Finite Element Analysis of Infant Skull Trauma using CT Images
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Finite Element Analysis of Infant Skull Trauma using CT Images ARNA ÓSKARSDÓ TTIR Master of Science Thesis Stockholm, Sweden 2012
Transcript of Finite Element Analysis of Infant Skull Trauma using CT Images
Finite Element Analysis of Infant Skull Trauma using CT Images
A R N A Ó S K A R S D Ó T T I R
Master of Science Thesis Stockholm, Sweden 2012
Finite Element Analysis of Infant Skull Trauma using CT Images
A R N A Ó S K A R S D Ó T T I R
Master’s Thesis in Scientific Computing (30 ECTS credits) Master Programme in Computer simulation
for Science and Engineering 120 credits Royal Institute of Technology year 2012
Supervisor at KTH was Svein Kleiven Examiner was Michael Hanke TRITA-MAT-E 2012:07 ISRN-KTH/MAT/E--12/07--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci
Finite element analysis of infant skull trauma using CT images
Abstract Some cases of infant skull fracture fall under the category of forensic study where it is not obvious whether the head trauma happened due to an accident or abuse. To be able to determine the cause of the head trauma with sufficient accuracy, biomechanical analysis using finite element modeling of the infant cranium has been established. By simulating the trauma, one may be able to obtain the fracture propagation of the skull and from it determine if the scenario narrative is plausible. Geometry of skull, sutures, scalp and brain of a 2 month old infant head was obtained using CT images and meshed using voxel hexahedral meshing. Simulation of an impact to the head from a fall of 0.82 m height, to a rigid floor, was carried out in the non-linear finite element program LS-Dyna. Two scenarios were simulated: an impact to the occipital-parietal bones and an impact to the right parietal bone. The fracture propagation was obtained using the Chang-Chang Composite Failure Model as a constitutive model for the skull bones. The amount of material parameters gathered in the present study to predict fracture of the infant skull has not been obtained before, to the best knowledge of author.
Validation of the models’ ability to show relatively correct fracture propagation was carried out by comparing the obtained fracture pattern from the parietal-occipital impact against published fracture patterns of infant PMHS skulls from a free fall onto a hard surface. The fracture pattern was found to be in good compliance with the published data. The fracture pattern in the parietal bone from the impact was compared against a fracture pattern from a previously constructed model at STH. The patterns of the models show some similarities but improvements to the model and further validations need to be carried out.
Finit elementanalys av skallbensfraktur hos småbarn med datortomografibilder
Sammanfattning Några skallskador hos spädbarn ger grund till kriminaltekniska studier där det inte är självklart om skallskadan skett på grund av en olycka eller misshandel. För att kunna fastställa orsaken till skallskadan med tillräcklig noggrannhet har biomekaniska analyser med finita element modeller av barns huvud genomförts. Genom att simulera traumat kan man kunna få sprickpropagering i skallbenet och från den avgöra om scenariot är rimligt. Geometrin för skallen, suturer, hårbotten och hjärnan hos ett 2 månader gammalt spädbarns huvud erhölls genom CT-bilder och Voxel hexahedermeshning. Simulering av påverkan på huvudet från ett fall på 0,82 m höjd mot ett hårt golv simulerades i det icke-linjära finita element programmet LS-Dyna. Två scenarier simulerades: ett islag mot nack-hjässbenet och ett mot det högra hjässbenet. Sprickpropagering simulerades med en Chang-Chang Composite konstitutiv frakturmodell för skallbenet. Den omfattande mängd materialparametrar som sammanfattades i denna studie för att prediktera skallbensfrakturer hos spädbarnets har, enligt författarens kännedom, inte erhållits tidigare.
Validering av modellernas förmåga att visa relativt korrekt sprickpropagering genomfördes genom att jämföra det erhållna frakturmönstret från simuleringarna med publicerade frakturmönster från spädbarn för fritt fall mot en hård yta mot nack-hjässbenet. Frakturmönstret befanns vara i god överensstämmelse med publicerade data. Brottmönstret i hjässbenet jämfördes med frakturmönstret från en tidigare konstruerad modell på KTH. Brottmönstren från modellerna visar vissa likheter men förbättringar av modellen och ytterligare valideringar måste genomföras.
Table of contents Introduction ................................................................................................................................... 1
1.1 Thesis overview ............................................................................................................. 1
1.2 Background ................................................................................................................... 2
1.3 Aim of the project ......................................................................................................... 4
1.4 Anatomy of an infant cranium ....................................................................................... 4
1.4.1 The skull bones ...................................................................................................... 5
1.4.2 Sutures and fontanelles .......................................................................................... 7
1.4.3 Dura mater ............................................................................................................. 7
Biomechanical & material properties of infant skull and suture ................................................... 9
2.1 Bone and suture elastic behavior ................................................................................... 9
2.1.1 Isotropic material properties ................................................................................ 10
2.1.2 Transversely isotropic material properties .......................................................... 12
2.2 Parameter selection...................................................................................................... 14
2.2.1 Parameters obtained from literature .................................................................... 15
2.2.2 Parameters estimations ........................................................................................ 18
Creating Finite Element mesh from CT data ............................................................................... 20
3.1 Introduction to computed tomography imaging .......................................................... 20
3.2 Image segmentation ..................................................................................................... 21
3.2.1 3D Slicer .............................................................................................................. 22
3.2.2 DeVIDE ............................................................................................................... 25
3.2.3 Matlab .................................................................................................................. 27
3.3 Hexahedral meshing .................................................................................................... 28
3.3.1 Dicer .................................................................................................................... 28
3.3.2 Mapping .............................................................................................................. 29
3.3.3 Mesh quality ........................................................................................................ 32
Finite element Modeling ............................................................................................................. 35
4.1 Finite element simulation in LS-Dyna ........................................................................ 35
4.1.1 Governing equation ............................................................................................. 35
4.1.2 Discretization: Hexahedral element and its shape function ................................. 36
4.1.3 Time integration .................................................................................................. 38
4.1.4 Matrix calculation................................................................................................ 39
4.1.5 Volume integration .............................................................................................. 41
4.1.6 Hourglass control ................................................................................................ 42
4.2 Simulation in LS-Dyna ................................................................................................ 42
4.2.1 Model preparation ............................................................................................... 43
4.2.2 Constitutive models ............................................................................................. 45
4.2.3 Contact constraints .............................................................................................. 49
Simulation results ........................................................................................................................ 51
5.1 Parietal-Occipital impact ............................................................................................. 51
5.2 Parietal impact ............................................................................................................. 56
Discussion ................................................................................................................................... 61
6.1 Material properties ...................................................................................................... 61
6.2 Image segmentation ..................................................................................................... 62
6.3 Models mesh ............................................................................................................... 62
6.4 Model simulation ......................................................................................................... 63
Conclusion ................................................................................................................................... 65
Appendix ..................................................................................................................................... 66
Constitutive model for the brain .......................................................................................... 66
References ................................................................................................................................... 67
Chapter 1
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Chapter 1
Introduction
Studies on cranial impact have been done for different scenarios such as head molding during birth, bike and car accidents and also in forensic cases. Head injuries caused by falls are in majority when it comes to infants and juveniles [1] showing the importance of understanding the biomechanics of their head to predict the occurring of the injury [2]. Forensic studies on infant skull fracture may be done to determine if a head trauma was caused by an accident or abuse. To be able to determine the cause of the head trauma with sufficient accuracy, biomechanical analysis using finite element (FE) modeling of the infant cranium can be established. A model of the cranium can be constructed from the patient specific computed tomography (CT) images and used to simulate the head trauma. By simulating the trauma, one may be able to obtain the fracture propagation of the skull and consequently determine if the scenario narrative is plausible. Biomechanical studies of this kind can also help to determine the injury mechanism of infant skull bones from different impact scenarios.
1.1 Thesis overview
The thesis covers the whole process of simulating an infant skull trauma from a fall. Patient specific CT image data of a 2 month old infant cranium is provided. The image data is processed to obtain the complex geometry of scalp, skull, sutures and brain and further processed to construct a hexahedral mesh in the (un-commercial) program Dicer, resulting in a model suitable for FE simulation. The patient specific model is simulated in the non-linear Finite Element program LS-Dyna to obtain an analysis of a mechanical impact to the cranium from a fall of a certain height. Results from an impact to the parietal bone and parietal and occipital bones are analyzed. Validation of the models ability to show relatively correct fracture propagation is carried out by comparing the obtained fracture pattern from the parietal-occipital impact against published fracture patterns of post mortem human subject (PMHS) infant skulls drop studies [3]. The fracture pattern in the parietal bone, from the impact, is compared against a fracture pattern from a previously constructed model at the School of Technology and Health (STH).
The work was carried out at the STH which is part of The Royal Institute of Technology, Stockholm. The image data was provided by Rättsmedicinalverket, Uppsala University, and the
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code of the meshing program, Dicer, was used with the permission of its author. The commercial program LS-Dyna was accessed at STH.
The thesis material is organized as follows. Chapter 1.2 covers previous studies found in literature relevant for the background of this study. Chapter 1.3 goes through the aim of the project and chapter 1.4 covers the anatomy of the human infant skull, sutures and dura mater. Chapter 2 covers the biomechanical and material properties of infant skull and suture. Chapter 2.1 goes through the elastic behavior of skull and sutures and 2.2 explains how parameters for the skull bones were obtained and estimated. Chapter 3.1 goes through the image processing and chapter 3.2 the hexahedral meshing. Chapter 4 covers the simulation process in LS-Dyna. Chapter 5 shows results from impact simulations, chapter 6 covers discussions and chapter 7 lists the conclusions.
1.2 Background
Studies on the biomechanics of fetal heads go back at least to the year 1888 when Murray et al. studied the effects of compression on fetal skull during labor [4]. Until the year 1980 few other studies were done on the topic of the biomechanics of head molding during labor, giving more qualitative then quantitative results [5]. The mechanical complexity of the fetal heads structure was well understood at this time and explained by Holland at al. in 1922 as follows: “The fetal head consists of a non-rigid-shell, of a peculiar shape, composed of a loosely-joint plates of liable bone jointed on a rigid base and strengthened by the attachment of dura mater and its system of septa: the contents of this shell are partly solid and partly fluid” [6] [5].
McPherson and Kriewall, 1980, were the first ones to do mechanical studies on tissue properties of the fetal cranium. They investigated the fiber orientation of specimens of varying locations from 6 cranial cadavers ranging in gestational age. They found that the fiber pattern of each bone of the skull were radially oriented from the center of ossification [5] versus homogeneous grain structure in adults [7]. The elastic modulus was also determined by them using three point bending test for specimens with fiber direction running parallel and perpendicular to the long axis. Elastic modulus was found to differ significantly between parallel and perpendicular fiber direction and also to increase with increased gestational age [5]. Further research by Kriewall at al. confirmed that the elastic modulus for fibers running parallel to the long axis had higher elastic modulus than fibers running perpendicular to it. They also studied the stiffness of the parietal bone by applying load to the inner surface of the eminence with the bone placed concave upon a tripod. High correlation was found between the bone stiffness and gestational age and bone stiffness and fetal weight [8].
Mazuchowski and Thibault at al. studied the mechanical properties of human skull and porcine sutures in 1997 and constructed a FE biofidelic model to simulate the deformation and flexibility of skull and sutures respectively and its biomechanical function during head impact. The cranial vault was simplified to a hemi-elliptical shell model with sutures of 5 mm constant thickness and anterior fontanels as a 15 mm square plate. The cranial cavity was assumed to be filled with brain. They concluded that geometry such as varying bone thickness and mechanical properties such as elastic modulus of oriented fibers must be taken into account when developing models of infant cranium. They were not successful in their testing of sutures but emphasized the importance of including sutures in the model as they increase the cranial
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tolerance to impact due to their flexibility [2] . A continuation of this research was done in 2000 using sutures from infant porcines by Margulies at al. [9]. They constructed two FE models with the size of a one month old infant head in the form of ellipsoid using shell elements to model the skull, sutures, fontanel and foramen magnum and brick elements for the brain. Cranial bone and sutures were assumed to have homogeneous properties. The models, one with parameters for adults and fused fontanels and sutures and another with infant porcine and human properties for sutures and cranium respectively were simulated for the contact problem in LS-Dyna3D. Their models were fixed at the base of the cranial vault and impact load applied to the parietal bone. They concluded that the structural change from the one layered compact bones in infants to the three layered cortical bones in adults were an important factor when determining the mechanical response of an impact to the head. Elastic modulus and ultimate stress were found to increase with age both for the human cranial bone and porcine sutures [9] [10] [11].
In 2006 Coats and Margulies addressed the hypothesis if some mechanical properties of the infant skull were bending and tensile rate dependent. Their studies on elastic modulus and ultimate stress of infant cranial bone and infant sutures were obtained from specimens of 23 cadavers, taken with fiber direction perpendicular to the long axis, of age less the one year old. The bending rate of 2.4 m/sec or higher in three point bending test were chosen to resemble a skull response to fall from low heights [12]. Earlier studies on elastic modulus of infant cranial
bones were done at a much lower rate, 6103.8 −⋅ m/s, to simulate the skull response to uterine contraction during labor [5] [9]. They came to the conclusion that strain rate of bending and tension test did not have significant effect on the elastic modulus and ultimate stress of bone and sutures for the sample age [12]. This was in contradiction to earlier studies done on human adult cortical bone [13].They hypothesized that future studies on specimens, taken with fiber direction parallel to the long axis, would show rate dependence [12]. They found that ultimate stress and elastic modulus were larger for parietal bones than occipital bone and both increased with age. No relation of elastic modulus and ultimate stress was found for age and strain rate for the human sutures. Ultimate strain for cranial bone was found to be significantly lower than for sutures and 23 times stiffer. They also reported that human sutures were not comparable to porcine sutures as was hypothesized in their earlier studies [9]. They concluded that the skull should not be considered homogeneous, as a result of their studies on biomechanical properties, and emphasized the need to use relevant age specific material properties for the skull and sutures to obtain accurate results when simulating skull traumas using computational models [12].
Coats and Margulies constructed a FE model from CT images of a 1.5 month old infant in 1997 and simulated an impact to the occipital bone from a low fall height. As their FE model was to analyze skull fractures of cranium, brain was assumed isotropic and homogeneous. They concluded that too stiff brain (large shear modulus) could increase the stress of the cranial bone. Suture geometry was also of importance and sutures width bigger than 10 mm showed decrease in maximum stress resulting in decreased fracture, emphasizing once again the importance of accuracy of geometry in this kind of fracture analysis. Their model with obtained material parameters and failure data showed promising results in being useful for mechanical analysis of impacts to infant’s cranium [14].
Weber et al did a study on fracture analysis of PMHS infant skulls of age from newborn to 9 month old from a free fall drop of 0.82 meter height to three different types of floor. He presented 15 fracture propagations of drops to the parietal-occipital region. He concluded that each fall of an infant from table-height might cause fracture in the skull bones [3].
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1.3 Aim of the project
The aim of this project is to generate a detailed FE model of an infant skull and sutures and study the skull fracture propagation from an impact from a fall of a certain height. The model is constructed using available CT images of an infant head of 2 month old age. It is of interest to evaluate the skull fracture propagation from models of skull with sutures using fine and coarse element mesh. The thesis aims to obtain the following:
• geometry of the infant skull, brain and scalp using CT images • detailed sutures from CT images of unknown Hounsfield unit (HU) values • fine and coarse hexahedral mesh of skull, sutures, brain and scalp • relevant material properties from literature and biomechanical relations for skull bones
and sutures • FE model of the head and simulation of its impact to a hard surface
• validation of the model ability to show relatively correct fracture propagation
If the model shows promising results of being a good estimation of fracture propagation one would like to construct models from CT images of infant heads from a range of age to have as comparison to future cases of skull traumas. The thesis aims to answer the following questions:
• Is it possible to obtain the sutures geometry from CT images? • Does fracture propagation change with mesh size? • Does fracture propagation resemble results of earlier constructed model and fractures
from Weber’s free fall study? • Does the method used in constructing the age specific model and simulating the skull
fracture demonstrate to be a good way to determine infant skull fracture?
1.4 Anatomy of an infant cranium
The geometry of the cranium varies between individuals in factors like age, gender, genetics and even environmental factors like sleeping position during the first years may influence its shape. The infant cranium is made up of the skull, sutures and fontanelles. The skull is formed from the following seven bones; two frontal bones, two parietal bones, two temporal bones and one occipital bone (Figure 1) in addition to facial bones and lower jaw. The bones of the skull are connected by four main sutures; metopic, coronal, lamboid and sagittal suture (Figure 4). The sutures intersect at two fontanelles which are spaces in between the bones of the infant skull. The posterior fontanelle is the junction of the parietal bones and occipital bone located at the back of the head and the anterior fontanelle is the junction of the frontal bones and parietal bones located at the top of the head [15].
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Figure 1: Cranium of 8 month old infant showing frontal, parietal, occipital and temporal bones. Geometry is
obtained from projects data and processed in the program Mimics from Materialize [16]
1.4.1 The skull bones
The skull can be divided into two parts: the neurocranium and the viscerocranium. The former one covers the brain and the latter one consists of lower face and jaw [15]. Here after the neurocranium will be referred to as skull. The structure of the infant and adult skull layer differs in composition. The infant skull has single layered compact bones (Figure 2) with varying porosity and fiber pattern radially oriented from the center of ossification [5] [8]. The adult skull layer consists of two flat compact bones and a spongy cancellous bone in the middle called the diploe (Figure 3). Bone thickness is not uniform throughout the skull [17] as can be seen in the plot of Figure 3 (right).
Figure 2: The left image shows a CT image of single layer compact frontal bone of 2 month old infant in sagittal
view. The right image shows a Hounsfield unit (HU) distribution of the profile line (green arrow) across the skull
bone in the left image. The smooth curve of the plot indicates that formation of spongy bone has not started.
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The two frontal bones are located at the superior part of the face just above the eye sockets. Each of the two bones ossifies intramembranously from the end of eight weeks of gestation and is separated by metopic sutures. The two bones are usually completely fused together during an infant first year of life. The frontal bone is one of the first bones to obtain the morphology of an adult bone (Figure 3). Each frontal bone ossifies from a center which is located at the eminence of the bone and can be visualized as a rounded elevation in the middle of it [18] [19]. The eminence of a frontal bone can be seen in Figure 3 approximately where the lower profile line crosses the bone.
The two parietal bones are separated by the sagittal suture and lie in between the lamboid and coronal sutures. They ossify intramembranously from the bone’s center which is located at the parietal eminence. In infants and children the eminence is easily noticeable as an elevation approximately at the bone center. The parietal bone is very thin in infant’s first years and is more easily breakable than the surrounding skull bones [18] [19].
The occipital bone is located in between the parietal bones and lambdoid sutures. The ossification of the occipital bone is more complex than for the previously mentioned bones of the skull. The occipital bone consists initially of 5 separate parts at early fetal stage with only the center area, called the interparietal part, ossifying intramembranously. The other parts develop from preformed cartilage. After five gestational months the interparietal part has fused with its supraoccipital part but fusion of all occipital bones is usually completed at the end of infants first year [19].
Bones of infants are more compliant than adult bones as they have not been fully ossified. Most skull bones are formed from intramembranous ossification. The bones start as membranous layer of connective tissue which takes part in the formation of cells called osteoblasts. The osteoblasts transfer calcium to the bone layer from the surrounding blood supply to build up cartilage. The cartilage matrix is composed of calcium and protein. The proteins are partly built up of collagen which is flexible but strong material. As the cartilage matrix continues to receive calcium the osteoblasts change into osteocytes of the compact bone layer. The original connective tissue, now surrounding the external of the bone layer is called periosteum [20] (Figure 6).
Figure 3: Beginning of formation of diploe layer in 8 month old frontal bone. Left image: Two profile lines cross
the CT image of the frontal bone (green) in sagittal view. Right image: HU distribution of the profile lines. The
dents in the plots indicate formation of spongy bone in-between two compact bones. The thickness of the bone
can also be estimated from the plot of each profile line, giving 2.27 mm and 3.17 mm respectively. The thickness
evaluation depends on the choice of the lower HU threshold.
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1.4.2 Sutures and fontanelles
Figure 4: Cranium of 2 month old infant showing coronal, metopic, lambdoid and sagittal sutures. (The geometry
is obtained from the project data and processed in the program Mimics [16]).
Ossification of the skull is incomplete at birth with sutures and fontanelles separating the individual bones (Figure 4). Sutures are membranous tissue built up of collagen fibers forming flexible joints which connect adjacent cranial bones [21] [22]. The morphology of sutures is easily distinguishable from the infant skull. Both interdigitated sutures and straight edged sutures exist [21] , for example between adjacent parietal bones and parietal and occipital bones respectively (Figure 4). The sutures and fontanelles are essential for molding of the head during normal birth [5] as well as following development of the growing and expanding brain [23]. The cranium can, because of the existence of sutures and fontanelles, be impacted from quasi-static or dynamic load without changing its shape significantly [2]. Mazuchowski et al. [2] did a study on the closing of sutures and fontanelles from MRI’s of infants of age from 30 week of gestation to 3 years old. He concluded that posterior fontanelles close between the age of 1 - 3 month old after birth and anterior fontanelles close somewhere in between the age of 12 to 24 month old [15] (Figure 5). Most sutures have grown together and fontanelles have closed by the age of three [2] [15] [24].
Figure 5: Infant age in months at which the fontanels start closing. Image is adapted from [2].
1.4.3 Dura mater
The dura mater is a membrane which surrounds the brain and the concave side of the skull with thickness around 0.3 mm at early age. The dura mater is divided into two parts, endosteal and meningeal dura mater. The endosteal dura mater lies closer to the concave skull and is attached
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around the bone margin, especially in children. The periosteum surrounds the sutures and the bone edges [22] (Figure 6). Hereafter the endosteal dura mater will be referred to as dura mater.
Figure 6: Schematic view of the interior of the head. Image is not to scale. Image is adapted from [22].
Chapter 2
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Chapter 2
Biomechanical & material properties of infant skull and suture
Infant skull response from an impact differs from the one of an adult skull. Material, geometrical properties and the structural composition of the skull affect the biomechanical behavior [9]. Few studies have been done to determine material properties of the infant skull and sutures and those which have been documented in literature are not always comparable due to difference in mechanical testing methods of the infants bone materials. The overall outcomes from these literatures of infant’s skull biomechanics are listed below:
• Fiber pattern of each bone of the infant skull is radially oriented from the center of ossification.
• Elastic modulus of infant skull bone is higher in the radial fiber direction from the center of ossification than perpendicular to the fiber direction.
• Elastic modulus and ultimate stress of bones increases with gestational age • The one layer bone structure of the infant skull has different mechanical responses from
an impact to a head than a three layered adult skull bone • Ultimate stress and elastic modulus is larger for the parietal bones than the occipital
bone • Age specific material data should be used in age specific models • Age and strain rate does not affect the elastic modulus and ultimate stress of infant
sutures.
2.1 Bone and suture elastic behavior
The skull bones of an infant are thin, have viscoelastic properties, and are nonhomogeneous with highly directional fiber orientation. The viscoelastic properties require knowledge of both viscous and elastic behavior of the material when undergoing deformation. Few studies have been done on the infant skull bones and none take viscoelasticity into account, so no parameters exist in literature for the viscoelasticity and nonhomogeneous properties, to the best knowledge
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of author. For this reason the infant skull bone will be simplified to be homogeneous and have only elastic properties.
Bones in general are considered to be anisotropic with many of them having a mineral orientation in the direction of maximum load. Each bone has therefore directional properties according to its unique function. The skull bone does not follow the same principles as described above. Its properties are not based on the load direction but much more likely on stiffness and damage tolerance [25]. The infant skull bone has anisotropic material properties with fiber patterns radially oriented from the center of ossification. This means that the bones stiffness varies in response to force applied from different directions as opposed to isotropic material where the stiffness is the same for all directions [25].
The adult sutures are considered anisotropic with complex collagen fiber arrangement [21]. Studies of infant suture properties are very limited and as in the FE simulations done in those studies [9] [14], infant sutures will be assumed to be linearly elastic, homogeneous and incompressible with same density as the dura mater.
2.1.1 Isotropic material properties
Due to limited data in literature, sutures are assumed to be linear elastic. Hooks law describes the linear behavior of a material under load as:
εσ E= (1)
with σ as stress, E as elastic modulus and ε as strain.
For material of a linear elastic solid the stress strain relation is often expressed using the stiffness tensor (elastic modulus tensor), C, or the elastic compliance tensor, S, [26] such that:
klijklij C εσ = 3,2,1,,, =lkji (2)
klijklij S σε = 3,2,1,,, =lkji (3)
For isotropic materials the stiffness tensor has two components, 11C and 12C , due to symmetric
properties in every direction:
=
0
0
0
111212
121112
121211
00000
00000
00000
000
000
000
C
C
C
CCC
CCC
CCC
C (4)
( )2
12110
CCC
−= (5)
The components of the stiffness matrix C and the compliance matrix S are obtained in terms of the engineering constants in equations (6) and (7) respectively:
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( ) ( ) ( )( ) klijjlikjkilijklEE
C δδνν
δδδδν 21112 −+
++−
= , (6)
( ) klijjlikjkilijkl EES δδνδδδδν +++=
2
1 (7)
Note that 111111 CC ≡ and 2211112212 CCC =≡ . The same applies for the elastic compliance
tensor S [26].
Expansion of equation (2) to matrix form is:
( )( )( )
( )
( )
−
−
−−
−−
−+=
12
13
23
33
22
11
12
13
23
33
22
11
2
2
2
2
2100000
02
210000
002
21000
0001
0001
0001
211
εεεεεε
ν
ν
νννν
νννννν
νν
σσσσσσ
E
(8)
with E as the elastic modulus, ν the Poisson’s ratio, 332211 σσσ == the stresses in x,y,z
direction respectively, 121323 σσσ == the shear stresses, 332211 εεε == the strains in x,y,z
direction respectively and 121323 εεε == the shear strains.
The above matrix can be expressed in a more convenient form using index notation:
−+
+= ∑
=
3
1211
kijkkijij
E δεν
νεν
σ }3,2,1{, ∈∀ ji , =
=else
jiifij 0
1δ
(9)
with the symmetry conditions:
jiijjiij εεσσ == , (10)
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2.1.2 Transversely isotropic material properties
The material structure of a tree is often used as an explanation of an anisotropic structure that has symmetrical properties. The tree structure can be simplified to having orthotropic material properties with orthogonal planes of symmetry. The stiffness matrix has then nine components as it has three perpendicular symmetry planes with the principal axis directions going axial, radial, and tangential (circumferential) (Figure 7).
Bone can be modeled as being orthotropic but is often simplified even further by assuming that it has a symmetry plane perpendicular to the fiber direction making it transversely isotropic. This reduces the parameters needed from nine, for orthotropic material, to five. For example, in long bones the fibers running perpendicular to the long axis can be assumed isotropic in its planes [25].
For infant skull bone it has been established that the elastic modulus parallel and perpendicular to the fiber direction is not the same. It is also known that the fiber direction is radially directed from the center of ossification. This means that the plane of the axial axis, representing the bone thickness, and the tangential axis is the plane of isotropy.
For transversely isotropic material of a linear elastic solid and isotropy in the 2nd-3rd plane, the stress strain relation can be represented using the stiffness matrix C:
=
55
55
44
222312
232212
121211
00000
00000
00000
000
000
000
C
C
C
CCC
CCC
CCC
C (11)
1312 CC = , 6655 CC = and ( )
22322
44CC
C−
= (12)
The compliance matrix, S, is a more convenient representation of the engineering constants E(elastic modulus), ν (Poisson’s ratio) and G(shear modulus) [26]:
Figure 7: Infant skull bone
representation. Principal axis directions
showing radial direction parallel to the
fiber direction in infant skull bones,
tangential direction perpendicular to the
fiber direction and axial direction in the
bone thickness.
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−−
−−
−−
=
12
12
23
32
23
1
31
3
23
21
21
3
13
2
12
1
100000
01
0000
001
000
0001
0001
0001
G
G
G
EEE
EEE
EEE
S
νν
νν
νν
(13)
The independent material constants that need to be obtained are 121221 ,,, νGEE and 23ν . The
plane symmetry gives 32 EE = , 3223 νν = and 3121 νν = . The shear modulus23G is calculated in
equation (14):
( )23
223 12 ν+
= EG (14)
The Poisson’s ratios are not symmetric but satisfy equation (15):
2
21
1
12
EE
νν = (15)
The components of the stiffness matrix (equation (11)) can be derived from the engineering constants of the compliance matrix:
( )Υ−= 223111 1 νEC (16)
( )Υ−== 211223322 1 ννECC (17)
( ) ( )Υ−=Υ−== 122312221232111312 νννννν EECC (18)
( )Υ−= 211223223 νννEC (19)
( )2
2344
12
E
vC
+= (20)
1255
1
GC = (21)
( ) ( )2321122232112 221
1
νννννν −−−=Υ (22)
Chapter 2
14
2.2 Parameter selection
In a patient specific model, as the one developed in this thesis, it is important to have accurate parameters in order to obtain reliable results. Obtaining correct parameters can be a hard task and much estimation needs to be done when there is limited data available either from material testing or from literature. Parameters needed for the FE element simulation of this thesis, to study the skulls biomechanical effects during an impact from a fall are obtained from literature and estimations. The biomechanical properties, elastic modulus, Poisson’s ratio and shear modulus are needed both for the skull and sutures. The additional parameters needed for the constitutive model used to predict failure (chapter 4.2.2) of the skull are longitudinal and transverse tensile strength, shear strength, transverse compressive strength, and normal and transverse shear strength.
The elastic modulus is a measure of the elastic deformation of the material when load is applied to it. It can be obtained from the slope of a stress-strain curve for each material (Figure 8). The shear modulus measures the resistance of a material to shear deformation. Higher value means that the material is more rigid to shear. The shear modulus is defined as the ratio of shear stress versus shear strain. The Poisson’s ratio is the ratio of lateral versus longitudinal strain in uniaxial tension. The value is dimensionless and ranges from -1 to 0.5 for stable materials and is equal to 0.5 for incompressible one. Tensile strength is the measure of materials strength in the form of how much tensile load it can withstand without failing. The tensile strength is also called ultimate strength and is defined as the maximum tensile force per unit cross sectional area of a specimen. A material deforms elastically under a slowly increasing tensile load up to a certain point and will return to its original size and shape if the load is removed. The material deforms permanently if the load is not removed at this point, which is called the yielding point. After the yielding point the material behavior depends on the material in question but finally it reaches its breaking point. The maximum of the stress-strain curve is known as the ultimate strength. Shear strength is the measure of the maximum shear stress which a material can withstand without failing. The shear stress acts parallel to the force direction and causes some portion of the material to shear with respect to another portion. Compressive strength is the measure of the maximum compressive load a material can take per cross sectional area without failing. For bones the compressive strength is usually greater than the tensile strength (Figure 9) [27] [28].
Chapter 2
15
Figure 8: Stress-strain profile of bone samples
taken at different orientations to the long axis of
compact femoral shaft. Image is adapted from
[29]
Figure 9: Failure strengths of anisotropic bone with
sample orientation perpendicular to the long axis.
Image is adapted from [29].
2.2.1 Parameters obtained from literature
Kriewall presented his data of elastic modulus, both parallel and perpendicular to the fiber direction, of parietal and frontal bone specimens obtained from a three point bending test [5] [8]. His specimens were taken from PMHS skulls of age from 24 to 40 weeks of gestation. He divided the results into two categories, preterm and term specimens. Data from 22 term specimens (age range 36 to 40 weeks of gestation) taken from parietal bones of 4 infant PMHS were presented. He found that the ratio of elastic modulus for specimens with the long axis parallel and perpendicular to the fiber direction was 4.2:1. Only data of 2 perpendicular specimens and 12 parallel specimens were presented for the frontal bones. The ratio parallel to perpendicular was found to be 1.8:1 [5].
Coats and Margulies tested the elastic modulus and ultimate stress for specimens with fiber direction perpendicular to the long axis and presented their data of parietal and occipital bone specimens from three point bending tests [12]. The specimens were taken from infant PMHS skulls of age from 21 weeks of gestation to 13 month old. They found significant influence of infant age on the elastic modulus. The data from Coats & Margulies was plotted and used as a basis to obtain age specific material parameters for the 2 month old infant skull model constructed in this thesis (Figure 10 to Figure 12). The ratio obtained by Kriewall was used to find the parallel elastic modulus for the skull bones (Table 1).
Chapter 2
16
Figure 10: Elastic modulus of parietal bone vs. age of specimens with fiber direction perpendicular to the long
axis. Data is obtained from [12]. Elastic modulus for parallel to the fiber direction is scaled by using ratio
difference stated in [5].
Figure 11: Elastic modulus of occipital bone vs. age of specimens with fiber direction perpendicular to the long
axis. Data is obtained from [12]. Elastic modulus for parallel fibers is scaled by using the ratio found in [5].
The plots presented in Figure 10 and Figure 11 show weak relationship between age and elastic modulus with few outliers which are highly influential on the linear regression line. The overall patterns indicate that the elastic modulus increases more before few weeks after birth than in the following months. None the less, due to limited data available, linear relationship is assumed.
As Coats and Margulies did not obtain parameters for the frontal bone and Kriewall did not obtain parameters for the occipital bone, the difference of their values of termed specimens (36 - 40 weeks gestation) from parietal bone was measured. It was found that their values differed by a factor of 2 for specimens taken perpendicular to the fiber direction. This value was used to scale down the elastic modulus of the frontal bone found by Kriewall for its value to be in compliance with the values obtained from Coats and Margulies. Assuming linear relation between age and elastic modulus, shown in Figure 10 and Figure 11, the elastic modulus
y = 8.48x + 433.3
0
200
400
600
800
1000
1200
1400
-05 00 05 10 15
Ela
stic
mo
du
lus
(MP
a)
Age in months
Parietal bone
E_perpendicular
y = 18.20x + 287.6
0
200
400
600
800
1000
1200
1400
-05 00 05 10 15
Ela
stic
mo
du
lus
(MP
a)
Age in months
Occipital bone
E_perpendicular
Chapter 2
17
obtained for parietal, occipital and frontal bone, for the 2 month old infant, are presented in Table 1.
Table 1: Elastic modulus for parietal, occipital and frontal bone both for parallel and perpendicular fiber
direction. Data is both obtained from [5] and [12] and estimated according to obtained ratios explained in the
text.
E_perpendicular (MPa)
E_parallel (MPa)
Parietal bone 450 1887
Occipital bone 324 1358
Frontal bone 1092 1929
Ultimate stress for parietal and occipital bone was obtained and plotted from data presented in [12]. From the graphs in Figure 12 linear relation may be assumed between elastic modulus and ultimate stress.
Figure 12: Ultimate stress vs. elastic modulus for parietal bone (left) and occipital bone (right). Data obtained
from [12].
The ultimate stress values for the 2 month old infant skull bones are listed in Table 2. The ultimate stress for the frontal bone is estimated from the linear relationship presented for the parietal bone in Figure 12 (left).
Table 2: Ultimate stress for parietal, occipital and frontal bone both for parallel and perpendicular fiber
direction. Data is both obtained from [12] and estimated according to obtained ratios explained in text
Ultimate stress_perp-endicular (MPa)
Ultimate stress_parallel (MPa)
Parietal bone 30 112 Occipital bone 15 45 Frontal bone 66 114
y = 0.06x + 3.82
0102030405060708090
0 500 1000 1500Ult
ima
te s
tre
ss M
Pa
Elastic modulus (MPa)
Parietal bone
E_perpendicular
y = 0.029x + 5.27
0
10
20
30
40
50
0 500 1000 1500Ult
ima
te s
tre
ss M
Pa
Elastic Modulus (MPa)
Occipital boneE_perpendicular
Chapter 2
18
The Poisson’s ratio is not known for infant skull bones and is therefore estimated to be the same as for an adult skull bone in radial compression with 19.012 =ν and in transverse compression
22.023 =ν [30].
The elastic modulus of the sutures is obtained from [12] and [14] as an average value for infants less the one years old, MPaEsuture 1.8= . The material is assumed to be nearly incompressible
with Poisson’s ratio 49.0=sutureν and density equal to that of dura mater,3
1130m
kgsuture =ρ [14]
[31].
2.2.2 Parameters estimations
Some parameters of the skull bones cannot be obtained from literature and need to be estimated. The parameters 1221 ,, νEE and 23ν are obtained from literature. Assumption is made that 32 EE = ,
3223 νν = and 3121 νν = due to the transverse isotropic symmetry plane. The Poisson’s ratio 21ν
is found using equation (15) and the shear modulus 23G is found using equation (14). Huber
[32] proposed that the shear modulus for an in-plane orthotropic material could be predicted by equation (23).
( )2112
2112
12 νν+=
EEG (23)
Craig and Summerscale [33] predicted from equation (23) that the bulk modulus could be estimated for a square symmetric material using equation (24).
( )3233112
3321
213 ννν+=
EEEK (24)
For transversely isotropic material the elastic modulus is equal within the plane of symmetry and the Poisson’s ratio 3121 νν = , thus equation (24) becomes:
( )3232112
3 221
213 ννν+=
EEK (25)
The values for the shear strength are not available and therefore need to be estimated according to best knowledge. One way is to scale the shear strengths according to the ratio of the shear modulus for a certain direction to the ratio of the elastic modulus and strength for one of the uniaxial directions. The shear strength is assumed to be symmetric. This would result in the estimation equation (26).
i
ultijultijij E
GGSσε ≈≈ (26)
The transverse compressive strength is estimated by scaling the transverse tensile strength according to ratio found between compact femoral shaft of transverse compressive strength and transverse tensile strength [29] (Figure 9). The bone should be stronger in compression which is in compliance with the obtained value. All the parameters needed for the constitutive model
Chapter 2
19
which is used to predict failure of the skull (Chapter 4.2.2), in the FE program LS-Dyna, are listed in Table 3.
Table 3: Parameters for constitutive model of skull bone fracture analysis for 2 month old infant
Abbrevi-ation
Parietal bone: 2 month
old
Occipital bone: 2 month
old
Frontal bone:
2 month
old
Unit
References, formulas and estimations
1E 1887 1358 1929 MPa
Elastic modulus parallel estimated from linear relation of parallel and perpendicular data from Kriewall and perpendicular data from Coats & Margulies 2006, [12], see Figure 10
2E 450 324 1092 MPa Elastic modulus perpendicular. Coats & Margulies 2006, [12], see Figure 10
12ν 0.19 0.19 0.19 − Poisson’s ratio. McElhaney et al, [30], from adult human cranial bone
21ν 0.045 0.045 0.11 − Equation (15)
23ν 0.22 0.22 0.22 − McElhaney et al, [30], from adult human cranial bone
23G 184.4 132.8 447.5 MPa Shear modulus transverse to plane of symmetry, equation (14)
12G 421.6 303.5 634.9 MPa Shear modulus in plane of symmetry is estimated using equation (23) , [34] [34].
ρ 2090 2090 2090 3/ mkg Density, Kriewall, [8], from infant of 42 week of gestational age
1S 111.9 44.5 114.4 MPa Longitudinal tensile strength, Coats & Margulies 2006, [12]. See Table 2
2S 29.0 14.6 66.4 MPa Transverse tensile strength, Coats & Margulies 2006, [12]. See Table 2
12S 25.0 9.9 37.7 MPa Shear strength. Estimated using equation (26)
2C 38.7 19.5 88.5 MPa
Transverse compressive strength, Scaling from “Frankel VH, Nordin M: Basic Biomechanics of the Skeletal System. Philadelphia, Lea & Febiger, 1980”
NS 29.0 14.6 66.4 MPa Normal shear strength, assumed to be equal to transverse tensile strength
23S 11.9 6 27.2 MPa Transverse shear strength, Estimated using equation (26)
31S 25.0 10 37.7 MPa Shear strength. Estimated using equation (26)
K 321.4 231.4 656.8 MPa Bulk modulus. Estimated using equation (25)
Chapter 3
20
Chapter 3
Creating Finite Element mesh from
CT data The infant head model, consisting of skull, suture, brain and scalp, is constructed from CT images of a 2 month old infant. For a realistic simulation to be carried out the scalp needs to be included as it may increase the contact area of head to plate [14]. The emphasis of the image processing is on obtaining relatively accurate and good representation of skull and sutures and approximate geometry of the scalp and brain. The chapter covers basic introduction to CT imaging and then goes through each step of the image segmentation process up to the point where a working model has been obtained. After a good representation of the head has been established, the model is meshed for it to be used in the FE simulation.
3.1 Introduction to computed tomography imaging
Computed tomography (CT) is an imaging technique which uses X-rays in the production of cross-sectional images of an object. The röntgen ray was discovered by Wilhelm Konrad Röntgen in 1895. A method to reconstruct the projection was developed by Johann Radon in 1917 and in 1972 the first CT scanner was developed by Godfrey N. Hounsfield and Dr. Allan M. Cormack [35]. The initial intensity of photons, I0, generated by the thin X-ray beam, builds up each cross sectional image by scanning the total field of view (FOV) by a set of lines from a number of angels and distances from the center. This is done by using an x-ray tube which emits radiation transverse through the patient
while rotating helically around him. The detector on the other side of the tube measures the residual radiation, I t. The linear attenuation coefficient µ is a function of both the photon
energy used during the scanning and the specific material in each pixel. For a homogeneous material and an x-ray beam consisting of single energy photons the residual energy is given by equation (27):
Figure 13: CT scanner rotating helically
around the object
Chapter 3
21
xt eII ∆−= µ
0 (27)
where ∆x is the length of the X-ray path through the material. The attenuation coefficient, µ ,
decreases, usually, with increased photon energy. The attenuation coefficient is used to calculate the Hounsfield unit (HU) of each pixel in the 2D slice of the FOV.
10002
2 ⋅−
=OH
OHHU
µµµ
(28)
where OH2µ is the attenuation coefficient of water. The HUs, representing the CT image, are in
the range of −1000 to 3095 HU, where -1000 HU is air and 0 HU is water. The HUs are a gray scale values with a range too wide for the eye to distinguish between its values in one window. Thus when the image is viewed in an imaging program, only a certain window level is selected by the observer depending on the HUs of the object of interest. All HUs below and above this range are set to black and white respectively (Figure 14 and Figure 15). Table 4 lists some relevant HU values for this project.
Figure 14: Window level setting adjusted for
visualization of bones
Figure 15: Window level setting adjusted for
visualization of brain
3.2 Image segmentation
The CT image data set used in this project is of a 2 month old infant head with image resolution of 512x512x327 pixels. The size of the pixels are 0.4063 mm in x and y direction and slice thickness 0.5 mm. The images were taken by Siemens Definition scanner with attenuation strength 120 KeVand tube current 165 mAs. The image data is compiled in a DICOM format which stands for Digital Imaging and Communications in Medicine, and is a format used worldwide to represent medical images and its information. The data sets are segmented and further processed in two medical imaging programs, 3D Slicer and DeVIDE, to obtain the geometry of skull, sutures, brain and scalp. 3D slicer is an open software for visualization and image analysis, allowing number of imaging formats for both importing and exporting data [36]. DeVIDE (The Delft Visualisation and Image processing Development Environment) is an open software under constant improvements. It offers pre-
Chapter 3
22
programmed modules for image processing but also has the option of implementation of new methods using Python or C++ language [37]. 3D slicer needs to run on a computer that has large RAM (4 GB or more) and a fast graphic card [36]. The same applies when working with large datasets in DeVIDE. It is important to find a proper segmentation method for the task at hand as it can be difficult to obtain an optimally segmented region with spatially accurate boundaries but still smooth enough to represent the correct morphology as noise and nearby objects may interfere. A range of segmentation methods exist such as pixel, edge and region based methods. Pixel-based segmentation is based on the gray value or HU scale which can be selectively chosen within some threshold interval. An edge-based method uses edges of an object detected by local discontinuities to separate the image into regions. A region-based method constructs homogeneous regions propagating from the origin of seed points placed by the user at regions of interest (ROI) [38].
3.2.1 3D Slicer
The dataset in DICOM format is imported into the 3D slicer where the sagittal, coronal and axial planes can be viewed (Figure 16). A pixel based method was chosen to obtain the model of the skull, sutures, brain and scalp using a threshold level to select the HU range which represents each part. The HU values used are based on values from literature (Table 4) as well as visual evaluation.
Figure 16: Window view in the 3D slicer environment. Axial, sagittal and coronal planes are shown in the three
vertical images on the right.
Chapter 3
23
Table 4: Table with HU of materials needed for segmentation of the infant head
Material HU Reference
calcium 100–300 [39]
bone 600 - 3071 [39]
Brain (white and gray matter)
15–35 (125keV), 30–40
[40] [39]
Sagittal sinus 50–68 (120keV) [41] [42]
Scalp (fat and soft tissue)
-200 – -30
-30 – 10
[43]
Collagen 250 [44]
The infant skull bone is compliant and contains both calcium and collagen. From literature, calcium has been found to be in the range of 100 to 300 HU, collagen around 250 HU and the lower threshold of an adult skull bone around 600 HU. Guided by the literature and visual evaluation, the infant skull bone was found to be within a threshold with a lower limit of 130 HU and an upper limit of 3071 HU, which is the maximum HU for this dataset (Figure 17). The lower jaw of the viscerocranium was separated from the skull by manual editing as it is not needed for the skull model (Figure 18).
Fat and soft tissue of a healthy adult head has been found in literature to be -100 to -30 HU and -30 to 10 HU respectively [43]. By visual evaluation, the range for an infant scalp, consisting of fat and soft tissue, was found to be in the range -100 to 15 HU (Figure 19).
Figure 17: Sagittal view with
thresholded skull area indicated by
color.
Figure 18: Coronal view of the jaw,
in pink, was separated from the
skull, yellow.
Figure 19: Fatty and soft tissue of
an infant scalp is shown in brown.
The white and gray matters of the brain are in between 15-35 HU (Figure 20) and 30-40 HU (Figure 21) respectively [40] [39]. For the FE model, a continuous area is needed for representation of the geometry of the brain. The higher threshold level was increased up to 63 HU, such that it would include vasculature giving a solid volume for the brain (Figure 22). There is still a missing gap between the brain and skull where the superior sagittal sinus lies
Chapter 3
24
(Figure 23). It was not possible to use thresholding or region growing methods for the sinus area as it is not only the sinus that is located in the gap but also partly dura mater and falx cerebri (Figure 6). These areas were therefore manually edited by going through every slice and selecting the regions missing for the completion of a solid brain part within the skull (Figure 24).
Figure 20: White matter of 15-35
HU
Figure 21: Gray matter of 35-40 HU
Figure 22: Continuous area of 15-
63 HU representing brain for the
FE model
Figure 23: Axial view of dorsal (top)
part of skull and brain showing also
the superior sagittal sinus crossing
the two hemispheres.
Figure 24: Axial view of solid brain
representation after manual editing
has been applied to include the
sagittal sinus.
Figure 25: Brain before and after
addition of sinus (left and right
respectively).
The sutures were the most difficult part to segment. No previously published HU values were found for this membranous tissue built up of collagen fibers. The connection of the sutures to the skull is also in continuous contact with the dura mater and periosteum (Figure 6) making it more difficult to obtain only the geometry for the sutures when segmented. By knowing the location of the sutures and fontanelles, which were easily visible after the segmentation of the skull (Figure 4), it was assumed that the HU range was somewhere in between 63 and 129 HU (Figure 26). This range of HU also captures some of the dura mater and periosteum which surrounds the inner (concave) and outer (convex) sides of the skull respectively as well as cartilage of the ears (Figure 27). Narrowing this HU interval resulted in a large discontinuity of the supposed suture material. Manual editing was applied to disconnect some well-connected and unwanted parts to the sutures. Further processing was carried out in DeVIDE.
Mask images of each part were saved in compressed Meta format as .mhd file and .zraw file. The .mhd file is a header containing information such as dimension and element size of the image data. The .zraw file contains compressed binaries of the image.
Chapter 3
25
Figure 26: Membranous tissue in the
range of 63-129 HU. Left gap is part
of metopic suture and right gap part
of coronal suture.
Figure 27: Top view of skull and
membranous tissue.
Figure 28: Sagittal view of skull,
membranous tissue and scalp.
Figure 29: Detailed view from Figure 24
showing where membranous tissue
covers the anterior fontanelle.
3.2.2 DeVIDE
The network in Figure 30 takes as input the geometry of the sutures and the network in Figure 31 takes as input the geometries of the brain, skull and scalp one at a time. All the image data is in compressed Meta format. A double threshold is used to isolate the masks of interest and set the values within the threshold range equal to 1 and the values outside the threshold equal to 0. The network in Figure 30 was constructed to continue the process of isolating the sutures from the dura mater and other tissues with the same HU values. The module fast marching is used to obtain the morphology of the sutures by selecting only pixels with continuous connection to the seed points place at ROI (Figure 33). Both networks use closing module with convolution kernel first with dilation followed by erosion. This closing filter is used to connect voxels of small gaps and discontinuities in the geometries. Figure 33 shows the geometry of the sutures obtained from the 3D slicer, Figure 34 shows the sutures after seed connectivity (fast marching) and Figure 35 shows the sutures after closing has been applied. Figure 36 shows the difference between original and final sutures volume. The large holes within the solid parts, which may still exist, are filled with the function imagefillholes (network in Figure 31) which checks if image regions of value zero have surrounding regions of value 1. In that case it treats it as a hole and changes its value from zero to 1. Mask of the brain with holes and filled holes is shown in Figure 37 and Figure 38 respectively. The same is done for both skull and scalp.
Chapter 3
26
Figure 30: Network for isolation of
sutures
Figure 31: Closing and
filling of holes for skull,
brain and scalp
Figure 32: Image mathematics performed
to prevent penetration of objects
Figure 33: Sutures from 3D slicer
Figure 34: Sutures after fast
marching
Figure 35: Sutures after closing
Figure 36: Original sutures are
shown in green and added voxels
in blue.
Figure 37: Brain (without sinus)
containing holes in volume.
Figure 38: Brain (without sinus) after
imagefillholes.
To prevent intersection of two objects a mathematical operation is performed for the sutures, skull and brain. The network in Figure 32 starts by subtracting the sutures from the brain using the module imagemathematics. Both objects have set of mask values equal to 1 such that voxels of intersections are equal to 0, the relative complement of sutures in brain is equal to 1 and the relative complement of brain in sutures is equal to -1. All values equal to -1 are set to 0 with the second imagemathematics module and then the process is repeated for the brain and the skull. The sutures are also subtracted from the skull leaving the sutures with its original volume and
Chapter 3
27
the brain and the skull with reduced volume equivalent to the expansion of voxels of the sutures into areas of the other objects. The results of sutures, skull, brain and scalp are all written to separate uncompressed Meta format.
Figure 39: Final volume of the brain
Figure 40: Final volume of the skull
Figure 41: Final volumes of the
skull, brain and sutures together
3.2.3 Matlab
The Meta images are imported into Matlab with the function metaImageRead.m from the Matlab program Slicer©. The function takes in datasets in Meta format and exports it as 3D arrays of voxel classification. Sutures and skull are added together to form one continuous geometry consisting of two parts, such that one part has voxels of value equal to 1 representing its volume and the second part with voxels of value equal to 2. The three datasets, skull with sutures, brain and scalp, now as 3D matrices, are saved in separate MAT files with the addition of an information file containing the size of the matrix and a second file containing voxel size. The three files within each MAT file are the inputs for the Matlab program Dicer (see Chapter 3.3.1) which is used for the meshing of the volumes.
For statistical purpose the segmented volumes are mapped to the original image to evaluate the number of voxels lying outside of its original HU region. The volumes for each object may have increased after the image processing of manual editing, closing and imagefillholes. A Matlab function, mask_vs_HU.m, was written which maps the mask to the original image to obtain the number of added voxels to the object.
Table 5: Voxel increase in geometries after image processing
Volume Number of voxels in original image
Number of voxels in mask
Percentage increase
Skull 1424407 1449366 1.8
Brain 6326495 6449044 1.9
Sutures 31094 51946 67.1
Chapter 3
28
From Table 5 it can be seen that volume increase of the brain and skull is very small and is probably mostly due to small holes and discontinuities which have been filled and fixed. The increase of the volume of the sutures is larger than expected. This could be a result from manual editing and inclusion of extra voxels during closing and imagefillholes filters. The sutures had discontinuities and contained many holes in the original image data due to its thin structure. The discontinuities and holes are mostly due to the resolution of the image data. The resolution may cause the average HU values on edges of two aligned anatomical structures to become skewed. Some small area belonging to the thin structure of sutures may therefore have been assigned to voxels of surrounding structures during the acquisition of the image data. The large increase of number of voxels after processing of the sutures data is party necessary to obtain continuous geometry. Manual editing might also have to be done with more care.
3.3 Hexahedral meshing
Many programs exist for meshing, but a wide range of them were developed for meshing of CAD (computer aided design) objects and are not suitable for meshing of complex geometries obtained from medical images (MRI, CT, etc.). A better image based approach is to create the mesh from the voxels of the image data. The mesh is then composed of brick elements with the geometric accuracy depending on the resolution of the image data [45].
3.3.1 Dicer
It is a great challenge to obtain a quality mesh for a complex geometry like the brain, skull, sutures and scalp for an application of FE method. To mesh the geometries, voxel hexahedral meshing was used. The segmented CT images are imported to the program “Dicer” [45] having been processed by 3D Slicer, DeVIDE and metaImageRead.m, to get the image data to a 3D array of voxel classification. The Dicer algorithm is based on voxel hexahedral meshing where every voxel of the image data becomes a hexahedral element.
The program, Dicer was chosen for the meshing of the head as it provides automatic hexahedral meshing for complex geometry in relatively short time. It has the option of merging planes to get larger elements, instead of resampling the image data, and it has error correction and smoothing algorithms [45]. The head geometry contains large number of voxels, it is constructed of more than one part with each part highly detailed and it has very thin morphology in some areas.
CT images have stair-stepped surfaces and the step size depending on the image resolution. The surface quality can be improved by applying smoothing. The term ‘smoothing’ refers to the moving of nodes of the mesh, maintaining a fixed connectivity, in order to improve the mesh quality [46]. When smoothing is applied to areas of only a few pixels in thickness some elements can become distorted [45]. The Dicer addresses this problem by having the option of merging planes together, such that two or more elements become one, and such that the boundaries of those elements follow better the morphology of the original volume when
Chapter 3
29
smoothened. Dicer has the option of choosing the smoothing coefficients for all boundary nodes and corner nodes of a value between 0 and 1, 0 resulting in no smoothing and 1 resulting in collapse of all nodes to a single point in space. For value in between 0 and 1, the new location of a boundary node is found from weighting the average of its original location and the centers of the final position of neighboring boundary nodes. This smoothing coefficient has to be selected with care to prevent element distortion [45]. To further prevent distortion of elements all internal nodes (nodes not lying on boundaries) are smoothened after all boundary nodes have been fixed in space [45].
Merging of planes is supposed to preserve the morphological details of the model better then resampling the image data to obtain lower resolution and less number of elements in a hexahedral mesh [45]. Dicer has the option of choosing how many planes should merge for each direction.
An estimate of the number of elements in the final mesh after merging was calculated from equation (29).
∏=
≈
zyxii
originalfinal
n
meshmesh
,,
(29)
With originalmesh
as the number of voxels in the original mesh and in as number of merging
planes in directioni . This is only an estimation as the complexity of the geometry does not allow merging of some element planes and thus the number of final elements was found to be somewhat larger. For example for a cuboid with even number of elements in each direction, merging 2 planes together in all directions reduces the number of elements in the final mesh to 12.5 % of the original mesh.
The Dicer program is a non-commercial program created in Matlab. The code was provided to the department of STH from the author of the program. Some problems in the code were fixed in order to make the program work. The Dicer has the option of writing the output file in Abaqus and Patran format, but after some checking their codes were not functioning correctly. The Abaqus output writer was fixed as the Abaqus format can be opened by many programs, for example with Hypermesh which was used to view the results of meshing, while working on fixing the Dicer. A function was then written, lsdynawrite.m, as an addition to the Dicer which writes the output of the mesh to LS-Dyna .k format and can therefore be directly imported to LS-Dyna Prepost (see Chapter 4.).
3.3.2 Mapping
The program Dicer does not have the option of merging planes for an object with more than one part. For the simulation of head impacts one needs to include the brain for skull-brain and suture-brain contact for realistic fracture results from skull impact. The interest is in fracture of the skull and stress-strain propagation through each element of the skull bones but not of the brain. If the skull, sutures and brain are meshed with the minimum elements size, the number of elements goes over 7 million for the whole model which is computationally heavy and will take long time to simulate in LS-Dyna, if at all possible. The brain has more than four times as many
Chapter 3
30
voxels as the skull and for this reason the mesh size of the brain is increased to reduce the number of elements in the model. Both fine and coarse mesh are wanted for the skull and sutures. The skull and sutures are therefore merged as one solid to obtain the coarse mesh and then afterwards, sutures are mapped to the solid.
Figure 42: The format of output file from centercoordinate.m
∑=
===
8
1
1 n
ijij c
ncc , zyxj ,,= (30)
The process of meshing and mapping is described by the flowchart in Figure 44. By meshing an object in Dicer with an element size corresponding to the image resolution, one obtains a detailed mesh with each part identity (pid) correctly assigned to the right element. The same object, but now with all parts as one, is meshed again with a chosen number of merged planes in each direction and smoothing values for boundary edges and corners. Next step is to map the correct pid from the fine mesh to the coarse mesh. A function has been constructed, centercoordinate.m, as an extension to Dicer which finds the center node (equation (30)) of each element in the coarse mesh, specified with an element identity (eid), and writes it to a .m file (Figure 42). Another function has been constructed, nodecoordinate.m, which finds the coordinate of each of the eight nodes belonging to each eid of the fine mesh and writes it to a .m file for each pid separately (Figure 43). The mapping is carried out by checking if a point from the coarse mesh is within the edges of an element from the fine mesh. This is done for all center nodes and elements in the coarse and fine mesh respectively.
A program, mapping.m, was constructed having three inputs: a fine mesh file for a specific part of the whole object, the whole coarse mesh file and a part identity (pid) to be assigned to the coarse mesh if a point is within an element. The Matlab function inhull.m is called within the mapping.m which carries out the search of a point in an element. The reason for only taking in one part of the object at a time is to make each run in a manageable time and observe the result. When mapping is completed each element of a column containing part number for the coarse mesh file should have a pid. In the case of more than two parts (p=2) one can map p-1 materials and assign the elements with no pid to pid = p. The .k file containing elements and nodes of the coarse mesh can now be written. The pid column is saved and loaded into the function lsdynawrite.m and the meshing is run again with same parameters for plane merging and smoothing but now the correct pid is given. It is important to comment out the drawing feature in the function dicer.m as the drawing takes extremely long time and is not in any case usable as the .k file can be opened and viewed very easily in LS-Dyna prepost or Hypermesh.
Figure 43: The format of output file from nodecoordinate.m
Chapter 3
31
Figure 44: Flowchart of functions applied during mapping
Voxel image
with varying
pid
Voxel img to binary img
allclasses=allclasses > 0
Mesh with element
size=pixel size
Mesh with merged planes
in x,y,z direction
Get coordinates for each
node:
nodecoordinate.m
Get center coordinates for
each element:
centercoordinate.m
Mapping.m
Run dicer with same
merging and smoothing
but new pid
pid.mat
Name of part
and pid
number
save('pid.mat', 'pid')
where pid=coarse(:,2)
Mesh ready
for LS-dyna
Part id mapping from fine mesh to coarse mesh mesh
Chapter 3
32
Number of elements in the fine and coarse mesh are given in Table 6.
Table 6: Number of elements in the fine and coarse models. The scalp is not included as its volume is reduced, in
further processing of the model, to only cover the part of the skull that suffers the impact.
Fine model
(number of elements)
Coarse model with 2 plane merging in all directions
(number of elements)
Skull 1450035 221585
Sutures 49945 8940
Brain (coarse mesh only) 824636 824636
3.3.3 Mesh quality
Mesh quality is an important factor when generating meshes for finite element calculations in relation to both accuracy and efficiency of the simulation. The mesh quality was evaluated by means of determinant of the Jacobian, ( )Jdet , at each integration point (Gauss points) of an
hexahedral element. To prevent distorted elements the ( ) 0det >J . Hypermesh gives the ratio of
minimum and the maximum ( )Jdet such that if the ratio equals to 1 then the element is perfectly
shaped [47].
The skull with sutures and brain was meshed with a different number of element merging and element smoothing. As the number of merging planes is increased the quality of the boundary elements becomes worse. This is because Dicer applies uniform merging throughout the volume resulting in small elements on the boundaries which share nodes with large elements next to them. The boundary elements can therefore become skewed with very small and large angles (<10° and >170°, respectively). This can partly be fixed with smoothing resulting in elimination of some small boundary elements. As the skull and sutures have very thin structure, merging of more than 2 planes in every direction does not result in larger elements. The brain thickness allows for higher number of merging planes but merging of more than 2 planes in every direction, with varying smoothing values, does not result in good boundary element quality (Figure 45). Two models were constructed for the simulation of the head impact, one with no merging of elements for the skull and sutures and merging of 2 planes in all directions for the brain, and the second model with merging of 2 planes in all directions for the skull, sutures and brain. Figure 45 shows a plot of minimum ( )Jdet versus smoothing coefficient for the skull and
brain of the two models.
Chapter 3
33
Figure 45: Minimum det(J) analysis for skull with sutures and brain. Skull with merging of 0 and 2 planes in all directions and brain with merging of 2 planes in all direction is plotted.
Smoothing value of 0.3 for the brain and 0.7 for coarse and fine skull is chosen for the two models as smoothing is assumed to increase the quality of the stress distribution. The models are analyzed further visually and quantitatively to guarantee sufficient mesh quality. Figure 46 to Figure 49 shows distribution of ( )Jdet , volume, aspect ratio and minimum angle, respectively, of
the skull surface from the coarse model. The sutures have been removed from the cranium and some internal elements are therefore visible where the sutures connect to the skull. Even if the minimum ( )Jdet is very small for the merged model it is important that most elements have
( )Jdet > 0.3. The maximum volume of an internal element in the coarse model is equal to 3-10106.60 m⋅ , and minimum element on the boundary, without smoothing, is equal to 3111025.8 m−⋅ . Smoothing causes the volume range to be larger and smaller than the values
given (Figure 47) but most boundary elements take a value around 310104 m−⋅ . The aspect ratio is calculated as the ratio of longest edge length versus shortest edge length of an element. For a perfectly shaped equilateral hexahedral element the aspect ratio is 1 (Figure 48). The skewness of the element can be calculated as:
°−°
°°−
90
90,
90
90max minmax θθ
(31)
with maxθ the maximum angle in an element and minθ the minimum angle. A value > 0.85 for a
hexahedral element results in a poor quality [48]. The maximum angle for the coarse mesh is around 170° and the minimum angle around 27° (Figure 49), resulting in few boundary elements having very poor quality elements. These few elements may give rise to inaccurate solution and slow convergence.
0,00
0,20
0,40
0,60
0,80
1,00
0 0,2 0,4 0,6 0,8 1
Min
imu
m J
aco
bia
n
Smoothing
Smooting vs Min det(J)
Brain merging(2 2 2)
Skull (merging 0 0 0)
Skull (merging 2 2 2)
Chapter 3
34
Figure 46: Coarse skull. Distribution of det(J) of the
skull and sutures in Hypermesh. Minimum det(J) is
8.92e-02 for the coarse mesh.
Figure 47: Coarse skull. Max element volume after
merging without smoothing should be 6.60e-010, and
minimum element volume 8.250e-011. Ununiform
merging and smoothing causes the range of element
size to be larger and smaller than the values given. The
fringe level represents volume (m3)
Figure 48: Aspect ratio of longest edge length versus
shortest edge length of an element. The fringe level
represents the lengths ratio.
Figure 49: Minimum angle. The fringe level represents
degrees ( ° )
Chapter 4
35
Chapter 4
Finite element Modeling LS-Dyna is a non-linear 3D simulation program that builds on the FE method for solving large deformations to static and dynamic problems. The input file (k-file) to the program is built up in a keyword format containing the specific data for each model part in the simulation.
The chapter goes through the governing equation of the FE method used in LS-Dyna and the setup of the k-file for the head model in LS-Prepost prior to simulation. LS-Prepost is a program for pre-processing the model by constructing the k-file and post-processing of the simulation results.
4.1 Finite element simulation in LS-Dyna
The purpose of using three-dimensional elements with FE modeling on the present complex structures is to give a possibility to investigate the detailed scenario of skull geometry when undergoing impact load. By using FE modeling the deformation of the patient specific model can be examined. For the present study 8-node hexahedral elements are used. A shape function is constructed by decomposing the domain into elements and then use simple piecewise polynomials for each element to obtain the approximation solution. By reducing the size of the elements the approximate solutions approaches the exact solution.
The LS-Dyna Theory manual [49] is used as guidance as well as [50], [51], [52], [53] for the FEM derivation in chapters 4.1.1 to 4.1.6.
4.1.1 Governing equation
An impact to the head can cause deformation of the skull which may result in material failure. To obtain this failure mechanism the head impact is simulated in LS-Dyna using the FEM. The program is often used for crash simulation as it has contact algorithm that allows gaps and sliding with friction between separately meshed objects. The kinematic behavior of the contact problem is computed by solving the momentum of the system known as the governing equation (32).
Chapter 4
36
( ) 0=+−− iiij afdiv ρρσ
(32)
with σ as the Cauchy stress tensor with components ijσ , ρ the density and if , ia , iu the
components of the body force vector per unit mass, the acceleration vector, and the displacement vector respectively, where ( )zyxji ,,, ∈ .
The boundary conditions are:
( )ttn iiij =⋅σ on tractionΓ (33)
( )txuu ii ,= on ntdisplacemeΓ
(34)
( ) 0=⋅− −+iijij nσσ on interiorΓ (35)
where interiorntdisplacemetractionV Γ∪Γ∪Γ=∂ . in is a unit outward normal of a boundary element
and it the components of the traction force vector at time t. The interior boundary conditions
assume that the contact discontinuities are equal to zero.
The weak formulation is obtained by multiplying equation (32) by a test function N and integrating over the volumeV :
( ) 0=+−⋅∇− ∫∫∫VVV
aNdVfNdVdVN ρρσ
(36)
Application of divergence theorem gives:
( ) ( ) 0
31
=Γ⋅−−Γ⋅−+−∇⋅ ∫∫∫∫∫Γ
−+
Γ
nNdNdnaNdVfNdVdVN
VVV
σσσρρσ
(37)
and implementation of the boundary conditions gives the weak formulation:
0
1
=Γ−+−∇⋅ ∫∫∫∫Γ
tNdaNdVfNdVdVN
VVV
ρρσ (38)
with 1Τ the traction boundaries and 3Τ the interior boundaries.
4.1.2 Discretization: Hexahedral element and its shape function
When constructing a FE problem one starts with choosing the element form for the problem in mind. 2D domains are often meshed with triangle or quadrilateral elements containing 3 and 4 nodes respectively and 3D domains with tetrahedron, wedge or hexahedral elements containing 4, 6 and 8 nodes respectively. Hexahedral elements are often chosen when modeling 3D solid geometry. An eight node hexahedral is a
Figure 50: Hexahedral element and it coordinate
system
Chapter 4
37
parallelepiped element often referred to as brick in the finite element framework. While the domain has x, y, z coordinate system the hexahedral element has ξ, µ, η coordinate system ranging from -1 to 1. The eight corner nodes, n, are numbered counter-clockwise first in the transverse face of an initial node and then in the parallel face as seen in Figure 50.
The element geometry can be described in matrix form:
=
8
2
1
821
821
821
1111
N
N
N
zzz
yyy
xxx
z
y
x
M
L
L
L
L
(39)
where jN , 8,,2,1 L=j is the shape function for a hexahedral element e of node j . The shape
functions are defined as first degree polynomials:
( )( )( )jjjejN ηηµµξξ +++= 111
8
1 (40)
with jjj ηµξ ,, taking the coordinate values of nodes 8,,2,1 L=j for element e:
Node 1 2 3 4 5 6 7 8
ξ -1 +1 +1 -1 -1 +1 +1 -1
µ -1 -1 +1 +1 -1 -1 +1 +1
η -1 -1 -1 -1 +1 +1 +1 +1
The displacement, u, of each element in the global coordinate system, x,y,z, is calculated from
the displacement within the local coordinate system jjj ηµξ ,, .The displacement interpolation
for the three global components are given in equation (41).
=
8
2
1
821
821
821
N
N
N
uuu
uuu
uuu
u
u
u
zzz
yyy
xxx
z
y
x
ML
L
L
(41)
The index notation is given in equation (42)
( )∑=
=8
1
,,j
jj Nuu ηµξ
(42)
The total nodal displacement vector eu of an element is:
[ ]8,882,22111 ,,,,,,, zyxzyxzyxeT uuuuuuuuuu K= (43)
Next step is to introduce a basis function as a set of shape functions from equation (44) and set
the test function to be the same as the basis function iN . By implementing them into the weak
formulation of equation (38) the approximate solution given in equation (45) is obtained.
Chapter 4
38
( )∑=
=8
1
,,j
jj Naa ηµξ (44)
where ji , are the node numbers.
0
1
8
1
=Γ−−∇⋅+ ∫∫∫∫∑Γ=
dtNdVfNdVNdVNNa i
V
ii
VV
ijj
j ρσρ
( )8,...,1=i
(45)
From equation (45) the discretization of the mass matrix, internal nodal forces vector and external nodal load vector are obtained:
∑∫∫=
==el
k
n
ke
eT
V
T NdVNNdVNM1
ρρ (46)
∑∫∫=
∇=∇=el
k
n
ke
eT
V
Tinternal dVNdVNF
1
σσ
(47)
∑∫∑∫∫∫==Γ
Γ+=Γ+=bel
k
el
k
n
kbe
beT
n
ke
eTT
V
Texternal tdNfdVNtdNfdVNF
11
ρ
(48)
Internal and boundary elements of the volume are given by equation (49)
Ueln
kke eV
1=
= and U
beln
kke be
1=
=Γ (49)
with eln as number of elements e, and beln as number of boundary elements be.
4.1.3 Time integration
The main solution method in LS-Dyna is based on an explicit time integration, by dividing the problem into number of time steps. The explicit time step is usually around 1 microsecond and the CPU time can therefore be large for domains with high number of elements.
From equation (45)-(48) the time integration can be derived from the equation of motion. The acceleration for every node i is found from the nodal vector forces:
( )ninternal
nexternalii
n FFMa −= −1 (50)
In order to have a fully explicit scheme the mass matrix, iiM , has to be lumped, i.e. to assume
that the distributed mass of the element is modeled as lumps at the nodes of the element. By using the central difference time integration, the velocity and displacement for the next time step are obtained:
Chapter 4
39
n
nnn
t
vva
∆−=
−+ 2121 (51)
21
121
+
++
∆−=
n
nnn
t
uuv
(52)
∆+∆+=+
++2
1211
nnnnn tt
vuu
(53)
with v as the global velocity vector, u as the global displacement vector and nt∆ the time step at
time n . The geometry of each element is then updated by adding the total displacement to the
initial global coordinates( ) Xzyx ∈,, :
101 ++ += nn uXX (54)
The simulation time interval is determined by selecting some termination time depending on the
specificity of the problem. The critical time stepet∆ of the simulation is computed by the time it
takes sound to pass through the smallest element in the model. Its value depends on the material properties as well as the element size.
c
Lt ee =∆ (55)
eL for solid elements is equal to the volume divided by the largest face area. The speed of
sound, c, for elastic material with constant bulk modulus is given by:
( )( )( )ρνν
ν211
1
−+−= E
c (56)
The explicit time integration is conditionally stable such that the time step has to be small enough for stability to hold.
4.1.4 Matrix calculation
The next step after discretization is to solve the relationship between nodal displacement and nodal force. The nodal forces are the external force propagation through each node in an element. After the element displacement has been found one can calculate the stresses and strains of the element from the stiffness matrix.
∑∫ ∑∫∑∫∑∫= ===
Γ++−=el
k
bel
k
el
k
el
k
n
ke
n
kbe
beT
n
ke
eT
n
ke
eT
eT tdNfdVNdVBNadVN
1 111
ρσρ
(57)
where a is the 24x1 nodal acceleration vector, the matrix B is the 6x24 strain displacement matrix and σ the 6x1 stress vector, f the 3x1 force load vector, t the 3x1 applied traction load
vector and N the 3x24 interpolation matrix.
The strain displacement matrix B is the derivatives of the interpolation matrix:
Chapter 4
40
( )ηµξ ,,
0
0
0
00
00
00
N
xz
yz
xy
z
y
x
NB
∂∂
∂∂
∂∂
∂∂∂∂
∂∂
∂∂
∂∂
∂∂
=∇= (58)
∂∂∂
∂∂
∂
z
Ny
Nx
N
i
i
i
(59)
To obtain the components in equation (59) needed to solve for B the Jacobian matrix, J , is calculated:
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
ηηη
µµµ
ξξξ
zyx
zyx
zyx
J (60)
and its inverse , 1−J , is used in equation (61) to obtain the components in equation (59).
∂∂∂
∂∂
∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
∂∂∂
∂∂
∂
η
µ
ξ
ηµξ
ηµξ
ηµξ
ei
ei
ei
ei
ei
ei
N
N
N
zzz
yzy
xxx
z
N
y
Nx
N
(61)
The center matrix in equation (61) is the inverse Jacobian matrix, 1−J . From the strain
displacement matrix B and the 24x1 element nodal displacement vector eu the strain ε can be
calculated from Bu=ε . The constitutive equation εσ C= , for example for sutures and skull, relates stress and strain by the 6x6 constitutive matrix C . The constitutive matrix was derived in equation (4) for isotropic material and equation (11) for transversely isotropic material.
The stress can therefore be derived from the relation of the constitutive matrix, the strain matrix and the nodal displacement vector:
e
xy
xz
yz
zz
yy
xx
CBu=
σσσσσσ
(62)
Finally the relationship between the nodal force and element displacement can be established
using the 24x24 element stiffness matrix eK :
Chapter 4
41
eeeinternal uKF = (63)
∫=
eV
eTe CBdVBK
(64)
( )e
volumeextF and ( )e
boundaryextF are the element vectors of the nodal forces representing external
forces applied to the volume nodes (equation (65)) and boundary nodes (equation (66)) respectively:
( ) ∫=
eV
eeTevolumeext fNF ρ (65)
( ) ∫=
beV
eeTeboundaryext tNF
(66)
The mass matrix has to be diagonalized in order to obtain the lumped mass matrix. The diagonal mass matrix is obtained by summing over the rows of the shape function as they sum to unity:
∫∫ ∑ === V
Ti
V jj
Tiii dVNdVNNM ρρ
8
1
(67)
4.1.5 Volume integration
The volume integration is approximated in equation (69) using Gaussian quadrature and numerical integration.
( )∑∑∑= = =
p
i
p
jijkijkk
p
kji Jg
1 1 1
detωωω (68)
where kji ωωω ,, are the weight factors, ijkg a function which is defined over the volume and p
the number of integration points [49].
Implementing equation (68) into the element stiffness matrix, K , of equation (64) gives:
( )∑∑∑= = =
=p
i
p
jijkijk
Tijkk
p
kji
e JCBBK1 1 1
detωωω (69)
The number of Gaussian points used in the approximation is denoted with ip for the local
directions. ijkB is abbreviation for ( )kjiB ηµξ ,, , and the Jacobian matrix ijkJ is abbreviation for
( )kjiJ ηµξ ,, . For a non-negative volume, the determinant of the Jacobian has to be positive.
One point integration is most often used in LS-Dyna to approximate the volume of an element. The advantage of one point integration is less CPU time than for full integration [49]. In the one
point integration the element stiffness matrix eK is evaluated at the elements center with weight
Chapter 4
42
factor 2=== kji ωωω . Equation (70) approximates the elements stiffness matrix with one point
integration:
( ) ( ) ( )( )0,0,0det0,0,00,0,08 JCBBK Te = (70)
4.1.6 Hourglass control
The disadvantage of one Gaussian point integration is the need to use hourglass control to reduce or eliminate the zero energy modes. The zero energy modes may cause nonphysical deformation with zero strain and no stress. A possible hourglass deformation of hexahedral element is shown in Figure 51. Hourglass modes should be minimized to obtain realistic results. It can be reduced by a good quality mesh and also by reducing the size of the elements as load to single points in coarse meshes can initiate hourglass modes transferring it to neighboring elements. Energy is supplied to the system due to the force needed to resist the formation of hourglass modes. This energy should not exceed 10% of the internal energy for reliable results. For non-regular elements or simulations of impact with high velocity, resulting in large deformation, hourglass modes can be eliminated by switching to the use of full integration as it does not result in hourglassing. The disadvantage here is that CPU time increases and element locking may arise resulting in overly stiff response [49].
Figure 51: Possible deformation of a hexahedral element using one point integration. Image is adapted from [49].
4.2 Simulation in LS-Dyna
From Dicer the element nodes and the node coordinates of each model part are written to a k-file which is the input file format for LS-Dyna. The k-files for the skull and sutures together and brain, scalp and contact plate are imported to LS-Prepost for further presetting. The model parts are adjusted such as segmenting the whole skull-bone to frontal, parietal and occipital bones, and rotated for the right global location for the impact to the plate. Constitutive models are chosen for each part, material properties are set and location for the center of anisotropy for each bone is assigned.
Chapter 4
43
The k-file is organized in a special way demonstrated in Figure 52. Each node of an element has its node identity (NID) and global coordinates (x,y,z). An element has its unique identity (EID)
followed by a part number (PID), which the element belongs to, and then has its nodes iN
(with )8,...,2,1(∈i for hexahedral element). Each part in the model is assigned a part identity
number (PID), a section identity (SID), which defines if elements are for example solid or shell, a material identity (MID), which relates the constitutive models to the elements, and hourglass identity (HGID) (see Chapter 4.1.6).
Figure 52: Organization of the keyword file. Image inspired from LS-Dyna manual [49]
4.2.1 Model preparation
The model parts are imported to LS-Dyna Prepost one after the other using offset import for nodes and elements such that each has its unique number. Next step is to manually segment the skull into two parietal bones, two frontal bones and one occipital bone. This is necessary for the implementation of anisotropy of the skull. The facial bones and lower part of skull are not segmented to parts but form one solid together. Figure 53 show the cranium before segmentation and Figure 54 after segmentation. The scalp is also reduced only to cover the part of the impact as it is considered not to contribute any protection to the skull except at location of the impact. This also reduces the number of elements in the whole model.
Chapter 4
44
Figure 53: The cranium before segmentation
Figure 54: The cranium after segmentation of the
skull bones
The model has to be rotated and translated such that its position to the plate is relative for a specific impact location to the head. Two impact positions are simulated for the head. The plate is located in the x-y plane and the head is located such that displacement due to velocity in z-direction causes either an impact to the right parietal bone (Figure 55 and Figure 56) or an impact to the parietal-occipital bones (Figure 57 and Figure 58 ).
The initial velocity is obtained from the law of conservation of energy which says that for an object of any mass, the velocity at which it hits the ground is:
ghvz 2= (71)
where zg is the gravitational acceleration of earth equal to 2/81.9 sm and h is the height of the
fall. The head models are given an initial velocity relevant to a fall from 82.0 meter height
resulting in velocity smvz 0110.4= .
Chapter 4
45
Figure 55: Relative position of head to plate for
impact to the parietal bone
Figure 56: Same impact location as in Figure 55 but
different view
Figure 57: Relative position of head to plate for
impact to the parietal-occipital bones
Figure 58: Same impact location as in Figure 57 but
different view
4.2.2 Constitutive models
The probability of a body tissue to be injured is directly related to the magnitude and direction of the applied stress as well as the area over which the force is distributed. Compressive stress, tensile stress and shear stress indicate the direction of the acting stress. An object both accelerates and/or deforms when a force is applied to it. The amount of deformation depends both on the force and the material properties of the object it acts on [27]. To determine the stress distribution in the head due to impact, constitutive models have to be selected to represent the biomechanics for each part in the model.
It is of interest to obtain the impact respond such as fracture propagation of the skull. The frontal, parietal and occipital bones were modeled using a Chang-Chang Composite Failure Model (material type 22 in LS-Dyna [49]) which is used for composite material such as the one
Chapter 4
46
with orthotropic or transversely isotropic fiber direction. The fiber direction of each element in the skull bones is locally transversely isotropic with the material principal directions determined from a point, P, in space and the global coordinate center of each element. The location of P in each bone represents the center of the ossification. An image representation of this is shown in Figure 59. The principal directions are calculated as follows: for an element the direction d is parallel to the global direction z . The radial direction ξ is a vector from the point P to the
center of an element. The tangential and thickness directions are calculated from cross productd×=ξµ and ξµη ×= respectively.
Figure 59: A skull bone sample showing the element which represents the center of ossification and an arbitrary
element showing the principal material axis for transversely isotropic elements. The location of P in each bone
represents the center of the ossification. Image is adapted from LS-Dyna Keyword manual [49].
The fracture of each bone is determined from using the four failure criteria of the Chang-Chang model. The parameters needed for the failure criteria are longitudinal and transverse tensile strength, shear strength, transverse compressive strength, normal and transverse shear strength. A description of how the parameters were obtained is explained in chapter 2.2 and listed in Table 3 for the frontal, parietal and occipital bone. The four failure criteria are given in equation (72) to (75) . A cracking failure occurs in the matrix of the material when the parameter
1>matrixF in equation (72). In that case the constants 12,122, νGE and 21ν are set to zero.
2
12
122
2
2
+
=
SSFmatrix
σσ (72)
A compression failure occurs if 1>compF in equation (73). In that case the constants122,νE and
12ν are set to zero.
2
12
12
2
22
12
22
12
2 122
+
−
+
=
SCS
C
SFcomp
σσσ (73)
A failure due to fiber breakage occurs if 1>fiberF in equation (74). In that case the constants
121221 ,,, νGEE and 21ν are set to zero.
Chapter 4
47
2
12
122
1
1
+
=
SSF fiber
σσ (74)
A delamination failure occurs if 1>delF in equation (75) [54].
2
31
312
23
232
3
+
+
=
SSSF
ndel
σσσ (75)
where 1σ , 2σ and 3σ , are the stresses in the fiber direction, the transverse direction and the
transverse-thickness direction respectively. The shear stress 12σ is in the fiber and transverse
plane, the shear stress 23σ is in the transverse and transverse-thickness plane and the shear
stress 31σ is in the transverse-thickness and fiber plane. 1S is the tensile strengths in the fiber
direction, 2S is the tensile strengths in the transverse direction, 2C is the compressive strengths
in the transverse direction and nS is the tensile strengths in the normal or thickness direction.
12S is the shear strength in fiber and transverse plane, 23S is the shear strength in the transverse
and transverse-thickness plane and 31S is the shear strength in the transverse-thickness and fiber
plane. Schematic drawing of the failure criteria is shown in Figure 60.
Figure 60: Schematic figure of an element failure due to fiber failure, matrix cracking, compression failure or
delamination failure.
For the facial bones together with the lower part of the skull, and the sutures, isotropic elastic constitutive model (material type 1 in LS-Dyna [49]) is used. This model has no failure criteria as fracture propagation is not assumed to reach this part of the cranium. Material properties needed are elastic modulus, E , Poisson’s ratio, ν , and density, ρ . Parameters for the bone and
sutures are given in Table 7 and Table 8. The isotropic elastic modulus for the facial bones and lower part of the skull was found as an average of the transversely isotropic elastic moduli from Table 3.
Chapter 4
48
Table 7: Material properties for the facial bones and
lower part of skull
Elastic bone material
Units Reference
E 1100 MPa
ν 0.19 [30]
ρ 2090 3/ mkg [8]
Table 8: Material properties for sutures
bbbbbbbbbbbbbbbb
Elastic suture material
Units Reference
E 8.1 MPa [14]
ν 0.45 [14] [31]
ρ 1133 3/ mkg [14] [31]
The brain was not of interest in this thesis as it acts mostly as contact constraint in the fracture analysis of the skull. For the completeness of the material description for the head model the constitutive model used for the brain will only be briefly presented and its parameters. Material suggestion and parameters for the brain were obtained from Kleiven et al. in his study on traumatic brain injury [55]. The brain undergoes a large elastic deformation under load. The constitutive material which best fits the brain response, was chosen to be hyperelastic with hydrostatic work term added to it. The Odgen Rubber material model (material type 77 in LS-Dyna [49]) was used for the constitutive model. The model for the brain is explained in more details in the appendix. All parameters for the brain model are given in Table 9.
Table 9: Parameters of the brain for the Ogden hyperelastic and linear viscoelastic model obtained from Kleiven
et al. [55]
1µ 53.8 Pa 1G 320 kPa 1β 610 s/1
2µ -120 Pa 2G 78 kPa 2β 510 s/1
1α 10.1 3G 6.2 kPa 3β 410 s/1
2α -12.9 4G 8.0 kPa 4β 310 s/1
ρ 1040 3/ mkg 5G 0.10 kPa 5β 210 s/1
ν 0.4999 6G 3.0 kPa 6β 110 s/1
For the scalp the same constitutive model is used as for the brain. The parameters for the scalp, given in Table 10, are obtained from Kleiven et al. [56].
Table 10: Parameters of the scalp for the Ogden hyperelastic and linear viscoelastic model
ρ 1133 3/ mkg 1µ 1.1 MPa 1G 275 kPa 1β 0.471 s/1
ν 0.49 1α 9 2G 458 kPa 2β 0.1067 s/1
Chapter 4
49
The contact plate was modeled as rigid (material type 30 in LS-Dyna [49]) with constrains for translation and rotation in all global directions such that it is fixed in space. Density, Poisson’s ratio and elastic modulus are listed in Table 11 for the rigid plate.
Table 11: Parameters for rigid contact plate
ρ 2000 3/ mkg
ν 0.22
E 15 GPa
4.2.3 Contact constraints
Contacts between separately meshed objects have to be set as a substitution for boundary conditions. As gaps initially exist between the boundaries of side by side objects (Figure 62), constraints are needed to prevent penetration during motion. To establish a good contact between side by side objects, the gaps between their boundaries have to be within some distance limit over the whole surface and the surfaces need to follow similar geometry. The surfaces are categorized into master and slave sides, where the slave surface usually follows the master surface for most contact types. Two contact types, “Tied surface to surface” and “Automatic surface to surface”, are used in the FEA of the head [50] [49] [57].
Figure 61: Coronal view of the head model
Figure 62: Detailed view from Figure 61. Gap between
the brain, skull and scalp is easily visible.
The contact “Tied surface to surface” is used for two contacts, between the skull bones, sutures and brain and the skull bones, sutures and scalp. The decision of which part is master and which is slave usually depends on stiffness and mesh size. The master surface is preferred to be the stiffer one and more coarsely meshed if relevant. At the beginning of the simulation the slave nodes are projected orthogonally toward the master surface. In case a master surface is reached within a given distance from the slave node, the slave node is moved to the surface of the
Chapter 4
50
master. This may alter the geometry of the relevant slave element but does not induce any stress [50] [49] [57].
The contact “Automatic surface to surface” is a symmetric contact such that both master and slaves nodes are checked for penetration making the definition of master and slave surfaces arbitrary. Penetration check is performed at every time step. At the time where penetration occurs the contact constraints act by applying force in proportion to the penetration depth to correct the slave node location [50] [49] [57].
Chapter 5
51
Chapter 5
Simulation results Simulation of an impact to the parietal-occipital bones and of an impact to the right parietal bone was carried out in LS-Dyna with the goal to see the fracture propagation due to fiber breakage, cracking failure in the matrix or compression failure. The reliability of the model has to be valuated with respect to the energy dissipation as strange energy fluctuation in the hourglass and total energy indicate unrealistic behavior. Results from simulation of the coarse mesh for the parietal-occipital impact from 0.82 m height is presented as well as the parietal impact from 0.82 m height. Unfortunately simulation of the fine mesh failed before giving any usable results. Possible explanation why the fine model did not work will be discussed in chapter 6.
Validation of the models’ ability to show relatively correct fracture propagation is carried out by comparing the results from the parietal-occipital impact to Weber’s study done on fracture propagation of infant PMHS skulls from a free fall of 0.82 m height onto a stone tile [3] [58]. The results from the parietal impact are compared to previously constructed model by Kleiven et al. [59] who simulated an impact to the parietal bone of a 4 month old infant head from a fall of 1.1 m height in LS-Dyna.
Coats & Margulies presented force-time contact of a 1.5 month old infant skull model from a fall of 0.3 m height to the parietal-occipital area to a rigid plate [14]. For the validation of the force-time contact for the parietal-occipital impact of 0.82 m height the model was also simulated for 0.3 m height.
5.1 Parietal-Occipital impact
Impact to the parietal-occipital bones was simulated from a fall of 0.82 m height resulting in velocity equal to 4.0 m/s. Termination time was set to 10 milliseconds (ms) but 5 ms was found to be enough time to simulate the impact and therefore the results are presented from simulation of 5 ms. CPU time for the simulation of 5 msek was approximately 50 hours due to decreasing step size throughout the simulation. Long CPU time may have been due to decreasing step size throughout the simulation as a result of deformation of elements.
Chapter 5
52
Figure 63: Parietal-occipital
impact
Figure 64: Energy-time curves for the parietal-occipital impact of 0.82 m
height. Max internal energy is 4.75 J
Figure 64 represents the energy dissipation of the model during the 5 ms simulation. The total energy is relatively steady throughout the simulation. Most kinetic energy has transformed to internal energy at time 4.6 ms. Maximum impact force of 1325 N is reached at time 4.1 ms for the 0.82 m fall to the parietal-occipital bones (Figure 67). Minimum velocity is 0.24 m/s at time 4.6 ms (Figure 47).
Figure 65: Force time contact curve for the parietal-
occipital impact of 0.82 m height. Maximum force is
1325 N at time 4.1 ms
Figure 66: Velocity time curve from the parietal-
occipital impact of 0.82 m height. Minimum
velocity is 0.24 m/s at time 4.6 ms
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
1
2
3
4
5
6
7
Ene
rgy
Time (sek)
Parietal-Occipital impact, coarse mesh
Internal energy
Kinetic energy
Total energy
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
200
400
600
800
1000
1200
1400
Time (sek)
For
ce (
N)
Parietal-Occipital impact, coarse mesh
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sek)
Vel
ocity
(m
/s)
Parietal-Occipital impact, coarse mesh
Chapter 5
53
Figure 67: Beginning of matrix cracking at time
2.8 msek. Anterior view of the skull showing
the convex side of the occipital and parietal
bones
Figure 68:
Failure range
from elastic =
1 to complete
failure = 0
Figure 69: Matrix cracking at time 2.8 msek.
View is of the concave side of the occipital
bone (inside of the bone)
The three bones impacted in the fall to the parietal-occipital region, shown in Figure 67, are the two parietal bones and the occipital bone. Figure 67 and Figure 69 show the beginning of the matrix breakage in the occipital bone from the fall at time 2.8 ms. At this time the hourglass energy is 15% of the internal energy so results have to be evaluated further for proving of significance. The breakage pattern seen in Figure 67 persists throughout the rest of the simulation as seen in Figure 70 and Figure 73 at minimum velocity at time 4.6 ms. The fringe level given in Figure 68 represents the failure ratio calculated from equations (72) - (74), with 1 representing elasticity and 0 representing complete failure.
Figure 70: Matrix cracking at time 4.6 ms from the
parietal-occipital impact from 0.82 m height
Figure 71: Matrix cracking at time 4.6 ms. View is
of the concave side of the occipital bone (inside of
the bone)
Chapter 5
54
Figure 72: Compressive failure at time 4.6 ms from
the parietal-occipital impact from 0.82 m height
Figure 73: Compressive failure at time 4.6 ms.
View is of the concave side of the occipital bone
(inside of the bone)
The initial force to the head is distributed between both of the parietal bones and the occipital bone at time of impact. The displacement of the head after impact is such that it rotates towards the occipital bone increasing its probability to fracture. Figure 70 and Figure 71 show the possible fracture propagation due to matrix cracking. Matrix cracking of an element is a result of transverse stress, 2σ ,
exceeding the ultimate transverse strength 2S or transverse
shear stress, 12σ , exceeding the ultimate shear strength
12S or contribution of both, 2σ , and, 12σ ,(see equations
(72) - (74)). This is in compliance with the elasticity of the infant skull bones with fiber strength being weaker in the transverse direction than the longitudinal direction due to the existence of radial fiber pattern from the center of ossification. Figure 72 and Figure 73 show fracture locations due to compressive failure. Location of the compression is in compliance with the bending of the bone to the rigid plate, at minimum velocity. This can be visualized from the cross sectional view in Figure 74, showing relevant part of the occipital bone before the impact, and in Figure 75 showing the same view as in Figure 74 but at the minimum velocity. Fiber failure does not occur from the parietal-occipital impact.
Figure 74: Cross section of the occipital
bone and the scalp before impact to
the rigid plate
Figure 75: Cross section of compression
failure of the occipital bone due to
impact to the rigid plate at minimum
velocity, at time 4.6 ms
Chapter 5
55
A simulation of the head model from a fall of 0.3 m height was carried out and the maximum force obtained compared against the maximum force Coats & Margulies obtained in their simulation of a 1.5 month old infant head model from a fall of 0.3 m height [14]. For the parietal-occipital head model of the 2 month old infant the maximum impact force is 750 N at time 4.4 ms (Figure 76). Coats & Margulies obtained maximum force of 555 N at time 4.5 ms. The forces in the two models are of the same order with the maximum force of 195 N higher for the 2 month old model than the 1.5 month old model. The model of the 1.5 month old infant head is 0.87 kg and the model of the 2 month old is 0.85 kg. The
higher contact force of the model of 2 month old infant can be related to a higher elastic moduli of the skull bones, possibly increased thickness in the skull bones and smaller sutures width compared to the Coats & Margulies model. The scalp in the Coats and Margulies model was modeled as linear elastic with Poisson’s ratio equal to 0.42 [14]. The scalp in the model of the 2 month old infant head was modeled as viscoelastic with Poisson’s ratio equal to 0.49. Higher Poisson’s ratio may increase the stiffness of the model. The sutures width is known to decrease with age and the skull bones get thicker and elastic modulus gets larger [14] [12]. Large sutures width is known to decrease the contact force [14].
The fracture pattern seen in Figure 70 and Figure 71 is comparable to one of the fracture pattern Weber obtained in his study [3] of a 4 month old PMHS infant skull dropped to the parietal-occipital region from 0.82 m height to a stone tile. Figure 70 is presented again in Figure 77 with possible fracture pattern shown by black lines. Figure 78 shows a sketch of the top view of the head from Weber’s study, with the fracture pattern drawn at the occipital bone.
Figure 76: Force-time contact curve for parietal-occipital
impact of 0.3 m height. Maximum force is 750 N at time
4.4 ms.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
100
200
300
400
500
600
700
800
Time (sek)
Fo
rce
(N
)
Parietal-Occipital impact, coarse mesh
Chapter 5
56
Figure 77: Possible fracture propagation of the
occipital bone of a 2 month old infant head model
from a fall of 0.82 m height to a rigid plate. View is
from the back of the skull.
Figure 78: Sketch of a fracture pattern (red) in the
occipital bone of 4 month old PMHS infant skull
dropped to the parietal-occipital region from 0.82 m
height to a stone tile in study done by Weber [3]. The image is adapted from a drawing of the skull
fracture presented in [3].The view is from the top of
the skull.
5.2 Parietal impact
Impact to the right parietal bone was simulated from a fall of 0.82 m height resulting in velocity equal to 4.0 m/s. Termination time was set to 10 ms but 5.5 ms was found to be enough time to simulate the impact. CPU time for the simulation of time 5.5 ms was approximately 60 hours. Long CPU time may have been due to decreasing step size throughout the simulation as a result of deformation of elements.
Figure 79: Parietal impact
Figure 80: Plot of internal, kinetic and total energy as function of time
Internal, kinetic and total energy is plotted as a function of time in Figure 80. Most kinetic energy has transformed to internal energy between time 3.4 and 3.6 ms. The total energy for the simulation is steady until time 3 ms where it shows a jump which may be a result of increased hourglass energy due to largely deformed elements at the location of fracture. It may also indicate a model error, possibly in the definition of the contacts. The hourglass energy is within
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
1
2
3
4
5
6
7
8
Time (sek)
En
erg
y
Parietal impact, coarse mesh
Internal energy
Kinetic energy
Total energy
Chapter 5
57
16% of the internal energy until time 3.4 ms and from that point forward it increases further. When the cranium hits the plate it rotates such that the minimum velocity is not reached until time 4.8 ms (Figure 82). Maximum impact force of 1469 N is reached at time 3.4 ms for the 0.82 m fall to the right parietal bone.
Figure 81: Force-time contact curve for the parietal
impact of 0.82 m height. Maximum force is 1469 N at
time 3.4 ms
Figure 82: Velocity-time curve from the parietal
impact of 0.82 m height. Minimum velocity is 0.25
m/s at time 4.8 ms
Figure 83: Beginning of a fiber breakage at
time 3.4 ms. View is of the concave side of the
right parietal bone (inside of the bone).
Orientation of the bone is as follows: D stands
for distal, V for ventral, P for posterior and A
for anterior
Figure 84:
Failure range
Figure 85: Fiber breakage at time 3.6 ms.
View is of the concave side of the right
parietal bone (inside of the bone).
Orientation of the bone is as follows: D
stands for distal, V for ventral, P for posterior
and A for anterior
The fringe level given in Figure 68 represents the failure ratio calculated from equations (72) - (74), with 1 representing elasticity and 0 representing complete failure. Figure 83 and Figure 85 show the beginning of fiber breakage in the parietal bone from a fall of 0.82 m height. At this time the hourglass energy is about 16% of the internal energy so results have to be evaluated further for proving of significance. The fracture pattern seen in Figure 85 persists throughout the rest of the simulation as seen in the images Figure 86 to Figure 91 at time (4.8 ms) of minimum velocity.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
500
1000
1500
Time (sek)
For
ce (
N)
Parietal impact, coarse mesh
Contact force
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-3
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sek)
Vel
ocity
(m
/s)
Parietal impact, coarse mesh
D
A P
V
D
A P
V
Chapter 5
58
Figure 86: Fiber breakage at time 4.8 ms from the
parietal impact from 0.82 m height. Orientation of
the bone is as follows: D stands for distal, V for
ventral, P for posterior and A for anterior
Figure 87: Fiber breakage at time 4.8 ms. View is of
the concave side of the right parietal bone.
Orientation of the bone is as follows: D stands for
distal, V for ventral, P for posterior and A for anterior
Figure 88: Matrix cracking at time 4.8 ms from the
parietal impact from 0.82 m height
Figure 89: Matrix cracking at time 4.8 ms. View is of
the concave side of the right parietal bone
Figure 90: Compressive failure at time 4.8 ms from
the parietal impact from 0.82 m height
Figure 91: Compressive failure at time 4.8 ms. View is
of the concave side of the right parietal bone
D
A P
V
D
P A
V
Chapter 5
59
Figure 86 and Figure 87 show possible fracture propagation due to fiber breakage. Fiber breakage of an element is a result of longitudinal stress, 1σ , exceeding the ultimate longitudinal
strength 1S or transverse shear stress, 12σ , exceeding the ultimate shear strength 12S or
contribution of both, 1σ , and, 12σ ,(see equations (72) - (74)). Further analysis on 1σ shows that
the longitudinal stress, for few elements, reaches 97 MPa at time 3.4 msek of maximum force (Figure 92) and 100 MPa at time 3.6 msek (Figure 93). Fracture due to fiber breakage is therefore likely to happen from the impact as the ultimate strength 1S is equal to 112 MPa.
Matrix and compressive failures are only present in small area from the parietal impact from 0.82 m height as can be seen in Figure 88 to Figure 91.
Figure 92: Longitudinal stress, σ1 in Pa, of the
right parietal bone at time 3.4 ms. View is of
the convex side of the bone. Maximum stress
is 97 MPa in an internal element at the
center of the bone.
Figure 93: Longitudinal stress, σ1 in Pa, of
the right parietal bone at time 3.6 ms. View
is of the concave side of the bone.
Maximum stress is 104 MPa in an internal
element close to the center of the bone.
The results from the parietal impact are compared to previously constructed model by Kleiven et al [59] who simulated an impact to the parietal bone of a 4 month old infant head from a fall of 1.1 m high in LS-Dyna. Fractures in both the 2 and 4 month old infant head models have fracture due to fiber breakage. The fracture patterns of the models show some similarities but because the impact to the 4 month old infant head is from a higher fall and ultimate stresses assigned in the Chang-Chang model are lower than for the 2 month old model, its fracture pattern is more profound. A drawing of the possible fracture pattern of the 2 month old infant head model is presented in Figure 94 and of the 4 month old infant head model in Figure 95.
Chapter 5
60
Figure 94: Possible fracture pattern at the right
parietal bone drawn from Figure 86 of a 2 month old
infant head model from a fall of 0.82 m height to the
right parietal bone. Orientation of the bone is as
follows: D stands for distal, V for ventral, P for
posterior and A for anterior
Figure 95: Possible fracture pattern at the right
parietal bone of a 4 month old infant head model
from a fall of 1.1 m height to the right parietal bone.
Orientation of the bone is as follows: D stands for
distal, V for ventral, P for posterior and A for anterior
D
P A
V
D
P A
V
Chapter 6
61
Chapter 6
Discussion
6.1 Material properties
Few material properties exist for the infant skull bones in literature. Material properties for the skull and sutures, obtained from literature in this thesis, are exclusively from Kriewall [5] [8] and Coats and Margulies [12] [14]. Other parameters are obtained from mechanical equations and estimations. The amount of material parameters gathered in the present study, to predict fracture of the infant skull from an impact from a fall, has not been obtained before, to the best knowledge of author.
Coats and Margulies [14] developed a model of 1.5 month old infant head and simulated an impact to the parietal-occipital region from fall of 0.3 m and 0.82 m height. To predict the bone failure they used an average principal stress of 3x3 element array throughout their simulation to compare against ultimate stress obtained from their bending tests of infant skull bones with fiber direction perpendicular to the long axis. The ultimate stress used for validation, in their study, was set equal to 9.4 MPa for the occipital bone and 27 MPa for the parietal bone. Kleiven et al. [59] constructed a model of a 4 month old infant head from CT images and simulated an impact to the parietal bone of 1.1 m high fall in LS-Dyna. They used the Chang-Chang failure constitutive model in LS-Dyna to predict the fracture. For this they used the same ultimate stress parameter, for the occipital, parietal and frontal bones, equal to 23 MPa for both longitudinal and transverse direction with respect to the fiber direction. The present model uses different parameters for the longitudinal and transverse ultimate stress for the occipital, parietal and frontal bones. Other specific parameters needed for the Chang-Chang model, such as shear strengths and compressive strength are estimated for each of the skull bones. These parameters are presented in Table 3. The infant skull bone is assumed to have higher elastic modulus parallel to the fiber direction than perpendicular to it. It should therefore be evident to assign higher ultimate stress value with higher elastic modulus. The values for the ultimate stress for the parallel fiber direction are obtained from the linear relation of perpendicular elastic modulus and ultimate stress presented in Coats and Margulies [12] and presented in Figure 12 in this thesis. The accuracy of the parameters obtained is best validated in this thesis by looking at the reliability of the fracture propagation and the fracture pattern from the simulations.
Further testing needs to be done to validate all of the parameters obtained in Table 3 and used in the Chang-Chang failure model in LS-Dyna. This could be done by constructing a base-line model simulation and then vary one parameter at a time for a new simulation. Observation on
Chapter 6
62
the change in the propagation of the fracture and its pattern would then be used to validate the significance of each parameter.
6.2 Image segmentation
The geometries obtained from the CT images for the skull, brain with sinus, and scalp are a good representations of each anatomical part. The skull thickness is possibly over estimated in the sense that it includes voxels of very low HU as a part of the calcified bone material. For the sake of continuity of the geometry and reduction of holes, due to thin bone structure in some areas, this lower HU level is required. The method to obtain the sutures is mostly based on manual editing where each image slice is edited to obtain only voxels that most likely represent the sutures. As periosteum and dura mater surrounds the convex and concave side respectively of both the skull bones and sutures it is difficult not to include it in the sutures thickness. This results in overly estimated thickness of the sutures. However, Coats and Margulies concluded that it is not considered to significantly affect the prediction of the fracture of the skull bone from an impact [14]. The sutures width is possibly couple of voxels wider than its original geometry, as the dura mater surrounds the bone and sutures edges. For continuous sutures layer from the bones, the dura mater needs to be included. This should not significantly affect the skull bone response to impact but overly wide sutures may decrease the maximum stress [14].
The sutures geometry is obtained using thresholding based segmentation method with the threshold interval obtained from visual evaluation. The obtained HU range is wide and not accurate enough to predict some optimal HU value range for the sutures. Even if the results are visually correct some voxels may be included that do not belong to the sutures. Further manual editing, after thresholding, is performed to separate periosteum and dura matter from the sutures to only obtain the sutures’ morphology. The process takes also relatively long time and should be optimized further by either finding a better segmentation method for the sutures or automatize the editing process in some way.
6.3 Models mesh
The program Dicer is good for meshing hexahedral elements from voxel classification but the merging feature is faulty in the way that if the number of voxels are not uniform in all directions, voxels on the boundaries are left unchanged or are only merged in some of the directions. This results in skewed elements and increased aspect ratio of an element. Smoothing is applied to eliminate some of the small boundary elements but it does not improve the mesh quality substantially. The element sizes of the coarse mesh, presented in Figure 47, are non-uniform with volume of the smallest element equal to 3% of the largest element. Even if this extreme only applies to a few elements in the mesh, this causes the time step in the simulation to
become very small (around s7102.1 −⋅ ) and decrease with the deformation of the elements
throughout the simulation (around s8105 −⋅ ). This results in substantially increased simulation CPU time.
Chapter 6
63
The skull bones have a complex geometry with less than 2.5 mm thickness in some regions such that merging of more than 2 planes in all directions is not possible. Simulations were therefore only carried out with models of mesh size from original voxel size (fine mesh) and from merging of 2 planes in all directions (coarse mesh). A mesh convergence study was therefore not optional. In addition the fine mesh model simulation failed, resulting in early termination, which may have been due to element connectivity problems, small holes and single element thickness in the geometry. The advantage of the coarse mesh is that merging of the planes increases connectivity and closes small holes in the geometry.
Dicer had not been used for meshing at the department of STH before this thesis and no information existed of its quality other than those given in the paper [45]. The output from the Dicer gave a mesh with distorted elements, to begin with, and very long meshing time of few hours depending on the number of elements. Substantial time went into going through the code to figure out how to fix these problems. At the time a solution was obtained and mesh quality was evident, time restrictions prevented the author to change to another meshing method. The Dicer is still in general a good meshing program for meshing of volumes from voxel classification that have more uniform geometry even though, as an opinion from the thesis author, some further work needs to be done on the program for the sake of improving element quality.
6.4 Model simulation
The model constructed from the CT images contains a good representation of the geometry of a head of a 2 month old infant. The mesh quality is a matter of discussion and is best validated by looking at the reliability of the fracture propagation from the simulation. The fracture propagation in the occipital bone from the parietal-occipital bone impact is in good compliance to Weber’s study of the same impact location of a 4 month old PMHS infant skull [58]. The fracture occurs in the matrix which is in compliance with the elasticity strength of the infant skull bone where the material strength is weaker transverse to the fiber direction. The total energy is constant throughout the simulation. The hourglass energy increase during the simulation is mainly after the initiation of the fracture and therefore does not diminish the significance of the initial fracture pattern indication at time 2.8 ms. A simulation of the head model from a fall from 0.3 m height was carried out and the maximum force obtained compared against the maximum force Coats & Margulies had previously obtained in their simulation of 1.5 month old infant head model from a fall of 0.3 m height [14]. Higher force contact for the 2 month old infant head model is in compliance with decreased suture width and higher elastic modulus with increased age. The maximum force from the simulation of 0.82 m fall can, from the above comparison, be assumed realistic.
The fracture in the right parietal bone from the parietal bone impact, from 0.82 m height, is mostly due to breakage in the fibers. The initial force to the head is only applied to the right parietal bone causing it to deform substantially at the time of impact. The displacement of the head after impact is such that it rotates along the right parietal bone increasing its probability to fracture even further. The total energy for the simulation shows a jump, before reaching maximum force, which may be a result of increased hourglass energy due to largely deformed elements at the location of fracture. It may also indicate a model error, possibly in the definition
Chapter 6
64
of the contacts. The deformation of the mesh at the location of fracture has to be investigated further. Mesh quality at this location might affect how the mesh rips apart and causes deformed elements, increased hourglass energy and disruption in the energy balance. It is however most likely due to the contact conditions. The kinetic energy does not get decomposed properly, as can be seen in Figure 80, indicating that the automatic surface-to-surface contact may be the cause of the total energy balance problem. To obtain more accurate and reliable results for the parietal impact, in terms of hourglass and total energy, alterations to the model needs to be done for future improvements of the model.
Appendix
65
Chapter 7
Conclusion
The model constructed of the 2 month old infant head consists of rather accurate representation of geometries of skull, suture, brain and scalp. Obtaining the sutures from CT images has shown to be possible using thresholding and manual editing. The HU range for the sutures is relatively large and further studies need to be done to obtain a more narrow and accurate HU range. Unfortunately it was not possible to compare fracture propagation of fine and coarse mesh as discontinuities in the fine mesh geometry caused the simulation to fail. By further improving the image processing methods used to obtain the geometries and the meshing method used, simulation of a fine model should be possible. The coarse mesh simulation of the parietal-occipital impact showed a fracture pattern that is in good compliance to one of the fracture pattern from Weber’s study. From this it can be concluded that the model, with obtained age specific material properties, shows to be promising for determining infant skull fracture from a fall of low heights. The model could therefore help in forensic cases to indicate whether a head trauma, resulting in skull fracture, was caused by an accident or abuse by simulating a specific scenario narrative. The fracture results from the parietal impact demonstrate though that further improvements need to be done to the model, in terms of mesh quality and contact conditions, for it to give results with sufficient accuracy.
Appendix
66
Appendix
Constitutive model for the brain
The Odgen Rubber material model (material type 77 in LS-Dyna [49]) was used for the constitutive of the brain. Hyperelastic material is fully reversible after large deformation with its stress derived from the strain energy potential. For a hyperelastic material with initially isotropic properties at unstressed mode the stored strain energy of the deformation can be written as a function of principal stretches W=W(λ1, λ2, λ3) [60]. A hydrostatic work term is added to the strain energy function W for the unconstrained (by incompressibility) material properties of the brain. The hydrostatic work is a function of relative volume, J, and bulk modulus, K [60].
22
321 )1(2
1)3
~~~( −+−++=∑ JKW
i i
i iii ααα λλλαµ
(76)
Where 3/1
~
Ji
iλλ = are the principal stretches independent of the volumetric effect and iα and iµ
are the Ogden constants. The second Piola-Kirchhoff stress tensor,ijS , corresponding to the
strain energy density is derived using:
)(2
1
jiijij E
W
E
WS
∂∂+
∂∂=
(77)
Where ijE is the Green’s strain tensor. In addition, rate effects are taken into account through
linear viscoelasticity by a convolution integral of the form:
∫ ∂∂
−=t
ijijklij d
EtGS
0
)( ττ
τν (78)
This stress is added to the stress tensor from equation (77) . The stress relaxation function, ijklG ,
is represented by six terms in a prony series, given by:
∑=
−=N
i
ti
ieGtg1
)( β (79)
where Gi represents the shear relaxation moduli, and βi the decay constants. Other parameters needed for the model are density,ρ , and Poisson’s ratio, ν . All parameters for the brain model
are given in Table 9.
References
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