PREDICTIONS OF RADIUS BENDING STRENGTH BY RADIUS STIFFNESS,
MINERAL, AND ULNA MECHANICAL PROPERTIES
______
A Thesis Presented to
The College of Arts and Sciences,
Ohio University
______
In Partial Fulfillment of the
Requirements for Graduation with Honors from the
College of Arts and Sciences with the degree of
Bachelor of Science in Biological Sciences
______
by
McKenzie L. Nelson
April 2017
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Table of Contents Acknowledgements……………………………….……………………...... 3 Abstract…………………………….……………….…………………...…4 1. Introduction…………………………………………...... 5 1.1 Bone Structure, Development, and Maintenance..………....…….5 1.2 Osteoporosis Manifestation and Diagnosis…………..……….…..8 1.3 Bone Biomechanics...…………………...……………………….….13 1.4 Importance of the Radius…..……………………………..…….….23 1.5 Specific Aims.………………..……………………………………....26 1.6 Hypotheses.…………..………………………………………....……26 2. Methods………………………………...... ………………….…....27 2.1 Specimens…………………………….....…………………….……..27 2.2 Experimental Design..………………………...... ……….…….32 2.3 Experimental Protocol………..……………………………….…...35 2.4 Data Analysis.………………………..……………………….……..48 2.5 Statistical Analysis.……………………..…………………….…….53 3. Results……………………………..…………………………….….….57 3.1 Data Exclusions……………..………………………………………57 3.2 Radius EI Corrections………..………………………….…………57 3.3 Univariate Analysis.…………...………………………….………...61 3.4 Bivariate Analysis in the Prediction of Radius Strength……….62 3.5 Bivariate Analysis in the Prediction of Radius EI………………69 4. Discussion…………………………………………………….….…...... 74 4.1 Main Findings………………………………………….…………….74 4.2 Significance in Context with Previous Research....……………..74 4.3 Strengths………………………….…………………………………..80 4.4 Weaknesses……………………………………………….……….….82 4.5 Future Research.……………………………………….………...….87 References………………………………………………..……….…….…89
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Acknowledgments
This project was made possible by funding from the Ohio Space Grant
Consortium and Dr. Anne B. Loucks of the Department of Biological Sciences at Ohio
University. I am grateful for Dr. Betty Sindelar of the Ohio University School of
Rehabilitation and Communication Studies, Division of Physical Therapy for allowing me to use her QMT system, the human tissue banks, Science Care and AdvancedMed, for providing specimens for this project, and to the donors and their families for selflessly giving their bodies to science. I am also thankful for the guidance of Dr.
Chris Griffin in completing this thesis.
I would further like to recognize the extensive work done by previous students in the lab that provided predictors used in this project: Emily Ellerbrock and Jennifer
Neumeyer for their analysis of MRTA data, Tyler Beck for his analysis of ulna QMT data, Gabrielle Hausfeld for her analysis of ulna bending strength predictors, and
Maureen Dean for her analysis of radius UD compressive strength and radius DXA scans. Without their efforts, this project would not have been possible. The support of my family and friends has additionally been crucial in the completion of this thesis, and I would like to thank them for providing me with the network I needed.
Finally, I have been very fortunate to work under Dr. Anne B. Loucks and Lyn
Bowman. I am deeply appreciative of Lyn’s patience in educating me on everything from biomechanics and statistical analysis to life lessons, and for his assistance in helping to develop the study protocol. I am also deeply indebted to Dr. Loucks for allowing me to work in her lab for the past two years, mentoring me, and giving me every opportunity to succeed in my project and educational pursuits. The challenges and education I received in this experience are unparalleled by any other experience in my college career. 4
Abstract
Osteoporosis is a systemic, skeletal disease characterized by decreased bone
strength that predisposes individuals to an increased risk of fracture. Unfortunately,
there is no clinical device able to measure bone strength. Instead, osteoporosis is
diagnosed on the basis of bone mineral density (BMD) as measured by dual-energy X-
ray absorptiometry (DXA). However, research has shown that BMD does not predict
fractures well. Bone strength has been shown to be predicted accurately by bone
stiffness (EI), but no clinical device measures bone stiffness either. Ohio University is
developing Mechanical Response Tissue Analysis (MRTA) to measure EI of the human
ulna in vivo. Accurate predictions of ulna bending strength are of limited clinical importance, however, unless the ulna is representative of other long bones. The radius
is of further clinical importance because a fracture of the radius often precedes and could
predict fractures at more serious sites such as the hip.
This project used cadaveric radius specimens, of which the ipsilateral ulna was
previously tested, to determine the accuracy with which radius bending strength was
predicted by various predictors from both mechanical testing of the radius and ulna and
DXA measurements of the radius. Mechanical testing methods included MRTA of the
ulna in vivo and quasistatic mechanical testing (QMT) of the ulna and radius ex vivo.
DXA measurements included scans of the standard UD and 1/3 sites of the radius.
Linear regression analyses revealed that ulna EI measured by MRTA is a more
accurate predictor of radius bending strength than BMD measurements, but is not the
most accurate predictor. Radius EI and BMC at the 1/3 site were the most accurate
predictors of radius bending strength, though not significantly different from each other.
The most accurate predictor of radius EI was BMC at the 1/3 site. 5
1. Introduction
1.1 Bone Structure, Development, and Maintenance
Bone is multifunctional. It offers mechanical support for locomotion,
protection of vital organs in the body, storage of exchangeable minerals for
homeostasis, and hematopoiesis (Burr and Allen, 2013). It therefore plays a crucial
role in viability. Bone is comprised of mineral, collagen, non-collagenous proteins
such as osteocalcin, cells, marrow tissue, and water (Burr and Allen, 2013). On a
macroscopic scale, bone tissue is of two distinct types: cortical bone and trabecular
bone. Cortical bone, or compact bone, is found in the shafts of long bones (the
diaphyses) and as a cortex or shell at the ends of long bones (metaphyses), around
vertebral bodies, and at the surface of the pelvis, skull, and other flat bones (Burr and
Allen, 2013). It is very dense, comprising 80% of bone mass, formed from
circumferential layers, or lamellae (Heller, 2004). Healthy cortical bone contains
Haversian canals that create a porosity of 3-5% (Burr and Allen, 2013). These canals,
each several millimeters long, are distributed throughout the length and thickness of
cortical bone tissue carrying nerves and capillaries. Volkmann’s canals are also
present, running transversely, connecting Haversian canals to each other and to the outside surface of the bone. (Figure 1) (Martin et al., 1998) 6
Figure 1: Composition of Cortical Bone Showing Lamellae, Haversian Canals, and
Volkmann’s Canals (Cowin and Cardoso, 2015)
Trabecular bone, also known as cancellous or spongy bone, makes up 80% of
the bone surface area and is found inside cuboidal bones like the vertebrae, flat bones,
and the ends of long bones (metaphyses). It is composed of interconnected rods and
plates of bone, known as trabeculae, creating a porosity of 50-90% that is filled with
marrow (Figure 2). The interconnected architecture offers strength while minimizing
the mass of the bone, as the trabeculae distribute weight-bearing stress to the stronger cortical bone. (Burr and Allen, 2013; Martin et al., 1998) 7
Figure 2: Composition of Trabecular Bone Showing Rods (Purple Arrows) and Plates (Blue
Arrows) (Burr and Allen, 2013)
As bones grow and are maintained, bone turnover must occur. In order to be sculpted in various ways in response to how the skeleton is loaded, some bone must be removed in certain areas and added to others. In growing children, this process is continuous and known as modeling. In older bone, the process, known as remodeling, is characterized by removal of underutilized bone and by removal and replacement of old, damaged bone (Martin et al., 1998)
In cortical bone, bone remodeling is accomplished through basic multicellular units (BMUs) which contain a few osteoclasts and many osteoblasts. When damage is present in cortical bone, osteoclasts are signaled to form and activate. As shown in
Figure 3, they begin resorbing bone perpendicular to the surface of a Haversian canal then turn and proceed for several millimeters parallel to the canal until they have passed through the damaged site. This process takes about four weeks. The osteoblasts then follow behind the osteoclasts and refill the resulting tunnel with new bone, taking 8
about three months. (Martin et al., 1998) The narrow residual tunnel becomes a new
Haversian canal, and as adults age, this remodeling repair process riddles the bone with additional porosity.
Figure 3: Basic Multicellular Unit (BMU) with Osteoclasts (Red) and Osteoblasts (Blue)
(Adapted from (Martin et al., 1998))
1.2 Osteoporosis Manifestation and Diagnosis
Osteoporosis is a systemic, skeletal disease characterized by decreased bone strength that predisposes individuals to an increased risk of fracture (NIH Consensus
Development Panel on Osteoporosis Prevention, 2001). Recent studies estimate that
10.2 million people in the United States ages 50 or older already have osteoporosis while an additional 43.3 million are at an increased risk for the disease due to low bone mass (Wright et al., 2014). In 2005, over two million osteoporotic fractures occurred in the United States, costing an estimated $17 billion. While this burden alone is rather large, as the population ages the burden will only increase as 9
osteoporosis becomes more prevalent. By 2025, fracture incidence and its financial cost is projected to increase by 50%. (Ensrud, 2013)
With age, the rate of bone formation tends to decreases due to less physical activity and sex steroid hormone production, which causes an imbalance between formation and resorption. This leads to an increase in bone loss and therefore a decrease in bone strength. In trabecular bone, osteoporosis leads to thinning of plates so that they become rods or disappear entirely. As rods thin, they may no longer connect to other trabeculae, and the structural strength becomes compromised (Figure
4). (Office of the Surgeon General, 2004)
Figure 4: Electron Micrographs of Trabecular Bone in Healthy and Osteoporotic Bone
(Office of the Surgeon General, 2004)
Aging is associated with a greater loss of cortical, rather than trabecular, bone with about 70% of all bone loss being due to cortical bone loss, because bone turnover is a surface phenomenon, and because the intracortical surface area of Haversian canals increases with age due to constant remodeling (Figure 5) while trabecular surface area decreases (Seeman, 2015). The resorption from BMUs also causes 10
thinning of the cortical shell as cortical bone is resorbed from the endosteal surface
outward (Figure 6). The extreme porosity of the cortical bone in osteoporosis is
described as trabecularization, because the normally compact, dense bone takes on a
chaotic trabecular-like appearance, without the load-bearing trabecular architecture, that greatly decreases its strength. The majority of bone loss characterized by this disease occurs in cortical bone. (Farr and Khosla, 2015; Zebaze et al., 2010)
Figure 5: Cortical Porosity in Women Aged 29 (A), 67 (B), and 90 (C) (Zebaze et al., 2010) 11
Figure 6: Cortical Porosity and Thinning in a 78 Year Old (A) and a 90 Year Old (D) (Zebaze
et al., 2010)
Another view of osteoporosis is that it is not merely a geriatric disease, but a pediatric disease due to a failure to acquire adequate bone mass in childhood (NIH
Consensus Development Panel on Osteoporosis Prevention, 2001). This failure to develop a strong skeleton during adolescence can be due to genetic abnormalities, poor nutrition including calcium, phosphorus, or Vitamin D deficiency, or a lack of weight-bearing exercise that would stimulate bone formation (Office of the Surgeon
General, 2004). Additionally, during longitudinal growth in adolescence, cortical bone experiences an increase in cortical thinning and porosity that has been found to peak during mid-puberty in both males and females. This is due in part to the increased demand for calcium during the adolescent growth spurt. Therefore, an increase in 12
calcium mobilization from cortical bone and an increase in remodeling to facilitate
rapid growth can cause some skeletal defects. These skeletal defects have been shown
to persist into adulthood as the increased porosity becomes permanent before further
deteriorating in the adult population. (Farr and Khosla, 2015)
Currently, osteoporosis is diagnosed by measuring bone mineral density
(BMD). BMD is normally determined by dual energy X-ray absorptiometry (DXA) in units of areal density (grams per square centimeter) or, in other words, as bone mineral content (BMC) over area, which does not account for depth. Therefore, bone size must be accounted for as DXA BMD is underestimated in small bones and overestimated in large bones. (Ensrud, 2013; Office of the Surgeon General, 2004) The International
Society for Clinical Densitometry (ISCD) recommends DXA scanning at only certain regions of interest (ROI) that include the lumbar spine, total hip, femoral neck, and the distal radius. The World Health Organization (WHO) classifies an osteoporotic diagnosis as a BMD T-score of 2.5 standard deviations or more below the mean for young, white adult women (NIH Consensus Development Panel on Osteoporosis
Prevention, 2001). For premenopausal women and men under 50 years of age, Z- scores rather than T-scores are preferred with 2 standard deviations below the mean suggesting a BMD below the expected mean. (Schousboe et al., 2013)
Unfortunately, DXA measurements of BMD are only accurate to ±1 T-score
(Blake and Fogelman, 2008). This could cause misdiagnosis of osteoporosis.
Furthermore, BMD has been shown to not predict fractures well. In a study of 163,979
postmenopausal women with no previous osteoporosis diagnosis, 96.5% of women 13
with an osteoporotic BMD T-score ≤ -2.5 did not fracture while 81% of fractures occurred in women with a BMD T-score > -2.5 (Siris et al., 2001).
Another problematic finding is that during old age, 80% of osteoporotic fractures occur in primarily cortical regions (Zebaze et al., 2010), but diagnostic decisions are made from BMD in primarily trabecular regions, and DXA is not able to distinguish between cortical and trabecular bone (Farr and Khosla, 2015).
Additionally, BMD values cannot be compared between systems, whether from the same or different manufacturer, without a cross calibration (Schousboe et al., 2013).
This could also cause difficulty in diagnosing osteoporosis when patients are scanned by more than one system during their diagnosis and treatment.
1.3 Bone Biomechanics
With the finding that BMD does not predict fractures well, other methods for predicting fractures are being explored. One explanation for why BMD is not a good predictor is that by normalizing for size (BMD = BMC/area), BMD loses the predictive influence of bone size. A larger bone would be expected to break under a larger force. In this respect, new diagnostic tools should account for the size of bone.
Mechanical Testing
Compression (which causes shortening and widening), tension (which causes lengthening and narrowing), shear, torsion (which causes angular distortion), and 14
bending are all ways that bone can experience a force or load through mechanical testing (Figure 7) (Bankoff, 2012).
Figure 7: Different Types of Forces on Bone (Bankoff, 2012)
A mechanical test can be performed as either a material test or a structural test.
Material tests determine the intrinsic properties of bone tissue, and are usually done on specimens of a standard size and shape. Stress and strain are plotted (Figure 8A), where stress is the force per unit area and strain is the change in length over the original length. (Njeh, 2004 ) The elastic modulus (E), or material stiffness, is defined as the slope of the elastic region of the stress vs. strain curve. In this region there is little irreversible bone deformation. However, beyond a particular load (the yield point), further loading causes the curve to become nonlinear, constituting the plastic region. In this region, bone deformation is irreversible. Material strength is then defined as the peak stress before the specimen fractures. (Sharir et al., 2008) 15
Figure 8A: Stress vs Strain Curve where E is Slope and the Failure Point is the Material
Strength (Sharir et al., 2008)
While material testing measures the mechanical properties of bone tissue, whole bones fracture as structures, not materials. A structural test accounts for the influence of geometry and architecture of the specimen in addition to its material properties, and, therefore, more closely reflects how real bones break. (Njeh, 2004 )
For structural tests, force and displacement are plotted (Figure 8B) and bone stiffness
(k) is measured (in units of force per displacement) as the slope of the elastic region of the force vs. displacement curve. Bone strength is then defined as the peak load before the specimen fractures (Fmax) for compression, tension, or shear testing, or by the maximum moment (Mmax) (Equation 1 where Fmax = peak load and L= length between supports) in bending or torsion testing. Other researchers have referred to the maximum moment before fracture as the fracture moment, or failure moment.
Mmax =Fmax * Eqn. 1 𝐿𝐿 4 16
Figure 8B: Force vs Displacement Curve where k is the Slope and the Failure Load is the
maximum force before fracture (Njeh, 2004 )
Bone stiffness (k) has previously been shown to be an accurate predictor of
bone strength (Borders et al., 1977; Fyhrie and Vashishth, 2000; Jurist and Foltz,
1977). For this project, bending stiffness (kb) is the focus. In a bending test, the
primary focus is on cortical bone because the central shaft of a long bone is comprised
entirely of cortical bone. As discussed, cortical bone provides the majority of the
strength inherent in bone whereas trabecular bone distributes any weight-bearing stress to the cortical bone, and the majority of bone loss in osteoporosis is within cortical bone. These qualities of cortical bone lends to the idea that it is most pertinent in the loss of strength and the cause of fracture. 17
The gold-standard method for measuring both bone stiffness and strength is
quasistatic mechanical testing (QMT), which must be conducted on excised bone. As
is explained further in section 2.3, the relationship between force and motion reflects
the elastic, viscous, and inertial effects in a mechanical test, but QMT allows for the
viscous and inertial effects to be negligibly small so that the measurements obtained
reflect only that of the elastic effects. In a bending test, QMT allows for the force and
bending displacement to be continuously measured. Bending stiffness (kb) is then
determined from the linear section of the force vs. displacement curve.
Measurements of kb are not appropriate for comparing bones to one another,
however, because for any given bone measured values of kb vary depending on the distance between supports. To compensate for this, a specific property of the bone
(independent of the distance between supports) is calculated. This is the flexural rigidity, or EI:
Eqn. 2 EI = kb* 3 𝐿𝐿 where L is the length between supports. 48
The E in EI is the elastic modulus, which is the inherent stiffness of the bone
material, as previously explained. (Sharir et al., 2008) The I in EI is the cross-sectional
moment of inertia of the bone structure, which is strongly dependent on the diameter
of the bone shaft:
4 4 I = π(do -di )/64 Eqn. 3 where do is the outer diameter of the bone shaft and di is the inner diameter. As can be
seen in Figure 9, bone tissue near the outer diameter of the bone contributes most 18
strongly to I and thereby to EI. This implies the outer diameter most strongly
contributes to the bending strength of the bone as well.
Figure 9: A 4th Order Function of Strength vs Diameter Showing that Strength is Most
Affected by Bone Tissue near the Outer Diameter of the Bone
Thus, EI is a measure of the stiffness of the bone structure (Hutchinson et al.,
2001). In previous studies, EI has been shown to predict bending strength very well. A study of 45 excised human ulna found a correlation coefficient of 0.958 between EI
and bending strength (Jurist and Foltz, 1977). Another study of 56 canine radii, ulnae,
and tibiae found a correlation coefficient of 0.962 between EI and bending strength
(Borders et al., 1977). 19
MRTA
Bone strength is the load at which the bone fractures and is, therefore, indicative of fracture risk. For this reason, a new technology known as mechanical response tissue analysis (MRTA) was invented to measure bone stiffness in vivo.
MRTA was originally developed in the 1980s at Stanford University to help NASA study the effects of space flight on the skeletal health of astronauts (Steele et al.,
1988). However, it was not successfully commercialized at that time. Together with other staff and students at Ohio University, Dr. Anne Loucks has been furthering the development of MRTA for predicting osteoporotic fractures.
MRTA is a non-invasive, radiation-free vibration analysis technique that measures ulna bending stiffness, calculates ulna EI, and estimates ulna bending strength. More than half of all cortical bone (non-vertebral) osteoporotic fractures occur in the arm (Ohsfeldt et al., 2006). Conveniently, the ulna is an ideal test specimen in vivo because its superficial location under a thin layer of tissue in the arm allows for accurate length measurements. Additionally, the ulna is simply supported by the humerus and can be easily supported at the wrist. However, in order for measurements of ulna EI to be clinically useful, they must be representative of other long bones in the body. Due to this, a major aim of this project was to determine how well MRTA measurements of ulna EI predict QMT measurements of radius bending strength.
During MRTA testing, a patient lies supine with an arm raised so that the upper arm is vertical and the forearm is horizontal. The wrist is supported by an adjustable 20
platform, and the elbow is in a fixed position. A probe then applies a load at the midpoint
of the ulna with randomly oscillating frequencies between 40 and 1200 Hz. Force and acceleration data are collected and analyzed to determine the complex forearm stiffness
(force/displacement) and compliance (displacement/force) frequency response functions (FRFs) (Figure 10). The forearm is then assumed to comprise a 7-parameter
mechanical skin-bone system (Figure 11). Solution of the differential equations of
motion for this model yields corresponding 7-parameter mathematical functions
(Equations 4A and 4B) that are fitted to the stiffness and compliance FRFs to quantify
the mechanical properties (i.e., damping, mass, and stiffness) of the skin and underlying
bone (Steele et al., 1988). As in QMT, this measurement of stiffness (Kb) is then used
to determine EI, which reflects the bone’s resistance to bending, independent of its
length. (Ellerbrock, 2014)
Figure 10: Example FRF of an Ulna in MRTA with Accelerance as the Top Waveform and
Compliance as the Bottom Waveform (Ellerbrock, 2014) 21
Figure 11: Seven-Parameter Mechanical Model of the Skin-Bone System (Steele et al., 1988)
Forearm Stiffness:
4 3 2 2 H(s) = F(s)/X1(s) = Ms (s + A3 s + A2 s + A1 s + A0) / (s + C1 s + C0) Eqn. 4A
Forearm Compliance:
2 4 3 2 Y(s) = X1(s)/F(s) = (s + C1 s + C0)/ Ms (s + A3 s + A2 s + A1 s + A0) Eqn. 4B
where A0 = Ks Kb / Ms Mb
A1 = [Kb (Bs + Bp) + Ks (Bb + Bp)] / Ms Mb
A2 = (Ks + Kb) / Mb + Ks / Ms + [Bs (Bb + Bp) + Bb Bp] / Ms Mb
A3 = (Bs + Bb) / Mb + (Bs + Bp) / Ms
C1 = (Bs + Bb) / Mb
C0 = (Ks + Kb) / Mb
Previous studies in Dr. Loucks’ lab have explored the accuracy of MRTA
measurements of ulna EI. Using fresh-frozen cadaveric human arms, MRTA measurements of ulna EI were made, followed by QMT measurements of ulna EI and bending strength (Figures 12 and 13). 22
Figure 12: MRTA Measurements on Cadaveric Arms
Figure 13: QMT Measurements on Cadaveric Ulnae
MRTA measurements of EI predicted bending moment with a coefficient of determination of R2 = 0.99 and a standard error of the estimate (SEE) of 6 Nm. These 23
results were not significantly different from the QMT results, for which EI predicted
bending moment with R2 = 0.99 and SEE = 4 Nm (Figure 14).
Figure 14: QMT Measurements of Strength Predicted by MRTA Measurements of Stiffness
1.4 Importance of the Radius
In order for stiffness measurements of the ulna to be clinically useful, they must
be representative of the stiffness and strength of other long bones in the body. The most
basic test of this would be for ulna EI to accurately predict the EI of the ipsilateral radius
from the same specimen. The radius is not well-suited for MRTA testing, however,
because the radius is not simply supported at the proximal end, so a well-controlled bending test cannot be performed.
Radius fractures are particularly important as they precede and often predict future fractures at other sites such as the hip and spine. One in five women have sustained a clinical fracture at the radius by the time they are 70 years old, but the 24
predictive strength of radius fractures does not only lie within the adult population.
Adolescents who experience a radius fracture also demonstrate an increased risk for fracture at other sites. (Lochmuller et al., 2002)
The high incidence of radius fractures has a large economic burden costing $385 million in 1995 alone (Ray et al., 1997). The distal radius in particular may be more vulnerable to fracture due to the thinness of the cortex at this point. During the modeling and remodeling processes of growth and maintenance, the increase in cortical porosity would be especially evident and could cause fragility sooner than in bones with thicker cortical regions. (Farr and Khosla, 2015)
Previous research has sought to standardize mechanical testing of bone, and, therefore, compared elastic modulus measurements from multiple skeletal sites in mice.
Three-point bending was chosen as the standardized testing method due to its simple, repeatable nature. Compared to the femur, humerus, metatarsal, and tibia, measurements of the radius were found to have lower error rates and variability. Additionally, the radius’ superior aspect ratio, or slenderness ratio, (defined as ¼ L/d where d is the diameter at the midshaft) resulted in measurements that were the closest to previously published measurements for mice as determined by ultrasonic measurements.
(Schriefer, 2005) The aspect ratio should be maximized in order to minimize the effects of shear stress. If the aspect ratio is too low, shear stresses may dominate over bending stresses in the testing. (Burr and Allen, 2013)
With the demonstrated utility of radius fractures as predictors of future osteoporotic fractures, it is important to determine if ulna mechanics predict radius 25
mechanics. The phenotype of an individual’s bones, such as being slender or robust, is consistent across skeletal sites with the radius robustness being significantly correlated with the femur (R2=0.60), tibia (R2=0.65), humerus (R2=0.74), second metacarpal
(R2=0.63), and third metacarpal (R2=0.55) (Schlecht, 2014). This suggests that deficits found at one site would be similar at others. (Figure 15) (Schlecht et al., 2014) In previous research, a test using the arms of 32 adult chacma baboons found that, on average, the midshaft of the radius is 10% stronger than the midshaft of the ulna, although 5 specimens were deviant in that the ulna was stronger than the radius.
(Mennen, 1989) The procedure did not use the gold-standard QMT method for fracturing, though, and used a pressure apparatus with a gauge to determine how much pressure was needed to fracture the bone. So, further testing is warranted to determine if the ulna mechanics do, indeed, predict radius mechanics, and if deviant specimens are a concern for measurements in human subjects.
Figure 15: Systemic Covariance Demonstrated in 4 Individuals (Schlecht et al., 2014) 26
1.5 Specific Aims
The specific aims for this project were:
1. To create 3-D models of 35 cadaveric human arms from CT scans,
2. To use the models to determine the clinical 1/3 site of the radius,
3. To drill holes through the 16% and 64% sites on the radii,
4. To perform QMT three-point bending tests to determine the bending
stiffness and bending strength of the radii, and
5. To perform statistical analyses to compare the accuracies with which ulna
EI and BMD of the radius at the 1/3 site predict the bending stiffness and
strength of the radii.
1.6 Hypotheses
This thesis tested the following null hypotheses:
H01: The Standard Error of the Estimate (SEE) in the regression of radius bending
strength (Mmax) with radius bending stiffness (EI) is not less than SEE in the regression
of radius bending strength with any other predictors (i.e., radius BMD and BMC at the
1/3 site, ulna EI, ulna bending strength (Mmax), ultradistal radius compressive strength,
and ultradistal radius BMD and BMC).
H02: The Standard Error of the Estimate (SEE) in regressions of radius EI with other
predictors (i.e., radius BMD and BMC at the 1/3 site, ulna EI, ulna bending strength
(Mmax), ultradistal radius compressive strength, and ultradistal radius BMD and BMC)
are not different from each other. 27
2. Methods
2.1 Specimens
Arm Donor Demographics
For the purpose of biomechanical testing, Dr. Loucks’ laboratory acquired 44
fresh-frozen, cadaveric human arms from two human tissue banks. Seven arm
specimens (C001-C007) were acquired from AdvancedMed in Las Vegas, Nevada and
thirty-seven arm specimens (C008-C044) were acquired from Science Care, Inc. in
Phoenix, Arizona. The arm donors were both male and female and ranged widely in age
(17-99 years) and body mass index (13-40 kg/m2). (Table 1, Figure 16) For the purpose
of previous projects in the laboratory, the first seven arm specimens (C001-C007) were
used to develop the experimental protocol for bending tests of ulna bones. For the
purpose of the current project, four radius bones (from arms C002-C005) were used to
develop the experimental protocol for bending tests of radius bone shafts. Radii were
not available from arm specimens C022, C027, C028, and C036, because those arms
had not been tested in any previous or ongoing projects. In addition, the radius bone
from arm specimen C001 had not been preserved in the original procedure and was,
therefore, not available for testing. Radius shafts from arm specimens C015 and C016
were not tested in this project, as these shafts were too short for the bending test
procedure. There were then 33 radius shaft specimens available for testing.
28
Table 1: Demographics and Clinical Information of Arm Donors (Part 1) (Dean, 2016) 29
Table 1: Demographics and Clinical Information of Arm Donors (Part 2) (Dean, 2016) 30
Donor Requirements and Exclusions
As previously described by others in the lab (Ellerbrock, 2014), prospective
donors with a medical history of bone cancer or cancers that are likely to have
metastasized to bone, such as bladder, breast, kidney, lung, melanoma, prostate, thyroid,
or uterine cancers, were excluded in order to minimize confounding effects on the bone
mechanics (National Cancer Institute, 2013). For the same reason, prospective donors
were excluded if they had previous injuries or fractures of the arm. Arm donors were
required to have been fresh-frozen within 10 days after death to avoid degradation of
bone stiffness. Bone stiffness degrades after the tenth day post-mortem at room or refrigeration temperature (Tennyson, 1972), however storage at freezing temperatures does not affect bone stiffness if the specimen is adequately hydrated (Sedlin, 1966).
This ensured that the stiffness of bone in the cadaveric human arms closely resembled that in living arms.
Certain exceptions were made to the donor requirements when it was believed that the donor was especially valuable to the project (Ellerbrock, 2014). One donor
(C024) with a medical history of lung cancer was accepted, because his especially large
BMI was expected to increase the range of strength and stiffness. Two donors with prostate cancer (C009 and C025) were also accepted, because many of the donors at the time had prostate cancer and these best fit into the BMI and age specifications. Lastly, two donors (C012 and C039) with a history of lymphatic cancer were accepted, because it is believed to metastasize to bone marrow rather than the actual bone (American
Society of Clinical Oncology, 2014). Overall, ulna bones from these arms did not 31
fracture in a deviant way. Of the donors with cancer histories, C024 fractured the most
distally, but 6 donors without cancer histories fractured more distally than C024,
suggesting it was not deviant (Hausfeld, 2015).
In order to ensure a wide range of bone strength and stiffness, donors were
requested to be men with a BMI greater than 25 kg/m2 and women with a BMI less than
25 kg/m2 (Figure 1). Additionally, donors were not accepted if they had infectious diseases such hepatitis B virus, hepatitis C virus, or human immunodeficiency virus.
The human tissue bank Science Care, Inc. used nucleic acid testing (NAT) on all specimens to evaluate for these infectious diseases in order to protect the safety of those
working in the laboratory. Nevertheless, the arms were treated as infectious, and Blood
Borne Pathogens protocol was followed at all times in handling the arms and specimens
therefrom (Ellerbrock, 2014).
Figure 16: BMI vs Age for all arm donors (Hausfeld, 2015)
32
Arm Storage
Arm specimens were shipped overnight in insulated boxes containing dry ice to ensure they remained frozen throughout transport to Dr. Loucks’ laboratory. Upon receipt, the arm specimens were placed in a -20°C freezer.
2.2 Experimental Design
Previous Data Collection
Before any mechanical tests were conducted on the arms, the frozen arms were scanned by computed tomography (CT) at Holzer Clinic in Athens, Ohio and then returned to the freezer. On subsequent mechanical testing days, an arm was removed from the freezer and allowed to thaw to room temperature. The upper arm was defleshed to facilitate fixation of the intact forearm during MRTA testing. Following the initial
MRTA test, the forearm was defleshed, ensuring to keep the elbow joint intact, and the hand was removed. The radius was sawn distal to the radial tuberosity to preserve the mechanical integrity of the elbow joint, and the distal portion was returned to the freezer for future research. The bare ulna was then subjected to mechanical testing by both
MRTA and QMT.
At a later date, the distal portions of the radii were removed from the freezer for
DXA scanning (Hologic, Discovery DXA System, Bedford, MA). Ultradistal (UD) and
1/3 site BMC and BMD measurements of the radii were obtained. Later, 10 mm long ultradistal specimens were cut from the distal end for compression testing in order to determine compressive strength. (Dean, 2016) All pieces of the radii were then returned 33
to the freezer for cold storage until the ultradistal specimens or shaft sections (Figure
17) were thawed and subjected to testing by QMT in compression and bending, respectively.
Between each step in the process of the various projects, bone specimens were returned to cold storage at -20°C. In order to reduce the loss of bone stiffness while in storage, the specimens were wrapped in gauze soaked in a calcium buffered 0.9% saline solution (Gustafson et al., 1996).
Figure 17: Boney Landmarks on the Radius and Location of Cuts (in Red) (Dean, 2016) 34
Figure 18: Previous Experimental Protocol
Current Project Design
The goal of this research was to compare the accuracies with which MRTA measurements of ulna bending stiffness and DXA measurements of radius BMD at the so-called 1/3 site predict QMT measurements of radius bending stiffness and strength. 35
These QMT measurements were made by subjecting each radius specimen to a
monotonic quasistatically increasing load in three-point bending at the 1/3 site until a fracture occurred. In order to do this, the 1/3 site of the radius needed to be identified on each radius shaft specimen.
2.3 Experimental Protocol
Avizo Modeling
According to the International Society for Clinical Densitometry, the region of
interest for DXA scanning in the forearm is the so-called 1/3 region of the radius
(Hans et al., 2008). However, because the length of the ulna is more conveniently measurable, this region actually comprises 1 cm on either side of 1/3 of the length of the ulna, measured from the distal end (Hologic®, 2012). To identify this location on radii, CT scans of each arm were used to create 3D computer models of the radius and ulna with Avizo image processing software (Visualization Sciences Group, SAS,
Merignac, France). Avizo models revealed that in all of the specimens the distal tip of the radius was some variable distance more distal than the ulna tip. This indicated that the standard 1/3 site for DXA imaging is not at 1/3 of the length of the radius.
Therefore, it was necessary to determine the proportion of the length of the radius at the standard 1/3 site.
In Avizo, each slice of the model was 0.5 mm thick. So, calculations were done in slices and then converted to millimeters and percentages of full length. The number of slices from the distal tip of the ulna to the slice at one-third of the length of 36
the ulna was calculated, and then the slice number at the distal tip of the radius was subtracted in order to determine the number of slices from the distal tip of the radius to the 1/3 site. This number of slices was then converted into millimeters and divided by the total length of the radius to obtain the percentage of radius length at 1/3 of the length of the ulna.
To use C008 as an example, the radius tip was at slice 4, the ulna tip at slice
23, and the 1/3 site on the ulna at slice 180. Therefore, the distance from the radius tip to the 1/3 site was 180-4= 176 slices or 88 mm. Since the length of the radius in arm
C008 was 218 mm, the 1/3 site of the radius was at 100×88/218= 40% of length of the radius. Indeed, the Avizo models and these calculations revealed that the 1/3 region of the radius was actually located, on average, at 40% of the length of the radius (Figure
19).
Figure 19: Avizo model of Forearm with an Orthoslice through the 1/3 Site 37
Specimen Preparation
In order for bending tests to be performed on asymmetric radius shaft specimens, the ends of each specimen need to be supported in a stable manner. Potting can be used to secure bone, but this requires dehydration of the bone and removal of any marrow and fat to allow the potting material to set (Keller and Liebschner, 2000).
This approach was rejected because it would affect the biomechanical properties of the bone. A simple support (Leppanen et al., 2006) was also ruled out because it did not allow for control over the load site of the specimen, as it could be free to rotate on its long axis before the load was applied. Pinning the bone, though, allowed for complete control of orientation without the damaging effects of potting on hydration.
Others have achieved simple support and prevented the rolling of specimens in
QMT bending tests by inserting pins horizontally through holes that were drilled perpendicular to the long axis of the specimen in the plane of bending, as shown in
Figure 20A (Roberts et al., 1996). With such pinning, however, the supporting reactive force upward at the ends of the specimen is applied over only the upper half of the cross section of a specimen, thereby doubling the shear stress when the load is applied downward. There was concern that such doubling of shear stress might fracture highly osteoporotic specimens at the pinning site in this project. Therefore, it was decided to prevent specimens from rolling by inserting pins vertically through holes that were drilled perpendicular to the long axis of the specimen in the plane of loading rather than bending. These pins continued vertically through holes in rollers under the ends of the specimen. By this approach, the reactive supporting shear stress was applied 38
against the entire cross section of the specimen and a doubling of the shear stress was avoided (Figure 20B).
Figure 20A: A Bending Specimen Pinned Horizontally (Roberts et al., 1996)
Figure 20B: A Bending Specimen Pinned Vertically
To achieve pinning, a hole was drilled near each end of each specimen starting at the medial interosseous crest and proceeding through the diameter using a drill press
(True Value, Model MM8050A, Chicago, Illinois) with a #46 drill bit (2.057 mm in 39
diameter) (Figure 21). The size of the hole was desired to be thin enough that it would
have a minimal effect on the biomechanical integrity of the specimen, but thick
enough that a steel pin of the same size going through the hole would be stiff enough
to withstand any stress the specimen would impose on it while under load. The
distance between these holes spanned the same 16%-64% percentage of each radius bone length centered at 40% of radius length.
Figure 21: Drilling of the Bone
Before drilling, specimens were measured and the drilling sites were marked
with an ink marker. The bone was placed into a 6” cross-slide vice (Palmgren, Model
30601B, Naperville, Illinois) that allowed for the bone to be moved into position under the drill press without altering the level or rotation (Figure 22). Foam was placed between the vice and the bone in order to minimize damage from clamping the bone in the vice. In the vice, the bone was rotated so that the interosseous crest was pointing 40
upward and centered. The bone was then leveled across the 40% site (Figure 23). An osteotome was used to nick the drilling sites to facilitate the start of drilling (Figure
24).
Figure 22: Bone in the Cross-Slide Vice
41
Figure 23: Leveling the Bone
Figure 24: Using the Osteotome (Red Arrow) to Prepare the Drilling Sites
After the first hole was drilled on the proximal end, a pin with string tied around it was inserted into the hole. The string was pulled taut across the 40% site of the bone to the distal drilling site to ensure that the drilling sites were in line with the load site (Figure 25). To prevent drilling from overheating and degrading the 42
mechanical properties of the bone tissue, the bones were drilled while frozen. They were then left overnight in containers with calcium buffered 0.9% saline to allow thawing to room temperature, because thawed bones better represent living tissue in vivo (Figure 26) (Njeh, 2004 ). After being left out overnight, the temperature of the saline was checked to ensure it was at room temperature.
Figure 25: Aligning the Drilling Site with the Load Site
Figure 26: Bones Thawing in Calcium Buffered 0.9% Saline Solution 43
QMT Bending Tests
Test Setup. For the purpose of projects in Dr. Loucks’ laboratory, the QMT test frame used (QTest-Elite, MTS Systems Corporation, Eden Prairie, MN) is in the laboratory of Dr. Betty Sindelar of Ohio University’s School of Rehabilitation and
Communication Studies, Division of Physical Therapy. For three-point bending tests, a knife-edge probe fitted into a guide was screwed into the 10 kN load cell of the
QMT test frame. A steel pin was inserted vertically through the hole at each end of the specimen and then through a hole in the freely rotating horizontal steel roller below.
Under load at the central 40% site, these rollers turned on semi-cylindrical mating surfaces machined into the tops of two supporting steel blocks, thereby effecting simple support of the ends of the specimen. Slots in the guide closely contained the heads of the vertical pins (Figure 27), thereby ensuring that the bone would not roll on its long axis when under load while leaving the pins free to turn with the steel rollers as the bone bent. The pin heads were wrapped in Teflon to minimize friction when in contact with the guide.
Figure 27: Pin Heads Encased in Guide 44
Before the load was applied, the bone was leveled across the central 40% site.
For any bones that were not level, small washers were placed on the pins between the bone and the steel roller to ensure the specimen was level (Figure 28). Additionally, a
V-shaped metal overlay was placed at the central 40% site of the specimen in order to prevent the knife-edge probe from digging into the bone and confounding the measurement of bending displacement. (Figure 29) The steel rollers, supporting blocks, steel pins, metal overlay, knife-edge probe and guide were created by a machinist in the Russ College of Engineering and Technology at Ohio University for the purpose of this project.
Figure 28: Washers for Leveling Purposes 45
Figure 29: QMT Setup
Loading Rate. In mechanical bending tests, the relationship between force and
motion reflects the elastic, viscous and inertial effects indicated by Equation 5:
F = kX + bẊ + mẌ Eqn. 5
where k = specimen bending stiffness, reflecting the elastic property of the specimen,
b = specimen damping, reflecting the viscous property of the specimen,
m = specimen mass, reflecting the inertial property of the specimen,
X = bending displacement,
Ẋ = bending velocity, and
Ẍ = bending acceleration. 46
The increasing viscous and inertial effects of strain rate on the relationship between force and displacement are illustrated in Figure 30.
Figure 30: Stress vs Strain Curves at Various Strain Rates (Martin et al., 1998)
In order to selectively measure k without cofounding the effects of b and m, quasistatic mechanical tests ensure that Ẋ and Ẍ are so small that viscous (bẊ) and inertial (mẌ) effects are negligibly small and Equation 5 reduces to Equation 6:
F = kX Eqn. 6
Repeated QMT measurements of bending stiffness pass through a transient of asymptotically increasing values before stabilizing around a mean value (Figure 31).
Therefore, to prep the specimen and the machine, the load was cycled at a rate of
0.1608 mm/min to 100N until the coefficient of variation for repeated bending stiffness and machine stiffness of the last five measurements was <1%. Then the load was monotonically increased at a rate of 0.6432 mm/min until the specimen fractured. 47
As seen in Equation 7, strain rate (ɛ’) increases with an increase in the ratio of D/L2
where D is specimen diameter, Sc is crosshead speed during the test, and L is length
(i.e., the distance between pins).
ɛ’ = Eqn. 7 6𝐷𝐷𝑆𝑆𝑐𝑐 2 For the radius specimen with the largest𝐿𝐿 strain rate (R035), the value of D/L2
-1 -1 was 0.0019 mm , and the strain rate at SC = 0.6432 mm/min was ɛ’ = 0.00012 s ,
which was an order of magnitude lower than the lowest strain rate in Figure 30. This
confirmed that our QMT tests were, indeed, quasistatic.
The computer connected to the QMT apparatus was equipped with Testworks
software (MTS Systems) that displayed a graph of force versus displacement during
testing (Figure 32). The fracture cycle was video recorded and viewed later to ensure
that there were no unexpected movements of the specimen or supports.
Figure 31: Asymptotical Increase in Stiffness for C038 48
Figure 32: Typical Testworks Force vs Displacement Curve
2.4 Data Analysis
The force (F) and displacement (x) data were transferred from the QMT
laboratory to Dr. Loucks’ computers by flash drive, and the data were entered into an
Excel file. In Excel, the force and displacement data were smoothed using a window of
eleven values that averaged the current value, the five previous values, and the next five
values to obtain Fs and xs. The incremental measured stiffness (kmss) was calculated as
the slope between sequential smoothed force and displacement data (ΔFs/Δxs), which
was also smoothed through a window of eleven to obtain kmssw.
The stiffness of the test frame (kTF) is not infinite, so any bending displacement that occurs during the test is partitioned between displacement of the test specimen and displacement of the test frame. As a result, the measured stiffness km underestimates 49
the stiffness of the specimen (ks). Specifically, it can be shown that the measured stiffness is
km = ks * kTF/(kTF + ks) Eqn. 8
where obviously kTF/(kTF + ks) < 1.
Algebraic rearrangement of Equation 8 and substituting kmssw for km reveals the actual specimen stiffness to be
ks = (kmssw * kTF) / (kTF – kmssw) Eqn. 8A
In order to obtain the stiffness of the test frame (kTF), measurements were
conducting on a steel block. Solid steel is stiffer than the test frame, and therefore any
displacement would be conferred to the test frame alone, allowing for the stiffness tested
to not be confounded by the steel block. Equations were then made to fit the stiffness
curve of the test frame at varying loads. (Figure 33)
Figure 33: Testworks’ Stiffness Curve for the Test Frame
As indicated in Figure 34 below, bone stiffness was non-linear. In accordance
with ASTM Standard 790 (ASTM), the stiffness (ks) of non-linear specimens was taken 50
as slope of the load-displacement curve at its inflection point. This corresponds to the peak stiffness before fracture (ksmax). The strength of the radius was taken to be the highest load before the fracture occurred, which was identified as the load when stiffness passed through zero.
Figure 34: Typical Graph Demonstrating a Fracture Cycle
EI was calculated through Equation 9.
EI = (ks * L3) / 48 Eqn. 9 where ks = ksmax, L = the length (in m) between the two steel pins (at 16% and 64% of bone length).
After the bones were fractured, the fracture zone was measured from the distal end with the minimum of the fracture zone being the most distal site with obvious damage and the maximum of the fracture zone being the most proximal site with obvious damage. These lengths were divided by the length of the entire radius, rather than of the shaft specimen, to get the fracture zone as an easily comparable percentage. 51
Data Exclusions
The fracture zones in the 33 radius shaft specimens tested are illustrated in
Figure 35. As is evident in the figure, three of the specimens broke in an anomalous
manner. C014 and C031 both underwent longitudinal rather than transverse fractures,
and C037 crushed in the primarily trabecular region less than 25% from the distal end
of the specimen (Schlenker and VonSeggen, 1976) rather than fracturing across the cortical shaft. The anomalous nature of the data for these specimens is evident in both the Testworks’ force versus displacement graphs and the fracture cycle stiffness curves calculated during data analysis (Figures 36, 37, 38). Therefore, to prevent the anomalous data from these specimens from confounding the statistical analysis of typical cortical fractures, the data from these specimens were excluded from the statistical analysis, leaving 30 specimens available for statistical analysis. 52
Figure 35: Fracture Location as a Percentage of the Radius
Figure 36: Fracture Cycle Graphs for C014
53
Figure 37: Fracture Cycle Graphs for C031
Figure 38: Fracture Cycle Graphs for C037
2.5 Statistical Analysis
Simple linear regressions were used to fit the simple linear model to dependent
(Y) and independent variables (X):
Yi = b1×Xi + b0 + εi
For these regressions, the following usual assumptions were made: 1) εi is a normally
2 2 distributed random error; 2) εi has a mean of 0 and a variance of σ or N(0, σ ); and 3)
the data are distributed with equal variance across the range of X. The simple linear
regression analyses were used to estimate the slope (b1), y-intercept (b0), the 54
coefficient of determination (R2), and σ. σ was estimated as the standard deviation of the residuals around the regression line, otherwise known as the standard error of the estimate (SEE):
σ = SEE = √[SSE/(n-k)]
n 2 where SSE is the sum of squares for error, i.e. SSE = Σ i=1(Yi - b1×Xi - b0) , n is the number of observations, and k is the number of predictors in the model (in this case 2: the slope and y-intercept). In each case, accuracy was quantified as SEE (Dean, 2016).
Linear regression analyses were performed to determine how accurately the dependent variable, radius shaft bending strength, was predicted by the independent variables:
1. Radius EI;
2. Radius BMD at the1/3 site (SEEDXA_BMD);
3. Radius BMC at the 1/3 site (SEEDXA_BMC);
4. Ulna EI measured by MRTA (SEEMRTA);
5. Ulna bending strength (SEEU_QMT);
6. Ultradistal radius compressive strength (SEECS);
7. Ultradistal radius BMD (SEEUD_BMD); and
8. Ultradistal radius BMC (SEEUD_BMC).
Then additional linear regression analyses were performed to determine how accurately the dependent variable, radius EI, was predicted by the independent variables:
1. Radius BMD at the1/3 region (SEEDXA_BMD); 55
2. Radius BMC at the 1/3 region (SEEDXA_BMC);
3. Ulna EI measured by MRTA (SEEMRTA)
4. Ulna bending strength (SEEU_QMT);
5. Ultradistal radius compressive strength (SEECS);
6. Ultradistal radius BMD (SEEUD_BMD); and
7. Ultradistal radius BMC (SEEUD_BMC).
The accuracies of pairs of predictors (SEE1 and SEE2) were then compared by
an F-statistic:
2 F = (SEE1/SEE2)
SEE1 was regarded to be significantly greater than SEE2 whenever F > F1-α(m,n),
where α is the tolerated Type 1 error rate 0.05, and m and n were the numbers of
degrees of freedom for SEE1 and SEE2, respectively (Dean, 2016). As explained in
Sections 2.1 and 2.4 above, in this project m = n = 44 – 5 – 4 – 2 – 3 - 2 = 28 and F1-
α(28,28) = 1.88. Thus, SEE ratios greater than λ = √1.88 = 1.37 were found to be
significantly different.
Two additional regression results were also of interest for the purpose of
determining the clinical error rate that will be explained in the Results section below.
These were the predicted values for the minimum and maximum values of the
predictor:
Y(Xmax) = b1×Xmax + b0
Y(Xmin) = b1×Xmin + b0
56
Statistical Power
It was determined that, ideally, a significant difference between two clinical predictors would be when there is double the number of type II errors in diagnosis. For this sample size, there is a power of 86% if this significant difference is present
(Carlberg, 2013).
57
3. Results
3.1 Data Exclusions
Upon examination of testing procedures, it was determined that certain bones
should be excluded on the principle that the test protocol had not yet been fully
developed during the testing of these specimens, and review of the video footage
showed clear anomalies. For specimens C008, C012, C026, C039, and C044, the metal
overlay was not yet used to prevent the knife-edge probe from digging into the bone.
The caused the measured displacement of the probe to be greater than actual bending displacement, which confounded measurements of kb and thereby EI. For specimens
C010 and C030, a rounded metal overlay was used, but this moved a significant amount during testing and, therefore, also confounded measurements of EI.
Additionally, DXA scans were not obtained of specimens C006 and C007.
Therefore, analyses that include BMC or BMD do not include any data from C006 and
C007. This left 23 specimens to be used in the analyses that did not include DXA measurements and 21 specimens to be used in the analyses that did include DXA measurements.
3.2 Radius EI Corrections
The equation for EI provided in section 1.3 is the Euler-Bernoulli theory for a
beam in pure bending and is the most commonly used for three-point bending tests.
However, when the span of the bone is short relative to the diameter (i.e. the aspect ratio
is too small), some of the measured displacement is due to shearing instead of pure 58
bending of the bone. Thus, bending is less, and resistance to bending (i.e., EI), is greater than displacement would suggest. So, when shear is no longer negligible and must be accounted for, Timoshenko beam theory (Equation 10) is used to account for the confounding effect of shear in the bending tests. (Kourtis et al., 2014)
EI = kb* (1 + 12 ) Eqn. 10 3 𝐿𝐿 𝐸𝐸𝐿𝐿 𝐼𝐼 2 48 𝐺𝐺𝑇𝑇 𝐴𝐴∗ 𝐿𝐿 where c = a shape correction factor (c =2 for a∗ thin𝑐𝑐 ∗-walled∗ hollow cylinder)
= ratio of the elastic and shear moduli = 4 (for human bone, 3.1 < < 5.2 𝐸𝐸𝐿𝐿 𝐸𝐸𝐿𝐿 𝐺𝐺𝑇𝑇 𝐺𝐺𝑇𝑇 (Kourtis et al., 2014))
A= area of the bone cross section at the point of load
L= length of the span across which the bone is supported
Note that the second term on the right hand side of Equation 10 is equal to
( )/ 12 = 12 = 6 + ( ( 4 )4/ ) 𝐸𝐸𝐿𝐿 𝐼𝐼 𝐸𝐸𝐿𝐿 𝜋𝜋 𝐷𝐷𝑂𝑂−𝐷𝐷𝐼𝐼 64 𝐷𝐷𝑂𝑂 2 𝐷𝐷𝐼𝐼 2 2 2 2 2 ∗ 𝑐𝑐 ∗ 𝐺𝐺𝑇𝑇 ∗ 𝐴𝐴∗ 𝐿𝐿 ∗ 𝑐𝑐 ∗ 𝐺𝐺𝑇𝑇 ∗ 𝜋𝜋 𝐷𝐷𝑂𝑂−𝐷𝐷𝐼𝐼 4 ∗ 𝐿𝐿 �� 𝐿𝐿 � � 𝐿𝐿 � � Eqn. 11 where DO and DI are the outer and inner diameters of the bone, respectively.
Since the inner diameter was needed in the calculation of this term, and only the outer diameter was available, an assumption had to be made about the inner diameter of the bone. At the upper limit of the completely osteoporotic extreme case, the inner diameters would be almost the same as the outer diameter. However, in a healthy case, the inner diameter is approximately half of the outer diameter. Thus, the values of this 59
term for the tested bones would have fallen between values for the extreme osteoporotic
and completely healthy cases in Equation 12:
Osteoporotic: 7.5 < 6 + < 12 :Healthy Eqn. 12 𝐷𝐷𝑂𝑂 2 𝐷𝐷𝑂𝑂 2 𝐷𝐷𝐼𝐼 2 𝐷𝐷𝑂𝑂 2 ∗ � 𝐿𝐿 � �� 𝐿𝐿 � � 𝐿𝐿 � � ∗ � 𝐿𝐿 � Thus, less shear occurs in osteoporotic bones than in healthy bones.
The inner diameters of the individual bones in this study could have been
determined by examining micro-computed tomographic images of the cross-sections of the bones. However, that work was not part of this study, so all of the bones in this study were assumed to be healthy in the following analytical results.
Calculations were done to compare the use of the osteoporotic case and healthy case for shear (Figure 39). Calculating the impact of shear on radius EI assuming all the bones were in the extreme osteoporotic condition did not change R2, SEE, or CER
values, but it did reduce the slope from 1.72 ± 0.03 to 1.59 ± 0.03, because it increased
EI values. This change in slope was not significantly different (p=0.36) (Figure 3).
Figure 39: Predictions of Radius Bending Strength by Radius EI with Shear Calculated
Assuming All Bones were in the Healthy Condition (Left)
or All in the Extreme Osteoporotic Condition (Right) 60
Though inner diameter measurements were not available, the BMD T-score for the 1/3 site was available. When the specimens with a T-score ≤ -2.5 were classified as osteoporotic, and the extreme osteoporotic case for shear calculations was applied to them alone, the R2 value still remained the same, but SEE increased from 3.2 Nm to 3.6
Nm and CER increased from 5% to 6%. In Figure 40, the specimens classified as osteoporotic are labeled in red. Those labeled in yellow are osteopenic bones with -2.5
< T-score ≤ -1.5. In the graph on the left, osteopenic bones had the healthy shear correction applied while in the graph on the right, osteopenic bones had the extreme osteoporotic shear calculation applied. This did not cause any significant change in the regression analysis.
Figure 40: Prediction of Radius Bending Strength by Radius EI with Shear Calculated under
the Healthy Bone Assumption for Radii with a T-score > -2.5 and under the Extreme
Osteoporotic Assumption for Radii with T-score ≤ -2.5 (Left), and Calculated under the
Healthy Bone Assumption for Radii with a T-score > -1.5 and under the Extreme Osteoporotic
Assumption for Radii with a T-score ≤ -1.5 (Right) (Yellow Data are Osteopenic and Red
Data are Osteoporotic Based on T-scores)
61
3.3 Univariate Analysis
Univariate analyses characterized the distributions of the variables quantified in this project. Variables measured by QMT included radius bending strength defined as maximum moment, radius EI, ulna bending strength defined as maximum moment, and radius ultradistal (UD) compressive strength. Ulna EI was measured by MRTA.
Variables measured by DXA included radius UD BMC, 1/3 BMC, UD BMD, and 1/3
BMD. The ratio of maximum to minimum indicates the amount of statistical leverage available for detecting a relationship, if one exists, between radius bending strength or
EI and its predictor. High ratios increase the likelihood of detecting a relationship.
Radius UD compressive strength had the most statistical leverage (Min/Max = 9.1), and, therefore, had the greatest probability of detecting a relationship. 1/3 BMD had the lowest leverage (Min/Max = 2.6), and, therefore, had the least probability. The physical dimensions of length between supports, outer diameter, and the ratio of length to outer diameter are also included, though not used in the regression analysis. (Table 2) 62
Variables Units Mean SD Maximum Minumum Max/Min Radius Bending Nm 34.7 15.6 68.0 11.6 5.9 Strength Radius EI Nm2 20.0 9.26 40.2 6.3 6.4 Ulna Bending QMT Nm 45.9 23.5 89.8 11.1 8.1 Strength Radius UD Compressive N 2908 1513 6424 707 9.1 Strength MRTA Ulna EI Nm2 36.0 20.2 81.6 11.2 7.3 UD BMC g 1.10 0.45 2.20 0.29 7.6 1/3 BMC g 1.85 0.61 3.04 0.80 3.8 Radius DXA UD BMD g/cm2 0.35 0.11 0.51 0.13 3.9 1/3 BMD g/cm2 0.65 0.16 0.87 0.34 2.6 Length mm 115 10.3 129.5 94 1.4 Dimmensions Outer Diameter mm 15.6 2.18 19.8 11.7 1.7 L/D N/A 7.50 0.97 8.97 5.37 1.7 O
Table 2: Summary of the Univariate Analyses Used in this Project
3.4 Bivariate Analysis in the Prediction of Radius Strength
Table 3 summarizes the bivariate analyses for the predictors of radius bending
strength. In regression analyses, the coefficient of determination (R2) cannot be
compared between variables that have a significant Y-intercept (ulna strength, radius
UD compressive strength, ulna EI, 1/3 BMC, and 1/3 BMD) and those that do not
(radius EI, UD BMC, and UD BMD), because R2 is calculated in a different manner in
each case. However, the standard errors of the estimate (SEE) can be compared as they
are calculated in the same manner in both cases. 63
Predictors of Radius Units R2 Y Int. ± SE p(b) Slope ± SE p(m) SEE Bending Strength
Radius EI Nm2 0.95 0 ± N/A NS 1.72 ± 0.030 2.2E-25 3.2 Nm Ulna Bending 0.84 7 ± 2.9 0.032 0.61 ± 0.057 5.3E-10 6.2 Nm QMT Strength Nm Radius UD Compressive 0.49 13 ± 5.1 0.017 0.007 ± 0.0016 1.1E-04 11.1 Nm Strength N
MRTA Ulna EI Nm2 0.83 9 ± 2.8 0.0030 0.71 ± 0.067 8.3E-10 6.4 Nm
UD BMC g 0.93 0 ± N/A NS 33 ± 1.0 1.6E-18 5.7 Nm
1/3 BMC g 0.95 (-8) ± 2.4 0.0025 24 ± 1.2 4.3E-14 3.4 Nm Radius DXA UD BMD g/cm2 0.92 0 ± N/A NS 105 ± 4.4 4.6E-16 7.5 Nm
1/3 BMD g/cm2 0.73 (-16) ± 7.2 0.040 80 ± 11 4.2E-07 7.8 Nm
Table 3: Bivariate Analyses for Predictions of Radius Strength
(In Y = mX + b, Y Int. = b and Slope = m)
64
Predictions of Radius Bending Strength by Mechanical Predictors
Figure 41 illustrates the relationships between radius bending strength and radius EI and UD radius compressive strength. Figure 42 illustrates the relationships between radius bending strength and ulna bending strength and ulna EI.
Figure 41: Prediction of Radius Bending Strength by Radius EI (Left) and by Radius UD
Compressive Strength (Right)
Figure 42: Prediction of Radius Bending Strength by Ulna Bending Strength (Left) and Ulna
EI (Right)
65
Predictions of Radius Bending Strength by Bone Mineral
Figure 43 illustrates the relationships between radius bending strength and BMC at the ultradistal (UD BMC) and one-third (1/3 BMC) sites. Figure 44 illustrates the relationships between radius bending strength and BMD at the ultradistal (UD BMD) and one-third (1/3 BMD) sites.
Figure 43: Prediction of Radius Bending Strength by UD BMC (Left) and 1/3 BMC (Right)
Figure 44: Prediction of Radius Bending Strength by UD BMD (Left) and 1/3 BMD (Right)
66
Accuracies of Radius Bending Strength Predictors
As described in section 2.5, the accuracies of two predictors were compared
2 through F-statistics, where F = (SEE1/SEE2) . For hypothesis H01, radius EI was used as
SEE2 in all cases, because it had the smallest SEE. Table 4 summarizes the findings. A
predictor was determined to be significantly less accurate than radius EI when F > F1-
α(m,n).
1/3 BMC was the only predictor not significantly less accurate than radius EI.
The SEEs of ulna bending strength, radius UD compressive strength, ulna EI, UD BMC,
UD BMD, and 1/3 BMD were all significantly larger than that of radius EI.
Predictors of Radius SEE (Nm) F F p Bending Strength 1-α(m,n) Radius EI 3.2 1.00 2.05 0.5 Ulna Bending 6.2 3.75 2.07 0.002 Strength QMT Radius UD Compressive 11.1 12.03 2.07 1.6E-07 Strength MRTA Ulna EI 6.4 4.00 2.07 0.001 UD BMC 5.7 3.17 2.10 0.006 Radius 1/3 BMC 3.4 1.13 2.13 0.4 DXA UD BMD 7.5 5.49 2.10 0.0002 1/3 BMD 7.8 5.94 2.13 0.0001
Table 4: Results of Testing Hypothesis H01 for the Accuracies of Predictors of Radius
Bending Strength
67
Clinical Error Rate (CER)
In many clinical conditions, physicians prefer to treat patients who fall below some value of a physiological parameter that cannot be measured. In such cases, physicians base their treatment decisions on the value of another parameter that is associated with the preferred parameter. Under the standard of care for osteoporosis, the preferred unmeasurable parameter is bone strength and the measurable parameter associated with it is BMD. (Hausfeld, 2015)
As Figure 45 shows, imperfect correlation between bone strength Y and predictor X causes false positive and false negative diagnoses. The clinical error rate
(CER) quantifies the proportion of these patients.
CER is determined by the
SEE of the predictor and the difference between the predicted values (Y) for the maximum and minimum values of the predictor (X).
(Hausfeld, 2015)
Figure 45: Derivation of the Clinical Error Rate (CER) (Hausfeld, 2015)
68
Table 5 lists the CER values for the predictors of radius bending strength. Radius
EI had the lowest CER with 5% and 1/3 BMC had a CER of 6%. Radius UD compressive strength had the highest CER with 28%.
Predictors of Radius Y (Xmax) Y (Xmin) SEE (Nm) CER Bending Strength Radius EI 69.1 10.8 3.2 5% Ulna Bending 61.8 13.8 6.2 13% Strength QMT Radius UD Compressive 58.0 17.9 11.1 28% Strength MRTA Ulna EI 66.9 17.0 6.4 13% UD BMC 72.6 9.6 5.7 9% Radius 1/3 BMC 65.0 11.2 3.4 6% DXA UD BMD 53.6 13.7 7.5 19% 1/3 BMD 53.6 11.2 7.8 18%
Table 5: Clinical Error Rates for Predictors of Radius Bending Strength
Summary for Predictors of Radius Bending Strength
In the F-statistic analysis of SEE, radius EI and 1/3 BMC were not significantly different from each other, and all other predictors were significantly less accurate in predicting radius bending strength. Radius EI and 1/3 BMC also had smaller CER values than the other predictors with radius EI being slightly lower.
69
3.5 Bivariate Analysis in the Prediction of Radius EI
Table 6 summarizes the bivariate analyses for the predictors of radius EI. As mentioned, the coefficient of determination (R2) cannot be compared, because radius
UD compressive strength, ulna EI, 1/3 BMC, and 1/3 BMD have a significant Y- intercept while ulna bending strength, UD BMC, and UD BMD do not. The standard errors of the estimate (SEE) are compared.
Predictors of Radius EI Units R2 Y Int. ± SE p(b) Slope ± SE p(m) SEE Ulna Bending Nm 0.91 0 ± N/A NS 0.42 ± 0.019 3.1E-16 4.8 Nm2 Strength QMT Radius UD Compressive N 0.52 7 ± 2.96 0.029 0.0045 ± 0.00091 6.6E-05 6.4 Nm2 Strength
MRTA Ulna EI Nm2 0.77 5 ± 1.9 0.010 0.40 ± 0.047 2.8E-08 4.5 Nm2
UD BMC g 0.92 0 ± N/A NS 19.0 ± 0.77 1.7E-16 4.1 Nm2
1/3 BMC g 0.91 (-5) ± 1.9 0.012 14.2 ± 0.97 9.5E-12 2.7 Nm2 Radius DXA UD BMD g/cm2 0.90 0 ± N/A NS 60 ± 3.1 1.9E-14 5.2 Nm2
1/3 BMD g/cm2 0.71 (-10) ± 4.4 0.041 47 ± 6.6 9.0E-07 4.9 Nm2
Table 6: Bivariate Analyses for Predictions of Radius EI
(In Y = mX + b, Y Int. = b and Slope = m)
70
Predictions of Radius EI by Mechanical Predictors
Figure 46 illustrates the relationships between radius and EI and ulna bending strength and radius UD compressive strength. Figure 47 illustrates the relationship between radius EI and ulna EI with the blue line being a 1:1 identity line.
Figure 46: Prediction of Radius EI by Ulna Bending Strength (Left) and by Radius UD
Compressive Strength (Right)
Figure 47: Prediction of Radius EI by Ulna EI
71
Predictions of Radius EI by Bone Mineral
Figure 48 illustrates the relationships between radius EI and radius UD and 1/3
BMC. Figure 49 illustrates the relationships between radius EI and radius UD and 1/3
BMD.
Figure 48: Predictions of Radius EI by UD BMC (Left) and 1/3 BMC (Right)
Figure 49: Predictions of Radius EI by UD BMD Left) and 1/3 BMD (Right)
72
Accuracies of Radius EI Predictors
For hypothesis H02, 1/3 BMC was used as SEE2 in all cases because it was the lowest SEE, and Table 7 summarizes the findings. A predictor was determined as significantly worse compared to 1/3 BMC when F > F1-α(m,n). Ulna strength, radius UD compressive strength, ulna stiffness, UD BMC, UD BMD, and 1/3 BMD all had an SEE significantly larger than that of 1/3 BMC.
2 Predictors of Radius EI SEE (Nm ) F F1-α(m,n) p Ulna Bending 4.8 3.16 2.08 0.005 Strength QMT Radius UD Compressive 6.4 5.62 2.11 0.0001 Strength MRTA Ulna EI 4.5 2.78 2.11 0.01 UD BMC 4.1 2.31 2.14 0.04 Radius 1/3 BMC 2.7 1.00 2.17 0.5 DXA UD BMD 5.2 3.71 2.14 0.003 1/3 BMD 4.9 3.29 2.17 0.006
Table 7: Results of Testing Hypothesis H02 for the Accuracies of Predictors of Radius EI
Clinical Error Rate (CER)
Table 8 demonstrates the CER values for the predictors of radius stiffness. 1/3
BMC had the lowest CER with 8%, while radius UD compressive strength, UD BMD, and 1/3 BMD had the highest CER values of 25%, 23%, and 20% respectively. 73
Predictors of Radius EI Y (Xmax) Y (Xmin) SEE (Nm2) CER Ulna Bending 37.7 4.7 4.8 15% Strength QMT Radius UD Compressive 35.9 10.2 6.4 25% Strength MRTA Ulna EI 37.6 9.5 4.5 16% UD BMC 41.8 5.5 4.1 11% Radius 1/3 BMC 38.2 6.4 2.7 8% DXA UD BMD 30.6 7.8 5.2 23% 1/3 BMD 30.9 6.0 4.9 20%
Table 8: Clinical Error Rates for the Predictors of Radius Stiffness
Summary for Predictors of Radius Stiffness
1/3 BMC had the best CER of 8%. In the F-statistic analysis of SEE, all predictors had an SEE that was significantly higher than that of 1/3 BMC. 74
4. Discussion
4.1 Main Findings
The main findings of this experiment were the rejections of both null hypotheses.
In the prediction of radius bending strength, radius EI was a significantly more accurate
predictor (i.e., had lower SEE values) than radius BMD at the 1/3 site, UD radius BMD,
UD radius BMC, UD radius compressive strength, ulna bending strength, and ulna EI.
However, radius EI was not significantly more accurate than radius BMC at the 1/3 site.
Additionally, radius BMC at the 1/3 site was a significantly better predictor of radius EI
than any other predictor. However, another important finding in this study was that
MRTA measurements of ulna EI were a better predictor of both radius bending strength
and radius EI than radius BMD at the 1/3 site, which is the clinical diagnostic criterion
for osteoporosis.
4.2 Significance in Context with Previous Research
Bending Stiffness as a Predictor of Bending Strength
As would be expected from previous studies (Borders et al., 1977; Hausfeld,
2015; Jurist and Foltz, 1977), radius EI was one of the most accurate predictors of radius bending strength based on SEE and CER. In the study of Borders et al., fresh canine radii, ulnae, and tibiae were tested by either three-point or four-point bending.
Regression analysis of a recreated digitization of their data (Figure 50) found their SEE to be 7.8 Nm, which is significantly and substantially larger than the 3.2 Nm found in the prediction of radius bending strength by radius EI in this study (p < 1E-07). The 75
lower SEE in this study might have resulted from the testing a single bone by a single
method, or from the choice of a different species.
To compare CER values, a plausible range of CER values was calculated. First,
a 95% CI was calculated for SEE (using QuickCalcs Confidence Interval of a SD
Calculator; GraphPad Software, Inc.). Using the 95% CI of the slope in the regression
analysis results, 95% CIs for Y(Xmax) and Y(Xmin) were also determined. Then
extreme values of dYmax and dYmin were calculated using Equations 13 and 14, and
Equations 15 and 16 were used to find extreme values of CER (i.e., CERmax and
CERmin).
dYmax = Y95+(Xmax) - Y95+(Xmin) Eqn. 13
dYmin = Y95-(Xmax) - Y95-(Xmin) Eqn. 14
CERmax = SEE95+ / dYmin Eqn. 15
CERmin = SEE95- /dYmax Eqn. 16
A plausible range of 4-8% was calculated for the 5% CER of QMT in this study. For
the data of Borders et al., the CER of QMT was found to be 7% with a range of 5-9%,
broadly overlapping the range of CER in this study. Thus, such a range might be
representative of the QMT method for bones in bending.
Another difference between this study and that of Borders et al. was the presence
a significant Y-intercept in their regression results (Figure 50). Such a finding is
conceptually problematic, because it implies that a non-zero moment is required to
break a bone of zero stiffness. Perhaps this, too, was the result of merging data from
tests of different bones by different methods. 76
Figure 50: Borders et al’s. Prediction of Bending Strength by EI
In Jurist and Foltz’s study, 45 embalmed human ulnae were tested in bending by
QMT at the standard 1/3 site. Regression analysis of a recreated digitization of their data
(Figure 51) showed their SEE to be 4.7 Nm, which is also significantly higher than SEE in this study (p = 0.01). The CER was found to be 8% with a plausible range of 7-11%, which overlaps only slightly with the 4-8% range of CER in this study. Like this study,
Jurist and Foltz tested the 1/3 site in human specimens. So, potentially higher CER in the study of Jurist & Foltz may have been due to their use of embalmed specimens, because embalming would have altered the mechanical properties of the bone (Wilke et al., 1996), possibly in a non-uniform manner. 77
Figure 51: Jurist and Foltz’s Prediction of Ulna Bending Strength by Ulna EI
A previous study in this lab compared human ulna EI measurements by MRTA and QMT as predictors of ulna bending strength at 50% of the bone length (Figure 52)
(Hausfeld, 2015). EI by MRTA predicted ulna bending strength with an SEE of 5.9 Nm, which is significantly larger than the SEE for this project (p < 0.001), and the CER was
7% with a range of 6-10%. EI by QMT predicted ulna bending strength with an SEE of
4.5 Nm, which is also significantly larger than the SEE of this project (p= 0.04).
However, the CER was 6% with range of 5-8%, which is entirely overlapped by the range of CER in this study.
Overall, radius EI was a significantly more accurate predictor of radius bending strength in this study than EI had been in previous studies of the radius and other bones.
To a greater or lesser extent, however, ranges of CER largely overlapped. 78
Figure 52: Hausfeld’s Prediction of Ulna Bending Strength by Ulna EI
DXA Predictions of Radius Bending Strength
Radius BMD at the 1/3 site is one of the clinical measurements for diagnosing
osteoporosis. As could be expected from previous work that showed radius BMD being a poor predictor of fractures (Siris et al., 2001), this study found that radius BMD is a
less accurate predictor of radius bending strength than either radius BMC or radius EI.
Clinically, this suggests that perhaps BMD should be replaced by BMC for diagnosing
osteoporosis, since DXA already produces BMC as well as BMD values. When the 95%
CI was calculated for 1/3 BMC, it was found to be 4-10% which overlaps with that of radius EI. Of additional importance is the finding that the clinically used 1/3 BMD measurement was a worse predictor than the MRTA measurement of ulna EI based on
CER (18% CER with a plausible range of 11-38% vs 13% CER with a plausible range 79
of 8-23% respectively), although the SEE values were not significantly different (p=
0.19). This warrants further study into MRTA as a predictor because it produces less
error than the clinically used measure.
A previous study by Lochmuller et al. also compared radius bending strength (as
max moment) with measurements from clinical densitometry. The mechanical testing
set-up was very similar to this study’s protocol, with the radius fixed at set proportions
on freely rotating supports. However, the load was centered at 33% of the radius length
rather than the clinical 1/3 site of the forearm, and the radius was oriented for antero-
posterior bending rather than medio-lateral bending (Figure 53). 119 specimens were tested and analyzed. Measurements of radius bending strength were correlated with
DXA measurements of BMD and BMC over the entire distal third of the radius, yielding correlation coefficients of r = 0.89 for BMC and 0.88 for BMD, which were not significantly different from one another (Lochmuller et al., 2002). Unfortunately,
Lochmuller et al. did not report results of regression analyses (slope and Y-intercept, R2
and SEE). Nor did they display their data explicitly to enable digitization, regression
analysis, and calculation of CER. So, the results of Lochmuller et al. cannot be
compared quantitatively to those in this study.
Figure 53: Lochmuller et al. Set-Up for Three-Point Bending on the Radius at 33%
(Lochmuller et al., 2002) 80
Predictions of Radius EI
With the expectation that radius EI would be the most accurate predictor of
radius bending strength, the most accurate predictor of radius EI was also sought,
because it cannot be measured in vivo. Radius BMC at the 1/3 site was the most accurate
predictor of radius EI and is clinically available from DXA scans. This would further
the argument for replacing BMD measurements with BMC for diagnositic purposes. Of
note is the finding that in the prediction of radius EI, MRTA measurements of ulna EI
were more accurate than radius BMD at the 1/3 site, which further supports the use of
ulna EI over radius BMD at the 1/3 site for osteoporosis diagnosis purposes.
4.3 Strengths
Donor Demographics
The wide range of both age and BMI, with age ranging from 17-99 years and
BMI ranging from 13.7-39.7 kg/m2, ensured that the radius bending strength and EI
values would also vary widely, which then increased the likelihood of finding a
relationship between the two variables, if one existed.
Gold-Standard Measurements
QMT measurements are the gold-standard reference method for measuring both bone strength and stiffness. Therefore, the use of QMT in this study gives credibility to the data obtained. Additionally, as was described in section 2.3, it was shown that the loading rate chosen conformed to quasistatic theory, which further validates the load
measurements obtained. Furthermore, the QMT procedure allowed for high 81
reproducibility because the set up was identical for each bone, and the loading and support sites on the specimens were a precise percentage of the bone length. Finally, while most biomechanical studies focus on material properties, this study focused on structural properties, since bones in vivo break as structures, not materials.
However when using QMT, it is pertinent to understand that while it is widely accepted as the most accurate and reproducible reference method for research purposes, it is not without sources of error. First, the 10kN load cell in the load frame is only calibrated by the manufacturer between 20% and 80% of its range. The majority of the specimens in this project fractured at less than 20% and, therefore, were tested in the uncalibrated range. In addition, the test frame, as already mentioned, is not infinitely stiff and, thereby, confounds measurements of the stiffness of specimens being tested.
Moreover, the stiffness of the test frame is not constant, and below 20% of full range its stiffness is highly nonlinear. For these reasons, the finite stiffness of the test frame was measured and fitted to a nonlinear function of load in this study to enable these sources of error to be corrected.
Determination of the 1/3 Site
The use of Avizo to create 3D computer models of the entire forearm constructed from CT images allowed the clinical 1/3 site in the forearm to be located on the radius.
The DXA scanning procedure defines the 1/3 site in relation to the ulna, so it was necessary to locate this site along the length of each radius. Previous researchers
(Lochmuller et al., 2002) have tested at 33% of the length of the radius, rather than at 82
the site on the radius that corresponds to the 1/3 site defined in the DXA scanning
procedure. The finding that the clinical 1/3 site was located at 40% of the radius length
permitted mechanical tests to be performed at the same site where DXA measurements
were obtained.
Additionally, the use of Avizo models allowed for precise determination of each
bone’s length. This then allowed the precise determination of where the bone would be
pinned and loaded. These Avizo models will also be useful for future students to analyze
relationships between radius bending strength and radius cross-sectional area, diameter, and porosity.
4.4 Weaknesses
Data Exclusions
In the process of developing the testing procedure, several anomalies occurred
that required data be excluded from the sample set. The smaller sample size reduced the statistical power of the study to find statistically significant differences between
predictors, described in section 2.5, to 74%. There was, therefore, a decreased ability to find a significant difference between radius EI and 1/3 BMC, if one exists.
Damage at the Loading Site
Five specimens were excluded due to the probe digging and cutting into the
bone, which could also cause indentation shown in Figure 54. In accordance with ASTM
Standard D790-03, in order to prevent significant indentation, the radius of the probe 83
should have been between 1.6 and 4 times the diameter of the bone (ASTM). This larger
diameter could also prevent the digging. The knife-edge probe had originally been chosen in order to emulate assumptions in the analysis of three-point bending test data where the load is assumed to be applied at a single point (rather than the broad area under a larger probe), and because it had not been a problem previously in bending tests of ulnae. As shown in Figure 55, though, a protective benefit of testing human bones in bending is that roughly 85% of the resulting stress comes from bending while only 15% comes from a combination of shear stresses (which were accounted for), indentation stresses (which would have been prevented by a wider probe), ovalization, and deformation at the supports. So, the magnitude of errors due to indentation in this study may have been small.
Figure 54: Indentation of a Long Bone during a Bending Test (Kourtis et al., 2014)
84
Figure 55: The Impact of Non-Bending Stresses in a Bending Test (Kourtis et al., 2014)
Individualized Shear Calculations
While the assumption that all the bones conformed to the healthy case of an inner diameter half that of the outer diameter was most convenient for this project, it is not realistic. In reality, all of the bones would fall somewhere between the green line
(healthy case) and red line (osteoporotic case) in Figure 56, which shows that the effect of shear is largest for healthy bones and would vary for bones between the healthy and osteoporotic cases. Each bone has a different ratio of inner to outer diameter. Ideally, the inner diameter of each bone at the location of fracture should be used with the outer diameter at the same location to produce individualized shear calculations. 85
Figure 56: Dependence of the Magnitude of Shear Stress on the Slenderness Ratio
Outliers
The results for ulna EI predicting radius EI were heavily influenced by two
outliers, C040 and C041, which are labeled in red in Figure 57. These two specimens
were not outliers, however, in the analysis of radius EI predicting radius bending
strength or for ulna EI predicting ulna bending strength. Based on BMD T scores for the 1/3 site, both of these bones should be considered under the healthy case for shear calculations in EI, as would be expected from their young age (38 and 23 respectively).
The stiffness vs displacement graph for C041 looks fairly smooth and flat over a broad range (except at a known artifact of the mechanical test frame at ~200N), which would suggest that the stiffness and strength measurements are correct (Figure 58 Left).
However, the curve for C040 is irregular (Figure 58 Right). The oscillating stiffness curve indicates discontinuities that might be attributable to rotation of the specimen, digging in of the probe, crushing of the bone at the supports, or partial fractures. The 86
stiffness was taken as the average across the three middle oscillations, because it was
not clear where the true values in this peak range were. If the peak stiffness of the
oscillations had been chosen, C040 would be even more of an outlier. At this time, the
cause of this oscillation is unknown. Further, the UD radius BMD T score of C040 was
osteoporotic (-2.75) which is inconsistent with its normal BMD T score at the 1/3 site
(0.015). If C040 is removed from the regression analysis, the R2 increases from 0.77 to
0.83 and the SEE decreases by 0.7 Nm2, however this is not statistically significant
(Figure 59).
Figure 57: Ulna Stiffness as a Predictor of Radius Stiffness Highlighting Outliers
Figure 58: Load vs Displacement Curve for C041 (Left) and C040 (Right) 87
Figure 59: Ulna Stiffness as a Predictor of Radius Stiffness Removing C040
4.5 Future Research
With the amount of data excluded from this data set, it would be useful to expand the data set with more cadaveric arms following the most recent protocol. The larger data set may help determine if a significant difference can be found between the accuracies with which radius EI and 1/3 radius BMC predict radius bending strength by increasing the power. Additionally, it would be beneficial to use µCT images to determine the inner diameter of each specimen in order to individualize the calculation of EI while accounting for shear stresses.
Future studies in the lab will be conducted on more cadaveric arms to obtain
MRTA measurements of ulnae that will be compared to radii, humeri, femora, and tibiae from the same specimens. From this, the extent to which the mechanical properties of the ulna are representative of other long bones of the body can be determined. In these future studies, it will be important to obtain DXA measurements prior to defleshing the 88
forearm as well as DXA measurements of the femoral neck. In the lab’s previous studies of ulna bending strength, DXA measurements were not used as a predictor. This oversight should be corrected in future studies, since DXA is the incumbent clinical technology. DXA measurements would allow for a more precise comparison of the accuracies of BMC and EI in predicting the bending strengths of other bones as well as the radius. It will also allow for a better comparison of prediction accuracy between
BMC and EI for leg bones, as BMC of leg bones would be expected to be a more accurate predictor of the bending strength of leg bones than would BMC of the forearm.
Furthermore, in the future studies, a new, robotic version of the MRTA device may improve the accuracy of EI measurements.
89
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