Hierarchical model for strain generalized streaming potential induced by the canalicular fluid flow of an osteon

Mathematically modeling fluid flow and fluid shear stress in the canaliculi of a loaded osteon

“…According to Wu et al [36], the osteon system is modeled as a hierarchical hollow annular cylinder structure (as shown in Fig. 1) and two types of boundary cases for the osteon wall are proposed (as shown in Fig.…”

“…According to Wu et al [36], the solutions of whole pore pressure distribution p m  =  p 0 m ( r ) e iωt ( is the angular frequency of loading, m  = 1 is for case I and m  = 2 is for case II) in the osteon wall is given by:where, is an elastic component of the stiffness tensor. E r and E z are radial and longitudinal Young modulus, μ r and μ z are radial and longitudinal drained Poisson’s ratio, I n and K n are the first kind and the second kind modified Bessel function of order n respectively, and c m is constant and can be found in Ref.…”

“…[33]. The constant C is given by Wu et al [1, 33–36]. where , μ and k are dynamic viscosity and intrinsic permeability, respectively.…”

“…According to the established canaliculi model in Wu et al [36], the solutions for pressure p′ and longitudinal velocity component u in the canaliculi are presented as: where, ρ is the fluid density, and μ is the dynamic viscosity. The canalicular radius is R and lengthened l ( l  =  b  −  a ).…”

“…A homogenization canaliculated porosity approach may avoid multiscale parameters of permeability. We specifically develop a canaliculi model on the basis of our previous osteon model [33–36] to link the external loads to the canalicular fluid velocity, pressure, flow rates, and shear stress. The model can also be used to analyze the structure of the proteoglycan matrix in the fluid space surrounding the osteocytic process in the canaliculus.…”