A Mathematical Framework to Study the Effects of Growth Factor Influences on Fracture Healing

J. theor. Biol. (2001) 212, 191}209 doi:10.1006/jtbi.2001.2372, available online at http://www.idealibrary.com on

A Mathematical Framework to Study the E4ects of Growth Factor In6uences on Fracture Healing ALICIA BAILOD N-PLAZA*

AND

MARJOLEIN C. H.

VAN DER

MEULEN

Sibley School of Mechanical & Aerospace Engineering, Cornell ;niversity, Ithaca, N> 14850, ;.S.A. (Received on 19 December 2000, Accepted in revised form on 10 June 2001)

During fracture healing, multipotential stem cells di!erentiate into specialized cells responsible for producing the di!erent tissues involved in the bone regeneration process. This cell di!erentiation has been shown to be regulated by locally expressed growth factors. The details of their regulatory mechanisms need to be understood. In this work, we present a twodimensional mathematical model of the bone healing process for moderate fracture gap sizes and fracture stability. The in#ammatory and tissue regeneration stages of healing are simulated by modeling mesenchymal cell migration; mesenchymal cell, chondrocyte and osteoblast proliferation and di!erentiation, and extracellular matrix synthesis and degradation over time. The e!ects of two generic growth factors on cell di!erentiation are based on the experimentally studied chondrogenic and osteogenic e!ects of bone morphogenetic proteins-2 and 4 and transforming growth factor-b-1, respectively. The model successfully simulates the progression of healing and predicts that the rate of osteogenic growth factor production by osteoblasts and the duration of the initial release of growth factors upon injury are particularly important parameters for complete ossi"cation and successful healing. This temporo-spatial model of fracture healing is the "rst model to consider the e!ects of growth factors. It will help us understand the regulatory mechanisms involved in bone regeneration and provides a mathematical framework with which to design experiments and understand pathological conditions.  2001 Academic Press

1. Introduction Fracture healing involves sequential tissue morphogenesis and a cascade of highly coordinated cellular events. In cases of moderate fracture gap sizes and fracture stability, bone healing involves the aggregation of pluripotent mesenchymal stem cells at the site of injury, and the formation by these cells of a stabilizing callus which subsequently chondri"es prior to ossi"cation. This bone regeneration process recapitulates embryonic skeletal development, when mesenchymal *Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/180191#19 $35.00/0

stem cells also aggregate to form the initial template of the skeleton which subsequently chondri"es and "nally ossi"es (Caplan, 1994; Ferguson et al., 1999; Vortkamp et al., 1998). This bone regeneration process recapitulates aspects of postnatal bone growth as well where cartilage formation also precedes ossi"cation. Fracture healing, therefore, constitutes a unique opportunity to investigate bone morphogenesis in an experimentally controllable environment. Moreover, approximately 5}10% of the 5.6 million fractures occurring annually in the United States develop into delayed unions or non-unions (Praemer et al., 1992). Understanding normal and  2001 Academic Press

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abnormal healing, and designing clinical strategies for the latter cases, are critical topics of research. In vivo and in vitro experiments have demonstrated that cell activity during bone tissue morphogenesis is initiated and tightly regulated by locally produced growth factors and matrix proteins. Additionally, absence or altered expression of a single growth factor or matrix protein may lead to dramatic skeletal and fracture healing aberrations in transgenic or knockout animals, and in patients su!ering from several pathological skeletal conditions, such as "brodysplasia ossi"cans progressiva and osteogenesis imperfecta (King et al., 1994; Kocher & Shapiro, 1998; Boskey et al., 1999; Virdi et al., 1999). These abnormalities demonstrate the crucial role of these factors for normal tissue morphogenesis. Moreover, growth factors such as bone morphogenetic proteins (BMPs) and transforming growth factor-betas (TGF-bs) are capable of inducing bone formation and accelerating healing when injected ectopically or administered exogenously to a fracture (Joyce et al., 1990a; Linkhart et al., 1996). These factors constitute promising therapeutic agents for complicated fractures, distraction osteogenesis and for other clinical skeletal conditions such as osteoporosis. Before their clinical use, which requires precisely determining growth factor dosage, treatment duration and delivery method, however, a better understanding of their regulatory mechanisms is needed. In this work, we examine the e!ects of growth factors on normal fracture healing and provide a mathematical framework to understand their regulatory mechanisms. 1.1. TISSUE DIFFERENTIATION DURING FRACTURE HEALING

Fracture healing involves tissue generation inside and surrounding the site of injury (Hulth, 1988). The production of the di!erent tissues is accomplished by distinct specialized cells which share the same origin (Caplan, 1994). Healing begins as undi!erentiated mesenchymal cells migrate from the surroundings and produce initial connective tissue around the fracture site, forming an initial stabilizing callus. The development of the callus is in#uenced by the size of the fracture gap and by the amount of mechanical

stability. Primary healing occurs in cases of extreme stability and negligible gap size. However, most cases, which involve moderate gap sizes and fracture stability, heal by secondary fracture healing. During this healing process the localized differentiation of the mesenchymal cells into cartilage and bone forming cells leads to the production of cartilage and bone tissue in the callus. In successful secondary healing, the initial connective tissue and cartilage are eventually entirely replaced by bone, leading to osseous union across the fracture gap and completion of the healing process. Conceptually, secondary fracture healing is generally divided into four temporal stages: (i) in#ammation, (ii) callus di!erentiation, (iii) ossi"cation, and (iv) remodeling [Fig. 1(a)}(d)]. In this work we focus on the "rst three stages of secondary healing involving cellular di!erentiation and tissue formation. The time-scale used to describe these stages is based on experimental studies on rats (Joyce et al., 1990b; Einhorn, 1998). The "rst stage encompasses all events from the initial bone rupture through the initial callus formation [Fig. 1(a)]. In#ammation occurs immediately after bone rupture, when blood emanates from the ruptured vessels and a hemorrhage quickly "lls the fracture gap space. Platelets and thrombotic factors produce a connective tissue matrix consisting of "brin and release multiple angiogenic and signaling factors, such as interleukin-6 (IL-6), interleukin-1 (IL-1), TGF-bs, insulin-like growth factors (IGFs), platelet-derived growth factors (PDGFs) and BMPs (Hulth, 1988; Barnes et al., 1999). Blood cells, including granulocytes, macrophages and lymphocytes, migrate to the fracture site. The macrophages digest the dead tissue and further promote the formation of a matrix, substrate necessary for the migration of mesenchymal cells. Mesenchymal cells originate from within the broken periosteum, the external tissue layer of the bone which is rich in vascularity, progenitor cells and growth factors; they also originate from the soft tissues tightly surrounding the bone, including the muscles. These cells replace the "brin matrix by a new connective tissue matrix and form the initial external callus. Similarly, in the marrow canal, mesenchymal cells originating from stromal cells in the marrow contribute to the formation of intramembranous woven bone in the canal.

GROWTH FACTORS AND FRACTURE HEALING

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FIG. 1. Schematic of a section through an intact long bone. Fracture healing stages: (a) in#ammation, a hematoma forms at the fracture gap and an initial fracture callus forms consisting mainly of mesenchymal cells; (b) callus di!erentiation, chondrous tissue forms adjacent to the fracture while intramembranous osseous tissue forms adjacent to the bone and distal to the fracture; (c) ossi"cation, osseous tissue replaces the cartilage via endochondral ossi"cation of the callus, and (d) remodeling, the original geometry of the bone is restored.

The second stage of fracture healing proceeds with the formation of bone and cartilage in distinct regions of the callus [Fig. 1(b)]. Along the bone, even within the "rst 24 hr, mesenchymal cells di!erentiate into osteoblasts which begin to actively synthesize intramembranous woven bone. By contrast, in the interior of the initial callus and adjacent to the fracture, at approximately day 7, mesenchymal cells di!erentiate into chondrocytes which synthesize cartilage. As healing progresses, the intramembranous ossi"cation front advances towards the center of the callus until 10}12 days of healing, and the chondrous callus grows in size mostly due to active mesenchymal cell di!erentiation into chondrocytes and chondrocytic proliferation (Sanberg et al., 1993). At 10}12 days post-fracture, ossi"cation of the cartilage callus begins, a process known as endochondral ossi"cation. The third healing stage begins with endochondral ossi"cation and ends at the point of

bone formation which will then continue to remodel. Endochondral ossi"cation involves a complex sequence of cellular events, coupling cartilage maturation and degradation, vascularity and osteogenesis. Prior to endochondral replacement, mature chondrocytes cease proliferation and, as one aspect of their terminal di!erentiation, they promote calci"cation of the extracellular matrix (ECM). This calci"ed ECM then becomes the sca!old for subsequent bone formation. Chondrocytes undergo apoptosis, and blood vessels grow into the space previously occupied by the chondrocytes. Osteoprogenitor cells associated with these vessels di!erentiate into osteoblasts which lay down bone on the calci"ed cartilage matrix. Once osteoblasts have surrounded themselves in bone they are removed by apoptosis (Olmedo et al., 1999) or become osteocytes or bone lining cells. Endochondral ossi"cation continues until all the cartilage has been replaced by bone, and an entirely bony bridge closes the fracture gap.

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Finally, the fourth stage of repair begins once the gap has ossi"ed and ends with the restoration of the original form of the bone [Fig. 1(d)]. During this last phase, the external callus is completely resorbed. In the fracture gap, osteoclasts and osteoblasts remodel the disorganized woven bone into transversely isotropic cortical bone, restoring the original tissue architecture. This last stage of bone remodeling will not be addressed in our model. 1.2. REGULATORY FACTORS

In fracture repair, the initiation and regulation of the healing response and tissue formation are largely due to the localized production, temporal release and di!usion of a number of cytokines and growth factors. Among these regulatory factors the most signi"cant are members of the TGF-b superfamily, such as TGF-b1, TGF-b2, BMP-2, BMP-3, BMP-4 and BMP-7, PDGF and acidic and basic "broblast growth factor-1 (FGF-1) and FGF-2 (Barnes et al., 1999; Linkhart et al., 1996; Sakou, 1998; Sanberg et al., 1993). These factors are temporally synthesized by di!erent cells and synergistically regulate a wide range of cell activities including migration, proliferation and di!erentiation. TGF-b is one of the most potent and wellstudied osteogenic growth factors involved in fracture healing. TGF-b is stored in the bone matrix, released during bone matrix resorption (Linkhart et al., 1996) and is expressed in the hematoma immediately after injury (Joyce et al., 1990b). This growth factor is produced by osteoblasts and induces osteogenesis in vitro by directing mesenchymal cell di!erentiation into osteoblasts, enhancing osteoblast proliferation and bone matrix synthesis, and directing angiogenesis, in a dosedependent manner (Joyce et al., 1990a; Bostrom & Anis, 1998; Saadeh et al., 1999). TGF-b also initiates intramembranous and endochondral bone formation when injected exogenously in laboratory animals (Joyce et al., 1990b). Recently, TGF-b1 has been shown to induce a dose-dependent production of vascular endothelial growth factor (VEGF), an angiogenic chemoattractant, by osteoblasts at the ossi"cation front (Saadeh et al., 1999). Moreover, dexamethasone, which inhibits TGF-b1-induced

production of VEGF in vitro, impairs fracture vascularization and healing in glucocorticoid treated patients (Saadeh et al., 1999). Together, these results suggest that, though the production of TGF-b1 and VEGF, osteoblasts are important regulators of angiogenesis, and that TGF-b1 may a!ect the angiogenic cascade, without which bone formation would not proceed (Saadeh et al., 1999). In vivo in the rat, TGF-b1 expression along the periosteal cortex is observed within the "rst 24 hr following fracture, and maps the region where intramembranous ossi"cation "rst appears (Joyce et al., 1990a). It is expressed intracellularly in osteoblasts and in proliferating mesenchymal cells next to the intramembranous bone. TGF-b1 also is found in the matrix during cartilage calci"cation and in areas of active endochondral ossi"cation. These results suggest, in accordance with the in vitro studies, that this growth factor activates the di!erentiation of mesenchymal cells into osteoblasts during intramembranous bone formation and is a key regulator of endochondral ossi"cation. Similarly, BMP-2 and -4 are among the most potent and well-studied chondrogenic growth factors involved in fracture healing. BMPs are produced by chondrocytes and induce cartilage formation by activating mesenchymal cell di!erentiation into chondrocytes, and by stimulating mitogenesis and matrix synthesis in a dose-dependent manner (Bostrom et al., 1995; Bostrom, 1998). BMP-2 and -4 expression occurs in the hematoma during fracture repair within the "rst 24 hr, just before mesenchymal cells di!erentiate into chondrocytes, suggesting that these growth factors are involved in activating the di!erentiation of mesenchymal cells into chondrocytes (Bostrom et al., 1995; Bax et al., 1999). BMP expression declines around 9}12 days after the initiation of healing as chondrocytic proliferation decreases and hypertrophy is initiated. 1.3. MODELING FRACTURE HEALING

Previous theoretical models of fracture healing have explored the correlation between the regions of cartilage and bone tissue formation, and the mechanical environment in the callus, determined by the amount of stability or

GROWTH FACTORS AND FRACTURE HEALING

motion between the bone fragments (Ament & Hofer, 2000; Blenman et al., 1989; Claes & Heigele, 1999; Gardner et al., 2000). The underlying hypothesis in these studies is that the level of mechanical stress or strain in di!erent regions of the callus determines where cartilage or bone tissue forms, by directing the di!erentiation of mesenchymal cells into chondrocytes or osteoblasts, and/or by a!ecting angiogenesis. Although in secondary fracture repair a variable mechanical and angiogenic environment may in#uence the development of cartilage and bone tissue, tissue di!erentiation is initiated and largely regulated by growth factors. This growth factor regulation has never been accounted for in previous fracture healing models. Furthermore, these previous models correlated tissue histology to the mechanical environment at discrete time points only. Therefore, the objective of this work was to develop a temporal fracture healing model able to predict the development of the di!erent tissues over time. Mathematical models of wound healing, tumor angiogenesis and morphogenesis have provided important insights into the regulatory roles of growth factors and the ECM (Dillon & Othmer, 1999; Olsen et al., 1995, 1997; Maini, 1993; Murray, 1993). Here we use the same approach to investigate bone healing. Our framework focuses on the activation of chondrocytic and osteoblastic di!erentiation by growth factors expressed in the callus. Cell migration, proliferation and di!erentiation, growth factor production and di!usion, and ECM synthesis and degradation are simulated over time at the fracture site. The model is then used to investigate the e!ects on the healing response of the initial concentrations of growth factors released; the time duration of this growth factor release, and parameters such as the production rate of these growth factors. Rate of change in: (1) mesenchymal density (number of cells ml\) (2) chondrocyte density (number of cells ml\) (3) (4) (5) (6) (7)

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2. Methods 2.1. MODEL SYNTHESIS AND BIOLOGICAL FRAMEWORK

The dependent variables in the model include cellular densities (number of cells ml\) for the three critical tissue-forming cells, mesenchyme, chondrocytes and osteoblasts. The variables also include the densities of two extracellular matrices (ECMs), connective/cartilage and bone. Connective/cartilage ECM represents both the connective tissue synthesized by mesenchymal cells and the cartilage synthesized by chondrocytes. This combination was chosen for model simplicity. Bone represents the tissue produced by the osteoblasts. Finally, the concentrations of two generic growth factors are also modeled, based on the chondrogenic e!ects of BMP-2/4 and the osteogenic e!ects of TGF-b1. These growth factors were selected because of their well-documented and potent chondrogenic and osteogenic e!ects in fracture healing. Mesenchymal cell migration into the callus is modeled based on a previous wound healing model by Olsen et al. (1997), considering the similar nature of the in#ammation response in skin and bone healing. The e!ects of matrix composition and density on cell migration and proliferation, and the regulation of cell di!erentiation and endochondral ossi"cation by di!using growth factors are modeled here. Speci"cally, our model focuses on the experimentally studied osteogenic and chondrogenic e!ects of TGF-b1 and BMP-2/4, respectively. These growth factors are assumed to di!use in the #uid component of the extracellular matrices, which is primarily water for the connective/cartilage matrix. Changes in cell and ECM composition, and in growth factor concentration, are regulated in the callus as follows [Fig. 2]:

Rate of change due to: " migration#mitosis!di!erentiation "mitosis#di!erentiation !endochondral replacement osteoblast density (number of cells ml\) "mitosis#di!erentiation!removal connective/cartilage ECM density (g ml\) "synthesis!degradation bone ECM density (g ml\) "synthesis!degradation osteogenic growth factor concentration (ng ml\) "di!usion#production!decay chondrogenic growth factor concentration (ng ml\) "di!usion#production!decay

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FIG. 2. Cell di!erentiation pathway. Under the in#uence of the osteogenic or chondrogenic growth factors (g and g ), @ A mesenchymal cells di!erentiate into osteoblasts or chondrocytes, respectively. Similarly, under the in#uence of the osteogenic growth factor and the connective/cartilage ECM matrix (m ), chondrocytes may be replaced into osteoblasts. A All the cells undergo mitosis, which is inhibited at high connective/cartilage ECM density for the mesenchyme and chondrocytes or at high bone ECM density (m ) for the @ osteoblasts. Mesenchymal cells synthesize connective/cartilage matrix; chondrocytes synthesize connective/cartilage matrix and chondrogenic growth factor, and osteoblasts synthesize bone matrix and osteogenic growth factor. Finally, osteoblasts undergo apoptosis or become osteocytes.

Mesenchymal cell migration is modeled based on the experimentally characterized behavior of random cell dispersal with cell}cell contract inhibition (Gruler & Bultmann, 1984; Bard & Hay, 1975). Moreover, the haptotactic, haptokinetic enhancing or inhibitory e!ects of ECM density and a non-homogeneous medium also are considered (Bard & Hay, 1975; Murray, 1993) as in Olsen et al. (1997). Cell proliferation is modeled using the logistic growth equation, where the rate of cell

division decreases linearly with cell density due to limitations in space, nutrient di!usion or resources (Murray, 1993). The cell population size stabilizes around a limiting density in accordance with experimental observations (Weinberg & Bell, 1985), with cell loss due to apoptosis balanced with new cell mitosis. Additionally, the enhancing or inhibitory e!ects of ECM at low and high densities, respectively, also are incorporated (Olsen et al., 1997). Mesenchymal cell di!erentiation into osteoblasts or chondrocytes is regulated by the concentration of the osteogenic and chondrogenic growth factors, respectively. The rate of di!erentiation is growth factor concentration dependent up to a maximum constant saturation level. This model is based on the experimental dose-dependent curves characterizing the response of cells to growth factors, and allows us to focus on growth factor dosages reported as chondrogenic and osteogenic for BMP and TGF-b (Barnes et al., 1999; Joyce et al., 1990b; Mayer et al., 1996). Endochondral ossi"cation is modeled as the localized degradation of cartilage and simultaneous formation of bone, occurring by the replacement of mature chondrocytes by osteoblasts, under the in#uence of TGF-b. In the model, chondrocyte maturation is related to cartilage ECM density as chondrocytes synthesize cartilage matrix at a constant rate and at high matrix density chondrocytes have essentially ceased proliferation. Although the details related to hypertrophic di!erentiation are not addressed in this model, we consider these matured chondrocytes to be the cells susceptible to bone replacement. Hence, the replacement of chondrocytes by osteoblasts is modeled to depend on the connective/cartilage ECM density and the concentration of the osteogenic growth factor. ECM synthesis and degradation depend on cell density and on cell and ECM density, respectively. During endochondral ossi"cation, osteoclasts from the invading vascular supply resorb the cartilage matrix. As the number of these cells increases together with the number of osteoprogenitor cells which di!erentiate into osteoblasts, the model assumes a correlation between osteoclast and osteoblast density and, for simpli"cation, does not model osteoclasts explicitly. Hence, connective/cartilage degradation during

GROWTH FACTORS AND FRACTURE HEALING

endochondral ossi"cation is dependent on osteoblast density in the model. A concentration-dependent rate of growth factor production by chondrocytes and osteoblasts is assumed, with a limiting level. Moreover, the production of the chondrogenic growth factor by chondrocytes ceases at hypertrophy, as experimentally described for BMP-2/4 (Bostrom et al., 1995). Finally, constant growth factor di!usion coe$cients and decay rates representing denaturation and tissue clearance by irreversible binding (Dinbergs et al., 1996), and a constant rate of osteoblast cell removal are assumed. 2.2. MATHEMATICAL FORMULATION

As described in Section 2.1, the primary model variables are the cell and matrix densities and the growth factor concentrations. These variables are labeled c , m and g , respectively, with subscripts V V V m, c and b indicating mesenchymal, cartilage and bone. These subscripts are consistent across all parameters. The variables are functions of two spatial and one temporal independent variables, x, y, and t. The cellular density (number of cells ml\) was modeled for mesenchymal cells, c "c (x, y, t), chondrocytes, c "c (x, y, t), and K K A A osteoblasts, c "c (x, y, t). The total matrix den@ @ sity (g ml\), m"m(x, y, t), consists of connective/cartilage matrix density, m "m (x, y, t), and A A bone matrix density m "m (x, y, t). Finally, @ @ growth factor concentrations (ng ml\) were modeled for the chondrogenic, g "g (x, y, t), A A and osteogenic, g "g (x, y, t), growth factors. @ @ As described in Section 2.1, the rates of change of cellular densities are regulated as follows: *c K" ) [D c !Cc m]#c [A !B c ] K K K K K K *t !F c !F c ,  K  K

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are the rates of proliferation for each cell type; B , B and B are determined from the limiting K A @ cell densities K , K and K : B "A /K , JK JA J@ K K JK B "A /K and B "A /K ; and F , F and F A A JA @ @ J@    are functions relating cell di!erentiation to growth factor concentrations. The haptotactic (h) and haptokinetic (k) speeds, and the rate of mesenchymal cell proliferation depend on the total ECM density as follows (Olsen et al., 1997): D F D" m, (K#m) F C I C" , (K #m) I A KM m. A " K (K #m) K Chondrocyte [eqn (2)] and osteoblast [eqn (3)] mitogenic parameters are modeled similarly: A AM m, A" A (K#m) A A @M m. A" @ (K#m) @ Mesenchyme di!erentiation into osteoblasts or chondrocytes is mediated by the osteogenic and chondrogenic growth factors and represented by concentration-dependent curves: >  g , F "  (H #g ) @  @

(1)

*c A"c [A !B c ]#F c !F c , A A A A  K  A *t

(2)

*c @"c [A !B c ]#F c #F c !d c , @ @ @ @  K  A @ @ *t

(3)

where D and C represent the haptotactic and haptokinetic cell migration speeds; A , A and A K A @

>  g. F"  (H #g ) A  A Finally, endochondral replacement of chondrocytes depends on connective/cartilage ECM density and on the concentration of osteogenic growth factor: > m  A ; g . F"  (B #m ) (H #g ) @  @ CA A

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The rate of change of connective/cartilage and bone extracellular matrix densities is the balance between synthesis and degradation which depends on the respective cell density: *m A"(P !Q m );(c #c )!Q m c , AQ AB A K A AB A @ *t *m @"(P !Q m )c , @Q @B @ @ *t

(4)

(5)

where P and P represent constants of connectAQ @Q ive/cartilage and bone matrix synthesis (s) and Q , Q and Q are constants of matrix degraAB AB @B dation (d). Finally, the rates of change of growth factor concentrations are de"ned as *g A" )[Dg g ]#Egc c !dgc g , A A A A *t

(6)

*g @" )[Dg g ]#Egb c !dgb g , @ @ @ @ *t

(7)

where Dg and Dg are di!usion coe$cients, A @ Eg and Eg are functions relating growth factor A @ production to growth factor concentration, and dg and dg are constants of decay. A @ The constants of growth factor production depend on growth factor concentration and on extracellular matrix: Gg g m A A ; , Eg " A (Hg #g ) (Kg #m) A A A Gg g @ @ . Eg " @ (Hg #g ) @ @ Non-dimensionalization of the parameters as described in the Appendix results in the following equations (where the tildes indicating nondimensionalized parameters have been omitted): *c K" ) [D c !Cc m]#A c [1!a c ] K K K K K K *t !F c !F c ,  K  K

(1)

*c A"A c [1!a c ]#F c !F c , A A A A  K  A *t

(2)

*c @"A c [1!a c ]#F c #F c !d c , (3) @ @ @ @  K  A @ @ *t *m A"P (1!i m );(c #c )!Q m c , (4) AQ A A K A AB A @ *t *m @"P (1!i m )c , @Q @ @ @ *t

(5)

*g A" ) [Dg g ]#Eg c !dg g , A A A A A A *t

(6)

*g @" ) [Dg g ]#Eg c !dg g . @ @ @ @ @ @ *t

(7)

The non-dimensional parameter values used were: DI "0.014; KI "0.25; CI "0.0034; F F I KI "0.5; AI "1.01; KI "0.1; AI "1.01; I KM K AM KI "0.1; a "1; AI "0.202; KI "0.1; a "1; A A @M @ @ >I "10; HI "0.1; >I "50; HI "0.1; >I "1000;      BI "1.5; HI "0.1; dI "0.1; PI "0.2; i "1; CA  @ AQ A QI "2; PI "2; i "1; DI g "0.005; A AB @Q @ DI g "0.005; GI g "50; HI g "1; KI g "0.1; @ A A A GI g "500; HI g "1; dI g "100; dI g "100. @ @ A @ 2.3. GEOMETRY, INITIAL AND BOUNDARY CONDITIONS

Assuming rotational symmetry about the marrow cavity, with no gradients along the circumferential direction, the problem is twodimensional. Moreover, assuming also symmetry about the fracture line, only one-quarter of the external callus and gap was modeled (Fig. 3). The model dimensions were obtained from experimental measurements of fracture calluses and are discussed in the Appendix. The length scale is based on the rat. Initially, all cellular densities in the callus are zero, except for a migrating mesenchymal cell front representing the soft tissues surrounding the bone (i.e. muscles) and the inner cambium layer of the periosteum. Although the proportion of cells originating from the two sources is not known, in this model we consider the inner layer of the periosteum a more important contributor to the initial mesenchymal cell pool than the surrounding soft tissues.

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FIG. 3. Model geometry. One-quarter of the callus is modeled by assuming symmetry about the fracture line and rotational symmetry about the marrow cavity. (a) Regions representing the external callus and the fracture gap. Initial conditions for (b) mesenchymal cell density representing the migrating cellular front from surrounding soft tissues and broken periosteum; (c) low extracellular matrix density representing the initial in#ammatory hematoma, and (d) sources of chondrogenic and osteogenic growth factors at the site of injury and at the cortex underneath the periosteum, respectively, corresponding to expression sites for BMP-2/4 and TGF-b1, respectively, at the onset of fracture healing: (n) external callus; ( ) fracture gap.

There is a low initial ECM density and there are two initial sources of osteogenic and chondrogenic growth factors, corresponding to the cortical layer beneath the broken periosteum and the hematoma, respectively [Fig. 3(d)]. As described, TGF-b expression along the periosteal cortex is observed within the "rst 24 hr and BMP has been reported in the hematoma at the onset of fracture healing. The initial concentrations of the chondrogenic and osteogenic growth factors were held as boundary conditions for a speci"ed number of healing days. The duration and magnitude of these boundary conditions were considered parameters in the model. Finally, no cellular movement or growth factor di!usion was permitted across the symmetry planes. The non-dimensional boundary conditions used were c - "0.5 and 1 representing the mesK GLGR enchymal cell fronts from the surrounding soft tissues and within the periosteum, respectively, and m "0.1, g - "20 and g - "20, repres@ GLGR GLGR A GLGR enting the in#ammatory conditions (Appendix). 2.4. NUMERICAL SIMULATIONS

An alternating direction implicit and explicit (for nonlinear terms) "nite di!erence method was used to solve the non-dimensionalized system of partial di!erential equations (Morton & Mayers, 1994). The mesh size was 0.02 cm for the x and y directions and the time step was 0.24 hr. The code was validated by comparing the solutions to reduced problems to the solutions of Olsen et al. (1997) and to solutions generated using the Matlab di!erential equation toolbox.

An extensive parametric study was performed to determine the sensitivity of the model to changes in a single or multiple parameters at a time. This parametric study was used also to propose ranges of values for a number of parameters which could not be estimated from the literature (Table 1). For these parameters, biologically feasible orders of magnitude estimates were "rst generated and large perturbations from these values were examined. The healing response predicted by the model is presented by reporting the spatial distribution of cell and ECM densities and growth factor concentrations in the callus at di!erent time points. These results correspond to parameter values for which large perturbations are necessary to significantly alter the solution. Deviations from the results presented due to large changes in speci"c parameters, however, also are reported and discussed. The percentage of the callus area occupied by cartilage or bone was calculated throughout the healing period by adding the areas of the mesh elements containing a minimum bone or cartilage density, respectively, greater than 0.011 g ml\. The average bone and cartilage density and growth factor concentration in the callus was calculated throughout healing by adding up the densities and concentrations from all mesh elements and dividing over the number of elements. Finally, peak mesenchymal, chondrocytic and osteoblast cell proliferation in the callus was determined by adding the proliferation rates from all the mesh elements for each cell type.

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TABLE 1 Parametric study values Parameter >  H  >  H  >  P @Q Q AB G EA G E@ g@ GLGR gA GLGR

Estimated range* 10}100 5}100 10}100 5}100 700}1500 0.2;10\}2;10\ 0.2;10\}2;10\ 0.2}1 0.2}1 1000}10 000 1000}10 000

Units

Non-dimensional value*

day\ ng ml\ day\ ng ml\ day\ (g ml\) day\ (cells ml\)\ (cells ml\)\ day\ (ng ml\) (g ml\) day\(cells ml\)\ (ng ml\) (cells ml\)\ day\ ng ml\ ng ml\

10}100 0.05}0.1 10}100 0.05}0.1 700}1500 0.2}2 0.2}2 20}100 200}1000 10}100 10}100

*See Appendix for further information.

3. Results The model simulates the progression of fracture healing as experimentally described (Fig. 4). Mesenchymal cells propagate from the surrounding soft tissues into the fracture region forming the initial soft callus. Exuberant mesenchymal callus formation is apparent by day 3. Di!erentiation of mesenchymal cells into chondrocytes and osteoblasts, adjacent to the fracture site and at the cortex, respectively, is initiated by this time point and completed by day 6. By day 7 there is a large growing chondrous callus adjacent to the fracture and an advancing intramembranous ossi"cation front along the cortex, corresponding to the di!used sites of expression of the chondrogenic and osteogenic growth factors, respectively (Fig. 5). Endochondral ossi"cation is advanced by day 12 as chondrocytes are progressively replaced by osteoblasts (Figs 4 and 6). Ossi"cation of the external callus and bone union is completed by 20 days. Mesenchymal cell proliferation in the callus peaks on day 4 and rapidly declines over the two following days. Chondrocyte proliferation peaks on day 5 and rapidly declines over the following 9 days. Osteoblast proliferation peaks on days 5 and 15. The early and late peaks for osteoblast proliferation correspond to active intramem-

branous and endochondral bone formation, respectively. The appearance of the "rst chondrocytes in the callus on day 3 corresponds to the time when mesenchymal cells have reached the region with di!used chondrogenic growth factor. The chondrogenic growth factor is distributed throughout the callus with peak concentration in the external callus on day 5 and in the fracture gap on day 9. The concentration of chondrogenic growth factor declines and is completely cleared by day 20. In contrast, the expression of the osteogenic growth factor is only localized to areas of active intramembranous and endochondral bone formation, and the concentrations of this growth factor are lower than those of the chondrogenic growth factor. Peak osteogenic growth factor concentration in the callus is on days 5 and 15, corresponding to the times of maximal rates of intramembranous and endochondral ossi"cation. These days of maximal expression of osteogenic growth factor are in exact agreement with days 5 and 15 reported to be of maximal expression of TGF-b1 in the rat callus (Joyce et al., 1990b). Multiple- and single-parameter parametric studies indicate that the model is insensitive to large perturbations in most parameter values. Cartilage tissue production is insensitive to large variations in the parameters associated with the

GROWTH FACTORS AND FRACTURE HEALING

201

FIG. 4. Numerical solutions for post-fracture cell densities. (a) Mesenchymal (day 1, 2, 3, 6), (b) chondrocyte (day 3, 7, 12, 20) and (c) osteoblast (day 3, 12, 14, 20) cell densities (;10 cells ml\) in the callus.

FIG. 5. Numerical solutions for post-fracture growth factor concentrations. (a) Chondrogenic (day 5, 8, 12, 20) and (b) osteogenic (day 5, 12, 15, 20) growth factor concentrations (;100 ng ml\) in the callus.

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FIG. 6. Numerical solutions for post-fracture ECM densities. (a) Connective/cartilage ECM (day 5, 8, 12, 20) and (b) bone ECM (day 5, 12, 15, 20) densities (;0.1 g ml\) in the callus.

chondrogenic growth factor. Similarly, intramembranous bone formation also is insensitive to parameters associated with the osteogenic growth factor. The rate of bone formation and cartilage degradation during endochondral ossi"cation, however, depends on the respective tissue formation and degradation constants (QI and PI , AB @Q Fig. 7). Moreover, endochondral ossi"cation in the model is sensitive to large changes in osteogenic growth factor production. Low values of osteogenic growth factor production lead to slow and incomplete endochondral ossi"cation of the external callus and to fracture non-union. In these cases, bone formation stops around day 8, with only about 10% of the external callus consisting of bone, and the fracture gap consisting entirely of cartilage (GI "200 and 350, Fig. 8). E@ Increased osteogenic growth factor production leads to normal healing (GI "500 and 1000, E@ Fig. 8). In these latter cases, the size of the chondrous external callus peaks around day 10, and is gradually replaced by bone. The external callus and fracture gap are almost fully ossi"ed by day 20 and contain minimal cartilage. The duration of growth factor expression in the hematoma, before production is initiated by the di!erentiated cells, also is an important

factor. A short duration of chondrogenic growth factor release leads to a minimal chondrous external callus, yet increasing the duration of this release two-fold leads to the expected large external chondrous callus. Similarly, a short duration of osteogenic growth factor release leads to impaired initiation of bone formation and no healing. Increasing the duration two-fold leads to successful ossi"cation and healing response (Fig. 8). Further increases of the time of growth factor release lead to no further enhancements in healing. Moreover, the magnitudes of the osteogenic and chondrogenic growth factor release do not a!ect the results. 4. Discussion The objective of this study was to develop a mathematical framework to explore the regulatory mechanisms of cell and growth factor interactions during fracture healing. We hypothesized that the qualitative nature of the results would be insensitive to important variations in a large number of parameters, but that select parameters would signi"cantly alter this response. The model successfully simulates the progression of the healing response. The stability of the model to perturbations in most parameters may capture the

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203

FIG. 7. Normalized cartilage and bone fractions (% area) in the callus (a, b) and average cartilage and bone matrix density (;0.1 g ml\) (c, d) throughout healing, for two levels of cartilage degradation (QI ) and bone formation (PI ) constants: AB @Q ) QI "0.2, PI "0.2; ( ) QI "0.2, PI "2; ( ) QI "2, PI "0.2; ( ) QI "2, PI "2. ( AB @Q AB @Q AB @Q AB @Q

intrinsic robustness of non-pathologic bone regeneration, which, despite immense environmental and biological variability within and across species, progresses histologically in a similar fashion in most animals, with substantial variations only in the time-scale. The initiation of chondrogenesis and intramembranous bone formation is directed by the localized expression and di!usion of the growth factors released upon injury. Considering growth factors with half-lives in the order of minutes and with rapid tissue clearance and denaturation (Dinbergs et al., 1996), the results from this model indicate that the duration of initial release into the hematoma needs to be su$ciently long to allow migrating pluripotential cells to reach their localized range of expression. Once cell di!erentiation is initiated, however, the production of cell di!erentiation growth factors by the specialized cells leads to robust cartilage and intramembranous bone tissue formation

throughout the callus. The need for su$cient quantities of growth factors such as BMP and TGF-b to be released at the site of injury for appropriate tissue regeneration may explain the lack of positive results from experiments where single injections or bolus administration of exogenous growth factors are used, instead of continuous infusions or sustained release administration (Barnes et al., 1999). This concept may also explain the slower healing response with aging and other conditions where the concentrations of growth factors stored in the bone, speci"cally TGF-b, decrease (Linkhart et al., 1996). Whether this decrease in stored growth factor concentration leads to a decrease in the dosage of the growth factors released upon injury, or in the duration during which these growth factors are released, needs to be examined. Endochondral ossi"cation involves coordinated cartilage maturation, and simultaneous

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FIG. 8. Normalized cartilage and bone fractions (% area) at 8 and 20 days post-fracture in the (a, c) external callus and the (b, d) fracture gap, for di!erent values of osteogenic growth factor production constant (GI g "200, 350, 500, 1000) and @ duration of release of osteogenic growth factor upon injury (2 days, 4 days). At 8 days the external callus has little bone and the fracture gap is entirely cartilage for all cases. At 20 days, low values of GI g (200, 350) and short duration of osteogenic growth @ factor release (2 days) leads to no additional bone formation in the callus and no healing. Higher values of GI g (*500) and @ a longer duration of osteogenic growth factor release (4 days) leads to normal healing with ossi"cation of the external and fracture callus: ( ) % cartilage area; (䊏) % bone area.

cartilage tissue degradation and bone tissue formation. In this model, the rates of cartilage turnover and bone formation a!ect the speed at which the ossi"cation front advances. Moreover, the model suggests that a minimum rate of osteogenic growth factor production is necessary for the initiation and progression of endochondral ossi"cation, but higher levels of production provide no further bene"ts. The time points of peak osteogenic growth factor concentration in the callus are consistent with those experimentally reported in the rat fracture callus (Joyce et al., 1990b). The results in this study indicating peak cartilage and bone matrix densities on day 10 and from day 20 onwards, respectively, are also close to the

reported time points for peak collagen II mRNA density on days 7 and 9, and for peak collagen I mRNA density on days 14 and 28, in the mouse callus (Hiltunen et al., 1993). The di!erences in the time points between the model and the latter experiment may re#ect species di!erences, as the parameters in this model were estimated not only from rat studies but also from in vitro and in vivo experiments using data from other species. In this model two extracellular matrices and three cell types are modeled. Although a larger number of matrix and cell types is present, the number of dependent variables and parameters in the model was limited to the most essential ones. The incorporation of a "brous matrix in the future, however, may be relevant when studying

GROWTH FACTORS AND FRACTURE HEALING

pathological conditions such as non-unions. Additionally, the model is inherently limited in its ability to capture certain details of the complex nature of growth factor biology. In particular, the model does not consider the di!erent natures of latent and active TGF-b, the e!ects of enzymes, or the complex interactions between receptors, extracellular matrix and growth factors. The e!ects of the mechanical environment in the callus were also not examined in this work. However, variations in the stability of the fracture and the motion between the bone fragments may in#uence cell migration, cell proliferation or matrix synthesis and the angiogenic response (Perren & Cordey, 1980; El Haj & Thomas, 1994). Moreover, studies have recently shown enhanced growth factor expression during increased mechanical stimulus during distraction osteogenesis (Sato et al., 1999) and di!erent responses in mechanically stable and unstable fractures to exogenous TGF-b2 (Critchlow et al., 1995). In particular, very unstable fractures have large gradients of stresses and strains along the line of the fracture, extending from the midline of the bone cortex to both the endosteal and periosteal sides. Cell di!erentiation and tissue development in this region may be sensitive to alterations in the mechanical environment. Future models need to incorporate the e!ects of the mechanical environment to understand unstable fractures. In summary, this work presents a model that successfully simulates tissue di!erentiation during fracture healing and gives insights into the regulatory roles of growth factors. Only individual factors were examined, so the next step is to explore the interaction between these and other growth factors, and between these growth factors and the mechanical environment. The development of experiments on rodent fracture and distraction osteogenesis models and on genetically altered mice, together with the development of a more comprehensive theoretical framework, will greatly improve our understanding of the healing process. The authors thank Dr Cornelia E. Farnum (College of Veterinary Medicine, Cornell University, U.S.A.) and Dr Steven H. Strogatz (Department of Theoretical & Applied Mechanics, Cornell University, U.S.A.)

205

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APPENDIX

E

The following scalings were chosen for nondimensionalization: t x y tJ " , xJ " , yJ " , ¹ ¸ ¸

c c cJ " K , cJ " A , K K A K JK JK

E

E

m c cJ " @ , mJ " A, A m @ K M JK

m g g mJ " @, gJ " A , gJ " @. @ m A g @ g M M M

Parameters were non-dimensionalized as follows: DI "D ¹/(¸m ); KI "K /m ; F F M F F M CI "C ¹/(¸m ); KI "K /m ; I I M I I M AI "A ¹/m ; KI "K /m ; BI "B K ¹; KM KM M K K M K K JK AI "A ¹/m ; KI "K /m ; BI "B K ¹; AM AM M A A M A A JK a "K /K ; A JK JA

AI "A ¹/m ; KI "K /m ; @M @M M @ @ M

BI "B K ¹; a "K /K ; @ @ JK @ JK J@ HI "H /g ;   M

>I "> ¹;  

>I "> ¹; HI "H /g ;     M

>I "> ¹; BI "B /m ; HI "H /g ;   CA CA M   M dI "d ¹; QI "Q K ¹; i "Q /P ; @ @ AB AB JK A AB AQ QI "Q K ¹; AB AB JK

QI "Q K ¹; i "Q /P ; @B @B JK @ @B @Q

E

DI g "Dg ¹/¸; DI g "Dg ¹/¸; A A @ @ GI g "Gg K ¹/(gm ); HI g "Hg /g ; A A JK M M A A M KI g "Kg /m ; A A M

GI g "Gg K ¹/g; @ @ JK M

HI g "Hg /g ; dI g "dg ¹; @ @ M A A

dI g "dg ¹. @ @

The following parameters were obtained from the literature, where we omit tildes on non-dimensional values for simplicity. E

E

A sensitive scale for extracellular matrix (ECM) density in the callus was chosen as m " M 0.1 g ml\, considering the typically high concentration of water in the tissue, with a density of 1 g ml\, and the presence of loose collagen "bers, fat and other substances (Olsen et al., 1995).

E

E

207

A typical growth factor concentration was chosen as g "100 ng ml\ based on osteoM genic and chondrogenic literature dosages (Joyce et al., 1990b). Typical temporal and length scales in fracture healing studies for the rat are ¹"1 day and ¸"3.5 mm. Microscopic study of leukocyte movement has measured a random locomotion di!usion constant of D"240 lm min\ (4E!8 cm s\) (Gruler & BuK ltmann, 1984). Studies on "broblasts have reported cell velocities yielding slower di!usion coe$cients of about 2}60 lm min\ (Friedl et al., 1998). As substrate conditions in in vitro studies may slow down cellular migration and growth factors act as chemoattractants to increase cell speeds during fracture healing, we chose the highest di!usion coe$cient to represent the speed of the mesenchymal cell pool. In our model of cell migration D(m)" D m(K#m)\ and C(m)" C (K #m)\. F F I I Therefore, the maximum rate of cell motility D occurs at an ECM density of m"K , yieldF ing D "2K D. Setting D(m)"C(m) at F F high ECM density, m"1 yields: C " I K D (K#m)\. K and K are chosen to be I F F F I positive, much smaller than m"1 and higher than m"m . GLGR A maximum mesenchymal population density of K "10 ml\ at the beginning of healing, JK based on a cell volume of about 10\ ml, is an order of magnitude estimate. This limiting cell density will then depend on ECM density: K "A /B (Fig. A1). JK K K We estimate a maximum proliferation rate of 5 cells day\. Assuming that the maximal cell proliferation occurs at the initial ECM density K "m , A "A m\ (2m ). L GLGR KM K?V GLGR GLGR We assume that proliferating chondrocytes in a fracture callus are approximately the same size as "broblasts, and model proliferation using the same parameters. Osteoblast proliferation rate, however, has been reported to be about 18% lower than that of "broblasts (Fedarko et al., 1995) and we set A accordingly. @M Quanti"ed rates of cell di!erentiation are not available, although they occur over a short period of time ((1 day). In the absence of data, the following range of the cell conversion

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A. BAILOD N-PLAZA AND M. C. H. VAN DER MEULEN

E

FIG. A1. At any point within the callus, extracellular matrix density limits the maximum number of cells present in an unit volume, as speci"ed by the limiting cell densities for mesenchymal cells (K ), chondrocytes (K ) and osteoJK JA blasts (K ). For example, at the onset of healing, for a low J@ initial ECM density (0.01 g ml\) more mesenchymal cells can exist (5;10 cells ml\); later, at a higher ECM density (0.1 g ml\) fewer mesenchymal cells can survive in the same volume (1;10 cells ml\).

E

E

E

constants is examined: 10}100 day\, for >  and > (Table 1). Also, as relatively low con centrations of growth factors have been shown to have osteogenic and chondrogenic e!ects, we examine a range of 5}100 ng ml\, for H  and H . We examined parametrically a range  of 700}1500 day\ for > (Table 1) and we let  H "H .   We set B "1.5 so that the endochondral ossiCA "cation parameter is not signi"cant for low ECM density. Literature values for the rate of osteoblast removal are not available and we perform a linear stability analysis to estimate this value. Osteoblast cell density is at steady state when eqn (3) is zero. This steady state is locally unstable if: A !2B c !d '0. At the beginning @ @ @ @ of healing A &1. At this time zero osteoblast @ density should be unstable as the population should grow in size initially, hence, d (1. As @ healing progresses, mesenchymal cell density and the concentration of chondrogenic growth factor become zero (c "0 and F "0). The K  steady state of eqn (3) corresponds to osteoblast densities where A !B c !d "0. The @ @ @ @ only non-trivial steady-state osteoblast density is c "(A !d )/B '0. Since all parameters @ @ @ @ are positive and at high ECM densities A &0.2, d (0.2. @ @ We let m and m be the maximal ECM K?V @ K?V and bone density values (globally stable steady states) at which *m/*t"0 and *m /*t"0, re@ spectively. In this case, m "P /Q and K?V AQ AB m - "P /Q . Appropriate non-dimension@Q @B @ K?V alized values are m "1 and m - "1. K?V @ K?V

E

E

E

Using estimates calculated by Olsen et al. (1997), we obtain Q "0.2 for the connectAB ive/cartilage matrix. A range of values for P @Q and Q are explored numerically (Table 1). AB The molecular weight of TGF-b and BMPs is about 25 kDa (Joyce et al., 1990a). In aqueous solution the molecular weight of a substance, M= , relates to its di!usion coe$cient, D , V V as follows: D "D (M= /M= ) V EJSAMQC EJSAMQC V (Vander et al., 1998), where D and EJSAMQC M= are the di!usion coe$cient and the EJSAMQC molecular weight of glucose. Approximating these parameters as 180 Da (Kutchai, 1993) and 0.67E\ cm s\, respectively, we estimate di!usion coe$cients for the growth factors in water as Dg "Dg "4.8;10\ cm s\. A @ However, the coe$cients for di!usion in other media may be much lower. In particular, in healthy carilage the di!usion coe$cient for solutes with a molecular weight of 24 kDa can be as low as 8% of the coe$cient in water, depending on the hydration of the cartilage (Foy & Blake, 2001). Therefore, our default di!usion coe$cient for the growth factors in the tissue was 15% of the estimated coe$cient in water. Using coe$cients 5}30% of the estimates in water did not a!ect the results. Again, with no experimental value for growth factor production constants Gg and Gg , A @ we explored a range of values numerically (Table 1). We let Hg "Hg "1, so the saturaA @ tion level for the production of growth factors occurs around typical growth factor concentration levels. Finally, we set the ECM density at which the production of chondrogenic growth factor begins to decrease with increasing ECM density to Kg "0.1. A Growth factors such as bFGF and TGF-b have short in vivo half-lives ((30 min, Co!ey et al., 1990; Dasch et al., 1989; Edelman et al., 1993). A half-life of 10 min corresponds to a decay constant of dg "dg "99.9 day\. A @ Model geometry, boundary and initial conditions Non-dimensionalized model dimensions are reported in Fig. A2. The proportions for the callus and the bone are based on measurements on sheep fracture calluses, used in previous fracture healing models (Claes & Heigele, 1999).

GROWTH FACTORS AND FRACTURE HEALING E

E

FIG. A2. Non-dimensionalized model geometry.

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We estimate a low ECM density representing the hematoma for the beginning of fracture healing of about 0.01 g ml\, yielding m " GLGR 0.1. We also estimate mesenchymal cellular densities for the cell front representing surrounding soft tissues and the inner layer of the broken periosteum as c - "0.5 and 1, K GLGR respectively. We consider a range of initial osteogenic growth factor concentrations of g - " @ GLGR 10}100 ng ll\"1000}10 000 ng ml\ as this was found to induce the transformation of mesenchymal cells to osteoblasts for TGF-b in vitro. The same range is examined for the chondrogenic growth factor.