the two muscles in which triadin expression has been demonstrated.

Triadin, a multiprotein family

Triadin was first believed to be a skeletal muscle specific protein, responsible of skeletal muscle EC coupling [START_REF] Caswell | Localization and partial characterization of the oligomeric disulfide-linked molecular weight 95,000 protein (triadin) which binds the ryanodine and dihydropyridine receptors in skeletal muscle triadic vesicles[END_REF]. In 1994, the presence of triadin in cardiac muscle was also demonstrated [START_REF] Peng | Structural diversity of triadin in skeletal muscle and evidence of its existence in heart[END_REF], as a 32kDa protein (theoretical MW). Nowadays, it is clear that the triadin gene can undergo multiple splicing, leading to a number of proteins. At least four isoforms are expressed in rodent skeletal muscles, named Trisk 95 (the 95kDa triadin first identified), Trisk 51, a 51kDa protein, Trisk 49 and Trisk 32, respectively 49kDa and 32kDa [START_REF] Marty | Cloning and characterization of a new isoform of skeletal muscle triadin[END_REF][START_REF] Vassilopoulos | Triadins are not triad-specific proteins: two new skeletal muscle triadins possibly involved in the architecture of sarcoplasmic reticulum[END_REF]. None of these isoforms is specific of a skeletal muscle type, and their pattern of expression is the same in fast or slow twitch muscles. Trisk 51 and Trisk 49 show an apparent molecular weight of 65 kDa in SDS-PAGE, and Trisk 32 migrates as a triplet centred on 37kDa. All the skeletal isoforms are truncated versions of Trisk 95, with a specific C-terminal end of variable length (figure 1). We have developed a number of antibodies, and depending on their epitope localization, in the common (N-terminal end) or in specific regions (C-terminal ends), these antibodies are either reactive on every triadin isoforms, or specific for one isoform (figure 2A). The antibody directed against the N-terminal end recognizes all triadins in all tissues and all species (mammalian) tested up to now, as this part is extremely conserved between all the spliced isoforms. Therefore this antibody allows the quantification of the relative expression level of the different isoforms in a given tissue. On the contrary, the antibodies specific of the Cterminal end of each isoform are only specific of a single isoform, and often specific of one species. Using antibodies against the common N-terminal part, we have observed that in rat skeletal muscle Trisk 95 and Trisk 51 are expressed at similar levels, each representing about 40% of the total triadin amount, Trisk 32 accounting for the remaining 20%. The situation is almost identical in mouse muscle. In adult human skeletal muscle, the major isoform is Trisk 51 (about 60% of all triadins), and Trisk 95 represents the remaining 40%. Trisk 32 is undetectable (figure 2B). The major cardiac triadin is CT1 [START_REF] Kobayashi | Identification of triadin 1 as the predominant triadin isoform expressed in mammalian myocardium[END_REF], but isoforms of higher molecular weight (CT2 and CT3) can be detected in cardiac muscle [START_REF] Guo | Biochemical characterization and molecular cloning of cardiac triadin[END_REF][START_REF] Kobayashi | Identification of triadin 1 as the predominant triadin isoform expressed in mammalian myocardium[END_REF][START_REF] Hong | Molecular cloning and characterization of mouse cardiac triadin isoforms[END_REF]. Trisk 32 is unambiguously identical to CT1, as demonstrated by cDNA cloning and by the reactivity of Trisk 32 specific antibodies with both skeletal and cardiac muscle. It is the only triadin isoform expressed in both muscles. The minor cardiac triadins detected with the anti-N-terminal antibody in heart homogenate are not recognized by the skeletal muscle specific antibodies of the same species, and are therefore cardiac muscle specific. As Trisk 32 is the only triadin isoform expressed in both muscles it could be renamed Trisk32/CT1, in order to clarify the nomenclature of the proteins, whereas the other tissue specific triadins should still be named Trisk (TRIadin Skeletal) or CT (Cardiac Triadin) respectively for the skeletal and cardiac specific triadins.

Localization of the triadin isoforms:

The question of the localization of triadin seems trivial, as this protein has been named "triadin" according to the specific triad localization of the first identified isoform, Trisk 95. Nevertheless, considering the identification of multiple isoforms, this question has to be carefully re-examined. Trisk 95 and Trisk 51 localization is restricted to the triad of skeletal muscle where they are associated with RyR. Therefore, these two triadin isoforms are full members of the calcium release complex [START_REF] Marty | Cloning and characterization of a new isoform of skeletal muscle triadin[END_REF][START_REF] Vassilopoulos | Triadins are not triad-specific proteins: two new skeletal muscle triadins possibly involved in the architecture of sarcoplasmic reticulum[END_REF]. The situation is much more complex for Trisk 32, which behaves differently in heart and in skeletal muscle. In heart, it is both colocalized and associated with the cardiac RyR. In skeletal muscle, the protein is mainly in the longitudinal sarcoplasmic reticulum [START_REF] Vassilopoulos | Triadins are not triad-specific proteins: two new skeletal muscle triadins possibly involved in the architecture of sarcoplasmic reticulum[END_REF], and therefore not associated with RyR. Interestingly, we have shown that in the longitudinal sarcoplasmic reticulum, Trisk 32 is associated with the IP3receptor. Nevertheless, a closer examination showed that in fact Trisk 32 is localized in the whole sarcoplamic reticulum, both in the terminal cisternae which are part of the triad and in the longitudinal reticulum which connects two adjacent triads. Immunoprecipitation experiments confirmed that in fact a low amount of RyR1 is associated with Trisk 32. This means that even though the majority of Trisk 32 is not at the triad, the low amount present in the triad is associated with RyR. The localization of the main triadin isoforms in skeletal muscle is schematized in figure 3.

Function of triadin : overexpression or Knock Out/Knock Down experiments:

In order to identify the function of triadin, a number of experiments were performed, based on the modification of triadin expression level, followed by the evaluation of the resulting calcium release by RyR. First, triadin (Trisk 95) was overexpressed in primary culture of rat skeletal myotubes, using adenovirus gene transfer, which resulted in a blocking of depolarization induced calcium release (Smida [START_REF] Rezgui | Triadin (TRISK 95) over-expression blocks excitation-contraction coupling in rat skeletal myotubes[END_REF]. On the other hand, experiments were performed on a calcium release complex completely or partially depleted of triadin. Mutant forms of RyR1 containing deletions in the triadin interacting domains (identified by [START_REF] Lee | Negatively charged amino acids within the intraluminal loop of ryanodine receptor are involved in the interaction with triadin[END_REF] were expressed in dyspedic myotubes, hence leading to suppression of triadin from the calcium release complex. In these myotubes, electrical depolarization was unable to induce calcium release from sarcoplasmic reticulum (Goonasekera et al, 2007), probably because of a reduction in the amplitude and kinetics of the calcium release. Similar results were obtained by reducing triadin expression with siRNA in C2C12 myotubes [START_REF] Wang | Altered stored calcium release in skeletal myotubes deficient of triadin and junctin[END_REF]. Lowering the amount of triadin resulted in a partial inhibition of the depolarization induced calcium release. The total deletion of triadin from the calcium release complex was performed by the production of a triadin KO mouse [START_REF] Shen | Triadins modulate intracellular Ca(2+) homeostasis but are not essential for excitation-contraction coupling in skeletal muscle[END_REF]. These experiments are more difficult to analyze, as compensatory reduction in calsequestrin expression is observed in the muscles of these mice. Nevertheless, in addition to the lower amount of stored calcium, a reduction in the depolarization induced calcium release was observed in myotubes which presented no reduction in calsequestrin. Therefore, the conclusion of all these experiments is that any modification in triadin expression levels, either by overexpression or by knock out/knock down, results in a common effect: reduction in depolarization-induced calcium release. Such apparent conflicting results underline the necessity of developing new approaches to understand triadin function.

Triadin expression in non muscle cells

To get insight into the function of triadin, we decided to use a different approach in order to reveal the intrinsic properties of triadin in a system devoided of its usual muscle partners. We expressed triadin in a non myogenic cell line, the COS-7 cells, and analyzed both the endoplasmic reticulum (ER) structure and the microtubule network, using specific antibodies [START_REF] Fauré | Triadin binding to the C-Terminal luminal loop of the ryanodine receptor is important for skeletal muscle excitation-contraction coupling[END_REF]. We observed that triadin induced drastic modifications of the ER structure, forming rope-like ER structures, associated with a massive reorganization of the microtubule network, the microtubules being bundled at the periphery of the cell. Such modifications were never observed when other ER-homing proteins, RyR for example, were overexpressed. In addition, these bundled microtubules are stable, resisting a depolymerization induced by nocodazole. Similar observations have already been made after overexpression of proteins involved in anchoring the ER to microtubules [START_REF] Vedrenne | Morphogenesis of the endoplasmic reticulum: beyond active membrane expansion[END_REF]. These different experiments thus point to a possible new function of triadin. Triadin could act as a protein connecting the reticulum (or the calcium release complex) to the microtubule network, and could also be involved in the stabilization of microtubules in muscle cells. This function would be a major one, since microtubules are known to depolymerize under high Ca 2+ concentration. If microtubules are indeed involved in triad organization, their depolymerization during each contraction would probably be dramatic. If triadin is involved in this structural function, it is probably not the only protein responsible for this anchoring/stabilization role, because triadin knock out mice muscles do not exhibit massive triad disorganization. Nevertheless, 30% of the triads show an abnormal orientation in the muscle of these triadin KO animals [START_REF] Shen | Triadins modulate intracellular Ca(2+) homeostasis but are not essential for excitation-contraction coupling in skeletal muscle[END_REF], and this proportion is confirmed in a triadin null mouse line we have developed in our team (unpublished data). This amount of misoriented triads was reproducibly observed in two different mouse lines, leading to a strong confirmation of the possible involvement of triadin in maintaining the triad structure. Therefore, these studies in a non muscle cell line have allowed us to unmask a specific intrinsic property of triadin, which is more difficult to visualize in muscle cells, perhaps because of other muscle proteins partially sharing similar function with triadin.

Conclusion

We have observed that triadin expression in COS-7 cells induces drastic modification in the organisation of the endoplasmic reticulum and microtubules. Therefore our current hypothesis is that triadin could be involved in the sarcoplamic reticulum structure in skeletal muscle, and could perform a regulatory function on RyR through this structural function. The overexpression as well as a down regulation of triadin in skeletal muscle cells would result in local sarcoplasmic reticulum disorganization, thereby induce an uncoupling between RyR and the facing DHPR. Nevertheless, additional studies need to be performed to confirm this hypothesis in skeletal muscle.

Figure 1 :

 1 Figure 1: The rat skeletal muscle triadin isoforms

Figure 2 :Figure 3

 23 Figure 2: A -The rat skeletal muscle isoforms, detected by an antibody in common region (Nter) or in specific Cterminal region of each isoform. B -The triadin isoforms expressed in different tissues (mouse heart, human skeletal muscle, mouse skeletal muscle, rat skeletal muscle) detected with an antibody against the common Nterminal part of all triadins, therefore able to react with all the triadins and used to quantify their relative amount. (H: heart; SK: skeletal muscle)

Acknowledgments:

The work of our team was supported by grants from Association Française contre les Myopathies (AFM) and GIS-Maladies Rares.