Journal of Biomechanics 43 (2010) 2765–2770

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Journal of Biomechanics

0021-92

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E-m

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Characterizing gait induced normal strains in a murine tibia corticalbone defect model

Jitendra Prasad n, Brett P. Wiater, Sean E. Nork, Steven D. Bain, Ted S. Gross

Department of Orthopaedics and Sports Medicine, University of Washington, Seattle, 325 Ninth Avenue, Box 359798, Seattle, WA 98104, USA

a r t i c l e i n f o

Article history:

Accepted 3 June 2010The critical role that mechanical stimuli serve in mediating bone repair is recognized but incompletely

understood. Further, previous attempts to understand this role have utilized application of externally

Keywords:

Bone healing

Cortical defect model

Finite element analysis

Strain characterization

Gait analysis

Inverse dynamics

90/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jbiomech.2010.06.030

esponding author. Tel.: +1 206 897 5609; fax

ail address: prasadj@u.washington.edu (J. Pra

a b s t r a c t

applied mechanical loads to study the tissue’s response. In this project, we have therefore endeavored

to capitalize on bone’s own consistently diverse loading environment to develop a novel model that

would enable assessment of the influence of physiologically engendered mechanical stimuli on cortical

defect repair. We used an inverse dynamics approach with finite element analysis (FEA) to first quantify

normal strain distributions generated in mouse tibia during locomotion. The strain environment of the

tibia, as previously reported for other long bones, was found to arise primarily due to bending and was

consistent in orientation through the stance phase of gait. Based on these data, we identified three

regions within a transverse cross-section of the mid-diaphysis as uniform locations of either peak

tension, peak compression, or the neutral axis of bending (i.e. minimal strain magnitude). We then used

FEA to quantify the altered strain environment that would be produced by a 0.6 mm diameter

cylindrical cortical bone defect at each diaphyseal site and, in an in situ study confirmed our ability to

accurately place defects at the desired diaphyseal locations. The resulting model will enable the

exploration of cortical bone healing within the context of physiologically engendered mechanical strain.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Bone’s mechanical environment has an important role inregulating the complex process of bone healing (Claes andHeigele, 1999; Isaksson et al., 2006). Not surprisingly, cellularand tissue responses associated with bone healing are sensitive toa variety of mechanical stimuli, which include principal strain,hydrostatic stress, deviatoric strain, shear strain, and fluid velocity(Claes and Heigele, 1999; Isaksson et al., 2009; Lacroix andPrendergast, 2002). However, the predominance of studies in thisarea has examined bone healing in the context of externallyapplied non-physiological mechanical stimuli, where non-phy-siological loading induced stimuli refer to those not engenderedby normal animal activities, but rather by an external loadingdevice.

In considering potential models to explore how mechanicalstimuli engendered by physiologic loading interact with biologicalhealing of bone defects, the mouse presents obvious potential forexploration of specific signaling pathways. As might be expected, anumber of fracture healing models originally developed in largeranimals (Connolly et al., 2003; Isaksson et al., 2009; Wang et al., 2007)

ll rights reserved.

: +1 206 897 5611.

sad).

have been recently implemented in mice. However, just as thesmall size of the murine skeleton challenged development of in vivo

bone healing models, quantification of physiologically induced bonestrains in mice (De Souza et al., 2005) has also proven challengingcompared with larger animals (Blob and Biewener, 1999; Demeset al., 2001; Gross et al., 1992; Rubin and Lanyon, 1982, 1984).Regardless of the stature of the animal, however, long bones aregenerally loaded in bending about a consistent plane during thestance phase of gait (Biewener and Dial, 1995; Main and Biewener,2004; Moreno et al., 2008; Rubin and Lanyon, 1982, 1984).Consequently, cortical regions of long bone diaphyses are consistentlyexposed to minimal normal strain (i.e. the neutral axis), tension orcompression (Demes et al., 2001, 1998; Gross et al., 1992; Liebermanet al., 2004; Mason et al., 1995).

In this study, our objective was to develop a mouse model thatwill enable exploration of how mechanical stimuli mediate bonehealing in the context of physiologically induced bone deforma-tion. To achieve this goal, we first used an inverse dynamics andfinite element analysis approach to quantify the normal strainsinduced in the mouse tibia during the stance phase of walking.We then investigated the effect of placing uni-cortical defectsthrough the diaphysis in three regions within the same diaphysealcross-section consistently exposed to distinct mechanical stimuliduring locomotion: (1) anterior cortex (tension), (2) posteriorcortex (compression), and (3) medial cortex (neutral axis) on

J. Prasad et al. / Journal of Biomechanics 43 (2010) 2765–27702766

normal strains induced by locomotion. We then hypothesized thatit would be possible to locate cortical bone defects such thatphysiological loading alone would expose healing bone to distincttypes of mechanical stimuli.

2. Methods

2.1. Inverse dynamics

Joint angles and hip crest height as a function of normalized time (during

stance phase) were derived from two previous studies (Akay et al., 2006; Leblond

et al., 2003). To model adult (16 week old) C57BL6/J female mice that have been

used throughout this study, the following segment lengths were used: hip (ilium)

5.00 mm, femur 14.95 mm, tibia 16.73 mm, foot (carpals and metacarpals)

7.23 mm, and toe (phalanges) 6.60 mm (Leblond et al., 2003; Lepicard et al.,

2006). Segments were assumed to be one-dimensional rigid links (bars) whose

relative motion (kinematics) was determined by the joint angles and hip crest

height. Murine hindlimb kinematics were then determined for 11 equidistant time

points—from initiation of the stance (phase 0.0) through the completion of the

stance (phase 1.0; Fig. 1a).

For this study, we considered a gait velocity of 0.3 m/s, representing the

maximal gait speed for caged mice (Neumann et al., 2009; Serradj and Jamon,

2009). Data for the horizontal (anteroposterior) and vertical ground reaction forces

generated during walking (corresponding to a mouse of 24 g weight) were directly

adapted from the work of Zumwalt et al. (2006). The horizontal ground reaction

force in the medial–lateral direction was neglected given the unavailability of 3-D

motion data, and our model therefore assumed that hind-limb motion was planar.

Using the kinematics and ground reaction force data, ankle moments were

calculated as a function of normalized time (i.e. stance phase). This moment, at

any point in time, was balanced by the calf muscle moment. The mean (7S.E.)

moment arm for the calf muscle at the ankle was experimentally determined by

measuring the horizontal distance between Achilles tendon attachment to

calcaneus and the center of proximal talus in separate C57 (n¼4) mice to be

1.2770.15 mm, which enabled determination of calf muscle force as a function of

normalized time. By transferring the ground reaction forces and the calf muscle

force to the distal end of tibia, resolved forces (longitudinal (or normal) and shear)

acting on tibia were determined (Fig. 1b).

2.2. Characterization and validation of gait induced normal strains

We first carried out ex vivo strain gage experiments on intact hindlimbs

(tibia and fibula) taken from a 16 week female C57 mice (n¼3). Single element

strain gages were attached to the anterior/lateral and medial/posterior cortices of

each specimen. Each specimen was then potted at the proximal end with the distal

tibia subjected to a range of normal (0.5–2 N in steps of 0.5 N) and shear static

loads (0.05 N–0.3 N in steps of 0.05 N). This range of end loads spanned the

magnitude of gait induced resolved forces for locomotion at 0.3 m/s. For each

specimen, the relationship between applied end load and induced normal strain

was linear within the range of assessed loads.

0 Stance Phase :

0.0

-5 0.1

0 2

-10

0.2

0.3

-15

Y (

mm

) 0.4

0.5

-20

0.5

0.6

-25

0.7

0.8-25

0.9

1 0-30-5 0 5 10 15

1.0

X (mm)

Fig. 1. Kinematics and kinetics of murine hindlimb during stance phase. Segment length

(P) are noted (a). Rigid body motion of the hindlimb was determined for 10 equal in

Rigid-body kinematics of the hindlimb and the ground reaction forces from the literatu

phase (b). The total force was resolved into axial force acting along the tibia long axis

The specimens were then scanned with 21 mm voxel size (SCANCO VivaCT 40).

Using an in-house computer program developed using Microsoft Visual Basic

2005, the mCT-scan images were transformed to a voxel-based FE model (made of

8-noded hexahedral elements). The meshing algorithms in the program had been

thoroughly debugged via comparison with commercial FE softwares (ABAQUS,

ANSYS, and Patran/Nastran). The material properties (Young’s modulus: 20 GPa

and Poisson’s ratio: 0.3) were derived from the literature for adult female C57 mice

(Akhter et al., 2004; Brodt et al., 1999). For each specimen, FE analysis was carried

out for each of the 11 stance phase time points using shareware FEM software

CalculiX (Dhondt and Wittig, 2008). Normal strains were resolved for the whole

bone, with mid-diaphyseal normal strains numerically quantified and compared to

the measured strain gage data (strain gage attachment sites identified via CT

imaging). These data were assessed qualitatively in the context of hindlimb

anatomy (to minimize soft tissue damage during surgery) to identify a transverse

diaphyseal region in which cortical defect sites could be placed in regions of cortex

exposed to distinct normal strains throughout the stance phase of gait (1.5 mm

proximal from the tibia–fibula junction).

2.3. Normal strain alterations due to cortical bone defects

Following verification of our ability to quantify tibia normal strain distribu-

tions in intact tibias, we used FEA to parametrically explore the effects of cortical

bone defects on normal strain magnitudes around the defect. A 0.6 mm-diameter

cylindrical bone defect through one cortical surface was created in silica at

diaphyseal locations continually exposed to 3 distinct types of mechanical stimuli:

(1) anterior cortex (the site of peak tension), (2) medial cortex (spanning the

medial neutral axis), and (3) posterior cortex (the site of peak compression).

The defects were 1.5 mm proximal from the tibiofibular junction in each of the

three tibia FE models. The chosen defect size (0.6 mm diameter) was within the

range of common drill-hole sizes (0.5–1.0 mm) found in the literature and

reflected a balance of surgical accessibility and size of the mouse tibia cortex

(Campbell et al., 2003; Kim et al., 2007).

For boundary conditions corresponding with peak induced strain during the

stance phase, two measures were assessed to define strain alterations arising from

the cortical defect at the time of peak strain. First, the strain concentration factor

was determined as the ratio of maximal induced normal strain around the defect

to the maximal induced normal strain for that cortical region in an intact tibia,

where the maximal strain was defined as the greatest centroidal strain (i.e. the

greatest average elemental strain) and was found to roughly correspond to the

97th percentile. Second, peak strain energy density was determined within a

0.1 mm ring of cortex surrounding each defect site. We then considered alterations

in total strain energy across the entire stance phase by integrating strain energy

density within the 0.1 mm ring surrounding each defect site across each of the 11

boundary condition solutions defined for the stance phase gait at 0.3 m/s and

these data were compared to those derived for the same cortical sites in an intact

tibia.

Last, to confirm surgical feasibility of our approach, we drilled 0.6 mm

diameter holes at each diaphyseal locations (n¼4 mice per location in a survival

surgery) using a micro-drill (Ideal Micro-Drill, model MD-1200, Braintree

Scientific). The hole placement was targeted 1.5 mm proximal to tibiofibular

junction. To quantify the actual hole location with respect to the target location,

Resolved Forces on Tibia

N l ShNormal Shear

1.2

0 8

Forc

e (

N)

0.4

0.00 0 2 0 4 0 6 0 8 10 0.2 0.4 0.6 0.8 1

Stance Phase

s for the hip (H), femur (F), tibia (T), foot (carpal and metacarpal; C), and phalange

tervals through the stance phase (0.0¼stance initiation, 1.0¼stance completion).

re were used to resolve the total force acting on the distal tibia through the stance

and shear force in anterio-posterior axis.

J. Prasad et al. / Journal of Biomechanics 43 (2010) 2765–2770 2767

we scanned the mouse tibia at a linear resolution of 10.5 mm (SCANCO VivaCT 40)

and used the mCT images to measure the actual distance of the defects from the

tibia-fibula junction. To quantify the effect of surgical variability upon induced

strains at each defect site, we then used FEA to determine strain concentration,

strain energy density, the location around the defect when the peak strain energy

density was observed (y, with y¼01 and 901 referring to posterior-most and

proximal-most locations), and total strain energy as described above. Data were

contrasted with those obtained via in silica modeling at each defect site.

Fig. 3. Normal strain distribution in tibia for medial, anterior, and posterior

defects. The holes were created in silica (i.e. in the FE models) to characterize the

strains. As detailed in the sectional (a) and top (b) views, the strain magnitude was

in general elevated by the cortical defect as compared with the intact bone.

However, the peak strains induced at the anterior and posterior defects

substantially exceeded the peak strain associated with the medial defects.

In vivo microCT imaging was used to assess the surgical placement variability of

each site (c).

3. Results

3.1. Characterization and validation of gait induced normal strain

environment

The mean (7SE) peak tensile and compressive strains at tibiadiaphyseal cross-section of interest (1.5 mm proximal to the tibia–fibula junction) were 321725 me (at anterior crest) and�368730 me (at posterior cortex; Fig. 2). The correspondingneutral axis was found to rotate minimally during the stancephase of gait, with a mean range of 20.970.51 (Fig. 2). FEApredicted peak strains were within 20.374.5 me of experimentallymeasured ex vivo strains across all attached strain gages.

3.2. Normal strain alterations due to cortical bone defects

In silica studies indicated that a 0.6 mm diameter bonedefect centered on the anterior cortex would elevate peak normalstrains from 321725 me to 15607300 me (strain concentrationfactor: 4.9; Fig. 3). When the defect was centered on the medialcortex peak, normal strains increased from �180715 me to�6707130 me (average strain concentration factor: 3.7).Similarly, a defect centered on the posterior cortex elevatednormal peak strain from �368730 me to �11987200 me (strainconcentration factor: 3.2).

Fig. 2. Representative FEA solution for the peak normal strain distribution of the

mouse tibia during walking. In general, the anterior and posterior cortices had

tensile and compressive strains, respectively (a). The tibial cross-section 1.5 mm

proximal from tibiofibular junction, locations of peak tension were found on the

anterior/lateral cortex (T) and peak compression on the posterior/medial

cortex (C; b). The neutral axis spanned the medial and lateral/posterior cortices

(solid line) at the time of maximal strain and minimally rotated through the stance

phase was found to rotate only 20.970.51 (bracketed by dotted lines; c).

In the in vivo surgical feasibility study, the mean distance forthe center of each defect sites from the 1.5 mm ideal proximal tothe tibia–fibula junction varied with the defect site. Defects wereplaced at the medial site within 0.1670.11 mm. Defect place-ment at the anterior and posterior sites was less accurate(0.6970.32 mm and 0.7070.14 mm, respectively). The net effectof defect site variation on induced normal strains duringlocomotion was to decrease peak strains by 32711%, 1872%and 1775%, for the anterior, medial, and posterior sites,respectively.

3.3. Strain energy distribution at the defect sites

Maximal strain energy density in the 0.1 mm-thick circumfer-ential volume surrounding each defect varied across sites butwas elevated at each site compared with those in intact bone(Fig. 4). The peak strain energy density at anterior defect(2.3970.15 mJ/mm3; y¼185751) was elevated 497% comparedwith same region in intact bone (0.4070.02 mJ/mm3;y¼250751). At the medial defect site, peak strain energydensity was elevated 74% compared with that in intact bone(0.8770.08 mJ/mm3; y¼335751 vs 0.3770.04 mJ/mm3;y¼3457101, respectively). At the posterior defect site, strainenergy density was elevated 131% (1.5070.30 mJ/mm3;y¼2507401 vs 0.6570.02 mJ/mm3; y¼2757101, respectively).

The relative increase of total strain energy (integrated within a1 mm cylindrical ring surrounding the defect) was similar acrossdefect sites (Fig. 5). Defects induced a 145% increase at theanterior site (37.575.3 nJ vs. 15.370.9 nJ), a 57% increase at themedial site (21.874.6 nJ vs. 13.972.9 nJ), and a 72% increase atthe posterior site (51.975.0 nJ vs. 30.171.4 nJ). When total

Strain Energy Density (Anterior)

Angle (Degrees)

Strain Energy Density (Posterior)Strain Energy Density (Medial)

2.0

1.5

1.0

0.5

0.0

2.0

1.5

1.0

0.5

0.00 90 180 270 360 0 90 180 270 360

2.5

2.0

1.5

1.0

0.5

0.00 90 180 270 360

Angle (Degrees) Angle (Degrees)

θ

Stra

in E

nerg

y D

ensi

ty (u

J/m

m3 )

Stra

in E

nerg

y D

ensi

ty (u

J/m

m3 )

Stra

in E

nerg

y D

ensi

ty (u

J/m

m3 )

Defect Intact

Defect IntactDefect Intact

Fig. 4. Strain energy density in the 0.1 mm thick ring encircling the anterior (b), medial (c), and posterior defects (d) as a function of circumferential angle (y), where 01

(or 3601), 901, 1801, and 2701, respectively, correspond to the posterior-most, proximal-most, anterior-most and distal-most portion of the defect (a). The peak strain

energy density increased around each defect compared with that of the intact bone. The relative elevation of strain energy density was least at the medial defect site (c).

J. Prasad et al. / Journal of Biomechanics 43 (2010) 2765–27702768

strain energy surrounding the defect was integrated through thestance phase, similar increases were observed at each defectsite (anterior: 18.871.7 vs. 7.671.0 nJ/cycle, 149%, p¼0.02;medial: 13.771.2 vs. 5.370.7 nJ/cycle, 158%, po0.02; posterior:20.371.8 vs. 12.070.7 nJ/cycle, 70%, p¼0.02).

4. Discussion and conclusion

We used an inverse dynamics approach complemented withFEA to estimate the normal strain distribution induced in themouse tibia during walking. The approach was validated viaex vivo strain gaging, with predicted strains found to closelyapproximate experimentally measured strains. Peak normalstrains at the mid-diaphysis were similar to those previouslyreported for mice at the gait speed chosen for modeling (De Souzaet al., 2005). As the gait induced strain in mouse tibia was causedpredominantly by anteroposterior bending, the neutral axis (ofbending) was found to be confined to a small region of the boneduring the stance phase of gait. Based on this information, threecandidate defect sites (at the same transverse section) wereidentified for their distinct strain environments (tension, com-pression, or minimal strain magnitude near the neutral axis). Aswas anticipated, cortical bone defects at each site amplified

continuum peak normal strain, peak strain energy density, totalstrain energy, and strain energy density integrated through thestance phase of gait, but the relative increase compared withintact tibiae varied with defect site.

Two assumptions in our model, our focus on model walking at0.3 m/s and restricting our analysis to 2-D kinematics, representclear simplifications of the complex loading to which long bonesare exposed during 24 h (Fowler et al., 2001; Judex and Carlson,2009). With regard to the first assumption, previous investigatorshave shown that daily mouse activities consist of lying (39.9%),sitting (27.8%), standing on all four limbs (11%), standing on twohind limbs (8.4%), walking (9.1%), running (0.1%), and climbing(3.7%; Carlson and Judex, 2007). Further, during spontaneouslocomotion mice demonstrate gait speeds ranging from 0.1 to0.3 m/s (Serradj and Jamon, 2009). During motion outside ofwalking, the morphology of the tibia will predominate, so thatincreased speed or in plane movements such as stepping up ordown should yield similarly disparate strain distributions aswere defined here. However, if significant torque was induced(e.g., rapid pivoting during a change of direction) such movementwould result in an altered gait induced strain distribution.To address this issue, our approach at strain quantification couldbe coupled with 24 h activity counts and quantification of3-D ground reaction forces during phenotypic activity. A more

Defect Intact Defect Intact

60 60

50 50

40 40

30 30

20

Stra

in E

nerg

y (n

J)

20

Stra

in E

nerg

y (n

J)

10 10

00.0

00.0 0.2 0.4 0.6 0.8 1.0

Stance Phase Stance Phase

Defect Intact

Defect Intact60

5020

40* #

30

20

Stra

in E

nerg

y (n

J)

10

10

ISE

(nJ/

cycl

e)

00.0 0.2 0.4 0.6 0.8 1.0

Stance Phase

0Anterior Medial Posterior

0.2 0.4 0.6 0.8 1.0

Fig. 5. Total strain energy surrounding the anterior (a), medial (b), and posterior (c) defects through the stance phase of gait compared to the same site in intact tibiae.

When integrated across the stance phase, integrated strain energy was significantly increased at each defect site (d, n, p¼0.02). Integrated strain energy at the medial defect

was significantly less than at either the anterior (#, po0.05) or posterior defects (po0.01).

J. Prasad et al. / Journal of Biomechanics 43 (2010) 2765–2770 2769

technical limitation with our model is associated with surgicalvariability in placing defects at the desired sites. We found thisvariability was the greatest at the anterior and posterior sites dueto bone curvature and more challenging surgical accessibility,respectively. The impact of this variability on the potential of themodel can be mitigated, as was done here, using d0 in vivo

microCT imaging to develop animal specific FEA in order toquantify defect site strain alterations for each mouse. Anotherlimitation is that the cortical defect is different from fracture withrespect to its mechanical cause and physical appearance, andthere was no evidence found in the literature that the type(tensile vs compressive) or the variation of strain around a holeaffects the healing process. However, the initial healing cascadefollowing a cortical defect is the same as that in fracture andtherefore the model will enable examination of how mechanicalstimuli influence early cellular events required for successful bonehealing (Uusitalo et al., 2005). Extrapolating from the literaturedescribing the interaction of mechanical stimuli and fracturehealing, it could be speculated that endochondral ossificationwould most likely be present at the posterior defect site due tohigh compressive strains, while the other two sites would tendtoward intramembranous ossification (Claes and Heigele, 1999;Isaksson et al., 2009). Additionally, periosteal callus volumes atanterior and posterior sites would be expected to be larger thanthat of the medial site due to larger strain magnitudes. In the caseof disuse, the three sites are expected to have same callus volumeand undergo intramembranous ossification (Claes and Heigele,

1999; Isaksson et al., 2006). It is our hope that this model willenable clarification of these hypotheses.

Despite these limitations, we believe this model has uniquepotential to explore the action of physiologically induced mechanicalstimuli and bone healing at a variety of levels. At the continuumlevel, the non-uniform normal strain environment of the tibia willenable determination of whether bone healing is influenced by acompressive or tensile strain environment (Claes and Heigele, 1999;Isaksson et al., 2006). The difference in strain magnitudes and strainenergy density at defect sites (particularly compared with themedial defect bridging the neutral axis) will enable exploration ofthe influence of strain magnitude upon healing (Claes and Heigele,1999; Isaksson et al., 2009). At a more cellular level, variations instrain and strain energy density around the periosteal surface of thedefect could be coupled with histological assays to define whetherthe initial cellular events crucial to successful defect healing areenhanced (or inhibited) by mechanical stimuli. Finally, as it ispossible to non-invasively load the murine tibia (Srinivasan et al.,2002; Sugiyama et al., 2008), it will also be possible to superimposeexogenous mechanical loading on the model in order to defineloading regimens that hold potential to enhance healing.

Conflict of interest statement

The authors do not have any conflict of interest regarding thecontent of this manuscript.

J. Prasad et al. / Journal of Biomechanics 43 (2010) 2765–27702770

Acknowledgements

This work was supported by the Sigvard T. Hansen, Jr.Endowed Chair, the Zimmer Fracture Biology Professorship, anda grant from Synthes, Inc.

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