## 8 Whole Bone Biomechanics

A primary function of the skeleton is to bear loads, supporting the body and allowing for motion. As such, the structural behavior of the entire bone is of interest, as are the determinants of this behavior. Three main factors determine whole bone mechanical behavior: tissue quantity, geometry, and material properties, all of which can be experimentally quantified (Fig. 8.1). Bone mass and geometry can be determined directly by measuring weight and size or by using imaging modalities such as computed tomography (CT), dual X-ray absorptiometry or microCT. Material properties can be determined from machined specimens of cancellous or cortical bone. Whole bone mechanical tests examine the culmination of all three contributing factors and the overall behavior of whole bone as a structure.

### 8.1 Overview of Testing Methods

The theory and methods presented in this chapter will focus on testing methodology and the resulting outcome measures of structural tests of whole bones. Whole bone mechanical testing is performed by applying a monotonically increasing load to a bone using fixtures in conjunction with a materials testing machine while recording the resulting displacement and applied load. The applied load can be axial (tension or compression), bending, or torsional. Relevant structural outcome measures such as stiffness and maximum load are calculated from the load-displacement data.

The load-displacement graphs for all whole bone testing methods have similar features, regardless of the loading mode (Fig. 8.2). Key parameters obtained from the data include the structural stiffness and yield, ultimate, and failure loads. Initially, load and displacement vary linearly, and the linear slope is related to the structural stiffness of the bone sample. The structural stiffness for a given loading mode again depends on the three factors bone mass, bone geometry, and the material properties of the bone tissue. If the sample is loaded beyond the linear portion, the sample yields and retains some permanent deformation. A precise definition of yield is difficult and therefore several methods have been suggested to determine the yield point. Because yield is associated with flattening of the load displacement curve, the yield load can be defined by the point where the slope of the linear portion of the curve decreases by 10%. 1 Yield can also be defined as the point where a line with 95% of the stiffness intersects the load-displacement curve. 2 Frequently, an offset yield point is defined by shifting the line of stiffness by a certain amount and determining the intersection between the shifted line and the load-displacement curve. The maximum load value attained during the test is called the ultimate load or somewhat colloquially the strength of the bone sample. The failure load is the load value prior to catastrophic failure of the sample, indicated by an abrupt drop in load with very little change in displacement. Depending on the specifics of the bone sample and testing mode, the yield, maximum, and failure load can overlap. Respective displacements of the yield, maximum, and failure load can also be reported by identifying the corresponding displacement at which the respective loads occur.

Mechanical tests on whole bone samples should ideally be performed immediately after harvesting. If immediate testing is not possible, bones can be wrapped in gauze soaked with phosphate-buffered saline, placed into sealed containers or evacuated plastic bags, and stored at −20°C or below. 3 Frozen samples must thaw in a hydrated state and equilibrate to the prescribed testing temperature prior to testing. To eliminate the effect of temperature and loading rate on mechanical outcomes, all samples should be tested at the same ambient temperature and load rate. The sampling rate of the displacement and load must be high enough such that a sufficient number of data points are available for the test duration.

The following sections review commonly used whole bone testing methods and associated outcome measurements. Key decisions to be made prior to testing whole bone samples include the loading mode, sample preparation, and specific test parameters including the loading rate, fixture geometries, and temperature.

### 8.2 Axial Loading

Axial loading tests are commonly performed in compression on bone specimens that would also physiologically experience axial loading such as whole vertebrae or lower limb bones. 4 In this testing mode, the applied load is a monotonic compressive load applied along the cranialcaudal axis of the bone. For example, in an axial compression test of a vertebral body, the loading fixture is composed of two circular loading platens with diameters approximately equal to the vertebral endplate (Fig. 8.3). To minimize bending moments during the test, the end-plates of the samples can be embedded in bone cement or shaved with a scalpel to make the loaded surfaces parallel to each other. A pin is secured to the lower platen and inserted through the vertebral foramen to consistently align the samples. The vertebral body is then secured at the caudal endplate with a thin layer of cyanoacrylate glue. The compressive load is applied at a constant load or displacement rate until the load drops abruptly, indicating failure. After failure, if loading is continued, the load will eventually increase indefinitely as the crushed vertebra is loaded again and compacted. During testing, the load and crosshead displacement are recorded.

The load-displacement curve represents the compressive load (*P*, in units of Newtons) plotted against the axial displacement (δ, in micrometers or millimeters). The compressive stiffness is the structure′s ability to resist deformation due to a compressive load and is often expressed as the ratio of applied load *P* and displacement δ, resulting in a value of compressive stiffness in N/mm. However, as the resistance of a sample strongly depends on the size of the sample, it is recommended to calculate the compressive stiffness by adjusting the load/displacement ratio by the specimen size:

where *P*/δ represents the slope of the load displacement curve and *h* is the height of the sample. The maximum load is often referred to as the compressive strength. Another factor that directly determines the compressive properties is the cross-sectional area of the specimen, which can be estimated by the cross-sectional area of the sample. However, as bone specimens are very rarely solid but contain trabecular bone or bone marrow and are furthermore irregularly shaped, the adjustment for cross-sectional area often is inadequate.

### 8.3 Bending Loading

Bending tests of whole bones are generally performed in three different bending configurations: cantilever bending, three-point bending, or four-point bending. For whole bones, these tests primarily measure the mechanical behavior of the cortical diaphysis and do not capture any contribution from cancellous bone tissue.

#### 8.3.1 Three-Point Bending

Three-point bending tests are typically performed on long bones with large length-to-width ratios such as the femur, humerus, radius, and tibia. 5 Though no bones have constant cross-sectional geometries or homogeneous and isotropic material properties, which are the underlying assumptions required for beam theory, analysis of bending tests of long bones are based on engineering beam theory. As the name suggests, three points contact the bone: two fixed supports on one side, and a single load point centered between the two supports on the opposing side (Fig. 8.4). As a rule, the span, or distance between the support points, is commonly half the bone length, but this can be increased if the bone ends are potted and stability of the bone on the supports is not compromised. The load point is actuated using a constant load or displacement rate. The bone must be consistently oriented in the loading fixture, such that the loads are always applied in the same anatomic plane, as the cross-sectional shape of the diaphysis rarely is axisymmetric.

The three-point bending configuration induces a moment on the bone that varies linearly along the length of the bone, but it is the moment at the center of the span we are interested in. Given the load (*P*) at any time during the test, the resulting moment at any point located between the supports (*x*) can be calculated using beam bending theory:

The moment ranges from zero at the supports to a maximum value at the center load point where *x* = *L*/2 and therefore the maximum moment *M _{max} *=

*PL*/4.

The outcome measures are analogous to those of other loading modes. The slope of the load-displacement curve is commonly reported in literature for bending tests (units of N/mm). However, this stiffness cannot be compared with other studies that use different span widths and is not the true resistance of the sample to bending. The bending stiffness should always be reported and accounts for testing geometry, intrinsic material stiffness, and specimen geometry. The slope of the linear portion of the moment-displacement curve (*P*/δ) is related to the bending stiffness:

where *L* is the span length. The yield, maximum, and failure moments are calculated using the respective load values and equation (8.2).

The deflection of any point along the beam depends on the applied load (*P*) and the position along the span length (*x*). Like the moment, the maximum deflection occurs at the midpoint of the span. Maximum deflection can be calculated using beam bending theory:

However, this approach requires estimating the Young′s modulus (*E*) of the bone sample and the moment of inertia (*I*), or geometric resistance to bending. The maximum displacement of the bone at midspan can be directly measured by the displacement of the actuator or from a linear variable displacement transformer (LVDT) placed at the bottom surface of the bone. When possible, the LVDT approach is recommended. While the addition of an LVDT can make testing more complicated, errors due to local crushing at the load point are eliminated.

In addition to material properties and microstructure of the bone tissue, the geometric parameter that influences bending outcome measures is the moment of inertia. The moment of inertia, also referred to as the second moment of area, takes into account the cross-sectional area and shape of the cross-section. The moment of inertia can be estimated by approximating the cortical cross-section to be a circular tube:

where *D _{o} *is the outer and

*D*is the inner diameter of the bone. The moment of inertia can also be estimated by approximating the cortical cross-section as an elliptical tube:

_{i}where *A* is the major outer diameter, *B* is the minor outer diameter, *a* is the inner major diameter, and *b* is the inner minor diameter. Approximating the cross-section as a hollow circle or ellipse is not ideal, and more accurate values of the moment of inertia can be directly calculated from CT or microCT scans of the bones prior to testing.