Hip fracture risk assessment using composite indices of femoral neck strength: the Rancho Bernardo study

Abstract

The objective of this work was to examine the prediction of incident hip fracture risk by composite indices of femoral neck strength, constructed from dual X-ray absorptiometry (DXA) scans of the hip. These indices integrate femoral neck size and body size with bone density, and reflect the structure’s ability to withstand axial compressive forces and bending forces, and the ability to absorb energy in an impact. We studied a prospective cohort of community-dwelling women, 55 years or older at baseline, average age 72 years, that was established in 1988, with follow-up until 1998. Higher baseline values of each composite strength index were associated with lower risk of incident hip fracture in this cohort. After adjusting for age and body mass index, relative risks associated with each standard deviation increment in the composite indices were 0.39 for compression strength (95% confidence interval 0.25–0.60), 0.43 for bending strength (95% confidence interval 0.27–0.70), and 0.34 for impact strength (95% confidence interval 0.20–0.58). Larger bone size relative to body size was associated with lower risk of incident hip fracture. After adjusting for age, body mass index, and bone density, each standard deviation increment in femoral neck width normalized by body weight was associated with incident hip fracture relative risk of 0.38 (95% confidence interval 0.20–0.70). Our results suggest that composite indices of femoral neck strength, constructed from DXA measurements, have the potential to improve hip fracture risk assessment. This needs further exploration in larger cohorts and in different population sub-groups.

Appendix: derivation of composite strength indices

Here, we derive composite indices of femoral neck strength using structural engineering principles. We look at the femoral neck’s ability to resist compressive loads (along its main axis) and bending loads, and its ability to absorb energy in an impact.

Some fractures of the femoral neck may be the result of compressive forces along the femoral neck axis. Basic concepts from strength of engineering materials show that if material properties are constant within a structure, then the ability of the structure to withstand axial compressive forces at any point is proportional to its cross-sectional area at that point. For instance, the compressive force needed to create unit axial strain is given by [23]:

$$ {\text{Load}}_{{{\text{compressive}}{\text{,unit}}}} {\text{ = Young's}}\;{\text{modulus}}\;{\text{of}}\;{\text{material}} * {\text{Cross-sectional}}\;{\text{area}}{\text{.}} $$

Similarly, if the ultimate axial stress that the material can tolerate in the elastic mode (material strength) is σ_lim, then the compressive force needed to cause failureis given by [24]:

$$ {\text{Compressive}}\;{\text{Failure}}\;{\text{Load}} = \lim * {\text{Cross - sectional}}\;{\text{area}}{\text{.}} $$

Young’s elastic modulus and material strength of bone are material properties that depend both on the quantity (mass) of mineral and on the distribution of mineral within bone [24]. Material properties are generally not uniform within whole bone, and differ for instance, between trabecular and cortical regions. Since details of bone composition are not readily available from DXA scans, we assume here that mineral mass in the femoral neck is largely in the cortex. This permits us to use macroscopic measurements of bone mass and size (from DXA) to gauge strength.

With this simplifying assumption, the femoral neck’s compressive strength (ability to withstand an axial compressive load) is proportional to the cross-sectional area of the mineralized cortex at its thinnest point, a purely structural factor. Although the cortex’s cross-sectional area is not directly obtainable from a DXA scan, it is related to the areal bone mineral density measured by DXA because of the following equality:

$$ \rho * {\text{CCSA}} = {\text{BMD}} * {\text{FNW}}, $$

where ρ is the cortex’s effective density (in g/cc), CCSA is the cortical cross-sectional area (in cm²), BMD is the projected areal bone density measured by DXA (in g/cm²) and FNW is the width of the femoral neck at its narrowest point in the DXA projection plane (in cm). This equality follows because the product on either side equals the total bone mineral content per unit-thickness cross-sectional slice (in g/cm). Although apparent density of whole bone (bone mass divided by whole bone volume) varies from person to person because of differences in porosity and soft tissue elements, the actual density of mineralized bone tissue varies relatively little, even in osteoporosis [21,25]. Since ρ is essentially constant, CCSA is directly proportional to BMD*FNW. Hence, the following potential measure of femoral neck compressive strength can be constructed from DXA scans:

$$ {\text{Strength}}_{{{\text{compressive}}}} = {\text{BMD}} * {\text{FNW}}. $$

With respect to fracture risk, absolute strength is not as important as strength relative to the size of the load: Structures that are exposed to larger loads need to be stronger than structures exposed to less heavy loads. The load borne by the femur during walking is proportional to body weight. Hence, we have the following composite index of the femoral neck’s compression strength, relative to its usual load:

$$ {\text{Compression Strength Index = }}\frac{{{\text{BMD*FNW}}}} {{{\text{Body Weight}}}}{\text{.}} $$

Some fractures of the femoral neck may be a result of bending forces. Again, basic concepts from strength of engineering materials show that if material properties are constant in a structure, then its ability to withstand a bending moment at any point is proportional to its section modulus at that point [23]. For instance, the bending force needed to create unit axial strain in the outermost fibers (where the strain is maximum) is given by:

$$ {\text{Load}}_{{{\text{bending}}{\text{,unit}}}} {\text{ = Young's elastic modulus of the material * }}\frac{{{\text{section modulus }}}} {{{\text{moment arm}}}}{\text{.}} $$

Similarly, if material strength is σ_lim, then the bending force needed to cause failure in the outermost fibers is given by:

$$ {\text{Bending}}\;{\text{Failure}}\;{\text{Load}} = {{\sigma }}_{{{\text{lim}}}} * {\text{ }}\frac{{{\text{section}}\;{\text{modulus}}}} {{{\text{moment}}\;{\text{arm}}}}. $$

While material strength and elastic modulus are both material properties, the second term that is common to both of the above expressions, the ratio of section modulus to moment arm, is a structural factor [24]. The section modulus of a structure at any point depends on the distribution of mass in its cross section at that point. Determining it exactly for a bone structure such as the femoral neck requires knowledge of the distribution of bone mineral in the femoral neck, which may be obtained either from three-dimensional computed tomography data or from the absorption profile along the neck width from either single photon absorptiometry [25] or DXA [21]. In the absence of such detailed information, the section modulus of the femoral neck can be estimated in the following fashion. Assuming that the cross-sectional shape of the femoral neck is approximately circular and that mineral mass is largely confined to its cortical rim, it follows that

$$ {\text{Section}}\;{\text{Modulus}} = \pi {\left( {{\text{R}}^{2} - {\text{r}}^{2} } \right)} * \frac{{{\text{R}}^{2} + {\text{r}}^{2} }} {{4{\text{R}}}}, $$

where R and r are the outer and inner radii of the mineralized annular cortex in cross section [26]. Notice that the first term in the above expression is the cross-sectional area of the mineralized cortical annulus. In the absence of direct measurement of the inner radius r, a crude approximation to the second term, which is appropriate for thin cortices, is obtained by using half of FNW for both R and r, so that the section modulus of the femoral neck at its thinnest point can be approximated by:

$$ {\text{Section}}\;{\text{Modulus}} = {\text{CCSA}} * \frac{{{\text{FNW}}}} {4}. $$

The earlier discussion showed that CCSA is proportional to BMD*FNW; therefore the section modulus of the femoral neck is proportional to BMD*FNW*FNW. Using this formula for section modulus, and using HAL as the moment arm, we obtain the following measure of the femoral neck’s bending strength (ability to withstand a bending load):

$$ {\text{Strength}}_{{{\text{bending}}}} {\text{ = }}\frac{{{\text{BMD * FNW * FNW}}}} {{{\text{HAL}}}}{\text{.}} $$

This is an absolute measure of bending strength. Since the bending forces borne by the femur during walking are proportional to body weight, we have the following composite index of the femoral neck’s bending strength, relative to its usual load:

$$ {\text{Bending Strength Index = }}\frac{{{\text{BMD*FNW}}^{{\text{2}}} }} {{{\text{HAL*Weight}}}}{\text{.}} $$

However, the vast majority of hip fractures in older men and women occur as a result of a fall [10]. The forces experienced in a fall depend not only on the weight of the person but also on the height of the fall [23]. In fact, because of the proportionality of individual body parts to overall body size, the potential energy to be absorbed in a fall from standing height is proportional to the product of body height and body weight.

It can be shown that the maximum energy that a cylindrical structure (such as the femoral neck) can absorb in axial impact before stresses reach the elastic limit σ_lim, is given by [23]:

$$ {\text{Maximum}}\;{\text{Compressive}}\;{\text{Energy}} = \frac{{\sigma ^{2}_{{\lim }} }} {{\text{Y}}} * {\text{Cross - sectional}}\;{\text{area}} * {\text{Length}}\;{\text{of}}\;{\text{structure}} $$

where Y is the Young elastic modulus of the material. Similarly, the maximum energy that a cylindrical structure with annular cross-section can absorb in bending mode before stresses reach the elastic limit σ_lim, is given by [23]:

$$ {\text{Maximum}}\;{\text{Bending}}\;{\text{Energy}} = \frac{{\sigma ^{2}_{{\lim }} }} {{\text{Y}}} * \frac{{{\text{Section}}\;{\text{Modulus}}}} {{6 * {\text{R}}}} * {\text{Moment}}\;{\text{arm}}. $$

Recalling that for the femoral neck, R equals FNW/2 and the section modulus is proportional CCSA*FNW, and using HAL for both the length of the structure and the moment arm, it follows that the maximum energy that the femoral neck can absorb in both compressive and bending modes is proportional to the same structural factor: CCSA*HAL. As before, CCSA can be estimated from DXA data as the product of BMD and FNW. Thus, the following potential measure of femoral neck (absolute) impact strength can be constructed from DXA scans:

$$ {\text{Strength}}_{{{\text{impact}}}} = {\text{BMD}} * {\text{FNW}} * {\text{HAL}}. $$

Since the potential energy in a fall from standing height is proportional to the product of body height and weight, we have the following composite index of the femoral neck’s impact strength, relative to the energy in a fall from standing height:

$$ {\text{Impact Strength Index = }}\frac{{{\text{BMD*FNW*HAL}}}} {{{\text{Height*Weight}}}}{\text{.}} $$