A mathematical framework to study the effects of growth factor influences on fracture healing

During fracture healing, multipotential stem cells differentiate into specialized cells responsible for producing the different tissues involved in the bone regeneration process. This cell differentiation has been shown to be regulated by locally expressed growth factors. The details of their regulatory mechanisms need to be understood. In this work, we present a two-dimensional mathematical model of the bone healing process for moderate fracture gap sizes and fracture stability. The inflammatory and tissue regeneration stages of healing are simulated by modeling mesenchymal cell migration; mesenchymal cell, chondrocyte and osteoblast proliferation and differentiation, and extracellular matrix synthesis and degradation over time. The effects of two generic growth factors on cell differentiation are based on the experimentally studied chondrogenic and osteogenic effects of bone morphogenetic proteins-2 and 4 and transforming growth factor-beta-1, respectively. The model successfully simulates the progression of healing and predicts that the rate of osteogenic growth factor production by osteoblasts and the duration of the initial release of growth factors upon injury are particularly important parameters for complete ossification and successful healing. This temporo-spatial model of fracture healing is the first model to consider the effects of growth factors. It will help us understand the regulatory mechanisms involved in bone regeneration and provides a mathematical framework with which to design experiments and understand pathological conditions.

Time-dependent effects of intermittent hydrostatic pressure on articular chondrocyte type II collagen and aggrecan mRNA expression

The normal loading of joints during daily activities causes the articular cartilage to be exposed to high levels of intermittent hydrostatic pressure. This study quantified effects of intermittent hydrostatic pressure on expression of mRNA for important extracellular matrix constituents. Normal adult bovine articular chondrocytes were isolated and tested in primary culture, either as high-density monolayers or formed aggregates. Loaded cells were exposed to 10 MPa of intermittent hydrostatic pressure at a frequency of 1 Hz for periods of 2, 4, 8, 12, and 24 hrs. Other cells were intermittently loaded for a period of 4 hrs per day for 4 days. Semiquantitative reverse transcription polymerase chain reaction assays were used to assess mRNA signal levels for collagen types II and I and aggrecan. The results showed that type II collagen mRNA signal levels exhibited a biphasic pattern, with an initial increase of approximately five-fold at 4 and 8 hrs that subsequently decreased by 24 hrs. In contrast, aggrecan mRNA signal increased progressively up to three-fold throughout the loading period. Changing the loading profile to 4 hrs per day for 4 days increased the mRNA signal levels for type II collagen nine-fold and for aggrecan twenty-fold when compared to unloaded cultures. These data suggest that specific mechanical loading protocols may be required to optimally promote repair and regeneration of diseased joints.

Mechanobiology of femoral neck structure during adolescence

Understanding femoral neck structure may be critical to preventing fractures at this site. We examined the correlates of changes in the femoral neck during adolescence. Dual energy x-ray absorptiometry measurements of proximal femora were made in 101 Caucasian youths (ages 9 to 26 years). Relationships were examined between developmental parameters (age, pubertal stage, height, body mass, lean mass, and fat mass) and femoral structure (bone mineral content, bone mineral density, neck width, cross-sectional area, and cross-sectional strength). Lean body mass was the best predictor of femoral neck structure, explaining 53-87 percent of the variance, and was independent of gender. Body mass only explained 51-79 percent of the variance. Previously we found body mass to be the strongest predictor of femoral mid-diaphyseal cross-sectional properties. These findings suggest that trabecular bone of the femoral neck may be more responsive to its mechanical environment than the cortical diaphysis. In addition, lean body mass may be a more reliable predictor of muscle loading than body mass.

BMP-5 deficiency alters chondrocytic activity in the mouse proximal tibial growth plate

The role of bone morphogenetic protein-5 (BMP-5) in regulating chondrocytic activity during endochondral ossification was examined in the mouse proximal tibial growth plate. Short ear mice homozygous for the SEA/Gn point mutation in the coding region for BMP-5 (King, J. A. et al. Dev Biol 166:112122; 1994) and heterozygous long ear littermates were examined at 5 and 9 weeks of age (n = 9/group, four groups). Animals were injected with oxytetracycline to estimate the rate of growth and with bromodeoxyuridine to identify proliferative chondrocytes. Age-related changes in chondrocytic stereological and kinetic parameters were compared by image analysis of 1-microm-thick growth plate sections. The number of proliferative chondrocytes did not vary with age in either genotype, but proliferative phase duration increased significantly (approximately 67%) with age in the long ear mice, whereas no change was detected in the short ear mice. The number of hypertrophic chondrocytes increased significantly (approximately 27%) in the short ears, whereas this number decreased significantly (approximately 40%) in the long ears. There was a small, but significant, increase in hypertrophic phase duration (approximately 45%) in short ear mice, but no change was detected in the long ears. These results indicate that BMP-5 deficiency prevents age-related decelerations in chondrocytic proliferation and initiation of hypertrophic differentiation, suggesting a role of BMP-5 in inhibiting these processes.

DXA-derived section modulus and bone mineral content predict long-bone torsional strength

Previous studies have used dual energy x-ray absorptiometry (DXA) scans to calculate the section modulus (Z) of adolescent and adult human femurs. The DXA-derived values of Z were assumed to be proportional to bone strength in bending and torsion. In this study we used dog (n 5), pig (n 4), and human (n 13) femurs covering a linear bone mineral content (BMCL) range of 0.91-6.1 g/cm. Using DXA scans, ex vivo torsional strength tests, and torsional finite element models, we assessed the validity of using the DXA-derived Z value as an indicator of strength. The correlation between BMCL and strength was r2 = 0.87 and the correlation between Z and strength was r2 = 0.86. Based on finite element results, the dog and pig section moduli were adjusted to be comparable to the human data based on cross-sectional shape and bone tissue shear strength differences. With these adjustments, the correlation between adjusted section modulus and measured strength did not improve (r2 = 0.87). These data indicate that DXA-derived section modulus can be used to predict strength over a wide range of bone sizes. However, a clear advantage of using DXA-derived section modulus rather than BMCL could not be found.

Stability of open-book pelvic fractures using a new biomechanical model of single-limb stance

OBJECTIVE

A new biomechanical model of single-limb stance was developed to test the stability of intact, injured, and internally fixed pelves.

DESIGN

Single-limb stance was simulated by applying muscle forces and body mass loading to cadaver pelves. We created a rotationally unstable "open-book" pelvic injury in nine embalmed pelves by dividing the ligaments of the pubic symphysis, pelvic floor, and anterior and interosseus sacroiliac joint. All pelves were devoid of gross structural abnormalities.

INTERVENTION

Two methods of internal fixation of the pubis symphysis were compared: (a) a curved six-hole 3.5-millimeter reconstruction plate across the superior pubic symphysis, and (b) the same six-hole 3.5-millimeter reconstruction plate plus a perpendicularly oriented four-hole 3.5-millimeter reconstruction plate placed across the anterior symphysis.

MAIN OUTCOME MEASUREMENTS

We measured vertical shear displacement at the public symphysis and horizontal displacement at the anterior sacroiliac joint. The results for the injured and fixed specimens were compared with each other and with the results for the intact specimens.

RESULTS

The injured unfixed specimens showed marked instability that was prevented by both methods of fixation of the pubic symphysis. No significant differences could be demonstrated between single and double plating of the disrupted pubic symphysis when using this single-limb stance model.

CONCLUSION

This model of single-limb stance suggests that a single symphyseal plate across the pubic symphysis can stabilize the open-book injury under short-term quasi-static loads.

Correspondence between theoretical models and dual energy x-ray absorptiometry measurements of femoral cross-sectional growth during adolescence

We have developed an analytical model of long bone cross-sectional ontogeny in which appositional growth of the diaphysis is primarily driven by mechanical stimuli associated with increasing body mass during growth and development. In this study, our goal was to compare theoretical predictions of femoral diaphyseal structure from this model with measurements of femoral bone mineral and geometry by dual energy x-ray absorptiometry. Measurements of mid-diaphyseal femoral geometry and structure were made previously in 101 Caucasian adolescents and young adults 9-26 years of age. The data on measured bone mineral content and calculated section modulus were compared with the results of our analytical model of cross-sectional development of the human femur over the same age range. Both bone mineral content and section modulus showed good correspondence with experimental measurements when the relationships with age and body mass were examined. Strong linear relationships were evident for both parameters when examined as a function of body mass.

Diaphyseal bone growth and adaptation: models and data

The cross-sectional growth and development of the long bone diaphysis is strongly influenced by in vivo mechanical loading. An analytical approach was developed to model these mechanobiologic influences. First, human growth under normal loading conditions was modeled. Our model predictions were validated by comparison to human data obtained during adolescence. Next, skeletal adaptation during growth under altered loading conditions was examined using an animal model. Rat hindlimb suspension experiments were performed and femoral adaptation to reduced loading during growth was measured and compared to normal controls. Then our model predictions of adaptation during growth were compared directly to our experimental data.

Body mass is the primary determinant of midfemoral bone acquisition during adolescent growth

To study the determinants of bone mass and structure during adolescence, we analyzed the femoral mid-diaphysis of 375 healthy adolescents and young adults, ages 9-26 years, from four ethnic cohorts (African-American, Asian-American, Caucasian, and Hispanic). Whole-body dual-energy X-ray absorptiometry (DXA) scans were used to determine diaphyseal length and mid-diaphyseal diameter of the left femur, as well as linear bone mineral content (BMCL) of a region at the mid-diaphysis. Cross-sectional geometric properties were estimated and used to calculate two structural strength indicators: the section modulus and the whole bone strength index. When the relationships between the bone measurements and age, pubertal group, height, or body mass were evaluated, all cross-sectional femoral measures correlated most strongly with body mass. Multiple regressions accounting for gender and ethnicity provided little additional predictive value over the simple regressions with body mass alone. Furthermore, accounting for all developmental parameters (age, pubertal group, body mass, lean body mass, calcium intake, physical activity level) as well as ethnicity and gender in a single saturated model also did not generally significantly improve the predictive results achieved using only body mass. Our results indicate that increases in midfemoral bone mass and cross-sectional properties during adolescence are primarily related to increases in mechanical loading as reflected by body mass.

Biomechanical comparison of posterior internal fixation techniques for unstable pelvic fractures

Early reduction and rigid fixation of unstable vertical shear pelvic fractures has been shown to decrease the incidence of late sequelae and facilitate early mobilization. The results of fixation of the posterior pelvic ring without anterior fixation are unknown. The purpose of this study was to perform a biomechanical comparison of the most frequently used techniques of posterior fixation for unstable pelvic sacroiliac dislocations in conjunction with ipsilateral rami fractures, i.e., an unstable vertical shear injury. The four methods of posterior fixation tested included sacroiliac (SI) screws, anterior SI plates, transiliac bars, and a combination of SI screws and transiliac bars. Six cadaveric pelvises were tested in axial compression and torsion on a biaxial servohydraulic testing machine. Compared to the intact pelvis, single posterior methods of fixation provided approximately 70-85% resistance to axial and torsional loading. By combining SI screws with transiliac bars, approximately 90% of intact pelvic stability was achieved. Our results suggest that rigid posterior fixation of sacroiliac dislocations alone may obviate the need for additional complex anterior surgical procedures to fix rami fractures.

Determinants of femoral geometry and structure during adolescent growth

Our goal was to understand developmental determinants of femoral structure during growth and sexual maturation by relating femoral measurements to gender and developmental factors (age, pubertal stage, height, and body mass). The bone mineral content of the femur was measured by dual energy x-ray absorptiometry in 101 healthy Caucasian adolescents and young adults, 9-26 years of age. After some simplifying assumptions had been made, cross-sectional geometric properties of the femoral midshaft were estimated. Two geometry-based structural indicators, the section modulus and whole bone strength index, were calculated to assess the structural characteristics of the femur. Femoral strength, as described by these structural indicators, increased dramatically from childhood through young adulthood. Regressions were performed between these femoral measurements and the developmental factors. Our data show that of age, pubertal stage, body mass, and height, body mass is the strongest predictor of femoral cross-sectional properties, and the correlation of body mass with femoral cross-sectional structure is independent of gender. A model including all four developmental factors and gender did not substantially increase the accuracy of predictions compared with the model with body mass alone. In light of previous research, we hypothesize that body mass is an indicator of in vivo loading and that this in vivo loading influences the cross-sectional growth of the long bones.

Hindlimb suspension diminishes femoral cross-sectional growth in the rat

Growth, functional adaptation, and torsional strength were examined in the femora of 39-day-old male Sprague-Dawley rats subjected to hindlimb suspension for 0, 1, 2, 3, or 4 weeks and were compared with measurements for age-matched control animals. Our goal was to understand the effect of reduced loading on the normal age-related changes in femoral properties during growth. The control animals exhibited growth-related increases in all geometric and torsional properties of the femur. The mean body mass and femoral length of the hindlimb-suspended rats were similar to those of the controls throughout the experiment. Over 4 weeks, the femoral cross-sectional and torsional measurements from the hindlimb-suspended rats demonstrated increases in comparison with the basal values (+33% cross-sectional area, +64% polar moment of inertia, +67% ultimate torque, and +181% torsional rigidity), but the age-matched controls showed significantly greater growth-related increases (+71% cross-sectional area, +136% polar moment of inertia, +127% ultimate torque, and +367% torsional rigidity). The differences in femoral structural strength between the hindlimb-suspended animals and the age-matched controls were attributable to differences in altered cross-sectional geometry.

Long bone geometry and strength in adult BMP-5 deficient mice

Bone morphogenetic proteins (BMPs) play a critical role in early skeletal development. BMPs are also potential mediators of bone response to mechanical loading, but their role in later stages of bone growth and adaptation has yet to be studied. We characterized the postcranial skeletal defects in mature mice with BMP deficiency by measuring hind-limb muscle mass and long bone geometric, material, and torsional mechanical properties. The animals studied were 26-week-old short ear mice (n = 10) with a homozygous deletion of the BMP-5 gene and their heterozygous control litter mates (n = 15). Gender-related effects, which were found to be independent of genotype, were also examined. The femora of short ear mice were 3% shorter than in controls and had significantly lower values of many cross-sectional geometric and structural strength parameters (p < 0.05). No significant differences in ash content or material properties were detected. Lower femoral whole bone torsional strength was due to the smaller cross-sectional geometry (16% smaller section modulus) in the short ear mice. The diminished cross-sectional geometry may be commensurate with lower levels of in vivo loading, as reflected by body mass (-8%) and quadriceps mass (-11%). While no significant gender differences were found in whole bone strength or cross-sectional geometry, males had significantly greater body mass (+18%) and quadriceps mass (+15%) and lower tibio-fibular ash content (-3%). The data suggest that adult female mice have a more robust skeleton than males, relative to in vivo mechanical demands. Furthermore, although the bones of short ear mice are smaller and weaker than in control animals, they appear to be biomechanically appropriate for the in vivo mechanical loads that they experience.

Developmental mechanics determine long bone allometry

Evolutionary and developmental factors responsible for the scaling relationships observed in animal skeletons are poorly understood. We have created a mathematical model for long bone cross-sectional development which incorporates both intrinsic growth and extrinsic, adaptive bone modeling in response to changes in bone mechanical strains during ontogeny. The model successfully simulates the developing morphology in individual animals and the bone geometric allometric relationships among adults across many species (range from mouse to elephant in size). Our results suggest that long bone scaling characteristics are not a result of intrinsic genetic factors but are the results of highly conserved, extrinsic biophysical processes whereby bone tissue strains modulate skeletal morphogenesis.

Improved method for analysis of whole bone torsion tests

Structural tests, such as whole bone torsion tests, have become widely accepted methods for assessing average bone material properties. To simplify interpretation of these tests, the nonuniform bone geometry is often analyzed as a tube with a constant cross section (prismatic) and the areal properties of the smallest bone section. This approach may not adequately represent the true torsional behavior of the cross section and does not account for any lengthwise variations in bone geometry. The errors introduced by these approximations are particularly significant when comparing bones of different sizes and geometries. In this paper, we examine the effects of approximating the cross-sectional torsional behavior and of neglecting lengthwise variations in bone geometry. We then present a simple, standardized procedure utilizing a FORTRAN computer program for accurate determination of material properties. We examine first simple idealized bone geometries and then a complex three-dimensional model of the femur from a 26-day-old male Sprague-Dawley rat. For these models, the conventional methods for interpreting torsion tests introduce errors of up to 42% in the shear modulus and up to 48% in the maximum shear stress; a straightforward extension of these methods reduces the errors to within 3%.

Age-related differences in cross-sectional geometry of the forearm bones in healthy women

Men exhibit age-related adaptive changes in long bone geometry, namely, endosteal resorption and periosteal apposition of bone, that help to preserve bone strength. It is not clear whether women undergo similar adaptive responses. To address this question, we assessed the bone mineral density and cross-sectional geometry of the radius and ulna at the one-third distal site by single photon absorptiometry and computed tomography (CT) in healthy young (n = 21, age 20-30 years) and older (n = 22, age 63-84 years) women. We used the CT data to compute the total subperiosteal, medullary, and cortical areas, as well as the maximum, minimum, and polar moments of inertia. We normalized the geometric parameters for bone length and performed comparisons using both the original and size-corrected data. Radial and ulnar bone mineral content and density were 20-30% lower in the older women (P < 0.0001). Ulnar width, total area, medullary area, and maximum and polar moment of inertia were greater in the older than in the younger women. Although we observed similar trends when we examined the radius data that were corrected for bone size, age-related differences in radial geometry were less pronounced and were not significant. We conclude that women undergo endosteal resorption and periosteal apposition of the ulna with age, thereby exhibiting an adaptive pattern that helps to preserve bone strength. The different behavior of these two bones suggests that local, rather than systemic, factors underlie this adaptation.

Mechanobiologic influences in long bone cross-sectional growth

We developed a computer model to simulate the interaction of biological and mechanobiological factors in the development of the cross-sectional morphology of long bones. The model incorporated a strong influence of biologically induced bone formation during early development. In addition, an assumed mechanical loading history during growth and development corresponding to age-related changes in body weight and muscle mass was applied. Based on the bone stress stimulus generated by the assumed loads, mechanically induced apposition and resorption rates were calculated at the periosteal and endosteal surfaces using a previously developed bone modeling theory. These methods successfully emulated the growth-related changes seen in long bone diaphyseal structure as well as changes observed in mature bones during aging. The simulations recreated the rapid increase in bone dimensions during development, stabilizing at maturity, and then the gradual, age-related subperiosteal expansion and cortical thinning. Throughout the growth, development, and aging simulations, the values of the bone radii, area, moments of inertia, and apposition rates corresponded well with measurements documented by other researchers.

Rabbit knee immobilization: bone remodeling precedes cartilage degradation

This study analyzed processes underlying osteoporosis and osteoarthrosis after short-term immobilization of the right hind limb of postadolescent (2.8 kg) and mature (4.0 kg) rabbits. After 3 weeks, the lateral posterior aspect of the lateral tibial plateau and the lateral femoral condyle of the immobilized limb exhibited prominent subchondral vascular eruptions. Femoral metaphyseal bone density decreased 27 and 18% in the immobilized limbs of postadolescent and mature rabbits, respectively. Calcein green fluorescence increased 1.9-fold (p less than 0.001) in the metaphyseal trabeculae of immobilized femurs. With immobilization, sulfate incorporation into femoral cartilage glycosaminoglycan increased, although total cartilage glycosaminoglycan and hydroxyproline levels were unchanged. Thymidine incorporation into DNA increased four- to fivefold in tibial and femoral cartilage of the immobilized limb. In this study, bone loss and remodeling preceded erosive cartilage degradation.

Effects of voluntary exercise on bone mineral content in rats

We used a voluntary running model to explore the relationship between average daily running distance and bone mineral status of rats. A total of 60 male Sprague-Dawley rats were randomly assigned at 6 weeks of age to a sedentary control group (n = 22) or to a group with unlimited access to a running wheel (n = 38). The running distance of exercising rats was monitored daily, and steady-state running levels ranged from 3.2 to 18.1 km/day. At the end of the experimental period, femora and tibiae were dissected and bone mineral content (BMC, g/cm) and bone mineral density (BMD, g/cm2) were measured by single-photon absorptiometry. Cross-sectional morphometry was examined by taking a transverse section of the femoral middiaphysis. Hindlimb percentage fat was significantly higher in controls than in runners (20.0 +/- 1.2 versus 11.1 +/- 0.6, p less than 0.001), and soleus mass was greater in runners than in controls (371 +/- 8.1 versus 320 +/- 0.8 mg, p less than 0.001). Femoral and tibial lengths, weights, and volumes were significantly higher in runners than in controls (p less than 0.005). BMC and BMD were higher in runners than in controls at all sites apart from the distal femur. Cross-sectional areas at the femoral midshaft were greater in running rats than in sedentary controls (6.26 +/- 0.1 versus 5.45 +/- 0.3 mm2, p less than 0.02), as was the polar moment of inertia (15.6 +/- 0.6 versus 12.7 +/- 0.2 mm4, p less than 0.05). No positive correlation was found between distance run and BMC, BMD, cross-sectional area, or polar moment of inertia.(ABSTRACT TRUNCATED AT 250 WORDS)