Analysis of avian bone response to mechanical loading—Part One: Distribution of bone fluid shear stress induced by bending and axial loading

Abstract

Mechanical loading-induced signals are hypothesized to be transmitted and integrated by a bone-connected cellular network (CCN) before reaching the bone surfaces where adaptation occurs. Our objective is to establish a computational model to explore how bone cells transmit the signals through intercellular communication. In this first part of the study the bone fluid shear stress acting on every bone cell in a CCN is acquired as the excitation signal for the computational model. Bending and axial loading-induced fluid shear stress is computed in transverse sections of avian long bones for two adaptation experiments (Gross et al. in J Bone Miner Res 12:982–988, 1997 and Judex et al. in J Bone Miner Res 12:1737–1745, 1997). The computed fluid shear stress is found to be correlated with the radial strain gradient but not with bone formation. These results suggest that the radial strain gradient is the driving force for bone fluid flow in the radially distributed lacunar-canalicular system and that bone formation is not linearly related to the loading-induced local stimulus.

Appendices

Appendices

Animal adaptation experiments by Gross et al. (1997) and Judex et al. (1997)

Two animal adaptation experiments, a turkey experiment (Gross et al. 1997) and a rooster experiment (Judex et al. 1997), were reported to study the correlation between mechanical loading parameters and bone formation. The functionally isolated right radius of ten adult turkeys (Gross et al. 1997) was subjected to a 4-week exogenous cyclic bending with a trapezoidal waveform (Fig. 3). Based on strain gauge data and finite element analysis, Gross et al. (1997) determined the bending-induced normal circumferential, radial and longitudinal strain gradients in the cross-section of the bone. The mid-diaphyseal cross-section of each loaded bone was divided into 24 equal angle sectors and the bending-induced bone formation within each sector was determined by comparing the loaded bone sections to the control sections of the contralateral limb (the left radius) (Fig. 12). In a separate experiment (Judex et al. 1997), six adult roosters were subject to an exercise model of bone adaptation, namely running 1,500 loading cycles/day on a treadmill for 3 weeks. Using the same method as in the turkey experiment, Judex et al. (1997) determined the loading-induced mechanical parameters (normal circumferential, radial and longitudinal strain gradients, longitudinal normal strain and peak strain rate) in the rooster mid-diaphyseal tarsometatarsus. The cross-section of the bone was divided into 12 equal angle sectors and the loading-induced bone formation within each sector was determined by comparing those sections to the corresponding sections of roosters not subject to loadings.

Fig. 12

A turkey bone cross-section with the 24 sectors from the Gross et al. (1997) adaptation experiment. CG circumferential strain gradient, RG radial strain gradient, LG longitudinal strain gradient, ɛ _i normal strain magnitude at surface i, D _ij linear distance between surfaces i and j. (Adapted from Gross et al. 1997)

Fourier series of the loading in Gross et al. (1997)

The loading variation is specified by a trapezoidal function f(t), 0<t<2t₁+t₂+2t₃, such that

$$ f(t) = \left\{ {\begin{array}{*{20}l} 0 & {0 < t < t_1 } \\ {k(t - t_1 )} & {t_1 < t < t_1 + t_3 } \\ F & {t_1 + t_3 < t < t_1 + t_3 + t_2 } \\ {k(t_1 + t_2 + 2t_3 - t)} & {t_1 + t_2 + t_3 < t < t_1 + t_2 + 2t_3 } \\ 0 & {t_1 + t_2 + 2t_3 < t < 2t_1 + t_2 + 2t_3 } \\ \end{array} } \right. $$

(22)

where k is the loading rate, F is the peak loading magnitude, and t₁, t₂, and t₃ are time durations shown in Fig. 9. A Fourier series representation which describes Eq. 22 as well as its periodic extensions to all other values of t is:

$$ f(t) = A_{0} /2 + {\sum\limits_{n = 1}^\infty }\,[A_{n} \cos (nwt) + B_{n} \sin (nwt)] $$

(23)

where w=2π/T _t , and T _t =2t₁+2t₃+t₂ is the loading period.

The Fourier constants are:

$$ A_0 = \frac{{2F(t_2 + t_3 )}} {{T_t }} $$

(24)

$$ \begin{aligned} A_n & = \frac{{2k}} {{T_t n^2 w^2 }}\{ \cos [nw(t_1 + t_3 )] - \cos (nwt_1 ) + \cos [nw(t_1 + t_2 + t_3 )] \\ & \quad - \cos [nw(t_1 + t_2 + 2t_3 )]\} \\ \end{aligned} $$

(25)

$$ B_n = 0 $$

(26)

Thus Eq. 23 can be written as:

$$ f(t) = \frac{{A_0 }} {2} + \sum\limits_{n = 1}^\infty \,A_n \cos (nwt). $$

(27)

Parameters in pore pressure solution

$$ A_{0n} = \frac{{AnN_0 }} {{iTF}}\frac{{K_0 (\sqrt {inT} R_{v2} ) - K_0 (\sqrt {inT} R_{v1} )}} {{I_0 (\sqrt {inT} R_{v1} )K_0 (\sqrt {inT} R_{v2} ) - I_0 (\sqrt {inT} R_{v2} )K_0 (\sqrt {inT} R_{v1} )}} $$

(28)

$$ B_{0n} = \frac{{AnN_0 }} {{iTF}}\frac{{I_0 (\sqrt {inT} R_{v1} ) - I_0 (\sqrt {inT} R_{v2} )}} {{I_0 (\sqrt {inT} R_{v1} )K_0 (\sqrt {inT} R_{v2} ) - I_0 (\sqrt {inT} R_{v2} )K_0 (\sqrt {inT} R_{v1} )}} $$

(29)

$$ A_{1n} = \frac{{AnN_m }} {{iTF}}R_{v2} \cos \,\gamma \frac{{R_{v1} K_1 (\sqrt {inT} R_{v2} ) - R_{v2} K_1 (\sqrt {inT} R_{v1} )}} {{R_{v2} I_1 (\sqrt {inT} R_{v1} )K_1 (\sqrt {inT} R_{v2} ) - R_{v2} I_1 (\sqrt {inT} R_{v2} )K_1 (\sqrt {inT} R_{v1} )}} $$

(30)

$$ B_{1n} = \frac{{AnN_m }} {{iTF}}R_{v2} \cos \gamma \frac{{R_{v2} I_1 (\sqrt {inT} R_{v1} ) - R_{v1} I_1 (\sqrt {inT} R_{v2} )}} {{R_{v2} I_1 (\sqrt {inT} R_{v1} )K_1 (\sqrt {inT} R_{v2} ) - R_{v2} I_1 (\sqrt {inT} R_{v2} )K_1 (\sqrt {inT} R_{v1} )}} $$

(31)

$$ A_{1n} = \frac{{AnN_m }} {{iTF}}R_{v2} \sin \gamma \frac{{R_{v1} K_1 (\sqrt {inT} R_{v2} ) - R_{v2} K_1 (\sqrt {inT} R_{v1} )}} {{R_{v2} I_1 (\sqrt {inT} R_{v1} )K_1 (\sqrt {inT} R_{v2} ) - R_{v2} I_1 (\sqrt {inT} R_{v2} )K_1 (\sqrt {inT} R_{v1} )}} $$

(32)

$$ B_{1n} = \frac{{AnN_m }} {{iTF}}R_{v2} \sin \gamma \frac{{R_{v2} I_1 (\sqrt {inT} R_{v1} ) - R_{v1} I_1 (\sqrt {inT} R_{v2} )}} {{R_{v2} I_1 (\sqrt {inT} R_{v1} )K_1 (\sqrt {inT} R_{v2} ) - R_{v2} I_1 (\sqrt {inT} R_{v2} )K_1 (\sqrt {inT} R_{v1} )}} $$

(33)

$$ A_3 = \frac{{K_0 (\gamma _1 ) - K_0 (\gamma _1 /q)}} {{I_0 (\gamma _1 /q)K_0 (\gamma _1 ) - I_0 (\gamma _1 )K_0 (\gamma _1 /q)}} $$

(34)

$$ B_3 = \frac{{I_0 (\gamma _1 /q) - I_0 (\gamma _1 )}} {{I_0 (\gamma _1 /q)K_0 (\gamma _1 ) - I_0 (\gamma _1 )K_0 (\gamma _1 /q)}} $$

(35)