Strain Gradients Correlate with Sites of Periosteal Bone Formation


  • Ted S. Gross,

    Corresponding author
    1. Musculo-Skeletal Research Laboratory, Department of Orthopaedics, State University of New York at Stony Brook, Stony Brook, New York, U.S.A.
    • Ted S. Gross, Ph.D. Department of Orthopaedic Surgery P.O. Box 670212 231 Bethesda Avenue University of Cincinnati Cincinnati, OH 45267–0212 U.S.A.
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  • Jonathan L. Edwards,

    1. Department of Orthopaedic Surgery, University of Cincinnati, Cincinnati, Ohio, U.S.A.
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  • Kenneth J. Mcleod,

    1. Musculo-Skeletal Research Laboratory, Department of Orthopaedics, State University of New York at Stony Brook, Stony Brook, New York, U.S.A.
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  • Clinton T. Rubin

    1. Musculo-Skeletal Research Laboratory, Department of Orthopaedics, State University of New York at Stony Brook, Stony Brook, New York, U.S.A.
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  • Presented, in part, at the 40th meeting of the Orthopaedic Research Society (New Orleans, LA, U.S.A.) in February, 1994.


We examined the hypothesis that peak magnitude strain gradients are spatially correlated with sites of bone formation. Ten adult male turkeys underwent functional isolation of the right radius and a subsequent 4-week exogenous loading regimen. Full field solutions of the engendered strains were obtained for each animal using animal-specific, orthotropic finite element models. Circumferential, radial, and longitudinal gradients of normal strain were calculated from these solutions. Site-specific bone formation within 24 equal angle pie sectors was determined by automated image analysis of microradiographs taken from the mid-diaphysis of the experimental radii. The loading regimen increased mean cortical area (±SE) by 32.3 ± 10.5% (p = 0.01). Across animals, some periosteal bone formation was observed in every sector. The amount of periosteal new bone area contained within each sector was not uniform. Circumferential strain gradients (r2 = 0.36) were most strongly correlated with the observed periosteal bone formation. SED (a scalar measure of stress/strain magnitude with minimal relation to fluid flow) was poorly correlated with periosteal bone formation (r2 = 0.01). The combination of circumferential, radial, and longitudinal strain gradients accounted for over 60% of the periosteal new bone area (r2 = 0.63). These data indicate that strain gradients, which are readily determined given a knowledge of the bone's strain environment and geometry, may be used to predict specific locations of new bone formation stimulated by mechanical loading.


Bone formation initiated by mechanical loading is predominantly confined to the portion of the skeleton that undergoes loading.(1,2) Site-specific bone formation is also evident within a given bone. Segments of cortex frequently demonstrate substantial surface modeling, while other sites within the same transverse cross-section remain quiescent.(3–5) Because bone formation is site-specific and the skeleton is sensitive to particular types of stimuli (e.g., strain magnitude,(6,7) strain rate,(8,9) strain frequency,(10) and the strain tensor(11)), it has been inferred that a spatial relation exists between specific mechanical stimuli and sites of bone formation.

Numerical models of bone adaptation illustrate this relation, as remarkably realistic trabecular morphologies have been constructed based upon a general strain stimulus derived from the loading environment of the bone. One parameter often used in this manner is strain energy density (SED), a scalar reflecting contributions from all stresses and strains acting at a given location.(12,13) In a simplified sense, these models suggest that bone mass increases where strain, or SED, is large, and decreases where this stimulus is small. However, preliminary attempts to relate the magnitude of a mechanical stimulus with specific sites of bone formation observed in an in vivo model have met with limited success.(8,14,15)

One explanation for the poor correlation between strain (or stress) magnitude and specific sites of bone formation is that the stimulus for adaptation may be a by-product of the mechanical environment (e.g., fluid flow or streaming potentials) rather than matrix deformation itself. Theoretical and experimental studies suggest that fluid flow is integral to the process by which bone perceives and responds to mechanical stimuli. Within this paradigm, bone cell populations respond to fluid flow induced by deformation of the tissue. Based on theoretical considerations, osteocytes have been proposed to be capable of directly sensing flow-induced shear stress.(16) Cell culture experiments indicate that osteocytes and osteoblasts are extremely responsive to pulsatile flow.(17,18) As well, fluid flow past charged surfaces induces streaming potentials, long hypothesized as a mechanism by which bone cells may derive physiologic signals from tissue deformations.(19,20) Interestingly, loading a bone in torsion (a condition in which fluid flow is small) precipitates little cellular activity as compared with loading the same bone in bending (which induces regions of high fluid flow).(21)

Figure FIG. 1.

Radiograph of right turkey wing illustrating a mechanically isolated right radius. External loads were applied via the transfixing Steinmann pins. The isolated diaphysis is 9 cm long and 0.8 cm wide.

If fluid flow participates in the pathway leading to the initiation of bone formation, it should be possible to correlate locations of high flow with locations of surface modeling. However, estimates of fluid flow within the tissue are highly dependent upon permeability constants and are difficult to validate experimentally. Strain gradients, in contrast, are readily determined given a knowledge of the bone's strain environment and the geometry of the bone. In the absence of substantial intramedullary pressure, pressure gradients are proportional to strain gradients within the bone.(22) These pressure gradients, in turn, directly influence fluid flow within the tissue. We have therefore hypothesized that peak magnitude strain gradients (as an indirect measure of maximal fluid flow) are spatially correlated with sites of periosteal bone formation. In this study, we examined this hypothesis by relating sites of periosteal bone formation observed in an in vivo model of bone adaptation with strain gradients induced by the daily loading regimen. As a comparison, we also correlated sites of bone formation with strain energy density, a scalar measure of stress/strain magnitude that has minimal relation to fluid flow.


In vivo model

Under general halothane anesthesia, 10 adult male turkeys underwent functional isolation of their right radii.(23) The distal metaphysis was directly approached through a skin incision, while portions of the brachialis, common digital extensor, and supinator medialis were mobilized to provide access to the proximal dorsal periosteum. A parallel-sided template was attached with towel clamps, and two parallel transverse osteotomies were performed at each of the proximal and distal metaphyses. Small (2–3 mm thick) cross-sections of bone were removed from both sites, and the exposed diaphyseal bone ends were freed of soft tissue. Stainless steel caps were filled with 5 cc of methyl-methacrylate and placed over the exposed bone ends. Parallel transcutaneous Steinmann pins were then passed through holes drilled in the radius and mating holes in the caps (Fig. 1). Turkeys were assigned to one of two pin orientations. The second pin orientation (“B”) was rotated 90° in the transverse plane as compared with the first orientation (“A”). The incisions were sutured, fixator clamps applied to the pins to prevent accidental loading, and the animal recovered from anesthesia. The left radius of each animal served as an intact contralateral control. The animals were group housed, with food and water administered ad libitum. All procedures were reviewed and approved by the State University of New York at Stony Brook Laboratory Animal Users Committee and met all federal guidelines for the care and welfare of laboratory animals.

The animals underwent 5 days of loading per week for 4 weeks. After removing the fixator clamps, 100 cycles of a trapezoidal loading waveform (2.5 s total period; 10.8 N peak load; 120 N/s load rate; peak load maintained for 1.8 s) was applied to the dorsal aspect of the preparation using a distending pneumatic cylinder with custom mating sleeves. The apparatus applied an axial load to the top of the distal pin, which, due to the large lever arm (13.2 cm), placed the radius primarily in bending (<1% axial tension). The magnitude of the applied axial load, and its resulting moment, were identical for both pin orientations.

Figure FIG. 2.

(A) Scanned microradiograph of radius mid-diaphysis cross-section following the 4 week loading regimen. Nonuniform new bone formation is evident around the periosteum (arrows). (B) An automated edge detection algorithm was used to define the boundaries of the original cortex (white border) and new periosteal bone (black border). (C) The original cortex (dark gray), new bone (light gray), and porosities (white) were then quantified using pixel summation.

Image analysis

Following sacrifice, thick sections (300 μm) were extracted from the midshaft of the experimental and contralateral control radii. These sections were ground to 125 μm, and microradiographs were taken of the experimental (right) and corresponding control sections (left). The microradiographs were enlarged (5×) and scanned at 150 dpi using a flatbed scanner (final resolution, 38 μm/pixel). This resolution was sufficient to identify a 200 μm diameter resorption cavity, but not a 50 μm diameter Haversian canal.

An automated edge detection algorithm was implemented to identify boundaries between the original cortex, new bone area, and intracortical porosities. The algorithm identified edges based upon pixel intensity variations(24) and was validated by comparing mean (±SE) areal determinations of the original cortex in the experimental radii with the cortical area from the contralalateral radii (25.8 ± 1.3 vs. 26.1 ± 1.2 mm2). The observed areal symmetry (1.2%) was similar to that previously reported for the turkey ulna using standard digitizing techniques.(25) Areal properties (original cortical area, area of porosity, periosteal new bone area) were determined using pixel summation (Fig. 2). Site-specific bone formation was determined by dividing the cross-section into 24 equal angle pie sectors using the centroid of the original cortex. The amount of periosteal new bone area within each sector was then determined by pixel summation. A nonparametric Wilcoxon test (p = 0.05) was used to determine whether the loading regimen significantly increased the periosteal envelope area.

Finite element analysis

Finite element meshes were developed for each animal based upon the geometry of the contralateral radius. Contact radiographs of cross-sections taken at 1 mm intervals through the length of the bone were backlit and directly scanned on a flatbed scanner at 600 dpi. Using grooves machined into the acrylic blocks in which the bones were embedded, the images were aligned, mirrored into a corresponding right radius image, and then used to generate mesh coordinates. Based on a series of parametric studies undertaken to optimize mesh density, each model contained 2088 20-node bricks arranged in 29 longitudinal layers, each consisting of 3 radial layers of 24 circumferential elements (each circumferential element corresponded to 1 of the 24 pie sectors). The longitudinal layers were coarsest at the bone ends (5 mm thick), with a finer resolution (1 mm) spanning the middle 2 cm of the diaphysis. Forces were applied to nodes corresponding to the loading pins such that the summed nodal forces (and their respective lever arms) were equivalent to the global force and bending conditions, and that the bending moment was equal at each node (i.e., nodes furthest from the centroid received the smallest applied force).

Orthotropic moduli were defined using a shape intrinsic cylindrical coordinate system (r, θ, z), with the z-axis corresponding to the longitudinal axis of the bone.(26) Longitudinal material properties were determined by ultrasound on mid-diaphyseal specimens taken from the radii of three age- and weight-matched animals.(27) Given the small cortical thickness of the radius (0.6 mm), only the longitudinal modulus could be measured directly (Ezz = 22.5 GPa). The remaining orthotropic moduli and Poisson ratios were derived from data reported for the turkey ulna.(28) The ulna moduli were expressed as a percentage of the ulna longitudinal modulus, and these values were used to establish radius values in proportion to the measured radius longitudinal modulus (Err = 10.5; Eθθ = 16.8; Grθ = 3.4; Grz = 4.3; Gθz = 6.6 GPa; υ = 0.28; υrz = 0.33; υθz = 0.26).

Figure FIG. 3.

A mid-diaphysis cross-section is shown divided into 24 equal angle pie sectors. One of the sectors is expanded into three dimensions to illustrate the strain gradient calculation. As an example, the radial gradient of normal strain (RG) was derived from the difference in normal strain magnitude at the periosteal and endocortical surfaces of the sector (εmath image − εmath image) divided by the linear distance between the two points (DR).

Model strain predictions were validated by comparing predicted surface strains (principal and coordinate) with those measured using triple rosette strain gages on two separate calibration bones (rosettes were attached on the dorsal, cranial, and caudal cortices of the midshaft and on the dorsal cortex 1.5 cm distal to the midshaft). When normalized to strain magnitude, mean predicted maximum principal strains (±SE) were within 9.6 (±2.4)% of those measured by strain gages. Mean predicted longitudinal normal strains (εzz) and surface shear strains (εθz) differed by 33 ± 34 με and 3 ± 45 με from experimental values. The correspondence between predicted and measured strains was achieved without altering material properties or boundary conditions.

Correlation between mechanical stimuli and bone formation

On the periosteal surface of each sector, strain gradients acting in the radial, circumferential, and longitudinal directions were determined by dividing the change in normal strain in a given direction by the linear distance between points of measure (Fig. 3). The SED (1/2τijεij) at a site bisecting each sector was obtained from the finite element solutions. To perform regressions on all 10 animals as one group, all strain and areal data from the 5 turkeys assigned to pin orientation “A” were rotated 90° to coincide with data obtained for pin orientation “B.” This rotation mimicked the rotation of strain environments achieved by the different pin orientations in the two loading groups. Site-specific areal data and strain parameter data were averaged across animals. Linear regression was then used to correlate the spatial location and magnitude of the periosteal new bone area with the numerically determined strain parameters.

Figure FIG. 4.

Mean mid-diaphyseal normal strain distributions (with 500 με isopleths) for the two loading pin orientations (“A” and “B”). The neutral axis of bending is noted by the dashed black line. Maximal engendered strains were similar for the two pin orientations, but the location of the neutral axis was rotated 89° clockwise in the second pin orientation. Dorsal (D) and caudal (CA) cortices are noted.


Induced strain environment

Loading the straight, cylindrical radius in bending induced a strain environment in which the neutral axis of bending passed near the section centroid. Maximum mid-diaphyseal compressive and tensile strains (±SE) averaged −1710 ± 100 με and 1550 ± 80 με, respectively. The mean orientation of the neutral axis (and therefore the locations of peak strain magnitude) differed by 89° for the two pin orientations (Fig. 4). Maximum circumferential and radial strain gradients were approximately twice the magnitude of peak longitudinal strain gradients. Sites of maximal circumferential strain gradients differed from the sites of maximal SED (Fig. 5).

Areal adaptation

The 4-week loading regimen induced substantial periosteal new bone formation in the experimental radii. No bone formation was observed in the unloaded, contralateral radii. Mean cortical area was significantly increased 32.3 ± 10.5% (p = 0.01). Across animals, some periosteal bone formation was observed in every sector. The mean amount of periosteal bone formation within each sector was not uniform but was focused in specific areas. In pin configuration “A,” the cranial/ventral cortex (sectors 4 through 9) accounted for 26% of the original cortical area, but was associated with 50% of the periosteal new bone area. In pin configuration “B,” the dorsal cortex (sectors 12 through 17; 24% of original cortical area) accounted for 57% of the periosteal new bone area (Fig. 6).

Figure FIG. 5.

(A) Mean (±SE) circumferential (CG), radial (RG), and longitudinal strain gradients (LG) and (B) SED for each of the 24 sectors. Maximal circumferential strain gradients (sectors 5 and 16) and maximal SED (sectors 11 and 24) were located in different portions of the cortex.

Figure FIG. 6.

Mean periosteal new bone area within each sector (±SE) for the two pin configurations. Animals loaded using pin configuration “A” demonstrated greatest periosteal new bone area on the cranial/ventral cortex (sectors 4 through 9). The location of maximal new bone area for animals exposed to pin configuration “B” was shifted to the dorsal cortex (12 through 17). This shift mimicked the rotation of the strain environments between the two pin configurations.

Correlation between mechanical stimuli and bone formation

Circumferential strain gradients (r2 = 0.36, F = 0.00) and radial strain gradients (r2 = 0.24, F = 0.02) were most strongly correlated with the observed periosteal bone formation. Longitudinal strain gradients (r2 = 0.08, F = 0.17) and SED (r2 = 0.01, F = 0.68) were poorly correlated with bone formation. Circumferential strain gradients predicted new bone area as well as radial and longitudinal gradients combined (r2 = 0.25, F = 0.05). The combination of circumferential, radial, and longitudinal strain gradients accounted for over 60% of the periosteal new bone area (r2 = 0.63, F = 0.00; Fig. 7).

Figure FIG. 7.

(A) Mean periosteal new bone area within each sector (±SE) averaged across all ten animals. Data from animals exposed to pin orientation “A” have been rotated to coincide with those from pin orientation “B.” (B) Mean (±SE) periosteal new bone area is then plotted versus the areal increase predicted by the regression containing the circumferential, radial, and longitudinal strain gradients (wide line). (C) The strong correlation between observed areal adaptation and that predicted by this regression is illustrated.


At the organ level, bone responds to mechanical loading in a dose–response manner (i.e., the greater the induced strain magnitude, the more prolific the bone hypertrophy). For a given cross-section, however, our data indicate that locations of maximal strain magnitude (as depicted by SED) were minimally related to the specific sites where bone formation was initiated. Instead, we found that peak magnitude strain gradients were spatially associated with locations of new bone formation. These results have implications toward the design of exercise regimens whose goal is to enhance the structural integrity of the skeleton. Because strain gradients are closely related to fluid flow, these data also support the thesis that fluid flow participates in the pathway leading to initiation of surface osteoblast bone formation.

As with any in vivo model, conclusions should be considered with respect to the limitations of the model and the variability of the data. Functional isolation of the radius necessitates an invasive surgery which may interfere with the process we seek to study. The absence of periosteal or endocortical bone formation when our model is subjected to disuse suggests that the surgery, by itself, does not initiate mid-diaphyseal bone formation.(23) As well, the absence of adaptive activity in the intact, contralateral radii indicates that the turkeys were not systemically influenced by surgery and daily handling. Two pin orientations were included in the study to control for the possibility that only portions of the cortex are capable of initiating bone formation in response to mechanical loading. Because the predominant area of periosteal bone formation rotated in tandem with the shift of strain environments, it is probable that the bone was responding to some aspect of the induced mechanical environment. Also, the daily strain environment of the turkey radius has not yet been defined. However, peak normal strains in a variety of animals and bones range primarily between 2000 με and 3000 με.(29) Within this range, the peak strains induced in this study (1700 με) would appear to be physiologic in magnitude.

Because of the fragile construct of the preparation, the loading regimen was limited to 4 weeks to ensure completion of the protocol. As a result, the observed periosteal bone formation represents an early, but representative, stage in an adaptive process that ultimately leads to consolidation of woven bone into lamellar bone at the same sites.(30) In this study, the mean coefficient of variation (CV = SD/mean) for periosteal expansion was 106%. This value is similar to the variability reported for other in vivo models of cortical bone adaptation. For example, the rat four point bending model (55%, 120%) and the canine osteotomy model (175%) bracket our data.(7,31,32) Few reports of site-specific bone formation are available in the literature, but the CV for our site-specific data (146%) is similar to that reported for the sheep forelimb model (175%).(8) From this, we conclude that the across animal variability observed in our model is similar to that reported for other in vivo models of bone adaptation.

We observed a highly nonuniform distribution of periosteal new bone area, with a substantial percentage of the periosteal expansion associated with locations of small strain magnitude. From a structural perspective, the affinity for periosteal bone formation to occur near sites of minimal strain is counterintuitive. In fact, many analytic and numerical models suggest that some aspect of strain (or stress) magnitude is the stimulus responsible for controlling adaptation in the skeleton, and predict that new bone will be deposited at the location of the peak stimulus. SED, in particular, has proven effective in this formulation.(12,33,34) Because SED is a scalar reflecting contributions from all stresses and strains acting at a single site (i.e., both normal and shear), it conveniently represents all the induced stimuli available to the cellular population. At the same time, this lack of specificity may explain the poor correlation of SED with specific locations of bone formation.

From a physiologic perspective, however, the strong spatial relation between strain gradients and induced bone formation is intriguing. Given the association between strain gradients and fluid flow, a number of processes could be hypothesized to serve as a mechanism underlying this correspondence (e.g., flow induced shear stress or streaming potentials).(16,19,20) These data do not indicate whether strain gradients initiate this pathway, nor do they indicate which cell population is responsible for perceiving this stimulus. Future efforts will seek to validate the relation between strain gradients and sites of bone formation under alternate loading conditions. Preliminary data in a nonsurgical model of bone adaptation suggests that strain gradients account for over 60% of the sites bone formation initiated by a treadmill running regimen.(35) In addition, molecular techniques such as reverse transcript polymerase chain reaction (PCR)(36,37) or differential display,(38) may prove particularly useful in investigating the mechanisms governing these tissue level observations.

Because current exercise regimens are quite limited in their ability to elevate bone mass once skeletal maturity is reached,(2,39) the potential relation between strain gradients and sites of bone formation may prove useful in optimizing exercise protocols. By aligning the loading environment of the bone appropriately, it may be possible to induce bone formation on the portions of the cortex that would most effectively enhance the structural integrity of the bone. Further, because the fluid flow that appears to underlie this process is influenced by the frequency content of the loading regimen, it may be possible to accentuate fluid flow by elevating loading frequency, while simultaneously reducing load magnitude.

In conclusion, the 4-week loading regimen stimulated site-specific periosteal new bone formation. The predominant areas of new bone formation were highly correlated with peak magnitude strain gradients. These data suggest that strain gradients may be used to predict specific locations of new bone formation stimulated by mechanical loading.


The authors gratefully thank Dr. Kelly Baker for the ultrasound material property measurements, Dr. Farshid Guilak for the initial development of the image analysis software, and Dr. Adrian Wilson for his comments on the modeling portions of the manuscript. This work was supported in part by NIH AR39278.