Multiscale Modelling of Proximal Femur Growth: Importance of Geometry and Influence...

Multiscale Modelling of Proximal Femur Growth: Importance of Geometry and Influence of Load

by

Priti Yadav

June 2017 Technical Reports from

Royal Institute of Technology Department of Mechanics

SE-100 44 Stockholm, Sweden

TRITA-MEK Technical report 2017:08 ISSN 0348-467X ISRN KTH/MEK/TR-17/08-SE ISBN: 978-91-7729-455-9

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen måndagen den 12 juni 2017 kl 10.00 i sal E3, Osquars backe 14, Kungliga Tekniska Högskolan, Stockholm.

© Priti Yadav 2017 Universitetsservice US-AB, Stockholm 2017

Multiscale Modelling of Proximal Femur Growth: Importance of Geometry and Influence of Load

Priti Yadav Dept. of Mechanics, Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract Longitudinal growth of long bone occurs at growth plates by a process called endochondral ossification. Endochondral ossification is affected by both biological and mechanical factors. This thesis focuses on the mechanical modulation of femoral bone growth occurring at the proximal growth plate, using mechanobiological theories reported in the literature. Finite element analysis was used to simulate bone growth.

The first study analyzed the effect of subject-specific growth plate geometry over simplified growth plate geometry in numerical prediction of bone growth tendency. Subject-specific femur finite element model was constructed from magnetic resonance images of one able- bodied child. Gait kinematics and kinetics were acquired from motion analysis and analyzed further in musculoskeletal modelling to determine muscle and joint contact forces. These were used to determine loading on the femur in finite element analysis. The growth rate was computed based on a mechanobiological theory proposed by Carter and Wong, and a growth model in the principal stress direction was introduced. Our findings support the use of subject- specific geometry and of the principal stress growth direction in prediction of bone growth.

The second study aimed to illustrate how different muscle groups’ activation during gait affects proximal femoral growth tendency in able-bodied children. Subject-specific femur models were used. Gait kinematics and kinetics were acquired for 3 able-bodied children, and muscle and joint contact forces were determined, similar to the first study. The contribution of different muscle groups to hip contact force was also determined. Finite element analysis was performed to compute the specific growth rate and growth direction due to individual muscle groups. The simulated growth model indicated that gait loading tends to reduce neck shaft angle and femoral anteversion during growth. The muscle groups that contributes most and least to growth rate were hip abductors and hip adductors, respectively. All muscle groups’ activation tended to reduce the neck shaft and femoral anteversion angles, except hip extensors and adductors which showed a tendency to increase the femoral anteversion.

The third study’s aim was to understand the influence of different physical activities on proximal femoral growth tendency. Hip contact force orientation was varied to represent reported forces from a number of physical activities. The findings of this study showed that all studied physical activities tend to reduce the neck shaft angle and anteversion, which corresponds to the femur’s natural course during normal growth.

The aim of the fourth study was to study the hypothesis that loading in the absence of physical activity, i.e. static loading, can have an adverse effect on bone growth. A subject-specific model was used and growth plate was modeled as a poroelastic material in finite element analysis. Prendergast’s indicators for bone growth was used to analyse the bone growth behavior. The results showed that tendency of bone growth rate decreases over a long duration of static loading. The study also showed that static sitting is less detrimental than

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static standing for predicted cartilage-to-bone differentiation likelihood, due to the lower magnitude of hip contact force.

The prediction of growth using finite element analysis on experimental gait data and person- specific femur geometry, based on mechanobiological theories of bone growth, offers a biomechanical foundation for better understanding and prediction of bone growth-related deformity problems in growing children. It can ultimately help in treatment planning or physical activity guidelines in children at risk at developing a femur or hip deformity.

Keywords: bone tissue modeling, deformity, biomechanics, osteogenic index, poroelastic material, octahedral shear stress, hydrostatic stress, fluid velocity, octahedral shear strain, MRI, walking, jumping, sedentation

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Flerskalig Modellering av Lårbenets Tillväxt: Betydelse av Geometri och Belastning.

Priti Yadav Institutionen för Mekanik, KTH SE-100 44 Stockholm, Sweden

Sammanfattning Tillväxten av ett ben i dess längsriktning sker i tillväxtplattorna genom så kallad endokondral ossifikation, som påverkas av både biologiska och mekaniska faktorer. Denna avhandling fokuserar på mekanisk modellering av lårbenets tillväxt kring lårbenshalsens tillväxtplatta, och baseras på mekanobiologiska teorier från litteraturen. Finit element-modellering har använts för att simulera bentillväxt.

Den första studien analyserade effekten av individ-specifik geometri hos tillväxtplattan, jämfört med en förenklad geometri, vid numerisk simulering av bentillväxt. En individ- specifik finit element-modell för lårbenet konstruerades från en magnetresonansundersökning av ett barn utan rörelsehinder. Gång-kinematik och kinetikdata erhölls från gånganalys, och analyserades vidare genom modellering av muskel-skelettsystemet för beräkning av kontaktkrafter över muskler och leder. Dessa data användes för att beräkna krafterna över lårbenet genom finit element-simulering. Tillväxthastigheten beräknades baserat på en mekanobiologisk teori enligt Carter och Wong, och en tillväxtmodell baserad på huvudspänningsriktningarna. Våra resultat stöder behovet av individ-specifik geometri och av en tillväxtmodell baserad på huvudspänningarna.

Den andra studien syftade till att illustrera hur olika muskelgruppers aktivering under gång påverkar tillväxten av övre delen av lårbenet hos barn utan rörelsehandikapp. Individ- specifika lårbensmodeller användes. Gång-kinematik och kinetik bestämdes för tre barn utan rörelsehinder, och kontaktkrafterna över muskel och led beräknades på liknande sätt som i första studien. Olika muskelgruppers bidrag till kontaktkrafterna över höftleden beräknades också. Finit element-simulering användes för att beräkna olika muskelgruppers påverkan på tillväxthastighet och tillväxtriktning. Tillväxtmodellen i simuleringarna indikerade, att belastning under gång tenderar att under tillväxt minska lårbenets anteversion och vinkeln mellan lårbenshals och lårbensskaft. De muskelgrupper som bidrar mest respektive minst till tillväxthastigheten är höftabduktorer och höftadduktorer. Aktiviteten i nästan alla muskelgrupper tenderade att minska anteversionen av lårbenet och vinkeln mellan lårbenshals och lårbensskaft. Detta gällde dock inte för höftextensorer och höftadduktorer, vilka visade en tendens att öka lårbenets anteversion.

Den tredje studiens syfte var att förstå hur fysisk aktivitet kan påverka tillväxten av övre delen av lårbenet. Riktningen på höftledens kontaktkrafter varierades för att representera olika fysiska aktiviteter. Studien visade att all fysisk aktivitet tenderar att minska vinkeln mellan lårbenshals och lårbensskaft, och att också minska anteversionen, vilket motsvarar lårbenets normala tilläxtriktning.

Den fjärde studien undersökte hypotesen att belastning utan samtidig rörelse, alltså statisk belastning, kan ha en hämmande effekt på bentillväxt. En individ-specifik modell användes, där tillväxtplattan modellerades som ett poroelastiskt material i en finit element-analys. Prender-

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gasts indikatorer för bentillväxt användes för att analysera tillväxten. Resultaten visade en tendens att bentillväxt minskar efter lång statisk belastning. Studien visar också att statiskt sittande är mindre skadligt för förväntad differentiering brosk-till-ben än statiskt stående, på grund av de mindre kontaktkrafterna i höften.

Förutsägelserna av tillväxt baserad på finit-elementbaserad simulering av experimentella gångdata och individ-specifk lårbensgeometri, och med inkluderande av mekano-biologiska teorier om ben-tillväxt, ger en biomekanisk grund för förståelse och förutsägelse av ben- deformiteter hos barn under tillväxt. De kan i det långa perspektivet ge bättre underlag för behandlingsplanering, och för rekommendationer om fysisk aktivitet hos barn med risk att utveckla lår eller höft-deformiteter.

Nyckelord: Benvävnadsmodellering, deformiteter, biomekanik, osteogeniskt index, poroelastiska material, oktahedral skjuvspänning, hydrostatisk spänning, flödeshastighet, oktahedral skjuvtöjning, MRI, gående, hoppande, stillasittande.

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Preface

This thesis studies the role of load and geometry on proximal femoral bone growth tendency by means of numerical modelling. The first part of the thesis presents a concise background, description of methods, results, conclusion and ideas for future work. The second part contains below mentioned four articles. The manuscripts are adjusted to adapt the present thesis format without changing any of the content.

Paper 1. Yadav P., Shefelbine S.J., Gutierrez-Farewik E.M. , 2016 “Effect of growth plate geometry and growth direction on prediction of proximal femoral morphology”, Journal of Biomechanics 49, 1613-1619.

Paper 2. Yadav P., Shefelbine S.J., Pontén E., Gutierrez-Farewik E.M., 2017 “Influence of muscle groups’ activation on proximal femoral growth tendency”, Biomechanics and Modeling in Mechanobiology, Accepted.

Paper 3. Yadav P., Gutierrez-Farewik E.M., 2017 “How can the load directions due to different physical activities affect proximal femoral growth tendency?”, Manuscript.

Paper 4. Yadav P., Gutierrez-Farewik E.M., 2017 “Modelling the effects of static load on proximal femoral growth behavior”, Manuscript.

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Division of work among authors

This project was initiated by Dr. Elena M. Gutierrez-Farewik (EGF) who is the main supervisor of the work. Prof. Anders Eriksson (AE) and Dr. Eva Pontén (EP) have acted as co-supervisor of this work. Dr. Sandra J. Shefelbine (SJS) is a co-author for Papers 1 and 2. Dr. Eva Pontén is a co-author for Paper 2. The context of the thesis was created by EGF and Priti Yadav (PY) in regular meetings, with clinical input from EP. PY has discussed the results and progress with co-authors in regular meetings.

Paper 1 The experiment data was collected by EGF and PY. PY processed the data and performed the FE simulations in ANSYS with the help from EGF and SJS. The paper was written by PY with help from EGF and SJS.

Paper 2 The experimental data was collected by EGF and PY. PY developed the FE model and performed the simulation in ANSYS with weekly discussion with EGF and occasional discussion with SJS and EP. The paper was written by PY with help from EGF, EP and SJS.

Paper 3 & 4 PY developed the FE model and performed the simulation in ANSYS with weekly discussions with EGF. The paper was written by PY with help from EGF.

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Contents

Abstract

iii

Sammanfattning v

Preface vii

Part I Overview

Chapter 1 Introduction 3

1.1 Motivation of the work 3

1.2 Femur 3

1.3 Longitudinal bone growth of long bones 4

1.4 Growth direction 9

1.5 Mechanical load on the bone 10

1.6 Finite element analysis of the femur bone 12

Chapter 2 Aims of the Thesis 15

Chapter 3 Methods 17

3.1 Data collection 17

3.2 Geometric model 18

3.3 Force computation during gait 19

3.4 Hip contact force during different physical activities and sedentation 22

3.5 Finite element analysis to predict growth tendency 22

3.6 Ethical consideration 23

Chapter 4 Results and Discussion 25

4.1 Effect of subject-specific geometry 25

4.2 Influence of growth direction 25

4.3 Effect of loading 25

4.4 Limitation of the current work 26

Chapter 5 Conclusion and Future work 29

5.1 Conclusion 29

5.2 Proposed future work 29

Acknowledgements 31

Bibliography 33

Part II Paper Paper 1. Effect of growth plate geometry and growth direction on prediction of 43

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proximal femoral morphology

Paper 2. Influence of muscle groups’ activation on proximal femoral growth 61 tendency

Paper 3 How can the load directions due to different physical activities affect 85 proximal femoral growth tendency?

Paper 4 Modelling the effects of static load on proximal femoral growth

behavior.

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Appendix A: Effect of muscle forces and HCF on proximal femoral growth plate 119 stresses

Appendix B: Sensitivity of direction cosine of femoral neck shaft axis for different 120 load settings

Appendix C: Value of constants considered in computation of specific growth rate 121

Appendix D: Contribution of different muscle groups to resultant HCF 122

Appendix E: Stresses, osteogenic index, and growth tendency due to different 123 Muscle groups

Appendix F: Muscle forces 128

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Part I Overview

Chapter 1 Introduction 1.1 Motivation of the work Growth of long bones is one of the fundamental features in a child’s development, as it affects both locomotor and psychological development (Anderson et al., 2013). Long bone geometry changes dramatically throughout the growth, from before birth to full maturity. Longitudinal growth of long bone occurs at cartilaginous growth plates located at the end of the bone, by the process called endochondral ossification. Longitudinal bone growth is influenced by biological and mechanical factors. The biological factors include the effect of genetics (Mankin et al., 2011), nutrition & hormone (Okonofua et al., 1991; Prentice et al., 2006). The mechanical factor is the force acting on the bone, which mainly depends on a child’s daily routine activities (Siebenrock et al., 2011; Wills, 2004). Growing children spend their time in either sedentary activity like watching TV, reading books, or sitting in classrooms, or in physical activities like walking, running, jumping, etc. During different activities, force direction, amplitude, frequency and duration vary. How these different parameters of loading can affect bone growth is still not clear.

In this thesis, an attempt has been made to examine how several aspects of the computational model and the load affect predicted growth in the femur. A better understanding of the mechanical influence on bone growth can help in treatment planning for children at risk of developing bone deformity problems.

1.2 Femur The femur is the most proximal bone in the lower extremity of the human body. It spans two joints and, with a highly complex arrangement of muscle and ligaments, influences the movement and stability in both the hip and knee. The femur’s morphology is mainly described by the following parameters (Arnold et al., 2001; Park et al., 2014):

o Neck-shaft axis: line connecting the femoral head center and the femoral neck base center. o Femoral shaft axis: line connecting the femoral neck base center and the attachment point

of the posterior cruciate ligament (which is easily identifiable in 3D models and is a reasonable approximation of knee joint center).

o Femoral neck plane: plane containing the neck shaft axis and the femoral shaft axis. o Condylar axis plane: plane passing through the femoral neck base center and the knee

joint center, parallel to the line joining the most posterior aspects of the medial and lateral condyles.

o Neck length: distance between the center of the femoral head and the intersection point of neck axis and femoral shaft axis.

o Neck-shaft angle: angle between the neck and femoral shaft axes.

3

4 Introduction

o Femoral anteversion angle: angle between femoral neck plane and condylar axis plane. The term anteversion generally implies that the femoral neck plane is torsioned anterior to the condylar axis plane, and retroversion implies the opposite.

In able-bodied individuals, the femoral anteversion and neck shaft angles high at birth approximately l40° and 150°, respectively, and get reduce to approximately to 15˚ and 120°, respectively, at skeletal maturity (Bobroff et al., 1999; Isaac et al., 1997; Jenkins et al., 2003; Schneider et al., 1997; Tamari et al., 2005).

The pediatric femur is made up of cortical bone, trabecular bone, bone marrow and growth plates. The growth plates are located at the femoral head region, just above the condyles, and near greater and lesser trochanters, as shown in Fig. 1.1.

Figure 1.1: A pediatric femur (http://www.handctr.com/growth-plate-fractures.html)

1.3 Longitudinal growth of long bones 1.3.1 Formation and growth of long bone

The formation and growth of long bone occur by a process called endochondral ossification, as shown in Fig. 1.2. In endochondral ossification, hyaline cartilage in the growth plates serves as a template to be completely replaced by new bone. The process can be divided into two parts:

Figure 1.2: Development of long bones from cartilage model (embryonic) to the formation of growth plate (http://www.slideshare.net/ahoward/anatomy-and-phisology-pp)

http://www.handctr.com/growth-plate-fractures.html)http://www.slideshare.net/ahoward/anatomy-and-phisology-pp

5 Bone formation and prenatal growth: The mesenchymal cells differentiate into cartilage cells (chondrocytes) to form the cartilage model for bone. Next, a perichondrium membrane appears that covers the cartilage model. After vascularization, the perichondrium becomes periosteum and contains a layer of osteoprogenitor (undifferentiated) cells. Later these cells become osteoblasts and secrete osteoids against the cartilage model to form the bone collar. In approximately the second or third month of fetal life, bone cell development and ossification ramps up and create the primary ossification center, which is in the middle of the diaphysis.

The chondrocytes in the primary ossification center are stimulated to proliferate and then go into hypertrophy, followed by matrix calcification and apoptosis. This results in inner cartilage deterioration and formation of a cavity within the bone. The vascular invasion of the hypertrophic cartilage forms the trabecular bone.

Post-natal longitudinal bone growth: Near or just after birth, secondary ossification centers appears in the ends of the long bones. The growth plate lies between the primary ossification center and secondary ossification center. The growth plate is responsible for the longitudinal growth of long bones. The growth plate histology has shown in Fig. 1.3 and discussed below.

Figure 1.3: Proximal tibial growth plate of a rat (Villemure and Stokes, 2009).

Based on growth plate chondrocyte functional and morphological state growth plate can be divided into four zones.

1. Reserve zone: this zone is closest to the epiphyseal end of the growth plate and contains the resting chondrocytes (stem cells). These chondrocytes are small, uniform, compactly located and rich in lipid and cytoplasmic vacuoles. The reserve zone cartilage produces morphogen, which plays a significant role in the column formation and the orientation of the proliferative chondrocytes (Abad et al., 2002). The depletion of stem cells in the reserve zone with age leads to growth plate senescence (Schrier, 2006). It has been reported that this zone is stiffer than the proliferative and hypertrophic zones (Sergerie et al., 2009).

2. Proliferative zone: this is the next zone towards the diaphysis. It contains slightly larger and flat chondrocytes, which actively store the calcium and produces the matrix. These chondrocytes divide and align into longitudinal columns parallel to the growth direction.

6 Introduction

These columns represent the clonal expansion of stem cells, and it may be for that reason that the chondrocytes within a column are more synchronized with each other than are chondrocytes in different columns (Farnum and Wilsman, 1993).

3. Hypertrophic zone: in this zone, chondrocytes stop dividing and begin to increase in volume. Volumetric enlargement of the hypertrophic chondrocytes is anisotropic; cellular height increases more than the width (Farnum et al., 2002). It has been reported that the enlargement of hypertrophic cells contributes most to the growth rate (Cooper et al., 2013).

4. Provisional calcification zone: this zone is closest to the diaphysis. In this zone, mineralization begins on the longitudinal collagen fibrils of the matrix followed by rapid hydroxyapatite formation as calcification starts (Shefelbine, 2002). The chondrocytes in this zone eventually undergo programmed cell death as the matrix around them becomes calcified, but the process by which hypertrophic chondrocytes die is still unclear (Adams and Shapiro, 2002; Roach et al., 2004; Roach and Clarke, 1999). The osteoblasts from the diaphysis penetrate into this zone and secrete bone tissue on the calcified cartilage. Thus, the zone of calcified matrix connects the epiphyseal plate to the diaphysis. A bone thus grows in length when osseous tissue is added to the diaphysis.

Growth at growth plates ceases when the primary and secondary ossification centers fuse, which is called epiphyseal closure (also called a fusion of epiphysis and diaphysis). The rate and extent of long bone growth are the combined effect of chondrocyte proliferation, matrix production and an increase in the height of chondrocytes in the growth direction during cell enlargement. However, there is an estimate that during longitudinal growth of long bones, approximately 60% of total growth is due to hypertrophy, 10% is due to chondrocyte proliferation, and the rest is due to matrix synthesis (Wilsman et al., 1996).

The proportion of growth occurring at each growth plate is not equal and changes with age. For example, the proportion of overall femur growth that occurs at the distal femoral growth plate varies from 60% at the age of seven to 90% at around skeletal maturity (Pritchett, 1992).

1.3.2 Factors influencing growth at growth plates

Biological factors: The effects of different biological factors on the zones of growth plates have been summarized in Table 1.1.

Mechanical factors: The mechanical load mostly affect the chondrocyte enlargement in hypertrophic zone (Stokes et al., 2002) and rate of proliferation in proliferation zone (Alberty et al., 1993).

The effect of mechanical forces on bone growth was first described in engineering terms by Hueuter and Volkmann. Their law states that increased static compressive load parallel to the direction of growth inhibits growth and reduced static compressive load or increased static tensile load promotes growth. This law was experimentally verified by many researchers

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(Arkin and Katz, 1956; Trueta and Trias, 1961). Arkin and Katz (1956), used plaster casts to restrict the hind limb of immature rabbits in a valgus position for six weeks. They observed that a valgus deformity was present after six weeks due to the regional variation in the growth rate. Trueta and Trias (1961) performed a similar experiment by applying a spring and clamp across the knee joint of immature rabbits. They reported that increase in static compressive force inhibits vascularization and ossification. Further, it has been reported that the magnitude of applied static or dynamic load determines the modulated growth rate (Lerner et al., 1998; Robling et al., 2001).

Table 1.1: Effect of different biological factors on different zones of growth plate (Gkiatas et al., 2015; Karimian, Chagin, & Sävendahl, 2012; Kronenberg, 2006; Lui & Baron, 2011; Maeda et al., 2007; Provot & Schipani, 2005; Robson, Siebler, Stevens, Shalet, & Williams, 2000; Van Der Eerden, Karperien, & Wit, 2003; Weise et al., 2001)

Zone Factors Effect

Growth hormones Stimulates the IGF-1 Glucocorticoids Inhibits chondrocyte proliferation Insulin-like growth factor 1 (IGF-1) Increases the rate of chondrocyte

proliferation and length of proliferative columns

Thyroid hormones triiodothyronine (T3) Inhibits chondrocyte clonal expansion and cell proliferation

Proliferative Zone Indian hedgehog (Ihh) Required for columnar chondrocyte differentiation

Parathyroid hormone-related peptide (PTHrP)

Increases chondrocytes proliferation and mitigates differentiation.

Fibroblast growth factors (FGFs) Stimulates cell proliferation and inhibits chondrocyte terminal differentiation

Bone morphogenic proteins (BMPs) Increases the Ihh production Glucocorticoids Inhibits chondrocyte hypertrophy

Hypertrophy Zone Thyroid hormones triiodothyronine (T3) Promotes hypertrophic

chondrocyte differentiation Indian hedgehog (Ihh) Prevent ectopic hypertrophy

Zone of calcification

Glucocorticoids Inhibits cartilage matrix synthesis

The Hueter-Volkmann law still forms the basis for clinical treatment to correct bone deformity problems during growth (Cherry, 1951; Herwig et al., 1987; Zuege et al., 1979). For example, casting and bracing influence bone growth direction and are used to correct clubfoot deformity (Desai et al., 2010). The Pavlik harness, used to correct hip dysplasia in newborns, prevents extension and adduction and ensures that the femoral head does not luxate from acetabular socket during growth (Atalar et al., 2007).

8 Introduction

The Hueter-Volkmann law illustrates the consequences of static loading on the bone. In normal conditions, most bones are subjected to intermittent or cyclic loads. Unequal growth across growth plates has been reported due to corresponding unequal cyclic compressive loading (Pauwels, 1980). Alberty et al. (1993), applied compression or distraction load once a day using an external fixation device across the distal femoral growth plate of immature rabbits. They found that distraction increased the height of proliferative and hypertrophic zones, whereas compression reduced the height of the same zones. The number of proliferating chondrocytes decreased after compressive loading. The effect of shear load and non-axial load on bone growth in animal studies has shown that shear stresses can modify the endochondral ossification (Moreland, 1980). The application of hydrostatic pressure was shown to suppress chondrocyte differentiation and thus maintain the cartilage in growth plate (Kubo et al., 1998).

Carter et al. (1987) introduced a theoretical mechanobiological model that incorporates the effect of multiaxial load on longitudinal growth. In their work, the multiaxial stress tensor due to cyclic loading was expressed by two scalar stress invariants: octahedral shear stress and hydrostatic compressive stress. Their theory states that cyclic or intermittent hydrostatic compressive stress inhibits growth while cyclic or intermittent octahedral shear stress accelerates growth. The mechanical growth rate 𝜀𝜀𝑚𝑚 can be estimated by the osteogenic index IO (Eq. 1.1), which is a linear sum of peak cyclic octahedral shear stress 𝜎𝜎𝑠𝑠 and cyclic hydrostatic stress 𝜎𝜎𝐻𝐻 throughout the load history (Stevens et al., 1999). Octahedral shear stress is always positive in nature and promotes growth. Hydrostatic stress can be positive (hydrostatic tensile stress) or negative (hydrostatic compressive stress), wherein hydrostatic tensile stress promotes growth and hydrostatic compressive stress inhibits growth. High values of the IO indicate stimulation of growth and ossification.

𝜀𝜀𝑚𝑚 = 𝐼𝐼𝑜𝑜 = 𝑎𝑎 ∙ 𝑚𝑚𝑎𝑎𝑥𝑥𝜎𝜎𝑠𝑠 + 𝑏𝑏 ∙ 𝑚𝑚𝑖𝑖𝑛𝑛𝜎𝜎𝐻𝐻 (1.1)

Coefficients a and b determine the relative influence of octahedral shear stress and hydrostatic stress on the IO and thus the mechanical growth rate. Previous studies have suggested that the b/a ratio should be between 0.3 and 1 to predict accurate endochondral ossification pattern (Carter and Wong, 1988; Wong and Carter, 1990).

The Carter and Wong theory has been used by many researchers to predict ossification patterns in long bones (Carter et al., 1998; Carter et al., 1987; Wong and Carter, 1990), the appearance and shape of the secondary site of ossification (Carter and Wong, 1988), ossification patterns in the sternum (Wong and Carter, 1988) and joint morphology (Heegaard et al., 1999). Shefelbine et al. (2004a; 2004b; 2002) used the Carter and Wong theory to predict the progression and shape of the growth front in children during growth. A similar study was performed by Carriero et al. (2011) to understand the effect of different gait patterns on the femoral morphology. In all these studies, generic femur and simplified growth plate geometry were used. Despite the simplified geometry assumption, the results of these studies indicate that muscle and joint forces acting on the bone may be a predictor for the

9 development of bone morphology, i.e. abnormal loading may lead to bone deformity in growing children.

Further, it has been observed that the computed osteogenic index differs for linear elastic and poroelastic material (Prendergast and Huiskes, 1996). However, the difference is not significant for moderate to high frequency-short duration load (Shefelbine, 2002). The use of the poroelastic material in numerical simulation has been mostly recommended for the cartilage body subjected to static load or load with low frequency (Carter and Wong, 2003). In poroelastic material, solid matrix is comprised of pores which are fluid filled. When an external load is slowly applied, the pore pressure changes due to the change in pore volume. The change in pore pressure leads to the fluid flow through the solid matrix. Prendergast et al., (1997) proposed an another mechanobiological theory and considered the fluid flow velocity (𝜗𝜗) and octahedral shear strain (𝛾𝛾) as the stimuli for tissue differentitation. In their theory, they used the stimulus factor (𝑓𝑓𝑠𝑠) to define the tissue differentiation stage (Betts and Müller, 2014).

𝑓𝑓𝑠𝑠 =

𝛾𝛾 + 𝜗𝜗 (1.2) 𝑎𝑎 𝑏𝑏

Where a and b are empirically derived constants varying for different tissue types.

1.4 Growth direction The mechanobiological theories can be used as a precursor to predict the growth rate, but none of the mechanobiological theories explicitly provide any idea about growth direction.

It has been reported in an experimental study that growth plate cartilage grows in the direction of deformation (Arkin and Katz, 1956). In that study, the authors found a valgus deformity at the proximal part of the tibia of immature rabbits, held in a valgus position for six weeks. The valgus position was maintained by applying a cast to the distal part of the tibia, and no direct external stresses were applied in the proximal region. The histological analysis of the proximal tibial growth plate showed the orientation of chondrocytes columns in the deformation direction. All previous studies of bone morphology prediction have considered the growth direction to be the average deformation of either the growth plate or the femoral neck axis (Carriero et al., 2011; Shefelbine et al., 2002; Shefelbine and Carter, 2004a, 2004b). Further, several authors have suggested that the topology of the growth plate may be “designed” to protect it from shear stresses (Ogden, 2000; Smith, 1962). The epiphyseal plate has a tendency to lie parallel to either maximum or minimum principal stress, which minimizes the shear stress between the epiphysis and diaphysis (Currey, 2002). Smith (1962) reported that the proximal femoral growth plate tends to lie parallel to principal tensile stress in the region of principal compressive stress and vice versa, hence practically free from shear stresses. Bright et al. (1974) also investigated that if a rat tibia is subjected to shear, the cartilaginous epiphyseal plate breaks at low forces. Hence, it may be a valid speculation that growth occurs in the direction of principal stress, i.e. minimum shear stress.

10 Introduction

1.5 Mechanical load on the bone The major forces acting on the bone are due to musculo-tendon forces and joint contact force. Estimating these forces on the bone is non-trivial; direct measurement can only be performed invasively, and non-invasive methods can only provide estimations.

Invasive methods: Musculo-tendon forces and joint contact forces can be measured directly by placing a force transducer on a tendon (Dennerlein, 2005) and joint (Hodge et al., 1986) respectively. However, direct measurements of musculoskeletal load in vivo requires invasive techniques (Dennerlein, 2005; Dennerlein et al., 1998; Fleming and Beynnon, 2004; Ravary et al., 2004; Schuind et al., 1992) and are often limited to minimally invasive measurements in superficial tendons such as the Achilles tendon (Finni et al., 1998; Komi, 1990).

Non-invasive methods: To overcome such difficulties and limitations of invasive methods, use of a non-invasive method, for example, ultrasonic techniques (Pourcelot et al., 2005) provide a good approach for tendon force estimation. In this technique, ultrasonic wave velocities in the tendons, which are related to tendon loading, are measured. The main limitation of this method is that it can be used only in immediately subcutaneous tendons.

Numerical simulation-based methods: The muscle forces and joint contact forces can be computed by simulating a captured or known motion using musculoskeletal models. A musculoskeletal model represents the human body from an anatomical and physiological perspective. The model definition mainly includes the details of bone geometry, joint definition, and musculo-tendon properties. The bone segments are modeled as rigid structures, and segment lengths are determined as the distance between the connected joint centers in a static condition. The joint definitions describe the relative movement of adjacent segment in relation to each other. The modelling parameters for musculoskeletal models are usually determined from experiments (Klein Horsman et al., 2007) or medical imaging (Hainisch et al., 2012). Most models must be scaled to fit the subject’s anthropometry, usually based on the 3D surface markers trajectories for a static posture, specific measurements, and the subject’s body weight. The musculo-tendon units are modeled as elements running from origin to insertion points through a defined path or wrapping surfaces. Each musculo-tendon unit is modeled with elements that represent muscle-specific properties such as fiber length, pennation angle, physiological cross-sectional area, tendon slack length, among others. These parameters, among others, are used to generate the force-generating capacity of the muscles through the use of a muscle-tendon actuator model. The Hill-type muscle-tendon actuator model (shown in Fig. 1.4) is most commonly used for muscle force computation (Ackland et al., 2012). In this model, the muscle is represented as an active contractile element (CE) in parallel with a passive elastic element and is in series with a tendon, represented by a non- linear elastic element. The joint kinematics, combined with the muscle-tendon attachment points, determine the length and shortening or lengthening velocity of the muscle-tendon actuator. These values, combined with descriptions of the musculo-tendon dynamics, can be used to determine the required muscle forces and activation. The estimation of muscle force is not straightforward due to their redundancy, i.e. there are a larger number of actuators than

11 there are degrees of freedom. This redundancy can be resolved by using an optimization- based method or experimental electromyography (EMG) to determine muscle activations.

Figure 1.4: Hill-type muscle-tendon actuator

EMG can be used to determine muscle activation, which further can be used in a musculo- tendon model to estimate musculo-tendon force (Erdemir et al., 2007), provided EMG information is collected for all muscles. This tends in practice to not be feasible; surface EMG is only useful for superficial muscles, and fine wire electrode EMG for deep muscles is an invasive method and thus challenging to collect in vivo.

Optimization models can also resolve the redundancy by introducing constraints into the equations, e.g., equality between joint torques obtained from inverse dynamics and the sum of moments generated by the muscle forces, and a strategy determined with a specific cost function, e.g., the minimum sum of squared muscle stresses. The static optimization algorithm with a cost function to minimize the sum of squared muscle stresses is the simplest optimization models because muscle control patterns are not required and, therefore, the muscle dynamics (i.e., the relationship between neuronal activation and muscle force production) are not modeled. In contrast, dynamic optimization problems may include muscle dynamics models, and, therefore, are time-dependent, less sensitive to measurement errors and computationally more expensive. Unless the purpose of the study requires muscle-control estimation, static optimization problems are usually favored because static and dynamic optimization models give similar results, with former being more computationally feasible, particularly if many musculo-tendons are modeled, and the model has many degrees of freedom (Anderson and Pandy, 2001).

Joint contact forces can be computed by performing the inverse dynamic analysis with computed muscle forces and external forces – including ground reaction force (GRF), inertial forces, and gravity – as input to the model. Previous studies have used inverse dynamic analysis and static optimization to study the effect of proximal femoral morphology (Lenaerts et al., 2008) and the influence of altered gait pattern (Carriero et al., 2014) on hip contact force (HCF). A similar approach was used to compute the contribution of individual muscles to overall hip (Correa et al., 2010) and knee (Sasaki and Neptune, 2010) joint contact force. To determine the GRF due to each muscle, the overall GRF was decomposed, based on the

12 Introduction

dynamic coupling concept that each muscle individually contributes to the acceleration of the center of mass and thus to the GRF.

Anderson and Pandy (2003) introduced a dynamic optimization-based method. In their work, vertical and fore-aft components of the GRF were decomposed. The results showed sharp discontinuities at certain intervals in the computed muscle contribution to the total GRF due to rigid foot-ground contact. This method has been used only for simulated gait data.

Liu et al. (2006), introduced another method based on perturbation analysis and used it with both simulated and experimental gait data (Liu et al., 2008, 2006). In this method, at each time instant of gait, a muscle force is perturbed by a very small amount, and the equations of motion are integrated. The limitation of this method is the high computation cost, which is proportional to the number of time steps in the gait cycle and the number of muscles used to actuate the model.

Lin et al. (2011) also developed a pseudo-inverse-based method to decompose the GRF. In this method, foot-ground contact was defined at five locations of the foot and for three different phases (heel-strike, foot flat and toe-off). A weighting function was used for smooth transition between the different phases, thus resolving the discontinuity in muscle contribution to overall GRF. The method is similar in concept to an optimization-based method except that this method provides an analytical solution to an indeterminate system with a weighting matrix. This method has been used to find the contribution of individual muscle forces to the overall GRF (Lin et al., 2011) and the center of mass acceleration (Lim et al., 2013). The results were similar to those computed with an optimization-based method. The pseudo- inverse based method was also found to be faster than the perturbation analysis-based method (Lin et al., 2011).

1.6 Finite element analysis of the femur bone Finite element (FE) methods have been used extensively to analyze the femur. The accuracy of any finite element analysis (FEA) depends on factors such as geometry, discretization (mesh), material property, and boundary conditions. Computed tomography (CT) scan data is widely used to develop realistic bone structures for FEA. However, due to the radiation involved in CT, its use is often limited to cadavers or to patients groups who already require a CT scan for medical reasons. The use of magnetic resonance imaging (MRI) has some benefits; for model development is not only safer but also has the capacity to distinct the boundary of bone and muscles, though the clarity of bone images is less in MRI than in CT.

The complexity of the material model depends on the problem definition, the restrictions in computing time, and the application. The growth plate, made of fibrous cartilage, contains approximately 80% fluid and 20% solid matrix (Mow et al., 1980), wherein most of the load is withheld by fluid. As such, if the rate of the load acting on the growth plate is high, then a model with single phase, linear elastic-nearly incompressible material is reasonable. However,

13 if the load rate is very low or even static, then it can be expected that fluid will slowly seep out and the load is instead transferred to the solid matrix. In this case, a poroelastic material model of the growth plate is more appropriate. The water content in cortical and trabecular bone is approximately 23% and 27% fluid, respectively (Oftadeh et al., 2015), and its porosity is much lower than that of cartilage. Bone tissue thus is commonly modeled as a linear elastic material for simulations with physiological loading conditions.

There are several methods reported in the literature that aims to define accurate boundary conditions for the femur in FEA (Duda et al., 1998; K Polgár et al., 2003a; 2003b; Tsai et al., 2013). In these studies, the load was applied at the femoral head, and the femur model was either constrained at mid-diaphysis or at the condyles. Speirs et al. (2007) proposed a different physiological boundary condition for the femur. In their model, the knee-center translation was constrained in all the three directions. The hip contact point was constrained in anterior- posterior and medial-lateral directions, and a node on the lateral epicondyle was constrained in the anterior-posterior direction. They applied joint contact force and all muscle forces and reported physiologically plausible deformation for the femur. It is, however, unclear whether a hip contact point constraint can be justified in relation to the in vivo situation, wherein the femoral head is not rigidly constrained to deflect in anterior-posterior and medial-lateral directions.

Further, the complexity of boundary conditions required for a reliable solution also depends on the problem. For example, if the objective is to determine the tissue stress at the femoral head region, then the results are not very sensitive to small changes in the distal boundary conditions.

Chapter 2 Aims of the Thesis The overall aims of this research are to develop a model to study growth at the proximal end of the femur in children, to study the influence of model geometry and growth direction model in computed results, and to study how simulated growth tendency is influenced by physical activity and sedentation.

The specific aims of the included studies are:

Study 1 • To develop a subject-specific finite element model of the femur and proximal femoral growth plate and evaluate its significance in growth prediction compared to a simplified growth plate model, computed using normal gait loading

• To investigate a new growth direction model, the principal stress direction (PSD) model, and compare the simulated growth to that from a femoral neck shaft axis deflection direction (FNDD) model.

Study 2 • To analyze how different muscle groups’ activation during able-bodied children’s normal gait influences proximal femoral growth tendency.

Study 3 • To understand how the direction of the HCF due to different physical activities (load frequency ≥ 1 Hz) affects the proximal femoral growth tendency.

Study 4 • To analyze the effect of static load on proximal femoral growth behavior.

15

Chapter 3 Methods

In this chapter, the methodology of the current work is summarized. Detailed descriptions of all the material and methods used in this thesis are mentioned in the appended papers in Part II of this thesis.

3.1 Data collection Subjects: The data used in this thesis is from a previous study (Yadav, 2015), where data was collected for three typically-developing children. The anthropometric data for all children is mentioned in Table 3.1.

Table 3.1: Anthropometric measurement data of participated subjects (Yadav, 2015)

Participated in Gender Age

(y) Body mass

(kg) Neck-shaft angle (deg)

Femoral anteversion (deg)

Subject 1 Studies 2, 3 & 4 Female 6 20.7 140 34

Subject 2 Studies 1 & 2 Male 7 23.8 136 33 Subject 3 Study 2 Female 11 49.5 138 18

Magnetic resonance imaging: MRI data of the lower limb for all subjects was collected with below mentioned specifications

o Imaging planes: Transverse plane (iliac crest to toes) and coronal plane (only for hip

joint) o Machine field strength: 3 T (Ingenia 3.0 T, Philips, Best, The Netherlands) o Contrast: T1 o Spin: turbo spin echo o Repetition time (TR) : 685 ms o Echo time (TE): 20 ms o Number of pixels: 500 x 448 o Slice thickness: 3 mm o Gap between slices: 3 mm

Gait analysis: 3D gait analysis was performed to collect the kinematics and GRF data. The parameters used in gait analysis are as follows:

o Motion capture system: 8 – camera (Vicon MX 40, Oxford, UK) o Number of force platform: 2 (Kistler, Winterthur, Switzerland) o Marker set: full body plug-in gait model was used with thirty-five reflective 9 – mm

markers

17

18 Methods

o Walking speed: self-selected, comfortable speed o Walkway length: 10 m o Sampling frequency for motion data: 100 Hz o Sampling frequency for GRF data: 1000 Hz

3.2 Geometric model 3.2.1 FEA geometric model

The MRI data was processed to construct the 3D subject-specific model of femur and growth plate (Materialise NV, Leuven, Belgium). The segmentation boundary for different parts of the bones (trabecular, cortical, growth plate and bone marrow) were finalized with the help of radiographer and online available MRI ATLAS (https://www.imaios.com, http://xrayhead.com/).

The growth plate was mainly visible in coronal (frontal) plane MRI data. The total number of available slices in the coronal plane were approximately 4 to 5. Hence the growth plate was created as the Boolean subtraction of proximal trabecular bone generated using coronal plane MRI from the proximal trabecular bone generated from transverse plane MRI.

The remaining femur (trabecular and cortical bones ) was constructed using transverse plane MRI data. The bone marrow was considered as the volume enclosed by the interior surface of the cortical bone, the inferior surface of the proximal trabecular bone and the superior surface of the distal trabecular bone.

Studies 1, 2 & 3: The proximal femoral morphology (neck shaft angle and femoral anteversion) was measured using 3D surface model (Arnold and Delp, 2001). The detailed description of the measurement method is provided in appended Paper 1(Yadav et al., 2016).

Study 1: For comparison with subject-specific growth plate model, a simplified growth plate was constructed of disk shape with constant thickness (Carriero et al., 2011) in the femoral neck region of the subject-specific femur.

Study 4: In this study, the only proximal femur was considered which included the proximal femoral growth plate (located in the femoral head region), proximal trabecular and cortical bones.

3.2.2 Musculoskeletal model for force computation

A generic musculoskeletal model (Gait model, SIMM) was scaled based on the static trial data collected during gait analysis (SIMM Motion Module, Musculographics Inc., Santa Rosa, CA, USA). The proximal femur of the scaled model was further modified to match each subject’s proximal femoral morphology namely neck shaft angle and femoral anteversion using the deform tool in SIMM 7.0. To ensure model similarity, femur geometry of the

http://www.imaios.com/http://www.imaios.com/

19 musculoskeletal models was visually verified by superimposing the MRI developed femur model to the deformed femur model.

The generated musculoskeletal model consisted of 41 body segments, 41 joints, 86 lower limb musculotendon actuators, 2 patellar ligaments (1 for each lower limb) and 40 degrees of freedom. The Hill-type model was used to define the force generating properties of the muscles. The different parameters (like, peak isometric muscle force, optimal muscle-fiber length, pennation angle, tendon slack length, etc.) of the muscle model were defined using default values as available in SIMM (Dynamics Pipeline, Musculographics Inc., Santa Rosa, CA, USA)

3.3 Force computation during gait 3.3.1 Muscle and hip joint force computation

The gait kinematics was computed using subject’s musculoskeletal model and recorded 3 D gait marker trajectories. The inverse dynamic analysis was performed followed by static optimization to compute the muscle forces (Dynamics Pipeline, Musculographics Inc., Santa Rosa, CA, USA). The optimization criterion was to minimize the cost function of the sum of squared muscle stresses. The computed muscle forces were constrained such they the resulting joint moment was equal to the joint moments computed using inverse dynamic analysis (Anderson and Pandy, 2001).

To compute the HCF, a second inverse dynamic analysis was performed by applying the computed muscle forces and the external forces (GRF, inertial forces, and gravity) to the subject’s musculoskeletal model.

3.3.2 Hip contact force due to different muscle groups

The HCF is mainly the vector sum of muscle forces and GRF. Due to the dynamic coupling, all the muscles can potentially accelerate all joints, irrespective of their location relative to different joints (Anderson and Pandy, 2003). Hence, all muscles contribute to the acceleration of the center of mass and so to GRF (based on Newton’s second law F = ma). As HCF is a function of GRF; all muscle contribute to HCF.

The contribution of different muscle groups to HCF was computed by GRF decomposition (contribution of different muscle groups to GRF) followed by inverse dynamics analysis with input as specific muscle group forces and associated GRF

Contribution of individual muscle to GRF: To compute muscle groups’ GRF contribution, pseudo-inverse induced acceleration analyses were (Dorn et al., 2012; Lin et al., 2011) performed, to decompose the GRF. The generalized equation of motion for the n degree of freedom skeleton with k musculo-tendon units is represented as:

𝜕𝜕

20 Methods

𝑴𝑴(𝒒𝒒)𝒒𝒒𝒒 = 𝑪𝑪(𝒒𝒒, 𝒒𝒒𝒒 ) + 𝑮𝑮(𝒒𝒒) + 𝑹𝑹(𝒒𝒒)𝒇𝒇𝑴𝑴 + 𝑬𝑬(𝒒𝒒)𝒇𝒇𝑬𝑬 (2.1) Where 𝒒𝒒, 𝒒𝒒𝒒 , 𝒒𝒒𝒒 are the vectors of generalized displacements, velocity, and accelerations respectively, 𝑴𝑴(𝒒𝒒) is an 𝑛𝑛 × 𝑛𝑛 system mass matrix, 𝑪𝑪(𝒒𝒒, 𝒒𝒒𝒒 ) is an 𝑛𝑛 × 1 generalized force vector consisting of centrifugal and coriolis forces, 𝑮𝑮(𝒒𝒒) is an 𝑛𝑛 × 1 generalized force vector due to gravity, 𝑹𝑹(𝒒𝒒) is 𝑛𝑛 × 𝑘𝑘 matrix of muscle moment arms; 𝒇𝒇𝑴𝑴 is a 𝑘𝑘 × 1 muscle force vector, 𝒇𝒇𝑬𝑬 is 3𝑚𝑚 × 1 vector of external reaction forces exerted on the foot by the ground via m foot-contact points as shown in Fig. 3.1, and 𝑬𝑬(𝒒𝒒) ia an 𝑛𝑛 × 3𝑚𝑚 matrix of linear partial velocities ( Lin et al., 2011; Yadav, 2015).

Figure 3.1: Foot-ground contact points during heel strike (Phase 1), foot flat (Phase 2 and 3) and toe off (Phase 4) of gait (Lin et al., 2011; Yadav, 2015).

The linear acceleration of the ith foot-contact point can be represented by Eq. (2), based on the assumption that the acceleration of the ith foot-contact point is zero whenever it comes in contact with the ground.

𝒂𝒂𝒊𝒊(𝒒𝒒, 𝒒𝒒𝒒 , 𝒒𝒒𝒒 ) = 𝑲𝑲𝒊𝒊(𝒒𝒒, 𝒒𝒒𝒒 ) + 𝑵𝑵𝒊𝒊(𝒒𝒒, 𝒒𝒒𝒒 )𝒒𝒒𝒒 = 𝟎𝟎 (2.2)

In Eq. (2.2), 𝑲𝑲𝒊𝒊 and 𝑵𝑵𝒊𝒊 are coefficient matrices and computed as

𝑲𝑲𝒊𝒊(𝒒𝒒𝒕𝒕, 𝒒𝒒𝒒 𝑡𝑡) = 𝒂𝒂𝒊𝒊(𝒒𝒒𝒕𝒕, 𝒒𝒒𝒒 𝒕𝒕, 0) (2.3)

𝑵𝑵𝒊𝒊(𝒒𝒒𝒕𝒕, 𝒒𝒒𝒒 𝒕𝒕) =

𝜕𝜕𝒂𝒂𝒊𝒊 (𝒒𝒒𝒕𝒕, 𝒒𝒒𝒒 𝒕𝒕) (2.4)

𝒒𝒒𝒕𝒕 and 𝒒𝒒𝒒 𝒕𝒕 are vectors containing the generalized displacements and velocities at each time step, t.

21 An individual muscle’s force contribution to the overall GRF was computed by solving the Eq. (2.1) and Eq. (2.2) with input as each muscle force in isolation. For example, to determine the contribution of muscle 𝛼𝛼 in GRF, Eq (2.1) and Eq (2.2) can be rewritten as

𝑴𝑴(𝒒𝒒)𝒒𝒒𝒒 𝛼𝛼 = 𝑭𝑭𝛼𝛼 + 𝑬𝑬(𝒒𝒒)𝒇𝒇𝑬𝑬𝛼𝛼 (2.5) 𝑾𝑾{𝑲𝑲𝒊𝒊(𝒒𝒒, 𝒒𝒒𝒒 ) + 𝑵𝑵𝒊𝒊(𝒒𝒒, 𝒒𝒒𝒒 )𝒒𝒒𝒒 𝛼𝛼} = 𝟎𝟎 (2.6)

Where 𝑾𝑾 is the weighting matrix which constraints the foor-contact point in a way to make it consistent with the actual movement of the foot during stance.

To compute the GRF, Eq. (2.5) and Eq. (2.6) was solved using a least-squares pseudo-inverse method (Eq. 2.7):

{

𝒒𝒒𝒒 𝛼𝛼 } = 𝑨𝑨+𝒃𝒃 (2.7) 𝒇𝒇𝑬𝑬𝛼𝛼

𝑴𝑴𝑛𝑛×𝑛𝑛 −𝑬𝑬 Where 𝑨𝑨+ is the Moore–Penrose pseudo-inverse of the matrix 𝑨𝑨 = 𝑾𝑾𝑾 [ 𝑾𝑾 ∙ 𝑵𝑵 𝟎𝟎3𝑚𝑚×3𝑚𝑚]

𝟎𝟎3𝑚𝑚×𝑛𝑛 𝑰𝑰3𝑚𝑚×3𝑚𝑚

𝑭𝑭𝛼𝛼 𝒃𝒃 = 𝑾𝑾𝑾 {−𝑾𝑾 ∙ 𝑲𝑲}

𝟎𝟎3𝑚𝑚×1

104𝑰𝑰𝑛𝑛×𝑛𝑛 𝑾𝑾𝑾 = [ 102𝑰𝑰3𝑚𝑚×3𝑚𝑚

𝑰𝑰3𝑚𝑚×3𝑚𝑚

] is a weighting matrix

Contribution of different muscles group to HCF: The contribution of following muscles groups to HCF was computed:

Case 1: All muscles Case 2: Hip flexors muscles Case 3: Hip extensors muscles Case 4: Hip adductors muscles Case 5: Hip abductors muscles Case 6: Knee extensors muscles

The inverse dynamic analysis was performed by applying the muscle forces of a specific muscle group and associated GRF. The direction and magnitude of muscle force and GRF were based on the computation mentioned in section 3.3.1 and 3.3.2 respectively.

22 Methods 3.4 Hip contact force during different physical activities and sedentation The literature was consulted to find the normalized HCF vector with respect to body weight during different physical activities and during sedentation (Stansfield et al. 2003; Lenaerts et al. 2008; Lewis et al. 2010; Modenese and Phillips 2012; Carriero et al. 2014; Wesseling et al. 2015; Bergmann et al. 2016; https://orthoload.com/database/). Further, the magnitude of HCF vector was scaled according to subject’s weight considered in Study 3 and 4.

3.5 Finite element analysis to predict growth tendency Mesh details: The transition zone was defined above and below the growth plate for a smooth transition of material property. Hexahedral (20 nodes, Solid 186 for linear elastic material and CPT 216 for poroelastic material, ANSYS Inc., Canonsburg, PA) elements were used to mesh the growth plate and transition zone volume due in order to:

o Avoid volumetric mesh locking for nearly incompressible material o Reduce the total number of elements o Helps in growth modelling

A hexahedral dominant mesh was created for the remainder of the femur

Material: The material for all the femoral parts was assumed as isotropic. The growth plate was modeled as linear elastic for load cases with load frequency equal or greater than 1 Hz (Studies 1 – 3), and it was modeled as poroelastic material for static load cases and load cases with load frequency lower than the 1 Hz (Study 4).

Loading: The HCF was applied as a distributed load over nodes of the femoral head surface, equivalent to an area of approximately 20 mm2, nearest the HCF’s line of action. Each muscle forces was applied as a point load (Studies 1 & 2).

Study 1: HCF and muscle forces were applied to the femur model. One gait cycle was discretized into nine load instances; five load instances correspond to initial contact, first peak, valley, second peak and foot off of the resultant HCF, and the remaining four load instances are the midpoints between these five.

Study 2: The loading was similar to Study 1, with 5 more load cases:

o All muscle load o Hip flexor load o Hip extensor load o Hip adductor load o Hip abductor load o Knee extensor load

23 For each load case, muscle forces and associated HCF were applied to the model. The nine discretized instances for each load case (2-6) were defined based on load case 1.

Studies 3 & 4: Only HCF was applied to the model.

Boundary Conditions: The condyles were constrained in all direction in Studies 1-3, and the distal end of the femoral shaft was constrained in all directions in Study 4. FEA was performed (ANSYS Inc., Canonsburg, PA) to determine stresses at the growth plate surfaces. Growth rate (Studies 1-3): The specific growth rate was computed as the sum of biological and mechanical growth rates. The biological growth rate was assumed as constant and estimated as described in the literature (Pritchett, 1992). The mechanical growth rate was computed based on Carter and Wong theory (Carter and Wong, 2003).

Growth direction computation: Study 1: In this study two growth directions were compared − principal stress direction (PSD) and femoral neck deformation direction (FNDD).

Study 2 & 3: Growth was simulated in the PSD.

The orthonormal thermal expansion analysis was performed to simulate the growth. For each element of the growth zone, a coordinate system was created such that, one of its axes was aligned with the computed growth direction. The coefficient of thermal expansion was defined 1 for the axis-direction aligned with growth direction and 0 for the remaining axes- directions. The computed specific growth rate was applied as the “temperature” for expansion. The expanded model was considered as the model after growth.

Stimulus factor (Study 4): A stimulus factor (𝑓𝑓𝑠𝑠) for cell behavior and differentiation likelihood over the distal growth plate surface elements was computed for each time step of load duration as suggestes in Prendergast mechano-regulation theory (Prendergast et al., 1997)

3.6 Ethical consideration The presented work was approved by the regional ethical review board in Stockholm, Sweden. All the children and their parents provided written informed consent.

Chapter 4 Results and Discussion 4.1 Effect of subject-specific geometry The findings of Study 1 showed different patterns of IO for the subject-specific growth plate vs. the simplified growth plate, in both their shape and in the locations of positive and negative IO. The subject-specific growth plate had an irregular shape and was located within the femoral head. The simplified growth plate, on the other hand, was shaped like a disk and located in the femoral neck, similar to a previous study (Carriero et al., 2011). The stresses and hence Io on the growth plate are thus sensitive to its shape and its location.

In this study, we have not analyzed whether the IO is more sensitive to the shape difference or to the location of the growth plate. However, we speculate that the use of generalized growth plate geometry for a particular age group population may suffice for reasonable accuracy, as the shape of the growth plate changes with age.

4.2 Influence of growth direction Results from Study 1 also showed that predictions of growth tendency, both magnitude, and direction, are sensitive to the growth direction model. The FNDD was evaluated based on experimental study findings from 1956 (Arkin and Katz, 1956). The PSD model was developed based on a previous finding that the growth plate tends to lie parallel to either maximum principal stress or to minimum principal stresses (Currey, 2002), and that cartilage material breaks at very low stresses when subjected to shear loading (Bright et al., 1974).

The FNDD was defined as the direction of average deflection of the neck shaft axis over the 9 load instances. The PDS was defined as the direction of highest absolute principal stress magnitude occurring during the load cycle.

In Study 1, the femur was modeled with linear elastic materials, thus stress and deformation fields will be the same. The difference in results is therefore due to the definitions of the growth direction models.

4.3 Effect of loading 4.3.1 Load during normal gait

Normal gait showed a tendency to reduce the NSA and FA. During gait, the hip abductors were found to contribute more than any other muscle group to the overall growth rate. All muscle groups showed a tendency to reduce the NSA and FA except the hip extensors and adductors, which tended to increase the FA.

4.3.2 Load during different physical activities (load frequency ≥ 1Hz)

25

26 Results and Discussion The feasible loading due to different physical activities showed a tendency to reduce both NSA and FA. However, in Study 3, some fictitious load samples were also considered. All load samples with anteriorly oriented HCF showed a tendency to increase the FA. For anteriorly oriented HCF, the NSA was found to be increased when the lateral component was higher than anterior component of HCF. The only inferiorly applied load showed a tendency to increase the NSA and FA.

4.3.3 Load during sedentation (static load and cyclic load of very low frequency)

The effect of the static load was analyzed using Prendergast mechanobiological theory (Prendergast et al., 1997), which considers the fluid velocity and octahedral shear strain as stimuli to predict the tissue differentiation status. In Study 4, the application of static load on femoral head showed a tendency to increase overall stimulus factor and so to resist the bone growth tendency. The study also showed that static sitting is less deleterious than static standing due to a lower magnitude of HCF. The application of cyclic load with low frequency showed a tendency to increase the stimulus factor, which is counterintuitive finding. This unreasonable result is suspected due to the consideration of constant permeability in the numerical simulation.

4.4 Limitation of current work In this thesis work, only the femoral head growth plate was considered. Predicted growth estimates the relative change of the femoral head center’s position. Morphological measurements are, however, based on the femoral neck axis, the condylar axis, and the femoral shaft axis definitions. The orientation of these axes may change with bone growth at the proximal, distal and the trochanteric growth plates. For more accurate prediction of overall femur morphological changes, all proximal and distal growth plates should be considered in the model.

Musculoskeletal model results are sensitive to bone and musculo-tendon geometry (Carbone et al., 2012; Scheys et al., 2011). In the current work, a deformed generic model was used for force computation. An MRI-based musculoskeletal model can be expected to increase accuracy but is currently still very time-consuming.

To compute the contributions of different muscle groups to the total HCF, the GRF was decomposed using pseudo-inverse induced acceleration analysis. In this analysis, the contact between the foot and the ground was assumed rigid, which is a simplified contact model. However, Anderson and Pandy (2003) reported that rigid-body contact is reasonable for force decomposition problems.

The included studies have low statistical power due to the small sample size, rendering statistical comparisons irrelevant, implying that the results are not generalizable. To form a firmer conclusion about bone growth, studies with larger sample sizes should be performed.

27 Our predictions of growth tendency correspond to the expected changes during growth, but the results are not validated; to accomplish this, experimental studies that follow up growing children over a long time period are warranted.

In this thesis, the considered mechanobiological theories can only provide an idea of where the growth is going to occur. However, in order to predict the actual growth, both growth rate and growth direction are required. In this thesis, the mechanical growth was estimated using an osteogenic index or stimulus factor which further depends on constants ‘a’ and ‘b.' The change in magnitude of these constants may change the results. In order to predict the accurate value of these constants, validated numerical simulation for the larger population is required.

In Study 4, to capture the effect of long duration static load, growth plate was modeled as poroelastic material. However, considered model was based on constant permeability. To capture the more realistic behavior of growth plate under static or slow loading, the use of the strain-dependent anisotropic poroelastic model is required.

Chapter 5 Conclusion and Future Work 5.1 Conclusion The work presented in this thesis elucidates the importance of load and geometry on proximal femoral growth tendency.

The work described in Study 1 is the first study to study proximal femoral growth tendency using model geometry from subject-specific femur and growth plate. It is also the first study to introduce the PSD in numerical simulation. The study showed that growth tendency is not only sensitive to geometrical details but also to the growth direction definition.

Though it is well known empirically that muscle activation and bone morphology are inextricably linked in growing children and that imbalanced muscle forces can induce bony deformities, no studies to the author’s knowledge have studied the influence of different muscle groups' activation to bone growth tendency. The work presented in this thesis (Study 2) provides an understanding of the muscle-bone mechanical relationship.

The results of these studies also suggest that for better prediction of bone growth morphology in growing children, model geometry should be subject-specific or at least generalized based on age.

The work presented in Study 3 showed that nearly all physical activities in the physiological range (and with load frequency ≥1 Hz) tend to reduce the NSA and FA. This has important clinical implications for therapists aiming to prevent or minimize expected secondary bony deformities in a patient population; all upright activity seems to be desirable.

In Study 4, the effect of static load on bone growth was analyzed. The results of this study showed the deleterious effect on bone growth due to static loading. However, the unexpected finding regarding low-frequency load that showed to retard the bone growth was suspected due to the consideration of simplified material model (poroelastic material with constant permeability)

The numerical prediction of bone growth using realistic load data and subject-specific geometry provides the basis to develop the better understanding of bone growth which can be utilized in treatment planning for children affected with or at risk of developing growing bone deformities.

5.2 Proposed future work 1. Development of a pediatric musculoskeletal model for accurate force computation: The currently available musculoskeletal models are based on American or European male adult

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30 Conclusion and Future Work populations. Immature bone anatomy is different from mature anatomy, which affects joint center definitions, making it somewhat unreliable to simply scale an adult model. Muscle attachment points, furthermore, can likewise be expected to differ. It is currently too time- consuming to create entirely individualized image-based musculoskeletal models, but a pediatric model, or even a few pediatric models for children of different age ranges, would likely improve the reliability of musculoskeletal modelling computations.

2. Macro- to micro- scale FEM for growth simulation to better understand growth mechanics: In the current thesis, growth is predicted based on the stress pattern in the growth plate tissue, but exactly how tissue stresses stimulate the cells within the tissue is still not well understood. Sub-modelling technique can be used to analyze the cellular level mechanics and can provide information as to how chondrocytes form columnar structures. The formation and orientation of columnar structure can help to determine the growth direction.

3. Development of growth model with a merger of biological and mechanical factors: In the current work, the total growth was assumed to be a linear sum of the biological and the mechanical growth rates. However, the biological and mechanical factors are interlinked to some extent. Both factors contribute to proliferation and hypertrophy of the chondrocytes, and growth mainly depends on their rate of proliferation and hypertrophy. A more accurate growth model may be achievable by studying the strain rate of chondrocytes in the proliferation zone and in the hypotrophy zone.

4. Validation of growth simulation: Validation of the growth predicted in this thesis work can be performed by predicting the growth over a long period and for a larger population, and comparing it to actual growth occurred in that population, as observed from medical images.

Acknowledgements I would like to express my deep sense of gratitude to my supervisor Dr. Elena M. Gutierrez- Farewik for her meticulous guidance, constant encouragement, and continuous support.

I am grateful to Dr. Sandra J. Shefelbine, for guiding me through the finite element analysis of bone growth. Her research experience and knowledge helped immensely to achieve the objectives of Studies 1 & 2.

I would also like to thank my co-supervisor Prof. Anders Eriksson and Dr. Eva Pontén for their valuable help and suggestions that helped me through this work. I thank Dr. Thröstur Finnbogason, Anna Lind for their help in MRI data collection. I also thank Dr. Kristina Löwing, Dr. Cecilia Lidbeck and Dr. Åsa Bartonek for helping me in gait data collection and sharing their knowledge. I also wish to thank children and their parents for voluntarily participating in this research work.

I wish to thank Dr. Arne Nordmark, Dr. Gunnar Tibert for their valuable suggestions and comments during group seminars.

I would like to acknowledge Swedish Research Council, Stiftelsen Promobilia and Norrbacka-Eugeniastiftelsen for providing funding for this research work.

Big thanks to Yang Zhou for being such a wonderful officemate. I also thank Zeinab Moradi Nour, Ashwin Vishnu, Giandomenico Lupo, Sagar Zade, Sudhakar Yogaraj, Krishnagowda, Ningegowda, Prabal Singh, Sembian Sundarapandian, Timea Kékesi, Ruoli Wang, Natalia Kosterina for their great company.

I would like to thank my former group mates, Erik Dijkstra, Marta Björnsdóttir, Ganesh Tamadapu, Amit Patil, Mikael Swarén, Nasseradeen Ashwear, Krishna Manda, Pau Mallol, and Huina Mao for their company, help and useful technical discussions.

I also thank my friends at Astrid Lindgrens Motorik Lab, Cecilia Lidbeck, Josefine Eriksson Naili, Elin Lööf, Michael Reimeringer, Marie Eriksson for always being helpful.

I would like to thank my parents and grandparents for their love, care, and support. Special thanks to my beloved father; you always encouraged me to recognize my potential and follow the dream hard.

Thanks to my two wonderful elder sisters and a smart brother; you all have always raised the bar higher to succeed in life.

Finally, many thanks to my love, Debasis for your unconditional love, support, understanding and always motivating me to keep the positive attitude in life.

31

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