The Development and Validation of a Finite Element Model of a Canine Rib For Use With a Bone Remodeling Algorithm.

Transcription

1 The Development and Validation of a Finite Element Model of a Canine Rib For Use With a Bone Remodeling Algorithm. A Thesis Presented to the Faulty of the College of Engineering California Polytehni State University San Luis Obispo In Partial Fulfillment of the Requirements for the Degree of Master of Siene in Engineering By Sott James Sylliaasen Deember 2010 i

2 2010 Sott James Sylliaasen ALL RIGHTS RESERVED ii

3 COMMITTEE MEMBERSHIP TITLE: The Development and Validation of a Finite Element Model of a Canine Rib For Use With a Bone Remodeling Algorithm. AUTHOR: Sott James Sylliaasen DATE SUBMITTED: Deember, 2010 COMMITTEE CHAIR: Dr. Sott Hazelwood, Assoiate Professor, Biomedial & General Engineering COMMITTEE MEMBER: Dr. Peter Shuster, Assoiate Professor, Mehanial Engineering COMMITTEE MEMBER: Dr. Robert Crokett, Diretor, General Engineering iii

4 Abstrat The Development and Validation of a Finite Element Model of a Canine Rib For Use With a Bone Remodeling Algorithm. By Sott James Sylliaasen Studies are urrently being performed to determine the effets of bisphosphonate treatments on the struture and density of bone tissue. One of the pathways for gaining a better understanding of the effets of this and other treatments involves reating a omputer simulation. Theory suggests that bone tissue struture and density are diretly related to the manner in whih the tissue is loaded. Remodeling is the proess in whih bone tissue is resorbed in areas of low stress distributions, and generated in areas of high stress distributions. Previous studies have utilized numerial methods and finite element methods to predit bone struture as a result of stress distributions within the tissues. The Finite Element method was hosen for this study. This study was done on a anine (beagle) rib. The goal of this study was to develop an FEA model of the rib that would be used in onjuntion with a bone remodeling algorithm, to model the behavior of the bone tissue. Appropriate boundary onditions, loads, and loading yles were determined from literature, and applied. Respiration was assumed as the dominating ativity; therefore the musles involved in respiration were the primary soure of the rib loading. The model also inluded an integrated UMAT sub-routine, whih utilized data from the FEA model to iterate bone tissue densities and strutures. The model losely predited the porosities of the bone tissue, when ompared to atual tissue samples, as well as what literature desribes. iv

5 Aknowledgements I would like to thank Dr. Sott Hazelwood of California Polytehni State University, San Luis Obispo for the opportunity to work on this projet. His patiene, guidane, and expertise were greatly appreiated throughout the entire span of this projet. Additionally I would like to thank my family, and fiané for their ontinuous support. v

6 Table of Contents List of Tables... ix List of Figures... x 1. Introdution... 1 Bones... 1 Bone Tissue... 2 Composition... 2 Bone Types... 2 Compat Bone... 3 Trabeular Bone... 5 Bone Damage and Repair... 6 Bone Cells... 6 Osteolasts Bone Resorbing Cells... 7 Osteoblasts and Osteoytes... 8 Bone Lining Cells... 9 Modeling and Remodeling Modeling Remodeling Osteoporosis Bisphosphonates Finite Element Modeling Bone Remodeling Simulations Goal Methods Overall Model Generation/Exeution Proedure Determination of Loading Conditions Respiration Pleural Pressure vi

7 Inspiration External Interostals Levatores Costarum Cranial Serratus Dorsalis Expiration Internal Interostals Summary of Loading Conditions Determination of Boundary Conditions The Motion of the Ribs Appliation of Loading/Boundary Conditions to FEA Model Generation of FEA Mesh Appliation of Loading/Boundary Conditions to FEA Mode FEA Model Parameters Boundary Conditions Load Steps Modifiation of UMAT Subroutine Parameters to Simulate Bone Adaptation for the Dog Rib Determining the Equilibrium Stimulus Validation of model by omparing to CT san results Obtaining Density/Porosity from Mimis Obtaining Porosity Values from Abaqus Statistial Analysis Results Visual Comparison of Model and Tissue Sample Quantitative Comparison of Model and Tissue Sample Disussion Model Limitations vii

8 Loading/Boundary Conditions Remodeling Algorithm Future Studies Conlusion Works Cited APPENDIX A UMAT SUBROUTINE viii

9 List of Tables Table 1 - Loading Conditions of 10th Rib During Inspiration and Expiration Table 2- Range of Remodeling Parameters Investigated Table 3 - Regions where Model Predited Tissue Densities Exeptionally Well (% Diff < 20%) Table 4 - Summary of Quadrant Based Porosity Comparisons of Bone Tissue Porosities (Cortial Only) ix

10 List of Figures Figure 1 - Typial Dog Skeleton (2)... 1 Figure 2 - The Struture of Long Bone (4)... 3 Figure 3 - Close up of osteonal bone in ross-setion. (4)... 4 Figure 4 - Trabeular bone struture in the distal end of a human femur. (4)... 5 Figure 5 - Multinulear osteolast (upper right) resorbing bone (lower left). CZ labels lear zones where the ell is sealed to the bone surfae; RB labels the ruffled border where enzymes are released to break down bone; B labels the alified bone matrix. (4)... 7 Figure 6 - Osteoblast forming bone. The dark area at the bottom is mineralized bone. Lighter material is osteoid produed by the rough endoplasmi retiulum of the ell. A portion of a proess of the ell protrudes into the lighter osteoid material. (4)... 8 Figure 7 - Network of Canniuli (Fine Dark Lines) (4)... 9 Figure 8 - Types of Bone Modeling (4) Figure 9 - Osteolasts (on the right) form the utting one of the BMU while osteoblasts (on the left) form new bone in the spae reated by the osteolasts. BMU is traveling to the right. (4) Figure 10 - Huiskes, et al. and Carter et al. Remodeling Algorithm Flow Charts Figure 11 - Thorai Cavity, levatores ostarum, retus thorais, ranial serratus dorsalis, and salenus musles. [2] Figure 12 - Insertion Points for Serratus Dorsalis Cranialis and Levatores Costarum and Terminology Figure 13 - Summary of Inspiration Fores x

11 Figure 14 - Diagram Depiting Head of Ribs [2] Figure 15 - Thorax (inlude referene) Figure 16 - Mesh of Rib Figure 17 - Cranial External Interostals Figure 18 - Caudal External Interostals Figure 19 - Caudal Internal Interostals (Loading Not Shown) Figure 20 - Cranial Internal Interostals (Loading Not Shown) Figure 21 - Cranial Serratus Dorsalis Figure 22 - Levatores Costarum Figure 23 - Pleural Pressure (Applied to Medial Surfaes of Rib, Varies Based on Inspiration or Expiration) Figure 24 - Dorsal Boundary Conditions (No Displaement or Translation Allowed. Rotation Allowed) Figure 25 - Ventral Boundary Conditions (No Displaement or Translation Allowed. Rotation Allowed) Figure 26 - Lateral Boundary Conditions (No Displaement Allowed. Rotation and Translation Allowed. Abdominal Musle Restritions) Figure 27 UMAT Subroutine Flow Chart [10] Figure 28 - Loations of Cross Setions for Comparison of Predited Porosities to Atual Figure 29 - Identifiation of Quadrants Figure 30 - Mimis Interfae, Graysale Values Listed as "Mean". Three loations where graysale measured highlighted in Yellow, Red and Purple xi

12 Figure 31 - Cross Setional Porosity Analysis of FEA Model Figure 32 - Longitudinal Cross Setion of Abaqus Model (SDV1 = Porosity %) Figure 33 - Horizontal Cross Setion (Mimis Left, Abaqus Right, SDV1 = Porosity %) Figure 34 - Horizontal Cross Setion (Mimis Left, Abaqus Right, SDV1 = Porosity %) Figure 35 - Comparison of Predited and Atual Porosities Based on Loation on Rib Figure 36 - % Differenes for Porosity Values by Quadrant xii

13 1. Introdution Bones The average anine body onsists of 321 bones (Figure 1) [1]. The main funtions of bone are to provide strutural support, struture for motion, protetion of vital organs, mineral storage, and hematopoiesis (the formation of red blood ells) [3]. Bone is a living, dynami tissue that is onstantly adapting to its mehanial environment, undergoing strutural hanges in order to optimize strength while minimizing weight. Bones are generally lassified in four separate ategories depending on their shape and loation [2]: long bones, short bones, flat bones, and irregular bones. As the name suggests, irregular bones are bones whose shapes and onfigurations do not fall into any of the other ategories. Long bones are found in the limbs, short bones are found in the ankle and wrist, while flat and irregular bones are harateristi of the skull and spine. Figure 1 - Typial Dog Skeleton [2] 1

14 Bone Tissue Composition Bone is omposed of hydroxyapatite mineral (70%), ollagen, mostly type-1 (18%), nonollagenous proteins and proteoglyans (2%), and water (10%) [2]. Collagen is a strutural protein, also found in tendons, ligaments, and skin, whih gives bone its tensile strength and flexibility. In addition, ollagen also provides sites for the nuleation of bone mineral rystals, omposed primarily of hydroxyapatite, Ca 10 (PO 4 ) 6 OH 2, whih gives bone its harateristi rigidity and ompressive strength [4]. Bone Types There are two main bone types, alled ortial (ompat) and trabeular (anellous) bone. Trabeular bone has a high porosity, 75% - 95%, whereas ompat bone has a muh lower porosity, 5% - 10% [4]. The omposition and struture of eah individual bone making up the human body is based on its speifi funtion, and what physiologial and mehanial environment it exists in. Trabeular bone is found in vertebrae, flat bones and the ends of long bones, and ompat bone is found in the shafts of long bones and around vertebrae. Figure 2 below illustrates the struture of long bone, with an outer shell of ompat bone surrounding an interior of trabeular bone. The enter of the bone ontains bone marrow, whih is responsible for the prodution of red blood ells and bone ells, as well as some immune system ells. 2

15 Figure 2 - The Struture of Long Bone [4] Compat Bone Due to its lower porosity whih results in higher density, ompat bone is the heavier of the two types of bone. Compat bone tissue is formed in layers approximately 5 µm thik alled lamellae whih are arranged differently in different parts of bone. Near the outer and inner surfae of bone, they are arranged irumferentially, while the ompat bone that exists in between these two layers is generally haraterized as osteonal bone. Osteonal bone, shown in Figure 2, and in more detail in Figure 3, onsists of ylindrial strutures formed by onentri lamellae, approximately 200 µm in diameter, 1 m long, and aligned with the long axis of the bone [4]. 3

16 Figure 3 - Close up of osteonal bone in ross-setion. [4] At the enter of eah osteonal struture lies a Haversian anal approximately 50 µm in diameter, and within eah anal exists blood vessels and nerves [4]. The funtion of the Haversian anals is to provide nutrients and remove wastes from the surrounding bone. Volkmann s anals run transversely between Haversian anals, onneting them to eah other and to the outside surfaes of the bone. The longitudinal alignment of the osteons and the orientation of the lamellae give ompat bone high stiffness and strength along the long axis of the bone, as well as good fatigue resistane. The lamellar struture of ompat bone gives the bone the ability to keep fatigue raks small and run in harmless diretions, thus improving fatigue resistane [4]. The struture essentially dissipates rak energy around the edge of the osteons, keeping the rak small, rather than allowing the rak to propagate entirely through the bone. The lamellar and osteonal struture of ompat bone allows it to ahieve an elasti modulus between 14.8 and 4

17 17.4 GPa, an ultimate strength between 133 and 195 GPa, and a frature toughness of MPa-m 1/2 [2,4]. Trabeular Bone Trabeular bone, also known as spongy or anellous bone, has a muh more porous struture than ompat bone. Interonneting struts that are 200 µm thik, alled trabeulae, form the struture of trabeular bone [4] (See Figure 4). The arrangement of these struts varies from highly organized to highly random. This omplex, interonneting struture gives bone the ability to dissipate and spread energy from impats, ating as a sort of shok absorber. In general, trabeular bone is haraterized with an elasti modulus of 272 ± 195 MPa and an ultimate strength of 2.54 ± 0.62 MPa [4]. Figure 4-4: Trabeular bone struture in in the distal end end of a of human a human femur. femur. [4] 5

18 Bone Damage and Repair Bones in the body are subjeted to two types of loading onditions whih an lead to failure, reep and fatigue. In reep, a load is applied ontinuously, suh as the ontinual loading of the vertebrae. Fatigue involves yli loading, suh as the yli loading of the femur during walking. The ontinual and yli loading of bones an lead to the reation and propagation of damage, in the form of miroraks. If miroraks ontinue to grow, they an eventually lead to failure of the bone, in the form of a frature. Fortunately, our body s remodeling mehanisms, desribed in the following setions, repair these miroraks and prevent them from growing and leading to failure. Bone Cells Earlier, it was mentioned that bone is a dynami tissue, whose struture is onstantly being modified to adapt to its biohemial and mehanial environment, in order to optimize strength with respet to weight. These proesses are arried out by bone ells of two ategories; bone ells that remove/resorb bone, and bone ells that form bone. The ooperation of these two ell types makes it possible for the skeletal system to undergo important strutural hanges and repairs. 6

19 Osteolasts Bone Resorbing Cells Osteolasts are multinulear ells whih resorb bone tissue. They are losely related to marophages, whih are responsible for removing harmful impurities from the tissues of the body. Bone resorption ours along the ruffled border of the ell and ours at a rate of tens of mirometers per day, utilizing a ombination of aids and enzymes, to demineralize the adjaent bone, and break down the remaining ollagen [4]. Figure 5 below shows a portion of an osteolast resorbing bone. Figure 5 - Multinulear osteolast (upper right) resorbing bone (lower left). CZ labels lear zones where the ell is sealed to the bone surfae; RB labels the ruffled border where enzymes are released to break down bone; B labels the alified bone matrix. [4] 7

20 Osteoblasts and Osteoytes Unlike osteolasts, osteoblasts are mononulear ells that aid in forming new bone tissue (Figure 6). Osteoblasts produe osteoid, whih is the organi portion of the bone matrix, onsisting of ollagen, nonollagenous proteins, proteoglyans, and water [4]. Muh of the organi omponents of osteoid are produed in the osteoblast s rough endoplasmi retiulum [4]. As more and more osteoid is produed, the osteoblasts beome engulfed in bone tissue, and beome osteoytes. Osteoytes are embedded in a omplex network of tunnels alled analiuli whih reate a vast ommuniation and transportation network between osteoytes (Figure 7). This omplex network has lead researhers to believe that osteoytes are important in the transportation of minerals and nutrients in and out of bone, and possibly in the sensing of mehanial stress [4]. Figure 6 - Osteoblast forming bone. The dark area at the bottom is mineralized bone. Lighter material is osteoid produed by the rough endoplasmi retiulum of the ell. A portion of a proess of the ell protrudes into the lighter osteoid material. [4] 8

21 Figure 7 - Network of Canniuli (Fine Dark Lines) [4] Bone Lining Cells Bone lining ells are former osteoblasts that were not engulfed in osteoid and lie inative on the external surfae of the bone tissue. Bone lining ells ommuniate with osteoytes via the ommuniation network desribed in the previous setion. Beause of their onnetion to the extensive network of anals, as well as their possession of hemial reeptors, they are believed to be responsible for mineral and nutrient transfer in and out of the bone, as well as sensing mehanial strain [4]. 9

22 Modeling and Remodeling The struture of skeletal tissue is modified via two methods: modeling and remodeling. Modeling involves the independent ations of osteolasts and osteoblasts to shape bones into the desired geometries. Remodeling utilizes the ombined ations of osteolasts and osteoblasts in groups alled basi multiellular units (BMU s) to maintain and modify the strutural integrity of the bone tissue, by simultaneously removing and replaing bone tissue, without hanging the bone s overall shape. Modeling ours rapidly during growth, and is greatly redued one skeletal maturity is reahed. Remodeling ours throughout life, but the rate at whih remodeling takes plae is redued when growth stops. Modeling Modeling is a proess whih takes plae during growth whih essentially sulpts bones into speifi shapes. The shape of eah bone is modified by osteolasts removing bone in one area while osteoblasts form bone in another. The rate and loations of these ativities are greatly influened by the stresses the bones are subjeted to. Thus, every individual has a skeleton that is essentially tailored to their speifi loading onditions. Examples of bone modeling inlude: metaphyseal modeling to redue bone diameter, diaphyseal modeling to modify bone diameter and alter urvature, and flat bone modeling to aommodate for brain growth in the skull. Metaphyseal modeling involves the reation of the 10

23 diaphysis by the ontinual removal of bone tissue from the periosteal surfae of the metaphysis to redue the shaft to the proper diameter (Figure 8a). Diaphyseal modeling to modify bone diameter involves the addition of bone to the exterior surfae of long bone shafts, while bone is removed from the interior of the shafts (Figure 8b). The urvature of the diaphysis is modified by removing and forming bone on the sides of the bone, ausing the ross setion of the bone to drift in relation to the ends of the bone (Figure 8) [4]. Another example of modeling involves the sulpting of the flat bones in the skull. As the brain grows, the flat bones of the skull must inrease in size as well as hange shape to aommodate. Figure 8 - Types of Bone Modeling [4] 11

24 Remodeling Remodeling involves the ooperation of osteolasts and osteoblasts in BMUs to repair bone, by removing old or damaged tissue and replaing it with newly formed tissue. The repair of mirosopi damage prevents the aumulation of fatigue damage, essentially reduing the likeliness of frature or other fatigue damage. It is also hypothesized that the ation of remodeling fine tunes the bone tissue to optimize mehanial effiieny, by reduing weight and optimizing strength [4]. Remodeling also aids in alium homeostasis, helping bones at as a alium reservoir for the body; removing alium from the reservoir via osteolasts, and adding alium to the reservoir via osteoblasts. As stated before, remodeling ours via the ooperation of osteolasts and osteoblasts in BMUs. BMUs are first ativated by hemial or mehanial signals, whih ause osteolasts to form. After the ativation and formation of the osteolasts, the BMU enters the resorption phase. Osteolasts begin removing bone tissue, forming a dith (on the bone surfae), or a tunnel (in ompat bone). Following the initiation of resorption, the osteoblasts begin differentiating from mesenhymal ells (a speifi type of stem ell) and begin the formation stage, in whih the resorbed tissue is replaed. The entire ARF (ativation, resorption, formation) lifeyle of BMUs takes about 4 months, with resorption taking about 3 weeks in humans, and formation/refilling taking about 3 months [4]. The exat ARF lifeyle in anines has not been speifially established in prior researh. Bone remodeling enompasses osteonal remodeling, trabeular remodeling, and endosteal and periosteal remodeling. Osteonal remodeling involves the tunneling of a BMU 12

25 whih forms a seondary osteon. The BMUs beome so isolated in the ortial tissue that in order to maintain vasular supply, they must not entirely fill in the resorbed area, leaving behind a Haversian anal when the proess is finished. In human adults, osteonal BMUs replae about 5% of ompat bone per year [4]. BMUs are omposed of about 10 osteolasts, whih form a utting one at the tip of the BMU, removing bone, followed by several hundred osteoblasts, whih replae the resorbed bone with new bone tissue [4]. BMUs tunnel at a rate of approximately 40 µm a day. In ortial bone the tunneling BMUs leave behind small ylindrial avities, whih are harateristi of the osteonal struture dithes on the surfae of trabeular bone, and tunnels through ompat bone. Figure 9 - Osteolasts (on the right) form the utting one of the BMU while osteoblasts (on the left) form new bone in the spae reated by the osteolasts. BMU is traveling to the right. [4] Trabeular remodeling also involves BMUs, but, the BMUs travel along the surfaes of the trabeulae, digging and refilling trenhes [4]. The bone turnover rate for adult humans is about 25% per year, but varies widely depending on the loation in the skeleton [4]. Researhers have observed that long bones expand radially with age, whih has led to speulation that remodeling ours on the endosteal and periosteal surfaes of bones [4]. In order for the diameters to grow, bone formation must exeed resorption on the periosteal surfae, but the lak of data on this issue has left it unertain. 13

26 Osteoporosis Osteoporosis is defined as the redution in bone mass and deterioration of its mirostruture, aompanied by an inreased suseptibility to frature [3]. Essentially, the porosity of the bone tissue inreases, leading to redued frature toughness. Over one million osteoporoti fratures our eah year in the United States, ourring most often in the spine, hip, and wrist [3]. Osteoporoti fratures are more ommon in women than in men, due to hormonal differenes between eah sex, and in part due to the fat that women live longer than men. Osteoporosis is aused by malfuntioning remodeling ations of the bone tissue an unbalane between the amount of bone removed by osteolasts and the bone replaed by osteoblasts, leading to the net removal of bone tissue [5]. This redution in bone tissue density redues the mehanial strength of the bone and inreases the suseptibility to frature. Current treatment options for subjets with osteoporosis inlude the use of nutritional supplements, suh as alium and vitamin D, to provide the body with suffiient nutrients and minerals to maintain bone tissue [5]. Resistane training is additionally used to inrease the loading to bone tissue to stimulate remodeling and improve frature resistane. In the past 10 years, bisphosphonate drugs have beome the most widely used drugs in the treatment of postmenopausal osteoporosis [5]. 14

27 Bisphosphonates The utilization of bisphosphonates (BPs) in the treatment of osteoporosis has shown remarkable promise. They have been shown to halve vertebral frature risk, and redue nonvertebral frature risk by 20-30% [5]. Bisphosphonate drugs work by inhibiting osteolasts, effetively reduing bone turnover. The tradeoff of using BP drugs is that by inhibiting the ation of osteolasts, they inhibit the ations of BMUs and the proess of remodeling. This results in the aumulation of mirodamage, whih may inrease frature risk. Redutions in bone toughness as well as redution in bone turnover rates have been onsistently doumented following bisphosphonate treatment, espeially in the vertebrae of animals. Dogs reeiving high doses of BPs showed inreased mirodamage and redued mehanial strength of bone tissue [5]. Beagles exposed to bisphosphonate treatments equal to or lesser than the doses used to treat postmenopausal women, showed no signifiant redution in vertebral bone toughness, while beagles exposed to higher doses experiened signifiant redutions in vertebral bone toughness [6]. Although extensive studies have been performed in the vertebrae of beagles validating the theory that the use of bisphosphonates redues bone toughness, studies involving the rib are onfliting, with one study showing no effet on rib toughness and another showing a signifiant redution [7,8]. A more reent study determined that although rib ortial bone experienes signifiant redutions in turnover following bisphosphonate treatment, only animals treated with doses above what are normally used to treat osteoporosis experiened a signifiant redution in bone toughness [6]. 15

28 Finite Element Modeling Bone Remodeling Simulations Finite element models of bone allow researhers to predit the effets of various treatments on the mehanial properties of the bone tissue itself. Finite element modeling involves the division of a speifi volume of the struture into elements, eah having its own geometry and mehanial properties [4]. Complex algorithms an be applied to the model to simulate the proesses of modeling, remodeling, and the utilization of pharmaeutials. Boundary onditions, suh as fores and onstraints are then applied to the struture and a omputer is used to predit the mehanial response of the tissue. If a remodeling algorithm is applied to the model, the model ould be used to predit the effets of the loading ondition or drug treatment algorithm on the remodeling state of the bone, in addition to the mehanial response of the tissue. Although FEA modeling an be used to predit the responses of omplex systems, it is not always entirely aurate. Sine many assumptions are made during the modeling proess and during the omputation of the results, one an expet that the results may not be an exat representation of what might happen in atuality. It is therefore important that the FEA model be onsistent with atual data, before it an be used to provide signifiant results. FEA models an be validated by a number of methods. The most ommonly used methods of validating FEA bone models involve omparing real speimens to the models. For example, if the researher reated a model of a femur and subjeted the model to three-point bending, he ould validate his model by subjeting a adaver femur to a three-point bend on a material tester, and ompare the results to those predited by the FEA model. Another 16

29 approah would be to ompare the histology of the real bone speimen to the struture and histology predited by the FEA model. Carter, et al, used an FEA model, to suessfully simulate the relationship between bone tissue density and struture to the loading history of a femur [23]. The model alulated bone density strutures that were similar to previously doumented samples. Similarly, Huiskes, et al, used FEM to model the effets of stress shielding by a medullary stem, suh as used in a Total Hip Arthroplasty, on the bone tissue density distribution in a femur [24]. He showed that the rigidity and bonding harateristis of the implant affet the resorption of the surrounding bone. The methods used by Huiskes and Carter required the use of a speifi bone remodeling algorithm. Figures 10 illustrates, in general, the algorithm loops used by eah to simulate bone remodeling under partiular loading onditions. In eah ase, loads, boundary onditions, and material properties were input into a finite element model. The stress and strain fields output by the model were used to predit a remodeling response by the tissue. The hanges in the tissue struture resulted in hanges in material properties, whih would be used as inputs in the next iteration. Huiskes used the FEM to determine strain energy density (SED), whih is essentially the produt of the loal stress and strain tensors [24]. The differene between the alulated SED and the homeostati SED, whih was determined at homogeneous onditions, was used to predit the remodeling response. The hange in material properties was determined from the 17

30 remodeling rate, and the apparent density was determined from the hange in the material properties. Carter s method was slightly different in that he utilized a stress stimulus as opposed to strain energy density, to determine the remodeling response [23]. The stress stimulus was derived from the effetive stress applied at a partiular number of yles, based on the ativity (walking, standing, et.). Additionally, Carter derived the hanging material properties from the apparent density, as opposed to Huiskes, who derived the apparent density from the hanging material properties. 18

31 Huiskes, et al. Apparent Density Material Properties Loads Boundary Conditions FEM Stress/Strain Fields/Tensors SED Strain Energy Density Differene Between Atual SED and Homogeneous SED Remodeling Response Carter, et al. Material Properties Loads Boundary Conditions FEM Initial Homogeneous Stress Fields Effetive Stress Atual Stress Stimulus Differene Between Atual and Attrator Stress Stimulus Remodeling Response Apparent Density Figure 10 - Huiskes, et al. and Carter et al. Remodeling Algorithm Flow Charts 19

32 Goal Experimental studies have been onduted to determine the short term effets of various antiremodeling agents on the mehanial properties and remodeling behavior of bone tissue. One study showed that the treatments affet osteoblasts and bone formation differently, depending on the predominant mehanism of formation, either modeling or remodeling [25]. The seond study showed that higher dosages of bisphosphonate treatments redue bone toughness [9]. Eah study was performed on skeletally mature beagles, whih were treated daily with various antiremodeling agents. The results of eah study were determined from rib tissues that were extrated one the treatment periods were onluded. The goal of this thesis is to develop a finite element model of the 10 th beagle rib, under normal and untreated loading onditions. Ultimately, the model will be utilized to further the experimental studies by prediting the long term effets of various bisphosphonate treatments on the mehanial properties, mirodamage aumulation, and bone turnover rate of the rib. As a first step in that proess, the model developed and validated in this projet will be the initial baseline model of the rib prior to bisphosphonate treatment. 20

33 In order to validate the model, a bone remodeling algorithm will be applied and the resulting struture/histology of the model will be ompared to that of an atual beagle speimen. Creating and validating the FEA model will involve the following objetives: 1) Researhing the proper natural loading onditions of the 10 th beagle rib. 2) Generating an FEA model of the rib. 3) Applying the remodeling algorithm to the model. 4) Validating the model by omparing struture/histology to atual rib. 21

34 2. Methods Overall Model Generation/Exeution Proedure As noted before, the goal of this thesis was to develop and validate an initial, baseline FEA model that ould aurately simulate the bone tissue properties of a dog rib. The method used was similar to the method used by Huiskes [24], desribed in the previous setion. The FEA model was used to alulate the strains and stresses throughout the bone tissue. The stress and strain values generated by the model was inputted into a user reated material subroutine (UMAT subroutine), to predit the struture of the resulting bone tissue. UMAT subroutines are essentially lines of ode that ABAQUS an use to simulate omplex tissues and materials. The following steps were taken to develop the FEA model. Determination of Loading/Boundary Conditions. Appliation of Loading/Boundary Conditions to FEA Model. Generation/Modifiation of Input file to be utilized in onert with UMAT subroutine. Modifiation of UMAT Subroutine parameters to simulate struture/histology. Validation of model by omparing to CT san results. Determination of Loading Conditions Musle insertion loations, fores, funtions, and ation angles all affet the stresses and strains exerted on the rib, thereby having a great deal of influene over the struture of the bone tissue. In order to reate an aurate model, the natural loading onditions of the dog rib were researhed utilizing various anine anatomy books. It was deided that the ativities of inspiration and expiration would be the most dominating ativities under normal onditions 22

35 that influened the rib; therefore, the musle groups that partiipate in eah ativity were investigated. The musle fore was determined based on the assumption that the maximal fore produed by a musle is proportional to its ross setional area. The area of ontat of eah musle group was estimated based on anatomy and the struture of the rib. Maximal stresses in musle tissue have been measured to be between 20 and 100 N/m 2 [4], and it was assumed that the musle tissue in this study produed 20 N/m 2, beause the dog whih is the primary fous of this study was small ompared to humans. A visual inspetion of the ranial, (See Figure 11 for definition of anatomial diretions for anines), portion of the rib reveals two separate areas on the bone whih ontain 2 to 3 and 7 to 9 peaks. These peaks indiate the insertion points of the levatores ostarum and the ranial serratus dorsalis musle groups [12], respetively. The area enlosing the entirety of the peaks was estimated as the area of musle insertion for eah group. Figure 11 - Canine Anatomi Diretions/Terms [2] 23

36 The external interostals span the spaes between ribs, from the insertion of the levatores ostarum, to the ostrahondral juntions, and are 4-5mm thik on large dogs [1]. The area of the external interostals musle insertion was estimated by using a thikness of 4mm, overing the extent of the rib. The internal interostals span the length of the rib as well, but lie medial to the external interostals. The musle insertion area for the internal interostals was estimated in a similar fashion, using a thikness of 2 mm (the thikness of the internal interostals is 2-3 mm in large dogs [1]). The magnitude of the fore values were distributed over the nodes on the FEA model in the areas that the musle group ats on, and then broken up into horizontal and vertial omponents. To simplify the model, the fores were assumed to be ating in two dimensions, as opposed to three dimensions. Respiration During inspiration, the diaphragm ontrats and is displaed audally into the abdomen, and the ribs are pulled forward, resulting in the expansion of the thorai avity and the filling of the lungs. The ribs are pulled forward by the external interostals, levatores ostarum, retus thorais, ranial serratus dorsalis, and the salenus musles (Figure 12) [11]. Expiration involves the opposite ation, the pulling of the ribs audally, towards the tail, and the relaxation of the diaphragm, thus reduing the volume of the thorai avity, resulting in the expulsion of air from the lungs. The musles involved in the movement of the ribs audally are the internal interostals and the audal serratus dorsalis. In addition to experiening the musle fores that draw the ribs ranially and audally, the ribs endure a ontinuous pleural pressure that resists 24

37 the motions of respiration. Pleural pressure is defined as the pressure of the avity surrounding the lungs. Pleural Pressure During inspiration, the pressure of the pleural avity inreases from -2mH 2 0 to approximately -4mH 2 0 [9]. This pressure resists the motion of inspiration and inreases in magnitude as the thorai avity expands and for this model, it is assumed that the pressure is applied to the entire medial surfae of the rib. Inspiration As stated before, the ribs are drawn forward by the external interostals, levatores ostarum, retus thorais, ranial serratus dorsalis, and the salenus musles, in order to failitate the expansion of the thorai avity (See Figure 12). The retus thorais and salenus musles only at on the first few ribs, and sine the goal of this thesis was to model the 10 th rib, these musle groups were ignored. In addition, the ations of the abdominal musles were assumed to be negligible. The remaining musle groups at on most of the ribs and therefore were inluded in the analysis. For a detailed list of the musle fores see Table 1, page

38 Figure 12 - Thorai Cavity, levatores ostarum, retus thorais, ranial serratus dorsalis, and salenus musles. [2] External Interostals The external interostals and internal interostals fill the spaes between the ribs (interostals spaes). The external interostals form the thiker outer layer of the interostal spaes, ranging from 4 to 5mm thik in large dogs [1]. They over the extent of the ribs, from the insertion of the levatores ostarum, to the ostohondral juntions. Their fibers originate from the audal border of eah rib, and run audoventrally to the ranial border of the next rib. In addition, their fibers run perpendiular to the fibers of the internal interostals, whih run ranioventrally [13]. Sine they run perpendiular to the internal interostals, it was assumed that they at at a 45 o to the surfae of the rib where they attah. In addition, the musles were assumed to at laterally at an angle of 35 o (See Figures 13,14). 26

39 Levatores Costarum The levatores ostarum insert into the ranial portion of the rib near the vertebrae and extend ranially to attah to the transverse proess of the adjaent rib. Their insertion points an typially be easily identified by 2-3 peaks near the dorsal portion of the rib (see Figures 13,14). The angle of ation of the levatores ostarum was measured from a drawing in Millers Anatomy [1] using a protrator. Figure 13 - Insertion Points for Serratus Dorsalis Cranialis and Levatores Costarum [2] 27

40 Figure 14 - Summary of Inspiration Fores Cranial Serratus Dorsalis The ranial serratus dorsalis musle group emerges from the supraspinous ligament and inserts into seven to nine peaks in the ranial borders of ribs three to ten [12]. (See Figures 13, 14) The angle of ation was estimated by using a protrator to measure a drawing from Millers Anatomy [1]. Expiration Expiration requires the use of less musle groups to be performed due to the inherent elastiity of the thorax and the magnitude of pleural pressure. As more and more air is drawn in, this resistane and pressure inreases. During expiration, these at to help redue the 28

41 volume of the thorai avity, and expel air. Due to the lessened need for musle interation, only a ouple musle groups are utilized during respiration; the internal interostals and audal serratus dorsalis. The audal serratus dorsalis exists only on the last ribs; ribs eleven through thirteen [12], so they were ignored in the development of this model. Internal Interostals The internal interostals lie medial to the external interostals and originate from the ranial border of one rib to the audal border of the rib next to it. The fibers run ranioventrally and run perpendiular to the fibers of the external interostals [13]. This musle group has a thikness of 2 to 3mm in large dogs, about half as thik as the external interostals. Following the assumptions made for the external interostals, the fibers of the internal interostals run at 45 o from the surfae of the rib, and instead of ating laterally at an angle of 35 o, they at medially, at 35 o. See figure 15 for a summary of the expiratory musle fores. Figure 15 Summary of Expiratory Fores 29

42 Summary of Loading Conditions The following table summarizes the loading onditions of the rib disussed in the prior setions. Notie that most of the loading from the musles ours during inspiration. Table 1 - Loading Conditions of 10th Rib During Inspiration and Expiration Rib Loading Condition Insertion Musle Fiber Diretion Area (mm 2 ) # Nodes Magnitude of Fore per Node (N/node) Vertial Component (N/node) Horizontal Component (N/node) External Interostals Inspiration Levatores Costarum Inspiration Cranial Serratus Dorsalis Inspiration Internal Interostals Expiration Pleural Pressure Both On ranial and audal rib surfaes. Superfiial to internal interostals. 2 to 3 peaks on raniodorsal portion of rib surfae. 7 to 9 peaks on ranial portion of rib surfae. On ranial and audal rib surfaes. Internal to external interostals. 45 o from fae - fibers run audoventrally 13.5 o from fae - fibers run audoventrally 13 o from fae - fibers run audoventrally 45 o from fae - fibers run ranioventrally Pleural pressure varies from -2mH 2 0 to -4mH 2 0 from beginning of inspiration to beginning of expiration, respetively. 30

43 Determination of Boundary Conditions The Motion of the Ribs The head of eah rib artiulates with ostal artiular surfaes on the bodies of vertebrae, and the heads of eah rib are held in plae against the surfaes by ligaments [10], (Figure 16). This mehanism allows the ribs to rotate ranially and audally during inspiration and expiration, respetively. To simulate this, the boundary onditions of the dorsal portion of the rib model were modeled fixed (no rotation, or displaement allowed). In anines, only the ostal artilages of the first nine ribs are fixed to the sternum, the ostal artilages of ribs ten through twelve are not attahed to the sternum, but unite to form the ostal arh. The thirteenth ribs end freely in musulature and are referred to as floating ribs. Sine the tenth rib is attahed to the ostal arh, the ventral portion of the rib was modeled similarly to the dorsal portion, fixed. In addition, boundary onditions that onstrain lateral translation of the outer surfae of the rib were added to simulate the effets of the abdominal musles that enase the thorax (Figure 17). Figure 16 - Diagram Depiting Head of Ribs [2] 31

44 Figure 17 - Thorax [2] Appliation of Loading/Boundary Conditions to FEA Model Generation of FEA Mesh The mesh utilized in this researh was provided by Dr. Hazelwood. The mesh was generated by the following proedure: A beagle rib was exised and run through a CT san. Mimis was used to generate a 3d model and surfae mesh from the CT san. The surfae mesh was imported into Abaqus. 32

45 The surfae mesh was then onverted to a 3d solid mesh in Abaqus, using 4 node tetrahedrons, generating a model onsisting of 4,252 elements and 1,160 nodes. Figure 18 - Mesh of Rib Appliation of Loading/Boundary Conditions to FEA Mode Abaqus was used to add the boundary and loading onditions speified in Table 1 to the provided mesh. The first step was to define the node and surfae sets that would ultimately represent the insertions and ation points of the various musle groups and boundary onditions. Surfaes were piked to define the areas that would be affeted by the pleural pressure, and node sets were piked to define the loations of musle fores and boundary onditions. 33

46 One the sets were defined, the speifi loads and boundary onditions were applied to the appropriate set The following figures (Figures 19-28) illustrate the loading and boundary onditions applied to the rib. Figure 19 - Cranial External Interostals Figure 20 - Caudal External Interostals 34

47 Figure 21 - Caudal Internal Interostals (Loading Not Shown) Figure 22 - Cranial Internal Interostals (Loading Not Shown) 35

48 Figure 23 - Cranial Serratus Dorsalis Figure 24 - Levatores Costarum 36

49 Figure 25 - Pleural Pressure (Applied to Medial Surfaes of Rib, Varies Based on Inspiration or Expiration, Pressure Applied Normal to Surfae) Figure 26 - Dorsal Boundary Conditions (FIXED) 37

50 Figure 27 - Ventral Boundary Conditions (FIXED) Figure 28 - Lateral Boundary Conditions (No Rotation or Displaement Allowed) 38

51 FEA Model Parameters Three separate load steps were reated and utilized with a UMAT subroutine. The UMAT subroutine is a Fortran program developed by Dr. Hazelwood whih utilizes the strains, generated by the loads and boundary onditions on the model, to predit the adaptation of the bone tissue to its mehanial environment after a speified number of days [15], (See Figure 29). The subroutine is alled by an input file in Abaqus to generate these results. In the model, eah loading ondition represents a different phase in the respiration yle. Essentially, the model simulates hanges in the mehanial properties and porosity of the bone tissue resulting from damage and disuse stimuli, whih are determined from the strain state and loading onditions of the bone. Damage is estimated from strain and the number of loading yles. Disuse exists when the stress stimulus is below a predefined equilibrium stimulus. The BMU ativation frequeny, a funtion of disuse and damage, is used to estimate the populations of refilling and resorbing bmu s, based on the resorption, reversal, and refilling time intervals of the BMU ativation frequeny history. Porosity values are alulated from the resorbing and refilling rates, and are used to alulate the resulting elasti modulus of the material. An equilibrium mehanial stimulus is set based on the natural loading onditions and strain state of the bone tissue. Any stimulus above the equilibrium stimulus generates damage, and therefore, ativates BMU s for repair. Any stimulus below the equilibrium generates a state of disuse, and BMU s are ativated to remove tissue. The FEA model alulates the strain energy density, whih is used to alulate the mehanial stimulus. The remodeling algorithm 39

52 uses the resulting mehanial stimulus, to predit the remodeling response of the tissue. The remodeling response of the tissue hanges the resulting porosity of the bone. Figure 29 UMAT Subroutine Flow Chart [10] It was assumed that anines breathe at a rate of 15 breaths/minute [14], so the model assumed the load steps were applied for a total of breaths/day. The UMAT subroutine was run to simulate 400 days of breathing until a steady state ondition was reahed by the remodeling parameters. 40

53 Boundary Conditions Dorsal and Ventral Ends of Rib FIXED (No Rotation, No Displaement) Lateral Surfae of Rib (No Rotation, No Displaement) Load Steps 1. Onset of Inspiration Beginning of Inspiration Cyle o External Interostals Cranial Surfae o External Interostals Caudal Surfae o Levatores Costarum o Cranialis Serratus Dorsalis o Pleural Pressure (-2mH20) 2. End of Inspiration After Full Breath Has Been Taken o External Interostals Cranial Surfae o External Interostals Caudal Surfae o Levatores Costarum o Cranialis Serratus Dorsalis o Pleural Pressure (-4mH20) 3. Onset of Expiration Canine Begins to Exhale o Internal Interostals Cranial Surfae o Internal Interostals Caudal Surfae o Pleural Pressure (-4mH20) Modifiation of UMAT Subroutine Parameters to Simulate Bone Adaptation for the Dog Rib The next step was to modify the UMAT subroutine to simulate adaptation of bone tissue for dog ribs, beause the original subroutine provided was developed to simulate human bone 41

54 tissue adaptation. Four parameters would have to be hanged to more losely simulate dog rib bone tissue adaptation. They are as follows: Number of loading yles The original UMAT subroutine was written to model a human femur. Sine the goal of the urrent model was of a dog rib and was used to simulate respiration, this value had to be hanged to the number of breaths a dog takes per day. Remodeling Periods The remodeling periods of dogs and humans are not the same. The remodeling periods for humans were determined from several histomorphometri studies [10]. The human remodeling periods were saled down, to shorter periods for anines. o For humans, the remodeling periods are: Tr (resorption) = 24 Days Ti (reversal) = 8 Days Tf (filling) = 64 Days o In this study, these periods were defined as: Tr (resorption) = 10 Days* Ti (reversal) = 5 Days* Tf (filling) = 44 Days* *Various numbers for remodeling periods were piked and evaluated in separate models. The periods whih reated the model that most losely mathed the natural bone porosity were utilized and are noted above. 42

55 Equilibrium Stimulus (φ 0 ) The equilibrium stimulus is the set point for bone remodeling. Stresses/strains and loading yles that produe a stimulus above or below this value ativate remodeling to adapt bone to its mehanial environment. It was assumed that the values of equilibrium stimulus for humans and dogs are not the same. See the following setion for the proedure of determining the equilibrium stimulus for the dog rib. Damage Coeffiient (Kd) A onstant utilized in the subroutine whih is alulated from the value of the equilibrium stimulus and other remodeling parameters from the dog rib. Determining the Equilibrium Stimulus The equilibrium stimulus and the damage oeffiient for the dog rib are unknown and had to be estimated for this analysis. They were alulated by the following formulas: (1) φ 0 = ε 4 * RL [10] (2) K d = D 0 *f a0 *A*F s /φ 0 [10] Where: D 0 = mm/mm 2 (Initial Damage Rate Coeffiient) [10] f a0 = BMUs/mm 2 /day (Initial Ativation Frequeny) [10] A = mm 2 [10] F s = 5 (Damage Removal Speifiity Fator) [10] ε = Strain 43

56 RL = Number of Loading Cyles The strain values used in the determination of the equilibrium stimulus for the dog rib were predited using a standard analysis in Abaqus with an assumed modulus of 17.4 Gpa [4]. The results predited strain values between 72 and -72 mirostrains. Values with magnitudes between 0 and 72 mirostrains were used to estimate the equilibrium stimulus and orresponding Kd value. These values were then used in the UMAT subroutine to generate models simulating the struture/porosity of the resulting rib bone tissue. The ideal result would be a struture that losely mathes the natural bone tissue, whih would indiate that the loading and boundary onditions, and model parameters are aurate. As stated in the introdution, bone generally has a higher porosity near the enter (trabeular bone), surrounded by bone of lower porosity (lamellar bone). Over 30 simulations were run in Abaqus, eah utilizing different equilibrium stimulus, orresponding Kd values, and remodeling periods, in an attempt to find a resulting model that losely math the predited struture. Table 2 on the following page lists the ranges of values investigated, and figure 30, on the following page illustrates the some of the strutures that were predited utilizing this method. The model in the far left of the figure generated an inonsistent and sporadi porosity struture. The model in the enter of the figure predited a very high porosity throughout the entire bone. The final model (Appendix A) that most losely mathed the atual struture of untreated bone, pitured in Figure 30 on the far right, utilized the following values: Strain Value 27 µstrain Equilibrium Stimulus pd Kd mm/mm 2 44

57 Table 2- Range of Remodeling Parameters Investigated Parameter Min Max Strain (με) 0 72 Equilibrium Stimulus (pd) E-13 kd (mm/mm 2 ) E+07 Figure 30 Remodeling Parameters Effets on Predited Bone Struture 45

58 Validation of model by omparing to CT san results In order to validate the model, the porosity values predited by the simulation were ompared to the density/porosity values omputed from the CT san. This involved alulating porosity values from graysale values in Mimis for ross setions of bone at seven different loations (Figure 31), and omparing those values to porosities predited by Abaqus at the same loations. Eah ross setion was divided into four equal quadrants, and eah setion was ompared on a quadrant by quadrant basis. It was deided to ompare porosity values of the ortial regions of bone tissue in eah quadrant to simplify the validation. The quadrants were divided about the enterline of eah ross setion in the manner illustrated in Figure 32. Quadrants 1 and 4 lay on the lateral portion of the bone, and quadrants 1 and 2 lie ranial to the rib. Figure 31 - Loations of Cross Setions for Comparison of Predited Porosities to Atual 46

59 Figure 32 - Identifiation of Quadrants Obtaining Density/Porosity from Mimis The graysale values reported from Mimis were onverted to apparent density by the following relationship: (3) ρ app =.001*HU [15] Where the graysale value is measured in Hounsfield Units and ρ app is measured in g/m 3. The graysale values are reported as the mean in Figure 29 on the following page. These values were then onverted into porosity using the relationship: (4) (ρ app - ρ m )/(ρ v ρ m -) = P v [4] 47

60 where ρ m is the density of the bone tissue, ρ v is the density of the voids or soft tissue, and P v is the porosity. During the validation proedure, values of ρ v = 0 g/m 3 and ρ m = 2 g/m 3 were used to alulate the porosity of the bone tissue. The use of eah of these values is disussed in the results and disussion setions of this report. Three porosity values per quadrant were averaged to produe the average porosity per quadrant for eah ross setion (See Figure 33). 48

61 Figure 33 - Mimis Interfae, Graysale Values Listed as "Mean". Three loations where graysale measured highlighted in Yellow, Red and Purple. 49

62 Obtaining Porosity Values from Abaqus Porosity values at three nodes per quadrant were identified in Abaqus using the view ut manager to view eah ross setion (See Figure 34). The loations/heights of the quadrants were idential to the loations in the Mimis model, and the nodes were taken from the same general area as the points measured in Mimis. An average value for eah quadrant for eah ross setion was determined. Figure 34 - Cross Setional Porosity Analysis of FEA Model 50

63 Statistial Analysis The predited tissue struture determined by the model was ompared qualitatively (visual inspetion) and quantitatively (omparison of porosity values). Cross-setions of the model were examined visually and ompared to ross-setions taken from idential loations of the atual rib CT data. The goal was for the model to have higher porosities towards the interior of the rib, mathing what is observed in dog ribs. To ompare the model quantitatively, the porosities obtained from speifi quadrants of the model were ompared to the porosities obtained from the atual rib in the same loations, using a T-test, assuming 95% onfidene, to indiate if a statistially signifiant differene existed. 51

64 3. Results Visual Comparison of Model and Tissue Sample Figure 35 depits a longitudinal view of the rib model generated by performing the analysis using the methods in the previous setion. On a ross-setion by ross-setion basis, the model predited fairly well the struture of the bone. Figures 33 and 34 illustrate the struture of bone tissue obtained from the rib sample side by side with the strutures predited by the FEA model. Figure 35 - Longitudinal Cross Setion of Abaqus Model (SDV1 = Porosity %) 52

65 Figures 36 and 37 are images of horizontal ross setions taken from the FEM side by side with CT data taken from the same loation. The FEM predits a high porosity around the enter of the rib, whih an be seen in the ross setion of the atual tissue and is onsistent with the struture of a typial medullary avity. The thikness of the ortial bone surrounding the enter of the rib appears to be fairly onsistent with that of the atual tissue as well. Figure 36 - Horizontal Cross Setion (Mimis Left, Abaqus Right, SDV1 = Porosity %) 53

66 Figure 37 - Horizontal Cross Setion (Mimis Left, Abaqus Right, SDV1 = Porosity %) Quantitative Comparison of Model and Tissue Sample As stated in the methods setion, the porosity values predited by the FEA model were ompared to the porosities omputed from the graysale values obtained from Mimis. Seven ross setion loations of eah set were ompared on a quadrant by quadrant basis. Statistial analysis suggests that the model predited the porosity of the resulting bone tissue relatively well. There were no statistially signifiant differenes between the majority of quadrants, 25 of 28, and ross-setions, 5 of 7. 54

67 The model predited the tissue densities exeptionally well, % Differenes <20%, in the following regions: Table 3 - Regions where Model Predited Tissue Densities Exeptionally Well (% Diff < 20%) Cross Setion Distane From Dorsal End (mm) Quadrant(s) 1 5 1, 3, , , 2, , , 2, 4 Statistially signifiant differenes between the model results and the CT san results existed in a few setions. In ross-setion 5, quadrant 4, the predited porosity from the model was 0.515±0.099 and the value obtained from Mimis was 0.184± Similarly in ross setion 6, quadrant 2, the FEA model predited a porosity of 0.463±0.104 and the value obtained from Mimis was 0.153±0.037; these values reate a perent differene of about 200%, but these values are still within reasonable limits given that porosity values in other ross-setion quadrants were statistially similar. Tables 4 and 5 on the following pages summarize the results of these omparisons. 55

68 % Differene (Cross SEtion Averages) Figure 38 was generated by averaging the predited and atual porosities for eah ross setion, and plotting the differene vs. the loation of the ross setion. It illustrates that the model onsistently predited porosities that were higher than existed in the atual tissue in the same loation. The model did very well in prediting the porosity of the tissue between 20 and 30 mm away from the dorsal end of the rib, and had differenes exeeding 50% throughout the rest of the tissue. The model had the most error at a loation 15 mm from the dorsal end, with an average % differene of 115%. 0% -20% % Differene Between Predited and Atual Bone Tissue Porosities % -60% -80% -100% -120% Distane from Dorsal End of Rib Figure 38 - Comparison of Predited and Atual Porosities Based on Loation on Rib 56

69 Figure 39 below illustrates how well the model predited porosities for eah quadrant. The perentages represented in the figure were generated by taking the average % differene for eah quadrant, aross all of the ross setions. It is apparent that the model struggled prediting aurate porosities in the third quadrant, audal-medially, and performed better in all other quadrants. Figure 39 - % Differenes for Porosity Values by Quadrant 57