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

Abstract

Background: Mechanical load-induced intraosseous pressure gradients may result in some fluid stimuli effects, such as fluid flow and fluid shear stress (FSS), which may enable bone cells to detect external mechanical signals. Interstitial bone fluid flow is known to occur in lacunar-canalicular porosity (PLC).

Methods: In order to characterize lacunar-canalicular fluid flow behavior, a hierarchical osteon system is developed. The osteon is modeled as a poroelastic annular cylinder with two types of impermeable boundary cases considered on its outer wall: one is elastic restrained (Case I), whereas the other is displacement confined (Case II). Analytical solutions such as canalicular fluid velocity, pressure, fluid flow rate (FFR), and shear stress are obtained.

Results: Results show that the amplitudes of FFR and FSS are proportional to strain amplitude and frequency. However, the key loading factor governing canalicular fluid flow behavior is the strain rate. The larger canalicular radius is, the larger amplitudes of FFR and FSS generalized, especially, the FSS amplitude is proportional to canalicular radius. In addition, both FFR and FSS amplitudes produced in case II are larger than those of case I.

Conclusion: Strain rate can be acted as a representative loading parameter governing the canalicular fluid flow behavior under a physiological state. This model can facilitate better understanding the load induced the fluid permeation in the PLC. The approach can also be used to analyze the structure of the proteoglycan matrix in the fluid space surrounding the osteocytic process in the canaliculus.

Keywords: Canaliculi; Fluid flow rates (FFR); Fluid shear stress (FSS); Osteon; Poroelasticity.

Fig. 1

The Hierarchical model for osteon system. a and b are the inner (Haversian canal surface) and outer osteon radius, respectively, and h is height

Fig. 2

The osteon outer wall in case I and case II are assumed to be elastic restrained (A) and displacement confined (B) respectively

Fig. 3

Fluid flow rate amplitude ($\left|Q\right|$, FFRA) and fluid shear stress amplitude ($\left|{\mathit{\tau }}_w\right|$, FSSA) as a function of the loading amplitude (ε _z0) and frequency (ω) at R = 5 × 10⁻⁷ m and k = 10^-18 m². a Fluid flow rate amplitude as a function of the loading amplitude. b Fluid shear stress amplitude as a function of the loading amplitude. c Fluid flow rate amplitude as a function of the frequency. d Fluid shear stress amplitude as a function of the frequency

Fig. 4

FFRA and FSSA as a function of the loading frequency at R = 5 × 10⁻⁷ m, k = 10^-18 m² and with the strain rate fixed at ${\dot{\mathit{\epsilon }}}_z$ = 0.0005, 0.001, and 0.003 s⁻¹. a FFRA as a function of the frequency for case I. b FSSA as a function of the frequency for case I. c FFRA as a function of the frequency for case II. d FFSA as a function of the frequency for case II

Fig. 5

History of strain loads at ω = 1Hz and ${\dot{\mathit{\epsilon }}}_z$ = 0.0005, 0.001, and 0.003 s⁻¹, the operator Re () gives the real part of the complex number

Fig. 6

Time responses of FFR (a Case I; c Case II) and FSS (b Case I; d Case II) at R = 5 × 10⁻⁷ m, k = 10^-18 m²

Fig. 7

Evolutions of FFRA (a, c) and FSSA (b, d) with canalicular radius at k = 10^-18 m²

Fig. 8

Effects of permeability with parameter values of R = 1 × 10⁻⁷ m, ${\dot{\mathit{\epsilon }}}_0=0.001$ s⁻¹. Case I (solid line); case II (dotted line). a FFRA as a function of the permeability. b FSSA as a function of the permeability