theoretical investigation of the effect of bending loads

Bulletin of the JSME

Journal of Biomechanical Science and EngineeringVol.11, No.2, 2016

© 2016 The Japan Society of Mechanical Engineers

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J-STAGE Advance Publication date : 6 June, 2016Paper No.15-00663

[DOI: 10.1299/jbse.15-00663]

*Department of Biomechanics, Institute for Frontier Medical Sciences, Kyoto University

53 Kawahara-cho, Shogoin, Sakyo-ku, Kyoto 606-8507, Japan

E-mail: [email protected]

**Department of Mechanical Engineering, Graduate School of Engineering, Osaka Prefecture University

1-1 Gakuen-cho, Naka-ku, Sakai-shi, Osaka 599-8531, Japan

Abstract An individual trabecula in a cancellous bone is a porous material consisting of a lamellar bone matrix and interstitial fluid in a lacuno-canalicular porosity. The flow of interstitial fluid created by the application of mechanical load to a bone is considered to stimulate osteocytes for regulating bone remodeling, as well as enhance the transport of signaling molecules. The purpose of this study is to investigate, based on poroelastic theory, the flow-induced stimuli given to the osteocytes embedded in an individual lamellar trabecula. A single trabecula was modeled as a two-dimensional poroelastic slab composed of multiple layers subjected to cyclic uniaxial and bending loading. To consider the spatial variations in material properties due to lamellar structure of the trabecula, each layer was assumed to have a different value of permeability. By analytically solving the diffusion equation obtained from poroelasticity, we developed a solution for the interstitial fluid pressure in the lacuno-canalicular porosity within the single trabecula. Based on the solution obtained, we demonstrated the distribution of seepage velocity across the trabecula and qualitatively assessed the mechanical stimuli given to the osteocytes. The results suggested that osteocytes close to the trabecular surfaces are normally exposed to larger flow stimuli than those located around the center of the trabecula, regardless of the loading conditions and the spatial variations in permeability. On the other hand, osteocytes around the center of the trabecula, particularly with relatively large inner permeability, are stimulated by the fluid flow when the bending load is more dominant than the uniaxial load. Our theoretical approach might provide a better understanding of the effect of the spatial variations in bone material properties on the flow-mediated cellular mechanotransduction and signal transport for bone remodeling.

Key words : Interstitial fluid flow, Trabecula, Lamellar structure, Osteocyte, Bending load, Poroelasticity

1. Introduction

A trabecula is an anatomical and functional unit of a cancellous bone, forming a three-dimensional network structure. The arrangement of the trabeculae is continually remodeled according to the mechanical environment to maintain the structural strength. The adaptive change in the trabecular structure is accomplished through osteoclastic bone resorption and osteoblastic bone formation on the trabecular surfaces (Parfitt, 1994; Tsubota et al., 2006; Sato et al., 2015), and their coordinated activity is believed to be orchestrated by mechanosensitive osteocytes embedded in the bone matrix (Tatsumi et al., 2007; Bonewald, 2011; Nakashima et al; 2011).

A typical individual trabecula is a rod-like porous material composed of a calcified bone matrix and interstitial fluid in a lacuno-canalicular porosity. When the trabecula is mechanically loaded, the fluid pressure gradient induced by the deformation of the bone matrix creates a flow of interstitial fluid around osteocytes. During the process of bone remodeling, the interstitial fluid flow is considered to work as a mechanical cue initiating an osteocytic response (Weinbaum et al., 1994; Burger and Klein-Nulend, 1999; Kameo et al., 2014), as well as enhance the transport of signaling molecules (Fritton and Weinbaum et al., 2009; Price et al., 2011; Ciani et al., 2014). The bone matrix of the

Yoshitaka KAMEO*, Yoshihiro OOTAO** and Masayuki ISHIHARA**

Theoretical investigation of the effect of bending loads on the interstitial fluid flow in a poroelastic lamellar trabecula

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Received 23 November 2015

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Kameo, Ootao and Ishihara, Journal of Biomechanical Science and Engineering, Vol.11, No.2 (2016)

© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/jbse.15-00663]

trabecula has a lamellar structure comprising mineral crystals and collagen fibers (Martin et al., 1998); thus, the mechanical properties of the bone matrix are generally different for each lamella, depending on the history of bone formation. Such heterogeneity of material properties can significantly affect the intensity of the flow stimuli given to osteocytes.

In order to analyze the behavior of a fluid-saturated porous medium in response to mechanical loading, Biot’s poroelastic theory has been widely used (Biot, 1941, 1955). Poroelastic modeling of a bone tissue allows us to quantitatively evaluate the interstitial fluid flow in a lacuno-canalicular porosity (Cowin, 1999). In most of the analytical studies, the bone tissue was regarded as a homogeneous poroelastic material with isotropic properties (Zhang and Cowin, 1994; Zeng et al., 1994; Kameo et al., 2008, 2009) or transversely isotropic properties (Rémond and Naili, 2005). The spatial gradients of material properties were considered in the poroelastic finite element analysis of an osteon, the elementary unit of cortical bone (Rémond et al., 2008). In our previous work, we developed an analytical solution for fluid pressure in a model of single lamellar trabecula with spatial variations in permeability, which is an important property that measures the ability of a porous material to transmit fluid (Kameo et al., 2016). This study addresses the interstitial fluid flow triggered by cyclic uniaxial loading only, even though an individual trabecula within a living body is normally subjected to cyclic loading that includes both uniaxial and bending components because of walking or running activity.

The aim of the present study is to investigate the flow-induced stimuli given to the osteocytes embedded in an individual lamellar trabecula under cyclic uniaxial and bending loading, considering the effect of heterogeneity of material properties. A single trabecula is modeled as a two-dimensional poroelastic slab composed of multiple layers, each of which has a different value of permeability. By extending our previous work (Kameo et al., 2016), we present an analytical solution for the steady-state response of interstitial fluid pressure to cyclic uniaxial and bending loading. Particularly, we focus on seepage velocity, which characterizes the macroscopic averaged flow in a lacuno-canalicular porosity, because it is closely associated with the flow stimuli given to osteocytes. Based on the solution obtained, we demonstrate the distribution of seepage velocity across the lamellar trabecula under various loading conditions.

2. Theoretical developments 2.1 Poroelastic model of the trabecula

As a model of a single lamellar trabecula, we considered a poroelastic slab of width L, consisting of n layers as shown in Fig. 1. This idealized two-dimensional model represents the longitudinal cross section of an individual cylindrical trabecula. The physical quantities for the i-th layer are indicated by the subscript i. Each layer, with a width li, is a homogeneous and isotropic porous medium, and has the same material properties except for the permeability ki. For convenience of description, a local coordinate xi for the i-th layer was introduced in addition to the global coordinate system (x, y), as shown in Fig. 1.

Under plane stress condition, a cyclic uniaxial load N(t) = N0sint and a cyclic bending moment M(t) = M0sint per unit thickness are applied to the trabecular model at time t = 0 through rigid and impermeable plates placed at the top and bottom. We assume that there are no shear stresses throughout the poroelastic slab, i.e., xy = 0, and that the slab edges x = 0, L are stress-free. Considering that the mechanical load is supported much more by the bone matrix than by the interstitial fluid (Weinbaum et al., 1994; Kameo et al., 2008), the stress field of the trabecular model is given by

0 0 sin2kk yy

N M Lx t

L I

, (1)

where I is the moment of inertia per unit thickness of the cross section, satisfying I = L3/12, and the Einstein summation convention for repeated subscripts is adopted. By substituting Eq. (1) into the diffusion equation obtained from poroelastic theory (Detournay and Cheng, 1993; Wang, 2000; Coussy, 2004), the fluid pressure within the i-th layer is governed by the following partial differential equation in dimensionless form:

2

2



*2 * ** *

*2 *

1 2 1cos

1i i

ii

xp pk t

x t

. (2)

When the slab edges at x = 0, L are drained, the initial and boundary conditions for the dimensionless fluid pressure can be expressed as follows:

* *0; 0 1 ~it p i n (3)

* *1 10; 0x p (4)

* *

* * * * * * * 11 1 1* *

1

, 0; , 1 ~ 1i ii i i i i i i

i i

p px l x p p k k i n

x x

(5)

* * *; 0n n nx l p . (6)

In Eq. (5), we have considered the continuity of fluid pressure and fluid flux at the boundaries of the layers. In Eqs. (2)–(6), we introduce the following dimensionless values:

* * *02

0 0

2* * 0

0 0 0 0

, , ,

3 2

6, , ,

i ii i

i i ii i

x c t px t p

N LMBL LL I

l k c MLl k

L k c c LN

, (7)

where k0 and c0 are representative values of the permeability and diffusion coefficient, respectively, and B is the Skempton coefficient. The last two parameters, and , are called the dimensionless loading frequency and the axial-bending loading ratio, respectively. In the following section, we omit the asterisk (*) from the dimensionless values to simplify the notation.

Fig.1 Model of a single lamellar trabecula subjected to cyclic uniaxial and bending loading. The trabecula is idealized as a

two-dimensional poroelastic slab of width L, consisting of n layers. This model represents the longitudinal cross section of an

individual cylindrical trabecula.

2.2 Solution for fluid pressure

The initial and boundary value problem described by Eqs. (2)–(6) can be analytically solved with the aid of the Laplace transform technique. By taking the Laplace transform of Eq. (2) with respect to dimensionless time under the

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initial condition given in Eq. (3), the fundamental solution for fluid pressure in the Laplace transformed domain is obtained as follows:

2 2

1 2 1cosh sinh

1i i i i ii i

xs sp A x B x

k k s

, (8)

where the tilde (~) signifies the Laplace transform, and Ai and Bi are unknown coefficients, both of which should be determined to satisfy the boundary conditions given in Eqs. (4)–(6). Substitution of Eq. (8) into Eqs. (4)–(6) leads to the following system of linear equations expressed in matrix form:

1 1

1

2 2

2 1

2

0

, 1 ~ 2kl

n n

n n

A f

B

a k l ns

A f

B f

. (9)

where the non-zero elements of the coefficient matrix kla and the column vector kf are given by

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2 ,2 1 2 ,2 2 ,2 1

2 1,2 1 2 1,2

2 1,2 2 1

2 ,2 1 2 ,2

1

2 1 1

2

1

cosh , sinh , 1

sinh , cosh ,

1 ~ 1

cosh , sinh

1

12

1 ~ 111

i i i i i i i ii i

i i i i i i i ii i

i i i

n n n n n nn n

i i i

n

a

s sa l a l a

k k

s sa k s l a k s l

k k

a k s i n

s sa l a l

k k

f

f k k i n

f

. (10)

By applying Cramer’s rule to the linear system represented by Eq. (9), the solution for fluid pressure in the Laplace transformed domain is obtained as follows:

2 2

1 2 1cosh sinh

1i i i i ii i

xs sp A x B x

k ks

, (11)

where is the determinant of the coefficient matrix kla , and iA and iB are the determinants of the matrix formed by replacing the (2i 1)-th and 2i-th columns, respectively, of the matrix kla by the column vector kf . The inverse Laplace transform of Eq. (11), by making use of the residue theorem, provides the solution for fluid pressure in the form of the summation of the steady-state part steady

ip and transient part transip :

steady transi i ip p p , (12)

where the first term is the sum of the residues corresponding to the simple poles s i and the second term is the sum of the residues corresponding to the roots of 0 . In this study, we focus only on the steady-state behavior of

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the interstitial fluid pressure. The steady-state response is more dominant than the transient response because the transient stage decays within the first cycle of the cyclic loading (Manfredini et al., 1999; Kameo et al., 2008, 2009). The steady-state solution steady

ip can be explicitly expressed as follows:

steady

1 2 1cosh sinh

1Im

i i i i s is is ii i i t

i

s i

xi iA x B x

k kp e

, (13)

where Im denotes the imaginary part of the complex number. In order to characterize the mechanical stimuli given to osteocytes via the interstitial fluid flow, we investigate the

seepage velocity in the poroelastic trabecular model, which represents the average fluid velocity in a lacuno-canalicular porosity. According to Darcy’s law, the x-component of the dimensionless seepage velocity vi for the i-th layer in steady state is derived as follows:

steady

2sinh cosh

1Im

i i i i s is is ii i i ti

i i ii s i

i iA x B x

k kpv k i k e

x

. (14)

2.3 Conditions for numerical analysis

The mechanical aspects of a fluid-saturated poroelastic material are characterized by seven independent material properties: the intrinsic permeability k0, fluid viscosity , shear modulus G, drained Poisson’s ratio , solid bulk modulus Ks, fluid bulk modulus Kf, and porosity . The intrinsic permeability is a particularly significant factor that influences the interstitial fluid flow, and its value is sensitive to the variations in a lacuno-canalicular morphology. Although there is generally a positive correlation between permeability and porosity (Kameo et al., 2010), in this study, we considered the spatial variations in only permeability for simplicity.

The seven material constants were determined from the previous reports (Smit et al., 2002; Beno et al., 2006) as follows: k0 = 1.1 × 1021 (m2), = 1.0 × 103 (Pa·s), G = 5.94 (GPa), = 0.325, Ks = 17.66 (GPa), Kf = 2.3 (GPa), and = 0.05. Among these constants, the bone permeability k0 is difficult to determine because of a complex three-dimensional structure of the lacuno-canalicular porosity with several hundred nanometers in diameter. According to the previous extensive research, the reported value of permeability based on theoretical and experimental methods ranged from 1026 to 1018 m2 (Zhang et al. 1998; Smit et al. 2002; Beno et al. 2006; Oyen 2008; Gailani et al. 2009; Kameo et al. 2010). The representative permeability we set above is the approximate median value. By using these seven values, the other poroelastic constants, the diffusion coefficient c0 and Skempton coefficient B, used in Eq. (7), can be derived as follows: c0 = 3.9 × 108 (m2/s), and B = 0.35. Assuming that an individual trabecula in vivo has a typical diameter of L = 200 m and is subjected to cyclic loading at 1–20 Hz owing to locomotion and posture maintenance (Weinbaum et al., 1994), the dimensionless loading frequency , defined by Eq. (7), is estimated to be between 6.5 and 130; thus, we set = 10 or 100. The axial-bending loading ratio for a single trabecula depends strongly on the location in the body rather than a type of daily activities. For example, the value of is expected to be different in the femur and in the vertebra. Considering that the axial-bending loading ratio can vary from zero (pure uniaxial loading) to infinity (pure bending loading) within a living body, we set ranging from 0 to 99.

The single trabecular model was assumed to be composed of six layers, each of which has the same width. While the variations in permeability inside an individual trabecula are extremely difficult to quantify, a cylindrical trabecula in the equilibrium state of bone remodeling can be expected to have an axisymmetric lamellar structure. Therefore, we assumed that the permeability in a single trabecula is linearly distributed along the radial direction with the average of k0 = 1.1 × 1021 (m2). To investigate the effect of the lamellar structure on the interstitial fluid flow, we considered two

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fundamental trabecular models, both of which have a symmetrical distribution of permeability about the central axis (x = L/2); in Case 1, *

1k = 0.5, *2k = 1.0, *

3k = 1.5, *4k = 1.5, *

5k = 1.0, and *6k = 0.5; in Case 2, *

1k = 1.5, *2k =

1.0, *3k = 0.5, *

4k = 0.5, *5k = 1.0, and *

6k = 1.5. In Case 1, the interstitial fluid around the center of the trabecula can move more rapidly than the fluid close to the trabecular surfaces, while Case 2 represents the reverse condition.

3. Results

In order to understand the behavior of the interstitial fluid in response to external cyclic loading, we investigated the seepage velocity inside the single trabecular model when the axial-bending loading ratio = 1. In this condition, the stress yy at the surface x = 0 is maintained as zero, as readily calculated from Eqs. (1) and (7). Figures 2 and 3 show the seepage velocity distributions in the steady state along the x-direction when the values of are 10 and 100, respectively, plotted for eight equal length phase points in a period. In both figures, part (a) corresponds to the result for Case 1, and part (b) corresponds to the result for Case 2. As shown in Fig. 2, when = 10, the fluid flow was observed in the whole region in Case 1 (Fig. 2a), while the interstitial fluid was locally trapped around x* = 0.6 in Case 2 (Fig. 2b). Under high loading frequency ( = 100), as shown in Fig. 3, the seepage velocity close to the surface x* = 1 was extremely high. In contrast to that, the seepage velocity in the region other than in the neighborhood of x* = 1 was comparable in magnitude to that observed under low frequency ( = 10). A comparison of part (a) with part (b) in Figs. 2 and 3 shows that the maximum seepage velocity, which was observed at x* = 1, was higher in Case 2 than in Case 1, regardless of the loading frequency.

We studied the dependence of seepage velocity distribution on the axial-bending loading ratio . Figures 4 and 5 show the contour maps of the amplitude of seepage velocity along the x-direction with respect to change in , plotted against log10 (1 + ), when the values of are 10 and 100, respectively. Moreover, in these figures, parts (a) and (b) correspond to the results for Cases 1 and 2, respectively. The profile of seepage velocity along the x-direction can be qualitatively categorized into two classes according to whether < 1 (log10 (1 + ) < 0.3) or not. When the uniaxial load is more dominant than the bending load ( << 1), the interstitial fluid around the center of the trabecula can hardly move, and the region with almost no flow enlarges with the increase in , as shown in Fig. 5. On the other hand, when the bending load is relatively dominant ( >> 1), the interstitial fluid can flow through the trabecula not only near the surfaces, but also around the center. If is constant, the seepage velocity around the center is higher in Case 1 than in Case 2, while the seepage velocity near the surfaces is higher in Case 2 than in Case 1.

Fig. 2 Distribution of seepage velocity in the steady state along x-direction when = 10: (a) Case 1 and (b) Case 2.

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© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/jbse.15-00663] 7

Fig. 3 Distribution of seepage velocity in the steady state along x-direction when = 100: (a) Case 1 and (b) Case 2.

Fig. 4 Contour maps of the amplitude of seepage velocity along x-direction with respect to change in ,

plotted against log10 (1 + ), when = 10: (a) Case 1 and (b) Case 2.

Fig. 5 Contour maps of the amplitude of seepage velocity along x-direction with respect to change in ,

plotted against log10 (1 + ), when = 100: (a) Case 1 and (b) Case 2.

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4. Discussion

We theoretically analyzed the interstitial fluid flow in the lacuno-canalicular porosity within a single lamellar trabecula, which was idealized as a two-dimensional multi-layered poroelastic slab subjected to cyclic uniaxial and bending loading. In order to consider the effect of variations in permeability on the flow stimuli given to osteocytes, we investigated the seepage velocity inside two trabecular models with a symmetrical distribution of permeability: one has a relatively large inner permeability (Case 1) and the other has a relatively large outer permeability (Case 2). The results indicated that the seepage velocity close to the trabecular surfaces is larger in Case 2, while the seepage velocity around the center of the trabecula is larger in Case 1.

In the previous study, the effect of spatial gradients of bone mechanical properties on the interstitial fluid flow was investigated through finite element analysis of an osteon, which was modeled as a poroelastic hollow cylinder (Rémond et al., 2008). The simulation results showed that permeability variations had no significant consequences on fluid velocity when the order of permeability was set to k0 ~ 1018 (m2), which is the largest value proposed by Beno et al. (2006). We also developed an analytical solution for a poroelastic problem to clarify the effect of variations in permeability on fluid pressure and seepage velocity in a single trabecula under cyclic uniaxial loading (Kameo et al., 2016). We found that the interstitial fluid flow can be remarkably influenced by permeability variations when the representative permeability is k0 ~ 1021 (m2), i.e., when the dimensionless loading frequency is sufficiently large. In the present study, we extended our previous solution by considering a bending load in addition to a uniaxial load to understand the role of bending moment imposed on an individual trabecula in vivo.

The microstructure of a lacuno-canalicular system and osteocyte network within it are believed to be variable. An experimental study using early mice showed that the osteocyte network formation in both cortical bone and cancellous bone was affected by mechanical loading (Sugawara et al., 2013). It was also reported that estrogen deficiency, caused by an ovariectomy, enhanced solute transport via interstitial fluid flow in a lacuno-canalicular porosity (Ciani et al., 2014). Furthermore, several experimental findings have suggested that osteocytes have an ability to remodel the perilacunar and pericanalicular matrix (Qing and Bonewald, 2009). These studies imply that an individual trabecula has a heterogeneous distribution of permeability owing to the morphological change in the lacuno-canalicular system, leading to variations in the flow stimuli given to osteocytes buried in the bone matrix. In this study, we focused on the seepage velocity as an indicator of the flow stimuli for the osteocytes. Investigating the distribution of the seepage velocity in a single trabecula enables us to qualitatively assess the flow-mediated cellular mechanotransduction and signal transport. The seepage flow with high velocity can activate osteocytes to produce signaling molecules and enhance their transport for bone remodeling. Our theoretical analysis indicated the possibility that the osteocytes in the neighborhood of the trabecular surfaces are normally exposed to larger flow stimuli than those located around the center of the trabecula. In particular, a single trabecula with relatively large outer permeability (Case 2) has the potential to produce a significant flow close to the surfaces. On the other hand, osteocytes around the center of the trabecula, particularly with relatively large inner permeability (Case 1), are stimulated by the fluid flow when the bending load is more dominant than the uniaxial load. These results imply that bone tissues can regulate the intensity of mechanical stimuli given to osteocytes and the transport of signaling molecules by altering the characteristics of lacuno-canalicular system according to the surrounding mechanical and biochemical environments.

One of the limitations of the present study lies in the idealization of a single trabecular morphology. We used a two-dimensional poroelastic slab for modeling a three-dimensional rod-like trabecula, for the sake of simplicity. Poroelastic theory shows that when a uniaxial compressive load with an intensity of 0 per unit cross-sectional area is suddenly applied to a two-dimensional poroelastic slab under plane stress condition, the instantaneous response is undrained everywhere in the slab and the fluid pressure can be derived as B0/3. This is equal to the instantaneous undrained fluid pressure in an axisymmetric poroelastic cylinder under the same loading condition. Considering that a poroelastic material subjected to high-frequency cyclic loading exhibits almost undrained behavior, these results suggest that the two-dimensional slab model is a good approximation for the three-dimensional cylindrical model even in the case of cyclic loading condition. In addition to using the slab model, we ignored the anisotropy of lacuno-canaliculr morphology and material properties of the bone matrix. For a more exact analysis of seepage velocity in the lacuno-canalicular porosity, it will be indispensable to construct a realistic trabecular model based on the assessment of microstructure. Because the seepage velocity means the average fluid velocity in a lacunao-canalicular porosity, this macroscopic mechanical property does not allow us to quantify the local fluid velocity in the immediate

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vicinity of each osteocyte. For quantitative evaluation of mechanical stimuli at the cellular scale, it is also essential to consider the microstructural features of the canaliculi, such as tortuosity and irregular wall surface (Kamioka et al., 2012). However, the results obtained in this study can provide novel information on the fundamental behavior of the interstitial fluid inside a lamellar trabecula under complex mechanical loading that includes both uniaxial and bending components. Our theoretical approach might provide a better understanding of the effect of the spatial variations in bone material properties on the flow-mediated cellular mechanotransduction and signal transport for bone remodeling.

Acknowledgements

This study was partially supported by a Grant-in-Aid for Young Scientists (B) (25820011) from the Japan Society for the Promotion of Science (JSPS).

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theoretical investigation of the effect of bending loads

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