growth factor controlled fetal bone growth · the nal length and shape of a bone are thus a result...

Growth factor controlled

fetal bone growth

a 2D �nite element model

Sietske Witvoet-Braam

BMTE 06.25

Master's thesis

Coach(es): C.C.v.Donkelaar

Supervisor: C.C.v.Donkelaar

Technische Universiteit EindhovenDepartment Biomedical EngineeringBone and Orthopaedic Biomechanics

Eindhoven, 30th May 2006

Abstract

Endochondral ossi�cation is a tightly regulated process through which bones grow andrepair. It involves growth of cartilage by proliferation, di�erentiation and matrix production,and subsequent mineralization of the cartilaginous matrix. Many growth factors and hormonesare involved in the regulation of bone growth. Brouwers et al. [7] has shown that a set of onlythree growth factors can regulate a stable growth plate. Ihh enhances the proliferation rateand the production of PTHrP. PTHrP inhibits the di�erentiation of the chondrocytes and thehypertrophy rate. Mineralization is stimulated by one additional growth factor, which can beeither MMP, which degrades the extracellular matrix around the hypertrophic chondrocytesor VEGF, which induces vascular ingrowth.

Mechanical forces also a�ect the growth and development of the bone. Compressive forcesby adjacent structures and tensile forces of attached muscles can deform the cartilage sca�oldduring growth. Mechanical loading can also a�ect the gene and protein expression of regulat-ing growth factors.

The �nal length and shape of a bone are thus a result of the growth factor controlled bonegrowth and the mechanical loading during growth. There are many growth disorders, like hipdysplasia, in which an abnormal mechanical loading can result in an abnormal geometry ofthe bone. However, it is not known how much the mechanical forces will in�uence the �nalshape of a bone.

The aim of this study is to establish a model that enables to quantitatively assess thecontributions of mechanical loading on the developing shape of a fetal long bone, when bonegrowth, by proliferation, matrix synthesis and hypertrophy, is growth factor controlled.

A histomorphological study was performed on embryonic mice third metatarsals to collectcharacteristic lengths of these elements at di�erent developmental stages. The initial stateof the simulation corresponds to the distal half of an 16.5 day old embryonic mice (E16.5)metatarsal. First, the parameters of the model were set, such that the characteristics of themodel would represent the E19.5 state after simulating bone growth for three days.

Using these parameters we performed some additional simulations. First, we started withan initial mesh representing approximately E14. In this simulation the model developed to ashape similar to the former initial state of E16.5.

Second, we simulated several disorders in which growth factor production was impaired.In all these simulations the zone height variations, in comparison to the normal development,corresponded with known in vivo disorder characteristics. Although the disorders all lead todwar�sm, total bone lengths were elongated in all simulations. This is most likely a short

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term e�ect.Finally, we applied mechanical compressive and tensile forces on the bone during growth.

It was shown that mechanical loading during growth can induce important changes to bonegeometry.

This model can be a useful tool to study the development of fetal long bones and the e�ectof growth factor synthesis, mechanical in�uences and their interactions.

Samenvatting

Endochondrale verbening is een nauwkeurig geregeld proces waardoor botten groeien enherstellen. Dit proces omvat kraakbeengroei door proliferatie, di�erentiatie en matrix pro-ductie, waarna de kraakbeenmatrix mineraliseert. Er zijn veel groeifactoren en hormonenbetrokken bij de regulatie van botgroei. Brouwers e.a. [7] hebben aangetoond dat een com-binatie van slechts drie groeifactoren een stabiele groeischijf kan reguleren. Ihh versterkt deproliferatie snelheid en de productie van PTHrP. PTHrP remt de di�erentiatie van chondro-cyten en de hypertro�e snelheid. Mineralisatie wordt gestimuleerd door een andere groeifactor.Dit kan zowel MMP zijn, dat de extracellulaire matrix rond de hypertrofe chondrocyten af-breekt, als VEGF, dat de ingroei van bloedvaten veroorzaakt.

Mechanische krachten beïnvloeden ook de groei en ontwikkeling van het bot. Drukkrachtendoor naastgelegen structuren en trekkrachten van aanhechtende spieren kunnen de kraakbeensca�old vervormen tijdens de groei. Mechanische belasting kan ook de gen- en eiwitexpressievan regulerende groeifactoren beïnvloeden.

De uiteindelijke lengte en vorm van een bot zijn dus het resultaat van de botgroei, gereg-uleerd door groeifactoren, en de mechanische belasting tijdens de groei. Er zijn veel groei-stoornissen, zoals heupdysplasie, waarbij een abnormale mechanische belasting kan resulterenin een abnormale geometrie van het bot. Het is echter onbekend hoe groot de invloed vanmechanische krachten is op de uiteindelijke vorm van een bot.

Het doel van deze studie is om een model op te stellen waarmee het mogelijk is om debijdrage van mechanische belasting op de ontwikkelende vorm van een foetaal bot te bepalen,in het geval matrixproductie, proliferatie en hypertro�e worden gereguleerd door groeifactoren.

Een histomorfologische studie is uitgevoerd op de derde middenvoetsbeentjes van muizen-embryo's om karakteristieke lengtes van deze elementen te verzamelen van verschillende ont-wikkelingsstadia. De begintoestand van de simulatie komt overeen met de distale helft vaneen metatarsus van een 16,5 dagen oude muizenembryo. Allereerst zijn de parameters vanhet model ingesteld, zodat de karakteristieken van het model een weergave zijn van de E19,5toestand na een botgroeisimulatie van drie dagen.

Met deze parameters hebben we enkele aanvullende simulaties uitgevoerd. Allereerst zijnwe begonnen met een begintoestand welke ongeveer E14 weergeeft. In deze simulatie ont-wikkelde het model tot een vorm die vergelijkbaar was met de eerdere begintoestand vanE16,5.

Daarna hebben we enkele stoornissen gesimuleerd waarin de groeifactor productie verzwaktwas. In al deze simulaties kwamen de zonehoogte veranderingen, vergeleken met de normale

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ontwikkeling, overeen met kenmerken van bekende in vivo stoornissen. Hoewel deze stoor-nissen allemaal leiden tot dwerggroei, was de totale botlengte in al deze simulaties juist groter.Hoogstwaarschijnlijk is dit een korte termijn e�ect.

Als laatste hebben we mechanische druk- en trekkrachten opgelegd op het bot tijdens degroei. Hieruit bleek dat mechanische belasting tijdens groei kan leiden tot belangrijke veran-deringen in de geometrie van het bot.

Dit model kan een bruikbaar hulpmiddel zijn om de ontwikkeling van foetale lange bottente bestuderen, rekening houdend met het e�ect van groeifactor productie, mechanische invloe-den en hun interactie.

Contents

1 Introduction 1

2 Theory 3

2.1 Anlage development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The growth plate, zones and function . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Regulation of bone growth through growth factors . . . . . . . . . . . . . . . . 6

2.4 In�uence of mechanical loading on bone growth . . . . . . . . . . . . . . . . . . 9

2.5 Disorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Histomorphological study 11

3.1 Materials and method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Model description 15

4.1 Initial mesh and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.2 Growth factor production . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.3 Growth factor regulations and zone transitions . . . . . . . . . . . . . . 18

4.2.4 Growth by proliferation, matrix synthesis and hypertrophy . . . . . . . 18

4.2.5 Di�usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Results 23

5.1 Growth and development E16.5 - E19.5 . . . . . . . . . . . . . . . . . . . . . . 23

5.1.1 Growth factor di�usion . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1.2 Bone growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1.3 Zone transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Growth and development E14 - E18 . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Impaired growth factor production . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 Mechanical loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusion and Recommendations 37

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Recommendations for further research . . . . . . . . . . . . . . . . . . . . . . . 38

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Bibliography 43

Appendices 49

A Extension of the 1D FE model to 2D 49

Chapter 1

Introduction

The skeletal development can be divided into two stages: skeletal patterning and ossi�ca-tion. In the �rst stage the skeletal elements are formed as soft tissue templates [22] and inthe second phase the anlagen ossify gradually and increase in length. Ossi�cation starts inthe center of the anlage and later secondary ossi�cation centers are formed in the centers ofthe epiphyses at both ends of the bone [35]. Between these regions of ossi�cation a cartilagi-nous disc, the growth plate, will remain. During the process of endochondral ossi�cation thechondrocytes undergo di�erent stages, which are morphological and biochemical unique [70].The transitions between these stages are tightly regulated by growth factors [26, 28, 48].

Mechanical forces a�ect the growth and development of the bone [11, 36, 42]. Compressiveforces by adjacent structures and tensile forces of attached muscles can deform the cartilagesca�old. Mechanical loading can also a�ect the gene and protein expression of regulatinggrowth factors [54, 74].

There are many growth disorders, like hip dysplasia, during which abnormal bone shapesdevelop. This di�erent shape is a result of abnormal loading of the developing bone [52].Although many research is done to bone growth during mechanical loading [53, 58, 66], it isstill not exactly known what the e�ect of mechanical loading is on the �nal shape of the bone.

The aim of this study is to establish a model that enables to quantitatively assess thecontributions of mechanical loading on the developing shape of a fetal long bone in growthfactor controlled bone growth.

Bone grows as a result of the combination of matrix production, proliferation and chondro-cyte maturation (hypertrophy) [32, 67]. To study the e�ects of mechanical loading on thesegrowth processes requires the possibility to relate them to the local mechanical conditions[57]. In addition, stresses and strains in the environment of mechanosensitive cells which ex-press regulatory growth factors are required. The FE method enables such evaluation of thedistributions of mechanical quantities.

Brouwers et al. [7] developed a 1D �nite element model of a developing fetal bone where allgrowth processes were growth factor controlled. The most obvious way to extend this model to2D is to develop a description where several of these 1D descriptions of di�erentiation are usedin parallel. This approach can be found in appendix A. However, although 2D di�usion canbe included, the columns with cells are only orientated in the longitudinal direction. Radialgrowth is impossible. Also, it included a static mesh, which was not able to deform due tomechanical forces or asymmetrical growth.

Therefore, we aim to follow a new approach in which the growth factor controlled, 2D

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2 CHAPTER 1. INTRODUCTION

temporo-spatial �nite element model is not static. To make the model more geometricallyrealistic, we will base it on the geometry of the distal half of a mouse metatarsal. A histo-morphological study on mice metatarsals will be performed to collect characteristic lengths ofthese elements at di�erent embryonic ages. Subsequently the model will be applied to threeconditions to evaluate the applicability. First, it will be extrapolated to a time period beyondthe period for which it was evaluated. Second, the e�ects of impaired growth factor productionwill be compared with corresponding knock-out mice. Finally, changes in development will beevaluated if mechanical forces are applied during the growth process.

In chapter 2 an overview is given of the composition and function of the growth plate,the regulation of growth factors and the in�uence of mechanical loading. Chapter 3 describesthe histomorphological study and shows the results. Chapter 4 describes the developed FEmodel and the parameter estimation. In chapter 5 the results of the di�erent simulations aredescribed and discussed. In chapter 6 the conclusions and recommendations can be found.

Chapter 2

Theory

2.1 Anlage development

In embryonic development, mesenchymal cells in the center of the limb bud aggregate anddi�erentiate into chondrocytes, forming a cartilage anlage [22, 37]. The mesenchymal cells atthe outside of the template di�erentiate in to a �brous layer, the perichondrium [22, 30]. Whenthe cartilaginous sca�old is formed, cells in the center of the anlage begin to di�erentiate intohypertrophic cells. Once fully di�erentiated, the cells die by apoptosis and vascular ingrowthwill occur [13]. Through vascular invasion chondroclasts and progenitors of osteoblasts arebrought in, which will break down and replace the cartilaginous extracellular matrix (ECM)by a bone ECM [44], this is called the primary ossi�cation center.

Later on, secondary ossi�cation centers will form in the epiphyses, at the ends of the bone[10]. In the secondary ossi�cations centers the same processes of chondrocyte di�erentiationtake place as in the �rst ossi�cation center. Between the ossi�cation centers a cartilaginousdisc, the growth plate, remains. The di�erentiation process of cartilaginous tissue into boneis called endochondral ossi�cation (see �gure 2.1). Besides in skeletogenesis, endochondralossi�cation also occurs in fracture healing [4, 69].

Figure 2.1: Endochondral ossi�cation [27]

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4 CHAPTER 2. THEORY

Figure 2.2: Overview of the di�erent zones of the growth plate [51]

2.2 The growth plate, zones and function

The growth plate consists of several distinct anatomic zones with unique morphology andbiochemical properties [70] (see �gure 2.2). From the epiphysis to the metaphysis these zonesare [24, 35]:

• The resting zone (RZ). These cells are small and round and randomly dispersed. Thiszone acts as a reserve zone of cells that can change into di�erentiating chondrocytes.The ratio of extracellular matrix to cell volumes is high and the extracellular matrixconsists mainly of collagen type II and proteoglycans [5].

• The proliferative zone (PZ). These cells are also small but �attened and once miner-alization has initiated and ossi�cation proceeds in one direction, the cells above theosseous border are arranged in columns [3, 10]. This zone contributes to longitudinalbone growth by proliferation and production of type II collagen and proteoglycans. Ithas been found that proliferative chondrocytes within one column show a regular patternof cell divisions, which is not synchrone with the adjacent columns. Columns next toeach other can also vary in the extend of proliferation [18]. In this zone one cell dividesapproximately 4 times [18, 67] and per column eight cells from the proliferative zone ofa rat growth plate are turned into early-hypertrophic cells per day [5].

• The early hypertrophic zone (eHZ), also called maturation zone. In this zone the cellsstop proliferating and start to di�erentiate. The cells produce Indian Hedgehog (Ihh)[25, 28]. The extracellular matrix that is synthesized by these cells consists mainly oftype II collagen [5].

• The hypertrophic zone (HZ). The cells in this zone continue to hypertrophy, and at theend of this zone, the total volume of the cell can be increased 5 till 10 times. Thisvolume increase is caused by swelling and synthesis of organelles [8, 17]. Wilsman etal. [71] showed that the increase in chondrocyte height contributes 44 − 59% to thebone length growth. Among other growth factors, the cells produce vascular endothelial

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growth factor (VEGF) and matrix metalloproteinases (MMP) and synthesize type Xcollagen [5, 48]. A chondrocyte spends approximately 48 hours in the growth plate tocomplete hypertrophy [18].

• Mineralized cartilage. The terminally di�erentiated cells become apoptotic and theirmatrix serves as a template for mineralization. Breur et al. [6] showed that the trans-formation of hypertrophic cells into metaphyseal bone occurred regardless of the size ofthe cells or the �nal matrix volume. There are several processes that will �nally lead tothis replacement of cartilage into bone. First, the ECM is degraded by MMP's and thehypertrophic cells die by apoptosis and are removed [44]. Both processes provide spacefor the ingrowth of vascular channels. Small blood vessels will grow from the perios-teum [13] and the metaphysis to the mineralized cartilage, stimulated by the presenceof VEGF in the hypertrophic zone. Second, matrix vesicles, which contain Ca2+ andenzymes which are responsible for the turnover of the matrix, are deposited into the sur-rounding extracellular matrix in the hypertrophic zone, especially at the septa betweenthe columns [5]. VEGF also acts as a chemoattractant of osteoclasts and osteoblasts.The �rst mineral consists mainly of poorly crystalline hydroxyappatite, the primaryspongiosa. By remodeling, this will be replaced by more mature bone, the secondaryspongiosa [19].

The bone growth rate of a growth plate is dependent on several parameters, like theproliferation rate, the �nal size of the hypertrophic chondrocytes and the matrix synthesisrate [17, 32, 24, 71]. In a stable growth plate the rate of cell production in the PZ is equal tothe rate of cell loss in the HZ and the number of cells per column is constant over the period ofa couple of days [6]. However, during the stages of development from foetus till puberty, thenumber of cells per column and the proportion of proliferative cells varies [1, 24, 49]. Growthrates of di�erent bones may vary by a factor of seven or more [6, 71] (see table 2.1).

The growth plate of a long bone has a discoidal shape. There are regional di�erencesbetween the zones in the central and the lateral parts of the growth plate [2, 38] (see �gure2.3). In the lateral parts the total height of the growth plate is approximal 10% larger thenin the central part. The height of the PZ is 20-30% larger in the lateral part than in thecentral part. The height of the HZ is 20% larger in the central part. Although these studiesincluded older and further developed long bones with secondary ossi�cation centers, it seemsreasonable that similar regional changes are present before secondary ossi�cation.

At skeletal maturity, the growth plates of larger mammals close and longitudinal growthceases. Growth plates of smaller rodents are maintained into old age, although structuralchanges in the morphology of these growth plates prevent further longitudinal growth after

Table 2.1: Relative contributions to daily total elongation at the chondro-osseous junction. [71]

proximal distal distal proximaltibia radius tibia radius

proliferative zone cell duplication 9% 10% 9% 7%contribution matrix synthesis 18% 16% 21% 21%

hypertrophic zone matrix synthesis 14% 16% 19% 28%contribution cellular enlargement 59% 58% 51% 44%

growth rate per day 396 µm 269 µm 138 µm 47 µm

6 CHAPTER 2. THEORY

Figure 2.3: Photomicrograph of growth plate showing the central region (C), lateral region (L),germinative zone (GZ), epiphyseal end (EP) and metaphyseal end (MP). Growth plate is divided intoresting (R) proliferative (P) and hypertrophic zone (H). Haematoxylin Eosin X 63. [2]

skeletal maturity [49].

Around the cartilage anlage is a �broblastic cell layer, the perichondrium. Besides en-dochondral ossi�cation, also perichondrial ossi�cation takes place, a process whereby boneis formed as a shaft around the cartilage anlage. When hypertrophy starts around day 13of mouse embryonic development (E13), perichondrial cells �anking the hypertrophic zonesstart to di�erentiate into osteoblast [30]. This part of the perichondrium is now called perios-teum. These osteoblasts express Cbfa1 which forms in the inner layer of the perichondrium,around the cartilaginous core, a mineralized structure, termed the bone collar (see �gure 2.2)[30, 60, 69].

2.3 Regulation of bone growth through growth factors

For adult men the di�erences in lengths between the left and right leg are only 0.3% forthe femur and 0.6% for the tibia [63]. Therefore, bone longitudinal growth must be a tightlyregulated process. For the orderly progression of bone growth, a maintained balance betweenproliferation and hypertrophy is required. Many di�erent growth factors are involved in thisprocess. It is generally accepted that proliferation and di�erentiation of the chondrocytes islargely coordinated by a negative-feedback loop of Indian Hedgehog (Ihh) and ParathyroidHormone relate Peptide (PTHrP) [33, 34, 35, 48] (see �gure 2.4).

Cells in the early hypertrophic zone produce Indian Hedgehog (Ihh). Ihh has a dualfunction in the growth regulation. First, Ihh stimulates the chondrocyte proliferation in aPTHrP-independent pathway, through activation of the transmembrane protein smoothened(Smo) [28, 30, 69, 74]. In Ihh−/− and Smo−/− mice it has been shown that the proliferationrate is reduced to a basal level that is only 50% of the normal proliferation rate [28]. Ihhwill also indirectly inhibit di�erentiation and hypertrophy, by stimulating PTHrP production[56]. PTHrP is expressed by the periarticular perichondrium cells [23, 31, 41]. These PTHrPexpressing cells are aligned around the head of the bone, as was shown by Karperien [29](�gure 2.5). It has been shown that Ihh expression resulted in increased levels of the Ihhtransduction pathway receptors, Ptc-1 and Gli [25, 26, 35]. So, with increasing levels of Ihh,the e�ect of Ihh on the PTHrP expression will increase by upregulation of the receptors.

It is assumed that PTHrP can di�use through the growth plate and it will bind to itsreceptor PPR which is expressed in the late proliferative and early hypertrophic chondrocytes

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Figure 2.4: The signaling pathway of PTHrP and Ihh. PTHrP is secreted by articular perichon-drium cells, overlaying the bone head. Ihh is secreted by early hypertrophic cells. 1) PTHrP inhibitsdi�erentiation 2) Ihh stimulates proliferation and 3) Ihh stimulated PTHrP synthesis [35].

Figure 2.5: PTHrP mRNA expression in pre-cartilage condensations of the extremities at day E12.5[29]

8 CHAPTER 2. THEORY

Figure 2.6: VEGF and MMP-13 are expressed by hypertrophic chondrocytes and lead to the miner-alization of cartilage [44].

[33]. When PTHrP is above a certain critical concentration it will keep chondrocytes in theproliferating pool, preventing them to di�erentiate into early hypertrophic cells [55]. Whileearly hypertrophic cells will still further di�erentiate into hypertrophic cells, this will resultin less early hypertrophic cells, and in a decrease in the Ihh concentration. As a result lessPTHrP is expressed, which permits the late proliferative chondrocytes that are not su�cientlystimulated by PTHrP to start maturation [32]. In this way this signaling pathway regulatesthe rate at which the cells leave the proliferative zone and irreversibly become hypertrophiccells. Another e�ect of PTHrP is inhibition of maturation processes of chondrocytes in theeHZ and HZ [23, 31].

Matrix metalloproteases (MMP's) are able to cleave a variety of ECM proteins, extra-cellular non-ECM proteins and cell surface proteins [44, 57, 60]. MMP-13 is produced bythe chondrocytes and osteoblasts in the lower hypertrophic zone and at the chondro-osseousjunction. MMP-9 is expressed by endothelial cells, osteoclasts and other cell types proximalto the chondro-osseous junction. MMP-9 and MMP-13 act synergetic and coordinate thedegradation of the ECM and the unmineralized septa of hypertrophic cells. It is thoughtthat the remodeling of ECM will induce an alternation in the environmental stress sensedby hypertrophic cells, leading to apoptosis [22]. Together with VEGF, which is produced byhypertrophic cells, MMP-13 induces several processes, like angiogenesis and attraction andactivation of osteoclasts and osteoblasts. These processes lead to the mineralization of thecartilage at the chondro-osseous junction [44] (see �gure 2.6). Based on this knowledge ascheme can be made presenting the expression locations of these growth factors and theire�ects (�gure 2.7).

It is not likely that the periarticular cartilage is still a relevant source of PTHrP aftersecondary ossi�cation, because the distance between the growth plate and the articular surfacehas increased. Besides, the secondary ossi�cation center (SOC) will in�uence the di�usion ofPTHrP. Tsukazaki et al. [64] showed that the expression level and localization of PTHrPwas di�erent before and after birth. Furthermore it has been shown that the expressionpatterns of Ihh and PPR in the growth plates change between the fetal and postnatal stages

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VEGF/MMP’s

Ihh

PTHrP

PZ

eHZ

HZ

BONE

Articularperichondrium

Figure 2.7: The position of the growth factors depicts its release site. Arrows between zones indicatecell di�erentiation. Solid lines represent inducing e�ects, dotted lines represent inhibiting e�ects [7].

of development [25, 70]. Therefore it is suggested that after secondary ossi�cation, anothersignalling pathway, probably through TGF-β, regulates the proliferation and di�erentiationof chondrocytes [5]. Alternatively, a local control loop involving PTHrP and Ihh may existwithin the growth plate itself [16].

Many more growth factors are involved in endochondral ossi�cation. The most prominentare shortly mentioned here. Fibroblast growth factors (FGF) can modulate the PTHrP-Ihhregulating feedback loop, by down-regulating the Ihh signalling pathway. FGF signaling alsonegatively regulates proliferation and di�erentiation [15, 37, 43, 48].

Transforming growth factor-beta (TGF-β) is produced by the perichondrial cells and is ac-tivated by Ihh. TGF-β acts on the perichondrial and periarticular cells to increase PTHrP syn-thesis [5]. TGF-β also inhibits chondrocyte di�erentiation in adolescence, when the PTHrP-Ihh feedback loop is no longer active [5]. TGF-β has also been shown to inhibit angiogenesisin growth plates [46].

Bone morphogenetic proteins (BMP's) are members of the TGF-β superfamily and areinvolved in the proliferation and maturation of chondrocytes [43]. Through misexpressionstudies it has been found that BMP receptors regulate the progression of chondrocyte di�er-entiation and may function as a intermediary relay that links Ihh signalling to the PTHrPpathway [56, 68]. BMP's seem to have opposite functions during chondrocyte developmentthan FGF [37, 48].

IGF-I is an autocrine factor which stimulates increased rates of cell division.

Most growth factors have their e�ects at distinct locations from where they are expressed.Because it takes time for the proteins to reach the site where they act, the regulation isdynamic in time and space.

2.4 In�uence of mechanical loading on bone growth

Besides growth factors, also mechano-biological factors in�uence skeletal ossi�cation [12,20]. Mechanical loading can in�uence growth factor controlled processes in two ways, namelythrough altering the expression of genes or proteins and by interacting with their transport[42, 61, 74]. Several studies showed the in�uences of muscular contractions [62] and di�erent

10 CHAPTER 2. THEORY

mechanical loadings, like cyclic or static compression, tension, torsion, or bending on thedevelopment of bone and during fracture healing [9, 11, 36, 39, 40, 54, 58, 59].

In general, it was found that longitudinal bone growth is retarded by compression and thatdistraction or reduced compression will stimulate longitudinal growth [12, 40]. This responseof the growth plate to mechanical loading was established in the Hueter-Volkmann law.

This �nding is also con�rmed with studies where incisions where made in the relativelysti� �brous perichondrium to reduce the tensional force of the perichondrium [20] and byincreasing the sti�ness of the perichondrium by DL-penicillamine [21]. These interventionslead to an increase and decrease in longitudinal growth, respectively.

Lerner et al. [36] suggested that in the embryonic phase biochemical factors are dominantover mechanical factors. They showed that mechanical stresses only play a modulating rolein the development of the growth plate. In experiments where mechanical stresses were re-moved, bone growth and shape development continued only at signi�cantly reduced rates. Inthe postnatal development, this trend seems to switch. Carter [11] assumed that biologicalfactors play an important role only in the initial phases of bone growth and ossi�cation andthat their in�uences disappear over time. Mechano-biological in�uences, however, remain im-portant throughout life.

2.5 Disorders

A lot of research is done to disorders of the growth plate. In most cases these disordersare caused by mutations leading to decreased or increased transcription of a growth factoror its receptor. For example, in transgenic mice, lacking either PTHrP or its receptor PPR,accelerated di�erentiation and premature hypertrophy are observed, leading to dwar�sm. Inhuman this disorder is called Blomstrand chondrodysplasia [33, 51, 56]. On the other hand,overexpression of PTHrP in mice resulted also in dwar�sm, because di�erentiation and min-eralization are delayed. In human, this mutation has been identi�ed as the cause of Jansen'smetaphyseal chondrodysplasia [31, 50, 56].

As mentioned also in section 2.3, Ihh−/− mice have a decreased proliferation rate, whichresults in a shorter proliferation zone. Furthermore an increase in the length of the HZ hasbeen seen, which is an e�ect of the decreased PTHrP expression [28, 33].

MMP-9−/− MMP-13−/− mice display an accumulation of hypertrophic chondrocytes,delayed trabecular bone and bone marrow cavity formation, and also a delay in secondaryossi�cation [5, 44].

Chapter 3

Histomorphological study

3.1 Materials and method

The mesh of the 2D FE model, is based on the morphology of embryonic mice metatarsals.Metatarsals are the long bones in the middle of the hind feet (�gure 3.1). Compared to otherlong bones, metatarsals are relatively small, fast developing bones and their growth platesconsist of clearly distinguishable zones.

For our histomorphological study we obtained dissected hind feet of E16.5, E17 and E19.5mice from dr M. Karperien, LUMC. The feet are embedded in para�ne and with a microtome(Leica, EG 1140C) approximately 40 slices of 6µm were cut in the transversal plane. Afterdewaxing and rehydrating the samples were stained with Hematoxylin Eosin, Safranin O/FastGreen, Masson trichrome or Picrosirius Red to obtain a broad set of di�erently stained slidesof the skeletal element. The samples were imaged with an inverted light microscope (Zeiss,Axiovert 200 M) at a magni�cation of 20x. A reference image of a cell counter plate with agrid of 1 mm x 1 mm was made, to measure the actual length of the elements. The image areacovered 546.9 by 406.0 µm (2088 x 1550 pixels). Overlapping images were assembled using anopen source photo editor (The Gimp 2.2.10).

From each set of slides the slide with the largest longitudinal section through the thirdmetatarsal was taken for data collection. The E17 and E19.5 elements where not sliced ex-actly parallel to the longitudinal axes, hence the images of several slides were combined toapproximate the total cross section of the element (�gure 3.3).

Figure 3.1: Schematic representation of the anatomy of the hind feet of mice [14].

11

12 CHAPTER 3. HISTOMORPHOLOGICAL STUDY

Figure 3.2: Schematic representation of the diverging columns in the E16.5 situation.

In each image the boundaries of the zones were determined, based on the di�erences instaining and/or cell size. The average length and width of each zone and the number ofcolumns were measured. For every zone, the number of cells was estimated.

3.2 Results

The metatarsals of E16.5 and E17 contained only resting, proliferating and hypertrophicchondrocytes. They did not di�er much in composition, and a length increase of approximately150 µm was measured. In the E19.5 metatarsal a relatively large mineralization area can beseen and the length of the metatarsal is doubled in comparison to the E16.5 sample. Fromthese three length measurements, we �nd a longitudinal growth rate of 286 µm per day duringthis stage of development. All measured characteristic lengths are summarized in table 3.1.

The width of the anlage decreases from the RZ to the HZ. In all three elements a constric-tion in width can be seen in the top part of the resting zone, in this region the perichondriumseems thicker. In the E19.5 anlage constrictions in width can also be seen around the PZ andthe mineralized area. The shape of the distal end of the metatarsal is �attened at the top androunded at the lateral sides. The proximal end has a more complex foot-like shape. For thisresearch we will focus on the distal end of the bone.

The cells of the RZ are small and lay randomly dispersed in the zone. The PZ cells arealso small and are arranged in columns. These columns are diverging in stead of aligned alongthe longitudinal axes of the bone (see �gure 3.2). As a result, there are more columns in thePZ then in de eHZ and HZ.

In the early hypertrophic and hypertrophic zones the cell heights increase gradually. Hun-ziker has shown, for 21 day old rat tibia, that cells in the PZ have an averaged height andwidth of 8±1 µm and 22±4 µm and cells in the hypertrophic zone are 31±2 µm high and29±9 µm wide on average [24]. Thus the cells will expand more in height than in width. Thiscorresponds with our observations.

The bone collar, a calci�ed ring, is observed around the hypertrophic zones. This ringis thicker around the terminally hypertrophic zone than around the early hypertrophic zone.The E17 slides were probably stained to long, resulting in a dark line in the PZ region (�gure3.3B). The number of cells in the PZ could therefore not been measured.

13

Figure 3.3: Scaled microscopic images of 3rd metatarsals of A) E16.5, stained with HE, B) E17,stained with Safranin O/Fast Green, C) E19.5, stained with Safranin O/Fast Green

Table 3.1: Characteristics of the distal half of embryonic mice metatarsals of E16.5, E17 and E19.5and the dimensions of the initial mesh, based on the E16.5 metatarsal, [µm]

age E16.5 E17 E19.5 FEM mesh

total bone length 857 1000 1710length distal half 443 499 908 450

height RZ 197 209 361 200width RZ 187 183 291 200# cells in RZ 539 470 885 480

height PZ 95 141 220 100width PZ 186 186 234 180# columns in PZ 16 15 16 12# cells in PZ 217 x x 240

height eHZ 48 43 54 52.5width eHZ 181 192 239 165# columns in eHZ 12 12 13 12# cells in eHZ 74 72 50 72

height HZ 103 106 73 97.5width HZ 150 172 211 150# columns in HZ 10 11 11 12# cells in HZ 76 72 55 72

height bone 0 0 200 0

Chapter 4

Model description

4.1 Initial mesh and boundary conditions

The initial mesh is a representation of the third metatarsal of a 16.5-days old embryonicmouse. Only the distal half of the anlage is modeled, and consists of 864 plain strain porepressure elements (CPE4P) (�gure 4.1). In the initial mesh every element corresponds to onecell and its surrounding extracellular matrix and is labeled with a cell type number. In theinitial mesh the elements of the RZ and PZ have heights of 5 µm. The heights of the elementsin the hypertrophic zones increase gradually (�gure 4.1).

The elements grow due to ECM production by the cells and by hypertrophy [6]. Thenumber of cells in the anlage increases due to proliferation. The number of elements in themesh remains equal and remeshing is not included in this model. Therefore, the size of someelements will become larger than physiological dimensions of the cells.

Twelve columns are aligned along the total length of the mesh. The number of columns isequal in every zone. The heights of the zones in the mesh correspond with our morphologicaldata. The number of cells per column in every zone is based on the total number of cells inthe zone divided by the number of columns in the mesh.

The bone collar is a broad ring around the anlage. The mesh is not axisymmetric but 2D,and the bone collar could not be simulated with ring elements. In stead, the function of thebone collar, to constrict the hypertrophic zones, is simulated with two-node linear displace-ment elements (T2D2). These line elements are orientated perpendicular to the longitudinalaxes of the bone (�gure 4.1). We assumed that around the HZ the sti�ness is 2 times higherthan around the eHZ. The Young's moduli were 5.e-2 and 2.5e-2 MPa, respectively, and weredetermined by trial and error. These values lead to a HZ width that was comparable with theresults of the morphological study.

Boundary conditions

The nodal displacements at the bottom plane of the mesh were con�ned in the vertical direc-tion. On all sides, except the bottom, which is the cross section through the bone, zero porepressure was prescribed, i.e. �uid can �ow in and out freely.

15

16 CHAPTER 4. MODEL DESCRIPTION

Figure 4.1: Initial mesh of the simulation, representing the distal half of a E16.5 third metatarsal.The height of the zone in µm is given between brackets. The articular perichondrium (APC) is alignedaround the bone end. Line elements (grey) are aligned along the rows in the hypertrophic zones.

4.2 Implementation

4.2.1 Mechanical properties

The tissues were assumed biphasic, consisting of a compressible solid part, representing theproteoglygans and collagens, and an incompressible �uid part, the cytoplasm and interstitial�uid. For the behavior of the solid part a compressible Neo-Hookean model was used [73].

σ = Kln(J)

JI+

G

J(F · FT − J2/3I), (4.1)

where J is the determinant of the deformation tensor F. The bulk (K) and shear (G) moduliare de�ned as

K =E

3(1− 2ν)(4.2)

G =E

2(1 + ν)(4.3)

where E is the Young's modulus and ν the Poisson's ratio. For each tissue type the perme-ability was set to 3e-2 mm4/Ns. The Young's modulus of immature cartilage is 4 MPa andthe Poisson ratio is 0.47 [12]. For newly formed bone a Young's modulus of 500 MPa and aPoisson's ratio of 0.3 is used [12]. For the cartilage elements a �uid fraction of 0.75 is assumed.

4.2.2 Growth factor production

In the present model, Ihh, PTHrP and MMP's are incorporated. Unlike the 1D model ofBrouwers et al. [7], MMP's were used as the mineralization regulating growth factor, in steadof VEGF, because vascular ingrowth is not included in the model.

Growth factor production is applied to the set of producing elements as a distributed �ux(DFLUX).

17

Elements increase in size due to matrix production or hypertrophy. This will result in lowerconcentrations if the production rates per element are constant. To maintain the concentra-tion levels during growth, the initial production rates are multiplied with a scaling factor,dependent on the area of the zones. Several measures were tested. The square root of thevolume increase of the zone was used as the scaling factor. With this factor concentrationlevels were maintained constant. The growth factor production rates of Ihh (aI) and MMP(aM ) per element than become

aI = a0I ·

√A(eHZ)

A0(eHZ)

and (4.4)

aM = a0M ·

√A(HZ+bone)

A0(HZ)

, (4.5)

where the zone volumes in the initial mesh serve as reference zone volumes (A0).It is known from literature [45, 75] that proliferation is stimulated at Ihh concentrations

ranging from 0.05 - 0.5 µM. Therefore, the concentration of Ihh in the eHZ is assumed to be0.5 µM. By trial and error the initial production rate for Ihh was de�ned (table 5.1).

Concentration levels of MMP's are not known, therefore the MMP production rate isnormalized.

In the FE model the articular perichondrium (APC) is not modeled as a separate structureor several elements. Instead, PTHrP is produced as a concentrated �ux (CFLUX) in the nodesat the outline of the bone head (�gure 4.1). PTHrP expression is stimulated by Ihh. Such anenzyme-substrate binding can be modeled with a Michaelis-Menten equation

a0P = b

cI

cI + KP, (4.6)

where aP is the synthesis rate of PTHrP, b is the maximal synthesis rate, cI is the localIhh concentration and KP is the concentration of Ihh at which half of the maximal PTHrPsynthesis rate is reached. This equation agrees with the fact that in absence of Ihh no PTHrPis produced. Ihh upregulates the expression of PTHrP through its receptors. A Hill equation;a Michaelis-Menten equation with all components raised to a power, accounts for this e�ect.With a Hill coe�cient of four, the resulting PTHrP production rate will have a sigmoid shapewith increasing Ihh concentration.

a0P = b

c4I

c4I + K4

P

(4.7)

For each APC node the production rate was calculated. The normal average Ihh concentrationin the APC nodes (KP ) during the simulation was 0.0188. From literature it was known thatPTHrP concentrations are in the range of 0.1 - 10 nM for the proliferating zone [75] andalkaline phosphatase expression, a marker for hypertrophy, is stimulated between 0.001 and0.1 nM PTHrP [75]. Using these data the maximum production rate of PTHrP was determinedby trial and error (table 5.1).

The production rate of PTHrP is scaled to account for growth of the periarticular peri-chondrium. The ratio of the actual length of the periarticular perichondrium and the initiallength,

aP = a0P ·

l(APC)

l0(APC)

, (4.8)

is used as the scaling factor for PTHrP production.


4.2.3 Growth factor regulations and zone transitions

As described in paragraph 2.3 the proliferation rate is dependent on the Ihh concentrationin the PZ. Proliferation rate depends also on the number of PZ cells, because in a largerproliferating zone a higher chance of proliferation exists at every moment. The measure forthe occurrence of proliferation is de�ned as

Plevel = a · nrPZ +nrPZ∑ c4

I

c4I + K4

pr,I

, (4.9)

where a is a scaling factor, such that without Ihh production a 50% decreased proliferation ratewill remain [28]. nrPZ is the number of PZ elements in each column, cI is the concentrationIhh and Kpr,I is the concentration Ihh at which half of the maximum proliferation rate isreached.

If Plevel is above the critical level (dI) (table 5.1), proliferation will occur in this column.Proliferation, is implemented as a transition of a RZ element to a PZ element. All RZ elementsin this column will enlarge to compensate for the transferred cell. The assumptions that percolumn 8 PZ cells will turn into eHZ cells per day [5, 18] and that in a stable growth plate thenumber of cells remain constant for a given time interval, will lead to the assumption that 8PZ cells per column will proliferate per day. Plevel will result under normal Ihh concentrationsin a proliferation rate of one cell per three hours per column.

The threshold value of PTHrP which determines the distance between the PTHrP source(the periarticular perichondrium) and the location where the cells stop proliferating and startto hypertrophy [55] is 0.1 nM [75]. If the PTHrP concentration in a PZ element adjacent tothe early hypertrophic zone is below this threshold concentration (dP ), this element will turninto an early hypertrophic element.

The size of the cell is taken as the criterion for the transition from eHZ to HZ, as wasproposed by Brouwers et al. in their 1D model of the growth plate [7].

MMP's degrade the extracellular matrix and induce apoptosis. Hence, bone formationthrough MMP's is dependent on the concentration of MMP's in the hypertrophic chondrocytesand the duration of MMP exposure on the ECM. Therefore, in the FE model the concentrationof MMP's in each hypertrophic cell is superimposed on the concentration for this cell in theprevious time step. No critical level of MMP concentration is known, it is only known fromour morphology data that mineralization will start between E17 and E19.5 and that the totalheight of the formed bone on E19.5 is approximately 200µm. The production rate of MMPand critical level (dM ) are adjusted to these data.

For all cell zone transitions the maximum number of cells per column that can turn intothe next cell type at the same moment is limited to one.

4.2.4 Growth by proliferation, matrix synthesis and hypertrophy

Growth as a result of matrix production, proliferation and hypertrophy was simulated byswelling of the tissue. Using the biphasic swelling model of Wilson et al. [72]. This biphasicmodel includes a description of osmotic swelling behavior. It assumes that electrolyte �ux canbe neglected in mechanical studies of charged materials. Hence, ion concentrations are alwaysin equilibrium. The swelling pressure is then given by

4π = RT√

c2F + 4c2

ext − 2RTcext (4.10)

19

where R is the gas constant, T the absolute temperature, cext is the external salt concentrationand cF the �xed charged density. When assuming that the external salt concentration andtemperature stay constant, the only non-constant in this equation is the amount of �xedcharges (cF ). This �xed charge density can be expressed as a function of the tissue deformation,as

cF = cF,0(nf,0

nf,0 − 1 + J) (4.11)

where nf,0 is the initial �uid fraction of the tissue, cF,0 the initial amount of negative chargesin the tissue and J the determinant of the deformation tensor.

Initially all negative charges were set to zero. Growth was induced by introducing a non-zero amount of �xed charges in the growing element. The amount of �xed charges dependedon proliferation, matrix synthesis and hypertrophy within the element, as computed in aseparate analysis described below. Due to this, �uid was attracted which caused the tissueto swell. The elements grew individually over two hours dependent on the amount of �xedcharges. Within these two hours equilibrium was reached. The tissue shape at the end of thesimulation was used as a stress free input for the next step, and the number of �xed chargeswas reset to zero. Hence, all stresses induced by growth were assumed to fully relax withintwo hours.

From our histomorphology study we observed that the daily increase in length of the bone,during this period of development, is approximately 286 µm per day. We assume that thedistal growth plate will contribute 50% of the total length growth. Wilsman et al. [71] havestudied the relative contribution of proliferation, matrix synthesis and hypertrophy to thelinear bone growth for several growth plates (table 2.1). As the growth contributions are notknown for the distal metatarsal growth plate, the percentages of the growth plate of the distaltibia was taken as a reference for our model.

As described above the tissue will grow due to introducing a non-zero amount of �xedcharges, cFx. Subscript x can be p,m or h corresponding to proliferation, matrix synthesis andhypertrophy. The values of cFx are determined in a 24h simulation including only proliferation,matrix synthesis or hypertrophy respectively (table 5.1). In these simulations the cFx weremanually adjusted, such that a length increase of 143 µm was obtained. cFp only has anon-zero value if Plevel is above the proliferation threshold value dI . cFp is added to the RZelements to compensate for the newly formed PZ cell.

We assumed that all cells produce the same amount of matrix independent on mechanicalconditions or growth factor concentrations. Hence, cFm is taken equal for all elements. Matrixsynthesis results in the presence of more matrix per element, however it is assumed that themechanical properties of the elements stay constant.

Hypertrophy is inhibited by PTHrP. The concentration PTHrP is decreasing from theearly hypertrophic to the hypertrophic cells, which will result in more hypertrophy in the HZcells than in the eHZ cells. The amount of added �xed charges to account for hypertrophy isgiven by

cFh = f ·Kh,P

Kh,P + 0.25 ·∑

nod=1:4

cP, (4.12)

Where f is the maximum growth rate, Kh,P is the average concentration of PTHrP in the


(e)HZ elements and cP is the nodal PTHrP concentration of the (e)HZ element. cFh is onlyadded to the (e)HZ elements.

4.2.5 Di�usion

The di�usion of the growth factors is described by Fick's Law in ABAQUS (MASS DIFFU-SION), using 864 two-dimensional di�usive elements (DC2D4). The concentrations of growthfactors are simulated as temperature pro�les, therefore it is not possible to produce severaldi�usion pro�les in the same simulation. For the three growth factors, three di�erent di�usionsimulations will be done after each other (�gure 4.2). Between the simulations, the relevantoutput was read as input for the next simulation.

In the physiological situation growth factors are continuously produced and degraded.However, this is not possible in the FE model. In stead a di�usion pro�le of each growth factoris calculated, which is assumed to be representative for the e�ective concentration pro�le ofthe growth factor. In these calculations concentrations from previous steps were neglected.This pro�le is calculated during a very short period, divided in 10 increments. The durationof this di�usion simulation is directly related to the distance that the produced growth factorswill cover. If the di�usion pro�le is calculated over a long period, the concentrations willaccumulate in the mesh, because the produced growth factors are not degraded. For a periodthat is too short, the growth factor may not di�use over the whole mesh. Because the meshis growing during the simulation, the period over which the di�usion pro�le is calculatedmust increase. However, this period is a �xed input parameter of the simulation. In steadof increasing the di�usion period, we adapted the di�usion coe�cient to the new mesh size.Obviously the di�usion coe�cient is a material parameter which cannot scale with tissue size.Hence, the increase of the di�usion coe�cient (D [m2/s]) is a trick to incorporate a longerperiod of di�usion. This new di�usion coe�cient is given by

D = D0 ·A

A0, (4.13)

where D0 is the initial di�usion coe�cient, A is the total area of the mesh and A0 is thetotal area of the initial mesh. Since D has no physical meaning, it can be chosen freely. Thevalue of D has been optimized for small computational time. The value of the initial di�usioncoe�cient was chosen such that the concentration pro�les correspond with the description ofthe growth factor distribution in a 0.5 sec. time period. For subsequent simulations aftergrowth, D is scaled with the surface (equation 4.13).

4.2.6 Simulation

The general simulation program is written in MATLAB. The bone growth simulations anddi�usion calculations are performed by ABAQUS.The simulated period of 3 days, from E16.5 till E19.5, is divided in periods of two hoursin which the elements grow, alternated with di�usion calculations (�gure 4.2). For the totalsimulation from E16.5 till E19.5 36 cycles are performed. It is assumed that during the twohours growth simulations, the growth factor concentrations remain constant. Between thesubsequent growth simulations, the element can change in cell type, dependent on previouslydescribed criteria.

21

Bone growth

• Proliferation

• Matrix synthesis

• Hypertrophy Diff

usi

on

Ihh

Diff

usi

on

PT

HrP

Diff

usi

on

MM

P’s

Zone transitionst = t + 2h

t = 0

If t = 72h

EXIT

Figure 4.2: Schematic representation of the simulation. Bone growth simulations of 2 hours areperformed, alternated with the calculation of di�usion pro�les. Cell zone transitions occur betweensubsequent growth simulations.

Chapter 5

Results

5.1 Growth and development E16.5 - E19.5

The parameters of the model were set as described in chapter 4 to simulate the normaldevelopment of the distal half of a E16.5 third metatarsal over three days. The result is theparameter set shown in table 5.1. The characteristics of the simulations using this parameterset are shown in this chapter.

5.1.1 Growth factor di�usion

The concentration pro�les of the growth factors at a certain moment are displayed in �gure5.1. Ihh is produced in the early hypertrophic zone. The highest concentration Ihh, in themiddle of the eHZ, is just above 0.5 µM, the value that was assumed for the concentrationof Ihh in the eHZ (paragraph 4.2.2). The Ihh concentration in the articular perichondrium isapproximately 0.018 µM, which corresponds to the parameter KP .

With this local Ihh concentration, PTHrP is produced at half the maximum rate, i.e. thenormal production level. PTHrP is produced by all nodes at the outline of the bone head.Because the lateral nodes of the articular perichondrium lay closer to the eHZ than the centralnodes, the Ihh concentration in the �rst nodes is higher. As a result, the production of PTHrPis higher in the lateral articular perichondrium. This results in a round PTHrP pro�le in thetop part of the mesh (�gure 5.1B). However, this pro�le will be �attened within a distanceof approximately 200 µm. The PTHrP concentration in the last PZ element is around thethreshold value 1.e-4 µM.

MMP's are produced in the HZ elements, and later also in the mineralized zone. Theconcentration is highest in the elements closest to the center of the bone. These elements willreach the threshold value �rst. Hence, mineralization will start in the center of the bone.

5.1.2 Bone growth

The growth parameters regulating matrix synthesis, proliferation and hypertrophy weremanually adjusted in a 24 h simulation to obtain 143 µm length increase similar to the mor-phology data, and the partial contributions as given by table 2.1. This resulted in a meshlength of 517.9 µm and 882.2 µm for the E17 and E19.5 situation, respectively. In the phys-iological situation the half length of the E17 and E19.5 elements were 500 µm and 855 µm

23

24 CHAPTER 5. RESULTS

Table 5.1: Parameters used in the model.parameter description value

aI production rate of Ihh 3.75

aP production rate of PTHrP 0.4e-6

KP constant for Ihh-dep. PTHrP production 0.0188

aM production rate of MMP 1

cFm cF increase by matrix synthesis 1.6e-2

cFp cF increase by proliferation 8.e-3

cFh cF increase by hypertrophy 2.2e-2

Kh,P constant for PTHrP-dep. hypertrophy 8.e-5

Kpr,I constant for Ihh-dep. proliferation 0.25

dvol threshold volume for eHZ - HZ transfer 2.05e-4

dI threshold value for proliferation 40

dP threshold concentration for PZ - eHZ transfer 1.e-4

dM threshold concentration for HZ - bone transfer 17.5

D0 initial di�usion coe�cient 0.02

Figure 5.1: Concentration pro�les of Ihh, PTHrP and MMP's, [µM].

25

Figure 5.2: Constrictions can be seen in the width of the bone (E19.5). The constrictions aroundthe hypertrophic zone are also observed in the results of the simulation, the constriction in the RZ wasnot seen in the simulation.

and thus the chosen growth parameters can simulate the longitudinal growth of the skeletalelement very well.

The maximum width of the mesh increased from 200.0 µm to 314.2 µm during the 72 hoursimulation. In the morphological data the bone width increased from 186 µm to 291 µm.

The initial mesh was tapered while in the E19.5 situation there are two constrictions inthe width of the mesh, above and below the terminally hypertrophic zone, as indicated byarrows in �gure 5.2. This corresponds to the shape of the bone at 19.5 embryonic days. Theconstriction in the RZ that was observed in the morphological study, was not observed inthese results.

5.1.3 Zone transitions

Initially the model starts with a stable growth plate including resting, proliferating, earlyhypertrophic and hypertrophic chondrocytes. The early hypertrophic zone is fully developedand is therefore producing a su�cient amount of Ihh to stimulate the PTHrP production in thearticular perichondrium and to keep the last PZ element in the proliferating zone. Initially onlythe lengths of the PZ and the hypertrophic zones increase due to proliferation and hypertrophy,respectively. After several hours an eHZ element becomes larger than the threshold value andturns into a HZ element. Now, less eHZ elements are present and the Ihh concentration drops.As a result, the PTHrP concentration also decreases, and its concentration in late PZ elementsdrops below the threshold concentration. This results in a transition of a PZ element into aneHZ element. During the �rst 30 hours of simulation, we see that the heights of the RZ andeHZ remained approximately equal and the heights of the PZ and HZ increased approximately1.5 and 3 times, respectively, by proliferation and hypertrophy (�gures 5.3 & 5.4).

From our histomorphological study we found that mineralization will start somewherebetween E17 and E19.5, and that the total length of mineralized cartilage would be around200µm at E19.5. We have chosen the threshold concentration of MMP for mineralization such,that mineralization will start around E18.0 (�gure 5.4). The zone heights of the simulation atE19.5 and the corresponding histological data are listed in table 5.2.


Figure 5.3: Resulting bone length and shape during simulation, with intervals of 12h.

16.5 17 17.5 18 18.5 19 19.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

embryonic age

zone

leng

th [m

m]

total lengthRZPZeHZHZmin. cartilage

Figure 5.4: Lengths of the di�erent zones during simulation.

27

Table 5.2: Final zone lengths, in µm, after simulating bone growth for 72h, compared to physiologicallengths of E19.5 metatarsal.

zone simulated length (72 h) physiological length (E19.5)

total bone 882.2 855

Resting zone 180.1 361

Proliferative zone 218.7 220

Early hypertrophic zone 35.2 54

Hypertrophic zone 101.5 73

Mineralized cartilage 346.6 200

Compared to the morphology data, the height of the RZ is lower in the model, the PZheight is almost equal in both situations and the early hypertrophic and hypertrophic zonesin the model are 17.4µm smaller and 27.6µm larger, respectively. The mineralization area isapproximately 146.6 µm higher in the simulation.

5.1.4 Discussion

The height of the zones is almost equal over the whole width of the growth plate and noobvious regional variations are observed. However, the transitions of a row of eHZ elements toHZ elements will often start in the center columns, as can also been seen in �gure 5.3 (t=60h).This leads to a higher HZ in the center area, as was also described by Miralles-Flores [38] andAbhaya [2].

In the �rst 12 hours after initiation of mineralization many hypertrophic elements willtransfer to mineralized cartilage and after this the mineralization rate is slowed down (�gure5.3 & 5.4). The cause of this fast mineralization rate is that there are hypertrophic elementsalready present in the initial mesh. At the start of the simulation these elements have a initialMMP concentration of zero. As a result, all these initial HZ elements will sense approximatelythe same amount of MMP's and will reach the threshold concentration at approximately thesame moment. Because the transitions are limited to one transition per two hour simulation,these elements will turn into mineralized cartilage in six successive steps. In the mean timesome eHZ elements have di�erentiated into new HZ elements, which will mineralize later, whentheir cumulative MMP concentration is also above the threshold value.

At the end of the simulation the mineralized zone is much larger than the zone height foundin the morphological study. It could be that initial mineralization is occurring later than wasassumed. Unknown parameters like initiation time of mineralization and mineralization ratecan be clari�ed if more morphological data is present. In this model a higher threshold or lessMMP production will result in delayed mineralization, but the hypertrophic zone will becomelarger due to prolonged hypertrophy and this will �nally lead to a higher mineralized zone.

The length of the RZ is approximately equal during the simulation, because each timeproliferation occurs, one element per column turns into a PZ element and all other elementscompensate by volume increase for the loss of this cell. However, in the physiological situationthe length of the RZ is increasing from 197 to 361 µm during the simulated period. Probablythe RZ cells synthesize more matrix than assumed in the model. Also RZ cells may proliferateas suggested by the higher number of cells in the RZ for the E19.5 situation than for the E16.5situation (table 3.1). Both would increase the height of the RZ in the model.

The other zone heights are comparable to the physiological zone heights and vary at most


Figure 5.5: Representation of the intial mesh without hypertrophic elements. The height of the zonein µm is given between brackets. The articular perichondrium (APC) is aligned around the bone end.

one cell height.

The results of the simulation are very sensitive to some of the chosen parameters. If forexample, the threshold volume for the transition of eHZ cells into HZ cells is chosen just 5%larger, the last formed HZ element would still be an eHZ element at day 19.5.

The length of the mesh is almost doubled during the simulation and the number of cells isincreased by proliferation. However, the number of elements in the mesh remains equal. Thisis a limitation for long simulations, because the larger elements will cause large variations inzone heights due to cell transitions.

The developed model represents a third metatarsal of a mouse, although this model canbe adapted to a growth plate of other fetal long bones by scaling the parameters.

The behavior of the solid part of the biphasic elements was described by a compressibleNeo-Hookean model. Although, the used values of the Young's modulus and Poisson's ratio,taken from literature [11], were determined for a linear elastic model. Resulting in a modelwith a higher sti�ness than the physiological sti�ness of the anlage. However, this had noin�uence on the development of the shape of the mesh, because all model parameters wereadjusted using these mechanical properties. Hence, for implementation of the right Young'smodulus and Poisson's ratio, the number of �xed charge densities, which are added to induceelement swelling, should be determined again.

5.2 Growth and development E14 - E18

Now that the parameters are set for the model, we can take the model one step further.Now, we will test if the chosen parameters will lead to the same stable growth plate if in theinitial mesh only RZ and PZ cells are present. This will correspond approximately with theE14 stage. A new initial mesh, consisting of only resting and proliferating cells, is made, byremoving the (early) hypertrophic elements from the E16.5 mesh. This new mesh consistsof 720 elements and has an initial height of 300 µm (�gure 5.5). A period of 100 hours wassimulated, i.e. 50 simulation cycles.

In the �rst step PZ cells transfer to the eHZ (�gure 5.6), because no Ihh is produced dueto the absence of eHZ cells, and thus no PTHrP is produced either. After 12 hours the eHZconsists of a normal number of elements (6 elements per column) and the Ihh concentrationhas increased to a normal level. Enough PTHrP is produced to keep the PZ elements in the

29

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Figure 5.6: Lengths of the di�erent zones during simulation.

proliferating zone. After 48 hours the �rst eHZ elements are large enough to turn into HZelements. As a consequence of the drop in Ihh production, new PZ elements turn into eHZelements. Gradually more and more eHZ elements become HZ elements. After 100 hoursmineralization initiates.

The bone lengthening is initially smaller than 143 µm per day, because growth is onlythe result of proliferation and matrix synthesis and not of hypertrophy, while hypertrophyaccounts for the majority of bone lengthening (see table 2.1). When the early hypertrophicand hypertrophic zones have developed, the longitudinal growth rate equals 130 µm (�gure5.6).

At t=60 h the length of the mesh is 455.5 µm, and equals the initial length of the E16.5mesh. The zone lengths (HZ = 56.8, eHZ = 49.6, PZ = 152, RZ = 197.3 µm) of the RZ andeHZ are comparable to the lengths of the corresponding zones in the E16.5 bone. The PZ islonger and the HZ is shorter in the model than in the histology.

Twelve hours later, the mesh length (507.0 µm) is slightly longer than the E17 bone.The zone lengths (HZ = 109, eHZ = 40, PZ = 162, RZ = 196 µm) are almost equal to thephysiological lengths of all zones.

Mineralization starts at t = 100 h. Compared to the t = 60 h and t = 72 h this situationwould present the E18.0 bone approximately, and according to our assumptions mineralizationis indeed initiated around this developmental age. We do not have histomorphological data ofthe zone lengths to compare with this situation.

After mineralization has initiated, the same two constrictions in width can be seen aboveand below the HZ as in the former simulation and in histology (�gures 5.2 and 5.7).

Starting with a model without hypertrophic cells, it will develop to the initial situation ofthe former mesh and from there it develops at approximately the same length and time scale.

The width of the mesh is larger than in the former simulation, because only the length ofthe initial mesh was adapted to an earlier stage. The mesh width does not correspond withthe physiological bone width.


Figure 5.7: Resulting bone length and shape during simulation.

At t=60 h the mesh size and zone heights correspond with the initial state of the formersimulation, but this mesh consists of less elements than the former simulation. As a result,the elements in this simulation are larger. This can cause larger variations in zone heights dueto cell transitions.

The simulations are only performed untill initiation of mineralization. However, it is ex-pected that the mineralization process is better simulated by this application than by theformer simulation, because all HZ elements contain a history of MMP exposure. To run thissimulation for a longer period, remeshing should be included.

For this smaller mesh the PTHrP pro�le is also almost �attened around the last PZelement. Around the transition area between PZ and eHZ the lateral PTHrP concentrationsare approximately 4.5% higher than the central concentrations. This will result in a highereHZ in the central region than in the lateral regions. The di�erence in height is one element(�gure 5.7).

PTHrP also inhibits hypertrophy. The elements in the center of a row of eHZ elements willtherefore hypertrophy more than the lateral elements in the same row. This will lead to anearlier transition of the inner eHZ cells to HZ cells. However, due to the very small di�erencesit is also possible that the volume of all elements is above the threshold and the whole row isturned into HZ elements. Elements turning earlier into HZ elements will be exposed longer toMMP than their neighboring elements and will therefore mineralize earlier.

Tanck et al. [62] observed that mineralization initiates in the center of the skeletal element.

31

Figure 5.8: Subsequent stages of in vivo mineralization of mouse metatarsals of E16 - E17. A: Thelighter center of the anlage is the hypertrophic zone. B: Mineralization (dark area) starts as a spherein the center of the rudiment. C: Extension of the mineralization to the periphery D: Extension of themineralization front towards the epiphyses [62].

This mineralized area extends laterally, forming a thin disc of mineralized tissue. This discwill grow towards the ends of the bone (�gure 5.8). In our model the di�erences in PTHrPconcentration between the lateral and central elements are small. Mineralization will �rstappear, either as a sphere in the center of the bone, or as a mineralized disc. This dependentson very small alterations of the parameters like PTHrP production rate or MMP thresholdconcentration. The observation of Tanck, that a clear central spot of mineralization occurs�rst, could not be con�rmed. However, it is shown that mineralization will certainly notinitiate in the lateral regions.

In the �rst step of the simulation PZ elements will turn into eHZ elements due to theabsence of Ihh. This is inherent to the �rst step with this regulation model consisting of onlyRZ and PZ elements, and is independent on the length of the initial mesh. However, in thephysiological situation hypertrophy of the chondrocytes will just initiate around E15/E16.Hence, there must be another factor that will prevent chondrocytes from becoming hyper-trophic. Ballock et al. [5] shows that TGF-β can act on the perichondrial and periarticularcells to increase the PTHrP synthesis and that TGF-β can also act directly on chondrocytesto inhibit hypertrophy.

5.3 Impaired growth factor production

One application of the model is to study e�ects of altered growth factor expression, suchas those observed in several pathologies in which the growth factors PTHrP, Ihh or MMP'sare involved. Therefore, we start with the mesh from paragraph 4.1 (E16.5) and perform threesimulations. In each of them the production rate of one of the growth factors is diminishedby 50%, similar to validation simulations of Brouwers et al. [7]. The results will be comparedto the phenotype of the transgenic mice, as described in paragraph 2.5.

In all simulations with decreased growth factor productions, a stable growth plate wasformed with di�erent zone heights compared to the normal situation (table 5.3). The simula-tions with 50% Ihh synthesis rate resulted in a decreased PZ length and increase in RZ lengthcompared to the simulation with normal production rate, due to less proliferation. Further-


Table 5.3: Final zone heights [µm] of the normal simulations and the simulations including impairedgrowth factor production. The direction of variation compared to the normal simulation was givenbetween brackets, if the variation was larger than 10%.

simulation total length RZ PZ eHZ HZ bone

normal production 878.1 180.1 218.8 35.3 101.5 342.4

50 % Ihh 895.3 (−) 210.1 (↑) 152.4 (↓) 50.8 (↑) 140.0 (↑) 342.0 (−)50 % PTHrP 909.1 (−) 173.8 (−) 205.4 (−) 42.4 (↑) 143.4 (↑) 344.1 (−)50 % MMP 982.2 (↑) 180.2 (−) 226.4 (−) 38.3 (−) 161.9 (↑) 376.4 (↑)

more the lengths of the eHZ and HZ zone were increased, as a result of the decreased PTHrPproduction. This corresponds with the observations in the Ihh−/− mice.

The simulations with 50% PTHrP synthesis resulted also in an increased length of the eHZand HZ, caused by increased hypertrophy. Furthermore, the distance of the end of the PZ tothe articular surface is decreased by 19.7 µm, because the threshold value for the transitionof PZ to eHZ elements is reached closer to the PTHrP source. These results are comparableto the observations in PTHrP −/− mice.

The simulations with a 50 % MMP production rate resulted in delayed bone formation (att=48 h in stead of t= 36 h, �gure 5.9) and a large increase in hypertrophic zone length. Theseresults are comparable with the characteristics of MMP−/− mice. During the simulation lessHZ cells have turned into bone (7 rows of elements in stead of 9). However, the �nal heightof the mineralized zone is higher than in the normal simulation, because the hypertrophic ele-ments were much larger due to a longer period of hypertrophy. The other zones are una�ectedby the decreased MMP production.

A decrease in growth factor production lead in all three simulations to growth plateswith di�erent zone heights compared to the normal production rates. All these variationscorrespond with known in vivo disorder characteristics.

However, there is one important di�erence with in vivo observations. In all cases ofimpaired production the �nal length of the bones is larger than the normal situation. However,bones of PTHrP−/−, Ihh−/− and MMP−/− mice are shortened. We suggest that this is ashort term e�ect. The simulations are only performed in a short period, the long term e�ectsare not predicted by the model. This can be con�rmed by the �nding of Van Donkelaar et al.[65]. They showed that altered growth factor expression, as a result of loading, will lead to ashort term overshoot, but a decrease in bone growth on the long term.

To continue simulations for longer times requires remeshing. It would be very interestingto evaluate whether growth retardation is preceded by increased growth.

5.4 Mechanical loading

Our �nal aim was to prove the principle that mechanical loading could a�ect growth andtherefore could determine the �nal shape of the bone.

During a simulation mechanical load was applied to the mesh, simulating the compressionapplied by neighboring bones, or the tension of attached muscles. On the nodes in the toprow of the mesh a high compressive force of 50 N in the negative y-direction was applied(CLOAD). To the outside nodes of the 10 highest rows a tensile force of 20

√2 N was applied

under an angle of (-)45 degrees to the positive y-axes (�gure 5.10).

33

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Figure 5.9: Zone lengths during the simulation with normal growth factor production and with im-paired production rates.


Figure 5.10: Bone shape and zone heights after 72 h growth simulation with applied compressive andtensile loads.

The shape of the head of the bone developed di�erent than in the non-loaded simulations.The sides of the bone head that where under tension show increased bone growth, while growthseems to be impaired in the center part that was compressed (�gure 5.11). This di�erencebetween lateral and central zone heights was only seen in the resting zone (�gure 5.10). Thetotal bone length was decreased compared to the non-loaded situation (�gure 5.12). Also thetotal height of the RZ and the mineralized zone were decreased. The compressive force on themesh was larger than the tensile force, this could explain the decreased bone lengthening.

This simulation shows that bone growth is in�uenced by mechanical loading. Hence themodel is also suitable to study the e�ect of mechanical loading on the development of theshape of the bone head.

The �ndings that growth was increased in the areas under tension and decreased in thecompressed areas correspond with the Hueter-Volkmann law. This law states that longitudinalbone growth is retarded by compression and distraction or reduced compression will stimulatelongitudinal growth [12, 40].

Stokes [58] found that the growth of rat tail vertebrals was sensitive for loads in the rangeof 2.5N. However, loads of this order did not deform the mesh (data not shown). The forcesthat were applied in the simulation are above this physiological range. The need for thesehigher load is probably caused by the high sti�ness of the model, due to the linear elasticmodel parameters that were used to describe the mechanical behavior of the model.

Mechanical loading can also in�uence the growth factor controlled process, because themechanical forces can alter the gene or protein expression and the transport of growth factors[42, 61, 74]. It is well possible to couple the growth factor production rate to local stressesor strains. Such extension would give ample possibilities to study the e�ects of mechanicalloading on growth. However, it must be mentioned that availability of suitable data in theliterature is very sparse.

35

Figure 5.11: A. Element volume increase over the total period of the loaded and unloaded simulations.Most volume increase was seen in the hypertrophic zones. In the whole RZ a small volume increase wasseen due to proliferation. B. In the lateral sides of the bone head (tension) a higher volume increasewas seen than in the central bone head (compression) in the unloaded simulation growth of the RZ wasuniform.

16.5 17 17.5 18 18.5 19 19.50

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Figure 5.12: Zone heights during the simulation including mechanical loading.

Chapter 6

Conclusion and Recommendations

6.1 Conclusion

The aim of this study was to establish a model that enables to quantitatively assess thecontributions of mechanical loading on the developing shape of a fetal long bone, when bonegrowth is growth factor controlled.

The developed model included bone growth by proliferation, matrix synthesis and hyper-trophy. The bone growth simulations were alternated with calculations of di�usion pro�les ofthe growth factors Ihh, PTHrP and MMP9 + MMP13.

The model parameters were scaled to the characteristics of embryonic mice metatarsalsfrom stage E16.5 till E19.5, observed in a histomorphological study. The model simulated thelongitudinal and radial bone growth very well. The growth factor regulation resulted in stablezone heights.

Although mineralization initiated around the assumed developmental stage, the mineral-ization process resulted in a higher mineralized zone than in our morphological observations.The �nal resting zone height predicted by the model was smaller than in our morphologicalobservations. This is probably caused by proliferation in this zone, or relatively more matrixsynthesis, which were not included in the model. The heights of the proliferative zone and thehypertrophic zones corresponded very well with our histomorphological study.

We have tested if these parameters will lead to a stable growth plate with the same char-acteristics starting at day E14, with an initial mesh without hypertrophic elements. In thissimulation the mesh develops till the E16.5 situation and from there to the initial mineraliza-tion at the correct length and time scale. At the corresponding moments the zone heights arealmost equal to the morphology of the E16.5 and E17 metatarsals. Hence, we can concludethat the model is able to simulate the development of fetal long bone at the correct time andlength scale from E14 till E18.

The model was also applied to several pathologies in�uencing the production rate of theregulating growth factors. Decreased growth factor productions lead to stable growth plateswith di�erent zone heights compared to the normal developing growth plate. The predictionsof the changes in zone heights correspond to the observations in knock-out mice. Total bonelength were larger than expected, which is most likely a short term e�ect.

Finally, the e�ects of compressive and tensile forces by adjacent bones or attached musclesis evaluated. It was found that the shape of the head of the bone develops di�erent than in thenon-loaded simulations. The sides of the bone head that where under tension show increased

37

38 CHAPTER 6. CONCLUSION AND RECOMMENDATIONS

bone growth, while growth seems to be impaired in the center part that was compressed.

These applications make this model a useful tool in studying the growth factor controlledgrowth and development of fetal bone and the in�uence of externally applied or local mechan-ical forces on growth and development.

6.2 Recommendations for further research

With the developed 2D model the period between initiation of hypertrophy and miner-alization could be simulated very well, resulting in comparable bone length, bone shape andzone heights. Because of the large gap between the ages of the hind feet that we obtainedfor our morphological study, we had to assume the moment of mineralization initiation andmineralization rate. This lead to some variations between the model and the actual lengthof the mineralized cartilage length, which could not be explained. If more morphologicaldata, around the initiation of mineralization and further development, can be analyzed, theparameters controlling mineralization can be adjusted, and perhaps provide better resultsregarding ossi�cation. With these parameters the simulations can also be extended to laterages.

Production and di�usion of growth factors are continuous processes. In this model growthfactor concentration pro�les are calculated every two hours. For each of these calculationsprevious growth factor concentrations are neglected. If it would become possible to performa bone growth simulation and simulations of multiple di�usion pro�les parallel to each other,we would recommend to implement this in the model. This will provide a better simulationof the growth factor concentration pro�les.

In the model cartilage was assumed biphasic. However, parameters of a linear elastic boneand cartilage model were used to describe the mechanical behavior of the model. This resultedin a higher sti�ness of the model than the physiological sti�ness of the anlage. Although thisdid not in�uence the results of the simulations, we would recommend for further research toimplement mechanical properties of a biphasic model. This requires rescaling of the modelparameters.

Cell division in the proliferative zone is one of the contributors of bone lengthening. Inour simulation the number of elements is kept constant and the actual increase in length byproliferation is compensated by an increase in the volume of the resting zone. Therefore eachelement in the RZ corresponds only in the beginning of the simulation to one chondrocyte,later an element can cover more than one cell. When a RZ element becomes a PZ element, thise�ect propagates to other zones. This compensation for proliferation is only a possible solutionfor short simulations. When the elements become much larger than the actual cell size of thecorresponding chondrocyte types, this will in�uence the zone transitions and growth factorproduction and will probable lead to an instable model. For longer simulations we wouldrecommend to implement remeshing of the model to maintain element sizes that correspondto actual cell sizes.

It is known that besides Ihh, also TGF-β can stimulate the production of PTHrP bythe periarticular cells and that TGF-β can also inhibit hypertrophy. For simulations of thedevelopment of fetal bone before hypertrophy is taken place, i.e. before E14, it could be usefulto add TGF-β as a regulating growth factor, or to implement a low basal PTHrP productionthat will prevent hypertrophy in small bones without Ihh expression.

The transition of early hypertrophic elements to hypertrophic elements is only based on

39

the cell size at this moment. Likely, it is triggered by a physical factor. Is is assumed that thehypoxia-inducible factor 1α (HIF1α) is involved in this cell zone transition. HIF1α is a keymolecule in the hypoxic response, regulating the expression of several growth factors, includingVEGF [47]. The growth plate is an avascular and hypoxic tissue with a typical outside-insidegradient of oxygenation. The HIF1α concentration is therefore inversely proportional to theoxygen pressure, which is highest in the vascularized regions and lowest around the eHZ/HZtransition area [48]. We would recommend further study to this growth factor and the oxygenpressure through the skeletal element. It would be challenging to incorporate the oxygenpressure gradient and growth factor activity through the bone as a critical level for terminalhypertrophy in the model.

Several studies are performed on the mechanical in�uence of the perichondrium on bonegrowth. In the developed �nite element model the perichondrium is not modeled as a distinctstructure. However, in a similar way as the bone collar is implemented in this model, therelative sti� perichondrium can also be included to study its mechanical in�uence on the bonegrowth rate.

In this model mechanical loading was applied to the bone to study the e�ects of loadingon the bone growth. Mechanical loading can also in�uence the growth factor regulation byaltering gene and protein expression. This model includes the possibility to couple the growthfactor production to local stresses or strains.

The �nite element model presented here was applied to the developmental process ofthird metatarsals of embryonic mice. However, simulation of other fetal long bones shouldalso be possible with this model. It is an intriguing question whether parameter values varysigni�cantly between long bones within an animal. This model could be applied to other bonesto evaluate this question and to study the relative importance of mechanical loading on thedevelopment of the �nal shape of the bone.

Much research is done to the changes in growth factor control after secondary ossi�cationhas initiated. With the used regulatory system it is not possible to simulate the initiationof secondary ossi�cation. More insight in the initiation process of the secondary ossi�cationcenter can provide challenging possibilities and on the long term it might be possible to extendthis model, such that besides fetal bone growth also postnatal growth can be simulated.

Acknowledgements

I would like to thank René van Donkelaar and Rik Huiskes for their support and insightduring this project. Thanks to Julienne Brouwers and Bram Sengers for their help with theSepran model. Furthermore I would like to thank Wouter Wilson and Hanna Isaksson forsharing their knowledge about ABAQUS. Finally, I would like to thank dr Marcel Karperienfrom the Leiden University Medical Center for providing the mice matatarsals and Petra Nij-man and Jasper Foolen, for helping me with the histomorphological study.

41

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Appendix A

Extension of the 1D FE model to 2D

A 1D �nite element model of a fetal femur was developed by Brouwers et al. [7]. In thismodel proliferation, hypertrophy and zone transitions were controlled by the growth factors,Ihh, PTHrP and VEGF. This model was extended to 2D by using 15 rows of this 1D descriptionin parallel.

This model provided insight in the two dimensional growth factor concentration pro�lesand the results were presented by a poster in the 52nd Annual Meeting of the OrthopaedicResearch Society, Chicago, Illinois (2006) (see below).

Although this model could simulate the 2D growth factor concentration pro�les, only lon-gitudinal bone growth was implemented. Radial growth was impossible, because the includedcolumns could not expand in that direction. Also, it included a static mesh, which was notable to deform due to mechanical forces or asymmetrical growth.

49

APPENDIX A. EXTENSION OF THE 1D FE MODEL TO 2D

/mhj

12

IntroductionFetal bone development originates form a cartilaginous anlage, which

mineralizes following a characteristic pattern (Fig 1):

1. Central chondrocytes hypertrophy2. Mineral formation occurs in the hypertrophic center

3. The mineralized area extends laterally, forming a disc

4. The mineralized disc grows proximally and distally

Major control mechanism of mineralization (Fig 2):

- PTHrP (expressed by periarticular perichondrium cells) keeps cells

proliferating and reduces rate of hypertrophy - Ihh (expressed in early hypertrophic zone (eHZ)) stimulates

proliferation and expression of PTHrP

- VEGF (expressed by hypertrophic cells) stimulates mineralization

and vascularization

AimWe aim to test the hypothesis that the typical spatial pattern of early

mineralization is determined by the location of PTHrP-expressing cells and the PTHrP-Ihh feed-back system.

MethodFE analyses:

- Anlage with distinct locations of PTHrP expression (Fig 3).

- Bone development controlled by PTHrP-Ihh system and VEGF.

protein transport via diffusion2. - Initial situation: Only the perichondrium, resting (RZ) and proliferative

zone (PZ) are present. Hypertrophic zones (eHZ and HZ) and

mineral have not yet formed.

Results- The shapes of the mineralized center, PZ and HZ depend on the

geometry of the PTHrP expressing perichodrium (Fig 4).

- The spatial distribution of PTHrP is responsible (Fig 5).

DiscussionIn vivo, PTHP-expressing perichondrium is round (Fig 6). This explains the

central mineralization (Fig 1 and Fig 4).

Shapes of the hypertrophic and mineralized zones follow the contours of the

periarticular perichondrium in simulations (Fig 4), in vivo (Fig 7, top) and in

vitro (Fig 7, bottom). A round perichondrium results in broader hypertrophic

zones in simulation and experiment.

Conclusion- The PTHrP-Ihh feedback system is able to drive and control

mineralization in the anlage.

- The location of PTHrP-expressing perichondrium determines the

appearance of the mineralized and hypertrophic zones. - We hypothesize that shape and location of the perichondrium may be

defined by external and internal mechanical effects. Such mechancial

effects are not included in these simulations.

References1 Tanck E, Blankevoort L et al, JOR 2000;18;613-619.

2 Brouwers JEM, van Donkelaar CC et al, J Biomech 2006 (on line).

3 Karperien M, Lanser P et al, Int J Dev Biol 1996;40;599-608.4 Kronenberg HM, Nature 2003;423;332-336.

Geometry of perichondrium determines

location of early mineralization in long bonesRené C.C. van Donkelaar, Sietske W. Witvoet-Braam, Rik Huiskes

Department of Biomedical Engineering, Eindhoven University of Technology, The Netherlands

Figure 1

Mineralization in 16-17 day fetal

mouse metatarsals

Bright area: hypertrophic zone

Dark area: mineralized zone.

(adapted from Tanck et al1).

Figure 6

PTHrP-mRNA (in situ hybridization)

in the distal tibial anlage (yellow

circle) of a 12.5 day old mouse

(adapted from Karperien et al3).

Figure 3

Initial geometries. PTHrP

expressing periarticular

perichondrium is represented by

dark elements.

Figure 4

Zonal organization in time

(horizontal) during simulations

with the two different initial

locations of PTHrP expression

as indicated in Fig 3.

Figure 5

Distributions of PTHrP and Ihh

(same time points as Fig 4).

Colors are chosen such that the

concentration gradients are best

visible in the eHZ region.

Figure 2

Schematic PTHrP-Ihh control loop:

- Ihh stimulates PTHrP expression,

-PTHrP keeps cells proliferating

(adapted from Kronenberg4)

High

LowIhh

PTHrP

RZ&PZ

eHZ

HZ

mineral

Simulation A

Simulation B

Simulation BSimulation A

Figure 7

Left: Simulated tissue distributions (see Fig 4).

Right: Tissue distributions in 17-day old mouse

metatarsal bones (top) and bones harvested at day 15,

cultured for 5 additional days (bottom)1.

50

growth factor controlled fetal bone growth · the nal length and shape of a bone are thus a result...

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