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Cellular accommodation and the response of bone to mechanical loading
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Journal of Biomechanics 38 (2005) 1838–1845 www.elsevier.com/locate/jbiomech www.JBiomech.com
Cellular accommodation and the response of bone to mechanical loading Jennifer L. Schriefera, Stuart J. Wardenb,c, Leanne K. Saxonc, Alexander G. Roblingd, Charles H. Turnerc,e, a
Department of Biomedical Engineering, Purdue School of Engineering and Technology, Indiana University Purdue University Indianapolis, Indianapolis, IN, USA b Department of Physical Therapy, School of Health and Rehabilitation Sciences, Indiana University, Indianapolis, IN, USA c Department of Orthopaedic Surgery, Indiana University School of Medicine, Indianapolis, IN, USA d Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA e Biomechanics and Biomaterials Research Center, Indiana University School of Medicine, Indianapolis, IN, USA Accepted 17 August 2004
Abstract Several mathematical rules by which bone adapts to mechanical loading have been proposed. Previous work focused mainly on negative feedback models, e.g., bone adapts to increased loading after a minimum strain effective (MES) threshold has been reached. The MES algorithm has numerous caveats, so we propose a different model, according to which bone adapts to changes in its mechanical environment based on the principle of cellular accommodation. With the new algorithm we presume that strain history is integrated into cellular memory so that the reference state for adaptation is constantly changing. To test this algorithm, an experiment was performed in which the ulnae of Sprague–Dawley rats were loaded in axial compression. The animals received loading for 15 weeks with progressively decreasing loads, increasing loads, or a constant load. The results showed the largest increases in geometry in the decreasing load group, followed by the constant load group. Bone formation rates (BFRs) were significantly greater in the decreasing load group during the first 2 weeks of the study as compared to all other groups (Po0:05). After the first few weeks of mechanical loading, the BFR in the loaded ulnae returned to the values of the nonloaded ulnae. These experimental results closely fit the predicted results of the cellular accommodation algorithm. After the initial weeks of loading, bone stopped responding so the degree of adaptation was proportional to the initial peak load magnitude. r 2004 Elsevier Ltd. All rights reserved. Keywords: Bone density; Biomechanics; Mechanical loading; Osteoporosis
1. Introduction The idea that bone structure and adaptation can be predicted by mathematical laws originated with Wolff and his contemporaries (Meyer, 1867; Wolff, 1892). It is an appealing idea and has been the genesis for many engineering studies of bone adaptation resulting in Corresponding author. Indiana University, 1120 South Dr., FH 115, Indianapolis, IN 46202, USA. Tel: +1-317-274-3226; fax: +1317-278-9568. E-mail address: [email protected] (C.H. Turner).
0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.08.017
algorithms to describe the process(es). These algorithms are based upon established mechanical engineering concepts like structural optimization or negative feedback control, and assume a mechanical strain setpoint or attractor state exists within bone tissue. For bone to adapt to a new mechanical loading state, bone cells must have some memory of their previous mechanical environment in order to determine that the new environment is different and requires a response. As Lanyon put it, bone adaptation is ‘‘error driven’’ (Lanyon, 1992). Bone cells must process loading information locally because bone tissue is poorly
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innervated and, unlike many mechanoreceptor cells, cannot rely on the central nervous system to integrate and distribute information about mechanical signals. Frost proposed that adaptive changes in bone shape occur when minimum effective strain (MES) thresholds are surpassed (Frost, 1983). Thus he assumes that bone cells are somehow programmed with the value of the MES which sets the threshold for a response. However, for the cellular response to be bone-specific the MES must vary from bone to bone. Furthermore, the MES may well be location dependent within bones. Most long bones are loaded in bending (Bertram and Biewener, 1988), creating a strain gradient across the bone section that has a value of zero along the neutral axis. To avoid excessive bone resorption at the neutral axis, the MES would have to vary considerably across the bone section. Each mechanosensitive cell must have some strain threshold above which a mechanical signal causes a cellular response. It is argued here that this threshold need not be a set value but is a product of local strain history. It is well known that many cell types, including osteoblasts, reorganize their cytoskeletons in response to a mechanical stimulus (Pavalko et al., 1998). Cytoskeletal reorganization in turn changes a cell’s mechanosensitivity, allowing the cell to accommodate to its strain environment. Also cellular mechanosensitivity may be altered through reworking the local environment around the cell (Rubin et al., 2002). Osteocytes are considered to be part of the mechanosensory apparatus in bone and any change in their extracellular microenvironment could change their mechanosensitivity. The extracellular environment of osteocytes does not remain constant as these cells actively form and remove layers of matrix on the surface of their lacunae (McKee and Nanci, 1996). It is assumed that when a strain threshold is surpassed the sensor cells will gradually accommodate to the new state, either by cytoskeletal reorganization or by changing the extracellular microenvironment. Using this definition of cellular accommodation, a setpoint could be determined simply by summation of the past history of daily strain stimuli (Turner, 1999). The bone formation (or resorption) response would then be proportional to the difference between the new strain stimulus and the ever changing setpoint. The temporal skeletal adaptive response will depend, to some extent, upon how the strain history is integrated into cellular memory. It is assumed that the memory of the cells’ previous loading state fades exponentially. An algorithm using the principle of cellular accommodation predicts a different time course of bone adaptation than do strain feedback algorithms. Consider three mathematical models for the adaptation of cancellous bone structure, starting with the generic
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model for adaptation of cancellous bone density: qr ¼ Bðf S0 Þ; (1) qt where r is density in g/cm3, t is time in weeks, B is a rate constant (g/cm3-wk), f is the strain stimulus function, and S 0 is the equilibrium strain stimulus (Cowin and Hegedus, 1976; Fyhrie and Schaffler, 1995). Eq. (1) does not account for the changes in trabecular surface area that accompany changes in cancellous density (Martin, 1984). Cellular activity is somewhat dependent upon the available surface area, i.e., trabecular remodeling activity is surface-driven. To account for this, the model can incorporate the effect of changing a specific surface (trabecular surface area normalized by volume) (Beaupre´ et al., 1990). This model takes the general form: qr ¼ BðrÞðf S 0 Þ; (2) qt where BðrÞ is a rate function that is proportional to the specific surface of the cancellous structure (specific surface is defined as a function of r) (Martin, 1984). Eq. (2) can be expanded further to correct for the effect of cellular accommodation: qr ¼ BðrÞðf F ðf; tÞÞ qt and
(3)
F ðf; tÞ ¼ S0 þ ðf S 0 Þð1 et=t Þ; where F takes the form of a relaxation function derived from the differential equation: dF ðf F Þ ¼ ; (4) dt t where t is the time constant describing the rate at which accommodation takes place. Cellular accommodation results from the propensity of F to approach f with time thus nulling the driving force for density change. There are certain characteristics of the algorithm incorporating cellular accommodation (Eq. (3)) that clearly distinguish it from the strain feedback algorithm (Eq. (1)). When bone is mechanically loaded over time with either progressively increasing or decreasing load magnitudes, the two algorithms predict extremely different results (Fig. 1). Presuming that the dose–response curve for mechanically induced bone formation is non-linear, as has been demonstrated experimentally (Turner et al., 1994), the response predicted by the cellular accommodation algorithm is path dependent. Path dependence, as the term is used here, means that the outcome from the algorithm is dependent upon the temporal sequence of the preceding mechanical loading events. Negative strain feedback algorithms (e.g. Eq. (1)) typically are not path dependent. Consequently, an experiment that varies the temporal sequence of loading
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bone bending model. Therefore, the purpose of this paper is to compare the experimental adaptation response of bone to changing mechanical loads to the predicted responses of the strain feedback and cellular accommodation algorithms.
2. Methods Fifty-three Sprague–Dawley female retired breeder rats were purchased from Charles River (Wilmington, MA). The rats were housed two per cage at Indiana University’s Laboratory Animal Resource Center, and provided standard rat chow and water ad libitum throughout the experiment. Once the animals arrived at the university they were given 2 weeks of acclimatization. All procedures were performed in accordance with the Institutional Animal Care and Use Committee guidelines of Indiana University. Animals were randomized to four loaded groups and two control groups (Fig. 2). In the loaded animals, the right ulna was loaded using the axial loading model described previously (Torrance et al., 1994). Compressive loads were applied as a haversine waveform 3 days per week for 360 cycles per day at 2 Hz under isofluraneinduced anesthesia (3.5% isoflurane delivered at 1.5 L/ min was used for induction and 2.5% isoflurane at 1.5 L/ min was used for maintenance). The left ulna served as a contralateral control. The progressively decreasing load group (n ¼ 10) was loaded with a peak force of 13.5 N for the first 5 weeks, 11.3 N for the next 5 weeks, and 9.0 N for the final five weeks (Fig. 2). The progressively increasing load group (n=11) received the loads in the opposite order compared to the decreasing load group, and the constant load group (n=10) received 11.3 N for the
Fig. 1. (A) Predicted results of the progressively decreasing load group and increasing load group in a long-term loading protocol according to the strain feedback algorithm. Despite the differences in the loading protocols, the two patterns of loading result in the same increase in cross-sectional area. (B) Predicted results of the decreasing load group and increasing load group in a long-term loading protocol according to the cellular accommodation algorithm. The predicted results show very different final cross-sectional areas for the two groups.
will distinguish cellular accommodation from negative feedback algorithms. To test this theory we performed an experiment investigating the adaptation of bone to decreasing peak mechanical loads, increasing peak mechanical loads, or constant peak loads in a rat model. To avoid the added complexities of bone surface area changes and trabecular orientation, this experiment focuses on cortical, rather than trabecular, bone adaptation using a long
Fig. 2. Experimental protocol. A and C indicate an alizarin or calcein injection was given on the Friday of the corresponding week, respectively. S indicates the Monday that sacrifice occurred. =decreasing load group, =increasing load group, =constant load group, =last 5 weeks group, AMC=age-matched control group, and BLC=baseline control group. Numbers for , , , and represent peak load magnitudes introduced to the right ulna in that week.
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entire 15 weeks. Thus, these three groups all received the same average load throughout the study. The last 5 weeks loaded group (n=9) rested for the first 10 weeks of the study and then received a load of 13.5 N for the final 5 weeks of the study. The purpose of the last 5 weeks group was to compare high magnitude loading at the end of the study to the increasing load group which received 10 weeks of prior loading. There were two control groups, an age-matched control (AMC; n=8) that was sacrificed on the same day as the loaded groups, and a baseline control (BLC; n=5) that was sacrificed on the first day of loading. At the end of weeks 1 and 2 as well as 11 and 12, the animals received intraperitoneal injections (7 mg/kg body weight) of calcein (Sigma, St. Louis, MO) (Fig. 2). At the end of weeks 6 and 7 and weeks 15 and 16 of the study the animals received intraperitoneal injections (25 mg/kg body weight) of alizarin complexone (Sigma, St. Louis, MO). Body mass measurements were collected once per week throughout the length of the study. All animals except the baseline controls were sacrificed 16 weeks after the start of loading by anesthetic inhalation and cervical dislocation. After sacrifice, the right and left ulnas of the animals were dissected free, cleaned, and stored in 70% EtOH. Bone mineral content (BMC) was measured from a single tomographic slice at the ulna midshaft using a Norland Medical Systems Stratec XCT Research SA+peripheral quantitative computed tomography scanner (pQCT; Stratec Electronics, Pforzheim, Germany), as described previously (Robling et al., 2002b). The ulnas were also scanned at the midshaft using a destop mCT (Scanco Medical AG, Bassersdorf, Switzerland) to produce images, as described previously (Robling et al., 2002a). The total area (mm2), and minimum (IMIN; mm4) and maximum (IMAX; mm4) second moments of area were calculated by importing the images into Scion Image v4.0.2 for Windows (Scion Corporation, Frederick, MD). Histologic sections were cut from plastic-embedded bones, as was reported previously (Hsieh and Turner, 2001). One midshaft slice was analyzed from each ulna using a Nikon Optiphot fluorescence microscope operating with a Bioquant digitizing system (R&M Biometrics, Nashville, TN). Bone formation rates (BFRs) on the periosteal surface were calculated as was previously described (Robling et al., 2001). To estimate the strain (me) applied to the ulna at each time point, strain was measured at the end of the experiment using a single element strain gauge (EA060015DJ-120; Measurements Group, Inc., Raleigh, NC) attached at the midshaft of cleaned ulnas. An axial load was applied to the ulnas, and the strain was measured at five different loads using a strain gauge conditioner and amplifier (2200 System; Measurements Group, Inc., Raleigh, NC). A dose–response relation-
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ship was discerned between load magnitude and ulna strain. A graph was then constructed plotting me/N versus c/IMIN, where c is 2/3 the maximum diameter on the IMIN plane. The slope of this graph (0.018 me/ N mm3) was used to predict strains at each time point. IMIN and strain were calculated at each time point by taking digital photographs of each histology slide. The inner label of each pair of fluorescent labels was then traced and analyzed using Scion Image v4.0. The largest diameter parallel to the IMIN axis was also measured. Using IMIN and c along with the slope from the previous strain calculations (0.018 me/N mm3) the strain at each time point was estimated using the equation: m c ¼ N 0:018 I MIN To test the slope of the line of me/N versus c/IMIN, linear regression analyses were used. Differences between right (loaded) and left (nonloaded) ulnas were tested for significance using paired t-tests. All geometrical data from the right limb was compared with that of the left using percent differences between the two limbs. Data from the histomorphometrical analyses were compared using the difference between the right and left limbs. Differences between the percentage change in geometry and change in histomorphometry data of each of the seven groups were tested for significance using a one-way analysis of variance (ANOVA). Significant ANOVAs were followed with Fisher’s protected least significant difference (PLSD) post hoc tests to detect differences between individual groups. All tests were two-tailed with a level of significance set at 0.05.
3. Results The animals in this study were growing, evident by a significant increase in body mass throughout the study. The animals started at an average body weight of 271 g, and ended at an average weight of 345 g, which is a 74 g (27%) increase throughout the 15 weeks. The largest percent change in BMC occurred in the decreasing load group followed by the constant load group (Fig. 3), and the percentage difference between the left and right ulnas of these two groups were significantly greater than all other groups (Po0:05). The progressively decreasing load group had the greatest change in IMIN, with a 74% increase in the right compared to left ulna, but the corresponding change in IMAX was only 10%. IMAX changed less than IMIN because the loading model produces the highest strains along the IMIN plane, not the IMAX plane. Percent change in IMIN in the progressively increasing load group was 17%, which was significantly less than in the
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Fig. 4. Representative images from histology sections of both a nonloaded and loaded ulna. In the nonloaded ulna only a single label is observed on the surface of the IMIN (mediolateral) plane while in the loaded ulna multiple labels are present in this area. Note the labeling in both ulnas in the IMAX (caudocranial) plane due to growth.
formation in the right ulna, as this was the start of loading for this group. In the progressively increasing load group there were no significant increases in bone formation with loading during these weeks. Throughout the course of the study bone formation rates in the left (control) ulnas decreased. The bones of the progressively decreasing load group and constant load group adapted to mechanical loading in the first few weeks of the study as evidenced by the bone formation rates. As a result of the increased crosssectional area in the decreasing load group and constant load group, the strain significantly decreased compared to all other groups (Po0:05) (Fig. 5). During weeks 11 and 15 the last five weeks group showed a significant decrease in strain compared to the decreasing load group, constant load group, and AMC group (Po0:05). Fig. 3. (A) Percent change in bone mineral content between the left and right ulnas in each group. The largest changes occurred in the groups that had the largest force applied at the start of loading. (B) Percent change in the minimum second moment of area in each of the groups. The IMIN plane was most affected by mechanical loading because this is the plane that experiences the largest strains. 1Po0.05 vs. the decreasing load group, 2Po0.05 vs. the increasing load group, 3 Po0.05 vs. the constant load group, 4Po0.05 vs. the last 5 weeks group, *Po0.05 vs. the age-matched control group, #Po0.05 vs. the baseline control group.
decreasing load group and the constant load group (Po0:05). BFRs were greatest in the first 2 weeks of loading (Fig. 4). The largest increase in the difference in BFR between the right and left ulnas during weeks 1 and 2 occurred in the progressively decreasing load group (Table 1). The right and left ulnas were also significantly different in the constant load group. During weeks 6 and 7 the left nonloaded ulnas had greater bone formation rates compared to the loaded ulnas in both the decreasing load group and constant load group (Po0:01). During weeks 11 and 12 the group loaded only for the last 5 weeks had significantly increased bone
4. Discussion The aim of this study was to compare the experimental results of a mechanical loading study to the predicted results of the strain feedback and cellular accommodation algorithms. The bone formation results in the mechanical loading experiment closely resemble the predicted results of the cellular accommodation algorithm. A simulation of a long bone loaded in bending predicted an increase in second moment of area (I) of 94% for the decreasing load group, but only a 13% increase for the increasing load group (see Appendix). This compares well to the measured changes in IMIN, which increased 74% in the decreasing load group and 17% for increasing load. A simulation of a strain feedback algorithm predicted the same change in bone structure regardless of whether the loads were progressively increased or decreased. These results demonstrate that the temporal order in which loading is applied influences the final bone structure as is predicted by the cellular accommodation algorithm. The largest changes occurred at the start of
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Table 1 Effect of mechanical loading on bone formation rates at four time points (mm3/mm2/year)
AMC
Label 1 (weeks 1–2)
Label 2 (weeks 6–7)
Left Right
341.6739.5 764.5774.9**
290.2719.8 209.8723.4**
Left Right
389.1733.7 448.0730.1
Left Right
Label 3 (weeks 11–12)
Label 4 (weeks 15–16)
76.1720.9 84.4715.4
55.9714.9 65.0715.6
267.7716.2 251.5717.4
113.9716.0 130.2714.9
93.5725.6 117.0720.7
350.7726.5 665.0754.7**
299.6725.8 243.1726.5**
156.8731.9 81.6713.2
58.3716.7 42.4710.7
Left Right
378.8740.4 360.5755.7
276.2719.5 237.2731.8
103.3720.4 240.7739.3**
38.0710.2 90.1718.0
Left Right
393.1735.6 394.8731.7
273.6746.2 237.9724.0
155.0717.1 161.9718.5
99.5721.6 68.7726.9
Values are mean7standard error. **Po0:01 (left vs. right comparison—paired t-test).
loading, so the groups that had the largest force applied at the start of the study saw the largest overall changes. After the initial weeks of the loading, bone stopped responding to mechanical loading. This explains why the decreasing load group had the largest response and the constant load group had the second largest response, since it had the second largest load applied at the start of the study. One limitation of this study is that the animals were growing throughout the course of the study, which is evident by significant weight gain and increases in the cross-sectional area of the left (nonloaded) ulnas. To account for this growth, comparisons between groups were made as a percentage difference between right and left ulnas. By expressing values as a percentage change the growth due to aging could be separated from the gains due to mechanical loading. However, the growth of the animals influenced the strain experienced by the ulna at each load. The accommodation occurring with 10 weeks of low magnitude loading was studied by comparing the progressively increasing load group and the group loaded only in the last 5 weeks. Both groups received 13.5 N during the final 5 weeks of the study. However, the last five weeks group had no prior loading before the 13.5 N while the increasing load group had been loaded at 9.0 N for 5 weeks and 11.3 N for 5 weeks. The bone formation rates were significantly increased with loading in the last 5 weeks group, while there was no significant effect of loading the increasing load group. This result suggests that 10 weeks of loading desensitized bone cells so no further adaptation response was possible in the last 5 weeks. The group with progressively increasing loading showed small gains in bone formation rates during the first 5 weeks of the study when it was loaded at 9.0 N, but after this initial period the bone did not respond to the increasing magnitudes of loads. This
result provides supporting evidence that bone responds in the initial weeks of loading, but over time the osteogenic response is diminished. The results of this study are not unique, as the theory of accommodation to loads is supported by other research. Previous studies found significant trabecular bone formation occurring in response to loading within the first week of loading, with significantly less in weeks 2–4 of loading (Kim et al., 2003). Another study using the same loading model used in this study showed no significant differences in percentage change in cortical area in ulnas that were loaded for 5 weeks compared to those loaded for 15 weeks (Saxon et al., 2004). Because the osteogenic response of bone to mechanical loading wanes after the initial weeks, the equations for cellular accommodation better fit the experimental data than those of the strain feedback algorithm. The mathematical models that use negative feedback theory to explain bone’s response to mechanical loading would suggest that different patterns of loading would result in the same changes in geometry at the end of a given time, provided that the cumulative load magnitude is the same (Fig. 1). However, as shown by the bone formation rates in the present study, all of the increases in bone formation occurred at the start of the loading protocol. This was especially evident by the finding that the increasing load group and the group receiving the same peak load in the last 5 weeks of the study had very different bone formation rates during that time. After the initial weeks of the loading protocol bone formation rates were equal or less than those in the nonloaded control ulnas. Therefore, the greatest influence on bone formation in this study was the initial magnitude of the applied load, not the long-term pattern of loading. This result is predicted by the algorithm of cellular accommodation.
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Skin Diseases (P01AR045218). The authors thank Dr. E. Polig for his helpful advice concerning the mathematical modeling of cellular accommodation.
Appendix A. Simulation of the adaptation of a long bone to bending The response of a long bone to externally applied bending loads was simulated using two bone adaptation algorithms: (1) the surface adaptation equation derived by Cowin and Hegedus (1976), dXðtÞ ¼ C ij ðS ij S0ij Þ dt
(A.1)
and (2) a surface adaptation equation with a cellular accommodation function F ðSij; tÞ; dXðtÞ ¼ C ij ðS ij F ðS ij ; tÞÞ dt
(A.2)
where X is the surface position at time t, Sij is the daily strain stimulus on the tissue, S 0ij is the equilibrium or reference strain stimulus, and C ij is a matrix of adaptation constants. The algorithms were simplified by assuming that the movement of the surface dX ðtÞ=dtis restricted to the direction normal to the bone surface, which makes it equal to the mineral apposition rate MAR. A relationship between the peak-to-peak amplitude of longitudinal strain and MAR (measured using histomorphometry) was estimated based upon experimental data from the rat tibia bending model (Turner et al., 1994). It was assumed that bending loads are applied to a functionally loaded limb. Therefore, Eq. (A.1) becomes MAR ¼ k½ð þ 0 Þ 0 ¼ k;
(A.3)
where k is a proportionality constant, is the absolute value of applied strain magnitude (we assume that tensile and compressive strains affect MAR equally, based upon experimental findings in the rat tibia bending model (Turner et al., 1994)), and 0 is the average strain magnitude from normal functional activity. Eq. (A.2) becomes Fig. 5. Percent changes in strain per force between (A) weeks 1 and 6, (B) weeks 6 and 11, and (C) weeks 11 and 15. The decrease in strain occurred at the start of the loading protocol. 1Po0.05 vs. the decreasing load group, 2Po0.05 vs. the increasing load group, 3 Po0.05 vs. the constant load group, 4Po0.05 vs. the last 5 weeks group, *Po0.05 vs. the age-matched control group.
MAR ¼ k½ð þ 0 Þ 0 ð1 et=t Þ ¼ k et=t ;
where t was assumed to be 3.4 weeks. The adaptation of cross-sectional shape was simplified by restricting the shapes of the periosteal and endosteal surfaces to ellipses. Therefore, the peak strains on the periosteal and endosteal surfaces are
Acknowledgements This work was supported by a USPHS grant from the National Institute of Arthritis, Musculoskeletal, and
(A.4)
p ¼
Mbp Mbe ; e ¼ ; EI EI
where I ¼ p4ðap bp3 ae be3 Þ; and
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M is the applied bending moment, E is the Young’s modulus, and a and b are the principal radii of the periosteal and endosteal surfaces (denoted by subscripts p and e). The initial dimensions of the cross-section were assumed to be ap ¼ bp ¼ 1:5 mm and ae ¼ be ¼ 0:75 mm: The Young’s modulus E was assumed to be 29.4 GPa. Consider that a long bone is subjected to cyclic bending loads following two distinct temporal loading sequences, illustrated by loading Cases A and B in Fig. 1. The magnitudes of bending loads were chosen to correspond to experimental design. As expected the strain feedback algorithm predicted the long bone’s second moment of area (I) would increase by the same amounts for Cases A and B, so the predicted outcome was unaffected by the sequence of applied loading. The cellular accommodation algorithm predicted an increase in second moment of area of 94% for Case A, but only a 13% increase in I for Case B. References Beaupre´, G.S., Orr, T.E., Carter, D.R., 1990. An approach for timedependent bone modeling and remodeling-theoretical development. Journal of Orthopaedic Research 8, 651–661. Bertram, J.E.A., Biewener, A.A., 1988. Bone curvature: sacrificing strength for load predictability? Journal of Theoretical Biology 131, 75–92. Cowin, S.C., Hegedus, D.H., 1976. Bone remodeling I: theory of adaptive elasticity. Journal of Elasticity 6, 313–326. Frost, H.M., 1983. A determinant of bone architecture: the minimum effective strain. Clinical Orthopaedics 175, 286–292. Fyhrie, D.P., Schaffler, M.B., 1995. The adaptation of bone apparent density to applied load. Journal of Biomechanics 28, 135–146. Hsieh, Y.-F., Turner, C.H., 2001. Effects of loading frequency on mechanically induced bone formation. Journal of Bone and Mineral Research 16, 918–924. Kim, C.H., Takai, E., Zhou, H., von Stechow, D., Mu¨ller, R., Dempster, D.W., Guo, X.E., 2003. Trabecular bone response to mechanical and parathyroid hormone stimulation: the role of mechanical microenvironment. Journal of Bone and Mineral Research 18, 2116–2125.
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Lanyon, L.E., 1992. The success and failure of the adaptive response to functional load-bearing in averting bone fracture. Bone 13 (Suppl.), S17–S21. Martin, R.B., 1984. Porosity and specific surface of bone. Critical Reviews in Biomedical Engineering 10, 179–222. McKee, M.D., Nanci, A., 1996. Osteopontin at mineralized tissue interfaces in bone, teeth, and osseointegrated implants: ultrastructural distribution and implications for mineralized tissue formation, turnover, and repair. Microscopy Research Techniques 33, 141–164. Meyer, G.H., 1867. Die architektur der spongiosa. Archiv fu¨r Anatomie, Physiologie und wissenschaftliche Medicin 34, 615–625. Pavalko, F.M., Chen, N.X., Turner, C.H., Burr, D.B., Atkinson, S., Hsieh, Y.-F., Qui, J., Duncan, R.L., 1998. Fluid shear-induced mechanical signaling in MC3T3-E1 osteoblasts requires cytoskeleton-integrin interactions. American Journal of Physiology 275, C1591–C1601. Robling, A.G., Burr, D.B., Turner, C.H., 2001. Recovery periods restore mechanosensitivity to dynamically loaded bone. The Journal of Experimental Biology 204, 3389–3399. Robling, A.G., Hinant, F.M., Burr, D.B., Turner, C.H., 2002a. Improved bone structure and strength after long-term mechanical loading is greatest if loading is separated into short bouts. Journal of Bone and Mineral Research 17, 1545–1554. Robling, A.G., Hinant, F.M., Burr, D.B., Turner, C.H., 2002b. Shorter, more frequent mechanical loading sessions enhance bone mass. Medicine and Science Sports and Exercise 34, 196–202. Rubin, C., Judex, S., Hadjiargyrou, M., 2002. Skeletal adaptation to mechanical stimuli in the absence of formation or resorption of bone. Journal of Musculoskeletal and Neuronal Interactions 2, 264–267. Saxon, L.K., Robling, A.G., Turner, C.H., 2004. A long period of rest between sessions of mechanical loading may improve bone strength. In: Transactions of the Orthopaedic Research Society. San Francisco, CA. Turner, C.H., 1999. Toward a mathematical description of bone biology: the principle of cellular accommodation. Calcified Tissue International 65, 466–471. Turner, C.H., Forwood, M.R., Rho, J., Yoshikawa, T., 1994. Mechanical loading thresholds for lamellar and woven bone formation. Journal of Bone and Mineral Research 9, 87–97. Torrance, A.G., Mosley, J.R., Suswillo, R.F., Lanyon, L.E., 1994. Noninvasive loading of the rat ulna in vivo induces a strain-related modeling response uncomplicated by trauma or periosteal pressure. Calcified Tissue International 54, 241–247. Wolff, J., 1892. Das Gesetz der Transformation der Knochen. Hirschwald, Berlin.
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Cellular accommodation and the response of bone to mechanical loading Jennifer L. Schriefera, Stuart J. Wardenb,c, Leanne K. Saxonc, Alexander G. Roblingd, Charles H. Turnerc,e, a
Department of Biomedical Engineering, Purdue School of Engineering and Technology, Indiana University Purdue University Indianapolis, Indianapolis, IN, USA b Department of Physical Therapy, School of Health and Rehabilitation Sciences, Indiana University, Indianapolis, IN, USA c Department of Orthopaedic Surgery, Indiana University School of Medicine, Indianapolis, IN, USA d Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA e Biomechanics and Biomaterials Research Center, Indiana University School of Medicine, Indianapolis, IN, USA Accepted 17 August 2004
Abstract Several mathematical rules by which bone adapts to mechanical loading have been proposed. Previous work focused mainly on negative feedback models, e.g., bone adapts to increased loading after a minimum strain effective (MES) threshold has been reached. The MES algorithm has numerous caveats, so we propose a different model, according to which bone adapts to changes in its mechanical environment based on the principle of cellular accommodation. With the new algorithm we presume that strain history is integrated into cellular memory so that the reference state for adaptation is constantly changing. To test this algorithm, an experiment was performed in which the ulnae of Sprague–Dawley rats were loaded in axial compression. The animals received loading for 15 weeks with progressively decreasing loads, increasing loads, or a constant load. The results showed the largest increases in geometry in the decreasing load group, followed by the constant load group. Bone formation rates (BFRs) were significantly greater in the decreasing load group during the first 2 weeks of the study as compared to all other groups (Po0:05). After the first few weeks of mechanical loading, the BFR in the loaded ulnae returned to the values of the nonloaded ulnae. These experimental results closely fit the predicted results of the cellular accommodation algorithm. After the initial weeks of loading, bone stopped responding so the degree of adaptation was proportional to the initial peak load magnitude. r 2004 Elsevier Ltd. All rights reserved. Keywords: Bone density; Biomechanics; Mechanical loading; Osteoporosis
1. Introduction The idea that bone structure and adaptation can be predicted by mathematical laws originated with Wolff and his contemporaries (Meyer, 1867; Wolff, 1892). It is an appealing idea and has been the genesis for many engineering studies of bone adaptation resulting in Corresponding author. Indiana University, 1120 South Dr., FH 115, Indianapolis, IN 46202, USA. Tel: +1-317-274-3226; fax: +1317-278-9568. E-mail address: [email protected] (C.H. Turner).
0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.08.017
algorithms to describe the process(es). These algorithms are based upon established mechanical engineering concepts like structural optimization or negative feedback control, and assume a mechanical strain setpoint or attractor state exists within bone tissue. For bone to adapt to a new mechanical loading state, bone cells must have some memory of their previous mechanical environment in order to determine that the new environment is different and requires a response. As Lanyon put it, bone adaptation is ‘‘error driven’’ (Lanyon, 1992). Bone cells must process loading information locally because bone tissue is poorly
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innervated and, unlike many mechanoreceptor cells, cannot rely on the central nervous system to integrate and distribute information about mechanical signals. Frost proposed that adaptive changes in bone shape occur when minimum effective strain (MES) thresholds are surpassed (Frost, 1983). Thus he assumes that bone cells are somehow programmed with the value of the MES which sets the threshold for a response. However, for the cellular response to be bone-specific the MES must vary from bone to bone. Furthermore, the MES may well be location dependent within bones. Most long bones are loaded in bending (Bertram and Biewener, 1988), creating a strain gradient across the bone section that has a value of zero along the neutral axis. To avoid excessive bone resorption at the neutral axis, the MES would have to vary considerably across the bone section. Each mechanosensitive cell must have some strain threshold above which a mechanical signal causes a cellular response. It is argued here that this threshold need not be a set value but is a product of local strain history. It is well known that many cell types, including osteoblasts, reorganize their cytoskeletons in response to a mechanical stimulus (Pavalko et al., 1998). Cytoskeletal reorganization in turn changes a cell’s mechanosensitivity, allowing the cell to accommodate to its strain environment. Also cellular mechanosensitivity may be altered through reworking the local environment around the cell (Rubin et al., 2002). Osteocytes are considered to be part of the mechanosensory apparatus in bone and any change in their extracellular microenvironment could change their mechanosensitivity. The extracellular environment of osteocytes does not remain constant as these cells actively form and remove layers of matrix on the surface of their lacunae (McKee and Nanci, 1996). It is assumed that when a strain threshold is surpassed the sensor cells will gradually accommodate to the new state, either by cytoskeletal reorganization or by changing the extracellular microenvironment. Using this definition of cellular accommodation, a setpoint could be determined simply by summation of the past history of daily strain stimuli (Turner, 1999). The bone formation (or resorption) response would then be proportional to the difference between the new strain stimulus and the ever changing setpoint. The temporal skeletal adaptive response will depend, to some extent, upon how the strain history is integrated into cellular memory. It is assumed that the memory of the cells’ previous loading state fades exponentially. An algorithm using the principle of cellular accommodation predicts a different time course of bone adaptation than do strain feedback algorithms. Consider three mathematical models for the adaptation of cancellous bone structure, starting with the generic
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model for adaptation of cancellous bone density: qr ¼ Bðf S0 Þ; (1) qt where r is density in g/cm3, t is time in weeks, B is a rate constant (g/cm3-wk), f is the strain stimulus function, and S 0 is the equilibrium strain stimulus (Cowin and Hegedus, 1976; Fyhrie and Schaffler, 1995). Eq. (1) does not account for the changes in trabecular surface area that accompany changes in cancellous density (Martin, 1984). Cellular activity is somewhat dependent upon the available surface area, i.e., trabecular remodeling activity is surface-driven. To account for this, the model can incorporate the effect of changing a specific surface (trabecular surface area normalized by volume) (Beaupre´ et al., 1990). This model takes the general form: qr ¼ BðrÞðf S 0 Þ; (2) qt where BðrÞ is a rate function that is proportional to the specific surface of the cancellous structure (specific surface is defined as a function of r) (Martin, 1984). Eq. (2) can be expanded further to correct for the effect of cellular accommodation: qr ¼ BðrÞðf F ðf; tÞÞ qt and
(3)
F ðf; tÞ ¼ S0 þ ðf S 0 Þð1 et=t Þ; where F takes the form of a relaxation function derived from the differential equation: dF ðf F Þ ¼ ; (4) dt t where t is the time constant describing the rate at which accommodation takes place. Cellular accommodation results from the propensity of F to approach f with time thus nulling the driving force for density change. There are certain characteristics of the algorithm incorporating cellular accommodation (Eq. (3)) that clearly distinguish it from the strain feedback algorithm (Eq. (1)). When bone is mechanically loaded over time with either progressively increasing or decreasing load magnitudes, the two algorithms predict extremely different results (Fig. 1). Presuming that the dose–response curve for mechanically induced bone formation is non-linear, as has been demonstrated experimentally (Turner et al., 1994), the response predicted by the cellular accommodation algorithm is path dependent. Path dependence, as the term is used here, means that the outcome from the algorithm is dependent upon the temporal sequence of the preceding mechanical loading events. Negative strain feedback algorithms (e.g. Eq. (1)) typically are not path dependent. Consequently, an experiment that varies the temporal sequence of loading
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bone bending model. Therefore, the purpose of this paper is to compare the experimental adaptation response of bone to changing mechanical loads to the predicted responses of the strain feedback and cellular accommodation algorithms.
2. Methods Fifty-three Sprague–Dawley female retired breeder rats were purchased from Charles River (Wilmington, MA). The rats were housed two per cage at Indiana University’s Laboratory Animal Resource Center, and provided standard rat chow and water ad libitum throughout the experiment. Once the animals arrived at the university they were given 2 weeks of acclimatization. All procedures were performed in accordance with the Institutional Animal Care and Use Committee guidelines of Indiana University. Animals were randomized to four loaded groups and two control groups (Fig. 2). In the loaded animals, the right ulna was loaded using the axial loading model described previously (Torrance et al., 1994). Compressive loads were applied as a haversine waveform 3 days per week for 360 cycles per day at 2 Hz under isofluraneinduced anesthesia (3.5% isoflurane delivered at 1.5 L/ min was used for induction and 2.5% isoflurane at 1.5 L/ min was used for maintenance). The left ulna served as a contralateral control. The progressively decreasing load group (n ¼ 10) was loaded with a peak force of 13.5 N for the first 5 weeks, 11.3 N for the next 5 weeks, and 9.0 N for the final five weeks (Fig. 2). The progressively increasing load group (n=11) received the loads in the opposite order compared to the decreasing load group, and the constant load group (n=10) received 11.3 N for the
Fig. 1. (A) Predicted results of the progressively decreasing load group and increasing load group in a long-term loading protocol according to the strain feedback algorithm. Despite the differences in the loading protocols, the two patterns of loading result in the same increase in cross-sectional area. (B) Predicted results of the decreasing load group and increasing load group in a long-term loading protocol according to the cellular accommodation algorithm. The predicted results show very different final cross-sectional areas for the two groups.
will distinguish cellular accommodation from negative feedback algorithms. To test this theory we performed an experiment investigating the adaptation of bone to decreasing peak mechanical loads, increasing peak mechanical loads, or constant peak loads in a rat model. To avoid the added complexities of bone surface area changes and trabecular orientation, this experiment focuses on cortical, rather than trabecular, bone adaptation using a long
Fig. 2. Experimental protocol. A and C indicate an alizarin or calcein injection was given on the Friday of the corresponding week, respectively. S indicates the Monday that sacrifice occurred. =decreasing load group, =increasing load group, =constant load group, =last 5 weeks group, AMC=age-matched control group, and BLC=baseline control group. Numbers for , , , and represent peak load magnitudes introduced to the right ulna in that week.
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entire 15 weeks. Thus, these three groups all received the same average load throughout the study. The last 5 weeks loaded group (n=9) rested for the first 10 weeks of the study and then received a load of 13.5 N for the final 5 weeks of the study. The purpose of the last 5 weeks group was to compare high magnitude loading at the end of the study to the increasing load group which received 10 weeks of prior loading. There were two control groups, an age-matched control (AMC; n=8) that was sacrificed on the same day as the loaded groups, and a baseline control (BLC; n=5) that was sacrificed on the first day of loading. At the end of weeks 1 and 2 as well as 11 and 12, the animals received intraperitoneal injections (7 mg/kg body weight) of calcein (Sigma, St. Louis, MO) (Fig. 2). At the end of weeks 6 and 7 and weeks 15 and 16 of the study the animals received intraperitoneal injections (25 mg/kg body weight) of alizarin complexone (Sigma, St. Louis, MO). Body mass measurements were collected once per week throughout the length of the study. All animals except the baseline controls were sacrificed 16 weeks after the start of loading by anesthetic inhalation and cervical dislocation. After sacrifice, the right and left ulnas of the animals were dissected free, cleaned, and stored in 70% EtOH. Bone mineral content (BMC) was measured from a single tomographic slice at the ulna midshaft using a Norland Medical Systems Stratec XCT Research SA+peripheral quantitative computed tomography scanner (pQCT; Stratec Electronics, Pforzheim, Germany), as described previously (Robling et al., 2002b). The ulnas were also scanned at the midshaft using a destop mCT (Scanco Medical AG, Bassersdorf, Switzerland) to produce images, as described previously (Robling et al., 2002a). The total area (mm2), and minimum (IMIN; mm4) and maximum (IMAX; mm4) second moments of area were calculated by importing the images into Scion Image v4.0.2 for Windows (Scion Corporation, Frederick, MD). Histologic sections were cut from plastic-embedded bones, as was reported previously (Hsieh and Turner, 2001). One midshaft slice was analyzed from each ulna using a Nikon Optiphot fluorescence microscope operating with a Bioquant digitizing system (R&M Biometrics, Nashville, TN). Bone formation rates (BFRs) on the periosteal surface were calculated as was previously described (Robling et al., 2001). To estimate the strain (me) applied to the ulna at each time point, strain was measured at the end of the experiment using a single element strain gauge (EA060015DJ-120; Measurements Group, Inc., Raleigh, NC) attached at the midshaft of cleaned ulnas. An axial load was applied to the ulnas, and the strain was measured at five different loads using a strain gauge conditioner and amplifier (2200 System; Measurements Group, Inc., Raleigh, NC). A dose–response relation-
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ship was discerned between load magnitude and ulna strain. A graph was then constructed plotting me/N versus c/IMIN, where c is 2/3 the maximum diameter on the IMIN plane. The slope of this graph (0.018 me/ N mm3) was used to predict strains at each time point. IMIN and strain were calculated at each time point by taking digital photographs of each histology slide. The inner label of each pair of fluorescent labels was then traced and analyzed using Scion Image v4.0. The largest diameter parallel to the IMIN axis was also measured. Using IMIN and c along with the slope from the previous strain calculations (0.018 me/N mm3) the strain at each time point was estimated using the equation: m c ¼ N 0:018 I MIN To test the slope of the line of me/N versus c/IMIN, linear regression analyses were used. Differences between right (loaded) and left (nonloaded) ulnas were tested for significance using paired t-tests. All geometrical data from the right limb was compared with that of the left using percent differences between the two limbs. Data from the histomorphometrical analyses were compared using the difference between the right and left limbs. Differences between the percentage change in geometry and change in histomorphometry data of each of the seven groups were tested for significance using a one-way analysis of variance (ANOVA). Significant ANOVAs were followed with Fisher’s protected least significant difference (PLSD) post hoc tests to detect differences between individual groups. All tests were two-tailed with a level of significance set at 0.05.
3. Results The animals in this study were growing, evident by a significant increase in body mass throughout the study. The animals started at an average body weight of 271 g, and ended at an average weight of 345 g, which is a 74 g (27%) increase throughout the 15 weeks. The largest percent change in BMC occurred in the decreasing load group followed by the constant load group (Fig. 3), and the percentage difference between the left and right ulnas of these two groups were significantly greater than all other groups (Po0:05). The progressively decreasing load group had the greatest change in IMIN, with a 74% increase in the right compared to left ulna, but the corresponding change in IMAX was only 10%. IMAX changed less than IMIN because the loading model produces the highest strains along the IMIN plane, not the IMAX plane. Percent change in IMIN in the progressively increasing load group was 17%, which was significantly less than in the
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Fig. 4. Representative images from histology sections of both a nonloaded and loaded ulna. In the nonloaded ulna only a single label is observed on the surface of the IMIN (mediolateral) plane while in the loaded ulna multiple labels are present in this area. Note the labeling in both ulnas in the IMAX (caudocranial) plane due to growth.
formation in the right ulna, as this was the start of loading for this group. In the progressively increasing load group there were no significant increases in bone formation with loading during these weeks. Throughout the course of the study bone formation rates in the left (control) ulnas decreased. The bones of the progressively decreasing load group and constant load group adapted to mechanical loading in the first few weeks of the study as evidenced by the bone formation rates. As a result of the increased crosssectional area in the decreasing load group and constant load group, the strain significantly decreased compared to all other groups (Po0:05) (Fig. 5). During weeks 11 and 15 the last five weeks group showed a significant decrease in strain compared to the decreasing load group, constant load group, and AMC group (Po0:05). Fig. 3. (A) Percent change in bone mineral content between the left and right ulnas in each group. The largest changes occurred in the groups that had the largest force applied at the start of loading. (B) Percent change in the minimum second moment of area in each of the groups. The IMIN plane was most affected by mechanical loading because this is the plane that experiences the largest strains. 1Po0.05 vs. the decreasing load group, 2Po0.05 vs. the increasing load group, 3 Po0.05 vs. the constant load group, 4Po0.05 vs. the last 5 weeks group, *Po0.05 vs. the age-matched control group, #Po0.05 vs. the baseline control group.
decreasing load group and the constant load group (Po0:05). BFRs were greatest in the first 2 weeks of loading (Fig. 4). The largest increase in the difference in BFR between the right and left ulnas during weeks 1 and 2 occurred in the progressively decreasing load group (Table 1). The right and left ulnas were also significantly different in the constant load group. During weeks 6 and 7 the left nonloaded ulnas had greater bone formation rates compared to the loaded ulnas in both the decreasing load group and constant load group (Po0:01). During weeks 11 and 12 the group loaded only for the last 5 weeks had significantly increased bone
4. Discussion The aim of this study was to compare the experimental results of a mechanical loading study to the predicted results of the strain feedback and cellular accommodation algorithms. The bone formation results in the mechanical loading experiment closely resemble the predicted results of the cellular accommodation algorithm. A simulation of a long bone loaded in bending predicted an increase in second moment of area (I) of 94% for the decreasing load group, but only a 13% increase for the increasing load group (see Appendix). This compares well to the measured changes in IMIN, which increased 74% in the decreasing load group and 17% for increasing load. A simulation of a strain feedback algorithm predicted the same change in bone structure regardless of whether the loads were progressively increased or decreased. These results demonstrate that the temporal order in which loading is applied influences the final bone structure as is predicted by the cellular accommodation algorithm. The largest changes occurred at the start of
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Table 1 Effect of mechanical loading on bone formation rates at four time points (mm3/mm2/year)
AMC
Label 1 (weeks 1–2)
Label 2 (weeks 6–7)
Left Right
341.6739.5 764.5774.9**
290.2719.8 209.8723.4**
Left Right
389.1733.7 448.0730.1
Left Right
Label 3 (weeks 11–12)
Label 4 (weeks 15–16)
76.1720.9 84.4715.4
55.9714.9 65.0715.6
267.7716.2 251.5717.4
113.9716.0 130.2714.9
93.5725.6 117.0720.7
350.7726.5 665.0754.7**
299.6725.8 243.1726.5**
156.8731.9 81.6713.2
58.3716.7 42.4710.7
Left Right
378.8740.4 360.5755.7
276.2719.5 237.2731.8
103.3720.4 240.7739.3**
38.0710.2 90.1718.0
Left Right
393.1735.6 394.8731.7
273.6746.2 237.9724.0
155.0717.1 161.9718.5
99.5721.6 68.7726.9
Values are mean7standard error. **Po0:01 (left vs. right comparison—paired t-test).
loading, so the groups that had the largest force applied at the start of the study saw the largest overall changes. After the initial weeks of the loading, bone stopped responding to mechanical loading. This explains why the decreasing load group had the largest response and the constant load group had the second largest response, since it had the second largest load applied at the start of the study. One limitation of this study is that the animals were growing throughout the course of the study, which is evident by significant weight gain and increases in the cross-sectional area of the left (nonloaded) ulnas. To account for this growth, comparisons between groups were made as a percentage difference between right and left ulnas. By expressing values as a percentage change the growth due to aging could be separated from the gains due to mechanical loading. However, the growth of the animals influenced the strain experienced by the ulna at each load. The accommodation occurring with 10 weeks of low magnitude loading was studied by comparing the progressively increasing load group and the group loaded only in the last 5 weeks. Both groups received 13.5 N during the final 5 weeks of the study. However, the last five weeks group had no prior loading before the 13.5 N while the increasing load group had been loaded at 9.0 N for 5 weeks and 11.3 N for 5 weeks. The bone formation rates were significantly increased with loading in the last 5 weeks group, while there was no significant effect of loading the increasing load group. This result suggests that 10 weeks of loading desensitized bone cells so no further adaptation response was possible in the last 5 weeks. The group with progressively increasing loading showed small gains in bone formation rates during the first 5 weeks of the study when it was loaded at 9.0 N, but after this initial period the bone did not respond to the increasing magnitudes of loads. This
result provides supporting evidence that bone responds in the initial weeks of loading, but over time the osteogenic response is diminished. The results of this study are not unique, as the theory of accommodation to loads is supported by other research. Previous studies found significant trabecular bone formation occurring in response to loading within the first week of loading, with significantly less in weeks 2–4 of loading (Kim et al., 2003). Another study using the same loading model used in this study showed no significant differences in percentage change in cortical area in ulnas that were loaded for 5 weeks compared to those loaded for 15 weeks (Saxon et al., 2004). Because the osteogenic response of bone to mechanical loading wanes after the initial weeks, the equations for cellular accommodation better fit the experimental data than those of the strain feedback algorithm. The mathematical models that use negative feedback theory to explain bone’s response to mechanical loading would suggest that different patterns of loading would result in the same changes in geometry at the end of a given time, provided that the cumulative load magnitude is the same (Fig. 1). However, as shown by the bone formation rates in the present study, all of the increases in bone formation occurred at the start of the loading protocol. This was especially evident by the finding that the increasing load group and the group receiving the same peak load in the last 5 weeks of the study had very different bone formation rates during that time. After the initial weeks of the loading protocol bone formation rates were equal or less than those in the nonloaded control ulnas. Therefore, the greatest influence on bone formation in this study was the initial magnitude of the applied load, not the long-term pattern of loading. This result is predicted by the algorithm of cellular accommodation.
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Skin Diseases (P01AR045218). The authors thank Dr. E. Polig for his helpful advice concerning the mathematical modeling of cellular accommodation.
Appendix A. Simulation of the adaptation of a long bone to bending The response of a long bone to externally applied bending loads was simulated using two bone adaptation algorithms: (1) the surface adaptation equation derived by Cowin and Hegedus (1976), dXðtÞ ¼ C ij ðS ij S0ij Þ dt
(A.1)
and (2) a surface adaptation equation with a cellular accommodation function F ðSij; tÞ; dXðtÞ ¼ C ij ðS ij F ðS ij ; tÞÞ dt
(A.2)
where X is the surface position at time t, Sij is the daily strain stimulus on the tissue, S 0ij is the equilibrium or reference strain stimulus, and C ij is a matrix of adaptation constants. The algorithms were simplified by assuming that the movement of the surface dX ðtÞ=dtis restricted to the direction normal to the bone surface, which makes it equal to the mineral apposition rate MAR. A relationship between the peak-to-peak amplitude of longitudinal strain and MAR (measured using histomorphometry) was estimated based upon experimental data from the rat tibia bending model (Turner et al., 1994). It was assumed that bending loads are applied to a functionally loaded limb. Therefore, Eq. (A.1) becomes MAR ¼ k½ð þ 0 Þ 0 ¼ k;
(A.3)
where k is a proportionality constant, is the absolute value of applied strain magnitude (we assume that tensile and compressive strains affect MAR equally, based upon experimental findings in the rat tibia bending model (Turner et al., 1994)), and 0 is the average strain magnitude from normal functional activity. Eq. (A.2) becomes Fig. 5. Percent changes in strain per force between (A) weeks 1 and 6, (B) weeks 6 and 11, and (C) weeks 11 and 15. The decrease in strain occurred at the start of the loading protocol. 1Po0.05 vs. the decreasing load group, 2Po0.05 vs. the increasing load group, 3 Po0.05 vs. the constant load group, 4Po0.05 vs. the last 5 weeks group, *Po0.05 vs. the age-matched control group.
MAR ¼ k½ð þ 0 Þ 0 ð1 et=t Þ ¼ k et=t ;
where t was assumed to be 3.4 weeks. The adaptation of cross-sectional shape was simplified by restricting the shapes of the periosteal and endosteal surfaces to ellipses. Therefore, the peak strains on the periosteal and endosteal surfaces are
Acknowledgements This work was supported by a USPHS grant from the National Institute of Arthritis, Musculoskeletal, and
(A.4)
p ¼
Mbp Mbe ; e ¼ ; EI EI
where I ¼ p4ðap bp3 ae be3 Þ; and
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M is the applied bending moment, E is the Young’s modulus, and a and b are the principal radii of the periosteal and endosteal surfaces (denoted by subscripts p and e). The initial dimensions of the cross-section were assumed to be ap ¼ bp ¼ 1:5 mm and ae ¼ be ¼ 0:75 mm: The Young’s modulus E was assumed to be 29.4 GPa. Consider that a long bone is subjected to cyclic bending loads following two distinct temporal loading sequences, illustrated by loading Cases A and B in Fig. 1. The magnitudes of bending loads were chosen to correspond to experimental design. As expected the strain feedback algorithm predicted the long bone’s second moment of area (I) would increase by the same amounts for Cases A and B, so the predicted outcome was unaffected by the sequence of applied loading. The cellular accommodation algorithm predicted an increase in second moment of area of 94% for Case A, but only a 13% increase in I for Case B. References Beaupre´, G.S., Orr, T.E., Carter, D.R., 1990. An approach for timedependent bone modeling and remodeling-theoretical development. Journal of Orthopaedic Research 8, 651–661. Bertram, J.E.A., Biewener, A.A., 1988. Bone curvature: sacrificing strength for load predictability? Journal of Theoretical Biology 131, 75–92. Cowin, S.C., Hegedus, D.H., 1976. Bone remodeling I: theory of adaptive elasticity. Journal of Elasticity 6, 313–326. Frost, H.M., 1983. A determinant of bone architecture: the minimum effective strain. Clinical Orthopaedics 175, 286–292. Fyhrie, D.P., Schaffler, M.B., 1995. The adaptation of bone apparent density to applied load. Journal of Biomechanics 28, 135–146. Hsieh, Y.-F., Turner, C.H., 2001. Effects of loading frequency on mechanically induced bone formation. Journal of Bone and Mineral Research 16, 918–924. Kim, C.H., Takai, E., Zhou, H., von Stechow, D., Mu¨ller, R., Dempster, D.W., Guo, X.E., 2003. Trabecular bone response to mechanical and parathyroid hormone stimulation: the role of mechanical microenvironment. Journal of Bone and Mineral Research 18, 2116–2125.
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