ABSTRACT
C. Oddou, T. Lemaire Laboratory of Multiscale Modelling and Simulation – Biomechanics, Faculty of Sciences & Technology, (MSME CNRS-8208), University Paris-Est, Cre´teil, France
J. Pierre Laboratory of Osteo-Articular Biomechanics and Biomaterials (B2OA CNRS7052), University Paris 12, Faculty of Sciences & Technology, Cre´teil Cedex, France
B. David Laboratory of Mechanics of Soils, Structures and Materials (MSSMat CNRS8579), E´cole Centrale Paris, Chaˆtenay-Malabry Cedex, France
T&F Cat#65416,
and
Parameter Notation Unit
Substrate overall length scale L m Porosity of the medium φ Length scale of the pore a m
Specific pore wall area SV ≈ 2φ a
m−1
Effective pore length of the sample
LP m
Tortuosity T = L 2 p / L2
Porous medium permeability K = φ(2a)2
96 × 1
T
= 1 6 × φ
S2V ×T
m2
Concentration of the nutrient (oxygen, . . . ) molecules
C mol×m−3
Michaelis-Menten constant KM mol×m−3 Cellular nutrient maximal
consumption rate VMax mol× cel−1 × sec−1
Rate of nutrient (oxygen, . . . ) consumption by unit area
RS mol×m−2 × sec−1
Surface density of the cells σcel cel×m−2 Maximum ratio of cell oxygen
consumption Rmax = σcel ×Vmax mol×m−2 × sec−1
Rate of nutrient (oxygen, . . . ) consumption by unit volume
RV =RS ×SV mol×m−3 × sec−1
T&F Cat#65416,
Parameter Notation Unit
Reference concentration of the nutrient
c0 mol×m−3
Diffusion coefficient of the nutrient solute
D m2 × s−1
Damko¨hler number Da= σcel ×VMax ×a
D× c0 Perfusion mean velocity u0 m× s−1 Density of the culture fluid ρ kg×m−3 Dynamic viscosity of the
culture fluid η Pa× sec
Reynolds number Re= ρ×a×u0/φ
η
Pe´clet number Pe= a×u0/φ
D
Viscous stresses τ = η× u0 a Pa
Debye length LD m Ionic force Ci mol×m−3 Vacuum permittivity ε0 C×V−1 ×m−1 Relative dielectric constant
of the solvent ε
Faraday constant F C×mol−1 Gas constant R J×K−1 ×mol−1 Absolute temperature T K Electric double-layer
potential ϕ V
Donnan osmotic swelling pressure
πD Pa
Oxygen molar concentration
cO2 mol×m−3
Oxygen binary diffusivity in water
DO2 m 2 × sec−1
Perfusion velocity u m× sec−1 Typical size of the solid
heterogeneities in the porous medium
h m
Fluid kinematic viscosity ν = nρ m 2 × sec−1
Dimensionless Michaelis constant
λM
Dimensionless oxygen flux at the frontier
(Continued)
T&F Cat#65416,
and
Parameter Notation Unit
Normal unit vector of a considered surface
n
Tangent unit vector to a considered surface
t
Identity tensor I Streaming potential ψ V Ionic binary diffusion
coefficient in water D± m2 × sec−1
Ionic flux density −→ J± mol×m−2 × sec−1
Poiseuille permeability tensor
κP m2 ×Pa−1 × sec−1
Osmotic permeability tensor
κC m5 ×mol−1 × sec−1
Electroosmotic permeability tensor
κE m2 × sec−1
Nondimensional number comparing electrical current and diffusion
Reference electric current density
I0 C×m−2 × sec−1
Length of the representative elementary volume
lCh m
Reference diffusion coefficient
D0 m2 × sec−1
Dean number Dn=Re×( a/rc
Curvature radius of the flow streamlines
rc m
Secondary transverse fluid velocity
ut m× sec−1
Characteristic oxygen diffusion velocity within a pore
uD ≈ DO2a m× sec−1
Interaction between fluid flow and living media is a complex matter, far from being completely understood: it is clear that not only cell and tissue
T&F Cat#65416,
metabolism is likely to be influenced by the transport phenomena of nutrient and waste products that are regulated by the flow, but it may also be directly affected by the various stresses generated by the fluid motion. Fluid flow within natural tissues plays important roles in morphogenesis, metabolism function, and pathogenesis. In the design of new biomaterials mimicking biological tissues, it is well recognized today that three-dimensional in vitro culture better recapitulates physiological cell environment. Indeed, fluid flow and solid strain that are imposed within tissue cultured in bioreactors not only affect cell nourishment but also exert on cells mechanical actions such as pressure effects, drag interactions, and viscous shear stresses (Chen 2008).
