- Email: [email protected]
Fluid shear stress in trabecular bone marrow due to low-magnitude high-frequency vibration
Journal of Biomechanics 45 (2012) 2222–2229
Contents lists available at SciVerse ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Fluid shear stress in trabecular bone marrow due to low-magnitude high-frequency vibration Thomas R. Coughlin, Glen L. Niebur n Tissue Mechanics Laboratory, Bioengineering Graduate Program, University of Notre Dame, IN 46556, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Accepted 12 June 2012
Low-magnitude high-frequency (LMHF) loading has recently received attention for its anabolic effect on bone. The mechanism of transmission of the anabolic signal is not fully understood, but evidence indicates that it is not dependent on bone matrix strain. One possible source of signaling is mechanostimulation of the cells in the bone marrow. We hypothesized that the magnitude of the fluid shear stress in the marrow during LMHF loading is in the mechanostimulatory range. As such, the goal of this study was to determine the range of shear stress in the marrow during LMHF vibration. The shear stress was estimated from computational models, and its dependence on bone density, architecture, permeability, marrow viscosity, vibration amplitude and vibration frequency were examined. Three-dimensional finite element models of five trabecular bone samples from different anatomic sites were constructed, and a sinusoidal velocity profile was applied to the models. In human bone models during axial vibration at an amplitude of 1 g, more than 75% of the marrow experienced shear stress greater than 0.5 Pa. In comparison, in vitro studies indicate that fluid induced shear stress in the range of 0.5 to 2.0 Pa is anabolic to a variety of cells in the marrow. Shear stress at the bonemarrow interface was as high as 5.0 Pa. Thus, osteoblasts and bone lining cells that are thought to reside on the endosteal surfaces may experience very high shear stress during LMHF loading. However, a more complete understanding of the location of the various cell populations in the marrow is needed to quantify the effects on specific cell types. This study suggests the shear stress within bone marrow in real trabecular architecture during LMHF vibration could provide the mechanical signal to marrow cells that leads to bone anabolism. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Bone Bone marrow Computational fluid dynamics Mechanobiology Shear stress Low magnitude high frequency loading
1. Introduction Low-magnitude high-frequency (LMHF) stimulation is anabolic to bone in animals (Rubin et al., 2002), but the mechanical signal and its mechanism of transmission have not been established. Rats subjected to 90 Hz vibration developed greater trabecular bone volume and thicker trabeculae than rats subjected to vibration at 45 Hz (Judex et al., 2007), even though the 90 Hz vibration induced lower strain in the bone. Trabecular bone volume also increases in non-weight bearing skeletal segments when subjected to LMHF loading (Garman et al., 2007). As such, matrix strain is not necessary to achieve the anabolic signal from LMHF stimulation, in contrast to exercise or unloading studies where strain is likely the dominant signal. In sheep, LMHF loading resulted in a 32% increase in trabecular bone volume, a decrease in trabecular spacing and an increase in trabecular number
n Corresponding author at: 147 Multidisciplinary Research Building Notre Dame, IN 46556, USA. Tel.: þ 1 574 631 3327; fax: þ1 574 631 21744. E-mail address: [email protected] (G.L. Niebur).
0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2012.06.020
(Rubin et al., 2001b). Together, the concentration of the anabolic effect of LMHF vibration in marrow rich trabecular bone, the ability of mechanical signals to influence bone marrow derived cells (Luu et al., 2009), and the lack of dependence on matrix strain, suggest that marrow cells may sense the mechanical signal from LMHF loading. Bone marrow is a cellular soft tissue found in the endosteal compartment of bone (Fig. 1). It is highly vascularized and forms the niche for a variety of cells, including mesenchymal and hematopoietic stem cells (MSCs and HSCs) (Gurkan and Akkus, 2008; Weiss, 1976). The cellular fates of MSCs and HSCs influence bone remodeling as osteoclasts are derived from the HSC lineage (Owen, 1980) and MSCs are multipotent cells that differentiate to connective tissue lines. Stem cell fate also affects bone marrow composition (Berg et al., 1998). For example, changes in fat content alter the bone marrow mechanical environment (Bryant et al., 1989). During LMHF loading, marrow cells could be subjected to inertially induced motion relative to the much more rigid bone causing shear stress within the fluid-like marrow. Shear stress induced by this motion during LMHF loading may, in turn, affect
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
2223
and (3) quantified the dependence of shear stress on bone density, architecture, permeability, marrow viscosity, vibration amplitude and vibration frequency.
2. Methods
Fig. 1. A histological section of bone marrow from a porcine cervical vertebra. Hematoxylin and eosin stain: (A) adipocytes, (B) trabecular bone (bar¼ 100 mm).
the marrow cell population. Shear stress in the range of 0.5 to 2.0 Pa increases MSC proliferation (Castillo and Jacobs, 2010), decreases adipogenesis in MSCs (Rubin et al., 2007), upregulates nitric oxide (NO) and prostaglandin release in osteoblasts (Johnson et al., 1996; Smalt et al., 1997), upregulates NO release in preosteoclasts (McAllister, 2000), increases prostaglandin production (Klein-Nulend et al., 1996), and upregulates intracellular calcium and gene expression in osteocytes (Li et al., 2012) and osteoblasts (Nauman et al., 2001). In the case of both steady and oscillatory flow, 1 Pa shear stress enhances the osteogenic phenotype (Arnsdorf et al., 2010; Case et al., 2011). The shear stress in the bone marrow during LMHF loading was previously estimated analytically using continuum level mixture theory (Dickerson et al., 2007). The resultant shear stress ranged from 0.5 to 5.0 Pa, increasing with volume fraction, vibration frequency and marrow viscosity. The analysis also predicted an increase in blood flow during LMHF loading. The shear stress in the marrow was determined from the Darcy permeability, estimated from drag theory. As such, it was not possible to determine detailed shear stress profiles in the marrow volume. Quantification of the shear stress throughout the bone marrow within real trabecular architectures during LMHF loading would provide insight into the potential for mechanical regulation of marrow cells. Moreover, the dependence of shear stress levels on trabecular architecture may elucidate site sensitivity of bone anabolism due to LMHF loading. While direct measurement of shear stress within the trabecular pore space is not possible with present technology, computational models of real morphologies can be used to simulate the loading and motion of the marrow during LMHF vibration. These results would ideally be used in conjunction with experiments designed to validate or refute the model assumptions. The goal of this study was to determine whether the magnitude of shear stress in bone marrow during LMHF loading of trabecular bone at amplitudes from 0.3 to 1.0 g and frequencies of 10 to 90 Hz is in the range that induces a physiological response from cells. Specifically, we (1) used finite element models to simulate the dynamics of bone marrow in the trabecular pore space during sinusoidal motion of the bone matrix, (2) quantified the statistical distribution of shear stress,
Computational fluid dynamics models were used to study shear stress in bone marrow with the assumption that marrow behaves as a homogeneous fluid. This approach does not capture specific cell–cell interactions, but rather homogenizes the behavior of the marrow as a viscous fluid. Although this approach has limitations, it is a common assumption (Bryant et al., 1989; Dickerson et al., 2007; Kafka, 1983; Ochoa et al., 1991). Five trabecular bone samples from five anatomic sites were studied to provide a range of architecture (Table 1). One sample was taken from a human lumbar spine, one each from the greater trochanter and neck of a human femur (obtained with informed consent through NDRI), and one from each of the medial and lateral femoral condyles of an ovine distal femur (obtained from another study under IACUC approval). The samples were prepared with their principal mechanical axes aligned with the scanning axis ¨ (Wang et al., 2004), and imaged by micro-CT (Scanco mCT-80, Bruttisellen, Switzerland) at 20 mm resolution. A 4 4 4.5 mm3 region was selected, Gaussian filtered and resampled by cubic interpolation to 35 mm resolution (Visualization Toolkit, Kitware, Clifton Park, NY). An earlier convergence study indicated that this resolution provided sufficiently small elements for mesh convergence in a steady fluid flow solution (Niebur et al., 2010). A 30 mm layer of fluid was added around the entire bone sample to allow continuity of flow from pores on the edges. The fluid flow was constrained on the faces perpendicular to the vibration direction to prevent flow across the boundaries, and there was no viscous drag for flow parallel to the wall (Fig. 2c). The marrow regions were discretized into tetrahedra using a marching cubes algorithm in VTK (Kitware, Clifton Park, NY), which were smoothed using Laplacian smoothing with heuristic checks to avoid inverted elements, and converted into finite element meshes (Fig. 2). The fluid elements were assigned incompressible Newtonian fluid properties with a density of 0.9 g/cm3 and viscosity of 400 mPa s (Bryant et al., 1989). The bone-marrow interface was assumed to be rigid, with a no-slip interface. For the baseline models, all nodes on the bone surface were assigned a 10 Hz sinusoidal velocity profile with a peak acceleration of 1 g. The velocity profile was first applied parallel to the principal fabric direction of the bone and then transverse to the principal direction to assess the effects of pore morphology relative to the direction of motion. The same velocity profile was imposed on the outer surfaces of the fluid volume lying parallel to the velocity, while the perpendicular faces had constant pressure boundary conditions to allow free flow. A transient, dynamic solution was performed in ADINA-Fs (Watertown, MA). The solution was carried out starting from 0 velocity and proceeded for at least two cycles. Womersley numbers (Wo), calculated at a frequency of 10 Hz, fluid density of 0.9 g/cm3, viscosity of 400 mPa s and with trabecular spacing as the characteristic dimension, were less than 1.0, which indicates that viscous effects – rather than inertial effects – dominate the flow. The Reynolds numbers (Re) were much less than 1.0, indicating that the assumption of laminar flow is reasonable (Table 2). A 3.6 3.6 4.4 mm3 sub-region in the center of the sample was analyzed to avoid artifacts from the edges (Fig. 2). Because the shear stress distribution was non-normal, the median, 25th, and 75th percentile values of the nodal results were considered. Data were analyzed at the end of the first half cycle, where the applied acceleration was a maximum, and the results were similar to those at subsequent acceleration peaks (Fig. 3). The permeability along the two vibration directions was calculated based on Darcy’s law (Eq. 1), where k is permeability, Q is flow, Dx is thickness, and DP is the pressure difference. A steady state flow problem was solved on the same mesh with a constant pressure differential of 13.9 Pa/mm. The volumetric flow rate was
Table 1 Sample information for the three human (greater trochanter, femoral neck and lumbar vertebra) and two ovine (femoral condyles) samples. Sample
Volume fraction (%)
Human greater trochanter Human lumbar vertebra Human femoral neck Ovine femoral condyle 1 Ovine femoral condyle 2
12.2 12.9 14.0 24.1 31.3
SMI
Trabecular spacing (mm)
1.34 0.66 2.02 1.62 0.48 0.15
0.88 0.76 0.46 0.36
Degree of anisotropy (DA) 1.8953 1.1369 1.7744 2.4311 2.0373
2224
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Fig. 2. (a) A uCT reconstruction of the human lumbar vertebral trabecular bone sample (20 mm resolution). (b) The marrow region was converted to a tetrahedral FE mesh. Meshes had at least 10 elements across pores. (c) A thin fluid layer was added beyond the edges of the trabecular bone sample. Fluid contacting the bone surfaces had an imposed sinusoidal velocity profile in one direction, the control volume faces parallel to the vibration (solid lines) allowed no flow perpendicular to the surface, but free flow parallel to the surface. Faces perpendicular to the flow (dashed lines) allowed free flow subject to the constraint that the same volume that exited at one face must enter through the opposite face (conservation of mass). The white box indicates the sub-region where the shear stress was analyzed. (d) Streamlines and velocity contours at the peak velocity of the marrow relative to the bone. normalized by the pressure drop and fluid viscosity to find the permeability. k¼
Q mDx DP
ð1Þ
The effects of frequency and viscosity were investigated parametrically in the lumbar vertebra sample. Bone marrow mechanical properties remain poorly characterized, and depend on composition (Bryant et al., 1989), anatomic site (Liney et al., 2007), age (Gurkan and Akkus, 2008; Moore and Dawson, 1990; Ricci et al., 1990) and bone density (Griffith et al., 2005; Yeung et al., 2005). Models were solved with viscosities of 50, 100, 200 and 400 mPa s to investigate the effects of this variability at both 10 and 30 Hz vibration. To examine the effect of frequency on shear stress, 10, 20, 30, 60, and 90 Hz sinusoidal vibrations were applied with a fluid viscosity of 400 mPa s. Finally, the shear stress was studied at amplitudes of 0.3 g, 0.6 g and 1.0 g in the lumbar vertebra and femoral neck samples to assess the effects of amplitude.
3. Results The magnitude of shear stress in the marrow was on the order of Pascals in all samples. The shear stress was highest just past the peak acceleration, and was near zero at the peak velocity (Fig. 3). The statistical distribution of shear stress was non-normal and skewed toward lower stress (Fig. 4). The majority of the marrow in the human trabecular bone samples (trochanter, lumbar vertebra and femoral neck) experienced shear stresses above 0.8 Pa at peak acceleration during axial vibration at 10 Hz and 1 g amplitude, and 95% of the marrow experienced shear stress above 0.5 Pa. Increasing
Table 2 The permeabilities for flow along the axial and transverse vibration directions, and dimensionless characteristic numbers of the flows. The Womersley numbers (Wo) were calculated for 10 Hz vibration along the principal orientation, and Reynolds numbers (Re) were based on the peak velocity of the endosteal surface. Both were calculated for a viscosity of 400 mPa s using trabecular spacing as the characteristic length. Wo is an indicator of the degree to which an oscillatory flow is dominated by inertial (Wo4 1) or viscous (Wo o 1) effects. Re is an indicator of turbulent (Re 4400) vs. laminar (Reo 400) flow. Sample
Axial permeability (mm2)
Transverse permeability (mm2)
Wo (–)
Re (–)
Human greater trochanter Human lumbar vertebra Human femoral neck Ovine femoral condyle 1 Ovine femoral condyle 2
0.00805
0.00388
0.25
0.23
0.00633
0.00669
0.33
0.31
0.00814
0.00340
0.29
0.27
0.00295
0.00338
0.17
0.16
0.00216
0.00040
0.14
0.13
the amplitude increased shear stress linearly in the samples tested (Fig. 5). The volume of marrow displaced across the ends of the bone was approximately 0.1% of the total marrow volume (Table 3).
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Shear stress, and the range of shear stress, increased with increasing permeability independent of vibration direction (Fig. 6a). The shear stress was negatively correlated with volume fraction (BV/ TV), and was lower for vibration transverse to the principal pore direction (Fig. 6b). Shear stress was positively correlated with
2225
trabecular spacing (Tb.Sp.) for both axial (po0.025, R2 ¼0.81) and transverse vibration (po0.035, R2 ¼0.75). Similarly, shear stress was marginally correlated with the structural model index (SMI), which describes the rod-like (SMI¼3) or plate-like (SMI¼0) nature of the bone (po0.055). However, combined with BV/TV neither of these parameters was significant when using multiple regression (p40.2). The effect of frequency on shear stress depended on viscosity. The shear stress increased with increasing viscosity at 30 Hz, but was insensitive to viscosity at 10 Hz (Fig. 7). Increasing the frequency from 10 to 30 Hz resulted in greater shear stress extending farther from the bone walls (Fig. 7), consistent with increasing viscous effects as Wo decreases. For a fixed viscosity of 400 mPa s, increasing the frequency from 10 to 90 Hz narrowed the range of shear stress, but had a minimal effect on the 25th percentile value (Fig. 8).
4. Discussion
Fig. 3. The 25th percentile, median and 75th percentile shear stress in the human femoral neck sample for the first 2.25 cycles of vibration at 10 Hz. The time is normalized to the period, T, of vibration. More than 75% of the marrow in this sample experienced shear stress above 0.5 Pa for over 60% of the cycle.
This study examined the potential for low-magnitude highfrequency dynamic loading of bone to induce mechanostimulatory stress levels within the marrow. When subjected to 1 g vibrations, the shear stress exceeded 0.5 Pa in 75% of the marrow for more than 65% of each cycle in the human bone samples, although these are overestimates of the actual shear stress due to the idealized boundary conditions applied. Shear stress above 0.5 Pa is mechanostimulatory to osteoblasts (Kapur et al., 2002; Klein-Nulend et al., 1996; Nauman et al., 2001), osteoclasts (McAllister, 2000) and MSCs (Castillo and Jacobs, 2010) in vitro. The shear stress increased with trabecular permeability and porosity. In general, more porous structures also exhibited elevated shear stress in a greater fraction of the marrow
Axial
O Transverse Fig. 4. (a) Distributions of shear stresses in the human femoral neck sample during axial and transverse vibration were non-normal. (b) Shear stress above the 25th percentile (0.75 Pa), extended well away from the bone-marrow interface. (c) The 75th percentile shear stress (1.5 Pa) was confined to the bone-marrow interface (white space represents bone tissue, and blue regions are at or below the respective threshold). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2226
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
space. The larger pores contained a greater marrow mass accelerating relative to the bone, resulting in greater velocity gradients and in turn larger shear stress. Thus, if the shear stress induced by inertial motion of the marrow is the mechanobiological signal in LMHF loading, then it would be most effective in locations with low bone density. The principal strength of this study was the quantification of marrow shear stress throughout real trabecular pore geometries. Current technology does not allow direct quantification of the bone marrow mechanical behavior. However, computational models can provide a reasonable estimate. To compensate for the wide range of reported properties for bone marrow, we used a range of viscosities to parametrically quantify the relative affects of each during LMHF vibration. Computation of the shear stress within the trabecular compartment in a variety of real architectures from multiple anatomic sites and species provided an estimate of how this signal may vary in different architectures, which was quantifiable by differences in the trabecular permeability. Some limitations of this study must be considered when interpreting the data. The primary limitation is the simplifying assumption that bone marrow is a homogeneous fluid. While bone marrow behaves in a fluid-like manner, it is heterogeneous, and the viscosity arises from phenomena such as cell adhesion that may cause it to behave as a Brinkman or Bingham like fluid. Moreover, red marrow in the pore space contains capillaries and other structural elements that affect the mechanical behavior.
Thus, capturing the true behavior would require multi-scale modeling that incorporates cell–cell adhesion, cell-fluid interaction and multiple cell geometries, rather than a homogenized viscosity (Forgacs, 1995; Preziosi et al., 2010). The marrow was
Fig. 5. The 25th, 50th and 75th percentile of shear stress in the marrow space for a viscosity of 400 mPa s. The shear stress increased with increasing amplitude in both samples. The bottom line is the 25th percentile stresses and the top is the 75th percentile stress.
Fig. 6. (a) The 25th, median and 75th percentile shear stress increased with increasing permeability along the vibration axis (p o0.0001, linear regression). (b) The 25th percentile shear stress during axial and transverse vibration at 10 Hz decreased with increasing volume fraction (p o 0.01, linear regression).
Table 3 The relative flow rate, percentage of the marrow volume that exited the bone and the median shear stress for axial vibration in the human lumbar spine and femoral neck samples at 30 Hz. The displaced volume of marrow is the integral of the volumetric flow rate over 1/2 cycle. The displaced volumes were on the order of 0.05 ml while the marrow volumes were 62 ml and 52 ml in the lumbar spine and femoral neck, respectively. Lumbar spine
Relative volumetric flow rate (ml/s) Volume of marrow entering and exiting the bone (ml) Fraction of marrow volume entering and exiting the bone (%) Median shear stress (Pa)
Femoral neck
0.3 g
0.6 g
1.0 g
0.3 g
0.6 g
1.0 g
1.38 0.015 0.024 0.31
2.76 0.029 0.047 0.62
4.59 0.049 0.079 1.04
1.25 0.013 0.025 0.33
2.5 0.027 0.051 0.68
4.04 0.043 0.082 1.10
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
30 Hz
50 mPas
10 Hz
2227
τ (Pa) Re = 0.82, Wo = 1.61
2
200 mPas
Re = 2.46, Wo = 0.93
1 Fig. 8. The bars show the 25th, 50th and 75th percentile of shear stress in the marrow space for a viscosity of 400 mPa s. The shear stress was in a narrower range at the higher frequencies, but the effect was minor.
Re = 0.21, Wo = 0.81
Re = 0.31, Wo = 0.33
Re = 0.10, Wo = 0.57
0
400 mPas
Re = 0.62, Wo = 0.47
Fig. 7. Shear stress in the marrow of a human lumbar vertebra sample at 10 and 30 Hz frequency loading and assumed viscosities of 50, 200 and 400 mPa s (white regions are bone and blue regions are below 0.5 Pa, the reported mechanostimulatory threshold). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
idealized as incompressible. The measured pressure in the marrow of a proximal femur at 2 BW of load was only 15 kPa (Downey et al., 1988), while the bulk modulus is over 300 MPa (Arramon and Cowin, 1997; Lim and Hong, 2000), indicating the volume change is less than 0.005%. The assumption of incompressibility required boundary conditions that allowed flow into and out of the bone, as no motion of the fluid would occur otherwise. While the marrow is actually entrapped and does not cross endosteal boundaries, the region of bone that was modeled was only a fraction of the trabecular bone volume in a whole bone, and the boundary conditions assumed the marrow flowed into or from the adjacent trabecular bone. The volume of marrow flowing across the boundary was less than 0.1% of the total marrow volume, which is consistent with the flow that has been assumed to occur in poroelastic models of trabecular bone (Arramon and Cowin, 1997; Cowin, 1999; Whyne et al., 2003). However, the boundary conditions applied do not account for the resistance to flow imposed by the ends of the bone and thus, the idealized boundary conditions resulted in an overestimation of the shear stress. Finally, we ignored the deformation of the bone that would
occur during loading. During LMHF loading, the applied velocity is much lower than the wave speed of bone, such that the induced bone strain would be small. However, activities of daily living may also induce marrow motion through deformation of the pores in the trabecular bone. Despite these limitations, the results provide important insight into the potential mechanobiology of LMHF loading and a reasonable first approximation of the system. MSC differentiation in a multicellular structure is dependent on the relative location of cells and corresponding stress. When MSCs seeded in a collagen matrix were subjected to loading, those experiencing low stress in the center of the construct differentiated into adipocytes, while those experiencing higher stresses at the edges of the matrix differentiated into osteoblasts (Ruiz and Chen, 2008). Assuming that the mechanostimulatory stress levels found in 2-D in vitro experiments apply to the 3-D in situ case studied here, the models predict that over 75% and as much as 95% of the marrow is subjected to stresses above a mechanostimulatory threshold in human trabecular bone when 1 g acceleration is applied. Only those cells near the center of the pore space are shielded. Cells at the bone-marrow interface, which are believed to be primarily quiescent osteoblasts and other bone lining cells (Castillo and Jacobs, 2010) would be subjected to the greatest stimulation. However, a more complete understanding of the spatial composition of marrow is needed to assign shear stress magnitudes to specific cell types. For example, at a cellular level the larger adipocytes (Fig. 1) may shield other cells from these stress levels. Future experiments could incorporate subject specific models to determine how the mechanical signals transmitted may differ in varying architectures and marrow compositions. For example, the anabolic response to vibration was dependent on age in mice, which suggests it may depend on marrow fat content (Lynch et al., 2010). LMHF vibration applied parallel to the principal pore direction results in higher shear stress than when applied perpendicular to the pores. In the case of transverse vibration, plates perpendicular to the fluid motion restrict the flow and there are fewer faces parallel to the flow to generate viscous drag. The permeability is dependent on both volume fraction and architecture. The relationships are complex, and depend on whether the architecture restricts flow by primarily viscous or inertial mechanisms
2228
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
(Nauman and Arroman, 2001). Consistent with this, the shear stress magnitudes and ranges were linearly correlated with architectural parameters, but no simple multiple regressions were found. The impact of this result is that the effects of LMHF vibration may be lower in sites where the trabecular fabric is not aligned with the vibration direction. In humans, vibrations applied in the inferior-superior direction would be misaligned with the principal trabecular orientation in the greater trochanter and femoral neck, resulting in lower shear. In contrast, trabeculae in the lumbar spine are aligned with the direction of vibration and the permeability is higher, which would result in higher shear stress over a greater portion of the marrow. Consistent with this result, postmenopausal women subjected to LMHF vibration for 1 year had a greater increase in BMD in the lumbar spine than in the greater trochanter or femoral neck (Rubin et al., 2003). However, it should be acknowledged that the experimental result could also be the result of biological differences in the cell populations at these sites. The weak dependence of shear stress on viscosity may appear to contradict a continuum poroelastic model, where the shear stress in the fluid depended strongly on viscosity (Dickerson et al., 2007). In poroelastic models flow is pressure driven, compared to the inertial flows in these models. Moreover, the shear stress in poroelasticity arises from a fluid drag term, which depends linearly on permeability and viscosity. In this study, higher marrow viscosity did increase the shear stress at the bone-marrow interface, but the effect was smaller in the remainder of the trabecular compartment. This reflects the transmission of shear stress from the wall across the open pore space, and the fact that shear stress goes to zero at some point in a fluid confined between two walls. The differences in the effect of permeability are likely due to the fact that the flow was driven by matrix strain in the previous study (Dickerson et al., 2007), compared to an inertially driven flow here. The lowest median shear stress magnitude was found at 30 Hz, with a slight increase up to 90 Hz frequency. Although the differences were small, this result is consistent with animal models, where bone growth due to LMHF loading was greater for 90 Hz vibration than at 45 Hz (Judex et al., 2007). In LMHF vibration, the marrow forms a harmonic oscillator within the pore space, and the response will likely differ between anatomic sites or species where the pore sizes and shapes – and hence the permeability – differ. Alternatively, the anabolic effect of increased frequency may be due to a threshold effect on shear stress, as higher frequencies result in more acceleration peaks and thus more times that the marrow shear stress reaches a mechanostimulatory threshold, or the frequency itself may be the factor (Duncan and Turner, 1995; Jacobs et al., 1998). In mice, there was not a consistent dose response to increased vibration amplitude (Christiansen and Silva, 2006). In contrast, our results indicate a monotonic increase in the shear stress with increasing amplitude, consistent with the linear relationship between force and the magnitude of acceleration of a fixed mass. Furthermore, our results indicate that the median shear stress is below 0.5 Pa for an amplitude of 0.3 g, which has previously been sufficient for an anabolic response (Rubin et al., 2003, 2001a). These results open the possibility that the effects of vibration are not related to shear stress in the marrow, or that the threshold for the shear stress is much lower than 0.5 Pa. Alternatively, 20% of the marrow in the lumbar spine model experienced shear stresses above 0.5 Pa during 0.3 g amplitude vibration, indicating that the anabolic signal may only be sensed by cells at the interface. This study provided an initial estimate of the range of shear stress that occurs in bone marrow during LMHF loading. Our data can provide input to fluid flow studies (Glossop and Cartmell, 2009; Kapur et al., 2002; Li et al., 2004; Nauman et al., 2001; Riddle et al., 2006; Sen et al., 2011), which quantify bone marrow
cellular response to shear stress. Applying shear stress in the marrow could provide improved insight into how cells of specific lineages function in vivo when subject to LMHF loading. Cell scale models that incorporate realistic cellular interactions could provide further insight by accounting for heterogeneity and spatial distribution of specific cell lineages in the marrow.
Conflict of interest statement The authors have no potential conflicts of interest.
Acknowledgements Science Foundation Ireland, E.T.S. Walton Program 07/W.I./ B1806, U.S. National Science Foundation CMMI-1100207. Human bone samples were obtained from the National Disease Research Interchange (NDRI). References Arnsdorf, E.J., Tummala, P., Castillo, A.B., Zhang, F., Jacobs, C.R., 2010. The epigenetic mechanism of mechanically induced osteogenic differentiation. Journal of Biomechanics 43, 2881–2886. Arramon, Y.P., Cowin, S.C., 1997. Hydraulic stiffening of cancellous bone. Forma 12 (3,4), 209–221. Berg, B.C.V., Malghem, J., Lecouvet, F.E., Maldague, B., 1998. Magnetic resonance imaging of the normal bone marrow. Skeletal Radiology 27, 471–483. Bryant, J., David, T., Gaskell, P.H., King, S., Lond, G., 1989. Rheology of bovine bone marrow. Proceedings of the Institution of Mechanical Engineers, Part H. Case, N., Sen, B., Thomas, J.A., Styner, M., Xie, Z., Jacobs, C.R., Rubin, J., 2011. S.a.o.f.f.p.a.s.o.p.C.T.I.-P., 2011. Steady and oscillatory fluid flows produce a similar osteogenic phenotype. Calcified Tissue International 88 (3), 189–197. Castillo, A.B., Jacobs, C.R., 2010. Mesenchymal stem cell mechanobiology. Current Osteoporosis Reports 8 (2), 98–104. Christiansen, B.A., Silva, M.J., 2006. The effect of varying magnitudes of wholebody vibration on several skeletal sites in mice. Annals of Biomedical Engineering 34 (7), 1149–1156. Cowin, S.C., 1999. Bone poroelasticity. Journal of Biomechanics 32 (3), 217–238. Dickerson, D.A., Sander, E.A., Nauman, E.A., 2007. Modeling the mechanical consequences of vibratory loading in the vertebral body: microscale effects. Biomechanics and Modeling in Mechanobiology 7 (3), 191–202. Downey, D.J., Simkin, P.A., Taggart, R., 1988. The effect of compressive loading on intraosseous pressure in the femoral head in vitro. The Journal of Bone and Joint Surgery American 70 (6), 871–877. Duncan, R.L., Turner, C.H., 1995. Mechanotransduction and the functional response of bone to mechanical strain. Calcified Tissue International 57, 344–358. Forgacs, G., 1995. Biological specificity and measurable physical properties of cell surface receptors and their possible role in signal transduction through the cytoskeleton. Biochemistry and Cell Biology 73, 317–326. Garman, R., Gaudette, G., Donahue, L.R., Rubin, C., Judex, S., 2007. Low-level accelerations applied in the absence of weight bearing can enhance trabecular bone formation. Journal of Orthopaedic Research 25, 732–740. Glossop, J.R., Cartmell, S.H., 2009. Effect of fluid flow-induced shear stress on human mesenchymal stem cells: differential gene expression of il1b and map3k8 in mapk signaling. Gene Expression Patterns 9 (5), 381–388. Griffith, J.F., Yeung, D.K.W., Antonio, G.E., Lee, F.K.H., Hong, A.W.L., Wong, S.Y.S., Lau, E.M.C., Leung, P.C., 2005. Vertebral bone mineral density, marrow perfusion, and fat content in healthy men and men with osteoporosis: dynamic contrast-enhanced mr imaging and mr spectroscopy. Radiology 236, 945–951. Gurkan, U.A., Akkus, O., 2008. The mechanical environment of bone marrow: a review. Annals of Biomedical Engineering 36 (12), 1978–1991. Jacobs, C.R., Yellowley, C.E., Davis, B.R., Zhou, Z., Cimbala, J.M., Donahue, H.J., 1998. Differential effect of steady versus oscillating flow on bone cells. Journal of Biomechanics 31 (11), 969–976. Johnson, D.L., McAllister, T.N., Frangos, J.A., 1996. Fluid lfow stimulates rapid and continuous release of nitric oxide in osteoblasts. American Journal of Physiology 271 (34), E205–E208. Judex, S., Lei, X., Han, D., Rubin, C., 2007. Low-magnitude mechanical signals that stimulate bone formation in the ovariectomized rat are dependent on the applied frequency but not on the strain magnitude. Journal of Biomechanics 40 (6), 1333–1339. Kafka, V., 1983. On hydraulic strengthening of bones. Biorheology 20 (6), 789–793. Kapur, S., Baylink, D.J., Lau, K.H.W., 2002. Fluid flow shear stress stimulates human osteoblast proliferation and differentiation through multiple interacting and competing signal transduction pathways. Bone 32, 241–251.
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Klein-Nulend, J., Semeins, C.M., Berger, E.H., 1996. Prostaglandin mediated modulation of transforming growth factor-beta metabolism in primary mouse osteoblastic cells. Journal of Cellular Physiology 168 (1), 1–7. Li, J., Rose, E., Frances, D., Sun, Y., You, L., 2012. Effect of oscillating fluid flow stimulation on osteocyte mrna expression. Journal of Biomechanics 45, 247–251. Li, Y.J., Batra, N.N., You, L., Meier, S.C., Coe, I.A., Yellowley, C.E., Jacobs, C.R., 2004. Oscillatory fluid flow affects human marrow stromal cell proliferation and differentiation. Journal of Orthopaedic Research 22, 1283–1289. Lim, T.H., Hong, J.H., 2000. Poroelastic properties of bovine vertebral trabecular bone. Journal of Orthopaedic Research 18 (4), 671–677. Liney, G.P., Bernard, C.P., Manton, D.J., Turnbull, L.W., Langton, C.M., 2007. Age, gender, and skeletal variation in bone marrow composition: a preliminary study at 3.0 T. Journal of Magnetic Resonance Imaging 26, 787–793. Luu, Y.K., Pessin, J.E., Judex, S., Rubin, J., Rubin, C.T., 2009. Mechanical signals as a non-invasive means to influence mesenchymal stem cell fate, promoting bone and suppressing the fat phenotype. Bonekey Osteovision 6 (4), 132–149. Lynch, M.A., Brodt, M.D., Silva, M., 2010. Skeletal effects of whole-body vibration in adult and aged mice. Journal of Orthopaedic Research 28 (2), 241–247. McAllister, T., 2000. Fluid shear stress stimulates prostaglandin and nitric oxide release in bone marrow-derived preosteoclast-like cells. Biochemical and Biophysical Research Communications 270 (2), 643–648. Moore, S.G., Dawson, K.L., 1990. Red and yellow marrow in the femur: age-related changes in appearance at mr imaging. Radiology 175, 219–223. Nauman, E., Arroman, Y.P., 2001. Bone Mechanics Handbook, second ed. CRC, Boca Raton, FL 2001. Nauman, E.A., Satcher, R.L., Keaveny, T.M., Halloran, B.P., Bikle, D.D., 2001. Osteoblasts respond to pulsatile fluid flow with short-term increases in PGE2 but no change in mineralization. Journal of Applied Physiology 90, 1849–1854. Niebur, G., Birmingham, E., McHugh, P. Direct computation of trabecular bone permeability. In: Computer Methods in Biomechanics and Biomedical Engineering Valencia, Spain; 2010. Ochoa, J.A., Heck, D.A., Brandt, K.D., Hillberry, B.M., 1991. The effect of intertrabecular fluid on femoral head mechanics. Journal of Rheumatology 18 (4), 580–584. Owen, M., 1980. The origin of bone cells in the postnatal organism. Arthritis Rheumatology 23 (10), 1073–1080. Preziosi, L., Ambrosi, D., Verdier, C., 2010. An elasto-visco-plastic model of cell aggregates. Journal of Theoretical Biology 262, 35–47. Ricci, C., Cova, M., Yang, Y.S., Rahmouni, A., Scott, W.W., Zerhouni, E.A., 1990. Normal age-related patterns of cellular and fatty bone marrow distribution in the axial skeleton: MRI imaging study. Radiology 177, 83–88. Riddle, R.C., Taylor, A.F., Genetos, D.C., Donahue, H.J., 2006. Map kinase and calcium signaling mediate fluid flow-induced human mesenchymal stem cell
2229
proliferation. American Journal of Physiology Cellular Physiology 290, C776–C784. Rubin, C., Recker, R., Cullen, D., Ryaby, J., McCabe, J., McLeod, K., 2003. Prevention of postmenopausal bone loss by a low-magnitude, high-frequency mechanical stimuli: a clinical trial assessing compliance, efficacy, and safety. Journal of Bone and Mineral Research 19 (3), 343–351. Rubin, C., Turner, A.S., Bain, S., Mallinckrodt, C., McLeod, K., 2001a. Anabolism: low mechanical signals strengthen long bones. Nature 412 (6847), 603–604. Rubin, C., Turner, A.S., Muller, R., Mittra, E., McLeod, K., Lin, W., Qin, Y.X., 2002. Quantity and quality of trabecular bone in the femur are enhanced by a strongly anabolic, noninvasive mechanical inter-vention. Bone and Mineral Research 17 (2), 349–357. Rubin, C., Xu, G., Judex, S., 2001b. The anabolic activity of bone tissue, suppressed by disuse, is normalized by brief exposure to extremely low-magnitude mechanical stimuli. FASEB Journal 15, 2225–2229. Rubin, C.T., Capilla, E., Luu, Y.K., Busa, B., Crawford, H., Nolan, D.J., Mittal, V., Rosen, C.J., Pessin, J.E., Judex, S., 2007. Adipogenesis is inhibited by brief, daily exposure to high-frequency, extremely low-magnitude mechanical signals. Proceedings of the National Academy of Sciences 104 (45), 17879– 17884. Ruiz, S.A., Chen, C.S., 2008. Emergence of patterned stem cell differentiation within multicellular structures. Stem Cells 26 (11), 2921–2927. Sen, B., Xie, Z., Case, N., Styner, M., Rubin, C.T., Rubin, J., 2011. Mechanical signal influence on mesenchymal stem cell fate is enhanced by incorporation of refractory periods into the loading regimen. Journal of Biomechanics 44, 593–599. Smalt, R., Mitchell, F.T., Howard, R.L., Chambers, T.J., 1997. Induction of no and prostaglandin e2 in osteoblasts by wall-shear stress but not mechanical strain. American Journal of Physiology–Endocrinology and Metabolism 273 (4), E751–758. Wang, X., Liu, X., Niebur, G.L., 2004. Preparation of on-axis cylindrical trabecular bone specimens using micro-CT imaging. Journal of Biomechanical Engineering 126, 122–125. Weiss, L., 1976. The hematopoietic microenvironment of the bone marrow: an ultrastructural study of the stroma in rats. Anatomical Records 186 (2), 161–184. Whyne, C.M., Hu, S.S., Lotz, J.C., 2003. Burst fracture in the metastatically involved spine: development, validation, and parametric analysis of a three-dimensional poroelastic finite-element model. Spine 28 (7), 652–660. Yeung, D.K.W., Griffith, J.F., Antonio, G.E., Lee, F.K.H., Woo, J., Leung, P.C., 2005. Osteoporosis is associated with increased marrow fat content and decreased marrow fat unsaturation: a proton mr spectroscopy study. Journal of Magnetic Resonance Imaging 22, 279–285.
Contents lists available at SciVerse ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Fluid shear stress in trabecular bone marrow due to low-magnitude high-frequency vibration Thomas R. Coughlin, Glen L. Niebur n Tissue Mechanics Laboratory, Bioengineering Graduate Program, University of Notre Dame, IN 46556, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Accepted 12 June 2012
Low-magnitude high-frequency (LMHF) loading has recently received attention for its anabolic effect on bone. The mechanism of transmission of the anabolic signal is not fully understood, but evidence indicates that it is not dependent on bone matrix strain. One possible source of signaling is mechanostimulation of the cells in the bone marrow. We hypothesized that the magnitude of the fluid shear stress in the marrow during LMHF loading is in the mechanostimulatory range. As such, the goal of this study was to determine the range of shear stress in the marrow during LMHF vibration. The shear stress was estimated from computational models, and its dependence on bone density, architecture, permeability, marrow viscosity, vibration amplitude and vibration frequency were examined. Three-dimensional finite element models of five trabecular bone samples from different anatomic sites were constructed, and a sinusoidal velocity profile was applied to the models. In human bone models during axial vibration at an amplitude of 1 g, more than 75% of the marrow experienced shear stress greater than 0.5 Pa. In comparison, in vitro studies indicate that fluid induced shear stress in the range of 0.5 to 2.0 Pa is anabolic to a variety of cells in the marrow. Shear stress at the bonemarrow interface was as high as 5.0 Pa. Thus, osteoblasts and bone lining cells that are thought to reside on the endosteal surfaces may experience very high shear stress during LMHF loading. However, a more complete understanding of the location of the various cell populations in the marrow is needed to quantify the effects on specific cell types. This study suggests the shear stress within bone marrow in real trabecular architecture during LMHF vibration could provide the mechanical signal to marrow cells that leads to bone anabolism. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Bone Bone marrow Computational fluid dynamics Mechanobiology Shear stress Low magnitude high frequency loading
1. Introduction Low-magnitude high-frequency (LMHF) stimulation is anabolic to bone in animals (Rubin et al., 2002), but the mechanical signal and its mechanism of transmission have not been established. Rats subjected to 90 Hz vibration developed greater trabecular bone volume and thicker trabeculae than rats subjected to vibration at 45 Hz (Judex et al., 2007), even though the 90 Hz vibration induced lower strain in the bone. Trabecular bone volume also increases in non-weight bearing skeletal segments when subjected to LMHF loading (Garman et al., 2007). As such, matrix strain is not necessary to achieve the anabolic signal from LMHF stimulation, in contrast to exercise or unloading studies where strain is likely the dominant signal. In sheep, LMHF loading resulted in a 32% increase in trabecular bone volume, a decrease in trabecular spacing and an increase in trabecular number
n Corresponding author at: 147 Multidisciplinary Research Building Notre Dame, IN 46556, USA. Tel.: þ 1 574 631 3327; fax: þ1 574 631 21744. E-mail address: [email protected] (G.L. Niebur).
0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2012.06.020
(Rubin et al., 2001b). Together, the concentration of the anabolic effect of LMHF vibration in marrow rich trabecular bone, the ability of mechanical signals to influence bone marrow derived cells (Luu et al., 2009), and the lack of dependence on matrix strain, suggest that marrow cells may sense the mechanical signal from LMHF loading. Bone marrow is a cellular soft tissue found in the endosteal compartment of bone (Fig. 1). It is highly vascularized and forms the niche for a variety of cells, including mesenchymal and hematopoietic stem cells (MSCs and HSCs) (Gurkan and Akkus, 2008; Weiss, 1976). The cellular fates of MSCs and HSCs influence bone remodeling as osteoclasts are derived from the HSC lineage (Owen, 1980) and MSCs are multipotent cells that differentiate to connective tissue lines. Stem cell fate also affects bone marrow composition (Berg et al., 1998). For example, changes in fat content alter the bone marrow mechanical environment (Bryant et al., 1989). During LMHF loading, marrow cells could be subjected to inertially induced motion relative to the much more rigid bone causing shear stress within the fluid-like marrow. Shear stress induced by this motion during LMHF loading may, in turn, affect
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
2223
and (3) quantified the dependence of shear stress on bone density, architecture, permeability, marrow viscosity, vibration amplitude and vibration frequency.
2. Methods
Fig. 1. A histological section of bone marrow from a porcine cervical vertebra. Hematoxylin and eosin stain: (A) adipocytes, (B) trabecular bone (bar¼ 100 mm).
the marrow cell population. Shear stress in the range of 0.5 to 2.0 Pa increases MSC proliferation (Castillo and Jacobs, 2010), decreases adipogenesis in MSCs (Rubin et al., 2007), upregulates nitric oxide (NO) and prostaglandin release in osteoblasts (Johnson et al., 1996; Smalt et al., 1997), upregulates NO release in preosteoclasts (McAllister, 2000), increases prostaglandin production (Klein-Nulend et al., 1996), and upregulates intracellular calcium and gene expression in osteocytes (Li et al., 2012) and osteoblasts (Nauman et al., 2001). In the case of both steady and oscillatory flow, 1 Pa shear stress enhances the osteogenic phenotype (Arnsdorf et al., 2010; Case et al., 2011). The shear stress in the bone marrow during LMHF loading was previously estimated analytically using continuum level mixture theory (Dickerson et al., 2007). The resultant shear stress ranged from 0.5 to 5.0 Pa, increasing with volume fraction, vibration frequency and marrow viscosity. The analysis also predicted an increase in blood flow during LMHF loading. The shear stress in the marrow was determined from the Darcy permeability, estimated from drag theory. As such, it was not possible to determine detailed shear stress profiles in the marrow volume. Quantification of the shear stress throughout the bone marrow within real trabecular architectures during LMHF loading would provide insight into the potential for mechanical regulation of marrow cells. Moreover, the dependence of shear stress levels on trabecular architecture may elucidate site sensitivity of bone anabolism due to LMHF loading. While direct measurement of shear stress within the trabecular pore space is not possible with present technology, computational models of real morphologies can be used to simulate the loading and motion of the marrow during LMHF vibration. These results would ideally be used in conjunction with experiments designed to validate or refute the model assumptions. The goal of this study was to determine whether the magnitude of shear stress in bone marrow during LMHF loading of trabecular bone at amplitudes from 0.3 to 1.0 g and frequencies of 10 to 90 Hz is in the range that induces a physiological response from cells. Specifically, we (1) used finite element models to simulate the dynamics of bone marrow in the trabecular pore space during sinusoidal motion of the bone matrix, (2) quantified the statistical distribution of shear stress,
Computational fluid dynamics models were used to study shear stress in bone marrow with the assumption that marrow behaves as a homogeneous fluid. This approach does not capture specific cell–cell interactions, but rather homogenizes the behavior of the marrow as a viscous fluid. Although this approach has limitations, it is a common assumption (Bryant et al., 1989; Dickerson et al., 2007; Kafka, 1983; Ochoa et al., 1991). Five trabecular bone samples from five anatomic sites were studied to provide a range of architecture (Table 1). One sample was taken from a human lumbar spine, one each from the greater trochanter and neck of a human femur (obtained with informed consent through NDRI), and one from each of the medial and lateral femoral condyles of an ovine distal femur (obtained from another study under IACUC approval). The samples were prepared with their principal mechanical axes aligned with the scanning axis ¨ (Wang et al., 2004), and imaged by micro-CT (Scanco mCT-80, Bruttisellen, Switzerland) at 20 mm resolution. A 4 4 4.5 mm3 region was selected, Gaussian filtered and resampled by cubic interpolation to 35 mm resolution (Visualization Toolkit, Kitware, Clifton Park, NY). An earlier convergence study indicated that this resolution provided sufficiently small elements for mesh convergence in a steady fluid flow solution (Niebur et al., 2010). A 30 mm layer of fluid was added around the entire bone sample to allow continuity of flow from pores on the edges. The fluid flow was constrained on the faces perpendicular to the vibration direction to prevent flow across the boundaries, and there was no viscous drag for flow parallel to the wall (Fig. 2c). The marrow regions were discretized into tetrahedra using a marching cubes algorithm in VTK (Kitware, Clifton Park, NY), which were smoothed using Laplacian smoothing with heuristic checks to avoid inverted elements, and converted into finite element meshes (Fig. 2). The fluid elements were assigned incompressible Newtonian fluid properties with a density of 0.9 g/cm3 and viscosity of 400 mPa s (Bryant et al., 1989). The bone-marrow interface was assumed to be rigid, with a no-slip interface. For the baseline models, all nodes on the bone surface were assigned a 10 Hz sinusoidal velocity profile with a peak acceleration of 1 g. The velocity profile was first applied parallel to the principal fabric direction of the bone and then transverse to the principal direction to assess the effects of pore morphology relative to the direction of motion. The same velocity profile was imposed on the outer surfaces of the fluid volume lying parallel to the velocity, while the perpendicular faces had constant pressure boundary conditions to allow free flow. A transient, dynamic solution was performed in ADINA-Fs (Watertown, MA). The solution was carried out starting from 0 velocity and proceeded for at least two cycles. Womersley numbers (Wo), calculated at a frequency of 10 Hz, fluid density of 0.9 g/cm3, viscosity of 400 mPa s and with trabecular spacing as the characteristic dimension, were less than 1.0, which indicates that viscous effects – rather than inertial effects – dominate the flow. The Reynolds numbers (Re) were much less than 1.0, indicating that the assumption of laminar flow is reasonable (Table 2). A 3.6 3.6 4.4 mm3 sub-region in the center of the sample was analyzed to avoid artifacts from the edges (Fig. 2). Because the shear stress distribution was non-normal, the median, 25th, and 75th percentile values of the nodal results were considered. Data were analyzed at the end of the first half cycle, where the applied acceleration was a maximum, and the results were similar to those at subsequent acceleration peaks (Fig. 3). The permeability along the two vibration directions was calculated based on Darcy’s law (Eq. 1), where k is permeability, Q is flow, Dx is thickness, and DP is the pressure difference. A steady state flow problem was solved on the same mesh with a constant pressure differential of 13.9 Pa/mm. The volumetric flow rate was
Table 1 Sample information for the three human (greater trochanter, femoral neck and lumbar vertebra) and two ovine (femoral condyles) samples. Sample
Volume fraction (%)
Human greater trochanter Human lumbar vertebra Human femoral neck Ovine femoral condyle 1 Ovine femoral condyle 2
12.2 12.9 14.0 24.1 31.3
SMI
Trabecular spacing (mm)
1.34 0.66 2.02 1.62 0.48 0.15
0.88 0.76 0.46 0.36
Degree of anisotropy (DA) 1.8953 1.1369 1.7744 2.4311 2.0373
2224
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Fig. 2. (a) A uCT reconstruction of the human lumbar vertebral trabecular bone sample (20 mm resolution). (b) The marrow region was converted to a tetrahedral FE mesh. Meshes had at least 10 elements across pores. (c) A thin fluid layer was added beyond the edges of the trabecular bone sample. Fluid contacting the bone surfaces had an imposed sinusoidal velocity profile in one direction, the control volume faces parallel to the vibration (solid lines) allowed no flow perpendicular to the surface, but free flow parallel to the surface. Faces perpendicular to the flow (dashed lines) allowed free flow subject to the constraint that the same volume that exited at one face must enter through the opposite face (conservation of mass). The white box indicates the sub-region where the shear stress was analyzed. (d) Streamlines and velocity contours at the peak velocity of the marrow relative to the bone. normalized by the pressure drop and fluid viscosity to find the permeability. k¼
Q mDx DP
ð1Þ
The effects of frequency and viscosity were investigated parametrically in the lumbar vertebra sample. Bone marrow mechanical properties remain poorly characterized, and depend on composition (Bryant et al., 1989), anatomic site (Liney et al., 2007), age (Gurkan and Akkus, 2008; Moore and Dawson, 1990; Ricci et al., 1990) and bone density (Griffith et al., 2005; Yeung et al., 2005). Models were solved with viscosities of 50, 100, 200 and 400 mPa s to investigate the effects of this variability at both 10 and 30 Hz vibration. To examine the effect of frequency on shear stress, 10, 20, 30, 60, and 90 Hz sinusoidal vibrations were applied with a fluid viscosity of 400 mPa s. Finally, the shear stress was studied at amplitudes of 0.3 g, 0.6 g and 1.0 g in the lumbar vertebra and femoral neck samples to assess the effects of amplitude.
3. Results The magnitude of shear stress in the marrow was on the order of Pascals in all samples. The shear stress was highest just past the peak acceleration, and was near zero at the peak velocity (Fig. 3). The statistical distribution of shear stress was non-normal and skewed toward lower stress (Fig. 4). The majority of the marrow in the human trabecular bone samples (trochanter, lumbar vertebra and femoral neck) experienced shear stresses above 0.8 Pa at peak acceleration during axial vibration at 10 Hz and 1 g amplitude, and 95% of the marrow experienced shear stress above 0.5 Pa. Increasing
Table 2 The permeabilities for flow along the axial and transverse vibration directions, and dimensionless characteristic numbers of the flows. The Womersley numbers (Wo) were calculated for 10 Hz vibration along the principal orientation, and Reynolds numbers (Re) were based on the peak velocity of the endosteal surface. Both were calculated for a viscosity of 400 mPa s using trabecular spacing as the characteristic length. Wo is an indicator of the degree to which an oscillatory flow is dominated by inertial (Wo4 1) or viscous (Wo o 1) effects. Re is an indicator of turbulent (Re 4400) vs. laminar (Reo 400) flow. Sample
Axial permeability (mm2)
Transverse permeability (mm2)
Wo (–)
Re (–)
Human greater trochanter Human lumbar vertebra Human femoral neck Ovine femoral condyle 1 Ovine femoral condyle 2
0.00805
0.00388
0.25
0.23
0.00633
0.00669
0.33
0.31
0.00814
0.00340
0.29
0.27
0.00295
0.00338
0.17
0.16
0.00216
0.00040
0.14
0.13
the amplitude increased shear stress linearly in the samples tested (Fig. 5). The volume of marrow displaced across the ends of the bone was approximately 0.1% of the total marrow volume (Table 3).
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Shear stress, and the range of shear stress, increased with increasing permeability independent of vibration direction (Fig. 6a). The shear stress was negatively correlated with volume fraction (BV/ TV), and was lower for vibration transverse to the principal pore direction (Fig. 6b). Shear stress was positively correlated with
2225
trabecular spacing (Tb.Sp.) for both axial (po0.025, R2 ¼0.81) and transverse vibration (po0.035, R2 ¼0.75). Similarly, shear stress was marginally correlated with the structural model index (SMI), which describes the rod-like (SMI¼3) or plate-like (SMI¼0) nature of the bone (po0.055). However, combined with BV/TV neither of these parameters was significant when using multiple regression (p40.2). The effect of frequency on shear stress depended on viscosity. The shear stress increased with increasing viscosity at 30 Hz, but was insensitive to viscosity at 10 Hz (Fig. 7). Increasing the frequency from 10 to 30 Hz resulted in greater shear stress extending farther from the bone walls (Fig. 7), consistent with increasing viscous effects as Wo decreases. For a fixed viscosity of 400 mPa s, increasing the frequency from 10 to 90 Hz narrowed the range of shear stress, but had a minimal effect on the 25th percentile value (Fig. 8).
4. Discussion
Fig. 3. The 25th percentile, median and 75th percentile shear stress in the human femoral neck sample for the first 2.25 cycles of vibration at 10 Hz. The time is normalized to the period, T, of vibration. More than 75% of the marrow in this sample experienced shear stress above 0.5 Pa for over 60% of the cycle.
This study examined the potential for low-magnitude highfrequency dynamic loading of bone to induce mechanostimulatory stress levels within the marrow. When subjected to 1 g vibrations, the shear stress exceeded 0.5 Pa in 75% of the marrow for more than 65% of each cycle in the human bone samples, although these are overestimates of the actual shear stress due to the idealized boundary conditions applied. Shear stress above 0.5 Pa is mechanostimulatory to osteoblasts (Kapur et al., 2002; Klein-Nulend et al., 1996; Nauman et al., 2001), osteoclasts (McAllister, 2000) and MSCs (Castillo and Jacobs, 2010) in vitro. The shear stress increased with trabecular permeability and porosity. In general, more porous structures also exhibited elevated shear stress in a greater fraction of the marrow
Axial
O Transverse Fig. 4. (a) Distributions of shear stresses in the human femoral neck sample during axial and transverse vibration were non-normal. (b) Shear stress above the 25th percentile (0.75 Pa), extended well away from the bone-marrow interface. (c) The 75th percentile shear stress (1.5 Pa) was confined to the bone-marrow interface (white space represents bone tissue, and blue regions are at or below the respective threshold). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
2226
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
space. The larger pores contained a greater marrow mass accelerating relative to the bone, resulting in greater velocity gradients and in turn larger shear stress. Thus, if the shear stress induced by inertial motion of the marrow is the mechanobiological signal in LMHF loading, then it would be most effective in locations with low bone density. The principal strength of this study was the quantification of marrow shear stress throughout real trabecular pore geometries. Current technology does not allow direct quantification of the bone marrow mechanical behavior. However, computational models can provide a reasonable estimate. To compensate for the wide range of reported properties for bone marrow, we used a range of viscosities to parametrically quantify the relative affects of each during LMHF vibration. Computation of the shear stress within the trabecular compartment in a variety of real architectures from multiple anatomic sites and species provided an estimate of how this signal may vary in different architectures, which was quantifiable by differences in the trabecular permeability. Some limitations of this study must be considered when interpreting the data. The primary limitation is the simplifying assumption that bone marrow is a homogeneous fluid. While bone marrow behaves in a fluid-like manner, it is heterogeneous, and the viscosity arises from phenomena such as cell adhesion that may cause it to behave as a Brinkman or Bingham like fluid. Moreover, red marrow in the pore space contains capillaries and other structural elements that affect the mechanical behavior.
Thus, capturing the true behavior would require multi-scale modeling that incorporates cell–cell adhesion, cell-fluid interaction and multiple cell geometries, rather than a homogenized viscosity (Forgacs, 1995; Preziosi et al., 2010). The marrow was
Fig. 5. The 25th, 50th and 75th percentile of shear stress in the marrow space for a viscosity of 400 mPa s. The shear stress increased with increasing amplitude in both samples. The bottom line is the 25th percentile stresses and the top is the 75th percentile stress.
Fig. 6. (a) The 25th, median and 75th percentile shear stress increased with increasing permeability along the vibration axis (p o0.0001, linear regression). (b) The 25th percentile shear stress during axial and transverse vibration at 10 Hz decreased with increasing volume fraction (p o 0.01, linear regression).
Table 3 The relative flow rate, percentage of the marrow volume that exited the bone and the median shear stress for axial vibration in the human lumbar spine and femoral neck samples at 30 Hz. The displaced volume of marrow is the integral of the volumetric flow rate over 1/2 cycle. The displaced volumes were on the order of 0.05 ml while the marrow volumes were 62 ml and 52 ml in the lumbar spine and femoral neck, respectively. Lumbar spine
Relative volumetric flow rate (ml/s) Volume of marrow entering and exiting the bone (ml) Fraction of marrow volume entering and exiting the bone (%) Median shear stress (Pa)
Femoral neck
0.3 g
0.6 g
1.0 g
0.3 g
0.6 g
1.0 g
1.38 0.015 0.024 0.31
2.76 0.029 0.047 0.62
4.59 0.049 0.079 1.04
1.25 0.013 0.025 0.33
2.5 0.027 0.051 0.68
4.04 0.043 0.082 1.10
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
30 Hz
50 mPas
10 Hz
2227
τ (Pa) Re = 0.82, Wo = 1.61
2
200 mPas
Re = 2.46, Wo = 0.93
1 Fig. 8. The bars show the 25th, 50th and 75th percentile of shear stress in the marrow space for a viscosity of 400 mPa s. The shear stress was in a narrower range at the higher frequencies, but the effect was minor.
Re = 0.21, Wo = 0.81
Re = 0.31, Wo = 0.33
Re = 0.10, Wo = 0.57
0
400 mPas
Re = 0.62, Wo = 0.47
Fig. 7. Shear stress in the marrow of a human lumbar vertebra sample at 10 and 30 Hz frequency loading and assumed viscosities of 50, 200 and 400 mPa s (white regions are bone and blue regions are below 0.5 Pa, the reported mechanostimulatory threshold). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
idealized as incompressible. The measured pressure in the marrow of a proximal femur at 2 BW of load was only 15 kPa (Downey et al., 1988), while the bulk modulus is over 300 MPa (Arramon and Cowin, 1997; Lim and Hong, 2000), indicating the volume change is less than 0.005%. The assumption of incompressibility required boundary conditions that allowed flow into and out of the bone, as no motion of the fluid would occur otherwise. While the marrow is actually entrapped and does not cross endosteal boundaries, the region of bone that was modeled was only a fraction of the trabecular bone volume in a whole bone, and the boundary conditions assumed the marrow flowed into or from the adjacent trabecular bone. The volume of marrow flowing across the boundary was less than 0.1% of the total marrow volume, which is consistent with the flow that has been assumed to occur in poroelastic models of trabecular bone (Arramon and Cowin, 1997; Cowin, 1999; Whyne et al., 2003). However, the boundary conditions applied do not account for the resistance to flow imposed by the ends of the bone and thus, the idealized boundary conditions resulted in an overestimation of the shear stress. Finally, we ignored the deformation of the bone that would
occur during loading. During LMHF loading, the applied velocity is much lower than the wave speed of bone, such that the induced bone strain would be small. However, activities of daily living may also induce marrow motion through deformation of the pores in the trabecular bone. Despite these limitations, the results provide important insight into the potential mechanobiology of LMHF loading and a reasonable first approximation of the system. MSC differentiation in a multicellular structure is dependent on the relative location of cells and corresponding stress. When MSCs seeded in a collagen matrix were subjected to loading, those experiencing low stress in the center of the construct differentiated into adipocytes, while those experiencing higher stresses at the edges of the matrix differentiated into osteoblasts (Ruiz and Chen, 2008). Assuming that the mechanostimulatory stress levels found in 2-D in vitro experiments apply to the 3-D in situ case studied here, the models predict that over 75% and as much as 95% of the marrow is subjected to stresses above a mechanostimulatory threshold in human trabecular bone when 1 g acceleration is applied. Only those cells near the center of the pore space are shielded. Cells at the bone-marrow interface, which are believed to be primarily quiescent osteoblasts and other bone lining cells (Castillo and Jacobs, 2010) would be subjected to the greatest stimulation. However, a more complete understanding of the spatial composition of marrow is needed to assign shear stress magnitudes to specific cell types. For example, at a cellular level the larger adipocytes (Fig. 1) may shield other cells from these stress levels. Future experiments could incorporate subject specific models to determine how the mechanical signals transmitted may differ in varying architectures and marrow compositions. For example, the anabolic response to vibration was dependent on age in mice, which suggests it may depend on marrow fat content (Lynch et al., 2010). LMHF vibration applied parallel to the principal pore direction results in higher shear stress than when applied perpendicular to the pores. In the case of transverse vibration, plates perpendicular to the fluid motion restrict the flow and there are fewer faces parallel to the flow to generate viscous drag. The permeability is dependent on both volume fraction and architecture. The relationships are complex, and depend on whether the architecture restricts flow by primarily viscous or inertial mechanisms
2228
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
(Nauman and Arroman, 2001). Consistent with this, the shear stress magnitudes and ranges were linearly correlated with architectural parameters, but no simple multiple regressions were found. The impact of this result is that the effects of LMHF vibration may be lower in sites where the trabecular fabric is not aligned with the vibration direction. In humans, vibrations applied in the inferior-superior direction would be misaligned with the principal trabecular orientation in the greater trochanter and femoral neck, resulting in lower shear. In contrast, trabeculae in the lumbar spine are aligned with the direction of vibration and the permeability is higher, which would result in higher shear stress over a greater portion of the marrow. Consistent with this result, postmenopausal women subjected to LMHF vibration for 1 year had a greater increase in BMD in the lumbar spine than in the greater trochanter or femoral neck (Rubin et al., 2003). However, it should be acknowledged that the experimental result could also be the result of biological differences in the cell populations at these sites. The weak dependence of shear stress on viscosity may appear to contradict a continuum poroelastic model, where the shear stress in the fluid depended strongly on viscosity (Dickerson et al., 2007). In poroelastic models flow is pressure driven, compared to the inertial flows in these models. Moreover, the shear stress in poroelasticity arises from a fluid drag term, which depends linearly on permeability and viscosity. In this study, higher marrow viscosity did increase the shear stress at the bone-marrow interface, but the effect was smaller in the remainder of the trabecular compartment. This reflects the transmission of shear stress from the wall across the open pore space, and the fact that shear stress goes to zero at some point in a fluid confined between two walls. The differences in the effect of permeability are likely due to the fact that the flow was driven by matrix strain in the previous study (Dickerson et al., 2007), compared to an inertially driven flow here. The lowest median shear stress magnitude was found at 30 Hz, with a slight increase up to 90 Hz frequency. Although the differences were small, this result is consistent with animal models, where bone growth due to LMHF loading was greater for 90 Hz vibration than at 45 Hz (Judex et al., 2007). In LMHF vibration, the marrow forms a harmonic oscillator within the pore space, and the response will likely differ between anatomic sites or species where the pore sizes and shapes – and hence the permeability – differ. Alternatively, the anabolic effect of increased frequency may be due to a threshold effect on shear stress, as higher frequencies result in more acceleration peaks and thus more times that the marrow shear stress reaches a mechanostimulatory threshold, or the frequency itself may be the factor (Duncan and Turner, 1995; Jacobs et al., 1998). In mice, there was not a consistent dose response to increased vibration amplitude (Christiansen and Silva, 2006). In contrast, our results indicate a monotonic increase in the shear stress with increasing amplitude, consistent with the linear relationship between force and the magnitude of acceleration of a fixed mass. Furthermore, our results indicate that the median shear stress is below 0.5 Pa for an amplitude of 0.3 g, which has previously been sufficient for an anabolic response (Rubin et al., 2003, 2001a). These results open the possibility that the effects of vibration are not related to shear stress in the marrow, or that the threshold for the shear stress is much lower than 0.5 Pa. Alternatively, 20% of the marrow in the lumbar spine model experienced shear stresses above 0.5 Pa during 0.3 g amplitude vibration, indicating that the anabolic signal may only be sensed by cells at the interface. This study provided an initial estimate of the range of shear stress that occurs in bone marrow during LMHF loading. Our data can provide input to fluid flow studies (Glossop and Cartmell, 2009; Kapur et al., 2002; Li et al., 2004; Nauman et al., 2001; Riddle et al., 2006; Sen et al., 2011), which quantify bone marrow
cellular response to shear stress. Applying shear stress in the marrow could provide improved insight into how cells of specific lineages function in vivo when subject to LMHF loading. Cell scale models that incorporate realistic cellular interactions could provide further insight by accounting for heterogeneity and spatial distribution of specific cell lineages in the marrow.
Conflict of interest statement The authors have no potential conflicts of interest.
Acknowledgements Science Foundation Ireland, E.T.S. Walton Program 07/W.I./ B1806, U.S. National Science Foundation CMMI-1100207. Human bone samples were obtained from the National Disease Research Interchange (NDRI). References Arnsdorf, E.J., Tummala, P., Castillo, A.B., Zhang, F., Jacobs, C.R., 2010. The epigenetic mechanism of mechanically induced osteogenic differentiation. Journal of Biomechanics 43, 2881–2886. Arramon, Y.P., Cowin, S.C., 1997. Hydraulic stiffening of cancellous bone. Forma 12 (3,4), 209–221. Berg, B.C.V., Malghem, J., Lecouvet, F.E., Maldague, B., 1998. Magnetic resonance imaging of the normal bone marrow. Skeletal Radiology 27, 471–483. Bryant, J., David, T., Gaskell, P.H., King, S., Lond, G., 1989. Rheology of bovine bone marrow. Proceedings of the Institution of Mechanical Engineers, Part H. Case, N., Sen, B., Thomas, J.A., Styner, M., Xie, Z., Jacobs, C.R., Rubin, J., 2011. S.a.o.f.f.p.a.s.o.p.C.T.I.-P., 2011. Steady and oscillatory fluid flows produce a similar osteogenic phenotype. Calcified Tissue International 88 (3), 189–197. Castillo, A.B., Jacobs, C.R., 2010. Mesenchymal stem cell mechanobiology. Current Osteoporosis Reports 8 (2), 98–104. Christiansen, B.A., Silva, M.J., 2006. The effect of varying magnitudes of wholebody vibration on several skeletal sites in mice. Annals of Biomedical Engineering 34 (7), 1149–1156. Cowin, S.C., 1999. Bone poroelasticity. Journal of Biomechanics 32 (3), 217–238. Dickerson, D.A., Sander, E.A., Nauman, E.A., 2007. Modeling the mechanical consequences of vibratory loading in the vertebral body: microscale effects. Biomechanics and Modeling in Mechanobiology 7 (3), 191–202. Downey, D.J., Simkin, P.A., Taggart, R., 1988. The effect of compressive loading on intraosseous pressure in the femoral head in vitro. The Journal of Bone and Joint Surgery American 70 (6), 871–877. Duncan, R.L., Turner, C.H., 1995. Mechanotransduction and the functional response of bone to mechanical strain. Calcified Tissue International 57, 344–358. Forgacs, G., 1995. Biological specificity and measurable physical properties of cell surface receptors and their possible role in signal transduction through the cytoskeleton. Biochemistry and Cell Biology 73, 317–326. Garman, R., Gaudette, G., Donahue, L.R., Rubin, C., Judex, S., 2007. Low-level accelerations applied in the absence of weight bearing can enhance trabecular bone formation. Journal of Orthopaedic Research 25, 732–740. Glossop, J.R., Cartmell, S.H., 2009. Effect of fluid flow-induced shear stress on human mesenchymal stem cells: differential gene expression of il1b and map3k8 in mapk signaling. Gene Expression Patterns 9 (5), 381–388. Griffith, J.F., Yeung, D.K.W., Antonio, G.E., Lee, F.K.H., Hong, A.W.L., Wong, S.Y.S., Lau, E.M.C., Leung, P.C., 2005. Vertebral bone mineral density, marrow perfusion, and fat content in healthy men and men with osteoporosis: dynamic contrast-enhanced mr imaging and mr spectroscopy. Radiology 236, 945–951. Gurkan, U.A., Akkus, O., 2008. The mechanical environment of bone marrow: a review. Annals of Biomedical Engineering 36 (12), 1978–1991. Jacobs, C.R., Yellowley, C.E., Davis, B.R., Zhou, Z., Cimbala, J.M., Donahue, H.J., 1998. Differential effect of steady versus oscillating flow on bone cells. Journal of Biomechanics 31 (11), 969–976. Johnson, D.L., McAllister, T.N., Frangos, J.A., 1996. Fluid lfow stimulates rapid and continuous release of nitric oxide in osteoblasts. American Journal of Physiology 271 (34), E205–E208. Judex, S., Lei, X., Han, D., Rubin, C., 2007. Low-magnitude mechanical signals that stimulate bone formation in the ovariectomized rat are dependent on the applied frequency but not on the strain magnitude. Journal of Biomechanics 40 (6), 1333–1339. Kafka, V., 1983. On hydraulic strengthening of bones. Biorheology 20 (6), 789–793. Kapur, S., Baylink, D.J., Lau, K.H.W., 2002. Fluid flow shear stress stimulates human osteoblast proliferation and differentiation through multiple interacting and competing signal transduction pathways. Bone 32, 241–251.
T.R. Coughlin, G.L. Niebur / Journal of Biomechanics 45 (2012) 2222–2229
Klein-Nulend, J., Semeins, C.M., Berger, E.H., 1996. Prostaglandin mediated modulation of transforming growth factor-beta metabolism in primary mouse osteoblastic cells. Journal of Cellular Physiology 168 (1), 1–7. Li, J., Rose, E., Frances, D., Sun, Y., You, L., 2012. Effect of oscillating fluid flow stimulation on osteocyte mrna expression. Journal of Biomechanics 45, 247–251. Li, Y.J., Batra, N.N., You, L., Meier, S.C., Coe, I.A., Yellowley, C.E., Jacobs, C.R., 2004. Oscillatory fluid flow affects human marrow stromal cell proliferation and differentiation. Journal of Orthopaedic Research 22, 1283–1289. Lim, T.H., Hong, J.H., 2000. Poroelastic properties of bovine vertebral trabecular bone. Journal of Orthopaedic Research 18 (4), 671–677. Liney, G.P., Bernard, C.P., Manton, D.J., Turnbull, L.W., Langton, C.M., 2007. Age, gender, and skeletal variation in bone marrow composition: a preliminary study at 3.0 T. Journal of Magnetic Resonance Imaging 26, 787–793. Luu, Y.K., Pessin, J.E., Judex, S., Rubin, J., Rubin, C.T., 2009. Mechanical signals as a non-invasive means to influence mesenchymal stem cell fate, promoting bone and suppressing the fat phenotype. Bonekey Osteovision 6 (4), 132–149. Lynch, M.A., Brodt, M.D., Silva, M., 2010. Skeletal effects of whole-body vibration in adult and aged mice. Journal of Orthopaedic Research 28 (2), 241–247. McAllister, T., 2000. Fluid shear stress stimulates prostaglandin and nitric oxide release in bone marrow-derived preosteoclast-like cells. Biochemical and Biophysical Research Communications 270 (2), 643–648. Moore, S.G., Dawson, K.L., 1990. Red and yellow marrow in the femur: age-related changes in appearance at mr imaging. Radiology 175, 219–223. Nauman, E., Arroman, Y.P., 2001. Bone Mechanics Handbook, second ed. CRC, Boca Raton, FL 2001. Nauman, E.A., Satcher, R.L., Keaveny, T.M., Halloran, B.P., Bikle, D.D., 2001. Osteoblasts respond to pulsatile fluid flow with short-term increases in PGE2 but no change in mineralization. Journal of Applied Physiology 90, 1849–1854. Niebur, G., Birmingham, E., McHugh, P. Direct computation of trabecular bone permeability. In: Computer Methods in Biomechanics and Biomedical Engineering Valencia, Spain; 2010. Ochoa, J.A., Heck, D.A., Brandt, K.D., Hillberry, B.M., 1991. The effect of intertrabecular fluid on femoral head mechanics. Journal of Rheumatology 18 (4), 580–584. Owen, M., 1980. The origin of bone cells in the postnatal organism. Arthritis Rheumatology 23 (10), 1073–1080. Preziosi, L., Ambrosi, D., Verdier, C., 2010. An elasto-visco-plastic model of cell aggregates. Journal of Theoretical Biology 262, 35–47. Ricci, C., Cova, M., Yang, Y.S., Rahmouni, A., Scott, W.W., Zerhouni, E.A., 1990. Normal age-related patterns of cellular and fatty bone marrow distribution in the axial skeleton: MRI imaging study. Radiology 177, 83–88. Riddle, R.C., Taylor, A.F., Genetos, D.C., Donahue, H.J., 2006. Map kinase and calcium signaling mediate fluid flow-induced human mesenchymal stem cell
2229
proliferation. American Journal of Physiology Cellular Physiology 290, C776–C784. Rubin, C., Recker, R., Cullen, D., Ryaby, J., McCabe, J., McLeod, K., 2003. Prevention of postmenopausal bone loss by a low-magnitude, high-frequency mechanical stimuli: a clinical trial assessing compliance, efficacy, and safety. Journal of Bone and Mineral Research 19 (3), 343–351. Rubin, C., Turner, A.S., Bain, S., Mallinckrodt, C., McLeod, K., 2001a. Anabolism: low mechanical signals strengthen long bones. Nature 412 (6847), 603–604. Rubin, C., Turner, A.S., Muller, R., Mittra, E., McLeod, K., Lin, W., Qin, Y.X., 2002. Quantity and quality of trabecular bone in the femur are enhanced by a strongly anabolic, noninvasive mechanical inter-vention. Bone and Mineral Research 17 (2), 349–357. Rubin, C., Xu, G., Judex, S., 2001b. The anabolic activity of bone tissue, suppressed by disuse, is normalized by brief exposure to extremely low-magnitude mechanical stimuli. FASEB Journal 15, 2225–2229. Rubin, C.T., Capilla, E., Luu, Y.K., Busa, B., Crawford, H., Nolan, D.J., Mittal, V., Rosen, C.J., Pessin, J.E., Judex, S., 2007. Adipogenesis is inhibited by brief, daily exposure to high-frequency, extremely low-magnitude mechanical signals. Proceedings of the National Academy of Sciences 104 (45), 17879– 17884. Ruiz, S.A., Chen, C.S., 2008. Emergence of patterned stem cell differentiation within multicellular structures. Stem Cells 26 (11), 2921–2927. Sen, B., Xie, Z., Case, N., Styner, M., Rubin, C.T., Rubin, J., 2011. Mechanical signal influence on mesenchymal stem cell fate is enhanced by incorporation of refractory periods into the loading regimen. Journal of Biomechanics 44, 593–599. Smalt, R., Mitchell, F.T., Howard, R.L., Chambers, T.J., 1997. Induction of no and prostaglandin e2 in osteoblasts by wall-shear stress but not mechanical strain. American Journal of Physiology–Endocrinology and Metabolism 273 (4), E751–758. Wang, X., Liu, X., Niebur, G.L., 2004. Preparation of on-axis cylindrical trabecular bone specimens using micro-CT imaging. Journal of Biomechanical Engineering 126, 122–125. Weiss, L., 1976. The hematopoietic microenvironment of the bone marrow: an ultrastructural study of the stroma in rats. Anatomical Records 186 (2), 161–184. Whyne, C.M., Hu, S.S., Lotz, J.C., 2003. Burst fracture in the metastatically involved spine: development, validation, and parametric analysis of a three-dimensional poroelastic finite-element model. Spine 28 (7), 652–660. Yeung, D.K.W., Griffith, J.F., Antonio, G.E., Lee, F.K.H., Woo, J., Leung, P.C., 2005. Osteoporosis is associated with increased marrow fat content and decreased marrow fat unsaturation: a proton mr spectroscopy study. Journal of Magnetic Resonance Imaging 22, 279–285.