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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.

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Abbreviations

a :

Radius of the osteocytic process

a 0 :

Radius of the fiber traversing the annular region between the osteocytic process and the canalicular wall

A :

Area of bone cross-section

b :

Radius of the canaliculus

B :

Skempton pore pressure parameter

c :

Diffusion coefficient in the differential equation governing the pore fluid pressure (=r 20 r )

E :

Longitudinal elastic modulus of bone

f(t):

Mechanical loading applied to bone

F :

Maximum magnitude of the mechanical loading f(t)

I :

Area moment of inertia for the Gross et al. (1997) experiment

I0, I1:

Modified Bessel functions of the first kind

I x , I y :

Area moment of inertia about the indicated axes for the Judex et al. (1997) experiment

k p :

Cell process channel scale Darcy law permeability constant for fluid flow through the mid-section of a cell process channel filled with transverse fibers

K0, K1:

Modified Bessel functions of the third kind

L :

Moment arm length for the Gross et al. (1997) experiment

L 0 :

Maximum value of the bone cross-section in a rectangular coordinates for the Judex et al. (1997) experiment

L c :

Distance between two osteocytes in the radial direction in the CCN

L d :

Distance between two osteocytes in the circumferential direction in the CCN

M x (t), M y (t):

Bending moments about the indicated axes for the Judex et al. (1997) experiment

p :

Pore fluid pressure

P :

Dimensionless pressure (=3p/(BTτ r ))

q :

Ratio of b to a

r, θ:

Polar coordinates

r 0 :

Radius of periosteal surface

r i :

Radius of endosteal surface

rv1, rv2:

Radii of any two vascular networks between which a bone lamina area is bounded

R :

Dimensionless radius (=r/r0)

s :

Shear stress acting on the surface of the cell process

t :

Time

T :

Dimensionless frequency parameter (=wr 20 /c)

x, y:

Rectangular coordinates

X, Y:

Dimensionless rectangular coordinates (=x/L0, y/L0)

γ:

Angle between the neutral axis and the x axis

γ1:

Dimensionless parameter \(( = b/\sqrt {k_p } )\)

Δ:

Open space between the transverse fibers in the channel between the osteocytic process and the wall of the canaliculus

ɛ zz :

Normal strain along z direction for the Judex et al. (1997) experiment

τ:

Dimensionless time (=ct/r 20 )

τ r :

Characteristic time of relaxation of the fluid pore pressure

σ zz :

Superposition of the three normal stresses

w :

Loading frequency

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Acknowledgements

The authors thank Dr. Ted Gross and Dr. Stefan Judex for providing the experimental data reported in Gross et al. (1997) and Judex et al. (1997), Dr. John Currey for providing turkey bone sections and Dr. Stephen Doty for histological examination of turkey bone sections. This study has been supported by NIH grant AR48699 and by PSC-CUNY grants 64429 and 65734.

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Correspondence to Stephen C. Cowin.

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
figure 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)

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Mi, L.Y., Fritton, S.P., Basu, M. et al. Analysis of avian bone response to mechanical loading—Part One: Distribution of bone fluid shear stress induced by bending and axial loading. Biomech Model Mechanobiol 4, 118–131 (2005). https://doi.org/10.1007/s10237-004-0065-4

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Keywords

  • Pore Pressure
  • Bone Cell
  • Strain Gradient
  • Fluid Shear Stress
  • Periosteal Surface